Increasing Accuracy and Efficiency for Regularized Navier-Stokes Equations

Abstract

In this article, a finite element scheme for the family of time relaxation models, that represent a regularization of Navier-Stokes equations, is developed, analyzed and numerically tested. The proposed finite element scheme combines three ideas: (i) the use of an incompressible filter, for better consistency outside the periodic domains, (ii) a second order accurate linearization for the nonlinear term, that allows to solve only one linear system per time step, and (iii) a stabilization in time term that compliments well the linearization. A complete numerical analysis of the scheme, that includes the computability of its numerical solutions, its stability, and velocity error estimates, is given. This is followed by numerical experiments that confirm the theoretical convergence rates and show the advantage of the proposed scheme.

A Review of Time Relaxation Methods

“…In the linear case (9) and (10), a similarity study was also conducted in [6] revealing that if χ is selected appropriately, one will not only observe an energy cascade, but potentially control that cascade by inducing a new effective Kolmogorov micro-scale for (9) and (10) that agrees with the filtering length δ. This was called a "perfect resolution," since in this case the extra dissipation perfectly balances the energy transfer from external power sources to the new cutoff scale while preventing a non-physical accumulation of energy there.…”

“…Further progress was made in [9], where the standard (P n , P n−1 ) finite element discretization implemented in [7] was augmented in two key ways. The first technique adjusts the differential filter discussed in Definition (1), requiring it to be locally divergence-free,…”

“…The implementations from [7,9,11] each perform the 2D flow about a full-step obstruction benchmark. This is an excellent experiment to test regularization techniques like TRMs, as it develops a highly-rotational flow, and on a coarse mesh a direct numerical simulation of the NSE produces an unacceptable amount of error.…”

“…This is an excellent experiment to test regularization techniques like TRMs, as it develops a highly-rotational flow, and on a coarse mesh a direct numerical simulation of the NSE produces an unacceptable amount of error. In this section we have collected and discussed the results from [7,9,11], summarizing a selection of their results.…”

“…We present and briefly discuss linear and nonlinear forms of time relaxation terms, and their effects on the transfer of kinetic energy from large scales of motion to small. This discussion continues in Section 4 where a brief survey of implementations utilizing finite element methods (FEM) is included, [7][8][9][10][11][12][13]. We will continue surveying these results by comparing the performance of the several TRM formulations and FEM techniques on the benchmark flow past a full step problem.…”