High Resolution Simulations of Gravity 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

Atmospheric dust storm Particles represent small mass fraction Large density ratio of particles/fluid Suspension mechanism? Coast of Africa (NASA)

Atmospheric dust storm Mars (NASA) Particles represent small mass fraction Large density ratio of particles/fluid Suspension mechanism? Coast of Africa (NASA)

Sandstorms

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

Turbidity current Turbidity current. Underwater sediment flow down the continental slope Can transport many km 3 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

Houston Study area Courtesy of F.A. Diegel Turbidite systems on the continental slope

Turbidity current (cont’d) Var Fan, off Nice coast, caused in 1979 by airport construction accident

Framework: Dilute flows Volume fraction of particles of O( ): 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

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  don’t resolve small scale flow field around each particle, but only the large fluid velocity scales (‘SGS model’)

Moderately dilute flows: Two-way coupling (cont’d) settling velocity effective density

Model problem Lock exchange configuration Dense front propagates along bottom wall Light front propagates along top wall

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)

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

Results: 3D turbidity current – Temporal evolution Necker, Härtel, Kleiser and Meiburg (2002a,b) DNS simulation (Fourier, spectral element, 7 x 10 7 grid points) turbidity current develops lobe-and-cleft instability of the front current is fully turbulent erosion, resuspension not accounted for

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)

Results: Sedimentation rate Global sedimentation rate as function of time early and late times are governed by different power law regimes

Results: Deposit profiles Comparison of transient deposit profiles with experimental data of de Rooij and Dalziel (1998) simulation reproduces experimentally observed sediment accumulation Experiment ___ Simulation

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

Results: 2D vs. 3D - Front velocity, suspended particle mass 3D front propagates slightly faster particles settle out faster in the 3D flow ____ 3D simulation D simulation

Results: 2D vs. 3D - Final deposit profile 2D and 3D simulations predict quantitatively similar deposit profiles

Results: Mixing of interstitial fluid 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 u s = 0 u s = 0.01 u s = 0.02

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

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

Simple model for scaling Late stages are dominated by two-layer structure, not by front 2 layer approximation of density field (Thorpe 1967)

Simple model – comparison with simulation Comparison of velocity at gate location with model results

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

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

Erosion, resuspension of particle bed (cont’d) based on experimentally measured correlation between shear stress at the surface of the bed and an effective resuspension flux

Erosion, resuspension of particle bed (cont’d) deposition outweighs erosion: decaying turbidity current erosion outweighs deposition: growing turbidity current

Erosion, resuspension of particle bed (cont’d) multiple, polydisperse flows feedback of deposit on subsequent flows formation of ripples, dunes etc.

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

Channelization by turbidity currents: A Navier-Stokes based linear instability mechanism Instability mechanism:

Strong density difference: Boussinesq vs. non-Boussinesq Momentum equation with Boussinesq approximation Non-Boussinesq momentum equation

Strong density difference large density contrast (non-Boussinesq): asymmetric fronts small density contrast (Boussinesq case): fronts are symmetric

Strong density difference Lowe, Linden and Rottman (2004) experiments confirm asymmetric fronts for large density contrast

Strong density difference for non-Boussinesq flows the lobe-and-cleft instability persists

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

Strong density difference density contours alone don’t allow us to determine nature of the flow

Strong density difference light front is approximately energy-conserving for all density ratios Light front velocity: Comparison with experimental data and theoretical value for energy-conserving (Benjamin) front

Strong density difference dense front behaves dissipatively for all density ratios Dense front velocity: Comparison with experimental data and theoretical values for a dissipative front without a bore

Strong density difference in the energy-conserving light front, the dissipation does not depend on  in the dissipative dense front, the dissipation varies with  Global dissipation within the dense and light fronts: ____ light front dense front

Gravity currents in stratified ambients generation of internal waves complex interaction of the current with the stratified ambient

Reversing buoyancy currents propagates along bottom over finite distance, then lifts off subsequently propagates along top

what forces and moments are exerted on the obstacle? steady vs. unsteady? erosion and deposition near the obstacle? Gravity currents may encounter underwater marine installations: Hazards posed by gravity and turbidity currents Constantinescu (2005)

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 Summary

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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