Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-09T04:31:54.719Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-Gaussian scalar statistics in homogeneous turbulence

Published online by Cambridge University Press:  26 April 2006

F. A. Jaberi ,
R. S. Miller ,
C. K. Madnia  and
P. Givi
F. A. Jaberi
Affiliation:
Department of Mechanical and Aerospace Engineering, State University of New York at Buffalo, Buffalo, NY 14260-4400, USA
R. S. Miller
Affiliation:
Department of Mechanical and Aerospace Engineering, State University of New York at Buffalo, Buffalo, NY 14260-4400, USA
C. K. Madnia
Affiliation:
Department of Mechanical and Aerospace Engineering, State University of New York at Buffalo, Buffalo, NY 14260-4400, USA
P. Givi
Affiliation:
Department of Mechanical and Aerospace Engineering, State University of New York at Buffalo, Buffalo, NY 14260-4400, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Results are presented of numerical simulations of passive scalar mixing in homogeneous, incompressible turbulent flows. These results are generated via the Linear Eddy Model (LEM) and Direct Numerical Simulation (DNS) of turbulent flows under a variety of different conditions. The nature of mixing and its response to the turbulence field is examined and the single-point probability density function (p.d.f.) of the scalar amplitude and the p.d.f.s of the scalar spatial-derivatives are constructed. It is shown that both Gaussian and exponential scalar p.d.f.s emerge depending on the parameters of the simulations and the initial conditions of the scalar field. Aided by the analyses of data, several reasons are identified for the non-Gaussian behaviour of the scalar amplitude. In particular, two mechanisms are identified for causing exponential p.d.f.s: (i) a non-uniform action of advection on the large and the small scalar scales, (ii) the nonlinear interaction of the scalar and the velocity fluctuations at small scales. In the absence of a constant non-zero mean scalar gradient, the behaviour of the scalar p.d.f. is very sensitive to the initial conditions. In the presence of this gradient, an exponential p.d.f. is not sustained regardless of initial conditions. The numerical results pertaining to the small-scale intermittency (non-Gaussian scalar derivatives) are in accord with laboratory experimental results. The statistics of the scalar derivatives and those of the velocity-scalar fluctuations are also in accord with laboratory measured results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 313 , 25 April 1996 , pp. 241 - 282
Copyright
© 1996 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

Andrews, L. C. & Shivamoggi, B. K. 1990 The gamma distribution as a model for temperature dissipation in intermittent turbulence. Phys. Fluids A 2, 105–110.Google Scholar
Anselmet, F., Gagne, Y., Hopfinger, E. J. & Antonia, R. A. 1984 High-order velocity structure functions in turbulent shear flows. J. Fluid Mech. 140, 63–89.Google Scholar
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
Ashurst, W. T., Chen, J.-Y. & Rogers, M. M. 1987a Pressure gradient alignment with strain rate and scalar gradient in simulated Navier-Stokes turbulence. Phys. Fluids 30, 3293–3294.Google Scholar
Ashurst, W. T., Kerstein, A. R., Kerr, R. M. & Gibson, C. H. 1987b Alignment of vorticity and scalar gradient with strain rate in simulated Navier-Stokes turbulence. Phys. Fluids 30, 2343–2353.Google Scholar
Batchelor, G. K. & Townsend, A. A. 1949 The nature of turbulence motion at large wave numbers. Proc. R. Soc. Lond. A 199, 534–550.Google Scholar
Belmonte, A., Tilgner, A. & Libchaber, A. 1994 Temperature and velocity boundary layers in turbulent convection. Phys. Rev. E 50, 269–279.Google Scholar
Brodkey, R. S. (ed.)1975 Turbulence in Mixing Operations. Academic.
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., Gagne, Y. & Hopfinger, E. J. 1990 Velocity probability density functions of high Reynolds number turbulence. Physica D 46, 177–200.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–Bernard convection. J. Fluid Mech. 204, 1–30.Google Scholar
Chen, H., Chen, S. & Kraichnan, R. H. 1989 Probability distribution of a stochastically advected scalar field. Phys. Rev. Lett. 63, 2657–2660.Google Scholar
Chen, S., Doolen, G., Herring, J. R., Kraichnan, R. H., Orszag, S. A. & She, Z. S. 1993 Far-dissipation range of turbulence. Phys. Rev. Lett. 70, 3051–3054.Google Scholar
Ching, E. S. C. & Tu, Y. 1994 Passive scalar fluctuations with and without a mean gradient: A numerical study. Phys. Rev. E 49, 1278–1282.Google Scholar
Christie, S. L. & Domaradzki, J. A. 1993 Numerical evidence for the nonuniversality of the soft/hard turbulence classification for thermal convection. Phys. Fluids A 5, 412–421.Google Scholar
Christie, S. L. & Domaradzki, J. A. 1994 Scale dependence of the statistical character of turbulent fluctuations in thermal convection. Phys. Fluids 6, 1848–1855.Google Scholar
Curl, R. L. 1963 Dispersed phase mixing: I. Theory and effects in simple reactors. AIChE J. 9, 175–181.Google Scholar
Dopazo, C. 1994 Recent developments in p.d.f. methods. In Turbulent Reacting Flows ed. (P. A. Libby & F. A. Williams), pp. 375–474. Academic.
Dopazo, C. & O'Brien, E. E. 1976 Statistical treatment of non-isothermal chemical reactions in turbulence. Combust. Sci. Tech. 13, 99–112.Google Scholar
Eswaran, V. & Pope, S. B. 1988 Direct numerical simulations of the turbulent mixing of a passive scalar. Phys. Fluids 31, 506–520.Google Scholar
Frankel, S. H. 1993 Probabilistic and deterministic description of turbulent flows with nonpremixed reactants PhD Thesis, Department of Mechanical and Aerospace Engineering, State University of New York at Buffalo, Buffalo, NY.
Frankel, S. H., Madnia, C. K. & Givi, P. 1993 Comparative assessment of closures for turbulent reacting flows. AIChE J. 39, 899–903.Google Scholar
Givi, P. 1989 Model free simulations of turbulent reactive flows. Prog. Energy Combust. Sci. 15, 1–107.Google Scholar
Givi, P. 1994 Spectral and random vortex methods in turbulent reacting flows. In Turbulent Reacting Flows ed. (P. A. Libby & F. A. Williams), pp. 475–572. Academic.
Givi, P. & Madnia, C. K. 1993 Spectral methods in combustion. In Numerical Modeling in Combustion ed. (T. J. Chung), pp. 409–452. Taylor & Francis.
Givi, P. & McMurtry, P. A. 1988 Non-premixed reaction in homogeneous turbulence: Direct numerical simulations. AIChE J. 34, 1039–1042.Google Scholar
Gollub, J. P., Clarke, J., Gharib, M., Lane, B. & Mesquita, O. N. 1991 Fluctuations and transport in a stirred fluid with a mean gradient. Phys. Rev. Lett. 67, 3507–3510.Google Scholar
Gurvich, A. S. & Yaglom, A. M. 1967 Breakdown of eddies and probability distributions for small-scale turbulence. Phys. Fluids Suppl. 10, S59–S65.Google Scholar
Hawthorne, W. R., Wedell, D. S. & Hottel, H. C. 1949 Mixing and combustion in turbulent gas jets. In 3rd Symp. on Combustion, Flames and Explosion Phenomena, pp. 266–288. The Combustion Institute, Pittsburgh, PA.
Heslot, F., Castaing, B. & Libchaber, A. 1987 Transitions to turbulence in helium gas. Phys. Rev. A 36, 5870–5873.Google Scholar
Hill, J. C. 1979 Simulation of chemical reaction in a turbulent flow. In Proc. Second R. F. Ruth Chemical Engineering Research Symposium, pp. 27–53. Ames, Iowa.
Holzer, M. & Pumir, A. 1993 Simple models of non-Gaussian statistics for a turbulently advected passive scalar. Phys. Rev. E 47, 202–219.Google Scholar
Holzer, M. & Siggia, E. D. 1994 Turbulent mixing of a passive scalar. Phys. Fluids 6, 1820–1837.Google Scholar
Hosokawa, I. & Yamamoto, K. 1989 Fine structure of a directly simulated isotropic turbulence. J. Phys. Soc. Japan 58, 20–23.Google Scholar
Jaberi, F. A. & Givi, P. 1995 Inter-layer diffusion model of scalar mixing in homogeneous turbulence. Combust. Sci. Tech. 104, 249–272.Google Scholar
Jaberi, F. A., Miller, R. S. & Givi, P. 1995 Conditional statistics in turbulent scalar mixing and reaction. AIChE J., In press.Google Scholar
Janicka, J., Kolbe, W. & Kollmann, W. 1979 Closure of the transport equation for the probability density function of turbulent scalar field. J. Nonequil. Thermodyn. 4, 47–66.Google Scholar
Jayesh & Warhaft, Z. 1991 Probability distribution of a passive scalar in grid-generated turbulence. Phys. Rev. Lett. 67, 3503–3506.Google Scholar
Jayesh & Warhaft, Z. 1992 Probability distribution, conditional dissipation, and transport of passive temperature fluctuations in grid-generated turbulence. Phys. Fluids A 4, 2292–2307.Google Scholar
Jimenez, J., Wray, A. A., Saffman, P. G. & Rogallo, R. S. 1993 The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255, 65–90.Google Scholar
Kerr, R. M. 1983 High-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. NASA TM 84407.
Kerr, R. M. 1985 High-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence. J. Fluid Mech. 153, 31–58.Google Scholar
Kerr, R. M. 1990 Velocity, scalar and transfer spectra in numerical turbulence. J. Fluid Mech. 211, 309–332.Google Scholar
Kerstein, A. R. 1988 A linear eddy model of turbulent scalar transport and mixing. Combust. Sci. Tech. 60, 391–421.Google Scholar
Kerstein, A. R. 1989 Linear eddy modelling of turbulent transport. II: Applications to shear layer mixing. Combust. Flame 75, 397–413.Google Scholar
Kerstein, A. R. 1990 Linear eddy modelling of turbulent transport. Part 3. Mixing and differential molecular diffusion in round jets. J. Fluid. Mech. 216, 411–435.Google Scholar
Kerstein, A. R. 1991 Linear-eddy modelling of turbulent transport. Part 6. Microstructure of diffusive scalar mixing fields. J. Fluid Mech. 231, 361–394.Google Scholar
Kerstein, A. R. 1992 Linear-eddy modelling of turbulent transport. Part 7. Finite-rate chemistry and multi-stream mixing. J. Fluid Mech. 240, 289–313.Google Scholar
Kerstein, A. R., & McMurtry, P. A. 1994a Low wave number statistics of randomly advected passive scalars. Phys. Rev. E 50, 2057–2063.Google Scholar
Kerstein, A. R., & McMurtry, P. A. 1994b Mean-field theories of random advection. Phys. Rev. E 49, 474–482.Google Scholar
Kimura, Y. & Kraichnan, R. H. 1993 Statistics of an advected passive scalar. Phys. Fluids A 5, 2264–2277.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301–305.Google Scholar
Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85.Google Scholar
Kraichnan, R. H. 1989 Closures for probability distributions. Bull. Am. Phys. Soc. 34, 2298.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon
Lane, B. R., Mesquita, O. N., Meyers, S. R. & Gollub, J. P. 1993 Probability distributions and thermal transport in a turbulent grid flow. Phys. Fluids A 5, 2255–2263.Google Scholar
Leonard, A. D. & Hill, J. C. 1988 Direct numerical simulation of turbulent flows with chemical reaction. J. Sci. Comput. 3, 25–43.Google Scholar
Leonard, A. D. & Hill, J. C. 1991 Scalar dissipation and mixing in turbulent reacting flows. Phys. Fluids A 3, 1286–1299.Google Scholar
Leonard, A. D. & Hill, J. C. 1992 Mixing and chemical reaction in sheared and nonsheared homogeneous turbulence. Fluid Dyn. Res. 10, 273–297.Google Scholar
Libby, P. A. & Williams, F. A. (ed.) 1980 Turbulent Reacting Flows. Topics in Applied Physics, vol. 44. Springer.
Libby, P. A. & Williams, F. A. (ed.) 1994 Turbulent Reacting Flows. Academic.
Madnia, C. K., Frankel, S. H. & Givi, P. 1992 Reactant conversion in homogeneous turbulence: Mathematical modeling, computational validations and practical applications. Theoret. Comput. Fluid Dyn. 4, 79–93.Google Scholar
McMurtry, P. A., Gansauge, T. C., Kerstein, A. R. & Krueger, S. K. 1993a Linear eddy simulations of mixing in a homogeneous turbulent flow. Phys. Fluids A 5, 1023–1034.Google Scholar
McMurtry, P. A. & Givi, P. 1989 Direct numerical simulations of mixing and reaction in a nonpremixed homogeneous turbulent flow. Combust. Flame 77, 171–185.Google Scholar
McMurtry, P. A., Menon, S. & Kerstein, A. R. 1993b Linear eddy modeling of turbulent combustion. Energy & Fuels 7, 817–826.Google Scholar
Metais, O. & Lesieur, M. 1992 Spectral large-eddy simulation of isotropic and stably stratified turbulence. J. Fluid Mech. 239, 157–194.Google Scholar
Miller, R. S. 1995 Passive scalar, magnetic field, and solid particle transport in homogeneous turbulence PhD Thesis, Department of Mechanical and Aerospace Engineering, State University of New York at Buffalo, Buffalo, NY.
Miller, R. S., Frankel, S. H., Madnia, C. K. & Givi, P. 1993 Johnson-Edgeworth translation for probability modeling of binary scalar mixing in turbulent flows. Combust. Sci. & Tech. 91, 21–52.Google Scholar
Miller, R. S., Jaberi, F. A., Madnia, C. K. & Givi, P. 1995 The structure and small-scale intermittency of passive scalars in homogeneous turbulence. J. Sci. Comput. 10, 151–180.Google Scholar
Miyawaki, O., Tsujikawa, H. & Uraguchi, Y. 1974 Turbulent mixing in multi-nozzle injector tubular mixers. J. Chem. Engng Japan 7, 52–74.Google Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics, Vol. 2. MIT Press.
Nomura, K. K. & Elghobashi, S. E. 1992 Mixing characteristics of an inhomogeneous scalar in isotropic and homogeneous sheared turbulence. Phys. Fluids A 4, 606–625.Google Scholar
O'Brien, E. E. 1980 The probability density function (p.d.f.) approach to reacting turbulent flows. In Turbulent Reacting Flows ed. (P. A. Libby & F. A. Williams), pp. 185–218. Springer.
Obukhov, A. M. 1962 Some specific features of atmospheric turbulence. J. Fluid Mech. 13, 77–81.Google Scholar
Pope, S. B. 1979 The statistical theory of turbulent flames. Phil. Trans. R Soc. Lond. A 291, 529–568.Google Scholar
Pope, S. B. 1982 An improved turbulent mixing model. Combust. Sci. Tech. 28, 131–145.Google Scholar
Pope, S. B. 1985 p.d.f. methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11, 119–192.Google Scholar
Pope, S. B. 1990 Computations of turbulent combustion: Progress and challenges. In Proc. 23rd Symp. (Intl) on Combustion, pp. 591–612. The Combustion Institute, Pittsburgh, PA.
Pumir, A. 1994 A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient. Phys. Fluids 6, 2118–2132.Google Scholar
Pumir, A., Shraiman, B. & Siggia, E. D. 1991 Exponential tails and random advection. Phys. Rev. Lett. 3, 2838–2840.Google Scholar
Rogers, M. M., Mansour, N. N. & Reynolds, W. C. 1989 An algebraic model for the turbulent flux of a passive scalar. J. Fluid Mech. 203, 77–101.Google Scholar
Rogers, M. M., Moin, P. & Reynolds, W. C. 1986 The structure and modeling of the hydrodynamic and passive scalar fields in homogeneous turbulent shear flow. Department of Mechanical Engineering TF-25, Stanford University, Stanford, CA.
Ruetsch, G. R. & Maxey, M. R. 1991 Small-scale features of vorticity and passive scalar fields in homogeneous-isotropic turbulence. Phys. Fluids A 3, 1587–1597.Google Scholar
Ruetsch, G. R. & Maxey, M. R. 1992 The evolution of small-scale structures in homogeneous-isotropic turbulence. Phys. Fluids A 4, 2747–2760.Google Scholar
Sano, M., Wu, X. Z. & Libchaber, A. 1989 Turbulence in helium gas free convection. Phys. Rev. A 40, 6421–6430.Google Scholar
She, Z. S. 1990 Physical model of intermittency in turbulence: Near dissipation range non-Gaussian statistics. Phys. Rev. Lett. 66, 600–603.Google Scholar
She, Z. S., Jackson, E. & Orszag, S. A. 1991 Structure and dynamics of homogeneous turbulence: Models and simulations. Proc. R. Soc. Lond. A 434, 101–124.Google Scholar
She, Z. S. & Orszag, S. A. 1991 Physical model of intermittency in turbulence: Inertial-range non-Gaussian statistics. Phys. Rev. Lett. 66, 1701–1704.Google Scholar
Siggia, E. D. 1994 High Rayleigh number convection. Ann. Rev. Fluid Mech. 26, 137–168.Google Scholar
Sinai, Y. G. & Yakhot, V. 1989 Limiting probability distributions of a passive scalar in a random velocity field. Phys. Rev. Lett. 63, 1962–1964.Google Scholar
Solomon, T. H. 1990 Transport and boundary layers in Rayleigh Bernard convection PhD Thesis, Department of Physics, University of Pennsylvania, Philadelphia.
Solomon, T. H. & Gollub, J. P. 1991 Thermal boundary layers and heat flux in turbulent convection: The role of recirculating flows. Phys. Rev. A 43, 6683–6693.Google Scholar
Tanaka, M. & Kida, S. 1993 Characterization of vortex tubes and sheets. Phys. Fluids A 5, 2079–2082.Google Scholar
Tavoularis, S., & Corrsin, S. 1981a Experiments in nearly homogenous 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 homogenous turbulent shear flow with a uniform mean temperature gradient. Part 2. The fine structure. J. Fluid Mech. 104, 349–367.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press
Thoroddsen, S. T. & Van Atta, C. W. 1992 Exponential tails and skewness of density-gradient probability density functions in stably stratified turbulence. J. Fluid Mech. 244, 547–566.Google Scholar
Tong, C. & Warhaft, Z. 1994 On passive scalar derivative statistics in grid turbulence. Phys. Fluids 6, 2165–2176.Google Scholar
Toor, H. L. 1975 The non-premixed reaction: A + B → Products. In Turbulence in Mixing Operations ed. (R. S. Brodskey), pp. 123–166. Academic.
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. 1970 Structure functions of turbulence in the atmospheric boundary layer over the ocean. J. Fluid Mech. 44, 145–159.Google Scholar
Vincent, A. & Meneguzzi, M. 1991 The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–20.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 Bernard convection. Phys. Rev. Lett. 63, 1965–1967.Google Scholar
Yamamoto, K. & Kambe, T. 1991 Gaussian and near exponential probability distributions of turbulence obtained from a numerical simulation. Fluid Dyn. Res. 8, 65–72.Google Scholar
Zocchi, G., Moses, E. & Libchaber, A. 1991 Coherent structures in turbulent convection: An experimental study. Physica A 166, 387–407.Google Scholar