Postulate of Statistical Independence for Decaying Reactants in Homogeneous Turbulence

Abstract

An independence hypothesis is proposed which predicts the evolution of the amplitude of Fourier modes of the concentration fluctuation field for a substance undergoing a second-order single-species isothermal reaction in homogeneous turbulence. The hypothesis is that the time history of such an amplitude is a product of its time history due to reaction and its time history due to turbulent mixing and diffusion. A sufficient condition for the hypothesis to be applicable is that the spectral decay due to reaction alone exhibits wavenumber similarity with a constant length scale. When the initial fluctuation intensity is high enough, the initial decay of the mean concentration (mean-square concentration) as predicted by the hypothesis gives a lower (upper) bound on the actual initial rate of decay. It is also demonstrated by numerical calculation of a particular initial isotropic concentration spectrum in the absence of turbulence that the independence hypothesis and the predictions of a valid closure for this problem differ by no more than an order of magnitude over the entire time history of decay for all possible values of the diffusivity and reaction rate. When turbulence is present, the long time evolution of the mean concentration and the mean-square concentrations as predicted by the hypothesis are in excellent agreement with previous computations employing direct interaction.