Equilibrium similarity solution of the turbulent transport equation along the centreline of a round jet

A novel similarity-based form is derived of the transport equation for the second-order velocity structure function of (δq)² along the centreline of a round turbulent jet using an equilibrium similarity analysis. This self-similar equation has the advantage of requiring less extensive measurements to calculate the inhomogeneous (decay and production) terms of the transport equation. It is suggested that the normalised third-order structure function can be uniquely determined when the normalised second-order structure function, the power-law exponent of q² and the decay rate constants of u² and v² are available. In addition, the current analysis demonstrates that the assumption of similarity, combined with an inverse relation between the mean velocity U and the streamwise distance x-x₀ from the virtual origin (i.e. U∝ (x-x₀)⁻¹ ), is sufficient to predict a power-law decay for the turbulence kinetic energy ( q² ∝ (x-x₀)ᵐ ), rather than requiring a power-law decay ( $m=-2$ ) as an additionalad hocassumption. On the basis of the current analysis, it is suggested that the mean kinetic energy dissipation rate, ε , varies as (x-x₀)ᵐ⁻² . These theoretical results are tested against new experimental data obtained along the centreline of a round turbulent jet as well as previously published data on round jets for 11\,000 ReD 184\,000 over the range 10 x/D 90 . For the present experiments, q² exhibits power-law behaviour with $m=-1.83$ . The validity of this solution is confirmed using other experimental data where q² follows a power law with -1.89 m -1.78 . The present similarity form of the transport equation for (δq)² is also shown to be closely satisfied by the experimental data.