M. Voßkuhle Particle Collisions in Turbulent Flows 2

Transcription

1 S Particle Collisions in Turbulent Flows Michel Voßkuhle Laboratoire de Physique 13 December 213

2 M. Voßkuhle Particle Collisions in Turbulent Flows 2

3 M. Voßkuhle Particle Collisions in Turbulent Flows 2

4 M. Voßkuhle Particle Collisions in Turbulent Flows 2

5 M. Voßkuhle Particle Collisions in Turbulent Flows 2

6 M. Voßkuhle Particle Collisions in Turbulent Flows 3 Particle Collisions in Turbulent Flows Outline Introduction Prevalence of sling/caustics/rum effect Multiple collisions KSvs. DNS

7 M. Voßkuhle Particle Collisions in Turbulent Flows 4 Rain formation and the droplet size distribution relative mass time [min] drop radius [µm] Lamb Meteorol. Monogr. (21);Shaw ARFM (23) f(a) t = nucleation/ condensation + collision/ coalescence

8 M. Voßkuhle Particle Collisions in Turbulent Flows 4 Rain formation and the droplet size distribution relative mass time [min] drop radius [µm] Lamb Meteorol. Monogr. (21);Shaw ARFM (23) f(a) t = nucleation/ condensation a a 2 a 2Γ(a,a )f(a )f(a )da Γ(a,a )f(a)f(a )da a 3 = a 3 a 3

9 M. Voßkuhle Particle Collisions in Turbulent Flows 4 Rain formation and the droplet size distribution relative mass Many Other Applications time [min] f(a) t = nucleation/ condensation a planetgrowthin drop radius [µm] protoplanetary disks diesel sprays Lamb Meteorol. Monogr. (21);Shaw ARFM (23) a 2 a 2Γ(a,a )f(a )f(a )da Γ(a,a )f(a)f(a )da a 3 = a 3 a 3

10 M. Voßkuhle Particle Collisions in Turbulent Flows 5 Introductory example: Kinetic theory Collision cylinder simplification: only one particle size collision rate for one particle R c = nπ(2a) 2 w a overall collision rate w t 2a N c = 1 2 n2 π(2a) 2 w Γ kin (a) Collision kernel Γ kin (a) = π(2a) 2 w

11 M. Voßkuhle Particle Collisions in Turbulent Flows 6 (Inertial) Particle collisions in Turbulent Flows finitedensity ρ p > ρ f finitesize < a η equations of motion: dx dt = V, dv dt = u(x,t) V τ p +G Maxey & Riley Phys. Fluids (1983) Gatignol J. méc. théor. appl. (1983) dimensionless quantity: Stokes number St = τ p τ = 2 ρ p a 2 K 9 ρ f η 2 DNS Navier Stokes equations periodic box gridpoints Re λ = 13 Kinematic Simulations synthetic turbulence efficient Fung et al. JFM (1992)

12 M. Voßkuhle Particle Collisions in Turbulent Flows 7 Determining the collision rate partofsimulationbox with particles

13 M. Voßkuhle Particle Collisions in Turbulent Flows 7 Determining the collision rate partofsimulationbox with particles divide into segments know which particles in which cell

14 M. Voßkuhle Particle Collisions in Turbulent Flows 7 Determining the collision rate partofsimulationbox with particles divide into segments know which particles in which cell consider only surrounding cells

15 M. Voßkuhle Particle Collisions in Turbulent Flows 8 Determining the collision rate extrapolation is inexact rather use interpolation Collision kernel N c (T) TV sys = N c = 1 2 n2 Γ(a)

16 M. Voßkuhle Particle Collisions in Turbulent Flows 9 Prevalence of the sling/caustics/rum effect

17 M. Voßkuhle Particle Collisions in Turbulent Flows 1 Saffman& Turner JFM(1956) St : particlesfollowflow R c = n w r (2a,Ω)Θ[ w r (2a,Ω)]dΩ average to obtain total collision rate 2a N c = 1 2 n2 1 2 w r(2a) dω a approximate w x (2a) = 2a u x / x assume Gaussian statistics with ( u x / x) 2 = ε/15ν Γ ST = ( 8π 1/2 15 ) (2a) 3 τ K

18 M. Voßkuhle Particle Collisions in Turbulent Flows 11 St Collision kernel(s) Saffman&TurnerJFM(1956) Γ ST = ( 8π 1/2 15 ) (2a) 3 τ K St Abrahamson Chem. Eng. Sci.(1975) Γ A = Γ kin withv rms = (η/τ K )f(st,re λ ) Γ A = 4 π(2a) 2 η τ K f(st,re λ )

19 M. Voßkuhle Particle Collisions in Turbulent Flows 11 St Collision kernel(s) Saffman&TurnerJFM(1956) Γ ST = ( 8π 1/2 15 ) (2a) 3 τ K Preferential concentration? St Γτ K /(2a) St Abrahamson Chem. Eng. Sci.(1975) Γ A = Γ kin withv rms = (η/τ K )f(st,re λ ) Γ A = 4 π(2a) 2 η τ K f(st,re λ ) Sling/caustics/RUM effect?

20 M. Voßkuhle Particle Collisions in Turbulent Flows 12 Preferential concentration mathematicallyexact Γ SC = 2π(2a) 2 g(2a) w r y(pixels) y(mm) radial distribution function g(r) g(r) (r/η) (D 2 3), r/η x(pixels) x(mm) Monchaux et al. Phys. Fluids (21) Sundaram & Collins JFM (1997) g(2a) Re λ = St

21 M. Voßkuhle Particle Collisions in Turbulent Flows 13 Sling/caustics/RUM effect Sling Caustics V t = t t > t faster particles overtake slower ones points in phase space multi-valued Wilkinson, Mehlig, & Bezuglyy PRL (26) X RUM Random Uncorrelated Motion two contributions to motion of inertial particles» smooth spatially correlated movement» random uncorrelated movement Simonin et al. Phys. Fluids (26) Reeks et al. Proc. FEDSM (26) particles slung by vortices Falkovich et al. Nature (22) Γ = Γ ST g(2a) +Γ A h S (St,Re λ ) a 3 a D 2 3 a 2

22 M. Voßkuhle Particle Collisions in Turbulent Flows 14 St = 2 9 ρ p a 2 ρ f η 2 ρ p /ρ f = 25 a = 2a ρ p /ρ f = 1 a = a ρ p /ρ f = 4 a = 1 2 a

23 M. Voßkuhle Particle Collisions in Turbulent Flows 14 St = 2 9 ρ p a 2 ρ f η 2 ρ p /ρ f = 25 a = 2a ρ p /ρ f = 1 a = a ρ p /ρ f = 4 a = 1 2 a Saffman& Turner Sling/caustics/RUM Γτ K /(2a) St Γτ K /[η(2a) 2 ] St Voßkuhle et al. arxiv: [physics.flu-dyn]

24 Γ = Γ ST g(2a) +Γ A h S (St,Re λ ) a 3 a D 2 3 a 2 Expected values for Γ(a )/Γ(2a ) = Γ(a /2)/Γ(a ) a 2 Γ(a )/Γ(2a ) = 1 4 a 3 Γ(a )/Γ(2a ) = Γ(a )/Γ(2a ) St Voßkuhle et al. arxiv: [physics.flu-dyn] M. Voßkuhle Particle Collisions in Turbulent Flows 15

25 Γ = Γ ST g(2a) +Γ A h S (St,Re λ ) a 3 a D 2 3 a 2 Expected values for Γ(a )/Γ(2a ) = Γ(a /2)/Γ(a ) a 2 Γ(a )/Γ(2a ) = 1 4 a 3 Γ(a )/Γ(2a ) = 1 8 a 3 a D 2 3 Γ(a )/Γ(2a ) = (1 2 ) D 2.3 Γ(a )/Γ(2a ) St Voßkuhle et al. arxiv: [physics.flu-dyn] M. Voßkuhle Particle Collisions in Turbulent Flows 15

26 M. Voßkuhle Particle Collisions in Turbulent Flows 16 Collision velocities Cumulative PDF of relative velocity w r F( w r ) = p( w r )dw r How many particle pairs have relative velocities smaller than w r? Cumulative distribution of the flux w r w r φ( w r ) = w r p( w r )dw r How many colliding particles have relative velocities smaller than w r? w r τ K /(2a) St = w r /u K

27 M. Voßkuhle Particle Collisions in Turbulent Flows 16 Collision velocities Cumulative PDF of relative velocity w r F( w r ) = p( w r )dw r How many particle pairs have relative velocities smaller than w r? Cumulative distribution of the flux w r w r φ( w r ) = w r p( w r )dw r How many colliding particles have relative velocities smaller than w r? w r τ K /(2a) St = w r /u K

28 M. Voßkuhle Particle Collisions in Turbulent Flows 16 Collision velocities Cumulative PDF of relative velocity w r F( w r ) = p( w r )dw r How many particle pairs have relative velocities smaller than w r? Cumulative distribution of the flux w r w r φ( w r ) = w r p( w r )dw r How many colliding particles have relative velocities smaller than w r? w r τ K /(2a) St = w r /u K

29 M. Voßkuhle Particle Collisions in Turbulent Flows 16 Collision velocities Cumulative PDF of relative velocity w r F( w r ) = p( w r )dw r How many particle pairs have relative velocities smaller than w r? Cumulative distribution of the flux w r w r φ( w r ) = w r p( w r )dw r How many colliding particles have relative velocities smaller than w r? w r τ K /(2a) St = w r /u K

30 M. Voßkuhle Particle Collisions in Turbulent Flows 16 Collision velocities Cumulative PDF of relative velocity w r F( w r ) = p( w r )dw r How many particle pairs have relative velocities smaller than w r? Cumulative distribution of the flux w r w r φ( w r ) = w r p( w r )dw r How many colliding particles have relative velocities smaller than w r? w r τ K /(2a) St = w r /u K

31 M. Voßkuhle Particle Collisions in Turbulent Flows 16 Collision velocities Cumulative PDF of relative velocity w r F( w r ) = p( w r )dw r How many particle pairs have relative velocities smaller than w r? Cumulative distribution of the flux w r w r φ( w r ) = w r p( w r )dw r How many colliding particles have relative velocities smaller than w r? w r τ K /(2a) St = w r /u K

32 M. Voßkuhle Particle Collisions in Turbulent Flows 17 Multiple collisions

33 M. Voßkuhle Particle Collisions in Turbulent Flows 18 Ghost collisions and the Saffman Turner theory flow is locally hyperbolic flow is persistent fluidelementcanpass through collision sphere repeatedly spurious ghost collisions Saffman & Turner JFM (1956) Brunk et al. JFM (1998) Andersson et al. EPL (27) Gustavsson, Mehlig, & Wilkinson NJP (28)

34 M. Voßkuhle Particle Collisions in Turbulent Flows 18 Ghost collisions and the Saffman Turner theory Saffman & Turner JFM (1956) Particles kept in flow may collide again Brunk et al. JFM (1998) Andersson et al. EPL (27) Gustavsson, Mehlig, & Wilkinson NJP (28)

35 M. Voßkuhle Particle Collisions in Turbulent Flows 19 Overestimation by Ghost collisions Γτ K /(2a) 3 Γ m /Γ Voßkuhle et al. PRE (213) St Γ GCA ghostcollisionapproximation Γ 1 onlyfirstcollisions Γ m multiplecollisions Γ GCA = Γ 1 +Γ m Ghost collisions overestimate collisionkernelbyupto2% Relative estimation error falls with Stokes number Consistent with previous results by Brunk et al. JFM (1998) and Wang et al. Phys.Fluids (1998)

36 M. Voßkuhle Particle Collisions in Turbulent Flows 2 Statistics of multiple collisions P(N c N c 1) Voßkuhle et al. PRE (213) St = N c Multiple collisions statistics follow P(N c N c 1) = β(st)α(st) N c Markovian interpretation: particle has probability α(st) to collide again Simple model Γ m = Γ 1 N c =2 βα N c = Γ 1 βα 2 1 α

37 M. Voßkuhle Particle Collisions in Turbulent Flows 2 Statistics of multiple collisions P(N c N c 1) Voßkuhle et al. PRE (213) Simple model St = Γ m = Γ 1 N c =2 N c Multiple collisions statistics follow P(N c N c 1) = β(st)α(st) N c Markovian interpretation: particle has probability α(st) to collide again 6 5 model 4 DNS 3 βα N c βα 2 2 = Γ α St Γ m τ K /(2a) 3

38 M. Voßkuhle Particle Collisions in Turbulent Flows 21 Distance between trajectories d(t 2 ) distance/2π St = t/t L distance/2a d(t 1 ) t/t L

39 M. Voßkuhle Particle Collisions in Turbulent Flows 22 Pair separation Jullien et al. PRL (1999) results for tracers starting with similar initial conditions Rast & Pinton PRL (211) Scatamacchia et al. PRL (212)

40 M. Voßkuhle Particle Collisions in Turbulent Flows 22 Pair separation Jullien et al. PRL (1999) results for tracers starting with similar initial conditions Rast & Pinton PRL (211) Scatamacchia et al. PRL (212)

41 M. Voßkuhle Particle Collisions in Turbulent Flows 23 Contact time distribution distance t 1 t 2 d c time

42 M. Voßkuhle Particle Collisions in Turbulent Flows 24 Contact time distribution distance t 1 t 2 d c time exponential tail for long time powerlawforshorttime independentofdistanced c P( t 1 ) forst = Voßkuhle et al. PRE (213) t 1 /T L d c = a/2 d c = a d c = 2a d c = 4a t 1 /T L

43 M. Voßkuhle Particle Collisions in Turbulent Flows 25 Contact time distribution distance t 1 t 2 d c time exponential tail persists power law vanishes independent of collision count P( t i ), i > 1 forst = Voßkuhle et al. PRE (213) t i /T L t 1 Nc >1 t 2 t 3 t t i /T L

44 M. Voßkuhle Particle Collisions in Turbulent Flows 25 Contact time distribution distance t 1 t 2 d c 1 1 time 1 P( t i ), i > 1 forst 1= (2a/σ) P( t σ/[2a]) Voßkuhle et al. PRE (213) t i /T L exponential tail persists Kinetic theory power law vanishes independent of collision count tσ/(2a) t 1 Nc >1 t 2 t 3 t t i /T L

45 M. Voßkuhle Particle Collisions in Turbulent Flows 25 Contact time distribution distance t 1 t 2 d c time exponential tail persists power law vanishes independent of collision count P( t i ), i > 1 forst = Voßkuhle et al. PRE (213) t i /T L t 1 Nc >1 t 2 t 3 t t i /T L

46 M. Voßkuhle Particle Collisions in Turbulent Flows 26 Contact time distribution Different St St =.5 St = 1. St = 2. St = t 1 /T L Voßkuhle et al. PRE (213) P( T 1 ) e κ t 1/T L t 1 T L κ ξ ξ St

47 M. Voßkuhle Particle Collisions in Turbulent Flows 27 Multiple collisions and sling/caustics/rum effect Collision velocities 5 4 wr c,i/uk 3 i = i = m St Voßkuhle et al. PRE (213) multiple collisions have small relative velocities multiple collisions stem from continuous collisions sling/caustics/rum effectdoesnotleadto multiple collisions

48 M. Voßkuhle Particle Collisions in Turbulent Flows 28 Kinematic Simulations vs. Direct Numerical Simulations

49 M. Voßkuhle Particle Collisions in Turbulent Flows 29 Kinematic Simulations u(x,t) = N k A n cos(k n x +ω n t)+b n sin(k n x +ω n t) n=1 A n k n = B n k n = A 2 n = B 2 n = E(k n ) k n E(k n ) k 5/3 n efficient highly turbulent flows widely used toy model Fung et al. JFM (1992)

50 M. Voßkuhle Particle Collisions in Turbulent Flows 3 Qualitativelythesamefor KS... ΓτK/(2a) Γ GCA Γ 1 Γ m St Voßkuhle et al. J. Phys.: Conf. Ser. (211) Γm/Γ St P(Nc Nc 1) N c TL P( t/tl) St = t/t L

51 M. Voßkuhle Particle Collisions in Turbulent Flows 3...butquantitativelyverydifferentfrom DNS Γ GCA τk/(2a) KS DNS St Voßkuhle et al. J. Phys.: Conf. Ser. (211) Γm/Γ KS DNS St P(Nc Nc 1) N c TL P( t/tl) St = t/t L

52 M. Voßkuhle Particle Collisions in Turbulent Flows 31 ω max ω rms

53 ω max ω max ω rms ω rms M. Voßkuhle Particle Collisions in Turbulent Flows 31

54 M. Voßkuhle Particle Collisions in Turbulent Flows 32 Acknowledgments Conclusion References A. Pumir E. Lévêque J. Bec M. Lance B. Mehlig M. Reeks M. Wilkinson Laboratoire de Physique

55 M. Voßkuhle Particle Collisions in Turbulent Flows 33 Acknowledgments Conclusion References Conclusions and perspectives St > : Sling/caustics/RUM effect dominates collision rates in turbulent flows Γ(a )/Γ(2a) St Γτ K/(2a) 3 Γ m/γ1 inertial particles may stay close for long times St P(N c Nc 1) St = N c » leads to multiple collisions t 1/T L» overestimation of the collision kernel M Voßkuhle et al. (213a). Multiple collisions in turbulent flows. In: Phys. Rev. E 88, p. 638 M Voßkuhle et al. (213b). Prevalence of the sling effect for enhancing collision rates in turbulent suspensions. In: ArXive-prints (July 213). arxiv: [physics.flu-dyn] M Voßkuhle et al. (211). Estimating the Collision Rate of Inertial Particles in a Turbulent Flow: Limitations of the Ghost Collision Approximation. In: J. Phys.: Conf. Ser. 318, p. 5224

56 M. Voßkuhle Particle Collisions in Turbulent Flows 34 Acknowledgments Conclusion References Abrahamson J (1975). Collision rates of small particles in a vigorously turbulent fluid. In: Chem. Eng. Sci. 3, pp Andersson B et al. (27). Advective collisions. In: EPL 8, p Brunk BK, Koch DL, Lion LW (1998). Turbulent coagulation of colloidal particles. In: J. Fluid Mech. 364, pp eprint: Falkovich G, Fouxon A, Stepanov MG (22). Acceleration of rain initiation by cloud turbulence. In: Nature 419, pp Fung JCH et al. (1992). Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes. In: J. Fluid Mech. 236, pp Gatignol R (1983). Faxen Formulae for a Rigid Particle in an Unsteady Non-uniform Stokes Flow. In: J. Méc. Théor. Appl. 1, pp Gustavsson K, Mehlig B, Wilkinson M (28). Collisions of particles advected in random flows. In: New J. Phys. 1, p Jullien M-C, Paret J, Tabeling P (1999). Richardson Pair Dispersion in Two-Dimensional Turbulence. In: Phys. Rev. Lett. 82, pp Lamb D (21). Rain Production in Convective Storms. In: Meteorol. Monogr. 28, pp Maxey MR, Riley JJ (1983). Equation of motion for a small rigid sphere in a nonuniform flow. In: Phys. Fluids 26, pp

57 M. Voßkuhle Particle Collisions in Turbulent Flows 35 Acknowledgments Conclusion References Monchaux R, Bourgoin M, Cartellier A (21). Preferential concentration of heavy particles: A Voronoï analysis. In: Phys. Fluids Rast MP, Pinton J-F (211). Pair Dispersion in Turbulence: The Subdominant Role of Scaling. In: Phys. Rev. Lett. 17, p Reeks MW, Fabbro L, Soldati A (26). In search of random uncorrelated particle motion (RUM) in a simple random flow field. In: Proc. 26 ASME Joint US European Fluids Engineering Summer Meeting. (July 17 2, 26). Miami, FL, pp Saffman PG, Turner JS (1956). On the collision of drops in turbulent clouds. In: J. Fluid Mech. 1, pp Scatamacchia R, Biferale L, Toschi F (212). Extreme Events in the Dispersions of Two Neighboring Particles Under the Influence of Fluid Turbulence. In: Phys. Rev. Lett. 19, p Shaw RA (23). Particle-Turbulence Interactions in Atmospheric Clouds. In: Annu. Rev. Fluid Mech. 35, pp Simonin O et al. (26). Connection between two statistical approaches for the modelling of particle velocity and concentration distributions in turbulent flow: The mesoscopic Eulerian formalism and the two-point probability density function method. In: Phys. Fluids 18, p Sundaram S, Collins LR (1997). Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations. In: J. Fluid Mech. 335, pp

58 M. Voßkuhle Particle Collisions in Turbulent Flows 36 Acknowledgments Conclusion References Voßkuhle M, Pumir A, Lévêque E (211). Estimating the Collision Rate of Inertial Particles in a Turbulent Flow: Limitations of the Ghost Collision Approximation. In: J. Phys.: Conf. Ser. 318, p Voßkuhle M et al. (213a). Multiple collisions in turbulent flows. In: Phys. Rev. E 88, p Voßkuhle M et al. (213b). Prevalence of the sling effect for enhancing collision rates in turbulent suspensions. In: ArXiv e-prints (July 213). arxiv: [physics.flu-dyn]. Wang L-P, Wexler AS, Zhou Y (1998). On the collision rate of small particles in isotropic turbulence. I. Zero-inertia case. In: Phys. Fluids 1, pp Wilkinson M, Mehlig B, Bezuglyy V (26). Caustic Activation of Rain Showers. In: Phys. Rev. Lett. 97, p

59 M. Voßkuhle Particle Collisions in Turbulent Flows 37 Supplementary information Kinematic Simulations Credits u(x,t)= N k n=1 A n cos(k n x+ω n t)+b n sin(k n x+ω n t) A n k n = B n k n = A 2 n= B 2 n= E(k n ) k n, E(k n ) k 5/3 n k 1 = 2π L, k N k = 2π η, k n= k 1 ( L η ) (n 1)/(N k 1) ω n = λ kne(k 3 n ), λ persistenceparameter efficient highly turbulent flows widely used Fung et al. JFM (1992) Go back...

60 M. Voßkuhle Particle Collisions in Turbulent Flows 38 Supplementary information Credits Bibliothèque municipale de Lyon. Photo: Vélo Vs in snow. others from the corresponding websites. Photos and logos on the acknowledgements page. Lukaschuk S. Photo: Clustering particles. Dept. Engineering, Univ. Hull. Some of the pictures are in the public domain and were taken from this website. Rain, Clover designed by Pavel Nikandrov, Computer, Newspaper designed by Yorlmar Campos, Handshake designed by Sam Garner, others in the public domain. Icons from The Noun Project.