Stability of wall-bounded compressible shear flows

Abstract

In this work, the linear stability of compressible shear flows in channels and pipes are studied. A steady fully-developed laminar flow of an ideal gas, driven in a channel and a pipe by a constant body acceleration, is considered as the base state. The base flow profiles, being functions of Mach number, are obtained through numerical solution of the Navier-Stokes equations under steady parallel-flow conditions. Small amplitude normal mode perturbations are added to this flow and temporally growing solutions are studied. The evolution equations for the normal modes, that form a linear eigenvalue problem, are solved numerically by Chebyshev-Pseudospectral method.

For both channel and pipe flows, the eigenspectra show presence of compressible higher modes that do not have a counterpart in the incompressible limit. These modes are categorised into two distinct families based on the variation of the real part of their wave-speed with stream-wise wave-number. Numerical studies show the dominant instability at finite Mach numbers to be due to the modes that show a monotonic increase in the real part of the wave-speed with wave-number. These modes become unstable at finite wave-numbers at Mach numbers above a critical value. We have extended the classical stability theorems to compressible flows in bounded domains. A new criteria for the existence of neutral modes is derived which is used to obtain the values of the critical Mach numbers for the stability of the higher compressible modes.

In the incompressible limit, a pipe flow is stable to all modal perturbations, but the channel flow is unstable to the Tollmien-Schlichting (T-S) mode. Numerical studies at finite Mach numbers show compressibility to have a stabilising effect on the T-S mode. The critical Reynolds numbers as a function of Mach number are obtained for the all the unstable modes in both channel and pipe flows. A universal scaling of the critical values is shown at high Mach numbers. The critical Reynolds numbers for three-dimensional disturbances are also calculated for a compressible channel flow. It is shown that oblique waves are more unstable than two-dimensional waves with the minimum critical Reynolds number appearing at a specific wave-angle corresponding to a particular Mach number.

Numerical calculations of the stability equations are also performed in the inviscid limit where the numerical contour of integration is suitably chosen to avoid the branch point singularity at the critical point. The inviscid limit of the dominant compressible modes in channel and pipe flows compared against the high Reynolds number viscous calculations reveal the instabilities to be viscous in nature. The instability in channel and pipe flow appear due to a change in the viscous wall layer due to the emergence of a critical point very close to the wall.

The unstable modes in channel and pipe flows are studied at high Reynolds numbers through an asymptotic analysis. The instabilities in compressible channel flow are categorised into a small wave-number mode, which is the finite Mach number extension of the T-S mode, and finite wave-number modes, which are the dominant compressible higher modes. The asymptotic analysis for the lower and upper branches of the stability curve are performed to obtain the scalings for the wave-number, wave-speed, as well as the wall layer scalings for viscous regularization. An adjoint-based procedure imposing the solvability condition on the first and second correction to the stability equations, is devised to obtain the leading order eigenvalues for the lower and upper branches at high Reynolds numbers. The same asymptotic analysis is performed for the finite wave-number modes of the compressible pipe flow as well.

We also study the stability of a compressible flow in a channel with compliant walls. The compliant walls are modelled as spring-backed plates that move in the direction normal to the flow due to the fluid stresses acting at the walls. Wall compliance introduces additional instabilities, referred to as FSI modes, in addition to the Tollmien-Schlichting and compressible higher modes. The numerical studies indicate flow compressibility to have a stabilising effect on the FSI modes, and wall compliance to have a stabilising role on the compressible higher modes. Both flow compressibility and wall compliance are observed to have a stabilising effect on the Tollmien-Schlichting mode. We also calculate the perturbation energy budgets for the different instabilities which allow us to differentiate the different mechanisms of destabilisation of these modes.