Study of the instability of the Poiseuille flow using a thermodynamic formalism

Abstract

The stability of the plane Poiseuille flow is analyzed using a thermodynamic formalism by considering the deterministic NavierStokes equation with Gaussian random initial data. A unique critical Reynolds number, Re c ≈ 2,332, at which the probability of observing puffs in the solution changes from 0 to 1, is numerically demonstrated to exist in the thermodynamic limit and is found to be independent of the noise amplitude. Using the puff density as the macrostate variable, the free energy of such a system is computed and analyzed. The puff density approaches zero as the critical Reynolds number is approached from above, signaling a continuous transition despite the fact that the bifurcation is subcritical for a finite-sized system. An action function is found for the probability of observing puffs in a small subregion of the flow, and this action function depends only on the Reynolds number. The strategy used here should be applicable to a wide range of other problems exhibiting subcritical instabilities.Poiseuille flow | subcritical transition | phase transition | statistical mechanics | free energy T he instability of shear flows, of which the Poiseuille flow is a canonical example, is among the most classical and most challenging problems in fluid mechanics, and a huge amount of effort has been devoted to it (1-13). The most definitive advance has been the recent experimental work by Avila et al. (9): By measuring the puff decaying and splitting times, they obtained an estimate for the critical Reynolds number, at around 2,040, for the 3D pipe flow. On the theoretical side, although the normal mode analysis in the linear stability theory of Poiseuille flow is regarded as being among the most important chapters of theoretical fluid dynamics and is associated with the names of OrrSommerfeld, Heisenberg, C. C. Lin, etc., it is generally accepted that normal mode analysis is insufficient in describing the instability of Poiseuille flow. If anything, Poiseuille flow should be regarded as an example of subcritical instability, perhaps the most well-known one.Currently, we still lack theoretical tools for tackling such instabilities. One interesting proposal has been to draw an analogy with phase transitions (13-18): The bifurcation diagram for subcritical instabilities resembles the phase diagram for firstorder phase transitions. However, to develop some quantitative tools out of this analogy, one also faces some serious issues. The first is that shear flow is well known to be a nonconservative and nonequilibrium system. More importantly, what we are interested in is the relative stability, i.e., the basins of attraction for laminar and turbulent states. This is opposite to the ergodic hypothesis that is the cornerstone of statistical mechanics. In fact, at least for finite-size systems undergoing subcritical instabilities, the complication arises precisely because the system is not ergodic. This casts serious doubt on the analogy with phase transitions in statistical mechanics.While philosophical issues exist, we p...

Linking dissipation, anisotropy, and intermittency in rotating stratified turbulence at the threshold of linear shear instabilities

“…Temporal dynamics should also be analyzed in terms of life-time of over-turning structures, as performed classically for example for pipe flows (see, e.g. 36,37 and references therein).…”

Nonlinear Nonmodal Stability Theory

“…This is sufficient to gauge the nonlinear stability of the reference state but, in practice, flows can be exposed to multiple, if not a continuous stream, of disturbances and how these interact to push the system to another attractor is more important. The optimization approach can be straightforwardly extended to consider multiple disturbances distributed across an interval and there is an interesting connection to be made with the continuously disturbed or 'noisy' situation (Freidlin & Wentzell 1998, Waugh & Juniper 2011, Wang et al 2015, Lecoanet & Kerswell 2017.…”

Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems

“…It is widely believed that turbulence [12][13][14][15] is chaotic. So, statistics is commonly used in the study of turbulence, and the direct numerical simulation (DNS) [16] , which solves the Navier-Stokes equations without averaging or approximation but with all essential scales of motion, has played an important role in turbulence statistics.…”