Lehmkuhl Barba, Oriol

Mira Martinez, Daniel

Silva, Lucas Gasparino F. da

Mira Martínez, Daniel

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Towards a low dissipation FE scheme for scaleresolving turbulent compressible flowsLucas Gasparino F. da Silva∗†, Oriol Lehmkuhl∗, Daniel Mira Martinez∗∗Barcelona Supercomputing Center, Barcelona, Spain†Universitat Polite`cnica de Catalunya, Barcelona, SpainE-mail: {lucas.gasparino, oriol.lehmkuhl, daniel.mira}@bsc.esKeywords—Compressible flow, CFD, LES, FEM, NumericalSimulation, Hyperbolic problems.I. EXTENDED ABSTRACTThis work focuses on developing a highly accurate FiniteElement numerical scheme for Large-Eddy simulation of tur-bulent compressible flows, where low numerical dissipationis required. The proposition is to develop a low-dissipationstabilizing term into an algorithm, implement it on a readily-available CFD code and test results to assess performance andsuitability.A. Discretisation basicsIn general, numerical solution of partial differential equationslike the Navier-Stokes set governing fluid flow involves someform of space-time discretisation procedure, where the physicaldomain is divided into a set of points called grid, or a set ofconnected subdomains referred to as elements or volumes. Ifthe problem is transient, then the continuous time must alsobe subdivided and discretised to allow for a local temporalsolution. In finite elements, the simplest form of discretisationprocedure is the Continuous Galerkin method, which yields acentral-like, 2nd order discrete solution to the problem.For convection dominated problems (those containing atransport-like term), direct application of a standard FE tech-nique often results in oscillatory, or fully unstable, solutions,and therefore some form of stabilization is required, generallyreferred to as artificial diffusion, as the added terms have thisexact behavior. Common stabilized methods are the StreamlineUpwind Petrov Galerkin method (SUPG) and its variations,which achieve a stable solution by effectively reducing dis-cretization order, ensuring the scheme remains monotonicitypreserving.Compared to a standard Galerkin discretization procedure,classical stabilized methods such as SUPG and nth Taylor-Galerkin are known to introduce strong diffusion terms intothe discrete Navier-Stokes equations, compatible with theirlow-order nature. Although this has minor influence overRANS and URANS models, Large-Eddy and Direct-Numericalsimulations (LES and DNS) are heavily affected by it, and oneis forced to move towards high-order central schemes, whichhave narrow stability ranges. In the compressible range, the hy-perbolic nature of the problem exacerbates this stability issues,and if shock waves form, or any other form of discontinuityexists, the method fails completely. As Finite Volume schemessolve this issue by means of Riemann solvers on the shock anduse of flux limiters that reduce the scheme’s order locally, theseare quite complex to implement in Finite Elements, and thusa different approach is necessary.B. Entropy Viscosity and uniqueness of solutionsResearch on stabilization of general hyperbolic problems hasled to the development of a low-dissipation nonlinear termnamed Entropy Viscosity, which, when used in a Finite Ele-ments context, behaves in a similar way to a FV flux-limiter,i.e., ensuring that 1st order solutions are employed only at highgradient regions, such as a shock or interface.This method is based on the answer of a purely mathematicalquestion: to find a unique solution for a general hyperbolicequation. As the strong form of this problem is not approach-able, a viscous solution is sought that satisfies a weak formof the reformulated conservation law. The true solution is thenfound by assuming the limiting case where viscosity vanishes.Such viscous solution is referred to as the entropy solution ofthe hyperbolic problem, as it must satisfy Kruzˇkov’s entropycondition.The mathematical theory described above lends itself well to aFinite Element discretisation procedure, as Kruzˇkov’s entropycondition is already stated in an ideal weak form. Moreover,it was found that the theory developed for 1D scalar casescan be directly extended to multi-dimensional hyperbolic (andquasi-hyperbolic) systems like the Euler and Navier Stokesequations, even if a nonlinear source term exists. This indicatesa possible candidate for a low-dissipation algorithm theory.C. Research stepsThe first part of this research project has focused on adaptingthe generic concept developed on hyperbolic equations tothe 1D Euler system describing inviscid compressible flows,as well as studying in practice how well it worked whencompared to other approaches. Following previous work al-ready developed by Guermond et. al. led to a unique imple-mentation in the FE context, which has shown remarkablygood performance overall. It is worth mentioning that thismethod lends itself well to explicit temporal discretization,and was only explored in this context. One interesting, andquite useful, property of the scheme is its capability forhandling nonlinear source terms, such as the ones appearingin combustion problems.6th BSC Severo Ochoa Doctoral Symposium40D. ConclusionThe initial success of this study encourages extendingthe method to 2D and 3D applications of viscous flows,where turbulence is allowed to develop in the latter case.Applications to be tested range from thermally affectedincompressible laminar flows to LES in subsonic regimeand, should these work as expected, chemically reacting andtransonic cases with weak shock formation. This later part isto be coupled with the in-house code Alya, a Finite-Elementsmultiphysics software. Should this research prove successful,the result will be a big step forward to its CFD capabilities incompressible turbulent ranges.II. ACKNOWLEDGMENTThe author would like to acknowledge the effort andsupport of his supervisors, Dr. Lehmkuhl and Dr. Mira, aswell as their invaluable contribution to this work.Lucas Gasparino received his BSc degree inMechanical Engineering from North UniversidadePaulista (UNIP), Sao Paulo, Brazil, in January 2016.He completed his MSc degree in Numerical Methodsand Finite Elements in Engineering Mechanics fromSwansea University, UK in September 2017. Since2018, he has been with the Computational Fluid Dy-namics group of Barcelona Supercomputing Center(BSC) as a PhD student in the CASE department.His experiences include Finite Element modelingand analysis of structures and structural componentsunder static and dynamic conditions, as well as computational analysis ofcomplex flows.6th BSC Severo Ochoa Doctoral Symposium41

Towards a low dissipation FE scheme for scale resolving turbulent compressible flows

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Towards a low dissipation FE scheme for scale resolving turbulent compressible flows

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