An empirical expression for on the axis of a slightly heated turbulent round jet

Abstract

Self-preservation analyses of the equations for the mean temperature and the second-order temperature structure function on the axis of a slightly heated turbulent round jet are exploited in an attempt to develop an analytical expression for$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$, the mean dissipation rate of$\overline{\unicode[STIX]{x1D703}^{2}}/2$, where$\overline{\unicode[STIX]{x1D703}^{2}}$is the temperature variance. The analytical approach follows that of Thiessetet al.(J. Fluid Mech., vol. 748, 2014, R2) who developed an expression for$\unicode[STIX]{x1D716}_{k}$, the mean turbulent kinetic energy dissipation rate, using the transport equation for$\overline{(\unicode[STIX]{x1D6FF}u)^{2}}$, the second-order velocity structure function. Experimental data show that complete self-preservation for all scales of motion is very well satisfied along the jet axis for streamwise distances larger than approximately 30 times the nozzle diameter. This validation of the analytical results is of particular interest as it provides justification and confidence in the analytical derivation of power laws representing the streamwise evolution of different physical quantities along the axis, such as:$\unicode[STIX]{x1D702}$,$\unicode[STIX]{x1D706}$,$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$,$R_{U}$,$R_{\unicode[STIX]{x1D6E9}}$(all representing characteristic length scales), the mean temperature excess$\unicode[STIX]{x1D6E9}_{0}$, the mixed velocity–temperature moments$\overline{u\unicode[STIX]{x1D703}^{2}}$,$\overline{v\unicode[STIX]{x1D703}^{2}}$and$\overline{\unicode[STIX]{x1D703}^{2}}$and$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$. Simple models are proposed for$\overline{u\unicode[STIX]{x1D703}^{2}}$and$\overline{v\unicode[STIX]{x1D703}^{2}}$in order to derive an analytical expression for$A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}$, the prefactor of the power law describing the streamwise evolution of$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$. Further, expressions are also derived for the turbulent Péclet number and the thermal-to-mechanical time scale ratio. These expressions involve global parameters that are most likely to be influenced by the initial and/or boundary conditions and are therefore expected to be flow dependent.