Ensemble mean closure using One-Dimensional Turbulence
Randall McDermott, University of Utah
One-Dimensional Turbulence (ODT) models the process of molecular transport (i.e. mixing) along a notional line of sight through a fully resolved flow field. The process of turbulent advection (i.e. stirring) is modelled with a stochastic sequence of instantaneous mapping events, which are mathematical constructs, but mimic many of the fundamental properties of real turbulence. The mapping conserves mass, momentum, and energy; it introduces no physical discontinuities in the flow field; it increases local scalar gradients, hence enhancing local molecular diffusion; it increases the number of scalar level crossings (the 1-d analog of increased surface area for scalar flux in 3-d); and it reduces length scales in a recursive manner, hence producing an energy cascade from large scales to small scales, consistent with classical turbulence theory.
Efforts are underway to incorporate ODT as a subgrid model in large-eddy simulation (LES). In LES the large scale, energy containing motions of a flow are simulated directly and the effect of the small scale motions is modelled. Being one-dimensional ODT has the advantage of remaining computationally tractable for flows of enormously high Reynolds number (e.g. atmospheric modelling and complex engineering design). This means that ODT provides a viable alternative where typical LES models fail, namely: wall bounded flows and flows where small scale turbulence/chemistry interactions are important.
In this paper we derive the form of the subgrid stress resulting from an ODT model in which all possible eddies are allowed to occur. To obtain this result we simplify the ODT model and envision eddy events to act upon the “mean” velocity field. That is, in the region of an eddy the velocity field is linearized to have a mean strain rate, S(y) = du/dy. We start with the simplest form of the ODT model (called “vanilla ODT”), and then begin adding levels of complexity by accounting for the redistribution of component energies based on a pressure-scrambling model (“neapolitan ODT”). Following this we add complexity to the physics in the time scale. Finally, the Reynolds number dependence is included by adding a viscous cut-off length scale.
An interesting consequence of the linearity simplification is that the ensemble ODT model begins to take the form of classical LES models (e.g. Smagorinsky). From here it is possible to deduce a theoretical value for the ODT rate constant, which otherwise must be obtained through empiricism.
Abstract Author(s): Randy McDermott, Alan Kerstein, and Philip J. Smith