Passive scalar statistics in a turbulent round jet: symmetry theory and direct numerical simulation

We analyse moment and probability density function (PDF) statistics of a passive scalar $\Theta$ at a Prandtl number of $Pr=0.71$ in a turbulent jet. For this, we conducted a direct numerical simulation at a Reynolds number of $Re=3500$ and, further, employed Lie symmetries applied to the multi-point moment equations, generalising recent work (Nguyen & Oberlack 2024 b under review with Flow Turbul. Combust. ) that focused on pure hydrodynamics. It is shown that the symmetry theory also provides highly precise results for free shear flows for all the quantities mentioned and statistical symmetries again play a key role. The scalar statistics are partly similar to the $U_z$ velocity statistics, and in particular, as in the above-mentioned work, a significant generalisation of the classical scalings has been derived so that a variation of the scaling laws solely controlled by the inflow is possible. An exponential behaviour of the scaling prefactors with the moment orders $m$ and $n$ for scalar and velocity is also discovered for any mixed moments. Instantaneous $\Theta$ -moments and mixed $U_z$ - $\Theta$ -moments exhibit a Gaussian distribution with variation of the scaled radius $\eta =r/(z-z_0)$ . Therein, the coefficient in the Gauss exponent is nonlinear with varying moment orders $m$ and $n$ . The scalar PDF statistics are clearly different from the velocity statistics, i.e. already deviate from the Gaussian distribution on the jet axis, as is observed for the $U_z$ statistics, and become clearly skewed and heavy tailed for increasing $\eta$ .