Jordi Vila Pérez defended his PhD thesis Low and high-order hybridised methods for compressible flows (pdf), supervised by Professors Antonio Huerta and Matteo Giacomini, on February 10, 2021, within the UPC doctoral program in Applied Mathematics. Currently, he is a Postdoctoral Associate at the Aerospace Computational Design Laboratory of MIT, where he is working on a project sponsored by the National Science Foundation under the supervision of Jaume Peraire and Ngoc Cuong Nguyen.
Thesis Summary
The aerospace community is challenged as of today for being able to manage accurate overnight computational fluid dynamics (CFD) simulations of compressible flow problems. Well-established CFD solvers based on second-order finite volume (FV) methods provide accurate approximations of steady-state turbulent flows, but are incapable to produce reliable predictions of the full flight envelope. Alternatively, promising high-order discretisations, claimed to permit feasible high-fidelity simulations of unsteady turbulent flows, are still subject to strong limitations in robustness and efficiency, placing their level of maturity far away from industrial requirements. In consequence, the CFD paradigm is immersed at this point into the crossroads outlined by the inherent limitations of low-order methods and the yet immature state of high-order discretisations. Accordingly, this thesis develops a twofold strategy for the high-fidelity simulation of compressible flows introducing two methodologies, at the low and high-order levels, respectively, based on hybridised formulations.
First, a new finite volume paradigm, the face-centred finite volume (FCFV) method, is proposed for the formulation of steadystate compressible flows. The present methodology describes a hybrid mixed FV formulation that, following a hybridisation process, defines the unknowns of the problem at the face barycenters. The problem variables, i.e. the conservative quantities and the stress tensor and heat flux in the viscous case, are retrieved with optimal first-order accuracy inside each cell by means of an inexpensive postprocessing step without need of reconstruction of the gradients. Hence, the FCFV solver preserves the accuracy of the approximation even in presence of highly stretched or distorted cells, providing a solver insensitive to mesh quality. In addition, the FCFV method is a monotonicity-preserving scheme, leading to non-oscillatory approximations of sharp gradients without resorting to shock capturing or limiting techniques. Finally, the method is robust in the incompressible limit and is capable of computing accurate solutions for flows at low Mach number without the need of introducing specific pressure correction strategies.
In parallel, the high-order hybridisable discontinuous Galerkin (HDG) method is reviewed in the context of compressible flows,
presenting an original unified framework for the derivation of Riemann solvers in hybridised formulations. The framework includes, for the first time in an HDG context, the HLL and HLLEM Riemann solvers as well as the traditional Lax-Friedrichs and Roe solvers. The positivity preserving properties of HLL-type Riemann solvers are displayed, demonstrating their superiority with respect to Roe in supersonic cases. In addition, HLLEM specifically outstands in the approximation of boundary layers because of its shear preservation, which confers it an increased accuracy with respect to HLL and Lax-Friedrichs.
An extensive set of numerical benchmarks of practical interest is introduced along this study in order to validate both the low and high-order approaches. Different examples of compressible flows in a great variety of regimes, from inviscid to viscous laminar flows, from subsonic to supersonic speeds, are presented to verify the accuracy properties of each of the proposed methodologies and the performance of the introduced Riemann solvers.
Highlighted publication: Vila-Pérez, J., Giacomini, M., Sevilla, R. and Huerta, A.: Hybridisable Discontinuous Galerkin Formulation of Compressible Flows. Archives of Computational Methods
in Engineering 28, 753-784 (2021). doi.