Theoretical and computational analysis of spectrally hyperviscous models of turbulent flow.

Computing turbulent flow is very difficult but forms the basis for computational ex- periments in Meteorology and Oceanography. To overcome the difficulty and complexity in turbulence computation, a spectrally hyperviscous version of Navier-Stokes equations (SHNSE) has been suggested(see [4] and the references contained therein). My PhD research has been focusing on the theoretical and computational analysis for the SHNSE. This dissertation bases on my research under the advise of Dr. Avrin and Dr. Deng [5,7,8,77]. The theoretical results that we obtained are the convergence of Galerkin solutions and the continuous dependence on data for the SHNSE [5], the estimates for the number of determining nodes and determining modes [8], and the inviscid limit ? ? 0 in the case of unforced turbulence [7]. Let uN denote the Galerkin solutions which approximate the solution u, and let wN = u - uN then by spectral decomposition, we have wN = PmwN + QmwN where Pm is the projection onto the first m eigenspaces of A = -? and Qm = I - Pm. For assumptions on ?m that compare well with those in previous results, the convergence of ?QmwN(t)?Hß depends linearly on key parameters (and on negative powers of ?m), which is reflective of Kolmogorov-theory predictions that in high wavenumber modes vis- cous (i.e. linear) effects dominate. Meanwhile ?PmwN(t)?Hß satisfies a more standard exponential estimate, but with fractional dependence on ?m. Similar results demonstrate continuous dependence on data. The estimates for the number of determining nodes of the three dimensional SHNSE are proportional to G3, where G denotes the Grashof number, comparing well with the exponential determining-node results for the two dimensional no-slip NSE. The estimates for the number of determining modes also compare well with previous results, and in par- ticular as long as a > 9/4 these estimates are less than those for the two dimensional space-periodic NSE. If a = 5/2, explicit non-exponential estimates for the dependence of the high-wavenumber modes on the determining modes can be obtained with improved computationally utility. We discuss these results in the context of physical and compu- tational experiments, and in terms of the potential of the SHNSE to reduce the number of degrees of freedom required for physical and computational experiments. We can use the pairing of the 3-D Euler system with spectrally applied hyperviscous terms and the SHNSE as a platform to study the inviscid limit ? ? 0 in the case of unforced turbulence. Let u? be the solution of the SHNSE, let u be the solution of the spectrally-hyperviscous Euler equations, and let w? = u - u?, then we will show that w? ? 0 strongly. The characteristic feature of our convergence methodology is its multiscale approach, which seems to optimize the results and yields behavior in each regime suggested by experiments and physical theory. The computational analysis implements the SHNSE for a periodic box by using pseudo- spectral methods, and numerical results obtained from large eddy simulations for the decaying turbulence are compared with those obtained by direct numerical simulation [77]. Numerical experiments are conducted to validate some of the theoretical properties of the SHNSE and to investigate optimal parameter choices. Numerical results indicate that the SHNSE model has strong potential to be a highly robust platform for studying turbulence which can retain spectral accuracy while significantly reducing the number of degrees of freedom needed for accurate simulation