Complete self-preservation on the axis of a turbulent round jet

Self-preservation (SP) solutions on the axis of a turbulent round jet are derived for the transport equation of the second-order structure function of the turbulent kinetic energy ( $k$ ), which may be interpreted as a scale-by-scale (s.b.s.) energy budget. The analysis shows that the mean turbulent energy dissipation rate, $\overline{{\it\epsilon}}$ , evolves like $x^{-4}$ ( $x$ is the streamwise direction). It is important to stress that this derivation does not use the constancy of the non-dimensional dissipation rate parameter $C_{{\it\epsilon}}=\overline{{\it\epsilon}}u^{\prime 3}/L_{u}$ ( $L_{u}$ and $u^{\prime }$ are the integral length scale and root mean square of the longitudinal velocity fluctuation respectively). We show, in fact, that the constancy of $C_{{\it\epsilon}}$ is simply a consequence of complete SP (i.e. SP at all scales of motion). The significance of the analysis relates to the fact that the SP requirements for the mean velocity and mean turbulent kinetic energy (i.e. $U\sim x^{-1}$ and $k\sim x^{-2}$ respectively) are derived without invoking the transport equations for $U$ and $k$ . Experimental hot-wire data along the axis of a turbulent round jet show that, after a transient downstream distance which increases with Reynolds number, the turbulence statistics comply with complete SP. For example, the measured $\overline{{\it\epsilon}}$ agrees well with the SP prediction, i.e. $\overline{{\it\epsilon}}\sim x^{-4}$ , while the Taylor microscale Reynolds number $Re_{{\it\lambda}}$ remains constant. The analytical expression for the prefactor $A_{{\it\epsilon}}$ for $\overline{{\it\epsilon}}\sim (x-x_{o})^{-4}$ (where $x_{o}$ is a virtual origin), first developed by Thiesset et al. (J. Fluid Mech., vol. 748, 2014, R2) and rederived here solely from the SP analysis of the s.b.s. energy budget, is validated and provides a relatively simple and accurate method for estimating $\overline{{\it\epsilon}}$ along the axis of a turbulent round jet.

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