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≡ (∂u/∂x)2p+1/[(∂u/∂x)2](2p+1)/2 to the hyperflatness F2p≡(∂u/∂x)2p/[(∂u/∂x)2] p, p=2,3,..., is constant independent of Taylor–Reynolds number Rλ, as is the ratio of the 2pth moment of one component of the vorticity Ω2p≡ω2px/(ω2x)p to F2p. A cutoff in a relevant time integration is then used to eliminate vortex-sheet-induced divergences in the integrals corresponding to ω2px, 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 Ω2p=Ω̂2pRλp/2−3/4, F2p=F̂2pRλp/2−3/4 and S2p+1=Ŝ2p+1Rλp/2−3/4, p=2,3,..., with explicit calculation of the numbers Ω̂2p, F̂2p, and Ŝ2p+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)].