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Turbulence kinetic energy transfers in direct numerical simulation of shock-wave–turbulence interaction in a compression/expansion ramp

Published online by Cambridge University Press:  03 February 2022

Niccolò Tonicello Open the ORCID record for Niccolò Tonicello [Opens in a new window] ,
Guido Lodato  and
Luc Vervisch
Niccolò Tonicello*
Affiliation:
CORIA - CNRS, Normandie Université, INSA de Rouen, Technopole du Madrillet, 76801 Saint-Etienne-du-Rouvray, France
Guido Lodato
Affiliation:
CORIA - CNRS, Normandie Université, INSA de Rouen, Technopole du Madrillet, 76801 Saint-Etienne-du-Rouvray, France
Luc Vervisch
Affiliation:
CORIA - CNRS, Normandie Université, INSA de Rouen, Technopole du Madrillet, 76801 Saint-Etienne-du-Rouvray, France
*
†Email address for correspondence: niccolo.tonicello@coria.fr
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Abstract

A direct numerical simulation is performed for a supersonic turbulent boundary layer interacting with a compression/expansion ramp at an angle $\alpha =24^{\circ }$, matching the same operating conditions of the direct numerical simulation by Priebe & Martín (J. Fluid Mech., vol. 699, 2012, pp. 1–49). The adopted numerical method relies on the high-order spectral difference scheme coupled with a bulk-based, low-dissipative, artificial viscosity for shock-capturing purposes (Tonicello et al., Comput. Fluids, vol. 197, 2020, 104357). Filtered and averaged fields are evaluated to study total kinetic energy transfers in the presence of non-negligible compressibility effects. The compression motions are shown to promote forward transfer of kinetic energy down the energy cascade, whereas expansion regions are more likely to experience backscatter of kinetic energy. A standard decomposition of the subgrid scale tensor in deviatoric and spherical parts is proposed to study the compressible and incompressible contributions in the total kinetic energy transfers across scales. On average, the correlation between subgrid scale dissipation and large-scale dilatation is shown to be caused entirely by the spherical part of the Reynolds stresses (i.e. the turbulent kinetic energy). On the other hand, subtracting the spherical contribution, a mild correlation is still noticeable in the filtered fields. For compressible flows, it seems reasonable to assume that the eddy-viscosity approximation can be a suitable model for the deviatoric part of the subgrid scale tensor, which is exclusively causing forward kinetic energy cascade on average. Instead, more complex models are likely to be needed for the spherical part, which, even in statistical average, provides an important mechanism for backscatter.

JFM classification

Compressible Flows: Compressible boundary layers Compressible Flows: Shock waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 935 , 25 March 2022 , A31
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© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Ackeret, J., Feldmann, F. & Rott, N. 1947 Investigations of compression shocks and boundary layers in gases moving at high speed. NACA Technical Memorandum 1113. National Advisory Committee for Aeronautics.Google Scholar
Adams, N.A. 2000 Direct simulation of the turbulent boundary layer along a compression ramp at ${Ma}= 3$ and ${Re}_\theta = 1685$. J. Fluid Mech. 420, 47–83.CrossRefGoogle Scholar
Adler, M.C., Gonzalez, D.R., Stack, C.M. & Gaitonde, D.V. 2018 Synthetic generation of equilibrium boundary layer turbulence from modeled statistics. Comput. Fluids 165, 127–143.CrossRefGoogle Scholar
Aluie, H. 2013 Scale decomposition in compressible turbulence. Phys. D: Nonlinear Phenom. 247 (1), 54–65.CrossRefGoogle Scholar
Andreopoulos, Y., Agui, J.H. & Briassulis, G. 2000 Shock wave–turbulence interactions. Annu. Rev. Fluid Mech. 32 (1), 309–345.CrossRefGoogle Scholar
Bookey, P., Wyckham, C., Smits, A. & Martín, M.P. 2005 New experimental data of STBLI at DNS/LES accessible Reynolds numbers. In 43rd AIAA Aerospace Sciences Meeting and Exhibit. AIAA Paper 2005-309.CrossRefGoogle Scholar
Cabot, W.H., Schilling, O. & Zhou, Y. 2004 Influence of subgrid scales on resolvable turbulence and mixing in Rayleigh–Taylor flow. Phys. Fluids 16 (3), 495–508.CrossRefGoogle Scholar
Chapelier, J.B. & Lodato, G. 2016 A spectral-element dynamic model for the large-eddy simulation of turbulent flows. J. Comput. Phys. 321, 279–302.CrossRefGoogle Scholar
Chen, S., Wang, J., Li, H., Wan, M. & Chen, S. 2019 Effect of compressibility on small scale statistics in homogeneous shear turbulence. Phys. Fluids 31 (2), 025107.Google Scholar
Comte-Bellot, G. & Craya, A. 1965 Ecoulement turbulent entre deux parois paralleles. Service de documentation scientifique et technique de l'armement.Google Scholar
Cook, A.W. & Cabot, W.H. 2004 A high-wavenumber viscosity for high-resolution numerical methods. J. Comput. Phys. 195 (2), 594–601.CrossRefGoogle Scholar
Dolling, D.S. 2001 Fifty years of shock-wave/boundary-layer interaction research: what next? AIAA J. 39 (8), 1517–1531.CrossRefGoogle Scholar
Dolling, D.S. & Murphy, M.T. 1983 Unsteadiness of the separation shock wave structure in a supersonic compression ramp flowfield. AIAA J. 21 (12), 1628–1634.CrossRefGoogle Scholar
Domaradzki, J.A., Liu, W. & Brachet, M.E. 1993 An analysis of subgrid-scale interactions in numerically simulated isotropic turbulence. Phys. Fluids A: Fluid Dyn. 5 (7), 1747–1759.CrossRefGoogle Scholar
Ducros, F., Ferrand, V., Nicoud, F., Weber, C., Darracq, D., Gacherieu, C. & Poinsot, T. 1999 Large-eddy simulation of the shock/turbulence interaction. J. Comput. Phys. 152 (2), 517–549.CrossRefGoogle Scholar
Eggels, J.G.M., Unger, F., Weiss, M.H., Westerweel, J., Adrian, R.J., Friedrich, R. & Nieuwstadt, F.T.M. 1994 Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment. J. Fluid Mech. 268, 175–210.CrossRefGoogle Scholar
Erlebacher, G., Hussaini, M.Y., Speziale, C.G. & Zang, T.A. 1990 Toward the large-eddy simulation of compressible turbulent flows. Tech. Rep. Institute for Computer Applications in Science and Engineering, Hampton VA.Google Scholar
Fernandez, P., Nguyen, N.C. & Peraire, J. 2018 a A physics-based shock capturing method for large-eddy simulation (preprint) arXiv:1806.06449.CrossRefGoogle Scholar
Fernandez, P., Nguyen, N.C. & Peraire, J. 2018 b A physics-based shock capturing method for unsteady laminar and turbulent flows. 43rd AIAA Aerospace Sciences Meeting Kissimmee, Florida. AIAA Paper 2018-0062.Google Scholar
Germano, M. 1986 Differential filters of elliptic type. Phys. Fluids 29 (6), 1757–1758.CrossRefGoogle Scholar
Green, J.E. 1970 Reflexion of an oblique shock wave by a turbulent boundary layer. J. Fluid Mech. 40 (1), 81–95.CrossRefGoogle Scholar
Helm, C.M. & Martín, M.P. 2016 New LES of a hypersonic shock/turbulent boundary layer interaction. In 54th AIAA Aerospace Sciences Meeting. AIAA Paper 2016-0346.CrossRefGoogle Scholar
Jameson, A. 2010 A proof of the stability of the spectral difference method for all orders of accuracy. J. Sci. Comput. 45 (1), 348–358.CrossRefGoogle Scholar
Jameson, A., Vincent, P.E. & Castonguay, P. 2012 On the non-linear stability of flux reconstruction schemes. J. Sci. Comput. 50 (2), 434–445.CrossRefGoogle Scholar
Jamme, S., Cazalbou, J.B., Torres, F. & Chassaing, P. 2002 Direct numerical simulation of the interaction between a shock wave and various types of isotropic turbulence. Flow Turbul. Combust. 68 (3), 227–268.CrossRefGoogle Scholar
Jiang, G. & Shu, C. 1996 Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126 (1), 202–228.CrossRefGoogle Scholar
Johnsen, E., Larsson, J., Bhagatwala, A.V., Cabot, W.H., Moin, P., Olson, B.J., Rawat, P.S., Shankar, S.K., Sjögreen, B., Yee, H.C. et al. 2010 Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. J. Comput. Phys. 229 (4), 1213–1237.CrossRefGoogle Scholar
Kawai, S., Shankar, S.K. & Lele, S.K. 2010 Assessment of localized artificial diffusivity scheme for large-eddy simulation of compressible turbulent flows. J. Comput. Phys. 229 (5), 1739–1762.CrossRefGoogle Scholar
Kerr, R.M., Domaradzki, J.A. & Barbier, G. 1996 Small-scale properties of nonlinear interactions and subgrid-scale energy transfer in isotropic turbulence. Phys. Fluids 8 (1), 197–208.CrossRefGoogle Scholar
Kerrebrock, J.L. 1956 The interaction of flow discontinuities with small disturbances in a compressible fluid. PhD thesis, California Institute of Technology.Google Scholar
Khani, S. & Waite, M.L. 2016 Backscatter in stratified turbulence. Eur. J. Mech. (B/Fluids) 60, 1–12.CrossRefGoogle Scholar
Klein, M., Sadiki, A. & Janicka, J. 2003 A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations. J. Comput. Phys. 186 (2), 652–665.CrossRefGoogle Scholar
Kokkinakis, I.W., Drikakis, D., Ritos, K. & Spottswood, S.M. 2020 Direct numerical simulation of supersonic flow and acoustics over a compression ramp. Phys. Fluids 32 (6), 066107.CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. URSS 30, 301–305.Google Scholar
Kopriva, D.A. & Kolias, J.H. 1996 A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys. 125 (1), 244–261.CrossRefGoogle Scholar
Kovasznay, L.S.G. 1953 Turbulence in supersonic flow. J. Aeronaut. Sci. 20 (10), 657–674.CrossRefGoogle Scholar
Laizet, S., Lamballais, E. & Vassilicos, J.C. 2010 A numerical strategy to combine high-order schemes, complex geometry and parallel computing for high resolution DNS of fractal generated turbulence. Comput. Fluids 39 (3), 471–484.CrossRefGoogle Scholar
Lee, S., Lele, S.K. & Moin, P. 1991 Direct numerical simulation and analysis of shock turbulence interaction. In 29th Aerospace Sciences Meeting. AIAA Paper 1991-523.CrossRefGoogle Scholar
Lee, S., Lele, S.K. & Moin, P. 1993 Direct numerical simulation of isotropic turbulence interacting with a weak shock wave. J. Fluid Mech. 251, 533–562.CrossRefGoogle Scholar
Lee, S., Lele, S.K. & Moin, P. 1997 Interaction of isotropic turbulence with shock waves: effect of shock strength. J. Fluid Mech. 340, 225–247.CrossRefGoogle Scholar
Lees, A. & Aluie, H. 2019 Baropycnal work: a mechanism for energy transfer across scales. Fluids 4 (2), 92.CrossRefGoogle Scholar
Li, X., Fu, D., Ma, Y. & Liang, X. 2010 Direct numerical simulation of shock/turbulent boundary layer interaction in a supersonic compression ramp. Sci. China Phys. Mech. Astron. 53 (9), 1651–1658.CrossRefGoogle Scholar
Li, J., Grube, N.E., Priebe, S. & Martín, M.P. 2013 a Analysis of the large eddy simulation of a shock wave and turbulent boundary layer interaction. In 43rd AIAA Fluid Dynamics Conference. AIAA Paper 2013-2734.CrossRefGoogle Scholar
Li, J., Grube, N., Priebe, S. & Martín, M.P. 2013 b LES study of shock wave and turbulent boundary layer interaction. In 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. AIAA Paper 2013-984.CrossRefGoogle Scholar
Liepmann, H.W. 1946 The interaction between boundary layer and shock waves in transonic flow. J. Aeronaut. Sci. 13 (12), 623–637.CrossRefGoogle Scholar
Liu, Y., Vinokur, M. & Wang, Z.J. 2006 Spectral difference method for unstructured grids I: basic formulation. J. Comput. Phys. 216 (2), 780–801.CrossRefGoogle Scholar
Livescu, D., Jaberi, F.A. & Madnia, C.K. 2002 The effects of heat release on the energy exchange in reacting turbulent shear flow. J. Fluid Mech. 450, 35–66.CrossRefGoogle Scholar
Livescu, D. & Li, Z. 2017 Subgrid-scale backscatter after the shock-turbulence interaction. In AIP Conference Proceedings, vol. 1793, Paper. 150009. AIP Publishing LLC.CrossRefGoogle Scholar
Lodato, G. 2019 a Characteristic modal shock detection for discontinuous finite element methods. Comput. Fluids 179, 309–333.CrossRefGoogle Scholar
Lodato, G. 2019 b Corrigendum to the article ‘Characteristic modal shock detection for discontinuous finite element methods’. Comput. Fluids 193, 104245.CrossRefGoogle Scholar
Lodato, G., Tonicello, N. & Pinto, B. 2021 Large-eddy simulation of bypass transition on a zero-pressure-gradient flat plate using the spectral-element dynamic model. Flow Turbul. Combust. 107, 845–874.CrossRefGoogle Scholar
Lodato, G., Vervisch, L. & Clavin, P. 2016 Direct numerical simulation of shock wavy-wall interaction: analysis of cellular shock structures and flow patterns. J. Fluid Mech. 789, 221–258.CrossRefGoogle Scholar
Lodato, G., Vervisch, L. & Clavin, P. 2017 Numerical study of smoothly perturbed shocks in the Newtonian limit. Flow Turbul. Combust. 99 (3), 887–908.CrossRefGoogle Scholar
Mahesh, K. & Lee, S. 1995 The interaction of an isotropic field of acoustic waves with a shock wave. J. Fluid Mech. 300, 383–407.CrossRefGoogle Scholar
Mahesh, K., Lele, S.K. & Moin, P. 1997 The influence of entropy fluctuations on the interaction of turbulence with a shock wave. J. Fluid Mech. 334, 353–379.CrossRefGoogle Scholar
Margolin, L.G. 2019 The reality of artificial viscosity. Shock Waves 29 (1), 27–35.CrossRefGoogle Scholar
Margolin, L.G. & Plesko, C. 2019 Discrete regularization. Evol. Equ. Control Theory 8 (1), 117–137.CrossRefGoogle Scholar
Matsubara, M. & Alfredsson, P.H. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168.CrossRefGoogle Scholar
Moin, P., Squires, K., Cabot, W. & Lee, S. 1991 A dynamic subgrid-scale model for compressible turbulence and scalar transport. Phys. Fluids 11 (3), 2746–2757.CrossRefGoogle Scholar
Moitro, A., Venkataraman, A.S. & Poludnenko, A. 2019 Kinetic energy backscatter in high-speed, compressible reacting turbulence. In APS Division of Fluid Dynamics Meeting Abstracts, pp. G05-007.Google Scholar
Moore, F.K. 1953 Unsteady oblique interaction of a shock wave with a plane disturbance. NACA Tech. Rep. 1165.Google Scholar
Morgan, B., Duraisamy, K., Nguyen, N., Kawai, S. & Lele, S.K. 2013 Flow physics and RANS modelling of oblique shock/turbulent boundary layer interaction. J. Fluid Mech. 729, 231–284.CrossRefGoogle Scholar
Morgan, B., Larsson, J., Kawai, S. & Lele, S.K. 2011 Improving low-frequency characteristics of recycling/rescaling inflow turbulence generation. AIAA J. 49 (3), 582–597.CrossRefGoogle Scholar
O'Brien, J., Urzay, J., Ihme, M., Moin, P. & Saghafian, A. 2014 Subgrid-scale backscatter in reacting and inert supersonic hydrogen–air turbulent mixing layers. J. Fluid Mech. 743, 554–584.CrossRefGoogle Scholar
Pan, S. & Johnsen, E. 2017 The role of bulk viscosity on the decay of compressible, homogeneous, isotropic turbulence. J. Fluid Mech. 833, 717–744.CrossRefGoogle Scholar
Pantano, C. & Sarkar, S. 2002 A study of compressibility effects in the high-speed turbulent shear layer using direct simulation. J. Fluid Mech. 451, 329–371.CrossRefGoogle Scholar
Passot, T. & Pouquet, A. 1987 Numerical simulation of compressible homogeneous flows in the turbulent regime. J. Fluid Mech. 181, 441–466.CrossRefGoogle Scholar
Persson, P.O. & Peraire, J. 2006 Sub-cell shock capturing for discontinuous Galerkin methods. 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, Jan. 9–12. AIAA Paper 2006-112.Google Scholar
Phillips, O.M. 1991 The Kolmogorov spectrum and its oceanic cousins: a review. Proc. R. Soc. Lond. Ser. A: Math. Phys. Sci. 434 (1890), 125–138.Google Scholar
Pinto, B. & Lodato, G. 2019 Synthetic freestream disturbance for the numerical reproduction of experimental zero-pressure-gradient bypass transition test cases. Flow Turbul. Combust. 103 (1), 25–54.CrossRefGoogle Scholar
Piomelli, U., Cabot, W.H., Moin, P. & Lee, S. 1991 Subgrid-scale backscatter in turbulent and transitional flows. Phys. Fluids A: Fluid Dyn. 3 (7), 1766–1771.CrossRefGoogle Scholar
Piomelli, U., Yu, Y. & Adrian, R.J. 1996 Subgrid-scale energy transfer and near-wall turbulence structure. Phys. Fluids 8 (1), 215–224.CrossRefGoogle Scholar
Pirozzoli, S. & Bernardini, M. 2011 a Direct numerical simulation database for impinging shock wave/turbulent boundary-layer interaction. AIAA J. 49 (6), 1307–1312.CrossRefGoogle Scholar
Pirozzoli, S. & Bernardini, M. 2011 b Turbulence in supersonic boundary layers at moderate Reynolds number. J. Fluid Mech. 688, 120–168.CrossRefGoogle Scholar
Pirozzoli, S., Bernardini, M. & Grasso, F. 2008 Characterization of coherent vortical structures in a supersonic turbulent boundary layer. J. Fluid Mech. 613, 205–231.CrossRefGoogle Scholar
Pirozzoli, S., Bernardini, M. & Grasso, F. 2010 Direct numerical simulation of transonic shock/boundary layer interaction under conditions of incipient separation. J. Fluid Mech. 657, 361–393.CrossRefGoogle Scholar
Pirozzoli, S. & Grasso, F. 2006 Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at $M= 2.25$. Phys. Fluids 18 (6), 065113.CrossRefGoogle Scholar
Pope, S.B. 2001 Turbulent Flows. Cambridge University Press.Google Scholar
Priebe, S. & Martín, M.P. 2012 Low-frequency unsteadiness in shock wave–turbulent boundary layer interaction. J. Fluid Mech. 699, 1–49.CrossRefGoogle Scholar
Priebe, S., Wu, M. & Martín, M.P. 2009 Direct numerical simulation of a reflected-shock-wave/turbulent- boundary-layer interaction. AIAA J. 47 (5), 1173–1185.CrossRefGoogle Scholar
Qiu, J. & Shu, C. 2004 Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys. 193 (1), 115–135.CrossRefGoogle Scholar
Qiu, J. & Shu, C. 2005 a Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method II: two dimensional case. Comput. Fluids 34 (6), 642–663.CrossRefGoogle Scholar
Qiu, J. & Shu, C. 2005 b Runge–Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput. 26 (3), 907–929.CrossRefGoogle Scholar
Rawat, P.S. & Zhong, X. 2010 On high-order shock-fitting and front-tracking schemes for numerical simulation of shock–disturbance interactions. J. Comput. Phys. 229 (19), 6744–6780.CrossRefGoogle Scholar
Ribner, H.S. 1953 Convection of a pattern of vorticity through a shock wave. NACA Tech. Rep. 2864.Google Scholar
Ribner, H.S. 1954 Shock-turbulence interaction and the generation of noise. NACA Tech. Rep. 1233.Google Scholar
Richardson, L.F. 1922 Weather Prediction by Numerical Process. Cambridge University Press.Google Scholar
Ringuette, M.J., Bookey, P., Wyckham, C. & Smits, A.J. 2009 Experimental study of a Mach 3 compression ramp interaction at ${Re}_{\theta }= 2400$. AIAA J. 47 (2), 373–385.CrossRefGoogle Scholar
Rogallo, R.S. & Moin, P. 1984 Numerical simulation of turbulent flows. Annu. Rev. Fluid Mech. 16 (1), 99–137.CrossRefGoogle Scholar
Salas, M.D. 1976 Shock fitting method for complicated two-dimensional supersonic flows. AIAA J. 14 (5), 583–588.CrossRefGoogle Scholar
Sarkar, S., Erlebacher, G. & Hussaini, M.Y. 1991 a Direct simulation of compressible turbulence in a shear flow. Theor. Comput. Fluid Dyn. 2, 291–305.CrossRefGoogle Scholar
Sarkar, S., Erlebacher, G., Hussaini, M.Y. & Kreiss, H.O. 1991 b The analysis and modelling of dilatational terms in compressible turbulence. J. Fluid Mech. 227, 473–493.CrossRefGoogle Scholar
Sayadi, T., Hamman, C.W & Moin, P. 2013 Direct numerical simulation of complete h-type and k-type transitions with implications for the dynamics of turbulent boundary layers. J. Fluid Mech. 724, 480–509.CrossRefGoogle Scholar
Settles, G.S. & Dodson, L.J. 1994 Supersonic and hypersonic shock/boundary-layer interaction database. AIAA J. 32 (7), 1377–1383.CrossRefGoogle Scholar
Settles, G.S., Fitzpatrick, T.J. & Bogdonoff, S.M. 1979 Detailed study of attached and separated compression corner flowfields in high Reynolds number supersonic flow. AIAA J. 17 (6), 579–585.CrossRefGoogle Scholar
Sidharth, G.S. & Candler, G.V. 2018 Subgrid-scale effects in compressible variable-density decaying turbulence. J. Fluid Mech. 846, 428–459.Google Scholar
Spalart, P.R. 1988 Direct simulation of a turbulent boundary layer up to $R\theta = 1410$. J. Fluid Mech. 187, 61–98.CrossRefGoogle Scholar
Sun, Y., Wang, Z.J. & Liu, Y. 2007 High-order multidomain spectral difference method for the Navier–Stokes equations on unstructured hexahedral grids. Commun. Comput. Phys. 2 (2), 310–333.Google Scholar
Tong, F., Tang, Z., Yu, C., Zhu, X. & Li, X. 2017 a Numerical analysis of shock wave and supersonic turbulent boundary interaction between adiabatic and cold walls. J. Turbul. 18 (6), 569–588.CrossRefGoogle Scholar
Tong, F., Yu, C., Tang, Z. & Li, X. 2017 b Numerical studies of shock wave interactions with a supersonic turbulent boundary layer in compression corner: turning angle effects. Comput. Fluids 149, 56–69.CrossRefGoogle Scholar
Tonicello, N., Lodato, G. & Vervisch, L. 2020 Entropy preserving low dissipative shock capturing with wave-characteristic based sensor for high-order methods. Comput. Fluids 197, 104357.CrossRefGoogle Scholar
Tonicello, N., Lodato, G. & Vervisch, L. 2021 LES of compression ramp using high-order dynamic SGS modeling. In AIAA Scitech 2021 Forum. AIAA Paper 2021-1947.CrossRefGoogle Scholar
Tonicello, N., Lodato, G. & Vervisch, L. 2022 Analysis of high-order explicit LES dynamic modeling applied to airfoil flows. Flow Turbul. Combust. 108, 77–104.CrossRefGoogle Scholar
Touber, E. 2010 Unsteadiness in shock-wave/boundary layer interactions. PhD thesis, University of Southampton.Google Scholar
Von Neumann, J. & Richtmyer, R.D. 1950 A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys. 21 (3), 232–237.CrossRefGoogle Scholar
Vreman, B., Geurts, B. & Kuerten, H. 1994 Realizability conditions for the turbulent stress tensor in large-eddy simulation. J. Fluid Mech. 278, 351–362.CrossRefGoogle Scholar
Vreman, B., Geurts, B. & Kuerten, H. 1995 A priori tests of large eddy simulation of the compressible plane mixing layer. J. Engng Maths 29 (4), 299–327.CrossRefGoogle Scholar
Wang, X., Chen, S., Wang, J., Li, H., Wan, M. & Chen, S. 2020 Effect of compressibility on the local flow topology in homogeneous shear turbulence. Phys. Fluids 32 (1), 015118.Google Scholar
Wang, J., Gotoh, T. & Watanabe, T. 2017 Scaling and intermittency in compressible isotropic turbulence. Phys. Rev. Fluids 2 (5), 053401.CrossRefGoogle Scholar
Wang, Z.J., Liu, Y., May, G. & Jameson, A. 2007 Spectral difference method for unstructured grids II: extension to the Euler equations. J. Sci. Comput. 32 (1), 45–71.CrossRefGoogle Scholar
Wang, B., Sandham, N.D., Hu, Z. & Liu, W. 2015 Numerical study of oblique shock-wave/boundary-layer interaction considering sidewall effects. J. Fluid Mech. 767, 526–561.CrossRefGoogle Scholar
Wang, J., Wan, M., Chen, S. & Chen, S. 2018 a Kinetic energy transfer in compressible isotropic turbulence. J. Fluid Mech. 841, 581–613.CrossRefGoogle Scholar
Wang, J., Wan, M., Chen, S., Xie, C. & Chen, S. 2018 b Effect of shock waves on the statistics and scaling in compressible isotropic turbulence. Phys. Rev. E 97 (4), 043108.CrossRefGoogle ScholarPubMed
Wu, X. 2017 Inflow turbulence generation methods. Annu. Rev. Fluid Mech. 49, 23–49.CrossRefGoogle Scholar
Wu, M. & Martín, M.P. 2007 Direct numerical simulation of supersonic turbulent boundary layer over a compression ramp. AIAA J. 45 (4), 879–889.CrossRefGoogle Scholar
Wu, M. & Martín, M.P. 2008 Analysis of shock motion in shockwave and turbulent boundary layer interaction using direct numerical simulation data. J. Fluid Mech. 594, 71–83.CrossRefGoogle Scholar
Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer. J. Fluid Mech. 630, 5–41.CrossRefGoogle Scholar
Xu, S. & Martín, M.P. 2004 Assessment of inflow boundary conditions for compressible turbulent boundary layers. Phys. Fluids 16 (7), 2623–2639.CrossRefGoogle Scholar
Yoshizawa, A. 1986 Statistical theory for compressible turbulent shear flows, with the application to subgrid modeling. Phys. Fluids 29 (7), 2152–2164.CrossRefGoogle Scholar
Zahr, M.J. & Persson, P.O. 2020 An r-adaptive, high-order discontinuous Galerkin method for flows with attached shocks. In AIAA Scitech 2020 Forum. AIAA Paper 2020-0537.Google Scholar
Zahr, M.J., Shi, A. & Persson, P.O. 2019 Implicit shock tracking using an optimization-based, $r$-adaptive, high-order discontinuous Galerkin method. J. Comput. Math. 410, 109385.Google Scholar
Zeman, O. 1990 Dilatation dissipation – the concept and application in modeling compressible mixing layers. Phys. Fluids A 2 (2), 178–188.CrossRefGoogle Scholar
Zhang, X. & Shu, C. 2010 On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229 (23), 8918–8934.CrossRefGoogle Scholar
Zhao, D. & Aluie, H. 2020 Energy cascades in Rayleigh–Taylor turbulence. J. Fluid Mech. 930, A29.CrossRefGoogle Scholar
Zhao, Z., Liu, N. & Lu, X. 2020 Kinetic energy and enstrophy transfer in compressible Rayleigh–Taylor turbulence. J. Fluid Mech. 904, A37.CrossRefGoogle Scholar
Zhu, X., Yu, C., Tong, F. & Li, X. 2017 Numerical study on wall temperature effects on shock wave/turbulent boundary-layer interaction. AIAA J. 55 (1), 131–140.CrossRefGoogle Scholar