1. Mitigation of Preferential Concentration due to electric charge in the dispersed phase
I m going to talk about my work in last 2 years. I m going to keep this fairly simple bcos its only the simple thing tht I understood and the complex things, I have left for future work.

2. Overview Introduction Simplifying assumptions / Numerical method
Measures of preferential accumulation
Stokes number dependence
Dependence on Re ?
Charged particle simulations
Conclusions
Numerical method – I would focus on particle assumptions
Lets put things in context: We sent a paper claiming that accumulation goes down with Re, and we got a review saying it doesn’t. So what is going on ?
If you feel I am going back and forth between slides too much and you feel lost, don’t hesitate to stop me and ask whats going on

3. The problem in physical space
Dispersed phase flows
continuous phase (fluid)
dispersed phase (particles)

4. Dispersed phase flows Particles: Fluid: One-way coupling
Particles do not influence fluid motion
Point particles
Particle wakes are not resolved
Particle diameter << kolmogorov length scale
Particle motion is governed only by “drag”
Gravitational force not modelled
Particle collisions not modelled (dilute suspension)
Particle density >> fluid density
Particles are “stochastic” for the purpose of charged particle simulations
Incompressible, homogeneous, isotropic
Stationarity obtained using artificial forcing
Be confident about each of the particle assumptions

5. Governing equations Particles: Fluid:
Modified Stokes drag law (Valid for Rep <= 800)
Large-scale forcing function added to maintain stationary turbulence
*Symbols have usual meanings

6. Numerical scheme (Fluid)
Fluid (pseudo-spectral method):
Particle:
(suppose)
Skip this fast !
dealiasing by 2/3rd rule
temporal discretization using RK3
stochastic forcing scheme* to sustain kinetic energy
*V. Eswaran and S.B. Pope, Computers and Fluids, Vol. 16(3), pp , 1988

7. Numerical Method (summary)
The turbulence is limited to homogeneous, isotropic case (HIT) in a periodic cube.
Particles are not resolved.
Force on particles is due to Stokes drag.
One way coupling between fluid and particles
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8. Simulation parameters
Stokes Number
Rayleigh Number
where,
Mono-sized particles
number of particles (Np):
particle stokes numbers
Stk :
Same charge on all particles (Ra = 0.8, γ = 0.05)
space charge densities (μC/m3): , 10, 25, 50, 100
Emphasize the importance of Stokes number. St -> 0 tracer particles. St -> inf, particles not responding to turbulence

9. Points to note -
All simulations for a given Re, are restarted from same fluid realisation.
Statistics are collected only after fluid has reached stationary state.
Particle distribution is assumed to reach stationary state when the positions are completely de-correlated from initial position.
Particle rms velocity
Turbulent kinetic energy

10. Evidence of preferential concentration
Network of high particle number density regions.
Showing here the 2 important parameters which govern this phenomenon.
Reλ = 24.24, St = 0.25
Reλ = 24.24, St = 4.00
*S. Scott, Ph.D. thesis, Imperial College London, 2006

11. Clustering at different scales
Clustering occurs broadly at 2 scales –
Dissipative scales
particles are centrifuged out of coherent eddies and accumulate in low-vorticity regions.
Inertial range
clustering is a multi-scale phenomenon.
Eddies larger than Kolmogorov length scale play a part in clustering.
Mention sweep-stick mechanism

12. Measures of Accumulation
Dissipative range measures
D ( Fessler et al., 1994 )
Dc ( Wang and Maxey, 1993 ) Dn
Inertial (multi-scale) measures
RDF ( Sundaram and Collins, 1997 ) D2
Fluid-particle correlation measures
<n’e’>, correlation between number density and enstrophy
ln, number density correlation length scale

13. D2 measure r
Correlation integral, C(r) : number of particles within range r of any given particle
D2 is slope of curve log( C(r) ) vs log( r )
D2 is equal to the spatial dimension for uniform distribution (equal to 3 for a 3D distribution)

14. D2 – probability to find 2 particles at a distance less than a given r: P(r) ~ rD2
Range pictures to the left and place underneath the bullet point text. Range captions to the bottom right hand corner of the picture. The image shown here is 7cm x 5cm.
D2 data for different cases compared to literature

15. Binning of particles h h – scale used for binning particles
Some are going to be void
h – scale used for binning particles
n – number density i.e no. of particles / bin volume
<nc> – mean number density i.e total particles / volume of cube

16. D, Dc : deviation from poisson distribution
Dc*, D** : Deviation from poisson distribution
D and dc have dependence on bin size which is annoying
Pc: probability of finding cells with given number of particles
k : number of particles in a cell
*L.P. Wang and M.R. Maxey, J. Fluid Mech., Vol. 256, pp , 1993
**J.R. Fessler, J.D. Kulick and J.K. Eaton, Phys. Fluids, Vol. 6(11), pp , 1994

17. ‘D’ measure of accumulation
Bin size used is corresponding to peak value of ‘D’.

18. <n’e’> – correlation between number density and enstrophy

19. Observations D and Dc measures clearly depend on bin-size
Dependence of Re is attributed to less number of ‘smaller particle structures’ at higher Re.
D2 measure looks at probability of finding particles in shells around a given particle
Shows nearly no dependence on Re
<n’e’> and ‘ln’ capture distribution of particle number density
Show dependence on Re

20. Destruction using Lorentz forces*
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*A. Karnik and J. Shrimpton, ILASS 2008, Sept 8-10, 2008

21. Particle position, fluid velocity
Network of high particle number density regions.
Showing here the 2 important parameters which govern this phenomenon.
Reλ = 24.24, St(k) = 1.0, Qv=5μC/m3
Reλ = 24.24, St(k) = 1.6, Qv=100μC/m3
*S. Scott, Ph.D. thesis, Imperial College London, 2006

22. Evidence of preferential concentration destruction
Network of high particle number density regions.
Showing here the 2 important parameters which govern this phenomenon.
Reλ = 24.24, St(k) = 1.0, Qv=5μC/m3
Reλ = 24.24, St(k) = 1.6, Qv=100μC/m3
*S. Scott, Ph.D. thesis, Imperial College London, 2006

23. Evidence of preferential concentration destruction
Network of high particle number density regions.
Showing here the 2 important parameters which govern this phenomenon.
Reλ = 24.24, St(k) = 1.0, Qv=5μC/m3
Reλ = 24.24, St(k) = 1.6, Qv=100μC/m3
*S. Scott, Ph.D. thesis, Imperial College London, 2006

24. Parametric study of bulk charge density levels
*St = 0.25 for all plots
Space charge density of µC/m3 is sufficient to destroy preferential accumulation
With increasing Reynolds number, greater charge density is required to significantly destroy accumulation

25. Effect of Stokes Numbers
Reλ = 24.2
Reλ = 81.1
Stokes number is defined differently – scaled by integral time scale
Charged particle systems continue to exhibit same trends with Reynolds and Stokes numbers as the charge-free case.

26. Schematic of a spray released from a charged injection atomizer
d0= 500 μm, Q0= 0.5 C/m3, θ = 45o
The charge level found in this study (50 μC/m3) corresponds to an area about 2 cm from the nozzle tip

27. Conclusions
Preferential accumulation is maximum at St ~ 1.0 based on kolmogorov scale, for all the measures used in this study.
While ‘ln’ shows clear dependence on Re, D2 is insensitive to Re.
Bulk charge density level of 50 μC/m3 is sufficient to significantly destroy preferential accumulation. This has been consistently observed using different sensors for preferential accumulation.
The required charge density level mentioned above is attainable within 2 cms from tip of a nozzle in practical charge injection atomizers.
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