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2009,21(4):541-549 DOI: 10.1016/S1001-6058(08)60182-1

EFFECT OF DISCHARGE RATIO ON FLOW CHARACTERISTICS IN 90o

EQUAL-WIDTH OPEN-CHANNEL JUNCTION*

ZHANG Ting College of Civil Engineering, Fuzhou University, Fuzhou 350108, China State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China, E-mail: zhangting@fzu.edu.cnXU Wei-lin, WU Chao

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China (Received January 15, 2008, Revised september 28, 2008) Abstract: For the 90° equal-width open-channel junction flow, the Reynolds averaged Navier-Stokes equations are solved while using the 3-D k model. The mean flow pattern and the secondary current are obtained. The model is validated by experimental data, and then applied to investigate the effect of the discharge ratio on the shape of separation zone shape, the cross-sectional mean flow angle and the contraction coefficient. The results are fairly close to those of the prior studies. The numerical modeling is both less time-consuming and less expensive to obtain the various flow parameters needed for engineering design. Key words: k turbulent model, open-channel junction, separation zone, secondary current 1. Introduction

Open-channel junction flow is often encountered in environmental engineering and hydraulic engineering while dealing with water and wastewater in treatment plants and irrigation systems. It generates theoretical interest due to a number of important flow phenomena involved. The interaction between the main and branch flows makes an increase in water depth upstream of channel junction, leading to flow deceleration and thus converting the fluid kinetic energy to potential energy. The branch flow causes the main flow to be entrained toward the opposite bank and generates an unstable separation streamline at the

* Project supported by the National Basic Research Program of China (973 Program, Grant No. 2007CB714705), the National Science Foundation of Fujian Province (Grant No. E0710013). Biography: ZHANG Ting (1977-), Male, Ph. D., Associate Professor

downstream corner of the junction. Downstream of the junction, the branch-side bank of the main channel restricts the supplementing of the entrained water, which results in lower pressure near the branch-side bank. Further downstream, this low pressure causes the separation streamline to deflect toward the branch-side bank, forming a 3-D separation zone (see Fig.1). Fig.1 Flow characteristics in open-channel junction[1]

Previous studies have focused on simplified

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mathematical approximations of different junction flow characteristics by physical models, 1-D theoretical analysis, or 1-D numerical solutions. Webber and Greated[2] implemented the method of conformal mapping to describe a theoretical flow pattern throughout the junction region and also included a method for estimating the relative energy loss across the junction. Modi et al.[3] calculated the size of the separation zone using a conformal mapping method under the assumption of inviscid and irrotational flow. Gurram et al.[4] provided a thorough background of the open-channel junction problem and a presentation of many junction characteristics. Among different numerical methods, the implicit finite-difference method has been widely used for the solution of 1-D unsteady open-channel flow problems[5-7]. On the basis of energy and continuity equations, Ni et al.[8] proposed a simple one-dimensional formulation to predict the transitional flow at an open-channel junction. Then an empirical relation between the junction losses, the junction angle, and the discharge ratio was suggested.

Using experimental data, Best and Reid[9] obtained the shape index of separation zone defined as the ratio of the width to the horizontal length of the separation zone. For the range 0.4 0.7Q , Ramamurthy et al.[10] investigated the moment transfer in terms of the pressure difference between the right and left banks of the branch channel and concluded that the cross-sectional mean flow angle, , is inversely related to discharge ratio, Q . Hager [11] and Kumar[12] derived an analytical solution of the coefficient of effective width, , at the separation zone and related the head loss coefficients to and Q . In addition to estimating the upstream/downstream depth ratio, Hsu et al.[13,14] studied the separation zone and the contraction of the flow as it passes through the separation zone. Energy and momentum correction coefficients were calculated at the downstream end of the separation zone. Their data also showed the depth-averaged angle of entry across the width of the branch channel flow. An energy loss coefficient, including eddy loss and friction loss, was given.

Detailed hydrodynamics of junction flow are found to be complex and there exist a number of parameters characterizing the flow physics. These include the size, shape, slope, angle between two combining channels, and the Froude and Reynolds numbers. Therefore, it is difficult to addressing the problem theoretically. Simplified theoretical or numerical models are less capable of taking some complex flow conditions into consideration such as the secondary flow and separation. More recently, some scholars[1,15] preformed extensive experimental studies of open channel junction flows for the purpose

of fully describing the general features and providing comprehensive data of the junction flow. On the other hand, 3-D free-surface flow CFD codes weres used to reproduce the flow characteristics by a number of researchers[16-21].

The purpose of this article is to adopt 3-D numerical method, which do not invoke many simplifying assumptions, which proves to be reliable and valuable, to obtain more detailed velocity and acceleration fields for engineering designs. The results are validated using experimental data of Weber et al.[1]. After the validation, it is easy to extend the range of simulated design data to meet the specific needs of other engineering devices that involve open-channel junction flows. 2. Numerical model 2.1 Governing equations

In the present study, the Reynolds-Averaged Navier-Stokes (RANS) equations are applied for the steady turbulent flow. The two-equation k model of Wilcox[22] is adopted for numerical simulation. The equations are transformed from Cartesian coordinates to generalized curvilinear coordinates necessary for the treatment of the free-surface boundary condition. The transformed equations can be written in conservation form and in generalized curvilinear coordinates as

Continuity equations:

1( ) =iiJ J U 0 (1)

Momentum equations:

1 11+ +m mii tm

U J J U U J gt Re

n

1+m

in

i

U J Px

= 0 (2)

k equation:

1 1*

1+ +m mtm

k

k J J U k J gt Re

n

+ =n

k G k 0 (3)

equation:

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1 1 +m tm

e

J J U J gt R

+ 1 mn

2+ =n Gk

0 (4)

where , is the mean velocity component along the

= 1, 2, 3i iU

ix direction, the superscript is the contravariant, m P is the piezometric

pressure, mng is the contravariant metric tensor component, J is Jacobian of the transformation, i is the computational curvilinear coordinates ( , , ), Re is the Reynolds number ( 0 0= /Re U H ), is the bulk fluid velocity, is the average water depth at the downstream end of the main channel,

0U 0H

t is the turbulence eddy viscosity ( = /t k ), and is the production term given by G

21= ( +2 j

jkit x xk k

UUG )i

k (5)

The five constants appearing in this model

assume their standard values: = 5/ 9 , = 3/ 40 , , , and . = 9 /100 = 2k = 2

2.2 Boundary conditions At the free surface, the normal gradients of ,

, u

v P , and k are set to be zero. The boundary condition for the vertical velocity, however, is determined by satisfying the free-surface kinematic condition, i.e.,

1 2= + +w U Ut

(6)

where = ( , , )x y t is the dimensionless water surface elevation. The free-surface elevation is determined using the dynamic condition , where is the pressure at the free surface. With the definition of piezometric pressure

= 0pp

P , the dynamic condition is written as

02 20 0

= + = 0 + =g HghP pU U 2Fr

(7)

where is the Froude number. Fr

At exit boundaries, the free-surface elevation is

prescribed and piezometric pressure P is fixed while all other variables are extrapolated from the interior of the solution domain. At inlet boundaries, known flow velocities and turbulence quantities are prescribed. At solid walls, the standard wall-function approach is used. 2.3 Numerical algorithm

Equations (1)-(4) can be typically written in general curvilinear coordinates for any dependent variable in the following form:

1 1+ ( ) =m mnm nJ J U J g

tS (8)

where represents , or iU k and is the source term. By integrating the above equation, one obtains the conservation equation for a control volume, expressed as

S

1 1d + ( m m

mV SV J n J U J g

tn

)d = dn VS S V (9)

A finite volume method is used to discretize the

model equations. The model employs the collocated-grid approach. The approximation of the convection term is handled by the deferred correction scheme, which combines the advantages of both the upwind difference scheme and the central difference scheme.

The pressure-velocity coupling is achieved using the SIMPLEC algorithm. It can be viewed as a predictor–corrector method and starts with a ‘‘predictor’’ step by assuming pressure field 0P is known and velocity field is predicted using Eq.(2) as

U

01 1 1+ + +I

I I I

UiP L iL PU U U m

P P P

PU A U QA A A

= 0m

ix (10)

The predicted U does not usually satisfy the

continuity equation. Therefore, the ‘‘corrector’’ step is initiated to modify both the pressure and velocity so that the following continuity and momentum equations are satisfied:

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1( )iiJ J U = 0 (11)

*1 1 1+ + +I

I I I

UiP L iL PU U U m

P P P

PU A U QA A A

= 0m

ix (12)

Equations (11) and (12) can be combined to

obtain the pressure correction equation for 0=P P P and then the new velocity is also

obtained. U

3. Model validations 3.1 Investigated subject and experimental data

The investigated subject is concerned with the sharp-edged, equal-width 90o junction (see Fig.1). The floor of the entire channel is horizontal. The origin “A” fixed at the bottom of the channel (Fig.1) is chosen to be , and . The coordinate system defined for this simulation has the positive

= 0x = 0y = 0z

x -axis oriented in the upstream direction of the main channel, and the positive -axis is upward in the vertical direction.

z

The experimental data of Weber[1] are selected for validation. The channel consists of a main channel of 21.946 m in length and a branch channel of 3.658 m in length. The branch channel is located 5.486 m downstream of the entrance of the main channel. Both channels have a width of 0.914 m, designated as W. The total combined flow rate is 0.170 mdQ 3/s and the downstream depth is held constant at 0.296 m. Therefore, the downstream bulk velocity is

, which gives a Froude number of 0.37 and a Reynolds number of 1.86 10

dH

= 0.628m / sdV5.

A total of six runs of experiments were conducted. In this study, the discharge ratio

is selected for validation as it generates a large separation zone and has a greater change in water-surface elevation. Therefore, the case is more challenging to calculate. is defined as the ratio of the upstream main channel flow to the combined total flow . All distances were normalized by the channel width, W , named as

= 0.250q

q

mQ

dQx , , and y z for

/x W , , and , respectively. The velocity components were normalized by the downstream average velocity, , called , and

/y W /z W

dV u v w for

, and , respectively. / du V / dv V / dw VThe simulation domain has been carefully chosen

to represent the inflow and outflow boundaries. It is noted that the free-surface elevation is determined numerically and is not known before the computation. Therefore, the main channel length upstream of the branch channel is lengthened to , and the branch channel length is prolonged to

, so that the fully developed flows are obtained near the downstream of the inlet boundaries. The length of main channel downstream of the branch channel is determined with 10W, where the flow depth is nearly constant

12 = 10.968mW

10 = 9.14mW

[1] (see Fig.1). Then, a grid independence study has been conducted to determine the appropriate number of grid points to be used by changing the total number of grid points. This grid is not fixed during computation due to the motion of the free surface. Instead, the grid distribution in the vertical direction is changing with the calculated free surface using a hyperbolic tangent stretching function given by Vinokur[23]. The nodes on the bed are fixed to facilitate the implementation of the wall function. The final grid chosen has a total of about 2 105 cells. A finer mesh used did not show an appreciable difference in the results. 3.2 Validation results

The distinctive characteristics of open-channel junction flow are a zone of separation immediately downstream of the junction and a descent in depth from the upstream contributing channels to the downstream channel. For a quantitative comparison, water surface elevation profiles at five transverse sections ( -direction) are shown in Fig.2 for the discharge ratio . The overall agreement between the prediction and experiment is fair and reasonable. It is seen that the surface elevation nearer to the branch channel (with less ) has more smart depression.

y= 0.250q

y

While a 2-D look at the junction flow provides an insight into the general flow patterns, a true model of an open-channel junction would have to be 3-D as the vertical velocity gradients and secondary currents are a significant aspect of this flow. Figure 3 shows the vector fields v - w for present calculation and experimental data at the cross section . It can be seen that the surface water approaches the opposite wall of junction, as a result of the lateral momentum being greater near the surface than that near the bed. Then, the surface water is deflected slightly downstream by the oncoming main channel water, and the flow travels downwards by the weight of the water itself. This motion creates a clockwise secondary current that is evident along the outer half of the channel ( ) when looking

= 1.333x

0.55 0.95y

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downstream. Overall, the numerical model predicts the flow pattern well and the position of vortex is also correctly predicted. Fig.2 Comparison of water surface elevation profiles

For a more complete comparison, the streamwise

velocity components, , at different cross sections in main channel are compared in Fig.4. It is seen that a good agreement between the predicted results and experimental data has been reached. In the upstream regions of main channel (at ), the velocity is evenly distributed across the section. Due to the momentum inflow of lateral channel and the influence of separation zone, the flow in the junction region shifts to the wall opposite to the junction and the flow distribution is quite non-uniform (see Figs.4.(b) and 4(c)). At the location of , it is apparent that higher velocities occur near the bed. This effect is attributed to the entrance angle of the lateral flow as will be discussed later. Downstream of the separation zone ( ), flow begins to recover from the

effect of separation and to distribute more evenly.

u

= 1.0x

= 0.25y

= 6.0x

Fig.3 Comparison of vector filed - v w

Fig.4 Comparison of streamwise velocity components at

different cross sections in main channel 4. Results and discussions of different discharge

ratios The results of previous studies indicated that the

characteristics of an open-channel junction flow, such as the shape index of separation zone, flow angle of lateral channel, contraction coefficients, and location of stagnation point, are quite dependent on the discharge ratio. The present study will focus on the effect of the discharge ratio on the flow characteristics to estimate the parameters needed for engineering applications. The simulation parameters and some results are shown in Table 1.

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Table 1 Simulation conditions and parameters

Case mQ (m3/s)

bQ (m3/s)

q Recirculation’s

width, D (m)

Recirculation’s length, L (m)

Shape index of separation

zone, /D L

Mean flow angle, cos

Contraction coefficient,

1 0.021 0.149 0.125 0.320 1.905 0.168 0.288 0.672

2 0.043 0.127 0.250 0.310 1.866 0.166 0.387 0.680

3 0.064 0.106 0.375 0.283 1.704 0.166 0.484 0.701

4 0.085 0.085 0.500 0.260 1.604 0.162 0.582 0.729

5 0.106 0.064 0.625 0.230 1.408 0.163 0.692 0.781

6 0.127 0.043 0.750 0.145 0.875 0.166 0.819 0.846

7 0.149 0.021 0.875 0.065 0.377 0.172 0.946 0.932

4.1 Mean flow pattern Figure 5 shows the calculated streamlines and

velocity patterns near the water surface for three different discharge ratios. The other cases are not shown here due to the similarity of results. It is seen that the size of the separation zone decreases as the discharge ratio increases. The results are consistent with the findings of Hsu et al.[13] and Weber et al.[1]. With decreasing discharge ratio, more flow energy loss occurs at the junction, as more lateral flow momentum needs to be turned into the main channel flow direction. More energy loss leads to a larger separation zone.

Additionally, it can be obviously seen from the streamlines that a “dividing line” develops between the two combining flows. It must be noted that the separation zone is strongly 3-D, and the “dividing line” will be at different stations at various vertical elevations. Therefore, this would be particularly true near the shear plane but not a shear line dividing the main and lateral flow. In experiments, the dye trajectory method is generally used to observe the location of this shear plane near the water surface for confirming the contracting section. Due to the strongly 3-D characteristic, it is difficult to determine the maximum flow constriction section. On the other hand, three-dimensional simulation can easily determine it. Compared the streamlines for different discharge ratios, Fig.5 shows that with the increase of discharge ratio, , the dividing line shifts to the left q

side of the main channel as a result that the contracted flow region in the vena contracta become smaller. The contraction coefficient will be discussed later. Fig.5 Streamlines and velocity pattern near water surface

As can be seen from the velocity distributions in Fig.5, lower velocities in the junction region are

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resulted from the reduced contraction of the main channel flow due to a smaller zone of separation. It is noted that the flow is separated immediately downstream of the junction as small positive velocity exists in the area. The velocity outside the separation zone is very high due to the separation, which could be a major concern in the design of erodible channel junction. Moreover, there is a low velocity region near the upstream corner of the branch channel entrance as shown in Fig.1. The minimum velocity point is the location of the stagnation point, which depends on the discharge ratio. As increasing, the location of the stagnation point migrates from the left bank corner of upstream main channel to the right bank corner of branch channel.

q

4.2 Shape index of separation zone Based on the simulation results (Fig.5 and Table

1), the surface shape index of separation zone, the ratio of the recirculation’s width, D , to its length, for various is shown in Fig.6. For validation, the values at the water surface observed by Hsu et al.

Lq

[13], Best and Reid[9] are also plotted in this figure. It is seen that although the size of separation zone decreases with the discharge ratio increasing, the shape index is approximately the same for various discharge ratios with a mean value of 0.166. This result correlates fairly well with the experimental data obtained by Hsu et al., who reported a mean value of 0.17. Whereas, the results observed by Best and Reid are somewhat scattered with a mean value of 0.19. The scatter in the data may be attributed to measurement difficulties associated with the unstable separation streamline and the three-dimensional nature of junction flow. Fig.6 Surface shape index of separation zone

4.3 Flow angle

The depth-averaged flow angles, , across the branch channel entrance for various discharge ratios are plotted in Fig.7. It reveals that large discharge ratios are associated with smaller depth-averaged flow angles and are less uniformly distributed across the

branch channel entrance. Flow deflection at the upstream corner of the branch channel entrance increases with the discharge ratio. In engineering applications, correct estimation of the cross-sectional mean flow angle at branch channel entrance affects the predicted values of momentum transfer, when 1-D mathematical model is used to solving junction flow. Therefore, the cross-section mean flow angles, , for various discharge ratios are calculated by taking an average of the depth-averaged flow angles across the branch channel entrance based on the simulation results (see Fig.8). It can be seen that the cross-section mean flow angles correlate fairly well with the discharge ratio, and it can be empirically expressed in terms of discharge ratios as cos = 0.1643 + 0.8711q (13)

The correlation coefficient . 2 = 0.996R Fig.7 Depth-average flow angle across branch channel entrance Fig.8 Cross-sectional mean flow angle versus q

The experimental data of Hsu et al.[13] and

Ramamurthy et al.[10] are also presented in Fig.8. In Hsu’s experiments, the local flow angle was determined through measured velocity vectors: eight velocity profiles were taken across the branch channel entrance with eight evenly spaced measuring points in

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each profile. Then, the result was calculated by taking an arithmetic average of the measured local angles, and they formulated the relationship as cos = 0.149 + 0.912q (14) The results of present study agree fairly well with the experimental data of Hsu et al.. It should be noted that Ramamurthy et al. obtained their data indirectly through measurement of pressure forces on sidewalls and the momentum flux of the branch channel. Therefore, their results are larger than the values of Hsu et al.. 4.4 Contraction coefficients

To develop a 1-D formulation of open-channel junction flow based on the energy and momentum theoretical models, another significant parameter is the contraction coefficient, , at the maximum flow constriction. In many junction flows, the contraction coefficient at the maximum flow constriction is difficult to be estimated due to the strongly three-dimensional characteristics at the separation zone. However, the desired solutions can be determined conveniently from the results of three- dimensional simulation. The variation of the contraction coefficient with the discharge ratio of the present study is shown in Fig.9 and Table 1. For comparison, the results of Hsu et al.[13] and Hager et al.[11] are also presented in the plot. It indicates that the simulated values of the present study agree well with the values obtained by experiments. It can be seen from Fig.9 that the contraction coefficients are simply a function of the discharge ratio. With the discharge ratio increasing, larger contraction coefficient occurs at the vena contracta as less lateral flow momentum is turned into the main channel flow direction. Fig.9 Contraction coefficients at vena contracta for various q

5. Conclusion

The 3-D two-equation turbulence model

faithfully reproduces the mean flow characteristics of 90o equal-width open-channel junction. The hyperbolic tangent stretching function is adopted to regenerate the mesh for obtaining the free-surface elevation. The numerical results agree well with the experimental data of Weber et al. The observed important hydrodynamic characteristics, such as the appreciable change in free-surface elevation and the secondary current, are reproduced faithfully. Then, the effect of discharge ratio on the flow characteristics is investigated for estimating the parameters needed in engineering applications. For more validation, the simulated results are also compared with previous studies and fairly agreement is reached. The numerical model can be used to conveniently obtain results such as depth and contraction coefficients, cross-sectional mean flow angles, and shape index of separation zone. Compared to experimental procedures, numerical modeling is less time-consuming and less expensive. References

[1] WEBER L. J., SCHUMATE E. D. and MAWER N.

Experiments on flow at a 90o open-channel junction[J]. J. Hydr. Engrg. , ASCE, 2001, 127(5): 340-350.

[2] WEBBER N. B., GREATED C. A. An investigation of flow behavior at the junction of rectangular channels[C]. Proc. Instn. Civ. Engrs. London: Thomas Telford Ltd., 1966, 34: 321-334.

[3] MODI P. N., DANDEKAR M. M. and ARIEL P. D. Conformal mapping for channel junction flow[J]. J. Hydr. Div., ASCE, 1981, 107(12): 1713-1733.

[4] GURRAM S. K, KARKI K. S. and HAGER W. H. Subcritical junction flow[J]. J. Hydr. Engrg., ASCE, 1997, 123(5): 447-455.

[5] YUAN Fei, REN Li-liang and YU Zhong-bo et al. A river routing model based on digital drainage networks[J]. Journal of Hydrodynamics, Ser. B, 2005, 17(4): 483-488.

[6] MU Jin-bin, ZHANG Xiao-feng. Real-time flood fore- casting method with 1-D unsteady flow model[J]. Journal of Hydrodynamics, Ser. B, 2007, 19(2): 150-154.

[7] ZHANG Ming-liang, SHEN Yong-ming. Study and application of steady flow and unsteady flow mathematical model for channel networks[J]. Journal of Hydrodynamics, Ser. B, 2007, 19(5): 572-578.

[8] NI Han-gen, LIU Ya-kun. Transitional flow in channel junctions[J]. Journal of Hydrodynamics, Ser. B, 2004, 16(4): 464 - 467.

[9] BEST J. L., REID I. Separation zone at open-channel junctions[J]. J. Hydr. Engrg., ASCE, 1984, 110(11): 1588-1594.

[10] RAMAMURTHY A. S., CARBALLADA L. B. and TRAN D. M. Combining open-channel flow at right-angled junctions[J]. J. Hydr. Engrg., ASCE, 1988, 114(12): 1449-1460.

[11] HAGER W. H. Transitional flow in channel junctions[J]. J. Hydr. Engrg., ASCE, 1989, 115(2): 243-259.

[12] KUMAR S. G. Transitional flow in channel junctions[J]. J. Hydr. Res., 1993, 31(5): 601-604.

549

[13] HSU C. C., WU F. S. and LEE W. J. Flow at 90o

equal-width open-channel junction[J]. J. Hydr. Engrg., ASCE, 1998, 124(2): 186-191.

[14] HSU C. C., LEE W. J. and CHANG C. H. Subcritical open-channel junction flow[J]. J. Hydr. Engrg., ASCE, 1998, 124(8): 847-855.

[15] BIRON P., BEST J. L. and ROY A. G. Effects of bed discordance on flow dynamics at open-channel confluences[J]. J. Hydr. Engrg., ASCE, 1996, 122(12): 676-682.

[16] WANG Li-yang, WU Ying-xiang and ZHENG Zhi-chu et al. Oil-water two-phase flow inside T-junction[J]. Journal of Hydrodynamics, 2008, 20(2): 147-153.

[17] HUANG J., WEBER L. J. and LAI Y. G.. Three- dimensional numerical study of flows in open-channel junctions[J]. J. Hydr. Engrg., ASCE, 2002, 128(3): 268-280.

[18] CZERNUSZENKO W., RYLOV A. Modeling of three- dimensional velocity field in open channel flows[J]. Journal of Hydraulic Research, 2002, 40(2): 135-143.

[19] SHABAYEK S., STEFFLER P. and HICKS F. Dynamic model for subcritical combining flows in channel junctions[J]. J. Hydr. Engrg., ASCE, 2002, 128(9): 821-828.

[20] RAMAMURTHY A. S., QU Jun-ying and ZHAI Chao. 3D simulation of combining flows in 90o rectangular closed conduits[J]. J. Hydr. Engrg., ASCE, 2006, 132(2): 214-218.

[21] DUAN Ya-li, LIU Ru-xun. Lattice Boltzmann simul- ations of triangular cavity flow and free-surface problems[J]. Journal of Hydrodynamics, Ser. B, 2007, 19(2): 127 -134.

[22] WILCOX D. C. Turbulence modeling for CFD[M]. 1st Ed., La Cacada, California USA: DCW Industries Inc., 1993.

[23] VINOKUR M. On one-dimensional stretching functions for finite-difference calculations[J]. J. Comput. Phys., 1983, 50(2): 215-234.