Extrapolated Shock Tracking: Bridging shock-fitting and embedded boundary methods

Abstract

We propose a novel approach to approximate numerically shock waves. The method combines the unstructured shock-fitting approach developed in the last decade by some of the authors, with ideas coming from embedded boundary techniques. The numerical method obtained allows avoiding the remeshing phase required by the unstructured fitting method, while guaranteeing accuracy properties very close to those of the fitting approach. This new method has many similarities with front tracking approaches, and paves the way to shock-tracking techniques truly independent on the data and mesh structure used by the flow solver. The approach is tested on several problems showing accuracy properties very close to those of more expensive fitting methods, with a considerable gain in flexibility and generality.

Extrapolated DIscontinuity Tracking for complex 2D shock interactions

“…Compared to its original version [1], the current algorithm is capable of dealing with several discontinuity-fronts which implies that the computational domain is split into several, disjoint, subdomains.…”

“…Specifically, centred finite difference formulae will be used when the downstream state is subsonic whereas upwind formulae are employed whenever the downstream state is supersonic. Further details are given in [1,40]. The unit vector n is conventionally assumed to be directed from the high entropy (downstream) side towards the low entropy (upstream) side of the discontinuity, see Fig.…”

“…This first transfer is required to update the flow variables along Γ U and Γ D . Following [37,1], this is achieved by first defining a map:…”

“…Within the region downstream of a shock, as well as for contact discontinuities, as discussed in the previous sections, the data computed by the flow solver must be corrected to account for the jump conditions. This fact has led us in the past to include the corrected data in the evaluation of the nodal gradients in these cases [1]. In the aforementioned reference, which did not account for discontinuity interactions, the gradient term of Eq.…”

“…This approach turns out to be best suited to vertex-centered CFD solvers, but it can be problematic if a different type of data representation is used. The eDIT approach, which has been described by the authors in [1] and is further developed in this paper, allows to overcome this limitation, because it avoids re-meshing by using ideas borrowed from immersed and embedded boundary methods developed to allow a flexible management of complex geometries. These two approaches rely on different philosophies, however.…”