A finite-time thermodynamics of unsteady fluid flows

Turbulent fluid has often been conceptualized as a transient thermodynamic phase. Here, a finite-time thermodynamics (FTT) formalism (Andresen, Salamon & Berry 1977) is proposed to compute mean flow and fluctuation levels of unsteady incompressible flows. The proposed formalism builds upon the Galerkin model framework which simplifies a continuum 3D fluid motion into a finite-dimensional phase-space dynamics and subsequently, into a thermodynamics energy problem. The Galerkin model consists of a velocity field expansion in terms of flow configuration dependent modes and of a dynamical system describing the temporal evolution of the mode coefficients. Each mode may be considered as a wave, parameterized by a wave number and frequency. In our FTT formalism, the mode is treated as one thermodynamic degree of freedom, characterized by an energy level. The dynamical system approaches local thermal equilibrium where each mode has the same energy if it is governed only by internal (triadic) mode interactions. However, in the generic case of unsteady flows, the full system approaches only partial thermal equilibrium with unequal energy levels due to strongly mode-dependent external interactions. In these interactions, large-scale modes typically gain energy from the mean flow while small-scale modes loose energy to the heat bath. The energy flow cascade from large to small scales is thus a finite-time transition phenomenon. The FTT model is first illustrated by a traveling wave governed by a 1D Burgers equation. It is then applied to two flow benchmarks: the relatively simple laminar vortex shedding which is dominated by 2 eigenmodes, and the homogeneous shear turbulence which has been modeled with 1459 modes.