Convergence Analysis of a Fully Discrete Family of Iterated Deconvolution Methods for Turbulence Modeling with Time Relaxation

We present a general theory for regularization models of the Navier-Stokes equations based on the Leray deconvolution model with a general deconvolution operator designed to fit a few important key properties. We provide examples of this type of operator, such as the (modified) Tikhonov-Lavrentiev and (modified) Iterated Tikhonov-Lavrentiev operators, and study their mathematical properties. An existence theory is derived for the family of models and a rigorous convergence theory is derived for the resulting algorithms. Our theoretical results are supported by numerical testing with the Taylor-Green vortex problem, presented for the special operator cases mentioned above. 1. Approximate Deconvolution for Turbulence Modeling Numerical simulations of complex flows present many challenges. The Navier-Stokes (NS) equations (NSE), given by the following is an exact model for the flow of a viscous, incompressible fluid, [1]. For turbulent flows (characterized by Reynold’s number ), it is infeasible to properly resolve all significant scales above the Kolmogorov length scale by direct numerical simulation. Thus, numerical simulations are often based on various regularizations of NSE, rather than NSE themselves. Accordingly, regularization methods provide a (computationally) efficient and (algorithmically) simple family of turbulence models. Several of the most commonly applied regularization methods include: where in each case and is an averaged velocity field , is pressure, and is the Bernoulli pressure. More details about these models can be found for instance in [2–5] and references therein. Although these regularization methods achieve high theoretical accuracy and perform well in select practical tests, those models do not provide a fully-developed numerical solution for decoupling the scales in a turbulent flow. In fact, results show that only time relaxation regularization truncates scales sufficiently for practical computations. Indeed, it is shown that time relaxation term for damps unresolved fluctuations over time [5, 6]. Note that the choice of is an active area of research and that solutions are very sensitive to variations in . Deconvolution-based regularization is also an active area of research obtained, for example, by replacing by in each (1.2)–(1.5) for some deconvolution operator . In [7], Dunca proposed the general Leray-deconvolution problem ( instead of ) as a more accurate extension to Leray’s model [8]. Leray used the Gaussian filter as the smoothing (averaging) filter , denoted above by overbar. In [9], Germano proposed the