A kinetic theory for the transport of small particles in turbulent flows
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Transcript of A kinetic theory for the transport of small particles in turbulent flows
Open Statistical Physics Open University,10/03/2010
A kinetic theory for the transport of small particles in turbulent flows
Michael W ReeksSchool of Mechanical & Systems Engineering,
Newcastle University
Open Statistical Physics Open University,10/03/2010
Environmental /industrial processes
• Mixing & combustion
• pollutant dispersion
• fouling / deposition
• clean up
• radioactive releases
• slurry /pneumatic conveying
• aerosol formation
Open Statistical Physics Open University,10/03/2010
Modelling Particle Flows
• Particle Tracking (Lagrangian)– track particles through a random flow by solving
particle equation of motion
• Two-Fluid Model (Eulerian)– Continuum equations for continuous (carrier flow) and
dispersed phase (particles)
– constitutive relations /closure approximations
– boundary conditions
Open Statistical Physics Open University,10/03/2010
Objectives
• application of formal closure methods in dilute flows to– derive continuum equations /constitutive relations for
the particle phase• compare with traditional heuristic approach• criteria for their validity
– incorporate the influence of turbulent structures on particle motion into continuum equations
• One particle dispersion• Two particle (pair) dispersion• Drift in inhomogeneous turbulence
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particle motion in plain vortex and straining flow
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Caustics – Mehlig & Wilkinson
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Settling in homogeneous turbulence Maxey 1988, Maxey & Wang 1992
fieldy velocitparticle ousinstantane theis ),(
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– Begin with particle equation of motion e.g. for gas-solid flows
– Separate particle velocities & aerodynamic forces into mean & fluctuating components
– Average over all realisations of the flow
iiijij
i uuxDt
D
Reynoldsstresses Inter-phase momentum
transferMass X
accel
,;
mean ighteddensity we particle /average ensemble
iiiiiii
ii
uuuu
Two-Fluid Model
timerelaxation particle drag Stokes;)( 1 ii
i udt
d
mean and fluctuating carrier flow velocities
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Kinetic/ PDF Approach
– analogous to the kinetic theory of gases– uses an equation analogous to the Maxwell-
Boltzmann Equation to derive the two-fluid equations for a dispersed flow
• mass-momentum and energy equations (c.f. RANS for continuous phase)
• constitutive relations
• boundary conditions
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Modelling of Particle Flows 11
Two PDF approaches
Apply closure to transport equations for •<W(,x,t)> particle phase space density•<P(,up,x,t)> up is the fluid velocity seen by the particle at x (Simonin approach)
)(diffusive
force drivingturbulent convective
WuWux
W
t
W
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Closure approximations for Reeks, Zaichek, Swailes, Minier
)(diffusive
force drivingturbulent convective
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t
W
WWtxu
,
flowcarrier theof gradients local upon the
depend and Lagrangian are ,
space-in xdiffusion yin velocitdiffusion
,0,, ,0,, txutxxtxutx
0s
0s
x
tx,,
Wu Wu
Wu
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Modelling of Particle Flows 13
iijiiji u
xDt
D
Body forceMass x acceleration
Turbulent stress
ijjiijp
Momentum Equation - PDF Approach
Equation of state
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Momentum equation as a diffusion equation
t
d
dC
C
x
u
Dt
D
x
1
sisturbophore
1
1
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15
Particle Reynolds stress transport eqns
forces by volume work of Rate
stressesshear by work of Rate
flux stress Reynolds
2 nmnmnm
i
mni
i
nmi
nmii
nm
uu
xx
xDt
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- Reynolds stresses depend on shearing of both phases- Requires closure for Reynolds stress flux
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16
Chapman Enskog Approximation
...
phase continuous for the Harlow &Daly with Compare
31
kjl
liD
skji
jil
lkkil
ljkjl
likji
x
kC
xxx
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Application- PDF solutions • Transport and deposition in turbulent boundary layers
Deposition velocity
k
particle relaxation time
Velocity distribution at a wall for particles settling under gravity in a turbulent flow with particla absorption at the wall
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Divergence of the particle velocity field
along a particle trajectory
•measures the change in particle concentration•zero for particles which follow an incompressible flow •non zero for particles with inertia
particlestreamlines
),(
),(stxYyp sy
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19
Application to kinetic approach
Gaussian are )( )(
,,
citydrift velo
0
tuandtif
WdsstxYtxuWx
WWu
p
t
p
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Pair dispersion and segregationTwo colliding spheres radii r1 , r2
r1
r2
n
Collision sphere
rg(r)
)()( Strrg
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Kinetic Equation for P(w,r,t)and moment equations
1 / /~)(,)( 222 KKr rrruru
citydrift velo
0
,,
ww
t
p
ijij
i
dsstxYtxux
u
uwwxDt
D
momentum
w = relative velocity between identical particle pairs, distance r apartΔu(r) = relative velocity between 2 fluid pts, distance r apart
Structure functions
Net turbulent Force mass Pu
wtrwPw
wr
Pw
t
P
),,(
convection β = St-1 , St=Stokes number
Probability density(Pdf)
0
wxt i
mass
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Kinetic Equation predictionsZaichik and Alipchenkov, Phys Fluids 2003
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Summary / Conclusions• Particle transport and segregation in a turbulent flow
– Kinetic / pdf approach (single particle transport)• Treatment of the dispersed particle phase as a fluid
– Continuum equations
– Constitutive relations
– Boundary conditions (perfectly / partially absorbing)
– Kinetic approach for particle pair transport• radial distribution function
• Role of compressibility in the formulation of a kinetic equation
– Net relative drift velocity between particle pairs
– enhancement local concentration of neighbouring particles
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Moments of particle number density
St=0.05 St=0.5
• Particle number density is spatially strongly intermittent• Sudden peaks indicate singularities in particle velocity field
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Influence of turbulent structures
tx
sstxX
dsstxXtxXtx
,t)x(x
,t)x( tdt
d
,t)x( t
p
t
ppp
pp
p
at time at wasparticle
thegiven that at timeposition particle theis , where
,vexp0,0,,
:issoln The
v
0
x,t
Xp,s
Consider the instantaneous concentration (x,t)derived froman initial concentration (x,0) and a particle velocity field vp(x,t).The conservation of mass equation is
Open Statistical Physics Open University,10/03/2010
Dispersion in a random compressible flow
t
ppp
t
ppp
p
t
ppp
t
pp
p
t
ppppp
ststttdsdsdssstxXtxD
tandtif
txdssstxXtx
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txtx
tandtif
dsstxXtxXtxtxtx
0
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0
p
0
0
p
p
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0
vvv,,v,v
Gaussian-non are )(v )(
,,,v,v
,,,,v
,,v
Gaussian are )(v )(
,exp0,0,,v,,v
Drift velocity(Maxey)
Diffusion tensor D (Taylor’s theory)
Net particle flux
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Segregation of inertial particles in turbulent flows
M. W. Reeks, R. IJzermans, E. Meneguz, Y.AmmarNewcastle University, UKM. Picciotto, A. SoldatiUniversity of Udine, It
‘Fractals, singularities, intermittency, and random uncorrelated motion’
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De-mixing of particles• Particles suspended in a turbulent flow do not mix but
segregate – depends upon the particles inertial response to:
• structure and persistence of the turbulence
• Important in mixing and particle collision processes – growth of PM10 and cloud droplets in the atmosphere
• the onset of rain.
• Presentation is about quantifying segregation– analysing statistics and morphology of the segregation
• using a Full Lagrangian Method (FLM)– use of compressibility to reveal
» fractal nature » intermittency» random uncorrelated motion
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particle motion in a vortex and straining flow
Stokes number St = τp/τf~1
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Segregation in isotropic turbulence
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Segregation - dependence on Stokes number St=τp/τf
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-4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
4
Segregation in counter-rotating vortices
Flow pattern translated randomly in space with finite life- time
Open Statistical Physics Open University,10/03/2010
Caustics – Mehlig & Wilkinson
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Compressibility of a particle flowFalkovich, Elperin,Wilkinson, Reeks
•zero for particles which follow an incompressible flow •non zero for particles with inertia•measures the change in particle concentration
Divergence of the particle velocity field along a particle trajectory
particlestreamlines
),(
),(stxYyp sy
Compressibility (rate of compression of elemental particle volume along particle trajectory)
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Measurement of the compressibility
ijj
ipij JJ
x
txxJ det ;
),(
,0
0,
Deformation of elemental volume
Compression - fractional change in elemental volume of particlesalong a particle trajectory
can be obtained directly from solving the particle eqns. of motion
Avoids calculating the compressibility via the particle velocity field
Can determine the statistics of ln J(t) easily.
The process is strongly non-Gaussian – highly intermittent
- xp(t),vp(t),Jij(t),J(t)) - Fully Lagrangian Method
Jdt
d
dt
dJJtxp ln,v 1
0
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Particle trajectories in a periodic array of vortices
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Deformation Tensor J
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Singularities in particle concentration
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Compressibility
Simple 2-D flow field of counter rotating vorticesKS random Fourier modes: distribution of scales, turbulence energy spectrum
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Moments of particle number density
• Along particle trajectory: particle number density n related to J by:
)(|)(| 1 tntJ || Jn || Jn
• Particle averaged value of is related to spatially averaged value:n
11 || Jnn
n
Trivial limits: ,10 n 11 n (equivalent to counting particles)
• Any space-averaged moment is readily determined, if J is known for all particles in the sub-domain
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Moments of particle number density
St=0.05 St=0.5
• Particle number density is spatially strongly intermittent• Sudden peaks indicate singularities in particle velocity field
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Comparison with analytical estimate
If St is sufficiently small: )exp( tn
),( St
•For first time, numerical support for theory of Balkovsky et al (2001, PRL): “ is convex function of ”.
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Random uncorrelated motion •Quasi Brownian Motion - Simonin et al•Decorrelated velocities - Collins •Crossing trajectories - Wilkinson •RUM - Ijzermans et al.• Free flight to the wall - Friedlander (1958)• Sling shot effect - Falkovich
Falkovich and Pumir (2006)
12
2L1L ),2(v),1(v)(
rrr
rrrRL
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Radial distribution function (RDF) g(r)
r
g(r)
)()( Strrg
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DNS: details of the code
• Statistically stationary HIT• Pseudo-spectral code• Grid 128x128x128• Re =65• Forcing is applied at the lowest wavenumbers
• NSE for an incompressible viscous turbulent flow:
• In a DNS of HIT, the solution domain is in a cube of size L, and: k
xik tet )(k,u)u(x,
• 100.000 inertial particles are random distributed at t=0 in a box of L=2• •Interpolation of the velocity fluid @ the particle position with a 6th order Lagrangian polynomial• Trajectories and equations calculated by RK4 method• Initial conditions so that volume is initially a cube
)0,u()0v( txt p
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Averaged value of compressibility vs time
Elena Meneguz 46
• Qualitatively the same trend with respect to KS
• We expect a different threshold value
0 0.1 0.2 0.3 0.4 0.5 0.6-2
-1
0
1
2
3
4
time
d<ln|J|>dt
St=0.7
St=1St=0.1
St=10
WHAT CAUSES THE POSITIVE VALUES???
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Moments of particle concentration
0 0.2 0.4 0.6 0.8 1 1.2 1.410
0
102
104
106
108
1010
time
St=1
alpha=0
alpha=2alpha=3
n
intermittency due to the presence of singularities in the pvf
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Turbulent Agglomeration
rr wnj 21
nrjK
rrj
rrr
cr
ccr
c
/)( areacollision
kernelCollison
at particles colliding ofcurrent )(
spherecollision of radius 21
Two colliding spheres radii r1 , r2
r1
r2
test particle
Saffman & Turner model
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22/12
222
)(),()(
15
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15 ;
22
rnrnrrKdt
rdn
rK
x
ur
x
u)σ(r
)σ(rrπwπrK
S
cS
cc
cπcrcS
Agglomeration in DNS turbulence L-P Wang et al. - examined S&T model•Frozen field versus time evolving flow field•Absorbing versus reflection Brunk et al. - used linear shear model to assess influence of persistence of strain rate, boundary conditions, rotation
n
Collision sphere
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Agglomeration of inertial particlesSundarim & Collins(1997) , Reade & Collins (2000): measurement of rdfs and impact velocities as a function of Stokes number St
)(),(4),( 221 StwStrgrrrK rcc Net relative velocity between colliding
spheres along their line of centresRDF at rc
)( crg
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Inertial collisions (RUM)
Ratio of the RMS of the relative velocity of colliding particles over the corresponding RMS of the relative fluid velocity; collision radius rc/ηk =0.1
r1
r2
rc=r1+r2
particle Stokes number St
2/12
2/12
w
w
rf
rc
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Probabalistic Methods
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Kinetic Equation and its Moment equationsZaichik, Reeks,Swailes, Minier)
1 / /~)(,)( 222 KKr rrruru
iijij
i uwwrDt
D
wwmomentum
w = relative velocity between identical particle pairs, distance r apartΔu(r) = relative velocity between 2 fluid pts, distance r apart
Structure functions
Net turbulent Force (diffusive)mass Pu
wtrwPw
wr
Pw
t
P
),,(
convection β = St-1
Probability density(Pdf)
0
wxt i
mass
1)(2
1 Strrg St
Open Statistical Physics Open University,10/03/2010
Kinetic Equation predictionsZaichik and Alipchenkov, Phys Fluids 2003
Open Statistical Physics Open University,10/03/2010
Dispersion and Drift in compressible flows (Elperin & Kleorin, Reeks, Koch & Collins, Reeks)
)(flux Drift )(flux Diffusive dD qqw
tdttrwtrtrwDt
D t
0
),(exp)0),((),(
•w(r,t) the relative velocity between particle pairs a distance r apart at time t•Particles transported by their own velocity field w(r,t) •Conservation of mass (continuity)
1/
1Stfor Only works
,(,
,(,)(,)(
22
0
0
KSt
t
idi
t
jiijj
ijDi
rg(r) ~r
tdttrwtrwq
tdttrwtrwrDr
rDq
Random variable
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Summary / Conclusions• Overview
– Transport, segregation, agglomeration dependence on Stokes number
– Use of particle compressibility d/dt(lnJ)
– Singularities, intermittency, fractals, random uncorrelated motion
– Measurement) and modeling of agglomeration• (RDF and de-correlated velocities
• PDF (kinetic) approach, diffusion / drift in a random compressible flow field
– New PDF approaches – statistics of acceleration points (sweep/stick mechanisms)(Coleman & Vassilicos)