Multiscale modeling of incompressible turbulent flows

Developing an effective turbulence model is important for engineering applications as well as for fundamental understanding of the flow physics. We present a mathematical derivation of a closure relating the Reynolds stress to the mean strain rate for incompressible flows. A systematic multiscale analysis 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 recovery of the Smagorinsky model for homogeneous turbulence validates our derivation. Another example is the channel flow, where we derive a simplified turbulence model using the asymptotic structure near the wall. Numerical simulations at two Reynolds numbers (Re's) using our model agrees well with both experiments and Direct Numerical Simulations of turbulent channel flow.