en

istr

ctuaatio. Inbuleuidid tsin

gas-droplet, liquid-particle and bubble-liquid ows, are widelyallurgical, aeronautical, astro-eering. The turbulence of uidplex pexistinuous propleteir dispersedbulencdisp

uctuation, and the larger the particle size, the weaker its turbu-lent uctuation. In the framework of two-uid models, Elghobashiet al. (1984) combined the gas k-e turbulence model with an alge-braic particle turbulence model (it is called by us a k-eAp model).Similar approaches have been taken by Melville and Bray (1979),Chen and Wood (1985), Mostafa and Mongia (1988), etc. All of

cases or in some regions of the ow eld, in contrast to theHinzeTchens theory, the particle uctuation is stronger thanthe uid turbulent uctuation, and the larger the particle size,

bility density function (PDF) approach. Due to the limitation ofthe length of this paper, in the following text only the two-phaseturbulence models developed by Zhou et al. will be reviewed.

For the effect ofwall on particle owbehavior a particle-wall col-lision theory accounting for the friction, restitution and wall rough-ness was proposed (Zhang and Zhou, 2005). For dense gas-particleows, both large-scale uctuation due to particle turbulence andsmall-scale uctuation due to inter-particle collision are taken intoaccount using a so-called USM-H theory (Yu and Zhou et al., 2005).

* Tel.: +86 10 62782231; fax: +86 10 62781824.

International Journal of Multiphase Flow 36 (2010) 100108

Contents lists availab

l

l seE-mail address: zhoulx@mail.tsinghua.edu.cn.phases. The turbulent uctuation of the dispersed phase will affectits mixing with the continuous phase, hence has important effecton the pressure drop, heat and mass transfer between two phases,collection efciency, ame stabilization, combustion efciency,pollutant formation, etc. There was less understanding to thebehavior of so-called particle/bubble/droplet turbulence until thesecond half of years 80 of the last century. Over a long period themost popular theory was the HinzeTchens particle-tracking-uid theory (Hinze, 1975), according to which the particle turbu-lent uctuation should be always weaker than the uid turbulent

turbulent kinetic energy, similar to that proposed by Zhou andHuang with only minor difference in the closure models of somephase interaction terms.

Later, it was found that the anisotropy of particle turbulence iseven greater than that of uid turbulence. A unied second-ordermoment (USM) theory, i.e., a theory of two-phase Reynolds stresstransport equations, was proposed (Zhou et al., 1994; Zhou andChen, 2001). On the other hand, a group of investigators, for exam-ple, Zaichik (2001), Reeks (1992), Simonin (1996), derived andclosed the particle Reynolds stress equations based on the proba-encountered in power, chemical, metnautical, nuclear and hydraulic engin(gas or liquid) itself is already a comdispersed multiphase ows with co-ticles, droplets, bubbles) and continmuch more complex. The particles, down strong uctuations leading to thmeanwhile the existence of the dischange (modication) of the uid turbulence interactions between the0301-9322/$ - see front matter 2009 Elsevier Ltd. Adoi:10.1016/j.ijmultiphaseow.2009.02.011henomenon. Turbulentg dispersed phase (par-hase (gas or liquid) ares or bubbles have theirpersion (diffusion), andphases will cause thee. There are strong tur-ersed and continuous

the stronger its turbulent uctuation. Instead of HinzeTchenstheory, a transport equation theory of particle turbulent kinetic en-ergy was proposed (Zhou and Huang, 1990), according to whichparticle turbulent uctuation depends on its own convection, dif-fusion, production due to mean motion and dissipation/productiondue to the effect of uid turbulence, and not only the effect of uidturbulence, as that predicted by the HinzeTchens theory. Subse-quently, Tu (1995) also proposed a transport equation of particleTurbulent dispersed multiphase ows, including gas-particle, tion. However, it was found by the present author that in someReview

Advances in studies on two-phase turbul

L.X. Zhou *

Department of Engineering Mechanics, Tsinghua University, Tsinghua Garden, Haidian D

a r t i c l e i n f o

Article history:Received 14 October 2008Received in revised form 6 February 2009Accepted 10 February 2009Available online 26 February 2009

Keywords:Dispersed owsMultiphase owsParticle turbulence

a b s t r a c t

Particle/droplet/bubble ubustion and pollutant form90 years of the last century20 years on two-phase turgas-particle and bubble-liqinter-particle collisions, utwo-uid RANS modeling u

1. Introduction

International Journa

journal homepage: www.ell rights reserved.ce in dispersed multiphase ows

ict, Beijing 100084, China

tion and dispersion are important to mixing, heat and mass transfer, com-n in dispersed multiphase ows, but are insufciently studied before thethis paper, the present author reports his systematic studies within nearlynce in dispersed multiphase ows, including particle uctuation in diluteows, particle-wall collision effect, coexistence of particle turbulence andurbulence modulation due to the particle wake effect and validation of theg large-eddy simulation.

2009 Elsevier Ltd. All rights reserved.

these approaches for the particle turbulence are based on the ideaof HinzeTchens particle-tracking-uid theory of particle uctua-

le at ScienceDirect

of Multiphase Flow

vier .com/locate / i jmulflow

MuFor the effect of particles on gas turbulence, including the particlewake effect, it was studied using both LES and RANSmodeling (Zengand Zhou et al., 2007). Also, to understand the instantaneous turbu-lence structures, large-eddy simulation of liquid-bubble ows wascarried out (Yang et al., 2002). Recently, the gas-particle ows arestudied using LES, in order to use LES statistical results for validatingthe USM closure models.

In the following text, we will give a brief description and exper-imental verication of the proposed models: (a) the two-phaseReynolds stress equation (unied second-order moment, USM)and two-phase turbulent kinetic energy equation (k-ekp) models;(b) the two-phase particle-wall collision model; (c) the USMmodelfor dense gas-particle ows (USM-Hmodel); (d) the gas turbulencemodication model accounting the particle wake effect; (e) a two-uid large-eddy simulation (LES) of gas-particle ows.

2. Particle uctuation and dispersion in dilute gas-particle ows

For predicting particle uctuation, Tchen rst considered thesingle-particle motion in a uid eddy, and afterwards Hinze usedthe Taylors statistical theory of turbulence to obtain the HinzeTchens model (Hinze, 1975) for the ratio of particle viscosity overgas viscosity or particle diffusivity over gas diffusivity as

mp=mT Dp=DT kp=k2 1 sr1=sT1;sr1 qsd2p=18l; sT k=e 1Thismodel can simplybedenotedas an Apmodel. It is used togetherwith the gas turbulence k-emodel, constituting a k-eAp model, andeven nowadays is widely adopted as particle dispersion models intwo-uid models in some commercial software. According to Eq.(1), the particle uctuation should be always smaller than the gasuctuation and the larger the particle size, the smaller the particleuctuation. However, in contrast to what predicted by the Ap model,the LDV and PDPA measurements show that the particle turbulenceintensity is larger than the gas one in thewhole ow eld of connedjets and in the reverse ow zones of recirculating and swirling ows,and the particle turbulence intensity increases with the increase ofthe particle size in a certain size range. Based on the concept of trans-port of particle turbulence, we started from the uid NS equationand instantaneous particle motion equation, using the Reynoldsexpansion and timeaveraging, derivedandclosedanenergyequationmodel of particle turbulence (kp model) (Zhou and Huang, 1990). In19901994 we proposed a two-phase Reynolds stress transportequation model, i.e., a unied second-order moment (USM) two-phase turbulence model (Zhou et al., 1994). Based on two-phaseinstantaneous momentum equations, using Reynolds expansionand time averaging, the uid and particle Reynolds stress equationsare derived and closed. In this case the governing equations for iso-thermal turbulent gas-particle ows, accounting for only the gravita-tional and drag forces, including uid and particle continuity,momentum and Reynolds stress equations, can be given as:

@q@t

@@xj

qVj 0 2@qp@t

@@xj

qpVpj 0 3@

@tqVi @

@xjqVjVi @p

@xi @sji

@xj Dqgi

qpsrp

Vpi Vi 4@

@tqpVpi

@

@xjqpVpjVpi qpgi

qpsrp

Vi Vpi 5@

@tqv iv j @

@xkqVkv iv j Dij Pij Gpij Pij eij 6

L.X. Zhou / International Journal of@

@tNpvpivpj @

@xkNpVpkvpivpj Dp;ij Pp;ij ep;ij 7where, Dij; Pij;Pij; eij are terms having the same meanings as thosewell known in single-phase uid Reynolds stress equations. Thenew source term for two-phase ows

Gp;ij Xp

qpsrp

vpiv j vpjv i 2v iv j

is a phase interaction term expressing the uid Reynolds stressproduction/destruction due to particle drag force. The transportequation of dissipation rate of uid turbulent kinetic energy fortwo-phase ows is:

@

@tqe @

@xkqVke @

@xcekevkv l

@e@xl

ekce1GGp ce2qe 8

where the new source term is Gp P

pqpsrp vpiv i v iv i.

Dp;ij; Pp;ij; ep;ij are the diffusion, production terms of particleReynolds stress and the production term due to uid turbulence,respectively

For a closed system, beside Eqs. (6)(8), the transport equationsof npvpi;npvpj;npnp;vpiv j;vpjv i also should be used. For example,the transport equations of two-phase velocity correlation vpiv jand particle turbulent kinetic energy are derived based on the uidand particle momentum equations and closed as:

@

@tvpiv jVkVpk @

@xkvpiv j

@@xk

memp @@xk

vpiv j

1qsrp

qpvpivpjqv iv jh

qqpvpiv ji vpkv j @Vpi

@xkvkvpi @Vj

@xk

ekvpiv jdij 9

@

@tNpkp @

@xkNpVpkkp @

@xkNpcsp

kpepvpkvpl

@kp@xl

PpNpep 10

where the last term on the right-hand side of Eq. (9) is closed byassumingthat thedissipationof two-phasevelocity correlation ispro-portional to the dissipation rate of the gas turbulent kinetic energy.

and ep 1srp vpiv i vpivpi 1Np Vi Vpinpvpi:

Eqs. (6)(10) constitute the unied second-order moment two-phase turbulence model. It is found that the k-ekp model is areduced form of the USM model in case of nearly isotropic turbu-lent gas-particle ows, which consists of the following expressionsand equations

v iv j 23kdijmt@Vi@xj

@Vj@xi

; vpivpj 23kpdijmp

@Vpi@xj

@Vpj@xi

11

npvpj mprp@Np@xj

; npvpj mprp@Np@xj

12

@

@tqk @

@xjqVjk @

@xj

lerk

@k@xj

GGpqe 13

@

@tqe @

@xjqVje @

@xj

lere

@e@xj

ek

ce1GGpce2qe 14

@Npkp@t

@@xk

NpkpVpk @@xk

Npmprp

@kp@xk

PpNpep 15

@

@tkpg VkVpk @

@xkkpg

@@xk

cske

2

ckpk2pep

!@kpg@xk

! 1qsrp

qpkpqkqqpkpg

12

v ivpk@vpi@xk

vpivk @v i@xk

1sekpg 16

The physical meanings of the USM and k-ek models are (1) the

ltiphase Flow 36 (2010) 100108 101p

particle turbulent uctuation is determined not only by the localgas turbulence as that given by the Ap model, but also by its own

convection, diffusion and production, so in some cases or someregions the particle turbulence may be stronger than the gas turbu-lence; (2) the particle turbulent uctuation is anisotropic and itsanisotropy may be stronger than that of gas turbulence. Fig. 1shows the simulation results of particle number density in wind-sand ows behind an obstacle (Laslandes and Sacre, 1998) usingboth k-ekp and k-eAp models and their comparison with experi-ments. It is seen that the k-eAp model based on the theory of par-

3. Particle-wall collision effect

It is well known that the particle-wall collisions are directlytreated in the Lagrangian discrete particle simulation. In theearly-developed EulerianEulerian or two-uid modeling of uid-particle ows, the particle-wall collision was not taken intoaccount, and zero normal particle velocity and zero normal gradi-ent of other particle variables at the wall are assumed as

Vpw 0@/p@y

w

0 17

This model is equivalent to the full reection condition withoutenergy loss in the Lagrangian approach, which is obviously not truein practical gas-particle ows where particle-wall collision playsimportant role. A particle-wall collision model in the frameworkof two-uid approach, taking the restitution, friction and wallroughness into account was proposed by the present author (Zhangand Zhou, 2005). For example, the particle number density, longitu-dinal velocity and longitudinal component of normal Reynoldsstresses at the walls are given as

1 1

1 V !

Fig. 1. Particle number density (after Laslandes and Sacre).

102 L.X. Zhou / International Journal of Multiphase Flow 36 (2010) 100108ticle tracking local gas turbulence, seriously under-predicts theparticle dispersion leading to more ununiform particle concentra-tion distribution, not observed in experiments, whereas the k-ekp model accounting for the convection of particle turbulence,much better predicts the particle dispersion, giving a more uniformparticle concentration distribution, in much better agreement withthe measurement results. Fig. 2 gives the predicted vertical normalReynolds stresses for the liquid and bubbles in bubble-liquid owsin a bubble column using a full second-order moment (FSM) modeland an algebraic stress model (ASM) (Zhou and Yang et al., 2002)and their comparison with the PIV measurement results, indicatinga good agreement. The results show that in the bubble column thedispersed phase turbulence-bubble turbulence is much strongerthan the liquid turbulence due to its higher inlet velocity, andthe liquid (with lower inlet velocity) turbulence is produced notonly by its own velocity gradient but also by the enhancementdue to bubble uctuation. The results also indicate that the anisot-ropy of bubble turbulence is stronger that of liquid turbulence (notshown here). These results are in contrast to what predicted previ-ously by some investigators who told us that bubbles always atten-uate liquid turbulence.Fig. 2. Vertical normal reynolds stNpb 2Np1 1 e 1 3p p12kp=3p 18Vpb Vp1 Vp1f 1 13a

02

19

upupb 13up1up1f3 a022 f 21 eg

13vp1vp11 e3f 2 a021 2f 2

23up1vp1f 3 a022e 3 13Up1Up1a

02f 21 e

13Vp1Vp13ef 2 a021 e 2ef 2 23Up1Vp1a

02f 1 2e20

where f, e and a denote the friction coefcient, restitution coef-cient and wall roughness, respectively, the capital alphabets Uand V denote time-averaged particle velocities and lower-casealphabets u and v denote particle uctuation velocities, the sub-script b denotes the values at the wall, and the subscript 1 denotesthe values in the near-wall grid nodes. These equations imply thatress ( Exp.; FSM; . . . ASM).

0.00 2 4

(m/s)wp(m/s)wp(m/s)wp(m/s)wp

.

0 2 4 0 2 4 0 2 4

mm

0

time

/s)2

Mu Exp x=3mm x=25mm x=52mm x=85

Fig. 3. Particle tangential

0.0

0.2

0.4

0.6

0.8

1.0

0 1(m/s)w p(m/s)w p

r/R

0 1 0 1 2(mw p(m/s)w p10.2

0.4

0.6

0.8

1.0

r/R

L.X. Zhou / International Journal ofthe particle number density,velocity components and Reynoldsstresses will change under the effect of particle-wall collision dueto friction, restitution and wall roughness, and not obey the lawof zero normal velocity and zero-gradient of other variables. Thewall roughness can lead to redistribution of particle Reynolds stresscomponents after particle-wall collision. The predicted particle tan-gential time-averaged velocity (Fig. 3) and RMS tangential uctua-tion velocity (Fig. 4) of swirling gas-particle ows measured bySommerfeld and Qiu (1991) show that the prediction results usingthe boundary condition bc 2, based on Eqs. (18)(20), give lowernear-wall particle tangential time-averaged and RMS uctuationvelocities due to the effect particle-wall collisions, in agreementwith those observed in experiments, whereas the prediction resultsusing the boundary condition bc 1, based on Eq. (17), not account-ing for the particle-wall collisions, give higher near-wall particletime-averaged and RMS uctuation velocities, not in agreementwith experimental results.

4. Coexistenceof particle turbulence and inter-particle collisions

In dense gas-particle ows there are both large-scale particleuctuations due to particle turbulence and small-scale particleuctuations due to inter-particle collisions. A USM-H two-phaseturbulence model for dense gas-particle ows was proposed bythe present author (Yu and Zhou et al., 2005). In this model thegas turbulence and particle large-scale uctuation are predictedusing the USM two-phase turbulence model, and the particlesmall-scale uctuation due to inter-particle collisions is predictedusing the particle pseudo-temperature equation H equation,given by Gidaspows kinetic theory (Gidaspow, 1994). This is nota simple superposition, since there are interaction terms in the par-

x=3mm x=25mm x=52mm x=85mm Exp.

Fig. 4. Particle RMS tangent bc1 bc2

2 4(m/s)wp

x=112mm

2 4(m/s)wp(m/s)wp

x=155mm

2 4(m/s)wp

x=195mm

2 4

x=315mm

-averaged velocity (m/s).

1 2(m/s)w p

1 2(m/s)w p

0.5 1.0(m/s)w p

0.5 1.0(m/s)w p

ltiphase Flow 36 (2010) 100108 103ticle Reynolds stress equations and the H equation. Some of theclosed USM-H model equations are:

The gas Reynolds stress equation

@ agqgmvgivgj

@t

@ agqgmVgkvgivgj

@xk Dg;ij Pg;ij Pg;ij eg;ij Gg;gp;ij 21

where Gg;gp;ij b vpivgj vgivpj 2vgivgj

The particle Reynolds stress equation

@ apqpmvpivpj

@t

@ apqpmVpkvpivpj

@xk Dp;ij Pp;ij Pp;ij ep;ij Gp;gp;ij 22

where Gp;gp;ij b vpivgj vpjvgi 2vpivpj

The equations of dissipation rate of turbulent kinetic energy forgas and particle phases:

@ agqgmeg

@t@ agqgmVgkeg

@xk

@@xk

Cgagqgmkgegvgkvgl

@eg@xl

egkg

ce1 Pg Gg;gp ce2agqgmegh i

23where Gg;gp 2b kgp kg

; ce3 1:8

@ apqpmep

@t

@ apqpmVpkep

@xk

@@xk

apqpmCdpkpepvpkvpl

@ep@xl

epkp

Cep;1 Pp Gp;gp Cep;2apqpmeph i

24where Gp;gp 2b kpg kp

x=112mm x=155mm x=195mmbc1 bc2

x=315mm

ial uctuation velocity.

0.0 0.2 0.4 0.6 0.8 1.0

1

2

3

4

5 USM

Parti

cle

Velo

city

(m

r/R

Fig. 6. Particle velocity.

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.10 0.15 0.20

Exp.

USM- USMk- -kP-

inp uu

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.10 0.15 0.20

Exp.

USM- USMk- -kP-

inp u

y/H

u

Fig. 7. Particle horizontal RMS uctuation velocity.

Exp.

USM- USMk- -kP-

0.6

0.8

1.0

MuThe two-phase velocity correlation equation:

@vpivgj@t

Vgk Vpk @vpivgj

@xk Dg;p;ij Pg;p;ij Pg;p;ij eg;p;ij

Tg;p;ij 25The particle pseudo-temperature transport equation:

32

@ apqpmH

@t

@ apqpmVpkH

@xk

24

35

@@xk

32apqpmvpkh CH

@H@xk

lp

@Vpk@xi

@Vpi@xk

@Vpi@xk

lpep Pp@Vpl@xl

np 23lp

@Vpl@xl

2 c 26

The interaction between the large-scale and small-scale particleuctuations is the third term on the right-hand side of Eq. (26),expressing the effect of the dissipation rate of particle turbulentkinetic energy on the particle pseudo-temperature. Figs. 5 and 6give the simulation results of particle volume fraction (Fig. 5) andparticle velocity (Fig. 6), respectively, for dense gas-particle owsin a downer, measured by Wang et al. (1992). It is seen that theUSM-H model, accounting for both particle turbulence and inter-particle collision, gives the particle volume fraction and velocitydistribution in best agreement with the measurement results. TheDSM-H model, neglecting particle turbulence, gives a high peak of

0.0 0.2 0.4 0.6 0.8 1.0

0.005

0.010

0.015

0.020

0.025

0.030

0.035 Experiment USMk-kP

DSM- USM

Parti

cle

vol

ume

frac

tion

r/R

Fig. 5. Particle volume fraction.

104 L.X. Zhou / International Journal ofparticle volume fraction near the wall and non-uniform particlevelocity distribution, not observed in experiments. The USM model,neglecting inter-particle collision, gives too uniform particle vol-ume fraction and velocity distributions, not in agreement withexperiments. The k-ekpH model, neglecting the anisotropy ofparticle turbulence, also over-predicts the non-uniformness of par-ticle velocity distribution. Figs. 7 and 8 show the simulation resultsof particle horizontal and vertical RMS uctuation velocities for hor-izontal gas-particle pipe ows measured by Kussin and Sommerfeld(2002). It is seen that the USM-H model can more properly predictthe anisotropy of particle RMS uctuation velocities-the axial com-ponent is greater than the vertical component, whereas the USMmodel over-predict this anisotropy and the k-ekpH modelentirely cannot predict this anisotropy.

5. Turbulence modulation in gas-particle ows with the particlewake effect

The problem of gas turbulence in gas-particle ows or, so-calledturbulence modulation from the single-phase turbulence, attractsmore and more attention in recent years. For dilute gas-particle6

7

8 Experiment

USM

k-kP

DSM-/s)

ltiphase Flow 36 (2010) 100108ows, various empirical and semi-empirical models have beenproposed. Up to now, in most of DNS, LES and RANS modeling, theparticles are treated as point sources. In the two-uid approach ofRANS modeling, the particle-source term in the gas Reynolds stressequation or the turbulent kinetic energy equation is the differencebetween the gas-particle velocity correlation and the gas Reynoldsstress. Owing to the fact that the former is always smaller than thelatter, the obtained source term is always negative, leading to the

0.0

0.2

0.4

0.00 0.05 0.10 0.15inp u

y/H

v

Fig. 8. Particle vertical RMS uctuation velocity.

Mu0.10

0.15 experiment m=0, U=13.4m/s experiment m=0.6,U=13.4m/s m=0, U=13.4m/s m=0.6,U=13.4m/s (wake effect) m=0.6,U=13.4m/s (no wake effect)

u / U

a

L.X. Zhou / International Journal ofdissipation of gas turbulence. In fact, particle has nite volume.Whengas passes over the particle, both thewake behind the particleand the vortex shedding should contribute to the velocity distur-bance and are considered as the sources of turbulence production.Since the large-size eddies are mainly responsible for the mecha-nism of particle enhancing gas turbulence, in our studies (Zeng andZhou et al., 2007)we at rst did the simulation of gas turbulent owspassing a single particle using both RANSmodeling with a Reynoldsstress equation turbulence model and LES with a Smagorinsky sub-grid scale stressmodel. The turbulence enhancement by the particleis studied under various particle sizes and relative gas velocities.Based on these simulation results, a turbulence enhancementmodel

-1.0 -0.5 0.0 0.5 1.00.00

0.05

r / R

-1.0 -0.5 0.0 0.5 1.00.00

0.05

0.10

0.15

experiment m=0, U=13.4m/s experiment m=3.4, U=10.7m/s m=0, U=13.4m/s m=3.4, U=10.7m/s (wake effect) m=3.4, U=10.7m/s (no wake effect)

u / U

r / R

b

-1.0 -0.5 0.0 0.5 1.00.00

0.05

0.10

0.15

experiment m=0, U=13.4m/s experiment m=0.5, U=13.1m/s m=0, U=13.4m/s m=0.5, U=13.1m/s (wake effect) m=0.5, U=13.1m/s (no wake effect)

u / U

r / R

c

Fig. 9. Air turbulence intensity ((a) with 0.5 mm particles; (b) with 1 mm particles;(c) with 0.2 mm particles).for the particlewake effect is proposed. Then, the proposedmodel istaken as a sub-model, incorporated into the two-phase owmodel,i.e., the second-order moment two-phase turbulence model and isused to simulate dilute gas-particle ows. The simulation resultsare compared with experimental results and the simulation resultsobtained by the two-phase owmodel not accounting for the parti-cle wake effect.

A turbulence enhancement model in the gas Reynolds stressequation based on the single-particle simulation is obtained as

Gpw cqpapV

2rel

srp27

where

srp qpd

2p

18lg1 Re2=3p =6; Rep

agqgdp ~Vg ~Vp

lg

Eq. (23) indicates that the turbulence enhancement due to the par-ticle wake effect is proportional to the particle size and the squire ofrelative velocity. The gas Reynolds stress equation with the particle-source term accounting for the particle wake effect is:

@ agqgmvgivgj

@t

@ agqgmVgkvgivgj

@xk Dg;ij Pg;ij Pg;ij eg;ij Gg;gp;ij Gpwdij 28

Fig. 9 gives the RMS gas uctuation velocities with different sizes ofparticles in vertical gas-particle pipe ows, measured by Tsuji et al.(1984). It is found that the results obtained using the modelaccounting for the particle wake effect are in much better agree-ment with the experimental results than those obtained using themodel not accounting for the particle wake effect in predictingthe following phenomena: 1 mm particles only enhance gas turbu-lence intensity, 0.5 mm particles enhance or attenuate gas turbu-lence at different locations, and 0.2 mm particles only attenuategas turbulence.

6. A two-uid LES of gas-particle ows and validation of theUSM two-phase turbulence model

Large-eddy simulation (LES) can give us the instantaneous tur-bulence structures and its statistical results can be used to validatethe RANS turbulence models. LES is used by us to validate the USMtwo-phase turbulence model. The ltered governing equations fora two-uid LES are given as

@

@takqk

@

@xjakqkVkj 0 k g;p 29

@

@tagqgVgi

@@xj

agqgVgiVgj

@pg@xj

@sg;sgs;ij@xj

@sg@xj

agqgsr

vpi vgi 30

@

@tapqpVpi

@

@xjapqpVpjVpi

@ss@xj

asqssr

Vgi Vpi @sp;sgs@xj

31

For particle-collision stress, neglecting the sub-grid scale particlestress and using the particle pseudo-temperature proposed byGidaspows kinetic theory (Gidaspow, 1994)

sp apqpH 1 2 1 e g0ap apppr Vpn odij 2aplpSp 32

ltiphase Flow 36 (2010) 100108 10532

@

@tapqpH

r apqsHVs r jprH cp 3bH 33

00

Mu0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.00.00 0.05 0.10 0.0 0.1

Uinu p

2

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.00.00 0.05 0.10

Y/H

0.0 0.1Uin

u p2

106 L.X. Zhou / International Journal ofThe Smagorinsky model (Smagorinsky, 1963) is used for the gassub-grid scale stress

sg;sgs;ij 2mTSij 13dijskk; mT C2sD

2 S

; Sij @Vi=@xj @Vj=@xi=2

S

2SijSij1=2 34Fig. 10 gives the LES and USM simulated particle RMS uctuationvelocities and their comparison with experimental results for back-ward-facing step gas-particle ows (Hishida and Maeda, 1991).Both of these modeling results are in agreement with the experi-mental results. It implies that the USM two-phase turbulence modelis validated by LES.

Fig. 11 shows the predicted particle axial RMS uctuation veloc-ity using LES and USM for axi-symmetric sudden-expansion gas-particle ows measured by Xu and Zhou (1999). Both modelingresults are in good agreement with experimental results. LESresults are somewhat better than the USM results.

Fig. 12 shows the axial component of gas-particle velocitycorrelation. The twomodels give the same trend in agreement with

0.0X/H=1.0 X/H=3.0

(a) streamwis

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.00.00 0.03

X/H=1.0

0.00 0.03

X/H=3.0

Uinv p

2

(b) lateral d

0.0X/H=1.0 X/H=3.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.00.00 0.03

X/H=1.0

Y/H

0.00 0.03

X/H=3.0

Uinv p

2

Fig. 10. Particle RMS uctuation velocit.2 0.0 0.1 0.2 0.0 0.1 0.2.2 0.0 0.1 0.2 0.0 0.1 0.2

ltiphase Flow 36 (2010) 100108experiments. The distribution of gas-particle velocity correlation issimilar to that of particle axial RMS uctuation velocity, but theformer is smaller than the latter. It is seen that LES results arecloser to the experimental results than the USM results. It impliesthat the USM model remains to be improved. There is still certaindiscrepancy between the LES results and experiments due to the 2-D LES and the shortcomings of the Smagorinsky SGS stress model,which also should be improved.

7. Conclusions

(1) The USM and k-ekp two-phase turbulence models can wellpredict stronger particle uctuation than the gas uctuationand stronger anisotropy of particle turbulence than that ofgas turbulence for some cases and in some regions of theow eld. These models are more reasonable than the tradi-tional HinzeTchens theory. However, in some cases, forexample, swirling gas-particle ows, the particle RMS uctu-ation velocities are still under-predicted.

X/H=5.0 X/H=8.0

e direction

0.00 0.03 0.06

X/H=5.0

0.00 0.03 0.06

X/H=8.0

irection

X/H=5.0 X/H=8.0

0.00 0.03 0.06

X/H=5.0

0.00 0.03 0.06

X/H=8.0

ies (m s1; j Exp., LES, . . .USM).

.3 0

L

RMS

eloc

Mu0.0

0.2

0.4

X=1.1 H X=3.4 H

EXP.

Fig. 11. Particle axial

0.0

0.2

0.4

0.6

0.8

1.00.0 0.1 0.0 0.1

X=1.1 H

r / R

X=3.4 H EXP.

Fig. 12. Gas-particle v0.6

0.8

1.00.0 0.1 0.2 0.3 0.0 0.1 0.2 0

r / R

L.X. Zhou / International Journal of(2) The particle-wall collision, including friction, restitution andwall roughness, has important effect on the near-wall parti-cle velocity and turbulence. It gives reduced particle velocityand particle turbulence owing to energy losses during colli-sion. The wall roughness will increase the longitudinal com-ponent of particle RMS uctuation velocity and reduce thenormal component, that is, leads to redistribution of normalcomponents of particle normal stresses near the wall.

(3) In dense gas-particle ows, there is interaction between thelarge-scale particle uctuation due to particle turbulenceand small-scale particle uctuationdue to inter-particle colli-sion. The former leads to enhancing particle dispersion,whereas the latterwill reduce particle large-scale uctuation.The USM-Hmodel can better predict these phenomena thanother models, neglecting either particle turbulence or inter-particle collision or the anisotropy of particle turbulence.

(4) The particle wake effect plays important role in the gas tur-bulence modulation. The proposed model can well predictdifferent behavior of different size range particles in turbu-lence modulation. However, the gas turbulence modulationin dense gas-particle ows remains to be further studied.

(5) The USM two-phase turbulence model is only preliminarilyvalidated by a two-uid LES. However, a more advancedtwo-phase sub-grid scale stress model remains to bedeveloped.

Acknowledgements

This studywas sponsored by the Projects of National Natural Sci-enceFoundationofChinaunder theGrants50736006and50606026.

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108 L.X. Zhou / International Journal of Multiphase Flow 36 (2010) 100108

Advances in studies on two-phase turbulence in dispersed multiphase flowsIntroductionParticle fluctuation and dispersion in dilute gas-particle flowsParticle-wall collision effectCoexistence of particle turbulence and inter-particle collisionsTurbulence modulation in gas-particle flows with the particle wake effectA two-fluid LES of gas-particle flows and validation of the USM two-phase turbulence modelConclusionsAcknowledgementsReferences