# Turbulence characteristics of 45° inclined dense jets

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## Abstract

In the present study, we performed an extensive laboratory investigation to quantify the turbulence characteristics of 45° inclined dense jets using Particle Image Velocimetry (PIV) over a wide range of Densimetric Froude Number. The objective was to provide benchmark data to guide high resolution turbulence numerical simulations for dense jets in the future. The PIV measurements were sampled at a relatively high frequency of 50 Hz, which enabled the analysis of second order turbulence statistics as well as the turbulence kinetic energy spectrum (including the inertial subrange) along the curvilinear jet trajectory, which has hitherto not been reported. The measurements showed that the spectral profile was flat near the discharge port with the potential core, since the Kelvin–Helmholtz shear-induced turbulence at the jet boundaries had not fully penetrated to the core. The spectral profile then evolved along the trajectory with progressive steepening towards the higher frequencies, and a fully-developed profile appeared beyond the terminal rise with a clearly identifiable inertial subrange for the energy cascade. In parallel, we also performed numerical simulations using the Large Eddy Simulations (LES) approach with the Dynamic Smagorinsky sub-grid model for the specific discharge conditions as in the experiments. The LES approach followed that of Zhang et al. (Environ Fluid Mech 16(1):101–121, 2016, J Hydro Environ Res 15:54–66, 2017) using GCI as the grid convergence criteria. The comparison showed that the time-averaged first order mixing characteristics of the inclined dense jet can be simulated reasonably well comparing to the experimental data. In terms of the turbulence kinetic energy spectrum, the low frequencies of the production range were also well captured by the simulations. However, the simulated transitional spectra towards the inertial subrange decayed substantially faster than the experiments. The discrepancy was attributed to the fact that the grid resolution was not sufficiently fine in the simulations (which were constrained by the available computational resources and time), such that stratified effects remained present inside the sub-grids producing additional turbulence energy that were not represented by the Dynamic Smagorinsky model. Thus, the numerical investigation showed that further improvement in sub-grid models that can incorporate the stratified effects would be desirable in the future for engineering simulations.

## Keywords

Inclined dense jet PIV Large Eddy Simulations (LES) Turbulence kinetic energy (TKE) Turbulence kinetic energy spectrum## List of symbols

*b*_{e}Velocity 1/e width

*C*_{0}Initial concentration

*C*_{m}Centerline concentration

*C*_{s}Smagorinsky constant

*D*Nozzle diameter

*Fr*Densimetric Froude number

*g*Gravitational acceleration

*g′*Reduced gravitational acceleration

*h*Port height

*H*Distance from the port to the water surface

*L*Distance from the port to the back and front vertical boundary

*L*_{M}Jet characteristic length scale

*L*_{ij}Resolved turbulent stress

*M*_{ij}Anisotropic part of the turbulent stress

*p*Pressure

*Q*_{j}SGS scalar flux or turbulent scalar flux

*r*Radial distance

*Re*Reynolds number

*s*Stream-wise distance from the nozzle

- \(\tilde{S}\)
Local strain rate

*S*_{Ct}Turbulent Schmidt number

- \(\widetilde{S}_{ij}\)
Rate of strain tensor for the resolved scale

*t*Time

*t*_{s}Run time of a simulation

*U*Stream-wise velocity

*U′*Fluctuation of stream-wise velocity

*U*_{0}Discharge velocity

*U*_{m}Jet velocity along the centerline

*U′*_{rms}Root mean square stream-wise velocity fluctuation

*u*_{i},*u*_{j}Velocity in

*i*,*j*direction, respectively*V*Radial velocity

*V′*Fluctuation of radial velocity

*V′*_{rms}Root mean square radial velocity fluctuation

*W*Distance from the nozzle to the left and right vertical boundary

*x*,*y*,*z*Cartesian Coordinates in the horizontal, lateral and vertical direction, respectively

*x*_{m}Horizontal location of centerline peak

- ∆
LES filter width

*ρ*Fluid density

*ρ*_{a}Ambient density

*ρ*_{b}Effluent density

*μ*Fluid viscosity

*μ*_{t}SGS eddy viscosity or turbulent eddy viscosity

*ϕ*Scalar concentration

*ε*Dissipation rate

- \(\tau_{ij}\)
SGS Reynolds stresses or Reynolds stresses

*τ*_{kk}Isotropic part of SGS stress

- Γ
Scalar diffusivity

- Γ
_{t} Turbulent dispersivity

## Notes

### Acknowledgements

The authors would like to thank the Nanyang Environment and Water Research Institute and the Interdisciplinary Graduate School at Nanyang Technological University, for the award of research scholarship to the first author.

## References

- 1.Celik IB, Ghia U, Roache PJ (2008) Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. J Fluids Eng 130(7):078001CrossRefGoogle Scholar
- 2.Chen, CJ, Rodi W (1980) Vertical turbulent buoyant jets: a review of experimental data. NASA STI/Recon Technical Report A 80Google Scholar
- 3.Christodoulou GC, Papakonstantis IG (2010) Simplified estimates of trajectory of inclined negatively buoyant jets. In: Proceedings of the 6th international symposium on environmental hydraulics vol
**1**, pp 165–170Google Scholar - 4.Christodoulou GC, Papakonstantis IG, Nikiforakis IK (2015) Desalination brine disposal by means of negatively buoyant jets. Desalin Water Treat 53(12):3208–3213CrossRefGoogle Scholar
- 5.Cipollina A, Brucato A, Grisafi F, Nicosia S (2005) Bench-scale investigation of inclined dense jets. J Hydraul Eng ASCE 131(11):1017–1022CrossRefGoogle Scholar
- 6.Crowe AT, Davidson MJ, Nokes RI (2016) Velocity measurements in inclined negatively buoyant jets. Environ Fluid Mech 16(3):503–520CrossRefGoogle Scholar
- 7.Crowe AT, Davidson MJ, Nokes RI (2016) Modified reduced buoyancy flux model for desalination discharges. Desalination 378:53–59CrossRefGoogle Scholar
- 8.Dejoan A, Leschziner MA (2005) Large eddy simulation of a plane turbulent wall jet. Phys Fluids 17(2):025102CrossRefGoogle Scholar
- 9.Drami D, Yacobi YZ, Stambler N, Kress N (2011) Seawater quality and microbial communities at a desalination plant marine outfall. A field study at the Israeli Mediterranean coast. Water Res 45(17):5449–5462CrossRefGoogle Scholar
- 10.Deshpande SS, Sathe MJ, Joshi JB (2009) Evaluation of local turbulence energy dissipation rate using PIV in jet loop reactor. Ind Eng Chem Res 48(10):5046–5057CrossRefGoogle Scholar
- 11.Fellouah H, Pollard A (2009) The velocity spectra and turbulence length scale distributions in the near to intermediate regions of a round free turbulent jet. Phys Fluids 21(11):115101CrossRefGoogle Scholar
- 12.Foucaut JM, Carlier J, Stanislas M (2004) PIV optimization for the study of turbulent flow using spectral analysis. Meas Sci Technol 14(6):1046–1058CrossRefGoogle Scholar
- 13.Germano M, Piomelli U, Moin P, Cabot WH (1991) A dynamic subgrid-scale eddy viscosity model. Phys Fluids A 3(7):1760–1765CrossRefGoogle Scholar
- 14.Gildeh HK, Mohammadian A, Nistor I, Qiblawey H (2015) Numerical modeling of 30 degrees and 45 degrees inclined dense turbulent jets in stationary ambient. Environ Fluid Mech 15(3):537–562CrossRefGoogle Scholar
- 15.Gruber MF, Johnson CJ, Tang CY, Jensen MH, Yde L, Hélix-Nielsen C (2011) Computational fluid dynamics simulations of flow and concentration polarization in forward osmosis membrane systems. J Membr Sci 379(1–2):488–495CrossRefGoogle Scholar
- 16.Heisenberg W (1948) On the theory of statistical and isotropic turbulence. Proc R Soc Lond A 195(1042):402–406CrossRefGoogle Scholar
- 17.Jasak H (2009) OpenFOAM: open source CFD in research and industry. Int J Nav Archit Ocean Eng 1(2):89–94Google Scholar
- 18.Jiang B, Law AWK, Lee JHW (2014) Mixing of 30 and 45 inclined dense jets in shallow coastal waters. J Hydraul Eng 140(3):241–253CrossRefGoogle Scholar
- 19.Kikkert GA, Davidson MJ, Nokes RI (2007) Inclined negatively buoyant discharges. J Hydraul Eng ASCE 133(5):545–554CrossRefGoogle Scholar
- 20.Kolmogorov AN (1941) The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl Akad Nauk SSSR 30:9–13Google Scholar
- 21.Lai CC, Lee JH (2012) Mixing of inclined dense jets in stationary ambient. J Hydro Environ Res 6(1):9–28CrossRefGoogle Scholar
- 22.Lai AC, Zhao B, Law AWK, Adams EE (2015) A numerical and analytical study of the effect of aspect ratio on the behavior of a round thermal. Environ Fluid Mech 15(1):85–108CrossRefGoogle Scholar
- 23.Law AWK, Wang H (2000) Measurement of mixing processes with combined digital particle image velocimetry and planar laser induced fluorescence. Exp Therm Fluid Sci 22(3):213–229CrossRefGoogle Scholar
- 24.Law AWK (2006) Velocity and concentration distributions of round and plane turbulent jets. J Eng Math 56(1):69–78CrossRefGoogle Scholar
- 25.Lilly DK (1992) A proposed modification of the germano-subgrid-scale closure method. Phys Fluids A Fluid Dyn 4(3):633–635CrossRefGoogle Scholar
- 26.Milione M, Zeng C (2008) The effects of temperature and salinity on population growth and egg hatching success of the tropical calanoid copepod, Acartia sinjiensis. Aquaculture 275(1):116–123CrossRefGoogle Scholar
- 27.Obukhov A (1941) Spectral energy distribution in a turbulent flow. Izv Akad Nauk SSSR Ser Geogr i Geofiz 5:453–466Google Scholar
- 28.Oliver CJ, Davidson MJ, Nokes RI (2008) k-ε Simulations of the initial mixing of desalination discharges. Environ Fluid Mech 8(5–6):617–625CrossRefGoogle Scholar
- 29.Palomar P, Lara JL, Losada IJ, Rodrigo M, Alvarez A (2012) Near field brine discharge modelling part 1: analysis of commercial tools. Desalination 290:14–27CrossRefGoogle Scholar
- 30.Papakonstantis IG, Christodoulou GC, Papanicolaou PN (2011) Inclined negatively buoyant jets 1: geometrical characteristics. J Hydraul Res 49(1):3–12CrossRefGoogle Scholar
- 31.Papakonstantis IG, Christodoulou GC, Papanicolaou PN (2011) Inclined negatively buoyant jets 2: concentration measurements. J Hydraul Res 49(1):13–22CrossRefGoogle Scholar
- 32.Papanicolaou PN (1984) Mass and momentum transport in a turbulent buoyant vertical axisymmetric jet. (Doctoral dissertation, California Institute of Technology)Google Scholar
- 33.Papanicolaou PN, Papakonstantis IG, Christodoulou GC (2008) On the entrainment coefficient in negatively buoyant jets. J Fluid Mech 614:447–470CrossRefGoogle Scholar
- 34.Pope S (2002) Turbulent flow. Cambridge University Press, CambridgeGoogle Scholar
- 35.Roberts PJW, Toms G (1987) Inclined dense jets in flowing current. J Hydraul Eng 113(3):323–341CrossRefGoogle Scholar
- 36.Roberts PJW, Ferrier A, Daviero G (1997) Mixing in inclined dense jets. J Hydraul Eng 123(8):693–699CrossRefGoogle Scholar
- 37.Shao D, Law AWK (2010) Mixing and boundary interactions of 30° and 45° inclined dense jets. Environ Fluid Mech 10(5):521–553CrossRefGoogle Scholar
- 38.Vafeiadou P, Papakonstantis IG, Christodoulou GC (2005) Numerical simulation of inclined negatively buoyant jets. In: Proceedings, 9th international conference on environmental science and technology, pp A1537–A1542Google Scholar
- 39.Wang H, Law AWK (2002) Second-order integral model for a round turbulent buoyant jet. J Fluid Mech 459:397–428Google Scholar
- 40.Westerweel J, Elsinga GE, Adrian RJ (2013) Particle image velocimetry for complex and turbulent flows. Annu Rev Fluid Mech 45:409–436CrossRefGoogle Scholar
- 41.Yimer I, Campbell I, Jiang LY (2002) Estimation of the turbulent Schmidt number from experimental profiles of axial velocity and concentration for high-Reynolds-number jet flows. Can Aeronaut Space J 48(3):195–200CrossRefGoogle Scholar
- 42.Zhang S, Jiang B, Law AWK, Zhao B (2016) Large eddy simulations of 45° inclined dense jets. Environ Fluid Mech 16(1):101–121CrossRefGoogle Scholar
- 43.Zhang S, Law AWK, Jiang M (2017) Large eddy simulations of 45° and 60° inclined dense jets with bottom impact. J Hydro Environ Res 15:54–66CrossRefGoogle Scholar