Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UF00075487/00001
## Material Information- Title:
- Vertical structure of estuarine fine sediment suspensions
- Series Title:
- UFLCOELTR-
- Creator:
- Ross, Mark Allen
University of Florida -- Coastal and Oceanographic Engineering Dept - Place of Publication:
- Gainesville Fla
- Publisher:
- Coastal and Oceanographic Engineering Dept., University of Florida
- Publication Date:
- 1988
- Language:
- English
- Physical Description:
- xix, 187 p. : ; 28 cm.
## Subjects- Subjects / Keywords:
- Estuaries ( jstor )
Modeling ( jstor ) Mud ( jstor ) Sediment transport ( jstor ) Sediments ( jstor ) Shear stress ( jstor ) Suspensions ( jstor ) Velocity ( jstor ) Viscosity ( jstor ) Waves ( jstor ) Civil Engineering thesis Ph. D Dissertations, Academic -- Civil Engineering -- UF Estuarine sediments ( lcsh ) Sedimentation and deposisiton ( lcsh ) Sediments (Geology) -- Analysis ( lcsh ) City of Gainesville ( local ) - Genre:
- bibliography ( marcgt )
technical report ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (p. 177-187).
- Funding:
- Sponsored by U.S. Army Engineers Waterways Experiment Station.
- General Note:
- Originally presented as the author's thesis.
- Statement of Responsibility:
- by Mark Allen Ross.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Mark Allen Ross. 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.
- Resource Identifier:
- 19294211 ( OCLC )
## UFDC Membership |

Full Text |

UFL/COEL TR/079 VERTICAL STRUCTURE OF ESTUARINE FINE SEDIMENT SUSPENSIONS
BY MARK ALLEN ROSS 1988 I -A REPORT DOCUMENTATION PAGE 1. Report No. 2. 3. Recipient's Accession No. 4. Title eod Subtitle 5. Report Date VERTICAL STRUCTURE OF ESTUARINE FINE SEDIMENT June, 1988 SUSPENSIONS 6. 7. Author(s) 8. Performing Organization Report No. Mark Allen Ross FL/COEL-TR/079 9. Performing Organization Name and Address i0. Project/Tsnk/Work Unit No. Coastal and Oceanographic Engineering Department University of Florida 11. Contract or crant No. 336 Weil Hall DACW 39-87-P-1064 Gainesville, Florida 32611 13. Type of Report 12. Sponsoring Organization Name and Address Technical U.S. Army Engineers Waterways Experiment Station Vicksburg, MS 39180-0631 14. 15. Supplementary Notes 16. Abstract Fine sediment suspension concentrations in estuaries vary with depth depending on sediment settling and mixing processes, which are in turn dependent on the turbulent flow field and the type of sediment. Two important phenomena, fluid mud and lutoclines, are characteristic of high concentration suspensions. Understanding the significance of these phenomena is of paramount importance to quantifying the mixing process and the rate of material transport advected with the prevailing currents. This research investigated the physical characteristics (vertical structure) of estuarine fine sediment suspension profiles within a comprehensive descriptive framework. Suspension related mechanisms of erosion, entrainment, diffusion (in the presence of buoyancy stabilization), advection, settling and deposition were examined in this context. A vertical mass transport model developed from functional relationships between the above processes was used to explain some of the important physical characteristics. Lutoclines, sharp steps (gradients) in the concentration profile, are regions where the local mixing rate is minimal. The mechanisms for their formation have been shown to be the non-linear relationships between 1) vertical diffusion and concentration gradient and 2) vertical settling and concentration. The effect of sediment is to furtuer stabilize the lutocline layer thereby making it much more persistent in high energy environments than other pychnocline (e.g., haloclines). Application of the vertical transport model to data from settling column tests, wave flume resuspension tests, and estuarine field investigations provided reasonable agreement for lutocline dynamics. Continued17. Originator's Key Words 18. Availability Statement Cohesive sediment Fine sediments Concentrated suspensions Fluid mud Concentration profiles Sediment transport Estuaries Sedimentation 19. U. S. Security Classif. of the Report 20. U. S. Security Classif. of This page 21. No. of Pages 22. Price 206 Fluid mud, a near-bed high concentration layer with negligible structural integrity, results from high bed erosion or fluidization rates relative to upward entrainment fluxes and from rapid deposition. Sensitive pore pressure and total pressure measurements made in a laboratory flume have been used to demonstrate that waves, for example, provide one mechanism for fluid mud formation by rapid destruction of effective stress in the sediment bed. The upper interface of the fluid mud layer, by definition a lutocline, represents a local maximum in net downward settling flux (i.e., maximum settling minus diffusive flux). The fluid mud layer thus forms (and grows) from rapid deposition whenever the depositional flux at the bed exceeds the rate at which the sediment can develop effective stress (usually very low). Fluid mud has been shown to be either horizontally mobile or stationary depending on the depth of horizontal momentum diffusion vertically downward into the high concentrat ion layer. Fluid mud tends to occur over a density range between 1.01 1.1 glcm but due to the dependence on hydrodynamic action near the bed a precise difinition cannot be made on the basis of density alone. VERTICAL STRUCTURE OF ESTUARINE FINE SEDIMENT SUSPENSIONS BY MARK ALLEN ROSS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988 ACKNOWLEDGEMENTS My deepest and most heartfelt appreciation is extended to my chairman, Dr. Ashish J. Mehta, Professor of Coastal Engineering. In his capacity as advisor, educator and friend, he has shown me many lofty values by example. My cochairman, Dr. Robert G. Dean, Graduate Research Professor, an individual of unmatched character and inspiration, receives a lion's share of my gratitude. Special thanks are extended to my committee members and teachers, Drs. Dave Bloomquist, Wayne Huber, Jim Kirby, and Dan Spangler, who served so patiently and were responsible for many fruitful ideas. My Ecuadorian research partner and friend, Eduardo Cervantes, proved to be a source of much assistance, insight and camaraderie. Honorable mention must be made of the tireless crew at the Coastal Engineering Laboratory especially Vernon Sparkman and Chuck Broward for their technical assistance. Helen Twedell of the Coastal Engineering Archives also was very helpful. Perhaps, most importantly for me is the great sense of honor to which I have been imbued by my family. The result of love, patience, encouragement and support shown by my beautiful wife and parents. Their belief in me never faltered. Finally, financial support for this work was derived from a research grant extended by the U.S. Army Engineers, Waterways Experiment Station, Contract No. DACW 39-87--P-l064. Particularly, technical and administrative assistance and input provided by Allen Teeter is gratefully acknowledged. TABLE OF CONTENTS page ACKNOWLEDGEMENTS .........................ii LIST OF TABLES ..........................vi LIST OF FIGURES ........................viii LIST OF SYMBOLS.........................xii ABSTRACT .............................xviii CHAPTER I INTRODUCTION ........................1 1.1 Problem Significance .................1 1.2 Objective and Scope. .................4 1.3 Outline of Presentation. ...............5 2 VERTICAL STRUCTURE OF SUSPENSIONS. ...........8 2.1 Introduction ....................8 2.2 Typical Concentration Profile .. ..........9 2.3 Problems Related to Defining The Bed .........12 2.3.1 Bed Formation Concepts. ...........13 2.3.2 Effective Stress .. .............16 2.4 Fluid Mud ......................20 2.4.1 Stationary Fluid Mud. ............22 2.4.2 Mobile Fluid Mud .. .............27 2.5 Lutoclines.....................30 3 TRANSPORT CONSIDERATIONS .. ...............34 3.1 Introduction....................34 3.2 Mass Conservation Equation .............35 3.3 Diffusive Transport ..................39 3.3.1 Turbulent Diffusion .............39 3.3.2 Gravitational Stabilization ..........43 3.4 Settling......................49 3.4.1 Free Settling. ...............50 3.4.2 Flocculation Settling ............52 3.4.3 Hindered Settling. .............54 3.5 Vertical Bed Fluxes... .. .. 3.5.1 Bed Erosion.... .. .. 3.5.2 Deposition... .. .. .. 3.6 Fluid Mud Entrainment.. .. .. 3.7 Horizontal Fluid Mud Transport 4 LABORATORY EXPERIMENTS. ............ .. .. 4.1 Introduction. ............ .. .. 4.2 Flume Study. ............... .. . 4.2.1 Objectives. ............ .. .. 4.2.2 Mud Characterization...... .. .. . 4.2.3 Equipment, Facilities and Techniques. 4.2.4 Summary of Test Conditions... .. .. 4.2.5 Results. ........... .. .. 4.2.6 Discussion. ............ .. .. 4.3 Settling Column Tests..... .. .. .. .. 4.3.1 Historical Approaches.... .. .. ... 4.3.2 Concentration Profile Approach .... 5 MODELING RESULTS AND DISCUSSION.... .. .. 5.1 Introduction. .......... .. .. 5.2 Settling. ........... .. .. 5.2.1 Quiescent Settling... .. ... 5.2.2 Turbulence-Enhanced Settling. 5.3 Wave Resuspension. ............... 5.4 Lutocline Evolution in Severn Estuary. 5.5 Fluid Mud Transport.... .. .. .. . 5.5.1 Wave Tank Fluid Mud Transport. 5.5.2 Avon River Fluid Mud Transport. 6 CONCLUSIONS AND RECOMMENDATIONS.... .. .. 6.1 Conclusions. ............ .. .. 6.2 Recommendations..... .. .. .. .. APPENDIX A DIMENSIONAL ANALYSIS OF TRANSPORT EQUATION. B DATA ON WAVE RESUSPENSION TESTS.... .. .. C MODEL SOURCE CODE. .......... .. .. REFERENCES. .................. .. . .113 .113 .113 .114 .126 .127 .130 .137 138 142 147 . .147 . .153 . .156 .159 .169 .177 . 76 . 76 76 . 76 77 78 . .85 . .86 . 99 . .100 . .101 . .103 LIST OF TABLES Table Fag e 2-1 Fluid Mud Definition by Density/Concentration. .....21 3-1 Summary of Coefficient Values for Turbulent Vertical Diffusion of Momentum in Continuously Stratified Flow 46 A-i Wave Data (Period, Length, Height and MWS Elevation), Run I..........................159 A-2 Visual Bed Elevations (cm), Run 1. ..........159 A-3 Wave-Averaged Bed Pressures (kPa), Run 1 .. ......160 A-4 Dynamic Pressure Amplitudes (0.1 kPa), Run I .. .....160 A-5 Sediment Bed Concentrations (gil), Run 1 .. ......160 A-6 Sediment Concentrations Station A (g/l), Run 1 .. ....161 A-7 Sediment Concentrations Station B (g/l), Run I .. ....161 A-8 Sediment Concentrations Station C (g/l), Run 1 .. ....161 A-9 Sediment Concentrations Station D (g/l), Run 1 .. ....161 A-10 Sediment Concentrations Station E (g/l), Run 1 .. ....162 A-il Wave Data (Period, Length, Height and MWS Elevation), Run 2..........................162 A-12 Visual Bed Elevations (cm), Run 2. ..........163 A-13 Wave-Averaged Bed Pressures (kPa), Run 2 .. .......164 A-14 Dynamic Pressure Amplitudes (0.1 kPa), Run 2 .. .....165 A-i5 Sediment Bed Concentrations (g/1), Run 2 .. .......165 A-16 Sediment Concentrations Station A (g/l), Run 2 .. ....166 A-i7 Sediment Concentrations Station B (g/l), Run 2 .. ....166 vi A-18 Sediment Concentrations Station C (g/1), Run 2 .. ....167 A-19 Sediment Concentrations Station D (g/1), Run 2 .. ....167 A-20 Sediment Concentrations Station E (g/1), Run 2 .. ....168 LIST OF FIGURES Figure Bag e 2-1 Typical Instantaneous Concentration and Velocity Profiles in High Concentration Estuarine Environments 10 2-2 Schematic Representation of Bed Formation Process ..14 2-3 Bed Formation Process According to Imai (1981). ......15 2-4 Definition Sketch of Bed Stress Terminology .. .....17 2-5 Effective Stress Profiles in a Settling/Consolidation Test (reprinted with permission from Been and Sills, 1986)...........................19 2-6 Mud Dynamic Viscosity Variation with Concentration 24 2-7 Bingham Yield Strength Variation with Concentration ..27 2-8 Settling Velocity Variation with Concentration Severn Estuary Mud (adapted from Mehta, 1986) .......28 2-9 Vertical Settling Flux Variation with Concentration (reprinted with permission from Ross et al., 1987) 29 2-10 Typical Suspended Concentration Profile Showing Multiple Lutocline Stability Over 10 min. Period (Kirby, 1986) ......................32 3-1 Diffusion Flux vs. Concentration Gradient .........49 3-2 Ratio C/C0 of Instantaneous to Initial Suspended Sediment Concentration Versus Time for Kaolinite in Distilled Water (after Mehta, 1973).................61 3-3 Simplified Description of Density Stratified Entrainment (after Narimousa and Fernando, 1987) 64 4-1 Grain Size Distribution of Hillsborough Bay Mud 78 4-2 Flume Configuration....................80 viii 4-3 Example of Pressure Gage Calibration. ..........83 4-4 Example of Wave Gage Calibration. ............84 4-5 Suspended Sediment Siphon Sampler ............85 4-6 Wave-Average Bed Pressures at Various Times for Run 1 87 4-7 Wave-Average Bed Pressures at Various Times for Run 2 88 4-8 Temporal Response of Effective Stress for Run 1 90 4-9 Temporal Response of Effective Stress for Run 2 ... 90 4-10 Structural and Visual Bed Elevations for Run 1. .. ...91 4-Il Structural and Visual Bed Elevations for Run 2. .. ...91 4-12 Concentration Versus I Pa Effective Stress Elevation. 92 4-13 Bed Concentration Variation With Time ..........94 4-14 Visual Bed Elevation Variation With Time for Run 1 .. 95 4-15 Visual Bed Elevation Variation With Time for Run 2 .. 95 4-16 Bed Dynamic Pressure Amplitudes With Time for Run 1 .96 4-17 Bed Dynamic Pressure Amplitudes With Time for Run 2 .96 4-18 Concentration Profiles at Station C for Run 1 .. ....98 4-19 Concentration Profiles at Station C for Run 2 .. ....98 4-20 Local Mean Settling Velocity as a Function of Time for Bentonite Clay and Alum in Water (adapted from Mclaughlin, 1958) ...................105 4-21 Scale Drawing of Settling Column .. ..........107 4-22 Grid Index used in the Settling Velocity Calculation Program..........................10 4-23 Settling Velocity Variation with Concentration of Tampa Bay Mud......................110 5-1 Settling Velocity and Flux Versus Concentration for Tampa Bay Mud ...................115 5-2 Model Simulated vs. Measured Settling Column Concentrations Initial Concentration, Co= 1 g/l 119 5-3 Model Simulated vs. Measured Settling Column Concentrations Initial Concentration, Co= 2 g/l 5-4 Model Simulated vs. Measured Settling Column Concentrations Initial Concentration, Co= 4 g/l 5-5 Model Simulated vs. Measured Settling Column Concentrations Initial Concentration, Co= 5.5 g/l 5-6 Model Simulated vs. Measured Settling Column Concentrations Initial Concentration, Co= 7 g/l 5-7 Model Simulated vs. Measured Settling Column Concentrations Initial Concentration, Co= 8 g/l 5-8 Model Simulated vs. Measured Settling Column Concentrations Initial Concentration, Co= 12 g/l. 5-9 Model Simulated vs. Measured Settling Column Concentrations Initial Concentration, Co= 17 g/l. 5-10 Conceptual Model of Concentration "Thinning" in Low Concentration Flocculation Settling.... .. .. ... 5-li Conceptual Model for Constant Settling in Moderate Concentration Range of Flocculation Settling .... 5-12 Simulated Field Settling of Parrett Estuary Suspensions. .................. .. 5-13 Model Simulated Versus Measured Concentrations -Run 1...................... .. 5-14 Model Simulated Versus Measured Concentrations -Run 2...................... .. 5-15 Model Simulated and Measured Lutoclines -- Severn Estuary...................... .. 5-16 Model Simulated and Measured (Kirby, 1986) Concentration Profiles 0900 hrs......... .. .. .. 5-17 Model Simulated and Measured (Kirby, 1986) Concentration Profiles 1100 hrs. ........ .. . 5-18 Model Simulated and Measured (Kirby, 1986) Concentration Profiles 1300 hrs. ........ .. . 5-19 Model Simulated and Measured (Kirby, 1986) Concentration Profiles 1530 hrs......... .. .. .. .119 .120 .120 .121 .121 .122 .122 .124 .124 .127 .129 .129 .133 .134 .134 .135 135 5-20 Model Simulated and Measured (Kirby, 1986) Concentration Profiles 1700 hrs. ...........136 5-21 Normalized Velocity Profiles -- Severn Estuary (data from Kirby, 1986) .................137 5-22 Total Fluid Mud Transport in Five Minutes -- Run 1 139 5-23 Non-Dimensional Bed Shear Stress (t;~) versus Wave Steepness (H/L0) (reprinted with permission from Dean, 1987)..........................140 5-24 Calculated Fluid Mud Velocity Profile -- Run I .. ....142 5-25 Measured Fluid Mud Concentration, Velocity and Horizontal Flux -- Avon River (data from Kendrick and Derbyshire, 1985) ...................144 5-26 Calculated and Measured Horizontal Fluid Mud Velocities ........................146 LIST OF SYMBOLS Symbol b Buoyancy jump across density interface b Body force per unit mass tensor C Sediment suspension concentration (mass/unit volume) Ca Concentration at upper fluid mud interface Cb Concentration at mobile/stationary fluid mud interface Cc Concentration at bed surface Cd Drag coefficient for sphere fall velocity Ceq Equilibrium concentration during deposition Ch Interference settling velocity concentration ChT Hindered settling (flux) concentration Ci Concentration of class i for deposition Cm Characteristic maximum concentration Css Steady state concentration after deposition CT Total concentration (sum of components) CO Initial concentration for settling; deposition C1 Cohesive (class) sediment concentration C2 Non-cohesive (class) sediment concentration C Time mean concentration C' Instantaneous concentration component about mean C' Non-dimensional concentration C/Cm d Equivalent spherical diameter of sediment grain D Molecular diffusivity d50 Sediment grain size diameter of 50% greater than fraction Eij Turbulent diffusivity components in i,j direction Em Turbulent momentum diffusion rate (eddy viscosity) E, Entrainment coefficient, ue/u* f Darcy-Weisbach friction factor Fb Vertical sediment bed flux (Fe+Fp) Fd Vertical sediment flux from diffusion Fe Vertical sediment flux from erosion Fp Vertical sediment flux at the bed from deposition Fpi Class i vertical sediment flux from deposition Fs Vertical sediment flux from settling g Acceleration of gravity h Water depth H Wave height Hb Breaking wave height Ho Deep water wave height k Wave number (271/L) K Turbulent mixing tensor Kn Local neutral mixing rate Ks Local mixing rate in presence of stratification Kx,y,z Turbulent mixing components (cartesian) Kx,y,z' Non-dimensional turbulent mixing components (cartesian) kI Flocculation settling velocity constant k2 Hindered settling velocity constant xiii 1 Mixing length scale of turbulence L Lutocline layer; wave length Lo Deep water wave length m Mass flux of sediment across bed boundary nI Flocculation settling velocity constant n2 Hindered settling velocity constant P Pressure variable used in the horizontal momentum equation P Relative Probability for deposition rate expression Pi Relative Probability for deposition (class i) Ph Hydrostatic pressure Ppw Pore water pressure P' Non-dimensional pressure P/yH q Mass flux vector R Reynolds' number of sediment grain (wsd/v) Ri Gradient Richardson number Ri* Bulk Richardson number (bh/u ) Riu Richardson number based on average velocity Rw Wave Reynolds number Rt Shear stress ratio, ty/to Sc Turbulent Schmidt number t Time variable to Characteristic time scale to Non-dimensional time t/to T Wave period T Stress tensor u Velocity component in x-direction xiv uo Characteristic velocity scale U Imposed velocity on the sheared turbid layer ue Entrainment rate ub Maximum near-bed orbital velocity ue Entrainment rate (dh/dt) Au velocity jump across stratified layer u* Friction velocity (/to/p) u Time mean velocity u' Non-dimensional velocity u/uo u' Instantaneous velocity component about mean v Velocity component in y-direction w Velocity component in z-direction ws Sediment settling velocity ws' Non-dimensional settling velocity Wsm Characteristic maximum settling velocity wso Richardson-Zaki reference settling velocity Wsol Stokes' settling velocity Wso2 Reference settling velocity for average floc size x Longitudinal (horizontal) cartesian coordinate direction x' Non-dimensional horizontal direction x/L y Lateral cartesian coordinate direction z Elevation variable (positive upwards) z' Non-dimensional veritical direction Za Upper fluid mud interface elevation Zb Mobile/stationary fluid mud interface zc Bingham plastic yield elevation xv Zc Bed elevation a Wave diffusivity constant as r,2 /.I ay Yield strength calculation constant a Viscosity/concentration constant a' Munk and Anderson constant ala2 Erosion rate constants aS Viscosity ratio, P2/4l Settling velocity constant Settling velocity constant for sediment class i Pe Exponential diffusivity constant (mass diffusivity) OH Holtzman constant OMA Munk and Anderson constant OOR Odd and Rodger constant ORM Rossby and Montgomery constant Oy Yield strength calculation constant OP Viscosity/concentration constant of Munk and Anderson constant mass diffusivity 08 Coefficient used in fluid mud calculations 8 Intermediate entrainment layer sc Similarity variable (Zc(t)/2V-t) 8' Sclfas &fm Mobile fluid mud thickness 6i Upper entrainment layer thickness ss Shear layer thickness; Similarity variable (z/2vv/7) Fluid shear rate (au/az) K von Karman constant (0.4) xvi p Density of water Pb Bulk density of suspension Po Fluid reference density for stratification Ps Granular density of dry sediment Pw Density of suspension fluid (water) PI Non-dimensional density p/pm PwP Dynamic viscosity of suspension fluid (water) I'm Dynamic viscosity of mud suspension Vm Kinematic viscosity of mud suspension (pm/p) VI Non-dimensional kinematic viscosity T Odd and Rodger peak gradient Richardson number C Munk and Anderson constant mass diffusivity Co Erosion rate constant a' Bed effective stress o Total stress; wave frequency (21T/T) Tb Applied (time-mean) bed shear stress tbm Critical bed shear stress for partial deposition tcd Critical bed shear stress for total deposition to Bed shear stress 'CS Bed shear strength for erosion 'Cxz Shear stress component acting in x-direction on z-face ty Yield strength of bed deposits x Log average of sediment concentration V Vector operator xvii Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy VERTICAL STRUCTURE OF ESTUARINE FINE SEDIMENT SUSPENSIONS By Mark Allen Ross August 1988 Chairman: Ashish J. Mehta Major Department: Civil Engineering Fine sediment suspension concentrations in estuaries vary with depth depending on sediment settling and mixing processes, which are in turn dependent on the turbulent flow field and the type of sediment. Two important phenomena, fluid mud and lutoclines, are characteristic of high concentration suspensions. Understanding the physical significance of these phenomena is of paramount importance to quantifying the mixing process and the rate of material transport advected with the prevailing currents. This research investigated the physical characteristics (vertical structure) of estuarine fine sediment suspension profiles within a comprehensive descriptive framework. Suspension related mechanisms of erosion, entrainment, diffusion (in the presence of buoyancy stabilization), advection, settling and deposition were examined in this context. A vertical mass transport model developed from functional relationships between the above processes was used to explain some of the important physical characteristics. Lutoclines, sharp steps (gradients) in the concentration profile, are regions where the local mixing rate is minimal. The mechanisms for their formation have been shown to be the non-linear relationships between 1) vertical diffusion and concentration gradient and 2) vertical xvii i settling and concentration. The effect of sediment settling is to further stabilize the lutocline layer thereby making it much more persistent in high energy environments than other pychnoclines (e.g., haloclines). Application of the vertical transport model to data from settling column tests, wave flume resuspension tests, and estuarine field investigations provided reasonable predictive agreement for lutocline dynamics. Fluid mud, a near-bed, high concentration layer with negligible structural integrity, results from high bed erosion or fluidization rates relative to upward entrainment fluxes and from rapid deposition. Sensitive pore pressure and total pressure measurements made in a laboratory flume have been used to demonstrate that waves, for example, provide one mechanism for fluid mud formation by rapid destruction of effective stress in the sediment bed. The upper interface of the fluid mud layer, by definition a lutocline, represents a local maximum in net downward settling flux (i.e., maximum settling minus diffusive flux). The fluid mud layer thus forms (and grows) from rapid deposition whenever the depositional flux at the bed exceeds the rate at which the sediment can develop effective stress (usually very low). Fluid mud has been shown to be either horizontally mobile or stationary depending on the depth of horizontal momentum diffusion vertically downward into the high concentration layer. Fluid mud tends to occur over a density range between 1.01 1.1 g/cm3 but due to the dependence on hydrodynamic action near the bed a precise definition cannot be made on the basis of density alone. xix CHAPTER 1 INTRODUCTION 1.1 Problem Significance Fine-grained, cohesive sediment is transported in suspension from fluvial and marine sources to depositional environments including navigation channels and harbors. This sediment affects water quality by transport of sorbed nutrients (or pollutants) and light penetration (Hayter, 1983). Shoaling is often one other critical issue. In the continental United States alone, the cost of maintenance dredging of coastal waterways, including estuarine ports and harbors, is approximately one-half billion dollars per year (Krone, 1987). Estimates of contaminant removal or dredging requirements are dependent upon a knowledge of the rates of horizontal transport of the suspended material over periods ranging from days to years. The accuracy of predictions, typically via numerical solutions of the sediment mass transport equation, is therefore strongly contingent upon an understanding of the structure of the vertical profile of sediment concentration and interaction with the turbulent flow field. Present day modeling of cohesive sediment transport is limited by knowledge of the fundamental transport processes of erosion, entrainment, settling, deposition and consolidation of these sediments. In particular, the dynamics of estuaries with relatively high concentration suspensions typical of macro-tidal (tidal range > 4 m) environments are 1 poorly understood (Parker, 1987). In this context engineers and other scientists are beginning to deal with the important question of fluid mud, loosely defined as a high concentration slurry transported in the form of a relatively thin suspension layer near the bed by the prevailing currents. At present, there are difficulties associated with measuring the slurry concentration and transport velocity. The result is that large errors often occur in calculating the associated horizontal flux of sediment transport over the water column. Fluid muds also occur in meso- (2-4 m) and micro-tidal (< 2 m) estuaries and along the open coasts where waves play a more important role than in macro-tidal environments. Wells and Kemp (1986) observed that waves traveling over nearshore mud shoals principally acted as an agent for softening and fluidizing the muddy bed. Maa and Mehta (1987) made similar observations in laboratory flume tests. In nearshore areas waves can thus significantly assist currents in transporting fluidized material to sites prone to sedimentation. Consequently, in micro-tidal waters the generation and transport of fluid mud is far more episodic than under macro-tidal conditions. Understanding the dynamics of fluid mud is central to the issue of understanding the response of the vertical concentration profile to hydrodynamic forcing by currents and waves. Unlike the boundary of beds composed of cohesionless material (e.g., sandy beds), the cohesive bed boundary is often poorly defined as it is not evident, e.g., from echo sounder data, at what depth the near-bed suspension ends and the bed begins. Parker (1986) noted ambiguities when lead lines, echo sounders or nuclear transmission or backscatter gauges are used to identify the 3 bed. In fact, Ross et al. (1987) noted that due to the dynamic nature of the cohesive bed boundary which responds significantly to hydrodynamic forcing, the density of the suspension by itself cannot be used either to identify the cohesive bed boundary or the fluid mud layer which occurs immediately above this boundary. An understanding of the interaction between the concentration (or density) profile with the flow field is critically important. Kirby (1986) recently published a summary of extensive field observations made in the Severn estuary, a macro-tidal estuary (maximum tidal range 14.8 m) on the west coast of England. Large mass transport rates via fluid mud generation regularly occur in this estuary. The dynamic interaction between the concentration field and flow field are further complicated by the extremely high concentrations. Surface concentrations reach 1000 mg/l easily, which may be compared with 20 mg/l in Florida's coastal waters (Mehta et al., 1984). A significant observation in the Severn was the generation of rather sharp gradients in concentration termed lutoclines, which rise and fall through the water column depending upon the flow condition. According to Kirby (1986), lutoclines, which are analogous to other types of pycnoclines, e.g., haloclines, seem to occur where the suspension concentration exceeds 500 mg/l. However, they differ from other pycnoclines by the added process of sediment settling. Sediment settling further supports stabilization and resulting high density gradients. For this reason lutoclines are much more persistent than for example haloclines in high energy environments. An example of a lutocline is the upper level of fluid mud within which concentrations can typically exceed 10,000 mg/l. 4 Above this level, lutoclines often show up as multiple "steps," which represent local complex imbalances between diffusive and settling fluxes. Kirby (1986) observed that lutoclines often are not simulated properly by numerical models with resulting errors in the estimates for the rates of mass transport. The aforementioned issues illustrate the strong need to examine the entire question of the vertical structure of concentration and its interaction with the flow field within a comprehensive framework. An attempt is made in this thesis to approach the problem via analysis of laboratory and field measurements within a descriptive framework for the vertical concentration structure. New definitions are proposed and the dynamics of the concentration profile are adduced through relatively simple mathematical models which are verified by laboratory and field data. The objectives and scope are accordingly as follows. 1.2 Objective and Scope The objectives of this study were to 1. Define the physical characteristics of fine sediment suspension profiles in estuaries including lutoclines, fluid mud, and the cohesive bed within a comprehensive descriptive framework. 2. Determine the important physical mechanisms and processes which influence these characteristics. 3. Develop simple but useful qualitative and quantitative descriptions for these processes which could be used in a predictive capacity to model suspensions in the prototype environment. To meet these objectives the scope of this research was as follows: 1. Laboratory tests were conducted, using natural estuarine 5 sediment, to measure the parameters important to cohesive bed and suspension profile definitions. 2. For the simple vertical structure model development, only vertical transport fluxes were considered. Analysis of turbulent diffusion was based on classical mixing length approximations and gradient Richardson number bouyancy stabilization relationships. Sediment settling velocity expressions were concentration dependent. 3 Horizontal transport in the fluid mud layer was calculated from consideration of momentum diffusion resulting from applied interfacial shear stress. 4. Verification of model applicability was limited to comparisons with selected field and laboratory data (e.g., time series concentration profiles). 1.3 Outline of Presentation The study is presented in the following order. Chapter 2 can be regarded as a description and definition chapter. Specific justification is presented for delineating processes influencing vertical suspended sediment structure. Physically based, qualitative definitions are given for lutoclines, stationary and mobile fluid mud layers, and bed elevation. This chapter also describes many of the complexities associated with defining the cohesive bed from theoretical and applied perspectives. Chapter 3 presents the theoretical development of the vertical transport and momentum diffusion models. For the transport model, the advection-diffusion equation is given and the individual terms are discussed. Entrainment, diffusion, settling and bed fluxes are addressed. For the fluid mud momentum diffusion model, theoretical formulations are presented with assumptions concerning rheological and temporal responses. 6 Chapter 4 presents the objectives, procedures and results of laboratory experiments with three specific themes: cohesive bed dynamics associated with wave-induced bed fluidization and delineation of the cohesive bed boundary; wave resuspension with emphasis on the evolution of the suspension profile with time; and settling velocity determination. The natural estuarine sediment used in the tests is characterized in Section 4.2.2. Historical approaches toward settling velocity determinations are discussed in the context of strengths and weaknesses. Section 4.3.2 presents an improved procedure for determining cohesive sediment settling velocity concentration relationships using settling columns. The application of the vertical transport model is presented in Chapter 5. An early attempt was made to progressively verify the individual routines in the model, before concurrent simulation. Thus, for example, the settling routine was first tested by reproducing quiescent (column) settling results (a nearly pure settling condition, see, for example, Yong and Elmonayeri, 1984, for diffusion in quiescent settling). Next, the diffusion, erosion and deposition routines were added and tested. Predictions of lutocline formations in field conditions together with fluid mud layer development in a wave-tank illustrating the model's ability to handle sediment fronts (i.e., sharp concentration gradients) are also shown in Chapter 5. The fluid mud horizontal transport model results also are included in Chapter 5. Steady and unsteady simulations of wave-tank data (presented in Chapter 4) and field data (published by Kendrick and Derbyshire, 1985) are shown. 7 Conclusions, recommendations for future research and miscellaneous closing comments are given in Chapter 6. Appendix A presents a dimensional analysis to determine the important terms in the transport equation. Appendix B contains the tabulated laboratory data taken during wave resuspension tests. Appendix C is a printout of the l-D vertical transport model developed for this study. CHAPTER 2 VERTICAL STRUCTURE OF SUSPENSIONS 2.1 Introduction Suspended sediment concentration in estuaries varies greatly with depth, the highest concentrations being usually found nearest the bed. Simply stated, this variance is because gravitational flux (associated with settling) counteracts mixing and prevents the sediment from becoming uniformly mixed as is the case with neutrally buoyant or dissolved constituents. In an equilibrium profile (profile not changing with time) the vertical flux associated with settling is everywhere equal to the vertical flux associated with upward diffusion (typically turbulent mixing). For sediment with a constant settling velocity and mixing based on a Prandtl/von Karman mixing length approximation, analytical solutions for the concentration profile follow the classical works of O'Brien (1933) and Rouse (1937). In a fully developed turbulent flow in the absence of significant density gradients the mixing rate, which is directly proportional to the eddy scale of turbulence, is smallest near the bed and increases upward reaching a maximum approximately at middepth (Schlichting, 1979). However, sediment in suspension can greatly increase the bulk density of the water. If high concentration (density) gradients develop, turbulent mixing will be greatly damped locally (Fischer et al., 1979). This has been well documented for stratified flows associated with dissolved salt- and temperature-induced density 9 gradients (Turner, 1973), but has been generally overlooked by classical solutions of vertical sediment transport (Rouse, 1937; Raudkivi, 1967). This chapter presents the physically based definitions for the vertical structure (or vertical characteristics) of fine-grained suspended sediment profiles. Much of the terminology and descriptions used have been liberally applied in broad contexts in previous related and unrelated studies. Most were disparate in their objective. The following pages will help clarify the usage. 2.2 Typical Concentration Profile Figure 2.1 shows a typical instantaneous concentration profile as might be observed in a high sediment load environment such as the Thames River (UK), San Francisco Bay or the Severn Estuary. While the values are assumed, they are representative of those commonly reported in the literature (Parker and Kirby, 1979). It is noted that there is a 4-5 order of magnitude range in concentration from water surface to bed surface. While most sediment transport models focus primarily on calibrations of the upper water column concentrations, the significance of neglecting the near bed layers should be obvious, but will be shown in detail. The largest layer is the mobile suspension layer which extends down to reference level, Za- This is the layer that is most often turbulent. It is also generally, dominated by pressure gradient driven flow associated with water surface elevation gradients resulting from tides and freshwater discharge. Concentrations in the upper mobile suspension 1'0 CONCENTRATION, C (mg/i) 1 2 3 4 5 6 10 10 10 10 10 10 U N < Mobile L LL Suspension 4 Lutocline4 Layer C,, im+ = 6 Mobile Fluid Mud WU ZB Stationary "Fluid" Mud z c 8 Bed 0 0.25 0.50 0.75 1.00 1.25 VELOCITY, u(m/s) Figure 2-1. Typical Instantaneous Concentration and Velocity Profiles in High Concentration Estuarine Environments 11 layer are usually 1-1000 mg/l but in rare cases exceed 10,000 mg/l (in the lowest portions) during extreme tides or storm conditions (Parker and Kirby, 1979). At various levels in the mobile suspension layer there can exist sharp increases in concentration which result from and further support local minima in mixing and upward vertical diffusion. These are termed lutocline layers which are one form of pycnoclines (regions of sharp density gradients). There can be multiple lutocline layers but more than 2-3 is rare. Such multiple layering in salinity or thermal structure is called finestructure (Posmentier, 1977). Below Za there is a sharp increase in concentration above 10,000 mg/l to 100-300 g/l. This is the so-called "fluid mud" layer defined in Section 2.4. Thus, Za represents a lutocline between upper column mobile suspensions and near-bed fluid mud. This is often mistaken as the bed on echo sounder records (Kirby, 1986). Depending on the rheological properties of the mud, the magnitude and duration of the applied interfacial (lutocline) shear stress, and/or the internal pressure gradients, a portion of this fluid mud layer is mobilized to flow in a direction with the applied force(s). The interface between the mobile and stationary mud suspensions is labeled reference level Zb. The symbol, Zb, will not necessarily be identifiable from concentration profiles but instead must be identified from accurate measurements of the velocity profile. Below the fluid mud layer at reference level Zc there exists a definable sediment interface below which the sediment exibits bed properties based on classical soil mechanical definitions. This is the cohesive bed elevation, above which only suspension occurs (discussed 12 in Section 2.3). Strictly speaking, the stationary fluid mud layer (ZbZc) may not necessarily behave as a fluid (i.e., not supporting shear stresses), but since it fits the general definition for fluid mud (i.e., near-bed, high-concentration layer) the terminology is nevertheless retained. A typical velocity profile is also shown in Figure 2-1 for reference. It is shown to be of almost logarithmic form in the mobile suspension layer--indicative of turbulent flow. Near the fluid mud layer turbulence is dampened out and there is a transition layer which, proceeding down with depth, gives way to a shear flow viscous layer. This is analogous to stratified flows of salts (Yih, 1980; Narimousa and Fernando, 1987) and is described further in Chapter 3. 2.3 Problems Related to Defining the Bed When trying to determine the bed elevation, Zc, to do so on the basis of concentration only is imprecise. As pointed out by Sills and Elder (1986) and in this report, bed properties (i.e., development of effective stess as defined in Section 2.3.2) can exist in concentrations as low as 70-80 g/l, depending primarily on the fluid suspension and bed dynamics (stress, strain and strain rate) at any particular time. Thus, under field conditions, a precise identification of the cohesive bed interface would not be possible without dynamical data (e.g., measurements to determine effective stress). Bed definition is also dependent on previous formation conditions, wave and current actions and consolidation properties of the particular sediment. These phenomena provide justification for a brief discussion of bed formation and 13 consolidation concepts followed by a subsection (2.3.2) on the concept of bed definition related to effective stresses. 2.3.1 Bed Formation Concepts To characterize the process of bed formation in a laboratory or field setting it is important to distinguish the mode of deposition. According to Parchure and Mehta (1985), in the laboratory, bed formation can be in the form of a "placed" or "deposited" bed. Placed beds are those developing from high concentration slurries. Deposited beds result from lower concentration, particle by particle deposition. Placed beds, therefore, are more uniform vertically, whereas deposited beds are nonuniform and dewater relatively rapidly. The specific character of each bed type is most pronounced earliest after formation, decaying with time until the properties are nearly indistinguishable. Because of the time scales involved, placed beds are probably more typical of laboratory conditions; however, rapid fluid mud deposition in an estuary would have similar characteristics. In the field, an alternative to considering the bed based on depositional mode is to examine in detail the physics of mud deposition and bed formation. A schematic representation of the bed formation process is shown below (Figure 2-2). There are basically two mechanisms responsible for bed formation: sedimentation (deposition) and consolidation. Sedimentation can be defined as the process by which particles or masses of particles leave suspension and settle onto the bed under gravity. Consolidation in a fully saturated environment results from particle framework (mineral Dilute Susenio IConcentrated Susenio Decreasing Increasing Rate of Soft-Rapidly Concentration Vertical Consolidating Movement Mud II, 1 Consolidated Bed Figure 2-2. Schematic Representation of Bed Formation Process skeleton) deformation under applied stress. The applied mechanical forces can be either due to net negative buoyancy (self-weight) or imposed overburden (surcharge) loading. Imai (1981) gave a description and graphical model of the bed formation processes. Figure 2-3 shows this description. The flocculation stage in Figure 2-3 actually includes the complex process of particle destabilization by doublelayer suppression in the presence of available cations and subsequent aggregation by interparticle collision and cohesion. The floc formation process takes place under settling conditions as pointed out by Krone (1962). The settling zone shown in Figure 2-3 would be more appropriately labeled hindered (or zone) settling. No further discussion of settling will be given here as the settling process is discussed further in Chapter 3. Between times t - "-. inTerroce o .Se ttlI~).: Soil Formation Line > Floccul ation Zone 0) .'..'.::: ..'' I Zone ..: .B Consolid ation Zone C __________________________0 tI t2 Time Figure 2-3. Bed Formation Process According to Imai (1981) and t2, sediment flocs settle to form a soft bed. The bed is continually built up by continuous deposition of these flocs but simultaneously undergoes dewatering and consolidation. During this time, bed properties begin to change with depth due to particle rearrangement and larger floc breakdown (Krone, 1962). After the settling stage, consolidation continues and the bed slowly begins to "harden" as depth-variation in bed properties (i.e., density, effective stresses, etc.) become more pronounced. This self-weight consolidation approaches a steady state condition exponentially. Since it is not within the scope of this study to discuss the details of consolidation, it will suffice to conclude this section by stating that a vast amount of geotechnical literature on 16 sedimentation/consolidation theories is available. The pioneering work by Terzaghi (1923) using one-dimensional finite strain theory now has evolved into complex multi-dimensional, non-linear finite strain theories. The reader is directed to the paper by Schiffman et al. (1986), which presents a noteworthy historical, theoretical, and applied account of one-dimensional sedimentation and consolidation. 2.3.2 Effective Stress Given that the porous medium (the cohesive bed) is a two-phase system consisting of a deformable mineral skeleton filled with an incompressible liquid (water), the effective stress, a', is defined as the difference between the total stress, a, and the pore water pressure, Ppw, at any point: = Ppw (2.1) Empirically, it is found to be the controlling parameter in determining soil strain, deformation and strength (Schiffman et al., 1986). Classically, one type of soil failure is defined as a "quick" condition in which the effective stress tends toward zero (Sowers, 1976). Another important parameter is excess pore pressure, Au. This is the difference between actual pore water pressure, Ppw (e.g., as measured by a manometer), and hydrostatic pressure, Ph- Under dynamic conditions, if the sum of excess pore pressure, Au, and hydrostatic pressure, Ph, approaches the total stress, a, liquefaction occurs (Perloff and Baron, 1976), Au + o ~(Liquefaction) (.) Figure 2-4 is a definition sketch for these terms. Water Surf, Mobile Suspension Ph 0 Fluid Mud Surfh Fluid Mud > \/C"Bed Surft Wu WL PRESSURE Figure 2-4. Definition Sketch of Bed Stress Terminology In a non-fluidized sandy (or any large porosity) sediment bed the effective stress everywhere is non-zero. The total pressure is the integral of the density profile over depth (acted on by gravity) and the pore pressure is everywhere hydrostatic. For finer sediments which are much less permeable, pore pressures easily increase to above hydrostatic. In the upper bed where the pore water pressure is equal to the total pressure, the sediment is in suspension and the water bears the weight of the sediment (increased bouyancy through higher bulk densities). When Au + Ph 4 a (2.2). 18 the pore pressure drops below the total vertical stress, there is particle interaction. Thus, a weak structure forms that is able to support some of the weight of the sediment. Therefore, the development of effective stress provides a fundamental distinction between suspension and structural bed, i.e., 0, =0 ; Pp = o Suspension 0, > 0 ; Pp < a Bed (2.3). The elevation of the (structured) bed, Zc in Figure 2-1, therefore, should be based on the development of effective stress below this elevation. Within the bed, the interaction between sediment flocs provides a resistance to erosion due to frictional and electrochemical bonding. A reduction in effective stress, therefore, leads also to the reduction in the yield strength and the critical shear stress for erosion. No effective stress means no inter-aggregate contact or friction. This important distinction was pointed out by Sills and Elder (1986). Been (1980) and Been and Sills (1981) made extensive laboratory measurements of the development of effective stress in quiescent settling/consolidation of fine sediments. A representative plot of measured effective stress profile is shown in Figure 2-5. In their experiments on Combwich mud using different initial concentrations, no unique concentration was found at which effective stress developed. The concentration range over which structural development occurred was between 80 and 220 g/l depending on the initial Pressure plotted above hydrostatic E BOO BOO0 a Total stress E E Pore pressure 400- 400 0 0 1.0 1 2 0.2 0-4 0-6 0 10 1?2 Density Mg/rn3 Pressure kN/rn2 Initial density 1.09 Mvg/rn3 4%hour profile Figure 2-5. Effective Stress Profiles in a Settling/Consolidation Test (reprinted with permission from Been and Sills, 1986) concentration of the slurry. One significant observation was that effective stress existed always in concentrations greater than 220 g/l. This observation seems to imply that structural phase development is dependent on sedimentation rate especially in low concentration quiescent conditions. It is noted in the following sections that structural phase development is also dependent on hydrodynamic agitation. This dependence is demonstrated in laboratory tests of wave erosion as described in Chapters 4 and 5. 2.4 Fluid Mud As stated in Chapter 1, fluid mud is defined as a near-bed, high density, cohesive sediment suspension layer (Ross et al., 1987). In 20 areas with large bed slopes, fluid mud loosened by currents or waves can flow down the slope by gravitational forces similar to mudslides and debris flow on hillsides (Odd and Rodger, 1986). For this reason, navigation channels and basins are especially vulnerable to this type of sedimentation. Many investigators have identified fluid mud in terms of a range of bulk density (or concentration) of the sediment-fluid mixture, as noted in Table 2-1. It should be pointed out that these investigations were, in general, disparate in terms of their aims, dealing with field observations or laboratory tests. Nevertheless, there seems to be some agreement amongst the proposed densities initially suggesting perhaps an approximate range of 1.03 to 1.20 g/cm3 (concentration range of 10 to 320 g/1). To provide a quantitative definition for fluid mud based on a discrete concentration range is not possible because, as pointed out in the previous section, the effect is not simply dependent on concentration but on the flow conditions and sediment settling properties. Therefore, the values given in Table 2-1, without qualifying the particular flow conditions and sediment settling behavior under which the ranges apply, are not amenable to developing a general definition applicable in all cases. All that can be deduced from the tabulated data is that fluid mud seems to occur within a rather wide concentration range of between 3 and 500 g/l (two orders of magnitude). Fluid mud can form during rapid erosion or deposition. During erosion, if initially the erosion rate greatly exceeds the turbulent entrainment rate, i.e., the rate at which sediment is mixed by turbulence 21 into the upper column mobile suspension layer, the near-bed high concentration further dampens turbulent mixing and the near-bed Table 2-1. Fluid Mud Definition by Density/Concentration =Conversion between density and concentration density of 2.65 g/cm3. based on assumed sediment suspension can be stabilized as a stratified flow. This effect is often the case in wave erosion (Maa and Mehta, 1987). This is discussed in Chapter 3 and later shown in laboratory wave resuspension tests documented in Chapter 4. During deposition, if the sediment deposition flux exceeds the rate of pore fluid transport upward (dewatering rate of the suspension), dense near-bed suspensions are formed that grow upward and only slowly consolidate (see Section 2.3.1). As shown in Figure 2.1, fluid mud can occur as a mobile or stationary suspension. This distinction was first made by Parker and Kirby (1977). Both conditions are discussed in the following sections. Density/Conc. Range Investigator(s) Bulk Dejsity Concentration (g/cm ) x 10 (mg/l) Inglis and Allen (1957) 1.03 1.30 10 480 Krone (1962) 1.01 .,,a 10 170 Wells (1983) 1.03 1.30 50 480a Nichols (1985) 1.003 1.20 3 320 Kendrick and Derbyshire (1985) 1.12 1.25a 200 400 2.4.1 Stationary Fluid Mud Within the fluid mud layer there are typically two distinct regions separated by a level below which no horizontal motion takes place. In the definition sketch (Figure 2-1) this was elevation Zb. For instance, this elevation might be considered to be the applicable elevation of the bottom boundary condition for a horizontal transport model. However, this level is quite sensitive to the rheological response of the mud from imposed stress (e.g., the lower extent of the vertical momentum diffusion resulting from an applied horizontal shear stress at the upper fluid mud interface). For the purpose of describing why this layer exists and how it is differentiated from the mobile layer above, several simple arguments are presented here and are more formally posed in Chapter 3 (Section 3.7). It is possible to estimate the stationary layer elevation, Zb, by making several simplifying assumptions. As a first approach, analogy can be made between flow in the fluid mud layer and unsteady couette flow development beneath an infinite plate moving with a constant velocity after being started from rest. A shear stress results on the upper fluid mud interface, elevation Za in the definition sketch (Figure 2-1), because of an imposed velocity in the upper column (mobile suspension). Momentum diffusion then occurs over a finite thickness, &fmP in the fluid mud layer. For a constant kinematic viscosity, vm of the mud, the temporal response (for relative time t after imposing the shear stress) of the mobile fluid mud layer thickness, 8fm' is Sfm = (2.4) -M (2.4) where 06 is a constant (Eskinazi, 1968). However, this leads to calculations of layer thicknesses of many meters after periods of minutes--unreasonably large values. Also, the viscosity of these high concentration layers is not constant but concentration (and thus depth) dependent (Krone,1962). Another approach is to consider the slurry to have a concentration dependent dynamic viscosity, pm(C). The density of the bulk suspension, Pb, is given by a simple linear relation to the (mass/unit volume) concentration, C, as (2.5) Pb Pw + C (1 w) Ps where Pw is the density of the suspending fluid (water) and ps is the granular density of sediment (typically 2.65 g/cm3). A summary of empirical relationships for dynamic viscosity variation with concentration is shown in Figure 2-6. The trend, in the fluid mud range (10 C 200 g/I), seems to be of an exponential or polynomial form: i.e., aoC Pm Pw 0 Pm Pw( + OP CP ) (2.6a) (2. 6b) where pm is the dynamic viscosity of mud suspension, pw is the dynamic 100 80 Krone(1963) *60 L 40 S20 Engelund-Zhaohul(1984) V) 2/ (Bentonite) 0 0 Delft Hyd. Lab (1985) > 8 Wu 6 I. (Kaolinite) LU cc 2 0 100 200 300 400 500 SUSPENDED SEDIMENT CONCENTRATION (g/I) Figure 2-6. Mud Dynamic Viscosity Variation with Concentration viscosity of clear (free from suspension) water, C is the concentration I of suspended sediment, a a, a and 0 are empirical coefficients. Engelund and Zhaohui (1984) proposed a relationship of the form of Eqn. (2.6b) for kaolinite suspensions. They found @1/a1 = 0.206 and a. = 1.68 for kaolinite concentration (in percent) and fresh water (pw = .001 Ns/m2). Equation 2.6b represents a truncated approximation of a power series expansion of pm(C). For a more general form for pm(C), the reader is directed to the discussion by Krone (1963). However, it must be pointed out that data published by Krone (1963) showed that multiple values for P, ap and aP are possible for a particular sediment, depending on the shear rate and degree of 25 aggregation. Therefore, caution must be advised concerning the validity of Eqn. 2.6 for more detailed application. By considering the mobile fluid mud as depth varying viscous Rayleigh flow (Stokes' first problem with variable viscosity; see Schlichting, 1979), a numerical solution of the flow and boundary layer thickness can be obtained. This is a non-steady state approach to determining the horizontal transport layer detailed in Section 3.7. An alternative approach to determining the mobile/stationary interface is considering the non-Newtonian rheological properties of high concentration suspensions. Past research has indicated that concentrations in the fluid mud range behave as Bingham plastics or pseudoplastics (Krone, 1963; Kirby and Parker, 1977; Faas, 1981; 1987; Nichols, 1985). Over short (tidal) time periods the designation of effective yield strengths may be appropriate. In this case, the data seem to suggest a concentration power law relationship. Figure 2-7 shows a very approximate linear (on log-log paper) relationship between yield strength, ty, and concentration through the data sets shown. The expression to relate this functional dependence is of the form, ay Ty = Oy C (2.7) where a y and ay are empirical constants. The data in Figure 2-7 suggest that 0y = 8.7 x 10-7 Pa (I Pa = 1 N/m2) and ay = 2.55. With this assumption it is possible to estimate the lower penetration distance (lower extent of horizontal motion), the mobile/stationary fluid mud layer interface, Zb, based solely on equating the applied bed shear to 10 1 Q San Francisco Bay Mud SWilmington Mud V Brunswick Mud SGrundite 1 02 Kaolinite Suspension I 110 L S11 E 10 -. 1 00 z // ty= yC~y 0LL 1 / ~~1d5 / / /1 I 10-2 103 101 /0 / I/ // / / 10 102 103 4 SEDIMENT CONCENTRATION, C(g/I) Figure 2-7. Bingham Yield Strength Variation with Concentration 27 the level of equal yield strength. However, field observations of fluid mud flows (Kendrick and Derbyshire, 1985) do not seem to support this approach (see Section 5.5). This is because flow occurs when the applied shear stress is less than the reported shear strength (Figure 2-7). This suggests that the behavior is more pseudoplastic than Bingham. Further discussion of possible means of determining the elevation, Zb, is described in Section 3.7. 2.4.2 Mobile Fluid Mud The mobile fluid mud layer as described in Section 2.2 is that part of the fluid mud layer which is advected along with the mobile suspension layer current. It may also be gravitational slump flow along a sloping bed (Kendrick and Derbyshire, 1985). The elevation, Za, (Figure 2-1) which defines the upper bound of the layer represents a local maximum in net downward vertical flux. The settling velocity of cohesive sediments varies with concentration in suspension, ws(C). Initially constant, the velocity rises with increasing concentration (due to flocculation) to a level where it becomes constant again then rapidly drops. An example of the settling characteristic of a natural estuarine sediment is shown in Figure 2-8. The point beyond which no further increase in settling velocity occurs has been termed "hindered settling" (Owen, 1970; Imai, 1980; Teeter, 1986a). For purposes that will become clear with the following arguments, it is important to distinguish hindered settling velocity from hindered settling flux. The details of free, flocculation and hindered settling velocity are discussed in Chapter 3 (Section 3.4). 28 100.0 kI = 0.513 E nl = 1.29 10.0- W=k C Wso= 2.6 mm/s 1s 1 k2 = 0.008 . fn2 = 4.65 1.0 a % 00 LUJ > 0.1~ 2 ~W WW30 (1-k2C)2 u '- 0.01 W 1-2T~ 0.001 1 1 0.01 0.10 1.0 10.0 100.0 CONCENTRATION, C (g/Q) Figure 2-8. Settling Velocity Variation with Concentration Severn Estuary Mud (adapted from Mehta, 1986) The vertical flux of sediment (mass per unit area per unit time) from settling, Fs, is the product of the local settling velocity and concentration as Fs(C) = ws(C) C (2.8) For the data of Figure 2-8 (source: Thorn, 1981) the vertical flux, Fs, is plotted against concentration, C, in Figure 2-9 below. From Figures 2-8 and 2-9 it is observed that the peak flux occurs at a much higher concentration (i.e., 20 g/l) than that at which the peak settling velocity occurs (i.e., 3 g/l). This is due to settling velocity being either constant or only slightly decreasing from the FM= 40 g/m-s CA= 2x104 mg/I C40 Fm E 40- M U2 LLT 20 103 10 4 CA 105 CONCENTRATION, C(mg/) Figure 2-9. Vertical Settling Flux Variation with Concentration (reprinted with permission from Ross et al., 1987) maximum (- 2-3 g/l) over a wide concentration band (2-10 g/l). The peak settling flux (i.e., 20 g/l) represents a more reasonable definition for hindered settling than that based on the peak settling velocity. Beyond this point, the actual vertical mass flux from settling diminishes rapidly with increasing concentration. The upper elevation, Za, of the fluid mud layer under settling conditions therefore occurs at the "hindered" (defined on the basis of flux) concentration. A discrete interface forms because the sediment accumulates at this level because the flux is increasing above and decreasing below this interface. After all the sediment in the upper suspension layer has settled onto the fluid mud layer, the interface settles according to the 30 interfacial settling region (shown in Figure 2-3). When the flow in the upper suspension layer is turbulent, diffusion and entrainment at the interface reduce the overall downward vertical flux and the interfacial concentration, Ca, drops from that given by pure settling conditions. Thus, Ca has a maximum value given by the hindered (flux) concentration. In the presence of mixing, the mobile fluid mud layer, Sfm = Zb Za, does not necessarily become thicker (by becoming more diffuse). Due to the sharp density gradients resulting from the high suspension concentrations, turbulent mass and momentum diffusion across the fluid mud layer is greatly damped. This results in a stable stratification, often termed buoyancy or gravitational stabilization (Fischer et al., 1979). In this case, upward entrainment, which is dependent on the degree of stratification and relative turbulent intensity (Yih, 1980) becomes the dominant mixing mechanism. Stratification development is discussed in greater detail in the next section. Mixing in the presence of gravitational stabilization is discussed in Section 3.3.2. 2.5 Lutoclines Lutoclines are defined as pronounced "steps" in the vertical concentration profile resulting from complex mixing-settling processes. The upper fluid mud interface, by this definition, is also a lutocline (shown as L1 in Figure 2-1). However, lutoclines can as well occur in the mobile suspension layer (shown as L2 and L3). Lutoclines have a vertical scale (distance between steps) dependent on the local vertical scale of turbulence (Posmentier, 1977). Therefore, only a limited number can exist and over limited periods. The origin of this term stems from 31 the Latin word lutum which means mud (Kirby, 1986). Lutoclines are analogous to other types of density stratification (pycnoclines) from sharp salinity gradients (haloclines) and temperature gradients (thermoclines) with the exception that suspended sediment exhibits settling independent of the fluid. They are easily recorded by high frequency echo sounders and are characteristically observed in high sediment (> 500 mg/l) environments. Figure 2-10 shows a typical suspended sediment profile showing the relative temporal stability of two lutoclines. The velocity data also shown in the figure together with the concentration profiles suggest turbulent, well mixed flow between lutoclines. The physics of lutocline genesis, growth, and decay is governed by the dynamic interaction between the counteracting processes of turbulent mixing and gravitational settling. Simply stated, lutoclines occur because sediment is heavier than water and it tries to settle out under quiescent conditions. Due to flocculation and hindered settling, fine sediment suspended at large concentration settles as a sharp interface, as opposed to concentration "thinning" (Bosworth, 1956). Turbulent eddies impinging on the interface exchange "parcels" of sediment-laden fluid. However, due to the potential energy difference of each "parcel" with its surroundings, they are returned to near origin levels with only modest mixing. This is in sharp contrast to the rapid mixing which takes place in the low density gradient regions (qualitatively defined below). Thus, the moderate mixing at the interface is counter-balanced by the sediment settling, and the interface remains stable. 32 Water surface* 0 ! I I I i 4- i , o 2 4 II I I I !I "1 o i i i i i 4 0 .5 15:4 5) 15:50 Time of profile 15:55 5 minute mean velocity .0 (g/l) 6 8 ' (M/s) Figure 2-10. Typical Suspended Concentration Profile Showing Multiple Lutocline Stability Over 10 min. Period (Kirby, 1986) One means of relating the relative magnitudes of gradients in kinetic energy, a(pu2/2)/az, to potential energy, 8(pgz)/az, is through the local gradient Richardson number defined as (2.9) Ri = R 8_E (au)-2 p 8z az where g is the acceleration of gravity, p is the fluid density, u is the horizontal velocity and z is the vertical coordinate direction (positive upwards). 33 Thus, the implications of equation 2.9 are Ri >- 0 High rate of kinetic energy dissipation relative to low potential energy gradient 4 Rapid Mixing Ri >> I Low rates of kinetic energy dissipation relative to high potential energy gradient 4 Minimal Mixing Ri < 0 Density gradient, ap/az, > 0 the system is unstably stratified 4 Overturning As an example, letting the local mixing rate in a neutrally stratified condition (no density stratification) be defined as Kn(Z) implying that it is variable with depth, the simplest relationship for mixing in the presence of stratification, based on Richardson number, is Ks(z) = Kn(Z) (I + Ri)-I (2.10) where Ks is the local mixing rate dampened by stratification. The limits on mixing meet the above requirements as it can be seen that Ks(z) -Kn(z) for Ri 4 0 and Ks(z) 0 if Ri This is then one means of quantifying the stratification dampened mixing. A more general form for the above equation, a review of literature, and a discussion of applicability of buoyancy stabilization are given in Section 3.3.2. CHAPTER 3 TRANSPORT CONSIDERATIONS 3.1 Introduction The physics related to the vertical structure of fine sediment suspension can be addressed by considering the important components of the advection-diffusion equation. This equation, simply an Eulerian conservation of sediment mass expression, relates the temporal changes in sediment concentration to the spatial gradients in fluxes. Simple arguments show that for the present purpose the important coordinate in the equation is the vertical, z (positive upwards from the bed), direction. Furthermore, gravitational forces which influence the diffusion and settling flux terms are responsible for the complex structure of lutoclines and fluid mud as defined in Chapter 2. Theoretical and rationally based relationships for settling velocity, neutral turbulent diffusivity, and buoyancy stabilization are presented in this chapter for explanation and predictive purposes. A simple one-dimensional numerical model, developed from these relationships and the advection-diffusion equation, is used to explain laboratory and field data presented later in this report. Finally, to distinguish the lower layer of mobile fluid mud, a simple numerical model based on momentum diffusion is developed to evaluate the dynamic and steady state characteristics of this layer and to estimate horizontal sediment transport rates. 35 3.2 Mass Conservation Equation In Cartesian coordinates (x, longitudinal; y, lateral; and Z, directed vertical upwards positive from the water surface), the instantaneous Eulerian conservation of mass equation for (scalar) sediment suspension concentrations C(x,y,z,t) (mass of sediment/volume of suspension) can be written as dt where q is the resultant mass flux vector (from diffusion and settling) and V is the vector operator. For Fickian molecular diffusion, the mean mass flux vector is qm -DVC (3.2) where it is assumed that the molecular diffusivity, D, is isotropic (Fischer et al., 1979). Since it is implausible to track particles in suspension on an instantaneous, infinitesimal scale, and because natural flows are typically turbulent, it is usual to express equation (3.1) in terms of time averaged values (e.g., time average velocity, u, and concentration, C) where the averaging time is sufficiently long to negate turbulent fluctuations but short enough to track longer period temporal behavior (Vanoni, 1975; McDowell and O'Connor, 1977). However, time averaging greatly increases the diffusive mass flux vector. Fortunately, as an 36 approximation, turbulent diffusion can be expressed analogous to Fickian diffusion in the form qt = K-VC (3.2a) where K is the turbulent mixing vector with Cartesian (x,y,z) coordinate components (KX,KY,Kz). Since turbulent mixing is much greater (2-8 orders of magnitude) than molecular diffusion, the latter is often neglected (McCutcheon, 1983). Simple perturbation analyses, i.e., letting the velocity (vector) and concentration (scalar) components be represented by mean (e.g., u,C ) and fluctuating values (e.g., u',C'), have been used to support this result mathematically. The reader is directed to Hayter (1983) or French (1985) for this derivation. The mass flux vector from settling is, simply, qs = Fs = -wsCj (3.3) where ws is the mean sediment settling velocity and j is the unit vector directed along the z axis. The resultant mass flux vector for suspended sediment is then approximated by q = qt + qs 0 < z < Zb (3.4) away from the boundaries (water surface z=O and bed surface, z=Zb). 37 For the purposes of considering vertical structure, only the vertical transport terms need to be evaluated. Horizontal gradients in concentration are (typically 3-4) orders-of-magnitude smaller than vertical gradients. ax az a y azi l << ICI << ' Non-dimensional scaling arguments have been used to determine the relative importance of the individual terms in Eqn. 3.1 This analysis is included for reference in Appendix A. For typical estuarine conditions (see Appendix A) horizontal and vertical advective fluxes and horizontal diffusive fluxes can be neglected for first order analysis. The governing equation for considering the vertical structure of fine suspended sediments is now reduced to C qz a ac at az = Cwsc + Kzz 0 < z < Zb (3.5) where qz is the resultant vertical flux from settling and vertical diffusion away from the boundaries (z=0 and z=Zb) shown by the bracketed. terms in Eqn. 3.5. The boundary conditions which must be imposed on Eqn. 3.5 are Bed Flux Boundary Condition. Application of Eqn. 3.5 at z = Zb requires that a bed flux term, Fb (mass of sediment per unit bed area per unit time), containing both erosion, Fe, and deposition, Fp, fluxes as Fb = Fe Fp (3.6) be defined. In addition, the diffusion and settling flux terms at the bed are zero. Thus qz(Zb,t) = Fb and ws = Kz = 0 at z = Zb is the appropriate bed boundary condition. Fb is dependent on sediment and hydrodynamic conditions. Section 3.5 presents a detailed discussion of bed fluxes (erosion and deposition) used in the vertical transport model. Surface Boundary Condition. The boundary condition at the water surface, z=0, is a no net flux boundary. This means that there is no net transport across the free surface and diffusion flux is always counterbalancing settling flux i.e.: qz(0,t) = (wsC) + [Kzaz] = 0 (3.7) The diffusion flux term Fd = [KzAz1 must include entrainment and gravitationally stabilized mixing. In the absence of well defined hydrodynamics (i.e., perhaps the results from a full turbulence model simulation), functional forms for the vertical turbulent diffusivity, Kz, based on first-order closure modeling using mixing length approximations can be used (McCutcheon, 1983). This assumes that the mass diffusivity can be related to the momentum diffusivity. Furthermore, due to differences in time scales, spatial variability, and kinetic energy dissipation, the functional forms for highly oscillatory currents (e.g., waves) are quite different from those for unidirectional flows. Stratification, in general, dampens turbulent mixing by the mechanisms described in Chapter 2. Through local gradient Richardson 39 number relationships of the Munk and Anderson (1948) form, buoyancy (gravitational) stabilization can be modeled. Stabilized diffusivity is treated separately in Section 3.3.2. On the subject of mixing and stratification it must be pointed out that surface waves can create interfacial waves which can build to breaking, thereby greatly enhancing interfacial mixing (Yih, 1970; Dean and Dalrymple, 1984). Due to the limited scope of this research and because this phenomenon was not observed in laboratory or field data for this study, no further discussion is provided. The reader can find additional information on this topic in Lamb (1945), Yih (1976), and Yih (1980). The settling flux (w5C) as written in Eqn. (3.5) allows for spatial variability in both unknowns, settling velocity and concentration. In general, for both cohesive and non-cohesive sediments, settling velocity is a function of concentration, ws(C). The settling behavior of cohesive and non-cohesive sediment is covered in Section 3.4. 3.3 Diffusive Transport 3.3.1 Turbulent Diffusion In turbulent flows mixing occurs mainly because the time-averaged products of the velocity and concentration fluctuations i.e., u1C', are non-zero. Through adequately measuring the simultaneous fluctuations in velocities and concentrations, turbulent mixing can be precisely quantified. Then, for predictive purposes, correlations to flow parameters such as bottom friction, mean velocity and pressure gradient are required. Reasonable success is beginning to be achieved in the area 40 of turbulence modeling (Zeman and Lumley, 1977; Sheng, 1983). However, in light of the difficulties in precise measurement of these fluctuations, verification poses difficulties. For fine sediment suspensions the turbulent diffusion of sediment mass, Ks, is approximately equal to that of the diffusion of momentum, Em. The turbulent Schmidt number, Sc (Daily and Harleman, 1966), which is the ratio of mass to momentum diffusivity is equal to one (Teeter, 1986b) as K c E (3.8) m In turbulent flows momentum diffusion is by Reynolds stress, Tij = -P ujuj, gradients where the time mean product of the velocity fluctuations is nonzero. For mass diffusion, the time mean product of the concentration and velocity fluctuation is nonzero analogously. This observation (Reynolds' analogy) allows the use of a wide body of literature on first-order closure modeling based on the coefficient of eddy viscosity, relating the Reynolds stress to mean velocity gradient as au. ij = -pEij x (3.9) where Eij is the ij component of the momentum diffusivity (eddy viscosity) tensor. It can be seen from Eqn. (2.9) that the eddy viscosity, in general, must be a function of mean shear rate and shear stress. It is also common to assume that turbulent diffusion is 41 isotropic (i.e., Eij = Eji = E, Fischer et al., 1979) in the absence of stratification. The most commonly applied expression of vertical variation in eddy diffusivity is the formulation given by Rouse (Vanoni, 1975). By following von Karman's assumptions of a linear shear stress distribution with depth leading to a logarithmic velocity profile, the following expression is found: z E(z) = Ku*z (1- -) (3.10) h where K is von Karman's constant, u* is the friction velocity (,o1t/p) and h is the flow depth. While this expression may be sufficient for describing turbulent-logarithmic uni-directional flows, it does not describe highly oscillatory flows such as under waves. Maa (1986), Kennedy and Locher (1972), and Hwang and Wang (1982) have reviewed currently popular expressions for diffusion coefficients under waves. There seems to be little consistency in the forms. One of the most promising expressions based on energy dissipation is that developed by Hwang and Wang (1982). Their model, applicable outside the wave boundary layer, is of the form sinh2kz E(z) = c*H2a 2sinh2kh (3.11) where a is a constant, H is the wave height (twice the amplitude), a is 42 the wave frequency (2nr/T, T = wave period), and k is the wave number (27r/L, L = wave length). Thimakorn (1984) found success using a coefficient similar to that given by Hwang and Wang (1982) to predict vertical profiles of natural clay concentration during resuspension in a wave flume. It should be pointed out that the concentrations reported were small (<1000 mg/l) and any buoyancy stability effects therefore were likely to be negligible. Next to the bed boundary layer effects greatly increase vertical mixing under waves due to the relatively large velocity gradients and shear (Neilson, 1979). Orbital particle trajectories are significantly altered from those predicted for example by linear wave theory (inviscid potential flow) because viscous (or turbulent) effects dominate. However, this layer is small (o/(2v) < 1 cm) and is often neglected (Maa, 1986). Further upward, the velocity amplitude gradients increase with distance above the bottom to a maximum at the surface. This is the basis for the Hwang and Wang (1982) form shown above. Maa (1986) conducted dye diffusion tests under waves which showed larger lateral spreading rates near the surface and immediately near the bottom. This is indicative of higher energy dissipation in those regions which would support the proposition of higher vertical mixing rates. In the presence of density stratification the form of the neutral diffusivity is not as important as the form of the stability coefficient (French, 1985), which provides the basis for a discussion of mixing in the presence of density stratification. 43 3.3.2 Gravitational Stabilization In the previous section, theoretical and empirical based expressions for the vertical turbulent diffusivity under current and waves were mentioned. In a continuously, stably stratified flow the vertical diffusion of both momentum and mass is inhibited by stratification, and significant modification of the turbulent diffusivity occurs. Furthermore, the diffusivity of momentum and mass are not affected in the same manner. In the presence of density stratification, the eddy viscosity (i.e., the turbulent momentum diffusity) is larger than the eddy diffusivity of heat and mass (French, 1985). Progress has been made towards estimating values and obtaining expressions for mass and momentum diffusion in a continuously stratified flow. However, it must be emphasized that at the present time an expression does not exist for either eddy viscosity or diffusivity which is considered universally valid. French (1985) provides a summary of several popular forms developed for uni-directional flow only. A brief review of those plus others is given here for the purpose of explaining vertical structure. Rossby and Montgomery (1935) first proposed an equation relating vertical eddy viscosity for stratified flow, Es, to the corresponding value for homogeneous or neutral conditions, En, of the form E E -I - (1 + ORMRi) (3.12) En where PRM is an empirical coefficient and Ri is the local gradient Richardson number, Eqn. (2.9). They assumed that the change in kinetic energy per unit mass in going from a neutral or unstratified condition to 44 a stably stratified condition is equal to the potential energy change due to displacement over the mixing length from the equilibrium position with a different density. Holzman (1943) suggested a somewhat different relationship E En (I PHRi) (3.13) where PH is a coefficient. Note the change in sign of the coefficients. Munk and Anderson (1948) proposed a generalized form of the Rossby and Montgomery (1935) and Holzman (1943) equations as E a MA En (I + OMARi) (3.14) where PMA and aMA are free coefficients. Kent and Pritchard (1957) also used a conservation of energy argument to develop an equation of the Munk and Anderson (1948) form; however, they argued that a=-2 on a theoretical basis. Delft Hydraulics Laboratory, (DHL) (1974), reported that the ratio of E. to En should decrease exponentially with increasing values of Ri or E -S Ri Es e (3.15) m where Pe is an empirical coefficient. Finally, Odd and Rodger (1978) used the original hypothesis of Rossby and Montgomery (1935) to define equations applicable for two specific cases: 45 Case 1. Stratified flow with a significant peak in the vertical profile of Ri at a distance z = z0 from the bottom boundary where T is the peak gradient Richardson number, then E E ( + O) n E - (1 + OR) En for T < 1 for T > I (3.16) (3.17) where BOR = a coefficient. Case 2. No significant peak exists in the vertical profile of Ri: then E s -I E- (1 + PoRRi) n E E n I + O0R) En for Ri I for Ri > I Equations (3.16) through (3.19) are applied throughout the vertical dimension, but near the boundaries, if Es > En then En is used. Note that Es/En = constant (not a function of depth) for all cases except conditions when Eqn. (3.18) applies. This is significantly different from the previously proposed forms (Eqns. 3.12 3.15) which are everywhere depth variable. (3.18) (3.19) 46 The problem with all the above methodologies is that, in general, they cannot be shown to be universally valid. Suggested values for some of the coefficients used in the above equations are summarized in Table 3-1 below. Table 3-1. Summary of Coefficient Values for Turbulent Vertical Diffusion of Momentum in Continuously Stratified Flow Equation_ a Source 3.12 2.5 --Nelson (1972) 5.05.0 DHL (1974) of 30.3 -- French and McCutcheon (1983) 3.14 10 -0.5 Munk and Anderson (1948) it 30 -0.5 DHL (1974) 3.16-3.19 0.31 0.747 French (1979) to0.062 0.379 French and McCutcheon (1983) 140-180 Odd and Rodger (1978) With regard to the data summarized in Table 3-1, the following should be noted: 1. Nelson (1972) used published oceanographic, atmospheric, and pipe flow data for his analysis, and the same was true of the analysis by the Delft Hydraulics Laboratory (1974). Thus, these investigators had no control over the quality of their data. 2. The data used by French (1979) were taken under laboratory conditions, but the flume used for these experiments had a small width-to-depth ratio, and the results may have been unduly affected by this fact. 3. Odd and Rodger (1978) used field data from a reach of tidal channel. Their data set is perhaps the best data presently 47 available regarding the turbulent vertical diffusion of momentum under stratified conditions. 4. French and McCutcheon (1983) used the Odd and Rodger (1978) data set for their analysis. The coefficients for Eqns. (3.16 3.19) used in their work differ from that of Odd and Rodger (1978) due to differences in the definition of reasonable fit. 5. In the past, Eqn. (3.12) has been the most commonly used method of estimating Es (Nelson, 1972). It is more theoretically justifiable than the methods of Odd and Rodger (1978), French and McCutcheon (1983) or French (1979). 6. Delft Hydraulics Laboratory (1974) concluded that when Ri < 0.7, the scatter of the data available is so great that no best-fit equation can be selected. A number of models for the eddy (mass) diffusivity in stratified flow have also been proposed. Most have been based on the results from* momentum diffusion; however, under stratified conditions, questions arise as to the applicability of this assumption (e.g., see Oduyemi, 1986). One of the most frequently used expression is of the form Ks c(l + O'Ri )-a' (3.20) Kn where Kn and Ks are the vertical mass diffusivities for homogeneous and stratified flows, respectively, and c, 0', and a' are coefficients. Munk and Anderson (1948) estimated that c = 1, a' = 1.5, and = 3.33. It is interesting to note that stratification apparently also acts to reduce the value of the turbulent transverse diffusion coefficient by 48 turbulence damping; however, the results presently available in this area (see, for example, Sumer, 1976) are inconsistent and are not relevant for vertical structure considerations. When gravitational stability is considered (e.g., by Eqn. 3.20), nonlinearity between diffusive flux, Fd, and vertical concentration gradient, C,z (note ,z denotes differentiation with respect to z), develops. Without regard to stabilization (Kz = Kn), by Fickian diffusion, the diffusive flux is linearly proportional to concentration gradient, Fd = Kz.C,z. However, from theoretical results presented for gravitational stabilization, the turbulent mass diffusion coefficient (Kz = Ks) was shown to be inversely proportional to the gradient Richardson number, Ri (given by Eqn. 2.9), to a power, a' > 1. The gradient Richardson number, of course, is directly proportional to the density gradient, p,,, which is a function of the concentration gradient because bulk density is a function of sediment concentration (Eqn. 2.5). The resulting dependence of diffusion flux to concentration gradient is therefore highly nonlinear. Figure 3-1 shows an example of the nonlinearity resulting between Fd and C,z using Eqn. 3.20 with Munk and Anderson values for stability coefficients (i.e., c = 1, a' = 1.5, and 0' = 3.33). For this case the flux initially increases with C,z reaches a maximum and then slowly decreases. For a given flux below the maximum two values of C,z (corresponding to the two roots) satisfy the equation. In the absence of settling, discontinuities in concentration profile (two distinct C,z's) are theoretically possible (because Fd,z = 0) and relatively stable as long as Fd is constant with time (e.g., near the bed during steady 4 to k L: 2x. a0 1 + 4.17 Ri? if 1 (+ 3.33 Ri)1. 5 0 0 0.0 0.2 0.4 0.6 0.8 Concentration Gradient, .C (kg/m2 az Figure 3-1. Diffusion Flux vs. Concentration Gradient erosion). For salinity concentrations in estuarine environments, this has been pointed out to be a likely cause of salinity finestructure (Postmentier, 1977). For suspended fine sediment, the nonlinearity in diffusion flux, Fd, (with C,z) has the effect of promoting lutocline growth and stability--in addition to the nonlinearity between settling flux, Fs, and concentration, C (pointed out in Section 2.4.2). The settling properties of estuarine fine sediment is presented in the following section (3.4). 3.4 Settling The predominant distinction between fine sediment suspensions and other density altering constituents (e.g., salt, temperature, etc.) is 50 that suspended sediment is negatively buoyant and settles independent of the suspending fluid which surrounds it. This counteracts mixing to the extent that under quiescent conditions partial or total clarification is possible only to be later well mixed again under high flow conditions. While the settling characteristics of non-cohesive sediments (e.g., sand) are reasonably well behaved, i.e., not so strongly dependent on concentration, salinity, etc., cohesive sediments are very sensitive to these variables. It is convenient to start by discussing the settling characteristics of individual particles and work into high concentration (>20,000 mg/l) settling suspensions. 3.4.1 Free Settling Free settling was defined in Chapter 2 as the concentration range over which individual settling sediment particles (both dispersed primary particles and aggregates) do not physically interfere with one another. For cohesive sediments, the upper concentration limit is in the range of 300-500 mg/l (Krone, 1962) but for non-cohesive sediments it is one to two orders-of-magnitude higher (McNown and Lin, 1952). Individual sediment particles settle at a terminal fall velocity which results in a force balance between form and skin friction (viscous) drag and net negative buoyancy. For a spherical particle of diameter, d, settling in a viscous fluid with kinematic viscosity, v, the settling velocity, w5, is =s 3 CD PW 12 (3.21) 51 where g is the acceleration of gravity, CD is the drag coefficient and p and pw are the sediment and fluid densities, respectively. The coefficient of drag, CD, is a function of the Reynolds' number of the sphere CR = wsd/v), but cannot be determined analytically for R > 1 (see, for example, Vanoni, 1975). In the viscous or Stokes' settling range (R < 0.1) the drag coefficient is given by CD = 24/R and the settling velocity is w gd 2 (P' (3.22) s 18v P Fine estuarial sediment in dispersed or quiescent conditions typically falls well within this range. Therefore, no further discussion of the deviations from Stokes settling will be presented here with one minor exception: fine estuarine sediment is not generally spherical. In dispersed form, cohesive size sediment is plate-like with a large surface area to volume ratio (Van Olphen, 1963). This results in a higher drag coefficient and slower settling velocity than spherical sediment of the same volume. Very fine (d < I pm) dispersed sediment may not settle at all due to the increased relative importance of Brownian motion. Aggregates, although irregularly shaped, are generally more spherical (and substantially larger than dispersed primary particles). For both particles, it is typical to define an "effective" particle diameter based on measured settling velocity and specific density (ps). A more thorough discussion of the effects of particle shape on settling velocity as well as other deviations from Stokes settling can be found in Vanoni (1975). 3.4.2 Flocculation Settling In the presence of small amounts of dissolved salts (< I ppt NaCI) cohesive sediment in suspension can flocculate greatly, thus changing the settling properties. Flocculation of cohesive sediment particles is the consequence of inter-particle collision and cohesion. Cohesion and collision are discussed in detail by Einstein and Krone (1962), Krone (1962), Partheniades (1964), O'Malia (1972), and Hunt (1980) and reviewed by Hayter (1983). Cohesion depends primarily on the mineral composition and the availability and charge of cations in the suspended fluid. Colloidal particles have both attractive and repulsive forces (Van Olphen, 1963). The attractive forces predominate when the coulombic repulsive forces are suppressed by sorbed cations near the particle surfaces. A measure of the relative cohesiveness of a particular colloidal sediment is the cation exchange capacity, CEC. A high CEC indicates a highly cohesive sediment. Montmorillinitic sediments have a higher CEC and thus are more cohesive than illitic or kaolinitic sediments with lower CECs. Collision intensity and frequency are dependent on three mechanisms: Brownian motion, fluid shearing, and differential settling. Brownian motion is the natural thermal agitation of the sediment particles in the suspending medium. Particle movement from Brownian motion is erratic, the collisions are weak and the resulting flocs are "fluffy" (of relatively low density and weakly bound). This motion becomes much less apparent as the floc size grows. Brownian motion in estuaries is the least significant collision mechanism of the three (Krone, 1962). Particle collision from fluid shearing, however, becomes much more 53 significant as the size of the flocs grows. The result is a greater intensity of collision and stronger flocs. Differential settling becomes increasingly more important as the distribution of the size widens. Under quiescent conditions e.g., at the time of slack water, with a natural non-uniform sediment this becomes the primary collision mechanism. The frequency of all three means of collision increases with increasing concentration. Two characteristics of flocculated sediment which differ from the dispersed form and which affect the settling velocity are particle density and shape. First, because of interstitial trapped water, the relative particle densities are reduced. This effect alone would lead to reduced settling velocity in the flocculated state. However, because of the larger size and more spherical shape, a decrease in viscous drag results. Since the reduction in drag is much more significant than the reduction in density, the settling velocity of the flocs are up to 4 orders of magnitude larger than dispersed particles (Bellessort, 1973). This can result in rapid sedimentation and shoaling in upper estuaries where flocculation (by introduction of dissolved salts) is first stimulated. Krone (1962) reasoned that the average (median, by weight) settling velocity of flocculating Mare Island Strait (San Francisco Bay) sediment for equal flocculation time was proportional to the sediment concentration raised to the 4/3 power, ws C4/3 (3.23) 54 His reasoning was based on consideration of collision probability and average floc size. He further supported this argument with data taken in settling column and flume studies. Burt (1986) used a general relationship for flocculation enhanced variation with concentration as w= k1 C (3.24) where k, depends on sediment composition and n, can vary from about 1 to 2. 3.4.3 Hindered Settling As the concentration of sediment in suspension increases beyond the flocculation settling range, the mean sediment settling velocity begins to drop. Aggregates are so closely spaced as to form a continuous network, and the interstitial fluid is forced to escape through smaller and smaller pore spaces. This is commonly termed "hindered settling" in the literature (Mehta, 1986, Lavelle and Thacker, 1978). However, the inadequacies of this definition were pointed out in Section 2.4.2. The pioneering work of Richardson and Zaki (1954) on the settling of uniform glass spheres resulted in a widely accepted relationship for the settling velocity as a function of concentration of the form, w= Wso (I k2C) (3.25) 55 where w50 is the initial or reference settling velocity, k2 is a coefficient which depends on the sediment composition and n2 5. The coefficient k2 can be considered to be the reciprocal of the hypothetical concentration where hindered settling gives way to primary, first-stage consolidation. This is typically in the neighborhood of 120-160 g/l (Mehta 1986, Einstein and Krone, 1962). For fine sand-coarse silt the reference velocity, w50, is given by Stokes' Law. For cohesive flocculated sediment the reference velocity, w50, is the maximum velocity of the flocculation range. Teeter (1986b) found that most natural fine bay sediments fit this relationship well. Lavelle and Thacker (1978) used an expression of this type in steady-state analysis of the high concentration data of Einstein and Chien (1955) for coarse-grained sediment. Including a term of (1-C)0 in the Rouse (1938) equation allowing for finite and reasonable concentrations at the bed (z=0), they found success in predicting the near-bed high concentration data of Einstein and Chein (1955). 3.5 Vertical Bed Fluxes The bed flux boundary condition for solution of Eqn. (3.5) plays a critical role in the evolution of the vertical suspension profile as the overall source and sink component of sediment mass in suspension. Bed fluxes can be either erosional or depositional. Both are discussed in the following paragraphs. It is important to point out that defining the elevation at which the erosion or deposition process takes place is, in itself, a formidable task. From a practical viewpoint, simultaneous continuous profiling of concentration, velocity and bed stresses 56 (pressures and shear) are required in the upper-bed to near bottom layers to define the interface elevation with time and hydrodynamic action. As was pointed out in Chapter 2, it is very important to distinguish the stationary bed material from the fluid mud layer. Additionally, erosion relationships developed for bed/mobile suspension interfaces may not be adequate for erosion and fluidization of the bed beneath a fluid mud layer. This is a possible limitation of the proposed erosion/deposition functions used in the vertical structure model and presented in the following subsections. 3.5.1. Bed Erosion Bed erosion occurs when the resultant hydrodynamic lift and drag forces on the sediment at or below the bed interface (Zc in Figure 2-1) exceed the resultant frictional, gravitational and physico-chemical bonding forces of the sediment grain or particle. Continuous interparticle contact ceases and individual or groups of aggregates become resuspended. There are two modes of erosion (Mehta, 1986), surface or particle by particle erosion and mass or bulk erosion. In surface erosion, individual particles break free of the bed surface as the hydrodynamic erosive force (i.e., instantaneous turbulent shear stress acting on the particle surface) applied to them exceeds the resultant gravitational, frictional and cohesive bed bonding force. Under mass erosion, failure occurs well below the bed surface resulting in large chunks of sediment being broken from the bed structure and, subsequently, resuspended. Bed fluidization is mass erosion where large structural breakdown occurs with 57 an initially minimum change in density. Surface erosion is more typical of low concentration, low energy environments while mass erosion occurs under higher flow and higher concentration conditions (Mehta, 1986). Surface waves and other highly oscillatory currents have a particularly pronounced influence on erosion in comparison with unidirectional currents. Because of the increased inertial forces (e.g., "added mass" drag) associated with a local change in linear momentum, the net entrainment force is much greater than with turbulent unidirectional flows. Much more significant is the effect bed "shaking" and "pumping" can have under highly oscillatory flows. "Shaking" or bed vibrations occur because of the oscillatory bed shear stress which is transmitted elastically (while at the same time damped) down through the bed. "Pumping" occurs from oscillatory fluid hydrostatic pressure at the bed which, given the low permeability of cohesive sediments, can lead to internal pore pressure build up and liquefaction, similar to earthquake failure of saturated terrigenous soils (Seed, 1976). This effect can cause destruction of effective stress in larger layers depending on the bed characteristics leading to mass erosion and fluid mud formation (Alishahi and Krone, 1964; Wells et al., 1978; and Maa and Mehta, 1987). The destruction of effective stress under waves is documented, perhaps for the first time, in laboratory measurements presented in Chapter 4. Erosion (particles leaving the bed surface) precedes scour (resulting decrease in bed elevation) which will continue under constant loading until the bed shear stress and the bed shear strength are equal. The bed shear strength is a function of the deposition and consolidation 58 history plus the physico-chemical characteristics of the sediment. The shear strength, in general, increases with depth into the bed. The rate of erosion (= flux of sediment from the bed), Fe, from surface erosion is linearly related to the "excess shear" stress, tb-ts, for spatially and temporally uniform bed properties (Kandiah, 1974) as F = a s) (3.26) Fe I Tl s where a1, is an empirical rate constant, tb is the applied (time-mean) bed shear stress and ts is the bed shear strength for erosion. For a given a,, which is related to the type of flow and sediment characteristics, the erosion rate, Fe, is constant. For non-uniform beds (e.g., soft, partially consolidated) the rate of erosion can be found by (Parchure 1984, Parchure and Mehta, 1983). F = co expfa2 [Tb cs(z)]1123 (3.27) where co and a2 are constants (determined empirically). Since ts increases with depth below bed, the erosion rate, Fe, decreases as scour proceeds. No currently unique expression exists for mass erosion since it must involve dynamic bed data (i.e., bed stresses and pressures) as well as imposed shear. For mass erosion under waves the practice is to increase the coefficients to account for the larger magnitude erosion. Maa (1986) showed success with this procedure and demonstrated that the coefficients 59 were as much as an order-of-magnitude larger for wave erosion than for what has been found for the uni-directional case. Under pure wave flow condition it is difficult to distinguish bed erosion from fluid mud entrainment. Even though wave erosion has a greater ability to break the bonding forces, without high momentum diffusion or turbulent entrainment rates the fluid mud may not become mobile. 3.5.2 Deposition Sediment particles or aggregates in suspension will redeposit on the bed if the bed shear stress drops below some critical threshold value, tcd" tcd is the shear stress below which all initially suspended sediment deposits eventually. In general, it takes lower turbulent bed shear stress to keep cohesive sediment in suspension than it does to erode it (i.e., tbm < ts)" tbm is the shear stress above which no deposition occurs and it is generally larger than the limit for total deposition, tbm > tcd, (Mehta, 1986). This is because after deposition interparticle bonding and orientation are time-dependent, as well as dependent on consolidation mechanics (e.g., overburden, etc. as discussed briefly in Chapter 2) and the critical shear stress for erosion increases with time. For a uniform sediment tcd = tbm" For uniform sediment, in a depositional environment (i.e., tb < tcd), the rate of sediment deposited (= flux of deposited sediment), Fp, on the bed is related to the average aggregate settling velocity, ws, the near-bed concentration in suspension, C, and the relative probability, P, that the sediment will stay on the bed as Fp = ws C P I b < tcd (3.28) 60 The probability, P, that the sediment will stick to the bed is related to the relative shear stress (Krone, 1962) as P = (- b (3.29) tcd As observed, this relationship indicates no deposition when tb tcd and rapid settling when the bed shear goes to zero (tb = 0). Krone (1962) and Mehta (1973) conducted deposition experiments under steady flows using natural estuarine sediments and commercial kaolinite. tcd was found to depend on sediment composition, varying from 0.04 to 0.15 N/m2. Mehta (1986) made the distinction for critical shear stress for deposition of non-uniform sediment. He pointed out that while deposition proceeds when tb < tbm, not all of the sediment in suspension deposits when tb > Tcd. This is illustrated by the data in Figure 3-2. Mehta (1986), in reanalyzing earlier data (Mehta, 1973), pointed out that even after long periods the ratio Ceq/Co of ultimate equilibrium concentration, Ceg, to initial concentration, Co, was only a function of tb (i.e., Ceq/Co = f(tb)), not of Co. This, then represents a fundamental distinction between cohesive and cohesionless sediment since for cohesionless sediment the equilibrium concentration, Ceq, is dependent on tb and independent of initial concentration, Co, (i.e., Ceq = f(tb)). For cohesionless sediment, the equilibrium concentration represents a balance between the rates of erosion and deposition, whereas for cohesive sediment simultaneous erosion and deposition did not occur under test conditions relative to U.0.6 0.4 0 S t I r t I 2 4 6 8 10 12 14 16 Is 2 TIME (hrs) Figure 3-2. Ratio C/Co of Instantaneous to Initial Suspended Sediment Concentration Versus Time for Kaolinite in Distilled Water (after Mehta, 1973). Figure 3-2. Thus Ceq, for cohesive soils was the steady state concentration, Css. For cohesive soils, winnowing (coarser material settling out first) is a likely cause of the variable steady state ratios, Css/Co (Mehta and Lott, 1987). Thus, the steady state concentration, Css, results in a suspension with a mean particle grain size finer than the original suspension. For modeling purposes, discretizing non-uniform suspended sediment into a finite number of classes, Ci, and treating erosion and deposition for each class separately would be one means of handling the winnowing (and resulting bed layering) phenomena. The vertical structure model considers independent settling and deposition of multiple classes of suspended 62 particles after it was found to be significant in settling column tests of natural bay sediment (see Section 4.3). For discretizing the nonuniformity of the deposition Eqn. 3.28 (originally developed for uniform sediment) is assumed to be valid as F = wsi Ci P i tb < tcdi (3.30) where the i subscripted variables must be defined for each class. 3.6 Fluid Mud Entrainment Once a fluid mud layer is formed, either from high erosion or deposition rates, entrainment of this high concentration sediment suspension can occur at the upper, mobile fluid mud interface (see Figure 2-1). Entrainment is markedly distinguised from bed erosion in that the sediment is already in suspension. Fluid mud entrainment results from interfacial instabilities and dissipation of kinetic energy with, as yet, limited theoretical analysis. However, because it is believed to behave analogously to two-layer density stratified flows associated with salt or temperature gradients, a relatively larger literary and theoretical base exists for these cases (Yih, 1980). Velocity shear at the interface accounts for the primary mixing mechanism. Unlike mixing in homogeneous or weakly stratified shear layers, strong stratification characterized by a high Richardson Number is composed of events such as interfacial wave generation and breaking, interchange of energy between waves and the mean flow, and local shearinstabilities (Narimousa and Fernando, 1987). 63 Kato and Phillips (1969) in laboratory experiments of entrainment of linearly stratified fluids found that the entrainment coefficient, E, = ue/u*, where ue is the entrainment rate (dh/dt) and u* is the friction velocity, decreased with increasing stratification. They found an inverse relation between entrainment rates and bulk Richardson Number, Ri* = Abh/u*2 in which Ab = g(p-po)/po is the buoyancy jump, p is the fluid density, po is a fluid reference density, g is the gravitational acceleration, and h is the average depth of mixed layer. They suggested E* Ri*,-/2. Other research indicates that it should be related to mean velocity, u, in the mixed (i.e., upper) layer (Price, 1979; Thompson, 1979) as u Ev __e = f(Riu)_4 (3.31) u where Riu = Abh/U2. Still other researchers (Phillips, 1977; Price, 1978; Narimousa and Fernando, 1987) showed supporting evidence for using the velocity jump, Au, across the interface defined as the difference between the mean flow velocities in each layer. Later researchers reasoned that the major portion of the energy for turbulent mixing at the density interface results from shear production at the entrainment zone itself and, therefore, Au is the significant velocity scale to obtain a measure of the energy dissipation rate. Narimousa and Fernando (1987) presented a graphical depiction of the entrainment process which is qualitatively descriptive enough to warrant reproduction here. Mixed layer W, Non-turbulent layer Figure 3-3. Simplified Description of Density Stratified Entrainment (after Narimousa and Fernando, 1987) Figure 3.3 shows the entrainment process based on experimental observations (Narimousa and Fernando, 1987). The upper turbulent layer of thickness, h, is well mixed and the lower layer is initially stationary. An intermediate entrainment layer, 6, separates the two layers and is the region characterized by high energy dissipation and buoyancy gradients. In the upper entrainment layer, the mean shearing rate, du(z)/dz, increases downward reaching a maximum at 6i, and then decreases as viscous dominant momentum diffusion penetrates deeper and deeper into the stationary layer. The shear layer thickness is shown as 6.The highest density gradients occur in the entrainment layer of thickness 6, which is inside the shear layer, 6s, where turbulence dampening is sufficient to eliminate turbulent penetration into the 65 lowest layer. The momentum diffusion (viscous) layer, of thickness t can be dynamic (growing with time) or relatively constant with respect to the interface. Also shown are the flattening of large eddies (with turbulent velocity components ul and w1) at the density interface and local scouring and internal waves of height Sw in the intermediate entrainment layer by the mixed layer eddies of mixing length scale, 1 (proportional to the mixed layer depth, h). As can be deduced from the number of characteristics in the above description, entrainment of density stratified flows of single phase fluids is, in itself, an interesting and challenging field. Add to this, particle settling associated with the two-phase sediment/fluid mixture and one can see that fluid mud entrainment deserves fundamental research. No effort has been made to distinguish fluid mud entrainment from general lutocline mixing in this research. Nevertheless, despite this limitation, reasonable success has been achieved in explaining the observed physical behavior of prototype and field vertical profiles, as shown in Chapter 5. Further research in fluid mud interfacial entrainment is required before a more refined understanding and usable results are obtained. 3.7 Horizontal Fluid Mud Transport Several approaches to solving for the horizontal transport of mobile fluid mud and the relative thickness of the mobile layer are available. These approaches are based on different simplifying assumptions concerning the rheological and temporal behavior of the fluid/sediment system. The solution approximations (for velocity profile in the fluid 66 mud layer) together with limitations are presented in order of increasing complexity, beginning with the analytical solution of viscous boundary layer development under an imposed shear stress. The following titles are given for solution approaches: A. Constant Viscosity Rayleigh Flow B. Constant Viscosity Unsteady Bingham Flow C. Variable Viscosity Steady Bingham Flow D. Variable Viscosity Rayleigh Flow E. Variable Viscosity Unsteady Bingham Flow Applicable solution techniques were applied to field and laboratory data, the results of which are presented in Chapter 5. The constitutive equations which govern fluid mud transport are the conservation of momentum (Cauchy's Equation) and mass (continuity) equations. The Cauchy Equation of motion written in tensor notation is (Malvern, 1969) du Pb + V-T (3.31) dt where p is the local fluid density u is the velocity vector, b is the body force per unit mass vector, T is the stress tensor, and V is the vector operator. The first term in Eqn. (3.31) is the time rate of change of momentum per unit volume. The other terms are the body force per unit volume and stress tensor gradient, respectively. For an incompressible viscous fluid, the conservation of horizontal momentum equation in Cartesian coordinates is 67 du -P a [ 8u+ a ,u, ( t= x+ T'ytay (3.32a) where P is the pressure. The dynamic viscosity, p, is assumed here to be isotropic but, in general, a function of concentration, p=p(C). Together with the continuity equation, 8u + a = 0 (3.32b) ax 0y T sufficient boundary and initial conditions (outlined below), the problem is said to be closed and formally defined. For a tractable solution to the horizontal flow problem, somewhat far reaching assumptions must be made. First, one-dimensional horizontal flow in the x direction is assumed (no v and w components in the velocity vector). Next, the assumption of lateral uniformity is made. Then, by continuity, the horizontal velocity component must only vary in the z direction, u = u(z,t). The third assumption is by far the most stringent. It is assumed, analogous to the laminar sub-layer next to a boundary (Schlichting, 1979) and the shear layer in a stratified fluid (Narimousa and Fernando, 1987), that the horizontal pressure gradient is much smaller than the vertical shear stress gradient, << 2axz (3.33a) ax az To more formally show the conditions under which this assumption is valid, scaling arguments are used to evaluate the relative magnitude of 68 the terms of Eqn. 3.32a for dynamic momentum diffusion into the fluid mud layer. First, defining non-dimensional (primed) variables as t u x z to uo L tu0~ ' PP' P = P- = (3.33b) yH po Vo where to and uo are the characteristic maximum time and velocity, L is the length scale of the estuary, 6 is the length scale of the fluid mud layer depth, H is the differential height of the water surface over L, y is the specific weight of mud, po and po are the characteristic mud density and dynamic viscosity. Substituting the above variables in Eqn. 3.32a and considering only vertical shear gives uo du' H 1 _P' [oUo] 8 ,u' [ d --+ [o] 1 (8' } (3.32c) to dt' poL p0L x p' po &2 P p' 8z where all terms not in brackets, [ ], are order 1. Multiplying to Eqn. 3.33c through by [-o] gives, uo du' gHto 1 8P' tooI 1 8 ,u'} [ ] -[ ] + [ z (3.32d) dt' L uo p 8x' Po p z ' Substituting typical numerical values for fluid mud layers in estuaries of g = 101 m/s2, H = 100 m, to = 103 s, L = 105 m, uo = 100 m/s, vo = o/Po = 10-4 m2/s, and 8 = 10-1 m, the order of magnitude of the pressure gradient term is (101 m/s2 )(100 m)(103 s) = [10- 1 (3.33d) (105 m)(100 m/s) The viscous shear term is (-4 m2/s(13s (10- m 2])(0O s = [10 1 (3.33e) (10 m) which, for the particular set of conditions, is two orders of magnitude greater than the pressure gradient magnitude. Hence, neglecting horizontal pressure gradients in Eqn 3.32a for qualitative understanding of the dynamic momentum diffusion depth is justified, albeit weakly. It must be emphasized that under fully developed steady flow, the order of magnitude of the viscous shear stress and horizontal pressure gradient terms are the same (since they are the only two non-zero terms in the equation). Under the above constraints the momentum equation becomes au 1 azau (3.34) at p az ~z (.4 The equation is now in a form in which analytical and simple numerical solutions are possible with careful specification of initial and boundary conditions and rheological behavior. 70 A. Constant Viscosity Rayleigh Flow. For the case of constant mud viscosity, pm, and unsteady shear flow, an analytical solution proposed by Stokes (see Schlichting, 1979) is appropriate. With the boundary conditions, 1) imposed velocity, U, at the upper interface u(z)=U @ z=Za (in Figure 2-1), and 2) U(z)=0 @ z4-. The solution for the horizontal flow velocity is (Eskinazi, 1968) u = U (I erfcSs) (3.35) where Ss is the similarity variable z/2'/7v and erfc is the complementary error function defined as erfc6s = 1 2 s(3.36) The penetration depth of the mobile fluid mud layer (Zb defined in Figure 2-1) can be found by considering the boundary layer thickness, defined by u/U = 0.01 which is 6 = 3.64 v (3.37) The inadequacy of this solution is that even for viscosities ten times higher than water (i.e., 10-5 m2/s), the predicted boundary layer thickness over several hours is too large; e.g., S = 0(3.64(10-5.104)1/2} = 0(3.64 m) (3.38) 71 Additionally, the approach does not adequately represent the rapid depth variation in concentration (i.e., increasing concentration with depth) of the fluid mud. B. Constant Viscosity Unsteady Bingham Flow. For the steady state flow of a Bingham plastic with constant viscosity (and constant yield strength), an analytic similarity solution has been presented by PhanThien (1983). He assumed a two layer system with properties as P2Y It I Ty (3.39) where j is the time rate of shearing, au, and y is the Bingham yield strength. Denoting the velocities in layer i (i=1,2) as ui(z,t) = 0 2v'7v-it Ui(Ss) Pi' (3.40) where to is the imposed shear stress, vi, pi are the kinematic and dynamic viscosities and Ss is the similarity variable. z (3.41) 8s =- Ui(&S) is the similarity solution of Eqn. 3.34 given as UI(S) = 8s- (l-R )M 8 -z2 1 _2 [ s 0s dz + 2 s] 6 dz 0oce d (3.42) and 8' -z2 -2 sis 1 s [6 e dz+-e ] U2 (S) = R { s- c Z 2(3.43) _ z, e dz c where Sc = zc(t)/2fvit a8 = 92/91 (a8 for an ideal Bingham fluid), 8 = 8s,/ag, SJ = Sc/~I/a, and R = t y/to. Additionally, since the velocity field is continuous at z = zc(t) and is represented as U2(8c) = a8 UI(Sc) (3.44) a value for 8c is determinable and consequently for zc(t). But, unfortunately, the inability to treat concentration variation with depth as was the case in approach A is still undesirable. C. Variable Viscosity Steady Bingham Flow. Neglecting the obvious error involved with omission of the pressure gradient term, for the steady flow from applied shear stress of an ideal Bingham plastic with concentration dependent yield strength and viscosity, the constitutive equation is to = P(z) to y (3.45) y = 0, u(z)=0 to < y In the region where the yield strength is exceeded, the flow velocity is analogous to Couette flow with depth varying viscosity. For the region where the shear strength exceeds the applied shear stress, no motion occurs (the mud behaves as a solid). For a solution to Eqn. 3.34 for this case, a concentration relationship for the viscosity and yield strength must be specified. For example, one approximation for the viscosity/concentration relationship, based on data presented in Chapter 2 (see Figure 2-6), is of the form a P()=PO+ OPC) (3.46) where pwis the viscosity of the suspending fluid (water), and andaP are empirical constants. A power law expression for yield strength, ry, (also presented in Chapter 2, Figure 2-7) is t:y = Py C (3.47) where ayand ayare empirical constants. The boundary conditions are 1) an imposed shear stress at the upper interface, z = Toj @ z = Za P and 2) no-slip at the lower yield elevation, u = 0 @ z = Zb. Additionally, for steady flow the depth varying shear stress, rty(z), is everywhere equal to the imposed interfacial shear stress, to) down to the stationary interface, zb, where t y = 'Co. This approach has been presented for comparison only since the aforementioned error would be appreciable under quasi-steady flows in estuaries. The absence of time-dependence is also a drawback to this approach. Depending on the depth of the fluid mud layer and the imposed shear stress, the velocity profile can take minutes to hours to reach steady state form. For imposed shear which is continuously changing, as is the case in tidal flows, steady flow is never reached. The last two approaches offer the most promise in providing for realistic spatial and temporal variability of horizontal momentum diffusion into a fluid mud layer. D. Variable Viscosity Rayleigh Flow. This approach describes unsteady flow of a fluid mud layer with depth varying viscosity initially subjected to horizontal motion at the upper interface, Za. The governing equation is still Eqn. (3.34) but no specification is made regarding the overall lower extent of the boundary layer, 6fm" In general, a numerical solution is warranted. An explicit, finite difference approximation (with j time and i direction index), for example, j+l = t Ati 1 u PN.- p (U- u (3.48) where pi= (Ii+I+ pi)/2, gives an easily obtainable solution path. The boundary conditions are those given in A. Additionally, proper concerns for numerical stability and convergence must be addressed (i.e., At Az2 p/2Pmax). The last approach offers the most realistic simulation of rheological and temporal variability of the approaches presented thus far. E. Variable Viscosity Unsteady Bingham Flow. A numerical solution of a form similar to Eqn. 3.48 above is employed for the region where the 75 mud is sheared. Additionally, the lower interface is tracked by considering the temporal response of the shear stress and yield strength at each layer. This is written Eqn. (3.48) for t(z,t) t y(Cit) u(z) = (.9 0 for t(z,t) < ty(Ct) The boundary conditions are the same as those of C. Comments concerning the numerical technique, stability and convergence mentioned in D, also apply for the shear layer here. With regard to Bingham plastic vs. Newtonian fluid (with viscosity which varies with concentration) behavior, the data in Figure 2-7 suggest yield strengths which are sufficiently large to preclude flow under mild bed shear stress (e.g., C = 100 g/l corresponds to ry = 0.1 N/m2). However, field and laboratory data used to verify the above approaches in Chapter 5 (Section 5.5) show evidence of relatively high flows under very mild imposed shear stress. For this reason, care must be taken in application of the above approaches using a functional relationship for yield strength such as Eqn. 3.47, that the empirical coefficients ay and Oy fit a particular sediment behavior. It is suggested that nonNewtonian pseudo-plastic behavior (where viscosity is a function of shear rate) may be a more reasonable model than Bingham plastic for fluid mud flows. However, no further supporting arguments or discussion are made in this report since application of the Newtonian models showed reasonable results. M CHAPTER 4 LABORATORY EXPERIMENTS 4.1 Introduction Laboratory experiments were conducted at the University of Florida's Coastal Engineering Laboratory. These experiments consisted of two flume tests and settling column tests. The flume tests were designed to evaluate the dynamical effects of wave action on a partially consolidated natural estuarine sediment bed. Bed erosion (by fluidization) and upper column suspension concentrations were measured. Settling tests were performed to obtain the concentration dependent settling properties of natural flocculating fine sediment. New settling column tests were devised to provide development and verification data needed for the vertical profile model. 4.2 Flume Study 4.2.1 Objectives The objectives of the wave flume study were as follows: 1. To use advanced pressure sensor instrumentation to measure and document the effective stress breakdown (fluidization) in a partially consolidated cohesive bed subjected to wave loading. 2. To observe, record and determine factors characterizing fluid mud formation and stability (during wave erosion) presented in Chapters 2 and 3. 77 3. To measure wave resuspension concentrations related to hydrodynamical data (i.e., wave height, water depth, fluid mud and bed thickness) for the purpose of verification of the descriptive vertical transport model. 4. Investigate the role of wave resuspension in the overall sediment transport process in the prototype setting. 4.2.2 Mud Characterization The estuarine sediment selected for use in the flume and settling column studies was mud from Tampa Bay, Florida. Collection was from a site adjacent to a Hillsborough Bay navigation channel. It was predetermined by a bay mapping study (City of Tampa, 1986) to be an area of predominately fine sediment (clay and fine silt) and relatively high sedimentation rates (0.3-1 in/year). Grain size distribution of dispersed free particles, obtained by standard ASTM hydrometer method, is shown in Figure 4-1. It can be seen that d50 = 2.6 pm, which indicates that -50% of the sediment sample was finer than the upper limit of clay size particles (2 pm). Furthermore, less than 10% by weight of the sediment sample was coarse silt to fine sand. The flocculated sediment was pumped into 55 gallon drums in the field then to washing and storage tanks in the laboratory. The sediment was then mixed and decanted several times to equilibrate with tap water until a slight background salinity (I ppt) remained. Details of this procedure can be found in Cervantes (1987). The slight salinity was sufficient to maintain the flocculated state of the cohesive (< 2OPm) particles. Characterization tests were conducted at the University of U. 0.010 0.r0 Equivalent Groin Size (mm) Figure 4-1. Grain Size Distribution of Hillsborough Bay Mud Florida Soils Science Laboratory. X-Ray diffraction revealed that the clay size fraction was primarily made up of montmorillinite (91%) and very small amounts of kaolinite (4%) and quartz (5%). A cation exchange capacity (CEC) test reported 197.2 meq/100g (an unrealistic and suspected erroneously high value). Percent organic carbon content, determined by standard combustion technique (e.g., ASTM 500'C incineration) indicated that 5% by weight of the sediment sample was of detrital (organic) origin. Chemical composition of the fluid (tap water) can be found in Dixit (1982). |

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