W4 Particles in Fluids 2015 - ETH Z · Particles in Fluids ... (Bernoulli‘s principle) Homogeneous flow Re

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

qQ

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.


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


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 transport by water

under water dunes in front of San Francisco bay


- 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




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W4 Particles in Fluids 2015 - ETH Z · Particles in Fluids ... (Bernoulli‘s principle) Homogeneous flow Re