Measuring segregation of inertial particles in turbulent flows by a Full Lagrangian approach E. Meneguz Ph.D. project: Rain in a box of turbulence Supervisor: M. W. Reeks PostDoc: R. H. A. IJzermans School of Mechanical and Systems Engineering UNIVERSITY OF NEWCASTLE 4 th IMS Workshop on Clouds and Turbulence Institute for Mathematical Sciences Imperial College London 23-25 March 2009
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Measuring segregation of inertial particles in turbulent flows by a Full Lagrangian approach
Measuring segregation of inertial particles in turbulent flows by a Full Lagrangian approach. 4 th IMS Workshop on Clouds and Turbulence Institute for Mathematical Sciences Imperial College London 23-25 March 2009. E. Meneguz Ph.D. project: Rain in a box of turbulence - PowerPoint PPT Presentation
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Measuring segregation of inertial particles in turbulent flows by a
Full Lagrangian approach
E. MeneguzPh.D. project: Rain in a box of turbulenceSupervisor: M. W. ReeksPostDoc: R. H. A. IJzermansSchool of Mechanical and Systems EngineeringUNIVERSITY OF NEWCASTLE
4th IMS Workshop on Clouds and TurbulenceInstitute for Mathematical Sciences
Imperial College London23-25 March 2009
Outline
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 2
• Introduction;
• Equations of motion of inertial particles and other equations;
• Compressibility of the Particle Velocity Field:
- MEPVF (Eulerian) - FLA (Lagrangian)
• Method
- flow field description - numerical simulations
• Results:
- Compressibility of the particle vel. field in both approaches - preliminary results in DNS of HIT - moments of the particle number density (comparison with theor. predictions)
• Conclusions and future developments
State of the art
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 3
• Importance of de-mixing of particles in turbulent flows: many environmental-industrial and statistical interest
• Preferential concentration (Crowe et al, 1993 and Maxey (1987); Sundaram and Collins (1997) and Wang et al. (1998);
• Recently studied from different viewpoints (Chen et al, 2006, Balkowsky et al, 2001, Sommerer & Ott (1993) and Wilkinson et al, 2005;2007)
• Random Uncorrelated Motion - “sling effect” (Falkovich et al. 2002) or “crossing trajectories effect” (Wilkinson et al. 2005)
• Fevrier et al. MEF as sum of two contributions: MEPVF and RUM; based on box-counting (EULERIAN);
• Osiptsov’s method (Reeks 2004, Healy and Young 2005) LAGRANGIAN
Equations of motion (Lagrangian frame)
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09
slide 4
• Particles/droplets: spherical, rigid, identical and heavy
• Dilute system
• Effect of gravity and Brownian motion not included
p
vxuv ),(1 t
Stdtd
pvx
dt
d p
px
uv
St
particle position
particle velocity
fluid velocity at the position of the particle
normalized particle relaxation number
Compressibility of the PVF
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 5
• Flow field incompressible:
• Preferential concentration: PVF compressible non-zero gradients in the particle number density
• Continuity equation:
• For sufficiently small Stokes number:
),( tn x
)( 2StSt uuuv
StQSt )( uuv
0 u
0 v
vv nn
tn
,ln vndtd
MEPVF and compressibility of PVF
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 6
• According to Fevrier et al. 2005: MEPVF = PVF + RUM
PVF =
jN
jj
jNt
1
1),( vxv
i-th cell
To be obtained from finite differences
),( txv
Lagrangian quantification of the compressibility
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 7
,ijij J
dtdJ ).(1
ijk
iik
ij JxuJ
StdtJd
2D EXAMPLE:
Evol. eqts.
nJJ ji /1)det( ,
Continuity equation and averaging over all particle trajectories:
v||ln Jdtd
j
iij x
txJ
,0
0 ),(
x
0x
,ln vndtd
Model of synthetic turbulent flow
2D carrier flow field (Babiano et al. 2000):
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 8
-4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
4
ywtAxtyx cos))sin(cos(),,( To study effect of RUM: ,0A 0w
Threshold value: St=0.25
Numerical methods (Lagrangian vs Eulerian)
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 9
• 10,000 inertial particles, uniformly distributed at t=0;• Periodic boundary conditions for particles;• Trajectories and equations for calculated by RK4 method;• Initial conditions:
(Volume is initially a cube).
jiJ ,,, ijjiJ
j
iji x
uJ
,
• 1,000,000 inertial particles, uniformly distributed at t=0;• Trajectories calculated by RK4 method;• Periodic boundary conditions for particles;• Divergence of PVF calculated using 2nd order finite difference scheme;• Numerical resolution varied between 102 and 602 cells.
Compressibility in time
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 10
St=0.05 St=0.2
St=0.5 St=2
Influence of numerical resolution
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 11
Lagrangian method corresponds to limiting case forinfinitely fine grid in Eulerian box-counting method
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 12
• Good agreement between Lagrangian and Eulerian method• Singularities seem to be detected better by Lagrangian method
St=1
Picciotto et al. 2005
Moments of particle number density
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 13
• 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
1 nn
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
n
11 || Jn
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 14
St=0.2 St=0.5
• Particle number density is spatially strongly intermittent• Sudden peaks indicate singularities in particle velocity field
Moments of particle number density
Comparison with analytical estimate
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 15
If St is sufficiently small:
In agreement with Balkovsky et al (2001, PRL)
)exp( tn ),( St
Trivial limits: ,10 n 11 n 0
Conclusions
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 16
• The Lagrangian and Eulerian method show good agreement in the compressibility of the PVF for a wide range of St;
• Singularities in the PVF can be detected by Lagrangian method but not by Eulerian method, due to finite grid size in the latter;
• Lagrangian method allows for determination of any space-averaged moment of the particle number density, in contrast with Eulerian which would have too limited spatial resolutions;
• The determination of moments of the particle number density have shown very high spatial intermittency due to RUM. • For first time, numerical support for theory of Balkovsky et al (2001, PRL): “ is convex function of ”.
Further developments
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 17
• Open question: high intermittency in particle number density in DNS?
• 3D DNS of stationary HIT for different St numbers
• pdf methods for two particle dispersion at higher Re numbers
Elena Meneguz ● 4th IMS Workshop on Clouds and Turbulence ● Imperial College London ● 24.03.09 slide 18