The effect of compressibility on the stability of wall-bounded Kolmogorov flow

Abstract

We extend the stability analysis of incompressible Kolmogorov flow, induced by a spatially periodic external force in an unbounded domain, to a compressible hardsphere gas confined between two parallel isothermal walls. The two-dimensional problem is studied by means of temporal stability analysis of a 'slip flow' continuumlimit model and the direct simulation Monte Carlo (DSMC) method. The neutral curve is obtained in terms of the Reynolds (Re) and Knudsen (Kn) numbers, for a given non-dimensional wavenumber (2πn) of the external force. In the incompressible limit (Kn, Kn Re → 0), the problem is governed only by the Reynolds number, and our neutral curve coincides with the critical Reynolds number (Re cr ) calculated in previous incompressible analyses. Fluid compressibility (Kn, Kn Re = 0) affects the flow field through the generation of viscous dissipation, coupling flow shear rates with irreversible heat production, and resulting in elevated bulk-fluid temperatures. This mechanism has a stabilizing effect on the system, thus increasing Re cr (compared to its incompressible value) with increasing Kn. When compressibility effects become strong enough, transition to instability changes type from 'exchange of stabilities' to 'overstability', and perturbations are dominated by fluctuations in the thermodynamic fields. Most remarkably, compressibility confines the instability to small (O(10 −3 )) Knudsen numbers, above which the Kolmogorov flow is stable for all Re. Good agreement is found between 'slip flow' and DSMC analyses, suggesting the former as a useful alternative in studying the effects of various parameters on the onset of instability, particularly in the context of small Knudsen numbers considered.

Nonlinear thermal effects in unsteady shear flows of a rarefied gas

“…Being a consequence of the form of solution in Eq. (39), this result will be further validated and discussed in Sec. VI.…”

“…In view of previous analyses of the stability of rarefied gas flows in other canonical configurations (e.g., Refs. [37][38][39]), it appears reasonable to expect that gas rarefaction stabilizes the system state, thus confining instability phenomena (and associated two-dimensional effects) to the limit of small Knudsen numbers. The 2ω-oscillatory amplitude functions of the O(Ma 2 ) vertical velocity v (2) (y) and density ρ (2)…”

Spatiotemporal dynamics in two-dimensional Kolmogorov flow over large domains

“…Further physics such as rotation, bottom friction and stratification (Lorenz (1972); Kazantsev (1998); Manfroi & Young (1999); Balmforth & Young (2002; Tsang & Young (2008, 2009)) can also be added to examine most commonly their effect on the inverse energy cascade in geophysical fluid dynamics. Recently compressible (Manela & Zhang (2012); Fortova (2013)), viscoelastic (Boffetta et al (2005); Berti & Boffetta (2010)) and even granular (Roeller et al (2009)) versions of Kolmogorov flow have been treated.…”

Effect of heat-flux boundary conditions on the Rayleigh-Bénard instability in a rarefied gas

“…The present work complements previous studies on the Rayleigh-Bénard instability in rarefied gases, all dedicated to the problem at isothermal walls conditions. Having demonstrated the impact of a change in the thermal boundary conditions on the system properties, it may be of interest to consider the effect of heat-flux conditions in other canonical stability problems, including the Taylor-Couette [26] or Kolmogorov [27] setups. As the dynamic and thermodynamic fluid descriptions are inevitably coupled at noncontinuum states, thermal conditions modifications may prove effective in monitoring these and other instability phenomena in rarefied gases.…”