Some Problems of Statistical Modeling of Intermittent Turbulent Flows and Their Solutions

Abstract

Some problems of statistical modeling of turbulent flows associated with the hydrodynamic effects of intermittency of various dynamic and scalar fields are identified, and a justification of their solutions is proposed. At first examines some problems of statistical modeling of large-scale (energy-containing) turbulence structure under the conditions of turbulent and nonturbulent fluid intermittency in the turbulent flow. This primarily applies to the problem of obtaining the differential equations of hydrodynamics for conditional statistical moments (conditional averages), i.e. for single-point statistical moments of each of the intermittent media of turbulent flow. To solve this problem is given a mathematically rigorous justification of conducting the operation of conditional averaging of the Navier-Stokes equations. As a result was obtained the system of differential equations for the conditional averages of turbulent and nonturbulent fluid. The main advantage of the obtained differential equations for the conditional averages is that these equations don’t contain the source terms. Therewith is given a physical substantiation of transformation process of nonturbulent fluid in turbulent, that occurs in a thin superlayer, which separates turbulent and nonturbulent fluid. Here also is given a justification of conducting the operation of total averaging of the partial derivates, which, as it turns out, is not a permutational and leads to a number of specific features in comparison with the unconditional averaging in RANS. Another problem associated with statistical modeling of inhomogeneous turbulent flows, i.e. flows with an inhomogeneous field of scalar substance. It is known that the dynamic field with turbulent fluid does not coincide with the inhomogeneous field of scalar substance. To solve this problem is proposed a method of calculating the statistical characteristics both of the turbulent fluid and the inhomogeneous field of a passive admixture concentration. It is shown, that the conditional averages of the passive admixture concentration are significantly different. In conclusion we consider the problem of statistical modeling of diffusional turbulent combustion of not pre-mixed chemically reacting gases, expressly the modeling by the known method PDF with involving the “reduced fuel concentration” as the passive impurity concentration. To solve this problem is given a justification of new differential equations, allowing to carry out calculations only of the conditional averages for the intersection region of the dynamic field of the turbulent fluid and the inhomogeneous field of the passive impurity concentration. Some results of such calculations, performed in the self-similar field of diffusion turbulent propane plume, are presented together with the experimental data available in literature.