Linear stability analysis of compressible pipe flow

The linear stability of a compressible flow in a pipe is examined using a modal analysis. A steady fully developed flow of a calorifically perfect gas, driven by a constant body acceleration, in a pipe of circular cross section is perturbed by small-amplitude normal modes and the temporal stability of the system is studied. In contrast to the incompressible pipe flow that is linearly stable for all modal perturbations, the compressible flow is unstable at finite Mach numbers due to modes that do not have a counterpart in the incompressible limit. We obtain these higher modes for a pipe flow through numerical solution of the stability equations. The higher modes are distinguished into an odd and an even family based on the variation of their wave-speeds with wave-number. The classical theorems of stability are extended to cylindrical coordinates and are used to obtain the critical Mach numbers below which the higher modes are always stable. The critical Reynolds number is calculated as a function of Mach number for the even family of modes, which are the least stable at finite Mach numbers. The numerical solution of the stability equations in the high Reynolds number limit demonstrates that viscosity is essential for destabilizing the even family of modes. An asymptotic analysis is carried out at high Reynolds numbers to obtain the scalings, and solutions for the eigenvalues in the high Reynolds number limit for the lower and upper branches of the stability curve.

Automated summary

The linear stability of compressible flow in a pipe is studied using modal analysis. A steady fully developed flow of a perfect gas in a circular pipe is perturbed by small-amplitude normal modes, and the temporal stability of the system is examined. The compressible flow is found to be unstable at finite Mach numbers due to modes that do not exist in the incompressible limit. The stability equations are numerically solved to obtain the higher modes of pipe flow, and the critical Mach numbers and Reynolds numbers are calculated to determine the stability boundaries.