Multi-Scale Gradient Expansion of the Turbulent Stress Tensor

Turbulent stress is the fundamental quantity in the filtered equation for large-scale velocity that reflects its interactions with small-scale velocity modes. We develop an expansion of the turbulent stress tensor into a double series of contributions from different scales of motion and different orders of space-derivatives of velocity, a Multi-Scale Gradient (MSG) expansion. We compare our method with a somewhat similar expansion —due to Yeo and Bedford, and to Leonard—that is based instead on defiltering. Our MSG expansion is proved to converge to the exact stress, as a consequence of the locality of cascade both in scale and in space. Simple estimates show, however, that the convergence rate may be slow for the expansion in spatial gradients of very small scales. Therefore, we develop an approximate expansion, based upon an assumption that similar or ‘coherent’ contributions to turbulent stress are obtained from disjoint subgrid regions. This Coherent-Subregions Approximation (CSA) yields an MSG expansion that can be proved to converge rapidly at all scales and is hopefully still reasonably accurate. As an important first application of our methods, we consider the cascades of energy and helicity in three-dimensional turbulence. To first order in velocity-gradients, the stress has three contributions: a tensile stress along principal directions of strain, a contractile stress along vortex lines, and a shear stress proportional to ‘skew-strain.’ While vortex-stretching plays the major role in energy cascade, there is a second, less scale-local contribution from ‘skew-strain’. For helicity cascade the situation is reversed, and it arises scale-locally from ‘skew-strain’ while the stress along vortex-lines gives a secondary, less scale-local contribution. These conclusions are illustrated with simple exact solutions of 3D Euler equations. In the first, energy cascade occurs by Taylor’s mechanism of stretching and spin-up of small-scale vortices due to large-scale strain. In the second, helicity cascade occurs by ‘twisting’ of small-scale vortex filaments due to a large-scale screw.