Nonequilibrium fluctuations and energy amplification in plane Couette flow
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
In this presentation we elucidate the contribution of vertical velocity fluctuations to the nonequilibrium energy amplification for a simple fluid undergoing plane Couette flow. We shall show how boundary conditions may be implemented for the solution of the stochastic Orr-Sommerfeld equation. The decay rates of the nonequilibrium vertical velocity fluctuations will be evaluated numerically, by exploiting several properties of the Airy functions. From these decay rates and associated hydrodynamic modes, a simple expression for the intensity of nonequilibrium vertical velocity fluctuations has been obtained. The corresponding autocorrelation function decays for both large and small wave numbers and exhibits a maximum for some nonzero wave number $q_m$. Our results show a connection between the concepts
of nonequilibrium enhancement of fluctuations 1 (used in statistical physics) and energy amplification of disturbances 2 (used in fluid mechanics).
Interestingly, the wave-number dependence of the decay rates shows
qualitative differences above and below a certain Reynolds number
$Re_c254$, close to the number usually
reported experimentally for the appearance of hydrodynamic
instabilities. However, so far this difference in the
wave-number dependence of the decay rates does not appear to have consequences for the nonequilibrium amplification of disturbances. Obviously, this point requires further study.
1) B.M. Law, R.W. Gammon, J.V. Sengers, Phys. Rev. Lett. 60
(1988) 1554.\\
2) B. Bamieh, M. Dahleh, Phys. Fluids 13 (2001) 3258.