Multiscale preferential sweeping of particles settling in turbulence
aa r X i v : . [ phy s i c s . f l u - dyn ] A p r This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Multiscale preferential sweeping of particlessettling in turbulence
Josin Tom and Andrew D. Bragg † Department of Civil and Environmental Engineering, Duke University, Durham, NC, USA(Received xx; revised xx; accepted xx)
In a seminal article, Maxey (1987, J. Fluid Mech., 174:441-465) presented a theoreticalanalysis showing that enhanced particle settling speeds in turbulence occur through thepreferential sweeping mechanism, which depends on the preferential sampling of thefluid velocity gradient field by the inertial particles. However, recent Direct NumericalSimulation (DNS) results in Ireland et al. (2016 b , J. Fluid Mech., 796:659–711) showthat even in a portion of the parameter space where this preferential sampling isabsent, the particles nevertheless exhibit enhanced settling velocities. Further, there areseveral outstanding questions concerning the role of different turbulent flow scales on theenhanced settling, and the role of the Taylor Reynolds number R λ . The analysis of Maxeydoes not explain these issues, partly since it was restricted to particle Stokes numbers St ≪
1. To address these issues, we have developed a new theoretical result, valid forarbitrary St , that reveals the multiscale nature of the mechanism generating the enhancedsettling speeds. In particular, it shows how the range of scales at which the preferentialsweeping mechanism operates depends on St . This analysis is complemented by resultsfrom DNS where we examine the role of different flow scales on the particle settlingspeeds by coarse-graining the underlying flow. The results show how the flow scales thatcontribute to the enhanced settling depend on St , and that contrary to previous claims,there can be no single turbulent velocity scale that characterizes the enhanced settlingspeed. The results explain the dependence of the particle settling speeds on R λ , andshow how the saturation of this dependence at sufficiently large R λ depends upon St .The results also show that as the Stokes settling velocity of the particles is increased, theflow scales of the turbulence responsible for enhancing the particle settling speed becomelarger. Finally, we explored the multiscale nature of the preferential sweeping mechanismby considering how particles preferentially sample the fluid velocity gradients coarse-grained at various scales. The results show that while rapidly settling particles do notpreferentially sample the fluid velocity gradients, they do preferentially sample the fluidvelocity gradients coarse-grained at scales outside of the dissipation range. This explainsthe findings of Ireland et al. , and further illustrates the truly multiscale nature of themechanism generating enhanced particle settling speeds in turbulence. Key words:
1. Introduction
The motion of inertial particles in turbulence settling under gravity is importantfor many environmental, biological and engineering multi-phase flows such as water † Email address for correspondence: [email protected]
J. Tom and A. D. Bragg droplets in clouds (Shaw 2003; Grabowski & Wang 2013), marine snow (Kiorboe 1997;Guseva et al. et al. η ), and heavy (particledensity much greater than fluid density) inertial particles, two important parametersare the particle Stokes number, St , and the Froude number, F r . The Stokes number, St ≡ τ p /τ η , provides a measure of the particle inertia, where τ p is the particle responsetime and τ η is the Kolmogorov timescale. The Froude number, F r ≡ a η /g , quantifies thestrength of the turbulence relative to gravity, where a η is the Kolmogorov accelerationscale and g is the magnitude of the gravitational acceleration. From St and F r thesettling parameter can be defined Sv ≡ St/F r ≡ τ p g/u η , that compares the Stokessettling velocity to the Kolmogorov velocity scale u η . Another important parameter isthe particle Reynolds number, Re p , that determines whether the drag force on the particleis a linear or non-linear function of the slip velocity between the particle and local fluidvelocity (Maxey & Riley 1983). It is typically assumed that the drag force is linear if Re p < . St ≪ Sv = O (1) and St = O (1). Theyalso confirmed the physical argument of Maxey (1987) that settling inertial particlespreferentially accumulate in low vorticity regions of the flow where fluid velocity alignswith the direction of gravity. They referred to this effect as the “preferential sweepingmechanism” or “fast-tracking”, and it has since been observed in several DNS (Bec et al. et al. b ; Rosa et al. et al. et al. et al. Re p >
1, a non-linear drag law should be used for calculating the drag forceon the particle (Clift et al. ultiscale preferential sweeping of particles settling in turbulence et al. (2016). However, Good et al. (2014) observed a significant reduction in theparticle settling velocity compared with the linear drag case, in certain regimes of
St, Sv .Another issue is whether in some cases it is possible for turbulence to reduce the particlesettling speed compared with the Stokes settling velocity. This scenario, referred to byNielsen (1993) as “loitering”, could occur when the particles spend more time in upwardmoving regions of the flow than downward moving regions. In their DNS study usingnon-linear drag, Wang & Maxey (1993) observed loitering only in a limited portion of theparameter space. Loitering has been observed in several experiments (Yang & Shy 2003;Kawanisi & Shiozaki 2008; Good et al. et al. (2019) did not show clear evidence of loitering. Good et al. (2014) wereonly able to observe loitering in their DNS when either the horizontal motion of theparticles was artificially eliminated, or else when a non-linear drag force was used forthe particles. However, as discussed earlier, other studies such as Rosa et al. (2016) havefound only a small effect of non-linear drag on the particle settling speeds, and they didnot observe loitering.These studies show that even for the relatively simple case of small particles settling inturbulence, the role of turbulence on the average particle settling speed is subtle, and anumber of issues remain to be solved. Moreover, there are other additional complexitiesthat can modify particle settling speeds in turbulence, including finite particle sizeeffects (Fornari et al. et al. St ≪
1, the theoretical analysis of Maxey (1987) gives little insight into which scales ofthe turbulence contribute to the particle settling speeds. Numerical studies, have, to alimited extent, considered the question of which flow scales contribute to the enhancedsettling. Wang & Maxey (1993) argued that their DNS results implied that the enhancedparticle settling speeds due to turbulence depends on the fluid r.m.s. velocity u ′ , whichis associated with the large scales of the turbulent flow. However, they also arguedthat this could be an artifact of the low Reynolds number of their DNS, and that inreal atmospheric flows where the scale separation is much larger, the particle settlingspeeds are likely to be only affected by a limited range of scales of the turbulence. The J. Tom and A. D. Bragg study of Yang & Lei (1998) considered the role played by different flow scales on theparticle settling speeds in turbulence using DNS and Large Eddy Simulations (LES).They concluded that the large scales of the flow play a key role and that the relevantfluid velocity scale determining the settling enhancement is u ′ and not u η . However, theyalso concluded that the settling enhancement depends on the particle inertia through τ p /τ η , and not τ p /τ L , where τ L is the integral timescale of the flow. Their argument isthat the preferential sweeping effect depends on the small scale clustering of the particles,and hence τ η , while the drag force on the particles depends mainly on the large scales,and hence u ′ . The recent study of Rosa et al. (2016) also argued that the relevant fluidvelocity scale determining the settling enhancement is u ′ .The success of “mixed scaling”, i.e. using a large scale quantity u ′ for the velocity scale,and a small scale quantity τ η for the time scale, used in previous studies (Yang & Lei1998; Good et al. u ′ really does dominate theparticle settling speeds, or whether there exists a range of velocity scales that governthe process. Indeed, as we shall point out in §
2, there are theoretical difficulties with theargument that the relevant turbulent velocity scale dominating the settling speed is u ′ .Moreover, it is not at all obvious that a single velocity scale should determine the settlingspeed, independent of St , since the interaction of an inertial particle with a given flowscale depends strongly on St . Motivated by the need to develop particle SubGrid Scale(SGS) models for LES of particles settling in turbulent flows, Rosa & Pozorski (2017)used numerical simulations to study the effect of the small scales of the turbulence onparticle settling speeds. Their results showed that scales smaller than a certain size didnot affect the particle settling speeds. However, they did not consider whether there existsan upper limit to the range of scales that affect the particle settling speeds, nor did theyprovide detailed information concerning how the range of velocity scales impacting thesettling speeds might depend on St or F r .In addition to these open issues, results in Ireland et al. (2016 b ) showed that significantenhancement of particle settling speeds due to turbulence are observed for St > O (1) and F r ≪
1, even though the DNS data showed that for St > O (1) and F r ≪ St ≪
1. The purpose of the present study then is to address these issuesand provide insight concerning which flow scales contribute to the modified particlesettling speeds due to turbulence. To this end, we develop a new theoretical frameworkfor analyzing the problem for arbitrary St , which is then used in conjunction with DNSdata for a range of Reynolds numbers to provide detailed insights into the multiscalemechanism leading to enhanced particle settling speeds in turbulence. ultiscale preferential sweeping of particles settling in turbulence
2. Theory
Background
We consider the settling of small ( d p /η ≪
1, where d p is the particle diameter), heavy( ρ p /ρ f ≫
1, where ρ p is particle density and ρ f is fluid density), spherical inertialparticles, that are one-way coupled to a statistically stationary isotropic turbulent flow.In the regime of a linear drag force on the particles, the particle equation of motionreduces to (see Maxey & Riley 1983)¨ x p ( t ) ≡ ˙ v p ( t ) = 1 Stτ η (cid:16) u ( x p ( t ) , t ) − v p ( t ) (cid:17) + g , (2.1)where x p ( t ) , v p ( t ) are the particle position and velocity vectors, respectively, u ( x p ( t ) , t )is the fluid velocity at the particle position, and g is the gravitational acceleration vector.As discussed earlier, some studies suggest that for Sv > O (10), nonlinear drag effectsare important for settling particle motion in turbulence (Good et al. Sv ≈
63, the effects of nonlinear drag are very small(Rosa et al. e z gives h v pz ( t ) i = h u z ( x p ( t ) , t ) i + Stτ η g, (2.2)since h ˙ v p ( t ) i = for this system. Equation (2.2) shows that the average particlevelocity may differ from the Stokes settling velocity Stτ η g only if h u z ( x p ( t ) , t ) i 6 = 0.Numerous studies, both numerical (Wang & Maxey 1993; Bec et al. et al. b ; Rosa et al. et al. et al. et al. h v pz ( t ) i 6 = Stτ η g ,implying h u z ( x p ( t ) , t ) i 6 = 0. Maxey (1987) developed a theoretical framework to explainhow h u z ( x p ( t ) , t ) i 6 = 0, even though the Eulerian average satisfies h u z ( x , t ) i = 0 for anisotropic flow. Essentially, the explanation is that particles with inertia do not uniformlysample the underlying fluid velocity field, and that gravity leads to a bias for inertialparticles to accumulate in regions of the flow where e z · u >
0. However, the analysis ofMaxey (1987) is restricted to St ≪ h u z ( x p ( t ) , t ) i . While this has not previously been systematically explored, one claim thathas been made in numerous studies (Wang & Maxey 1993; Yang & Lei 1998; Good et al. et al. et al. h u z ( x p ( t ) , t ) i depends on u ′ , thelarge scale fluid velocity. However, this seems unlikely because if h u z ( x p ( t ) , t ) i dependson u ′ , then even if Sv = O (1), h u z ( x p ( t ) , t ) i /u η → ∞ as R λ → ∞ since u ′ /u η ∼ R / λ (Pope 2000). Yet this seems to lead to a contradiction since h v pz ( t ) i /u η = h u z ( x p ( t ) , t ) i /u η + Sv, (2.3)so that for Sv = O (1), the particle settling becomes irrelevant to the mean motion of theparticle in the limit R λ → ∞ , yet the settling is supposed to be the very thing responsiblefor h u z ( x p ( t ) , t ) i 6 = 0. Clearly then new insight is needed to understand which scales of theturbulent flow influence h u z ( x p ( t ) , t ) i , and we now develop a new theoretical frameworkto provide such insight. J. Tom and A. D. Bragg
Theoretical framework for arbitrary St In this section we develop a theoretical framework for considering the behavior of h u z ( x p ( t ) , t ) i for arbitrary St , and for revealing which scales of motion contribute to h u z ( x p ( t ) , t ) i 6 = 0.In the analysis of Maxey (1987), attention was restricted to St ≪
1, for which it ispossible to approximate v p ( t ) as a field v p ( t ) = (cid:16) u ( x , t ) − Stτ η [ a ( x , t ) − g ] (cid:17)(cid:12)(cid:12)(cid:12) x = x p ( t ) + O ( St ) , (2.4)where x is a fixed point in space (unlike x p ( t )), and a ≡ ∂ t u +( u ·∇ ) u is the fluid acceler-ation field. Using this field representation, Maxey was able to construct an expression for h u z ( x p ( t ) , t ) i using a continuity equation for the instantaneous particle number density.However, when St > O (1), this field approximation for v p ( t ) fundamentally breaks downdue to the formation of caustics in the particle velocity distributions, wherein particlevelocities at a given location become multivalued (Wilkinson & Mehlig 2005). Therefore,a quite different approach to that employed by Maxey (1987) must be sought in order toanalyze h u z ( x p ( t ) , t ) i for arbitrary St .We begin by noting that for homogeneous turbulence we may write h u z ( x p ( t ) , t ) i = ̺ − D u z ( x , t ) δ ( x p ( t ) − x ) E , (2.5)where δ ( · ) is a Dirac distribution, and ̺ ≡ h δ ( x p ( t ) − x ) i is the Probability DensityFunction (PDF) of x p ( t ). Here, h·i is an ensemble average over all possible realizations ofthe system, and this includes not only an average over all realizations of u , but also anaverage over all initial particle positions x p (0) = x and velocities v p (0) = v . From (2.5)it follows that for a homogeneous turbulent flow, h u z ( x p ( t ) , t ) i = h u z ( x , t ) i = 0 if x p ( t )is uncorrelated with u z ( x , t ). This occurs for St ≫ t = 0 (Bragg et al. b ), sincetheir spatial distribution remains constant and uniform ∀ t due to incompressibility. Infact, h u z ( x p ( t ) , t ) i 6 = 0 can only occur if δ ( x p ( t ) − x ) both fluctuates in time and is alsocorrelated with u z ( x , t ).To proceed with an analysis of h u z ( x p ( t ) , t ) i that applies for arbitrary St , we introducethe averaging decomposition h·i = hh·i x , v u i u (Bragg et al. a ), where h·i x , v u denotesan average over all initial particle positions x p (0) = x and velocities v p (0) = v fora given realization of the fluid velocity field u , and h·i u denotes an average over allrealizations of u . Introducing this decomposition to (2.5) we have h u z ( x p ( t ) , t ) i = ̺ − DD u z ( x , t ) δ ( x p ( t ) − x ) E x , v u E u = ̺ − D u z ( x , t ) ϕ ( x , t ) E u , (2.6)where ϕ ( x , t ) ≡ D δ ( x p ( t ) − x ) E x , v u , (2.7)and also ̺ ≡ h ϕ ( x , t ) i u . The reason for introducing this averaging decomposition is thatit will allow us to introduce a particle velocity field that is valid for arbitrary St , unlike(2.4) that is only valid for St ≪ ϕ is given by ∂ t ϕ + ∇ · (cid:16) ϕ V ( x , t ) (cid:17) = 0 , (2.8) ultiscale preferential sweeping of particles settling in turbulence V ( x , t ) ≡ D v p ( t ) E x , v x , u . (2.9)The particle velocity field V ( x , t ) differs fundamentally from the particle velocity fieldused in Maxey (1987), namely (2.4), since V ( x , t ) does not presume that v p ( t ) is uniquelydetermined for x p ( t ) = x in a given realization of u . Rather, V ( x , t ) is constructed asan average over different particle trajectories (each corresponding to different x , v )satisfying x p ( t ) = x in a given realization of u . We also emphasize that both ϕ and V are turbulent fields, in general, since they depend upon the evolution of the particularrealization of u to which they correspond.The solution to (2.8) may be written formally as ϕ ( x , t ) = ϕ ( X (0 | x , t ) ,
0) exp − Z t ∇ · V ( X ( s | x , t ) , s ) ds ! , (2.10)where ˙ X ( t ) ≡ V ( X ( t ) , t ), and the notation s | x , t denotes that the variable is measuredat time s along a trajectory satisfying X ( t ) = x . Note that ∇ · V ( X ( s | x , t ) , s ) is to beunderstood as ∇ · V ( X ( s | x , t ) , s ) ≡ ∇ · V ( y , s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = X ( s | x ,t ) such that the operator ∇ · {} acts on the spatial coordinate of the field V and not onthe trajectory end-point coordinate x .For simplicity, we will take ϕ ( X (0 | x , t ) ,
0) = 1 / D , corresponding to particles thatare initially uniformly distributed throughout the volume D of the system (this wasalso assumed in Maxey (1987)). Further, we note that for a statistically stationary,homogeneous system, ̺ D = 1 ∀ t when ϕ ( X (0 | x , t ) ,
0) = 1 / D . Using this initial conditionfor all realizations of u , we may then insert (2.10) into (2.6) and obtain h u z ( x p ( t ) , t ) i = * u z ( x , t ) exp − Z t ∇ · V ( X ( s | x , t ) , s ) ds !+ u . (2.11)Before proceeding, we note that in the regime St ≪ V ( x , t ) and obtain V ( x , t ) = D u ( x p ( t ) , t ) − Stτ η [ a ( x p ( t ) , t ) − g ] E x , v x , u + O ( St )= u ( x , t ) − Stτ η [ a ( x , t ) − g ] + O ( St ) , (2.12)and inserting this into (2.11) we obtain essentially the same result as Maxey (1987) h u z ( x p ( t ) , t ) i ≈ * u z ( x , t ) exp Stτ η Z t (cid:16) S ( x p ( s | x , t ) , s ) − R ( x p ( s | x , t ) , s ) (cid:17) ds !+ u , (2.13)where S and R are the second invariants of the fluid strain-rate and rotation-ratetensors, respectively. The interpretation of (2.13) given by Maxey (1987) is that h u z ( x p ( t ) , t ) i > S − R > u z > St ≪
1, suggests more generally that
J. Tom and A. D. Bragg h u z ( x p ( t ) , t ) i > V ( x , t ) is compressible, and also when there existsa correlation between regions where ∇ · V < u z >
0. The argument givenby Wang & Maxey (1993) for this correlation is essentially connected to the fact thatsettling particles typically approach the turbulent vortices from above as they fallthrough the flow, and they are swept around the vortices due to the centrifuge effect. Asupplementary argument to theirs is that the settling particles tend to follow the “pathof least resistance”. In particular, downward moving particles prefer to move around thedownward moving side of vortices in the flow since on this side they experience a weakerdrag force (“less resistance”) than they would if they were to fall around the upwardmoving side of the vortice.Unlike the St ≪ St > O (1), V ( x , t ) depends non-locally in time uponthe fluid velocity field. Indeed, using the formal solution to (2.1) we may write (ignoringinitial conditions) V ( x , t ) = Stτ η g + 1 Stτ η Z t e − ( t − s ) /Stτ η D u ( x p ( s ) , s ) E x , v x , u ds, (2.14)so that V ( x , t ) depends on u at earlier times along the particle trajectory, and is in thissense temporally non-local. Due to this non-locality, there need not exist a correlationbetween ∇ · V and the local properties of the flow. When ∇ · V is uncorrelated with thefluid velocity field then from (2.11) we have h u z ( x p ( t ) , t ) i = * u z ( x , t ) exp − Z t ∇ · V ( X ( s | x , t ) , s ) ds !+ u = * u z ( x , t ) + u * exp − Z t ∇ · V ( X ( s | x , t ) , s ) ds !+ u = 0 . (2.15)Thus it is not merely clustering of the particles (related to ∇ · V <
0) that is requiredfor h u z ( x p ( t ) , t ) i >
0, but that the clustering be correlated in some way with thefluid velocity field. In view of this it is essential to make a distinction between twophenomena, namely particle clustering and preferential concentration. As emphasized inBragg et al. (2015), these are distinct: clustering refers to non-uniformity of the particlespatial distribution, irrespective of any correlation the distribution may have with thefluid flow field. In contrast, preferential concentration describes the situation where thespatial distribution of the particles is not only non-uniform, but is also correlated to thelocal properties of the flow i.e. the particles cluster in specific regions of the flow, sothat the particles preferentially sample the flow field. Recent results have shown thatfor settling particles this distinction is particularly important, since settling inertialparticles can strongly cluster in the dissipation range of turbulence, despite the fact thattheir spatial distribution is entirely uncorrelated with the dissipation range propertiesof the turbulence (Ireland et al. b ). Indeed, while Ireland et al. (2016 b ) showedthat the particle clustering (measured by the Radial Distribution Function) becomesmonotonically stronger at progressively smaller scales for all St , in § St . This can occur because when St > O (1), the mechanismthat causes the particle clustering is not the centrifuge mechanism and the associatedpreferential sampling of the flow field discussed in Maxey (1987), but rather a non-localmechanism that does not depend upon the particle interaction with the local fluid velocity ultiscale preferential sweeping of particles settling in turbulence et al. et al. et al. St > O (1), the clusteringof settling particles may not be correlated with the local flow, and therefore analyzingthe particle settling velocities conditioned on the local particle concentration no longerprovides a meaningful test of the validity of the preferential sweeping mechanism. It isinteresting to note that Rosa & Pozorski (2017) observed that particle settling velocitiesincrease with increasing local particle concentration for low inertia particles, but anopposite and weaker trend was observed for high inertia particles. This reversal in trendwas attributed to the loitering effect and the ineffectiveness of the preferential sweepingmechanism for higher inertia particles. While this interpretation in terms of loitering maybe valid, the differing behavior observed for low and high St may be simply a reflectionof the fact that while the settling velocity conditioned on concentration is an appropriatemeasure of settling velocity enhancement for St ≪
1, it is not for St > O (1), as explainedbefore. 2.3. Multiscale insight
Having constructed an expression for h u z ( x p ( t ) , t ) i in (2.11) that is valid for arbitrary St , we now develop the result further in order to gain insight into the multiscale natureof the problem. To do this, we introduce the coarse-graining decompositions u z ( x , t ) = e u z ( x , t ) + u ′ z ( x , t ) and V = e V + V ′ , where e u z and e V denote the fields u z and V coarse-grained on the length scale ℓ c ( St ), while u ′ z ( x , t ) ≡ u z ( x , t ) − e u z ( x , t ) and V ′ ≡ V − e V are the “sub-grid” fields. Inserting these decompositions into (2.11) we obtain h u z ( x p ( t ) , t ) i = *e u z ( x , t ) exp − Z t (cid:16) ∇ · e V ( X ( s | x , t ) , s ) + ∇ · V ′ ( X ( s | x , t ) , s ) (cid:17) ds !+ u + * u ′ z ( x , t ) exp − Z t (cid:16) ∇ · e V ( X ( s | x , t ) , s ) + ∇ · V ′ ( X ( s | x , t ) , s ) (cid:17) ds !+ u . (2.16)We now consider Taylor Reynolds number R λ → ∞ , and choose the coarse-graininglength scale ℓ c ( St ) to be a function of St , i.e. ℓ c ( St ). To do this, we define the scale-dependent Stokes number St ℓ ≡ τ p /τ ℓ , where τ ℓ is the eddy-turnover timescale at scale ℓ . We then define ℓ c ( St ) through St ℓ c = γ , where γ is a constant such that γ ≪ ℓ > ℓ c ( St ) corresponds to flow scales at which the effects of theparticle inertia are negligibly small, and the effects of particle inertia are only felt atscales ℓ < ℓ c ( St ). Since τ ℓ is a non-decreasing function of ℓ in homogeneous turbulence,it follows that ℓ c ( St ) is a non-decreasing function of St . In order to illustrate more clearlythe connection between ℓ c and St , we may use K41 to derive their relationship for thecase where ℓ c lies in the inertial range. From K41 we have that for ℓ in the inertial range0 J. Tom and A. D. Bragg τ ℓ ∼ h ǫ i − / ℓ / , and then using the definition of ℓ c ( St ) we obtain ℓ c ( St ) ∼ η ( St/γ ) / .This then shows how as St is increased, ℓ c ( St ) also increases.According to the definition of ℓ c ( St ), the particle clustering at scales ℓ > ℓ c ( St ) isnegligible and therefore exp − Z t ∇ · e V ( X ( s | x , t ) , s ) ds ! ≈ , (2.17)such that the contribution in (2.16) associated with fluctuations of the particleconcentration field ϕ ( x , t ) arises only from the sub-grid contribution exp( − R t ∇ ·V ′ ( X ( s | x , t ) , s ) ds ). Further, since γ ≪
1, significant deviations of ∇ · V ′ from zerowill only occur at scales ℓ ≪ ℓ c ( St ), and therefore ∇ · V ′ should be uncorrelated with e u z ( x , t ), under the standard assumption that widely separated flow scales in turbulenceare uncorrelated. This assumption, together with (2.17), reduces (2.16) to the result h u z ( x p ( t ) , t ) i ≈ * u ′ z ( x , t ) exp − Z t ∇ · V ′ ( X ( s | x , t ) , s ) ds !+ u . (2.18)Since the RHS of this result only contains the sub-grid fields, it shows that the particlesettling speeds are not affected by every scale of the turbulent flow. Instead, only scalesof size ℓ < ℓ c ( St ) contribute to the enhanced settling due to turbulence. The physicalmechanism embedded in (2.18) is a multiscale version of the original preferential sweepingmechanism described by Maxey (1987) and Wang & Maxey (1993). In particular, accord-ing to (2.18), h u z ( x p ( t ) , t ) i > ℓ < ℓ c ( St ).2.4. Implications of result
A number of interesting and important implications and predictions follow from (2.18),which we now discuss.2.4.1.
The scales of motion that influence the particle settling speed
The result in (2.18) shows that the turbulent flow scales that contribute to the enhancedparticle settling speeds are those with size less than ℓ c ( St ), while scales of size greaterthan or equal to ℓ c ( St ) make a negligible contribution. Since ℓ c ( St ) is a non-decreasingfunction of St then it follows that increasingly larger scales contribute to the enhancedsettling speeds as St is increased.One important implication of this is that there cannot, on theoretical grounds, be anysingle flow scale that determines h u z ( x p ( t ) , t ) i . This is in contrast to previous work (e.g.Wang & Maxey 1993; Yang & Lei 1998; Good et al. et al. h u z ( x p ( t ) , t ) i is u ′ , whichis associated with the large scales of the flow. We would argue that the results in thoseprevious studies were strongly affected by the fact that the R λ values they consideredwere such that u ′ /u η was not very large.According to (2.18), the flow scales that determine h u z ( x p ( t ) , t ) i depend essentiallyupon St , and when St ≪
1, we expect ℓ c ( St ) = O ( η ) so that h u z ( x p ( t ) , t ) i depends on u η , not u ′ . On the other hand, when 1 ≪ St ≪ τ L /τ η (where τ L is the integral timescale ofthe flow), the velocity scale dominating h u z ( x p ( t ) , t ) i will correspond to those of inertialrange eddies that have velocities much greater than u η , but much less than u ′ (when R λ → ∞ ). ultiscale preferential sweeping of particles settling in turbulence U ( ℓ ) / u η ℓ/η L / η L / η ℓ c ( S t ) / η ℓ c ( S t ) / η R λ Figure 1: Plot to illustrate how ℓ c ( St ) determines the R λ dependence of h u z ( x p ( t ) , t ) i .The values of R λ shown correspond to R λ = [29 , , , Influence of R λ on the particle settling speed Another implication of (2.18) concerns the influence of R λ on h u z ( x p ( t ) , t ) i . Accordingto K41 (Pope 2000), the velocity scale associated with an eddy of size ℓ , i.e. U ( ℓ ), growsas U ( ℓ ) ∝ ℓ in the dissipation range, U ( ℓ ) ∝ ℓ / in the inertial range, and for ℓ > L we have U ( ℓ ) = u ′ , where L is the integral length scale of the flow. Further, using K41we also have U ( ℓ > L ) /u η ∝ R / λ . In figure 1, we plot U ( ℓ ) /u η for four different valuesof R λ (these curves are plotted using the curve fit for U ( ℓ ) in Zaichik & Alipchenkov(2009)), and we denote the integral length scales for these flows as L , L , L , L , where L < L < L < L (we assume η is the same for each flow for simplicity of thediscussion). Also indicated using dashed lines are two values of ℓ c ( St ) corresponding toStokes numbers St and St where St < St so that ℓ c ( St ) < ℓ c ( St ). An importantpoint is that when normalized by the Kolmogorov scales, U ( ℓ ) is independent of R λ when ℓ < L . Indeed a consequence of K41 is that when considering any two turbulent flows, U ( ℓ ) /u η will be independent of R λ up to scales where ℓ is of the order of the integrallengthscale of the flow that has the smallest R λ .Now, let us consider how h u z ( x p ( t ) , t ) i /u η would vary across these four R λ values for St and St . For St , since ℓ c ( St ) < L then h u z ( x p ( t ) , t ) i /u η should be independent of R λ since U ( ℓ ) /u η is the same for each of the flows until ℓ > L . On the other hand, for St , h u z ( x p ( t ) , t ) i /u η will differ for the smallest two values of R λ since ℓ c ( St ) > L . However,for the largest two values of R λ , h u z ( x p ( t ) , t ) i /u η should be the same since ℓ c ( St ) < L .In other words, the R λ dependence of h u z ( x p ( t ) , t ) i /u η for St will saturate once theintegral lengthscale of the flow, L ( R λ ), exceeds ℓ c ( St ).More generally, h u z ( x p ( t ) , t ) i /u η will only depend on R λ while ℓ c ( St ) > L ( R λ ). Forfinite St , ℓ c ( St ) is always finite, and therefore for R λ → ∞ , the R λ dependence of h u z ( x p ( t ) , t ) i /u η will always saturate for all finite St .2.4.3. Influence of Froude number on the scales governing particle settling speeds
Usually, preferential concentration at scale ℓ is said to be strongest when St ℓ = O (1).However, in the presence of gravity, this needs to be nuanced. In particular, we shouldinstead say that preferential concentration at scale ℓ is strongest when Stτ η = O ( T ℓ ),where T ℓ is the eddy turnover timescale seen by the particle , which depends upon St and2 J. Tom and A. D. Bragg
F r , and, like τ ℓ , is a non-decreasing function of ℓ . For a given St and ℓ , T ℓ decreaseswith decreasing F r , reflecting the fact that the faster a particle falls through the flow,the faster the fluid velocity along its trajectory will decorrelate. This then means thatfor a given St , as F r is decreased, one has to go to larger flow scales in order to observe
Stτ η = O ( T ℓ ). Now suppose we re-define ℓ c ( St ) using T ℓ instead of τ ℓ , by defining ℓ c ( St )through Stτ η /T ℓ c = γ , and again take γ ≪
1. With this definition we observe that forfixed St , ℓ c ( St ) increases with decreasing F r , i.e. as
F r is decreased, one has to go tolarger values of ℓ in order to satisfy Stτ η /T ℓ = γ . Through (2.18), this then means thatas F r is decreased, the scales contributing to the particle settling enhancement increase.
3. Testing the arguments using DNS data
We will test the theoretical arguments presented in the previous section using DNSdata spanning a range of R λ , St and F r .3.1.
DNS Details
Our DNS dataset is identical to that in Ireland et al. (2016 a , b ), and we therefore referthe reader to that paper for the details of the DNS. Here we just give a brief summary. Weperform a pseudo-spectral DNS of isotropic turbulence on a three-dimensional triperiodiccube of length L with N grid points. The Navier-Stokes equation for an incompressiblefluid was solved, with the pressure term eliminated using the divergence-free conditionfor the velocity field. ∇ · u = 0 (3.1) ∂ u ∂t + ω × u + ∇ Pρ f + u ! = ν ∇ u + f (3.2)where u is the fluid velocity, ω is the voticity, P is the pressure, ρ f is the fluid density, ν is the kinematic viscosity and f is a large-scale forcing term that is added to makethe flow field statistically stationary. Deterministic forcing was applied to wavenumberswith magnitude k = √
2. A detailed description of the numerical methods used can befound in Ireland et al. (2013). The gravity term in the Navier-Stokes equation is preciselycancelled by the mean pressure gradient and so it has no dynamical consequence on theturbulent flow field. When using periodic boundary conditions, particles can artificiallyre-encounter the same large eddy as they are periodically looped through the domain, ifthe time it takes the settling particles to traverse the distance L is smaller than the largeeddy turnover time, i.e if L /Stτ η g < O ( τ L ). In all of our simulations, the domain lengths L are chosen to satisfy L /Stτ η g > τ L , thereby suppressing the artificial periodicity effectsfor settling particles (a systematic study of this was presented in Ireland et al. (2016 b )).In order to analyze the same dynamical system we considered in the theory, we trackinertial particles governed by (2.1), where particle are acted upon by both gravityand linear drag force. We assume that the particle loading is dilute and hence interparticle interactions and two-way coupling can be neglected (Elghobashi & Truesdell1993; Sundaram & Collins 1997). An eight-point B-spline interpolation scheme (with C continuity) based on the algorithm in van Hinsberg et al. (2012) was used to compute thefluid velocity at the particle position, u ( x p ( t ) , t ). We consider R λ = 90 ,
230 and 398 (notethat we refer to the simulations with R λ = 224 , ,
237 nominally as having R λ ≈ ν ) andtwo F r that are representative of the conditions in clouds,
F r = 0 .
052 for a cumulus ultiscale preferential sweeping of particles settling in turbulence Simulation I II III IV V VI R λ
90 224 230 237 398 398
F r ∞ ∞ L π π π π π πN ν h ǫ i L L/η u ′ u ′ /u η τ L τ L /τ η k max η N p Table 1: Flow parameters in DNS of isotropic turbulence where all dimensionalparameters are in arbitrary units and all statistics are averaged over the duration ofthe run ( T ). R λ ≡ u ′ λ/ν ≡ k/ p / ν h ǫ i is the Taylor microscale Reynolds Number, F r is the Froude number, λ is the Taylor microscale, L is the domain size, N is the number ofgrid points in each direction, ν is the fluid kinematic viscosity, h ǫ i ≡ ν R κ max κ E ( κ ) dκ isthe mean turbulent kinetic energy dissipation rate, κ is the wavenumber in Fourier space, E is the energy spectrum, L ≡ (3 π/ k )( R κ max ( E ( κ ) /κ ) dκ ) is the integral length scale, η ≡ ( ν / h ǫ i ) / is the Kolmogorov length scale, u ′ ≡ p k/ k is the turbulent kinetic energy, u η ≡ ( h ǫ i ν ) / is the Kolmogorov velocityscale, τ L ≡ L/u ′ is the large-eddy turnover time, τ η ≡ p ν/ h ǫ i is the Kolmogorov timescale, k max = p N/ N p is the number ofparticles per Stokes number used in the simulation. The grid spacing is kept constant asthe domain size is increased, and so the small-scale resolution, k max η , is approximatelyconstant between the different domain sizes. The viscosity and forcing parameters werekept the same when increasing domain size, and thus both small-scale and large-scaleflow parameters are held approximately constant.clouds and F r = 0 . et al. b ). Fourteen different particle classes are simulated with St ∈ [0 , R λ = 237 and F r = 0 . et al. (2018), while the rest is from Ireland et al. (2016 a , b ).3.2. Testing Methodology
It would be very difficult to directly test (2.18) owing to the practical difficultyin constructing the field V ′ ( x , t ) (and a simple evolution equation for V ′ ( x , t ) is notavailable). However, the insights and predictions from the theoretical analysis can betested by computing h u ′ z ( x p ( t ) , t ) i for various coarse-graining length scales, and analyzinghow the coarse-graining affects the results for varying St, R λ and F r . Strictly speaking, h u ′ z ( x p ( t ) , t ) i does not actually correspond to (2.18), but rather corresponds to (2.18) with ∇ · V ′ replaced by ∇ · V . However, according to the arguments and definitions leadingto (2.18), these two quantities are asymptotically equivalent since the coarse-graininglength scale ℓ c is defined in such a way that ∇ · V ′ ≈ ∇ · V since | ∇ · e V | ≪ | ∇ · V ′ | .4 J. Tom and A. D. Bragg
Therefore, comparing h u ′ z ( x p ( t ) , t ) i with the implications and predictions following from(2.18) is appropriate.To take full advantage of our existing large-database on inertial particle motion inisotropic turbulence (see Ireland et al. a , b ), we compute h u ′ z ( x p ( t ) , t ) i from ourexisting DNS data via postprocessing. To do this we take our DNS data for u z ( x , t )and the particle positions x p ( t ) at a number of different times, for multiple St , R λ and F r . We then apply a sharp spectral cut-off at wavenumber k F to u z ( x , t ), and from thisobtain the sub-grid field through u ′ z ( x , t ) ≡ u z ( x , t ) − e u z ( x , t ) = X k k k >k F ˆ u z ( k , t ) e i k · x , (3.3)where here and throughout, f ( · ) denotes the coarse-grained field, while ( · ) ′ denotes thesub-grid field. We then interpolate u ′ z ( x , t ) to the positions of inertial particles x p ( t )using an eight-order B-spline interpolation scheme to obtain u ′ z ( x p ( t ) , t ). The values of u ′ z ( x p ( t ) , t ) are then averaged over all the particles (with a given St ) and over multipletimes to obtain h u ′ z ( x p ( t ) , t ) i . This process is then repeated for multiple k c in order toexamine the effect of the coarse-graining and how its effect depends on St, R λ and F r .In order to relate the spectral cut-off wavenumber k F to a physical space filtering scalewe define ℓ F ≡ π/k F (Eyink & Aluie 2009).By considering the results of h u ′ z ( x p ( t ) , t ) i for various St, ℓ F , F r and R λ and comparingthem with those for h u z ( x p ( t ) , t ) i , we can test the predictions of the theory regardingwhich flow scales contribute to h u z ( x p ( t ) , t ) i .
4. Results and discussion
The scales of motion that influence the particle settling speed
The theoretical analysis predicts that the range of scales that contribute to h u z ( x p ( t ) , t ) i should monotonically increase as St increases. To test this prediction,in figure 2, we plot the ratio h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i for various filtering length scales ℓ F , and various R λ . For St ≪ ℓ c ( St ) = O ( η ), and so filtering out scales ℓ F > O ( η ) would have little effect, since the particle settling speed is only affected by thescales less than ℓ c ( St ). Hence, for St ≪
1, we expect h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i ≈ St = O (1) we would have ℓ c ( St ) > O ( η ), and so filtering out scales ℓ F > O ( η ) would have a strong effect, since these scales make a strong contribution to theparticle settling speed. Hence, for St = O (1), we expect h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i ≪ ℓ F , h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i should decrease with increasing St , reflecting the fact that h u z ( x p ( t ) , t ) i is affectedby increasingly larger scales as St is increased. The results in figure 2 confirm thisprediction. They also reveal how sensitive h u z ( x p ( t ) , t ) i is to scales ℓ ≫ η , even when St = O (0 . h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i asa function of ℓ F /η for various St . For St . . ℓ F is increased, h u ′ z ( x p ( t ) , t ) i approaches a constant value, implying that there do indeed exist scalesbeyond a certain size that have a negligible effect on the particle settling velocity, aspredicted by the theory. However, for St = O (1), we do not see such an asymptote, andtheir settling velocity is significantly affected by the largest scales in the flow. In orderto understand why this is the case, we will now estimate ℓ c ( St ).Recall that in the derivation of our theoretical result we prescribed the parameter γ to have the asymptotic value γ ≪
1. However, we may estimate a value for γ from ultiscale preferential sweeping of particles settling in turbulence -1 St h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i ℓ F /η = 12 . R λ = 90 ℓ F /η = 31 . R λ = 90 ℓ F /η = 125 . R λ = 90 ℓ F /η = 12 . R λ = 230 ℓ F /η = 31 . R λ = 230 ℓ F /η = 125 . R λ = 230 ℓ F /η = 12 . R λ = 398 ℓ F /η = 31 . R λ = 398 ℓ F /η = 125 . R λ = 398 Figure 2: DNS results for h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i as a function of St , for F r = 0 . ℓ F /η . The circles correspond to R λ = 90, the triangles to R λ ≈ R λ = 398. The dashed lines correspond to ℓ F /η = 125 . ℓ F /η = 31 .
4, and the dotted lines to ℓ F /η = 12 . ℓ F /η h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i St = 0 . St = 0 . St = 0 . St = 0 . St = 0 . St = 0 . St = 1 Increasing St Figure 3: DNS data for h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i as a function of ℓ F /η for different St ,and F r = 0 . R λ = 398.our DNS. To do this, we note that for St = 0 . h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i ≈ ℓ F /η &
600 and hence ℓ c ( St = 0 . ≈ η . Then using the K41 result discussed earlier, ℓ c ( St ) ∼ η ( St/γ ) / and using ℓ c ( St = 0 . ≈ η , we obtain the estimate γ ≈ × − .Using this value, in figure 4 we plot ℓ c ( St ) ∼ η ( St/γ ) / , where we have taken R λ → ∞ so that the inertial range scaling ℓ c ( St ) ∼ η ( St/γ ) / applies for all St > .
05. Usingthis estimated behavior we find that for St = 1, ℓ c ( St ) /η ≈ . × . In our DNS at R λ = 398, the ratio of the integral length scale L to η is L/η = 4 . × which is twoorders of magnitude smaller than ℓ c ( St ) /η for St = 1. This then explains why in ourDNS we do not observe a saturation of h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i for St = O (1) as ℓ F isincreased. An important conclusion that follows from this is that while the settling speedof particles is dominated by a restricted range of scales, namely scales of size less than6 J. Tom and A. D. Bragg -1 St ℓ c ( S t ) / η ℓ c ( St ) /η ∝ St / Figure 4: Plot illustrating how ℓ c ( St ) /η grows with St . ℓ c ( St ), this range can actually be quite large. Indeed, figure 4, and the results in figure 3indicate that even for St = 0 . ℓ c ( St ) /η ≈ . × such that their settling speeds areaffected by scales much larger than those in the dissipation range.These findings also explain why in many previous numerical and experimental studies,it was found that h u z ( x p ( t ) , t ) i had a strong dependence on u ′ , the large scale fluidvelocity scale. We have argued that on theoretical grounds, h u z ( x p ( t ) , t ) i cannot becharacterized by a single flow scale since the range of scales contributing to h u z ( x p ( t ) , t ) i depend on St . However, in these previous works R λ was sufficiently small so thatthe particle settling speeds were significantly affected by all scales, and as a result h u z ( x p ( t ) , t ) i was found to have a strong dependence on u ′ . In contrast, in natural flowswhere R λ is much larger, this would not be the case. In the atmosphere, typical valuesare η = O ( mm ) and L = O (100 m ) (Shaw 2003; Grabowski & Wang 2013), and togetherwith the results in figure 4 this implies we would have, for example, ℓ c ( St = 0 . ≈ . m and ℓ c ( St = 1) ≈ m . Consequently, for St
1, the large scale fluid velocities in theatmosphere, characterized by u ′ , would play no role in the particle settling. Nevertheless,the estimate ℓ c ( St = 1) ≈ m shows that the range of atmospheric flow scales that maycontribute to the enhanced settling speeds due to turbulence is quite large. This meansthat for St = O (1), particle settling in the atmosphere may be strongly influenced bynon-ideal effects such as flow inhomogeneity, anisotropy, and stratification (noting thatthe Ozmidov scale is greater than or equal to O ( m ) in the atmosphere (Riley & Lindborg2012)).So far we have emphasized that as St is increased, larger scales contribute to theparticle settling since ℓ c ( St ) is a non-decreasing function of St . However, in reality,as St is increased, not only do larger scales contribute to the particle settling, butsmaller scales begin to contribute less. This is because the preferential sweeping effectat any scale is only effective when St ℓ ≪ St ℓ ≫
1. Let us define b ℓ c ( St ) as thescale below which the preferential sweeping mechanism is not effective, so that scales ℓ < b ℓ c ( St ) correspond to scales at which St ℓ ≫
1. Then, the scales at which thepreferential sweeping mechanism would operate are b ℓ c ( St ) ℓ < ℓ c ( St ), and both b ℓ c ( St )and ℓ c ( St ) are non-decreasing functions of St . While our theoretical analysis could beextended to also include the lower limit scale b ℓ c ( St ), we have chosen not to do so since itwould render the theoretical result much more complicated. Furthermore, our principle ultiscale preferential sweeping of particles settling in turbulence -1 St h u ′ z ( x p ( t ) , t ) i ℓ F /η = 12 . R λ = 90 ℓ F /η = 31 . R λ = 90 ℓ F /η = 125 . R λ = 90 ℓ F /η = ∞ , R λ = 90 ℓ F /η = 12 . R λ = 230 ℓ F /η = 31 . R λ = 230 ℓ F /η = 125 . R λ = 230 ℓ F /η = ∞ , R λ = 230 ℓ F /η = 12 . R λ = 398 ℓ F /η = 31 . R λ = 398 ℓ F /η = 125 . R λ = 398 ℓ F /η = ∞ , R λ = 398 Figure 5: DNS results for h u ′ z ( x p ( t ) , t ) i as a function of St , for F r = 0 . ℓ F /η . The circles correspond to R λ = 90, the triangles to R λ ≈ R λ = 398. The solid lines correspond to ℓ F /η = ∞ , the dashed lines to ℓ F /η = 125 .
6, the dash-dot lines to ℓ F /η = 31 .
4, and the dotted lines to ℓ F /η = 12 . ℓ c ( St ).Nevertheless, understanding the minimum flow scales affecting the settling process is keyto the development of particle SubGrid Scale (SGS) models for LES of particles settlingin turbulent flows. The effect of the smallest scales of the turbulence on particle settlingspeeds was investigated by Rosa & Pozorski (2017) using DNS and their results showedthat scales smaller than a certain size did not affect the particle settling speeds. Althoughtheir data does not provide enough information to determine exactly how this “cut-offscale” depends on St , their data is consistent with our theoretical prediction that b ℓ c ( St )is a non-decreasing function of St .4.2. Influence of R λ on the particle settling speed As discussed in § R λ is increased, the range of scales in the turbulent flowincreases. According to our theoretical analysis, when St ≪ ℓ c ( St ) is small enoughso that the particles are not influenced by the additional scales introduced by increasing R λ . Consequently, h u z ( x p ( t ) , t ) i should not vary with R λ for St ≪
1. However, as St is increased, so also does ℓ c ( St ), and for sufficiently large St , this allows the particlesto feel the effects of the additional flow scales introduced by increasing R λ . As a result, h u ′ z ( x p ( t ) , t ) i can depend on R λ as St is increased. The results in figure 5 confirm thispicture and show that without filtering (i.e. ℓ F /η = ∞ ), h u ′ z ( x p ( t ) , t ) i = h u z ( x p ( t ) , t ) i issignificantly enhanced with increasing R λ when St & .
2, whereas it is almost insensitiveto R λ when St . . ℓ F /η arefiltered out, so that the range of scales contained in u ′ z is the same for each R λ , the R λ dependence of h u ′ z ( x p ( t ) , t ) i is dramatically suppressed. Indeed, for St & . R λ is entirely suppressed for ℓ F /η values considered. This confirms that the strongeffect of R λ on h u z ( x p ( t ) , t ) i is principally due to the enhanced range of scales availablefor the particles to preferentially sample as R λ is increased. Recall that increasing R λ leads to two distinct effects, namely an increased range of flow scales, and enhanced8 J. Tom and A. D. Bragg
100 150 200 250 300 350 40010 R λ A ( R λ , S t ) / A ( R λ = , S t ) St = 0 . St = 0 . St = 0 . St = 0 . St = 0 . St = 1 Increasing St Figure 6: DNS results for A ( R λ , St ) / A ( R λ = 90 , St ) as a function of R λ , for various St ,and for F r = 0 . ℓ F (so that each flow of differing R λ has the same rangeof scales), the effect of enhanced intermittency still remains. It can be seen in figure 5that for ℓ F /η .
6, the curves collapse for St & .
4, while there is a residual effect of R λ for St . .
3, which must be due to intermittency. That the effect of intermittency isonly apparent for small St is consistent with previous works which show that the effectof flow intermittency on inertial particle motion in turbulence is mainly confined to small St , while larger St particles filter out the effects of intermittent fluctuations in the flow(Bec et al. et al. R λ more clearly, in figure 6, we plot A ( R λ , St ) / A ( R λ = 90 , St )where A ( R λ , St ) ≡ h u z ( x p ( t ) , t ) i /u η , as a function of R λ and for different values of St .In agreement with the theoretical analysis, as St is increased, A ( R λ , St ) / A ( R λ = 90 , St )becomes increasingly sensitive to R λ . The analysis leads us to expect that for any St ,the ratio A ( R λ , St ) / A ( R λ = 90 , St ) will eventually saturate at sufficiently large R λ andthe R λ at which saturation occurs would increase with St . Our data is consistent withthis, however, we do not have enough R λ data points to be conclusive, and data at larger R λ is required to observe the saturation for St = O (1). Again, this is because in orderto observe the saturation we must consider values of R λ for which ℓ c ( St ) < L ( R λ ), andour DNS does not satisfy this for R λ
398 and St = O (1).4.3. Influence of Froude number on the scales governing particle settling speeds
Another prediction of the theory is that for a given St , as F r is decreased, ℓ c ( St )increases meaning that larger scales become responsible for the behavior of h u z ( x p ( t ) , t ) i .In figure 7, we plot h u ′ z ( x p ( t ) , t ) i for F r = 0 . F r = 0 . R λ ≈ St & .
1, the results confirm the prediction, since they show that h u z ( x p ( t ) , t ) i ismore strongly affected by filtering as F r is decreased, which is equivalent to saying that h u z ( x p ( t ) , t ) i is affected by increasingly larger scales as F r is decreased.To show this more clearly, in figure 8 we plot the ratio h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i asa function of ℓ F /η , for different F r , St and for R λ ≈ F r is decreased, the ratio decreases for a given St and ℓ F /η . This confirms the prediction ofthe theory as it indicates that as F r is decreased, larger scales become responsible for the ultiscale preferential sweeping of particles settling in turbulence -1 St h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i ℓ F /η = 12 . F r = 0 . ℓ F /η = 31 . F r = 0 . ℓ F /η = 125 . F r = 0 . ℓ F /η = 12 . F r = 0 . ℓ F /η = 31 . F r = 0 . ℓ F /η = 125 . F r = 0 . Figure 7: DNS results for h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i as a function of St for F r = 0 . F r = 0 . R λ ≈ ℓ F /η =125 .
6, the dashed lines to ℓ F /η = 31 .
4, and the dash-dot lines to ℓ F /η = 12 . ℓ F /η h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i St = 0 . St = 0 . St = 0 . St = 0 . St = 0 . St = 0 . Figure 8: DNS data for h u ′ z ( x p ( t ) , t ) i / h u z ( x p ( t ) , t ) i as a function of ℓ F /η for different St ,and F r = 0 . F r = 0 .
052 (open symbols), and R λ ≈ et al. (2017). In particular, as the particles settle faster due to gravity, the scalesat which the clustering mechanisms (such as the preferential sampling of the filteredvelocity gradient field, see Bragg et al. (2015)) become active move to larger scales.4.4. Scale dependence of preferential sweeping
So far, we have tested the predictions following from our theoretical analysis regard-ing the scales that contribute to enhanced particles settling speeds in turbulence. Wenow turn to consider in more detail the multiscale nature of the preferential sweepingmechanism. Recall that the preferential sweeping mechanism involves the idea thatthe enhanced particle settling is associated with the tendency of inertial particles to0
J. Tom and A. D. Bragg -15 -10 -5 0 5 10 1510 -8 -6 -4 -2 ν e Q / h ǫ i P ( e Q , t ) St = 0 St = 0 . St = 0 . St = 1 (a) -15 -10 -5 0 5 10 1510 -8 -6 -4 -2 ν e Q / h ǫ i St = 0 St = 0 . St = 0 . St = 1 (b) -6 -4 -2 0 2 410 -10 -8 -6 -4 -2 ν e Q / h ǫ i P ( e Q , t ) St = 0 St = 0 . St = 0 . St = 1 (c) -6 -4 -2 0 2 410 -10 -8 -6 -4 -2 ν e Q / h ǫ i St = 0 St = 0 . St = 0 . St = 1 (d) Figure 9: DNS data for P ( e Q , t ) and different St . Plots (a),(b) are for ℓ F /η = 0, whileplots (c),(d) are for ℓ F /η = 66 .
1. Plots (a),(c) are for
F r = ∞ , while plots (b),(d) arefor F r = 0 . ℓ < ℓ c ( St ), such that the preferential sweeping mechanism is multiscalein general, and does not only involve the small scales as in the St ≪ Q p ( t ) ≡ S ( x p ( t ) , t ) − R ( x p ( t ) , t ), where S and R are the second invariantsof the strain-rate S ≡ ( ∇ u + ∇ u ⊤ ) / R ≡ ( ∇ u − ∇ u ⊤ ) / Q p ( t ) along fluid and inertial particle trajectoriesprovides a clear way to consider preferential sampling since their statistics can only differif the inertial particles are both non-uniformly distributed, and if their distribution iscorrelated to the local flow, i.e. if they exhibit preferential concentration. However, inorder to consider how the particles preferential sample the flow at different scales wemust instead consider the coarse-grained quantity e Q p ( t ) ≡ e S ( x p ( t ) , t ) − e R ( x p ( t ) , t ),where e S ≡ e S : e S , e R ≡ e R : e R , and e S , e R denote S , R coarse-grained on the scale ℓ F . ultiscale preferential sweeping of particles settling in turbulence -1 St h e Q p ( t ) i / q h [ e Q p ( t ) ] i − h e Q p ( t ) i ℓ F /η = 0 ℓ F /η = 66 . ℓ F /η = 157 ℓ F /η = 1257 ℓ F /η = 0 ℓ F /η = 66 . ℓ F /η = 157 ℓ F /η = 1257 Figure 10: DNS data for h e Q p ( t ) i / q h [ e Q p ( t )] i − h e Q p ( t ) i as a function of St for different ℓ F /η , for F r = ∞ (filled symbols) and F r = 0 .
052 (open symbols), and R λ = 398.In figure 9, we plot the PDF of e Q p ( t ), namely P ( e Q , t ) ≡ h δ ( e Q p ( t ) − e Q ) i , (4.1)where e Q is the sample-space variable. The results are shown for R λ = 398, F r = ∞ and F r = 0 . St and ℓ F /η . The results show, as expected, that therole of St is different at different scales, which is because the behavior of P ( e Q , t ) at anyscale depends upon St ℓ , not St . The results also show the strong effect of gravity on thepreferential sampling, which is to suppress it. For example, in figure 9 (c), correspondingto the no gravity case, we see that preferential sampling is strongest for St = 1. However,the results in in figure 9 (d) show that when gravity is active, the preferential samplingfor St = 1 is very weak. The suppression of preferential sampling due to gravity at anyscale is because as F r is decreased, the particles fall through the flow faster, whichin turn reduces the interaction time between the particles and flow eddies, therebycausing the centrifuging mechanism to be less efficient. We emphasize, however, thatthis does not mean that their clustering is diminished by gravity. Indeed it has beenshown using DNS that for St &
1, clustering is actually enhanced by gravity (Bec et al. et al. b ). This is a reflection of the distinction between clustering andpreferential concentration and the mechanisms responsible for each, as discussed in § § h e Q p ( t ) i / q h [ e Q p ( t )] i − h e Q p ( t ) i . For homogeneous turbulence, this quantity is zero when measured along the trajectoriesof particles that do not preferentially sample the flow. The results in figure 10 show thatwith or without gravity, the maximum value for this quantity weakens as ℓ F is increased.This then implies that the maximum preferential sampling decreases with increasingscale. As ℓ F is increased, we also see that the peak value of the curve shifts to larger St .2 J. Tom and A. D. Bragg -1 St h e Q p ( t ) i / q h [ e Q p ( t ) ] i − h e Q p ( t ) i ℓ F /η = 0 ℓ F /η = 157 ℓ F /η = 0 ℓ F /η = 157 ℓ F /η = 0 ℓ F /η = 157 Figure 11: DNS data for h e Q p ( t ) i / q h [ e Q p ( t )] i − h e Q p ( t ) i as a function of St for ℓ F /η = 0(filled symbols) and ℓ F /η = 157 (open symbols) at R λ ≈ F r = ∞ (blacksolid lines), F r = 0 . F r = 0 .
052 (blue dash-dot lines).This can be explained by noting that we would expect the preferential sampling at anyscale to be maximum for St ℓ = O (1), and as ℓ is increased, the value of St for which St ℓ = O (1) moves to larger St .The results in figure 10 show that gravity significantly suppresses the preferentialsampling at all scales, and the peak of the curves occurs at much lower St than in the nogravity case. This latter point can be understood by the fact that since the eddy turnovertimescale seen by the particle T ℓ decreases with decreasing F r (for fixed St ), then in orderto observe τ p /T ℓ = O (1) (at which one would expect the strongest preferential sampling)one has to go to smaller τ p than in the case without gravity.In figure 11, we show results corresponding to figure 10 but now for R λ ≈
230 and forthree different values of
F r in order to further check the trends based on
F r observedin figure 10. The results confirm the trends observed in figure 10, showing that as
F r is decreased, the preferential sampling is systematically suppressed, and the St value atwhich the preferential sampling is strongest shifts to smaller values.Finally, we pointed out earlier that results in Ireland et al. (2016 b ) showed that h u z ( x p ( t ) , t ) i > St > O (1) and F r ≪
1, even though for St > O (1)and F r ≪ ℓ F /η = 0 indeed show that for St > O (1) and F r = 0 . St and F r , h u z ( x p ( t ) , t ) i issignificantly positive. However, the results in figures 10 and 11 for ℓ F /η > St > O (1) and F r = 0 .
052 do notpreferentially sample the fluid velocity gradient field, they do preferentially sample thecoarse-grained fluid velocity gradient field, i.e. h e Q p ( t ) i / q h [ e Q p ( t )] i − h e Q p ( t ) i becomesfinite for St > O (1) and F r = 0 .
052 as ℓ F /η is increased. This confirms the picturepresented by our theoretical analysis that as St is increased, the scales responsible forthe enhanced particle settling via preferential sweeping become larger. ultiscale preferential sweeping of particles settling in turbulence
5. Conclusions
In this paper, we have considered the multiscale nature of the mechanism leading to theenhanced settling speeds of small, heavy particles in isotropic turbulence. The traditionalexplanation of Maxey (1987) is that enhanced particle settling in turbulence occursbecause inertial particles preferentially sample the fluid velocity gradient field, exhibitinga tendency to be swept around the downward moving side of vortices. This mechanism isknown as the preferential sweeping mechanism (Wang & Maxey 1993). However, recentresults have raised questions about the completeness of this explanation. Moreover, thereare several outstanding questions concerning the role of different turbulent flow scales onthe enhanced settling, and the role of the Taylor Reynolds number R λ . The theoreticalwork of Maxey (1987) is not able to answer these questions due to its restriction toparticles with Stokes numbers St ≪ St . The theory utilizes a decomposition ofthe ensemble averaging operator for the system that allows us to construct a result thatinvolves a particle velocity field that is well defined for all St , unlike the particle velocityfield of Maxey (1987) that is only valid for St ≪
1. Coarse-graining decompositions arealso used in the theory in order to provide insight into the role of different scales inthe turbulent flow on the particle settling speeds. Our theoretical result shows that theparticle settling speeds are only affected by scales of the flow with size ℓ < ℓ c ( St ), where ℓ c ( St ) is the lengthscale beyond which the effects of particle inertia are asymptoticallysmall. Since ℓ c ( St ) is a non-decreasing function of St , our theory shows shows that as St is increased, increasingly larger scales contribute to the enhanced particle settlingdue to the turbulence. In other words, the preferential sweeping mechanism operates onprogressively larger scales as St is increased. Several new insights and predictions followfrom our theoretical analysis, which were then tested using DNS data.First, our theoretical analysis predicts that the range of scales contributing to theenhanced particle settling depends upon St , and that as a result, there can be nosingle turbulent flow scale that characterizes the enhanced settling. This is contraryto several previous experimental and numerical works that claim on the basis of theirdata that the enhanced settling speeds depend on the r.m.s velocity of the turbulence u ′ (associated with the large scales of the flow). Therefore, even though previous studieshave pointed to certain aspects of the multiscale nature of the problem (e.g. that thesettling speeds depend on u ′ and τ η ), they concluded that the relevant turbulent velocityscale determining the particle settling speed is the same for any St . However, accordingto our analysis, the fluid velocity scale of the turbulence that dominates the settlingenhancement depend essentially on St . The DNS results confirmed this prediction,showing that as St is increased, progressively larger scales of the flow contribute tothe enhanced settling. However, while it is true that only scales with size ℓ < ℓ c ( St )contribute, our estimates show that ℓ c ( St ) is larger than might be expected, such thateven for St = O (0 . η contributeto the enhanced settling.Second, our theoretical analysis predicts that the settling velocity of the particle willonly be influenced by R λ when the integral length scale of the flow L , is smaller than ℓ c ( St ). Once L > ℓ c ( St ), the R λ dependence saturates because the particles are notaffected by the additional scales of the flow that are introduced by increasing R λ . When St ≪ ℓ c ( St ) is relatively small and so the particle settling speed should show a weakdependence on R λ . However, when St = O (1), ℓ c ( St ) can be large enough for the particlesto feel the effects of the additional flow scales introduced by increasing R λ . Our DNS4 J. Tom and A. D. Bragg results confirmed this prediction, and also provided evidence consistent with the ideathat for any St , the particle settling speeds become independent of R λ for sufficientlylarge R λ . Other DNS results also confirmed that the dominant effect of R λ on the particlesettling speeds is through the scale separation in the flow that increases with increasing R λ , rather than effects of intermittency. However, we did observe evidence of effects ofintermittency on the settling speeds for St . . St and R λ , as the Froudenumber F r is decreased, ℓ c ( St ) increases, such that the faster the particles settle,the larger the scales that contribute to their enhanced settling. This is essentially aconsequence of the fact that settling reduces the correlation timescale of the flow seenby the particles. Our DNS results confirmed this picture except for St . .
1, where theopposite behavior was observed in some cases. We are unsure as to the explanation forthis.Finally, we used our DNS data to examine the preferential sampling of the flow by theparticles at different scales. The preferential sampling of the flow is part of the preferentialsweeping mechanism, and our analysis suggests that the preferential sampling of the flowshould occur at different scales depending on St and F r . To examine this we computedthe statistics of the difference between the second invariants of the coarse-grained (atscale ℓ F ) strain-rate and rotation-rate tensors evaluated at the positions of the inertialparticles x p ( t ). The results showed that the strongest preferential sampling at any scale ℓ F is associated with increasingly larger St as ℓ F is increased. Moreover, for a given St ,there is an optimum range of scales where the preferential sampling is strongest. As F r is decreased, the preferential sampling is suppressed, which is again due to the fact thatsettling reduces the correlation timescale of the flow seen by the particles, reducing theability of local flow structures to modify the spatial distribution of the particles. When St = O (1) and F r ≪
1, the particles sample the fluid velocity gradient field uniformly,yet they exhibit enhanced settling speeds due to the turbulence. This observation, whichappears to contradict the traditional preferential sweeping mechanism, is explained byour theory as being due to the fact that for St = O (1) and F r ≪
1, the scales at whichthe preferential sweeping mechanism operate do not lie in the dissipation range, but atlarger scales. The DNS results confirm this since they show that while particles with St = O (1) and F r ≪ ℓ F outside the dissipation range.The authors wish to thank Mohammadreza Momenifar for providing some of the dataused in this paper, as well as routines for producing some of the plots. This workused the Extreme Science and Engineering Discovery Environment (XSEDE), whichis supported by National Science Foundation grant number ACI-1548562 (Towns et al. REFERENCESAliseda, A., Cartellier, A., Hainaux, F. & Lasheras, J. C.
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