Kinematic Diffusion of Scalar Quantities in Turbulent Velocity Fields

From Kraichnan's direct interaction approximation the normal mode equations are set up for a scalar quantity diffusing kinematically under a turbulent velocity field which is statistically homogeneous and stationary. It is demonstrated: (i) that the mean scalar field responds only to the symmetric part of the velocity turbulence tensor; (ii) that the Kraichnan equation describing the normal mode behaviour is a singular nonlinear integral equation; (iii) that for velocity turbulence which is switched on and off infinitely rapidly the normal modes of the mean scalar field decay in time at a rate which is always greater than that obtaining in the absence of the turbulent velocity field. The motivation underlying these calculations is the problem of particle transport in turbulent astrophysical situations such as the interstellar medium. In such cases the effective Reynolds number is normally large compared with unity, so that expansion approximations for small Reynolds number are apparently not completely free of error.

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Global energy fluxes in turbulent channels with flow control

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.749 ◽

2018 ◽

Vol 857 ◽

pp. 345-373 ◽

Cited By ~ 7

Author(s):

Davide Gatti ◽

Andrea Cimarelli ◽

Yosuke Hasegawa ◽

Bettina Frohnapfel ◽

Maurizio Quadrio

Keyword(s):

Reynolds Number ◽

Shear Stress ◽

Energy Dissipation ◽

Velocity Field ◽

Flow Control ◽

Reynolds Shear Stress ◽

Power Input ◽

Energy Fluxes ◽

Mean Velocity ◽

The Mean

This paper addresses the integral energy fluxes in natural and controlled turbulent channel flows, where active skin-friction drag reduction techniques allow a more efficient use of the available power. We study whether the increased efficiency shows any general trend in how energy is dissipated by the mean velocity field (mean dissipation) and by the fluctuating velocity field (turbulent dissipation). Direct numerical simulations (DNS) of different control strategies are performed at constant power input (CPI), so that at statistical equilibrium, each flow (either uncontrolled or controlled by different means) has the same power input, hence the same global energy flux and, by definition, the same total energy dissipation rate. The simulations reveal that changes in mean and turbulent energy dissipation rates can be of either sign in a successfully controlled flow. A quantitative description of these changes is made possible by a new decomposition of the total dissipation, stemming from an extended Reynolds decomposition, where the mean velocity is split into a laminar component and a deviation from it. Thanks to the analytical expressions of the laminar quantities, exact relationships are derived that link the achieved flow rate increase and all energy fluxes in the flow system with two wall-normal integrals of the Reynolds shear stress and the Reynolds number. The dependence of the energy fluxes on the Reynolds number is elucidated with a simple model in which the control-dependent changes of the Reynolds shear stress are accounted for via a modification of the mean velocity profile. The physical meaning of the energy fluxes stemming from the new decomposition unveils their inter-relations and connection to flow control, so that a clear target for flow control can be identified.

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