1. Application of Computer Algebra in Fluid Dynamics

Stability problems in Mechanics and Fluid Dynamics lead necessarily to the investigation of eigenvalue problems for nonlinear partial differential equations. By the standard procedure of the treatment of that kind of problems the linear stability analysis ends up with a system of homogeneous ordinary differential equations with homogeneous boundary conditions at the two end point of the interval. One way to get the lowest eigenvalue is an analytical approach of the eigenvalue problem introducing power series expansions at both the boundaries of the interval of the independent variable. The well-known Lanczos method for approximating the lowest eigenvalue gives a powerful means to determine the lowest real eigenvalue with high accuracy. As an example the Orr-Sommerfeld equation is to be treated to get an approximation of the lowest eigenvalue for the plane Poiseuille flow and for the linear stability analysis of flow over a flat plate (boundary layer flow).

(U-c)(F´´- a^2 F)-F U´´ = -i/(aRe)(F´´´´- 2a^2 F´´ + a^4 F)

Another important field of application of the computer algebra is the asymptotic expansion of solutions for unstationary supersonic flows around slender bodies under acceleration or deceleration. The attached shock wave has a curvature which is in the first approximation linearly related to the curvature of the contour of the body in the nose point of the profile and the instantaneous acceleration of the body. Higher order approximations are necessary to estimate the validity of the first term in the asymptotic expansion. Numerical calculations are carried out for this problem for the flow around plane and axisymmetric bodies on the basis of the Euler equations and the compressible Navier-Stokes equations.

Special interest is payed to the flow behind a plane shock wave approaching a ramp, where a Mach stem is established, and a vortical motion behind the reflected shock evolves. A fine resolution of the non uniform grid is introduced to analyse the flow field in the corner point when the shock has passed this region. For the reason of comparison an analytical approach of the flow behind the shock wave is carried out using the computer algebra for an asymptotic expansion of the solution in the vicinity of the corner point.

For the basic equations of fluid dynamics several investigations were carried out in the past concerning their invariant character against coordinate transformations of special type, like translation in space and time, rotation around axes, and invariance against Galilean transformation. Modelling of flows in fluid dynamics on the other side has led to a great variety of systems of equations, the invariant character of which is not investigated rigorously. The aim of the computer algebraic analysis is to ensure that model equations obey the same laws of transformations as the basic systems of equations prescribe.