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

Mirco Ciallella, Mario Ricchiuto, Renato Paciorri, Aldo Bonfiglioli

TL;DR

The paper tackles the difficulty of accurately simulating shocks in compressible flows by introducing Extrapolated Shock-Tracking (eST), a method that unifies shock-fitting concepts with embedded-boundary ideas to avoid remeshing. eST uses two surrogate boundaries and Taylor-based extrapolation to couple upstream and downstream flow fields while explicitly tracking the shock surface, enabling near-shock-fitting accuracy on general solvers and meshes. The authors validate eST on multiple 2D problems (including quasi-1D nozzle, planar source flow, and blunt-body hypersonics) and demonstrate favorable accuracy in the shock-downstream region, along with a meaningful cost-to-accuracy trade-off and successful hybrid fits for shock-interaction scenarios. The work advances shock-tracking by delivering a framework that is largely independent of the data structure of the underlying solver, opening avenues for broader adoption across CFD codes and future 3D extensions with moving and interacting discontinuities.

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 re-meshing 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 Shock Tracking: Bridging shock-fitting and embedded boundary methods

TL;DR

The paper tackles the difficulty of accurately simulating shocks in compressible flows by introducing Extrapolated Shock-Tracking (eST), a method that unifies shock-fitting concepts with embedded-boundary ideas to avoid remeshing. eST uses two surrogate boundaries and Taylor-based extrapolation to couple upstream and downstream flow fields while explicitly tracking the shock surface, enabling near-shock-fitting accuracy on general solvers and meshes. The authors validate eST on multiple 2D problems (including quasi-1D nozzle, planar source flow, and blunt-body hypersonics) and demonstrate favorable accuracy in the shock-downstream region, along with a meaningful cost-to-accuracy trade-off and successful hybrid fits for shock-interaction scenarios. The work advances shock-tracking by delivering a framework that is largely independent of the data structure of the underlying solver, opening avenues for broader adoption across CFD codes and future 3D extensions with moving and interacting discontinuities.

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 re-meshing 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.
Paper Structure (25 sections, 24 equations, 29 figures, 2 tables)

This paper contains 25 sections, 24 equations, 29 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Generalities
  3. Unstructured shock-fitting algorithm
  4. Shifted-boundary method
  5. Extrapolated Shock-Tracking
  6. Geometrical setting
  7. Cell removal around the shock front
  8. Computation of the tangent and normal unit vectors
  9. Solution update using the CFD solver
  10. Solution transfer from/to the shock to/from the surrogates
  11. First transfer: from the surrogate boundaries to the shock
  12. Shock calculation
  13. Second transfer: from the shock to the surrogate boundaries
  14. Shock displacement
  15. Re-interpolation of nodes crossed by the shock
  16. ...and 10 more sections

Figures (29)

  • Figure 1: Examples of fitted discontinuities on unstructured meshes.
  • Figure 2: The SBM: the surrogate and actual boundaries, and the distance vector $\vec{d}$.
  • Figure 3: The computational-mesh is obtained by removing those cells of the background mesh that are crossed by the shock-mesh.
  • Figure 4: Test needed to check whether point $P_{i+1}$ belongs to the domain of dependence of $P_i$.
  • Figure 5: The solution update is performed using the computational mesh.
  • ...and 24 more figures