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Inertial gravity current produced by the drainage of a cylindrical reservoir from an outer or inner edge

Published online by Cambridge University Press:  04 July 2019

Marius Ungarish Open the ORCID record for Marius Ungarish [Opens in a new window] ,
Lailai Zhu Open the ORCID record for Lailai Zhu [Opens in a new window]  and
Howard A. Stone Open the ORCID record for Howard A. Stone [Opens in a new window]
Marius Ungarish*
Affiliation:
Department of Computer Science, Technion – Israel Institute of Technology, Haifa 32000, Israel
Lailai Zhu
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Linné Flow Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, Stockholm, SE-10044, Sweden
Howard A. Stone
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: unga@cs.technion.ac.il
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Abstract

We consider the time-dependent flow of a fluid of density $\unicode[STIX]{x1D70C}_{1}$ in a vertical cylindrical container embedded in a fluid of density $\unicode[STIX]{x1D70C}_{2}~({<}\unicode[STIX]{x1D70C}_{1})$ whose side boundary is suddenly removed and the fluid drains freely from the edge. We show that in the inertial–buoyancy regime (large initial Reynolds number) the flow is modelled by the shallow-water equations and bears similarities to a gravity current released from a lock (the dam-break problem) driven by the reduced gravity $g^{\prime }=(1-\unicode[STIX]{x1D70C}_{2}/\unicode[STIX]{x1D70C}_{1})g$. This formulation is amenable to an efficient finite-difference solution. Moreover, we demonstrate that similarity solutions exist, and show that the flow created by the dam break approaches the predicted self-similar behaviour when the volume ratio ${\mathcal{V}}(t)/{\mathcal{V}}(0)\approx 1/2$ where $t$ is time elapsed from the dam break. We considered two cases of drainage: (i) outward from the outer boundary in a full-radius reservoir; and (ii) inward from the inner radius in an annular-shaped reservoir. For the first case the similarity solution is expressed analytically, while the second case is more complicated and requires a numerical solution. In both cases ${\mathcal{V}}(t)/{\mathcal{V}}(0)$ decays like $t^{-2}$, but the details are different. The similarity solutions admit an adjustable virtual-origin constant, which we determine by matching with the finite-difference solution. The analysis is valid for both Boussinesq and non-Boussinesq systems, and a wide range of geometric parameters (inner and outer radii, and height). The importance of the neglected viscous terms increases with time, and eventually the inertial–buoyancy model becomes invalid. An estimate for this occurrence is also provided. The predictions of the model are compared to results of direct numerical simulations; there is good agreement for the position of the interface and for the averaged radial velocity, and excellent agreement for ${\mathcal{V}}(t)/{\mathcal{V}}(0)$. A box model is used for estimating the effect of a partial (over a sector) dam break. This study is an extension of the work for a rectangular reservoir of Momen et al. (J. Fluid Mech., vol. 827, 2017, pp. 640–663). We demonstrate that there are some similarities, but also significant differences, between the rectangular and the cylindrical reservoirs concerning the velocity, shape of the interface and rate of drainage, which are of interest in applications. The overall conclusion is that this simple model captures very well the flow field under consideration.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Gravity currents Geophysical and Geological Flows: Shallow water flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 874 , 10 September 2019 , pp. 185 - 209
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Copyright
© 2019 Cambridge University Press 

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References

Acton, J. M., Huppert, H. E. & Worster, M. G. 2002 Two-dimensional viscous gravity currents flowing over a deep porous medium. J. Fluid Mech. 440, 359–380.Google Scholar
Anderson, D. A., Tannehill, J. C. & Pletcher, R. M. 1984 Computational Fluid Mechanics and Heat Transfer. Hemisphere.Google Scholar
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.Google Scholar
Bonometti, T., Ungarish, M. & Balachandar, S. 2011 A numerical investigation of constant-volume non-Boussinesq gravity currents in deep ambient. J. Fluid Mech. 673, 574–602.Google Scholar
Childs, H. et al. 2012 VisIt: an end-user tool for visualizing and analyzing very large data. In High Performance Visualization–Enabling Extreme-Scale Scientific Insight, pp. 357–372. Chapman & Hall/CRC.Google Scholar
Hogg, A. J., Baldock, T. E. & Pritchard, D. 2011 Overtopping a truncated planar beach. J. Fluid Mech. 666, 521–553.Google Scholar
Martin, J. C. & Moyce, W. J. 1952a An experimental study of the collapse of fluid columns on a rigid horizontal plane, in a medium of lower, but comparable, density. Phil. Trans. R. Soc. Lond. A 244, 325–334.Google Scholar
Martin, J. C. & Moyce, W. J. 1952b An experimental study of the collapse of liquid columns on a rigid horizontal plane. Phil. Trans. R. Soc. Lond. A 244, 312–324.Google Scholar
Momen, M., Zheng, Z., Bou-Zeid, E. & Stone, H. A. 2017 Inertial gravity currents produced by fluid drainage from an edge. J. Fluid Mech. 827, 640–663.Google Scholar
Monismith, S. G., Mcdonald, N. R. & Imberger, J. 1993 Axisymmetric selective withdrawal in a rotating stratified fluid. J. Fluid Mech. 249, 287–305.Google Scholar
Morton, K. W. & Mayers, D. F. 1994 Numerical Solutions of Partial Differential Equations. Cambridge University Press.Google Scholar
Ray, W. J. & Raba, G. W. 1991 Behavior of welded steel water-storage tank. J. Struct. Engng 117, 61–79.Google Scholar
Roenby, J., Bredmose, H. & Jasak, H. 2016 A computational method for sharp interface advection. R. Soc. Open Sci. 3 (11), 160405.Google Scholar
Simpson, J. E. 1997 Gravity Currents in the Environment and the Laboratory. Cambridge University Press.Google Scholar
Ungarish, M. 2009 An Introduction to Gravity Currents and Intrusions. Chapman & Hall/CRC Press.Google Scholar
Ungarish, M. 2010 The propagation of high-Reynolds-number non-Boussinesq gravity currents in axisymmetric geometry. J. Fluid Mech. 643, 267–277.Google Scholar
Ungarish, M. & Huppert, H. E. 2000 High Reynolds number gravity currents over a porous boundary: shallow-water solutions and box-model approximations. J. Fluid Mech. 418, 1–23.Google Scholar
Weller, H. G., Tabor, G., Jasak, H. & Fureby, C. 1998 A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys. 12 (6), 620–631.Google Scholar
Zgheib, N., Bonometti, T. & Balachandar, S. 2015 Dynamics of non-circular finite-release gravity currents. J. Fluid Mech. 783, 344–378.Google Scholar