Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-19T00:05:42.812Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential tails and skewness of density-gradient probability density functions in stably stratified turbulence

Published online by Cambridge University Press:  26 April 2006

S. T. Thoroddsen  and
C. W. Van Atta
S. T. Thoroddsen
Affiliation:
Applied Mechanics and Engineering Sciences, 0411, University of California, San Diego, La Jolla, CA 92093, USA
C. W. Van Atta
Affiliation:
Applied Mechanics and Engineering Sciences, 0411, University of California, San Diego, La Jolla, CA 92093, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We have measured the probability density functions (PDFs) of density fluctuations and of density-gradient fluctuations in decaying stratified turbulence, using a thermally stratified wind tunnel. The turbulence was generated by passing the flow through a biplanar grid at the entrance to the test section. The linear mean vertical temperature gradient could be adjusted to produce different stratification strengths. The PDFs of the density-gradient fluctuations exhibit extended exponential tails, while those for the density fluctuations are nearly Gaussian. As the turbulence decays away from the grid the exponential tails of the density gradient PDFs become steeper and the central rounded part of the distribution widens. The tail steepness scales approximately as $Re \lambda{-\frac{1}{2}}$. Buoyancy forces are not the cause of the exponential tails, since when normalized in r.m.s. units the behaviour of the tails is independent of stratification strength. The vertical temperature gradients ∂θ/∂z (measured using two cold wires) show a strong positive skewness close to the grid where the turbulence is most vigorous. This skewness is not caused by non-Boussinesq effects and is present for all stratification strengths. We propose a simple phenomenological model (similar to that of Budwig et al. 1985), based on stirring of fluid parcels advected in the mean gradient, to explain the presence of this skewness. The skewness observed by other researchers and their interpretations are discussed in the context of this model. The buoyancy flux PDF also shows strong exponential tails and is very strongly skewed. Both of these properties are consistent with joint-Gaussian statistics of the vertical velocity and temperature fluctuations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 244 , November 1992 , pp. 547 - 566
Copyright
© 1992 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antonia R. A., Hopfinger E. J., Gagne, Y. & Anselmet F. 1984 Temperature structure functions in turbulent shear flows Phys. Rev. A 30, 2704–2707.Google Scholar
Antonia, R. A. & Van Atta C. W. 1978 Structure functions of temperature fluctuations in turbulent shear flows. J. Fluid Mech. 84, 561–580.Google Scholar
Balachandar, S. & Sirovich L. 1991 Probability distribution functions in turbulent convection Phys. Fluids A 3, 919–927.Google Scholar
Benzi R., Biferale L., Paladin G., Vulpiani, A. & Vergassola M. 1991 Multifractality in the statistics of the velocity gradients in turbulence. Phys. Rev. Lett. 67, 2299–2302.Google Scholar
Budwig R., Tavoularis, S. & Corrsin S. 1985 Temperature fluctuations and heat flux in grid-generated isotropic turbulence with streamwise and transverse mean-temperature gradients. J. Fluid Mech. 153, 441–460.Google Scholar
Castaing B., Gunaratne G., Heslot F., Kadanoff L., Libchaber A., Thomae S., Wu X.-Z., Zaleski, S. & Zanetti G. 1989 Scaling of hard thermal turbulence in Rayleigh-Bénard convection. J. Fluid Mech. 204, 1–30.Google Scholar
Ching E. S. C. 1991 Probabilities for temperature differences in Rayleigh-Bénard convection Phys. Rev. A 44, 3622–3629.Google Scholar
Gerz, T. & Schumann U. 1991 Direct simulation of homogeneous turbulence and gravity waves in sheared and unsheared stratified flows. In Turbulent Shear Flows 7 (ed. F. Durst), pp. 27–45. Springer.
Gibson C. H., Friehe, C. A. & McConnell S. O. 1977 Structure of sheared turbulent fields. Phys. Fluids Suppl. 20, S156–S167.Google Scholar
Gibson C. H., Stegen, G. R. & Williams R. B. 1970 Statistics of the fine structure of turbulent velocity and temperature fields measured at high Reynolds number. J. Fluid Mech. 41, 153–167.Google Scholar
Goldhirsch, I. & Yakhot A. 1990 Probability distribution of a passive scalar: a soluble example Phys. Fluids A 2, 1303–1305.Google Scholar
Gollub J. P., Clarke J., Gharib M., Lane, B. & Mesquita O. N. 1992 Fluctuations and transport in a stirred fluid with a mean gradient. Phys. Rev. Lett. 67, 3507–3510.Google Scholar
Haugdahl, J. & Lienhard V. J. H. 1988 A low-cost, high-performance cold-wire bridge. J. Phys. E: Sci. Instrum. 21, 167–170.Google Scholar
Jensen M. H., Paladin, G. & Vulpiani A. 1992 Shell model for turbulent advection of passive scalar fields Phys. Rev. A 45, 7214–7221.Google Scholar
Kraichnan R. H. 1974 Convection of a passive scalar by a quasi-uniform random straining field. J. Fluid Mech. 64, 737–762.Google Scholar
Kraichnan R. H. 1990 Models of intermittency in hydrodynamic turbulence. Phys. Rev. Lett. 65, 575–578.Google Scholar
Lienhard V, J, H. & Van Atta, C. W. 1989 Thermally stratifying a wind tunnel for buoyancy influenced flows. Exps Fluids 7, 542–546.Google Scholar
Lienhard V, J. H. & Van Atta, C. W. 1990 The decay of turbulence in thermally stratified flow. J. Fluid Mech. 210, 57–112.Google Scholar
Métais, M. & Lesieur M. 1992 Spectral large-eddy simulation of isotropic and stably stratified turbulence. J. Fluid Mech. 239, 157–194.Google Scholar
Papoulis A. 1972 Probability, Random Variables, and Stochastic Processes. McGraw-Hill.
Peterka, J. A. & Cermak J. E. 1975 Wind pressures on buildings - probability densities. J. Structural Div. ASCE 101, (ST6). 1255–1267.Google Scholar
Pumir A., Shraiman, B. & Siggia E. D. 1991 Exponential tails and random advection. Phys. Rev. Lett 66, 2984–2987.Google Scholar
She Z.-S. 1991 Physical model of intermittency in turbulence. Near-dissipation-range non-Gaussian statistics. Phys. Rev. Lett. 66, 600–603.Google Scholar
Sinai, Ya. G. & Yakhot V. 1989 Limiting probability distributions of a passive scalar in a random velocity field. Phys. Rev. Lett. 63, 1962–1964Google Scholar
Sirovich L. 1987 Turbulence and the dynamics of coherent structures. Part II. Symmetries and transformations. Q. Appl. Maths 45, 573–582.Google Scholar
Sreenivasan K. R., Antonia, R. A. & Danh H. Q. 1977 Temperature dissipation fluctuations in a turbulent boundary layer. Phys. Fluids 20, 1238–1249.Google Scholar
Sreenivasan, K. R. & Tavoularis S. 1980 On the skewness of the temperature derivative in turbulent flows. J. Fluid Mech. 101, 783–795.Google Scholar
Tavoularis S., Bennett, J. C. & Corrsin S. 1978 Velocity-derivative skewness in small Reynolds number, nearly isotropic turbulence. J. Fluid Mech. 88, 63–69.Google Scholar
Tavoularis, S. & Corrsin S. 1981a Experiments in nearly homogeneous turbulent shear flow with a uniform mean temperature gradient. Part 1. J. Fluid Mech. 104, 311–347.Google Scholar
Tavoularis, S. & Corrsin S. 1981b Experiments in nearly homogeneous turbulent shear flow with a uniform mean temperature gradient. Part 2. The fine structure. J. Fluid Mech. 104, 349–367.Google Scholar
Taylor G. I. 1938 Production and dissipation of vorticity in a turbulent fluid Proc. R. Soc. Lond. A 165, 15–23.Google Scholar
Tennekes, H. & Lumley J. L. 1972 A First Course in Turbulence. MIT Press.
Thoroddsen, S. T. & Van Atta C. W. 1992 Experiments on anisotropy of density gradients in stably stratified turbulence. J. Fluid Mech. (to be submitted).Google Scholar
Townsend A. A. 1947 The measurement of double and triple correlation derivatives in isotropic turbulence. Proc. Camb. Phil. Soc. 43, 560.Google Scholar
Van Atta, C. W. & Antonia, R. A. 1980 Reynolds number dependence of skewness and flatness factors of turbulent velocity derivatives. Phys. Fluids 23, 252–257.Google Scholar
Van Atta, C. W. & Chen, W. Y. 1968 Correlation measurements in grid turbulence using digital harmonic analysis. J. Fluid Mech. 34, 497–515.Google Scholar
Van Atta, C. W. & Chen, W. Y. 1970 Structure functions of turbulence in the atmospheric boundary layer over the ocean. J. Fluid Mech. 44, 145–159.Google Scholar
Warhaft, Z. & Lumley J. L. 1978 An experimental study of the decay of temperature fluctuations in grid-generated turbulence. J. Fluid Mech. 88, 659–684.Google Scholar
Yakhot V. 1989 Probability distributions in high-Rayleigh-number Bénard convection. Phys. Rev. Lett. 63, 1965–1967.Google Scholar
Yamamoto, K. & Hosokawa I. 1988 A decaying isotropic turbulence pursued by the spectral method. J. Phys. Soc. Japan 57, 1532–1535.Google Scholar