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Direct numerical simulation of stationary homogeneous stratified sheared turbulence

Published online by Cambridge University Press:  01 March 2012

D. Chung  and
G. Matheou
D. Chung*
Affiliation:
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
G. Matheou
Affiliation:
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
*
†Email address for correspondence: dchung@jpl.nasa.gov
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Abstract

Using direct numerical simulation, we investigate stationary and homogeneous shear-driven turbulence in various stratifications, ranging from neutral to very stable. To attain and maintain a stationary flow, we throttle the mean shear so that the net production stays constant for all times. This results in a flow that is characterized solely by its mean shear and its mean buoyancy gradient, independent of initial conditions. The method of throttling is validated by comparison with experimental spectra in the case of neutral stratification. With increasing stratification comes the emergence of vertically sheared large-scale horizontal motions that preclude a straightforward interpretation of flow statistics. However, once these motions are excluded, simply by subtracting the horizontal average, the underlying flow appears amenable to the standard methods of turbulence analysis. It is shown that a direct acknowledgement of the confining influence of the periodic simulation box can lead to a meaningful physical interpretation of the large scales. Once an appropriate confinement scale is identified, many features, including horizontal spectra, flux–gradient relationships and length scales, of stratified sheared turbulence can be readily understood, both qualitatively and quantitatively, in terms of Monin–Obukhov similarity theory. Finally, the similarity-theory framework is used to interpret the scaling of the vertical diapycnal diffusivity in stratified turbulence.

JFM classification

Geophysical and Geological Flows: Ocean processes Turbulent Flows: Stratified turbulence Turbulent Flows: Turbulence simulation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 696 , 10 April 2012 , pp. 434 - 467
Copyright
Copyright © Cambridge University Press 2012

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References

1. Balmforth, N. J., Llewellyn Smith, S. G. & Young, W. R. 1998 Dynamics of interfaces and layers in a stratified turbulent fluid. J. Fluid Mech. 355, 329–358.Google Scholar
2. Batchelor, G. K., Canuto, V. M. & Chasnov, J. R. 1992 Homogeneous buoyancy-generated turbulence. J. Fluid Mech. 235, 349–378.Google Scholar
3. Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 1645–1651.Google Scholar
4. Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343–368.CrossRefGoogle Scholar
5. Brucker, K. A., Isaza, J. C., Vaithianathan, T. & Collins, L. R. 2007 Efficient algorithm for simulating homogeneous turbulent shear flow without remeshing. J. Comput. Phys. 225, 20–32.Google Scholar
6. Businger, J. A., Wyngaard, J. C., Izumi, Y. & Bradley, E. F. 1971 Flux-profile relationships in the atmospheric surface layer. J. Atmos. Sci. 28, 181–189.Google Scholar
7. Champagne, F. H. 1978 The fine-scale structure of the turbulent velocity field. J. Fluid Mech. 86, 67–108.CrossRefGoogle Scholar
8. Corrsin, S. 1958 Local isotropy in turbulent shear flow. NACA Tech. Rep. 58B11.Google Scholar
9. Diamessis, P. J. & Nomura, K. K. 2004 The structure and dynamics of overturns in stably stratified homogeneous turbulence. J. Fluid Mech. 499, 197–229.CrossRefGoogle Scholar
10. Dillon, T. M. 1982 Vertical overturns: a comparison of Thorpe and Ozmidov length scales. J. Geophys. Res. 87, 9601–9613.CrossRefGoogle Scholar
11. Dougherty, J. P. 1961 The anisotropy of turbulence at the meteor level. J. Atmos. Terr. Phys. 21, 210–213.CrossRefGoogle Scholar
12. van Driest, E. R. 1956 On turbulent flow near a wall. J. Aero. Sci. 23, 1007–1011.Google Scholar
13. Ellison, T. H. 1957 Turbulent transport of heat and momentum from an infinite rough plane. J. Fluid Mech. 2, 456–466.Google Scholar
14. Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455–493.CrossRefGoogle Scholar
15. 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
16. Gargett, A. E., Osborn, T. R. & Nasmyth, P. W. 1984 Local isotropy and the decay of turbulence in a stratified fluid. J. Fluid Mech. 144, 231–280.Google Scholar
17. Gerz, T., Schumann, U. & Elgobashi, S. E. 1989 Direct numerical simulation of stratified homogeneous turbulent shear flows. J. Fluid Mech. 200, 563–594.CrossRefGoogle Scholar
18. Gerz, T. & Yamazaki, H. 1993 Direct numerical simulation of buoyancy-driven turbulence in a stably stratified fluid. J. Fluid Mech. 249, 415–440.Google Scholar
19. Gibson, C. H. 1986 Internal waves, fossil turbulence, and composite ocean microstructure spectra. J. Fluid Mech. 168, 89–117.CrossRefGoogle Scholar
20. Godeferd, F. S. & Cambon, C. 1994 Detailed investigation of energy transfers in homogeneous stratified turbulence. Phys. Fluids 6, 2084–2100.Google Scholar
21. Godoy-Diana, R., Chomaz, J.-M. & Billant, P. 2004 Vertical length scale selection for pancake vortices in strongly stratified viscous fluids. J. Fluid Mech. 504, 229–238.CrossRefGoogle Scholar
22. Gonzalez-Juez, E. D., Kerstein, A. R. & Shih, L. H. 2011 Vertical mixing in homogeneous sheared stratified turbulence: a one-dimensional-turbulence study. Phys. Fluids 23, 055106.CrossRefGoogle Scholar
23. Gregg, M. C. 1987 Diapycnal mixing in the thermocline: a review. J. Geophys. Res. 92, 5249–5286.Google Scholar
24. Herring, J. R. & Métais, O. 1989 Numerical experiments in forced stably stratified turbulence. J. Fluid Mech. 202, 97–115.Google Scholar
25. Högstrom, U. 1988 Non-dimensional wind and temperature profiles in the atmospheric surface layer: a re-evaluation. Boundary-Layer Meteorol. 42, 55–78.CrossRefGoogle Scholar
26. Holt, S. E., Koseff, J. R. & Ferziger, J. H. 1992 A numerical study of the evolution and structure of homogeneous stably stratified sheared turbulence. J. Fluid Mech. 237, 499–539.CrossRefGoogle Scholar
27. Holzer, M. & Siggia, E. D. 1994 Turbulent mixing of a passive scalar. Phys. Fluids 6, 1820–1837.Google Scholar
28. Howell, J. F. & Sun, J. 1999 Surface-layer fluxes in stable conditions. Boundary-Layer Meteorol. 90, 495–520.CrossRefGoogle Scholar
29. Isaza, J. C. & Collins, L. R. 2009 On the asymptotic behaviour of large-scale turbulence in homogeneous shear flow. J. Fluid Mech. 637, 213–239.Google Scholar
30. Isaza, J. C., Warhaft, Z. & Collins, L. R. 2009 Experimental investigation of the large-scale velocity statistics in homogeneous turbulent shear flow. Phys. Fluids 21, 065105.Google Scholar
31. Itsweire, E. C., Helland, K. N. & Van Atta, C. W. 1986 The evolution of grid-generated turbulence in a stably stratified fluid. J. Fluid Mech. 162, 299–338.CrossRefGoogle Scholar
32. Itsweire, E. C., Koseff, J. R., Briggs, D. A. & Ferziger, J. H. 1993 Turbulence in stratified shear flows: implications for interpreting shear-induced mixing in the ocean. J. Phys. Oceanogr. 23, 1508–1522.2.0.CO;2>CrossRefGoogle Scholar
33. Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40, 169–184.CrossRefGoogle Scholar
34. Jacobitz, F. G., Sarkar, S. & Van Atta, C. W. 1997 Direct numerical simulations of the turbulence evolution in a uniformly sheared and stably stratified flow. J. Fluid Mech. 342, 231–261.Google Scholar
35. Kaimal, J. C., Wyngaard, J. C., Izumi, Y. & Coté, O. R. 1972 Spectral characteristics of surface-layer turbulence. Q. J. R. Meteorol. Soc. 98, 563–589.Google Scholar
36. Kaltenbach, H.-J., Gerz, T. & Schumann, U. 1994 Large-eddy simulation of homogeneous turbulence and diffusion in stably stratified shear flow. J. Fluid Mech. 280, 1–40.Google Scholar
37. Keller, K. H. & Van Atta, C. W. 2000 An experimental investigation of the vertical temperature structure of homogeneous stratified shear turbulence. J. Fluid Mech. 425, 1–29.Google Scholar
38. Laval, J.-P., McWilliams, J. C. & Dubrulle, B. 2003 Forced stratified turbulence: successive transitions with Reynolds number. Phys. Rev. E 68, 036308.Google Scholar
39. Lilly, D. K. 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci. 40, 749–761.2.0.CO;2>CrossRefGoogle Scholar
40. Lin, J.-T. & Pao, Y.-H. 1979 Wakes in stratified fluids. Annu. Rev. Fluid Mech. 11, 317–338.CrossRefGoogle Scholar
41. Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.CrossRefGoogle Scholar
42. Lumley, J. L. 1964 The spectrum of nearly inertial turbulence in a stably stratified fluid. J. Atmos. Sci. 21, 99–102.Google Scholar
43. Lumley, J. L. 1967 Similarity and the turbulent energy spectrum. Phys. Fluids 10, 855–858.Google Scholar
44. Lumley, J. L. & Panofsky, H. A. 1964 The Structure of Atmospheric Turbulence. Interscience.Google Scholar
45. Métais, O. & Herring, J. R. 1989 Numerical simulations of freely evolving turbulence in stably stratified fluids. J. Fluid Mech. 202, 117–148.CrossRefGoogle Scholar
46. Oakey, N. S. 1982 Determination of the rate of dissipation of turbulent energy from simultaneous temperature and velocity shear microstructure measurements. J. Phys. Oceanogr. 12, 256–271.Google Scholar
47. Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 83–89.2.0.CO;2>CrossRefGoogle Scholar
48. Park, Y.-G., Whitehead, J. A. & Gnanadeskian, A. 1994 Turbulent mixing in stratified fluids: layer formation and energetics. J. Fluid Mech. 279, 279–311.CrossRefGoogle Scholar
49. Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.Google Scholar
50. Piccirillo, P. & Van Atta, C. W. 1997 The evolution of a uniformly sheared thermally stratified turbulent flow. J. Fluid Mech. 334, 61–86.CrossRefGoogle Scholar
51. Praud, O., Fincham, A. M. & Sommeria, J. 2005 Decaying grid turbulence in a strongly stratified fluid. J. Fluid Mech. 522, 1–33.Google Scholar
52. Pumir, A. 1996 Turbulence in homogeneous shear flows. Phys. Fluids 8, 3112–3127.Google Scholar
53. Riley, J. J. & de Bruyn Kops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15, 2047–2059.Google Scholar
54. Riley, J. J. & Lelong, M.-P. 2000 Fluid motions in the presence of strong stable stratification. Annu. Rev. Fluid Mech. 32, 613–657.CrossRefGoogle Scholar
55. Riley, J. J. & Lindborg, E. 2008 Stratified turbulence: a possible interpretation of some geophysical turbulence measurements. J. Atmos. Sci. 65, 2416–2424.Google Scholar
56. Rogallo, R. S. 1981 Numerical experiments in homogeneous turbulence. NASA Tech. Rep. 81835.Google Scholar
57. Saddoughi, S. G. & Veeravalli, S. V. 1994 Local isotropy in turbulent boundary layers at high Reynolds number. J. Fluid Mech. 268, 333–372.CrossRefGoogle Scholar
58. Schumacher, J. 2004 Relation between shear parameter and Reynolds number in statistically stationary turbulent shear flows. Phys. Fluids 16, 3094–3102.Google Scholar
59. Shih, L. H., Koseff, J. R., Ferziger, J. H. & Rehmann, C. R. 2000 Scaling and parameterization of stratified homogeneous turbulent shear flow. J. Fluid Mech. 412, 1–20.CrossRefGoogle Scholar
60. Shih, L. H., Koseff, J. R., Ivey, G. N. & Ferziger, J. H. 2005 Parameterization of turbulent fluxes and scales using homogeneous sheared stably stratified turbulence simulations. J. Fluid Mech. 525, 193–214.CrossRefGoogle Scholar
61. Smith, L. M. & Waleffe, F. 2002 Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech. 451, 145–168.CrossRefGoogle Scholar
62. Smyth, W. D. & Moum, J. N. 2000 Length scales of turbulence in stably stratified mixing layers. Phys. Fluids 12, 1327–1342.Google Scholar
63. Smyth, W. D., Moum, J. N. & Caldwell, D. R. 2001 The efficiency of mixing in turbulent patches: inferences from direct simulations and microstructure observations. J. Phys. Oceanogr. 31, 1969–1992.2.0.CO;2>CrossRefGoogle Scholar
64. Spalart, P. R., Moser, R. D. & Rogers, M. M. 1991 Spectral methods for the Navier–Stokes equations with one infinite and two periodic directions. J. Comput. Phys. 96, 297–324.Google Scholar
65. Sreenivasan, K. R. 1991 On local isotropy of passive scalars in turbulent shear flows. Proc. R. Soc. Lond. A 434, 165–182.Google Scholar
66. Staquet, C. & Godeferd, F. S. 1998 Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 1. Flow energetics. J. Fluid Mech. 360, 295–340.Google Scholar
67. Stillinger, D. C., Helland, K. N. & Van Atta, C. W. 1983 Experiments on the transition of homogeneous turbulence to internal waves in a stratified fluid. J. Fluid Mech. 131, 91–122.CrossRefGoogle Scholar
68. Stretch, D. D. & Venayagamoorthy, S. K. 2010 Diapycnal diffusivities in homogeneous stratified turbulence. Geophys. Res. Lett. 37, L02602.Google Scholar
69. Tang, W., Caulfield, C. P. & Kerswell, R. R. 2009 A prediction for the optimal stratification for turbulent mixing. J. Fluid Mech. 634, 487–497.Google Scholar
70. Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
71. Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–20.CrossRefGoogle Scholar
72. Waite, M. L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. Fluid Mech. 517, 281–308.Google Scholar
73. Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.CrossRefGoogle Scholar
74. Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.Google Scholar
75. Wyngaard, J. C. & Coté, O. R. 1972 Cospectral similarity in the atmospheric surface layer. Q. J. R. Meteorol. Soc. 98, 590–603.CrossRefGoogle Scholar