Spectral transfer and bispectra for turbulence with passive scalars

Abstract

We examine the statistical mechanisms by which energy and scalar variance are cascaded to small scales for isotropic, three-dimensional turbulence. Two avenues are explored: (i) the traditional transfer function (defined by the nonlinear cascade that gives the time rate of change of the energy spectrum), and (ii) the bispectrum (the elementary triple-point correlation, averaged over directions perpendicular to three co-linear observation points). Our tools are direct numerical simulations (DNS), and the statistical theory of turbulence, here in the form of the test field model (TFM) (Kraichnan 1971). Comparison of the results indicates a fairly good quantitative agreement between DNS and the TFM at large Prandtl numbers (Pr ≥ 0.25), but substantial disagreement at lower Pr, where the transfer to small scales becomes too small. This disparity we trace to the Markovian aspect of the TFM; the more fundamental direct interaction approximation (DIA) (Kraichnan 1959) compares more favourably to DNS as Pr → 0. For Pr ∼ 1, we compare DNS and TFM bispectra for velocity and scalar fields in both Fourier and physical space. The physical space representation of bispectra serves as a useful means of discriminating between velocity and scalar transfer.