Effect of Schmidt number on the velocity-scalar cospectrum in isotropic turbulence with a mean scalar gradient

We consider transport of a passive scalar by an isotropic turbulent velocity field in the presence of a mean scalar gradient. The velocity-scalar cospectrum measures the distribution of the mean scalar flux across scales. An inequality is shown to bound the magnitude of the cospectrum in terms of the shell-summed energy and scalar spectra. At high Schmidt number, this bound limits the possible contribution of the sub-Kolmogorov scales to the scalar flux. At low Schmidt number, we derive an asymptotic result for the cospectrum in the inertial-diffusive range, with a -11/3 power law wavenumber dependence, and a comparison is made with results from large-eddy simulation. The sparse direct-interaction perturbation (SDIP) is used to calculate the cospectrum for a range of Schmidt numbers. The Lumley scaling result is recovered in the inertial-convective range and the constant of proportionality was calculated. At high Schmidt numbers, the cospectrum is found to decay exponentially in the viscous-convective range, and at low Schmidt numbers, the -11/3 power law is observed in the inertial-diffusive range. Results are reported for the cospectrum from a direct numerical simulation at a Taylor Reynolds number of 265, and a comparison is made at Schmidt number order unity between theory, simulation and experiment.