3d scour model

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Three-Dimensional Numerical Model with Free Water Surfaceand Mesh Deformation for Local Sediment Scour

Xiaofeng Liu1 and Marcelo H. García, M.ASCE2

Abstract: A numerical model for local sediment scour with free surface and automatic mesh deformation is constructed and numericalresults are compared with experimental results. For the turbulence closure, the two equation k–� model is used. Two interfaces �water andair, water and sediment� in the domain are captured with different approaches. The free surface of the flow is simulated with the volumeof fluid �VOF� scheme, which is an Eulerian approach. A new method for the VOF scheme is proposed to reduce the computational timewhile retaining relative accuracy. The behavior of the water-sediment interface �bed� is captured with a moving-mesh method, which is aLagrangian approach. The flow field is coupled with sediment transport �both bed load and suspended load� using a quasi-steady approach.Good results have been obtained using the current model. The flow field is similar to the one observed in experiments. Scour patterns aresimilar to the experimental data as well. Large computational times are needed for morphological simulation and parallel computation isused to accelerate this process. The results obtained from the model are promising.

DOI: 10.1061/�ASCE�0733-950X�2008�134:4�203�

CE Database subject headings: Scour; Numerical models; Free surfaces; Deformation; Waves; Sediment.

Introduction

Local scour problems have been studied for many years bothexperimentally and numerically. In this paper, a new numericalmodel is proposed to simulate the scour process in the presence ofa free surface and allowing for automatic mesh deformation. Inthe scour problem, there are at least two sharp interfaces: water-air and water-sediment. In this work, these two interfaces arecaptured using different methods. The water-air interface is cap-tured with an Eulerian method, namely the volume of fluid �VOF�method. The water-sediment interface behavior is captured withthe Lagrangian method, namely the moving-mesh method �Liuand García 2006�.

In numerical simulations of open channel flow, free water sur-face is usually replaced by a rigid lid. This is valid only if the freesurface does not change too much along the channel. For rapidlychanging water surface �e.g., hydraulic jump�, the rigid lid ap-proximation will introduce nonphysical errors. There are manysurface tracking or capturing methods available to simulate thefree surface, e.g., the marker and cell �MAC� method, the VOFmethod, and the level set method �LSM�. The MAC method is

1Research Assistant, Ven Te Chow Hydrosystems Laboratory, Dept. ofCivil and Environmental Engineering, Univ. of Illinois at Urbana andChampaign, 205 N Mathews Ave., Urbana, IL 61801 �correspondingauthor�. E-mail: [email protected]

2Chester and Helen Siess Professor, and Director, Ven Te ChowHydrosystems Laboratory, Dept. of Civil and Environmental Engineering,Univ. of Illinois at Urbana and Champaign, 205 N Mathews Ave.,Urbana, IL 61801. E-mail: [email protected]

Note. Discussion open until December 1, 2008. Separate discussionsmust be submitted for individual papers. To extend the closing date by onemonth, a written request must be filed with the ASCE Managing Editor.The manuscript for this paper was submitted for review and possiblepublication on June 6, 2006; approved on September 4, 2007. This paperis part of the Journal of Waterway, Port, Coastal, and OceanEngineering, Vol. 134, No. 4, July 1, 2008. ©ASCE, ISSN 0733-950X/
2008/4-203–217/$25.00.
JOURNAL OF WATERWAY, PORT, COASTA

J. Waterway, Port, Coastal, Ocean

based on a Lagrangian approach and a set of marker particleswhose position at any time step are used to reconstruct the inter-face. The VOF method is based on the Eulerian point of viewinstead. The widely used VOF method was first proposed by Hirtand Nicholls �1981� which used a donor-acceptor formulation.The VOF method has been used by several researchers to capturethe free water surface when the scour process is investigated.Among many others, Richardson and Panchang �1998� used com-mercial code Flow–3D, which implemented the VOF method onstructured mesh, to simulate the flow field around the scour holes.The flow field compared well with the experiments and the par-ticle tracking in the resulting free surface flow field revealed someof the characteristics of the scour process. The LSM divides thedomain into grid points which hold the value of level set function�Sethian 1996�. The contour of zero level set gives the interface.The level set method has a more rigorous mathematical basis andit is easier to calculate the curvature on the surface which isimportant for the surface tension force. In this paper, a high res-olution VOF method proposed by Ubbink and Issa �1999� is cho-sen to track the free surface. The compressive interface capturingscheme for arbitrary meshes �CICSAM� scheme treats the wholedomain as the mixture of two liquids. Volume fraction of eachliquid is used as the weighting factor to get the mixture proper-ties, such as density and viscosity.

In the scour problem, the bed deformation needs to be coupledwith the transient flow field. Brørs �1999� used a structured gridto model the scour under pipelines. The grid on the bed and thecorresponding grids above were moved according to bed eleva-tion changes. This method for structured grids is similar to manu-ally moving each grid point at each time step and is not efficient.Another alternative is to use the so called blocking method. Thedownside of the method is that the accuracy depends on the meshsize around the interface.

In unstructured grids, it is more complicated to move the gridpoints on the bed and in the domain. Arbitrary movement of thepoints may invalidate the grid. In this paper, an automatic grid
movement algorithm is used to avoid these problems. There have
L, AND OCEAN ENGINEERING © ASCE / JULY/AUGUST 2008 / 203

Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

been many mesh deformation methods developed in the pastyears, such as spring analogy and Laplacian operator smoothing.The latter is chosen in this paper because it is more robust andeasy to implement in the unstructured grid for a complex domain.The Laplacian operator smoothing method is used here to solvethe Laplacian equation with the deformation condition at theboundary. With this approach, the inner grid deforms smoothlyand the computations are stable.

Governing Equations

Fluid Flow Model

The governing equations for the fluid flow are the Reynolds-averaged Navier–Stokes equations

� · u = 0 �1�

��u

�t+ � · ��u u� − � · �� + �t�S� = − �p + �g + �K

��

��2�

where u�velocity vector field; p�pressure field; �t�turbulenteddy viscosity; and ��volume fraction function for the two fluidsdefined by

� = �0 volume occupied by air

1 volume occupied by water�3�

S�strain rate tensor defined by S=1 /2��u+�uT�; ��surfacetension; and K�surface curvature. Surface tension is not as im-portant in the scour problem but it is included for completeness.The density � and viscosity � in the domain are given by

� = ��1 + �1 − ��2 �4�

� = ��1 + �1 − ��2 �5�

The volume fraction � is transported by the fluid velocity field.The equation for the volume fraction scalar � is

��

�t+ � · �u�� = 0 �6�

Numerical diffusion will spread out the sharp interface be-tween water and air. A compressive interface capturing scheme isused to resharpen the interface. Details about the present freesurface modeling algorithm and CICSAM scheme can be found inUbbink and Issa �1999�.

Turbulence Model

Turbulence in the fluid flow model is estimated by the conven-tional k–� two-equation model �Rodi 1993�

�t = C��k2

��7�

�k+ � · �uk� =

1� ��t � k� + 2

�t ��u�2 − � �8�
�t � �k �
204 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINE


��

�t+ � · �u�� =

1

�� t

��

� �� + 2C1�t

��u�2

�

k− C2

�2

k�9�

where k�turbulent kinetic energy; and ��turbulent energy dissi-pation rate. The constants in Eqs. �7�–�9� take the values given inTable 1.

For wave driven orbital motion, the two equation turbulencemodels �such as the k−� model� will become unstable undersome conditions. From the writers’ experience, when the wave isstrong or the wave is breaking, instabilities will be observed.Mayer and Madsen �2000� carried out analytical stability analysisto investigate the erroneous behavior of the turbulence model �thek−� model in their case� under waves. It turns out that the tur-bulent eddy viscosity will grow unbounded and therefore causethe model to become unstable. It seems that no general purposeturbulence model has been developed especially for unsteadywave driven flows. Most models rely on the tuning of empiricalparameters. For the numerical code used in this research, the val-ues of turbulence parameters are bounded when they grow out ofrange �e.g., k and � become negative�. Fortunately, for all the testcases considered in this work, instability of the turbulence modeldid not arise. The modified two-equation turbulence model forwave flows proposed by Mayer and Madsen �2000� or othermodified models can be used when the wave is strong or evenbreaking and the instabilities are expected.

Sediment Transport Model

The scour problem is caused by the erosion movement of sedi-ment. In this numerical model, both bed load and suspended loadare considered. The Exner equation is used to update the bedelevation �García 1999�.

Bed-Load TransportThere are many bed-load transport rate formulas in the literature.Most of them relate bed-load transport rate to shear stress. Theformula of Engelund and Fredsøe �1976� is chosen for the currentmodel

q* = �18.74�� − �c��1/2 − 0.7�c1/2� if � �c

0 otherwise�10�

where q*�dimensionless bed-load transport rate known as theEinstein number which is given by

q* =q0

Rgdd

Here q0�bed-load transport rate for flat bed; and � and�c�Shields number and the critical Shields number for initiationof motion, respectively, where � is defined as

� =b

�gRd

Here b�bed-shear stress calculated with the fluid flow model;��density of water; g�gravitational acceleration; R�submergedspecific gravity of sediment �=1.65 for quartz�; and d�sediment

Table 1. Constants in k–� Model

C� C1 C2 �k ��

0.09 1.44 1.92 1.0 1.3

grain diameter.

ERING © ASCE / JULY/AUGUST 2008

Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

The critical Shields number for particular sediment is alwaysgiven for flat bed and it should be adjusted according to the localslope of the bed and local shear force direction. The criticalShields number for a flat bed, �c0, is a function of flow intensityand sediment properties. �c0 is given as a constant in most nu-merical models. The approach used here to account for the slopeeffect is similar to that used by Roulund et al. �2005�. It consid-ered the slope effect and followed the approach suggested byEngelund and Fredsøe �1976� to use the flow velocity at the par-ticle position and the steepest slope to adjust the critical Shieldsnumber. In this research, the shear stress vector, instead of veloc-ity near the bed, is used. Fig. 1 shows the scheme employed forthe adjustment of the bed-shear stress, where n�surface normalvector; R�steepest-slope unit vector; and �wall shear stressvector acting on the bed. The critical Shields number is adjustedaccording to

�c = �c0�cos �1 −sin2 � tan2 �

�s2 −

cos � sin �

�s� �11�

where ��angle between the velocity vector and the bed steepestslope direction; ��slope angle of the bed; and �s�static frictioncoefficient and has the value of 0.63 for this paper. From Eq. �11�,when the wall shear stress tries to move sediment upslope, thecritical Shields number �c increases, and vice versa.

Eq. �10� only gives the total bed load. The bed-load transportrates in different directions are given by

qi = q0i

��− C�q0�

�

�xi, i = 1,2 �12�

where �bed elevation; and C�constant in the range of 1.5–2.3which is used to reflect the slope effect on the sediment flux�Brørs 1999�.

Suspended LoadThe governing equation for suspended load is a standard convec-tion diffusion equation. Sediment must be fine enough to ignorethe inertial effect of the particle

Table 2. Sediment Entrainment Models

Authors Model

García and Parker �1991� E= AZu5 / 1+ A / 0.3Z

van Rijn �1984� E=0.015d50�1.5

/ ZbD

Smith and McLean �1977� E= 0.65�0�� / �c −1� / 1+�0�

Fig. 1. Slope effect on sediment transport



�c

�t+ � · �u − vs

g

�g� �c = � · ��t�c� �13�

Here c�sediment concentration; u�fluid velocity vector;vs�sediment fall velocity; and �t�diffusivity which is taken asthe same value as turbulence eddy viscosity.

Bed Morphology Model—Exner EquationBed elevation changes are based on the continuity of sediment.The Exner equation, which describes the sediment continuity, is

�

�t= 1/�1 − n��− � · qb + D − E� �14�

where �bed elevation; n�porosity of the bed; qb�bed-loadtransport rate vector whose components are given by Eq. �12�;D�deposition rate; and E�entrainment rate. The deposition rateD at the bed is

D = vscb �15�

where cb�sediment concentration very near the bed. In the cur-rent model, the concentration at the nearest cell center is used.The entrainment rate E, unlike the deposition rate D, needs someempirical model. E can be written in dimensionless form as

E =E

vs�16�

Many models are available in the literature for estimating the

entrainment rate of sediment into suspension E. García and Parker�1991� performed a detailed comparison of eight such relationsagainst data. Three of the many existing sediment entrainmentmodels are listed in Table 2. The entrainment rate is given at somereference level Zb very near the bed to avoid singularity. Thereference levels in García and Parker �1991� and van Rijn �1984�are both 5% of the water depth from bed while in Smith andMcLean �1977�, the reference level is defined as

Zb = 26.3� �

�c− 1�d + ks �17�

where ks�equivalent roughness height of the bed.All of these three empirical models in Table 2 perform well

against the data �García and Parker 1991�. In this paper, themodel in van Rijn �1984� is used. Some simpler models for E alsoexist in the literature. Brørs �1999� and Liang et al. �2005� usedthe concept of turbulence diffusivity and set the vertical sedimententrainment flux as E=�t��C /�y�. This formula will simplify thenumerical model while still giving reasonable results.

Mesh Deformation Solver—Laplacian Equations

As mentioned before, the automatic mesh deformation is imple-mented by a Laplacian smooth operator. The governing equation

Parameters

A=1.3�10−7 Zu= U*s / vsRep

0.6

D*

=d50�R−1 / g�2�1/3 �= �� / �c −1� Zb=0.05H

1� �0=2.4�10−3

u5

*0.3

� / �c −


Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

for the mesh motion equation is the Laplacian equation �Jasak andTuković 2007�. Let v be the grid motion velocity field, then theequation becomes

� · �� v� = 0 �18�

where ��diffusion coefficient chosen to control the mesh motion.� can be a constant or a variable defined by other properties in thedomain. The selection of variable � depends on the specific meshmotion problem and needs objective judgment. Jasak and Tuković�2007� suggested several possibilities to set the � value based ondistance �linear, quadratic, and exponential� from some boundaryor mesh characteristics �orthogonality and skewness�. Thesemethods all have their merits and shortcomings. In this work, � isset as a nonzero constant for simplicity. The boundary conditionfor Eq. �18� is given by the Exner equation. After each time stepof bed elevation change, the bed grid motion velocity is known.When the grid motion velocity field is solved, the grids in thewhole domain can be moved based on the following equation:

xk+1 = xk + v�t �19�

Here, xk+1 and xk�grid position vectors at time level k+1 and k,respectively, and �t�time step.

Fig. 2 is a simple test case for the mesh deformation compo-nent of the numerical model. The bottom of a rectangular flume�with unstructured grid� is “vibrating” in a sinusoidal mode with aperiod T=0.8 s. Mesh grids at t=0, T /4, T /2, and 3T /4 areshown. In the real cases, the bed will move in a more complicatedway.

Numerical Simulation Schemes and Procedure

The governing equations presented in the previous section form astrongly coupled system of equations. The solution of this systemof equations requires some effort because even for each singlecomponent �fluid, free surface, or sediment� it is a difficult prob-lem. Also the time scales of the flow field and sediment transportare so largely different that the whole system is very stiff. Con-sequently, numerical instabilities can be expected to arise. Theemphasis of this research is not on the numerical details but onbuilding a numerical model for the local scour problem and veri-

Fig. 2. Mesh deformation of vibrating flume bottom

fication of the numerical solution of the model. So numerical



schemes for each component of this system are described onlybriefly. Several references about these numerical schemes are pro-vided for the interested readers.

The code used here is the open source computational fluiddynamics �CFD� code OpenFOAM �OpenCFD 2005�. It is freelyavailable through the internet. OpenFOAM is primarily designedfor problems in continuum mechanics. It uses the tensorial ap-proach and object oriented techniques �Weller et al. 1998�. Open-FOAM provides a fundamental platform to write new solvers fordifferent problems as long as the problem can be written in ten-sorial partial differential equation form. Here, the flow field issolved by the adaption of the original turbulence solver for in-compressible fluid. The sediment transport equation and bed de-formation are added to the fluid flow solver to form a new solvercalled FOAMSCOUR. The core of this code is the finite volumediscretization of the governing equations. Almost all kinds of dif-ferential operators possible in a partial differential equation, suchas temporal derivative, divergence, Laplacian operator, curl, etc.,can be discretized in the code. The finite volume details of thecode can be found in Jasak �1996�. The next section is a briefintroduction of the numerical scheme used in FOAMSCOUR.

Numerical Scheme for Flow Field

The numerical solution of the Navier–Stokes equation for incom-pressible fluid flow imposes two main problems �Jasak 1996�: thenonlinearity of the momentum equation and the pressure-velocitycoupling. For the first problem, two common methods can beused to deal with it. The first is to solve a nonlinear algebraicsystem after the discretization. This will require a lot of compu-tational effort. The other is to linearize the convection term in themomentum equation by using the fluid velocity in previous timesteps which satisfies the divergence-free condition. The lattermethod is used in this research. For pressure-velocity coupling,many schemes exist, such as the semi-implicit method for pres-sure linked equation �SIMPLE� �Patankar 1981� and pressure im-plicit splitting of operators �PISO� �Issa 1986�. The PISO schemeis used in this code. For the k−� turbulence model equations,although k and � equations are coupled together, they are solvedby a segregated approach, which means they are solved one at atime. This is also the usual approach used in most CFD codes.

For the free surface, the CICSAM scheme is easy to code andcaptures the free surface fairly well. But the problem with thisscheme is that it is very slow. The time step required is so smallthat the computation usually takes a very long time even for asmall case. This is partially caused by the fact that the control cellwhich has the largest Courant number usually is in the air phase.In our case, the two phases are water and air. The density of wateris almost 1,000 times larger than that of air and the velocitymagnitude in the air phase is far greater than in the water phase.A modified approach is proposed in this paper analogous to thoseused in other CFD codes. The aim is to relax the time step re-quirement while capturing the free surface. In the modified ap-proach, the volume occupied by the air phase can be called void.In the void region, the velocity is set to zero and the pressure isset to atmospheric pressure. This will not upset the computationsbecause of the large density ratio between water and air. Theinertia force associated with air can thus be neglected. The thresh-old value of � for the void �i.e., below what value of � it can beviewed as void� needs some consideration. The classical dambreak problem is carried out to investigate the effect of the thresh-old value. As in Fig. 3�a�, a column of water is released at time
t=0 s and an obstacle is located in front of the water column. Fig.

Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

3 shows the contour of the VOF scalar � which is the volumefraction of water. The results at t=0.3 s for different thresholdvalues �0.01 and 0.1� are plotted together with the originalscheme �threshold value equals 0�. The result from thresholdvalue 0.01 is very close to the original scheme while that fromthreshold value 0.1 causes more damping of the water flow. Thisis because high threshold values imply that a larger portion of theflow field is set to zero velocity and the water needs to consumemore energy to go through such an area. In this paper, a thresholdvalue of 0.01 is used for all computations and almost half of thecomputational time is reduced.

Numerical Scheme for Sediment Transport

The bed load and suspended load in the sediment transport aresolved separately. Since the shear stress on the bottom is obtainedfrom the flow solver, using Eq. �10� to calculate the bed load isstraightforward. The suspended load equation �Eq. �13�� is aconvection-diffusion equation and the sediment is transportedpassively by the flow field. The finite volume method is also usedto solve this equation.

After the sediment load �namely bed load and suspended load�is calculated, the Exner equation is solved to change the bedelevation. The Exner equation is a two-dimensional �2D� equationalthough the bed is in 3D space. This is because the verticalcoordinate z is the variable to be solved and it is no longer anindependent variable. The bed elevation change is due to the di-vergence of sediment transport flux together with erosion anddeposition of sediment on the bed surface. This simple 2D equa-tion can usually be solved by the finite difference method using astructured grid when the bottom has a regular shape �Van Beekand Wind 1990; Olsen and Melaaen 1993, 1999; Brørs 1999�. Forcomplex domains, such as a natural river bottom or in the pres-ence of objects �bridge piers, break water etc.� �Cataño Loperaand García 2006�, the finite volume or finite-element method canbe used with unstructured grids. In this research, two meshes areused. One is for the fluid flow solver and the other is for the

Fig. 3. Test cases for modified CICSM scheme: �a� initial state; �b�critical �=0; �c� critical �=0.01; and �d� critical �=0.1

bottom Exner equation. The Exner equation is solved on the 2D



mesh using the finite volume method. In Fig. 4, it is shown thatthe coupled problem is solved through mapping between the fluidmesh bottom boundary and sediment 2D mesh. After the flowsolver and turbulence solver are finished, the shear stress andother parameters are available. Those flow parameters are mappedfrom the 3D fluid bottom boundary to the 2D bed sediment mesh.On the 2D sediment mesh, the Exner equation is solved and thenew bed elevation is calculated. Then the bed elevation is mappedback from 2D sediment mesh to 3D fluid mesh. The 3D fluidmesh is deformed according to this as a boundary condition. Theuse of separate meshes for the flow field and the Exner equationkeep the concept clear and easy to manage in the code. Mappingback and forth between these two meshes could lead to loss ofsome accuracy when averaging and interpolation are needed. Butwhen the mesh is sufficiently fine or the mapping is almost one toone, the accuracy is not a problem.

Numerical Scheme for Automatic Mesh Deformation

The Laplacian equation in Eq. �18� needs to be solved on the 3Dmesh to get the movement of each node. The finite-elementmethod �FEM� which is implemented in the original OpenFOAMpackage�, the finite volume method, as well as other numericalmethods can be used. In the original OpenFOAM package, eachcell of the 3D mesh is decomposed into tetrahedral elements andthe new refined mesh is used to solve the Laplacian equation.Details of the FEM implementation can be found in OpenCFD�2005� and Jasak and Tuković �2007�. This built-in FEM methodis very slow, especially when the 3D mesh is very big. Numericaltests show that the decomposition time is almost half that of thetotal mesh motion solution step. Also the decomposition makesthe new mesh much larger �usually several times the originalmesh size� and therefore makes the computation even longer. Inthis paper, some simplifications are made according to the uniquecharacter of the problem under consideration. For the scour prob-lem in this paper, the specific characteristic is that the sand bedboundary movement is in the vertical direction �deposition orerosion�. This makes the problem relatively simpler. A new finitevolume solver for the Laplacian equation is written and it is usedon the 3D fluid mesh. No mesh decomposition is used and themesh movement is solved at the cell centers. After those centervalues are obtained, they are interpolated to the grid points to

Fig. 4. Mapping between 3D and 2D bed mesh

perform the actual mesh motion.


Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Boundary Conditions

There are basically five boundary types of the domain: inlet, out-let, atmosphere, walls, and sand bed. For sediment transport, theboundary conditions for suspended sediment concentration andbed-load feeding rate also need to be specified.

Inlet and OutletFor the inlet boundary, the velocity vector is specified. Pressure isset to be zero normal gradient �p /�u=0 to be consistent withvelocity condition. Turbulence quantities such as k and � are alsoset as constant �5% turbulence intensity is used to set the turbu-lence level�. The outlet condition is chosen as zero gradient for allquantities except for the pressure. At the outlet, the dynamic pres-sure Pdynamic=1 /2�u2, instead of total pressure �Ptotal�, is specifiedas zero normal gradient. The hydrostatic pressure Pstatic is sub-tracted from the total pressure. This makes it easy to specify thepressure boundary condition since the free surface will changeand the hydrostatic pressure will also change at the outlet. Detailsof the outlet free surface condition can be found in OpenCFD�2005�.

AtmosphereThe top boundary of the domain is the atmosphere and total pres-sure is set to zero. k and � at the top boundary will depend on theflow direction relative to the boundary. When the flow goes out ofthe domain, then the zero gradient condition is used. When theflow goes into the domain, a 1% turbulence intensity is used tospecify k and �. The small amount of air turbulence will not affectthe water flow field too much since the water is much heavierthan air.

Walls and Sand BedAt the surfaces of objects �such as pile, sluice gate, and flumewalls�, wall boundary conditions are used. No-slip condition u=0 is set for velocity with zero normal gradient for pressure. Forthe turbulence quantities, it will depend on the smoothness/roughness of the wall. For smooth walls, the value of turbulent

Fig. 5. Flow c



kinetic energy k has to be specified, while for rough and transi-tional cases, normal gradient of k is set equal to zero. For the testcases in this paper, the walls are all rough and a zero normalgradient condition is used for k. Test cases using the zero normalgradient k condition give numerical results which are in betteragreement with experiments �Roulund et al. 2005�. The bedboundary is almost the same as the walls. One exception is thatthe vertical velocity is set to be the same as the bed elevationchange velocity. This adjustment is particularly important in theearly stages of scour and the scour rate is very high. In the laterstages, the scour is almost at equilibrium and the vertical velocityof the movable bed is very small compared to that of the flowfield. At this stage, it is reasonable to set the bed fluid velocityequal to zero.

Sediment Boundary ConditionsAt the inlet, the suspended sediment concentration c is set to zerowhich corresponds to clear water coming into the domain. At theoutlet, zero gradient is applied for c. At the atmosphere boundary,c is set to zero since no sediment particle will go through thewater surface. For the solid walls, the flux of sediment throughthese surfaces is zero. The flux of suspended sediment qs �includ-ing convective and diffusive fluxes� is defined as

qs = �u − vs

g

�g� �c − �t�c �20�

So the condition for zero flux of suspended sediment through thewall is expressed as

qs · n = 0 �21�

At the sand bed, the suspended sediment flux is specified as thenet upward normal flux

qs · n = E − D �22�

where D is defined in Eq. �15� and E is calculated by using theempirical model by van Rijn �1984� as shown in Table 2.

f computation
hart o

Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Bed-load boundary conditions are defined similarly. At theinlet, bed load is set to zero which corresponds to zero feedingrate because of the concrete bed before flow hits the sand bed �seethe “Test Case” section�. At the outlet, zero gradient is applied. Atthe wall surfaces, the bed-load flux is set to zero as for suspendedload, i.e.

qb · n = 0 �23�

Simulation Flow Chart

The flow chart for the simulation process is plotted in Fig. 5. Aquasi-steady approach is used in the model. In this approach, it isassumed the time scale of bed change is far larger than that offlow field and the time steps for flow field and morphologicalcalculations are different. When the flow field is being calculated,the domain boundaries are fixed. Only at the sediment transportstep are the domain boundaries adjusted. A scheme similar to themethod proposed by Liang et al. �2005� is used here. During thesimulation, m time steps of flow field simulation are carried outbetween the morphological steps. This approximation is onlyvalid when the bed changes are very slow when compared to theflow changes. Through numerical experiments, it is found that atthe beginning of the local scour process, the bed changes veryrapidly and most of the scour occurs in this stage. In order toincrease the accuracy, smaller values of m are used at the begin-ning of the simulation. After the flow field and bed change reachrelatively steady state, larger values of m can be used to acceler-ate the computation. In the test cases considered here, m=1 isused at the initial stage of simulation and m gradually increases to100 when the whole system tends to equilibrium. There are somedifference between the current time marching scheme and the onein Liang et al. �2005�. Liang et al. �2005� only use k, where k�m, time steps between morphological steps and assume thatfrom time step k+1 to m, the flow field does not change much.

Fig. 6. Turbulent wall jet scour schematic view

Fig. 7. Turbulent wall



The flow field from time step k is used to perform the morpho-logical calculation. This will further reduce the computationaltime.

Model Verification and Applications

In this section, applications of turbulent wall jet scour �2D case�and wave scour around a large vertical circular cylinder �3D case�are carried out using FOAMSCOUR and the numerical results arecompared with experiments. In the turbulent wall jet scour pro-cess, the jet flow and momentum diffusion are of interest as bothhave an impact on the scour hole development. In wave scouraround a big pile, the water wave and its interaction with the pileand bed sediment are important. The coupled problems of flowfield and morphodynamics usually take a very long time for com-putation. Parallel computation with domain decomposition is usedto accelerate the process. OpenFOAM comes with parallel com-puting support and FOAMSCOUR easily uses these modules toimplement both flow field solver and sediment transport solver.All computations were done on the National Center for Super-computing Applications �NCSA� Tungsten Xeon Cluster at theUniversity of Illinois at Urbana-Champaign.

Wall Jet Scour Test Case

The experimental data of turbulent wall jet scour by Chatterjeeand Ghosh �1980� and Chatterjee et al. �1994� are used. Thisexperiment investigated the flow field and sediment transport dueto a submerged wall jet. The schematic view of the experiment isshown in Fig. 6. The jet is formed by a small opening under asluice gate and flows over a rigid apron onto an erodible bed. Thedetails of the experiment can be found in the original papers.Run2 of the experiment is chosen as a typical run to be simulated.The bed material is sand with d50=0.76 mm, submerged specificgravity 1.65, porosity 0.43, and angle of repose 29°. The flowcondition for this run is as follows. The apron length is 0.66 mand the jet inlet velocity is 1.56 m /s. The downstream waterdepth is controlled to be 0.291 m by the outlet weir. In the ex-periment, the jet inlet velocity is caused by the difference betweenupstream and downstream water depth. The corresponding waterdepth difference in Run2 is 0.118 m. But in this simulation, theconditions upstream of the sluice gate are not modeled. Equiva-lently, velocity is specified at the jet inlet. Although the test case

field and free surface
jet flow

Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

is 3D, the flume side walls have little effect on the main region ofthe flow and the scour profile is almost uniform in the spanwisedirection.

In Fig. 7, free surface and velocity magnitude contours at theequilibrium state are plotted. When the jet flow starts, it induceswater waves in the domain. After the flow calms down, the freesurface of water keeps an almost steady position. The free surfaceat the equilibrium state is almost flat except in the region ofmaximum scour. The velocity magnitude contours show that thejet flow comes into the domain through the small opening andafter momentum diffusion and water entrainment it leaves thedomain through free fall over the outlet weir.

The stream trace in the region of jet opening and scour hole atequilibrium state is plotted in Fig. 8. The flow pattern is moreclearly shown. The jet induces strong flow recirculation and justabove the jet inlet, there are some small circulation areas whichare isolated from the main flow. Inside the scour hole, the streamtrace lines show another circulation pattern. These flow patternsare closely related to the scour process. When the water leaves thejet orifice, the main physical processes are the momentum diffu-sion and entrainment of water from above into the jet flow. As aresult, the magnitude of the maximum velocity and also the ve-locity distribution will be changed. This change is a complicatedfunction of flow domain geometry and sediment properties �Chat-terjee and Ghosh 1980�. In Fig. 9�a�, the maximum horizontalvelocity normalized by jet inlet velocity is plotted against thenondimensionalized x distance where B0 is the jet opening width.The wall jet momentum diffusion characteristics found in the ex-periment are reproduced well by the numerical simulation. As inthe experiment, the diffusion process is slower in the rigid apronregion than in the erodible bed region. In the near jet openingregion, the numerical results are very close to the experimentaldata. But in the far downstream region, the momentum diffusionis overestimated by the numerical simulation, i.e., the maximumvelocity is smaller and the velocity distribution in the verticaldirection is more uniform. This may come from the numericaldiffusion nature of the schemes used in this test case.

Fig. 8. Turbulent w
all jet flow stream trace
Wall jet flow near the jet opening can be divided into two



Fig. 9. Turbulent wall jet characteristic: �a� jet diffusion along x axis;�b� velocity distribution at x=0.6 m


Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

regions. One is the inner boundary layer region and the other isthe outer shear layer region. When the boundary layer meets theshear layer, the potential core will disappear and the jet flow isfully developed �Rajaratnam 1976�. The horizontal velocity atlocation x=0.6 m downstream of the jet opening is plotted in Fig.9�b�. In Fig. 9�b�, � is defined as y /�, where � is the verticaldistance from the wall to the location of the maximum horizontalvelocity. Nondimensionalized velocity is plotted as a function of�. From Fig. 9�b�, the numerical simulation predicted the velocitydistribution fairly well.

Fig. 10 shows the mesh development from the initial flat bedcondition to the equilibrium configuration. The unstructured meshis composed of three layers of fine quadrilateral cells in the

Fig. 10. Turbulent wall jet scour mesh deformation: �a� initial; �b�equilibrium

Fig. 11. Wa



boundary layer on the bottom and triangle cells elsewhere. Withthe change of bottom elevation due to scour and deposition, thecomputational domain gradually deforms while the unstructuredmesh remains valid. There is a small artificial step at the transitionpoint between the concrete apron and the sand bed. It has a depthof 5 mm. This small step is used to improve the mesh qualitybecause the scour hole will cause the cells in front of the transi-tion point to become vertically stretched. Inside the small step,several mesh cells are introduced and it makes the simulationstable.

Scour profiles from numerical simulation at different times areplotted in Fig. 11 as solid lines together with the experimentalresults. The time it requires to get to equilibrium state is almost1 h for this test case. Fig. 12 shows the time development of themaximum scour and deposition. At equilibrium state, according tothe numerical simulation results, the maximum scour depth isabout 0.63 m and the maximum deposition depth is about 0.4 m.These fit well with the observations made in the experiment.From Fig. 12, the scour process occurs very rapidly in the earlystage while at a later stage, the profile changes very slowly. Themaximum scour position and the peak of the sediment depositiondune move very slowly downstream.

In Fig. 13, the Shields number at equilibrium state is plotted.Chatterjee and Ghosh �1980� gave an estimation of critical shearstress along the bed using von Karman’s integral theorem. Butthere is a discrepancy between this estimation and the criticalshear stress using Shields method. This discrepancy could be dueto the nonhydrostatic pressure distribution, nonuniformity of thegrain size, or the pressure differential recorded by the Prestontube used in the experiment. Thus in Fig. 13, only numericalresults of shear stress are plotted. The Shields number �shearstress� decreases with the distance from the jet opening. This is inagreement with the experiment. The maximum Shields numberoccurs near the transition point between the concrete apron andthe sand bed. At equilibrium state, the Shields numbers along thebed are all below the critical Shields number which means that nosediment is moving.

Wave Scour around Large Vertical Circular Cylinder

The scour process around vertical pile caused by the wave is morecomplicated because of the 3D nature of the problem. The free

cour profile
ll jet s

Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

surface of water wave and 3D scour hole around the pile make ita good test case for the application of the solvers presented above.The test case uses the experimental data obtained by Sumer andFredsøe �2001�. The schematic view of the experiment is shownin Fig. 14. Test Case 7 of the experiment is used in this paper. Inthis case, the wave has a period of 3.5 s, wave height of 6.4 cm,

Fig. 12. Wall jet maximum scour and deposition development withtime: �a� maximum scour; �b� maximum deposition

Fig. 13. Bed-shear stress distribution at equilibrium for wall jet scour



wave length of 6.79 m, and Keulegan–Carpenter �KC� numberof 0.61. The pile has a diameter of 1 m and the sand hasd50=0.2 mm.

The mechanism causing the scour around a vertical pile can bedifferent under different conditions. Flow regimes change whendimensionless number KC is increased. The KC number is de-fined as

KC =UmT

D�24�

where Um�maximum undisturbed orbital velocity at the bed.When the pile is slender, the scour is primarily caused by vortexshedding �KC�O�100�� and by horseshoe vortex �KCO�100�� Sumer et al. 1992, 1993�. For this case, the free sur-face is of secondary importance and can be replaced by a rigid lid.Numerical simulations using the rigid-lid approximation capturedthe vortex shedding and horseshoe vortex well and gave goodresults �Roulund et al. 2005�. When the pile diameter is bigenough, the scour is caused by the combined effect of the phase-resolved component of the wave flow and the steady streamingdue to the presence of the large cylinder �Sumer and Fredsøe2001�. For this case, the modeling of the free water surface isvery important. Without the free surface, the steady streamingcannot be correctly simulated and the scour process cannot becaptured.

The waves in the experiment are generated by a piston typewave maker and at the end of the wave tank, there is a waveabsorber to eliminate wave reflection. The details of the experi-ment can be found in the original paper. In the numerical simu-lations the wave can be generated generally by two approaches:moving mesh �Aliabadi et al. 2003� and wave boundary condition�Mayer et al. 1998�. For the moving mesh approach, the pistonmovement is simulated by moving the boundary of the computa-tional domain. For the wave boundary condition approach, themesh is fixed but the boundary condition on the piston part isgiven by wave theory. The second approach is adopted in thispaper. A time varying velocity profile is imposed at the piston

Fig. 14. Wave scour around large vertical circular cylinder scheme�Reprinted with permission from Sumer and Fredsøe 2001�

boundary to generate the waves


Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

U = fr�t��U�y�sin��t + �� 25�

where U�velocity vector at the boundary; ��wave frequency;��phase; and U�y� is imposed by the type of piston. In the testcase, the relative depth h /L and wave steepness H /L are about1/17 and 1/106, respectively. The linear wave maker theory ap-plies and for the piston type wave maker in this case, U�y�=constant �Dean and Dalrymple 1991�. For finite amplitude wavecases, the nonlinear effect is important and higher order termsneed to be added. fr�t��“ramp” function to start up the piston.fr�t� has the form

fr�t� = t

T−

1

�sin��

1

T� for 0 � t � T

1 for t T� �26�

where T�wave period.The wave absorber is simulated using a damping zone �or

sponge layer�. In this damping zone, wave energy is artificiallydissipated and wave reflection is minimized. The water depth andfluid velocity in the damping zone is modified at each time stepaccording to the method proposed in Mayer and Madsen �2000�.Two blocks of meshes are used in the computation. One is for thewave tank and the other is for the damping zone which has thelength of twice the wave length. The damping zone block hasrelatively coarse mesh to reduce the computational time. Thecoarser mesh in the damping zone also has the effect of numericaldissipation which will make the wave absorber even better. Nu-merical simulation for Test 1 of the rigid bed experiments inSumer and Fredsøe �2001� is carried out to test the wave absorberalgorithm. Wave profiles in one period are shown in Fig. 15 andthe reflection coefficient is about 10% which is in the range oflaboratory experiment setups.

It is shown by the experiment that when a large vertical cyl-inder is subjected to a progressive wave, the combined effect ofincident wave, reflected wave, and diffracted wave will createphase-resolved flow and steady streaming. Steady streaming isvery important for the mass transport near the bed and thereforethe scour process. Steady streaming due to the wave and inducedby the presence of the cylinder are simulated and compared withthe experiment. For steady streaming due to the wave, Test 1 ofthe rigid bed experiment in Sumer and Fredsøe �2001� is simu-lated. Fig. 16 shows the period-averaged velocity profile at thelocation of the cylinder center �without the cylinder in the wave

Fig. 15. Wave profiles in one period for Test 1

tank�. The numerical velocity profile is obtained by averaging the



velocity over 30 wave periods. The return flow pattern caused bythe wave drift and confined wave tank environment is well cap-tured. The steady streaming near the bed is also captured despitethe fact that the streaming velocity is lower than the experiment.For steady streaming due to the presence of the cylinder, Test 1 ofthe rigid bed experiment in Sumer and Fredsøe �2001� is simu-lated with the cylinder in the wave tank. Fig. 17 shows the periodaveraged velocity components at point P which is very near thecylinder against the measurement. The radial velocity Ur is in theoutward direction over the phase interval from about �t=150 to360° which is in agreement with the experiment. The tangentialvelocity U� shows a steady streaming in the opposite direction ofwave propagation since the negative tangential velocity experi-ences a longer period of time.

Fig. 18 shows an instantaneous free surface of the wave. Aprobe point �x=5.7 m, y=0.2 m, z=5.3 m� is also indicated in thefigure. Fig. 19 shows the pressure change with time at the probelocation. During the first several time periods, the wave is underdevelopment and the pressure has some noise. But when the waveis fully developed, the pressure is undergoing periodical changesas expected.

Fig. 20 shows the equilibrium state of the scour around the

d bed experiment in Sumer and Fredsøe �2001�

Fig. 16. Period-averaged velocity profile at location of cylindercenter in undisturbed flow

of rigi


Eng. 2008.134:203-217.



Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

pile. The numerical result is comparable with the experimentaldata although the maximum scour depth obtained with the nu-merical simulation is slightly higher than the experiment result.As in the experiment, the deposition zone is on the side of the pileand the scour zone is in the front and in the back. Although thescour pattern is not identical to that of experimental results, theyare very similar. Fig. 21 shows the scour around the periphery ofthe cylinder base. Only half of the periphery is shown because ofsymmetry. From this figure, the scour and deposition patternsaround the pile are seen to agree well with the experiment exceptfor the amplitude. This may be caused by the fact that some otherfactors in the experiment are not included in the numerical model.Further research is needed to make the model more accurate.

Discussion

Effect of Free Surface

For the rigid-lid approach, the free surface is replaced with animaginary horizontal frictionless plane. In reality, even when thefree surface is almost flat and will not change with time, a rea-sonable position for the free surface is not easy to define. This isbecause locally �such as free surface around a bridge pier� the freesurface may change very abruptly and the rigid-lid approach sim-ply cannot capture these features. On the other hand, for the sub-merged wall jet case, the water depth in the flow domain is muchbigger than the jet opening and the free surface changes veryslowly. This makes the rigid-lid approximation a reasonable one.

wave around pile

Fig. 19. Pressure probe at point �x=5.7 m,y=0.2 m,z=5.3 m�

Fig. 17. Period-averaged velocity components at point P forKC=1.1, D /L=0.15, z=0.4 from bed: �a� measurement point; �b�radial velocity; and �c� tangential velocity

Fig. 18. Free surface of


Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

For scour around slender piles under waves, the rigid-lid assump-tion is also valid since the most important mechanism here is thevortex shedding and horseshoe vortex. However, for the wave-induced scour around large piles, it is unreasonable to use therigid-lid approximation since the dominant mechanism here is thephase-resolved component of the wave flow and the steadystreaming.

Fig. 20. Wave scour around large vertical circular cylinder: �a� exp

Fig. 21. Scour at periphery of cylinder base



Interface Capturing Techniques

In the current numerical model, the two moving surfaces �waterfree surface and moving bed� are captured by different ap-proaches, namely Eulerian and Lagrangian approaches. The freesurface is solved by the CICSAM VOF scheme, which is a Eule-rian approach. The moving bed is captured by the mesh deforma-tion method, which is a Lagrangian approach. The scour test casesin the paper are relatively simple. For more complicated prob-lems, if there are objects interacting with the fluid phase and thebed, the moving mesh technique will not be appropriate. Theinterface between the object, water, and bed is so complicated andchanges with time. Simply moving the grid point along the inter-face will not be feasible. For such cases, the Eulerian approach�VOF or LSM� can be used to implicitly track the interface. Thiscould provide a new research direction in the scour problem andis being explored by the writers.

Limitation of Mesh Deformation Approach

As mentioned in the previous section, the mesh deformation ap-proach is hard to implement with the presence of interaction be-tween moving objects and bed or when the boundary movementis irregular. Also when the amplitude of the boundary movementis big enough, the mesh may be highly distorted and the deterio-rated mesh quality will make the computation hard to converge,eventually leading to unstable solutions. In order to avoid these

ntal data; �b� numerical result; and �c� 3D view of numerical result
erime
problems, the dynamic mesh approach should be used. In the


Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

dynamic mesh approach, mesh cells can be split or merged whennecessary. But comparing them with mesh deformation, the dy-namic mesh approach is more difficult. A combination of thesetwo approaches can be promising for scour problem where thescour pattern is complicated and the scour/deposition depth isvery large.

Conclusions

The numerical model FOAMSCOUR for local scour with freesurface and automatic mesh deformation is proposed. The turbu-lence model used is the simple two equation k−� model. Otherturbulence models, even large eddy simulation, can be used toimprove the accuracy of the fluid flow field simulation. All thesemodels are readily implemented in OpenFOAM. The free surfaceis modeled by the VOF method while the scour process is mod-eled by the moving mesh method. These two methods for movingboundaries �free surface and bed� are coupled together. Eachmethod has its merits and shortcomings. The Eulerian approachcan be used to capture the complex scour profiles. Flow field iscoupled with sediment transport �both bed load and suspendedload� using a quasi-steady approach. Parallel computations areused to reduce the CPU time which is usually tremendously largefor morphological simulation �Parker and García 2006�.

Numerical simulations for turbulent wall jet scour and wavescour around the large vertical cylinder are carried out to comparethem with experiments. Good results have been obtained usingthe proposed modeling approach. Velocity field and other flowfield characteristics compare fairly well with experimental obser-vations. The maximum scour depths and local scour profile fit theexperimental data well. Further research is needed to investigatethe effect of the turbulence model for free surface waves �espe-cially for near breaking and breaking waves� and to study thepossibility of using a Eulerian approach for morphologicalmodeling.

Acknowledgments

This research was supported by the Coastal Geosciences Programof the Office of Naval Research, Grant No. N00014-05-1-0083.The writers gratefully thank Professor B. M. Sumer for his help-ful discussion and suggestions. The computational resources pro-vided by the National Center for Supercomputing Applications�NCSA� at University of Illinois at Urbana-Champaign are grate-fully acknowledged.

Notation

The following symbols are used in this paper:C�, C1, C2 � coefficients for turbulence model;

c � sediment concentration;cb* � equilibrium sediment concentration at bed;D

* � particle size parameter;D, E � sediment deposition rate and erosion rate;

d � sediment grain diameter;

E � dimensionless sediment erosion rate;fr � ramp function for wave generation;

g, g � gravity vector, gravity constant;
K � surface curvature;


k � turbulence kinetic energy;ks � equivalent roughness height;n � surface normal vector;n � sand porosity;p � pressure;

q* � dimensionless bed-load transport rate;qs, qb � suspended load and bed-load flux vector;q0, qi � dimensional bed-load transport flux,

dimensional bed-load transport flux component;R � steepest slope;R � submerged specific gravity of sediment;S � strain rate tensor;T � motion period or wave period;t � time;

u � fluid velocity vector;v � mesh motion velocity;

vs � sediment fall velocity;xk � grid point position at time step k;

x, y, z � Cartesian coordinates �y denotes verticalcoordinate�;

Zb � reference level;� � fluid volume fraction;� � slope angle of the bed;� � mesh motion diffusion coefficient;� � turbulence energy dissipation rate; � bed elevation;

�, �c, �c0 � Shields number, critical Shields numbers forsloped, and flat bed;

�, �t � fluid dynamic viscosity, turbulence viscosity;�s � static friction coefficient;

� � dimensionless vertical distance in wall jetscour case;

�, �1, �2 � mixture density, fluid density, air density;� � surface tension coefficient;

�k, �� coefficients for turbulence model;b, i � bed-shear stress, bed-shear stress component;

� � wave phase;� � angle between velocity vector and bed

steepest slope;� � dimensionless excess bed-shear stress; and� � wave frequency.

References

Aliabadi, A., Abedi, J., Zellars, B., Bota, K., and Johnson, A. �2003�.“Simulation technique for wave generation.” Commun. Numer. Meth-ods Eng., 19�5�, 349–359.

Brørs, B. �1999�. “Numerical modeling of flow and scour at pipelines.” J.Hydraul. Eng., 125�5�, 511–523.

Cataño Lopera, Y. A., and García, M. H. �2006�. “Burial of short cylin-ders induced by scour under combined waves and currents.” J. Water-way, Port, Coastal, Ocean Eng., 132�6�, 439–449.

Chatterjee, S. S., and Ghosh, S. N. �1980�. “Submerged horizontal jetover erodible bed.” J. Hydr. Div., 106�11�, 1765–1782.

Chatterjee, S. S., Ghosh, S. N., and Chatterjee, M. �1994�. “Local scourdue to submerged horizontal jet.” J. Hydraul. Eng., 120�8�, 973–992.

Dean, R. G., and Dalrymple, R. A. �1991�. Water wave mechanics forengineers and scientists, World Scientific, River Edge, N.J.

Engelund, F., and Fredsøe, J. �1976�. “A sediment transport model forstraight alluvial channels.” Nord. Hydrol., 7, 293–306.

García, M. H. �1999�. “Sedimentation and erosion hydraulics.” Hydraulicdesign handbook, Chap. 6, McGraw-Hill, Inc., New York.

García, M. H., and Parker, G. �1991�. “Entrainment of bed sediment into


Eng. 2008.134:203-217.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Har

yo A

rmon

o on

10/

01/1

5. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

suspension.” J. Hydraul. Eng., 117�4�, 414–435.Hirt, C. W., and Nicholls, B. D. �1981�. “Volume of fluid �vof� method for

dynamics of free boundaries.” J. Comput. Phys., 39�1�, 201–225.Issa, R. I. �1986�. “Solution of the implicitly discretised fluid flow equa-

tions by operator-splitting.” J. Comput. Phys., 62�1�, 40–65.Jasak, H. �1996�. “Error analysis and estimation for the finite volume

method with application to fluid flows.” Ph.D. thesis, Imperial Col-lege of Science, Technology and Medicine, London.

Jasak, H., and Tuković, Z. �2007�. “Automatic mesh motion for the un-structured finite volume method.” Transactions of Faculty of Me-chanical Engineering and Naval Architecture, Univ. of Zagreb,Crotia, 30�6�, 1–18.

Liang, D., Cheng, L., and Li, F. �2005�. “Numerical modeling of flow andscour below a pipeline currents. Part II: Scour simulation.” CoastalEng., 52�1�, 43–62.

Liu, X., and García, M. H. �2006�. “Numerical simulation of local scourwith free surface and automatic mesh deformation.” Proc., EWRI

World Water and Environmental Congress, ASCE, Reston, Va.Mayer, S., Garapon, A., and Sorensen, L. S. �1998�. “A fractional step

method for unsteady free-surface flow with application to non-linearwave dynamics.” Int. J. Numer. Methods Fluids, 28�2�, 293–315.

Mayer, S., and Madsen, P. �2000�. “Simulation of breaking waves in thesurf zone using a Navier–Stokes solver.” Proc., Int. Conf. in Coastal

Engineering, ASCE, Reston, Va.Olsen, N. R. B., and Melaaen, M. C. �1993�. “Three-dimensional calcu-

lation of scour around cylinders.” J. Hydraul. Eng., 119�9�, 1048–1054.

Olsen, N. R. B., and Melaaen, M. C. �1999�. “Three-dimensional numeri-cal flow modeling for estimation of maximum local scour depth.” J.Hydraul. Res., 36�9�, 579–590.

OpenCFD. �2005�. “OpenFOAM: The open source computational fluiddynamics �cfd� toolbox.” �http:://www.OpenFoam.org �Dec. 10,2006�.

Parker, G., and García, M. H. �2006�. “River, coastal and estuarine mor-
phodynamics.” Proc., RCEM 2005, Taylor & Francis Group, London.


Patankar, S. V. �1981�. Numerical heat transfer and fluid flow, McGraw-Hill, New York.

Rajaratnam, N. �1976�. Turbulent jet, Elsevier Scientific, New York.Richardson, J. E., and Panchang, V. G. �1998�. “Three-dimensional simu-

lation of scour-inducing flow at bridge piers.” J. Hydraul. Eng.,124�5�, 530–540.

Rodi, W. �1993�. Turbulence models and their application in hydraulics—A state of the art reviews, 3rd Ed., International Association for Hy-draulic Research, Delft, Balkema, Rotterdam, The Netherlands.

Roulund, A., Sumer, B. M., Fredsøe, J., and Michelsen, J. �2005�. “Nu-merical and experimental investigation of flow and scour around acircular pile.” J. Fluid Mech., 534, 351–401.

Sethian, J. A. �1996�. Level set methods: Evolving interfaces in geometry,fluid mechanics, computer vision, and material science, CambridgeMonographs on Applied and Computational Mathematics, CambridgeUniversity Press, Cambridge, U.K.

Smith, J. D., and McLean, S. �1977�. “Spatially averaged flow over awavy surface.” J. Geophys. Res., 83, 1735–1746.

Sumer, B. M., Christiansen, N., and Fredsøe, J. �1993�. “Influence ofcross section on wave scour around piles.” J. Waterway, Port,Coastal, Ocean Eng., 119�5�, 477–495.

Sumer, B. M., and Fredsøe, J. �2001�. “Wave scour around a large verticalcircular cylinder.” J. Waterway, Port, Coastal, Ocean Eng., 127�3�,125–134.

Sumer, B. M., Fredsøe, J., and Christiansen, N. �1992�. “Scour around avertical pile in waves.” J. Waterway, Port, Coastal, Ocean Eng.,118�1�, 15–31.

Ubbink, O., and Issa, R. I. �1999�. “A method for capturing sharp fluidinterfaces on arbitrary meshes.” J. Comput. Phys., 153, 26–50.

Van Beek, F. A., and Wind, H. G. �1990�. “Numerical modeling of ero-sion and sedimentation around offshore pipelines.” Coastal Eng., 14,107–128.

van Rijn, L. C. �1984�. “Sediment transport. Part II: Suspended loadtransport.” J. Hydraul. Eng., 110�11�, 1613–1641.

Weller, H. G., Tabor, G., Jasak, H., and Fureby, C. �1998�. “A tensorialapproach to computational continuum mechanics using object-
oriented techniques.” Comput. Phys., 12�6�, 620–631.

Eng. 2008.134:203-217.

3d scour model

Documents

level set method lsm

blocking method

cell mac method

mac method isdomain

thefree surface

rapidlychanging water

grid points

urbana andchampaign