Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-13T04:07:33.797Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  26 November 2019

H. Jane Bae Open the ORCID record for H. Jane Bae [Opens in a new window] ,
Scott T. M. Dawson Open the ORCID record for Scott T. M. Dawson [Opens in a new window]  and
Beverley J. McKeon Open the ORCID record for Beverley J. McKeon [Opens in a new window]
H. Jane Bae*
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA91125, USA
Scott T. M. Dawson
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA91125, USA Mechanical, Materials and Aerospace Engineering Department, Illinois Institute of Technology, Chicago, IL60616, USA
Beverley J. McKeon
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA91125, USA
*
†Email address for correspondence: hjbae@caltech.edu
Get access Rights & Permissions [Opens in a new window]

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.

JFM classification

Compressible Flows: Compressible boundary layers Turbulent Flows: Compressible turbulence Turbulent Flows: Turbulence modelling
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A29
Copyright
© 2019 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alizard, F., Robinet, J.-C. & Filliard, G. 2015 Sensitivity analysis of optimal transient growth for turbulent boundary layers. Eur. J. Mech. (B/Fluids) 49, 373–386.CrossRefGoogle Scholar
Bernardini, M. & Pirozzoli, S. 2011 Wall pressure fluctuations beneath supersonic turbulent boundary layers. Phys. Fluids 23 (8), 085102.CrossRefGoogle Scholar
Bitter, N. & Shepherd, J.2014 Transient growth in hypersonic boundary layers. In 7th AIAA Theoretical Fluid Mechanics Conference, Atlanta, GA, AIAA Paper 2014-2497.Google Scholar
Bradshaw, P. 1974 The effect of mean compression or dilatation on the turbulence structure of supersonic boundary layers. J. Fluid Mech. 63 (3), 449–464.CrossRefGoogle Scholar
Brun, C., Boiarciuc, M. P., Haberkorn, M. & Comte, P. 2008 Large eddy simulation of compressible channel flow. Theor. Comput. Fluid Dyn. 22 (3-4), 189–212.CrossRefGoogle Scholar
Cebeci, T. & Bradshaw, P. 2012 Physical and Computational Aspects of Convective Heat Transfer. Springer Science & Business Media.Google Scholar
Christov, C. I. 1982 A complete orthonormal system of functions in L 2(-∞, ∞) space. SIAM J. Appl. Maths 42 (6), 1337–1344.CrossRefGoogle Scholar
Chu, B.-T. 1965 On the energy transfer to small disturbances in fluid flow (Part I). Acta Mech. 1 (3), 215–234.CrossRefGoogle Scholar
Coleman, G. N., Kim, J. & Moser, R. D. 1995 A numerical study of turbulent supersonic isothermal-wall channel flow. J. Fluid Mech. 305, 159–183.CrossRefGoogle Scholar
Coles, D. 1964 The turbulent boundary layer in a compressible fluid. Phys. Fluids 7 (9), 1403–1423.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Del Alamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135–144.CrossRefGoogle Scholar
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, 419–445.CrossRefGoogle Scholar
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, 245–267.CrossRefGoogle Scholar
Duan, L., Choudhari, M. M. & Zhang, C. 2016 Pressure fluctuations induced by a hypersonic turbulent boundary layer. J. Fluid Mech. 804, 578–607.CrossRefGoogle Scholar
Duan, L. & Martin, M. P. 2011 Direct numerical simulation of hypersonic turbulent boundary layers. Part 4. Effect of high enthalpy. J. Fluid Mech. 684, 25–59.CrossRefGoogle Scholar
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.Google Scholar
Ekoto, I. W., Bowersox, R. D. W., Beutner, T. & Goss, L. P. 2008 Supersonic boundary layers with periodic surface roughness. AIAA J. 46 (2), 486–497.CrossRefGoogle Scholar
Erm, L. P. & Joubert, P. N. 1991 Low-Reynolds-number turbulent boundary layers. J. Fluid Mech. 230, 1–44.CrossRefGoogle Scholar
Ffowcs Williams, J. E. & Maidanik, G. 1965 The Mach wave field radiated by supersonic turbulent shear flows. J. Fluid Mech. 21 (4), 641–657.CrossRefGoogle Scholar
Gaviglio, J. 1987 Reynolds analogies and experimental study of heat transfer in the supersonic boundary layer. Intl J. Heat Mass Transfer 30 (5), 911–926.CrossRefGoogle Scholar
Grosch, C. E. & Orszag, S. A. 1977 Numerical solution of problems in unbounded regions: coordinate transforms. J. Comput. Phys. 25 (3), 273–295.CrossRefGoogle Scholar
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, 1–33.CrossRefGoogle Scholar
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, 426–438.CrossRefGoogle Scholar
Hanifi, A., Schmid, P. J. & Henningson, D. S. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8 (3), 826–837.CrossRefGoogle Scholar
Huang, P. G., Coleman, G. N. & Bradshaw, P. 1995 Compressible turbulent channel flows: DNS results and modelling. J. Fluid Mech. 305, 185–218.CrossRefGoogle Scholar
Illingworth, S. J., Monty, J. P. & Marusic, I. 2018 Estimating large-scale structures in wall turbulence using linear models. J. Fluid Mech. 842, 146–162.CrossRefGoogle Scholar
Jeun, J., Nichols, J. W. & Jovanović, M. R. 2016 Input–output analysis of high-speed axisymmetric isothermal jet noise. Phys. Fluids 28 (4), 047101.CrossRefGoogle Scholar
Jiménez, J. & Hoyas, S. 2008 Turbulent fluctuations above the buffer layer of wall-bounded flows. J. Fluid Mech. 611, 215–236.CrossRefGoogle Scholar
Jiménez, J., Hoyas, S., Simens, M. P. & Mizuno, Y. 2010 Turbulent boundary layers and channels at moderate Reynolds numbers. J. Fluid Mech. 657, 335–360.CrossRefGoogle Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.CrossRefGoogle Scholar
Kistler, A. L. 1959 Fluctuation measurements in a supersonic turbulent boundary layer. Phys. Fluids 2 (3), 290–296.CrossRefGoogle Scholar
Konrad, W. & Smits, A. J. 1998 Turbulence measurements in a three-dimensional boundary layer in supersonic flow. J. Fluid Mech. 372, 1–23.CrossRefGoogle Scholar
Kovasznay, L. S. G. 1953 Turbulence in supersonic flow. J. Aero. Sci. 20 (10), 657–674.CrossRefGoogle Scholar
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), 121–144.CrossRefGoogle Scholar
Lagha, M., Kim, J., Eldredge, J. D. & Zhong, X. 2011 A numerical study of compressible turbulent boundary layers. Phys. Fluids 23 (1), 015106.CrossRefGoogle Scholar
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), 997–1012.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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, 38–70.CrossRefGoogle Scholar
Mack, L. M.1984 Boundary-layer linear stability theory. AGARD Report No. 709, Part 3. NASA Jet Propulsion Laboratory.Google Scholar
Maeder, T. 2000 Numerical Investigation of Supersonic Turbulent Boundary Layers, vol. 394. ETH Zurich.Google Scholar
Malik, M., Alam, M. & Dey, J. 2006 Nonmodal energy growth and optimal perturbations in compressible plane Couette flow. Phys. Fluids 18 (3), 034103.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Martín, M. P. 2007 Direct numerical simulation of hypersonic turbulent boundary layers. Part 1. Initialization and comparison with experiments. J. Fluid Mech. 570, 347–364.CrossRefGoogle Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.CrossRefGoogle Scholar
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), 692–701.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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, 275–316.CrossRefGoogle Scholar
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), 218–226.CrossRefGoogle Scholar
Modesti, D. & Pirozzoli, S. 2016 Reynolds and Mach number effects in compressible turbulent channel flow. Intl J. Heat Fluid Flow 59, 33–49.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
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, 431–442.CrossRefGoogle Scholar
Morkovin, M. V. 1962 Effects of compressibility on turbulent flows. In Mécanique de la Turbulence (ed. Favre, A.), pp. 367–380. CNRS.Google Scholar
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, 969–984.CrossRefGoogle Scholar
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), 393–413.CrossRefGoogle Scholar
Özgen, S. & Kırcalı, S. A. 2008 Linear stability analysis in compressible, flat-plate boundary-layers. Theor. Comp. Fluid Dyn. 22 (1), 1–20.CrossRefGoogle Scholar
de Pando, M. F., Schmid, P. J. & Sipp, D. 2014 A global analysis of tonal noise in flows around aerofoils. J. Fluid Mech. 754, 5–38.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Phillips, O. M. 1960 On the generation of sound by supersonic turbulent shear layers. J. Fluid Mech. 9 (1), 1–28.CrossRefGoogle Scholar
Pirozzoli, S. & Bernardini, M. 2011 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., 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), 530–545.CrossRefGoogle Scholar
Poggie, J., Bisek, N. J. & Gosse, R. 2015 Resolution effects in compressible, turbulent boundary layer simulations. Comput. Fluids 120, 57–69.CrossRefGoogle Scholar
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, 59–69.CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
Rowley, C. W., Colonius, T. & Murray, R. M. 2004 Model reduction for compressible flows using POD and Galerkin projection. Physica D 189 (1-2), 115–129.CrossRefGoogle Scholar
Roy, C. J. & Blottner, F. G. 2006 Review and assessment of turbulence models for hypersonic flows. Prog. Aerosp. Sci. 42 (7-8), 469–530.CrossRefGoogle Scholar
Schlatter, P. & Örlü, R. 2010 Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116–126.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2000 Stability and transition in shear flows. Applied Mathematical Sciences, vol. 142. Springer Science & Business Media.Google Scholar
Schmidt, O. T., Towne, A., Rigas, G., Colonius, T. & Brès, G. A. 2018 Spectral analysis of jet turbulence. J. Fluid Mech. 855, 953–982.CrossRefGoogle Scholar
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), 369–397.CrossRefGoogle Scholar
Sharma, A. S. & McKeon, B. J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Simens, M. P., Jiménez, J., Hoyas, S. & Mizuno, Y. 2009 A high-resolution code for turbulent boundary layers. J. Comput. Phys. 228 (11), 4218–4231.CrossRefGoogle Scholar
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), 617–635.CrossRefGoogle Scholar
Spina, E. F. & Smits, A. J. 1987 Organized structures in a compressible, turbulent boundary layer. J. Fluid Mech. 182, 85–109.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.Google Scholar
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, 187–213.CrossRefGoogle Scholar
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.Google Scholar
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, 821–867.CrossRefGoogle Scholar
Trettel, A. & Larsson, J. 2016 Mean velocity scaling for compressible wall turbulence with heat transfer. Phys. Fluids 28 (2), 026102.CrossRefGoogle Scholar
Van Driest, E. R. 1951 Turbulent boundary layer in compressible fluids. J. Aero. Sci. 18 (3), 145–160.CrossRefGoogle Scholar
Walz, A. 1969 Boundary Layers of Flow and Temperature. MIT press.Google Scholar
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, 237–270.CrossRefGoogle Scholar
Wilson, R. E. 1950 Turbulent boundary-layer characteristics at supersonic speeds-theory and experiment. J. Aero. Sci. 17 (9), 585–594.CrossRefGoogle Scholar
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), 617–627.CrossRefGoogle Scholar
Yeh, C.-A. & Taira, K. 2019 Resolvent-analysis-based design of airfoil separation control. J. Fluid Mech. 867, 572–610.CrossRefGoogle Scholar
Young, N. 1988 An Introduction to Hilbert Space. Cambridge University Press.CrossRefGoogle Scholar
Zare, A., Jovanović, M. R. & Georgiou, T. T. 2017 Colour of turbulence. J. Fluid Mech. 812, 636–680.CrossRefGoogle Scholar
Zhang, C., Duan, L. & Choudhari, M. M. 2018 Direct numerical simulation database for supersonic and hypersonic turbulent boundary layers. AIAA J. 56 (11), 4297–4311.CrossRefGoogle Scholar
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.Google ScholarPubMed
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, 392–420.CrossRefGoogle Scholar