A conservative degree adaptive HDG method for transient incompressible flows

TL;DR

This paper introduces a conservative projection method to improve degree adaptive HDG simulations of transient incompressible flows, significantly reducing numerical artifacts and improving accuracy in aerodynamic quantities like drag.

Contribution

A novel conservative projection technique for degree adaptive HDG methods that enhances accuracy without significant computational overhead.

Findings

The conservative projection removes non-physical oscillations in drag calculations.

Standard degree adaptive methods can produce inaccurate pressure fields in wake regions.

The new method achieves better agreement with reference solutions in transient flow simulations.

Abstract

Purpose: This study aims to assess the accuracy of degree adaptive strategies in the context of incompressible Navier-Stokes flows using the high order hybridisable discontinuous Galerkin (HDG) method. Design/methodology/approach: The work presents a series of numerical examples to show the inability of standard degree adaptive processes to accurate capture aerodynamic quantities of interest, in particular the drag. A new conservative projection is proposed and the results between a standard degree adaptive procedure and the adaptive process enhanced with this correction are compared. The examples involve two transient problems where flow vortices or a gust needs to be accurately propagated over long distances. \noindent \textbf{}Findings:polynomials with a lower degree. Due to the coupling of velocity-pressure in incompressible flows, the violation of the incompressibility constraint leads to inaccurate pressure fields in the wake that have a sizeable effect on the drag. The new conservative projection proposed is found to remove all the numerical artefacts shown by the standard adaptive process. Originality/value: This work proposes a new conservative projection for the degree adaptive process. The projection does not introduce a significant overhead because it requires to solve an element-by-element problem and only for those elements where the adaptive process lowers the degree of approximation. Numerical results show that with the proposed projection non-physical oscillations in the drag disappear and the results are in good agreement with reference solutions.