Limiting probability distributions of a passive scalar in a random velocity field

The stretching of line elements, surface elements and wave vectors by a random, isotropic, solenoidal velocity field in D dimensions is studied. The rates of growth of line elements and (D – 1)-dimensional surface elements are found to be equal if the statistics are invariant to velocity reversal. The analysis is applied to convection of a sparse distribution of sheets of passive scalar in a random straining field whose correlation scale is large compared with the sheet size. This is Batchelor's (1959) κ−1 spectral regime. Some exact analytical solutions are found when the velocity field varies rapidly in time. These include the dissipation spectrum and a joint probability distribution that describes the simultaneous effect of Stretching and molecular diffusivity κ on the amplitude profile of a sheet. The latter leads to probability distributions of the scalar field and its space derivatives. For a growing κ−1 range at zero κ, these derivatives have essentially lognormal statistics. In the steady-state κ−1 regime at κ > 0, intermittencies measured by moment ratios are much smaller than for lognormal statistics, and they increase less rapidly with the order of the derivative than in the κ = 0 case. The κ > 0 distributions have singularities a t zero amplitude, due to a background of highly diffused sheets. The results do not depend strongly on D. But as D → ∞, temporal fluctuations in the stretching rates become negligible and Batchelor's (1959) constant-strain dissipation spectrum is recovered.

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Dissipation fluctuations of a passive scalar advected by a random velocity field

Physical Review E ◽

10.1103/physreve.54.2610 ◽

1996 ◽

Vol 54 (3) ◽

pp. 2610-2615 ◽

Cited By ~ 2

Author(s):

Victor Yakhot

Keyword(s):

Velocity Field ◽

Passive Scalar ◽

Random Velocity

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Passive scalar advected by a very irregular random velocity field

Physical Review E ◽

10.1103/physreve.56.2279 ◽

1997 ◽

Vol 56 (2) ◽

pp. 2279-2281 ◽

Cited By ~ 3

Author(s):

D. Gutman

Keyword(s):

Velocity Field ◽

Passive Scalar ◽

Random Velocity

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THEORY OF RANDOM ADVECTION IN TWO DIMENSIONS

International Journal of Modern Physics B ◽

10.1142/s0217979296001033 ◽

1996 ◽

Vol 10 (18n19) ◽

pp. 2273-2309 ◽

Cited By ~ 4

Author(s):

M. CHERTKOV ◽

G. FALKOVICH ◽

I. KOLOKOLOV ◽

V. LEBEDEV

Keyword(s):

Velocity Field ◽

Probability Distribution ◽

Probability Distributions ◽

Classical Problem ◽

Passive Scalar ◽

Two Dimensions ◽

Change Of Variables ◽

The Matrix ◽

Two Dimensional Flow ◽

Non Gaussian

The steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described at scales less than the correlation length of the flow and larger than the diffusion scale. The probability distribution of the scalar is expressed via the probability distribution of the line stretching rate. The description of the line stretching can be reduced to the classical problem of studying the product of many matrices with a unit determinant. We found a change of variables which allows one to map the matrix problem into a scalar one and to prove thus a central limit theorem for the statistics of the stretching rate. The proof is valid for any finite correlation time of the velocity field. Whatever be the statistics of the velocity field, the statistics of the passive scalar in the inertial interval of scales is shown to approach Gaussianity as one increases the Peclet number Pe (the ratio of the pumping scale to the diffusion one). The first n < ln (Pe) simultaneous correlation functions are expressed via the flux of the squared scalar and only one unknown factor depending on the velocity field: the mean stretching rate. That factor can be calculated analytically for the limiting cases. The non-Gaussian tails of the probability distributions at finite Pe are found to be exponential.

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