MODELING SEDIMENT SUSPENSION BY WAVES OVER A PLANE BED

Bai Yuchuan^{1, 2}, Zeng Qian¹ and Jiang Changbo¹

¹Institute of Sediment on River and Coast Engineering Tianjin University,

Tianjin 300072,China

Tel: 022-27401747, E-mail: Ychbai@tju.edu.cn

²The State Key Laboratory of Estuarine and Coastal Research

East China Normal University, Shanghai 200062, China

Abstract: N-S equations and TUMMAC method are used to simulate evolution of a wave train and to trace free-surface movement. With the  model to described turbulence effect for wave propagating and sediment transporting. The turbulence intensity and sediment concentration have been resolved. Some results from these simulations for the time-averaged vertical suspended sediment concentration profiles are compared with that measured in a wave flame. This simulation makes it possible to examine in detail the mechanism of sediment entrainment, vortex formation and convection. This work contributes to knowledge about wave-sediment interactions and to understanding of sediment transport mechanics.

Keywords: sediment, wave, turbulence  model

1  INTRODUCTION

The magnitude and direction of wave-induced sand transport over a bed is related strongly to the local hydrodynamic conditions and to the grain size of sediments comprising the bed. These factors in turn influence sediment grain size distribution in suspension for graded sediment beds and bed formation such as the bed ripple.

To date, many numerical studies have been carried out to clarify wave transformation on the plane bottom. Most of the numerical models were based on Boussinesq equation, which performed well in shallow water. Madsen et al^[1] firstly presented a method for extending the validity of the Boussinesq equation to deeper water. And then the use of Boussineq equation was also extended as far as surf zone by Shaffer et al^[2] , Karambas et al^[3] , and Kabilling et al^[4]. They have used some appropriate additional terms to express the momentum mixing, the researching results showed that the Bussiinesq equation can simulate wave transformation in the surf zone basically. However, the deformation process in the generation and disappearance of the vortex is caused by non-potential flow motion after the inrush of water mass, the viscous fluid forces should be considered in the numerical analysis. At present, Shibayama et al^[6] , Huang et al^[7] and Kishi et al^[8] have used two-dimensional Navier-Stokers equations and some simple turbulence models to simulate the deformation of water wave.

In the present study, a numerical simulation system based on the  model has been developed for analyzing wave transformation on the plane bottom. The marker and cell method (MAC) has been improved to calculate the deformation of free water surface, and a detailed comparison between the experiment and numerical results is discussed.

2  HYDRODYNAMIC MODEL

2.1  Navier—Stokes equations for two-dimensional flow in a vertical plane

                                                       (1)

                                           (2)

                                          (3)

where  and  components of current in a Cartesian coordinate,  is the vertical coordinate,  is time,  is the density of water and  is the gravitational acceleration, is the eddy viscosity,  which being decided by the  turbulence model.

2.2  Turbulence model

To resolve the turbulence, the  model is used, which consists of a transport equation for turbulent energy  and dissipation of the turbulent kinetic energy .

                            (4)

                               (5)

in which is j-component of the mean velocity,  is the j-component of fluctuating velocity,  is the kinematics viscosity,  is the Cartesian coordinates, and  is time. From  and , the eddy viscosity is calculated.

                                                        (6)

Following the concept of Boussineq’s eddy viscosity, the Reynolds stress term  is obtained as following:

                                                (7)

The closure coefficient are given as =1,  ,  ,  , , .

This model is actually similar to the conventional  model, but it improves the latter in two respects: (1). the model is known to perform better in flows with strong adverse pressure gradients. (witcox^[9], 1993), which is exactly the case for wave boundary. (2). It is possible to integrate through the viscous sublayer without dubious damping-functions, thus avoiding the use of wall-function( Putel and Youn^[10],1995)

Further more, the boundary condition at the bed for  and  are simple and make it possible to account for both smooth and rough bed conditions. The boundary at the bed adopted in the present work are (1) the no-slip condition for the velocities, (2). =0 condition of  and (3) the following relationship for (J. Fredsφe^[11],1999).

                                                        (8)

Here, the parameter  is related to the grain-roughness Reynolds number,  

 ,  for                                                (9)

,   for                                               (10)

in which  is Nikuradse’s equivalent roughness height (grain roughness), and  is the local friction velocity. (Patel and Yoon^[11], 1995) have shown that this relation is valid for a flat-wall for values of  of up to about 4000.

The  model has previously been used with success to calculate the flow over dune in rivers ( Yoon and Patel, 1996) and wave plus current over a ripple-covered bed((Jφrgen Fredsφ,et al ,1999).

2.3  Model for tracking the free surface

Tracking the free surface and modeling wave crawling up the slope, the modified MAC method (TUMMAC) is used. The tracking points in free surface are decided by the following formula:

                                                        (11)

The  describes the new free surface,  ( )is the velocity of a tracking point. With moving of the fluid, the distribution of the marking points will become uneven, so the new marking points should be redetermined for even distribution and the points will not regarded as the marking points in next time. As shown in Fig.1, the new marking points 、 and will be 、 and , the velocities of which are solved out by the nine-points interpolating formula obtained by Taylor expansion.

                                   (12)

In above formula, the meaning of signs can be seen in Fig.2. The formula of solving is similar.

When solving the grid pressure and the marking points moving velocity in free surface, if a velocity outside the grid is needed, a linear extrapolating formula is generally used. Ten situations can be divided for any known and unknown velocity points, seeing the Fig.3. Using ● to represent a known velocity point and ◆ to represent a unknown velocity point，as the fifth case, the unknown velocity can be obtained by formula (13).

 Fig.1  Re-determination of marking points

 Fig.2   Interpolating diagram of nine points

                                                 (13)

The extrapolating process is progressed along the sketch of the free surface, as showing in Fig.4. Firstly the velocity of the nearest layer from the free surface will be solved, then extrapolate again until the pressure of the free surface and the velocity of marking point could be satisfied. In Fig4. the numbers represent the cases with which is followed. The method for solving  is in same way.

Fig.3  Relationship between known and unknown velocities

Fig.4  The process of extrapolation for velocity

3  SEDIMENT TRANSPORT MODEL

The concentration of suspended sediment is calculated using the well-know convection-diffusion equation which states that the temporal change in suspended sediment concentration, , is due to the convective transport of suspended sediment and to diffusive processes. Thus

                                        (14)

whereby  and  are the horizontal and vertical coordinate,  is the sediment settling velocity. The exchange of the sediment per unit of time, , can be estimated by considering the difference between the rate at which sediments enter suspension, defined by a pick-up function  and the rate at which sediments settle back to the bed are defined as . Thus

                                                    (15)

in water surface, the sediment flax is equal zero, like following:

，                                                               (16)

The sediment quantity being picked up from the bed is equal to the sediment Pick-up rate , which is an outer variable to be determined by the local hydraulic conditions, especially turbulence of tidal and wave boundary layer at the location concerned (e.g. O’Connor et al,1988; Nadaoka, K ,1991). This relation can be written as follow:

，                                                                    (17)

4  NUMERICAL SOLUTION AND RESULTS

The numerical model, the equations are solved by the finite difference. The forward time difference method is used to discretize the time derivative. The convection terms are discretized by the central difference method, and further more, in order to ensure numerical stability and decrease numerical dispersion, the donor-cell method has been used. To track free surface locations through the wave propagating and breaking, the TUMMAC method and the irregular stars method are used. The Marker and Cell Method (MAC) was originally developed by Harlow et al ^[19] and then some modifications have been made to improve the numerical accuracy and ability to deal the free surface, such as SMAC Method (Viecell^[20]), SUMMAC(CHAN^[21]), TUMMAC (Miyata^[22]-[23]), the details of these improvements are given in these literatures.

The test case is employed to simulate the sediment concentration under second order Storkes wave action, the flow boundary conditions are given as following:

Fig.5  Vertical distribution of the sediment concentration

where  is the wave height, is the radian frequency,  is the wave number and d is the water depth. At the bottom, the boundary conditions at the bed are summarized:

The initial condition for the mean flow is treated as still water with no wave or no current motion, so the initial velocities are given zero, and the initial pressure fields are given as stationary water pressure.

Fig. 5 shows the vertical distribution of the sediment concentration under the wave at different cross section, the solid line represents under wave crest and the dotted line represents under wave trough. Fig. 6 shows the typical velocity field. Fig.7 shows the distribution of calculated sediment concentration in the space certain time. From the Fig.7 the sediment concentration vary with wave progress. As the wave crest passing, the sediment concentration is high, and with the increasing distance above the bottom, the sediment concentration become small. in addition, there is a slow decay in the concentration under trough.

Fig.6  The typical velocity under wave action

Fig. 7  The typical sediment concentration contour under wave action

5  CONCLUSION

A mathematical model has been developed to describe the space and time variations in the concentration of suspended sediment under action of wave. The free surface is traced by using TUMMAC method, the model reveals the real phenomena of wave propagation, and it is easy to consider the wave nonlinearity. In this paper, it is the first step using the turbulence model to study the wave boundary layers with free surface, further studies will be conducted about the dynamic of wave transforming and sediment transport in the surfer zone.

 Acknowledgements

This is research was supported by the National Natural Science Foundation of China under contracts no. 59809006 and No. 59890200, and also by the Science foundations of Tianjin Municipality under contract No.983702011 and The state key laboratory of E.stuarine & coastal Research under contract No.98001.we are also thankful for help of Prof. Zhao .Z .D.

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