Jun 15, 2000 - an optimum vortex Reynolds number such that the residence time of the particle is ...... Hunt, J. C. R., T. R. Auton, L. Sene, N.H. Thomas, and.
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. C6, PAGES 14,261-14,272, JUNE 15, 2000
Residence time of inertial particles in a vortex J. C. H. Fung Department of Mathematics, Hong Kong University of Scienceand Technology,Hong Kong
Abstract. The residencetime of a particle within a spatial domain is the total time it spends within this domain. We study the residencetime of small dense particles in a spreadingline vortex. Trajectorieshave been calculated for small denseparticlesreleasednear a spreadingline vortex. The resultsshowthat particles releasedfar away from the vortex center will not be trapped by it, but particles released
close to it will remain
there
for a considerable
time.
This
residence
time
is shownto be a functionof the particleinertia rp, its fall velocityVT, and the vortex'sReynoldsnumberRe,. The resultsof simulationsalsoshowthat there is an optimum vortex Reynolds number such that the residencetime of the particle is maximized within this simulated spreadingline vortex. These findings are expected to be applicable to the couplingbetween the small-scaleturbulent flow structures and the motion of the suspendedparticles.
1.
Introduction
Suspended particles occur in many natural turbulent flows; in the atmosphere the residence times of various aerosols,such as man-made pollutants and water droplets falling under gravity, determine how long these aerosolsremain suspendedin air. A turbulent flow is not completely incoherent, i.e., consistingonly
of thesevortical structuresis 3•/-5r/, where •/is the Kolmogorov length scale and their circulation F increases
with the vortex Reynoldsnumber Rev. This discovery has generated considerableexcitement in the turbulence community. One reasonfor this interest is that being strong and therefore presumablyalecoupledfrom the influence of other flow components, the behavior of the worms should be relatively easy to understand.
of white noise;localizedcoherentmotion(eddiesor vortices) exists in the turbulent flow. The most impor- Moreover,WangandMaxey[1993]showedthat the partant flow structures in connectionwith suspendedpar-
ticle are vortices[Yule,1980; Chungand Thoutt,1988; Fung, 1993, 1997, 1998; Chan and Fung, 1999; Davila
and Hunt, 2000]becausesuchvorticesare able to trap particlestemporarily and carry them along over considerable distances,eventually flinging them out as they start to decay. Recent large-scale numerical simulations of turbu-
lence,for examplethoseof VincentandMeneguzzi [1991] Ruetschand Maxey [1991],and others,indicatestrong organizedstructure in the form of long slendervortices ("worms")are presentamongthe small scalesof many turbulent flows. These vortices tend to be organized in tubes or ribbons of more or less elliptical crosssection. Their radii appear to scalewith the Kolmogorov microscale, and their lengths scale with the order of the integral scale of the flow. These vortex structures tend to persist over a long period of time, even when their length reachesthe integral scaleof the flow. These features have also been confirmed by direct numerical simulations of isotropic turbulence by Jim•nez et
al. [1993],who alsoshowedthat the typicalcoresize• Copyright2000 by the AmericanGeophysical Union. PaperNumber2000JC000260. 0148-0227/00/2000JC000260509.00
ticle motion, in particular, the settling rate of particles, dependsstrongly on the small-scalevortical structures of the turbulent flow field, which occupy a small fraction of the fluid volume and provide a small fraction of the dissipation. Their results demonstrate that the maximum increaseof particle settling rate occurswhen the inertial parameters are comparable to the flow Kolmogorovscales.This Kolmogorovscalingindicatesthat the small-scaleflow dynamics plays a significant role in certain aspects of particle transport in fully developed turbulence. The study indicates that the structural
view of turbulent
flows can contribute
to the de-
velopmentof multiphase flow modeling, in addition to the more commonly used statistical view. These smallscale vortical structures can be modeled by a Burgers
vortex [Jim•nez et al., 1993]or a spreadingline vortex [Vassilicos and Nikiforakis,1997],both of them incompressiblesolutions of the Navier-Stokes equation. Given the important role that the small scalesplay in the interaction between particles and turbulence and the prominence of vortex-like structures among these small scales,we investigatethe motion of small inertial particles in a spreadingline vortex here. The paper is organized as follows. In section 2 the decay of a line vortex is considered,which leads to the diffusion of a localized vorticity distribution or to a spreadingline vortex. In section3 the equation of mo-
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FUNG' RESIDENCE
TIME OF INERTIAL
PARTICLES IN A VORTEX
tion for an inertial particle is described. In section4
_ -
the simulation method and the results of the simula-
tion of particlemotionin a steadyline vortexand in an unsteadyspreadingline vortex are presented.
2. Spreading Line Vortex
I I -exp ---r02 ,
(2b)
2•rr
wherero- œ(t)/L is a functionof the initial coresize and viscosity(or vortexReynoldsnumber).The corresponding velocityfieldin Cartesiancoordinates becomes
We considera vortex filamentin an incompressible fluidin which,initially,the vorticityiszeroeverywhere,
-- Y [1-exp (-r•)], (3a)
ux -- -2•-r2
except on the axis of the vortex filament with small core
sizeœ0and with strengthF at t - O. Physically, this
_ x /.2 uy2•rr•. [1-exp(-r-•) ] . (3b)
means that for small times, there is a narrow vortex core near the axis, and the rest of the flow field is irro-
tational. Vorticity is here diffusedradiallyawayfrom Note that the center of vortex is chosen to be at the an initial concentration on a line, and we usecylindri- originand L=/F is the characteristic turnovertime of cal coordinates andtakethe z axisalongthe axisof the the vortex over a characteristiclength scaleL. filament. Let ur and u0 be the radial and circumfer-
In the caseof moderatevortex Reynoldsnumbersthe ential velocity componentsof the vortex, respectively; vorticity is diffusedradially away from the initial conthen ur = 0, and the vorticity is governedby centration;hencethe coresizeœ(t)of the vortexwill increasewith time. The velocitydistributionof a spreading line vortex with different core sizesat different valr•rr ' ues of t are shownin Figure I as a function of r. The where most obviousfeature is that as r increases,uo increases
0( Ot = " •(02( Or • +-
(la)
to a certain maximum
• = Or+ --'
(lb)
The boundaryconditionis uo(r -> oo,t > 0) -> 0. The solution[seeBatchelor,1967]of (1)is
•
value and then decreases to zero
as r --• oo. Note that for large valuesof r (>> r0) the velocity distribution has the same form at all t. The value r - rm for which the maximum uo can be found is given by the solution of
1 + 2p- ep,
1-exp-œ2(t )
wherep - (rm/ro)•'. Notealsothat the maximumvewhereœ2 (t) - œ•+ 4pt;œ(t)is definedasthe coresizeof locity uo is shiftedaway from the vortex centeras the
the vortex at time t with an initial size of œ0,and • is coresizeof the vortex (r0) increases. the kinematic viscosityof the fluid. The dimensionless In this paper we considerthe motion of a particlein
quantityF/p is knownas the vortexReynoldsnumber, a vort.ex flowin two cases:(1) a casewith a verylarge based on the circulation of the vortices. This vortex ReynoldsnumberwhereF/• >> 1 andœ(t)cantherefore ReynoldsnumberRev scalesapproximatelyas Rev , be takento be time-independent and (2) a casewith ,, •/2 [Jimdnez et al., 1993;Marcuet al., 1995] 20t•ex , differentmoderatevortex Reynoldsnumbersarrivedat where Rex is the Reynolds number formed with the Taylor microscale. Note that near r = 0 the motion is a rigid body rotation. The streamlinesare circles,and far away from the vortex, the velocity field decaysas
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1/r. The intensityof the vortex decreases with time as the core spreadsradially outward. In the limit of large vortex Reynoldsnumbers,where
0.1 0.75
F/v >> 1, advectionaroundthe vortex is so fast that a fluid element may complete many turns around the
vortexbeforeœ(t)changesappreciably.In this paperwe considerfirst the casewhereF/• >> 1 and cantherefore take a referencelengthL to be a time-independentcharacteristiclengthscaleof the vortexflow [seeTing and Klein, 1991]and F/L to be the characteristicvelocity. The nondimensional
B
flow field has the form o.oo
ur
--
O,
1 1- exp- œ2(t)/L2 27rr
(2a)
o.o
0.5
1.0
1.5
2.0
Figure 1. The velocity distribution associatedwith a spreadingline vortex of different coresizes. The number indicates the core size of the vortex.
FUNG:
RESIDENCE
TIME
OF INERTIAL
PARTICLES
by changingthe valuesof the viscosityv, hencethe core size of the vortex
xvill be a function
3. Equation of Motion
IN A VORTEX
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VT : 6•rat•F mpgL..• Wo F/L '
of time.
(6a)
where wo - mpg/6•ral• (the settlingvelocityin still
for Inertial
fluid) and the dimensionless inertia parameter
Particle
mpF
The motion of small spherical particles in a fluid is gOVt•rnoc] hy parti_c!einort.ia; gravity: and drag force
w0 F2
w0F
r.p= 6•ra/uL 2"'•F/LgLa= L2 '
(6b)
(from the relative motion betweenthe particle and the fluid). Assume that the particle concentrationsare which measuresthe dimensionlessresponsetime of the low, so that particle interactions can be neglected, and particle motion to changesin the velocity of the surthat the particles are sufficiently small relative to the rounding fluid. The importance of particle inertia in a smallestlengthscalein the turbulence(the Kolmogorov given situation is determinedby the value of rp: when lengthscaler/ - (v3/s)¥ lmmin theloweratmo- rp is small, the inertial term is small, and the particle respondsrapidly to changesin the surroundingflow sphere),so that the relative motion approachinga parfield, whileif the valueof rp is large,the particleinertia ticle is nearly a uniform flow. If, also, the particle time has a strong influence on the particle motion. Even if constantrp is short relative to the shortesttimescales we assume a force linear in velocity, solving the equa-
ofthevelocity field(r - (v/s)«• 0.08sin theatmo- tion of motion is still a complex problem becauseof the
sphere),then the fluid force on the particle is simply a nonlineardependence of U(Xp,t) on the particleposiStokesdrag force, and we can approximately detersnine tion. the equationof motion [Maxey,1987]as In this study, we find that particle inertia and the dv vortex Reynolds number have some important effect on mp • - 6•ral• [U(Xp, t) - v(t)]+ rapg, (4) the residencetime of the particle. Physically, the effect of particle inertia is to limit the responseof the particle whereu is the fluid flow field and rr•p,a, /• and g are to the rapid changesin the local fluid velocity and hence the particle mass,particle radius, fluid viscosity,and to produce a phase lag between the two.
gravitational acceleration constant, respectively. Asthe particle movesalong its trajectory Xp(/) in the flow fieldu(x, t), its velocitywill changecontinuously in response to the changes in the localfluidvelocityU(Xp,t). Equation (4) is a simplifiedversionof the more general equationof motion [Maxey and Riley, 1983; Hunt et al., 1988].Previousexperience indicatesthat Stokes draggivesqualitativelysimilarresultsassomenonlinear empirical draglaw [WangandMaxey,1993;Fung,1998] if the particle Reynoldsnumber is small. A Stokes drag is assumedhere. Despite the restrictionsthat havebeenimposed,(4) is applicableto many different problems,suchas aerosolsin gasesand small particles in water. (For typical solidparticlesin air, for exam-
4.
Simulation
Method
and
Results
The motion of a particle is found by numerically solv-
ing the equationof motion (5) usingthe fourth-order Runge-Kutta method with an adaptive step size control. This method will automatically adjust the step size to control the truncation error with a prescribed tolerance. For each simulation a flow field is generated as described in section 2. The particles are released
with initial velocityv(0) = (0, VT). Before discussingthe general results corresponding to the parametric study of the residence time of the particle inside the trapping region, which is a function
ple,VT•< 0.2ms-1, corresponding to particleradiiless of the threegoverningparameters•p, VT, and v, we will discusshow to definethe trapping regionof a spreading than 50/•m.) The equation is scaledby the characteristicvelocity line vortex and show that it is a function of VT and r0. and the characteristiclength scaleof the flow, namely,
F/L and L. Nondimensional variablesare introduced as follows'
4.1.
Particle
4.1.1.
X* X V* --
, Xp 1}*tL2 r'
Xp-, vL F'
uL U* ------. F
Motion
Particles
without
inertia.
In the limit
of a very large vortex Reynolds number the core size of the vortex can be regarded as a constant in time, i.e., œ(t)= œ0,and someusefulresultscan be obtained by consideringthe simplified casein which the particle
inertia tends to zero, i.e., rp --> O. In this casethe
Thescaled formof (4), withtheasterisks suppressed, is fluid drag forceon the particle balancesat every instant dv
Tp-•--U(Xp, t)--V(t)q-VT,
the force due to gravity; that is, the net force on the particle is zero. This implies that the right-hand side
(5) of (5) vanishes,that the particleis everywheremoving
wherethe two governingdimensionless parametersare vertically at its settling velocity relative to the ambient the scaleddimensionless Stokessettlingvelocityfor still fluid velocity,and that its trajectoryXp(t) : (x,y) is fluid determined by the solution of
14,264
FUNG'
RESIDENCE
TIME
OF INERTIAL
PARTICLES -IJ•Llzlll
-
t) + vr
IN A VORTEX i I I I I • I i I J I I i I•i
• III
I I I I I i tl i I i i i i i 1
1.00
0.75
and the specificationof its initial position. Therefore the velocity componentsof the particle are
•
dx dt
dy dt
=
ux
(8a)
=
Uy- VT.
(8b) • -0.2
Followingthe work of Maxey[1987],a "particlevelocity field" can be defined that is incompressibleand can be expressedin terms of a "particle stream function,"
•p - • +
-0.50
•
-0.75-
•
-1.00-
where •) is the stream function of fluid velocity. The trajectory of a particle is a curve along which •)p is a constant. However, in this case,• cannot be expressed analytically. Another useful way to view the motion of the particles is to follow the paths of a line of particles initially distributed uniformly far above the vortex with some particles releasedclosedto the center of the vortex. Figure 2 shows such particle trajectories. The obviousfeature is that for VT < max(u0), there is a possibility of permanent particle suspension,with particles executing closed orbits about a point on the x axis. Regions where particles exhibit endlessclosedorbits are called trapping regions. Any particle already in this region of closed orbits will stay there indefinitely, and any particle outside that region will never enter it. The trajectories more remote from the center of the vortex are not closed. Particles placed on one of these trajectories are rapidly swept through the vortex with no possibility of suspension. The largest closed orbit, or the strea'mlinewhich separatesthe region of particle trapping from the rest of the flow, is calledthe bounding
I•1 I I •'• I I I I I I I i i'• i i i Ii i i i i • i i i i i i i i i"• i i i i i i -0.50 0.00 0.50 1.00
Figure 2. The trajectoryfieldfor VT < max(u0). Thedashedlinerepresents the bounding trajectory;the crossrepresentsthe center of the vortex; B is the sad-
dle point;and C is the lowestpointof the bounding trajectory.
for x. Equation (9) has two parameters,the settling velocity VT and the core radius r0. Alternatively, the
solutionsto (9) canbe plottedversusuovalues(seeFigure 1). The differentvelocitydistributioncurvescorrespond to different values of core radius r0 of the vortex, as indicated. Such a velocity distribution curve inter-
sectedwith a constantu0 line yieldsthe x (or radial) positionof the equilibriumlocations(sincewheny: 0, u• = uo), of whichtherearenoneif VT > max(u0),one if VT = max(u0),or twoif Vr < max(u0).Notethat we are only interestedin the larger valueof the intersection point (the point B in the caseVr < max(uÜ))sincewe trajectory (the dashedline in Figure 2), and it starts want to find the largestclosedorbit, i.e., the bounding at the saddlepoint B on the particle trajectory [see trajectory. Maxey and Corrsin,1986]. A similarparticlepath in For illustration purposes,let us study the solutions a Rankine vortex was alsoobtainedby Nielsen[1984, for Vr = 0.3 and r0 : 0.1, in which case Vr < Figure 3]. In two dimensionsthe existenceof sucha max(u0) m 1.0. From Figure 1 one can seethat the trapping region containing closed trajectories implies curve of the velocity distribution with r0 = 0.1 interthe presenceof equilibrium points at which the local sectsline uo = 0.3 at two points (A and B). Hence flow velocity is equal and oppositeto the falling veloc- there are two solutionsfor x, from which one can nuity, i.e., U(Xp,t)q-VT : 0, SOthat a particlethere mericallycomputethe coordinatesof thesetwo equilibwould remain at rest. Therefore, by setting the particle rium pointsA and B using(9). Both the equilibrium velocityv(t): (0, 0) one can then solve(8) given (3) pointsA (locatedat smallradiusin the proximityof the for all the locations where equilibrium is achieved for centerof vortex)and B (locatedfar awayfrom centerof the particle, stably or not. From (8a) the y location vortex) are saddlepointsthat are unstable.Note that of the equilibrium points are always located at y = 0, boththesepointsare locatedat the upflowregionof the and from (8b) the corresponding x coordinatescan be vortex. The boundingtrajectory, which separatesthe obtained by numerically solving regionof particle trapping from the rest of the vortex (dashedlinein Figure2) startsat a saddlepointB. For i 1-exp--w -VT-O (9) smallervaluesof VT the trappingregionis larger(since x
r5
the pointB is fartherawayfromthe vortexcenter).In
FUNG: RESIDENCE TIME OF INERTIAL PARTICLES IN A VORTEX I
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Lv, versusthe coresizeg(t). From Figure4a onecansee
_
that despite the core size of the vortex increasingwith time (seeFigure 1) the sizeof trapping regiondecreases and shifts to the right with time. These closed trajectories are similar in many ways
1.0_
_
_
to those found by Nielsen [1984] in his study of os_
_
•:• 0.0
_
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-0.5-
_
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_
-1.0-
_
cillating flow over sand ripples. At certain times in the wave cycle a recirculating vortex forms behind the ripple, and sand particle streamlines in the vortex are
closed.The simplifieddrag-buoyancyequation(8) does not accountfor the drift of particles causedby inertia effectsand cannot predict the continuoustrapping of fresh particles by a vortex, which is observedin exper-
iments[Yule,1980]. To determinethe existenceof this trapping of particlesevenwith inertia, (5) with inertia effects must be used.
_
-0.5
0.0
0.5
1.0
1.5
In order to obtain a feeling for typical particle sizes that might be trapped by the stretchedBurgers-likeor spreading line vortices observed at the small scales of
Figure 3a. The size of the trapping regiondecreases turbulentflows,Marcu et al. [1995]obtaineda criterion as the settling velocity of the particle (VT) increases for the maximum radius of the particle, from 0.1 to 0.3 with g0 - 0.1. The crossrepresentsthe center of the vortex.
the limit Iq, tendsto zeroall particleswouldbe trapped and suspended insidethe vortex. Note that no trapping regionexistswhen V:r = max(u0). From (9) we can concludethat the shapeand size of the trappingregionof the vortex,Lv (definedasthe horizontallengthof the trappingregion(the dashedline in Figure 2) when y = 0) is a functionof the settling velocity V•- and the core size of the vortex r0. We will showthat the residencetime of the particle is a function
V/"yRex' I dtrap --107
(10)
that might be trapped. Here -y - pp/pf is the density ratio betweenthe particle and its surroundingfluid. For
a typical laboratoryexperiment[Browneet al., 1983], with Rex - 190 and V - 0.16 ram, (10) givesdtrap• 4 ttm for water dropletsin air with "7- 1000or dtrap• 71 pm for sand particles in water with -/ - 2.65. A
typicalatmospheric flow[Wyngaard, 1992],onthe other hand,is characterized by Rex ..• 104 and • - 1 mm, giving dtrap•0 3.5 ttm for coal particlesin air with -y833 and dtrap• 3.1 pm for water dropletsin air with
of V•- and r0 too. Similarly,the heightof the bounding ff - 1000.
trajectory, Ln, can also be defined as the vertical distance between the center of the vortex and the lowest
point of the bounding trajectory, i.e., the point C in
Figure2 (seealsoFigure8c). Figure3a showsthe shapesand sizesof the bounding trajectorieswith three differentvalues0.1, 0.2, and 0.3 for the settling velocity V•- of the particle and with a fixed value of the core sizeg(t) = g0 = 0.1. Figure 3b showsthe corresponding sizesof the trapping region, Lv, versusV•-, showingthat the size of the trapping regionbecomessmaller as VT increases(the point B movescloserto the centerof the vortex). The trapping regiondisappearsat Vr = max(u0).
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For a finite vortex ReynoldsnumberF/y the core sizeof the vortex increases with time sincethe vorticity is diffused radially outward. Therefore the size of the 0.00 ' ' ' ' I ' 0.00 0.25 0.50 0.75 1.00 trapping region changeswith time too for a fixed value of the settling velocity of the particle. Figure 4a shows the shapesand sizesof the trapping regionswith three Figure 3b. Plot of the sizeLv of the trapping region
differentvaluesfor the vortex coresizesg(t) = 0.1, 0.2, as a functionof VT corresponding to Figure 3a. Note and 0.3 and with a fixed value of V•- - 0.3. Figure 4b that when VT -- max(u0), the trapping regiondisapshowsthe correspondingsizesof the trapping region, pears.
14,266
FUNG:
RESIDENCE
TIME
OF INERTIAL
lllllll]lJlllljlllllillfllff]llllllllli
0.4
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region. However, when the particles are releasedinside the trapping region, they remain there for a considerable time and then slowly spiral outward and eventually
-
leavethe trapping region(Figures5b, 5d, and 5f). An
--
approximate analytical account of this processwithout particle inertia in a Rankine vortex was also given by
_
0.1
-
Nielsen[1984]. In all of thesecases,no new particles
• -
can enter the trapping region. It seemsperhaps counterintuitive that the higher the inertia of the particle,
' _ -
the quickerthe particle leavesthe trappingregion. This can be easily explained when we considerthe effect of particle inertia on movingobjects' particleswith inertia do not turn easily. The outward inertial centrifugal force acting normal to the trajectory is alwayslarger
•
thantheinward radialdragforce.Hence theparticle is
_
0.2 0.0
e(t) - 0.3
IN A VORTEX
_ __
0.3-
PARTICLES
-
_
_
driven by the flow away from the vortex center. Therefore one of the effectsof particle inertia is to destroy closedorbit, and this will likely have an important ef-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 fect on the residencetime of the particle. Figures5a, 5c, and 5e showthat outsidethe trapping Figure 4a. The sizeof the trappingregiondecreases region, there is a critical dividing trajectory; particles asthe coresizeof the vortex (g(t)) increases from 0.1 to released on one side of it pass on the downflowside of 0.3 with a fixed value of VT -- 0.3. The cross represents the center of the vortex. the vortex where the particle is being accelerated;particles releasedon the othersidepasson the upfiow side where the particle is being decelerated.The separation of particle trajectories on the downflowside of the vorAs far as the vortices observed at the small-scale
tur-
bulent air flow are concerned,only small particles with sizes of 4-71 •m can be trapped in a typical labora-
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tory experiment(Rex = 190), and thosewith sizes•4
/•m canbe trappedin atmospheric flow (Rex = 104). Therefore one would expect that only small particles with sizes -• 4 /•m might be trapped by this spreading line vortex, which is observedat the small scale of turbulent
0.70
0.60
0.50
flows. 0.40-
4.1.2.
Particles
with
inertia.
Without parti-
cle inertia and with VT < max(u0), particlescan be trappedin the trappingregionpermanently,that is the residencetime is infinitely long. In the presenceof particle inertia the motion of a particle is determined by (5). Sometypicaltrajectorieswith differentinertia pa-
rametersrp for a very largevortexReynoldsnumber Re• -+ c• (in this casethe coresizeof the vortexcan
0.30-
0.20-
0.10-
0.00
r[ 0.00
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be regardedas constantin time; hencethe trapping region is constantin time too) are shownin Figure 5. Figure 4b. Plot of the size Lv of the trapping region In none of these trajectories do we see a permanently againstg(t) corresponding to Figure4a. Note that the closedorbit correspondingto a trapping particle even trappingregionsuddenlyvanisheswhenthe coresizeof when the particles are initially located in the trapping the vortex increases to - 0.314.
Figure 5.
The particle trajectoriesobtained by solving(5) with fixed valuesof VT - 0.3 and
e(t) - e0 - 0.1 and with threedifferentvaluesof the inertiaparameters:(a) and (b) rp - 0.004, (c) and (d) rp - 0.04, and (e) and (f) rp - 0.4. The dashedline representsthe bounding trajectory and the crossrepresentsthe center of the vortex. Figures 5a, 5c, and 5e show the trajectoriesof particlesinitially releasedoutsidethe trapping region,and figures5b, 5d, and 5f showthose of particles initially releasedat the center of the vortex. Note that the trappir•g region is independent of the inertia of the particle.
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I
[
[
I
I
0.75
14,268
FUNG' RESIDENCE TIME OF INERTIAL PARTICLES IN A VORTEX I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
•
I
•
I
I
I_
1.00(
sizeof the trappingregiondecreases with time. Figure 6a showsthe trajectories of particles releasedfar away above a growingvortex center with • - 0.0007.
Thegeneral featureissimilarto thatofFigure5a. Figure 6b showsthe trajectory of a particle releasedat the centerof the samegrowingvortex. The particlenot only showsan outward spiral motion but alsoshiftsto the right sincethe trapping regionalso drifts to the
0.50
right and becomes smalleras time increases (seealso
0.00
Figure 4). There is also a slow drift to the left that Nielsen[1984]predicted.
Thesetrappingfeaturesarenotlimitedto thisspreadinglinevortex.Forexample, Nielsen[1984]andPerkins andHunt [1995]considered the velocityfieldof a Rankinevortexandshowed that thereareregionswherepar-
-0.50
ticlesfollowa closedtrajectory.
-1'00 t [
[
i
0.30_
i I [ -0.50
]
[
[ 'l [ 0.00
[
[
[
I [ 0.50
[
[
t
I [ 1.00
[
[
[
I [ 1.50
4.2.
Residence Time of the Particle
The residence time Tp of a particlecanbe definedas the total time for whichthe particleremainstrapped
i
(b)
insidethe trapping regionif releasedat the centerof a vortex. The sizeof the trappingregionis a functionof
0.20-
the particle properties and the size of the vortex. Our results show that it dependson the three dimension-
_
0.10-
lessparameters, particleinertia•-p,settlingvelocityof particleVT,andvortexReynolds numberRe In thissection wewillconsider theeffectof (1) varying thesettling velocity oftheparticlewhilekeeping •-pand
0.00
-0.10
-0.20
Re• constant, (2) varying •-pwhilekeeping Vr andRe•
-0.30
constant,and (3) varyingthe vortexReynoldsnumber whilekeeping•-pand Vr constant.
-0.40-
4.2.1. Fixed values of •-pand Rev. The residence timesoftheparticleTpinsidethetrappingregion
-0.50-
of the vortex havebeencomputedfor differentvalues
-0.60_
-0.70i
-0.25
[
'
[
I
0.00
•
•
[
'
I
0.25
]
[
[
[
I
0.50
'
]
[
]
I
0.75
Figure 6. The particletrajectorieswith fixedvalues of VT = 0.3, •-p= 0.004,œ0= 0.1, and • = 0.0007in a spreading vortexwithcoresizechanging withtime. The dashedlinerepresents the bounding trajectoryat time t - 0 with an initial coresizeœ0- 0.1, and the cross represents the centerof the vortex. (a) The particles are initiallyreleased outsidethe trappingregion,and (b) the particleis initiallyreleasedat the centerof the
of VT for Re• --• oo (• --• 0) and for threedifferent fixedvaluesof •-p(seeFigure7). The mostobvious feature is that the residence time decreases monotonically
lOOO
_
I I I I I , I I I I i I I I I I I I I I I I I I I I I I
_
500 f 00'004 .006 0.009 100_
_
5O_
vortex.
10-
rex is verymuchreducedasthey passroundthe vortex; simulationresultsshowthat particle accelerationsare large,and thereis considerable interparticleshear.Notice that there is a regionof spacebelowthe vortexthat
the particlescannotenter,and this regionof spacebecomeslargerasthe inertia of the particleincreases from 0.004 to 0.4.
r • [ , I [ [ [ ] i [ [ [ , i [ [ [ , i , [ [ [ i ] , [ ]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
v':,-,
FigureI. Linearlogplotof theresidence timeTp For a moderatevortex Reynoldsnumber the core of the particle versusVT with three different valuesof sizeof the vortexincreases with time. However,the inertialparameters•-p- 0.004,0.006,and 0.009.
FUNG'
RESIDENCE
TIME
OF INERTIAL
with VT. This is becausethe particle cuts through the vortex fasterfor larger valuesof ]fT. This monotonicde-
PARTICLES
IN A VORTEX
14,269
range. This situationcan be clearlyseenin Figure 8b with two particleinertia parametersof •-p= 0.008and
creaseis easilyanticipated fromthe resultis - g/wof•2 •-p- 0.0085.A particlewith •-p= 0.0085clearlyhasa Nielsen[1984•equation(57)]forthe spiralling timescale largerradiusof circulationthan onewith •-p= 0.008. of a particle in a solid core vortex with angular fre-
quencyf•. Also, note that the larger the value of (higherinertia),the smallerthe valueof Tp for all fixed valuesof VT sincethe centrifugalforceincreases(or the
particlespiralsout faster)as •-pincreases.The same simulations as above were also carried out with moder-
Illllllll]llllllllllllllll[111
o.o t (a)
15.0 \"\T pdecreases Tp
ate vortex Reynolds numbers (i.e., • • 0),and similar
resultsfor Tp werealsoobserved.
4 2 2. Fixedvalueof VTand/•½• Theinfluence of •-pon the residence time of the particleinsidethe
10.0 \
ß
rangesof •-pin Figure8a. For example,as •-pincreases from 0.0075to 0.0085,Arc(the dashedline) remainsun-
•.///•! ß
ß -
•.•
-
remains constant
ß
_
0.0
-
ß
5.0-_ N•decreases J' -' '
Re7 -• cx•(• -• 0) is alsoinvestigated(Figure8a) The residence time Tp (solidline) no longerdecreases monotonically as•-pincreases. Tpnowoscillates with •-p. This can be explainedby countingthe numberof turns, Arc,the particle makesor, equivalently,by countinghow many times the trajectory of the particle cuts acrossthe positivex axis beforeleavingthe trapping region. From Figure 8a we can see that whenever the residencetime Tp decreases, Arcdecreases too, but whenever the residencetime Tp increases,Arcremainsconstant. This can be explainedby looking at two typical
.
_
trapping region for a fixed value of VT -- 0.3 and for
_-
increases
•.i
-
, , , •/ •, , , i , , , , i , •, , i , , , •
0.004
0.005
0.006
0.007
0.008
0.009
0.01
rp
o.o- (b)
0.20..'"' ......... ............... .......................... 0.10
..'
0.00
'...
!
.-'
-0.10
changeat 5 while the residencetime increases.This is
becausethe particlespiralsout of the trappingregion faster as •-p increasesand at the sametime, the distancebetweenthe particletrajectorywith •-p- 0.0085 and the vortex center becomeslarger comparedwith that for •-p- 0.008, while both of them makethe same numberof turns (Arc- 5) insidethe trapping region beforethey escape.Note alsothat the largerthe radius of the circulationof the particle aroundthe vortex center, the longer it will take for the particle to complete
-0.20
-0.30 .... -0.40
-0.50
-0.60
.. '..
-0.70-•
-0.25
a turn (smallerspiralingrate) sincethe motionof the particle will be sloweras u0 decreases with increasing r (for r > rm). Thereforethe particlewill spendmore time insidethe trappingregionas •-pincreases in this
0.00
0.10-
(c) Two particle trajectorieswith two differentinertial
parameter•-pobtainedby solving(5) with fixedvalues of V2• = 0.3 and g(t) = g0 = 0.1 in a vortex. The particlesare releasedat the centerof the vortex, which is insidethe trapping region. The dashedline represents the boundingtrajectory and the crossrepresentsthe center of the vortex. In Figure 8b the solid line is for
•-p- 0.008,and the dottedline is for •-p- 0.0085. In Figure8c the solidline is for rp - 0.009,andthe dotted line is for •-p- 0.0085.
0.75
..................-...-......:
_
thealternative residence timeT; oftheparticle.(b)and
0.50
_
0.20-
Figure 8. (a) The residence time of the particle(solid line) and the number of turns Nc the particle makes insidethe trappingregion(dashedline) as functionsof •-pin a steadyvortex. The dash-dottedline represents
0.25
_
o.o o-
.•,
.,,.,'
'-..'..:,
.... ,•
.•
'...-.
:';'
".... •• '.
:1
::,
•,,
.... ...;
/ ,"
-0.10
-0.20-
-0.30ß
-0.40_
..
-0.50-
'...
_
".,
-0.60-
". _
;
-0.70-
-0.25
0.00
0.25
0.50
0.75
14,270
FUNG'
RESIDENCE
TIME
OF INERTIAL
However,asrp increases from 0.0085to 0.009,Nc decreasesfrom 5 to 4, and the residencetime decreases
too. This is becausethe particlewith rp = 0.009makes one lessturn, comparedto the one with rp = 0.0085 • inside the trapping region, before it escapes(seeFig- • ure 8c). One may ask whether this oscillation of residence time may be merely a result of our specificdefinition
of the residencetime sincewe use the trapping region of inertialessparticle for the residencetime calculation of finite inertia particles. Therefore an alternative defi-
PARTICLES
t(a) Tp increases 30.0111 IIlll IIllIIillIIIIIII•l'''•i
25.0
•"-•
_•./•'•\.
Tp decreases
•
-
20.0-
!
/.i-..,
_
• 15.0•
•
-
10.0-
,-•
Nc remains constar•t ß
_ .
.
. .
_
-.
ß
-
.
,,-.i
.
¸
rn
.
5.0
Nc decreases
nitionoftheresidence timeT; isthetimeit takesfora particle to settle a distance 1.5Lb below the vortex cen-
IN A VORTEX
.
.
.
0.0
ter (seeFigure8c), whereLh is the heightof the bound-
i
i
i
i
i
i
i
i
i
0.005
0.004
]
i
i
i
0.006
i
i
i
i
0.007
i
]
i
i
i
i
0.008
i
i
i
i
0.009
i
0.01
ing trajectory that was defined in section4.1. This definition is taken from Figure 8c, which showsthat the
larger inertia particle, after escapingfrom the trapping region,will moveup more beforeit settlespermanently.
In Figure8athecomputed alternative residence timeT; (dash-dottedline) of the particle is alsoplotted. It is clearthat a similaroscillatoryresultwith rp is obtained and that the locationsof the crest and trough occur at
moreorlessthoseofTp,butT; tendsto havelargervaluespurely becausethe particle will take a longertime to fall below 1.5Ln, comparedwith just getting out of the bounding trajectory. Therefore we can concludethat the oscillatorybehaviorof the residencetime Tp with
rp is genuineandnot an artifactof ourdefinitionof the residence time.
In the case of a moderate vortex Reynolds number
with • - 0.001 the relationshipsbetween•-p,the res-
idencetimeTp, andthe alternative residence timeT; are shown in the Figure 9a. Figure 9a showssimilar featuresto those of the steady vortex. The reasonfor the oscillatingresidencetime is similar to that of the
0.50_
0.20_
0.10_
0.00_
-0.10_
-0.20-
-0.50-0.40-
-0.50-
-0.60-0.70i
[
i
,
[
[
[
i
,
0.00
-0.25
[
i
[
,
]
0.25
[
[
,
,
,
0.50
[
0.75
steadyvortex: the particlewith the larger•-pmakesa slightly bigger spiral from the center of the vortex or makes one lessturn before it escapesfrom the trapping region(seeFigures9b and 9c). _
0.20-
.. - ......
0.10-
Figure 9. The residence time of the particle(solid line) and the numberof turns Nc the particlemakes insidethe trapping region (dashedline) versus•-p in a spreadingvortex with VT = 0.3, œ0 = 0.1, and • = 0.001. The dash-dotted line representsthe alter-
'
' / .:i" ;!
0.00-
''
..'
•% i•! • %
-0.10-
•
-..
'.
..
'-.........
\
.: .:.' ..':
..'
'...
..'
."
....'
..'
"-
.:
,,:
..'
.."
...'
..'
,' ....................
,½.4,-.'
'...
-0.20-
nativeresidence timeT; of the particle.(b) and(c) The correspondingtwo particle trajectories with two
-0.50 ........
differentinertialparameter•-pobtainedby solving(5)
-0.40
with fixed values of VT = 0.3, œ0= 0.1, and • .- 0.001
in a spreadingvortex. The particlesare releasedat the
-0.50
center of the vortex, which is insidethe trapping region. The dashedline representsthe boundingtrajectory,and the crossrepresentsthe centerof the vortex. In Figure
-0.60-
9b the solidline is for •-p= 0.008 and the dottedline is for rp = 0.0075. In Figure9c the solidline is for •'p= 0.008,andthe dottedlineis for •-p= 0.0085.
-0.70 [
-0.25
[
[
[
I
0.00
'
[
i
[
0.25
X
i
i
i
'
I
0.50
[
i
[
[
I
0.75
,
FUNG:
RESIDENCE
TIME
OF INERTIAL
i•illlllillli•lliitllilllJll]
5.
/'\ /t .."i il
;i
PARTICLES
IN A VORTEX
14,271
Conclusion
This paper deals with particle entrapment within ambient vortices and provides various characteristics for particle-vortex interaction, including the residence time. The motivation for this study came from recent achievementsin turbulence theory showingthat three-
\\ •\ • ...
•
..
iil • •••'"'• ........... ß •1
\
......
ent elongated vortices among the small scalesof many turbulent flows that live for quite a few Taylor scale• • %.... based eddy turnover times and that their scalesare of • .. ..... the order of Taylor microscale. These vortices are modeled as a spreading line vortex. Trajectories have been calculated for denseparticles 0.0 0.0005 0.001 0.0015 0.002 0.0025 0.003 releasednear a steady vortex and a spreading line vortex. These show that particles released well away from Figure 10. Theratiooftheresidence timeofthepar- the vortex will not be trapped by it, but particles reticle versusv in a spreadingline vortexwith VT = 0.3, leased closeto it will be temporarily trapped inside the œ0-- 0.1, and three differentvaluesof particleinertia rp- 0.003(solid), 0.006(dashed), and0.009(dotted). trapping region of the vortex. We have shownthat the size of the trapping region is .:
• ß
....
•
'"'..... ...............
determinedby the coreradiusof the vortex (the vortex Reynoldsnumber) and the settling velocity VF of the particle. Numerical simulation results show that as time 4.2.3. Changing the vortex Reynolds number. Of interest is the dependenceof the residencetime Tp increases,the vorticity is diffusedoutward, showingthat on the vortex Reynoldsnumber F/v. By varying the the core size of the vortex increases and the extent of valuesof the viscosityu of the fluid the vortex Reynolds the trapping region becomessmaller. Moreover, as VT numbers can be changed. The influence of the vortex increases,the trapping region also becomessmaller. We have also shown that the residence time of the Reynoldsnumber on the residencetime of the particle for a fixed value of VT = 0.3 and three different values particle dependson three parameters,namely,•-p,Vz, of rp is shownin Figure 10. RTp is the ratio of the and v. When varying one of the parameterswhile keepresidence time Tp(u) for a particularvalueof viscosity ing the other two fixed, different phenomena are observed.For fixed valuesof •-pand mthe residencetime to that of the steadycase(Tp(u-• 0)), i.e., of the particle decreasesmonotonicallyas Vz increases. This is becausewhen the force due to gravity increases, = the particle will get out of the trapping region faster. As u increasesfrom zero, all these curvesincreasefrom However, for the case with fixed values of Vz and m I to a maximum of around u = 0.0003. As u is increased the results show that the residencetime of the particle further, the residence time decreases. Some results of oscillateswith rp. this sort may be expected since as the vortex grows, For the caseof fixed valuesof rp and l/St an optithe sizeof the trappingregiondecreases with time (see mal vortex's Reynolds number is found such that the Figure4); onewouldexpectthat the particleshouldget residencetime of the particle is maximized in a spreadout of the trapping region faster. However, as the vortex ing line vortex. Of course,the length of the residence growsto a larger coresize,u0 decreases with time (see time is related to the interaction of particles with flow Figure 1) so that the rate of spirallingoutwardof the vorticity field, in particular, the lifetime of these inparticle decreasestoo. For small values of u < 0.001 the tense vortical regions. As the intensevortical regions results suggestthat the rate of decreaseof the trapping are a feature of the small-scale,dissipationrange flow region is smaller than the rate at which the particle is dynamics[Sheet al., 1990],the lengthof residence time spiralling outward; hence the residencetime increases. is likely to be longest when the particle parameters are If u increasesfurther, i.e., the vortex grows faster with comparableto the dissipationrange scales,namely, the tizne, the size of the trapping region decreasesfaster. Kolmogorovscale. This is consistentwith the finding Thereforethe particle will escapefrom the trapping re- of Wangand Maxey [1993]that the maximumincrease gion faster; hence the residencetime should decrease. of particle settling rate occurs when the particle paFigure 10 suggeststhat there is a critical value of vor- rametersare madecomparable to the flowKolmogorov tex Reynoldsnumber(about u = 0.0003) suchthat the scales.
residencetime of the particle can be maximized in the spreadingline vortex.
The resultsshowthat the small-scale flow dynamicsplay an importantrole in particleresuspension and
14,272
FUNG: RESIDENCE TIME OF INERTIAL
transport in multiphase flows. This is in contrast with the traditional approach that the large-scale, energycontainingfluid motionsdominate the transport of particles, in particular, the dispersionand suspensionprocesses. This does not invalidate the traditional approach but implies that the traditional approach may not be adequate for many aspects of multiphase flow processes.
Many questions remain to be explored. For example, the vortices are assumed to be strictly horizontal and two-dimensional in the present model, whereas in fact, the tube-like structuresare very rarely strictly horizontal, and the three-dimensionalityof the vortex may significantlymodify the trajectory of the particles and hence affect the residence time. Also, note that the coherent structures discussedin this paper pertain to
PARTICLES IN A VORTEX
velocity of particles in homogeneousisotropicturbulence, J. Geophys.Res., 105, 27,905-27,917, 1998. Hunt, J. C. R., T. R. Auton, L. Sene, N.H. Thomas, and R. Kowe, Bubble motions in large eddiesand turbulent flows, in Proceedingsof the Conferenceon Transient Phenomena in Multiphase Flow, edited by N.H. Afgan, Hemisphere, Bristol, England, U.K., 1988. Jim•nez, J., A. A. Wary, P. G. Saffman, and R. S. Rogallo, The structure of intense vorticity in isotropic turbulence, J. Fluid Mech., 255, 65-90, 1993.
Marcu, B., E. Meiburg, and R. K. Newton, Dynamics of heacy particles in a Burgurs vortex, Phys. Fluids, 7, 400410, 1995.
Maxey, M. R., The motion of small spherical particles in a cellular flow field, Phys. Fluids, $0, 1915-1928, 1987. Maxey, M. R., and S. Corrsin, The gravitational settling of aerosolparticles in randomly oriented cellular flow field, J. Atmos. $ci., J3, 1112-1134, 1986. Maxey, M. R., and J. J. Riley, Equation of motion for a a "normal" three-dimensional turbulent flow such that small rigid spherein a non-uniform flow, Phys. Fluids, 26 883-889, 1983. their scalesare of the order of Taylor microscale,i.e., Nielsen, P., On the motion of suspended sand particles, too small to be resolvedgenerally in geophysicalsimJ. GeophysRes., 89, 616-626, 1984. ulations. The vortices treated in this paper are not Perkins, R. J., and J. C. R. Hunt, The interaction between the relatively large structures known to geophysicists heavy particles and an isolated vortex, paper presentedat 2nd International Conferenceon Multiphase Flow, Kyoto, as "eddies," and this paper should not be overextended Japan, 1995. to eddies-inducedsediment transport and related resiRuetsch, G. R., and M. R. Maxey, Small-scalefeature of vordencetime. However,the presentfindingsare expected ticity and passsivescalar fields in homogeneousisotropic to be applicableto the couplingbetweenthe small scales turbulence, Phys. Fluids, 3, 1587-1597, 1991. of a turbulent flow, which can be modeled as a collec- She, Z. S., J. Jackson,and S. A. Orszag,Intermittent vortex structures in homogeneousisotropic turbulence, Nature, tion of a vortex tube and the motion of inertial particles $JJ, 266-268, 1990. suspendedin such a flow. Ting, L., and R. Klein, IZiscous IZorticaI Flows: Lecture Notes in Physics, vol. 374. Springer-Verlag,New York, Acknowledgments. Special thanks are extended to 1991. Miss W.Y. Ng and Mr. W.C. Wong who assistedwith the Vassilicos,J. C., and J. C. H. Fung, The self-similar topol-
early phasesof this study. The support of the Hong Kong ResearchGrant Council is gratefully acknowledged. References
ogy of passive interfaces advected by two-dimensional turbulent-like flows, Phys. Fluids, 7, 1970-1998, 1995. Vassilicos,J. C., and N. Nikiforakis, Flamelet-vortex interaction and the Gibson scale, Combust. Flame, 109, 293-302, 1997.
Batchelor,G. K., An Introductionto Fluid Dynamics,615 pp.. Vincent, A., and M. Meneguzzi, The spatial structure and Cambridge Univ. Press, 1967. statistical properties of homogeneousturbulence, J. Fluid Browne, L. W. B., R. A. Antonia, and N. Rajagopalan, The Mech., 225, 1-25, 1991. spatial derivative of temperature in a turbulent flow and Wang, L. P., and M. R. Maxey, Settling velocity and conTaylor's hypothesis, Phys. Fluids , 26, 1222-1227, 1983. centration distribution of heavy particlesin homogeneous Chan, C. C., and J. C. H. Fung, The change in settling isotropic turbulence, J. Fluid Mech., 256, 27-68, 1993. velocity of inertial particles in cellular flow, Fluid Dyn. Wyngaard, J. C., Atmosphericturbulence, Annu. Rev. Fluid Res., 25, 257-273, 1999. Chung, J. N., and T. R. Thoutt, Simulation of particle dispersion in an axisymmetric jet, J. Fluid Mech., 186, 199222, 1988. Davila, J., and J. C. R. Hunt, Settling of particles near vortices and in turbulence, J. Fluid Mech., in press, 2000. Fung, J. C. H., Gravitational settling of particles and bubbles in homogeneousturbulence, J. Geophys. Res., 98, 20,287-20,297, 1993.
Mech., 2•, 205-233, 1992. Yule, A. J., Investigations of eddy coherencein jet flows, in The Role of Coherent Structures in Modeling Turbulenceand Mixing, edited J. Jinenez, pp. 188-207, SpringerVerlag, New York, 1980.
J. Fung, Department of Mathematics, Hong Kong Universityof Scienceand Technology,Clear Water Bay, Hong
Fung,J. C. H., Gravitational settlingofsmallspherical par- Kong. ticlesin unsteady cellularflowfields,J. Aerosol $ci.,28,
753-787, 1997. (ReceivedNovember9, 1998; revisedJuly 5, 1999;; Fung, J. C. H., The effect of nonlineardrag on the settling acceptedNovember 24, 1999.)
