A conjectured hierarchy of length scales in a generalization of the Navier-Stokes-α equation for turbulent fluid flow
Eliot Fried
Department of Mechanical, Aerospace, and Structural Engineering
Washington University - St. Louis
We present a continuum-mechanical formulation and generalization of the Navier-Stokes-α theory based on a general framework for fluid-dynamical theories with gradient dependencies. Our flow equation involves two additional problem-dependent length scales α and β. The first of these scales enters the theory through the internal kinetic energy, per unit mass, α2|D|2, where D is the symmetric part of the gradient of the filtered velocity. The remaining scale is associated with a dissipative hyperstress which depends linearly on the gradient of the filtered vorticity. When α and β are equal, our flow equation reduces to the Navier-Stokes-α equation. In contrast to the original derivation of the Navier-Stokes-α equation, which relies on Lagrangian averaging, our formulation delivers boundary conditions. For a confined flow, our boundary conditions involve an additional length scale l characteristic of the eddies found near walls. Based on a comparison with direct numerical simulations for fully-developed turbulent flow in a rectangular channel of height 2h, we find that α/β ∼ Re0.470 and l/h ∼ Re-0.772, where Re is the Reynolds number. The first result, which arises as a consequence of identifying the internal kinetic energy with the turbulent kinetic energy, indicates that the choice α = β required to reduce our flow equation to the Navier-Stokes-α equation is likely to be problematic. The second result evinces the classical scaling relation η/L ∼ Re-3/4 for the ratio of the Kolmogorov microscale η to the integral length scale L. The numerical data also suggests that l ≤ β. We are therefore led to conjecture a tentative hierarchy, l ≤ β < α , involving the three length scales entering our theory.