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Theoretical model of continuous inertial gravity currents including a jump condition

Safir Haddad1,*, Samuel Vaux1,†, Kevin Varrall2,‡, and Olivier Vauquelin2,§

  • 1Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PSN-RES, SA2I, LIE, Cadarache, 13115 Saint-Paul-lez-Durance, France
  • 2Aix-Marseille Université (AMU), Laboratoire IUSTI, CNRS UMR 7343, 5 Rue Enrico Fermi, 13453 Marseille Cedex, France

  • *safir.haddad@irsn.fr
  • †samuel.vaux@irsn.fr
  • ‡kevin.varrall@univ-amu.fr
  • §olivier.vauquelin@univ-amu.fr
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Phys. Rev. Fluids 7, 084802 – Published 19 August, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.084802

Abstract

This paper examines the theoretical modeling of a steady horizontal gravity current involving miscible fluids. The main objective is to determine the longitudinal evolution of the current characteristic quantities: its mean velocity U, its height h, its mean density ρ, and the local Richardson number Ri=gΔρh/ρU2, which characterizes the flow regime. The theory developed by Ellison and Turner [J. Fluid Mech. 6, 423 (1959)] for Boussinesq gravity currents is first extended to the general non-Boussinesq case. In this theoretical approach, the differential equations derived from the conservation equations reveal a mathematical singularity which no longer allows them to be solved when the current passes from a supercritical to a subcritical regime, i.e., when the Richardson number reaches unity. To circumvent this problem, we propose to introduce a jump condition into the model which leads to a sudden transition from a supercritical to a subcritical regime when needed. The jump location is set to satisfy the boundary condition at the exit. Numerical simulations are carried out using a large-eddy simulation code in order to obtain reference results. These results are first used to select a suitable entrainment model among those proposed in the literature. The simulations are then compared with the theoretical model. In the case of a supercritical current without regime change, the agreement between simulation and theory is good. In the case of a supercritical current turning subcritical before the exit, the discontinuity imposed in the model by the jump is clearly abrupt in comparison with the physical reality, but it allows us to reproduce the nonmonotonic evolution of the velocity, height, and Richardson number of the current and to obtain an acceptable estimation of these quantities.

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References (58)

  1. D. R. Fitzjarrald, Katabatic wind in opposing flow, J. Atmos. Sci. 41, 1143 (1984).
  2. G. A. Valentine, Stratified flow in pyroclastic surges, Bull. Volcanol. 49, 616 (1987).
  3. E. J. Hopfinger, Snow avalanche motion and related phenomena, Annu. Rev. Fluid Mech. 15, 47 (1983).
  4. E. Meiburg and B. Kneller, Turbidity currents and their deposits, Annu. Rev. Fluid Mech. 42, 135 (2010).
  5. M. R. Flynn and P. F. Linden, Intrusive gravity currents, J. Fluid Mech. 568, 193 (2006).
  6. R. E. Britter and J. E. Simpson, A note on the structure of the head of an intrusive gravity current, J. Fluid Mech. 112, 459 (1981).
  7. T. von Kármán, The engineer grapples with nonlinear problems, Bull. Amer. Math. Soc. 46, 615 (1940).
  8. T. B. Benjamin, Gravity currents and related phenomena, J. Fluid Mech. 31, 209 (1968).
  9. D. Hoult, Oil spreading on the sea, Annu. Rev. Fluid Mech. 4, 341 (1972).
  10. J. E. Simpson and R. E. Britter, The dynamics of the head of a gravity current advancing over a horizontal surface, J. Fluid Mech. 94, 477 (1979).
  11. H. E. Huppert, The propagation of two-dimensional and axisymmetric viscous gravity currents at a fluid interface, J. Fluid Mech. 121, 43 (1982).
  12. A. Dai, C. E. Ozdemir, M. I. Cantero, and S. Balachandar, Gravity currents from instantaneous sources down a slope, J. Hydraul. Eng. 138, 237 (2012).
  13. Z. Borden, E. Meiburg, and G. Constantinescu, Internal bores: An improved model via a detailed analysis of the energy budget, J. Fluid Mech. 703, 279 (2012).
  14. M. A. Khodkar, M. M. Nasr-Azadani, and E. Meiburg, Partial-depth lock-release flows, Phys. Rev. Fluids 2, 064802 (2017).
  15. M. Ungarish, An Introduction to Gravity Currents and Intrusions (CRC Press, New York, 2009).
  16. M. R. Chowdhury and F. Y. Testik, A review of gravity currents formed by submerged single-port discharges in inland and coastal waters, Environ. Fluid Mech. 14, 265 (2014).
  17. T. H. Ellison and J. S. Turner, Turbulent entrainment in stratified flows, J. Fluid Mech. 6, 423 (1959).
  18. B. R. Morton, G. Taylor, and J. S. Turner, Turbulent gravitational convection from maintained and instantaneous sources, Proc. R. Soc. London A 234, 1 (1956).
  19. K. Lofquist, Flow and stress near an interface between stratified liquids, Phys. Fluids 3, 158 (1960).
  20. G. Parker, M. Garcia, Y. Fukushima, and W. Yu, Experiments on turbidity currents over an erodible bed, J. Hydraul. Res. 25, 123 (1987).
  21. T. van Kessel and C. Kranenburg, Gravity current of fluid mud on sloping bed, J. Hydraul. Eng. 122, 710 (1996).
  22. C. J. Dallimore, J. Imberger, and T. Ishikawa, Entrainment and turbulence in saline underflow in Lake Ogawara, J. Hydraul. Eng. 127, 937 (2001).
  23. R. L. Alpert, Turbulent ceiling-jet induced by large-scale fires, Combust. Sci. Technol. 11, 197 (1975).
  24. G. H. Jirka, Turbulent Buoyant Jets and Plumes, edited by W. Rodi, Vol. 6 (Pergamon Press, Oxford, 1982), pp. 69–119.
  25. M. Princevac, H. J. Fernando, and C. D. Whiteman, Turbulent entrainment into natural gravity-driven flows, J. Fluid Mech. 533, 259 (2005).
  26. G. C. Christodoulou, Interfacial mixing in stratified flows, J. Hydraul. Res. 24, 77 (1986).
  27. H. J. Fernando, Turbulent mixing in stratified fluids, Annu. Rev. Fluid Mech. 23, 455 (1991).
  28. Q. Guo, Y. Z. Li, H. Ingason, Z. Yan, and H. Zhu, Theoretical studies on buoyancy-driven ceiling jets of tunnel fires with natural ventilation, Fire Safety J. 119, 103228 (2021).
  29. D. Bonn, A. Andersen, and T. Bohr, Hydraulic jumps in a channel, J. Fluid Mech. 618, 71 (2009).
  30. R. Dasgupta, G. Tomar, and R. Govindarajan, Numerical study of laminar, standing hydraulic jumps in a planar geometry, Eur. Phys. J. E 38, 45 (2015).
  31. M. Dhar, G. Das, and P. K. Das, Planar hydraulic jumps in thin film flow, J. Fluid Mech. 884, A11 (2020).
  32. V. T. Chow, Open-Channel Hydraulics, McGraw-Hill Civil Engineering Series (McGraw-Hill, New York, 1959).
  33. F. Henderson, Open Channel Flow, Macmillan Series in Civil Engineering (Macmillan, New York, 1966).
  34. L. Gu and G. A. Lawrence, Analytical solution for maximal frictional two-layer exchange flow, J. Fluid Mech. 543, 1 (2005).
  35. J. P. Kunsch, Critical velocity and range of a fire-gas plume in a ventilated tunnel, Atmos. Environ. 33, 13 (1998).
  36. L. Hu, R. Huo, Y. Li, H. Wang, and W. Chow, Full-scale burning tests on studying smoke temperature and velocity along a corridor, Tunnell. Underground Space Tech. 20, 223 (2005).
  37. W. Chow, Y. Gao, J. Zhao, J. Dang, C. Chow, and L. Miao, Smoke movement in tilted tunnel fires with longitudinal ventilation, Fire Safety J. 75, 14 (2015).
  38. W. Chow, Y. Gao, J. Zhao, J. Dang, and N. C. Chow, A study on tilted tunnel fire under natural ventilation, Fire Safety J. 81, 44 (2016).
  39. D. L. Wilkinson and I. R. Wood, A rapidly varied flow phenomenon in a two-layer flow, J. Fluid Mech. 47, 241 (1971).
  40. C.-S. Yih and C. R. Guha, Hydraulic jump in a fluid system of two layers, Tellus 7, 358 (1955).
  41. A. Regev, S. Hassid, and M. Poreh, Density jumps in smoke flow along horizontal ceilings, Fire Safety J. 39, 465 (2004).
  42. A. Regev, S. Hassid, and M. Poreh, Calculation of entrainment in density jumps, Environ. Fluid Mech. 6, 407 (2006).
  43. N. Najafpour, M. Sarnie, B. Firoozabadi, and H. Afshin, Theoretical and experimental investigation of density jump on an inclined surface, Sci. Iran. 21, 1655 (2014).
  44. J. R. Dormand and P. J. Prince, A reconsideration of some embedded Runge-Kutta formulae, J. Comput. Appl. Math. 15, 203 (1986).
  45. F. Nicoud and F. Ducros, Subgrid-scale stress modelling based on the square of the velocity gradient tensor, Flow Turbul. Combust. 62, 183 (1999).
  46. C. H. Bruneau and P. Fabrie, Effective downstream boundary conditions for incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids 19, 693 (1994).
  47. C. H. Bruneau and P. Fabrie, New efficient boundary conditions for incompressible Navier-Stokes equations: A well-posedness result, ESAIM: M2AN 30, 815 (1996).
  48. C. H. Bruneau, Boundary conditions on artificial frontiers for incompressible and compressible Navier-Stokes equations, ESAIM: M2AN 34, 303 (2000).
  49. X. Zhou, K. H. Luo, and J. J. Williams, Large-eddy simulation of a turbulent forced plume, Eur. J. Mech. B 20, 233 (2001).
  50. R. Mehaddi, S. Vaux, F. Candelier, and O. Vauquelin, On the modelling of steady turbulent fountains, Environ. Fluid Mech. 15, 1115 (2015).
  51. S. Vaux, R. Mehaddi, O. Vauquelin, and F. Candelier, Upward versus downward non-Boussinesq turbulent fountains, J. Fluid Mech. 867, 374 (2019).
  52. W. H. Hager, R. Bremen, and N. Kawagoshi, Classical hydraulic jump: Length of roller, J. Hydraul. Res. 28, 591 (1990).
  53. H. G. Hornung, C. Willert, and S. Turner, The flow field downstream of a hydraulic jump, J. Fluid Mech. 287, 299 (1995).
  54. G. L. Richard and S. L. Gavrilyuk, The classical hydraulic jump in a model of shear shallow-water flows, J. Fluid Mech. 725, 492 (2013).
  55. Z. Borden and E. Meiburg, Circulation based models for Boussinesq gravity currents, Phys. Fluids 25, 101301 (2013).
  56. M. A. Khodkar, M. M. Nasr-Azadani, and E. Meiburg, Gravity currents propagating into two-layer stratified fluids: Vorticity-based models, J. Fluid Mech. 844, 994 (2018).
  57. S. W. Bauer and W. H. Graf, Free overall as flow measuring device, J. Irrig. Drain. Div. 97, 73 (1971).
  58. W. H. Graf and M. S. Altinakar, Hydraulique Fluviale, Vol. 16 (Presses Polytechniques et Universitaires Romandes, Lausanne, 2000).

Outline

Information

Phys. Rev. Fluids 7, 084802– Published 19 August, 2022
  • Received 26 December 2021
  • Accepted 25 July 2022
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