Settling of Small Particles in Homogeneous Turbulence: Settling Velocity Enhancement by TwoWay Coupl

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Settling of Small Particles in Homogeneous
TurbulenceSettling Velocity Enhancement by
Two-Way Coupling

• T. Bosse, L. Kleiser (ETHZ), E. Meiburg (UCSB)

• Motivation
• One-way coupled flows
• - particle-laden mixing layer
• Two-way coupled flows
• - particles settling in homogeneous turbulence
• - dynamics of a suspension drop
• Summary

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Motivation

• Particle-air interaction influences
• Growth / Amplification
• Front velocity
• Deposition
• Runout length

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Motivation

• Turbidity current Sediment flow
• down the continental slope
• Repeated turbidity currents in the
• same region can lead to the
• formation of hydrocarbon
• reservoirs
• Effective settling rate determines
• properties of sediment layer
• - particle layer thickness distribution
• - particle size distribution

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

Other applications water/air quality, dust
storms, cloud dynamics, medical devices, spray
combustion, industrial processes . . .
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Dilute flows

• Volume fraction of particles of O(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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Very dilute flows One-way coupling

• Small mass fraction of heavy particles (dusty
  gas, dilute spray)
• particles move independently of each other
• particles have negligible effect on the fluid
  motion
• can first solve for fluid motion, afterwards for
  particle dynamics
• Particle dynamics is governed by balance of
• particle inertia
• viscous drag force
• gravity
• added mass, lift forces, pressure gradients in
  the fluid, and
• Basset history term can be neglected

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Very dilute flows One-way coupling (contd)

• Three physically relevant time scales
• aerodynamic response time of particle ta
• characteristic time of flow field tf
• particle settling time ts

Two dimensionless parameters govern particle
motion
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Very dilute flows One-way coupling (contd)
Continuity and momentum equation for single-phase
fluid
Solve ODE for each particle (Maxey and Riley
1983)
Stokes drag gravity
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Example Particle laden mixing layer

• Martin and Meiburg (1994)

• small particle inertia, weak gravitational
  effects particles follow fluid motion
• no local accumulation of particles
• clear and particle laden fluid mix through
  entrainment

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Particle laden mixing layer (contd)

• Martin and Meiburg (1994)

• intermediate particle inertia, weak
  gravitational effects particles are ejected
• from vortex centers
• optimal ejection of particles with intermediate
  Stokes number (Crowe et al.)
• local accumulation of particles in bands midway
  between vortex\centers

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Particle laden mixing layer (contd)

• Martin and Meiburg (1994)

• intermediate particle inertia, strong
  gravitational effects sedimenting particles
• are ejected by vortices
• organization of the particle concentration field
  into sedimenting bands

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Particles settling in homog. turbulence One-way
coupling

• Maxey (1987), Wang and Maxey (1993)
  simulation
• - analyze cellular flows and isotropic
  turbulence under one-way coupling
• - particles accumulate in regions of low
  vorticity and high strain
• - increase in mean settling velocity as
  compared to Stokes velocity because
• of preferential sweeping towards regions
  of downward fluid velocity

inertial bias
preferential sweeping

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Particles settling in homog. turbulence Two-way
coupling

• Aliseida et al. (2002), Yang and Shy (2005)
• - wind tunnel/closed container experiments,
  spray droplets/solid particles
• - fluid is accelerated downwards in regions of
  high particle concentration,
• which leads to enhanced settling
• - large discrepancy between the two studies
  w.r.t. magnitude of this effect

inertial bias
preferential sweeping

collective particle drag

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Dilute, two-way coupled flows

• Suspended particles occupy small volume fraction,
  but have O(1) mass fraction, strong particle
  inertia
• each particle locally exerts force on the fluid
  (equal and
• opposite to the fluid force acting on
  the particle)
• volume coupling can still be neglected
• Suspension dynamics can be described by
• incompressible continuity equation
• Navier-Stokes equation plus additional force term
• set of ODEs for each particles location,
  velocity

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Dilute, two-way coupled flows Governing equations

• Scaling with Taylor microscale l and rms-velocity
  u

inverse drag force
Dimensionless parameters
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Dilute, two-way coupled flows (contd)

• As will be seen, results suggest that it is
  preferable to scale the particle
• equation with Kolmogorov scales

Dimensionless parameters
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Simulation approach

• Fluid equations
• Fourier pseudospectral method, RK/CN time
  stepping
• Turbulence forcing procedure according to
  Eswaran Pope (1988)
• Particles
• Lagrangian tracking
• Coupling terms
• Trilinear interpolation between particle and
  grid point locations
• Steps
• Fluid only Run simulation until statistically
  stationary
• Add particles with random spatial distribution,
  Stokes setting velocity
• Run with one-way coupling until statistically
  stationary
• Turn on two-way coupling
• Run until statistically stationary

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Simulation approach Related work

• For dilute flows with many particles, several
  variations of force coupling
• Lagrangian-based Navier-Stokes approaches
  (Elghobashi et al.,
• Eaton et al., Walther and Koumoutsakos,
  Lohse et al., etc.)
• Stokeslet-based simulations (Nitsche and
  Batchelor 97, Machu
• et al. 01)
• Multipole expansions (Maxey and Patel 01)
• For O(10-100) particles
• DNS (Joseph, Glowinski et al.)
• Force coupling method (Maxey and Dent 98)
• For dense particle loading
• Two-fluid simulations (Drew 83, Crowe et al.
  96)
• closure assumptions needed

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Results One-way coupling
Validation against Wang Maxey (1993)
WM,
Settling velocity enhancement most pronounced for

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Results One-way coupling (contd)
Temporal evolution of particle concentration
distribution

• Large particle-free regions emerge
• Regions of high particle concentration grow
• Regions of moderate particle concentration
  decrease
• Good agreement with Wang Maxey (1993)

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Results Two-way coupling
Correlation between particle volume fraction and
vorticity magnitude
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Results Two-way coupling (contd)
Settling velocity enhancement

• Two-way coupling effects increase with particle
  volume fraction
• Increase in settling velocity noticeable above
  volume fraction O(10-5)

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Results Two-way coupling (contd)
Particle concentration distribution

• Small particle volume fractions probability
  functions not affected by
• two-way coupling
• Larger particle volume fractions fewer
  particle-free regions

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Results Two-way coupling (contd)
Enhancement of particle settling velocity

• Enhancement due to two-way coupling above
  volume fractions O(10-6)
• Above volume fractions O(5 x 10-5), turbulence
  properties are modified, so
• that the settling velocity enhancement
  increases less than linearly

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Results Two-way coupling (contd)
If turbulence properties are kept constant by
adjusting forcing

• - - Rel adjusts itself
• _____ Rel kept constant

• Nearly linear increase in settling velocity
  with volume fraction

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Results Two-way coupling (contd)
Mechanism of settling velocity enhancement
Vertical fluid velocity as function of particle
volume fraction
Settling velocity enhancement as function of
particle volume fraction

• Downward fluid velocity increases in regions of
  high particle concentration
• Increased downward fluid velocity enhances
  particle settling velocity

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Results Comparison with experiments
Comparison with Aliseda et al. (2002)
(
)
experiment,
two-way,
two-way,
one-way
Comparison with Yang Shy (2005)
)
(
two-way,
?
?
one-way,
?
experiment,

• simulations underpredict two-way coupling
  effects measured by Aliseida et al. (02)
• simulations overpredict two-way coupling effects
  measured by Yang and Shy (05)

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Results Comparison with experiments

• Potential reasons for discrepancies
• experiments particle size distribution,
  simulation monodisperse particles
• simulations
• - match turbulence Re number, but other
  turbulence parameters may be
• somewhat different
• - low order interpolation may cause some
  errors, but a few per cent at most
• experiments
• - particles may induce mean downward fluid
  motion in the windtunnel test section

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Summary

• One-way coupling
• Successful validation against Wang Maxey, JFM
  (1993)
• Strongest particle-fluid interaction for Stokes
  numbers around unity
• Large inhomogeneities in particle distribution,
  correlation between vorticity and particle
  concentration
• Two-way coupling
• Particle settling velocity enhancement
  found for
• Monotonic increase of with particle
  volume fraction, relation roughly linear if
  microscale Reynolds number kept constant
• Turbulence modification sets in for
  particles have dissipative effect on
  turbulent carrier fluid
• Collective particle drag responsible for
  additional settling velocity enhancement compared
  to one-way coupling
• Comparison with experiment
• Still significant differences between numerical
  and experimental results
• Further research necessary

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Numerical simulation of a suspension drop

• Mitts (1996)

Bosse (2002)
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Numerical simulation of a suspension drop
Red 0.01
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Numerical simulation of a suspension drop
Red 1
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Numerical simulation of a suspension drop
Red 300
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Summary

• Challenges in the simulation of particle laden
  flows
• different parameter ranges dominated by different
  physical
• mechanisms
• large variety of numerical approaches
  (Lagrangian, Eulerian,
• two-fluid, statistical, hybrid.)
• two-way coupling between fluid and particles
  momentum,
• volume, thermal, chemical
• interaction between suspension and bed particle
  deposition,
• erosion, sorting