Similarity of energy structure functions in decaying homogeneous isotropic turbulence

An equilibrium similarity analysis is applied to the transport equation for 〈(δq)²〉 (≡〈(δu)²〉 + 〈(δv)²〉 + 〈(δw)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 x½. 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 (~〈q²〉½ 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 〈(δq)²〉 and, to a lesser extent, 〈(δu)(δq)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 〈(δu)(δq)²〉 is in reasonable agreement with measurements. Kolmogorov-normalized distributions of 〈(δq)²〉 and 〈(δu)(δq)²〉 collapse only at small r. Assuming homogeneity, isotropy and a Batchelor-type parameterization for 〈(δq)²〉, it is found that Rλ may need to be as large as 10⁶ before a two-decade inertial range is observed.