High Resolution Flow Modelling for Meteorological Purposes

This paper describes methods used in creation of a high-resolution mathematical model for describing flow in complex geometry areas. In the future it should be used mainly for airflow calculations in urban areas. In the second part we present some preliminary results of test experiments. At this stage, our model describes the 2D laminar incompressible flow but extension to full 3D approach is straightforward. Selected verification examples are the lid driven cavity flow and the flow around a square cylinder at low and moderate Reynolds numbers. Introduction In this contribution, we present a newly developed model of flow within the complicated geometry areas (e.g. urban areas). Some of methods we used, like high-resolution schemes for example [Harten, 1983], are not widely used in meteorological models. For spatial discretization we used a finite volume framework. A MUSCL type of reconstruction [van Leer, 1979] is used for dependent variables together with the Runge-Kutta scheme for discretization of inviscid fluxes. For the implementation of the geometrically complicated boundary condition we used a so-called direct forcing variant [Kim et al., 2001] of the immersed boundary method (hereinafter abbreviated as IBM, see [Peskin, 1982]). At this moment the model is tested as the laminar 2D one. There are several possibilities how to implement turbulence and some of them will be mentioned later. At this moment also the impact of vertical temperature stratification has not yet been studied. The organization of this contribution is as follows. In the first part we present a description of the used numerical methods. In the second part we mention implementation of the IBM and in the third part of this contribution we describe preliminary results. Conclusion together with outlooks dealing with the future developments of the described basis of the model is a content of the final part. Numerical methods Governing equations for unsteady incompressible viscous laminar and isothermal flow in the nondimensional form write ∂~u ∂t + div (~u⊗ ~u) = −gradp+ 1 Re div grad~u+ ~ f (1) div ~u = q (2) where indexes i, j = 1, 2, 3, t is time, u stands for flow velocity vector, p is a pressure, f and q are artificial momentum and mass source terms that result from the immersed boundary method utilization (see [Kim et al., 2001], more details will by given in the subsequent sections), Re stands for the Reynolds number. All dependent and independent variables have been non-dimensioned with a standard way [Ferziger, Perić, 1997] and in the following text all variables are used in the non-dimensional form. As mentioned earlier, at this moment the model deals with the laminar flow thus no turbulence parametrization has been used. The same statement is true for vertical temperature stratification the impact of it is also not involved into the model equations and the stratification is supposed to be neutral. Time-integration method used is based on the fractional step approach [Brown et al., 2001] where a pseudo-pressure is used to correct the velocity field so that the continuity equation is satisfied at each time step. This method uses the TVD Runge-Kutta 2nd order temporal scheme [Shu, Osher, 1988] for advection terms of (1) and the Crank-Nicolson implicit scheme for viscous terms. Spatial discretization uses the staggered finite volume (FV) method [Harlow, Welch, 1965]. The whole computational domain is discretized using the non-equidistant Cartesian grid. In the staggered approach there is not a problem with the pressure gradient terms evaluation but the most problematic terms are the non-linear advection ones that can create non-physical wiggles in the areas where high gradients occur. Therefore we used their discretisetion a method, that belongs to the class of so-called high-resolution methods [Harten, WDS'07 Proceedings of Contributed Papers, Part III, 150–155, 2007. ISBN 978-80-7378-025-8 © MATFYZPRESS