On the linear stability of compressible plane Couette flow

Abstract

91-11267 ABSTRACTThe linear stability of compressible plane Couette flow is investigated. The correct and proper basic velocity and temperature distributions are perturbed by a small amplitude normal mode disturbance. The full small amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in sene detail. It is found that instability can occur, although the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wavespeed of the disturbances approaches the velocity of either of the walls, and these regimes are also analyzed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.

Decomposition of the temporal growth rate in linear instability of compressible gas flows

“…We would like to stress that the goal of this section is not a comprehensive study of the linear stability of these flows, since this has been carried out before (see e.g. Duck, Erlebacher & Hussaini 1994;Zhong 1998 andParras &Le Dizès 2010), but rather to exemplify the more profound physical understanding of instability mechanisms that can be gained from the proposed decomposition of the temporal growth rate ω i .…”

An asymptotically stable compact upwind-biased finite-difference scheme for hyperbolic systems

“…The base flow is parallel between an isothermal moving wall and an adiabatic wall at rest as investigated in [10]. The Mach number is M 1 = 2 and the viscosity of the base flow is determined according to SutherlandÕs law with Su/T 1 = 110.4K/220.8K for an isothermal wall temperature of T W = 1 (lengths are referred to the wall distance, time to the reference length divided by the velocity of the isothermal wall, pressure to the density times the square of the velocity at this wall and all other variables to their isothermal wall values).…”

“…Further parameters are a Prandtl number of Pr = 0.72 and a ratio of specific heats c = 1.4. In the inviscid limit the mode considered c lin = 1.143770 is neutrally stable for a = 2, see [10]. The eigensolution has been calculated with a shooting method [10] using the RungeKutta scheme as applied for time discretisation and 981 equidistant grid points.…”

“…The viscous eigensolution ðc lin = 1.135676Ài AE 8.089441 · 10 À3 ) has been determined by a single-domain Chebychev collocation method [15] using 256 Chebychev points. Table 3 Simulation of an inviscid neutral linear-stability eigensolution in plane Couette flow, c lin = 1.143770 for a = 2, M 1 = 2 [10] jc r,DNS À c r,lin j RMS c r;DNS jc i,DNS À c i,lin j RMS c i,en DNS /2 In Tables 3 and 4 the computed eigensolutions for a disturbance amplitude of A = 10 À5 (based on the maximum of the streamwise velocity) and a resolution of 50 · 50 grid points are shown. They are determined for the first time step and for the time interval from t = 0 to t = 1000 without using intermediate points of the interval.…”

Complex-grid spectral algorithms for inviscid linear instability of boundary-layer flows

“…They showed that the combination of a complex grid and the spectral method is essential for the inviscid linear analysis of their trailing vortex model problem. Duck et al [7] also found that the same approach for the instability equations of compressible plane Couette flow delivers accurate results (private communication, 2000) although no discussion of this issue using a spectral method was presented in [7].…”