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A statistical model of fluid-element motions and vertical diffusion in a homogeneous stratified turbulent flow

Published online by Cambridge University Press:  20 April 2006

H. J. Pearson ,
J. S. Puttock  and
J. C. R. Hunt
H. J. Pearson
Affiliation:
Department of Applied Mathematics and Theoretical Physics. University of Cambridge, Silver Street, Cambridge CB3 9EW Present address: Smith Associates Consulting System Engineers Ltd, 45–47 High Street, Cobham, Surrey KT11 3DP, U.K.
J. S. Puttock
Affiliation:
Department of Applied Mathematics and Theoretical Physics. University of Cambridge, Silver Street, Cambridge CB3 9EW Present address: Shell Research Ltd, Thornton Research Centre, P.O. Box 1, Chester CH1 3SH, U.K.
J. C. R. Hunt
Affiliation:
Department of Applied Mathematics and Theoretical Physics. University of Cambridge, Silver Street, Cambridge CB3 9EW
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Abstract

The vertical displacements Z(t) of fluid elements passing through a source z = 0 at t = 0 in a horizontal mean flow with stably stratified statistically stationary turbulence (with buoyancy frequency N and velocity time-scale T), under the action of random pressure gradients and damping by internal wave motions, are investigated by a model Langevin-like equation, and by a general Lagrangian analysis of the displacements, of the density flux and of the energy of fluid elements. Solutions for the mean-square displacement $\overline{Z^2}(t)$, the mean-square velocity $\overline{w^2}$, and the autocor-relation of the velocity are calculated in terms of the spectrum [Fcy ](s) of the pressure gradient. We use model equations for the momentum of fluid elements and for the exchange of density fluctuations between fluid elements, taking the elements’ diffusion timescale to by γ−1 times the buoyancy timescale N−1, where γ is a measurable parameter.

In the case of moderate-to-strong stable stratification (i.e. NT [gsim ] 1), we find the following.

  1. (i) When there is no change of the fluid elements’ density (γ = 0), the mean-square displacement $\overline{Z^2}$ of marked particles ceases to grow when t [gsim ] N−1, and its asymptotic value is proportional to $\overline{w^2}/N^2$, where the constant of proportionality, $\overline{\zeta^2}(\infty)$, is O(1) and a decreasing function of (NT)−1. This result is also shown to be a general consequence of the finite potential and kinetic energy in the stationary turbulence.

  2. (ii) If there is a small diffusive interchange of density between fluid elements (i.e. γ [Lt ] 1), the marked particles’ mean-square displacement has a slow linear growth (i.e. $\overline{Z^2}\sim\overline{w^2}/N^2(1+O(\gamma^2)tN) $).

  3. (iii) Such molecular processes must also dilute the initial concentration of contaminants (e.g. dye or smoke) in those fluid elements that diffuse above the limit in (i).

  4. (iv) The mean-square fluctuation of density is proportional to the product of the asymptotic mean-square displacement of marked particles and the square of the mean density gradient $(-{\cal G})\;({\rm i.e.}\overline{\rho^{\prime 2}}={\cal G}^2\overline{Z^2}=\frac{1}{2}\overline{\zeta^2}(\infty ,\gamma = 0){\cal G}^2\overline{w^2}/N^2)$.

  5. (v) The flux Fρ of density in a turbulent flow can be expressed exactly as the sum of two terms, the first $\frac{1}{2}(d\overline{Z^2}/dt)\,{\cal G}$, being caused by the growth of the displacements of fluid elements, and the second $\overline{\dot{Z}(\rho + \rho^{\prime})}$ being caused by the mixing between fluid elements. In stable flows, it is shown that the second element is dominant, and $F_{\rho}\sim\gamma\overline{w^2}N\rho_og $ while the first is smaller by O(γ).

Previous laboratory and field measurements of $\overline{w^2}, \overline{Z^2}, \overline{\rho^{\prime 2}} $ and Rw(t) are discussed in some detail and shown to be consistent with this model.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 129 , April 1983 , pp. 219 - 249
Copyright
© 1983 Cambridge University Press

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