Budgets of turbulent kinetic energy, Reynolds stresses, and dissipation in a turbulent round jet discharged into a stagnant ambient

Abstract

This paper presents a set of stereoscopic particle image velocimetry (SPIV) measurements of a turbulent round water jet (jet exit Reynolds number $Re=2679$ and turbulent Reynolds number $Re_{T}=113$) discharged into an initially stationary ambient. The data were taken on the jet centerplane and at non-dimensional downstream distances $x/D=27−37$ ($x$= axial coordinate and $D$= orifice diameter), where the jet turbulence had evolved into a self-preserving state. Budgets of turbulent kinetic energy k and individual components of the Reynolds stress tensor $R_{ij}$ are extracted from the velocity measurements and compared to recent experimental data of an air jet ($x/D=30,Re=140,000$) and direct numerical simulation data ($x/D=15,Re=2000$). The comparison reveals that the datasets are consistent with each other but that the turbulent transport of energy $\overline{u^2_i}$ appears to differ between the present low-$Re$ water jet and the high-$Re$ air jet. Nonetheless, the non-dimensional profile of turbulent dissipation rate ${\overline{\epsilon}}$, obtained as the closing term (balance) of the k-budget, is very similar in all studies. The commonly used Lumley’s model for pressure–velocity correlation (pressure transport term in k-budget) is evaluated using the instantaneous pressure field computed from the time-resolved planar velocity data. We find that Lumley’s model is deficient in the jet core $|r/b_{g}|<0.3$ ($r$= radial coordinate and $b_{g}$= Guassian half-width), while performing adequately away from it. Finally, the present data are used to compute terms appearing in the exact transport equation of ${\overline{\epsilon}}$. Combining both the k and ${\overline{\epsilon}}$ budgets, model coefficients in the commonly used two-equation $k−{\overline{\epsilon}}$ turbulence closure model are evaluated from the present data. If a fixed set of model coefficients is to be employed in a jet simulation, the following values of the model coefficients are recommended to optimize predictions for the mean flow field, for k, and for ${\overline{\epsilon}}$- $C_{1\epsilon} = 1.2, C_{2\epsilon} = 1.6, C_{\mu } = 0.11, \sigma \_{k} = 1.0$ and $\sigma_{\epsilon} = 1.3$.