Similarity of energy structure functions in decaying homogeneous isotropic turbulence

An equilibrium similarity analysis is applied to the transport equation for $\langle(\delta q)^{2}\rangle$ ( ${\equiv}\,\langle(\delta u)^{2}\rangle + \langle(\delta v)^{2}\rangle + \langle(\delta w)^{2}\rangle$ ), the turbulent energy structure function, for decaying homogeneous isotropic turbulence. A possible solution requires that the mean energy $\langle q^{2}\rangle$ decays with a power-law behaviour ( $\langle q^{2}\rangle\,{\sim}\,x^{m}$ ), 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 $R_{\lambda}$ ( ${\sim}\,{\langle q^{2}\rangle}^{1/2} \lambda/\nu$ ); $R_{\lambda}$ should decay as $x^{(m+1)/2}$ when $m . The solution is tested at relatively low $R_{\lambda}$ against grid turbulence data for which $m \simeq -1.25$ and $R_{\lambda}$ decays as $x^{-0.125}$ . Although homogeneity and isotropy are poorly approximated in this flow, the measurements of $\langle(\delta q)^{2}\rangle$ and, to a lesser extent, $\langle(\delta u)(\delta q)^{2}\rangle$ , 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 $\langle(\delta u)(\delta q)^{2}\rangle$ is in reasonable agreement with measurements. Kolmogorov-normalized distributions of $\langle(\delta q)^{2}\rangle$ and $\langle(\delta u)(\delta q)^{2}\rangle$ collapse only at small $r$ . Assuming homogeneity, isotropy and a Batchelor-type parameterization for $\langle(\delta q)^{2}\rangle$ , it is found that $R_{\lambda}$ may need to be as large as $10^{6}$ before a two-decade inertial range is observed.