Incompressible fluid turbulence at large Reynolds numbers: Theoretical basis for the t−1 decay law and the form of the longitudinal correlation function

Abstract

Approximately valid for large values of the time t, a formal solution to the Hopf Φ equation is obtained here as an asymptotic power series in t−1. This approximate solution is directly applicable to grid-generated isotropic homogeneous turbulence at large Reynolds numbers during the initial (inertial-force dominated) period of decay; thus, the solution accounts for the observed t−1 decay law and the fact that the longitudinal correlation function f is independent of t. It is observed that the longitudinal correlation function measured by Frenkiel, Klebanoff, and Huang is consistent with the theoretical asymptotic behavior f = (const)r−3 as r→∞ and fitted by the expression f = [1+0.770(r/M)]−3, where M is the grid mesh length and the separation distance r is greater than the Taylor microscale (10νt)1/2. Interestingly enough, this form for the longitudinal correlation function is shown to be derivable from a variational principle.