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

- 641 Downloads
- 2 Citations

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

## Keywords

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

### Acknowledgements

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 https://data.gulfresearchinitiative.org ( https://doi.org/10.7266/N7RJ4GZM). The authors thank John Charonko at the Los Alamos National Laboratory for providing his code for pressure field computation.

## References

- 1.Adrian R, Christensen K, Liu ZC (2000) Analysis and interpretation of instantaneous turbulent velocity fields. Exp Fluids 29(3):275–290CrossRefGoogle Scholar
- 2.Bewley G, Chang K, Bodenschatz E (2012) On integral length scales in anisotropic turbulence. Phys Fluids 24:061,702CrossRefGoogle Scholar
- 3.Calluaud D, David L (2004) Stereoscopic particle image velocimetry measurements of a flow around a surface-mounted block. Exp Fluids 36:33–61CrossRefGoogle Scholar
- 4.Chan S, Lee J (2016) A particle tracking model for sedimentation from buoyant jets. J Hydraul Eng ASCE 142(5):04016001CrossRefGoogle Scholar
- 5.Charonko J, King C, Smith B, Vlachos P (2010) Assessment of pressure field calculations from particle image velocimetry measurements. Meas Sci Technol 21:105,401CrossRefGoogle Scholar
- 6.Darisse A, Lemay J, Benaissa A (2013) Ldv measurements of well converged third order moments in the far field of a free turbulent round jet. Exp Therm Fluid Sci 44:825–833CrossRefGoogle Scholar
- 7.Darisse A, Lemay J, Benaissa A (2015) Budgets of turbulent kinetic energy, Reynolds stresses, variance of temperature fluctuations and turbulent heat fluxes in a round jet. J Fluid Mech 774:95–142CrossRefGoogle Scholar
- 8.Dimotakis PE (2000) The mixing transition in turbulent flows. J Fluid Mech 409:69–98CrossRefGoogle Scholar
- 9.Eckstein A, Vlachos P (2009) Assessment of advanced windowing techniques for digital particle image velocimetry (dpiv). Meas Sci Technol 20:075,402CrossRefGoogle Scholar
- 10.Ganapathisubramani B, Lakshminarasimhan K, Clements N (2007) Determination of complete velocity gradient tensor by using cinematographic stereoscopic piv in a turbulent jet. Exp Fluids 42:923–939CrossRefGoogle Scholar
- 11.Hewitt VJ (ed) (2005) Prediction of turbulent flows. Cambridge University Press, CambridgeGoogle Scholar
- 12.Huang H, Fiedler H, Wang J (1993) Limitation and improvement of piv part ii: particle image distortion, a novel technique. Exp Fluids 15:263–273CrossRefGoogle Scholar
- 13.Hussein H, Capp S, George W (1994) Velocity measurements in a high-Reynolds-number, momentum-conserving, axisymmetric, turbulent jet. J Fluid Mech 258:31–75CrossRefGoogle Scholar
- 14.Lai C (2015) An analysis of bubble plumes in unstratified stagnant water. Ph.D. thesis, Texas A&M UniversityGoogle Scholar
- 15.Lee J, Chu V (2003) Turbulent jets and plumes—a Lagrangian approach. Kluwer, DordrechtCrossRefGoogle Scholar
- 16.Liu X, Katz J (2006) Instantaneous pressure and material accelerations measurements using a four exposure piv system. Exp Fluids 41:227–240CrossRefGoogle Scholar
- 17.Lumley JL (1978) Computational modeling of turbulent flows. Adv Appl Mech 18:123–176CrossRefGoogle Scholar
- 18.Mansour NN, Kim J, Moin P (1988) Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J Fluid Mech 194:15–44CrossRefGoogle Scholar
- 19.Panchapakesan N, Lumley J (1993) Turbulence measurements in axisymmetric jets of air and helium. Part 1: Air jet. J Fluid Mech 246:197–223CrossRefGoogle Scholar
- 20.Parker K, Ellenrieder K, Soria J (2005) Using stereo multigrid dpiv (smdpiv) measurements to investigate the vortical skeleton behind a finite-span flapping wing. Exp Fluids 39:281–298CrossRefGoogle Scholar
- 21.Pope SB (2000) Turbulent flows. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- 22.Raffel M (2007) Particle image velocimetry: a practical guide, 2nd edn. Springer, BerlinGoogle Scholar
- 23.van Reeuwijk M, Salizzoni P, Hunt G, Craske J (2016) Turbulent transport and entrainment in jets and plumes: a dns study. Phys Rev Fluids 1(7):074,301CrossRefGoogle Scholar
- 24.Scarano F (2002) Iterative image deformation methods in piv. Meas Sci Technol 13:R1–R19CrossRefGoogle Scholar
- 25.Scarano F, Riethmuller L (2000) Advances in iterative multigrid piv image processing. Exp Fluids 29(Supplement 1):S051–S060Google Scholar
- 26.Shademan M, Balachandar R, Roussinova V, Barron R (2016) Round impinging jets with relatively large stand-off distance. Phys Fluids 28:075,107CrossRefGoogle Scholar
- 27.Socolofsky S, Lai C (2017) Three-dimensional stereoscopic particle image velocimetry data at the centerplane of a single-phase turbulent round jet. Distributed by Gulf of Mexico Research Initiative Information and Data Cooperative (GRIIDC). Texas A&M University, College Station. https://doi.org/10.7266/N7RJ4GZM
- 28.Soloff S, Adrian R, Liu ZC (1997) Distortion compensation for generalized stereoscopic particle image velocimetry. Meas Sci Technol 8(8):1441–1454CrossRefGoogle Scholar
- 29.Soria J (1996) An investigation of the near wake of a circular cylinder using a video-based digital cross-correlation particle image velocimetry technique. Exp Therm Fluid Sci 12:221–233CrossRefGoogle Scholar
- 30.Speziale C (1995) A review of Reynolds stress models for turbulent shear flows. Tech. rep., NASA Langley Research CenterGoogle Scholar
- 31.Taub G, Lee H, Balachandar S, Sherif S (2013) A direct numerical simulation study of high order statistics in a turbulent round jet. Phys Fluids 25:115,102CrossRefGoogle Scholar
- 32.Terashima O, Onishi K, Sakai Y, Nagata K, Ito Y (2014) Simultaneous measurements of all three velocity components and pressure in a plane jet. Meas Sci Technol 25:055,301CrossRefGoogle Scholar
- 33.Thiesset F, Antonia R, Djenidi L (2014) Consequences of self-preservation on the axis of a turbulent round jet. J Fluid Mech 748:R2CrossRefGoogle Scholar
- 34.Wang H, Law AK (2002) Second-order model for a round turbulent buoyant jet. J Fluid Mech 459:397–428Google Scholar
- 35.Westerweel J (2005) Universal outlier detection for piv data. Exp Fluids 39:1096–1100CrossRefGoogle Scholar
- 36.Wieneke B (2005) Stereo-piv using self-calibration on particle images. Exp Fluids 39(2):267–280CrossRefGoogle Scholar
- 37.Willert C, Gharib M (1991) Digital particle image velocimetry. Exp Fluids 10(4):181–193CrossRefGoogle Scholar