A high accuracy minimally invasive regularization technique for navier–stokes equations at high reynolds number

Abstract

A method is presented, that combines the defect and deferred correction approaches to approximate solutions of Navier–Stokes equations at high Reynolds number. The method is of high accuracy in both space and time, and it allows for the usage of legacy codes a frequent requirement in the simulation of turbulent flows in complex geometries. The two‐step method is considered here; to obtain a regularization that is second order accurate in space and time, the method computes a low‐order accurate, stable, and computationally inexpensive approximation (Backward Euler with artificial viscosity) twice. The results are readily extendable to the higher order accuracy cases by adding more correction steps. Both the theoretical results and the numerical tests provided demonstrate that the computed solution is stable and the accuracy in both space and time is improved after the correction step. We also perform a qualitative test to demonstrate that the method is capable of capturing qualitative features of a turbulent flow, even on a very coarse mesh. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 814–839, 2017

A Defect-Deferred Correction Method for Fluid-Fluid Interaction

“…As noted in [21], since AV is introduced for all scales, it produces too diffusive approximation. The one considered herein employs SAV stabilization, associated with the VMS method, as a predictor step which introduces AV only for small scales.…”

“…On the other hand, we can roughly predict possible outcomes based on our expertise with one domain flows. The setup on the top domain has been extensively used for a qualitative assessment of fluid flows, see, e.g., [21,35,36]. By choosing Re = 100, two vortices start to develop behind the cylinder, they then separate into the flow, and then, a vortex street forms.…”

“…We refer to the reader [14,20,19] for comprehensive analytical insight. The predictor-corrector type algorithms considered in [21,22] employ AV for stabilization of all scales. In this approach, the over-diffusive effect of the first step (AV based defect step) is subtracted via a correction step.…”

“…This discretization idea proved to be the key idea for unconditional stability and higher accuracy without extra computational cost. The defect-deferred correction method with an artificial diffusion step for (1.1)-(1.6) was studied in [3].As noted in [21], since AV is introduced for all scales, it produces too diffusive approximation. The one considered herein employs SAV stabilization, associated with the VMS method, as a predictor step which introduces AV only for small scales.…”

Defect-deferred correction method for the two-domain convection-dominated convection–diffusion problem

“…In order to create an unconditionally stable, second order accurate in both space and time, partitioned time stepping method, we apply the combined defect and deferred correction techniques to the data-passing scheme of [15]. The combination of the defect and deferred correction was introduced and 1 successfully tested in [2] in application to the one-domain Navier-Stokes equations.…”