Numerical Simulation of Irregular Wave Overtopping ...

China Ocean Eng., Vol. 26, No. 1, pp. 153 − 166 © 2012 Chinese Ocean Engineering Society and Springer-Verlang Berlin Heidelberg DOI 10.1007/s13344-012-0011-7, ISSN 0890-5487

Numerical Simulation of Irregular Wave Overtopping Against A Smooth Sea Dike* GUO Xiao-yu (郭晓宇)a, b, WANG Ben-long (王本龙) a, b and LIU Hua (刘 a

桦) a, b, c, 1

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China

b

Key Laboratory of Hydrodynamics of MOE, Shanghai Jiao Tong University, Shanghai 200240, China

c

State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (Received 20 July 2011; received revised form 18 August 2011; accepted 9 October 2011)

ABSTRACT Based on the filtered Navier-Stokes equations and Smagorinsky turbulence model, a numerical wave flume is developed to investigate the overtopping process of irregular waves over smooth sea dikes. Simulations of fully nonlinear standing wave and regular wave's run-up on a sea dike are carried out to validate the implementation of the numerical wave flume with wave generation and absorbing modules. To model stationary ergodic stochastic processes, several cases with different random seeds are computed for each specified irregular wave spectrum. It turns out that the statistical mean overtopping discharge shows good agreement with empirical formulas, other numerical results and experimental data. Key words: overtopping discharge; irregular wave; numerical wave flume; Navier-Stokes equations

1. Introduction In coastal engineering, wave overtopping is considered as an important issue for the design and upgrading of seawalls. In the last decades, many numerical investigations on wave overtopping have been carried out. Unfortunately, most work is focused on regular waves. In fact, regular waves never occur in real sea state and the real waves are usually random and irregular. Therefore, it is more appropriate to use irregular wave train as input for overtopping investigation. By comparing general computational fluid dynamics solvers, numerical simulation of irregular wave encounters two additional difficulties. The first problem is the setup of effective wave generating and absorbing modules. Absorbing reflected wave from computational domain is very important for irregular wave sequence, especially for a long time simulation. The other difficulty is the computational efficiency. For irregular wave, the simulation time should be long enough to fulfill the conditions for a stationary ergodic stochastic process. Hence, a strong challenge is posed to numerical solver which should be robust and stable when violent free surface deformation, breaking and air entrainment occur in the numerical wave flume. So far, the numerical wave tank can be classified into two categories based on the potential flow * This project was financially supported by the National Natural Science Foundation of China (Grant No. 10972138), the Natural Science Foundation of Shanghai Municipality (Grant No. 11ZR1418200), Key Project of Science and Technology Commission of Shanghai Municipality (Grant No. 09231203402) and Key Doctoral Programme Foundation of Shanghai Municipality (Grant No. B206). 1 Corresponding author. E-mail: [email protected]

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GUO Xiao-yu et al. / China Ocean Eng., 26(1), 2012, 153 − 166

theory and the Navier-Stokes equations. Boo (1994) developed a numerical wave tank and simulated irregular wave using the high order boundary element method, then Boo (2002) studied irregular wave forces acting on a vertical truncated circular cylinder. Wei et al. (1999) simulated the propagation of regular and random wave with Boussinesq-type wave model using source function method. Zhan et al. (2002) developed a numerical modeling of multi-directional irregular wave based on the Boussinesq-type equations. Avgeris and Prinos (2004) adopted the source function method and simulated the propagation of irregular wave train over submerged breakwaters using the Boussinesq-type model. Li, Yu and Zhang (2004) developed a 2D numerical wave tank using

σ-transformation based on potential flow and calculated the irregular wave forces acting on a vertical wall. Ning et al. (2008) developed a 3D numerical wave tank based on high order boundary element method for the propagation of regular wave and irregular wave. Although there are many advantages for the methods based on potential flow theory and the boundary element method to study the propagation of nonlinear water waves, it is hard to predict the complicated flow field relating violent deformation of the free surface and wave breaking with this approach. With rapid development of computer hardware and computational methods, there has been a great progress in numerical simulation of interactions between nonlinear waves and coastal structures with the Navier-Stokes equations. Soliman (2003) investigated the influences of seawall slope, wave type and crest freeboard on overtopping discharges of irregular wave based on the RANS equations and k − ε turbulence model. Chen et al. (2004) developed a numerical wave tank based on the solution of time dependent σ -transformed Navier-Stokes equations and simulated the propagation of irregular wave. Li, Troch and de Rouch (2004) simulated the interaction between irregular waves and vertically fixed impermeable barrier in front of breakwater. Lara et al. (2006) carried out a numerical investigation on the random wave interaction with submerged structure. Li (2008) presented a 3D model for regular and irregular wave propagation based on the spatial fixed σ -transformed Navier-Stokes equations and discussed the computational cost. Liang et al. (2010) applied commercial software FLUENT to implement numerical simulation of irregular wave propagation and then a detailed study on local characteristics of predetermined irregular wave train was shown (Liang et al., 2011). Although numerical wave flume based on the Navier-Stokes equations has been developed in various forms for a certain years, numerical simulation of overtopping over sea dike for irregular wave seems still in its infancy. The aim of the present study is to demonstrate the development and application of numerical wave flume to reproduce irregular wave trains as well as overtopping processes over sea dike. Furthermore, a general approach is proposed to simulate the stationary ergodic stochastic processes when the spectrum parameters of irregular wave are predetermined, which is a common situation in practical applications. To this end, the present study is, therefore, followed as: First, the procedures of generating and absorbing irregular waves in the numerical model are described and verified. Next, the propagation of irregular wave trains over a flat bottom is simulated and compared with analytical results. Finally, reliable prediction of irregular wave overtopping discharge over sea dikes is investigated.


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2. Numerical Methods 2.1 Governing Equations Incompressible and viscous flows are considered. One set of equations is used to describe the behavior of two phase fluids together, which includes the filtered Navier-Stokes equations and continuity equation: ∂ ( ρ ui ) ∂t

+

∂ ( ρ u j ui ) ∂x j

=−

∂p ∂ + ∂xi ∂x j

⎛ ∂u ⎜⎜ ( μ + μ t ) i ∂x j ⎝

⎞ ⎟⎟ + ρ g i , ⎠

∂ui = 0. ∂xi

(1) (2)

For two dimensional flow, xi (i=1,2) represents the Cartesian coordinate system; t is the time;

ui is the

filtered velocity component; ρ is the fluid density; μd and μ t are molecular dynamic viscosity and turbulence viscosity, respectively; gi is the gravitational acceleration in the i-direction. According to the volume fraction function F, the density and viscosity of the mixture fluid can be expressed as general form:

f = Ff w + (1 − F ) f a .

(3)

The subscripts w and a stand for the variables of water and air, respectively. To estimate the small-scale turbulence generated during wave breaking, Smagorinsky turbulence model (Smagorinsky, 1963) is employed which is widely applied in the flow simulations related to the free surface waves (Li, Troch and de Rouch, 2004; Hieu and Tanimoto, 2006). This model is closely connected to the strain rate and the grid size. The eddy viscosity μ t is determined from the strain rate:

μ t = ρ ( Cs Δ ) 1 ⎛ ∂u ∂u j ⎞ in which, Sij = ⎜ i + ⎟, 2 ⎜⎝ ∂x j ∂xi ⎟⎠

2

2 Sij Sij ,

(4)

Δ is a characteristic length scale of the small eddies depending on the

mesh size and given by Δ = min(Δx, Δy ) , and Cs is the Smagorinsky constant which is chosen as 0.075. 2.2 Two Phase Flow Model The volume of fluid (VOF) algorithm is used to track the interface between air and water. For air-water flow, VOF method introduces function F to define water region. Function F is referred to as volume fraction of a cell occupied by water which can be evaluated as F = Vw / Vc , where Vw is the volume of water inside a cell and Vc is the cell volume. Hence, interface is established further: in air ⎧0 ⎪ (5) F = ⎨0 < F < 1 across the interface ⎪1 in water ⎩ The algorithm for tracking the free surface consists of two parts: interface reconstruction algorithm and interface advection algorithm. At each time step, the volume fraction function F is used to

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approximately reconstruct the interface with certain geometric solutions and the reconstructed interface is then used to evolve new volume fraction forward in time with solution of an advection equation. The two dimensional transport equation for F is given by:

∂u j ∂F ∂ ( Fu j ) + =F . ∂t ∂x j ∂x j

(6)

The algorithm for solving this equation follows the procedure of Pillod and Pucket (2004). 2.3 Method of Solution The governing equations are discretized by the finite volume method on staggered grids. Velocity components are evaluated at the cell sides, while scalar quantities are evaluated at the cell center. The convective terms in momentum equation are discretized by flux-limiter van Leer scheme and the diffusion term is discretized with the central difference scheme. The Navier-Stokes equations are solved with the pressure correction procedure in form of the modified SIMPLE-type method. The resultant linear equation sets are solved by FGMRES iteration method (Saad, 1993). 2.4 Numerical Wave Generation and Absorbing Method Analytical relaxation approach is adopted for wave generation and absorbing in numerical wave flume. This method was originally presented by Mayer et al. (1998). Wang and Liu (2005) extended the analytical relaxation wavemaker approach and applied it to the numerical study of water wave based on the high order Boussinesq equations. In the past several years, the relaxation approach was formulated as an external force, i.e. source terms in the RANS equations, and implemented in commercial package Fluent with UDF by Zhou et al. (2005). Some applications of the numerical wave flume in wave-structure interactions were reported, such as Lu et al. (2007), Liu et al. (2007) and Sun (2010). Instead of using the source terms in the momentum equations, we apply the original relaxation approach in the solution of the Navier-Stokes equations and implement wave generation and absorption by updating velocity directly at each time step, which can be expressed as: uM = CuC + (1 − C ) uI , vM = CvC + (1 − C ) vI ,

(7)

in which the velocity variables with the subscript C are the computed value from the Navier-Stokes solver, variables with subscript I denote the incoming wave velocity. C = C ( x) is the relaxation function related to spatial coordinate. In general, the numerical wave flume is composed of four parts: wave maker zone, relaxation zone, work zone and sponger zone. In the wave maker zone, smoothly varying function C is assured that the velocities at right boundary are incoming wave velocity. At two ends of the wave maker zone, function C satisfies:

[C ]

x min

= 1,

[C ]

x max

=0.

(8)

In the relaxation zone, function C is chosen for the purpose of eliminating the waves reflected from work zone. Function C satisfies:

[C ]

x min

= 0,

[C ]

x max

=1.

(9)


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In the sponger zone, function C is used to absorb the right-going waves out of work zone to prevent wave reflection at the boundary, which satisfies:

[C ]

x min

=1 ,

[C ]

x max

=0.

(10)

2.5 Initial and Boundary Conditions At the initial time, still water is assumed in the whole computational domain and the pressure is given by the hydrostatic pressure. At the surface of solid structure, no-slip wall boundary condition is imposed by use of one type of immerse boundary method, namely discrete forcing approach which is similar as what has been proposed by Mohd-Yusof (1997). The other kind of computational domain boundaries are symmetric boundaries to satisfy non-penetrating condition, as illustrated in Fig. 1. The purpose of using non-penetrating condition for bottom is to eliminate unnecessary viscous damping of wave energy during wave propagation along the work zone. The sponger zone at the right end is not shown in this setup, but is reserved in other simulations.

Fig. 1. Setup of boundary conditions and computational domain.

3. Calibration of Numerical Wave Flume 3.1 Standing Wave A standing wave train is simulated to validate the efficiency of the numerical approach for implement of the wave maker zone and the relaxation zone. The sponger zone is removed and the right boundary of work zone is set as smooth vertical wall. Consequently, the right going wave train will be perfectly reflected. The superposition of the right going incoming waves and the left going reflection waves forms a spatial uniform nonlinear standing wave train. The wave parameters are wave length 2 m, water depth 0.5 m, and wave height 0.05 m. The solution of the stream function wave theory is used as the input signal for wave generation. As shown in Fig. 2, it is noticed that the amplitude of standing wave is 2 times that of the incident wave, indicating that the reflection waves from the work zone are perfectly absorbed. The relaxation zone ensures that no second reflection is in the wave flume. Viscous force is switched off in this simulation. 3.2 Run-up of Regular Waves on A Sea Dike To further validate the accuracy of the numerical wave flume, regular wave run-up over a sea dike is simulated following the same conditions as Li Troch and de Rouch (2004). The schematic of the sea dike is shown in Fig. 3. Two wave gauges WG2 and WG3 take the same locations as Li Troch and de Rouch (2004). The wave parameters are: water depth d =0.7 m in front of the sea dike, wave height 0.16


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irregular wave train. For convergence tests, three grids are compared in horizontal direction: 100, 300 and 400 of grid nodes within a wavelength respectively. For these three meshes, vertical resolutions take the same value with the minimum grid size of Δy = 0.006 m. It is found that simulation with 300 grid nodes per wavelength of the significant wave can well describe the irregular waves, as shown in Fig. 7 for the case of N = 60 . We take 300 grid nodes per wavelength in the following computation for overtopping studies.

Fig. 6. Comparison of time series of surface elevation at G3 point for 300 nodes per wavelength L, N=60. Solid line is the computed curve and dashed-dot line is the input signal for wave maker. (From top to bottom: origin frequencies, splitting maximum frequency)

Fig. 7. Time series of surface elevation at different locations. Solid line is the computed curve and dashed-dot line is the input signal for wave maker. (From top to bottom, G1, G2 and G3)


161

4. Irregular Wave Overtopping over Simple Sea Dike 4.1 Empirical Formulas of Overtopping Discharges for Irregular Wave During last several decades, many researches have been conducted to estimate overtopping discharge under the effect of irregular waves. The empirical formulas proposed by Owen (1980) and van der Meer and Janssen (1995) are chosen for comparison and validation of the numerical model. Owen (1980) proposed a relationship between overtopping discharge Q and crest freeboard Rc based on an

extensive series of model tests for a range of seawall designs subject to different random wave climates, which can be recalled for an impermeable smooth straight seawall: ⎛ Rc Q = A exp ⎜ − B ⎜ gH sTm Tm gH s ⎝

⎞ ⎟, ⎟ ⎠

(11)

where Q is the overtopping discharge per unit width, H s is the significant wave height, Tm is the spectrum deep water mean period, g is the acceleration of gravity, coefficients A and B depend on the sea wall profile. Besley (1999) studied the empirical coefficients A and B for estimating overtopping discharge over impermeable smooth straight seawall in UK and suggested that A = 0.00939 and B = 21.6 .

Van der Meer and Janssen (1995) made a distinction between breaking and non-breaking waves for overtopping discharge calculation on straight impermeable sloped seawall. For breaking wave: ξ p < 2 ⎛ 1 = 0.06 exp ⎜ −5.2 R ⎜ ξp γ r γ bγ hγ β gH s ⎝ R 2πH s tan α . R = c , ξp = , Sp = H sξ p gTp0 2 Sp tan α

Q

3

⎞ ⎟⎟ ; ⎠

(12) (13)

For non-breaking wave: ξ p > 2 ⎛ ⎞ R 1 = 0.2 exp ⎜ −2.6 c (14) ⎟, ⎜ H s γ r γ b γ h γ β ⎟⎠ gH s ⎝ ξ p is the surf-similarity parameter; H s is the significant wave height; α is the slope angle; Tp0 is the Q

3

spectrum deep water peak period; γ b , γ h , γ r and γ β are the coefficients considering the influence of a berm, shallow foreshore, roughness and the angle of wave attack, respectively. All these coefficients are ranging from 0.5 to 1.0, which means that when maximizing overtopping, the coefficients should be 1.0, that is the case for no berm, no shallow foreshore, smooth slope and head-on wave. Van der Meer’s formula is the most commonly method used for calculation of overtopping discharge and is recommended for the design of sea dikes in Germany (Soliman, 2003). 4.2 Irregular Wave Overtopping over Simple Seawall As stationary ergodic stochastic processes, the theoretical overtopping discharge should be irrelevant to the irregular wave sequences when the wave spectrum is specified. However, finite time span is used


162

in numerical simulations which violate the stationary condition for random processes. Furthermore, if only one irregular wave sequence of long time span is simulated, there is only one set random seeds for each wave components in the wave sequence. The ergodic condition could not be satisfied as well. Taking these two aspects into account, we expect to model the stochastic processes in multiple irregular wave sequences with different random seeds as well as long time span for each sequence. The critical problem is that huge computational resources are necessary. With some preliminary trial tests, we found the statistical values calculated with 5~8 irregular wave sequences appear reasonable similar magnitude. For each sequence, there are about one hundred waves in the wave train. With these experiences, each of computational cases calculates overtopping discharges by averaging the overtopping discharges of six different wave train series. If we have enough processors, these six runs can be carried out independently. All the simulations are run in Dell T5400 work stations with dual quad-core Xeon processors. Irregular wave overtopping is simulated by use of the same cases as those used by Soliman (2003). The configuration of computational domain is shown in Fig. 1, where the slope of simple sea dike is 1:2, and the water depth is 8 m. The computed cases are listed in Table 1. Given significant wave height, spectrum peak period and water depth, wave train can be established according to inverse Fourier transformation from JONSWAP spectrum. Based on the experiences on simulating irregular wave propagation, equi-energy divided method is used to discretize the energy spectrum into 60 components in frequency domain. A fine mesh with 300 grid nodes per spectrum peak wavelength is used to guarantee sufficient grid resolution for different wave modes. Table 1

Configuration of sea dikes and irregular wave parameters

Parameter

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

Rc (m)

1.0

1.0

1.0

2.0

2.0

2.0

2.0

Hs (m)

0.79

1.23

1.73

0.86

1.29

1.75

2.34

Tp (s)

3.51

4.43

5.13

3.51

4.37

5.20

6.04

Tm (s)

3.05

3.85

4.46

3.05

3.80

4.52

5.25

The incoming irregular wave sequences at the foot of sea dike with the same specified spectrum and corresponding time series of overtopping histories for Case 1 are shown in Fig. 8. Except for the first 30 s in each run, the overtopping discharge shows a good correlation with the wave amplitude. It takes around 30 s for the wave front propagating from the wave maker region to the sea dike; therefore there is no overtopping in this period. Obviously, the large waves will result in significant overtopping discharge and other relative small waves make no contribution. The total overtopping discharge depends heavily on how many large waves are included in the irregular wave sequence. Taking statistical analysis over multiple independent runs will eliminate this kind deviation and make the predicted overtopping discharge much more reliable. These runs follow the same energy spectrum but with different random phases. For Case 7, in which there is significant difference among each runs, the values of overtopping discharge are listed in Table 2 for 7 runs. As can be observed, the population variance is not a small value compared with the mean value, which should be expected since phase lags may result in different

164


It should be pointed out that the mean overtopping discharge could not give any information for the instantaneous overtopping discharge, which could reach dozens of times of the mean overtopping discharge. More importantly, these large waves do induce the failure of the sea dike in reality. Predicting the maximum instantaneous overtopping discharge and corresponding velocity distribution are out of the scope of this paper. To investigate the influence of sea dike crest freeboard, non-dimensional overtopping discharges versus ratio Rc / H s are shown in Fig. 9, in which the mean value of the computed overtopping discharge is linked by the solid line and, for the same case, the maximum and the minimum of overtopping discharges are presented as the error bar. The overtopping discharges can be reasonably predicted for the cases of relatively large overtopping discharge. However, for extreme small overtopping discharge, there are some discrepancies. Usually, these small overtopping discharges are out of interest for engineering purpose.

Fig. 9. Comparison among present results, BWNM, van der Meer formula and van der Meer’s lab. data.

The relationship between dimensionless freeboard and overtopping discharges of the present model is depicted in Fig. 10. By comparing the values computed by Eq. (11) and Eq. (14), it can be found that the difference between the numerical results and empirical formula increases as dimensionless freeboard decreases. Nevertheless, we could conclude that the numerical approach works well for prediction of overtopping discharge over a simple sea dike under the action of irregular waves.

Fig. 10. Comparison between the computed overtopping discharges and empirical design formulas.

5. Conclusions A two-dimensional numerical model is developed to study generation, propagation and overtopping of irregular waves over a smooth sea dike. A nonlinear standing wave is simulated in the numerical wave flume to validate the capability of wave generating and absorbing modules. Based on a


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two-parameter energy spectrum, an irregular wave train is generated in the numerical wave flume with an improved discretization approach for a spectrum in frequency domain. It is found that numerical wave flume is efficient and stable. With the in-house developed code for the numerical computation, irregular wave overtopping discharge is investigated numerically. From the perspectives of physical significance of stationary ergodic stochastic processes, a robust overtopping discharge prediction approach is proposed for which multiple random wave sequences for a single wave spectrum are used to calculate statistically the mean value. A series of cases are simulated and overtopping discharges are compared with experimental data, empirical formulas and numerical results available. Good agreement implies that the proposed numerical approach could predict reliable overtopping discharge over a simple sea dike. References Avgeris, T. V. K. and Prinos, P., 2004. Boussinesq modeling of wave interaction with porous submerged breakwaters, Proc. 29th Int. Conf. Coastal Eng., 604~616. Besley, P., 1999. Overtopping of Seawalls: Design and Assessment Manual, Hydraulics Research Wallingford, Report W178. Boo, S.Y., 1994. A numerical wave tank for nonlinear irregular wave by 3D HOBEM, Int. J. Offshore Polar Eng., 4(4): 265~272. Boo, S.Y., 2002. Linear and nonlinear irregular waves and forces in a numerical wave tank, Ocean Eng., 29(5): 475~493. Chen, Y. P., Li, Z. W. and Zhang, C. K., 2004. Development of a fully nonlinear numerical wave tank, China Ocean Eng., 18(4): 501~514. Hieu, P. D. and Tanimoto, K., 2006. Verification of a VOF-based two-phase flow model for wave breaking and wave-structure interactions, Ocean Eng., 33(11-12): 1565~1588. Lara, J. L., Garcia, N. and Losada, I. J., 2006. RANS modeling applied to random wave interaction with submerged permeable structures, Coast. Eng., 53(5-6): 395~417. Li, B., 2008. A 3-D model based on Navier-Stokes equations for regular and irregular water wave propagation, Ocean Eng., 35(17-18): 1842~1853. Li, B. X., Yu, Y. X. and Zhang, N. C., 2004. Development of 2-D numerical random wave tank and its application, Marine Science Bulletin, 23(5): 1~9. (in Chinese) Li, T. Q., Troch, P. and de Rouck, J., 2004. Wave overtopping over a sea dike, J. Comput. Phys., 198(2): 686~726. Liang, X. F., Yang, J. M., Li, J., Xiao, L. F. and Li, X., 2010. Numerical simulation of irregular wave-simulation irregular wave train, Journal of Hydrodynamics, Ser. B., 22(4): 537~545. Liang, X. F., Yang, J. M., Li, J. and Li, X., 2011. A numerical study on local characteristics of predetermined irregular wave trains, Ocean Eng., 38(4): 651~657. Liu, Y. N., Guo, X. Y., Wang, B. L. and Liu, H., 2007. Numerical simulation of wave overtopping over seawalls using the RANS equations, Journal of Hydrodynamics, Ser. A., 22(6): 682~688. (in Chinese) Lu, Y. J., Liu, H., Wu, W. and Zhang, J. S., 2007. Numerical simulation of two-dimensional overtopping against seawalls armored with artificial units in regular waves, Journal of Hydrodynamics, Ser. B., 19(3): 322~329. Mayer, S., Garapon, A. and Sorensen, L. S., 1998. A fractional step method for unsteady free-surface flow with applications to nonlinear wave dynamics, Int. J. Numer. Methods Fluids, 28(2): 293~315.

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