MATHEMATICAL MODEL OF SUSPENDED SEDIMENT  DIFFUSED TRANSPORT WITH INTERACTIONS OF WAVE AND TIDE CURRENT*

Zhu Zhixia^{1, 2}, Han Qiwei² and Ding Pingxing³

Ph.D.  Assistant Professor

^１Dept. Of Harbor&Coastal Engineering, Shanghai Jiao Tong University,

1954 Hua Shan Road,Shanghai 200030, China

Tel.：+862162933143，Fax：+862162933160, E-mail：zxzhu@mail.sjtu.edu.cn

²Department of Sediment Engineering, China Institute of Water Resources

and Hydropower Research, Beijing 100044, China

³State Key Laboratory of Estuarine and Coastal Research, East China

Normal University, Shanghai 200062, China

Abstract: Based on the refraction-diffraction combined numerical model of random wave in tidal current field under the condition being appropriate to a mild topography on a large scale, the shallow water current equations under the effect of waves and the suspended sediment diffusion equation with interactions of wave and tidal current, the 2-D sediment mathematical model with interactions of wave and tidal current has been set up. The paper utilizes this model in the northwestern sea zone of the Bohai gulf to carry out the numerical simulations of the dredged materials’ suspended sediment diffused transport. The results are satisfactory, which have had enormous reliable scientific bases for engineering organization to make a decision.

Keywords: refraction-diffraction combined, suspended sediment diffusion, interactions of wave and tidal current, mathematical model

1  INTRODUCTION

In estuary and seacoast area, wave and tidal current are existing at the same time and interacting. They both are the most important dynamic factors of the sediment motion. Hanson brought forward sediment mathematical model for the first time in 1956, but it hadn't been developed until 1970s because it restricted by the condition of calculation. Since 1990s, the study of sediment mathematical model in estuary and seacoast has been entered a new period and have obtained rich study achievements[1],[2],[3]. But because of the actions of the primary dynamic factors—wave and tidal current are very complex, there aren’t practical theoretic models for interactions of wave and tidal current. So it is the popular and key problems for coastal scientists and research workers for all countries.

In this paper, according to the characteristics of the engineering sediment problem and utilizing both the wave and tidal current coupling mathematical model[4] in a large area of mild topography and the diffusion equation of suspended sediment with interactions of wave and tidal current, we have established the 2-D sediment mathematical model with interactions of wave and tidal current. This model has been successfully applied to engineering practice.

2  BASIC EQUATIONS

2.1  2-D shallow water current equations under the function of waves

                                                                       (1)

                                            (2)

                                            (3)

Where z = water level, the distance between free surface and base level; h = water depth, the distance between free surface and seabed, h=z+z₀; z₀ = seabed surface elevation, the distance between base level and seabed surface; u, v are the horizontal velocity components in the x and y directions, respectively; f = Coriolis parameter; g = gravity acceleration, g＝9.81m/s² ; ρ= water density ;  are components of wind stress  in the x and y directions, respectively;  are components of bottom shear stress  with interactions of wave and tidal current in the x and y directions, respectively; = horizontal eddy viscosity coefficient; ； ； ； is radiation stress.

2.2  Basic equations of wave field computation under the function of tidal current

                                                           (4)

                               (5)

                                                                 (6)

Where  = the angle of wave direction and x axis; = the angle of tidal current direction and x axis; 、 、 are wave energy , wave number and wave frequency of wave component;  = multiplication of wave celerity and wave group celerity of wave component, That is  ;   = wave number of wave component; U = tidal current velocity;  = operator of horizontal gradient.

2.3  Suspended sediment diffusion equation

                                     (7)

Where P and D are sediment yield of suspending and settling on the seabed surface respectively, which relate with sediment concentration in the condition of sediment carrying saturation, suspending sediment concentration C and sediment fall velocity W_s.

Because sediment transport produced by tidal current velocity is larger than natural diffusion of sediment, sediment yield of suspending can be ignored and equation (7) may be simplified as following:

                                                 (8)

Equation(8) is the 2-D suspended sediment diffusion equation with interactions of wave and tidal current because of sedimentation[5].

3  MATHEMATICAL MODLE

3.1  Confirmation of basal parameter

In the paper, we select improved Wen’s theory spectrum[6] as target spectrum, which not only reflect developed and fully developed spectra of deep and shallow water waves, but also the developed spectrum form is consistent with Jonswap’s spectrum and the fully developed spectrum is consistent with P-M spectrum. There are extensive representations. Improved forms of parameters of Wen’s theory spectrum are:

                                                           (9a)

                                                            (9b)

Where, peak factor of spectrum P is defined as following:

                                                      (10)

Where, the relations of peak frequency ω₀, zero order moment m₀ and significant wave factors are as following:

                                                      (11)

Depth factor  and the function of depth factor m are defined as following:

                                                    (12)

Where,  can be changed from 0 to 0.5. When , equation(9) represents energy spectrum of deepwater wave; when , equation(9) represents wave energy spectrum when waves begin to break.

Bottom shear stress  with the interaction of wave and tidal current can be defined as the following[7]:

                                   (13)

Where  （u_w，v_w） = bottom orbital horizontal velocities under wave trough; C_s = Chezy coefficient, ; n = Manning’s roughness coefficient; B = affective coefficient of the interactions of wave and tidal current, where B=0.359；f_w = the bottom friction coefficient[7]. According to the document [8], we can obtain fall velocity of sediment W_s :

                                         （14）

Where  are coefficients relating with sediment concentration. We select ； are coefficients relating with tidal current velocity. We select ； = influence coefficient of salinity. When salinity equal to 30‰, ； = sediment fall velocity in still water. We can calculate the fall velocity of single sediment particle according to Stokes formula.

3.2  Method of calculation

There are two steps for the wave and tidal current coupled calculation. First of all, select a different method of calculation, according to its feature. In condition of mild topography on a large scale, the refraction-diffraction combined numerical model of random wave in unsteady and uneven tidal current field is solved by method of finite difference with iterated, while the 2-D tidal current numerical model under the wave function is solved by the joint method of both locality finite element and finite difference[9]. Then, reached by iterating calculation. The detailed iterating process is that: (1) The numerical calculation of 2-D wave field without tidal current; (2) The numerical calculation of 2-D tidal current under the wave function by wave elements gotten from first step; (3) Substitute the solution of the numerical calculation of second step to numerical model of wave field, and calculation wave field again; (4) Then, we proceed with numerical model of 2-D tidal current under the wave function by wave constituents. We can get satisfactory result of wave and tidal current field after several iterating calculations.

The suspended sediment diffused transport with interactions of wave and tidal current are calculated as following. First, we can calculate wave and tidal current field. Then we adopt the joint method of both locality finite element and finite difference to calculate suspended sediment diffused transport.

 4  EXAMPLE OF THE APPLICATION[9]

Combining with the engineering problem concerning the disposal of dredged silt of near sea zone of Tianjin New Harbor, we apply this model to carry out the suspended sediment diffused transport under the functions of the different silt settling incipient time and the different silt settling way. The calculation domain locates at northwestern sea zone of the Bohai gulf. The distance between south and north is 38km and there is 54km from east to west.

According to the construction method of dredged silt settling of outside channel of Tianjin New Harbor, we proceed with numerical calculation of suspended sediment diffused transport after dredged silt settling during construction. We carried out the numerical calculations of suspended sediment diffused transport of both one boat dredged silt (silt settling quantity in one time is 3000m³, which need two minutes) under the function of the different silt settling incipient time and dredged silt settling of construction course (silt settling quantity in one time is 3000m³, which need two minutes. A boat of dredged silt was settled every 2 hours. The dredged silt was settled five days continually and then stop two days). Where, figure 1 represents the suspended sediment diffused transport in incipient time between high and low tidal stand after settling a boat of dredged silt from 1st hour to sixth

Fig. 1  Suspended sediment diffused transport

hour. In the figure, the sediment concentration in the contour line of the most outer side is 5×10^–5 kg/m³ (Approach to zero). The increment of sediment concentration in every contour is 0.002 kg/m³ from outside to inside. The highest sediment concentration is S_max.

The above results are in accordance with the conclusion† and the rules of the suspended sediment diffused transport after the construction of dredged silt settling of outside channel of Tianjin New Harbor. So they can be used for project predication.

5  CONCLUSION

Wave and tidal current are main dynamic factors leading to the evolution of the coastal zone. Numerical simulation is the powerful tool for studying the suspended sediment diffused transport with interactions of wave and tidal current. In this paper, using the suspended sediment diffusion equation and the wave and tide current coupling mathematical model under the condition being appropriate to a mild topography on a large scale, the 2-D mathematical model of suspended sediment is established. In the modal, the bottom shearing stress with interactions of wave and tidal current, radiation stress, eddy viscosity and Coriolis force’s influence are considered. The coupling computation of wave and tidal current is fulfilled. The result is in accordance with reality on the basis of the technology of dealing with moving water boundary. Combining with the engineering problem concerning the disposal of dredged silt of near sea zone of Tianjin New Harbor, we apply this model to carry out the suspended sediment diffused transport under the functions of the different silt settling incipient time and the different silt settling way. This model supplies scientific criterion for engineering organization to make a decision, so it has a great applied prospect.

References

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