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

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


• 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


• 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


• 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


,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):


-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)


• 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


St=0.05 St=0.2

St=0.5 St=2

Influence of numerical resolution


Lagrangian method corresponds to limiting case forinfinitely fine grid in Eulerian box-counting method

Resolution:102 cells ----202 cells ----602 cells ----d<ln|J|>/dt ----

St=0.2

Compressibility in DNS of HIT


• 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


• 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


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


If St is sufficiently small:

In agreement with Balkovsky et al (2001, PRL)

)exp( tn ),( St

Trivial limits: ,10 n 11 n 0

Conclusions


• 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


• 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


Thank you

for your attention

measuring segregation of inertial particles in turbulent flows by a full lagrangian approach

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