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TECHNICAL PAPERS

# Direct Numerical Simulations of Planar and Cylindrical Density Currents

[+] Author and Article Information
Mariano I. Cantero1

Department of Civil and Environmental Engineering,  University of Illinois at Urbana-Champaign, Urbana, IL 61801mcantero@uiuc.edu

S. Balachandar

Department of Theoretical and Applied Mechanics,  University of Illinois at Urbana-Champaign, Urbana, IL 61801s-bala@uiuc.edu

Marcelo H. García

Department of Civil and Environmental Engineering,  University of Illinois at Urbana-Champaign, Urbana, IL 61801mhgarcia@uiuc.edu

James P. Ferry

Center for the Simulation of Advanced Rockets,  University of Illinois at Urbana-Champaign, Urbana, IL 61801jferry@uiuc.edu

1

To whom correspondence should be addressed.

J. Appl. Mech 73(6), 923-930 (Dec 19, 2005) (8 pages) doi:10.1115/1.2173671 History: Received May 24, 2005; Revised December 19, 2005

## Abstract

The collapse of a heavy fluid column in a lighter environment is studied by direct numerical simulation of the Navier-Stokes equations using the Boussinesq approximation for small density difference. Such phenomenon occurs in many engineering and environmental problems resulting in a density current spreading over a no-slip boundary. In this work, density currents corresponding to two Grashof (Gr) numbers are investigated ($105$ and $1.5×106$) for two very different geometrical configurations, namely, planar and cylindrical, with the goal of identifying differences and similarities in the flow structure and dynamics. The numerical model is capable of reproducing most of the two- and three-dimensional flow structures previously observed in the laboratory and in the field. Soon after the release of the heavier fluid into the quiescent environment, a density current forms exhibiting a well-defined head with a hanging nose followed by a shallower body and tail. In the case of large Gr, the flow evolves in a three-dimensional fashion featuring a pattern of lobes and clefts in the intruding front and substantial three-dimensionality in the trailing body. For the case of the lower Gr, the flow is completely two dimensional. The dynamics of the current is visualized and explained in terms of the mean flow for different phases of spreading. The initial phase, known as slumping phase, is characterized by a nearly constant spreading velocity and strong vortex shedding from the front of the current. Our numerical results show that this spreading velocity is influenced by Gr as well as the geometrical configuration. The slumping phase is followed by a decelerating phase in which the vortices move into the body of the current, pair, stretch and decay as viscous effects become important. The simulated dynamics of the flow during this phase is in very good agreement with previously reported experiments.

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## Figures

Figure 1

Sketch of a density current in cylindrical configuration showing the main features of the flow. The dashed line shows the initial condition; this is a cylindrical region of denser fluid located at the center of the domain. After the release a density current develops.

Figure 2

Three-dimensional planar current in lock-exchange configuration for Gr=1.5×106 and Sc=0.71. Flow visualized by an isosurface of density ρ̃=0.5. At t̃=0 the left half of the domain has ρ̃=1 and the right half ρ̃=0. The flow starts as two-dimensional forming the head of the current (t̃=5), then the flow turns three dimensional (t̃=10, 15, and 20) developing the lobes and clefts observed in experiments.

Figure 3

Mean flow of planar current in lock-exchange configuration for Gr=1.5×106 and Sc=0.71. Flow visualized by density contours. The flow starts symmetrically, but this symmetry is lost as the flow becomes three dimensional. Observe also the beginning of vortex pairing at t̃=15 in the front advancing to the right.

Figure 4

Front velocity in the slumping phase as a function of Gr number. Planar refers to the planar lock-exchange configuration and Cylindrical to the finite volume release in cylindrical configuration. The open square is the outcome of the simulation by Härtel, Meiburg, and Necker (22) in good agreement with our results.

Figure 5

Three-dimensional cylindrical current for Gr=1.5×106 and Sc=1. Flow visualized by an isosurface of density ρ̃=0.25. The figure shows only one quarter of the simulation domain. At t̃=0 the cylindrical region has ρ̃=1 and everywhere outside it ρ̃=0. The flow starts as two dimensional, but soon after it develops three-dimensional instabilities at the front. The flow becomes completely three dimensional eventually.

Figure 6

Maximum value of ρ̃(ρ̃max) over time. Planar refers to the planar lock-exchange configuration and Cylindrical to the finite volume release in cylindrical configuration, both for Gr=1.5×106. The value of ρ̃max in the current is related to the front velocity. While ρ̃max=1 the current moves at approximately constant velocity.

Figure 7

Lobe and cleft instability in a cylindrical current. (a): Visualization of the front in a laboratory experiment for Gr∼108 and Sc=700 using the same geometrical configuration of the numerical simulations. (b): Numerical result for Gr=1.5×106 and Sc=1.

Figure 8

Three-dimensional cylindrical current for Gr=105 and Sc=1. Flow visualized by surface of density ρ̃=0.25. The figure shows only one quarter of the simulation domain. For this case, the pattern of lobes and clefts is not observed.

Figure 9

Mean flow of cylindrical current for Gr=1.5×106 and Sc=1. Flow visualized by density contours. The main vortex structures are indicated in the figure. The dynamic of the vortical structures is in complete agreement with the experimental results of Alahyari and Longmire (24) for Gr∼107.

Figure 10

Mean flow of cylindrical current for Gr=105 and Sc=1. Flow visualized by density contours. For this case the flow presents weak vortex structures.

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