ai et al,2010_non hydro static 3d

8/8/2019 Ai Et Al,2010_non Hydro Static 3d

1/17

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids (2010)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.2317

A new fully non-hydrostatic 3D free surface flow model for waterwave motions

Congfang Ai,, Sheng Jin and Biao Lv

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024,

Peoples Republic of China

SUMMARY

A new fully non-hydrostatic model is presented by simulating three-dimensional free surface flow on avertical boundary-fitted coordinate system. A projection method, known as pressure correction technique,is employed to solve the incompressible Euler equations. A new grid arrangement is proposed under a

horizontal Cartesian grid framework and vertical boundary-fitted coordinate system. The resulting modelis relatively simple. Moreover, the discretized Poisson equation for pressure correction is symmetric andpositive definite, and thus it can be solved effectively by the preconditioned conjugate gradient method.Several test cases of surface wave motion are used to demonstrate the capabilities and numerical stabilityof the model. Comparisons between numerical results and analytical or experimental data are presented.It is shown that the proposed model could accurately and effectively resolve the motion of short waveswith only two layers, where wave shoaling, nonlinearity, dispersion, refraction, and diffraction phenomenaoccur. Copyright q 2010 John Wiley & Sons, Ltd.

Received 13 October 2009; Revised 1 February 2010; Accepted 6 February 2010

KEY WORDS: fully non-hydrostatic; free surface; three-dimensional; boundary-fitted coordinate; waves;projection method

1. INTRODUCTION

As the computational power advanced, three-dimensional (3D) non-hydrostatic free surface model

for simulating water waves was developed and became more popular [1 4]. Compared with 3Dshallow water model [5, 6], non-hydrostatic model can resolve short wave motions, buoyancy-drivenflows, and flows occurring in steep bottom gradients. Wave dispersion is the main characteristic

of short waves, and is mainly caused by a non-hydrostatic pressure distribution. To efficiently and

accurately simulate water wave motions, non-hydrostatic effects must be concerned.

A number of models for the simulation water waves have been applied for coastal engineering-

type applications. Depth-averaged models based on integrating over the water depth reduce the

dimension of computational domains, including shallow water equation model, mild slope equation

model, and Boussinesq-type equation model. Shallow water equation model based on hydrostaticpressure distribution is not able to resolve short waves motions. Boussinesq equation model [7] is anextension of shallow water equation model, and together with mild slope equation model [8], theyare suited for the modeling of wave shoaling, nonlinearity, dispersion, refraction, and diffraction.

To date, many efforts have been sought to obtain the highly nonlinear and dispersive models [9].They can be applied to large amplitude waves in relatively deep water, but complicated discretized

Correspondence to: Congfang Ai, State Key Laboratory of Coastal and Offshore Engineering, Dalian University ofTechnology, Dalian 116024, Peoples Republic of China.

E-mail: [email protected]

Copyright q 2010 John Wiley & Sons, Ltd.


2/17

C. AI, S. JIN AND B. LV

methods and large computational expense are the key problems. Although many depth-averaged

models can be successfully applied to simulate short wave motions, they are unable to predict the

vertical flow structure. 3D non-hydrostatic model as a unified model is capable of resolving this

short wave motions successfully.

One of the main issues on developing non-hydrostatic models is applying the pressure boundary

condition precisely on the free surface. Many efforts have been sought to impose zero pressure

boundary condition at the free surface. In early non-hydrostatic models, hydrostatic pressuredistribution at the top layer is proposed by Casulli [10] and a large number of vertical layers arenecessary to simulate short wave motions. Stelling and Zijlema [1] proposed an edge-based compactdifference scheme known as the Keller-box scheme for the approximation of vertical gradient of the

non-hydrostatic pressure at the cell faces of the vertical grids. Such a definition of non-hydrostatic

pressure term allows zero pressure boundary condition to be imposed straightforward at the free

surface. Their satisfactory results show that with only two vertical layers their model is capable

of predicting short wave motions in an accurate and efficient manner. Subsequently, Zijlema and

Stelling [3] changed the numerical algorithm to pressure correction technique and employed avertical boundary-fitted coordinate system. Their results demonstrated the accuracy, robustness,

and efficiency of the model. Yuan and Wu [2] proposed an integral method to remove the top-layerhydrostatic distribution assumption. Their method is built upon a standard staggered grid system

and integrates the vertical momentum equation from the center of top layer to the free surface

to impose zero pressure boundary condition precisely. As a result, using a very small number of

vertical layers, the model is capable to simulate accurately a range of free surface flow problems.

Non-hydrostatic models with accurate pressure boundary condition on the free surface are usually

called fully non-hydrostatic model. The models concerned above are all fully non-hydrostatic

model except Casulli [10].For free surface flow simulations, the other difficulty is to numerically capture the moving

boundary. Several methods to simulate this boundary have been successfully incorporated in the

NavierStokes equations, such as marker and cell (MAC) method [11], the volumes of fluid (VOF)method [12, 13], level set method [14], and LagrangianEulerian method [15]. All of these methodsare limited by high computational expense and strict stability requirement. On the other hand, in

many environmental stratified flows, the free surface elevation can be defined as a single-valued

function of horizontal positions and can be calculated by integrating the mass conservation equation

to the water column and kinematic free surface boundary condition. This method can effectivelytrack the free surface motions with relatively small computational expense and have been widely

applied in a wide variety of problems.

For the numerical solution of 3D non-hydrostatic models, almost all of the computational time is

spent in resolving the Poisson equation for the pressure correction, since the overall efficiency of the

numerical code will depend on its performance. In many 3D non-hydrostatic models, the Poisson

systems are non-symmetric, and to raise models efficiency various iterative solution methods are

applied for the solution of the Poisson equation. To solve non-symmetric systems, Krylov subspace

methods, including bi-conjugate gradient method (BiCGSTAB) and generalized minimal residual

method (GMRES), are very popular. Chen [16] deployed BiCGSTAB solver with a preconditioningprocedure to solve resulting matrix systems. Zijlema and Stelling [1] also employed this solverand combined with preconditioners based on the incomplete LU decompositions to solve their

Poisson systems. Lee et al. [17] and Bingham et al. [18] employed the GMRES iterative methodto solve their resulting systems. To further optimize the preconditioning in terms of efficiencyand robustness, Engsig-Karup [19] employed the multi-grid method as their GMRES solverspreconditioner.

If the discretization of the Poisson equation results in linear system of equations with a symmetric

positive definite matrix, the well-known preconditioned conjugate gradient method could be applied.

Casulli [10] employed hydrostatic approximation at the top layer and constructed a symmetricand positive definite Poisson matrix. As a result, 1620 vertical layers were employed to obtain

reasonable results. In Lin and Lis model [20], the resulting Poisson matrix is also symmetric andpositive definite, and incomplete Cholesky conjugate gradient method was employed to solve this

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2010)

DOI: 10.1002/fld


3/17

NON-HYDROSTATIC 3D FREE SURFACE FLOW

matrix, but results showed that the model with 20 vertical layers was capable of resolving water

wave motions.

In this paper, a new fully non-hydrostatic, 3D free surface model is developed. The model

is built on a vertical boundary-fitted coordinate system. The numerical algorithm is based on

pressure correction method, which solves the 3D Euler equations in two major steps. First, inter-

mediate velocity field is achieved by means of solving the momentum equations that contain the

non-hydrostatic pressure at the previous time level. In the second step, the pressure correction,which is the difference between the new and old non-hydrostatic pressure, is computed by the

discretized Poisson equation. This equation is symmetric and positive definite, and is obtained

by the combination of the discretized continuity and the discretized momentum equation with

the pressure correction terms. In this step, a conjugated gradient method with symmetric Gauss

Seidel preconditioning procedure is used to solve the Poisson equation. Then, through the pressure

correction, to result in a divergence-free velocity field, the intermediate velocity is corrected.

A new grid arrangement under a horizontal Cartesian grid framework and vertical boundary-fitted

coordinate system is presented. Following Stellings work, the pressure is defined at the edge of

the layer. This definition guarantees that the zero pressure boundary condition at the free surface

can be approximated very accurately, which enables the model to simulate water wave motions

with only a few vertical layers. Vertical velocity is defined at the layer center whereas most other

models, such as

[1 4

], are edge based. Such definition simplifies the discretization of the vertical

momentum equation compared with Keller-box scheme and results in symmetric and positive defi-

nite Poisson equation. In the following section, the governing equations together with boundary

conditions are described. Numerical approximation for the non-hydrostatic model is explained.

Finally, the model is validated by several examples, including standing wave in closed basin,

solitary wave propagation over a long channel, wave propagation over a submerged bar, and wave

transformation over an elliptic shoal on sloped bottom, to demonstrate the accuracy and efficiency

of the developed model.

2. GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

2.1. Governing equations

For non-hydrostatic free surface water flow, the governing equations are the 3D incompressible

Euler equations, which are based on the conservation of mass and momentum. By splitting the

pressure into hydrostatic and non-hydrostatic ones, p=g(z)+q, the following equations areobtained:

*u

*x+ *v*y

+ *w*z

=0 (1)

*u

*t+ *u

2

*x+ *uv

*y+ *uw

*z=g *

*x *q*x

(2)

*v

*t +*uv

*x +*v2

*y +*vw

*z =g*

*y *q

*y(3)

*w

*t+ *uw

*x+ *vw

*y+ *w

2

*z=*q

*z(4)

where t is the time; u, v, and w are the velocity components in the horizontal x , y, and vertical

z directions, respectively; the normalized pressure p is defined as the pressure divided by a constant

reference density; is the free surface elevation; q is the non-hydrostatic pressure component; and

g is the gravitational acceleration.


DOI: 10.1002/fld


4/17


2.2. Boundary conditions

Boundary conditions are required at all the boundaries of a 3D domain, including the free surface

and the bottom. At the moving free surface , the kinematic boundary condition is

*

*t+u *

*x+v *

*y=w|z= (5)

In addition, atmospheric pressure is assumed at free surface elevation, giving q|z==0.For the impermeable bottom surface z=h(x,y), the kinematic bottom boundary condition is

u *h*x

v *h*y

=w|z=h (6)

Using kinematic boundary conditions (5) and (6) in the integrated form of the continuity equation (1)

over the water column, the free surface equation is obtained:

*

*t+ **x

hu dz+ *

*y

hv dz=0 (7)

At the inflow, the normal velocity component is specified by either analytical solutions or linear

wave theory. Furthermore, tangential velocity components are set to zero. At the outflow boundary,a combination of the Sommerfeld radiation condition and a sponge layer technique is implemented

to minimize wave reflection into the computational domain. For solid walls, the impermeability

condition is specified, i.e. velocity normal to the wall is zero, whereas a zero normal gradient for

the tangential velocity components is specified.

3. NUMERICAL METHOD

3.1. Spatial discretization

The 3D physical domain is bounded by the moving free surface, z=, and the bottom, z=d,and is discretized as a 2D structured horizontal grid with several horizontal layers. In this system,

the 3D grid projection on horizontal plane forms a rectilinear grid system, which has a set of Nxand Ny cells in x , y directions, respectively. In the vertical direction, boundary-fitted grid system

is employed and the physical domain is divided into Nz layers.

Take i , j , and k as the grid indexes in the x , y, and z directions, respectively. A 3D cell with

center at (i, j,k) is bounded by the intersection of the water column between the bottom and

the free surface with the horizontal levels zk1/2 and the horizontal grid lines xk1/2 and yk1/2.In the vertical grid system, the number of vertical layers Nz across the whole domain is constant

and the layer thickness hk is allowed to vary with respect to both the horizontal locations and time.

A new grid arrangement under a horizontal Cartesian grid framework and vertical boundary-

fitted coordinate system is presented. Figure 1 shows the definition of main variables on the xzplane. The 2D variables (free surface and bottom elevation) are defined at the geometric center of

the 2D horizontal mesh cells, which is denoted by (i, j). The non-hydrostatic pressure component

is stored at the cell faces, which are denoted by (i, j,k+1/2). The horizontal velocity componentsu and v are stored at the center of the cell faces (i+1/2, j,k) and (i, j +1/2,k). On the contrary,the vertical velocity component w is stored at the cell centers (i, j,k) rather than at the cell faces

(i, j,k+1/2).

3.2. Numerical algorithms

The governing equations (1)(4) are first integrated over a vertical layer to get the semi-discretized

equations based on the vertical boundary-fitted coordinate system, and then, the pressure correction

method is applied for solving the equations in two major steps. In the first step, the interme-

diate velocity field is achieved by means of solving the momentum equations that contain the


DOI: 10.1002/fld


5/17


6/17


It should be emphasized that there have two major differences in momentum equations from

Zijlema and Stellings layer integrated equations. First, in the momentum equations (9) and (10),

the horizontal gradient of the non-hydrostatic pressure is assumed to be independent of the depth,

and so, integration over layer k has no effect on it. Second, unlike Zijlema and Stellings integrated

algorithm applied for vertical momentum equation, the integration applied in this paper is also

over layer k.

3.2.2. Intermediate velocity field calculation. The momentum equations containing the non-

hydrostatic pressure at the preceding time level are solved to get the intermediate velocity

field.

In a control volume centered at (i+1/2, j,k), Equation (9) can be expressed as follows:

*(hu )i+1/2,j,k*t

+ *(hu u)i+1/2,j,k*x

+ *(huv)i+1/2,j,k*y

+i+1/2,j,k+1/2ui+1/2,j,k+1/2i+1/2,j,k1/2ui+1/2,j,k1/2

+gh i+1/2,j,k*

*xi+1/2,j,k+hi+1/2,j,k

*q

*x i+1/2,j,k=0 (13)

Multiplying Equation (8) with ui+1/2,j,k and subtracting the result from Equation (9) yields thefollowing equations for u, details can be found in Reference [21].

hi+1/2,j,k*ui+1/2,j,k

*t+ *(hu u)i+1/2,j,k

*x+ *(huv)i+1/2,j,k

*y

ui+1/2,j,k*(hu )i+1/2,j,k

*x+ *(hv)i+1/2,j,k

*y

+i+1/2,j,k+1/2(ui+1/2,j,k+1/2ui+1/2,j,k)i+1/2,j,k1/2(ui+1/2,j,k1/2ui+1/2,j,k)

+gh i+1/2,j,k*

*x

i+1/2,j,k+hi+1/2,j,k*q

*x

i+1/2,j,k=0 (14)This equation can be discretized by finite volume method over a control volume centered at

(i +1/2, j,k). This gives

hni+1/2,j,kxy(un+1/2i+1/2,j,kuni+1/2,j,k)/t

+y[(hu )ni+1,j,k(uni+1,j,kuni+1/2,j,k)(hu)ni,j,k(uni,j,kuni+1/2,j,k)]+x(hv)ni+1/2,j+1/2,k(uni+1/2,j+1/2,kuni+1/2,j,k)x(hv)ni+1/2,j1/2,k(uni+1/2,j1/2,kuni+1/2,j,k)

+xyni+1/2,j,k+1/2(uni+1/2,j,k+1/2uni+1/2,j,k)xyni+1/2,j,k1/2(uni+1/2,j,k1/2uni+1/2,j,k)+gh ni+1/2,j,ky(ni+1,j ni,j)+hni+1/2,j,ky[(qni+1,j,k+1/2+qni+1,j,k1/2)(qni,j,k+1/2+qni,j,k1/2)]/2=0 (15)

where (hu)i,j,k is the flux through the left side of this control volume, and is obtained by averaging

the fluxes through the faces at (i 1/2, j,k) and (i+1/2, j,k). Similar algorithm can also beapplied for other fluxes, such as hv and hw.


DOI: 10.1002/fld


7/17


The u with the over bar notation indicates that it must be obtained by means of interpolation

method. For this paper, a combination of the first-order upwind scheme and the second-order

central differencing scheme are employed to get it, e.g.

uni,j,k= (uni1/2,j,k+uni+1/2,j,k)/2sgn((hu )ni,j,k)(uni+1/2,j,kuni1/2,j,k)/2 (16)where controls the amount of blending between the first-order upwind scheme and the second-

order central differencing scheme. A value of about 0.1 is accurate enough and is recommended.Equation (15) can be simplified as follows:

un+1/2i+1/2,j,k= Adec(u)+ fu1ni+1,j + fu2ni,j + fu3qni+1,j,k+1/2+ fu4qni+1,j,k1/2

+ fu5qni,j,k+1/2+ fu6qni,j,k1/2 (17)where Adec(u) is the advection term and could be expressed as

Adec(u)= fa1uni+1/2,j,k+ fa2uni+3/2,j,k+ fa3uni1/2,j,k+ fa4uni+1/2,j+1,k+ fa5uni+1/2,j1,k+ fa6uni+1/2,j,k+1+ fa7uni+1/2,j,k1 (18)

Similar discretizing procedure can be applied to Equations (10) and (11). The simplified discretizedequations for Equations (10) and (11) can be expressed as follows:

vn+1/2i,j+1/2,k= Adec(v)+ fv1ni,j+1+ fv2ni,j + fv3qni,j+1,k+1/2+ fv4qni,j+1,k1/2

+ fv5qni,j,k+1/2+ fv6qni,j,k1/2 (19)

wn+1/2i,j,k =Adec(w)+ fw1qni,j,k+1/2+ fw2qni,j,k1/2 (20)

where Adec(v) and Adec(w) are the advection terms for v and w momentum equations and

can be obtained by the method as expression (16). The coefficients fa , fu , fv, and fw in

Equations (17)(20) as a function of known values at the previous time can be easily obtained.

3.2.3. Non-hydrostatic component calculation. In the second step of calculations, the new velocity

field un+1i+1/2,j,k, vn+1i+1/2,j,k, and w

n+1i,j,k are computed by correcting the intermediate values after

including the non-hydrostatic correction term.

un+1

i+1/2,j,k= Adec(u)+ fu1ni+1,j + fu2ni,j + fu3qn+1i+1,j,k+1/2+ fu4qn+1i+1,j,k1/2+ fu5qn+1i,j,k+1/2+ fu6qn+1i,j,k1/2 (21)

vn+1i,j+1/2,k= Adec(v)+ fv1ni,j+1+ fv2ni,j

+ fv3qn+

1i,j+1,k+1/2+ fv4q

n+

1i,j+1,k1/2+ fv5q

n+

1i,j,k+1/2+ fv6q

n+

1i,j,k1/2 (22)

wn+1

i,j,k=Adec(w)+ fw1qn+1i,j,k+1/2+ fw2qn+1i,j,k1/2 (23)

Here, subtracting Equations (17)(20) from Equations (21)(23), the following equations for

correcting the intermediate velocity field are obtained as follows:

un+1i+1/2,j,k= un+1/2i+1/2,j,k+ fu3qi+1,j,k+1/2+ fu4qi+1,j,k1/2+ fu5qi,j,k+1/2+ fu6qi,j,k1/2 (24)


DOI: 10.1002/fld


8/17


vn+1i,j+1/2,k= vn+1/2

i,j+1/2,k+ fv3qi,j+1,k+1/2+ fv4qi,j+1,k1/2

+ fv5qi,j,k+1/2+ fv6qi,j,k1/2 (25)

w

n

+1

i,j,k=wn

+1/2

i,j,k + fw1

qi,j,k+1/2+ fw2

qi,j,k1/2 (26)where q =qn+1qn is the non-hydrostatic pressure correction term.

In the definition of velocities illustrated in Figure 1, the discretized continuity equation (1)

above the bottom cell is taken to be

(un+1i+1/2,j,k+un+1i+1/2,j,k1)(un+1i1/2,j,k+un+1i1/2,j,k1)2x

+(vn+1i,j+1/2,k+vn+1i,j+1/2,k1)(vn+1i,j1/2,k+vn+1i,j1/2,k1)

2y

+wn+1

i,j,kwn+1

i,j,k1hi,j,k1/2 =0, k=2,3, . . . ,Nz (27)

where hi,j,k1/2= (hni,j,k1+hni,j,k)/2For k=1 and by setting wn+1i,j,1/2=0, the finite difference approximation of Equation (1) in a

half bottom cell is

un+1

i+1/2,j,1un+1i1/2,j,1x

+vn

+1i,j+1/2,1vn+1i,j1/2,1

y+

wn+1

i,j,1

hi,j,1/2=0 (28)

where hi,j,1/2=hni,j,1/2The Poisson equation for the non-hydrostatic pressure correction term q is derived by substi-

tuting the expressions for the new velocities from (24)(26) into (27) and (28), respectively. Then,the Poisson equation is symbolically written as:

Aq =B (29)Due to the use of the vertical boundary-fitted coordinate system, A is a sparse coefficient matrix

with the dimension of (Nx NyNz)(Nx NyNz); q is a vector of the calculated non-hydrostaticpressure correction; and B is a known vector related to the intermediate velocities. The sparse

matrix A contains 10 non-zero diagonals at bottom cells and contains 15 non-zero diagonals at

other cells; moreover, it is symmetric and positive definite. Thus, Equation (29) can be solved

efficiently by preconditioned conjugated gradient method. In this paper, the symmetric Gauss

Seidel preconditioning method [17] is employed as the preconditioner.Once the non-hydrostatic pressure correction is obtained, the corresponding velocity field is

readily determined from Equations (24) to (26).Finally, the new free surface elevation is obtained by the finite volume approximation of

Equation (7) as follows:

n+1i,j

ni,jt

+ 1x

Nz

k=1hni+1/2,j,ku

n+1i+1/2,j,k

Nzk=1

hni1/2,j,kun+1i1/2,j,k

+ 1y

Nz

k=1hni,j+1/2,kv

n+1i,j+1/2,k

Nzk=1

hni,j1/2,kvn+1i,j1/2,k

=0 (30)


DOI: 10.1002/fld


9/17


On the other hand, the vertical velocity k+1/2 relative to layer level zk+1/2 can be obtained bythe discrete form of Equation (8) as follows:

hn+1i,j,khni,j,kt

+ 1x

(hni+1/2,j,kun+1i+1/2,j,khni1/2,j,kun+1i1/2,j,k)

+ 1y

(hni,j+1/2,kvn+1i,j+1/2,khni,j1/2,kvn+1i,j1/2,k)

+n+1i,j,k+1/2n+1i,j,k1/2=0, k=1,2, . . . ,Nz 1 (31)

where

hn+1i,j,k= (n+1i,j +di,j)/Nz (32)

Considering 1/2=0, all the velocities k+1/2 (k=1,2, . . . ,Nz 1) can be calculated by (31).

4. NUMERICAL EXPERIMENTS

In order to evaluate the accuracy and capability of the non-hydrostatic model developed in this

paper, the model is applied to four test cases of short wave motions: (1) standing wave in closed

basin, (2) 2D solitary wave propagation on both flat and uneven bottoms, (3) wave propagation

over a submerged bar, and (4) wave transformation over an elliptic shoal on a sloped bottom.

4.1. Standing wave in closed basin

To verify the models accuracy for simulating linear wave dispersion over arbitrary water depths,

the standing wave in a closed basin of constant depth is considered. The closed basin has a length

of L with a constant depth h =10 m. Initially, the following free surface profile is taken:

(x, t=0)=a cos(kx)cos

2

Tt, 0xL (33)

where a is the amplitude of the standing wave; k=2/L is the wave number; and T is the waveperiod. Here, the wavelength equals the length of the basin. To generate various kh values, we

change the length of the basin, but maintain the condition of infinitesimal waves, by adjusting the

wave amplitude. The horizontal grid size x =L/20 and the time step t=T/40 are used in thecomputations.

Figure 2 shows the normalized wave celerity versus kh . In this figure, the quantities c and

c0=

gh are the wave phase velocity and the long wave celerity, respectively. =|ccomputedcexact|/cexact is the relative error. c is specified by the following linear dispersion relation:

c=gk tanh(kh) (34)It can be seen from Figure 2 that the present model with two vertical layers can accurately predict

linear dispersive waves up to kh7 with a relative error of at most 0.5%. Hence, only two layers

are therefore taken in the following numerical tests.

To specifically show the present models accuracy with two vertical layers, the test for a deep-

water case (kh =2) is considered. In this test, a=0.05 is employed. Figure 3 compares thecomputed free surface elevation with the analytical solutions at x =5.25m. It is clear that over10 wave periods, the fully non-hydrostatic model accurately predicts both the amplitude and phase

of the wave.


DOI: 10.1002/fld


10/17


Figure 2. Normalized wave celerity versus kh for linear dispersion. Present model with two layers (circles),exact (solid line), and relative error (dashed line).

Figure 3. Comparisons of the free surface elevation at x =5.25m between the numericalresults and analytical solutions.

4.2. Solitary wave propagation over a long channel

The solitary wave is a nonlinear wave of finite amplitude that consists of a single displacement

of water above the mean water level. Neglecting bottom friction and viscosity, solitary wave

propagating over a flat bottom maintains a permanent form without changing its shape and velocity.

Many non-hydrostatic models [1,20,22,23] have been applied to simulate this type of waveproblems to verify their models accuracy and efficiency.

In this test, the analytical form of solitary wave whose wave crest is located at 80 m on a constant

depth of 10 m is set as an initial condition. The wave amplitude is 2 m and the computational

domain is from 0 to 1000 m. The uniform horizontal grid size is 1 m and two vertical layers are

taken. The time step oft=0.05s is chosen. At the outflow boundaries (x =1000m), a radiationcondition is imposed to minimize wave reflection.

Figure 4 shows the comparison of the calculated and analytical free surface elevation at the

locations of x =200, 400, and 600 m. With only two vertical layers, good agreements are obtainedby the present model. At x

=600m, the passing wave detected is conserved up to 98% of the

incident wave in terms of wave amplitude, indicating the accuracy of the developed model.In Figure 5, the comparisons of horizontal and vertical velocities between the computation and

theory on the center of the surface layer are depicted at t=20 and 50s. Fair comparisons areobtained. The agreement in Figure 5 for the vertical velocity indicates that the dynamic pressure

has been well simulated by our model.

4.3. Wave propagation over a submerged bar

In this example, we aim to examine the performance of our non-hydrostatic solver in modeling

wave deformation over uneven bottoms. Propagation of waves over the bar involves the generation

of bound higher harmonics through the increase of nonlinearity at the shoaling stage, and the


DOI: 10.1002/fld


11/17


Figure 4. Comparisons of numerical results (solid lines) and analytical solutions (circles) at three locationsfor solitary wave propagation.

transformation of wave patterns due to the release of higher harmonics at the deshoaling stage.

Many researchers have paid a great deal of efforts to investigate this process using physical

experiments [2426] or numerical simulations [13, 20, 23, 27, 28].In this study, the experimental setup by Nadaoka et al. [25] has been selected and depicted

in Figure 6. The wave flume has a length of 30 m. The still water depth is 0.3 m and is reduced

to 0.1 m at the submerged bar. The upward and downward slopes of the bar are 1:20 and 1:10,

respectively. A sinusoidal velocity distribution based on the linear theory is imposed at the inflow

(left) boundary, where an incident wave condition with wave height H0=2.0cm and wave periodT0 =1.5s is used. At the outflow (right) boundary, a 5-m sponge layer with a radiation boundaryis applied. Free surface elevations are measured using wave gauges at seven different stations.

In addition, the velocity fields are measured at the depth of z=0.02m, 0.16m, and 0.26 mat station 7 to examine wave decomposition on the downward slope.


DOI: 10.1002/fld


12/17


Figure 5. Comparisons of u and w at the center of the surface layer between numerical results (solidlines) and analytical solutions (circles) at t=20 and 50 s (from left to right).

Figure 6. Sketch of the experimental setup and measurement stations of free surface elevations (stations17) and velocities (circles) according to Nadaoka et al.

To discretize the computational domain, the equidistant grid spacing of 0.0125 m along the

x -direction and only two vertical layers are used. The time step is taken as 0.005 s.

Comparisons of the free surface elevation at the six measured stations between numerical results

and experimental data are plotted in Figure 7. Figure 8 shows the excellent comparison of

velocity fields between model results and experimental measurements at the three different depths

at station 7.

Figure 7 indicates that the model correctly simulates the shoaling phenomenon at stations 2 and 3.

As the wave riding over the bar, the development of higher frequency components at locations

4 and 5 is also well predicted. The release of generated higher frequency components occurring

behind the bar at locations 6 and 7 is also well simulated. The overall good agreements in free

surface elevation and velocity fields indicate that the model using only two vertical layers can

accurately simulate wave shoaling, nonlinearity, and dispersion phenomena as well as the vertical

flow structure.


DOI: 10.1002/fld


13/17


Figure 7. Comparisons of the free surface elevation between numerical results (solid lines) and experimentaldata (circles) at six wave gauge stations.

4.4. Wave transformation over an elliptic shoal on a sloped bottom

In the last example, the model is aimed to simulate refraction and diffraction phenomenon caused by

wave propagation over a 3D uneven bottom. We compare the numerical results with experimental

data from Berkhoff et al. [29]. Figure 9 shows the experimental setup for the bathymetry ofthe elliptic shoal. Let (x ,y ) be the slope-oriented coordinates, which are related to the (x,y)coordinate system by means of rotation over 20. Then, the boundary of the shoal is given by

x

4

2+

y

3

2=1 (35)

whereas the thickness of the shoal is

d=0.3+0.5

1

x

5

2

y

3.75

2(36)

The bed elevation is given by

zb =min[0.45,max(0.10,0.450.02(5.84+y))]+d (37)The incoming wave with a wave height H0 =4.64cm and a wave period T0 =1.0s is specifiedat the lower boundary y=10m based on linear wave theory. At the end of the computational


DOI: 10.1002/fld


14/17


Figure 8. Comparisons of u and w between numerical results (solid lines) andexperimental data (circles) at Station 7.

domain, y=20 m, a sponge layer coupled with a radiation boundary is also employed to minimizewave reflection.

In the horizontal plane, the grid spacing is set to x =0.1m and y =0.05m. Having shownthe accuracy and efficiency of the model, two layers are also employed in vertical direction.

The total grids in computation are therefore 2006002. The time step is taken as 0.01 s and thetotal simulation time is up to 34 s reaching a stationary wave field shown in Figure 10.

Figure 11 shows the comparison of normalized wave heights at six sections between the

numerical results and experimental data. The computed wave height is obtained by averaging over

four wave periods (i.e. from t=30 to 34 s) once a steady solution is achieved. In Sections 3 and5, the model has satisfactorily simulated the focusing effect and the maximum normalized wave

height is about 2.2 and 1.8, respectively. In other sections, the model results are generally close to

the experimental data. The computations were carried out on Intel Core 2 Q9400 processor with

4GB internal memory. The total CPU time per time step required for the present model was about2.6 s. Overall, these results indicate that the newly developed model using only two vertical layers

can effectively resolve the nonlinear effects of wave refraction and diffraction over a 3D irregular

bottom.

5. CONCLUSIONS

In this paper, a new fully non-hydrostatic model that solves the Euler equations on the vertical

boundary-fitted coordinate system has been developed. The governing equations is integrated over


DOI: 10.1002/fld


15/17


Figure 9. Bottom configuration corresponding to the experimental setup of Berhoff et al. [29].

Figure 10. 2D view of stationary wave field at the end of simulation time.

a layer to get the semi-discretized equations, which have been solved by the pressure correction

technique. A new grid arrangement has been proposed under a horizontal Cartesian grid framework

and a vertical boundary-fitted coordinate system, which rends the relatively simple resulting model.

More importantly, the discretized Poisson equation for pressure correction is symmetric and positive

definite, which can be solved effectively by the preconditioned conjugate gradient method.

The developed model is validated by a series of numerical tests that include standing short

wave in closed basin, solitary wave propagation over a long channel, wave propagation over a

submerged bar, and wave transformation over an elliptic shoal on sloped bottom. It is found that

the model with only two vertical layers is capable of accurately resolving linear dispersion over


DOI: 10.1002/fld


16/17


Figure 11. Comparison of normalized wave heights between numerical results (solid line) and experimentaldata (circles) at six sections.

water depths up to kh =7 and can accurately predict essential wave phenomena, such as waveshoaling nonlinearity, dispersion, refraction, and diffraction. Moreover, the model is shown to well

resolve vertical velocity profiles of solitary wave propagating over a long channel and nonlinear

wave deformation over a submerged bar.

In the near future, the validated model with wetting and drying algorithm can be easily applied

to study wave breaking of wave propagation in the surf zone. Meanwhile, it will also be coupled

with a turbulence closure scheme, and then applied to study 3D wavestructure interactions.

REFERENCES

1. Stelling GS, Zijlema M. An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow

with application to wave propagation. International Journal for Numerical Methods in Fluids 2003; 43:123.

2. Yuan H, Wu CH. An implicit three-dimensional fully non-hydrostatic model for free-surface flows.International

Journal for Numerical Methods in Fluids 2004; 46:709733.

3. Zijlema M, Stelling GS. Further experiences with computing non-hydrostatic free-surface flows involving water

waves. International Journal for Numerical Methods in Fluids 2005; 48:169197.

4. Badiei P, Namin M, Ahmadi A. A three-dimensional non-hydrostatic vertical boundary fitted model for free-surface

flows. International Journal for Numerical Methods in Fluids 2008; 56:607627.


DOI: 10.1002/fld


17/17


5. Casulli V, Cheng RT. Semi-implicit finite difference methods for three-dimensional shallow water flow.

International Journal for Numerical Methods in Fluids 1992; 15:629648.

6. Cheng RT, Casulli V, Gartner JW. Tidal, residual, intertidal mudflat (TRIM) model and its applications to

San Francisco Bay, California. Estuarine Coastal Shelf Science 1993; 36:235280.

7. Madsen PA, Fuhrman DR, Wang Benlong. A Boussinesq-type method for fully nonlinear waves interacting with

rapidly varying bathymetry. Coastal Engineering 2006; 53:487504.

8. Berkhoff JCW. Mathematical models for simple harmonic linear water waves: wave refraction and diffraction.

Ph.D. Thesis, Delft Technical University of Technology, Delft, 1976.9. Fuhrman DR, Bingham HB. Numerical solutions of fully non-linear and highly dispersive Boussinesq equations

in two horizontal dimensions. International Journal for Numerical Methods in Fluids 2004; 44:231255.

10. Casulli V. A semi-implicit finite difference method for non-hydrostatic, free-surface flows.International Journal

for Numerical Methods in Fluids 1999; 30:425440.

11. Harlow FH, Welch JE. Numerical calculation of time-dependent viscous incompressible flow.Physics of Fluids

1965; 8:21822189.

12. Shen YM, Ng CO, Zheng YH. Simulation of wave propagation over a submerged bar using the VOF method

with a two-equation k-epsilon turbulence modeling. Ocean Engineering 2004; 31:8795.

13. Hur DS, Mizutani N. Numerical estimation of the wave forces acting on a three-dimensional body on submerged

breakwater. Coastal Engineering 2003; 47:329345.

14. Chen HC, Yu K. CFD simulations of wave-current-body interactions including greenwater and wet deck slamming.

Computers and Fluids 2009; 38:970980.

15. Hodges BR, Street RL. On simulation of turbulent nonlinear free-surface flows.Journal of Computational Physics

1999; 151:425457.

16. Chen X. A fully hydrodynamic model for three-dimensional, free-surface flows. International Journal forNumerical Methods in Fluids 2003; 42:929952.

17. Lee JW, Teubner MD, Nixon JB, Gill PM. A 3-D non-hydrostatic pressure model for small amplitude free-surface

flows. International Journal for Numerical Methods in Fluids 2006; 50:649672.

18. Bingham HB, Zhang Haiwen. On the accuracy of finite-difference solutions for nonlinear water waves.Journal

of Engineering Mathematics 2007; 58:211228.

19. Engsig-Karup AP, Bingham HB, Lindberg O. An efficient flexible-order model for 3D nonlinear water waves.

Journal of Computational Physics 2009; 228:21002118.

20. Lin P, Li C. A -coordinate three-dimensional numerical model for surface wave propagation. International

Journal for Numerical Methods in Fluids 2002; 38:10451068.

21. Zijlema M, Stelling GS. Efficient computation of surf zone waves using the nonlinear shallow water equations

with non-hydrostatic pressure. Coastal Engineering 2008; 55:780790.

22. Namin MM, Lin B, Falconer RA. An implicit numerical algorithm for solving non-hydrostatic free-surface flow

problems. International Journal for Numerical Methods in Fluids 2001; 35:341356.

23. Choi DU, Wu CH. A new efficient 3D non-hydrostatic free-surface flow model for simulating water wave

motions. Ocean Engineering 2006; 33:587609.24. Beji S, Battjes JA. Experimental investigation of wave propagation over a bar. Coastal Engineering 1993;

19:151162.

25. Nadaoka K, Beji S, Nakakawa Y. A fully-dispersive nonlinear wave model and its numerical solutions. In

Proceedings of the 24th International Conference on Coastal Engineering (ASCE), Kobe, Japan, 1994; 427441.

26. Ohyama T, Kiota W, Tada A. Applicability of numerical models to nonlinear dispersive waves.Coastal Engineering

1995; 24:297313.

27. Yuan H, Wu C H. Fully nonhydrostatic modeling of surface waves.Journal of Engineering Mechanics 2006;

132:447456.

28. Young C, Wu CH. An efficient and accurate non-hydrostatic model with embedded Boussinesq-type like equations

for surface wave modeling. International Journal for Numerical Methods in Fluids 2009; 60:2753.

29. Berkhoff JCW, Booy N, Radder AC. Verification of numerical wave propagation models for simple harmonic

linear water waves. Coastal Engineering 1982; 6:255279.


DOI: 10 1002/fld

ai et al,2010_non hydro static 3d

Documents