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A unified temperature transformation for high-Mach-number flows above adiabatic and isothermal walls

Published online by Cambridge University Press:  10 November 2022

Peng E.S. Chen ,
George P. Huang ,
Yipeng Shi ,
Xiang I.A. Yang Open the ORCID record for Xiang I.A. Yang [Opens in a new window]  and
Yu Lv Open the ORCID record for Yu Lv [Opens in a new window]
Peng E.S. Chen
Affiliation:
Department of Mechanics and Engineering Science, Peking University, Beijing 100871, PR China
George P. Huang
Affiliation:
Mechanical and Materials Engineering, Wright State University, Dayton, OH 45435, USA
Yipeng Shi
Affiliation:
Department of Mechanics and Engineering Science, Peking University, Beijing 100871, PR China
Xiang I.A. Yang*
Affiliation:
Mechanical Engineering, Pennsylvania State University, PA 16802, USA
Yu Lv*
Affiliation:
State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

The mean velocity follows a logarithmic scaling in the surface layer when normalized by the friction velocity, i.e. a velocity scale derived from the wall-shear stress. The same logarithmic scaling exists for the mean temperature when one normalizes the temperature with the friction temperature, i.e. a scale derived from the wall heat flux. This temperature normalization poses challenges to adiabatic walls, for which the wall heat flux is zero, and the logarithmic temperature scaling becomes singular. This paper aims to establish a temperature transformation that applies to both isothermal walls and adiabatic walls. We show that by accounting for the diffusive flux, both the Van Driest transformation and the semi-local transformation (and other transformations alike) apply to adiabatic walls. We also show that the classic Walz equation works well for adiabatic walls because it models the diffusive flux, albeit in a rather crude way. For validation/testing, we conduct direct numerical simulations of supersonic Couette flows at Mach numbers $M=1$, 3 and 6, and various Reynolds numbers. The two walls are adiabatic, and a source term is included to cancel the aerodynamic heating in the domain. We show that the adiabatic wall data collapse onto the same incompressible logarithmic law of the wall like the isothermal wall data.

JFM classification

Turbulent Flows: Compressible turbulence Aerodynamics: High-speed flow Turbulent Flows: Turbulence modelling
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 951 , 25 November 2022 , A38
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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