Coastal Engineering 65 (2012) 1–10
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Coastal Engineering journal homepage: www.elsevier.com/locate/coastaleng
Improving the efficiency and accuracy of a nonhydrostatic surf zone model Scott F. Bradford Research Scientist, Image Science and Applications Branch, Code 7261, Naval Research Laboratory, Washington, DC 20375, United States
a r t i c l e
i n f o
Article history: Received 24 June 2011 Received in revised form 7 February 2012 Accepted 8 February 2012 Available online 22 March 2012 Keywords: Navier–Stokes Surf zone Wave propagation Breaking wave
a b s t r a c t A previously developed model for simulating breaking surf zone waves is improved to yield more accurate and computationally efficient predictions. The model employs a sigma coordinate transformation in the vertical direction and solves the Reynolds averaged Navier–Stokes equations in a fractional step manner with the pressure split into hydrostatic and nonhydrostatic components. The hydrostatic equations are first solved with an approximate Riemann solver and the velocity field is then corrected to be divergence free by including the nonhydrostatic pressure. The previous model required a cumbersome modification to accurately predict surf zone wave heights. In this paper, a simpler alteration is presented that is shown to also yield accurate surf zone wave height predictions. In addition, other opportunities for improving model accuracy, robustness, and computational efficiency are also investigated including the discretization of advection terms and the selective neglect of potentially insignificant lateral viscous terms in the governing equations. Published by Elsevier B.V.
1. Introduction Many numerical models based on the σ-coordinate transformed Navier–Stokes equations have been shown to realistically simulate nonlinear wave transformation (Bradford, 2005; Li and Fleming, 2001; Lin and Li, 2002; Mahadevan et al., 1996). These models typically require far fewer cells in the vertical direction than the Marker-And-Cell (MAC) and Volume-Of-Fluid (VOF) based models to accurately resolve the free surface and do not require tedious free surface reconstruction to apply boundary conditions. These factors yield a more computationally efficient model that makes large scale coastal simulations more feasible. But σ-coordinate based models lack the ability to capture relatively small scale flow features such as wave overturning and splashing. More recently, σ-coordinate based models have been successfully applied to simulate breaking surf zone waves. Zijlema and Stelling (2008) proposed a model that uses a bottom and free surface conforming grid similar to the σ coordinate system. Bradford (2011) adapted a previously developed σ-coordinate model (Bradford, 2005) to successfully simulate surf zone waves as well. However, this model required a modification to prevent the overprediction of wave heights within the surf zone. While the modification worked well, it required the delineation of the surf zone from the rest of the domain. In this paper, an alternative modification is proposed that eliminates this requirement. In addition, other opportunities for improving model accuracy, robustness, and computational efficiency are also investigated. For example, the proposed model uses linear variable reconstruction at
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cell faces to obtain second order spatial accuracy of advection terms. Lateral gradients are needed for the flow depth, velocities, and turbulence quantities, while vertical gradients are also needed for the velocities and turbulence variables. Slope limiting of the computed gradients is used to prevent the development of spurious oscillations, which is computationally expensive. The choice of limiter as well as reverting to first order accuracy in various directions and for different flow variables is investigated. Also, the strategic neglect of potentially insignificant lateral viscous terms in the governing equations is also explored. Finally, the number of vertical cell layers is varied to determine the resulting effects on model accuracy. 2. Hydrostatic equations The model is fully described by Bradford (2005, 2011) and therefore only a brief outline is presented here. The model first solves the incompressible, Reynolds averaged, Navier–Stokes equations for hydrostatic flow. The governing equations are transformed vertically from z to σ space as, σ¼
z−h D
ð1Þ
where h is the free surface elevation and D is the total water depth. Furthermore, the equations are transformed from x and y space via a curvilinear transformation to ξ and η space such that the grid size in each of the ξ and η directions is one. In this coordinate system the momentum equations are ∂U ∂ ∂ ∂ ∂U þ ðF−Fv Þ þ ðG−Gv Þ þ H−Hv −νlσσ ¼S ∂t ∂ξ ∂η ∂σ ∂σ
ð2Þ
2
S.F. Bradford / Coastal Engineering 65 (2012) 1–10
where U = (Du Dv Dw) T and u, v, and w are the velocities in the x, y, and z directions, respectively. ν is the eddy viscosity, which is computed with the Mellor–Yamada 2.5 level turbulence model (Galperin et al., 1988; Mellor and Yamada, 1982). This is a two equation model that predicts the turbulence kinetic energy (k) and a turbulence length scale (l). In addition, 2
2
lσσ ¼ σ x þ σ y þ
1 D2
ð3Þ
where σx and σy are the vertical grid transformation metrics. The fluxes are defined as, 0
1 0 1 1 2 1 2 0 1 Wu B DUu þ 2 gD ξx C B DVu þ 2 gD ηx C B C B C F¼B 1 2 C G¼B 1 2 C H ¼ @ Wv A @ DUv þ gD ξy A @ DVv þ gD ηy A Ww 2 2 DUw DVw
The finite volume approach is used to discretize the governing equations. The domain is divided into hexagonal computational cells indexed with j, k, l and all dependent variables are defined at cell centers. ξ is in the direction of contiguous j indices, while η and σ are in the directions of k and l indices, respectively. Advection and hydrostatic pressure terms are integrated in time using an explicit predictor-corrector method to obtain second order temporal accuracy. The primitive form of the inviscid, hydrostatic flow equations is used to compute the predictor values as nþ1=2
n
¼ Dj;k −
Dj;k
in Δt h U Δξ D þ V Δη D þ D ξx Δξ U þ ξy Δη U þ ηx Δξ V þ ηy Δη V 2 j;k
ð4Þ
where g is gravity, ξx, ξy, ηx, and ηy are the lateral grid transformation metrics, and the computational velocities are U ¼ uξx þ vξy V ¼ uηx þ vηy W ¼ D σ t þ uσ x þ vσ y þ w ¼ Dσ t þ W
3. Numerical solution
ð5Þ
with σt representing the temporal vertical grid transformation metric. The lateral viscous terms are 0
0 1 1 ∂u ∂u ∂u ∂u ∂u ∂u B νD ∂ξ lξξ þ ∂η lξη þ ∂σ lξσ C B νD ∂ξ lξη þ ∂η lηη þ ∂σ lησ C B B C C B C B C ∂v ∂v ∂v ∂v ∂v ∂v B C B C Fv ¼ B νD lξξ þ lξη þ lξσ C Gv ¼ B νD lξη þ lηη þ lησ C B C B C ∂ξ ∂η ∂σ ∂ξ ∂η ∂σ C C B B @ A @ A ∂w ∂w ∂w ∂w ∂w ∂w lξξ þ lξη þ lξσ lξη þ lηη þ lησ νD νD ∂ξ ∂η ∂σ ∂η ∂σ 0 1 ∂ξ ∂u ∂u B νD ∂ξ lξσ þ ∂η lησ C B C B C ∂v ∂v B C Hv ¼ B νD lξσ þ lησ C B C ∂ξ ∂η C B @ A ∂w ∂w νD l þ l ∂ξ ξσ ∂η ησ
ð6Þ
ð11Þ
nþ1=2
¼ uj;k;l −
n
nþ1=2
¼ vj;k;l −
uj;k;l
n
vj;k;l
nþ1=2
wj;k;l
n
in Δt h UΔξ u þ VΔη u þ WΔσ u þ g ξx Δξ h þ ηx Δη h 2 j;k;l ð12Þ in Δt h UΔξ v þ VΔη v þ WΔσ v þ g ξy Δξ h þ ηy Δη h 2 j;k;l ð13Þ
¼ wj;k;l −
in Δt h UΔξ w þ VΔη w þ WΔσ w 2 j;k;l
where n is the current time level and Δt is the time increment. Computation of the predictor values requires several cell average gradients of h, D, U , V , u, v, w, k, and l. These are slope limited to prevent spurious numerical oscillations from developing near flow discontinuities. Bradford (2005, 2011) used the Double Minmod limiter, which is defined for the ξ gradient of an arbitrary variable ϕ as
Δξ ϕ ¼
8
0 sign δj−1=2 min jδj−1=2 þ δjþ1=2 j; 2jδj−1=2 j; 2jδjþ1=2 j 2 : 0 otherwise
where ξ2x
ð15Þ ξ2y
η2x
lξξ ¼ þ lξη ¼ ξx ηx þ ξy ηy lηη ¼ þ lξσ ¼ ξx σ x þ ξy σ y lησ ¼ ηx σ x þ ηy σ y
η2y
ð7Þ
where δj − 1/2 = ϕj, k, l − ϕj − 1, k, l and δj + 1/2 = ϕj + 1, k, l − ϕj, k, l. The most dissipative choice is the Minmod limiter, which is defined as (
The source term for constant water density is 1 0 ∂κ ∂κ −g þ ξ η x B ∂η x C B C ∂ξ C S¼B ∂κ ∂κ B −g C þ ξ η @ y y A ∂ξ ∂η 0
Δξ ϕ ¼
∂ DV ∂D ∂ DU þ þ ¼0 ∂t ∂ξ ∂η
ð9Þ
where the depth averaged velocities are U ¼ ∫−1 Udσ
0
V ¼ ∫−1 Vdσ
sign δj−1=2 min jδj−1=2 j; jδjþ1=2 j
δj−1=2 δjþ1=2 > 0
0
otherwise
ð16Þ
ð8Þ
where κ = zb(h − zb/2) and zb denotes the bottom elevation, which is defined at the bottom cell corner nodes. This particular form of S balances the hydrostatic thrust terms in F and G for still water and will therefore not initial movement over a sloping bed. The depth integrated incompressibility constraint is used to compute the flow depth and is defined as
0
ð14Þ
ð10Þ
Setting the gradients to zero yields first order accuracy. With Δh estimated, the depth gradients are computed as ΔD = Δh − Δzb where Δzb is computed from the bottom elevation without slope limiting. The gradients are also used for variable reconstruction at the cell faces to obtain second-order spatial accuracy (Van Leer, 1979). The reconstructed data are used to compute F and G using the approximate Riemann solver developed by Roe (1981) and presented by Bradford (2005). H is computed from the reconstructed data using the upwind method. Lateral diffusion is integrated with the explicit Euler method, while vertical diffusion is integrated with the implicit Euler method to enhance model stability. Bradford (2004) showed through a Von Neumann analysis that this approach is more accurate, and computationally efficient than using the Crank–Nicholson method because of the fractional step nature of the model. All viscous terms are
S.F. Bradford / Coastal Engineering 65 (2012) 1–10
3
Fig. 1. Surface elevation and (hxn)2 − (hxn + 1)2 for a wave near incipient breaking for a plunging breaker.
discretized with space-centered differences, which yields the follow~ ing equation for the hydrostatic velocity field U, ~ −Un U nþ1=2 nþ1=2 n n j;k;l j;k;l þ Fjþ1=2;k;l −Fj−1=2;k;l −Fvjþ1=2;k;l þ Fvj−1=2;k;l þ Δt nþ1=2 nþ1=2 n n Gj;kþ1=2;l −Gj;k−1=2;l −Gvj;kþ1=2;l þ Gvj;k−1=2;l þ ! nþ1=2 nþ1=2 ~ ~ Hj;k;lþ1=2 −Hj;k;l−1=2 U j;k;lþ1 −U j;k;l −ðνlσσ Þj;k;lþ1=2 þ Δσ Δσ ! ~ ~ −U U j;k;l nþ1=2 j;k;l−1 ðνlσσ Þj;k;l−1=2 ¼ Sj;k;l Δσ
Fig. 3. Contours of w in m/s for spilling breakers. Upper plot σx and σy are updated and lower plot they are not updated.
ð17Þ nþ1
ðDvÞ
where Δσ is the vertical grid spacing. Writing Eq. (17) for each cell yields a tridiagonal system of equations that is solved with the Thomas algorithm. 4. Nonhydrostatic pressure The hydrostatic velocity field is corrected to account for nonhydrostatic effects in a fractional step manner by including the dynamic pressure p, in the Navier–Stokes equations. The nonhydrostatic velocities are written at cell faces as nþ1
ðDuÞ
~ ¼ Du−D
nþ1
Δt
nþ1 ∂p ∂p ∂p ξx þ ηx þ σx ∂ξ ∂η ∂σ
nþ1
w
nþ1 ∂p ∂p ∂p nþ1 ~ ¼ Dv−D Δt ξy þ ηy þ σy ∂ξ ∂η ∂σ
~ ¼ w−
nþ1 Δt ∂p Dnþ1 ∂σ
ð19Þ
ð20Þ
where p is the nonhydrostatic pressure component. Space centered differences are used to discretize the pressure gradients. Note that the equations are integrated in time with the first order accurate implicit Euler method. Bradford (2011) found that using the second order accurate Crank–Nicholson method yielded no discernible improvement in the model predictions for the test cases considered here.
ð18Þ
Fig. 2. Comparison of predicted water levels with experimental data (symbols) for spilling breakers. Solutions with σx and σy updated (solid line) and not updated (dashed line).
Fig. 4. Comparison of predicted water levels with experimental data (symbols) for plunging breakers. Solutions with σx and σy updated (solid line) and not updated (dashed line).
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S.F. Bradford / Coastal Engineering 65 (2012) 1–10
The final velocities are forced to simultaneously satisfy the incompressibility constraint by substituting Eqs. (18)–(20) into the discrete form of the incompressibility constraint given as, nþ1
nþ1
nþ1
nþ1
ðDU Þjþ1=2;k;l −ðDU Þj−1=2;k;l þ ðDV Þj;kþ1=2;l −ðDV Þj;k−1=2;l þ
nþ1 W nþ1 j;k;lþ1=2 −W j;k;l−1=2
Δσ
¼0
ð21Þ
~ are computed by simple arithmetic Face values of DŨ, DV~ , and W averaging of adjacent cell-average values. Assembling Eq. (21) for each cell in the domain yields a banded matrix for p that is solved with the preconditioned biconjugate gradient scheme (Press et al., 1992). With p known, the nonhydrostatic velocities at the n + 1 time level can then be computed from Eqs. (18)–(20). 5. Previous wave breaking treatment
Fig. 5. Contours of w in m/s for plunging breakers. Upper plot σx and σy are updated and lower plot they are not updated.
~ is needed at the free surface prior to computing p and boundary W conditions are required to estimate it. Ghost cells are defined adjacent to the free surface, but above the computational domain. Boundary conditions presented by Bradford (2011) are used to define u, v, and scalars in the ghost cells that yield the desired boundary fluxes. The kinematic free surface boundary condition (KFBC) is used to specify the hydrostatic value of w in the ghost cell as, ~ j;k;n þ1 ¼ 2 ht þ u ~ j;k;n ~ j;k;n hx þ v~ j;k;n hy −w w σ σ σ σ
Fig. 6. Comparison of predicted water levels with experimental data (symbols) for (a) spilling breakers and (b) plunging breakers. Solutions including lateral diffusion (solid line) and neglecting lateral diffusion (dashed line).
ð22Þ
Fig. 7. Comparison of predicted water levels with experimental data (symbols) for (a) spilling breakers and (b) plunging breakers. Solutions with second order advection for all variables (solid line), first order turbulence advection (long dashed line), and first order turbulence and w advection (short dashed line).
S.F. Bradford / Coastal Engineering 65 (2012) 1–10
where nσ is the number of cells in the vertical direction, ht is the time derivative of h, and hx and hy are the free surface gradients. Eq. (22) is not used as a boundary condition for the nonhydrostatic value of w, which instead is computed from Eq. (20) once p is known. The application of Eq. (22) requires values for ht, hx, and hy. For computational efficiency, Bradford (2005) computed them once at the beginning of each time step, which worked well for nonbreaking and weakly nonlinear breaking waves. However, this approach was later found to over predict w along the front face of highly nonlinear breaking waves, which caused a corresponding over prediction of wave heights in the surf zone. ~ in Eq. (9) to recompute ht prior Bradford (2011) proposed using U ~ at the free surface. This yielded accurate wave to computing W heights inside the surf zone, but caused an under prediction of wave heights outside the surf zone. Therefore it was proposed that ht be computed at the beginning of the time step outside the surf zone and updated using the hydrostatic velocity field inside the surf zone only. This method yielded accurate wave height predictions, but
5
added the complication of differentiating the surf zone from the rest of the domain. 6. Proposed wave breaking treatment The previous model used D n + 1, σxn, and σyn to compute the pressure Poisson matrix coefficients. Examination of Eqs. (18)–(20) reveals that this is not numerically consistent and instead σxn + 1, and σyn + 1 should be used. This inconsistency is the actual cause of the over prediction of wave heights in the surf zone as will now be demonstrated. To see why the over prediction occurs, consider a cell at the free surface. For a two-dimensional simulation, the Poisson matrix coefficient (denoted a) for the pressure in this cell is
a¼
2D 3 þ 2h2x þ ½ð1−Δσ Þhx þ Δσ zbx 2 þ Δx2 DΔσ 2
ð23Þ
Fig. 8. Comparison of predicted mean velocity and turbulence profiles with experimental data (symbols) for spilling breakers at (a) at (x − xb)/db = 1.332 (b) (x − xb)/db = 4.397 (c) (x − xb)/db = 10.528 and (d) (x − xb)/db = 16.709. Note that the subscript b denotes the location of initial wave breaking. Solutions with second order advection for all variables and updating σx and σy (solid line), second order advection for all variables and not updating σx and σy (long dashed line), and first order turbulence and w advection and updating σx and σy (short dashed line).
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S.F. Bradford / Coastal Engineering 65 (2012) 1–10
where Δx denotes the lateral grid spacing and zbx is the bottom gradient in the x direction. At the front face of a breaking wave, the model often yields |hx| = O(1), which is substantially greater than the bed slope that is typically of O(0.1) or less. Therefore, for the purposes of this analysis, zbx may be neglected and Eq. (23) may be approximated as a≈
2 D Δz 2 3 þ 3h þ 2 x Δx Δz2
ð24Þ
where Δz = DΔσ. In the simulations performed in this paper, Δz/Δx varies from approximately 1.67 in the deepest water to 0 at the shoreline. At initial breaking, the ratio is roughly equal to 1. Fig. 1 shows h and the difference, (hxn) 2 − (hxn + 1) 2, for a wave near the initial breaking location in a plunging breaker case. Similar results were obtained for a spilling breaker case but for the sake of brevity are not presented. This figure shows that along the front face of the wave, |hxn| is significantly smaller than |hxn + 1|. Eq. (24) shows that
using hxn instead of hxn + 1 results in the under prediction of a by approximately 10% along the front of the wave. If a is under predicted, the computed p in the surface cell will be over predicted. The over prediction of pressure at the surface results in the over prediction of w at the surface as well, which is now illustrated. Eq. (20) may be written for w at the free surface as nþ1
ws
~s þ ¼w
2Δtpnþ1 s Dnþ1 Δσ
ð25Þ
where ws is the vertical surface velocity and ps is the cell-centered pressure in the surface cell. Since ps > 0 along the front face of the wave, this equation shows that an over prediction of ps causes a corresponding over prediction of ws and excessive upward flux out of the cell. Incompressibility dictates that the excessive upward flux out of a cell along the front face of the wave must be compensated for by reducing the outward flux through the downstream face or increasing the inward flux through the lower and/or upstream faces. Either
Fig. 9. Comparison of predicted mean velocity and turbulence profiles with experimental data (symbols) for plunging breakers at (a) at (x − xb)/db = − 3.247 (b) (x − xb)/db = 3.571 (c) (x − xb)/db = 12.987 and (d) (x − xb)/db = 16.883. Note that the subscript b denotes the location of initial wave breaking. Solutions with second order advection for all variables and updating σx and σy (solid line), second order advection for all variables and not updating σx and σy (long dashed line), and first order turbulence and w advection and updating σxand σy (short dashed line).
S.F. Bradford / Coastal Engineering 65 (2012) 1–10
scenario results in the over prediction of the wave height as shown by Eq. (9). Therefore, the approach proposed in the present study is to update the values of hx and hy everywhere in the domain after computing D n + 1 and prior to computing p. These gradients are needed to compute σxn + 1 and σyn + 1, which are updated at the same time. Zijlema and Stelling (2005) also update the vertical grid spacings prior to computing the dynamic pressure. ht is kept frozen at the old time level until after p is computed. The results of the new approach are now illustrated in two surf zone simulations involving spilling and plunging breakers.
7. Simulation of wave breaking on a plane beach Ting and Kirby (1994) performed laboratory experiments with breaking waves on a plane beach with 1:35 slope and still water depth, d = 0.4 m. The channel is 30 m long and the slope begins at x = 14 m from the incoming wave boundary. Spilling and plunging breakers were investigated and the spilling waves were defined by 2 s period and 0.125 m height. Eighth order stream function wave theory (Dalrymple, 1974) is used to model the incoming waves and the model was run for 40 s with lateral grid spacing, Δx = 0.0375 m, Δt = 0.0075 s, and the number of vertical layers, nσ = 8. Fig. 2 compares the computed hmax −h, h, and hmin −h with experimental measurements. Note that hmax is the crest elevation, hmin is the trough elevation, and h is the mean surface elevation. Also shown are the predictions obtained by updating σx and σy prior to assembling the pressure Poisson matrix. In both cases, all gradients are computed with the Double Minmod limiter. This figure shows that updating σx and σy greatly improves the wave height predictions in the surf zone. Fig. 3 compares contour plots of w and velocity vectors for the two cases and shows that not updating σx and σy prior to computing p yields slightly larger magnitudes of w as shown by the contour legend. Not updating the grid metrics also yields a greater wave height in the surf zone (compare the furthest right wave in each case.) For the plunging breaker simulations, the incoming wave height is 0.128 m, the period is 5 s, and eighth order wave theory is again used to define the incoming waves. The same grid spacing and time step size were used in this case. Fig. 4 compares numerical predictions with experimental data for hmax −h, h, and hmin −h. Once again, the predictions obtained by updating σx and σy are also shown to more closely match the experimental wave heights in the surf zone. Fig. 5 compares contour plots of w and velocity vectors for the two simulations. This figure shows that not updating σx and σy yields significantly larger downward values of w along the back side of the breaking wave in addition to a much higher wave height in the surf zone. An additional modification to the model is to neglect Fv, Gv, and Hv, which may be expected to be small compared to lateral advection from boundary layer scaling arguments (Mellor and Yamada, 1982). Fig. 6a compares predicted wave heights for the spilling breaker case. In both cases, σx and σx are updated prior to computing p. Fig. 6b shows the plunging breaker case. In both instances, neglecting lateral diffusion yields very little change in predicted waves heights indicating that in terms of this metric the simplification is justified and actually slightly improves wave height predictions within the surf zone. This may be due to the large velocity gradients along the front face of a breaking wave, which cause additional momentum dissipation in this region when the diffusion terms are included in the model. The velocity gradients are smaller along the back of the wave and therefore momentum is not as heavily affected by including the diffusion terms. The enhanced dissipation along the front of the wave decreases the local velocity, which causes additional fluid to accumulate in the wave and produces the larger wave height predictions observed in Fig. 6.
7
One additional benefit of this simplification is the increased robustness of the model. The explicit integration of the lateral diffusion terms imposes a numerical stability limit on Δt, which is removed when these terms are neglected. However, the real benefit of this modification is improved computational efficiency of the model. Neglecting the lateral diffusion terms reduced the required computational time by roughly 21% in the two cases simulated here. All remaining results are obtained by neglecting lateral diffusion. As previously mentioned, the use of slope limiters in data reconstruction is computationally expensive and cost cutting measures in this area are now explored. Model tests not presented here revealed that the Double Minmod limiter should be used when computing lateral gradients of h, u, v, U , and V . Using the Minmod limiter or reverting to first order accuracy by setting the gradients to zero introduced severe numerical dissipation that caused the rapid decay of wave heights in the simulations. The choice of limiter for the vertical gradients of u and v was found to have no significant impact on wave height predictions. However, reverting to first order accuracy in the vertical direction yielded some solution degradation. Therefore, in all remaining simulations, the Double Minmod limiter is used for computing the gradients of h, u, v, U , and V . For the turbulence variables k and l, the choice of limiter was found to only slightly impact model predictions. Fig. 7a compares predictions of surface elevations for the second order (Double Minmod) and first order discretization of turbulence advection for the spilling breaker case. Fig. 7b shows the same comparison for the plunging breaker case. In both cases, the first order predictions actually better
Fig. 10. Comparison of predicted water levels with experimental data (symbols) for (a) spilling breakers and (b) plunging breakers. Solutions with nσ = 4 (long dashed line), nσ = 8 (solid line), and nσ = 16 (short dashed line).
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S.F. Bradford / Coastal Engineering 65 (2012) 1–10
Fig. 11. Comparison of predicted mean velocity and turbulence profiles with experimental data (symbols) for spilling breakers at (a) at (x − xb)/db = 1.332 (b) (x − xb)/db = 4.397 (c) (x − xb)/db = 10.528 and (d) (x − xb)/db = 16.709. Note that the subscript b denotes the location of initial wave breaking. Solutions with nσ = 4 (long dashed line), nσ = 8 (solid line), and nσ = 16 (short dashed line).
match the experimental data. It appears counterintuitive that the theoretically more dissipative first order method actually yields greater wave heights around the outer surf zone than the second order method. This may be due to the grid skew in this region that causes the second order method with its larger computational stencil to be more dissipative than the first order method. The first order treatment of turbulence advection yields a 4 percent savings in computational effort. Also shown in Fig. 7 are predictions obtained by treating advection for w with the first order upwind method. For the spilling breaker case, this yields even greater wave heights, while not much change is introduced for the plunging breaker case. Fig. 8 compares the predictions of the mean velocity, u, and mean turbulence kinetic energy, k, with experimental data as a function of depth at four stations in the surf pzone ffiffiffiffiffiffi for the spilling breaker case. The results are normalized by c ¼ gd. The methods include not updating the vertical grid metrics and treating all advection terms with second order accuracy, updating the grid metrics and treating all advection terms with second order accuracy, and updating the grid metrics with the first order discretization of w and turbulence
advection. In terms of velocity, all three methods yield fairly similar results, while there is slightly more difference in the predictions of turbulence. Fig. 9 shows the same results for the plunging breaker case. In this case, the predictions are very similar for both velocity and turbulence. The previous results were all obtained with nσ = 8. Predictions are now presented with 4 and 16 vertical layers in an attempt to identify the minimal number of layers needed to obtain satisfactory predictions. These results are all obtained by updating σx and σy and treating w and turbulence advection with the first order upwind method. Fig. 10a shows surface elevation levels for the spilling breaker case, which reveals that the nσ = 4 solution slightly underpredicts wave heights in the outer surf zone. Increasing nσ to 16 yields a better match with experimental data in this region than the nσ = 8 solution. Fig. 10b shows the corresponding results for the plunging breaker case. In this case, the predictions are less sensitive to changes in nσ, but the nσ = 16 case yields the closest match to the measurements. Fig. 11 compares the corresponding results for u and k in the spilling breaker case. For u, all of the predictions are relatively close, but the nσ = 16 yields the closest match to the measurements near the
S.F. Bradford / Coastal Engineering 65 (2012) 1–10
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Fig. 12. Comparison of predicted mean velocity and turbulence profiles with experimental data (symbols) for plunging breakers at (a) at (x − xb)/db = − 3.247 (b) (x − xb)/ db = 3.571 (c) (x − xb)/db = 12.987 and (d) (x − xb)/db = 16.883. Note that the subscript b denotes the location of initial wave breaking. Solutions with nσ = 4 (long dashed line), nσ = 8 (solid line), and nσ = 16 (short dashed line).
bed. Fig. 12 compares results for the plunging breaker case and once again the nσ = 16 predictions of u generally match the measurements the most closely, particularly near the bed. The results near the free surface do not differ much when increasing nσ from 8 to 16, which indicates that the model would benefit most from near bed vertical refinement of the grid rather than globally increasing nσ. 8. Summary and conclusions The proposed nonhydrostatic surf zone model is complex, which introduces several opportunities for making it more accurate, computationally efficient, and robust. In this paper, three primary modifications were investigated. First, it was shown that it is critical to update not only D, but also the free surface gradients and the vertical grid transformation terms, σx and σy, prior to computing the dynamic pressure, p. There is a significant phase difference between the surface gradients (as well as σx and σy) at the old and new time levels. In regions outside the surf zone, the surface gradients are sufficiently small so that failing to update them does not introduce an appreciable
error in the model predictions. However, along the front face of a breaking wave, the surface gradients are large enough that the phase lag causes a significant underprediction of the pressure Poisson coefficient for surface cells. This causes an over prediction of the surface p, w, and wave height. Despite this modification's significant effect on wave height predictions, a more modest effect on computed mean u and k was observed. This proposed modification is much more satisfactory than the one presented by Bradford (2011) because it requires no explicit identification of the surf zone location to implement. The second modification that was examined is the neglect of lateral diffusion in the governing equations. This was shown to modestly improve wave height predictions, but more importantly it trims approximately 21% from the required computational time. No adverse effects of this simplification were observed in the predictions of mean velocity and turbulence kinetic energy. The final area of investigation was variable reconstruction prior to the calculation of advective fluxes. It was found to be crucial to treat the lateral gradients of h, u, v, U , and V with second order accuracy.
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Moreover, the Double Minmod limiter yielded much less numerical dissipation than the Minmod limiter. The choice of limiter for vertical gradients of these variables appears not to matter much. Both the Double Minmod and Minmod limiters yielded similar results. For w, k, and l, setting the gradients to zero yielded more accurate wave heights with little impact on u and k depth profiles. Finally, the number of vertical layers was varied from 4 to 16 and it was observed that using just 4 layers yielded wave height predictions nearly as accurate as the higher resolution solutions. However, 8 or more layers are needed to obtain accurate profiles of u particularly near the bed and free surface. However, selective grid refinement near the bed and free surface could be used to minimize the total number of vertical cells required to obtain satisfactory solution accuracy. The proposed model provides a compromise between including the most significant surf zone physics while maximizing computational efficiency in order to make larger scale coastal applications feasible. The model requires no empirical wave breaking parameters and therefore no numerical tuning is needed. However, the model is still computationally expensive and would require additional efforts such as parallelization to make large scale simulations realistically possible. References Bradford, S.F., 2004. Stability and accuracy of a semi-implicit Godunov scheme for mass transport. International Journal for Numerical Methods in Fluids 45, 365–389. Bradford, S.F., 2005. Godunov-based model for nonhydrostatic wave dynamics. Journal of Waterway Port, Coastal, and Ocean Engineering 131, 226–238.
Bradford, S.F., 2011. Nonhydrostatic model for surf zone simulation. Journal of Waterway Port, Coastal, and Ocean Engineering 137, 163–174. Dalrymple, R.A., 1974. A finite amplitude wave on a linear shear current. Journal of Geophysical Research 79, 4498–4504. Galperin, B., Kantha, L.H., Hassid, S., Rosati, A., 1988. A quasi-equilibrium turbulent energy model for geophysical flows. Journal of Atmospheric Science 45, 55–62. Li, B., Fleming, C.A., 2001. Three-dimensional model of Navier–Stokes equations for water waves. Journal of Waterway Port, Coastal, and Ocean Engineering 127, 16–25. Lin, P., Li, C.W., 2002. A σ-coordinate three-dimensional numerical model for surface wave propagation. International Journal for Numerical Methods in Fluids 38, 1045–1068. Mahadevan, A., Oliger, J., Street, R., 1996. A nonhydrostatic mesoscale ocean model. Part II: numerical implementation. Journal of Physical Oceanography 26, 1881–1900. Mellor, G.L., Yamada, T., 1982. Development of a turbulence closure model for geophysical fluid problems. Reviews Geophysics and Space Physics 20, 851–875. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992. Numerical Recipes in Fortran. Cambridge University Press, New York. Roe, P.L., 1981. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics 43 (2), 357–372. Ting, F.C.K., Kirby, J.T., 1994. Observation of undertow and turbulence in a laboratory surf zone. Coastal Engineering 24, 51–80. Van Leer, B., 1979. Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov's method. Journal of Computational Physics 32 (1), 101–136. Zijlema, M., Stelling, G.S., 2005. Further experiences with computing non-hydrostatic free-surface flows involving water waves. International Journal for Numerical Methods in Fluids 48, 169–197. Zijlema, M., Stelling, G.S., 2008. Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure. Coastal Engineering 55, 780–790.
