Numerical evidence of anomalous energy dissipation in incompressible Euler flows: towards grid-converged results for the inviscid Taylor-Green problem. (English) Zbl 1515.76125
This research work focuses on global quantities such as the temporal evolution of the kinetic energy, avoiding geometrical complexities in visualisation and striving towards a clearer indication of singular behaviour. The present method might be computationally less demanding to resolve the kinetic energy than the vorticity in numerical simulations. The theory and analysis have been presented by number of computational experiments.
Reviewer: Prabhat Kumar Mahanti (Saint John)
MSC:
| 76M99 | Basic methods in fluid mechanics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76B99 | Incompressible inviscid fluids |
| 76F99 | Turbulence |
Keywords:
computational method; Navier-Stokes equations; turbulence theory; kinetic energy evolution; vorticityReferences:
| [1] | Agafontsev, D.S., Kuznetsov, E.A. & Mailybaev, A.A.2015Development of high vorticity structures in incompressible 3D Euler equations. Phys. Fluids27 (8), 085102. · Zbl 1326.65136 |
| [2] | Ainsworth, M.2004Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys.198 (1), 106-130. · Zbl 1058.65103 |
| [3] | Alzetta, G., et al.2018The deal.II library, version 9.0. J. Numer. Maths26 (4), 173-184. · Zbl 1410.65363 |
| [4] | Arndt, D., Bangerth, W., Davydov, D., Heister, T., Heltai, L., Kronbichler, M., Maier, M., Pelteret, J.-P., Turcksin, B. & Wells, D.2021The deal.II finite element library: design, features, and insights. Comput. Maths. Applics.81, 407-422. · Zbl 1524.65002 |
| [5] | Arndt, D., Fehn, N., Kanschat, G., Kormann, K., Kronbichler, M., Munch, P., Wall, W.A. & Witte, J.2020 ExaDG: high-order discontinuous Galerkin for the exa-scale. In Software for Exascale Computing - SPPEXA 2016-2019 (ed. H.-J. Bungartz, S. Reiz, B. Uekermann, P. Neumann & W.E. Nagel), pp. 189-224. Springer. |
| [6] | Beale, J.T., Kato, T. & Majda, A.1984Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys.94 (1), 61-66. · Zbl 0573.76029 |
| [7] | Boratav, O.N. & Pelz, R.B.1994Direct numerical simulation of transition to turbulence from a high-symmetry initial condition. Phys. Fluids6 (8), 2757-2784. · Zbl 0845.76065 |
| [8] | Brachet, M.E.1991Direct simulation of three-dimensional turbulence in the Taylor-Green vortex. Fluid Dyn. Res.8 (1-4), 1-8. |
| [9] | Brachet, M.E., Meiron, D.I., Orszag, S.A., Nickel, B.G., Morf, R.H. & Frisch, U.1983Small-scale structure of the Taylor-Green vortex. J. Fluid Mech.130, 411-452. · Zbl 0517.76033 |
| [10] | Brachet, M.E., Meneguzzi, M., Vincent, A., Politano, H. & Sulem, P.L.1992Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows. Phys. Fluids A4 (12), 2845-2854. · Zbl 0775.76026 |
| [11] | Brenier, Y., De Lellis, C. & Székelyhidi, L.2011Weak-strong uniqueness for measure-valued solutions. Commun. Math. Phys.305 (2), 351-361. · Zbl 1219.35182 |
| [12] | Brenner, M.P., Hormoz, S. & Pumir, A.2016Potential singularity mechanism for the Euler equations. Phys. Rev. Fluids1, 084503. |
| [13] | Brown, D.L.1995Performance of under-resolved two-dimensional incompressible flow simulations. J. Comput. Phys.122 (1), 165-183. · Zbl 0849.76043 |
| [14] | Buckmaster, T., De Lellis, C. & Székelyhidi, L. Jr.2016Dissipative Euler flows with Onsager-critical spatial regularity. Commun. Pure Appl. Maths69 (9), 1613-1670. · Zbl 1351.35109 |
| [15] | Buckmaster, T., De Lellis, C., Székelyhidi, L. Jr. & Vicol, V.2018Onsager’s conjecture for admissible weak solutions. Commun. Pure Appl. Maths72 (2), 229-274. · Zbl 1480.35317 |
| [16] | Buckmaster, T., Masmoudi, N., Novack, M. & Vicol, V.2021 Non-conservative \(H^1/2\)-weak solutions of the incompressible 3D Euler equations. arXiv:2101.09278. |
| [17] | Buckmaster, T. & Vicol, V.2019Nonuniqueness of weak solutions to the Navier-Stokes equation. Ann. Maths189 (1), 101-144. · Zbl 1412.35215 |
| [18] | Buckmaster, T. & Vicol, V.2020Convex integration and phenomenologies in turbulence. EMS Surv. Math. Sci.6 (1), 173-263. · Zbl 1440.35231 |
| [19] | Burgers, J.M.1948A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech.1, 171-199. |
| [20] | Bustamante, M.D. & Brachet, M.2012Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem. Phys. Rev. E86, 066302. |
| [21] | Campolina, C.S. & Mailybaev, A.A.2018Chaotic blowup in the 3D incompressible Euler equations on a logarithmic lattice. Phys. Rev. Lett.121, 064501. |
| [22] | Chalmers, N., Agbaglah, G., Chrust, M. & Mavriplis, C.2019A parallel hp-adaptive high order discontinuous Galerkin method for the incompressible Navier-Stokes equations. J. Comput. Phys. X2, 100023. · Zbl 07785505 |
| [23] | Chapelier, J.-B., De La Llave Plata, M. & Renac, F.2012 Inviscid and viscous simulations of the Taylor-Green vortex flow using a modal discontinuous Galerkin approach. In 42nd AIAA Fluid Dynamics Conference and Exhibit, AIAA Paper 2012-3073. |
| [24] | Cheskidov, A., Constantin, P., Friedlander, S. & Shvydkoy, R.2008Energy conservation and Onsager’s conjecture for the Euler equations. Nonlinearity21 (6), 1233-1252. · Zbl 1138.76020 |
| [25] | Cichowlas, C., Bonaïti, P., Debbasch, F. & Brachet, M.2005Effective dissipation and turbulence in spectrally truncated Euler flows. Phys. Rev. Lett.95, 264502. |
| [26] | Cichowlas, C. & Brachet, M.-E.2005Evolution of complex singularities in Kida-Pelz and Taylor-Green inviscid flows. Fluid Dyn. Res.36 (4-6), 239-248. · Zbl 1153.76393 |
| [27] | Constantin, P., Weinan, E. & Titi, E.S.1994Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys.165 (1), 207-209. · Zbl 0818.35085 |
| [28] | Coppola, G., Capuano, F. & De Luca, L.2019Discrete energy-conservation properties in the numerical simulation of the Navier-Stokes equations. Appl. Mech. Rev.71 (1), 010803. |
| [29] | Daneri, S., Runa, E. & Székelyhidi, L.2021Non-uniqueness for the Euler equations up to Onsager’s critical exponent. Ann. PDE7 (1), 1-44. · Zbl 1478.35171 |
| [30] | De Lellis, C. & Székelyhidi, L. Jr.2013Dissipative continuous Euler flows. Invent. Math.193 (2), 377-407. · Zbl 1280.35103 |
| [31] | De Lellis, C. & Székelyhidi, L. Jr.2014Dissipative Euler flows and Onsager’s conjecture. J. Eur. Math. Soc.16 (7), 1467-1505. · Zbl 1307.35205 |
| [32] | Diperna, R.J. & Majda, A.J.1987Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys.108 (4), 667-689. · Zbl 0626.35059 |
| [33] | Drivas, T.D. & Eyink, G.L.2019An onsager singularity theorem for Leray solutions of incompressible Navier-Stokes. Nonlinearity32 (11), 4465. · Zbl 1428.35282 |
| [34] | Drivas, T.D. & Nguyen, H.Q.2019Remarks on the emergence of weak Euler solutions in the vanishing viscosity limit. J. Nonlinear Sci.29 (2), 709-721. · Zbl 1448.76050 |
| [35] | Dubrulle, B.2019Beyond Kolmogorov cascades. J. Fluid Mech.867, P1. · Zbl 1415.76248 |
| [36] | Duchon, J. & Robert, R.2000Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity13 (1), 249-255. · Zbl 1009.35062 |
| [37] | Eggers, J.2018Role of singularities in hydrodynamics. Phys. Rev. Fluids3, 110503. |
| [38] | Eyink, G.L.1994Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer. Physica D78 (3), 222-240. · Zbl 0817.76011 |
| [39] | Eyink, G.L.2008Dissipative anomalies in singular Euler flows. Physica D237 (14), 1956-1968. · Zbl 1143.76403 |
| [40] | Eyink, G.L. & Sreenivasan, K.R.2006Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys.78, 87-135. · Zbl 1205.01032 |
| [41] | Fehn, N., Heinz, J., Wall, W.A. & Kronbichler, M.2021High-order arbitrary Lagrangian-Eulerian discontinuous Galerkin methods for the incompressible Navier-Stokes equations. J. Comput. Phys.430, 110040. · Zbl 07506519 |
| [42] | Fehn, N., Kronbichler, M., Lehrenfeld, C., Lube, G. & Schroeder, P.W.2019High-order DG solvers for under-resolved turbulent incompressible flows: a comparison of \(L^2\) and \(H\)(div) methods. Intl J. Numer. Meth. Fluids91 (11), 533-556. |
| [43] | Fehn, N., Munch, P., Wall, W.A. & Kronbichler, M.2020Hybrid multigrid methods for high-order discontinuous Galerkin discretizations. J. Comput. Phys.415, 109538. · Zbl 1440.65135 |
| [44] | Fehn, N., Wall, W.A. & Kronbichler, M.2017On the stability of projection methods for the incompressible Navier-Stokes equations based on high-order discontinuous Galerkin discretizations. J. Comput. Phys.351, 392-421. · Zbl 1380.65204 |
| [45] | Fehn, N., Wall, W.A. & Kronbichler, M.2018aEfficiency of high-performance discontinuous Galerkin spectral element methods for under-resolved turbulent incompressible flows. Intl J. Numer. Meth. Fluids88 (1), 32-54. · Zbl 1415.76451 |
| [46] | Fehn, N., Wall, W.A. & Kronbichler, M.2018bRobust and efficient discontinuous Galerkin methods for under-resolved turbulent incompressible flows. J. Comput. Phys.372, 667-693. · Zbl 1415.76451 |
| [47] | Fernandez, P., Nguyen, N.C. & Peraire, J.2018 On the ability of discontinuous Galerkin methods to simulate under-resolved turbulent flows. arXiv:1810.09435. |
| [48] | Fischer, P. & Mullen, J.2001Filter-based stabilization of spectral element methods. C. R. Acad. Sci. I332 (3), 265-270. · Zbl 0990.76064 |
| [49] | Flad, D. & Gassner, G.2017On the use of kinetic energy preserving DG-schemes for large eddy simulation. J. Comput. Phys.350, 782-795. · Zbl 1380.76019 |
| [50] | Frigo, M. & Johnson, S.G.2005The design and implementation of FFTW3. Proc. IEEE93 (2), 216-231. |
| [51] | Fu, G.2019An explicit divergence-free DG method for incompressible flow. Comput. Meth. Appl. Mech. Engng345, 502-517. · Zbl 1440.76060 |
| [52] | Gibbon, J.D.2008The three-dimensional Euler equations: where do we stand?Physica D237 (14), 1894-1904. · Zbl 1143.76389 |
| [53] | Grafke, T., Homann, H., Dreher, J. & Grauer, R.2008Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods. Physica D237 (14), 1932-1936. · Zbl 1143.76515 |
| [54] | Grauer, R., Marliani, C. & Germaschewski, K.1998Adaptive mesh refinement for singular solutions of the incompressible Euler equations. Phys. Rev. Lett.80, 4177-4180. |
| [55] | Guzmán, J., Shu, C.-W. & Sequeira, F.A.2016H(div) conforming and DG methods for incompressible Euler’s equations. IMA J. Numer. Anal.37 (4), 1733-1771. · Zbl 1433.76085 |
| [56] | Hou, T.Y. & Li, R.2008Blowup or no blowup? The interplay between theory and numerics. Physica D237 (14), 1937-1944. · Zbl 1143.76390 |
| [57] | Isett, P.2017 Nonuniqueness and existence of continuous, globally dissipative Euler flows. arXiv:1710.11186. · Zbl 1367.35001 |
| [58] | Isett, P.2018A proof of Onsager’s conjecture. Ann. Maths188 (3), 871-963. · Zbl 1416.35194 |
| [59] | Josserand, C., Pomeau, Y. & Rica, S.2020Finite-time localized singularities as a mechanism for turbulent dissipation. Phys. Rev. Fluids5, 054607. · Zbl 0948.76098 |
| [60] | Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K. & Uno, A.2003Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Phys. Fluids15 (2), L21-L24. · Zbl 1185.76191 |
| [61] | Karniadakis, G.E., Israeli, M. & Orszag, S.A.1991High-order splitting methods for the incompressible Navier-Stokes equations. J. Comput. Phys.97 (2), 414-443. · Zbl 0738.76050 |
| [62] | Karniadakis, G.E. & Sherwin, S.J.2005Spectral/HP Element Methods for Computational Fluid Dynamics, 2nd edn. Oxford University Press. · Zbl 1116.76002 |
| [63] | Kerr, R.M.1993Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids A5 (7), 1725-1746. · Zbl 0800.76083 |
| [64] | Kerr, R.M.2013Bounds for Euler from vorticity moments and line divergence. J. Fluid Mech.729, R2. · Zbl 1291.76073 |
| [65] | Kerr, R.M. & Hussain, F.1989Simulation of vortex reconnection. Physica D37 (1), 474-484. |
| [66] | Kida, S.1985Three-dimensional periodic flows with high-symmetry. J. Phys. Soc. Japan54 (6), 2132-2136. |
| [67] | Kolmogorov, A.N.1991The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Proc. R. Soc. Lond. A434 (1890), 9-13. · Zbl 1142.76389 |
| [68] | Krais, N., Schnücke, G., Bolemann, T. & Gassner, G.J.2020Split form ALE discontinuous Galerkin methods with applications to under-resolved turbulent low-Mach number flows. J. Comput. Phys.421, 109726. · Zbl 1437.65142 |
| [69] | Krank, B., Fehn, N., Wall, W.A. & Kronbichler, M.2017A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow. J. Comput. Phys.348, 634-659. · Zbl 1380.76040 |
| [70] | Kronbichler, M. & Kormann, K.2019Fast matrix-free evaluation of discontinuous Galerkin finite element operators. ACM Trans. Math. Softw.45 (3), 29:1-29:40. · Zbl 1486.65253 |
| [71] | Kuzzay, D., Saw, E.-W., Martins, F.J.W.A., Faranda, D., Foucaut, J.-M., Daviaud, F. & Dubrulle, B.2017New method for detecting singularities in experimental incompressible flows. Nonlinearity30 (6), 2381. · Zbl 1369.76049 |
| [72] | Lamballais, E., Dairay, T., Laizet, S. & Vassilicos, J.C.2019 Implicit/explicit spectral viscosity and large-scale SGS effects. In Direct and Large-Eddy Simulation XI (ed. M.V. Salvetti, V. Armenio, J. Fröhlich, B.J. Geurts & H. Kuerten), pp. 107-113. Springer. |
| [73] | Larios, A., Petersen, M.R., Titi, E.S. & Wingate, B.2018A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization. Theor. Comput. Fluid Dyn.32 (1), 23-34. |
| [74] | Lions, P.-L.1996Mathematical Topics in Fluid Mechanics. Volume 1: Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, vol. 3. Oxford University Press. · Zbl 0866.76002 |
| [75] | Luo, G. & Hou, T.Y.2014Potentially singular solutions of the 3D axisymmetric Euler equations. Proc. Natl Acad. Sci. USA111 (36), 12968-12973. · Zbl 1431.35115 |
| [76] | Manzanero, J., Ferrer, E., Rubio, G. & Valero, E.2020Design of a Smagorinsky spectral vanishing viscosity turbulence model for discontinuous Galerkin methods. Comput. Fluids200, 104440. · Zbl 1519.76095 |
| [77] | Mckeown, R., Ostilla-Mónico, R., Pumir, A., Brenner, M.P. & Rubinstein, S.M.2018Cascade leading to the emergence of small structures in vortex ring collisions. Phys. Rev. Fluids3, 124702. |
| [78] | Moffatt, H.K.2019Singularities in fluid mechanics. Phys. Rev. Fluids4, 110502. |
| [79] | Morf, R.H., Orszag, S.A. & Frisch, U.1980Spontaneous singularity in three-dimensional inviscid, incompressible flow. Phys. Rev. Lett.44, 572-575. · Zbl 1404.76030 |
| [80] | Moura, R., Mengaldo, G., Peiró, J. & Sherwin, S.2017a An LES setting for DG-based implicit LES with insights on dissipation and robustness. In Spectral and High Order Methods for Partial Differential Equations (ed. M.L. Bittencourt, N.A. Dumont & J.S. Hesthaven), pp. 161-173. Springer. · Zbl 1388.76101 |
| [81] | Moura, R.C., Mengaldo, G., Peiró, J. & Sherwin, S.J.2017bOn the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES/under-resolved DNS of Euler turbulence. J. Comput. Phys.330, 615-623. · Zbl 1378.76036 |
| [82] | Murugan, S.D., Frisch, U., Nazarenko, S., Besse, N. & Ray, S.S.2020Suppressing thermalization and constructing weak solutions in truncated inviscid equations of hydrodynamics: lessons from the Burgers equation. Phys. Rev. Res.2, 033202. |
| [83] | Onsager, L.1949Statistical hydrodynamics. Il Nuovo Cimento6, 279-287. |
| [84] | Orlandi, P.2009Energy spectra power laws and structures. J. Fluid Mech.623, 353-374. · Zbl 1157.76342 |
| [85] | Orlandi, P., Pirozzoli, S. & Carnevale, G.F.2012Vortex events in Euler and Navier-Stokes simulations with smooth initial conditions. J. Fluid Mech.690, 288-320. · Zbl 1241.76326 |
| [86] | Pearson, B.R., Krogstad, P.-Å. & Van De Water, W.2002Measurements of the turbulent energy dissipation rate. Phys. Fluids14 (3), 1288-1290. |
| [87] | Pelz, R.B.2001Symmetry and the hydrodynamic blow-up problem. J. Fluid Mech.444, 299-320. · Zbl 1002.76095 |
| [88] | Pelz, R.B. & Gulak, Y.1997Evidence for a real-time singularity in hydrodynamics from time series analysis. Phys. Rev. Lett.79, 4998-5001. |
| [89] | Piatkowski, S.-M.2019 A spectral discontinuous Galerkin method for incompressible flow with applications to turbulence. PhD thesis, Ruprecht-Karls-Universität Heidelberg. |
| [90] | Ray, S.S., Frisch, U., Nazarenko, S. & Matsumoto, T.2011Resonance phenomenon for the Galerkin-truncated Burgers and Euler equations. Phys. Rev. E84, 016301. |
| [91] | Saw, E.-W., Kuzzay, D., Faranda, D., Guittonneau, A., Daviaud, F., Wiertel-Gasquet, C., Padilla, V. & Dubrulle, B.2016Experimental characterization of extreme events of inertial dissipation in a turbulent swirling flow. Nat. Commun.7, 12466. |
| [92] | Schroeder, P.W.2019 Robustness of high-order divergence-free finite element methods for incompressible computational fluid dynamics. PhD thesis, Georg-August-Universität Göttingen. |
| [93] | Schroeder, P.W. & Lube, G.2018Divergence-free \(H\)(div)-FEM for time-dependent incompressible flows with applications to high Reynolds number vortex dynamics. J. Sci. Comput.75 (2), 830-858. · Zbl 1392.35210 |
| [94] | Shu, C.-W., Don, W.-S., Gottlieb, D., Schilling, O. & Jameson, L.2005Numerical convergence study of nearly incompressible, inviscid Taylor-Green vortex flow. J. Sci. Comput.24 (1), 1-27. · Zbl 1161.76535 |
| [95] | Sreenivasan, K.R.1998An update on the energy dissipation rate in isotropic turbulence. Phys. Fluids10 (2), 528-529. · Zbl 1185.76674 |
| [96] | Sulem, C., Sulem, P.L. & Frisch, H.1983Tracing complex singularities with spectral methods. J. Comput. Phys.50 (1), 138-161. · Zbl 0519.76002 |
| [97] | Taylor, G.I.1935Statistical theory of turbulence. Proc. R. Soc. Lond. A151 (873), 421-444. · JFM 61.0926.02 |
| [98] | Taylor, G.I. & Green, A.E.1937Mechanism of the production of small eddies from large ones. Proc. R. Soc. Lond. A158 (895), 499-521. · JFM 63.1358.03 |
| [99] | Thalabard, S., Bec, J. & Mailybaev, A.A.2020From the butterfly effect to spontaneous stochasticity in singular shear flows. Commun. Phys.3 (1), 122. |
| [100] | Wiedemann, E.2017 Weak-strong uniqueness in fluid dynamics. arXiv:1705.04220. · Zbl 1408.35158 |
| [101] | Winters, A.R., Moura, R.C., Mengaldo, G., Gassner, G.J., Walch, S., Peiro, J. & Sherwin, S.J.2018A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations. J. Comput. Phys.372, 1-21. · Zbl 1415.76461 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
