Caustics & collision velocities in turbulent aerosols

Transcription

1 Caustics & collision velocities in turbulent aerosols B. Mehlig, Göteborg University, Sweden based on joint work with M. Wilkinson and K. Gustavsson. log Ku log St

2 Formulation of the problem Dynamics of particles suspended in random incompressible (mixing) flows. Spherical particles of size amove independently. Force on particles: drag given by Stokes law: ṙ = v, v = γ(u(r,t) v). No gravity. Flow velocity field u(r,t) random function with appropriate statistics. log Ku Dimensionless parameters St = 1 Ku = u 0τ γτ η St Stokes number, Ku Kubo number, η, τ, u 0 length, time, velocity scales. log St

3 Caustics Michael Wilkinson 1 and Bernhard Mehlig 2 1 Faculty of Mathematics and Computing, The Open University, Walton Hall, Milton Keynes, MK7 6AA, England. 2 Physics and Engineering Physics, Gothenburg University/Chalmers, Gothenburg, Sweden. (Dated: August 3, 2004) Falkovich, Fouxon & Stepanov, Nature 419 (2002)151 Wilkinson & Mehlig, Europhys. Lett. 71 (2005) 186 Networks of caustics can occur in the distribution of particles suspended in a randomly moving gas. These can facilitate coagulation of particles by bringing them into close proximity, even in cases where the trajectories do not coalesce. We show that the long-time morphology of these caustic patterns is determined by the Lyapunov exponents λ 1, λ 2 of the suspended particles, as well as the rate J at which particles encounter caustics. We develop a theory determining the quantities J, λ 1, λ 2 from the statistical properties of the gas flow, in the limit of short correlation times. One-dimensional model ẍ = γ(u(x, t) ẋ). v Aerosols are usually unstable systems, in that the suspended particles eventually coagulate. Understanding the process giving rise to this coagulation, and determining the time scale over which it occurs are important questions in describing any aerosol system. If the gas 0.82 phase does not have macroscopic motion, the coagulation may be effected by diffusion of the suspended particles, or (if the suspended particles are of a volatile substance) 0.8 by Ostwald ripening. The coagulation process can be greatly accelerated if the aerosol undergoes macroscopic internal motion. Ultrasound, for example, has been used 0.78 PSfrag replacements to accelerate coagulation in aerosols [1]. Turbulent flow could also play a role in the coagulation of suspended particles; this could be relevant in the coalescence of visible a b t c moisture into rain droplets [2]. If suspended particles are simply advected in an incompressible 0.01 flow, their density remains constant. Iner tial effects are therefore required for coagulation, unless the flow exhibits significant compressibility. In earlier work [3, 4] we discussed the motion of inertial particles in a random velocity field. We showed that there is a phase where the trajectories of the particles coalesce, so that arbitrarily 0.78 small 0.8 particles 0.82 coagulate. In the limit x x where the correlation x time τ of the flow approaches zero, this path-coalescing phase only exists when the velocity field is predominantly potential flow (such as the flow due to sound waves) [4]. Turbulent fluid flow is expected to be predominantly solenoidal, and it is of interest to find alternative mechanisms of coagulation which operate outside the path-coalescence phase. Here we describe an alternative mechanism facilitating coagulation, illustrated in Fig. 1: we show the distribution of particles suspended in a randomly moving gas (the equations of motion and statistics of the flow field are xi(t) v This singularity (`catastrophe ) is a caustic. Implications for collision rates. `Sling effect v PSfrag replacements PSfrag replacements a b c b a PSfrag replacements c c FIG. 1: Distribution of inertial particles suspended in domly moving fluid (blue corresponds to lowest density to highest). The initial distribution is a random scatt large panel shows caustics at short time. Panels (a)-( the long-time behaviour. In all cases, the region is t square, the mean particle density is , m = there is potential flow (with parameters ξ = 0.03, σ δt = 0.05, see text). Main panel: γ = 0.53, t = γ = 1.18, t = 500, (b): γ = 0.72, t = 125, (c): γ t = 125. The three cases correspond to: (a) λ 2 < λ 1 λ 1 > 0, λ 1 + λ 2 < 0, and (c) λ 1 > 0, λ 1 + λ 2 > 0, see of Fig. 1 show the distribution of particles after time, in three different cases: part (a) shows th coalescence phase where the trajectories conden points. Parts (b) and (c) show two cases where t copyright A. W. Peet no path coalescence, but a steady state with sig inhomogeneities of density due to caustics: thes very different morphologies, depending on the par a b

4 Caustics in two spatial dimensions Equation of motion: ṙ = v, v = γ(u(r,t) v). Left: particle velocities in x- y -plane (arrows). Location of caustics ( ). Right: particle number-density: black (high), white (low). Matrix Z of particle-velocity gradients with elements Z ij = v i / r j. Matrix J of infinitesimal deformations ( J ij = r i (t)/ r j (0)). As caustic is formed: det J 0 and tr Z.

5 Dynamics of Z Equation of motion: ṙ = v, v = γ(u(r,t) v). Dynamics of matrix of particle-velocity gradients Z (dimensionless variables): Ż = Z Z 2 + A. A is matrix of fluid-velocity gradients with elements A ij = u i / r j. In co-moving coordinate system n j (t) =O(t)n j (0) for j =1,...3, n 1 (t) separation vector between two close particles: Ż = Z Z 2 +[Z, O Ȯ]+A, Z 12 with Z ij(t) =n i (t) Z(t) n j (t),.... Large excursions. Algebraic tails in distribution of elements of Z, P (Z 11 ) Z Z 11 Mehlig and Wilkinson, Phys. Rev. Lett. 92 (2004) Duncan et al., Phys. Fluids 19 (2007)

6 Caustic activation One-dimensional model ẍ = γ(u(x, t) ẋ). Limit Ku 0, St ( 2 Ku 2 St constant): onedimensional Kramers escape problem. Wilkinson & Mehlig, Phys. Rev. E 68 (2003) At finite Ku, two-dimensional Kramers problem, variables z = v/ x and A = u/ x (Ornstein- Uhlenbeck process). Bray & McKane, Phys. Rev. Lett. (1989) Wilkinson, J. Stat. Phys. (2010) WKB-theory: J /γ exp( S) with (6 S 2 ) 1 St large, (96 Ku 2 St 2 ) 1 St small. Falkovich, Fouxon & Stepanov, Nature 419 (2002)151 Wilkinson, Mehlig & Bezuglyy, Phys. Rev. Lett. 97 (2006) Gustavsson & Mehlig, arxiv: S St 2 S St 1

7 Multi-valued particle velocities Commonly employed hydrodynamic approach for particle density (r,t)+[ v(r,t)](r,t)=0 t (and corresponding theory for correlation functions) cannot be used for St > 0, because particle velocities are multi-valued. Phase-space approach required. Lyapunov exponents: Duncan, Mehlig, Östlund & Wilkinson, Phys. Rev. Lett. 95 (2005) Kinetic phase-space approach: Reeks, Phys. Fluids A 3 (1991) Zaichik & Alipchenkov, Phys. Fluids 15 (2003) Gustavsson, Mehlig, Wilkinson & Uski, Phys. Rev. Lett. 101 (2008) Random uncorrelated motion: Simonin et al., Phys. Fluids 18 (2006) v xi(t) x (r,t) v t x

8 Phase-space description Aim: compute phase-space distribution ρ( x, v) of separations x and relative velocities v. - one-dimensional model (compressible): ẋ = v, v = γ(u(x, t) v) - `white-noise limit ( Ku 0 ) - single spatial and temporal scales η and τ Fractal clustering in phase space v x Caustics cause large relative velocities v = v 1 v 2 x = x 1 x 2 =0. Distribution of v at x =0? at small separations x

9 Summary of results Gustavsson & Mehlig, Phys. Rev. E (2011) Distribution ρ( v, x) for 0 < x 1. ρ( v, x) 1 2 v µ c 2 3 v x µ c z x v c v D 2 is equal to phase-space correlation dimension.

10 Moments of v One spatial dimension. Moments of v m p ( x) = d v v p ρ( v, x) ρ( v, x) 1 2 v µ c 2 Smooth and singular regions contribute. As x 0 m p ( x) x p+µ c 1 b p + c p x p µ c+1 z x 3 v c v Compare interpolation formula v v adv +e 1/(62) v gas Wilkinson, Mehlig & Bezuglyy, Phys. Rev. Lett. 97 (2006) Exponent µ c equals phase-space correlation dimension D 2.. Numerics shows: c 1 exp[ 1/(6 2 )]

11 Bec, Biferale, Cencini, Lanotte, & Toschi, J. Fluid Mech. (2010) Relative velocity structure functions Moments of relative velocities caustic activation m p (r) r p+d C p (St) r d 1 At small r, caustic contribution dominates: m p (r) r d 1 ( p =1, 2,.. ). Structure functions ( d 2 =min{d, D 2 } ) volume element Gustavsson & Mehlig (2010) Cencini & Bec, talk at COST meeting (2009) S (p) (r) = m p(r) m 0 (r) rp+d 2 1 min{d,d 2 } + C p (St)r d min{d,d 2} S (2) r D (r) 2 <d 1 const. D 2 >d Scaling ansatz: 0 S (p) (r) r ζ p ts Find power laws in ρ(v r,r) v r D 2 d 1 (for D 2 <d+1). Gustavson & Mehlig, Phys. Rev. E (2011) ζ1 3 d 2 ζ St 10 1

12 Higher dimensions ( d =2, 3 ) Data for d =2, 3. Left: m p (r). Right: S 2 (r) =m 2 (r)/m 0 (r) ( m 0 (r) =dn/dr ). p =0 p =1 p =2 p =2 d =2, Ku 0. Cencini & Bec, talk at COST (2008) Bec et al., JFM (2010) r d 2 1 r d 1 d =3 Gustavsson, Meneguz, Reeks & Mehlig, arxiv: , finite Ku, kinematic simulation Salazar & Collins, JFM (2012)

13 Collision rate Sundaran & Collins, J. Fluid Mech. 335 (1997) 335 Wang, Wexler & Zhou, J. Fluid Mech. 415 (2000) 117 Collision rate (particle radius a, particle-number density n 0 ) but re-collisions R n 0 dv r v r ρ(v r, 2a) θ( v r ) Collision rate determined by clustering and relative velocities. Common formulation: R n 0 a 2 g(2a) S (1) (2a) Gustavsson, Mehlig & Wilkinson (2008) Andersson, Gustavsson, Mehlig & Wilkinson (2007) log R clustering (pair correlation function) relative velocities (asymm. structure function). Here: S(St) R R caustic activation e adv +e S(St) R kin a D 2 a 2 log St Falkovich, Fouxon & Stepanov, Nature 419 (2002)151 Wilkinson, Mehlig & Bezuglyy, Phys. Rev. Lett. 97 (2006) Abrahamson, Chem. Eng. Science 30 (1975) 1371

14 Conclusions -Caustics in turbulent aerosols give rise to large collision velocities (sling effect, random uncorrelated motion). log Ku log St - Activated behaviour of caustic formation J/γ exp[ S(St)]. 1 1 with S = for, and for. 96Ku 2 St 2 St 1 S = 18Ku 2 St > 1 St - Relative velocity-distribution in turbulent flows (nota bene: restrictions) ρ(v r,r) v r D 2 d 1 - Universal form at small separations (power law): moments of relative velocity: m p (r) r p+d C p (St) r d 1

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