The evolution of surfaces in turbulence

Abstract

There are several phenomena that can be described to advantage in terms of surfaces within a turbulent fluid. Examples are: turbulent mixing (particularly at high Schmidt number); turbulent premixed flames; and, turbulent diffusion flames. These phenomena can (under appropriate conditions) be analyzed in terms of material surfaces, propagating surfaces, and constant-property surfaces, respectively. Deterministic and probabilistic equations are developed for the evolution of the local properties of these surfaces. The local geometry of regular surfaces is described by the surface element properties: position; normal to the surface; principal curvatures and directions; and, fractional area increase. Exact evolution equations for these properties are derived which reveal the effects of various processes—straining, and surface propagation, for example. For material surfaces and simple propagating surfaces these equations are closed with respect to surface properties:, that is, given the velocity field, the equations can be solved from specified initial conditions. The circumstances that can lead to a breakdown of regularity of an initially regular surface are determined. The fundamental one-point Eulerian probabilistic descriptor of a regular surface is the surface density function. From this can be determined the expected surface-to-volume ratio and the joint probability density function of the surface properties. An exact evolution equation for the surface density function is derived and discussed. For material surfaces and simple propagating surfaces the only unknowns in this equation are statistics of the velocity field. These statistics can be modelled (via Langevin equations, for example) and then the surface-density-function equation can be solved by a Monte Carlo method.