Title: A conjectured hierarchy of length scales in a generalization of the Navier–Stokes- equation for turbulent fluid flow Speaker: Professor Eliot Fried, McGill University, Montreal Canada Date/Time: Monday, January 26, 2009 at 1:30 – 2:30 pm Location: CSRI Building/Room 90 (Sandia NM) Brief Abstract: The direct numerical simulation (DNS) of high Reynolds number turbulence provides a formidible computational challenge, even with access to state-of-the-art supercomputers. For this reason, there remains a strong interest in alternative methods, based on Reynolds averaging and filtering, that resolve only large-scale motions while modeling small-scale motions. Although they reduce computational costs, the additional dissipation associated with such methods often lead to artificially sluggish flows. An alternative method that reduces the severity of this problem is provided by simulations based on the Navier–Stokes-a(NS-a) equation, which arises from Lagrangian averaging. Aside from the density r and the kinematic viscosity n of the fluid, the NS-a equation involves an additional parameter a > 0 carrying dimensions of length. In the context of Lagrangian averaging, a is the statistical correlation length of the excursions taken by a fluid particle away from its phase-averaged trajectory. More intuitively, a can be viewed as the characteristic length of the smallest eddies that the model is capable of resolving. We present a continuum-mechanical formulation leading to a generalization of the NS-a equation. That generalization involves two additional length scales. The first of these scales, a, enters the theory through the specific internal kinetic energy a2|D|2, with D being the symmetric part of the gradient of the filtered velocity. The remaining scale, b, is associated with a dissipative hyperstress which depends linearly on the gradient of the filtered vorticity. Setting b equal to a reduces our flow equation to the NS-a equation. In contrast to Lagrangian averaging, our formulation delivers boundary conditions. For confined flows, these involve an additional length scale l characteristic of the eddies found near walls. A comparison with DNS results for fully-developed turbulent flow, with Reynolds number Re, in a rectangular channel of height 2h yields a/b~ Re-0.470 and l/h ~ Re−0.772. The first result, which arises as a consequence of identifying the internal kinetic energy with the turbulent kinetic energy, suggests that the choice a = b leading to the NS-a equation might be problematic. The second result resembles the classical scaling relation h/L ~ Re−3/4 for the ratio of the Kolmogorov microscale h to the integral length scale L. The data also suggests that l£ b. These results lead to a tentative hierarchy, l£ b < a, involving the three length scales entering our theory. Studies of Lundgren’s strained spiral-vortex model and simulations of periodic turbulence support the inequality b < a. CSRI POC: Richard Lehoucq, (505) 845-8929 |