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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Authors: | B J Legg M R Raupach |
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Institution: | (1) CSIRO Division of Environmental Mechanics, Canberra, Australia;(2) Present address: Physics Department, Rothamsted Experimental Station, Harpenden, Herts, UK |
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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
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 + t) = aw(t) + b
wE + (1 – a)T
L
(
wE
2)/ z, where w(t) is the vertical velocity at time t, 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(– t/T
L), and b = (1 – a
2)1/2. This equation can be used for inhomogeneous turbulence in which the mean wind speed,
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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Keywords: | |
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