Term Project
ME 608: Computational Methods in Heat, Mass and Momentum Transfer
Spring 2006, Purdue University
COONTROL VOLUME FINITE ELEMENT METHOD FOR THE 2DIMENSIONAL
CONVECTIONDIFFUSION EQUATION
Chandrashekhar Varanasi
Graduate Student
Department of Mechanical Engineering
Purdue University
[email protected]
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
convectiondiffusion
equation
in
a
square
geometry. The governing equations have been discretized and
the resulting set of linear equations has been solved using the
GaussSeidel 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 2dimensional convection diffusion equation.
Application to the complete NavierStokes 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
bydirection upwinding in finite volume methods is a
seemingly simple solution for orthogonal meshes, but is prone
to falsediffusion. 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.
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 Fall '10
 NA
 Numerical Analysis, C. Varanasi

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