Passive scalar in turbulent flows

The theory of mixing of a passive scalar concentration field subject to advection and diffusion in a high Reynolds number turbulent flow is based on the works of Obukhov (1949) and Corrsin (1951). Consider a statistically stationary state with a large-scale source of scalar fluctuations in the case when both Pe and Re are large. The 

A stationary state with constant average concentration is obtained when S is a source/sink term with zero mean. In this case the fluctuating component of the scalar field 0(x,t) = C(x, t) — (C) also satisfies the above equation and can be characterized by the variance (02). To obtain an equation for the variance we multiply the equation by 0 and integrate over the whole domain 

In analogy with Kolmogorov s assumption of the finite energy dissipation rate, Obukhov and Corrsin assumed that as the diffusivity is reduced to zero the gradients increase in such a way that the rate of dissipation of the scalar variance has a finite non-zero limit 

Thus analogously to the inertial scale of turbulence, the statistical properties of the scalar fluctuations in the inertial-convective range, i.e. in a range of scales below the forcing scale where both diffusion and viscosity are negligible, can only depend on the dissipation rate ee, the energy dissipation rate e, and on the length scale. Thus the only dimensionally correct form of the second order scalar structure function of the concentration fluctuations is 

This is consistent with our assumption that Id V only when u/D 1. The ratio of the diffusivity of momentum and of the diffusivity of concentration defines the Schmidt number Sc = u/D. This description also applies to temperature fluctuations in turbulent flows, when the temperature field can be approximated as a passive scalar. In that case the diffusion coefficient is replaced by the heat conduction coefficient n and the non-dimensional ratio Pr = u/n is known as the Prandtl number. 

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Warhaft, Z. (2000). Passive scalars in turbulent flows. Annual Reviews of Fluid Mechanics 32, 203-240. 

Figure 2.23 Schematic power spectrum of the passive scalar in turbulent flows in the case of large Schmidt number.

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