LI Zhi-wei, HUAI Wen-xin, QIAN Zhong-dong, ZENG Yu-hong, YANG Zhong-hua

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China, E-mail: lizhiwei0114@163.com

NUMERICAL STUDY OF FLOW AND DILUTION BEHAVIOR OF RADIAL WALL JET*

LI Zhi-wei, HUAI Wen-xin, QIAN Zhong-dong, ZENG Yu-hong, YANG Zhong-hua

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China, E-mail: lizhiwei0114@163.com

(Received January 6, 2010, Revised March 20, 2010)

The radial wall jet is a flow configuration that combines the radial jet and the wall jet. This article presents a simulation of the radial wall jet by applying the transition Shear-Stress Transport ( SST) model. Tanaka’s experimental data are used for validation. The computed velocity profiles agree well with the experimental ones. The distributions of the velocity on cross-sections show a similarity in the main region and the profiles are different with those of the free radial jet or the wall jet, because the presence of the wall limits the expansion of the jet. By introducing the equivalent nozzle width, the maximum velocity decays and the half-width distributions are normalized, respectively. In addition to compare the flow field with experiments, this paper also analyzes the dilution effect of radial wall jets in terms of the concentration distributions. The concentrations on the wall keep constant within a certain distance from the nozzle. And the concentration distributions also show a similarity in the main region. Both the decays of the maximum concentration and the distributions of the concentration half-width fall into a single curve, respectively. The dilution effect of radial wall jets is thus verified.

radial wall jet, velocity profile, concentration distribution, concentration half-width

1. Introduction

The jet is a fluxion that flows from one region to another in specific export forms and has wide applications in hydraulic and hydro-power engineering, aerospace engineering, environment engineering and other related fields. It can make fluid mix with each other substantially, especially, the radial jet and the wall jet, whose dilution effect is more effective. The radial jet is a flow directed along the radial direction after discharging from an exit, while the wall jet is a jet spreading out over a plane or curved surface. There are quite a number of studies in this field. For the radial jet, we may quote Tanaka and Tanaka[1], Guo and Sharp[2], Hunt and Ingham[3], Song and Abraham[4], Krejci and Kosner[5], and Li et al.[6], for the wall jet, we have Law and Herlina[7], Fureby and Grinstein[8], Dejoan and Leschziner[9], Fan et al.[10], Huai et al.[11]. However, the radial wall jet, as a flow configuration that combines the above two jets, has not been well investigated. Tanaka and Tanaka[12]studied it through changing the nozzle opening and discharge velocity by experiments. In this article, a comprehensive numerical simulation of the radial wall jet is carried out, covering most cases of the conditions in Tanaka’s research, and the calculated results of the velocity field are in good agreement with those in their experiments. Furthermore, a species transport equation is added to the model and the dilution effect of the radial wall jet is analyzed in terms of spreading of tracer substance. Subsequently, it is found that the concentration distributions and the concentration half-width are related with the distance from the wall or the jet exit, and the corresponding relationships are obtained.

Fig.1 Sketch of model and coordinate system

2. Mathematical model and calculation method

class (i.e., the gradients perpendicular at the boundary are chosen to be equal to zero).

(3) On the axis-symmetry line:

When r =0, ?3 .5 m ≤r ≤3.5 m .

(4) Wall boundary:

When y=0, D0/2 ≤r≤415m.

The no-slip condition and the acrylic wall are assumed. Because the first near-wall node is placed at y+≤ 1, the wall boundary conditions for u , v , k, Care treated in the same way as the ones in the enhanced wall function. The value ofω is specified as:

2.3 Computational method

From Tanaka’s[12]experimental data, we select 10 cases and list them in Table 1. The Finite Volume Method (FVM) is used to discretize the governing equations. The Semi-Implicit Method for Pressure-Linked Equations Consistent (SIMPLEC) algorithm is used for the pressure-velocity coupling. We use the bounded second order upwind based discretization for all equations. The calculation is considered convergent when the residual is less than 1×10-5for the governing equations.

Table 1 Calculation conditions

3. Results and discussions

3.1 Velocity characteristics

In this section, we analyze the velocity distributions on certain cross-sections and the decay of the maximum velocity along the radial direction.

Fig.2 Distributions of velocity

Fig.3 velocity distribution in main region

The velocity profile of the radial wall jet varies more slowly than the planar turbulent wall jet (Classical Wall Jet (CWJ)) obtained by Verhoff (1963)

Fig.4 Maximum velocities distribution along r-axis

Fig.5 distribution of maximum velocity in r-axis

3.1.3 Distributions of half-width

The distributions of velocity half-width b on some sections are plotted in Fig.6. These values fall on the same line in the main region ( x/ L≥ 12),which can be fitted into the curve as follows:

This result agrees with what obtained by Bakke (1957) as follows:

Fig.6 Growth of half-width of velocity in r-direction

But the calculated result is different from that obtained by Tanaka. The reason is that the x in this article is the distance from the nozzle exit while that in Tanaka’s is the radial distance from the virtual origin.

Fig.7 Lateral distributions of concentrations

3.2 Dilution characteristics

3.2.1 Concentrations on cross-sections

The distributions of dilutions (C /C0) on some sections are shown in Fig.7. It is shown that the maximum concentration keeps constant in some distance from the wall in the near-field (i.e., the concentration at the wall has not decayed). After some distance from the jet outlet, the maximum concentrations begin to decay. But the spread widthenlarges with the increase of the distance from the jet exit. All results fall on a curve shown in Fig.8, if C/ Cmis used as the normalized parameter. From this curve the relationship can be obtained as follows:

3.2.2 Decay of the maximum concentration

Figure 7 shows that the maximum values gradually decrease after some distance from the jet exit. In this region, the maximum values appear at the wall. This region is defined as the concentration main-region. The normalized maximum concentration Cm/C0against x/ L is plotted in Fig.9. They are different for different nozzle openings. Replacing the exit width L by the equivalent nozzle widthA, all the maximum values would lie on the same curve expressed as:

Fig.8 concentration distributions in main region

Fig.9 Maximum concentration distribution along r-axis

As a reference, the maximum concentration decays of a radial free jet by Li[13]is also plotted as the dotted line in Fig.10.

Fig.10 Decays of maximum concentration in r-direction

Fig.11 Concentration boundary-layer thickness variations

Fig.12 Decays of concentration boundary-layer thickness

3.2.3 Concentration boundary layer

From results in Fig.7, it is discovered that the concentrations keep the maximum value within a certain range from the wall in the near-field of concentration. This range is defined as the concentration boundary layer and its thickness is represented by a parameter δc. The concentration boundary layer thickness is defined as the distance from the wall or the boundary to the point where the external concentration reaches 99% of the local maximum. Figure 11 shows the distributions of the concentration boundary layer thickness normalized by nozzle width L, which shows that they are related withD0/L. They assume a more gentle variationwith the increase of D0/L. The results are rearranged in Fig.12 by replacing X /A with x/ L, in which data points lie on a single curve when X /A≥0.1and the corresponding relationship can be expressed as

3.2.4 Distributions of concentration half-width

In this section, the concentration half-widthbcis defined in a similar way as the velocity half-width. It is equal to the ordinate where C=Cm/2. Figure 13 shows the distributions of the concentration half-width bc/L against the parameter x/L. It can be seen that the bc/L lies around a single curve whenx/ L≥15 and the following relationship can be obtained:

Fig.13 Growth of concentration half-width

Fig.14 Variations of bc? δcalong r-directon

With the new dimensionless variables (bc? δc)/A and x /A, all data lie around a single line as shown in Fig.14, where the semi-logarithmic coordinates are used. The results are in good agreement with the following relationship:

For reference, the concentration half-width growths of a radial free jet (δc=0) are also plotted in Figs.13 and 14, respectively.

4. Conclusions

In this article, the Transition SST model is used to simulate the radial wall jets. The results of the flow field are in good agreement with those of Tanaka’s experiments. And the dilution effect of the radial wall jets is evaluated. The following conclusions are obtained:

(1) The velocity distribution on cross-sections shows a self-similarity in the main region and its profile is expressed by formula (9), as is different from that of the CWJs.

(2) The decay of the maximum velocity and the growth of the half-width are proportional to the distance along the radial direction, which can be expressed by formulas (11) and (12), respectively.

(3) The concentration profile also shows a similarity after a certain distance from the nozzle and the corresponding relationship is expressed by Eq.(14).

(4) The concentration boundary-layer thickness is related with the distance along the x-axis. Its decay follows well with Eq.(16).

(5) The distributions of the maximum concentration and the concentration half-width fall into a single curve, respectively, by using the equivalent nozzle width, as expressed by formulas (15) and (17).

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10.1016/S1001-6058(09)60103-7

* 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 No. 10972163).

Biography: LI Zhi-wei (1985- ), Male, Ph. D. Candidate

HUAI Wen- xin, E-mail: wxhuai@whu.edu.cn