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- Permanent Link:
- https://ufdc.ufl.edu/UFE0022613/00001
## Material Information- Title:
- Significance of Probabilistic Parameterization in Cohesive Sediment Bed Exchange
- Creator:
- Letter, Joseph
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2009
- Language:
- english
- Physical Description:
- 1 online resource (428 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Coastal and Oceanographic Engineering
Civil and Coastal Engineering - Committee Chair:
- Mehta, Ashish J.
- Committee Co-Chair:
- Sheppard, Donald M.
- Committee Members:
- Dean, Robert G.
Thieke, Robert J. Jaeger, John M. - Graduation Date:
- 8/8/2009
## Subjects- Subjects / Keywords:
- Aggregation ( jstor )
Mathematical variables ( jstor ) Particle size classes ( jstor ) Sediment transport ( jstor ) Sediments ( jstor ) Shear strength ( jstor ) Shear stress ( jstor ) Silts ( jstor ) Simulations ( jstor ) Velocity ( jstor ) Civil and Coastal Engineering -- Dissertations, Academic -- UF cohesive, deposition, erosion, exclusive, probabilistic, sedimentation, simultaneous, turbulence City of Miami ( local ) - Genre:
- Electronic Thesis or Dissertation
born-digital ( sobekcm ) Coastal and Oceanographic Engineering thesis, Ph.D.
## Notes- Abstract:
- SIGNIFICANCE OF PROBABILISTIC PARAMETERIZATION IN COHESIVE SEDIMENT BED EXCHANGE The primary issue addressed in this study is whether the decades-old paradigm of exclusive erosion or deposition in turbulent flow has legitimacy based on physical principles within cohesive sediment dynamics. The exclusive paradigm assumes that sediment exchange condition at the bed-water interface is either erosion, deposition or neither, but never both. In contrast, the more recently espoused simultaneous exchange paradigm admits the possibility of erosion and deposition of cohesive sediment occurring at the same time. The exclusive paradigm is, in part, the result of early attempts to understand basic cohesive sediment transport behavior based on inferred data in laboratory apparatuses such as flumes averaged over time and space. The time scale of averaging is longer than the time scale of turbulence and the spatial dimension is scaled by water depth in the apparatus. Bed sediment exchange has been deduced primarily from the increase or reduction in the suspended sediment concentration within the water column, rather then from difficult to record observations of particle movement very close to the bed surface. The net result of averaging will be positive, negative or zero sediment flux at the bed surface, but not both positive and negative. With the inclusion of greater details in newer mathematical models, such as particle size distributions and flocculation sub-models, the bed exchange algorithms have required revision. Numerical modelers have found the need to use the simultaneous approach to replicate observed sedimentation rates in the field environment. The numerical sediment transport tool developed for this research has been shown to be capable of simulating several processes critical for simulation of bed exchange. These processes include aggregation and disaggregation dynamics, stochastic effects in bed exchange and aggregation/disaggregation, hindered settling, attainment of a depositional or erosional equilibrium concentration for a fixed shear stress, and floc spectrum features documented by field experimentation. Observations made during development and application of the numerical tool are: ? The effects of a probabilistic treatment of the key variables are more pronounced for erosion than for deposition. These variables include current velocity, bottom shear stress, floc shear strength, critical shear stresses for erosion and deposition, internal shear and settling velocity. ? Probabilistic effects are amplified through the flocculation model over the effects that occur through bed exchange alone. ? For a given shear stress the flocculation model will tend toward an equilibrium distribution of particle sizes. ? The probabilistic treatment results in a broader floc distribution spectrum than occurs with use of mean-valued variables. ? Deposition or erosion will be initiated sooner and transition from one to the other more gradual in response to changing shear stress when a probabilistic treatment is used compared to a mean-valued treatment. The differential timing will be a function of the standard deviations of the probabilistic variables and the rate of change of the shear stress. ? The use of the exclusive paradigm with a floc size distribution can perform as well as a simultaneous treatment with a single particle size. ? A simulation was performed of a flume test by Parchure and Mehta (1985) designed to evaluate the exclusive versus simultaneous paradigm by diluting the concentration of a flume suspension that had achieved an equilibrium concentration from bed erosion. If the exclusive paradigm was valid, the concentration at the end of dilution should remain constant. If the concentration began to rise after dilution was ceased, then the simultaneous paradigm would be an explanation. The flume concentration did rise after the dilution stopped, but at a very low rate of erosion. The numerical model was able to replicate the flume behavior with the correct rate of rise after the end of dilution by using the exclusive paradigm with a probabilistic treatment of the variables. ? The appropriate use of either the exclusive or continuous paradigm appears to be dictated by the level of temporal and spatial averaging used in the development of empirical data and in the formulation of the variables in the analysis. ? Empirical coefficients developed for mean-valued analysis may require adjustment when used in a probabilistic treatment. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2009.
- Local:
- Adviser: Mehta, Ashish J.
- Local:
- Co-adviser: Sheppard, Donald M.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28
- Statement of Responsibility:
- by Joseph Letter.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Letter, Joseph. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 2/28/2010
- Classification:
- LD1780 2009 ( lcc )
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PAGE 1 1 SIGNIFICANCE OF PROBABILISTIC PARAMETERIZATION IN COHESIVE SEDIMENT BED EXCHANGE By JOSEPH VINCENT LETTER, JR. A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Joseph Vincent Letter, Jr. PAGE 3 3 To my wife Linda PAGE 4 4 ACKNOWLEDGMENTS I gratefully acknowledge a numbe r of individuals who have b een instrumental in allowing me to continue my lifelong pursuit of education. I am most profoundly grateful to my professors at the University of Floridas Civil and Coasta l Department, some of whom have been teaching me since 1973. In particular, I am indebted to Drs. Ashish Mehta, Robert Dean and Max Sheppard who have each sustained the quality of the educational experience at the University of Florida for me. I am also grateful to Dr. R obert Thieke, an exceptional teacher who recognizes the imperative of learning by getting your hands wet, challenges his students and guarantees a continuing level of educational exce llence in the department. I am also grateful to Dr. Tian-Jian Hsu for his guidance and suggestions and to Dr. John Jaeger for his willing participation on my committee in spite of my distance le arning over the past several years. I am especially grateful to Dr. Mehta for his to lerance of an older student set in his ways. I appreciate his constant willingness to accommodat e a wandering path to this culmination. In particular, I acknowledge the valu e of his guidance and council through the procedural maze of the University. I acknowledge the management of the Coasta l and Hydraulics Labor atory (CHL) at the U.S. Army Engineer Research and Development Center (ERDC) for their financial support and commitment to higher education and an innovative work environment that encourages higher learning. I am grateful to my coworkers at ERDC for their friendship and encouragement. Special thanks are offered to the field data collect ion staff for their assistance in analysis of the San Francisco data. I am especially grateful to my parents, Joseph and Helen Letter, for their unfailing love and support throughout my life. Nothing is impo ssible for me in their eyes. I especially PAGE 5 5 acknowledge their patience and underst anding of my educational priori ties at a time in their lives when moments spent with them are so very precious. I am profoundly grateful to my wife, Linda for her love, support and patience and for mowing the lawn by herself on far too many o ccasions during the pursuit of my education. Finally, I acknowledge the patien ce and understanding of the rest of my family at all the times when I missed those family gatherings because I was in my cave. PAGE 6 6 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ........11 LIST OF FIGURES.......................................................................................................................13 NOTATION...................................................................................................................................23 ABSTRACT...................................................................................................................................35 INTRODUCTION.........................................................................................................................38 1.1 Need for Research.......................................................................................................... ...38 1.2 Objectives and Tasks........................................................................................................43 1.3 Approach................................................................................................................... ........44 1.3.1 Examination of Governing Equations.................................................................... 45 1.3.2 Revision of Aggregation/Disaggregation Model for Probabilistic Fluctuations.... 45 1.3.3 Effects of Suspended Sediment on Turbulence ......................................................46 1.3.4 Development of a Probabilistic Bed-Exchange Model.......................................... 46 1.3.5 Application of the Probabilistic Formulation to Selected Test Cases.................... 46 1.4 Scope...................................................................................................................... ...........46 1.5 Presentation Outline....................................................................................................... ...47 COHESIVE SEDIMENT TRANSPORT......................................................................................48 2.1 Estuarine Cohesive Sediment Properties.......................................................................... 48 2.1.1 Estuarine Sediment.................................................................................................48 2.1.2 Fine Sediment Classification.................................................................................. 49 2.1.2.1 By size..........................................................................................................49 2.1.2.2 By shape.......................................................................................................50 2.1.2.3 By composition............................................................................................ 50 2.1.2.4 By electrochemical properties...................................................................... 51 2.1.3 Characterizing Aggregates.....................................................................................52 2.1.3.1 Primary particle distribution......................................................................... 52 2.1.3.2 Order of aggregation.................................................................................... 54 2.1.3.3 Floc size spectra........................................................................................... 55 2.1.3.4 Fall velocity.................................................................................................. 55 2.1.3.5 Floc density.................................................................................................. 55 2.1.3.6 Floc shearing strength.................................................................................. 56 2.1.3.7 Fractal dimension......................................................................................... 58 2.1.4 Bulk Properties of Cohesive Sediments................................................................. 61 2.2 Cohesive Sediment Transport in Steady Uniform Flow................................................... 61 2.2.1 Aggregation Processes............................................................................................ 62 2.2.1.1 Brownian motion.......................................................................................63 PAGE 7 7 2.2.1.2 Differential settling......................................................................................64 2.2.1.3 Shear.............................................................................................................64 2.2.1.4 Salinity.........................................................................................................65 2.2.2 Fall Velocity........................................................................................................... 65 2.2.3 Hindered Settling....................................................................................................70 2.2.4 Deposition and Erosion.......................................................................................... 76 2.3 Cohesive Sediment Transport in Unsteady Nonuniform Flow......................................... 82 2.3.1 Aggregation Processes............................................................................................83 2.3.2 Settling Velocity..................................................................................................... 84 2.3.3 Bed Exchange.........................................................................................................85 2.4 Governing Equations........................................................................................................ 86 2.5 Applicability of Existing Knowledge to Field Conditions...............................................88 2.6 Biological Influences........................................................................................................88 PROBABILISTIC DESCRI PTION OF COHESIVE SEDIMENT TRANSPORT.................... 113 3.1 Conceptual Framework................................................................................................... 113 3.2 Particle Definitions....................................................................................................... ..116 3.3 Shear Stress.....................................................................................................................117 3.4 Aggregation and Disaggregation.................................................................................... 122 3.5 Floc Density............................................................................................................... .....123 3.6 Floc Strength...................................................................................................................123 3.7 Bed Exchange.................................................................................................................125 3.8 Summary of Probabilistic Treatment.............................................................................. 130 MULTI-CLASS DEPOSITION WITH AGGREGATION.........................................................158 4.1 Introduction............................................................................................................... ......158 4.2 Sediment Transport......................................................................................................... 159 4.2.1 Sediment Size Classes..........................................................................................159 4.2.2 Sediment Transport Equation............................................................................... 161 4.2.3 Settling Velocities................................................................................................164 4.2.4 Vertical Mixing....................................................................................................165 4.2.5 Bed Exchange.......................................................................................................169 4.2.6 Aggregation Processes..........................................................................................174 4.2.7 Disaggregation Processes..................................................................................... 179 4.2.7.1 Shear-Induced Disaggregation................................................................... 179 4.2.7.2 Collision-Induced Disaggregation.....................................................................181 4.3 Hydrodynamics...............................................................................................................185 4.4 Solution Method.............................................................................................................186 4.5 Probabilistic Representation........................................................................................... 188 4.7 Analytical Test Cases.....................................................................................................189 4.7.1 Stokes first problem..............................................................................................190 4.7.2 Couette flow problem...........................................................................................194 4.7.3 Stokes second problem.........................................................................................196 4.7.4 von Karman Mixing Length Velocity Profile for Fully Rough Flow.................. 198 PAGE 8 8 SEDIMENT TRANSPORT AND DEPOSITION EXPERIMENTS.......................................... 230 5.1 Kynch (1952) Sedimentation Theory ............................................................................. 230 5.2 Krone (1962) Flume De position Experiments................................................................ 235 5.2.1 Settling Tests with Variable Shear ....................................................................... 235 5.2.2 Settling Test with Tagged Sediments................................................................... 235 5.3 Mehta 1973 Flume Results............................................................................................. 236 5.4 Parchure and Mehta (1985) Dilution Test...................................................................... 237 5.5 Parchure and Mehta (1985) Erosion Test....................................................................... 237 5.6 Sanford and Halka (1993) Data Set................................................................................ 238 METHOD APPLICATION......................................................................................................... 252 6.1 Preamble................................................................................................................... ......252 6.1 Kynch (1952) Quiescent Deposition Test.......................................................................254 6.2 Mehta 1973 Flume Deposition Tests.............................................................................. 256 6.3 Krone (1962) Flume Deposition Tests........................................................................... 263 6.4 Parchure and Mehta (1985) Dilution Test...................................................................... 264 6.5 Parchure and Mehta (1985) Erosion Test....................................................................... 270 6.6 Sanford and Halka (1993) Field Datasets....................................................................... 271 6.7 Krone (1962) Tagged-Sediment Settling Test................................................................274 SUMMARY AND CONCLUSIONS..........................................................................................320 7.1Summary..........................................................................................................................320 7.2 Conclusions....................................................................................................................324 7.3 Recommendations...........................................................................................................325 DEVELOPMENT OF GOVERNING EQ UATIONS FOR UNSTEADY AND NONUNIFORM SEDIMENT TRANSPORT...................................................................... 327 A.1 Hydrodynamics..............................................................................................................327 A.1.1 Continuity Equation.............................................................................................328 A.1.2 Momentum Equations.......................................................................................... 328 A.2 Sediment Transport Equation........................................................................................329 A.2.1 Continuity Equation for Sediment Laden Fluid.................................................. 332 A.2.2 Continuity Equation with Differen tial Sediment Particle Velocity..................... 333 A.3 Turbulence .....................................................................................................................335 A.3.1 Continuity Equation.............................................................................................337 A 3.1.1 Variable density case with CTD................................................................ 337 A 3.1.2 Homogeneous case with CTD................................................................... 338 A 3.1.3 Variable density case with MTD............................................................... 338 A.3.2 Turbulence Effects on Continuity Equation........................................................339 A.3.2.1 CTD method..............................................................................................340 A.3.2.2 Homogeneous flow....................................................................................341 A.3.2.3 MTD method............................................................................................. 341 A.3.3Viscous Stresses...................................................................................................342 PAGE 9 9 A.3.4Turbulent Interact ion with Viscosity....................................................................343 A.3.4.1 CTD me thod..............................................................................................343 A.3.4.2 Homogeneous case....................................................................................344 A.3.4.3 MTD method............................................................................................. 344 A.3.5 Mean Momentum Equations............................................................................... 344 A.3.5.1 CTD approach...........................................................................................344 A.3.5.2 Homogeneous case....................................................................................345 A.3.5.3 MTD approach..........................................................................................346 A.3.6 Instantaneous Fluctuat ing Momentum Equations............................................... 346 A. 3.6.1 CTD approach..........................................................................................346 A.3.6.2 Homogeneous Case...................................................................................347 A 3.6.3 MTD approach..........................................................................................347 A.3.7 Large Scale Velocity Field Stresses.................................................................... 348 A.3.7.1 CTD method..............................................................................................348 A.3.7.2 MTD Method.............................................................................................349 A.3.8 Mean Velocity Reynolds Stresses.......................................................................349 A.3.8.1 CTD approach...........................................................................................349 A.3.8.2 Homogeneous form...................................................................................351 A.3.8.3 MTD method............................................................................................. 351 A.3.9 Reynolds Stresses................................................................................................ 352 A.3.9.1 CTD method..............................................................................................352 A.3.9.2 Homogeneous flow....................................................................................352 A.3.9.3 MTD method............................................................................................. 352 A.3.10 Forcing Terms in Reynolds Stress Equations.................................................... 353 A.3.10.1 CTD method............................................................................................353 A.3.10.2 Homogeneous flow..................................................................................353 A.3.10.3 MTD method........................................................................................... 353 A.3.11 Reynolds Stresses Revisited.............................................................................. 354 A.3.11.1 CTD method............................................................................................354 A.3.11.2 Homogeneous flow..................................................................................355 A.3.11.3 MTD method........................................................................................... 356 A.4 Turbulent Kinetic Energy.............................................................................................. 357 A.4.1 CTD Method........................................................................................................ 357 A.4.2 Homogeneous Flow.............................................................................................359 A.4.3 MTD Method.......................................................................................................359 A.5 Rate of Turbulent Energy Dissipation........................................................................... 360 A.6 Sediment Transport Equation........................................................................................378 DIMENSIONLESS ANALYSIS................................................................................................. 386 B.1 Continuity Equation.......................................................................................................389 B.2 Momentum Equation...................................................................................................... 390 B.4 Dissipation Equation......................................................................................................392 B.5 Sediment Transport........................................................................................................399 PARTICLE SIZE DISTRIIBUT ION PLOTTING ANALYSIS.................................................401 PAGE 10 10 OUTLINE OF KEY SEDIMENT SUBROUTINES IN COMPUTATIONAL MODEL........... 407 D.1 AGGFLUX....................................................................................................................407 D.2 FALLVEL......................................................................................................................409 D.3 BEDXCHG....................................................................................................................410 LIST OF REFERENCES.............................................................................................................414 BIOGRAPHICAL SKETCH.......................................................................................................428 PAGE 11 11 LIST OF TABLES Table page Table 2-1. Size classification and general cohesive characteristics..............................................50 Table 2-2. Properties of exam ple clay minerals............................................................................ 51 Table 2-3 Comparison of shear strength versus excess density for locales................................... 58 Table 3-1. Variables impor tant to cohesive sedime nt transport processes................................. 132 Table 3-2. Parame ters in the shear st ress distributions shown in Figure 3-3..............................133 Table 3-3. Summ ary of Monte Carlo simulation of the Krone flume deposition experiment shear stress distributions.................................................................................................. 133 Table 3-4. Statistical parameters for example probability distribution curves in Figures 3-8, 3-9 and 3-10.....................................................................................................................133 Table 3-5. Coefficients of Equation 3-12 for fits to data sets in Figure 2-7 and the norm alized deviation between the fit and the data...........................................................134 Table 3-6. Probabilistic treatm en t of significant CST variables................................................. 134 Table 4-1. Standardk model coefficients for high Reynolds number flow........................... 200 Table 4-2. Summary of ex ample flocculation model.................................................................. 200 Table 4-3. Summary of boundary condition specifications. ....................................................... 200 Table 4-4. Simulation conditions for special laminar flow problem s......................................... 200 Table 4-5. Error measures for the S tokes first problem for varying time and num ber of cells.. 201 Table 4-6. Error measures for the Couette pr oblem for varying time and number of cells. ....... 201 Table 4-7. Error measures for the Stokes second problem for varying time and number of cells.......................................................................................................................... ........201 Table 5-1. Summary of Kynch settling column data and graphical analysis (after M ehta, 2007)................................................................................................................................239 Table 5-2. Estimation of the concentra tion profiles based on the intersection of characteristic lines with vertical profiles at specific time s (after Mehta, 2007).............. 239 Table 6-1. Parameters comm on to all simulated tests................................................................. 276 Table 6-2. Parameters used in the Kynch (1952) deposition tests. ............................................. 276 PAGE 12 12 Table 6-3 Parameters used in the Mehta (1973) deposition tests............................................... 277 Table 6-4 Parameters us ed in the Me hta (1973) deposition sensitivity test s.............................. 278 Table 6-5 Parameters used in the Krone (1962) deposition tests................................................ 279 Table 6-6 Parameters used in the Pa rchure and Mehta (1985) dilution tests.............................. 280 Table 6-7 Param eters used in the Pa rchure and Mehta (1985 ) erosion tests..............................281 Table 6-8 Pa rameters used in the Sanf ord and Halka (1993) tidal tests for 1989...................... 282 Table 6-9 Param eters used in the Sanf ord and Halka (1993) tidal tests for 1990...................... 283 Table 6-10 Parameters used in the Sanf ord and Halka (1993) tidal tests f or 1991.................... 284 Table 6-11 Parame ters used in the Sanford and Halka (1993) tidal tests for calibration........... 285 Table A-1. Comparison of continuity equation f orms for me thod of turbulence decomposition.................................................................................................................. 381 Table A-2. Comparison of momentum e quation forms for method of turbulence decomposition .................................................................................................................. 382 Table A-2. Comparison of momentum e quation forms for method of turbulence decomposition (continued)............................................................................................... 383 Table A-2. Comparison of momentum e quation forms for method of turbulence decomposition (concluded) .............................................................................................. 384 Table A-3 Terms in the TKE equation....................................................................................... 385 Table B-1 Coefficients in the di ssipation equation for the standard kmodel.......................... 398 Table C-1. Input data for floc size analysis................................................................................. 401 Table C-2. Example calculations for devel opment of the volumetric concentration................. 404 PAGE 13 13 LIST OF FIGURES Figure page Figure 2-1. Example particle size distributi ons, showing the MIT si ze classification. The flocculated San Francisco Bay sediment ha s a size distribution comparable to beach sand........................................................................................................................... .........92 Figure 2-2. Example of size distributions for dispersed pa rticles and for flocculated sediments in suspension expressed as a volume fraction. The sym bols are the measurements for San Francisco Bay sediment s. The lines are the fits of Equation 23 distributions (adapted from Kranck and Milligan, 1992)............................................... 93 Figure 2-3. Conceptual model of order of a ggregate flocculation processes (adapted from Krone, 1963)......................................................................................................................94 Figure 2-4. Effects of the set tling decay term, K, on the partic le size distribution spectra in volume fraction (adapted from Kranck and Milligan, 1992). ............................................ 95 Figure 2-5. Measured particle size distributions in San Franci sco Bay: a) dispersed grain distributions and b) flocculated distri bution (from Kranck and Milligan, 1992; reprinted with permission)................................................................................................. 96 Figure 2-6. Dispersed particle distributions for nearby bed samples for San Francisco Bay site (solid) and for suspended sedim ents in San Pa blo Strait. All distributions had a slope, m near zero (from Kranck and Millig an, 1992; reprinted with permission).......... 96 Figure 2-7. Relationship of shear strength of flocs to the exce ss density of the flocs (data from Krone, 1963). The data values are from multiple harbors; cu rve fits are shown for each harbor, for all o f the data combined. A curve fit with the exponent on the density of 2.5 (Partheniades, 1993) is shown in red..........................................................97 Figure 2-8. Effect of salinity on settling velocity (adapted from Krone,1962). Final peak floc size was estimated from Stokes Law us ing the final peak settling velocity............... 98 Figure 2-9. Comparison of Equati ons 2-17 and 2-18 for the drag coefficient as a function of the particle Reynolds number............................................................................................ 99 Figure 2-10. Comparison of Equations 2-26 and 2-31 for settling velocity versus floc diam eter (adapted fr om Winterwerp, 1999)..................................................................... 100 Figure 2-11. Settling velocity versus floc diameter from Chesapeake Bay and Tamar (UK) estuary (adapted from Winterwerp, 1999). ...................................................................... 101 Figure 2-12. Settling velocity versus floc diameter from VIS, Ems and Ems (adapted from W interwerp, 1999).................................................................................... 102 PAGE 14 14 Figure 2-13. Comparison of fit curves for individual data sets of fall velocity with a single fit to all data.....................................................................................................................103 Figure 2-14. Fractal dime nsi on from Equation 2-19 used in Equation 2-31 as plotted in Figure 2-10; dfc = 8000 microns, Dfc= 2.6, d5 0 = 2 microns............................................ 104 Figure 2-15. Data comparison of model for sett ling velocity with a power law for the fr actal dimension (from Khelifa and Hill, 2006, reprinted with permission)............................. 105 Figure 2-16. Effects of variab le fractal dimension on the eff ective (excess) density as a function of floc diameter (f rom Khelifa and Hill, 2006). ................................................ 106 Figure 2-17. Variation of settling velocity wi th suspended sediment concentration. Results based on field and laboratory tests using sedim ent from Cleveland Bay, Australia (adapted from Wolanski et al., 1992)............................................................................... 107 Figure 2-18. Schematic diagram showing the de pendence of settling velo city on floc size in the flocculation settling range (adapted from Teeter, 2001)............................................ 108 Figure 2-19. Comparison of Equation 2-46 with settling da ta for varying initial concentration and shear rate (adapted from Teeter, 2001). ............................................. 108 Figure 2-20. Paradox of simultaneous versus exclusive erosion and deposition: a) bed deposition, b) bed erosion................................................................................................ 109 Figure 2-21. Number of hours re quired to settle a distance of one m eter as a function of particle size, based on the Equa tion 2-29 curve in Figure 2-10.......................................110 Figure 2-22. Conceptual mode l of the feedback between a ggregation and disaggregation with flow conditions (from Maggi, 2005; repr inted with permission). Top: shows the temporal variation in the num ber of fl ocs within a unit volume. As aggregation occurs the number of particles is redu ced. With disaggregation the number of particles increases. Bottom: shows the fl oc size distribution (FSD), which reflects the aggregation/disaggregation cycling........................................................................... 111 Figure 2-23. Relationship between the moda l floc diameter and shear stress and concentration (from Dyer 1989)..................................................................................... 112 Figure 3-1. Conceptual vi ew of CST processes.......................................................................... 135 Figure 3-2. Fit of Equation 3-5 to sample data set (data from Obi, et al., 1996)........................ 136 Figure 3-3. Shear stress dist ributions used by Wi nterwerp and van Kesteren (2004) developed fro m analysis of Petit (1999). Parameters for the fits were developed by a two-parameter error minimization algorithm in applying Equation 3-7.......................... 137 PAGE 15 15 Figure 3-4. Effect of normally distributed velocity distribution on the shear stress distribution based on a Monte Carlo simulation of Equation 3-10. The stochastic velocity distribution is c onfirmed against the analyti cal normal distribution..................138 Figure 3-5. Transform ation of a mean 0.113 m/ s normally distributed velocity with a standard deviation of 0.022 m/s to the shear stress distribution of Winterwerp and van Kesteren.....................................................................................................................139 Figure 3-6. Transformation of a mean 0.134 m/ s normally distributed velocity with a standard deviation of 0.0243 m/s to the shear stress distribution of W interwerp and van Kesteren.....................................................................................................................140 Figure 3-7. Transformation of a mean 0.152 m/ s normally distributed velocity with a standard deviation of 0.026 m/s to the shear stress distribution of Winterwerp and van Kesteren.....................................................................................................................141 Figure 3-8. Transformation of 0.5 m/s mean velo city to shear stress for varying standard deviation in the normal velocity distribu tion. The solid symbols are the velocity distributions. The open symbols are the sh ear stress distributions with the sam e symbol as the associated velocity distribution................................................................. 142 Figure 3-9. Transformation of 1.0 m/s mean velo city to shear stress for varying standard deviation in the normal velocity distribu tion. The solid symbols are the velocity distributions. The open symbols are the sh ear stress distributions, with the same symbol as the associated velocity distribution................................................................. 143 Figure 3-10. Transformation of 2.0 m/s mean ve locity to shear stress for varying standard deviation in the normal velocity distribu tion. The solid symbols are the velocity distributions. The open symbols are the sh ear stress distributions, with the same symbol as the associated velocity distribution................................................................. 144 Figure 3-11. Ratio of the standa rd deviation of velocity to th e mean velocity plotted against the ratio of the standard deviation of shear stress to the m ean shear stress.....................145 Figure 3-12. Data from Krone (1963) relating the floc strength to the order of aggregation of sediments from a number of harbors........................................................................... 146 Figure 3-13. Data from Krone (1963) showing the effects of concentr ation on shear strength of flocs from a variety of harbors.................................................................................... 147 Figure 3-14. Variation in shear strength of naturally deposited cohesive bed as a function of the mean shear streng th (data from Ar ulanandan, et al., 1980). The line is a regression fit to the logtransforme d variables................................................................ 148 Figure 3-15. The probability of erosion wh en both the shear stress and the bed shear strength with respect to erosion are repr esented probabilistica lly. The CDF for the shear strength is used in the integral in Equation 3-18 in conjunction with the PDF for the shear stress............................................................................................................149 PAGE 16 16 Figure 3-16. Integration of a probability distribution of shear stress fo r the case of a singlevalued bed shear strength with respect to er osion. The PDF for the shear strength is a delta function and the CDF is a heavyside function..................................................... 150 Figure 3-17. Shear stress standard deviation, b, effect on the probability of erosion for the case of s = 1 Pa and s = 0.25 Pa.................................................................................... 151 Figure 3-18. Shear strength standard deviation, s, effect on the probability of erosion for the case of s = 1 Pa and b /b = 0.30.............................................................................152 Figure 3-19. Effect of the variable s on the probability of er osion for the case of s = 25% and b /b= 0.30................................................................................................................ 153 Figure 3-20. Variation of erosion flux with sh ear stress for data in Long Island Sound (after Wang, 2003). Variation is in part due to differing tidal conditions and wave energy, with the largest erosion associated with storms............................................................... 154 Figure 3-21. The variation of the probability of erosion and deposition with the m ean shear stress for s = 0.5 Pa, s = 25% and b = 30%................................................................. 155 Figure 3-22. Probability threshold greater than 0.5 (0.75) used for definition of critical shear stresses leads to a critical shear stress or erosion g reater than the critical shear stress for deposition...................................................................................................................156 Figure 3-23. Probability threshold less than 0.5 (0.33) used for definition of critical shear stresses leads to a critical shear stress or erosion less than the critical shear stress for deposition. ........................................................................................................................157 Figure 4-1. Comparison of E quations 4-32 and 4-33 for eros ion rate with data from Partheniades (1965), using the coefficient Cb = 0.8459 kg/m3 in Equation 4-32, Cb = 0.112 kg/m3 in Equation 4-33, w=1030 kg/m3, fi =1, and the variables developing the probability of erosion s =0.55 Pa, s=0.25 Pa and b = 0.3 Pa......................................202 Figure 4.2. Collision frequency for a particle di ameter of 10.6 microns with variable second particle diam eter. The flow conditions for this case are a flow depth of 0.3048 m, with a depth-averaged velocity of 0.142 m/ s. The probabilisti c settling velocity cases assumed a 30 percent standard deviation in the settling velocity...........................203 Figure 4-3. Computer progr am COHPROB flow chart (MAIN) for phase 1........................204 Figure 4-4. Phase 2 computer program flow chart for COHPRO B (MAIN); spin up of the hydrodynamic model....................................................................................................... 205 Figure 4-5. Phase 3 (sediment transport) flow chart of COHPROB (M AIN)......................... 206 Figure 4-6. Self-similar velocity distributi on solution for the Stokes first problem .................... 207 Figure 4-7. Results of 80-cell resolution over doma in for simu lation of Stokes first problem. 207 PAGE 17 17 Figure 4-8. Results of 40-cell resolution over domain for simu lation of Stokes first problem. 208 Figure 4-9. Results of 20-cell resolution over domain for simu lation of Stokes first problem. 208 Figure 4-10. Results of 10-cell resolution over domain for simulation of Stokes first problem........................................................................................................................ ....209 Figure 4-11. Nondimensional velocity distri bution for 80 cell simulation of Stokes first problem........................................................................................................................ ....209 Figure 4-12. Nondi mensional velocity distri bution for 40-cell simulation of Stokes first problem........................................................................................................................ ....210 Figure 4-13. Nondi mensional velocity distri bution for 20-cell simulation of Stokes first problem........................................................................................................................ ....210 Figure 4-14. Nondi mensional velocity distri bution for 10-cell simulation of Stokes first problem........................................................................................................................ ....211 Figure 4-15. Effects of s uspended sediment concentration of 20 kg/m3 on Stokes first problem solution. The clear symbols are for no sediment and the blackened symbols are for the sediment-laden case........................................................................................ 212 Figure 4-16. Effects of suspended sediment concentration of 100 kg/m3 on Stokes first problem solution. The clear symbols are for no sediment and the blackened symbols are for a sediment concentration of 100 kg/m3................................................................213 Figure 4-17. Analytical velocity distribution solution of the Couette flow problem for various nondimensional tim e scales, ts............................................................................214 Figure 4-18. Comparison of simulation of C ouette flow problem with 80 cells to the analytical solution............................................................................................................215 Figure 4-19. Com parison of simulation of C ouette flow problem with 40 cells to the analytical solution............................................................................................................216 Figure 4-20. Com parison of simulation of C ouette flow problem with 20 cells to the analytical solution............................................................................................................217 Figure 4-21. Com parison of simulation of C ouette flow problem with 10 cells to the analytical solution............................................................................................................218 Figure 4-22. Effects of 20 kg/m3 suspended sediment concentration on the Couette flow problem. The time scale for the sediment laden flow was computed with the clear water viscosity to show the effects.................................................................................. 219 PAGE 18 18 Figure 4-23. Effects of 100 kg/m3 suspended sediment concen tration on the Couette flow problem. The time scale for the sediment laden flow was computed with the clear water viscosity to show the effects.................................................................................. 220 Figure 4-24. Analytical solution of Stokes s econd problem (Equation 4-91). The dashed red envelopes are the bounding curve for the amplitude of the damped harmonic oscillation. ........................................................................................................................221 Figure 4-25. Simulation with 80 cells of St okes second problem. Symbols are the mo del results and the lines are the analytical solution at the appropriate phases....................... 222 Figure 4-26. Simulation with 40 cells of St okes second problem. Symbols are the mo del results and the lines are the analytical solution at the appropriate phases....................... 223 Figure 4-27. Simulation with 20 cells of St okes second problem. Symbols are the mo del results and the lines are the analytical solution at the appropriate phases....................... 224 Figure 4-28. Simulation with 10 cells of St okes second problem. Symbols are the mo del results and the lines are the analytical solution at the appropriate phases....................... 225 Figure 4-29. Effects of 20 kg/m3 suspended sediment concentration on the results of Stokes second problem for the 40-cell test case. The value of for the sediment laden flow was computed with the clear water viscosity to show the effects.................................... 226 Figure 4-30. Effects of 100 kg/m3 suspended sediment concen tration on the results of Stokes second problem for the 40-cell test case. The value of for the sediment laden flow was computed with the clear water viscosity to show the effects.................. 227 Figure 4-31. Temporal development of velo city profile using von Karman mixing length test case. ..................................................................................................................... ......228 Figure 4-32. Comparison of simulated fully de veloped velocity profile to the analytical solution, using 80 cells over water colum n. Also plotted is the shear stress distribution over the water column from the simulation. ................................................. 229 Figure 5-1. Kynch (1952) settling test development. a) initia l uniform dilute suspension in a quiescent setting column, b) the sec ondary lutocline (S) settles at a rate ws while the isopycnal interface defining the primary lutocline (P) rises from the bed at a rate wp, c) isopycnal primary lutocline meets th e secondary lutocline, and d) the final deposit concentration Cf and height hf are reached after a peri od of hindered settling....240 Figure 5-3. Kynch (1952) settli ng test evolution of the seco ndary lutocline elevation. Application of the method of characteristics to es timate the suspended sediment concentrations (after Mehta, 2007)..................................................................................242 Figure 5-4. Estimated concentration profile s using the Kynch graphical method based on the method of characteristics. ..........................................................................................243 PAGE 19 19 Figure 5-5. Deposition test re sults from Krone (1962)............................................................... 244 Figure 5-6. Results of the tagged sedi ment experim ent of Krone (1962). .................................. 245 Figure 5-7. Results from Meht a (1973) showing the effects of shear stress on the relative concentration. Initial co ncentrations were 1.0 kg/m3......................................................246 Figure 5-8. Results of the flow volume replacement experiment by Parchure (1985)............... 247 Figure 5-10. Suspended sediment concentrati on and shear stress during monitoring exercise on 5 January 1989 (from Sanford and Halka, 1993)........................................................ 249 Figure 5-11. Suspended sedim ent concentrati on and shear stress during monitoring exercise on 2 February 1990 (from Sanford and Halka, 1993)......................................................250 Figure 5-12. Suspended sediment concentrati on and shear stress during monitoring exercise on 15 January 1991 (from Sanford and Halka, 1993)......................................................251 Figure 6-1. Example of fall velocity estimates from video analysis of in situ sediments in San Francisco Bay (dat a from Sm ith, 2007). ................................................................... 286 Figure 6-2. Results of simulation of the Kync h (1952) test case using a hindered settling exponent of m =1 in Equation 2-40..................................................................................287 Figure 6-3. Results of simulation of the Kync h (1952) test case using a hindered settling exponent of m =2 in Equation 2-40. .................................................................................288 Figure 6-5. Example representa tion of shear strength of floc s, critical shear stress for erosion and critical shear stress for depositi on as functions of pa rticle size. Shear strength defined by Equation 3-11, with Bf = 1200 Pa and Df = 2.6. Critical shear stresses for erosion and deposition are defined by Equation 2-53. The critical shear stress for deposition is based on d0 = 0.01 Pa, dref =0.1 microns and = 0.5................. 290 Figure 6-7. Specification development for Mehta (1973) for use of the mean values in the classical erosion and depos ition exclusive paradigm....................................................... 292 Figure 6-9. Simulations of Mehta (1973) experi mental tests for shear stresses of 0.25 Pa, 0.40 Pa, 0.60 Pa and 0.85 Pa, and using clas sical erosion probabilities with sim ultaneous deposition................................................................................................... 294 Figure 6-11. Simulations of Mehta (1973) test cases for shear stress of 0.40 Pa using combinations of probabilistic versus mean-valued depositional and erosion treatment, with either simultaneous or exclusive erosion/deposition.............................. 296 Figure 6-12. Schematic representation of a probabilistic representation for the Mehta 1973 0.25 Pa test. The shaded area is the zone of deposition from the average value analysis. The differences are conceptual on ly since the displayed range of values is only +/1 standard deviation for each variable (ce, cd, b).......................................... 297 PAGE 20 20 Figure 6-13. Particle size distribution comp arison between the mean value simulation and the probabilistic simulation for the 0.25 Pa Me hta (1973) test. Both simulations used the same sediment characteristics and critical shear stresses and the exclusive paradigm....................................................................................................................... ...298 Figure 6-14 Simulation using the classical ex clusive bed exchange processes using mean values for the Krone (1962) deposition te st with a shear stress of 0.0305 Pa. One sensitivity simulation was made with an added supplem ental internal shear of 100 Hz.....................................................................................................................................299 Figure 6-15. Simulation using the classical be d exchange processes using mean values for the Krone (1962) deposition test with a shear stress of 0.0415 Pa..................................300 Figure 6-16. Sim ulation using the classical be d exchange processes using mean values for the Krone (1962) deposition test with a shear stress of 0.0515 Pa..................................301 Figure 6-17. Illustration of the change in depositional response in the Krone (1962) recirculating flume tests when concentr ations fall below approximately 0.3 kg/m3.......302 Figure 6-18. Model simulation to test the d ilution rate for the Pa rchure and Mehta (1985) dilution test.......................................................................................................................303 Figure 6-19. Initial particle size concentra tion distribu tion for bed initialization for the Parchure and Mehta (1985) dilution experiment. The si mulation started with no sediment in suspension and then eroded th e bed to an equilibrium concentration.......... 304 Figure 6-20. Parchure and Meht a (1985) dilution test results with bed exchange included, with the classical excess shear stres s exclusive formulation and an exclusive simulation using probabilistic trea tment of the key parameters...................................... 305 Figure 6-21. Variation of floc size dist ribution during the Parc hure and Mehta (1985) dilution test.......................................................................................................................306 Figure 6-22. Initial cohesive bed particle concentration fo r the Parchure and Mehta (1985) erosion test................................................................................................................... ....307 Figure 6-23. Results of simulation of an er osion test (Parchure a nd Mehta, 1985) with a progressive increase in shear stress using the exclusive erosion/deposition and me an values...............................................................................................................................308 Figure 6-24. Effects of switching from cla ssically exclusive mean-value calibrated bed exchange to an exclusive/probabilistic treatm ent without parameter adjustments.......... 309 Figure 6-25. Variation in water depth during Sanford and Halka (1993) field test.................... 310 Figure 6-26. Results of calibrating continuous deposition for use on an average shear stress and using the probabilistic model of bed exchange: 5 January 1989.............................. 311 PAGE 21 21 Figure 6-27. Results of calibrating continuous deposition for use on an average shear stress and using the probabilistic model of bed exchange: 2 February 1990............................ 312 Figure 6-28. Results of calibrating continuous deposition for use on an average shear stress and using the probabilistic model of bed exchange: 15 January 1991. ........................... 313 Figure 6-29. Evolution of floc size distribution during num eric al simulation of the Sanford and Halka (1993) data set of 5 January 1989. Both tests used the continuous deposition bed model. The black distributi ons are for the use of the average bottom shear stress, while the red curves are for the probabilistic shear stress formulation. (hours refer to Figure 6-26) .............................................................................................. 314 Figure 6-30. Evolution of floc size distribution during numeric al simulation of the Sanford and Halka (1993) data set of 5 January 1989. Both tests used the exclusive erosiondeposition bed model. The black distributi ons are for the use of the average bottom shear stress, while the red curves are for the probabilistic shear stress formulation. (hours refer to Figure 6-26).............................................................................................. 315 Figure 6-31. Results of simulations showing the effects of the c ontinuous deposition and probabilistic bed exchange treatment 5 Janua ry 1989. All other model variables are held the same.................................................................................................................. ..316 Figure 6-32. Results of simulations showing the effects of the c ontinuous deposition and probabilistic bed exchange treatm ent 2 Febr uary 1990. All other model variables are held the same.................................................................................................................. ..317 Figure 6-33. Results of simulations showing the effects of the c ontinuous deposition and probabilistic bed exchange treatm ent 15 Ja nuary 1991. All other model variables are held the same.................................................................................................................. ..318 Figure 6-34. Results of simulation of the Krone gold-tagged sedim ent experiment using the sim ultaneous erosion and deposition tr eatment of the mean variables............................ 319 Figure D-1. Particle si ze distribution for exam ple video analysis.............................................. 406 PAGE 22 22 PAGE 23 23 NOTATION af = coefficient in settling velocity equation (m/s) an = scale factor for exponent in the hindered settling equation (-) A = coefficient on diameter squared in Stokes settling law (m-1s-1) A = a generic variable or constant (varies) bf = hindered settling coefficient in settling velocity equation (kg/m3) bn = exponent in the equation for exponent in the hindered settling equation (-) Bf = floc strength scale factor (Pa) ci = concentration of indexed cohesive sediment class i (kg/m3) csi = concentration of indexed silt sediment class i (kg/m3) c1 = coefficient on production term in turbulent dissipation equation (-) c2 = coefficient on dissipation term in turbulent dissipation equation (-) c3 = coefficient on buoyancy term in turbulent dissipation equation (-) c = scaling coefficient on relationship of eddy viscosity to turbulence parameters (-) C = total sediment concentration, mass per unit volume (kg/m3, mg/l) CD = coefficient of drag (-) CL = concentration of sediment below the lutocline (kg/m3) Co = initial or referen ce concentration (kg/m3) d = sediment or particle diameter (m, or m ) df = floc diameter (m, or m ) dfi = floc diameter for class i (m, or m ) dfc = reference floc diameter (m, or m ) di = indexed diameter (m, or m ) PAGE 24 24 dp = primary grain diameter (m, or m ) d50 = median grain diameter (m, or m ) D = general collection term for dissipative te rms in a generic balance equation (varies) D = used as indicator of total derivative (-) D = depositional flux (kg m-2s-1) DB = diffusion associated with Brownian motion (m2/s) Df = fractal dimension (-) Dfc = reference fractal dimension (-) Di = dispersion coefficien t in tensor notation (m2/s) Di = depositional flux for sediment class i (kg m-2s-1) Dm = molecular diffusivity (m2/s) Dt = turbulent diffusivity (m2/s) Dti = sediment class i dependent turbulent dispersion coefficient (m2/s) Dx = dispersion coefficient in x-direction (m2/s) Dy = dispersion coefficient in y-direction (m2/s) Dz = dispersion coefficient in z-direction (m2/s) e = exponential constant = 2.718281828 (-) E = expected value of any variable (varies) E = erosion flux (kg m-2s-1) Ei = erosion flux for class i (kg m-2s-1) Ef = floc erosion flux (kg m-2s-1) Ev = energy required to disperse unit volume of aggregate (N-m, joules) fA = probability density function of variable A (units of A-1) PAGE 25 25 fc = friction coefficient (-) f= damping factor for rate of turbulent kinetic energy dissipation near bottom (-) f= damping factor for turbulent viscosity near bottom (-) FA = cumulative probability function of the variable A (-) FC = combined buoyancy and cohesive forces in mobility analysis (N) FC = collision diameter function (-) Fi = applied forces per unit volume on the fluid parcel (N/m3) Fi = depositional flux for sediment class i (kg/m-s) vis iF= applied viscous forces on the fluid parcel (Pa) FL = hydrodynamic lift (N) FP = coefficient of relative depth of penetration during particle collision (-) Fs = force of shear (Pa) Fs = local settling flux (kg m-2s-1) g = acceleration of gravity (m/s2) gi = acceleration of gravity tensor (m/s2) G = internal shear (Hz) GiA = gain of flocs to class i due to aggregation (-) GiB = gain of flocs to class i due to shear breaking of flocs (-) GiC = gain of flocs to class i due to collision breaking of flocs (-) h = local water depth (m) hL = height of lutocline (m) H = Heaviside function (-) K = decay coefficient for settling in partial size distribution model (s/m) PAGE 26 26 K = relational coefficient for floc shear strength (varies) Kf = relational coefficient for shear strength as a function of excess density (varies) k = counting index (-) k = turbulent kinetic energy (m2/s2) kb = Boltznmanns constant (ergs/oK) kf = correlation constant for computing shear strength based on concentration (Pa kg-5/2m15/2) ks = roughness height (m) LiA = loss of flocs from class i due to aggregation (-) LiB = loss of flocs from class i due to shear breaking of flocs (-) LiC = loss of flocs from class i due to collision breaking of flocs (-) m = exponent in particle size distribution model (-) m = modal factor in the shear st ress distribution model (Pa) mEq = milliequivalents mf = hindered settling exponent (-) mi = mass of particles in class i (kg) mi =i-th moment of the sh ear stress distribution (mi) m1 = number of primary grains in primary floc (-) m2 = size increase on flocculation order increase (-) M = erosion rate constant (kg/m2/s) Mclass = number of cohesive size classes (-) Mi = i-th moment of floc distribution (m3) ML = mass of suspended sediment below lutocline (kg/m2) n = fluid porosity, volume of fluid per unit volume (-) n = aggregation number for higher fl oc, order of aggregate = n-1 (-) PAGE 27 27 ni = number concentration of particles of class i (m-3) nf = exponent for concentration e ffects on flocculation settling (-) N = number of aggregate bonds (-) Nclass = number of noncohesive size classes (-) Nf = number of primary grains in a floc (-) Nij = number of new flocs created by aggregation (-) Nz = number of cells in the vertical (-) p = pressure, force per unit area (Pa) P = probability (-) P = general collection term for production terms in a ge neric balance equation (varies) Pd = probability of deposition (-) Pe = probability of erosion (-) q = extraction rate (m3/s/m2) Q = concentration scale for pa rticle size distribution (ppm) R-squared = sum of regression squared residuals (-) Rews = particle size Reynolds number (-) Ret = turbulence Reynolds number (-) Rij = Reynolds stresses (m2/s2) Rijk = third order moment Reynolds stresses (m3/s3) s = skewness factor in the shear stress distribution function (Pa) s = specific gravity of sediment particles (-) S = salinity of the fluid, pr actical salinity units (psu) S = slope of ener gy grade line (-) PAGE 28 28 Si = net source of sediment into size class i (kg/m3/s) Sij = deviator stress tensor (Pa) S0 = Reference salinity (psu) t = time (s) tij = kinematic Reynolds mean stress tensor (m2/s2) ts = dimensionless time scale (-) t = hydrodynamic mean flow time scale (s) T = temperature, degrees Celsius unless otherwise specified (oC) TL = turbulence eddy timescale (s) TP = particle response to turbulence timescale (s) Tss = process time scale (s) u, v, w = Cartesian components of water velocity in x, y, z directions, respectively (m/s) ui = Cartesian components of water velocity using tensor notation (m/s) uip = effective sediment velocity for total se diment concentration for tensor direction i (m/s) uid = differential sediment velocity relative to fl uid velocity for total sediment concentration for tensor direction i (m/s) umi = sediment particle velocity for se diment class m for tensor direction i (m/s) *u = shear velocity (m/s) Vi = volume of particle in class i (m3 or m3) wp = rate of rise of primary lutocline (m/s) ws = sediment particle fall velocity (m/s) wseff = effective settling velocity (m/s) wsm = peak flocculation settling velocity at the beginning of hindered settling (m/s) PAGE 29 29 x,y,z = Cartesian coordinates, z vertical (m) xi = Cartesian coordinates in tensor notation (m) y = probability density distri bution of shear stress (Pa-1) Greek Symbols = generic scaling coefficient (varies) = scale factor for fractal dimension (-) a = aggregation efficiency (-) c = efficiency of collision-induced floc breakage (-) d = disaggregation efficiency (-) t = time scale scaling factor (-) 1 = coefficient in flux coupling function (-) 1 = mass factor coefficient (kg/m3) 2 = flux limiting concentration in flux coupling function (kg/m3) 2 = area shape factor = class size progression factor (-) = fractal dimension exponent (-) = drag force asymmetry factor (-) B = collision frequency due to Brownian motion (-) D = collision frequency due to differential settling (-) d = class size progression factor for particle diameter (-) m = class size progression fact or for particle mass (-) = collision frequency due to shear (-) PAGE 30 30 = natural logarithm of the Reynolds number, ln(Re) (-) = boundary layer thickness (m) = exponent in critical shear stress for deposition relationship (-) ij = Kronecker delta, equals 1 when i = j; zero otherwise (-) 1 = scaling coefficient for depositional flux coupling (-) 2 = bounding concentration control for depositional flux coupling (kg/m3) f = excess floc density (kg/m3) s = excess sediment grain density (kg/m3) t = time step of numerical simulation (s) z = computational discretization cell depth (m) = dissipation rate of turbulent kinetic energy (m2/s3) = erosion flux (kg/m2/s) i = voids ratio of floc (-) = kinematic bulk viscosity of the fluid (m2/s) = dimensionless distance (-) = generic vari able (varies) = dynamic bulk viscosity of the fluid (kg/m/s2) = von Karman coefficient (-) = Taylor microscale (m) i = aggregation mass di stribution factor (-) 0 = Kolmogorov eddy length scale (m) 1 = coefficient of flocculation effect s of shear on settling velocity (s) PAGE 31 31 2 = coefficient of disaggregation eff ects of shear on se ttling velocity (s2) 3 = decay coefficient of shear effects on sett ling velocity with increasing concentration (-) = dynamic shear viscosity of the fluid (kg/m/s2) = mean shear stress in shear stress distribution function (Pa) = kinematic fluid viscosity (m2/s) T = kinematic turbulent fluid viscosity (m2/s) = representation of generic turbulent variable = ratio of circular circum ference to its diameter (-) = fluid composite density, mass per unit volume (kg/m3) f = floc density, mass per unit volume (kg/m3) s = density of sediment particles or flocs, mass per unit volume (kg/m3) w = clear fluid density, mass per unit volume (kg/m3) = normalized autocorrelation of th e turbulent velocity perturbation (-) b = standard deviation of the bottom shear (Pa) c = correlation scale between turbulent viscosity and concentration diffusivity (-) ce = standard deviation of the crit ical shear stress for erosion (Pa) cd = standard deviation of the critical shear stress for deposition (Pa) k, = standard deviations of turbulence variables (m2/s2, m2/s3) ij = viscous stress tensor, force per unit area (Pa) L = standard deviation of the lift force on a particle (N) T = correlation scale between turbulent viscosity and concentration diffusivity (-) PAGE 32 32 v = standard deviation of the velocity (m/s) = correlation scale between turbulent viscos ity and diffusivity of dissipation rate (-) = standard deviation of the shear stress (Pa) = shear stress, force per unit area (Pa) b = bottom shear stress, force per unit area (Pa) d = critical shear stress for deposition, force per unit area (Pa) 0 d = reference critical shear stress for deposition (Pa) e = critical shear stress for er osion, force per unit area (Pa) f = shear strength of flocs, force per unit area (Pa) ij = shear stress in the i-normal f ace acting in the j direction, (Pa) ijk = shear stress in the i-normal f ace acting in the j direction (Pa) y = yield stress of flocs (Pa) = volumetric concentration (-) f = volumetric concentration of floc (-) s = volumetric concentration of sediment grains (-) p = volumetric concentration of primary aggregates (-) pna = volumetric concentration of first order flocs (-) = space filling volumetric concentration (-) = floc density factor for nonuniform primary particles (-) =angular frequency (s-1) =dummy of inte gration (varies) PAGE 33 33 Subscripting i, j, k, or m = Cartesian directi on tensor indices (-) i, j, k = particle size class indices (-) k = spatial discreti zation reference (-) = representation of generic turbulent variable = temporal average of generic variable = mass-weighted temporal average of generic variable = turbulent perturbation from th e conventional temporal average = turbulent perturbation from th e mass-weighted temporal average = non-dimensional generic variable ACRONYMS ASM algebraic second-moment closure model CCA cluster-cluster aggregation CEC cation exchange capacity CST cohesive sediment transport DLA diffusion-limited aggregation DLCCA diffusion limited cluster-cluster aggregation DSM differential second-moment closure model EVM eddy viscosity/diffusivity model FSD floc size distribution MIT Massachusetts Institute of Technology RANS Reynolds-averaged Navier-Stokes RLCCA reaction limited cluster-cluster aggregation PAGE 34 34 SAR sodium adsorption ration SMC second-moment closure model TKE turbulent kinetic energy USACE United States Army Corps of Engineers PAGE 35 35 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SIGNIFICANCE OF PROBABILISTIC PARAMETERIZATION IN COHESIVE SEDIMENT BED EXCHANGE By Joseph Vincent Letter, Jr. August 2009 Chair: Ashish Jayant Mehta Major: Civil and Oceanographic Engineering The primary issue addressed in this study is whether the decades old paradigm of exclusive erosion or deposition in turbulent fl ow has legitimacy based on physical principles within cohesive sediment dynamics. The exclusive paradigm assumes that sediment exchange condition at the bed-water interface is either eros ion, deposition or neither, but never both. In contrast, the more recently espoused simultaneous exchange paradigm admits the possibility of erosion and deposition of cohesive sediment occurring at the same time. The exclusive paradigm is, in part, the result of early attempts to understand basic cohesive sediment transport behavior based on inferred da ta in laboratory apparatuses such as flumes averaged over time and space. The time scale of averaging is longer than the time scale of turbulence and the spatial dimension is scaled by water depth in the apparatus. Bed sediment exchange has been deduced primarily from the increase or reduction in the suspended sediment concentration within the water column, rather th en from difficult to record observations of particle movement very close to the bed surface. The net result of averaging will be positive, negative or zero sediment flux at the bed surface, but not both positive and negative. PAGE 36 36 With the inclusion of greater details in newer mathematical models, such as particle size distributions and flocculation sub-models, the be d exchange algorithms have required revision. Numerical modelers have found the need to use the simultaneous approach to replicate observed sedimentation rates in th e field environment. The numerical sediment transport tool develope d for this research has been shown to be capable of simulating several processes critical for simulation of bed exchange. These processes include aggregation and disaggregation dynamics stochastic effects in bed exchange and aggregation/disaggregation, hinde red settling, attainment of a depositional or erosional equilibrium concentration for a fixed shear stre ss, and floc spectrum features documented by field experimentation. Observations made during development and application of the numerical tool are: The effects of a probabilistic treatment of the key variables are more pronounced for erosion than for deposition. These variables include current velocit y, bottom shear stress, floc shear strength, critical shear stresses fo r erosion and deposition, internal shear and settling velocity. Probabilistic effects are amplified through the flocculation model ove r the effects that occur through bed exchange alone. For a given shear stress the flocculation model will tend toward an equilibrium distribution of particle sizes. The probabilistic treatment result s in a broader floc distributi on spectrum than occurs with use of mean-valued variables. Deposition or erosion will be initiated sooner and transition from one to the other more gradual in response to changing shear stress when a probabilistic treatment is used compared to a mean-valued treatment. The differential timing will be a function of the standard deviations of the proba bilistic variables and the rate of change of the shear stress. The use of the exclusive paradigm with a floc size distribution can perform as well as a simultaneous treatment with a single particle size. A simulation was performed of a flume test by Parchure and Mehta (1985) designed to evaluate the exclusive versus simultaneous paradigm by diluting the concentration of a flume suspension that had achieved an equilibr ium concentration from bed erosion. If the PAGE 37 37 exclusive paradigm was valid, the concentration at the end of dilution should remain constant. If the concentration began to rise after dilution was ceased, then the simultaneous paradigm would be an explanatio n. The flume concentration did rise after the dilution stopped, but at a very low rate of erosion. The numerical model was able to replicate the flume behavior with the correct ra te of rise after the end of dilution by using the exclusive paradigm with a probab ilistic treatment of the variables. The appropriate use of either the exclusive or continuous paradigm appears to be dictated by the level of temporal and spatial averaging used in the development of empirical data and in the formulation of th e variables in the analysis. Empirical coefficients developed for mean-val ued analysis may require adjustment when used in a probabilistic treatment. PAGE 38 38 CHAPTER 1 INTRODUCTION 1.1 Need for Research The state of knowledge in cohesive sediment transport (CST) is the result of an evolving philosophical approach to CST research. The pr imary motivation for research has been driven by the need to better evaluate impacts of cohesi ve sediments on maintenance of estuarine harbors and on sensitive environmental areas (US Ar my Engineer Committee on Tidal Hydraulics, 1963). Research needs have expanded in the last several decades to include cohesive sediments as a valuable resource in coastal restoration and erosion control. Coasta l Louisiana is nearing a crisis over accelerating land loss a ssociated with global eustatic s ea level rise (Day, et. al., 1995), aggravated by local subsidence of land mass due to subsurface sediment compaction, tectonic down-warping and oil and gas withdrawals (G agliano, 1981; Boesch, et. al., 1994; Day, et. al., 2000). Management of sediment supplies of th e Mississippi and Atchafal aya Rivers has been identified as critical to mitigating la nd loss (DeLaune, et. al., 1992; USACE, 2004). Estuarine sedimentation is complex on a numbe r of levels (Dyer, 1989). Estuaries are characterized by significant variab ility in sediment supply and local transport and deposition. A typical estuary may have a wide range of sediment types, from coarse sand to clay, depending on the local sediment supply and the hydrodynamic e nvironment. These sediment mixing zones vary depending on the temporal variability of hydrodynamics, and therefore in sediment supply. As river discharge and tidal c onditions vary, the character of the local sediment supply and transport processes may vary significantly. A rive r delta is a vivid example of fluctuating spatial deposition characteristics between clays at low flows that depo sit a broad expanse of shallow pro-delta clays to sands at high river discharges resulting in the formation of deltaic lobes PAGE 39 39 (Roberts, 1997). Strong variation in tide range during the spri ng neap monthly cycle may result in substantial variation in sediment mob ility, dominating the morphology of the estuary. Cohesive sediment processes often dominate estuarine sedimentat ion. Much of the research in sediment transport for non-cohesive sediment has focused on transport under steady flows, attempting to develop equilibrium transpor t relationships for each flow condition. Much of estuarine cohesive research has focused on the rates of erosion or deposition for a given flow condition (Krone, 1962; Partheniades 1962) rather than equilibrium conditions, which are elusive for cohesive transport. Most cohesive sedime nt research has also been under uniform steady flow conditions. Research has attempted to define macroscale hydrodynamic conditions, characterized by current velocity and bottom sh ear stresses under which erosion and deposition will occur. This paradigm of seeking conditions of either erosion or deposition has had a strong influence on the conceptual model of cohesive be d exchange with suspended sediments. As long as the mathematical treatment of the processe s remained macroscale in spatial and temporal discretization, for example focusing on the tota l suspended concentration, the macroscale incorporation of a mean settling velocity is c onsistent with a bulk treatment of erosion and deposition. Within steady uniform flume test conditions these bulk representations of bed exchange fit the macroscale variables well and reinforce the conceptual model of exclusive erosion and deposition. As treatment of the cohesive sedimentation has evolved to the discreti zation of the particle size distribution and the associ ated variation in the settling velocities within complex flocculation models, the paradigm of exclusive erosion or depositi on has been challenged. If the differential response of varying particle and fl oc sizes to hydrodynamics within a flocculation PAGE 40 40 model is to be consistently formulated with the bottom bed exchange, then it may require the admissibility of simultaneous erosion and deposi tion. This need is further stressed by the differential response of size classes to temporal and spatial variations in hydrodynamics in estuaries. The general processes that need to be addressed further in order to better deal with estuarine cohesive sediment transport are: Sediment aggregation (floc growth) Sediment disaggregation (floc breakup) Sediment settling velocity Deposited sediment consolidation Fluid mud formation Sediment resuspension Shear strength of settled bed The growth and breakup of flocs directly impact settling velocities through changes in both size and density of the flocs. This extends to the bed consolidation rate by affecting the initial structure of the bed and the rate of dewa tering. Resuspension of sediment from an unconsolidated bed is partially due to breakup of flocs in the bed as well as simple reentrainment. The application of modest shear to a partially consolidat ed bed can strengthen the bed by shifting the flocs in a manner that reinforces inter-floc bonds within the bed. When the further complexities of unstea dy and non-uniform flows are added these additional processes may need to be addressed: Sediment sorting due to spatia l variability in hydrodynamics Differential mobility of various sediment classes PAGE 41 41 Fluid mud flows Wave-induced effects Modifications to each of the above processes due to temporal and spatial gradients in both hydrodynamics and sediment concentration Field observations have provided the motivation for historic al research. Each field data set, however, is constrained in its usefulness to the processes that were monitored and the accuracy of those measurements. The complexity of environmental conditions experienced in field monitoring programs makes comparis ons between data sets highly complex. Laboratory testing can simplify data set comparisons by controlling the conditions of the tests. Laboratory data are also constrained by the accuracy of measurements and by the processes that were monitored. Extremely accurate data sets monitored historically will not, however, be useful for evaluating process interactions if one of the critic al processes was not monitored. The basic connections between the real world and the abilit y to scientifically study CST processes are the conceptual models that have be en developed for each of the processes. These models are typically cast in mathematical term s developed from first principles and basic analyses of the monitored data, from both the field and laboratory. The conceptual models are then useful in evaluating engineering aspects of CST and in the design of new field or laboratory testing. Early numerical models of c ohesive sediment transport were relatively simplistic in conceptual framework of CST processes. These simplifications were in part the result of limitations placed on these models of the temporal and spatial scales that could reasonably be addressed and efficiently incorporated into num erical models. For the last few decades a combination of increased computational power and sophistication in co mputational techniques PAGE 42 42 has lead to fewer simplifications in the formul ation of estuarine CST models (conceptual and numerical). Each of the proce sses can now be represented in a more descriptive and interactive manner at smaller spatial and temporal scales (McAnally and Mehta, 2001). These more complex numerical models now play a significant role in guiding the direction of new research. New research is typically aimed at improving specific process conceptual models. The numerical models can be tested to identify the sensitivity of the model primary results to each of the component process descriptions. The trad e-offs between model sensitivity and critical knowledge gaps can then be used to optimize additional research. If a particular process can only be estimated within a given ac curacy, but the sensitiv ity of other primary processes is small, then improving that secondary process may by given a lower priority. Toorman (1993) refers to this class of numerical models as virtual laboratories. Even with strides in computational capabiliti es, engineering applica tions of CST numerical models must still utilize paramete rizations of the basic processes to a fair degree. When regional spatial scales and seasonal to annual hydrol ogical time scales are required to evaluate morphodynamic issues, the CST processes must be bulked to more manageable relationships. Within classical steady-state flume experime nts, the macroscale bulking paradigm dealt with the mean flow conditions, averaging out turbulence. Erosion and deposition rates were often estimated by the change in the suspended se diment concentration, which was essentially an averaging technique. The measurement of turbul ent fluctuations in velocity and bottom shear stresses within a flume show that erosion or de position with time scales less than the turbulent frequency may both occur for nominally steady-state conditions. Measurements are generally incomplete to full y define all processes for a given data set, whether field or laboratory data. This study inte nds to make use of existing data sets that PAGE 43 43 incorporate monitoring of as ma ny of the processes as reasonab le and limited additional field observations in a highly complex estuarine setting. The earliest conceptual models for cohesive se diment erosion and/or deposition have been simplistic, but have served th e science well (K rone, 1962; Ariathurai, 1974). These involve threshold values for both erosion and deposition based on the bulk properties of the sediment; a shear stress threshold for deposition above which deposition will not occur, and a critical shear stress for erosion based on the bul k strength of the sediment in the bed, below which erosion will not occur. The fact that these bulk thresholds naturally tend to separate, with the deposition threshold less than the erosiona l threshold, supports the simple conceptual model and has been used by many to argue that erosion and deposition should not occur simultaneously (Partheniades, 1971). This inference is arguabl y valid when discussing the processes using bulk sediment properties, particularly when compared against the time-averaged bottom shear stress. However, there has been a recurri ng debate on the issue that stem s from a basic intuitive sense that the real world is just not that simple (Sanford and Halk a, 1993; Lau and Krishnappan, 1994; Winterwerp, 2007) The exclusive process (erosion or depos ition) view evolved partly from the use of these bulk sediment parameters and mean shear stresses derived from mean velocity values. Now that most research sediment transport models for CST incorporate multiple floc sizes, many of these bulking assumptions are no longer seen as viable (Winterwerp, 2007). Based on the limitations and research needs de scribed above, the following objectives and tasks were defined. 1.2 Objectives and Tasks This research investigates the importance of stochastic properties to cohesive sediment transport processes, including aggregation and disaggregation, settling, deposition, erosion and turbulent mixing. The incorpora tion of a probabilistic treatment of the primary variables is PAGE 44 44 targeted to study the influence of turbulence and heterogeneity of sediment properties on bed exchange. The objectives of this research are to: Develop and test probabilisti c representations of selected hydrodynamic variables and sediment properties. Evaluate their significance to aggregation and disaggrega tion processes along with bed interaction. Investigate their sensitivity and interactions. The primary tasks performed to achieve these objectives are: Identify significant variables for incorpor ation into a probabili ty function, based on a thorough review of the literature. Develop a conceptual model of cohesive sedi ment transport processe s that incorporates probabilistic repr esentations of the variables. Define the applicability of the new conceptu al model by numerical testing against field and laboratory data with sensitivity testing to identify the most critical variables for full probabilistic sp ecification. Determine the appropriate means and value of parameterization of the new conceptual models of aggregation and disaggregation into engineering scale numerical models. Assess the needs for future research. 1.3 Approach To accomplish the objectives, a careful and thorough examination was carried out of the governing equations for flow and sediment tran sport. Terms included in the equations are interaction between turbulence closure in hydr odynamics and sediment concentration, and aggregation processes in a multiple particle-size-class CST model. A literature review was conducted to define the current st ate of the science in cohesive sedimentation. The current state of the art in discrete particle aggregation modeling was modified to incorporate additional effects arising from interaction of probabilistic terms. The first step was to evaluate the importance of PAGE 45 45 differential particle response to a probabilistic parameterization on sediment aggregation and disaggregation. Once aggregation has proceeded a nd increased concentrations evolve near the bottom, the influence of sediment concentration on damping vertical mixi ng could be addressed. A bed exchange formulation was developed based on integration of joint probability functions weighting the erosion threshold. Research was performed to compile field and laboratory data appropriate for testing the revised algorithms. In general, the data of interest for this re search include particle and floc size distributions, settlin g velocity distribution by floc size, total suspended sediment concentration, turbulent velocity measurements, wate r depth, shear strength of sediment deposits, erosion rate estimates, salinity of the fluid, organic content and bul k densities of the bed material. 1.3.1 Examination of Governing Equations The governing equations for the hydrodynamics are the Navier-Stokes equations. The turbulence closure for the momentum equations when time averaging is required was handled in two ways: use of Boussinesq approximation of the Reynolds stresses and the turbulent kinetic energy (TKE) transport equation () with a damping equation for the TKE (). The sediment transport equation is the advective-diffusion e quation with turbulent mixing and source/sink terms due to deposition and associated with a ggregation processes. These equations were examined using revised interaction terms inco rporating probabilistic variable interactions. 1.3.2 Revision of Aggregation/Disaggregatio n Model for Probabilistic Fluctuations The effects of the probabilistic representation of specific variables of importance to the flocculation model of aggregation and disa ggregation were incor porated and tested. A sediment particle suspended within a tu rbulent hydrodynamic flow field will be buffeted around by turbulent fluctuations in the velocities in three dime nsions. The response of the particle to these fluctuations will depend on th e frequency and magnitude of the turbulence, the PAGE 46 46 size and density of the sediment aggregates and the viscosity of the fluid. When the sediment suspension is made up of a variety of aggregate sizes and densities, the responses will vary and the potential exists for differential turbulent particle responses that can enhance particle collisions. 1.3.3 Effects of Suspended Sediment on Turbulence Particle collisions increase with sediment c oncentration and can eventually reach the point where they significantly extract momentum from th e turbulence. This effect combined with the feedback effects of flow blockage, similar to hi ndered settling, can signif icantly dampen vertical mixing due to the turbulence. These effects are handled by the incorporation of a two-parameter turbulence closure model. 1.3.4 Development of a Probabilistic Bed-Exchange Model The conventional bed exchange model was m odified to incorpor ate the effects of probabilistic variables on the erosion and deposition. The sensitivity of the probabilistic formulation in conjunction with the discretized particle size di stribution was evaluated using both the exclusive and simultaneous conceptual models of erosion/deposition. 1.3.5 Application of the Probabilistic Fo rmulation to Selected Test Cases The probabilistic model was applied to select te st cases in an attemp t to provide insight into when the use of probabilistic paramete rization of certain variables provides added performance. 1.4 Scope The goal of the research described here will be to attempt development of improvements in the conceptual models of cohesive sediment aggr egation and bed exchange processes as typically encountered within estuarine waters. The de velopment of numerical modeling tools will be performed simply to provide a mechanism to test the revised algorithms. PAGE 47 47 1.5 Presentation Outline The research is reported in the following presentation outline. Chapter 1 has presented the introduction includ ing motivation, research needs, objectives of the present research, approach and the scope of present research. Chapter 2 presents an overview of the current state of CST processe s in the context of bot h steady uniform flow assumptions and discusses the specific constraint s imposed by those assumptions, as well as the limited knowledge available for complex hydrodynamic situations. Chapter 3 presents the development of a probabilistic description for sediment properties and hydrodynamics for CST in fully generalized unsteady nonuniform hydrodyna mics along with the mathematical approach for incorporating the effects. Chapter 4 pres ents the implementation of the probabilistic treatment and details of the numerical model. The effects of new terms on multi-class deposition with aggregation are presented in Chapter 4. Chap ter 5 details field and laboratory test cases that have been compiled for evaluation. Applications of the new methodology to the test cases are presented in Chapter 6. Finall y, the conclusions and recommendatio ns of further research are discussed in Chapter 7. PAGE 48 48 CHAPTER 2 COHESIVE SEDIMENT TRANSPORT 2.1 Estuarine Cohesive Sediment Properties The general properties of estuar ine sediments will be discusse d to a level appropriate for the research topic presented. 2.1.1 Estuarine Sediment Cohesive sediments in estuarine waters are unique in their ability to morph from single mineral particles to flocs in appropriate flow conditions. These flocs have a distribution of sizes, from the smallest tightly packed zero-order units to the largest units limited by the turbulent flow shear. When torn apart by higher shear or collis ions, the larger flocs may morph back to their zero-order constituents. If a suitable chemical dispersing agent is now added and the suspension is centrifuged, the zero-ord er flocs will revert to the primary particle state. They are easily mobilized into suspension by current and wave energy levels found in estuaries. Cohesive sediments may transition back and forth between the zero-order flocs a nd various larger floc sizes innumerable times before becoming a more permanent bed deposit when transported out of the higher energy reaches of the estuary. Cohesive sediments ultimately find their way to quiescent areas and settle out of the water column to create mo re permanent deposits. Fine particles tend to migrate to the recesses of es tuaries, along the shorelines and in wetlands, making them very critical to many ecological systems. Cohesive sediments have historically been described by bulk properties of the deposited sediment, such as bulk density, wa ter content and plasticity, which serve as indicato rs of erosion resistance. General sediment size fractions of sand, silt and clay within the bottom sediments can provide a strong indication of the expected transport beha vior of the bulk sediment. PAGE 49 49 Sea salinity provides the electrolytes necessary to restrict the electros tatic repulsive forces sufficiently to allow the van der Waal forces to create a net attraction between cohesive sediment particles. The saline environment of estuarie s is a catalyst for aggreg ation and can lead to dramatic sedimentation within the estuary. 2.1.2 Fine Sediment Classification There are several ways fine sediments are classified. These include sediment size, shape, mineral composition, rheologic properties, elect rochemical properties, and by the sediment transport behavior. Floc properties are genera lly a reflection of hydrodyna mic flow conditions. Therefore, most investigators fo cus on the dispersed particle di stribution when classifying the sediment. 2.1.2.1 By size The size distribution of fine cohe sive sediments has been most effectively classified by the MIT scale as shown in Table 2-1. Particles in suspension less than a tenth of a micron in size comprise a sol and when concentrations are sufficiently high are essentially dominated by particle cohesion. Fine, medium and coarse clays lie between 0.1 and 2 microns, and cohesion is very important in their behavior. As the size classes become increasingly larger, the importance of cohesion diminishes. Coarse silt is practic ally cohesionless and sands are cohesionless. Size, as discussed in Table 2-1, refers to the unflocculated primary particle size distribution, because the floc si ze distribution can be a function of the flow field. Example distributions of cohesive sediments are shown in Figure 2-1. The figure presents for comparison the dispersed particles distributions for Sa n Francisco Bay mud, Maracaibo Bay mud, and kaolinite (Mehta, 1973). San Fran cisco Bay dispersed grain and floc distributions are also shown. The suspended floc size distribution is comp arable to that of beach sand. Figure 2-1 also presents the MIT size classification boundaries. PAGE 50 50 One notable feature of Figure 2-1 is the vari ation in the size dist ribution both for the dispersed and the flocculated sediment. These dist ributions will need to be explicitly addressed in analysis of the basic CST processes. Table 2-1. Size classification and general cohesive characteristics Size (m) Classification Level of Cohesion <0.1 Sol Cohesion dominates 0.1 to 2 Fine, medium, and coarse clay Cohesion very important 2 to 20 Fine and medium silt Cohesion important 20 to 40 Coarse silt Cohesion increasingly important with decreasing size 40 to 62.5 Coarse silt Practically cohesionless > 62.5 Sand and coarser Cohesionless 2.1.2.2 By shape The shape of the primary particles has a str ong influence on the cohesive properties of the sediment. Clay minerals are plate-like structur es, which result in a very large surface area to volume (or mass) ratio, amplifying the influenc e of electrochemical forces relative to gravitational effects. Shape is a property of th e basic mineralogy of the sediment grains. Some clay minerals form needles and tubes. The sh ape of the minerals can affect the degree of cohesion through the impact on the surface to volum e ratio and electrochemical forces. Shape has a role in interactions with flow for noncohesive particles. Ho wever, for cohesive sediments, since their dispersed particles ar e incorporated within flocs, th e shape of the flocs, not the dispersed particles, influe nce settling and deposition. 2.1.2.3 By composition The mineral composition of fine cohesive sediments influences their degree of cohesion. The crystalline structure of various clay minerals is a combination of layers of silica, oxygen, hydroxyls, and alumina (or iron or magnesium), with possible adsorbed water layers. The specific arrangement defines the cl ass of clay mineral. The general properties of the primary PAGE 51 51 clay minerals discussed in the literature are s hown in Table 2-2. The variation in the mineral content of the sediments within a particular estuary is not consider ed in the current research. In addition, the content of organic material can sign ificantly affect the cohesion of the sediment mixture. Table 2-2. Properties of example clay minerals Mineral Parameter kaolinite Montmorillonite Illite chlorite Specific gravity 2.60-2.68 2.20-2.70 2.64-3.10 2.60-2.96 Plate diameter (m) 0.1-4 0.01-0.1 0.003-0.3 1.0-4.0 Plate thickness (m) 0.05-2 0.01 0.02 0.03 Specific surface area 103-104 104-105 104-105 104 Cation exchange capacity (mEq/100g) 3.0-15.0 80-120 10.0-40.0 20-50 Source (Mehta, 2007) 2.1.2.4 By electrochemical properties The degree of cohesion is related to the strengt h of the electrochemical interactions of the mineral grains. A measure of the strength is th e cation exchange capacity (CEC), which reflects the affinity of the clay minerals to exchange io ns with the solute. The CEC of clay sediment grains varies with the mineral composition and a number of other factors, including the size of the particles, availability of i ons, valence of the ions and organic content. The range of CEC for four example clays is presented in Table 2-2. The electrostatic charges on the surface of the primary cohesive particles in dionized water will generally cause repulsion of the grains when they come into close proximity. Natural fresh water contains sufficient ions in solution to alter the electrochemical potential field around the particles and allow some particles to get close enough for the van der Waals attractive forces to cause cohesion. When the sediment enters the estuarine environment, there are significantly more ions available from the ocean salinity to dramatically increase the probability of cohesion. Adhesion occurs when particles that collide are bonded together by at tractive van der Waals PAGE 52 52 forces or biological material. Th e efficiency of adhesion is the fr action of colliding particles that are bonded. The efficiency of adhesion in estuaries is normally taken as unity. Clay mineral particle plate-like surfaces are negatively charged. The positive ions in the solution are attracted to the su rface of the mineral and create the double layer of water around the particle. An inner layer of water containing positively charged ions adheres to the particle and moves with the particle. The edge of the inner layer is calle d the slipping plane, since when in motion through the fluid the particle will have a shear along that surfac e. The double layer of water contains preferential posi tive ions, so that the mineral double layer acts like a flat plate condenser. The concentration of ions in the solution influe nces the thickness of the double layer. At higher ionic concentrations, the double layer is compressed reduci ng the strength of the repulsive forces, resulting in a greater chance that the van der Waals attraction will do minate and result in particle cohesion. 2.1.3 Characterizing Aggregates 2.1.3.1 Primary particle distribution The primary particle distribution of the sedi ments in suspension is the building block for the cohesive aggregates. The primary particle distribution has been proposed as a self-similar distribution for a particular syst em (Kranck and Milligan, 1992), whereby the basic shape of the dispersed or flocculated size spectra is retained as the total concentration varies and the mean floc size changes with time. Kranck and Mi lligan proposed a spectral shape that is a combination of a distribution for the fine end of the spectra and a decay shape at the coarser end of the spectrum. The shape for the fine e nd of the spectrum is shown in Equation 2-1. ()m oCdQd (2-1) PAGE 53 53 Co(d) is the particle size distribution (measur ed as volume of particles per volume of suspension), Q is a coefficient scaled with th e total suspended concentration, d is the particle diameter and m is an exponentia l coefficient. This was comb ined with the analytical timedependent solution for a well-mixed susp ension with settling (Equation 2-2). 0() s w t hCtCe (2-2) The variables are: t is time, ws is the settling velocity, and h is the water depth. The expression for fall velocity was substituted and the term t/h was viewed as a settling decay term, K to yield Equation 2-3. 2()expmCdQdKAd (2-3) The term Ad2 represents the fall velocity, where A includes all of the terms from Stokes settling law, 18 f w wg A g is gravitational acceleration, f is the floc density, w is the fluid density, and is the kinematic viscosity of the fluid. This basic spectral shape agreed with the observed distribution from data collected in San Francisco Bay for both the dispersed and flocculated sediment (see Figure 2-2). The dispersed spectra were de veloped from a Coulter counter analysis of the disaggre gated sediment and the flocculated spectra from the analysis of in-situ photographs taken by a plankton camera. The curve fit to the dispersed spectrum of Equation 2-3 was reported by Kranck and Milligan to have the coefficients m = 0.022, K = 3.53 and Q = 1.32. They reported the flocculated spectrum curve fit has m = 2.72, K = 0.081 and Q = 0.0001. Careful evaluation of the equation and thei r data suggest two issues. First, they stated that the units of the coefficient K are s/cm and fitting of the curves to their data suggests that they used the diameter of the particles in microns for the initial concentration. Within the exponential function, the diameter needs to be in centimeters in order for the value of K indicated. Secondly, PAGE 54 54 in the fitting of the curves to the flocs and the grains, they would have us ed the full grain density, in error, for the floc density curv e in order for the stated value of K (0.081) to match the data. For each of these curves they apparen tly used the grain density of 2.65 g/cm3 and 1.0 g/cm3 for the fluid, giving a value of A = 8029 1/(gs), since with the K values cannot match with a lower relative density ratio. The modal floc size for the flocculated sediment s is around 400 microns. 2.1.3.2 Order of aggregation Krone (1963) developed a conceptual mode l of flocculation based on the order of aggregation. He hypothesized that when prim ary mineral particles aggregate, their bonds and mineral arrangement are established and will not significantly change on successive higher order aggregations. Primary aggregates are formed from the cohesion of mineral grains, forming strong bonds. Primary aggregates do not compress, retaining their density. Primary or zerothorder aggregates (pa), collide and combine to form first order aggregates (p2a). In the notation (p n a), the order of aggregation is (n-1). First order aggregates then combine to create second order aggregates (p3a) and so on. The strength of the flocs gets w eaker as the order of aggregation increases. The progression of orde rs of aggregation is shown schematically in Figure 2-3. It was assumed that aggregation be yond primary aggregates requires intermeshing of the lower order aggregates. Krone then made th e assumption that the ratio of the increment in voids to the volumes of the preceding aggregation is the same as the ratio of voids created during the preceding aggregation to th e preceding aggregate volume ( i+1 = i/(i + 1) ), where i is the voids ratio for flocs of (i-1)-th order of aggregation ( pia ). This fractal assumption leads to a relationship for the volume fraction as a functio n of the primary aggreg ate volume fraction, the order of aggregation and the void ration of the primary (first order) flocs (See Equation 2-4). 111pnapn (2-4) PAGE 55 55 The volume fraction of primary aggregates isp the voids ratio of a primary aggregate is1 and the order of aggregation is (n -1). 2.1.3.3 Floc size spectra Figure 2-4 illustrates the effects of th e parameter K in Equation 2-3 on the size distribution. Increasing the valu e of K causes the spectra to fall off more quickly as the particle size increases. The value of Q is scaled by the total concentration. The basic slope, m of the spectra on the fine end was c onsistently near zero for all of the San Francisco Bay samples collected (see Figure 25). The consistency in the grain spectra is believed to be characteristic of the primary sedime nts in the entire bay system. The grain spectra for bottom sediments nearby the profiling locati on and in the San Pablo Strait south of the deployment showed a similar shape (Figure 2-6). 2.1.3.4 Fall velocity Fall velocity is another way to characterize cohesive sediment, since it is readily measurable. It can be measured either on si te for in situ conditions or in a laboratory environment for the dispersed particle conditio ns. The size distribution can then be inferred from the fall velocity based on Stokes law. 2.1.3.5 Floc density Floc density is inversely related to floc size. The relationship is consistent with Krones order of aggregation and has been shown to be fractal in nature (Krane nburg, 1994). Primary sediment grains have the basic mineral density. As primary sediment grains flocculate into small flocs, if the floc size is defined as some enclosing volume, the density of the volume will be lower than the mineral density because water will fill the voids between the primary grains. As smaller flocs aggregate into larger flocs, the re lative fraction of voids within the now larger PAGE 56 56 enclosing volume increases and the floc density is further reduced. The fractal relationship between floc excess density and floc dimension is shown in Equation 2-5 (Kranenburg, 1994). 3 f D p fs fd d (2-5) The variable f is excess density of the floc, (f w), s is excess density of the primary particles (s w), s is sediment mineral density, dp is the primary sediment grain diameter, df is the floc diameter and Df is the fractal dimension. The fractal dimension for excess density varies with the flocculation environment and the controls on flocculation. 2.1.3.6 Floc shearing strength The shearing strength of flocs is also inversel y proportional to the floc size; with the floc structure becoming more fragile as the floc size increases. Krone ( 1963) developed a rotating concentric cylinder viscometer and related an inferred Bingham yield (shear) strength of the suspended flocs, f to concentration for San Francisco Bay sediment by Equation 2-6. 5/2 fKC (2-6) The units-dependent coefficient K was rela ted to the order of aggregation and the suspended sediment concentration (see Equation 2-7). 5/2 112pna v psN KE n (2-7) N is the number of simultaneous particle aggregate bonds ruptured, Ev is the energy required to disperse a unit volume of particle aggregate, and the other variables as previously defined. Kranenburg (1944) argued that because the shear strength and breaking of flocs is controlled by the weakest bonds within the floc, the force required to break a floc should be PAGE 57 57 independent of floc size. Since the force on th e floc at rupture is pr oportional to the shear strength times the exposed area of the floc, it can be assumed that the shear strength is proportional to the inverse of the par ticle dimension squared (Equation 2-8). 2 21 constantf f sf fFd d (2-8) Combining Equations 2-5 and 2-8 results in Equation 2-9, expressing the shear strength based on the excess floc density. 2 3 2 3 2constantf f D D s ff sd (2-9) Based on the data of Krone, Partheniades (199 3) proposed that the fl oc strength is related to the excess density of the flocs (Equation 2-10). 5 2 fffK (2-10) Since the terms in the parentheses in Equatio n 2-9 equals a constant, we can equate the exponents on the excess floc density in Equations 2-9 and 2-10: 5/2 = 2/(3Df) implies that the fractal dimension is Df = 2.2. Shear strength data from Krone for various ha rbors are presented in Figure 2-7 against the excess density of the flocs. The figure shows the reasonableness of Pa rtheniades 5/2 exponent ( Df = 2.2). Linear regression of the log-transformed data sets shows a variation in the exponent and the fractal dimension. These are summarized in Table 2-3. Regressions for each of the individual har bor data sets have a very good fit (high Rsquares), with a minimum R-squared value of 0.952. Within any of the data sets, the relationship between excess density and floc strength is predictable, but there remains a large scatter in the data when handled as one dataset, with an R-squared value of only 0.559. This means that for an PAGE 58 58 accurate estimation of the shear strength for a specific system, data collection and analysis may be required. Table 2-3 Comparison of shear strength versus excess density for locales Data set Regression exponent (EQ, 2-9) R-squared Fractal dimension (Df) All data 1.91 0.559 1.95 Wilmington 1.89 0.952 1.94 Brunswick 2.70 0.998 2.26 Gulfport 2.05 0.976 2.02 San Francisco 2.86 0.967 2.30 White River 2.02 0.974 2.01 2.1.3.7 Fractal dimension The fractal dimension of cohesive sedime nts enters relationships between several descriptive variables as a function of a spatial sc ale, typically the particle or floc size. The fractal description of aggregation starts with th e consideration of the pr imary aggregation of a number, m1, of primary mineral grains into a primary aggregate. Then it is assumed that m1 of these primary aggregates combine, structurally si milar in arrangement to the arrangement of the mineral grains in the primary aggregate, to form a first order aggregate. The number of primary grains in the first order aggregate will then be m1m1. If the size of the ne wly formed first order aggregate is a factor, m2, larger in size than the size of a primary aggregate, and subsequent levels of aggregation repeat the same structural si milarity and size scaling, we will have the total number of primary grains in a floc related to the order of aggregation or it s size. The number of primary particles in a floc, Nf, will be (m1)n, where n is the order of aggregation. The size of the floc compared to the size of the primary floc will be df /dp = (m2)n. The fractal dimension, Df, can be introduced as shown in Equation 2-11. f D p f fd N d (2-11) PAGE 59 59 This can be expressed as a relationship between m1 and m2 as in Equation 2-12. 2 12f f f D D n D p f nn f ppdm d Nm m dd (2-12) Equation 2-12 can be converted to a definition of the fractal dimension (Equation 2-13). 1 2ln() ln() ln() lnf f f pN m D m d d (2-13) This fractal dimension is associated with th e linear scale of the flocs. Other fractal dimensions could be developed for surface area or volume. Alternatively, other relationships can be expressed as functions of th e linear fractal dimension. The ratio of the volume fraction of flocs to the volume fraction of the mineral grains in the floc can be derived (Equation 2-14). 3 f D ff spd d (2-14) The ratio of the floc densities to the primary grain densities is expr essed in Equation 2-5, scaling at the power of (3-Df). The fractal relationship for the floc strength is shown in Equation 2-9, scaling with the excess density to the power 2/ ( 3-Df), or to the inverse square of the floc diameter. As will be shown later, the floc settling velocity scales to the power of (Df 1 ). Khelifa and Hill proposed (2006) that it may not be reasonable to expect that the fractal dimension remains a constant over the full spec trum of floc dimensions experienced in the literature. Their derivation will be summarized here. The definition of floc density is extended (Equation 2-15) to acknowledge that the primary sediment grains are polysized, with a primary particle distribution defined by k sizes di ( i=1,k ). 3 1 3 k i i fs fd d (2-15) PAGE 60 60 The equivalent spherical diameter of the floc, df, for monosized primary particles of diameter d is defined in Equation 2-16. 1/ f D fddk (2-16) They extended this definition to polysized pr imary particles as the summation over each of the grain sizes raised to the fractal dimension, and then taking the fractal root of the sum (see Equation 2-17). 1/ 1 f f D k D fi idd (2-17) A fractal dimension that varies with floc si ze was proposed in the form of a power law as shown in Equation 2-18. This takes into account the observed changing structure of the flocs as they grow. f fd D d (2-18) The coefficients and were defined from two limiting cases First, the fractal dimension should approach a maximum value of 3 when the floc size approaches the primary particle size. The fractal dimension s hould reach a lower value Dfc when the floc size reaches some characteristic value dfc. Applying these end va lues to Equation 2-18 we get Equation 2-19, where the primary particle, dp, replaces d. log(/3) log(/)3fc f cpD dd f f pd D d (2-19) This variation in fractal dimension will be incorporated into the density and settling velocity later. It generally improves the rang e of agreement of the models with observations. PAGE 61 61 2.1.4 Bulk Properties of Cohesive Sediments Bulk properties of cohesive sediments are useful in the ability to infer other characteristics of the sediment that can be used in engin eering applications, including development of parameters for more applied numerical methods. These include general size class percentages (sand, silt, and clay percentages) organic content, mineral content, fall velocity and bulk erosion rates in conjunction with shear strength of bed deposits. These parameters do not provide information useful in evaluating aggregation processes, but provide general guidance for expected behavior. The bulk bed exchange properties will be discussed later. 2.2 Cohesive Sediment Transpor t in Steady Uniform Flow The current state of the knowledge in CST is applicable to steady uniform flow. The sedimentation processes will be discussed in this section in the context of their interaction with macroscale hydrodynamics that are steady and uniform. Steady flow conditions are loosel y defined as flows that change very slowly with time. But slowly is a relative term. A time scale can be defined that reflects processes over the dominant spatial scale of the system of intere st. That time scale may be defined differently, depending on the specific processes of inte rest. For CST, the logical time scale, Tss, is a factor times the ratio of the depth to a representative settling velocity, h/ws. An indicator for steady state would be that 1 t Tss where t is a measure of how slowly the flow conditions are changing. Ultimately, a tolerance for change need s to be defined for the primary forcing. For example, if the mean value of a critical variable, has a tolerance of before the flow conditions are considered unst eady, then the time scale t can be defined as the tolerance divided by the time derivative of the critic al variable (see Equation 2-20). PAGE 62 62 t t (2-20) The steady-state criterion can then be expressed as Equation 2-21. sw t h (2-21) The tolerance of steady-state fluctuations is associated with an acceptable change in the conditions of interest. The value of should reflect how stringent the need for a steady-state condition. For example, assuming is 10 and the depth is 1 m with a fall velocity of 10-4 m/s, Tss would be approximately 28 hours. That would be the time scale for comparing the observed changes in the system, for example, in a large test flume. For a field scale of 10 m deep the value of Tss would be 277 hours. 2.2.1 Aggregation Processes Flocculation results from electrostatic c ohesion and electrochemical adhesion between sediment grains, small flocs and organic matte r. Flocculation pro cesses are affected by temperature, salinity, ionic content, pH, mineral content, organic conten t, the total suspended concentration and turbulence in the water column. Aggregation processes can be divided into two general categorie s (Winterwerp, 1999, Vicsek, 1992); diffusion limited a ggregation (DLA) or cluster-clus ter aggregation (CCA). DLA occurs in extremely dilute concentrations, limite d by the number of aggregates to interact. CCA occurs when there are many clusters of aggregates available to adhere. CCA can be further divided into two classes. In diffusion limited CCA (DLCCA), the probability of cluster adhesion approaches unity, so the only factor to limit the aggregation is the number of collisions, assumed to be controlled by diffusion. When the probabil ity of aggregation is much less than unity, the PAGE 63 63 process is reaction limited (RLCCA ). The particles can bounce into one another very frequently, but the probability of adhesion limits the aggr egation rate. The fract al dimension for DLCCA typically ranges from 1.7 to 1.8. RLCCA has a typical fractal dimension between 1.9 and 2.1. The presence of organic material, which increases the probability of adhesion, will effectively lower the fractal dimension of the flocs. When very high-suspended concentrations are encountered, the interaction between particle s affects the fractal dimension. The distance between flocs becomes comparable to the floc di mension itself and clusters pack very closely together at a fractal dime nsion approaching 3. But the floc cl usters themselves have a structure of fractal dimension about 2.0, resulting in an ov erall macroscale fractal di mension of 2.6 to 2.8. Marine snow has a fractal dimension around 1.4, suggesting they are formed in a DLA environment. The processes that bring particles close e nough to create cohesion are related to the interactions of the particles with the fluid. Th ese are the modes of aggregation. The modes for steady uniform flow are Brownian motion, differential settling, and turbulent shear. Biogenic aggregation can also be an im portant factor in certain environments (Andersen, 2001; Hill, 1998); on the continental shelf or in wetlands, areas where turbulen ce is not as great a factor. 2.2.1.1 Brownian motion Brownian motion is the collision of water mol ecules with sediment particles as the water molecules move erratically due to thermal energy. Brownian motion is only of importance at the smallest particle sizes (less than 0.5 m) (Hunt, 1982). Water molecule collisions with the sediment become less significant as the size of the particle increases because the mass of the particle increases, making the momentum exchange sm all relative to the inertia of the sediment. In addition, at larger scales, th e number of molecular collisions in creases and tends to average to PAGE 64 64 a zero net impact because of th eir random nature. Brownian mo tion may be important in the initiation of flocculation at small scales in quies cent waters. Brownian motion is insignificant for aggregation in most estuarine waters and is often neglected in aggregation models (Maggi, 2005). 2.2.1.2 Differential settling Differential settling creates differential particle velocities that allow larger particles to overtake the smaller particles. The effects of larg er spherical particles on fluid flow lines in the return flow helps to move the smaller particles away from the falling larger particle. However, large voids within the open structure of flocs diminish the effects on the flow, increasing the probability of a collision with the slower settling smaller flocs compared to the case of solid spherical particles. As the concentration increases, the influence of differential settling increases due to the increased probability of encounters and because the effects of the particles on the return flow are constricted by the proximity of other particles. Ev en without aggregation, larger particles can pull smaller particle s down with them as they se ttle (Teeter, 2001). Hunt (1982) found that differential settling is the most co mmon mechanism for collisions for floc sizes greater than 50-60 microns. 2.2.1.3 Shear Either laminar or turbulent shear can induce pa rticle collisions by the differential transport perpendicular to the velocity gradient. In lamina r flow, the differential ve locity is the component of velocity in the primary flow direction. Colli sions occur because faster moving particles in the positive velocity gradient direction overtake slower moving particles down gradient. In turbulent flows, the differential velocities can be three-dimensional due to differences in inertial response of different size particles to the turbulent fluctua tions as well as the shear effects from the mean PAGE 65 65 flow. Because the magnitude of the velocity differe ntial is a function of th e size of the particles, shear has a significant impact on collisions of flocs in the 2 to 10 micron range (Hunt, 1982). The relative importance of the modes of a ggregation varies with the sediment size spectrum and the hydrodynamic conditions. In a quiescent settling column, shear will not be important, while in a highly turbulent tidal bore, Brownian motion will no t be important. Very large flocs in relatively quiescent waters, such as the deep ocean, tend to be formed by differential settling (Lick, et. al., 1993; Hill, et. al., 2001). 2.2.1.4 Salinity Salinity in estuaries provides the ions to compress the double layer, allowing for increased efficiency of aggregation. Krone (1962) inve stigated the effects of increasing salinity on flocculation in one-liter graduate d cylinders at four different se diment concentrations. Figure 28 summarizes his findings, with the settling velo city showing dependence both on salinity and on the initial suspended concentration. Between 1 and 2 ppt, the fall velocity increases rapidly for all concentration levels. Median floc size incr eased for the final equilib rium settling velocity with the initial concentration. Above a salinity of 10 ppt the eff ects of salinity are minor and are fully realized above 15 ppt where the ma ximum fall velocity has been realized. 2.2.2 Fall Velocity Flocs settle and deposit at rates several orders of magnitude faster than the constituent particles in the flocs. The settling velocity of flocs is influenced by the floc size, density, concentration and the level of tur bulence. Free settling velocity is assumed at concentrations low enough that the effect of individual flocs fa lling through the water does not impact adjacent particle trajectories. Free settling is estim ated from Stokes settling law, developed from a balance between the drag force and the buoyant weight of the particle, presented in Equation 222, for quadratic drag. PAGE 66 66 4 3s Dg wd C (2-22) The drag coefficient is a function of the par ticle fall velocity Reynolds number, defined by Equation 2-23. Res s wwd (2-23) It is assumed that the drag coefficient for spheres is appropriate for flocs, since there are no data for flocs. This is the reason why Stokes diameter is used. The variation of the drag coefficient for a spherical particle was developed by Clift, et. al. (1978) ov er a series of ranges ofRe s w. These are shown in Equation 2-24. 20.820.5 0.6305 1.64351.12420.1558243 1Re; Re0.01 Re16 24 10.1315Re;0.01Re20 Re 24 10.1935Re;20Re260 Re 10 ; 260Re1500ss s ws s s ws s s sww w w w D w w wC (2-24) The variable is defined as the natural logarithm of the Reynolds number, lnResw. An alternative single equation recommended by Graf (1971) is shown in Equation 2-25. The comparison between Equations 2-24 and 2-25 is presented in Figure 2-9, indicating that the single equation gives reasonable agreement. 0.68724 10.15Re Re800 Rews s sDw wCf o r (2-25) PAGE 67 67 When the effects of flocculation on partic le density (Equation 2-5) are included in Equation 2-24 and the drag coefficient from Equation 2-25 is used we get Equation 2-26. 31 0.68718 10.15Reff w sDD s spfg wd d (2-26) This was also developed by Winterwerp (1999) who added the weighting factor where is a shape factor for non-spherical part icle effects in the gravitational force and in the drag force. This fall velocity is for free settling with a variable drag coefficient and with the effects of floc density included. The effect of concentration on the fall velocity is indirect. As the concentration of all sizes increases, the probability of flocculation increa ses. As flocculation progresses, the mean fall velocity of the sediment increases through the de pendence of the fall velocity with particle size in Equation 2-26. Khelifa and Hill (2006) developed a polydisperse version of Equation 2-5, still based on a constant fractal dimension (see Equa tion 2-27) for a floc comprised of k primary particles of arbitrary dimension di. 3 1 3/ 1 f fk i i fs D k D i id d (2-27) Mean variables were then introduced (Equation 2-28). 3 11 3 f kk D ii ii fdd mandm kk (2-28) These can be introduced into Equati on (2-27) to yield Equation (2-29). (3)/ffDD fsk (2-29) PAGE 68 68 The variable is defined as 3/ 3/ f D fmm Khelifa and Hill showed that for monosized particles =1 and this reduced back to Equation 2-5. The formulation of Equation 2-29 is intractable since the number of particles within a floc is not generally kn own. It was therefore proposed that a median primary particle diameter be used as representative of the primary particles, so that a more usable form is obtained (Equation 2-30). (3) 50fD f fsd d (2-30) This equation is their proposed model for the excess density of flocs that takes into account the distribution of polydisperse primary particles and, with the use of Equation 2-19, a variable fractal dimension, with d replaced by d50. Incorporating the revised excess density of E quation 2-29 into Equation 2-26 results in the settling velocity relationship is shown in Equation 2-31. 31 50 0.68718 10.15Reff w sDD s sfg wd d (2-31) Khelifa and Hill recommend that when using d50 to represent the primary particle distribution that the value of should be unity. Sensitivity of the can be included with the shape factor terms and into a single coefficient for implementation and calibration to observed data, particularly since th e effects of particle shape was not incorporated into Equations 2-28, which define Data summarized by Winterwerp (1999) are presented in Figure 2-10 with fits of Equations 2-26 and 2-30. There is significant sc atter in the data, and in order to make a reasonable fit using Equation 2-26 a fractal dimension of 2.6 is needed. The curve in the figure from Equation 2-31 used the Khelifa & Hill parameters: Dfc = 2.1, dfc = 8000 microns and d50 = 2 PAGE 69 69 microns. Because of the scatter, it is unrealistic to expect a single set of parameters for a theoretical relationship to be universally applicable and provide a good fit to a composite dataset as shown in Figure 2-10. Separate datasets from four of the estu aries in Figure 2-10 are presented in Figures 2-11 and 2-12 with their own independent curve fits. Most of the individual datasets can be reasonably fit with a theoretical curve, when the coefficients are adjusted to that estuary. The variation in the fit curves is presented in Figu re 2-13 along with the composite curve fit for comparison. The co mposite curve is essentially the same curve as used to fit the Ems estuary (The Netherlands) 1990 dataset. The sp read of the data valu es biases the settling velocity curve to a steeper slope (higher fractal dime nsion). It is of particular interest that the Ems estuary has such a significant difference between the two datasets. The variation in the fractal dimension of Equation 2-31 fit in Figure 2-10 is shown in Figure 2-14. The fractal dime nsion is 3 at the primary d50 diameter of 2 microns and decreases by the power of -0.043 ( in Equation 2-18) until the fractal dimension is 2.1 at a floc size of 8,000 microns. Differences in the primary particle distribu tions, as well as salinity regime and tidal energy, may explain the wide scatter in the data. The data reinforce the point that accurate representation of the CST processe s in a particular estuarine system requires site specific data collection and analysis. The mo tivation behind the variable fractal dimension was the tendency for the settling velocity to be over-predicted at la rger floc sizes with a constant fractal dimension (see Figure 2-15). There may be a correlation between the larger floc sizes and higher concentrations, for which hindered settling may have a role. The effect of the variable fractal dimension on the excess density as a function of floc size is i llustrated in Figure 2-16. The behavior above 100 microns has been observed by several inve stigators who have displayed PAGE 70 70 constant fractal curves for multiple data sets that essentially have a pivot in the slope of those data just above 100 microns. 2.2.3 Hindered Settling As the suspended sediment concentration become s large, the settling particles effect on the fluid being displaced becomes significant. The displaced fluid begins to interact with the displaced fluid from nearby particles as they settle. Wolanski, et. al.(1989) proposed an empirical formula for hindered settling as shown in Equation 2-32. 22f f n sf m fC wa Cb (2-32) The coefficient af is a scaling factor, nf is the flocculation settling exponent, bf the hindered settling coefficient and mf the hindered settling exponent. The coefficients fit to the Krone (1962) San Francisco Bay data are: af = 0.048, nf = 0.40, bf = 25 kg/m3 and mf =1.0. These values were, however, generally outliers for the coefficients developed for 24 data sets from various systems around the world. It is noted that if the expone nts are not selected such that nf = 2 mf, then the units of af will not be the same as ws. For the Krone values above the units of af are (m/s) (kg/m3)1.6. Consider deposition in a quiescent fluid, fo r example a settling co lumn. Applying the results from Equation (A-26) for the continuity equation for a sediment-lad en fluid to sediment depositing, we have Equation 2-33. 1ii d ifiuuC x x (2-33) This equation states that the divergence of the velocity field is in balance with the changing volume of sediment in suspension, the divergen ce in the sediment volume. The divergence in the sediment volume has the same meaning as for th e fluid. Both the sediment and the fluid are PAGE 71 71 incompressible, so the divergen ce of one is offset by the dive rgence in the other with the opposite sign. A settling column initially mixed uniformly with sediment will upon initiation of settling have a downward volume flux of sediment that must be balanced by an upward water volume flux. This equation takes the same form as a comp ressible flow equation because with a change in the sediment concentration there is a change in the volume of fluid in a unit volume. When the water flux is balanced around the control volume the gain or loss in water volume will be matched by the loss or gain of sediment volume. The differential sediment velocity idu is measured relative to the water. This is assume d to be the reference particle settling velocity relative to a static fluid (see Equation 2-34). idpisuuuw (2-34) The variable up is the particle velocity. The fluid velocityiuis referenced to the sides of the fixed reference frame (settling column). The fall velocity relative to the fluid is s w. Inserting Equation 2-34 into 2-33 yields Equation 2-35. 1is ifiuwC x x (2-35) If the fall velocity is a property of the sediment and does not change with xi then we have Equation 2-36. is ifiuw C x x (2-36) Integrating with respect to xi we get Equation 2-37. is fC uw (2-37) PAGE 72 72 The constant of integration is zero so that in the limit as C approaches zero the return flow of the water approaches zero. Equation 2-37 can now be combined with Equation 2-34 to solve for the particle velocity relative to the fixed reference frame (Equation 2-38). 1psis fC uwuw (2-38) The particle velocity is positive upward a nd it is an effective settling velocity, -wseff. The relative reduction in fall velocity can be expressed as given in Equation 2-39. 1seff s fw C w (2-39) The ratio C/f is the volume fraction of the flocs f, so the correction for the return flow is (1f). Winterwerp (1999) derived a similar expression and included the effects of buoyancy and viscosity (see Equation 2-40). *11 12.5m seff s s fw w (2-40) The variable *1m is the return flow effect, where is set to the min{1,f }, to account for cases of consolidating fluid mud where f can exceed one. The exponent m is introduced to account for the nonlinear effects of return fl ow interaction between particles (pressure distributions, drag coefficients and added mass), particle collisions and interactions. Winterwerp advocates using m = 1. The term 1 s is a correction for buoyancy effects and 12.5f is the correction for viscosity (Einstein, 1911). The effects of shear on settling velocity we re developed by Winterwerp, et. al. (2006) taking into account aggregation a nd floc breakup (see Equation 2-41). PAGE 73 73 1 9/8 1/2 1/2 3 4240 3/8 3/8expfD qq sf fkh CC wkkkd D (2-41) The coefficients k2, k3, k4, and q are sediment dependent coefficients, C is the total concentration, h is the water depth and is the shear stress. Within Equation 2-41 units of the coefficients are a function of the exponents, making comparisons between specific data sets difficult. Teeter (2001) found that in order to simulate the partic le size distribution observed by Kranck and Milligan (1992), it was necessary to couple the inte ractions between grain classes during settling. When discretizing the sediment sizes into successive classes, the depositional flux was coupled to the next higher size class when the probability for deposition for the class, Pdi, was less than 0.05 (see Equation 2-42). 11 120.05ii idi iCF FP C (2-42) Fi is the depositional flux for the i-th sediment class, Ci is the suspended mass concentration of the i-th class, 1 is a coefficient that contro ls the proportion of the flux and 2 limits the flux of smaller grains as Ci+1 tends toward zero. This e ffect is analogous to a sinking ship drawing huge masses of water down with it. Wolanski et al. (1992) determ ined the settling velocity ws of suspended sediment in Cleveland, Australia, in two ways. The first me thod involved measurements of vertical profiles of concentration at different times during a dr edged material disposal operation from a hopper dredger. In the second instance the same sedime nt tested in a laborat ory settling column in which vertically oscillating rings (Figure 2-17 inset), nearly flush with the inner wall, were used to generate flow shear during the settling process. The depth-mean shear rate can be controlled PAGE 74 74 by setting the amplitude (and frequency) of oscillation. Concentration (C) profiles at different times were measured. Equation 2-43 along with an estimate of the diffusion coefficient Dz, initial condition of constant concentration at th e start of the test, surf ace condition of zero net sediment flux and bed condition of zero resuspensi on were solved numerically to obtain the plots shown in Figure 2-17. The settling velocity in E quation 2-43 is taken as a positive value with the downward flux associated with the negative sign on the term. 0szCCC wD tzzz (2-43) Two noteworthy effects are the difference between observations under different levels of agitation in water, and the difference between fi eld and laboratory measurements. Field data indicate higher settling velocitie s when the weather was calm (minor wave activity) than under higher wave action (with a swell period of about 15 s and sea peri od of 3.5 s). The slowing of settling by waves was attributed to the formation of a lutocline that contained large, more slowly settling aggregates than when the conditions were calm. A similar behavior is seen in the laboratory tests, in which settli ng was more hindered when the ri ngs were oscillated than under quiescent condition. Differences between field and laboratory results indicate the effects of aggregation processes that are not easily scaled, and in turn highlight the importance of field observations. In Figure 2-17 the effect of shear rate on the settling velocity appear s to be important over the entire range of concentr ations, although above ~10 kg m-3 there is the onset of a trend of convergence of the curves. This trend is the re sult of the role of high concentration, which damps turbulence and leads to in creasingly congruent hindered se ttling of the sediment mass. Schematically, one would expect the actual trends to appear as shown by the dotted curves in Figure 2-18. Teeter (2001), however, assumed a simpler model in the flocculation settling range PAGE 75 75 ( Cf PAGE 76 76 Teeter found that the values of nf decreased with increasing floc size. The relationship as shown in Equation 2-47 is used here, with the exp onent considered negative to agree with Teeter. nb fn pd na d (2-47) 2.2.4 Deposition and Erosion The exchange of cohesive sediment at the wa ter-bed interface is a function of the settling velocity of particles in suspension and the turbulent mixing that resuspends particles upward that have either reached the near bed by settling or by erosion from the bed by the bed shear stress. The level of the turbulent mixi ng can be characterized by the mean shear stress on the bottom. Erosion has been observed in several modes in laboratory and fiel d experiments (Mehta and Partheniades, 1982; Amos, et. al., 1993), controlled by the sh ear stress and the vertical structure of the bed sediment. The strength of th e bed to erosion is a refl ection of the history of previous bed exchange. A recently deposited bed will tend to have a stratified structure because of the lack of sufficient time for consolidation. Older beds that may have been exposed by previous erosion will be compacted and genera lly stronger, with a more uniform vertical structure. At low shear stresses partic le surface erosion can occur when the inter-particle bond strength for primary particles or flocs adhere d to the bed are broken. The response for a stratified bed, given sufficient time, will be asym ptotic in suspended concentration as the bed erodes down to an equilibrium between the shear stress and the bed strength (Mehta and Partheniades, 1982). Amos et. al. (1993) further defined supply limited asymptotic type I surface erosion as type Ia surface erosion. At intermediate sh ear stresses when the asymptotic behavior is unclear whether due to limited supply or a ba lance between deposition and erosion Amos et. PAGE 77 77 al. (1993) classified it as type Ib surface erosion, acknowledging the possibility of simultaneous erosion and deposition. Mehta and Partheniades (1992) classified a sediment response with linear increase in conc entration, with no apparent limit in supply as would accompany an unstratified bed, as type II surface erosion. At higher shear stresses the erosion patte rns can become more chaotic and nonlinear without any clear trends toward equilibrium. Tolhurst et. al. (2009) refer to this as transitional erosion. At even higher shear stresses the sh ear load may be transferred into the bed and a weaker layer in the bed can fail causing the mass of sediment a bove it to become entrained in the flow. This mode of erosion has been called type II bulk erosion Given sufficient time Tolhurst et. al. argue that even type II surface erosi on would eventually reach an equilibrium. In special conditions, particularly in wave environments, sediment can be eroded through the pulsing action of the waves to create a fluid mud layer near the bed. When the waves attenuate the fluid mud layer wi ll generally redeposit. Anot her mode on erosion is the entrainment of fluid mud upward into suspension. Fluid mud has the ability to absorb wave energy through increased viscosit y of the fluid mud layer itself, which can very efficiently dissipate the oscillatory wave en ergy. The present research focu ses primarily on surface particle erosion, which is the most common mode of erosion within the estuarine environment. Tolhurst et. al. ( 2009) discuss the fact that there is e ssentially no true cr itical entrainment shear stress, given that evidence of winnowing of small particles from the bed has been observed for very low shear stresses (Mehta and Partheniades, 1982; Black, 1991) This opens the question as whether such entrainm ent can occur during deposition. The assignment of a critical shear stress for erosion, therefore, requires th e specification of a minimum sediment entrainment rate. No standardization of the threshold for significant bed flux has been identified. PAGE 78 78 There is usually a stark contrast between obs erved erosion in field experiments compared to laboratory experiments, where the flume be ds are carefully constr ucted through controlled deposition or reforming of poured slurry. The fiel d is inherently heterogeneous in bed structure and biological stabilization. Tolhurst, et. al. (2000) conducted comparisons on four different insitu erosion devices in a field experiment and documented orders of magnitude differences in estimated erosion rate between devices. They a ttributed the variance to the differences in device size, the heterogeneity of the bed and di fferences in the device deployment time. The simplified governing equation for the vertic al distribution of suspended sediment in uniform flow can be expressed as in Equation 2-43. The boundary condition at the bottom of the water column can be written as in Equation 2-48. szC wCDDE z (2-48) The variable E is erosion and D is deposition. The sign of D is positive because ws is defined to have a positive value. The erosion, E is also positive, since the concentration gradient is typically negative. Many inve stigators have asserted there is exclusively either erosion or deposition (Mehta and Partheni ades, 1975; Lau and Krishnappan, 1993). The classical cohesive model for bed interaction treats erosion and deposition as mutua lly exclusive processes. The decision for whether deposition, erosion or no exchange occurs is based on the threshold shear stress, e and d, the critical shear stresses for erosion and deposition, respectively. It is generally acknowledged that e > d. When the bottom shear stress, b, is less than d deposition will occur at a rate of the settling flux times a probability of deposition (Equation 2-49). DsDPwC (2-49) The probability of deposition was estimated by Krone (1962) as given in Equation (2-50). PAGE 79 79 1 0b D bd d D bdPf o r Pf o r (2-50) When the bottom shear stress lies between th e critical shear stress for deposition and erosion, then the classical cohesi ve bed model has no bed exchange. When the bottom shear stress exceeds the critical shear stress for eros ion, the erosion is estimated by Equation (2-51) (K andiah, 1974; Ariathurai, 1974). 1b be eEM for (2-51) M is the rate of erosion coefficient, in units of kg/m2/s and is an empirical exponent. M is normally developed from analys is of bottom samples from th e study site. An alternative formulation (Mehta and Parchure, 2000 ), which works well for stratified sediments characteristic of newly deposited beds ( type I surface erosion; Mehta and Partheniades, 1982) is shown in Equation 2-52, which differentiates between a floc erosion rate, Ef, and the bulk erosion rate, E lnbe be fE for E (2-52) The linear form of Equation 2-51 has been re ported to work better for compacted beds (type II surface erosion) and the exponential form of Equation 2-52 works better for type I surface erosion (Tolhurst, et. al., 2009). The critical shear stress for erosion has been related to the shear strength of the flocs. The floc strength is the strength required to break the floc apart, while the critical shear stress for erosion is that required to disl odge the particle from the bed. (Mehta, 2007) relates the critical shear stress for erosion to the yield strength, which can be ap proximated as the floc strength PAGE 80 80 (Equation 2-53) and the critical shear stress for deposition is assumed to be a power function of the relative floc size (class-referenced as i). 00.256 1.6 0.289 1.6ei yi yi ei yi yi fi did p f orPa f orPa d d (2-53) Otsubo and Muraoka (1988) developed a similar relationship for ei with a single equation to the power of 0.6 and a coefficient of 0.27, whic h is close to the two-part equations in Equation 2-53. The relationship for the cri tical shear stress for deposition in Equation 2-53 is similar to a form developed by Mehta and Lott (1987), based on the relative fall velo cities. If the fall velocities are computed as a func tion of the floc sizes, then thes e are functionally equivalent. Partheniades (1962) developed a probabil istic estimate of the erosion flux, of a cohesive bed, modifying the stochastic approach of Ei nstein (1950) for noncohesive sediments using the statistics of the velocity fluctuations. He de rived an integral equati on method that he could evaluate after assuming that the lift force is no rmally distributed, leading to the evaluation of error functions (Equation 2-54). 111111 1 2 22CC bLL bLLFF kerf erf kk (2-54) Winterwerp (2007) advocated us ing a highly complex asymmetric shear stress distribution based on Petit (1999) arguing that the normal di stribution assumed by Pa rtheniades overlooked the importance of the skewness on erosion. The formulation of Petit, however, is cumbersome to use. PAGE 81 81 The conventional bed exchange theory for noncohesive sediments permits simultaneous erosion and deposition at the bed. This approach has been advocated by several investigators for use with cohesive sediments as well (Winterw erp, 1999; Sanford and Ha lka, 1993). Winterwerp advocates the inclusion of the settling flux as a continuous process with a probability of one applied. In order to account for the rates of sedimentation in a num ber of European estuaries, the depositional flux was required without c onstraint by a threshold shear stress. Net deposition in the simultaneous models is a balance between settling flux and vertical turbulent mixing near the bed. As larger and weaker flocs settle toward the high shear zone near the bed, they may be torn apart and reentraine d into the upper water colu mn. The argument for exclusiveness in the processes is partly based on th e logic that if a floc is strong enough to settle through the shear layer near the bed and become attached to the bed, then it will be strong enough to withstand the shear stresses to erode it until the shear increases. The seeming paradox of simultaneous or exclusiv e erosion and/or depos ition is illustrated in Figure 2-20. The question can be posed; if the net flux across an arbitrary boundary some distance, above the permanent water bed interface is either a net downward (deposition) or a net upward (erosion), is the interaction at the be d interface either all erosion or all deposition or can it be both erosion and deposition with the net at the water bed interface equaling the net at the arbitrary boundary at ? The logical step is to allow to approach zero and argue that in the limit the two conditions should match, implying th at the exclusive model should fail. But the argument can still be made that as approaches zero the net fl ux over the upper boundary should asymptote to the exclusive erosion or deposition. The argument for exclusivity would be that as approaches zero, one or the other of the erosion or deposition terms will vanish at the upper surface as appropriate to the overa ll net flux at the bed interface. PAGE 82 82 A more pragmatic line of logic is to consider the temporal variation in the bottom shear stress exerted on the bed. Winterwerp (2004) pr oposed the use of a probability distribution of the shear stress because of the influence of near-bed turbul ent bursts and sweeps that are common in estuarine scale hydrodynamics. Even in relatively homogeneous steady flow conditions, some significant variability remains in the turbulen ce at time scales shorter than what would be required to av erage to a steady-state hydrodynamic c ondition. In addition, he argues that the asymmetry in the shear stress distributi on can be critical to the estimation of either erosion or deposition potential. This will be discussed further in the next chapter. 2.3 Cohesive Sediment Transport in Unsteady Nonuniform Flow In unsteady nonuniform flow, all of the processe s discussed previously will continue to be important. The CST processes will now likely be put into a continuous state of being out of equilibrium. Unsteady and nonuniform hydrodyna mics will now introduce additional factors into the processes. The importance of these additional effects will vary for different cases. Most studies assume the time and space scales of tidal va riations are large compar ed to the time scales for CST processes. Figure 2-21 presents the inverse of the fall velocity, converted to hours, based on the settling velocity curve for Equation 231 in Figure 2-10. The parameters used in Equation 2-31 were d50 = 2 microns, Dfc = 2.1 and dfc = 8 mm. This represents the length of time required for the particle of a gi ven size to settle one meter. Assuming that one meter is an appropriate scale for CST processe s in the vertical, and that the tidal time scales of significant changes in currents and water dept h is three hours, then the settli ng of particle diameters smaller than 25 microns will be significantly affected by tidal energy. The minimum in the curve corresponds to the maximum fall velocity. Because the time step involved in most numer ical CST calculations is controlled by the fastest process, the time steps are on the order of seconds to minutes. As long as the parameters PAGE 83 83 that are affected by the changing hydrodynamics are updated in each time step, the changes in the impacts will be realized. However, addi tional terms may need to be added to the calculations. 2.3.1 Aggregation Processes Aggregation processes from steady uniform flow conditions will continue to be important in an unsteady nonuniform environment. Unsteady fl ows include inertial e ffects associated with accelerating or decele rating flows and changes in water depth. The larger more massive particles will be slower to respond to accelerating than the smaller less massive particles, creating additional differential velocities. These could augment either aggrega tion or disaggregation. Maggi (2005) presents a con ceptual description of an example time evolution of aggregation and disaggregation fo r the case of a fixed steady state shear condition. Figure 2-22 shows the case of steady-state hydrodynamics, but serves to con ceptualize the feedback between the aggregation, disaggregation and the flow conditions. The in itial state has the floc size distribution (FSD) that is well out of equilibrium with the sh ear condition. The FSD has a maximum floc size that can genera lly withstand the shear condition. Flocs larger than that size have a high likelihood of eventual disaggregation. Flocs sma ller have a high likelihood of aggregation. The steady-state shear condition is an average condition, with some degree of variability in shear. The plot at the top in th e figure shows the temporal variation in the number of flocs within a unit volume. As aggregation oc curs, the number of particles is reduced. With disaggregation the number of partic les increases. The plot at the bottom shows the concurrent FSD, which reflects the aggreg ation/disaggregation cycling It could be possible that steadystate equilibrium may never be achieved. The relationship between the modal floc diamet er within the FSD, th e shear stress and the concentration was presented by Dyer (1989) and is shown in Figure 2-23. At very low shear PAGE 84 84 stresses the modal floc diameter increases dramatically with increasing concentration. The lower shear has a positive effect on flocculation. As the shear stress increases, the destructive effects of shearing begins to control the maximum floc size. At higher shear stresses, an increase in concentration increases the likelihood of particle s collisions that create disaggregation and the modal floc diameter is reduced even further. McAnally (1999) developed an aggregation model that included three-body collisions (as recommended by Burban, et. al., 1989). In an attempt to replicate Dyers Figure (2-23), the model signif icantly over predicted floc sizes for low-shear conditions. The over-prediction of floc sizes at low shear stresse s could arise from either overpredicting the aggreg ation or under-predic ting the disaggregation. At low shear rates it is more likely that the aggregation is over-predicted. The balance between aggreg ation associated with differential settling and with shear could be a factor in the over prediction, since at the no-shear boundary the model should be using mainly differential settling. The inclusion of three-body collisions could also be a factor. Burban et. al. (1989) tested the effects of sh ear and concentration and concluded that the mean floc diameter decreased with increasing con centration. The shears te sted were all greater than 0.1 Pa, assuming thatbG all above the low shear peak in Figure 2-23. With the conceptual picture in Figure 2-22 of the FSD and the aggr egation/disaggregation processes oscillating about stea dy-state equilibrium, consider th e impact of continually moving the target equilibrium associated with varying tidal conditions in an estuary. The target equilibrium condition as char acterized by the shear will be constantly changing. 2.3.2 Settling Velocity The particle settling velocity is normally assumed to be independent of the hydrodynamics. The concept of settling velocity is based on e quilibrium between the gr avitational and drag PAGE 85 85 forces on the particle as it falls through quiescent water, reaching a terminal velocity. It varies with the fluid density, particle size, particle density and viscosity. The assumption is made throughout the science that the se ttling velocity can be linearly superimposed on the current velocity field as a vector addition. Because of that assumption, the settling velocity, as used, will not be affected by either uns teady or nonuniform hydrodynamics. Ho wever, variations in the shape of the particles and effec tive density for flocs leads to so me variability in the settling velocity for a nominally defined particle size. 2.3.3 Bed Exchange Effects of unsteady and nonuniform hydrodyna mics on bed exchange will only be potentially seen in the erosion rates, based on the arguments above about the setting velocity. Because the theory behind whether a particle or floc is disengaged from the sediment bed is based on the instantaneous hydrodynamic lift a nd drag and the proper ties of the particle it is reasonable to assert that the onl y way that the erosion can be a ffected is through the turbulence characteristics within the bottom boundary laye r. If changing hydrodynami cs has an impact on the frequency and intensity of turbulent bursts and sweeps then it may influence that erosion rate. That influence would have to be handled in a statistical manner to be tractable (Butterfield, 1993). The other mechanism for having variable hydrodynamics influence bed exchange is through the cumulative effects of gradients in space and time to create localized perturbations in either hydrodynamics or the suspended sediment concentration field, which can then have second order effects on the CST processes. Such an effect is the phenomenon of a turbidity maximum. Tidal sorting leads to mixtur es of cohesive and noncohesi ve sediment for which the mobility of the noncohesive sediments becomes re lated to the fraction of cohesive sediment PAGE 86 86 present in the bed. Mehta a nd Lee (1994) developed a relationship (Equation 2-55), taking the effects of cohesion into account. 12 3 3 12tan/tan tan tanca a cs a sw a swF gd gd (2-55) The force of cohesion, Fc, adds an additional component to be overcome to have sediment mobility. If Fc vanishes, then Equation 2-55 takes on the form of the original Shields relationship. The coefficients 1, 2 and 3 are related to shape factors and ratios of lift to drag coefficients and a is the angle to the critical contact point for pivoting for particle entrainment. For the case of no cohesive force the angle a can be estimated form Equation 2-56. 1cos 1a sd z k d k (2-56) where z* is the average level of the bottom of the al most moving particles. A value of -0.02 for z* is recommended. 2.4 Governing Equations The governing equations for CST in unstea dy and non-uniform flows is presented in Equation 2-57n for the suspended sediment con centration of a single sediment class (class number n). The derivation of this equation in tensor notati on can be found in Appendix A (Equation A-164n). The subscript n is placed on the equation number to denote that there are multiple equations, one for each size class. E quation 2-57n is time averaged after turbulent decomposition into the mean and fluctuating components. 3 3 jsn n jsnn njn n n nmt j j jjjjwc wc uc cc SDD txxxxx (2-57n) PAGE 87 87 The index j indicates the coordinate components 1 through 3, with 3 being the vertical coordinate. The velocity component, ujn, admits the possibility that the particle velocity may be different from the fluid velocity. j3 is the Kronecker delta function which equals one when j = 3 and zero otherwise. This equation incorporat es the temporal change in sediment class concentration, the spatial advective gradients, pa rticle settling, turbulent diffusion, a turbulent correlation term between the settling velocity and the concentration and a source/sink term. The source/sink term incorporates the aggregation an d disaggregation processes. The diffusion term Dm is molecular diffusivity and Dtj turbulent diffusivity. The governing equation for the total concentr ation is presented in Equation (2-58) 33 111 11 NNN njnjsnnjsn n nnn jjj NN n nm t j nn jjC uc wc wc txx x c SDD xx (2-58) Strictly speaking, the total c oncentration equation is redundant. If all of the individual classes are conserved by virtue of the proper specification of the source/sink terms, then the total will have to be conserved. The total concen tration equation imposes that balance in the interaction terms, including interaction with the bed as a boundary condition. The boundary condition at the water surface is a no-flux condi tion. These boundary conditions are shown in Equation 2-59n. 3 333 3 33 30ws bn nnmt xz n nnmt nn xzc uwcDD x c uwcDDED x (2-59n) PAGE 88 88 The bed boundary condition is applied to each si ze class. The issue of the simultaneous versus the exclusive erosion and deposition rema ins unresolved. However, the formulation of the size distribution of the bed exchange is a ffected by whether the exclusive or simultaneous theory is used due to required resolution of in consistencies in the relationships between floc strength and the implementation of bulk critical shear stresses. 2.5 Applicability of Existing Knowledge to Field Conditions Steady-state CST processes have been applied directly to real world problems with dramatic unsteady and nonuniform features with reasonable success. Some difficulties have been observed in obtaining sufficient vertical mixi ng of suspended sediment into the upper water column when applying the steady-state developed processe s (Czernuszenko, 1998) Extreme variability in the documented CST vari ables for specific sites indicates that the use of existing theories requires site-sp ecific data for calibration of the models. 2.6 Biological Influences Most sediments in natural waters are influen ced by biological activity. Organisms, either living or deceased, can have an influence on sediment processes. Deceased organisms, both plant and animal, decay and provide natural or ganic matter (NOM) that becomes mixed with mineral sediment particles to form complex struct ures (Paterson, 1997). A wide range of biota produces cell exudates, which are of great importance in sediment stabilization. These exudates, as well as fecal pellets, are also NOM. Live organisms also disturb th e settled bed, which is their habitat, resulting in bioturbation. The effects of biotur bation can be either beneficial or detrimental to stability of the be d with regard to erosion. Adsorbed organic matter can affect clay particles by influenci ng interparticle bonds (Bennett, et. al., 1991). Flum e tests by Dennett et. al. (1998) with kaolinite showed that increasing organic matter levels (% carbon) from 0.0 to 0.12% caused increasing resistance to PAGE 89 89 erosion. This was explained by a change in the predominant type of associations between platelike clay particles. Without organics the particles are mainly e dge-to-face (E-F) bonds in settled suspensions, resulting in a loose structure with la rge water content. The positively charged edges are preferentially associated w ith the negatively charged faces of the clay minerals. With increased organic content the large negativel y charged macromolecules of NOM associate mainly with the positively charged particle ed ges. The number of face-to-face (F-F) particle associations increases, with a reduction in the po rosity and water content of the flocs. Initial erosion rates, defined from the slope of the concentration curve at initiation of erosion, and critical shear stresses were lowest for the no-or ganic tests and highest fo r intermediate organic levels (0.006 to 0.009% carbon) where the struct ure was a mixture of f ace-to-face and edge-toface, creating a braced structure. At higher organic contents the edge-to-face structures began to break down and the initial erosion rate and the critical shear stress both increased. Duck and McManus (1991) conducted labor atory experiments confirming the influence of organic matter on settling veloc ity via enhanced aggregation. Settling tests were replicated after removal of organic matter from the sample s by oxidation with hydrogen peroxide. Settling tests were performed in natural water and di spersing solution for samples with and without organics. The results showed that without or ganics the accelerated settling associated with aggregation was not observed as it wa s in the sediments with organics. Organic matter also may contribute to more th an 60% of the volume of mud particles in situ (Greiser, et. al., 1996). Jarvis, et. al. (2005) conducted experiments on the growth, breakup and regrowth of flocs with NOM. They found that flocs that contain poly mer chains, which were broken during floc breakup, did not reform to the pr evious floc sizes since polymer destruction is not reversible. However, flocs formed from physical/chemical bonds were found to replicate PAGE 90 90 floc sizes in growth, breakup and regrowth. This finding implies that organic contributions to floc size will be more significant in low shear en vironments, such as lakes, reservoirs and the continental shelf than in high en ergy coastal and estuarine waters. The presence of living organisms in the benthi c zone leads to impacts on both cohesive and noncohesive sediment processes. A biofilm co ating on noncohesive mineral grains can make them behave cohesively and contribute to aggregation. Biological activity of living organisms can ha ve a negative or positive impact on the stability of the bed sediments. Biogenic stabilization results in a decrease in erodibility of sediment from the bed surface (Paterson and Da born, 1991). As mentioned bioturbation of the bed can also have either a positive or negative impact on stability (Paterson, 1997). Bioturbation creates challenges in analysis of depositional events for geologists when the biodiffusivity and organism density are sufficient to destroy the st ratigraphic record (Wheat craft and Drake, 2003). Benthic diatoms, which produce large amounts of polymer (mucilage), are particularly important (Holland et. al., 1974) in cohesive sediment processe s. These polymers form a stabilizing cohesive framework bridging between sediment part icles and diatoms (Paterson, 1997). Cyanobacteria and blue-green algae are also of major importa nce to bed stability (Wetherbee et al., 1998). Growth of bacteria, al gae and diatoms at the bed surface, filling voids and reducing hydraulic roughness can result in biosta bilization of the bed. The net effect is a reduction in the shear stress at th e bed and reducing erodibility. Olafsson and Paterson (2004) found in laboratory testing that the effects of biogenesis can be both physical and chemical. They showed that shear strength in the upper centimeter of the bed increased with the bed density of tube-bu ilding chironomid larvae, which build their tubes with silk that spreads over the bed surface adjacent to the tubes. PAGE 91 91 Tolhurst, et. al. (2008) inves tigated the formation of mi crophytobenthic biofilm in the laboratory over a 45-day test pe riod. They found that the er osion threshold was positively correlated with the water content, chlorophy ll a and colloidal carbohydrate within the upper 2 mm of the bed. However, the bulk density was highly variable. Lumborg, et. al. (2006) studied biogenic effects on an intertid al mud flat, finding that the cohesive sediment dynamics was more cont rolled by benthic biology than by physical parameters. Haubois, et. al. (2005) describe th e significance of motile epipelic diatoms to move vertically within the bed layer in intertidal mud flats, effec tively distributing the biological influence over a deeper zone. Clay particles with organic coatings have a high adsorption capacity for metals and hydrophobic organic contaminants (D ennett, et al., 1998) and play an important role in their transport and fate. The added variability of the degree of influence of biota and organics as a stabilizing effect on erosion and an influence on aggregation of both cohesive and noncohesive mineral particles makes cohesive sediment processes presently mo re unpredictable and variable, both in space and time. The variation in time is most significan t on a seasonal scale. However, the spatial variability makes the use of bulk parameters to characterize the behavior of such complex interactions a crude approximation. PAGE 92 92 Figure 2-1. Example particle size distributi ons, showing the MIT size classification. The flocculated San Francisco Bay sediment ha s a size distribution comparable to beach sand. PAGE 93 93 Figure 2-2. Example of size dist ributions for dispersed particles and for flocculated sediments in suspension expressed as a volume fraction. The symbols are the measurements for San Francisco Bay sediments. The lines ar e the fits of Equation 2-3 distributions (adapted from Kranck and Milligan, 1992). PAGE 94 94 Figure 2-3. Conceptual model of order of aggregate flocculation processes (adapted from Krone, 1963). Particle aggregate pa (p1a) Particle aggregate-aggregate paa Particle aggregate-aggregate-aggregate paaa Particle aggregate-aggregate-aggregate -a gg re g ate p aaaa (p 4a ) (p2a) (p2a) (p2a) (p2a) (p3a) (p3a) (p3a) (p4a) PAGE 95 95 Figure 2-4. Effects of the set tling decay term, K, on the particle size distribution spectra in volume fraction (adapted from Kranck and Milligan, 1992). PAGE 96 96 Figure 2-5. Measured particle size distributions in San Franci sco Bay: a) dispersed grain distributions and b) flocculated distri bution (from Kranck and Milligan, 1992; reprinted with permission). Figure 2-6. Dispersed particle distributions for nearby bed samp les for San Francisco Bay site (solid) and for suspended sediments in San Pa blo Strait. All dist ributions had a slope, m near zero (from Kranck and Milliga n, 1992; reprinted with permission). PAGE 97 97 Figure 2-7. Relationship of shear strength of flocs to the excess density of the flocs (data from Krone, 1963). The data values are from multip le harbors; curve fits are shown for each harbor, for all of the data combined. A curve fit with the exponent on the density of 2.5 (Partheniades, 1993) is shown in red. PAGE 98 98 Figure 2-8. Effect of salinity on settling velocity (adapted from Krone,1962). Final peak floc size was estimated from Stokes Law using the final peak settling velocity. PAGE 99 99 Figure 2-9. Comparison of Equati ons 2-17 and 2-18 for the drag coefficient as a function of the particle Reynolds number. PAGE 100 100 Figure 2-10. Comparison of Equations 2-26 and 2-31 for settling ve locity versus floc diameter (adapted from Winterwerp, 1999). PAGE 101 101 Figure 2-11. Settling velocity versus floc diameter from Chesapeake Bay and Tamar ( UK) estuary (adapted from Winterwerp, 1999) PAGE 102 102 Figure 2-12. Settling velocity versus floc diameter from VIS, Ems and Ems (adapted from Winterwerp, 1999). PAGE 103 103 Figure 2-13. Comparison of fit curves for individual data sets of fall velocity with a single fit to all data. PAGE 104 104 Figure 2-14. Fractal dimension fr om Equation 2-19 used in Equati on 2-31 as plotted in Figure 210; dfc = 8000 microns, Dfc= 2.6, d5 0 = 2 microns. PAGE 105 105 Figure 2-15. Data comparison of model for settling velocity with a power law for the fractal dimension (from Khelifa and Hill, 2006, reprinted with permission). PAGE 106 106 Figure 2-16. Effects of variable fractal dimension on the eff ective (excess) density as a function of floc diameter (from Khelifa and Hill, 2006). PAGE 107 107 Figure 2-17. Variation of settling velocity with suspended sediment concentration. Results based on field and laboratory tests using se diment from Cleveland Bay, Australia (adapted from Wola nski et al., 1992). PAGE 108 108 Figure 2-18. Schematic diagram showing the depe ndence of settling veloc ity on floc size in the flocculation settling range (a dapted from Teeter, 2001). Figure 2-19. Comparison of Equation 2-46 with settling data for varying initial concentration and shear rate (adapt ed from Teeter, 2001). PAGE 109 109 Figure 2-20. Paradox of simultaneous versus exclusive erosion and deposition: a) bed deposition, b) bed erosion. Permanent Bed Near bed Deposition wsC zC D z E=0 D wsC zC D z E + D = net D ??????? Exclusive Simultaneous a) Permanent Bed Near bed Erosion wsC zC D z E D=0 wsC zC D z E + D = net E ??????? Simultaneous Exclusive b) PAGE 110 110 Figure 2-21. Number of hours requ ired to settle a distance of one meter as a function of particle size, based on the Equation 2-29 curve in Figure 2-10. PAGE 111 111 Figure 2-22. Conceptual model of the feedback between aggreg ation and disaggregation with flow conditions (from Maggi, 2005; reprinte d with permission). Top: shows the temporal variation in the number of floc s within a unit volume. As aggregation occurs the number of particles is redu ced. With disaggregation the number of particles increases. Bottom: shows the floc size distribution (FSD ), which reflects the aggregation/disaggregation cycling. PAGE 112 112 Figure 2-23. Relationship between the modal floc diameter and shear stress and concentration (from Dyer, 1989). PAGE 113 113 CHAPTER 3 PROBABILISTIC DESCRIPTION OF COHESIVE SEDIMENT TRANSPORT 3.1 Conceptual Framework This chapter identifies those variables for c onsideration in the evaluation of the influence of probabilistic representation within CST processes related to bed exchange. One way to view CST processes is from the degree of uncertainty in the processes. Uncertainty can result from lack of adequate knowl edge of the processes or ability to measure them accurately (epistemic uncertainty), or from the random nature of the processes themselves (natural uncertainty) (M erz and Thieken, 2005). Epistemic uncertainty can be reduced through research and improvements in data collection tec hniques. Natural uncerta inty cannot be reduced, just better sensed. Improving the estimate of the natural variability, by definition, is epistemic uncertainty reduction. For CST processes, uncertainty in the size di stribution of the primary particles would be classified as epistemic uncertainty. The actual size distribution is a real entity that we are limited in estimating by measurement and analysis proce dures. Uncertainty in bottom shear stress is clearly controlled by natural uncertainty. The be st possible instrumentation will not change the fact that turbulence is a random process. Hybrid uncertainty arises from the interact ion of epistemic uncertainty with natural uncertainty. An example is the settling velocity of a particular particle size. The size of the particle is something that we could define w ith acceptable uncertainty, but the process of its falling through a fluid introduces some level of natural uncertainty, associated uncertainty in the drag coefficient for higher particle Reynolds num bers and interference be tween particles. The level of uncertainty in the fall velocity for a spherical grain is relatively small. However, variation in the shape increases the uncertainty. Flocs of different density, shape and fractal PAGE 114 114 dimension introduce even more uncertainty. Th e configuration in which cohesive particles combine in flocs is influenced by the detail s of the double layer around each particle. The variation in the size and shape of the particles introduces variabili ty in the cohesive bonds. The fractal dimension of sediment is a reflection of th e statistical tendency seen in a large number of sediment flocs. The details for an individual floc will deviate about the mean tendency. The uncertainty could be quantified as the variance in observed fa ll velocities for a particular equivalent size particle. Settl ing velocity contains both epis temic and natural uncertainty. In resolving the complexity of CST proce sses, a closer look at uncertainty in each component process is warranted. Figure 3-1 presents a simplified view of sediment transport processes, depicting the pathways of an individual sediment grai n as it transits through various states and locations. The sediment encounters various processes during the transitions between floc conditions in suspension, depos ition to and erosion from the bed. Table 3-1 lists the concepts of importance to sediment transport. The entities of interest are the water and the sediment, both having their associated properties. Properties are further divided into the basic properties: size shape a nd density for the sediment and temperature, conductivity and salinity for water. Density could be considered a basic property of water, but it is defined by the other properties (salinity, te mperature and suspended matter). A much longer list of water quality parameters could be listed, but are not pertinent here. Any uncertainty in each of these basic properties is epistemic, sin ce these properties are definitive tangible values. Table 3-1 also indicates where sediment and wate r may be located as they factor into CST. Water can be significant when its properties within the floc interstices vary from the ambient conditions within the flow. The sediment properties of significance depend on whether the sediment is in suspension or in the bed. PAGE 115 115 Water and sediment take on additional propert ies based on state properties, in the sense that they are in a condition that creates the prop erty. The fall velocity of the sediment comes from the state of settling, while the internal sh ear of water is created by the state of turbulent flow. State properties, it is assumed, are the resu lt of stochastic processe s and their uncertainty is natural. Primary processes within sediment transport are also listed in Table 3-1. With the exception of cases of laminar se ttling in quiescent conditions, a ll of the other processes are driven by the turbulence of th e flow. These processes are therefore considered stochastic. Bulk properties of both water and sediment are included in Table 3-1. Water can be characterized by its density and viscosity. Th e flow can be characterized by the level of turbulence (turbulent kinetic ener gy and turbulent dissipation). Th e sediment is characterized by critical shear stress for transport for sediment a nd the bed shear strength w ith respect to erosion, the floc strength, a threshold sh ear stress for deposition and the fractal dimension of cohesive sediment. The source of uncertainty for bulk prop erties can be both natural and epistemic. How to best deal with these variables within CST an alysis, in part, depends on the origins of the uncertainty. The primary cohesive sediment tran sport variables will be considered in greater detail. It is of value to clarify the definitions used herein of probabilistic variables and stochastic variables. Probabilistic repr esentation of variables defi nes the range and frequency of occurrence of specific values of the variable. The distributio n of those probabilities may represent fixed known values, such as the grain size distribution. A probabilistic distribution does not necessarily provide any insi ght into the processes that cont ribute to that distribution. Stochastic variables have a random characteristic. Their statistical (probability) distribution may PAGE 116 116 resemble the distribution of nonstochastic variables, but only when the duration of temporal averaging is sufficiently long to reach a stationary distribution. Stochastic variables may have significant variability when the time span is relatively short. In this chapter the focus is on use of pr obabilistic frequency distributions for both stochastic and non-stoc hastic variables. The variables can be represented in one of several ways in the analysis: The variable can be assumed to be a constant. The variable can be assumed to have a st ationary statistical distribution for fixed discretized values. The variable can be assumed to have a stationary st atistical distribution, with a continuous analytical specification that is easily integrated. The variable can have a non-sta tionary statistical di stribution, with e xplicitly discretized values, whose probability changes with time. 3.2 Particle Definitions The particle size distribution is a basic property of the sedi ment and requires explicit discretization for CST analysis. By incorporatin g size classes into the analysis, much of the probabilistic character is handled. If there is significant variability of the properties within a single size class, it can be addressed by increasing the number of size classes. The particle size classes need not distingui sh between flocculate d and non-flocculated sediments, primarily because their distributions do not normally overlap significantly (see Figure 2-5). Larger particles and smaller flocs within the same size range exhibi t significantly different densities. Some thought was given to the possibility of differentiating between the two. Teeter (2001) included separate classes for cohesive and noncohesive sediments, without interaction within aggregation and disaggregation. The definition of two classes of sediment is straightforward. However, in order to keep a distinction between particle s and flocs the particle PAGE 117 117 size distributions within the flocs would have to be tr acked through the aggregation and disaggregation processes. That effort is not de emed feasible at this time. Aggregation and disaggregation conceptual models used today can move sediment mass between size classes as appropriate, but do not track the variable particle sizes making up the flocs. 3.3 Shear Stress The shear stress to which sediment is expos ed is the primary property affecting bed exchange and overall CST processes. It cont rols both erosion and deposition and, through its relation to the internal shear, influences flocculati on. Furthermore, turbulence in the flow field is the most significant stochastic property among the variables in Table 3-1. As mentioned in Chapter 2, Partheniades (1965) developed an appr oach that handled the bottom lift force with a Gaussian probability distribution. The erosi on flux was developed by integrating the probability distribution over limits defined by a balance between the lift for ce on the sediment particles and the restraining forces of buoyant weight and c ohesion in an analogous approach to threshold motion in noncohesive sediment. The derivation fo llowed the approach of Einstein (1950) and a probability of erosion, Pe, was introduced as an integral of the normally distributed lift force (or shear stress) fluctuati on. The erosion flux, was estimated as shown in Equation 3-1. 21 2 11 1 2c LLL c LLLF F e F F M PM ed (3-1) The erosion rate constant, M is usually developed from laboratory erosion experiments for specific sediment and has units of kg/m2/s The combined forces of buoyant weight and cohesion are included in the force variable Fc.. The hydrodynamic lift is FL, and the variable L is the standard deviation of the nor malized lift fluctuation variable /LLFF the ratio of the PAGE 118 118 fluctuating lift force to the time-averaged mean lift. The value of L was estimated from measurements to be approximately 0.5. Sharma (1973) reported the variation in the relative turbulent intensity for air (defined in Equation 3-2), the ratio of the root mean square (RMS) turbulent ho rizontal velocity fluctuations to the mean velocity, to range between 14% to 16%. The relative turbulent intensity is a normalized standard deviation. 2u Relativeturbulentintensity u (3-2) Sharma also reported the normalized boundary shear stress standard deviation (replacing velocities with shear stresses in Equation 3-2) to have a range of 18% to 30% for air. He performed a numerical evaluation of the shear stress distribu tion, wherein he assumed that the equation for the mean shear stress as a power function of the mean velocity was also applicable to the instantaneous velocity fluctuations to esti mate the instantaneous shear stress. He showed that the estimated shear stress fluctuations were 43% to 93% higher than the shear stress variations measured by a flush mounted hot film probe. From this he co ncluded that the shear stress response to velocity fluctuations (at 1 mm from the boundary) is either not instantaneous or that there may be a better correlation closer to the boundary. The mean shear stress is computed by Equation 3-3, given the probability distribution function, y() of the instantaneous bed shear stress yd (3-3) The normalized standard deviation is computed via Equation 3-4, with the coefficients of skewness and kurtosis, via Equati ons 3-5 and 3-6, respectively. PAGE 119 119 21 y d (3-4) 3 3 31my d (3-5) 4 4 41my d (3-6) Rather than using the Gaussian distribution, Winterwerp and van Kesteren (2004) used an asymmetric probability distribution of the shear stress, y(),developed by Petit (1999) as shown in Equation 3-7. They argued that the process of turbulent ejections of vortices and subsequent sweeps downward into the bed create asymmetric sh ear stress distributions. The importance of these bursts of shear to bed interaction may be significant. 2 22() 2 21() (;,,)1 2 2mms m bmms ymserf e m ms (3-7) The variable is the mean value and m and s are shape variables for the distribution. The higher moments of this asymmetric distribution are computed by Equation 3-8. ;,,n n M ymsd (3-8) The parameters in the distribution fu nction are defined in Equation 3-9. 1 1 3 3 11231 8124 2M mMMMM (3-9) and 2 4 32 3 3 1123121 223 44 2sMM MMMM An example asymmetric distribution used by Winterwerp and van Ke steren (2004) was fit to a distribution of Equation 3-7 using = 0.03, m=1.01 and s=0.56 (presented in Figure 3-2, data from Obi, et al., 1996). In addition, they developed shear stress distributions for application PAGE 120 120 to three sedimentation experiments of Krone ( 1962) as shown in Figure 3-3. The distribution curves were fit to Equation 3-7 using a multi-pa rameter optimization algorithm to minimize the error in the fit to determine the parameters m and s given the mean, The parameters used in the fitting of Equation 3-7 to these data are presented in Table 3-2. A skewed shear stress distribution can be attributed to the nonlinear transformation of the velocity to a shear stress. A normally distributed velocity record will yield a skewed shear stress. Assume a normal distribution for the velocity To illustrate the transformation a Monte Carlo simulation was performed using the velocity cumulative distributi on function (CDF). The CDF spans the full range of velocity values fr om accumulative probability of 0 for the minimum to a cumulative probability of 1 for the maximu m. Random numbers were generated between 0 and 1 to select an associated velocity value fr om the velocity CDF. The shear stress was then computed for that velocity base d on Equation 3-10, assuming, as Sharma (1973) did, that it is valid for the fluctuating shear stress es from the fluctuating velocity. 22cfu f (3-10) The variables in Equation 3-10 are the friction coefficient, fc, the fluid density, f, and the velocity, u. The simulation used ten million random ve locity values and 200 partitions of both velocity and shear stress. The results of one si mulation are presented in Figure 3-4 for a velocity distribution with a 0.5 m/s mean velocity and a st andard deviation of 0.09 m/s. The variables used in Equation 3-10 were selected as 0.005 for fc and 1,027 kg/m3 for f. The Monte Carlo reconstitution of the inpu t normal distribution is plotted agains t the analytical normal distribution function to confirm the velocity distribution. The resulting shear stress distribution has a mean value of 0.650 Pa, and a standard deviation of 0.232 Pa. The normalized shear stress standard deviation is 35.4%. The coefficient of skewness is 0.535 and the coefficient of kurtosis is 3.372. PAGE 121 121 For comparison, a normal distribution has a skewness of 0 and a kurtosis of 3.0. The shear stress distribution is skewed toward hi gher shear stresses and the distribution is more peaked than a normal distribution. Monte Carlo simulations were repeated for the three experiments of Krone (1962) for which Winterwerp and van Kester en (2004) developed shear stress distributions. Results of these simulations are presented in Figures 3-5, 3-6 and 3-7 for the flume tests with mean velocities of 0.113 m/s, 0.134 m/s and 0.152 m/s. re spectively. The figures compare the assumed normal velocity distributions for each test with the stochastic velocity PDF generated by the Monte Carlo simulation and compare the Monte Ca rlo simulation shear stress distribution with the shear stress distributions pres ented by Winterwerp and van Kesteren. Standard deviations of the velocity distributions were adjusted in th e Monte Carlo simulations to get a match of the shear stress distribution curves of Winterwerp and van Kesteren. The parameters for each of the Krone test curves are summarized in Table 3-3. The effects of the standard deviation of the velocity di stribution on the shear stress distribution are summarized in Figures 3-8, 3-9 and 3-10 for mean velocities of 0.5 m/s, 1.0 m/s and 2.0 m/s, respectively The statistics of these sample dist ributions are presente d in Table 3-4. As expected, increasing the velocity standard de viation increases the standard deviation of the shear stress. The modal shear stress is reduced wi th increasing velocity standard deviation. The normalized standard deviation for the shear stress is amplified compared to the normalized standard deviation of the velocity distribution as shown in Figure 311. The slope of the trend in the Monte Carlo simulation data matches the exponent (2) on the shear stress-velocity relationship in Equation 3-10. Using Equation 3-10 for the transformation correlates the velocity and the shear stress and results in greater amp lification of the variance measured by Sharma PAGE 122 122 (1973). However, as Sharma concluded the spat ial separation in his measurements may account for the reduced correlation. This analysis illustrates that a specific shear stress distribution need not be explicitly specified if the velocity distri bution can be estimated and the sh ear stress relationship to the velocity is known. The bottom shear stress will be tested for the importance of probability distribution on the accuracy and sensitivity of simulations of CST processes. 3.4 Aggregation and Disaggregation Variables that enter into the aggregation processes include the concentration of the individual size classes, the inte rnal turbulent shear and the settling velocity of each floc size class. The aggregation model accounts for the variation in floc size and concentration explicitly. Internal shear can be developed as a probabilistic dist ribution if the bed shear is treated as a distribution. Collision probabilities are also rela ted to the number concentration of particles and their sizes. The settling velocity of flocs for a specific eq uivalent spherical diam eter may vary with floc density and shape, resulting in differences in their response to th e hydrodynamics. Specific measurements of settling will include both natural variations as well as errors in the sampling process itself. When large numbers of sediment particles are tested, the natural variability will tend to average itself out. Variations could be important when addressi ng differential settling related aggregation. However, if fluctuations in apparent settling velociti es are associated with turbulence, then the effects can be included in to the shear-induced aggregation. Hwang (1985) showed that sediment response to oscillating flows was to reduce the effective fall velocity, through inhibiting the attainment of terminal settling velocity. Variation in settling velocity may contribute to the comple xity of bed exchange. The settling velocity will be tested probabilistically. PAGE 123 123 3.5 Floc Density Floc density, for the majority of studies, is an inferred parameter. For example, it is often estimated from the particle size and observed settli ng velocity using Stokes law. This parameter is best handled as a static fractal property of the basic flocculation model and will not be treated probabilistically, in part, because much of its impact is realized through the fall velocity. 3.6 Floc Strength Floc strength is a function of the mineral ch aracteristics and environmental factors that influence the fractal structure of the flocs. C onditions under which the flocs form influence the strength of the flocs. Jarvis, et al. (2005) offers a review of met hods for estimating floc strength. Methods are either macroscopic or microscopic, with hi storical methods focusing on the macroscopic relationship between applied shear and the sh ear rate. New techniques measure the energy required to break individual floc s by tensile or compressive for ces. However, many microscopic techniques damage the flocs in the process of measurement. Relating the maximum floc size to floc strength based on fractal characteristics of flocs is a co mmon assumption in microscopic methods. The primary macroscopic method assumption is th at the strength of the flocs is manifested in the viscosity of the water-sedi ment mixture. Measurements of the viscosity can be related to the strength of flocs through documenting the thresholds for transitions between the consistent slopes (of the stress versus shear rate) with increasing shear rate. Reductions in the slope of the stress-shear rate plots have b een assumed to be due to the pr ogressive breaking of the highest order of aggregation flocs. The data have been interpreted as indicating the shear strength based on the order of aggregation as shown in Figure 312. Data from Krone (1963) based on applied PAGE 124 124 shear through a concentric cylinder viscometer showed a consistent relationship between the concentration and the floc strength (see Equation 2-6). The 5/2 exponent in Equation 2-6 was found to be consistent among different sediment types, while the proporti onality coefficient, kf, varied from site to site. The data are presented in Figure 3-13 along with Equati on 2-6 using a value of kf = 2.83x10-6 Pa/(kg/m3)5/2. This relationship is awkward to us e because of the units of kf. Arulanandan, et al. (1980) developed the re lationship between the erosion flux constant and the bed shear strength with respect to erosion for undisturbed settled beds within a flume. In the development of the relations hip they documented the range in the estimate of the critical shear stresses for each test. Those data provide in sight into the variability in the shear stresses. Those ranges are assumed to be approximately tw o standard deviations, and a coefficient of variability is estimated and plot ted against the mean shear stress in Figure 3-14. As the shear strength increases the variability decreases. The trend is a reflection of effects of consolidation. As sediments consolidate the floc structures collapse and the w eakest primary particle bonds are broken, resulting in higher strengt h. Consolidation essentially pr ogressively lowers the order of aggregation of the flocs that se ttle to the bed. It is inferre d that the largest magnitude of variability, associated with the lower shear st rengths are a result of a greater range in the strengths of the bonds within the flocs. The floc strength was defined by McAnally (199 9) as a modification of the development of Kranenburg (1994) as shown in Equation 3-11. 2/3 f D f ff wB (3-11) PAGE 125 125 The coefficients in this equation used in fitting the data of Figure 2-7 are presented in Table 3-5. The table includes the average norm alized residuals of the data values for each project site. The differences be tween the fit and the data were normalized by the fit value. The mean and standard deviations of those normalized re siduals are listed in Ta ble 3-5. The standard deviations ranged between 50% and 100% of the mean normalized residuals. The floc strength exhibits consid erable variation and uncertainty It will be evaluated as a probabilistic variable in the numerical evaluation. 3.7 Bed Exchange The estimation of the probability of erosion using a probability distribution for both the shear stress and the sediment strength is presented in Figure 3-15. For multiple variables with probability distributions, the probability that a part icular combination of values of the variables will occur can be estimated by the products of the individual probabilities for their values, defining their joint probability. The validity of the joint probability approach is based on the assumption that the variables are statistically independent. Figure 315 shows a probability distribution of shear stre ss that includes negative values in the lower tail of the distribution. Negative shear is associated with negative turb ulent velocity fluctuations due to complex turbulent processes. The potential for erosion exists for both positive and negative shear. The negative shear can be evaluated by taking the shear strength distri bution and folding it over the origin and integrating the negative shear tail. The modified form for the probability of erosion is presented in Equation 3-12, assuming that the proba bility of erosion is th e probability that the bed shear stress is greater than the shear streng th. This assumes that sediment mobility will result if the instantaneous shear stress is grea ter than the shear strength of the bed. If the instantaneous shear stress is less than the shear strength of the bed no erosion will occur. The probability distribution of the shear strength can be viewed as the variation in strength over a unit PAGE 126 126 surface area of the bed. Each le vel of instantaneous shear stress, if assumed uniform over the unit area of bed, may cause sediment mobility over some fraction of the bed area based on the fractional area for which the shea r stress exceeds the distributed shear strength. As the shear stress increases a larger fraction of the bed will experience erosion, until when the minimum shear stress is greater than the maximum shear strength area of erosion wi ll be the entire bed surface, or the probability of eros ion is 1. Note that the probability of erosion does not address the magnitude of the erosive flux, only whether eros ion is likely to occur. If the shear stresses are dramatically reduced the probability of er osion will approach zero. If the shear stress distribution is bounded, it will re ach 0 when the maximum shear st ress is less than the minimum shear strength. 00 00b sb sb beee ssbb ssbbpppfdfd fdfd (3-12) Alternatively, Equation 3-12 could be written as shown in Equation 3-13. 0 0bs s s bseee bbss bbsspppfdfd fdfd (3-13) Both of these equations can be expressed as shown in Equation 3-14, using a Heaviside step function. A Heaviside function takes a value of 0 if the ar gument is negative and a value of 1 if the argument is positive and a va lue of 0.5 for an argument of zero. PAGE 127 127 00 00s seeebsbssb bsbssbPPPffHdd ffHdd (3-14) The most convenient form for the probability of erosion comes from the recognition that the integral over the shear strength within the brackets in Equation 3-12 is the CDF function for the shear strength. This simplifies the probability of erosion to Equation 3-15. 0 0() ()sb sbeeebbbbbbPPPFfdFfd (3-15) This form is presented graphically in Figure 3-15. The CDF for the shear strength (which only includes 0 s values) is applied in a symmetric ma nner about the origin to facilitate integrating both the positive and negative shear st ress contributions to th e erosion probability. The PDF for the bottom shear stress is taken as it s signed value on each side of the origin to be consistent with the folding of the shear strength distribution. The nega tive contribution to the probability is usually insignificant because it mu st exceed the (hypothetically) negative value of the minimum shear strength before it makes a ny contribution and nega tive shear stresses contributions tend to be in the tails of both distributions, where the probabilities are both low. Physically, that means it does not happen that of ten and, when it does, the erosive flux will be small. The probability is made up of positive and negative shear stress contributions associated with turbulent velocity fluctuations that ch ange direction between downstream and upstream. The area under the shear stress probability distri bution that contributes to the probability of erosion is the product of the probability density distribution for the shear stress and the cumulative probability distribution for the shear stre ngth. This is compared to the equivalent PAGE 128 128 integral for a single-valued shear strength in Figu re 3-16, where the CDF of the shear strength is a heavyside step function as show n in Equation 3-16. The PDF of a single-valued shear strength is a Dirac delta func tion at the value of s. 00 001eeebbssbbbssbPPPfHddfHdd (3-16) As the mean shear stress increases, the inte grals of the probability functions will approach the value 1. The sensitivity of the varying shear stress to the changes in the variables b, the standard deviation of the bed shear stress, s, the standard deviation of the bed shear strength, and s are presented in Figures 3-17, 3-18 and 3-19, respectively. The selection of the standard deviation of the shear stress was based on the rang e of the ratio the standard deviation to the mean shear stress observed in Table 3-4. The m ean shear strength and standard deviation values were selected based on the data from Figure 3-14, with the standard deviation of the higher side so as to facilitate the demonstration of the effects. The greatest influence is the change in the mean shear strength. The sensitivity to changes in the standard deviation of the shear stress is a variation in the spread of the probability distri bution about the fixed 0.5 probability at a fixed shear stress. The greater the variability in the shear stress the greater the spread in the probability of erosion. The variation of the probabi lity with either of the standard deviations is for a steeper probability transition for the smaller st andard deviations. With a larger variation in either of the variables there is an increased region of overlap in their distributions. This results in a reduction in the probability of erosion for mean shear stresses greater than the mean strength and an increase in the probability for mean shear stresses lower than the mean strength. With small standard deviations, the onset of erosion o ccurs over a narrow range of shear stress. In the extreme case of t = s =0 the onset of erosion would be a dr amatic bed failure as the shear stress PAGE 129 129 reached the shear strength. The larger standard deviation would result in a gradual increase in bed erosion as the shear stress increased through the range of bed strength. Figure 3-20 presents in situ erosion flux data versus shear stresses co llected in Long Island Sound (Wang, 2003) during a twelve day period in December 1983 covering a variety of tidal and wave conditions. Hydrographic and suspended sediment data were collected from a bottommounted instrument array with sensors positi oned at 1 m above the bottom. The array was equipped with a two-axis electr omagnetic current meter, a compass, two water temperature sensors, a conductivity probe, and tw o red-light transmissometers wi th a 10-cm path length. The tidal conditions at the site have a mean tide range of approximately 2 m, with peak tidal velocities between 0.25 and 0.5 m/s. These data illustrate the scatter in field data, with the erosion flux varying by an order of magnitude for a given shear stress. Wang considered the temporal variation in the proce sses as contributing to the variability because there was little horizontal variation in the suspe nded sediment concentration. The presence of loose recently deposited sediment from the previous slack wa ter biased the erosion flux during subsequent erosion events. These data are shown primarily to illustrate a case where temporal variability leads to significant uncertainty in the processes when horizontal variability is small. The application of the reverse logic of the probability of erosion to the probability of deposition results in Equation 3-17. 0 01() 1()eb ebddd bbb bbbPPPFfdFfd (3-17) The integral in Equation 3-17 yi elds the logical finding that Pd = 1 Pe. The probability that the shear stress is greater than the bed strength plus the probability that it is less than it equals one. An example plot of the two curves for the case of e=0.25, b=0.3, e = 0.5 Pa is shown in Figure 3-21. The bed shear strength with respect to erosion and the shear threshold for PAGE 130 130 deposition are normally viewed as single-valued shear stress levels for a given sediment, with the value for erosion greater than the value fo r deposition (Krone, 1962; Partheniades, 1962; Ariathurai, 1974). If the criter ion for defining a threshold value is a 50% probability for either erosion or deposition, then the thresholds for erosion and deposition based on Figure 3-21 would be the same shear stress (approximately 0.57 Pa). If the same probability is to be applied to both erosion and deposition in estimating the threshold, then the deposition thres hold will be less than the erosion threshold only if the probability criteria are greater than 0.5 (see Figure 3-22). If it were less than 0.5, then the erosion threshold would be less than the deposition threshold (see Figure 3-23). The processes model will be tested for insights and sensitivity to a probabilistic formulation of the shear stress and the erosion strength with re gard to bed exchange. This research will evaluate the response of the model to use of probability di stributions for the shear stress and the erosion strength w ith regard to bed exchange separately and in combination. 3.8 Summary of Probabilistic Treatment The approach used in the model for the treatm ent of probabilistic variables is summarized in Table 3-6. The nature of the primary variables is summarized along with the method of treatment in the model. The discretization of th e variables is also list ed. Variables that are affected by the local hydrodynamics and concentrat ion will require computat ion for each cell in the water column. Variables associated with pa rticle sizes require development by class. The mineral density is a specified uniform constant. The current velocity, floc shear strength and the settling velocity have a normalized stochastic st andard deviation specifie d, then the stochastic probability distribution developed for a normal dist ribution with the specified mean and standard deviation. Those distributions maybe associated with cells in the vertical over the water column or for size classes. The bottom shear stress stoc hastic probability distribution is developed from PAGE 131 131 a transformation of the velocity distribution. The TKE, its dissipation rate and the vertical velocity distribution are solved from their governing equations. Many of the remaining variables are probabilistic as a result of dependence on the sp ecified probabilistic variables. However, the expected values are the reported values. PAGE 132 132 Table 3-1. Variables important to cohesive sedi ment transport processes Entity Basic property Location State property Processes Bulk property Water Temperature Flow Internal shear Momentum transfer Density Conductivity Floc interstices Velocity Viscosity Salinity Bed TKE TKE Dissipation Cohesive Sediment Grain size Suspen sion Floc size Aggregation Df Grain density Bed Floc density Disaggregation d Grain shape Settling e Hindered settling Shear strength Erosion Entrainment Deposition Consolidation PAGE 133 133 Table 3-2. Parameters in the shear stress distributions shown in Figure 3-3 Velocity (m/s) Mean shear stress (Pa) M s 0.113 0.0305 0.00828 0.00797 0.134 0.0415 0.01146 0.01069 0.152 0.0515 0.01459 0.01320 Table 3-3. Summary of Monte Carlo simulation of the Krone fl ume deposition experiment shear stress distributions Mean velocity (m/s) Velocity Standard deviation (m/s) Reported mean shear stress (Pa) Simulated mean shear stress (Pa) Shear stress standard deviation (Pa) Coefficient of skewness Coefficient of kurtosis 0.113 0.0220 0.0305 0.0293 0.0111 0.570 3.43 0.134 0.0243 0.0415 0.0410 0.0145 0.533 3.38 0.152 0.0260 0.0515 0.0526 0.0176 0.502 3.32 Table 3-4. Statistical parameters for example pr obability distribution curves in Figures 3-8, 3-9 and 3-10 Velocity (m/s) Shear stress (Pa) Normalized u v Skewness Kurtosis /vu / / /vu 0.5 0.06 0.638 0.154 0.360 3.170 0.12 0.241 2.011 0.5 0.07 0.642 0.180 0.419 3.232 0.14 0.280 2.003 0.5 0.08 0.646 0.206 0.478 3.301 0.16 0.319 1.993 0.5 0.09 0.655 0.232 0.535 3.372 0.18 0.354 1.968 1.0 0.12 2.514 0.613 0.362 3.167 0.12 0.244 2.032 1.0 0.14 2.527 0.716 0.421 3.230 0.14 0.283 2.024 1.0 0.16 2.542 0.820 0.480 3.303 0.16 0.323 2.016 1.0 0.18 2.560 0.924 0.538 3.381 0.18 0.361 2.005 2.0 0.24 10.258 2.454 0.336 3.069 0.12 0.239 1.994 2.0 0.28 10.293 2.836 0.341 2.955 0.14 0.276 1.968 2.0 0.32 10.309 3.182 0.320 2.805 0.16 0.309 1.929 2.0 0.36 10.297 3.489 0.287 2.662 0.18 0.339 1.882 PAGE 134 134 Table 3-5. Coefficients of Equation 3-12 for fits to data sets in Figure 2-7 and the normalized deviation between the fit and the data Normalized residuals of fit Project Bf (Pa) Df Mean Standard deviation Wilmington District 45.3 1.94 0.216 0.199 Brunswick Harbor 726.6 2.26 0.056 0.028 Gulfport Channel 155.0 2.02 0.165 0.116 San Francisco Bay 113.1 2.30 0.244 0.133 White River 139.1 2.01 0.165 0.111 Table 3-6. Probabilistic treatm ent of significant CST variables Parameter Nature of variable Treated by code Discretization Mineral density Constant Defined constant uniform Particle/floc size Probabilistic Explicit probabilistic by class Floc density Fractal Fr actally defined by class Settling velocity Probabilist ic stochastic Specified stochastic by class and cell Concentration Deterministic Solved governing equations by class and cell Floc shear strength Probabilistic Specified stochastic by class Shear strength of bed Probabilistic Derived stochastic by class Velocity Probabilistic stochas tic Specified stochastic by cell TKE Deterministic Solved governing equations by cell TKE dissipation Deterministic Solv ed governing equations by cell Eddy viscosity Probabilistic stochas tic Resultant stochastic by cell Bottom shear stress Probabilistic stocha stic Derived stochastic ay bottom Local shear Probabilistic stocha stic Derived stochastic by cell Internal shear rate Probabilistic st ochastic Derived stochastic by cell Aggregation Probabilistic stochastic Resultant stochastic by class, cell Disaggregation Probabilistic stochastic Resultant stochastic by class, cell PAGE 135 135 Figure 3-1. Conceptual view of CST processes. Shear stress Velocity Shear Primary particles Disaggregation Turbulent mixing Entrainment Erosion Deposition Shearing Hindered settling Settling Aggregation Primary aggregate Consolidation PAGE 136 136 Figure 3-2. Fit of Equation 3-5 to sample data set (data from Obi, et al., 1996). PAGE 137 137 Figure 3-3. Shear stress distri butions used by Winterwerp and van Kesteren (2004) developed from analysis of Petit (1999). Parameters for the fits were developed by a twoparameter error minimization algor ithm in applying Equation 3-7. PAGE 138 138 Figure 3-4. Effect of normally distributed velo city distribution on the shear stress distribution based on a Monte Carlo simulation of Equa tion 3-10. The stochastic velocity distribution is confirmed against th e analytical normal distribution. PAGE 139 139 Figure 3-5. Transformation of a mean 0.113 m/s normally distributed velocity with a standard deviation of 0.022 m/s to the shear stress distributi on of Winterwerp and van Kesteren. PAGE 140 140 Figure 3-6. Transformation of a mean 0.134 m/s normally distributed velocity with a standard deviation of 0.0243 m/s to the shear stre ss distribution of Winterwerp and van Kesteren. PAGE 141 141 Figure 3-7. Transformation of a mean 0.152 m/s normally distributed velocity with a standard deviation of 0.026 m/s to the shear stress distributi on of Winterwerp and van Kesteren. PAGE 142 142 Figure 3-8. Transformation of 0.5 m/s mean ve locity to shear stress for varying standard deviation in the normal velocity distributi on. The solid symbols are the velocity distributions. The open symbols are the sh ear stress distributions with the same symbol as the associated velocity distribution. PAGE 143 143 Figure 3-9. Transformation of 1.0 m/s mean ve locity to shear stress for varying standard deviation in the normal velocity distributi on. The solid symbols are the velocity distributions. The open symbols are the shear stress distributi ons, with the same symbol as the associated velocity distribution. PAGE 144 144 Figure 3-10. Transformation of 2.0 m/s mean ve locity to shear stress for varying standard deviation in the normal velocity distributi on. The solid symbols are the velocity distributions. The open symbols are the shear stress distributi ons, with the same symbol as the associated velocity distribution. PAGE 145 145 Figure 3-11. Ratio of the standard deviation of velocity to the m ean velocity plotted against the ratio of the standard deviation of sh ear stress to the mean shear stress. PAGE 146 146 Figure 3-12. Data from Krone ( 1963) relating the floc strength to the order of aggregation of sediments from a number of harbors. PAGE 147 147 Figure 3-13. Data from Krone (1963) showing the effects of c oncentration on shear strength of flocs from a variety of harbors. PAGE 148 148 Figure 3-14. Variation in shear strength of natu rally deposited cohesive bed as a function of the mean shear strength (data from Arulanandan, et al., 1980). The lin e is a regression fit to the log-transformed variables. PAGE 149 149 Figure 3-15. The probability of erosion when both the shear stress and the bed shear strength with respect to erosion are represented probabilistically. The CDF for the shear strength is used in the integral in Equation 3-18 in conjunction with the PDF for the shear stress. PAGE 150 150 Figure 3-16. Integration of a probability distri bution of shear stress for the case of a singlevalued bed shear strength with respect to er osion. The PDF for the shear strength is a delta function and the CDF is a heavyside function. PAGE 151 151 Figure 3-17. Shear stress standard deviation, b, effect on the probability of erosion for the case of s = 1 Pa and s = 0.25 Pa. PAGE 152 152 Figure 3-18. Shear strength standard deviation, s, effect on the probability of erosion for the case of s = 1 Pa and b /b = 0.30. PAGE 153 153 Figure 3-19. Effect of the variable s on the probability of er osion for the case of s = 25% and b /b= 0.30. PAGE 154 154 Figure 3-20. Variation of erosion flux with sh ear stress for data in Long Island Sound (after Wang, 2003). Variation is in part due to differing tidal conditions and wave energy, with the largest erosion associated with storms. PAGE 155 155 Figure 3-21. The variation of the probability of erosion and deposition with the mean shear stress for s = 0.5 Pa, s = 25% and b = 30%. PAGE 156 156 Figure 3-22. Probability threshold greater than 0.5 (0.75) used for definition of critical shear stresses leads to a critical shear stress or erosion greater than the critical shear stress for deposition. PAGE 157 157 Figure 3-23. Probability threshold less than 0.5 (0.33) used for definition of critical shear stresses leads to a critical shear stress or erosion less than the critical shear stress for deposition. PAGE 158 158 CHAPTER 4 MULTI-CLASS DEPOSITION WITH AGGREGATION 4.1 Introduction This chapter details the overall implementation of the basic concepts from Chapter 2 and the special features associated with the probabi listic treatment developed in Chapter 3 into a numerical procedure. The first topics addressed within the chapter address the sediment transport model. These include: : Sediment size class definition Discretization of the sedi ment transport equation Settling velocities by class Vertical mixing of sediment Bed exchange Aggregation Disaggregation This chapter also discusses the implementati on of the hydrodynamics within the numerical model. A general description of the numerical solution method a nd probabilistic integration is discussed. A general description of the numerical program is outlined. A series on analytical test cases is presente d to close this chapter that illustrates the accuracy and consistency of the numerical impl ementation of the governing equations. These test cases are presented here because each one ha s an analytical solution that can be compared with the model results. PAGE 159 159 4.2 Sediment Transport 4.2.1 Sediment Size Classes The model has optional specification of the sedi ment classes either by particle diameter or by particle mass. If sediment interaction is viewed as a pure deterministic calculation for single sized particles for each class with pr ecise mass balances during aggregation and disaggregation, then specifying the sediment cl asses by mass would be the best method of size class specification. Defining the particle sizes su ch that the particle masses are integer multiples of the smallest mass class will lead to results that guarantee precise mass conservation. However, if the class concentrations are viewed as the result of a distribution of particles within the size range for the class, then explicit acc ounting for individual floc interactions becomes inconsistent and the mass conservation must be a ssociated with the concentrations rather than numbers of particles. The classes can be de fined by particle diamet er and mass conservation incorporated directly into the flocculation model. If the floc size is used to define size classes fractional flocs can be processed by the aggregati on model If the deterministic approach is used, then the calculations are essentially dealing with integer math for the individual flocs and flocs cannot get combined precisely unless they create masses for which a larger class exists. Therefore, sediment classes were defined by a logarithmic progression of grain/floc diameter as in Equation 4-1. The ratio of successive size classes is a constant, and the difference between the logarithms of successive size classes is a constant. The subscript i refers to the sediment size class, so that the i-th class size will be a factor times the previous size class, i-1. 1 1 idi imidd ormm (4-1) PAGE 160 160 The coefficient of the progression, is defined by Equation 4-2 based on the minimum particle size, dmin, maximum particle size, dmax, and the number of sediment size classes, Mclass. If the size classes are defined based on the mass of the particles, then dmin and dmax are the sizes associated with particle masses mmin and mmax, the range of mass to be represented. Equations 41 and 4-2 give the appearance of a fractal represen tation. If the number of size classes is large enough, careful selection of the si ze range can result in a value of that provides advantageous features, taking on a fractal char acter. The difficulty arises from the differing character of aggregation and disaggregation. If disaggregation was simply the reverse process of aggregation, the selectio n would be trivial and = 2 would lead to reason able results. However, aggregation is not a simple reversible process, with collisional disaggregation resulting in fractional breakage products on the order of 3/16 the original mass (see Section 4.6.2). Note that for nontrivial specifications of the particle size classes, Mclass will always be greater than 1. 1/(1) 1/(1) max max min min;class classMM dmdm dm (4-2) As long as a reasonable number of sediment classes is used, the method of defining the sediment classes should not have an impact on the results. For example, if the fractional masses associated with collisional breakage of flocs are to be accommodated in the size distribution we would require that m is on the order of 1/(1-3/16) =1.23. Note that to use Equation 4-2 the number of classes must be greater than one. The size classes will span the combined range of disaggregated grains and aggregates, allowi ng the aggregation model to rearrange the distribution based on the processe s. To span a size range of 1 to 1000 microns would span a particle mass of approximately 1.8x10-15 kg to 5.4x10-6 kg. This would require approximately 100 size classes for m =1.23. The differentiation between the dispersed grain size distribution PAGE 161 161 and small flocs composed of the smallest of grai ns cannot be directly addressed within a single size class progression as described above. The total suspended sediment concentration is defined as the sum of the concentrations of the individual size classes (Equation 4-3). 11class classMN ii iiCcc (4-3) Separate Nclass silt classes are included in Equation 4-3, which would have its own size distribution. The silt sizes (2 to 62 microns) are assumed to be noncohesive. In subsequent equations silt classes will not be explicitly separated. 4.2.2 Sediment Transport Equation The starting point for development of the se diment transport mode l is the conservation equation for the mean concentration of each sediment size class (Equation 4-4i). 3 3 jsii jsni iji i i imt j j jjjjwc wc uc cc SDD txxxxx (4-4i) All variables have been previ ously defined. Equation 4-4i is three-dimensional and applies to any point in the flow system in its present form. The advective velocity, iju, is subscripted with the particle class i as well as the coordinate j, admitting the possibility of differential particle and fluid velocities. Both the mean and turbulent fluctuating settling velocities, s iw and s iw, are functions of the size class. The diffusion is at this point separated into molecular and turbulent mixing coefficients. The vertical dimension is defined by the x3 coordinate. The boundary conditions are mass fluxes along all bo undaries. The conventional water surface (z = zws) boundary condition balances vertic al diffusion with the net ver tical advective movement of PAGE 162 162 sediment. At the bottom of the water column (z = zb) the difference in that balance equals the net erosion from the bed. These are expressed in Equation 4-5. 3 30ws bi siimt zz i s iimt ii zzc uwcDD z c uwcDDED z (4-5) If the vertical water velocity is assumed to be zero then the surface boundary condition matches the settling flux with the diffusive fl ux. The lateral inflow/outflow boundaries must have either an inflow (or outflow) flux or a diffusive flux. The most common diffusive flux boundary is zero, achieved by setting the concentrati on gradient to zero. Lateral walls have a total flux set to zero, since the advective flux w ill be zero, then the diffusive flux will also be zero. The source/sink term, Si, in Equation 4-4i includes interactions between sediment classes. Size class i can gain particles from aggregation of sm aller particles into this class or from disaggregation of larger flocs that have class i particles as a remnant of floc breakup. Class i can lose particles due to disaggregation by shear breaking of flocs or from collisions of flocs. These categories of interactions are delineated as shown in Equation 4-6. iiAiAiBiBiCiCSGLGLGL (4-6) The terms in Equation 4-6 represent gains ( G ) and losses ( L ) of particles in class i associated with the processes of aggregation ( A ), floc breakage ( B ) due to shear and due to collisions ( C ). Each of these terms will be discussed in section 4.2.7. Collision terms are not present in all floccu lation models (Maggi, 2005). In estuarine environments many assume that all collisions result in aggregation (Friedlander, 2000). McAnally (1999) and Lick and Lick (1988) include collisions as a disaggregation potential. PAGE 163 163 Burban, et al. (1998) found that fluid shear had no significant influence on disaggregation in the range of 100 to 600 s-1. They also concluded that three-body collisions were necessary to account for the reduced median floc size as the concen tration increases. They acknowledged that floc breakage from collisions may be an indirect result of the shear. The governing equation for the to tal suspended concentration, C comes from the summation of the Mclass equations to obtain Equation 4-7. 33 11 11 1 1class class class class class classMM MM jsiijsi i iji ii ii jjj M M i i imt j i jjwc wc cuc txxx c SDD xx (4-7) 3 j is the Dirac delta function (= 0 for j=1,2, and =1 for j=3). The advect ive velocity has been assumed to be the same for the fluid and sediment so that the velocity moves outside the summation on the advective term. A representative bulk settling ve locity can be defined based on concentration-weighted class settling velocities as shown in Equation 4-8. 11 1class class classMM s ii sii in s M i iwcwc w C c (4-8) It is asserted that the net effect of aggregation and di saggregation over all floc size classes has to balance, so that 10classM i iS This and Equation 4-8 can be substituted into Equation 4-7 to yield Equation 4-9. 3 3 1classM jsi i js i j mt j jjjjwc wC uC CC DD txxxxx (4-9) PAGE 164 164 The remaining summation term in Equation 4-9 will vanish if the settling velocity is taken as a constant, with no variability (0siw ), which is not a reasonable assumption. If the sediment concentration is to be treated only as an expected value the effects of this remaining term can be incorporated into the tur bulent diffusion coefficient (Equation 4-10). 3 1classM tj s nj n n n jC Dw cu c x (4-10) Many of these assumptions may never be i nvoked because the settling and diffusion are simulated for each separate size class rather than for the total concentration, which is basically a summation step once the individual size class concentrations are known. The primary transport mechanisms for sedime nt movement in the vertical are through gravitational settling and vertical diffusion. In some extreme cases of complex flows, such as around bridge piers or in macrotidal estuaries, ve rtical accelerations of flow can move sediments vertically through the vertical co mponent of advective velocity. 4.2.3 Settling Velocities Settling velocities have been discussed in Chapter 2. The progression of estimates of settling velocities begins in low concentrations where the free se ttling velocity is a function of particle size and density. As th e concentration increases above the limit for free settling (around 0.05 to 0.3 kg/m3), flocculation increases because of the larger numbers of particles in suspension, causing larger flocs with increased settling veloc ity up to the upper limit of flocculation settling (8-13 kg/m3). Beyond this peak concentra tion the rate of settling is progressively hindered with increasing concen tration through fluid mud concentrations. Particle fall velocities are coded in the mode l based on a 3-step computation. These steps are: PAGE 165 165 Free settling velocity is com puted based on Equation 2-22, which is the Stokes fall velocity using a coefficient of drag from Equation 2-25. Effects of flocculation are then included based on Equation 2-26, which accounts for the density effects through the fractal dimension and the floc size. Effects of concentration and internal shear are then in cluded through Equations 2-40, 245 and 2-46. 4.2.4 Vertical Mixing Vertical mixing in the sediment model is handled as a turbulent mixing coefficient proportional to the turbulent eddy viscosity. The estimation of turbulent eddy viscosity is from either an assumed logarithmic profile or by application of a k turbulence model over the water column. Vertical distribution of eddy viscosity for a logarithmic profile is estimated from the shear velocity, as in Equation 4-11. *1tz uz h (4-11) An alternative estimation of the mixing was developed by the application of a kmodel. The model was based on the standard high-Reynolds number kmodel (Hanjalic, 2004; Speziale, 1998; Winterwerp, 1999) with adjustment s for the effects of suspended sediment (Hsu, et. al., 2007) and for localized low Re ynolds number flows (Hanjalic, 2004). The governing equations for the turb ulent kinetic energy per unit mass, k, and the turbulent kinetic energy dissipation rate, are given in Equations 4-12 and 4-13. 3 jj ii TT Tj jjkjjijTjuk u uu kkg txxxxxxx (4-12) PAGE 166 166 1 2 3321jj ii T T j jjj i j T j Tjuu uu c txxxkxxx g cc kxk (4-13) These comprise the standard high-Reynolds number form of the equations with coefficients as provi ded in Table 4-1. When the k model is used the turbulent eddy visc osity is defined (E quation 4-14) as a function of k and 2 Tk c (4-14) In uniform flow Equations 4-12 and 4-13 simplify to Equations 4-15 and 4-16. 2 TT T Tkkug tzzzz (4-15) 2 2 1321TT T kTug ccc tzzkzkzk (4-16) If the flow is at steady state, where the vertic al turbulence structure is stabilized, the time terms are omitted and the equations reduce to Equations 4-17 and 4-18. 2 TT T Tkug zzzz (4-17) 2 2 1321TT T kTug ccc zzkzkzk (4-18) Boundary conditions for the equations are given in Equation (4-19). PAGE 167 167 23 ** 00; 0 0; 0bb s s s szz zz zz zz zz zzuu k kz c k kor or zz (4-19) A correction for the presence of suspended sediment in Equation 4-15 was proposed by Hsu, et. al. (2007), as shown in Equation 4-20. This form ignores the advective tr ansport of k. 21 1 2 11T T T Cp Lk k tzz usk sg zzT T (4-20) The volume concentration of sediment, and the specific gravity of the sediment, s, are used to approximate the effects of two-phase flow features. The corresponding dissipation equation is given in Equation 4-21. 2 2 1231 1 2 11T k T T CpLtzz us k cccsg kzkkzTT (4-21) Hsu, et al. (2007) modified the bottom boundary condition for the k -equation (Equation 4-20) from a Dirichlet (specified k ) to a Neumann condition ( 0bzzk z ) due to the complexities of the sediment-turbulence interacti ons near the bottom, which make the definition of a concentration an over-c onstraint. The bottom boundary condition for the dissipation rate was specified as a Dirichlet condition ( 3/43/2bzzCk z ). The specific gravity, s is computed via Equation 4-22, taking into ac count the density of the flocs as well as the silt concentration. PAGE 168 168 ,, 1, 1, f ijfijssij iMclass iNsilt j wjcc s C (4-22) The index j in Equation 4-22 corresponds to the j -th cell in the vertical, indicating that the specific gravity of the composite se diment mixture will be a functi on of the mix of floc sizes and silt concentrations. In the estuarine environment, there are periods close to slack water for which the highReynolds number assumption used for Equations 4-12 through 4-21 may not be satisfied. The coefficients in Table 4-1 are assumed valid for wall Reynolds numbers above 100. The wall Reynolds number may be defined as in Equati on 4-23, using the distance above the bottom, z, as the length scale. 1/2Ret tkz (4-23) Corrections for low Reynol ds numbers within the kmodel are described by Hanjalic (2004), which involve adding damping terms in Equation 4-14 (Equation 4-24). 2 Tk cf (4-24) The damping term, f, is defined by Equation 4-25. 23.4 exp 1Re/50tf (4-25) In addition, a damping term, f is added to the dissipation term in the turbulent dissipation equation, as shown in Equation 4-26. PAGE 169 169 2 2 1231 1 2 11T k T T CpLtzz us k cc fcsg kzkkzTT (4-26) The dissipation damping factor is defined by Equation 4-27. 210.3expRetf (4-27) The variable Tp is the particle response time and was estimated by Hsu, et. al. (2007) as in Equation 4-28, using Stokes law with a correction for hindered settling. 3 21 18sf p wd T (4-28) The variable TL is the turbulent eddy time scale a nd is estimated from Equation 4-29. 6Lk T (4-29) The conventional argument is th at these corrections are needed in the near bottom layer where viscous effects become more significant. However, Hanjalic (200 4) points out that the need arises more from the inadequacies of th e linear eddy viscosity m odel used in the stand ard kmodel and the lack of isotropy in the turbulence near a wall. 4.2.5 Bed Exchange The simulation model incorporates options for the specification of the deposition and erosion fluxes at the bed interface. These in clude decisions on whether erosion and deposition are allowed to occur simultaneously or exclusively one or the other, whether to incorporate the probabilistic variables in the estim ations of probabilities of eros ion and/or deposition and which variables to treat probabilistically. PAGE 170 170 When deposition and erosion at the bed ar e handled in the model as simultaneous processes, they are treated as described in Section 3.7, based on the comparison of the sediment shear strength and the shear stress at the bed. Each individual sedi ment class is treated separately. The depositional flux is straightforward for individual floc size classes from the class settling velocity and concentration. The erosiv e flux associated with in dividual floc sizes is more of a challenge. When flocs settle into contact with the bed, th ey start to lose their characteristics and begin to coalesce into more pe rmanent contact with other sediment within the bed. Immediately after depositi on their erosive resistance is likely very similar to the floc strength of resistance to shearing disaggregation. Krone (1963) s uggested that if flocs deposit to the bed without breakage by the shear stresses near the bottom, the bed surface will initially have flocs of one order of aggregati on higher than the settling flocs. With continued deposition the the weight of flocs over the deposited layer can reach the point where the stress exceeds the shear strength of the deposited flocs and the layer will begin to crush at the lower layer. Krone reasoned that this crushing of flocs into the be d is a key mechanism in development of erosion resistant shoal material in variable flows. The treatment of the erosive flux by Partheni ades (1965) was partly based on concepts used by Einstein for noncohesive sediments. The primary development starts with the statement (Equation 4-30) that the number of sediment particles eroded pe r unit time per unit area of the bed, N can be expressed as the probability of eros ion divided by the area of the bed occupied by a single particle and by a characte ristic time scale of erosion, tb. 2 2 e bP N dt (4-30) The coefficient 2 is an area shape factor. Einstein assumed that the time scale is proportional to d/ws, the ratio of the particle size to the settling velocity of the particle. For PAGE 171 171 noncohesive particles the focus is in the near-bed zone, where the majority of mobilized particles fall quickly back to the bed. He related the be dload transport to the pr obability of how quickly the particles return to th e bed, correlated to the distance trav eled while saltating along the bed. For cohesive transport much of the resistance to mobility comes from the cohesive bonds. Once overcome, the particles are likely to travel signif icant distances before returning to the bed. The distances above and along the bed take on the scale of the water de pth. The focus of the time scale for bed interaction in cohesive transport becomes the time scale on th e particle entrainment itself. For most cohesive transport models th at address the vertical variation the analogous particle saltation is modeled explicitly by the simulation. The expression in Equation 4-30 was develope d for either uniform particle size or a representative size. If a variation in particle sizes is considered, every term in the equation will become particle size dependent (except for 2 if spheres are considered.). Equation 4-31 is a modification of Equation 4-30 for a particle size among others that occupies a fraction of the bed, fi. That fraction is applied to the number rather than the area of the particle. 2 2 ei ii ibiP Nf dt (4-31) The mass flux is essentially the number for each class multiplied by the mass of each size class particle. It is reasonable to expect that the time response will be dependent on the intensity of the erosive force in addition to the particle size. An alternat ive choice for a time scale would be */bititdu where t is a scaling factor. If the particle mass, mi = 1 di 3, is incorporated into Equation 4-30 we obtain an estimate for th e erosive mass flux for a single class (Equation 432). 1* 2 iiiieibiei tu mNfPCufP (4-32) PAGE 172 172 The result of this expression is that the basic erosion rate coefficient, Cb, is independent of the particle size if the scaling factor t is independent size. However, with *u the erosion flux would be nonlinear in b This retains an empirical coefficient Cb, with units of concentration which is still likely to be dependent on the c ohesive properties of the sediment (e.g. mineral composition, cation exchange capac ity, organic material). If t is dependent on the particle density, then the coefficient Cb would be different for each size class ( Cbi). Prooijen and Winterwerp (2009, in publication) po inted out that if the erosion rate constant is a function of the shear stress, then it needs to be incorporated within the integral of Equation 3-16. With this approach the integral is no longer the development of a probability of erosion, but an integral of the incremental contribu tions to the total erosion. This modification is shown in Equation 4-33. 1/2 00 00 1/21bbbssb bi i bbbssbfHdd Cf E fHdd (4-33) When the shear stress and shear strength in Equation 4-33 are both represented probabilistically the erosion efficiency will invo lve a double integral of the probabilities after expressing both *u and as functions of b. This functional relationship is pr esented against the erosion data of Partheniades (1965) in Figure 4-1, assuming a class inde pendent erosion concentration, Cb, of 0.859 kg/m3, using the coefficient Cb =0.8585 kg/m3, w=1000 kg/m3, and fi=1. The variables us ed in developing the probability of erosion distribu tion (via Equation 3-16) were s=0.55 Pa, s=0.25 Pa and b=0.3 Pa. Also included in Figure 4-1 is the linear ex cess shear stress formulati on of Ariathurai (1974), PAGE 173 173 using a value of ce=0.32 Pa and an eros ion rate constant M =0.03125 kg/m2/hr/Pa (Equation 251). The derivation of Partheni ades essentially assumed thatbt is independent of the flow conditions, which leads to an expression that places an upper bound on the erosive flux at M (see Equation 3-1), since the probabi lity of erosion should never be greater than unity if the probability integration approach is used. The form of Equation 4-32 continues to monotonically increase erosion flux with increasi ng shear stress. Also plotted in Figure 4-1 is the form of the equation if the probability of erosi on is set to 1 over the full range of shear stress. This provides the asymptote toward which the erosion converges as the probability increases toward one. The effect of taking the time constant inside the inte gral on the erosion rate (Equation 4-33) is also plotted in Figure 4-1, using a seawater density (1030 kg/m3) and a value of Cb = 0.112 kg/m3. When the probability of deposition becomes small for a particular size class, but the next larger size class is experienci ng deposition, there is a tendenc y for coupling of the smaller sediment class to the larger because of wake effects of the larger flocs (Teeter, 2001; Dent, 1999). The coupling of size classe s used by Teeter (2001) is inco rporated as an option (Equation 4-34). 11 12 ii i icF F c (4-34) The depositional flux for the ith sediment class is Fi, the concentration ci and the coefficients 1 and 2 are controlling parameters for the coupling. Note that there is an implicit assumption that 1 iidd Equation 4-34 was applied when th e probability of deposition for the size class was less than 0.05. The preliminary de velopment here was to apply the equation in a cascading manner from the largest size class down to the next higher class. However, that method would have no flux correction if the concentration of he next higher class were small, but PAGE 174 174 there could be significant flux from even larger sizes. Therefor e, the application of Teeters flux correction has been revised (Equation 4-35) to a pply the summation of fluxes of larger classes. In addition, the correction has b een revised to include the fluxes of both the cohesive sediments and the silt classes into the correction. 1,,, 1,, 1 1 ,, ,,2 ,,2 1 1 MclassNsilt MclassNsilt cicksk sick sk kikj kjki ci si MclassNsilt MclassNsilt cksk cksk kikj kjkicFF cFF FF cc cc (4-35) The size class j in the summations of Equation 4-35 refers to the smallest size class of other sediment type (coh esive or silt) with a gr eater settling velocity For the assumption of simultaneous deposition and erosion, the depositional flux is always included. Erosion is only eviden t when it exceeds the depositional flux. 4.2.6 Aggregation Processes The aggregation module incorpor ated within the m odel is based on the standard collision frequency approach utilized by a number of researchers (Saffman & Turner, 1956; Broadway, 1978; Hunt, 1980; McCave, 1984; Tsai & Hwa ng, 1995, McAnally, 1999; Parshukov, 2001, Maggi, 2005, etc.). The basi s for the aggregation module is that the frequency of collision is a function of the number of sediment particles within the suspension and forces that tend to bring the particles close enough to collide. Once a collision occurs there is an assumed efficiency of aggregation, a probability the particles will adhere to one another to form a larger floc. In addition, a probability is developed for disaggrega tion, whereby the collision causes the flocs to break apart into smaller flocs. The number of particles within a partic ular size class, defined from Equations 4-1 and 4-2, will be the class co ncentration divided by the estimated mass of the PAGE 175 175 particles within the class. The mass of the particles within size class i, mi, is defined by Equation 4-36, using Equation 2-5 for the floc size, dfi, and an equivalent spherical volume, Vi. 3 3366fD fi p fi iiiwfi ws fiddd mV d (4-36) The number concentrat ion of particles, ni, in suspension size class i will be the concentration divided by the particle mass (Equation 4-37). i i ic n m (4-37) Contributors to collisions of sediment pa rticles and flocs incl ude Brownian motion, laminar and turbulent shear and differential settli ng. The assumption is made that these effects are linear and can be added together (Maggi, 2005). The number of ne w flocs formed per unit volume by collisions between the two mass classes i and j is defined by an efficiency term times the cumulative probabilities of collision from Br ownian motion, velocity shear and differential settling, times the product of th e numbers of particles in each cl ass per unit volume. This is expressed in Equation 4-36, with the laminar and turbulent shear contributions lumped together. ,,,ijkaBijTijkDijkikjkNn n (4-38) The collision frequencies, are indexed to a spatial location by the k-index and the particle classes by i and j. The subscripts B, T and D correspond to Brownian motion, turbulence and differential settling, respectively. The collision frequency for Brownian motion is not indexed to a spatial location because it is assumed to be uniform throughout the flow field. The relative significance of the three mechanisms will be evaluated in connection with the influence of the probabilistic variables on thes e contributors to a ggregation processes. PAGE 176 176 The efficiency, a, is the fraction of collisions that will result in aggreg ation. Its value ranges between 0 and 1. It has been proposed that a bove a reference salinity of approximately 2 psu (practical salinity units) the aggregation efficiency equals 1, since in estuarine waters it has been observed that essentially all collisions result in aggr egation (Krone, 1962; McAnally, 1999). McAnally proposed a functional relationship for a as presented in Equation 4-39. 0 0 01aS f orSS S f orSS (4-39) It was recommended that S0 is approximately 2 psu, so for most areas within an estuary a is 1. The collision frequencies are computed as shown in Equations 4-40, 4-41 and 4-42: Brownian motion: 2 ,2 3ij Bc Bij ijdd kTF dd (4-40) Turbulent shear: 3 ,4 3k TijkijG dd (4-41) Differential settling: 2 2 ,4c D ijk ijsiksjkF ddww (4-42) Terms in Equation 4-40 are Boltzmanns constant, 16o1.3805410erg/KBk the average temperature, oinKT, an empirical collision diameter function, Fc, the dynamic viscosity, and the class particle diameters, di and dj. The collision diameter function, Fc, varies between 0 and 1 and is a correction factor to account for particle collisions having some meshing together before aggregati on can occur; a slight glancing ma y not result in aggregation. The turbulent shear collision efficiency is a function of the internal shear rate, G, and the particle PAGE 177 177 diameters. For the differential settling freque ncy the settling velocities are associated with locations in space because of the dependence of th e settling velocity on concentration and shear. An example comparison of the significance of each process for a fir st particle of 10.6 microns relative to a second part icle of varying size is presen ted in Figure 4.2. The first particle size class at 10.6 microns was the closest to 10 microns from a size class distribution spanning 0.1 microns to 1000 microns using 60 size classes. That fi rst particle class illustrates the relative importance of the contributions to th e aggregation frequency over the defined range. The differential settling collision frequency is relatively constant for second particle sizes smaller than 10 microns, but increases by eight orders of magnitude between 10 and 1000 microns. For a second particle size near 10 microns the collis ion frequency for differential settling becomes very small, approaching zero for 10.6 microns, wher e both particles settle at the same rate. The collision frequency for Brownian motion is small over the full range of second particle diameters, with the smallest frequency at the ma tching size of 10.6 microns. It is only important relative to the other modes of collision belo w 0.5 microns. For the example case shown, turbulence is significant at all size classes relative to Brownian motion and differential settling. For a different flow condition (e.g ., quiescent settling) the turbul ence might not be as dominant. The effects of using a probabil ity distribution for the settling velocities are to eliminate the zero contribution well at 10.6 mi crons. This implies that particles of the same effective spherical diameter will likely make a contribution to aggregation from differential settling. This may be interpreted as a result of the lack of perfect association between particle mass, water content, particle density, effec tive size and ultimately settling velocity. For large differences in particle size there is no difference in the contribution between using the mean settling rate and a probabilistic re presentation. PAGE 178 178 The computation of the internal shear is dependent on whether the k-model is invoked or a profile assumed. For using the k-model the shear is comp uted via Equation 4-43. G (4-43) This is applied locally from the solved field of When a logarithmic profile is assumed the shear is estimated based on the computed sh ear distribution from the profile Equation 4-44. 3 *1 u z G zh (4-44) Aggregation fluxes are the mass transfer rate s between sediment size classes as a result of aggregation of particles. These fluxes for an arbitrary size class, i, can be either a mass loss (LijA) due to flocs in class i combining with other pa rticles of size class j to form larger flocs, or mass gains (Gj,iA) from the aggregation of smaller particle sizes (j and ) that combine to form flocs of size class i. The total aggregation fl uxes to and from size class i is the summation over all particle interactions within the flocculati on model. The total aggr egation gains for class i is the result of the summati on shown in Equation 4-45. iA jiA jGG (4-45) The specific aggregation fluxes (Gj,iA) are determined from the rate of particle aggregation, based on Equation 4-38, and the partic le masses. This relationship is presented in Equation 4-46. jiAjjiijGNmmmmm (4-46) The combined masses of particles of class j and result in approximately the mass of size class i The parameter i is the distribution factor for the combined mass to size class i The distribution factor is developed from linear interpolation between log-transformed particle masses as shown in Equation 4-47. PAGE 179 179 1 1 1 1 1log()log() log()log() log()log() log()log()ji i ii iji ij i iimmm mm f ormmmm mmm mm (4-47) The loss of mass due to aggregation, LiA, from class i is also based on the rate of particle aggregation and the class i particle mass (see Equation 4-48), the summation of flux loses to all other size classes that have experienced aggregation with class i iAijAiji jjLLNm (4-48) The results of particle collisions can be either aggregation or disaggregation of the colliding particles. When two particle s of known size combine to form a new larger aggregate, the combined size may not precisely match the mass of a larger size class. This introduces the need to deal with the number concentrations as real nu mbers, interpreted as an expected value for the integer number of particles per un it volume. This frees the numerical analysis from dealing with special cases of integer mathematics. It does require the use of brea kup distribution functions (Lick and Lick, 1988) to distribut e the products of the disaggrega tion to lower size classes in a manner that both conserves mass and is based on observed behavior. 4.2.7 Disaggregation Processes 4.2.7.1 Shear-Induced Disaggregation When the local shear stress in the water column exceeds the shear strength of the floc, the floc is broken into smaller flocs based on a br eakup distribution function such that the mass of the original floc is conserved. The shear strength of the flocs is assumed to be a function of the density as defined in Equation 3-11. PAGE 180 180 The local shear stress is assumed to be linear from a maximum at the bottom to zero at the surface. The relationship is given in Equation 449 for the case of using an assumed velocity profile. 2 *11wbzz u hh (4-49) The value *u can be estimated via Equation 4-50, from the depth-averaged velocity, avgu, and a Darcy-Weisbach friction factor, f. 1/2 *8avgf uu (4-50) If the k-model is used the shear stress can be com puted from the local dissipation rate and the vertical turbulent-mean velocity gradient as shown in Equation (4-51). u z (4-51) The breakage fluxes can also be either a loss or again for a specific size class. Mass loss from a class results when flocs within that size class are broken apart. The remnants of that breakage become mass gains to the appropriate sma ller size class. This analysis assumes that floc breakage results in only two remnant particles, of masses jm and 1jm, where varies between 0 and 1, but logically would be closer to 0.5. McAnally (1999) recommends a value of 3/16. The resulting remnant mass is then distributed to the closest existing mass classes consistently with Equation 4-47, usingjm and 1jm, rather than mj + mk. The shear breakage loss fluxes can then be ex pressed as shown in Equation 4-52. di iBiLC t (4-52) PAGE 181 181 The efficiency for shear induced breakage is assumed to apply to all sediments in suspension, since no collisions are needed to have breakage. The tim e scale, t, is assumed to be defined as the ratio */zu. The efficiency of the disaggregation has b een assumed to be proportional to the excess shear stress, with a time cons tant proportional to the shear velocity (see Equation 4-53). 0 f f dd A (4-53) The variable A is an adjustment coefficient and 0 is the Kolmogorov eddy length scale. When the shear stress, is below the strength, f, of the floc no disaggregation will occur and as the shear stress increases beyond the strength of th e floc, the rate of di saggregation will increase proportionally. The mass flux gains to size class i resulting from shear breaking of flocs can be expressed as the summation of all classes broken th at yield remnants of mass in class i (see Equation 4-54). d iBijBiijj jj aGGNm (4-54) The variable i in Equation 4-54 represents all fractions that yiel d breakage byproducts within size class i (i.e, both and 1terms). 4.2.7.2 Collision-Induced Disaggregation Disaggregation as a result of collisions between particles is defined based on a modification of the approach of McAnally (1999) The decision for breakage due to collision is based on the estimation of the stress of the collisions according to Equation 4-55, which estimates the shear stress on particle k during a two-body collision with particle i. McAnally (1999) extended Equation 4-55 to three-body collisions, arguing th at three-body collisions more PAGE 182 182 frequently result in breakage. The hydrodynamic interaction of a th ree-body collision was schematized by Clercx and Schram (1992) as a sequence of two two-body collisions, enabling consideration of three-b ody collisions as implicitly accounted for over time intervals larger than the breakup time scale. 2 28ik ikk pkikikumm Fdddmm (4-55) The variable Fp is a coefficient of the relative dept h of inter-particle collision, which acknowledges that flocs are not hard spheres, but rather can deform upon collision, its value ranging from 0 to 0.5, with a value of 0.1 used. The collision velocity, ui is estimated by McAnally (1999) based on Equation (4-56). 3 2 215b ik ik i siskkT forBrowianmotion dd dd u forinternalshear ww fordifferentialsettling (4-56) For isotropic turbulence the definition of the turbulence dissipation can be simplified (Equation 4-57). The Taylor microscale length, is defined as shown in Equation 4-58 (Tennekes and Lumley, 1973), where Sij is the deviator stress tensor. 2215ijiju SS z (4-57) 2 2 2u u z (4-58) PAGE 183 183 This scale length is representative of the ener gy transfer from large to small scales. The Taylor microscale can be can be used with the shear velocity to estimate the dissipation rate as shown in Equation 4-59. 2 2 *1515u u z (4-59) If the shear, G, is assumed to be approximated by *u we obtain Equation 4-60. 15 G (4-60) If the factor of 15 is ignored in Equation 460, then Equation 4-43 is obtained. The shear term in Equation 4-60 will underestimate the shear if the additional 2/ is also incorporated under the radical, since the conventiona l approach has been to ignore the 1 15, to obtain 4-43 (Maggi, 2005). Floc breakage logic is based on the relative values of the i ndividual particle shear strength and the collision shear. If the shear strengths of both particles involve d in the collision are greater than the collision shear stress, aggr egation may occur based on the efficiency of aggregation (Equation 4-39). If onl y one of the two particles is st ronger than the collision shear, the stronger floc will steal a fr action of the weaker floc. McAn ally (1999) argued that the most likely fraction to transfer was 3/16 based on the angl e of collision. If bot h of the particles are weaker than the collision shear, then the outcome will be three particles, with essentially the two stolen parts of the two original flocs combining to form the third floc. Again, the most likely fractions were assumed to be 3/16 from each original floc. The collision breakage mass fluxes can be expre ssed similarly to the shearing fluxes. The collision disaggregation efficiency is defined by Equation 4-61. PAGE 184 184 0, f iijifi Ci ijid A (4-61) The collision breakage mass flux losses from class i due to collisions with particles of class j is based on the dissaggregation efficiency ap plied to those collisions that do not result in aggregation (see Equation 4-62). 1Ci iCijCAiji jj ALLNm (4-62) This suggests that when the aggregation e fficiency approaches one, as suggested for estuaries, that the significance of co llision-based disaggr egation vanishes. The disaggregation gains for size class i from collision disaggregation when particles of classes j and collide is defined by Equation 4-63. ,1Cj iCjiCAjijjjji jj AGG Nmmmmm (4-63) The variables take the same meaning as described previously. The fractions of the original flocs, j and are the appropriate values associated with smaller or larger byproducts of collisions as necessary to result in the mass of the class i flocs. The distribution fractions i are based on the linear interpolation in log-transformed space (see Equation 4-47). For the calculation of the gains and losses of sediment mass for aggregation and disaggregation, the gains and losses for each process must balance. That is, 0; 0; 0iAiA iBiB iCiC iiiGLGLGL An example accounting of the net balances in the computational cells over the ver tical is presented in Table 4-2. The breakage balances precisely, while the net aggregation is accurate to six orders of magnitude smaller (O(10-9) than the aggregation flux itself (O(10-3))). PAGE 185 185 4.3 Hydrodynamics Hydrodynamics are evaluated by the use of the Reynolds-averaged Navier-Stokes momentum equations (Equation 4-64). 11tuu uu p ji ii g i txxxx jijj (4-64) Adding corrections for effects of sediment concentration on the momentum equations emulating the effects form two-pha se flow, again based on Hsu, et. al. (2007), results in Equation 4-65. 001 11 2.343 1.89 33 1 111 1s w e tr w wuu u p ji i g i txx ji uu ii xxx jjj (4-65) The volume concentration at th e space-filli ng threshold, 0, is included in a manner that greatly amplifies the e ffects of viscosity as 0 For the case of unif orm flow Equation 4-65 simplifies to Equation 4-66. 1111tr wwup u txzz (4-66) The turbulent viscosity is estimated by the revised form of Equa tion 4-24, taking the sediment concentration into accoun t, as shown in Equation 4-67, 21Tk cf (4-67) PAGE 186 186 In addition, Hsu, et. al. also included the effects of the sediment stresses on the momentum through an added relative visc osity term, defined by Equation 4-68. 2.34 3 01re (4-68) The boundary condition for the momentum equation at the water surface is a no-stress boundary, which results in a zero gradient boundary condition (Equation 4-69). 0szzu z (4-69) The bottom boundary condition for the mean-velocity momentum equation is the bottom shear stress, which leads to a veloc ity gradient condition (Equation 4-70). 0 b z tu z (4-70) In applying Equation 4-66 the pressure term is converted, using the hydrostatic pressure assumption into Equation 4-71. 22 4/3 h Un ggSg www x Rp x (4-71) The new variables in Equa tion 4-71 are the bed slope, S, the depth-averaged mean velocity, U, and the hydraulic radius, R. This formulation can be viewed as either the case of a horizontal bed with a pressure gradient app lied or as a uniform flow case with no pressure gradient but with a gravitational force component along the bed slope of the system. 4.4 Solution Method The equations for the flow and sediment trans port are solved by a simple finite difference discretization of each of the e quations over evenly spaced cells through the water column. The PAGE 187 187 vertical dimension of each cel l is therefore the water depth divided by the number of cells, zh z N. The general form of most of the governing equations in the unifor m flow case takes the form of Equation 4-72. jjABPD txx (4-72) The time rate of change of a generic variable less the diffusion of the variable balances the production and dissipation, or other forcing. Other transformations of the variable can be lumped into the production or dissipation. The governing equation is discretized as shown in Equation 4-73. The temporal time step is a forward difference for the temporal deriva tive and the remaining terms in the equation are averaged over the time step. 112nnnn iiiiLL t (4-73) The loading terms, L, are expressed as s hown in Equation 4-74. 11 111111 11,,, ,,,,nnnnnn iiiii nnnnnnn iiiiiLLV LLVV (4-74) The right hand side of Equation 4-73 at the time level n involves the tri-diagonal variables of n and other generic variables, Vn, at time level n variables which contribute to terms in the right-hand-side of the matrix equations. The loading term above at the new time level n+1 will involve the implicit variables for the variable as the unknowns, but with a mixture of other variables at both the time levels n and n+1, depending on whether those variables have been PAGE 188 188 solved for at the new time level. Equation 4-75 presents the diffusion term for the new time level, with the coefficients A evaluated with the most currentl y available information (time level n for the predictor step and time level n+1 for the corrector step). Each of the primary variables ui, ki, i, and cij are solved in succession and then an addi tional corrector sweep of solutions is performed to update the new time level solutions. ''11''11 11 11 11 22nnnnnnnn kkkkkkkk n kAA AA L zzz (4-75) The coefficient matrix of Equation 4-73 is tr i-diagonal and a standard tri-diagonal matrix inversion algorithm was used to solve the matrix The boundary conditions applied at the bottom and the surface were either Dirichlet or a Neum ann condition, depending on the specific variable being solved. The boundary conditions applied for each of the governing equations are summarized in Table 4-3. 4.5 Probabilistic Representation The representation of variables probabilistical ly leads to the issue of how variables are combined within the governing equations. The probability of one vari able exceeding another was addressed in Chapter 3 (Equations 3-13 thro ugh 3-16). When two variables are multiplied together, the products can be developed based on rules of integration. If both variables are expressed probabilistically then th e product can also be represented as a probability distribution. For most of the calculations developed within the model the goal from the products of the variables is the expected value of the product. Assume that two variables A and B are to be incorporated as a product, when each is represented by a probability distribution. The expected value of the product is shown in Equation 4-76 for both double and triple products. PAGE 189 189 ()() ()() ()()() ()()()() ()()() ()A B nmnm AB nmpnmp ABC B ABABpAfA pBfB EABfAfBABdAdB EABCfAfBfCABCdAdBdC PABfAdAfBdBFBfBdB (4-76) The variables A B and C are arbitrary. The probability distribution functions (PDFs) of those variables are fA, fB and fC. The cumulative distribution function (CDF) for A is FA. In products of many variables, if one of the variab les is not represented probabilistically, it is treated as a constant and moves out side the integrals. All the va riables with a tilde are dummies of integration. 4.6 Program Outline The overall flow logic of the computer program is presented in Figures 4-3 through 4-5. The flow logic for the MAIN por tion of program COHPROB is presented in Figure 4-3 for the preliminary initialization phase, in Figure 4-4 for the hydrodynamic spin-up phase and in Figure 4-5 for the simulation phase for sediment transport. The details of select ed critical subroutines are flow-charted in Appendix D. The first phase of the program includes data initialization, specification of constants, data input and preliminary setup calls to certain subroutines to fully develop parametric input data specification. The first phase also includes several special problem subroutines to perform simulations of analytical test cases. 4.7 Analytical Test Cases A series of analytical cases were simulated within the computational program to provide validation of the numerical procedures used. These cases are separate from the sediment cases described in Chapter 5, which are based on measured data. The laminar flow test cases described PAGE 190 190 here each have analytical solutions that can be used to calculate the accuracy of the simulation using error estimates. Table 4-4 summarizes th e laminar flow problems tested and pertinent simulation parameters. 4.7.1 Stokes first problem Stokes first problem (Schlichting, 1968) i nvolves the instantaneous acceleration of an infinite flat plate from rest to a fixed velocity, u0, in the plane of the plate. The gradients parallel to the plate vanish and the pressure is a ssumed uniform within the domain. The governing equation for the movement simplifies to a one-d imensional equation (Equation 4-77) with the zdirection taken perpendicular to the plate. 2 2uu t z (4-77) The boundary conditions are: 00for all for 0 for 0, for 0 uzt uuzt (4-78) The viscosity of the fluid is assumed to be uniform within the domain. Equation 4-77 is reduced to an ordinary differential equation by the substitution of a nondimensional distance shown in Equation 4-79. 2 z t (4-79) Using the assumption that u is a function of given in Equation 4-80, the ordinary differential Equation 4-81 is derived. 0() uuf (4-80) 20 f f (4-81) PAGE 191 191 The boundary conditions for Equation 4-81 are f = 1 at = 0 and f = 0 at The solution is given in Equation 4-82, which involves the complimentary error function. 0uuerfc (4-82) The complimentary error function is given in Equation 4-83. 22 exp() erfc d (4-83) The non-dimensionalization of the distance from the wall via equation 4-68 results in the velocity distribution being self-similar. The boundary layer grows proportionally with the square root of the product of the kinematic viscosity a nd time. The analytical solution (Equation 4-82) is presented in Figure 4-6 as the nondimensional velocity, u/u0, versus the nondimensional distance, Stokes first problem was simulated within the m odel for spatial dimensions and velocity of the plate that would result in a low Reynolds number (below 500), making the laminar flow assumption reasonable. Strictly speaking lamina r flow assumption is valid when the Reynolds number is 0.01. The velocity distribution and the boundary layer evolve with time. The domain of the analytical problem is an semi-infinite half space; however, the numerical model must be bounded. The boundary condition applied at the limit of the model domain away from the moving wall was chosen to be a zero gradient boundary. A zero flow specification was also tested. The limit of the validity of the simulation is reached when the velocity at the boundary becomes nontrivial, and the zero flux condition makes interpretation of when that limit occurs more straightforward. The depth of the domain was chosen as 0.1 m, with a wall velocity of 0.01 m/s. The kinematic viscosity for all of the laminar flow problems was 1.12 x 10-6 m2/s. The model used a 0.1 second time step. PAGE 192 192 The boundary conditions for the discretized dom ain are given in Equation 4-84. The wall (z=0) boundary condition at the movi ng wall is a Dirichlet specifica tion for the velocity of the wall converted to a stress condition. The bounda ry condition at the open boundary is a no-flux condition. 0 0 0(1,) ()() at z = 0 /2 0f o r a l l tz zhuut u tt zz u z (4-84) The variable h is the size of the domain simulated in the model and has no significance to the problem other than to lim it the duration of the validity of the approximate boundary condition there. The number of cells, N and the domain size, h, define the grid cell size (z = h/N ). The model was simulated with a number of spatial resolutions to evaluate the sensitivity of the accuracy of the solution. The results of the simulations using 80, 40, 20 and 10 cells across the domain, respectively, are presented in Fi gures 4-7 through 4-10, pr esenting the normalized velocity versus the distance from the wall fo r three times after the initiation of the wall movement. These figures illustrate that the relative resolution becomes a function of the boundary layer thickness, which is proportional to t The resolution for the 40-cell case after 90 seconds is comparable to the 80-cell case at 24 seconds. The 10-cell case resolution at 198 seconds is comparable to the 20-cell case after 24 seconds. These observati ons are a direct result of the self-similarity of the evolving velocity distribution. The times 24, 90 and 198 seconds were selected for comparison because they result in uniform incremental boundary layer thicknesses. PAGE 193 193 Although the model simulated the domain 0.1 m across, the useable domain was limited to about half that distance to minimize the effects of the finite domain and the approximate no-flux boundary condition at z=h The limitation on the domain essentially translates into a limit on the time length of a valid simulation. The varying resolutions are presented using the nondimensional distance from the wall in Figures 4-11,4-12 and 4-13. These figures confir m the self-similarity of the velocity profiles generated by the model. The 80-cell and 40-ce ll simulations very accurately reproduced the analytical distribution The 20cell simulation slightly underestimated the ve locities for the 24 second profile. The 10-cell simulation underpredicted the velocities at all three analyzed time levels, but with the error becoming smaller with time as the effective resolution increases. The errors in the nondimensional velocity dist ribution were estimated by differencing the simulated velocity (divided by u0) and the analytical solution. These error estimates were then analyzed to obtain a standard deviation of the errors. These error standard deviations are summarized in Table 4-5 for the Stokes first prob lem. Several observations can be made of the relative errors. The 80-cell results are consiste ntly more accurate than the lower resolution simulations. However, as time progresses, the error increases for the 80-cell simulation. This may be associated with the finite bound on the domain and the artificial boundary condition. The lower resolution simulations had the error d ecrease as the simulation progressed, primarily due to the poor representation with less relative resolution early in the simulation when the boundary layer was small relative to the grid cell size. Thes e error estimates are only comparable within the context of this approximation of this analytical problem and have little meaning for comparing between test problems. PAGE 194 194 The effects of sediment concentration on th e solution were tested by defining a uniform concentration and defining the viscosity by Equati on 4-68. The simulation results are presented in Figures 4-15 and 4-16, for c oncentrations of 20 and 100 kg/m3 corresponding to the fluid mud density range. The presence of the sediment incr eases the effective viscosity and accelerates the evolution of the boundary layer. The 20 kg/m3 concentration increased the boundary layer growth rate by approximately 4 percent, while the 100 kg/m3 concentration increased the growth rate on he order of 16 percent. 4.7.2 Couette flow problem Couette flow is a modification of the Stoke s first problem by imposing a stationary boundary at a distance, h from the moving wall. The moving wall is instantaneously accelerated to the constant speed, u0, but the velocity at the opposite wa ll is zero. The governing equation remains Equation 4-77. The boundary conditions are: 00for all for 0 for 0, for 0 0 for for all t uzt uuzt uzh (4-85) The analytical solution for the Couette flow probl em is given in Equation 4-86, a series solution based on a Laplace transform. 11 00 0 11 1122 1 ()(2)(2) (4)(4)......nnu erfcnerfcn u erfcerfcerfc erfcerfc (4-86) where 1 is the dimensionless distance between the two walls (Equation 4-87). 12 h t (4-87) PAGE 195 195 The analytical solution (Equati on 4-86) is presented in Figure 4-17 for a series of nondimensional times, ts, defined by Equation 4-88. 42st t h (4-88) The steady-state solution for Couette flow is a lin ear variation of velocity between the two plates. The numerical model hydrodynamic component was applied to the Couette problem with varying level of spatial resolution. The boundary condition at the moving wall is the same as used in Stokes first problem (Equation 4-84). The zero-flow specification at the opposite wall was converted into an applied stress, dependent on the velocity in the ce ll adjacent to the wall (Equation 4-89). 0 0 0(1,) ()() at z = 0 /2 (,) 0()() /2z zh zh zhuut u tt zz uuNt utt zz (4-89) At steady-state conditions (t ) the stress will be the same magnitude at the walls but with opposite sign. The viscosity and time step were the same as for the Stokes first problem. The results of the application of the model to the Couette problem are presented in Figures 4-18 through 4-21 for numbers of cells of 80, 40, 20 and 10, respectively. The numerical model results are plotted as symbols against the analytical solution, which are the lines in the plots, for several nondimensional time scales The 80-cell and 40-cell results very accurately replicate the analytical results. The 20-cell resolution shows some deviation from the analytical solution at the earlier time scales, but the accuracy is good for larger time scales. The 10-cell results show even greater error for the earlier time scales, but agree well for dimensionless time scales of 1.0 and greater. PAGE 196 196 It is interesting to note that for the earlier time scales of the Couette flow problem, before the fixed opposite wall has a significant stress, the simulation is essentially Stokes first problem. The conclusions about the relative resolution ne ar the moving wall apply during the early portion of the Couette flow problem. The standard deviations of the error between the simulate d and analytical results are summarized in Table 4-6 for the Couette flow probl em. These error measures show that as the resolution increases the error is reduced and as th e solution evolves toward steady state the errors are generally reduced. The effects of suspended sediment concentration on the Couette flow problem were simulated with concentr ations of 20 and 100 kg/m3. The results are presented in Figures 4-22 and 4-23. The effective time scale was computed for the sediment simulations using clear water viscosity in order to see the eff ects of the sediment. If the effective viscosity with the sediment were used, the curves would be the same. This also shows that the presence of sediment accelerates the development of the velocity profile. 4.7.3 Stokes second problem Stokes second problem is an extension of the first problem to include a harmonic oscillation of the moving plate. The governing equation remains Equation 4-77 and the boundary conditions are given in Equation 4-90: Th e effect of suspended sediment is greatest early in the simulation before the velocities r each the opposite wall, during which the solution is essentially the Stokes first problem. 0(0,)cos() at 0 for time (,)0 at for all time ututzt utzt (4-90) The solution of Equation 4-77 with the boundary conditions of Equation 4-90 is given in Equation 4-91. PAGE 197 197 0(,)exp()cos() uztukztkz (4-91) The dispersion equation, relating k and for Equation 4-90 as a so lution of Equation 4-26 is given in Equation 4-92. 2 k (4-92) The analytical solution (Equati on 4-91) is presented in Figu re 4-24. The solution is a damped harmonic oscillation, with the bounding amp litude of the harmonic plotted as the dashed red lines in Figure 4-24. In applying the numerical model to Stokes second problem the no-slip boundary condition at the moving wall is again converted to a stre ss boundary condition as given in Equation 4-93. 0 0 0()(/2,) ()() at z = 0 /2zutuzt u tt zz (4-93) where u0( t ) is the oscillating velocity at the wall and u(z /2 ,t ) is the velocity in the center of the cell adjacent to the wall. The boundary condition at the open boundary is the same as for the Stokes first problem, a no-stress condition (see Equation 4-84). The frequency of the oscillation used in simulating Stokes second problem was carefully selected in conjunction with the size of the simulated domain such that the solution is substantially damped at the open boundary. The period of oscillation was chosen to be 288 seconds, giving an angular frequency of 0.021816616 s-1.The size of the simulated domain was retained at 0.1 m, as for the Stokes first probl em. The time step for the simulation was 0.1 s and the clear water kinematic viscosity 1.12 x 10-6 m2/s. The results of the Stokes second problem si mulations are presented in Figures 4-25 through 4-28 for spatial discreti zation of 80, 40, 20 and 10 cells, respectively. Ignoring the 10cell case, the agreement is very good with the excepti on of the 20-cell case very close to the PAGE 198 198 oscillating wall. The case with only 10 cells was simulated and the results presented but the results were so poorly resolved that they shoul d be discounted. The effective resolution is practically about half of the st ated resolution because of the need to put the open boundary far from the oscillating wall. The sensitivity of the Stokes second problem to sediment c oncentrations was tested by simulations with 20 and 100 kg/m3 suspended sediment concentra tions. Those tests used 40-cell resolution. The results are presented in Figures 4-29 and 4-30. For thes e plots the values of for the sediment runs were computed using the clear water viscosity to see the impact of the suspended sediment. The effect of the sediment through increased viscosity, is to stretch the velocity distribution away from th e oscillating wall by a factor of th e square root of the ratio of the viscosities. The relative effects are the same as seen in the other laminar flow test cases. 4.7.4 von Karman Mixing Length Velocity Profile for Fully Rough Flow The hydrodynamic setting for most of the sedime ntation analyses addressed within this work are open channel flows in a fully rough condition (**Re/70 s ku ). A model simulation was made to evaluate the velocity profile generated using the von Karman mixing length which results in the turbulent eddy viscosit y distribution shown in Equation 4-94. *1tz uz h (4-94) The hydrodynamic equation (Equation 4-66) was solved using a pressure gradient calculated via Equation 4-71. The depth-averaged velocity used in Equation 4-71 is the final equilibrium velocity, not the time evolving actual mean velocity. The initial condition was set at no flow [ u( z ,0) = 0]. The velocity profile evolved over time until it reached an equilibrium profile. The evolution of the profile over nondi mensional time (Equation 4-88) is presented in Figure. 4-31. After a nondimensional time ts =0.73, the velocity distribution remained PAGE 199 199 unchanged. The equilibrium velocity distributi on from the model is compared (Figure 4-32) with the classical analytical logarithmic profile given in Equation 4-95 to evaluate the performance of the hydrodynamic component. The simulation used a water depth of 1 m, a depth-averaged velocity of 0. 5 m/s and the roughness height wa s 0.1 m. The estimated shear velocity was 0.0424 Pa. *0112 9 7 ln1ln1suzz uzk (4-95) The differences between the model results and the analytical profile give a standard deviation of the error of 0.00267 m/s. Also presented in Figure 4-32 is the vertical distribution of the local shear stress at equilibrium. The shear stress profile is linear, as expected. PAGE 200 200 Table 4-1. Standardk model coefficients for high Reynolds number flow. C C1 C2 C3 k 0.09 1.44 1.92 1.2 1 1.3 Table 4 2. Summary of exam ple flocculation model. Aggregation (kg/m3/s) Breakage (kg/m3/s) Vertical cell Elevation (m) Gains Losses Net Gains Losses Net 1 0.015 7.54E-03 7.54E-03 9.31E-10 4.04E-05 4.04E-05 0 2 0.046 6.96E-03 6.96E-03 -4.19E-09 1.33E-05 1.33E-05 0 3 0.076 6.76E-03 6.76E-03 -1.86E-09 7.89E-06 7.89E-06 0 4 0.107 6.64E-03 6.64E-03 -4.19E-09 5.55E-06 5.55E-06 0 5 0.137 6.56E-03 6.56E-03 -1.86E-09 4.24E-06 4.24E-06 0 6 0.168 6.49E-03 6.49E-03 5.12E-09 3.39E-06 3.39E-06 0 7 0.198 6.44E-03 6.44E-03 0.00E+00 2.79E-06 2.79E-06 0 8 0.229 6.38E-03 6.38E-03 -2.79E-09 2.32E-06 2.32E-06 0 9 0.259 6.32E-03 6.32E-03 -8.38E-09 1.92E-06 1.92E-06 0 10 0.29 6.25E-03 6.25E-03 0.00E+00 1.48E-06 1.48E-06 0 Table 4-3. Summary of bounda ry condition specifications. Problem Bottom boundary condition Surface boundary condition Hydrodynamics: Stokes first problem u(0) = u0 no stress Hydrodynamics: Couette flow problem u(0) = u0 u( h) = 0 Hydrodynamics: Stokes second problem u(0) = u0 cos(wt) no stress Hydrodynamics:Logarithmic profile open channel flow u(0) = 0 no stress Turbulent kinetic energy k (0) = 2 */ uc No flux Turbulent kinetic energy dissipation (0) = 3 *0/ ukz No flux Suspended sediment concentration specified flux no flux Table 4-4. Simulation conditions fo r special laminar flow problems. Special Problem Domain size (m) u0 (m/s) Frequency (1/s) time step (sec) Stokes first problem 0.1 0.01 0.1 Couette flow problem 0.1 0.01 0.1 Stokes second problem0.1 0.01 0.021816616 0.1 PAGE 201 201 Table 4-5. Error measures for the Stokes first problem for varying time and number of cells. Nondimensional error standard deviation Time(s) Number of cells 24 (=0.01) 90 (=0.02) 198 (=0.03) 10 0.0385 0.0121 0.00538 20 0.00823 0.00202 0.000933 40 0.00146 0.000313 0.000538 80 6.02 E-10 8.27 E-6 0.00014 Table 4-6. Error measures for the Couette problem for varying time and number of cells. Nondimensional error standard deviation Number of cells Time scale, ts 80 40 20 10 0.25 0.000414 0.000906 0.005432 0.031066 0.5 0.001245 0.001528 0.002894 0.009549 1.0 0.000697 0.000649 0.000408 0.001633 1.5 0.000943 0.000921 0.000688 0.000278 2.0 0.000454 0.000394 0.000366 0.000176 Table 4-7. Error measures for the Stokes second problem for varying time and number of cells. Nondimensional error standard deviation Phase, t Number of cells 0 /2 3 /2 10 0.03193 0.09237 0.31791 0.28498 20 0.00702 0.00950 0.00587 0.00910 40 0.00300 0.00282 0.00160 0.00243 80 0.00265 0.00164 0.00164 0.00152 PAGE 202 202 Figure 4-1. Comparison of Equati ons 4-32 and 4-33 for erosion rate with data from Partheniades (1965), using the coefficient Cb = 0.8459 kg/m3 in Equation 4-32, Cb = 0.112 kg/m3 in Equation 4-33, w =1030 kg/m3 fi =1, and the variables deve loping the probability of erosion s =0.55 Pa, s=0.25 Pa and b = 0.3 Pa PAGE 203 203 Figure 4.2. Collision frequency for a particle diameter of 10.6 microns with variable second particle diameter. The flow conditions fo r this case are a flow depth of 0.3048 m, with a depth-averaged velocity of 0.142 m/s. The probabilistic se ttling velocity cases assumed a 30 percent standard de viation in the settling velocity. PAGE 204 204 Figure 4-3. Computer pr ogram COHPROB flow chart (MAIN) for phase 1. COHPROB (MAIN) Phase 1 Code initialization INITIALIZE CONSTANTS INPUT GETVEL SRENGTH IVEL=1? GRID INITCON BEDUPDATE(1) DENSITY(0) HYDRO(0,1) TURBKE(0) VSHEAR(0) CLASS EDDYVIS(0,2) BEDXCHG FALLVEL(1) EROSRATE Phase 2 STOKES1(IT) COUETTE(IT) STOKES2(IT) PAGE 205 205 Figure 4-4. Phase 2 computer program flow chart for COHP ROB (MAIN); spin up of the hydrodynamic model. COHPROB (MAIN) Converged? TURBKE(-1) Phase 3 IKE=1 ? yes yes no Phase 2 Hydrodynamic initialization Converged? TURBKE(2) yes no TURBKE(1) Converged? yes no EDDYVIS(0,2) Converged? yes no EDDYVIS(0,2) TRIDIAG TRIDIAG TRIDIAG TURBKE(2) TURBKE(1) TRIDIAG TRIDIAG TURBKE(1) TRIDIAG HYDRO(0,1) TRIDIAG_U TURBKE(2) TRIDIAG HYDRO(0,2) TRIDIAG_U PAGE 206 206 Figure 4-5. Phase 3 (sediment transport) flow chart of COHPROB (MAIN). COHPROB (MAIN) AGGFLUX(IT) OUTPUT BEDUPDATE(1) DENSITY(IT) VSHEAR(0) EDDYVIS(0,2) BEDXCHG(IT) SEDCOH(IT) FALLVEL(IT) END Phase 3 Sediment transport simulation OUTPUT GETVEL IVEL=1? IFLOC=1? FLUXCORR IFLUX=1? TRIDIAG IGOLD=1 ? SEDCOHG(IT) TRIDIAG SEDSILT(IT) TRIDIAG IKE =2? TURBKE(IT) TRIDIAG HYDRO(1,IT) TRIDIAG_U Time steps (1,NTIME) Predictor/ corrector (1,2) no Predictor/ corrector (1,2) PAGE 207 207 Figure 4-6. Self-similar velocity distribut ion solution for the Stokes first problem. Figure 4-7. Results of 80-cell resolution over domain for simu lation of Stokes first problem. PAGE 208 208 Figure 4-8. Results of 40-cell resolution over domain for simu lation of Stokes first problem. Figure 4-9. Results of 20-cell resolution over domain for simu lation of Stokes first problem. PAGE 209 209 Figure 4-10. Results of 10-cell resolution over domain for simu lation of Stokes first problem. Figure 4-11. Nondimensional velocity distributio n for 80 cell simulation of Stokes first problem. PAGE 210 210 Figure 4-12. Nondimensional velocity distributio n for 40-cell simulation of Stokes first problem. Figure 4-13. Nondimensional velocity distributio n for 20-cell simulation of Stokes first problem. PAGE 211 211 Figure 4-14. Nondimensional velocity distributio n for 10-cell simulation of Stokes first problem. PAGE 212 212 Figure 4-15. Effects of suspended sediment concentration of 20 kg/m3 on Stokes first problem solution. The clear symbols are for no sediment and the blackened symbols are for the sediment-laden case. PAGE 213 213 Figure 4-16. Effects of suspended sediment concentration of 100 kg/m3 on Stokes first problem solution. The clear symbols are for no sediment and the blackened symbols are for a sediment concentration of 100 kg/m3. PAGE 214 214 Figure 4-17. Analytical velocity distribution solution of the C ouette flow problem for various nondimensional time scales, ts. PAGE 215 215 Figure 4-18. Comparison of simulation of Couette flow problem with 80 cells to the analytical solution. PAGE 216 216 Figure 4-19. Comparison of simulation of Couette flow problem with 40 cells to the analytical solution. PAGE 217 217 Figure 4-20. Comparison of simulation of Couette flow problem with 20 cells to the analytical solution. PAGE 218 218 Figure 4-21. Comparison of simulation of Couette flow problem with 10 cells to the analytical solution. PAGE 219 219 Figure 4-22. Effects of 20 kg/m3 suspended sediment concentration on the Couette flow problem. The time scale for the sediment laden flow was computed with the clear water viscosity to show the effects. PAGE 220 220 Figure 4-23. Effects of 100 kg/m3 suspended sediment concen tration on the Couette flow problem. The time scale for the sediment laden flow was computed with the clear water viscosity to show the effects. PAGE 221 221 Figure 4-24. Analytical solution of Stokes s econd problem (Equation 4-91). The dashed red envelopes are the bounding curve for the amplitude of the damped harmonic oscillation. PAGE 222 222 Figure 4-25. Simulation with 80 cells of Stokes second problem. Symbols are the model results and the lines are the analytical solution at the appropriate phases. PAGE 223 223 Figure 4-26. Simulation with 40 cells of Stokes second problem. Symbols are the model results and the lines are the analytical solution at the appropriate phases. PAGE 224 224 Figure 4-27. Simulation with 20 cells of Stokes second problem. Symbols are the model results and the lines are the analytical solution at the appropriate phases. PAGE 225 225 Figure 4-28. Simulation with 10 cells of Stokes second problem. Symbols are the model results and the lines are the analytical solution at the appropriate phases. PAGE 226 226 Figure 4-29. Effects of 20 kg/m3 suspended sediment concentration on the results of Stokes second problem for the 40-cell test case. The value of for the sediment laden flow was computed with the clear water viscosity to show the effects. PAGE 227 227 Figure 4-30. Effects of 100 kg/m3 suspended sediment concentr ation on the results of Stokes second problem for the 40-cell test case. The value of for the sediment laden flow was computed with the clear water viscosity to show the effects. PAGE 228 228 Figure 4-31. Temporal development of velocity profile using von Karman mixing length test case. PAGE 229 229 Figure 4-32. Comparison of simulated fully develope d velocity profile to the analytical solution, using 80 cells over water column. Also plotte d is the shear stress distribution over the water column from the simulation. PAGE 230 230 CHAPTER 5 SEDIMENT TRANSPORT AND DEPOSITION EXPERIMENTS There are several laboratory flume experiment s by various investigators that have been recognized as key to the development of CST theory In addition, because of their frequent use as benchmark cases for testing new developments in the literature they remain valuable data sets for use in new research. Some of these test cases are briefly documented and used in simulations documented in Chapter 6. 5.1 Kynch (1952) Sedimentation Theory The sedimentation theory of Kynch (1952) provides a general description of the development of a sediment deposit during settlin g in quiescent conditions. Conservation of sediment mass is combined with the method of characteristics to provide insights into densification during settling. The theory provides a good test case for the simulation model. The description here is based on lect ure notes of Mehta (2007). The theory deals with the case of a quiescent settling column that initially has a uni formly mixed dilute sediment suspension at a concentration below the level of hindered settling (see Figure 5-1). As the sediment begins to settle at the free settling rate, sediment near the bottom of the column will accumulate and begin to affect, through hindered settling, the subsequent sediment settling rate approaching the bed. The transition location within the water colu mn where free settling tr ansitions to hindered settling will begin to rise upward from the bed at a speed designated as wp, creating the primary lutocline (labeled P in Figure 5-1). Water at the surface will become sediment free as the settling sediment cannot be replaced from above. The interface between the clear wa ter and the settling suspension is a secondary lutocline, which falls downward at the free settling rate, ws. This secondary lutocline settling PAGE 231 231 rate is assumed to be equal to the free settling rate of the sediment that is located at the lutocline. Figure 5-2 presents a schematized representation of the time evolution of the sediment deposit. The governing equation for the conservation of sediment mass is given in Equation 5-1. 0s F c tz (5-1) The settling flux can be defined as s s F wc Equation 5-1 can be revised using the chain rule for differentiation as shown in Equation 5-2. 0s F cc tcz (5-2) At the primary lutocline the ra te of rise can be defined as ()pswcFc leading to Equation (5-3), which is a statement that the to tal derivative of the sediment concentration is zero. 0pccDc w tzDt (5-3) The conservation of the sediment concentrati on within the z-t domain can be expressed as shown in Equation 5-4, using the fundamental statement of the total derivative. 0czcDc ttzDt (5-4) Comparison of Equations 5-3 a nd 5-4 leads to the definition pwzt which is the characteristic velocity of the rising primary lutocline, P Below the primary lutocline settling is hindered, but it is unclear how the concentration may vary. However, if the rate of rise of the primary lutocline were constant the average concentration belo w the lutocline would need to remain constant. During the unhindered settling phase, the supply of sediment to the sediment deposit below the lutocline can be estimated as the sum of the free settling flux 0 swc and the PAGE 232 232 sediment overtaken by the rising lutocline 0 pwc. This is based on the observation that if the upper portion of the suspension between the prim ary and secondary lutoclines is assumed to settle at a constant velocity, then the concentration will remain constant at 0c until it encounters the primary lutocline. Therefore, the rate of change in the total mass of sediment below the primary lutocline, ML, can be expressed as shown in Equation 5-5. 0L ps M wwc t (5-5) The total mass below the lutocline can be defined as the average concentration times the height of the lutocline, hL. Equation 5-5 can be expressed as Equation 5-6. 0LLLL LLpschch hcwwc ttt (5-6) If wp is assumed to be a constant duri ng the unhindered se ttling phase, then L phwt and by definition L p h w t With these equalities, Equation 56 can be converted to Equation 5-7. 0 ps L L pww c ct c tw (5-7) The right hand side of Equation 5-7 is constant if it is assumed that wp is constant. Therefore, the only way for this to hold is if L c is constant and equal to the right hand side of Equation 5-7. Equation 5-6 is valid even if wp is not a constant, however, only when the concentration above the primar y lutocline is constant at0c. The mean concentration is constant below the primary lutocline if the characterist ics for all concentrations below the lutocline originate at the bottom (z =0) at the initiation of settling ( t =0) and each has a constant slope to the PAGE 233 233 characteristic for that concentration, czt assuring the proportionality among the concentrations. If any arbitrary characteristic is extended to the point of in tersection with the secondary lutocline, at a height of h it will define a time, *t, at point Q in Figure 52. The concentration between the primary and secondary lutoclines is assumed to be constant, c which leads to an approximate conservation statement shown in Equa tion 5-8. This is approximate because it would be exact if the two lutoc lines moved linear until they met and the concentration above the primary lutocline remained at the initial concentration. *00()s ps M cwwtch A (5-8) This is a statement that if the two lutoclines transited th rough the water column at fixed rates, then the two transit hei ghts would equal the depth of the column. The total sediment mass is the initial concentratio n times that water depth. Since the equalities shown in Equation 5-9 hold, they can be introduced into Equation 5-8 to yield Equation 5-10. *;spdh h ww dt t (5-9) 00 *dh chchtch dt (5-10) The boundary between unhindered and hindered settling is assumed to occur when the falling secondary lutocline is no longer linear. The sediment below the primary lutocline begins the consolidation process soon after the deposit has formed and will affect the nonlinearity of the evolution of the primary lutocline such that the rise is not linear. PAGE 234 234 The conditions for each of the settling zones delineated in Figure 5-1 can be summarized as follows: Unhindered settling zone: 0;sdh wcc dt (5-11) Hindered settling zone: 00 *()ch ct h (5-12) Settled zone: 00() 0f f f sch ctc h hh dh w dt (5-13) From the settling column test data of Kynch (1952), the estimated settling velocity as a function of the lutocline eleva tion was developed by Mehta (2007) and is summarized in Table 5-1. The settling velocity is taken as the slope of the secondary lutocline. The estimated concentration profiles based on the method of char acteristics are summarized in Figure 5-3 and Table 5-2. At each of the data values defining the falling secondary lutocline in Figure 5-3, the characteristic lines were connected back to the origin. The tangent lines at each of the data values define dh/ dt which then define h* via Equation 5-10 and the concentration for the characteristic line is defined by Equation 5-12. Figure 5-3 has the concentrations computed at each data point on the lutocline curve, which then apply along the associated characteristic line. The estimated concentration profiles at hours 5, 15 and 40 are summarized in Table 5-2 for the elevations where the characteristics cross the prof iles. The resulting vertic al profiles are plotted PAGE 235 235 in Figure 5-4. The profiles were integrated to estimate the total mass in suspension, yielding 1.257, 1.263 and 1.243 kg/m2, for the hours 5, 15 and 40 profiles. These compare well with the initial condition mass of 1.25 kg/m2. 5.2 Krone (1962) Flume Deposition Experiments Krone (1962) reported results of investigations in a recirc ulating flume 30.5 m long and 0.9 m wide with a level bed. Flow was genera ted by a propeller pump a ttached to a variable speed motor. The flume had an active deposi tional length of 27.4 m, with the upper 3.1 m containing baffling to dissipate entrance turbulence. Measurements performed in the flume were the temporal evolution of suspended sediment concentrations from the return flow at the upstream end of the flume and inferred flow conditions. 5.2.1 Settling Tests with Variable Shear The primary tests of interest from the fl ume study were the deposition tests in flowing water. The tests were conducted in water with a configured sodium chloride and calcium chloride solution to the proporti on of monovalent and divalent cat ions found in seawater, at a concentration of 17 ppt. The focus of these te sts was initial concentrations below 0.3 kg/m3, typical of those found in San Francisco Bay. Te sts were run with variable flows and initial concentrations. Some of the test results are presented in Figure 5-5. Of particular interest is the change in the slopes of the con centration curves at around 0.3 kg/m3. Above that concentration the deposition is more rapid, and below the slope with time is constant, representing exponential decay. This behavior has been explained as the result of flocculation effects on the settling velocity at the higher concentrations. 5.2.2 Settling Test with Tagged Sediments Another valuable test conducted by Krone was an experiment that tagged a portion of the suspended sediment with gold-198. This test was conducted in a 0.3 m wide and 10.7 m long PAGE 236 236 flume. The mean velocity in the test was repor ted to be 0.085 m/s, and the bottom shear stress was estimated to be 0.02 Pa. The results of this test are presented in Figure 5-6. The figure suggests that the tagged sediment had deposited at a more rapid ra te than the total concentration of sediment. Krone concluded that there ha d to be an interchange between the bed and suspension during transport, even in a depositional environment. Krone indicated that only a fraction of the susp ended sediment was labeled. Figure 5-6, reproduced after Krone (1962), indicates that at hour 0 the tagge d sediment had a concentration of 1.3 kg/m3 and shortly after the initiation of the test (at approximately 0.1 hour) the total suspended sediment concentration was 1.4 kg/m3. This apparent discrepancy is explained if the initial concentr ation of gold-labeled sediment in suspension, although not 100 percent of the sediment, serves to tag the init ial sediment mixture. For example, if only one percent of the sediment was initially labeled at the beginning of the te st and after some time the concentration of labeled sediment drops to one half percent, then the initial tagged sediment concentration dropped to 50 percent. 5.3 Mehta 1973 Flume Results The deposition tests of Mehta (1973) were co nducted in a rotating annular flume consisting of a circular channel, 0.2 m wide and 0.45 m deep with a mean diameter of 1.5 m. The solution was driven by a rotating upper lid, with the ability for the flume itself to rotate in the opposite direction to minimize secondary cu rrents. Experiments were carried out on a variety of mud, including San Francisco Bay mud. Mud was added to the flume with a high flow velocity to mix it thoroughly before the shear stress was reduced below the critical level for deposition. Testing evaluated the relationship between shear stress and deposition. Some of the results are summarized in Figure 5-7. The results show an equilibrium concentration during deposition that is a fixed percentage of the initia l concentration that is related to the shear stress. This is counter PAGE 237 237 to what Krone (1962) had found, where no equilibrium was ever reached. Mehta explained this as an effect of the particle si ze distribution of the source mud. If the same mud is used for all tests, then if a fraction of the sediment is a very fine non-cohesive that resists deposition for a given shear stress level, then the same percenta ge will remain in suspension for the same shear stress, independent of the initial concentration. 5.4 Parchure and Mehta (1985) Dilution Test Parchure and Mehta (1985) conducted an expe riment in the same flume as the Mehta experiments to help evaluate the question of simulta neous versus exclusive erosion and/or deposition. A deposited bed of co mmercial kaolinite was flocculated in tap water of a total salt concentration of 278 ppm. The settled bed wa s then eroded for a period of 120 hours with a bottom shear stress of 0.2 Pa in a flow depth 0f 0.26 m. The rate of erosion decreased rapidly during the first few hours as softer layers were eroded exposing deeper deposited sediments with greater shear strength. The concentration reached an equilibrium value of 3.85 kg/m3 early and remained constant throughout the remainder of the test. After 120 hours, the volume of the flume was slowly replaced with sediment-free wa ter over a period of 4 hours without disturbing the bed. At the end of the 4 hours of sediment removal the concentratio n within the flume had dropped to 0.03 kg/m3. Flow conditions were maintain ed for an additional 24 hours during which the concentration slightly increased to 0.1 kg/m3. The results of the variation in concentration during the fluid re placement are shown in Figure 58. Parchure discounted the small increase in concentration at the end of the test and concluded that, for the conditions tested, the erosion was independent of su spended sediment concentration. 5.5 Parchure and Mehta (1985) Erosion Test Parchure and Mehta (1985) also conducted an erosion test on a deposited bed subjected to progressive increases in the shear stress. The go al of the testing was to evaluate the vertical PAGE 238 238 structure of the bed shear strength The initial concentration was zer o, with an initial shear stress of 0.1 Pa. Subsequent shear stresses were incr eased by 20% over the previous shear stress. Each shear stress was maintained for an hour, before the next increase. The progression of shear stresses was: 0.1, 0.12, 0.144, 0.173, 0.207, 0.249, 0.299, and 0.358 Pa. The time evolution of the suspended sediment concentration was monitored. The results of one of the test series are presented in Figure 5-9. Within the first five steps in the shear stress the erosion had reached an approximate equilibrium by the end of the hour. For the higher shear stresses one hour may not have been sufficient to fully reach equilibrium. 5.6 Sanford and Halka (1993) Data Set Sanford and Halka (1993) presented a series of field deployments to measure the resuspension and deposition of placed dredged material at several disposal sites in the Chesapeake Bay. Their attempts to numerically si mulate the observed data led them to conclude that for using a single grain size of sediment that it was required to use the simultaneous erosion and deposition paradigm. The data sets were collected at three cons ecutive years, 1989-1991, with a bottom-moored tripod equipped with a pr ofiling capability to monitor currents, salinity, temperature and turbidity. From the collected data, shear stresses were estimated and suspended sediment loads calculated over single semidiurnal tidal periods. Also collected was the water surface elevation, from which the local water depth was estimated. The sediment loads were converted to average concentrations and are presented in Figures 5-10 through 5-12. The data from the 1989 and 1991 monitoring were following a dredging disposal, while the 1990 monitoring was at a disposal site where the material had consolidated for about a year. These field sites provide a means of testing the probabilistic formulation in a dynamic mode when transient conditions are important. PAGE 239 239 Table 5-1. Summary of Kynch settling column da ta and graphical analysis (after Mehta, 2007). Time, hours Secondary lutocline elevation, m Settling velocity, m/s h*, m 0 0.25 9.30E-06 0.25 1 0.22 3.80E-06 0.232 2 0.205 3.70E-06 0.231 3 0.192 3.40E-06 0.229 4 0.18 3.20E-06 0.226 5 0.17 3.00E-06 0.224 6 0.16 2.60E-06 0.216 8 0.146 2.00E-06 0.203 10 0.134 1.40E-06 0.181 15 0.116 7.00E-07 0.152 20 0.105 3.90E-07 0.133 25 0.099 1.70E-07 0.114 30 0.097 1.10E-07 0.109 35 0.096 7.80E-08 0.106 40 0.095 5.60E-08 0.104 Table 5-2. Estimation of the concentration prof iles based on the intersect ion of characteristic lines with vertical profiles at sp ecific times (after Mehta, 2007). Time = 5 hours Time = 15 hours Time = 40 hours Elevation h* C Elevation h* C Elevation h* C (m) (m) (kg/m3) (m) (m) (kg/m3) (m) (m) (kg/m3) 0.170 0.224 5.6 0.130 0.216 5.8 0.090 0.203 6.2 0.068 0.181 6.9 0.116 0 0.039 0.152 8.2 0.116 0.152 8.2 0.027 0.133 9.4 0.078 0.133 9.4 0.02 0.114 11 0.059 0.114 11 0.016 0.109 11.6 0.048 0.109 11.6 0.014 0.106 11.8 0.041 0.106 11.8 0.095 0 0.012 0.104 12 0.035 0.104 12 0.095 0.104 12 0.009 0.095 13.2 0.028 0.095 13.2 0.076 0.095 13.2 0 0.095 13.2 0 0.095 13.2 0 0.095 13.2 PAGE 240 240 Figure 5-1. Kynch (1952) settling test development. a) initial uniform dilute suspension in a quiescent setting column, b) t he secondary lutocline (S) settles at a rate ws while the isopycnal interface defining the primary lutocline (P ) rises from the bed at a rate wp, c) isopycnal primary lutocline meets the secondary lutocline, and d) the final deposit concentration Cf and height hf are reached after a period of hindered settling. h a) b) c) d) ws w p C0 w s w p hs( t ) hp( t ) C ( t ) S S P S,P h f h p ( t ) C ( t ) C f PAGE 241 241 Figure 5-2. Schematic representation of the Kynch (1952) settling test with th e use of the method of characteristics. h0 h* h ws t* t wp PAGE 242 242 Figure 5-3. Kynch (1952) settling test evolution of the secondary lutocline elevation. Application of the method of characteristics to estimate the suspended sediment c oncentrations (after Mehta, 2007). PAGE 243 243 Figure 5-4. Estimated concentration profiles using the Kynch graphical method based on the method of characteristics. PAGE 244 244 Figure 5-5. Deposition test results from Krone (1962). PAGE 245 245 Figure 5-6. Results of the tagged se diment experiment of Krone (1962). PAGE 246 246 Figure 5-7. Results from Mehta (1973) showin g the effects of shear stress on the relative concentration. Initial concentrations were 1.0 kg/m3. PAGE 247 247 Figure 5-8. Results of the flow volume replacement experiment by Parchure (1985). PAGE 248 248 Figure 5-9. Step erosion test se ries of Parchure and Mehta (1985) PAGE 249 249 Figure 5-10. Suspended sediment concentration and shear stress during monitoring exercise on 5 January 1989 (from Sanford and Halka, 1993). PAGE 250 250 Figure 5-11. Suspended sediment concentration and shear stress during monitoring exercise on 2 February 1990 (from Sanford and Halka, 1993). PAGE 251 251 Figure 5-12. Suspended sediment concentrati on and shear stress during monitoring exercise on 15 January 1991 (from Sanford and Halka, 1993) PAGE 252 252 CHAPTER 6 METHOD APPLICATION 6.1 Preamble This chapter describes the application of the computational procedures described in Chapter 4 to the sediment transport test cases identified in Chapter 5. The overall program outline was presented in section 4.6 of Chapter 4. A detailed description of the primary sediment transport subroutines is presente d in Appendix D, with identifica tion of the equations used for each of the calculations. The hydrodynamic compone nts of the model were tested in Chapter 4 on the classical analytical cases. With the exception of the Kynch deposition test all simulations performed incorporated the flocculation model. Because the concentration levels are relatively low for most of the test cases, sensitivity testing showed that the TKE model added limited value to these simulations and complicated the interpretation of the primary testing. In addition, the computational burden of the probabilistic bed exchange, the flocculation model and the TKE model proved to be unmanageable. Therefore, the simulations re ported here do not incor porate the TKE model. The probabilistic treatment, wh en invoked, includes the representation of the bottom shear stress, the floc strength, the critical shear stress for erosion, the critical shear stress for deposition, the local shear in the water column the internal shear and the fall velocity as probability distributions. The probability dens ity for velocity is assumed to be normally distributed, with a standard devi ation assumed to be 20 percent of the mean velocity, which is based on a typical ratio of turbul ence intensity to mean velocity (Sharma, 1973). The probability distribution for the bottom shear stress is developed from the tr ansformation of the velocity distribution as discussed in secti on 3.3. The mean shear stress versus mean velocity relationship has been assumed to be valid for the instantaneous shear stress versus the instantaneous velocity. PAGE 253 253 The relationship is assumed to be of the form B A u The coefficients A and B were varied for each experiment simulated based on information from the original studies when available. Floc shear strengths were also assumed to be normally distributed about the mean values for each size class defined by Equation 3-11. The standard deviations for the shear strengths were assumed to be 20 percent of the mean valu es, based on the variability in the shear strength data of Krone (1963).. Settling velocities were also assumed to be normally distributed about the size class mean settling velocity developed from the equations in s ections 2.2.2, 2.2.3 and detailed in section D.2 of Appendix D. The standard deviation of the settling velocities was assumed to be 30 percent of the mean value. This value is consistent with estimates of distributi on of settling velocities developed from analysis of in situ video images collected in San Francisco Bay (see Figure 6-1). A standard deviation of 30 percent fits the data well in the middle floc sizes between 80 to 150 microns, where there are sufficient particles analy zed for statistical accur acy and less effects of scatter due to residual fluid moti on within the settling chamber, which causes scatter at smaller sizes. The probability density distributions for the loca l shear stress in the water column and the internal shear rate were assumed to be proportion al to the probability distribution for the bottom shear stress, since each of these variables is de rived from the nonlinear dependence on velocity. Any arbitrary variable, R whose distribution is assumed to be proportional to the bottom shear stress distribution woul d satisfy the relation: ()()/bb p dfRpdfR The distributions all used a numerical integration based on a discretization of the PDFs, using 101 values for discretization, with truncation of the tails of the distributions at plus and PAGE 254 254 minus three standard deviations from the mean value, which will represent 99.8 percent of the total probability of a normal distribution.. Model parameters common to all simulations conducted are presented in Table 6-1. The specific variables developed for each test wi ll be presented in Tables 6-2 through 6-12 6.1 Kynch (1952) Quiescent Deposition Test The Kynch (1952) deposition test described in section 5.1 was simulated as a test of settling in quiescent conditions and for the abil ity to replicate the theoretical treatment of hindered settling derived from a la boratory test case. No information was available from the literature on the particle size distri bution of the physical test data se t. A single size silt particle of 4.31 microns was selected to ma tch the free settling velocity obser ved in the data. The flow in the model was quiescent and the initia l suspended concentration was 5 kg/m3. The simulation duration was 40 hours. The simulation used a space-filling concentration of 13.2 kg/m3, which was the maximum concentration in the physical da ta. For quiescent conditions, vertical diffusion was set to Brownian diffusion (Equation 6-1) in the absence of flow. 3b BkT D d (6-1) Processes of importance to the Kynch test ar e accumulation of near bed material and the upward movement of the primary lutocline. In order to capture these processes, bed exchange was turned off within the model. Neither deposition downward out of the bottom cell, nor reentrainment from the bed (outside the model domain) was permitted. This allowed the high concentration of suspended sediment to expe rience hindered settling, resulting in the development of the primary lutocline. The model simulation coefficients for the Kynch test simulations are presented in Table 62. The model used 40 uniform cells over the water depth, with no depth-averaged velocity or PAGE 255 255 shear stress. The number of cells in the vertical was set based on sensitivity testing to insure adequate resolution to resolve the lutoclines. Th e concentration effect on settling velocity used the coefficients fr om Equation 2-46 of 1 = 320 s, 2 =75 s2, and 3 = 0.8. These values were used by Teeter (2001; personal communication). Because silt was simulated the flocculation model was turned off, primarily because the only contribution to the test would have been from Brownian motion, and this would have been minor. The results of a test using the exponent m =1 in the hindered settling equation of Winterwerp (1999; Equation 2-40) are presented in Figure 6-2. After 5 hours, the model suspended sediment in the upper portion of th e water column had dropped nearly uniformly downward, forming a diffuse secondary lutoclin e that lacked a sharp interface. This was apparently the result of numerical diffusion, si nce the assigned numerical term was computed to be the Brownian diffusion (order 10-8 m2/s). Below the diffuse upper lutocline the model concentration remained constant at the initial con centration over almost half of the water column. Near the bottom of the column the primary lutocl ine had begun to form as a sharp interface in the model results. After 15 hours the general location of both the pr imary and the secondary lutocline were in approximately the same location as in the test data. However, the model secondary upper lutocline had been further diffused. After 40 hours the model generally matched the final concentration profile, bu t retained some diffusion at th e merged primary and secondary lutoclines. One significant feature of the test data is th at the concentration at the secondary lutocline increased above the original con centration early in the test (hour 5). This observation suggests that the suspension is experiencing hindered settling even at the initia l concentration. In order to PAGE 256 256 increase the hindered settling within th e model the hindered settling exponent m was increased to 2. The results of that test simulation are pr esented in Figure 6-3. The model results show increased effects of hindered settlin g, with the location ra ised in the water column where at hour 5 the concentration is increased above the initial value (from 0.09 m for the m= 1 simulation to 0.13m above the bed for the m= 2 simulation). For the m =2 test the orie ntation of the concentration gradient above the primary lutocline aligns much more closely with the test data. The results of these simulations demonstrated that the model is capable of simulating the hindered settling process with qualitative agreement. Without more precise information about the actual sediment size distri bution of the settling column, fu rther numerical testing was not performed. Furthermore, for the quiescent conditi ons of this test case it was not felt to be of significant value to test the probabilistic formulati on of the model. The pr obabilistic features of the model are derived from the influence of tur bulence on the processes, which is missing in this test case. 6.2 Mehta 1973 Flume Deposition Tests Mehta (1973) documented that the percentage of sediment remaining in suspension at equilibrium for a given bulk suspended sedime nt sample is independent of the initial concentration for a fixed shear stress. When the sediment concentrations are relatively dispersed, such that they do not impact the level of turbulence, then the flows ability to sustain the particles in suspension is related to the indi vidual particle characteristics rather than the properties of the total suspension. So if the initial concentrati on is doubled, but the fraction of a particular size particle is fixed, then the final pe rcentage of the particles capable of remaining in suspension will be fixed compared to the total sedi ment initially in suspension. When the shear stress changes, the fraction of sediment capable of staying in suspension will change based on the size distribution of the sediment and the critic al shear stress for deposition. If the initial PAGE 257 257 concentration is sufficiently high, then the sedime nt concentration will begi n to affect turbulence and alter the suspending capacity of the flow. The model was first simulated using the cla ssical excess shear stress formulas for both deposition and erosion, as de scribed by Equations 2-50 and 2-51 in section 2.2.4. Model simulation specifications for the Mehta(1973) simulations are presen ted in Tables 6-3 and 6-4. The testing was performed on four different shea r stresses: 0.25, 0.40, 0.60 and 0.85 Pa. The flow depth was assumed to be 0.305 m. The c onversion of the mean velocity specified for the flow was converted to a shear stress by the relationship de veloped by Mehta (1973) for the original flume (Equation 6-2, in units of Pa and m/s)). 0.94 0.956bu (6-2) The over-bar on the velocity refers to the depth-averaged velocity. The calibration procedure used by Mehta (1973) for Equation 6-2 involved profiling the velocity in the flume with a miniature propeller probe attached to the upper ring of the rotating flume. With the known differential velocity, V between the top and bottom fl ume rotating surfaces and the probe velocity, up, the velocity relative to the bed at th e measurement elevation is the differential velocity less the probe velocity, u = V up. The velocity profiling was performed without sediment in suspension due to interference of sediment particles with the velocity probe. Profiling was performed for two flume depths (0.16 m and 0.23 m) and for multiple differential velocities. The bed shear stress was developed from a regression of direct measurements of shear stress with strain gages to the differentia l velocity. Shear stress was regressed against the computed depth-averaged velocities from the measured profiles to obtain Equation 6-2. The mean velocities that gave the four shear stress es above were 0.24, 0.40, 0.61, and 0.88 m/s. PAGE 258 258 Model simulations used only 5 cells over the wa ter depth because the concentrations were vertically almost uniform in the model. The shear stress profile was developed using a Karman mixing length linear profile. Sediment classes we re defined for 60 cohesive classes between 0.1 and 2000 microns and for 10 silt classes between 4 and 60 microns. The initialization of the floc distribution was limited to the first 40 size classe s (0.1 to 70 microns). A fractal dimension of 2.2 was used, which corresponds to strong estuarin e flocs (Winterwerp, 1999). As discussed in section 2.1.3.6, analysis by Part heniades (1963) of Krones data showed the shear strength to vary with the excess density to the 5/2 power, wh ich is effectively a fractal dimension of 2.2 {5/2=2/(3Df)}. Figure 6-4 presents a comparison of the shear strength developed from Equation 3-11 with the Krone (1963) data. Two curves for Equa tion 3-11 are presented; one using a scaling coefficient, Bf = 600 Pa with a fractal dimension of 2. 2 and another with a scaling coefficient Bf = 1200 Pa and a fractal dimension of 2.5. These s how that varying the fract al dimension requires rescaling the equation to fit Krones data. In the laboratory tests the same sediment was used in each test case with varying shear stress. Therefore, the sediment properties n eed to be established and not changed between simulations for each test case. Calibrating the model required careful evaluation of the defined critical shear stresses for erosion and deposition ( by Equations 2-53 in section 2.2.4) relative to the range of tested shear stress es. Figure 6-5 presents an exam ple variation of the mean floc shear strength, and mean critical shear stresses for erosion and deposition. When the functional value of the critical shear stress for deposition be comes larger than the cr itical shear stress for erosion, it should take on the erosion threshold value. PAGE 259 259 If a 0.085 Pa shear stress is assumed, (prese nted in Figure 6-6 for the example in Figure 65), then for use of a classica l probability of depos ition (Equation 2-50), de position would occur over a floc size range from approximately 2 to 70 microns. Above 70 microns erosion would occur. The critical shear stresses and shear strength used in simulation of the Mehta (1973) deposition experiments for the use of mean valu e treatment of the processes is presented in Figure 6-7, with the two extreme bottom shear st resses denoted by the horizontal lines at the appropriate shear stress values. The critical sh ear stress for erosion was specified from the floc shear strength with Bf = 1800 Pa and a fractal dimension of 2.2. The critical shear stress for deposition is from Equation 2-53, with d0 = 0.03 Pa, dref = 0.1 microns and exponent = 0.8. In implementing Equation 2-53, it was discovered th at the two equations intersect at a yield strength of 1.27 Pa, rather than at 1.6 Pa. In order to avoid a discontinuity in the critical shear stress for erosion the threshold of 1.27 Pa wa s used between the two parts of Equation 2-53. At the lower shear stress (0.25 Pa) of the four Mehta tests the mean-value exclusive paradigm using a classical probability of depos ition indicates deposition could occur over the floc size range of 1.5 to 15 microns. Erosion of flocs would occur fo r flocs larger than 15 microns. At a shear stress of 0.85 Pa there is es sentially no opportunity fo r deposition, and flocs larger than 8 microns would potentially erode. For intermediate shear stresses between 0.25 and 0.85 Pa, the floc size window fo r deposition shrinks with increasi ng shear stress above 0.25 Pa, and the minimum floc-size for erosion beco mes smaller with increasing shear stress. The initial silt fraction in suspension was set at 40 percent, which was the percent of the silt fraction in the commercial kaolinite used in the original flume investigation. The flocculation model used an aggreg ation efficiency of 1.0, disaggr egation efficiency of 0.75 and PAGE 260 260 collision disaggregation efficiency of 0.5. McAnally (1999) reported a range of effective aggregation efficiencies of 0.005 to 0.70 from numerical collision model experiments. Winterwerp (1999) proposed that these co efficients are basically empirical. The initial suspended sediment concentration was specified for each size class as shown in Figure 6-8. The model was not very sensitive to the initial specification of the sediment size distribution. The model makes no distinction be tween initially present primary particles and flocs, essentially assuming a single primary particle size. Consequently, the numerical flocculation model redistributes the size to approach equilibrium between aggregation and disaggregation for the given shear stress. The results of the model for a classical shear stress treatment are pres ented in Figure 6-9, with use of the simultaneous deposition paradi gm. The model sediment size concentration distribution in suspension initially was adjust ed, along with the critic al shear stresses for deposition and erosion to obtain r easonable agreement with the original flume data. Attempts at calibration of the model using the exclusive paradigm were not very successful, with difficulty in obtaining equilibrium concentrations as low as the data. The model reproduces the asymptotic trend towa rd an apparent equilibrium concentration, dependent on the shear stress. The initial sedi ment reductions within the first hour were not precisely replicated, particularly for the two inte rmediate shear stress cases. The complexity of the processes and the number of degrees of freedom in the model specification make precise agreement difficult. The analysis of sediment size concentration distributi on prior to initiation of settling in light of the composite effective se ttling velocity could provide some guidance on improving the replica tion of the results. PAGE 261 261 Figure 6-10 provides a compar ison of the exclusive and simultaneous paradigm simulations for the 0.25 Pa test. Simultaneous deposition reduces the equilibrium concentration compared to the exclusive simulation by increasing the depositional flux to better match the flume data. The sensitivity of probabilistic versus mean-v alue representation and the exclusive versus simultaneous bed exchange are illustrated in Figure 6-11 for the 0.60 Pa test. The same coefficients used in the replication of the result s in Figure 6-9 were used with the probabilistic treatment of the primary variables. The result s clearly show that the increased variability introduced by the probabilistic treatment leads to increased erosion potential. The use of simultaneous deposition sets the equilibrium con centration lower for both the mean value and the probabilistic approach. If calibration has been perform ed using one choice of averaging/deposition paradigm comb ination and the combination is ch anged, then to replicate the results will require recalibrati on of the empirical coefficients. Figure 6-12 presents a conceptual illustration of the impact of using probabilistic variables in the analysis. The variations plotted about the mean shear values represent one standard deviation of each variable. The model performs the calculation on three standard deviations. The critical thresholds for deposition will be th e intersection of the minimum bottom shear stress with the largest critical shear stre ss for deposition. The effect of the variance in each variable is to widen the span of floc sizes for which depos ition may occur, while allowing an overlap in the deposition and erosion sizes when the exclus ive paradigm in used in conjunction with probabilistic variables. The span of deposition for the mean -value analysis is shown as the shaded area. Figure 6-12 offers the concep t using only a single standard deviation. PAGE 262 262 The evaluation of the effect of using probabilistic variables by inspection of Figure 6-12 would initially imply that additional deposition sh ould occur. However, with the expansion of the variability in the variables to three standa rd deviations, the overl ap between erosion and deposition covers a wider range of floc sizes Over time, erosion will dominate deposition because of the effects of aggregation, which crea tes a flux of sediment mass toward larger flocs, out of the deposition range of floc sizes. Figure 6-13 presents a comparison of the floc size distributio n at the end of simulations using the exclusive paradi gm for the mean value and probabilist ic treatment of the variables. The distributions are potted for both surface and bo ttom cells in the model for each simulation. The results show that the probabilistic method yields a broader floc spectrum, which results in higher overall suspended sediment concentration than the mean value simulation. The mean value simulation has an abrupt dr op in concentration on the lower si de of the spectrum associated with the single-valued floc size threshold for deposition. The probabilistic spectrum has a more gradual a transition. In comparing the two spectrums in Figur e 6-13 with the characteristics spectra developed by Kranck and Milligan (1992 ) in Figures 2-2 and 2-5, the probabilistic spectrum shape has the same skewed spectrum with th e modal floc size larger than the mean floc size. The simulated spectrum with use of averag e values of the variables is more symmetric. The relative variation between su rface and bottom distributions is maintained for both averaging methods, implying that the differences are contro lled by the mean stresses in the water column. The primary observation from the simulati on of the Mehta 1973 tests is that the simultaneous deposition paradigm seems to work better than the exclusive erosion or deposition method. One advantage of the simultaneous pa radigm is the simplification of parameter specification, as the critical shea r stress for deposition no longer pl ays a role in the model. The PAGE 263 263 probabilistic treatment of variables complicates the exclusive paradigm by making the influence of the aggregation processes different relative to the mean valued approach. This makes the development of general guidelines for paramete r adjustment between the two methods more difficult. 6.3 Krone (1962) Flume Deposition Tests The flume deposition tests of Krone (1962), as described in section 5.2.1, were simulated for three shear stress experiments. The model sp ecifications for these simulations are presented in Table 6-5. The simulations used averaged variables (shear stresses, shear strengths and settling velocities) and the clas sical probabilities of erosion a nd deposition based on excess shear stress (Equations 2-50 and 2-51). Simulation results are presented in Figures 6-14 through 6-16 for the three shear stress tests, with 0.0305, 0.0415 and 0.0515 Pa. The model is able to track the settling for each of the test ca ses for the first 50 hours of the simulations. However, the flume data has a long-t erm trend that was not sa tisfactorily replicated in the numerical simulations. After about fifty hour s in the flume tests there is s change in the depositional trend. The trend can be seen in Figure 6-17 The plot shows that the linear trends on a log plot indicate exponential decay with a break in the decay rate at around 50 hours into each test. The inability of the numerical model to rep licate the trend may be associated with the effects of the recirculating pumps altering the floc size distribution in the flume. Adding additional shear in the numerical model tended to increase flo cculation and increase deposition rather than decrease it. Figure 6-14 presents the effects of adding 100 Hz supplemental shear to the 0.0305 Pa Krone deposition test. Krone di d not measure floc sizes, but estimated the maximum floc size to be between 20 and 30 micr ons based on the estimated thickness of the laminar boundary layer near the bed. Krone ha d estimated the sheari ng within the pumping PAGE 264 264 system of the flume to be in excess of 220 to 950 Hz in the walls of the re turn lines for the range of shear tests being considered here. The peak floc size within the distribution in the numerical experiments for the flume experiments was in the range of about 40 to 50 microns. Winterwerp (2004) also had difficulty in rep licating the Krone depositional experiments. He incorporated a time varying critical shea r stress for erosion in combination with the continuous deposition paradigm th at could not fully account for the persistence of the flume sediments to remain in suspension. He questioned the possible accuracy of the bottom shear stress measurement estimates. The primary change in depositional behavior when the concentration falls below 0.3 kg/m3, suggests that some transition occurred in the aggregation. That change could be related to the interaction of the suspende d sediments within the pumping system causing some lubricating effect that changed as the concentration fell. One aspect of the Krone depositional experime nts that makes them different from other laboratory tests is the basic time scale of the tests. While most of the annular flume experiments for deposition at flow depths and shear stresse s comparable too the Krone tests have test durations on the order of 10 hours, the Krone tests have time scales of tens to hundreds of hours. The primary differences are in the re circulating system for the flume. The primary conclusion from the simulations of the Krone deposition test s is that extensive sensitivity analysis of probabilistic treatment and the exclusive versus continuous deposition paradigm are not warranted. 6.4 Parchure and Mehta (1985) Dilution Test The simulation of the volume removal test by Parchure and Mehta (1985 ), as described in section 5.4, was performed by assuming that th e sediment mass extract ed over the four-hour PAGE 265 265 period is taken uniformly over the flume. The expre ssion for the time rate of change of the depthaveraged concentration in a unit horizontal area of the flume due to an extraction rate per unit area of the flume, q, is presented in Equation 6-3. This e xpression assumes that the effects of the extraction on the suspended sediment concentrati on are uniform within th e flume, both laterally and vertically. Longitudinal uniformity is assured in the annular flume. CqC th (6-3) An analytical solution for Equation 6-3 is know n, as shown in Equa tion 6-4. With the water flow depth known (0.26 m), the observed d ilution of 0.00789 (0.03/3.8) and the duration of the test, the approximate withdrawal discharge can be estimated as 0.00008765 m3/s/m2. 0exp qt hCC (6-4) The model was tested with no bed exchange to confirm the extraction dilution formulation and prove that it can match the solution given in E quation 6-4. Results of the dilution rate test are presented in Figure 6-18. For this test th e initial concentration was prescribed as the equilibrium (3.85 kg/m3) developed in the original flume te st by eroding a settled bed prior to initiation of extraction of water/sediment mixture. With no bed exchange the numerical solution should match the analytical solution of Equation 6-3, which is also included in Figure 6-18. This shows that the analytical solution describes the fl ume data very well and also that the numerical model very precisely matche s the analytical solution. The model was then simulated with bed exchange turned on. The specifications for the Parchure and Mehta dilution simulations are presented in Table 6-6. For this test the initial suspended sediment concentration was set to zero and an initial particle size distribution was specified within the bed. The initial bed concen tration distribution is pr esented in Figure 6-19, PAGE 266 266 which shows a uniform concentration below 70 microns, with no sediments larger than 70 microns. The total bed mass was adjusted until the erosion of the bed dur ing the initial spin up of the model approached the desire d initial concentration (3.85 kg/m3) as an equilibrium. The model was run for 4 hours, having reached near e quilibrium in less than an hour, and then the dilution begun at hour 4 and continued until hour 8. The model simulation continued for a total simulation period of 15 hours. This test was simu lated with the classical exclusive deposition or erosion model based on excess shear stress (E quations 2-50 and 2-51) and also with the probabilistic treatment of the bed ex change described in Chapter 4. The results of these tests are presented in Figure 6-20. Both of the model simulations quickly reached the desired initial equilibrium concentration. During the first 2 hours of the dilution phase the two model tests follow the test data very closely. However, near the end of the dilution, the classical treatment continues to follow the analytical solution, while the probabi listic treatment deviates slightly, with some apparent additional erosion from the bed. Afte r the cessation of dilution the classical simulation remains nearly constant at the final concentration after dilution of 0.03 kg/m3. The concentration does rise slightly to 0.0304 kg/m3 by the end of the simulation at hour 15. The probabilistic treatment, however, exhibits more significant erosion after the end of the dilution phase, increasing the suspended concentration to 0.066 kg/m3 by the end of the simulation. This relatively minor erosion is in agreement with an increase in the flume test to 0.1 kg/m3 after an additional 24 hours after the end of the dilution phase. The probabilistic treatment of the bed exchange is in both qualitative a nd quantitative agreement with the flume data with regard to the persisting slight erosion. The classical treatment does not capture that feature. PAGE 267 267 Lau and Krishnappan (1994) also conducted ex periments in a rotating annular flume designed to evaluate the question of simultaneous erosion and deposition. Instead of conducting erosion to equilibrium, as Parchure and Mehta di d, they conducted deposition experiments that reached an apparent equilibrium concentration. After equilibrium was reached they replicated the Parchure and Mehta experiment (1985), sl owly removing sediment-laden water from the flume, replacing it with clear water. They monitored the floc size distribution periodically during each phase of the test. Th ey identified three features of th e experimental results that they reasoned could be used to judge the question of simultaneous erosion and deposition. First was the change in particle size di stribution during the depositional period be tween initiation of the deposition and when equilibrium concentration was reached. They argued that changes in the floc size distribution that showed reductions in all size classes means that all size classes must have experienced settling. They argued that th is is contradicts the conceptual model of Lick (1982) that fine fractions never settle, largest frac tions deposit quickly and intermediate fractions experience both deposition and reentrainment. The Lick model was interpreted to lead to a shifting of the particle size dist ribution to the smaller fractions. Another explanation could be that smaller size flocs can undergo aggregation, converting to larger flocs before settling, thus maintaining a proportional distribution. The second feature of their results was the change in particle size distribution during dilution. The Partheniades, et al. (1968) c onceptual model would expect that no further deposition will occur after equilibrium has been reached and that no further erosion should occur upon dilution. The Partheniades et al. model for the process relies on the condition that equilibrium concentrations are reached because the sediment supply from the bed for erodible sediments has been exhausted and that all materi al capable of depositing has settled out. The PAGE 268 268 particle size distribution should simply dilute, maintaining its proportionality. The Lick model would again result in a redistribution of the si zes, as the finer fractions would not be replaced, while the intermediate sizes could be replaced by erosion, resulting in a coarsening of the distribution. Their size distribution evolution supported the Pa rtheniades et al. model, with the D50 remaining essentially constant and the cla ss concentrations each essentially diluted. The final feature that was used to judge the processes was the evolution of the total concentration during dilution. If the simultaneous erosion and deposition model is correct then the diluted concentration should be replaced progre ssively by erosion that would result in a new equilibrium concentration. The Partheniades et al. model would simply dilute the concentration; which was also born out in their results. The tests by Lau and Krishnappan were an im provement with the monitoring of the size class distribution but still re lied on inference of the net de position from the suspended concentration. Their logic in judging the simu ltaneous question relies, in part, on the assumption of a static equilibrium condition for aggregati on and disaggregation during their tests. In addition, their conclusions are based on the assump tion of a constant rese rvoir of all particles sizes from the bed for potential erosion, w ithout any winnowing during the test. The Lau and Krishnappan particle size distribut ion evolution during the initial di lution is not consistent with the observations of Kranck and Milligan that the distribution varies prim arily on the large floc size end of the distribution, resulting in the mean and modal flocs size being dependent on the total concentration. The deposit ion phase prior to dilution, wh en the shear stress was reduced should have experienced a reduction in the mean floc size, according to th e Kranck and Milligan model. PAGE 269 269 The changes in the floc size distribution during the present numerical simulation are presented in Figure 6-21. Comparison between the average shear stress simulation and the probabilistic shear stress simulati on at the initiation of dilution ( hour 4) shows sharply different distributions. The probabilistic simulation results in significantly larger flocs, with a modal size of around 300 microns, compared to approximately 70 microns for the average shear stress simulation. The initial floc size distributi on during the Lau and Kr ishnappan depositional equilibrium test had a modal floc size of only 5 microns. The modal floc size at the end of dilution in the present simulations dropped dramatic ally in both the aver age and the probabilistic treatments, with the probabilistic peak size (15 microns) slightly smaller than the peak for the average shear stress simulation (18 microns). However, after dilu tion the probabilistic distribution retained its broader distribution compared to th e average shear stress. The distribution at the end of the simulation (hour 15) had not cha nged for the average shear stress method, plotted as points to show that the dist ributions overlay one another. The probabilistic distribution, however, shows an expa nsion of the distribution into larg er floc sizes. Note that this behavior is consistent with th e Krank and Milligan (1992) model of floc distribution evolution. The simulations of the Parchure and Mehta ( 1985) dilution experiment showed that the probabilistic treatment of the pr imary variables was able to re plicate the concentration rebound after the dilution was stopped. This winnowing erosion was not s een in the simulation using the mean values of the variables. The comparison of these numerical results with the results of the Lau and Krishnappan laboratory floc size data suggest that the fl oc size distribution evolution may depend on the nature of the process that led to the initial concentration prior to dilution. The Parchure and Mehta test appr oached equilibrium by erosion prior to dilution, while the Lau and Krishnappan test approached equilibrium by deposition prior to dilution. The Lau and PAGE 270 270 Krishnappan tests used higher shear stresses (0.2 5 to 0.37 Pa) than the Parchure and Mehta test (0.2 Pa). The numerical model floc size distri bution evolution follows the observations of the Kranck and Milligan conceptual model. 6.5 Parchure and Mehta (1985) Erosion Test The step-erosion test of Parchure and Mehta (19 85) described in section 5.5 was chosen to evaluate the models ability to address progressive erosion for stepped incr eases in shear stress. Table 6-7 presents the specifications for these simulations. The model specification for the bed was a particle size distribution of the sediment in the bed combined with the total mass of sediment in the erodible bed layer. The distribu tion for the cohesive sediment mass in the initial bed is presented in Figure 6-22. The silt fraction was 10 percent of the total mass and was uniformly distributed over the silt classes. Th e model initialization of the bed was the primary calibration tool to match the incremental eros ion for each shear stress. Because the shear strength and critical shear stress for erosion are determined in the model as a function of particle size, the distribution of erosion re sistance in the bed is directly co rrelated to the size distribution. The shear strength and critical shear stress for er osion are inversely proportional to the particle size (Equations 2-9 and 2-53). The results of th e model simulation using average shear stresses are presented in Figure 6-23. The simulation wa s able to capture the incremental erosion and asymptotic behavior by the adjustme nt of the erosion rate constant. To illustrate the difference in the response between the classical treatment of the erosion and the probabilistic, again the simulation was rep eated, keeping the erosion rate constants fixed. A comparison of the two simulations is included in Figure 6-24. This clearly illustrates the need for development of erosion rate information that is consistent with the shear stress treatment. The dramatic initial jump in the concentration during the first shear level (0.1 Pa) hour is an indication that the first improvement when ap plying the probabilistic approach would be the PAGE 271 271 analysis of the bottom shear stress distribution plot ted against the critical shear stress levels for each shear level. Note that afte r the first major erosion event in the first hour, the progression of erosion for the probabilistic approa ch is not dramatically different from the data, if the initial offset is removed. 6.6 Sanford and Halka (1993) Field Datasets The three data sets from Sanford and Halka ( 1993) described in section 5.6 were simulated within the process model to reevaluate the conclu sion of the authors regarding the need for using a continuous deposition model to accurately simu late the observed erosion and deposition in a tidal estuarine environment. The simulations c onducted in the evaluation of these datasets are summarized in Table 6-8 through 6-11. Tables 6-8 through 6-10 present the simulations for the three years of data collection used to identify sensitivity of bed exchange treatment. Table 6-11 presents the adjustments in the basic coefficien ts to obtain the calibrations presented for each data period simulation. The simulations performed differed from th e original Sanford and Halka numerical application in the number of size classes used. They used a single size class. Rather than specifying particle size, they used settli ng velocity, ranging from 0.00009 to 0.00014 m/s for their three experiments. The current modeling application used 40 size classes for cohesive sediment ranging from 0.1 to 1000 microns, with a fractal dimension of 2.2. The model had 10 silt size classes between 2 and 60 microns, compri sed 40 percent of the bed material and of the initial suspended concentrations. The water dept hs during each of the three deployments are as shown in Figure 6-25. The number of size classe s was limited for these tests because of the computational burden of the probabilistic approach and the flocculation model. PAGE 272 272 The Sanford and Halka numerical analysis evaluated three bed exchange models. One was the classical exclusive linear treatment, usi ng Equations 2-50 and 2-51. Sanford and Halka referred to the exclusive treatment as the cohesive model. They also considered the continuous deposition model combined with the linear clas sical erosion model. The third model they evaluated was continuous depositi on with a power law representa tion for erosion (Equation 6-5) bE (6-5) The coefficients were developed empirically during model application. The values developed for were 5.9, 0.69 and 7.5 mg/cm2/h, for the 1989, 1990 and 1991 data, respectively. The corresponding values of the exponent, were 1.4, 4 and 1.3. Sanford and Halka came to the conclusion that the continuou s paradigm is required to model the field concentrations accurately, but that either the li near or power law erosion equations worked well over the range of conditions they evaluated. Model simulations in the current research we re performed separately for the use of the classical exclusive erosion or deposition model and the continuous deposition model. Each of these bed models was also tested with both the average shear stress in the bed exchange equations and with the probabilistic treatment of shear stresses and settling velocities. It was confirmed from the tests that the best perfor mance of the models in matching the observed suspended sediment variation was with the contin uous deposition model. This was true for both the classical average shear stresse s and for the probabilistic bed exchange. The quality of the calibrations is compared for the two shear stress treatments with the c ontinuous deposition model in Figures 6-26 through 6-28 for the consecuti ve years of deployment. The parameter adjustments that were made for calibration are pr esented in Table 6-11. These adjustments were as simple as changing the initial concentration or adjusting the erosion rate constant. PAGE 273 273 The comparisons show that the probabilisti c treatment captures the variability in the concentrations much better than the average sh ear stresses. In addition, the timing of the occurrence of extremes in the co ncentration variation is more in phase for the probabilistic method. The relative response of the probabilistic ve rsus the average shear stress method can be better understood by inspecting the changes in the floc size distribution during the simulation. Figure 6-29 presents the floc dist ributions for the two shear met hods for continuous deposition at the times of the extremes in the concentration variability for the 1989 cas e. The distributions show three characteristic regions of the distribution, generally the same for both formulations. Below 1 micron the rate of loss of the finest part icles is associated with aggregation to larger sizes, since the settling velocity of the finest pa rticles is very low. At sizes between 1 and 10 microns the initial distribution is progressive ly being reduced for both methods. This is associated with the continuous deposition, in a ra nge of floc sizes where the shear stresses are not sufficient to re-entrain the sediment. A bove 10 microns the distribu tion rises and falls in conjunction with erosion and deposition of the larger flocs. The concentration is at a minimum near hour 14, a maximum at hour 17 and th en at another minimum at hour 21. Another feature of the floc si ze distribution comparison was that the probabilistic approach results in higher modal volume fractions, with the m odal fraction at larger fl oc sizes. In addition, the range of floc sizes associated with the modal rise is broader for the pr obabilistic method. All of these features are a di rect result of the higher shear stresses associated with the upper tail of the shear stress distribution. The floc size distribution evolution for th e simulations using the excusive erosiondeposition model is presented in Figure 6-30. The exclusive model shows that for both the PAGE 274 274 average and the probabilistic methods the flocculati on end of the distribution is more prominent. Concentrations in the middle range of floc sizes (1-10 microns ) are reduced compared to the simultaneous distributions (Figure 6-29). The net e ffect is that the exclus ive treatment results in a shift to larger median particle size over th e duration of the simulation for both shear stress treatments. To illustrate the relative significance of the probabilistic treatment of the processes and of the use of continuous deposition, a group of simu lations were made where the only changes in the model specifications were the choice of continuous deposition versus exclusive erosion or deposition and whether the probabilistic method is used. This gave four simulations for each of the Sanford and Halka test cases. These are presented in Figu res 6-31 through 6-33. Note that the simultaneous-probabilistic simulations in these tests were not the same as those presented in Figures 6-26 through 6-28 where other parameters were calibrated to improve the comparison to the observations. The simulations on each of the test cases reflect the same overall results. The lowest concentrations are associated with the simultaneous/average simulations. This follows from the less variable shear stresses and the continuous deposition. In contrast, the highest concentrations are associat ed with the exclusive treatment combined with probabilistic shear stresses. That combinati on maximizes erosion and minimizes deposition. 6.7 Krone (1962) Tagged-Sediment Settling Test Attempts were made to design a numerical simulation that could address the exclusive versus simultaneous paradigm question raised by the Krone (1962) gold-labeled experiment described in section 5.2.2. A se parate class of cohesive sediment was included in the numerical code, with identical cohesive properties as the pr imary cohesive sediment. The interactions of the two sediments within the aggregation mode l were handled by simple mass ratios between interacting classes. A sample simulation using the simultaneous deposition and erosion is PAGE 275 275 presented in Figure 6-34. The re sults have not been calibrated well to the flume data. However, there is a slight difference between the apparent deposition rate of the tagged sediment and the total concentration. That difference is associat ed with initial bed erosion differences at the beginning of the simulation. These results provided no insights relative to the paradigm. 6.9 Concluding Comments These simulations reiterate the often procla imed need for development of sediment transport parameters that are site specific. In addition, it is apparent that the methods applied for the analysis may have to be taken into account when field and laboratory observations are to be utilized. This is particularly true for the probabilistic treatment of the variables, which most standardized field and laboratory an alyses are not designed to support. Specific data that would improve the understand ing of the bed exchange processes, which are not currently available, are; Erosive fluxes from the bed by specific particle s size distribution. This would improve the bed exchange models directly and would provide information on the similarity of the form of that distribution for various sites. If self -similar patterns are observed, then historical bulk erosion estimates could be used in a distributed manner. Measurements or indicators of spatial heterogeneity of bed fluxes. This data would help to develop statistical models of variability that can be incorporated into the models of probabilistic interaction. Floc spectra that differentiate between primary mineral particles and flocs, with the ability to estimate the order of aggregation of the flocs. Much of the variability in settling velocities may be related to the ratio of prim ary mineral grains to floc characteristics. Process monitoring in the near-bed zone to indentify the gross and net bed fluxes over some relatively large domain. This would be the final determination of the exclusive versus simultaneous paradigm. However, it is recognized that measuring exclusive deposition or erosion once is a necessary c ondition to argue in favor of the exclusive paradigm, but it is not a suffici ent condition to exclude the po ssibility of simultaneous bed exchange. PAGE 276 276 Table 6-1. Parameters common to all simulated tests. Parameter Units Value Water density kg/m3 1025 Cohesive mineral density kg/m3 2650 Silt density kg/m3 2650 Shear effects coefficient, 1 s 320 Shear effects coefficient, 2 s2 75 Shear effects coefficient, 3 0.8 Free settling concentration, Cf kg/m3 0.06 Hindered settling concentration, Ch kg/m3 0.66 Table 6-2. Parameters used in the Kynch (1952) de position tests. Test simulation Variable or parameter Units Kynch-1 Kynch-2 Water depth m 0.25 0.25 Number of cells 40 40 Number of noncohesive classes 3 3 Minimum cohesive size, Dmin micron 4.3 4.3 Maximum cohesive size, Dmax micron 4.32 4.32 Number of silt classes 0 0 Floc strength coefficient, Bf Pa 100 100 Maximum settling velocity m/s 0.0000164 0.0000164 Cohesive settling velocity factor 1.0 1.0 Reference particle size, Dref micron 4.3 4.3 Fractal dimension, Df 2.2 2.2 Aggregation efficiency 0 0 Disaggregation efficiency 0 0 Collision breakage efficiency 0 0 Initial concentration kg/m3 5.0 5.0 Bottom shear stress Pa 0.0 0.0 Supplemental internal shear s-1 0.0 0.0 Probabilistic treatment no no Cohesive deposition none none Clay erosion scale factor kg/m3 0.0 0.0 Hindered settling exponent 1 2 PAGE 277 277 Table 6-3 Parameters used in the Mehta (1973) deposition tests. Test simulation Variable or parameter Units M25-C1 M40-C1 M60-C1 M85-C1 Water depth m 0.305 0.305 0.305 0.305 Number of cells 5 5 5 5 Number of noncohesive classes 60 60 60 60 Minimum cohesive size, Dmin micron 0.1 0.1 0.1 0.1 Maximum cohesive size, Dmax micron 2000 2000 2000 2000 Number of silt classes 10 10 10 10 Minimum silt size micron 4 4 4 4 Maximum silt size micron 60 60 60 60 Initial silt fraction in bed 0.2 0.2 0.2 0.2 Floc strength coefficient, Bf Pa 1800 1800 1800 1800 Silt mobility coefficient 1.05 1.05 1.05 1.05 Critical shear stress for deposition scale factor, cd0 Pa 0.03 0.03 0.03 0.03 Critical shear stress for deposition exponent, 0.8 0.8 0.8 0.8 Maximum settling velocity m/s 0.0002 0.0002 0.0002 0.0002 Cohesive settling velocity factor 4.0 4.0 4.0 4.0 Silt settling velocity factor 1.0 1.0 1.0 1.0 Reference particle size, Dref micron 0.1 0.1 0.1 0.1 Fractal dimension, Df 2.2 2.2 2.2 2.2 Aggregation efficiency 1.0 1.0 1.0 1.0 Disaggregation efficiency 0.75 0.75 0.75 0.75 Collision breakage efficiency 0.25 0.25 0.25 0.25 Initial concentration kg/m3 5.0 5.0 5.0 5.0 Initial silt fraction in suspension 0.4 0.4 0.4 0.4 Bottom shear stress Pa 0.25 0.40 0.60 0.85 Supplemental internal shear s-1 0.0 0.0 0.0 0.0 Probabilistic treatment No No No No Cohesive deposition Sa Sa Sa Sa Silt deposition Sa Sa Sa Sa Clay erosion scale factor kg/m3 1.0 1.0 1.0 1.0 Silt entrainment scale factor kg/m3 1.0 1.0 1.0 1.0 Hindered settling exponent 1 1 1 1 a S = simultaneous; PAGE 278 278 Table 6-4 Parameters used in the Mehta (1973) deposition sensitivity tests. Test simulation Variable or parameter Units M25-C2 M40-C2 M40-P1 M40-P2 Water depth m 0.305 0.305 0.305 0.305 Number of cells 5 5 5 5 Number of noncohesive classes 60 60 60 60 Minimum cohesive size, Dmin micron 0.1 0.1 0.1 0.1 Maximum cohesive size, Dmax micron 2000 2000 2000 2000 Number of silt classes 10 10 10 10 Minimum silt size micron 4 4 4 4 Maximum silt size micron 60 60 60 60 Initial silt fraction in bed 0.2 0.2 0.2 0.2 Floc strength coefficient, Bf Pa 1800 1800 1800 1800 Silt mobility coefficient 1.05 1.05 1.05 1.05 Critical shear stress for deposition scale factor, cd0 Pa 0.03 0.03 0.03 0.03 Critical shear stress for deposition exponent, 0.8 0.8 0.8 0.8 Maximum settling velocity m/s 0.0002 0.0002 0.0002 0.0002 Cohesive settling velocity factor 4.0 4.0 4.0 4.0 Silt settling velocity factor 1.0 1.0 1.0 1.0 Reference particle size, Dref micron 0.1 0.1 0.1 0.1 Fractal dimension, Df 2.2 2.2 2.2 2.2 Aggregation efficiency 1.0 1.0 1.0 1.0 Disaggregation efficiency 0.75 0.75 0.75 0.75 Collision breakage efficiency 0.25 0.25 0.25 0.25 Initial concentration kg/m3 5.0 5.0 5.0 5.0 Initial silt fraction in suspension 0.4 0.4 0.4 0.4 Bottom shear stress Pa 0.25 0.40 0.60 0.85 Supplemental internal shear s-1 0.0 0.0 0.0 0.0 Probabilistic treatment No No Yes Yes Cohesive deposition Exclusive Exclusive Sa Sa Silt deposition Exclusive Exclusive Sa Sa Clay erosion scale factor kg/m3 1.0 1.0 1.0 1.0 Silt entrainment scale factor kg/m3 1.0 1.0 1.0 1.0 Hindered settling exponent 1 1 1 1 a S = simultaneous; PAGE 279 279 Table 6-5 Parameters used in the Krone (1962) deposition tests. Test simulation Variable or parameter Units K305-C1 K415-C1 K515-C1 K305-C2 Water depth m 0.305 0.305 0.305 0.305 Number of cells 5 5 5 5 Number of noncohesive classes 60 60 60 60 Minimum cohesive size, Dmin micron 0.1 0.1 0.1 0.1 Maximum cohesive size, Dmax micron 2000 2000 2000 2000 Number of silt classes 10 10 10 10 Minimum silt size micron 2 2 2 2 Maximum silt size micron 60 60 60 60 Initial silt fraction in bed 0.2 0.2 0.2 0.2 Floc strength coefficient, Bf Pa 1200 1200 1200 1200 Critical shear stress for deposition scale factor, cd0 Pa 0.02 0.02 0.02 0.02 Critical shear stress for deposition exponent, 0.45 0.45 0.45 0.45 Silt mobility coefficient 1.05 1.05 1.05 1.05 Maximum settling velocity m/s 0.00001 0.00001 0.00001 0.00001 Cohesive settling velocity factor 1.0 1.0 1.0 1.0 Silt settling velocity factor 1.0 1.0 1.0 1.0 Reference particle size, Dref micron 0.1 0.1 0.1 0.1 Fractal dimension, Df 2.6 2.6 2.6 2.6 Aggregation efficiency 0.75 0.75 0.75 0.75 Disaggregation efficiency 0.75 0.75 0.75 0.75 Collision breakage efficiency 0.25 0.25 0.25 0.25 Initial concentration kg/m3 0.72 0.93 0.50 0.72 Initial silt fraction in suspension 0.4 0.4 0.4 0.4 Bottom shear stress Pa 0.0305 0.40 0.60 0.0305 Supplemental internal shear s-1 0.0 0.0 0.0 100.0 Probabilistic treatment No No No No Cohesive deposition ExclusiveExclusiveExclusive Exclusive Silt deposition ExclusiveExclusiveExclusive Exclusive Clay erosion scale factor kg/m3 1.0 1.0 1.0 1.0 Silt entrainment scale factor kg/m3 1.0 1.0 1.0 1.0 Hindered settling exponent 1 1 1 1 PAGE 280 280 Table 6-6 Parameters used in the Pa rchure and Mehta (1985) dilution tests. Test simulation Variable or parameter Units PMD-C1 PMD-P1 Water depth m 0.26 0.26 Number of cells 5 5 Number of noncohesive classes 60 60 Minimum cohesive size, Dmin micron 0.1 0.1 Maximum cohesive size, Dmax micron 2000 2000 Number of silt classes 10 10 Minimum silt size micron 4 4 Maximum silt size micron 60 60 Initial silt fraction in bed 20 20 Floc strength coefficient, Bf Pa 1200 1200 Critical shear stress for deposition scale factor, cd0 Pa 0.1 0.1 Critical shear stress for deposition exponent, 0.5 0.5 Silt mobility coefficient 1.05 1.05 Maximum settling velocity m/s 0.0002 0.0002 Cohesive settling velocity factor 4.0 4.0 Silt settling velocity factor 1.0 1.0 Reference particle size, Dref micron 0.1 0.1 Fractal dimension, Df 2.2 2.2 Aggregation efficiency 1.0 1.0 Disaggregation efficiency 0.75 0.75 Collision breakage efficiency 0.25 0.25 Initial concentration kg/m3 0.0 0.0 Initial silt fraction in suspension 0 0 Bottom shear stress Pa 0.2 0.2 Supplemental internal shear s-1 0.0 0.0 Probabilistic treatment No Yes Cohesive deposition Exclusive Exclusive Silt deposition Exclusive Exclusive Clay erosion scale factor kg/m3 1.0 1.0 Silt entrainment scale factor kg/m3 1.0 1.0 Hindered settling exponent 1 1 PAGE 281 281 Table 6-7 Parameters used in the Pa rchure and Mehta (1985 ) erosion tests. Test simulation Variable or parameter Units PME-C1 PME-P1 Water depth m 0.26 0.26 Number of cells 10 10 Number of noncohesive classes 60 60 Minimum cohesive size, Dmin micron 0.1 0.1 Maximum cohesive size, Dmax micron 2000 2000 Number of silt classes 10 10 Minimum silt size micron 2 2 Maximum silt size micron 20 20 Initial silt fraction in bed 0.1 0.1 Floc strength coefficient, Bf Pa 100 100 Critical shear stress for deposition scale factor, cd0 Pa 0.02 0.02 Critical shear stress for deposition exponent, 0.14 0.14 Silt mobility coefficient 1.1 1.1 Maximum settling velocity m/s 0.004 0.004 Cohesive settling velocity factor 1.0 1.0 Silt settling velocity factor 1.0 1.0 Reference particle size, Dref micron 0.1 0.1 Fractal dimension, Df 2.2 2.2 Aggregation efficiency 1.0 1.0 Disaggregation efficiency 0.75 0.75 Collision breakage efficiency 0.25 0.25 Initial concentration kg/m3 0.0 0.0 Initial silt fraction in suspension 0 0 Bottom shear stress Pa 0.2 0.2 Supplemental internal shear s-1 0.0 0.0 Probabilistic treatment No Yes Cohesive deposition Exclusive Exclusive Silt deposition Exclusive Exclusive Clay erosion scale factor kg/m3 1.0 1.0 Silt entrainment scale factor kg/m3 1.0 1.0 Hindered settling exponent 1 1 PAGE 282 282 Table 6-8 Parameters used in the Sanf ord and Halka (1993) tidal tests for 1989. Test simulation Variable or parameter Units SH-SA1 SH-SP1 SH-EA1 SH-EP1 Water depth m Variable Variable Variable Variable Number of cells 3 3 3 3 Number of noncohesive classes 40 40 40 40 Minimum cohesive size, Dmin micron0.1 0.1 0.1 0.1 Maximum cohesive size, Dmax micron1000 1000 1000 1000 Number of silt classes 10 10 10 10 Minimum silt size micron2 2 2 2 Maximum silt size micron60 60 60 60 Initial silt fraction in bed 0.4 0.4 0.4 0.4 Floc strength coefficient, Bf Pa 2000 2000 2000 2000 Critical shear stress for deposition scale factor, cd0 Pa 0.02 0.02 0.02 0.02 Critical shear stress for deposition exponent, 0.5 0.5 0.5 0.5 Silt mobility coefficient 1.05 1.05 1.05 1.05 Maximum settling velocity m/s 0.001 0.001 0.001 0.001 Cohesive settling velocity factor 1.0 1.0 1.0 1.0 Silt settling velocity factor 5.0 5.0 5.0 5.0 Reference particle size, Dref micron0.1 0.1 0.1 0.1 Fractal dimension, Df 2.2 2.2 2.2 2.2 Aggregation efficiency 1.0 1.0 1.0 1.0 Disaggregation efficiency 0.5 0.5 0.5 0.5 Collision breakage efficiency 0.5 0.5 0.5 0.5 Initial concentration kg/m3 0.015 0.015 0.015 0.015 Initial silt fraction in suspension 0.4 0.4 0.4 0.4 Bottom shear stress Pa Variable Variable Variable Variable Supplemental internal shear s-1 0.0 0.0 0.0 0.0 Probabilistic treatment No Yes No Yes Cohesive deposition Sa Sa Exclusive Exclusive Silt deposition Sa Sa Exclusive Exclusive Clay erosion scale factor kg/m3 0.004 0.004 0.004 0.004 Silt entrainment scale factor kg/m3 0.004 0.004 0.004 0.004 Hindered settling exponent 1 1 1 1 a S = simultaneous; PAGE 283 283 Table 6-9 Parameters used in the Sanf ord and Halka (1993) tidal tests for 1990. Test simulation Variable or parameter Units SH90-SA1SH90-SP1SH90-EA1 SH90-EP1 Water depth m Variable Variable Variable Variable Number of cells 3 3 3 3 Number of noncohesive classes 40 40 40 40 Minimum cohesive size, Dmin micron 0.1 0.1 0.1 0.1 Maximum cohesive size, Dmax micron 1000 1000 1000 1000 Number of silt classes 10 10 10 10 Minimum silt size micron 2 2 2 2 Maximum silt size micron 60 60 60 60 Initial silt fraction in bed 0.4 0.4 0.4 0.4 Floc strength coefficient, Bf Pa 1200 1200 1200 1200 Critical shear stress for deposition scale factor, cd0 Pa 0.02 0.02 0.02 0.02 Critical shear stress for deposition exponent, 0.5 0.5 0.5 0.5 Silt mobility coefficient 1.05 1.05 1.05 1.05 Maximum settling velocity m/s 0.001 0.001 0.001 0.001 Cohesive settling velocity factor 5.0 5.0 5.0 5.0 Silt settling velocity factor 2.0 2.0 2.0 2.0 Reference particle size, Dref micron 0.1 0.1 0.1 0.1 Fractal dimension, Df 2.2 2.2 2.2 2.2 Aggregation efficiency 1.0 1.0 1.0 1.0 Disaggregation efficiency 0.5 0.5 0.5 0.5 Collision breakage efficiency 0.5 0.5 0.5 0.5 Initial concentration kg/m3 0.009 0.009 0.009 0.009 Initial silt fraction in suspension 0.4 0.4 0.4 0.4 Bottom shear stress Pa Variable Variable Variable Variable Supplemental internal shear s-1 0.0 0.0 0.0 0.0 Probabilistic treatment No Yes No Yes Cohesive deposition Sa Sa Exclusive Exclusive Silt deposition Sa Sa Exclusive Exclusive Clay erosion scale factor kg/m3 0.01 0.01 0.01 0.01 Silt entrainment scale factor kg/m3 0.01 0.01 0.01 0.01 Hindered settling exponent 1 1 1 1 a S = simultaneous; PAGE 284 284 Table 6-10 Parameters used in the Sanf ord and Halka (1993) tidal tests for 1991. Test simulation Variable or parameter Units SH91-SA1SH91-SP1SH91-EA1 SH91-EP1 Water depth m Variable Variable Variable Variable Number of cells 3 3 3 3 Number of noncohesive classes 40 40 40 40 Minimum cohesive size, Dmin micron 0.1 0.1 0.1 0.1 Maximum cohesive size, Dmax micron 1000 1000 1000 1000 Number of silt classes 10 10 10 10 Minimum silt size micron 2 2 2 2 Maximum silt size micron 60 60 60 60 Initial silt fraction in bed 0.4 0.4 0.4 0.4 Floc strength coefficient, Bf Pa 1800 1800 1800 1800 Critical shear stress for deposition scale factor, cd0 Pa 0.02 0.02 0.02 0.02 Critical shear stress for deposition exponent, 0.5 0.5 0.5 0.5 Silt mobility coefficient 1.05 1.05 1.05 1.05 Maximum settling velocity m/s 0.001 0.001 0.001 0.001 Cohesive settling velocity factor 5.0 5.0 5.0 5.0 Silt settling velocity factor 2.0 2.0 2.0 2.0 Reference particle size, Dref micron 0.1 0.1 0.1 0.1 Fractal dimension, Df 2.2 2.2 2.2 2.2 Aggregation efficiency 1.0 1.0 1.0 1.0 Disaggregation efficiency 0.5 0.5 0.5 0.5 Collision breakage efficiency 0.5 0.5 0.5 0.5 Initial concentration kg/m3 0.009 0.009 0.009 0.009 Initial silt fraction in suspension 0.4 0.4 0.4 0.4 Bottom shear stress Pa Variable Variable Variable Variable Supplemental internal shear s-1 0.0 0.0 0.0 0.0 Probabilistic treatment No Yes No Yes Cohesive deposition Sa Sa Exclusive Exclusive Silt deposition Sa Sa Exclusive Exclusive Clay erosion scale factor kg/m3 0.004 0.004 0.004 0.004 Silt entrainment scale factor kg/m3 0.004 0.004 0.004 0.004 Hindered settling exponent 1 1 1 1 a S = simultaneous; PAGE 285 285 Table 6-11 Parameters used in the Sanford and Halka (1993 ) tidal tests for calibration. Test simulation Variable or parameter Units SH89-SP2 SH91-SP2 SH91-SP2 Water depth m Variable Variable Variable Number of cells 3 3 3 Number of noncohesive classes 40 40 40 Minimum cohesive size, Dmin micron 0.1 0.1 0.1 Maximum cohesive size, Dmax micron 1000 1000 1000 Number of silt classes 10 10 10 Minimum silt size micron 2 2 2 Maximum silt size micron 60 60 60 Initial silt fraction in bed 0.4 0.4 0.4 Floc strength coefficient, Bf Pa 1800 1800 1800 Critical shear stress for deposition scale factor, cd0 Pa 0.02 0.02 0.02 Critical shear stress for deposition exponent, 0.5 0.5 0.5 Silt mobility coefficient 1.05 1.05 1.05 Maximum settling velocity m/s 0.001 0.001 0.001 Cohesive settling velocity factor 5.0 5.0 5.0 Silt settling velocity factor 2.0 2.0 2.0 Reference particle size, Dref micron 0.1 0.1 0.1 Fractal dimension, Df 2.2 2.2 2.2 Aggregation efficiency 1.0 1.0 1.0 Disaggregation efficiency 0.5 0.5 0.5 Collision breakage efficiency 0.5 0.5 0.5 Initial concentration kg/m3 0.01 0.024 0.009 Initial silt fraction in suspension 0.4 0.4 0.4 Bottom shear stress Pa Variable Variable Variable Supplemental internal shear s-1 0.0 0.0 0.0 Probabilistic treatment Yes Yes Yes Cohesive deposition Sa Sa Sa Silt deposition Sa Sa Sa Clay erosion scale factor kg/m3 0.004 0.004 0.008 Silt entrainment scale factor kg/m3 0.004 0.004 0.01 Hindered settling exponent 1 1 1 a S = simultaneous; PAGE 286 286 Figure 6-1. Example of fall velo city estimates from video analysis of in situ sediments in San Francisco Bay (data from Smith, 2007). PAGE 287 287 Figure 6-2. Results of simulation of the K ynch (1952) test case using a hindered settling exponent of m =1 in Equation 2-40. PAGE 288 288 Figure 6-3. Results of simulation of the K ynch (1952) test case using a hindered settling exponent of m =2 in Equation 2-40. PAGE 289 289 Figure 6-4. Effects of fractal dimension and shear strength co efficient on density difference versus floc shear strength (based on Equation 3-11), with comparison with data from Krone (1963). PAGE 290 290 Figure 6-5. Example representati on of shear strength of flocs, cr itical shear stress for erosion and critical shear stress for deposition as f unctions of particle size. Shear strength defined by Equation 3-11, with Bf = 1200 Pa and Df = 2.6. Critical shear stresses for erosion and deposition are defined by Equa tion 2-53. The critical shear stress for deposition is based on d0 = 0.01 Pa, dref =0.1 microns and = 0.5. PAGE 291 291 Figure 6-6. Relationship between mean shear st rength, critical shear stresses for erosion and deposition, and bottom shear stress as a function of particle size. PAGE 292 292 Figure 6-7. Specification development for Meht a (1973) for use of the mean values in the classical erosion and deposition exclusive paradigm. PAGE 293 293 Figure 6-8. Distribution of th e initial suspended sediment c oncentration for the Mehta (1973) deposition tests. PAGE 294 294 Figure 6-9. Simulations of Mehta (1973) experimental tests for sh ear stresses of 0.25 Pa, 0.40 Pa, 0.60 Pa and 0.85 Pa, and using classical erosion probabilities with simultaneous deposition. PAGE 295 295 Figure 6-10. Comparison of simulations using the simultaneous paradigm with the exclusive paradigm for the 0.25 Pa test using the mean values in calculations. PAGE 296 296 Figure 6-11. Simulations of Mehta (1973) test cases for shear stress of 0.40 Pa using combinations of probabilistic versus mean -valued depositional and erosion treatment, with either simultaneous or exclusive erosion/deposition. PAGE 297 297 Figure 6-12. Schematic representation of a probabilistic representation for the Mehta 1973 0.25 Pa test. The shaded area is the zone of deposition from the average value analysis. The differences are conceptual only since th e displayed range of values is only +/1 standard deviation for each variable (ce, cd, b). PAGE 298 298 Figure 6-13. Particle size distribution compar ison between the mean value simulation and the probabilistic simulation for the 0.25 Pa Mehta (1973) test. Both simulations used the same sediment characteristics and critical shear stresses and the exclusive paradigm. PAGE 299 299 Figure 6-14 Simulation using the classical exclusive bed exchange processes using mean values for the Krone (1962) deposition test with a shear stress of 0.0305 Pa. One sensitivity simulation was made with an added supplemental internal shear of 100 Hz. PAGE 300 300 Figure 6-15. Simulation using the classical bed exchange processes using mean values for the Krone (1962) deposition test with a shear stress of 0.0415 Pa. PAGE 301 301 Figure 6-16. Simulation using the classical bed exchange processes using mean values for the Krone (1962) deposition test with a shear stress of 0.0515 Pa. PAGE 302 302 Figure 6-17. Illustration of the change in deposit ional response in the Kr one (1962) recirculating flume tests when concentrations fall below approximately 0.3 kg/m3. PAGE 303 303 Figure 6-18. Model simulation to test the d ilution rate for the Pa rchure and Mehta (1985) dilution test. PAGE 304 304 Figure 6-19. Initial particle size concentration distribution for bed initia lization for the Parchure and Mehta (1985) dilution experiment. The simulation started with no sediment in suspension and then eroded the bed to an equilibrium concentration. PAGE 305 305 Figure 6-20. Parchure and Mehta (1985) dilution test re sults with bed exchange included, with the classical excess shear stress exclusive formulation and an exclusive simulation using probabilistic treatment of the key parameters. PAGE 306 306 Figure 6-21. Variation of floc size distribu tion during the Parchure a nd Mehta (1985) dilution test. PAGE 307 307 Figure 6-22. Initial cohesive bed particle concentration fo r the Parchure and Mehta (1985) erosion test. PAGE 308 308 Figure 6-23. Results of simulation of an er osion test (Parchure a nd Mehta, 1985) with a progressive increase in shear stress using the exclusive erosion/deposition and mean values. PAGE 309 309 Figure 6-24. Effects of switching from classicall y exclusive mean-value calibrated bed exchange to an exclusive/probabilistic treatm ent without parameter adjustments. PAGE 310 310 Figure 6-25. Variation in water depth during Sanford and Halka (1993) field test. PAGE 311 311 Figure 6-26. Results of calibrating continuous deposition for use on an average shear stress and using the probabilistic model of bed exchange: 5 January 1989. PAGE 312 312 Figure 6-27. Results of calibrating continuous deposition for use on an average shear stress and using the probabilistic model of bed exchange: 2 February 1990. PAGE 313 313 Figure 6-28. Results of calibrating continuous deposition for use on an average shear stress and using the probabilistic model of bed exchange: 15 January 1991. PAGE 314 314 Figure 6-29. Evolution of floc size distribution during numerical simulation of the Sanford and Halka (1993) data set of 5 January 1989. Both tests used the continuous deposition bed model. The black distributions are for the use of the average bottom shear stress, while the red curves are for the probabilisti c shear stress formulation. (hours refer to Figure 6-26) PAGE 315 315 Figure 6-30. Evolution of floc size distribution during numerical simulation of the Sanford and Halka (1993) data set of 5 January 1989. Both tests used th e exclusive erosiondeposition bed model. The black distributi ons are for the use of the average bottom shear stress, while the red curves are for the probabilistic shear stress formulation. (hours refer to Figure 6-26). PAGE 316 316 Figure 6-31. Results of simulations showing the effects of the c ontinuous deposition and probabilistic bed exchange treatment 5 Janua ry 1989. All other model variables are held the same. PAGE 317 317 Figure 6-32. Results of simulations showing the effects of the c ontinuous deposition and probabilistic bed exchange treatment 2 Febr uary 1990. All other model variables are held the same. PAGE 318 318 Figure 6-33. Results of simulations showing the effects of the c ontinuous deposition and probabilistic bed exchange treatment 15 Janua ry 1991. All other model variables are held the same. PAGE 319 319 Figure 6-34. Results of simulation of the Kr one gold-tagged sediment experiment using the simultaneous erosion and deposition tr eatment of the mean variables. PAGE 320 320 CHAPTER 7 SUMMARY AND CONCLUSIONS 7.1Summary The primary issue addressed within the research here is whether the paradigm of exclusive erosion or deposition has scien tific legitimacy within the phy sics of CST. The exclusive paradigm assumes that bed exchange condition is either erosion, deposition or neither, but never both. In contrast, the simultaneous paradigm admits the possibility of both occurring at the same time. Data supporting the exclusive paradigm tend to come from laboratory experiments (Ariathurai, et al., 1977; Partheniades, 1965; Lau and Kris hnappan, 1994), while data supporting the simultaneous model come primarily from fiel d experiments (Lick, 1982; Bedford, et al., 1987; Sanford and Halka, 1993; Wint erwerp and van Kesteren, 2004) The original exclusive paradigm is, in part, the result of early atte mpts to understand basic cohesive sediment transport combined with limitations of data collection and analysis procedures. Early research collected sediment deposition and erosion information by inference. The analysis procedures have been highly innova tive in developing basi c cohesive sediment processes information from other measurements (for example, floc shear strength, densities and orders of aggregation from viscometers). However, a lack of technology to measure erosion and/or deposition at the turbulen ce time scale leaves analysis met hods still with significant level of inference based on averagi ng over longer time scales. Be d exchange has been inferred primarily from the increase or reduction in the suspended sediment concentration within the flume, which is an averaging over time (and over the domain of the measuring apparatus such as a flume). The net result of averaging will be positive, negative or zero, but not both positive and PAGE 321 321 negative. Settling velocities are also normally inferred from depositiona l devices that provide averaged data. Mathematical conceptual models developed of CST were based on correlations between the time-averaged inferred data and bulk propert ies of the sediment. The result of these conceptual models is that averaged variables pr edict averaged responses. Until recently, this has not been a problem since most numerical models are also based on time-averaged equations applied over discretized spatial domains that al so involve averaging (e.g ., cell face averaged in finite difference). The earliest conceptual mathem atical models of bed exchange have generally been appropriate for the numeri cal treatment (Ariathurai, 1974). With the development of greater details in conceptual models, such as particle size distributions and flocculation sub-models, the bed exchange algorithms have required revision. The first logical step has been to extend the bulk equations to the individual particle size classes, which has shown the need for recalibration of the coefficients characterizing bulk equations. The next logical step (used here in) is to define the relative be havior between sediment classes and apply bulk scaling over all classes. Th ere is a knowledge gap be tween the theoretical framework of the processes and the data collection and monitoring technology to support the theory, still largely related to averaging issues. The present m odeling effort has shown that the time step required to su pport the flocculation model can be no longer than 1 second. Use of larger times steps generally resulted in the c ondition where the fraction of a class concentration being transferred to other clas s sizes by the aggregation/disagg regation fluxes approaches 1.0 and the numerical scheme becomes unstable. The variability of conditions in field experi ments typically takes em pirical relationships derived from highly controlled la boratory experiments well outside the limits of calibration. PAGE 322 322 Numerical modelers have found the need to use the simultaneous approach to replicate real world depositional behavior (Sanford and Halka, 1994; Winterwerp, 2007). In response to this, the next level of complexity to be incorporated into the conceptual and numerical models is the explicit incorporation of stochastic effects into the cohesive processes (Prooijen and Winterwerp, 2009). The logical arguments for use of the exclusive paradigm are strong within the appropriate context. When the question of simultaneous exchange is applied to a single sediment particle the exclusive paradigm obviously must be applied. However, the numbers of particles within a suspension require the application of statistical measures (e ven if just the mean). If the scales of concern are at an engineering level, for exam ple seasonal or yearly sedimentation within a harbor, then the exclusive view may be appropr iate simply based on the averaging involved. The characteristic difference between values of the critical shear stress for erosion and the critical stress for deposition has been one st rong argument for the exclusive paradigm. The difference is the result of the disparity between the binding cohesive force needed to be overcome to dislodge cohesive part icles from the bed and the shear force required to retain the same size particle in suspension (Partheniades, 19 65, 1971). When a wide range of particle sizes is considered the difference in stresses for the bulk behavior of cohesi ve sediment becomes less defined (Tolhurst, 2009). Arguments for the simultaneous erosion and de position approach are all associated with the statistical variation in the variables. Sediment properties within the bed are usually heterogeneous and the erosive forces are stochastic The particle size distribution combined with turbulence fluctuations offers a clear conceptual visualization of the potential for simultaneous erosion and deposition in terms of some sizes u ndergoing erosion while ot hers are depositing. A PAGE 323 323 visualization of a unit area of the bed with a variety of irregular ities exposed to turbulent flow should lead one to admit the probability of both erosion and deposition within the unit area at some time. The conclusion, however, may change with varying the units of the area. A single particle will truly either erode or deposit or stay where it is. But engineering analysis cannot be performed at the single grain scale, so si multaneous erosion and deposition is almost unavoidable, unless the defined variables are th e result of averaging over time and space. Winnowing of fine sediments is an example of a simultaneous effect. Even at very low flow velocities, there are very weak primary particle bonds in the be d that can eventually be broken and particles eroded flake by flake. This is a result of interaction of the very weak end of the floc strength spectrum with the hi gh end of the shear stress spectrum. The probabilistic treatment incl udes a distribution of most of the prim ary variables in cohesive sediment transport. These include cu rrent velocity, bottom shear stress, floc shear strength, critical shear stresses for erosion and deposition, internal (flow) shear and settling velocity. In the present analysis the current ve locity, shear strength and settling velocities are assumed to be normally distributed. The shear stress variables are generated from transformations of the current velocity distribu tion. The primary effects of the implementation are realized in the aggregation model and in bed exchange. The application to bed exchange is accomplished through numerical integration of the appropriate products of probability density functions and their interactions. This allows for the investigation of the effects of various bed exchange paradigms explicitly within the numerical model. The numerical sediment transport tool develope d for this research has been shown to be capable of: Simulating aggregation and disaggregation, re lated to interparticle collisions due to Brownian motion, internal sheari ng, and differential settling. PAGE 324 324 Incorporating stochastic eff ects into bed exchange and a ggregation/disaggregation through effects on differential settling velocities and in terparticle collision-re lated floc breakage associated with the probabilistic treatment of the floc strength. Addressing hindered settling by using previously developed theory (Winterwerp, 1999) within the formulation of probabilistic settling velocity. Reaching a depositional equilibriu m concentration for a fixed shear stress. This is not a significant new finding, but it provides some va lidation that the probabilistic treatment produces certain fundamental behavior. Reaching an erosional equilibrium concentration for a fixed shear stress. This is a validation of another fundamental behavior fo r layered type I surf ace erosion (Mehta and Partheniades, 1982). Exhibiting the floc spectrum features docum ented by Kranck and Milligan (1992). The numerical results show that er osion and deposition o ccur primarily on the larger end of the particle size spectra and that th e shape of the spectra at the lo west particles sizes tends to remain consistent during tidal va riation (see Figures 6-29 and 6-30). 7.2 Conclusions A number of observations have been made during the development and application of the research numerical tool. Some observations of interest are: The effects of a probabilistic treatment of the key variables are more pronounced for erosion than for deposition. The erosion rate is controlled by the interaction between the shear stress distribution and the spectrum of critical shear stress for erosion. The deposition rate is controlled by the settling ve locity spectrum, with the probability of reentrainment, essentially erosion. The erosion effects, involving the interaction of two spectra, can be viewed as an amplif ication of the probabilistic effect. Probabilistic effects are amp lified through the flocculation model. The probabilistic variables when incorporated into the existing collision frequency formul ations, result in a greater collision frequency for differential setting and particle collision breakage. This increases the interparticle si ze-class mass fluxes, which assists is providing a sediment supply to a wider range of si ze classes for bed exchange. For a given shear stress the flocculation model will tend toward an equilibrium distribution, confirming a basi c CST behavior (Kranck, 1973; Kranck and Milligan, 1992). The probabilistic treatment results in a broader floc distribution spectrum. This broadening of the spectrum is the result of increased inter-class mass exchanges associated with nondiscrete values in the aggregation model. The effect is analogous to adding diffusion to the floc spectrum. Without the probabilistic treatment, the resulting spectra normally have a PAGE 325 325 sharp boundary at the size clas ses bounding those discrete thresholds. The probabilistic results are more intuitively acceptable. A probabilistic treatment facilitates a faster response in the erosion and deposition to changes in the hydrodynamics. This is associated with the interactions of the tails of the probability distributions. If the shear stress is clearly below the threshold for erosion with an oncoming increase in shear stress due to changing hydrodynamics, the mean-value analysis will not exhibit erosi on until the mean value of th e shear stress rises above the mean shear strength of the bed. With a probabilistic treatment, when the upper tail of the shear stress distribution inter acts with the lower tail of th e shear strength distribution erosion will be initiated. Si milar logic can be used for deposition if the mean threshold criteria are used. The use of the exclusive paradigm with a floc size distribution can perform as well as a simultaneous treatment with a single particle size. The use of size distribution will, by definition, add a distribution of settling velocities and critical sh ear stresses. So within the distribution, there can be eros ion of larger flocs while depo sition occurs at intermediate floc sizes (see Figure 6-12). The rebound of the concentration at the end of the dilution test (Par chure and Mehta, 1985) can be replicated using the probabilistic treatment. This implies that the equilibrium that was achieved prior to dilution may have been the result of a balance between small residual erosion and deposition. The time step required for accurate simulati on of the aggregation/disaggregation submodel is limited by the magnitude of the inter-class mass fluxes compared to the concentration of the size class. Changing from an application using averaged variables to one that incorporates a probabilistic treatment may require recalibra tion of the empirical coefficients. The calibration changes for the Sanford and Halk a (1994) field tests included changing the erosion rate coefficient by a factor of 2 in one test, and by simply changing the initial condition for the starting concentration in the other two tests. Empirical coefficients developed based on mean values of the parameters will require some adjustment when applied with a probabilistic treatment. It is expected that the variability between the empirical values will be not more severe than the natural variability of the mean-valued coefficien ts between specific sites. 7.3 Recommendations Many of the key features of cohesive processe s, such as variability in settling rates and the transition to erosion from deposition and from erosion to deposition over a range of shear stresses, can be captured with a properly resolved particle size distribution combined with an PAGE 326 326 aggregation model. If however, rapid changes in sediment response are important, then use of a probabilistic treatment can improve the model perf ormance. The definition of rapid change is relative to the ability of the analysis to adequate ly track changes in the sediment transport. The Sanford and Halka (1993) example showed that ev en with the simultaneous deposition paradigm, the concentrations in the field observations fell more rapidly than could be simulated with the mean value of the variables. The probabilistic treatment allows for the largest fall velocities within the probability di stribution to initiate depo sition (or not be reentrained if the continuous paradigm is used) sooner. The initiation of er osion will, similarly, occur sooner on increasing shear stress. The following future research is recommended for consideration: Development of more rigorous methods for a pplication of existing empirical coefficients such as erosion rate constants and threshold shear stresses to analysis that addresses individual size classes. Research to invest igate the development of class based empirical coefficients is needed. Evaluation of the effects of a probabilistic treatment of variables within the aggregation model. A more thorough sensitiv ity analysis is needed for the changes in the performance of the aggregation model when the probabilistic treatment is used. Th e current application impacts the performance through differentia l settling and interp article collision disaggregation. Effects of probabilistic treatment in the hi ndered settling region. When hindered settling begins to become important, it is reasonable to expect that the probability distribution of settling velocities will become much narrower. The influence in the current application is a linear scaling with reduction in the mean settling rate. The nonlinear effects need further research. Development of more efficient approach to incorporate probabilis tic effects without explicit integration of the probability distributions. This is needed to make the effects available to engineering tools. This research was recently addressed by van Prooijen and Winterwerp (2009), with an analytical treatme nt of the probability densities, which they then simplified into a polynomial. Furt her research is needed in this area. PAGE 327 327 APPENDIX A DEVELOPMENT OF GOVERNING EQUATI ONS FOR UNSTEADY AND NONUNIFORM SEDIMENT TRANSPORT The motivation for this dissertation is the evaluation of nonlinear effects introduced by unsteady and nonuniform hydrodynamics on the sediment transport processes. The governing equations for sediment transport are linear with regard to the sediment concentration in the transport and diffusion terms within the formulati on. Any nonlinearity in the sediment transport equation is introduced through the dependence of the effective diffusion coefficient on turbulent mixing and the dependence of the fall velocity on the sediment concentration. Careful investigation of the developmen t of the turbulent mo mentum exchange and mixing processes in the presence of suspended sediment is needed to deal with these nonlinea rities in the sediment transport equation. The temporal and spatial varia tions in density associated with the suspended sediment concentration influence the balance of momentum As the density of a composite sedimentladen fluid parcel increases, the momentum will increase when the velocity remains constant. In order to develop the governing equations for sedi ment transport the mixing process driven by the hydrodynamics must first be addressed. Th erefore, the derivation of the governing hydrodynamic equations will start with what appe ars to be compressible flow conditions, allowing to evaluate the influen ce of the changing density on the overall momentum balance. The rigorous development for compressible flow will be a valuable tool in understanding the sediment transport processes. A.1 Hydrodynamics We will develop the basic governing hydrodynamic equations in the presence of suspended sediment and then impose the conventional cons traints to simplify the derivation to the case PAGE 328 328 without suspended sediment within the fluid. This will help in developing an understanding of the interpretation of specific terms in the system of equations. The governing equations for compressible hydrodynamics in three dimensions in the conservative form using tensor notation are: A.1.1 Continuity Equation The basic universally applicable co ntinuity equation is Equation A-1. 0j ju tx (A-1) In Equation A-1 the density is a function of temperature, pressure and all constituent variables. A.1.2 Momentum Equations ji i i juu u F tx (A-2i) where vis iii i ij vis i jp FgF x F x ij is the viscous stress tensor and the viscous forces are derived from the gradients in the stresses. The stress tensor is 2 3j ik ijij j iku uu x xx (A-3ij) Note that the subscript i in the equation label for Equations A-2i corresponds to the i index for the equation and is in essence three equations, i=1,2,3 for the three Cartesian coordinates. In Equations A-3ij both i and j span 1 to 3, independe ntly, resulting in nine PAGE 329 329 equations. The k indices in Equations A-3ij imply a summation of paired values of k from 1 to 3. The last partial derivative in Equation A-3 with dual k indices is the divergence of the velocity field. Also note that the visc ous stress tensor is dependent on the dynamic viscosity. If we represent the dynamic viscosity as the product of the density and the kinematic viscosity, then vis iF will include terms involving the density gradients. A.2 Sediment Transport Equation If the sediment in suspension can be characteri zed by a series of size or mass classes, then the transport conservation equation can be developed for each class. The sediment transport for a sediment class m is given generally by 3mjismm m m mmj j jjuwc cc SDD txxx (A-4m) where there are m=1,M equations for each of M sediment size classes. The equations are written at this point with the pa rticle advective velocity mju separate for each size class, and with diffusion coefficients mjD for each size class. Also, the generalized source term mS is by size class. The total source term for sediment class m will be the summation of the interactions with all other size classes. 1(1)M mmiim iSS (A-5m) At this point we need to define the total suspended sediment concentration as a summation of the concentrations of the sediment classes as in Equation A-6. 1M m iCc (A-6) PAGE 330 330 The questions to be evaluated are how does mjD interact with equations (A-2i) and C with Equation A-1 and Equations A-2i. The equations for conservation of momentum are a statement that the changes in the momentum are affected by changes in the fluid velocity and density in response to pressure gradients, gravity and viscous transfer of moment um with adjacent fluid of different momentum. The focus of attention here is that the fluid mo mentum is dependent on changes in the composite density of the fluid, regardless of the contri bution, whether from te mperature, salinity, or suspended sediment gradients. For sediment tr ansport problems the com posite fluid density is primarily affected by the suspended sediment. So the density in the equations above must be a combination of fluid and sediment densities weig hted by the porosity as shown in Equation A-7. (1)wsnn (A-7) where n is the porosity of the wate r parcel (volume of water per unit volume of water-sediment mixture). This can be developed further by defini ng an equation of state for the fluid that related w to water temperature, salinity and pressure, although the pre ssure correction is negligible for shallow estuarine conditions. The sediment density is assumed to be independent of temperature. This assumption may not be comp letely accurate if the sediment particles are defined as flocs that are made up of minerals an d interstitial water, which will be affected by temperature and salinity. The porosity of the fluid mixture for a given sediment mass concentration will be dependent on the effective de nsity of the collection of sediment particles and flocs. If the mi xture is all sand then s will be the mineral density, the specific gravity of sand times the density of water. If the sediment is cohesive, the size and density of the flocs will vary. The porosity is defined form the sediment concentration and sediment density in Equation A-8. PAGE 331 331 1 s C n (A-8) where s is the effective sediment density. The eff ective density is defined as the density that when used with the total sediment concentratio n yields the correct total effective volume of sediment in the mixture. Thus 1 s M m m smC c (A-9) where s m is the density associated with the sediment size class m. Inserting equation (A-8) into equation (A-7) yields 1w sC C (A-10) or alternatively 1w w s C (A-11) In the computation of the composite fluid density with the effects of the suspended sediment it is more convenient to work directly from the mass concentration of the suspended sediment rather than to attempt to address the size of the flocs and the effective floc density. That attention can be deferred to th e particle interaction analysis. At this point it is sufficient to state that the composite density is a function of salinity, temperature, pressure and suspended sediment (Equation A-12). (,,,,) s sSTpC (A-12) This density can be developed from any a ppropriate equation of state (e.g., EOS 90). PAGE 332 332 A.2.1 Continuity Equation for Sediment Laden Fluid Now we will derive the continuity equation for the fluid in the presence of suspended sediment mass based on the following assumptions The volume of fluid within the unit volume of fluid and sediment is n, the porosity. Ther efore, the mass of fluid within the control volume is wn The flux of fluid mass across a contro l volume face perpendicular to the xi-axis is iwnu Following a conventional control volume de velopment we obtain for the fluid mass in the control volume the governing equation. 0wwi innu tx (A-13) This equation is universally valid for any sedime nt-laden fluid. Expanding these terms yields 0ww i wiwwi iiiu nn nn unu ttxxx (A-14) If the incompressible assumption is i nvoked for the clear fluid density, then 0ww itx and the fluid continuity equation becomes 0i wwwi iiu nn nu txx (A-15) Returning to the variable density case, inserting equation (A-8) into equation (A-14) results after reorganizing 21wiwwiwsis sisisiuu Cu CC C txtxtx (A-16) This statement acknowledges that with time the relative volume of water and the volume and density of the sediment within the control volume are changing. This change is either a source or PAGE 333 333 sink of water or sediment and they are direc tly and inversely related. Assuming that the sediment mineral density is consta nt, then equation (A-16) becomes 1wiwwiwi s isisiuu Cu CC C txtxx (A-17) As the concentration of sediment in the control volume rises, the total volume and mass of water will be reduced by a proportion of approximately 1s wsC If the sediment concentration is negligible compared to the sediment density then 1 n and the fluid continuity equation becomes equation (A-1) with replaced byw If the clear fluid density is assumed to be c onstant, then equation (A-17) reduces further to 1ii isiuCuC x tx (A-18) This states that the divergence of the velocity field vanishes only if the conservation of sediment mass in the control volume is atta ined when using the fluid velocity field. A.2.2 Continuity Equation with Differential Sediment Particle Velocity For the suspended sediment within the cont rol volume we will now acknowledge that the sediment particle velocity need not be precisely the same as the fluid velocity. This is may be particularly true in an accelerat ing flow field or a gravitationa l field. Define the sediment particle velocity components as ipiiduuu (A-19i) This defines the sediment velocity as the flui d velocity plus a differential component. The characteristics of this difference will be dealt with in the momentum equations. For now we will address the impacts on the sedi ment continuity equation. PAGE 334 334 The mass of sediment within the unit control volume is by definition C The unit flux of sediment across the xi-coordinate face is ipuC Using similar logic for the sediment fraction of the control volume fluxes and storage we get th e sediment mass conserva tion equation, ignoring diffusion, as 0ip iuC C tx (A-20) Expanding the particle velocity we get the sediment equation 0ii d ii d iiiiCCuCu uCuC txxxx (A-21) We can obtain an alternate form of the fluid c ontinuity equation by combining equation (A-21) and equation (A-17) to give 10wiwwiid sisiiuuu C C C txxx (A-22) This final fluid continuity e quation accounts for the conventi onal fluid continuity equation for the fraction of the control volume occupied by the fluid and the change in the fractional volume occupied by the fluid due to the net diverg ence of the sediment from the control volume. The two individual contributions are not each zero because the fraction of the fluid in the control volume changes in time as the sediment concentration changes. Notice that the first grouping in equation (A-2 2), the conventional continuity equation, will only manifest itself alone when there is no sediment in the control volume. When C = 0 equation (A-22) reduces to equation (A-1) withw replacing The basic density of the fluid, w changes with temperature and salinity in estuarine waters. If it is assumed that the temporal cha nges in the density are negligible the continuity equation becomes PAGE 335 335 10iwwiid sisiiuuu C C C xxx (A-23) Next, if the spatial gradients in clear fluid density are small compared to the changes due to sediment effects, with using some identities we can simplify to 1ii d isiuuC x x (A-24) If the particle velocities in the horizontal are then assumed to be equal to the fluid velocity, then 0idu and several terms can be canceled an d the continuity e quation reduces to 0i iu x (A-25) So the conclusion is that only if the clear fl uid is assumed incompressible and the sediment particles are transported by the fluid velocity exactly does the continuity equation reduce to the statement that the divergence of the velocity field vanishes. A.3 Turbulence The practical impact of turbulence on both hydrodynamics and sediment transport is that the temporal evolution of the details of flow and sediment concentration fields becomes unpredictable after a relatively short period of time. There are a variety of explanations for the lack of predictability. Most of these relate to the lack of th e proper physics of the initiation of turbulence and eddy formation. However, from a pragmatic point of view, even if the physics of turbulence were known because the scale of in itiation and given the current computational capability, it would be impractical to simulate details of turbulence for typical estuarine problems. Consequently, turbulence has classically been ad dressed from a statistical perspective. The philosophy (Bernard, et. al. 1998) with regard to turbulence is the development of statistical PAGE 336 336 measures from a number (an ensemble) of turb ulent time series associated with the same macroscale conditions (discharge, depth, pressu re gradient, etc.). The main dependent parameters are the mean values of each turbulent variable. The turbulent fluctuations of all variables ar e normally represented in the classical manner, where the parameters are represented as the sum of their mean and fluctuating parts, Reynolds decomposition. iii iiiuuuppp TTTSSSccc (A-26) The classical method can be called conventi onal turbulent decom position (CTD), where the variables are directly averaged or integrated The overbar is the st andard averaging of the ensemble and the single prime the perturbation from that mean. In the development of the governing equatio ns for turbulent comp ressible flow Canuto (1997) presented an ensemble averaging based on Farve filtering for the velocity component that is a mass-weighted averaging yielding unique properties. The generic chaotic variable is decomposed either by CTD into or into a mass-weighted average and its perturbation. (A-27) The mass weighted ensemble average is written (A-28) This definition leads to the following relations: 0 (A-29a) 0 (A-29b) (A-29c) PAGE 337 337 0 (A-29d) (A-29e) (A-29f) 0 (A-29g) Alternative approaches to the derivation of th e governing equations for turbulent flow will now be explored. Three approaches are of in terest. First is the conventional turbulence decomposition with variable density in space and time with a chaotic turbulent component. Secondly is the simplification of the CTD approach for homogene ous flow, where the density is assumed constant. Finally, a mass-weighted tu rbulent decomposition (M TD) averaging of the velocity components in a variable density situati on is considered. Each of these derivations will be presented for the purpose of better understanding the contributions of each of the terms in the resulting equations for tur bulent sediment-laden flow. A.3.1 Continuity Equation The continuity equation will first be addre ssed for variable density without explicit consideration of the se diment concentration. A 3.1.1 Variable density case with CTD With variable density the continuity equati on when using CTD results in the continuity equation 0jj j j j jj j j juuuuu txttxxxx (A-30) Averaging results in an equation for the mean density. PAGE 338 338 0jj jjuu txx (A-31) or alternatively 0,jj j jj juu DD u Dtxx Dttx (A-32) and for the fluctuating density, subtracti ng equation (A-31) from (A-30) gives 0jjjj jjjjuuu u txxxx (A-33) This can be converted to an equation for / 0jj jjuu D Dtxx (A-34) A 3.1.2 Homogeneous case with CTD The continuity equation for homogeneous turbul ent flow reduces to two equations, one for the mean velocity components and one for the turbulent components: 0i iu x (A-35a) 0i iu x (A-35b) A 3.1.3 Variable density case with MTD Revisiting the continuity equation using the mass-weighted turbulence decomposition for the velocity with conventional decomposition for density yields 0jj j j jj j juuuu txttxxx (A-36) Temporal averaging yields PAGE 339 339 0j ju tx (A-37) Or alternatively 0,j j j ju DD u Dtx Dttx (A-38) For steady-state conditions this expre sses the conservation of the mass flux j u Subtracting equation (A-37) from (A-36) yields an equation for 0jj jjuu D Dtxx (A-39) This can also be converted to an equation for / 0j ju D Dtx (A-40) The MTD equations are somewhat more compact, with fewer terms. A.3.2 Turbulence Effects on Continuity Equation Now the effects of sediment suspension in tu rbulent flow on the continuity equation are addressed. Define the primary variab le turbulent decompositions as follows ipiid iii ipipip idipiipi idipi idipiuuu uuu uuu uuuuu uuu uuu CCC (A-41i) The turbulent perturbation in the density of th e fluid is assumed to be primarily dependent on the suspended sediment concentration. The te mperature and salinity fl uctuations are assumed to be negligible at the time and space scales of the turbulence. PAGE 340 340 Also note that it can be shown that th e turbulent decomposition of the sediment concentration passes linearly to the class sizes as follows. The mean sediment concentration can be defined by averaging equation (A-6) 11MM mm iiCcc (A-42) So by definition 1111MMMM mmmmm iiiiCCCccccc (A-43) Therefore, assuming that w and s are constants, the density perturbations associated with suspended sediment are defined as 1 1w w s w sC C (A-44) A.3.2.1 CTD method If the variations in the clear fluid density ar e ignored the sediment water mixture continuity Equation A-24 with conventional turbulent decompositi on results in the instantaneous equation 1 0ii ididid id isiiiiuu uCuCuCuC xxxxx (A-45) On averaging we get 1 0ii di d isiiuuCuC xxx (A-46) Subtracting Equation A-46 from Equation A-45 yields the instantaneous equation PAGE 341 341 1 0i idid id id isiiiiu uCuCuCuC xxxxx (A-47) A.3.2.2 Homogeneous flow The homogeneous case will essentially be the case of C = 0, in which case equation (A-46) reverts back top equation (A36a) and (A-47) to (A-36b). A.3.2.3 MTD method Applying the mass-weighted decomposition to the velocity field gives i iiii ip ipipipip ipiid idipiipi id idipi idipiu uuuu u uuuu uuu uuuuu u uuu uuu CCC (A-48) Equation A-24 is independent of the method of turbulence decomposition. This becomes on the use of the MTD method 11ipi ipi ii iisisiuuCuuC uu xxxx (A-49) Time-averaging results in 1iiidid iisiiuuuCuC xxxx (A-50) This is similar to the CTD method result (E quation A-46) except th at the concentration perturbation in the averaged term with the veloci ty fluctuation difference is replaced by the full concentration, C. PAGE 342 342 These various forms of the continuity equations are summarized in Table A-1 A.3.3Viscous Stresses Expanding the viscous term in equation (A-2i) 2 3ij j vis ik i ij jjji ku uu F x xxxx (A-51i) where is the bulk viscosity. This is normally only important sediment (two-phase flow) the term can become important when the divergen ce is significant (Brady, Khair and Swaroop, 2006). The difficulty is that the bulk viscosity is difficult to measure. The bulk viscosity is hypothesized to be similarly dependent on the density as the shear viscosity. Define the kinematic bulk viscosity For example, 3 11121 11 111231 12 12 21 3 1 13 3122 2 33u uuuuu u xx xxxx uu xx u u xx The viscous stress comes from the gradients in these stresses 123 123ij vis iii i jF x xxx expanding for the u1 equation gives for constant density PAGE 343 343 2 22 2 13 3 1112 1 12 1 22 1231 11213 2 222 3 121 22 212313 222 111 222 1232 2 3 3visu uuu F xxxxxxxxx u uuu xxxxxx uuu xxxx 3 12 1123u uu xxx If the flow is incompressible with constant dens ity the dilatation term vanishes, then we get 222 111 1 222 123visuuu F x xx If the density is not constant then we get 2 22 2 13 3 1112 1 12 1 22 1231 11213 22 3 112 1212 2 111232122212 2 3 1 2 3visu uuu F xxxxxxxxx u uuuuuuu x xxxxxxxxxx 2 2 33 11 2 313331 222 3 111 12 222 123 1123 3 112 1 11123223 1 2 3 uu uu xxxxxx u uuuuu xxxxxxx u uuu u xxxxxxx 3 21 1331u uu x xxx A.3.4Turbulent Interaction with Viscosity A.3.4.1 CTD method Inserting the conventional turbulence decompos ition into the viscous stress term yields, upon averaging 2 3 j j ii jiji ij vis i jj kk ij kkuu uu x xxx F xx uu xx (A-52i) PAGE 344 344 A.3.4.2 Homogeneous case If the flow is assumed hom ogeneous, then equation (A-52i) reduces back to the form of equation (A-51i) above with the averaged valu es replacing the full variables 2 3j vis ik iij jjiku uu F x xx x (A-53i) The implication is that the turbulence has no direct effect on the vi scous dissipation unless the fluid density also has a turbulent component; then the nonlinear effect s make a contribution. A.3.4.3 MTD method If the mass-weighted turbulent decomposition is used we obtain 2 3 j j ii j iji vis i j kk ij kkuu uu x xxx F x uu xx (A-54i) This is the same form as the one above for the CTD approach except that the full density is used in the product of the turbulent velocity grad ients rather than the de nsity perturbation. With the MTD approach, the equation reduces to the sa me homogeneous form as above because if the density is constant, since it can be taken insi de the spatial derivatives of the velocity perturbations and then terms will vanish by definition for the MTD averaging method. A.3.5 Mean Momentum Equations A.3.5.1 CTD approach The instantaneous equation from the conven tional turbulence decomposition and variable density is developed by inserting the equatio ns (A-7) into equations (A-2i) to yield PAGE 345 345 jijiji iiii jjj jijijijiji jjjjj vis iii iiuuuuuu uuuu ttttxxx uuuuuuuuuu xxxxx pp ggF xx (A-55i) With the conventional turbulence decomposition we obtain the Reynolds form of the NavierStokes equation with variable density by averaging jijijiji ii i jjjjuuuuuuuu uu F ttxxxx (A-56i) where vis ii i ip FgF x (A-57i) The viscous contribution for the variable density case using CTD was addressed by equation (A-52i). A.3.5.2 Homogeneous case Assuming constant density results in the following variables to vanish: 0jj jjjjuu ttxxxx The constant density momentum equation for the homogeneous case reduces to: 11jiji vis i ii jjiuuuu u p gF txxx (A-58i) The viscous contribution for the constant density case using CTD was addressed by equation (A-53i). PAGE 346 346 A.3.5.3 MTD approach Introducing the mass-weighted decompositi on variables for the velocity and the conventional decomposition for the remaining variab les into the momentum equations results in the instantaneous turbulent momentum equation (Equation A-59i). jijiji iiii jjj jijijijiji jjjjj vis iii iiuuuuuu uuuu ttttxxx uuuuuuuuuu xxxxx pp ggF xx (A-59i) For getting the most simplification from the MTD equations the density is retained within the derivatives. Some terms will then vanish on the temporal averaging. If the momentum equations are averaged over the time scale used to define the averaged values defined above we obtain for the MTD jiji vis i ii jjiuuuu up gF txxx (A-60i) The viscous contribution for the variable density case using MTD was addressed by Equation A-54i. A.3.6 Instantaneous Fluctuat ing Momentum Equations A. 3.6.1 CTD approach Now subtracting Equation A-56i from Equation A-55i we obtain an equation for the instantaneous turbulent fluctuations using the CTD approach PAGE 347 347 j ijiji iii jjj jijijiji i jjjj jijijiji ii jjjjuuuuuu uuu tttxxx uuuuuuuu u xxxxt uuuuuuuu FF xxxx (A-61i) with visvis iiiii ip FFgFF x (A-62i) and 2 3 22 33 2 3jj j ii i jijjii j ik ij visvis ji k ii j kk ij ij kk k ijuu u uuu xxxxxx u uu xxx FF x uu xx u kx (A-63i) A.3.6.2 Homogeneous Case Invoking constant density in equations (A-61i) yields the homogeneous case equation for the fluctuating momentum jijijiji i ii jjjjuuuuuuuu u FF txxxx (A-64i) A 3.6.3 MTD approach Subtracting Equation (A-60i) from equation (A-59i) we obtain an equation for the instantaneous turbulent fluctuat ions using the MTD approach PAGE 348 348 jijijijiji iii jjjjj jijiji visvis iii jjjiuuuuuuuuuu uuu tttxxxxx uuuuuu p gFF xxxx (A-65i) A.3.7 Large Scale Velocity Field Stresses A.3.7.1 CTD method Using the CTD method the la rge-scale velocity field u kinematic Reynolds stress tensor can be defined as ijijtuu Multiplying Equation A-56i for iu by j u and again by reversing the i and j we obtain after summing jk iik ikik i jjjjjj kkkk jjk jkjk jk j iiiiii kkkk vis vis jjijiiijij ijuu uuuuuuuu uuuuuu ttxxxx uuuuuuuuuu uuuuuu ttxxxx pp uuguFuuguF xx (A-66ij) In the large-scale equations it is common at this point to ignore the viscous terms (Canuto, 1997). However, because of the treatm ent of the effects of mo lecular viscosity on the dissipation of energy these terms are retained. Simplifying we obtain ij ij jkj iki ki j kkk kj jk ki ik jijij i i j kkkkuuuuuuu uuu uu u txtxtx uu uu uuuuuFuF xxxx (A-67ij) In this equation the viscous terms are retained within the F terms. If the homogeneous assumption is made this reduces to the constant density case: PAGE 349 349 ij ij jk ik kjij i i j kkkuuuu uuuuFuF txxx (A-68ij) A.3.7.2 MTD Method Utilizing the mass-weighted turbulence d ecomposition yields a somewhat simpler expression. Multiplying Equation A-60i for iuby j u and again by reversing the i and j we obtain after summing ij ij jk ik kji kkk vis jjiji i vis iijij juuuu uuu txxx p uuguF x p uuguF x (A-69ij) Defining vis ii i ip FgF x (A-70i) we get ij ij jk ik kjij i i j kkkuuuu uuuuFuF txxx (A-71ij) A.3.8 Mean Velocity Reynolds Stresses A.3.8.1 CTD approach If we construct and equation by taking the sum of j u Equations A-2i for iu and iu Equations A-2j for j u we obtain jj k ik i j ijij ii j kkuu u uu u uuuuuFuF ttxx or reorganizing PAGE 350 350 ijijk k ij jiij kkuuuuu u uu uFuF txtx The brackets contain the full continuity equation, which vanishes to yield ijijk j iij kuuuuu uFuF tx (A-72ij) This equation can now be averaged to yield ijijuuuu tt i juu t ijijijijuuuuuuuu tttt ijkijk kkuuuuuu xx ijk kuuu x ijk kuuu x i jk kuuu x ijkikjjki kkk jikkijijkijk jij i kkkkuuuuuuuuu xxx uuuuuuuuuuuu uFuF xxxx Consolidating the equation and using the dynamic fo rm of the Reynolds stre ss tensor definitions ijijuu and ijkijkuuu we obtain ijkijij ij ij ij k kk ijkikjjki kkk jikijkijk jiij kkkuuuuuuuuu u txtttx uuuuuuuuu xxx uu uFuF xxx (A-73ij) Using equation (A-71ij) the terms within the bracket in equati on (A-73ij) can be replaced to yield PAGE 351 351 ij ijijk k kk ijjiijkikjjki kkk j ik ik jk ij jiijjiij kkku txx uuuuuuuuuuuuu ttxxx u uu uFuFuFuF xxx (A-74ij) A.3.8.2 Homogeneous form If the constant density assumption is invoked equation (A-74ij) simplifies to ij ijijk k kk j ik ik jk ij jiijjiij kkku txx u uu uFuFuFuF xxx (A-75ij) A.3.8.3 MTD method Utilizing the mass-weighted turbulence d ecomposition yields a somewhat simpler expression. Expanding equation (A-72ij) using the mass-weighted turbulence decomposition yields ijijuuuu tt 0ijijuuuu tt 0jiuu t 0ijkijk kkuuuuuu xx 0ijk kuuu x 0jik kuuu x 0kij kuuu x 0kijjikijkijk jiij kkkkuuuuuuuuuuuu uFuF xxxx (A-76ij) Using equation (A-70ij) we obtain ij ijijk k kk j ki ij ik jk jiijjiij kkku txx u uu uFuFuFuF xxx (A-77ij) PAGE 352 352 The form of the MTD equation above is the same form as the homogeneous case for CTD method of decomposition (A-74ij) A.3.9 Reynolds Stresses A.3.9.1 CTD method The form of the equations for the Reynolds st resses as velocity perturbation correlations can be developed from the CTD equati ons for shear stresses (equations A-74ij). Defining the Reynolds stresses as 1ijijR and 1ijkijkR ij ijijk k kk ijjiijkikjjki kkk j i ik jk jiijjiij kkRRR u txx uuuuuuuuuuuuu ttxxx u u RRuFuFuFuF xx (A-78ij) A.3.9.2 Homogeneous flow The Reynolds stress equation with the c onstant density assumption reduces to ij ijijk k kk j i ik jk jiijjiij kkRRR u txx u u R RuFuFuFuF xx (A-79ij) A.3.9.3 MTD method The form of the equations for the Reynolds st resses as velocity perturbation correlations can be developed from the MTD equati ons for shear stresses (equations A-77ij). ij ijijk k kk j i ik jk jiijjiij kkRRR u txx u u R RuFuFuFuF xx (A-80ij) PAGE 353 353 The MTD method yields a Reynolds stress equation that has the same form as the homogeneous equation for the CTD method. A.3.10 Forcing Terms in Reynolds Stress Equations A.3.10.1 CTD method The forcing terms, involving F, on the right hand side of equation (A-78ij) can be expanded as j iijjiijji jiij ij j kj k ik ik jiji kkkkpp uFuFuFuFuuugug xx uuuu x xxx (A-81ij) A.3.10.2 Homogeneous flow With the constant density assumption equation (A-81ij) reduces to jiijjiij j kj k ik ik jijiji ijkkkkuFuFuFuF pp uuuuuu x xxxxx (A-82ij) A.3.10.3 MTD method With MTD averaging 1jiijjiij j i ji ijij jk jk ik ik jiji kkkkpppp uFuFuFuFuuuu xxxx uuuu xxxx (A-83ij) Notice that the form of the MTD method does not have the gravitati onal buoyancy terms, but rather additional terms involving the mean pressure gradients. These represent fluctuations in mass due to the mass averaging of the velocity field. There will be differences in the viscous PAGE 354 354 terms as well, but the buoyancy terms are important for high-concentration sediment suspensions that create density stratification. A.3.11 Reynolds Stresses Revisited The insertion of the forcings expansions yi elds revised Reynolds equations as follows A.3.11.1 CTD method ij ijijk k kk ijjiijkikjjki kkk j i ik jk j i jiij kkij jk ik ji kkRRR u txx uuuuuuuuuuuuu ttxxx u u pp R Ruuugug xxxx uu xx j k ik ji kkuu x x (A-84ij) This equation can be rewritten as ij ijijijijijTGijDR DBPDTT Dt (A-85ij) Where the terms are defined as: total derivative ijij ij k kDRRR u Dttx ijD diffusion tensor 12 3ij ijkijkikjjkiikjjki kDRupuuuu x (A-86ij) TT = temporal transformations ijji Tuuuu T tt (A-87ij) TG = transformation from spatial gradients in density fluctuations PAGE 355 355 ijkikjjki G kkkuuuuuuuuu T xxx (A-88ij) ij = production from mean velocity gradients j i ijikjk kku u RR x x (A-89ij) Bij = buoyancy production ijjiij B ugug (A-90ij) ij = pressure-velocity correlation transformations 2 3ijj i ijk ijk p pp uuu x xx (A-91ij) PD = pressure dilatation term 22 33k ku PDppd x (A-92ij) where the dilatation is k ku d x ij = dissipation tensor j i ijikjk kku u x x (A-93ij) A.3.11.2 Homogeneous flow With the constant density assumption the temporal and gradient transformation terms vanish as well as the pressure-dilatation term, to simplify to ij ijijijijDR D Dt (A-94ij) PAGE 356 356 A.3.11.3 MTD method The equation becomes by inserting Equation A-83ij into Equation A-80 1ij ij ijk j i ki k j k kk kk jiji ijij jk jk ik ik jiji kkkkRRRu u uRR txxxx p ppp uuuu x xxx uuuu xxxx (A-95ij) This can be generalized as ij ijijijijijijDR DBPD Dt (A-96ij) Where the terms are now defined as total derivative ijij ij k kDRRR u Dttx (A-97ij) ijD diffusion tensor 12 3ij ijkijkikjjkiikjjki kDRupuuuu x (A-98ij) ij = production from mean velocity gradients j i ijikjk kku u RR x x (A-99ij) Bij = buoyancy production 1ij j i ij p p Buu x x (A-100ij) ij = pressure-velocity correlation transformations PAGE 357 357 2 3ijj i ijk ijk p pp uuu x xx (A-101ij) PD = pressure dilatation term 22 33k ku PDppd x (A-102) where the dilatation is now k ku d x (A-103) ij = dissipation tensor j i ijik jk kku u x x (A-104ij) The MTD method results in the elimination of the temporal and gradient transformation terms. The buoyancy term has changed to an equi valent buoyancy that is a function of the mean pressure gradient. When stratification develops the buoyancy can be manifested in the gradients of pressure. If the hydrostatic assumption is made the MTD buoyancy term matches that of the CTD method directly. A.4 Turbulent Kinetic Energy The turbulent kinetic energy e quation is obtained from the trace of the Reynolds stress equation, that is for i=j. A.4.1 CTD Method The turbulent kinetic energy for the CTD method is defined as 11 22ii iiuu kR Taking one half of the trace of equation (A-84ii) yields the following equation (avoiding confusion by not using k as an index) PAGE 358 358 11 22iij iijiji ii j jj jj ij ij i ij i iii i j ij jRu u u u u u uu kk u txxtxx u p Ruuguu x xxx (A-105) This can be abbreviated as 11 () 22iiiiii TGDk Dk BPDTT Dt (A-106) With the terms defined as total derivative j j Dkkk u Dttx ()Dk diffusion tensor 111 () 23iijjijiiji jDk Rupuu x (A-107) TT = temporal transformations ii Tuu T t (A-108) TG = transformation from spatial gradients in density fluctuations 1 2iijiji G jjuuuuuu T xx (A-109) ij = production from mean velocity gradients 2i ij ij j u R x (A-110) Bii = buoyancy production 2ii ii B ug (A-111) ii = pressure-velocity correlation transformations PAGE 359 359 2 3iii i p u x (A-112) PD = pressure dilatation term 11 33j ju PDppd x (A-113) where the dilatation is j j u d x = rate of dissipation of TKE 11 2i iiij j u x (A-114) A.4.2 Homogeneous Flow For the case of homogeneous flow the tu rbulent kinetic energy equation becomes 11 2iij ij ij i ji ji i i jjjijjR kkup uRuuu txxxxxx (A-115) Or summarizing 1 () 2iiiiDk Dk PD Dt (A-116) A.4.3 MTD Method The TKE equation using the MTD met hod of turbulence decomposition gives 1 2 1iij i ji j jjj ij ij iiii iijjR u kk uR txxx pp uuuu x xxx (A-117) This can be summarized as PAGE 360 360 11 () 22iiiiijDk Dk BPD Dt (A-118) 111 () 23iijjijiiji jDk Rupuu x (A-119) ij = production from mean velocity gradients 2i ii ik ku R x (A-120ij) Bij = buoyancy production 2ii i i p Bu x (A-121) ij = pressure-velocity correlation transformations 1 3iii j ij p p uu x x (A-122ij) PD = pressure dilatation term 11 33k ku PDppd x (A-123) where the dilatation is now k ku d x (A-124) ij = trace of the dissipation tensor 2i iiij j u x (A-125ij) A.5 Rate of Turbulent Energy Dissipation The rate of turbulent energy dissipation developed in the governing equation for the transport of turbulent kine tic energy was defined as PAGE 361 361 111 2 23j ii k i iiij ij jj ik ju uu uu x xx xx 2 12 3j ii iiiii jjijjjjj jj j iii ijijji kikiki ij kjkjkju uuuuuuu xxxxxxxx uuu uuu xxxxxx uuuuuuu xxxxxx ik jku x x (A-126) The conventional definition of the dissipation rate developed for incompressible flow is ii s j juu x x (A-127) This dissipation has been referred to as the solenoidal dissipation ra te (Canuto, 1997). This is the first term in equation (A-126). The remaining terms within the first grouping of terms are associated with the fluctuations in density and the final group of terms is associated with the dilatation of the turbulent velocity field. We can define generally the to tal dissipation of TKE as the sum of three contributions s dc (A-128) The terms are the solenoidal, dilatative and the de nsity fluctuation contributions. The density is subscripted c because we are considering the dens ity variations of the fluid parcel associated with the sediment concentration. The governing equation for the trans port of the rate of dissipati on can be developed by assuming that equation A-61i is a nonlinear operator iNu and constructing the equation ( Speziale and So, 1998) PAGE 362 362 20i i jju Nu xx (A-129) The challenge is whether the additional dissipative terms will manifest themselves from this constructed equation. Inspecting equation (A-61i) will lead to some simplification. The previously averaged terms that were subtracted from the instantaneous equation to form the equation above will average out when multiplied by a turbulent vari able, so remove them now and combine the density decomposition terms to remain compact. This leaves j ijijiji ii jjjj visvis iii iuuuuuuuu uu ttxxxx p gFF x (A-130i) visvis ii iii ip FFgFF x (A-131i) and similarly with the viscous terms, remove terms that will average out. 22 33jj ii visvis jiji ii j kk ij ij kkuu uu xxxx FF x uu xx (A-132i) So we will start with the nonlinear equation ()iNu defined as PAGE 363 363 22 33jijijiji ii jjjj jj ii jiji i ij kk ij ij kkuuuuuuuu uu ttxxxx uu uu xxxx p g xx uu xx (A-133i) The dissipation equation will be formed as 2i i mmu Nu xx This results in 2 22 33jijijiji ii jjjj ij j ii mm jiji i ij kk ij ij kkuuuuuuuu uu ttxxxx uu u uu xx xxxx p g xx uu xx (A-134) The first term on the LHS 22 2222 2 2iiii i mm mm iiii i mmmmm ii iiii i i mmmmmm mmuuuu u xxtxxtt uuuu u xxtxtxtxt uuuuuuu u txxxtxxxtxxt (A-135) PAGE 364 364 The second term on the LHS 22 2222 2 2222iii i mm mm iiii i mmmmm iiiiiii i mm m i mmmmuuu u xxxxtt uuuu u xxtxtxt u t xt uuuuuuu u xxtxxtxxtx 22222 22m iiii mmmm ii ii mm mm x t uuuu xxtxxttt uu uu xxtxxt (A-136) PAGE 365 365 The third term on LHS 2 222 2ji j ii i ji i j mmjmmjjj j i ijji mjmjmj jjj ii ii mmjmjmj j i j mjuu u uu u uuuu xxxxxxxx u u uuuu xxxxxx uuu uu uu xxxxxxx u u u xx 2 22222 2 2 2 2ii j mjmj jj ii iij jmmmjmj ii jiji mjm mjm j i ii j j jm mm muu uu u xxxx uu uu uu u xxxxxxx uu uuuu xxxxxx u u u xxx xxx 2222 22 222 2 2jj i ii jmmj jj ii ii mjm mjm j ii ii ii jj mmjmmjmmj j i j i jm m i mu uu uu u uu xxxx uu uu uu xxxxxx u uuuuuu uu xxxxxxxxx xx u u x x 22ii i jj mjj mmjuu u uu xxxxxx (A-137) PAGE 366 366 The fourth term on the LHS 2 222 2ji j ii i ji i j mmjmmjjj j i ijji mjmjmj jjj ii ii mmjmjmj j i j mjuu u uu u uuuu xxxxxxxx u u uuuu xxxxxx uuu uu uu xxxxxxx u u u xx 2 2 2222 222 2ii j mjmj j ii ii ijj i mmjmmj mmj jjj iiii ii jmmmjmmjm i juu u xxxx u uuuu uuu u xxxxxxxxx uuu uuuu uu xxxxxxxxx u u 222j ii ii i j jmmmjmmjmu uuuuu u xxxxxxxxx (A-138) PAGE 367 367 The fifth term on the LHS 2 222 2ji j ii i ji i j mmjmmjjj j i ijji mjmjmj jjj ii ii mmjmjmj j i j mjuu u uu u uuuu xxxxxxxx u u uuuu xxxxxx uuu uu uu xxxxxxx u u u xx 2 22222 222 2ii j mjmj jj iii i ii j jmmjmmmjm ii i i jiji j mjmmjmmjm iuu u xxxx uu uuuu uuu xxxxxxxxx uuuu uuuu u xxxxxxxxx u 2222 222 222jjj iii i i mmjmmjmmj jjj ii i i ii mmjmmjmmj iiiii jj jmmjmmuuu uuu u u xxxxxxxxx uuu uuu u uu xxxxxxxxx uuuuu uu xxxxxxx 22222j i jmm j iiiiii jj jmmmjmmjmu u xx u uuuuuu uu xxxxxxxxx (A-139) PAGE 368 368 The sixth term on the LHS 2 222 2ji j ii i ji i j mmjmmjjj j i ijji mjmjmj jjj ii ii mmjmjmj j muu u uu u uuuu xxxxxxxx u u uuuu xxxxxx uuu uu uu xxxxxxx u u x 2 22222 222 2j iii j jmjmj jj ii iij jmmjmmj j iii ji ji i mjmmjmmmju uu u xxxxx uu uu uuu xxxxxxx u uuu uu uu u xxxxxxxxx 2 222 222 22jj j ii ii mmjjmmj j ii i i i ijj mmjmmjmmj jj iiii j mmjmmjuuu uu uu xxxxxxx u uu uu u uuu xxxxxxxxx uu uuuu u xxxxxx jx (A-140) PAGE 369 369 Now the RHS 22 22 33 22i i jj ii i jiji mm j kk ijij kk ii mmimp g x uu uu u RHS xxxx xx x uu xx upu xxxxx 22 22 33 22 2i m jj ii jiji i mmj kk ij ij kk ii i mmimm i jj i mmg uu uu xxxx u xxx uu xx upu g xxxxx u u xx u xx 222 33 22 2jj i ijji kk ij ij jkjk ii i mmimm i mmu u xxxx uu xxxx upu g xxxxx u xx 2 2 2 2 2 222 33 2 3jj ii jjjijji jj ii jjjijji kk ij ij jk jk juu uu xxxxxxx uu uu xxxxxxx uu xxxx x 2 3kk ijij kj kuu xx x (A-141) PAGE 370 370 2 2 22 2 23 23 22 222 2ii i mmimm jj ii mjjjmjmjijmi jj ii mjmjmjimji i mj i mupu RHS g xxxxx uu uu xxxxxxxxxxxx uu uu xxxxxxxxxx u xx u x 2 22 23 23 22 2222 33 2 3jj i jjmjmjijmi jj ii mjmjmjimji kk ijij mjkjmk ij muu u xxxxxxxxxx uu uu xxxxxxxxxx uu xxxxxx x 22 32 3 22 33 22 33kk ij jkjmk kk ijij mjkjmk kk ij ij mjk mjkuu xx xxx uu xxxxxx uu xxx xxx (A-142) PAGE 371 371 2 222 2 23 2222 222 222ii i mmimm j iiiii mmjjmjmjmmji j ii i i i mjmimmjmmjupu RHS g xxxxx u uuuuu xxxxxxxxxxxx u uuu uu xxxxxxxxxx 23 2 2 22 23 2222 222 22jj iii i mmjimmjimmjj jj iii i mjmjmmjimjmi iii mmjmuu uuuu xxxxxxxxxxxx uu uuuu xxxxxxxxxxxx uuu xxxx 2 2 3 222 2 22 22 33 22 22 33j ii mjmmji j i mmji ikik ij ij mmjk mjmk ik ij mmjku uu xxxxxx u u xxxx uuuu xxxxxxxx uu xxxx 22 322 22 33 22 22 33ik ij mjmk ikik ij ij mmjk mjmk ikik ij ij mmjk mmjkuu x xxx uuuu xxxxxxxx uuuu x xxxxxxx (A-143) PAGE 372 372 22 2 2 22 222 23 22 22222 222 22iii i i mmimm jmmj jj ii i i mjmjimmjmimj ii ii jmmmjmupuuu RHS g xxxxxxxxx uu uu u u xxxxxxxxxxxx uu uu xxxxxx 2 2 3 22 222 222 22 222 222j i jimm j iiiii mjimmjmjmjmj iiiii jjmmjjmmjmju u xxxx u uu uu u xxxxxxxxxxxx uuuuu xxxxxxxxxxx 22 2 222 223 22 222 3222 222 2 j mi jjj iii mjmi jmmijmmi ii ii ii mmjmmj mmj ii mmju x x uuu uuu xxxxxxxxxxxx uuuuuu xxxxxxxxx uu xxx 22 22 2 33 22 22 2222 22 22 22 33jj ii mmjimmji jj ii mmjimmji ik ik ij ij mjmk jmmkuu uu xxxxxxxx uu uu xxxxxxxx uu uu xxxx xxxx 2 22 22 22 222 22 33 22 22 33 2 2 3ik ik ij ij mmjk mjmk ik ik ij ij mmjk mjmk i ij mmjuu uu xxxx xxxx uuuu xxxx xxxx uu xxx 3 22 2 3ki k ij km m j kuu xxxxx (A-144) PAGE 373 373 22 2 2 22 222 23 22 22222 222 22iii i i mmimm jmmj jj ii i i mjmjimmjmimj ii ii jmmmjmupuuu RHS g xxxxxxxxx uu uu u u xxxxxxxxxxxx uu uu xxxxxx 2 2 3 22 222 22 22 2 22 222 22 2j i jimm j iiiii mjimmjmjmjmj jj ii jjmjmimjmi j i jmmu u xxxx u uuuuu xxxxxxxxxxxx uu uu xxxxxxxxxx u u xxxx 2 2 22 2 2222 22 22 22 22 22222 22 222j ii i ijmmimmj iii ii mmjj mjmj jj ii i i mjmjmmjimmju uu u xxxxxxx uu uu xxxxxxxx uu uuu u xxxxxxxxxxxx 22222 22 22 22 2 222 22 22 33 22 2 33i iiiii ij mimj mimj ik ik ij ij mjmk jmmk ik ij mmjkuuuu xxxxxxxxxx uuuu xxxx xxxx uu xxxx 3 2 22 22 22 222 22 22 33 22 22 33 2ik ij mjmk kiik ijij kmmjmmjk kiik ijij mkmjmjmkuu xxxx uu uu xxxx xxxx uu uu xxxx xxxx 22 22 33 222 22 33 22 22 33ki ki ij ij jkmm jkmm ki ki ij ij mjkm mjkmuu uu xxxx xxxx uu uu x xxx xxxx (A-145) PAGE 374 374 Or reorganizing 22 2 22 2 22 2222 22 22iii i mmimmjjj j iiii mjmjmmjji j iiii mjmjimmjupu RHS g xxxxxxxx u uuuu xxxxxxxxx u uuuu xxxxxxxx 22 22 22 2 22 222 222j ji jj iiii mjjimmjmji jjj iiii mjmjijmmijmmu xx uu uuuu xxxxxxxxxx uuu uu u u xxxxxxxxxxxx 222 2 222 22 22 2 22 22 22 22222 22 22i iiiiii mmjmmjmjmj jj iii mmjimmjiij ii ii mimj mix uuuuuu xxxxxxxxxx uu uu xxxxxxxxxx uuuu xxxxxxx 22 2 322 2 22 2 3mj ikikik mjmkjmmkmmjk ikkiik mjmkkmmjmmjk ij kii mkmjmx uuuuuu xxxxxxxxxxxx uuuuuu xxxxxxxxxxxx uuu xxxxx 22 233 kki jmkjkmm kikiki jkmmmjkmmjkmuuu xxxxxxx uuuuuu x xxxxxxxxxxx (A-146) PAGE 375 375 Combining the density terms on the RHS for equation compactness 22 2 2 2 22 222 22 2iiii i mmimmjijjj jj iiii mmjjimjmji j ii mmjjiupu RHS g x xxxxxxxxx uu uuuu xxxxxxxxxx u uu xxxxx 2 2 2 2 2 22 2 2 2 22 2222 222 222j ii mjjim jj iiiii mjmjijmmimmj j iii mjmjmmjiu uu xxxxx uu uu u uu xxxxxxxxxxxx u uuu xxxxxxxx 22 32 2 22 2 2 22 2 3ii mimj ikikik mmjkjmmkmjmk ikkiki ij mmjkkmmjmkmj ki jkmuu xxxx uuuuuu xxxxxxxxxxxx uuuuuu xxxxxxxxxxxx uu xxx 23 ki ki mjkmmmjkmuuuu x xxxxxxxx (A-147) For homogeneous incompressible conditions this simplifies to 222 2 222iii i mmi mjmj juu u p RHS x xx xxxx x (A-148) PAGE 376 376 Assembling the pieces of the dissipation equation using the CTD method 2 222 222jjjj jj jj jj i ii ii j j i mm jj mjmm ii i i mm mmuuuu txxtxx uu uu u uu u u u xxtxxxxxx uuu u xxtxxt 2 2 2 22222 2i i mm ii ii ii ii mmmmmm mm jj j iiiiii ji mjmmmjmmjmmu u xxt uuuuuuuu txxxtxxxtxxt uuu uuuuuu uu xxxxxxxxxxx 2 222 2i j jjj iiiii i mmjjmmmmj jj iiiiii ijij jmmmmmmmjm i j mmu x uuu uuuuu u xxxxxxxxx uu uuuuuu uuuu xxxxxxxxxx u u xx 22 22jj iiiiiii jjii jjmmjmjmj ii iii i mjmmmjmmj j iiii i mmjjmmuu uuuuuuu uuuu xxxxxxxxx uuuuu u xxxxxxxxx u uuuuu xxxxxx 2 22 22ii i mjmmmj j iiiiii ii mmjmjjmjm jj ii ii jmmmmmjuu u xxxxxx u uuuuuu uu xxxxxxxxx uu uu uu xxxxxxx 2 22 2ii ii mm jj j ii i i ij jmmmjmmmjmmj jj iiii j mjmjmmjuu uu xx uuu uuuu uu xxxxxxxxxxxx uu uuuu u xxxxxxx 2 22j iii j jmmmm j iiii j mmjmmju uuu u x xxxx u uuuu uR H S xxxxxx (A-149) PAGE 377 377 Now assuming homogeneous incompressible flow the dissipation equation can be greatly simplified. All terms with a ,,,ii iiiuu or txxx vanish, leaving 2 2222 2222 222jj iiii ii j j j mmjjmmmjm j iii iii mmjmmij mjmjuu uuuuuu uu txxxxxxxxxx u uuup uu xxxxxxxxxxx (A-150) Dividing through by 2 2222 2 2222 1 222jj iiii ii j j j mmjjmmmjm j iiiiii mmj mmijmjmjuu uuuuuu uu txxxxxxxxxx u uuup uu xxxxxxxxxxx (A-151) This equation is equivalent to the equation pres ented for incompressible flow in Speziale and So (1998). 2 22 2 222 22 2 2jj iiki i iikkjjk iikii k kmmjkj ki i k kmmkmm ii kmkmuu uuuu u txxxxxxx uuuuu u xxxxxx uu u p u x xxxxx uu xxxxx 2 i (A-152) PAGE 378 378 Expanding the Speziale deri vatives in parentheses 2 2 2 222 22 22jj iiki i iikkjjk iikii k kmmjkj kk kmm mmk kii ii k kmmkmmuu uuuu u txxxxxxx uuuuu u xxxxxx uu pp xxxxxx uuuuu uu xxxxxx 2 22 2 2 22ii k mkm ii kmkmiuu x xx uu xxxxx (A-153) This is exactly the same equation. Thes e results were confir med by developing the equation from the original homogeneous equation. The various forms of the momentum equati on and its components are summarized in Table A-2. The terms in the TKE equation are summarized in Table A-3. A.6 Sediment Transport Equation The sediment conservation equation for sedi ment within size class n is provided in Equation A-154n. 3njisnn nn nm j jjuwc cc SD txxx (A-154n) The diffusion coefficient Dm is molecular diffusion at this stage of development, and is coordinate independent. If we insert the conventional turbulen t decomposition into the equation njnnjnnjnnjn nn snnsnn j jjjjj snnsnnnn nnm jj jjjucucucuc ccwcwc ttxxxxxx wcwc cc SSD xxxxx (A-155n) PAGE 379 379 If the total sediment concentration is the sum of the individual class concentrations 1M n nCc (A-156) And we decompose the concentrations in to the mean and fluctuating components nnnccc (A-157n) Then the average of the to tal concentration will be 11MM nn nnCcc (A-158) And the turbulent fluctuation in the total concentration will be 1111MMMM nnnnn nnnnCCCccccc (A-159) The identities in (A-153) and (A-155) allows the summation of equation (A-152n) for all classes to yield 1111 1111 111NNNN njn njn njn njn nnnn jjjj NNNN snn snn snn snn nnnn jjjj NNN nn nnm nnn jjjCC ucucucuc ttxxxx wcwcwcwc xxxx cc SSD xxx (A-160) The summation on the advective terms does not pa ss through to the concentrations because of the size dependent particle velocities. Similarly, the size dependent diffusion coefficients keep the summation from operating on the sediment class concentrations in the diffusion terms. The temporal averaged version of Equation A-155n is presented in Equation A161n. PAGE 380 380 3 3jsn n jsnn nj njn nn jjjj n nm jjwc wc uc uc c txxxx c SD xx (A-161n) The temporally averaged total concentra tion equation (A-160) b ecomes Equation A-162. 3 111 3 111NNN njn nj jsnn n nnn jjj NNN n jsn n m n nnn jj jC ucuc wc txxx c wcSD xx x (A-162) The turbulence correlation term between the veloc ity and the concentration can be expressed as being proportional to the mean gradients (see Equation A-163) nj nn tj j jjuc c D x xx (A-163) The turbulent diffusivity, Dtj, may be different in the coordina te directions. However, it is assumed to be independent of sediment size cl ass. The class time averaged equation becomes Equation (A-164n) 3 3jsn n jsnn njn n jjj n nmt j jjwc wc uc c txxx c SDD xx (A-164n) The total concentration conservatio n equation becomes Equation (A-165) 3 11 3 111NN njn jsnn nn jj NNN n jsn n mtj n nnn j jjC uc wc txx c wcSDD x xx (A-165) PAGE 381 381 Table A-1. Comparison of continuity equati on forms for method of turbulence decomposition Equation Homogeneous case Conventional turbulence decomposition Mass-weighted turbulence decomposition Instantaneous continuity 0ii iiuu xx 0jjj jjjjuuu Ddu Dtdtxxxx 0jjj jjjuuu ttxxx Temporally averaged continuity 0i iu x 0jj jjuu txx 0j ju tx Relative density fluctuation 0i iu x 0jj jjuu D Dtxx 0j ju D Dtx Instantaneous sedimentladen 0ii iiuu xx 1 0ii ididid id isiiiiuu uCuCuCuC xxxxx 11ipi ipi ii iisisiuuCuuC uu xxxx Sedimentladen averaged 0i iu x 1 0ii di d isiiuuCuC xxx 1iiidid iisiiuuuCuC xxxx PAGE 382 382 Table A-2. Comparison of momentum equation forms for method of turbulence decomposition Equation Homogeneous case Conventional turbulence d ecomposition Mass-weighted turbulence decomposition Mean viscous stresses 2 3 u u j i xx vis ji F i x u j k ij x k 2 3 u u j i xx ji u u j vis i F i xxx jji uu kk ij xx kk 2 3 uu uu j j ii x xxx j iji vis F i x uu j kk ij xx kk Momentum (timeaveraged) 11 uuuu u jiji i txx jj p vis gF ii x i uu uuuu j i iiii ttttx j uuuuuuuu j ijijiji xxxx j jjj uuuuuu jijiji xxx jjj pp vis ggF iii xx ii jiji i jj vis ii iuuuu u txx p gF x PAGE 383 383 Table A-2. Comparison of momentum equation forms for method of turbulence decomposition (continued) Equation Equation Equation Equation Turbulent Velocity jiji i jj jiji ii jjuuuu u txx uuuu F F xx j iji iii jj jijijiji jjjj jijiji i jjj jiji ii jjuuuu uuu tttxx uuuuuuuu xxxx uuuuuu u xtxx uuuu FF xx jiji iii jj jijijiji jjjj jiji jj visvis iii iuuuu uuu tttxx uuuuuuuu xxxx uuuu xx p gFF x Mean velocity Reynolds stresses ij ijijk k kk j ik ik jk ij kkk jiijjiiju txx u uu x xx uFuFuFuF ij ijijkijji k kk ijkikjjki kkk j ik ik jk ij kkk jiijjiijuuuu u txxtt uuuuuuuuu xxx u uu xxx uFuFuFuF ij ijijk k kk j ki ij ik jk kkk jiijjiiju txx u uu x xx uFuFuFuF PAGE 384 384 Table A-2. Comparison of momentum equation forms for method of turbulence decomposition (concluded) Equation Homogeneous case Conventional turbulence d ecomposition Mass-weighted turbulence decomposition Turbulent Reynolds Stresses ij ijijijijDR D Dt ij ij ijijijijTGijDR D Dt BPDTT ij ij ijijijijijDR D Dt BPD Turbulent kinetic energy () 1 2iiiiDk Dk Dt PD () 11 22iiiiij TGDk Dk Dt BPDTT () 11 22iiiiijDk Dk Dt BPD PAGE 385 385 Table A-3 Terms in the TKE equation Term Normal Incompressible development Variable density development Rate of TKE dissipation ii j juu x x 11ii ii h jj jjuuuu xxxx Mechanical production due to mean velocity gradients i j i j u uu x 1iii ij jiji j jjuuu uuuuuu x xx Turbulent diffusion of TKE i ji j u uu x 1iii jijiji j jjuuu uuuuuu x xx Molecular diffusion of TKE 2 2 j k x 11i i j jjju k u x xxx Diffusion from pressure fluctuations 1i i p u x 1i i p u x Temporal transformations 1 2ii iiiiiiuu uuukuuu ttttt Buoyancy production 1iiug Mean dilatation transformation 1jjj ii iiii jjjuuu uuuuuu x xx Ttransformation from turbulent dilatation 1 j jjj iiii ii ii j jjjuuuu uuuuuuuu x xxx Production from mean density gradients 1jiiiijiij j jjuuuuuuuuu x xx Transformation from turbulent density gradients 1iijjiiiijjii j jjjuuuuuuuuuuuu x xxx PAGE 386 386 APPENDIX B: DIMENSIONLESS ANALYSIS Dimensionless analysis defines representative scales for each of the variables in the governing equation so that the re lative significance of each term in the equation can be weighed against the other terms in the equa tion. The intent of defining nondimensional scales is to result in scaled terms in the equation that involve gr oupings of nondimensional va riables that approach a scale of unity. The significance of each diffe rential term is then defined in the resulting nondimensional coefficient groups, whic h are comprised of the scales. The primary instantaneous vari ables of interest are nondimens ionalized by the following scales in Equation B-1. The unde rscoring of a variable with a tid al (~) will indicate that the variable is nondimensionalized. 2,,,, / ,,,ii ii f id id fsf x utp xutp hUhUU uC uC W (B-1) These lead to the ensemble-averaged nondi mensional variables in Equation B-2. 2,,,,ii d ii d f sffupuC upu C UUW (B-2) The velocity scale U is assumed to be a representa tive magnitude appropriate to the specific case. The scaling of the differential particle velocity, uid, is taken as a representative fall velocity of the sediment, Ws; for example, the Stokes fall velo city. The turbulent perturbation variables would nondimensionalize as defined in Equation B-3. ,,,id i id i s ufuu C uu C W (B-3) These variables vanish on ensemble averaging, just as the dimensional variables vanish. PAGE 387 387 The turbulent velocity deviation scale u is assumed to be representative of the turbulent velocity fluctuations, for example, the standa rd deviation of the turbulent fluctuations ( 2uk). The scalar magnitude of the derivatives of a pr imary variable is based on the constancy of the scales as shown in Equation B-4. iii iiiuUuUu x hxhx (B-4) When ensemble averaging is performed on the products of fluctuating variables the timeaverage of the product will not ne cessarily scale by the product of the variable scales. For example, an appropriate scale for the term iu will not be u but rather an independent scale associated with the net turbulen t mass flux. Call that turbulent flux tQ where it is hypothesized that in general t uQ If the terms in the product are the same variable, e.g. iiuu then by definition the ensemble average will be the variance of iu Consequently, if the two variables to be averaged are perfectly correlated, then the en semble average is the product of the standard deviations of the two variables. If the two variables are less than perfectly correlated, then the product of standard deviatio ns is an upper bound on the ensemble average. If they are not correlated at all, then the ensemble average will approach zero. This is illustrated by the development of the identity for the scaling of the nondimensional turbulent corr elation in Equation B-5. 11 1tT tT u tt tT uu tuud tud t TT udtu T (B-5) PAGE 388 388 This would be appropriate for scaling iu only if u is close to unity, which is the root of the problem. In a turbulent flow where the mean value of the perturbation terms is by definition zero, the mean of th e product of nondimensional variable s should be much less than unity. Therefore, assuming that thes e scaling values for the second order ensemble averaged terms are for now unknown but unique to each te rm, we introduce the nondimensional scaling shown in Equation B-6. 0,,, ,ji id i id i ji ttt cm jii i jii i t Euu uC u uC u uu QQQ uuu pu uuu pu Q (B-6) Note that since 2 *ij jiuuu it is appropriate to use 2 *t mfQu, scaling for the turbulent momentum flux. It ma y be possible to extend the use of *u and select *tQu and 3 *t EfQu, representing the scales for turbulent ma ss flux and turbulent energy flux. The term 0 is a representative scale value for the turbulen t pressure-velocity correla tion. For the context being developed here the pr essure fluctuations will result from fluctuations in the density of the fluid associated primarily with the suspended sediment concentration and the turbulent velocity fluctuations. Furthermore, when the ensemble average i nvolves terms of mixed derivatives a key assumption must be invoked relativ e to the appropriate scale. That is, for example, each of the following terms are assumed to have the same appropriate nondimensional scale PAGE 389 389 ,,i i i iiiu u u x xx These should all scale by a common scale of tQ h This commonality of scaling for the mixe d derivatives is invoked, in part, due to the ability to move the spatial derivatives outside the ensemble integral. So the power of the above assumption is that the nondimensional scaling for the ensemble averaged terms simplifies down to the co rrelations among the primary variables. B.1 Continuity Equation Inserting the nondimensional equa lities above into the full co ntinuity Equation A-1 without turbulent decomposition we obtain Equation B-7. 0j ju tx (B-7) The scaling terms cancel each ot her out, leaving little addition al information. However, inserting the nondimensional terms into the deri ved continuity equation (A-46) accounting for the impact of the presence of suspended sediment we obtain. 0t f isidcid isisfiuWuCQuC xUxWx (B-8) The nondimensional term s W U is simply the ratio between the differential velocity of the sediment relative to the fluid and the velocity of the fluid. The ratio between the fluid density and the sediment floc density appears along with the velocity ratio. The second imbedded nondimensional term t c s fQ W can be interpreted as the ratio between the turbulent flux of the sediment relative to the fluid and the flux of fluid relative to the sediment. PAGE 390 390 B.2 Momentum Equation Inserting the nondimensional va riables into Equations A-56i we obtain: 2 2 *//tt iij i ffj i jj t jif ji i jjuuu u UUu u QU Q hUthUthxhx uuuuu UQ F hxhx B-9i) where: 2 22 222 3f ifi i t fj j ii ji ji t j f kk ij kkp U Fg hx Uuu Q uu hxxhxx x U Q uu hxhx B-10i) Dividing through Equations B-9i and B-10i by 2fU h yields Equation B-11i. 2 22 t ijii jiji jfjj ji i ji t j ii jifj juuuu uuuu Q txUtxx p uu gh u UxxU u Q uu xxUx xhU 2 3j i t kk ij kfku x Q uu xUx B-11i) The nondimensional terms that results are: PAGE 391 391 1ehUR where Re is the Reynolds number, the ratio of advective momentum transport to molecular dissipation of momentum. 221i rgh UF where Fr is Froude number, the ratio of inertia to gravitational acceleration. *u U is the ratio of turbulent intensity to mean velocity. t fQ U is the ratio of turbulent density flux to advective flux. If the scaling for the turbulent density flux, tQ is taken to be *u then we would get t fQ U, where f is ratio of the density fluctuations to the fluid density. B.3 TKE Equation The TKE equation introduces several additional nondimensiona l scales. Inserting the nondimensional terms into Equatio n A-105 yields Equation B-12. 23 ** 2 2 01 2 1 2iij ff j jj t iiiijiji jj f i ij jR uUu kk u htxhx uuuuuuuu UQ htxx uU u R hx 2 2 f ij t iiii ijp u uQguu hx hx (B-12) Dividing through Equation B-12 by 2 fuU h results in Equation B-13. PAGE 392 392 0 2 *1 2 11 2 1iij j jj t iiiijiji fjj i iji jfiR kk u txx uuuuuuuu Q utxx p Q u Ru xUux 2 *1t ij i ii f ejgh uu uURx (B-13) B.4 Dissipation Equation The rate of dissipation of TKE, (Equation A-146) is scaled similarly resulting in the nondimensional Equation B-14. PAGE 393 393 22 1 2t jjjj jj f jj t jj i i mm jfj t i fuuuu Q txtxUxx uu Q u u xxtxUx Q u Ux 22 2 2 221 111t iii ii mmfmmmm ttt ii iiii fm mfmmm mfQ uuu uu xtUxxtxxt QQQ uuuuuu UtxxUxtxxxtU 2 2 22ii mm jjj iiiiiii ji mjmmmjmmjmmj j iii i mmjjuu x xt uuu uuuuuuu uu xxxxxxxxxxxx u uuu u xxxx 222 2jj ii mmmmj jj iiii ii iji i jj i jmmmmmmmjm muu uu xxxxx uu uuuu uu uuuu uu u xxxxxxxxxxx 21 2jj iiiiiiii jjjii mmjjmmjmjmj t iii i fmjmmm juu uuuuuuuu uuuuu xxxxxxxxxxx Q uuu u Uxxxxx u 2 2 2 21 1 2ii jmmj t iiiiiii i fmmjjmmmjmmmj j muu xxxx Q uuuuuuu u Uxxxxxxxxxxxx u Q x 2 21 11 2tt iiiiii ii fmjjmfmjjm tt j i i jfmmfQ uuu uuu uu UxxxxUxxxx u QQ u u xUxxU 2 2 211 2 1 2i i mm tt j ii ii mjfmfm t j ii ij fj m mm j mmu u xx u QQ uu uu xxUxUx u Q uu uu Uxxxxxxx 2 2111 2 1 2jj ii mjmmj ttt jj iiii j mfjmfjmfmj t fuu uu xxxxx uu QQQ uuuu u xUxxUxxUxx Q u U 21 22 2j iii j jmmmm tt j ii iiii jj fmjmfmmjmmju uu u xxxxx u QQ uuuuuu uuR H S UxxxUxxxxxx (B-14) PAGE 394 394 And the RHS results for Equation B-14 are given in Equation B-15. 2 2 222 2 222 11 2 2 1t ii i mmifrmm j ii ii ejijjjemjjim ii mmj eQ up u RHS g xxxUFxx u uu R xxxxxRxxxxx uu xxx R 2 22 2 222 1jj ii jimjmji j ii mmjji jj ii i mjmjijmmi euu uu xxxxxxx u uu xxxxx uu uu u xxxxxxxxx R 2 2 2 2222 32 2222 21 2 3ii mmj j iiiii mjmjmmjimimj ikik mmjkjmmk i mj ij euu xxx u uuuuu xxxxxxxxxxxx uuuu xxxxxxxx u xx R 2 2 22 2 231 1kik mkmmjk t ki fjkmm ki ki t kmmjmkmj f ki ki jkmmmjkmuuu xxxxxx Q uu Uxxxx uuuu xxxxxxxx Q U uuuu xxxxxxxx (B-15) Here, as before, if the scaling for the turbulent density flux is taken to be *tQu then we would get t fQ U, where f is ratio of the density fluctuations to the fluid density. Returning to the mean momentum equa tion and dropping second order terms in and leaves Equation B-16. PAGE 395 395 2 21 12 3iji ji jjir j ik ij jejikp uuu uu txxxF u uu xRxxx (B-16) However, for the TKE equation none of the terms can be dropped as second order. Expressing Equation B-13 with the nondimensi onal bulk variables gives Equation B-17. 0 22 *11 22iij ii iijiji j jj jj ii ij i jfiRuuuuuuuu kk u txxtxx p ug h Ru xUuxU 1ij ii ejuu Rx (B-17) If the turbulent pressure velocity correlation,0 is scaled by 3 *fu we get Equation B-18. 211 22 1iij ii iijiji j jj jj ii ij i i jiRuuuuuuuu kk u txxtxx p ug h Ruu xxU ij i eju Rx (B-18) In the dissipation equation, using the scaling for the turbulent mass flux, tQ as *u only a few terms drop out leavi ng Equations B-19 and B-20. PAGE 396 396 22 22 222 222jj j i i jjm mj ii i i ii mm mm mm ii ii i mm mmmuuu u u txtxxxtx uu u u uu xxtxxtxxt uu uuu txxxtxx 2 2 22 2ii i mm m jjj iiiii ii ji mjmmmjmmjmmj jjj ii ii i i mmjjmmmmjuuu xtxxt uuu uuuuuuu uu xxxxxxxxxxxx uuu uuuuu u xxxxxxxxx 222 2jj iiii ii iji i jj i jmmmmmmmjm m iiii ii jjji mmjjmmjuu uuuu uu uuuu uu u x xxxxxxxxxx uuuuuu uuuu xxxxxxx 22 222jj ii i mjmj ii iii i mjmmmjmmj j iiii iii i mmjjmmmjmmuu uu u xxxx uuuuu u xxxxxxxxx u uuuuuuu u xxxxxxxxxx 22 22mj j iii iii ii mmjjm mjjm jj ii ii jm mm mmxx u uuuuuu uu xxxxxxxxx uu uu uu xxxxxx 2 22 2ii ii jmm jjj iiii ij jmmmjmmmjmmj jj iiii j mjmjmmuu uu xxx uu u uuuu uu xxxxxxxxxxxx uu uuuu u xxxxxxx 222j j iii ii jj jmmmm mjmu uuuuu uu R H S xxxxxxxx (B-19) And on the RHS of Equation B-19 (Equation B20) no terms can be dropped as second order. PAGE 397 397 2 2 22 2 2 222 11 2 2 1ii i mmirmm j ii ii ejijjjemjjim j ii mmjj eupu RHS g xxxFxx u uu R xxxxxRxxxxx u uu xxxx R 2 22 2 222 1j ii imjmji j ii mmjji jj iiii mjmjijmmim eu uu xxxxxx u uu xxxxx uu uu u u xxxxxxxxxx R 2 2 2 2222 32 2222 21 2 3i mj j iiiii mjmjmmjimimj ikiki mmjkjmmkmjm ij eu xx u uuuuu xxxxxxxxxxxx uuuuuu xxxxxxxxxxx R 22 22 23 k k ikki mmjkjkmm ki ki kmmjmkmj ki ki jkmmmjkmx uuuu xxxxxxxx uuuu xxxxxxxx uuuu xxxxxxxx (B-20) The standard form of the k model has essentially the fully derived k -equation intact. However, because of the complexity of the fully derived -equation, even for the homogeneous case, the standard form utilized for the conservation equation for is essentially empirical. It takes the same general form as the k equation: total derivative equals a source term plus a diffusive term and a decay term. And sin ce the dissipation equati on is empirical, the k equation PAGE 398 398 is also typically developed with empirical coe fficients (Speziale, 1998; Hanjalic, 2004). The general form is as shown in Equation B-21. 3 1 2 3321jj ii TT Tj jjkjjijTj jj ii T T jjj jij T j Tjuk u uu kkg txxxxxxx uu uu c txxxkxxx g cc kxk (B-21) The empirical coefficients in the dissipation equation are presented in Table B-1. Table B-1 Coefficients in the dissipation equation for the standard kmodel. c c1 c2 c3 c4 k 0.09 1.44 1.92 0.8 0.33 1.0 1.3 0.9 The nondimensionalized version s of the standa rd k-e model are given in Equation B-22. 2 3 22 2 ** 2 1 22 *1T jj ki i TT j jjj i j Tj T j Ti jjjuk u uu Ug kk txUhxhuxxxUux u cUu txUhxhukx 2 33 2 2 *1 1j i ij T j Tju u xx g cc Uukxk (B-22) These dimensionless numbers within the brac kets in Equation B-22 are versions of a turbulent Reynolds number. The first term, T kUh is the ratio of the advection of turbulent energy and turbulent diffusion of tu rbulent energy. The first term in the dissipation equation is PAGE 399 399 essentially the same ratio. The term 2 *TU hu can be expressed as 2 2 *11T t eU hUuR where t e R is a turbulent Reynolds number, in that the turbul ent eddy viscosity is used rather than the molecular viscosity. Also note that when the ed dy viscosity dominates ov er molecular viscosity. 1T k t eUhR The buoyancy term has led to another nondimensional number, 2 *T Tg Uu This term can be rearranged to be *111reFR where *e Tuh R The form of Equation B-22 can be restated as Equation B-23. 2 3 22* 2 1 2211 11 1jj ii j tt jejejijrej jj ii tt jejejiuk u uu kk txRxRxxxFRx uu cuu txRxRkxxx 2 3 32 *1 1j j rejc c FRkxk (B-23) B.5 Sediment Transport Nondimensionalizing Equation A-156 for sedi ment transport yields Equation B-24. 11 2 11t NN f c njnnjn nn jj NN ffnj n n nn jjU CQ uc uc htxhx UD c S hxhx (B-24) Dividing through Equation B-24 by f U h yields Equation B-25. PAGE 400 400 11 1 1t NN N N nj cn njn njnn nn n n j fj j jD CQ c uc ucS txUx xhUx (B-25) If the turbulent sediment flux,t cQ is scaled by *cu we obtain Equation B-26. 11 1 1NN N N nj n njn C njnn nn n n j jjjD Cc uc ucS tx x xhUx (B-26) Where a new nondimensional term, C has been introduced whic h is the ratio of the turbulent sediment concentration scale and the freshwater density, c f . PAGE 401 401 APPENDIX C PARTICLE SIZE DISTRIIBUT ION PLOTTING ANALYSIS This appendix provides a detailed step-by-step description of the analysis steps in the development of the particle size distribu tions presented in Chapters 2 and 6. A particle size distribution has been developed from the analysis of a video capture of in situ suspended cohesive flocs. The density of the flocs has been estimated from the observed settling velocity computed from Stokes settling la w. The data are tabulated below. Construct a particle size distribution plot based on the volumetric con centration (in ppm) using the distribution density based on the lo garithm of the particle size. Assume the following: the clear fluid density is 31025/wkgm the mineral density is 32650/skgm and the spacing of the size classes in log space is uniform 10.075051iidLogdLogdLogd. What is the total concentration? Table C-1. Input data for floc size analysis Equivalent spherical diameter (ESD) Number of flocs per unit volume Estimated floc density Equivalent spherical diameter (ESD) Number of flocs per unit volume Estimated floc density (microns) (# /m3) (kg/m3) (microns) (# /m3) (kg/m3) di dNi f di dNi f 32.830 0 219.698 1854349 1052.9 39.023 0 261.143 1348618 1050.5 46.384 7754551 1092.4 310.405 505732 1062.9 55.134 27815237 1179.3 368.960 168577 1038.9 65.534 32703976 1184.5 438.562 0 77.897 32872553 1135.2 521.293 0 92.592 27478083 1098.0 619.630 0 110.058 18037760 1076.2 736.519 0 130.820 13823330 1082.9 155.498 10114632 1060.4 184.831 5563047 1082.2 PAGE 402 402 Step-by-step development: refer to Tables 1 and 2 below. This analysis was performed with Microsoft Excel. Columns A, B and C are the input data provided. Column D: The volume of the floc ESD in m3. 36id V with di converted to meters Column E: The lower boundary diameter for th is size class, taken as the diameter resulting from averaging the logarithms of th e adjacent size classes as follows: 1(log10()log10())/210iidd iLd Column F: The upper boundary diameter for this size class, taken as the diameter resulting from averaging the logarithms of the adjacent size classes. Note that the lower boundary for class i is the upper boundary for class i-1. 1(log10()log10())/210iidd iHd Column G: The difference, in microns, between the upper and lower class boundaries. iiHiLddd Column H: Vlow, the volume of the equivale nt spherical diameter diL, the lower bound. Use same equation as for column D above. Column I: Vhigh, the volume of the equiva lent spherical diameter diH. Use same equation as for column D above. Column J: The differential individual particle volume spanned by this size class, dVpi. This is computed as Vhigh Vlow. Column K: Dcon is the differential sediment concentrati on within this size class. It is the pr4oduct of the number of flocs (Column B) tim es the average floc density (Column C) times the individual fl oc volume (Column D). ifipiDcondNV Units = (1/ m3)*(kg/m3)*(m3) Column L: nd, the probability density for the dist ribution by particle diameter. The generic units of nd are number of particles (of size di) per unit volume of solution per unit span of the particle diameter. In this case: #/m3/micron. The value is computed from: /diindNd Column B/ Column G PAGE 403 403 Column M: nv, the probability density for the dist ribution by particle diameter. The generic units of nv are number of particles (of size di) per unit volume of solution per unit span of the particle volume. In this case: #/m3/m3. The value is computed from: /vipindNdV Column B/ Column J Note that columns L and M are not necessary to compute the volumet ric distribution, but are useful in understand ing the distribution. Column N: dVi, is the differential sediment volume within this size class. It is computed as the product of the particle number density (column B) times the volume of each floc in this class (column D). The units of dVi are dimensionless (m3/m3), volume per volume. iipidVdNV Column O: i pidV dLogd This is the equivalent probabi lity density expressed per unit span of the logarithm of the particle size rather than per span of the particle size. This value is computed as the ratio of the differential sediment volume in each class (column N) divided by the uniform differential span of each size class used to define the classes 10.075051iidLogdLogdLogd as given. 0.075051ii pidVdV dLogd The units of i pidV dLogd are per log micron, or vol ume per volume per log micron. Column P: This is just conve rts the value in column O into units of ppm (parts per million volumetric concentration density) Total concentration: the sum of column K provides the estimate of the total concentration in kg/m3. Ctot = 0.140 kg/m3 PAGE 404 404 Table C-2. Example calculations for deve lopment of the volumetric concentration. A B C D E F G H I J K Given Given Given Eq. 5-9 F-E Eq. 5-9 Eq. 5-9 I-H di dNi density Volume, Vpi di low di high di Vlow Vhigh dVpi Dcon distribution microns #/m3 kg/m3 m3 microns microns microns m3 m3 m3 kg/ m3 1 32.830 0 1.85E-14 2 39.023 0 3.11E-14 35.792 42.544 6.752 2.40089E-14 4.03E-14 1.63116E-14 0 3 46.384 0.043071 1092.4 5.23E-14 42.544 50.570 8.026 4.03205E-14 6.77E-14 2.73936E-14 2.45844E-12 4 55.134 0.154494 1179.3 8.78E-14 50.570 60.110 9.540 6.77141E-14 1.14E-13 4.60046E-14 1.59879E-11 5 65.534 0.181648 1184.5 1.47E-13 60.110 71.449 11.339 1.13719E-13 1.91E-13 7.72599E-14 3.17077E-11 6 77.897 0.182584 1135.2 2.47E-13 71.449 84.927 13.478 1.90979E-13 3.21E-13 1.2975E-13 5.12977E-11 7 92.592 0.152622 1098 4.16E-13 84.927 100.948 16.021 3.20729E-13 5.39E-13 2.17902E-13 6.96544E-11 8 110.058 0.100187 1076.2 6.98E-13 100.948 119.991 19.043 5.3863E-13 9.05E-13 3.65943E-13 7.52615E-11 9 130.820 0.076779 1082.9 1.17E-12 119.991 142.626 22.635 9.04573E-13 1.52E-12 6.14563E-13 9.7469E-11 10 155.498 0.05618 1060.4 1.97E-12 142.626 169.531 26.905 1.51914E-12 2.55E-12 1.03209E-12 1.17285E-10 11 184.831 0.030899 1082.2 3.31E-12 169.531 201.512 31.981 2.55123E-12 4.28E-12 1.73329E-12 1.10555E-10 12 219.698 0.0103 1052.9 5.55E-12 201.512 239.526 38.014 4.28452E-12 7.2E-12 2.91089E-12 6.02137E-11 13 261.143 0.007491 1050.5 9.32E-12 239.526 284.710 45.185 7.19541E-12 1.21E-11 4.88853E-12 7.33763E-11 14 310.405 0.002809 1062.9 1.57E-11 284.710 338.419 53.708 1.20839E-11 2.03E-11 8.20977E-12 4.67536E-11 15 368.960 0.000936 1038.9 2.63E-11 338.419 402.259 63.840 2.02937E-11 3.41E-11 1.37875E-11 2.55833E-11 16 438.562 0 4.42E-11 402.259 478.141 75.883 3.40812E-11 5.72E-11 2.31546E-11 0 17 521.293 0 7.42E-11 478.141 568.339 90.197 5.72357E-11 9.61E-11 3.88857E-11 0 18 619.630 0 1.25E-10 568.339 675.551 107.212 9.61214E-11 1.61E-10 6.53045E-11 0 19 736.519 0 2.09E-10 0 Sum 1.0 7.77603E-10 PAGE 405 405 Table C-2. Example calculations for developmen t of the volumetric con centration (concluded) A L M N O P Q R Given Corrected Corrected di Dcon dNi nd nv dVi dV/dLOGd dV/dLOGd (microns) (kg/ m3) (#/m3) (#/ (m3/m3) (1/log( m)) (ppm/log( m)) 1 32.830 2 39.023 0.0000E+00 0 0 0 0 0 0 3 46.384 4.4264E-04 7754681 966221 2.83079E+20 4.0519E-07 5.3989E-06 5.4 4 55.134 2.8785E-03 27815702 2915758 6.04618E+20 2.4408E-06 3.2522E-05 32.5 5 65.534 5.7089E-03 32704522 2884153 4.23298E+20 4.8196E-06 6.4217E-05 64.2 6 77.897 9.2358E-03 32873102 2438935 2.53353E+20 8.1357E-06 0.0001084 108.4 7 92.592 1.2540E-02 27478542 1715150 1.26103E+20 1.1421E-05 0.00015218 152.2 8 110.058 1.3550E-02 18038061 947212 4.92912E+19 1.2591E-05 0.00016776 167.8 9 130.820 1.7548E-02 13823561 610698 2.2493E+19 1.6204E-05 0.00021591 215.9 10 155.498 2.1115E-02 10114801 375935 9.80011E+18 1.9912E-05 0.00026532 265.3 11 184.831 1.9905E-02 5563140 173950 3.20952E+18 1.8392E-05 0.00024507 245.1 12 219.698 1.0841E-02 1854380 48781 6.3704E+17 1.0296E-05 0.00013719 137.2 13 261.143 1.3211E-02 1348640 29847 2.75874E+17 1.2575E-05 0.00016756 167.6 14 310.405 8.4179E-03 505740 9416 6.16012E+16 7.9196E-06 0.00010552 105.5 15 368.960 4.6059E-03 168580 2641 1.22268E+16 4.4334E-06 5.9072E-05 59.1 16 438.562 0.0000E+00 0 0 0 0 0 0 17 521.293 0.0000E+00 0 0 0 0 0 0 18 619.630 0.0000E+00 0 0 0 0 0 0 19 736.519 0.0000E+00 0 0 0 0 Sum 0.140 PAGE 406 406 Figure D-1. Particle size distribution for ex ample video analysis. This example analysis is intended to dem onstrate the steps involved in the analysis of the San Francisco Bay video imaging analysis for the determination of the particle size distribution. PAGE 407 407 APPENDIX D OUTLINE OF KEY SEDIMENT SUBROUTINES IN COMPUTATIONAL MODEL This appendix provides a deta iled description of the comput ational steps and equations used in several of the key subroutines within the research com putational program. The primary subroutines detailed here are those that develop the sedime nt properties a nd interaction processes. These subroutines are the impl ement the aggregation/disaggregation model (subroutine AGGFLUX), the development of se ttling velocities (subroutine FALLVEL) and the bed exchange (subroutine BEDXCHG). The genera l architecture of the computational program was outlined in Chapter 4. The development of the sediment transport over the simulation time involves calling each of these subr outines during each time step. D.1 AGGFLUX Subroutine AGGFLUX computes the mass fluxe s between size classes associated with aggregation, disaggregation due to both hydrodynamic shear and floc breakage associated with particle-particle collisions. Th e logical flow of the subroutine is described in the following program sequence, with reference to the equations used. The logic below is repeated for each spatia l discretization over the water depth and for each cohesive sediment class. 1. The current values of the mass fluxes are stored as the values for the previous time step, and the current fluxes initialized to zero in prep aration for being updated during this pas through the subroutine. 2. The following computations are performed for each cohesive size class: a. Collision frequencies are developed ba sed on Equations 4-40 through 4-42, for Brownian motion, turbulent shear and differential settling. The collision frequencies are developed for all parings of size classes. b. If a probabilistic treatment is used for differential settling, Equation 4-42 is integrated over the two probability distributions for the two class settling velocities. This is PAGE 408 408 accomplished via a subroutine for numeri cal integration of the probability distributions for each possible pairing of size classes. c. The numbers of flocs formed per second are computed based on Equation 4-38 for each pairing of size classes. 3. A check is made of the summation of all possible floc formations involving each size class, to ensure that there are sufficient numbers of flocs present to satisfy the total. Adjustments are made to all floc formation rates constraine d by a criterion for time step reduction. If the adjustment fraction is less than the constrai ning criterion then the program stops, flagging the need to reduce the time step. The limiting factor used was 0.75. 4. Aggregation losses, LiA, from each of the classes involve d in the floc aggregation are computed based on Equation 4-48. 5. Aggregation gains, GiA, to each size class are computed by Equation 4-45 and 4-46, using the distribution factors deve loped by Equation 4-47. 6. Floc breakage due to local shear stresse s was computed based on Equation 4-53. 7. If probabilistic treatment is used, then Equa tion 4-53 in integrated over the two probability distributions for and fi. 8. Disaggregation breakage loss flux, LiB, is then computed usi ng Equation 4-52 for each size class. 9. Disaggregation breakage gains, GiB, are summed from the distribution of the remnants of the breakage losses, using a two-fragment assumpti on with masses 3/16 and 13/16 of the original floc. This is given in Equation 4-54. 10. Collision related disaggregation is developed fr om the interparticle shearing computed from Equation 4-55. 11. The collision efficiency of floc breakage is computed from Equation 4-61, with either the mean value form for classical treatment or with an integrated probabilistic treatment of the two shear stress distributions. 12. Collision disaggregation losses, LiC, are estimated based on Equation 4-62. 13. Collision dissaggregation gains, GiC, are developed by summing up the redistribution of the fragments of the disaggregation losses, using the two-fragment 3/16-13/16 redistribution of the floc masses. All of the aggregation flux terms (gains and losses) become contributions to the governing equations for each sediment size class. Those eq uations are solved in subroutines SEDCOH and SEDCOHG for the cohesive sediment classes and the tagged sediment classes, respectively. PAGE 409 409 D.2 FALLVEL Subroutine FALLVEL computes the settling veloci ties for every cell over the water depth and for every sediment size class, both cohesi ve and noncohesive, for each model time step. The requirement each time step comes from the evolu tion of the sediment concentration field, which in turn affects the floccula tion and hindered settling effect s on the settling velocities. The computational steps in the subroutine are: 1. Settling velocities for the current time level are stored as the old time level. 2. The free settling velocities for all particle classes (both cohe sive and noncohesive) are solved iteratively using Equations 2-25 and 2-26. 3. The exponents, nf(d) from Equations 2-44 and 2-46 are calculated by solving Equation 2-44 to yield max() ln () lnsfi s fii f hwd w nd C C The variables wsmax, Cf and Ch are specified as input, allowing for the calculation of the exponent for each cohesive size class. 4. If C > Cf then the effects of concentration and internal shear on the setting velocities for each cohesive size class are computed based on Equation 2-46. 5. If f CC the concentration and shear corrected settling velocity remains wsf. 6. The effects of hindered settling are then appl ied to the concentration and shear corrected cohesive settling velocities ba sed on Equation 2-40. This ap plication is applied without condition on concentration since at low concentr ations the correction vanishes explicitly. This correction is applied for each size cla ss for every computati onal cell in the water column. 7. The effects of hindered settling are then also ap plied to the concentration and shear corrected noncohesive free settling velocities based on Equation 2-40. PAGE 410 410 D.3 BEDXCHG The primary treatment of the focus of the research here is de veloped in subroutine BEDXCHG, where the exchange fluxes for er osion (EROS and EROSG) and deposition (DEP and DEPG) are developed for each size class, for both tagged and unt agged cohesive sediments. The silt bed fluxes (ENTR and SETS ) are also computed in the subroutine for each silt size class. These flux terms are then used in the governing equations for each size cl ass as source and sink terms for the bottom computational cell. The units of the erosion terms are kg m-2 s-1, which is the actual erosive flux. The units on the de position terms are m/s, since these variables are multiplied by the concentration in the bottom co mputational cell to define the deposition flux. The bed exchange is specified by two general op tions. The first is the decision of whether to treat the primary variables as mean values or probabilistically. The second option is whether to treat erosion and deposition as exclusive or allow for simultaneous erosion and deposition. For sensitivity testing and analys is, the option for either no deposition and/or no erosion can be invoked. The logical flow of subroutine BEDXCHG can be summarized as follows. 1. The current bed exchange variables are assigned to the values for the previous time step, and the current fluxes initialized to zero. 2. For each cohesive size class, i the following computations are executed for the mean value option. a. For the case when b < ei, i. If b < di, then the probability of deposition is estimated as 1b Di diP Otherwise, PDi = 0 ii. If the option for simultaneous erosion and deposition is used, then set PDi = 1 iii. If the option for no deposition is chosen, set PDi = 0 PAGE 411 411 iv. For the case when the shear stress is below the deposition threshold, set the erosion flux to zero (EROSi =0; EROSGi=0). v. Set the deposition variable, DEPi= PD wsc, where wsc is the aggregation and hindered settling adjusted fall velocity for the bottom cell for sediment class i estimated from Equations 2-26, 2-40 and 2-46 b. When b > ei, i. The probability of erosion is estimated as 1b ei eiP ii. The erosion rate for this size class is then estimated as*ibcieiCufP based on the empirical coefficient, Cbc (kg/m3), the shear velocity and the fraction of the total bed mass associated with this size class, fi. iii. If the option of no erosion is invoked, then set0i iv. The only difference between the applica tion of the erosion logic for tagged and untagged cohesive sediment of the same size class is use of the appropriate fraction of the bed mass. v. EROS = i. EROSG = iG, the erosion rates for the untagged and tagged sediment. 3. For each cohesive size class, i the following computations are executed for the probabilistic option. a. The probability of deposition is calculated from the two probability curves for b and di, based on Equation 4-76 for PDi = P (b < di). b. If applying simultaneous erosi on and deposition, then set PDi = 1 c. If the option for no deposition is chosen, set PDi = 0 d. Set the deposition variable, DEPi= PDi wsi, where wsi is the aggregation and hindered settling adjusted fall velocity for the bottom cell for sediment class i estimated from Equations 2-26, 2-40 and 2-46 e. Calculate the erosion potential integral, Ei, for this size class from Equation 4-33, which incorporates the shear stress depende nt time response inside the probability integral. f. The erosion for the untagged and tagged sedi ments, if being simulated, differs only by the relative fractions of the cl ass concentrations in the bed. g. For the option of no erosion, se t the erosion flux to zero (EROSi =0; EROSGi=0). PAGE 412 412 The exclusive paradigm for the probabilistic treatment involves the use of the probability of deposition in modifying the deposition flux. For the simultaneous paradigm the probability of deposition is set to 1. 4. The silt critical shear stresses for mobility, csi, are determined by size class based on Equation 4-55. 5. For the use of mean variables th e silt bed exchange logic is: a. If the bottom shear stress is less than the threshold shear stress (b < csi) then: i. The probability of deposition is computed as: 1b Di csiP ii. If applying simultaneous erosi on and deposition, then set PDi = 1 iii. If the option for no deposition is chosen, set PDi = 0 iv. The deposition velocity is defined as SETS = wsi PDi. The silt settling velocity wsn is corrected for hindered settling. v. The erosion flux is set to zero. b. If the bed shear stress is larger than the threshold shear stress (b > csi) then: i. The normalized excess shear stress is computed as: 1b ei csiP ii. If the simultaneous option for erosion and deposition is selected then the deposition velocity is set to the settling velocity for this silt class, with a correction for hindered settling. (SETS = wsi) iii. If the exclusive erosion or deposition paradigm is selected, set SETS = 0. iv. The erosion flux is calculated as ENTR = Cbs wsi fi Pei c. For a probabilistic treatment for silt size classes: i. The probability of deposition is calculated from a numerical integration of the two probability curves to yield PDi = P (b < csi) based on Equation 4-76. ii. If applying simultaneous erosi on and deposition, then set PDi = 1 iii. If the option for no deposition is chosen, set PDi = 0 iv. The deposition velocity is defined as: SETS = wsi PDi. PAGE 413 413 v. The expected value of the normalized excess shear stress, Pei, is computed as the numerical integration of the double integral of the excess shear stress using the probability distribution functi ons for the bottom shear stress and the critical shear stress for mobility of this silt class. (Equation 4-76) vi. The erosion flux is computed as: ENTR = Cbs wsi fi Pei 6. The erosion flux terms are checked against the mass of material for each size class in the bed plus the depositional flux for the size class for the case of simultaneous erosion and deposition. If the computed flux exceeds the available sediment it is limited to that mass divided by the time step. a. EROS(i) = MIN{EROS(i), mass in bed of class i/ t + DEP(i)}. b. EROSG(i) = MIN{EROSG(i), mass in bed of class i/ t + DEPG(i)}. c. 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Emery, was an employee of the University of Florida for several decades as a technician in the Botany Department. He grew up in Jacksonville, Lake Worth, Tampa and, Miami, Florida. He graduated from Miami No rland High School in 1968. He graduated from Georgia Institute of Technology in 1972 with a Bach elor of Civil Engineering. He received a Master of Engineering degree in 1973 from the University of Florida in Coastal and Oceanographic Engineering. On March 16, 1974, he married Linda Marie Arlotta, the love of his life. He extended his gradua te studies in 1979 through 1980 at the University of Miami in Ocean Engineering. He returned to the Univer sity of Florida in 2000 to continue graduate studies in Civil and Coastal Engineering. He has been employed by the U. S. Army Engineer Research and Development Center (formerly known as the U.S. Army Corps of En gineers Waterways Experiment Station) in Vicksburg, Mississippi since 1974. He is currently Group Lead er, Long Waves Group, Estuarine Engineering Branch, Coastal and Hydraulics Laboratory. |