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The effect of stable thermal stratification on turbulent boundary layer statistics

Published online by Cambridge University Press:  11 January 2017

Owen Williams Open the ORCID record for Owen Williams [Opens in a new window] ,
Tristen Hohman ,
Tyler Van Buren ,
Elie Bou-Zeid  and
Alexander J. Smits
Owen Williams*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA William E. Boeing Department of Aeronautics and Astronautics, University of Washington, Seattle, WA 98195, USA
Tristen Hohman
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Tyler Van Buren
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Elie Bou-Zeid
Affiliation:
Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USA
Alexander J. Smits
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: ojhw@uw.edu
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Abstract

The effects of stable thermal stratification on turbulent boundary layers are experimentally investigated for smooth and rough walls. For weak to moderate stability, the turbulent stresses are seen to scale with the wall shear stress, compensating for changes in fluid density in the same manner as done for compressible flows. This suggests little change in turbulent structure within this regime. At higher levels of stratification turbulence no longer scales with the wall shear stress and turbulent production by mean shear collapses, but without the preferential damping of near-wall motions observed in previous studies. We suggest that the weakly stable and strongly stable (collapsed) regimes are delineated by the point where the turbulence no longer scales with the local wall shear stress, a significant departure from previous definitions. The critical stratification separating these two regimes closely follows the linear stability analysis of Schlichting (Z. Angew. Math. Mech., vol. 15 (6), 1935, pp. 313–338) for both smooth and rough surfaces, indicating that a good predictor of critical stratification is the gradient Richardson number evaluated at the wall. Wall-normal and shear stresses follow atmospheric trends in the local gradient Richardson number scaling of Sorbjan (Q. J. R. Meteorol. Soc., vol. 136, 2010, pp. 1243–1254), suggesting that much can be learned about stratified atmospheric flows from the study of laboratory scale boundary layers at relatively low Reynolds numbers.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Turbulent Flows: Stratified turbulence Turbulent Flows: Turbulent boundary layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 812 , 10 February 2017 , pp. 1039 - 1075
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© 2017 Cambridge University Press 

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References

Adrian, R. J., Meinhart, C. D. & Tomkins, C. 2000 Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54.Google Scholar
Arya, S. P. S.1968 Structure of stably stratified turbulent boudary layer. PhD thesis, Colorado State University.Google Scholar
Arya, S. P. S. 1975 Buoyancy effects in a horizontal flat plate boundary layer. J. Fluid Mech. 68, 321–343.CrossRefGoogle Scholar
Arya, S. P. S. & Plate, E. J. 1969 Modeling of the stably stratified atmospheric boundary layer. J. Atmos. Sci. 26, 656–665.2.0.CO;2>CrossRefGoogle Scholar
Aubertine, C. D. & Eaton, J. K. 2005 Turbulence development in a non-equilibrium turbulent boundary layer with mild adverse pressure gradient. J. Fluid Mech. 532, 345–364.Google Scholar
Blackadar, A. K. 1962 The vertical distribution of wind and turbulent exchange in neutral atmosphere. J. Geophys. Res. 67, 3095–3103.CrossRefGoogle Scholar
Bou-Zeid, E., Higgins, C., Huwald, H., Meneveau, C. & Parlange, M. B. 2010 Field study of the dynamics and modelling of subgrid-scale turbulence in a stable atmospheric surface layer over a glacier. J. Fluid Mech. 665, 480–515.Google Scholar
Castillo, L. & George, W. K. 2001 Similarity analysis for turbulent boundary layer with pressure gradient: outer flow. AIAA J. 39 (1), 41–47.Google Scholar
Chung, D. & Matheou, G. 2012 Direct numerical simulation of stationary homogeneous stratified sheared turbulence. J. Fluid Mech. 696, 434–467.CrossRefGoogle Scholar
Clauser, F. H. 1954 Turbulent boundary layers in adverse pressure gradients. J. Aero. Sci. 21, 91–108.Google Scholar
Connelly, J. S., Schultz, M. P. & Flack, K. A. 2006 Velocity-defect scaling for turbulent boundary layers with a range of relative roughness. Exp. Fluids 40, 188–195.Google Scholar
Demarco, G., Puhales, F., Acevedo, O. C., Costa, F. D., Avelar, A. C. & Fisch, G. 2015 Dependence of turbulence-related quantities on the mechanical forcing for wind tunnel stratified flow. Amer. J. Environ. Engng 5 (1A), 15–26.Google Scholar
van Driest, E. R. 1951 Turbulent boundary layer in compressible fluids. J. Aero. Sci. 18, 145–160.Google Scholar
van Driest, E. R. 1956 On turbulent flow near a wall. J. Aero. Sci. 23, 1007–1011.Google Scholar
Flack, K. A., Schultz, M. P. & Connelly, J. S. 2007 Examination of a critical roughness height for outer layer similarity. Phys. Fluids 19.Google Scholar
Flack, K. A., Schultz, M. P. & Shapiro, T. A. 2005 Experimental support for Townsend’s Reynolds number similarity hypothesis on rough walls. Phys. Fluids 17, 035102.Google Scholar
Flores, O. & Riley, J. J. 2011 Analysis of turbulence collapse in the stably stratified surface layer using direct numerical simulation. Boundary-Layer Meteorol. 139, 241–259.Google Scholar
Grachev, A., Andrea, E., Fairall, C., Guest, P. & Persson, P. 2008 Turbulent boundary layer measurements in the stable atmospheric boundary layer during SHEBA: ten years after. Acta Geophys. 56 (1), 142–166.CrossRefGoogle Scholar
Grachev, A., Andreas, E., Fairall, C., Guest, P. & Persson, P. 2007 SHEBA flux-profile relationships in the stable atmospheric boundary layer. Boundary-Layer Meteorol. 124, 315–333.Google Scholar
Grachev, A., Andreas, E., Fairall, C., Guest, P. & Persson, P. 2015 Similarity theory based on the Dougherty–Ozmidov length scale. Q. J. R. Meteorol. Soc. 141, 1845–1856.CrossRefGoogle Scholar
Grachev, A., Fairall, C., Persson, P., Andreas, E. & Guest, P. 2005 Stable boundary layer scaling regimes: the SHEBA data. Boundary-Layer Meteorol. 116, 201–235.CrossRefGoogle Scholar
Hama, F. R. 1954 Boundary-layer characteristics for rough and smooth surfaces. Trans. Soc. Naval Archit. Mar. Engng 62, 333–358.Google Scholar
Huang, H. T., Fiedler, H. E. & Wang, J. J 1993 Limitation and improvement of PIV. II – particle image distortion, a novel technique. Exp. Fluids 15, 168–174.Google Scholar
Huang, J. & Bou-Zeid, E. 2013 Turbulence and vertical fluxes in the stable atmospheric boundary layer. Part I: a large-eddy simulation study. J. Atmos. Sci. 70, 1513–1527.Google Scholar
Jambunathan, K., Ju, X. Y., Dobbins, B. N. & Ashforth-Frost, S. 1995 An improved cross-correlation technique for particle image velocimetry. Meas. Sci. Technol. 6 (5), 507.CrossRefGoogle Scholar
Jimenez, J. 2004 Turbulent flows over rough walls. Annu. Rev. Fluid Mech. 36, 173–196.Google Scholar
Kaimal, J. C., Wyngaard, J., Izumi, Y. & Coté, O. R. 1972 Spectral characteristics of surface layer turbulence. Q. J. R. Meteorol. Soc. 98, 563–589.Google Scholar
Katul, G. G., Porporato, A., Shah, S. & Bou-Zeid, E. 2014 Two phenomenological constants explain similarity laws in stably stratified turbulence. Phys. Rev. E 89 (2), 023007.Google Scholar
Li, Q., Bou-Zeid, E., Anderson, W., Grimmond, S. & Hultmark, M. 2016 Quality and reliability of les of convective scalar transfer at high Reynolds numbers. Intl J. Heat Mass Transfer 102, 959–970.Google Scholar
Ligrani, P. M. & Moffat, R. J. 1986 Structure of transitional rough and fully rough turbulent boundary layers. J. Fluid Mech. 162, 69–98.CrossRefGoogle Scholar
Lu, S. S. & Willmarth, W. W. 1973 Measurements of the structure of the Reynolds stress in a turbulent boundary layer. J. Fluid Mech. 60, 481.Google Scholar
Mahrt, L. 1998 Stratified atmospheric boundary layers and breakdown of models. J. Theor. Comput. Fluid Dyn. 11, 263–279.Google Scholar
Mahrt, L. 2014 Stably stratified atmospheric boundary layers. Annu. Rev. Fluid Mech. 46, 23–45.Google Scholar
McKeon, B. J., Li, J. D., Jiang, W., Morrison, J. F. & Smits, A. J. 2004 Further observations on the mean velocity distribution in fully developed pipe flow. J. Fluid Mech. 501, 135–147.Google Scholar
Miles, J. W. & Howard, L. N. 1964 Note on heterogenous shear flow. J. Fluid Mech. 20 (2), 331–336.Google Scholar
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, 163–187.Google Scholar
Morkovin, M. V. 1961 Effects of compressibility on turbulent flows. In Mécanique de la Turbulence (ed. Favre, A. J.), pp. 367–380. CNRS.Google Scholar
Nicholl, C. I. H. 1970 Some dynamical effects of heat on a turbulent boundary layer. J. Fluid Mech. 40 (2), 361–384.Google Scholar
Nieuwstadt, F. T. M. 1984 The turbulent structure of the stable, nocturnal boundary layer. J. Atmos. Sci. 41, 2202–2216.Google Scholar
Nieuwstadt, F. T. M. 1985 A model for the stationary, stable boundary layer. In Turbulence and Diffusion in Stable Environments (ed. Hunt, J. C. R.), pp. 149–179. Clarendon Press, Oxford University Press.Google Scholar
Nikuradse, J.1933 Laws of flow in rough pipes. NACA Tech. Rep. TM 1292.Google Scholar
Nogueira, J., Lecuona, A. & Rodriguez, P. A. 1999 Local field correction PIV: on the increase of accuracy of digital PIV systems. Exp. Fluids 27 (2), 107–116.Google Scholar
Ogawa, Y., Diosey, K., Uehara, K. & Ueda, H. 1985 Wind tunnel observation of flow and diffusion under stable stratification. Atmos. Environ. 19, 65–74.Google Scholar
Ogawa, Y., Diosey, P. G., Uehara, K. & Ueda, H. 1982 Plume behavior in stratified flows. Atmos. Environ. 16 (6), 1419–1433.Google Scholar
Ohya, Y. 2001 Wind-tunnel study of atmospheric stable boundary layers over a rough surface. Boundary-Layer Meteorol. 98, 57–82.CrossRefGoogle Scholar
Ohya, Y., Neff, D. E. & Meroney, R. N. 1997 Turbulence structure in a stratified boundary layer under stable conditions. Boundary-Layer Meteorol. 83, 139–161.Google Scholar
Ohya, Y. & Uchida, T. 2003 Turbulence structure of stable boundary layers with a near-linear temperature profile. Boundary-Layer Meteorol. 108, 19–38.Google Scholar
Ohya, Y. & Uchida, T. 2004 Laboratory and numerical studies of the convective boundary layer capped by a strong inversion. Boundary-Layer Meteorol. 112, 223–240.Google Scholar
Panton, R. L. 1990 Scaling turbulent wall layers. Trans. ASME J. Fluids Engng 112 (4), 425–432.Google Scholar
Perry, A. E., Schofield, W. H. & Joubert, P. N. 1969 Rough wall turbulent boundary layers. J. Fluid Mech. 37 (2), 383–413.Google Scholar
Piat, J.-F. & Hopfinger, E. J. 1981 A boundary layer topped by a density interface. J. Fluid Mech. 113, 411–432.Google Scholar
Plate, E. J. & Arya, S. P. S. 1969 Turbulence spectra in a stably stratified boundary layer. Radio Sci. 4 (12), 1163–1168.Google Scholar
Prasad, A. K., Adrian, R. J., Landreth, C. C. & Offutt, P. W. 1992 Effect of resolution on the speed and accuracy of particle image velocimetry interrogation. Exp. Fluids 13 (2–3), 105–116.Google Scholar
Puhales, F. S., Demarco, G., Martins, L. G. N., Acevedo, O. C., Degrazia, G. A., Welter, G. S., Costa, F. D., Fisch, G. F. & Avelar, A. C. 2015 Estimates of turbulent kinetic energy dissipation rate for a stratified flow in a wind tunnel. Physica A 431, 175–187.CrossRefGoogle Scholar
Scarano, F. 2002 Iterative image deformation methods in PIV. Meas. Sci. Technol. 13 (1), R1–R19.Google Scholar
Schlichting, H. 1935 Hauptaufsätze. turbulenz bei wärmeschichtung. Z. Angew. Math. Mech. 15 (6), 313–338; (Translation by Cedrick Ansorge, 2014 – https://www.researchgate.net/publication/264508137_Turbulence_under_Stratification_H_Schlichting).CrossRefGoogle Scholar
Schlichting, H. 1979 Boundary-Layer Theory. McGraw-Hill.Google Scholar
Schumann, U. & Gerz, T. 1995 Turbulent mixing in stably stratified shear flows. J. Appl. Meteorol. 34 (1), 33–48.Google Scholar
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, 494–539.Google Scholar
Smits, A. J. & Dussage, J.-P. 2005 Turbulent Shear Layers In Supersonic Flow, 2nd edn. Springer.Google Scholar
Smits, A. J., Matheson, N. & Joubert, P. N. 1983 Low Reynolds number turbulent boundary layers in zero and favourable pressure gradients. J. Ship Res. 27, 147–157.Google Scholar
Sorbjan, Z. 2010 Gradient-based scales and similarity laws in the stable boundary layer. Q. J. R. Meteorol. Soc. 136, 1243–1254.Google Scholar
Sorbjan, Z. 2012 The height correction of similarity functions in the stable boundary layer. Boundary-Layer Meteorol. 142, 12–31.Google Scholar
Stull, R. B. 1988 An Introduction to Boundary Layer Meteorology, vol. 13. Springer.Google Scholar
Taylor, G. I. 1931 Effect of variation in density on the stability of superimposed streams of fluid. Proc. R. Soc. Lond. A 132, 499.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flows. Cambridge University Press.Google Scholar
Westerweel, J. & Scarano, F. 2005 Universal outlier detection for PIV data. Exp. Fluids 39 (6), 1096–1100.Google Scholar
Williams, O. J. H.2014 Density effects on turbulent boundary layer structure: from the atmosphere to hypersonic flow. PhD thesis, Princeton University.Google Scholar
Wu, Y. & Christensen, K. T. 2006 Population trends of spanwise vortices in wall turbulence. J. Fluid Mech. 568, 55–76.Google Scholar
Zagarola, M. V. & Smits, A. J. 1998a Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 33–79.Google Scholar
Zagarola, M. V. & Smits, A. J.1998b A new mean velocity scaling for turbulent boundary layers. In ASME Paper No. FEDSM98-4950.Google Scholar
Zilitinkevich, S. S., Elperin, T., Kleeorin, N., Rogachevskii, I. & Esau, I. 2013 A hierarchy of energy- and flux-budget (EFB) turbulence closure models for stably-stratified geophysical flows. Boundary-Layer Meteorol. 146, 341–373.Google Scholar