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- Permanent Link:
- https://ufdc.ufl.edu/UFE0021804/00001
## Material Information- Title:
- Modeling Sediment Transport in the Sheet Flow Layer Using a Mixture Approach
- Creator:
- Hesser, Tyler J
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2007
- Language:
- english
- Physical Description:
- 1 online resource (66 p.)
## Thesis/Dissertation Information- Degree:
- Master's ( M.S.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Coastal and Oceanographic Engineering
Civil and Coastal Engineering - Committee Chair:
- Slinn, Donald N.
- Committee Members:
- Hsu, Tian-Jian
- Graduation Date:
- 12/14/2007
## Subjects- Subjects / Keywords:
- Bed sheets ( jstor )
Coastal engineering ( jstor ) Diameters ( jstor ) Modeling ( jstor ) Particle interactions ( jstor ) Sediment transport ( jstor ) Sediments ( jstor ) Velocity ( jstor ) Viscosity ( jstor ) Waves ( jstor ) Civil and Coastal Engineering -- Dissertations, Academic -- UF flow, mixture, model, sediment, sheet, transport - Genre:
- Electronic Thesis or Dissertation
born-digital ( sobekcm ) Coastal and Oceanographic Engineering thesis, M.S.
## Notes- Abstract:
- Due to the existence of sheet flow during storm events, numerically quantifying sediment transport during sheet flow conditions is an important step in understanding coastal dynamics. Traditional methods for modeling sediment transport require solving separate equations for fluid and particle motion. We have chosen an alternate approach that assumes a system containing sediment particles can be approximated as a mixture having variable density and viscosity that depend on the local sediment concentration. Here, the interactions are expressed through the mixture viscosity and a stress-induced diffusion term. There are five governing equations that describe the flow field - the mixture continuity and momentum equations, and a species continuity equation for the sediment. The addition of a bed-stiffness coefficient to simulate particle pressures in the bed has increased the consistency of the model results. This model which was developed for CROSSTEX has shown promising comparisons to Horikawa laboratory data demonstrating the effectiveness of the mixture model in simulating sediment transport in the sheet flow layer. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (M.S.)--University of Florida, 2007.
- Local:
- Adviser: Slinn, Donald N.
- Statement of Responsibility:
- by Tyler J Hesser.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Hesser, Tyler J. 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.
- Classification:
- LD1780 2007 ( lcc )
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The flow is driven by an external oscillating force, F, that approximates the velocity field of a surface gravity wave propagating over a seabed. The forcing equation is defined as 27w 27t F = pfU, cos t T T (2-9) where Uo and T are the amplitude and period of the oscillation, respectively. The sediment continuity equation (Eq. 2-10) describes how the sediment moves within the mixture. aC cuj at +x 9t 9xy acwt aNj Oz 9x (2-10) where Wt is the settling velocity and N is the diffusive flux of sediment described below in this section. Using laboratory experiments, Richardson and Zaki (1954) reported the settling velocity can be calculated as a function of sediment concentration by (2-11) where Wto is the settling velocity of a single particle in a clear fluid. The variable q in Eq. 2.2.1 is dependent on the particle Reynolds number, Rep, defined as (2-12) Rep- dpf I Wto I-If where d is the grain size diameter. The emperical constant q is then defined by Richardson and Zaki (1954) as 4.35Re-0.03 4.35Re-o.10 2.39 when 0.2 < Rep < 1, when 1 < Rep < 500, when 500 < Rep Wt Wto (t C), The particle pressure represents the normal force that opposes the net forces on the particle (Fig. 2-2). Integranular particle-particle and fluid-particle stresses both become very important in high concentration regions, while these stresses can be somewhat neglected in dilute regions. The particle pressure is implemented in the model using a concentration dependent bed stiffness coefficient, G(C). The particle pressure is solved for by APp,' + = -C. O1 F(t) Bp G(C) [u (uA u + F + A DA + g) (2-24) where C, 10 is the average concentration in the x-direction at a given y and z position, and G(C) is a function of the form G(C) = 0 when C < 0.43, G(C) C8 when 0.43 < C < 1.0. The G(C) function is modeled after Jenkins and Hanes (1998) calculations of particle pressure with respect to boundary l-iv-r height and viscosity relationship. Penko and Slinn (2006) tested the bed stiffness coefficient on many different cases and developed an eighth power exponential function as seen in Figure 2-3. Bp is a constant that was tested with values ranging from 0.0to0.2. The optimal bed response value was found to be B = 0.1. Higher values were overly rigid and values close to Bp = 0.0 could not hold bedformssush as sand ripples. The function starts allowing for the forces to be opposed when the concentration is greater than 30 percent by volume. However, the bed is never completely rigid due to the pore pressure and grain shape allowing water between the grains. The pu from Eq. 2-23 is used to implement the fluid pressure and solve for the momentum at the n + 1 time step. A =-APf n+ (2-25) At S= 30 S I 0 0.5 U/Uo S= 90 0 0.5 U/Uo = 150 U/Uo U/Uo Figure 3-3. HWK horizontally averaged velocity profiles through phases. 0=0 S S S 0 U/Uo S= 60 U/Uo = 120 -Model, no PP, 8 x 4 x 8 cm -Model, no PP, 4 x 2 x 4 cm * HWK exp. Figure 4-1. Comparisons of H, between two different sized domains of the model without particle pressure and HWK data set. The 8x4x8 cm domain is way off compared to the smaller domain. Im the figure to the right the model results have been shifted 30 degrees in time. increases the maximum viscosity from 18.8 g cm-ls-1 to 28.8 g cm s8-1. In Figure 4-13 there does not appear to be much of a change in the sheet flow li. -r thickness between the two different viscosity model runs. This result indicates that the viscosity formulation is not a sensitive variable in the overall model response. The particle diameter or d50 inputted into the model is a single value, while in nature or in laboratory experiments, the sediment tends to have a spectrum of grain sizes. SedMix 3D is a monodisperse system based on a uniform grain size, where as HWK data sets come from a polydisperse system with a nonuniform grain size. One possible solution to the phase lead was to input a larger grain size. The idea is the larger grains should be slower to be suspended into the water column and settle out more rapidly. Equation ?? shows that the sediment diffusion is strongly sensitive to the particle diameter. In preliminary tests, using the version of the model without particle pressure, the phase lead was decreased from 0.3 seconds to 0.15 seconds by increasing the particle diameter, d50 from 0.2 mm to 0.32 mm. However, with the improved model, including particle pressure in the bed 1 I r, this phase improvement is no longer realized for this experiment. In Figure 4-14 the d50 is increased from 0.02 cm to 0.032 cm, which means increasing the fall velocity from 2.6 cm s-1 to 3.2 cm s-1. Here, changing the sediment diameter does not fix the phase lead problem. However, the thickness of the sheet flow lI-v-r is thinner as would be expected with an increase in the size of the sediment particles. 2.2.3 Boundary and Initial Conditions The model is initialized with a raised bed to allow a disturbance in the flow. All the grid cells with sediment are packed to maximum concentration or C = 1.0, and the flow initially at rest with velocities equal to zero. Figure 2-5 is a snap shot of the initial flow conditions. Horizontal boundary conditions are periodic in the x and the y directions. This is equivalent to examining a small region under a long wave that approximately feels a uniform horizontal pressure gradient that oscillates in time. At the top of the domain a free slip boundary condition is used for the u and v velocities and a zero gradient boundary condition is used for the diffusion coefficient, D. The concentration, C, and the vertical velocity, w, both equal zero at the top of the domain. The bottom boundary condition is no-slip so u = v = w = 0. The pressure boundary conditions can be seen later in the the pressure section of this chapter. The concentration field and diffusion coefficient both have a no flux condition at the bottom. An initial averaged concentration profile can be seen in Figure 2-5. Boundary conditions are summarized in Table 2-1. 2.2.4 Input Parameters Initializing a run requires specific input parameters which allows the model to simulate many different flow conditions. The sediment particle diameter is the dso, and the initial settling velocity is for a single particle at the diameter of the d0o. The size of the domain must be tall enough to prevent sediment from reaching the top of the domain and long/wide enough to allow sediment motion to fully develop. The amount of the domain initially filled with sediment must also be determined to avoid motion of the sediment mixture or scouring at the bottom of the domain. Experimental specific variables are also needed such as the free stream velocity and the period of oscillation. From these input parameters, the rest of the flow conditions can be solved such as the time-steps for the run, the non-dimensional parameters listed above, and the particle Reynolds number. 5.3 Summary of Contributions ............... ........ 60 5.4 Future Research ............... .............. 60 REFERENCES ...... ........... .................. .. 62 BIOGRAPHICAL SKETCH ................. . . 66 MODELING SEDIMENT TRANSPORT IN THE SHEET FLOW LAYER USING A MIXTURE APPROACH By TYLER J. HESSER A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2007 Similarly, the mixture momentum equation is found by adding momentum conservation equations for the individual phases 9a,, a,9, ,,, aPM 9aij aPp + a / -/ I + O M + i + + a (2-6) at dxj dxi dxj dxi where PM is the mixture pressure, ij is the stress tensor, F is the external driving force as described in Eq. 2-9, g is the gravitational constant, and Pp is the particle pressure. Assuming the fluid-sediment mixture is a Newtonian fluid, Bagnold (1954) and Bird et al. (2002) have shown that nij can be approximated by ui + uj 2 uk Oxj Oxi (27) where p is the mixture viscosity, which is a function of sediment concentration as determined by Leighton and Acrivos (1987). Hunt et al. (2002) performed experiments similar to Bagnold (1954) to determine the effect of sediment concentration on the viscosity of a mixture. The variable viscosity used in SedMix 3-D from Leighton and Acrivos (1987), Eq. 2-8, is plotted against Hunt's experiments in Figure 2-1(b). The effects of the high concentration of particles are parameterized with a bulk viscosity. [= f C ,- j2 (2-8) Cp C In Eq 2-8, pf is the fluid viscosity and Cp is the maximum packing concentration. Subia et al. (1998) gives a range of 0.52 to 0.74 for the maximum packing concentration of sediment particles depending on the shape and size. For this research the maximum packing concentration is set to a value of 0.64 which approximates close random packing. REFERENCES Ahilan, R. V., Sleath, J. F. A., Mar. 1987. Sediment transport in oscillatory flow over flat beds. Journal of Hydraulic Engineering-ASCE 113 (3), 308-322. 1.3.2 Ahmed, A., Sato, S., 2001. Investigation of bottom boundary l v-r dynamics of movable bed by using enhanced piv technique. Coastal Engineering Journal 43 (4), 239-258. 1.3.1 Ahmed, A., Sato, S., 2003. A sheetflow transport model for .-i-ii, ii: oscillatory flows part i: uniform grain size sediments. Coastal Engineering 45 (C3), 321-337. 1.3.1, 1.3.2 Asano, T., Sep. 1995. Sediment transport under sheet-flow conditions. Journal of Waterway Port Coastal And Ocean Engineering-ASCE 121 (5), 239-246. 1.3, 1.3.1 Bagnold, R., 1946. Motion of waves in shallow water: Interaction between waves and sand bottoms. Proceedings of the Royal Society of London, Series A 187, 1-15. 1.3.1 Bagnold, R., 1954. Experiments on a gravity-free dispersion of large solid spheres in a newtonian fluid under shear. Proceedings of the Royal Society of London, Series A 225, 219-232. 1.3.2, 2.2.1, 2.2.1 Bird, R. B., Stewart, W. E., Lightfoot, E. N., 2002. Transport Phenomena. John Wiley & Sons, New York. 2.2.1 Calantoni, J., Holland, K. T., Drake, T. G., Sep. 2004. Modelling sheet-flow sediment transport in wave-bottom boundary 1-.-r i using discrete-element modelling. Philosophical Transactions of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 362 (1822), 1987-2001. 1.3.2 Dili i i M., Kioka, W., Mar. 2000. Long waves and the change in cross-shore sediment transport rates on a sheet flow dominated beach. Coastal Engineering Journal 42 (1), 87-110. 1.3.1 Dili iiii M., Moriya, T., Watanabe, A., Mar. 2001. A representative wave model for estimation of nearshore local transport rate. Coastal Engineering Journal 43 (1), 1-38. 1.3.2 Dil ii ii M., Watanabe, A., Nov. 1998. Transport rate under irregular sheet flow conditions. Coastal Engineering 35 (3), 167-183. 1.3.1 Dohmen-Janssen, C. M., Hanes, D. M., 2002. Sheet flow dynamics under monochromatic nonbreaking waves. Journal of Geophysical Research 107. 1.3.1 Dohmen-Janssen, C. M., Kroekenstoel, D. F., Hassan, W. N., Ribberink, J. S., Jun. 2002. Phase lags in oscillatory sheet flow: experiments and bed load modelling. Coastal Engineering 46 (1), 61-87. 1.3.2 driven flows. The study focused more on the suspension of sediments than the higher concentration transports. 1.4 Research Problem Studying the sheet flow l-v.-r regime is a difficult task because of the small thickness and high concentrations. Recently more field and laboratory experiments have began to develop advance techniques for measuring concentrations and transport rates in this region. However, the high concentration region at the bottom of the sheet flow l1v r is still a difficult area to collect accurate data. It is the goal of this research to develop a numerical model and calibrate it to past lab experiments, so it may be capable of advancing the current understanding of the region. pressure into the model, results are less variable based on the domain size. For this reason, the comparison between model and experimental results are done utilizing the particle pressure in the model, or version 2.0. Due to the experimental results presented by Horikawa et al. (1982), the half wave period is broken into six phases. The phases used are 0, 30, 60, 90, 120, 150 degrees where 180 degrees is the point of flow reversal and 90 degrees is the phase of maximum flow velocity. During initial calibration of the model, a trend became apparent in the results. The results were predicting the sheet flow lIv.r thickness 30 degrees ahead of the laboratory observations. Determining the effect of certain variables on this model data was an important step in the calibration of SedMix 3-D. The viscosity at the maximum packing concentration was tested to determine if a higher maximum viscosity was needed. The model is a mono-dispersed (single grain size) system that does not allow for the affect of larger or smaller grain diameters as seen in natural sediment distributions. One technique for testing the model sensitivity to sediment parameterizations is to input the d5o as a larger value than used in the laboratory experiments being tested. Due to the computational expense to run the model with a larger domain or smaller grid -p ii.. the extent of domain sizes and grid spacings are tested to save time with future use of the model. =0 0 =30 20 20 - Model velocity 15 HWK exp. 15 - 10 10 - \ 5* 0 0 C ~ 1 U/Uo 0 = 60 U/Uo 0=120 U/Uo U/Uo 0 = 90 U/Uo 0=150 U/Uo Figure 4-9. Horizontally averaged velocity profile comparisons between the model and HWK for each phase of the flow. 0 30 60 90 Phase (degrees) Figure 4-4. 120 150 Model with PP - -- -- Model without PP HWK exp. i . i . . . . O 30 60 90 Phase (degrees) 120 150 The particle pressure is able to correct the domain sensitivity from the original model, but it does not seem to affect the smaller and previously more accurate domain size. The comparison of Hs between the model with particle pressure and without particle pressure appear to be very similar, and with a good relationship to HWK. 0 I- The model is capable of calculating accurate results at a grid spacing of 0.625 cm, which runs to completion in approximately two di,-- on a single processor computer. The lower resolution runs allow alterations to the model to be made without waiting two weeks or more for results. Further improvements to the model can now be reviewed in an eighth of the time. 5.2.1 Phase Lead Before the focus of this research turned to the sensitivity of the model, one of the original goals was to understand and eliminate the phase lead of the model. SedMix 3D picks up sediment too quickly which shifts the sheet flow liv-r thickness plots 30 degrees or 0.3 seconds ahead of the Horikawa et al. (1982) data set. A few tests have been completed in hopes of finding an answer to this issue. Laboratory data and field data both use real sand which is a polydisperse system of many different grain sizes in the sediment. However, the model allows only one grain size to be inputted, so it is classified as a monodisperse system. In order to test the affects of the grain size on model results, the diameter of particles in the model was increased from 0.2 mm to 0.32 mm. The results do not correct the phase lead problem. The larger diameter run only decreases in magnitude of the sheet flow 1-ivr thickness and does not change the time dependent bed response. The second variable that was tested was the maximum viscosity of the mixture, which is correlated to the maximum packing concentration of the model. The maximum viscosity in the model is approximately half the value recommended by Hunt et al. (2002), so the viscosity was increased from 18.8 g cm-ls8- to 28.8 g cmls-1 This was accomplished by decreasing the maximum packing concentration from 0.615 to 0.612. The results of the new maximum packing viscosity are not significantly different compared to the maximum viscosity originally inputted. Through both the particle diameter and maximum viscosity tests, no solution to the 30 degree phase lead has been determined at this time. 8 6 4 2 - 0 30 60 90 120 150 Phase (degrees) Figure 3-1. HWK sheet flow liv-,r thickness. individual sediment particle, and applies these forces to transport the sediment in sheet flow. Calantoni et al. (2004) noticed the results of the model were closer to laboratory data when a non-spherical particle was utilized instead of a spherical particle. Another approach to modeling sediment transport in the sheet flow regime is to look at two different materials occupying the same space as a continuum or mixture (Drew, 1983). McTigue (1981) developed a mixture approach based on the equations presented by Drew (1983) in order to study the sediment transport over a flat bottom. The model was employ, ,1 to develop a better understanding of the turbulent diffusion required for the modeling of sediment transport. Hagutun and Eidsvik (1986) also utilized the equations for a mixture model presented by Drew (1983) to study the transport under oscillatory flows. However, the focus of this model was in the lower concentration region, so the particle-particle interactions were ignored for the model runs. Applying diffusion and viscosity equations developed by Leighton and Acrivos (1986) and Leighton and Acrivos (1987) respectively, Nir and Acrivos (1990) developed a mixture approach to modeling sediment transport on inclined surfaces. The research dove deeper into the strengths and weaknesses of the mixture approach, which aided in the building of the current model. An affordable way to study sediment transport due to stresses from a fluid without a full wave flume is to use a Couette apparatus, which creates stresses as fluid flows around a the centrally located sediment. Phillips et al. (1992) developed a mixture approach to modeling the sediment transport in a Couette flow. The model consists of Newtonian equations with a variable viscosity, and a diffusion equations for the shear induced particle migration. The diffusion equation, similar to the one in the present model, adds different components of the diffusion to achieve the overall diffusion. Aspects of the diffusion include; diffusion due to spatial variation in viscosity, diffusion due to spatial variations in collisions, and Brownian diffusion. Adding on to this research, Subia et al. (1998) applied the model developed by Phillips et al. (1992) to study a broader range of flows. The model results quantified how sediment transport occurs in pipe flows and in piston Copyright 2007 by Tyler J. Hesser where U, is the obital wave velocity and v is the fluid viscosity and can often be used to identify when the flow will transition from laminar to turbulent flow. Sheet flow is characterized by a highly concentrated region of sediment due to turbulent shear stress which erodes away ripples making a plane bed. Sheet flow transport has a large affect on the overall sediment transport in a region. However, sheet flow events tend to occur with large storm waves or shallower depths. The inception of sheet flow from a rippled bed has been studied in the past, and researchers have formulated equations to quantify this change. Manohar (1955) relates the inception of sheet flow to the mobility number, T, and the Reynolds number, Re, as seen in (Eq. 1-5). ( Re1/2) 2000 (1-5) On the other hand, Komar and Miller (1975) relate the inception of sheet flow to the Shields parameter, 0, and the Reynolds number, Re, as seen in (Eq. 1-6). (ORe1/3) =4.4 (1 6) A relationship between the fluid shear stress and the inception of sheet flow may be another mechanism to predicting the development of the sediment transport. Because of the dynamic nature of sediment transport, it has been difficult to quatify these transitional points from saltation and sedimentation over rippled beds to sheet flow. 1.3 Literature Review Sheet flow conditions are the dominant sediment transport mechanism during storms, but due to the difficulty of measurements in this region descriptive techniques are limited to qualitative estimates (Asano, 1995). Field experiments and models have alv--x depended on each other for advancement to occur. In order to understand numerical models, the laboratory and field experiments, which provide calibration for models, must be explored. to the stabilizing force on a sediment particle under waves. It is a measure of a sediment particle's tendency to move due to wave action. (aw)2 ( ) (s 1)gd where a is the wave orbital excursion (defined in Eq. 1-2), w is the radial frequency (27/T), s is the specific gravity of the sediment (s = 2.65 for quartz), and d is the grain size diameter. The wave orbital excursion is UoT a (12) 27 where Uo is the free stream velocity and T is the wave period. The shape of the wave obital velocity is an important parameter in the cross-shore sediment transport under breaking and nonbreaking waves (Hsu and Hanes, 2004). A second parameter used to measure incipient motion is the Shields parameter (Eq. 1-3). AT ,ii: researchers have found relationships between the type of motion present and the value of the Shields parameter. The Shields parameter, 2 0 = 7* (1-3) (s l)gd where u, is the friction velocity ( /rp), T is the bed shear stress, and p is the density, is the balance between disturbing and stabalizing forces on sand grains in the bed (Nielsen, 1992). The critical Shields parameter is used to determine the point when sediment will start to move based on the flow conditions. The Reynolds number (Eq. 1-4) is the ratio of inertial forces to viscous forces Re Ud (1-4) V S Model, = 18.8 g/cm*s - - Model, = 28.8 g cm*s S HWK exp. Figure 4-13. Affect of increasing the maximum viscosity from 18.8 g cm ls-1 to 28.8 g cm- s 1 12 12 12 12 Model, d = 0.02 cm - Model, d = 0.032 cm 10 10 HWKexp. 8 8 - 6- 6 - 0 30 60 90 120 150 0 30 60 90 120 150 Phase (degrees) Phase (degrees) Figure 4-14. Sensitivity of the sheet flow lw1 r thickness calculations to changing the particle diameter from 0.2 mm to 0.32 mm. CHAPTER 1 INTRODUCTION 1.1 General Introduction The transition of waves from deep water to shallow water is defined by the interaction of the wave with the sea bed. As waves become larger or depths become shallower the interaction increases until sediment is transported along the bottom. Wave interaction with the bed causes sediment motion in the following v--i,- saltation over flat beds, saltation and suspension over rippled beds, and sheet flow under high bed shear stress. The boundary l1-v-r between water and sediment is a very small l-v-r which makes it difficult to deploy gauges to measure this area without interacting with the flow. Because of this, field data has been difficult to collect and models of the region have been slow to develop. In the past ten years more advanced techniques are being developed for non obtrusive measurements of the flow parameters in this region. Sediment transport under waves can affect many visible aspects of the beaches populated by tourists from around the world. Sand bar migration, accretion, and erosion can be affected at the smallest level with the amount and direction of small scale sediment transport. Sheet flow generally occurs under larger storm waves, so the majority of the sediment transport occurs during these sheet flow events. Understanding the amount of sediment picked up and transported by a given set of waves allows larger scale processes to become more predictable. A model is only as good as the initial and boundary conditions and the input parameters. Hence, larger spacial models have a hard time being accurate if smaller scale models do not accurately predict the amount of sediment entrained by a given wave. A three-dimensional live-bed model has been developed which is capable of simulating all ranges of wave conditions to evaluate the type and amount of sediment transport that occurs. More specifically, sheetflow conditions from laboratory experiments can be replicated to calibrate the model. Once calibration is completed, the model will aid Nielsen, P., 1992. Coastal Bottom Boundary L -i rs and Sediment Transport. World Scientific, Singapore. 1.2 Nir, A., Acrivos, A., 1990. Sedimentation and sediment flow on inclined surfaces. Journal of Fluid Mechanics 212, 139-153. 1.3.2, 2.2.1, 2.2.1 Penko, A., Slinn, D., 2006. Modeling sand ripple evolution under wave and current boundary l v,-_-. In: Eos Trans. AGU. Vol. 87 of Ocean Sciences Meeting Supplement, Abstract OS44N-02. 2.2, 2.4, 4.1 Phillips, R. J., Ai-i2...-- R. C., Brown, R. A., Graham, A. L., Abbott, J. R., Jan. 1992. A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Physics Of Fluids A-Fluid Dynamics 4 (1), 30-40. 1.3.2, 2.2.1 Ribberink, J. S., Jul. 1998. Bed-load transport for steady flows and unsteady oscillatory flows. Coastal Engineering 34 (1-2), 59-82. 1.3.2 Ribberink, J. S., Al-Salem, A. A., Jul. 1995. Sheet flow and suspension of sand in oscillatory boundary lv. rs. Coastal Engineering 25 (3-4), 205-225. 1.3.1 Richardson, J., Zaki, W., 1954. The sedimentation of a suspension of uniform spheres under conditions of viscous flow. Chemical Engineering 3, 65-73. 2.2.1, 2.2.1 Rijn, L. v., 1984. Sediment transport, part i: Bed load transport 110 (10), 1431-1456. 1.2 Savage, S., McKeown, S., 1983. Shear stresses developed during rapid shear of concentrated suspensions of large spherical particles between concentric cylinders. Journal of Fluid Mechanics 127, 453-472. 1.3.2 Sawamoto, M., Yamashita, T., 1987. Sediment transport in sheet flow regime. Coastal Sediments, 415-423. 1.3.1 Slinn, D., Hesser, T., Burdick, G., 2006. Modeling sediment transport in oscillatory boundary l V-.-i~ using a mixture approach. In: Eos Trans. AGU. Vol. 87 of Ocean Science Meeting Supplement, Abstract OS44N-01. 2.2, 5.1 Smyth, W., 2003. Secondary kelvin-helmholtz instability in weakly stratified shear flow. Journal of Fluid Mechanics 497, 67-98. 4.1 Subia, S. R., Ingber, M. S., Mondy, L. A., Altobelli, S. A., Graham, A. L., Oct. 1998. Modelling of concentrated suspensions using a continuum constitutive equation. Journal Of Fluid Mechanics 373, 193-219. 1.3.2, 2.2.1 4.3 Model Sensitivity As stated previously, SedMix 3D was originally unable to accurately predict the sheet flow lI--r thickness for the domain size of 8 x 4 x 8 cm because of the large scale eddy features that penetrated too deeply into the sediment 1 .-r. Inputting the particle pressure into the model stiffened the resistance of the bed and enabled the sensitivity to domain changes to be reduced, as seen in Figure 4-11. The two model outputs are similar to each other, but both are still off from the HWK data set by a phase shift of 30 degrees. The magnitude of the larger domain remains within one standard deviation from the smaller domain. Understanding the grid spacing required for accurate results is important because SedMix 3D requires long computational times to run through entire periods. On average a 4 x 2 x 4 cm run with a delta or minimum grid spacing of 0.312 cm requires around two weeks to run three wave periods at 3.6 second periods. An understanding of the sensitivity of SedMix 3D to the grid spacing could speed up the time required to complete the model runs. It can be seen in Figure 4-12 when delta becomes greater than 0.625 cm, the calculation of the sheet flow li--r thickness becomes inaccurate. However, when delta is less than 0.625 cm, the calculated results are within one standard deviation of the higher resolution run. Model runs with a grid resolution of 0.625 cm require two di-4 to complete which allows for test runs to be examined in a shorter period of time. In this way, modifications to the model can be tested with the lower resolution case before running with the higher resolution for final production runs. 4.4 Phase Lead Nearly all of the model experiments had a phase lead compared to the lab results.We tested two variables to determine the model response in attempts to reduce the phase lead. The maximum viscosity, fmax, is a potentially important factor in the way the mixture responds. In order to alter the maximum viscosity in the model, the maximum packing concentration, Cp, was decreased from 0.615 to 0.612. The decrease in Cp two runs di- t1 i-, lI in this figure have the same dimensions and grid p i1:- so only the affect of particle pressure is represented. The affects of the particle pressure on the model are seen in the reduction of the sheet flow li.-,-r thickness to a normal thickness. The sheet flow lI-v.-r thickness in this figure is the average of the values from three oscillatory periods. The standard deviation from the mean is also di -i '1 in the figure to represent the differences in thickness values between periods. The physical change in the transport of sediment can be seen in Figure 4-3 where the still frame on the left is version 1.0 and the one on the right is version 2.0. The lack of particle pressure in the old model allowed large Kelvin-Helmholtz roll ups to form that caused thicker sheet flow l-'.-ir to develop. In the still frame on the right, the roll ups are not present which means the particle pressure is resisting the penetration of the fluid vorticies into the bed. These figures demonstrate the ability of the particle pressure to remove sensitivity to domain size changes from the model. However, as stated earlier SedMix 3D was found to have considerable prediction skill before the particle pressure wsa added for the 4 x 2 x 4 cm domain. It can be seen in Figure 4-4 the particle pressure did not change the accuracy of the model that was already present. The particle pressure did not fix the phase lead previously seen in the model, but it computes similar values to the old model for the smaller domain. Another example of this similarity can be seen in Figure 4-5 where the vertical concentration profiles are plotted against each other and the HWK data set. Several limitations that were present in the initial model have not been addressed by the particle pressure, and the model with the particle pressure preserves the good features of matching the laboratory concentration results of version 1.0. In order to further test the model, all the calibration tests were replicated with the new model including the particle pressure. 4.2 Current Model The three dimensional, live bed characteristics of SedMix 3D allow for many features of the flow to be computed and studied. In Figure 4-6, the characteristics of the flow can 0=0 I I 3 0. 01 0.2 0.3 0.4 C/Cm U/Uo 0 = 60 10 * -* 5- * * 0.1 0.2 03 09 C/Cm U/Uo S= 30 0.1 0.2 0.3 or] ol 'OS C/Cm U/Uo 0 = 90 0.1 02 C/Cm U/Uo = 120 0.1 0.2 0.3 0.4 C/Cm U/Uo = 150 0.1 0.2 0.3 0.4 C/Cm U/Uo Figure 3-4. HWK sediment fluxes through phases. 0.3 0.4 10 * 5- * 0 * LIST OF TABLES Table Page 2-1 Model Boundary Conditions ............... .......... .. 31 3-1 Run Conditions ............... .............. .. 37 to be a valuable tool in the understanding of sediment transport in the boundary 1liv. regime. 2.2.1 Governing Equations The five governing equations for the mixture model include the mixture continuity, mixture momentum, and sediment continuity equations. The mixture continuity equation combines the fluid and sediment species continuity equations O(1 C)pf (1 '- C)pfuf + =o (2 1) OCpa s Cpsuxj cp + ac(2 2) at + xj where C is the volumetric sediment concentration, pf is the fluid density, and ps is the sediment density. The fluid and sediment velocity are represented by uf, and us. The definition of mixture density and mixture momentum are p = (1 C)pf + Cp, (2-3) pUj (1 C)pffj + CpsUsj (2-4) where p and uj are the mixture density and mixture velocity. The mixture density is a linear function relating the concentration of sediment in the mixture to the variable density as seen in Figure 2-1(a). Combining equations Eq. 2-1 and Eq. 2-2 with reference to Eq. 2-3 and Eq. 2-4 produces the mixture continuity equation. + 0 (2-5) at Oxj I dedicate this thesis to my family. My parents for the guidance they have provided me. My brother for leading the way and albv-- ~ being there with words of support. Xena, Dutch, and Dodger ah--,i-b there for humor and relaxation. Also, to all my friends who have become family, and Ryan for albv--i being there to push me on good di,-- and lift me back up on bad. existing models in order to better quantify the sediment transport in the cross-shore direction. Data collection techniques have progressed to the point that laboratory measurements in the upper portions of the sheet flow lI-. -r produce accurate data. However, the lower ranges of the sheet flow l1.,-vr, or higher concentration region, is still to complicated for most measurement techniques to accurately sample. The model results coupled with existing laboratory data should advance the understanding of transport for the entire sheetflow 1 ,- '. 1.2 Background Large scale processes such as erosion and accretion begin with sediment particles and their interactions with fluid motion. Sediment transport is the effect of fluid motion interacting with sediment particles. The type of motion that occurs is dependent on characteristics of the flow and the sediment. When flow initiates, the sediment particles start to roll or slide leading to small amounts of sediment transport. Increasing the flow velocity and shear stresses causes saltation to occur that leads to suspension of the sediment (Rijn, 1984). The bed shape can be linked to the type of transport occurring under a given flow condition. A flat bed is normally apparent under slower flows with rolling and sliding of particles and beginning stages of saltation. As the flow increases, ripples can form with some saltation and suspension of sediment particles. Sheet flow is a combination of all types of transport occurring under high levels of shear stress and flow velocity. These types of transport transitions are directly related to the shear stress of the fluid, and the particle diameter of the sediment. Important parameters when examining sediment transport and sheet flow dynamics include the mobility number, i, wave orbital excursion, a, Shields' parameter, 0, and Reynolds number, Re. The mobility number (Eq. 1-1) is a ratio of the disturbing force SEDTRAN92 was updated by Li and Amos (2001) to SEDTRAN96 which includes more rigorous calibration, and additional sediment transport algorithms. SEDTRAN96 is tested to compute sediment transport during both waves and current events. Both these models have the ability to calculate near bed velocities and shear stresses along with sediment transport for both cohesive and noncohesive sediments. Malarkey and Davies (1998) developed a different variation on Grant and Madsen (1979) model by adding a time varying eddy viscosity. This addition helped solve some of the initial problems present in the original model by working through some non-linearities that were apparent. Malarkey et al. (2003) developed a quasi-steady, one-dimensional model including the capability to quatify unsteady sheet-flow. The focus on this project was the near bed area, and its ability to track erosion and deposition in relation to the amount of sediment in the sheet flow I-.-r. The model used empirical formulas for the sheet flow 1 ,-v-r thickness and for bed roughness to help in the calculation of near bed transport. The quasi-steady model predicts the sediment transport based on the instantaneous reaction of sediment to changes in the velocity of the fluid. However, a phase lag could be present in the transport of sand which would not be picked up by these quasi-steady models (Dohmen-Janssen et al., 2002). For this reason, Dohmen-Janssen et al. (2002) developed a semi-unsteady model to quatify the time dependent changes in the sediment transport. The net sediment transport was found to be over-predicted by the quasi-steady model developed by Ribberink (1998), and the semi-unsteady model gave a better prediction due to the ability to pick up the phase lag. The movement of sediment along the sea floor is the result of the interactions between water and sand. One approach to modeling these interactions is to develop an understanding of the fluid and the sediment independently. This approach is commonly called the two phase approach [eg. Dong and Zhang (1999); Hsu et al. (2003b,c); Hsu and Hanes (2004); Liu and Sato (2005b)]. Two phase flow modeling can be very difficult due to the complicated interactions between particles and between fluid and particles (Dong 0=0 Model - - Model shifted 30 degrees HWK exp. C/Cm 0 =60 C/Cm 0 =120 C/Cm 0 =30 0.5- C/Cm 0 =90 C/Cm 0 =150 * 0 C/Cm Figure 4-8. Vertical concentration profile comparisons between the model and HWK for each phase of the flow. In this mixture model, the diffusion flux approximated by Nir and Acrivos (1990) is emploiv, 1 in Eq. 2-10. Leighton and Acrivos (1986), Nir and Acrivos (1990), and Phillips et al. (1992) reported that the sediment diffusion depends on collision frequency, the spatial variation of viscosity, and Brownian diffusion such that N = N, + N, + NB (2-13) where N, is the flux due to collisions, N, is the flux due to the variation of viscosity, and NB is the flux to due Brownian diffusion. Leighton and Acrivos (1986) and Nir and Acrivos (1990) developed the expression for diffusive flux under the assumption that the flux is dominated by collisions. It includes a variable diffusion coefficient that is a function of particle size, concentration, mixture stresses, and is given by aC N = D j (2-14) where D =- d2(C) (2-15) Oxj and where 3(C) is a dimensionless coefficient empirically determined by Leighton and Acrivos (1986). This is given by (C) = aC2 + 8.8C (2-16) where a is an empirical constant. Leighton and Acrivos (1986) observed a as approximately 1/3. Sensetivity tests with our mixture model indicate that best fits to the present laboratory data sets are achieved with a = 0.4. Table 2-1. Model Boundary Conditions Bottom ac=0 6z )Oz U 0 v= 0 w 0 P= 0 Top C=o D 0-o az au 0 =z av 0 wz w 0 a (pw)* az At CHAPTER 3 EXPERIMENTAL PLAN 3.1 Experimental Cases Horikawa et al. (1982) performed a laboratory experiment in an oscillatory flow tank in order to quantify the transport inside the sheet flow lV v-r. A motor-driven 35 mm camera employ, -1 to capture the concentration in the upper flow, and the lower flow was captured with an electro-resistance sediment concentration detector. The physical properties of the experimental cases examined are shown in Table 3-1. Horikawa et al. (1982) reported the non-dimensional concentration, (C ) as a function of height, z, in millimeters and phase, 0, shown here in Figure 3-2. In Figure 3-3 and Figure 3-4, Horikawa's results for the velocity and sediment flux can be seen as a function of z and 0. The sheet flow 1I,--r thickness, Hs, is another quanity that can be compared to the laboratory data, and is defined as the 1, -r for which 0.05 < ( c) < 0.95. As seen in Figure 3-1 the sheet flow l-\ v-r thickness follows the pick up and depostion of sediment through the phases of an oscillatory flow. The thickness of the sheet flow lI,-i-r at different phases in an oscillatory flow can be compared between model results and Horikawa's laboratory data. SedMix 3D is a three dimensional model, so the results must be horizontally averaged in order to directly compare to the experimental results. 3.2 Model Tests During the initial analysis of model results, the model's calculations were promising for a specific set of initial conditions. As calibration of the model continued and initial conditions where change, changes in domain size appeared to be affecting the results. Specifically in the larger domain sizes, the sheet flow l .v?, grows to a thickness greater than seen in experimental results. Further test of other variables helped lead to the conclusion that a force was missing from the momentum equation. This force, as described in C'! lpter 2, is the particle pressure. The particle pressure adds a stiffness to the bed which holds back the over development of the sheet flow l1 -.-r. After adding the particle CHAPTER 5 SUMMARY 5.1 Conclusions The original model developed by Slinn et al. (2006) was capable of predicting characteristics of sheet flow liv. r sediment transport for specific conditions. Once these conditions were changed, the model was unable to calculate the correct sheet flow lIv.,r thicknesses or vertical concentration profiles. However, the addition of the particle pressure into the model, in the form of a bed stiffness coefficient, stabilizes the model with respect to the changing input parameters. The model now accurately calculates the magnitude of the sheet flow lI- -r thickness and vertical concentration profiles given by Horikawa et al. (1982). The horizontally averaged velocity profiles and the flux profiles outputted by the model appear to have the same profile through the phases. The model does appear to predict the pick up too quickly which causes the model to have a 30 degree phase lead on experimental data. Understanding the phase lead allows future improvement of the model to be targeted and corrections to be made in the post processing that minimize the affects of it in relation to the accuracy of the model. 5.2 Model Sensitivity The focus of this research turned early on to understanding the sensitivity of the model in hopes that future research will not be limited to specific input parameter ranges. The model originally was unable to produce accurate results under specific conditions. The size of the domain was the variable that most affected the results of the model. After the introduction of the particle pressure into the model, the domain no longer appears to affect the results. The sheet flow lIv.-r thickness is within one standard deviation of the Horikawa results. SedMix 3D is a sophisticated model and approximates the magnitude of the sheet flow lI- -r thickness, but the computation time needed for a three period run is very large. For this reason, it is important to understand the affects of grid spacing on model results. C&P Momentum -- & Velocity Figure 2-4. Ghost points di i -- <-4 The i r.--- ird grid with concentrations and pressures calculated at the points and velocities calculated at the arrows. The outer most points or the gray area represents the ghost points. t= 0.000 s x/ \Y C/Cm 0.98 0.91 0.84 0.77 0.70 0.63 0.56 0.49 0.42 0 .. 0.35 0.28 0.21 0.14 0.07 Figure 2-5. Initial conditions in 3-D grid with for pick up once the flow starts. the cross shaped raised section which allows - - - - - - 60 90 Phase (degrees) S Model Average -Model o Model +1o 0 HWK exp */ 30 60 90 Phase (degrees) 120 150 Figure 4-7. Comparison of sheet flow l-,-r thickness between model output and HWK data set. The standard deviation is also plotted. 2.2.2 Non-dimensionalizing Non-dimensional parameters are utilized in the calculations for the mixture model. The physical parameters are non-dimensionalized by using the following where the carat indicates a dimensionless parameter d w = c W = tlWtol d P1 Pf Uj Iwtol Substituting in for the scaled variables, Eq. 2-5, Eq. 2-6, and Eq. 2-10, become 9t+ O =0, at0 oi (2-17) + 9i at0 a ac + ac0 ataxi + + F6 Ri~i3 + -, Rep, xj xij acwt, a D ac 0+ D gij respectively, and where DJ =3(C) |au, aix ( -(1 -)dg Ri = "I *, and (2-18) (2-19) and (2-20) (2-21) be seen in three dimensions at six phases of a wave period. The flow is initiated at the 0 degree phase, and the pick up of sediment starts to occur at the 30 degree phase. The maximum velocity of a wave occurs at the 90 degree phase, and the largest sheet flow l?--r thickness for HWK experiments occurs during the 120 degree phase. The values of the sheet flow li-v-r thickness produced by the model and from the HWK experiments can be seen in Figure 4-7. The model appears to compute the correct magnitude of the maximum sheet flow l.T-r thickness from HWK, but the phase is off. The model has a 30 degree phase lead on the laboratory data which can be seen in the right had plot of Figure 4-7. More details will be given about the phase lead later in this chapter. The model does a good job of predicting the sheet flow l1v.-r thickness of the flow, but there are other characteristics that also should be checked. Comparisons between the vertical sediment concentration profiles of the model and the HWK data set are seen in Figure 4-8. The model results are a reasonable fit to the laboratory data, but the red dashed line, that represents the model results 0.3 seconds later, fits the data more accurately. The phase shifted vertical concentration profiles tend to be more accurate than the non shifted profiles through the first four phases. However, the non shifted profiles are more accurate in the last two phases. The model is able to predict the deposition phases of the flow more accurately, but the model predicts the pick up phase 30 degrees to early. The horizontally averaged velocity profiles of the flow are plotted against results from HWK in Figure 4-9. Multiplying the horizontally averaged velocity profiles with the vertical concentration profiles gives the flux of the mixture. In Figure 4-10, the blue line is the flux calculated by the model, while the red line is the flux calculated using 30 degrees shifted concentration profiles. In most of the phases, no significant difference seems to be present, but it appears that the differences that do exist result in the shifted flux being closer to the HWK data set. The phase average seen in Figure 4-10 is the average over three periods, or six samples. This checks for consistent results through all six samples. o 0 -. * 5* sO s 5 III 0 30 60 90 Phase (degrees) 120 150 - Model with PP - --- Model without PP 0 HWK exp. P0 30 60 90 Phase (degrees) Figure 4-2. Comparisons H, between the model with particle pressure and the model without particle pressure. As seen, the particle pressure model is much closer to the actual values of HWK. C/Cm 0.98 0.91 0.84 0.77 0.70 0.63 0.56 0.49 0.42 0.35 0.28 0.21 0.14 0.07 AK C/Cm 0.98 0.91 0.84 0.77 0.70 0.63 0.56 0.49 0.42 0.35 0.28 0.21 0.14 0.07 Figure 4-3. The physical difference between the particle pressure model and the model without particle pressure can be seen in this figure. On the right, there is a large roll up which contributes to the large sheet flow l-v- r thicknesses while on the left the roll up is inhibited by the particle pressure. 0 - * 5 n II '*^ 120 150 Scotian Shelf. Correlating the video images with wave and current data enhanced the understanding of the transition between ripples and sheet flow. 1.3.2 Past Models Due to the complexity of imput parameters and assumptions made by researchers, experimental data is the driving force for developing more advanced models. Difficulties with quantifying assumptions for physical processes in models has led to the predictive capability being mostly estimates (Calantoni et al., 2004). In the study of sheet flow sediment transport there are many different modeling approaches attempting to solve the same problem. Approaches such as transport models, both quasi-steady models and semi-unsteady models, two-phase models, and continuum models continue to add understanding to the sheet flow regime. Transport models developed by Ribberink (1998), Dili. 'iii i et al. (2001), and Ahmed and Sato (2003) are designed to quantify the sediment transport under waves. Ribberink (1998) developed a quasi-steady model based on the concepts first introduced by A1. i,- r-Peter and Mueller (1948) for steady flow. This model calculates net sediment transport based on the instantaneous Shields parameter. Dil, ii.i ' et al. (2001) proposed a semi-unsteady model based on the transport of uniform sediment under .i-vmmetric oscillations. Included in the model is a value for the thickness of the moving li---r which is described in this research as the sheet flow l1-ir thickness. Ahmed and Sato (2003) advanced the model by D1il .iii et al. (2001) by adding a new relationship for the moving li-v-r thickness based on the Shields parameter and sediment flow acceleration. Early in the development of models, Grant and Madsen (1979) designed a model for wave-current motion over rough beds using a eddy viscosity model. Li and Amos (1995) updated the eddy viscosity model by incorporating sediment transport solvers from past research based on the type of problem being solved. The model, SEDTRAN92, allows the user to pick one of seven algorithms based on the conditions. The algorithms picked can solve for bed load transport, suspended load transport, or cohesive transport. Dong, P., Z!i i.- K. F., Mar. 1999. Two-phase flow modelling of sediment motions in oscillatory sheet flow. Coastal Engineering 36 (2), 87-109. 1.3.2 Drake, T., Calantoni, J., 2001. Discrete particle model for sheet flow sediment transport in the nearshore. Journal of Geophysical Research 106 (C9), 19,859-19,868. 1.3.2 Drew, D. A., 1983. Mathematical-modeling of 2-phase flow. Annual Review of Fluid Mechanics 15, 261-291. 1.3.2 Grant, W., Madsen, 0., 1979. Combined wave and current interaction with a rough bottom. Journal of Geophysical Research 84, 1797-1808. 1.3.2 Hagutun, K., Eidsvik, K., 1986. Oscillating turbulent boundary 1-.-r with suspended sediments. Journal of Geophysical Research 91 (Cl1), 13,045-13,055. 1.3.2 Hassan, W. N., Ribberink, J. S., Sep. 2005. Transport processes of uniform and mixed sands in oscillatory sheet flow. Coastal Engineering 52 (9), 745-770. 1.3.1 Horikawa, K., Watanabe, A., Katori, S., 1982. Sediment transport under sheet flow condition. Coastal Engineering n/a, 1335-1352. 1.3.1, 3.1, 3.2, 4.1, 5.1, 5.2.1, 5.3 Hsu, T. J., Hanes, D. M., May 2004. Effects of wave shape on sheet flow sediment transport. Journal of Geophysical Research-Oceans 109 (C5), C05025. 1.2, 1.3.2 Hsu, T.-J., Jenkins, J. T., Liu, P. L.-F., 2003a. On two-phase sediment transport: Dilute flow. Journal of Geophysical Research 108, n/a. 1.3.2 Hsu, T. J., Jenkins, J. T., Liu, P. L. F., Aug. 2004. On two-phase sediment transport: sheet flow of massive particles. Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 460 (2048), 2223-2250. 1.3.2 Hsu, T. W., Chang, H. K., Hsieh, C. M., 2003b. A turbulence model of wave-induced sheet flow. Journal of Coastal Research 19 (1), 12-23. 1.3.2 Hsu, T. W., Chang, H. K., Hsieh, C. M., 2003c. A two-phase flow model of wave-induced sheet flow. Journal of Hydraulic Research 41 (3), 299-310. 1.3.2 Hunt, M., Zenit, R., Campbell, C., Brennen, C., 2002. Revisiting the 1954 suspension experiments of R. A. Bagnold. Journal of Fluid Mechanics 452, 1-24. 2.2.1, 2-1, 5.2.1 Jenkins, J., Hanes, D., 1998. Collisional sheet flows of sediment driven by a turbulent fluid. Fluid Mechanics 370, 29-52. 1.3.2, 2.4 Komar, P., Miller, M., 1975. The initiation of oscillatory ripple marks and the development of plane-bed at high shear stresses under waves. Journal of Sedimentary Petrology 45 (3), 697-703. 1.2 Laney, C. B., 1998. Computational C -1vii ii"nics. Cambridge University Press, New York. 2.3 C/Cm ....C/Cm 098 098 091 091 084 084 077 077 070 070 -, 063 0 63 0 56 056 049 049 042 042 035 -, 035 028 1, 028 0 21 021 S014 014 50 150 007 50 150 007 100 200 100 200 t= 4.200 s T I t= 4.500 s 077 0077 070 070 021 021 063 063 056 056 007 007 100 200 100 200 z z t= 4.800s IT\ | t= 5.100 S I S Cm .. C Cm 1098 098 091 091 S084 084 S077 077 070 ,'' 070 063 98 063 -,028 028 0 5621 021 Figure 4-6. Evolution of the flow through the phases of an oscillatory wave. Figure 4 6. Evolution of the flow through the phases of an oscillatory wave. CHAPTER 2 METHODOLOGY 2.1 Model Approach/Characteristics The two phase model approach requires independent equations for the fluid and sediment with closure assumptions used to represent the fluid-particle and the particle-particle interactions. Fluid-particle interactions are generally accounted for with lift and drag forces. Dilute flows tend to neglect the partcle-particle interactions. Dense flows cannot neglect particle-particle interactions so closure schemes similar in form to fluid stress relationships have been developed. A minimum of eight governing equations is required for development of a two phase flow model. SedMix 3D is an alternative approach to modeling sediment transport employing a fluid-sediment mixture instead of representing the sediment and fluid phases independently. A variable mixture density and viscosity are calculated depending on the local volumetric concentration. A mixture viscosity and a stress induced diffusivity represent these variable functions. Five governing equations are required for the mixture, three conservation of momentum equations, a sediment concentration equation, and a Poisson equation for the pressure field. Advancements in the understanding of stress induced diffusion have allowed the mixture approach to be a possible technique when modeling sediment transport. 2.2 Physics Slinn et al. (2006) developed a live-bed, three-dimensional, turbulent wave bottom boundary l.-r mixture model which was later improved by Penko and Slinn (2006). This model has previously been applied to the development and decay of ripples in oscillatory waves. SedMix 3-D is a finite difference model which solves for time dependent oscillating and steady currents on a three dimensional live bed. Utilizing a control volume approach on a i I,.-.-. red grid, the model is second order accurate in space and third order in time. Physical interactions in the two-phase system such as fluid-particle and particle-particle interactions are approximated using a variable viscosity and density. Leighton, D., Acrivos, A., 1986. Viscous resuspension. C'!I. dI11 I Engineering Science 41 (6), 1377-1384. 1.3.2, 2.2.1, 2.2.1, 2.2.1, 2.2.1 Leighton, D., Acrivos, A., Apr. 1987. Measurement of shear-induced self-diffusion in concentrated suspensions of spheres. Journal of Fluid Mechanics 177, 109-131. 1.3.2, 2.2.1, 2-1 Li, L., Sawamoto, M., 1995. Multi-phase model on sediment transport in sheet-flow regime under oscillatory flow. Coastal Engineering Japan 38 (2), 157-178. 1.3.2 Li, M. Z., Amos, C. L., May 1995. Sedtrans92 a sediment transport model for continental shelves. Computers & Geosciences 21 (4), 533-554. 1.3.2 Li, M. Z., Amos, C. L., Jun. 1999. Field observations of bedforms and sediment transport thresholds of fine sand under combined waves and currents. Marine Geology 158 (1-4), 147-160. 1.3.1 Li, M. Z., Amos, C. L., Jul. 2001. Sedtrans96: the upgraded and better calibrated sediment-transport model for continental shelves. Computers & Geosciences 27 (6), 619-645. 1.3.2 Liu, H., Sato, S., 2005a. Laboratory study on sheetflow sediment movement in the oscillatory turbulent boundary 1.w -r based on image analysis. Coastal Engineering Journal 47, 21-40. 1.3.1 Liu, H., Sato, S., 2005b. Modeling sediment movement under sheetflow conditions using a two-phase flow approach. Coastal Engineering 47 (4), 255-284. 1.3.2 Malarkey, J., Davies, A., 1998. Modelling wave-current interactions in rough turbulent bottom boundary 1 i-..--. Ocean Engineering 25 (2-3), 119-141. 1.3.2 Malarkey, J., Davies, A. G., Li, Z., Jun. 2003. A simple model of unsteady sheet-flow sediment transport. Coastal Engineering 48 (3), 171-188. 1.3.2 Manohar, M., 1955. Mechanics of bottom sediment movement due to wave action. Tech. rep., Technical Memo 75, Beach Erosion Board, U.S. Army Corps of Engineers, Washington, D.C. 1.2 McLean, S. R., Ribberink, J. S., Dohmen-Janssen, C. M., Hassan, W. N., M Ci 2001. Sand transport in oscillatory sheet flow with mean current. Journal of Waterway, Port, Coastal & Ocean Engineering 127 (3), 141-151. 1.3.1 McTigue, D., 1981. Mixture theory for suspended sediment transport. Journal of the Hydraulics Division-ASCE 107 (6), 659-673. 1.3.2 A iv r-Peter, E., Mueller, R., 1948. Formulas for bed-load transport. In: Proc. IAHR, Stockholm. 1.3.2 5.3 Summary of Contributions The direction of this research project has taken many twists and turns through the two years of work. The original goal was to finish up calibration on the model and compare it to data from the CROSSTEX laboratory experiment. Due to the problems with the model as described earlier in the paper, these goals have not been able to be reached. The focus of this research was to solve the problems with SedMix 3D so future research can continue without sensitivity problems. Diagnosing the trouble spots in the model took some time, and during this time the sensitivity tests for grid spacing were carried out. These tests allowed the studies using the particle pressure to occur in much less time. The particle pressure, originally developed and tested with SedMix 3D for sand ripples, needed to be tested for sheet flow. Minor adjustments allowed the bed stiffness coefficient to work for the sheet flow regime. Once the particle pressure was tested and found to be working, comparisons to Horikawa et al. (1982) were carried out. Finally the attempts at finding a relationship between the maximum viscosity or the particle diameter to the phase lead were completed. No solution has yet been found for the problem, but more understanding of the model has been achieved through the tests. 5.4 Future Research The field of small scale sediment transport or boundary li.-r dynamics is a constantly developing and progressing field. There are so many questions still to be answered, and many quality models attempting to find answers. SedMix 3D can fit in with these other numerical models with potential to advance the field. The next obvious step with this model is to compare results against additional laboratory experiments with different flow parameters. The goal of the future is to adapt the forcing function to allow irregular waves in the hope of calculating values of net sediment transport instead of just for oscillatory waves. Advancements to the mixture approach in the future could allow it 1.3.1 Past Field and Laboratory Experiments One of the first laboratory experiments concerning small scale sediment transport is Bagnold (1946). He employ, l an oscillating sand tray in water to artificially reproduce an oscillating water waves. Specifically, Bagnold (1946) attempted to measure the affect of waves on ripple growth and decay. The development of water flumes and wave tanks has since advanced the study of sediment transport. Horikawa et al. (1982) was one of the first sheet flow 1,- -r transport laboratory experiments. More details on this experiment are available in ('! Ilpter 3. Another early sheet flow laboratory experiment used a U-tube to simulate oscillatory wave conditions was Sawamoto and Yamashita (1987). Through this experiment the theory of 1.5 power relation between bed shear stress and sediment transport rate was confirmed with laboratory data. As technology developed the U-tube experiments were advanced with better measurement techniques and the ability to run both sinusoidal and .i-vmmetric waves. Ribberink and Al-Salem (1995) performed laboratory experiments in the attempt to study the vertical structure of boundary l~-v-r flow in the sheet flow regime. The experiment was run on .i-vmmetric and symmetric waves with an electro-resistance probe for measuring sediment concentrations. A three l,- rv transport system was discovered in the results consisting of a pick up 1 i--r, an upper sheet flow l1 -r, and a suspension l i-.-r. Using two painted light plastic particles, Asano (1995) studied the sediment transport rate by following the painted particles with a high resolution camera. The experiment attempted to quantify the inception of sheet flow. Results agreed with previous research of the mobility number raised to the 1.5 power. As more laboratory data sets were performed, different details of sheet flow sediment transport were able to be studied. Dil iiii and Watanabe (1998) used a loop-shape oscillatory/steady flow water tunnel capable of running .i-vii,,. I i. sawtoothh shape) and nonlinear (high narrow crest with a shallow wide trough) waves. D1ii iii I and Kioka (2000) continued on with these experiments CHAPTER 4 RESULTS 4.1 Original Model The mixture model, SedMix 3D version 1.0, was found to be an accurate model in comparisons with the Horikawa et al. (1982) or HWK data set. During the process of calibration, the sensitivity of the model to input changes was tested, and during these tests an additional level of complexity was found. The size of the domain was raised from 4 x 2 x 4 cm to 8 x 4 x 8 cm ito test for domain sensitivity. The calculated sheet flow l-i--r thickness for the larger domain was twice as large as the smaller domain. The smaller domain in version 1.0 was suppressing the Kelvin-Helmholtz billows (Smyth, 2003) from firmii,:. but the larger domain could not suppress the billows. The K-H billows in the larger domains created larger sheet flow thicknesses than experimental observed by Horikawa. In Figure 4-1, the comparisons between the two domain sizes and the HWK data set are plotted. The change in the size of the calculated sheet flow l-i--r thickness can be seen in both the unshifted and 30 degrees shifted plots. The 30 degrees shifted results are di-pl i-- I1 for easy comparisons between the model and experimental data. The model tends to pick up and deposit sediment 30 degrees ahead of the laboratory data. Theories for correcting the phase lead are still in the process of being tested. This problem forced all the progress of calibrating the model to be halted until a solution for the larger than expected sheet flow l-i--r thicknesses could be found. At the same time as the problems described above were discovered, Penko and Slinn (2006) were working on incorporating particle pressure into SedMix 3D to study ripples. A bed stiffness coefficient is the numerical implementation of the particle pressure in the model, as described in C'! lpter 2. The particle pressure was then incorporated into the sheet flow version of SedMix 3D, creating version 2.0, in order to test the effects of the particle pressure on faster flows. The change in the sheet flow l-v-r thickness between the model without particle pressure and with particle pressure can be seen in Figure 4-2. The Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science MODELING SEDIMENT TRANSPORT IN THE SHEET FLOW LAYER USING A MIXTURE APPROACH By Tyler J. Hesser December 2007 ('!C ,i: Donald N. Slinn Major: Coastal Engineering Due to the existence of sheet flow during storm events, numerically quantifying sediment transport during sheet flow conditions is an important step in understanding coastal dynamics. Traditional methods for modeling sediment transport require solving separate equations for fluid and particle motion. We have chosen an alternate approach that assumes a system containing sediment particles can be approximated as a mixture having variable density and viscosity that depend on the local sediment concentration. Here, the interactions are expressed through the mixture viscosity and a stress-induced diffusion term. There are five governing equations that describe the flow field the mixture continuity and momentum equations, and a species continuity equation for the sediment. The addition of a bed-stiffness coefficient to simulate particle pressures in the bed has increased the consistency of the model results. This model which was developed for Crosstex has shown promising comparisons to Horikawa laboratory data demonstrating the effectiveness of the mixture model in simulating sediment transport in the sheet flow l -i ,-r. ACKNOWLEDGMENTS I acknowledge the support of the Office of N i,. Research for funding of the CrossTex project, advisory committee for their guidance through the thesis process, and my fellow graduate students and office peers for their support and advice. Table 3-1. Conditions for Horikawa et al. (1982) laboratory experiment Condition HWK Particle diameter, d50 (cm) 0.02 Settling velocity, Wto (cm/s) 2.6 Fluid density, pf (g/cm3) 1.0 Particle density ps (g/cm3) 2.66 Fluid viscosity pf (g/cm/s) 0.0131 Maximum concentration, C, 0.6 Maximum packing fraction, Cp 0.615 Initial bed height, Hb (cm) 0.9 Period, T (s) 3.6 Free stream velocity, Uo (cm/s) 127 Simulation time (s) 10.8 Gravity Figure 2-2. The relationship between the particle pressure force and all other forces on a sediment particle. Cave Figure 2-3. A plot of the equation employ, 1- for the bed stiffness coefficiant. The model is not affected until the concentration reaches a minimum value of 0.48. 2.3 Numerics A control volume approach on a three dimensional -1 I.-.- red grid is applied as seen in Figure 2-4. The circles represent the location of concentration and pressure points while the arrows represent the location of velocity and momentum points. The shaded areas around the outside are ghost points used in the model. Spatial derivatives are calculated with a one-sided differencing scheme at the cell faces. The node values are found by taking the arithmetic mean of the four surrounding cell faces. This process gives the spacial derivatives a second order accuracy in space. A 3rd order Adams-Bashforth scheme is employ. -I to advance the concentration and momentum equations in time. However, explicit Euler and 2nd order Adams-Bashforth schemes are utilized as starting methods prior to the use of the 3rd order Adams-Bashforth. A projection method advances the pressure in time, with fractional time steps between the pressure and advection schemes. Techniques are required to solve the sediment continuity equation in order to ensure mass conservation, solution stability, and the propagation of the bed height as particles settle out. A harmonic mean acts as a flux limiter (Laney, 1998) to ensure the propagation of the bed height. A smoothing diffusion coefficient ensures the concentration gradient across three grid cells does not become to steep for the stability of the model 2.4 Pressure The projection method advances the pressure with fractional time stepping. The momentum equation is first calculated using the 3rd order Adams-Bashforth discretization scheme. Values for pu are found from the nonlinear, diffusion, forcing, and gravity terms. u = AB3(-uA u + 0 + Fbi -, .) (2-22) At Oxj The value of pu, where ^ represents a fractional time step, from Eq. 2-22 is emploi- -l in the next fractional time step to include particle pressure. PU pU AP 1-23) At P 2 (223) at 0=0 Model with PP ---- Model without PP HWK exp. C/Cm 0 =60 * C C/Cm 0/ 0=120 C/Cm 0=30 C/Cm 0 =90 - *' S5 1 C/Cm 0=150 C/Cm Figure 4-5. The comparisons of the vertical concentration profiles between the model with particle pressure and without particle pressure also shows good agreement through all phases of the flow. 0=0 Model flux Model flux, phase shift HWK exp. 0.2 0.3 0.4 C/Cm u/Uo S= 60 C/Cm u/Uo = 120 C/Cm u/Uo -0 S=30 0.2 0.3 0.4 C/Cm u/Uo 0 =90 C/Cm u/Uo 0 =150 C/Cm u/Uo Figure 4-10. Comparisons of sediment fluxes for the model versus HWK. The red line is the model flux calculated using a 30 degree phase shifted concentration profile. LIST OF FIGURES Figure Page 2-1 Mixture density and mixture viscosity. ............. .... 32 2-2 Forces on a sediment particle. .................. ......... 33 2-3 Bed stiffness coefficiant .............. .......... 33 2-4 SI .-.- .rd grid .................. .................. .. 34 2-5 Initial conditions . .................. ............ 34 3-1 HWK sheet flow iv.--r thickness ................ . ...... 38 3-2 HWK vertical concentrations. .................. ......... .. 39 3-3 HWK horizontally averaged velocity profiles. .............. 40 3-4 HWK sediment fluxes. ............... ........... .. 41 4-1 Old model sheet flow l-iv. r thickness. .................. .... 47 4-2 Particle pressure affect on large domain. ................. .. 48 4-3 3-D image of large domain. ............... ......... .. 48 4-4 Particle pressure affect on small domain. ............... .... .. 49 4-5 Particle pressure affect on concentration profiles. .... . . 50 4-6 Evolution of the flow. .................. .............. .. 51 4-7 Sheet flow l-v r thickness of model versus HWK. ............. .. 52 4-8 Vertical concentration profiles of model versus HWK. . . 53 4-9 Horizontally averaged velocity profiles of model versus HWK. . ... 54 4-10 Sediment fluxes of model versus HWK. ............. .. .. 55 4-11 Sensitivity of the model to domain size changes. ................. 56 4-12 Sensitivity of the model to grid size changes ................ .. .. 56 4-13 Affect of altering the maximum viscosity on H. ................ .. 57 4-14 Sensitivity to particle diameter changes. ............ .. .. 57 Taking the divergence of Eq. 2-25 and rearranging to solve for pressure yields apn+1 At A + -A(pu), + A, ,, (2-26) axj From the mixture continuity equation, (Eq. 2-17) it can be seen that Apu equals the partial of p with respect to time. Opu apy pw Op ( (2-27) ax ay az at Substituting (2-27) into 2-26 therefore reduces the approximation of pressure at the n + 1 time step to a computable Poisson equation of the form a2p n+1 P2 p n+1 2 p2+1] 1 [p a(pu) a(pv) 9(pw) S+ + + ax + ay + j (2-28) OX2 6y2 9z2 At t Ox y z The numerical boundary condition at the top of the domain applied for the approximation of Eq. 2-28 is (2 29) az At while the bottom boundary condition for pressure is constant because there is no flow through the boundary. 1.8 1.7 1.6- 1.5- -1.4 - 1.3- 1.2 1.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Sand Concentration (cm3/cm3) (a) 104 Hunt et al. S- L&A, Cp = 0.61 -- L&A, Cp = 0.615 103 0 > 102 Ul) 101 100 10-4 10-3 10-2 101 100 Sand Concentration (cm /cm3) (b) Figure 2-1. (a) The mixture density variation versus the sediment concentration. (b) Hunt et al. (2002) viscosity versus the Leighton and Acrivos (1987) viscosity calculations as sediment concentration changes. TABLE OF CONTENTS ACKNOW LEDGMENTS ................................. LIST O F TABLES . . . . . . . . . . LIST OF FIGURES . . . . . . . . . A B ST R A C T . . . . . . . . . . CHAPTER 1 INTRODUCTION .................................. 1.1 General Introduction 1.2 Background ..... 1.3 Literature Review . 1.3.1 Past Field and 1.3.2 Past Models 1.4 Research Problem Laboratory Experiments 2 M ETHODOLOGY .................................. 2.1 M odel Approach/Cl i ':teristics ........................ 2.2 P hysics . . . . . . . . . . 2.2.1 Governing Equations .......................... 2.2.2 Non-dimensionalizing .......................... 2.2.3 Boundary and Initial Conditions .................... 2.2.4 Input Param eters .. .. .. .. ... .. .. .. .. ... .. . 2.3 Numerics ...... 2.4 Pressure ....... 3 EXPERIMENTAL PLAN 3.1 Experimental Cases 3.2 Model Tests ..... 4 RESULTS ................ ................ . Original Model . Current Model . Model Sensitivity . Phase Lead ..... 5 SUMMARY ........ 5.1 Conclusions ..... 5.2 Model Sensitivity . 5.2.1 Phase Lead . . . . ..............................: BIOGRAPHICAL SKETCH Pursuing my education by learning more about the ocean has alv--,v- been a dream of mine. When I was in high school in Alpharetta, GA, I was drawn to the ocean by my love for scuba diving. I took any opportunity to travel out to the beach just to get a glimpse of the waves. I wanted to know more about the scientific aspect of the sea, so I decided to attend Coastal Carolina University for my undergraduate education. My in i, r at CCU was marine science, and I was soon drawn to the physical side of marine science. I feel in love with physics and mathematics, and I decided I would pursue my undergraduate research in coastal science. I participated in field experiments for two years including the beach evolution research and monitoring project or BERM along with learning how to use side scan sonars and other scientific research equipment. My time at CCU helped me decide to continue my studies and get a graduate degree in coastal engineering. The University of Florida was my top choice for graduate school because of the quality of education I knew I would receive here. I spent my first year in graduate school working as a teaching assistant under the guidance of Dr. Robert Theike in the hydrodynamics class. In May of 2005, I started working for Dr. Donald Slinn performing numerical simulations of small scale sediment transport. After two and a half years of working with numerical code, I have learned a lot in both computational fluid dynamics and coastal sciences. I am looking to continue my studies to earn a Ph.D. in coastal engineering. and Zhang, 1999). A method for quantifying fluid-particle interactions developed by Li and Sawamoto (1995) was employ, l in the model developed by Dong and Zhang (1999). Another qualitative assumption must be used for the particle-particle interactions in the higher concentration regions, so a relationship developed by Ahilan and Sleath (1987) was utilized. The turbulence closure applied in the model was Prandtl's mixing length theory as proposed by Li and Sawamoto (1995). Another approach to only model the particle to particle interactions is to look more in depth at the dilute regions instead of the highly concentrated region (Hsu et al., 2003a). Using the large scale Reynolds stresses to account for the fluid-sediment interactions and a k c turbulence closure scheme, Hsu et al. (2003a) developed a numerical model comparable to experimental results for dilute flows. Liu and Sato (2005b) developed a version of the two phase model using a similar assumption for turbulence closure as Dong and Zhang (1999). However, the fluid-particle interactions were assumed to be a direct relation to the drag force, the added mass force, and the vertical-directional lift force. The particle-particle interactions were assumed through relationships between the proposed ideas by Bagnold (1954) for linear relationships between stresses and Savage and McKeown (1983) formula for intergranular stress. One of the difficulties which must be overcome when using a two phase flow model is the qualification of the particle-particle interactions. Hsu et al. (2004) developed a model for the transport of massive particles where the particle-particle interactions become very important. These collision dominated flows were described by Jenkins and Hanes (1998) to be a function of granular temperature and the particle stress. Adding complexity to the model, Hsu et al. (2004) added closure for the fluid turbulent suspension from Hsu et al. (2003a) for the lower concentration regions. A more direct approach to solving the particle-particle interactions was undertaken by Drake and Calantoni (2001) and Calantoni et al. (2004). A discrete particle model was designed to study the transport of sediment starting at the particle level. The model calculates all the forces on an -- Model, domain 4 x 2 x 4 cm - - Model, domain 8 x 4 x 8 cm 0 HWK exp. T Figure 4-11. Sensitivity of the model to domain size changes. Figure 4-12. Sensitivity of the model to grid size changes. by testing the affects of long wave components on the transport in the sheet flow l'iv.r regime. The advancement of experimental studies on the sheet flow 1-iv-r allowed more realistic field conditions to be observed. McLean et al. (2001) performed laboratory experiments on sheet flow 1i-v-r motion under oscillatory waves with the addition of a current. The study was performed in an oscillatory water tunnel using a pair of conductivity sensors to measure concentration changes. The conductivity sensors were further developed by Hassan and Ribberink (2005) in order to measure time-depended concentrations and particle velocities in the sheet flow 1l -v-r. Using the sensors in a large oscillating water tunnel Hassan and Ribberink (2005) attempted to gain a better understanding of the size-selective sediment transport. Ahmed and Sato (2001) developed a PIV (Particle Image Velocimeter) technique for studies in the sheet flow regime which Ahmed and Sato (2003) later carried out under .,-mmetric oscillations. The PIV system was improved and added to existing measurement techniques by Liu and Sato (2005a). In an oscillatory flow water tunnel, the sheet flow liv.r sediment transport was studied using both a High Speed Video Camera and a PIV system. The sediment movement was recorded with the High Speed Video Camera and the horizontal and vertical velocities were measured with the PIV. As scientist continue to gain more understanding of the processes behind the transport of sediment during sheet flow conditions in oscillatory water tunnels, the next step was to apply this knowledge to full scale waves. Dohmen-Janssen and Hanes (2002) used the large-scale wave flume with natural sand to observe sheet flow transport underneath full scale waves. A MTA, Multiple Transducer Array, an ADV, Acoustic Doppler Velocimeter, and a CC' I, Conductivity Concentration Meter, quantified characteristics of the sediment such as bed level, flow velocity and sediment concentration. Li and Amos (1999) made an attempt at obtaining sediment transport data in field conditions by deploying a video camera and a S4 wave-current data profiler on the S0=0 0 = 30 5 5> 0.5 1 C/Cm C/Cm 0 0 = 60 ] 0 = 90 5 5-0 *I 0 * ; ----- 'r* ---- -i-- --- ^ ---- * o.0 0.5 0 * -5 -5 - C/Cm C/Cm ]o 0 =120 0 0 =150 0 * 0 * * 0.5 1 I 0 C/Cm C/Cm Figure 3-2. HWK vertical concentration profiles through phases. |

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PAGE 1 MODELINGSEDIMENTTRANSPORTINTHESHEETFLOWLAYERUSINGAMIXTUREAPPROACHByTYLERJ.HESSERATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2007 1 PAGE 2 Copyright2007byTylerJ.Hesser 2 PAGE 3 Idedicatethisthesistomyfamily.Myparentsfortheguidancetheyhaveprovidedme.Mybrotherforleadingthewayandalwaysbeingtherewithwordsofsupport.Xena,Dutch,andDodgeralwaysthereforhumorandrelaxation.Also,toallmyfriendswhohavebecomefamily,andRyanforalwaysbeingtheretopushmeongooddaysandliftmebackuponbad. 3 PAGE 4 ACKNOWLEDGMENTSIacknowledgethesupportoftheOceofNavalResearchforfundingoftheCrossTexproject,advisorycommitteefortheirguidancethroughthethesisprocess,andmyfellowgraduatestudentsandocepeersfortheirsupportandadvice. 4 PAGE 5 TABLEOFCONTENTS ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 1.1GeneralIntroduction .............................. 10 1.2Background ................................... 11 1.3LiteratureReview ................................ 13 1.3.1PastFieldandLaboratoryExperiments ................ 14 1.3.2PastModels ............................... 16 1.4ResearchProblem ................................ 20 2METHODOLOGY .................................. 21 2.1ModelApproach/Characteristics ........................ 21 2.2Physics ...................................... 21 2.2.1GoverningEquations .......................... 22 2.2.2Non-dimensionalizing .......................... 26 2.2.3BoundaryandInitialConditions .................... 27 2.2.4InputParameters ............................ 27 2.3Numerics .................................... 28 2.4Pressure ..................................... 28 3EXPERIMENTALPLAN .............................. 35 3.1ExperimentalCases ............................... 35 3.2ModelTests ................................... 35 4RESULTS ....................................... 42 4.1OriginalModel ................................. 42 4.2CurrentModel ................................. 43 4.3ModelSensitivity ................................ 45 4.4PhaseLead ................................... 45 5SUMMARY ...................................... 58 5.1Conclusions ................................... 58 5.2ModelSensitivity ................................ 58 5.2.1PhaseLead ............................... 59 5 PAGE 6 5.3SummaryofContributions ........................... 60 5.4FutureResearch ................................. 60 REFERENCES ....................................... 62 BIOGRAPHICALSKETCH ................................ 66 6 PAGE 7 LISTOFTABLES Table Page 2{1ModelBoundaryConditions ............................. 31 3{1RunConditions. ................................... 37 7 PAGE 8 LISTOFFIGURES Figure Page 2{1Mixturedensityandmixtureviscosity. ....................... 32 2{2Forcesonasedimentparticle. ............................ 33 2{3Bedstinesscoeciant. ............................... 33 2{4Staggardgrid. ..................................... 34 2{5Initialconditions. ................................... 34 3{1HWKsheetowlayerthickness. ........................... 38 3{2HWKverticalconcentrations. ............................ 39 3{3HWKhorizontallyaveragedvelocityproles. .................... 40 3{4HWKsedimentuxes. ................................ 41 4{1Oldmodelsheetowlayerthickness. ........................ 47 4{2Particlepressureaectonlargedomain. ...................... 48 4{33-Dimageoflargedomain. ............................. 48 4{4Particlepressureaectonsmalldomain. ...................... 49 4{5Particlepressureaectonconcentrationproles. .................. 50 4{6Evolutionoftheow. ................................. 51 4{7SheetowlayerthicknessofmodelversusHWK. ................. 52 4{8VerticalconcentrationprolesofmodelversusHWK. ............... 53 4{9HorizontallyaveragedvelocityprolesofmodelversusHWK. .......... 54 4{10SedimentuxesofmodelversusHWK. ....................... 55 4{11Sensitivityofthemodeltodomainsizechanges. .................. 56 4{12Sensitivityofthemodeltogridsizechanges. .................... 56 4{13AectofalteringthemaximumviscosityonHs. .................. 57 4{14Sensitivitytoparticlediameterchanges. ...................... 57 8 PAGE 9 AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceMODELINGSEDIMENTTRANSPORTINTHESHEETFLOWLAYERUSINGAMIXTUREAPPROACHByTylerJ.HesserDecember2007Chair:DonaldN.SlinnMajor:CoastalEngineeringDuetotheexistenceofsheetowduringstormevents,numericallyquantifyingsedimenttransportduringsheetowconditionsisanimportantstepinunderstandingcoastaldynamics.Traditionalmethodsformodelingsedimenttransportrequiresolvingseparateequationsforuidandparticlemotion.Wehavechosenanalternateapproachthatassumesasystemcontainingsedimentparticlescanbeapproximatedasamixturehavingvariabledensityandviscositythatdependonthelocalsedimentconcentration.Here,theinteractionsareexpressedthroughthemixtureviscosityandastress-induceddiusionterm.Therearevegoverningequationsthatdescribetheoweldthemixturecontinuityandmomentumequations,andaspeciescontinuityequationforthesediment.Theadditionofabed-stinesscoecienttosimulateparticlepressuresinthebedhasincreasedtheconsistencyofthemodelresults.ThismodelwhichwasdevelopedforCrosstexhasshownpromisingcomparisonstoHorikawalaboratorydatademonstratingtheeectivenessofthemixturemodelinsimulatingsedimenttransportinthesheetowlayer. 9 PAGE 10 CHAPTER1INTRODUCTION1.1GeneralIntroductionThetransitionofwavesfromdeepwatertoshallowwaterisdenedbytheinteractionofthewavewiththeseabed.Aswavesbecomelargerordepthsbecomeshallowertheinteractionincreasesuntilsedimentistransportedalongthebottom.Waveinteractionwiththebedcausessedimentmotioninthefollowingways:saltationoveratbeds,saltationandsuspensionoverrippledbeds,andsheetowunderhighbedshearstress.Theboundarylayerbetweenwaterandsedimentisaverysmalllayerwhichmakesitdiculttodeploygaugestomeasurethisareawithoutinteractingwiththeow.Becauseofthis,elddatahasbeendiculttocollectandmodelsoftheregionhavebeenslowtodevelop.Inthepasttenyearsmoreadvancedtechniquesarebeingdevelopedfornonobtrusivemeasurementsoftheowparametersinthisregion.Sedimenttransportunderwavescanaectmanyvisibleaspectsofthebeachespopulatedbytouristsfromaroundtheworld.Sandbarmigration,accretion,anderosioncanbeaectedatthesmallestlevelwiththeamountanddirectionofsmallscalesedimenttransport.Sheetowgenerallyoccursunderlargerstormwaves,sothemajorityofthesedimenttransportoccursduringthesesheetowevents.Understandingtheamountofsedimentpickedupandtransportedbyagivensetofwavesallowslargerscaleprocessestobecomemorepredictable.Amodelisonlyasgoodastheinitialandboundaryconditionsandtheinputparameters.Hence,largerspacialmodelshaveahardtimebeingaccurateifsmallerscalemodelsdonotaccuratelypredicttheamountofsedimententrainedbyagivenwave.Athree-dimensionallive-bedmodelhasbeendevelopedwhichiscapableofsimulatingallrangesofwaveconditionstoevaluatethetypeandamountofsedimenttransportthatoccurs.Morespecically,sheetowconditionsfromlaboratoryexperimentscanbereplicatedtocalibratethemodel.Oncecalibrationiscompleted,themodelwillaid 10 PAGE 11 existingmodelsinordertobetterquantifythesedimenttransportinthecross-shoredirection.Datacollectiontechniqueshaveprogressedtothepointthatlaboratorymeasurementsintheupperportionsofthesheetowlayerproduceaccuratedata.However,thelowerrangesofthesheetowlayer,orhigherconcentrationregion,isstilltocomplicatedformostmeasurementtechniquestoaccuratelysample.Themodelresultscoupledwithexistinglaboratorydatashouldadvancetheunderstandingoftransportfortheentiresheetowlayer.1.2BackgroundLargescaleprocessessuchaserosionandaccretionbeginwithsedimentparticlesandtheirinteractionswithuidmotion.Sedimenttransportistheeectofuidmotioninteractingwithsedimentparticles.Thetypeofmotionthatoccursisdependentoncharacteristicsoftheowandthesediment.Whenowinitiates,thesedimentparticlesstarttorollorslideleadingtosmallamountsofsedimenttransport.Increasingtheowvelocityandshearstressescausessaltationtooccurthatleadstosuspensionofthesediment Rijn 1984 .Thebedshapecanbelinkedtothetypeoftransportoccurringunderagivenowcondition.Aatbedisnormallyapparentunderslowerowswithrollingandslidingofparticlesandbeginningstagesofsaltation.Astheowincreases,ripplescanformwithsomesaltationandsuspensionofsedimentparticles.Sheetowisacombinationofalltypesoftransportoccurringunderhighlevelsofshearstressandowvelocity.Thesetypesoftransporttransitionsaredirectlyrelatedtotheshearstressoftheuid,andtheparticlediameterofthesediment.Importantparameterswhenexaminingsedimenttransportandsheetowdynamicsincludethemobilitynumber,,waveorbitalexcursion,a,Shields'parameter,,andReynoldsnumber,Re.ThemobilitynumberEq. 1{1 isaratioofthedisturbingforce 11 PAGE 12 tothestabilizingforceonasedimentparticleunderwaves.Itisameasureofasedimentparticle'stendencytomoveduetowaveaction.=a!2 s)]TJ/F15 11.955 Tf 11.956 0 Td[(1gd{1whereaisthewaveorbitalexcursiondenedinEq. 1{2 ,!istheradialfrequency=T,sisthespecicgravityofthesediments=2:65forquartz,anddisthegrainsizediameter.Thewaveorbitalexcursionisa=UoT 2{2whereUoisthefreestreamvelocityandTisthewaveperiod.Theshapeofthewaveobitalvelocityisanimportantparameterinthecross-shoresedimenttransportunderbreakingandnonbreakingwaves HsuandHanes 2004 .AsecondparameterusedtomeasureincipientmotionistheShieldsparameterEq. 1{3 .ManyresearchershavefoundrelationshipsbetweenthetypeofmotionpresentandthevalueoftheShieldsparameter.TheShieldsparameter,=u2 s)]TJ/F15 11.955 Tf 11.955 0 Td[(1gd{3whereuisthefrictionvelocityp ,isthebedshearstress,andisthedensity,isthebalancebetweendisturbingandstabalizingforcesonsandgrainsinthebed Nielsen 1992 .ThecriticalShieldsparameterisusedtodeterminethepointwhensedimentwillstarttomovebasedontheowconditions.TheReynoldsnumberEq. 1{4 istheratioofinertialforcestoviscousforcesRe=Uwd {4 12 PAGE 13 whereUwistheobitalwavevelocityandistheuidviscosityandcanoftenbeusedtoidentifywhentheowwilltransitionfromlaminartoturbulentow.Sheetowischaracterizedbyahighlyconcentratedregionofsedimentduetoturbulentshearstresswhicherodesawayripplesmakingaplanebed.Sheetowtransporthasalargeaectontheoverallsedimenttransportinaregion.However,sheetoweventstendtooccurwithlargestormwavesorshallowerdepths.Theinceptionofsheetowfromarippledbedhasbeenstudiedinthepast,andresearchershaveformulatedequationstoquantifythischange. Manohar 1955 relatestheinceptionofsheetowtothemobilitynumber,,andtheReynoldsnumber,Re,asseeninEq. 1{5 .Re1=2=2000{5Ontheotherhand, KomarandMiller 1975 relatetheinceptionofsheetowtotheShieldsparameter,,andtheReynoldsnumber,Re,asseeninEq. 1{6 .Re1=3=4:41{6Arelationshipbetweentheuidshearstressandtheinceptionofsheetowmaybeanothermechanismtopredictingthedevelopmentofthesedimenttransport.Becauseofthedynamicnatureofsedimenttransport,ithasbeendiculttoquatifythesetransitionalpointsfromsaltationandsedimentationoverrippledbedstosheetow.1.3LiteratureReviewSheetowconditionsarethedominantsedimenttransportmechanismduringstorms,butduetothedicultyofmeasurementsinthisregiondescriptivetechniquesarelimitedtoqualitativeestimates Asano 1995 .Fieldexperimentsandmodelshavealwaysdependedoneachotherforadvancementtooccur.Inordertounderstandnumericalmodels,thelaboratoryandeldexperiments,whichprovidecalibrationformodels,mustbeexplored. 13 PAGE 14 1.3.1PastFieldandLaboratoryExperimentsOneoftherstlaboratoryexperimentsconcerningsmallscalesedimenttransportis Bagnold 1946 .Heemployedanoscillatingsandtrayinwatertoarticiallyreproduceanoscillatingwaterwaves.Specically, Bagnold 1946 attemptedtomeasuretheaectofwavesonripplegrowthanddecay.Thedevelopmentofwaterumesandwavetankshassinceadvancedthestudyofsedimenttransport. Horikawaetal. 1982 wasoneoftherstsheetowlayertransportlaboratoryexperiments.MoredetailsonthisexperimentareavailableinChapter 3 .AnotherearlysheetowlaboratoryexperimentusedaU-tubetosimulateoscillatorywaveconditionswas SawamotoandYamashita 1987 .Throughthisexperimentthetheoryof1.5powerrelationbetweenbedshearstressandsedimenttransportratewasconrmedwithlaboratorydata.AstechnologydevelopedtheU-tubeexperimentswereadvancedwithbettermeasurementtechniquesandtheabilitytorunbothsinusoidalandasymmetricwaves. RibberinkandAl-Salem 1995 performedlaboratoryexperimentsintheattempttostudytheverticalstructureofboundarylayerowinthesheetowregime.Theexperimentwasrunonasymmetricandsymmetricwaveswithanelectro-resistanceprobeformeasuringsedimentconcentrations.Athreelayertransportsystemwasdiscoveredintheresultsconsistingofapickuplayer,anuppersheetowlayer,andasuspensionlayer.Usingtwopaintedlightplasticparticles, Asano 1995 studiedthesedimenttransportratebyfollowingthepaintedparticleswithahighresolutioncamera.Theexperimentattemptedtoquantifytheinceptionofsheetow.Resultsagreedwithpreviousresearchofthemobilitynumberraisedtothe1.5power.Asmorelaboratorydatasetswereperformed,dierentdetailsofsheetowsedimenttransportwereabletobestudied. DibajniaandWatanabe 1998 usedaloop-shapeoscillatory/steadyowwatertunnelcapableofrunningasymmetricsawtoothshapeandnonlinearhighnarrowcrestwithashallowwidetroughwaves. DibajniaandKioka 2000 continuedonwiththeseexperiments 14 PAGE 15 bytestingtheaectsoflongwavecomponentsonthetransportinthesheetowlayerregime.Theadvancementofexperimentalstudiesonthesheetowlayerallowedmorerealisticeldconditionstobeobserved. McLeanetal. 2001 performedlaboratoryexperimentsonsheetowlayermotionunderoscillatorywaveswiththeadditionofacurrent.Thestudywasperformedinanoscillatorywatertunnelusingapairofconductivitysensorstomeasureconcentrationchanges.Theconductivitysensorswerefurtherdevelopedby HassanandRibberink 2005 inordertomeasuretime-dependedconcentrationsandparticlevelocitiesinthesheetowlayer.Usingthesensorsinalargeoscillatingwatertunnel HassanandRibberink 2005 attemptedtogainabetterunderstandingofthesize-selectivesedimenttransport. AhmedandSato 2001 developedaPIVParticleImageVelocimetertechniqueforstudiesinthesheetowregimewhich AhmedandSato 2003 latercarriedoutunderasymmetricoscillations.ThePIVsystemwasimprovedandaddedtoexistingmeasurementtechniquesby LiuandSato 2005a .Inanoscillatoryowwatertunnel,thesheetowlayersedimenttransportwasstudiedusingbothaHighSpeedVideoCameraandaPIVsystem.ThesedimentmovementwasrecordedwiththeHighSpeedVideoCameraandthehorizontalandverticalvelocitiesweremeasuredwiththePIV.Asscientistcontinuetogainmoreunderstandingoftheprocessesbehindthetransportofsedimentduringsheetowconditionsinoscillatorywatertunnels,thenextstepwastoapplythisknowledgetofullscalewaves. Dohmen-JanssenandHanes 2002 usedthelarge-scalewaveumewithnaturalsandtoobservesheetowtransportunderneathfullscalewaves.AMTA,MultipleTransducerArray,anADV,AcousticDopplerVelocimeter,andaCCM,ConductivityConcentrationMeter,quantiedcharacteristicsofthesedimentsuchasbedlevel,owvelocityandsedimentconcentration. LiandAmos 1999 madeanattemptatobtainingsedimenttransportdataineldconditionsbydeployingavideocameraandaS4wave-currentdataproleronthe 15 PAGE 16 ScotianShelf.Correlatingthevideoimageswithwaveandcurrentdataenhancedtheunderstandingofthetransitionbetweenripplesandsheetow.1.3.2PastModelsDuetothecomplexityofimputparametersandassumptionsmadebyresearchers,experimentaldataisthedrivingforcefordevelopingmoreadvancedmodels.Dicultieswithquantifyingassumptionsforphysicalprocessesinmodelshasledtothepredictivecapabilitybeingmostlyestimates Calantonietal. 2004 .Inthestudyofsheetowsedimenttransporttherearemanydierentmodelingapproachesattemptingtosolvethesameproblem.Approachessuchastransportmodels,bothquasi-steadymodelsandsemi-unsteadymodels,two-phasemodels,andcontinuummodelscontinuetoaddunderstandingtothesheetowregime.Transportmodelsdevelopedby Ribberink 1998 Dibajniaetal. 2001 ,and AhmedandSato 2003 aredesignedtoquantifythesedimenttransportunderwaves. Ribberink 1998 developedaquasi-steadymodelbasedontheconceptsrstintroducedby Meyer-PeterandMueller 1948 forsteadyow.ThismodelcalculatesnetsedimenttransportbasedontheinstantaneousShieldsparameter. Dibajniaetal. 2001 proposedasemi-unsteadymodelbasedonthetransportofuniformsedimentunderasymmetricoscillations.Includedinthemodelisavalueforthethicknessofthemovinglayerwhichisdescribedinthisresearchasthesheetowlayerthickness. AhmedandSato 2003 advancedthemodelby Dibajniaetal. 2001 byaddinganewrelationshipforthemovinglayerthicknessbasedontheShieldsparameterandsedimentowacceleration.Earlyinthedevelopmentofmodels, GrantandMadsen 1979 designedamodelforwave-currentmotionoverroughbedsusingaeddyviscositymodel. LiandAmos 1995 updatedtheeddyviscositymodelbyincorporatingsedimenttransportsolversfrompastresearchbasedonthetypeofproblembeingsolved.Themodel,SEDTRAN92,allowstheusertopickoneofsevenalgorithmsbasedontheconditions.Thealgorithmspickedcansolveforbedloadtransport,suspendedloadtransport,orcohesivetransport. 16 PAGE 17 SEDTRAN92wasupdatedby LiandAmos 2001 toSEDTRAN96whichincludesmorerigorouscalibration,andadditionalsedimenttransportalgorithms.SEDTRAN96istestedtocomputesedimenttransportduringbothwavesandcurrentevents.Boththesemodelshavetheabilitytocalculatenearbedvelocitiesandshearstressesalongwithsedimenttransportforbothcohesiveandnoncohesivesediments. MalarkeyandDavies 1998 developedadierentvariationon GrantandMadsen 1979 modelbyaddingatimevaryingeddyviscosity.Thisadditionhelpedsolvesomeoftheinitialproblemspresentintheoriginalmodelbyworkingthroughsomenon-linearitiesthatwereapparent. Malarkeyetal. 2003 developedaquasi-steady,one-dimensionalmodelincludingthecapabilitytoquatifyunsteadysheet-ow.Thefocusonthisprojectwasthenearbedarea,anditsabilitytotrackerosionanddepositioninrelationtotheamountofsedimentinthesheetowlayer.Themodelusedempiricalformulasforthesheetowlayerthicknessandforbedroughnesstohelpinthecalculationofnearbedtransport.Thequasi-steadymodelpredictsthesedimenttransportbasedontheinstantaneousreactionofsedimenttochangesinthevelocityoftheuid.However,aphaselagcouldbepresentinthetransportofsandwhichwouldnotbepickedupbythesequasi-steadymodels Dohmen-Janssenetal. 2002 .Forthisreason, Dohmen-Janssenetal. 2002 developedasemi-unsteadymodeltoquatifythetimedependentchangesinthesedimenttransport.Thenetsedimenttransportwasfoundtobeover-predictedbythequasi-steadymodeldevelopedby Ribberink 1998 ,andthesemi-unsteadymodelgaveabetterpredictionduetotheabilitytopickupthephaselag.Themovementofsedimentalongtheseaooristheresultoftheinteractionsbetweenwaterandsand.Oneapproachtomodelingtheseinteractionsistodevelopanunderstandingoftheuidandthesedimentindependently.Thisapproachiscommonlycalledthetwophaseapproach[eg. DongandZhang 1999 ; Hsuetal. 2003b c ; HsuandHanes 2004 ; LiuandSato 2005b ].Twophaseowmodelingcanbeverydicultduetothecomplicatedinteractionsbetweenparticlesandbetweenuidandparticles Dong 17 PAGE 18 andZhang 1999 .Amethodforquantifyinguid-particleinteractionsdevelopedby LiandSawamoto 1995 wasemployedinthemodeldevelopedby DongandZhang 1999 .Anotherqualitativeassumptionmustbeusedfortheparticle-particleinteractionsinthehigherconcentrationregions,soarelationshipdevelopedby AhilanandSleath 1987 wasutilized.TheturbulenceclosureappliedinthemodelwasPrandtl'smixinglengththeoryasproposedby LiandSawamoto 1995 .Anotherapproachtoonlymodeltheparticletoparticleinteractionsistolookmoreindepthatthediluteregionsinsteadofthehighlyconcentratedregion Hsuetal. 2003a .UsingthelargescaleReynoldsstressestoaccountfortheuid-sedimentinteractionsandak)]TJ/F22 11.955 Tf 12.861 0 Td[(turbulenceclosurescheme, Hsuetal. 2003a developedanumericalmodelcomparabletoexperimentalresultsfordiluteows. LiuandSato 2005b developedaversionofthetwophasemodelusingasimilarassumptionforturbulenceclosureas DongandZhang 1999 .However,theuid-particleinteractionswereassumedtobeadirectrelationtothedragforce,theaddedmassforce,andthevertical-directionalliftforce.Theparticle-particleinteractionswereassumedthroughrelationshipsbetweentheproposedideasby Bagnold 1954 forlinearrelationshipsbetweenstressesand SavageandMcKeown 1983 formulaforintergranularstress.Oneofthedicultieswhichmustbeovercomewhenusingatwophaseowmodelisthequaticationoftheparticle-particleinteractions. Hsuetal. 2004 developedamodelforthetransportofmassiveparticleswheretheparticle-particleinteractionsbecomeveryimportant.Thesecollisiondominatedowsweredescribedby JenkinsandHanes 1998 tobeafunctionofgranulartemperatureandtheparticlestress.Addingcomplexitytothemodel, Hsuetal. 2004 addedclosurefortheuidturbulentsuspensionfrom Hsuetal. 2003a forthelowerconcentrationregions.Amoredirectapproachtosolvingtheparticle-particleinteractionswasundertakenby DrakeandCalantoni 2001 and Calantonietal. 2004 .Adiscreteparticlemodelwasdesignedtostudythetransportofsedimentstartingattheparticlelevel.Themodelcalculatesalltheforcesonan 18 PAGE 19 individualsedimentparticle,andappliestheseforcestotransportthesedimentinsheetow. Calantonietal. 2004 noticedtheresultsofthemodelwereclosertolaboratorydatawhenanon-sphericalparticlewasutilizedinsteadofasphericalparticle.Anotherapproachtomodelingsedimenttransportinthesheetowregimeistolookattwodierentmaterialsoccupyingthesamespaceasacontinuumormixture Drew 1983 McTigue 1981 developedamixtureapproachbasedontheequationspresentedby Drew 1983 inordertostudythesedimenttransportoveraatbottom.Themodelwasemployedtodevelopabetterunderstandingoftheturbulentdiusionrequiredforthemodelingofsedimenttransport. HagutunandEidsvik 1986 alsoutilizedtheequationsforamixturemodelpresentedby Drew 1983 tostudythetransportunderoscillatoryows.However,thefocusofthismodelwasinthelowerconcentrationregion,sotheparticle-particleinteractionswereignoredforthemodelruns.Applyingdiusionandviscosityequationsdevelopedby LeightonandAcrivos 1986 and LeightonandAcrivos 1987 respectively, NirandAcrivos 1990 developedamixtureapproachtomodelingsedimenttransportoninclinedsurfaces.Theresearchdovedeeperintothestrengthsandweaknessesofthemixtureapproach,whichaidedinthebuildingofthecurrentmodel.AnaordablewaytostudysedimenttransportduetostressesfromauidwithoutafullwaveumeistouseaCouetteapparatus,whichcreatesstressesasuidowsaroundathecentrallylocatedsediment. Phillipsetal. 1992 developedamixtureapproachtomodelingthesedimenttransportinaCouetteow.ThemodelconsistsofNewtonianequationswithavariableviscosity,andadiusionequationsfortheshearinducedparticlemigration.Thediusionequation,similartotheoneinthepresentmodel,addsdierentcomponentsofthediusiontoachievetheoveralldiusion.Aspectsofthediusioninclude;diusionduetospatialvariationinviscosity,diusionduetospatialvariationsincollisions,andBrowniandiusion.Addingontothisresearch, Subiaetal. 1998 appliedthemodeldevelopedby Phillipsetal. 1992 tostudyabroaderrangeofows.Themodelresultsquantiedhowsedimenttransportoccursinpipeowsandinpiston 19 PAGE 20 drivenows.Thestudyfocusedmoreonthesuspensionofsedimentsthanthehigherconcentrationtransports.1.4ResearchProblemStudyingthesheetowlayerregimeisadiculttaskbecauseofthesmallthicknessandhighconcentrations.Recentlymoreeldandlaboratoryexperimentshavebegantodevelopadvancetechniquesformeasuringconcentrationsandtransportratesinthisregion.However,thehighconcentrationregionatthebottomofthesheetowlayerisstilladicultareatocollectaccuratedata.Itisthegoalofthisresearchtodevelopanumericalmodelandcalibrateittopastlabexperiments,soitmaybecapableofadvancingthecurrentunderstandingoftheregion. 20 PAGE 21 CHAPTER2METHODOLOGY2.1ModelApproach/CharacteristicsThetwophasemodelapproachrequiresindependentequationsfortheuidandsedimentwithclosureassumptionsusedtorepresenttheuid-particleandtheparticle-particleinteractions.Fluid-particleinteractionsaregenerallyaccountedforwithliftanddragforces.Diluteowstendtoneglectthepartcle-particleinteractions.Denseowscannotneglectparticle-particleinteractionssoclosureschemessimilarinformtouidstressrelationshipshavebeendeveloped.Aminimumofeightgoverningequationsisrequiredfordevelpmentofatwophaseowmodel.SedMix3Disanalternativeapproachtomodelingsedimenttransportemployingauid-sedimentmixtureinsteadofrepresentingthesedimentanduidphasesindependently.Avariablemixturedensityandviscosityarecalculateddependingonthelocalvolumetricconcentration.Amixtureviscosityandastressinduceddiusivityrepresentthesevariablefunctions.Fivegoverningequationsarerequiredforthemixture,threeeconservationofmomentumequations,asedimentconcentrationequation,andaPoissonequationforthepressureeld.Advancementsintheunderstandingofstressinduceddiusionhaveallowedthemixtureapproachtobeapossibletechniquewhenmodelingsedimenttransport.2.2Physics Slinnetal. 2006 developedalive-bed,three-dimensional,turbulentwavebottomboundarylayermixturemodelwhichwaslaterimprovedby PenkoandSlinn 2006 .Thismodelhaspreviouslybeenappliedtothedevelopmentanddecayofripplesinoscillatorywaves.SedMix3-Disanitedierencemodelwhichsolvesfortimedependentoscillatingandsteadycurrentsonathreedimensionallivebed.Utilizingacontrolvolumeapproachonastaggeredgrid,themodelissecondorderaccurateinspaceandthirdorderintime.Physicalinteractionsinthetwo-phasesystemsuchasuid-particleandparticle-particleinteractionsareapproximatedusingavariableviscosityanddensity. 21 PAGE 22 2.2.1GoverningEquationsThevegoverningequationsforthemixturemodelincludethemixturecontinuity,mixturemomentum,andsedimentcontinuityequations.Themixturecontinuityequationcombinestheuidandsedimentspeciescontinuityequations@)]TJ/F22 11.955 Tf 11.956 0 Td[(Cf @t+@)]TJ/F22 11.955 Tf 11.955 0 Td[(Cfufj @xj=0{1@Cs @t+@Csusj @xj{2whereCisthevolumetricsedimentconcentration,fistheuiddensity,andsisthesedimentdensity.Theuidandsedimentvelocityarerepresentedbyuf,andus.Thedenitionofmixturedensityandmixturemomentumare=)]TJ/F22 11.955 Tf 11.956 0 Td[(Cf+Cs{3uj=)]TJ/F22 11.955 Tf 11.955 0 Td[(Cfufj+Csusj{4whereandujarethemixturedensityandmixturevelocity.ThemixturedensityisalinearfuntionrelatingtheconcentrationofsedimentinthemixturetothevariabledensityasseeninFigure 2{1a .CombiningequationsEq. 2{1 andEq. 2{2 withreferencetoEq. 2{3 andEq. 2{4 producesthemixturecontinuityequation.@ @t+@uj @xj=0{5 22 PAGE 23 Similarly,themixturemomentumequationisfoundbyaddingmomentumconservationequationsfortheindividualphases@ui @t+@uiuj @xj=)]TJ/F22 11.955 Tf 10.494 8.088 Td[(@PM @xi+@ij @xj+Fi1)]TJ/F22 11.955 Tf 11.955 0 Td[(gi3+@PP @xi{6wherePMisthemixturepressure,ijisthestresstensor,FistheexternaldrivingforceasdescribedinEq. 2{9 ,gisthegravitationalconstant,andPPistheparticlepressure.Assumingtheuid-sedimentmixtureisaNewtonianuid, Bagnold 1954 and Birdetal. 2002 haveshownthatijcanbeapproximatedbyij=@ui @xj+@uj @xi)]TJ/F15 11.955 Tf 13.15 8.088 Td[(2 3@uk @xk{7whereisthemixtureviscosity,whichisafunctionofsedimentconcentrationasdeterminedby LeightonandAcrivos 1987 Huntetal. 2002 performedexperimentssimilarto Bagnold 1954 todeterminetheeectofsedimentconcentrationontheviscosityofamixture.ThevariableviscosityusedinSedMix3-Dfrom LeightonandAcrivos 1987 ,Eq. 2{8 ,isplottedagainstHunt'sexperimentsinFigure 2{1b .Theeectsofthehighconcentrationofparticlesareparameterizedwithabulkviscosity.=f1:5CCp Cp)]TJ/F22 11.955 Tf 11.955 0 Td[(C2{8InEq 2{8 ,fistheuidviscosityandCpisthemaximumpackingconcentration. Subiaetal. 1998 givesarangeof0.52to0.74forthemaximumpackingconcentrationofsedimentparticlesdependingontheshapeandsize.Forthisresearchthemaximumpackingconcentrationissettoavalueof0.64whichapproximatescloserandompacking. 23 PAGE 24 Theowisdrivenbyanexternaloscillatingforce,F,thatapproximatesthevelocityeldofasurfacegravitywavepropagatingoveraseabed.TheforcingequationisdenedasF=fUo2 Tcos2 Tt{9whereUoandTaretheamplitudeandperiodoftheoscillation,respectivel.ThesedimentcontinuityequationEq. 2{10 describeshowthesedimentmoveswithinthemixture.@C @t+@Cuj @xj=)]TJ/F22 11.955 Tf 10.494 8.088 Td[(@CWt @z+@Nj @xj{10whereWtisthesettlingvelocityandNisthediusiveuxofsedimentdescribedbelowinthissection.Usinglaboratoryexperiments, RichardsonandZaki 1954 reportedthesettlingvelocitycanbecalculatedasafunctionofsedimentconcentrationbyWt=Wt0)]TJ/F22 11.955 Tf 11.955 0 Td[(Cq{11whereWt0isthesettlingvelocityofasingleparticleinaclearuid.ThevariableqinEq. 2.2.1 isdependentontheparticleReynoldsnumber,Rep,denedasRep=dfjWt0j f{12wheredisthegrainsizediameter.Theempericalconstantqisthendenedby RichardsonandZaki 1954 as q=4:35Re)]TJ/F21 7.97 Tf 6.586 0 Td[(0:03pwhen0:2 PAGE 25 Inthismixturemodel,thediusionuxapproximatedby NirandAcrivos 1990 isemployedinEq. 2{10 LeightonandAcrivos 1986 NirandAcrivos 1990 ,and Phillipsetal. 1992 reportedthatthesedimentdiusiondependsoncollisonfrequency,thespatialvariationofviscosity,andBrowniandiusionsuchthatN=Nc+N+NB{13whereNcistheuxduetocollisions,Nistheuxduetothevariationofviscosity,andNBistheuxtodueBrowniandiusion. LeightonandAcrivos 1986 and NirandAcrivos 1990 developedtheexpressionfordiusiveuxundertheassumptionthattheuxisdominatedbycollisions.Itincludesavariablediusioncoecientthatisafunctionofparticlesize,concentration,mixturestresses,andisgivenbyNj=Dj@C @xj{14whereDj=d2Cj@ui @xj{15andwhereCisadimensionlesscoecientempiricallydeterminedby LeightonandAcrivos 1986 .ThisisgivenbyC=C21+1 2e8:8C{16whereisanempiricalconstant. LeightonandAcrivos 1986 observedasapproximately1=3.Sensetivitytestswithourmixturemodelindicatethatbesttstothepresentlaboratorydatasetsareachievedwith=0:4. 25 PAGE 26 2.2.2Non-dimensionalizingNon-dimensionalparametersareutilizedinthecalculationsforthemixturemodel.Thephysicalparametersarenon-dimensionalizedbyusingthefollowingwherethecaratindicatesadimensionlessparameter ^xj=xj d^t=tjWtoj d^C=C Cm^= f^= f^uj=uj jWtoj^Wt=Wt jWtojSubstitutinginforthescaledvariables,Eq. 2{5 ,Eq. 2{6 ,andEq. 2{10 ,become@^ @^t+@^^uj @^xj=0;{17@^^ui @^t+@^^ui^uj @^xj=)]TJ/F22 11.955 Tf 10.494 8.087 Td[(@^PF @^xi+1 Rep@^ij @^xj+^Fi1)]TJ/F22 11.955 Tf 11.955 0 Td[(Rii3+@^PP @^xi;{18and@^C @^t+@^C^uj @^xj=)]TJ/F22 11.955 Tf 10.494 8.088 Td[(@^C^Wt @^z+@ @^xj^Dj@^C @^xj!;{19respectively,andwhere^Dj=Cj@^ui @^xjj;{20andRi=)]TJ/F15 11.955 Tf 13.023 0 Td[(^dg jW2toj:{21 26 PAGE 27 2.2.3BoundaryandInitialConditionsThemodelisinitializedwitharaisedbedtoallowadisturbanceintheow.Allthegridcellswithsedimentarepackedtomaximumconcentrationor^C=1:0,andtheowinitiallyatrestwithvelocitiesequaltozero.Figure 2{5 isasnapshotoftheinitialowconditions.Horizontalboundaryconditionsareperiodicinthexandtheydirections.Thisisequivalenttoexaminingasmallregionunderalongwavethatapproximatelyfeelsauniformhorizontalpressuregradientthatoscillatesintime.Atthetopofthedomainafreeslipboundaryconditionisusedfortheuandvvelocitiesandazerogradientboundaryconditionisusedforthediusioncoecient,D.Theconcentration,C,andtheverticalvelocity,w,bothequalzeroatthetopofthedomain.Thebottomboundaryconditionisno-slipsou=v=w=0.Thepressureboundaryconditionscanbeseenlaterinthethepressuresectionofthischapter.Theconcentrationeldanddiusioncoecientbothhaveanouxconditionatthebottom.AninitialaveragedconcentrationprolecanbeseeninFigure 2{5 .BoundaryconditionsaresummarizedinTable 2{1 .2.2.4InputParametersInitializingarunrequiresspecicinputparameterswhichallowsthemodeltosimulatemanydierentowconditions.Thesedimentparticlediameteristhed50,andtheinitialsettlingvelocityisforasingleparticleatthediameterofthed50.Thesizeofthedomainmustbetallenoughtopreventsedimentfromreachingthetopofthedomainandlong/wideenoughtoallowsedimentmotiontofullydevelop.Theamountofthedomaininitiallylledwithsedimentmustalsobedeterminedtoavoidmotionofthesedimentmixtureorscouringatthebottomofthedomain.Experimentalspecicvariablesarealsoneededsuchasthefreestreamvelocityandtheperiodofoscillation.Fromtheseinputparameters,therestoftheowconditionscanbesolvedsuchasthetime-stepsfortherun,thenon-dimensionalparameterslistedabove,andtheparticleReynoldsnumber. 27 PAGE 28 2.3NumericsAcontrolvolumeapproachonathreedimensionalstaggeredgridisappliedasseeninFigure 2{4 .Thecirclesrepresentthelocationofconcentrationandpressurepointswhilethearrowsrepresentthelocationofvelocityandmomentumpoints.Theshadedareasaroundtheoutsideareghostpointsusedinthemodel.Spatialderivativesarecalculatedwithaone-sideddierencingschemeatthecellfaces.Thenodevaluesarefoundbytakingthearithmeticmeanofthefoursurroundingcellfaces.Thisprocessgivesthespacialderivativesasecondorderaccuracyinspace.A3rdorderAdams-Bashforthschemeisemployedtoadvancetheconcentrationandmomentumequationsintime.However,explicitEulerand2ndorderAdams-Bashforthschemesareutilizedasstartingmethodspriortotheuseofthe3rdorderAdams-Bashforth.Aprojectionmethodadvancesthepressureintime,withfractionaltimestepsbetweenthepressureandadvectionschemes.Techniquesarerequiredtosolvethesedimentcontinuityequationinordertoensuremassconservation,solutionstability,andthepropagationofthebedheightasparticlessettleout.Aharmonicmeanactsasauxlimiter Laney 1998 toensurethepropagationofthebedheight.Asmoothingdiusioncoecientensurestheconcentrationgradientacrossthreegridcellsdoesnotbecometosteepforthestabilityofthemodel2.4PressureTheprojectionmethodadvancesthepressurewithfractionaltimestepping.Themomentumequationisrstcalculatedusingthe3rdorderAdams-Bashforthdiscretizationscheme.Valuesfor^uarefoundfromthenonlinear,diusion,forcing,andgravityterms.^u)]TJ/F22 11.955 Tf 11.955 0 Td[(un t=AB3)]TJ/F22 11.955 Tf 9.299 0 Td[(uu+@ij @xj+Fi1)]TJ/F22 11.955 Tf 11.955 0 Td[(gi3{22Thevalueof^u,where^representsafractionaltimestep,fromEq. 2{22 isemployedinthenextfractionaltimesteptoincludeparticlepressure.^^u)]TJ/F15 11.955 Tf 15.379 0 Td[(^u t=)]TJ/F15 11.955 Tf 9.299 0 Td[(Ppn+1 2{23 28 PAGE 29 TheparticlepressurerepresentsthenormalforcethatopposesthenetforcesontheparticleFig. 2{2 .Integranularparticle-particleanduid-particlestressesbothbecomeveryimportantinhighconcentrationregions,whilethesestressescanbesomewhatneglectedindiluteregions.Theparticlepressureisimplementedinthemodelusingaconcentrationdependentbedstinesscoecient,GC.TheparticlepressureissolvedforbyPpn+1 2=)]TJ/F15 11.955 Tf 11.964 3.022 Td[(Cx10Ft)]TJ/F22 11.955 Tf 11.955 0 Td[(BpGC[un)]TJ/F15 11.955 Tf 11.956 0 Td[(uu+F+D~u+g{24whereCx10istheaverageconcentrationinthex-directionatagivenyandzposition,andGCisafunctionoftheform GC=0whenC<0:43;GC=C8when0:43 PAGE 30 TakingthedivergenceofEq. 2{25 andrearrangingtosolveforpressureyieldst@Pn+1 @xi=)]TJ/F15 11.955 Tf 9.299 0 Td[(ui+^^ui:{26Fromthemixturecontinuityequation,Eq. 2{17 itcanbeseenthatuequalsthepartialofwithrespecttotime.)]TJ/F22 11.955 Tf 10.494 8.088 Td[(@u @x)]TJ/F22 11.955 Tf 13.15 8.088 Td[(@v @y)]TJ/F22 11.955 Tf 13.151 8.088 Td[(@w @z=@ @t{27Substituting 2{27 into 2{26 thereforereducestheapproximationofpressureatthen+1timesteptoacomputablePoissonequationoftheform@2Pn+1 @x2+@2Pn+1 @y2+@2Pn+1 @z2=1 t24@ @t+@^^u @x+@^^v @y+@^^w @z35{28ThenumericalboundaryconditionatthetopofthedomainapplyiedfortheapproximationofEq. 2{28 is@^P @^z=^^w t{29whilethebottomboundaryconditionforpressureisconstantbecausethereisnoowthroughtheboundary. 30 PAGE 31 Table2{1.ModelBoundaryConditions TopBottom C=0@C @z=0@D @z=0@D @z=0@u @z=0u=0@v @z=0v=0w=0w=0@P @z=w tP=0 31 PAGE 32 a bFigure2{1.aThemixturedensityvariationversusthesedimentconcentration.b Huntetal. 2002 viscosityversusthe LeightonandAcrivos 1987 viscositycalculationsassedimentconcentrationchanges. 32 PAGE 33 Figure2{2.Therelationshipbetweentheparticlepressureforceandallotherforcesonasedimentparticle. Figure2{3.Aplotoftheequationemployedforthebedstinesscoeciant.Themodelisnotaecteduntiltheconcentrationreachesaminimumvalueof0.48. 33 PAGE 34 Figure2{4.Thestaggardgridwithconcentrationsandpressurescalculatedatthepointsandvelocitiescalculatedatthearrows.Theoutermostpointsorthegrayarearepresentstheghostpoints. Figure2{5.Initialconditionsin3-Dgridwiththecrossshapedraisedsectionwhichallowsforpickuponcetheowstarts. 34 PAGE 35 CHAPTER3EXPERIMENTALPLAN3.1ExperimentalCases Horikawaetal. 1982 performedalaboratoryexperimentinanoscillatoryowtankinordertoquantifythetransportinsidethesheetowlayer.Amotor-driven35mmcameraemployedtocapturetheconcentrationintheupperow,andthelowerowwascapturedwithanelectro-resistancesedimentconcentrationdetector.ThephysicalpropertiesoftheexperimentalcasesexaminedareshowninTable 3{1 Horikawaetal. 1982 reportedthenon-dimensionalconcentration,C Cmasafunctionofheight,z,inmillimetersandphase,,shownhereinFigure 3{2 .InFigure 3{3 andFigure 3{4 ,Horikawa'sresultsforthevelocityandsedimentuxcanbeseenasafunctionofzand.Thesheetowlayerthickness,Hs,isanotherquanitythatcanbecomparedtothelaboratorydata,andisdenedasthelayerforwhich0:05C Cm0:95.AsseeninFigure 3{1 thesheetowlayerthicknessfollowsthepickupanddepostionofsedimentthroughthephasesofanoscillatoryow.ThethicknessofthesheetowlayeratdierentphasesinanoscillatoryowcanbecomparedbetweenmodelresultsandHorikawa'slaboratorydata.SedMix3Disathreedimensionalmodel,sotheresultsmustbehorizontallyaveragedinordertodirectlycomparetotheexperimentalresults.3.2ModelTestsDuringtheinitialanalysisofmodelresults,themodel'scalculationswerepromisingforaspecicsetofinitialconditions.Ascalibrationofthemodelcontinuedandinitialconditionswherechange,changesindomainsizeappearedtobeaectingtheresults.Specicallyinthelargerdomainsizes,thesheetowlayergrowstoathicknessgreaterthanseeninexperimentalresults.Furthertestofothervariableshelpedleadtotheconclusionthataforcewasmissingfromthemomentumequation.Thisforce,asdescribedinChapter 2 ,istheparticlepressure.Theparticlepressureaddsastinesstothebedwhichholdsbacktheoverdevelopmentofthesheetowlayer.Afteraddingtheparticle 35 PAGE 36 pressureintothemodel,resultsarelessvariablebasedonthedomainsize.Forthisreason,thecomparisonbetweenmodelandexperimentalresultsaredoneutilizingtheparticlepressureinthemodel,orversion2.0.Duetotheexperimentalresultspresentedby Horikawaetal. 1982 ,thehalfwaveperiodisbrokenintosixphases.Thephasesusedare0,30,60,90,120,150degreeswhere180degreesisthepointofowreversaland90degreesisthephaseofmaximumowvelocity.Duringinitialcalibrationofthemodel,atrendbecameapparentintheresults.Theresultswerepredictingthesheetowlayerthickness30degreesaheadofthelaboratoryobservations.DeterminingtheeectofcertainvariablesonthismodeldatawasanimportantstepinthecalibrationofSedMix3-D.Theviscosityatthemaximumpackingconcentrationwastestedtodetermineifahighermaximumviscositywasneeded.Themodelisamono-dispersedsinglegrainsizesystemthatdoesnotallowfortheaectoflargerorsmallergraindiametersasseeninnaturalsedimentdistributions.Onetechniquefortestingthemodelsensitivitytosedimentparameterizationsistoinputthed50asalargervaluethanusedinthelaboratoryexperimentsbeingtested.Duetothecomputationalexpensetorunthemodelwithalargerdomainorsmallergridspacing,theextentofdomainsizesandgridspacingsaretestedtosavetimewithfutureuseofthemodel. 36 PAGE 37 Table3{1.ConditionsforHorikawaetal.1982laboratoryexperiment ConditionHWK Particlediameter,d50cm0.02Settlingvelocity,Wt0cm/s2.6Fluiddensity,fg=cm31.0Particledensitysg=cm32.66Fluidviscosityfg/cm/s0.0131Maximumconcentration,Cm0.6Maximumpackingfraction,Cp0.615Initialbedheight,Hbcm0.9Period,Ts3.6Freestreamvelocity,U0cm/s127Simulationtimes10.8 37 PAGE 38 Figure3{1.HWKsheetowlayerthickness. 38 PAGE 39 Figure3{2.HWKverticalconcentrationprolesthroughphases. 39 PAGE 40 Figure3{3.HWKhorizontallyaveragedvelocityprolesthroughphases. 40 PAGE 41 Figure3{4.HWKsedimentuxesthroughtphases. 41 PAGE 42 CHAPTER4RESULTS4.1OriginalModelThemixturemodel,SedMix3Dversion1.0,wasfoundtobeanaccuratemodelincomparisonswiththe Horikawaetal. 1982 orHWKdataset.Duringtheprocessofcalibration,thesensitivityofthemodeltoinputchangeswastested,andduringthesetestsanadditionallevelofcomplexitywasfound.Thesizeofthedomainwasraisedfrom4x2x4cmto8x4x8cmitotestfordomainsensitivity.Thecalculatedsheetowlayerthicknessforthelargerdomainwastwiceaslargeasthesmallerdomain.Thesmallerdomaininverision1.0wassuppressingtheKelvin-Helmholtzbillows Smyth 2003 fromforming,butthelargerdomaincouldnotsuppressthebillows.TheK-HbillowsinthelargerdomainscreatedlargersheetowthicknessesthanexperimentalobservedbyHorikawa.InFigure 4{1 ,thecomparisonsbetweenthetwodomainsizesandtheHWKdatasetareplotted.Thechangeinthesizeofthecalculatedsheetowlayerthicknesscanbeseeninboththeunshiftedand30degreesshiftedplots.The30degreesshifedresultsaredisplayedforeasycomparisonsbetweenthemodelandexperimentaldata.Themodeltendstopickupanddepositysediment30degreesaheadofthelaboritorydata.Theoriesforcorrectingthephaseleadarestillintheprocessofbeingtested.Thisproblemforcedalltheprogressofcalibratingthemodeltobehalteduntilasolutionforthelargerthanexpectedsheetowlayerthicknessescouldbefound.Atthesametimeastheproblemsdescribedabovewerediscovered, PenkoandSlinn 2006 wereworkingonincorporatingparticlepressureintoSedMix3Dtostudyripples.Abedstinesscoecientisthenumericalimplementationoftheparticlepressureinthemodel,asdescribedinChapter 2 .TheparticlepressurewasthenincorporatedintothesheetowversionofSedMix3D,creatingversion2.0,inordertotesttheeectsoftheparticlepressureonfasterows.ThechangeinthesheetowlayerthicknessbetweenthemodelwithoutparticlepressureandwithparticlepressurecanbeseeninFigure 4{2 .The 42 PAGE 43 tworunsdisplayedinthisgurehavethesamedimensionsandgridspacing,soonlytheaectofparticlepressureisrepresented.Theaectsoftheparticlepressureonthemodelareseeninthereductionofthesheetowlayerthicknesstoanormalthickness.Thesheetowlayerthicknessinthisgureistheaverageofthevaluesfromthreeoscillatoryperiods.Thestandarddeviationfromthemeanisalsodisplayedintheguretorepresentthedierencesinthicknessvaluesbetweenperiods.ThephysicalchangeinthetransportofsedimentcanbeseeninFigure 4{3 wherethestillframeontheleftisversion1.0andtheoneontherightisversion2.0.ThelackofparticlepressureintheoldmodelallowedlargeKelvin-Helmholtzrollupstoformthatcausedthickersheetowlayerstodevelop.Inthestillframeontheright,therollupsarenotpresentwhichmeanstheparticlepressureisresistingthepenetrationoftheuidvorticiesintothebed.Theseguresdemonstratetheabilityoftheparticlepressuretoremovesensitivitytodomainsizechangesfromthemodel.However,asstatedearlierSedMix3Dwasfoundtohaveconsiderablepredictionskillbeforetheparticlepressurewsaaddedforthe4x2x4cmdomain.ItcanbeseeninFigure 4{4 theparticlepressuredidnotchangetheaccuracyofthemodelthatwasalreadypresent.Theparticlepressuredidnotxthephaseleadpreviouslyseeninthemodel,butitcomputessimilarvaluestotheoldmodelforthesmallerdomain.AnotherexampleofthissimilaritycanbeseeninFigure 4{5 wheretheverticalconcentrationprolesareplottedagainsteachotherandtheHWKdataset.Severallimitationsthatwerepresentintheinitialmodelhavenotbeenaddressedbytheparticlepressure,andthemodelwiththeparticlepressurepreservesthegoodfeaturesofmatchingthelaboratoryconcentrationresultsofversion1.0.Inordertofurthurtestthemodel,allthecalibrationtestswerereplicatedwiththenewmodelincludingtheparticlepressure.4.2CurrentModelThethreedimensional,livebedcharacteristicsofSedMix3Dallowformanyfeaturesoftheowtobecomputedandstudied.InFigure 4{6 ,thecharacteristicsoftheowcan 43 PAGE 44 beseeninthreedimensionsatsixphasesofawaveperiod.Theowisinitiatedatthe0degreephase,andthepickupofsedimentstartstooccuratthe30degreephase.Themaximumvelocityofawaveoccursatthe90degreephase,andthelargestsheetowlayerthicknessforHWKexperimentsoccursduringthe120degreephase.ThevaluesofthesheetowlayerthicknessproducedbythemodelandfromtheHWKexperimentscanbeseeninFigure 4{7 .ThemodelappearstocomputethecorrectmagnitudeofthemaximumsheetowlayerthicknessfromHWK,butthephaseiso.Themodelhasa30degreephaseleadonthelaboratorydatawhichcanbeseenintherighthadplotofFigure 4{7 .Moredetailswillbegivenaboutthephaseleadlaterinthischapter.Themodeldoesagoodjobofpredictingthesheetowlayerthicknessoftheow,butthereareothercharacteristicsthatalsoshouldbechecked.ComparisonsbetweentheverticalsedimentconcentrationprolesofthemodelandtheHWKdatasetareseeninFigure 4{8 .Themodelresultsareareasonablettothelaboratorydata,butthereddashedline,thatrepresentsthemodelresults0.3secondslater,tsthedatamoreaccurately.Thephaseshiftedverticalconcentrationprolestendtobemoreaccuratethanthenonshiftedprolesthroughtherstfourphases.However,thenonshiftedprolesaremoreaccurateinthelasttwophases.Themodelisabletopredictthedepositionphasesoftheowmoreaccurately,butthemodelpredictsthepickupphase30degreestoearly.ThehorizontallyaveragedvelocityprolesoftheowareplottedagainstresultsfromHWKinFigure 4{9 .Multiplyingthehorizontallyaveragedvelocityproleswiththeverticalconcentrationprolesgivestheuxofthemixture.InFigure 4{10 ,thebluelineistheuxcalculatedbythemodel,whiletheredlineistheuxcalculatedusing30degreesshiftedconcentrationproles.Inmostofthephases,nosignicantdierenceseemstobepresent,butitappearsthatthedierencesthatdoexistresultintheshifteduxbeingclosertotheHWKdataset.ThephaseaverageseeninFigure 4{10 istheaverageoverthreeperiods,orsixsamples.Thischecksforconsistantresultsthroughallsixsamples. 44 PAGE 45 4.3ModelSensitivityAsstatedpreviously,SedMix3Dwasoriginallyunabletoaccuratelypredictthesheetowlayerthicknessforthedomainsizeof8x4x8cmbecauseofthelargescaleeddyfeaturesthatpenetratedtoodeeplyintothesedimentlayer.Inputtingtheparticlepressureintothemodelstienedtheresistanceofthebedandenabledthesensitivitytodomainchangestobereduced,asseeninFigure 4{11 .Thetwomodeloutputsaresimilartoeachother,butbotharestillofromtheHWKdatasetbyaphaseshiftof30degrees.Themagnitudeofthelargerdomainremainswithinonestandarddeviationfromthesmallerdomain.UnderstandingthegridspacingrequiredforaccurateresultsisimportantbecauseSedMix3Drequireslongcomputationaltimestorunthroughentireperiods.Onaveragea4x2x4cmrunwithadeltaorminimumgridspacingof0.312cmrequiresaroundtwoweekstorunthreewaveperiodsat3.6secondperiods.AnunderstandingofthesensitivityofSedMix3Dtothegridspacingcouldspeedupthetimerequiredtocompletethemodelruns.ItcanbeseeninFigure 4{12 whendeltabecomesgreaterthan0.625cm,thecalculationofthesheetowlayerthicknessbecomesinaccurate.However,whendeltaislessthan0.625cm,thecalculatedresultsarewithinonestandarddeviationofthehigherresolutionrun.Modelrunswithagridresolutionof0.625cmrequiretwodaystocompletewhichallowsfortestrunstobeexaminedinashorterperiodoftime.Inthisway,modicationstothemodelcanbetestedwiththelowerresolutioncasebeforerunningwiththehigherresolutionfornalproductionruns.4.4PhaseLeadNearlyallofthemodelexperimentshadaphaseleadcomparedtothelabresults.Wetestedtwovariablestodeterminethemodelresponseinattemptstoreducethephaselead.Themaximumviscosity,max,isapotentiallyimportantfactorinthewaythemixtureresponds.Inordertoalterthemaximumviscosityinthemodel,themaximumpackingconcentration,Cp,wasdecreasedfrom0.615to0.612.ThedecreaseinCp 45 PAGE 46 increasesthemaximumviscosityfrom18.8gcm)]TJ/F21 7.97 Tf 6.586 0 Td[(1s)]TJ/F21 7.97 Tf 6.587 0 Td[(1to28.8gcm)]TJ/F21 7.97 Tf 6.586 0 Td[(1s)]TJ/F21 7.97 Tf 6.587 0 Td[(1.InFigure 4{13 theredoesnotappeartobemuchofachangeinthesheetowlayerthicknessbetweenthetwodierentviscositymodelruns.Thisresultindicatesthattheviscosityformulationisnotasensitivevariableintheoverallmodelresponse.Theparticlediameterord50inputtedintothemodelisasinglevalue,whileinnatureorinlaboratoryexperiments,thesedimenttendstohaveaspectrumofgrainsizes.SedMix3Disamonodispersesystembasedonauniformgrainsize,whereasHWKdatasetscomefromapolydispersesystemwithanonuniformgrainsize.Onepossiblesolutiontothephaseleadwastoinputalargergrainsize.Theideaisthelargergrainsshouldbeslowertobesuspendedintothewatercolumnandsettleoutmorerapidly.Equation??showsthatthesedimentdiusionisstronglysensitivetotheparticlediameter.Inpreliminarytests,usingtheversionofthemodelwithoutparticlepressure,thephaseleadwasdecreasedfrom0.3secondsto0.15secondsbyincreasingtheparticlediameter,d50from0.2mmto0.32mm.However,withtheimprovedmodel,includingparticlepressureinthebedlayer,thisphaseimprovementisnolongerrealizedforthisexperiment.InFigure 4{14 thed50isincreasedfrom0.02cmto0.032cm,whichmeansincreasingthefallvelocityfrom2.6cms)]TJ/F21 7.97 Tf 6.587 0 Td[(1to3.2cms)]TJ/F21 7.97 Tf 6.587 0 Td[(1.Here,changingthesedimentdiameterdoesnotxthephaseleadproblem.However,thethicknessofthesheetowlayeristhinneraswouldbeexpectedwithanincreaseinthesizeofthesedimentparticles. 46 PAGE 47 Figure4{1.ComparisonsofHsbetweentwodierentsizeddomainsofthemodelwithoutparticlepressureandHWKdataset.The8x4x8cmdomainiswayocomparedtothesmallerdomain.Imtheguretotherightthemodelresultshavebeenshifted30degreesintime. 47 PAGE 48 Figure4{2.ComparisonsHsbetweenthemodelwithparticlepressureandthemodelwithoutparticlepressure.Asseen,theparticlepressuremodelismuchclosertotheactualvaluesofHWK. Figure4{3.Thephysicaldierencebetweentheparticlepressuremodelandthemodelwithoutparticlepressurecanbeseeninthisgure.Ontheright,thereisalargerollupwhichcontributestothelargesheetowlayerthicknesseswhileonthelefttherollupisinhibitedbytheparticlepressure. 48 PAGE 49 Figure4{4.Theparticlepressureisabletocorrectthedomainsensitivityfromtheoriginalmodel,butitdoesnotseemtoaectthesmallerandpreviouslymoreaccuratedomainsize.ThecomparisonofHsbetweenthemodelwithparticlepressureandwithoutparticlepressureappeartobeverysimilar,andwithagoodrelationshiptoHWK. 49 PAGE 50 Figure4{5.Thecomparisonsoftheverticalconcentrationprolesbetweenthemodelwithparticlepressureandwithoutparticlepressurealsoshowsgoodagreementthroughallphasesoftheow. 50 PAGE 51 Figure4{6.Evolutionoftheowthroughthephasesofanoscillatorywave. 51 PAGE 52 Figure4{7.ComparisonofsheetowlayerthicknessbetweenmodeloutputandHWKdataset.Thestandarddeviationisalsoplotted. 52 PAGE 53 Figure4{8.VerticalconcentrationprolecomparisonsbetweenthemodelandHWKforeachphaseoftheow. 53 PAGE 54 Figure4{9.HorizontallyaveragedvelocityprolecomparisonsbetweenthemodelandHWKforeachphaseoftheow. 54 PAGE 55 Figure4{10.ComparisonsofsedimentuxesforthemodelversusHWK.Theredlineisthemodeluxcalculatedusinga30degreephaseshiftedconcentrationprole. 55 PAGE 56 Figure4{11.Sensitivityofthemodeltodomainsizechanges. Figure4{12.Sensitivityofthemodeltogridsizechanges. 56 PAGE 57 Figure4{13.Aectofincreasingthemaximumviscosityfrom18.8gcm)]TJ/F21 7.97 Tf 6.587 0 Td[(1s)]TJ/F21 7.97 Tf 6.587 0 Td[(1to28.8gcm)]TJ/F21 7.97 Tf 6.586 0 Td[(1s)]TJ/F21 7.97 Tf 6.586 0 Td[(1. Figure4{14.Sensitivityofthesheetowlayerthicknesscalculationstochangingtheparticlediameterfrom0.2mmto0.32mm. 57 PAGE 58 CHAPTER5SUMMARY5.1ConclusionsTheoriginalmodeldevelopedby Slinnetal. 2006 wascapableofpredictingcharacteristicsofsheetowlayersedimenttransportforspecicconditions.Oncetheseconditionswerechanged,themodelwasunabletocalculatethecorrectsheetowlayerthicknessesorverticalconcentrationproles.However,theadditionoftheparticlepressureintothemodel,intheformofabedstinesscoecient,stabilizesthemodelwithrespecttothechanginginputparameters.Themodelnowaccuratelycalculatesthemagnitudeofthesheetowlayerthicknessandverticalconcentrationprolesgivenby Horikawaetal. 1982 .Thehorizontallyaveragedvelocityprolesandtheuxprolesoutputtedbythemodelappeartohavethesameprolethroughthephases.Themodeldoesappeartopredictthepickuptooquicklywhichcausesthemodeltohavea30degreephaseleadonexperimentaldata.Understandingthephaseleadallowsfutureimprovementofthemodeltobetargetedandcorrectionstobemadeinthepostprocessingthatminimizetheaectsofitinrelationtotheaccuracyofthemodel.5.2ModelSensitivityThefocusofthisresearchturnedearlyontounderstandingthesensitivityofthemodelinhopesthatfutureresearchwillnotbelimitedtospecicinputparameterranges.Themodeloriginallywasunabletoproduceaccurateresultsunderspecicconditions.Thesizeofthedomainwasthevariablethatmostaectedtheresultsofthemodel.Aftertheintroductionoftheparticlepressureintothemodel,thedomainnolongerappearstoaecttheresults.ThesheetowlayerthicknessiswithinonestandarddeviationoftheHorikawaresults.SedMix3Disasophisticatedmodelandapproximatesthemagnitudeofthesheetowlayerthickness,butthecomputationtimeneededforathreeperiodrunisverylarge.Forthisreason,itisimportanttounderstandtheaectsofgridspacingonmodelresults. 58 PAGE 59 Themodeliscapableofcalculatingaccurateresultsatagridspacingof0.625cm,whichrunstocompletioninapproximatelytwodaysonasingleprocessorcomputer.Thelowerresolutionrunsallowalterationstothemodeltobemadewithoutwaitingtwoweeksormoreforresults.Furtherimprovementstothemodelcannowbereviewedinaneighthofthetime.5.2.1PhaseLeadBeforethefocusofthisresearchturnedtothesensitivityofthemodel,oneoftheoriginalgoalswastounderstandandeliminatethephaseleadofthemodel.SedMix3Dpicksupsedimenttooquicklywhichshiftsthesheetowlayerthicknessplots30degreesor0.3secondsaheadofthe Horikawaetal. 1982 dataset.Afewtestshavebeencompletedinhopesofndingananswertothisissue.Laboratorydataandelddatabothuserealsandwhichisapolydispersesystemofmanydierentgrainsizesinthesediment.However,themodelallowsonlyonegrainsizetobeinputted,soitisclassiedasamonodispersesystem.Inordertotesttheaectsofthegrainsizeonmodelresults,thediameterofparticlesinthemodelwasincreasedfrom0.2mmto0.32mm.Theresultsdonotcorrectthephaseleadproblem.Thelargerdiameterrunonlydecreasesinmagnitudeofthesheetowlayerthicknessanddoesnotchangethetimedependentbedresponse.Thesecondvariablethatwastestedwasthemaximumviscosityofthemixture,whichiscorrelatedtothemaximumpackingconcentrationofthemodel.Themaximumviscosityinthemodelisapproximatelyhalfthevaluerecommendedby Huntetal. 2002 ,sotheviscositywasincreasedfrom18.8gcm)]TJ/F21 7.97 Tf 6.586 0 Td[(1s)]TJ/F21 7.97 Tf 6.586 0 Td[(1to28.8gcm1s)]TJ/F21 7.97 Tf 6.587 0 Td[(1.Thiswasaccomplishedbydecreasingthemaximumpackingconcentrationfrom0.615to0.612.Theresultsofthenewmaximumpackingviscosityarenotsignicantlydierentcomparedtothemaximumviscosityoriginallyinputted.Throughboththeparticlediameterandmaximumviscositytests,nosolutiontothe30degreephaseleadhasbeendeterminedatthistime. 59 PAGE 60 5.3SummaryofContributionsThedirectionofthisresearchprojecthastakenmanytwistsandturnsthroughthetwoyearsofwork.TheoriginalgoalwastonishupcalibrationonthemodelandcompareittodatafromtheCROSSTEXlaboratoryexperiment.Duetotheproblemswiththemodelasdescribedearlierinthepaper,thesegoalshavenotbeenabletobereached.ThefocusofthisresearchwastosolvetheproblemswithSedMix3Dsofutureresearchcancontinuewithoutsensitivityproblems.Diagnosingthetroublespotsinthemodeltooksometime,andduringthistimethesensitivitytestsforgridspacingwerecarriedout.Thesetestsallowedthestudiesusingtheparticlepressuretooccurinmuchlesstime.Theparticlepressure,originallydevelopedandtestedwithSedMix3Dforsandripples,neededtobetestedforsheetow.Minoradjustmentsallowedthebedstinesscoecienttoworkforthesheetowregime.Oncetheparticlepressurewastestedandfoundtobeworking,comparisonsto Horikawaetal. 1982 werecarriedout.Finallytheattemptsatndingarelationshipbetweenthemaximumviscosityortheparticlediametertothephaseleadwerecompleted.Nosolutionhasyetbeenfoundfortheproblem,butmoreunderstandingofthemodelhasbeenachievedthroughthetests.5.4FutureResearchTheeldofsmallscalesedimenttransportorboundarylayerdynamicsisaconstantlydevelopingandprogressingeld.Therearesomanyquestionsstilltobeanswered,andmanyqualitymodelsattemptingtondanswers.SedMix3Dcantinwiththeseothernumericalmodelswithpotentialtoadvancetheeld.Thenextobviousstepwiththismodelistocompareresultsagainstadditionallaboratoryexperimentswithdierentowparameters.Thegoalofthefutureistoadapttheforcingfunctiontoallowirregularwavesinthehopeofcalculatingvaluesofnetsedimenttransportinsteadofjustforoscillatorywaves.Advancementstothemixtureapproachinthefuturecouldallowit 60 PAGE 61 tobeavaluabletoolintheunderstandingofsedimenttransportintheboundarylayerregime. 61 PAGE 62 REFERENCES 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BIOGRAPHICALSKETCHPursuingmyeducationbylearningmoreabouttheoceanhasalwaysbeenadreamofmine.WhenIwasinhighschoolinAlpharetta,GA,Iwasdrawntotheoceanbymyloveforscubadiving.Itookanyopportunitytotravelouttothebeachjusttogetaglimpseofthewaves.Iwantedtoknowmoreaboutthescienticaspectofthesea,soIdecidedtoattendCoastalCarolinaUniversityformyundergraduateeducation.MymajoratCCUwasmarinescience,andIwassoondrawntothephysicalsideofmarinescience.Ifeelinlovewithphysicsandmathematics,andIdecidedIwouldpursuemyundergraduateresearchincoastalscience.IparticipatedineldexperimentsfortwoyearsincludingthebeachevolutionresearchandmonitoringprojectorBERMalongwithlearninghowtousesidescansonarsandotherscienticresearchequipment.MytimeatCCUhelpedmedecidetocontinuemystudiesandgetagraduatedegreeincoastalengineering.TheUniversityofFloridawasmytopchoiceforgraduateschoolbecauseofthequalityofeducationIknewIwouldreceivehere.IspentmyrstyearingraduateschoolworkingasateachingassistantundertheguidanceofDr.RobertTheikeinthehydrodynamicsclass.InMayof2005,IstartedworkingforDr.DonaldSlinnperformingnumericalsimulationsofsmallscalesedimenttransport.Aftertwoandahalfyearsofworkingwithnumericalcode,Ihavelearnedalotinbothcomputationaluiddynamicsandcoastalsciences.IamlookingtocontinuemystudiestoearnaPh.D.incoastalengineering. 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