Spectral/hp element methods’ linear mechanism of (apparent) energy transfer in Fourier space: insights into dispersion analysis for implicit LES. (English) Zbl 07605584
Summary: In recent years, different dispersion-diffusion (eigen)analyses have been developed and used to assess various spectral element methods (SEMs) with regards to accuracy and stability, both of which are very important aspects for under-resolved computations of transitional and turbulent flows. Not surprisingly, eigenanalysis has been used recurrently to probe the inner-workings of SEM-based implicit LES approaches, where numerical dissipation acts alone in lieu of a subgrid model. In this study we present and discuss an intriguing linear mechanism that causes energy transfer across Fourier modes as seen in the energy spectrum of SEM computations. Despite its linear nature, this mechanism has not been considered in eigenanalyses so far, possibly due to its connection to the often overlooked multiple eigencurves feature of periodic eigenanalysis. As we unveil the mechanism in the simplified context of linear advection, we point out how its effects might take place in actual turbulence simulations. In particular, we highlight how taking it into account in eigenanalysis can improve dissipation estimates in wavenumber space, potentially allowing for a superior correlation between dissipation estimates and energy spectra measured in SEM-based eddy-resolving turbulence computations.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 76Mxx | Basic methods in fluid mechanics |
| 76Fxx | Turbulence |
Keywords:
spectral element methods; continuous Galerkin; discontinuous Galerkin; dispersion/diffusion analysis; implicit LESReferences:
| [1] | Moura, R. C.; Sherwin, S. J.; Peiró, J., Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods, J. Comput. Phys., 298, 695-710 (2015) · Zbl 1349.76131 |
| [2] | Mengaldo, G.; Moura, R. C.; Giralda, B.; Peiró, J.; Sherwin, S. J., Spatial eigensolution analysis of discontinuous Galerkin schemes with practical insights for under-resolved computations and implicit LES, Comput. Fluids, 169, 349-364 (2017) · Zbl 1410.76103 |
| [3] | Moura, R. C.; Aman, M.; Peiró, J.; Sherwin, S. J., Spatial eigenanalysis of spectral/hp continuous Galerkin schemes and their stabilisation via DG-mimicking spectral vanishing viscosity for high Reynolds number flows, J. Comput. Phys., 406, Article 109112 pp. (2020) · Zbl 1453.76153 |
| [4] | Peiró, J.; Moxey, D.; Turner, M.; Mengaldo, G.; Moura, R. C.; Jassim, A.; Taylor, M.; Sherwin, S. J., Spectral/hp element methods for under-resolved DNS: paving the way to industry-relevant simulations, ERCOFTAC Bull., 116, 5-13 (2018) |
| [5] | Moura, R. C.; Peiró, J.; Sherwin, S. J., Under-resolved DNS of non-trivial turbulent boundary layers via spectral/hp CG schemes, (Garcia-Villalba, M.; Kuerten, H.; Salvetti, M. V., Direct and Large-Eddy Simulation XII (2020), Springer), 389-395 |
| [6] | Moura, R. C.; Sherwin, S. J.; Peiró, J., Eigensolution analysis of spectral/hp continuous Galerkin approximations to advection-diffusion problems: insights into spectral vanishing viscosity, J. Comput. Phys., 307, 401-422 (2016) · Zbl 1352.65362 |
| [7] | Mengaldo, G.; De Grazia, D.; Moura, R. C.; Sherwin, S. J., Spatial eigensolution analysis of energy-stable flux reconstruction schemes and influence of the numerical flux on accuracy and robustness, J. Comput. Phys., 358, 1-20 (2018) · Zbl 1381.76129 |
| [8] | Moura, R. C.; Fernandez, P.; Mengaldo, G.; Sherwin, S. J., Viscous diffusion effects in the eigenanalysis of (hybridisable) DG methods, (Sherwin, S. J.; Moxey, D.; Peiro, J.; Vincent, P. E.; Schwab, C., Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2018 (2020), Springer), 371-382 · Zbl 1484.65230 |
| [9] | Moura, R. C.; Cassinelli, A.; da Silva, A. F.C.; Burman, E.; Sherwin, S. J., Gradient jump penalty stabilisation of spectral/hp element discretisation for under-resolved turbulence simulations, Comput. Methods Appl. Mech. Eng., 388, Article 114200 pp. (2022) · Zbl 1507.76118 |
| [10] | Moura, R. C.; Mengaldo, G.; Peiró, J.; Sherwin, S. J., On 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 (2017) · Zbl 1378.76036 |
| [11] | Bergmann, M.; Gölden, R.; Morsbach, C., Numerical investigation of split form nodal discontinuous Galerkin schemes for implicit LES of a turbulent channel flow, (6th European Conference on Computational Mechanics / 7th European Conference on Computational Fluid Dynamics (ECCOMAS/ECFD 2018) (2018)) |
| [12] | Moura, R. C.; Peiró, J.; Sherwin, S. J., Implicit LES approaches via discontinuous Galerkin methods at very large Reynolds, (Salvetti, M.; Armenio, V.; Fröhlich, J.; Geurts, B.; Kuerten, H., Direct and Large-Eddy Simulation XI (2019), Springer), 53-59 |
| [13] | Fernandez, P.; Moura, R. C.; Mengaldo, G.; Peraire, J., Non-modal analysis of spectral element methods: towards accurate and robust large-eddy simulations, Comput. Methods Appl. Math., 346, 43-62 (2019) · Zbl 1440.76057 |
| [14] | Mengaldo, G.; Moxey, D.; Turner, M.; Moura, R. C.; Jassim, A.; Taylor, M.; Peiró, J.; Sherwin, S. J., Industry-relevant implicit large-eddy simulation of a high-performance road car via spectral/hp element methods, SIAM Rev., 63, 4, 723-755 (2021) |
| [15] | Moura, R. C.; Sherwin, S. J.; Peiró, J., Modified equation analysis for the discontinuous Galerkin formulation, (Kirby, R. M.; Berzins, M.; Hesthaven, J. S., Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2014 (2015), Springer), 375-383 · Zbl 1421.76212 |
| [16] | Moura, R. C.; Mengaldo, G.; Peiró, J.; Sherwin, S. J., An LES setting for DG-based implicit LES with insights on dissipation and robustness, (Bittencourt, M.; Dumont, N.; Hesthaven, J., Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2016 (2017), Springer), 161-173 · Zbl 1388.76101 |
| [17] | Manzanero, J.; Rubio, G.; Ferrer, E.; Valero, E., Dispersion-dissipation analysis for advection problems with nonconstant coefficients: applications to discontinuous Galerkin formulations, SIAM J. Sci. Comput., 40, 2, A747-A768 (2018) · Zbl 1453.65337 |
| [18] | Alhawwary, M.; Wang, Z. J., Fourier analysis and evaluation of DG, FD and compact difference methods for conservation laws, J. Comput. Phys., 373, 835-862 (2018) · Zbl 1416.65252 |
| [19] | Alhawwary, M. A.; Wang, Z. J., A study of DG methods for diffusion using the combined-mode analysis, (Proceedings of the AIAA Scitech 2019 Forum (AIAA Paper 2019-1157). Proceedings of the AIAA Scitech 2019 Forum (AIAA Paper 2019-1157), San Diego, USA (2019)) |
| [20] | Tonicello, N.; Lodato, G.; Vervisch, L., A comparative study from spectral analyses of high-order methods with non-constant advection velocities, J. Sci. Comput., 87, 3, 1-38 (2021) · Zbl 1483.65173 |
| [21] | Manzanero, J.; Ferrer, E.; Rubio, G.; Valero, E., Design of a Smagorinsky spectral vanishing viscosity turbulence model for discontinuous Galerkin methods, Comput. Fluids, 200, Article 104440 pp. (2020) · Zbl 1519.76095 |
| [22] | Chapelier, J. B.; de la Llave Plata, M.; Lamballais, E., Development of a multiscale LES model in the context of a modal discontinuous Galerkin method, Comput. Methods Appl. Math., 307, 275-299 (2016) · Zbl 1436.76021 |
| [23] | Flad, D.; Gassner, G., On the use of kinetic energy preserving DG-schemes for large eddy simulation, J. Comput. Phys., 350, 782-795 (2017) · Zbl 1380.76019 |
| [24] | Winters, A. R.; Moura, R. C.; Mengaldo, G.; Gassner, G. J.; Walch, S.; Peiró, J.; Sherwin, S. J., A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations, J. Comput. Phys., 372, 1-21 (2018) · Zbl 1415.76461 |
| [25] | de la Llave Plata, M.; Lamballais, E.; Naddei, F., On the performance of a high-order multiscale DG approach to LES at increasing Reynolds number, Comput. Fluids, 194, Article 104306 pp. (2019) |
| [26] | Vincent, P. E.; Castonguay, P.; Jameson, A., Insights from von Neumann analysis of high-order flux reconstruction schemes, J. Comput. Phys., 230, 22, 8134-8154 (2011) · Zbl 1343.65117 |
| [27] | Trefethen, L. N.; Trefethen, A. E.; Reddy, S. C.; Driscoll, T. A., Hydrodynamic stability without eigenvalues, Science, 261, 5121, 578-584 (1993) · Zbl 1226.76013 |
| [28] | Schmid, P. J., Nonmodal stability theory, Annu. Rev. Fluid Mech., 39, 129-162 (2007) · Zbl 1296.76055 |
| [29] | Kerswell, R. R., Nonlinear nonmodal stability theory, Annu. Rev. Fluid Mech., 50, 319-345 (2018) · Zbl 1384.76022 |
| [30] | Eckhardt, B.; Schneider, T. M.; Hof, B.; Westerweel, J., Turbulence transition in pipe flow, Annu. Rev. Fluid Mech., 39, 447-468 (2007) · Zbl 1296.76062 |
| [31] | Kawahara, G.; Uhlmann, M.; Van Veen, L., The significance of simple invariant solutions in turbulent flows, Annu. Rev. Fluid Mech., 44, 203-225 (2012) · Zbl 1352.76031 |
| [32] | Sundaram, P.; Suman, V. K.; Sengupta, A.; Sengupta, T. K., Effects of free stream excitation on the boundary layer over a semi-infinite flat plate, Phys. Fluids, 32, 9, Article 094110 pp. (2020) |
| [33] | Sengupta, A.; Sundaram, P.; Sengupta, T. K., Nonmodal nonlinear route of transition to two-dimensional turbulence, Phys. Rev. Res., 2, 1, Article 012033 pp. (2020) |
| [34] | Sengupta, T. K.; Suman, V. K.; Sundaram, P.; Sengupta, A., Analysis of pseudo-spectral methods used for numerical simulations of turbulence, WSEAS Trans. Comput. Res., 10, 9-24 (Jan 2022) |
| [35] | Le Roux, D. Y.; Eldred, C.; Taylor, M. A., Fourier analyses of high-order continuous and discontinuous Galerkin methods, SIAM J. Numer. Anal., 58, 3, 1845-1866 (2020) · Zbl 1448.65166 |
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