Environmental Fluid Mechanics

, Volume 19, Issue 2, pp 349–377 | Cite as

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

  • Chris C. K. LaiEmail author
  • Scott A. Socolofsky
Original Article


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 \(\sigma _\epsilon = 1.3\).


Turbulent round jets \(k-{\overline{\epsilon }}\) models Energy budgets Dissipation 



This research was made possible by a Grant from The Gulf of Mexico Research Initiative to the Gulf Integrated Spill Research (GISR) Consortium. Data are publicly available through the Gulf of Mexico Research Initiative Information & Data Cooperative (GRIIDC) at ( The authors thank John Charonko at the Los Alamos National Laboratory for providing his code for pressure field computation.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.P-23, Physics DivisionLos Alamos National LaboratoryLos AlamosUSA
  2. 2.Zachry Department of Civil EngineeringTexas A&M UniversityCollege StationUSA

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