Atmospheric Research 142 (2014) 45–56

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Orientation statistics and settling velocity of ellipsoids indecaying turbulence

C. Siewert⁎, R.P.J. Kunnen 1, M. Meinke, W. SchröderInstitute of Aerodynamics, RWTH Aachen University, Wüllnerstrasse 5a, 52062 Aachen, Germany

a r t i c l e i n f o

⁎ Corresponding author. Tel.: +49 2418094820.E-mail address: c.siewert@aia.rwth-aachen.de (C. S

1 Current address: Fluid Dynamics Laboratory, DePhysics, Eindhoven University of Technology, P.O. Box 51Netherlands.

0169-8095/$ – see front matter © 2013 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.atmosres.2013.08.011

a b s t r a c t

Article history:Received 28 February 2013Received in revised form 15 August 2013Accepted 20 August 2013

Motivated by applications in technology as well as in other disciplines where the motion ofparticles in a turbulent flow field is important, the orientation and settling velocity of ellipsoidalparticles in a spatially decaying isotropic turbulent flow are numerically investigated. Withrespect to cloud microphysics ellipsoidal particles of various shapes are interpreted asarchetypes of regular ice crystals, i.e., plates and columns approximated by oblate and prolateellipsoids. The motion of 19 million small and heavy ellipsoidal particles is tracked by aLagrangian point-particle model based on Stokes flow conditions. Five types of ellipsoids ofrevolution such as prolates, spheres, and oblates are considered. The orientation and settlingvelocity statistics are gathered at six turbulence intensities characterized by the turbulentkinetic energy dissipation rate ranging from 30 to 250 cm2s−3. It is shown that the preferentialorientation of ellipsoids is disturbed by the turbulent fluctuations of the fluid forces andmoments. As the turbulence intensity increases the orientation probability distributionbecomes more and more uniform. That is, the settling velocity of the ellipsoids is influencedby the turbulence level since the drag force is dependent on the orientation. The effect is morepronounced, the longer the prolate or the flatter the oblate is. The theoretical settling velocitybased on the orientation probability of the non-spherical particles is smaller than that found inthe simulation. The results show the existence of the preferential sweeping phenomenon alsofor non-spherical particles. These two effects of turbulence on the motion of ellipsoids changethe settling velocity and as such the swept volume, that is expected to result in modifiedcollision probabilities of ellipsoid-shaped particles.

© 2013 Elsevier B.V. All rights reserved.

Keywords:TurbulenceEllipsoidsOrientation distribution functionSettling velocityDirect numerical simulation

1. Introduction

During the last decades the influence of in-cloud turbu-lence on the formation of precipitationwas a field of intensiveresearch (Devenish et al., 2012). The collision probabilities ofcloud particles are described physically by collision kernels(Beheng, 2010). One major effect of turbulence is the modi-fication of the collision kernel by modifying the relativevelocities of the cloud particles (Grabowski andWang, 2013).Whereasmany investigations about the collision probabilities

iewert).partment of Applied3, 5600MBEindhoven,

ll rights reserved.

in warm clouds have been conducted, only relatively little isknown about mixed-phase clouds where supercooled waterdrops and ice particles coexist (Pinsky and Khain, 1998).This is mainly due to the complex shapes of ice particlesdepending on the temperature and the available supersatu-ration (Pruppacher and Klett, 1997). Due to the diverseshapes of the ice particles, velocities and relative velocitiesdepend on their orientation, which is why the orientation ofnon-spherical particles in turbulent atmospheric flow waspart of investigations which are mentioned in the following.However, the findings of these studies vary regarding theorientation behavior. Using dimensional arguments Cho et al.(1981) concluded that turbulence intensities typically foundin clouds cannot destroy the preferential orientation. This issupported by the orientation model of Klett (1995). Both

46 C. Siewert et al. / Atmospheric Research 142 (2014) 45–56

studies conclude that ellipsoids are in a stable orientation withtheir long axes horizontal as measured under quiescentconditions (Newsom and Bruce, 1994). On the other hand,Krushkal and Gallily (1988) concluded from their numericalresults that the preferential orientation effect decreases atincreasing turbulent dissipation energy. The measurements ofNewsom and Bruce (1998) support this conclusion.

Also inmore technical applications Fan andAhmadi (1995a)aswell as Olson (2001) numerically studied themotion of fibersin a modeled isotropic turbulent field. However, they did notconsider gravity. Thus, these studies only provide time scalesfor turbulent translation and rotation of fibers.

Recently, the motion of rods and fibers was analyzed byseveral research groups. Parsheh et al. (2005) measured theorientation of fibers in a planar contraction. Shin and Koch(2005) numerically investigated the motion of fibers inisotropic turbulence. Wilkinson et al. (2009) investigated theorientation of fibers experimentally and numerically. Parsa et al.(2011) reported measured rod alignment in 2D flows. Pumirand Wilkinson (2011) determined the orientation statistics ofrods in turbulence. Parsa et al. (2012) compared experimentalresults with numerical findings. What all these studieshave in common is that they investigated the motion andorientation of fibers at the same density as the carrierflow. That is, the particle translation motion is the same asof fluid tracers. The studies found that fibers rotate likematerial lines and are more aligned with the direction of thevorticity vector than with the principal axis of shear. Thus, itcan be concluded that the knowledge about the Lagrangianproperties of turbulence is extended towards the rotationaldegree of freedom.

However, the presence of inertia and gravity results in aneven more complex behavior. For spherical particles severaladditional mechanisms are known. For example it was firstshown by Wang and Maxey (1993) that particles with masssediment faster in a turbulent flow than in quiescent conditions.This result is due to the fact that if the time scale of the particlesand the turbulent eddies match, the particles preferentiallychoose the downward-moving side of the eddy such that theyexperience a downward fluid motion on average. Dávila andHunt (2001) predicted theoretically that this effect of preferen-tial sweeping should bemaximal if the so-called particle Froudenumber is unity. This was confirmed numerically by Ayala et al.(2008) and Siewert et al. (2013). The higher velocity increasesthe swept volume which alters the relative velocities betweenpairs of particles and as such the collision probabilities, whichmight promote rain formation.

The proof of the existence of the preferential sweepingeffect for non-spherical particles is not trivial. The drag and assuch the sedimentation velocity of a non-spherical particledepend on its orientation even in quiescent conditions. Ingeneral, the drag might even be unknown, although the mostcommon ice particle shapes, namely hexagonal plates andcolumns, can be reasonably well approximated by oblate andprolate ellipsoids (Pruppacher and Klett, 1997), for whichthe drag is known. Anyway the orientation will be influencedby the presence of turbulence. Therefore, the orientationunder the influence of turbulence must be known before thesettling velocities are investigated. The results of Mallier andMaxey (1991) investigating non-spherical particles in lami-nar cellular flow suggest that the preferential sweeping effect

might also exist for non-spherical particles. However, Gavzeet al. (2012) have recently shown by analyzing the governingequation of the rotational motion of prolate ellipsoids thatthe time scales the particle needs to reach a stable orientationare larger than the shear time scale of the turbulent flow.Therefore, a Lagrangian simulation of ellipsoids in a turbulentflow is required.

Until today all the numerical studies have utilized flowmodels, i.e., a turbulence parameterization to study the effectof isotropic turbulence on the motion of heavy non-sphericalparticles. In contrast, the current study is the first to investigatethe motion of heavy ellipsoids using a direct numerical simu-lation of isotropic turbulence, i.e., a time-dependent simulationwithout any turbulence modeling of the flow phase. Addition-ally, to the authors' knowledge, this is the first numerical studyof the motion of plate-like particles in any turbulent flow.While themotion of elongated particles like rods and fibers in aturbulent flow have been the subject of many investigations,only a few theoretical studies considered disk- or plate-likeshaped particles, e.g. Klett (1995). With the present methodit is possible to investigate the existence of the preferentialsweeping effect for non-spherical particles.

This study is mainly attributed to themotion of ice particlesin turbulent clouds, i.e., all parameters are chosen to matchcloud conditions as closely as possible. However, the generalsetup and the results also are of interest for many otherscientific and engineering fields where the orientation ofnon-spherical particles plays a role in , e.g., rheology, aerosolscience, papermaking, drag reduction in turbulent multiphasepipe flows and so forth.

This paper is organized as follows. First, the governingequations of the motion of ellipsoids in the limit of Stokesflow are presented in general form. Afterwards, the setup forthis investigation is briefly introduced including the descrip-tion of the types of ellipsoids. Then, the orientation statisticsof ellipsoids in turbulent flow are presented. Resulting fromthese statistics the theoretical sedimentation velocities aredetermined and compared with ensemble-averaged veloci-ties to get an insight into the preferential sweeping effectof non-spherical particles. At the end a physical interpreta-tion for the occurrence of the preferential orientation andsweeping is given.

2. Equations of motion for ellipsoids

The numerical method used to simulate the motion ofsmall and heavy ellipsoids is similar to that used by Fan andAhmadi (1995b) and subsequent investigators (Marchioliet al., 2010; Mortensen et al., 2008a; Zhang et al., 2001).They investigated the deposition rates of fibers in turbulentchannel flows. However, the method is generalized to dealwith all kinds of ellipsoids of revolution similar to Gallily andCohen (1979) as well as Krushkal and Gallily (1988). Hence,the equations of motion are presented in detail.

2.1. Kinematics

The half-axes of an ellipsoid are denoted by a, b, and c.The coordinate system is defined such that it is fixed atthe particle center of mass and rotates with the particle,such that, the axes of the coordinate system denoted by a hat

Fig. 1. Ellipsoidal coordinate system. Coordinates in the ellipsoidal coordinatesystem bx;by;bz� �

can be converted into the inertial coordinate system (x, y, z)by two steps. First, a rotation around the ellipsoid center of mass determinedby the axes of rotation e and the angle of rotation Ω. Second, a translationdetermined by with the position vector of the ellipsoid center of mass xe.

47C. Siewert et al. / Atmospheric Research 142 (2014) 45–56

always coincide with the principal axes of the ellipsoid. Theorientation of the ellipsoid, i.e., the particle fixed coordinatesystem relative to the inertial system, is tracked by quater-nions because of their numerical superiority over the Eulerangles (Fan and Ahmadi, 1995b). Using the sketch shown inFig. 1, the four quaternions �1, �2, �3, η are related to the axesof rotation e and the angle of rotation Ω

�1�2�3

0@ 1A ¼ e sinΩ2

� �; η ¼ cos

Ω2

� �: ð1Þ

Thus, the coordinates given in the particle fixed coordi-nate system ( bx;by;bz ) can be converted into the inertialcoordinate system (x, y, z) by a translation operation usingthe position of the particle center of mass xe and a rotationoperation using the rotation matrix (Goldstein, 1980)

A ¼1−2 �

22 þ �

23

� �2 �1�2 þ �3ηð Þ 2 �1�3−�2ηð Þ

2 �2�1−�3ηð Þ 1−2 �23 þ �

21

� �2 �2�3 þ �1ηð Þ

2 �3�1 þ �2ηð Þ 2 �3�2−�1ηð Þ 1−2 �21 þ �

22

� �0BBB@

1CCCA: ð2Þ

The orientation of an ellipsoid changes over time due toits angular velocities bω by

d�1dtd�2dtd�3dtdηdt

0BBBBBBBBBBB@

1CCCCCCCCCCCA¼ 1

2

þη −�3 þ�2þ�3 þη −�1−�2 þ�1 þη−�1 −�2 −�3

0BB@1CCA ωbx

ωbyωbz

0@ 1A: ð3Þ

To prevent error accumulation the quaternions are normal-ized each time step to fulfill (Mortensen et al., 2008a)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�21 þ �22 þ �23 þ η2

q¼ 1: ð4Þ

The position of the center of mass xe depends on thevelocity v:

dxe

dt¼ v: ð5Þ

2.2. Dynamics

In the limit of very small and heavy particles the translationand rotation equations of motion can be significantly simpli-fied. In this study, the ellipsoid half-axes a, b, and c are at leastone order of magnitude smaller than the Kolmogorov scale ηk.Nevertheless, the ellipsoids are large and heavy enough toneglect the influence of Brownian motion (Foss et al., 1989).The density of the ellipsoids ρe is nearly three orders ofmagnitude larger than the fluid density ρf. In these limits, thetranslation acceleration only depends on the gravity g, theellipsoidmassm ¼ ρe

43πabc, and the hydrodynamic drag force F

(Brenner, 1964). Under creeping flow or Stokes conditions thehydrodynamic drag force depends only on the fluid viscosityv, the fluid density ρf, the particle resistance tensor bK , andthe difference of the fluid velocity at the particle position u andthe particle velocity v:

dvdt

¼ gþ Fm

¼ gþ νρ f

mA−1bKA u−vð Þ: ð6Þ

Note that the particle resistance tensor is a dimensionalquantity. To extend the range of the Stokes approximationtowards higher particle Reynolds numbers Rep = |v|a/ν,several non-linear drag laws were proposed for sphericalparticles (Clift, 1978). Using perturbation methods forces(Brenner and Cox, 1963) and moments (Cox, 1965) ofparticles of arbitrary shapes have been generalized by anexpansion in the Reynolds number. However, as pointed outby Chester (1990): “although the evaluation for ellipsoids isstraightforward in principle, it is not so in execution”. So, tothe authors' knowledge, a non-linear correction for generalellipsoids is not available in the literature. However, sinceRep b 1 it seems reasonable to use the Stokes approximationin this investigation.

The resistance tensor of an ellipsoid bK is a diagonal matrixin its principal axes. A first solution attempt can be found inOberbeck (1876):

bK ¼ Ekbx bxkby bykbz bz

0B@1CA ¼ 16πabcE

1χ0 þ a2α0

1χ0 þ b2β0

1χ0 þ c2γ0

0BBBBBBBB@

1CCCCCCCCA: ð7Þ

The quantity E denotes the identity matrix. The shapeparameters χ0, α0, β0, and γ0 are given by Brenner (1964) bythe following integrals

χ0 ¼ abcZ ∞

0

dλΔ

α0 ¼ abcZ ∞

0

dλa2 þ λ� �

Δ

β0 ¼ abcZ ∞

0

dλb2 þ λ� �

Δ

γ0 ¼ abcZ ∞

0

dλc2 þ λ� �

Δ

with Δ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ λ� �

b2 þ λ� �

c2 þ λ� �q

:

ð8Þ

Fig. 2. Domain setup. At the lower side inlet synthetic turbulence u′ is addedto the constant mean flow U opposite to gravity g. The six statistic volumesare located in the upper part of the domain.

48 C. Siewert et al. / Atmospheric Research 142 (2014) 45–56

The rotational motion of an ellipsoid about its principalaxes is governed by

dωbxdtdωbydtdωbzdt

0BBBBBBBB@

1CCCCCCCCA¼

ωbyωbz Iby−IbzIbx

ωbz ωbx Ibz−IbxIby

ωbx ωby Ibx−IbyIbz

0BBBBBBBBBBB@

1CCCCCCCCCCCAþ

TbxIbxTbyIbyTbzIbz

0BBBBBBBBBB@

1CCCCCCCCCCA: ð9Þ

The moments of inertia of an ellipsoid about its principalaxes constitute a diagonal matrix

Ibx 0 0

0 Iby 0

0 0 Ibz

0BB@1CCA ¼ m

5E

b2 þ c2

a2 þ c2

b2 þ a2

0B@1CA ¼ 4

15πρeabcE

b2 þ c2

a2 þ c2

b2 þ a2

0B@1CA:

ð10Þ

The hydrodynamic torques T for ellipsoids in general shearflow in the limit of Stokes flow are given in the classical paperby Jeffery (1922). Although it is only a leading-order approx-imation, the formulation is widely used, since only for specialcases more accurate torques are known, e.g., nearly sphericalparticles from perturbation methods Cox (1965) or very longfibers from the slender body theory (Schiby and Gallily, 1980;Shin et al., 2009). Hence, the most general approximation ofJeffery (1922) is used here:

TbxTbyTbz

0B@1CA ¼ ρ fν

163

πabc

b2 þ c2

b2β0 þ c2γ0

c2 þ a2

c2γ0 þ a2α0

a2 þ b2

a2α0 þ b2β0

0BBBBBBBBBB@

1CCCCCCCCCCA

b2−c2

b2 þ c2τbz by þ ωbz by−ωbx� �

c2−a2

c2 þ a2τbx bz þ ωbx bz−ωby� �

a2−b2

a2 þ b2τby bx þ ωby bx−ωbz� �

0BBBBBBBBB@

1CCCCCCCCCA:

ð11Þ

The torque components depend on the fluid shear stressand the fluid vorticity in the particle fixed coordinate system

τbz by ¼ 12

∂ubz∂by þ

∂uby∂bz

!τbx bz ¼ 1

2

∂ubx∂bz þ

∂ubz∂bx

!τby bx ¼ 1

2

∂uby∂bx þ ∂ubx

∂by !

ωbz by ¼ 12

∂ubz∂by −

∂uby∂bz

!ωbx bz ¼ 1

2

∂ubx∂bz −

∂ubz∂bx

!ωby bx ¼ 1

2

∂uby∂bx −

∂ubx∂by

!:

ð12Þ

In brief, 13 coupled partial differential equations have tobe solved to describe the motion of an ellipsoid, three for theposition, four for the orientation, and three for the translationand the rotation velocities.

3. Setup

3.1. Flow field

The flow field used for this investigation is the same as inKunnen et al. (2013). The influence of the particles on the

flow is neglected because of the small volume loading. Hence,the flow field can be calculated independent of the type ofsuspended particles. The computational domain is an elon-gated box (Fig. 2).

The inlet is at the lower side, where turbulence isgenerated and advected by a constant mean flow velocityU = |U| = 1.5 m/s to the top, which is opposite to the gravity|g| = 9.81 ms−2. The Reynolds number Re = UL/ν = 80,000is based on the mean flow velocity U, the length of the domainL, and the viscosity ν = 1.5 × 10−5 m2/s. All the values arechosen to be representative for cloud turbulence (Devenishet al., 2012; Pruppacher and Klett, 1997; Siebert et al.,2006). Note, however, that the length scale and as such theReynolds number are restricted by the numerical capacity dueto the computational effort, i.e., an acceptable overall computingtime.

To simulate this flow the domain contains about 53 mil-lion Cartesian cells at a cell length of 0.00125 Δx/L. Theresolution is chosen such that all turbulent scales down to theKolmogorov length are resolved such that a direct numericalsimulation of the flow field is performed. The full compress-ible Navier–Stokes equations are solved using a finite volumesolver (Hartmann et al., 2008). Since the flow problem isessentially incompressible the Mach number is set to 0.1.At the outflow boundary, the pressure is kept constant andthe streamwise velocity and density gradients are zero. Onthe sidewalls, free-slip boundary conditions are used andthe wall-normal pressure and density gradients are zero. Asponge layer is imposed on all boundaries except the inflow.In these layers the flow is increasingly forced towards theprescribed mean velocity U = (U,0,0) (Hartmann et al.,2008). They act as a smooth transition region such that the

49C. Siewert et al. / Atmospheric Research 142 (2014) 45–56

domain can be bounded without major effects on the internalflow. Synthetic turbulence is generated at the inlet accordingto the formulation of Batten et al. (2004), which is anextension of the method of Kraichnan (1970). The turbulenceevolves according to the Navier–Stokes equations and reachesa physical state after an initial transition region (Siewert et al.,2013). Since neither mean shear nor perturbations arepresent except at the inflow, the turbulence decays due toviscous damping. Hence, the flow field consists of isotropicspatially decaying turbulence which mimics grid-generatedwind tunnel turbulence (e.g. Bateson and Aliseda, 2012; Bordáset al., 2013). Under these conditions the turbulent kineticenergy dissipation rate � satisfies

� tð Þ ¼ �0tt0

� �n−1ð13Þ

with n = −1.3 (Pope, 2000). The temporal evolution can berelated to a spatial dependence using the constant mean flowvelocity U. Fig. 3 shows � as a function of x/L in a log–log plot.Near the inlet, � is almost constant and then further down-stream the turbulent flow reaches a physical state where �

decays with the theoretical exponent of −2.3. The turbulentkinetic energy dissipation rate is important for the transportof small and heavy particles since it is proportional to thelocal shear rates. An advantage of the presented setup is thatthe dependence on � can be investigated within one singlecomputation since the particles are advected through thedecaying turbulence field. For this reason, the statistics aregathered in six different volumes distributed consecutively inthe x-direction, see Fig. 2. The streamwise locations are chosen

Fig. 3. The turbulent energy dissipation rate � as a function of the streamwisecoordinate x/L in a log–log plot. Additionally, the scaling for decayingisotropic turbulence is plotted with x−2.3 (Pope, 2000). The symbols showthe locations of the statistic volumes, see also Fig. 2. The symbols have thesame meaning throughout the paper.

to match dissipation rates relevant for in-cloud turbulence(Devenish et al., 2012). The streamwise extent of the volumesis a tradeoff between the streamwise changing turbulenceproperties within the volumes and the computing time neededto obtain accurate statistics, see Kunnen et al. (2013) for moredetails. It has to be kept in mind that streamwise decayingturbulence is considered and the given dissipation rates arestreamwise averages within the statistic volumes. As indicatedin Fig. 3 the averaged dissipation rates within the statisticvolumes are 250 to 30 cm2s−3. Note that the Reynolds numberdependence cannot be checked since it is restricted by thecomputational effort to values of approximately Reλ = 20based on the Taylor length scale. It is expected that higherReynolds numbers also affect the result, particularly due tointermittency. Nevertheless, Rosa et al. (2011) recently reportedsaturation of the collision probabilities of spherical particles atrelatively modest Reynolds numbers.

3.2. Particle phase

Only ellipsoids at the same length a and b of the semiaxesare investigated. These are called spheroids or ellipsoids ofrevolution. The third half-axis c points in the bz direction.Using the aspect ratio β = c ∕ a, prolates are ellipsoids atβ N 1, the special case β = 1 is a sphere, and defines β b 1 anoblate shape. The shapes of the different types of ellipsoidsare visualized in Fig. 4. The solutions of the shape factorintegrals (Eq. (8)) for the special case of an ellipsoid ofrevolution with c = aβ are listed in Table 1.

After the flow field has reached a statistical stationarystate, i.e., it is independent from the initial conditions, theellipsoids are constantly released at random positions atthe inflow with their velocity equal to the local instanta-neous fluid velocity. The orientation is chosen randomly:all four quaternions are chosen from a standard normaldistribution and normalized according to Eq. (4). Five typesof ellipsoids are considered whose properties are listed inTable 2.

They are distinguished by the aspect ratio β ranging from4.0 to 0.25. The lengths of the semi-axes are chosen suchthat the mass is equivalent to a sphere at radius 30 μm.The density ratio of the spheroid to the fluid is ρe/ρf = 992(Pruppacher and Klett, 1997). A constant number of 19 mil-lion ellipsoids, equally distributed among the five types ofspheroids, is advanced in time. A predictor–corrector methodis used for the time integration. The necessary fluid velocitiesaswell as the derivatives at the position of the spheroid centerare interpolated by a tri-cubic least square method. The timestep is the same as for the fluid solution, i.e., 5.5 × 10−5 s,which is a factor 200 smaller than the translation time scaleof the spherical particle. Thus, it is expected that also theellipsoidal motions are accurately resolved. After an initialtime period of 1 × L ∕ U the statistics are gathered over atime span of 5 × L ∕ U. To ensure converged statistics, thevalues are additionally ensemble averaged over all ellipsoidsof a specific type in the region of interest. As explained inSection 3.1 the six regions of interest are distinguished by thestreamwise averaged turbulence intensity or dissipation rate,respectively (see Fig. 3). Due to the inertia the particles mightmemorize the turbulent vortices they were in contact before.

a) Prolate ( = 4)β b) Sphere ( = 1)β c) Oblate ( = 0.25)β

Fig. 4. Different types of ellipsoids of revolution or spheroids. The third semiaxis c axis is always aligned with the bz of the particle fixed coordinate system.

50 C. Siewert et al. / Atmospheric Research 142 (2014) 45–56

Hence, the results might not be solely dependent on the localmean dissipation rate.

4. Results

4.1. Orientation distribution function (ODF)

To present the orientation of an ellipsoid in a turbulentflow it is convenient to introduce a spherical coordinatesystem (Fig. 5). The angle between the direction of the thirdsemiaxis c and the x axis in the inertial system is the polarangle θ, such that

cos θð Þ ¼ cxcj j : ð14Þ

The angle between the projection of c into the y–z planeand the y axis is the azimuth angle ϕ, such that

tan ϕð Þ ¼ czcy

: ð15Þ

The orientation of a type of ellipsoid can now be character-ized by a θ − ϕ joint probability density function Ψ(θ,ϕ), also

Table 1Solution of the shape factor integrals (Eq. (8)) for ellipsoids of revolutionwith a = b and c = aβ for the three cases of prolates (β N 1), spheres (β = 1)and oblates (β b 1). Similar expressions can be found in Happel and Brenner(1965, chaps. 5–11).

β N 1 β = 1 0 b β b 1

χ0 − a2βffiffiffiffiffiffiffiffiffiffiffiffiffiβ2−1

q κ 2a2к a2βffiffiffiffiffiffiffiffiffiffiffiffiffi1−β2

q π−κð Þ

α0 = β0β2

β2−1þ β

2ffiffiffiffiffiffiffiffiffiffiffiffiffiβ2−1

q 3 κ23κ − β

2ffiffiffiffiffiffiffiffiffiffiffiffiffi1−β2

q 3 κ−π þ 2βffiffiffiffiffiffiffiffiffiffiffiffiffi1−β2

q� �

γ0 − 2

β2−1− βffiffiffiffiffiffiffiffiffiffiffiffiffi

β2−1q 3 κ

23κ 1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1−β2q 3 βκ−πβ þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffi1−β2

q� �

к logβ−

ffiffiffiffiffiffiffiffiffiffiffiffiffiβ2−1

qβ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ2 þ 1

q !

12arctan βffiffiffiffiffiffiffiffiffiffiffiffiffi

1−β2q !

Table 2Definition of the five types of spheroids used in this study. The aspect ratio βranges from 4 to 0.25. The length of the semiaxis is chosen such that all typesof spheroids have the same mass.

β [−] a [μm] c [μm]

4.0 19.44 77.762.0 24.49 48.991.0 30.86 30.860.5 38.88 19.440.25 48.99 12.25

,

known as orientation distribution function. The ODF fulfills theidentityZ 2π

0

Z π

0Ψ θ;ϕð Þsin θð Þdθdϕ ¼ 1: ð16Þ

Fig. 6 is colorcoded byΨ(θ,ϕ) over θ andϕ for the spheroidtype with β = 0.25 at a dissipation rate of 50 cm2s−3. It canbe seen that the orientation is not uniformly distributedΨ≠ 1

4πð Þ but intermediate values of θ are more likely.Additionally, it can be seen that Ψ is independent of ϕ and

symmetric in θ. The independence of ϕ is expected becauseellipsoids of revolution are considered. The symmetry regardingθ is due to the fore-aft symmetry of the ellipsoids. Although notexplicitly shown, these two statements hold for all types ofellipsoids and all dissipation rates. Hence, the definition of theODF Ψ is simplified asZ π=2

0Ψ θð Þsin θð Þdθ ¼ 1: ð17Þ

In Fig. 7 the calculated ODFs for all five types of ellipsoidsare plotted. Additionally, the dependence on the dissipationrate is depicted. The θ bins are chosen such that theycorrespond to equal surface areas on the orientation spheresuch that the statistical accuracy is maximized. Hence, thedata point density increases at increasing θ. It can be seenin Fig. 7(a) and (b) that low values of θ are more likely forprolate ellipsoids. Thus, they are preferentially aligned withgravity or the mean flow direction, respectively. As statedin the Introduction there is no study concerning isotropic

Fig. 5. Spherical coordinate system to evaluate the orientation of an ellipsoidin the inertial system using the angles ϕ and θ.

51C. Siewert et al. / Atmospheric Research 142 (2014) 45–56

turbulence to directly compare to, however, the results seem tobe consistent with themeasurements of Bernstein and Shapiro(1994) and the numerical analyses of Zhang et al. (2001).They found that in a turbulent channel flow the fibers arepreferentially aligned with the mean flow direction. This willbe further discussed in Subsection 4.3. The spherical particlesdo not show a preferential orientation as expected (Fig. 7(c)).For oblate particles high values of θ are more likely (Fig. 7(d)and (e)). It can also be seen that the preferential orientation isstronger at more extreme values of β by comparing Fig. 7(a)with (b) and Fig. 7(d) with (e). Additionally, the influenceof the turbulence intensity, respectively or dissipation rate isshown. At higher dissipation rates the preferential alignment isweaker and the probability distribution tends to be distributeduniformly. This trend was also measured by Bernstein andShapiro (1994) and Newsom and Bruce (1998), but seemsto be in contrast to the predictions of Cho et al. (1981) andKlett (1995). In the latter investigations it is concluded thatturbulence intensities in the range of the current values cannotdestroy the preferential orientation. Klett (1995) showedthat the average tilt angle of a column with an aspect ratioof two changes by approximately 2°, comparing the non-turbulent case with the � = 100 cm2s−3 case. However, theresults of these studies do not necessarily contradict eachother. That is, since Bernstein and Shapiro (1994) presentedODFs and Klett (1995) only showed averaged orientation

Fig. 6. Orientation distribution function Ψ(ϕ,θ) as contour plot over ϕ and θfor the ellipsoids with β = 0.25 at a dissipation rate of 50 cm2s−3.

angles. While the ODFs are obviously dependent on thedissipation rate (see Fig. 7), the changes of the mean ofthe angle θ are relatively small (see Fig. 8). For a prolate withβ = 2, the difference between the mean of the angle θ ofthe � = 30 cm2s−3 and � = 150 cm2s−3 case is also onlyapproximately 2°.

4.2. Settling velocities

The settling velocity of a non-spherical particle depends onits orientation even in stagnant air because of the orientationdependent drag force. Only for the sphereβ = 1 the translationresistance tensor (Eq. (7)) is isotropic and can be invertedunambiguously. The terminal velocity vt is then often expressedvia a particle response time τp

vt;Sphere ¼ τpg ¼ 2ρea2

9ρ f vg: ð18Þ

To define an equivalent ellipsoidal response time differentformulations have been introduced in the literature. Shapiroand Goldenberg (1993) defined the equivalent responsetime under the assumption of isotropic particle orientationleading to

vt;Shapiro¼τeq;Shapiro g¼2πaβ1

kbx bx þ 1kbyby þ 1

kbz bz !

2ρea2

9ρ fνg: ð19Þ

Fan and Ahmadi (1995b) averaged the translation tensorresulting in

vt;Fan¼τeq;Fang¼18πaβ1

kbx bx þ kby by þ kbz bz !

2ρea2

9ρ fνg: ð20Þ

In general, the settling velocity of a spheroid in a fixedorientation in a stagnant fluid, i.e., the solution of Eq. (6) bybalancing gravity and drag force for a fixed orientation angleθ (see Fig. 5), can be written as

vt θð Þ ¼ 6πaβ1

kbz bz−kbx bx� �cos2 θð Þ þ kbx bx

0@ 1A2ρea2

9ρ fνg: ð21Þ

Thereby the rotation matrix A is composed of the intrinsicEuler angles as defined in Fig. 5 and the resulting equation issimplified by exploiting kbx bx ¼ kby by for ellipsoids of revolu-tion. Hence, Eq. (21) reflects the independence of the resultsfrom the angle ϕ and the symmetry regarding the angle θ asmentioned already in Section 4.1. Using the ODFΨ(θ) definedin Eq. (17) and depicted in Fig. 7 the theoretical meanturbulent settling velocity can be calculated as

vt;ODF �ð Þ ¼Z π=2

0Ψ θ; �ð Þvt θð Þsin θð Þdθ: ð22Þ

In Table 3 the values of the different velocities definedin Eqs. (18) to (22) are listed. In the first two rows, theminimum and maximum settling velocities are reportedcorresponding to the θ values of 0 and π/2. Then, thevelocities defined by Shapiro and Goldenberg (1993) and

a) b)

c)

d) e)

Fig. 7. Orientation distribution function for all five types of spheroids over θ. Additionally, the dependence of the ODF on the dissipation rate � is depicted.

52 C. Siewert et al. / Atmospheric Research 142 (2014) 45–56

Fan and Ahmadi (1995b) are given. Next, the theoreticalsettling velocities using Eq. (22) are calculated for the lowestand highest investigated turbulence dissipation rates. Finally,the x-velocities found in the simulation are time andensemble averaged and the mean flow velocity is subtracted.The resulting ensemble-averaged turbulent settling velocitiesvx;turb �ð Þ are also given for the lowest and highest dissipationrates. The comparison of all these values shows that vt,Fanis always smaller than vt,Shapiro and they are both always

smaller than vt;ODF. Hence, considering the parameter rangeof this study, the equivalent response time defined byShapiro and Goldenberg (1993) fits better. This is especiallytrue for higher turbulence intensities since the spheroids aremore randomly orientated. Finally, it should be pointedout that the settling velocities of all ellipsoids are smallerthan those of the sphere with the equivalent mass. This isimportant since several cloud microphysics schemes treat iceparticles as spheres (Gavze et al., 2012).

Fig. 8. The dependence of the mean of the angle θ on the dissipation rate � forall investigated types of spheroids.

Fig. 9. Ratio of the ensemble-averaged x-direction velocity vx;turb and thetheoretical mean settling velocity vt;ODF for all five β values and the sixdissipation rates �.

53C. Siewert et al. / Atmospheric Research 142 (2014) 45–56

In Fig. 9 the ensemble-averaged velocity vx;turb �ð Þ iscompared with the corresponding vt;ODF �ð Þ for all considereddissipation rates.

It can be seen that vx;turb � ¼ 250cm2s−3� �

is significantlylarger than the corresponding vt;ODF � ¼ 250cm2s−3

� �calcu-

lated from the orientation based drag force. The increase iscomparable with the one found by Ayala et al. (2008) forspherical particles. Thus, in addition to the effect that thedrag and as such the velocity of an ellipsoid dependent onits orientation depends on the orientation, there is alsopreferential sweeping, making the velocity in the directionof gravity dependent on the turbulence intensity. Thepreferential orientation effect slows down the spheroids ina turbulent flow. In contrast, the preferential sweepingeffect accelerates the spheroids and turns out to bedominant. These combined effects change the relativevelocities of the particles in turbulent flows. As such theaggregation probability of regular ice crystals as well as theriming probability of regular ice crystals and supercooleddroplets is expected to be changed in the presence of cloudturbulence.

Table 3Settling velocity for all investigated types of spheroids.

[cm/s] β = 4 β = 2 β = 1 β = 0.5 β = 0.25

vt(θ = 0) 13.64 14.37 13.73 12.04 9.97vt(θ = π/2) 10.60 12.55 13.73 13.75 12.68vt,Shapiro 11.61 13.15 13.73 13.18 11.78vt,Fan 11.45 13.10 13.73 13.13 11.63vt;ODF � ¼ 30cm2s−3

� �11.78 13.27 13.73 13.35 12.13

vt;ODF � ¼ 250cm2s−3� �

11.62 13.19 13.73 13.26 11.92vx;turb � ¼ 30cm2s−3

� �11.92 13.30 13.72 13.37 12.22

vx;turb � ¼ 250cm2s−3� �

12.09 13.45 13.91 13.50 12.34

4.3. Physical interpretation

After the presentation of the ellipsoidal motion under theinfluence of gravity and isotropic turbulence some physicalinterpretation is given on the coherence of the preferentialorientation and the preferential sweeping. For this purposethe orientation of the ellipsoids is investigated for threedifferent cases: without turbulence but with gravity, withturbulence but without gravity, and the already studied casewith turbulence and gravity.

In quiescent laminar air, or in laminar air moving with aconstant uniform velocity U the ellipsoids would not rotatewhile falling under the influence of gravity since the torquesbased on creeping flow conditions depend only on the fluidvelocity gradient tensor.

The case without gravity but with a turbulent backgroundflow is more complex. At first glance, a uniformly distributed,random orientation of the ellipsoids would be expected sincefromanEulerian point of view the fluid velocity gradient tensorin isotropic turbulence provides a uniform distribution whoseaverage is zero. This also holds for the Lagrangian statistics offluid tracers. The recent numerical investigations of turbulentchannel flow suggest that random orientation can also beexpected for inertial particles. For instance, Mortensen et al.(2008b) found that the orientation becomes more and moreisotropic going from the wall region to the core region of aturbulent channel where the turbulence gets more and moreisotropic. However, for spherical particles with inertia itwas shown that these particles show effects like clustering inan isotropic turbulent flow (Ayala et al., 2008). The particlesare expected to have a higher probability of presence inregions of low vorticity and high strain. Hence, there couldexist a preferential orientation of ellipsoids in a turbulentflow without the influence of gravity due to the inertial bias.However, following the theoretical arguments of Dávila andHunt (2001) preferential sweeping and preferential clustering

54 C. Siewert et al. / Atmospheric Research 142 (2014) 45–56

are only weakly correlated for a particle Froude number Fp N 1.In this study, the particle Froude number of the sphericalparticle is Fp ∼ 9. On the other hand, also non-sphericalparticles are expected to cluster, although the effect should beweaker than at spherical particles due to the variable drag forcedepending on the orientation. Thus, an effect of the preferentialclustering on the preferential orientation cannot be completely

a)

c)

b)

Fig. 10. Orientation distribution function of spheroids with β = 4, 2, and 1over θ in the turbulent flow without gravity. Additionally, the dependence othe ODF on the dissipation rate � is depicted.

f

ruled out. Therefore, an additional simulation without gravitybut with isotropic turbulence is conducted. The results areshown in Fig. 10. Comparing Fig. 10 with Fig. 7 the ellipsoidorientation is found to be random for all investigated ellipsoidsand dissipation rates without gravity. Hence, it is likely thatthe alignment of the ellipsoids with the mean flow directionfound in studies of turbulent channel flows mentioned inSubsection 4.1 is not due to the mean flow itself but the higherstreamwise velocity fluctuations, i.e., the anisotropy of theturbulent flow, which was also suggested by Mortensen et al.(2008a) and Marchioli et al. (2010). It can be concluded thatclustering has no effect on the orientation of the ellipsoids, atleast for the particle mass examined in this study.

However, in the investigated case the ellipsoids are subjectto both gravity and turbulence. From the orientation statisticsit is now clear that gravity has a pronounced impact. Sincethe underlying mechanisms are very complex and could beinfluenced by many different parameters like the particleinertia, the translation and rotation resistance, the vortextime and length scales etc., we cannot conclusively explainthe underlying mechanisms of preferential orientation at themoment. However, we want to give a plausible explanation,on why the randomizing effect of the turbulent fluctuations isbroken and a preferential orientation effect exists. Giventhat we have proven the preferential settling of ellipsoids inthe last subsection, it is reasonable to assume that ellipsoidsalso preferentially sample the downward-moving sides ofthe turbulent eddies (Dávila and Hunt, 2001). Thereby theellipsoids predominantly encounter a certain subset of velocitygradients (the shear induced by a turbulent eddy in its directvicinity) that tends to orient their longest axis parallel togravity. When the turbulence has a higher dissipation rate �

this effect is reduced given that more and smaller eddiescontribute to the local velocity gradient, partially destroyingthe preferential orientation but enhancing preferential settling.In conclusion, preferential orientation and preferential settlingare both thought to be driven by the specific interactions ofgravity and turbulence with the ellipsoids.

5. Conclusion

The motion of ellipsoids which in this study are interpretedas archetypes of regular ice crystals under the influence ofgravity and isotropic turbulence has been studied by a DNSof spatially decaying isotropic turbulence. Within this flowthe motions of 19 million ellipsoids of revolution are trackedby a Lagrangian point model. Five types of ellipsoids atconstant mass are considered, among them prolates, spheres,and oblates. Using a one-way coupling the forces andmomentson the ellipsoids are evaluated assuming creeping flowconditions. Statistics have been determined for six turbulenceintensities characterized by turbulent kinetic energy dissipa-tion rates ranging from 30 to 250 cm2s−3. First, the orientationprobabilities have been investigated. A preferential orientationregarding to the direction of gravity is found. It is increasinglypronounced for both larger and smaller aspect ratios. Thehigher the dissipation rate is the closer the orientationprobability is to a uniform distribution. Thus, the turbulenceinfluences the settling velocity of ellipsoids, since their drag isdependent on their orientation. Additionally, the settlingvelocities are found to be higher than those calculated from

55C. Siewert et al. / Atmospheric Research 142 (2014) 45–56

the orientation probabilities. Hence, the existence of thepreferential sweeping effect was shown for non-sphericalparticles. Just like spheres, non-spherical particles do fall faster,the higher the turbulence intensities are. They might preferen-tially choose the downward-moving sides of the turbulenteddies. These two turbulence effects change the settling velocityof ellipsoids. It is likely that this will affect the mean radialrelative velocity, which is generally determined in large part bythe differential settling velocities of two particles, although theconnection might not be linear (Grabowski and Wang, 2013).Thereby, also the collision probability of non-spherical particlessuch as ice crystals is expected to change due to the turbulencein clouds.

Acknowledgments

The funding of this project under grant number SCHR 309/39 in the framework of the priority program METSTROEM bythe Deutsche Forschungsgemeinschaft is gratefully acknowl-edged. The authors are grateful for the computing resourcesprovided by the High Performance Computing Center Stuttgart(HLRS). We would like to thank Klaus Beheng for his helpfulcomments. We also thank the reviewers for their commentswhich improved the manuscript considerably.

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