Beaugendre, Héloïse

Bridel-Bertomeu, Thibault

Nauleau, Florent

Tierny, Julien

Vivodtzev, Fabien

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HAL Id: hal-03434853https://hal.inria.fr/hal-03434853Submitted on 22 Nov 2021HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Topological Data Analysis for numerical methodcomparisons of 2D turbulent flowsFlorent Nauleau, Thibault Bridel-Bertomeu, Fabien Vivodtzev, HéloïseBeaugendre, Julien TiernyTo cite this version:Florent Nauleau, Thibault Bridel-Bertomeu, Fabien Vivodtzev, Héloïse Beaugendre, Julien Tierny.Topological Data Analysis for numerical method comparisons of 2D turbulent flows. Journée Visu2021, Jun 2021, Journée virtuelle, France. ￿hal-03434853￿Journée Visu 2021Topological Data Analysis for numerical method comparisons of 2Dturbulent flowsFlorent Nauleau*CEA, FranceThibault Bridel-Bertomeu†CEA, FranceFabien Vivodtzev‡CEA, FranceHéloı̈se Beaugendre§Université de Bordeaux, FranceJulien Tierny¶CNRS, Sorbonne Université, LIP6, FranceFigure 1: (left) Test cases matrix heat map and (right) developed Kelvin-Helmholtz instability at t = 5 obtained for the different testcases - visualization of the local enstrophy E .1 INTRODUCTIONDirect numerical simulations of two-dimensional inviscid mixinglayers have been conducted to help understanding the different struc-tures composing 2D turbulence and the interactions between thevortices over time. Because of its intrinsic multiscale nature, nu-merically simulating turbulence stemming from the Euler equationscan be a very challenging task. Common workaround for this chal-lenge are the increase of degrees of freedom, the use of a high-orderreconstruction scheme (see e.g. [6, 12, 15]) and/or the use of a low-dissipation Riemann solver (see e.g. [18]) for finite-volume fluxes.However, there are many choices available to researchers and anyof these ingredients increases the walltime of the simulation. There-fore, a thorough study of what each brings to the resolution of 2Dturbulence needs to be done in order to find the best combinationyielding the best resolution with the least computational time.1.1 ContributionsIn this ongoing work we investigate a topological data analysis ap-proach to help the scientist choose the best combination of numericalmethods to simulate turbulence. To do so, we look into the capabili-ties of WENO-Z [1,3,4,9] and TENO reconstruction schemes of 5th-and 7th-order [6–8] combined with different Riemann flux solvers(AUSM-up, HLL) [11, 18, 19]. Evaluating and comparing thosevarious approaches can be very costly and it uses time not spentstudying the targeted physics, i.e. here the behavior of 2D inviscidturbulent flows. Moreover comparing the geometrically complexshapes produced over time by a Kelvin-Helmholtz instability can be*e-mail: florent.nauleau@cea.fr†e-mail: thibault.bridel-bertomeu@cea.fr‡e-mail: fabien.vivodtzev@cea.fr§e-mail: heloise.beaugendre@math.u-bordeaux.fr¶e-mail: julien.tierny@sorbonne-universite.frvery tedious. This is why we offer in the present study a compari-son procedure built on a topological abstraction of the features ofthe flows that eases, if not suppresses, the difficulty of comparingturbulent structures.1.2 Numerical simulation of a Kelvin-Helmholtz instabil-ityIn this study, the two-dimensional compressible unsteady Eulerequations for inviscid flows are solved. The system can be writtenas (see e.g. [13]):Ut +Fx +Gy = 0, (1)where the subscripts indicate differentiation, U is the vector of con-servative dimensionless variables and F and G represent the inviscidfluxes in x− and y−direction respectively. Those vectors are definedas follow:U = ρρuρvρE , F = ρuρu2 + pρuv(ρE + p)u , G = ρvρuvρv2 + p(ρE + p)v .(2)In the above expressions, t denotes the time and x and y are theCartesian coordinates. ρ denotes density, u and v denote the x− andy−direction velocity components, E denotes the specific total energyand p denotes the static pressure. The aforementioned mathematicalmodel is described as it is implemented in the in-house code HYPE-RION (HYPERsonic vehicle design with Immersed bOuNdaries)whose primary capabilities as a massively parallel structured solverusing immersed boundary conditions have already been discussed byBridel-Bertomeu [2]. The present study uses only regular Cartesiangrids with constant grid spacings in both directions of space, ∆x and∆y, and will not rely on any immersed boundary condition duringthe computations presented later. This being said, the finite-volumemethod [10, 18, 20] is then employed for space discretization of thecompressible Euler equations (1).The 2D turbulence at the heart of this study is generated using aKelvin-Helmholtz instability (see [14] for a complete description of1Journée Visu 2021Figure 2: (left) Segmentation by merge tree and (right) segmentationby ascending Morse-Smale complex on the local enstrophy E of aKelvin-Helmholtz instability at t = 5.the initialization) simulated with high-order low-dissipation WENO-Z and TENO reconstruction schemes of 5th- and 7th-order - seefigure 1 for representations of the flow at t = 5. The numerical fluxesbetween the cells are obtained using either the HLL or AUSM-upRiemann solvers. To emulate turbulence in a infinite medium, allboundary conditions are set as periodic. One common measureof turbulence in two dimensions that we will rely on is the localenstrophy E , defined locally as the square of the flow vorticity, i.e.E = 0.5 |∇×u|2.1.3 Topological Data AnalysisTopological Data Analysis (TDA) is a recent set of techniques [16]which focus on structural features in data. Thanks to advancedconcepts such as persistent homology [5], TDA provides tools forthe multi-scale representation of the structural features of interestsuch as the turbulence in a flow. It also introduces distance metricsin between topological abstractions to compare such structures. Inthis work, we used several established techniques, readily availablein the “Topology ToolKit” (TTK) [17].2 SIMULATION COMPARISON2.1 TDA pipelineWe first compute the enstrophy E to emphasize the features of theturbulence located around vortices. Then we compute the persistencediagram for each simulation and group all diagrams. We use the per-sistence diagram distance matrix (according to the L2-Wassersteindistance) to extract the topological distance of the enstrophy per-sistence between all the runs. The test cases matrix heat map canthen be generated (see left part of figure 1): it gives a matrix repre-sentation from black (topologically close) to red (topologically far)of the aforementioned distances between each and every cases. Tocomplete the pipeline we use a dimension reduction filter to clearlyidentify groups that are shown on figure 3.2.2 InterpretationAs shown on figure 3, using a dimension reduction procedure, wecould distinguish 3 groups in the matrix heat map. The blue squarerepresents the runs done with WENO-Z of 5th- and 7th-order andthe HLL solver, the green square represents the runs done withWENO-Z of 5th- and 7th-order and the AUSM-up solver and withTENO of 5th- and 7th-order and the HLL solver. The last red squaregroups together the runs done with the TENO of 5th- and 7th-orderand the AUSM-up solver. These 3 groups we obtain using thetopological analysis are consistent with the simple visual analysisof the results. Indeed, the HLL is more dissipative and leads tomore difficulties in capturing the turbulence than the AUSM-upsolver. What is more, the TENO schemes are by definition more fitto capture turbulence than the WENO-Z schemes [6] but they arealso more computationally expensive. It is interesting to see thatthere are no topological differences between the results obtainedwith the same reconstruction schemes but at different orders - theFigure 3: (left) Matrix of distances between the persistence diagramsof all numerical methods and (right) the corresponding planar view,obtained by multi-dimensional scaling.7th-order schemes are more computationally expensive however. Weobserve on the other hand that the Riemann solver seems to be ofutmost importance to capture the turbulence properly: contrary tochanging the order of the reconstruction, changing the Riemannsolvers had a marked impact on the end result.3 CONCLUSION AND FUTURE WORKIn this work, we showed that established methods from TDA couldbe used to compare numerical methods in CFD. Based on this com-parison we are going to focus on the 5th-order TENO scheme incombination with the AUSM-up solver and further investigate theturbulence more precisely while minimizing the computation time.The ratio between a 7th-order and a 5th-order TENO scheme is in-deed approximately 1.3 when it comes to computational time. ThisTDA pipeline is now going to be used as a preliminary setup tofurther investigations. Coming soon we will work with the 5th-orderTENO and the AUSM-up solver to extract vortices out of the data.We will use the ascending Morse-Smale segmentation and the mergetree segmentation to try and understand the energy transfer occur-ring between the small and the big structures in two-dimensionalturbulence - see figure 2.4 ACKNOWLEDGMENTSThis work is partially supported by the European Commission grantERC-2019-COG ”TORI” (ref. 863464).REFERENCES[1] R. Borges, M. Carmona, B. Costa, and W. S. Don. An improvedweighted essentially non-oscillatory scheme for hyperbolic conser-vation laws. Journal of Computational Physics, 227(6):3191–3211,2008.[2] T. Bridel-Bertomeu. Immersed boundary conditions for hypersonicflows using eno-like least-square reconstruction. Computers & Fluids,215:104794, 2021.[3] M. Castro, B. Costa, and W. S. Don. High order weighted essentiallynon-oscillatory weno-z schemes for hyperbolic conservation laws. Jour-nal of Computational Physics, 230(5):1766–1792, 2011.[4] W.-S. Don and R. Borges. Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. Journal of Compu-tational Physics, 250:347–372, 2013.[5] H. Edelsbrunner and J. Harer. Computational Topology: An Introduc-tion. AMS, 2009.[6] L. Fu, X. Y. Hu, and N. A. Adams. A family of high-order targeted enoschemes for compressible-fluid simulations. Journal of ComputationalPhysics, 305:333–359, 2016.[7] L. Fu, X. Y. Hu, and N. A. Adams. Targeted eno schemes with tai-lored resolution property for hyperbolic conservation laws. Journal ofComputational Physics, 349:97–121, 2017.[8] L. Fu, X. Y. Hu, and N. A. Adams. A new class of adaptive high-ordertargeted eno schemes for hyperbolic conservation laws. Journal ofComputational Physics, 374:724–751, 2018.[9] F. Hu. A note on weno-z scheme. Applied Mathematics and Computa-tion, 396:125886, 2021.2Journée Visu 2021[10] R. J. LeVeque et al. Finite volume methods for hyperbolic problems,vol. 31. Cambridge university press, 2002.[11] M.-S. Liou. A sequel to ausm, part ii: Ausm+-up for all speeds. Journalof computational physics, 214(1):137–170, 2006.[12] X.-g. Liu, S. Osher, and T. Chan. Weighted essentially non-oscillatoryschemes. Journal of computational physics, 115(1):200–212, 1994.[13] K. Masatsuka. I do Like CFD, vol. 1, vol. 1. Lulu. com, 2013.[14] O. San and K. Kara. Evaluation of riemann flux solvers for wenoreconstruction schemes: Kelvin–helmholtz instability. Computers &Fluids, 117:24–41, 2015.[15] C.-W. Shu and S. Osher. Efficient implementation of essentiallynon-oscillatory shock-capturing schemes, ii. In Upwind and High-Resolution Schemes, pp. 328–374. Springer, 1989.[16] J. Tierny. Topological Data Analysis for Scientific Visualization.Springer, 2018.[17] J. Tierny, G. Favelier, J. A. Levine, C. Gueunet, and M. Michaux.The Topology ToolKit. IEEE TVCG, 2017. https://topology-tool-kit.github.io/.[18] E. F. Toro. Riemann solvers and numerical methods for fluid dynamics:a practical introduction. Springer Science & Business Media, 2013.[19] E. F. Toro, M. Spruce, and W. Speares. Restoration of the contactsurface in the hll-riemann solver. Shock waves, 4(1):25–34, 1994.[20] J. A. Trangenstein. Numerical Solution of Hyperbolic Partial Differen-tial Equations. Cambridge University Press, 2007.3

Topological Data Analysis for numerical method comparisons of 2D turbulent flows

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Topological Data Analysis for numerical method comparisons of 2D turbulent flows

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