Max Gunzburger

Two “Better” Ways to Solve the Navier-Stokes Equations

The facetious and self-serving title refers to novel approaches towards solving the Navier-Stokes equations (NSEs) in two settings. The first is a fractional Laplacian-based closure with exponent s for turbulent flows that preserves the Kolmogorov -5/3 scaling exponent except for one specific value of s for which the power law of the energy spectrum in the inertial range has a correction in the Kolmogorov scaling exponent which corresponds to Richardson’s particle pair-distance superdiffusion of a fully-developed homogeneous turbulent flow. The second approach involves spectral viscosity and filtered hierarchical finite element methods for the Navier-Stokes equations. Here we consider applying high-pass filters to hyperviscosity regularization and a regularization due to Ladyzhenskaya. For such approaches, well posedness is proved and as are the well posedness of both spectral and finite element methods. For the latter, we also derive error estimates. Note that the Smagorinsky model is a special case of Ladyzhenskaya regularization. [One or the other of these works is joint with Nan Jian, Eunjun Lee, Yuki Saka, Catalin Trenchea, Xiaoming Wang, and Fefei Xu.]