Investigation of the order of accuracy of the discontinuous Galerkin method in combination with an immersed boundary condition

Kurzfassung

The aerospace industry is interested in scale-resolving simulations (SRS) with shortened time-to-solution. Work is underway at the DLR to develop a rapid SRS process chain, which combines: 1) automatic mesh generation, 2) immersed imposition of boundary conditions, 3) use of wall models, and 4) the discontinuous Galerkin spectral element method (DGSEM). This fourth ingredient caters to the requirements of large eddy simulation (LES), as it is capable of resolving a wide range of wavenumbers with minimal numerical dispersion and dissipation, while being both robust and efficient [2]. The present work represents a first attempt at combining a high-order discontinuous Galerkin (DG) discretization with automatically generated meshes by means of an immersed boundary method (IBM), in the context of the CFD software by ONERA, DLR, Airbus (CODA^1) [3]. Efforts towards extending the split-form DGSEM in CODA to support h/p-nonconforming setups are ongoing. The meshing software Cassiopée [1] is used to generate a block-uniform Cartesian grid around a closed shape (the solid), which employs iterative octree refinement towards the boundary of the solid such that the smallest cell size ends up being (approximately) equal to a user-specified value. Here, no attempt is made to have the mesh boundary conform to the solid boundary; see Figure 1. This mesh is converted to single-block unstructured before being imported into CODA. Cassiopée is also responsible for generating a donor and a wall point for each immersed face integration point. The imposition of an immersed slip boundary condition (the Euler equations are the initial focus of this work) is closely related to the approach in [4] and consists of two steps. The first step is a (high order) reconstruction of the state vector at the donor point. The second step is the interpolation of each of the state quantities, between a donor point and its associated wall point, to the location of their corresponding immersed face integration point. For the normal component of momentum, which is zero at the wall, a linear interpolation is used for simplicity. Density, tangential momentum and total energy, whose gradients in normal direction should be zero at the wall, are obtained via constant interpolation (i.e. each takes the same value at all three points). The spatial discretization is a modal DG scheme with orthonormal polynomial basis functions of up to a specified total degree in 3D physical space. The numerical flux at faces is computed using the Riemann solver of Roe without entropy fix. The discrete equations are solved with a linearized implicit Euler scheme. The linear problem in each nonlinear solver iteration is solved using the GMRES method with a lines-implicit Jacobi preconditioner. All solutions are converged in the sense of each equation’s residual being reduced by a factor of 10e-10. A subsonic, inviscid flow around a circular cylinder of unit diameter has been computed on a family of meshes like the one shown in Figure 1, for DG polynomial degrees from 0 to 3. Figure 2 summarizes the obtained results. There are two main findings: first, the combination of IBM and DG does achieve the design order of (approximately) 2 in the linear basis function case; second, the design order is not retained as the degree of the discretization increases (accuracy remains bound to 2nd order), even under the ideal circumstances represented by this model problem. The use of linear interpolation in the normal momentum component is hypothesized to be the cause of this loss of order, as this process introduces a 2nd order error independent of the reconstruction order – precisely what the data seems to suggest. In order to test this hypothesis, the linear interpolation will be replaced with a quadratic one for the case of degree 2. Measuring 3rd order of accuracy would then confirm the hypothesis. Future work would, in that case, consist on generalizing the higher-order interpolation to any discretization degree. Should this not be the case, an alternative explanation of these observations would need to be found. ^1CODA is the computational fluid dynamics (CFD) software being developed as part of a collaboration between the French Aerospace Lab ONERA, the German Aerospace Center (DLR), Airbus, and their European research partners. CODA is jointly owned by ONERA, DLR and Airbus. [1] C. Benoit, S. Péron, and S. Landier, “Cassiopee: A CFD pre- and post-processing tool,” Aerospace Science and Technology, vol. 45, pp. 272–283, Sep. 2015, doi: 10.1016/j.ast.2015.05.023. [2] D. Flad and G. Gassner, “On the use of kinetic energy preserving DG-schemes for large eddy simulation,” Journal of Computational Physics, vol. 350, pp. 782–795, Dec. 2017, doi: 10.1016/j.jcp.2017.09.004. [3] T. Leicht et al., “DLR-Project Digital-X - Next Generation CFD Solver ‘Flucs,’” in Deutscher Luft- und Raumfahrtkongress 2016, Feb. 2016. [Online]. Available: https://elib.dlr.de/111205/ [4] S. Péron, C. Benoit, T. Renaud, and I. Mary, “An immersed boundary method on Cartesian adaptive grids for the simulation of compressible flows around arbitrary geometries,” Engineering with Computers, vol. 37, no. 3, pp. 2419–2437, Jul. 2021, doi: 10.1007/s00366-020-00950-y.