NUMERICAL SIMULATION OF HORIZONTAL BUOYANT WALL JET*

HUAI Wen-xin, LI Zhi-wei, QIAN Zhong-dong, ZENG Yu-hong, HAN Jie

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China, E-mail: wxhuai@whu.edu.cn

PENG Wen-qi

Department of Water Environment, China Institute of Water Resources and Hydropower Research, Beijing 100038, China

(Received June 14, 2009, Revised August 16, 2009)

NUMERICAL SIMULATION OF HORIZONTAL BUOYANT WALL JET*

HUAI Wen-xin, LI Zhi-wei, QIAN Zhong-dong, ZENG Yu-hong, HAN Jie

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China, E-mail: wxhuai@whu.edu.cn

PENG Wen-qi

Department of Water Environment, China Institute of Water Resources and Hydropower Research, Beijing 100038, China

(Received June 14, 2009, Revised August 16, 2009)

This article applies the realizable k-ε model to simulate the buoyant wall jet and gives the results of cling length, centerline trajectory and temperature dilutions at certain sections. The comparison between the numerical results and Sharp’s experimental data indicates that the model is effective in estimating velocity distribution and temperature dilutions. The velocity profiles at the cental plane and z-plane both show a strong similarity at certain distance from the nozzle, and the distributions of velocity and temperature dilutions also exhibit a similarity along the axial direction at centerline in the near-field. Based on the results, the article gives the corresponding relationships between the distance and the dilutions of velocity and temperature, which is useful in predicting the behavior of the wall buoyant jet.

buoyancy, wall jet, cling length, velocity profile, temperature dilution

1. Introduction

Many industrial heat wastewaters, which are discharged into the receiving water after primary treatment through pipelines, can be considered as jets. Recently, single or multi-circular jets[1-3]have been frequently applied to coastal sewage disposal projects, while their dilution of the near field is less effective than that of wall buoyant jets. Because the wall buoyant jets have a larger mixing zone than circular jets, it can effectively improve the initial dilution of the water flow field. A complete theoretical system about buoyant jets has been formed[4-7], while the one about wall jets is still being studied[8-12]. In this article, we take into account the effect of buoyancy and wall boundary on the jet. The Coanda effect[13]of buoyant wall jets can increase the mixing zone between wastewater and ambient water so that the pollution in the near field can be effectively diluted. Sharp[14,15]gave the relations between the cling length and densimetric Froude number through experimental research and theoretical analysis. Combining with Sharp’s experimental data and theoretical results, this paper has comprehensively studied the flow and temperature field. The results agree well with Sharp’s experiments in cling length, centerline trajectory and temperature dilutions. Furthermore, the present article obtains the distributions of velocity and temperature dilution and then gives the corresponding relationships between the distributions and thedistance from the nozzle.

2. Mathematical model and calculation method

2.1 Governing equations

A wall buoyant jet with the initial velocity U0, the temperature T0, the densityρ0, the Reynolds number Re0and the nozzle diameterD, is discharged into the static ambient water with an infinite depth, the temperature Taand the density ρ0. Generally, we should use the Boussinesq approximation, which states that density difference only works on the gravitational component in the fluid with small density variation. The sketch of model and coordinate system is shown in Fig.1. The governing equations in the Cartesian coordinate system are given as follows (where u, v, w are respectively the components of mean velocity in thex, y, z directions):

(1) Continuity equation

Fig.1 Sketch of model and coordinate system

(2) Momentum equations

(3) Temperature equation

(4) k-equation

(5) ε-equation

where p is the fluid pressure,g the acceleration of gravity, k the turbulence kinetic energy,ε the turbulence dissipation rate, Gkthe generation of turbulence kinetic energy due to the mean velocity gradients andPtr the Prandtl number, Prt=0.85,νe=ν+νt, ν=μ/ ρ, νt=μe/ρ, μ is dynamic viscosity of water, and μ=Cρk2/ε the turbulent viscosity ,tμ Cμ=1/(A0+ASU∗k/ε) the function of the mean strain and rotation rates, where A0=4.04,in static ambient water, the angular velocity of reference frame ωk=0, soScthe Schmidt number of heat transfer, Sc=0.7, C2=1.9, C3=1.44, ΔT=T0-Ta,α the thermal expansion coefficient of water, which changes with water temperature. Here α is determined by Batchelor’s function[16]:

where T is the temperature of local water.

2.2 Boundary conditions

This study only computes half of wall jet domain due to the symmetry of the nozzle.

(1) Inlet boundary

(2) Outlet boundary

Regarding the flow at outlet section as a fully developed turbulent flow leads to the second class of boundary conditions for u, v, w, k,ε, T, i.e., the gradients perpendicular to the boundary are zero.

(3)On the plane of symmetry

As y=0, v=0,

(4) Wall boundary

As z=0, w=0 (i.e., no-slip condition) and adiabatic wall condition are adopted. u, v, k,ε are assumed to be standard wall functions[17].

2.3 Computational method

The incompressible steady flow is considered in this article. Referring to the experimental data given by Sharp[14], we selected 25 cases and listed them in Table 1. The Finite Volume Method (FVM) combined with structured grids is applied to discretize the governing equations. And SIMPLEC scheme is used for the coupling between pressure and velocity. The computation is convergent as the residual is smaller than 1×10-6for the energy equation and 1×10-5for the other governing equations.

3. Cling length

The region near the wall can not provide enough fluid for the entrainment of the jet, so the region will have lower pressure than that on the outer side of the jet. The difference of pressure causes the jet cling to the wall and slip for some distance before it leaves the floor. As the velocity decreases, the suction pressure becomes lower and the buoyancy dominates the flow, which will leave the wall when the buoyancy forces overcome the pressure difference. Therefore, the wall buoyant jet can be divided into three regions (Fig.2): initial region, wall jet region and free jet region. The initial region is the region where the maximum velocity keeps constant, and the one behind the wall jet region is the free jet region, which behaves similarly to the free jet. According to the position of the jet lifting off, the wall jet region is further divided into Regions Ι and ΙΙ. The region between the end of initial region and the location of the jet lifting off at central plane is named as wall jet Region Ι. And the other region of the wall jet region is wall jet Region ΙΙ. In the wall jet Region Ι, the distribution of velocity and dilution exhibit similarity. For the convenience of describing the wall buoyant jets, we define the initial region and wall jet Region Ι as the near-field.

Table 1 Calculational conditions

Fig.2 Regions of flow

4. Trajectory of centerline

The centerline trajectory given by numerical computation, experiments and theoretical solutions of Sharp and Abraham at the central plane in the free jet region are shown in Fig. 4. And the computed results are close to experimental result. The comparisons indicate that the wall buoyant jet clings to the floor and slips for some distance before it lift off, and theposition that the centerline leaves wall is closer to the nozzle. The distance between the nozzle and the position where the trajectory leaves the floor is L′ and is determined by L′=2DFd.

Fig.3 Length of cling

Fig.4 Comparison of centerline trajectories

Fig.5 Comparison of temperature dilutions

5. Temperature dilutions

The temperature dilution can be defined as S=(T0-Ta)/(T-Ta), which is similar to the definition for the dilution of thermal buoyant jets in co-flowing shallow water[19]. T0is the initial jet temperature, Tais the ambient water temperature, and T is the temperature on the numerical mesh node. Figure 5 shows the temperature dilutions along the centerline trajectory for several cases at the cross-section z/ D=35 and z/ D=70. The comparisons between experimental results and theoretical solutions of the dilutions can be found in Ref.[12]. Figure 6 shows the temperature dilution profiles of Case 3 at horizontal plane of z/ D=70 and z/ D =100. The profiles appear to be pear-shaped, and the other cases also have the similar shape. The conclusion agrees well with that of Sharp’s[15].

Fig.6 Dilution contours at horizontal sections (z/ D=70, z/ D =100) of Case 3

In view of the facts mentioned above, the cling length, trajectory of centerline and temperature dilutions agree well with Sharp’s experimental results and theoretical solutions. Other numerical results obtained by realizable k-ε model will be given next.

6. Characteristics of dilutions

The contours of temperature dilutions at central plane (y=0) (Fig.7) illustrates that the dilutions increase with the increasing distance and are related to the jet outlet diameterD and the densimentric Froude number Fd.

Fig.7 Contours of temperature dilutions at central section for Cases 4, 6 and 11

Temperature dilutions of some cases in the near-field change along the centerline (Fig.8), and obey the same discipline in the zone of wall jet Region Ⅰ. The article gives the best-fit curve in this region as:

The dilution almost keeps constant at the distance of 0＜x＜5D, which can be defined as the boundary of the initial jet region.

Fig.8 Dilutions change along trajectory in the near-field

Fig.9 Velocity profiles in central plane in near field for Cases 2 and 10

7. Characteristics of velocity

7.1 Velocity profiles in central plane

Velocity profiles in the central plane in the near-field for Cases 2, 3, 10 and 12 are exhibited in Figs.9 and 10. Both figures take um/umoas the abscissa and z/ zm/2as the ordinate, where umis the velocity component in thex direction at z of central plane, umois the maximum velocity of umand its ordinate iszm. zm/2is the velocityhalf-height, that is, the ordinate of um=umo/2. The velocity profiles for Cases 2 and 10 (Fig.9) exhibit similarity after 5D, which indicates that the range of initial region is 0＜x＜5D . Figure 10 shows the velocity distributions of Cases 3 and 12 as x/(DFd) equals the some values. As x＜2DFd, the velocity profiles for different cases behave similarly to each other. Furthermore, the profiles yield the position where centerline leaves the wall is L′=2DFd. The velocity profiles exhibit a strong similarity in the range of 5D＜x＜2DFd, and other cases also have the same shape. The empirical equation (Eq.(10)) given by Verhoff (1963) for planar turbulent wall jet[18]is also plotted in the Figs.9 and 10, which bothare in good agreement and illustrate that velocity profiles are the same as that of the one for planar turbulent wall jet in the wall jet Region Ι.

Fig.10 Velocity profiles in central plane in near field for Cases 3 and 12

7.2 Velocity distribution in z-plane

Figures 11 and 12 give the velocity profiles at the z-plane with dimensionless coordinate u/ umoand y/ b, where u is the velocity component at y in the plane ofz=zm, and b is velocity-half-weight, that is, the ordinate of u=umo/2. Both the figures indicate that the velocity profiles ofx/ D or x/(DFd) behave similarly when x＞5D and also show the similarity not only in the wall jet region but also in the free jet region. The article fits the distribution into the following function:

Fig.11 Velocity profiles in the z-piane of Case 1

Fig.12 Velocity profiles in the z-piane of Case 9

Fig.13 Decay of centerline velocity in the near-field

7.3 Decay of centerline velocity

Figure 13 shows the decay of centerline velocity in the near-field of several cases. We use U0/umoas theordinate andas the abscissa because cling length is related to the densimetric Froude number Fd. The figure shows that the centerline velocities have a similarity, and the other cases are similar. We fit the profile to the function

8. Conclusions

The buoyant wall jets can broaden the interface between polluted water and receiving water due to the Coanda effect and enhance the entrainment and turbulence in receiving water to form an intense dilution zone. So we can use it to rapidly improve near-field dilution for waste water discharging and to enhance the dilution effect of the sewage diffuser.

This article uses the realizable k-ε model to simulate the wall buoyant jet and compares the numerical results of the cling length, centerline trajectory and dilutions with Sharp’s experimental data and theoretical results. The main conclusions are reached as follows:

(1) Cling length is proportional to the outlet diameter D and initial densimetric Froude number Fd(i.e., L/ D=3.2Fd).

(2) Temperature dilutions are related to the distance from outlet x, outlet diameter D and densimetric Froude number Fd. And the decay of temperature dilutions along with the axis in the wall jet Region Ι. fits to the Eq.(9).

(3) Velocity profiles on central surface are similar to that of planar turbulent wall jet, and are in good agreement with the classical wall jet curve.

(4) Velocity profile exhibits excellent similarity in the z-plane after certain distance (x＞5D), and fits to Gaussian shape. Function is the Eq.(11),

(5) The decay of centerline velocity is related to the outlet diameter D and densimetric Froude number Fd, and fits to the Eq.(12).

[1] XU Jing-lei, SHA Jiang and LIN Chun-feng et al. PIV experimental research of instantaneous flow characteristics of circular orifice synthetic jet[J]. Journal of Hydrodynamics, Ser. B, 2007, 19(4): 453-458.

[2] YIN Zhao-qin, ZHANG Hong-jun and LIN Jian-zhong. Experimental study on the flow field characteristics in the mixing region of twin jets[J]. Journal of Hydrodynamics, Ser. B, 2007, 19(3): 309-313.

[3] FAN Jing-yu, ZHANG Yan and WANG Dao-zeng. Experimental study on the vortex formation and entrainment characteristics for a round transverse jet in shallow water[J]. Journal of Hydrodynamics, 2009, 21(3): 386-393.

[4] LI Wei, HUAI Wen-xin. Theory and application of buoyant jets[M].Beijing: Science Press, 1997(in Chinese).

[5] KUANG C. P., LEE J. H. W. Effect of control on stability and mixing of vertical plane buoyant jet in confined depth[J]. Journal of Hydraulic Research, 2002, 39(4): 375-391.

[6] BALASUBRAMANIAN V., JAIN S. C. Horizontal buoyant jets in quiescent shallow water[J]. Journal of the Environmental Engineering Division, 1978, 104 (4): 717-730.

[7] HUAI Wen-xin. Numerical study of stability of horizontal round buoyant jets in quiescent shallow water [J]. Journal of Hydrodynamics. Ser. A, 2002, 17(1): 39-45(in Chinese).

[8] LAW A. W. K., HERLIN A. An experiment study on turbulent circular wall jets[J]. Journal of Hydraulic Engineering, 2002, 128(2): 161-174.

[9] JIN Ming, GUO Zi-zhong. A simple model for calculation of turbulent wall jet[J]. Advances in Hydrodynamics, 1987, 2(4): 79-88(in Chinese).

[10] VILLAFRUELA J. M., CASTRO F. and PARRA M. T. Experimental study of parallel and inclined turbulent wall jets[J]. Experimental Thermal and Fluid Science, 2008, 33(1): 132-139.

[11] EAD S. A., RAJARATNAM N. Plane turbulent wall jets in shallow tailwater[J]. Journal of Engineering Mechanics, 2002, 128(2): 143-155.

[12] FARUQUE M. A. A., SARATHI P. and BALACHANDAR R. Clear water local scour by submerged three-dimensional wall jets: Effect of tailwater depth[J]. Journal of Hydraulic Engineering, 2006. 132(6): 575-580.

[13] SOBEY R. J., KEANE R. D. Horizontal round buoyant jet in shallow water[J]. Journal of Hydraulic Engineering. 1988, 114(8): 910-929.

[14] SHARP J. J. The use of a buoyant wall jet to improve the dilution of a submerged outfall[J]. Proc. Instn. Civ. Engrs (London), Part 2, 1975, 59(2): 527-534.

[15] SHARP J. J., VYAS B. D. The buoyant wall jet[J]. Proc. Instn. Civ. Engrs. (London), Part 2, 1977, 63(2): 593-611.

[16] BATCHELOR G. K. An introduction to fluid dynamics[M]. Cambridge: Cambridge University Press, 2000.

[17] HUAI Wen-xin. 3-D mathematical modeling and its calculational method for heated water jets in co-flowing stream[J]. Journal of Wuhan Univversity of Hydrdulic and Electric Engineering, 1997, 30(4): 1-5(in Chinese).

[18] YU Chang-zhao. Turbulent jets[M]. Beijing: Higher Education Press, 1993(in Chinese).

[19] HUAI Wen-xin. Behavior of near field for 3-dimensional buoyant jets in shallow co-flowing receiving waters[J]. Journal of Hydrodynamics. Ser. A, 1998, 13(1): 79-87(in Chinese).

10.1016/S1001-6058(09)60028-7

* Project supported by the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20070486021), the State Water Pollution Control and Management of Major Special Science and Technology (Grant No. 2008ZX07104-005) and the National Natural Science Foundation of China (Grant Nos. 50679061, 10972163, 50709025).

Biography: HUAI Wen-xin (1963-), Male, Ph. D., Professor