Hou, Thomas Y.; Yang, Danping; Ran, Hongyu Multiscale analysis and computation for the three-dimensional incompressible Navier-Stokes equations. (English) Zbl 1248.76037 Multiscale Model. Simul. 6(2007), No. 4, 1317-1346 (2008). Summary: 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. Cited in 5 Documents MSC: 76D05 Navier-Stokes equations for incompressible viscous fluids 35Q30 Navier-Stokes equations 76F05 Isotropic turbulence; homogeneous turbulence 76F65 Direct numerical and large eddy simulation of turbulence 76M20 Finite difference methods applied to problems in fluid mechanics 76M50 Homogenization applied to problems in fluid mechanics Keywords:multiscale analysis; turbulence modeling; three-dimensional Navier-Stokes equations PDFBibTeX XMLCite \textit{T. Y. Hou} et al., Multiscale Model. Simul. 6, No. 4, 1317--1346 (2008; Zbl 1248.76037) Full Text: DOI