Two-point similarity in the round jet

The governing equations for the two-point velocity correlations in the far field of the axisymmetric jet are examined and it is shown that these equations can have equilibrium similarity solutions for jets with finite Reynolds number that retain a dependence on the growth rate of the jet. The two-point velocity correlation can be written as the product of a scale that depends on the downstream position of the two points and a function that only depends on the similarity variables. Physically, this result implies that the turbulent processes producing and dissipating energy at the different scales of motion, as well as transferring energy between the different scales of motion, are in equilibrium as the flow evolves downstream. A particularly interesting prediction from the analysis is that the two-point similarity solutions depend only on the separation distance between the points in the streamwise similarity coordinate (i.e. ν = ξ′ - ξ), that is, the logarithm of the streamwise coordinate itself (i.e. ξ = ln x₁, where x₁ is measured from a virtual origin). Thus, the measures of the turbulence are homogeneous in the streamwise similarity coordinate. The predictions from the similarity analysis forthe streamwise two-point velocity correlation were compared with combined hot-wire and LDA measurements on the centreline of a round jet at a Reynolds number of 33000, and with two-point velocity correlations computed from PIV measurements in a round jet at a Reynolds number of 2000 performed by Fukushima et al. In both cases, the measured two-point velocity correlations in the streamwise direction collapsed when they were scaled in the manner predicted by the similarity analysis. The results provide further evidence that the equilibrium similarity hypothesis does describe the development of the flow in fully developed turbulent round jets and that the two-point correlations are statistically homogeneous in the streamwise similarity coordinate.