High Resolution Simulations of Gravity and Turbidity Currents

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High Resolution Simulations ofGravity and
Turbidity Currents

• Eckart Meiburg
• UC Santa Barbara

• Motivation
• Governing equations / computational approach
• Results
• - particle driven gravity currents
• - gravity currents down a slope
• - non-Boussinesq gravity currents
• - gravity currents with erosion and
  resuspension
• Summary and outlook

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Atmospheric dust storm

• Particles represent small
• mass fraction
• Large density ratio of
• particles/fluid
• Suspension mechanism?

Coast of Africa (NASA)
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Atmospheric dust storm

• Particles represent small
• mass fraction
• Large density ratio of
• particles/fluid
• Suspension mechanism?

Coast of Africa (NASA)
Mars (NASA)
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Sandstorms
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Avalanche

• Non-Boussinesq
• Formation
• Growth / Amplification
• Front velocity
• Particle-particle interaction
• Erosion / Resuspension
• Deposition
• Influence of bottom topography
• Runout length

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Turbidity current

• Underwater sediment flow down
• the continental slope
• Can transport many km3 of
• sediment
• Can flow O(1,000)km or more
• Often triggered by storms or
• earthquakes
• Repeated turbidity currents in the
• same region can lead to the
• formation of hydrocarbon
• reservoirs
• Properties of turbidite
• - particle layer thickness
• - particle size distribution
• - pore size distribution

• Turbidity current.
• http//www.clas.ufl.edu/

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Turbidite systems on the continental slope
Houston
Study area
Courtesy of F.A. Diegel
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Turbidity current (contd)
Var Fan, off Nice coast, caused in 1979 by
airport construction accident
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Framework Dilute flows

• Volume fraction of particles of O(10-2 - 10-3)
• particle radius particle separation
• particle radius characteristic length scale of
  flow
• coupling of fluid and particle motion primarily
  through
• momentum exchange, not through
  volumetric effects
• effects of particles on fluid continuity equation
  negligible

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Moderately dilute flows Two-way coupling

• Mass fraction of heavy particles of O(10), small
  particle inertia (e.g., sediment transport)
• particle loading modifies effective fluid density
• particles do not interact directly with each
  other
• Suspension dynamics can be described by
• incompressible continuity equation
• variable density Navier-Stokes equation
  (Boussinesq)
• conservation equation for the particle
  concentration field
• ? dont resolve small scale flow field around
  each particle,
• but only the large fluid velocity
  scales (SGS model)

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Moderately dilute flows Two-way coupling
(contd)
effective density
settling velocity
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Model problem
Lock exchange configuration
Dense front propagates along bottom
wall Light front propagates along top wall
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Past work

• Benjamin (1968)
• shape of energy-conserving gravity current head
• height of energy-conserving current ½ height
  of lock
• Shallow water theory
• Rottman Simpson, Keller Chyou
• Particle-driven currents
• Bonnecaze Huppert, Garcia Parker
• Non-Boussinesq currents
• Groebelbauer et al., Lowe et al., Birman
  Meiburg
• Numerical simulations
• Klemp et al., Härtel, Meiburg et al.,
  Balachandar et al.
• Reviews
• Simpson (1997), Rottman and Linden (2001)

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Numerical method

• Fourier spectral method in the streamwise and
  spanwise
• directions
• sixth order compact finite difference method or
  spectral
• element method in the vertical direction
• third order Runge-Kutta time stepping
• mostly equidistant grids
• up to 70 million grid points

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Results 3D turbidity current Temporal evolution
DNS simulation (Fourier, spectral element, 7x107
grid points)

• Necker, Härtel, Kleiser and Meiburg (2002a,b)

• turbidity current develops lobe-and-cleft
  instability of the front
• current is fully turbulent
• erosion, resuspension not accounted for

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Results 3D turbidity current - Frontal
instability

• Lobe-and-cleft instability (Simpson 72)
• Instability wavelength and growth rate agree
  with linear theory
• by Härtel et al. (2000)

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Results Sedimentation rate
Global sedimentation rate as function of time

• early and late times are governed by different
  power law regimes

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Results Deposit profiles
Comparison of transient deposit profiles with
experimental data of de Rooij and Dalziel
(1998)

• - - - Experiment
• ___ Simulation

• simulation reproduces experimentally observed
  sediment accumulation

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Results 2D vs. 3D simulations

• early times good agreement between 2D and
  spanwise averaged 3D results
• late times spanwise instabilities lead to more
  rapid decay of 3D flow

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Results 2D vs. 3D - Front velocity, suspended
particle mass

• ____ 3D simulation
• - - - - 2D simulation

• 3D front propagates slightly faster
• particles settle out faster in the 3D flow

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Results 2D vs. 3D - Final deposit profile

• 2D and 3D simulations predict quantitatively
  similar deposit profiles

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Results Mixing of interstitial fluid

• us 0
• us 0.01
• us 0.02

• higher particle settling velocity results in
  better mixing of interstitial fluid
• reason by the time the more rapidly settling
  particles have been deposited,
• there is still sufficient kinetic energy
  left to effect thorough mixing

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Gravity current moving down a slope

• Gravity current moving down a slope for Re
  4,000.

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Front velocity

• Beyond a certain angle, the front velocity
  reaches a quasisteady state, then jumps to larger
  value

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Simple model for scaling
Late stages are dominated by two-layer structure,
not by front
2 layer approximation of density field (Thorpe
1967)
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Simple model comparison with simulation
Comparison of velocity at gate location with
model results
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Results Bottom wall shear stress

• wall shear stress distribution reflects
  spanwise and streamwise flow structures
• allows prediction as to where particle bed
  erosion may occur

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Erosion, resuspension of particle bed

• Experimentally determined correlation by Garcia
  Parker (1993) evaluates resuspension flux at the
  particle bed
• surface as function of
• bottom wall shear stress
• settling velocity
• particle Reynolds number
• Here we model this resuspension as diffusive flux
  from the
• particle bed surface into the flow

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Erosion, resuspension of particle bed (contd)

• based on experimentally measured correlation
  between shear stress at the
• surface of the bed and an effective
  resuspension flux

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Erosion, resuspension of particle bed (contd)
deposition outweighs erosion decaying turbidity
current
erosion outweighs deposition growing turbidity
current
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Erosion, resuspension of particle bed (contd)

• multiple, polydisperse flows
• feedback of deposit on subsequent flows
• formation of ripples, dunes etc.

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Channelization by turbidity currents A
Navier-Stokes based linear instability mechanism
Focus on cross-section behind the front

• evaluate base flow from numerical simulations
  or simplified analytical model
• in order to obtain U(z), C(z)
• linearize 3D flow around 1D base state, obtain
  eigenvalue problem

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Channelization by turbidity currents A
Navier-Stokes based linear instability mechanism
Instability mechanism
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Strong density difference Boussinesq vs.
non-Boussinesq
Momentum equation with Boussinesq approximation
Non-Boussinesq momentum equation
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Strong density difference

• small density contrast (Boussinesq case)
  fronts are symmetric

• large density contrast (non-Boussinesq)
  asymmetric fronts

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Strong density difference

• Lowe, Linden and
• Rottman (2004)

• experiments confirm asymmetric fronts for large
  density contrast

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Strong density difference

• for non-Boussinesq flows the lobe-and-cleft
  instability persists

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Strong density difference

• Theory based on two-layer shallow water equations
  (Lowe, Rottman and Linden 2004)
• different types of flows are possible, with or
  without bore
• front velocities and spatial distribution of
  dissipation rates decide which solution forms in
  reality
• obtain detailed information on dissipation from
  simulations

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Strong density difference

• density contours alone dont allow us to
  determine nature of the flow

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Strong density difference
Light front velocity Comparison with
experimental data and theoretical value for
energy-conserving (Benjamin) front

• light front is approximately energy-conserving
  for all density ratios

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Strong density difference
Dense front velocity Comparison with
experimental data and theoretical values for a
dissipative front without a bore

• dense front behaves dissipatively for all
  density ratios

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Strong density difference
Global dissipation within the dense and light
fronts

• ____ light front
• - - - - dense front

• in the energy-conserving light front, the
  dissipation does not depend on ?
• in the dissipative dense front, the dissipation
  varies with ?

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Gravity currents in stratified ambients

• generation of internal waves
• complex interaction of the current with the
  stratified ambient

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Reversing buoyancy currents

• propagates along bottom over finite distance,
  then lifts off
• subsequently propagates along top

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Hazards posed by gravity and turbidity currents
Gravity currents may encounter underwater marine
installations
Constantinescu (2005)

• what forces and moments are exerted on the
  obstacle?
• steady vs. unsteady?
• erosion and deposition near the obstacle?

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Summary

• high resolution three-dimensional simulations of
  gravity currents
• detailed information regarding sedimentation
  dynamics, energy
• budgets, mixing behavior, dissipation
• important differences between 2D and 3D
  simulation results
• current extension to gravity currents flowing
  down a slope, more
• complex geometries, erosion and
  resuspension, intrusions,
• reversing buoyancy, submarine structures
• non-Boussinesq currents light front is
  energy-conserving,
• dense front is dissipative

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Acknowledgments

• National Science Foundation, NASA, BHP Billiton
  Petroleum
• V. Birman, F. Necker, C. Härtel, L. Kleiser, J.
  Martin,
• F. Blanchette, B. Hall, E. Gonzales, M.
  Strauss, B. Kneller,
• M. Glinsky

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University of California at Santa Barbara

• Founded 1944
• 20,000 students
• 5 Nobel Prizes since 1997
• Reputation for outstanding scientific research
  and
• interdisciplinary collaboration

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Mechanical and Environmental Engineering

• 500 undergrads
• 85 graduate students, 50 of them
  international
• 30 faculty members, 10 members of the NAE

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Research Areas

• Computational Science and Engineering
• Dynamics, Control, and Robotics
• Fluids and Thermal Transport
• Microscale and Nanoscale Engineering
• Solid Mechanics, Materials, and Structures