On the Lundgren–Townsend model of turbulent fine scales

Abstract

The strained-spiral vortex model of turbulent fines scales given by Lundgren [Phys. Fluids 25, 2193 (1982)] is used to calculate vorticity and velocity-derivative moments for homogeneous isotropic turbulence. A specific form of the relaxing spiral vortex is proposed modeled by a rolling-up vortex layer embedded in a background containing opposite signed vorticity and with zero total circulation at infinity. The numerical values of two dimensionless groups are fixed in order to give a Kolmogorov constant and skewness which are within the range of experiment. This gives the result that the ratio of the ensemble average hyperskewness S2p + 1[equivalent] ([partial-derivative]u/[partial-derivative]x)2p + 1/[([partial-derivative]u/[partial-derivative]x)2](2p + 1)/2 to the hyperflatness F2p[equivalent]([partial-derivative]u/[partial-derivative]x)2p/[([partial-derivative]u/[partial-derivative]x)2] p, p=2,3,..., is constant independent of Taylor–Reynolds number Rlambda, as is the ratio of the 2pth moment of one component of the vorticity Omega2p[equivalent]omega^2p_x/(omega²_x)p to F2p. A cutoff in a relevant time integration is then used to eliminate vortex-sheet-induced divergences in the integrals corresponding to omega^2p_x, p=2,3,..., and an assumption is made that the lateral scale of the spiral vortex in the model is the geometric mean of the Taylor and the Kolmogorov microscales. This gives Omega2p=Omega-hat 2pR _lambda ^{p/2 - 3/4}, F2p=F-hat 2pR _lambda ^{p/2 - 3/4} and S2p + 1=S-hat 2p + 1R _lambda ^{p/2 - 3/4}, p=2,3,..., with explicit calculation of the numbers Omega-hat 2p, F-hat 2p, and S-hat 2p + 1. The results of the model are compared with experimental compilation of Van Atta and Antonia [Phys. Fluids 23, 252 (1980)] for F4 and with the isotropic turbulence calculations of Kerr [J. Fluid Mech. 153, 31 (1985)] and of Vincent and Meneguzzi [J. Fluid Mech. 225, 1 (1991)].

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