Stochastic representation of the Reynolds transport theorem: revisiting large-scale modeling

We explore the potential of a formulation of the Navier-Stokes equations incorporating a random description of the small-scale velocity component. This model, established from a version of the Reynolds transport theorem adapted to a stochastic representation of the flow, gives rise to a large-scale description of the flow dynamics in which emerges an anisotropic subgrid tensor, reminiscent to the Reynolds stress tensor, together with a drift correction due to an inhomogeneous turbulence. The corresponding subgrid model, which depends on the small scales velocity variance, generalizes the Boussinesq eddy viscosity assumption. However, it is not anymore obtained from an analogy with molecular dissipation but ensues rigorously from the random modeling of the flow. This principle allows us to propose several subgrid models defined directly on the resolved flow component. We assess and compare numerically those models on a standard Green-Taylor vortex flow at Reynolds 1600. The numerical simulations, carried out with an accurate divergence-free scheme, outperform classical large-eddies formulations and provides a simple demonstration of the pertinence of the proposed large-scale modeling

Modeling the Pollution of Pristine Gas in the Early Universe

We conduct a comprehensive theoretical and numerical investigation of the pollution of pristine gas in turbulent flows, designed to provide new tools for modeling the evolution of the first generation of stars. The properties of such Population III (Pop III) stars are thought to be very different than later generations, because cooling is dramatically different in gas with a metallicity below a critical value Z_c, which lies between ~10^-6 and 10^-3 solar value. Z_c is much smaller than the typical average metallicity, , and thus the mixing efficiency of the pristine gas in the interstellar medium plays a crucial role in the transition from Pop III to normal star formation. The small critical value, Z_c, corresponds to the far left tail of the probability distribution function (PDF) of the metallicity. Based on closure models for the PDF formulation of turbulent mixing, we derive equations for the fraction of gas, P, lying below Z_c, in compressible turbulence. Our simulation data shows that the evolution of the fraction P can be well approximated by a generalized self-convolution model, which predicts dP/dt = -n/tau_con P (1-P^(1/n)), where n is a measure of the locality of the PDF convolution and the timescale tau_con is determined by the rate at which turbulence stretches the pollutants. Using a suite of simulations with Mach numbers ranging from M = 0.9 to 6.2, we provide accurate fits to n and tau_con as a function of M, Z_c/, and the scale, L_p, at which pollutants are added to the flow. For P>0.9, mixing occurs only in the regions surrounding the pollutants, such that n=1. For smaller P, n is larger as mixing becomes more global. We show how the results can be used to construct one-zone models for the evolution of Pop III stars in a single high-redshift galaxy, as well as subgrid models for tracking the evolution of the first stars in large cosmological simulations.Comment: 37 pages, accepted by Ap

Algorithms and Models for Turbulence Not at Statistical Equilibrium

Standard eddy viscosity models, while robust, cannot represent backscatter and have severe difficulties with complex turbulence not at statistical equilibrium. This report gives a new derivation of eddy viscosity models from an equation for the evolution of variance in a turbulent flow. The new derivation also shows how to correct eddy viscosity models. The report proves the corrected models preserve important features of the true Reynolds stresses. It gives algorithms for their discretization including a minimally invasive modular step to adapt an eddy viscosity code to the extended models. A numerical test is given with the usual and over diffusive Smagorinsky model. The correction (scaled by $10^{-8}$ ) does successfully exhibit intermittent backscatter.Comment: 18 pages, 2 figure

Large Eddy Simulations (LES) and Direct Numerical Simulations (DNS) for the computational analyses of high speed reacting flows

The principal objective is to extend the boundaries within which large eddy simulations (LES) and direct numerical simulations (DNS) can be applied in computational analyses of high speed reacting flows. A summary of work accomplished during the last six months is presented