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Resolvent-based study of compressibility effects on supersonic turbulent boundary layers

  • H. Jane Bae (a1), Scott T. M. Dawson (a1) (a2) and Beverley J. McKeon (a1)

Abstract

The resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) is applied to supersonic turbulent boundary layers to study the validity of Morkovin’s hypothesis, which postulates that high-speed turbulence structures in zero-pressure-gradient turbulent boundary layers remain largely the same as their incompressible counterparts. Supersonic zero-pressure-gradient turbulent boundary layers with adiabatic wall boundary conditions at Mach numbers ranging from 2 to 4 are considered. Resolvent analysis highlights two distinct regions of the supersonic turbulent boundary layer in the wave parameter space: the relatively supersonic region and the relatively subsonic region. In the relatively supersonic region, where the flow is supersonic relative to the free-stream, resolvent modes display structures consistent with Mach wave radiation that are absent in the incompressible regime. In the relatively subsonic region, we show that the low-rank approximation of the resolvent operator is an effective approximation of the full system and that the response modes predicted by the model exhibit universal and geometrically self-similar behaviour via a transformation given by the semi-local scaling. Moreover, with the semi-local scaling, we show that the resolvent modes follow the same scaling law as their incompressible counterparts in this region, which has implications for modelling and the prediction of turbulent high-speed wall-bounded flows. We also show that the thermodynamic variables exhibit similar mode shapes to the streamwise velocity modes, supporting the strong Reynolds analogy. Finally, we demonstrate that the principal resolvent modes can be used to capture the energy distribution between momentum and thermodynamic fluctuations.

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Corresponding author

Email address for correspondence: hjbae@caltech.edu

References

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Alizard, F., Robinet, J.-C. & Filliard, G. 2015 Sensitivity analysis of optimal transient growth for turbulent boundary layers. Eur. J. Mech. (B/Fluids) 49, 373386.
Bernardini, M. & Pirozzoli, S. 2011 Wall pressure fluctuations beneath supersonic turbulent boundary layers. Phys. Fluids 23 (8), 085102.
Bitter, N. & Shepherd, J.2014 Transient growth in hypersonic boundary layers. In 7th AIAA Theoretical Fluid Mechanics Conference, Atlanta, GA, AIAA Paper 2014-2497.
Bradshaw, P. 1974 The effect of mean compression or dilatation on the turbulence structure of supersonic boundary layers. J. Fluid Mech. 63 (3), 449464.
Brun, C., Boiarciuc, M. P., Haberkorn, M. & Comte, P. 2008 Large eddy simulation of compressible channel flow. Theor. Comput. Fluid Dyn. 22 (3-4), 189212.
Cebeci, T. & Bradshaw, P. 2012 Physical and Computational Aspects of Convective Heat Transfer. Springer Science & Business Media.
Christov, C. I. 1982 A complete orthonormal system of functions in L 2(-, ) space. SIAM J. Appl. Maths 42 (6), 13371344.
Chu, B.-T. 1965 On the energy transfer to small disturbances in fluid flow (Part I). Acta Mech. 1 (3), 215234.
Coleman, G. N., Kim, J. & Moser, R. D. 1995 A numerical study of turbulent supersonic isothermal-wall channel flow. J. Fluid Mech. 305, 159183.
Coles, D. 1964 The turbulent boundary layer in a compressible fluid. Phys. Fluids 7 (9), 14031423.
Cook, D. A., Thome, J., Brock, J. M., Nichols, J. W. & Candler, G. V.2018 Understanding effects of nose-cone bluntness on hypersonic boundary layer transition using input–output analysis. In 2018 AIAA Aerospace Sciences Meeting, AIAA Paper 2018-0378.
Dawson, S. T. M. & McKeon, B. J. 2019 Studying the effects of compressibility in planar Couette flow using resolvent analysis. In AIAA SciTech, p. 2139.
Del Alamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135144.
Duan, L., Beekman, I. & Martin, M. P. 2010 Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature. J. Fluid Mech. 655, 419445.
Duan, L., Beekman, I. & Martin, M. P. 2011 Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number. J. Fluid Mech. 672, 245267.
Duan, L., Choudhari, M. M. & Zhang, C. 2016 Pressure fluctuations induced by a hypersonic turbulent boundary layer. J. Fluid Mech. 804, 578607.
Duan, L. & Martin, M. P. 2011 Direct numerical simulation of hypersonic turbulent boundary layers. Part 4. Effect of high enthalpy. J. Fluid Mech. 684, 2559.
Dwivedi, A., Gs, S., Candler, G. V., Nichols, J. W. & Jovanovic, M.2018 Input–output analysis of shock boundary layer interaction. In AIAA 2018 Fluid Dynamics Conference, AIAA Paper 2018-3220.
Ekoto, I. W., Bowersox, R. D. W., Beutner, T. & Goss, L. P. 2008 Supersonic boundary layers with periodic surface roughness. AIAA J. 46 (2), 486497.
Erm, L. P. & Joubert, P. N. 1991 Low-Reynolds-number turbulent boundary layers. J. Fluid Mech. 230, 144.
Ffowcs Williams, J. E. & Maidanik, G. 1965 The Mach wave field radiated by supersonic turbulent shear flows. J. Fluid Mech. 21 (4), 641657.
Gaviglio, J. 1987 Reynolds analogies and experimental study of heat transfer in the supersonic boundary layer. Intl J. Heat Mass Transfer 30 (5), 911926.
Grosch, C. E. & Orszag, S. A. 1977 Numerical solution of problems in unbounded regions: coordinate transforms. J. Comput. Phys. 25 (3), 273295.
Guarini, S. E., Moser, R. D., Shariff, K. & Wray, A. 2000 Direct numerical simulation of a supersonic turbulent boundary layer at Mach 2.5. J. Fluid Mech. 414, 133.
Hadjadj, A., Ben-Nasr, O., Shadloo, M. S. & Chaudhuri, A. 2015 Effect of wall temperature in supersonic turbulent boundary layers: a numerical study. Intl J. Heat Mass Transfer 81, 426438.
Hanifi, A., Schmid, P. J. & Henningson, D. S. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8 (3), 826837.
Huang, P. G., Coleman, G. N. & Bradshaw, P. 1995 Compressible turbulent channel flows: DNS results and modelling. J. Fluid Mech. 305, 185218.
Illingworth, S. J., Monty, J. P. & Marusic, I. 2018 Estimating large-scale structures in wall turbulence using linear models. J. Fluid Mech. 842, 146162.
Jeun, J., Nichols, J. W. & Jovanović, M. R. 2016 Input–output analysis of high-speed axisymmetric isothermal jet noise. Phys. Fluids 28 (4), 047101.
Jiménez, J. & Hoyas, S. 2008 Turbulent fluctuations above the buffer layer of wall-bounded flows. J. Fluid Mech. 611, 215236.
Jiménez, J., Hoyas, S., Simens, M. P. & Mizuno, Y. 2010 Turbulent boundary layers and channels at moderate Reynolds numbers. J. Fluid Mech. 657, 335360.
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145183.
Kistler, A. L. 1959 Fluctuation measurements in a supersonic turbulent boundary layer. Phys. Fluids 2 (3), 290296.
Konrad, W. & Smits, A. J. 1998 Turbulence measurements in a three-dimensional boundary layer in supersonic flow. J. Fluid Mech. 372, 123.
Kovasznay, L. S. G. 1953 Turbulence in supersonic flow. J. Aero. Sci. 20 (10), 657674.
Laderman, A. J. & Demetriades, A. 1974 Mean and fluctuating flow measurements in the hypersonic boundary layer over a cooled wall. J. Fluid Mech. 63 (1), 121144.
Lagha, M., Kim, J., Eldredge, J. D. & Zhong, X. 2011 A numerical study of compressible turbulent boundary layers. Phys. Fluids 23 (1), 015106.
LeHew, J., Guala, M. & McKeon, B. J. 2011 A study of the three-dimensional spectral energy distribution in a zero pressure gradient turbulent boundary layer. Exp. Fluids 51 (4), 9971012.
Lobb, R. K., Winkler, E. M. & Persh, J.1955 NOL hypersonic tunnel No. 4, results 7: experimental investigation of turbulent boundary layers in hypersonic flow. Tech. Rep. Naval Ordnance Lab, White Oak, MD.
Luhar, M., Sharma, A. S. & McKeon, B. J. 2014 On the structure and origin of pressure fluctuations in wall turbulence: predictions based on the resolvent analysis. J. Fluid Mech. 751, 3870.
Mack, L. M.1984 Boundary-layer linear stability theory. AGARD Report No. 709, Part 3. NASA Jet Propulsion Laboratory.
Maeder, T. 2000 Numerical Investigation of Supersonic Turbulent Boundary Layers, vol. 394. ETH Zurich.
Malik, M., Alam, M. & Dey, J. 2006 Nonmodal energy growth and optimal perturbations in compressible plane Couette flow. Phys. Fluids 18 (3), 034103.
Malik, M., Dey, J. & Alam, M. 2008 Linear stability, transient energy growth, and the role of viscosity stratification in compressible plane Couette flow. Phys. Rev. E 77 (3), 036322.
Martín, M. P. 2007 Direct numerical simulation of hypersonic turbulent boundary layers. Part 1. Initialization and comparison with experiments. J. Fluid Mech. 570, 347364.
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336382.
Metzger, M. M. & Klewicki, J. C. 2001 A comparative study of near-wall turbulence in high and low Reynolds number boundary layers. Phys. Fluids 13 (3), 692701.
Moarref, R., Jovanović, M. R., Tropp, J. A., Sharma, A. S. & McKeon, B. J. 2014 A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization. Phys. Fluids 26 (5), 051701.
Moarref, R., Sharma, A. S., Tropp, J. A. & McKeon, B. J. 2013 Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels. J. Fluid Mech. 734, 275316.
Mochizuki, S. & Nieuwstadt, F. T. M. 1996 Reynolds-number-dependence of the maximum in the streamwise velocity fluctuations in wall turbulence. Exp. Fluids 21 (3), 218226.
Modesti, D. & Pirozzoli, S. 2016 Reynolds and Mach number effects in compressible turbulent channel flow. Intl J. Heat Fluid Flow 59, 3349.
Monokrousos, A., Bottaro, A., Brandt, L., Di Vita, A. & Henningson, D. S. 2011 Nonequilibrium thermodynamics and the optimal path to turbulence in shear flows. Phys. Rev. Lett. 106 (13), 134502.
Monty, J. P., Hutchins, N., Ng, H. C. H., Marusic, I. & Chong, M. S. 2009 A comparison of turbulent pipe, channel and boundary layer flows. J. Fluid Mech. 632, 431442.
Morkovin, M. V. 1962 Effects of compressibility on turbulent flows. In Mécanique de la Turbulence (ed. Favre, A.), pp. 367380. CNRS.
Morra, P., Semeraro, O., Henningson, D. S. & Cossu, C. 2019 On the relevance of Reynolds stresses in resolvent analyses of turbulent wall-bounded flows. J. Fluid Mech. 867, 969984.
Owen, F. K., Horstman, C. C. & Kussoy, M. I. 1975 Mean and fluctuating flow measurements of a fully-developed, non-adiabatic, hypersonic boundary layer. J. Fluid Mech. 70 (2), 393413.
Özgen, S. & Kırcalı, S. A. 2008 Linear stability analysis in compressible, flat-plate boundary-layers. Theor. Comp. Fluid Dyn. 22 (1), 120.
de Pando, M. F., Schmid, P. J. & Sipp, D. 2014 A global analysis of tonal noise in flows around aerofoils. J. Fluid Mech. 754, 538.
Patel, A., Peeters, J. W. R., Boersma, B. J. & Pecnik, R. 2015 Semi-local scaling and turbulence modulation in variable property turbulent channel flows. Phys. Fluids 27 (9), 095101.
Peltier, S. J., Humble, R. A. & Bowersox, R. D. W. 2016 Crosshatch roughness distortions on a hypersonic turbulent boundary layer. Phys. Fluids 28 (4), 045105.
Phillips, O. M. 1960 On the generation of sound by supersonic turbulent shear layers. J. Fluid Mech. 9 (1), 128.
Pirozzoli, S. & Bernardini, M. 2011 Turbulence in supersonic boundary layers at moderate Reynolds number. J. Fluid Mech. 688, 120168.
Pirozzoli, S., Bernardini, M. & Grasso, F. 2008 Characterization of coherent vortical structures in a supersonic turbulent boundary layer. J. Fluid Mech. 613, 205231.
Pirozzoli, S., Grasso, F. & Gatski, T. B. 2004 Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at M = 2. 25. Phys. Fluids 16 (3), 530545.
Poggie, J., Bisek, N. J. & Gosse, R. 2015 Resolution effects in compressible, turbulent boundary layer simulations. Comput. Fluids 120, 5769.
Ringuette, M. J., Wu, M. & Martin, M. P. 2008 Coherent structures in direct numerical simulation of turbulent boundary layers at Mach 3. J. Fluid Mech. 594, 5969.
Rosenberg, K., Symon, S. & McKeon, B. J. 2019 The role of parasitic modes in nonlinear closure via the resolvent feedback loop. Phys. Rev. Fluids 4, 052601(R).
Rowley, C. W., Colonius, T. & Murray, R. M. 2004 Model reduction for compressible flows using POD and Galerkin projection. Physica D 189 (1-2), 115129.
Roy, C. J. & Blottner, F. G. 2006 Review and assessment of turbulence models for hypersonic flows. Prog. Aerosp. Sci. 42 (7-8), 469530.
Schlatter, P. & Örlü, R. 2010 Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116126.
Schmid, P. J. & Henningson, D. S. 2000 Stability and transition in shear flows. Applied Mathematical Sciences, vol. 142. Springer Science & Business Media.
Schmidt, O. T., Towne, A., Rigas, G., Colonius, T. & Brès, G. A. 2018 Spectral analysis of jet turbulence. J. Fluid Mech. 855, 953982.
Shahab, M. F., Lehnasch, G., Gatski, T. B. & Comte, P. 2011 Statistical characteristics of an isothermal, supersonic developing boundary layer flow from DNS data. Flow Turbul. Combust. 86 (3–4), 369397.
Sharma, A. S. & McKeon, B. J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196238.
Sharma, A. S., Moarref, R. & McKeon, B. J. 2017 Scaling and interaction of self-similar modes in models of high Reynolds number wall turbulence. Phil. Trans. R. Soc. Lond. A 375 (2089), 20160089.
Sillero, J. A., Jiménez, J. & Moser, R. D. 2014 Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to 𝛿+ ≈ 2000. Phys. Fluids 26 (10), 105109.
Simens, M. P., Jiménez, J., Hoyas, S. & Mizuno, Y. 2009 A high-resolution code for turbulent boundary layers. J. Comput. Phys. 228 (11), 42184231.
Sipp, D. & Marquet, O. 2013 Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. 27 (5), 617635.
Spina, E. F. & Smits, A. J. 1987 Organized structures in a compressible, turbulent boundary layer. J. Fluid Mech. 182, 85109.
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.
Tichenor, N. R., Humble, R. A. & Bowersox, R. D. W. 2013 Response of a hypersonic turbulent boundary layer to favourable pressure gradients. J. Fluid Mech. 722, 187213.
Towne, A., Lozano-Durán, A. & Yang, X. I. A.2019 Resolvent-based estimation of space–time flow statistics. arXiv:1901.07478 [physics. flu-dyn]; J. Fluid Mech. to appear.
Towne, A., Schmidt, O. T. & Colonius, T. 2018 Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J. Fluid Mech. 847, 821867.
Trettel, A. & Larsson, J. 2016 Mean velocity scaling for compressible wall turbulence with heat transfer. Phys. Fluids 28 (2), 026102.
Van Driest, E. R. 1951 Turbulent boundary layer in compressible fluids. J. Aero. Sci. 18 (3), 145160.
Walz, A. 1969 Boundary Layers of Flow and Temperature. MIT press.
Williams, O. J. H., Sahoo, D., Baumgartner, M. L. & Smits, A. J. 2018 Experiments on the structure and scaling of hypersonic turbulent boundary layers. J. Fluid Mech. 834, 237270.
Wilson, R. E. 1950 Turbulent boundary-layer characteristics at supersonic speeds-theory and experiment. J. Aero. Sci. 17 (9), 585594.
Yang, X. I. A. & Lv, Y. 2018 A semi-locally scaled eddy viscosity formulation for LES wall models and flows at high speeds. Theor. Comput. Fluid Dyn. 32 (5), 617627.
Yeh, C.-A. & Taira, K. 2019 Resolvent-analysis-based design of airfoil separation control. J. Fluid Mech. 867, 572610.
Young, N. 1988 An Introduction to Hilbert Space. Cambridge University Press.
Zare, A., Jovanović, M. R. & Georgiou, T. T. 2017 Colour of turbulence. J. Fluid Mech. 812, 636680.
Zhang, C., Duan, L. & Choudhari, M. M. 2018 Direct numerical simulation database for supersonic and hypersonic turbulent boundary layers. AIAA J. 56 (11), 42974311.
Zhang, Y.-S., Bi, W.-T., Hussain, F., Li, X.-L. & She, Z.-S. 2012 Mach-number-invariant mean-velocity profile of compressible turbulent boundary layers. Phys. Rev. Lett. 109 (5), 054502.
Zhang, Y.-S., Bi, W.-T., Hussain, F. & She, Z.-S. 2014 A generalized Reynolds analogy for compressible wall-bounded turbulent flows. J. Fluid Mech. 739, 392420.
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Resolvent-based study of compressibility effects on supersonic turbulent boundary layers

  • H. Jane Bae (a1), Scott T. M. Dawson (a1) (a2) and Beverley J. McKeon (a1)

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