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Symbolic Manipulation and Supersonic Transition

G. Erlebacher
Journal / Anthology

Winter Annual Meeting of the American Society of Mechanical Engineers Dallas, TX, USA, 19901125-19901130, on Symbolic Computations and Their Impact on Mechanics
Year: 1990
Volume: 205
Page range: 285-299

Progress in the understanding of the physical mechanisms that underlie supersonic and hypersonic transition phenomena is greatly impeded by the nonlinearity of the Navier-Stokes equations. The convective terms in the momentum equations are cubic nonlinearities, and the temperature dependent viscosity and thermal conductivity greatly complicate the structure of the viscous stress and heat flux terms. When performing series expansions of the flow variables, these nonlinearities quickly render manual computations impractical, particularly if many such calculations become necessary. In this paper, we describe the use of the Mathematica symbolic manipulation software to efficiently compute matrix elements which arise in weak nonlinear theory. Only the nonlinear terms are treated herein. The algorithm is efficient in that intermediate terms which are usually computed and carried along are never encountered, thus speeding the algorithm by over an order of magnitude. The symbolic code handles arbitrary systems of equations with the restriction that its coefficients be polynomial in the independent variables. Fully nonlinear terms must be Taylor expanded to the appropriate order before the equations are input to the code. We also present an alternate package for derivatives and a package to generate useful Fortran code from Mathematica in a readable form.

*Engineering > Aeronautical and Astronautical Engineering