Nonlocal Turbulence Modeling Close to Criticality Involving Kolmogorov’s Dissipation Microscales

Abstract

In zero-equation turbulence modeling the Reynolds shear stress is usually assumed to be an order of magnitude larger than the viscous shear stress. Therefore, this kind of modeling is only accurate for flows at Reynolds numbers Re well above criticality, Re_c. New nonlocal and fractional turbulence models, as e.g. the Difference-Quotient Turbulence Model (DQTM), are based on a self-similar cascade of turbulent eddies of diameters which are related to Lévy flights of corresponding jump sizes. The smaller the Reynolds number is, the smaller will be the number of eddy classes and their total momentum transfer by eddy motion. By tending from above to a Reynolds number close to its critical value, only a single class of eddies with the largest eddy diameter remains. In this special case the diameter of these remaining eddies is identical to the Kolmogorov dissipation length that now equals the characteristic overall length scale of the flow domain. Below criticality even these largest eddies diminish and the flow lacks eddies of all sizes and, therefore, is laminar. Based on these ideas and Kolmogorov’s microscales the DQTM is generalized. This new extension leads to a vanishing Reynolds shear stress at criticality. The derived nonlocal multiplicative correction term in the DQTM is our main result and can be easily split off and combined with other turbulence models.