Similarity of energy structure functions in decaying homogeneous isotropic turbulence

An equilibrium similarity analysis is applied to the transport equation for (δ q)² ( ≡\,(δ u)² + (δ v)² + (δ w)² ), the turbulent energy structure function, for decaying homogeneous isotropic turbulence. A possible solution requires that the mean energy q² decays with a power-law behaviour ( q²\,~\,xᵐ ), and the characteristic length scale, which is readily identifiable with the Taylor microscale, varies as x1/2 . 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 Rλ ( ~\, q²1/2 λ/ν ); Rλ should decay as x(m+1)/2 when $m < -1$ . The solution is tested at relatively low Rλ against grid turbulence data for which m -1.25 and Rλ decays as x-0.125 . Although homogeneity and isotropy are poorly approximated in this flow, the measurements of (δ q)² and, to a lesser extent, (δ u)(δ q)² , satisfy similarity reasonably over a significant range of r/λ , 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 (δ u)(δ q)² is in reasonable agreement with measurements. Kolmogorov-normalized distributions of (δ q)² and (δ u)(δ q)² collapse only at small r . Assuming homogeneity, isotropy and a Batchelor-type parameterization for (δ q)² , it is found that Rλ may need to be as large as 10⁶ before a two-decade inertial range is observed.