Journal of Colloid and Interface Science285(2005)
性感诱惑167–178
/locate/jcis CFD simulation of shear-induced aggregation and breakage in turbulent
Taylor–Couetteflow
Liguang Wang∗,R.Dennis Vigil,Rodney O.Fox
Department of Chemical Engineering,Iowa State University,2114Sweeney Hall,Ames,IA50011-2230,USA
Received15August2004;accepted27October2004
Available online23December2004
Abstract
An experimental and computational investigation of the effects of localfluid shear rate on the aggregation and breakage of∼10µm latex spheres suspended in an aqueous solution undergoing turbulent Taylor–Couetteflow was carried out.First,computationalfluid dynamics (CFD)simulations were performed and theflowfield predictions were validated with data from particle image velocimetry experiments. Subquently,the quadrature method of moments(QMOM)was implemented into the CFD code to obtain predictions for mean particle size that account for the effects of local shear rate on the aggregation and breakage.The predictions were then compared with experimental data for latex sphere aggregates(using an in situ optical imaging method).Excellent agreement between the CFD-QMOM and experimental results was obrved for two Reynolds numbers in the turbulent-flow regime.
2004Elvier Inc.All rights rerved.
Keywords:Computationalfluid dynamics;Aggregation;Breakage;Taylor–Couetteflow;QMOM
1.Introduction
Aggregation and breakage occur in many process such
as precipitation,crystallization,paration,and reaction in
multiphaflow.Due to the difficulties in describing the evo-
lution of the particle size distribution(PSD)and the incom-
plete understanding of the aggregation and breakage mech-
anism,including the role of hydrodynamics,simulation of
such process is complicated.In our previous paper[1],
computationalfluid dynamics(CFD)was coupled with the
population balance equations(PBE)to describe the evolu-
tion of the particulate pha and this approach was success-
ful in simulating the aggregation and breakage process in
我和书的故事400字laminar Taylor–Couetteflow.The success of the method is
primarily due to the ability of CFD to obtain the shear rate
field which is important to determine local aggregation and
breakage rates.However,it should be pointed out that the
spatial inhomogeneity of the shear rate is often neglected in
*Corresponding author.
E-mail address:lgwang@iastate.edu(L.Wang).
the investigation of shear-induced aggregation and breakage
process[2–5].As shown in[1],significant error is likely
introduced when only the spatial average shear is ud in
calculating aggregation and breakage rates.
Several methods have been developed to solve popula-
tion balances which consist of ts of nonlinear integro-
differential equations[6].The discretized population bal-
ance(DPB)approach is bad on the discretization of the
internal ,particle size)[7–9].The main ad-
vantage of the DPB method is that the PSD is calculated
directly.However,a large number of scalars are usually re-
quired to get reasonably accurate results.Therefore,it is
computationally prohibitive to implement the DPB method
in a detailed CFD code.Another method to solve PBEs is the
Monte Carlo method[10].This stochastic approach is also
not computationally tractable within a CFD framework be-
cau of the large number of scalars required.More recently,
the quadrature method of moments(QMOM)has been ud
to solve the PBE for particle aggregation and breakage phe-
nomena[11].The closure problem arising from the aggre-
gation and breakage terms for the conventional method of
0021-9797/$–e front matter 2004Elvier Inc.All rights rerved.
doi:10.1016/j.jcis.2004.10.075
168L.Wang et al./Journal of Colloid and Interface Science 285(2005)167–178
moments [12]has been successfully solved with the u of a quadrature approximation,where weights and abscissas of the quadrature approximation can be found by the product-difference algorithm [13].The QMOM approach requires only a small number of scalars (i.e.,4–6lower order mo-ments of the PSD)and hence is suitable for u with CFD.A Taylor–Couette device consists of two concentric cylin-ders.The hydrodynamic instabilities inside the gap between the two cylinders have been studied for more than a cen-tury [14].For a fixed outer cylinder and a Newtonian work-ing fluid,it is well known that as the angular velocity of the inner cylinder increas from rest,the flow undergoes a ries of transitions that give ri to flow states including laminar vortex flow,wavy vortex flow,modulated wavy vor-tex flow,and turbulent vortex flow.The flow transitions occur at specific values of the azimuthal Reynolds numbers,Re ,defined as (1)Re =
ωr i d
ν
,
where ωis the inner cylinder angular velocity,r i is the radius of the inner cylinder,d =r 0−r i is the annular gap width,and νis the kinematic viscosity.The critical azimuthal Reynolds number at which Taylor instability occurs,Re c ,depends upon the specific geometry (i.e.,cylinder radii and length)of the flow device ud [15].Conquently,it is convenient to define an azimuthal Reynolds number ratio R (=Re /Re c )to parameterize the flow.As were found in [16],the flow is in the laminar vortex flow regime when 1<R 5.5,and in the wavy vortex flow regime when 5.5<R 18,and in the turbulent vortex flow regime when R >18for our apparatus.We have also found that in the wavy vortex flow regime,the local shear rate varies with time due to az-imuthal waves prent in the flow and this phenomena must be taken into account in the simulation of aggregation and breakage process [1].Therefore,turbulent intermittence (causing fluctuations in ε,hence in the local shear rate)is accounted for in this study when the flow is in the turbulent vortex flow regime.
Although a comprehensive understanding of the physical mechanisms governing particle aggregation and breakup in a sheared suspension is lacking,the incorporation of local shear and particle morphology into PBEs result in signifi-cant improvements in the prediction of particle size distrib-utions [1].Since the CFD and PBE approach was successful in the simple laminar Taylor–Couette flow in [1],one im-mediately wonders if the same approach can be extended to more comple
x turbulent Taylor–Couette flow.Moreover,becau most industrial process are operated in turbulent flow,this is a question with significant practical implications.Hence in this work,we attempt to answer the question by us-ing the following approach.First,CFD is ud to predict the velocity and turbulent dissipation rate in a Taylor–Couette reactor and the turbulence model is validated with particle image velocimetry (PIV)data.Next,QMOM is ud to solve the PBE for the aggregation and breakage process using
appropriate kinetic expressions that incorporate information concerning the particle morphology (fractal dimension)and the local shear.The resulting predictions for the evolution of the mean particle size are then compared with measured val-ues obtained from the in situ optical image capturing system described in our previous work [1].
2.Theory
中国封建社会2.1.General aggregation-breakage equation
The general Reynolds-averaged form for the aggregation-breakage equation [6,12,17,18]in a clod system is given by
∂n(L ;t)
∂t + U ·∇n(L ;t)−∇· D T ∇n(L ;t) =L 22
L
α((L 3−λ3)1/3,λ)β((L 3−λ3)1/3,λ)n((L 3−λ3)1/3;t)n(λ;t)
(L 3−λ3)2/3dλ−n(L ;t)
∞
α(L,λ)β(L,λ)n(λ;t)dλ
(2)
+
∞ L
a(λ)b(L |λ)n(λ;t)dλ−a(L)n(L ;t),where n(L ;t)is the Reynolds-averaged number-density
function in terms of the particle size (note that this length-bad PBE is derived [1]from the volume-b
ad PBE by assuming v =L 3,hence L does not necessarily reprent particle length except for the specific ca of cubic particles), U is the Reynolds-averaged mean velocity,D T is the tur-bulent diffusivity,α(L,λ)is the collision efficiency between particles with size L and λ,β(L,λ)is the length-bad ag-gregation kernel that describes the frequency that particles with size L and λcollide,a(L)is the volume-bad break-age kernel that is the frequency of breakage of a particle of volume L ,and b(L |λ)is the fragment distribution func-tion.The terms on the right-hand side of Eq.(2)reprent birth and death of particles due to aggregation and breakup,respectively.
2.2.Quadrature method of moments In this work,the moments are defined as (3)
m α=
∞
n(L)L αdL,
where α=0,1,...,5.In the QMOM,a quadrature approxi-mation for the moments of the size distribution is employed
L.Wang et al./Journal of Colloid and Interface Science 285(2005)167–178169
as follows [11]:
(4)
m α=
∞
n(L)L α
dL ≈
N i =1
w i L αi ,
where N is the order of the quadrature formulae.The weights,w i ,and abscissas,L i ,are then found by solving the nonlinear system of algebraic equations that result from application of the quadrature approximation to Eq.(4)for k =0,...,2N −1via u of the product-difference (PD)al-gorithm.The resulting clod transport equation for the m αis [1]
∂m α(x ,t)
失其所与∂t + U ∂m α(x ,t)∂x i −
∂∂x i D T ∂m α(x ,t)∂x i
=12 i w i j αij βij w j L 3i +L 3j α/3−L αi −L α
j (5)+ i
微平台推广>借调工作总结a i ¯
b (α)i
w i −
i
小黄鱼刺多吗
a i w i L αi ,where m αis the Reynolds-averaged moment in terms of the particle size L ,αij is the collision efficiency between parti-cles with size R i and R j ,βij is the aggregation kernel that describes the frequency that particles with size R i and R j collide,a i is the breakage kernel that is th
e frequency of
breakage of a particle of size R i ,and ¯b
i is the fragment distribution function.Note that Eq.(5)is derived from the volume-bad PBE such that it can be applied for arbitrary fractal dimension of the flocs so long as particle density is constant [1].Also note that the abscissas L i calculated by the PD algorithm do not necessarily reprent particle length except for the specific ca of cubic particles,as we previ-ously mentioned.2.3.Aggregation kernel
In turbulent flows,the ratio between particle size and the Kolmogorov scale,η(=ν3/ε)1/4,is crucial.When particles are smaller than the Kolmogorov scale (as is the ca in this work)the aggregation kernel can be approximated [19–21]by
(6)βij =β(L i ,L j )=1.294 ε
ν 1/2(R c ,i +R c ,j )3,where εis the turbulent dissipation rate,and R c ,i is the col-lision radius of an i -sized particle.R c ,i can be calculated by [1,3,22,23](7)
R c ,i =2−3/D f R 1−3/D f
L 3/D f
i
,where D f is the mass fractal dimension and R 0is the radius of the primary particle.Note that since εis determined by the flow field and D f can be measured experimentally,Eq.(6)contains no adjustable parameters.Note that the aggregation due to Brownian diffusion is neglected in this work becau perikinetic aggregation is much slower than orthokinetic ag-gregation in our system [1].Aggregation caud by particle
continuous movements or Brownian motion is called periki-netic aggregation,on the other hand,aggregation brought
about by fluid motion is known as orthokinetic aggregation.2.4.Fragmentation
2.4.1.Power-law breakage kernel
One form of the breakage rate kernel that has found ap-plication to a wide variety of fragmentation phenomena is the power law distribution [2,24–26]:(8)
a i =b(2R c ,i )γ.
The value of the breakage exponent is commonly taken as
γ=2.The breakage rate coefficient b has the following form for shear-induced fragmentation [24]:(9)
b =b G y ,
where y is a constant inverly proportional to the floc strength and b is determined empirically.In this work,y =1.85is ud for latex spheres,as was determined from our previous investigation under laminar-flow conditions [1].The constant b is assumed to be independent of Reynolds number,and will be determined by fitting the CFD simu-lation to the experimental data.A more detailed discussion about the u of power-law breakage kernels can be found in [21,25–27].
2.4.2.Fragment distribution
A symmetric binary fragmentation function
(10)
b(L |λ)=
2,if L =λ/21/3
,
0otherwi
is ud in this work,resulting in ¯b (α)i =2(3−α)/3L αi
.Such a breakup function was also ud successfully for latex parti-cle aggregation-breakage in [1,4,28].2.5.Collision efficiency
For suspended particles in water,the inter-particle forces include the attractive van der Waals force,the repulsive electrical double layer force and the repulsive hydrody-namic force.The latter aris mainly from the distortion of the fluid between approaching particles.When particles ap-proach each other they do not follow rectilinear paths,as assumed in Smoluchowski’s equation [29].The collision ef-ficiency,α,defined as the ratio of the actual frequency that particles collide and stick together to the theoretical aggre-gation frequency given by Smoluchowski’s equation [29],is then less than unity.
No rigorous analysis accounting for hydrodynamic inter-actions between particles in turbulent flow has been carried out thus far.Since the collisions between small particles in the smaller eddies can b
e treated as collisions in simple laminar shear flow,a first approximation for turbulent ag-gregation is to multiply Eq.(6)by the collision efficiency
170L.Wang et al./Journal of Colloid and Interface Science 285(2005)167–178
computed for simple laminar shear flow (with G =(ε/ν)1/2)to obtain the actual collision frequency.
The collision efficiencies for solid spherical particles in simple laminar shear flow can be derived from calculations of the relative particle trajectories [30–32].The collision ef-ficiency in simple laminar shear flow is a function of the flow number
(11)
F l =6πµG(R i +R j )3
8A ∗
,where the Hamaker constant A ∗ud in this work is 3.5×10−20J [23].For collisions between equally sized particles,(12)
αij =k αF l −0.18
(10<F l <105).
The pre-factor k αis a function of primary particle size (k αis 0.79,0.87,and 0.95,respectively,when R i =R j =2,1,and 0.5µm in [30]).
The collision efficiencies computed by Eq.(12)can be applied to aggregation between solid spheres (i.e.,to the doublet formation stage).On the other hand,for encounters between flocs,the only attractive interaction that is typically considered is that between the two nearest primary particles,since other particles in the flocs are parated by too large a distance for the attraction force to be significant.Defining the flow number as [23]
(13)
F l =
6πµG(R i +R j )3
8A ∗2R i R j R 0(R i +R j )
,the collision efficiency can be calculated in a fashion simi-lar to the solid particle model.This model is known as the
impermeable flocs model in [33].The constant k α=0.3is ud in this work by minimizing the differences between the simulation and experiment results.It was found that the fi-nal aggregate size in the turbulent regime is independent of monomer size and is instead controlled by the Kolmogorov scale η[34].Therefore,it is assumed that if 2(R i +R j )>0.5η,the collision efficiency α(R i ,R j )is zero,which is equivalent to tting a maximal particle size for the aggrega-tion and breakage process.Another model,known as the shell–core model reported in [33],is validated for the aggre-gation of flocs with lower fractal dimension (ramified flocs).The shell–core model envisions the aggregate has a solid core and permeable outer shell allowing fluid to penetrate,resulting in the decline of repulsive hydrodynamic force be-tween approaching particles.
2.6.Instantaneous turbulence dissipation rate
2.6.1.Turbulence frequency
As we pointed out in [1],a Lagrangian time ries for εcan be extracted from direct numerical simulation (DNS)[35]and are highly intermittent.In other words,fluctuations in εcan be many times larger than the mean value ε predicted by a turbulence model.Since the ag-gregation and breakage rates depend on the instantaneous value of G (i.e.,b G y =b ∗εy/2),the value of the exponent
(i.e.,y )will play a crucial role in determining how aggrega-tion and breakage scales with Reynolds number.Thus,the
CFD model should take into account the fluctuations in εand their dependence on Reynolds number in order to accu-rately predict particle size distributions in turbulent flow,at least for process that occur on time scales shorter than the fluctuation time scale.
For a fixed turbulence kinetic energy k ,in the prent pa-per we model the Lagrangian turbulence frequency ω∗(t)(ω∗=ω/ ω ,ω≡ε(t)/k )using the stretched-exponential form [36]
dω∗(t)=C 1 1−(w
∗(t))γ1 (ω∗(t))1+γ1
ω∗(t)dt (14)+ω∗(t)
2C 1(1+C ω
ω∗(t))3(γωωω∗(t)) 1/2dW (t),where ω∗(t) =1by definition,and γω,C ω,and γ1are model parameters controlling the shape of the PDF of ω.The parameters γωand C ωare equal to 10/9and 0.35,re-spect
ively [36].In particular,γ1determines the decay rate of the PDF for large ωand is known to decrea slowly with Reynolds number,and a suitable power-law fit is given by [37]
(15)
γ1=1.24Re −0.26
1
,where,using standard isotropy relations,we obtain (16)
Re 1≡k √ν ε
.
The constant C 1can be fit to the DNS data as [37](17)
C 1=2.54Re 0.577
1
.The white noi dW (t)=√
,where w is a standard nor-mal random number.(See [38]for a discussion on numerical methods for generating dW (t).)
2.6.2.Lagrangian PDF models
In the Lagrangian PDF model,the fluid-particle variables are reprented by notional particles.The notional particle position vector X (t)evolves by a modeled velocity as [39]
d X = U(X ,t)
+∇ΓT (X ,t) dt
(18)+
2ΓT (X ,t) 1/2dW x (t).Note that the turbulent diffusivity ΓT (X ,t)is modeled by (19)
ΓT =C µSc T k 2
,
where C µ=0.09and Sc T =0.7.This model can be ud to describe the position of a solid particle in t
urbulent Taylor–Couette flow.
The particle transport equations in physical space are cast in the cartesian coordinate frame corresponding to (z,y,x ).However,for an axisymmetric domain,the Navier–Stokes equations are solved using a cylindrical coordinate frame
L.Wang et al./Journal of Colloid and Interface Science 285(2005)167–178171
corresponding to (z,r,θ).Hence a random walk correction must be applied.In particular,the corresponding x compo-nent of the random walk is modeled as
(20)dx =
2ΓT dt.Since the cartesian y component is the radial projection of the random walk in cylindrical coordinates,the following correction in the y coordinate of the particles is ud:
(21)r =
y 2+x 2,where r is the corrected y position.The corrected position is always further away from the axis than the original particle position.The notional particles are thus kept from reaching the axis.This correction algorithm is applied after normal particle tracking has been completed.
The PDF simulation algorithm is as follows:(1)The fluid dynamics (2D,axisymmetric)of the Taylor–
Couette reactor are simulated by using FLUENT (e Section 4.1for details),and the grid positions and cor-responding velocities,k and ε ,are written into a data file for a pair of co-rotating Taylor vortices in the middle of the reactor.
(2)Initially particles are evenly distributed in the computa-tional domain (900particles per cell).This number of particles is sufficiently large so that any further increa will not improve the simulation statistics significantly.Every particle contains information concerning its posi-tion,X ,and turbulence frequency ω∗.A value ω∗=1is ud for the initial conditions of particles.
(3)The particle position and ω∗are governed by Eqs.(18)
and (14),respectively.The time increment dt =0.0002is specified in the simulations to ensure that the turbu-lence frequency is sufficiently accurate.Periodic bound-ary conditions are applied along the z -axis and station-ary wall boundary conditions are applied along the r -axis.
attending(4)After applying Eq.(18),the cell mean ω∗n i is calcu-lated as
ω∗n
i (t)=1N i N i
n =1
ω∗(t)n
(22)
for all X (t +dt)in the i th cell ,
where N i is the number of particles in the i th cell.
2.6.
3.Typical Lagrangian time ries and mean correlation A typical time ries for R =34found from the La-grangian PDF models is shown in Fig.1.It is clear that the normalized quantity ω∗(=ε(t)/ ε )fluctuates with time and the instantaneous εcan surge up to approximately 10times ε .Such intermittency in εshould be taken into ac-count in the CFD models.Since the intermittency time scale is much smaller than the aggregation-breakage process in this work,it is prohibitive to simulate the process by
using
Fig.1.Typical Lagrangian time ries of normalized turbulence frequency ω∗at R =
34.
Fig.2.The relationship between k n R
and n .the transported PDF methods to account for the intermit-tency.However,the relationship between εn and ε n de-termined from the transported PDF models can be incorpo-rated into the CFD models to account for the intermittency.
In every computational cell i ,we can obtain the values of k n i (t)for
(23)
ω∗n i =k n i (t) ω∗ n i ,
where ω∗ n i =1.Similarly,k i (t)is found to satisfy (24)
εn i =k n i (t) ε n i .
For every time step,we can obtain an average value k n R of k n i (t)over the entire computational domain.It is found
that k n R
changes little in the interested range of R (34 R 220).Hence,here we report the k n R
for R =34.When n =0.41,k n R =0.94.(βij
is proportional to G 0.82,as we combine Eqs.(6)and (13).)When 0.4 n 1.7,the val-ues of k n R at various n are shown in Fig.2.The value of k n R
for this study is then equal to 0.984for n =0.925.How-ever,note that the Reynolds number investigated is fairly small (R is just above the transition to turbulent vortex flow).For example,at R =34,2.32 Re 1 7.56.When R is larger,we would expect a larger effect of intermittency on the aggregation-breakage process.
172L.Wang et al./Journal of Colloid and Interface Science 285(2005)167–178
3.Experimental methods
The Taylor–Couette cell is horizontally mounted on an optic table [1].The inner cylinder has radius of 3.49cm and is constructed of polished stainless steel.The transpar-ent outer cylinder is constructed from precision glass Pyrex tubing and has an inside radius of 4.86cm.The length of the cylinder is 43.2cm and the annular gap width between cylinders is 1.37cm.For this apparatus,Re c =82.8.A shaft on the inner cylinder is rotated by a precision motor with a digital controller.As we have discusd in [1],orthokinetic aggregation is dominant under the current experimental con-ditions.
Surfactant-free polystyrene latex spheres ud in this work were purchad from the Interfacial Dynamics Corpo-ration (batch 9321).The spheres,which are suspended in a stabilizing solution,have a diameter of 9.6µm ±7.4%.The spheres are completely washed using nanopure water before u in the experiments.The resulting particle number density is measured using a Coulter Counter (Multisizer 3,Beckman Coulter).The latex stock sample was placed in an ultrasonic bath (V1A,Tekmar sonic disruptor)for 2.5min prior to aggregation in order to break up any doublets and triplets in the suspension.A suspension of latex spheres was then mixed with a sodium chloride solution (using nanop-ure water)so that the concentration of NaCl in the resulting suspension was 1.29M.To avoid dimentation and radial migration effects,the density of the solution was adjusted by adding sodium chloride until it matched the density of the latex particles,ρ=1.055g /cm 3.The visc
osity of the fluid is 1.097×10−3kg /m s,as measured using a Canon–Fenske viscometer.As a result of the prence of the salt in the solution,the electric double layer was reduced ap-proximately to the Debye–Hückel length of 10−8m and the electric double layer force is negligible compared with van der Waals force and hydrodynamic repulsive force.All ex-periments were carried out at 24.0◦C.After the reactor is filled with the particle suspension,the rotational speed of the inner cylinder is slowly incread to the desired value.Then an in situ optical particle image capturing system is ud to record the particle size distribution information at different times [1].
Once all images were recorded,an off-line image analy-sis procedure was applied to extract aggregate information using the image tools in Matlab 6.1.After removal of aggre-gates straddling the image border,the projected perimeter and area of the remaining aggregates were calculated.In this work,circle equivalence diameter L D was ud,which was calculated from the cross-ctional projection area A of the aggregate by L D =2√
A/π.After the PSD was obtained,d 43(=m 4/m 3)and d 10(=m 1/m 0)were ud to charac-terize the mean particle size evolution of the aggregation process,where the j th moment is defined as
(25)
m j =
∞
n(L D )L j
D dL,
where n(L D )is the number-density function.The relation-ship between the projected area A and perimeter P of an aggregate is given by [40](26)
A ∝P 2/D pf ,
where the perimeter is related to the projected area of the ag-gregate by the projected fractal dimension D pf .Note that this definition implies that D pf ranges between a value of 1(cor-responding to a Euclidean aggregate)to 2(corresponding to a linear aggregate).Values of D pf near 2are character-istic of highly ramified floc structures.Spicer et al.[40]ud Eq.(26)to characterize polystyrene sphere flocs and found values of D pf ranging from 1.1to 1.4.The projected area fractal dimension,D pf ,is directly related to the surface frac-tal dimension,D s [41].However,there is no straightforward relationship between D pf and the mass fractal dimension D f of a floc [42].Neverthel
ess,measurement of D pf is straight-forward,provides comparable insight into particle morphol-ogy,and circumvents pitfalls associated with obtaining mass fractal dimension measurements,such as the need to perform ex situ light scattering experiments.
4.Results and discussion 4.1.Validation of turbulence model
Particle image velocimetry (PIV)was ud to capture the velocity field in a meridional plane (r –z plane).Readers can find more information about PIV measurements in [1,16].Here,the experimental data are ud to validate the turbu-lence model in a commercial CFD code FLUENT 6.1.The experimental results were obtained by averaging from 200to 800instantaneous velocity vector fields at the Reynolds number ratio (34 R 220)investigated in this work.
The dimensions of the simulated reactor geometry are r i =3.49cm,r 0=4.76cm and length L =43.2cm.For this apparatus,Re c =84.5.The obtained number of vortex pairs in the simulation is 14,which is the same as appeared in the experiment.A 2D steady-state simulation in the meridian plane of the reactor under the hypothesis of axial symmetry is carried out.Different meshes were tested in order to find a grid-independent solution.The final mesh has 20grids in the radial direction and 680in the axial direction for a total of 13,600nodes.All grids are uniformly distributed.The con-ver
gence criteria is 10−6.Further increasing the grid density in either direction yields little improvement in the final solu-tions.Different turbulence models and near-wall treatments were tested in FLUENT and the comparison showed that de-pending on R ,veral combinations of turbulence models and near-wall treatments give good agreement with the ex-perimental data.However,the Reynolds stress model (RSM)with standard wall functions gives the most satisfying results and will be ud to model the flow field.
In Fig.3,the mean velocity vector field of axial veloc-ity u z and radial velocity u r at R =160are compared
