Interaction between different scales of turbulence over short times

Abstract

The distortion by large-scale random motions of small-scale turbulence is investigated by examining a model problem. The changes in energy spectra, velocity and vorticity moments, and anisotropy of small-scale turbulence are calculated over timescales short compared with the timescale of small-scale turbulence by applying rapid distortion theory with a random distortion matrix for different initial conditions: irrotational or rotational, and isotropic or anisotropic large-scale turbulence with or without mean strain, and isotropic or anisotropic small-scale turbulence. We have obtained the following results: (1) Irrotational random strains broaden the small-scale energy spectrum and transfer energy to higher wavenumbers. (2) The rotational part of the large-scale strain is important for reducing anisotropy of turbulence rather than transferring energy to higher wavenumbers. (3) Anisotropy in small-scale turbulence is reduced by large-scale isotropic turbulence. The reduction of anisotropy of the velocity field depends on the initial value of the velocity anisotropy tensor of the small-scale velocity field u i defined by $\overline{u_iu_j}/\overline{u_lu_l}-\frac{1}{3}\delta_{ij}$ , and also on the anisotropy of the distribution of the energy spectrum in wavenumber space. The reduction in anisotropy of the vorticity field ω i depends only on the vorticity anisotropy tensor. (4) The pressure-strain correlation is calculated for the change in Reynolds stress of the anisotropic small-scale turbulence. The correlation is proportional to time and depends on the difference between the velocity and wavenumber anisotropy tensors. These results (which are exact for small time) differ significantly from current turbulence models. (5) The effect of large-scale anisotropic turbulence on isotropic small-scale turbulence is calculated in general. Results are given for the case of axisymmetric large scales and are compared with the observed behaviour of small-scale turbulence near interfaces. (6) When a mean irrotational straining motion is applied to turbulence with distinct large-scale and small-scale components in their velocity field, the large-scale irrotational motions combine with the mean straining to increase further the anisotropy of the vorticity of the small scales, but the large-scale rotational motions reduce the small-scale anisotropy. For isotropic straining motion, the latter is weaker than the former. After the mean distortion ceases, both kinds of large-scale straining tend to reduce the anisotropy. This also has implications for modelling the rate of reduction of anisotropy.