Numerical Linear Stability Analysis for Compressible Fluids

The Rayleigh-Bénard problem and the Taylor-Couette problem are two well-known stability problems that are traditionally treated with linear stability analysis. In the vast majority of these stability calculations the fluid is considered to be incompressible [Cha61, DR81]. Only with this assumption and simplification is possible to conduct a linear stability analysis analytically.

In order to calculate the stability limits of a compressible fluid by use of a linear stability analysis therefore in this work a numerical linear stability analysis is presented. The numerical stability analysis is based upon the equations of balance for mass, momentum and energy that are completed with the constitutive equations by Navier-Stokes and Fourier. The algorithm allows to calculate the regions of stability for arbitrary one-dimensional and stationary basic states.

This numerical stability analysis is used to calculate the stability region for the Rayleigh-Bénard problem. The main result is that the critical Rayleigh number does not have a constant value, as calculations involving the Boussinesq approximation suggest misleadingly, but that the value of the critical Rayleigh number depends strongly on the thickness of the fluid layer. Furthermore, an empirically found relationship between the critical Rayleigh number and the thickness of the fluid layer is presented (14). Its efficiency is successfully verified with the results of the numerical linear stability analysis. The results for the critical Rayleigh number show clearly that the compressibility of a fluid must not be neglected in the stability analysis of the Rayleigh-Bénard problem.

Secondly, the more complicated Taylor-Couette problem is treated with the numerical linear stability analysis. In contrast to the traditional stability analysis by Taylor [Tay23], the fluid is considered to be compressible and includes the temperature as a field variable. The effectiveness of the numerical linear stability analysis is manifested by the good agreement of the comparison with experimental results. In addition to that, temperature effects are studied and are compared with experiments.