Three Dimensional Wakes of Freely Falling Planar Polygons

Experiments in Fluids (2019) 60:114 https://doi.org/10.1007/s00348-019-2760-z

RESEARCH ARTICLE

Three dimensional wakes of freely falling planar polygons

Luis Blay Esteban1 · John Shrimpton1 · Bharathram Ganapathisubramani1

Received: 25 February 2019 / Revised: 6 June 2019 / Accepted: 6 June 2019 / Published online: 18 June 2019 © The Author(s) 2019

Abstract The wake characteristics of various thin particles with identical material properties but diferent frontal geometries (disks, hexagonal plates and square plates) are examined by means of three dimensional measurements of the instantaneous velocity feld. The reference particle is a circular disk that lies within the Reynolds number—dimensionless moment of inertia ∗ domain ( Re − I ) corresponding to the futtering regime, as defned by Willmarth et al. (Phys Fluids 7:197–208, 1964). Hexagonal and square plates are manufactured to have the same frontal area and material properties of the reference particle. Three dimensional trajectories obtained from high-speed imaging show that disks preferably adopt a quasi-2D oscillatory descent; i.e. ‘planar zig-zag’, whereas particles with less circularity adopt three dimensional trajectories more frequently; i.e. ‘transitional’ and ‘spiral’ descent. The wake behind free-falling disks is found to be a succession of hairpin vortices shed of at every turning point linked by a pair of counter rotating vortices that grow downstream from the leading edge of the disk. In contrast, square plates describing ‘spiral’ descent show an almost time-independent wake morphology with large-scale vortex shedding around the entire perimeter of the particle. The large-scale wake structures of hexagonal plates resemble either the disks’ or the squares’ depending on the falling regime that they adopt. Finally, we compare the dimensionless vorticity distribution in the wake of the particles and found that this also depends on the falling style that the particle adopts during the descent. Graphic abstract

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1 Introduction wake. Also, they showed that the frequency and the ampli- tude of the oscillations are independent of when they are The physical mechanisms involved in the free fall of par- made non-dimensional with the appropriate length and ( ∕ − ) ticles in a viscous media have been investigated in many velocity scale, D and p f 1 gD , respectively. studies across a broad range of disciplines, from earth Zhong et al. (2011) focused on the falling style of thin science to engineering problems. Examples of these phe- disks with very low dimensionless moment of inertia, cor- nomena in natural environments include seed dispersal responding to a region of the phase diagram associated with (Sabban and van Hout 2011), the deposition of volcanic the futtering regime. They showed that the descent paths ash (Wilson and Huang 1979), the falling of rain droplets of these particles exhibit a transition from ‘planar zig-zag’ and ice crystals (List and Schemenauer 1971; Jayaweera to ‘spiral’ motion that is in fact determined by the dimen- 1972; Kajikawa 1992), and sedimentation processes as in sionless moment of inertia and the Reynolds number. They Feng et al. (1994). Industrial processes include the genera- also carried fow visualization and showed that the change tion of pollutants by combustion processes, as described in path style comes with a drastic change in the turbulent in Jones et al. (2014). Fundamental research of free falling structures present in the wake. Lee et al. (2013) extended particles has focused on wake structures, vortex shedding the work of Zhong et al. (2011) investigating the transition mechanisms and body-to-fuid interactions with the objec- between these two falling modes. They also performed pla- tive of understanding the characteristics of the paths of the nar Particle Image Velocimetry (PIV) measurements on the falling objects. particle wake, reafrming the fndings previously observed The frst phase diagram that characterised the falling qualitatively by Zhong et al. (2011). Interestingly, a study regime of disks was defned experimentally by Willmarth focused on the landing position distribution of disks fall- et al. (1964). They mapped three diferent falling regimes; ing under diferent descent modes (Heisinger et al., 2014) ∗ i.e. steady, futtering and tumbling, and discussed that the showed that a disk (with a given Re and I ) can follow both phase diagram held for disks as long as the aspect ratio types of motion; i.e. ‘planar zig-zag’ and ‘spiral’ at difer- = D∕h ( ), defned as (where h is the thickness of the ent realizations. This suggests that the falling motion of a disk and D the diameter), was small. Thus, they were given particle is not unique and therefore should be exam- able to predict the falling style of a given disk once the ined through statistical analysis. Reynolds number (Re) and the dimensionless moment of Several direct numerical simulations studies, as Auguste I∗ inertia ( ) were known. The Reynolds number defned as et al. (2013) and Chrust et al. (2013), have focused on the Re = V D∕ V z , where z is the average settling velocity descent style of thin disks for small to moderate Reynold and ⟨is the⟩ kinematic ⟨viscosity⟩ of the fuid. The dimen- numbers ( Re < 300 ) and found remarkable diferences in I∗ = I ∕ D5 I sionless moment of inertia as p f , where p the dynamics of disks for diferent values. However, these is the mass moment of inertia about the diameter of the modes are only present at moderate Reynolds number and disk and f is the density of the fuid. Following a similar will not be further discussed in this study. approach, Smith (1971) constructed the corresponding Fewer studies are available for other planar particles phase diagram for quasi 2D rectangular plates. than disks and rectangular plates. Early investigations were Field et al. (1977) identifed a new falling regime that carried out by List and Schemenauer (1971) on the steady lies in between the futtering and tumbling regimes. This fall of planar irregular particles relative to a reference disk. regime was labelled as ‘chaotic’ due to the apparent unpre- These included hexagonal plates, broad-branched crystals dictable particle motion during the descent. This new fall- and stellar crystals. In addition to an increase in drag coef- ing style has been investigated in more detail during the fcient with Reynolds number associated with an increase in past decades, especially for quasi 2D rectangular plates, perimeter (due to viscous efects), they also observed small as in Belmonte et al. (1998), Andersen et al. (2005a, b). self-induced oscillations during the fall. These secondary Fernandes et al. (2007) performed a numerical investi- motions were qualitatively smaller for the irregular particles gation on the oscillatory motion of axisymmetric bodies than for the case of the reference disk. Jayaweera (1972) 80 < Re < 330 1.5 <�<20 in the range of and and found also studied the steady free fall of several planar particles diferent body motion and wake characteristics as a func- and showed that the terminal velocity of star-shape particles tion of the body aspect ratio . They showed that as was up to 25% smaller than the one for the equivalent disk. increases, the axial symmetry of the wake and the wake In the early 90s, Kajikawa (1992) studied the falling descent oscillations appeared at a smaller critical Reynolds num- of ‘snow crystals’, showing that dendritic crystals with large �<6 ber. They distinguished between thick ( ) and thin internal ventilation fall following a stable descent over a �>6 ( ) bodies based on the comparison between the shed- larger Re range compared to simple hexagonal plates. This ding frequency of a freely falling disk with a fxed body result was later confrmed by Vincent et al. (2016), showing

1 3 Experiments in Fluids (2019) 60:114 Page 3 of 12 114 that the internal ventilation of disks improves the stability motion during all parts of the descent to within 2% of the of the falling motion. particle diameter. In each frame the dark particle projec- Despite these isolated studies, a gap still remains on how tion is recorded onto the white background and the posi- the wake of planar irregular particles difer from the char- tion of the particle centre of mass was obtained by locating acteristic wake of a disks. In here, we describe and com- the geometric centre of each particle projection. The image pare the motion of a free falling disk with a square and a processing was performed using an in-house script devel- hexagonal planar particle with same frontal area and mate- oped in MATLAB, as in Esteban et al. (2018) and Esteban rial properties. The surrounding fow is also investigated et al. (2019). The method to obtain the particle position by means of volumetric velocimetry (3D-3C) so that the from the grey-scale image is a threshold-based method, coupling between vortex shedding and particle motion can where raw images are converted into black and white be addressed. images after applying a user-defned intensity threshold. For the accuracy on the particle location we rely on a stable light intensity during all the trajectories. Commercial 2 Methods and experimental setup LED panels connected to a stable DC power supply were used to back illuminate the complete feld of view of both To investigate how the frontal geometry of a planar par- cameras. The measured trajectories were smooth, and a ticle afects the settling dynamics and wake characteris- polynomial flter of 3rd order and frame length of 5 points tics we manufactured planar particles with three diferent was used to flter out high-frequency noise. geometries. The reference particle is a disk with diameter A set of releases for a sphere falling in air was per- = = = D 30 mm , thickness h 2 mm and density p 1.2g/ formed to establish limitations on the accuracy associated 3 cm that lies within the futtering regime, with a measured with the suction mechanism as well as the camera align- ≈ Reynolds number Re 1370 and dimensionless moment of ment. The variance in the landing position was interpreted ∗ = × −3 inertia I 5 10 . Then, keeping the particle thickness as the uncertainty of the complete system. This was found (h) and frontal area ( A p ) constant we vary the geometry of to be two orders of magnitude smaller than the sphere the perimeter to a square and a hexagonal shape. Particles diameter, in accordance with the uncertainty typically ± were laser cut within a precision of 0.5 mm and painted found in the literature for similar drop mechanisms, (Heis- black to facilitate the image processing. inger et al. 2014) among others. A square grid was placed In water, the planar particles were released from about inside the water tank to measure the camera magnifcation; two disk diameters below the water surface so that entry and to account for image distortion from the lens in the and surface efects were avoided. The particles were always interrogation area but this was found to be negligible. released using an active suction mechanism with a suc- To build a baseline for comparison with the square and tion cup of 10 mm diameter that can accommodate all three hexagonal particles, a frst set of drops for the circular disk geometries. The angle of the particle prior to the release was performed. The frst section of the particle descent is was carefully adjusted to zero. To control the particle angle, known to be infuenced by the initial transient dynamics, the suction cup was mounted fushed into a submerged fat and the determination of the distance at which the parti- plate. Therefore, when the particle was being held by the cle reaches a ‘steady state’ is non-trivial (this term refers suction cup the upper surface of the particle was parallel to the natural falling style of the particle once this is not and in contact to the plate. The alignment of the submerged infuenced by initial conditions). Heisinger et al. (2014) ∗ − plate was carefully performed before flling the tank using I = 3 × 10 3 ◦ showed experimentally that for disks with a digital angle within an accuracy of 0.1 . This initial tilt and Re ≈ 1200 a vertical distance of about 6.5 D from the angle accuracy is sufcient to have descent trajectories unaf- release point was required to achieve a steady state. Here fected by this parameter, as shown in Lee et al. (2013) for the reference disk has similar dimensionless parameters − ∗ ∗ −3 thin disks within the same Re I region. as in Esteban et al. (2018) ( Re ≈ 1370 , I ≈ 5 × 10 ), and The water tank where the particles were released is an therefore we use the same distance of 7 D as the start of the open glass tank of 0.45 m square cross section and 0.5 steady state of all particles tested in this study. m high. To record the particle descent, two JAI-GO5000 The bottom of the tank also infuences the dynamics cameras were used. One camera captured the frontal view of the particles due to hydrodynamic interactions and of the descent motion, whereas the other captured the pla- therefore we do not process the particle trajectory once it − nar ( X Y ) motion of the particle through a mirror at 45° reaches a distance of 2 D from the tank bottom. Thus, we located underneath the water tank. The cameras were both use trajectory sections that go from 7 D from the top to 2 D focussed on the middle plane of the tank to minimise the from the bottom, and these are not infuenced neither by lens curvature distortion. The trajectories were recorded initial nor end conditions. at 60 fps and this was sufcient to resolve the translational

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Particles are always released in water at room temperature from the reference plane where the three views overlap. The = 3 = × −6 2∕ f 0.998 g/cm and 1.004 10 m s . The waiting reader is referred to Pothos et al. (2009) for a more detailed time between successive drops was 20 min , corresponding explanation of the imaging and processing principles of this to more than 600 times the particle time-scale of the oscilla- technique. tory motion. This waiting time was verifed to be sufcient to In these experiments we performed a pre-calibration have quiescent fow by visualizing tracer particles with the a before each set of drops. This was done using a backlighted 2 volumetric 3-component velocimetry system (V3V camera calibration target with dimensions of 140 × 140 mm with system from TSI). a Cartesian grid of circular dots with a distance of 2 mm in The water media was seeded with 55 μm polycrystalline both directions. A traverse system was used to move the cali- particles. Then, using the same approach, another set of 15 bration target to diferent positions along the camera system trajectories was recorded at 7.25 fps and these were synchro- optical axis. The maximum distance between these planes nised with the V3V camera system. The volumetric meas- was 2 mm and a total number of 30 planes were recorded. urements of the instantaneous velocity feld were obtained The de-warping error is also obtained from the calibra- for a vertical location where the particle trajectory was tion. This is a measure of the standard deviation of grid dot steady (> 10D from the release point). This camera system position error after de-warping and was found to be ≤ 0.3 consists of three 2048 × 2048pixels 12-bit frame-straddle pixels. Considering that tracer displacements were limited to CCD cameras aligned in a coplanar triangle pattern. This less than 8 pixels, the spatial uncertainty in the fow veloci- confguration allowed us to map a feld of view (FOV) of metry is about 4%. Also, the magnifcation of the cameras 140 mm × 140 mm × 60 mm , as sketched in Fig. 1. A syn- was compared to the pinhole camera at each calibration chroniser by TSI was used as an external trigger and con- plane and this is used to adjust the pixel size in each of the nected to the laser and the camera system. A 200mJ/pulse three views during the triplet search. double-pulsed laser (Bernoulli-PIV Litron) was used to A node volume of 10 mm 3 was defned with a 75% of illuminate the FOV. The two-frame double exposure image node volume overlap in the velocity interpolation process. pairs were set at 7.25 Hz and image triplets were analysed The smoothing factor is set such as velocity information via Insight 4G software from TSI. The 3D imaging principle from neighbouring nodes do not contribute to the current of this system is a multi-view photo-grammetry technique. node volume. The FOV of the V3V system was set after a The FOV of the three cameras intersect to form the camera vertical distance of 7 D from the release point to assure the system mapping region. Thus, any seeding particle inside steady state of the particle. For each particle, 15 trajecto- the FOV is recorded from three diferent angles creating ries were recorded and then fully processed and reviewed the basis for multi-view stereo vision. In the image plane, to assure the robustness of the results. During these obser- the three particle images form the coplanar camera arrange- vations, particles were found to describe both zig-zag and ment form a triangle, which centre determines the x and y three dimensional descents but we always found that similar coordinates of the particle. The defocusing of the triangle particle trajectories shared the same wake characteristics. formed is the basis to obtain the location in the z coordinate 3 Determination of non‑dimensional parameters

Five dimensional quantities are generally used to defne the non-dimensional parameters that characterise the falling style of disks and quasi 2D rectangular plates; and these quantities are: the characteristic length, the thickness and the density of the particle; and the density and kinematic viscosity of the fuid. These dimensional quantities are com- bined to form three dimensionless numbers that determine the settling behaviour of the freely falling particle. These ∗ are the dimensionless moment of inertia ( I ), the Reynolds number (Re) and the aspect ratio of the particle ( ). The planar particles of this study have a nearly constant aspect Fig. 1 Sketch of the experimental setup. Two high-speed cameras ratio, ≈ 0.07 . Therefore, we believe the dynamics of the record the trajectory of the particle (front view and bottom view fall to be independent of this parameter. However, the par- through a mirror at 45°). The V3V system acquires triplets of images that are later used to reconstruct the velocity feld around the falling ticles have severe diferences in their frontal geometry and particle the particle characteristic length cannot be fully captured by

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A the equivalent particle diameter. Thus, we use the approach Table 1 Geometric characteristics of the particles; p refers to D P proposed in Esteban et al. (2018) to obtain a new particle frontal area, c to diameter of circumscribed disk, to perimeter of frontal geometry and Q to isoperimetric quotient, defned characteristic length-scale from which to obtain an equiva- Q = A ∕P2 as 4 p . Mean descent velocity ( V ) and peak to peak V z lent Reynolds number and dimensionless moment of inertia. velocity ( z pp ) along the descent. The (z|s) ⟨notation⟩ in the hexagons Thus, the dimensionless moment of inertia is defned as stands for particles describing zig-zag and spiral, respectively Disk Hexagons (z|s) Square I p I∗ = 5 (1) A 2 f p (cm ) 4.34 4.39 4.41 D c (cm) 2.35 2.6 2.97 = ∕ P where D c Q , with D c the diameter of the circum- (cm) 7.4 7.8 8.4 = ∕ 2 Q scribed circle and Q 4 Ap P is the isoperimetric quo- 1 0.91 0.79 tient. Similarly, the Reynolds number is defned as, Mass (g) 1.07 1.01 1 V ⋅ −1 ± ± ± ± z (mm s ) 58 2 56 3 | 54 2 55 3 V = z ⟨V ⟩ ⋅s−1 Re . (2) z p p (mm ) 110 94 | 78 67 ⟨ ⟩ The advantage of using this defnition for the particle char- = ∕ acteristic length-scale, D c Q is that the characteristic length ( ) of the reference disk is still its true diameter and ∗ therefore the ( Re − I ) phase diagram defned by Field et al. (1977) is not altered. The Reynolds number and dimensionless moment of inertia for disks, hexagons and squares studied in here appear to be within the futtering regime originally defned by Field et al. (1977) (Fig. 2). The dimensionless numbers of these particles can be found in Table 2.

4 Results

In this section, results on the trajectory characteristics and fow visualization of the wake behind the planar particles are presented. First, we introduce the trajectory characteristics ∗ obtained from the high-speed cameras and the diferences Fig. 2 Re − I phase diagram for thin circular disks as in Esteban between frontal geometries are compared. Second, vorticity et al. (2018). The solid and broken lines defne the regimes and sub- iso-surfaces and contours in the wake of the particles are regimes found in Field et al. (1977) and Zhong et al. (2011), respec- shown for trajectories that are representative of the motion tively. The colour gradient is a qualitative measure of the likelihood of any particle to describe either ‘planar zig-zag’ or ‘spiral’ motion, of each geometry. where white represents pure ‘planar zig-zag’ motion and dark red ‘spiral’ motion. A more detailed analysis on the statistics of the parti- 4.1 Trajectory characteristics cle trajectory can be found in Esteban et al. (2018)

Two views of the particle descent are acquired syn- deviations from the idealised complete turn of 180 deg in chronously so that the 3D position of the particle can planar velocity direction. Similarly, ‘transitional’ motion be obtained, as in Fig. 3. We observe that diferent tra- refers to trajectories with the same oscillatory pattern as in jectories show strong diferences in the relevance of the ‘planar zig-zag’ motion but with a superimposed constant out-of-plane motion, as observed experimentally and in angular velocity about the Z−axis. The particle trajectory simulations for disks with diferent dimensionless inertia in the X − Y plane resembles a rhodonea curve, as shown ∗ I [Zhong et al. 2011, 2013; Zhong and Lee 2012; Chrust in Zhong et al. (2011) for disks with small dimensionless ∗ et al. 2013; Auguste et al. 2013 among others] and experi- moment of inertia I . Trajectories describing highly 3D mentally for planar polygon particles (Esteban et al. 2018). motion are named ‘spiral’ motion; the X − Y footprint of Thus, in here ‘planar zig-zag’ motion refers to trajectories these trajectories shows a particle describing a sequence with two trajectory sections that repeat periodically; glid- of circles with a fnite ofset between consecutive loops. ing sections where the particle oscillates within a X − Y These trajectories also show a much steadier descent plane and turning sections where the particle shows small

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Fig. 3 Reconstructed trajectory section of planar particles describ- motion, b hexagon under ‘transitional’ motion, c, (i) hexagon under ing the three regimes proposed in Esteban et al. (2018); a, (i) disk ‘spiral’ motion and c, (ii) square under ‘spiral’ motion under ‘planar zig-zag’ motion, a, (ii) hexagon under ‘planar zig-zag’ velocity. This is captured in the peak to peak velocity We observe a diference in descent velocity between hexa- (V z p p ) shown in Table 1. gons describing ‘zig-zag’ and ‘spiral’. The descent velocity The variation in the mean descent velocity measured from is about 5% higher for the particles describing ‘zig-zag’ the experimental data for the three geometries can be motion but this result comes from a small number of drops. explained in terms of the variations in mass and surface area We believe that a more thorough experiment (multiple pla- Ap . From a force balance between gravity forces and aero- nar particles with diferent Reynolds numbers) should be dynamic forces, the average descent velocity V z is done to establish a more robust relationship between falling ∕ expected to scale with Mg A p . If one computes the⟨ non-⟩ dimensional descent velocity for each geometry using the measured physical properties of the particles they are within Q Table 2 Values of the particle Reynolds number (Re) based on the 1% of each other. This suggests that variations in are V mean fall velocity ( z ) and the characteristic length-scale ( ), and unlikely to cause variations in the mean descent velocity of I∗ dimensionless moment⟨ of⟩ inertia based on the length-scale ( ) the particles. This is consistent with our previous observations where Q plays an important role in determining the Disk Hexagon Square I∗ −3 −3 −4 non-dimensional inertia of the object and not necessarily the 5.01 ⋅ 10 1.95 ⋅ 10 5.13 ⋅ 10 Re descent velocity (or the corresponding Reynolds number). 1370 1450 1620

1 3 Experiments in Fluids (2019) 60:114 Page 7 of 12 114 style and mean descent velocity, as performed in Tam (2015) create a vortex loop elongating in the direction of the body for elastic plates. movement to form hairpin-like structures, but as the planar Square plates show a peak-to-peak velocity along the symmetry is broken, disks shed a vortex chain from the outer trajectory (V z pp ) approximately 30% smaller than disks, edge, forming a helicoidal vortex wrapping around the wake showing that the descent motion of these particles is much region. steadier than that observed in disks. To the best knowledge of the authors, this is the frst In Lee et al. (2013), they described how a non-uniform experimental study showing three dimensional three com- lift distribution over the disk surface can create a lift- ponent fow felds in the wake of planar particles. Also, it induced torque, leading to the disruption of the original shows the strong connection between particle descent style planar motion. Following the same reasoning they showed and wake characteristics for the futtering sub-modes identi- ∗ how the dimensionless moment of inertia of the particle ( I ) fed in Zhong et al. (2011) (‘planar zig-zag’, ‘transitional’ might either stabilise or destabilise the system depending and ‘spiral’ motion) but extended to other planar geometries. ∗ ∗ on its relative magnitude. Therefore, if I < Icrit the system ∗ ∗ instabilities grow at every turning point, while if I > Icrit 4.2.1 Disks the system becomes more stable. The non-uniform lift distribution explained in Lee et al. (2013) occurs naturally for We found severe differences in the wake of a particle particles with non-axisymmetric frontal geometries. We describing ‘planar zig-zag’ motion depending on its loca- hypothesize that this mechanism is the reason why as the tion relative to the turning point. The wake behind a disk particle circularity reduces the likelihood of having a planar describing ‘planar zig-zag’ motion shows characteristic trajectory also does, as shown in Esteban et al. (2018) and vortex structures that repeat during the particle descent at confrmed here. every gliding and turning section. As the disk describes a In the following section, the strong diferences in the fall- gliding section of the trajectory, a pair of counter-rotating ing dynamics between particle trajectories are investigated vortices progressively forms in the downstream direction by means of volumetric fow analysis of the wakes. from both sides of the symmetry plane of the disk aligned with the gliding motion. These vortices remain primarily 4.2 Wake characteristics aligned with the motion of the disk, as shown in Fig. 4. On the other hand, as the disk approaches a turning event, it The efect of vorticity in the fow behind the particle is decelerates and increases angle of attack. When the angle one of the key features to understand the path instability of of attack becomes too high the disk experiences stall and a planar particles. Disks freely falling in viscous media have recirculation zone, whose predominant vorticity component been extensively investigated during the last 60 years, as in is normal to the direction of motion, is formed at the leading Willmarth et al. (1964), Field et al. (1977), Fernandes et al. edge of the disk (shown in red in the iso-contours). Then, the (2005), Zhong et al. (2011) or Lee et al. (2013) among oth- particle planar velocity reduces progressively together with ers. Fernandes et al. (2005) showed experimentally that the the descent velocity up to the turning point where it becomes phase diference between the velocity and the inclination of zero and the disk motion reverses direction. At this point, the the body axis greatly difers for the irrotational theory esti- recirculation zone at the leading edge has grown in size and mation, showing that vortical efects in the wake were cru- detaches forming a hairpin-like vortex. The development of cial to understand body dynamics. A few years later, Zhong the vorticity structures on the circular disk leading edge near ∗ et al. (2011) showed that thin disks lying in the Re − I the turning point resembles the leading edge vortex observed domain corresponding to futtering motion exhibit three in fapping foils, where this structure is also shed into the diferent descent styles that are associated with a change in wake after the motion is reversed. These measurements con- ∗ their dimensionless moment of inertia ( I ); i.e. ‘planar zig- frm the wake structures visualized by Zhong et al. (2011) zag’, ‘transitional’, and ‘spiral’. They used fuorescence dye with fuorescence dye. On the other hand, these measure- to visualized vortex patterns corresponding to the ‘planar ments do not show the second recirculation zone found in zig-zag’ and ‘spiral’ descent. This showed that in each cycle Lee et al. (2013) on the lower surface of the particle. Lee of the zig-zag motion a pair of hairpin vortices were shed et al. (2013) showed that the both recirculation zones co- into the wake, whereas for the spiral descent they observed existed at a similar relative location of the trajectory and a helicoidal vortex evolution. Similarly, Lee et al. (2013) eventually merged together and detached also forming a performed fow visualizations with fuorescence dye and pla- hairpin-like vortex. We believe that the lower recircula- nar PIV measurements for disks describing a ‘zigzag-spiral- tion zone might form and develop during short times and zigzag’ intermittence fnding that the characteristic turbulent the limited time resolution of the system used in here ( 7.25 structures in the wake of the particle go hand by hand with fps) might not be sufcient to capture this wake dynamics. their descent mode. Thus, disks describing zig-zag descent Similarly, we do not capture secondary hairpins in the wake

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Fig. 4 a Reconstructed trajectory section of a disk describing ‘planar zig-zag’ motion with vorticity iso-surfaces of the wake behind the disk at dif- ferent locations relative to the turning point. b Contour plots of the vorticity magnitude in the X − Y plane at a distance A of p from the upper surface √ of the particle at the same locations. The iso-surfaces have a −3 −1 magnitude of 2.5 × 10 s and the contours are also saturated −3 −1 at 2.5 × 10 s . Yellow and blue contours represent positive and negative X−vorticity, black and red Y−vorticity, green and pink Z−vorticity

of the particle. We believe this is due to the low vorticity generate vortical structures with diferent dominant compo- magnitude that these structures carry relative to the counter- nents. Taken together, these diferent vorticity components rotating vortices that grow from the leading edge of the disk, give rise to a large-scale vortex structure in the wake during as described in Zhong et al. (2013). the particle fall. This is best visualized by examining the The complete wake region of the disk now includes two contours in the wake (shown in separate contour plots on the rows of hairpin-like vortices evenly spaced in the vertical right) taken at one characteristic length scale ( Ap ) behind √ direction that carry vorticity in opposite directions linked the particle. These contours clearly show that there is really by a pair of counter rotating vortices. The overall structure no diference in the strength of vorticity or a preference to of the wake is also in good agreement with results from which side of the particle this vorticity is present. It almost numerical simulations of free-falling thin disks at moderate seems like the vorticity has rotational invariance about the Reynolds numbers, as in Churst et al. (2013). A closer look vertical axis, despite the fact that the particle itself is not to the vortical structures shed of at every extreme point of axisymmetric. This invariance in the vorticity strength is the oscillatory motion shows that these are responsible for a refection of the type of motion followed by the square the regions of strong downwash in the wake of the disk. The plates. When the square describes a section of the trajectory magnitude of the velocity in the region dominated by the with any of its corners aligned with the direction of motion, downwash is of the same order of magnitude of the descent a pair of counter-rotating vortices forms from both sides velocity observed along the disk trajectory and this also of the corner, propagating downstream (as in the leading agrees with previous planar particle image velocimetry data edge of a delta wing). However, instabilities in the particle in Zhong et al. (2013). wake due to small misalignments of the leading corner with the lateral motion or instabilities originated at other particle 4.2.2 Square plates sharp corners quickly realign the particle to a diferent orientation. In this new orientation, the same phenomenon is The wake characteristics behind a square plate falling under repeated. Therefore, the descent of a square particle becomes ‘spiral’ motion are almost unafected by the relative position a series of reorientations all along its trajectory. This process of the particle along the trajectory, exhibiting nearly continu- results in the breaking down of individual vortical structures ous shedding of vortical structures along the perimeter of and gives rise to a wake with nearly rotational invariance. the particle. The vortex structures are signifcantly diferent Further time-resolved measurements with higher spatial from those observed in the planar motion of a disk. Here, resolution are necessary to examine the fner details of this vortex structures with diferent dominant vorticity compo- wake formation. nents are formed indistinctly during the descent of the particle, as in Fig. 5a. Each of the particle corners perceive the fow with a diferent relative angle of incidence and therefore

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Fig. 5 a Reconstructed trajectory section of a square describing ‘spiral’ motion with vorticity iso-surfaces of the wake behind the square at diferent locations. b Contour plots of the vorticity magnitude in the X − Y plane at a distance A of p from the upper surface √ of the particle at the same locations. The iso-surfaces have a −3 −1 magnitude of 2.5 × 10 s and the contours are also saturated −3 −1 at 2.5 × 10 s . Yellow and blue contours represent positive and negative X−vorticity, black and red Y−vorticity, green and pink Z−vorticity

4.2.3 Hexagons is weaker and therefore not as clearly observed in the iso- surfaces. However, the contour plots taken downstream of Freely falling hexagons describe both types of trajectories; the particle in Fig. 6b seem to show that the vorticity is i.e. ‘planar zig-zag’ and ‘spiral’ motion. When the hexagon preferentially distributed along the perimeter of the particle describes ‘planar zig-zag’ motion, the wake behind the par- just as in the case of the disks. As the hexagon describes a ticle has some similarities to the wake behind disks under gliding section of the trajectory, two large counter-rotating ‘planar zig-zag’, as seen in Fig. 6a. The vortex structures vortices appear on both sides of the symmetry plane of the shown for disks seem to be present in the wake of the hexa- hexagon aligned with the motion, yet they are substantially gon but the strength of these structures for the hexagons altered by the vorticity generated on the other edges of the

Fig. 6 a Reconstructed trajectory section of a hexagon describing ‘planar zig-zag’ motion with vorticity iso- surfaces of the wake behind the hexagon at diferent locations relative to the turning point. b Contour plots of the vorticity magnitude in the X − Y plane A at a distance of p from the √ upper surface of the particle at the same locations. The iso-surfaces have a magni- −3 −1 tude of 2.5 × 10 s and the contours are also saturated at −3 −1 2.5 × 10 s . Yellow and blue contours represent positive and negative X−vorticity, black and red Y−vorticity, green and pink Z−vorticity

1 3 114 Page 10 of 12 Experiments in Fluids (2019) 60:114 geometry. It is hypothesised that when the particle reaches mean descent velocity. The velocity felds are fltered prior a turning point, the non-uniform pressure distribution along to computing vorticity magnitude using a Gaussian flter the particle geometry is the reason why the particle rotates with kernel size of half of the diameter of the reference cir- about an axis perpendicular to the particle surface and it is cular disk. realigned towards a diferent plane of motion. Thus, the hex- The distribution of the vorticity magnitude in Fig. 8a agon starts a new cycle, following a ‘planar zig-zag’ motion shows a progressive decrease in high-vorticity events in but contained in a new oscillatory plane. However, when the the wake of the particle as this becomes less axisymmet- hexagon describes ‘spiral’ descent, see Fig. 7a), the wake ric, where black corresponds to disks, green to hexagonal of the particle resembles the wake shown for square plates, particles and red to squares. To evaluate the connection with similar vorticity distribution along the particle descent. between falling style and the signature of vorticity mag- This is also clear in the contour plots, shown in Fig. 7b), nitude, we separate the trajectories of the hexagonal par- obtained downstream of the particle and the similarities with ticles according to the descent style; i.e. ‘planar zig-zag’ the wake of a square are clear. and ‘spiral’ motion. Thus, we now obtain the distribution Thus, particles with diferent frontal geometry; i.e. disks of vorticity magnitude for the hexagonal particles with and hexagons; and hexagons and squares, exhibit qualita- similar trajectory style and wake morphology. Figure 8b) tively similar wake structures when falling under the same shows the non-dimensional vorticity magnitude of these mode. However, the modes of secondary motion for the two types of trajectories (green dashed line for hexagons same conditions are not equally probable for all particles, in ‘planar zig-zag’ and green dotted line for hexagons in but depend on the particle geometry. ‘spiral’) together with disks and squares. One can observe To have a more quantitative comparison between the that the PDF for hexagonal particles describing ‘planar wakes of the particles, Fig. 8 shows the probability density zig-zag’ motion becomes nearly identical to the one corre- function (PDF) of the non-dimensional vorticity magnitude sponding to disks, whereas the distribution in the vorticity = ∕ = 2 + 2 + 2 ( ̃� �V z Ap , where ) on the wake magnitude in the wake of hexagonal particles describing √ x y z of the n-sided polygons investigated. Since the vorticity in ‘spiral’ motion resembles the one measured for square the wake of particles has diferent components and direction, particles. examining the distribution of vorticity magnitude provides This suggests that the vorticity magnitude in the wake a way to compare these across all cases. The three dimen- of planar particles with diferent frontal geometry might be sional vorticity magnitude felds used for this task contain a well predicted once the falling style is known; and this at fuid domain that includes two particle oscillations about the the same time can be approximated using the isoperimetric

Fig. 7 a Reconstructed trajectory section of a hexagon describing ‘spiral’ motion with vorticity iso-surfaces of the wake behind the hexagon at diferent locations. b Contour plots of the vorticity magnitude in the X − Y plane at a distance A of p from the upper surface √ of the particle at the same locations. The iso-surfaces have a −3 −1 magnitude of 2.5 × 10 s and the contours are also saturated −3 −1 at 2.5 × 10 s . Yellow and blue contours represent positive and negative X−vorticity, black and red Y−vorticity, green and pink Z−vorticity

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(a) (b)

Fig. 8 a Probability density function of the non-dimensional vorti- disks, the red solid line to squares, the green solid line to hexagonal city magnitude in the wake of disks, squares and hexagonal plates. planes prior to the trajectory classifcation, whereas the green dash b Probability density function of the non-dimensional vorticity mag- line and the green dotted line correspond to hexagonal plates falling nitude in the wake of disks, squares and hexagon plates grouped in ‘planar zig-zag’ and ‘spiral’ motion, respectively according to the descent style. The black solid line corresponds to quotient of the frontal area, as discussed in Esteban et al. the normal direction of motion. Thus, when the motion of (2018). the disk is reversed the recirculation zone on the upper surface detaches forming a hairpin-like vortex structure that is shed into the wake. 5 Conclusion In contrast, squares were found to describe ‘spiral’ motion more often, with a wake dominated by large-scale shedding This work investigated the efect of the particle edge geom- of vorticity along the entire periphery of the particle. The etry on the free-falling motion and associated wake charac- authors believe that the lack of a preference direction in the teristics of planar particles in a viscous media. The reference vorticity leads to the characteristic three dimensional tra- particle was chosen to be a disk that lies within the Reyn- jectory of these particles. Relatively strong vortical struc- ∗ olds - dimensionless moment of inertia domain ( Re − I ) tures are observed to the sides of a sharp corner when this is corresponding to the ‘planar zig-zag’ sub-mode found by aligned with the particle planar motion. However, other vor- Zhong et al. (2011) within the futtering regime originally tical structures with diferent dominant components quickly defned by Willmarth et al. (1964). The material properties realign the particle to a diferent orientation breaking the and the frontal area of the planar particles were maintained trajectory periodicity along the descent. constant and the isoperimetric quotient (a measure of the Hexagons exhibit ‘planar zig-zag’, ‘transitional’ and ‘spi- particle circularity) was varied by altering frontal geometry ral’ motions. When these undergo a ‘planar zig-zag’ motion of the particle to diferent n-sided polygons; i.e. hexagons their wake is qualitatively similar to the wake of a circular and squares. disk and for ‘spiral’ motions the wake exhibits large-scale High-speed imaging and three dimensional three compo- vortex shedding as seen for square particles. nent measurements of the instantaneous velocity feld were The probability density function of the non-dimensional used to characterise the particle trajectory and the associated vorticity magnitude in the wake of the particles shows that wake. Disks were found to describe ‘planar zig-zag’ trajec- n-sided planar particles not only share similar descent style tories most of the time. These trajectories are characterised and wake morphology but also have wakes with comparable by a sequence of gliding–turning sections. The near wake vorticity strength. However, these observations are limited to of disks during the gliding phase is comprised by a pair of large-scale structures and noticeable diferences in the wake counter-rotating vortices that grow from the leading edge of of these particles might occur at smaller scales than half of the disk in the downstream direction carrying most of the particle diameter (kernel size of the Gaussian flter). vorticity in the stream-wise direction. In contrast, the near Further time-resolved measurements would be of great wake during the turning section is characterised by leading interest since these would provide further details of the edge fow separation whose predominant vorticity is along

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Afliations

Luis Blay Esteban1 · John Shrimpton1 · Bharathram Ganapathisubramani1

* Luis Blay Esteban Bharathram Ganapathisubramani [email protected] [email protected] John Shrimpton 1 University of Southampton, Southampton, UK [email protected]

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