Multiscale Modeling and Computation of 3D Incompressible Turbulent Flows

In the first part, we present a mathematical derivation of a closure relating the Reynolds stress to the mean strain rate for incompressible turbulent flows. This derivation is based on a systematic multiscale analysis that expresses the Reynolds stress in terms of the solutions of local periodic cell problems. We reveal an asymptotic structure of the Reynolds stress by invoking the frame invariant property of the cell problems and an iterative dynamic homogenization of large- and small-scale solutions. The Smagorinsky model for homogeneous turbulence is recovered as an example to illustrate our mathematical derivation. Another example is turbulent channel flow, where we derive a simplified turbulence model based on the asymptotic flow structure near the wall. Additionally, we obtain a nonlinear model by using a second order approximation of the inverse flow map function. This nonlinear model captures the effects of the backscatter of kinetic energy and dispersion and is consistent with other models, such as a mixed model that combines the Smagorinsky and gradient models, and the generic nonlinear model of Lund and Novikov.;Numerical simulation results at two Reynolds numbers using our simplified turbulence model are in good agreement with both experiments and direct numerical simulations in turbulent channel flow. However, due to experimental and modeling errors, we do observe some noticeable differences, e.g., root mean square velocity fluctuations at Retau = 180.;In the second part, we present a new perspective on calculating fully developed turbulent flows using a data-driven stochastic method. General polynomial chaos (gPC) bases are obtained based on the mean velocity profile of turbulent channel flow in the offline part. The velocity fields are projected onto the subspace spanned by these gPC bases and a coupled system of equations is solved to compute the velocity components in the Karhunen-Loeve expansion in the online part. Our numerical results have shown that the data-driven stochastic method for fully developed turbulence offers decent approximations of statistical quantities with a coarse grid and a relatively small number of gPC base elements.