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Markov-chain simulation of particle dispersion in inhomogeneous flows: The mean drift velocity induced by a gradient in Eulerian velocity variance

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Abstract

The Langevin equation is used to derive the Markov equation for the vertical velocity of a fluid particle moving in turbulent flow. It is shown that if the Eulerian velocity variance Σ wE is not constant with height, there is an associated vertical pressure gradient which appears as a force-like term in the Markov equation. The correct form of the Markov equation is: w(t + δt) = aw(t) + bΣ wEζ + (1 − a)T L ∂(Σ wE 2)/∂z, where w(t) is the vertical velocity at time t, ζ a random number from a Gaussian distribution with zero mean and unit variance, T L the Lagrangian integral time scale for vertical velocity, a = exp(−δt/T L), and b = (1 − a 2)1/2. This equation can be used for inhomogeneous turbulence in which the mean wind speed, Σ wE and T L vary with height. A two-dimensional numerical simulation shows that when this equation is used, an initially uniform distribution of tracer remains uniform.

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References

  • Arnold, L.: 1974, ‘Stochastic Differential Equations: Theory and Applications’, Wiley-Interscience, New York.

    Google Scholar 

  • Bradley, E. F., Antonia, R. A., and Chambers, A. J.: 1981, ‘Turbulence Reynolds Number and the Turbulent Kinetic Energy Balance in the Atmospheric Surface Layer’, Boundary-Layer Meteorol. 21, 183–197.

    Google Scholar 

  • Counihan, J.: 1975, ‘Adiabatic Atmospheric Boundary Layers: A Review and Analysis of Data from the Period 1880–1972’, Atmos. Environ. 9, 871–905.

    Google Scholar 

  • Csanady, G. T.: 1973, Turbulent Diffusion in the Atmosphere, D. Reidel Publ. Co., Dordrecht, Holland.

    Google Scholar 

  • Durbin, P. A.: 1980, ‘A Random Flight Model of Inhomogeneous Turbulent Dispersion’, Phys. Fluids 23, 2151–2153.

    Google Scholar 

  • Hall, C. D.: 1975, ‘The Simulation of Particle Motion in the Atmosphere by a Numerical Random-Walk Model’, Quart. J. Roy. Meteorol. Soc. 101, 235–244.

    Google Scholar 

  • Hinze, J. O.: 1975, Turbulence, 2nd ed., McGraw-Hill, New York.

    Google Scholar 

  • Hunt, J. C. R. and Weber, A. H.: 1979, ‘A Lagrangian Statistical Analysis of Diffusion from a Ground-Level Source in a Turbulent Boundary Layer’, Quart. J. Roy. Meteorol. Soc. 105, 423–443.

    Google Scholar 

  • Legg, B. J.: 1982, ‘Turbulent Dispersion from an Elevated Line Source: Markov-chain Simulations of Concentration and Flux Profiles’, Quart. J. Roy. Meteorol. Soc. (submitted).

  • Monin, A. S. and Yaglom, A. M.: 1971, Statistical Fluid Mechanics: Mechanics of Turbulence, Volume 1, The MIT Press, Cambridge, Massachusetts.

    Google Scholar 

  • Raupach, M. R. and Shaw, R. H.: 1982, ‘Averaging Procedures for Turbulent Flow in Canopies’, Boundary-Layer Meteorol. 22, 79–90.

    Google Scholar 

  • Raupach, M. R. and Thom, A. S.: 1981, ‘Turbulence in and Above Plant Canopies’, Ann. Rev. Fluid Mech. 13, 97–129.

    Google Scholar 

  • Reid, J. R.: 1979, ‘Markov-chain Simulations of Vertical Dispersion in the Neutral Surface Layer for Surface and Elevated Releases’, Boundary-Layer Meteorol. 16, 3–22.

    Google Scholar 

  • Tennekes, H. and Lumley, J. L.: 1972, A First Course in Turbulence, The MIT Press, Cambridge, Massachusetts.

    Google Scholar 

  • Thompson, R.: 1971, ‘Numeric Calculation of Turbulent Diffusion’, Quart. J. Roy. Meteorol. Soc. 97, 93–98.

    Google Scholar 

  • Wang, M. C. and Uhlenbeck, G. E.: 1945, ‘On the Theory of Brownian Motion II’, Rev. Mod. Phys. 17, 323–341.

    Google Scholar 

  • Wilson, J. D.: 1980, ‘Turbulence Measurements in a Corn Canopy and Numerical Simulation of Particle Trajectories in Inhomogeneous Turbulence’, Ph.D. Thesis, University of Guelph.

  • Wilson, N. R. and Shaw, R. H.: 1977, ‘A Higher-Order Closure Model for Canopy Flow’, J. Appl. Meteorol. 16, 1198–1205.

    Google Scholar 

  • Wilson, J. D., Thurtell, G. W., and Kidd, G. E.: 1981a, ‘Numerical Simulation of Particle Trajectories in Inhomogeneous Turbulence, II: Systems with Variable Turbulent Velocity Scale’, Boundary-Layer Meteorol. 21, 423–441.

    Google Scholar 

  • Wilson, J. D., Thurtell, G. W., and Kidd, G. E.: 1981b, ‘Numerical Simulation of Particle Trajectories in Inhomogeneous Turbulence, III: Comparison of Predictions with Experimental Data for the Atmospheric Surface Layer’, Boundary-Layer Meteorol. 21, 443–463.

    Google Scholar 

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Author notes

  1. B. J. Legg

    Present address: Physics Department, Rothamsted Experimental Station, Harpenden, Herts, UK

Authors and Affiliations

  1. CSIRO Division of Environmental Mechanics, Canberra, Australia

    B. J. Legg & M. R. Raupach

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  1. B. J. Legg
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  2. M. R. Raupach
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Legg, B.J., Raupach, M.R. Markov-chain simulation of particle dispersion in inhomogeneous flows: The mean drift velocity induced by a gradient in Eulerian velocity variance. Boundary-Layer Meteorol 24, 3–13 (1982). https://doi.org/10.1007/BF00121796

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  • DOI: https://doi.org/10.1007/BF00121796

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