Skip to main content

Mixed convection in turbulent channels with unstable stratification

  • Sergio Pirozzoli (a1), Matteo Bernardini (a1), Roberto Verzicco (a2) (a3) and Paolo Orlandi (a1)

We study turbulent flows in pressure-driven planar channels with imposed unstable thermal stratification, using direct numerical simulations in a wide range of Reynolds and Rayleigh numbers and reaching flow conditions which are representative of fully developed turbulence. The combined effect of forced and free convection produces a peculiar pattern of quasi-streamwise rollers occupying the full channel thickness, with aspect ratio considerably higher than unity; it has been observed that they have an important redistributing effect on temperature and momentum, providing for a substantial fraction of the heat and momentum flux at bulk Richardson numbers larger than $0.01$ . The mean values and the variances of the flow variables do not appear to follow Prandtl’s scaling in the free-convection regime, except for the temperature and vertical velocity fluctuations, which are more directly affected by wall-attached turbulent plumes. We find that the Monin–Obukhov theory nevertheless yields a useful representation of the main flow features. In particular, the widely used Businger–Dyer flux-profile relationships are found to provide a convenient way of approximately accounting for the bulk effects of friction and buoyancy, although the individual profiles may have wide scatter from the alleged trends. Significant deviations are found in direct numerical simulations with respect to the commonly used parametrization of the momentum flux in the light-wind regime, which may have important practical impact in wall models of atmospheric dynamics. Finally, for modelling purposes, we devise a set of empirical predictive formulae for the heat flux and friction coefficients, which are within approximately $10\,\%$ standard deviation from the numerical results in a wide range of flow parameters.

Corresponding author
Email address for correspondence:
Hide All
Abe, H. & Antonia, R. A. 2009 Near-wall similarity between velocity and scalar fluctuations in a turbulent channel flow. Phys. Fluids 21, 025109.
Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503537.
Ahlers, G., He, X., Funfschilling, D. & Bodenschatz, E. 2012 Heat transport by turbulent Rayleigh–Bénard convection for Pr ≃ 0. 8 and 3 × 1012Ra⩽1015 : aspect ratio 𝛾 = 0. 50. New J. Phys. 14, 103012.
Armenio, V. & Sarkar, S. 2002 An investigation of stably stratified turbulent channel flow using large-eddy simulation. J. Fluid Mech. 459, 142.
Avsec, D. 1937 Sur les formes ondulées des tourbillons en bandes longitudinales. C. R. Acad. Sci. Paris 204, 167169.
Avsec, D. & Luntz, M. 1937 Tourbillons thermoconvectifs et électroconvectifs. La Météorologie 31, 180194.
Bénard, H. & Avsec, D. 1938 Travaux récents sur les tourbillons cellulaires et les tourbillons en bandes. Applications à l’astrophysique et à la météorologie. J. Phys. Radium 9 (11), 486500.
Bergman, T. L., Lavine, A. S., Incropera, F. P. & DeWitt, D. P. 2011 Introduction to Heat Transfer. Wiley.
Bernardini, M., Pirozzoli, S. & Orlandi, P. 2014 Velocity statistics in turbulent channel flow up to Re 𝜏 = 4000. J. Fluid Mech. 742, 171191.
Brown, R. A. 1980 Longitudinal instabilities and secondary flows in the planetary boundary layer: a review. Rev. Geophys. 18 (3), 683697.
Businger, J. A., Wyngaard, J. C., Izumi, Y. & Bradley, E. F. 1971 Flux-profile relationships in the atmospheric surface layer. J. Atmos. Sci. 28 (2), 181189.
Cebeci, T. & Bradshaw, P. 1984 Physical and Computational Aspects of Convective Heat Transfer. Springer.
Chung, D. & Matheou, G. 2012 Direct numerical simulation of stationary homogeneous stratified sheared turbulence. J. Fluid Mech. 696, 434467.
Clever, R. M. & Busse, F. H. 1991 Instabilities of longitudinal rolls in the presence of Poiseuille flow. J. Fluid Mech. 229, 517529.
Clever, R. M. & Busse, F. H. 1992 Three-dimensional convection in a horizontal fluid layer subjected to a constant shear. J. Fluid Mech. 234, 511527.
Deardorff, J. W. 1970 Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci. 27 (8), 12111213.
Deardorff, J. W. 1972 Numerical investigation of neutral and unstable planetary boundary layers. J. Atmos. Sci. 29 (1), 91115.
Domaradzki, J. A. & Metcalfe, R. W. 1988 Direct numerical simulations of the effects of shear on turbulent Rayleigh–Bénard convection. J. Fluid Mech. 193, 499531.
Dyer, A. J. 1974 A review of flux-profile relationships. Boundary-Layer Meteorol. 7 (3), 363372.
Esau, I., Davy, R., Outten, S., Tyuryakov, S. & Zilitinkevich, S. 2013 Structuring of turbulence and its impact on basic features of Ekman boundary layers. Nonlinear Process. Geophys. 20, 589604.
Fukui, K. & Nakajima, M. 1985 Unstable stratification effects on turbulent shear flow in the wall region. Intl J. Heat Mass Transfer 28 (12), 23432352.
Fukui, K., Nakajima, M. & Ueda, H. 1991 Coherent structure of turbulent longitudinal vortices in unstably-stratified turbulent flow. Intl J. Heat Mass Transfer 34 (9), 23732385.
Gage, K. S. & Reid, W. H. 1968 The stability of thermally stratified plane Poiseuille flow. J. Fluid Mech. 33 (01), 2132.
Garai, A., Kleissl, J. & Sarkar, S. 2014 Flow and heat transfer in convectively unstable turbulent channel flow with solid-wall heat conduction. J. Fluid Mech. 757, 5781.
Garcia-Villalba, M. & del Álamo, J. C. 2011 Turbulence modification by stable stratification in channel flow. Phys. Fluids 23 (4), 045104.
Haines, D. A. 1982 Horizontal roll vortices and crown fires. J. Appl. Meteorol. 21 (6), 751763.
Hamman, C. & Moin, P. 2015 Thermal convection from a minimal flow unit to a wide fluid layer. Bull. Am. Phys. Soc. 60 (21). See
Hanna, S. R. 1969 The formation of longitudinal sand dunes by large helical eddies in the atmosphere. J. Appl. Meteorol. 8 (6), 874883.
Hill, G. E. 1968 On the orientation of cloud bands. Tellus 20 (1), 132137.
Iida, O. & Kasagi, N. 1997 Direct numerical simulation of unstably stratified turbulent channel flow. Trans. ASME J. Heat Transfer 119 (1), 5361.
Jiménez, J., Wray, A. A., Saffman, P. G. & Rogallo, R. S. 1993 The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255, 6590.
Johansson, C., Smedman, A.-S., Högström, U., Brasseur, J. G. & Khanna, S. 2001 Critical test of the validity of Monin–Obukhov similarity during convective conditions. J. Atmos. Sci. 58 (12), 15491566.
Kader, B. A. 1981 Temperature and concentration profiles in fully turbulent boundary layers. Intl J. Heat Mass Transfer 24, 15411544.
Kader, B. A. & Yaglom, A. M. 1990 Mean fields and fluctuation moments in unstably stratified turbulent boundary layers. J. Fluid Mech. 212, 637662.
Katul, G. G., Li, D., Chamecki, M. & Bou-Zeid, E. 2013 Mean scalar concentration profile in a sheared and thermally stratified atmospheric surface layer. Phys. Rev. E 87 (2), 023004.
Kays, W. M., Crawford, M. E. & Weigand, B. 1980 Convective Heat and Mass Transfer. McGraw-Hill.
Kerr, R. M. 1996 Rayleigh number scaling in numerical convection. J. Fluid Mech. 310, 139179.
Khanna, S. & Brasseur, J. G. 1997 Analysis of Monin–Obukhov similarity from large-eddy simulation. J. Fluid Mech. 345, 251286.
Khanna, S. & Brasseur, J. G. 1998 Three-dimensional buoyancy-and shear-induced local structure of the atmospheric boundary layer. J. Atmos. Sci. 55 (5), 710743.
Kim, J., Moin, P. & Moser, R. D. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.
Kuettner, J. P. 1959 The band structure of the atmosphere. Tellus 11 (3), 267294.
Kuettner, J. P. 1971 Cloud bands in the earth’s atmosphere: observations and theory. Tellus 23 (4–5), 404426.
Lee, M. & Moser, R. D. 2015 Direct simulation of turbulent channel flow layer up to Re 𝜏 = 5200. J. Fluid Mech. 774, 395415.
Li, D. & Bou-Zeid, E. 2011 Coherent structures and the dissimilarity of turbulent transport of momentum and scalars in the unstable atmospheric surface layer. Boundary-Layer Meteorol. 140 (2), 243262.
Mal, S. 1930 Forms of stratified clouds. Beitr. Phys. Atmos. 17, 4068.
Mizushina, T., Ogino, F. & Katada, N. 1982 Ordered motion of turbulence in a thermally stratified flow under unstable conditions. Intl J. Heat Mass Transfer 25 (9), 14191425.
Monin, A. S. & Obukhov, A. M. 1954 Basic laws of turbulent mixing in the surface layer of the atmosphere. Contrib. Geophys. Inst. Acad. Sci. USSR 151, 163187.
Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. J. 2000 Turbulent convection at very high Rayleigh numbers. Nature 404, 837840.
Obukhov, A. M. 1946 Turbulence in an atmosphere with inhomogeneous temperature. Trudy Geofiz. Inst. AN SSSR 1, 95115.
Orlandi, P., Bernardini, M. & Pirozzoli, S. 2015a Poiseuille and Couette flows in the transitional and fully turbulent regime. J. Fluid Mech. 770, 424441.
Orlandi, P., Pirozzoli, S. & Bernardini, M. 2015b Influence of wall roughness and thermal coductivity on turbulent natural convection. Bull. Am. Phys. Soc. 60 (21). See
Pabiou, H., Mergui, S. & Benard, C. 2005 Wavy secondary instability of longitudinal rolls in Rayleigh–Bénard–Poiseuille flows. J. Fluid Mech. 542, 175194.
Panofsky, H. A., Tennekes, H., Lenschow, D. H. & Wyngaard, J. C. 1977 The characteristics of turbulent velocity components in the surface layer under convective conditions. Boundary-Layer Meteorol. 11 (3), 355361.
Papavassiliou, D. V. & Hanratty, T. J. 1997 Interpretation of large-scale structures observed in a turbulent planet Couette flow. Intl J. Heat Fluid Flow 18, 5569.
Park, S.-B. & Baik, J.-J. 2014 Large-eddy simulations of convective boundary layers over flat and urbanlike surfaces. J. Atmos. Sci. 71, 18801892.
Patton, E. G., Sullivan, P. P., Shaw, R. H., Finnigan, J. J. & Weil, J. C. 2014 Atmospheric stability influences on coupled boundary layer and canopy turbulence. J. Atmos. Sci. 73, 16211647.
Paulson, C. A. 1970 The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer. J. Appl. Meteorol. 9 (6), 857861.
Pirozzoli, S., Bernardini, M. & Orlandi, P. 2014 Turbulence statistics in Couette flow at high Reynolds number. J. Fluid Mech. 758, 327343.
Pirozzoli, S., Bernardini, M. & Orlandi, P. 2016 Passive scalars in turbulent channel flow at high Reynolds number. J. Fluid Mech. 788, 614639.
Prandtl, L. 1932 Meteorologische anwendung der strömungslehre. Beitr. Phys. Atmos. 19, 188202.
Rao, K. G. 2004 Estimation of the exchange coefficient of heat during low wind convective conditions. Boundary-Layer Meteorol. 111 (2), 247273.
Rao, K. G. & Narasimha, R. 2006 Heat-flux scaling for weakly forced turbulent convection in the atmosphere. J. Fluid Mech. 547, 115135.
Scagliarini, A., Einarsson, H., Gylfason, Á. & Toschi, F. 2015 Law of the wall in an unstably stratified turbulent channel flow. J. Fluid Mech. 781, R5.
Scagliarini, A., Gylfason, Á. & Toschi, F. 2014 Heat-flux scaling in turbulent Rayleigh–Bénard convection with an imposed longitudinal wind. Phys. Rev. E 89 (4), 043012.
Shah, S. K. & Bou-Zeid, E. 2014 Direct numerical simulations of turbulent Ekman layers with increasing static stability: modifications to the bulk structure and second-order statistics. J. Fluid Mech. 760, 494539.
Shishkina, O., Stevens, R. J. A. M., Grossmann, S. & Lohse, D. 2010 Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution. New J. Phys. 12 (7), 075022.
Sid, S., Dubief, Y. & Terrapon, V. 2015 Direct numerical simulation of mixed convection in turbulent channel flow: on the Reynolds number dependency of momentum and heat transfer under unstable stratification. In Proceedings of the 8th International Conference on Computational Heat and Mass Transfer, Istanbul, Turkey, p. 190.
Stevens, R. J. A. M., Lohse, D. & Verzicco, R. 2011 Prandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection. J. Fluid Mech. 688, 3143.
Stull, R. B. 2012 An Introduction to Boundary Layer Meteorology, vol. 13. Springer Science & Business Media.
Wyngaard, J. C. 1992 Atmospheric turbulence. Annu. Rev. Fluid Mech. 24 (1), 205234.
Wyngaard, J. C., Coté, O. R. & Izumi, Y. 1971 Local free convection, similarity, and the budgets of shear stress and heat flux. J. Atmos. Sci. 28 (7), 11711182.
Young, G. S., Kristovich, D. A. R., Hjelmfelt, M. R. & Foster, R. C. 2002 Rolls, streets, waves, and more: a review of quasi-two-dimensional structures in the atmospheric boundary layer. Bull. Am. Meteorol. Soc. 83 (7), 9971001.
Zilitinkevich, S. J., Hunt, J. C. R., Esau, I. N., Grachev, A. A., Lalas, D. P., Akylas, E., Tombrou, M., Fairall, C. W., Fernando, H. J. S., Baklanov, A. A. et al. 2006 The influence of large convective eddies on the surface-layer turbulence. Q. J. R. Meteorol. Soc. 132, 14261456.
Zonta, F. & Soldati, A. 2014 Effect of temperature dependent fluid properties on heat transfer in turbulent mixed convection. Trans. ASME J. Heat Transfer 136 (2), 022501.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 5
Total number of PDF views: 255 *
Loading metrics...

Abstract views

Total abstract views: 413 *
Loading metrics...

* Views captured on Cambridge Core between 25th May 2017 - 14th August 2018. This data will be updated every 24 hours.