Nov 16, 2006 - A new mathematical model of overtopping against seawalls with ... Comparing the computational value of overtopping over the seawall with the.
Third Chinese-German Joint Symposium on Coastal and Ocean Engineering National Cheng Kung University, Tainan November 8-16, 2006
Numerical simulation of violent breaking and overtopping against seawalls Hua Liu Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai
Abstract A new mathematical model of overtopping against seawalls with artificial units in regular waves was presented. The 2D numerical wave flume, based on the Reynolds averaged Navier-Stokes equations and the standard k-epsilon turbulence model VOF, was developed to calculate the turbulent flows with the free surface, in which the Volume of Fluid method was used to handle with the large deformation of the free surface. A series of physical model experiments were carried out in the same condition of the numerical simulation to determine the drag coefficient of the porous media model in terms of the overtopping discharge. Comparing the computational value of overtopping over the seawall with the experimental data, the values of the effective drag coefficient was calibrated for the layers of dolosse at different locations along the seawalls.
1 Introduction Violent breaking and overtopping occurs when water waves break against seawalls throwing water and spray over the top. Violent overtopping events can be extremely dangerous with people, cars and facilities behind seawalls in coasts. Waves generated by storms, either locally or offshore, exacerbated by high tides and low air pressure lead to the disruption of road services, flooding, structural damage and occasionally loss of life. For the design and maintenance works of coastal structures, reliable predictions of wave overtopping are required. Traditionally, physical model experiment approaches have been used to evaluate the overtopping over some seawalls for specific projects and a large data set obtained from numerous physical model tests can be found in literature. Development, refinement and applications of the Numerical Wave Tanks have been recognized as one of most intensively studied topics in the field of coastal
and ocean engineering. The Navier-Stokes equations, a coupled system of partial differential equations describing the conservation of mass, momentum and energy in a fluid, can be used to describe wave overtopping events if a proper approach of tracking the free surface is adopted. There is significant progress in the development and application of the RANS based numerical wave tanks, including the numerical schemes and algorithm of the Navier-Stokes equations, the numerical approaches for the violent deformation of the free surface and the of numerical and methods, and the specific method for setting up the wave generation and absorbing boundaries. Based on the RANS equations and VOF method, Lin & Liu (1998) developed a numerical model for breaking waves in coastal surf zone by the use of the k-epsilon turbulence model. Arikawa and Shimosako(2003) performed numerical simulation of propagation of the nonlinear wave trains by ose of a two dimensional numerical wave flume, in which the Raynolds averaged Navier Stokes equations for flows in porous media are solved and the VOF method is applied to track the free surface. The effects of overtopping flows on the structures behind the seawall are studied, while the artificial blocks are considered on the seawall. Zhou et al (2005) developed a new module of wave generation and absorbing for the numerical wave flume based on the commercial CFD tool FLUENT. The relaxation approach of wave generation and absorbing can be referred to Wang & Liu(2005) for details. Because it is hard to implement the numerical algorithm for describing exact geometric profiles of artificial blocks in the numerical wave flume, development and validation of an efficient approach for considering effects of dolosse on wave runnup and overtopping against seawalls or breakwaters need to be studied and which motivates the present research into the practical application of the numerical wave flumes. Development and application of the numerical wave flume for overtopping against seawalls are presented in the paper.
2 Mathematical Formulations Considering flows of an incompressible fluid with the free surface, the governing equations are the continuity equation and the Reynolds averaged Navier-Stokes equations as follows： ∂u i =0 ∂xi ∂ρ ui ∂ρ ui u j ∂p ∂ + =− + ∂t ∂x j ∂xi ∂x j
(1) ⎡ ⎛ ∂u ∂u j 2 ∂u ⎞ ⎤ ∂ − δ ij l ⎟ ⎥ + − ρ ui′u ′ j + ρ fi + Fi ⎢ μ ⎜⎜ i + ⎢⎣ ⎝ ∂x j ∂xi 3 ∂xl ⎟⎠ ⎥⎦ ∂x j
(
)
(2)
in which t is time and xi (i = 1,2) denotes coordinates in two dimensional case, u i (i = 1,2) the velocity components, ρ the water density, μ the dynamic viscosity,
f i the volume force of the fluid, Fi is the additional source terms. The standard kepsilon model is adopted to compute the Reynolds stress in equation (2).The free surface is computed by the VOF method. The porous media model is used to simulate energy dissipation of flows in the layer of artificial units on seawall slope. It is proposed that the porous media model can been implemented by define a special source term as following：
Fi =
1 v C D ui u 2
(3)
v in which， Fi is the source term in equation (3), C D is the drag coefficient and u the velocity vector. It is a key issue to implement wave generation and wave absorbing in the specified boundaries. In the paper, the relaxation approach is developed to derive the additional terms in equation (2). The relaxation algorithm for velocity and pressure within the wave generation zone and the sponge layer zone can be written as,
ui M = CuiC + (1 − C )ui I PM = CPC + (1 − C ) PI
(4) (5)
in which the subscript C stands for the computed value, the subscript I denotes the incoming wave, C = C (x) is the relaxation function。The detailed procedure of deriving the additional source term based on Euler equations can be referred to Zhou et al (2005). The additional source term is implemented by the UDF of Fluent. The computational domain of the numerical wave flume and overtopping against seawalls are shown in Fig. 1.
wavemaker
y
Symmetry
Symmetry
Symmetry
etr mm y S
Symmetry
sponger e mm y S
try
Symmetry
Fig. 1 Sketch of numerical wave flume with seawall
3 Validation of Numerical Wave Flume A validation computation of a regular wave train was carried out in numerical wave flume of uniform water depth. The sponge layer is defined on the right end of the computational domain. The snapshots of the free surface are presented in Fig. 2. It is confirmed that the relaxation algorithm works well for both wave generation zone and sponge layer.
η(m)
x(m) Fig. 2 Wave surface elevation for progressive waves in different time (wave height H=0.132m，water depth d=0.45m，wave period T=1.357s)
η(m)
x(m) Fig. 3 Wave surface elevation for standing waves in different time (wave height H=0.132m, water depth d=0.45m, wave period T=1.357s)
In order to check the efficiency of the relaxation algorithm for dissipation of the reflected wave caused by a structure in the numerical flume, a sponge layer is defined in the front of the wave generation zone while the vertical wall is specified at the right end of the flume. Snapshots of the free surface of a standing wave are presented in Fig. 3. The numerical results demonstrate that the reflection wave is dissipated well within the sponge layer near the wave maker.
4 Model Calibrations In order to determine the drag coefficient of the porous media, the physical model experiment was carried out in the wave flume at Hydrodynamics Laboratory at Shanghai Jiao Tong University. The length of the wave flume is 60m and the width is 0.8m. There is a hydro-servo irregular wave maker in the wave flume. The surface elevation is measured by the wave gauges. The water of overtopping is collected by a specially designed tank behind the seawall. To generate regular waves of large wave amplitude, a transient bottom of gentle slope is set up between the wave maker and the model, as shown in Fig. 4. Five cases are designed for the experiments in terms of different location of the artificial units. The measured data of the rate of overtopping are compared. The porous media layer defined numerical wave flume covers the domain occupied by the artificial units along the slope of seawalls. Given the drag coefficient of the porous media, the rate of overtopping against the seawall with artificial units can be obtained by the numerical wave flume. Four patterns of the layout of the artificial units on the seawall are shown in Fig. 5. As the drag coefficient is predetermined, the overtopping can be computed numerically, as shown in Fig. 6, in which the symbols are the computed values of overtopping and the solid lines are the fitting curves of experimental Fig. 4 Profile and setup of artificial units data obtained by the least square method.
Fig. 5 The sketch of the porous media zone
Fig. 6 The curve of the overtopping flux versus the drag coefficient
7 Conclusions Based on the porous media model, a numerical model is developed to simulate the wave overtopping against seawalls armoured artificial units. Comparing the numerical experiments and the corresponding physical experiment in a wave flume, the effective drag coefficient of the porous media model was calibrated. Validations and applications of the numerical wave flume will be presented in the conference.
8 Acknowledgements The author would like to appreciate Prof. YS He, Prof. TQ Hu, Prof. LP Xue for their support in conducting the research and Mr. QJ Zhou, Mr. JS Zhang and Dr. BL Wang for carrying out the numerical computations and physical model experiments. The financial supports from the NSFC（10572093 and 10172058） and the Doctorial Program Foundation of MOE of China (2000024817) are appreciated.
9 References Arikawa T. and KI Shimosako. Numerical Simulations of Hydraulic Overflow Pressure Acting on Structures Behind the Seawall. Coastal Structure, 147(50), 606-618, 2003. Lin, P. and P. L.-F. Liu. A Numerical Study of Breaking Waves in the Surf Zone. Journal of Fluid Mechanics, 359: 239-264, 1998. Roger JM., et al. Flow Through Porous Media, Academic Press New York and London, 91-108, 1969. Wang BL. and H. Liu. Higher Order Boussinesq-type Equations for Water Waves on Uneven Bottom. Applied Mathematics and Mechanics, 26(6):714-722, 2005. Zhou QJ, BL. Wang, YM. Lan and H. Liu. Numerical Simulation of Wave Overtopping over Seawalls. Chinese Quarterly of Mechanics, 26(4): 629-633, 2005. (in Chinese)
