##
**Removal of a dense bottom layer by a gravity current.**
*(English)*
Zbl 1486.76029

Summary: We investigate the removal of a dense bottom layer by a gravity current, via Navier-Stokes simulations in the Boussinesq limit. The problem is governed by a dimensionless thickness parameter for the bottom layer, and by the ratio of the density differences between bottom layer, gravity current and ambient fluids. A quasisteady gravity current forms that propagates along the interface and displaces some of the dense bottom fluid, which accumulates ahead of the gravity current and forms an undular bore or a series of internal gravity waves. Depending on the ratio of the gravity current front velocity to the linear shallow-water wave velocity, we observe the existence of different regimes, characterized by small-amplitude waves or by a train of steep, nonlinear internal waves. We develop a semiempirical model that provides reasonable estimates of several important flow properties. We also formulate a more sophisticated, self-contained model based on the conservation principles for mass and vorticity that does not require empirical closure assumptions. This model is able to predict such quantities as the gravity current height and the internal wave or bore velocity as a function of the governing dimensionless parameters, generally to within approximately a 10% accuracy. An energy budget analysis provides information on the rates at which potential energy is converted into kinetic energy and then dissipated, and on the processes by which energy is transferred from the gravity current fluid to the dense and ambient fluids. We observe that the energy content of thicker and denser bottom layers grows more rapidly.

### MSC:

76D50 | Stratification effects in viscous fluids |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76M20 | Finite difference methods applied to problems in fluid mechanics |

86A05 | Hydrology, hydrography, oceanography |

### Keywords:

quasisteady gravity current; internal wave; stratified flow; Navier-Stokes equations; Boussinesq approximation; conservation law; finite difference scheme
PDF
BibTeX
XML
Cite

\textit{R. Zhu} et al., J. Fluid Mech. 916, Paper No. A30, 23 p. (2021; Zbl 1486.76029)

Full Text:
DOI

### References:

[1] | Benjamin, T.B.1968Gravity currents and related phenomena. J. Fluid Mech.31 (2), 209-248. · Zbl 0169.28503 |

[2] | Biegert, E., Vowinckel, B. & Meiburg, E.2017aA collision model for grain-resolving simulations of flows over dense, mobile, polydisperse granular sediment beds. J. Comput. Phys.340, 105-127. · Zbl 1376.76069 |

[3] | Biegert, E., Vowinckel, B., Ouillon, R. & Meiburg, E.2017bHigh-resolution simulations of turbidity currents. Prog. Earth Planet. Sci.4 (1), 33. |

[4] | Borden, Z. & Meiburg, E.2013aCirculation based models for Boussinesq gravity currents. Phys. Fluids25 (10), 101301. · Zbl 1287.76073 |

[5] | Borden, Z. & Meiburg, E.2013bCirculation-based models for Boussinesq internal bores. J. Fluid Mech.726, R1. · Zbl 1287.76073 |

[6] | Britter, R.E. & Simpson, J.E.1981A note on the structure of the head of an intrusive gravity current. J. Fluid Mech.112, 459-466. |

[7] | Carvalho, L., et al.2020The sundowner winds experiment (SWEX) pilot study: understanding downslope windstorms in the Santa Ynez Mountains, Santa Barbara, California. Mon. Weather Rev.148 (4), 1519-1539. |

[8] | Cheong, H.-B., Kuenen, J.J.P. & Linden, P.F.2006The front speed of intrusive gravity currents. J. Fluid Mech.552, 1-11. · Zbl 1087.76022 |

[9] | De Rooij, F., Linden, P.F. & Dalziel, S.B.1999Saline and particle-driven interfacial intrusions. J. Fluid Mech.389, 303-334. · Zbl 0954.76092 |

[10] | Flynn, M.R. & Linden, P.F.2006Intrusive gravity currents. J. Fluid Mech.568, 193-202. · Zbl 1178.76108 |

[11] | He, Z., Zhao, L., Lin, T., Hu, P., Lv, Y., Ho, H.-C. & Lin, Y.-T.2017Hydrodynamics of gravity currents down a ramp in linearly stratified environments. J. Hydraul. Engng143 (3), 04016085. |

[12] | He, Z., Zhao, L., Zhu, R. & Hu, P.2019Separation of particle-laden gravity currents down a slope in linearly stratified environments. Phys. Fluids31 (10), 106602. |

[13] | Holyer, J.Y. & Huppert, H.E.1980Gravity currents entering a two-layer fluid. J. Fluid Mech.100 (4), 739-767. · Zbl 0444.76086 |

[14] | Khodkar, M.A., Nasr-Azadani, M.M. & Meiburg, E.2016Intrusive gravity currents propagating into two-layer stratified ambients: vorticity modeling. Phys. Rev. Fluid1 (4), 044302. · Zbl 1429.76065 |

[15] | Kundu, P.K. & Cohen, I.M.2001Fluid Mechanics, pp. 214-277. Elsevier. |

[16] | Lattemann, S. & Amy, G.2013Marine monitoring surveys for desalination plants—a critical review. Desalination Water Treat.51 (1-3), 233-245. |

[17] | Lowe, R.J., Linden, P.F. & Rottman, J.W.2002A laboratory study of the velocity structure in an intrusive gravity current. J Fluid Mech.456, 33-48. · Zbl 0987.76508 |

[18] | Mehta, A.P., Sutherland, B.R. & Kyba, P.J.2002Interfacial gravity currents. II. Wave excitation. Phys. Fluids14 (10), 3558-3569. · Zbl 1185.76252 |

[19] | Monaghan, J.J.2007Gravity current interaction with interfaces. Annu. Rev. Fluid Mech.39, 245-261. · Zbl 1296.76013 |

[20] | Nasr-Azadani, M.M., Hall, B. & Meiburg, E.2013Polydisperse turbidity currents propagating over complex topography: comparison of experimental and depth-resolved simulation results. Comput. Geosci.53, 141-153. |

[21] | Nasr-Azadani, M.M. & Meiburg, E.2011Turbins: an immersed boundary, Navier-Stokes code for the simulation of gravity and turbidity currents interacting with complex topographies. Comput. Fluids45 (1), 14-28. · Zbl 1429.76018 |

[22] | Necker, F., Härtel, C., Kleiser, L. & Meiburg, E.2005Mixing and dissipation in particle-driven gravity currents. J. Fluid Mech.545, 339-372. · Zbl 1085.76559 |

[23] | Ooi, S.K., Constantinescu, G. & Weber, L.J.20072D large-eddy simulation of lock-exchange gravity current flows at high Grashof numbers. J. Hydraul. Eng.133 (9), 1037-1047. |

[24] | Ouillon, R., Meiburg, E., Ouellette, N.T. & Koseff, J.R.2019aInteraction of a downslope gravity current with an internal wave. J. Fluid Mech.873, 889-913. · Zbl 1419.76133 |

[25] | Ouillon, R., Meiburg, E. & Sutherland, B.R.2019bTurbidity currents propagating down a slope into a stratified saline ambient fluid. Environ. Fluid Mech.19 (5), 1143-1166. |

[26] | Samothrakis, P. & Cotel, A.J.2006aFinite volume gravity currents impinging on a stratified interface. Exp. Fluids41 (6), 991-1003. |

[27] | Samothrakis, P. & Cotel, A.J.2006bPropagation of a gravity current in a two-layer stratified environment. J. Geophys. Res.-Oceans111, C01012. |

[28] | Simpson, J.E.1982Gravity currents in the laboratory, atmosphere, and ocean. Annu. Rev. Fluid Mech.14 (1), 213-234. · Zbl 0488.76107 |

[29] | Smith, C., Hatchett, B. & Kaplan, M.2018Characteristics of sundowner winds near Santa Barbara, CA, from a dynamically downscaled climatology: environment and effects aloft and offshore. J. Geophys. Res.-Atmos.123 (23), 13-092. |

[30] | Sutherland, B.R., Kyba, P.J. & Flynn, M.R.2004Intrusive gravity currents in two-layer fluids. J. Fluid Mech.514, 327-353. · Zbl 1067.76021 |

[31] | Tanimoto, Y., Ouellette, N.T. & Koseff, J.R.2020Interaction between an inclined gravity current and a pycnocline in a two-layer stratification. J. Fluid Mech.887, A8. · Zbl 1460.76158 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.