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 $\langle ({\it\delta}q)^{2}\rangle$ 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 $\langle q^{2}\rangle$ and the decay rate constants of $\langle u^{2}\rangle$ and $\langle v^{2}\rangle$ 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 $x-x_{0}$ from the virtual origin (i.e. $U\propto (x-x_{0})^{-1}$ ), is sufficient to predict a power-law decay for the turbulence kinetic energy ( $\langle q^{2}\rangle \propto (x-x_{0})^{m}$ ), rather than requiring a power-law decay ( $m=-2$ ) as an additional ad hoc assumption. On the basis of the current analysis, it is suggested that the mean kinetic energy dissipation rate, $\langle {\it\epsilon}\rangle$ , varies as $(x-x_{0})^{m-2}$ . 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 $11\,000\leqslant \mathit{Re}_{D}\leqslant 184\,000$ over the range $10\leqslant x/D\leqslant 90$ . For the present experiments, $\langle q^{2}\rangle$ exhibits power-law behaviour with $m=-1.83$ . The validity of this solution is confirmed using other experimental data where $\langle q^{2}\rangle$ follows a power law with $-1.89\leqslant m\leqslant -1.78$ . The present similarity form of the transport equation for $\langle ({\it\delta}q)^{2}\rangle$ is also shown to be closely satisfied by the experimental data.

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