Macro and micro issues in turbulent mixing

Numerical prediction of turbulent mixing can be divided into two subproblems: to predict the geometrical extent of a mixing region and to predict the mixing properties on an atomic or molecular scale, within the mixing region. The former goal suffices for some purposes, while important problems of chemical reactions(e.g. flames) and nuclear reactions depend critically on the second goal in addition to the first one. Here we review recent progress in establishing a conceptual reformulation of convergence, and we illustrate these concepts with a review of recent numerical studies addressing turbulence and mixing in the high Reynolds number limit. We review significant progress on the first goal, regarding the mixing region, and initial progress on the second goal, regarding atomic level mixing properties. New results concerning non-uniqueness of the infinite Reynolds number solutions and other consequences of a renormalization group point of view, to be published in detail elsewhere, are summarized here.The notion of stochastic convergence(of probability measures and probability distribution functions) replaces traditional pointwise convergence. The primary benefit of this idea is its increased stability relative to the statistical "noise" which characterizes turbulent flow. Our results also show that this modification of convergence, with sufficient mesh refinement, may not be needed. However, in practice, mesh refinement is seldom sufficient and the stochastic convergence concepts have a role.Related to this circle of ideas is the observation that turbulent mixing, in the limit of high Reynolds number, appears to be non-unique. Not only have multiple solutions been observed(and published) for identical problems, but simple physics based arguments and more refined arguments based on the renormalization group come to the same conclusion.Because of the non-uniqueness inherent in numerical models of high Reynolds number turbulence and mixing, we also include here numerical examples of validation. The algorithm we use here has two essential components. We depend on Front Tracking to allow accurate resolution of flows with sharp interfaces or steep gradients(concentration or thermal), as are common in turbulent mixing problems. The higher order and enhanced algorithms for interface tracking, both those already developed, and those proposed here, allow a high resolution and uniquely accurate description of sample mixing problems. Additionally, we depend on the use of dynamic subgrid scale models to set otherwise missing values for turbulent transport coefficients, a step that breaks the non-uniqueness.