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Numerical Solution of the Navier-Stokes Equations at High Reynolds Numbers

Abstract

In this work, we are working with the unsteady Navier-Stokes equations in the stream function-vorticity formulation. In order to show that thenumerical schemes used are able to handle highReynolds numbers, we are reporting results for the well known problem of the un-regularized driven cavity. We are dealing with Reynolds numbers in the range of 7500 ≤Re ≤ 50000. The results shown here are obtained using two numerical schemes, the first one, is based on a fixed point iterative process (see [1]) applied to the elliptic nonlinear system that results after time discretization. The second scheme (see [2], [3]) which, as we are going to show in the results, is faster than the first one, solves the transport type equation appearing in the Stream function-vorticity formulation of the Navier-Stokes equations using matrixes A and B; the first one resulting from the discretization of the Laplacian term appearing in the equation, and the second one resulting from the discretization of the advective term. Both schemes, for this problem, have been robust enough to handle such high Reynolds numbers, but the second one has proved to be much faster, especially for high Reynolds numbers. In [4] it has already been said that even though turbulence is a tri-dimensional phenomenon, two-dimensional flows at high Reynolds numbers give some clues of transition to real turbulence.

Blanca Bermúdez, W. Fermín Guerrero S., A. Rangel-Huerta

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