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A generalized Reynolds analogy for compressible wall-bounded turbulent flows

Published online by Cambridge University Press:  20 December 2013

You-Sheng Zhang ,
Wei-Tao Bi ,
Fazle Hussain  and
Zhen-Su She
You-Sheng Zhang
Affiliation:
State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China Institute of Applied Physics and Computational Mathematics, Beijing 100094, China
Wei-Tao Bi*
Affiliation:
State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China
Fazle Hussain
Affiliation:
Department of Mechanical Engineering,Texas Tech University, Lubbock, TX 79409-1021, USA
Zhen-Su She
Affiliation:
State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China
*
†Email address for correspondence: weitaobi@pku.edu.cn
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Abstract

A generalized Reynolds analogy (GRA) is proposed for compressible wall-bounded turbulent flows (CWTFs) and validated by direct numerical simulations. By introducing a general recovery factor, a similarity between the Reynolds-averaged momentum and energy equations is established for the canonical CWTFs (i.e. pipes, channels, and flat-plate boundary layers that meet the quasi-one-dimensional flow approximation), independent of Prandtl number, wall temperature, Mach number, Reynolds number, and pressure gradient. This similarity and the relationships between temperature and velocity fields constitute the GRA. The GRA relationship between the mean temperature and the mean velocity takes the same quadratic form as Walz’s equation, with the adiabatic recovery factor replaced by the general recovery factor, and extends the validity of the latter to diabatic compressible turbulent boundary layers and channel/pipe flows. It also derives Duan & Martín’s (J. Fluid Mech., vol. 684, 2011, pp. 25–59) empirical relation for flows at different physical conditions (wall temperature, Mach number, enthalpy condition, surface catalysis, etc.). Several key parameters besides the general recovery factor emerge in the GRA. An effective turbulent Prandtl number is shown to be the reason for the parabolic profile of mean temperature versus mean velocity, and it approximates unity in the fully turbulent region. A dimensionless wall temperature, that we call the diabatic parameter, characterizes the wall-temperature effects in diabatic flows. The GRA also extends the analysis to the fluctuation fields. It recovers the modified strong Reynolds analogy proposed by Huang, Coleman & Bradshaw (J. Fluid Mech., vol. 305, 1995, pp. 185–218) and explains the variation of the temperature–velocity correlation coefficient with wall temperature. Thus, the GRA unveils a generalized similarity principle behind the complex nonlinear coupling between the thermal and velocity fields of CWTFs.

JFM classification

Compressible Flows: Compressible boundary layers Turbulent Flows: Compressible turbulence Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 739 , 25 January 2014 , pp. 392 - 420
Copyright
©2013 Cambridge University Press 

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References

Bradshaw, P. 1977 Compressible turbulent shear layers. Annu. Rev. Fluid Mech. 9, 33–52.CrossRefGoogle Scholar
Brun, C., Petrovan Boiarciuc, M., Haberkorn, M. & Comte, P. 2008 Large eddy simulation of compressible channel flow. Theor. Comput. Fluid Dyn. 22 (3), 189–212.Google Scholar
Busemann, A. 1931 Handbuch der Experimentalphysik, vol. 4. Geest und Portig.Google Scholar
Cebeci, T. & Smith, A. M. O. 1974 Analysis of Turbulent Boundary Layers. Academic.Google Scholar
Chi, S. W. & Spalding, D. B. 1966 Influence of temperature ratio on heat transfer to a flat plate through a turbulent boundary layer in air. In International Heat Transfer Conference, 3 RD, Chicago, Illinois, pp. 41–49.Google Scholar
Crocco, L. 1932 Sulla trasmissione del calore da una lamina piana a un fluido scorrente ad alta velocita. L Aerotecnica 12, 181–197.Google Scholar
Debiève, J. F. 1976 Contribution à l’étude du comportement d’un écoulement compressible turbulent $(M= 2. 3)$ soumis à des gradients élevés de vitesse et de pression. Doctorat de spécialité Univ. d’Aix Marseille II.Google Scholar
Debieve, J.-F, Gouin, H. & Gaviglio, J. 1982 Momentum and temperature fluxes in a shock wave-turbulence interaction. Struct. Turbul. Heat Mass Transfer 1, 277–296.Google Scholar
Dong, M. & Zhou, H. 2010 The improvement of turbulence modelling for the aerothermal computation of hypersonic turbulent boundary layers. Sci. China G: Phys. Mech. Astron. 53 (2), 369–379.Google Scholar
Duan, L. 2011 DNS of hypersonic turbulent boundary layers. PhD thesis, Princeton University.Google Scholar
Duan, L., Beekman, I. & Martín, M. P. 2010 Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature. J. Fluid Mech. 655, 419–445.Google Scholar
Duan, L., Beekman, I. & Martín, M. P. 2011 Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number. J. Fluid Mech. 672, 245–267.Google Scholar
Duan, L. & Martín, 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
Gatski, T. B. & Bonnet, J. P. 2009 Compressibility, Turbulence and High Speed Flow. Elsevier.Google 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.Google Scholar
Ghosh, S., Foysi, H. & Friedrich, R. 2010 Compressible turbulent channel and pipe flow: similarities and differences. J. Fluid Mech. 648, 155–181.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.Google Scholar
Hopkins, E. J. & Inouye, M. 1971 An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers. AIAA J. 9 (6), 993–1003.Google Scholar
Howarth, L. 1953 Modern Developments in Fluid Dynamics: High Speed Flow, vol.1. Clarendon.Google Scholar
Huang, P. G., Coleman, G. N. & Bradshaw, P. 1995 Compressible turbulent channel flows: DNS results and modelling. J. Fluid Mech. 305, 185–218.Google Scholar
Jiang, L.-Y. & Campbell, I. 2008 Reynolds analogy in combustor modelling. Intl J. Heat Mass Transfer 51 (5), 1251–1263.Google Scholar
Kays, W. M. 1994 Turbulent Prandtl number – where are we? J. Heat Transfer 116, 284–295.CrossRefGoogle Scholar
Laderman, A. J. 1978 Effect of wall temperature on a supersonic turbulent boundary layer. AIAA J. 16 (7), 723–729.Google 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, 121–144.Google Scholar
Li, X.-L., Fu, D.-X. & Ma, Y.-W. 2006 Direct numerical simulation of a spatially evolving supersonic turbulent boundary layer at $Ma= 6$ . Chin. Phys. Lett. 23 (6), 1519–1522.Google Scholar
Li, X., Ma, Y. & Fu, D. 2001 DNS and scaling law analysis of compressible turbulent channel flow. Sci. China A: Maths 44 (5), 645–654.Google Scholar
Maeder, T., Adams, N. A. & Kleiser, L. 2001 Direct simulation of turbulent supersonic boundary layers by an extended temporal approach. J. Fluid Mech. 429, 187–216.Google Scholar
Morkovin, M. V. 1962 Effects of compressibility on turbulent flows. In Mecanique de la Turbulence (ed. Favre, A.), pp. 367–380. CNRS.Google 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.Google Scholar
Pirozzoli, S. & Bernardini, M. 2011 Turbulence in supersonic boundary layers at moderate Reynolds number. J. Fluid Mech. 688, 120–168.Google Scholar
Pirozzoli, S., Grasso, F. & Gatski, T. B. 2004 Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at $M$ . Phys. Fluids 16 (3), 530–545.Google Scholar
Reynolds, O. 1874 On the extent and action of the heating surface of steam boilers. Manchester Lit. Phil. Soc. 14, 7–12 (reproduced in Reynolds, O. 1961 On the extent and action of the heating surface of steam boilers. Intl J. Heat Mass Transfer 3 (2), 163–166)..CrossRefGoogle Scholar
Rubesin, M. W. 1990 Extra compressibility terms for favre-averaged two-equation models of inhomogeneous turbulent flows. NASA STI/Recon Tech. Rep. N 90, 23701.Google Scholar
She, Z.-S., Chen, X., Wu, Y. & Hussain, F. 2010 New perspective in statistical modelling of wall-bounded turbulence. Acta Mechanica Sin. 1–15.Google Scholar
She, Z. S., Wu, Y., Chen, X. & Hussain, F. 2012 A multi-state description of roughness effects in turbulent pipe flow. New J. Phys. 14 (9), 093054.CrossRefGoogle Scholar
Sheshagir, K. & Paranjpe, P. A. 1969 Skin friction in compressible turbulent boundary layers. AIAA J. 7 (4), 793–796.Google Scholar
Smith, D. R. & Smits, A. J. 1993 Simultaneous measurement of velocity and temperature fluctuations in the boundary layer of a supersonic flow. Exp. Therm. Fluid Sci. 7 (3), 221–229.Google Scholar
Smits, A. J. & Dussauge, J. P. 2006 Turbulent Shear Layers in Supersonic Flow. Springer.Google Scholar
Sons, J. 2005 A critical assessment of Reynolds analogy for turbine flows. J. Heat Transfer 127 (5), 472–485.Google Scholar
Spalding, D. B. & Chi, S. W. 1964 The drag of a compressible turbulent boundary layer on a smooth flat plate with and without heat transfer. J. Fluid Mech. 18, 117–143.Google Scholar
Spina, E. F., Smits, A. J. & Robinson, S. K. 1994 The physics of supersonic turbulent boundary layers. Annu. Rev. Fluid Mech. 26 (1), 287–319.Google Scholar
Stalker, R. J. 2005 Control of hypersonic turbulent skin friction by boundary-layer combustion of hydrogen. J. Spacecr. Rockets 42 (4), 577–587.Google Scholar
Van Driest, E. R. 1951 Turbulent boundary layer in compressible fluids. J. Aeronaut. Sci. 18 (3), 145–160; (Also available in ‘Van Driest, E. R. 2003 Turbulent boundary layer in compressible fluids. J. Spacecr. Rockets 40 (6), 1012–1028’).Google Scholar
Walz, A. 1962 Compressible Turbulent Boundary Layers, pp. 299–350 CNRS.Google Scholar
Walz, A. 1966 Strömungs-und Temperaturgrenzschichten. Braun (translation in Boundary Layers of Flow and Temperature, MIT Press, 1969).Google Scholar
Wang, Y.-Z. 2012 A multi-layer theory for mean field prediction in compressible channel flow (in Chinese). PhD thesis, Peking University.Google Scholar
White, F. M. 2006 Viscous Fluid Flow. McGraw-Hill.Google Scholar
Whitfield, D. L. & High, M. D. 1977 Velocity-temperature relations in turbulent boundary layers with nonunity Prandtl numbers. AIAA J. 15 (3), 431–434.Google 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.CrossRefGoogle ScholarPubMed