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Optimal transient growth in compressible turbulent boundary layers

Published online by Cambridge University Press:  30 March 2015

F. Alizard ,
S. Pirozzoli ,
M. Bernardini  and
F. Grasso
F. Alizard*
Affiliation:
Laboratoire DynFluid, Arts & Métiers ParisTech et CNAM, 151 Boulevard de l’Hôpital, 75013 Paris, France
S. Pirozzoli
Affiliation:
Dipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza Università di Roma, Via Eudossiana 18, 00184 Rome, Italy
M. Bernardini
Affiliation:
Dipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza Università di Roma, Via Eudossiana 18, 00184 Rome, Italy
F. Grasso
Affiliation:
Laboratoire DynFluid, Arts & Métiers ParisTech et CNAM, 151 Boulevard de l’Hôpital, 75013 Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

The structure of zero-pressure-gradient compressible turbulent boundary layers is analysed using the tools of optimal transient growth theory. The approach relies on the extension to compressible flows of the theoretical framework originally developed by Reynolds & Hussain (J. Fluid Mech., vol. 52, 1972, pp. 263–288) for incompressible flows. The model is based on a density-weighted triple decomposition of the instantaneous field into the contributions of the mean flow, the organized (coherent) motions and the disorganized background turbulent fluctuations. The mean field and the eddy viscosity characterizing the incoherent fluctuations are here obtained from a direct numerical simulation database. Most temporally amplified modes (optimal modes) are found to be consistent with scaling laws of turbulent boundary layers for both inner and outer layers, as well as in the logarithmic region, where they exhibit a self-similar spreading. Four free-stream Mach numbers are considered: $\mathit{Ma}_{\infty }=0.2$, 2, 3 and 4. Weak effects of compressibility on the characteristics length and the orientation angles are observed for both the inner- and the outer-layer modes. Furthermore, taking into account the effects of mean density variations, a universal behaviour is suggested for the optimal modes that populate the log layer, regardless of the Mach number. The relevance of the optimal modes in describing the near-wall layer dynamics and the eddies that populate the outer region is discussed.

JFM classification

Compressible Flows: Compressible Flows Turbulent Flows: Compressible turbulence Turbulent Flows: Turbulent boundary layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 770 , 10 May 2015 , pp. 124 - 155
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© 2015 Cambridge University Press 

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References

del Alamo, J. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.Google Scholar
del Alamo, J., Jiménez, J., Zandonade, P. & Moser, R. 2006 Self-similar vortex clusters in the turbulent logarithmic region. J. Fluid Mech. 561, 329–358.Google Scholar
Alizard, F., Robinet, J.-C. & Filliard, G. 2015 Sensitivity analysis of optimal transient growth for turbulent boundary layers. Eur. J. Mech. Fluids 49, 373–386.CrossRefGoogle Scholar
Alizard, F., Robinet, J.-C. & Guiho, F. 2013 Transient growth in a right-angled streamwise corner. Eur. J. Mech. Fluids 719, 406–430.Google Scholar
Bernardini, M. & Pirozzoli, S. 2011a Inner/outer layer interactions in turbulent boundary layers: a refined measure for the large-scale amplitude modulation mechanism. Phys. Fluids 23, 061701.Google Scholar
Bernardini, M. & Pirozzoli, S. 2011b Wall pressure fluctuations beneath supersonic turbulent boundary layers. Phys. Fluids 23, 052102.CrossRefGoogle Scholar
Butler, K. & Farrel, B. 1992 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids 5, 774–777.CrossRefGoogle Scholar
Cantwell, B. & Coles, D. 1983 An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder. J. Fluid Mech. 136, 321–374.Google Scholar
Chu, B. T. 1965 On the energy transfer to small disturbances in fluid flow (Part 1). Acta Mechanica 1, 215–234.CrossRefGoogle Scholar
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large-scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94.Google Scholar
Duan, L., Beekman, I. & Martin, M. 2011 Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number. J. Fluid Mech. 672, 245–267.Google Scholar
Elsinga, G., Adrian, R., van Oudheusden, B. & Scarano, F. 2010 Three-dimensional vortex organization in a high-Reynolds-number supersonic turbulent boundary layer. J. Fluid Mech. 644, 35–50.Google Scholar
Farrell, B. & Ioannou, P. 1993 Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5, 1390–1400.Google Scholar
Ganapathisubramani, B., Clemens, N. & Dolling, D. 2006 Large-scale motions in a supersonic turbulent boundary layer. J. Fluid Mech. 556, 271–282.CrossRefGoogle Scholar
Guegan, A., Huerre, P. & Schmid, P. 2007 Optimal disturbances in swept Hiemenz flow. J. Fluid Mech. 578, 223–232.Google Scholar
Hamilton, J., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–349.Google Scholar
Hanifi, A., Schmid, P. & Henningson, D. 1996 Transient growth in compressible boundary layer. Phys. Fluids 8, 826–837.Google Scholar
He, J., Kazakia, T. & Walker, J. 1995 An asymptotic two-layer model for supersonic turbulent boundary layers. J. Fluid Mech. 285, 159–198.CrossRefGoogle Scholar
Hutchins, N. & Marusic, I. 2007 Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.Google Scholar
Hwang, Y. & Cossu, C. 2010a Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 51–73.Google Scholar
Hwang, Y. & Cossu, C. 2010b Self-sustained process at large scales in turbulent channel flow. Phys. Rev. Lett. 105, 044505.Google Scholar
Hwang, Y. & Cossu, C. 2011 Self-sustained processes in the logarithmic layer of turbulent channel flows. J. Fluid Mech. 23, 061702.Google Scholar
Jiménez, J. 2013 How linear is wall-bounded turbulence? Phys. Fluids 25, 110814.CrossRefGoogle Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359.CrossRefGoogle Scholar
Kim, K. C. & Adrian, R. J. 1999 Very large-scale motion in the outer layer. Phys. Fluids 11, 417–422.Google Scholar
Kim, H., Kline, S. & Reynolds, W. 1971 The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech. 50, 133–160.Google Scholar
Kim, J. & Lim, J. 2000 A linear process in wall-bounded turbulent shear flows. Phys. Fluids 12 (8), 1885–1888.Google Scholar
Kline, S., Reynolds, W., Schraub, F. & Runstadler, P. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773.Google Scholar
Lagha, M., Kim, J., Eldredge, J. & Zhong, X. 2011 A numerical study of compressible turbulent boundary layers. Phys. Fluids 23, 015106.Google Scholar
Landahl, M. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.Google Scholar
Lee, M., Kim, J. & Moin, P. 1990 Structure of turbulence at high shear rate. J. Fluid Mech. 216, 561–583.CrossRefGoogle Scholar
Malik, M. 1990 Numerical methods for hypersonic boundary layer stability. Phys. Fluids 86, 376–413.Google Scholar
Malik, M., Meheboob, A. & Dey, J. 2006 Nonmodal energy growth and optimal perturbations in compressible plane Couette flow. Phys. Fluids 18, 034103.Google Scholar
Marusic, I. & Heuer, W. 2007 Reynolds number invariance of the structure inclination angle in wall turbulence. Phys. Rev. Lett. 99, 114504.Google Scholar
Marusic, I., Monty, J., Hultmark, M. & Smits, A. J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.CrossRefGoogle Scholar
Moarref, R., Sharma, A., Tropp, J. & McKeon, B. 2013 Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels. J. Fluid Mech. 734, 275–316.CrossRefGoogle Scholar
Monkewitz, P., Chauhan, K. & Nagib, H. 2007 Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers. Phys. Fluids 19, 115101.Google Scholar
Morkovin, M. 1961 Effects of compressibility on turbulent flows. In Mécanique de la Turbulence (ed. Favre, A.), pp. 367–380. CNRS.Google Scholar
Panton, R. 2001 Overview of the self-sustaining mechanisms of wall turbulence. Prog. Aerosp. Sci. 37, 341–383.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. 2013 Probing high-Reynolds-number effectis in numerical boundary layers. Phys. Fluids 25, 021704.Google 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.Google Scholar
Pirozzoli, S., Bernardini, M. & Grasso, F. 2010 On the dynamical relevance of coherent vortical structures in turbulent boundary layers. J. Fluid Mech. 648, 325–349.Google Scholar
Pujals, G., Garci-Villalba, M., Cossu, C. & Depardon, S. 2009 A note on optimal transient growth in turbulent channels flows. Phys. Fluids 21, 01519.CrossRefGoogle Scholar
Reshotko, E. 1976 Boundary-layer stability and transition. Annu. Rev. Fluid Mech. 8, 311–349.CrossRefGoogle Scholar
Reynolds, W. & Hussain, K. 1972 The mechanics of an organized wave in turbulence shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech. 52, 263–288.Google Scholar
Ringuette, M. J., Wu, M. & Martín, M. 2008 Coherent structures in direct numerical simulation of turbulent boundary layers at Mach 3. J. Fluid Mech. 594, 59–69.Google Scholar
Schmid, P. & Henningson, D. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Sidje, R. 1998 EXPOKIT: a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24, 130–156.CrossRefGoogle Scholar
Smith, M. & Smits, A. 1995 Visualization of the structure of supersonic turbulent boundary layers. Exp. Fluids 18, 288–302.Google Scholar
Smits, A., McKeon, B. & Marusic, I. 2011 High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43, 353–375.Google Scholar
Spina, E., Smits, A. & Robinson, S. 1994 The physics of supersonic turbulent boundary layers. Annu. Rev. Fluid Mech. 26, 287–319.CrossRefGoogle Scholar
Theodorsen, T.1952 Mechanism of turbulence. In Proceedings of Second Midwestern Conference on Fluid Mechanics, Ohio State University, Columbia, OH, USA, pp. 1–19.Google Scholar
Tomkins, C. & Adrian, R. 2003 Spanwise structure and scale growth in turbulent boundary layers. J. Fluid Mech. 490, 37–74.Google Scholar
Townsend, A. 1976 The Structure of Turbulent Shear Flow, vol. 2. Cambridge University Press.Google Scholar
Van Driest, E. 1951 Turbulent boundary layer in compressible fluids. J. Aero. Sci. 18, 145–160.Google Scholar