Inhomogeneous distribution of water droplets in cloud turbulence

Abstract

We solve the problem of spatial distribution of inertial particles that sediment in turbulent flow with small ratio of acceleration of fluid particles to acceleration of gravity g. The particles are driven by linear drag and have arbitrary inertia. The pair-correlation function of concentration obeys a power-law in distance with negative exponent. Divergence at zero signifies singular distribution of particles in space. Independently of particle size the exponent is ratio of integral of energy spectrum of turbulence times the wavenumber to g times numerical factor. We find Lyapunov exponents and confirm predictions by direct numerical simulations of Navier-Stokes turbulence. The predictions include typical case of water droplets in clouds. This significant progress in the study of turbulent transport is possible because strong gravity makes the particle's velocity at a given point unique.PACS numbers: 47.10. Fg, 05.45.Df, 47.53.+n Inhomogeneity of distribution of water droplets in clouds, caused by air turbulence, is significant factor in the formation of rain [1][2][3]. Due to inhomogeneity droplets collide and coalesce more often speeding up the formation of larger drops. It rains when the drops get so large as to reach the ground without evaporating on the way.Turbulence-induced inhomogeneities of transported quantities could seem contradicting to the well-known mixing of turbulence producing uniform distribution [4]. Indeed mixing dominates on larger scales. On smaller scales turbulence produces highly irregular spatial structures. Similarly to ordinary centrifuges the rotating turbulent vortices push the inertial droplets out [5] causing very strong non-uniformities in droplets distribution to accumulate with time. Those occur at the typical scale of the vortices (the Kolmogorov scale) which is much smaller than the typical scale of the flow. Thus formation of inhomogeneities of particles by turbulence is small-scale phenomenon, often disregarded due to finite resolution of instruments. Since it is at those scales that droplets collide then the study of the inhomogeneities is necessary to predict rain formation.Much progress was obtained in the study of nonuniform distributions of inertial particles in the flow when gravity is negligible [1][2][3][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In the regime where the inertia is not too large, which is where turbulence is most relevant in the rain formation process [3], the paircorrelation function of concentration was demonstrated to obey a power-law with negative exponent signifying singular (fractal) structure formed by particles in space. Furthermore the leading order term for the exponent at small inertia was obtained for Navier-Stokes turbulence without model-dependent assumptions on the statistics * Electronic address: itzhak8@gmail.com † Electronic address: clee@yonsei.ac.kr of turbulence [1,6]. It depends on statistics of turbulence via the ratio of time integral of correlation function of laplacian of pressure divided by...

Clustering of particles in turbulence due to phoresis

“…whose definition [61] in the case of weak compressibility reduces to the ratio of logarithmic growth rates of infinitesimal volumes δV and areas δA of particles [40]. The simplest definition of δA is found considering the area of a triangle formed by three particles.…”

“…This factor is a "proper correlation": if n 0 (x) is larger in certain regions of space then particles will tend to go to that region independently of the behavior of other particles so the product n(x)n(x + r) will be larger there trivially. Our derivation holds for arbitrary weakly compressible flow so that it can be used for all the phoretic phenomena described in the previous Section, inertial particles in turbulence at small Stokes or Froude numbers [20,40] or other cases. The reasons why turbulence increases the probability of two particles to get close can be understood from the fact that on average the divergence of velocity in the particle's frame is negative ∇ · v[t, x(t, x)] < 0.…”

“…Preferential concentration is well-studied in the case of inertial particles where it plays an important role in a wide range of phenomena including aerosols spreading in the atmosphere [35,36], planetary physics [37], transport of materials by air or by liquids [38], liquid fuel combustion engines [39], rain formation in liquid clouds [7,8,20,40] and many more. Inertia, in the case of small particles, produces a small but finite difference between the particle's velocity and the local velocity of the fluid.…”

“…In the case of turbulence the "boundary" on which inertial particles concentrate becomes very complex and time-dependent but it still has zero volume being multi-fractal [7,9,20]. The statistics of preferential particle concentration in turbulent flows obeying the incompressible Navier-Stokes equations can be described theoretically in the universal framework of weakly compressible flows [7,20,40], for not too heavy particles much smaller than the Kolmogorov lengthscale. The complete statistics of the particle concentration where fluctuations are non-negligible (small-scale turbulence) depend on the statistics of turbulence through a single parameter ∆, which provides the scaling exponent of the power-law correlations of the particle concentration.…”

Inertial particles distribute in turbulence as Poissonian points with random intensity inducing clustering and supervoiding

“…For a long time the particles' flow description was known to hold for weakly inertial particles [19]. It was found recently that the flow description can be introduced in the case of strong gravity as well [8]. The difference between the particles' and the fluid flow is caused by the combined effect of inertia and gravity that separate the particles from the local fluid.…”