mech7220-numerical - 3 Numerical Methods for Convection The...

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Unformatted text preview: 3 Numerical Methods for Convection The topic of computational fluid dynamics (CFD) could easily occupy an entire semester indeed, we have such courses in our catalog. The objective here is to examine some of the basic features of solving, via numerical means, the DEs which describe convectivediffusive transport. We will restrict the analysis to 2D, laminar, and incompressible problems. The notes will not describe, in any depth, topics that are commonly dealt with in a numerical methods course i.e., solution of ODEs and solution of linear equations and which can be implemented using the black box routines in matlab or mathematica. Rather, we will concentrate on the issues involved in casting the DEs into a numerical form. 3.1 The stream functionvorticity formulation The incompressible continuity and momentum equations appear as u = 0 (1) D u Dt =- 1 P + 2 u + f (2) in which 2 represents, in this case, the vector Laplacian : 2 u = ( u ) | {z } =0- u (3) in which the incompressible continuity equation was applied. Together, these equations represent S +1 equations, where S is the spacial dimension of the problem (i.e., 1D, 2D, etc.). Unknowns are the S components of the velocity u and the pressure P ; these are sometimes known as the primitive variables to the problem. Obtaining a solution in primitive variables is challenging, in large part due to the nature of the pressure dependence on the problem. Observe that the pressure appears only in the gradient operator, and in this sense nowhere for this incompressible model does the absolute value of the pressure affect the problem. That is, P + C , where C is an arbitrary constant, will always satisfy the incompressible momentum equations. The fact that the pressure appears in a firstorder spacial derivative also make difficult the specification of boundary conditions for the pressure: one can not arbitrarily set values of pressure on all the boundaries, as this will overconstrain the problem. Often the pressure field is not of interest in itself. For such cases the pressure can be eliminated from the problem by introduction of the vorticity vector . The vorticity is defined as the curl of the velocity; = u (4) The vorticity transport equation is obtained by taking the curl of the momentum equation. Since the curl of the gradient of any scalar is always zero, i.e., P = 0 (5) the pressure field is identically eliminated from the resulting equation. The convective part in the material derivative can be reduced by using the vector identity: ( u ) u = 1 2 ( u u )- u u = 1 2 ( u u )- u (6) and since the curl of a gradient is identically zero, the first term is eliminated when the curl of the momentum equation is taken. Using Eq. ( 3 ), the curl of 2 u will equal 2 , because the divergence of a curl is also identically zero. Therefore, the vorticity transport equation...
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This note was uploaded on 09/24/2011 for the course MECH 7220 taught by Professor Staff during the Fall '10 term at Auburn University.

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mech7220-numerical - 3 Numerical Methods for Convection The...

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