A spectral study of differential diffusion of passive scalars in isotropic turbulence

Abstract

In a companion paper, Ulitsky & Collins (2000) applied the eddy-damped quasi-normal Markovian (EDQNM) turbulence theory to the mixing of two inert passive scalars with different diffusivities in stationary isotropic turbulence. Their paper showed that a rigorous application of the EDQNM approximation leads to a scalar covariance spectrum that violates the Cauchy–Schwartz inequality over a range of wavenumbers. The violation results from the improper functionality of the inverse diffusive time scales that arise from the Markovianization of the time evolution of the triple correlations. The modified inverse time scale they proposed eliminates this problem and allows meaningful predictions of the scalar covariance spectrum both with and without a uniform mean gradient.

This study uses the modified EDQNM model to investigate the spectral dynamics of differential diffusion. Consistent with recent DNS results by Yeung (1996), we observe that whereas spectral transfer is predominantly from low to high wavenumbers, spectral incoherence, being of molecular origin, originates at high wavenumbers and is transferred in the opposite direction by the advective terms. Quantitative comparisons between the EDQNM model and the DNS show good agreement. In addition, the model is shown to give excellent estimates for the dissipation coefficient defined by Yeung (1998).

We show that the EDQNM scalar covariance spectrum for two scalars with different molecular diffusivities can be approximated by the EDQNM autocorrelation spectrum for a scalar with molecular diffusivity equal to the arithmetic mean of the first two scalars. The result is exact for the case of an isotropic scalar and is shown to be a very good approximation for the scalar with a uniform mean gradient. We then exploit this relationship to derive a simple formula for the correlation coefficient of two differentially diffusing scalars as a function of their two Schmidt numbers and the turbulent Reynolds number. A comparison of the formula with the EDQNM model shows the model predicts the correct Reynolds number scaling and qualitatively predicts the dependence on the Schmidt numbers.

To investigate the degree of local versus non-local transfer of the scalar covariance spectrum, we divided the energy spectrum into three ranges corresponding to the energy-containing eddies, the inertial range, and the dissipation range. Then, by conditioning the scalar transfer on the energy contained within each of the three ranges, we have determined that the transfer process is dominated first by local interactions (local transfer) followed by non-local interactions leading to local transfer. Non-local interactions leading to non-local transfer are found to be significant at the higher wavenumbers. This result has important implications for defining simpler spectral models that potentially can be applied to more complex engineering flows.