Thermodynamically consistent large-eddy simulation models

Abstract

Filtered budgets for anelastic turbulence and a general expression of the turbulent sensible heat flux are derived for a multicomponent fluid with an arbitrary equation of state. A family of subgrid-scale closures is then found under the constraint of consistency with (i) the first and second laws of thermodynamics and (ii) invariance with respect to irrelevant thermodynamic constants. A similar family of fully compressible models is also constructed heuristically. These models predict turbulent kinetic energy, assume down-gradient closures for three-dimensional turbulent fluxes and impose certain relationships between the closures for the turbulent fluxes of heat, matter, entropy, and the work of buoyancy forces. A key finding is the explicit derivation of the local rate of entropy production in the filtered model. Positive entropy production is guaranteed whenever the turbulent diffusions of heat and composition are positive and no cross-diffusion occurs. Cross-diffusivities are admissible provided their magnitude is within the bounds of an explicit criterion. The filtered model is invariant under a wider class of transformations than the unfiltered model. Furthermore, in the special case of a single turbulent diffusivity, an arbitrary conservative variable can be prognosed while ignoring its precise relationship to entropy. These findings show that down-gradient closures are consistent with the first and second law of thermodynamics even when they lead to a turbulent sensible heat flux up the temperature gradient. Indeed, while molecular conduction/diffusion is spontaneous and energy-conserving, stratified turbulent mixing is driven by mechanical turbulence and enabled by the consumption of turbulent kinetic energy.