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.
© 2012 Elsevier Inc. Received 15 March 2012. Received in revised form 15 August 2012. Accepted 16 August 2012. Available online 1 September 2012. This research was in part supported by a NSF Grant DMS-0908546 and by an AFOSR MURI Grant FA9550-09-1-0613. We would like to thank Professor Olivier Pironneau for many stimulating and inspiring discussions. His comments and suggestions have played an essential role in this work. We also thank Dr. Daniel Chung for providing the DNS code of turbulent channel flow.