Instability of High Mach Number Poiseuille (Channel) Flow: Linear Analysis and Direct Numerical Simulations

Abstract

Poiseuille flow is a prototypical wall-bounded flow in which many fundamental aspects of fluid physics can be analyzed in isolation. The objective of this research is to establish the stability characteristics of high-speed laminar Poiseuille flow by examining the growth of small perturbations and their subsequent breakdown toward turbulence. The changing nature of pressure is considered critical to the transformation from incompressible to compressible behavior. The pressure-velocity interactions are central to the present investigation.

The study employs both linear analysis and temporal direct numerical simulations (DNS) and consists of three distinct parts. The first study addresses the development and validation of the gas kinetic method (GKM) for wall-bounded high Mach number flows. It is shown that sustaining the Poiseuille flow using a body force rather than pressure-gradient is better suited for accurate numerical simulations. Effect of uniform and non-uniform grids on the simulation outcomes is examined. Grid resolution and time-step convergence studies are performed over the range of Mach numbers of interest. The next study establishes the stability characteristics at very high and very low Mach number limits. While stability at low Mach number limit is governed by the well-established Orr-Sommerfeld analysis, the pressure-released Navier-Stokes equation is shown to accurately characterize stability at the infinite Mach number limit. A semi-analytical stability evolution expression is derived. It is shown that the GKM numerical approach accurately captures the low and high Mach number solutions very precisely. The third study examines the critical effects of perturbation orientation and Mach number on linear stability, and investigates the various stages of perturbation evolution toward turbulent flow. This study can break into two parts. In the first part, an initial value linear analysis is performed to establish the self-similar scaling of pressure and velocity perturbations. The scaling then is confirmed with DNS. Based on analytical and numerical results, regions of stability and instability in the orientation space are established. Compressibility is shown to strongly stabilize streamwise perturbations. However, span-wise modes are relatively unaffected by Mach number. The multiple stages of temporal perturbation evolution are explained. The manner of Tollmien-Schlichting (TS) instability suppression due to compressibility is also described. In the second part, the progression from linear to nonlinear to preliminary stages of breakdown is examined. It is shown that nonlinear interactions between appropriate oblique perturbation mode pairs lead to span-wise and streamwise modes. The streamwise modes rapidly decay and span- wise perturbations are ultimately responsible for instability and breakdown toward turbulence. Overall, the studies performed in this research lead to fundamental advances toward understanding transition to turbulence in wall-bounded high-speed shear flows. Such an understanding is important for developing transition prediction tools and flow control strategies.