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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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