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.
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.
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.
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.
