1 Statistics of volumes, swept by spheroidal particles, in a turbulent flow.

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1 1 Statistics of volues, swept by spheroidal particles, in a turbulent flow. B. Grits*, M. Pinsky, and A. Khain Institute of Earth Science, The Hebrew University of Jerusale 1. INTRODUCTION Collisions between sall non-spherical ice crystals deterine the foration of larger crystals and snowflakes in clouds. Collisions of nonspherical ice crystals and water drops give rise to the foration of graupel in clouds. As a whole, particle collisions are the key processes deterining particles size spectru evolution. As such they are of crucial iportance for understanding and appropriate description of ice icrophysics and ice precipitation. At the sae tie investigation of these processes is far fro being copleted due to: a) difficulties in estiation of hydrodynaic forces and torques acting on particles; b) realistic representation of turbulent field, characteristic for cloudy conditions. Due to these difficulties there are only a few theoretical investigations on droplets collisions in a turbulent ediu at present (e.g. De Aleida, 1976 and 1979; Grover and Pruppacher, 1985; Pinsky et al. 1999). All of the indicate that turbulence increases collision rate of cloud droplets several ties. It is natural therefore to expect analogous influence of atospheric turbulence on ice crystals. However no theoretical investigations on ice crystals collisions were reported in literature up to now (Pruppaher and Klett, 1997). The case of sall Stokes nuber, or Stokesian, particles siplifies significantly the collision proble. In this study we represent a novel approach for evaluation of Stokesian nonspherical particles collision statistics together with preliinary results. Prolate and oblate spheroids were chosen as an exaple of nonspherical particles. Strongly elongated spheroids can odel needle-like ice crystals, while oblate spheroids plate-like hexagonal pris, one of the ost abundant ice crystals for *Boris Grits, The Departent of the Atospheric Sciences, The Hebrew University of Jerusale, Givat Ra 91904, Jerusale, Israel. Tel: (2) ; Eail: bgrits68@yahoo.co 2. STOKESIAN PARTICLES COLLISION The rate of collisions is deterined by collision kernel that describes the probability of collisions between two particles per unit tie. It is accepted to represent this probability as the product of collision probability without hydrodynaic interaction (hereafter HDI) between particles and collision efficiency that takes into account the effects of HDI. The rate of collisions with and without HDI is deterined by the volue, swept (hereafter referred to as SV) by colliding particles in their relative otion per unit tie correspondingly with and without HDI. In this study we liit ourselves by the case without HDI. For turbulent field representation an approach, elaborated by Pinsky et al. (2004, 2005), was eployed. According to this approach, turbulent field is represented by a set of non-correlated saples of turbulent field. Each saple (hereafter, eleentary volue) can be assigned to a certain point of a turbulent flow at a certain tie oent. The scales of eleentary volue are deterined as those within which Lagrangian acceleration of air velocity, A, and tensor of velocity strains, S, can be considered unifor in space and invariable in tie. As was shown by Pinsky et al. (2005), these scales are of order Kologorov length and tie scales. Siilar statistical representation of a turbulent flow as a set of sall volues (packets) within which energy dissipation rate, velocity shears and particle concentration were assued unifor was used recently by Koch and Pope (2002). Let us introduce the volue, within which HDI is iportant. Siple scale considerations show that the scales of HDI volue lie within eleentary volue for ass uivalent radius 1µ r 20µ. This is the particles size range for which one can regard collisions as occurring at constant A and S. To siulate large nuber o f collisions long series (50.000) of S and A pairs (field realizations) were produced. These pairs were generated by eans of statistical generators, reproducing Largangian accelerations and shears with probability distribution functions (PDFs) found in cloud easureents and laboratory

2 2 experients for high Reynolds nubers (Antonia et al., 1981; Belin et al., 1997; Hill and Thoroddsen, 1997; Kholyansky et al., 2001; La Porta et al., 2001; Voth et al., 2002). Another iportant feature of our approach is that we approxiate particle otion in eleentary volues with the help of approxiate analytical solution. This solution allows obtaining particle orientation and velocities (translation and angular) probability distribution functions (PDFs). This is a generalization of an approach used by Pinsky et al. (2005) for investigation of sall cloud droplets collisions. The knowledge on these PDFs allows in its turn calculation of SV as the integral over all possible collision variants properly weighted. Estiation of SV for generated series of A and S produce SV series. Mean SV was estiated by averaging over these series. SV PDF was obtained by calculation series histogras. 3. ANALYTICAL SOLUTION Equations, describing spheroid otion in the creeping flow approxiation, are (for, exaple, Broday et al., 1998): dv ρ f = K( u v) g(1 ) e 3, (6) dt ρ p dw di I + w= Q( R u) Ω w. (7) dt dt u is a fluid velocity; K, Q and Ω are tensors of particle resistance to translation and rotation. These uations iply: a) sall particle Reynolds nuber approxiation; b) assuption on constant S. We apply the ethod of successive approxiations (Korn and Korn, 1968), using solution in the non-inertial liit ( 0 ) as a starting point. 3.1 Translation velocity The approxiate solution is: v= u+ Kv( e) Gi( e), (8) i= 1 Matrix K v and vectors { G i} are functions of spheroid orientation, e. Consuently spheroid velocity deviation fro air velocity, v' = v u, is also function of its orientation only. This solution provides relatively siple recurrent relation for vectors { G. i} Let s define Stokes nuber to be St τ t τ f. 3 Here τ t 10 s characterizes particle adjustent tie to fluid translation, andτ 1 Sh is a characteristic tie of fluid velocity variation (Sh - characteristic shear). The truncation error is of order ( St) k, k being first neglected ter in (8). For exaple, oitting ters of order i 2 for St = 0.04 (corresponds to r = 20µ ) one obtains truncation error less then 0. 2%. The iportant consuences of (8) are that v ' is defined basically by the Lagrangian acceleration of the fluid and increases approxiately as a 2. Both these conclusions iply St Angular velocity The approxiate solution is: w= Kw( e) ξ ( e) + Ri( e). (9) i= 1 ξ is the spheroid angular velocity in the noninertial liit. Matrix K w and vectors ξ and are all functions of spheroid orientation. { R i } Consuently spheroid angular velocity is also deterined by orientation. A ain there exists. The truncation recurrent relation for vectors { g R i } error in (9) scales again as ( St. If exact solutions of (6) and (7) are bounded (stable behavior), then any arbitrary initial velocity approaches quickly (on tie scale of τ t ) to a solution, shown scheatically in fig. 1 by a dashed line (liit solution). Solutions (8) and (9) do not describe transient processes but converge directly to this solution. We therefore refer to the as liit solutions. Main inaccuracy when using (8) and (9) arises rather due to the difference between liit and exact solutions than due to truncation procedure. This difference however is sall for sall St. If exact solutions of (6) and (7) are unbounded (unstable behavior), solutions (8) and (9) diverge. It eans that they can not describe the so-called sling effects. Such events however are rare for Stokesian particles in atosphere (Pinsky et al., 2005). To check solutions (8) and (9), we copared the with nuerical solution of (6) and (7) in several situations: a) siple shear flow; b) Poiseuille flow; c) realization of turbulent filed. In all the cases the difference did not exceed 0.1%. ) k f

3 3 Fig. 1 Stable and unstable velocity scenarios. Liit solution is shown by dashed line. Fig. 2 SV PDFs for oblate spheroids with r 15µ and 10µ 4. RESULTS We calculated ean SV and SV PDF for a wide range of turbulent field paraeters: Re and ε 0.1 λ ( Re λ - Taylor scale based Reynolds nuber; ε - energy dissipation rate). This range covers cloudy turbulence fro stratifor up to deep cuulus clouds. The calculations were perfored for spheroids of different sizes (1µ r 20µ ) and different aspect ratios ( 0.05 β 20 ). Here we present soe of the results. Fig. 2 shows SV PDF (histogra) for pair of oblate spheroids with radii 15µ and 10µ 3 ( Re = λ 10, ε = 0.05s ). Fig. 3 shows ean SV for the pair of prolate spheroids with uivalent radii 15µ and 10µ 3 ( Re = 20 λ 10 ). Fig 4 shows the sae results for oblate spheroids of the sae ass. SV is noralized everywhere by SV in a pure gravity case. Figures 5 and 6 give analogous results for spheroids with uivalent radii 2µ and 1µ. Fig. 7 and 8 show SV for prolate and oblate spheroids of siilar size: r 1 = 15, r2 = 15µ δ r µ. Reynolds nuber in this 3 case is 20 10, dissipation rate 0.05 s 2 3. δ varies between 0.2µ and 1µ. The r results anifest that: - SV PDF differs significantly fro Gaussian due to enhanced large SV part of distribution. - turbulence agnifies ean SV for several tenth of percent coparing with the pure gravity case; the effect enlarges with flow intensity. - ean SV for spheroids is higher than that for droplets of the sae ass; the effect enlarges with aspect ratio deviation fro unity, being ore pronounced in case of elongated spheroids. - influence of turbulence on ean SV becoes especially large for sall particles (of order 1µ ) and particles of siilar size. The proposed ethod incorporates three ain approxiations: a) sall particle Reynolds nuber approxiation; b) assuption on constant A and S during collision; c) approxiation of otion uations solution by eans of the liit solution. While it is not possible to evaluate error due to the second approxiation, one can check the validity of the first and third approxiations. To check sall Reynolds approxiation we evaluated characteristic value of spheroid Re in all realizations and calculated the part, in which Re turned out to be larger than 0.1. While for r = 15µ spheroid accuracy turned out to be sufficient in all cases, for r = 20µ error becoes large for intensive turbulent field. For r = 2 all realizations resulted in Re < 0.1. µ To check the third approxiation we copared nuerical and liit solutions and calculated the part of realizations, in which this difference exceeded 5%. Again r = 15µ spheroid provided sufficient accuracy. Accuracy for r = 20µ spheroid was sufficient in the case of oblate spheroid, but in the case of elongated for and intensive turbulent field it becae low. For r = 2µ spheroid accuracy was again excellent in all cases.

4 4 Fig. 3 Mean SV for the pair of prolate spheroids ( r are 15µ and 10µ ) as a function of dissipation rate. Fig. 6 The sae as in fig. 3 for oblate spheroids with r 2µ and 1µ. Fig. 4 The sae as in fig. 3 for oblate spheroids of the sae asses. Fig. 7 Mean SV for the pair of prolate spheroids ( r are 15µ and 15µ δ r ) as a function of δ r. Fig. 5 The sae as in fig. 3 for prolate spheroids with r 2µ and 1µ. Fig. 8 The sae as in fig. 7 for oblate spheroids of the sae asses.

5 5 As a whole, together with the restriction due to assuption on constant field paraeters, r 20µ ay be regarded as the upper liit for the validity of the ethod proposed. It is worthy noting, however, that the errors due to both sall Reynolds and eployent of liit solution reduce quickly with decreasing ε and 2 3 for ε = 0.01 s, for exaple, ain liitation coes fro the assuption on constant A and S. 5. CONCLUSIONS A novel approach is elaborated for investigation of Stokesian non-spherical particles collisions in a turbulent field, characteristic for cloudy conditions. Turbulent field is represented as a large set of sall field saples (eleentary volues), within which field paraeters ay be regarded constant in space and tie. Approxiate analytical solution of spheroids otion uations in a general shear flow is found and eployed for description of spheroid otion in eleentary volues. This solution allows obtaining spheroid orientation and velocities PDFs. The statistics of SV (PDF and ean value) is estiated by eans of averaging over SV series, calculated for generated series of eleentary volues. This approach ay be applied for estiation of collision statistics both with and without HDI. In the present, first, stage the case without HDI was regarded. SV PDF and ean value were calculated for the wide range of turbulent field paraeters, representing turbulence intensity fro stratifor up to deep cuulus clouds, and for different spheroids sizes and aspect ratios (fro strongly elongated up to plate-like fors). The results indicate that turbulence enhances SV significantly coparing with the pure gravity case. The effect increases with turbulence intensity and aspect ratio deviation fro unity, and becoes especially pronounced in case of siilar and icron size spheroids. Analysis shows that for spheroids ass uivalent radii 1µ r 20µ the approach proposed ay be regarded valid. It is worthy noting however that this range depends on turbulent field intensity and for cal enough 2 3 turbulence (e.g. ε = s ) it ay be wider. ACKNOWLEDGEMENTS The study was conducted under support of the Israel Acadey of Science Foundation grant 173/03) and The Israel Ministry of Science (Geran-Israel collaboration in Water Resources, grant WT 040). REFERENCES Antonia, R.A., Chabers A.J., and Satyaprakash B.R. 1981: Reynolds nuber dependence of high order oents of the streawise turbulent velocity derivative, Boundary-layer Met. 21, Belin F, Maurer J, Tabeling P. and Willaie H., 1997: Velocity gradient distributions in fully developed turbulence: An experiental study, Phys. Fluids, 9, Broday D., Fichan M., Shapiro M. and Gutfinger C., 1998: Motion of spheroidal particles in vertical shear flows. Phys. Fluids 10 (1), 86. De Aleida, F. C., 1976: The collisional proble of cloud droplets oving in a turbulent environent Part I: A ethod of solution. J. Atos. Sci., 33, , 1979: The collisional proble of cloud droplets oving in a turbulent environent. Part II: Turbulent collision efficiencies. J. Atos. Sci., 36, Grover, S. N., and H. R. Pruppacher, 1985: The effect of vertical turbulent fluctuations in the atosphere on the collection of aerosol-particles by cloud drops. J. Atos. Sci., 42, Hill R. J. and S. T. Thoroddsen, 1997: Experiental evaluation of acceleration correlations for locally isotropic turbulence, Phys. Rev. E, 55, Kholyansky M., Tsinober A. and Yorich S., 2001: Velocity derivatives in the atospheric surface layer at Re λ =10 4, Phys. Fluids 13, Koch, D.L. and S.B. Pope, 2002: Coagulationinduced particle-concentration fluctuations in hoogeneous, isotropic turbulence. Phys. Fluids, 14, n.7, La Porta A., G.A. Voth, A. M. Crawford, J. Alexander and E. Bodenschatz, 2001: Fluid particle accelerations in fully developed turbulence, Nature, 409, Maxey M.R., 1987: The gravitational settling of aerosol particles in hoogeneous turbulence and rando flow fields. J. Fluid Mech. 174, 441. Monin, A. S. and A. M. Yaglo, 1975: Statistical fluid echanics: Mechanics of turbulence, v.2, MIT Press.. Pinsky M.B., Khain A.P., and Shapiro M., 1999: Collisions of sall drops in a turbulent flow. Pt.1: Collision efficiency: proble forulation and preliinary results. J. Atos. Sci. 56, Pinsky M., Shapiro M., Khain A. and Wirzberger H., 2004: A statistical odel of strains in

6 6 hoogeneous and isotropic turbulence, Physica D, 191, Pinsky M.B.,. Khain A.P., Grits B. and Shapiro M., 2005: Collisions of cloud droplets in a turbulent flow. Part 3. Relative droplet fluxes and swept volues. J. Atos. Sci (in press). Pruppacher H.R. and Klett J.D., 1997: Microphysics of clouds and precipitation. Oxford Press. Voth, G.A., A.La Porta, A. M. Crawford, J. Alexander and E. Bodenschatz, 2002: Measureents of particle accelerations in fully developed turbulence, J. Fluid Mech, 469,