Transformation invariance of the longitudinal velocity correlation in grid-generated turbulence at high Reynolds numbers

Abstract

For grid Reynolds numbers from 12800 to 81000, the Frenkiel-Klebanoff-Huang data for approximately isotropic homogeneous grid-generated turbulence shows that the longitudinal correlation function is given by the simple empirical expression f = [1 + (r/2L)]⁻³, where r(≥ 0.01M) is the separation distance between two points in the fluid flow and L = L(t) is the integral scale. It follows that the longitudinal velocity correlation 〈u₁(x + re,t)u₁(x,t)〉 = u²f with e = (1,0,0) is invariant under the separation-distance time-contraction transformations $r\rightarrow [r+(1 - \lambda)2L], t\rightarrow\lambda^{\frac{5}{2}}t$ for all positive parameter values λ ≤ 1. Conversely, if the longitudinal correlation function is prescribed to have the form f = J (r/L(t)), then the indicated transformation invariance holds if and only if [Fscr ](ξ) = (1 + ½ξ)⁻³. It is also shown that a Gaussian normal probability distribution at t = 0 and the Kármán-Howarth equation for all t > 0 are compatible with the transformation invariance and associated expression for f.