HIGH RESOLUTION SIMULATION OF TURBULENT COLLISION-COALESCENCE OF CLOUD DROPLETS. Hossein Parishani

Transcription

1 HIGH RESOLUTION SIMULATION OF TURBULENT COLLISION-COALESCENCE OF CLOUD DROPLETS by Hossein Parishani A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering Spring 2014 c 2014 Hossein Parishani All Rights Reserved

2 UMI Number: All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI

3 HIGH RESOLUTION SIMULATION OF TURBULENT COLLISION-COALESCENCE OF CLOUD DROPLETS by Hossein Parishani Approved: Suresh G. Advani, Ph.D. Chair of the Department of Mechanical Engineering Approved: Babatunde A. Ogunnaike, Ph.D. Dean of the College of Engineering Approved: James G. Richards, Ph.D. Vice Provost for Graduate and Professional Education

4 Signed: I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Lian-Ping Wang, Ph.D. Professor in charge of dissertation Signed: I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Ajay Prasad, Ph.D. Member of dissertation committee Signed: I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Louis Rossi, Ph.D. Member of dissertation committee Signed: I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Kausik Sarkar, Ph.D. Member of dissertation committee Signed: I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Wojciech W. Grabowski, Ph.D. Member of dissertation committee

5 ACKNOWLEDGEMENTS First and foremost, I would like to thank my advisor Dr. Lian-Ping Wang for his extremely valuable expertise and guidance. Over the course of my PhD program, Dr. Lian-Ping Wang helped me through every step of my research and supported me wholeheartedly in the pursuit of scientific research. Special thanks are extended to Dr. Wojciech W. Grabowski, for giving me the chance to work under his mentor-ship and for his insightful input on my dissertation. I thank Drs. Ajay Prasad, Louis Rossi and Kausik Sarkar for serving on my dissertation committee and for commenting on this dissertation. Their comments and hints very much enlightened me in this research and significantly added to the quality of this work. I owe a debt of gratitude to Dr. Orlando Ayala and Dr. Bogdan Rosa, my close friends and officemates, for all the productive discussions we shared during the collaborative research we have carried out toward this dissertation. I extend my sincere gratitude to Dr. Hui Gao, Ms. Xiaoyan Shi and Ms. Queming Qiu for the memorable times we all had in SPL 325. From half a world away, I would like to thank my parents and siblings for all the patience and support they offered to me along this path. This research would not have been possible without their continuous love and support. Finally, I would like to acknowledge the financial support by the National Science Foundation (NSF) as well as the computational support by the National Center for Atmospheric Research (NCAR). Most of the simulations were conducted using the Bluefire and Yellowstone machines at NCAR. I am grateful for the additional computing resources provided by the Scientific Computing Division at NCAR. Some local computations were performed on UD supercomputing facilities (Mills and Chimera). iv

6 TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES NOMENCLATURE ABSTRACT viii ix xiv xxii Chapter 1 INTRODUCTION Atmospheric Clouds: Importance and Formation Droplet Growth: The Size Gap Problem Mechanisms Explaining Fast Growth Rate and Broad Size Spectrum Condensational phase Collision-coalescence phase Motivation and Objectives PROBLEM FORMULATION AND SIMULATION METHOD Droplet and Turbulence Properties in Clouds Methodology Background Air Turbulence Droplet-droplet hydrodynamic interaction Droplet tracking Collision statistics Summary of The Solution Method PARALLEL IMPLEMENTATION AND SCALABILITY ANALYSIS OF THE HYBRID DNS METHOD Introduction v

7 3.2 Parallel Implementation Using 2D DD and A Complexity Analysis Elemental Times in The Complexity Analysis t comp and t copy t comm and t startup Background turbulent flow Interpolation The cell-index and linked list algorithm Disturbance flow velocities Advancing particle equation of motion Post-processing droplet statistics Periodicity and non-overlapping conditions Performance Analysis Scalability Complexity analysis study Summary EFFECTS OF GRAVITY ON THE ACCELERATION AND PAIR STATISTICS OF INERTIAL PARTICLES SUSPENDED IN HOMOGENEOUS ISOTROPIC TURBULENCE Numerical Method Results and Discussion Particle acceleration statistics Radial relative velocity for monodisperse particle pairs Radial relative velocity for bidisperse particle pairs Asymptotic Analysis of Effect of Gravity on Particle Acceleration Variance The first approach: gravity without inertia The first approach: inertial particles without gravity The first approach: combining gravity and particle inertia The second approach: gravity and inertia together Summary and Conclusions vi

8 5 GROWTH WITH TURBULENT COLLISION-COALESCENCE Numerical Method Effect of Turbulent Collision-Coalescence Effect of Average Flow Dissipation Rate Effect of Local Flow Dissipation Rate Effect of Flow Reynolds Number Discussion SUMMARY AND FUTURE WORK Summary and Main Conclusions Future Work BIBLIOGRAPHY vii

9 LIST OF TABLES 1.1 Droplet size range and the relevant growth mechanisms Kolmogorov scales of a typical turbulent cloud Turbulence characteristics of the flow Characteristics of cloud droplets (ν = 0.17cm 2 /s, ρ = g/cm 3, ρ w = 1.0 g/cm 3 ) Parameter setting and average flow statistics. Except the last 5 rows which show dimensionless values, the other parameters are in arbitrary units Estimated elemental times on Stampede (or Yellowstone) for the complexity analysis Estimated elemental times on Stampede for the complexity analysis Flow parameters in arbitrary units except R λ and k max η which are dimensionless Relative contribution of different terms in Equation (5.6) with respect to gravitational term (R λ = 143, ν = 0.17cm 2 /s, ρ = g/cm 3, ρ p = 1.0 g/cm 3 ) Characteristics of cloud droplets (ǫ = 400cm 2 /s 3, ν = 0.17cm 2 /s, ρ = g/cm 3, ρ p = 1.0 g/cm 3 ) viii

10 LIST OF FIGURES 1.1 An updraft causes cloud droplets to form and grow in three stages [Ackerman and Knox, 2001] Our problem consists of simulation of background air turbulence (vorticity isosurface on the left panel) and droplet tracking (right panel) in a 3-D periodic cube. These two figures should be overlaid to give a realistic view of turbulent collision-coalescence, but here they are separated for demonstration of the solution method. Droplet sizes and number density are not on scale A superposition of Stokes flows has been considered to model droplet hydrodynamic interaction. Here, a single spherical droplet is shown falling with velocity v p Two spatial domain decompositions. Left: 8 subdomains in a 1D decomposition, Right: 16 subdomains in a 2D decomposition. The fluid nodes and droplets in each subdomain are assigned to an individual processor. In this figure, 4 and 8 subdomains are used for demonstration purpose, though usually a higher number of divisions is used A sketch showing 16 subdomains in 2D DD. Left panel shows fluid nodes and droplets in the physical space and each subdomain is assigned to an individual processor. X, Y and Z are the three directions of the coordinate system and g points in the gravity direction. Right panel shows how the domain is represented in the spectral space, noting the order of the wave number coordinate system due to the transposes for FFT calculations The measured computation and copy time as a function of array size (or number of elements in the array). The test runs were performed using double precision. (a) Computational time; (b) Copy time. The lines correspond to the theoretical expressions and the markers correspond to actual measurements taken from Stampede ix

11 3.4 The communication time between a number of processors as a function of transmitted data size Two different communication strategy for fluid velocity interpolation. Red lines show the layers of grid velocity data to be communicated. Left panel: Indirect communication. First, all subdomains communicate to right and left subdomains. The data of the corner subdomains is extracted and padded to the beginning and the end of local buffer matrix in the north and south subdomains. The buffered data is communicated form north and south subdomains altogether. Right panel: Direct communication. All data is communicated to and from 8 neighboring subdomains directly Shaded regions around a subdomain 9 denote all halo regions in 2D domain decomposition. All particle data in these regions are made available for the subdomain 9 via communications from 8 neighboring subdomains. Arrows indicate the direction of data communication Relative percentages of the wall-clock time spent on each subroutine for two cloud microphysics parameter settings: (a) 512 3, 45 µm - 50 µm, 730,000 particles (LWC 2.2 g/m 3 ), (b) , 20 µm - 30 µm, 16,500,000 particles (LWC 1.0 g/m 3 ). Here LWC indicates the liquid water content Execution time as a function of P for a DNS flow grid with 730,000 droplets of 45 and 50 µm size: (a) flow simulation and disturbance flow velocities, (b) interpolation, (c) cell-index and linked list method, (d) particle motion, (e) post-processing of droplet statistics, and (f) periodicity and non-overlapping conditions The average number of iterations for the disturbance flow velocities for this case was 10. The lines correspond to the prediction from the complexity analysis and the markers denote the measured wall-clock time taken from Stampede Comparison between the complexity analysis and measured wall-clock time for two cloud microphysics settings: (a) 512 3, 45 µm - 50 µm, 730,000 particles (LWC 2.2 g/m 3 ), (b) , 20 µm - 30 µm, 16,500,000 particles (LWC 1.0 g/m 3 ) Wall-clock time when running the code on Kraken for the case: 512 3, 45 µm - 50 µm, 730,000 particles (LWC 2.2 g/m 3 ). The complexity analysis of same case run in Stampede is also included for reference.. 70 x

12 3.11 Sensitivity of the wall-clock time on different cloud microphysics parameters. (a) Liquid water content (LWC), monodisperse system of a=25 µm, and ; (b) Turbulent flow resolution, monodisperse system of a=25 µm, and LWC = 1 g/m 3 ; (c) Particle size, monodisperse, , and LWC = 1 g/m 3 ; and (d) Bidisperse case, a 1 = 25 µm, , and LWC = 1 g/m Theoretical percentage of the wall-clock time per subroutine for two cloud microphysics cases from Figure (a) , 25 µm (monodisperse), LWC = 1 g/m 3. (b) , 5 µm (monodisperse), LWC = 1 g/m Study of possible improvements to the code. (a) Number of parallel particle realizations to balance the flow simulation. Base case: 25 µm (monodisperse), LWC = 1 g/m 3 ; (b) FFT wall-clock time for the turbulent flow resolution at grid; Study of possible improvements to the code. (a) Theoretical wall-clock time for t flow and t particle to determine when t flow < t particle (Case: , LWC = 1 g/m 3, P = 8196). (b) Theoretical wall-clock time using two dt levels, and asymptotic approximation to track the particle, here dt flow > dt particle (Case: , 5 µm (monodisperse), LWC = 1 g/m 3 ); Maximum (or minimum) number of processors allowed based on different considerations as a function of flow resolution. (a) Base case: 25 µm (monodisperse), LWC = 1 g/m 3 ; (b) Base case: 5 µm (monodisperse), LWC = 1 g/m The pdfs of normalized particle acceleration are compared against the results of Bec et al. [2006]. Legend: D denotes the deterministic forcing; S is for the stochastic forcing The pdfs of normalized particle acceleration obtained using two different forcing methods. Legend: D denotes the deterministic forcing; S is for the stochastic forcing The rms of particle acceleration for non-sedimenting setup. The three directions are averaged. Legend: D denotes the deterministic forcing; S is for the stochastic forcing The rms value of particle acceleration as a function of particle Stokes number. Sedimenting droplets. Legend: D denotes the deterministic forcing; S is for the stochastic forcing xi

13 4.5 PDF of the vertical acceleration for 3 different values of g PDF of the horizontal acceleration for 3 different values of gravitational acceleration. Only the side of positive acceleration is shown as the PDF for horizontal acceleration is symmetric From left to right w g, w s and w a represent contributions to relative velocity from gravity, fluid shear and particle differential acceleration respectively PDF of the normalized relative velocity due to the shear term w s,11. σ is the rms value of the plotted parameter PDF of the normalized relative velocity due to the relative acceleration term w a,11. σ is the rms value of the plotted parameter The ratio w s / w a for monodisperse cases as a function of particle radius, showing that the shear term is dominant for small particles, while the differential acceleration term becomes dominant for large particles PDF of w s,12 (bidisperse, a 2 = 60µm and St 2 = 2.28) PDF of w a,12 (bidisperse, a 2 = 60µm and St 2 = 2.28, a 1 = 30, 40 and 50µm) The ratio of the shear contribution (w s ) to the gravity term (w g ) for bidisperse particles, with a 2 = 60µm. DNS: current study, Estimate: estimation of Grabowski and Wang [2013] The ratio of the differential acceleration term to the gravity term for bidisperse particles, with a 2 = 60µm. DNS: current study, Estimate: estimation of Grabowski and Wang [2013] The value of w s with gravity relative to that without gravity, for bidisperse particles. a 2 = 60µm The ratio of wa with gravity to that without gravity. a 2 = 60µm The correlation coefficient between the gravity term and the differential acceleration term. a 2 = 60µm xii

14 4.18 Theoretical prediction for particle acceleration rms in comparison with DNS measurements. Legend: D denotes the deterministic forcing; S is for the stochastic forcing Schematic of a sedimenting particle acceleration due to interaction with an eddy. Red arrows show the acceleration (a) in the vertical plane, (b) in the horizontal plane Size distribution at 25% coalescence in comparison with the results of Reade and Collins [2000a] for non-sedimenting droplets. i is the number of coalesced droplets, namely r/r 1 = 3 i where r 1 is the initial droplet radius Effect of fluid turbulence on the evolution of average droplet radii of sedimenting particle pairs. The initial collected droplet radius is 10µm. The collector radius is: (a) 20µm, (b) 30µm, (c) 40µm, (d) 50µm, (e) 60µm. The y axis is normalized using the radius of the collected droplet Effect of fluid turbulence on the evolution of average droplet radii of sedimenting particle pairs. The initial collected droplet radius is 20µm. The collector radius is: (a) 30µm, (b) 40µm, (c) 50µm, (d) 60µm. The y axis is normalized using the radius of the collected droplet Effect of dissipation rate on the evolution of average droplet radii for different particle pairs at R λ = 143. The initial collected droplet radius is 20µm. The collector radius is: (a) 30µm, (b) 40µm, (c) 50µm, (d) 60µm. The y axis is normalized using the radius of the collected droplet Average values of local St for different particle pairs at R λ = 143 and ǫ = 400cm 2 /s 3. St 1 is the initial Stokes number for the smallest droplets in the domain. The collected droplet radius is 10µm. The collector radius is: (a) 20µm, (b) 30µm, (c) 40µm, (d) 50µm, (e) 60µm Effect of R λ on the evolution of average droplet radii for different particle pairs at ǫ = 400cm 2 /s 3. The collected droplet radius is 10µm and the collector radius is: (a) 20µm, (b) 30µm, (c) 40µm, (d) 50µm, (e) 60µm. The y axis is normalized using the radius of the collected droplet xiii

15 NOMENCLATURE a a k or a (k) a c Droplet radius, in µm or cm k-th droplet radius, in µm or cm Critical droplet radius for particle clustering, in µm or cm a f Local fluid acceleration, in cm/s 2 a p Local particle acceleration, in cm/s 2 a f Local fluid acceleration component, in cm/s 2 a p Local particle acceleration component, in cm/s 2 a η Fluid acceleration at Kolmogorov scale, in cm/s 2 a rms Acceleration variance, in cm/s 2 a 0 C D L Empirical rescaling parameter for fluid acceleration Non-dimensional truncation radius Lagrangian fluid velocity correlation D (k) Drag force on the k-th droplet, in g cm/s 2 Du (Du ) 2 Dt Turbulent flow acceleration, in cm/s 2 d int dt Dt dt flow dt part d (mk) d (mk) i Turbulent flow acceleration variance, in cm 2 /s 4 Droplet-droplet average spacing, in cm Time step, in s Flow time step, in s Particle time step, in s Separation vector between m-droplet and k-droplet, d (mk) in cm i-th component of the droplet separation vector, in cm xiv

16 E 12 Collision efficiency between droplets 1 and 2 E g 12 Collision efficiency between droplets 1 and 2 for the hydrodynamically-gravitational case E 11 Collision efficiency between droplets 1 and 1 FLOP Floating point operation F p Particle Froude number as defined in Davila and Hunt (2001) F p,λ F(r) Particle Froude number defined using Taylor scales Total viscous dissipation contained within a spherical region of radius r around a droplet, in g cm 2 /s 3 f(r) f(re p ) Longitudinal two-point velocity correlation Non-linear drag coefficient f(x, t) Random forcing term, f(x, t) in cm/s 2 g(r) Transverse two-point velocity correlation g 12 Radial distribution function between droplets 1 and 2 g g 12 Geometric radial distribution function g NO 12 Radial distribution function under non-overlap condition g 12 (r 1, r 2 ) Radial distribution function within a shell with inner radius r 1 and outer radius r 2 g Gravitational acceleration vector, g = 980 cm/s 2 HI k x, k y, k z k max L L f Hydrodynamic interactions Three components of the wave number in Fourier space Maximum turbulent wave number, in cm 1 Computational domain size, in cm Integral length scale of turbulence, in cm LWC Liquid water content, in g/m 3 m p N c Droplet mass, in g Average geometric collision rate per unit volume, in s 1 cm 3 xv

17 N pair N p N pk n k n pairs NHI or No HI P(w) P(w r ) or P wr P P y P z P(y) Total number of pairs detected Total number of droplets Total number of k-droplets Average k-th droplet number concentration, in cm 3 Total number of distinct droplet pairs No hydrodynamic interactions between droplets Probability density of the vector relative velocity Probability density of the radial relative velocity Total number of subdomains (processors) Number of decompositions in y direction Number of decompositions in z direction Probability of collision of two droplets initially separated by a horizontal distance y P(θ; R o ) Probability of collision of two droplets initially separated by a radial distance R o and an angle θ R R = a 1 + a 2 R λ R o R ij Re p r r Separation distance vector of droplets at contact, R in cm Geometric collision radius or collision sphere radius, in cm Taylor-microscale Reynolds number Droplet-droplet initial radial separation distance, in cm Two-point fluid velocity correlation Droplet Reynolds number Droplet separation vector, r in cm Global mass-averaged droplet radius for growing droplets, in cm r 1024 Mass-averaged droplet radius for 1024 largest droplets of the domain, in cm r r r 1 and r 2 Droplet radius, in cm Radial distance, in cm Inner and outer radius, in cm xvi

18 S Vapor supersaturation St Droplet Stokes number St 1024 St k Average local droplet Stokes number for 1024 largest droplets of the domain k-th droplet Stokes number S c Turbulent effective collision cross section, in cm 2 Sv Droplet non-dimensional settling velocity Sv k k-th droplet non-dimensional settling velocity T e T L t t comm t comp t copy t startup t FLOW t FFT t INTERP t HEAD,LIST U U i U (m) U u u Large-eddy turnover time of turbulence, in s Lagrangian integral time of turbulence, in s Time, in s Time to communication a single word from one processor to another, in s Time to perform one floating point (FLOP) compution, in s Time to copy one word from memory to memory, in s Startup/latency time during communication, in s Wall clock time to compute the flow part, in s Wall clock time to compute one 3D FFT, in s Wall clock time to interpolate fluid velocity to particle centers, in s Wall clock time to find the neighbor particles and list them, in s Turbulent flow velocity vector, U in cm/s i-th component of the turbulent flow velocity, in cm/s Turbulent flow velocity vector at m-droplet, U in cm/s Fluid flow velocity vector as seen by the droplet, U in cm/s r.m.s. fluctuation turbulent flow velocity, in cm/s Disturbance flow velocity vector, u in cm/s xvii

19 u charac u i u i (k) u S V or V p V (k) or V p (k) V p (k) Some characteristic velocity of the flow, in cm/s i-th component of the disturbance flow velocity, in cm/s i-th component of the disturbance flow velocity on the k-th droplet, in cm/s Stokes disturbance flow vector, u S in cm/s Droplet velocity vector, V in cm/s k-th droplet velocity vector, V (k) in cm/s Modified k-th droplet velocity vector due to HI, V p (k) in cm/s V s Volume of a spherical shell, in cm 3 V B or V box Volume of the computational domain, in cm 3 v k v p v pnld v pk Kolmogorov velocity scale, in cm/s Droplet settling velocity in a stagnant fluid, in cm/s Modified droplet settling velocity in a stagnant fluid using non-linear drag, in cm/s Modified k-droplet terminal velocity along the vertical direction due to HI, in cm/s v (k) r k-droplet velocity component along the separation vector r, v v (k) or v k v r (k) v i (k) in cm/s r.m.s. of a droplet velocity, in cm/s r.m.s. of the k-droplet velocity, in cm/s Turbulent radial component of the k-th droplet velocity, in cm/s Turbulent i-th component of the k-th droplet velocity, in cm/s (v i(1) v i (2) ) Particle velocity covariance, in cm 2 /s 2 (v i (1) ) 2 Mean-square particle velocity, in cm 2 /s 2 v pturb Average turbulent droplet settling velocity, in cm/s xviii

20 w w w w g w s w a w r w r w r (r 1, r 2 ) w g r w HI r w NHI r w NO r x Y f Y Y (k) y y c Droplet relative velocity vector, w in cm/s Average of the relative velocity, in cm/s Particle terminal velocity in stagnant fluid, in cm/s Contribution to radial relative velocity from gravity, in cm/s Contribution to radial relative velocity from fluid shear, in cm/s Contribution to radial relative velocity from differential acceleration, in cm/s Droplet radial relative velocity, in cm/s Average of the radial relative velocity, in cm/s Average of the radial relative velocity within a shell with inner radius r 1 and outer radius r 2, in cm/s Geometric droplet radial relative velocity, in cm/s Droplet radial relative velocity with hydrodynamic interactions included, in cm/s Droplet radial relative velocity with no hydrodynamic interactions considered, in cm/s Droplet radial relative velocity under non-overlap condition, in cm/s Position vector, x in cm Fluid Lagrangian location vector, Y f in cm Instantaneous droplet location vector, Y in cm Instantaneous k-th droplet location vector, Y (k) in cm Horizontal offset of two droplet centers, in cm Horizontal offset of two droplet centers that result in a grazing trajectory, in cm xix

21 GREEK SYMBOLS α Angle between the relative velocity of the droplets and gravity α Effective hydrodynamic-interaction distance factor δ Small fraction of the collision sphere radius R δ 1 and δ 2 Inner and outer small fractions of the radial distance r δ ij Delta Kronecker ǫ Turbulent dissipation rate, in cm 2 /s 3 ǫ Spatially averaged turbulent dissipation rate, in cm 2 /s 3 ǫ(r, θ) Local dissipation rate, in cm 2 /s 3 ε Relative error for an iterative method η E η G η T η Turbulent enhancement factor on the collision efficiency Turbulent enhancement factor on the geometric collision kernel Total turbulent enhancement factor Kolmogorov length scale, in cm µ Dynamic air viscosity, in g/(cm s) ν Kinematic air viscosity, in cm 2 /s λ Taylor microscale of turbulence, in cm λ D ω φ φ k Longitudinal Taylor-type microscale of fluid acceleration, in cm Turbulent flow vorticity vector, ω in s 1 Total drople volume fraction Volume fraction of droplet group k ρ Air density, in g/cm 3 ρ ga Cross correlation coefficient between w g and w a ρ w Water density, in g/cm 3 xx

22 σ Standard deviation of the radial relative velocity, in cm/s σ 2 Variance of the droplet radial relative velocity, cm 2 /s 2 τ k τ p or τ (p) τ pnld τ pk θ ε Kolmogorov time scale, in s Droplet inertial response time, in s Modified droplet inertial response time using non-linear drag, in s k-th droplet inertial response time, in s Droplet-droplet relative angle, in rad or degrees Tolerance parameter Γ Collision kernel, in cm 3 /s Γ 12 Collision kernel between droplets 1 and 2, in cm 3 /s Γ g 12 Geometric collision kernel, in cm 3 /s Γ vort Circulation around a vortical region, in cm 2 /s W Particle relative settling velocity, in cm/s x DNS grid spacing, in cm t Time interval, in s v p Differential terminal velocity, in cm/s xxi

23 ABSTRACT In this dissertation we investigate the effects of turbulence, inertia and gravitational settling on the dynamics of colliding cloud droplets. This study is motivated by the open question in cloud microphysics concerning the fast growth of droplets in warm clouds through the size range from 10 to 50 µm in radius, for which neither the diffusional growth nor the gravitational collision-coalescence is effective. In order to simultaneously compute the turbulent flow field and the droplet dynamics with a sufficient domain size (a cubic box of O(1m 3 ) in volume), we developed a massively parallel direct numerical simulation (DNS) code using 2D domain decomposition (2D DD). We simulate the flow using a pseudo spectral DNS code and track individual droplets whose motion is affected by Stokes drag, gravity and local dropletdroplet hydrodynamic interactions. The hydrodynamic interactions are represented analytically by Stokes disturbance flows that are coupled to the background air turbulence (Wang et al. [2005a]). To better understand the scalability of this parallel hybrid DNS code, we measured the wall clock time of the code under different simulation conditions and developed a theoretical complexity analysis to predict the execution time for any problem size and any number of processors. The resulting analysis is found to be in good agreement with the measured timing data for all the problem sizes and numbers of processors tested. The execution time scales with the number of processors almost linearly before the performance saturates due to excessive communication latency. The complexity analysis is used to estimate the maximum number of processors below which the linear scalability could be sustained, demonstrating that our code is likely to perform well on Petascale machines for large problem sizes. xxii

24 In the second part of the dissertation, the newly developed 2D DD code is used to study the dynamics of droplets in a turbulent flow up to flow Taylor Reynolds number of R λ = 143. The effects of gravity on the inertial particle acceleration was studied. We found that the gravity plays a very important role in particle acceleration statistics: a) A peak value of particle acceleration variance appears in both the horizontal and vertical directions at a particle Stokes number of about 1.2, at which the particle horizontal acceleration clearly exceeds the fluid-element acceleration; b) Gravity suppresses extreme acceleration events both in the vertical and horizontal directions by reducing the particleeddy interaction time, and thus effectively enhances the inertial filtering mechanism. A theory was developed to explain the effects of gravity and turbulence on the horizontal and vertical acceleration variance of droplets at small particle Stokes numbers. We found analytically that gravity affects particle acceleration variance both in horizontal and vertical directions, resulting in an increase in particle acceleration variance in both directions. Furthermore, the effect of gravity on the horizontal acceleration variance is found to be stronger than that in the vertical direction, in agreement with our DNS results. By decomposing the radial relative velocity of the particles into three parts: the gravitational term, the shear term and the differential acceleration term, we were able to evaluate separately their contributions. For monodisperse particles, our results show that the presence of gravity does not have a significant effect on the shear term. On the other hand, gravity broadens the pdf tails of the acceleration term due to both the inertial bias and the preferential sweeping effect when encountering a vortical structure. For bidisperse cases, we found that gravity can decrease the shear term slightly by dispersing particles into vortices where fluid shear is relatively low. We also found that the differential acceleration term is positively correlated with the gravity term, and this correlation is stronger when the difference in colliding particle radii becomes smaller. Through DNS data, we found that the ratio of the differential acceleration to the gravity term is between 30% to 60%. This implies that the enhancement factor due to turbulence relative to the gravity term, is 1.04 to 1.17, consistent with the results xxiii

25 in previous studies. We also confirm that the scaling analysis of Grabowski and Wang [2013] provides a reasonable estimate of the magnitude of the shear and differential acceleration terms. In the third part, the size distribution of the growing cloud droplets under cloud conditions was studied. Two average radii were measured for the droplets: the average of droplet radii over whole domain (the average radius) and the average radius for the 1024 largest droplets in the domain (the lucky-droplet average radius). For an initially bidisperse system, the turbulence contributes significantly via acceleration and coupling mechanisms. The growth rate of lucky droplets was found to be significantly affected by turbulence. For instance, for 10 60µm pair the growth rate of the lucky-droplet average radius is increased by a factor of by turbulence (R λ = 143 and ǫ = 400cm 2 /s 3 ). This increase is considerably larger than that of the averaged radius. The effect of flow dissipation rate is significant when the difference in droplet sizes becomes small. An increase of the the dissipation rate from 400cm 2 /s 3 to 1000cm 2 /s 3 caused the growth rate of the lucky-droplet average radius to increase by 67% for the 20 30µm pair. By measuring the local values of Stokes number along the droplet trajectory, we found that the lucky droplets sample up to 35% lower local dissipation rates leading to a smaller values of local Stokes numbers. The highest decrease in the local dissipation rate (= 35%) is observed for the 10 20µm pair where the the droplet inertia is low and consequently the droplet clustering is weak. Finally, we study the effect of turbulent Reynolds number on the growth rate of the droplets. We found that the flow Reynolds number has a negligible effect on the growth rate of average droplet radius. However, the growth rate of the lucky-droplet average radius is significantly affected indicating that the lucky droplets interact with the large-scales flow. A Froude number based on the Taylor scale is introduced to help interpret this interaction. We found that this interaction is stronger when the particle Froude number F p,λ is of order unity. In summary, the results obtained in this dissertation provide insights into the contributions of the different scales of turbulent air motion on the initiation and development xxiv

26 of rain droplets from cloud droplets through the collision-coalescence mechanism. xxv

27 Chapter 1 INTRODUCTION Interactions of inertial particles and fluid turbulence have attracted a great deal of attention due to their relevance to many industrial applications and natural processes. The nature of the interactions may depend on flow and particle characteristics, such as flow dissipation rate and Reynolds number that determine the flow time and length scales, particle volume concentration, particle size, density ratio, etc. A few specific examples to be described below show that particle-laden turbulent flows cover a broad range of applications from nano-size particles in study of aerosols to mm-size particles or even larger in sediment transport. Motivated by our interest in the formation and growth of water droplets in clouds, our main focus will be on a dilute suspension of small and heavy particles in a turbulent flow where gravity, particle inertia and turbulent flow together determine the dynamics and evolution of these particles. Pollutant formation, control and filtering are of importance to global warming and human health. Intra et al. [2010] studied the performance of electrostatic precipitators used to collect and filter particulate emissions resulted from biomass combustion. On the other hand, a better theoretical understanding of the dispersion, evaporation and coalescence of spray droplets can help design more efficient combustion process and cleaner engines [Sornek et al., 2000; Nijdam et al., 2006]. Planktons in ocean water may be treated as small particles whose contact rate is significantly affected by water turbulence; which is vital for marine life [Schmitt and Seuront, 2008; Jumaras et al., 2009]. In coastal problems, sediment transport plays a role in deepsea landslide, structure formation and bed erosion [Seminara, 2010]. 1

28 Powder production, mixing and separation are also a good example of a dilute particulate flow. In a recent numerical study, Radeke et al. [2010] used a massively parallel GPU platform to simulate a large-scale powder mixer. Pratsinis [1998] studied the particle growth in flame aerosol synthesis of ceramic powders. Flame aerosol technology is used worldwide for large-scale production of ceramic material such as Titania, fumed silica and alumina. Achieving a certain powder size distribution and a certain shape of agglomerate is of high interest for powder manufacturers. In another numerical study, Yu et al. [2008] simulated the particle size, particle morphology, particle collision rate and flame temperature in the process of TiO 2 nanoparticle synthesis. Fluidized beds and cyclone separators are among industrial devices which have been widely used in chemical engineering as reactors or as sizing and separation devices. Gasification of biomass fuel in a fluidized bed is, for instance, modeled by Gomez-Barea and Leckner [2010]. Cortes and Gil [2007] modeled the gas and particle flow inside cyclone separators. In atmospheric physics, clouds are a suspension of water droplets in air turbulence. Understanding the physics of a cloud requires a broad knowledge of particulate flows [Shaw, 2003; Grabowski and Wang, 2013]. Since we will focus on the dynamics of cloud droplets relevant to rain formation in this dissertation, the cloud and rain droplets shall be specifically highlighted in the following sections. 1.1 Atmospheric Clouds: Importance and Formation Formation of clouds and precipitation is a fundamental aspect of weather and climate. Clouds have a significant effect on absorbing and reflecting solar and planet radiation. Short-wave solar radiation is largely reflected back to space on the top of a cloud (albedo effect), while long-wave solar radiation could be absorbed and re-emitted by a cloud (greenhouse effect). Hence, clouds are important in regulating Earth s energy balance and its average temperature. In another balancing effect, clouds are responsible in redistributing excess heat from the equator toward the poles. This effect takes place by water evaporation at equator followed by rise and transfer of humid air 2

29 (cloud to be) toward poles. Cloud altitude and its optical thickness 1 are two main parameters influencing how effective a cloud could redistribute heat on the Earth surface [Pruppacher and Klett, 1997]. Clouds also act as water carriers transporting large amounts of water from oceans, where the majority of water is stored, to different parts of mainland. Clouds form a critical link in the hydrological cycle by precipitating water back to the Earth surface. One of the main forms of precipitation caused by clouds is rain which along with clouds themselves is the subject of this research. In meteorology, the presence of clouds in the sky is the signal indicating possible changes in the local weather. Color, density, shape, degree of coverage, light reflectivity and the cloud altitude are the main characteristics on which clouds could be classified. Further, the type of clouds is indicative of on-going atmospheric processes. For instance, presence of stratus clouds (gray sheet-like clouds which cover large portion of sky) in a region indicates that air in that region has been lifted up slowly [Pruppacher and Klett, 1997]. To understand how clouds form in the atmosphere, consider a moist pack (or parcel) of air close to the Earth surface. This pack of air could initially be warmed up either directly by surface heat transfer or indirectly by solar radiation. As a result its density will decrease and consequently the pack will tend to rise. At higher altitude, lower temperature will cause the water vapor to condensate on certain aerosol particles known as the cloud condensation nuclei (CCN), and cloud droplets form. At this point, the temperature of the air parcel should have dropped below the dew point temperature. Aerosol particles (dust and salt particles, smoke, pollutant agglomerates, volcanic ash, etc.) are the main sources of CCN [Pruppacher and Klett, 1997]. This process in atmospheric physics is labeled as nucleation and involves the aerosol particles or CCN of typical size O(0.1µm) [Rogers and Yau, 1989; Rauber et al., 2003]. Aerosols are provided by natural sources such as volcanoes, forest fires or sea salt, as well as from 1 Optical thickness is the measure of how much light a cloud can intercept which is determined by cloud droplet size and total water content. 3

30 human activities, for instance air pollution. Further condensation of water vapor on newly formed droplets is referred to as the condensational growth. When droplets reach the size of 5 to 15µm, they become visible in the form of a cloud. The further growth above 15µm is mostly achieved by collision-coalescence, the process where two droplets collide and merge to form a droplet of combined mass. If cloud droplets grow even further in size, they become too heavy to stay suspended. Suspension of cloud droplets is essentially caused by upward air motion (called an updraft). Depending on the updraft air velocity there will be a limiting droplet size above which the updraft drag force could not overcome the gravitational force acting in the opposite direction. The fall of these liquid droplets toward the Earth surface is known as warm rain. The term warm rain is used as opposed to cold rain in which ice crystals are also involved either in the stage of forming cloud droplets or in the stage of fall. It is evident that ambient air temperature, somewhere in the process, must fall below the freezing point of water in order to initiate cold rain. Statistically, as Lau and Wu [2003] indicated, warm rain accounts for 31% of the total planetary rainfall and roughly 72% of the total rainfall in the tropical regions. In this research we will focus our attention on warm rain formation (or ice-free precipitation). 1.2 Droplet Growth: The Size Gap Problem There are three phases of formation and growth before drops reach to precipitable diameter of O(0.1mm 1mm) [Pruppacher and Klett, 1997], namely: 1. nucleation or CCN activation 2. water vapor diffusion and condensation 3. droplet 2 collision and coalescence. Activation stage is the initiation of water droplet by condensation of water vapor on aerosol particles (salt, dust, smoke, etc.). Aerosol particles are not needed for water vapor to condensate but the presence of aerosols will significantly ease up the activation 2 The terms particles and droplets may be used interchangeably. Small cloud droplets remain spherical and behave like solid particles as far as the viscous drag is concerned [Pruppacher and Klett, 1997]. 4

31 stage [Pruppacher and Klett, 1997]. Activation or nucleation mainly takes place at the cloud base where droplets are driven by an upward airflow. In the next stage, the droplet will continue to grow by absorbing more water vapor from ambient. Diffusion of water vapor on the droplet surface is driven by the local vapor pressure gradient and temperature gradient. At this stage, the important parameters which govern the growth rate are: Ambient supersaturation: Water vapor diffusion will be more effective at higher air supersaturation. Solute effect: Any substance dissolved in the water will decrease its saturation vapor pressure. Curvature effect: A curved surface has a higher vapor pressure than that of a flat surface which will yield a higher equilibrium vapor pressure at the droplet surface. An analysis of mass balance on the surface of water droplet leads to the conclusion that the rate of change of droplet radius is inversely proportional to the droplet radius [Pruppacher and Klett, 1997]. That is, the growth rate by condensation will be slower for larger droplets. Consequently, the condensational growth becomes increasingly slow, for instance it would take 1h for a droplet to grow by diffusion from 15µm to 50µm in radius (more details can be found in Pruppacher and Klett [1997]). As a result, for droplets of radius 15µm or larger, the condensational growth becomes less effective when compared to growth by collision-coalescence. Another consequence of the droplet growth rate being inversely proportional to droplet radius is the narrowing in the droplet size distribution as the mean droplet size increases, if the condensational growth were to be considered alone. During the third phase of growth, droplets which have grown to a few tens of microns and are moving in a gravitational field, will collide with each other leading to the merging of colliding droplets (a process known as coalescence). For coalescence to take place, the kinetic energy of the colliding droplets must overcome the surface energy barrier allowing two separate surfaces to unite into one. Under gravity alone, larger droplets fall faster than smaller ones, resulting in the larger droplets overtaking the smaller droplets and sweeping them down leading to the growth of the larger droplet. This process is called gravitational collision-coalescence and is known to be 5

32 effective for droplet radii larger than 40µm in radius [Pruppacher and Klett, 1997; Grabowski and Wang, 2013]. During the first two stages of droplet growth, droplets experience upward motion under the effect of an updraft. In contrast, during the third stage, the droplet terminal velocity could become comparable to the updraft air velocity, with larger droplets having a terminal velocity that may exceed the air updraft velocity. Figure 1.1 shows a schematic of droplet growth and precipitation in an updraft. With the growth mechanisms mentioned so far (condensation and gravitational collision-coalescence), we proceed to explain size evolution and growth of a poly-disperse suspension of droplets. The condensational growth of droplets in this system will be effective until they reach the size of 15µm at which point the relevant time-scale is O(10) minute per µm of growth of droplet radius. Since gravitational coalescence mechanism is effective only for droplets larger than 40µm in radius (partly due to very low collection efficiency), practically the growth of droplets will stop at radii 15µm or it becomes extremely slow to grow further by condensation. This will cause the size spectrum of droplets to be restricted to a narrow region around 15µm. On the other hand, gravitational collision-coalescence operates solely on the fact that different droplet Figure 1.1: An updraft causes cloud droplets to form and grow in three stages [Ackerman and Knox, 2001]. 6

33 sizes have different sedimentation velocities so that larger droplets can sweep smaller ones in their way down relative to small droplets. Therefore, the gravitational collisioncoalescence may not take place effectively for a suspension of droplets near 15µm in radius. The models based on the gravitational coalescence alone predict a time interval of 1h for droplets to grow from 20µm to 100µm in radius [Pruppacher and Klett, 1997]. Now we briefly review what has been observed in real clouds. As early as 1958, it was observed that the initiation time for a warm rain cloud be less than half an hour [Squires, 1958; Rauber et al., 2007]. In recent years, several researchers observed by radar that warm rain could be formed in cumulus clouds within about minutes [Szumowski et al., 1997; Knight et al., 2002]. More recently, in one of the largest warm rain field projects, Rauber et al. [2007] flew three heavily instrumented research aircrafts in cumulus clouds to study rain initiation by obtaining high resolution data from clouds. Consistent with data analysis of Rauber et al. [2007], the warm-rain initiation time is believed to be of the order of minutes. That is, often it takes less than half an hour for a cloud droplet to grow to the size of a raindrop [Rogers and Yau, 1989; Rauber et al., 2003]. However, we should mention that there are also debates regarding the difficulties of interpreting radar data to obtain rain initiation and ending time [Knight et al., 2002; Rauber et al., 2003]. Since the initiation time predicted by current models based on gravitational collision-coalescence alone is not consistent with what have been observed, the problem of explaining the droplet growth in the size range from 10µm to 40µm in radius is often referred to as the size gap problem. The size gap problem essentially poses the following question: How could the cloud droplets grow as quickly as often observed, particularly in the radius range from 10 to 40µm for which neither the condensation nor the gravitational collision-coalescence mechanism is effective?. It should be noted that the precise boundary for the size gap differs significantly in the published literature. In this study, we will be focusing on the radius range from 10µm to 60µm. 7

34 1.3 Mechanisms Explaining Fast Growth Rate and Broad Size Spectrum Over the last half century after Squires [1958], researchers in different fields of physics, engineering and applied mathematics attempted to identify and model physical processes which explain the rapid growth of cloud droplets. This effort is mainly focused on the size gap range. The efforts in identifying growth mechanisms could be divided into two camps. The first concerns the mechanisms that could enhance activation and condensational phase of the growth. The second group of studies addresses the processes affecting the collision-coalescence growth of droplets. These efforts are outlined in the following paragraphs Condensational phase Among the efforts in the first camp, Beard and Ochs [1993] provided a thorough overview of the mechanisms triggering warm rain and elaborated the contribution of each mechanism via physical interpretation without being involved in cumbersome mathematics. The mechanisms and contributors can be categorized as following: 1. Ambient humidity: Randomness or intermittency of the humidity field (the socalled stochastic condensation) is thought to affect the growth rate [Belayev, 1961; Levin and Sedunov, 1967; Mazin and Smirnoff, 1983]. Another way for humidity to affect droplet growth is the favorable condensation on a few droplets that fall through patches of high supersaturation in mixed regions [Beard and Ochs, 1993]. Also mixing of parcels having different supersaturation histories could broaden droplet size spectrum [Cooper, 1989]. Cooper [1989] provided theoretical explanation on how turbulent motions may broaden droplet spectrum provided turbulence is accompanied by dry-air entrainment leading to vertical supersaturation fluctuation. In his study, Cooper [1989] provided an analytical expression to predict the enhanced growth rate of droplets. 2. Air turbulence: Vaillancourt et al. [2001] and Vaillancourt et al. [2002] introduced a microscopic approach to model the condensational droplet growth. Taking into account the spatial non-uniformity of droplet size and positions, their results showed that condensation alone does not lead to a significant broadening of droplet size spectrum. On the other hand, in a more recent study, Lanotte et al. [2009] investigated the dynamics and growth of up to several million droplets in homogeneous isotropic turbulence and argued that a significant spectrum broadening could be possible if their DNS results at low flow Reynolds number is extrapolated to the flow Reynolds numbers in real clouds. In similar studies [Bartlett and Jonas, 1972; 8

35 Mazin and Smirnoff, 1983; Shaw et al., 1998; Celani et al., 2005; Paoli and Shariff, 2009] the turbulent condensation of droplets is investigated via direct numerical simulation and stochastic modeling. The latter studies also suggest that turbulent velocity, vapor and temperature fluctuations cause a significant increase in mean and variance of droplet radius. On the other hand, Grabowski and Vaillancourt [1999] argued that the importance of turbulence in diffusional growth of cloud droplets is exaggerated. They specifically referred to the study of Shaw et al. [1998] and mentioned at least two reasons to support their argument. First, they mention that laboratory results could not be directly generalized to atmospheric flows because of the much weaker intensity of turbulence in the atmospheric clouds. Second, gravity was not included in the model of Shaw et al. [1998], which leads to unrealistically longer droplet-turbulence interaction time. 3. Super-terminal fall velocity: Recently, there are observations which confirm droplets could fall at speeds higher than their terminal velocity (so-called super-terminal velocity). Montero-Martinez et al. [2009] have investigated the super-terminal fall of droplets and observed that droplets could accelerate to higher than 1.3 times their theoretical terminal velocity. The super-terminal phenomenon takes place when a droplet breaks up producing several smaller droplets which will move at the same speed as parent droplet. Therefore, the smaller droplets resulted from break-up will have velocity magnitudes much larger than their own terminal velocity. Montero-Martinez et al. [2009] showed that depending on precipitation rate and droplet diameter, this increase could even exceed 10 times the droplet terminal velocity. This will lead to higher growth rate since super-terminal droplets could sweep a higher volume on their way down. 4. Electric charge: Accumulation of electric charge on cloud droplets and rain droplets can affect the attractive and repulsive forces between them. This topic is studied thoroughly in Harrison and Ambaum [2008] and found to enhance cloud formation. The charges carried by droplets will reduce the minimum supersaturation at which haze droplets start to grow. Here it should be mentioned that droplet charging usually happens near the outer edge of clouds and not everywhere in the cloud [Harrison and Ambaum, 2008]. 5. Giant and ultra-giant aerosols: For a long time, the presence of giant and ultragiant aerosol particles (or cloud condensation nuclei, CCN) is thought to explain the formation of large cloud droplets [Johnson, 1982; Lowenstein et al., 2010]. In a recent study, Lowenstein et al. [2010] have observed in their field experiments that giant and ultra-giant aerosols indeed accelerate the growth of larger size droplets. 6. Entrainment and mixing: Additional CCN could be entered to a cloud by dry air entrainment and mixing via the base, top and the sides of a cloud. This topic is investigated by Derksen et al. [2009] and their results show that activation of 9

36 newly entrained CCN accompanied by vertical fluctuation and supersaturation will broaden the size spectrum. A trivial argument is that entrained CCN will increase the CCN concentration and consequently will lead to higher number of activation sites for droplet growth. Introduction of new CCN through entrainment and mixing will broaden droplet size spectrum toward smaller size droplets which might affect the growth rate. 7. Pre-existing clouds: In several studies [Beard and Ochs, 1993; Pinsky and Khain, 1997; Rauber et al., 2003, 2007], pre-existing clouds have been shown to act as local beds for early appearance of large droplets. Pre-existing clouds would accelerate the size growth mainly by the fact that they contain droplets which have already reached certain droplet size on which further growth could take place Collision-coalescence phase In the second camp of efforts, investigation is focused on mechanisms which may accelerate droplet collision-coalescence. These investigations stress the importance of air turbulence and its derivative effects. This could be understood qualitatively when we observe that equal-size droplets would never have a chance to grow by collision and coalescence in quiescent air since they have a zero relative terminal velocity. On the other hand, since turbulence can generate a non-zero relative velocity for droplets, same-size droplets are able to collide in a turbulent air. It should be added that atmospheric air motion in general and specifically air motion in clouds is naturally turbulent. Thereby, there is a likelihood for air turbulence to partially address the size-gap problem. In the following sections, this possibility is explored in details, which opens a new host of questions in cloud physics. A great deal of theoretical, experimental and numerical efforts have been devoted to study droplet-turbulence interaction. Shaw [2003] provided an excellent in-depth review on particle-turbulence interaction elaborating the mechanisms and their contribution to droplet growth in atmospheric clouds. Falkovich et al. [2002] demonstrated that air turbulence substantially accelerates the appearance of large droplets leading to earlier trigger of rainfall. They introduced a sling mechanism which essentially produces jets of droplets detaching from air vortical structure. These jets, then, collide with 10

37 neighboring droplets and lead to higher coalescence rate, consequently causing early appearance of large droplets in a suspension. Also, in the latest overview of in-situ and laboratory measurements, Siebert et al. [2010] showed that cloud droplets could gain accelerations higher than that of gravity leading to highly non-local droplet-droplet and droplet-turbulence interactions. Below, we summarize the effects of air turbulence on cloud droplets and related topics in a more categorized manner: 1. Local turbulent shear: Ever-changing turbulent viscous drag acting on droplets will produce a relative fluctuating motion for droplets. The initial steps toward explaining the turbulence effects on droplets were taken by Arenberg [1939] followed by other researchers [Gabilly, 1949; East and Marshall, 1954]. In a pioneering work, Saffman and Turner [1956] (referred to as ST56 hereafter) developed a rigorous theoretical formulation to quantify the effect of turbulent shear on relative fluctuating motion of zero-inertia and weak-inertia droplets. High shear and acceleration magnitude in turbulence leads to higher relative velocity of droplets compared to gravitational sedimentation. Qualitatively speaking, turbulent shear and air velocity fluctuation will cause relative velocity fluctuation for two near-by droplets which will increase their collision-coalescence probability. Later, ST56 theory was extended for weak-inertia particles [Hu and Mei, 1997; Wang et al., 1998b; Dodin and Elperin, 2002]. In chapter 4, we measure the contribution from turbulent shear to relative terminal velocity of the particles. Our results show that that the turbulence indeed increases the relative velocity and consequently causes an increase in the collision rate of droplets. 2. Preferential sweeping: An important feature of droplet-vortex interaction is that, in a gravitational field, droplets prefer to fall in the regions with downward fluid motion. In other words, during an interaction with a spinning vortex, droplets tend to avoid the side of the vortex which has upward flow motion and prefer the downward-flow side of a vortex. This will give a curvy trajectory for particles and will lead droplets fall at speeds significantly higher than their theoretical terminal velocity. This phenomenon is called preferential sweeping (or preferential sedimentation) and is studied by [Maxey, 1987; Wang and Maxey, 1993; Davila and Hunt, 2001]. It should be noted that preferential sweeping could yield a higher relative velocity between droplets as well, which consequently will increase the collision rate of droplets. 3. Droplet clustering: Inertial clustering of cloud droplets due to turbulent fluid is probably the most significant and visually observable effect of turbulence on droplets. Sundaram and Collins [1997] studied the clustering of non-sedimenting droplets and observed that finite-size droplets tend to accumulate (cluster) in 11

38 low vorticity regions of the flow. The vortical structures act as centrifuges forcing the droplets out of the vortex core, therefore droplets end up in low vorticity regions outside the vortices. In earlier studies Wang and Maxey [1993] and Squires and Eaton [1991] have also observed that heavy particles tend to collect on the peripheries of vortices. Since local collision rate of droplets scales with the square of local number density, clustering will lead to a higher average collision rate for droplets. This effect is labeled as the preferential concentration of particles and has been described theoretically through an enhancement factor (radial distribution function g 12 ) introduced by Sundaram and Collins [1997]. There is also some limited experimental evidence of cloud droplet clustering obtained by in-situ field measurements. Using a Fast Forward-Scattering Spectrometer Probe (FFSSP), Baker [1992] found that droplet spatial distribution might significantly deviate from the Poisson distribution for random fluctuations. In another experiment, Uhlig et al. [1998] found deviations from Poisson statistics using the holographic droplet and aerosol recording system (HODAR). Furthermore, using an ACTOS (Airborn Cloud Observation System) instrumented balloon, Lehmann et al. [2007] measured droplet number density in cumulus clouds and observed that droplet clustering could take place even in weakly turbulent clouds. 4. The sling effect: A strong vortex in turbulent air can shoot the droplets out of curved streamlines of turbulent flow and produce a jet of droplets. This concept is known as the sling effect and is shown to significantly accelerate the droplet growth rate [Falkovich et al., 2002; Falkovich and Pumir, 2007]. Droplet inertia plays an important role in this process. Later, Ducasse and Pumir [2009] quantified this phenomenon and reported that the sling effect can be explicitly responsible for 50% of the total collision rate. However, we should note that this result is obtained using a synthetic model of turbulent flow which is a superposition of finite number of Fourier modes. Therefore the flow might not fully represent the Navier-Stokes system and consequently the reported result might not be accurate. 5. Turbulence-gravity interplay: As Woittiez et al. [2009] noted, interplay between gravity and turbulence has a significant effect on collision-coalescence of droplets and the simple superposition of the effects of gravity alone and turbulence alone will not yield a realistic behavior. In their DNS study, Woittiez et al. [2009] have found that for monodisperse droplets gravity will decrease the collision rate compared to nonsedimenting case because gravity decreases droplet-turbulence interaction time. However, for a bidisperse distribution, gravity increases the collision rate because droplets will have a higher relative sedimentation velocity in gravitational field. In chapter 4, we will show that there is a strong coupling between gravity and turbulence. Our DNS results show that that the turbulence-gravity coupling 12

39 can increase particle acceleration variance both in the horizontal and vertical directions. This increase leads to higher probabilities of caustics (i.e., the sling effect) and consequently a faster growth rate for droplets. 6. Local pressure: Another parameter which affects the coalescence rate is pressure variation at different layers of clouds or at different locations along droplet trajectory. Beard et al. [2001] carried out experiments at lower atmospheric pressures of 745mb and 545mb which are the typical air pressure for medium and high altitude clouds. Their findings clearly show that reduced ambient pressure gives rise to higher coalescence events. 7. High acceleration magnitude: In the context of droplet-turbulence interaction, most of the studies assume gravity and the viscous drag are the dominant forces governing droplet equation of motion. Also the fluid acceleration magnitude is usually assumed to be much less than gravitational acceleration. However, recently there are numerical and experimental evidences [Voth et al., 2002; Shaw, 2003; Toschi and Bodenschatz, 2009; Volk et al., 2008] showing that the local fluid and droplet acceleration might be significant (up to 2 or 3 orders of magnitude higher than the gravitational acceleration). In our simulations, we include inertia term in addition to drag and gravity. Therefore, the inertia effect together with drag force captures the effect of turbulent fluid acceleration. 8. Effect of turbulence on hydrodynamic interaction: When two droplets approach each other, in close enough distances, droplet-droplet hydrodynamic interaction causes repulsive force which prevents droplets from colliding. Even without turbulence, hydrodynamic interaction will reduce the collision rate of droplets thereby reducing the growth rate. Fluid turbulence will weaken the effect of hydrodynamic interaction forces on droplets by increasing the relative motion of droplets. So far, only a few studies have been carried out to quantify the simultaneous effects of turbulence and hydrodynamic interaction on droplet dynamics [Pinsky et al., 1999; Wang et al., 2005b, 2008]. Table 1.1 summarizes the various growth mechanisms for different size range of droplets in cloud physics. 1.4 Motivation and Objectives Reliable weather and climate prediction at both local and global scales depends on our understanding of both microphysical processes (i.e., nuclei activation, condensational growth, collision-coalescence growth, drop breakup, etc.) and the small-scale cloud dynamics (i.e., entrainment, mixing, multiscale turbulent transport, and thermodynamics). As a basic component of climate, clouds are a turbulent suspension 13

40 Table 1.1: Droplet size range and the relevant growth mechanisms Droplet radii Dominant growth mechanism(s) Consistency (µm) with observation CCN activation + Condensational growth + Humidity + Partially Turbulence + Super-terminal velocity + Giant aerosols Entrainment & Mixing + Electric charge Turbulence: includes local shear, preferential sedimentation, Partially clustering, sling effect, high acceleration + weakening HDI interactions + Local pressure variation + Gravity-turbulence interplay + Collisional coalescence 60 O(1000) Gravitational sedimentation + Collisional coalescence + Yes Turbulence of drops and ice particles that exhibits strong aerodynamic and thermodynamic interphase couplings. The interactions between the dynamics and microphysics within clouds are complex multiscale, multiphase processes [Pruppacher and Klett, 1997; Shaw, 2003; Bodenschatz et al., 2010]. In this regards, mathematical and numerical tools are constantly being developed to improve the accuracy and reliability of weather models. In warm (ice-free) clouds, turbulent collision-coalescence of cloud droplets affects the droplet growth rate and size distribution and is a necessary step for precipitation formation [Wang and Grabowski, 2009]. Motivated by the effect of turbulence on droplet growth, in this research we shall quantify turbulent collision-coalescence of cloud droplets via numerical methods and mathematical models. In recent years, direct numerical simulation (DNS) of small-scale air turbulence has been applied to study turbulent collision-coalescence of cloud droplets [Franklin et al., 2007; Ayala et al., 2008a; Wang et al., 2008], and also the growth of cloud droplets by diffusion of water vapor [Vaillancourt et al., 2002; Lanotte et al., 2009]. In DNS of turbulent collision-coalescence of cloud droplets, the first step is to resolve turbulent air 14

41 motion at dissipation-range scales (mm to cm scales) and a limited range of inertialsubrange. This is carried out by solving incompressible Navier-Stokes equations through a pseudo-spectral or other highly accurate numerical methods to obtain a velocity field over a cubic domain periodic in 3D. In the next step, droplets are added to the domain and tracked by solving their equation of motion. Droplets can interact either with the carrier-phase turbulent fluid or with other droplets. It should be mentioned that there are a large number of variables involved (turbulence intensity, gravity, droplet size, dissipation rate, etc.). Also, a wide range of length and time scales co-exist in the problem. For instance, length scales being from Kolmogorov length on the order of mm to cloud scale on the order of km render the problem highly multi-scale. In the last few years, our group led by Professor Wang has made a significant contribution to this area of cloud micro physics [Wang et al., 2005a; Wang et al., 2005b, 2006, 2007, 2008, 2009; Wang and Grabowski, 2009]. Collision statistics (collision kernel, radial relative velocity, droplet preferential concentration and collision efficiency) have been studied rigorously through theories and numerical experiments. The aim of my research is to extend the current work and results to larger problem size and also to incorporate new elements of physics. The current limitation of DNS to low-flow Reynolds numbers (Taylor microscale Reynolds number R λ < 1000), relative to atmospheric turbulence, makes it impossible to directly address the effect of larger-scale turbulent eddies on droplet-scale processes. Recent measurements indicate that R λ could vary from 5000 in stratocumulus clouds to in cumulus clouds [Siebert et al., 2006, 2010]. Since the turbulent collisioncoalescence of droplets is governed mostly by small-scale turbulence, it is not necessary to match the DNS flow Reynolds numbers to those in real cloud. Still, current DNS do not cover all relevant flow scales. Therefore, we must increase R λ in numerical simulations which in turn requires highly scalable parallel computations. The aim of this dissertation is to develop an efficient parallel MPI (Message Passing Interface) solver which paves the way toward a higher level of parallelization and a more complete description of relevant 15

42 physical scales in DNS. To summarize, we focus on understanding and quantifying the effects of turbulence on the collision rate of cloud droplets using rigorous numerical and theoretical approaches. This is carried out using direct numerical simulations (DNS) of the turbulent flow field and incorporating the motion and collisional interactions of droplets. Also, particle tracking and collision detection is treated using domain decomposition (to make parallel implementation feasible). Following Ayala et al. [2007], hydrodynamic interactions of droplets in a turbulent flow will be implemented through 2D domain decomposition enabling us to extend their work to a larger computational domain size, a wider range of turbulence parameters and a larger number of droplets. Furthermore, the evolution of droplet size spectrum is studied directly in a one-step simulation approach combining the collision-coalescence and growth of droplets. The remaining chapters of the dissertation are arranged as follows. Chapter 2 provides the details of problem formulation and simulation method. In chapter 3, parallel implementation and scalability of our numerical method is discussed. In chapter 4, the effect of gravity on the acceleration and pair statistics of inertial particles is studied. Using a one-step approach, in chapter 5, we investigate the droplet growth under different air turbulence characteristics. Finally chapter 6 summarizes the main conclusions and future work. 16

43 Chapter 2 PROBLEM FORMULATION AND SIMULATION METHOD 2.1 Droplet and Turbulence Properties in Clouds In this section we describe the general characteristics of small water droplets (radii 10 to 60µm) in turbulent air. Regarding air turbulence in clouds, we begin with the following important observations: Since the local volume fraction of droplets (dispersed phase) in atmospheric clouds is typically on the order of 10 6 (or mass loading on the order of 10 3 ), it is usually assumed that air turbulence (continuum phase) will not be affected by the presence of droplets. This is the main reason we will refer to cloud turbulence as the background air turbulence. It has been argued that at core regions of a cumulus cloud, air turbulence can be considered homogeneous and isotropic. Vaillancourt and Yau [2000] mentioned two reasons to support this argument. First, vertical stratification of air can be neglected on scales smaller than 100 m (typical integral length scale of atmospheric turbulence). This will allow the behavior of air turbulence in vertical direction be similar to the other two horizontal directions. Second, at distances sufficiently far from mixing regions of clouds, the small-scale flow is primarily driven by nonlinear energy transfer from larger scales. It should be noted that in the cloud core, in the absence of other energy transfer mechanisms, the only available source of energy to sustain the small scale turbulence is the interaction with large scale flow. The small scale (dissipation-range) motions of turbulence governs the collision rate of cloud droplets [Shaw, 2003]. The small scale features of turbulence are characterized by Kolmogorov length, velocity and time scales which are defined in Table 2.2. The Kolmogorov scales are solely determined by ǫ (average dissipation rate in turbulent flow) and ν (kinematic viscosity of air). Radar observations [Jonas, 1996; Vaillancourt and Yau, 2000] and in-situ measurements [Siebert et al., 2006] suggest that ǫ could vary from 1 to 1000 cm 2 /s 3 in cumulus and stratocumulus clouds. In this research, 17

44 we choose three intermediate values of dissipation rate which are listed in Table 2.1 along with the Kolmogorov scales of the flow (air viscosity ν = 0.17 cm 2 /s). Besides the Kolmogorov scales, Table 2.2 summarizes other important metrics of a turbulent flow. Here we emphasize the significance of the less clear metrics in few words. λ is the transverse Taylor microscale marking the length scale of the velocity gradients of the flow. Furthermore, in our study we will consider a turbulent flow with zero mean velocity and we define u as the r.m.s. fluctuation velocity of the turbulence. Also the eddy turn over time T e defines the spin-time of an eddy and L f is the integral length scale above which the two-point velocity correlation decays to zero. In a homogeneous isotropic turbulence the relevant Reynolds number is Taylormicroscale Reynolds number R λ defined based on the length scale λ. Direct numerical simulations of turbulence are limited to a narrow range of R λ and our numerical method (DNS) is not an exception. The highest values of R λ achievable in DNS is typically on the order of 10 2 which is about two orders of magnitude smaller than the R λ in real clouds [Shaw, 2003]. Since the dynamics of cloud droplets is governed mainly by the dissipation-range fluid motion, it is assumed that dissipation rate ǫ plays the main role in turbulent collision-coalescence and that R λ is of secondary importance. From Table 2.1, we note that the cloud droplets are one to two orders of magnitude smaller than the Kolmogorov eddies. Therefore we assume that the cloud droplets move under the effect of local fluid motion (i.e. drag force), gravitational body force and droplet inertia. As long as droplets are far away from each other, their motion will be Table 2.1: Kolmogorov scales of a typical turbulent cloud ǫ τ k η υ k (cm 2 /s 3 ) (s) (cm) (cm/s)

45 Table 2.2: Turbulence characteristics of the flow Metric Definition Significance Kolmogorov length scale η = Kolmogorov time scale τ k = Kolmogorov velocity scale v k = ( ν 3 /ǫ) 1/4 size of the smallest vortices in turbulence ( ν/ǫ) 1/2 time scale of a vortex of size η ( νǫ) 1/4 typical velocity of fluid element in a vortex of size η Taylor microscale length λ (15νu 2 /ǫ) 1/2 inertial-range length scale Turbulent Reynolds number R λ = ( ) 15 u 2 v k turbulence intensity Turbulent fluctuation r.m.s u <U U> 3 mean turbulent fluctuation velocity Large-eddy turnover time T e u 2 /ǫ typical spin time for a large eddy Integral length scale L f u T e large-scale flow decorrelation length independent of other droplets. However, if droplets approach each other in distances comparable to their size, the disturbance flow generated by a droplet will affect the motion of neighboring droplets. The viscous forces exerted by the local disturbance flows around a droplet will push other droplets away, modifying their trajectories. This is known as droplet-droplet local hydrodynamic interactions which will be discussed in For heavy droplets in a gravitational field g, under the assumption of small Re p, the velocity V(t) of a droplet is governed by: dv(t) dt = U (Y(t), t) V(t) τ p + g, (2.1) 19

46 where U (Y(t), t) is the fluid velocity, seen by droplet [Maxey and Riley, 1983]. The Stokes inertial response time of the droplet, τ p, is defined by: τ p 2ρ wa 2 9ρν. (2.2) where a is droplet radius, ρ w is droplet density and ν and ρ are fluid kinematic viscosity and density respectively. Particle response time τ p is the characteristic time for the particle to react to changes in the local fluid acceleration. It should be added that the full equation of motion of droplets has several additional terms [Maxey and Riley, 1983], but since density of water droplets (ρ w ) is much larger than air density ρ, it could be simplified to (2.1). The importance of particle Reynolds number in validity of Eqn. (2.1) is discussed in Clift et al. [1978]. If particle Reynolds number is large, then vortex shedding and wake effects will cause extensive disturbances in the flow yielding to modification of the background turbulent flow field. On the other hand, small Re p will guarantee that local vortex shedding and wake will not distort the background turbulence. In our study, droplet Reynolds number Re p is defined as: Re p = 2av p ν (2.3) where v p = τ p g is the terminal velocity of the droplet. The numerical values of Re p are listed in Table 2.3. When droplet size is close to a = 40µm or larger, Re p becomes on the order of one which requires Eqn. (2.1) be modified (due to the nonlinear drag). However, here we use the linear Stokes drag, since in earlier studies of [Ayala et al., 2008a; Wang et al., 2008] the effects of the nonlinear drag have already been considered. A quantitative scaling of droplet characteristics is presented in Table 2.3 in comparison with characteristic scales of dissipation-range air turbulence. In the interplay 20

47 Table 2.3: Characteristics of cloud droplets (ν = 0.17cm 2 /s, ρ = g/cm 3, ρ w = 1.0 g/cm 3 ) a ǫ =10 cm 2 /s 3 ǫ =100 cm 2 /s 3 ǫ =400 cm 2 /s 3 (µm) τ p (s) v p ( cm s ) Re p F p,k St Sv a/η St Sv a/η St Sv a/η between gravity, droplet inertia and vortex structures, three parameters have a significant role on the dynamics of droplets: Stokes number St, dimensionless sedimentation number Sv and droplet Froude number F P. These three dimensionless numbers are defined by: St = τ p τ k, Sv = v p v k, F p = τ 2 pv p (2.4) Γ vort where Γ vort is the circulation of the vortices interacting with a droplet. Essentially, F p is a derived parameter and is dependent on St and Sv. This dependence is discussed in the following paragraphs. It should be highlighted that a range of time and length scales co-exist in a turbulent flow. Therefore, quite a few dimensionless parameters could be defined. However, Kolmogorov scales of the turbulence are the most relevant reference scales in our study. Here we summarize the importance and significance of each dimensionless parameter in the context of cloud micro-physics: Stokes number St is a key parameter determining the nature of droplet response to the fluid motion. If a droplet reacts instantaneously to the local fluid acceleration (small Stokes number), then it behaves as a tracer which will always follow the fluid streamlines. On the other hand, if the particle response is slow (large Stokes numbers), then the droplet will not feel the fluid acceleration thereby following a completely different trajectory compared to the local fluid element. A strong interaction between the flow and the particles in a turbulent flow occurs when the Stokes number is on the order of one [Wang and Maxey, 1993]. 21

48 Dimensionless sedimentation number Sv plays an important role in determining the relative importance of turbulence and gravity [Squires and Eaton, 1991; Wang and Maxey, 1993; Davila and Hunt, 2001]. If Sv is larger than one, it indicates that gravity dominates the droplet motion since the particle sedimentation through an eddy will be so fast that droplet will not have enough time to interact with that eddy. On the other hand, small Sv indicates that sedimentation does not play a significant role in droplet dynamics. Particle Froude number F p has a little more subtle interpretation. Since in a turbulent cloud, droplets are mostly interacting with small eddies, let us consider F p for Kolmogorov scale eddies for which Γ vort v k η = ν yielding F p,k = τp3 g 2. ν Davila and Hunt [2001] demonstrated that when F p,k is on the order of unity (for instance for 20µm droplets in Table 2.3), droplets are significantly affected by turbulence yielding the maximum increase in sedimentation velocity. This was confirmed in direct numerical simulation of Ayala et al. [2008a]. Particle Froude number F p is the ratio of droplet response time to droplet-eddy interaction time. Therefore, the highest increase in sedimentation is observed when droplet response time is on the order of droplet-eddy interaction time. It is worth mentioning that F p,k is not an independent dimensionless variable since F p,k = St Sv 2. However in general F p could be independent of St and Sv. Finally, for water droplets in a turbulent cloud, from Table 2.3 we observe that the Stokes number St is typically less than one while the nondimensional settling number Sv is typically larger than one. Therefore, consistent with several other studies [Vaillancourt and Yau, 2000; Woittiez et al., 2009], we conclude that the gravitational effect is always important for cloud droplets. In other words, to yield a physically meaningful results, gravity cannot be neglected. Particle Froude number F p,k changes by several orders of magnitude but, independent of flow dissipation rate, marks the radius of 20µm as the point where small-scale eddies of the flow have the maximum interactions with droplets. More importantly, in the range of droplet size and dissipation rate we have considered, a/η takes the maximum value of 0.1 which confirms that the dynamics of droplets are mainly determined by Kolmogorov scale dynamics of fluid turbulence. However, for larger size droplets or for vigorously turbulent flows, inertial-subrange physics of turbulence could affect cloud droplet dynamics. The range of St, Sv and F p,k magnitudes in Table 2.3 indicates that the relative importance of droplet inertia, turbulence and gravity varies significantly in cloud physics. 22

49 For ǫ < 10 cm 2 /s 3, turbulence effect is weak and gravity dominates the motion of droplets. On the other hand for ǫ > 400 cm 2 /s 3, domination of gravity will start to fade and turbulent inertia effect will play the most important role. For intermediate values of ǫ, in which both effects of turbulent particle inertia and gravity are important, the relative importance of the competing forces is not fully understood. 2.2 Methodology To quantify the turbulent collision-coalescence of cloud droplets, we simulate the flow and droplet motion in a periodic cube which is representative of the core of a cloud in which air parcel is considered to be adiabatic (or quasi-adiabatic). The adiabatic assumption eliminates heat transfer from the study allowing us to focus on momentum and mass transfer of the problem. We mention in passing that processes happening at the edges of a cloud could not be considered adiabatic mainly due to a considerable entrainment and also external radiative heat transfer. To provide a visual picture of our problem of interest, Figure 2.1 shows an snapshot of air vorticity isosurfaces (on the left) on which the droplets (right panel) should be overlaid. In other words, our approach (and consequently our numerical scheme) could be divided into two layers. The first layer is the the simulation of background isotropic homogeneous turbulence which is governed by incompressible Navier-Stokes equations. The second layer concerns the influence of the flow on the droplets and also droplet tracking. These will be elaborated in the next few sections Background Air Turbulence The first step in study of turbulent collision-coalescence of droplets is to numerically solve time dependent Navier-Stokes system: U t = U ω ( P ρ U2) + ν 2 U + f(x, t), (2.5) for an incompressible flow with continuity equation as: 23

50 Figure 2.1: Our problem consists of simulation of background air turbulence (vorticity isosurface on the left panel) and droplet tracking (right panel) in a 3-D periodic cube. These two figures should be overlaid to give a realistic view of turbulent collision-coalescence, but here they are separated for demonstration of the solution method. Droplet sizes and number density are not on scale. U(x, t) = 0 (2.6) Here ω U is the vorticity vector, P is the pressure, ρ is fluid density (air density in our simulation) and ν is fluid kinematic viscosity (ν of air in our simulation). Thereby, we assume a homogeneous isotropic turbulence in a cubic and 3D-periodic domain, and solve Eqns.(2.5) and (2.6) for air velocity filed U(x, t). It should be noted that (2.5) is derived from the more familiar form of Navier-Stokes equations: by applying the vector identity: U t + ( U ) U = P ρ + ν 2 U + f(x, t) (2.7) ( U 2 ) ( ) = U U + U ω. (2.8) 2 24