Physics of the interac/ons between iner/al par/cles and ...
Transcript of Physics of the interac/ons between iner/al par/cles and ...
Physicsoftheinterac/onsbetweeniner/alpar/clesandturbulence
AlbertoAlisedaDepartmentofMechanicalEngineering
UniversityofWashington([email protected])
StokesFlowaroundasphericalpar/cle
θ
φ
r
U∞
With boundary conditions:
No slip condition at the surface of the sphere, r=R
Where is the axisymmetric Laplacian operator in spherical coordinates
StokesFlowaroundasphericalpar/cle:OseenCorrec/on
StokesFlowaroundasphereisbasedonaflawedassump/on:Iner/altermisnegligibleforRe=VR/ν<<1Forlargevaluesofr,theiner/altermisnotnegligible:Re*=Vr/ν≈1,thatisforr=R/Re,weneedtoregularizetheproblem.
θ
φ
r>>R è Re ≈ 1
U∞
R
TerminalVelocityofaSphericalPar/cleinS/llFluid
Vt
Drag=1/2ρfV2tCDπD2/6
Weight=mpg
ForStokesFlow,CD=24/Re:
!! =118!!!!!
(!!!!− 1)!
TerminalVelocityofaSphericalPar/cleinS/llFluid
Vt
Drag=1/2ρfV2tCDπD2/6
Weight=mpg
WithOseen’scorrec/on,CD=(24/Re)(1+(3/16)Re):
TerminalVelocityofaSphericalPar/cleinLaminar(Horizontal)Flow
Vt
Drag=1/2ρfV2tCDπD2/6
Weight=mpgUslip=U∞-Vpx
!!!!!"!" = −(!! −!!)!
+ (12!! 0− !!" !! − !! !!!!!/4!
!! =118!!!!!
(!!!!− 1)!
TerminalVelocityofaSphericalPar/cleinLaminar(Horizontal)Flow
Vt Uslip=U∞-Vpx
!!!!!"!" = 0!Forthevelocitytoreach
terminalvalue:
IfRe<<1andStokesflowapplies,thatisifthen:
“Equation of Motion of a small rigid spherical particle in a non-uniform flow”. Maxey, M. and Riley, J., Physics of Fluids, 1983
Particle dynamics in non-uniform flow: the modified Basset-Boussinesq-Oseen equation
Two Fluid Equations (Eulerian-Eulerian) vs Particle Tracking Equations (Eulerian-Lagrangian)
u is the unpertubed fluid velocityv is the actual fluid velocity influenced by the presence of the spherical particleY(t) is the position of the center of the sphere
u(x,y,z,t)
V(t)
v(x,y,z,t)
Faxen, H., Ark Mat. Astron. Fys, 1922 (in Maxey & Riley, PoF 1983)
Particle dynamics in non-uniform flow: the modified Basset-Boussinesq-Oseen equation
TwoFluidEqua/ons(Eulerian-Eulerian)vsPar+cleTrackingEqua+ons(Eulerian-Lagrangian)
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Faxen Forces introduce the effect of curvature in the underlying flow in the calculation of the stresses on the surface of the sphere
Particle dynamics in non-uniform flow: the modified Basset-Boussinesq-Oseen equation
Particle Tracking Equations (Eulerian-Lagrangian)
Equations of motion for the fluids are expressed in an Eulerian Frame: Cons. of mass, cons. of Momentum, no slip velocity on the surface of the sphere, velocity of the fluid (v) is equal to the unperturbed velocity of the fluid as the distance to the particle tends to infinity
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Particle dynamics in non-uniform flow: the modified Basset-Boussinesq-Oseen equation
Exact calculation of forces on the sphere, based on the PERTURBED flow, v, which is unknown. The key to this problem is to approximate the forces based on the UNPERTURBED flow velocity, u, which is known.
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Particle Tracking Equations (Eulerian-Lagrangian)
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Particle dynamics in non-uniform flow: the modified Basset-Boussinesq-Oseen equation
Two Fluid Equations (Eulerian-Eulerian) vs Particle Tracking Equations (Eulerian-Lagrangian)
The motion of the spherical particle is forced by:1. Buoyancy2. Unperturbed flow stresses on the surface of the spherical particle3. Added Mass4. Stokes Drag due to particle to unperturbed fluid velocity slip5. History or Basset force, due to the history of vorticity shed by the
particle in its wake
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1 2 3 4
5
0
€
d V
dt =ρ p −ρ f( )ρ p + 1
2ρ f( ) g + 3
2ρ f
ρ p + 12ρ f( )
D u Dt
+18 fdρ fν f
ρ p + 12ρ f( )⋅d2
u −
V & ' (
) * + +
CL2
ρ f
ρ p + 12ρ f( )
u − V &
' (
) * + × ω
+9ρ fν
12
π1
2 ρ p + 12ρ f( )d
1t−τ
ddτ u −
V & ' (
) * + dτ
0
t∫
Whenwegrouptheaddedmasstogetherwiththeledhand-side(forthepar/cleaccelera/on)andthefluidstresses(forthefluidaccelera/on)anddividetheMReqbythemassofthesphere,ρpπd3/6,plusthefluidaddedmass,1/2ρfπd3/6,weget:
Particle dynamics in non-uniform flow: the modified Basset-Boussinesq-Oseen equation
0+3
4
5
2+3
1
A note about the scales of turbulence
λ:Taylor’smicroscale
η:Kolmogorov’smicroscale
Richardson’s 1921 poem: “Big whirls have little whirls That feed on their velocity and little whirls have littler whirls and so on to viscosity”
48
0
0 0
V 0 0 - b 0
V m 0 D -
O 8 - 0
0
I I
103
10'
10'
E(K)I1 Eo
1 OD
lo-'
lo-? C
M . R. Wells and D . E . Stock
; 1
FIGURE 17. Longitudinal one-dimensional energy spectrum : 0, x / M = 15; 0, 30; 0 , 45; V, 60; 0 , 75; H, 90.
The longitudinal and lateral power spectra plotted in Kolmogoroff coordinates are shown in figures 17 and 18 respectively. The coordinates are non-dimensionalized using Eo = (eV5)k, k = 27cf/u and 7 = ($/€)a, where e and f are the turbulent eddy dissipation and frequency respectively. In the region of high wavenumber the spectral profiles were found to collapse to a single curve.
A check of the spectral estimates was made by comparing the results of two methods of calculating the eddy dissipation per unit mass. First assuming homogeneous turbulence and a constant mean velocity, the turbulent kinetic-energy equation reduces to (Hinze 1959, p. 75)
dq2 2 dt _- - - -€,
then making the transformation d - - d -- - u- dt dx
gives odp 2 dx €=- - - -
By assuming q2 = 3,d2, where pf2 = ur2 = d2 = wT2, e may be calculated by fitting q2 versus x- xo and differentiating. The dissipation may also be calculated directly from the measured spectral-energy function (Hinze 1959)
r m
Settling velocity and concentration distribution of particles 35 103
102
10'
100
lo-'
10-3
1 0-4
A
A
I I I I I I l l 1 I I I I I I l l 1 I I I I I I l l 1 I I I I I I T
10-3 10-2 lo-' loo 10'
417 FIGURE 2. The one-dimensional energy spectra for the simulated flows at various Re, under Kolmogorov scaling: 0, 323, Re, = 21; ., 483, Re, = 31; 0, 643, Re, = 43; e, 963, Re, = 62; A, 483 and use of the second forcing scheme, Re, = 30. The line is a curvefit for all the measured data taken from Comte-Bellot & Corrsin (1971) for a flow behind a 2-in grid.
Wells & Stock (1983), both of which studied the dispersion of heavy particles in a grid- generated turbulence, and found that all the experimental curves agree very well for ky 2 0.01. Such universality of the energy spectrum is reproduced by the simulations, although this is somewhat surprising as the flow Reynolds numbers are not high. The flow Reynolds numbers for all the above experimental measurements are comparable to those in our simulations. As a side note, George (1992) recently suggested a different scaling using h and u' for the energy spectrum rather than Kolmogorov scaling for self- preserving decaying turbulence. For our forced stationary turbulence, we found, however, the Kolmogorov scaling collapses data much better than the suggested Taylor scaling.
Finally, the velocity derivative skewness and flatness are shown in table 1. The velocity derivative skewness in our simulations is in the range -0.55 to -0.40, which compares very well with the experimental range -0.5 to - 0.3 for Re, < 100 (Van Atta & Antonia 1980). The flatness also agrees with the experimental value of about 4. In summary, although the simulated flows are contaminated by the artificial forcing and
A note about the scales of turbulence
L:integrallengthscale
η:Kolmogorov’smicroscale
A note about the scales of turbulenceThera/oofL(integrallengthscale)toη(Kolmogorov’smicroscale)dependsontheReynoldsnumber:L/η≈Reλ3/4TheresultsfromMaxeyandRileyarevalidwhen D/η<<1andRep=DVt/νf<1
The dynamics of heavy particles �in a homogeneous and isotropic turbulence�
depends on : - the droplet inertia, characterized by the Stokes number: where:
- the terminal velocity, characterized by the ratio: where:
VStokesUKolmogorov
VSt = τ p.g€
St =τpτk
τ p =(ρp ρair).dp
2
18ν air
τk =νε
$ % & ' ( ) 12
Uk = ν ⋅ ε( )14
Governingparameters
= 1.2 ms
= 12 ms
= 1.2 cm/s
= 3.5 cm/s
For a 10 micron particle in 0.1 m2/s3 air turbulence
Heavy Particles Bubbles
Particle dynamics in non-uniform flow: a model for the interaction of
particles with turbulence
g
€
dVdt
=u(Y ,t) −V (t)( )
τ p
€
ρ p
ρ f>1
Heavy particles are centrifuged away from the vortex core, and thus tend to accumulate on the outer zone of the eddies, in regions of low vorticity and high strain.
The inertial bias mechanism
Turbulence-inducediner/aldynamics:Preferen/alaccumula/on
Preferen/alaccumula/onbytheiner/albiasmechanism
1!c"# suggests that the regions with little or no particles areon the order of 1/10 the box length, making the size of these
regions on the order of 10$.
III. MODELING CONSIDERATIONS
A. Radial distribution function
Consider a canonical ensemble of systems, each of vol-
ume V , containing N indistinguishable particles of diameter,
%, and density, &p . For such an ensemble, the joint probabil-ity that each of the N particles lie within volumes dx1 cen-
tered at x1 ,. . . , through dxN centered at xN is defined as
P !N "!x1 ,. . . ,xN"dx1 .. .dxN , !7"
where the standard normalization applies, i.e.,
!V
¯!V
P !N "!x1 ,. . . ,xN"dx1¯dxN!1. !8"
The two-particle distribution function is then obtained by
integrating out the dependence on the remaining particles
P !2 "!x1 ,x2"'!V
¯!V
P !N "!x1 ,. . . ,xN"dx3¯dxN . !9"
The two-particle radial distribution function is then defined
as32,33
g!x1 ,x2"!N!N"1 "
n2P !2 "!x1 ,x2", !10"
where n'N/V . For a statistically homogeneous and isotro-
pic volume, particle positions can be expressed in terms of a
relative separation distance, r'"x1"x2", and P (2)(x1 ,x2) re-duces to P (2)(r)/V to give the working definition of g(r)
used in this study
g!r "!N!N"1 "
n2VP !2 "!r ". !11"
As the rdf is near unity for a uniformly distributed system, it
is convenient to define a residual rdf !rrdf" as
h!r "'g!r ""1. !12"
A physical interpretation of g(r) is the number of par-
ticle centers located in a spherical shell between r and r
#dr about a central particle divided by the expected number
of particles given a uniformly distributed particle field.
Based on the definition of the rdf shown in Eq. !11" and theintegral relationship given in Eq. !8", it is easy to show thatthe rrdf must satisfy the following integral constraint34
n!V
h!r "dr!"1. !13"
B. Parametric dependence
Isotropic turbulence is characterized by the fluid density,
&, kinematic viscosity, v , turbulence intensity U!, and ki-netic energy dissipation rate, (. In dimensionless terms, thisreduces to the turbulent Reynolds number, defined here in
terms of the Taylor microscale
Re)'U!2!15
v(. !14"
For a monodisperse suspension, the particle phase introduces
three additional variables, viz., the particle density &p , diam-eter %, and total number N. In terms of dimensionless vari-ables, these can be expressed as the volumetric loading *'+%3N/6V , nondimensional size parameter %̂'%/$ and
particle Stokes number St ,see Eq. !1"#. This implies that themost general form of the rdf in isotropic turbulence can be
expressed functionally as
g! r̂;Re) ,* ,%̂ ,St", !15"
where r̂'r/$ is the dimensionless independent variable and
the variables after the semicolon are the dimensionless pa-
rameters.
C. Simplifying assumptions
The large parameter space shown in Eq. !15" wouldmake it difficult to interpret and correlate the results from the
numerical simulations. It is, therefore, advantageous to con-
sider the sensitivity of the rdf to each of the parameters, and
search for simplifications where applicable.
FIG. 1. 2d slices of ghost-particle simulations at: !a" St!0.0; !b" St!0.2;!c" St!0.7; !d" St!1.0; !e" St!2.0; and, !f" St!4.0. Dots correspond toparticle center locations.
2533Phys. Fluids, Vol. 12, No. 10, October 2000 Effect of preferential concentration on turbulent collision rates
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75
M.R. Maxey. The motion of small spherical particles in a cellular flow field. Phys. Fluids,
30(4):1915–1928, 1987a.
M.R. Maxey. The gravitational settling of aerosol particles in homogeneous turbulence and
random flow fields. J. Fluid Mech., 174:441–465, 1987b.
M.R. Maxey and S. Corrsin. Gravitational settling of aerosol particles in randomly oriented
cellular flows. J. Atmospher. Sci., 43:1112–1134, 1986.
M.R. Maxey and J.J. Riley. Equation of motion for a small rigid sphere in a nonuniform
flow. Phys. Fluids, 26(4):883–889, 1983.
M. S. Mohamed and J. C. LaRue. The decay power law in grid-generated turbulence. J.
Fluid Mech., 219:195–214, 1990.
M.B. Pinsky and A.P. Khain. Turbulence e�ects on the collision kernel. I: Formation of
velocity deviations of drops falling within a turbulent three-dimensional flow. Q. J. R.
Meteorol. Soc., 123:1517–1542, 1997.
M.B. Pinsky, A.P. Khain, and M. Shapiro. Turbulence e�ects on droplet growth and size
distribution in clouds - A review. J. Aerosol Sci., 28:1177–1214, 1997.
M.B. Pinsky, A.P. Khain, and M. Shapiro. Collisions of small drops in a turbulent flow.
Part I: Collision e⇥ciency. problem formulation and preliminary results. J. Atmospher.
Sci., 56(15):2585–2600, 1999.
M.B. Pinsky, A.P. Khain, and M. Shapiro. Stochastic e�ects of cloud droplet hydrodynamic
interaction in a turbulent flow. 53(1-3):131–169, 2000.
M.B. Pinsky, M. Shapiro, and A.P. Khain. Collision e⇥ciency of drops in a wide range of
Reynolds numbers: E�ect of pressure on spectrum evolution. J. Atmospher. Sci., 58(17):
742–764, 2001.
S.B. Pope. Turbulent Flows. Cambridge University Press, Cambridge, 2006.
Walter C. Reade and Lance R. Collins. E�ect of preferential concentration on turbulent
collision rates. Physics of Fluids, 12(10):2530–2540, 2000.
Alterna/veviewontheiner/albiasmechanism:preferen/alaccumula/oninhighstrainregions
Eddies, stream^, and Convergence Zones 203
FIGURE 1 ~ . or E zones for slow reactions.
Showing how reacting species (A,B) tend to react in C zones for fast
FIGURE 1B. Showing how particles tend to concentrate in the streaming S zones.
“Demixing”ofheavypar/clesinaplanarmixinglayer
Wang,L.P.Maxey,M.Burton,T.D.andStock,D.E.(1992)Chao/cDynamicsofPar/cleDispersioninFluids.PhysicsofFluidsA4(8)
vonicity VOtTkiV
fluids
I I I , I / i-‘--i
particles
Fc-ssqj 2 $ 7
/d / i,7&-” I
FIG. 21. Evolution of vorticity distribution for the shear layer and material lines for fluids and heavy particles.
Chao/ctrajectoriesinasimplecellularflow
Wang,L.P.Maxey,M.Burton,T.D.andStock,D.E.(1992)Chao/cDynamicsofPar/cleDispersioninFluids.PhysicsofFluidsA4(8)
-40 -30 -20 -10 0 10 20 30 _ (a) Xl 7
b M t? ‘0 3-t
b
E(f) 7;
: 2
b e
“0 .+ P ;il 0 ii 4; 6 ;I
(b) f
FIG. 5. (a) A typica trajectory of a particle in the chaos region with St=5 and R=0.3. Initial particle location is (1.6,1.6). Dashed lines mark the cell boundaries. (b) The power spectra for the particle velocities in the x, and x2 directions for the same condition.
almost all the points have a positive average exponential growth rate. We expect that all the points will have a sim- ilar, single asymptotic growth rate when t-t CO, indicating that the long-term behavior is unique and independent of the initial location. Further evidence is presented later when the ergodicity of the system is discussed.
and u2 spectrum as expected. The broadband frequency contribution indicates the motion is chaotic. The power spectrum was obtained by taking a long time series of par- ticle velocity (with 2048X50 data points) at a sampling rate of 2.5 samples per time unit. The VFFTPK library on Cray-YMP at the Pittsburgh Supercomputing Center (PSC) was used to derive the Fourier coefficients.
A typical trajectory for the chaotic motion is shown in Figure 6 shows a typical particle trajectory when the Fig. 5. The parameters used in this calculation are shown motion is periodic. It will be referred to as case II and the in Fig. 3 as point I. This set of parameters will be referred parameters are shown as point II in the parameter space of to as case I in later discussions. The initial particle location Fig. 3. In this case the particle travels along a 45” zigzag ( 1.6,1.6) is near the center of the cell. The particle moves curve. The power spectrum has a strong peak at the fun- along an outward spiral curve for some initial time due to damental frequency and its harmonics. The period is about inertial (centrifugal) force. The long-time motion is very 8.64 time units. In contrast with the periodic motion of irregular. The centrifugal effect does not allow the particle fluid elements where the orbit is closed streamlines, the to stay in any cell for long. Actually the particle seldom periodic motion of the aerosol particles follows an open visits the center of the cell where the vorticity is strong. At trajectory. In other words, the particle inertia destroys times the particle cuts through the cell boundary and en- closed periodic orbits and in turn can generate open peri- ters a new cell. This process repeats nonperiodically. Also odic orbits. It is noted that by open periodic orbits we do shown are the power spectra of the particle’s velocity at not distinguish (y, +2m7~, y,+2nr) from (y,, y2), where long time. There is no difference between the u1 spectrum m and n are arbitrary integers.
(a) by------- x1
5-5 2 4 6
lb) f FIG. 6. (a) A typical periodic trajectory of a particle with St=5 and R=O.l. Initial particle location is (1.6,1.6). (b) The power spectra for the particle velocities in the x1 and x2 directions for the same condition.
1793 Phys. Fluids A, Vol. 4, No. 8, August 1992 Wang et al. 1793
computed and the spatial complexity of the ABC flow, no relation between diffusivity and fractal dimension was found.
Some preliminary effort has been made to study La- grangian motion of particles in fully turbulent flows and shear mixing layers using dynamical systems tools. Wang et a1.9 used a finite number of random Fourier modes to represent unsteady, three-dimensional turbulent flow and computed the fractal dimension for the attractor of heavy particles. They found that the fractal dimension increased linearly with increasing particle diffusivity for a wide range of particle inertia and settling velocities. This suggests that the notion of chaos may be useful for studying particle dispersion and mixing in real turbulence. GaiZin-Calvo and Lasheras” used a steady Stuart vortex solution of the Eul- erian equations to represent a plane, free-shear layer and showed that heavy particles can be suspended in the layer and have periodic, quasiperiodic, or chaotic orbits. When the motion is chaotic, the attractor in physical space has a finite height, indicating some degree of transverse disper- sion.
Most of the above studies are still of a preliminary nature. They raise many issues that need further, detailed investigation. Questions that come to mind include the fol- lowing: Why does the appearance of chaos give rise to dispersion’? Does the occurrence of chaos guarantee disper- sion? What measure based on chaotic dynamics is corre- lated to the dispersion coefficient? In this paper we address these issues by examining the Lagrangian motion of small, heavy particles in the steady two-dimensional cellular flow. Chaotic orbits are shown to exist and the resulting disper- sion of particle clouds are found to have many of the fea- tures of turbulent dispersion. Correlations between the fractal dimension, the effective dispersion rate, and the ef- fective mixing efficiency are explored. Finally particle dis- persion and mixing in an evolving free-shear layer is briefly discussed in a similar context.
II. AEROSOL PARTICLES IN A CELLULAR FLOW
In this section we examine the Lagrangian motion of aerosol particles in a cellular flow. Such a flow arises in thermal convection with free-slip boundaries and has also been used to represent the transport effects of small-scale turbulent eddies on a passive scalar field.” Although the same flow was considered by C&anti et a1.,5 the equation of motion for particles adopted here is more general than in their work and a different particle parameter range is con- sidered. Maxey and his co-workers, in a series of papers,W2,13 have studied the motion of small spherical and nonspherical particles in the same flow. Their main focus was on particle suspension and possible change in the particle mean settling rate. Our focus is on the chaotic motion of small spherical particles and the relationship between chaos, dispersion, and mixing.
A. Dynamical system
The steady, two-dimensional, and incompressible cel- lular flow is given in terms of streamfunction $ in fixed coordinates by
b.0000 3.i416 6.:
x1
FIG. 1. Streamlines in the cellular flow field. The increment in the values of streamfunction is 0.1.
4= sin x1 sin x2. (1) All the variables are assumed to be normalized by a char- acteristic length scale L and a velocity scale Ua of the flow.12 Typical streamlines are shown in Fig. 1. The flow extends with periodic repetition in both the x1 and x2 di- rections. The maximum flow velocity occurs on the cell boundaries, and there are stagnation points at the center of each cell and at the corners. The orbits for fluid elements are streamlines, thus the fluid motion for almost all initial conditions is periodic and regular with a period changing from 2~ near the center to infinity near cell boundaries. Equation ( 1) is a solution to the inviscid Euler equation for steady flow. The pressure is minimum at the center of the cell and maximum at the corners. The pressure gradi- ent provides the force for the rotational motion of the fluid.
We restrict our study to small spherical particles in the aerosol range as defined by Maxey,12 that is, the ratio of the particle density to the fluid density is larger than 2, pdpf> 2. The particle equation of motion in a steady flow is given in a nondimensional form by12
dv u[Y(t),tl --v dt St +Ru*Vu+; RveVu. (2)
In these equations, v(t) and y(t) are particle velocity and location; u(x,t) is Eulerian flow velocity field, and, for the cellular flow, it is independent oft. The terms on the right- hand side of Eq. (2) represent Stokes drag, the pressure gradient force on the particle (more generally, the fluid force on an equivalent element from the undisturbed flow field), and the added mass effect. The effect of gravitational settling is not considered in the present paper. (One may assume that gravity is aligned in the direction normal to the two-dimensional plane and thus does not affect the motion in the plane.) Equation (2) is a simplified version
1790 Phys. Fluids A, Vol. 4, No. 8, August 1992 Wang et al. 1790
Experimental Setup. �Heavy particles in isotropic turbulence.
Volume Fraction ranging from 8.10-6 to 8.10-5
0
0.1
0.2
0.3
0.4
0.5
0 10 20 30 40 50 60 70
Pdf(D)
Diameter (microns)
St=1
SMD
D10
Disperse Phase Characteristics D10 ≈ 10µm
D32 ≈ 25µm
Continuous Phase Characteristics Decaying homogeneous isotropic turbulence.
2000
2500
3000
3500
4000
4500
5000
1000 1200 1400 1600 1800 2000 2200
Linear decay of the turbulent intensity
(U/u')2
x (mm)
Linear regression. R=0.9908
10-7
10-6
10-5
0.0001
0.001
0.0001 0.001 0.01 0.1 1
Power Spectrum
E11
(m2/s)
k*eta
slope = -5/3
u'<U >
=v'
<U >≈ 2% ;Reλ =100;η = 200µm >> D32
2.5 mAtomizers
BlowerHoneycomb,
Grids
Flow
20 cmxSpray
g
Aliseda,A.,Hainaux,F.,Cartellier,A.,Lasheras,J.C.(2002)Effectofpreferen/alconcentra/onontheseslingvelocityofheavypar/clesinhomogeneousisotropicturbulence.J.FluidMech.468:77-105
Quantifying the Deviation from Randomness
0
0.005
0.01
0.015
0.02
0.05
0.1
0.15
0.2
0.25
10 100
2 4 6 8 10 30 50
D1
D2
Box size (pixels)
Box size (η)
D2 =σ −σ Rµ
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3 3.5 4
P(c/c0)
Random Distribution
c/co
Compute the pdf of the concentration in boxes of different
sizes
Comparing the difference with a random distribution a typical
length scale of the clusters is exhibited.
D1= Pc(C)− Pcr (C)( )∑
2
In the presence of gravity, settling particles are preferentially swept to the downward side of eddies, and thus spent more time in regions where the vertical velocity of the fluid is negative.
The preferential sweeping mechanism
g€
dVdt
=u(Y ,t)−V (t)+VSt( )
τ p
Turbulence-inducediner/aldynamics:Enhancedsesling
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Enhancedseslingpreferen/alsweepingmechanism
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Enhancedseslingpreferen/alinDirectNumericalSimula/onofHomogeneousIsotropicTurbulenceSettling velocity and concentration distribution of particles 45
(a) 0.6
0.5
0.4
@Vl> o.3 vk
0.2
0.1
0 1
0.04 i" 0 1 2 3 4
%/rk
FIGURE 7. The increase (<AV,)) in the particle mean settling velocity as a function of r,/rk for a fixed terminal velocity, W = u,. The error bar represents the statistical uncertainty. (a) Normalized by Kolmogorov velocity scale uk; (b) normalized by r.m.s. fluid velocity u'. 0, 323, Re, = 21; ., 483, Re, = 31 ; 0, 648, Re, = 43; 0, 963, Re, = 62; a, 4g8 and use of the second forcing scheme, Re, = 30. For the 323 simulations, the size of the symbol is made the same as the error bar.
value of 7,/7k at which (AY,)/v, is greatest is consistently within the interval of 0.5 < 7,/7, < 1 and is centred around 0.8 as the Reynolds number Re, is varied. By contrast, the ratio of T,/7, varies from 5.3 to 15.8, so that if the data were presented in terms of the ratio of 7, to T,, which is also representative of Lagrangian integral timescales, then there would be a much greater variation in the location of the peak increase with changing Re,.
These results are also shown in figure 7 where <A 4) is normalized by u'. We see that the quantitative difference among the simulated resulted for different Re, with the first
Wang,L.P.Maxey,M.(1993)Seslingvelocityandconcentra/ondistribu/onofheavypar/clesinhomogeneousisotropicturbulence.J.FluidMech.256:27-68
42 L.-P. Wang and M . R. Maxey
FIGURE 5. Normalized particle concentration (left-hand side) and flow scalar-vorticity field (right- hand side) on an (xl, x,)-slice (x, = 27L/48) at the 5 consecutive time frames. The first frame is at t = 0 when the concentration is uniform. The time interval is 0.018 or about twice the Kolmogorov timescale. The mesh size is 483, Re, = 31, and particle parameters are 7, = 7k, W = vk.
becomes non-uniform as time increases. We present in figure 5 the evolution of a typical normalized concentration field, C/ (C), and the scalar-vorticity field on a slice from the three-dimensional simulation box at five consecutive time frames during the initial development period. The normalized scalar-vorticity Q is defined as (wi wi)i/ ((wiwi);) , where wi is the flow vorticity. Throughout this paper, the grey scale is
42 L.-P. Wang and M . R. Maxey
FIGURE 5. Normalized particle concentration (left-hand side) and flow scalar-vorticity field (right- hand side) on an (xl, x,)-slice (x, = 27L/48) at the 5 consecutive time frames. The first frame is at t = 0 when the concentration is uniform. The time interval is 0.018 or about twice the Kolmogorov timescale. The mesh size is 483, Re, = 31, and particle parameters are 7, = 7k, W = vk.
becomes non-uniform as time increases. We present in figure 5 the evolution of a typical normalized concentration field, C/ (C), and the scalar-vorticity field on a slice from the three-dimensional simulation box at five consecutive time frames during the initial development period. The normalized scalar-vorticity Q is defined as (wi wi)i/ ((wiwi);) , where wi is the flow vorticity. Throughout this paper, the grey scale is
Interaction of inertial droplets with turbulence leads to: – Increased settling velocity
(preferential sweeping and collective behaviour/clustering)
– Increased local concentration (preferential accumulation)
EnhancedSeslingResults
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0 1 2 3 4 5
St=0.03
St=0.05
St=0.07
St=0.11
St=0.19
St=0.55
St=1.10
• The settling velocity of a particle inside a cluster can be up to 30% of u’ higher than outside a cluster (for a particle of St~1 that represents 200% of VSt).• The settling velocity increases linearly with the local concentration, independent of the St number.
Seslingvelocityenhancedbyclustering
<Vz(St |CC0) > − < Vz(St) >
u'
Local Concentration C/Co
Bubble dynamics
Preferential accumulation and sweeping also occur when bubbles interact with the vorticity field of a turbulent flow.
Because bubbles are less dense than the surrounding fluid, they are attracted to the vortex cores, where pressure is minimum.
BBO equation for bubbles:
dVdt
=u(Y, t)−V (t)+VSt( )
τ B+3Du
Dt
ρpρ f
<<1
Preferen/alsweepingforbubbles:stableequilibriumpoint
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Buoyancy
PressureGradient/AddedMass
Drag
Resultant
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Downloaded 21 Oct 2009 to 128.95.34.219. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp
Preferen/alsweeping:comparisonofbubbleandpar/clebehaviour
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1804 Phys. Fluids A, Vol. 4, No. 8, August 1992 Wang et al. 1804
Bubblesaccumulateneartheirequilibriumpoint(highvor/city)
HeavyPar/clesarecentrifugedawayfromvortexcoresandconcentrateinregionsofhighstrain)
Experimental Setup
Re1 m = 2- 6 105
Reθ = 0.7- 2 103
Initial Bubble Void fraction
Φ= 1 - 5 %
Reλ = 200
Bubble Stokes number
St = 0.1 – 3
Preferential accumulation of bubbles
The instantaneous concentration of bubbles was found to be highly non-random.The deviation from randomness was maximum at a length scale equal to 20 times the Kolmogorov length scale of the turbulence.
DC = (P(c)−PPoisson(c))2c=1
Nb∑
0 10 20 30 40 Size of the measuring window (in Kolmogorov lengths)
Rise velocity of the bubbles
-0.55
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
0.1 0.15 0.2 0.25 0.3 0.35 0.4
Vz-V
St/V
St
Stokes number
from PDPA measurements
-0.4
-0.3
-0.2
-0.1
0
0.1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Rise velocity. U=0.43, C=8.5 10- 4
Vz-V
t/u'
Vz-V
t/u' (lower bound)
Vz-V
t/u' (upper bound)
Vz-V
t/u'
Stokes number
from PDF evolution
Flow Visualization. �Horizontal Cut.
• Horizontal cuts show large inhomogeneities in the bubble concentration.• Significant vertical vorticity created by the mean horizontal shear is apparent.
Bubble clustering due to the turbulent structures
The preferred length scale for accumulation is of the order of 100 wall units (δ+ = ν / uτ )
324 A. Aliseda and J. C. Lasheras
0.02
0.03
0.04
0.05
0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0 100 200 300 400 500
Dsu
m =
Σ(P
– P
pois
son)
2
Dsi
gma
= (σ
– σ
pois
son)
/λ
Window size, [δ+]
Figure 19. Bubble accumulation as a function of the length scale. Indicators defined inequations (3.2) and (3.3). (i) Dsigma = (σ − σpoisson)/λ; (ii) Dsum = Σ(P − Ppoisson)2.
the turbulent structures present in the flow. The algorithm used to identify bubbleclustering by the turbulence consists of the following steps. First, each image is dividedinto square windows of a certain size. The number of bubbles within each of these non-overlapping windows is counted and recorded. With the information correspondingto all the windows covering every image taken under a given condition, we cancompute the density function for the probability of finding a number of bubblesP (nb) in each one of these windows. If the bubbles were passive scalars, their spatialconcentration due to the random stirring of the turbulence would correspond to aPoisson distribution.
Ppoisson(n) =e−λ λn
n!, (3.1)
where λ is the mean number of particles per window, Nb/Nw . By comparing theobserved probability density function with the theoretical Poisson PDF resulting froma purely random process, it is then possible to quantify how the bubble concentrationfield deviates from randomness. By repeating this algorithm for different windowsizes, the dependency of this deviation on the length scale can be determined. Twodifferent ways of comparing the PDFs were employed to quantify the extent of bubbleclustering in the boundary layer. Their definitions, due to Wang & Maxey (1993a)and Fessler, Kulick & Eaton (1994) respectively, are as follows:
Dsum =Nb∑
n=1
(P (nb) − Ppoisson(nb))2, (3.2)
Dsigma =σ − σpoisson
λ, (3.3)
where P (n) is the probability of finding n bubbles in a window, and σpoisson is thestandard deviation of the Poisson distribution.
The values of these quantities, computed for many different window sizes areplotted in figure 19. Both indicators of preferential accumulation reach a maximum
Velocity field.�PIV and Particle tracking measurements.
Two sets of measurements: �- 1. Streamwise Horizontal laser plane: U, V velocity components.�- 2. Streamwise Vertical laser plane: U, W velocity components.
1 2
Streamwise and Spanwise Velocity Profiles Re = 1920
The streamwise velocity profile U is typical of a turbulent boundary
layer.
The spanwise velocity profile V is that of a convergent flow.