Skip to main content
SpringerLink
Log in

Reynolds flux modeling; new numerical insights into inclined dense jets

  • Original Article
  • Published:
Environmental Fluid Mechanics Aims and scope Submit manuscript

Abstract

This paper aims to provide a more detailed representation of the scalar flux modeling (SFM) approach for modeling turbulent inclined negatively buoyant jets. The SFM approach addresses the limitations of eddy viscosity models in terms of the mean concentration field and turbulent scalar flux within the context of Reynolds-averaged Navier–Stokes modeling. In this study, the contribution of the involved terms in the transport equation of the turbulent scalar flux vector was evaluated, and the geometrical and mixing parameters of concentration and velocity of 45\(^\circ\) inclined negatively buoyant jets were verified. The SFM approach, along with the required modification for momentum flux modeling, was implemented in OpenFOAM v6. Results showed that the SFM approach can accurately predict mixing parameters due to the complex interactions between different turbulence contributors of the flow involved in the model. In comparison to simpler approaches, such as gradient-type models, that only correlate the gradient of the scalar field with turbulence, the SFM approach’s capability to predict mixing parameters is significantly higher.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Data availibility

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Notes

  1. https://docs.alliancecan.ca/wiki/Cedar.

Abbreviations

\(\mathbf {b_{ij}}\) :

Reynolds stress anisotropy

\({\textbf {D}}\) :

Nozzle diameter

\({\textbf {Fr}}\) :

Densimetric Froude number

\(\mathbf {G/G_{ij}}\) :

Buoyancy production

\({\textbf {g}}\) :

Gravitational acceleration

\({\textbf {k}}\) :

Turbulent kinetic energy

\(\mathbf {P/P_{ij}}\) :

Pressure/stress production

R :

Ratio of dynamic to scalar time scales

S :

Salinity

\(\mathbf {Sc_{t}}\) :

Turbulent Schmidt number

\(\mathbf {S_m}\) :

Centerline peak dilution ratio

\(\mathbf {S_r}\) :

Return point dilution ratio

\(\mathbf {\overline{S}_{ij}}\) :

Mean strain rate

\(\varvec{S_\phi }\) :

Source term

\(\mathbf {U_0}\) :

Initial jet velocity

\(\mathbf {\overline{u'_i u'_j}}\) :

Reynolds stress tensor

\(\mathbf {\overline{u'_i \phi '}}\) :

Reynolds flux vector

\(\mathbf {\overline{W_{ij}}}\) :

Mean rate of rotation tensor

\(\mathbf {X_m}\) :

Horizontal distance of the centerline peak to the origin

\(\mathbf {X_r}\) :

Horizontal distance of the return point to the origin

\(\mathbf {Y_m}\) :

Vertical distance of the centerline peak to the origin

\(\mathbf {Y_t}\) :

Terminal rise height

\(\varvec{\beta }\) :

Saline contraction coefficient

\(\varvec{\Gamma }\) :

Molecular diffusivity

\(\varvec{\delta }\) :

Kronecker delta

\(\varvec{\epsilon , \epsilon _\phi }\) :

Dissipation rate of turbulent kinetic energy

\(\varvec{\nu }\) :

Molecular viscosity

\(\varvec{\nu _t}\) :

Eddy viscosity

\(\varvec{\Pi _{ij}}\) :

Pressure-strain correlation

\(\varvec{\pi _{i\phi }}\) :

Pressure-scalar correlation

\(\varvec{\rho _0}\) :

Jet density

\(\varvec{\rho _a, \rho _r}\) :

Water density

\(\varvec{\tau _{ij}}\) :

Stress tensor

\(\varvec{\Phi }\) :

Scalar quantity, mean scalar quantity

\(\varvec{\phi '}\) :

Scalar fluctuation

References

  1. Roberts PJ, Salas HJ, Reiff FM, Libhaber M, Labbe A, Thomson JC (2010) Marine wastewater outfalls and treatment systems

  2. Roberts PJ, Ferrier A, Daviero G (1997) Mixing in inclined dense jets. J Hydraul Eng 123(8):693–699

    Article  Google Scholar 

  3. Zeitoun M, Reid R, McHilhenny W, Mitchell T (1972) Model studies of outfall systems for desalination plants. Office of Saline Water, US Department of the Interior, Washington DC

  4. Pincince AB, List EJ (1973) Disposal of brine into an estuary. J Water Pollut Control Fed 1:2335–44

    Google Scholar 

  5. Cipollina A, Brucato A, Grisafi F, Nicosia S (2005) Bench-scale investigation of inclined dense jets. J Hydraul Eng 131(11):1017–1022

    Article  Google Scholar 

  6. Kikkert GA, Davidson M, Nokes R (2007) Inclined negatively buoyant discharges. J Hydraul Eng 133(5):545–554

    Article  Google Scholar 

  7. Papakonstantis IG, Christodoulou GC, Papanicolaou PN (2011) Inclined negatively buoyant jets 1: geometrical characteristics. J Hydraul Res 49(1):3–12

    Article  Google Scholar 

  8. Papakonstantis IG, Christodoulou GC, Papanicolaou PN (2011) Inclined negatively buoyant jets 2: concentration measurements. J Hydraul Res 49(1):13–22

    Article  Google Scholar 

  9. Shao D, Law AW-K (2010) Mixing and boundary interactions of 30 and 45 inclined dense jets. Environ Fluid Mech 10(5):521–553

    Article  Google Scholar 

  10. Lai CC, Lee JH (2012) Mixing of inclined dense jets in stationary ambient. J Hydro Environ Res 6(1):9–28

    Article  Google Scholar 

  11. Oliver C, Davidson M, Nokes R (2013) Removing the boundary influence on negatively buoyant jets. Environ Fluid Mech 13(6):625–648

    Article  Google Scholar 

  12. Oliver C (2012) Near field mixing of negatively buoyant jets

  13. Bashitialshaaer R, Larson M, Persson KM (2012) An experimental investigation on inclined negatively buoyant jets. Water 4(3):720–738

    Article  Google Scholar 

  14. Jiang B, Law AW-K, Lee JH-W (2013) Mixing of 30 and 45 inclined dense jets in shallow coastal waters. J Hydraul Eng 140(3):241–253

    Article  Google Scholar 

  15. Abessi O, Roberts PJ (2015) Effect of nozzle orientation on dense jets in stagnant environments. J Hydraul Eng 141(8):06015009

    Article  Google Scholar 

  16. Abessi O, Roberts PJ (2015) Effect of nozzle orientation on dense jets in stagnant environments. J Hydraul Eng 141(8):06015009

    Article  Google Scholar 

  17. Crowe A, Davidson M, Nokes R (2016) Velocity measurements in inclined negatively buoyant jets. Environ Fluid Mech 16(3):503–520

    Article  Google Scholar 

  18. Abessi O, Roberts PJ (2015) Dense jet discharges in shallow water. J Hydraul Eng 142(1):04015033

    Article  Google Scholar 

  19. Papakonstantis IG, Tsatsara EI (2018) Trajectory characteristics of inclined turbulent dense jets. Environ Process 5(3):539–554

    Article  Google Scholar 

  20. Papakonstantis IG, Tsatsara EI (2019) Mixing characteristics of inclined turbulent dense jets. Environ Process 6(2):525–541

    Article  Google Scholar 

  21. Crowe AT (2013) Inclined negatively buoyant jets and boundary interaction. Ph.D. thesis, University of Canterbury

  22. Tahmooresi S, Ahmadyar D (2022) Application of Reynolds flux modeling in CFD simulation of 45\(^\circ\) inclined negatively buoyant jets. Environ Fluid Mech 1:55–86

    Article  Google Scholar 

  23. Mohammadian A, Gildeh HK, Nistor I (2020) CFD modeling of effluent discharges: a review of past numerical studies. Water 12(3):856

    Article  Google Scholar 

  24. Doneker RL, Jirka GH (2007) Cormix user manual: a hydrodynamic mixing zone model and decision support system for pollutant discharges into surface waters. US Environmental Protection Agency, Washington DC

    Google Scholar 

  25. Robinson D, Wood M, Piggott M, Gorman G (2016) Cfd modelling of marine discharge mixing and dispersion. J Appl Water Eng Res 4(2):152–162

    Article  Google Scholar 

  26. Papakonstantis IG, Christodoulou GC (2020) Simplified modelling of inclined turbulent dense jets. Fluids 5(4):204

    Article  Google Scholar 

  27. Crowe A, Davidson M, Nokes R (2016) Modified reduced buoyancy flux model for desalination discharges. Desalination 378:53–59

    Article  Google Scholar 

  28. Morton B, Taylor GI, Turner JS (1956) Turbulent gravitational convection from maintained and instantaneous sources. Proc R Soc Lond Ser A Math Phys Sci 234(1196):1–23

    Google Scholar 

  29. Fan LN (1967) Turbulent buoyant jets into stratified or flowing ambient fluids

  30. Wang H, Law AW-k (2002) Second-order integral model for a round turbulent buoyant jet. J Fluid Mech 459:397–428

    Article  Google Scholar 

  31. Jirka GH (2004) Integral model for turbulent buoyant jets in unbounded stratified flows part i: single round jet. Environ Fluid Mech 4(1):1–56

    Article  Google Scholar 

  32. Yannopoulos P (2006) An improved integral model for plane and round turbulent buoyant jets. J Fluid Mech 547:267–296

    Article  Google Scholar 

  33. Oliver C, Davidson M, Nokes R (2013) Predicting the near-field mixing of desalination discharges in a stationary environment. Desalination 309:148–155

    Article  Google Scholar 

  34. Yannopoulos P, Bloutsos A (2012) Escaping mass approach for inclined plane and round buoyant jets. J Fluid Mech 695:81–111

    Article  Google Scholar 

  35. Bloutsos AA, Yannopoulos PC (2020) Revisiting mean flow and mixing properties of negatively round buoyant jets using the escaping mass approach (EMA). Fluids 5(3):131

    Article  Google Scholar 

  36. Papakonstantis IG, Papanicolaou PN (2022) On the computational modeling of inclined brine discharges. Fluids 7(2):86

    Article  Google Scholar 

  37. Jirka GH (2008) Improved discharge configurations for brine effluents from desalination plants. J Hydraul Eng 134(1):116–120

    Article  Google Scholar 

  38. Vafeiadou P, Papakonstantis I, Christodoulou G (2005) Numerical simulation of inclined negatively buoyant jets. In: The 9th international conference on environmental science and technology, September, pp 1–3

  39. Oliver C, Davidson M, Nokes R (2008) k-\(\varepsilon\) predictions of the initial mixing of desalination discharges. Environ Fluid Mech 8(5–6):617

    Article  Google Scholar 

  40. Kheirkhah Gildeh H, Mohammadian A, Nistor I, Qiblawey H, Yan X (2015) CFD modeling and analysis of the behavior of 30 and 45 inclined dense jets-new numerical insights. J Appl Water Eng Res 4(2):112–127

    Article  Google Scholar 

  41. Kheirkhah Gildeh H, Mohammadian A, Nistor I, Qiblawey H (2015) Numerical modeling of 30 and 45 degree inclined dense turbulent jets in stationary ambient. Environ Fluid Mech 15(3):537–562

    Article  Google Scholar 

  42. Tahmooresi S, Ahmadyar D (2021) Effects of turbulent Schmidt number on CFD simulation of 45\(^\circ\) inclined negatively buoyant jets. Environ Fluid Mech 21(1):39–62

    Article  Google Scholar 

  43. Ramezani M, Abessi O, Firoozjaee AR (2021) Effect of proximity to bed on 30\(^\circ\) and 45\(^\circ\) inclined dense jets: a numerical study

  44. Zhang S, Jiang B, Law AW-K, Zhao B (2015) Large eddy simulations of 45 inclined dense jets. Environ Fluid Mech 16(1):101–121

    Article  Google Scholar 

  45. Zhang S, Law AW-K, Jiang M (2017) Large eddy simulations of 45 and 60 inclined dense jets with bottom impact. J Hydro Environ Res 15:54–66

    Article  Google Scholar 

  46. Jiang M, Law AW-K, Lai AC (2019) Turbulence characteristics of 45 inclined dense jets. Environ Fluid Mech 19(1):27–54

    Article  Google Scholar 

  47. Jiang M, Law AW-K, Song J (2019) Mixing characteristics of inclined dense jets with different nozzle geometries. J Hydro Environ Res 27:116–128

    Article  Google Scholar 

  48. Jiang M, Law AW-K (2018) Mixing of swirling inclined dense jets—a numerical study. J Hydro Environ Res 21:118–130

    Article  Google Scholar 

  49. Vafa N, Azadi A, Firoozabadi B (2021) Effect of numerical schemes on advection terms for large eddy simulation of inclined dense jets. Environ Fluid Mech 21:1227–52

    Article  Google Scholar 

  50. Palomar P, Lara J, Losada I (2012) Near field brine discharge modeling part 2: validation of commercial tools. Desalination 290:28–42

    Article  Google Scholar 

  51. Rodi W (1980) Turbulence models and their application in hydraulics. Germany: University of Karlsruhe,

  52. Millero FJ, Poisson A (1981) International one-atmosphere equation of state of seawater. Deep Sea Re Part A Oceanogr Res Pap 28(6):625–629

    Article  Google Scholar 

  53. Younis BA, Speziale CG, Clark TT (2005) A rational model for the turbulent scalar fluxes. Proc R Soc A Math Phys Eng Sci 461(2054):575–594

    Google Scholar 

  54. Batchelor G (1949) Diffusion in a field of homogeneous turbulence i. Eulerian analysis. Aust J Chem 2(4):437–450

    Article  Google Scholar 

  55. Daly BJ, Harlow FH (1970) Transport equations in turbulence. Phys Fluids 13(11):2634–2649

    Article  Google Scholar 

  56. Hanjalić K, Launder B (2011) Modelling turbulence in engineering and the environment: second-moment routes to closure. Cambridge University Press, Cambridge

    Book  Google Scholar 

  57. Launder B (1976) Heat and mass transport, chapter 6 in turbulence, ed. by p. bradshaw. Topics in Applied Physics 12

  58. Launder B (1989) The prediction of force field effects on turbulent shear flows via second-moment closure. In: Advances in turbulence 2, pp. 338–358. Springer

  59. Hossain M, Rodi W (1982) A turbulence model for buoyant flows and its application to vertical buoyant jets. In: Turbulent buoyant jets and plumes, pp 121–178. Elsevier

  60. Monin A (1965) On the symmetry properties of turbulence in the surface layer of air. Isv Atmos Ocean Phys 1:45–54

    Google Scholar 

  61. Launder B (1975) On the effects of a gravitational field on the turbulent transport of heat and momentum. J Fluid Mech 67(3):569–581

    Article  Google Scholar 

  62. Gibson M, Launder B (1978) Ground effects on pressure fluctuations in the atmospheric boundary layer. J Fluid Mech 86(3):491–511

    Article  Google Scholar 

  63. Gibson M, Launder B (1976) On the calculation of horizontal nonequilibrium turbulent shear flows under gravitational influence. J Heat Trans 98:81–87

    Article  Google Scholar 

  64. Speziale CG, Sarkar S, Gatski TB (1991) Modelling the pressure-strain correlation of turbulence: an invariant dynamical systems approach. J Fluid Mech 227:245–272

    Article  Google Scholar 

  65. Greenshields CJ (2018) Openfoam user guide version 6. The OpenFOAM Foundation 237

  66. Roberts PJ, Toms G (1987) Inclined dense jets in flowing current. J Hydraul Eng 113(3):323–340

    Article  Google Scholar 

  67. Malin M, Younis B (1990) Calculation of turbulent Buoyant plumes with a Reynolds stress and heat flux transport closure. Int J Heat Mass Transf 33(10):2247–2264

    Article  Google Scholar 

  68. Shao D, Law AW-K (2009) Turbulent mass and momentum transport of a circular offset dense jet. J Turbul 10(10):40

    Article  Google Scholar 

  69. Cresswell R, Szczepura R (1993) Experimental investigation into a turbulent jet with negative buoyancy. Phys Fluids A 5(11):2865–2878

    Article  Google Scholar 

Download references

Acknowledgements

This research was enabled in part by support provided by WestGrid (www.westgrid.ca) and Compute Canada Calcul Canada (www.computecanada.ca).

Funding

The authors have no relevant financial or non-financial interests to disclose.

Author information

Author notes

  1. Abdolmajid Mohammadian, Ioan Nistor and Hossein Kheirkhah Gildeh have contributed equally to this work.

Authors and Affiliations

  1. Department of Civil Engineering, University of Ottawa, 75 Laurier Ave E, Ottawa, ON, K1N 6N5, Canada

    Sina Tahmooresi, Abdolmajid Mohammadian, Ioan Nistor & Hossein Kheirkhah Gildeh

Authors
  1. Sina Tahmooresi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdolmajid Mohammadian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ioan Nistor
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hossein Kheirkhah Gildeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors whose names appear on the submission made substantial contributions to the conception or design of the work.

Corresponding author

Correspondence to Sina Tahmooresi.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tahmooresi, S., Mohammadian, A., Nistor, I. et al. Reynolds flux modeling; new numerical insights into inclined dense jets. Environ Fluid Mech 23, 551–577 (2023). https://doi.org/10.1007/s10652-023-09919-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10652-023-09919-z

Keywords

Search

Navigation