CHEN Jian-gang, ZHANG Jian-min, XU Wei-lin, WANG Yu-rong

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China, E-mail: chenjg@yeah.net

SCALE EFFECTS OF AIR-WATER FLOWS IN STILLING BASIN OF MULTI-HORIZONTAL SUBMERGED JETS*

CHEN Jian-gang, ZHANG Jian-min, XU Wei-lin, WANG Yu-rong

State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China, E-mail: chenjg@yeah.net

(Received April 5, 2010, Revised October 16, 2010)

A series of experiments were carried out on multi-horizontal submerged jets with four different model scales of 1:36, 1:57, 1:80, 1:200. In routine tests, scale effects have to be considered, due to complex vortex structure and strong air entrainment in stilling basin. Our focus was laid on measuring and analyzing the time-averaged pressure distribution, water depth and closed-to-bed velocity in the stilling basin of multi-horizontal submerged jets. The experiments results show that the model scale has but a slight effect on the time-averaged hydraulic characteristics in the stilling basin of multi-horizontal submerged jets, which indicates that the results of time-averaged hydraulic characteristics for a normal pressure model are reliable. However, the scale effects of air entrainment, fluctuation pressure and vortex structure call for further investigations in order to make the results of experiments serve as scientific references for practical engineering.

multi-horizontal submerged jets, Xiangjiaba hydropower station, scale effects, experimental investigation, hydraulic characteristics

1. Introduction

In physical model experiments, it is impossible to satisfy all the similarity criteria. Generally, the Froude similitude is adopted in the experimental investigation. Therefore, scale effects have to be considered in extrapolating the experimental results to the prototype to ensure the reliability of the prototype hydropower project designed based on the model experiment. There are three issues related with the scale effects: changes of flow pattern, variations of surface tension and changes in environmental pressures. It has been shown that[1-3]the roughness similitude conditions can often be ignored when the difference of the water surface profile in the spillway between prototype and model is small, otherwise certain compensation and amendments should be made to minimize the scale effects caused in this respect.

A hydraulic jump stilling basin was adopted in Xiangjiaba Hydropower Station, as the ski-jump energy dissipater with a serious atomization. However, the velocity near the slab of stilling basin reaches 40 m/s because of high water head and large flood discharge, which poses a serious threat to the hydraulic structure. According to the requirements of flood discharge and energy dissipation, the multi-horizontal submerged jets were proposed as a new type of energy dissipater, which was much studied with some useful conclusions.

Rajanamam and Subramanya[4], Hager[5]and Ohtsu et al.[6]Zeng[7], Liu and Ni et al.[8], Fu et al.[9]and Monhamed[10]performed a series experimental and numerical studies of the characteristics of hydraulic jump stilling basin with abrupt drops or symmetric expansions. Katakam[11]studied the characteristics of hydraulic jump stilling basin with abrupt drops and sudden enlargement. It is found that the relative energy loss in this stilling basin is larger than that in a sudden enlargement type basin or a abrupt drop type basin. Zhang et al.[12,13]derived atheoretical formula for the sequent depth and energy dissipation ratio for submerged hydraulic jumps in a stilling basin. Deng et al.[14,15]and Li et al.[16,17]performed a series of experimental studies of the characteristics of multi-horizontal submerged jets. It is found that the flow patterns are fairly stable with respect to the change of the tail water level under the condition of a large aspect ratio in stilling basin. Sun et al.[18]found that the contracted orifice width would result in a stable flow pattern in stilling basin, increase the energy loss and reduce the atomization. Huang et al.[19]studied the effect of drop height on the velocity near the slab in stilling basin. Yang et al.[20,21], Gao et al.[22], Li et al.[23]and Chen et al.[24]studied the velocity field in stilling basin by numerical simulations. It is found that there is no perforative vertical whirl in stilling basin and the vortex would dissociate on the slab, instead of fixing at a place. These results provide some important guidance for practical engineering applications.

In this article, the characteristics of the hydraulic jumps in stilling basin of multi-horizontal submerged jets are studied with four models of different scales. In order to identify the effects of scales and related laws, this article considers the following four different scales: 1:200, 1:80, 1:57, 1:36.

2. Theoretical analysis

2.1 Froude similitude

In routine model tests, gravity is considered as the leading force, and the Froude similitude is to be satisfied, which means Frp=Frm, that is,v2p/gpHp= vm2/gmHm, where Fr is Froude number,g is gravity and H is the water depth.

In model tests, gp=gm,thenVp=. It is not easy to satisfy both Froude similitude and Reynolds similitude, as it is impossible to select a kind of appropriate liquid for model tests when the model is not large enough. Usually, the same liquid is utilized for model experiments as in prototype, which will lead to viscosity dissimilarity. When the flow is in the region of quadratic resistance law, namely, the Reynolds number is large enough, the viscous force can be ignored. Meanwhile, experimental results always contain some errrors due to measurement, calculation or other influencing factors. The theoretical relationship between 1/λLand Vmis shown in Fig.1. Curves for different values ofVpshown in Fig.1 indicate that for a given prototype flow velocity, the flow velocity in the model experiment increases as 1/λLincreases. Assuming that velocity is about 45 m/s in the practical project, the velocity in the model tests is, respectively, about 3.2 m/s, 5.0 m/s, 6.0 m/s, 7.5 m/s under the four scales of 1:200, 1:80, 1:57 and 1:36. In the smooth chute, the velocity is large enough to satisfy the air entrainment similitude, which is about 5 m/s - 7 m/s, while in the stilling basin of multi-horizontal submerged jets, the flow is strongly turbulent and the air entrainment is significant, for the interaction between the jets in the crest over flowing orifices and mid-discharge orifices is very strong. The phenomena of air entrainment in the stilling basin can be seen clearly in the Figs.2 and 3.

2.2 Pressure similarity conditions

In order to have a complete similarity between the flows in model and prototype, the Froude number and Euler number in models must be equal to those in prototype, respectively. The surface tension and elastic forces of water flow can be ignored when theReynolds number is larger than the critical value and the model is designed according to Froude similitude. According to the pressure similarity conditions, Eup=Eum, pp/ρpv2p=pm/ρmvm

2, where Eu is Euler number,ρ is the density of water, v is the velocity of the liquid.

The liquid in model experiments is the same as in prototype experiments, thus, ρp=ρm, and from the Froude similitude, v2=λv2. Dividing the pressure

pLmon the slab of stilling basin into the time-average pressure and the fluctuating pressure, the Froude similitude can still be satisfied when the time-average pressure satisfy the geometric scale condition and the flow velocity satisfy the condition v2p=λLvm

2.

According to the relation Eup=Eum, thus,therefore, the time-averaged pressure satisfies the Froude similitude law, whereandstand for the time-averaged pressure in prototype and model, respectively. There are many researches for the similitude of fluctuating pressures, but with no consensus, that is,stand for the fluctuating pressure in prototype and model, respectively. When ε=1, the fluctuating pressure satisfies the Froude similitude, then, or else, the fluctuating pressure can not satisfy the Froude similitude, thenSome studies indicate that the fluctuating pressure is not consistent with Froude similitude, where the range of εis from 1/2 to 2/3, while other results show that the fluctuating pressure in the main body of the hydraulic jump is in line with Froude similitude, namely ε=1,For a given model scale and experimental value, the theoretical relationship between the pressures in model and prototype is shown in Fig.4. Curves for different values ofε shown in Fig.4 indicate that the model experimental value differs when the value of ε is different at the same measuring point. Model tests according to the practical project were carried out with the scale 1:36, and the fluctuating pressure on the slab of stilling basin is 0.3 m water column. If ε=0.6 and 1.0, respectively, the fluctuating pressure in prototype will be 2.6 m water column and 10.8 m water column.

The liquid in model experiments is the same as in prototype experiments, thus, ρm=ρp, and to satisfy the Froude similitude, v2p=λLvm

2, pp=λLpm. Then the following expression is obtained,

where λLis the model scale, λL=Lp/Lm, β is the air entrainment concentration, and L is the length, where the subsript m stands for model and p stands for prototype.

Theoretical curves shown in Fig.5 for relationship between 1/λLand βmcan be derived from calculations of different air concentration with βp=0.10, 0.20, 0.30, 0.40, 0.50, respectively. It can be seen from Fig.5 that for any given air concentration, the value ofβmincreases gradually as the model scale increases. Assuming that the air concentration in the practical project is about βp=0.50, the air concentration in the model with scale 1:36 should be about 5.6 times of that in the model with scale 1:200.

3. Experimental setup

The experimental setup, shown in Fig.6, was constructed in the State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University. The model was designed based on Xiangjiaba Hydropower Station according with Froude similitude. The test section consists of flood discharging section, head-water reservoir, stilling basin and downstream reach. Flood discharging section includes 6 crest overflowing orifices and 5 mid-discharge orifices, which are alternately arranged. The chute width of the crest overflowing orifices and mid-discharge orifices is 8.0 m and 6.0 m, respectively. The stilling basin is 228.0 m long and 108.0 m wide. The elevation of stilling basin slab is 245.0 m. The discharge is done by a pressure conduit with diameter of 0.15 m. At the end of the downstream channel, the tail water can be adjusted by a moveable gate. Further downstream, the discharge is measured by a rectanglar measuring weir. The accuracy of the discharge measurement is approximately 2%. A HD-4B PC velocity meter is used to measure the velocity. The fluctuating pressure is measured by CYG1 505 sensors made in China. The software used in the dynamic signal analysis system is TST5 000. In this article, four typical operating conditions are investigated, including flooddischarge through crest overflowing orifices, mid-discharge orifices and joint discharge of crest overflowing orifices and mid-discharge orifice (see Table 1).

4. Analysis of experimental results

4.1 Water depth in stilling basin

The water depth in stilling basin for four different runs is shown in Fig.7, where X is the distance from drop to the measuring points, S is the height from the slab to the outlet of the mid-discharge orifice, H is the water depth in stilling basin, and Hmaxis the maximum water depth in stilling basin. It can be seen from Fig.7 that the water depth in the three different models differs little when X/S＜9, among which the maximum error is less than 5%. The water depth varies slightly when X/S＞12. In the same operating condition, the variation of the water depth in stilling basin shows a consistent trend except that the water depth in the small model is less than that in the large model, which indicates that the change of geometric scaling ratio has not resulted in great variations of water depth.

According to the theoretical analysis, assuming that the time-averaged air concentration is 40%, the order of size of the air concentration of the four different models is C1:200＜C1:80＜C1:57＜C1:36. The aerated water in the foreside of stilling basin is strongly turbulent and the free water surface has a great random fluctuation, which makes it difficult to identify the testing point exactly. This is why the water depth in the foreside of stilling basin sees a great difference. At the end of stilling basin, the water flow is smooth and the wave is small, thus experimental values are more stable.

4.2 Pressure distribution on the slab of stilling basin

The pressure distribution on the slab of stilling basin in three typical runs is shown in Fig.8, where P is the pressure on the slab and Pmaxis the maximum pressure on the slab. It can be seen from Fig.8 that the pressure distribution has a certain variation at the foreside of stilling basin, where the maximum error is about 4.22 m water column at the same measuring point; while the values tend to be uniform at the end of stilling basin. The variations of the pressure distribution for different model scales show the same trend in the same operating conditions, and the lateral time-averaged pressure has an even distribution, while the variation is significanlt along the stilling basin, showing the trend of being low in the foreside and high at the end, which is consistent with the water depth distribution in the stilling basin.

The pressure difference in the foreside of stilling basin is mainly due to the fact that with the increase of the model scale, the water velocity increases and the surface is broken violently. The air entrainment and the turbulent intensity in the foreside of the stilling basin in different models are different, therefore, the scale effect on the turbulent intensity and the energy dissipation between model and prototype is different in different models and the time-averaged pressure is different in the foreside of stilling basin.

According to the aforementioned theoretical analysis, the pressure inconsistency for scales frommodel to prototype and the measuring errors lead to the differences in the pressure value. Assuming that the fluctuating pressure in a certain measuring point is 0.06 × 9.81 kPa, if ε=0.6 and 1, respectively, the values of the fluctuating pressure in prototype will be 0.83 × 9.81 kPa and 4.80 × 9.81 kPa, respectively, which means that the results differ greatly due to different values of ε.

4.3 Velocity near the slab in stilling basin

For the safety of the slab, a low close-to-bed velocity can alleviate the scour to the slab. The velocity distribution is shown in Fig.9, where V stands for the close-to-bed velocity at different measuring points and Vmaxis the maximum close-to-bed velocity. It can be seen that the velocity varies with model scales, with the maximum error of less than 5 m/s at the same measuring point. The points of the maximum positive and negative velocity are located almost at the same measuring point in the stilling basin and the velocity distribution shows the same trend. The scale has only a slight effect on the flow velocity.

Meanwhile, experimental results cannot be exactly accurate due to measurement, calculation or other influencing factors. The theoretical relationship between 1/λLand Vmis shown in Fig.1. Curves for different values ofVPshown in Fig.1 indicate that for a given prototype flow velocity, the flow velocity in model experiments increases as 1/λLincreases.

According to theoretical analysis, the differnt model velocity as comparing with that of prototype is due to the viscosity that has not been taken into consideration in the model experiment based on Froude similitude. When the flow is in the region of quadratic resistance law, namely, the Reynolds number is large enough, the effect of viscous force can be ignored, on the other hand, during the experiments, the water in stilling basin is severely turbulent and the location of the maximum velocity is not fixed, while the experimental value is the instantaneous maximum at a fixed point, which is much less important than the instantaneous maximum near the slab. This is the main reason why the flowvelocity is different in different scales. Furthermore, due to different model scales, the aeration and the turbulent intensity in the foreside of stilling basin will reduce the flow velocity near the slab. Theoretically, the smaller the model scale is, the higher the flow velocity will be. However, with the decrease of the model scale, the flow velocity becomes smaller, which means a greater scale effect. Therefore, the above two factors offset each other which leads to a slight difference of the flow velocity near the slab.

4.4 Energy dissipation characteristics

The energy dissipation ratio in the models with different scales in the same runs is shown in Table 2. It is indicated that the testing results of energy dissipation ratio see a slight variation (from 0.55% to 2.50%) among different model scales, which means that the scale effect on energy dissipation ratio is not significant. It can be seen that the results of model experiments are reliable and the energy dissipater design of multi-horizontal submerged jets is reasonable. Therefore, the results can be applied to practical engineering. As the energy dissipation ratio is calculated based on the time-averaged velocity, in which the fluctuating characteristics of the flow in the stilling basin are not considered. The scale effects on air entrainment and fluctuating characteristics are complex, which calls for further studies.

5. Conclusions

A series of experiments were carried out on multi-horizontal submerged jets with four different model scales of 1:36, 1:57, 1:80, 1:200. The characteristics of the hydraulic jumps in stilling basin of multi-horizontal submerged jets are obtained and the following conclusions are drawn:

(1) When the air concentration in prototype is fixed, the greater the model scale is, the higher the air concentration in model is required. In the four different models, for small inflow Reynolds numbers, the air entrainment rate is small. Experimental observation shows that there are certain scale effects in the small hydraulic jumps in terms of void fraction and bubble count rate.

(2) Time-average pressure is influenced slightly by the scales, while the fluctuating pressure is influenced more than the time-average pressure. The main reason is that the air entrainment and the turbulence significantly influence the characteristics of vortices in the region of air entrainment and turbulence.

(3) When Froude similitude conditions are satisfied, the time-averaged water depth, pressure, velocity and energy dissipation ratio would be slightly influenced by the scale effect, while the distribution and variation trend of each hydraulic factor are consistent.

The time-averaged water depth, pressure andvelocity distribution in the stilling basin of multi-horizontal submerged jets are analyzed through four different routine model experiments. The experimental results show that the time-averaged water depth, pressure, velocity and energy dissipation ratio are slightly influenced by the model scales, so are the distribution and the variation trend of each hydraulic factor. Experimental results show that the model scale has a slight effect on the time-averaged hydraulic characteristics in stilling basin of multi-horizontal submerged jets. The results of model experiments can provide usefulReferencesfor practical engineering. The air entrainment may slightly affect the structure of vertical vortices and vortices with horizontal axis. Furthermore, the effect of the model scale on vortex structures and turbulent intensity calls for further studies.

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10.1016/S1001-6058(09)60117-7

* Project supported by the National Key Basic Research Program of China (973 Program, Grant No. 2007CB714105), the Science Foundation of Ministry of Education of China (Grant No. 2008108111) and the Program for New Century Excellent Talents in University (Grant No. NCET-08-0378).

Biography: CHEN Jian-gang (1982-), Male, Ph. D. Candidate

ZHANG Jian-min,

E-mail: jmzhangscu@263.net