TURBULENCE MODELING AS AN ILL-POSED PROBLEM

by Iuliana, Stanculescu

This thesis is concerned with the derivation and mathematical analysis of new turbulence models, based on methods for solving ill-posed problems. Turbulence causes the formation of eddies of many different length scales. Small, unresolved scales have deterministic roles in the statistics of the resolved scales. The main problem of computational turbulence is to accurately represent the effect of the unknown small scales upon the observable large scales. This is really just another ill-posed problem and the work in this thesis shows that excellent turbulence models do come from standard methods for ill-posed problems. Large Eddy Simulation (LES) exploits this decoupling of scales in a turbulent flow: the larger unsteady turbulent motions are directly represented, while the effects of the smaller scale motions are modeled. This is achieved by introducing a filtering operation, which depends on a chosen averaging radius. Once an averaging radius and a filtering process is selected, an LES model can be developed and then solved numerically. One of the most interesting approaches to generate LES models is via approximate deconvolution or approximate/asymptotic inverse of the filtering operator. Herein, we develop an abstract approach to modeling the motion of large eddies in a turbulent flow and postulate conditions on a general deconvolution operator that guarantee the existence and uniqueness of strong solutions of Approximate Deconvolution Models. We also introduce new deconvolution operators which fit in this abstract theory. The Accelerated van Cittert algorithm and the Tikhonov regularization process are two methods for solving ill-posed problems that we adapt to turbulence. We study the mathematical properties of the resulting deconvolution operators. We also study a new family of turbulence models, the Leray-Tikhonov deconvolution models, which is based on a modification (consistent with the large scales) of the Tikhonov regularization process. We perform rigorous numerical analysis of a computational attractive algorithm for the considered family of models. Numerical experiments that support our theoretical results are presented.