Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-11T01:46:56.925Z Has data issue: false hasContentIssue false
  • English
  • Français

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: xzy48@psu.edu, lvyu@imech.ac.cn
†Email addresses for correspondence: xzy48@psu.edu, lvyu@imech.ac.cn
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abe, H., Kawamura, H. & Matsuo, Y. 2004 Surface heat-flux fluctuations in a turbulent channel flow up to $Re_\tau = 1020$ with $Pr= 0.025$ and 0.71. Intl J. Heat Fluid Flow 25 (3), 404–419.CrossRefGoogle Scholar
Alcántara-Ávila, F., Hoyas, S. & Pérez-Quiles, M.J. 2021 Direct numerical simulation of thermal channel flow for. J. Fluid Mech. 916, A29.CrossRefGoogle Scholar
Bergman, T.L., Incropera, F.P., Lavine, A.S. & DeWitt, D.P. 2011 Introduction to Heat Transfer. John Wiley & Sons.Google Scholar
Bradshaw, P. & Huang, G.P. 1995 The law of the wall in turbulent flow. Proc. R. Soc. Lond. A 451 (1941), 165–188.Google Scholar
Brun, C., Boiarciuc, M.P., Haberkorn, M. & Comte, P. 2008 Large eddy simulation of compressible channel flow. Theor. Comput. Fluid Dyn. 22 (3), 189–212.CrossRefGoogle Scholar
Chen, P.E.S., Lv, Y., Xu, H.H.A., Shi, Y. & Yang, X.I.A. 2022 LES wall modeling for heat transfer at high speeds. Phys. Rev. Fluids 7 (1), 014608.CrossRefGoogle Scholar
Coleman, G.N., Kim, J. & Moser, R.D. 1995 A numerical study of turbulent supersonic isothermal-wall channel flow. J. Fluid Mech. 305, 159–183.CrossRefGoogle Scholar
Duan, L., Beekman, I. & Martin, M.P. 2010 Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature. J. Fluid Mech. 655, 419–445.CrossRefGoogle Scholar
Griffin, K.P., Fu, L. & Moin, P. 2021 Velocity transformation for compressible wall-bounded turbulent flows with and without heat transfer. Proc. Natl Acad. Sci. USA 118 (34), e2111144118.CrossRefGoogle ScholarPubMed
Howarth, L. 1948 Concerning the effect of compressibility on laminar boundary layers and their separation. Proc. R. Soc. Lond. A 194 (1036), 16–42.Google Scholar
Huang, P.C., Coleman, G.N. & Bradshaw, P. 1995 Compressible turbulent channel flows: DNS results and modelling. J. Fluid Mech. 305, 185–218.CrossRefGoogle Scholar
Incropera, F.P. & DeWitt, D.P. 1990 Introduction to Heat Transfer, 2nd edn. John Wiley.Google Scholar
Kader, B.A. 1981 Temperature and concentration profiles in fully turbulent boundary layers. Intl J. Heat Mass Transfer 24 (9), 1541–1544.CrossRefGoogle Scholar
Kasagi, N., Tomita, Y. & Kuroda, A. 1992 Direct numerical simulation of passive scalar field in a turbulent channel flow. J. Heat Transfer 114 (3), 598–606.CrossRefGoogle Scholar
Kawai, S. & Larsson, J. 2012 Wall-modeling in large eddy simulation: length scales, grid resolution, and accuracy. Phys. Fluids 24 (1), 015105.CrossRefGoogle Scholar
Kays, W.M. & Crawford, M.E. 1980 Convective Heat and Mass Transfer, 2nd edn. McGraw-Hill.Google Scholar
Kim, J. & Moin, P. 1989 Transport of passive scalars in a turbulent channel flow. In Turbulent Shear Flows 6 (ed. J.-C. André et al.), pp. 85–96. Springer.CrossRefGoogle Scholar
Lee, M. & Moser, R.D. 2015 Direct numerical simulation of turbulent channel flow up to $Re_\tau \approx 5200$. J. Fluid Mech. 774, 395–415.CrossRefGoogle Scholar
Lee, M. & Moser, R.D. 2018 Extreme-scale motions in turbulent plane Couette flows. J. Fluid Mech. 842, 128–145.CrossRefGoogle Scholar
Li, X., Fu, D. & Ma, Y. 2008 Direct numerical simulation of hypersonic boundary layer transition over a blunt cone. AIAA J. 46 (11), 2899–2913.CrossRefGoogle 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.Google Scholar
Lozano-Durán, A. & Jiménez, J. 2014 Effect of the computational domain on direct simulations of turbulent channels up to $Re_\tau = 4200$. Phys. Fluids 26 (1), 011702.CrossRefGoogle Scholar
Marusic, I., Monty, J.P., Hultmark, M. & Smits, A.J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.CrossRefGoogle Scholar
Modesti, D., Bernardini, M. & Pirozzoli, S. 2015 High-Reynolds-number effects on turbulent scalings in compressible channel flow. Proc. Appl. Math. Mech. 15 (1), 489–490.CrossRefGoogle Scholar
Modesti, D. & Pirozzoli, S. 2016 Reynolds and Mach number effects in compressible turbulent channel flow. Intl J. Heat Fluid Flow 59, 33–49.CrossRefGoogle Scholar
Modesti, D. & Pirozzoli, S. 2019 Direct numerical simulation of supersonic pipe flow at moderate Reynolds number. Intl J. Heat Fluid Flow 76, 100–112.CrossRefGoogle Scholar
Modesti, D., Pirozzoli, S. & Grasso, F. 2019 Direct numerical simulation of developed compressible flow in square ducts. Intl J. Heat Fluid Flow 76, 130–140.CrossRefGoogle Scholar
Morkovin, M.V. 1962 Effects of compressibility on turbulent flows. Méc. Turbul. 367 (380), 26.Google Scholar
Oliver, T.A., Malaya, N., Ulerich, R. & Moser, R.D. 2014 Estimating uncertainties in statistics computed from direct numerical simulation. Phys. Fluids 26 (3), 035101.CrossRefGoogle Scholar
Patel, A., Boersma, B.J. & Pecnik, R. 2016 The influence of near-wall density and viscosity gradients on turbulence in channel flows. J. Fluid Mech. 809, 793–820.CrossRefGoogle Scholar
Patel, A., Boersma, B.J. & Pecnik, R. 2017 Scalar statistics in variable property turbulent channel flows. Phys. Rev. Fluids 2 (8), 084604.CrossRefGoogle Scholar
Patel, A., Peeters, J.W.R., Boersma, B.J. & Pecnik, R. 2015 Semi-local scaling and turbulence modulation in variable property turbulent channel flows. Phys. Fluids 27 (9), 095101.CrossRefGoogle Scholar
Pirozzoli, S. & Bernardini, M. 2011 a Direct numerical simulation database for impinging shock wave/turbulent boundary-layer interaction. AIAA J. 49 (6), 1307–1312.CrossRefGoogle Scholar
Pirozzoli, S. & Bernardini, M. 2011 b Turbulence in supersonic boundary layers at moderate Reynolds number. J. Fluid Mech. 688, 120–168.CrossRefGoogle Scholar
Pirozzoli, S., Bernardini, M. & Orlandi, P. 2014 Turbulence statistics in Couette flow at high Reynolds number. J. Fluid Mech. 758, 327–343.CrossRefGoogle Scholar
Pirozzoli, S., Bernardini, M. & Orlandi, P. 2016 Passive scalars in turbulent channel flow at high Reynolds number. J. Fluid Mech. 788, 614–639.CrossRefGoogle Scholar
Pirozzoli, S., Grasso, F. & Gatski, T.B. 2004 Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at $M = 2.25$. Phys. Fluids 16 (3), 530–545.CrossRefGoogle Scholar
Smits, A.J. & Dussauge, J.-P. 2006 Turbulent Shear Layers in Supersonic Flow. Springer.Google Scholar
Tennekes, H. & Lumley, J.L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Trettel, A. & Larsson, J. 2016 Mean velocity scaling for compressible wall turbulence with heat transfer. Phys. Fluids 28 (2), 026102.CrossRefGoogle Scholar
Van Driest, E.R. 1951 Turbulent boundary layer in compressible fluids. J. Aeronaut. Sci. 18 (3), 145–160.CrossRefGoogle Scholar
Van Driest, E.R. 1956 The Problem of Aerodynamic Heating. Institute of the Aeronautical Sciences.Google Scholar
Volpiani, P.S., Bernardini, M. & Larsson, J. 2018 Effects of a nonadiabatic wall on supersonic shock/boundary-layer interactions. Phys. Rev. Fluids 3 (8), 083401.CrossRefGoogle Scholar
Volpiani, P.S., Bernardini, M. & Larsson, J. 2020 a Effects of a nonadiabatic wall on hypersonic shock/boundary-layer interactions. Phys. Rev. Fluids 5 (1), 014602.CrossRefGoogle Scholar
Volpiani, P.S., Iyer, P.S., Pirozzoli, S. & Larsson, J. 2020 b Data-driven compressibility transformation for turbulent wall layers. Phys. Rev. Fluids 5 (5), 052602.CrossRefGoogle Scholar
Walz, A. 1969 Boundary Layers of Flow and Temperature. MIT Press.Google Scholar
Wan, T., Zhao, P., Liu, J., Wang, C. & Lei, M. 2020 Mean velocity and temperature scaling for near-wall turbulence with heat transfer at supercritical pressure. Phys. Fluids 32 (5), 055103.Google Scholar
Wei, T., Fife, P., Klewicki, J. & McMurtry, P. 2005 a Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows. J. Fluid Mech. 522, 303–327.CrossRefGoogle Scholar
Wei, T., Fife, P., Klewicki, J. & McMurtry, P. 2005 b Scaling heat transfer in fully developed turbulent channel flow. Intl J. Heat Mass Transfer 48 (25–26), 5284–5296.CrossRefGoogle Scholar
Wolfshtein, M. 1969 The velocity and temperature distribution in one-dimensional flow with turbulence augmentation and pressure gradient. Intl J. Heat Mass Transfer 12 (3), 301–318.CrossRefGoogle Scholar
Yang, X.I.A. & Abkar, M. 2018 A hierarchical random additive model for passive scalars in wall-bounded flows at high Reynolds numbers. J. Fluid Mech. 842, 354–380.CrossRefGoogle ScholarPubMed
Yang, X.I.A., Chen, P.E.S., Hu, R. & Abkar, M. 2022 Logarithmic-linear law of the streamwise velocity variance in stably stratified boundary layers. Boundary-Layer Meteorol. 183, 199–213.CrossRefGoogle Scholar
Yang, X.I.A., Meneveau, C., Marusic, I. & Biferale, L. 2016 Extended self-similarity in moment-generating-functions in wall-bounded turbulence at high Reynolds number. Phys. Rev. Fluids 1 (4), 044405.CrossRefGoogle Scholar
Yang, X.I.A., Pirozzoli, S. & Abkar, M. 2020 Scaling of velocity fluctuations in statistically unstable boundary-layer flows. J. Fluid Mech. 886, A3.CrossRefGoogle Scholar
Yang, X.I.A. & Lv, Y. 2018 A semi-locally scaled eddy viscosity formulation for LES wall models and flows at high speeds. Theor. Comput. Fluid Dyn. 32 (5), 617–627.CrossRefGoogle Scholar
Yang, X.I.A., Urzay, J., Bose, S. & Moin, P. 2018 Aerodynamic heating in wall-modeled large-eddy simulation of high-speed flows. AIAA J. 56, 731–742.CrossRefGoogle Scholar
Yao, J. & Hussain, F. 2020 Turbulence statistics and coherent structures in compressible channel flow. Phys. Rev. Fluids 5 (8), 084603.CrossRefGoogle Scholar
Yu, M. & Xu, C.-X. 2021 Compressibility effects on hypersonic turbulent channel flow with cold walls. Phys. Fluids 33 (7), 075106.CrossRefGoogle Scholar
Yu, M., Xu, C.-X. & Pirozzoli, S. 2019 Genuine compressibility effects in wall-bounded turbulence. Phys. Rev. Fluids 4 (12), 123402.CrossRefGoogle Scholar
Yu, M., Xu, C.-X. & Pirozzoli, S. 2020 Compressibility effects on pressure fluctuation in compressible turbulent channel flows. Phys. Rev. Fluids 5 (11), 113401.CrossRefGoogle Scholar
Zhang, B.-Y., Huang, W.-X. & Xu, C.-X. 2021 A near-wall predictive model for passive scalars using minimal flow unit. Phys. Fluids 33 (4), 045119.CrossRefGoogle Scholar
Zhang, C., Duan, L. & Choudhari, M.M. 2018 Direct numerical simulation database for supersonic and hypersonic turbulent boundary layers. AIAA J. 56 (11), 4297–4311.CrossRefGoogle 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
Zhou, A., Pirozzoli, S. & Klewicki, J. 2017 Mean equation based scaling analysis of fully-developed turbulent channel flow with uniform heat generation. Intl J. Heat Mass Transfer 115, 50–61.CrossRefGoogle Scholar
Zhu, Y., Lee, C., Chen, X., Wu, J., Chen, S. & Gad-el Hak, M. 2018 Newly identified principle for aerodynamic heating in hypersonic flows. J. Fluid Mech. 855, 152–180.CrossRefGoogle Scholar