Efficient Nonhydrostatic Modeling of Surface Waves from Deep to Shallow Water. Chin H. Wu1; Chih-Chieh Young2; Qin Chen3; and Patrick J. Lynett4. Abstract: ...
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Chin H. Wu1; Chih-Chieh Young2; Qin Chen3; and Patrick J. Lynett4 Abstract: An efficient nonhydrostatic model with the embedded Boussinesq-type like equations at the free surface is presented to simulate surface waves from shallow to deep water. The new free-surface treatment yields an accurate expression of vertical distribution of nonhydrostatic pressure. This approach is free of the irrotational flow assumption. Two locations of reference velocities obtained through interpolations of nonhydrostatic velocities are used to optimize frequency dispersion for a wide range of wave conditions. Accuracy and efficiency of the model is critically tested against analytical solutions and experimental data. Overall, the present nonhydrostatic model using a small number of vertical layers 共i.e., 2–4兲 is capable of resolving dispersive wave motions with various effects. DOI: 10.1061/共ASCE兲WW.1943-5460.0000032 CE Database subject headings: Surface waves; Wave dispersion; Boussinesq equations; Currents; Wave velocity; Shallow water; Deep water. Author keywords: Non-hydrostatic; Implicit method; Free-surface waves; dispersion, Boussinesq-type equations; Sheared current, reference velocity.
Introduction Efficient and accurate modeling of free-surface waves is important to many coastal and ocean engineering design problems 共e.g., Mei and Liu 1993兲. As a wave travels from the deep-water to shallow-water regions, its transformation is dominated by the combined effects of dispersion, shoaling, refraction, and diffraction. Nonlinear effects can further redistribute wave energy. A large-scale current field also affects wave propagation. Especially when highly sheared currents are present, the wave motion becomes much more complicated and the wave field may no longer be irrotational. For the past several decades, continuous effort has been devoted to developing unified models that can accurately and efficiently predict wave propagation with various effects 共see, e.g., Wu 2001; Liu and Losada 2002兲. The Boussinesq-type wave equations, one kind of the depthintegrated models, have been prevailing due to their computational efficiency. The basic idea is to consider a certain degree of nonhydrostatic effects attributed to the vertical acceleration of fluids while integrating the conservation laws over the water depth. For example, by assuming that both nonlinearity and frequency dispersion are weak and in the same order of magnitude, 1
Dept. of Civil and Environmental Engineering, Univ. of Wisconsin, Madison, WI 53706 共corresponding author兲. E-mail: chinwu@ engr.wisc.edu 2 Dept. of Civil Engineering, National Taiwan Univ., Taipei 10617, Taiwan. 3 Dept. of Civil and Environmental Engineering, Louisiana State Univ., Baton Rouge, LA 70803. 4 Dept. of Civil Engineering, Texas A&M Univ., College Station, TX 77843. Note. This manuscript was submitted on April 9, 2009; approved on August 12, 2009; published online on February 12, 2010. Discussion period open until August 1, 2010; separate discussions must be submitted for individual papers. This paper is part of the Journal of Waterway, Port, Coastal, and Ocean Engineering, Vol. 136, No. 2, March 1, 2010. ©ASCE, ISSN 0733-950X/2010/2-104–118/$25.00.
Peregrine 共1967兲 derived the classical Boussinesq equations for uneven bottoms. Continuous progress has been made to remove the original constraints of dispersion and nonlinearity, resulting in various sets of new Boussinesq-type equations 共BTEs兲 关see, e.g., Madsen and Sørensen 共1992兲; Nwogu 共1993兲; Wei et al. 共1995兲; Gobbi et al. 共2000兲; Madsen et al. 共2002兲; Lynett and Liu 共2004a兲; Bingham et al. 2009, etc.兴. While the latest Boussinesq-type wave equations are capable of simulating highly nonlinear and dispersive waves, considerable efforts are also required to tackle several challenging issues, including 共1兲 assumptions of irrotational flow 共Chen et al. 1998; Shen 2001兲; 共2兲 difficulties in describing vertical flow structure and wave breaking 共Kennedy et al. 2000; Chen et al. 2000; Veeramony and Svendsen 2000兲; and 共3兲 complexities and stabilities due to higher-order derivatives of mathematical equations and numerical schemes 共Lynett and Liu 2004a兲. Nonhydrostatic modeling becomes feasible due to the rapid growth of computing power in recent years. Based upon the Navier-Stokes equations where nonhydrostatic pressure is directly included and nonlinear effects are taken into account, nonhydrostatic models are capable of simulating nonlinear dispersive short waves where the vertical motion is comparable to the horizontal motion 共Marshall et al. 1997兲. Additionally, similar to the multilayer Green-Naghdi models 共Green and Naghdi 1976; Choi 2003; Percival et al. 2008兲 and the velocity-based boundaryintegral model 共Nwogu 2009兲, nonhydrostatic models without imposing an irrotational flow assumption can be used to investigate waves interacting with shear and rotational flows 共Yuan and Wu 2006兲. Recently, one of the advances in nonhydrostatic modeling is the use of a small number of vertical layers, similar to the multilayer BTEs 共Lynett and Liu 2004a兲, to efficiently predict wave propagation and transformation. The essence of these efficient nonhydrostatic models is to address the top-layer nonhydrostatic pressure by the employment of 共1兲 a Keller-box scheme under an edged-based grid system 共Stelling and Zijlema 2003; Zijlema and Stelling 2008兲; 共2兲 an integration method 共Yuan and Wu 2004a; Choi and Wu 2006; Young et al. 2007; Young et al.
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2009兲; 共3兲 an interpolation method 共Badiei et al. 2008; Cea et al. 2009兲; 共4兲 an extrapolation method 共Bradford 2005; Li 2008兲; and 共5兲 a Boussinesq-type approach 共Young and Wu 2009兲 under a staggered grid system. For example, using an analytical-based pressure distribution with the reference velocity concept 共Nwogu 1993兲, Young and Wu 共2009兲 shows that a two-layer nonhydrostatic model 共NHM兲 with the embedded BTE 共i.e., NHM-BTE兲 can resolve linear wave dispersion up to the dimensionless water depth Kh ⬃ 10. Increasing vertical layers further facilitates highly dispersive wave modeling. In comparison with traditional NavierStokes-based nonhydrostatic models, numerical tests indicate that NHM-BTE dramatically reduces computational costs due to the small number of required vertical layers. While these results are promising, there are two main issues in the use of a single location for the reference velocity in NHM-BTE 共Young and Wu 2009兲. First, under a staggered grid system, interpolation 共or extrapolation兲 errors can easily be introduced during the matching step between NHM and BTE. Second, it is recognized that parameter optimization for different water depths somehow limits practical application involving a wide range of wave conditions. The purpose of this paper is to present an efficient NHM for simulating surface waves from shallow to deep water by addressing the issues found in Young and Wu 共2009兲. Specifically, we aim to use two locations of the references velocities under a staggered grid system to optimize the linear wave dispersion property. A modified top-down resolving 共MTDR兲 method, different from a top-layer control 共TLC兲 technique 共Young and Wu 2009兲, is proposed here to tackle the issue of parameter optimization for a range of water depths. Accuracy and efficiency of the model is critically tested against analytical solutions and experimental data. Particular attention is paid to the model capability in resolving waves interacting with sheared currents and a wide range of wave conditions under different degree of dispersion by the employment of a fixed set of optimized parameters. In the following, the “Nonhydrostatic Model” section presents the governing equations and the boundary conditions. The “Numerical Methods” section describes the numerical methods, including the general discretization, the modified free-surface treatment for the toplayer pressure, and the overall algorithm of the implicit nonhydrostatic solver. Numerical experiments including linear progressive waves, wave-current interaction, and wave-structure interaction are then illustrated in the “Numerical Experiments” section. Finally, summary and conclusions are given in the “Conclusions” section.
w w w 2w 2w 2w w P +u −g+ 2 + 2 + 2 +v +w =− x y z x y z t z 共4兲 where u, v, and w = velocity components in the x, y, and z 共pointing upward with the original at the still water level兲 directions, respectively; P = normalized pressure, i.e., pressure divided by a constant reference density; g = gravitational acceleration constant; and = kinematic viscosity which is set to zero for an inviscid fluid. Boundary Conditions Different types of boundary conditions are supplied in solving the governing equations of the nonhydrostatic model. At the free surface, z = , the kinematic boundary condition is
+u +v = w兩z= x y t
In addition, the dynamic condition demands that at the free surface the normal stress is equal to the atmospheric pressure, i.e., P = Pa = 0, and the tangential stress is zero. At the impermeable bottom, z = −h, the kinematic boundary condition is u
Governing Equations The governing equations describing the free-surface wave motions are unsteady, incompressible, three-dimensional 共3D兲 Navier-Stokes equations
u v w + + =0 x y z
共1兲
u u u 2u 2u 2u P u +u +v +w =− + 2 + 2 + 2 x y z x y z t x
共2兲
v v v 2v 2v 2v v P +u +v +w =− + 2 + 2 + 2 x y z x y z t y
共3兲
h h + v + w兩z=−h = 0 x y
共6兲
At the inflow boundary, either analytical solutions or laboratory conditions are specified. Meanwhile, a ramp-function fr = tanh共t / 2T兲, where T is the wave period, is applied at the inflow boundary to avoid unwanted waves generated by impulse motions. To minimize wave reflection at the outflow boundary, two methods are used. First, at the immediate end of the computational domain, we add a sponge layer technique 共Yuan and Wu 2004a兲 that essentially contain the additional artificial terms
冋冉 冊 册
共7a兲
冋冉 冊 册
共7b兲
␥
x − xo LSPx
2
zb − z u=0 zb − z f
and ␥
Nonhydrostatic Model
共5兲
y − yo LSPy
2
zb − z v=0 zb − z f
to the right-hand side of the horizontal momentum Eqs. 共2兲 and 共3兲, respectively, where xo and y o = starting positions of the sponge layer; LSPx and LSPy = length of the sponger layer along the x and y directions; zb and z f denote the positions at the bottom and free surface, respectively; and ␥ = damping coefficient and is set to ⫺2.0 in all the examples in this paper. At the end of the sponge layer the Sommerfeld radiation boundary condition
⌽ ⌽ ⌽ + C cos + C sin =0 x y t
共8兲
is used, where = angle between the normal direction of the boundary and the direction of wave propagation; C = wave celerity, and ⌽ denotes u and v here. Finally, at solid lateral walls, the impermeable condition is again specified, i.e., the velocity normal to the wall is zero.
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Fig. 1. 共a兲 Computational grids 共solid frames兲 for the NHM; 共b兲 virtual grids and the location of the reference velocity 共dashed frames兲 n+1 n+1 n+1 n+1 n+1 ¯ i−1/2,j ¯ i+1/2,j i,j = Bfu + Cfu + Df¯vi,j−1/2 + Ef¯vi,j+1/2 + Ff 共14兲
Numerical Methods Discretization The computational domain is discretized by N1, N2, and N3 cells in the x, y, and z directions with the grid index i, j, and k, respectively. Nonuniform grid spacing, ⌬xi, ⌬y j, and ⌬zk along each direction can be set. To represent the moving free surface as well as the irregular bathymetries, an effective partial-cell method 共Pacanowski and Gnanadesikan 1998兲 is used. For example, the thickness of top-layer cells can be expressed as ⌬zi,j,N3 = i,j + ⌬zN3. Fig. 1共a兲 shows the Cartesian staggered grid arrangement of the x-z plane. The same staggered grid arrangement is also applied to the y-z plane. An implicit Crank-Nicolson scheme is applied to discretize the governing Eqs. 共1兲–共4兲, respectively, yielding n+1 ញ n+1 ញ¯un+1 + Dw ញ¯vn+1 + Ew ញ¯vn+1 ¯ i,j w = Bw¯ui−1/2,j + Cw i+1/2,j i,j−1/2 i,j+1/2
共9兲
ញ¯un+1 + Cx ញ¯un+1 + Dx ញ¯un+1 + Ex ញ ¯Pn+1 + Fx ញ ¯Pn+1 = Gx Bx i−1/2,j i+1/2,j i+3/2,j i,j i+1,j 共10兲
ញ¯vn+1 + Cy ញ¯vn+1 + Dy ញ¯vn+1 + Ey ញ ¯Pn+1 + Fy ញ ¯Pn+1 = Gy By i,j−1/2 i,j+1/2 i,j+3/2 i,j i,j+1 共11兲 n+1 n+1 n+1 n+1 n+1 ¯ i−1/2,j ¯ i+1/2,j Pn+1 + Czu + Dz¯vi,j−1/2 + Ez¯vi,j+1/2 + Fz i,j,k = Pi,j,k+1 + Bzu
共12兲 n+1 ¯ui⫾1/2,j ,
n+1 ¯vi,j⫾1/2 ,
where the single overbar of the velocity field and n+1 ¯ i,j w and Gx 共or Gy兲 represents a column vector and the double ញ , Cw ញ , . . ., etc.兲 denotes a twooverbar of capital symbols 共e.g., Bw dimensional 共2D兲 matrix. The superscript index n denotes time discretization. To ensure mass conservation, integrating the continuity Eq. 共1兲 over the water depth and using Kinematic Boundary Conditions 共5兲 and 共6兲 give the conservative form of freesurface equation
+ t x
冕
udz + y −h
The discretized form of Eq. 共13兲 is
冕
−h
vdz = 0
共13兲
where Bf, Cf, Df, and Ef = coefficient column vectors. One can refer to Yuan and Wu 共2004a兲 for detailed discretization procedure and matrices which are not shown here for brevity. Free-Surface Treatment In the discretized vertical momentum Eq. 共12兲, the pressure at lower layers is related the one at upper layers and its adjacent horizontal velocities. Therefore, a top-layer pressure is needed to explicitly express the pressure at lower layers. Difficulties to express the top-layer pressure under a staggered grid system have been recognized and documented 共Casulli 1999; Stelling and Zijlema 2003; Yuan and Wu 2004b兲. In this paper, we address the issue of using a single reference location in the BTEs 共Young and Wu 2009兲. Two locations, i.e., zRU for the horizontal reference velocity and zRW for the vertical reference velocity, are employed to obtain the analytical-based pressure profile at the top freesurface layer. Then, linear wave dispersion of the model’s property can be optimized by tuning the reference velocities. Detailed description of the free-surface treatment is presented below. Three steps are performed to determine the pressure distribution at the top layer. First, the vertical profiles of horizontal velocity components are obtained using the Taylor series expansion at the reference location, zRU = i,j − ␣⌬zi,j,N3, where ␣ is a free parameter. The horizontal velocity profiles under a virtual grid system 关see the dashed line in Fig. 1共b兲兴 are
冏冉 冏冉 冊 冏冉 冏冉 冊
u共z兲VG ⬇ uVG兩z=zRU + 共z − zRU兲 · +
1 · 共z − zRU兲2 · 2
u z2 2
v共z兲VG ⬇ vVG兩z=zRU + 共z − zRU兲 ·
+
1 · 共z − zRU兲2 · 2
v z2 2
冊冏 冏 冊冏 冏
u z
VG z=zRU
共15兲
VG z=zRU
v z
VG z=zRU
共16兲
VG z=zRU
A continuous condition between the reference velocity and the nonhydrostatic velocity is imposed, e.g.,
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uVG兩z=zRU = ui+1/2,j,kRU = f u共ui+1/2,j,N3,ui+1/2,j,N3−1, . . .兲
冏冉 冊 冏 冉 冊 冋冉 冊 u z
u = z z=zRU
VG
i+1/2,j,kRU
u z
= fu
, i+1/2,j,N3
冉 冊 u z
,... i+1/2,j,N3−1
A = uVG兩z=zRU − zRU ·
C = vVG兩z=zRU − zRU ·
册
E = wVG兩z=zRW − zRW ·
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and
冏冉 冊 冏 冉 冊 冋冉 冊 2u z2
2u z2
=
VG z=zRU
i+1/2,j,kRU
u z2 2
= fu
冉 冊
2u , 2 i+1/2,j,N3 z
,... i+1/2,j,N3−1
and F =
册
where f u = interpolation operator; the first and second vertical derivatives of the horizontal velocity at each layer are obtained by the central difference approximation. Similar procedure is also performed to the reference velocity v and its derivatives along the y direction in Eq. 共16兲. In contrast to the conventional Boussinesq approach, the horizontal velocity profiles are obtained without invoking the irrotational flow assumption in the present study. Second, we obtain the vertical velocity profile by substituting Eqs. 共15兲 and 共16兲 into the continuity Eq. 共1兲 and integrating the resulting equation from the reference location zRW = i,j − ⌬zi,j,N3 关see the dashed-dotted line in Fig. 1共b兲兴, where  is a second free parameter, to an arbitrary location z. The vertical velocity profile, therefore, is w共z兲VG ⬇ wVG兩z=zRU + 共z − zRW兲
zRU v + y z ⫻
冏冉
−
冊冏 VG
冏冉
u zRU u v + − x x z y
−
1 + 关z2 − 2z · zRU − zRW共zRW − 2zRU兲兴 2 z=zRU
2u 2v − x z y z
冊冏
共17兲
VG z=zRU
Similarly, continuous condition is imposed by interpolating the neighboring velocities from the nonhydrostatic model. The final step is to obtain the analytical form of pressure profile. Integrating the vertical momentum Eq. 共4兲 from an arbitrary z to the free surface and applying the Leibniz’s rule with the use of free-surface kinematic boundary condition 共5兲 and free-surface pressure condition give
P = g共 − z兲 + t
冕
z
wdz + x
冕
z
uwdz + y
冕
− z 2兲 ·
冉
vwdz − w 兩z . 2
z
冊
E E E 1 +A· +C· + E · F + · 共2 2 t x y
冊
F F E F E +A· +B· +C· +D· + F2 , t x x y y 共19兲
where
,
冏冉 冊 冏 v z
−
B=
VG z=zRU
,
D=
VG z=zRU
冏冉 冊 冏 u z
VG z=zRU
冏冉 冊 冏 v z
VG z=zRU
u zRU u v zRU v + + − x x z y y z
u zRU u v zRU v + + − x x z y y z
冊冏
冊冏
VG z=zRU
VG z=zRU
Eq. 共19兲, based upon the reference velocity concept, enables the NHM to accurately evaluate the top-layer pressure without any approximation, i.e., Pi,j,N3 = P共zⴱ兲VG, where zⴱ = i,j − 共⌬zi,j,N3 / 2兲 represents the level at the center of the top-layer moving cell. Discretizing Eq. 共19兲 and inserting Eqs. 共9兲 and 共14兲 yields the top-layer pressure equation in terms of horizontal velocities n+1 n+1 n+1 n+1 n+1 ¯ i−1/2,j ¯ i+1/2,j Pi,j,N = Btu + Ctu + Dt¯vi,j−1/2 + Et¯vi,j+1/2 + Ft 共20兲 3
where Bt, Ct, Dt, and Et = coefficient column vectors. Implicit Nonhydrostatic Solver Substituting the analytical form of pressure profile of Eq. 共20兲, based upon the reference velocity concept, for the top-layer and pressures at lower layers of Eq. 共12兲 into the discretized horizontal momentum Eqs. 共10兲 and 共11兲, gives a system matrix
冋
Auu Auv Avu Avv
册冋 册 冋 册 n
u v
n+1
=
Ru Rv
n
共21兲
where Anuu, Anuv, Anvu, and Anvv = block coefficient matrices with a dimension of 共N3N1N2兲 ⫻ 共N3N1N2兲; 关un+1 , vn+1兴T = unknown vector of horizontal velocities; and 关Rnu , Rnv兴T = known vector at n time step. Since no banded feature exists in Eq. 共21兲, a domain decomposition method 共Wu and Yuan 2007兲 with an iteration procedure is applied to decompose the matrix system of Eq. 共21兲 into a series of vertical 2D, i.e. Anu j · un+1 = Rn+ⴱ j uj
共22a兲
with a dimension of 共N3N1兲 ⫻ 1 in for the horizontal velocity un+1 j each x-z plane 共j = 1 , ¯ , N2兲, and = Rvn+ⴱ , Avni · vn+1 i i
Substituting the analytical form of velocity profiles in Eqs. 共15兲–共17兲 into Eq. 共18兲 yield the vertical distribution of pressure
冉
−
u z
冏冉
共18兲
P共z兲VG = 共 − z兲 · g +
冏冉
冏冉 冊 冏
共22b兲
with a dimension of 共N3N2兲 ⫻ 1 in for the horizontal velocity vn+1 i each y-z plane 共i = 1 , ¯ , N1兲. Note that Aun , a sub-block matrix of j Anuu, is a block tridiagonal matrix with a dimension of 共N3N1兲 ⫻ 共N3N1兲; Similarly, each Anv , a sub-block matrix of Anvv, is also i a triblock tridiagonal matrix with a dimension of 共N3N2兲 ⫻ 共N3N2兲. With the superscript n + ⴱ as the temporal time step, the n n n+ⴱ subvectors, Run+ⴱ and Rn+ⴱ vi , are the known values of Ru − Auvv j n n+ⴱ n and Rv − Avuu , respectively. The advantage of the implicit algorithm is that the resulting block tridiagonal matrix can be directly solved 共Keller 1974兲. Convergence of all vertical 2D problems would be the solution of horizontal velocity at the updated time step. After that, the vertical velocity component, freesurface elevation, and pressure are updated by back-substituting the solved horizontal velocity component.
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nonhydrostatic pressure are much stronger near the free surface. In this paper, considering the validity of the BTEs 共Young and Wu 2009兲, we further propose a MTDR method, i.e.
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冦 Fig. 2. Sketch of a vertical 2D numerical wave tank
Numerical Experiments In this section, to examine the efficiency and accuracy of the NHM based upon the modified free-surface treatment, four numerical experiments are conducted. First 2D linear progressive waves would be used to demonstrate the model’s capability in resolving linear wave dispersion from shallow to deep water. Afterwards, waves interacting with sheared currents would be used to examine the model that is free from irrotational flow assumption. The last two examples, periodic waves propagation over a 2D submerged bar and a 3D semicircular shoal, are used to show model’s ability in resolving wave transformation due to bathymetry effects under a wide range of nondimensional water depth conditions. Linear Progressive Waves We test the model using linear progressive waves with various dimensionless water depths Kh, ranging from shallow-water to extremely deep-water conditions. Properties of wave frequency dispersion and wave amplitude predicted by the model using different vertical layers are assessed. Fig. 2 depicts a 2D numerical wave tank with a length L = 10, where is wavelength. The water depth is h = 1 m with an initially flow field at still. At the inflow boundary, the analytical solution of horizontal velocity for the free-surface displacement 共x , t兲 = a · cos共Kx − t兲 is specified, i.e. u共x,z,t兲 = a ·
gK cosh关K共h + z兲兴 · · cos共Kx − t兲 cosh共Kh兲
共23兲
where a = wave amplitude; K = 2 / = wave number; and = angular frequency that is determined by the linear dispersion relation 2 = gK tanh共Kh兲. At the end of the tank, additional two wavelengths, i.e., LSP = 2, served as a sponge layer and a Sommerfeld radiation boundary condition 共RBC兲 are adopted to avoid wave reflection. To examine the effects of frequency dispersion, we vary the incident wave period from T = 6.487 s to T = 0.506 s, yielding a dimensionless water depth from Kh = 0.314 = 0.1 共shallow-water condition兲 to Kh = 15.70= 5 共extremely deep-water condition兲. For all the incident wave conditions, a small and constant wave steepness aK = 0.001 is used to satisfy a linear wave condition. In the model, the computational domain is horizontally discretized by a uniform spacing ⌬x = / 20. In the vertical z direction, two, three, and four layers are employed. To determine a set of suitable thickness for each vertical layer, Yuan and Wu 共2006兲 suggested a top-down resolving method. The concept is to maintain a finer resolution 共i.e., K⌬z = 0.5兲 from the top layer down to the one just above the coarse bottom layer since the effects of
冉
冊
Kh k⬎1 , N3 K⌬zc = Kh − 共N3 − 1兲K⌬z f , k = 1
K⌬z f = min 1.0,
冧
共24兲
The time step is determined by setting the Courant number Cr = 共c · ⌬t兲 / ⌬x = 0.5, where the wave phase speed c = / K = 冑g / K tanh共Kh兲. The total simulation time is 20 wave periods. For linear wave dispersion properties ranging from Kh = 0.314 to Kh = 15.70, two free parameters ␣ and , for the reference velocities u and w, respectively, are determined with the increments of 0.01 to minimize the phase error 共e.g., ⬍0.1%兲 in the model. It should be noticed that different parameter optimization, similar to that in Lynett and Liu 共2004b兲 and Young and Wu 共2009兲, can be obtained for certain situations 共e.g., the number of vertical layers or various water depths兲. Nevertheless, a fixed set of parameter optimization is highly desirable for practical purposes. In this paper, based upon the phase error minimization analysis the parameters ␣ = 0.5 and  = 1.05 are selected for all numerical testing cases. Fig. 3 shows the comparison of free-surface displacement time series at x = 3 between analytical solutions and model results for four dimensionless water depths. For Kh = 1.571= 0.5 共an intermediate-water condition兲, the model using two, three, or four vertical layers accurately predicts the wave propagation and wave amplitude, consistent with those predicted by the NHM using an integral method at the top layer 共Yuan and Wu 2006兲. Excellent results for shallower-water condition are also found but not shown here for brevity. For Kh = 3.14 共a deep-water condition兲, results predicted by three- and four-layer models well match the analytical solution with both errors in wave speed, c = 共cmodel − cexact兲 / cexact, and wave amplitude, a = 共Hmodel − Hexact兲 / Hexact, less than 0.1%. Nevertheless, the results calculated by the twolayer model underestimate the wave speed and amplitude, i.e., c = −3.2% and a = −3.5%. For the cases of Kh = 9.42 and 15.70 共extremely deep-water conditions兲, the wave becomes more dispersive and the vertical distribution of horizontal velocity components varies sharply at the upper portion of the water column. The two-layer model significantly underpredicts the wave speed and amplitude. The three-layer model dramatically improves the results, yielding slightly underestimated wave speed and amplitude. Excellent agreement is found between the four-layer model results and the analytical solutions. Overall unlike many other NSEbased models 共Mayer et al. 1998; Casulli 1999; Zhou and Stansby 1998; Li and Fleming 2001; Namin et al. 2001; Lin and Li 2002; Koçyigit et al. 2002; Chen 2003; Anthonio and Hall 2006兲 that require 10–20 vertical cells to satisfactorily calculate dispersive waves, the present model using a small vertical number of layers 共i.e., 2–4兲 is capable of simulating wave motions from shallowwater to extremely deep-water conditions. Fig. 4 quantifies the relative errors of wave speed and amplitude versus Kh. For a given tolerance error bound, i.e., 兩c兩 ⱕ 1% and 兩a兩 ⱕ 1%, a two-layer model is accurate up to Kh ⬃ 2.0. After that the performance of a two-layer model dramatically deteriorates. A three-layer model can accurately resolve linear wave dispersion and amplitude up to Kh ⬃ 6.28. Under extremely deep-water conditions, the errors of wave speed and amplitude in a three-layer model are roughly c ⬃ −2.0% and a
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Fig. 3. Comparison of the predicted free-surface elevation time series at x = 3 for 共a兲 Kh = 1.57; 共b兲 3.14; 共c兲 9.42; and 共d兲 15.70. Analytical solution 共solid lines兲, two-layer model 共dashed lines兲, three-layer model 共dash-dotted lines兲, and four-layer model 共circles兲.
⬃ −3.0%. For a four-layer model, the wave characteristics including wave speed and amplitude are precisely resolved up to Kh ⬃ 15.70 共c ⬍ 0.1% and a ⬍ 0.1%兲. Three main points are discussed here. First, in comparison with the results in the NHM of Yuan and Wu 共2006兲 based upon an integral method, the present NHM model with the embedded Boussinesq-type like equations dramatically improves accuracy in capturing linear wave dispersion, indicating the success of the proposed free-surface treatment in this paper. Second, while twolayer model of Young and Wu 共2009兲 can well resolve wave dispersion up to Kh ⬃ 10.0, the issue of parameter optimization for different water depths is recognized. The present NHM with the embedded BTEs using two reference velocities does not need to further tune the parameter. The MTDR method provides an effective method to determine the thickness of each vertical layer for a wider range of wave conditions. Third, increasing vertical layers effectively improves the accuracy of the model. In other words, both three-layer and four-layer models can be used to simulate large-scale wave propagation from the deep ocean toward the shoreline with a little higher computational cost, i.e., 1.5 and 2.25 times the CPU time of a two-layer model, respectively. Overall, the predicted results with error analysis in this example clearly demonstrate the efficiency and accuracy of the present model in resolving linear wave dispersion.
Wave-Current Interaction The second test case is to examine the model’s capability in describing shear 共rotational兲 currents interacting with coexistent waves in a vertical 2D plane 共see Fig. 5兲. We focus on a linear dispersive wave interacting with a uniform current or a linear shear current. The velocity profile of the current therefore can be expressed as Uc共z兲 = U fs + ⍀ · z, where U fs represents current velocity at the free surface, taken positive in the wave-propagating direction, and ⍀ is vorticity, yielding a linear shear current or giving a uniform current by setting ⍀ as zero. The analytical solutions of free-surface elevation and horizontal velocity for wave component 共Tsao 1959兲 can be described as 共x,t兲 = a ·
gK共 − U fsK兲 · cos共Kx − t兲 关gK − ⍀共 − U fsK兲兴
共25兲
gK cosh关K共h + z兲兴 · · cos共Kx − t兲 cosh共Kh兲
共26兲
and u共x,z,t兲 = a ·
where a = wave amplitude under a no-current condition; = apparent wave angular frequency; and K = apparent wave num-
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Fig. 4. Accuracy analysis: 共a兲 relative phase error c; 共b兲 relative amplitude error a under different Kh. Analytical solution of normalized wave speed 共solid lines兲, two-layer model 共dashed lines兲, threelayer model 共dash-dotted lines兲, and four-layer model 共dash-dotdotted lines兲.
ber determined by the linear Doppler-shifted dispersion relation 共Peregrine 1976兲, i.e. 共 − U fsK兲2 = 关gK − ⍀ · 共 − U fsK兲兴 · tanh共Kh兲
共27兲
In the model, the same numerical tank in the test case 1 except for a longer tank length L = 35 m is used. At the inflow boundary, we impose an incident wave with an amplitude a = 0.01 m and a period T = 1.5 s interacting with four different current conditions: 共1兲 a periodic wave without current 共U fs = 0 m / s and ⍀ = 0 s−1兲; 共2兲 a wave on an opposing uniform current 共U fs = −0.2 m / s and ⍀ = 0 s−1兲; 共3兲 a wave on a weak opposing sheared current 共U fs = −0.2 m / s and ⍀ = −0.75 s−1兲; and 共4兲 a wave on a strong opposing sheared current 共U fs = −0.2 m / s and ⍀ = −1.50 s−1兲. At the outflow boundary, a sponge layer with a length of 5 m in conjunction with a RBC is applied to avoid wave reflection while allowing the current to pass through. The computational domain is discretized by 800 cells in the x direction, giving ⌬x = 0.05 m
Fig. 5. Sketch of a vertical 2D numerical wave tank for wave-current interaction
共 / 55– / 80 for all four current conditions兲. In the z direction, three vertical layers with the MTDR arrangement are chosen to resolve the vertical flow structure since Kh = 1.57– 2.24. The same parameters, i.e., ␣ = 0.5 and  = 1.05, for the reference velocities are used. The total simulation time is 30T with a ⌬t = 0.005 s, yielding Cr= 0.18– 0.26. Fig. 6 shows the spatial profile of free-surface displacements for all the conditions. In comparison with the Condition 共1兲 of a wave interacting with no-current, the opposing uniform current in the Condition 共2兲 shortens the wavelength and amplifies the wave amplitude. Figs. 6共c and d兲 show that the negative vorticity in weakly and strongly sheared current conditions 共3兲 and 共4兲 counteracts the effects caused by the opposing current. Consequently the shortened wavelength is lengthened and the increased wave amplitude is decreased. Overall the model results 共open circles in Fig. 6兲 are in excellent agreement with analytical solutions, demonstrating the present model’s capability in simulating waves interacting with uniform or linear sheared currents. To illustrate the feature of the model in resolving vorticity, we also demonstrate the results based upon the free-surface treatment with an irrotational flow assumption which is physically incorrect for waves interacting with sheared currents. The filled circles in Fig. 6 clearly show that the model involving an irrotational flow assumption, i.e., u / z = w / x, in Eq. 共15兲 well predicts the wave amplitude and phase speed of the no shear Conditions 共1兲 and 共2兲 but yields slight/apparent phase shifts in weakly/strongly sheared currents in the Condition 共3兲/共4兲. In contrast, the NHM without the irrotational flow assumption 共open circles兲 is capable of resolving correct wave phase speed in sheared currents. Fig. 7 shows the vertical distribution of horizontal velocity under the wave crest on a strong opposing sheared current in the condition 共4兲. An obvious phase shift is predicted by the model with irrotational flow assumption that is usually employed in the BTEs 共Liu and Losada 2002; Kirby 2003兲. On the other hand, the three-layer model without imposing irrotational flow assumption can faithfully resolve the vertical flow structure. In comparison with the NHM of Yuan and Wu 共2006兲 that requires five vertical layers to simulate wave-current interactions, the present model using fewer layers can well resolve the interactions between waves and currents, indicating the advantage of the new freesurface treatment in this study. Wave Propagation over a Submerged Bar The third numerical experiment is to examine the model’s capability in resolving fundamental wave propagation, nonlinear higher harmonic generation due to shoaling, and the release of high-frequency free waves generated from waves past a submerged bar 共Beji and Battjes 1993; Luth et al. 1994; Dingemans 1994; Nadaoka et al. 1994; Ohyama et al. 1995兲. In this study, we use the experiments setup by Beji and Battjes 共1993兲. Fig. 8 shows the wave flume with a length of 30 m and a still water depth of 0.4 m that is reduced to 0.1 m at the top of the submerged bar. The upward and downward slopes of the bar are 1:20 and 1:10, respectively. Free-surface elevations were measured at seven stations. The model is applied to two incident wave conditions: Case 共I兲 is a longer wave with wave period T = 2.0 s and a wave height H = 2.0 cm, giving the dimensionless water depth Kh = 0.68 and wave steepness aK = 0.017; and Case 共II兲 is a shorter wave with T = 1.25 s and H = 2.5 cm, yielding Kh = 1.23 and aK = 0.038. At the outflow boundary, the beach that was used for absorbing waves is replaced by a combination a 5-m-long sponge layer and
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Fig. 6. Comparison of the predicted spatial free-surface displacement for linear waves interacting with 共a兲 no current; 共b兲 an opposing uniform current; 共c兲 a weak opposing sheared current; and 共d兲 a strong opposing sheared current. Analytical solution 共solid lines兲, model without the irrotational flow assumption 共open circles兲, and model with the irrotational flow assumption 共filled circles兲.
a RBC. The computational domain is discretized by 1,200 uniform cells in the x direction, giving ⌬x = 0.025 m. In the vertical direction, the top-layer cell is set higher than the top of the submerged obstacle and the bottom-cell size is determined by an effective partial-cell approach 共Yuan and Wu 2004a兲 for the irregular topography. For comparison, both two-layer and threelayer models are considered here. The cell sizes from the top layer to the bottom layer are 0.08 and 0.32 m for the two-layer model and 0.04, 0.04, and 0.32 m for the three-layer model. The reference velocities are kept the same as those in the previous examples. The time step is determined by the Courant number Cr = 0.25. Fig. 9 compares the free-surface elevation time series between model results and experimental data at six stations 共from Stations 2–7兲 for Case 共I兲. As the incident wave propagates onto the upward slope at Station 2, wave becomes steeper due to shoaling effects. At the top of the bar, i.e., from Stations 3–5, the wave steepens further and the higher bounded harmonics are created due to nonlinearity. After the wave passes the downward slopes of
the bar, the generated higher harmonics are released as free waves. The surface displacements 共dashed lines兲 predicted by the two-layer well agree with the experimental data except for the higher harmonics at Stations 6 and 7, which are consistent with the reliable dispersion range 共i.e., Kh ⬍ 2兲 of the two-layer model shown in the linear progressive wave example. In contrast, the three-layer model 共solid lines in Fig. 9兲 simulates correct wave shoaling and captures the release of these free higher harmonic waves, demonstrating the model’s capability in predicting nonlinear and dispersive 共e.g., Kh ⬃ 6.28兲 wave characteristics over uneven bottoms. Fig. 10 compares the spatial evolution of the first, second, third, and fourth harmonics by performing the fast Fourier transform over the time series of surface displacement at each grid point. Excellent comparison between the three-layer model results and experimental data further supports the accuracy of the model. As the incident wave propagates onto the upward slope, i.e., x = 6 – 12 m, the first harmonic remains the same but the amplitude of higher harmonic increases, indicating the effects of shoaling.
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Fig. 7. Comparison of the predicted horizontal velocity profiles for linear waves interacting with a strong opposing sheared current. Analytical solution 共solid lines兲, model without the irrotational flow assumption 共open circles兲, and model with the irrotational flow assumption 共filled circles兲.
Fig. 8. Sketch of the experimental setup and measurement stations by Beji and Battjes 共1993兲 112 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING © ASCE / MARCH/APRIL 2010
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Fig. 9. Comparison of the free-surface displacement at six measurement stations for Case 共I兲 with wave period T = 2.0 s and a wave height of H = 2.0 cm. The experimental data 共circles兲, two-layer model 共dashed lines兲, and three-layer model 共solid lines兲.
Fig. 10. Comparison of the harmonic amplitudes for Case 共I兲 with wave period T = 2.0 s and a wave height of H = 2.0 cm. The experimental data 共circles兲 and three-layer model 共solid lines兲. JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING © ASCE / MARCH/APRIL 2010 / 113
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Fig. 11. Same as Fig. 8 except for Case 共II兲 with wave period T = 1.25 s and a wave height of H = 2.5 cm
Once the wave rides on the top of the bar, i.e., x = 12– 14 m, significant amount of energy from the first harmonic transfers to the higher harmonics. Behind the submerged bar, the release of higher harmonic free waves takes place and the oscillations in third and fourth harmonics occurs 共Madsen and Sørensen 1992; Gobbi and Kirby 1999兲. For Case 共II兲, Fig. 11 shows the surface displacement time series. The two-layer and three-layer model results are in good agreement with the experimental data. While the incident wave is more dispersive, a smaller Ursell number Ur= a2h−3 = 0.822, in comparison with Ur= 2.133 in Case 共I兲, results in weaker development of higher harmonics. Fig. 12 shows similar shoaling, wave steepening, and decomposition along the spatial evolution of harmonics. Different from Case 共I兲, the primary harmonic maintains approximately the same and the gain of the second harmonic is apparent over the submerged bar. A relatively small amount of the third and fourth harmonics is generated and released as free waves, explaining the little difference of the surface displacements predicted by the two-layer and three-layer models. Overall, the results predicted by a three-layer NHM with the proposed free-surface treatment well match the experimental data. In contrast to other NSE-type models 共Mayer et al. 1998; Casulli 1999; Zhou and Stansby 1998; Li and Fleming 2001; Lin and Li 2002; Chen 2003兲 that usually require 10–20 vertical cells, the NHM using a relatively small number of vertical layers 共i.e., two to three layers兲 effectively predicts wave transformation over irregular bottoms. Furthermore, similar to those models based upon the higher-order BTEs 共Agnon et al. 1999; Gobbi and Kirby 1999; Madsen et al. 2003兲 or the multilayer BTEs 共Lynett and Liu 2004a,b; Hsiao et al. 2005兲, the present NHM has the capabilities
in simulating a wide range of dispersive and nonlinear waves, overcoming the limitations of the traditional Boussinesq equations 共Peregrine 1967兲. Wave Propagation over a Semicircular Shoal For the last example, the model is applied to 3D nonlinear refraction-diffraction waves. Whalin 共1971兲 carried out a series of experiments on wave refraction over a semicircular sloping topography 共see Fig. 13兲. The still water depth at the incident end is hi = 0.4572 m and the bottom topography can be approximated by h共x,y兲
冦
0.4572 0 ⱕ x ⱕ 10.67 − G = 0.4572 + 共10.67 − G − x兲/25 10.67 − G ⱕ x ⱕ 18.29 − G 0.1542 x ⱖ 18.29 − G
冧
共28兲
where G共y兲 = 关y共6.096− y兲兴1/2 , 0 ⱕ y ⱕ 6.096. Due to refraction effect, the focusing of water waves would lead to a focal region, in which energy was transferred to the higher harmonics. Three different incident conditions are considered here: 共1兲 wave amplitude aI = 0.97 cm and wave period TI = 1 s, giving aIKI = 0.041 and KIhi = 1.922; 共2兲 wave amplitude aII = 0.75 cm and wave period TII = 2 s, yielding aIIKII = 0.012 and KIIhi = 0.735; and 共3兲 wave amplitude aIII = 0.68 cm and TIII = 3 s, resulting in aIIIKIII = 0.007 and KIIIhi = 0.468. The corresponding Ursell numbers Ur = a2h−3 for these three cases are therefore 0.227, 1.198, and 2.676, respectively.
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Fig. 12. Same as Fig. 9 except for Case 共II兲 with wave period T = 1.25 s and a wave height of H = 2.5 cm
In the model, the length and the width of the computational domain are 36.576 and 6.096 m, respectively. The inflow horizontal velocity is specified according to the experimental conditions above. The combination of a 6.576-m-long sponge layer and a RBC is used at the outflow boundary. For the lateral walls, fully reflective and free-slip boundary conditions are employed. The computational domain is discretized by a set of 360⫻ 60 horizontal grids, giving ⌬x = ⌬y = 0.1016 m. In the z direction, three vertical layers are chosen to cover the full range of wave conditions. The cell sizes from top layer to bottom layer are 0.06, 0.06, and 0.3372 m. All the reference velocities are maintained the same as
Fig. 13. Bottom configuration for periodic waves over a semicircular shoal, according to the experimental setup of Whalin 共1971兲
in the previous examples. The time step 0.02 s is chosen for a total simulation time 40 s to achieve a stationary wave condition. Fig. 14 compares the predicted harmonics with the experimental data. Especially for TI = 1 s, the dispersive degree is out of the range of the conventional Boussinesq equations 共Peregrine 1967; Rygg 1988兲. As shown in Fig. 14共a兲, the three-layer model faithfully predicts the evolution of the first and second harmonics along the centerline, in spite of the data scattering. After shoaling, the primary waves become steeper and then focus roughly at x ⬃ 22.45 m. The apparent oscillations in the second harmonic explain that the free second harmonic is released on the top of the shelf due to the abrupt change of the topography. Similar phenomena were also found in those results based upon the BTEs 共Madsen and Sørensen 1992; Chen and Liu 1995; Beji and Nadaoka 1996; Zhao et al. 2004; Bingham et al. 2009兲, the nonlinear mild slope equations 共Tang and Ouellet 1997兲, and the nonlinear Laplace equation 共Li and Fleming 1997兲. Note that the third harmonic is plotted here for the reference comparison while no available experimental data were reported. Overall, the good agreement indicates the three-layer nonhydrostatic model’s capability in resolving relatively deep-water wave transformation over 3D irregular topographies, consistent with the results in all previous examples. For TII = 2 s and TIII = 3 s, these incident waves have weaker dispersive degrees but a higher nonlinearity 共i.e., Ur= 1.198 and Ur= 2.676, respectively兲. Fig. 14共b兲 indicates that the predicted
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Fig. 14. Comparison of the harmonic amplitudes for 共a兲 TI = 1 s; 共b兲 TII = 2 s; and 共c兲 TIII = 3 s. Symbols for the experimental data and lines for the three-layer model results: the first harmonic 共filled circles, solid lines兲, the second harmonic 共empty circles, dashed lines兲, and the third harmonic 共dash-dot-dotted lines兲.
harmonics for the Case 共II兲 are in excellent agreement with the experimental data. As the waves propagate over the semicircular shoal, nonlinear effects from wave-bottom and wave-wave interactions become more evident, leading to the rapid growth of the second and third harmonics at the expense of the primary wave component 共Janssen et al. 2006兲. For the Case 共III兲, reasonably good comparison between the model results and the laboratory data are shown in Fig. 14共c兲. Interestingly, the second-harmonic amplitude behind the focusing region, i.e., x ⬎ 20 m, becomes even larger than that of the fundamental harmonic. Fig. 15 further compares the fully developed 3D free-surface elevation for these three conditions. For TI = 1 s, the incident waves are linear at the deeper portion of the tank. As the linear wave propagates onto the semicircular shoal, the wave becomes steeper due to shoaling. Meanwhile, refraction takes place, yielding wave focusing at x ⬃ 22.45 m along the centerline. After the focusing, wave energy gradually spreads out due to diffraction. For TII = 2 s, the relatively longer wave with a weaker dispersive degree focuses earlier at x ⬃ 18.63 m, suggesting the importance of higher harmonics. Diffraction effect makes wave energy spread more obviously. For TIII = 3 s, a nearly shallow-water condition, the combined refraction-diffraction over the semicircular shoal becomes more complicated due to significant nonlinearity. Wave focusing and diffraction takes place within one wavelength, different from the slowly varied wave fields in the previous cases. Overall, the present NHM using three vertical layers well resolve
Fig. 15. Perspective view of fully developed instantaneous wave field for 共a兲 TI = 1 s; 共b兲 TII = 2 s; and 共c兲 TIII = 3 s
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dispersive wave transformation, e.g., shoaling, refraction, diffraction, and focusing processes, over 3D uneven bottom, demonstrating the accuracy and efficiency of the model.
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Conclusions In this paper, an efficient NHM for simulating surface waves from deep to shallow water is developed. The basic concept is to obtain an accurate expression of nonhydrostatic pressure distribution at the top layer by matching the reference velocities with the ones under the NHM grid system. Different from a single location for the reference velocity in NHM-BTE 共Young and Wu 2009兲, two locations of the references velocities under a staggered grid system are employed to optimize the linear wave dispersion property. The present NHM with the embedded Boussinesq-type like equations does not need to further tune the parameter for the reference locations. Furthermore the MTDR method, different from a TLC technique 共Young and Wu 2009兲, provides a robust way to determine the thickness of each vertical layer for a wider range of wave conditions Accuracy of the modified model is examined through four free-surface wave examples. Results of linear progressive wave example show that the two-, three-, and four-layer models can well resolve phase speed up to Kh = 2.0, 6.28, and 15.70 with a given error tolerance of c ⱕ 0.01, respectively. Particular attention is then paid to simulate wave-current interaction. Free from the irrotational flow assumption, the three-layer model can correctly resolve the free-surface displacement and the velocity profile of linear dispersive waves interacting with linear shear currents, which cannot be easily obtained by the BTEs equations. Finally, the last two examples, i.e., wave propagation over 2D and 3D irregular topographies, show the three-layer model’s capability in predicting nonlinear dispersive wave transformation, including shoaling, refraction, diffraction, and focusing processes. Overall, with the proposed free-surface treatment, this new type of nonhydrostatic models using a few vertical layers 共e.g., 2–4兲 is able to effectively and accurately resolve large-scale 共e.g., 10 ⫻ 10 km2兲 wave propagation from deep water toward the shore and waves interacting with arbitrary sheared currents 共Yao and Wu 2005兲. Progress on this application will be reported later.
Acknowledgments The research is partly supported by Grant No. NSF-OCE0628560, “The carbon balance of Lake Superior: Modeling lake processes and understanding impacts of regional carbon cycle and the Wisconsin Coastal Management Program,” and Grant No. AD089091-009.23, “Development of a real-time wave climate observation system in the Apostle Islands.” We also thank National Science Council of the Republic of China, Taiwan under the Grant No. 096-2917-I-002-036 for the second writer and the U.S. Office of Naval Research under the Award No. N00014-071-0955 for the third writer.
References Agnon, Y., Madsen, P. A., and Schäffer, H. 共1999兲. “A new approach to high order Boussinesq models.” J. Fluid Mech., 399, 319–333. Anthonio, S. L., and Hall, K. R. 共2006兲. “High-order compact numerical schemes for non-hydrostatic free surface flows.” Int. J. Numer. Meth-
ods Fluids, 52, 1315–1337. Badiei, P., Namin, M., and Ahmadi, A. 共2008兲. “A three-dimensional non-hydrostatic vertical boundary fitted model for free-surface flows.” Int. J. Numer. Methods Fluids, 56共6兲, 607–627. Beji, S., and Battjes, J. A. 共1993兲. “Experimental investigation of wave propagation over a bar.” Coastal Eng., 19, 151–162. Beji, S., and Nadaoka, K. 共1996兲. “A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth.” Ocean Eng., 23, 691–704. Bingham, H. B., Madsen, P. A., and Fuhrman, D. R. 共2009兲. “Velocity potential formulations of highly accurate Boussinesq-type models.” Coastal Eng., 56, 467–478. Bradford, S. F. 共2005兲. “Godunov-based model for nonhydrostatic wave dynamics.” J. Waterway, Port, Coastal, Ocean Eng., 131共5兲, 226– 238. Casulli, V. 共1999兲. “A semi-implicit finite difference method for nonhydrostatic, free-surface flows.” Int. J. Numer. Methods Fluids, 30, 425–440. Cea, L., Stelling, G., and Zijlema, M. 共2009兲. “Non-hydrostatic 3D free surface layer-structured finite volume model for short wave propagation.” Int. J. Numer. Methods Fluids, 61共4兲, 382–410. Chen, Q., Kirby, J. T., Dalrymple, R. A., Kennedy, A. B., and Chawla, A. 共2000兲. “Boussinesq modeling of wave transformation, breaking, and runup, II: 2D.” J. Waterway, Port, Coastal, Ocean Eng., 126共1兲, 48– 56. Chen, Q., Madsen, P. A., Schaffer, H. A., and Basco, D. R. 共1998兲. “Wave-current interaction based on an enhanced Boussinesq approach.” Coastal Eng., 33, 11–39. Chen, X. 共2003兲. “A fully hydrodynamic model for three-dimensional, free-surface flows.” Int. J. Numer. Methods Fluids, 42共9兲, 929–952. Chen, Y., and Liu, P. L.-F. 共1995兲. “Modified Boussinesq equations and associated parabolic models for water wave propagation.” J. Fluid Mech., 288, 351–381. Choi, D. Y., and Wu, C. H. 共2006兲. “A new efficient 3D non-hydrostatic free-surface flow model for simulating water wave motions.” Ocean Eng., 33共5–6兲, 587–609. Choi, W. 共2003兲. “Strongly nonlinear long gravity waves in uniform shear flows.” Phys. Rev. E, 68, 026305. Dingemans, M. W. 共1994兲. “Comparison of computations with Boussinesq-like models and laboratory easurements.” Delft Hydraulics Rep. No. H1684.12, Delft, The Netherlands. Gobbi, M. F., and Kirby, J. T. 共1999兲. “Wave evolution over submerged sills: tests of a high-order Boussinesq model.” Coastal Eng., 37, 57– 96. Gobbi, M. F., Kirby, J. T., and Wei, G. 共2000兲. “A fully nonlinear Boussinesq model for surface waves. II. Extension to O关共kh兲4兴.” J. Fluid Mech., 405, 181–210. Green, A. E., and Naghdi, P. M. 共1976兲. “Derivation of equations for wave-propagation in water of variable depth.” J. Fluid Mech., 78, 237–246. Hsiao, S. C., Lynett, P., Hwung, H. H., and Liu, P. L.-F. 共2005兲. “Numerical simulations of nonlinear short waves using a multilayer model.” J. Eng. Mech., 131共3兲, 231–243. Janssen, T. T., Herbers, T. H. C., and Battjes, J. A. 共2006兲. “Generalized evolution equations for nonlinear surface gravity waves over twodimensional topography.” J. Fluid Mech., 552, 393–418. Keller, H. B. 共1974兲. “Accurate difference methods for nonlinear twopoint boundary value problems.” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 11共2兲, 305–320. Kennedy, A. B., Chen, Q., Kirby, J. T., and Dalrymple, R. A. 共2000兲. “Boussinesq modeling of wave transformation, breaking, end runup. I: 1D.” J. Waterway, Port, Coastal, Ocean Eng., 126共1兲, 39–47. Kirby, J. T. 共2003兲. “Boussinesq models and applications to nearshore wave propagation, surf zone processes and wave-induced currents.” Advances in Coastal Modeling, 67, 1–41. Koçyigit, M. B., Falconer, R. A., and Lin, B. 共2002兲. “Three-dimensional numerical modeling of free surface flows with non-hydrostatic pressure.” Int. J. Numer. Methods Fluids, 40共9兲, 1145–1162. Li, B. 共2008兲. “A 3-D model based on Navier-Stokes equations for regu-
JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINEERING © ASCE / MARCH/APRIL 2010 / 117
J. Waterway, Port, Coastal, Ocean Eng. 2010.136:104-118.
Downloaded from ascelibrary.org by SOUTHERN CALIFORNIA UNIVERSITY on 05/28/14. Copyright ASCE. For personal use only; all rights reserved.
lar and irregular water wave propagation.” Ocean Eng., 35共17–18兲, 1842–1853. Li, B., and Fleming, C. A. 共1997兲. “A three dimensional multigrid model for fully nonlinear water waves.” Coastal Eng., 30, 235–258. Li, B., and Fleming, C. A. 共2001兲. “Three-dimensional model of NavierStokes equations for water waves.” J. Waterway, Port, Coastal, Ocean Eng., 127, 16–25. Lin, P., and Li, C. W. 共2002兲. “A -coordinate three-dimensional numerical model for surface wave propagation.” Int. J. Numer. Methods Fluids, 38, 1045–1068. Liu, P. L.-F., and Losada, I. J. 共2002兲. “Wave propagation modeling in coastal engineering.” J. Hydraul. Res., 40共3兲, 229–240. Luth, H. R., Klopman, G., and Kitou, N. 共1994兲. “Project 13G: Kinematics of waves breaking partially on an offshore bar; LDV measurements for waves with and without a net onshore current.” Delft Hydraulics Rep. No. H1573, Delft, The Netherland. Lynett, P., and Liu, P. L.-F. 共2004a兲. “A two-layer approach to wave modeling.” Proc. R. Soc. London, Ser. A, 460共2049兲, 2637–2669. Lynett, P., and Liu, P. L.-F. 共2004b兲. “Linear analysis of the multi-layer model.” Coastal Eng., 51, 439–454. Madsen, P. A., Bingham, H. B., and Liu, H. 共2002兲. “A new Boussinesq method for fully nonlinear waves from shallow to deep water.” J. Fluid Mech., 462, 1–30. Madsen, P. A., Bingham, H. B., and Schaffer, H. A. 共2003兲. “Boussinesqtype formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis.” Proc. R. Soc. London, Ser. A, 459, 1075–1104. Madsen, P. A., and Sørensen, O. R. 共1992兲. “A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry.” Coastal Eng., 18, 183–204. Marshall, J., Hill, C., Perelman, L., and Adcroft, A. 共1997兲. “Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling.” J. Geophys. Res., 102, 5733–5752. Mayer, S., Garapon, A., and Sørensen, L. S. 共1998兲. “A fractional step method for unsteady free-surface flow with applications to non-linear wave dynamics.” Int. J. Numer. Methods Fluids, 28, 293–315. Mei, C. C., and Liu, P. L.-F. 共1993兲. “Surface waves and coastal dynamics.” Annu. Rev. Fluid Mech., 25, 215–240. Nadaoka, K., Beji, S., and Nakakawa, Y. 共1994兲. “A fully-dispersive nonlinear wave model and its numerical solutions.” Proc., 24th Int. Conf. on Coastal Engineering ASCE, ASCE, Reston, Va., 427–441. Namin, M., Lin, B., and Falconer, R. 共2001兲. “An implicit numerical algorithm for solving non-hydrostatic free-surface flow problems.” Int. J. Numer. Methods Fluids, 35, 341–356. Nwogu, O. G. 共1993兲. “Alternative form of Boussinesq equations for nearshore wave propagation.” J. Waterway, Port, Coastal, Ocean Eng., 119, 618–638. Nwogu, O. G. 共2009兲. “Interaction of finite-amplitude waves with vertically sheared current fields.” J. Fluid Mech., 627, 179–213. Ohyama, T., Kiota, W., and Tada, A. 共1995兲. “Applicability of numerical models to nonlinear dispersive waves.” Coastal Eng., 24, 297–313. Percival, J. R., Cotter, C. J., and Holm, D. D. 共2008兲. “A Euler-Poincare framework for the multilayer Green-Nagdhi equations.” J. Phys. A: Math. Theor., 41, 344018. Peregrine, D. H. 共1967兲. “Long waves on a beach.” J. Fluid Mech., 27, 815–827. Peregrine, D. H. 共1976兲. “Interaction of water waves and currents.” Adv.
Appl. Mech., 16, 9–117. Rygg, O. B. 共1988兲. “Nonlinear refraction-diffraction of surface waves in intermediate and shallow water.” Coastal Eng., 12, 191–211. Shen, C. Y. 共2001兲. “Constituent Boussinesq equations for waves and currents.” J. Phys. Oceanogr., 31, 850–859. Stelling, G., and Zijlema, M. 共2003兲. “An accurate and efficient finitedifference algorithm for non-hydrostatic free-surface flow with application to wave propagation.” Int. J. Numer. Methods Fluids, 43共1兲, 1–23. Tang, Y., and Ouellet, Y. 共1997兲. “A new kind of nonlinear mild-slope equation for combined refraction-diffraction of multifrequency waves.” Coastal Eng., 31, 3–36. Tsao, S. 共1959兲. “Behavior of surface waves on linearly varying flow.” Tr. Mosk. Fiz-Tech. Inst. Issled. Mekh. Prikl. Mat., 3, 66–84. Veeramony, J., and Svendsen, I. A. 共2000兲. “The flow in surf-zone waves.” Coastal Eng., 39共2–4兲, 93–122. Wei, G., Kirby, J. T., Grilli, S. T., and Subramanya, R. 共1995兲. “A fully nonlinear Boussinesq model for surface waves. I. Highly nonlinear unsteady waves.” J. Fluid Mech., 294, 71–92. Whalin, R. W. 共1971兲. “The limit of applicability of linear wave refraction theory in convergence zone.” Research Rep. No. H-71-3, U.S. Army Corps of Engineers, Waterways Experiment Station, Vicksburg, Miss. Wu, C. H., and Yuan, H. 共2007兲. “Efficient non-hydrostatic modelling of surface waves interacting with structures.” Appl. Math. Model., 31, 687–699. Wu, T. Y. 共2001兲. “A unified theory for modeling water waves.” Adv. Appl. Mech., 37, 1–88. Yao, A., and Wu, C. H. 共2005兲. “Incipient breaking of unsteady waves on sheared currents.” Phys. Fluids, 17共8兲, 082104. Young, C. C., and Wu, C. H. 共2009兲. “An efficient and accurate nonhydrostatic model with embedded Boussinesq-type like equations for surface wave modeling.” Int. J. Numer. Methods Fluids, 60共1兲, 27–53. Young, C. C., Wu, C. H., Kuo, J. T., and Liu, W. C. 共2007兲. “A higherorder -coordinate non-hydrostatic model for nonlinear surface waves.” Ocean Eng., 34共10兲, 1357–1370. Young, C. C., Wu, C. H., Liu, W. C., and Kuo, J. T. 共2009兲. “A higherorder non-hydrostatic sigma model for simulating non-linear refraction-diffraction of water waves.” Coastal Eng., 56共9兲, 919–930. Yuan, H., and Wu, C. H. 共2004a兲. “An implicit 3D fully non-hydrostatic model for free-surface flows.” Int. J. Numer. Methods Fluids, 46共7兲, 709–733. Yuan, H., and Wu, C. H. 共2006兲. “Fully nonhydrostatic modeling of surface waves.” J. Eng. Mech., 132共4兲, 447–456. Yuan, H. L., and Wu, C. H. 共2004b兲. “A two-dimensional vertical nonhydrostatic sigma model with an implicit method for free-surface flows.” Int. J. Numer. Methods Fluids, 44共8兲, 811–835. Zhao, M., Teng, B., and Cheng, L. 共2004兲. “A new form of generalized Boussinesq equations for varying water depth.” Ocean Eng., 31, 2047–2072. Zhou, J. G., and Stansby, P. K. 共1998兲. “An arbitrary Lagrangian-Eulerian 共ALES兲 model with non-hydrostatic pressure for shallow water flows.” Comput. Methods Appl. Mech. Eng., 178, 199–214. Zijlema, M., and Stelling, G. S. 共2008兲. “Efficient computation of surf zone waves using the nonlinear shallow water equations with nonhydrostatic pressure.” Coastal Eng., 55, 780–790.
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