Numerical Linear Stability Analysis for Compressible Fluids

Andreas S. Bormann ()
Additional contact information
Andreas S. Bormann: Technische Universität Berlin, Institut für Verfahrenstechnik

A chapter in Analysis and Numerics for Conservation Laws, 2005, pp 93-105 from Springer

Abstract: Summary 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.

Keywords: Rayleigh Number; Linear Stability Analysis; Boussinesq Approximation; Outer Cylinder; Compressible Fluid (search for similar items in EconPapers)
Date: 2005
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-540-27907-5_5

Ordering information: This item can be ordered from
http://www.springer.com/9783540279075

DOI: 10.1007/3-540-27907-5_5

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().