Modeling Sediment Transport in the Sheet Flow Layer Using a Mixture Approach

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Modeling Sediment Transport in the Sheet Flow Layer Using a Mixture Approach
Hesser, Tyler J
Place of Publication:
[Gainesville, Fla.]
University of Florida
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1 online resource (66 p.)

Thesis/Dissertation Information

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:


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
Electronic Thesis or Dissertation
born-digital ( sobekcm )
Coastal and Oceanographic Engineering thesis, M.S.


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 )
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Includes vita.
Includes bibliographical references.
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Thesis (M.S.)--University of Florida, 2007.
Adviser: Slinn, Donald N.
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by Tyler J Hesser.

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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


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


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


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


Rep- dpf I Wto

where d is the grain size diameter. The emperical constant q is then defined by Richardson

and Zaki (1954) as




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


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)

S= 30

0 0.5


S= 90

0 0.5


= 150

U/Uo U/Uo

Figure 3-3. HWK horizontally averaged velocity profiles through phases.






S= 60


= 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


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







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.


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.

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.

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


=0 0 =30
20 20 -

Model velocity
15 HWK exp. 15 -

10 10 -

\ 5*

0 0
C ~ 1


0 = 60





0 = 90




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.




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


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

a (12)

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,

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)

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.


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


I I 3 0.
01 0.2 0.3 0.4

C/Cm U/Uo

0 = 60


5- *

0.1 0.2 03

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


5- *


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.


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


- - Model shifted 30 degrees
HWK exp.


0 =60


0 =120


0 =30



0 =90


0 =150

* 0


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

N = D j (2-14)


D =- d2(C) (2-15)

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




U 0

v= 0

w 0

P= 0



D 0-o

au 0

av 0

w 0

a (pw)*
az At


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


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.


Momentum --
& Velocity

Figure 2-4.

Ghost points

i --


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

0 .. 0.35

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


= c

W =


Substituting in for the scaled variables, Eq. 2-5, Eq. 2-6, and Eq. 2-10, become

9t+ O =0,
at0 oi


+ 9i
at0 a

ac + ac0

+ + F6 Ri~i3 + -,
Rep, xj xij

acwt, a D ac
0+ D gij

respectively, and where

DJ =3(C) |au,

( -(1 -)dg
Ri =
"I *,







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 -.


sO s



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.




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 -


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.

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.


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


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



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.


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


Figure 2-2. The relationship between the particle pressure force and all other forces on a
sediment particle.


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)

Model with PP
---- Model without PP
HWK exp.


0 =60

* C






0 =90

- *'
S5 1




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.


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



0.2 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


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

At A + -A(pu), + A, ,, (2-26)
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 (
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.





-1.4 -




0 0.1 0.2 0.3 0.4 0.5 0.6
Sand Concentration (cm3/cm3)


Hunt et al.
S- L&A, Cp = 0.61
-- L&A, Cp = 0.615


> 102



10-4 10-3 10-2 101 100
Sand Concentration (cm /cm3)


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.


ACKNOW LEDGMENTS .................................

LIST O F TABLES . . . . . . . . . .

LIST OF FIGURES . . . . . . . . .

A B ST R A C T . . . . . . . . . .


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 .......


3.1 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

. . . .



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


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


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


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

0 *
; ----- 'r* ---- -i-- --- ^ ----


o.0 0.5

-5 -5 -
C/Cm C/Cm

]o 0 =120 0 0 =150


0 *

0.5 1
I 0

C/Cm C/Cm

Figure 3-2. HWK vertical concentration profiles through phases.

Full Text




Copyright2007byTylerJ.Hesser 2


Idedicatethisthesistomyfamily.Myparentsfortheguidancetheyhaveprovidedme.Mybrotherforleadingthewayandalwaysbeingtherewithwordsofsupport.Xena,Dutch,andDodgeralwaysthereforhumorandrelaxation.Also,toallmyfriendswhohavebecomefamily,andRyanforalwaysbeingtheretopushmeongooddaysandliftmebackuponbad. 3


ACKNOWLEDGMENTSIacknowledgethesupportoftheOceofNavalResearchforfundingoftheCrossTexproject,advisorycommitteefortheirguidancethroughthethesisprocess,andmyfellowgraduatestudentsandocepeersfortheirsupportandadvice. 4


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


5.3SummaryofContributions ........................... 60 5.4FutureResearch ................................. 60 REFERENCES ....................................... 62 BIOGRAPHICALSKETCH ................................ 66 6


LISTOFTABLES Table Page 2{1ModelBoundaryConditions ............................. 31 3{1RunConditions. ................................... 37 7


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


AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceMODELINGSEDIMENTTRANSPORTINTHESHEETFLOWLAYERUSINGAMIXTUREAPPROACHByTylerJ.HesserDecember2007Chair:DonaldN.SlinnMajor:CoastalEngineeringDuetotheexistenceofsheetowduringstormevents,numericallyquantifyingsedimenttransportduringsheetowconditionsisanimportantstepinunderstandingcoastaldynamics.Traditionalmethodsformodelingsedimenttransportrequiresolvingseparateequationsforuidandparticlemotion.Wehavechosenanalternateapproachthatassumesasystemcontainingsedimentparticlescanbeapproximatedasamixturehavingvariabledensityandviscositythatdependonthelocalsedimentconcentration.Here,theinteractionsareexpressedthroughthemixtureviscosityandastress-induceddiusionterm.Therearevegoverningequationsthatdescribetheoweldthemixturecontinuityandmomentumequations,andaspeciescontinuityequationforthesediment.Theadditionofabed-stinesscoecienttosimulateparticlepressuresinthebedhasincreasedtheconsistencyofthemodelresults.ThismodelwhichwasdevelopedforCrosstexhasshownpromisingcomparisonstoHorikawalaboratorydatademonstratingtheeectivenessofthemixturemodelinsimulatingsedimenttransportinthesheetowlayer. 9


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


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


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


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


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


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


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


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


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


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


drivenows.Thestudyfocusedmoreonthesuspensionofsedimentsthanthehigherconcentrationtransports.1.4ResearchProblemStudyingthesheetowlayerregimeisadiculttaskbecauseofthesmallthicknessandhighconcentrations.Recentlymoreeldandlaboratoryexperimentshavebegantodevelopadvancetechniquesformeasuringconcentrationsandtransportratesinthisregion.However,thehighconcentrationregionatthebottomofthesheetowlayerisstilladicultareatocollectaccuratedata.Itisthegoalofthisresearchtodevelopanumericalmodelandcalibrateittopastlabexperiments,soitmaybecapableofadvancingthecurrentunderstandingoftheregion. 20


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


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


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


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

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


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


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


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


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

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


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


a bFigure2{1.aThemixturedensityvariationversusthesedimentconcentration.b Huntetal. 2002 viscosityversusthe LeightonandAcrivos 1987 viscositycalculationsassedimentconcentrationchanges. 32


Figure2{2.Therelationshipbetweentheparticlepressureforceandallotherforcesonasedimentparticle. Figure2{3.Aplotoftheequationemployedforthebedstinesscoeciant.Themodelisnotaecteduntiltheconcentrationreachesaminimumvalueof0.48. 33


Figure2{4.Thestaggardgridwithconcentrationsandpressurescalculatedatthepointsandvelocitiescalculatedatthearrows.Theoutermostpointsorthegrayarearepresentstheghostpoints. Figure2{5.Initialconditionsin3-Dgridwiththecrossshapedraisedsectionwhichallowsforpickuponcetheowstarts. 34


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


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


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


Figure3{1.HWKsheetowlayerthickness. 38


Figure3{2.HWKverticalconcentrationprolesthroughphases. 39


Figure3{3.HWKhorizontallyaveragedvelocityprolesthroughphases. 40


Figure3{4.HWKsedimentuxesthroughtphases. 41


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


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


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


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


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


Figure4{1.ComparisonsofHsbetweentwodierentsizeddomainsofthemodelwithoutparticlepressureandHWKdataset.The8x4x8cmdomainiswayocomparedtothesmallerdomain.Imtheguretotherightthemodelresultshavebeenshifted30degreesintime. 47


Figure4{2.ComparisonsHsbetweenthemodelwithparticlepressureandthemodelwithoutparticlepressure.Asseen,theparticlepressuremodelismuchclosertotheactualvaluesofHWK. Figure4{3.Thephysicaldierencebetweentheparticlepressuremodelandthemodelwithoutparticlepressurecanbeseeninthisgure.Ontheright,thereisalargerollupwhichcontributestothelargesheetowlayerthicknesseswhileonthelefttherollupisinhibitedbytheparticlepressure. 48


Figure4{4.Theparticlepressureisabletocorrectthedomainsensitivityfromtheoriginalmodel,butitdoesnotseemtoaectthesmallerandpreviouslymoreaccuratedomainsize.ThecomparisonofHsbetweenthemodelwithparticlepressureandwithoutparticlepressureappeartobeverysimilar,andwithagoodrelationshiptoHWK. 49


Figure4{5.Thecomparisonsoftheverticalconcentrationprolesbetweenthemodelwithparticlepressureandwithoutparticlepressurealsoshowsgoodagreementthroughallphasesoftheow. 50


Figure4{6.Evolutionoftheowthroughthephasesofanoscillatorywave. 51


Figure4{7.ComparisonofsheetowlayerthicknessbetweenmodeloutputandHWKdataset.Thestandarddeviationisalsoplotted. 52


Figure4{8.VerticalconcentrationprolecomparisonsbetweenthemodelandHWKforeachphaseoftheow. 53


Figure4{9.HorizontallyaveragedvelocityprolecomparisonsbetweenthemodelandHWKforeachphaseoftheow. 54


Figure4{10.ComparisonsofsedimentuxesforthemodelversusHWK.Theredlineisthemodeluxcalculatedusinga30degreephaseshiftedconcentrationprole. 55


Figure4{11.Sensitivityofthemodeltodomainsizechanges. Figure4{12.Sensitivityofthemodeltogridsizechanges. 56


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


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


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


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


tobeavaluabletoolintheunderstandingofsedimenttransportintheboundarylayerregime. 61


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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. 66