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Direct numerical simulations of turbulent Ekman layers with increasing static stability: modifications to the bulk structure and second-order statistics

Published online by Cambridge University Press:  11 November 2014

Stimit K. Shah  and
Elie Bou-Zeid
Stimit K. Shah
Affiliation:
Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08540, USA
Elie Bou-Zeid*
Affiliation:
Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08540, USA
*
†Email address for correspondence: ebouzeid@princeton.edu
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Abstract

Direct numerical simulations of stably stratified Ekman layers are conducted to study the effect of increasing static stability on turbulence dynamics and modelling in wall-bounded flows at three moderate Reynolds numbers. The flow field is analysed by examining the mean profiles of wind speed, potential temperature and momentum flux, as well as streamwise velocity and temperature spectra. The maximum stabilizing buoyancy flux that a flow can sustain while remaining fully turbulent is found to depend on the Reynolds number. The flows with the highest Reynolds number display a relatively well-developed inertial range and logarithmic layer, and are found to bear similarities to much higher-Reynolds-number flows like the ones encountered in the atmospheric boundary layer. In particular, the near-wall mean profiles follow the Monin–Obukhov similarity theory. However, several flow features, such as the critical Richardson number and the stress–strain alignment, are found to maintain significant dependence on the Reynolds number. The budgets of turbulence kinetic energy (TKE), vertical velocity variance, momentum and buoyancy fluxes, and temperature variance are analysed. The results indicate that the effect of stability on turbulence is first directly manifested in the vertical velocity variance budget, and results in damping of vertical motions. This then leads to a reduction in the downward transport of horizontal momentum components towards the surface, and consequently to a decrease in the shear production term in the TKE budget: changes in the vertical profile of TKE shear production with increasing Richardson number are significant and have a larger impact on TKE than direct buoyancy destruction. The reduction in vertical velocity variance also results in significant drops in the production terms in the momentum flux, buoyancy flux and temperature variance budgets. Various assumptions and parameters related to low-order turbulence closures are investigated. The results suggest that the vertical velocity variance is a more appropriate parameter than the full TKE on which to base eddy-diffusivity and viscosity models.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Turbulent Flows: Stratified turbulence Turbulent Flows: Turbulence simulation
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Papers
Information
Journal of Fluid Mechanics , Volume 760 , 10 December 2014 , pp. 494 - 539
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© 2014 Cambridge University Press 

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References

Abe, H. & Kawamura, H. 2002 A study of turbulence thermal structure in a channel flow through DNS up to $\mathit{Re}_{{\it\tau}}=640$ with $\mathit{Pr}=0.025$ and 0.71. In Proceedings of 9th European Turbulence Conference on Advances in Turbulence, Barcelona: CIMNE (ed. Castro, I. P. et al. ), pp. 399–402.Google Scholar
Albertson, J. D. & Parlange, M. B. 1999a Natural integration of scalar fluxes from complex terrain. Adv. Water Resour. 23 (3), 239–252.Google Scholar
Albertson, J. D. & Parlange, M. B. 1999b Surface length scales and shear stress: implications for land–atmosphere interaction over complex terrain. Water Resour. Res. 35, 2121–2132.Google Scholar
André, J. C., de Moor, G., Lacarrère, P. & Du Vachat, R. 1978 Modeling the 24-h evolution of the mean and turbulent structures of the planetary boundary layer. J. Atmos. Sci. 35, 1861–1883.2.0.CO;2>CrossRefGoogle Scholar
Andren, A. 1995 The structure of stably stratified atmospheric boundary layers: a large-eddy simulation study. Q. J. R. Meteorol. Soc. 121 (525), 961–985.Google Scholar
Andrén, A. & Moeng, C. H. 1993 Single-point closures in a neutrally stratified boundary layer. J. Atmos. Sci. 50, 3366–3379.2.0.CO;2>CrossRefGoogle Scholar
Ansorge, C. & Mellado, J. P. 2014 Global intermittency and collapsing turbulence in a stratified planetary boundary layer. Boundary-Layer Meteorol. 153 (1), 89–116.Google Scholar
Arya, S. P. S.1968 Structure of stably stratified turbulent boundary layer. PhD thesis, Colorado State University, Fort Collins.Google Scholar
Arya, S. P. S. 1975 Buoyancy effects in a horizontal flat-plate boundary layer. J. Fluid Mech. 68, 321–343.Google Scholar
Beckers, M., Verzicco, R., Clercx, H. J. H. & Van Heijst, G. J. F. 2001 Dynamics of pancake-like vortices in a stratified fluid: experiments, model and numerical simulations. J. Fluid Mech. 433, 1–27.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.CrossRefGoogle Scholar
Bou-Zeid, E., Meneveau, C. & Parlange, M. 2004 Large-eddy simulation of neutral atmospheric boundary layer flow over heterogeneous surfaces: blending height and effective surface roughness. Water Resour. Res. 40 (2), W02505.CrossRefGoogle Scholar
Bou-Zeid, E., Meneveau, C. & Parlange, M. 2005 A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows. Phys. Fluids 17 (2), 025105.CrossRefGoogle Scholar
Bou-Zeid, E., Parlange, M. & Meneveau, C. 2007 On the parameterization of surface roughness at regional scales. J. Atmos. Sci. 64 (1), 216–227.Google Scholar
Brost, R. A. & Wyngaard, J. C. 1978 A model study of the stably stratified planetary boundary layer. J. Atmos. Sci. 35, 1427–1440.2.0.CO;2>CrossRefGoogle Scholar
Brutsaert, W. 1982 Evaporation into the Atmosphere: Theory, History, and Applications. Riedel.Google Scholar
Cai, X. M. & Steyn, D. G. 1996 The von Kármán constant determined by large eddy simulation. Boundary-Layer Meteorol. 78, 143–164.Google Scholar
Caldwell, D. R. & Van Atta, C. W. 1970 Characteristics of Ekman boundary layer instabilities. J. Fluid Mech. 44, 79–95.Google Scholar
Caldwell, D. R., Van Atta, C. W. & Helland, K. N. 1972 A laboratory study of the turbulent Ekman layer. Geophys. Fluid Dyn. 3 (1), 125–160.Google Scholar
Canuto, V. M. 2002 Critical Richardson numbers and gravity waves. Astron. Astrophys. 384, 1119–1123.Google Scholar
Chung, D. & Matheou, G. 2012 Direct numerical simulation of stationary homogeneous stratified sheared turbulence. J. Fluid Mech. 696, 434–467.Google Scholar
Coleman, G. N. 1999 Similarity statistics from a direct numerical simulation of the neutrally stratified planetary boundary layer. J. Atmos. Sci. 56, 891–900.Google Scholar
Coleman, G. N., Ferziger, J. H. & Spalart, P. R. 1990 A numerical study of the turbulent Ekman layer. J. Fluid Mech. 213, 313–348.Google Scholar
Coleman, G. N., Ferziger, J. H. & Spalart, P. R. 1992 Direct simulation of the stably stratified turbulent Ekman layer. J. Fluid Mech. 244, 677–712.CrossRefGoogle Scholar
Conzemius, R. & Fedorovich, E. 2007 Bulk models of the sheared convective boundary layer: evaluation through large eddy simulations. J. Atmos. Sci. 64 (3), 786–807.Google Scholar
Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Deardorff, J. W. 1970 A three-dimensional numerical investigation of the idealized planetary boundary layer. Geophys. Fluid Dyn. 1 (3–4), 377–410.Google Scholar
Deardorff, J. W. 1972 Numerical investigation of neutral and unstable planetary boundary-layers. J. Atmos. Sci. 29 (1), 91–115.Google Scholar
Dyer, A. J. 1974 A review of flux-profile relationships. Boundary-Layer Meteorol. 7, 363–372.CrossRefGoogle Scholar
Faller, A. J. 1963 An experimental study of the instability of the laminar Ekman boundary layer. J. Fluid Mech. 15, 560–576.Google Scholar
Fernando, H. J. S. & Weil, J. C. 2010 Whither the stable boundary layer? Bull. Am. Meteorol. Soc. 91 (11), 1475–1484.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
Galperin, B., Sukoriansky, S. & Anderson, P. S. 2007 On the critical Richardson number in stably stratified turbulence. Atmos. Sci. Lett. 8 (3), 65–69.Google Scholar
García-Villalba, M. & Del Álamo, J. C. 2011 Turbulence modification by stable stratification in channel flow. Phys. Fluids 23 (4), 045104.Google Scholar
Garg, R. P., Ferziger, J. H., Monismith, S. G. & Koseff, J. R. 2000 Stably stratified turbulent channel flows. I. Stratification regimes and turbulence suppression mechanism. Phys. Fluids 12, 2569–2594.Google Scholar
Gioia, G., Guttenberg, N., Goldenfeld, N. & Chakraborty, P. 2010 Spectral theory of the turbulent mean-velocity profile. Phys. Rev. Lett. 105 (18), 184501.Google Scholar
Hattori, H., Houra, T. & Nagano, Y. 2007 Direct numerical simulation of stable and unstable turbulent thermal boundary layers. Intl J. Heat Fluid Flow 28 (6), 1262–1271.Google Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.Google Scholar
Hoyas, S. & Jiménez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to $\mathit{Re}_{{\it\tau}}=2003$ . Phys. Fluids 18 (1), 011702.Google Scholar
Huang, J. & Bou-Zeid, E. 2013 Turbulence and vertical fluxes in the stable atmospheric boundary layer. Part 1: A large-eddy simulation study. J. Atmos. Sci. 70 (6), 1513–1527.Google Scholar
Huang, J., Bou-Zeid, E. & Golaz, J.-C. 2013 Turbulence and vertical fluxes in the stable atmospheric boundary layer. Part 2: A novel mixing-length model. J. Atmos. Sci. 70 (6), 1528–1542.Google Scholar
Hunt, J. C. R., Kaimal, J. C. & Gaynor, J. E. 1985 Some observations of turbulence structure in stable layers. Q. J. R. Meteorol. Soc. 111 (469), 793–815.Google Scholar
Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40 (1), 169–184.Google Scholar
Jiménez, J.1998 The largest scales of turbulent wall flows. Tech Rep., Center for Turbulence Research, Annual Research Briefs.Google Scholar
Jiménez, M. A. & Cuxart, J. 2005 Large-eddy simulations of the stable boundary layer using the standard Kolmogorov theory: range of applicability. Boundary-Layer Meteorol. 115, 241–261.Google Scholar
Kaimal, J. C. & Finnigan, J. J. 1994 Atmospheric Boundary Layer Flows: Their Structure and Measurement. Oxford University Press.CrossRefGoogle Scholar
Kaimal, J. C., Wyngaard, J. C., Izumi, Y. & Coté, O. 1972 Spectral characteristics of surface-layer turbulence. Q. J. R. Meteorol. Soc. 98, 563–589.Google Scholar
Katul, G. G., Koning, A. & Porporato, A. 2012 Mean velocity profile in a sheared and thermally stratified atmospheric boundary layer. Phys. Rev. Lett. 107, 268502.Google Scholar
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.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, 023007.Google Scholar
Kondo, J., Kanechika, O. & Yasuda, N. 1978 Heat and momentum transfers under strong stability in the atmospheric surface layer. J. Atmos. Sci. 35, 1012–1021.Google Scholar
Kosović, B. & Curry, J. A. 2000 A large eddy simulation study of a quasi-steady, stably stratified atmospheric boundary layer. J. Atmos. Sci. 57, 1052–1068.Google Scholar
Kumar, V., Kleissl, J., Meneveau, C. & Parlange, M. B. 2006 Large-eddy simulation of a diurnal cycle of the atmospheric boundary layer: atmospheric stability and scaling issues. Water Resour. Res. 42, 6.Google Scholar
Kunkel, G. J., Allen, J. J. & Smits, A. J. 2007 Further support for Townsend’s Reynolds number similarity hypothesis in high Reynolds number rough-wall pipe flow. Phys. Fluids 19 (5), 055109.Google Scholar
Lettau, H. H. 1979 Wind and temperature profile prediction for diabatic surface layers including strong inversion cases. Boundary-Layer Meteorol. 17, 443–464.Google Scholar
Li, D., Katul, G. G. & Bou-Zeid, E. 2012 Mean velocity and temperature profiles in a sheared diabatic turbulent boundary layer. Phys. Fluids 24 (10), 105105.Google Scholar
Lumley, J. L. 1967 Similarity and the turbulent energy spectrum. Phys. Fluids 10, 855–858.Google Scholar
Mahrt, L. 1999 Stratified atmospheric boundary layers. Boundary-Layer Meteorol. 90, 375–396.Google Scholar
Marlatt, S. W., Waggy, S. B. & Biringen, S. 2010 Direct numerical simulation of turbulent Ekman layer: turbulent energy budgets. J. Thermophys. Heat Transfer 24, 544–555.Google Scholar
Marlatt, S. W., Waggy, S. B. & Biringen, S. 2011 Direct numerical simulation of the turbulent Ekman layer: evaluation of closure models. J. Atmos. Sci. 69, 1106–1117.Google Scholar
Mason, P. J. & Derbyshire, S. H. 1990 Large-eddy simulation of the stably-stratified atmospheric boundary layer. Boundary-Layer Meteorol. 53, 117–162.Google Scholar
Mason, P. J. & Thomson, D. J. 1987 Large-eddy simulations of the neutral-static-stability planetary boundary layer. Q. J. R. Meteorol. Soc. 113, 413–443.Google Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.Google Scholar
Mironov, D. & Fedorovich, E. 2010 On the limiting effect of the Earth’s rotation on the depth of a stably stratified boundary layer. Q. J. R. Meteorol. Soc. 136 (651), 1473–1480.Google Scholar
Miyashita, K., Iwamoto, K. & Kawamura, H. 2006 Direct numerical simulation of neutrally stratified turbulent Ekman boundary layer. J. Earth Simul. 6, 3–15.Google Scholar
Moeng, C.-H. & Sullivan, P. P. 1994 A comparison of shear- and buoyancy-driven planetary boundary layer flows. J. Atmos. Sci. 51, 999–1022.Google Scholar
Monin, A. S. & Obukhov, A. M. 1954 Basic laws of turbulent mixing in the surface layer of the atmosphere. Tr. Akad. Nauk SSSR Geophiz. Inst. 24, 163–187 (translation to English by John Miller, 1959).Google Scholar
Monin, A. & Yaglom, A. 1975 Mechanics of Turbulence, Statistical Fluid Mechanics, vol. 1. MIT Press.Google Scholar
Morris, K., Handler, R. & Rouson, D. 2011 Intermittency in the turbulent Ekman layer. J. Turbul. 12, 12.CrossRefGoogle Scholar
Nadeau, D. F., Pardyjak, E. R., Higgins, C. W., Fernando, H. J. S. & Parlange, M. B. 2011 A simple model for the afternoon and early evening decay of convective turbulence over different land surfaces. Boundary-Layer Meteorol. 141 (2), 301–324.Google Scholar
Nieuwstadt, F. T. M. 1984 The turbulent structure of the stable, nocturnal boundary layer. J. Atmos. Sci. 41, 2202–2216.Google 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
Orszag, S. 1971 On the elimination of aliasing in finite-difference schemes by filtering high-wavenumber components. J. Atmos. Sci. 28 (6), 1074.Google Scholar
Orszag, S. A. & Pao, Y. H. 1974 Numerical computation of turbulent shear flows. Adv. Geophys. 18, A225.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Porté-Agel, F., Meneveau, C. & Parlange, M. 2000 A scale-dependent dynamic model for large-eddy simulation: application to a neutral atmospheric boundary layer. J. Fluid Mech. 415, 261–284.Google Scholar
Remmler, S. & Hickel, S. 2012 Spectral structure of stratified turbulence: direct numerical simulations and predictions by large eddy simulation. Theor. Comput. Fluid Dyn 27, 319–336.Google Scholar
Riley, J. J. & de Bruyn Kops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15, 2047–2059.Google Scholar
Schlichting, H. 1979 Boundary Layer Theory. McGraw-Hill (translated by J. Kestin).Google Scholar
Shah, S. & Bou-Zeid, E.2012 Direct and large-eddy simulations of stable boundary layers: turbulent kinetic energy and flux budgets. In Proceedings of the 20th Symposium on Boundary Layers and Turbulence/18th Conference on Air–Sea Interaction. Westin Copley Place, Boston, MA, 9–13 July 2012.Google Scholar
Shingai, K. & Kawamura, H. 2004 A study of turbulence structure and large-scale motion in the Ekman layer through direct numerical simulations. J. Turbul. 5, 13.Google Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations. Mon. Weath. Rev. 91 (3), 99–164.Google Scholar
Spalart, P. R., Coleman, G. N. & Johnstone, R. 2008 Direct numerical simulation of the Ekman layer: a step in Reynolds number, and cautious support for a log law with a shifted origin. Phys. Fluids 20 (10), 101507.Google Scholar
Spalart, P. R., Coleman, G. N. & Johnstone, R. 2009 Retraction: Direct numerical simulation of the Ekman layer: a step in Reynolds number, and cautious support for a log law with a shifted origin [Phys. Fluids 20, 101507 (2008)]. Phys. Fluids 21 (10), 109901.Google Scholar
Stoll, R. & Porté-Agel, F. 2008 Large-eddy simulation of the stable atmospheric boundary layer using dynamic models with different averaging schemes. Boundary-Layer Meteorol. 126, 1–28.Google Scholar
Stull, R. B. 1988 An Introduction to Boundary Layer Meteorology. Kluwer.Google Scholar
Sullivan, P. P. & Patton, E. G. 2011 The effect of mesh resolution on convective boundary layer statistics and structures generated by large-eddy simulation. J. Atmos. Sci. 68 (10), 2395–2415.Google Scholar
Tatro, P. R. & Mollo-Christensen, E. L. 1967 Experiments on Ekman layer instability. J. Fluid Mech. 28, 531–543.Google Scholar
Taylor, J. R. & Sarkar, S. 2008a Direct and large eddy simulations of a bottom Ekman layer under an external stratification. Intl J. Heat Fluid Flow 29 (3), 721–732.Google Scholar
Taylor, J. R. & Sarkar, S. 2008b Stratification effects in a bottom Ekman layer. J. Phys. Oceanogr. 38, 2535.Google Scholar
Tennekes, H. H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.Google Scholar
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Van de Wiel, B. J. H., Moene, A. F., Jonker, H. J. J. & Clercx, H. J. H. 2011 The collapse of turbulence in the atmospheric boundary layer. J. Phys.: Conf. Ser. 318 (3), 032037.Google Scholar
Venayagamoorthy, S. K. & Stretch, D. D. 2010 On the turbulent Prandtl number in homogeneous stably stratified turbulence. J. Fluid Mech. 644, 359–369.Google Scholar
Webb, E. K. 1970 Profile relationships: the log-linear range, and extension to strong stability. Q. J. R. Meteorol. Soc. 96, 67–90.Google Scholar
Wyngaard, J. C. 2010 Turbulence in the Atmosphere. Cambridge University Press.Google Scholar
Wyngaard, J. C. & Coté, O. R. 1972 Cospectral similarity in the atmospheric surface layer. Q. J. R. Meteorol. Soc. 98 (417), 590–603.Google Scholar
Yamamoto, G. 1975 Generalization of the KEYPS formula in diabatic conditions and related discussion on the critical Richardson number. J. Met. Soc. Japan 53, 189–195.Google Scholar
Zhou, B. & Chow, F. K. 2011 Large-eddy simulation of the stable boundary layer with explicit filtering and reconstruction turbulence modeling. J. Atmos. Sci. 68, 2142–2155.Google Scholar
Zikanov, O., Slinn, D. N. & Dhanak, M. R. 2003 Large-eddy simulations of the wind-induced turbulent Ekman layer. J. Fluid Mech. 495, 343–368.Google Scholar
Zilitinkevich, S. & Mironov, D. V. 1996 A multi-limit formulation for the equilibrium depth of a stably stratified boundary layer. Boundary-Layer Meteorol. 81, 325–351.Google Scholar