An efficient 3D particle transport model for use in stratified flow

Random‐walk models are a versatile tool for modelling dispersion of both passive and active tracers in turbulent flow. The physical and mathematical foundations of stochastic Lagrangian models of turbulent diffusion have become more and more solid over the years. An important aspect of these types of models that has not received much attention is the behaviour of the particles near boundaries. Often, a simple stochastic, numerical scheme is used. Because turbulent mixing in the vertical direction is much more complicated than in the two horizontal directions, it is in the vertical direction that a simple numerical scheme, such as the Euler scheme, may cause problems. In this paper our main goal is the development of an efficient 3D particle transport model that can be used in stratified flow. For this type of situation the vertical direction is of special interest. First, a closer look is taken at some considerations that should be regarded when choosing a numerical scheme. Specifically schemes are investigated that can be used in the vertical direction, where the diffusion coefficient is varying in that direction. Experiments are performed regarding the accuracy of different numerical schemes in various situations. The behaviour of the particles near an impermeable layer interface is investigated. The stochastic Heun and Runge–Kutta schemes turn out to be very attractive for this type of model. For the simulation of the transport of various physical quantities, such as salinity, heat, silt, oxygen, or bacteria, different types of models are available. In this case we will take a closer look at the modelling of the transport of pollutants from point sources (either instantaneous or continuous transport). For this purpose a 3D particle transport model has been developed that is especially suited for stratified situations such as can be found in estuaries. The main idea is to use a simple numerical scheme for the horizontal directions and a higher‐order method for the vertical direction. The results play an important role in making specific choices for this type of particle transport model. Copyright © 2005 John Wiley & Sons, Ltd.     

Implementation and performance of the time integration of a 3D numerical transport model

The total solution of a three-dimensional model for computing the transport of salinity, pollutants, suspended material (such as sediment or mud), etc. in shallow seas involves many aspects, each of which has to be treated in an optimal way in order to cope with the tremendous computational task involved. In this paper we focus on one of these aspects, i.e. on the time integration, and discuss two numerical solution methods. The emphasis in this paper is on the performance of the methods when implemented on a vector/parallel, shared memory computer such as a Cray-type machine. The first method is an explicit time integrator and can straightforwardly be vectorized and parallelized. Although a stabilizing technique has been applied to this method, it still suffers from a severe time step restriction. The second method is partly implicit, resulting in much beter stability characteristics; however, as a consequence of the implicitness, it requires in each step the solution of a large number of tridiagonal systems. When implemented in a standard way, the recursive nature would prevent vectorization, resulting in a very long solution time. Following a suggestion of Golub and Van Loan, this part of the algorithm has been tuned for use on the Cray C98/4256. On the basis of a large-scale test problem, performance results will be presented for various implementations.     

A conservative 2D model of inundation flow with solute transport over dry bed

In this paper, a transient 2D coupled vertically averaged flow/transport model is presented. The model deals with all kind of bed geometries and guarantees global conservation and positive values of both water level and solute concentration in the transient solution. The model is based on an upwind finite volume method, using Roe's approximate Riemann solver. A specific modification of the Riemann solver is proposed to overcome the generation of negative values of depth and concentration, that can appear as a consequence of existing wetting/drying and solute advance fronts over variable bed levels, or by the generation of new ones when dry areas appear. The numerical stability constraints of the explicit model are stated incorporating the influence of the flow velocity, the bed variations and the possible appearance of dry cells. Faced to the important restriction that this new stability condition can impose on the time step size, a different strategy to allow stability using a maximum time step, and in consequence a minimum computational cost is presented. Copyright © 2006 John Wiley & Sons, Ltd.