Skip to main content
×
×
Home

Direct numerical simulation of turbulent slope flows up to Grashof number $Gr=2.1\times 10^{11}$

  • M. G. Giometto (a1), G. G. Katul (a2), J. Fang (a3) and M. B. Parlange (a1)
Abstract

Stably stratified turbulent flows over an unbounded, smooth, planar sloping surface at high Grashof numbers are examined using direct numerical simulations (DNS). Four sloping angles ( $\unicode[STIX]{x1D6FC}=15^{\circ },30^{\circ },60^{\circ }$ and $90^{\circ }$ ) and three Grashof numbers ( $\mathit{Gr}=5\times 10^{10},1\times 10^{11}$ and $2.1\times 10^{11}$ ) are considered. Variations in mean flow, second-order statistics and budgets of mean- (MKE) and turbulent-kinetic energy (TKE) are evaluated as a function of $\unicode[STIX]{x1D6FC}$ and $Gr$ at fixed molecular Prandtl number $(Pr=1)$ . Dynamic and energy identities are highlighted, which diagnose the convergence of the averaging operation applied to the DNS results. Turbulent anabatic (upward moving warm fluid along the slope) and katabatic (downward moving cold fluid along the slope) regimes are identical for the vertical wall set-up (up to the sign of the along-slope velocity), but undergo a different transition in the mechanisms sustaining turbulence as the sloping angle decreases, resulting in stark differences at low $\unicode[STIX]{x1D6FC}$ . In addition, budget equations show how MKE is fed into the system through the imposed surface buoyancy, and turbulent fluctuations redistribute it from the low-level jet (LLJ) nose towards the boundary and outer flow regions. Analysis of the TKE budget equation suggests a subdivision of the boundary layer of anabatic and katabatic flows into four distinct thermodynamical regions: (i) an outer layer, corresponding approximately to the return flow region, where turbulent transport is the main source of TKE and balances dissipation; (ii) an intermediate layer, bounded below by the LLJ and capped above by the outer layer, where the sum of shear and buoyant production overcomes dissipation, and where turbulent and pressure transport terms are a sink of TKE; (iii) a buffer layer, located at $5\lessapprox z^{+}\lessapprox 30$ , where TKE is provided by turbulent and pressure transport terms, to balance viscous diffusion and dissipation; and (iv) a laminar sublayer, corresponding to $z^{+}\lessapprox 5$ , where the influence of viscosity is significant. $(\cdot )^{+}$ denotes a quantity rescaled in inner units. Interestingly, a zone of global backscatter (energy transfer from the turbulent eddies to the mean flow) is consistently found in a thin layer below the LLJ in both anabatic and katabatic regimes.

Copyright
Corresponding author
Email address for correspondence: mgiometto@civil.ubc.ca
Footnotes
Hide All

Department of Civil and Environmental Engineering, Monash University, Clayton, VIC 3800, Australia.

Footnotes
References
Hide All
Abkar, M., Bae, H. J. & Moin, P. 2016 Minimum-dissipation scalar transport model for large-eddy simulation of turbulent flows. Phys. Rev. Fluids 1 (4), 041701.
Albertson, J. D. & Parlange, M. B. 1999a Natural integration of scalar fluxes from complex terrain. Adv. Water Resour. 23 (3), 239252.
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 (7), 21212132.
Arduini, G., Staquet, C. & Chemel, C. 2016 Interactions between the nighttime valley-wind system and a developing cold-air pool. Boundary-Layer Meteorol 161 (1), 4972.
Axelsen, S. & Dop, H. 2009a Large-eddy simulation of katabatic winds. Part 1: comparison with observations. Acta Geophys. 57 (4), 803836.
Axelsen, S. L. & Dop, H. 2009b Large-eddy simulation of katabatic winds. Part 2: sensitivity study and comparison with analytical models. Acta Geophys. 57 (4), 837856.
Bou-Zeid, E., Meneveau, C. & Parlange, M. B. 2005 A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows. Phys. Fluids 17 (2), 025105.
Bou-Zeid, E., Overney, J., Rogers, B. D. & Parlange, M. B. 2009 The effects of building representation and clustering in large-eddy simulations of flows in urban canopies. Boundary-Layer Meteorol. 132 (3), 415436.
Burkholder, B., Fedorovich, E. & Shapiro, A. 2011 Evaluating subgrid-scale models for large-eddy simulation of turbulent katabatic flow. In Qual Reliab Large-eddy Simulations II, pp. 149160. Springer.
Burkholder, B., Shapiro, A. & Fedorovich, E. 2009 Katabatic flow induced by a cross-slope band of surface cooling. Acta Geophys. 57 (4), 923949.
Burns, P. & Chemel, C. 2014 Evolution of cold-air-pooling processes in complex terrain. Boundary-Layer Meteorol. 150 (3), 423447.
Burns, P. & Chemel, C. 2015 Interactions between downslope flows and a developing cold-air pool. Boundary-Layer Meteorol. 154 (1), 5780.
Calaf, M., Meneveau, C. & Meyers, J. 2010 Large eddy simulation study of fully developed wind-turbine array boundary layers. Phys. Fluids 22 (1), 015110.
Calaf, M., Parlange, M. B. & Meneveau, C. 2011 Large eddy simulation study of scalar transport in fully developed wind-turbine array boundary layers. Phys. Fluids 23 (12), 126603.
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 2006 Spectral Methods. Springer.
Chamecki, M., Meneveau, C. & Parlange, M. B. 2009 Large eddy simulation of pollen transport in the atmospheric boundary layer. J. Aero. Sci. 40, 241255.
Chemel, C., Staquet, C. & Largeron, Y. 2009 Generation of internal gravity waves by a katabatic wind in an idealized alpine valley. Meteorol. Atmos. Phys. 103 (1–4), 187194.
Chorin, A. J. 1968 Numerical solution of the Navier–Stokes equations. Math. Comput. 22 (104), 745762.
Chow, F. K., Weigel, A. P., Street, R. L., Rotach, M. W. & Xue, M. 2006 High-resolution large-eddy simulations of flow in a steep Alpine valley. Part I: Methodology, verification, and sensitivity experiments. J. Appl. Meteorol. Climatol. 45 (1), 6386.
Chu, P. C. 1987 An instability theory of ice-air interaction for the formation of ice edge bands. J. Geophys. Res. 92 (C7), 69666970.
Defant, F. 1949 Zur theorie der hangwinde, nebst bemerkungen zur theorie der berg- und talwinde. Arch. Meteorol. Geophys. Bioklimatol. Ser A 1, 421450.
Denby, B. 1999 Second-order modelling of turbulence in katabatic flows. Boundary-Layer Meteorol. 92 (1), 6598.
Doran, J. C. & Horst, T. W. 1981 Velocity and temperature oscillations in drainage winds. J. Appl. Meteorol. 20 (4), 361364.
Egger, J. 1985 Slope winds and the axisymmetric circulation over Antarctica. J. Atmos. Sci. 42 (17), 18591867.
Fedorovich, E. & Shapiro, A. 2009 Structure of numerically simulated katabatic and anabatic flows along steep slopes. Acta Geophys. 57 (4), 9811010.
Fedorovich, E. & Shapiro, A. 2017 Oscillations in Prandtl slope flow started from rest. Q. J. R. Meteorol. Soc. 143 (703), 670677.
Fernando, H. J. S. 2010 Fluid dynamics of urban atmospheres in complex terrain. Annu. Rev. Fluid Mech. 42 (1), 365389.
Fernando, H. J. S., Pardyjak, E. R, Di Sabatino, S., Chow, F. K., De Wekker, S. F., Hoch, S. W., Hacker, J., Pace, J. C., Pratt, T., Pu, Z. et al. 2015 The MATERHORN: Unraveling the intricacies of mountain weather. Bull. Am. Meteorol. Soc. 96 (11), 19451967.
Giometto, M. G., Christen, A., Meneveau, C., Fang, J., Krafczyk, M. & Parlange, M. B. 2016 Spatial characteristics of roughness sublayer mean flow and turbulence over a realistic urban surface. Boundary-Layer Meteorol. 160 (3), 425452.
Giometto, M. G., Grandi, R., Fang, J., Monkewitz, P. A. & Parlange, M. B. 2017 Katabatic flow: a closed-form solution with spatially-varying eddy diffusivities. Boundary-Layer Meteorol 162 (2), 307317.
Grachev, A. A., Leo, L. S., Sabatino, S. D., Fernando, H. J. S., Pardyjak, E. R. & Fairall, C. W. 2016 Structure of turbulence in katabatic flows below and above the wind-speed maximum. Boundary-Layer Meteorol. 159 (3), 469494.
Greuell, J. W., Broeke Van den, M. R., Knap, W., Reijmer, C., Smeets, P. & Struijk, I.1994 PASTEX: a glacio-meteorological experiment on the Pasterze (Austria). Tech. Rep., Institute for Marine and Atmospheric Research, University, Utrecht.
Grisogono, B. & Axelsen, S. L. 2012 A note on the pure katabatic wind maximum over gentle slopes. Boundary-Layer Meteorol. 145 (3), 527538.
Grisogono, B., Jurlina, T., Večenaj, Ž. & Güttler, I. 2014 Weakly nonlinear Prandtl model for simple slope flows. Q. J. R. Meteorol. Soc. 141, 883892.
Grisogono, B., Kraljevic, L. & Jericevic, A. 2007 The low-level katabatic jet height versus Monin Obukhov height. Q. J. R. Meteorol. Soc. 133, 21332136.
Grisogono, B. & Oerlemans, J. 2001a A theory for the estimation of surface fluxes in simple katabatic flows. Q. J. R. Meteorol. Soc. 127, 27252739.
Grisogono, B & Oerlemans, J 2001b Katabatic flow: analytic solution for gradually varying eddy diffusivities. J. Atmos. Sci. 58 (21), 33493354.
Grisogono, B. & Oerlemans, J. 2002 Justifying the WKB approximation in pure katabatic flows. Tellus 54 (5), 453462.
Gutman, L. N. 1983 On the theory of the katabatic slope wind. Tellus 35A, 213218.
Güttler, I., Marinović, I., Večenaj, Ž & Grisogono, B. 2016 Energetics of slope flows: linear and weakly nonlinear solutions of the extended Prandtl model. Front. Earth Sci. 4 (July), 113.
Haiden, T. & Whiteman, C. D. 2005 Katabatic flow mechanisms on a low-angle slope. J. Appl. Meteorol. 44 (1), 113126.
Hang, C., Nadeau, D. F., Gultepe, I., Hoch, S. W., Román-Cascón, C., Pryor, K., Fernando, H. J. S., Creegan, E. D., Leo, L. S. & Silver, Z. 2016 A case study of the mechanisms modulating the evolution of valley fog. Pure Appl. Geophys. 173 (9), 120.
Higgins, C. W., Parlange, M. B. & Meneveau, C. 2003 Alignment trends of velocity gradients and subgrid-scale fluxes in the turbulent atmospheric boundary layer. Boundary-Layer Meteorol. 109 (1), 5983.
Hultmark, M., Calaf, M. & Parlange, M. B. 2013 A new wall shear stress model for atmospheric boundary layer simulations. J. Atmos. Sci. 70 (11), 34603470.
Iida, O., Kasagi, N. & Nagano, Y. 2002 Direct numerical simulation of turbulent channel flow under stable density stratification. Intl J. Heat Mass Transfer 45, 16931703.
Israeli, M. & Orszag, S. A. 1981 Approximation of radiation boundary conditions. J. Comput. Phys. 41 (1), 115135.
Jensen, D. D., Nadeau, D. F., Hoch, S. W. & Pardyjak, E. R. 2016 Observations of near-surface heat-flux and temperature profiles through the early evening transition over contrasting surfaces. Boundary-Layer Meteorol. 159 (3), 567587.
Kavavcic, I. & Grisogono, B. 2007 Katabatic flow with Coriolis effect and gradually varying eddy diffusivity. Boundary-Layer Meteorol. 125 (2), 377387.
Kravchenko, A. G. G. & Moin, P. 1997 On the effect of numerical errors in large eddy simulations of turbulent flows. J. Comput. Phys. 131 (2), 310322.
Kumar, V., Kleissl, J., Meneveau, C. & Parlange, M. B. 2006 Large-eddy simulation of a diurnal cycle of theatmospheric boundary layer: atmospheric stabilityand scaling issues. Water Resour. Res. 42, W06D09.
Lehner, M., Whiteman, C. D., Hoch, S. W., Jensen, D., Pardyjak, E. R., Leo, L. S., Di Sabatino, S. & Fernando, H. J. S. 2015 A case study of the nocturnal boundary layer evolution on a slope at the foot of a desert mountain. J. Appl. Meteorol. Climatol. 54 (4), 732751.
Lu, H. & Porté-Agel, F. 2010 A modulated gradient model for large-eddy simulation: application to a neutral atmospheric boundary layer. Phys. Fluids 22 (1), 015109.
Mahrt, L. 1982 Momentum balance of gravity flows. J. Atmos. Sci. 39 (12), 27012711.
Mahrt, L. 1998 Stratified atmospheric boundary layers and breakdown of models. Theor. Comput. Fluid Dyn. 11, 263279.
Mahrt, L. 2013 Stably stratified atmospheric boundary layers. Annu. Rev. Fluid Mech. 46 (July), 2345.
McNider, R. T. 1982 A note on velocity fluctuations in drainage flows. J. Atmos. Sci. 39 (7), 16581660.
Menold, E. R. & Yang, K. 1962 Asymptotic solutions for unsteady laminar free convection on a vertical plate. J. Appl. Mech. 29 (1), 124126.
Monti, P., Fernando, H. J. S. & Princevac, M. 2014 Waves and turbulence in katabatic winds. Environ. Fluid Mech. 14 (2), 431450.
Monti, P., Fernando, H. J. S., Princevac, M., Chan, W. C., Kowalewski, T. A. & Pardyjak, E. R. 2002 Observations of flow and turbulence in the nocturnal boundary layer over a slope. J. Atmos. Sci. 59 (17), 25132534.
Moser, R. D., Kim, J. & Moin, P. 1999 Direct numerical simulation of turbulent channel flow up to Re = 590. Phys. Fluids 11 (4), 1113.
Nadeau, D. F., Pardyjak, E. R., Higgins, C. W., Huwald, H. & Parlange, M. B. 2013a Flow during the evening transition over steep Alpine slopes. Q. J. R. Meteorol. Soc. 139 (672), 607624.
Nadeau, D. F., Pardyjak, E. R., Higgins, C. W. & Parlange, M. B. 2013b Similarity scaling over a steep alpine slope. Boundary-Layer Meteorol. 147 (3), 401419.
Oerlemans, J. 1998 The atmospheric boundary layer over melting glaciers. In Clear Cloudy Bound Layers, pp. 129153. Royal Netherlands Academy of Arts and Sciences.
Oerlemans, J., Björnsson, H., Kuhn, M., Obleitner, F., Palsson, F., Smeets, C. J. P. P., Vugts, H. F. & Wolde, J. D. 1999 Glacio-meteorological investigation on Vatnajokull, Iceland, summer 1996: an overview. Boundary-Layer Meteorol. 92 (1), 324.
Oldroyd, H. J., Katul, G. G., Pardyjak, E. R. & Parlange, M. B. 2014 Momentum balance of katabatic flow on steep slopes covered with short vegetation. Geophys. Res. Lett. 41 (13), 47614768.
Oldroyd, H. J., Pardyjak, E. R., Higgins, C. W. & Parlange, M. B. 2016a Buoyant turbulent kinetic energy production in steep-slope katabatic flow. Boundary-Layer Meteorol. 161 (3), 405416.
Oldroyd, H. J., Pardyjak, E. R., Huwald, H. & Parlange, M. B. 2016b Adapting tilt corrections and the governing flow equations for steep, fully three-dimensional, mountainous terrain. Boundary-Layer Meteorol. 159 (3), 539565.
Orszag, S. A. 1969 Numerical methods for the simulation of turbulence. Phys. Fluids 12 (12), II250.
Orszag, S. A. 1970 Transform method for the calculation of vector-coupled sums: application to the spectral form of the vorticity equation. J. Atmos. Sci. 27 (6), 890895.
Orszag, S. A. & Pao, Y. H. 1975 Numerical computation of turbulent shear flows. In Advances in Geophysics, vol. 18, pp. 225236. Elsevier.
Parish, T. R. 1992 On the role of Antarctic katabatic winds in forcing large-scale tropospheric motions. J. Atmos. Sci. 49 (15), 13741385.
Parish, T. R. & Bromwich, D. H. 1998 A case study of Antarctic katabatic wind interaction with large-scale forcing. Mon. Weath. Rev. 126 (1), 199209.
Parmhed, O., Oerlemans, J. & Grisogono, B. 2004 Describing surface fluxes in katabatic flow on Breidamerkurjo kull, Iceland. Q. J. R. Meteorol. Soc. 130, 11371151.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Porté-Agel, F., Meneveau, C. & Parlange, M. B. 2000 A scale-dependent dynamic model for large-eddy simulation: application to a neutral atmospheric boundary layer. J. Fluid Mech. 415, 261284.
Prandtl, L. 1942 Führer durch die Strömungslehre. Vieweg & Sohn.
Princevac, M., Hunt, J. C. R. & Fernando, H. J. S. 2008 Quasi-steady katabatic winds on slopes in wide valleys: hydraulic theory and observations. J. Atmos. Sci. 65 (2), 627643.
Rampanelli, G., Zardi, D. & Rotunno, R. 2004 Mechanisms of up-valley winds. J. Atmos. Sci. 61 (24), 30973111.
Renfrew, I. A. 2004 The dynamics of idealized katabatic flow over a moderate slope and ice shelf. Q. J. R. Meteorol. Soc. 130 (598), 10231045.
Renfrew, I. A. & Anderson, P. S. 2006 Profiles of katabatic flow in summer and winter over Coats Land, Antarctica. Q. J. R. Meteorol. Soc. 132 (616), 779802.
Rotach, M. W. & Zardi, D. 2007 On the boundary-layer structure over highly complex terrain: Key findings from MAP. Q. J. R. Meteorol. Soc. 133 (625), 937948.
Schumann, U. 1990 Large-eddy simulation of the up-slope boundary layer. Q. J. R. Meteorol. Soc. 116 (493), 637670.
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.
Shapiro, A. & Fedorovich, E. 2004 Prandtl number dependence of unsteady natural convection along a vertical plate in a stably stratified fluid. Intl J. Heat Mass Transfer 47 (22), 49114927.
Shapiro, A. & Fedorovich, E. 2005 Natural convection in a stably stratified fluid along vertical plates and cylinders with temporally periodic surface temperature variations. J. Fluid Mech. 546, 295311.
Shapiro, A. & Fedorovich, E. 2007 Katabatic flow along a differentially cooled sloping surface. J. Fluid Mech. 571, 149175.
Shapiro, A. & Fedorovich, E. 2008 Coriolis effects in homogeneous and inhomogeneous katabatic flows. Q. J. R. Meteorol. Soc. 134 (631), 353370.
Shapiro, A. & Fedorovich, E. 2014 A boundary-layer scaling for turbulent katabatic flow. Boundary-Layer Meteorol. 153 (1), 117.
Sharma, V., Calaf, M., Lehning, M. & Parlange, M. B. 2016 Time-adaptive wind turbine model for an LES framework. Wind Energy 19 (5), 939952.
Sharma, V., Parlange, M. B. & Calaf, M. 2017 Perturbations to the spatial and temporal characteristics of the diurnally-varying atmospheric boundary layer due to an extensive wind farm. Boundary-Layer Meteorol. 162 (2), 255282.
Skyllingstad, E. D. 2003 Large-eddy simulation of katabatic flows. Boundary-Layer Meteorol. 106 (2), 217243.
Smeets, C. J. P. P., Duynkerke, P. G. & Vugts, H. F. 1997 Turbulence characteristics of the stable boundary layer over a mid-latitude glacier. Part 1: a combination of katabatic and large-scale forcing. Boundary-Layer Meteorol. 87, 117145.
Smeets, C. J. P. P., Duynkerke, P. G. & Vugts, H. F. 2000 Turbulence characteristics of the stable boundary layer over a mid-latitude glacier. Part 2: pure katabatic forcing conditions. Boundary-Layer Meteorol. 97, 73107.
Smith, C. M. & Skyllingstad, E. D. 2005 Numerical simulation of katabatic flow with changing slope angle. Mon. Weath. Rev. 133 (11), 30653080.
Temam, R. 1968 Une methode d’approximation de la solution des equations de Navier–Stokes. Bull. Soc. Maths France 96, 115152.
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow. Cambridge University Press.
Weigel, A. P., Chow, F. K., Rotach, M. W., Street, R. L. & Xue, M. 2006 High-resolution large-eddy simulations of flow in a steep alpine valley. Part II: flow structure and heat budgets. J. Appl. Meteorol. Climatol. 45 (1), 87107.
Whiteman, C. D. 1990 Observations of thermally developed wind systems in mountainous terrain. Atmos Process over complex terrain, Meteor. Monogr. 23 (45), 542.
Zardi, D. & Serafin, S. 2015 An analytic solution for time-periodic thermally driven slope flows. Q. J. R. Meteorol. Soc. 141, 19681974.
Zardi, D. & Whiteman, C. D. 2013 Diurnal mountain wind systems. In Mt Weather Res Forecast, pp. 35119. Springer.
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? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 15
Total number of PDF views: 171 *
Loading metrics...

Abstract views

Total abstract views: 405 *
Loading metrics...

* Views captured on Cambridge Core between 22nd September 2017 - 18th August 2018. This data will be updated every 24 hours.