2005/12/10 by Gregory L. Eyink, GREGORY L. EYINK · 2 citations
Engineering · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Fluid Dynamics and Turbulent Flows #Fluid dynamics and aerodynamics studies #nlin.CD
paper · pdf · doi:10.1017/s0022112005007895
31 pages, 3 figures
arxiv created 2005/12/10 · openalex publication_date 2006/02/08 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 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 three-dimensional Euler equations. In the first, energy cascade occurs by Taylor's mechanism of stretching and spin-up of small-scale vortices owing to large-scale strain. In the second, helicity cascade occurs by ‘twisting’ of small-scale vortex filaments owing to a large-scale screw.