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Review Article

Modeling Gravity and Turbidity Currents: Computational Approaches and Challenges

[+] Author and Article Information
Eckart Meiburg

Department of Mechanical Engineering,
University of California,
Santa Barbara, CA 93106
e-mail: meiburg@engineering.ucsb.edu

Senthil Radhakrishnan

Department of Mechanical Engineering,
University of California,
Santa Barbara, CA 93106
e-mail: senthil@engineering.ucsb.edu

Mohamad Nasr-Azadani

Department of Mechanical Engineering,
University of California,
Santa Barbara, CA 93106
e-mail: mmnasr@engineering.ucsb.edu

1Corresponding author.

Manuscript received January 13, 2015; final manuscript received July 8, 2015; published online July 27, 2015. Assoc. Editor: Herman J. H. Clercx.

Appl. Mech. Rev 67(4), 040802 (Jul 27, 2015) (23 pages) Paper No: AMR-15-1008; doi: 10.1115/1.4031040 History: Received January 13, 2015

In this review article, we discuss recent progress with regard to modeling gravity-driven, high Reynolds number currents, with the emphasis on depth-resolving, high-resolution simulations. The initial sections describe new developments in the conceptual modeling of such currents for the purpose of identifying the Froude number–current height relationship, in the spirit of the pioneering work by von Kármán and Benjamin. A brief introduction to depth-averaged approaches follows, including box models and shallow water equations. Subsequently, we provide a detailed review of depth-resolving modeling strategies, including direct numerical simulations (DNS), large-eddy simulations (LES), and Reynolds-averaged Navier–Stokes (RANS) simulations. The strengths and challenges associated with these respective approaches are discussed by highlighting representative computational results obtained in recent years.

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Figures

Grahic Jump Location
Fig. 1

Idealized gravity current models introduced by (a) von Kármán [18] and (b) Benjamin [19]

Grahic Jump Location
Fig. 2

Model predictions for the Froude number as a function of the fractional current height α for gravity currents: Benjamin's model (solid line), circulation model (dashed-dotted line), and von Kármán model (solid circle)

Grahic Jump Location
Fig. 3

Schematic of a lock-exchange configuration. The lock region to the left initially contains a fluid of higher density ρ1, which is separated from the lighter fluid of density ρ2 to the right by a partition. The hydrostatic pressure difference across the partition generates a gravity current along the bottom wall when the partition is removed. Gravity acts in the vertical direction −y.

Grahic Jump Location
Fig. 5

Nondimensional particle deposit profiles as function of the streamwise coordinate. Results are shown for times t = 7.3 and t = 10.95, along with the final profile (t) after all particles have settled out. Solid line: two-dimensional simulation, dashed line: experimental data of de Rooij and Dalziel [76]. In both cases Re = 10,000, us = 0.02, l = 0.75, and H = d = 2.0 (Reprinted with permission from Necker et al. [57]. Copyright 2002 by Elsevier).

Grahic Jump Location
Fig. 6

(a) Time history of potential energy Ep and kinetic energy k. (b) Time history of the dissipation Ed due to the resolved motion, and the dissipation Es due to the unresolved Stokes flow around the individual particles. All energy components are normalized by the initial potential energy Ep0. Solid (dashed) lines indicate turbidity (gravity) current results. The dotted-dashed line in (a) gives the sum of Ep, k, Ed and Es for the turbidity current. Re = 2240. Initial parameters: L = 23, l = 1, H = 2, d = 2, and W = 2 (Reprinted with permission from Necker et al. [77]. Copyright 2005 by Cambridge University).

Grahic Jump Location
Fig. 7

“Structure of a particle-driven gravity current visualized by isosurfaces of concentration at t = 0 (a), t = 2 (b), t = 8 (c), and t = 14 (d). Results obtained from a 3D simulation for a Reynolds number of Re = 2240 and a dimensionless settling velocity of us = 0.02. In all cases, an isovalue of 0.25 is employed.” Note that at t = 14 the concentration in the rear part of the channel has almost dropped to zero. Initial parameters: L = 23, l = 1, H = 2, d = 2, and W = 2 (Reprinted with permission from Necker et al. [57]. Copyright 2002 by Elsevier).

Grahic Jump Location
Fig. 8

“Instantaneous shape of Boussinesq density currents (ρ1/ρ2 = 1.01) at time t = 7. Isolines of (ρ − ρ2)/(ρ1 − ρ2) = 0.05, 0.25, 0.5, 0.75, and 0.95 are shown.” Initial parameters: l = x0, H = 10, d = h0 = 1, L = 37.5 for λ = 18.75 and L = 12.5 for all other λ values (Reprinted with permission from Bonometti et al. [79]. Copyright 2011 by Cambridge University).

Grahic Jump Location
Fig. 9

“Time evolution of the front velocity for planar currents from three-dimensional simulations. The plot also includes experimental data from two of the lower Reynolds number experiments by Marino et al. [81] with l = x0 = 1 and d = h0 = 1. Also included are the theoretical predictions for all phases of spreading. The viscous phase predictions are for Re = 8950, x0 = 1 and h0 = 1” (Reprinted with permission from Cantero et al. [80]. Copyright 2007 by Cambridge University).

Grahic Jump Location
Fig. 10

“Visualization of the flow structure in the near-wall region of a turbulent gravity current for Re = 87,750. (a) Vertical vorticity contours near the bottom wall; (b) streamwise velocity contours showing the high- and low-speed streaks in a plane located at about 11 wall units from the bottom wall. The light and dark vorticity contours in (a) correspond to ωy = 2ub/h and ωy = −2ub/h, respectively” (Reprinted with permission from Ooi et al. [66]. Copyright 2009 by Cambridge University).

Grahic Jump Location
Fig. 4

Full-depth, lock-exchange Boussinesq gravity current at Re = 1 225. The flow is visualized at different times t by means of the three-dimensional density isosurface ρ = 0.5, along with density contours in the side-plane. Initial parameters: L = 23, l = 10, H = 2, d = 2, and W = 3 (Reprinted with permission from Härtel et al. [56]. Copyright 2000 by Cambridge University).

Grahic Jump Location
Fig. 11

Dominant unstable eigenfunction modes for a turbidity current propagating over a plain erodible sediment bed. Shown is a plane normal to the main flow direction. “The solid and dashed lines depict positive and negative concentration perturbation contours, respectively. Streamlines of the transverse perturbation velocity field are superimposed, with arrows denoting the flow direction. In the top frame, gray shading reflects the perturbation u-velocity, with lighter areas indicating positive values, and darker areas negative values. The middle frame shows perturbation shear ∂u/∂z through gray shading, with lighter areas indicating positive values and darker areas negative values. The shape of the interface perturbation is shown in the bottom frame” (Reprinted with permission from Hall et al. [86]. Copyright 2008 by Cambridge University).

Grahic Jump Location
Fig. 19

Turbulence statistics as a function of the wall-normal coordinate for different settling velocities: case 6 (us = 2.3 × 10−2), case 7 (us = 2.5 × 10−2), case 8 (us = 3 × 10−2), case 9 (us = 3.5 × 10−2), case 10 (us = 5 × 10−2). (a) Streamwise component (c) vertical component (Reprinted with permission from Cantero et al. [121]. Copyright 2009 by the American Geophysical Union).

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Fig. 17

“Problem setting: the flow in the channel is driven by the excess density of the water-sediment mixture. Since the channel is open at both ends to the tank, there is no net pressure gradient acting on the flow. The flow represents an idealized case of a turbidity current in which ambient water entrainment is not allowed in the channel [121]” (Reprinted with permission from Cantero et al. [121]. Copyright 2009 by the American Geophysical Union).

Grahic Jump Location
Fig. 18

Turbulence statistics as a function of the wall-normal coordinate for different settling velocities: case 0 (us = 0), case 1 (us = 5 × 10−3), case 2 (us = 10−2), case 3 (us = 1.75 × 10−2), case 4 (us = 2 × 10−2), case 5 (us = 2.125 × 10−2). (a) Streamwise component (c) vertical component (Reprinted with permission from Cantero et al. [121]. Copyright 2009 by the American Geophysical Union).

Grahic Jump Location
Fig. 12

Gravity current interacting with a localized seamount. The development of the current structure is visualized by a sediment concentration contour for a flat seafloor (FL—top row), a shallow seamount (B1—middle row), and a taller seamount (B2—bottom row). While the current primarily flows over the shallow seamount, it mostly flows around the taller seamount. “The shading in the bottom plane indicates the magnitude of the wall shear stress, while the vertical plane to the right depicts the concentration field in the symmetry plane z = 1.5.” Initial parameters: L = 38, l = 1, H = d = 2, and W = 3 (Reprinted with permission from Nasr-Azadani and Meiburg [55]. Copyright 2014 by Cambridge University).

Grahic Jump Location
Fig. 13

Gravity current interacting with a bottom-mounted square cylinder. Temporal evolution of the concentration (left) and vorticity (right) fields. “Instantaneous streamlines in the laboratory reference frame are superimposed onto the concentration fields.” Initial parameters: L = 24, l = 9, H = d = 1 (Reprinted with permission from Gonzalez-Juez et al. [110]. Copyright 2009 by Cambridge University).

Grahic Jump Location
Fig. 14

Temporal evolution of the drag (thick solid line) (a) and lift (thick solid line) (b) experienced by a bottom-mounted square cylinder during the interaction with a gravity current. “Also shown in (a) are the pressure forces on the upstream Fw (dashed line) and downstream Fe (dashed-dotted line) faces. The viscous drag and lift components (thin solid line) are much smaller than the pressure components” (Reprinted with permission from Gonzalez-Juez et al. [110]. Copyright 2009 by Cambridge University).

Grahic Jump Location
Fig. 15

“Temporal evolution of the drag (a) and lift (b) in the experiments by Ermanyuk and Gavrilov [108] (solid line, squares), three-dimensional LES results (dashed line), and two-dimensional results (dashed-dotted line).” While the two-dimensional simulation overpredicts the force fluctuations, the three-dimensional simulation accurately captures the nature of the Kelvin–Helmholtz vortices and the vortex shedding process, so that it yields good force predictions (Reprinted with permission from Gonzalez-Juez et al. [110]. Copyright 2009 by Cambridge University).

Grahic Jump Location
Fig. 16

Gravity current interacting with a circular cylinder mounted above a wall. “Spanwise vorticity fields at z/h = 0.5 and different times for Re = 9000, D/h = 0.1, and G/h = 0.03: (a) t/(h/V) = 8.5, (b) and (c) t/(h/V) = 9.2, (d) t/(h/V) = 9.9, and (e) and (f) t/(h/V) = 16.3. The cylinder is located at x/h = 9 − 9.1. The region near the cylinder in (b) and (e) is enlarged in (c) and (f), respectively. Note the formation of a jet of dense fluid in the gap in (b) and (c), and the subsequent plunge of the current downstream of the cylinder in (d),” Initial parameters: L = 28, l = 12, H = 2.5, and d = 1 (Reprinted with permission from Gonzalez-Juez et al. [113]. Copyright 2010 by Cambridge University).

Grahic Jump Location
Fig. 20

Swirling strength isosurface λci = 22 for us = 0.026. At this value of the settling velocity, strong streamwise and hairpin vortices are visible near the bottom wall, indicating the presence of vigorous turbulence (Reprinted with permission from Shringarpure et al. [122]. Copyright 2012 by Cambridge University).

Grahic Jump Location
Fig. 21

Swirling strength isosurface λci = 22 for us = 0.0265. At this slightly larger value of the settling velocity, the turbulence is strongly damped, so that the presence of hairpin vortices is greatly reduced (Reprinted with permission from Shringarpure et al. [122]. Copyright 2012 by Cambridge University).

Grahic Jump Location
Fig. 22

Front velocity of a dense, non-Boussinesq current as function of the density ratio γ. Theoretical predictions are plotted as lines, while experimental and simulation data are represented by symbols. The theoretical prediction of Rotunno et al. [136] is shown as a solid line, and the theoretical prediction of Lowe et al. [135] is displayed as a dash-dotted line. Diamonds: experimental results of Lowe et al. [135]; squares: experimental data of Gröbelbauer et al. [137]; asterisks: results of Keller and Chyou [138]; triangles: simulation data of Birman et al. [128]; circles: simulation data of Bonometti et al. [139]; plus symbols: simulation data of Rotunno et al. [136]. Figure based on data from Lowe et al. [135] and Rotunno et al. [136].

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