A conservative degree adaptive HDG method for transient incompressible flows

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 accurately 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.

Findings

The lack of robustness and accuracy of standard degree adaptive processes is linked to the violation of the free-divergence condition when projecting a solution from a space of polynomials of a given degree to a space of 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.