Nonmodal energy growth and optimal perturbations in compressible plane Couette flow

Abstract

Nonmodal transient growth studies and an estimation of optimal perturbations have been made for the compressible plane Couette flow with three-dimensional disturbances. The steady mean flow is characterized by a nonuniform shear rate and a varying temperature across the wall-normal direction for an appropriate perfect gas model. The maximum amplification of perturbation energy over time, G max , is found to increase with increasing Reynolds number Re, but decreases with increasing Mach number M. More specifically, the optimal energy amplification G opt ͑the supremum of G max over both the streamwise and spanwise wave numbers͒ is maximum in the incompressible limit and decreases monotonically as M increases. The corresponding optimal streamwise wave number, ␣ opt , is nonzero at M = 0, increases with increasing M, reaching a maximum for some value of M and then decreases, eventually becoming zero at high Mach numbers. While the pure streamwise vortices are the optimal patterns at high Mach numbers ͑in contrast to incompressible Couette flow͒, the modulated streamwise vortices are the optimal patterns for low-to-moderate values of the Mach number. Unlike in incompressible shear flows, the streamwise-independent modes in the present flow do not follow the scaling law G͑t /Re͒ϳRe 2 , the reasons for which are shown to be tied to the dominance of some terms ͑related to density and temperature fluctuations͒ in the linear stability operator. Based on a detailed nonmodal energy analysis, we show that the transient energy growth occurs due to the transfer of energy from the mean flow to perturbations via an inviscid algebraic instability. The decrease of transient growth with an increasing Mach number is also shown to be tied to the decrease in the energy transferred from the mean flow ͑Ė 1 ͒ in the same limit. The sharp decay of the viscous eigenfunctions with increasing Mach number is responsible for the decrease of Ė 1 for the present mean flow.

Nonlinear stability, bifurcation and vortical patterns in three-dimensional granular plane Couette flow

“…Another avenue in the search for streamwise vortices in granular shear flow is to examine the optimal perturbations (that are responsible for maximum amplification of perturbation energy) using non-modal stability analyses (Malik et al 2006(Malik et al , 2008. The present work should relations of Lun et al (1984):…”

“…Although the incompressible plane Couette flow (Drazin & Reid 1981) is known to be linearly stable at all Reynolds numbers (but there are finite-amplitude solutions that bifurcate from Re = ∞), its compressible counterpart (i.e. the plane Couette flow of a gas of elastic hard spheres) supports acoustic instabilities at supersonic Mach numbers (Malik, Alam & Dey 2006;Malik, Dey & Alam 2008), provided the Reynolds number is very large (Re > 10 5 ). The compressibility of the fluid is primarily responsible for these instabilities, and the underlying family of instability modes are called Mack modes (Mack 1984).…”

On the longitudinal optimal perturbations to inviscid plane shear flow: formal solution and asymptotic approximation

“…Henningson (1998) for boundary layers and by Malik, Alam & Dey (2006) for nonisothermal plane Couette flow. Still, to the authors' knowledge, the problem of the optimal perturbations in the case of inviscid incompressible flow with an arbitrary profile has not been further analysed in the literature.…”

Transient dynamics of perturbations in astrophysical disks

“…The matrix method to search for optimal perturbations has been used in many studies on stability of laboratory flows (see, for example, [45], [47], [116], [53], [117], [54]) and in astrophysical papers [51], [52], [118]. Here, we elucidate it by a simple semi-analytical study [113], where the eigenvector basis 8 is calculated in the WKB approximation in a geometrically thin and barotropic quasi-Keplerian torus with free boundaries.…”