1. Inertial Particle Segregation and Deposition in Large-Eddy Simulation
P20 WG3 COST MEETING
Technische Universiteit Eindhoven
Inertial Particle Segregation and
Deposition in Large-Eddy Simulation
of Turbulent Channel Flow (TCF)
C. Marchioli§, M.V. Salvetti§§ and A. Soldati §*
§ Dipartimento di Energetica e Macchine & Centro Interdipartimentale
di Fluidodinamica e Idraulica, Università di Udine
§§ Dipartimento di Ingegneria Aerospaziale, Università di Pisa
*Dept. Fluid Mechanics, International Center for Mechanical Sciences, Udine
October 29-30, 2009, Eindhoven, the Netherlands

2. Issues in Lagrangian tracking of particles in LES flow fields
Motivation
Issues in Lagrangian tracking of particles in LES flow fields
In LES only the filtered velocity field is simulated.
How particles of different inertia
react to the lack of SGS scales?
For fixed inertia, which physical
phenomena and/or statistical
quantities are more sensitive?
Is a closure model needed in the
particle motion equations (p.m.e.)
to reintroduce the effects of the
filtered SGS scales?
Which characteristics should this
closure model have?
Large-Eddy Simulation of a modern Pratt & Whitney gas turbine combustor (Mahesh et al. 2005, Moin & Apte 2005).

3. Premise: Physics of Turbulent Particle Dispersion in TCF
Physical Background: from DNS-based Eulerian-Lagrangian studies
we know that dispersion can be envisioned as a multi-step process

4. Rule of Thumb for Maximum Segregation/Deposition
Particles have to scale with wall flow scales
Red: high stream-
wise velocity
Purple particles:
to the wall
Blue: low stream-
Blue particles:
off the wall
In the context of industrial modelling of turbulent dispersed flows:
… is an accurate quantification of this effect currently available? (practical issue)‏
… will Large Eddy Simulation be able to capture this effect? (modelling issue)‏

5. Use of LES for Turbulent
Dispersed Flow (Ret=150)
DNS
(1283)
LES
(643)
“Different” structures (few in the center for LES) and “different” particle distribution (more homogeneous in LES, weaker interaction with vortices)

6. Use of LES for Turbulent
Dispersed Flow (Ret=150)
DNS
LES
NEAR-WALL
REGION
1D streamwise spectra (energy associated
with frequency) for turbulent channel flow
at shear Reynols numbers Ret=150.
LES
DNS
CHANNEL
CENTERLINE
Qualitative “differences” correspond to
quantitative differences (in the energy
spectra for instance):
Any filter will prevent particles from
being exposed to small scales which
can modify their local behavior (e.g.
segregation, dispersion…)
Inaccurate particle dispersion will bring
errors into subsequent particle motion
and fluid motion
Qualitative visualization of instantaneous particle distribution superposed to the flow structure already highlights differences between DNS and LES, in particular as far as the capability of capturing the microscale interaction between the phases which is responsible for all macroscale phenomena discussed in the sixth slide.
However, trying to be more quantitative, we can retrieve differences in the statistics: here we show the 1D energy spectra for DNS and LES both in the near wall region and in the channel centerline. The characteristic frequency with which particles with different Stokes number interact with turbulence is also shown. The energy of the flow is lower in LES fields than in DNS, the range of frequencies shifts toward higher values in the channel centerline (this means that the scales filtered by LES correspond to higher frequencies to which particles are no longer exposed: this affects also those particles with “low-frequency” that are not much affected by LES filtering of “high frequencies” in the near-wall region).

7. No SGS model needed in the p.m.e. SGS model needed in the p.m.e.
Also Note: Literature Background on
Need for SGS Modeling in the P.M.E.
Importance of SGS fluid turbulence on particle motion (previous studies)
Armenio et al., PoF (1999)
Effects of SGS terms on dispersion statistics and one-point statistics on heavy particles are negligible for “well-resolved” LES.
Fede & Simonin, PoF (2006)
Effects of SGS terms on preferential concentration and on collisions are significant only for St<5 (in HIT).
Shotorban & Mashayek, J. Turb. (2006) Kuerten & Vreman, PoF (2005)
Effects of SGS terms on particle statistics and on preferential concentration are significant especially for small particles.
Kuerten, Phys. Fluids (2006)
Large effects of SGS terms on turbophoresis.
NO GENERAL AGREEMENT!!
Particle segregation (Sp) versus
Particle Stokes number (tFfp/tK)
in HIT (Fede & Simonin, PF, 2006)
No SGS model needed in the p.m.e.
SGS model needed in the p.m.e.

8. Outline of the Presentation
Physical Problem and Modelling Approach
Part 1: A-priori tests at Ret=150
(cfr. Marchioli et al., PoF, 2008)
1.1 Effect of filtering on concentration and segregation
1.2 Effect of SGS modeling in the p.m.e.
1.2.1 Fractal Interpolation
Part 2: A-posteriori tests at Ret=150 and 300
(cfr. Marchioli et al., PoF, 2008; ACME, 2008)
2.1 Effect of filtering on concentration and segregation
2.2 Effect of SGS modeling in the p.m.e.
2.2.1 Fractal Interpolation
2.2.2 Approximate Deconvolution
Part 3: Scaling issues (if time permits) MA ANCHE NO DIREI
Discussion and Conclusions

9. Physical Problem and Modelling Approach
Point-particle LES/DNS – Flow domain and flow field calculation
Time-dependent 3D turbulent gas flow field at shear Reynolds number: Re=uh/=150, 300
Channel size: Lx x Ly x Lz = 4ph x 2ph x 2h
Pseudo-spectral LES/DNS: Fourier modes (1D FFT) in the homogeneous directions (x and y)‏, Chebyschev coefficients in the wall-normal direction (z)‏
Dynamic Smagorinsky SGS model
Time intergration: Adams-Bashforth (convective terms), Crank-Nicolson (viscous terms)

10. Pointwise-particle LES/DNS – Lagrangian particle tracking
Physical Problem and
Modelling Approach
Pointwise-particle LES/DNS – Lagrangian particle tracking
One-way coupling
Particle wall-collisions: fully elastic
Time-integration: 4th-order Runge-
Kutta scheme
Fluid velocity interpolation: 6th-order
Lagrange polynomials‏
Particle Stokes number, St = p/ f
Influence of Inertia:
Particle Time Scale, p = dp2 p/18 
Flow Time Scale, f = L/U = /u
Coupled to the flow solver is a Lagrangian particle tracking routine, which solves for the following particle motion equation

11. PART 1 A-Priori Tests

12. Summary of Simulations
for A-Priori Tests
Pointwise-particle DNS/filtered-DNS Database
DNS
Filtered DNS
Filter
Ret=150
128x128x129
64x64x129
Top-hat
Cut-off
32x32x129
16x16x129
GRIDS St
tp [s ]
dp+
dp [m ]
0.2 1 5 25
125
0.227
1.133
5.660
28.32
141.5
0.068
0.153
0.342
0.765
1.71
9.1
20.4
45.6
102.6
228
Effects of neglecting SGS velocity fluctuations for: different particle statistics and phenomena, different particle sizes, different widths and types of filtering
STOKES
TOTAL TRACKING TIME: t+=1200 (dt+=0.045)

13. 1.1 A-Priori Tests without SGS Modelling in the P.M.E.
(Well-known) effects of filtering on the fluid velocity fluctuations
R.m.s of fluid streamwise velocity
R.m.s of fluid wall normal velocity
Filtering
Filtering
DNS +
Red: cut-off
filtered 32x32
filtered 64x64 •
filtered 16x16
Blue: top-hat

14. 1.1 A-Priori Tests without SGS Modelling in the P.M.E.
Effects of filtering on the particle velocity fluctuations: underestimation for all the considered particle sets, and depending on particle inertia.
R.m.s of particle wall-normal velocity CF
St=1
St=5
St=25
DNS +
filtered 32x32 (CF=4)
Red: cut-off
filtered 64x64 (CF=2) •
filtered 16x16 (CF=8)
Blue: top-hat

15. 1.1 A-Priori Tests without SGS Modelling in the P.M.E.
In alternativa alle due slide precedenti (stessi risultati, solo cut-off)

16. 1.1 A-Priori Tests without SGS Modelling in the P.M.E.
Effects of filtering on the particle velocity fluctuations
Mean particle TKE (normalized to the DNS value)
OBSERVATIONS:
significant reduction of particle velocity fluctuations also at a resolution typical of LES (32x32)
sensitivity to filter width and type
sensitivity to filtering is higher for smaller particles but it is significant also for large ones (St=25). Only for p+=125 we found small effects of filtering.
St=25
Red: cut-off +
St=5 o
Blue: top-hat
St=1

17. 1.1 A-Priori Tests without SGS Modelling in the P.M.E.
Effects of filtering on the particle concentration (filter: 32x32 – CF=4)
St=0.2
St=1
St=5
St=25
Significant underestimation of turbophoresis for large particles

18. 1.1 A-Priori Tests without SGS Modelling in the P.M.E.
Effects of filtering on the particle concentration (CF=2 vs CF=4)
DNS
DNS
Cut-off
Top-hat
Cut-off
Top-hat
CF=4
CF=2
Larger underestimation of turbophoresis on coarser grids

19. 1.1 A-Priori Tests without SGS Modelling in the P.M.E.
In alternativa alle due slide precedenti (stessi risultati, forma diversa…)
St=1
St=5
St=25

20. 1.1 A-Priori Tests without SGS Modelling in the P.M.E.
Regular distribution
Random distribution
Clustered Distribution
Particle segregation parameter (or maximum deviation from randomness)
Sp = (s-sp)/l
with l = average number of particles per cell, s = standard deviation of the PDF

21. 1.1 A-Priori Tests without SGS Modelling in the P.M.E.
near wall (0<z+<5)
centerline
CONCLUSION: Particle preferential concentration is always underestimated
-> Can SGS modelling in P.M.E. improve the situation?

22. 1.2 A-Priori Tests with SGS Modelling in the P.M.E.
Candidate SGS models for the P.M.E.
Inverse Filtering or Approximate Deconvolution (Kuerten, 2005; Kuerten, 2006; Shotorban and Mashayek, 2005)
Fractal Interpolation (Scotti & Meneveau, LES of single-phase homogeneous isotropic turbulence)
Stochastic models in a Lagrangian context (Wang & Squires 1996, Fukagata et al. 2006, Elhami Amiri et al., 2006)

23. 1.2 A-Priori Tests with SGS Modelling based on F.I.
Our approach: fractal interpolation (Scotti & Meneveau, 1999)
Basic idea: reconstruct the velocity field from the filtered one by iteratively applying an affine mapping procedure
The characteristics of the reconstructed signal depend on two stretching parameters, which are related to the fractal dimension of the signal.
6th reconstruction
6th reconstruction
2nd reconstruction
d=±0.794
d=±0.5
1st reconstruction

24. 1.2 A-Priori Tests with SGS Modelling based on F.I.
Lagrangian Particle Tracking
Fractal Interpolation
DNS
Filtering
Expected advantages:
no need of defining/apply/invert a filter
possibility of reconstructing the velocity at/close the particle position (reduction of interpolation error in LES)
easy to be applied (also in 3D)
Present implementation:
Interpolation in the homogeneous directions up to the DNS resolution
Stretching parameters:
constant in the whole domain and for all the velocity components, equal to the values proposed in (Scotti & Meneveau, 1999) for homogeneous isotropic turbulence (d=±0.795)
constant on the horizontal planes equal to the values obtained from the fractal dimension of the DNS velocity signals following Keshevarzi et al., CSF (2005).

25. 1.2 A-Priori Tests with SGS Modelling based on F.I.
Fractal interpolation with constant stretching parameters (d=±0.794)
R.m.s of the particle wall-normal velocity
St=1
St=5
St=25
DNS +
filtered 32x32
Red: cut-off
filtered 64x64 •
filtered 16x16
Blue: top-hat
no significant improvement for particle statistics and concentration

26. 1.2 A-Priori Tests with SGS Modelling based on F.I.
Sensitivity to the stretching parameters
R.m.s of the particle streamwise velocity
R.m.s of the particle normal velocity
St=1
St=1 +
Red: computed (local) stretching parameters
filtered 32x32 •
filtered 16x16
Blue: constant stretching parameters
no significant improvement of more realistic stretching parameters

27. 1.2 A-Priori Tests with SGS Modelling based on F.I.
What is the problem?
If the filtered velocity signal at subsequent nodes is almost aligned, the fractal interpolation can not add fluctuations (no fractal shape to be copied)
DNS/filtered/reconstructed u instantaneous velocity signal in the streamwise direction (y+=119, z+=). Coarsening factor 8.
Possible remedies:
to use a variable stencil for fractal interpolation (not always 3 subsequent nodes);
deconvolution first and, then, fractal interpolation

28. PART 2 A-Posteriori Tests

29. Summary of Simulations for A-Posteriori Tests
Pointwise-particle LES/DNS – Database
DNS
LES
Ret=150
128x128x129
32x32x65
64x64x65
Ret=300
256x256x257
64x64x129
GRIDS St
Ret=150
0.2, 1, 5, 25, 125
Ret=300
1, 4, 5, 20, 25, 100
STOKES
TOTAL TRACKING TIME: t+=1800 (dt+=0.045/0.03)

30. 2.1 A-Posteriori Tests without SGS Modelling in the P.M.E.
Effects of filtering on the fluid velocity fluctuations
R.m.s of fluid streamwise velocity
R.m.s of fluid wall normal velocity
TOGLIERE?

31. 2.1 A-Posteriori Tests without SGS Modelling in the P.M.E.
Effects of filtering on the particle velocity fluctuations: underestimation for all the considered particle sets, and depending on particle inertia.
R.m.s of particle wall-normal velocity
St=1
St=5
St=25
In the well resolved LES the particle velocity fluctuations are well predicted. Nevertheless…
TOGLIERE?

32. 2.1 A-Posteriori Tests without SGS Modelling in the P.M.E.
Effects of filtering on the particle concentration at Ret=150
St=25
St=5
St=1
Significant underestimation of
turbophoresis for large particles
(PROFILI PUBBLICATI SU POF 2008) TOGLIERE?

33. 2.1 A-Posteriori Tests without SGS Modelling in the P.M.E.
Effects of filtering on particle concentration at Ret=150
St=25
St=5
St=1
Significant underestimation
of turbophoresis (still)
(PROFILI PUBBLICATI SU ACME 2008) TOGLIERE?

34. 2.1 A-Posteriori Tests without SGS Modelling in the P.M.E.
Effects of filtering on particle segregation at Ret=150
near wall (0<z+<5)
centerline
As in a priori tests, large underestimation of particle segregation for all particle sets (but for St=125), also in the well resolved LES.
TOGLIERE?

35. 2.1 A-Posteriori Tests without SGS Modelling in the P.M.E.
Particle segregation and near-wall accumulation in LES fields at Ret=150
Filtered fluid
velocity!! c c
From a practical viewpoint, all these differences eventually lead to inaccuracies in the quantitative prediction of quantities such as deposition rates and concentration, which are of interest in engineering applications. See for instance particle concentration: the qualitative behavior is captured by LES even with no SGS modeling, however the peak of concentration at the wall is severely underpredicted. It thus seems necessary to introduce some kind of closure in the p.m.e. to fill this gap.
OBSERVATION: particle segregation
and near-wall concentration are
underpredicted when the filtered
fluid velocity is used in the p.m.e.
CONCLUSION: IT’S NECESSARY TO MODEL THE EFFECT OF SGS VELOCITY FLUCTUATIONS ON PARTICLES BY INTRODUCING SOME KIND OF CLOSURE IN THE P.M.E.!!! AGAIN...
???LASCIARE SOLO QUESTA AL POSTO DELLE 6 SLIDE PRECEDENTI???

36. 2.2.1 A-Posteriori Tests with SGS Modelling based on F.I.
R.m.s of the particle wall-normal velocity at Ret=150
St=1
St=5
St=25
Fine LES
Coarse LES
No significant improvement for particle statistics and concentration

37. 2.2.1 A-Posteriori Tests with SGS Modelling based on F.I.
Effects of filtering on the particle concentration at Ret=150
Fine LES
Coarse LES
St=1
Significant underestimation of turbophoresis for large particles

38. 2.2.1 A-Posteriori Tests with SGS Modelling based on F.I.
Effects of filtering on the particle concentration at Ret=150
Fine LES
Coarse LES
St=5
Significant underestimation of turbophoresis for large particles

39. 2.2.1 A-Posteriori Tests with SGS Modelling based on F.I.
Effects of filtering on the particle concentration at Ret=150
Fine LES
Coarse LES
St=25
Significant underestimation of turbophoresis for large particles

40. 2.2.2 A-Posteriori Tests with SGS Modelling based on A.D.
Lagrangian tracking of particles in LES fields with Approximate Deconvolution (A.D.)
LES
Approximate Deconvolution
Filtered fluid
Velocity
Deconvolved
fluid vel.
The filtered velocity is defined by the convolution product between the instantaneous fluid velocity and the kernel filter G. Assuming that G has an inverse G-1, the inverse operator G-1 can be expanded as an infinite series of filter operators or approximated by truncating the series at some N.
Lagrangian Particle Tracking
(top-hat filter)

41. 2.2.2 A-Posteriori Tests with SGS Modelling based on A.D.
Root Mean Square of particle wall-normal velocity at Ret=150 **
DNS vs LES vs LES+Deconvolution (LES grid is 64x64x129)‏
St=1
St=25
Approximate deconvolution reintroduces the correct amount of fluid
velocity fluctuations -> good estimation of particle velocity fluctuations!
However...
** References for results at Ret=150: Marchioli et al., Phys. Fluids (2008); Marchioli et al., Acta Mech. (2008)‏

42. 2.2.2 A-Posteriori Tests with SGS Modelling based on A.D.
Particle instantaneous concentration at Ret=150 **
St=1
St=25
St=1
General trend in the results: still an underestimation of the near wall particle concentration (although the agreement with DNS is improved)‏
Is this trend confirmed at Ret=300?
** References for results at Ret=150: Marchioli et al., Phys. Fluids (2008); Marchioli et al., Acta Mech. (2008)‏

43. 2.2.2 A-Posteriori Tests with SGS Modelling based on A.D.
Root Mean Square of particle wall-normal velocity at Ret=300
DNS vs LES vs LES+Deconvolution (for particles)‏
Well-resolved LES (128x128x129 grid) Coarse LES (64x64x129 grid)
Deconvolution improves the agreement with DNS when applied to a well-resolved LES
Deconvolution overpredicts fluctuations when applied to a coarse LES
St=4

44. 2.2.2 A-Posteriori Tests with SGS Modelling based on A.D.
Root Mean Square of particle wall-normal velocity at Ret=300
DNS vs LES vs LES+Deconvolution (for particles)‏
Well-resolved LES (128x128x129 grid) Coarse LES (64x64x129 grid)
Deconvolution improves the agreement with DNS when applied to a well-resolved LES
Deconvolution overpredicts fluctuations when applied to a coarse LES
St=20

45. 2.2.2 A-Posteriori Tests with SGS Modelling based on A.D.
Particle instantaneous concentration at Ret=300
DNS vs LES vs LES+Deconvolution (for particles)‏
Well-resolved LES (128x128x129 grid) Coarse LES (64x64x129 grid)
Deconvolution improves the agreement with DNS (particularly
when applied to a well-resolved LES)
St=4
Why does deconvolution increase the agreement with DNS despite overpredicting vel. Fluctuations. We think it’s a sort of error compensation: overprediction compensates the inability of LES to provide accurate quantitative representation of the physical mechanisms leading to deposition at the wall.

46. 2.2.2 A-Posteriori Tests with SGS Modelling based on A.D.
Particle instantaneous concentration at Ret=300
DNS vs LES vs LES+Deconvolution (for particles)‏
Well-resolved LES (128x128x129 grid) Coarse LES (64x64x129 grid)
Deconvolution improves the agreement with DNS (particularly
when applied to a well-resolved LES)
St=20

47. 2.2.2 A-Posteriori Tests with SGS Modelling based on A.D.
Segregation parameter in the near-wall region at Ret=300
DNS vs LES vs LES+Deconvolution (for all particle sets)‏
Coarse LES (64x64x129 grid)
General trend in the results: SAME AS SEEN AT Ret=150!! That is still an underestimation of the near wall particle segregation (although the agreement with DNS is improved)

48. METTERE FIGURA PER CAMPO
DECONVOLUTO
DNS
(1283)
LES
(643)
“Different” structures (few in the center for LES) and “different” particle distribution (more homogeneous in LES, weaker interaction with vortices)

49. Conclusions on Capabilities of Lagrangian Tracking in LES Fields
A SGS closure model is needed in the particle motion equation
A SGS closure model based on approx deconvolution can recover the correct level of velocity fluctuations for both fluid and particles (particularly with a well-resolved LES)
However, this does not warrant: to reproduce the subgrid turbulence effects on a quantitative basis, and to have a quantitative accurate prediction of near-wall accumulation and local particle segregation
Therefore, accurate subgrid closure models for particles may require also information about the higher-order moments of the velocity fluctuations
We look at the higher-order moments because we think that additional information on the flow field at the sub-grid level is required.
OPEN ISSUE: how to incorporate additional information on the flow field at the SGS level into the P.M.E.?
Possible solution: non-Gaussian stochastic Lagrangian models cf. Guingo & Minier, Phys. Fluids 2008; Chibbaro & Minier, J. Aerosol Sci. 2008

50. PART 3 Scaling issues

51. But if you want to use LES, you want to know the influence of Re...
This is a 1D streamwise spectrum (energy associated with frequency) for turbulent channel flow, computed at z+=25 for two different shear Reynolds numbers, Ret
Any filter will prevent particles from being exposed to small scales which can modify their local behavior, segrega- tion, dispersion…
Ret=300 w
32x32
Cut-off
LES
DNS
Ret=150 w
64x64
Cut-off
LES
DNS
Inaccurate particle dispersion will bring errors into subsequent particle motion and
fluid motion
We need to solve for the smallest relevant scales…

52. But if you want to use LES, you want to know the influence of Reynolds...
Influence of the Stokes number and of the Reynolds number
on local particle segregation (our preliminary results)
Z<0.15H (near-wall region) ??
Z=H (channel centerline) ??
Particle segregation (Sp) versus particle Stokes number (St=tFfp/tF) in
turbulent channel flow: Ret=150 versus Ret=300.

53. Influence of Reynolds: Scaling of Particle Concentration
Possible scaling:
where h refers to the higher Reynolds number simulation (Ret=300)
l refers to the lower Reynolds number simulation (Ret=150)
For instance: Stl=0.2 should scale like Sth=1 roughly. and so on…
The particle velocity fluctuations should be proportional to the fluid velocity fluctuations within the range of Reynolds number considered,
provided that the proposed Ret scaling of preferential concentration is suitable.

54. Reynolds Number Scaling of Particle Concentration
Stl=0.2 vs Sth=1

55. Reynolds Number Scaling of Particle Concentration
Stl=1 vs Sth=5

56. Reynolds Number Scaling of Particle Concentration
Stl=5 vs Sth=25

57. But if you want to use LES, you need to know the influence of Re...
This is a 1D streamwise spectrum (energy associated with frequency) for turbulent channel flow, computed at z+=25 for two different shear Reynols numbers, Ret
Any filter will prevent particles from being exposed to small scales which can modify their local behavior, segrega- tion, dispersion…
Ret=300 w
32x32
cut-off
LES
DNS
Ret=150
64x64 w
cut-off
DNS
LES
Inaccurate particle dispersion will bring errors into subsequent particle motion and
fluid motion
We need to solve for the smallest relevant scales…

58. But if you want to use LES, you need to know the influence of Re...
Influence of the Stokes number and of the Reynolds number
on local particle segregation (our preliminary results)
Z<0.15H (near-wall region)
Z=H (channel centerline)
Particle segregation (Sp) versus particle Stokes number (St=tFfp/tF) in
turbulent channel flow: Ret=150 versus Ret=300.

59. “Low-order” vs “high-order” filter for approximate deconvolution
Filtered fluid velocity
(from LES)
Deconvolved
fluid velocity
Particle concentration
Low-order
(top-hat)
filter
High-order
(discrete)
G(k)
Dk/p ^
We change the Gs in the equation above!
However, not much changes compared to LES without SGS modeling in the p.m.e.!

60. A Closure Model for PME based on Approximate Deconvolution
Skewness of particle wall-normal velocity at Ret=300
DNS vs LES vs LES+Deconvolution (for particles)‏
Well-resolved LES (128x128x129 grid) Coarse LES (64x64x129 grid)
Deconvolution does not modify the third-order moments
St=20

61. A Closure Model for PME based on Approximate Deconvolution
Flatness of particle wall-normal velocity at Ret=300
DNS vs LES vs LES+Deconvolution (for particles)‏
Well-resolved LES (128x128x129 grid) Coarse LES (64x64x129 grid)
Deconvolution does not modify the fourth-order moments
St=20

62. * Stokes Number: St=p/f Flow Time Scale: f=/u
Modelling Approach (2) Pointwise-particle DNS - Lagrangian Particle Tracking
Equation of motion for
the (heavy) particles *
* Stokes Number: St=p/f Flow Time Scale: f=/u

63. Modelling Approach (3) Pointwise-particle DNS - Lagrangian Particle Tracking
Kolmogorov scales:
length scale 1.6 < k+ < 3.6 (k,avg+ =2)‏
time scale < k+ < 13 (k,avg+ =4)‏
Non-Dimensional Kolmogorov Time Scale, +,
vs Wall-Normal Coordinate, z+
St/k+ ~ O(10)‏
dp+/k+ ~ O(1) [In principle, it should be << 1!]

64. Numerical Methodology (4) Pointwise-particle DNS - Lagrangian Particle Tracking
Further Relevant Simulation Details:
Point-particle approach: local flow distortion is assumed negligible.
This approach requires: particle radius << characteristic length scale of flow
particle radius << particle separation
…yet it allows: particle time scale >> characteristic time scale of flow
One-way coupling: dilute flow condition is assumed (particles represent
small mass fraction).
Particle-wall collisions: fully elastic (particle position and velocity at impact
and time of impact are recorded for post-processing!)‏
Particle-particle collisions: neglected.
Fluid velocity interpolation: 6th-order Lagrangian polynomials
Statistically-developing condition for particle concentration
St/k+ >> 1
dp+/k+ << 1

65. Methodology Filtering
Filtering applied only in the homogeneous directions
Cut-off and top-hat filters exactly applied in the wave-number space
Filter widths corresponding to grid resolutions:
64x129x64
32x129x32
16x129x16 E k
Cut-off filter
Energy spectrum
Resolved scales
Top-hat filter
Smooth filter: it subracts energy also from the resolved scales
SGS scales
Cutoff wave number

66. Approximate deconvolution or filtering inversion
Basic idea: to recover the energy subtracted from the resolved scales by smooth filters. E k
Energy spectrum
Resolved scales
Filtered spectrum
SGS scales
Cutoff wave number
Methodology: exact filter inversion or approximated filter inversion by truncated series (e.g. van Cittert expansion).
SGS fluctuations are not retrieved