Similarity of energy structure functions in decaying homogeneous isotropic turbulence

An equilibrium similarity analysis is applied to the transport equation for <(deltaq)(2)> (equivalent to <(deltau)(2)> + <(deltav)(2)> + <(deltaw)(2)>), the turbulent energy structure function, for decaying homogeneous isotropic turbulence. A possible solution requires that the mean energy <q(2)> decays with a power-law behaviour (<q(2)> similar to x(m)), and the characteristic length scale, which is readily identifiable with the Taylor microscale, varies as x(1/2). This solution is identical to that obtained by George (1992) from the spectral energy equation. The solution does not depend on the actual magnitude of the Taylor-microscale Reynolds number R-lambda (similar to <q(2)>(1/2)lambda/nu); R-lambda should decay as x((m+1)/2) when m < -1. The solution is tested at relatively low R-lambda against grid turbulence data for which m similar or equal to -1.25 and R-lambda decays as x(-0.125). Although homogeneity and isotropy are poorly approximated in this flow, the measurements of <(deltaq)(2)> and, to a lesser extent, <(deltau)(deltaq)(2)>, satisfy similarity reasonably over a significant range of r/lambda, where r is the streamwise separation across which velocity increments are estimated. For this range, a similarity-based calculation of the third-order structure function <(deltau)(deltaq)(2)> is in reasonable agreement with measurements. :Kolmogorov-normalized distributions of <(deltaq)(2)> and <(deltau)(deltaq)(2)> collapse only at small r. Assuming homogeneity, isotropy and a Batchelor-type parameterization for <(deltaq)(2)>, it is found that R-lambda may need to be as large as 10(6) before a two-decade inertial range is observed.