Reactive Tracers in Turbulence Models

Abstract

This thesis characterizes the nature of inaccuracies in the description of two tracers, one passive and one reactive (with a particular choice of model) within Reynolds Averaged Navier Stokes simulations (RANS) and Large Eddy Simulations (LES). Both tracers are taken to be passive with respect to the fluid dynamics and the reactive tracer is assumed to undergo a growth reaction governed by Fisher's equation. To begin, the Navier-Stokes equations and the partial differential equation governing the reactive tracer evolution are Reynolds averaged and spatially filtered to obtain the governing equations for each of the turbulence models. The procedure is applied to a generalized polynomial reaction function and can be extended to other sufficiently smooth non-polynomial reactions. The Reynolds averaged and filtered reaction equations are analyzed using a simplified, zero dimensional toy model. A one dimensional toy model is used to illustrate how a non-linear reaction term, advection, and diffusion each influence the spectral distribution of a reactive tracer. To consider the effect of Reynolds averaging, an ensemble of 50 two-dimensional direct numerical simulations is run. Within each simulation, the reactive tracer is subjected to mixing induced by a Rayleigh-Taylor instability. A posteriori Reynolds averaging is applied to the ensemble data to evaluate the discrepancies between the mean system and the dynamics of the ensemble members. A two-dimensional toy model with specified velocities is used to illustrate the effect of spatial filtering. Further, the dependence of the sub-filter-scale flux and reaction terms on the cutoff wavenumber of a low-pass filter and the reaction rate is evaluated. To investigate a system with a larger range of spatial scales, a posteriori LES is applied to data from a three-dimensional simulation of a reactive tracer subjected to turbulence induced by a Rayleigh-Taylor instability. Various filter choices are applied and the sub-filter-scale terms are quantified. Given the scope of this thesis, discussions of the findings and their implications for modelling can be found at the end of chapters 5 and 6. This thesis concludes with a broader discussion of the findings and highlights avenues for future work.