1. Introduction
In the study of frozen hydrometeors, disk-like particles can be used to reproduce the settling dynamics of ice plates and dendrite crystals at laboratory scale (Willmarth et al. Reference Willmarth, Hawk and Harvey1964; Locatelli & Hobbs Reference Locatelli and Hobbs1974; Kajikawa Reference Kajikawa1992; Garrett & Yuter Reference Garrett and Yuter2014; Li et al. Reference Li, Guala and Hong2023; Singh et al. Reference Singh, Pardyjak and Garrett2023; Li et al. Reference Li, Guala and Hong2024; Tinklenberg et al. Reference Tinklenberg, Guala and Coletti2023, Reference Tinklenberg, Guala and Coletti2024). The terminal velocity of hydrometeors is reached when the drag and the weight forces acting on the particles are balanced. The drag force on disk-like settling particles is defined by
$F_d=({1}/{2})\rho C_D A_s V_t^2$
, based on the terminal velocity
$V_t$
, the fluid density
$\rho$
, the projected solid drag area
$A_s$
(determined by its orientation with respect to the vertical direction coincident with the settling velocity) and the drag coefficient
$C_D$
. The latter accounts for the viscous shear stress between the fluid and the particle, which dominates at low Reynolds number
$Re=V_tD/\nu$
, (where
$\nu$
is the kinematic viscosity) and the pressure gradient established between the leading and trailing surfaces of the particle, which represents a form drag contribution. This depends on the particle orientation and wake, and thus on
$Re$
. In the study of the drag of disk-like or nearly spherical frozen hydrometeors, such as plates or graupels, it is typically assumed that the nominal frontal area for a particle of diameter
$D$
is
$A=\pi D^2/4$
. However, this is not always a correct assumption. When assuming a nominal frontal area,
$C_D$
should include effects of preferential orientation of settling hydrometeors and unsteadiness due to fluttering and/or tumbling motions, both contributing to temporal fluctuations of the frontal area. This implies a strong coupling between the snow’s rotational and translational dynamics, especially for anisotropic particles that tend to be rotationally unstable. In the limit of a thin disk, the diameter to thickness aspect ratio
$\chi =D/h$
and the solid-to-fluid density ratio
$\tilde {\rho }=\rho _s/\rho$
, are the two parameters controlling the inertia ratio
$I^*=(\pi /64) \tilde {\rho }/\chi$
. Here,
$I^*$
represents the moment of inertia of the disk normalised by a term
$\rho D^5$
, proportional to the moment of inertial of a circumscribed sphere of surrounding fluid (Willmarth et al. Reference Willmarth, Hawk and Harvey1964);
$I^*$
plays a key role in the falling style and rotational dynamics of a settling disk. In particular, in the non-Stokesian, fully nonlinear drag regime relevant to frozen hydrometeors, that often settle at the terminal velocity
$V_t$
. This leads to
$Re\gt 100$
(Tinklenberg et al. Reference Tinklenberg, Guala and Coletti2023), under which the Jefferey solution for the viscous torque does not apply (Jeffery Reference Jeffery1922). The stabilising effect induced by increasing rotational inertia,
$I^*$
, has been correlated with the decrease of the rotational Strouhal number,
$Str=fD/V_t$
, which is a dimensionless form of the disk tumbling frequency,
$f$
, and is proportional to the tip speed ratio of the rotating disk (Willmarth et al. Reference Willmarth, Hawk and Harvey1964; Auguste et al. Reference Auguste, Magnaudet and Fabre2013; Tinklenberg et al. Reference Tinklenberg, Guala and Coletti2023).On the other hand, inertia can also trigger oscillatory motions in falling particles. Frozen hydrometeors are usually associated with values of
$I^*$
orders of magnitude larger than in typical laboratory experiments in water, due to the much higher solid-to-fluid density ratio (Stout et al. Reference Stout, Westbrook, Stein and McCorquodale2024). This has been indicated as the cause of oscillations of non-spherical particles falling in air at
$Re$
as low as 5 (Bhowmick et al. Reference Bhowmick, Seesing, Gustavsson, Guettler, Wang, Pumir, Mehlig and Bagheri2024), which contrasts with observations in water.
When approaching realistic snow morphologies during natural snowfall, an additional complication must be accounted for. Due to riming, the snow crystal morphology can vary widely not only in the overall shape of the drag area, but also in how the internal structure evolves. This affects the particle density and porosity (Locatelli & Hobbs Reference Locatelli and Hobbs1974; Garrett & Yuter Reference Garrett and Yuter2014; Li et al. Reference Li, Guala and Hong2023). The presence of pores (and in general openings) within the hydrometeor structure allows the flow to penetrate it during its descent. This permeability has not been directly considered in studies of frozen hydrometeors, but was systematically investigated in recent studies of rising and falling thin plates, indicating a profound influence on the falling style (Vagnoli et al. Reference Vagnoli, Zampogna, Camarri, Gallaire and Ledda2023; Sánchez-Rodríguez & Gallaire Reference Sánchez-Rodríguez and Gallaire2025).
Frozen hydrometeor density has been observed to widely vary depending on morphological types, wind, air temperature and humidity conditions, and still represents an area of active research (Magono & Lee Reference Magono and Lee1966; Locatelli & Hobbs Reference Locatelli and Hobbs1974; Brandes et al. Reference Brandes, Ikeda, Thompson and Schönhuber2008; Li et al. Reference Li, Guala and Hong2023; Singh et al. Reference Singh, Pardyjak and Garrett2023). Irregularities in the nominal area and solidity of snow plates and dendrites have been accounted by List & Schemenauer (Reference List and Schemenauer1971) and Böhm (Reference Böhm1989) in the formulation of a modified drag coefficient curve
$C_D=f(Re,\,A_R)$
, which incorporates the ratio
$A_R$
of the crystal cross-sectional solid area
$A_s$
to that of a circumscribed disk
$A$
. The Böhm (Reference Böhm1989) drag model incorporates the development of shear layers from the particle edges, assuming a flat, horizontal falling style, i.e. with the largest nominal area exposed to the relative flow.
More recently, a variety of laboratory studies aimed to analyse the area ratio effect more systematically. Vincent et al. (Reference Vincent, Shambaugh and Kanso2016) considered annular disks settling, i.e. disks with a singular centred hole, and compared their laboratory findings with solid equivalents. As the central hole size is increased, they find strengthening stabilisation due to the development of an additional counter-rotating vortex ring at the hole edge, which acts in opposition to the vortex ring shed at the outer edge. This was confirmed in the recent laboratory experiments of Zhang et al. (Reference Zhang, Bi and Wei2023). The stabilising effect forces the tumbling behaviour of the corresponding solid disk to be dampened into a fluttering motion, in particular for
$d_i/d_o \approx 0.3 \pm 0.1$
, where
$d_i$
is the inner hole diameter and
$d_o$
is the outer diameter of the disk, corresponding to an area ratio of
$\simeq 0.9$
. Even if the single hole shape is not representative of dendrites, the physical connection between disk permeability and rotational stability is of interest. Westbrook & Sephton (Reference Westbrook and Sephton2017) and McCorquodale & Westbrook (Reference McCorquodale and Westbrook2021a
) approach this problem with direct relevance to frozen hydrometeors through the use of 3D-printed analogues of complex ice particles. Their laboratory experiments confirm the robustness of the model proposed by Böhm (Reference Böhm1989), and later modified by Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010), except in the case of very thin and flat particles. While these experiments were able to match the particle Reynolds number, the settling dynamics in liquids did not preserve
$I^*$
, thus limiting the range of falling styles and the coupled rotation–translation that frozen hydrometeors may experience in air (Kajikawa & Okuhara Reference Kajikawa and Okuhara1997). At the field scale, Auer & Veal (Reference Auer and Veal1970) and Vázquez-Martín et al. (Reference Vázquez-Martín, Kuhn and Eliasson2021) suggested that ice plates are among the shape groups with the lowest area ratios, and in turn exhibit reduced velocities, highlighting the importance of this parameter, along with particle anisotropy, in drag force predictive models. Li et al. (Reference Li, Guala and Hong2024) measured the shape and density of natural snow particles and identified events with dominant morphological types: dendrites and plates were again observed to have a higher drag coefficient with respect to the Böhm (Reference Böhm1989) model. Both results indicate that plates and dendrites experience higher drag than approximately round particles, such as graupel, with the same nominal diameter. This suggests that a flat preferential orientation characteristic of fluttering, as opposed to tumbling, is more likely to occur.
Measurements during natural snow fall defines the parameter space of interest, but faces another additional complication represented by the effect of ambient turbulence. Particle–turbulence interactions cover a range of ubiquitous and intricate mechanisms that depend on particle size, (an)isotropy, inertial properties and on the ambient turbulent scales (Voth & Soldati Reference Voth and Soldati2017; Brandt & Coletti Reference Brandt and Coletti2022). Preferential sweeping induces high velocity paths along the downward side of vortical motions where (usually spherical sub-Kolmogorov) particles tend to settle faster than in quiescent flows (Wang & Maxey Reference Wang and Maxey1993; Aliseda et al. Reference Aliseda, Cartellier, Hainaux and Lasheras2002; Garrett et al. Reference Garrett, Fallgatter, Shkurko and Howlett2012; Ferrante & Elghobashi Reference Ferrante and Elghobashi2003; Li et al. Reference Li, Abraham, Guala and Hong2021b
). Loitering, cross-flow-induced drag or other nonlinear drag effects drive particles along convoluted or high flow resistance paths, reducing their average settling velocity (Singh et al. Reference Singh, Pardyjak and Garrett2023; Bagheri & Bonadonna Reference Bagheri and Bonadonna2016; Tinklenberg et al. Reference Tinklenberg, Guala and Coletti2024). In addition, particle rotational instabilities, governing the statistical occurrence of tumbling and fluttering, were observed to be amplified by the fluctuating velocities of ambient turbulent flows, even if the tumbling frequency was primarily controlled by the disk inertial properties
$I^*$
(Tinklenberg et al. Reference Tinklenberg, Guala and Coletti2024). Both preferential orientation and the particle angular velocity are inferred to modulate the particle wake and the drag coefficient. Therefore, experimental studies of plate crystals and dendrites settling in a laboratory set-up should include (i) a large number of realisations in ambient turbulence, so that the cumulative effect of different falling styles can be statistically captured in the settling velocity ensemble averages, and (ii) a similarly large sample in quiescent flow, so that the effect of turbulence can be disentangled from the variability of falling styles and orientations.
We call attention to the lack of studies of complex particle geometries performed in air turbulence. Besides field data collected in the atmospheric surface layer, there is little focus on realistic particle geometries, including permeability and anisotropic features as both area and aspect ratios, and the interaction with ambient turbulence and wind of varying intensity. This is particularly relevant for the prediction of the terminal velocities of snow plates and dendrites in atmospheric turbulence, which present the challenges of being thin, perforated and prone to tumbling. In our current laboratory study, we control the disk material properties and geometries, thus limiting the uncertainty associated with the particle density, size and weight, which are challenging to measure during natural snowfalls, along with their direct effect on the terminal velocity. Our set-up allows us to explore particles with size, density and fall speed comparable to differently rimed snow dendrites and plates (Locatelli & Hobbs Reference Locatelli and Hobbs1974) in turbulent air flows comparable to low–moderate wind conditions during snowfalls in the atmospheric surface layer (Nemes et al. Reference Nemes, Dasari, Hong, Guala and Coletti2017; Li et al. Reference Li, Guala and Hong2024). The main questions we address in this work are on the stabilising effect of disk perforations under quiescent and ambient turbulent flows, and on the relevance of the area ratio in the prediction of the particle drag and terminal velocity. We eventually hope to shed some light on the coupling between tumbling and falling motions for solid and perforated disks, and better assess the snow particle drag modelling approach proposed by Böhm (Reference Böhm1989) and Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) under well controlled laboratory conditions.
In our ad hoc designed laboratory facility, we cover the relevant region in the
$I^*-Re$
parameter space (Kajikawa & Okuhara Reference Kajikawa and Okuhara1997) and generate root mean square (r.m.s.) velocity fluctuations
$u'$
consistent with atmospheric flows, reaching a Reynolds number
$Re_\lambda =u'\lambda /\nu \simeq 400$
, where
$\lambda$
is the Taylor micro-scale (Carter et al. Reference Carter, Petersen, Amili and Coletti2016). We vary the disk thickness, area ratios and perforation geometry, and quantify their effect on the fall speed, drag coefficient and rotational kinematics, extending the dimensionless parameter space investigated by Tinklenberg et al. (Reference Tinklenberg, Guala and Coletti2023, Reference Tinklenberg, Guala and Coletti2024).
The paper is organised as follows. In the following section (§ 2) we describe the disk inertial properties along with snow plate and dendrite drag models, and we provide an overview of the experimental set-up. Section 3 is devoted to the experimental results on the disk translational dynamics (average settling velocity, drag coefficient and trajectory angle) and rotational dynamics (orientation and tumbling) in quiescent and turbulent conditions. Our experimental observations are compared with field-scale results in the discussion (§ 4). Conclusions follow.
2. Methodology
2.1. Disk properties and fabrication
The particles utilised in the laboratory experiments are thin disks of both solid and perforated geometries. Figure 1(
$a$
) shows images of every disk type considered. All of the solid disks are made of polyethylene terephthalate (PET), with a density of
$\rho _d=1380$
kg m−3. Disks with nominal diameters of 0.3 mm, 0.5 mm, 1 mm, 2 mm and 3 mm (as referenced in table 1) are manufactured for use as commercial glitter, all with a thickness of
$h =$
50 µm. Their settling dynamics is described in Tinklenberg et al. (Reference Tinklenberg, Guala and Coletti2024). Thicker disks of
$h \approx 100$
µm, are manufactured with a commercial laser cutter out of PET sheets. Two classes of such disks are produced: a solid geometry shown in figure 2(
$a$
) and a four-quadrant, perforated geometry shown in figure 2(
$b$
). Both are approximately 3 mm in diameter; we refer to them as 3 mm
$^*$
and 3 mm
$^{\times }$
, respectively. The single hole (3 mm
$^{\circ }$
) disks were selected to match the open area of the four quadrant (3 mm
$^{\times }$
) disks. The laser cutting method which produced the 3 mm
$^{\times }$
did not warrant sufficient precision in the case of the 3 mm
$^{\circ }$
, in particular with respect to the centre location of the single hole. The 3 mm
$^{\circ }$
were thus commercially manufactured as sequins, with a measured density of 2454 kg m-
$^3$
, and chosen for the similar diameter and open area ratio of the 3 mm
$^{\times }$
.
Images of all disks studied. (
$a$
) From left to right,
$D$
increasing from 0.3 to 3 mm
$^{\circ }$
. (
$b$
) Microscope images of 0.3 mm and of (
$c$
) 0.5 mm disks.
Disk properties including the measured mean disk diameter
$D$
, the measured mean disk thickness
$h$
, both listed with
$\pm$
one standard deviation (
$\sigma$
). Also included are the diameter to thickness aspect ratio
$\chi =D/h$
, the density ratio
$\tilde {\rho }=\rho _d/\rho _f$
, the disk area ratio
$A_R = A_s/A$
, the Galileo number
$Ga=U_gD/\nu$
, where
$U_g=\{2|\tilde {\rho }-1|gh\}^{1/2}$
is the gravitational velocity, and the inertia ratio
$I^*=(\pi /64)\tilde {\rho }/\chi$
. To calculate
$I^*$
of the perforated disk, we use the geometry on which the laser-cutter manufacturing is based. The settling velocity number,
$Sv_L = V_{t,0}/u'$
, is listed for the both turbulence conditions tested as
$Sv_L(w)$
and
$Sv_L(s)$
, referring to weaker and stronger turbulence, respectively (see table 2 for the flow statistics).
Image groupings of laser cut and perforated disks showing the variation within the populations of the (
$a$
) 3 mm
$^*$
, (
$b$
) 3 mm
$^{\times }$
and (
$c$
) 3 mm
$^{\circ }$
.
(a) Turbulence facility diagram including global coordinate system and basic dimensions. (b) Definition of the disk trajectory angle
$\phi$
. (c) Representation of the disk orientation vector
$\boldsymbol {\hat {p}}$
and its vertical component
$p_y$
. (d) Definition of the disk solid area
$A_s$
, nominal diameter
$D$
, and thickness
$h$
.
Dimensions of a subset of 30 disks per diameter are directly measured and yield the mean and standard deviations listed in table 1, along with the key physical parameters for each disk type. The diameters are obtained by imaging disks lying on a tray, while the thicknesses are obtained by tightening calliper teeth on stacks of various numbers of disks and calculating the average thickness in each stack, confirming the specification from the vendor. The 0.3 mm and 0.5 mm disks are hexagonal in shape (see figure 1
$(b,c)$
) and we take
$D$
as the diameter of the circumscribing circle, as is common in the hydrometeor literature (Böhm Reference Böhm1989; Heymsfield & Westbrook Reference Heymsfield and Westbrook2010).
The area ratio is defined as
$A_R = A_s/A$
, where
$A_s$
is the actual solid area of each disk geometry and
$A=\pi D^2/4$
is the solid disk area of the minimum circumscribing circle of
$A_s$
(see figure 3). As the disk orientation is unknown and often rapidly varying during tumbling or fluttering motions, we consider all projected areas always corresponding to the maximum drag area of a flat (or horizontal) falling disk. For the sake of notation clarity, we recall that the solid area
$A_s$
is defined as
$A_e$
in Böhm (Reference Böhm1989) and as
$A$
in Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010). Here,
$A_R$
is measured for a subset of 100 disks for 3 mm
$^*$
, 3 mm
$^{\times }$
and 3 mm
$^{\circ }$
by imaging the individual disks with a calibration length in the frame. Binarisation of the image allows the solid disk pixels to be identified and compared with the area of the minimum encompassing circle of each. These values are also summarised in table 1. Note that
$A_R = 1$
for all the solid disk types. Disk geometric and kinematic variables, along with a basic schematics of the experimental apparatus and reference system, are illustrated in figure 3.
Falling style parameter space of
$I^*$
versus
$Ga$
with disks in the current study placed among data from the literature. Solid black lines show the falling mode boundaries identified by Auguste et al. (Reference Auguste, Magnaudet and Fabre2013). Dashed black lines indicate upper and lower boundaries of the region of bistability found by Lau et al. (Reference Lau, Huang and Xu2018).
2.2. Disk kinematics phase space
The pioneering laboratory work of Willmarth, Hawk & Harvey (Reference Willmarth, Hawk and Harvey1964) established the use of the inertia ratio
$I^* = I/(\rho _f D^5)$
, normalising the disk moment of inertia
$I$
(around the central axis parallel to its diameter) with a scaling quantity
$\rho _f D^5$
proportional to the moment of inertia of a surrounding fluid sphere. This formula is used to calculate the inertia ratio for the perforated geometries. For a solid thin disk of thickness
$h$
, the inertia is derived as
$I^*=(\pi /64) \tilde {\rho }/\chi$
, with the aspect ratio
$\chi =D/h$
assumed much larger than unity (Field et al. Reference Field, Klaus, Moore and Nori1997; Auguste, Magnaudet & Fabre Reference Auguste, Magnaudet and Fabre2013; Lau, Huang & Xu Reference Lau, Huang and Xu2018). The parameter
$I^*$
has been used to predict the different falling styles in a parameter space along with the Galileo number
$Ga=U_g D/\nu$
(Chrust, Bouchet & Dušek Reference Chrust, Bouchet and Dušek2013; Moriche, Uhlmann & Dušek Reference Moriche, Uhlmann and Dušek2021; Moriche et al. Reference Moriche, Hettmann, García-Villalba and Uhlmann2023; Tinklenberg et al. Reference Tinklenberg, Guala and Coletti2023). The convenience of using
$Ga$
rather than
$Re$
stems from the a priori definition of the gravitational velocity
$U_g=\{2|\tilde {\rho }-1|gh\}^{1/2}$
(where
$g$
is the gravitational acceleration) which is a reasonable approximation of
$V_t$
if the drag coefficient
$C_D$
is of order unity (and
$Re$
independent). Figure 4 depicts the
$I^*-Ga$
regime map based on several of the above mentioned studies, with values summarised in table 1. Also indicated are the cases considered in the present paper, which are designed to be representative of plate crystals falling in the atmosphere. However, Vincent et al. (Reference Vincent, Shambaugh and Kanso2016) noted that the falling mode transitions of solid disks do not predict the behaviour transitions of perforated geometry disks.
Turbulent flow statistics for the forcing conditions investigated. Kolmogorov scales of velocity
$u_{\eta }$
, length
$\eta$
and time
$\tau _{\eta }$
; Taylor microscale
$\lambda$
; integral scales of (r.m.s.) velocity
$u'$
, length
$L$
and time
$\tau _L$
; dissipation
$\varepsilon$
; and Reynolds numbers
$Re_{\lambda } = u' \lambda /\nu$
and
$Re_L = u' L/\nu$
. Weaker turbulence corresponds to G6 forcing, with grids in place to diffuse the jets, while stronger turbulence corresponds to B6 forcing, as described in Carter et al. (Reference Carter, Petersen, Amili and Coletti2016).
2.3. Experimental apparatus and turbulence properties
Turbulence is generated via two vertical panels facing each other at 1.81 m apart in an enclosed chamber. Each panel has 128 nozzles distributed in a planar array of 8 x 16 (see figure 3
a). All the nozzles are connected to a pressurised air line at 700 kPa and attached to computer-controlled solenoid valves. These are actuated independently to induce each jet in a randomised sequence following the sun-bathing algorithm proposed by Variano & Cowen (Reference Variano and Cowen2008). A region of approximately homogeneous turbulence with negligible mean flow is formed in the central section of the facility of size
$O(1$
m3). The intensity of the turbulence fluctuations is varied by controlling the mean actuation time of the jets, and reduced by adding square-mesh grids in front of the jet arrays. The random-jet forcing is run for sufficient time to allow the homogeneous region of turbulence to reach a statistically stationary state and is sustained throughout the duration of each experiment. The flow characterisation of the facility is described in detail in Carter et al. (Reference Carter, Petersen, Amili and Coletti2016) and Carter & Coletti (Reference Carter and Coletti2017, Reference Carter and Coletti2018). It has been used extensively to study particle turbulence interaction for the cases of small spherical particles (Petersen et al. Reference Petersen, Baker and Coletti2019; Berk & Coletti Reference Berk and Coletti2021; Hassaini & Coletti Reference Hassaini and Coletti2022; Hassaini et al. Reference Hassaini, Petersen and Coletti2023) and for the settling of solid disks in quiescent and turbulent flows (Tinklenberg et al. Reference Tinklenberg, Guala and Coletti2023, Reference Tinklenberg, Guala and Coletti2024).
We operated the turbulent chamber with and without the square-mesh grids, corresponding to the nominal definitions of ‘weaker turbulence’ and ‘stronger turbulence’, as summarised in table 2, and consistent with the solid disk experiments by Tinklenberg et al. (Reference Tinklenberg, Guala and Coletti2024). The dissipation rate
$\varepsilon$
is higher than typical levels in clouds (Grabowski & Wang Reference Grabowski and Wang2013), but
$\eta$
is comparable to that encountered in the atmospheric surface layer through which hydrometeors precipitate (see, e.g. the field studies by Li et al. Reference Li, Lim, Berk, Abraham, Heisel, Guala, Coletti and Hong2021a, where
$\eta \sim 0.5$
mm, or values of
$\eta = O(1$
mm) reported by Nemes et al. Reference Nemes, Dasari, Hong, Guala and Coletti2017 and Shaw Reference Shaw2003). The r.m.s. velocity fluctuations
$u'$
yield realistic values of the settling velocity number
$Sv_L$
(see, e.g. Nemes et al. Reference Nemes, Dasari, Hong, Guala and Coletti2017 and Li et al. Reference Li, Lim, Berk, Abraham, Heisel, Guala, Coletti and Hong2021a
, where
$Sv_L=0.60-10.7$
), which is expected to be influential for the settling dynamics (Wang & Maxey Reference Wang and Maxey1993; Good et al. Reference Good, Ireland, Bewley, Bodenschatz, Collins and Warhaft2014; Byron et al. Reference Byron, Einarsson, Gustavsson, Voth, Mehlig and Variano2015; Fornari et al. Reference Fornari, Picano and Brandt2016a
,
Reference Fornari, Picano, Sardina and Brandtb
; Petersen et al. Reference Petersen, Baker and Coletti2019; Brandt & Coletti Reference Brandt and Coletti2022).
2.4. Experimental methods and disk image analysis
The facility operation, imaging techniques and post-processing methods were described in Petersen et al. (Reference Petersen, Baker and Coletti2019) and Tinklenberg et al. (Reference Tinklenberg, Guala and Coletti2023, Reference Tinklenberg, Guala and Coletti2024). Those are briefly summarised here. With the homogeneous region of turbulence in a statistically stationary state, a sieve shaker positioned above the 3 m long chute leading into the chamber releases disks at a chosen volume fraction of
$\Phi _V = O(10^{-5})$
. The disks fall for approximately 1 m inside the chamber before reaching the camera field of view.
The imaging plane is positioned vertically in the centre of the chamber, illuminated by a 3 mm thick laser sheet. Images are captured via a high-speed CMOS camera (Phantom VEO 640) operated at 4300 Hz. A 105 mm Nikon lens collects images at 11.4 pixels
${\textrm{mm}}^{-1}$
in a 11.2 cm
$\times$
8.4 cm field of view. To account for any run-to-run variability and increase statistical convergence, five experimental runs each lasting approximately 10 s are performed at minimum for each disk type and flow condition. This number of runs captures statistics of
$O(10^3-10^4)$
individual disk trajectories for each case, with typical trajectory lengths of
$O(10^2)$
frames.
Binarising the images allows for the disks to be identified and filtered based on their intensity, size and sharpness. The image intensity gradient obtained by Sobel approximation identifies blurry objects to be removed. Disks are individually fit with an ellipse, resulting in sub-pixel-accurate centroid location and major axis orientation. The ellipse fits are used to reconstruct disk orientations in three dimensions. An orientation vector,
$\boldsymbol {\hat {p}}$
, along the axis of rotational symmetry of the disk, can then be obtained following Baker & Coletti (Reference Baker and Coletti2022). Finally, the centroid trajectories and orientation progressions are convolved with a Gaussian kernel in time to reach the smoothed translational and rotational disk motions.
2.5. Thin disk drag models
A semi-empirical formula has been proposed by Böhm (Reference Böhm1989) and Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010), among others, to predict the settling velocity of frozen hydrometeors. The model is inspired by the early work of Abraham (Reference Abraham1970) on the drag of a spherical particle based on an inviscid term
$C_0$
and a
$Re$
-dependent contribution representing the boundary layer developing from the edge of the sphere, namely
$C_D^{sph}=C_0(1+ ({\delta _0}/{Re^{1/2}}))^2$
, where
$\delta _0$
is a fitting constant.
Böhm (Reference Böhm1989) and Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) extended the model of Abraham (Reference Abraham1970) to generic snow particles, in fact assuming a master drag equation for frozen hydrometeors
$C_{DM}=C_D^{sph}$
with a new set of fitting parameters. They introduced the Best, or Davies, number
$X=C_{DM} Re^2$
to eliminate the settling velocity from the equation, imposing the equilibrium of drag and weight forces and only accounting for inertial and geometrical snow properties in the estimate of
$X$
(see step 1 in table 3). The area ratio
$A_R$
was introduced to include different snowflake geometries while maintaining the ability of the formulation and parameters of the master drag model
$C_{DM}$
to describe specific snow particle drag coefficients precisely as
$C_D=A_R^{-n} C_{DM}=A_R^{-n} C_0(1+ ({\delta _0}/{Re^{1/2}}))^2$
. Ongoing efforts are still devoted to fitting or altering the model parameters
$C_0$
,
$\delta _0$
and
$n\gt 0$
to achieve a universal dependency of the area ratio for a variety of complex snow morphologies (needles, dendrites, stellars and aggregates) and to collapse all the re-scaled, particle-specific drag coefficients on the master curve (McCorquodale & Westbrook Reference McCorquodale and Westbrook2021a
,
Reference McCorquodale and Westbrookb
). With all disk properties measured, the step 1 equation in table 3 can be inverted to obtain the settling Reynolds number (step 2), and eventually the settling velocity
$V_t$
(step 3). Note that Böhm (Reference Böhm1989) and Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) use different coefficients
$C_0$
and
$\delta _0$
and a different power-law exponent
$n=0.75-0.5$
to account for the area ratio, but share the same master drag equation. The differences in the formulation and in the fitting parameters are attributed primarily to different types of data sets considered in the validation, including snowflakes in air and artificial snow particles settling in liquids (see also Böhm Reference Böhm1992; Mitchell Reference Mitchell1996; Khvorostyanov & Curry Reference Khvorostyanov and Curry2002, Reference Khvorostyanov and Curry2005; Mitchell & Heymsfield Reference Mitchell and Heymsfield2005, among many others).
Step by step procedure for the estimate of the terminal velocity
$V_t$
of a solid or perforated disk, given
$\mu$
,
$\rho _f$
,
$m$
,
$A_R$
and
$D$
, as originally proposed by Böhm (Reference Böhm1989). The parameters
$C_0$
and
$\delta _0$
are defined in the master drag equation
$C_{DM}=C_0(1+({\delta _0}/{Re^{1/2}}))^2$
. Updated
$C_0$
,
$\delta _0$
coefficients and area ratio power-law dependency
$A_R^{1-n}$
provided by Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) are listed for comparison.
It is, however, important to stress that the effect of air turbulence has never been explicitly included – in terms of loitering, cross-flow-induced drag, preferential sweeping or induced variability in falling styles. Therefore, the resulting
$V_t$
estimates from all formulations summarised in table 3 are generally associated with quiescent flow conditions (Nemes et al. Reference Nemes, Dasari, Hong, Guala and Coletti2017; Li et al. Reference Li, Abraham, Guala and Hong2021b
; Singh et al. Reference Singh, Pardyjak and Garrett2023; Li et al. Reference Li, Guala and Hong2024). Our apparatus provides the opportunity to compare these models under controlled quiescent and turbulent flow conditions, reasonably consistent density ratios and without the uncertainties in particle size and mass associated with natural snowfalls.
Empirically derived velocity estimation method results from Böhm (Reference Böhm1989), Mitchell (Reference Mitchell1996) and Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010). The terminal velocities are listed with
$\pm$
one standard deviation. Errors shown in parentheses below each velocity estimate are computed in comparison with the measured
$V_t$
in quiescent air in the current study.
3. Results
3.1. Mean settling velocity and drag coefficient
We first apply the models of Böhm (Reference Böhm1989) and Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) to the current disks and compare with the velocities measured in quiescent air. Summarised in table 4, we find that applying the model and coefficients of Böhm (Reference Böhm1989) using the area ratio for each disk geometry yields an average error compared with the measured velocities of −36.7 % (the model underestimates the measured velocities in quiescent conditions). If we use the same
$C_0$
and
$\delta _0$
coefficients estimated by Böhm (Reference Böhm1989) but ignore the area ratio of the perforated disks, thus treating them as solid disks (
$A_s=A$
), we are in fact adopting the approach by Mitchell (Reference Mitchell1996), still leading to a significant -34.2 % underestimation of the settling velocity. The adjusted coefficients and dependency on the area ratio of Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) improves the average estimated velocity error to -19.9 %. The latter model seems to better capture the drag of larger disks falling at higher
$Re$
and the effect of permeability for the perforated disk geometries.
Drag coefficient versus Reynolds number curves plotted from measurements in quiescent air, weaker turbulence and stronger turbulence. Data included in grey symbols from Willmarth et al. (Reference Willmarth, Hawk and Harvey1964), Jayaweera & Mason (Reference Jayaweera and Mason1965), Jayaweera & Cottis (Reference Jayaweera and Cottis1969) and McCorquodale & Westbrook (Reference McCorquodale and Westbrook2021 a). The empirically derived drag curve from Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) is shown by the solid black line and the dashed black line is re-scaled for the area ratio of our perforated disks. The inset focuses on an axis range around the data points from the perforated disk measurements. Data from other publications digitised using WebPlotDigitizer (Rohatgi Reference Rohatgi2021).
The disk drag coefficients
$C_{D} = 2mg/\rho _f A_s V_t^2$
were estimated from the projected, solid disk area
$A_s$
assuming flat falling and the averaged measured settling velocity
$V_t$
. The latter is defined as the average of the vertical component of the particle velocity. The area ratio has a unitary value
$A_R=1$
for the solid disks, whose drag coefficient can be directly compared with the master drag curve, but the perforated geometries require rescaling. To compare the experimental results and the theoretical predictions, we plot the particle-specific drag coefficients
$C_D$
as the function of Reynolds number in figure 5, and scale the predictive curve(s) from the master equation. Inset in figure 5 is a zoomed in view on the perforated disks. The grey symbols are from low-density-ratio laboratory studies of disks, summarised by McCorquodale & Westbrook (Reference McCorquodale and Westbrook2021
a). The solid black curve following Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) for the solid disks is given by
$C_D=C_{DM}=C_0(1+\ ({\delta _0}/{Re^{1/2}}))^2$
. The dashed curve is the re-scaled prediction for perforated geometries (
$C_D=C_{DM} A_R^{-1/2}$
) using the power-law dependency of the area ratio proposed by Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010), which best reproduced the average settling velocity.
Our disk data, including those of the 3 mm
$^{\times }$
and 3 mm
$^{\circ }$
perforated geometries, align quite well with Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010). At low
$Re$
, non-tumbling solid disks are well captured by the master curve. With increasing
$Re$
and emerging variability in the disk falling styles, deviations are observed. Overestimated drag coefficients are associated with the above underestimated settling velocities. There are two possible reasons for this: (i) the orientation of a disk in our analysis is assumed flat in the definition of the projected drag area
$A_s$
, implying that any change in orientation, e.g. due to fluttering, contributes to reducing the drag force and increasing the settling velocity; (ii) when tumbling occurs, depending on both
$Re$
and
$I^*$
, the near wake, form drag and
$C_D$
are affected, generally resulting in a decrease in drag, as seen, e.g. when comparing the drag of a flat disk (
$C_D\sim 1$
) with the one of a sphere (
$C_D\sim 0.5$
) resembling a tumbling disk along with a tumbling mass of entrapped fluid. This follows what was found in the quiescent experiments of Tinklenberg et al. (Reference Tinklenberg, Guala and Coletti2023): low-density-ratio experiments, even when matching
$Re$
of frozen hydrometeors in the atmosphere, may be slightly biased towards stable flat falling due to relatively lower
$I^*$
values. Rotationally unstable disks in air flows are expected to exhibit a more random orientation and a reduced drag area, on average, resulting in a higher settling velocity, as observed here. This difference is especially apparent starting at
$Re \approx 100$
in figure 5, above which unsteady wakes, fluttering and tumbling are much more likely to occur, in particular for the higher-density-ratio cases (Willmarth et al. Reference Willmarth, Hawk and Harvey1964; List & Schemenauer Reference List and Schemenauer1971; Bréon & Dubrulle Reference Bréon and Dubrulle2004; Noel & Chepfer Reference Noel and Chepfer2004). We thus infer that the more chaotic falling styles affecting frozen hydrometeors in the atmosphere must be incorporated into the calibration of predictive models. The model by Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) incorporates both laboratory and field data of rimed and unrimed plates and dendrites into the calibration, which can be assumed to include some ambient turbulence, in addition to various falling styles. The formulation of Böhm (Reference Böhm1989) compares with previous models and measurements from Locatelli & Hobbs (Reference Locatelli and Hobbs1974) for aggregates, but seems to include only laboratory data in a liquid tank, hence at low density ratio, for plate crystals up to
$Re=100$
. The above differences may partially explain the better agreement between measured and predicted settling velocities using the Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) model for quiescent flow conditions.
Mean vertical velocity
$V_t$
plotted by disk geometry in each flow condition.
3.2. Phenomenology of perforated versus solid disk settling in turbulent flows
Figure 6 shows the mean vertical velocity plotted by disk geometry and forcing condition. Both the solid disks (3 mm
$^*$
) and perforated disks (3 mm
$^{\times }$
and 3 mm
$^{\circ }$
) experience a velocity reduction on average due to ambient turbulence. This was also observed for smaller diameter disks compiled in Tinklenberg et al. (Reference Tinklenberg, Guala and Coletti2024). The higher settling velocity observed for the 3mm
$^{\circ }$
disk is primarily due to the increased material density, as predicted by all models in table 4. Particle settling variability is illustrated in the form of error bars and quantified by the standard deviation of all instantaneous points from all experimental runs, divided by the number of experimental runs. This method yields marginally smaller errors for both of the perforated geometries as compared with the solid particle, indicating that the perforated disks experience less variability in vertical velocity, in particular in quiescent flow. As discussed in Tinklenberg et al. (Reference Tinklenberg, Guala and Coletti2024), the observed fluctuation of settling velocity results from the spatio-temporal variability of the disk orientations and the associated drag and wake. Thus, we infer that such a reduced variability could be associated with a more consistent falling dynamics.
Falling style, including preferential orientation and tumbling, has been shown to greatly influence the drag on plates at high density ratios (McCorquodale & Westbrook Reference McCorquodale and Westbrook2021b ; Tinklenberg et al. Reference Tinklenberg, Guala and Coletti2023) and may explain the reduced variability in vertical velocity. In particular, the transition from tumbling to fluttering, discussed in Vincent et al. (Reference Vincent, Shambaugh and Kanso2016) for the single hole coin, is inferred to increase the disk rotational stability and decrease the variability in orientation of our perforated disks.
Evidence of the effect of disk permeability and turbulence on preferential settling orientation is shown in figure 7, with the probability density functions PDFs of instantaneous
$|p_y|$
(see figure 3
c for the local orientation-dependent reference system). Plate geometries, if rotationally stable, are expected to settle predominantly with their maximum frontal area perpendicular to the fall speed direction (
$p_y=1$
) in the limit of an infinitely thin disk – a limit which may not hold in the current scenario (Willmarth et al. Reference Willmarth, Hawk and Harvey1964; Sassen Reference Sassen1980; Matrosov et al. Reference Matrosov, Reinking, Kropfli, Martner and Bartram2001; Chrust et al. Reference Chrust, Bouchet and Dušek2013; Wang Reference Wang2021). This is observed in all experiments, in particular for the solid disk under quiescent conditions.
Increasing turbulence intensity tends to homogenise the orientation of disks that are prone to tumbling: the high-probability peak associated with steady falling of the solid disks in quiescent air is less pronounced, and resembles the distributions of perforated geometries with a broader hump (
$0.75\lt p_y\lt 1$
), pointing at increased fluttering. The lesser variation in the perforated trends between the different turbulent conditions indicates that turbulence does not significantly change the orientation of perforated disks. This suggests that the observed appreciable variation in drag coefficients is due to the disks’ rotational dynamics and their horizontal slip velocity, rather than to their variability in the projected area.
The PDFs of the modulus of the instantaneous disk orientation vector with respect to the vertical,
$|p_y|$
. Plotted at each forcing level for the (
$a$
) 3 mm
$^*$
, (
$b$
) 3 mm
$^{\times }$
and (
$c$
) 3 mm
$^{\circ }$
disks. Here,
$|p_y| \sim 0$
indicates an edge-on orientation and
$|p_y| \sim 1$
indicates a flat-falling orientation.
3.3. Trajectory angles and lateral motions
We focus here on the turbulence effects on the disk drag in a broad perspective by quantifying how much the disks move laterally under a variety of falling styles. The underlying mechanisms include the rotational-induced lift discussed in Tinklenberg et al. (Reference Tinklenberg, Guala and Coletti2023) and the cross-flow-induced drag effect discussed in Bagheri & Bonadonna (Reference Bagheri and Bonadonna2016) and Tinklenberg et al. (Reference Tinklenberg, Guala and Coletti2024). The trajectory angle
$\phi$
within the Particle Tracking Velocimetry observation plane, as defined in figure 3(b), is chosen as the key metric for disk lateral dispersion. The average
$\phi$
values are plotted in figure 8 for each flow condition, while the corresponding PDFs are included in figure 9. The stabilising effect of permeability is first observed in quiescent flow, with the solid disk exhibiting the largest angles, and the single hole disks falling more vertically. Overall, turbulence enhances the lateral dispersion of the disks, with the 3 mm
$^{\times }$
following a similar trend in the mean as the 3 mm
$^*$
. The 3 mm
$^{\circ }$
disks, on the other hand, appear to be swept laterally much less by turbulence, especially under the strong forcing.
Average two-dimensional projected trajectory angles plotted for each forcing case with lines for each disk type.
Distributions of two-dimensional projected trajectory angles off the vertical,
$|\phi |$
, plotted for each disk type in (
$a$
) quiescent air, (
$b$
) weaker turbulence and (
$c$
) stronger turbulence. Colours correspond to those assigned by disk type as in figure 8.
This behaviour is likely due to multiple concurring factors. Firstly, the single hole allows for greater permeability of the flow through the single perforation as compared with the smaller perforations of the four quadrant geometry. This is despite the same open area ratio between the two perforated geometries, suggesting that the single parameter of
$A_R$
cannot fully encapsulate the overall dynamics of perforated plates (this may contribute to the settling velocity variability of snow dendrites). Secondly, the larger density of the 3 mm
$^{\circ }$
results in greater rotational inertia
$I$
and thus increased
$I^*$
for the same nominal area, also leading to increased stabilisation of rotational motions. A disk which is tumbling less and more horizontally oriented is, in fact, reducing the exposed area to horizontal turbulent gusts, thus limiting the lateral motion of the disk. Hence, rotationally stabilised disks are expected to experience less cross-flow-induced drag and fall more vertically in turbulent flows. We infer that competing mechanisms contribute to explain the reduced
$C_D$
of the perforated disks: more stably falling, perforated disks offer a larger frontal area to the settling motion and experience a reduced lateral dispersion, while solid disks are more inclined to tumble, which reduces the projected area and makes them more susceptible to lateral wind gusts. With increasing turbulence, the drag coefficient increases, suggesting that cross-flow-induced drag becomes dominant. In fact, the 3 mm
$^{*}$
and 3 mm
$^{\times }$
disks under strong turbulence have a similar
$C_D$
and distribution of
$\phi$
. It remains to be understood what controls the disks’tumbling rate and what are the specific roles played by turbulence, moment of inertia and perforation geometry.
3.4. Falling styles and rotation rates
The method adopted in Tinklenberg et al. (Reference Tinklenberg, Guala and Coletti2023, Reference Tinklenberg, Guala and Coletti2024) to determine falling style was based on the angular excursion along the trajectory (
$\Delta p_y$
), and as such relied on having long enough trajectories captured within the field of view to detect an entire 360
$^{\circ }$
rotation for tumbling disks. Due to the faster descent of the thicker 3 mm disks of current focus, less instantaneous frames are captured along a single trajectory. Hence, there is a strong likelihood that even if a disk is tumbling, the entire 360
$^{\circ }$
progression will not be sampled. This will cause an under-detection of tumbling trajectories, biasing the sample towards fluttering and oscillating disk motions. Using the instantaneous values of
$p_y$
, the sign change is sufficient to detect the overturning of a disk, indicating tumbling. Comparatively, a fluttering disk will have oscillating values of
$p_y$
, but the sign will not change. This method is applied to the disks of focus in this work to obtain falling style percentages shown in figure 10. The 1 mm and 2 mm disks tumble naturally in the quiescent case. Their tumbling is slightly hindered under weak turbulence, and more significantly in the stronger turbulence case. This is due to intermittent air gusts disrupting the regular rotation and the wake shedding, leading to more intense fluttering. The 3 mm and 3 mm
$^*$
disks see the opposite effect, where the tumbling is enhanced by the turbulence. The different behaviour between the 1 and 2 mm disks and the 3 and 3 mm
$^*$
disks is due to the different range of
$Ga$
and
$I^*$
. The 3 mm disks tumble more intermittently in quiescent air, and the turbulence enhances the existing instabilities, causing tumbling to occur more frequently for otherwise fluttering or stable falling disks. As the perforations significantly stabilise the 3 mm
$^{\times }$
and 3 mm
$^{\circ }$
, the turbulence has a minimal effect on the falling style in these cases, with more persistent fluttering as opposed to tumbling.
Histograms of trajectory falling style percentages, separated by disk size. Red bars represent non-tumbling and blue represents tumbling. Dark to light shading indicates flow conditions going from quiescent air, to weaker turbulence and stronger turbulence, respectively. Percentage values are listed in white text for both the non-tumbling and tumbling populations on their respective bars.
For solid disks, it has been shown that the angular velocity and falling style are correlated (Tinklenberg et al. Reference Tinklenberg, Guala and Coletti2023). The presence of a strongly bimodal distribution of the trajectory-averaged angular velocity
$\omega_t$
in the case of the 3 mm
$^*$
shown in figure 11(
$a$
) confirms previous results, with the fluttering mode characterised by a lower
$\omega _t$
. This, however, is no longer the case for either of the perforated geometries. Despite a residual bimodality for the 3 mm
$^{\circ }$
, figure 11(
$b,c$
) show that the angular velocity for the tumbling solid disks, with a peak above 200 rad s-1 is significantly reduced by the perforation. We utilise the identification methods used for figure 11 to quantify the conditional tumbling and non-tumbling angular velocities averages, yielding figure 12. The distinct bimodality of the solid geometries is clear in this view. The near halving of the peak angular velocity measured for tumbling 3mm
$^{\times }$
and 3mm
$^{\circ }$
disks provides direct evidence of the combined stabilising effect of the perforation, reducing the rotational inertia, and of the permeability stabilising the wake, in both quiescent and turbulent air.
Trajectory-averaged angular velocity distributions for all flow conditions for the (
$a$
) 3 mm
$^*$
, (
$b$
) 3 mm
$^{\times }$
and (
$c$
) 3 mm
$^{\circ }$
disks.
Overall, turbulence has a stronger impact on the angular velocities of non-tumbling particles, while tumbling particles possess sufficient angular momentum to maintain their rotational motion. The 3 mm
$^*$
disks are somewhat of a transition point between the distinct separation of tumbling versus non-tumbling for the thinner 1 mm, 2 mm and 3 mm disks, and the more stable perforated geometries. Fluttering of the 3 mm
$^*$
disks becomes less distinguishable from the tumbling as the turbulence is strengthened, resulting in amplified oscillations without always reaching a full rotation. Indeed, according to the classification by Lau et al. (Reference Lau, Huang and Xu2018), the 3 mm
$^*$
disks fall in the bistable region of the parameter space where steady falling and tumbling coexist.
Median angular velocities separated by falling style, where red lines represent non-tumbling disk populations for each disk size and blue lines represent tumbling.
In order to better quantify the stabilising effects of disk perforation, which are marginally reducing drag and more significantly reducing the angular velocity as compared with the solid disks, we estimate the rotational Strouhal number
$Str = fD/V_t$
. Here, this is based on the peak frequency of the velocity signal, corresponding to twice the frequency of a full tumbling motion (
$f = \omega _p/\pi$
, with
$\omega _p$
representing the mode of the
$\omega _t$
distribution). The Strouhal number is plotted as a function of the inertia ratio in figure 13. As the terminal velocities of the perforated disks are comparable to their solid equivalent, the dampened angular velocity contributes to reduce
$Str$
by
$\sim 50\,\%$
. This results in a deviation from the relation
$Str \sim {I^*}^{-0.5}$
, which was proposed by Fernandes et al. (Reference Fernandes, Risso, Ern and Magnaudet2007) and observed to fit well to our solid disks. Surprisingly, the 3mm
$^{\times }$
disks, which most closely resemble snow crystals, exhibit a much stronger deviation, as compared with the single hole disks, which are predominantly stabilised by their density, mass and moment of inertia. This again means that disks with comparable area ratio
$A_R$
may still manifest very distinct settling kinematics that are not fully described by the inertia ratio
$I^*$
, even when the perforation is included in the estimate of the disk moment of inertia. This suggests that turbulence plays a significant role in the bistable regime, but with a more marginal effect when the disk falling style is well defined by the material properties and
$I^*$
. The perforation geometry, however, when associated with a different permeability and regardless of
$A_R$
, manifests a significant effect in the dampening of
$\omega _t$
and in the deviation from the
$Str-I^*$
dependency.
Strouhal number
$Str$
plotted as a function of
$I^*$
, with respective disk geometry shown along the abscissa. Light coloured disks represent those with
$h=50$
µm and darker represent
$h\approx 100$
µm.
4. Discussion
The parameter space to study the coupled settling and rotational kinematics of snow plates and dendrites is multidimensional and expected to include the additional effects of the ambient flow. Figure 14 is inspired by the early work of Kajikawa & Okuhara (Reference Kajikawa and Okuhara1997), where a limited number of more or less rimed snow plates have been mapped in the inertia ratio (
$I^*$
) and Reynolds number (
$Re$
) phase space. In their original contribution, the authors investigated how increased riming, affecting the particle density and aspect ratio, leads to different falling style, classified as (i) rotation or spiral (R-S), affine to our tumbling definition and quantification, and (ii) swing or non-rotation (S-N), which includes fluttering and flat falling. As opposed to the predictable Galileo number, which is based on the gravitational velocity
$U_g$
,
$Re=V_tD/\nu$
features the actual average settling velocity and thus incorporates the correct particle drag and turbulence effects. Our solid (D
$\geqslant$
1 mm) and perforated disks in figure 14 are coloured based on the observed tumbling percentage, and can thus be compared with the R-S classification. Laboratory and field data exhibit fairly consistent settling kinematics and, to a lesser extent, tumbling characteristics in a range of
$100\lt Re\lt 500$
and
$I^*\gt 0.7$
. With increasing mass (along with terminal velocity, and
$Re$
) and permeability, the disks are more rotationally stable, while with increasing ambient turbulence and drag, disks are more keen to tumble. Based on this scattered laboratory and field data, we cannot develop a predictive model, but only speculate about natural conditions favouring snow particle tumbling. Thin and light disks are likely to feel small eddies in turbulent air flows and develop a wake able to sustain autorotation, providing they are large or dense enough to reach
$Re\gt 100$
. Natural candidates in such a phase space are relatively large, unrimed plates, and to a lesser extent, snow dendrites, which may, however, be stabilised by their own permeability. The snow dendrite and plate density (Locatelli & Hobbs Reference Locatelli and Hobbs1974; Li et al. Reference Li, Guala and Hong2023) is somewhat smaller than the solid disks tested here, and their typical size, ranging from 0.1 to 1 mm, is only partially overlapping with our disks in a range where tumbling quantification is challenging even at laboratory scale (Tinklenberg et al. Reference Tinklenberg, Guala and Coletti2024). Results from McCorquodale & Westbrook (Reference McCorquodale and Westbrook2021b
) with snow analogues in quiescent liquids reproduce the range of
$Re$
, but are limited to a low range of
$I^*$
up to 0.01. It thus appears that the phase space
$0.2\lt I^*\lt 0.7$
and
$50\lt Re\lt 300$
is meteorologically relevant and worth exploring statistically in future experiments. Transition between regimes may occur around
$Re \simeq 100$
and
$I^* \simeq 0.5$
, bordering the chaotic regime as suggested by Field et al. (Reference Field, Klaus, Moore and Nori1997) and Jayaweera & Mason (Reference Jayaweera and Mason1965), which may explain the overlap between rotational (R-S) and swing/flat (S-N) crystals falling in Kajikawa & Okuhara (Reference Kajikawa and Okuhara1997). In addition, with different levels of turbulent forcing in the above regimes, the associated marginal stability curves may need to be redefined, which again emphasises the importance of controlled laboratory experiments with correct particle shapes, fluid density ratios and a range of quiescent to turbulent forcing conditions. Finally, we stress that the observed scatter in figure 14, can be interpreted in view of the
$Str$
and
$I^*$
deviation exhibited by the 3mm
$^{\times }$
disks in figure 13 and the 3 mm disks in figure 10. Accordingly, when the disk falling style is in a bistable transitioning regime, turbulence intensity has a significant effect; however, when the disks are tumbling, the permeability, rather than the size of the perforation or the turbulence, plays a key role. Since turbulence and permeability are not explicitly accounted for, both the tumbling frequencies in figure 14 and the drag coefficients in figure 5 exhibit significant variability that propagates to the settling velocity.
Inertia ratio
$I^*$
and Reynolds number
$Re$
scatter plot of currently investigated solid and perforated disks under the three forcing regimes. The symbol colour coding is set to quantify the tumbling percentage of the disk as indicated in the histograms of figure 10. The perforated disks 3mm
$^{\times }$
and 3mm
$^{\circ }$
are highlighted by
$\bigoplus$
and
$\bigcirc$
overlapped symbols. Our results are overlapped with field data of more or less rimed plates and dendrites as classified by Kajikawa (Reference Kajikawa1992) into R-S (spiral or rotation) and S-N (swing and non-rotation).
5. Conclusion
We have reported on an experimental study of thin, mm-sized solid and perforated disks falling in air. The considered range of non-dimensional parameters,
$\chi =5-60$
,
$Re = 10-663$
and
$I^* \approx O(1)$
, is relevant to the settling of snow plate crystals and dendrites in the atmosphere. Still, to our knowledge, it has only been recently investigated in laboratory experiments by Tinklenberg et al. (Reference Tinklenberg, Guala and Coletti2024). Therein, we did not attempt to broadly scan the parameter space, but rather expanded the number of realisations for each considered case in order to achieve the statistical description of a settling process characterised by coexisting falling styles and high variability, even in quiescent flows (Lau et al. Reference Lau, Huang and Xu2018; Tinklenberg et al. Reference Tinklenberg, Guala and Coletti2023). The dynamics of solid disks is compared here with those with perforated geometries in both quiescent air and in two homogeneous turbulence flows with different
$Re_\lambda$
. Two types of perforated disks, characterised by four quadrant and single hole geometries with matching area ratios and solidity, were investigated to assess the specific influence of the perforation shape and size on the settling kinematics, as well as the predictive ability of currently available models to estimate the terminal velocity
$V_t$
. Using laser illumination and high-speed imaging, we have gathered and analysed the objects translational and rotational kinematics based on thousands of trajectories obtained in repeated runs with volumetric concentration of
$O(10^{-6})$
comparable to natural snowfalls (Li et al. Reference Li, Lim, Berk, Abraham, Heisel, Guala, Coletti and Hong2021a
).
The main results of our investigation can be summarised as follows:
-
(i) When
$Re \geqslant 100$
, turbulence consistently reduces the fall speed of perforated disks, as compared with quiescent flow results, confirming the results of Tinklenberg et al. (Reference Tinklenberg, Guala and Coletti2024) on solid disks and supporting the statistical predominance of cross-flow-induced drag enhancement over preferential sweeping in all the conditions investigated. -
(ii) Perforated disks are stabilised in their descent motion, and therefore tumble less frequently and at a reduced angular velocity. This result is robust in both quiescent air and homogeneous turbulence. For the solid disk, we see a strongly bimodal distribution of the angular velocity, which directly correlates with the falling style of the disks. For example, the peak tumbling angular velocity for the solid disks is reduced by
$50\,\%$
for the perforated geometries, even when comparing disks with the same diameter and thicknesses (3mm
$^{\times }$
perforated disk versus the solid 3mm
$^*$
). However, different perforation geometries lead to specific reductions in the Strouhal number, as highlighted by observed deviations from the functional dependency on the rotational inertial properties,
$I^*$
, characteristic of solid disks (figure 13). This points again at the key role of permeability, rather than the area ratio or turbulence, in settling the tumbling dynamics of perforated disks in the conditions investigated. -
(iii) The area ratio
$A_R$
is an important parameter in determining the fall speed, and needs to be appropriately taken into account in settling velocity models as inspired by Böhm (Reference Böhm1989). However,
$A_R$
cannot entirely describe the disk drag and settling dynamics. This is because the disk permeability may vary for the same
$A_R$
and the associated rotational effects may contribute to the observed differences in the
$C_D(Re)$
curves in figure 5 and in the average settling velocity. -
(iv) Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) reasonably reproduces the fall speed of model hydrometeors in quiescent air, although it shows a systematic under-prediction. The latter is even more marked in Böhm (Reference Böhm1989) and Mitchell (Reference Mitchell1996), which mostly rely on laboratory experiments in liquids. In this respect, previous laboratory studies of
$Re$
-analogous plate settling in viscous liquids yield a distinctly different
$C_D$
plateau, which deviates from the present data at
$Re \approx 100$
. In the extension to atmospheric precipitation, this again highlights the importance of high-density-ratio experiments under varying turbulence intensities to reproduce the variability in falling styles and tumbling rate of high inertia ratio particles such as snow plates and dendrites. -
(v) The good agreement between our results under turbulence and the Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) model confirms that ambient turbulence, which clearly modulates the field data and impacts their fitting constant, should be explicitly parameterised in hydrometeor fall speed and drag models.
In summary, the disk settling velocity and tumbling behaviour reveal a strong coupling between falling styles, particle inertial properties, turbulence levels and flow permeability. It is inferred that ambient turbulence increases the disk drag coefficient mostly through cross-flow-induced (nonlinear) drag. Perforation effects are partially captured by the area ratio parameterisation of the terminal velocity in Böhm (Reference Böhm1989), Heymsfield & Westbrook (Reference Heymsfield and Westbrook2010) and McCorquodale & Westbrook (Reference McCorquodale and Westbrook2021a ); however, permeability strongly influences the disk tumbling rate, which has a secondary, although non-negligible, effect on the drag coefficient.
An improved definition of the surface porosity and permeability of frozen hydrometeors, or generally falling objects, would be of great interest in a modelling study to develop a more comprehensive understanding of hole placement and the correlation to stabilising wake structures developed. For instance, in the case of fluffy snow flake aggregates, the many small perforations may not only reduce the effective density but perhaps randomise the wake and the associated vortex shedding, with an increased stabilising effect. Other applications in literature with similar implications include the flight of dandelions (Cummins et al. Reference Cummins, Seale, Macente, Certini, Mastropaolo, Viola and Nakayama2018; Qiu et al. Reference Qiu, He and Bao2020), leakiness of bristled insect wings (Ford et al. Reference Ford, Kasoju, Gaddam and Santhanakrishnan2019; Lee et al. Reference Lee, Lee and Kim2020; O’Callaghan & Lehmann Reference O’Callaghan and Lehmann2023) and pipe flow through a perforated plate or screen (Bayazit et al. Reference Bayazit, Sparrow and Joseph2014).
The effects of turbulence and permeability could potentially be decoupled when the disk falling style is robustly predicted. Unfortunately, when disks lay in the bistable
$I^*-Ga$
(or
$I^*-Re$
) phase space, all the above variables are intertwined, settling velocity prediction remains very challenging, and large variability is expected in the field measurements. This is amplified when a range of snow morphologies and sizes characterise the same snowfall event, as observed by Li et al. (Reference Li, Guala and Hong2023, Reference Li, Guala and Hong2024). The variability in snow morphology may lead to the simultaneous occurrence of various particle–turbulence interaction mechanisms, featuring e.g. graupel-like particles experiencing preferential sweeping, clustering and settling velocity enhancement (Nemes et al. Reference Nemes, Dasari, Hong, Guala and Coletti2017; Li et al. Reference Li, Lim, Berk, Abraham, Heisel, Guala, Coletti and Hong2021a
,
Reference Li, Abraham, Guala and Hongb
), along with plate-like crystals affected by loitering, cross-flow-induced drag and settling velocity reduction (Garrett & Yuter Reference Garrett and Yuter2014; Singh et al. Reference Singh, Pardyjak and Garrett2023). We thus infer that the thick tails, skewness and other non-Gaussian features observed in settling velocity distributions from field data may be caused not only by a range in size and snow morphologies, but also by the different effects that turbulence exerts on specific subsets of particles according to various interaction mechanisms.
Funding
This work was supported by the US National Science Foundation (Physical and Dynamic Meteorology program) through grant NSF-AGS-1822192 and by the National Aeronautics and Space Administration (Weather and Atmospheric Dynamics program) through grant 80NSSC20K0912.
Declaration of interests
The authors report no conflict of interest.
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