Phenomenological model for turbulent mixing

4 Because it implies that (1.28) will be purely convective, this range of scalar wavenumbers is called the convective 

5 In the discussion that follows, we will assume that 1 Sc so that a high Reynolds number suffices to imply the existence of an inertial range for the turbulence and a convective range for the scalar. 

In a fully developed turbulent flow, the rate at which the size of a scalar eddy of length l,P decreases depends on its size relative to the turbulence integral scale Lu and the Kolmogorov scale rj. For scalar eddies in the inertial sub-range (rj l Lu), the scalar mixing rate can be approximated by the inverse of the spectral transfer time scale defined in (2.68), p. 42 8 

A Lagrangian description of the velocity field can be used to find the location X(f) of the fluid element at time 0 t that started at X(0). In the Lagrangian description, (3.3) implies that the scalar field associated with the fluid element will remain unchanged, i.e., p(X(t), t) = / (X(0), 0). 

equivalently, the turbulence field will change the scalar energy spectrum. 

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