High Schmidt number scalars in turbulence: Structure functions and Lagrangian theory
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We demonstrate the existence of Batchelors viscous-convective subrange using direct numerical simulation (DNS) results to confirm the logarithmic dependence of the scalar structure function on the separation for the scalar field generated by stationary isotropic turbulence acting on a uniform mean scalar gradient. From these data we estimate the Batchelor constant B5. By integrating a piecewise continuous representation of the scalar variance spectrum we calculate the steady-state scalar variance as a function of Reynolds number and Schmidt number. Comparison with DNS results confirms the Re1 behavior predicted from the spectral integration, but with a coefficient about 60% too small. In the large Reynolds number limit the data give a value of 2.5 for the mechanical-to-scalar time scale ratio. The dependence of the data for the scalar variance on Schmidt number agrees very well with the spectral integration using the values of the Batchelor constant estimated from the structure function. We also carry out an exact Lagrangian analysis of the scalar variance and structure function, explicitly relating the Batchelor constant to the Lyapunov exponent for the separation of pairs of fluid particles within the turbulence dissipation subrange. Our results, particularly for the scalar variance, illustrate explicitly the singular nature of the zero diffusivity limit. For finite values of the Schmidt number and Reynolds number the viscous-convective subrange contribution to the variance can be significant even at moderate values of the Reynolds number.
