Editorial Type:
Article
Article Type:
Research Article

Gravitational Collision of Small Nonspherical Particles: Swept Volumes of Prolate and Oblate Spheroids in Calm Air

Ehud Gavze aInstitute of Earth Science, Hebrew University of Jerusalem, Jerusalem, Israel

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Alexander Khain aInstitute of Earth Science, Hebrew University of Jerusalem, Jerusalem, Israel

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Online Publication:
18 May 2022
Print Publication:
01 Jun 2022
DOI:
https://doi.org/10.1175/JAS-D-20-0336.1
Page(s):
1493–1514
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Abstract

The aggregation rate of ice crystals depends on their shape and intercrystal relative velocity. Unlike spherical particles, the nonspherical ones can have various orientations relative to the gravitational force in the vertical direction and can approach each other at many different angles. Furthermore, the fall velocity of such particles could deviate from the vertical direction velocity. These properties add to the computational complexity of nonspherical particle collisions. In this study, we derive general mathematical expressions for gravity-induced swept volumes of spheroidal particles. The swept volumes are shown to depend on the particles’ joint orientation distribution and relative velocities. Assuming that the particles are Stokesian prolate and oblate spheroids of different sizes and aspect ratios, the swept volumes were calculated and compared to those of equivalent volume spheres. Most calculated swept volumes were larger than the swept volumes of equivalent spherical particles, sometimes by several orders of magnitude. This was due to both the complex geometry and the side drift, experienced by spheroids falling with their major axes not parallel to gravity. We expect that the collision rate between nonspherical particles is substantially higher than that of equivalent volume spheres because the collision process is nonlinear. These results suggest that the simplistic approach of equivalent spheres might lead to serious errors in the computation of the collision rate.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Alexander Khain, alexander.khain@mail.huji.ac.il

Abstract

The aggregation rate of ice crystals depends on their shape and intercrystal relative velocity. Unlike spherical particles, the nonspherical ones can have various orientations relative to the gravitational force in the vertical direction and can approach each other at many different angles. Furthermore, the fall velocity of such particles could deviate from the vertical direction velocity. These properties add to the computational complexity of nonspherical particle collisions. In this study, we derive general mathematical expressions for gravity-induced swept volumes of spheroidal particles. The swept volumes are shown to depend on the particles’ joint orientation distribution and relative velocities. Assuming that the particles are Stokesian prolate and oblate spheroids of different sizes and aspect ratios, the swept volumes were calculated and compared to those of equivalent volume spheres. Most calculated swept volumes were larger than the swept volumes of equivalent spherical particles, sometimes by several orders of magnitude. This was due to both the complex geometry and the side drift, experienced by spheroids falling with their major axes not parallel to gravity. We expect that the collision rate between nonspherical particles is substantially higher than that of equivalent volume spheres because the collision process is nonlinear. These results suggest that the simplistic approach of equivalent spheres might lead to serious errors in the computation of the collision rate.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Alexander Khain, alexander.khain@mail.huji.ac.il
Keywords: Clouds; Cloud microphysics; Ice crystals; Ice loss/growth; Cumulus clouds

1. Introduction

Collisions between cloud particles lead to precipitation and affect the radiation properties of clouds. The collision rate of cloud drops is typically calculated by assuming each collision is between two spheres falling vertically at different terminal velocities (gravitational coagulation). The probability of a collision between two spherical drops of radius r1 and r2 in calm air is proportional to the swept volume. This in turn is the product of their geometrical cross section π(r1 + r2)2 and their relative velocity |υ2 − υ1| (Pruppacher and Klett 1997). The relative velocity between droplets depends on their mass (radius) and has been investigated in depth. The definition of the swept volume in turbulent flows is different from that in calm air, because the relative velocity between approaching droplets has a random value. Moreover, droplets in turbulent flow can collide not only in the vertical direction, but also at a certain angle (Pinsky et al. 2000). As a result, collision kernels in turbulent flows might be substantially larger (up to a factor 10) than in calm air, as was shown in previous studies (Khain and Pinsky 1995; Pinsky et al. 2000, 2007, 2008; Ayala et al. 2007, 2008; Wang et al. 2008).

The increase in the collision kernel in turbulent flows leads to a significant acceleration of raindrop formation (Benmoshe et al. 2012; Seifert et al. 2010), and to a decrease in first radar echo height. Additionally, turbulence increases precipitation in large mesoscale convective systems (Lee and Baik 2016).

One of the main gaps in cloud microphysics knowledge is related to ice–ice and ice–water collisions. Collisions between ice crystals give rise to aggregate formation (snow) and lead to charge separation and lightning formation (Hashino and Tripoli 2007; Formenton et al. 2013). Aggregates dominate stratiform clouds that cover large parts of the globe and determine winter precipitation (Hosler et al. 1957; Kajikawa and Heymsfield 1989). Collisions between ice crystals and cloud droplets determine the formation of graupel and hail embryos. In turn, graupel and hail contribute to the precipitation rate in deep convective clouds (Mitchell 1988; Khain and Pinsky 2018). The high sensitivity of cloud microphysics and precipitation to the calculated rate of collisions, among ice crystal and between ice crystals and drops, was shown by Benmoshe and Khain (2014).

Observations show that small ice crystals begin producing aggregates quite rapidly in stratiform clouds, convective clouds anvils and stratiform zones of squall lines (Field and Heymsfield 2003). Nevertheless, the maximum aggregation rate occurs in the dendritic area (around T = −10° to −15°C), which could be related to these branch-type crystals ability to aggregate. Connolly et al. (2012) and Phillips et al. (2015) showed that the peak in sticking efficiency is near −15°C, when ice particles branches interlock. However, even above the dendritic growth zone, radar reflectivity from stratiform clouds could reach 15–20 dBZ (Fan et al. 2015), which indicates that aggregation between columnar or platelike crystals could also be efficient. In Arctic clouds aggregates might form at temperatures lower than the typical dendrite formation temperature (Fan et al. 2011). Moreover, the collisions between ice crystals might create the first aggregates that trigger fast aggregation in the dendritic region of the deep stratiform cloud layers forming behind squall lines (Shpund et al. 2019).

Despite the importance of these collisions in the formation of microphysical cloud structures, the collision rates between ice crystals, used in cloud models, have significant inaccuracies because of the many complications associated with the nonspherical shape of ice crystals. Fall velocity of nonspherical particles is closely related to their orientation. Nonspherical particles fall in numerous orientations, and not necessarily parallel to the gravitational acceleration vector. The particles can rotate during their fall, especially in the presence of wind shear. Therefore, like spherical particles in a turbulent flow, they may collide due to differences in both their vertical and horizontal velocities.

These factors make the calculation of swept volumes of nonspherical particles a challenging mathematical problem. Calculations of collision rates among ice crystals and between ice crystals and water droplets in current cloud models are greatly simplified to sidestep this difficulty. Typically, ice crystals are treated as “equivalent volume spheres” so that the collision rate can be calculated as if it were between spherical droplets (Hall 1980). Some studies have attempted to take the nonsphericity of ice crystals into account. In these cases, the ice crystals orientation was assumed to be with the maximum cross section perpendicular to the gravity and independent of time (Khain and Pinsky 1995; Wang 2002; Khain and Pinsky 2018).

A more accurate approach is to take into account the nonsphericity of ice crystals. The spheroidal approximation is commonly used to approximate ice particle shape, because it allows for the geometric representation of two particle length scales instead of one. Spheroids have been used in the development of numerical models of ice crystal growth processes (Chen and Lamb 1999; Hashino and Tripoli 2007; Harrington et al. 2013) and they have been used to represent snow aggregates (Stein et al. 2015; Hogan et al. 2012; Sulia et al. 2021) though aggregate shapes may be better approximated by ellipsoids (Dunnavan et al. 2019; Dunnavan 2021). Jiang et al. (2019) tested the assumption that ice particle aggregates (snowflakes) are well represented by oblate spheroids by applying ellipsoid fits to aggregate images. The resulting ellipsoids had shapes closer to prolate spheroids rather than to oblate spheroids. It was found that the most probable orientation of the maximum dimension of the retrieved ellipsoids was not in the horizontal plane.

Siewert et al. (2014a), performing direct numerical simulations (DNS), tracked the orientations of spheroids of several aspect ratios in several turbulence intensities. In the absence of gravity the orientation distribution function was uniform. However, preferential orientations were found if gravity was applied. Both prolate and oblate spheroids tended to align with their longer axis parallel to gravity. This alignment increased with increased deviation of the spheroid’s shape from spherical. Increasing the kinetic energy dissipation rate led to a more uniform distribution. Siewert et al. (2014b) investigated the collision rate and the motion of ellipsoids of different aspect ratios, settling due to gravity, in direct numerical simulated isotropic turbulence. The simulated collision rates were much higher than those of spherical particles of the same mass and volume. The authors argued that these higher rates were due to the fact that the drag force depends on the orientation. Consequently the settling velocity differences were higher and could remain so until contact if particle inertia was high enough. Jucha et al. (2018) and Naso et al. (2018) simulated the collision of small oblate spheroids and of spheroids with droplets in isotropic turbulence generated by DNS. They investigated the relative effects of turbulence intensity and gravitational settling on the orientations and on the collision rate.

As was mentioned above, ice particle orientation plays an important role in their motion and interaction. Polarimetric measurements showed that the polarimetric signatures of relatively small ice crystals could be obtained when assuming that the standard deviation of their orientation could reach up to 40° (Ryzhkov et al. 2011). Garrett et al. (2015) analyzed photographs of nearly 73 000 snowflakes in free fall and determined the aspect ratios and orientations of aggregates, moderately rimed particles, and graupel. (The aspect ratio was defined as the ratio of the shortest to longest dimension.) The analysis of Garrett et al. (2015) indicates that the range of orientation angles is much broader than has been indicated by previous observational and theoretical studies.

Despite these observational and DNS findings, the microphysical cloud models use a simplified assumptions about the orientations and fall velocities of ice crystals. Abraham (1970) developed a theory for the drag on spheres extending from the viscous regime of low particle Reynolds numbers (Re) to the regime of high Reynolds number (but still laminar). Böhm (1989, 1992a) extended the theory of Abraham (1970) to nonspherical particles, to porous particles, and to aggregates. Böhm (1992b,c) applied this theory to the study of the collision kernels and of the riming and aggregation of drops on nonspherical ice particles. He assumed that a spheroid moves in the direction of maximum drag. This implies that in calm air the large axis of the spheroid is perpendicular to gravity. Consequently, the fall velocity is always parallel to gravity. Wang and Ji (2000) also assumed a horizontal fixed orientation of the falling ice crystals, (see their previous paper; Wang and Ji 1997). The underlying assumption in the computation of the drag was that it is in the direction of the flow, or, equivalently, that particles in the gravitational field settle in the direction of gravity. Besides, in these studies, the orientations of the colliding particles were prescribed initially. Böhm (1992a) obtained very good agreement with experiments for particles falling parallel to gravity. Jensen and Harrington (2015) calculate riming growth of ice particles approximated by spheroids assuming that crystals fall in the vertical direction perpendicular to the maximum ice particle cross section. Such an assumption is valid for comparatively large crystals with Reynolds number Re > 1 (Klett 1995), but is certainly not justified for particles falling with small Reynolds numbers.

In our study we calculate the swept volume of small spheroids with Re < 1. We assume that the spheroids may have arbitrary orientations, an assumption that is valid for crystals with Re < 1. In this case the swept volume may be considerably higher due to side drift of the particles (Siewert et al. 2014a,b). Theoretically the orientations of Stokesian spheroids of small inertia, embedded in general linear shear flows, converge to either fixed points or to limit cycles. In the special case of simple shear they converge to periodic solutions named Jeffery orbits (Gavze et al. 2012, 2016a,b). The convergence time scale may sometimes be on the order of the shear time scale but may sometimes be much longer: it depends 1) on the relation between the vorticity and the eigenvectors of the strain tensor and 2) on the spheroid’s shape. The rate of convergence to these attractors is important to consideration of the orientations in time–space varying flow fields. This rate is equally relevant to the understanding of orientation distributions of spheroidal particles, smaller than Kolmogorov scale and of small inertia, in turbulent flows.

In the present paper we develop a theoretical framework for calculating the mean swept volume of an ensemble of pairs of nonspherical particles. We further develop a method to calculate the swept volume of two spheroidal particles of arbitrary orientations and aspect ratios. To make the problem tractable we focus on small spheroidal particles falling in a quiescent air. Such particles could serve as adequate models for columnar and platelike ice crystals.

2. Definition of the swept volume and the normal plane

In the following sections we deal with the collisions of small nonspherical Stokesian particles in calm air. The nonspherical particles are represented by spheroids that possess three symmetry axes, the rotation axis and two equal equatorial axes, thus forming a rotational symmetry with respect to the rotation axis. We designate the half length of the rotation axis by C and the half lengths of the two equatorial axes by A. We assume that the following conditions hold:

  1. During the collision process the colliding particles’ orientations remain unchanged.1

  2. The relative velocities between the colliding particles are determined by their terminal velocities v∞. They are not necessarily oriented in the vertical direction since the terminal velocities of nonspherical particles may have different orientations.

  3. The particle ensemble is large enough for the probability distribution function (pdf) of each particle type’s rotation axis orientation to be defined.

The collision rate is calculated with the aid of the so-called swept volume of a collecting particle, which contains all the collected particles that collide with the collecting particle in a unit time. The product of the collected particles concentration and the swept volume yields the number (or mass) of the particles collected during a unit time. The swept volume is the product of the relative velocity between two particles and the cross section on a plane normal to the relative velocity, through which the collecting particle collides with the collected particle [defined mathematically below in Eq. (2)]. We shall call this plane the normal plane and the area of this cross section the collecting area. The definition of the normal plane is illustrated in Fig. 1.

Fig. 1.
Fig. 1.

Two-dimensional depiction of the normal plane. The normal plane (dashed line) is perpendicular to the vector of the relative velocity Δv(1, 2) = v∞(1) − v∞(2), the vector difference between the terminal velocities of the spheroids. The approaching particles’ mutual location is depicted in a space coordinate system (see text for details).

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

In the following we shall make use of three coordinate systems. The first is the space coordinate system, defined by the orthogonal unit vectors x^, y^, and z^ (Fig. 2a). It is fixed in space with z^ parallel to gravity but pointing in the opposite direction. The second system, the body coordinate system, is attached to the spheroid and defined by the orthogonal unit vectors e1, e2, and e3, where e3 is parallel to the spheroid axis of rotation and e1 is in the z^–e3 plane (see Figs. 2a and 2b). The relation between the two coordinate systems is given by the elevation angle θ (with respect to the gravity direction) and the azimuthal angle ϕ. The elevation angle θ is the angle between the vectors z^ and e3; ϕ is the angle between x^ and p(e3), where p(e3) is the projection of e3 on the x^–y^ plane. The third coordinate system is attached to the normal plane and is defined below.

Fig. 2.
Fig. 2.

(a) The space coordinate system x^, y^, z^ and the body coordinate system e1, e2, e3 are related to the spheroid. The spheroid and its projection are marked by blue and gray, respectively. e3 is parallel to the spheroid axis of rotation, and e1 is in the z^–e3 plane. p(e3) is the projection of e3 on the x^–y^ plane. The angles θ and ϕ determine the orientation of the body coordinate system with respect to the space system. (b) (left) Projections of the spheroid in on the z^–p(e3) plane and (right)projections of the e2 and e3 axes on the x^–y^ plane. (c) Characteristic dimensions of (left) prolate and (right) oblate spheroids. A is half the length of the equatorial axes, and C is half the length of the rotation axis.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

The relative velocity of spherical particles settling due to gravity is always in the gravity direction, and the collecting area is a circle formed by the sum of the two particles radii. The situation is more complicated when dealing with nonspherical particles since the settling velocities, and therefore their relative velocities, have components parallel and perpendicular to gravity. Therefore, the normal plane is not necessarily perpendicular to gravity. Let S(1, 2) ≡ S(A1, C1, A2, C2, θ1, θ2, ϕ1, ϕ2) be the collecting area of two spheroidal particles. (Ai, i = 1, 2 is the half length of the equatorial axes of the ith spheroid, Ci is the half length of the rotation axis.) Let Δv(1, 2) be the relative terminal velocity (Fig. 1),
Δv(1,2)≡Δv(A1,C1,A2,C2,θ1,θ2,ϕ1,ϕ2)=v∞(1)−v∞(2).
Here v∞(1) and v∞(2) are the terminal velocities of spheroid 1 and spheroid 2. Then the swept volume SV(1, 2) is
SV(1,2)≡SV(A1,C1,A2,C2,θ1,θ2,ϕ1,ϕ2)=S(1,2)‖Δv(1,2)‖
(∥v∥ denotes the norm of a vector v). In the following we drop the arguments A1, C1, A2, and C2 for brevity. Equation (2) is valid since, by definition, the collecting area is in the plane perpendicular to Δv(1, 2). A list of symbols is given in Table 1.
Table 1

List of symbols.
Table 1
Let Π1,2(θ1, θ2, ϕ1, ϕ2) be the joint probability density that the collector and the collected particles be found at the orientations (θ1, θ2, ϕ1, ϕ2); then the average swept volume with respect to all possible values of azimuth angles (ϕ1, ϕ2) for a given pair of elevation angles (θ1, θ2) is
SV¯ϕ≡SV¯(θ1,θ2)=∫02π∫02πΠ1,2(θ1,θ2,ϕ1,ϕ2)SV(θ1,θ2,ϕ1,ϕ2)dϕ1 dϕ2.
Expression (3) is the most general one. It is necessary to know the orientation joint probability density and to calculate the collecting area and relative velocity between the colliding particles in order to use it.

3. A local coordinate system relative to the normal plane

Since the swept volume is related to the area projected by the colliding spheroids on the normal plane it is convenient to work with a coordinate system within which one of the axes is normal to this plane. Let li, i = 1, 2 be a unit vector in the symmetry axis direction of the ith spheroid (Fig. 3); li coincides with the base vector e3 of the ith spheroid body coordinate system. For the sake of convenience, we use index i = 1 for the collecting particle and the index i = 2 for the collected one. In the space coordinate system, the components of the vector li have the following representation:
li=(sin⁡θi cos⁡ϕisin⁡θi sin⁡ϕicos⁡θi).
where θi and ϕi are the angles shown in Fig. 2. We define Cartesian and polar coordinate systems relative to the normal plane (Fig. 3). We define the unit vector e˜3 to be perpendicular to the normal plane (i.e., parallel to the relative velocity Δv of the colliding particles). We introduce two orthogonal unit vectors in the normal plane, e˜1 and e˜2, i.e., they are orthogonal to each other and orthogonal to e˜3 (see Fig. 3).
Fig. 3.
Fig. 3.

Mutual location of colliding spheroids in a coordinate system related to the normal plane. The spheroids, marked blue and yellow, are the collecting and collected particles, respectively. The gray ellipses are their projections on the normal plane. a1, b1, a2, and b2 are the semiaxes of the projected ellipses [see Eq. (8)].

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

We define now the corresponding polar coordinates of the two spheroids (with respect to the normal plane system):
(li·e˜3)=cos⁡θ˜i,(li·e˜1)=sin⁡θ˜i cos⁡ϕ˜i,(li·e˜2)=sin⁡θ˜i sin⁡ϕ˜i.
We therefore get
θ˜i=arccos⁡(li·e˜3),ϕ˜i=arctan⁡[(li·e˜2)(li·e˜1)].
The angles θ˜i and ϕ˜i are shown in Fig. 3.
For the sake of convenience we may choose the direction of e˜1 to be parallel to the projection of l1 on the normal plane, i.e., parallel to p(e3). With this choice the unit vectors are
e˜3=Δv||Δv||, e˜1=l1−(l1·e˜3)e˜3‖l1−(l1·e˜3)e˜3‖, e˜2=e˜3×e˜1
and ϕ˜1=0. Here (·) and (×) are the scalar and the vector products, respectively. Table 2 summarizes the coordinate systems we use.
Table 2

The coordinate systems and related vectors.
Table 2

4. Computation of the collecting area on the normal plane

Consider a spheroid with half axis lengths (A, A, and C) (see Fig. 2). To calculate the swept volume we need to calculate the collecting area S(1, 2), which is the area on the normal plane through which particle 2 will be collected by particle 1 (Fig. 4; see section 2). The collecting area S(1, 2) depends on the projections S1 and S2 of the colliding spheroids on the normal plane and on Δϕ˜=ϕ˜1−ϕ˜2 (Figs. 3 and 4). Typically one of the two colliding particles is called the collector (usually, but not necessarily, the larger particle), and its counterpart is called the collected particle. The collector and the collected will be called particle 1 and particle 2, respectively. The projections of the collector and the collected spheroids are ellipses. Their areas, S1=S(A1,C1,θ˜1) and S2=S(A2,C2,θ˜2), respectively, are determined by the following formula, derived from Vickers [1996, Eq. (A1)]:
S(Ai,Ci,θ˜i)=πaibi, ai=Ai2 cos⁡2θ˜i+Ci2 sin⁡2θ˜i, bi=Ai,
where i = 1, 2; ai and bi are the projections of the semiaxes Ai and Ci, respectively, on the normal plane. The collecting area S(1, 2) may be computed by moving the projection of the collected particle around the projection of the collecting particle while keeping the orientation of the two projections constant and one point of contact. This point of contact is designated as the vector u (Fig. 5). The collecting area is the region surrounded by the path of center of the projection of the collected particle. The collecting particle in the figure is colored gray while the collected one is in pink. The axes (ai, bi) of the two ellipses are defined in Eq. (8). For each point of contact u, we designate the position of the center of the collected particle projection on the normal plane by the two dimensional vector w = (w1, w2) (see Fig. 5 and the corresponding notations). The closed set of all points w = (w1, w2) creates the perimeter of the collecting area. The collecting area, computed numerically from the set of points w = (w1, w2) by using Green theorem,2 is
S(1,2)≡S(θ1,θ2,ϕ1,ϕ2)=12∮Γw1 dw2−w2 dw1.
(The integration is performed counterclockwise, Γ is the set of points w that comprise the perimeter of the collecting area.) Details of the computation of w and of the collecting area S(1, 2) are given in appendixes A and B, respectively.
Fig. 4.
Fig. 4.

The collecting area S(1, 2) for two colliding particles is the fraction of the normal plane, bounded by the dashed line. The ellipses are the projections of the collecting (blue) and collected (green) spheroids on the normal plane. The angle between the projections is Δϕ˜, defined in Fig. 3. The dashed line is the path of the center of the projection of the collected spheroid, obtained when moving it around the projection of the collecting spheroid.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

Fig. 5.
Fig. 5.

The positions of the points of contact u, and the center of the collected particle w = (w1, w2) that determines the boundary of the collecting area S(1, 2). The collecting particle is colored gray while the collected one is in pink. The axes (ai, bi) of the two ellipses are defined in Eq. (8).

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

Note that according to Eq. (9) the swept volume depends on the vector w only. It follows from Eqs. (A3) and (A5) in appendix A, and Fig. 5, that w does not dependent on which of the two colliding particles is considered the collector. Similarly, the swept volume has the same property.

The swept volume SV(θ1, θ2, ϕ1, ϕ2) can now be computed by substituting the collecting area from Eq. (9) into Eq. (2). The azimuth angle average SV¯ϕ(θ1,θ2) is computed from Eq. (3). (We have again dropped the arguments A1, C1, A2, and C2 for convenience.)

We have so far made no assumptions concerning the orientation distribution of the spheroids. Equation (3) could be simplified under some symmetry assumptions. According to the first and second conditions in section 2, the orientation pdf of any two particles are independent so that their joint pdf is the product of the two individual pdfs:
Π1,2(θ1,θ2,ϕ1,ϕ2)=Π1(θ1,ϕ1)Π2(θ2,ϕ2).
Equation (10) is in particular true assuming that the distributions of ϕ1 and ϕ2 are uniform, in which case Π1=Π2=1/(2π). This assumption is valid in calm air if the initial azimuth orientation distributions of the spheroids are uniform and if hydrodynamic interactions are not taken into account (and also in horizontally isotropic turbulent flow, which is not dealt with here). In this case the swept volume depends on Δϕ and one of the azimuth angles, say, ϕ1 could be fixed, ϕ1=ϕ10 so that Eq. (3) could be rewritten as
SV¯ϕ(θ1,θ2)=12π∫02πSV(θ1,θ2,ϕ10,ϕ2)dϕ2.
Equation (11) will serve us below for the comparison with swept volumes of spheres. To compute SV, we need to determine the relative velocity between approaching particles. To this end, we introduce the terminal fall velocities of the spheroids in the following section.

5. Terminal velocity of Stokesian spheroidal particles

a. The translation tensor of Stokesian spheroids

In the following we assume that the particles’ Reynolds number is less than one and that we can treat them as Stokesian (Batchelor 1967; Happel and Brenner 1973). In application to cloud ice crystals, those with a maximum size below about 100–150 μm could be considered as Stokesian. For Stokesian particles there exists a linear relationship between the drag force F and the resulting particle velocity v. This relationship is manifested in the translation tensor K, which depends on the particle geometry and on the fluid viscosity (Happel and Brenner 1973)
F=K·v.
For spherical particles K is isotropic, i.e., F and v are parallel, and Eq. (12) reduces to the well-known Stokes law. For nonspherical particles K is in general not isotropic but is always symmetric. The terminal velocity of nonspherical particles, like ice crystals, is directed, in general, at some angle to the drag force (and to external force like gravitation), resulting in a fall velocity having both vertical and horizontal components. This is the consequence of K not being isotropic. In general there may exist a coupling between translational and rotational motions, i.e., translational motion may cause rotation, and vice versa, but no such coupling exists for spheroids (see, for example, Kim and Karrila 1991). Due to its symmetry the tensor K includes nine components of which six are different. The symmetry of K implies that there always exists a coordinate system in which K is diagonal, i.e., it contains only three independent components. This means that the friction force depends on three independent parameters (which depend on the body shape) and on the particle orientation. In the case of spheroids, the tensor K is diagonal when the axes of the body coordinate system are directed along the symmetry axes of the spheroid (Fig. 2). If, in addition, the rotation symmetry axis (there is one and only one for spheroids) is directed parallel to the third unit vector e3 (see Fig. 2), then K, written in body coordinates, has the following form:
K=(K⊥000K⊥000K∥).
Here, the symbol ∥ designates the direction of the rotation axis e3 and ⊥ the perpendicular plane. We define the aspect ratio Φ=C/A, where C is the half length of the rotation axis and A is the equatorial radius (Fig. 2). The elements of K are shown in Table 3, where μ is the dynamic viscosity. For prolate spheroids the elements of K are given in Happel and Brenner [1973, Eqs. (5-11.18), (5-11.19), (5-11.22), (5-11.23), (5-11.29)] and for oblate in Happel and Brenner [1973, Eqs. (5-11.20), (5-11.21), (5-11.24), (5-11.25)]. These expressions can also be found in Fuchs (1964), Happel and Brenner (1973, 222–226), Gallily and Cohen (1979), Chwang and Wu (1975), and Holmstedt et al. (2016).
Table 3

The elements of the translation tensor K.
Table 3
The translation tensor K (Table 3) could be nondimensionalized in two forms: as KA, with respect to the equatorial radius A, and as KV, with respect to the volume V, such that
K/μ=KA·A=(34π)1/3KV·V1/3,A=[3V/(4πΦ)]1/3.
The value of KA is obtained from Table 3 by dividing K by A. The relationship between KA and KV, as obtained from Eq. (14), is
KV=KA1Φ1/3.
It follows that KV → ∞ when Φ ≫ 1 for prolate spheroids, and when Φ ≪ 1 for oblate spheroids, when the volume is kept constant. Therefore, if a spheroid volume is kept constant, then at both limits of very long or very short rotation axes, the resistance to translation, described by the translation tensor, becomes infinitely large. (Note that we consider Stokesian particles of small Reynolds numbers. Accordingly, we do not consider such limiting cases in which K is too large. Nevertheless, the asymptotic behavior of relation 15 remains valid within a reasonable range of Re.)

b. Fall velocities parallel to gravity

The fall (terminal) velocities are derived from Eq. (12), where the drag F is equal to the gravity force. We first consider the case in which one of the principal axes of the spheroid is parallel to the gravity vector. The terminal velocity in this case would also be parallel to gravity:
υ⊥∞=VpρpgK⊥, υ∥∞=VpρpgK∥, Vp=4π3 A2C,
where Vp is the particle volume, ρp is the density, υ⊥∞ is the terminal velocity when the rotation axis is perpendicular to gravity, and υ∥∞ when it is parallel. We then define the nondimensional terminal velocity, normalized by the (scalar) terminal velocity υs∞, of a sphere of equal volume [Lamb 1945, p. 599; Batchelor 1967, Eq. (4.9.20); Pruppacher and Klett 1997]:
v¯¯∞=v∞/υs∞,υs∞=29 ρgμ V2/3(34π)2/3,
then
υ¯¯⊥∞=6πK⊥V, υ¯¯∥∞=6πK∥V.
The asymptotic expressions for very elongated particles (prolate) when Φ ≫ 1 and very flat (oblate) when Φ ≪ 1 when the volume is kept constant are shown in Table 4. In the case of prolate spheroids the nondimensional terminal velocity decreases to zero almost as Φ−2/3 for very elongated particles (large Φ). In the case of oblate spheroids in the limit of very thin flat particles (small Φ) and constant volume, the nondimensional terminal velocity decreases to zero as Φ1/3. Figure 6 shows the elements of the nondimensional translation tensor KV and the nondimensional terminal velocities v¯¯∞ as a function of the aspect ratio for constant volume.
Fig. 6.
Fig. 6.

(a) Elements of the nondimensional translation tensor KV and (b) the nondimensional terminal velocity v¯¯∞ as a function of the aspect ratio for constant particle volume. “par” indicates that the rotation axis is parallel to gravity, and “perp” that it is perpendicular.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

Table 4

Asymptotic values of terminal velocities when the symmetry axes are parallel to gravity and the volume is kept constant.
Table 4

It is observed that the nondimensional fall velocity of prolate spheroids decreases to zero faster than that of oblate spheroids. The ratio between the asymptotic fall velocities of vertically and horizontally oriented particles is 1.5–2 for both prolate and oblate spheroids. Most spheroids fall slower than the equivalent volume spheres (corresponding to Φ = 1), except for prolate spheroids oriented parallel to gravity whose aspect ratio Φ is no larger than about five.

Let us now consider the difference between the fall velocities of two spheroids defined by their density ρi and volume Vi. Let η be the ratio between the two volumes, η = V1/V2. It follows from Eq. (17)
Δv∞=29 gμ ρ2(34π V2)2/3[ρ1ρ2 η2/3v¯¯1∞−v¯¯2∞].
Figure 7 shows the absolute value of the nondimensional fall velocity difference [η2/3v¯¯1∞−v¯¯2∞] between two spheroids, for vertical and horizontal orientations, for volume ratios η = 1.02, 2, and 5, and for densities ρ1 = ρ2, as a function of the aspect ratio of the two spheroids. In these orientations the fall velocities are parallel to gravity. (We chose to show the absolute value of the velocity difference for the sake of comparison with the swept volume, shown below.) The aspect ratio range is from Φ = 10−2 to 102. This range covers the typical range of pristine crystals aspect ratios, from 0.1 to 10 (Pruppacher and Klett 1997), and even a wider range of aspect ratios, beginning from 0.02 (Naso et al. 2018; Connolly et al. 2012; Wang 2002). The gray lines in the figure are the geometrical counter diagonal (bottom-left to upper-right corner) and leading diagonal (top-left to bottom-right corner). These lines correspond to zero velocity difference in case of identical and identically oriented particles. The zones, separated by blue lines of zero velocity difference, are numbered. The figures reveal a certain symmetry in the fields of relative velocity. The deviation from the symmetry is related to the fact that prolate and oblate velocities at the same Φ and 1/Φ values are different (see Fig. 6b).3 Maximum velocity differences occur between spherical particles and spheroids with maximum aspect ratios. This corresponds to the results shown in Fig. 6b.
Fig. 7.
Fig. 7.

Absolute values of the normalized terminal velocity differences η2/3υ¯¯1∞−υ¯¯2∞ for different volume ratios η; θ = 0 for the parallel orientation and θ = π/2 for perpendicular. The different zones, numbered 1–4, are separated by strips of low terminal velocity difference.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

The aspect ratio effect on the relative velocity is better illustrated in Fig. 8, which shows cross sections from Fig. 7. Shown are normalized fall velocities differences between a sphere, Φ2 = 1, and spheroids whose aspect ratios change from Φ1 = 10−2 to 102 (Fig. 8a) and similarly for a sphere Φ1 = 1 and spheroids (Fig. 8b). The elevation angles are θ1 = θ2 = 0 (i.e., the rotation axis is parallel to gravity) and the volume ratios are η = V1/V2 = 1.02, 2, and 5. Figure 8 shows in detail the magnitude and the correct sign of the velocity difference in Fig. 7. (Note that it is immaterial, which is the collector and which is the collected spheroid.) Figure 8a shows the velocity difference between particle 1 (spheroid), the larger, and particle 2 (sphere). At aspect ratio Φ1 = 1 the two particles are spheres. The first sphere falls faster because its mass is larger (relative velocity is positive). If the aspect ratio is large (prolate) or small (oblate), the spheroidal particle falls slower than the sphere. Accordingly, the positive velocity difference is replaced by a negative one. Figure 8b shows the velocity difference when the particle with the larger volume (particle 1) is a sphere and the particle with the smaller volume is a spheroid. In this case the sphere falls almost always faster, and the velocity difference is actually positive. It is apparent from the η = 1.02 curve that spheres fall faster than spheroids of the same volume, except for a very small range of prolate aspect ratios close to spherical, in accord with Fig. 6. In the first column of Fig. 7 (Figs. 7a,d,g,j), which correspond to η = 1.02, the velocity difference in zones 1 and 3 is negative while it is positive in zones 2 and 4. The negative values in zones 1 and 3 are apparent in Fig. 8a (θ1 = θ2 = 0), where negative values are seen almost everywhere, as discussed above. The strictly positive values in zones 2 and 4 are in regions where the spheroid shape, of a slightly higher volume, is close to spherical. In Figs. 7b, 7e, 7h, and 7k (the second column of Fig. 7), which correspond to η = 2, the negative velocity differences in zones 1 and 3 have decreased while the positive velocity differences in zones 2 and 4 have expanded: this is because particles with twice the volume fall faster. In the right column (Figs. 7c,f,i,l), which correspond to η = 5, i.e., to a large volume difference, the negative velocity difference zones have decreased even further. Zone 1 has almost disappeared in Figs. 7c and 7f and so has zone 3 in Figs. 7i and 7l. This high volume ratio leads to a very homogeneous velocity difference in most of the aspect ratios range, irrespective of the orientations.

Fig. 8.
Fig. 8.

Cross sections of the normalized fall velocities differences between a spheroid of variable aspect ratio and a sphere (Φ = 1). (a) The sphere is particle 2; (b) the sphere is particle 1. The volume of the spheroid is larger in (a) and smaller in (b). The elevation angles are θ1 = θ2 = 0; the volume ratios are η = V1/V2 = 1.02, 2 and 5. This figure corresponds to the top row in Fig. 7. In (a) Φ1 > 1 corresponds to zone 1 while Φ1 < 1 to zone 3. In (b) Φ2 > 1 corresponds to zone 2 while Φ2 < 1 to zone 4. The zeroes here (marked with short vertical lines where the curves cross the red line) are located on the blue strips in Fig. 7.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

c. Fall velocities of arbitrarily oriented spheroids

Next we consider the case of an arbitrarily oriented rotation axis and compute the particle velocities in the space coordinate system. Equation (20) defines the transformation matrices from the space fixed to the body coordinate systems, where θ is the elevation angle and ϕ is the rotation about the z^ axis (see Fig. 2):
Tθ=(cos⁡ θ0−sin⁡ θ010sin⁡ θ0cos⁡ θ), Tϕ=(cos⁡ ϕsin⁡ ϕ0−sin⁡ ϕcos⁡ ϕ0001).
Due to sedimentation the terminal velocity v∞ in the space coordinate system is determined from the following equation:
TθϕTKTθϕv∞=ρpVpg, Tθϕ=TθTϕ,
where K here is in the body coordinate system, Eq. (13), and TT is the transposed transformation matrix. Performing the matrix multiplication and solving Eq. (21) for the fall velocities we get
(υ1∞υ2∞υ3∞)=−ρpVpg[12 cos⁡ϕ sin⁡2θ(1K∥−1K⊥)12 sin⁡ϕ sin⁡2θ(1K∥−1K⊥)1K⊥ sin⁡2θ+1K∥ cos⁡2θ]
(Happel and Brenner 1973, 203–205). The elements of K in Eq. (22) are defined in Table 3 and Eq. (15); υ∞ is the fall velocity in the space coordinate system. This relationship gives rise to a settling velocity, which is not parallel to gravity. In the case of spherical particles, where K⊥ = K∥ = 6πμr, we get the well-known expression of Eq. (17).
By differentiating the fall velocity magnitude, one finds that the maximum is achieved at θ = 0, i.e., the vertical position, and the minimum at θ = π/2, i.e., the horizontal position. We define the drift angle λ, with respect to the gravity direction:
λ=arctan⁡[−υ1∞υ3∞]
(ϕ = 0 for simplicity). Figure 9 presents the drift velocity direction in terms of the drift angle. Figure 10 shows the drift angle λ as a function of the aspect ratio Φ for various spheroid orientations θ. As expected, λ is maximal at θ ≈ 45° and equals to zero at θ = 0° and 90°. The drift angle λ ranges between 10° and 15° and it is antisymmetric with respect to θ = 0°. The drift angles of prolate and oblate spheroids are in opposite directions for the same elevation angle θ but the magnitudes are different. The effect of the side drift is to increase the relative velocity between spheroids. Indeed, Siewert et al. (2014b) found in their DNS simulations that the collision rate between spheroids that settle due to gravity in turbulent flows was much higher than the rate of collision between equivalent spheres of the same volume and mass.
Fig. 9.
Fig. 9.

A prolate spheroid drift velocity direction where θ is the elevation angle and λ is the drift angle.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

Fig. 10.
Fig. 10.

The drift angle λ as a function of Φ: prolate, Φ > 1; oblate, Φ < 1. λ is antisymmetric with respect to the elevation angle θ = 0.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

6. Swept volumes of spheroids of different aspect ratios

a. Design of the computational tests

In this section we present computational tests of the azimuthal mean normalized swept volumes SV¯ϕ(θ1,θ2) of pairs of spheroids with different aspect ratios. The mean was computed with respect to the azimuth angle ϕ of one of the spheroids [see Eq. (11)]. To analyze the effect of the nonsphericity of the spheroids, the swept volumes were normalized by the swept volume of equivalent volume spheres [Eq. (25)]. As before, the volume ratios were η = 1.02, 2, and 5. The swept volume of an equivalent volume sphere is given as
SVeq=π(r1+r2)2Δυ∞,
where ri are the equivalent volume spheres’ radii, ri = [(3/4π)Vi]1/3. Taking into account expression (17) for the terminal velocity of a Stokesian spherical particle, Eq. (24) can be rewritten:
SVeq=29(34)4/31π1/3 gμ(V11/3+V21/3)2|ρ1V12/3−ρ2V22/3|.
After introducing the ratio between volumes V1 and V2 in terms of the parameter η > 0, i.e., V1 = ηV2 we get
SVeq=29 gμ πr24(1+η1/3)2|ρ1η2/3−ρ2|.
In the examples shown below the aspect ratios of both spheroids are allowed to change but their volumes are kept constant. The terminal velocities are given in Eq. (22). The elements of K are obtained from Table 3, and the equatorial radius A is given in Eq. (14). The calculations were performed for nearly equal volumes of the colliding spheroids (η = 1.02), for moderate volume ratio (η = 2) and for large volume ratio (η = 5). We separate our results into two groups. In the first group the spheroids fall vertically, i.e., the relative velocities are parallel to gravity. The corresponding elevation angles θ1 and θ2 take the discrete values of 0 (parallel to gravity) and π/2 (perpendicular to gravity). In the second group θ1 or θ2 or both assume the value of π/4 so that Δv ∦ g and λ ≠ 0. In all the presented cases the densities of the particles are equal, i.e., ρ1 = ρ2. The mutual orientations of the colliding spheroids in the tests are illustrated in Fig. 11.
Fig. 11.
Fig. 11.
Fig. 11.

The colliding particles orientations in the tests. (I) A test with various vertical fall velocities; (II) a test with nonzero drift velocity and various horizontal relative velocities. Φ > 1 indicates prolate spheroids, and Φ < 1 indicates oblate spheroids. Index 1 indicates the collector (a larger volume); index 2 indicates the collected particles. The figure illustrates situations when one of the colliding particles has an aspect ratio Φ = 1, i.e., it is spherical. For the sake of analysis, the aspect ratios plane is divided into four regions, as annotated in the top-left panel of (a).

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

The actual swept volumes are obtained by multiplying the normalized swept volumes by the swept volumes of the equivalent spheres [Eq. (25)]. Since these quantities are independent of the aspect ratio, figures showing the actual swept volumes will look the same except for a multiplicative constant, which depends on the volume fraction η and on the radius r2 of the collected particle. Taking, for example, that radius to be r2 = 50 μm, then, in order to convert the normalized swept volumes, the multiplicative constants for η = (1.02, 2, 5) are SVeq(η) = (1.3 × 10−4, 7.1 × 10−3, 3.4 × 10−2) cm3.

b. Relative velocity parallel to gravity

1) Nearly equal volumes: η = 1.02

Figure 12 presents the azimuthal average of the normalized swept volumes for spheroid pairs, of volume ratio η = 1.02. The aspect ratios range between 0.01 and 100. The swept volumes are normalized by the swept volume of equivalent volume spheres [Eq. (25)]. As seen in Fig. 12, the normalized mean swept volumes can exceed values of 100. These high values are obtained because the swept volume of a pair of equivalent volume spheres of nearly equal volumes, used for the normalization, is very close to zero.

Fig. 12.
Fig. 12.

The azimuthal means of the swept volumes, normalized by equivalent volume spheres. Volume ratio η = V1/V2 = 1.02. υ∞∥ g.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

The swept volume of a spheroid pair is the outcome of the collecting area and the fall velocity joint effects. The structure of the swept volume fields in Fig. 12 resembles that of the relative normalized fall velocities in Fig. 7a, 7d, 7g, and 7j. It is characterized by small narrow strips of low values close to the diagonals that result from very low relative velocities. These strips define four separate zones of high values designated as zones 1, 2, 3, and 4 on the plots. In zone 1 spheroid 1 is prolate, in zone 2 spheroid 2 is prolate, in zone 3 spheroid 1 is oblate, and in zone 4 spheroid 2 is oblate. The other spheroid in each zone assumes all aspect ratios. It may be deduced from Fig. 8 (for θ1 = θ2 = 0) that zones 1 and 3 (Fig. 8a) correspond to zones of negative relative velocities whereas zones 2 and 4 (Fig. 8b) correspond to positive relative velocities. Unlike the relative velocity, these swept volume zones lack the symmetry of the fall velocities and may roughly be divided into two pairs of high and low values thus demonstrating the effect of the collecting area. The highest values in the different orientations are obtained in the following aspect ratio ranges:

  • In Fig. 12a, zones 3 and 4: Both spheroids are in the vertical position.

  • In Fig. 12b, zones 2 and 3: The first spheroid is in the vertical position while the second is in the horizontal position.

  • In Fig. 12c, zones 1 and 4: The first spheroid is in the horizontal position and the second in the vertical position.

  • In Fig. 12d, zones 1 and 2: Both spheroids are in the horizontal position.

In all four cases the highest values of the swept volume were obtained in the zones where both spheroids have their largest dimension normal to gravity, i.e., the rotation axis in the case of prolate spheroids and both radial axes in the case of oblate spheroids. Such a configuration results in the largest collecting area. In these zones the highest swept volume were obtained between a sphere and a spheroid, either extremely prolate or extremely oblate, in which case the relative velocity is high. As already mentioned above, the aspect ratios of pristine crystals typically range from 0.1 to 10 (Pruppacher and Klett 1997) but may even be as small as 0.02 (Naso et al. 2018; Connolly et al. 2012; Wang 2002). Note in this connection that the increase in the swept volume is large even at these aspect ratios and that many ice crystals formed by secondary nucleation, like breakup by collisions or ice splinter formation by drop explosion due to freezing, can have larger aspect ratios than the regular primary ice crystals, (Rangno 2008).

In all plots the counter diagonal (bottom-left to the upper-right corner) represents interactions of spheroids having the same aspect ratio. Therefore, if the orientations and volumes of the spheroids were the same it would serve as an axis of reflection. This is nearly the case in Figs. 12a and 12d where the orientations are the same and the volumes are only slightly different. For the same reason, Figs. 12b and 12c are an almost exact reflection of each other about this axis.

2) Moderate volume ratio: η = 2

Figure 13 presents the azimuthal average of the normalized swept volumes of spheroid pairs of volume ratio η = 2. The aspect ratios range between 0.01 and 100. The normalized swept volumes values are lower than in the case of η = 1.02 because the swept volumes of the equivalent volume spheres, used for the normalization, are much larger. The roles of the relative velocities and of the collecting area in this case are similar to those when η = 1.02. The structure of Fig. 13 resembles that of the relative fall velocities in Figs. 7b, 7e, 7h, and 7k. Zones 1 and 3 are surrounded by arcs of low values, which correspond to the arcs of zero relative fall velocities in Fig. 7. These zones correspond to negative relative velocities, whereas zones 2 and 4 correspond to positive relative velocities (see Fig. 8). As in the case of η = 1.02, the collecting area effect is apparent in the largest swept volume values. These values are obtained in zones where the largest dimension of the spheroid is perpendicular to gravity so that the collecting area is the largest.

Fig. 13.
Fig. 13.

The azimuthal means of the swept volumes normalized by equivalent volume spheres. Volume ratio η = V1/V2 = 2. The case υ∞∥ g.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

3) Large volume ratio: η = 5

Figure 14 presents the azimuthal average of the normalized swept volumes of spheroid pairs, of volume ratio η = 5. The aspect ratios range between 0.01 and 100. The normalized swept volume values are lower than when η = 2 because the normalizing equivalent sphere swept volumes are larger. A dual, but somewhat more complicated, effect of the relative velocity and the collecting area is observed. As already stated the relative velocity dominated by the large spheroid is much more uniform and barely sensitive to the orientation, except for when Φ1 ≈ 102 (zone 1) and Φ1 ≈ 10−2 (zone 3), in the right column of Fig. 7, where it approaches zero. This approach to zero has the largest effect on the swept volume in zone 3 of Figs. 14a and 14b and zone 1 of Figs. 14c and 14d. The reduced relative velocity acts against the collecting area tendency to increase the swept volume in these zones when η = 1.02 and 2 (see Figs. 12 and 13). In general the swept volume structure is relatively uniform, similar to the structure of the relative velocity.

Fig. 14.
Fig. 14.

The azimuthal means of the swept volumes normalized by equivalent volume spheres. Volume ratio η = V1/V2 = 5. The case υ∞∥ g.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

c. Relative velocity not parallel to gravity

Here we present azimuthal averages of the normalized swept volumes of pairs of spheroids with a relative velocity that is not parallel to gravity. Here too the swept volumes are normalized by the swept volume of equivalent volume spheres [Eq. (25)]. Since equivalent spheres have no horizontal relative velocity component, we observe an increase in the normalized swept volumes, especially for small volume ratios. The effect is illustrated in Figs. 15–17.

Fig. 15.
Fig. 15.

The azimuthal means of the swept volumes normalized by equivalent volume spheres. Volume ratio η = V1/V2 = 1.02. The case Δv ∦ g.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

Fig. 16.
Fig. 16.

The azimuthal means of swept volumes normalized by equivalent volume spheres. Volume ratio η = V1/V2 = 2. The case Δv ∦ g.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

Fig. 17.
Fig. 17.

The azimuthal means of the swept volumes normalized by equivalent volume spheres. Volume ratio η = V1/V2 = 5. The case Δv ∦ g.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

1) Nearly equal volumes: η = 1.02

Figure 15 presents the azimuthal mean of the normalized swept volumes of spheroid pairs of volume ratio η = 1.02. The aspect ratios range between 0.01 and 100. A comparison with the Fig. 12 shows a significant increase in the normalized swept volumes for colliding spheroids of nearly the same volumes. Most values, except for the central zone where the shapes of the spheroids are close to spherical, are close to 1000. As was mentioned above this effect is related to the fact that the swept volume of effective spheres of similar volumes is close to zero.

2) Moderate volume ratio: η = 2

Figure 16 presents the azimuthal mean of the normalized swept volumes of spheroid pairs, of volume ratio η = 2 and aspect ratios range between 0.01 and 100. The fields of normalized volumes at η = 2 (Fig. 16) resemble those in Fig. 13. However, the minimum values do not fall to zero because of the horizontal component’s effect of the relative velocity.

3) Large volume ratio: η = 5

The azimuthal averages of the normalized swept volumes of colliding spheroids with large volume ratios of η = 5 (Fig. 17) resemble those in Fig. 14. There are, however, significant differences. First, the normalized swept volumes in the presence horizontal components of the relative velocity never fall below ≈0.5. Second, the swept volumes in the case of Δv ∦ g are more symmetric with respect to the line Φ2 = 1 (cf. Fig. 17c and Fig. 14c). This, again, is related to the existence of significant horizontal relative velocity of spheroids with high aspect ratios.

7. Discussion and conclusions

In this work we developed a method to compute gravity-induced swept volumes of spheroidal particles in calm air. In clouds, prolate spheroids can be associated with columnar crystals, while oblate spheroids with plates and dendrites. The analytic expressions for the swept volumes are suitable for any particle size, and the numerical realization is simple and computationally efficient. Terminal velocities and swept volumes were computed for Stokesian particles, which correspond to ice crystals up to about 100–150 μm. The azimuthal averaged swept volumes were shown to be dependent on particle mass, aspect ratios and the elevation angles of the colliding particles. To investigate the role of these parameters, the swept volumes were normalized and compared with swept volumes of equivalent volume spheres that have identical density and volumes as the corresponding spheroids.

The significant variability in the swept volumes is related to variability in the aspect ratios and orientations. Most swept volumes, normalized to the corresponding swept volumes of equivalent spheres, are larger than 1 and can be as large as 1000. The horizontal component of the relative velocity between the spheroids increases the normalized swept volumes. In this study we considered spheroidal particles of small Reynolds numbers whose fall velocity is not necessarily parallel to gravity. Note that in microphysical models it is usually assumed that ice crystals of any shape fall parallel to gravity. Such an assumption may be wrong, not only for small Stokesian particles, but also for larger particles due to nonspherical shape, oscillations performed in larger Re, and an uneven distribution of mass. An additional horizontal velocity component may increase the collision rates. The DNS computation of Siewert et al. (2014b) exhibits a significant increase in the collision rate of spheroids relative to that of spheres of equal volume. Therefore, we conclude that utilization of the effective spheres assumption when computing collisions between spheroids could lead to a substantial underestimation of the aggregation rate, particularly if ice crystals are modeled as spheres. This underestimation is especially important because of the nonlinear feedback in the collision process: the increase in the collision kernel leads to a nonlinear increase in the collision rate.

In this study we present a solution to the first problem related to collisions of nonspherical particles, namely, calculation of the swept volume of nonspherical particles in calm air. In addition to the swept volume, a complete calculation of the collision kernel requires the determination of the collision efficiency. Unlike spherical drops, collision efficiency between ice crystals and between ice crystals and drops is expected to depend on their aspect ratios and their orientation distribution. The general problem of collision of nonspherical particles requires consideration of additional two important mechanisms: hydrodynamic interaction between colliding particles and the turbulence effects on collisions. Problems of hydrodynamic interaction and turbulent effects in the case of nonspherical particles are much more complicated than those in spherical droplets because the rotation of the particles should be taken into account in addition to their translational motions.

The calculation of the swept volume, presented in this study, is the first important step toward the investigation of collisions between nonspherical particles in clouds. The next steps will include the incorporation of hydrodynamic interactions and the effects of turbulence.

1

Change of orientations of Stokesian particles in calm air may occur due to hydrodynamic interactions. These interactions become important when the distance separating the two particles is on the order of a few times the size of the particles. The change of the orientations may be neglected if 1) the relative velocity of the particles is large enough so that the time they spend in the vicinity of each other is short enough and 2) the relaxation times of the two particles are long enough. We are aware that this condition might be a little too stringent. See section 7 for a further consideration of this point.

2

For a 2D domain in the (x, y) plane, bounded by a curve L, define F = x, G = −y. Then S=(1/2)∬(∂F/∂x−∂G/∂y)dx dy=∮Lx dy−y dx.

3

To gain some understanding of this asymmetry note first that the terminal velocity, when parallel to gravity, is inversely proportional to the corresponding elements of the translation tensor. For parallel and perpendicular orientations the drag increases with increasing Φ and 1/Φ thus showing a direct proportionality to the surface area (at constant volume the surface area is minimal for a sphere), which is reasonable for Stokes flow where viscosity plays the main role. On the other hand, it is seen in Fig. 6a that the drag is also proportional to the cross-section area, perpendicular to the motion, a behavior typical of pressure, which balances the viscous forces. Thus, for example, the drag on an oblate spheroid with its rotation axis parallel to the motion, is larger than that on an oblate spheroid when its rotation axis is perpendicular to the motion. An opposite behavior is observed for the prolate spheroid. It is therefore the interplay between these two mechanisms that determines the actual drag.

Acknowledgments.

The study was supported by the Israel Science Foundation (Grant 2027/17) and the U.S. Department of Energy (Grants DE-FOA-0001638, DE-AC05-76RL01830 81). We thank Kaushal Gianchandani for his help and valuable suggestions.

APPENDIX A

Computation of w—The Collecting Area Boundary

Referring to Fig. 5 we define a local coordinate system whose two axes are located in the normal plane and are parallel to the major and minor axes of the projection of the collecting particle (an ellipse). These two axes are e˜1 and e˜2, defined above. Let the projection of the collecting particle be located at the origin of this coordinate system, such that u is the contact point with the projection of the collected particle (another ellipse) and y = u − w is the vector connecting w and u. Similarly we define a coordinate system whose axes are parallel to the major and minor axes of the collected particle projection. Let T be the transformation matrix from the coordinate system of the collecting particle to the coordinate system of the collected particle:
T=(cos⁡Δϕ˜sin⁡Δϕ˜−sin⁡Δϕ˜cos⁡Δϕ˜).
The equations of the two ellipses (the projected spheroids) in the main coordinate system, i.e., the coordinate system of the collecting particle projection, are
Ψ1=uTE1u=Ψ2=yTTE2TTy=1,
where
E1=(1a12001b12), E2=(1a22001b22), y=u−w.
The superscript T denotes the transpose of a vector or matrix. At the point of contact, the normals to the circumferences of the two ellipses are parallel; therefore,
α∇Ψ1=∇Ψ2
for some scalar α that has to be determined. Computing the gradients, using Eqs. (A2) and (A3) we get
αE1u=TE2TT(u−w)⇒y=u−w=αT(E2)−1TTE1u,
from which we get the equation for w:
w=L(u)u,L(u)=[I−α(u)N],
N=T(E2)−1TTE1×[1a12(a22 cos⁡2 Δϕ˜+b22 sin⁡2 Δϕ˜)−12b12(a22−b22)sin⁡ 2Δϕ˜−12a12(a22−b22)sin⁡ 2Δϕ˜1b12(a22 sin⁡2 Δϕ˜+b22 cos⁡2 Δϕ˜)],
where I is the identity matrix. In particular, we get the expression for the distance of the center of the collected particle projection from the large axis of the collecting particle projection, expressed in terms of the main coordinate system (see Fig. 5),
w2=L2juj=L21u1+L22u2.
To determine α, substitute the expression for y, Eq. (A5) in Eq. (A2):
Ψ2=1=yTTE2TTy =α2uTE1T(E2)−1TTTE2TTT(E2)−1TTE1u=α2uTMu,M=E1T(E2)−1TTE1=E1T(E2)−1(E1T)T=E1N.
We relied here on the fact that E1 and E2 are symmetric and TTT = I:
M=E1N=[1a14[a22 cos⁡2Δϕ˜+b22 sin⁡2Δϕ˜]−12a12b12(a22−b22)sin⁡2Δϕ˜−12a12b12(a22−b22)sin⁡2Δϕ˜1b14[a22 sin⁡2Δϕ˜+b22 cos⁡2Δϕ˜]].
We therefore have
α2(u)=1uTMu.
Since M is a symmetric matrix we get
uTMu=M11u12+M22u22+2M12u1u2.
In particular, at the end points u1 = ±a1, u2 = 0 or u2 = ±b1, u1 = 0, the points on the intersection of the circumference of the projected collecting particle with its major axes (see Fig. 5) we get, using the relation u22=b12(1−u12/a12),
uTMu|u1=±a1=M11a12, α=±1a1M11,uTMu|u2=±b1=M22b12, α=±1b1M22.
Note that we should always choose the negative value for α as ∇Ψ1 and ∇Ψ2 are in opposite directions. Note also that α depends on u and on Δϕ˜, but that α(−u) = α(u). The collecting area S(1,2) =S(a1,b1,a2,b2,Δϕ˜) is determined by the points w, the locations of the centers of the collected ellipse as it is moved around the collecting ellipse while keeping Δϕ˜ constant (see Figs. 4, 5). These locations are determined by Eq. (A6), namely,
w1=L11u1+L12u2,w2=L21u1+L22u2.

The computation order is as follows:

  1. Compute θ˜i, ϕ˜i and Δϕ˜ from Eq. (6).

  2. Compute ai and bi from Eq. (8).

  3. Choose a point u on the boundary of the collecting particle projection on the normal plane.

  4. Compute N from Eq. (A7).

  5. Compute M from Eq. (A10).

  6. Compute α from Eqs. (A11) and (A13).

  7. Compute L from Eq. (A6).

  8. Compute w from Eq. (A14).

APPENDIX B

Computation of the Collecting Area

The collecting area is the area bounded by the line formed by the centers w of the collected particle projection on the normal plane. Let Γ be the set of the points w. The domain bounded by Γ is clearly convex. The points (w1, w2) ∈ Γ are obtained from Eq. (A14) for all u1 ∈ (−a1, a1) and u2=±b11−(u1/a1)2. Note that Eq. (A14) is nonlinear due to the term α that appears in L. The collecting area is computed numerically from the set of points (w1, w2) using Green theorem:
S(1,2)=12∮Γw1 dw2−w2 dw1
(the integration is performed counterclockwise).
Referring to Fig. B1, let P be the leftmost position of w, formed when the point of contact of the two spheroids is on the x axis; let Q be the point on the right-hand side of the collecting particle when w lies on the x axis; and let R be the reflection of P, i.e., R = −P. The vectors P, Q, and R are expressed in terms of the main coordinate system. w is above the x axis when it moves from P to Q, and below the x axis as it moves from Q to R. The components of u at the end points are
u1(P)=−a1, u2(P)=0,  u1(R)=a1, u2(R)=0.
Therefore, using Eq. (A8),
w1min⁡=w1(P)=P1=L11u1(P)+L12u2(P)=−L11a1,w2(P)=P2=L21u1((P)+L22u2(P)=−L21a1,w(R)=R=−P=−w(P),w1max⁡=w1(R)=R1=L11a1.
For point Q, Q2=0=L21u1+L22u2⇒u2=−(L21/L22)u1. Substituting in the ellipse equation and using the linear relation between Q and u again, as in Eq. (B3), we get u on the circumference of the collecting ellipsoid projection, corresponding to w = Q:
u1(Q)=a1b1L22a12L212+b12L222, u2(Q)=−a1b1L21a12L212+b12L222,w1(Q)=Q1=L11u1+L12u2=a1b1a12L212+b12L222 det(L).
The area we seek is
12S12=∫−L11(a1)a1L11(a1)a1|w2(x)| dx=∫P1Q1w2 dx−∫Q1R1w2 dx.
Fig. B1.
Fig. B1.

Calculation of the collecting area.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-20-0336.1

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Journal of the Atmospheric Sciences

  • Abraham, F. F., 1970: Functional dependence of drag coefficient of a sphere on Reynolds number. Phys. Fluids, 13, 2194–2195, https://doi.org/10.1063/1.1693218.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ayala, O., W. W. Grabowski, and L.-P. Wang, 2007: A hybrid approach for simulating turbulent collisions of hydrodynamically-interacting particles. J. Comput. Phys., 225, 51–73, https://doi.org/10.1016/j.jcp.2006.11.016.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ayala, O., B. Rosa, and L.-P. Wang, 2008: Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 2. Theory and parameterization. New J. Phys., 10, 075016, https://doi.org/10.1088/1367-2630/10/7/075016.

    • Crossref
    • Export Citation

  • Batchelor, G. K., 1967: An Introduction to Fluid Dynamics. Cambridge University Press, 615 pp.

    • Export Citation

  • Benmoshe, N., and A. Khain, 2014: The effects of turbulence on the microphysics of mixed-phase deep convective clouds investigated with a 2-D cloud model with spectral bin microphysics. J. Geophys. Res. Atmos., 119, 207–221, https://doi.org/10.1002/2013JD020118.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Benmoshe, N., M. Pinsky, A. Pokrovsky, and A. Khain, 2012: Turbulent effects on the microphysics and initiation of warm rain in deep convective clouds: 2-D simulations by a spectral mixed-phase microphysics cloud model. J. Geophys. Res., 117, D06220, https://doi.org/10.1029/2011JD016603.

    • Search Google Scholar
    • Export Citation

  • Böhm, H. P., 1989: A general equation for the terminal fall speed of solid hydrometeors. J. Atmos. Sci., 46, 2419–2427, https://doi.org/10.1175/1520-0469(1989)046<2419:AGEFTT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Böhm, H. P., 1992a: A general hydrodynamic theory for mixed-phase microphysics. Part I: Drag and fall speed of hydrometeors. Atmos. Res., 27, 253–274, https://doi.org/10.1016/0169-8095(92)90035-9.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Böhm, H. P., 1992b: A general hydrodynamic theory for mixed-phase microphysics. Part II: Collision kernels for coalescence. Atmos. Res., 27, 275–290, https://doi.org/10.1016/0169-8095(92)90036-A.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Böhm, H. P., 1992c: A general hydrodynamic theory for mixed-phase microphysics. Part III: Riming and aggregation. Atmos. Res., 28, 103–123, https://doi.org/10.1016/0169-8095(92)90023-4.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Chen, J.-P., and D. Lamb, 1999: Simulation of cloud microphysical and chemical processes using a multicomponent framework. Part II: Microphysical evolution of a wintertime orographic cloud. J. Atmos. Sci., 56, 2293–2312, https://doi.org/10.1175/1520-0469(1999)056<2293:SOCMAC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Chwang, A. T., and T. Y.-T. Wu, 1975: Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for stokes flows. J. Fluid Mech., 67, 787–815, https://doi.org/10.1017/S0022112075000614.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Connolly, P., C. Emersic, and