A Mathematical Theory for LES Convergence

Practical simulations of turbulent processes are generally cutoff, with a grid resolution that stops within the inertial range, meaning that multiple active regions and length scales occur below the grid level and are not resolved. This is the regime of large eddy simulations (LES), in which the larger but not the smaller of the turbulent length scales are resolved. Solutions of the fluid Navier-Stokes equations, when considered in the inertial regime, are conventionally regarded as solutions of the Euler equations. In other words, the viscous and diffusive transport terms in the Navier-Stokes equations can be neglected in the inertial regime and in LES simulations, while the Euler equation becomes fundamental.For such simulations, significant new solution details emerge as the grid is refined. It follows that conventional notions of grid convergence are at risk of failure, and that a new, and weaker notion of convergence may be appropriate. It is generally understood that the LES or inertial regime is inherently fluctuating and its description must be statistical in nature. Here we develop such a point of view systematically, based on Young measures, which are measures depending on or indexed by space time points. In the Young measure (ξ)x,t, the random variable ξ of the measure is a solution state variable, i.e., a solution dependent variable, representing momentum, density, energy and species concentrations, while the space time coordinates, x, t, serve to index the measure.Theoretical evidence suggests that convergence via Young measures is sufficiently weak to encompass the LES/inertial regime; numerical and theoretical evidence suggests that this notion may be required for passive scalar concentration and thermal degrees of freedom. Our objective in this research is twofold: turbulent simulations without recourse to adjustable parameters (calibration) and extension to more complex physics, without use of additional models or parameters, in both cases with validation through comparison to experimental data.