Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-09T19:16:20.279Z Has data issue: false hasContentIssue false
  • English
  • Français

A stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence, and its application to the problem of concentration variance

Published online by Cambridge University Press:  26 April 2006

D. J. Thomson
D. J. Thomson
Affiliation:
Meteorological Office, London Road, Bracknell, Berkshire, RG12 2SZ
Get access Rights & Permissions [Opens in a new window]

Abstract

A new stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence is proposed. The model is three-dimensional and its formulation takes account of recent improvements in the understanding of one-particle models. In particular the model is designed so that if the particle pairs are initially well mixed in the fluid, they will remain so. In contrast to previous models, the new model leads to a prediction for the particle separation probability density function which is in qualitative agreement with inertial subrange theory. The values of concentration variance from the model show encouraging agreement with experimental data. The model results suggest that, at large times, the intensity of concentration fluctuations (i.e. standard deviation of concentration divided by mean concentration) tends to zero in stationary conditions and to a constant in decaying turbulence.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 210 , January 1990 , pp. 113 - 153
Copyright
© 1990 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

Anand, M. S. & Pope, S. B. 1985 Diffusion behind a line source in grid turbulence. Turbulent Shear Flows, vol. 4. Springer.
Batchelor, G. K. 1952 Diffusion in a field of homogeneous turbulence. II. The relative motion of particles. Proc. Camb. Phil. Soc. 48, 345–362.Google Scholar
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Batchelor, G. K. 1959 Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5, 113–133.Google Scholar
Batchelor, G. K., Howells, I. D. & Townsend, A. A. 1959 Small-scale variation of convected quantities like temperature in turbulent fluid. Part 2. The case of large conductivity. J. Fluid Mech. 5, 134–139.Google Scholar
Batchelor, G. K. & Townsend, A. A. 1956 Turbulent diffusion. In Surveys in Mechanics (ed. G. K. Batchelor & R. M. Davies), pp. 352–399. Cambridge University Press.
De Baas, A. F., Van Dop, H. & Nieuwstadt, F. T. M. 1986 An application of the Langevin equation for inhomogeneous conditions to dispersion in a convective boundary layer. Q. J. R. Met. Soc. 112, 165–180.Google Scholar
Durbin, P. A. 1980 A stochastic model of two-particle dispersion and concentration fluctuations in homogeneous turbulence. J. Fluid Mech. 100, 279–302.Google Scholar
Durbin, P. A. 1982 Analysis of the decay of temperature fluctuations in isotropic turbulence. Phys. Fluids 25, 1328–1332.Google Scholar
Egbert, G. D. & Baker, M. B. 1984 Comments on paper ‘The effect of Gaussian particle-pair distribution functions in the statistical theory of concentration fluctuations in homogeneous turbulence’ by B. L. Sawford (Q. J. April 1983, 109, 339-353). Q. J. R. Met. Soc. 110, 1195–1199.Google Scholar
Fackrell, J. E. & Robins, A. G. 1982 Concentration fluctuations and fluxes in plumes from point sources in a turbulent boundary layer. J. Fluid Mech. 117, 1–26.Google Scholar
Gifford, F. A. 1982 Horizontal diffusion in the atmosphere: A Lagrangian dynamical theory. Atmos. Environ. 16, 505–512.Google Scholar
Hanna, S. R. 1981 Lagrangian and Eulerian time-scale relations in the daytime boundary layer. J. Appl. Met. 20, 242–249.Google Scholar
Haworth, D. C. & Pope, S. B. 1986 A generalised Langevin model for turbulent flows. Phys. Fluids 29, 387–405.Google Scholar
Lamb, R. G. 1981 A scheme for simulating particle pair motions in turbulent fluid. J. Comp. Phys. 39, 329–346.Google Scholar
Lee, J. T. & Stone, G. L. 1983 The use of Eulerian initial conditions in a Lagrangian model of turbulent diffusion. Atmos. Environ. 17, 2477–2481.Google Scholar
Lesieur, M. 1987 Turbulence in fluids. Dordrecht: Martinus Nijhoff.
Ley, A. J. & Thomson, D. J. 1983 A random walk model of dispersion in the diabatic surface layer. Q. J. R. Met. Soc. 109, 867–880.Google Scholar
Lin, C. C. & Reid, W. H. 1963 Turbulent flow. Theoretical aspects. Handbuch der Physik, vol. 8/2. Springer.
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics, vol. 1. MIT Press, Cambridge, Mass.
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics, vol. 2. MIT Press, Cambridge, Mass.
Newman, G. R., Launder, B. E. & Lumley, J. L. 1981 Modelling the behaviour of homogeneous scalar turbulence. J. Fluid Mech. 111, 217–232.Google Scholar
Novikov, E. A. 1963 Random force method in turbulence theory. Sov. Phys., J. Exp. Theor. Phys. 17, 1449–1454.Google Scholar
Obukhov, A. M. 1959 Description of turbulence in terms of Lagrangian variables. Adv. Geophys. 6, 113–116.Google Scholar
Pasquill, F. & Smith, F. B. 1983 Atmospheric Diffusion, 3rd edn. J. Wiley & Sons.
Pope, S. B. 1985 Pdf methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11, 119–192.Google Scholar
Pope, S. B. 1987 Consistency conditions for random walk models of turbulent dispersion. Phys. Fluids 30, 2374–2379.Google Scholar
Saffman, P. G. 1960 On the effect of the molecular diffusivity in turbulent diffusion. J. Fluid Mech. 8, 273–283.Google Scholar
Sawford, B. L. 1982 Lagrangian Monte Carlo simulation of the turbulent motion of a pair of particles. Q. J. R. Met. Soc. 108, 207–213.Google Scholar
Sawford, B. L. 1983 The effect of Gaussian particle-pair distribution functions in the statistical theory of concentration fluctuations in homogeneous turbulence. Q. J. R. Met. Soc. 109, 339–353.Google Scholar
Sawford, B. L. 1984 The basis for, and some limitations of, the Langevin equation in atmospheric relative dispersion modelling. Atmos. Environ. 18, 2405–2411.Google Scholar
Sawford, B. L. 1985 Lagrangian statistical simulation of concentration mean and fluctuation fields. J. Clim. Appl. Met. 24, 1152–1166.Google Scholar
Sawford, B. L. & Hunt, J. C. R. 1986 Effects of turbulence structure, molecular diffusion and source size on scalar fluctuations in homogeneous turbulence. J. Fluid Mech. 165, 373–400.Google Scholar
Schuss, Z. 1980 Theory and applications of stochastic differential equations. J. Wiley & Sons.
Stapountzis, H., Sawford, B. L., Hunt, J. C. R. & Britter, R. E. 1986 Structure of the temperature field downwind of a line source in grid turbulence. J. Fluid Mech. 165, 401–424.Google Scholar
Sykes, R. I., Lewellen, W. S. & Parker, S. F. 1984 A turbulent-transport model for concentration fluctuations and fluxes. J. Fluid Mech. 139, 193–218.Google Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. Lond. Math. Soc. (2), 20, 196–211.Google Scholar
Thomson, D. J. 1986a A random walk model of dispersion in turbulent flows and its application to dispersion in a valley. Q. J. R. Met. Soc. 112, 511–530.Google Scholar
Thomson, D. J. 1986b On the relative dispersion of two particles in homogeneous stationary turbulence and the implications for the size of concentration fluctuations at large times. Q. J. R. Met. Soc. 112, 890–894.Google Scholar
Thomson, D. J. 1987 Criteria for the selection of stochastic models of particle trajectories in turbulent flows. J. Fluid Mech. 180, 529–556.Google Scholar
Thomson, D. J. 1988 Random walk models of turbulent diffusion. PhD thesis, Brunel University.
Townsend, A. A. 1954 The diffusion behind a line source in homogeneous turbulence. Proc. R. Soc. A224, 487–512.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.
Uberoi, M. S. & Corrsin, S. 1952 Diffusion of heat from a line source in isotropic turbulence. National Advisory Committee for Aeronautics, Tech. Note 2710.
Van Dop, H., Nieuwstadt, F. T. M. & Hunt, J. C. R. 1985 Random walk models for particle displacements in inhomogeneous unsteady turbulent flows. Phys. Fluids 28, 1639–1653.Google Scholar
Warhaft, Z. 1984 The interference of thermal fields from line sources in grid turbulence. J. Fluid Mech. 144, 363–387.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