Theory and Simulation of Passive Scalar Mixing in the Presence of a Mean Scalar Gradient
<p>The turbulent mixing of a passive scalar in the presence of a mean scalar gradient was investigated using theory and simulation. 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...
| Summary: | <p>The turbulent mixing of a passive scalar in the presence of a mean scalar gradient was investigated using theory and simulation. 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 use an argument of Batchelor, Howells, and Townsend (1959) to derive a new asymptotic result for the cospectrum in the inertial-diffusive range, with a -11/3 power law wavenumber dependence. A comparison is made with results from large-eddy simulation at low Schmidt number.</p>
<p>The sparse direct-interaction perturbation (SDIP) was used to calculate the cospectrum for a range of Schmidt numbers. The Kolmogorov type 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. The stretched-spiral vortex model was used to calculate the cospectrum, and asymptotic expressions were found for the contribution to the cospectrum from the axial velocity in the vortex structures. Results are reported for the cospectrum from a direct numerical simulation at a Taylor Reynolds number of 265, and a comparison is made of results for the cospectrum from the SDIP, the stretched-spiral vortex model, simulation, and experiment.</p>
<p>The stretched-spiral vortex model was also used to derive expressions for the modal time correlation functions of the velocity and scalar fields. These expressions were evaluated numerically and asymptotically. Winding by the vortex core is shown to lead to an inertial timescale, and movement of the vortex structures by the large scale flow leads to a sweeping timescale. The velocity and scalar modal time correlation functions were calculated in the direct numerical simulation. They coincide for large enough wavenumber, and are found to collapse to universal forms when a sweeping timescale is used.</p> |
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