Turbulence Batchelor spectrum

For mixtures with 5c 1 the classical turbulence theory for scalars fields predicts that the scalar spectrum has a shape similar to for k < kx- For kx < k < kx, the so-called Batchelor spectrum [4] is found for which Ec k) oc k, and for k > kx, the scalar spectrum falls rapidly towards zero due to the effect of scalar dissipation by molecular diffusion, as illustrated in Fig 7.9. The Batchelor [4] wave number is of the order kx 27r(e/j/D ) /. Molecular... 

Since b > 0 the power spectrum is steeper than k l as a consequence of the chemical decay, as has been verified numerically by Nam et al. (1999). Just like the Batchelor spectrum, the spectral form (6.21) is expected to hold for any smooth velocity field that generates chaotic advection. Thus, it remains valid even in the absence of an inertial range, as is the case for unsteady laminar flows or in the enstrophy cascade range of two-dimensional turbulence, which is relevant for geophysical flows. 

G.K. Batchelor. Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids, 12 233, 1969. 

Figure 2-13 Spectrum of velocity [E(n)] and temperature or concentration [r(n)] fluctuation wavenumbers (m ) in the equilibrium range of homogeneous isotropic turbulence for the case of large Sc or Pr (modified from Batchelor, 1959). In the Batchelor scale, k is either the thermal diffusivity (k/pCp) or the molecular diffusivity (Dab)-...

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