A Study on Application of Discrete Vortex Method to Coastal Circulation

 

 

 

 

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Keita FURUKAWA*

 

 

 

INTRODUCTION

              To implement environmental measures in enclosed inner bay, modeling tools for macro and micro scale environmental impact assessment are required.  A discrete vortex method can be an effective method to simulate multi scale flow such as eddies behind island, headlands and narrow channel at inlet of a bay.  Nevertheless, a discrete vortex method has been developed as one of direct simulation methods for Navier-Stokes equation.  The method should be extended to facilitate 3-D effects of shallow coastal water and modeling of several hundred meters to kilo meters sized scale (meso-scale) eddy in a bay.  Firstly, the model parameter fitted for small scale (several centi-meters to meter) flow.  Secondly, the model was modified to simulate meso-scale coastal circulation.

 

FORUMURATION OF DISCRETE VORTEX METHOD

              A viscous vortices model is introduced for reduction of a calculation time and simplification of the model.  The vortices model is expressed as;

 

 


i‚Pj

 

 

where,  is distance from center of vortices,  is tangential velocity,  is total circulation of represented vortices,  is diffusion scale parameter that has same dimension of a diffusion coefficient, and  is elapsed time after the vortices released into the flow field.  Way of parameter fitting and modification of the method are presented at following section.

 

MODEL EXAMINATION BY BASIC HYDRAULIC MODEL TEST

              Discrete vortex methodfs parameters were tuned for small scale basic hydraulic model test such as a wake formulation behind an inclined flat plate (Figure 1).  These parameters are formulated by Reynolds number using far field flow velocity as velocity scale and plate size as scale length.

 

 

 

 

 

 

 

 

 

 


Figure 1: One of test case for tuning of discrete vortex method.

 

              The second case is wake generation by series of steps on a wall.  The most upstream step generate eddies that transported to downstream.  If the eddies interact with successive eddies generated from second – third steps is tested by numerical model tests using the discrete vortex method.  Boundaries can be set by simple sequence of boundary vortices put on the boundary surface.

              Velocity attenuation process due to multi objects set on flow field is studied as third case.  Simulated wakes from objects are interacted each other, and makes patchy pattern of water retardant area and accelerated jet flow area.  By detail velocity information, it is enables to determine total drag force acting on objects.

 

MODEL ALTERNATION FOR SIMULATE COASTAL CIRCULATION

              The discrete vortex model is based on 2-D Navier-Stokes equation.  Thus, only two dimensional effects are modeled.  Furthermore, parameter fitting based on Reynolds number gives too small time step to solve large scale phenomenon.  The model was altered to use in meso-scale (100m-10km) coastal circulation.

              Bottom frictional effects are separately modeled, and additional terms are added in advection terms of vortex transport equation.  The model parameters were adjusted by using diffusion scale.  The extended vortex discrete method successfully models asymmetry wake circulation behind Rattary Island, Australia (Figure 2).

              The extended method also checked to model jet flow generated by small channel between island and mouth of bay (Figure 3).  The discrete vortex method show equivalent performance to full 3-D model code.

Figure 2: Wake behind Rattray Island, Australia.  (a) observed current pattern at ebb tide, (b) corresponding current pattern modeled by original discrete vortex method, (c) corresponding current pattern modeled by extended discrete vortex method that considers bottom frictional effects.

 

 

 

 

 

 

 

 

 

Figure 3: Flow around a channel between Palm islands.  Left) observed current pattern at flood tide by VHF radar.  Right) calculated current pattern by discrete vortex method.

 

 

 

CONCLUSION

              The discrete vortex method presented by this paper shows time efficient and high accuracy performance to calculate coastal circulation such as wake, eddy street, and vortex shedding.  Tested range of Reynolds number is 1000 – 80000.

              The method is extended to simulate coastal circulation but restricted for semi-3D calculation (barotropic flow).  It is big restriction of using the method.  Nevertheless, if the user understand advantage and disadvantage of the method, the discrete vortex method can be a powerful tool to assess coastal circulation.

 

* (National Institute for Land and Infrastructure Management)