cvfem - Term Project ME 608: Computational Methods in Heat,...

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Term Project ME 608: Computational Methods in Heat, Mass and Momentum Transfer Spring 2006, Purdue University COONTROL VOLUME FINITE ELEMENT METHOD FOR THE 2-DIMENSIONAL CONVECTION-DIFFUSION EQUATION Chandrashekhar Varanasi Graduate Student Department of Mechanical Engineering Purdue University vvc@purdue.edu ABSTRACT The class of numerical methods known as control volume finite element method (CVFEM) has been studied in this work. The method has been applied to the solution of the 2- dimensional convection-diffusion equation in a square geometry. The governing equations have been discretized and the resulting set of linear equations has been solved using the Gauss-Seidel method. The method has been found to yield satisfactory results for stringent test cases. However, care needs to be taken in formulating and implementing the method, as it is particularly sensitive to geometric parameters. Also, extending the method to 3 dimensions is a fairly complicated task. INTRODUCTION The CVFEM technique was introduced by Baliga and Patankar in 1980 [1]. The method was initially applied to the solution of the 2-dimensional convection diffusion equation. Application to the complete Navier-Stokes equations followed ([2] and [3]). Finite element methods use meshes that are capable of meshing complicated geometries. This is an attractive feature for any numerical method. Finite volume methods, on the other hand, are based on the physically appealing idea of flux balance over a control volume, and are hence perfect for the formulation of discrete equations for transport processes, since the governing equations are a statement of conservation. Using a scheme that emulates this idea is certainly a promising candidate for any numerical method used for fluid flow and heat transfer phenomena. The CVFEM seeks to combine these two aspects of finite volume and finite element methods. FEM methods typically use linear interpolation functions over elements, which lead to spurious oscillations in the numerical solution for high Peclet number flows. Thus, a suitable interpolation over the element needs to be developed that mimics the idea of upwinding. Direction- by-direction upwinding in finite volume methods is a seemingly simple solution for orthogonal meshes, but is prone to false-diffusion. An exponential interpolation function has been used in this work to interpolate for convective terms. This leads to a formulation that closely resembles “stream- wise upwinding”, as the direction of the flow field dictates the value of the variable, and not the sign of the face velocity. The CVFEM thus aims at developing a method that is able to handle complex geometries while basing the formulation of discrete equations on ideas of flux balance, using exponential interpolation functions to impose ideas of upwinding. NOMENCLATURE
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This note was uploaded on 12/29/2011 for the course ME 608 taught by Professor Na during the Fall '10 term at Purdue University-West Lafayette.

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cvfem - Term Project ME 608: Computational Methods in Heat,...

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