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

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...

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Bibliographic Details
Published inJournal of fluid mechanics Vol. 951
Main Authors Chen, Peng E.S., Huang, George P., Shi, Yipeng, Yang, Xiang I.A., Lv, Yu
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 25.11.2022
Subjects
Adiabatic
Adiabatic flow
Aerodynamic heating
Couette flow
Direct numerical simulation
Fluctuations
Fluid flow
Friction
Heat
Heat flux
Heat transfer
JFM Papers
Law of the wall
Logarithms
Mach number
Mathematical models
Reynolds number
Scaling
Shear stress
Surface layers
Temperature
Transformations (mathematics)
Velocity
Viscosity
Wall shear stresses
compressible turbulence
high-speed flow
turbulence modelling
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Summary: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.
ISSN:0022-1120
1469-7645
DOI:10.1017/jfm.2022.860