Four problems of fluid flow and heat transfer were designed in which non-orthogonal, boundary-fitted grids were to be used. These are proposed to serve as test cases for testing new solution methods. This paper presents solutions of the test problems obtained by using a multigrid finite volume method with grids of up to 320 x 320 control volumes. Starting from zero fields, iterations were performed until the sum of the absolute residuals had fallen seven orders of magnitude, thus ensuring that the variable values did not change to six most significant digits. By comparing the solutions for successive grids at moderate Reynolds and Rayleigh numbers, the discretization errors were estimated to be lower than 0 1 YO. The results presented in this paper may thus serve for comparison purposes as bench-mark solutions.* This kind of error analysis was inspired by Ferziger.I3
SUMMARYIn the numerical solutions of fluid flow problems in moving co-ordinates, an additional conservation equation, namely the space conservation law, has to be solved simultaneously with the mass, momentum and energy conservation equations. In this paper a method of incorporating the space conservation law into a finite volume procedure is proposed and applied to a number of test cases. The results show that the method is efficient and produces accurate results for all grid velocities and time steps for which temporal accuracy suffices. It is also demonstrated, by analysis and test calculations, that not satisfying the space conservation law in a numerical solution procedure introduces errors in the form of artificial mass sources. These errors can be made negligible only by choosing a sufficiently small time step, which sometimes may be smaller than required by the temporal discretization accuracy.
SUMMARYAn existing two-dimensional method for the prediction of steady-state incompressible flows in complex geometry is extended to treat also compressible flows at all speeds. The primary variables are the Cartesian velocity components, pressure and temperature. Density is linked to pressure via an equation of state. The influence of pressure on density in the case of compressible flows is implicitly incorporated into the extended SIMPLE algorithm, which in the limit of incompressible flow reduces to its well-known form. Special attention is paid to the numerical treatment of boundary conditions. The method is verified on a number of test cases (inviscid and viscous flows), and both the results and convergence properties compare favourably with other numerical results available in the literature.
SUMMARYIn this paper a method is presented that can be used for both the Lagrangian and the Eulerian solution of the Navier-Stokes equations in a domain of arbitrary shape, bounded by boundaries which move in any prescribed time-varying fashion. The method uses the integral form of the governing equations for an arbitrary moving control volume, with pressure and Cartesian velocity components as dependent variables. Care is taken to also satisfy the space conservation law, which ensures a fully conservative computational procedure. Fully implicit temporal differencing makes the method stable for any time step.A detailed description is provided for the discretization in two dimensions, with a collocated arrangement of variables. Central differences are used to evaluate both the convection and diffusion fluxes. The well known SIMPLE algorithm is employed for pressure-velocity coupling. The resulting algebraic equation systems are solved iteratively in a sequential manner. Results are presented for a flow in a channel with a moving indentation; they show favourable agreement with experimental observations.
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