×

On reference solutions and the sensitivity of the 2D Kelvin-Helmholtz instability problem. (English) Zbl 1442.76050

Summary: Two-dimensional Kelvin-Helmholtz instability problems are popular examples for assessing discretizations for incompressible flows at high Reynolds number. Unfortunately, the results in the literature differ considerably. This paper presents computational studies of a Kelvin-Helmholtz instability problem with high order divergence-free finite element methods. Reference results in several quantities of interest are obtained for three different Reynolds numbers up to the beginning of the final vortex pairing. A mesh-independent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations. A theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self-organization of 2D turbulence. Possible sources of perturbations that arise in almost any numerical simulation are discussed.

MSC:

76D17 Viscous vortex flows
76D05 Navier-Stokes equations for incompressible viscous fluids
76E20 Stability and instability of geophysical and astrophysical flows
76F20 Dynamical systems approach to turbulence
76F65 Direct numerical and large eddy simulation of turbulence
76M10 Finite element methods applied to problems in fluid mechanics

Software:

Netgen
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Kolmogoroff, A. N., The local structure of turbulence in incompressible viscous fluid for very large Reynold’s numbers, C. R. (Doklady) Acad. Sci. URSS (N.S.), 30, 301-305 (1941) · Zbl 0025.37602
[2] Kolmogoroff, A. N., Dissipation of energy in the locally isotropic turbulence, C. R. (Doklady) Acad. Sci. URSS (N.S.), 32, 16-18 (1941) · Zbl 0063.03292
[3] Moser, R. D.; Kim, J.; Mansour, N. N., Direct numerical simulation of turbulent channel flow up to \(R e_\tau = 590\), Phys. Fluids, 11, 4, 943-945 (1999) · Zbl 1147.76463
[4] Rodi, W.; Ferziger, J. H.; Breuer, M.; Pourquié, M., Status of large eddy simulation: results of a workshop, J. Fluids Eng., 119, 2, 248-262 (1997)
[5] Lesieur, M.; Staquet, C.; Le Roy, P.; Comte, P., The mixing layer and its coherence examined from the point of view of two-dimensional turbulence, J. Fluid Mech., 192, 511-534 (1988)
[6] John, V., An assessment of two models for the subgrid scale tensor in the rational LES model, J. Comput. Appl. Math., 173, 1, 57-80 (2005) · Zbl 1107.76040
[7] Yang, L.; Badia, S.; Codina, R., A pseudo-compressible variational multiscale solver for turbulent incompressible flows, Comput. Mech., 58, 6, 1051-1069 (2016) · Zbl 1398.76042
[8] Gravemeier, V.; Wall, W. A.; Ramm, E., Large eddy simulation of turbulent incompressible flows by a three-level finite element method, Internat. J. Numer. Methods Fluids, 48, 10, 1067-1099 (2005) · Zbl 1070.76034
[9] Schroeder, P. W.; Lube, G., Divergence-Free \(H\)(div)-FEM for time-dependent incompressible flows with applications to High Reynolds number vortex dynamics, J. Sci. Comput., 75, 2, 830-858 (2018) · Zbl 1392.35210
[10] John, V.; Linke, A.; Merdon, C.; Neilan, M.; Rebholz, L. G., On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev., 59, 3, 492-544 (2017) · Zbl 1426.76275
[12] Van Groesen, E., Time-asymptotics and the self-organization hypothesis for 2D Navier-Stokes equations, Physica A, 148, 1-2, 312-330 (1988) · Zbl 0678.76020
[13] Schneider, K.; Farge, M., Numerical simulation of a mixing layer in an adaptive wavelet basis, C. R. Acad. Sci. Paris Sér. IIB, 328, 3, 263-269 (2000) · Zbl 1006.76069
[14] Lesieur, M., Turbulence in Fluids (2008), Springer Netherlands · Zbl 1138.76004
[15] Michalke, A., On the inviscid instability of the hyperbolic-tangent velocity profile, J. Fluid Mech., 19, 4, 543-556 (1964) · Zbl 0129.20302
[16] Boersma, B. J.; Kooper, M. N.; Nieuwstadt, F. T.M.; Wesseling, P., Local grid refinement in large-eddy simulations, J. Engrg. Math., 32, 2-3, 161-175 (1997) · Zbl 0911.76052
[17] Ahmed, N.; Chacón Rebollo, T.; John, V.; Rubino, S., Analysis of a full space-time discretization of the Navier-Stokes equations by a local projection stabilization method, IMA J. Numer. Anal., 37, 3, 1437-1467 (2017) · Zbl 1407.76059
[18] Vreman, B.; Geurts, B.; Kuerten, H., Large-eddy simulation of the turbulent mixing layer, J. Fluid Mech., 339, 357-390 (1997) · Zbl 0900.76369
[19] Griebel, M.; Koster, F., (Málek, J.; Nečas, J.; Rokyta, M., Adaptive Wavelet Solvers for the Unsteady Incompressible Navier-Stokes Equations. Adaptive Wavelet Solvers for the Unsteady Incompressible Navier-Stokes Equations, Advances in Mathematical Fluid Mechanics (2000), Springer, Berlin, Heidelberg), 67-118 · Zbl 0985.35062
[20] Burman, E., Interior penalty variational multiscale method for the incompressible Navier-Stokes equation: Monitoring artificial dissipation, Comput. Methods Appl. Mech. Engrg., 196, 41-44, 4045-4058 (2007) · Zbl 1173.76332
[21] Iannelli, P.; Denaro, F. M.; De Stefano, G., A deconvolution-based fourth-order finite volume method for incompressible flows on non-uniform grids, Internat. J. Numer. Methods Fluids, 43, 4, 431-462 (2003) · Zbl 1032.76614
[22] Burman, E.; Ern, A.; Fernández, M. A., Fractional-step methods and finite elements with symmetric stabilization for the transient Oseen problem, ESAIM: M2AN, 51, 2, 487-507 (2017) · Zbl 1398.76097
[23] Nägele, S.; Wittum, G., Large-eddy simulation and multigrid methods, Electron. Trans. Numer. Anal., 15, 152-164 (2003) · Zbl 1201.76093
[24] Oñate, E.; Valls, A.; García, J., Computation of turbulent flows using a finite calculus-finite element formulation, Internat. J. Numer. Methods Fluids, 54, 6-8, 609-637 (2007) · Zbl 1128.76038
[25] Doering, C. R.; Gibbon, J. D., Applied Analysis of the Navier-Stokes Equations (1995), Cambridge University Press · Zbl 0838.76016
[26] Gargano, F.; Sammartino, M.; Sciacca, V., High Reynolds number Navier-Stokes solutions and boundary layer separation induced by a rectilinear vortex, Comput. Fluids, 52, 73-91 (2011) · Zbl 1271.76066
[27] Ayala, D.; Protas, B., Maximum palinstrophy growth in 2D incompressible flows, J. Fluid Mech., 742, 340-367 (2014) · Zbl 1328.76020
[28] Ayala, D.; Protas, B., Vortices, maximum growth and the problem of finite-time singularity formation, Fluid Dyn. Res., 46, 3, 031404 (2014) · Zbl 1310.76058
[29] Clercx, H. J.H.; van Heijst, G. J.F., Dissipation of coherent structures in confined two-dimensional turbulence, Phys. Fluids, 29, 11, 111103 (2017)
[30] Gunzburger, M. D., Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms, 871 (1989), Academic Press, Inc. Boston
[31] Ern, A.; Guermond, J.-L.., Theory and Practice of Finite Elements (2004), Springer New York
[32] Zeidler, E., Nonlinear Functional Analysis and its Application III: Variational Methods and Optimization (1985), Springer-Verlag New York
[33] Davidson, P. A., Turbulence: An Introduction for Scientist and Engineers (2004), Oxford University Press · Zbl 1061.76001
[35] Lehrenfeld, C.; Schöberl, J., High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows, Comput. Methods Appl. Mech. Engrg., 307, 339-361 (2016)
[36] Lederer, P. L.; Schöberl, J., Polynomial robust stability analysis for \(H(\operatorname{div})\)-conforming finite elements for the Stokes equations, IMA J. Numer. Anal., 38, 4, 1832-1860 (2018) · Zbl 1462.65192
[37] Schroeder, P. W.; Lehrenfeld, C.; Linke, A.; Lube, G., Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier-Stokes equations, SeMA J. (2018), (in press). URL https://doi.org/10.1007/s40324-018-0157-1
[38] Boffi, D.; Brezzi, F.; Fortin, M., Mixed Finite Element Methods and Applications (2013), Springer-Verlag Berlin Heidelberg · Zbl 1277.65092
[39] Di Pietro, D. A.; Ern, A., Mathematical Aspects of Discontinuous Galerkin Methods (2012), Springer-Verlag Berlin Heidelberg · Zbl 1231.65209
[40] Ascher, U. M.; Ruuth, S. J.; Wetton, B. T.R., Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32, 3, 797-823 (1995) · Zbl 0841.65081
[41] Ascher, U. M.; Ruuth, S. J.; Spiteri, R. J., Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25, 2-3, 151-167 (1997) · Zbl 0896.65061
[42] John, V., Finite Element Methods for Incompressible Flow Problems (2016), Springer, Cham · Zbl 1358.76003
[43] Gassner, G. J.; Beck, A. D., On the accuracy of high-order discretizations for underresolved turbulence simulations, Theor. Comput. Fluid Dyn., 27, 3-4, 221-237 (2013)
[44] Fehn, N.; Wall, W. A.; Kronbichler, M., Robust and efficient discontinuous Galerkin methods for under-resolved turbulent incompressible flows, J. Comput. Phys., 372, 667-693 (2018)
[45] Schöberl, J., NETGEN An advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Sci, 1, 1, 41-52 (1997) · Zbl 0883.68130
[46] Beck, A. D.; Flad, D. G.; Tonhäuser, C.; Gassner, G.; Munz, C.-D., On the influence of polynomial de-aliasing on subgrid scale models, Flow Turbul. Combust., 97, 2, 475-511 (2016)
[47] Kopriva, D. A., Stability of overintegration methods for nodal discontinuous Galerkin spectral element methods, J. Sci. Comput., 76, 1, 426-442 (2017) · Zbl 1404.65174
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.