Simulation of swirling gas–particle flows using a nonlinear

k–e–kp two-phase turbulence model

L.X. Zhou *, H.X. Gu

Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

Received 1 January 2002; received in revised form 1 August 2002; accepted 2 August 2002

Abstract

The linear k–e–kp two-phase turbulence model is rather simple, but it cannot predict the anisotropic turbulence of strongly swirling gas–

particle flows. The second-order moment, two-phase turbulence model can better predict strongly swirling flows, but it is rather complex.

Hence, the algebraic Reynolds stress expressions are derived based on two-phase Reynolds stress equations, and then the nonlinear

relationships of two-phase Reynolds stresses and two-phase velocity correlation with the strain rates are obtained. These relationships,

together with the transport equations of gas and particle turbulent kinetic energy and the two-phase correlation turbulent kinetic energy

constitute the nonlinear k–e–kp turbulence model. The proposed model is applied to simulate swirling gas–particle flows. Predictions give

the two-phase time-averaged velocities and Reynolds stresses. The prediction results are compared with phase Doppler particle anemometer

(PDPA) measurements and those predicted using the second-order moment model. The results indicate that the nonlinear k–e–kp model has

modeling capability nearly equal to that of the second-order moment model, but the former can save much computation time.

D 2002 Elsevier Science B.V. All rights reserved.

Keywords: Gas–particle flows; Turbulence model; Nonlinear model

1. Introduction

Recently, remarkable strides have been made in devel-

oping nonlinear k–e models for single-phase flows. The

nonlinear stress–strain relationships can be obtained by

different approaches: the generalized Cayley–Hamilton

formula and the invariant principle [1,2], the renormaliza-

tion group (RNG) theory [3], the two-scale DIA method [4]

and the algebraic Reynolds stress model [5]. In the nonlinear

stress–strain model, the equivalent eddy viscosity is aniso-

tropic. Therefore, the nonlinear model can simulate the

anisotropic turbulence of swirling flows and separating

flows. The nonlinear k–e model has been validated in

simple flows (homogeneous shear flows and channel flows)

and has been applied to simulate complex flows (backward-

facing step flows, swirling flows) [1,6]. Comparison with

experiments shows that it does make significant improve-

ment over the standard isotropic k–e model and is almost as

economical as the linear k–e model.

As for the two-phase turbulence model in developing

two-fluid models of turbulent gas–particle flows, several

years ago, the two-phase Reynolds stress or unified second-

order moment (USM) and k–e–kp models were proposed by

the first author of this paper based on Reynolds averaging

and a closure method similar to that used in single-phase

turbulence modeling [7,8]. These models are used to simu-

late swirling gas–particle flows [9,10]. The second-order

moment models of only particle turbulence were also devel-

oped by Refs. [11,12] based on the PDF transport equation.

It has been found in application that the k–e–kp model is

rather simple and can simulate well nonswirling and weakly

swirling gas–particle flows. However, for strongly swirling

flows, the USM model should be better, but the USM model

is rather complex and is not convenient for engineering

application. A best compromise between the applicability

and simplicity is either an implicit algebraic two-phase

Reynolds stress model, or a nonlinear k–e–kp two-phase

turbulence model, i.e. an explicit algebraic two-phase Rey-

nolds stress model. Since the algebraic Reynolds stress

models frequently cause some divergence problems due to

lack of diffusion terms in the momentum equation, partic-

ularly in 3-D flows, a nonlinear k–e–kp two-phase turbu-

0032-5910/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved.

PII: S0032 -5910 (02 )00153 -5

* Corresponding author. Tel.: +86-10-6278-2231; fax: +86-10-6278-

1824.

E-mail address: zhoulx@mail.tsinghua.edu.cn (L.X. Zhou).

www.elsevier.com/locate/powtec

Powder Technology 128 (2002) 47–55

lence model is preferred. This is because the momentum

equations and k, e, kp equations have the same form for both

linear and nonlinear k–e–kp models, so it is easier to obtain

convergent results.

In this paper, algebraic two-phase Reynolds stress

expressions are derived based on two-phase Reynolds stress

equations, and then the nonlinear relationships of two-phase

Reynolds stresses and two-phase velocity correlation with

the strain rates are obtained. The nonlinear k–e–kp model is

applied to simulate swirling gas–particle flows. Comparison

is made between the predictions using the nonlinear k–e–kpmodel, the USM model and the PDPA measurement data to

assess the proposed model.

2. The two-phase Reynolds stress equations and

algebraic stress expressions

Previously, a second-order moment, two-phase turbu-

lence model [7,8] was proposed. The second-order moment,

two-phase turbulence model was used to simulate swirling

gas–particle flows [9,10]. In the second-order moment

model, the closed form of the two-phase Reynolds stress

equations is:

B

BtðqvivjÞ þ

B

BxkðqVkvivjÞ ¼ Dij þ Pij þ Gpij þ Pij � eij

ð1Þ

B

BtðNpvpivpjÞ þ

B

BxkðNpVpkvpivpjÞ ¼ Dp;ij þ Pp;ij þ ep;ij ð2Þ

where Dij, Pij, Pij, eij are terms having the same meanings as

those in single-phase Reynolds stress equations, thus, we

have

Dij ¼B

Bxkcsk

evkv1

Bvivj

Bxk

� �;

Pij ¼ �q vivkBVj

Bxkþ vjvk

BVi

Bxk

� �;

Pij ¼ Pij;1 þ Pij;2; Pij;1 ¼ �c1ek

q vivj �2

3dijk

� �;

Pij;2 ¼ �c2 Gij �2

3dijG

� �; eij ¼

2

3dije;

G ¼ �qvivkBVi

Bxk

and the source term in case of two-phase flows

Gp;ij ¼Xp

qp

srpðvpivj þ vpjvi � 2vivjÞ

is the gas Reynolds stress production/destruction due to the

particle drag force. The transport equation of dissipation rate

of gas turbulent kinetic energy is:

B

BtðqeÞ þ B

BxkðqVkeÞ ¼

B

Bxcek

evkv1

BeBx1

� �

þ ek½ce1ðGþ GpÞ � ce2qe� ð3Þ

where the new source term is

Gp ¼Xp

qp

srpðvpivi � viviÞ

Dp,ij, Pp,ij, ep,ij are the diffusion, production terms of particle

Reynolds stress and the production/destruction term due to

gas turbulence, respectively, and they can be given as

follows:

Dp;ij ¼B

BxkNpc

sp

kp

epvpkvp1

B

Bx1ðvpivpjÞ

� �

Pp;ij ¼ �ðVpknpvpj þ NpvpkvpjÞBVpi

Bxk� ðVpknpvpi

þ NpvpkvpiÞBVpj

Bxkþ npvpjgi þ npvpigj

ep;ij ¼1

srpNpðvpivj þ vpjvi � 2vpivpjÞ þ ðVi � VpiÞnpvpj�

þ ðVj � VpjÞnpvpi�

For a closed system, besides Eqs. (1)–(3), the transport

equations of npvpi; npvpj; npnp; vpivj; vpjvi are also used.

For example, the transport equation of vpjvi based on our

closure method is:

B

BtðvpivjÞ þ ðVk þ VpkÞ

B

BxkðvpivjÞ

¼ B

Bxkðme þ mpÞ

B

BxkðvpivjÞ

� �þ 1

qsrp

� ½qvpivpj þ qvivj � ðq þ qpÞvpivj�

� vpkvjBVpi

Bxkþ vkvpi

BVj

Bxk

� �� e

kvpividij ð4Þ

When using this full second-order moment model for a

three-dimensional flow, it is necessary to solve 26 differ-

ential equations, including six gas Reynolds stress equa-

tions, six particle Reynolds stress equations, nine two-phase

L.X. Zhou, H.X. Gu / Powder Technology 128 (2002) 47–5548

velocity correlation equations, one dissipation rate equation

for gas turbulent kinetic energy, three particle diffusion mass

flux equations and one equation for the mean square value

of particle number density fluctuation.

In order to reduce the computation time and simulta-

neously retain the anisotropic features of the turbulence

model, as that done in the single-phase turbulence model, an

algebraic two-phase stress model is obtained by simplifying

the stress transport equations. Neglecting the convection and

diffusion terns in (Eqs. (1),(2) and (4), the algebraic expres-

sions of two-phase Reynolds stresses and two-phase veloc-

ity correlation can be obtained as:

vivj ¼ ð1� kÞ 23kdij þ k

k

evivk

BVj

Bxkþ vjvk

BVi

Bxk

� �

þ k

c1qeðvpivj þ vivpj � 2vivjÞ ð5Þ

vpivpj ¼ � srp2

vpivpkBVpj

Bxkþ vpjvpk

BVpi

Bxk

� �

þ 1

2ðvivpj þ vpivjÞ ð6Þ

vpivj ¼ � qsrpq þ qp

vpivkBVj

Bxkþ vjvpk

BVpi

Bxk

� �

þ qq þ qp

vivj þqp

q þ qp

vpivpj �qsrp

q þ qp

1

sevpividij

ð7Þ

where vivj and vpivpj are gas phase and particle phase

Reynolds stresses, respectively. vpivj is the two-phase

velocity correlation. The last term on the right-hand-side

of the expression for vpivj is the dissipation term. It is

assumed that the dissipation is an isotropic one, which is

proportional to the summation of the normal components

divided by a time scale. This time scale can be either the gas

turbulence scale k/e, or the particle relaxation time srp, or theeddy interaction time accounting for the crossing trajectory

effect, se =min[k/e, srp]. Predictions indicate that the last oneis better than other time scales [13].

3. The nonlinear k–e–kp two-phase turbulence model

It has been found that the above stated implicit algebraic

stress model, in which the unknown stresses exist on both

two sides of Eqs. (5)–(7), frequently causes divergent

problems, in particular in 3-D flows. Therefore, in devel-

oping the single-phase turbulence model, the nonlinear k–emodel, or in other words, the explicit algebraic stress model

is preferred. To construct a nonlinear k–e–kp two-phase

turbulence model, we can transform Eqs. (5)–(7) into the

following explicit form, on the right-hand side of which

there are no terms containing the unknowns vpivj; vpivpj;vivj.

After transformation, we have

vivj ¼ Að1Þdij þ Að2Þ vivkBVj

Bxkþ vjvk

BVi

Bxk

� �

þ Að3Þ vpivpkBVpj

Bxkþ vpjvpk

BVpi

Bxk

� �

þ Að4Þ vpivkBVj

Bxkþ vjvpk

BVpi

Bxk

�

þ vivpkBVpj

Bxkþ vpjvk

BVi

Bxk

�

vpivpj ¼ Bð1Þdij þ Bð2Þ vivkBVj

Bxkþ vjvk

BVi

Bxk

� �

þ Bð3Þ vpivpkBVpj

Bxkþ vpjvpk

BVpi

Bxk

� �

þ Bð4Þ vpivkBVj

Bxkþ vjvpk

BVpi

Bxk

�

þ vivpkBVpj

Bxkþ vpjvk

BVi

Bxk

�

vpivj ¼ Cð1Þdij þ Cð2Þ vivkBVj

Bxkþ vjvk

BVi

Bxk

� �

þ Cð3Þ vpivpkBVpj

Bxkþ vpjvpk

BVpi

Bxk

� �

þ Cð4Þ vivpkBVpj

Bxkþ vpjvk

BVi

Bxk

� �

þ Cð5Þ vpivkBVj

Bxkþ vjvpk

BVpi

Bxk

� �

where A(1)–A(4), B(1)–B(4), C(1)–C(5) are all functions

of k, e, kp, kpg, qp, q and srp

Að1Þ ¼ ð1� kÞ 23k �

2qp

c1q2

3kpg

Bð1Þ ¼ Cð1Þ ¼ ð1� kÞ 23k � srp

ekþ

2qp

c1q

� �2

3kpg

Að2Þ ¼ Bð2Þ ¼ Cð2Þ ¼ �kk

eAð3Þ ¼ �

qpMsrpq

Bð3Þ ¼ � srp2q

ðq þ qp þ 2MqpÞ

Cð3Þ ¼ �qpsrp

q1

2þM

� �

Að4Þ ¼ �Msrp Bð4Þ ¼ �srp M þ 1

2

� �

L.X. Zhou, H.X. Gu / Powder Technology 128 (2002) 47–55 49

Cð4Þ ¼ �srp M þqp

2ðq þ qpÞ

!

Cð5Þ ¼ �srp M þqp

2ðq þ qpÞþ q

q þ qp

!

M ¼kqp

c1qesrpsrp ¼

d2pqp

18l1þ Re

2=3p

6

!�1

Rep ¼A!vp �!vAdp=m

The variables k, e, kp and krg will be determined by the

governing equations, where k is the gas turbulent kinetic

energy, e is its dissipation rate, kp is the particle turbulent

kinetic energy and kpg is defined by kpg ¼ vivpi.

We can write the expressions of vmvk ; vpmvpk ; vpmvk ;vmvpk ; ðm ¼ i; jÞ in a similar form, then these expressions

are iterated into the equations of vivj; vpivpj; and vpivj .

Finally, obtained expressions should take the following

form:

vivj ¼ Cdij þ OBVm

Bxn

� �þ O

BVm

Bxn

� �2 !

þ OBVm

Bxn

� �3 !

þ : : : : : : þ OBVm

Bxn

� �l!

where {C} is a constant

For practical application, we must cut the series to a

certain term. For single-phase flows, it is suggested that a

quadratic form with appropriate coefficients can predict

turbulent shear flows and swirling flows quite well. Here,

the obtained relationships are written to quadratic terms of

the strain rates. In fact, it is a perturbation method. The

resulting nonlinear stress–strain rate relationships written to

quadratic power terms of the strain rates are:

vivj ¼ G1dij þ G2

BVi

Bxjþ BVj

Bxi

� �þ G3

BVpi

Bxjþ BVpj

Bxi

� �

þ G4

BVi

Bxk

BVj

Bxkþ BVk

Bxj

� �þ BVj

Bxk

BVi

Bxkþ BVk

Bxi

� �� �

þ G5

BVpi

Bxk

BVpj

Bxkþ BVpk

Bxj

� ��

þ BVpj

Bxk

BVpi

Bxkþ BVpk

Bxi

� ��

þ G6

BVi

Bxk

BVpj

Bxkþ BVpk

Bxj

� �þ BVj

Bxk

BVpi

Bxkþ BVpk

Bxi

� �� �

þ G7

BVpi

Bxk

BVj

Bxkþ BVk

Bxj

� �þ BVpj

Bxk

BVi

Bxkþ BVk

Bxi

� �� �

þ G8

BVi

Bxk� BVpi

Bxk

� �BVj

Bxk� BVpj

Bxk

� �� �ð8Þ

vpivpj ¼ P1dij þ P2

BVi

Bxjþ BVj

Bxi

� �þ P3

BVpi

Bxjþ BVpj

Bxi

� �

þ P4

BVi

Bxk

BVj

Bxkþ BVk

Bxj

� �þ BVj

Bxk

BVi

Bxkþ BVk

Bxi

� �� �

þ P5

BVpi

Bxk

BVpj

Bxkþ BVpk

Bxj

� ��

þ BVpj

Bxk

BVpi

Bxkþ BVpk

Bxi

� ��

þ P6

BVi

Bxk

BVpj

Bxkþ BVpk

Bxj

� ��

þ BVj

Bxk

BVpi

Bxkþ BVpk

Bxi

� ��þ P7

BVpi

Bxk

BVj

Bxkþ BVk

Bxj

� ��

þ BVpj

Bxk

BVi

Bxkþ BVk

Bxi

� ��þ P8

BVi

Bxk� BVpi

Bxk

� ��

� BVj

Bxk� BVpj

Bxk

� ��ð9Þ

vpivj ¼ T1dij þ T2BVi

Bxjþ BVj

Bxi

� �þ T3

BVpi

Bxjþ BVpj

Bxi

� �

þ T4BVj

Bxiþ BVpi

Bxj

� �þ T5

BVi

Bxk

BVj

Bxkþ BVk

Bxj

� �

þ T6BVj

Bxk

BVi

Bxkþ BVk

Bxi

� �þ T7

BVpi

Bxk

BVpj

Bxkþ BVpk

Bxj

� �

þ T8BVpj

Bxk

BVpi

Bxkþ BVpk

Bxi

� �þ T9

BVpi

Bxk

BVj

Bxkþ BVk

Bxj

� �

þ T10BVpj

Bxk

BVi

Bxkþ BVk

Bxi

� �

þ T11BVi

Bxk

BVpj

Bxkþ BVpk

Bxj

� �

þ T12BVj

Bxk

BVpi

Bxkþ BVpk

Bxi

� �

þ T13BVi

Bxk

BVpj

Bxk� BVj

Bxk

� �þ BVpj

Bxk

BVi

Bxk� BVpi

Bxk

� �� �

þ T14BVj

Bxk

BVpi

Bxk� BVi

Bxk

� �þ BVpi

Bxk

BVj

Bxk� BVpj

Bxk

� �� �ð10Þ

whereallofG1–G8,P1–P8,T1–T14arefunctionsofk,e,kp,kpg,kpg, qp, q and srp. These coefficients are determined by the

following expressions, for example:

G1 ¼ Að1Þ; G2 ¼ Að2ÞAð1Þ þ Að4ÞCð1Þ;

G3 ¼ Að3ÞBð1Þ þ Að4ÞCð1Þ;

G4 ¼ Að1ÞðAð2ÞAð2Þ þ Að4ÞCð2ÞÞþ Cð1ÞðAð2ÞAð4Þ þ Að4ÞCð5ÞÞ;

L.X. Zhou, H.X. Gu / Powder Technology 128 (2002) 47–5550

G5 ¼ Bð1ÞðAð3ÞBð3Þ þ Að4ÞCð3ÞÞþ Cð1ÞðAð3ÞBð4Þ þ Að4ÞCð5ÞÞ;

G6 ¼ Bð1ÞðAð2ÞAð3Þ þ Að4ÞCð3ÞÞþ Cð1ÞðAð2ÞAð4Þ þ Að4ÞCð4ÞÞ;

G7 ¼ Að1ÞðAð3ÞBð2Þ þ Að4ÞCð2ÞÞ þ Cð1ÞðAð3ÞBð4Þ

þ Að4ÞCð4ÞÞ;

G8 ¼ 2Cð1ÞAð4ÞðCð4Þ � Cð5ÞÞ

where all of A(1)–A(4), B(1)–B(4), C(1)–C(5) are func-

tions of k, e, kp, kpg, qp, q and srp.

Að1Þ ¼ ð1� kÞ 23k �

2qp

c1q2

3kpg;

Bð1Þ ¼ Cð1Þ ¼ ð1� kÞ 23k � srp

ekþ

2qp

c1q

� �2

3kpg;

Að2Þ ¼ Bð2Þ ¼ Cð2Þ ¼ �kk

e; Að3Þ ¼ �

qpMsrpq

;

Bð3Þ ¼ � srp2q

ðq þ qp þ 2MqpÞ;

Cð3Þ ¼ �qpsrp

q1

2þM

� �; Að4Þ ¼ �Msrp;

Bð4Þ ¼ �srp M þ 1

2

� �; Cð4Þ ¼ �srp M þ

qp

2ðq þ qpÞ

!;

Cð5Þ ¼ �srp M þqp

2ðq þ qpÞþ q

q þ qp

!;

M ¼kqp

c1qesrpsrp ¼

d2pqp

18l1þ Re

2=3p

6

!�1

;

Rep ¼A!V p �!VAdp=m

The expressions for coefficients P1–P8, T1–T14 are

obtained in a similar manner. The variables k, e, kp and

kpg are determined by the governing equations given in the

following section, where k is the gas turbulent kinetic

energy, e is its dissipation rate, kp is the particle turbulent

kinetic energy and kpg is defined by kpg ¼ vivpi.

The transport equations of k, e, kp and kpg can be obtainedfrom Eqs. (1)–(4) by putting i= j:

B

BtðqkÞ þ B

BxkðqVkkÞ

¼ B

Bxkqcs

k

evkv1

Bk

Bx1

� �� qvivk

Bvi

Bxk� qe

þqp

srpð2kpg � 2kÞ ð11Þ

B

BtðqeÞ þ B

BxkðqVkeÞ

¼ B

Bxkqce

k

evkv1

BeBx1

� �

þ ek

ce1 �qvivkBvi

Bxkþ

qp

srpð2kpg � 2kÞ

� �� ce2qe

� �ð12Þ

B

BtðqpkpÞ þ

B

BxkðqpVpkkpÞ

¼ B

Bxkqpckp

kp

epvpkvpl

Bkp

Bx1

� �

� qpvpivpkBvpi

Bxkþ

qp

srpð2kpg � 2kpÞ ð13Þ

B

BtðkpgÞ þ ðVk þ VpkÞ

B

BxkðkpgÞ

¼ B

Bxkcsk

evkv1 þ ckp

kp

epvpkvpl

� �Bkpg

Bx1

� �

þ 1

qsrpðqpkp þ qk � ðq þ qpÞkpgÞ

� 1

2vivpk

Bvpi

Bxkþ vpivk

Bvi

Bxk

� �� 1

sekpg ð14Þ

The algebraic expressions (8)–(10) together with the

differential equations (Eqs. (11)–(14)) constitute the non-

linear k–e–kp two-phase turbulence model. It can be seen

that in comparison with the USM model, in this model the

Fig. 1. Swirl chamber.

L.X. Zhou, H.X. Gu / Powder Technology 128 (2002) 47–55 51

number of differential equations is reduced more than six

times (from 26 to 4) in simulating 3-D flows.

4. Simulation results and discussion

The nonlinear k–e–kp (NKP) model was used to simu-

late swirling gas–particle flows measured by Sommerfeld

and Qiu [14] using phase Doppler particle anemometer

(PDPA). The adopted grid nodes are 20� 43, but a fine

grid system of 40� 86 was also used, and it gives the same

results. The differential equations are integrated in the

computational cells to obtain finite difference equations

using a hybrid scheme. The FDEs are solved using the

SIMPLEC algorithm. The criterion for convergence is the

summation of residual mass sources less than 10� 4 for both

Fig. 2. Gas axial velocity (m/s, s = 0.47).

Fig. 3. Gas tangential velocity (m/s, s= 0.47).

Fig. 4. Axial velocity of 45-Am particles (m/s, s = 0.47).

L.X. Zhou, H.X. Gu / Powder Technology 128 (2002) 47–5552

gas and particle phases. A computer code called NKP-2 is

developed. Running a case in the Pentium-2 PC takes about

8 min. Using USM code it takes about 15 min. Comparison

is made between the predictions using the nonlinear k–e–kp(NKP) model, the USM model, and the PDPA measure-

ments. The geometrical configuration and the sizes of the

chamber are shown in Fig. 1. The particle size is 45 Am. The

inlet flow parameters are the same as those in experiments.

Figs. 2 and 3 give the predicted gas axial and tangential

velocities using both NKP and USM models. There is

almost no difference between these two models, and all of

them give results in good agreement with measurements.

For the axial and tangential velocities of 45-Am particles

(Figs. 4 and 5), in most regions of the flow field, the

difference between two model predictions is small and both

of them are in good agreement with experiments. Figs. 6 and

7 show the predicted gas axial and tangential fluctuation

velocities. Both NKP and USM models underpredict the

measured values, and the USM predictions are somewhat

but not much better than the NKP predictions. Figs. 8 and 9

Fig. 6. Gas axial fluctuation velocity (m/s, s = 0.47).

Fig. 5. Tangential velocity of 45-Am particles (m/s, s = 0.47).

Fig. 7. Gas tangential fluctuation velocity (m/s, s = 0.47).

L.X. Zhou, H.X. Gu / Powder Technology 128 (2002) 47–55 53

give the predicted axial and tangential fluctuation velocities

of 45-Am particles. The particle fluctuation velocities are

also underpredicted using both two models. Except for the

axial fluctuation velocity profile in the third section, the

difference between the two model predictions is smaller

than that for the gas phase. Both models predict that the gas

and particle axial fluctuation velocities are larger than the

tangential ones. This is in qualitative agreement with the

experimental results. In general, the NKP model can predict

what the USM model can predict, but the former can save

almost 50% computation time for a 2-D flow with small

geometrical sizes. Keeping in mind that in engineering

applications the accuracy of predicting the two-phase aver-

aged velocities is more important, one can consider that the

NKP model can be used instead of the USM model.

The discrepancy between predicted two-phase fluctua-

tion velocities (using either of two models) and measured

ones might be caused by the following reasons. First, in the

PDPA measurements each size group of particles has a

certain size range, so the measured gas or particle fluctua-

tion velocity includes the effect of particle size range. Next,

the particle–wall interaction may increase the particle

turbulence, which is not taken into account in either model.

Finally, the closure model of dissipation rate for the gas

turbulent kinetic energy in two-phase flows and the gas–

particle velocity correlation needs to be further improved.

5. Conclusions

(1) The NKP model has a capability nearly equal to that of

the USM model in simulating the two-phase averaged

velocities and fluctuation velocities of swirling gas–

particle flows.

(2) Both NKP andUSMmodels can give predicted two-phase

averaged velocities in good agreement with those

measured.

(3) Both of these models can simulate anisotropic two-phase

turbulence, but underpredict the two-phase fluctuation

velocities.

(4) The nonlinear k–e–kp model has no problem of conver-

gence encountered in the algebraic stress model.

(5) The NKP computer code is easier to obtain by modifying

a KP computer code.

(6) In 2-D flows in small geometries, the NKPmodel can save

about 50% of the computation time of the USM model.

However, in 3-D flows with large geometrical sizes, the

NKP model can save much more computation time.

Fig. 9. Tangential fluctuation velocity of 45-Am particles (m/s, s = 0.47).

Fig. 8. Axial fluctuation velocity of 45-Am particles (m/s, s = 0.47).

L.X. Zhou, H.X. Gu / Powder Technology 128 (2002) 47–5554

Acknowledgements

This study was supported by the Special Funds for Major

State Basic Research G-1999-0222-08, PRC.

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