Theory for the effect of fluid inertia on the orientation of a small particle settling in turbulence
K. Gustavsson, M. Z. Sheikh, D. Lopez, A. Naso, A. Pumir, B. Mehlig
TTheory for the effect of fluid inertia on theorientation of a small spheroid settling in turbulence
K. Gustavsson , M. Z. Sheikh , D. Lopez , A. Naso ,A. Pumir , and B. Mehlig Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden Univ. Lyon, ENS de Lyon, Univ. Claude Bernard, CNRS, Laboratoire de Physique,F-69342, Lyon, France Univ. Lyon, Ecole Centrale de Lyon, Univ. Claude Bernard, CNRS, INSA de Lyon,Laboratoire de M´ecanique des Fluides et d’Acoustique, F-69134, Ecully, France
Abstract.
Ice crystals settling through a turbulent cloud are rotated by turbulentvelocity gradients. In the same way, turbulence affects the orientation of aggregatesof organic matter settling in the ocean. In fact most solid particles encountered inNature are not spherical, and their orientation affects their settling speed, as wellas collision rates between particles. Therefore it is important to understand thedistribution of orientations of non-spherical particles settling in turbulence. Here westudy the angular dynamics of small prolate spheroids settling in homogeneous isotropicturbulence. We consider a limit of the problem where the fluid torque due to convectiveinertia dominates, so that rods settle essentially horizontally. Turbulence causes theorientation of the settling particles to fluctuate, and we calculate their orientationdistribution for prolate spheroids with arbitrary aspect ratios for large settling numberSv (a dimensionless measure of the settling speed), assuming small Stokes number St(a dimensionless measure of particle inertia). This overdamped theory predicts thatthe orientation distribution is very narrow at large Sv, with a variance proportionalto Sv − . By considering the role of particle inertia, we analyse the limitations of theoverdamped theory, and determine its range of applicability. Our predictions are inexcellent agreement with numerical simulations of simplified models of turbulent flows.Finally we contrast our results with those of an alternative theory predicting that theorientation variance scales as Sv − at large Sv. a r X i v : . [ phy s i c s . f l u - dyn ] M a r rientation of a small spheroid settling in turbulence
1. Introduction
The settling of particles in turbulence is important in a wide range of scientific problems.An example is the settling of small ice crystals in turbulence, a process that is consideredin the context of rain formation from cold cumulus clouds [1, 2, 3]. Further examples arethe settling of small aggregates of organic matter (‘marine snow’) [4], and the dynamicsof swimming microorganisms [5, 6, 7] in the turbulent ocean.The settling of spherical particles in turbulence has been intensively studied. Maxeyand collaborators [8, 9, 10] found that turbulence increases the settling speed of smallspherical particles. This pioneering work has led to many experimental and numericalstudies, using direct numerical simulation (DNS) of turbulence, and it is a questionof substantial current interest [11, 12]. An important question is how frequentlyparticles collide as they settle in turbulence [13, 14]. The collision rate is influencedby spatial inhomogeneities in the particle-number density due to the effect of particleinertia. There is substantial recent progress in understanding this two-particle problem[15, 16, 17, 18, 19]. The conclusion is that settling may increase or decrease spatialclustering of spherical particles, and that it tends to decrease the relative velocities ofnearby particles because settling reduces the frequencies of ‘caustics’, singularities inthe inertial-particle dynamics [15].Most solid particles encountered in Nature and in Engineering are not spherical,yet less is known about the settling of non-spherical particles in turbulence, and theirsettling depends in an essential way on their orientation. In a fluid at rest the orientationof a slowly settling non-spherical particle is determined by weak torques induced by theconvective inertia of the fluid - set in motion by the moving particle. For a single,isolated particle in a quiescent fluid this effect is well understood [20, 21, 22, 23]:convective fluid inertia due to slip between the particle and the fluid velocity causesnon-spherical particles to settle with their broad side first. For axisymmetric rods, forexample, symmetry dictates that the angular dynamics has two fixed orientations: eitherthe rod is aligned with gravity (tip first) or perpendicular to gravity. At weak inertia,only the latter orientation is stable, so that the rod settles with its long edge first. Butwhen there is turbulence, then turbulent vorticity and strain exert additional torquesthat cause fluctuations in the orientations of the settling crystals [1, 24].To understand the angular motion of a non-spherical particle settling in turbulenceis in general a very complex problem, because there are many dimensionless parametersto consider. There is particle shape (shape parameter Λ), and the effect of particleinertia is measured by the Stokes number St. The importance of settling is determinedby Sv, a dimensionless measure of the settling speed. The significance of fluid inertiais quantified by two Reynolds numbers, the particle Reynolds number Re p (convectiveinertia due to slip between particle and fluid velocity), and the shear Reynolds numberRe s (convective inertia due to fluid-velocity gradients). The nature of the turbulentvelocity fluctuations is determined by the Taylor-scale Reynolds number Re λ .If the particles are so small that they just follow the flow and that any inertial rientation of a small spheroid settling in turbulence p = Re s = 0), then the angular dynamicsof small crystals in turbulence is well understood [25, 26, 27, 28, 29, 30, 31, 32, 33, 7, 34].The particle orientation responds to local vorticity and strain through Jeffery’s equation[25]. The effect of particle inertia is straightforward to take into account [35], but therole of fluid inertia is more difficult to describe, even in the absence of settling. Incertain limiting cases fluid-inertial effects are well understood. The most importantexample is that of a small neutrally buoyant (Re s = St) spheroid moving in a time-independent linear shear flow, so that the centre-of-mass of the particle follows the flow(Re p = 0). Neglecting inertial effects (Re s = 0) and angular diffusion, the angulardynamics degenerates into a one-parameter family of marginally stable orbits, the so-called Jeffery orbits [25]. Fluid inertia breaks this degeneracy and gives rise to certainstable orbits [36, 37, 38, 39]. Much less is known when Re p is not zero. Candelier, Mehlig& Magnaudet [40] recently showed how to compute the effect of a small slip upon theforce and torque on a non-spherical particle in a general linear time-independent flow,by generalising Saffman’s result [41, 42] on the lift upon a small sphere in a shear flow,valid in the limit where Re p (cid:28) √ Re s (cid:28) s [48]. Butexperiments and numerical simulations of slender particles settling in a vortex flow [49]and in turbulence [50] show that convective inertial torques due to settling can make aqualitative difference to the orientation distribution.In this paper we therefore consider the effect of the convective inertial torques onthe orientation of small spheroids settling in turbulence. Following Ref. [49], our modelassumes that the hydrodynamic torque is approximately given by the sum of Jeffery’storque and the convective inertial torque in a homogeneous, time-independent flow. Fornearly spherical particles this convective torque was calculated by Cox [20], and forslender bodies by Khayat & Cox [21]. Their results were generalised to spheroids witharbitrary aspect ratios in Ref. [22].Our goal is to analyse how the turbulent-velocity fluctuations affect the orientationdistribution of a prolate spheroid settling through turbulence. We assume that theparticles are small enough so that convective-inertia effects due to the fluid-velocitygradients are negligible, that inertial effects on the centre-of-mass motion are small(small St and Re p ), but that the settling number Sv is large enough so that the fluid-inertia torque dominates the angular dynamics.We find an approximate theory for the angular distribution of settling spheroidsusing a statistical model [51, 45] for the turbulent fluctuations. The theory is valid forlarge Sv and small St, in the overdamped limit, and its predictions are in excellent rientation of a small spheroid settling in turbulence − in the limit of large settlingnumber Sv, for small enough Stokes number St, and the theory determines how the pre-factor depends on the shape of the spheroid. In the slender-body limit, the Sv − -scalingof the variance was also found in Ref. [54] using an approach equivalent to ours.We contrast our results with a theory for the orientation variance derived by Klett[24] for nearly spherical particles. This theory predicts that the variance is proportionalto Sv − . At first sight this may appear to be at variance with the overdamped theory,but we show that the overdamped approximation breaks down into several differentregimes when particle inertia begins to matter. At very large values of Sv, when thetime scale at which the fluid-velocity gradients decorrelate is the smallest time scaleof the inertial dynamics, our numerical simulations show a Sv − -scaling, as suggestedby Klett’s theory. But the theory is difficult to justify because it neglects particleinertia in the centre-of-mass dynamics. We show that translational particle inertia hasa significant effect upon the angular dynamics, so that it must be taken into account assoon as the overdamped approximation breaks down.The remainder of this paper is organised as follows. In Section 2 we describe ourmodel: the approximate equations of motion and the statistical model for the turbulent-velocity fluctuations. In Section 3 we show results of numerical simulations of our model.We describe how and why the results differ from those in Refs. [43, 44, 45, 46, 47], andexplain the intuition behind our theory for small St and large Sv. The overdampedtheory is described in Section 4. Section 5 discusses the effect of particle inertia, andSection 6 contains our conclusions as well as an outlook.
2. Model
Newton’s equations of motion for a single non-spherical particle read: m p ˙ v p = f + m p g , ˙ x p = v p , (1a) m p dd t (cid:2) I p ( n ) ω p (cid:3) = τ , ˙ n = ω p ∧ n . (1b)Here g is the gravitational acceleration with direction ˆ g = g / | g | , x p is the position ofthe particle, v p its centre-of-mass velocity, m p the particle mass, and the dots denotetime derivatives. We assume that the particle is axisymmetric, so that its orientation ischaracterised by the unit vector n along the symmetry axis of the particle. The angularvelocity of the particle is denoted by ω p , and I p ( n ) is its rotational inertia tensor perunit-mass in the lab frame. For a spheroid, the elements of I p ( n ) are given by [55]( I p ) ij ( n ) = I ⊥ ( δ ij − n i n j ) + I (cid:107) n i n j , I ⊥ = 1 + λ a ⊥ , I (cid:107) = 25 a ⊥ , (2)where λ ≡ a (cid:107) /a ⊥ is the aspect ratio of the spheroid, 2 a (cid:107) is the length of the symmetryaxis, and 2 a ⊥ is the diameter of the spheroid. Prolate spheroids correspond to λ > rientation of a small spheroid settling in turbulence λ < f and torque τ on theparticle. In the Stokes approximation, unsteady and convective inertial effects areneglected. In this creeping-flow limit [55], the force and torque upon the spheroidare linearly related to the slip velocity W ≡ v p − u , to the angular slip velocity ω p − Ω ,and to the fluid strain S : (cid:34) f (0) τ (0) (cid:35) = 6 πa ⊥ µ (cid:34) A C H (cid:35) u − v p Ω − ω p S . (3)Here µ is the dynamic viscosity of the fluid, u ≡ u ( x p , t ) is the undisturbed fluid velocityat the particle position x p , Ω ≡ ∇ ∧ u is half the vorticity of the undisturbed fluid-velocity field at the particle position, and S is the strain-rate matrix, the symmetricpart of the matrix of the undisturbed fluid-velocity gradients (its antisymmetric partis denoted by O ). The tensors A , C , and H are translational and rotational resistancetensors. Their forms are determined by the shape of the particle. Eq. (3) shows that thetensor A relates the hydrodynamic force f (0) to the slip velocity W . For an axisymmetricparticle with fore-aft symmetry the tensor takes the form A ij ≡ A ⊥ ( δ ij − n i n j ) + A (cid:107) n i n j . (4)The resistance coefficients A ⊥ and A (cid:107) depend on the shape of the particle. For aspheroid, they are given by [55]: A ⊥ = 8( λ − λ [(2 λ − β + 1] , A (cid:107) = 4( λ − λ [(2 λ − β − , (5)with β = ln[ λ + √ λ − λ √ λ − . For a sphere one has A ⊥ = A (cid:107) = 1, so that f (0) simplifies to the usual expression forStokes force on a sphere moving with velocity v p through a fluid with velocity u .In the creeping-flow limit, the steady slip velocity W of a spheroid subject to agravitational force m p g is obtained by setting the acceleration ˙ v p to zero in Eq. (1a): W (0) = τ p (cid:104) A − ⊥ ( − nn T ) + A − (cid:107) nn T (cid:105) g . (6)Here is the unit matrix, and τ p ≡ (2 a (cid:107) a ⊥ ρ p ) / (9 νρ f ) is the particle response time inStokes’ approximation with kinematic viscosity ν = µ/ρ f , fluid-mass density ρ f , andparticle-mass density ρ p . The slip velocity depends on the orientation n of the particle.For an axisymmetric particle with fore-aft symmetry, the rotational resistancetensors take the form: C ij ≡ C ⊥ ( δ ij − n i n j ) + C (cid:107) n i n j and H ijk = H (cid:15) ijl n k n l . (7)Here (cid:15) ijl is the anti-symmetric Levi-Civita tensor, and we use the Einstein summationconvention: repeated indices are summed from 1 to 3. For a spheroid, the rotational rientation of a small spheroid settling in turbulence C ⊥ = 8 a (cid:107) a ⊥ ( λ − λ [(2 λ − β − , C (cid:107) = − a (cid:107) a ⊥ ( λ − β − λ , (8) H = − C ⊥ λ − λ + 1 . Expressions (3) to (8) determine the hydrodynamic force and torque in the creeping-flowlimit. Fluid-inertia effects are neglected in f (0) and τ (0) .There are two distinct corrections when fluid-inertia effects are weak but notnegligible, due to the undisturbed fluid-velocity gradients, S and O , and due to theslip velocity W . The former are parameterised by the shear Reynolds number Re s , thelatter by the particle Reynolds number Re p :Re s = sa ν and Re p = W (0) ⊥ aν . (9)Here a = max { a ⊥ , a (cid:107) } is the largest dimension of the particle, and W (0) ⊥ is an estimateof the slip velocity: the magnitude of the velocity of a small slender spheroidal particlesettling under gravity in a quiescent fluid with its symmetry axis perpendicular togravity. From Eq. (6) we see that W (0) ⊥ = τ p g/A ⊥ . In the definition of Re s , the parameter s is a characteristic shear rate. In turbulence it is on average of the order s ∼ τ − where τ K is the Kolmogorov time τ K = (cid:0) (cid:104) Tr SS T (cid:105) (cid:1) − / ∼ ( ν/ E ) / . (10)Here the average (cid:104)· · · (cid:105) is over Lagrangian fluid trajectories, and E is the turbulentdissipation rate per unit mass. This yields the estimate [48] Re s ∼ ( a/η K ) , where η K = √ ντ K ∼ ( ν / E ) / (11)is the Kolmogorov length [56]. Thus the shear Reynolds number is small for smallparticles.Now consider the effect of convective inertia. Following Ref. [49] we assume thatthe torque on the particle is given by the sum of Jeffery’s torque and the instantaneousconvective-inertia torque in a homogeneous flow. This approximation can be strictlyjustified for a steady linear flow in the limit √ Re s (cid:28) Re p (cid:28)
1. In this limit thesingular perturbation problem that determines the fluid-inertia torque simplifies: theSaffman length ( ∝ Re − / s ) is much larger than the Oseen length ( ∝ Re p − ). This impliesthat the leading convective-inertial corrections to the torque are those correspondingto a quiescent fluid, and a similar argument can be made for the convective-inertiacontribution to the force. While there is no general theory explaining how the convective-inertia contributions to the force and the torque are affected by spatial inhomogeneitiesin time-dependent flows, the results of Ref. [49] show that the simple model used here cansuccessfully explain important features of the orientation distribution of rods settling ina vortex flow. rientation of a small spheroid settling in turbulence Figure 1.
Geometrical shape factors. (a) Shape factor F ( λ ) in Eq. (13). The datashown are obtained by evaluating Eqs. (4.1) and (4.2) in Ref. [22]. (b) Shape factor A ( λ ) defined in Eq. (25), as a function of the particle aspect ratio λ . The leading-order inertial force correction reads for a spheroid moving in a quiescentfluid [57, 21]: f (1) = − (6 πa ⊥ µ ) Re p WW (0) ⊥ (cid:2) A − I ( ˆ W · A ˆ W ) (cid:3) A W , (12)with W = | W | and ˆ W = W /W . For a spheroid, the leading-order inertial contributionto the torque was calculated in Ref. [22]: τ (1) = F ( λ ) µa Re p W W (0) ⊥ ( n · ˆ W )( n ∧ ˆ W ) . (13)The shape factor F ( λ ) is given in Ref. [22]. It is also shown in Fig. 1(a).Combining Eqs. (1), (2), (3) with Eqs. (12,13) yields the equations of motion forour model. We use the Kolmogorov time τ K and the Kolmogorov length η K to de-dimensionalise the equations of motion, x (cid:48) = x /η K , t (cid:48) = t/τ K , v (cid:48) = v τ K /η K , ω (cid:48) = ω τ K .This gives (after dropping the primes):˙ x p = v p , (14a)˙ v p = 1St (cid:104) − (cid:16) + aη K W (cid:2) A − I ( ˆ W · A ˆ W ) (cid:3)(cid:17) A W + Svˆ g (cid:105) , (14b)˙ n = ω p ∧ n , (14c)˙ ω p = 1St (cid:2) I − C ( Ω − ω p ) + I − H .. S + A (cid:48) ( n · W )( n ∧ W ) (cid:3) (14d)+ Λ( n · ω p )( ω p ∧ n ) . Eqs. (14) have four independent dimensionless parameters:Λ = λ − λ + 1 , aη K , St = τ p τ K , Sv = gτ p τ K η K . (15)Here Λ is the shape parameter that appears in Jeffery’s equation, and Sv is the settlingnumber [58], a dimensionless measure of the settling speed. It is proportional to theparticle size squared, a , just as the Stokes number. rientation of a small spheroid settling in turbulence I − C ] ij = C ⊥ I ⊥ ( δ ij − n i n j ) + C (cid:107) I (cid:107) n i n j , [ I − H ] ijk = − C ⊥ Λ I ⊥ (cid:15) ijl n k n l , (16)as well as A (cid:48) = 56 π F ( λ ) max( λ, λ + 1 . (17)The Reynolds number Re p does not appear explicitly in Eqs. (14) because we de-dimensionalised the equations of motion with the Kolmogorov scales τ K and η K . Ifwe use an estimate of the slip velocity instead (such as W (0) ⊥ ), then Re p features inthe dimensionless equations of motion. The latter convention is used in Refs. [21, 22],and more generally in perturbative calculations of weak inertial effects on the motionof particles in simple flows [59, 41, 42, 40]. These two different choices must lead toequivalent equations of motion, but our scheme has the advantage that it emphasises thedifferent roles played by f (1) and τ (1) for small particles in turbulence. Eq. (14b) showsthat the fluid-inertia contribution to the force, f (1) , is multiplied by the dimensionlessprefactor a/η K . This means that f (1) makes only a small contribution for small enoughparticles, which we do not expect to qualitatively change the results derived below. Inthe following we therefore neglect this contribution (although it could be taken intoaccount in simulations and theory).More importantly, the fluid-inertia contribution to the torque in Eq. (14d) hasno such factor. The fluid-inertia torque is of the same order as the Jeffery torque.This implies that the fluid-inertia contribution to the torque is potentially much moresignificant than the fluid-inertia contribution to the force. At large Sv in particularthe particle settles rapidly, so that W is large. In this limit one therefore expects thefluid-inertia torque τ (1) to dominate over Jeffery’s torque τ (0) , so that the inertial torquecannot be neglected (as was done in Refs. [43, 44, 45, 46, 47]). It is argued in Ref. [60]that the orientation bias predicted in Refs. [43, 44, 45] can possibly be observed insmall-Re λ flow, but not at high Re λ .In the following we neglect the contribution from f (1) . At the same time we assumethat the settling speed is so large that the fluid-inertia torque τ (1) dominates the angulardynamics. If there was no flow, the particles would settle with their broad side first inthis limit. The question is how turbulent fluctuations modify the orientation distributionof the settling particles. In our theory we use a statistical model [51] to represent the turbulent fluctuations.We model the incompressible homogeneous and isotropic turbulent fluid-velocity field u ( x , t ) as a Gaussian random function with correlation length (cid:96) , correlation time τ ,and rms magnitude u (here and in Section 2.3 we write the equations in dimensionalform because we want to make explicit how these scales are related to the Kolmogorov rientation of a small spheroid settling in turbulence u ( x , t ) in three spatialdimensions (3D) as u = N ∇ ∧ A . (18)The components A j of the vector field A are Gaussian random functions with meanzero, (cid:104) A j ( x , t ) (cid:105) = 0, and with correlation functions (cid:104) A i ( x , t ) A j ( x (cid:48) , t (cid:48) ) (cid:105) = δ ij (cid:96) u exp (cid:16) − | x − x (cid:48) | (cid:96) − | t − t (cid:48) | τ (cid:17) . (19)We choose the normalisation N = 1 / √ u = (cid:112) (cid:104)| u | (cid:105) . Below we also quoteresults for a two-dimensional (2D) version of this model. In this case we take u = N (cid:34) ∂ A − ∂ A (cid:35) (20)with N = 1 / √
2, and where ∂ j represents the derivative with respect to the spatialcoordinate x j . As the equation of motion for the 2D model we use Eq. (14) with n andthe translational dynamics constrained to the flow plane.The statistical model has an additional dimensionless parameter, the Kubo number[51] Ku = u τ /(cid:96) . In the limit of large Ku the Gaussian random function u ( x , t ) modelssmall-scale fluid-velocity fluctuations in the dissipative range of homogeneous isotropicturbulence. Evaluating Eq. (10) in the statistical model gives (Section 5.1 in Ref. [51]): ττ K = √ d + 2 Ku , (21)where d is the spatial dimension. The spatial correlation length (cid:96) satisfies (cid:96) = (cid:104) u (cid:105) / (cid:104) ( ∂ u ) (cid:105) , which defines the Taylor length scale [56] in turbulence. The lengthscale (cid:96) is related to the Kolmogorov length by [56, 61] (cid:96)η K = C (cid:112) Re λ , (22)where C is a constant of order unity. The ratio (cid:96)/η K (or alternatively the Taylor-scale Reynolds number Re λ ) constitutes a sixth dimensionless parameter of the model,in addition to the Kubo number and the four parameters listed in Eq. (15). In allstatistical-model simulations described in this paper we set Ku = 10 and (cid:96)/η K = 10,and we determine the parameters τ and (cid:96) of the statistical model from Eqs. (21,22).The statistical model is constructed to approximate the dissipative-rangefluctuations of 3D turbulence [51]. We note that the predictions of the 2D and3D statistical models are essentially similar, but the two-dimensional model iseasier to analyse, and it can be simulated more accurately. Two-dimensional andthree-dimensional turbulence, by contrast, exhibit significantly different fluid-velocityfluctuations. rientation of a small spheroid settling in turbulence Figure 2.
Distribution of n g = n · ˆ g obtained by numerical simulations of Eqs. (14)for the three-dimensional statistical model. (a) Disk-like particles with aspect ratio λ = 0 .
1, Sv = 4 .
5, St = 0 .
022 (red, ◦ ), St = 0 .
22 (green, (cid:3) ), St = 2 . (cid:5) ). (a)Rod-like particle with λ = 5, Sv = 45, St = 0 .
22 (magenta, (cid:77) ) and St = 2 . (cid:79) ). To demonstrate the robustness of our theory we also compare its predictions to resultsof numerical simulations using a different model for the turbulent flow, namely theKinematic-Simulation (KS) model [52]. The KS model has been shown to reproducequalitatively many features of turbulent transport, and it provides a convenient wayto represent a flow with a wide range of spatial scales, such as turbulence, albeit in asimplified manner. In short, we discretise Fourier space in geometrically spaced shells,up to a largest wavenumber. The largest and smallest length scales of the flow are L and η , respectively. The total number of shells is denoted by N k . We choose thecharacteristic wave vector in shell n as: k n = k ( L/η ) ( n − / ( N k − . In each cell, we pickone wave vector, k n . The flow is then simply constructed as a sum of Fourier modes: u ( x , t ) = N k (cid:88) n =1 a n cos( k n · x + ω n t ) + b n sin( k n · x + ω n t ) . (23)The Fourier coefficients are chosen so that k n · a n = k n · b n = 0 (incompressibility),and with magnitude a n = b n = E ( k n )∆ k n , where E ( k n ) = E k − / n represents theKolmogorov spectrum [56]. The frequency ω n in Eq. (23) is taken to be ω n = (cid:112) k n E ( k n ). Further details about the implementation of this model for u ( x , t ) canbe found in Ref. [53].
3. Orientation distributions
Figure 2 shows orientation distributions obtained by numerical simulations of Eqs. (14)for the three-dimensional statistical model described in Section 2.2. Shown aredistributions of n g = n · ˆ g for a range of different Stokes numbers. We see that theparticles settle with their broadside approximately aligned with gravity. For rods this rientation of a small spheroid settling in turbulence n ⊥ ˆ g , so that n g = 0, and for disks n (cid:107) ˆ g , so that n g = 1. These are thestable orientations for prolate and oblate particles settling in a quiescent fluid [21, 22].Compare the distributions in Fig.2 to those shown in Fig. 1 of Ref. [45]. There, bycontrast, the rods tend to settle tip first, and disks tend to settle edge first. The reasonfor the difference is that the effect of the fluid-inertia torque was neglected in Ref. [45],whereas in the present work we choose parameters where this torque dominates theangular dynamics.When the Stokes number is small we expect that the vector n spends most of itstime close to a stable fixed point of the angular dynamics. But we expect that theturbulent velocity gradients modify the fixed point, so that it is no longer simply n g = 0(rods) or n g = 1 (disks). Since the turbulent velocity gradients change as functionsof time, the fixed-point orientation becomes time dependent too. In the overdampedlimit (small Stokes numbers) we expect that the particle orientation follows the fixed-point orientation quite closely. This allows us to derive a theory for the orientationdistribution in this limit, described in the following Section.
4. Overdamped limit
The model (14) is very difficult to analyse in general. Therefore, to simplify the analysis,we consider a limit of the problem where the relaxation time of n is much faster than thetime scale on which the gradients change as the particle moves through the flow. Thiscorresponds to the overdamped limit of the problem, St → W (0) ( n ), Eq. (6). In this limit we find: W = W (0) ( n ) , (24a) ω p = Ω + Λ( n ∧ S n ) + A Sv n g ( n ∧ ˆ g ) , (24b)with n g = n · ˆ g , as defined in Section 3. The overdamped equation for the dynamics ofthe vector n corresponding to Eq. (24b) reads˙ n = O n + Λ[ S n − ( n · S n ) n ] + A Sv n g (ˆ g − n g n ) . (24c)To simplify the notation we introduced the parameter A = A (cid:48) I ⊥ A (cid:107) A ⊥ C ⊥ . (25)Fig. 1(b) shows how A depends on the particle-aspect ratio λ . rientation of a small spheroid settling in turbulence Figure 3.
Angular dynamics of a settling particle in two spatial dimensions. Shownis the angle φ ( t ) obtained by simulation of Eqs. (14) (red), and the analytically exactresult for the stable fixed point φ ∗ ( t ) (blue). (a) St = 0 .
1, (b) St = 0 .
05, (c) St = 0 . λ = 5. The three simulations were performed with thesame initial conditions and for the same realisation of the function u ( x , t ) in the 2Dstatistical model. We consider the 2D model first because it is much easier to analyse than the three-dimensional model. We assume that the gravitational acceleration points into the ˆ e -direction, and define φ to be the angle (0 ≤ φ < π ) between n and this axis, so that n g = n · ˆ g = cos φ . For prolate particles ( λ > >
0) the overdampedangular dynamics (24c) becomes in two spatial dimensions: dd t φ = Ω + Λ[ S cos(2 φ ) − S sin(2 φ )] + | A | Sv sin(2 φ ) . (26)This two-dimensional overdamped equation of motion for the angular dynamics isessentially equivalent to model M2 in Ref. [49], used there for simulations of theangular dynamics of rods settling in a two-dimensional vortex flow. Apart from thefact that Ref. [49] considers a different flow, it describes small cylindrical particles withslightly different resistance tensors, and it approximates the n -dependence of the settlingvelocity.Equation (26) shows that the fluid-inertia torque has the same angular dependenceas the S -component of the strain, but in general the sign may differ. When S >
0, thestrain tends to align the rod with ˆ e , the direction of gravity. The fluid-inertia torqueacts against alignment with this direction. To quantify this statement, consider thefixed points of the angular dynamics (26). In the limit | A | Sv → ∞ the inertial torquedominates the angular dynamics, so that the fluid-velocity gradients do not matter. Inthis limit the fixed points are φ ∗ = 0 and φ ∗ = π/
2. For a prolate particle ( λ > φ ∗ = 0 is unstable while φ ∗ = π/ φ ∗ is stable the rod settles with its broadside first. For an oblate particle the stabilities are reversed [22].What is the effect of the turbulent flow? In general this question is difficult toanswer. But if the angle φ relaxes much more quickly than the fluid-velocity gradientschange along the particle path, then the problem becomes tractable. Assuming that thegradients are constant, we can find exact expressions for the two fixed points of Eq. (26), rientation of a small spheroid settling in turbulence λ > π/ | A | Sv is large: φ ∗ = π − B | A | Sv − B B A Sv ) + . . . (27)Here B ij are the elements of the matrix B = O + Λ S . Eq. (27) shows how the fixed-point orientation changes as the turbulent velocity gradients evolve. We expect thatthe orientation of a settling rod follows these fixed-point orientations closely in theoverdamped limit, provided that its angular relaxation time is smaller than the timescale on which the flow (and thus φ ∗ ) changes. We now analyse the angular dynamicsof the settling particles in this ‘persistent limit’ [63].Fig. 3 shows examples of how the fixed point φ ∗ ( t ) of the angular dynamicsfluctuates as the particle settles through the turbulent flow and encounters differentfluid-velocity gradients. The data are obtained by numerical simulation of the 2D modeldescribed in Section 2, for small Stokes numbers. Also shown is the instantaneous angle φ ( t ) obtained in these simulations. We see that the orientation dynamics follows thefixed point φ ∗ quite closely when St is small.In the overdamped limit the relaxation time τ φ of the angular dynamics (in unitsof τ K ) is given by the inverse of the stability exponent σ of the fixed point φ ∗ . From(26) we find to first order in ( | A | Sv ) − that σ ∼ −| A | Sv . This gives τ φ ∼ | σ − | = 1 | A | Sv . (28)When Sv is large, the fluid-velocity gradients seen by the settling particle change at thesettling time scale τ s , the time it takes a particle settling with an angle φ = π/ (cid:96)τ s = 1 τ K (cid:96)A ⊥ τ p g = (cid:96)η K A ⊥ Sv . (29)We therefore conclude that the persistent limit requires: τ φ τ s = 1 A ⊥ | A | Sv η K (cid:96) (cid:28) . (30)This indicates that the persistent approximation works in the overdamped limit when | A | Sv is large enough. In the opposite limit, for small values of Sv, the settling timescale τ s is larger than the Lagrangian time scale, so that the fluid-velocity gradientschange at the Lagrangian time scale, of order unity in units of τ K . Hence we mustdemand τ φ (cid:28) | A | Sv (cid:29) . (31)In the persistent limit, the overdamped angular dynamics (26) responds so rapidly thatthe orientation of the particle follows the instantaneous fixed point of the dynamicalsystem (26) quite closely. In this case the orientation distribution of the settling particleis determined by the distribution of φ ∗ , and thus by the distribution of fluid-velocity rientation of a small spheroid settling in turbulence Figure 4.
Orientation distributions for the two-dimensional statistical model. (a)Distribution of angle φ = acos( n g ) obtained from numerical simulation of the dynamics(14) (markers) and the limiting theory for small Stokes numbers, Eq. (33) (solid lines).Parameters: Sv = 22, St = 0 . λ = 3 (red, ◦ ), λ = 5 (green, (cid:3) ), λ = 7 . ♦ ), λ = 10 (magenta, (cid:77) ). (b) Same, but for different Stokes numbers. Parameters: λ = 5, and St = 0 .
022 (green, (cid:3) ), St = 0 .
22 (red, (cid:79) ), St = 22 (dark green, (cid:63) ). gradients encountered by the particle, through Eq. (27). This distribution may differfrom the distribution of fluid-velocity gradients at a fixed spatial position (preferentialsampling [51]). But in the overdamped limit preferential sampling of the fluid-velocitygradients is expected to be weak. We have checked that it is negligible for data shownin this paper.If we consider only the leading correction in Eq. (27), then the orientationdistribution is determined by the distribution P B ( B ) of B : P ( φ ) = (cid:90) ∞−∞ d B P B ( B ) δ (cid:16) φ − π B | A | Sv (cid:17) = P B (cid:2) ( π − φ ) | A | Sv (cid:3) . (32)In the two-dimensional statistical model the distribution P B ( B ) is Gaussian withvariance σ B = (2 + Λ ). This means that the distribution of φ is Gaussian too: P ( φ ) = e − ( φ − π/ σ φ (cid:113) πσ φ , (33)with variance σ φ = 18 2 + Λ ( | A | Sv ) . (34)Eq. (32) shows that the distribution of φ simply reflects that of the fluid-velocitygradients, in the overdamped and persistent limit. The corresponding distribution of n g = n · ˆ g is: P ( n g ) = 1sin φ P ( φ ) = exp (cid:2) − (acos( n g ) − π/ / (2 σ φ ) (cid:3)(cid:113) πσ φ (cid:112) − n g . (35) rientation of a small spheroid settling in turbulence Figure 5.
Orientation distribution for the three-dimensional statistical model. Sameconventions and parameters as in Fig. 4. (a) P ( n g ) in the overdamped limit. (b) Same,but for different Stokes numbers. Fig. 4 shows that Eqs. (33) and (34) agree well with results of simulations of theoverdamped dynamics in two spatial dimensions, provided that St is small enough [panel(a)]. When the Stokes number becomes larger [panel (b)], the distribution is much widerthan predicted by the overdamped theory.
In this Section we show how to obtain the distribution of n g = n · ˆ g for the three-dimensional statistical model, in the same overdamped and persistent limit consideredabove. The calculation is analogous to the one described in Section 4.1. Let p = n − n g ˆ g .Using p = 1 − n g we express the equation of motion (24c) of n g as˙ n g = ˆ g · ˙ n = ˆ g · O n + Λ[ˆ g · S n − ( n · S n ) n g ] + A Sv n g (1 − n g )= O gp +Λ[(1 − n g ) S gp + n g (1 − n g ) S gg − n g S pp ] + A Sv n g (1 − n g ) . (36)Here the subscripts g and p denote contractions with ˆ g and p . In the limit of | A | Sv → ∞ , n ∗ g = 0 is the stable fixed point for prolate particle ( λ > | A | Sv ) − as in Section 4.1, of the form n ∗ g ∝ / ( | A | Sv ) + . . . . We obtain to leadingorder: n ∗ g = ˆ g · B p | A | Sv . (37)Assuming that the orientation of p is uncorrelated from the fluid-velocity gradients, weobtain for the variance of the distribution of n g : σ n g = (cid:104) B (cid:105)(cid:104)| p | (cid:105) ( | A | Sv ) ≈ σ B ( | A | Sv ) , (38)where σ B is the variance of the distribution of B (the gravitational acceleration pointsin the ˆ e -direction). We also used that p = 1 − n g ≈
1. This is a good approximation rientation of a small spheroid settling in turbulence n g is small for prolate particles. Assuming that p and the fluid-velocity gradients are uncorrelated implies that the distribution of n g isGaussian in the statistical model: P ( n g ) = 1 √ πσ n g exp (cid:16) − n g σ n g (cid:17) , (39)and the variance evaluates to σ n g = 1( | A | Sv ) . (40)Figure 5 shows results for the distribution of n g from simulations of the three-dimensionalstatistical model. Panel (a) shows results for small Stokes numbers, the parameters arethe same as in Fig. 4(a). Also shown are the results of the theory, Eqs. (39) and (40).In this case St is small enough and Sv large enough so that the theory works very well.Panel (b) shows the orientation distribution for different Stokes numbers, to demonstratehow the theory fails when the Stokes number becomes larger. The behaviour is similarto that described in Section 4.1: the distribution widens as St increases.Eq. (38) says that the variance of the distribution of n g is inversely proportionalto the fourth power of Sv, σ n g ∝ Sv − , for large values of the settling number providedthat the Stokes number is small enough. In Fig. 6(a) this prediction is compared withresults of simulations of the three-dimensional statistical model. Shown is the varianceof n g as a function of Sv, for two Stokes numbers. When the Stokes number is small wesee that the prediction (40) works well for large Sv, as expected. Fig. 6(b) shows thekurtosis β = (cid:104) n g (cid:105) / (cid:104) n g (cid:105) , measuring the flatness of the distribution P ( n g ). As predictedby the theory, the kurtosis approaches the Gaussian limit ( β = 3) for large settlingnumbers, at small enough Stokes numbers.When Sv → and β → , indicating that the persistentapproximation fails because Eq. (31) is no longer satisfied. In this limit the distributionof n g becomes uniform and independent of the Stokes number, because the angulardynamics is isotropic when gravitational settling is weak. Fig. 6(c) shows results for thevariance from numerical simulations using the KS model (Section 2.3), for three differentvalues of the Stokes number. The results are very similar to those obtained using thestatistical model [Fig. 6(a)]. There is good agreement with the overdamped theory,Eq. (38), at large Sv for small enough St. We determined σ B from the KS simulations,so there are no fitting parameters in Fig. 6(c). The good agreement shows that theoverdamped theory is robust, insensitive to the details of the spectrum of the velocityfluctuations. Fig. 6 also shows numerical data for larger values of St. For small Sv thismakes little difference, the distribution is uniform. For larger Sv the numerical resultsfirst follow Eq. (38) or (40). But as Sv increases further, the overdamped theory startsto fail, the earlier the larger the Stokes number. This indicates that particle inertiabegins to become important. rientation of a small spheroid settling in turbulence Figure 6.
Width of the orientation distribution. (a) Variance of n g from simulationsof the three-dimensional model, as a function of Sv, for two values of the Stokes number:St = 0 .
022 (red, ◦ ) and St = 0 .
22 (green, (cid:3) ). Also shown is the theory for large Sv,Eq. (40), solid line, and the result for a uniform distribution, (cid:104) n g (cid:105) = (dashed line).(b) Kurtosis β = (cid:104) n g (cid:105) / (cid:104) n g (cid:105) . Same parameters as in panel (a). The overdampedtheory (Section 4.2) gives a Gaussian distribution with kurtosis equal to β = 3 (solidline). For a uniform distribution, β = (dashed line). (c) Results for σ n g from KSfor St = 0 .
025 (blue, (cid:5) ), 0 . (cid:77) ), and 0 . (cid:79) ). Also shown is the theory,Eq. (38), solid line, as well as the uniform limit (dashed line).
5. Effect of particle inertia
We saw in the previous Section that the overdamped theory breaks down at large Sv. Tounderstand when and why the overdamped theory fails one must check the full inertialdynamics. We analyse the 2D statistical model first.
Consider the angular dynamics in the absence of flow, to estimate the time scales thatare important for the angular dynamics. When u = 0, the dynamics of the phase-spacecoordinate z ≡ ( v p x , v p y , φ, ω ) has the stable fixed point z ∗ = (Sv /A ⊥ , , π/ , e . The stability matrix follows from Eq. (14): J ≡ ∂ ˙ z ∂ z = 1St − A ⊥ − A (cid:107) A (cid:107) − A ⊥ A ⊥ Sv 00 0 0 St0 − A (cid:48) A ⊥ Sv + A (cid:48) A ⊥ Sv − C ⊥ I ⊥ , (41)where A (cid:48) was defined in Eq. (17). The relaxation time following from Eq. (41) is givenby τ φ = max( − / (cid:60) σ i ), the maximal stability time of J . Here (cid:60) σ i denotes the real partof the i -th eigenvalue of J . One eigenvalue of this matrix is σ = − A ⊥ / St. We havecomputed the other eigenvalues numerically and analytically in limiting cases. We findthat the time scale τ φ interpolates between Eq. (28) for small St and ∼ St /A ⊥ for largeSt, for a fixed value of Sv. If we fix St, by contrast, then we find that the time scale τ φ interpolates between Eq. (28) for small Sv and ∼ St /A ⊥ for large Sv.We expect that the overdamped approximation fails when the inertial estimatefor the relaxation time of the angular dynamics, τ φ ∼ St /A ⊥ , becomes larger than the rientation of a small spheroid settling in turbulence Figure 7.
Variance (cid:104) δφ (cid:105) for the two-dimensional statistical model. (a) Results ofnumerical simulations as a function of Sv for λ = 5, St = 0 . (cid:3) ), St = 0 . (cid:5) ), St = 0 .
4, (magenta, (cid:77) ). Also shown: theory from Section 4.1, Eqs. (33) and(34), thick solid black line; condition | A | Sv = A ⊥ / St for the overdamped theory tofail [Eq. (42)], vertical dashed lines; condition (44) for the white-noise limit, verticaldash-dotted lines; large-Sv scaling (43), thick black dashed line; uniform distributionat small Sv, horizontal black dashed line. (b) Results as a function of the particleaspect ratio λ for Sv = 25, St = 0 . (cid:3) ), and St = 0 . (cid:77) ). overdamped estimate Eq. (28). This means that the overdamped approximation requires | A | Sv (cid:28) A ⊥ / St . (42)Conversely, when Eq. (42) is not satisfied then particle inertia matters, so that the over-damped approximation must fail [Fig. 6(a)]. For a quantitative comparison, Fig. 7(a)shows numerical results for the variance of the orientation distribution obtained fromsimulations of the two-dimensional model. We see that the overdamped approximationbreaks down for values of Sv larger than ∼ (cid:112) A ⊥ / ( | A | St), as predicted by Eq. (42). Weobserve that the variance decreases more slowly as Sv increases further.Fig. 7(a) also reveals that there is yet another, asymptotic regime at very largevalues of Sv – so large that it is difficult to achieve small Re p at the same time (Section 6).It is nevertheless of interest to analyse this regime, because it reveals the ingredientsthat a theory describing effects of particle inertia must contain. Fig. 7(a) suggests that (cid:104) n g (cid:105) ∼ c Sv (43)for very large values of Sv. Our simulations indicate that the prefactor c depends upon (cid:96)/η K , St, and upon λ (not shown). We surmise that this regime describes particlessettling so rapidly that the settling time scale τ s is the smallest time scale in the system.This cannot hold unless τ φ ∼ St /A ⊥ is much larger than τ s , and this crossover occurs atSv St A ⊥ η K (cid:96) ∼ . (44)We expect Eq. (43) to be accurate for values of Sv much larger than those given byEq. (44). This condition is also shown in Fig. 7, and we see that the large-Sv regime rientation of a small spheroid settling in turbulence W = W (0) ( n ) (assumed in the overdamped theory of Section 4).This means that particle inertia is expected to modify the angular dynamics in at leasttwo ways. Firstly, it introduces the time derivative d d t δφ into the angular dynamics.Secondly, the fluctuations of the torque change because W (cid:54) = W (0) ( n ) when particleinertia matters. This is discussed in Section 5.2.Fig. 7(b) shows how the variance of δφ depends on particle shape, for fixed Sv andSt. There are four regimes. First, in the limit λ → ∞ the distribution is uniform andindependent of the Stokes number. In this regime the dynamics is overdamped [condition(42)], but the persistent approximation fails because Eq. (31) is not satisfied. Second, forintermediate aspect ratios, both conditions are satisfied, so that the theory [Eqs. (33)and (34)] is accurate. Third, at λ becomes smaller, the overdamped approximationbreaks down. In this regime particle inertia must be taken into account. Fourth, as λ → λ = 1) the orientation distribution is uniform, but already for λ ∼ .
05 there is strong alignment.
Klett [24] proposed a theory for the orientation variance of nearly spherical particlessettling in turbulence, including particle inertia in the angular dynamics. He uses thatthe orientation variance is very small for large values of Sv. This suggests to expand theequations of motion in small deviations of the angle φ = acos( n · ˆ g ) from its equilibriumvalue: φ = φ ∗ + δφ where φ ∗ = π for prolate particles. Klett assumes that W = W (0) ( n )[Eq. (6)] and expands the angular dynamics for nearly spherical particles in δφ .We can derive an equation of motion consistent with his by expanding Eqs. (14) toleading order in δφ , assuming that W = W (0) ( n ), and retaining only the leading termsin ( | A | Sv ) − . In this way we obtain for a prolate particle of arbitrary aspect ratio inthree spatial dimensions:d d t δφ + C ⊥ I ⊥ St dd t δφ + C ⊥ I ⊥ St | A | Sv δφ = − C ⊥ I ⊥ St ˆ g · B p . (45)When we expand the geometrical coefficients in Eq. (45) for small Λ we find that theprefactors of the terms on the l.h.s. of this equation are almost identical, in this limit,to those in Eq. (17) of Ref. [24]. Slight discrepancies arise in the δφ -term because we usethe expression for the inertial torque from Ref. [22], while Klett uses the form obtainedby Cox [20] (the relative error of the prefactors is of the order of 10 − [22]). At any rate,Eq. (45) is simply a damped driven harmonic oscillator, with implicit solution δφ ( t ) = C ⊥ Ω I ⊥ St (cid:90) t d t e C ⊥ ( t − t ) / (2 I ⊥ St) sin[Ω ( t − t )] ˆ g · B ( t ) p . (46) rientation of a small spheroid settling in turbulence = [ C ⊥ / (2 I ⊥ St)] (cid:112) | A | Sv I ⊥ St /C ⊥ −
1. Note that we discarded terms relatedto the initial angle, because they cannot be important for the steady-state variance of δφ in the limit of large Sv. Squaring Eq. (46) and averaging over realisations of theturbulent fluctuations in the statistical model we obtain for large Sv (cid:104) δφ (cid:105)∼ c Sv , (47)where c is a function of (cid:96)/η K , St, and of the aspect ratio λ . We neglected a Sv − contribution to (cid:104) δφ (cid:105) because it is exponentially suppressed. Eq. (47) fails to describethe large-Sv behaviour (43), shown as the thick black dashed line in Fig. 7. Thismeans that Eq. (45) cannot be used to estimate the large-Sv width of the orientationdistribution, or to compute deviations from the overdamped theory.Which approximation causes Eq. (45) to fail? Since the variance is small for largeSv, δφ remains small at all times. Therefore we see no reason to doubt that the small-angle expansion is valid. This leads us to conclude that the assumption W = W (0) ( n )breaks down, in agreement with our conclusions in the previous Section. To checkthis, we artificially imposed the constraint W = W (0) ( n ) in simulations of the two-dimensional statistical model. The resulting large-Sv variance follows Eq. (47), andthus fails to give the correct scaling, Eq. (43). This demonstrates that it is importantto allow W to deviate from W (0) ( n ) when particle inertia matters.Klett’s theory is difficult to justify from first principles because it assumes that W = W (0) ( n ). However, he obtains that (cid:104) δφ (cid:105) ∝ Sv − , assuming that the fluid-velocitygradients on the r.h.s. of Eq. (45) are just white noise in time. In view of Eq. (44)it is possible that a first-principles theory may yield just that. But fluctuations of W − W (0) ( n t ) yield additional time-dependent terms in the angular equation of motionthat are expected to change the properties of the noise driving the angular dynamics,resulting in a different prediction for the orientation variance. More importantly, Fig. 7demonstrates that (cid:104) δφ (cid:105) ∝ Sv − applies only in the unphysical limit of very large Sv, andthat particle inertia causes a complex parameter dependence of the orientation varianceat smaller values of Sv, with a number of different regimes to consider.
6. Conclusions
Convective fluid inertia affects the orientation of a small axisymmetric particle settlingin a turbulent flow. In Refs. [43, 44, 45, 46, 47] this effect was neglected. Here weconsidered a limit of the problem where it is dominant, but where turbulent fluctuationsstill matter. Our goal was to compute the distribution of orientations of a spheroid inturbulence, to work out how the torques due to convective fluid inertia and due to theturbulent velocity gradients affect the orientation distribution. In general the angulardynamics of the settling particle is very complicated. Here we looked at a limit in whichthe problem becomes tractable: we assumed small Stokes number (a dimensionlessmeasure of particle inertia) and large settling number (dimensionless settling speed).For small Stokes numbers the dynamics is overdamped. For large values of the settling rientation of a small spheroid settling in turbulence − for large Sv, and it determines how the prefactor depends on aspect ratio λ of theparticle. In the limit λ → ∞ the variance was computed in Ref. [54].We demonstrated that the overdamped theory breaks down at finite Stokesnumbers, when the settling number exceeds a threshold determined by St. In thisregime particle inertia matters. Klett [24] proposed a theory for the orientationvariance for nearly spherical particles, taking into account particle inertia in the angulardynamics. His theory assumes that this dynamics is driven by the fluid-velocity gradientsexperienced by the settling particle, and that these gradients are uncorrelated in time sothat diffusion approximations can be applied. Klett’s theory predicts that the varianceis proportional to Sv − , and we do observe this scaling for very large Sv, so large thatthe settling time is the smallest time scale of the inertial dynamics. But to derive atheory from first principles it is necessary to take into account particle inertia not onlyin the angular dynamics but also in the centre-of-mass motion, resulting in additionalfluctuating terms in the angular equation of motion that are expected to change theorientation variance. More importantly, our simulations also show that particle inertiagives rise to a complex dependence of the orientation variance on particle shape, on theStokes number, and upon the settling number. When the variance is small, it may bepossible to derive a theory for the variance using small-angle approximations. But thisremains a question for the future.Here we applied our theory only to prolate particles. It is of interest to consideroblate particles too, because flat disks and slender rods have qualitatively differentshape factors (Fig. 1). We therefore expect that the effect of particle inertia on theangular dynamics of flat disks can be quite different from that on slender rods. Also, weconsidered only the leading order in the inverse settling number, but the overdampedtheory allows us to take into account higher-order corrections in this parameter. Suchcorrections change the relation between the fixed points of the angular dynamics andthe fluid-velocity gradients experienced by the particle. This modifies the form of thedistribution of n g , and it may explain the overshooting seen in Fig. 6(b) at moderatevalues of Sv, but the details remain to be worked out.Here we analysed a limit of the problem where the fluid-inertia torque dominates theangular motion. In Refs. [43, 44, 45, 46, 47], by contrast, this torque was neglected. Thequestion is thus whether one can find regions where inertial torque does not dominate. rientation of a small spheroid settling in turbulence λ is small. In a veryturbulent flow, when Re λ is large, the torque induced by fluid inertia is always dominant.More precisely, when the ratio of the correlation length over the Kolmogorov length islarge, (cid:96)/η K ∝ Re λ / (cid:29)
1, then the only possible orientation bias corresponds to non-spherical particles settling with their broad sides down, the limit considered here.The experiments measuring the orientations of rods settling in a vortex flowdescribed in Ref. [49] are performed in the overdamped limit. In the future we intend toapply the theory outlined in Section 4 to spheroids settling in a two-dimensional vortexflow, using the fact that the fixed points of the angular dynamics can be found explicitlyas functions of the fluid-velocity gradients in two spatial dimensions. We will analyse theeffect of particle shape by considering the angular dynamics of flat disks settling in suchflows. Figure 1 indicates that the behaviour could be quite different from that of rods,because the shape factors are so different. This two-dimensional system is well suited tostudy the effects of finite Stokes numbers in more detail, because the two-dimensionaldynamics is much simpler than the three-dimensional turbulent dynamics.The overdamped theory [Eq. (38)] assumes that Sv is large, and that St issmall enough. Since Sv = St gτ /η K = St / Fr, this requires some discussion. HereFr = η K / ( gτ ) is the Froude number [58]. We conclude that the Froude number must besmall for the overdamped theory to work quantitatively. In turbulence Fr ∼ E / / ( gν / )where E is the dissipation rate per unit mass. Using ν ∼ − m s − and g = 10 m s − we find that Fr ranges from 0 .
002 at E = 1 cm s − to 0 . E = 1000 cm s − . So werequire modest values of the dissipation rate per unit mass, E , for the theory to workquantitatively. This is the limit where gravity dominates over the turbulent fluctuations,the limit we intended to describe.In the future it is necessary to address possible shortcomings of our model whichapproximates the inertial contributions to force and torque by those for a homogeneoussteady flow. Even in the steady case it remains an open question how to model thetorque when Re p and √ Re s are of the same order, even if both dimensionless numbersare small. Furthermore, turbulent flow is unsteady. While it is common practice touse steady approximations for the instantaneous force and torque (as we do here)it is not known how to compute contributions to the torque due to unsteadiness forgeneral inhomogeneous flows. We expect that the methods presented in Ref. [40] can begeneralised to treat at least spatially linear, unsteady flows. Finally, to justify our modelfor the inertial torque it is necessary that Re p is small. At the same time we assumedthat Sv is large. From the definitions (9) and (15) of these dimensionless numbers wesee that Re p = ( a/η K )(Sv /A ⊥ ). To satisfy both requirements we must therefore assumethe particles to be much smaller than the Kolmogorov length. Since η K ∼ ( ν / E ) / thiscondition is more easily met when E is small. In the slender-body limit, Khayat & Cox[21] obtained an improved approximation for the inertial torque, valid for larger Re p ,which was tested in Ref. [49] and was found to agree better with the experiments atlarger Re p . But corresponding corrections for other particle shapes are not yet known. rientation of a small spheroid settling in turbulence Acknowledgments
BM and AP thank E. Guazzelli for enlightening discussions. KG and BM were supportedby the grant
Bottlenecks for particle growth in turbulent aerosols from the Knut andAlice Wallenberg Foundation, Dnr. KAW 2014.0048, and in part by VR grant no.2017-3865. AP and AN acknowledge support from the IDEXLYON project (ContractANR-16-IDEX-0005) under University of Lyon auspices. Computational resources wereprovided by C3SE and SNIC, and PSMN. [1] H. R. Pruppacher and J. D. Klett.
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