Scale-by-scale energy budget on the axis of a turbulent round jet

The well-known Kolmogorov relationship between second- and third-order structure functions is an asymptotic result, unlikely to be satisfied in turbulent flows encountered in the laboratory at moderate values of the Reynolds number. The main reason for this is the persistent effect of inhomogeneities arising from a number of different sources. The identification of these sources and the quantification of their effects is an important, although complex task. The present paper focuses on the similarity region of a turbulent round jet, where the major source of inhomogeneity is associated with the streamwise decay of turbulent energy. We derive a transport equation for ⟨ Δ u i Δ u i ⟩ (where Δ u i is the velocity increment of the velocity component u i ) which, in the limit of large separations, reduces to the one-point energy budget on the jet axis and within a limited radial region around it. The equation is shown to be satisfied by experimental data to an acceptable level of accuracy. In particular, ⟨ Δ u 1 Δ u i Δ u i ⟩ can be estimated, via the equation, to within 15%.