Post on 09-May-2020
Particles in Fluids
• Sedimentation
• Fluidized beds
• Size segregation under shear
• Pneumatic transport
• Filtering
• Saltation
• Rheology of suspensions
Sandstorm
Fluidized Bed
Incompressible Navier-Stokes equation:
Equation of motion of fluid
1( )
vv v p v
t
0v
and are velocity and pressure field of the fluid, and μ its density and dynamic viscosity.
)(xv
( )p x
0
( ) constv
t
x x xx y z
y y yx y z
z z zx y z
v v vv v v
x y z
v v vv v v
x y z
v v vv v v
x y z
Initial and Boundary Conditions of NS eqs.
0 0 0 0
0 0
102( ) ,
( , ) ( ) , ( , ) ( )
( , ) v ( ) , ( , ) p ( )
vv v p v v
t
v x t V x p x t P x
v t t p t t
( , ) ( , )v x t p x t
velocity field, pressure field
viscosity
Reynolds number Re
Vh
ReV is characteristc velocityh is characteristic lengthμ is dynamic viscosity
Re << 1 is the Stokes limit (laminar flow)
Re >> 1 is turbulent limit (Euler equation)
Solvers for NS equation
• Penalty method with MAC
• Finite Volume Method (FLUENT)
• Turbulent case: k-ε model or spectral method
CFD = Computational Fluid Dynamics
Classical fluid solvers
Solving the differential equations numerically.
Discrete fluid solvers
• Lattice Gas Automata (LGA)
• Lattice Boltzmann Method (LBM)
• Dissipative Particle Dynamics (DPD)
• Smooth Particle Hydrodynamics (SPH)
• Stochastic Rotation Dynamics (SRD)
• Direct Simulation Monte Carlo (DSMC)
One particle in fluid
particlevv
fluid
e.g. pull sphere through fluid
particlev
Γ
no-slip condition:
create shear in fluid : exchange momentum
movingboundary condition
Drag force
AdFD
jiij ij
j i
vvp
x x
drag force
stress tensor
η = μ is static viscosity
(Bernoulli‘s principle)
Homogeneous flow
Re << 1 Stokes law:
FD = 6π η R v(exact for Re = 0)
R
v
Re >> 1 Newton‘s law: FD = 0.22π R2v2
general drag law:
CD is the drag coefficient
22
Re8 DD CF
R is particle radius, v is relative velocity
Drag coefficient CD
Reynolds number Re = Dv/μ
Re
Inhomogeneous flow
In velocity or pressure gradients: Lift forcesare perpendicular to the direction of the external flow,
important for wings of airplanes.
when particle rotates: Magnus effectimportant for soccer
lift force:
CL is „lift coefficient“
Many particles in fluids
•The fluid velocity field followsthe incompressible NavierStokes equations.
• Many industrial processesinvolve the transport of solidparticles suspended in a fluid.The particles can be sand,colloids, polymers, etc.
•The particles are dragged bythe fluid with a force:
simulating particles moving in a sheared fluid
22
Re8 DD CF
Stokes limit
hydrodynamic interaction between the particles
ij
jjiiji vrrMv
matrixmobility
)(
for Re = 0 mobility matrix exact
Stokesian Dynamics (Brady and Bossis)
invert a full matrix only a few thousand particles
Numerical techniques
Calculate stress tensor directly by evaluating the gradients of the velocity field
through interpolation on the numerical grid,e.g. using Chebychev polynomials (Kalthoff et al.).
Method of Fogelson and Peskin:Advect markers that were placed in the particle and then put springs between
their new an their old position.These springs then pull the particle.
1
2
Sedimentation
• Sedimentation ist the descent ofparticles in a fluid due to the action ofgravity.
• The interaction between the particlesand the fluid is given by the conditionthat the velocity of the fluid on theentire surface of each particle is equalto the velocity of this particle.
• Measure settling velocity, i.e. velocityof the upper front.
• If particles are of different speciesthen one has several fronts.
• Open question: size dependence ofthe density fluctuations.
Sedimentation
comparing experiment and simulation
Glass beadsdescendingin silicon oil
using penaltymethod withMAC grid
Sedimentation velocity
1954
settling velocity vS = v0 (1-Φ)5
Φ = volume fraction of particles
Sedimentation of platelets
Oblate ellipsoids descendin a fluid under the actionof gravity.
This has applications inbiology (blood), industry(paint) and geology (clay).
Shear flow
A.J.C. Ladd, J.Fluid Mech. 1994
using LBM(= LatticeBolzmannMethod)
• Assume spherical particles
• Use molecular dynamics for the simulation
• DLVO potentials describe the dominant
particle-particle interaction:
- screened Coulomb potential
(ions / counterions), repulsive
- Van-der-Waals-attraction for
short distances
• Hertz force for overlapping particles
• Lubrication force ~
Simulating clay
• Al2O3 interaction potentials
Shear flow
Shear viscosity
Viscosity at small vs
Sheared Blood
Porous Media
Calculation of fluid motion
FLUENT
We solve the incompressible Navier-Stokes equation with a commercial discrete volume solver on an adaptive triangulated mesh
We create a two-dimensional realization of a porous medium by placing randomly disks that do not overlap (RSA).
Flow through porous medium
Flow through a porous medium
Color represents the absolute value of the fluid velocity.
important foroil recovery,filtration andfluidized beds
Flow through porous medium
Macroscopic permeability
2
3
0 )1(
K
K
12/20 hK
Flux depends onpressure drop as:
We verify the law ofKozeny – Carman:
where:
is the reference value for an empty channel.
pK
p
ε is the porosity = void fraction
Darcy
Stokes number:
uinertial impaction
direct interception
diffusion
Massive tracerparticles of diameter dp,
velocity up
and density pare released.
Filtration
δ D
x
y
u
St : = D and St 0: = 0
= capture efficiency
Trajectories of particles
St = 2.0610-4
St = 8.1210-3
Trajectories
porosity = 0.7, Stokes number St = 0.1
colours code the absolute value of the velocity
add the action of gravity
Dependence on gravity
Capture efficiency
−4 −3 −2 −1 0 1log10[St/(ε−εc)]
−4
−3
−2
−1
0lo
g 10(δ
/D)
ε=0.85ε=0.90ε=0.95
−4 −3 −2 −1 0log10(St−Stc)
−2
−1
0
log 10
(δ/D
)
0.5
1.0
)(~ cStSt
Without gravitywe find a criticalStokes number
Stc = 0.2096
with 0.5
with gravity
without gravity
Non-captured particles
is the penetration depth
is fraction of non-captured particles.
for 1 it is easyto show that:
= D/4(1-) = D2/4(1-)
typical decay:
we find moregenerally:
/)( xex
ε = 0.7
Aeolian Sand transport
Transport by Saltation
even on a wet surface
The Mechanism of Saltation
h
Windh
• Grains are drawn from the ground and accelerated by the wind. With moreenergy they impact again against the surface and eject a splash of newparticles. In this way more and more grains saltate until saturation is reacheddue to momentum conservation.
Ralph A. Bagnold
Dependence on grain size
typical grain diametersfor saltation on earth:
100 – 300 m
wind tunnel measurements
Bagnold and Chepil
Wind Channel in Aarhus
Measuring the wind velocity in channel
Schematic saltating trajectory
θ>
up
>>
ux(y) y
x
mobile wall at top
= ejection angle up = particle velocityux(y) = wind velocity profile
The turbulent air flow
*
0
lnx
u zu z
z
logarithmic velocity profile of the horizontalcomponent of the velocity as function of height y:
0.4 is the von Kármán constant
z0 is the roughness length
Solve it with k- model using FLUENT.
viscosity of air: η = 1.789510-5 kg m-1 s-1
density of air: = 1.225 kg m-3
a commercial finite volume solver on an adaptive
triangulated mesh
Types of transient behaviour
0.0 0.5 1.0 1.5 2.0 2.5x(m)
0.00
0.02
0.04
0.06
0.08
y(m
)
u* > utu* < ut
threshold velocity ut 0.35
ppDp guuF
dt
ud
1force on particle:
Steady state
saturated flux qs
Saturated flux
0.5 1.0 1.5 2.0u* (m/s)
0.001
0.01
0.1
1.0
q s (k
g m
−1 s−
1 )
simulationEq. (7)Lettau−LettauBagnold
0.01 0.1 1.0 10u*−ut
0.001
0.01
0.1
1.0
10
qs
2.0
2* )( ts uuaq
Lettau and Lettau:(1978)
fit of solid line:
Bagnold (1941):
3*uqs
Wind velocity profile
0.0 0.5 1.0 1.5ux(0)−ux(q) (m/s)
0.0
0.2
0.4
0.6
0.8
1.0
y(m
)
0 10 20 30 40[ux(0)−ux(q)]/q
0.0
0.2
0.4
0.6
0.8
1.0
y(m
)
q=0.010q=0.015q=0.020q=0.030q=0.043
u* = 0.51
collapse whennormalizing
with flux q
difference between
disturbed andundisturbedvelocity profile
Height of saltation layer
0.3 0.4 0.5 0.6 0.7 0.8u* (m/s)
0.00
0.05
0.10
0.15
y max
(m
)
0.35
ut
ymax height ofmaximum lossof velocity
ut = 0.330.01
linear increasewith u*
becomes zero at:
BobFest, Duke University, Oct.12-13, 2013
Splash in 3d
θ = 0.15 in 3d
BobFest, Duke University, Oct.12-13, 2013
Saltons jumping on the soft bed
Role of collisions: saltons
θ = 0.90e = 0.7
Contribution to the flux
Planet Mars
Earth Mars
[Greeley and Iversen (1985)]u*t 0.2 m/s u*t 2.0 m/s
1.8 10 kg/sm 1.1 10 kg/sm
(Viscosity of CO2 at C)
- u* on Earth is 0.4 m/s and on Mars, Pathfinder Mission 1997 found u*
close to threshold. Further, it has been found that the angle of the slip face of martian dunes is the same as of terrestrial dunes.
2sm 81.9g 2sm 71.3g3
air mkg 225.1 3air mkg 02.0
3grain mkg 2650 3
grain mkg 3200
m 250 d m 600 d
Parameters on Mars
Saltation on Mars
2
3*
* *
1 1fl t ts
u uq C u
g u u
White (1979) Greenley et al. (1996)
C = 18 for EarthC = 2.9 for Mars
C = 19 d / lv
lv = (μ2/g)1/3
Saltation on Mars
0
ss
s
q
2
0 *23
227.3 fls t t
dq u u u
g
Saltation on Mars
*
*
12 3 /
salt v t
salt v t
v
L t u u
H t u u
t g
Length Lsalt and height Hsalt of saltation trajectory
Dune velocities
Particle mixing in turbulent channel flow
• Consider a channel through which particles are dragged by a turbulent flow.
• Do not calculate the whole fluid field, but rather use a stochastic approach to model the influence of the turbulent velocity field on the particle movement.
• The particle density is constant throughout the system.• Concentrate on a small region in the center of the channel, which means
we ignore the effects of the walls.• Study the mixing of two types of spherical particles.
Particle mixing in turbulent channel flow
Fluid velocity inside the channel: )()( tuutu t
mean fluid velocity intrinsic fluctuations
Use the empirical drag law to couple the particle movement to the fluid velocity )(tu
Acceleration of tracer particles in fully developed turbulence
N. Mordant et al., Physica D 193 (2004), 245–251
Experimental measurements of the probability density function (pdf) of the acceleration of
tracer particles show a clearly non‐Gaussian behavior.
La Porta et al., Nature 409 (2001), 1017
Measured trajectory of a tracer particle in a turbulent water flow at Reynolds number Re=63’000.
Intrinsic velocity fluctuations
We study the influence of the intrinsic velocity fluctuations on the mixing.
The mean fluid velocity is kept constant:
The fluctuating part is determined by using the stochastic model introduced by A. M. Reynolds (Phys. Rev. Lett. 91 (2003), 084503) that reproduces well the experimentally observed distributions and autocorrelations of velocities
and accelerations of tracer particles in fully developed turbulence.
The model calculates a time series for the velocity of a tracer particle. For every “real” particle we generate a tracer particle and evolve it in time. The velocity of the tracer particle then gives us a stream line and
the “real” particle is dragged into the direction of this line.
.constu
)(tut
Particle mixing in turbulent channel flow
At different positions xs along the channel we make a vertical slice and measure the relative particle densities of every type of particles (see inset). From the differences of these densities we
calculate the density differences μ(y).
)()(1)( 21 yyy
Particle mixing in turbulent channel flow
Particle transport by water
under water dunes in front of San Francisco bay
Particle transport by water
- Transport Mechanisms:
- 1. Creep – rolling and sliding of grains on the soil
- 2. Saltation – hops of grains near the soil
- 3. Sheet Flow – completely mobile sand bed, grains moving in granular sheets
- 4. Suspension – turbulent lift forces overcome gravity, particles can travel very long distances
Particle transport by water
Particle transport by water
Particle transport by water