Multiscale Analysis and Computation for the Three-Dimensional Incompressible Navier–Stokes Equations
In this paper, we perform a systematic multiscale analysis for the three-dimensional incompressible Navier–Stokes equations with multiscale initial data. There are two main ingredients in our multiscale method. The first one is that we reparameterize the initial data in the Fourier space into a formal two-scale structure. The second one is the use of a nested multiscale expansion together with a multiscale phase function to characterize the propagation of the small-scale solution dynamically. By using these two techniques and performing a systematic multiscale analysis, we derive a multiscale model which couples the dynamics of the small-scale subgrid problem to the large-scale solution without a closure assumption or unknown parameters. Furthermore, we propose an adaptive multiscale computational method which has a complexity comparable to a dynamic Smagorinsky model. We demonstrate the accuracy of the multiscale model by comparing with direct numerical simulations for both two- and three-dimensional problems. In the two-dimensional case we consider decaying turbulence, while in the three-dimensional case we consider forced turbulence. Our numerical results show that our multiscale model not only captures the energy spectrum very accurately, it can also reproduce some of the important statistical properties that have been observed in experimental studies for fully developed turbulent flows.
©2008 Society for Industrial and Applied Mathematics. Received by the editors February 7, 2007; accepted for publication (in revised form) September 20, 2007; published electronically March 26, 2008. This work was supported in part by the NSF under grant DMS-0073916 and ITR grant ACI-0204932. The work of this author [D.P.Y.] was supported in part by the National Basic Research Program of the People's Republic of China under grant 2005CB321703 and under the NSFC grants 10571108 and 10441005. We thank Professors George Papanicolaou, Olivier Pironneau, and Dale Pullin for many stimulating discussions and helpful suggestions regarding this work. We also express our gratitude to the referees for their valuable comments on our original manuscript and for suggesting a better norm to use in our adaptive multiscale algorithm.