Toward Building a Virtual Laboratory to Investigate Rainfall Microphysics at Process Scales

Lihui Ji aDepartment of Civil and Environmental Engineering, University of Illinois Urbana–Champaign, Urbana, Illinois

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Ana P. Barros aDepartment of Civil and Environmental Engineering, University of Illinois Urbana–Champaign, Urbana, Illinois

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Abstract

A 3D numerical model was built to serve as a virtual microphysics laboratory (VML) to investigate rainfall microphysical processes. One key goal for the VML is to elucidate the physical basis of warm precipitation processes toward improving existing parameterizations beyond the constraints of past physical experiments. This manuscript presents results from VML simulations of classical tower experiments of raindrop collisional collection and breakup. The simulations capture large raindrop oscillations in shape and velocity in both horizontal and vertical planes and reveal that drop instability increases with diameter due to the weakening of the surface tension compared with the body force. A detailed evaluation against reference experimental datasets of binary collisions over a wide range of drop sizes shows that the VML reproduces collision outcomes well including coalescence, and disk, sheet, and filament breakups. Furthermore, the VML simulations captured spontaneous breakup, and secondary coalescence and breakup. The breakup type, fragment number, and size distribution are analyzed in the context of collision kinetic energy, diameter ratio, and relative position, with a view to capture the dynamic evolution of the vertical microstructure of rainfall in models and to interpret remote sensing measurements.

Significance Statement

Presently, uncertainty in precipitation estimation and prediction remains one of the grand challenges in water cycle studies. This work presents a detailed 3D simulator to characterize the evolution of drop size distributions (DSDs), without the space and functional constraints of laboratory experiments. The virtual microphysics laboratory (VML) is applied to replicate classical tower experiments from which parameterizations of precipitation processes used presently in weather and climate models and remote sensing algorithms were derived. The results presented demonstrate that the VML is a robust tool to capture DSD dynamics at the scale of individual raindrops (precipitation microphysics). VML will be used to characterize DSD dynamics across scales for environmental conditions and weather regimes for which no measurements are available.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ana P. Barros, barros@illinois.edu

Abstract

A 3D numerical model was built to serve as a virtual microphysics laboratory (VML) to investigate rainfall microphysical processes. One key goal for the VML is to elucidate the physical basis of warm precipitation processes toward improving existing parameterizations beyond the constraints of past physical experiments. This manuscript presents results from VML simulations of classical tower experiments of raindrop collisional collection and breakup. The simulations capture large raindrop oscillations in shape and velocity in both horizontal and vertical planes and reveal that drop instability increases with diameter due to the weakening of the surface tension compared with the body force. A detailed evaluation against reference experimental datasets of binary collisions over a wide range of drop sizes shows that the VML reproduces collision outcomes well including coalescence, and disk, sheet, and filament breakups. Furthermore, the VML simulations captured spontaneous breakup, and secondary coalescence and breakup. The breakup type, fragment number, and size distribution are analyzed in the context of collision kinetic energy, diameter ratio, and relative position, with a view to capture the dynamic evolution of the vertical microstructure of rainfall in models and to interpret remote sensing measurements.

Significance Statement

Presently, uncertainty in precipitation estimation and prediction remains one of the grand challenges in water cycle studies. This work presents a detailed 3D simulator to characterize the evolution of drop size distributions (DSDs), without the space and functional constraints of laboratory experiments. The virtual microphysics laboratory (VML) is applied to replicate classical tower experiments from which parameterizations of precipitation processes used presently in weather and climate models and remote sensing algorithms were derived. The results presented demonstrate that the VML is a robust tool to capture DSD dynamics at the scale of individual raindrops (precipitation microphysics). VML will be used to characterize DSD dynamics across scales for environmental conditions and weather regimes for which no measurements are available.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ana P. Barros, barros@illinois.edu

1. Introduction

Quantifying the evolution of hydrometeor populations from aerosol activation during cloud formation until raindrops hit the ground has been a long-standing challenge. In warm rain, large cloud droplets can grow by collecting small droplets (collision–coalescence), which fall out when their fall speed exceeds the cloud updraft. Each droplet has the probability of colliding with another droplet, at which point they either coalesce into a single large drop or break up into multiple small drops (Testik and Barros 2007). In both Eulerian and Lagrangian drop size distribution (DSD) models, the collision is represented by a stochastic kernel (e.g., Low and List 1982a,b; Hu and Srivastava 1995; Seifert et al. 2005; Prat and Barros 2007; Duan et al. 2019; Unterstrasser et al. 2020), and the terminal velocity that is directly related to drop size is essential to predict collision probability and outcomes. Collision probability increases with the difference in fall speed between the two colliding raindrops, which reduces the time scale for droplets to catch up and interact. The probability of collisional breakup increases with the difference in terminal velocity due to stronger collision kinetic energy (CKE).

A variety of experiments to measure and parameterize raindrop fall velocities in quiescent air and at steady state (terminal velocity) have been conducted since the 1940s (Serio et al. 2019). Gunn and Kinzer (1949) measured raindrop terminal velocities for over 1500 droplets, with the diameter ranging from 0.1 to 5.8 mm. Guzel and Barros (2001) measured fall velocity by acoustic emission testing (AET), finding that the measured velocity depends on falling height up to 13 m in their experiments, beyond which the drops reached terminal velocity. Ong et al. (2021) simulated fall velocities of droplets ranging from 0.025 to 0.5 mm, which fills an important gap for small raindrops in the literature.

Quantifying realistic raindrop terminal velocities remains challenging due to the effects of turbulence and shape oscillations after individual collisions (e.g., Saha and Testik 2023). For example, Ren et al. (2020) found that air turbulence could decrease the effective terminal velocity of droplets by shortening the wake recirculation region. Testik et al. (2006) visualized raindrops falling from a 14-m rain tower with high-speed imaging and reported a random lateral drift caused by periodic shape oscillations. They also measured a 10% smaller terminal velocity for a 1.9-mm drop compared with previous experiments, explained by additional drag force induced by shape oscillations. The shape oscillations primarily originate from the asymmetric vortex in the wake region, which could be formulated by spherical harmonic perturbations (Lamb 1932; Rayleigh 1879). With drop size increase, the amplitude of the oscillations increases, and the transverse oscillation mode transitions to a mixture of transverse and axisymmetric modes (Beard and Kubesh 1991), thus adding uncertainty to measurements of terminal velocity.

Predicting collision outcomes is another challenge to produce the correct DSD in simulations (List and Gillespie 1976; Saleeby et al. 2022; Prat et al. 2008, 2012; Wilson and Barros 2014; D’Adderio et al. 2015). Magarvey and Geldart (1962) were among the first to visualize coalescence and breakup with two cameras vertically separated 1 m apart. McTaggart-Cowan and List (1975) classified breakups in their experiments into three major breakup types: filament, also known as neck (27%), sheet (55%), and disk (18%) breakups. Building on this work, Low and List (1982a,b, hereafter LL82) proposed parameterizations of the fragment size distribution of collision breakups that are still used in precipitation models. Barros et al. (2008, hereafter B08) replicated LL82 and analyzed collision breakups by high-speed collision images as shown in Fig. 1. Their data were analyzed by Testik et al. (2011) in the We–p framework proposed by Testik (2009), where We is the Weber number [We = ρl(d2/2)(ΔV)2/σ] and p is the diameter ratio (p = d2/d1), ρl is the density, σ is the surface tension, ΔV is the velocity difference between two drops, and d is the drop diameter with subscripts 1 and 2 denoting the larger and smaller drops, respectively. The collision outcome is dominated by coalescence at small We and p, shifting toward sheet type and then disk type as We and p increase. They further parameterized the separation between coalescence and breakup, and the separation between bounce and coalescence, expressed by {S1=p2We/[6(1+p3)]=1} and {S2=We/[6(1+p3)]}, respectively. The frequency of different collision mechanisms in B08 was very different from LL82 with filament breakup being the most frequent mechanism instead of sheet breakup as in LL82, because the size of small drops was much smaller in B08, demonstrating the importance of the We–p framework for predicting the type of collisional breakup mechanism. Nevertheless, because of the permanent information lost by projecting the 3D scene on a 2D camera frame, the bias persists in measuring velocities, diameters, and shapes, classifying breakup types, and counting droplet numbers.

Fig. 1.
Fig. 1.

High-speed image sequence of breakup types: (a) filament (neck), (b) sheet, and (c) disk. There are five frames in each case with a time interval of 2 ms. The figure is adapted from B08.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

Explicit numerical simulation of raindrops is a robust alternative to tower experiments. Besides the potential to reproduce 3D structures, the computational approach can be adapted easily to new environments by simple modification of boundary conditions. Premnath and Abraham (2005) simulated binary collisions with a multi-relaxation-time multiphase flow lattice Boltzmann model and discussed collision outcomes in the context of the size-averaged Weber number and the Ohnesorge number. Beheng et al. (2006) and Schlottke et al. (2010) predicted collisions of a variety of drop pairs using direct numerical simulation (DNS) and the volume of fluid (VOF) method to capture liquid–air interfaces. Their results were analyzed by Straub et al. (2010) to estimate coalescence efficiency and breakup fragment size distributions.

In this work, we simulate drop dynamics with similar algorithms (DNS and VOF), but with a focus on raindrop dynamics such as velocity oscillation and shape distortion to reveal the uncertainties in free-falling and collision processes. For this purpose, a new solver was developed on OpenFOAM (The OpenFOAM Foundation 2021) to simulate collisional processes. OpenFOAM is an open-source computational fluid dynamics (CFD) platform with many code libraries. The “interFoam” solver in OpenFOAM is able to solve multiphase fluid dynamics using the VOF method applied to the Navier–Stokes equation for momentum transport of the mixture of water and air phases (Damian 2012). In VOF, different immiscible fluids are treated as one fluid, with physical properties determined as the volume-weighted average of fluid components [Eqs. (105)–(107) in Damian 2012]. The new solver “raindropFoam” presented here was developed by adding mesh motion functionality to interFoam, thus facilitating 3D simulation of raindrop free-falling and collisions. The raindropFoam is the computational engine behind the virtual microphysics laboratory (VML), a 3D simulator of raindrop processes.

The paper is organized as follows. Section 2 introduces the new solver raindropFoam and its computational setting, including domain, mesh, physical properties, and initial and boundary conditions for the experiments presented here; a hierarchical clustering algorithm to calculate fragment number and size is introduced and discussed. In section 3, the speed and velocity evolution over time of various drop sizes in free-falling experiments are examined. In addition, the five major collision outcomes in the VML are visualized: coalescence, break (disk, sheet, and filament), and slip (a newly defined breakup category). The outcome type, fragment number, and drop sizes are analyzed statistically in the context of collision kinetic energy, diameter ratio, and relative position in section 4. Discussion and conclusions are presented in sections 5 and 6, respectively. In particular, the occurrence of secondary coalescence and breakup is discussed along with its impact introducing uncertainty in the outcome analysis. The effects of diameter and relative position on collision outcomes are discussed, and a new parameterization framework is proposed to incorporate distributions of the relative spatial position of colliding pairs in natural rainfall.

2. Methodology

a. Lagrangian tracking algorithm

In both free-falling and collision experiments, considerable fall distance is required for raindrops to reach a pseudo–steady state under gravity and drag forces. Guzel and Barros (2001) measured raindrops falling from heights (∼1–10 m) three orders of magnitude larger than the scale of raindrop diameters (∼mm). This large-scale gap poses challenges for numerical models to resolve collision processes explicitly. Because the droplets are moving under the pull of gravity and only near-drop flow fields can affect their trajectory, a Lagrangian formulation is suitable so that the near-field resolving grid can move with the drop, which has the benefit of reducing the computational overhead. Using the dynamic mesh functionality (Jasak 2009) in OpenFOAM, a new solver raindropFoam was developed to track falling raindrops in a Lagrangian framework. At each time step, the position of the droplet center was calculated by averaging all 3D coordinates of water points and passed to the mesh motion solver to update the grid coordinates. The volume fraction of each fluid component is solved by the transport equation with interface compression to redistribute mass from low water fraction cells to high water fraction cells to reduce the thickness of the mixed layer [Eq. (108) in Damian 2012]. By implementing raindropFoam, the length of the sliding grid domain was reduced to a centimeter scale. Specifically, in the simulation of a 1-mm drop, the domain height was only 4 cm, even if the distance required to reach terminal velocity is 2.5 m. In the free-falling experiments, the mesh moved in three directions such that the droplet was permanently fixed in the center of the mesh domain. For collision experiments, the mesh only moved vertically to minimize horizontal numerical oscillations.

b. Computational setup

For the free-falling experiments, raindrops of diameter 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0 mm were simulated, and the initial velocities of drop and air were set to be 0 m s−1. For the collision experiments, drop pairs were selected to be the combinations of the following: 0.4, 0.8, 1.2, 1.6, 2.0, 2.4, 2.8, 3.2, 3.6, and 4.0 mm to yield 45 pairs total. The drop diameter is linearly spaced because of its linear relationship with the Weber number, which measures the relative strength of the collision force that governs breakup against the surface tension force that governs coalescence processes. The initial velocity of air was set to be 0 m s−1, while the initial velocity of drops was set as terminal velocities given by the empirical formula parameterized from Gunn and Kinzer (1949) by Lhermitte (1990), as follows in Eq. (1):
Ut=923[1(ρa,0ρa,z)e(6.8d24.88d)],
where d is the diameter (cm), Ut is the terminal velocity (cm s−1), and ρa,z and ρa,0 are the z-altitude and ground air density (Lhermitte 1990). The top drop had a larger diameter such that it could catch up and collide with the lower drop.

The domain size depends on the initial water-phase dimension D, which is the minimum diameter of a sphere that can cover all droplets under translation and rotation. The term D = d for a single-diameter d drop and D = d1 + d2 for a two-drop system. The raindrop free-falling experiments were initially carried out with domain length L, width W, and height H corresponding to 10D, 10D, and 20D, respectively. The results exhibited abnormal numerical oscillations, especially for larger drops, which were attributed to boundary effects. To address this issue, the domain size was increased to 20D × 20D × 20D, thus larger than that used for direct numerical simulation of water droplets by Ren et al. (2020). However, the domain size remained unchanged for the collision experiments because of the shorter time scales. In addition, boundary artifacts are negligible compared with the energy burst during collision processes. A smaller domain enables a finer mesh to capture the curvature of distorted droplets.

The coordinate system was constructed with the origin at the center of the domain’s top surface, and the x, y, and z axes pointing right, front, and upward, respectively. In free-falling experiments, the droplet was placed in the center of the domain (Fig. 2a). In collision experiments, the bottom drop is on the right side of the top drop and the initial condition for the relative position is represented by a parameter shift ratio (SRR0+) defined in Eq. (2):
SR=horizontalcenterdistanceD/2=(x1x2)2+(y1y2)2D/2,
where (x1, y1) and (x2, y2) are the centers of the top and bottom droplets, respectively. Six distinct initial conditions were set with SR = 0, 0.25, 0.5, 0.75, 1, and 1.25 per drop pair, for a total of 270 collision cases with 45 drop pairs to examine the impact of the varying collision geometry on the breakup outcomes. Note that the relative position of the drops at collision (SRcol) varies from the initial SR value due to velocity oscillations and shape distortion. Because of the short time scales and travel distances, we take the initial SR condition as a representative indicator of the range of actual collision geometries SRcol for all simulations with the same initial conditions. The y coordinate for both drops was set at 0, the z coordinate was placed such that the vertical distance of drop centers was D, and the middle point between centers was aligned with the vertical axis center (Figs. 2b,c). To ensure that most of the drop system is in the highest resolution region, as shown in Fig. 3, the left and right boundaries are determined by the maximum distance on the x axis away from the origin between the largest and smaller drops.
Fig. 2.
Fig. 2.

(a) Free-falling case setting, D = d with the center of the droplet at (0, 0, −20D). (b) Collision case setting, D = d1 + d2, with the center of the droplets at z = −10D, and the left and right boundaries of the drop system are determined by the distance of the larger and smaller drops away from the origin [(x, y) = (0, 0)] on the x axis. (c) Enlarged diagram for a collision case.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

Fig. 3.
Fig. 3.

(a) Local refinement demonstration (dividing one cell into eight cells) and (b) mesh size in different regions. Level 0: initial mesh constructed on the whole domain with the cell number of 100 × 100 × 200; level 1: refine mesh of the box region 0.5L × 0.5W × 0.5H in the center of the level 0 region; level 2: refine mesh of the box region 0.2L × 0.2W × 0.2H in the center of the level 1 region; level 3: refine mesh of the box region 0.1L × 0.1W × 0.1H in the center of the level 2 region. The minimum cell size produced is (L/800) × (W/800) × (H/1600). The total cell number is 5.542 × 106.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

Collisions simulated in the VML are different from the Free Surface 3D (FS3D) simulations by Schlottke et al. (2010) and Straub et al. (2010). First, instead of setting the same diameters as in LL82, where the drop diameters are irregularly distributed due to insufficient flexibility in the laboratory setting, the drop diameters in VML are evenly spaced, resulting in an evenly distributed diameter ratio to uniformly sample the We–p space, and the VML cases include all combinations of selected drop radius, covering the full range of diameter ratios, which is essential to prevent outcome bias. Second, instead of prescribing the shape ratio, all drops are initialized as spherical in VML, and subsequently, they undergo shape distortion and velocity oscillations following physical laws, thus consistent with reality in tower experiments. Third, VML enables investigating collision cases with SR > 1, when drop shape distortion is large enough to result in contact between two drops. The maximum SR in FS3D simulations is 0.95, and it is assumed that collision at SR > 1 has minor effects on DSD evolution. However, the VML simulations reveal that collision at SR = 1.25 has nontrivial outcomes, including coalescence and spontaneous breakup (see section 5d for discussion).

To capture the drop surface curvature, the mesh was refined locally by OpenFOAM’s “refineMesh” utility, which divides each cell into eight subgrids within the selected region. Three levels of refinement were implemented, producing four levels of resolution, as illustrated in Fig. 3. Compared to the equidistant staggered grid used in FS3D (Ren et al. 2020; Schlottke et al. 2010; Straub et al. 2010; Beheng et al. 2006), the locally refined grid improves numerical accuracy without a sharp computational cost increase to reduce the cell size across the whole domain. Furthermore, the locally refined grid has the potential to simulate more complex problems such as multidrop interactions. The setting enables all water points in collision cases with SR ≤ 1 to lie in the highest resolution region.

The physical properties for standard temperature and pressure conditions (20°C and 101 325 Pa) are summarized in Table 1.

Table 1.

Summary of physical properties used in simulations.

Table 1.

The boundary condition of the water-phase fraction was “inletOutlet,” which switches between a zero gradient when the fluid flows outward and a zero value when the fluid flows inward because there was no water outside of the domain in the simulation cases. The velocity boundary condition was “pressureInletOutletVelocity,” which assigns a zero-gradient boundary for the outflow and a face-normal velocity component for the inflow. The pressure boundary condition was “totalPressure,” which calculates pressure according to Bernoulli’s equation [P=Patm(1/2)|u|2], where P, Patm, and u are boundary pressure, atmospheric pressure, and the velocity vector, respectively. The time step was initially set at as a tenth of a millisecond (0.0001 s) and adjusted at each time step to meet the Courant condition based on the maximum Courant number of unity.

Both free-falling and collision experiments were conducted on the Illinois Campus Cluster with 128 processors, determined by a series of parallelization tests.

c. Outcome analysis

Retrieving the number and size distribution of fragments produced in collision breakups is the primary objective of the simulations toward comparing against results from laboratory tower experiments. The model outputs a sequence of field values (pressure, velocity, and water fraction) at different times. In each frame, first the 3D coordinates of all water points are extracted according to a water fraction threshold of 0.8. The water points are upscaled to match the coarsest resolution of water-occupied mesh cells. Then, the points are clustered into groups, each representing a droplet. Hierarchical clustering with single linkage is effective in collision experiments because prior information (maximum mesh size) is available to set the cutoff distance (Wittek 2014). The volume of each droplet is calculated by multiplying the mesh cell volume by the number of water points. The outcome fragment number of a collision case is determined by the maximum droplet number of all frames. The appendix further describes the methodology and presents illustrative examples.

3. Results and assessment against laboratory experiments

a. Free-falling experiments

The velocity evolution of a 3-mm-diameter droplet in x, y, and z directions is shown in Figs. 4a–c as an example. The droplet accelerated until it reached pseudoequilibrium between gravity and drag forces, determined by the time when the droplet reached the maximum fall velocity. At equilibrium, the droplet did not experience gravity-driven acceleration and entered a pure oscillation stage (Fig. 4c). The results show that instead of a fixed value, the terminal velocity is characterized by a mean velocity and upper and lower deviation limits. The droplet also exhibited nonzero lateral speed oscillations (Figs. 4a,b), with a magnitude of approximately 0.2 m s−1, about 3% of the terminal velocity, and close to the 2% of terminal velocity oscillations observed by Beard and Kubesh (1991). These results indicate interactions among the dynamical effects due to the changing droplet shape and the droplet fall velocity. In the classical experiments, the shape is quantified by the axis ratio α¯=υ/h, where υ and h are the vertical and horizontal chords obtained from 2D imagery. In the 3D VML, α¯ is defined as follows:
α¯=cab=zmaxzmin(xmaxxmin)(ymaxymin),
where a, b, and c are the maximum difference in the coordinates x, y, and z of water points, the points with water fraction larger than the threshold (0.8). The 3Dα¯ is equivalent to α¯ in the 2D image taken from an azimuth angle between the x and y axes.
Fig. 4.
Fig. 4.

Simulation outputs for a 3-mm droplet. (a)–(c) Velocities in x, y, and z directions of the entire time. (d) Vertical velocity oscillation after reaching pseudoequilibrium with the mean magnitude of 6.56 m s−1 represented by the horizontal dashed line and upper and lower magnitude deviation limits of 0.49 and 0.25 m s−1, respectively. This is an enlargement of (c). (e) Drop center vertical coordinates (−20D = −0.06 m initially). (f) Drop axis ratio α_ time evolution. The vertical dashed line represents the time when the drop reached pseudoequilibrium, which is 0.77 s in this case, corresponding to a fall distance of 2.83 m.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

After time 0.75 s and before time 0.77 s (onset of the oscillation stage), the shape of the drop changes rapidly from nearly round (α¯1) to oblate because the high fall speed induces a strong drag force that deforms the drop and reduces the fall speed. The surface tension force is stronger before the transition (larger surface area) than in the oscillation stage. A larger drag force is needed to distort the round shape. Therefore, the velocity at the transition point is larger than the velocity in the oscillation stage. Shape distortion flattens the bottom of drops, which increases the drag force and in turn amplifies shape distortion. The cycle intensifies the shape change at the transition point, as shown in Fig. 4f. Note that the rapid shape change could be caused in part by numerical error due to insufficient resolution in time and space. The drop velocity–shape interactions persisted during the entire fall process and were dominant when the drop entered the pseudo-steady-state stage because the net force (gravity–drag) was close to zero.

Abnormal behavior was found in the simulation of 0.5- and 1.0-mm droplets. The vertical velocity of the 0.5-mm droplet did not reach the pseudoequilibrium stage, and the horizontal velocities of the 0.5- and 1.0-mm droplets were larger than expected. These results are artifacts explained by numerical errors tied to boundary and resolution effects. Increasing the domain size by multiples of the drop diameters may not eliminate these artifacts because the terminal velocity (Fig. 6) does not scale linearly with the raindrop diameter. Indeed, the horizontal velocities of 0.5- and 1.0-mm droplets increased in one direction instead of oscillating back and forth, causing prolate shapes and higher terminal velocity. Such oscillations can also be caused by insufficient grid resolution.

Simulations of free fall for different mesh domain depths of the 3.5-mm drop are compared in Fig. 5. After increasing domain size and mesh resolution, the unidirectional horizontal drift (up to 1.5 m s−1) changed to oscillations with only 0.2 m s−1 magnitude, resulting in a smoother vertical acceleration and smaller terminal velocity. This suggests that by further expanding the domain and increasing the resolution, the terminal velocities of 0.5- and 1.0-mm drops could decrease to approach the results from past measurements (e.g., Guzel and Barros 2001; Testik and Barros 2007).

Fig. 5.
Fig. 5.

The comparison between two different simulations for a 3.5-mm droplet. (a)–(c) Velocities in x, y, and z directions with mesh domain size 10D × 10D × 20D and uniform mesh size 200 × 200 × 400. (d)–(f) Velocities in x, y, and z directions of the current simulation with domain size 20D × 20D × 40D and locally refined mesh (Fig. 3).

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

Figure 6 shows the velocities at the pseudoequilibrium stage for drops of different diameters for comparison with classical experiments and parameterizations. The terminal velocity increases with the diameter with a decaying rate. For droplets of diameter 2.5, 3.0, 3.5, and 4.0 mm, the terminal velocities are close to measurements from a 4.65-m fall height by Guzel and Barros (2001). However, the falling distances to reach terminal velocities are 2.44, 2.85, 3.24, and 3.13 m, respectively, shorter than for measurements in classical tower experiments. Note that the experiments reported by Guzel and Barros (2001) relied on a laser system placed at a specific position to measure fall speed; thus, the experimental measurement height is not the same as the height needed to reach steady state. Compared to the empirical equation proposed by Lhermitte (1990), the VML simulation underestimates the velocities of larger drops, which can be explained by the differences between the computational and natural environments, the velocity, and the pressure of surrounding air, specifically the fact that the idealized at rest conditions imposed in the VML never occur in reality. Horizontal velocity perturbations could impose pressure on the side of the droplet, resulting in a more prolate shape and thus higher fall velocity.

Fig. 6.
Fig. 6.

The fall velocity (m s−1) vs diameter (mm). Red dots with error bars represent the mean fall velocity and upper–lower range. The experimental measurements of falling heights of 4.65, 3.95, 3.25, and 2.73 m by Guzel and Barros (2001) and the measurements by Gunn and Kinzer (1949) and its parameterization proposed by Lhermitte (1990) are shown as comparison. Note that results of 0.5 and 1 mm indicate large numerical errors (refer to the discussion in section 3a).

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

The droplet shape varies for different sizes, which has been verified experimentally (Andsager et al. 1999; Beard et al. 1991; Bringi et al. 1998; Chandrasekar et al. 1988; Jones 1959; Kubesh and Beard 1993; Sterlyadkin 1988; Pruppacher and Beard 1970) and parameterized by Andsager et al. (1999) and Beard and Chuang (1987). The VML droplets are initialized as spheres, and then, they deform as velocity increases. The shape evolution of a 3-mm-diameter droplet is depicted in Fig. 7. The drop has an oblate shape (α¯<1) with a flat bottom at terminal velocity. The flat raindrop bottom occurs due to strong upward air pressure. Because asymmetric wake oscillations persist as the drop falls, the droplet shape oscillates, resulting in shape difference in the front view (section X) and side view (section Y).

Fig. 7.
Fig. 7.

The time evolution of droplet 3D view and cross sections of the 3-mm-diameter droplet. Sections X, Y, and Z refer to the cross sections at the x, y, and z centers of the domain.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

The shape distortion and oscillation magnitude vary with drop size. The raindrop cross-sectional shapes at pseudoequilibrium are shown in Fig. 8. Smaller drops (D = 0.5, 1.0, and 1.5 mm) are nearly spherical. As raindrop size increases, surface tension becomes relatively weak compared with air drag. The axis ratios α¯ of different raindrop sizes are shown in Fig. 9 with comparison against classical experimental measurements and parameterizations. The α¯ decreases with the raindrop diameter, which implies that raindrops tend to become flatter due to increased terminal velocity. The shape oscillation intensity increases with drop size. The oscillation range covers most field measurements. The VML undershoots α¯ at 4 mm, and the more oblate shape causes the underestimation of the fall velocity.

Fig. 8.
Fig. 8.

The droplet 3D view and cross sections of drops in the pseudoequilibrium with diameters ranging from 0.5 to 4.0 mm. Sections X, Y, and Z refer to the cross sections at the x, y, and z centers of the domain.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

Fig. 9.
Fig. 9.

The VML axis ratio α_ at different diameters is shown by red dots with the red bars representing the range of variability of the axis ratio in the drop’s oscillation stage. The VML results are plotted against results from classical experiments and parameterizations as per the legend. Note that results of 0.5 and 1 mm indicate large numerical errors indicated by abnormally low variability in axis ratio (refer to the discussion in section 3a).

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

b. Collision experiments

Five collision outcomes have been identified in the VML simulations: coalescence, disk breakup, sheet breakup, filament breakup, and slip, as shown in Fig. 10. Coalescence prevails when the relative velocity is small, or the bottom droplet is small, both leading to a small Weber number and small kinetic energy. Disk breakup prevails when the momentum is strong enough to deform the shape instantly, forming a cone structure and then breaking up into many pieces. Sheet breakup occurs when a larger drop hits a smaller one with moderate momentum energy between disk breakup and coalescence. A small SR is preferential for sheet breakup because it alleviates the collision energy burst and reinforces shear stresses to form a sheet shape. Filament breakup and slip occur when the SR is large. In the filament breakup, a bridge forms between two drops that subsequently break up into small drops, which typically account for a small portion of the total mass. In the case of slip, the two drops experience contact momentarily and then separate recovering their original shapes. Whether the bridge can form depends on the contact area of two drops, which is characterized by the diameter ratio p. Lower p indicates a large diameter difference and thus a smaller portion of contact area over the total surface area, which hinders bridge formation and favors the slip outcome (see section 5c for discussion). The definitions, collision characteristics, and outcome drop number of coalescence, disk breakup, sheet breakup, filament breakup, and slip, are summarized in Table 2.

Fig. 10.
Fig. 10.

The example outputs of different collision outcomes. (a) Coalescence, in the collision between 0.8-mm (top) and 0.4-mm (bottom) droplets with initial fall velocities of 3.2 and 1.7 m s−1 and position at SR = 0.5. (b) Disk-type breakup, in the collision between 2.4-mm (top) and 1.2-mm (bottom) droplets with initial fall velocities of 7.3 and 4.6 m s−1 and position at SR = 0. (c) Sheet-type breakup, in the collision between 2.0-mm (top) and 1.2-mm (bottom) droplets with initial fall velocities of 6.6 and 4.6 m s−1 and position at SR = 0. (d) Filament/neck-type breakup, in the collision between 3.2-mm (top) and 2.0-mm (bottom) droplets with initial fall velocities of 8.3 and 6.6 m s−1 and position at SR = 0.75. (e) Slip, in the collision between 2.8-mm (top) and 1.6-mm (bottom) droplets with initial fall velocities of 7.8 and 5.7 m s−1 and position at SR = 1. Each panel has a different scale as indicated by the blue scale bar. The scenes are scaled to include all droplets for each panel. Please see sections 2c and 5b for details on the drop counting.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

Table 2.

Summary of definitions, collision characteristics, and outcome drop number of coalescence, disk breakup, sheet breakup, filament breakup, and slip outcomes.

Table 2.

4. Parameterization of collision outcomes

The coalescence, disk breakup, sheet breakup, and filament breakup outcomes are analyzed in the context of the Weber number and diameter ratio p phase space, as shown in Fig. 11. The results agree overall with Testik et al. (2011) with regard to the separation between coalescence and breakup {S1=p2We/[6(1+p3)]}. However, there is some disagreement at a high Weber number. We propose an adjustment to the separation curve parameterization as follows:
Sadj,1=(We+20)(4p/30.25)3/430=1.
The proposed separation curve has a steep slope at p ≈ 0.15. Therefore, the outcome will always be coalescence regardless of collision kinetic energy if the diameter ratio is lower than the threshold of 0.15, corresponding to coalescence by collection as a Bergeron-type seeder–feeder process, whereby a rain droplet falling through haze or fog collects all the droplets in their trajectory (e.g., Wilson and Barros 2014; Barros and Lettenmaier 1993).
Fig. 11.
Fig. 11.

The scatterplot of coalescence, disk breakup, sheet breakup, and filament breakup outcomes in the We–p diagram. The coalescence, sheet breakup, disk breakup, and filament breakup are identified as per the legend. The symbols of different SR values are moved to avoid overlapping. For each drop pair, the deviations from We–p are shown via the layout diagram in the legend, which are (−0.02, 0), (−0.02/3, 5), (0.02/3, 5), (0.02, 0), (−0.02/3, 5), and (−0.02/3, −5), corresponding to SR = 0, 0.25, 0.5, 0.75, 1, and 1.25, respectively. The dashed and solid lines represent the separation curve between coalescence and breakup in Testik et al. (2011) and Eq. (4), respectively. The slip breakup and no-collision outcomes are not shown because they are highly independent on SR instead of We–p (see section 5d for discussion).

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

The separation between coalescence and rebound bounce in Testik et al. (2011) is not found in VML experiments because the VML initial setting does not meet the bounce prerequisite, which is the existence of both perpendicular and tangential relative velocities. Beard and Ochs (1983) and Ochs and Beard (1984) identified two types of bounce mechanisms when the air film between the two colliding drops is not broken: a rebound bounce when the drops pull back due to surface tension effects and a grazing bounce when the drops slip past each other. Overall, the probability of rebound bounce collisions is very small, although it can have an impact on the dynamic evolution of raindrop size distributions (Prat et al. 2012). Testik et al. (2011) reported that fewer than 2% of cases could be classified as rebound bounce in tower experiments (B08). In the VML simulations, grazing bounce is referred to as slip and it can take place over the entire We–p space.

The distribution of fragment numbers (FN) produced from the collision of each drop pair averaged according to breakup outcome type is shown in Fig. 12. A total of 45 diameter pairs were simulated and analyzed, spanning a wider drop size range compared with LL82. FN is influenced by both the diameter of each drop and the differences in diameter. Drop instability increases with diameter due to the weakening of surface tension compared with the body force. Diameter differences are tied to drop velocity differences and therefore kinetic collision energy. Larger diameter or diameter differences favor larger FN. Disk breakup tends to have the largest FN because of the strongest collision energy burst and shape distortion among all breakup types. The filament breakup typically produces up to seven drops because only a limited number of drops can be generated from the thin water bridge between the two colliding drops. Sheet breakup produces medium FN, in between disk and filament breakups, because the connection area is larger than filament breakup, and the collision energy is smaller than disk breakup.

Fig. 12.
Fig. 12.

(a) Average FN over each colliding drop pair. Only breakup outcomes are considered. The 10 drop pairs in the LL82 collision experiments are labeled as marked as blue “x.” (b) Distribution of FN in sheet (2–12), disk (5–17), and filament (2–7) breakups color-coded as red, blue, and green, respectively.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

The fragment sizes are analyzed next toward developing a physical parameterization of collisional breakup. LL82 conducted 100 repetitions of each collision experiment to obtain 100 replicates of FN, which were subsequently averaged. The VML is deterministic producing the same results for the same initial conditions and numerical formulation and setup; thus, perturbation experiments toward determining an ensemble would be required that are computationally prohibitive and therefore were not pursued. Further, we hypothesize that the different results of the repeated experiments in LL82 are caused by collisions of the same drop pair at different relative positions (see section 5c for discussion). Therefore, the FN average of the 100 replicates is the weighted average of the drop size distribution over the SR distribution. However, SR distribution is not static because the DSD varies in time and space. Parameterizing fragment sizes by breakup type is an alternative, which enables good generalization to natural rainfall. Fragment diameter fraction (FDF) is defined in Eq. (5) to normalize cases with different diameters:
FDFf=(Vf/Vtotal)1/3,
where Vf is the volume of a fragment and Vtotal is the total volume of all fragments. The FDF distributions of disk, sheet, and filament breakups are shown in Fig. 13. In disk breakup, the FDF distribution is Gaussian-like because it typically involves normal collisions with evenly distributed forces. For the filament breakup, the FDF distribution is discontinuous, peaking at small and large diameter fractions, respectively, small around 0.1 and large around 1. The drops originating from the water bridge (filament) occupy a small portion of the total volume (left peak), while most of the initial drops remain unchanged (right peak). Sheet breakup’s normal and shear stress lies between filament and disk breakups. As a result, the distribution peaks at unity but with a lower frequency compared with filament breakup. Also, the FDF distribution of sheet breakup was skewed to the left due to stronger shear force compared with disk breakup. Overall, the fragment size distributions in the VML agree well with LL82 (Fig. 13). Because the VML distribution is averaged over all collision outcomes for each breakup type, it is more dispersed compared with the LL82 distribution, which represents single cases of fixed diameter pairs.
Fig. 13.
Fig. 13.

(a)–(c) FDF distribution of sheet, disk, and filament breakups in VML, respectively. The distribution is calculated based on all cases of each type. (d)–(f) FDF distribution of sheet, disk, and filament breakups in LL82, respectively. The distribution is calculated from the ensemble average of one collision case repeated over 100 times. The diameters of small and large drops in LL82 are 0.18 and 0.36 cm for all cases. The bin size of FDF is 0.05 for all panels.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

5. Discussion

a. Secondary coalescence and breakup

After a collision event when two drops have separated or broken up into fragments, coalescence and breakup can happen again among the fragments, as illustrated in Fig. 14. The secondary coalescence occurs when two already separated droplets collide again and coalesce into a single drop. If the large drop is on top, it has a larger terminal velocity that enables it to catch up and collide with the second drop (Fig. 14a). If the smaller drop is above the larger drop, it can still accelerate due to low air resistance by staying in the wake region of the large drop (as pointed out by Woods and Mason 1965). Consequently, the smaller drop might catch up with the larger drop, producing secondary coalescence (Fig. 14b). In addition, the drop can spontaneously break up after the separation due to instability (Fig. 14c). To our knowledge, it is the first time that a spontaneous breakup is simulated. The instability could come from shape distortion and air turbulence created by surrounding drops. These are rare occurrences which have been observed in tower experiments only rarely (B08).

Fig. 14.
Fig. 14.

(a) Secondary coalescence after filament breakup resulting from the collision between 2.8-mm (top) and 2.0-mm (bottom) droplets with initial fall velocities of 7.8 and 6.6 m s−1 and SR = 0.75. (b) Secondary coalescence after slip between 4.0-mm (top) and 2.8-mm (bottom) droplets with initial fall velocities of 8.8 and 7.8 m s−1 and SR = 0.75. (c) Secondary breakup after slip between 3.6-mm (top) and 3.2-mm (bottom) droplets with initial fall velocities of 8.6 and 8.3 m s−1 and SR = 1.25. (d) Secondary breakup (circled in red) and secondary coalescence (circled in green) after disk breakup resulting from the collision between 4.0-mm (top) and 2.0-mm (bottom) droplets with initial fall velocities of 8.8 and 6.6 m s−1 and position SR = 0. The scenes are scaled to include all droplets for each panel as indicated by the blue scale bar. The time sequences are presented at nonuniform time intervals to highlight frames with secondary coalescence and breakup. Please see sections 2c and 5b for details on the drop counting.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

b. Uncertainty of outcome classification and fragment analysis

Secondary breakup and coalescence add uncertainty to the end time point of a collision event. For example, the cases shown in Fig. 14a can be classified as filament breakup followed by coalescence if examined separately frame by frame or slip if comparing only the first and last frames of the event. In VML analysis, the secondary coalescence and breakup are treated as individual events so that the collision outcome is classified and parameterized only based on the primary collision. This approach is consistent with the governing physical mechanisms. For example, in Fig. 14b, classifying the outcome as filament instead of coalescence is consistent with the first mechanism because the drop fragments originate from the “filament” breakup that connected the initial drop pair. Additionally, this approach is consistent with the record of real experiments for which the timespan for image sequencing is insufficient to observe secondary collisions. The timespan is limited by the camera’s field of view (FOV), approximately 10 ms in LL82 and 20 ms in Testik et al. (2011) (Fig. 1), typically shorter than the time scales of secondary coalescence and breakup.

The fragment number and the corresponding size distribution are determined based on the frame with the largest fragment number to minimize the impact of secondary coalescence and breakup among breakup outcomes. However, this approach could underestimate FN when some drops become undetectable after entering the coarse mesh region due to large numerical diffusion that reduces water fraction below the threshold, especially in the case of disk breakup when water is splashed horizontally. Also, the clustering algorithm is subject to error when drop pairs are too close in the fine mesh region such that the distance is smaller than the coarse gridcell size, and consequently, the algorithm would only count them as one drop (see the appendix for algorithm details). This error can affect the differentiation between slip and no-collision cases, as well as the fragment number and size calculation of breakup outcomes. The classification problem is resolved by manually checking the high-resolution image for each case. For fragment analysis, the fragments tend to separate further until the distance exceeds the maximum cell size so that the algorithm can cluster water points effectively.

c. The effect of drop diameter on breakup outcome

The diameter of drops affects the breakup outcome through velocity and shape. The velocity difference between two drops determines collision energy (We), and as a result, it characterizes the boundary between disk and sheet-type breakups. In the two cases shown in Figs. 10b and 10c, the only difference is the radius of the top droplet, corresponding to relative velocity ΔV of 2.7 and 2.0 m s−1. The outcome differs dramatically. The large-ΔV case produced 10 fragments in disk breakup, while the small-ΔV case produced only three fragments in sheet breakup.

In addition, drop size determines the magnitude of shape distortion that in turn affects the contact area in the collision. This mechanism becomes important at grazing positions, for which case whether the bridge can form and break into droplets is essential to differentiate between filament and slip. As shown in Fig. 15, the initial position (SR = 1) is the same for all three cases, but the drop diameter increases from Figs. 15a–c. Consequently, due to the increase in the contact area, the bridge between the two drops forms, and the thickness increases. The filament breakup happens when the bridge is thick enough to produce nontrivial droplets. This mechanism is verified by inspecting the We–p space for SR = 1, as shown in Fig. 16. The outcome shifts from slip to filament as the diameter ratio p increases, which implies that the diameters of the drop pair become more similar, such that the contact area portion of the total surface area increases, favoring the bridge formation and thus filament breakup.

Fig. 15.
Fig. 15.

The transition from slip to filament breakups. (a) Slip without bridge formation, in the collision between 2.4- and 1.2-mm droplets with initial fall velocities of 7.3 and 4.6 m s−1 and position at SR = 1. (b) Slip with bridge formation, in the collision between 3.6-mm (top) and 2.0-mm (bottom) droplets with initial fall velocities of 8.6 and 6.6 m s−1 and position at SR = 1. (c) Filament breakup, in the collision between 3.6-mm (top) and 2.4-mm (bottom) droplets with initial fall velocities of 8.6 and 7.3 m s−1 and position at SR = 1. Each panel has a different scale as indicated by the blue scale bar. The scenes are scaled to include all droplets for each panel. Please see sections 2c and 5b for details on the drop counting.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

Fig. 16.
Fig. 16.

The collision outcomes as a function of SR in We–p space with (a)–(f) corresponding to SR = 0, 0.25, 0.5, 0.75, 1, and 1.25, respectively.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

d. Collision geometry

The SR defines the initial collision geometry condition (section 2b), and it plays a governing role in determining the outcomes of drop–drop interactions. Collision outcomes for different SR values are scattered in the We–p diagram shown in Fig. 16 and accompanied by the outcome frequency histogram in Fig. 17. At low SR, collision kinetic energy is spread more evenly across the contact surface, favoring coalescence at low CKE and disk-type breakup at high CKE. As SR increases, shear stress alleviates the normal collision momentum, which reduces the energy burst intensity. Instead of collapsing as in disk breakup, the drops form a transitional shape and stabilize because of losing smaller drops in sheet breakup. Filament breakup occurs at higher SR (0.5, 0.75, and 1) because it requires the formation of a narrow channel between the two drops, arising from shape distortion caused by strong shear stress. The shear stress is analyzed with ezx = ∂Uz/∂x, which is linearly related to the maximum shear stress under dimensional analysis (other velocity gradient components are approximately 0). The ezx distributions of different SR values for one drop pair are shown in Fig. 18. The ezx is evenly distributed at SR = 0, resulting in disk breakup. When SR > 0, ezx exhibits an uneven distribution with the peak moving rightward. In addition, the maximum value of ezx increases with SR, indicating a sharper shear stress peak. As a result, the outcome shifts to sheet breakup (SR = 0.25 and 0.5) and then filament breakup (SR = 0.75 and 1). Slip prevails at SR = 1 and can occur at SR = 1.25 when the drops come into contact due to shape distortion and horizontal oscillation including air turbulence created by the interacting drop pairs. Figure 19 analyzes an example case of slip at SR = 1.25. When two drops approach, the air velocity in the gap between them increases due to the mutual blockage (Fig. 19a). This lowers the air pressure in the gap (Fig. 19d), leading to deformation of the drops as they move closer to each other, which is observed after 1 ms (e.g., the left protrusion of the smaller drop in Figs. 19b,e). When the two drops are close enough, surface tension forces further pull them together, forming a transitional drop (Figs. 19c,f). Most drop pairs do not collide at SR = 1.25 when the horizontal separation is sufficient to prevent the contact.

Fig. 17.
Fig. 17.

Simulation outcome frequency for different SR values. Six outcomes, coalescence, disk breakup, sheet breakup, filament breakup, slip breakup, and no-collision outcome, are included in six scenarios of SR = 0, 0.25, 0.5, 0.75, 1, and 1.25, respectively.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

Fig. 18.
Fig. 18.

The ezx distribution of different SR values in the collision between 4.0-mm (top) and 2.0-mm (bottom) droplets with initial fall velocities of 8.8 and 6.6 m s−1 shown in the xz-plane cross section. (a) SR = 0 resulting in disk breakup. (b) SR = 0.25 resulting in disk breakup. (c) SR = 0.5 resulting in disk breakup. (d) SR = 0.75 resulting in disk breakup. (e) SR = 1 resulting in disk breakup. Initially, the large drop is on the left of the small drop.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

Fig. 19.
Fig. 19.

Velocity and pressure profiles in the slip outcome between 4.0- and 3.2-mm droplets with initial fall velocities of 8.8 and 8.3 m s−1 and position at SR = 1.25. (a)–(c) Velocity profile at 11, 12, and 13 ms. (d)–(f) Pressure profile at 11, 12, and 13 ms. Note high pressure (red) is nearby droplets. In all panels, water droplets are colored white, and the velocity direction is represented by vectors, with black arrows indicating air velocity and silver arrows indicating water velocity.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

These findings imply the necessity of expanding the drop sweeping volume associated with SR ∈ [0, 1] in traditional stochastic collision models. Because of shape and velocity oscillations, two drops with an initial SR > 1 still have a chance to collide, resulting in slip and coalescence. The slip outcome can be neglected because drops mostly maintain the original diameter. However, coalescence is essential to DSD evolution. Furthermore, a secondary breakup can occur after slip at SR > 1 (Fig. 14c). Extending SR over 1 in the parameterization improves model accuracy.

To account for the DSD variation, we incorporate the SR in the parameterization of collision outcomes. The proposed SR-weighted coalescence and breakup kernels are shown in the equations below:
C(r1,r2)=01.25P(SR)C(SR,r1,r2)SRP(SR)C(SR,r1,r2)B(r1,r2)=01.25P(SR)B(SR,r1,r2)SRP(SR)B(SR,r1,r2),
where C and B are the classical coalescence and breakup kernels used in stochastic collision models (Pruppacher and Klett 1978; McTaggart-Cowan and List 1975; McFarquhar 2004; McFarquhar and List 1993; Prat and Barros 2007) and P is the probability of SR. Measuring the relative location of drop pairs in individual rain events to obtain the actual distribution of SR for different rainfall regimes is impractical, and explicit simulation would require a significant upgrade in computational capabilities (e.g., quantum computing). Presently, one approach to infer realistic SR distributions is to rely on a Bayesian physical–statistical framework to optimize distribution parameters against disdrometer measurements of DSDs by modifying the coalescence and breakup kernels in existing stochastic rainshaft models (e.g., Prat et al. 2008; Arulraj and Barros 2021) as per Eq. (6). The parameterized SR distribution can be generalized by solving the inverse problem using data for different precipitation regimes. This approach has been successfully implemented for other geophysical problems (e.g., Berliner 2003; von Toussaint 2011) and will be pursued in future work.

e. Recapitulation of abbreviations and symbols

The abbreviations and symbols frequently used in the article are summarized in Table 3.

Table 3.

Summary of abbreviations and symbols.

Table 3.

6. Conclusions

In summary, the VML is robust to simulate drop-falling and drop–drop interactions in time and space. The free-falling simulation reveals that “uncertainty” in disdrometer measurements of raindrop velocities is rooted in physics. Both shape and velocity oscillations persist in the terminal stage. More prolate shapes experience higher vertical acceleration with a smaller horizontal cross-sectional area and thus smaller drag force. The collision simulations reproduced realistic laboratory experiments well. Examination of the VML simulations enabled the identification and visualization of spontaneous breakup, and secondary coalescence and breakup for the first time to our knowledge in the peer-review literature. This begs the questions of whether and how chains of collisions that modify the momentum of the DSD impact its evolution in 3D and what are time scales and rainfall intensity at which such interactions must be considered (e.g., Prat and Barros 2009). An alternative parameterization of the boundary between coalescence and breakups in We–p space was proposed to separate coalescence and breakup outcomes [Eq. (4)].

Analysis of the fragment number and size distributions from collisional breakup in the VML for disk, sheet, and filament categories revealed the significance of relative position at colliding drop pairs (SR) in determining the type of breakup and transitions from one category to another (Fig. 11). The maximum SR for collision is larger than one because drop shape and velocity oscillate continuously. A modification of the coalescence and breakup kernels in stochastic rainfall models is proposed to account for SR effects [Eq. (6)].

This work shows that the VML can be used in lieu of tower experiments to investigate rainfall processes. The next steps are to develop a general parameterization of the number and size distribution of drop collision outcomes with the Weber number, diameter ratio, and relative position and to investigate the fingerprints of collision chains on the evolution of the shape of the DSD such as the competition between coalescence and breakup processes as described by Wilson and Barros (2014) that can have a significant impact on the vertical microstructure of rainfall. The ultimate purpose of the VML is to conduct end-to-end simulations of aerosol–cloud–rainfall microphysical processes, the implementation of which is ongoing. Nevertheless, despite the significant progress in our ability to simulate these processes, the spatial and temporal resolutions are still insufficient to capture micrometer-scale physics.

Acknowledgments.

This research was funded by NASA Grant 80NSSC19K0685 with the second author. The authors are grateful to three anonymous reviewers for their insightful comments and suggestions.

Data availability statement.

To reproduce results, the CFD software OpenFOAM v9 is available from https://openfoam.org/version/9/. The raindropFoam solver can be accessed from GitHub https://github.com/lihui-ji/Raindrop-Simulation/tree/main/raindropFoam_internal. All simulation results are available upon request to the corresponding author.

APPENDIX

Hierarchical Clustering Algorithm

Before clustering, the 3D water points are upscaled to the lowest resolution in the water region by the nearest neighbor method. Fig. A1 demonstrates the upscaling of four different levels, depending on the smallest resolution region that covers all water points in the last frame of the collision case between 2.8 and 1.6 mm at SR = 0.5, and the upscaling level is determined to be level 1 (Fig. A1b), corresponding to mesh dimension of 0.11 mm × 0.11 mm × 0.11 mm and volume of 1.31 × 10−3 mm−3.

Fig. A1.
Fig. A1.

The four levels of upscaling of collision outcome between 2.8-mm (top) and 1.6-mm (bottom) droplets with initial fall velocities of 7.8 and 5.7 m s−1, respectively, and SR is 0.5. Note that the drop boundary expanded after upscaling. The frame is the last frame when the collision has been completed, sixteen milliseconds from the start of the simulation.

Citation: Journal of the Atmospheric Sciences 81, 6; 10.1175/JAS-D-23-0121.1

The upscaled water point coordinates are passed to the hierarchical cluster algorithm to determine fragment number and size. The clustering algorithm iteratively merges the two nearest clusters until only one cluster presents, consisting of all initial points. The distance between clusters C1 and C2 is defined in Eq. (A1), which is the minimum distance between two points P1 and P2 belonging to clusters C1 and C2.
d(C1,C2)=minP1C1,P2C2P1P22
By analyzing the distance of merged clusters in each iteration, the cluster number can be determined by the point where the distance exceeds the cutoff distance, which is set to be the maximum mesh size of the water region. The computation of drop number counting was done with SciPy Python Library (Virtanen et al. 2020). Table A1 is an exemplary output of hierarchical clustering for the collision outcome shown in Fig. A1.
Table A1.

The exemplary output of the hierarchical clustering algorithm for the collision outcome shown in Fig. A1.

Table A1.

The iterations terminate when the distance exceeds the maximum cell size (0.1D = 0.44 mm), resulting in three clusters, containing 155, 116 (271–155), and 7791 (8062–271) points each. By multiplying by cell volume, the volumes of three droplets are 0.206, 0.154, and 10.370 mm3 and the diameters are 0.73, 0.67, and 2.71 mm.

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    • Search Google Scholar
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  • Low, T. B., and R. List, 1982a: Collision, coalescence and breakup of raindrops. Part I: Experimentally established coalescence efficiencies and fragment size distributions in breakup. J. Atmos. Sci., 39, 15911606, https://doi.org/10.1175/1520-0469(1982)039<1591:CCABOR>2.0.CO;2.

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  • Low, T. B., and R. List, 1982b: Collision, coalescence and breakup of raindrops. Part II: Parameterization of fragment size distributions. J. Atmos. Sci., 39, 16071619, https://doi.org/10.1175/1520-0469(1982)039<1607:CCABOR>2.0.CO;2.

    • Search Google Scholar
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  • Magarvey, R. H., and J. W. Geldart, 1962: Drop collisions under conditions of free fall. J. Atmos. Sci., 19, 107113, https://doi.org/10.1175/1520-0469(1962)019<0107:DCUCOF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • McFarquhar, G. M., 2004: A new representation of collision-induced breakup of raindrops and its implications for the shapes of raindrop size distributions. J. Atmos. Sci., 61, 777794, https://doi.org/10.1175/1520-0469(2004)061<0777:ANROCB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • McFarquhar, G. M., and R. List, 1993: The effect of curve fits for the disdrometer calibration on raindrop spectra, rainfall rate, and radar reflectivity. J. App. Meteor., 32, 774782, https://doi.org/10.1175/1520-0450(1993)032%3C0774:TEOCFF%3E2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • McTaggart-Cowan, J. D., and R. List, 1975: Collision and breakup of water drops at terminal velocity. J. Atmos. Sci., 32, 14011411, https://doi.org/10.1175/1520-0469(1975)032<1401:CABOWD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ochs, H. T., III, and K. V. Beard, 1984: Laboratory measurements of collection efficiencies for accretion. J. Atmos. Sci., 41, 863867, https://doi.org/10.1175/1520-0469(1984)041<0863:LMOCEF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Ong, C. R., H. Miura, and M. Koike, 2021: The terminal velocity of axisymmetric cloud drops and raindrops evaluated by the immersed boundary method. J. Atmos. Sci., 78, 11291146, https://doi.org/10.1175/JAS-D-20-0161.1.

    • Search Google Scholar
    • Export Citation
  • Prat, O. P., and A. P. Barros, 2007: A robust numerical solution of the stochastic collection–breakup equation for warm rain. J. Appl. Meteor. Climatol., 46, 14801497, https://doi.org/10.1175/JAM2544.1.

    • Search Google Scholar
    • Export Citation
  • Prat, O. P., and A. P. Barros, 2009: Exploring the transient behavior of ZR relationships: Implications for radar rainfall estimation. J. Appl. Meteor. Climatol., 48, 21272143, https://doi.org/10.1175/2009JAMC2165.1.

    • Search Google Scholar
    • Export Citation
  • Prat, O. P., A. P. Barros, and C. R. Williams, 2008: An intercomparison of model simulations and VPR Estimates of the vertical structure of warm stratiform rainfall during TWP-ICE. J. Appl. Meteor. Climatol., 47, 27972815, https://doi.org/10.1175/2008JAMC1801.1.

    • Search Google Scholar
    • Export Citation
  • Prat, O. P., A. P. Barros, and F. Y. Testik, 2012: On the influence of raindrop collision outcomes on equilibrium drop size distributions. J. Atmos. Sci., 69, 15341546, https://doi.org/10.1175/JAS-D-11-0192.1.

    • Search Google Scholar
    • Export Citation
  • Premnath, K. N., and J. Abraham, 2005: Simulations of binary drop collisions with a multiple-relaxation-time lattice-Boltzmann model. Phys. Fluids, 17, 122105, https://doi.org/10.1063/1.2148987.

    • Search Google Scholar
    • Export Citation
  • Pruppacher, H. R., and K. V. Beard, 1970: A wind tunnel investigation of the internal circulation and shape of water drops falling at terminal velocity in air. Quart. J. Roy. Meteor. Soc., 96, 247256, https://doi.org/10.1002/qj.49709640807.

    • Search Google Scholar
    • Export Citation
  • Pruppacher, H. R., and J. D. Klett, 1978: Microphysics of Clouds and Precipitation. D. Reidel Publishing Company, 714 pp.

  • Rayleigh, L., 1879: On the capillary phenomena of jets. Proc. Roy. Soc. London, 29, 7197, https://doi.org/10.1098/rspl.1879.0015.

  • Ren, W., J. Reutzsch, and B. Weigand, 2020: Direct numerical simulation of water droplets in turbulent flow. Fluids, 5, 158, https://doi.org/10.3390/fluids5030158.

    • Search Google Scholar
    • Export Citation
  • Saha, R., and F. Y. Testik, 2023: Assessment of OTT parsivel2 raindrop fall speed measurements. J. Atmos. Oceanic Technol., 40, 557573, https://doi.org/10.1175/JTECH-D-22-0091.1.

    • Search Google Scholar
    • Export Citation
  • Saleeby, S. M., B. Dolan, J. Bukowski, K. V. Valkenburg, S. C. van den Heever, and S. A. Rutledge, 2022: Assessing raindrop breakup parameterizations using disdrometer observations. J. Atmos. Sci., 79, 29492963, https://doi.org/10.1175/JAS-D-21-0335.1.