The numerical simulation was based on the commercial software ANSYS CFX. The flow passage of the pump turbine was from the draft tube inlet to the volute outlet as shown in Figure 2. The commercial software ICEMCFD was used to generate the grid. Tetrahedral and hexahedral grids were used in this study to ensure the computational quality and save computational cost. A grid independence check was conducted as shown in Figure 2a. The residual value of predicted head at the global best efficiency point under *α* = 14 degrees was chosen as the criterion and a residual less than 0.001 was accepted. The schematic diagram of the final grid is shown in Figure 2b. The grid number of each component is shown in Table 2. The total grid element number was 9,668,956. *Water* **2022**, *14*, x FOR PEER REVIEW 6 of 19

**Figure 2.** Fluid domain and grid. (**a**) Grid independence check; (**b**) the schematic map of the final **Figure 2.** Fluid domain and grid. (**a**) Grid independence check; (**b**) the schematic map of the final grid.

pump was used to maintain the fluid circulation of the test system.

Model test is an important part of this study. As shown in Figure 3, the model scale pump turbine unit was installed in the test section. The upstream and downstream were

During the model test, the energy characteristics were measured at first. The parameters such as measured physical quantities and apparatus uncertainty are given in Table 3. Based on the constant speed test method, the working condition was changed by adjusting the valve and matching the guide vane opening angle, and the head *H*, flow rate

rectification, joint regulation of cavitation with vacuum pump and so on. The supply

grid.

*3.3. Model Test*

**Table 2.** Grid number of each component.

A multi reference frame model was used in this study. The runner was set as the rotating domain, and the rest of the parts were stationary domains. The reference pressure was set to 1 Atm. The boundary conditions are given as follows. The inlet boundary was set at the inlet of draft tube with a given velocity. The outlet boundary was set at the volute outlet, and the given outlet static pressure was 0 Pa. The wall boundaries were no-slip wall type. The interfaces between rotor and stators were set as the stage type. The minimum number of iterations was set to 300, the maximum number of iterations was set to 1000 and the convergence criterion was RMS residual less than 1 × 10−5 .

#### *3.3. Model Test*

Model test is an important part of this study. As shown in Figure 3, the model scale pump turbine unit was installed in the test section. The upstream and downstream were storage tank and cavitation tank, respectively, which can play the role of water storage, rectification, joint regulation of cavitation with vacuum pump and so on. The supply pump was used to maintain the fluid circulation of the test system.

During the model test, the energy characteristics were measured at first. The parameters such as measured physical quantities and apparatus uncertainty are given in Table 3. Based on the constant speed test method, the working condition was changed by adjusting the valve and matching the guide vane opening angle, and the head *H*, flow rate *Q*, rotational speed *n*, shaft torque *M* and guide vane opening angle *α* were recorded. The values of power *P* and efficiency *η* can be calculated to determine the operating condition point of the unit. Among them, the head *H* was measured by the differential pressure sensor, and the static pressure *ps* and the pressure difference *psd* between inlet and outlet were measured. The calculation formula is as follows:

$$H = \frac{p\_{\rm so} - p\_{\rm si}}{\rho \mathbf{g}} = \frac{p\_{\rm sd}}{\rho \mathbf{g}} \tag{13}$$

where *ρ* is density and *g* is the acceleration of gravity. Flow rate *Q* was measured by electro-magnetic flow meter. The measurement of power *P* needs to consider torque *M* and rotational speed *n*, where *n* was measured by rotating speed encoder and *M* was measured by torque meter. Power *P* can be calculated by:

$$P = \frac{2\pi n}{60}M\tag{14}$$

where the unit of *n* is [r/min]. The efficiency *η* can be calculated by:

$$\eta = \frac{\rho \text{gQH}}{P} \tag{15}$$

The guide vane opening *α* was measured by using the angular displacement sensor. The above test parameters were transmitted to the data acquisition terminal for analog-todigital conversion and post-processing.

(**a**)

*Q*, rotational speed *n*, shaft torque *M* and guide vane opening angle *α* were recorded. The values of power *P* and efficiency *η* can be calculated to determine the operating condition point of the unit. Among them, the head *H* was measured by the differential pressure sensor, and the static pressure *ps* and the pressure difference *psd* between inlet and outlet

> *so si sd ppp H= =* ρ

where *ρ* is density and *g* is the acceleration of gravity. Flow rate *Q* was measured by electro-magnetic flow meter. The measurement of power *P* needs to consider torque *M* and rotational speed *n*, where *n* was measured by rotating speed encoder and *M* was meas-

> 2 60 *n P= M* π

*gQH = P* ρ η

The guide vane opening *α* was measured by using the angular displacement sensor. The above test parameters were transmitted to the data acquisition terminal for analog-

 ρ*g g*

− (13)

(15)

(14)

were measured. The calculation formula is as follows:

ured by torque meter. Power *P* can be calculated by:

to-digital conversion and post-processing.

where the unit of *n* is [r/min]. The efficiency *η* can be calculated by:

**Figure 3.** Test rig and apparatus. (**a**) The schematic map of the test rig and apparatus; (**b**) the on-site view of the test rig. **Figure 3.** Test rig and apparatus. (**a**) The schematic map of the test rig and apparatus; (**b**) the on-site view of the test rig.

**Table 3.** Measured physical quantities and apparatus uncertainty. **Table 3.** Measured physical quantities and apparatus uncertainty.

Figure 4 is the experimental–numerical comparison of the energy performance in pump mode. Five different guide vane opening angles were analyzed, including the head *H* and efficiency *η*. For the experimental data, the *Q-H* envelop curve and *Q-η* envelop curve are shown. The *Q-η* envelop curve is drawn based on the best efficiency point (*QBEP*) of each guide vane opening angle. The operation of the prototype pump turbine will follow this curve in order to have a higher efficiency. The head values *H* of these best efficiency points (*QBEP*) were recorded as the *Q-H* envelop curve. For each *Q-H* curve, its BEP point matches with the *Q*-*η* envelop curve and *Q-H* envelop curve. Generally, as shown by the *Q-H* envelop curve, *H* increases with a decrease in *Q*. The *Q-η* envelop curve has a peak (best efficiency point) as indicated in Figure 4. The global best efficiency point is

Rosemount 1151AP ±0.1%

sensor

Tail-water pressure Absolute pressure

Guide vane angle Angular displace-

**4. Comparison of Energy Performance**

under *α* = 14 degrees.

#### **4. Comparison of Energy Performance**

Figure 4 is the experimental–numerical comparison of the energy performance in pump mode. Five different guide vane opening angles were analyzed, including the head *H* and efficiency *η*. For the experimental data, the *Q-H* envelop curve and *Q-η* envelop curve are shown. The *Q-η* envelop curve is drawn based on the best efficiency point (*QBEP*) of each guide vane opening angle. The operation of the prototype pump turbine will follow this curve in order to have a higher efficiency. The head values *H* of these best efficiency points (*QBEP*) were recorded as the *Q-H* envelop curve. For each *Q-H* curve, its BEP point matches with the *Q*-*η* envelop curve and *Q-H* envelop curve. Generally, as shown by the *Q-H* envelop curve, *H* increases with a decrease in *Q*. The *Q-η* envelop curve has a peak (best efficiency point) as indicated in Figure 4. The global best efficiency point is under *α* = 14 degrees. *Water* **2022**, *14*, x FOR PEER REVIEW 9 of 19

**Figure 4.** Experimental–numerical comparison of energy performance in pump mode. **Figure 4.** Experimental–numerical comparison of energy performance in pump mode.

In the analysis of CFD prediction results, the guide vane opening angle *α* from 18 degrees to 10 degrees was analyzed. With the decrease in the guide vane opening angle *α*, the flow rate becomes smaller and smaller. This shows the good flow-adjusting ability of the guide vane. The individual *Q-H* curves match the *Q-H* envelop curve well. Head *H* increases with the decrease in flow rate *Q* with local variations. For *Q-η* curves, each individual curve has a local peak. This means that the efficiency increases and decreases with the decrease in the flow rate. Each guide vane opening angle has a best operation range. The efficiencies of the CFD-predicted results are higher than the experimental data. This is reasonable because of the ignorance of the mechanical efficiency in the experiment. The individual *Q-η* curves have different laws because of the different guide vane openings. Detailed analyses are given in the following sections. CFD prediction will help the internal flow field analysis for the understanding of the relationship between the flow pattern and In the analysis of CFD prediction results, the guide vane opening angle *α* from 18 degrees to 10 degrees was analyzed. With the decrease in the guide vane opening angle *α*, the flow rate becomes smaller and smaller. This shows the good flow-adjusting ability of the guide vane. The individual *Q-H* curves match the *Q-H* envelop curve well. Head *H* increases with the decrease in flow rate *Q* with local variations. For *Q-η* curves, each individual curve has a local peak. This means that the efficiency increases and decreases with the decrease in the flow rate. Each guide vane opening angle has a best operation range. The efficiencies of the CFD-predicted results are higher than the experimental data. This is reasonable because of the ignorance of the mechanical efficiency in the experiment. The individual *Q-η* curves have different laws because of the different guide vane openings. Detailed analyses are given in the following sections. CFD prediction will help the internal flow field analysis for the understanding of the relationship between the flow pattern and guide vane opening angle.

guide vane opening angle. To have a better understanding of the drop in the head when the flow rate is relatively small, the head stagnation point and head valley point are defined for each *α* con-To have a better understanding of the drop in the head when the flow rate is relatively small, the head stagnation point and head valley point are defined for each *α* condition. For a *Q-H* curve, the slope is generally negative but one or multiple positive slope regions can

dition. For a *Q-H* curve, the slope is generally negative but one or multiple positive slope regions can be found. With a decrease in the flow rate, the point that *H* stops the increment

rate continually decreases, *H* will rise again, the valley point is defined as the head valley point and the flow rate is denoted as *QVAL*. The values of *QSTA* and *QVAL* are shown in Figure 5a,b. Both *QSTA* and *QVAL* decrease with a decrease in *α*. This means that the unstable flow rate of the guide vane decreases synchronously with the decrease in the guide vane opening angle. A detailed analysis of the best efficiency *ηBEP* and flow rate of best efficiency points *QBEP* is shown in Figure 5c,d. The value of *ηBEP* increases within *α* = 10~14 degrees and decreases within *α* = 14~18 degrees. The value of *QBEP* generally increases from *α* = 10 degrees to *α* = 18 degrees, with an exception at *α* = 16 degrees. This means that the optimal matching opening angle of the guide vane increases synchronously with the increase in

be found. With a decrease in the flow rate, the point that *H* stops the increment is defined as the head stagnation point and the flow rate is denoted as *QSTA*. If the flow rate continually decreases, *H* will rise again, the valley point is defined as the head valley point and the flow rate is denoted as *QVAL*. The values of *QSTA* and *QVAL* are shown in Figure 5a,b. Both *QSTA* and *QVAL* decrease with a decrease in *α*. This means that the unstable flow rate of the guide vane decreases synchronously with the decrease in the guide vane opening angle. A detailed analysis of the best efficiency *ηBEP* and flow rate of best efficiency points *QBEP* is shown in Figure 5c,d. The value of *ηBEP* increases within *α* = 10~14 degrees and decreases within *α* = 14~18 degrees. The value of *QBEP* generally increases from *α* = 10 degrees to *α* = 18 degrees, with an exception at *α* = 16 degrees. This means that the optimal matching opening angle of the guide vane increases synchronously with the increase in the flow rate. Therefore, the optimal matching and undesirable matching of the guide vane and impeller need further analyses. *Water* **2022**, *14*, x FOR PEER REVIEW 10 of 19 the flow rate. Therefore, the optimal matching and undesirable matching of the guide vane and impeller need further analyses.

**Figure 5.** Variation in typical conditions of different attack angles *α*. (**a**) *QSTA*. (**b**) *QVAL*. (**c**) *ηBEP*. (**d**) *QBEP*. **Figure 5.** Variation in typical conditions of different attack angles *α*. (**a**) *QSTA*. (**b**) *QVAL*. (**c**) *ηBEP*. (**d**) *QBEP*.

#### **5. Analysis of Internal Flow 5. Analysis of Internal Flow**

The analysis of the internal flow in the pump turbine includes five parts with five different guide vane opening angles. They are defined as the large guide vane opening angle (18 degrees), medium-large guide vane opening angle (16 degrees), medium guide vane opening angle (14 degrees), medium-small guide vane opening angle (12 degrees) and small guide vane opening angle (10 degrees), and are discussed in the following sections. The analysis of the internal flow in the pump turbine includes five parts with five different guide vane opening angles. They are defined as the large guide vane opening angle (18 degrees), medium-large guide vane opening angle (16 degrees), medium guide vane opening angle (14 degrees), medium-small guide vane opening angle (12 degrees) and small guide vane opening angle (10 degrees), and are discussed in the following sections.

#### *5.1. Large Guide Vane Opening Angle (18 Degrees) 5.1. Large Guide Vane Opening Angle (18 Degrees)*

Figure 6 shows the situation of the guide opening angle *α* of 18 degrees. Figure 6a, respectively, shows the flow–head *Q-H* and flow–efficiency *Q-η* Curve, where *QBEP* is approximately 0.415 m3/s and *QVAL* is approximately 0.339 m3/s. Through the analysis of the Figure 6 shows the situation of the guide opening angle *α* of 18 degrees. Figure 6a, respectively, shows the flow–head *Q-H* and flow–efficiency *Q-η* Curve, where *QBEP* is approximately 0.415 m3/s and *QVAL* is approximately 0.339 m3/s. Through the analysis of

flow energy loss proportion inside the volute, stay vane, guide vane, runner and draft tube as shown in Figure 6b, it is found that, when the flow rate is at the *QBEP* point, the loss

ner is close to that inside the guide vane. The loss proportion of the stay vane and volute is relatively small. The loss in the draft tube is less than 1%, which is very small. When the flow rate decreased to *QVAL*, the loss proportion of guide vane increased significantly to 53.9%, which was higher than the sum of other components. It can be seen from the *Spro*

the flow energy loss proportion inside the volute, stay vane, guide vane, runner and draft tube as shown in Figure 6b, it is found that, when the flow rate is at the *QBEP* point, the loss proportion of the guide vane is approximately 40.8%. The loss proportion inside the runner is close to that inside the guide vane. The loss proportion of the stay vane and volute is relatively small. The loss in the draft tube is less than 1%, which is very small. When the flow rate decreased to *QVAL*, the loss proportion of guide vane increased significantly to 53.9%, which was higher than the sum of other components. It can be seen from the *Spro* contour and *v* vectors at the *QBEP* point that there is only a small amount of guide vane passages with an obvious loss increase, and that the intensity is not high. This is related to the local secondary flow structures. At the *QVAL* point, there are approximately seven channels of the guide vane with significant loss. The local intensity is very high, which is also related to the secondary flow in the guide vane channels, and it causes the flow blockage in the downstream components. *Water* **2022**, *14*, x FOR PEER REVIEW 11 of 19 contour and *v* vectors at the *QBEP* point that there is only a small amount of guide vane passages with an obvious loss increase, and that the intensity is not high. This is related to the local secondary flow structures. At the *QVAL* point, there are approximately seven channels of the guide vane with significant loss. The local intensity is very high, which is also related to the secondary flow in the guide vane channels, and it causes the flow blockage in the downstream components.

**Figure 6.** Performance curve and flow pattern at 18° attack angle. (**a**) *Q-H* and *Q-η* curve; (**b**) proportion of flow energy loss; (**c**) VAL *Spro*; (**d**) VAL *v*; (**e**) BEP *Spro*; (**f**) BEP *v*; VAL: local valley of *Q-H* curve; BEP: best efficiency point. **Figure 6.** Performance curve and flow pattern at 18◦ attack angle. (**a**) *Q-H* and *Q-η* curve; (**b**) proportion of flow energy loss; (**c**) VAL *Spro*; (**d**) VAL *v*; (**e**) BEP *Spro*; (**f**) BEP *v*; VAL: local valley of *Q-H* curve; BEP: best efficiency point.

Figure 7 shows the situation of the guide opening angle *α* of 16 degrees. Figure 7a, respectively, shows the flow–head *Q-H* and flow–efficiency *Q-η* curve, where *QBEP* is

*5.2. Medium-Large Guide Vane Opening Angle (16 Degrees)*

#### *5.2. Medium-Large Guide Vane Opening Angle (16 Degrees)*

Figure 7 shows the situation of the guide opening angle *α* of 16 degrees. Figure 7a, respectively, shows the flow–head *Q-H* and flow–efficiency *Q-η* curve, where *QBEP* is approximately 0.339 m3/s and *QVAL* is approximately 0.327 m3/s. As shown in Figure 7b, it is found that, when the flow rate is at the *QBEP* point, the loss proportion of the guide vane is approximately 51.8%, which is the largest. The loss proportion inside the runner is the second largest. The loss proportion of the stay vane and volute is smaller. The loss in the draft tube is also less than 1%, which is very small. When the flow rate decreased to *QVAL*, the loss proportion of the guide vane became 52.2%, which was almost unchanged. It can be seen from the *Spro* contour and *v* vectors at the *QBEP* point and *QVAL* point that secondary flow structures are found in guide vane channels. Flow blockage in the downstream components can be also observed. Based on the comparison between the *QBEP* point and *QVAL* point, the intensity of the flow energy loss at the *QVAL* point is much stronger. This is why head *H* drops suddenly. *Water* **2022**, *14*, x FOR PEER REVIEW 12 of 19 approximately 0.339 m3/s and *QVAL* is approximately 0.327 m3/s. As shown in Figure 7b, it is found that, when the flow rate is at the *QBEP* point, the loss proportion of the guide vane is approximately 51.8%, which is the largest. The loss proportion inside the runner is the second largest. The loss proportion of the stay vane and volute is smaller. The loss in the draft tube is also less than 1%, which is very small. When the flow rate decreased to *QVAL*, the loss proportion of the guide vane became 52.2%, which was almost unchanged. It can be seen from the *Spro* contour and *v* vectors at the *QBEP* point and *QVAL* point that secondary flow structures are found in guide vane channels. Flow blockage in the downstream components can be also observed. Based on the comparison between the *QBEP* point and *QVAL* point, the intensity of the flow energy loss at the *QVAL* point is much stronger. This is why head *H* drops suddenly.

**Figure 7.** Performance curve and flow pattern at 16◦ attack angle. (**a**) *Q-H* and *Q-η* curve; (**b**) proportion of flow energy loss; (**c**) VAL *Spro*; (**d**) VAL *v*; (**e**) BEP *Spro*; (**f**) BEP *v*; VAL: local valley of *Q-H* curve; BEP: best efficiency point.

#### *5.3. Medium Guide Vane Opening Angle (14 Degrees)* **Figure 7.** Performance curve and flow pattern at 16° attack angle. (**a**) *Q-H* and *Q-η* curve; (**b**) proportion of flow energy loss; (**c**) VAL *Spro*; (**d**) VAL *v*; (**e**) BEP *Spro*; (**f**) BEP *v*; VAL: local valley of *Q-H*

Figure 8 shows the situation of the guide opening angle *α* of 14 degrees. Figure 8a, respectively, shows the flow–head *Q-H* and flow–efficiency *Q-η* curve, where *QBEP* is approximately 0.363 m3/s and *QVAL* is approximately 0.267 m3/s. As shown in Figure 8b, it is found that when the flow rate is at the *QBEP* point, the loss proportion of the guide vane is approximately 36.9%. The loss proportion inside the runner is close to that inside the guide vane. The loss proportion of the stay vane and volute is smaller than that in the runner and guide vane. The loss in the draft tube is still much smaller. When the flow rate decreased to *QVAL*, the loss proportion of the guide vane increased significantly to 69.7%, which was higher than the sum of other components. The proportion of loss in the runner becomes lower. It can be seen from the *Spro* contour and *v* vectors at the *QBEP* point that only few guide vane passages have a relatively strong energy loss and that the intensity is not that high. At the *QVAL* point, there are approximately eight channels of the guide vane with a significant flow energy loss with very high intensity. The sudden increase in *Spro* also indicates the mismatching of the guide vane and runner incoming flow. The bad flow regime will cause flow blockage in the stay vane and volute. curve; BEP: best efficiency point. *5.3. Medium Guide Vane Opening Angle (14 Degrees)*  Figure 8 shows the situation of the guide opening angle *α* of 14 degrees. Figure 8a, respectively, shows the flow–head *Q-H* and flow–efficiency *Q-η* curve, where *QBEP* is approximately 0.363 m3/s and *QVAL* is approximately 0.267 m3/s. As shown in Figure 8b, it is found that when the flow rate is at the *QBEP* point, the loss proportion of the guide vane is approximately 36.9%. The loss proportion inside the runner is close to that inside the guide vane. The loss proportion of the stay vane and volute is smaller than that in the runner and guide vane. The loss in the draft tube is still much smaller. When the flow rate decreased to *QVAL*, the loss proportion of the guide vane increased significantly to 69.7%, which was higher than the sum of other components. The proportion of loss in the runner becomes lower. It can be seen from the *Spro* contour and *v* vectors at the *QBEP* point that only few guide vane passages have a relatively strong energy loss and that the intensity is not that high. At the *QVAL* point, there are approximately eight channels of the guide vane with a significant flow energy loss with very high intensity. The sudden increase in *Spro* also indicates the mismatching of the guide vane and runner incoming flow. The bad flow regime will cause flow blockage in the stay vane and volute.

*Water* **2022**, *14*, x FOR PEER REVIEW 13 of 19

**Figure 8.** Performance curve and flow pattern at 14° attack angle. (**a**) *Q-H* and *Q-η* curve; (**b**) proportion of flow energy loss; (**c**) VAL *Spro*; (**d**) VAL *v*; (**e**) BEP *Spro*; (**f**) BEP *v*; VAL: local valley of *Q-H* curve; BEP: best efficiency point. **Figure 8.** Performance curve and flow pattern at 14◦ attack angle. (**a**) *Q-H* and *Q-η* curve; (**b**) proportion of flow energy loss; (**c**) VAL *Spro*; (**d**) VAL *v*; (**e**) BEP *Spro*; (**f**) BEP *v*; VAL: local valley of *Q-H* curve; BEP: best efficiency point.

Figure 9 shows the situation of the guide opening angle *α* of 12 degrees. Figure 9a,

found that, when the flow rate is at the *QBEP* point, the loss proportion of the guide vane is approximately 47.5%, which is the largest. The loss proportion inside the runner, stay vane and volute is similar. The loss in the draft tube is still less than 1%. When the flow rate decreased to *QVAL*, the loss proportion of the guide vane increased significantly to 65.6%, which was higher than the sum of other components. The proportion of loss in the volute becomes obviously lower. It can be seen from the *Spro* contour and *v* vectors at the *QBEP* point that the loss is relatively low and the flow regime is smooth in all of the guide vane channels. At the *QVAL* point, some guide vane channels have a high flow energy loss with a high intensity. This is related to the secondary flow in the guide vane, and the

*5.4. Medium-Small Guide Vane Opening Angle (12 Degrees)*

downstream components are influenced.

(**a**) (**b**)

#### *5.4. Medium-Small Guide Vane Opening Angle (12 Degrees)* curve; BEP: best efficiency point.

Figure 9 shows the situation of the guide opening angle *α* of 12 degrees. Figure 9a, respectively, shows the flow–head *Q-H* and flow–efficiency *Q-η* curve, where *QBEP* is approximately 0.315 m3/s and *QVAL* is approximately 0.219 m3/s. As shown in Figure 9b, it is found that, when the flow rate is at the *QBEP* point, the loss proportion of the guide vane is approximately 47.5%, which is the largest. The loss proportion inside the runner, stay vane and volute is similar. The loss in the draft tube is still less than 1%. When the flow rate decreased to *QVAL*, the loss proportion of the guide vane increased significantly to 65.6%, which was higher than the sum of other components. The proportion of loss in the volute becomes obviously lower. It can be seen from the *Spro* contour and *v* vectors at the *QBEP* point that the loss is relatively low and the flow regime is smooth in all of the guide vane channels. At the *QVAL* point, some guide vane channels have a high flow energy loss with a high intensity. This is related to the secondary flow in the guide vane, and the downstream components are influenced. *5.4. Medium-Small Guide Vane Opening Angle (12 Degrees)*  Figure 9 shows the situation of the guide opening angle *α* of 12 degrees. Figure 9a, respectively, shows the flow–head *Q-H* and flow–efficiency *Q-η* curve, where *QBEP* is approximately 0.315 m3/s and *QVAL* is approximately 0.219 m3/s. As shown in Figure 9b, it is found that, when the flow rate is at the *QBEP* point, the loss proportion of the guide vane is approximately 47.5%, which is the largest. The loss proportion inside the runner, stay vane and volute is similar. The loss in the draft tube is still less than 1%. When the flow rate decreased to *QVAL*, the loss proportion of the guide vane increased significantly to 65.6%, which was higher than the sum of other components. The proportion of loss in the volute becomes obviously lower. It can be seen from the *Spro* contour and *v* vectors at the *QBEP* point that the loss is relatively low and the flow regime is smooth in all of the guide vane channels. At the *QVAL* point, some guide vane channels have a high flow energy loss with a high intensity. This is related to the secondary flow in the guide vane, and the downstream components are influenced.

**Figure 8.** Performance curve and flow pattern at 14° attack angle. (**a**) *Q-H* and *Q-η* curve; (**b**) proportion of flow energy loss; (**c**) VAL *Spro*; (**d**) VAL *v*; (**e**) BEP *Spro*; (**f**) BEP *v*; VAL: local valley of *Q-H*

*Water* **2022**, *14*, x FOR PEER REVIEW 14 of 19

(**e**) (**f**)

**Figure 9.** Performance curve and flow pattern at 12° attack angle. (**a**) *Q-H* and *Q-η* curve; (**b**) proportion of flow energy loss; (**c**) VAL *Spro*; (**d**) VAL *v*; (**e**) BEP *Spro*; (**f**) BEP *v*; VAL: local valley of *Q-H* curve; BEP: best efficiency point. **Figure 9.** Performance curve and flow pattern at 12◦ attack angle. (**a**) *Q-H* and *Q-η* curve; (**b**) proportion of flow energy loss; (**c**) VAL *Spro*; (**d**) VAL *v*; (**e**) BEP *Spro*; (**f**) BEP *v*; VAL: local valley of *Q-H* curve; BEP: best efficiency point.

vane becomes bad and causes an increase in the flow energy loss.

Figure 10 shows the situation of the guide opening angle *α* of 10 degrees. Figure 10a,

proximately 0.243 m3/s and *QVAL* is approximately 0.183 m3/s. As shown in Figure 10b, it is found that, when the flow rate is at the *QBEP* point, the loss proportion of the guide vane is approximately 54.8%, which is the largest. The loss proportion inside the runner and stay vane is similar and lower than that in the guide vane. The loss in the volute is much lower. The loss in the draft tube is less than 1%, which is very low. When the flow rate decreased to *QVAL*, the loss proportion of the guide vane increased significantly to 70.5%, which was higher than the sum of other components and completely dominant. The proportion of loss in the stay vane and volute becomes obviously lower. It can be seen from the *Spro* contour and *v* vectors at *QBEP* point that the loss is relatively low and the flow regime is smooth in all of the guide vane channels. At the *QVAL* point, the flow regime in the guide

*5.5. Small Guide Vane Opening Angle (10 Degrees)*

#### *5.5. Small Guide Vane Opening Angle (10 Degrees)*

Figure 10 shows the situation of the guide opening angle *α* of 10 degrees. Figure 10a, respectively, shows the flow–head *Q-H* and flow–efficiency *Q-η* curve, where *QBEP* is approximately 0.243 m3/s and *QVAL* is approximately 0.183 m3/s. As shown in Figure 10b, it is found that, when the flow rate is at the *QBEP* point, the loss proportion of the guide vane is approximately 54.8%, which is the largest. The loss proportion inside the runner and stay vane is similar and lower than that in the guide vane. The loss in the volute is much lower. The loss in the draft tube is less than 1%, which is very low. When the flow rate decreased to *QVAL*, the loss proportion of the guide vane increased significantly to 70.5%, which was higher than the sum of other components and completely dominant. The proportion of loss in the stay vane and volute becomes obviously lower. It can be seen from the *Spro* contour and *v* vectors at *QBEP* point that the loss is relatively low and the flow regime is smooth in all of the guide vane channels. At the *QVAL* point, the flow regime in the guide vane becomes bad and causes an increase in the flow energy loss. *Water* **2022**, *14*, x FOR PEER REVIEW 16 of 19

**Figure 10.** Performance curve and flow pattern at 10° attack angle. (**a**) *Q-H* and *Q-η* curve; (**b**) proportion of flow energy loss; (**c**) VAL *Spro*; (**d**) VAL *v*; (**e**) BEP *Spro*; (**f**) BEP *v*; VAL: local valley of *Q-H* curve; BEP: best efficiency point. **Figure 10.** Performance curve and flow pattern at 10◦ attack angle. (**a**) *Q-H* and *Q-η* curve; (**b**) proportion of flow energy loss; (**c**) VAL *Spro*; (**d**) VAL *v*; (**e**) BEP *Spro*; (**f**) BEP *v*; VAL: local valley of *Q-H* curve; BEP: best efficiency point.

The stagnation and drop in head *H* with the decrease in flow rate *Q* is related to the mismatching of the guide vane flow angle. In order to have a better understanding of this

most important change from the *QBEP* point to *QVAL* point is the decrease in the flow rate.

**6. Discussion**

#### **6. Discussion**

The stagnation and drop in head *H* with the decrease in flow rate *Q* is related to the mismatching of the guide vane flow angle. In order to have a better understanding of this mis-matching, Figure 11 is schematically drawn with two conditions of *QBEP* and *QVAL*. The most important change from the *QBEP* point to *QVAL* point is the decrease in the flow rate. *Water* **2022**, *14*, x FOR PEER REVIEW 17 of 19

**Figure 11.** Schematic map of the mismatching of guide vane flow angle. **Figure 11.** Schematic map of the mismatching of guide vane flow angle.

The most important analysis method of turbomachinery is the velocity triangle analysis. If we draw the velocity triangle at the runner trailing-edge in pump mode, the rotational linear velocity *u* can be determined at a specific rotational speed. The direction of relative velocity *w* can be determined because the runner blade is fixed. The meridional component of absolute velocity *vm* can be calculated by [32]: The most important analysis method of turbomachinery is the velocity triangle analysis. If we draw the velocity triangle at the runner trailing-edge in pump mode, the rotational linear velocity *u* can be determined at a specific rotational speed. The direction of relative velocity *w* can be determined because the runner blade is fixed. The meridional component of absolute velocity *vm* can be calculated by [32]:

$$v\_m = \frac{Q}{A} \tag{16}$$

$$\dots \quad \dots \quad \dots$$

where *A* is the flow passing area where *vm* is analyzed.

where *A* is the flow passing area where *vm* is analyzed. When the flow rate is *QBEP*, which represents a good flow condition, the absolute flow angle between the runner and guide vane, defined as *αicf*, will be almost equal to the guide vane installation angle *αgv*. When the flow rate *Q* decreases, *vm* will become smaller. Because the direction of *w* is almost unchanged, *αicf* will be smaller, as indicated. The absolute When the flow rate is *QBEP*, which represents a good flow condition, the absolute flow angle between the runner and guide vane, defined as *αicf*, will be almost equal to the guide vane installation angle *αgv*. When the flow rate *Q* decreases, *vm* will become smaller. Because the direction of *w* is almost unchanged, *αicf* will be smaller, as indicated. The absolute difference ∆*α* between *αgv* and *αicf* can be defined as:

$$
\Delta a = \left| a\_{\rm icf} - a\_{\rm \mathcal{S}^{\mathcal{D}}} \right| \tag{17}
$$

Δ − αα α *= icf gv* (17) Therefore, Δ*α* will be larger. This means that the good matching between the flow angle and guide vane installation angle will be broken when the flow rate decreases. If this mismatching becomes strong, the flow regime in guide vane channels will be very bad, leading to a 'stall' situation. This is why the stagnation of and drop in head *H* occur. This mis-matching will also occur when the flow rate increases. This is why the efficiency Therefore, ∆*α* will be larger. This means that the good matching between the flow angle and guide vane installation angle will be broken when the flow rate decreases. If this mismatching becomes strong, the flow regime in guide vane channels will be very bad, leading to a 'stall' situation. This is why the stagnation of and drop in head *H* occur. This mis-matching will also occur when the flow rate increases. This is why the efficiency curve has an optimal region. For the large flow rate conditions, there is no positive *Q-H* slope problem.

curve has an optimal region. For the large flow rate conditions, there is no positive *Q-H*

limitation will also make the originally estimated incoming flow direction upstream to the guide vane deflect to some extent. This phenomenon is not the same in different turbomachinery [33]. It will be strong in a pump turbine and will cause a change in velocity triangle components. As for the effect of the slip phenomenon on the *Q-H* instability of the pump turbine, we will pay special attention to it in future research. In general, the mismatching of the guide vane flow angle is especially important in small flow rate con-

ditions, which is extremely off-design.

slope problem.

In addition, the slip phenomenon of the blade channel caused by the blade number limitation will also make the originally estimated incoming flow direction upstream to the guide vane deflect to some extent. This phenomenon is not the same in different turbomachinery [33]. It will be strong in a pump turbine and will cause a change in velocity triangle components. As for the effect of the slip phenomenon on the *Q-H* instability of the pump turbine, we will pay special attention to it in future research. In general, the mismatching of the guide vane flow angle is especially important in small flow rate conditions, which is extremely off-design.

#### **7. Conclusions**

In this study, the flow energy loss and *Q-H* stability in a reversible pump turbine in pump mode was discussed for five different guide vane opening angles. These opening angle conditions cover the main conditions during daily operation in pump mode. The mismatching of the guide vane and runner incoming flow will induce an increase in flow energy loss in the guide vane and downstream components. It will cause a drop in head *H* and instability in the *Q-H* relationship. This situation occurs at both the large guide vane opening and small guide vane opening. With a decrease in the flow rate, the angle difference between the runner outlet flow and guide vane becomes bigger and bigger in order to generate the positive slope *Q-H* region. If this positive slope region overlaps a specific head, the operation stability will be influenced.

In the current pump turbine operation, the selection of the on-cam operation curve depends on the judgment of the high-efficiency envelop. However, the *Q-H* unstable region of the pump turbine in pump mode is also very important. As discussed in this study, if only considering high efficiency, the unit may face the risk of an unstable operation. Based on the research results of this paper, we can determine a scheme of selecting the on-cam envelop curve that takes into account both efficiency and stability without changing the hydraulic design. Our future research will compare and discuss the advantages and disadvantages of the selection of the on-cam operation scheme.

Generally, this study provides a reference for the solution of engineering problems, mainly serving the stable operation of the unit. This study is of great significance for the technical improvement in large-capacity medium and high head pumped storage power stations.

**Author Contributions:** Methodology, Z.W.; software, W.Y.; validation, R.T.; formal analysis, W.Y.; investigation, D.Z.; resources, W.Y.; writing—original draft preparation, W.Y.; writing—review and editing, R.T.; visualization, D.Z.; supervision, Z.W.; project administration, W.Y.; funding acquisition, R.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Open Research Fund Program of State Key Laboratory of Hydroscience and Engineering (No. sklhse-2022-E-01).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **Improvement of the Flow Pattern of a Forebay with a Side-Intake Pumping Station by Diversion Piers Based on Orthogonal Test Method**

**Chen Zhang 1 , Haodi Yan 2 , Muhammad Tahir Jamil 2,\* and Yonghai Yu 2**

**Abstract:** The flow analysis of the forebay of a lateral intake pumping station with asymmetrical operating pumps was carried out with a realizable k-ε turbulent model and SIMPLEC (Semi Implicit Method for Pressure Linked Equations Consistent) algorithm. The Pressure Inlet boundary condition was adopted and the pressure between the top surface and the bottom surface was linear with the height of the inlet section. The Mass Flow Outlet boundary condition was also adopted to ensure the accuracy and precision of the CFD (Computational Fluid Dynamics) simulation. The diversion pier was selected as the optimization strategy based on the flow parameters. The layout of the diversion piers was designed with four parameters which are the relative length, relative height, width, and straight-line distance of the piers' tail. Each parameter had three values. Based on the orthogonal test, nine groups of the numerical simulation on different layouts of diversion piers were analyzed with the uniformity of axial flow velocity and weighted average angle of the flow velocity of the inlet cross-section of each pump, reducing the number of tests from 64 (43 ) groups to 9 groups, improving work efficiency. The results show that the diversion piers had a significant adjustment of uniformity of axial flow velocity and weighted average angle of flow velocity. After optimization of the forebay, the uniformity of axial flow velocity of intake of No.1 pump was 80.26% and the weighted average angle of flow velocity was 77.68◦ . The above values of the No.2 pump were 98.74% and 87.84◦ , respectively. The values of the No.4 pump were 93.41% and 77.28◦ . The results of numerical simulation, which was carried out to estimate the rectification effect under the operation combination of the No.1, No.3, and No.4 pumps, showed that the uniformity and the angle of the No.1 pump were 92.65% and 72.66◦ , respectively, the uniformity and the angle of No.3 pump were 94.54% and 85.14◦ , and the uniformity and the angle of the No.4 pump were 75.81% and 78.21◦ . This research proves that the orthogonal test method, in a reasonable and convenient way, can be applied in hydraulic optimization for a lateral intake pumping station.

**Keywords:** flow pattern; orthogonal test method; lateral intake; CFD numerical simulation; diversion pier

#### **1. Introduction**

In the forebay of a lateral intake pumping station, vortices, spiral flows, and large-scale reversed flows occurred, directly affecting the pump's inlet conditions, and resulting in a decline in the pump's energy performance and steam erosion performance. These flow patterns also cause vibration in the unit and may even jeopardize its safety. Therefore, rectification measures must be developed for the lateral intake pumping stations [1–3]. In order to improve flow performance, some researchers have studied three-dimensional flows in pump or pumping station intakes [4–8]. Several studies have been conducted on the flow and the alteration of the flow pattern in the forebay of the pumping stations. Kadam P. et al. [9] studied the flow field of the inlet building of a pumping station using

**Citation:** Zhang, C.; Yan, H.; Jamil, M.T.; Yu, Y. Improvement of the Flow Pattern of a Forebay with a Side-Intake Pumping Station by Diversion Piers Based on Orthogonal Test Method. *Water* **2022**, *14*, 2663. https://doi.org/10.3390/w14172663

Academic Editor: Anargiros I. Delis

Received: 30 July 2022 Accepted: 26 August 2022 Published: 28 August 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

physical model tests along with numerical simulation techniques. They also evaluated the flow characteristics of the forebay and the inlet pipe and discovered that the poor intake flow pattern was caused by the high diffusion angle of the forebay and small inundation depth. To improve the flow in the forebay of the pumping station, Liu, C. et al. [10,11] used division piers that study and validate conclusions numerically and experimentally. Feng X. [12] used the rectifier sill as a rectifying measure for the forward inflow forebay of the pumping station. Li J. et al. [13] simulated flow patterns in the forebay and suction sump of the Tianshan pumping station through a finite element analysis approach. Xu, B. et al. [14] studied the influence of the length of the diversion piers on the flow pattern in an asymmetric combined sluice-pump station project based on the CFD numerical simulation. Finally, Xia, C. et al. [15] added the inverted T-shaped diversion piers in the forebay to significantly improve the uniformity of the internal flow velocity distribution.

Luo, C. et al. [16] added the sill to rectify the flow pattern in the forebay and analyzed the general rules of the position and height of the sill. Yu, Y. et al. [17] studied the influence of the flow deflector on the flow pattern in the forebay of the pumping station and obtained the reasonable layout parameters of the flow deflector based on the physical model test and numerical simulation. According to a study by Luo, C. et al. [18], opening the diversion pier can lower the lateral and axial flow velocity as well as the oblique flow area of the surface layer of the flow surface in front of the pier head, enhancing navigational safety. The orthogonal test method is a scientific approach to testing a design based on the analysis of multi-component and multi-level test situations. It significantly reduces the number of tests while allowing for the investigation of the influence of each factor level on evaluation indices through the creation of orthogonal test tables and ranges. The influence of each factor on the head of the pump was investigated by Wang, W. et al. [19] using an orthogonal test design with four factors, three levels, and numerical simulation calculation. J. Zhou [20] obtained the influence of each factor on the uniformity of axial flow velocity and the weighted average angle of velocity by the length, width, radian, and relative height of the orthogonal test design of five factors and four levels. The study by Xu, B. et al. [21] showed that the orthogonal test method has a significant effect on the parameter design for the rectification measures of pumping stations.

The orthogonal test method can optimize the hydraulic design of the diversion piers. Numerous studies combining physical model testing and numerical simulation have been carried out for the rectification measures for the forebay of pumping stations. These studies show that the results of the CFD numerical simulation are essentially similar to those of the physical model testing and that the only approach to enhance the flow pattern of the forebay of pumping stations is through the use of the CFD numerical simulation method [22–26]. As a result, in this paper, the CFD numerical simulation method will be used to analyze the flow pattern of the forebay of a specific pumping station and to improve the design parameters of the diversion pier. The analysis method and procedure are shown in Figure 1.

The novelty of this study lies in the object of flow analysis, which includes not only the forebay with side-intake of the pumping station but also asymmetrical operating pumps and nine groups of numerical simulation on the various layouts of diversion piers investigated using the orthogonal test with uniform axial flow velocity and weighted average angle of the flow velocity at the inlet cross-section of each pump. This study is divided into five sections: study area and method, typical sections in calculation domain and evaluation indexes of flow pattern in the forebay, design of diversion piers based on the orthogonal test, CFD analysis of Pumps 1, 3, and 4, and conclusion. This study provides valuable resources for related topics of interest in pumping station engineering.

of 1:8, as shown in Figure 2.

#### **Figure 1.** Analysis flowchart. **2. Study Area and Method**

#### *2.1. Study Area*

**2. Study Area and Method**  *2.1. Study Area*  There are four pumps in the certain lateral inlet pumping station in Shandong prov-There are four pumps in the certain lateral inlet pumping station in Shandong province, China, among which, the rated flow of the No.1 or No.4 pump is 2.6 m3/s and the rated flow of the No.2 or No.3 pump is 5.4 m3/s. The inlet cross-section of every pump is 3.2 m high and 4.85 m wide. The water depth of the forebay before the slope is 4 m, the water

ince, China, among which, the rated flow of the No.1 or No.4 pump is 2.6 m3/s and the rated flow of the No.2 or No.3 pump is 5.4 m3/s. The inlet cross-section of every pump is

water depth of the forebay behind the slope is 5.4 m, and the bottom slope has a gradient

Two main pump operation schemes are the combination of No.1 pump, No.2 pump, and No.4 pump and the combination of No.1 pump, No.3 pump, and No.4 pump. The rectification measures with diversion piers of the forebay under the combination of

*Water* **2022**, *14*, 2663 4 of 14

depth of the forebay behind the slope is 5.4 m, and the bottom slope has a gradient of 1:8, as shown in Figure 2. of pumps 1, 3, and 4 will be tested.

pumps 1, 2, and 4 will be mainly studied. The effectiveness of the rectification measures

**Figure 2.** The plane layout of the certain lateral intake pumping station (EL: m, OTH: mm:). (Scale 1:200).

**Figure 2.** The plane layout of the certain lateral intake pumping station (EL: m, OTH: mm:). (Scale 1:200). *2.2. Study Method*  FLUENT is used to numerically simulate the flow pattern of the forebay. The flow Two main pump operation schemes are the combination of No.1 pump, No.2 pump, and No.4 pump and the combination of No.1 pump, No.3 pump, and No.4 pump. The rectification measures with diversion piers of the forebay under the combination of pumps 1, 2, and 4 will be mainly studied. The effectiveness of the rectification measures of pumps 1, 3, and 4 will be tested.

#### governing equations are the continuity equation and the Navier-Stokes equation. LES *2.2. Study Method*

(Large eddy simulation) and RANS (Reynolds averaged Navier-Stokes) are two simulation methods of turbulence. LES can capture large-scale effects and coherent structures in unsteady and nonequilibrium processes and RANS will be incapable of action, but the computation load will be bigger, and consumption time will be longer for LES. The simulation accuracy by RANS can meet the requirement. The realizable k-ε turbulence model for RANS was suitable for the simulation of the flow in the forebay with side-intake of the pumping station [3]. The Reynolds number in the forebay is 2.86 × 105 greater than 3 × 104, which shows that the turbulence is fully developed. The SIMPLEC algorithm is adopted [27]. FLUENT is used to numerically simulate the flow pattern of the forebay. The flow governing equations are the continuity equation and the Navier-Stokes equation. LES (Large eddy simulation) and RANS (Reynolds averaged Navier-Stokes) are two simulation methods of turbulence. LES can capture large-scale effects and coherent structures in unsteady and nonequilibrium processes and RANS will be incapable of action, but the computation load will be bigger, and consumption time will be longer for LES. The simulation accuracy by RANS can meet the requirement. The realizable k-ε turbulence model for RANS was suitable for the simulation of the flow in the forebay with side-intake of the pumping station [3]. The Reynolds number in the forebay is 2.86 × 105 greater than 3 × 104 , which shows that the turbulence is fully developed. The SIMPLEC algorithm is adopted [27].

#### 2.2.1. Boundary Conditions 2.2.1. Boundary Conditions

The side inlet cross-section is used as the inlet of the calculation domain, and the pressure inlet boundary condition was adopted. The pressure on the top of the inlet section is 8 kPa. The pressure at the bottom of the inlet section is 40 kPa. The pressure between the top surface and the bottom surface is linear with the height of the inlet section. For the free surface of the forebay, the shear stress generated by the air on the water surface and heat exchange can be ignored, and the treatment of the symmetrical boundary with the surface of the forebay was adopted according to the rigid-lid assumption. The outlet of the discharge pipe of every pump was used as the outlet boundary. The Mass Flow Outlet The side inlet cross-section is used as the inlet of the calculation domain, and the pressure inlet boundary condition was adopted. The pressure on the top of the inlet section is 8 kPa. The pressure at the bottom of the inlet section is 40 kPa. The pressure between the top surface and the bottom surface is linear with the height of the inlet section. For the free surface of the forebay, the shear stress generated by the air on the water surface and heat exchange can be ignored, and the treatment of the symmetrical boundary with the surface of the forebay was adopted according to the rigid-lid assumption. The outlet of the discharge pipe of every pump was used as the outlet boundary. The Mass Flow Outlet

boundary condition was then adopted. The mass flow of the No. 1 pump was 2600 kg/s,

2600 kg/s. All the solid parts were set to the wall according to the wall function method.

boundary condition was then adopted. The mass flow of the No.1 pump was 2600 kg/s, the mass flow of the No.2 pump was 5400 kg/s, and the mass flow of the No.4 pump was 2600 kg/s. All the solid parts were set to the wall according to the wall function method. *Water* **2022**, *14*, 2663 5 of 14

#### 2.2.2. Grid Independence Verification 2.2.2. Grid Independence Verification

The computing domain is meshed by Fluent Meshing, including a tetrahedral grid in complex parts and hexahedral mesh for non-complex parts. In order to improve the calculation speed and accuracy of the computational domain, grid independence verification should be done. The number of mesh is selected from 180,000, 320,000, 570,000, 800,000, 1.41 million, 3.33 million, and 4.93 million. Head loss, *hf* , was used as an evaluation indicator for grid independence verification. The computing domain is meshed by Fluent Meshing, including a tetrahedral grid in complex parts and hexahedral mesh for non-complex parts. In order to improve the calculation speed and accuracy of the computational domain, grid independence verification should be done. The number of mesh is selected from 180,000, 320,000, 570,000, 800,000, 1.41 million, 3.33 million, and 4.93 million. Head loss, *hf*, was used as an evaluation indicator for grid independence verification.

$$h\_f = \frac{(P\_{\rm in} - P\_{\rm out})}{\rho g} \tag{1}$$

where *Pin* is the total pressure of the inlet; *Pout* is the total pressure of the outlet; *ρ* is the density of water; and *g* is the gravity acceleration. where *Pin* is the total pressure of the inlet; *Pout* is the total pressure of the outlet; *ρ* is the density of water; and *g* is the gravity acceleration.

According to Figure 3, after the number of mesh exceeds 800,000, the head loss of the calculation domain changes little and its head loss does not exceed 2%. Therefore, the number of mesh was chosen as 800,000. According to Figure 3, after the number of mesh exceeds 800,000, the head loss of the calculation domain changes little and its head loss does not exceed 2%. Therefore, the number of mesh was chosen as 800,000.

**Figure 3.** Head loss at different numbers of mesh. **Figure 3.** Head loss at different numbers of mesh.

#### **3. Typical Sections in Calculation Domain and Evaluation Indexes of Flow Pattern in Forebay 3. Typical Sections in Calculation Domain and Evaluation Indexes of Flow Pattern in Forebay**

The uniformity of axial flow velocity, au, is shown in Equation (2).

*V n V V* =

 =− ×

In the formula, ai is the axial velocity of each mesh, m/s; is the average axial ve-

The weighted average angle of flow velocity, *θ,* is shown in Equation (3); *θ* = 90°

1

*i*

As shown in Figure 4, three horizontal sections are taken in the upper, middle, and lower layers of the calculation domain as typical sections, which are called horizontal section I (Z = 1.4 m), horizontal section II (Z = 0.1 m), and horizontal section III (Z = −1 m). The elevation of the bottom of the forebay, as indicated in Figure 1, is Z = 0. As shown in Figure 4, three horizontal sections are taken in the upper, middle, and lower layers of the calculation domain as typical sections, which are called horizontal section I (Z = 1.4 m), horizontal section II (Z = 0.1 m), and horizontal section III (Z = −1 m). The elevation of the bottom of the forebay, as indicated in Figure 1, is Z = 0.

The leftmost section of the forebay at length direction is Y = 0. Cross-section I (Y = 49 m) is selected at the inlet passage of every pump to calculate the uniformity of axial flow velocity and weighted average angle of flow velocity. The uniformity of axial flow velocity (UAFV) and weighted average angle of flow velocity (WAAFV) on the cross-section I can

1 / 100% ai a / 1 *n*

2

( )− (2)

locity of the section, m/s; and *n* is the number of mesh.

quantitatively indicate the inlet condition of pumps.

au

=

In the formula, ti is the transverse velocity of each mesh, m/s.

∑ ai ቀ90° − ୲୧

∑ ୟ୧

ୟ୧ ቁ

(3)

**Figure 4.** Typical sections.

**Figure 4.** Typical sections. **4. Design of Diversion Piers Based on Orthogonal Test Method**  *4.1. Dimension Design of Diversion Piers*  The leftmost section of the forebay at length direction is Y = 0. Cross-section I (Y = 49 m) is selected at the inlet passage of every pump to calculate the uniformity of axial flow velocity and weighted average angle of flow velocity. The uniformity of axial flow velocity (UAFV) and weighted average angle of flow velocity (WAAFV) on the cross-section I can quantitatively indicate the inlet condition of pumps.

As shown in Figure 5, there are three diversion piers: 1 (outer), 2 (middle), and 3 The uniformity of axial flow velocity, *V*au, is shown in Equation (2).

$$V\_{\rm au} = \left(1 - \sqrt{\sum\_{i=1}^{n} \left(V\_{\rm ai}/V\_{\rm a} - 1\right)^{2}/n}\right) \times 100\% \tag{2}$$

and the length of the arc segment of diversion piers 2 and 3 were determined by relative length which is the ratio between the lengths of the two straight-line segments or two arc In the formula, *V*ai is the axial velocity of each mesh, m/s; *V*a is the average axial velocity of the section, m/s; and *n* is the number of mesh.

segments, L1/L2 = l1/l2. C is the length of the straight-line segment at the tail of diversion pier 3, and the length of the straight-line segment at the tail of diversion piers 1 and 2 may The weighted average angle of flow velocity, *θ,* is shown in Equation (3); *θ* = 90◦ indicates that the inlet condition of the pump is very good.

$$\theta = \frac{\sum \, V\_{\text{ai}} \left( 90^{\circ} - \arctan \frac{V\_{\text{ii}}}{V\_{\text{ai}}} \right)}{\sum \, V\_{\text{ai}}} \tag{3}$$

0.9 m. In the formula, *V*ti is the transverse velocity of each mesh, m/s.

#### **4. Design of Diversion Piers Based on Orthogonal Test Method**

#### *4.1. Dimension Design of Diversion Piers*

As shown in Figure 5, there are three diversion piers: 1 (outer), 2 (middle), and 3 (inner) in consideration of the lateral intake forebay. The diversion piers are designed by arc section and the straight-line segment at the head and tail. Diversion piers 2 and 3 are designed with a straight segment at the head L1 = 4.5 m with the arc segment l1 = 6.7 m in diversion pier 1 as the reference length. The length of the straight-line segment at the head and the length of the arc segment of diversion piers 2 and 3 were determined by relative length which is the ratio between the lengths of the two straight-line segments or two arc segments, L1/L2 = l1/l2. C is the length of the straight-line segment at the tail of diversion pier 3, and the length of the straight-line segment at the tail of diversion piers 1 and 2 may be determined to keep the ends of three diversion piers in the same cross-section of the forebay. Relative height is h = H1/H2, where H1 is the height of the diversion pier and H2 is the water depth of the forebay. B is the width of the diversion pier which is smaller than 0.9 m.

*Water* **2022**, *14*, 2663 7 of 14

**Figure 5.** Dimension design of the diversion piers. **Figure 5.** Dimension design of the diversion piers.

#### *4.2. Factors and Results Based on Orthogonal Test Method*

*4.2. Factors and Results Based on Orthogonal Test Method*  The relative length, the width of the diversion pier, the length of the straight-line segment at the tail of the diversion pier, and the relative height are the design parameters for the design of the diversion pier. The uniformity of axial flow velocity and the co-angle of the weighted average angle of flow velocity (CWAAFV) were taken as the evaluation indexes for the orthogonal test results, and the orthogonal test table ଽሺ3ସሻ was selected according to the specified factors and the appropriate amount of the level. The level of each factor and the orthogonal test results are shown in Tables 1 and 2. A, B, C, and D The relative length, the width of the diversion pier, the length of the straight-line segment at the tail of the diversion pier, and the relative height are the design parameters for the design of the diversion pier. The uniformity of axial flow velocity and the co-angle of the weighted average angle of flow velocity (CWAAFV) were taken as the evaluation indexes for the orthogonal test results, and the orthogonal test table *L*9 3 4 was selected according to the specified factors and the appropriate amount of the level. The level of each factor and the orthogonal test results are shown in Tables 1 and 2. A, B, C, and D represent the relative length, the width of the diversion pier, the length of the straight-line segment at the tail of the diversion pier, and the relative height, respectively.

2 1.3 0.4 1.2 0.8 3 1.4 0.5 1.4 1

1 1 1 1 1 85.665 7.398 2 1 2 2 2 88.528 22.473 3 1 3 3 3 79.698 37.819 4 2 1 2 3 76.714 33.429

6 2 3 1 2 80.727 14.701 7 3 1 3 2 73.546 25.101 8 3 2 1 3 82.402 31.221 9 3 3 2 1 77.980 13.649

Ki (i = 1,2,3) of factor I (I = A,B,C,D) for UAFV or CWAAFV in Table 3 is the sum of

three UAFV or CWAAFV values for level i and factor I in Table 2, and then ki = Ki/3, =

**A B C D UAFV (%) CWAAFV (°)**

**Test Number Factors Results of Test**

represent the relative length, the width of the diversion pier, the length of the straight-line **Table 1.** Three levels with four factors.

**Table 2.** Orthogonal test results.

*4.3. Analysis of Orthogonal Test Table of the Diversion Piers*

**Table 2.** Orthogonal test results.

#### *4.3. Analysis of Orthogonal Test Table of the Diversion Piers*

Ki (i = 1,2,3) of factor I (I = A,B,C,D) for UAFV or CWAAFV in Table 3 is the sum of three UAFV or CWAAFV values for level i and factor I in Table 2, and then ki = Ki/3, *R* = {max(*k*1, *k*2, *k*3) − min(*k*1, *k*2, *k*3)}. The *R* value can reflect the strength of the factor on the result, that is, the larger the R value, the greater the influence of the factor on the result. It can be seen from Table 3 that the influence of UAFV is: A > B > C > D, and the order of the design factors of the diversion piers is: the relative length, the width of the diversion pier, the length of the straight-line segment at the tail and the relative height of the diversion pier. It can also be seen from Table 3 that the influence of CWAAFV is: D > C > A > B, and the order of the design factors of the diversion piers is the relative height of the diversion pier, the length of the straight-line segment at the tail, the relative length, and the width of the diversion pier.

**Table 3.** Range analysis table.

The trend diagram of UAFV is shown in Figure 6. It can be seen from Figure 5 that the relative length of the diversion pier and the length of the straight-line segment at the tail and the relative height of the diversion pier increases, and the UAFV value decreases, but when the width of the diversion pier (0.3 m, 0.4 m, or 0.5 m) is 0.4 m, the UAFV is maximized. So, the test scheme when UAFV is optimal is: A1B2C1D1.

ሼmaxሺ1, 2, 3ሻ − minሺ1, 2, 3ሻሽ. The *R* value can reflect the strength of the factor on the result, that is, the larger the R value, the greater the influence of the factor on the result. It can be seen from Table 3 that the influence of UAFV is: A > B > C > D, and the order of the design factors of the diversion piers is: the relative length, the width of the diversion pier, the length of the straight-line segment at the tail and the relative height of the diversion pier. It can also be seen from Table 3 that the influence of CWAAFV is: D > C > A > B, and the order of the design factors of the diversion piers is the relative height of the diversion pier, the length of the straight-line segment at the tail, the relative length, and the

The trend diagram of UAFV is shown in Figure 6. It can be seen from Figure 5 that the relative length of the diversion pier and the length of the straight-line segment at the tail and the relative height of the diversion pier increases, and the UAFV value decreases, but when the width of the diversion pier (0.3 m, 0.4 m, or 0.5 m) is 0.4 m, the UAFV is

The trend diagram of CWAAFV is shown in Figure 7. When the relative length reaches 1.3 or the width of the diversion pier is 0.3 m or the length of the straight-line segment at the tail is 1.0 m or the relative height of the diversion pier is 0.6 m, CWAAFV

> **UAFV CWAAFV A B C D A B C D**

K1 253.890 235.925 248.794 246.071 67.690 65.927 53.319 33.821 K2 239.867 253.356 243.222 242.801 60.904 66.469 69.551 62.274 K3 233.928 238.405 235.670 238.814 69.970 66.168 75.695 102.469 k1 84.630 78.642 82.931 82.024 22.563 21.976 17.773 11.274 k2 79.956 84.452 81.074 80.934 20.301 22.156 23.184 20.758 k3 77.976 79.468 78.557 79.605 23.323 22.056 25.232 34.156 R 6.654 5.810 4.375 2.419 3.022 0.181 7.458 22.883

maximized. So, the test scheme when UAFV is optimal is: A1B2C1D1.

values are minimized. So, the optimal scheme for CWAAFV is: A2B1C1D1.

width of the diversion pier.

**Table 3.** Range analysis table.

**Range Analysis**

**Figure 6. Figure 6.**  The trend of UAFV with three levels for each factor. The trend of UAFV with three levels for each factor.

The trend diagram of CWAAFV is shown in Figure 7. When the relative length reaches 1.3 or the width of the diversion pier is 0.3 m or the length of the straight-line segment at the tail is 1.0 m or the relative height of the diversion pier is 0.6 m, CWAAFV values are minimized. So, the optimal scheme for CWAAFV is: A2B1C1D1. *Water* **2022**, *14*, 2663 9 of 14

**Figure 7.** The trend of CWAAFV with three levels for each factor. **Figure 7.** The trend of CWAAFV with three levels for each factor.

#### *4.4. Analysis of Optimal Result 4.4. Analysis of Optimal Result*

For UAFV, the selection scheme is A1B2C1D1, and for CWAAFV, the selection scheme is A2B1C1D1. The comprehensive balance method is adopted, that is, according to the influence degree of each factor on the evaluation index, the advantages and disadvantages of each factor level for the evaluation index are calculated, and the most favorable scheme is selected. The optimal scheme, A1B2C1D1, of diversion pier design is obtained by calculating with the comprehensive balance method. Based on the numerical simulation, the flow pattern for the optimal scheme is shown in Figure 8. For the No.1 pump, the UAFV was 80.26% and the WAAFV was 77.68°. For the No.2 pump, the UAFV was 98.74% and the WAAFV was 87.84°. For the No.4 pump, the UAFV was 93.41% and the WAAFV was 77.28°. For UAFV, the selection scheme is A1B2C1D1, and for CWAAFV, the selection scheme is A2B1C1D1. The comprehensive balance method is adopted, that is, according to the influence degree of each factor on the evaluation index, the advantages and disadvantages of each factor level for the evaluation index are calculated, and the most favorable scheme is selected. The optimal scheme, A1B2C1D1, of diversion pier design is obtained by calculating with the comprehensive balance method. Based on the numerical simulation, the flow pattern for the optimal scheme is shown in Figure 8. For the No.1 pump, the UAFV was 80.26% and the WAAFV was 77.68◦ . For the No.2 pump, the UAFV was 98.74% and the WAAFV was 87.84◦ . For the No.4 pump, the UAFV was 93.41% and the WAAFV was 77.28◦ .

Horizontal section Ⅰ Local amplification of horizontal section Ⅰ

**Figure 7.** The trend of CWAAFV with three levels for each factor.

For UAFV, the selection scheme is A1B2C1D1, and for CWAAFV, the selection scheme is A2B1C1D1. The comprehensive balance method is adopted, that is, according to the influence degree of each factor on the evaluation index, the advantages and disadvantages of each factor level for the evaluation index are calculated, and the most favorable scheme is selected. The optimal scheme, A1B2C1D1, of diversion pier design is obtained by calculating with the comprehensive balance method. Based on the numerical simulation, the flow pattern for the optimal scheme is shown in Figure 8. For the No.1 pump, the UAFV was 80.26% and the WAAFV was 77.68°. For the No.2 pump, the UAFV was 98.74% and the WAAFV was 87.84°. For the No.4 pump, the UAFV was 93.41% and

*4.4. Analysis of Optimal Result*

the WAAFV was 77.28°.

**Figure 8.** Flow pattern for the optimal scheme of the diversion piers for operating pumps 1, 2, and 4. **Figure 8.** Flow pattern for the optimal scheme of the diversion piers for operating pumps 1, 2, and 4.

and the direction of the flow velocity at the cross-section I is basically parallel to the axial

Figure 9 is a vector diagram of the flow velocity in the domain. With the increase in the trip, the water flow velocity gradually decreases, the backflow at the outlet of the di-

direction, showing that the diversion pier has a good rectification effect.

Figure 9 is a vector diagram of the flow velocity in the domain. With the increase in the trip, the water flow velocity gradually decreases, the backflow at the outlet of the diversion piers gradually disappears, the flow velocity distribution is gradually uniform, and the direction of the flow velocity at the cross-section I is basically parallel to the axial direction, showing that the diversion pier has a good rectification effect. *Water* **2022**, *14*, 2663 11 of 14

**Figure 9.** The velocity vector of the approach diagram. **Figure 9.** The velocity vector of the approach diagram. the flow pattern in the forebay of the pumping station is calculated by the same CFD

#### **5. CFD Numerical Test Analysis of Pumps 1, 3, and 4** method, and the streamline diagram of each typical section is shown in Figure 10. By the calculation, the UAFV of the No.1, No.3, and No.4 pumps were 92.65%,

**5. CFD Numerical Test Analysis of Pumps 1, 3, and 4**  In order to test the rectification effect of the optimal scheme designed in the orthogonal test of the diversion piers under the pump operation scheme of pumps 1, 3, and 4, the flow pattern in the forebay of the pumping station is calculated by the same CFD method, and the streamline diagram of each typical section is shown in Figure 10. In order to test the rectification effect of the optimal scheme designed in the orthogonal test of the diversion piers under the pump operation scheme of pumps 1, 3, and 4, the flow pattern in the forebay of the pumping station is calculated by the same CFD method, and the streamline diagram of each typical section is shown in Figure 10. 94.54%, and 75.81%, respectively. The WAAFV of pumps 1, 3, and 4 were 72.66°, 85.14°, and 78.21°, respectively. Therefore, it can be proved that the optimal scheme of the diversion piers designed by the orthogonal test method is more suitable for different pump startup modes under lateral water intake.

**Figure 10.** *Cont*.

Horizontal section Ⅰ Local amplification of horizontal section Ⅰ

**Figure 10.** Flow pattern for the optimal scheme of the diversion piers for operating pumps 1, 3, and 4. **Figure 10.** Flow pattern for the optimal scheme of the diversion piers for operating pumps 1, 3, and 4.

**6. Conclusions**  The CFD numerical simulation approach was used to study the flow pattern in the forebay and to improve the design parameters of the diversion pier of a pumping station. The forebay of a lateral intake pumping station with asymmetrical working pumps was By the calculation, the UAFV of the No.1, No.3, and No.4 pumps were 92.65%, 94.54%, and 75.81%, respectively. The WAAFV of pumps 1, 3, and 4 were 72.66◦ , 85.14◦ , and 78.21◦ , respectively. Therefore, it can be proved that the optimal scheme of the diversion piersdesigned by the orthogonal test method is more suitable for different pump startup modes under lateral water intake.

#### subjected to flow analysis using a realizable k-ε Turbulent model, the SIMPLEC method, Pressure Inlet boundary condition, where the pressure between the top surface and the **6. Conclusions**

bottom surface is linear with the height of the inlet section, and Mass Flow Outlet boundary condition. Based on the flow parameters, the diversion pier was chosen as the optimization strategy. Nine groups of the numerical simulation on various layouts of diversion piers were examined using the orthogonal test method, with the uniform axial flow velocity and weighted average angle of the flow velocity of each pump's inlet cross-section. This provided a reference for related pumping station engineering. The following conclusions were drawn: (1) The parameters of diversion piers of the orthogonal test design having four factors and three levels are different, and the uniformity of axial flow velocity and the weighted average angle of the flow velocity of the inlet passages for the pumps are also different. The CFD numerical simulation approach was used to study the flow pattern in the forebay and to improve the design parameters of the diversion pier of a pumping station. The forebay of a lateral intake pumping station with asymmetrical working pumps was subjected to flow analysis using a realizable k-ε Turbulent model, the SIMPLEC method, Pressure Inlet boundary condition, where the pressure between the top surface and the bottom surface is linear with the height of the inlet section, and Mass Flow Outlet boundary condition. Based on the flow parameters, the diversion pier was chosen as the optimization strategy. Nine groups of the numerical simulation on various layouts of diversion piers were examined using the orthogonal test method, with the uniform axial flow velocity and weighted average angle of the flow velocity of each pump's inlet

cross-section. This provided a reference for related pumping station engineering. The following conclusions were drawn:

(1) The parameters of diversion piers of the orthogonal test design having four factors and three levels are different, and the uniformity of axial flow velocity and the weighted average angle of the flow velocity of the inlet passages for the pumps are also different. With a comprehensive balance method, the optimal combination of diversion pier parameters can be selected as A1B2C1D1. The uniformity of axial flow velocity and the weighted average angle of flow velocity after selecting the optimal combination are greatly improved.

(2) After selecting the optimal scheme through the orthogonal test design of the diversion piers, the results showed that a small range of reflux phenomena appears in the outlet position of the diversion piers. However, with the increase in the water flow approach, the backflow area gradually disappears, and the water flow is relatively straight. Because the flow of the No.2 pump unit is greater than that of the No.1 pump unit and No.4 pump unit, the velocity distribution of the cross-section of the forebay shows a trend of middle-high and low on both sides.

(3) Under the pump operation scheme of pumps 1, 3, and 4, the rectification effect of the optimal scheme designed in the orthogonal test of the diversion piers under the pump operation scheme of pumps 1, 2, and 4 has been well tested. It can be proved that the size of the diversion pier designed by the orthogonal test method has a high application value in the study of the lateral intake pumping station.

RANS with a realizable k-ε turbulent model was used in this study. The LES turbulent model will be a good approach for higher accuracy and fine analysis of the vortex field. The diversion piers in the paper combined with the sill is likely obtain a better rectification effect, according to the relevant references. Sediment deposition in the front of the sill, however, needs to be taken into consideration; it also necessitates further investigation.

**Author Contributions:** Conceptualization, C.Z. and H.Y.; methodology, C.Z.; software, C.Z.; validation, H.Y., Y.Y., and M.T.J.; formal analysis, C.Z.; investigation, C.Z.; resources, Y.Y.; data curation, H.Y.; writing—original draft preparation, M.T.J.; writing—review and editing, Y.Y.; visualization, C.Z.; supervision, Y.Y.; project administration, Y.Y.; funding acquisition, Y.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research was supported by the National Key Research and Development Program (2019YFC0409000).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **Analysis of Hydraulic Performance and Flow Characteristics of Inlet and Outlet Channels of Integrated Pump Gate**

**Chuanliu Xie 1,\*, Weipeng Xuan 1 , Andong Feng 1 and Fei Sun 2**

**Abstract:** The integrated pump gate structure can improve the shortcomings of traditional asymmetric pumping stations with large floor space, but its internal flow mechanism is not clear, which affects its efficient, stable, and safe operation. In order to reveal its internal fluid flow characteristics, numerical simulations based on the N-S equation with the SST *k*-*ω* turbulence model are used in this paper, and experimental validation is carried out. The test results yielded an efficiency of 60.50% near the design flow condition, corresponding to a flow rate of 11.5 L/s, a head of 2.7569 m, a hydraulic loss of 0.064 m in the inlet channel, and a hydraulic loss of 1.337 m in the outlet channel. The integrated pump gate has a uniform inlet water flow pattern, less undesirable flow pattern, and a large backflow vortex in the outlet water. This paper reveals the internal flow characteristics of its integrated pump gate inlet and outlet water, and the research results can provide some reference for the design, theoretical analysis, and application of similar integrated pump gates.

**Keywords:** integrated pump gate; inlet channel; outlet channel; flow pattern; hydraulic performance

#### **1. Introduction**

In recent years, the frequent occurrence of extreme rainfall weather has posed a serious challenge to the flood control and drainage capacity of cities and towns. In the original pumping station construction, the asymmetric arrangement form is usually used, that is, the combination of sluice gate and pumping station, the gate is set in the middle of the river cross-section direction, and the pump room is set next to the sluice gate. Usually, the area of the water barrier sluice is small, and the pump chambers on both sides cannot pass through the water when no flooding work is needed, resulting in low water exchange efficiency inside and outside the river. At the same time, the construction of a large area of land, long construction period, high economic costs are the traditional pumping station disadvantages. Therefore, the integrated pump and gate device came into being. The integrated pump and gate device is a pumping station that combines a pumping station and a water barrier sluice into a whole arrangement, in which the pump sluice can be used as a water barrier structure instead of a traditional sluice gate, and can provide support for the pump. The pump is arranged on the gate, so there is no need to build a fixed pump room, so that the gate and the pump room are combined into a whole device, which greatly improves the flooding capacity of the river, significantly increases the vitality of the water body in the river channel, and has the advantages of occupying a small area of land compared to traditional pumping stations, short project cycle, lifting and handling, and automatic control, etc., which can also significantly reduce the construction cost, land cost, and alteration cost, and so on, while improving the efficiency of water exchange.

The integrated pump gate consists of sluice gate, gate pump, clapper gate, water stop structure, opening and closing mechanism, etc. Relevant scholars have analyzed its application characteristics and technical advantages, and compared it with traditional pump gate in terms of floor space, circulation, economy, construction cycle, etc. Considering

**Citation:** Xie, C.; Xuan, W.; Feng, A.; Sun, F. Analysis of Hydraulic Performance and Flow Characteristics of Inlet and Outlet Channels of Integrated Pump Gate. *Water* **2022**, *14*, 2747. https:// doi.org/10.3390/w14172747

Academic Editors: Ran Tao, Changliang Ye and Xijie Song

Received: 28 July 2022 Accepted: 30 August 2022 Published: 2 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

that integrated pump gates play a significant role in controlling drainage within urban areas and managing black smelly water bodies [1]. Some scholars have discussed their design points and concluded that in urban and rural river management, integrated pump gates will become the mainstream and where the trend of development lies [2]. Some scholars have explored their structures, such as the design of horizontal axial submersible pumps suitable for pump gates [3], to investigate the effects of pump form, installation number, overhang height, pump spacing, and flapper angle on the performance of integrated pump gates [4]. Some other scholars have conducted numerical calculations and model experimental studies on pump gates to derive the stress conditions, flow conditions, and gate vibration characteristics of vertical surface-hole integrated pump gates and horizontal surface-hole integrated pump gates under different operating conditions [5,6]. There is always a negative pressure zone directly below the bottom edge of the flat bottom gate [7], suggesting the two-phase flow and pump gate characteristics in the pump gate pool [8], testing the energy characteristics of the ecological gate pump during bi-directional operation, and resulting in a error of ±2% with the numerical simulation, which verified the reasonableness of the numerical simulation, and finally predicted the optimal installation angle of the vane by numerical simulation [9]. These research results provide basic data for subsequent pump gates selection and other aspects, and also provide reference for subsequent pump gates design and research. urban areas and managing black smelly water bodies [1]. Some scholars have discussed their design points and concluded that in urban and rural river management, integrated pump gates will become the mainstream and where the trend of development lies [2]. Some scholars have explored their structures, such as the design of horizontal axial submersible pumps suitable for pump gates [3], to investigate the effects of pump form, installation number, overhang height, pump spacing, and flapper angle on the performance of integrated pump gates [4]. Some other scholars have conducted numerical calculations and model experimental studies on pump gates to derive the stress conditions, flow conditions, and gate vibration characteristics of vertical surface-hole integrated pump gates and horizontal surface-hole integrated pump gates under different operating conditions [5,6]. There is always a negative pressure zone directly below the bottom edge of the flat bottom gate [7], suggesting the two-phase flow and pump gate characteristics in the pump gate pool [8], testing the energy characteristics of the ecological gate pump during bi-directional operation, and resulting in a error of ±2% with the numerical simulation, which verified the reasonableness of the numerical simulation, and finally predicted the optimal installation angle of the vane by numerical simulation [9]. These research results provide basic data for subsequent pump gates selection and other aspects, and also provide reference for subsequent pump gates design and research. At present, the integrated pump gate has formed a certain scale of industrial system

application characteristics and technical advantages, and compared it with traditional pump gate in terms of floor space, circulation, economy, construction cycle, etc. Considering that integrated pump gates play a significant role in controlling drainage within

*Water* **2022**, *14*, x FOR PEER REVIEW 2 of 22

At present, the integrated pump gate has formed a certain scale of industrial system and user demand, but the related research is less, and the flow pattern of the front pool and inlet and outlet water channels is not clear, which limits the further development of the integrated pump gate technology, so it is necessary to carry out further related research, and the research results of this paper have certain theoretical significance and engineering value. and user demand, but the related research is less, and the flow pattern of the front pool and inlet and outlet water channels is not clear, which limits the further development of the integrated pump gate technology, so it is necessary to carry out further related research, and the research results of this paper have certain theoretical significance and engineering value.

#### **2. Numerical Simulation Model and Method 2. Numerical Simulation Model and Method**  *2.1. Pump Gate Modeling*

#### *2.1. Pump Gate Modeling* In accordance with a project pump station design, impeller wood mold diagram,

In accordance with a project pump station design, impeller wood mold diagram, pump gate structure diagram, and other applications of SolidWorks software were used to establish an integrated pump gate model, imported into ANSYS Workbench software [10] in the geometry module for adjustment and boundary condition naming; where model impeller diameter *D* = 60 mm, rotation rate *n* = 6692 r/min, the gate length is 188 mm, width is 60 mm and height is 299 mm, integrated pump gate design flow rate *Qd* = 11.5 L/s, design head *H* = 2.7569 m. Three-dimensional geometric model of integrated pump gate as shown in Figure 1. pump gate structure diagram, and other applications of SolidWorks software were used to establish an integrated pump gate model, imported into ANSYS Workbench software [10] in the geometry module for adjustment and boundary condition naming; where model impeller diameter *D* = 60 mm, rotation rate *n* = 6692 r/min, the gate length is 188 mm, width is 60 mm and height is 299 mm, integrated pump gate design flow rate *Qd* = 11.5 L/s, design head *H* = 2.7569 m. Three-dimensional geometric model of integrated pump gate as shown in Figure 1.

**Figure 1.** *Cont*.

100

*Water* **2022**, *14*, x FOR PEER REVIEW 3 of 22

**Figure 1.** Three-dimensional geometric model of integrated pump gate. (**a**) Integrated gate general assembly; (**b**) Inlet channel; (**c**) Outlet channel; (**d**) Gate; (**e**) Impeller; (**f**) Guide vane. **Figure 1.** Three-dimensional geometric model of integrated pump gate. (**a**) Integrated gate general assembly; (**b**) Inlet channel; (**c**) Outlet channel; (**d**) Gate; (**e**) Impeller; (**f**) Guide vane. **Figure 1.** Three-dimensional geometric model of integrated pump gate. (**a**) Integrated gate general assembly; (**b**) Inlet channel; (**c**) Outlet channel; (**d**) Gate; (**e**) Impeller; (**f**) Guide vane.

#### *2.2. Mesh Division 2.2. Mesh Division 2.2. Mesh Division*

According to the three-dimensional structure of the integrated pump gate, the water overflow part of it is extracted, and the numerical simulation area of the integrated pump gate includes: open inlet channel, gate, impeller, guide vanes, and open outlet channel. In the calculation of this paper, the integrated pump gate is a table-hole type, and in order to better retain the characteristics of the pump gate surface, the model is divided by unstructured mesh, and the number of geometric model meshes is changed under the design flow condition (*Qd* = 11.5 L/s) for mesh-independent analysis. After the grid number reaches 3.03 million, the efficiency of the pump gate basically does not change with the increase of the grid number [11], and this paper determines how to use the grid number of 3.03 million for the subsequent numerical simulation work. Mesh division of each calculation component as shown in Figure 2. The grid-independence results are shown in Figure 3. According to the three-dimensional structure of the integrated pump gate, the water overflow part of it is extracted, and the numerical simulation area of the integrated pump gate includes: open inlet channel, gate, impeller, guide vanes, and open outlet channel. In the calculation of this paper, the integrated pump gate is a table-hole type, and in order to better retain the characteristics of the pump gate surface, the model is divided by unstructured mesh, and the number of geometric model meshes is changed under the design flow condition (*Qd* = 11.5 L/s) for mesh-independent analysis. After the grid number reaches 3.03 million, the efficiency of the pump gate basically does not change with the increase of the grid number [11], and this paper determines how to use the grid number of 3.03 million for the subsequent numerical simulation work. Mesh division of each calculation component as shown in Figure 2. The grid-independence results are shown in Figure 3. According to the three-dimensional structure of the integrated pump gate, the water overflow part of it is extracted, and the numerical simulation area of the integrated pump gate includes: open inlet channel, gate, impeller, guide vanes, and open outlet channel. In the calculation of this paper, the integrated pump gate is a table-hole type, and in order to better retain the characteristics of the pump gate surface, the model is divided by unstructured mesh, and the number of geometric model meshes is changed under the design flow condition (*Qd* = 11.5 L/s) for mesh-independent analysis. After the grid number reaches 3.03 million, the efficiency of the pump gate basically does not change with the increase of the grid number [11], and this paper determines how to use the grid number of 3.03 million for the subsequent numerical simulation work. Mesh division of each calculation component as shown in Figure 2. The grid-independence results are shown in Figure 3.

**Figure 2.** Mesh division of each calculation component.

**Figure 3.** Grid-independent analysis. **Figure 3.** Grid-independent analysis.

*2.3. Control Equations, Boundary Conditions, and Calculation Methods*

*2.3. Control Equations, Boundary Conditions, and Calculation Methods*  In the numerical simulation of the integrated pump gate, the fluid is treated as a three-dimensional incompressible viscous turbulent flow, without considering heat ex-In the numerical simulation of the integrated pump gate, the fluid is treated as a threedimensional incompressible viscous turbulent flow, without considering heat exchange. Boundary condition settings as shown in Table 1.

change. Boundary condition settings as shown in Table 1. **Table 1.** Boundary condition settings.

Static interfaces None In this paper, the SST *k*-*ω* [12] turbulence model is used to calculate the flow characteristics inside the integrated pump gate, which combines the advantages of the standard *k*-*ε* model [13] and the standard *k*-*ω* model [14] and captures the flow in the boundary In this paper, the SST *k*-*ω* [12] turbulence model is used to calculate the flow characteristics inside the integrated pump gate, which combines the advantages of the standard *k*-*ε* model [13] and the standard *k*-*ω* model [14] and captures the flow in the boundary layer better by using automatic functions in the boundary layer, while the finite element-based finite volume method is used to solve the problem.

layer better by using automatic functions in the boundary layer, while the finite elementbased finite volume method is used to solve the problem. In order to evaluate the hydraulic performance of the integrated pump gate after numerical simulation, the hydraulic losses of the inlet and outlet channels, the uniformity of the axial velocity distribution in the characteristic section, the weighted average angle of the velocity in the characteristic section and the energy characteristics of the pump gate In order to evaluate the hydraulic performance of the integrated pump gate after numerical simulation, the hydraulic losses of the inlet and outlet channels, the uniformity of the axial velocity distribution in the characteristic section, the weighted average angle of the velocity in the characteristic section and the energy characteristics of the pump gate were introduced as evaluation bases, and the performance characteristics of the integrated pump gate were evaluated by five quantitative indicators.

were introduced as evaluation bases, and the performance characteristics of the integrated pump gate were evaluated by five quantitative indicators. In this paper, the total energy difference between the inlet channel of the integrated In this paper, the total energy difference between the inlet channel of the integrated pump gate and the outlet of the outlet channel is defined as the head of the device and is expressed by the following equation [15]:

$$H\_{het} = \left(\frac{\int \mathcal{P}\_2 u\_l ds}{\rho Qg} + H\_2 + \frac{\int u\_2^2 u\_{l2} ds}{2Qg}\right) - \left(\frac{\int \mathcal{P}\_1 u\_l ds}{\rho Qg} + H\_1 + \frac{\int u\_1^2 u\_{l1} ds}{2Qg}\right) \tag{1}$$

ρ

 where the first term on the right side of the equation is the total pressure at the outlet of where the first term on the right side of the equation is the total pressure at the outlet of the outlet channel and the second term is the total pressure at the inlet of the inlet channel.

the outlet channel and the second term is the total pressure at the inlet of the inlet channel. Where *Q* is the flow rate (m3/s), *H*1, *H*2 are the inlet and outlet section elevations of the integrated pump gate (m), *s*1, *s*2 for the integrated pump gate inlet and outlet section, *u*1, *u*2 for the integrated pump gate inlet and outlet water channel section flow rate at each Where *Q* is the flow rate (m3/s), *H*1, *H*2 are the inlet and outlet section elevations of the integrated pump gate (m), *s*1, *s*2 for the integrated pump gate inlet and outlet section, *u*1, *u*2 for the integrated pump gate inlet and outlet water channel section flow rate at each point (m/s), *ut*1, *ut*2 are the normal components of flow velocity (m/s) at each point of the inlet and outlet channel sections of the integrated pump gate, *P*1, *P*2 are the static pressure

ρ (Pa) at each point of the inlet and outlet sections of the integrated pump gate, *g* is the acceleration of gravity (m/s2 ).

The efficiency of the integrated pump gate is calculated as [16,17]:

$$\eta = \frac{\rho g Q H\_{\text{net}}}{T\_p w} \tag{2}$$

where *Tp* is the torque (N-m), *ω* is the rotational angular speed of the impeller. The hydraulic loss *hf* is calculated as [18,19]:

$$h\_f = E\_1 - E\_2 = \left(\frac{P\_1}{\rho g} - \frac{P\_2}{\rho g}\right) + (Z\_1 - Z\_2) + \left(\frac{\mu\_1^2}{2g} - \frac{\mu\_2^2}{2g}\right) \tag{3}$$

Among them:

$$E\_1 = \frac{P\_1}{\rho g} + Z\_1 + \frac{\mu\_1^2}{2g};\ E\_2 = \frac{P\_2}{\rho g} + Z\_2 + \frac{\mu\_2^2}{2g}.$$

where *E*1, *E*2 are the total energy at the inlet and outlet of the open flow channel, *P*1, *P*2 are the static pressure at the inlet and outlet of the open flow channel (Pa), *Z*1, *Z*2 are the height of the open flow channel inlet and outlet (m), *u*1, *u*2 are the open flow channel inlet and outlet velocity (m/s).

The uniformity of flow velocity distribution is calculated as [20]:

$$V\_{\rm u} = \left\{ 1 - \frac{1}{\overline{\upsilon\_a}} \sqrt{\left[ \sum\_{i=1}^n (v\_{ai} - \overline{v\_a})^2 \right] / n} \right\} \times 100\% \tag{4}$$

where, *Vu* is the uniformity of axial flow velocity distribution in the characteristic section (%), *vai* is the axial velocity of each calculation unit (m/s), *n* is the number of calculation units. The velocity weighted average angle is calculated as [21]:

$$\overline{\theta} = \frac{\sum u\_{ai} \left[ \Re^{\diamond} - \arctan \left( \frac{u\_{ti}}{u\_{ai}} \right) \right]}{\sum u\_{ai}} \tag{5}$$

where *uti* is the horizontal velocity (m/s) of each unit in the characteristic section of the flow channel. *uai* is the axial velocity (m/s) of each calculation unit.

#### **3. Numerical Simulation Results and Analysis**

*3.1. Hydraulic Performance Results and Analysis*

The pressure and torque are extracted from the numerical simulation result file, and the pump gate head is calculated according to equation (1), and the pump gate efficiency is calculated according to equation (2) to obtain the energy performance of the integrated pump gate, as shown in Table 2. The energy performance curve of the integrated pump gate is plotted as shown in Figure 4.

**Table 2.** Numerical simulation of the energy performance of an integrated pump gate.

*Water* **2022**, *14*, x FOR PEER REVIEW 6 of 22

**Figure 4.** Pump gate energy performance curve. **Figure 4.** Pump gate energy performance curve.

The calculation results in Table 2 and Figure 4 show that the flow-efficiency curve of the pump gate is approximately quadratic when the inlet water flow is *Q* = 0.5~15.5 L/s, and the head of the pump gate gradually decreases from 7.24 m to 0.34 m, and the efficiency of the pump gate is 60.50% near the design flow condition, the corresponding flow is 11.5 L/s and the head is 2.7569 m. When the flow rate is 10.5~12.5 L/s (0.91~1.09 *Qd*), the pump gate is in the high efficiency zone, and the efficiency of the pump gate is around 59~60%. When the flow rate is 5.5~7.5 L/s (0.48~0.65 *Qd*), the pump gate is located near the saddle area, and the operation of the pump gate is not stable at this time, so it is recommended to avoid operating in this flow range. The calculation results in Table 2 and Figure 4 show that the flow-efficiency curve of the pump gate is approximately quadratic when the inlet water flow is *Q* = 0.5~15.5 L/s, and the head of the pump gate gradually decreases from 7.24 m to 0.34 m, and the efficiency of the pump gate is 60.50% near the design flow condition, the corresponding flow is 11.5 L/s and the head is 2.7569 m. When the flow rate is 10.5~12.5 L/s (0.91~1.09 *Qd* ), the pump gate is in the high efficiency zone, and the efficiency of the pump gate is around 59~60%. When the flow rate is 5.5~7.5 L/s (0.48~0.65 *Qd* ), the pump gate is located near the saddle area, and the operation of the pump gate is not stable at this time, so it is recommended to avoid operating in this flow range. the pump gate is approximately quadratic when the inlet water flow is *Q* = 0.5~15.5 L/s, and the head of the pump gate gradually decreases from 7.24 m to 0.34 m, and the efficiency of the pump gate is 60.50% near the design flow condition, the corresponding flow is 11.5 L/s and the head is 2.7569 m. When the flow rate is 10.5~12.5 L/s (0.91~1.09 *Qd*), the pump gate is in high efficiency zone, and the efficiency of the pump gate is around 59~60%. When the flow rate is 5.5~7.5 L/s (0.48~0.65 *Qd*), the pump gate is located near the saddle area, and operation of the pump gate is not stable at this time, so it is recommended to avoid operating in this flow range.

The calculation results in Table 2 and Figure 4 show that the flow-efficiency curve of

13.5 (1.17 *Qd*) 1.7245 54.87 14.5 (1.26 *Qd*) 1.0426 43.58

13.5 (1.17 *Qd*) 1.7245 54.87 14.5 (1.26 *Qd*) 1.0426 43.58

#### *3.2. Analysis of Internal Flow in Inlet Channel 3.2. Analysis of Internal Flow in Inlet Channel*

*3.2. Analysis of Internal Flow in Inlet Channel*  An axial section was created using the impeller rotation axis as a reference to explore the internal flow characteristics of the inlet flow channel before the pump gate. The sche-An axial section was created using the impeller rotation axis as a reference to explore the internal flow characteristics of the inlet flow channel before the pump gate. The schematic diagram of the axial section is shown in Figure 5. An axial section was created using the impeller rotation axis as a reference to explore the internal flow characteristics of the inlet flow channel before the pump gate. The schematic diagram of the axial section is shown in Figure 5.

**Figure 5.** Schematic diagram of the axial section of the open inlet channel. **Figure 5.** Schematic diagram of the axial section of the open inlet channel.

**Figure 5.** Schematic diagram of the axial section of the open inlet channel. 3.2.1. Inlet Channel Streamline and Axial Flow Velocity Distribution

3.2.1. Inlet Channel Streamline and Axial Flow Velocity Distribution 3.2.1. Inlet Channel Streamline and Axial Flow Velocity Distribution The streamline and axial flow velocity distribution clouds of the open inlet channel The streamline and axial flow velocity distribution clouds of the open inlet channel under different flow conditions in the section are drawn, as shown in Figure 6.

under different flow conditions in the section are drawn, as shown in Figure 6.

under different flow conditions in the section are drawn, as shown in Figure 6.

The streamline and axial flow velocity distribution clouds of the open inlet channel

**Figure 6.** Cloud diagram of inlet channel streamline and axial flow velocity distribution under different working conditions. **Figure 6.** Cloud diagram of inlet channel streamline and axial flow velocity distribution under different working conditions.

Through the analysis of Figure 6, it is concluded that under different flow conditions, the streamline of open inlet channel is smooth, contraction is reasonable, and the streamline is regular, which can provide better water inlet conditions for the impeller. The slope of the open inlet channel is not easy to be too large, as too large will not only lead to aggregation of water flow, but also very easy to form vortex at the bottom, hence should be avoided; the figure can be found in the inlet channel of the slope of the decline is reasonable, no bad flow conditions. Through the analysis of Figure 6, it is concluded that under different flow conditions, the streamline of open inlet channel is smooth, contraction is reasonable, and the streamline is regular, which can provide better water inlet conditions for the impeller. The slope of the open inlet channel is not easy to be too large, as too large will not only lead to aggregation of water flow, but also very easy to form vortex at the bottom, hence should be avoided; the figure can be found in the inlet channel of the slope of the decline is reasonable, no bad flow conditions.

Under each flow condition (*Q* = 8.5~14.5 L/s), the flow velocity increases in the direction from the inlet of the inlet channel to the inlet of the pump gate; at the top of the pump gate structure, there is a low velocity zone because the water flow receives the constraint of the pump gate; the water flow gathers at the inlet of the pump because of the negative pressure created when the pump rotates, and the flow velocity reaches the maximum value here, which is greater than 0.6 m/s. The flow velocity at the inlet of the pump gate varies in a uniform gradient. Under the design condition (*Q* = 11.5 L/s), the velocity field is the most uniform in the inlet channel, and under low and high flow conditions, the high and low velocity areas are mixed more obviously. Under each flow condition (*Q* = 8.5~14.5 L/s), the flow velocity increases in the direction from the inlet of the inlet channel to the inlet of the pump gate; at the top of the pump gate structure, there is a low velocity zone because the water flow receives the constraint of the pump gate; the water flow gathers at the inlet of the pump because of the negative pressure created when the pump rotates, and the flow velocity reaches the maximum value here, which is greater than 0.6 m/s. The flow velocity at the inlet of the pump gate varies in a uniform gradient. Under the design condition (*Q* = 11.5 L/s), the velocity field is the most uniform in the inlet channel, and under low and high flow conditions, the high and low velocity areas are mixed more obviously.

The local area from the slope of the inlet channel to the inlet of the pump gate is extracted separately as a research object, and the streamline and pressure distribution clouds at the inlet of the pump gate of the open inlet channel under different flow conditions are drawn in the section, as shown in Figure 7. The local area from the slope of the inlet channel to the inlet of the pump gate is extracted separately as a research object, and the streamline and pressure distribution clouds at the inlet of the pump gate of the open inlet channel under different flow conditions are drawn in the section, as shown in Figure 7.

It can be concluded from Figure 7 that with the increase of flow rate (*Q* = 8.5~14.5 L/s), the static pressure in the inlet channel is gradually increasing, and under the design condition (*Q* = 11.5 L/s), the pressure at the bottom of the slope to the pump inlet is approximately the same, and the pressure only starts to decrease in front of the pump gate impeller inlet, under the high flow rate and low flow rate conditions, the pressure gradient near the pump gate inlet is large and the pressure gradually decreases, the pressure is lowest at the inlet of the pump gate.

#### 3.2.2. Hydraulic Loss of Inlet Channel

According to the Formula (3) to calculate the inlet channel hydraulic loss, drawing different flow conditions in the inlet channel hydraulic loss curve, as shown in Figure 8.

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(**a**) *Q* = 8.5 L/s (0.74 *Qd*) (**b**) *Q* = 9.5 L/s (0.83 *Qd*) (**c**) *Q* = 10.5 L/s (0.91 *Qd*)

**Figure 7.** Cloud diagram of streamline and pressure distribution at the inlet of pump gate under different working conditions. **Figure 7.** Cloud diagram of streamline and pressure distribution at the inlet of pump gate under different working conditions. According to the Formula (3) to calculate the inlet channel hydraulic loss, drawing different flow conditions in the inlet channel hydraulic loss curve, as shown in Figure 8.

3.2.2. Hydraulic Loss of Inlet Channel

**Figure 8. Figure 8.** Hydraulic loss curve of inlet channel under different working conditions. Hydraulic loss curve of inlet channel under different working conditions.

**Figure 8.** Hydraulic loss curve of inlet channel under different working conditions.

The analysis of Figure 8 shows that the hydraulic loss of the inlet channel *hf* is positively correlated with the flow rate *Q*, which approximately satisfies the quadratic function, and the hydraulic loss is the smallest when the inlet flow rate is 8.5 L/s (0.74 *Qd* ), which is 0.039 m, and the largest when the flow rate is 14.5 L/s (1.26 *Qd* ), which is 0.100 m. The larger the flow rate, the larger the hydraulic loss, and in this type of pump station, the hydraulic loss of the inlet and outlet channels is a decisive factor in the efficiency of the pump gate. The calculation results show that the average level of hydraulic loss of the inlet channel is about 6 cm, which is in line with the conventional theory and design.

#### *3.3. Analysis of Internal Flow in Outlet Channel 3.3. Analysis of Internal Flow in Outlet Channel 3.3. Analysis of Internal Flow in Outlet Channel*  An axial section is created with the impeller rotation axis to observe the flow charac-

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*Water* **2022**, *14*, x FOR PEER REVIEW 9 of 22

An axial section is created with the impeller rotation axis to observe the flow characteristics in the outflow channel. The schematic diagram of the axial section is shown in Figure 9. An axial section is created with the impeller rotation axis to observe the flow characteristics in the outflow channel. The schematic diagram of the axial section is shown in Figure 9. teristics in the outflow channel. The schematic diagram of the axial section is shown in Figure 9.

The analysis of Figure 8 shows that the hydraulic loss of the inlet channel *hf* is positively correlated with the flow rate *Q*, which approximately satisfies the quadratic function, and the hydraulic loss is the smallest when the inlet flow rate is 8.5 L/s (0.74 *Qd*), which is 0.039 m, and the largest when the flow rate is 14.5 L/s (1.26 *Qd*), which is 0.100 m. The larger the flow rate, the larger the hydraulic loss, and in this type of pump station, the hydraulic loss of the inlet and outlet channels is a decisive factor in the efficiency of the pump gate. The calculation results show that the average level of hydraulic loss of the inlet channel is about 6 cm, which is in line with the conventional theory and design.

tively correlated with the flow rate *Q*, which approximately satisfies the quadratic tion, and the hydraulic loss is the smallest when the inlet flow rate is 8.5 L/s (0.74 *Qd*

which is 0.039 m, and the largest when the flow rate is 14.5 L/s (1.26 *Qd*), which is 0.100 m. The larger the flow rate, the larger the hydraulic loss, and in this type of pump station,

hydraulic loss of the inlet and outlet channels is a decisive factor in the efficiency of pump gate. The calculation results show that the average level of hydraulic loss of the

inlet channel is about 6 cm, which is in line with the conventional theory and design.

The analysis of Figure 8 shows that the hydraulic loss of the inlet channel *hf* is posi-

**Figure 9.** Schematic diagram of axial section of open outlet channel. **Figure 9.** Schematic diagram of axial section of open outlet channel. **Figure 9.** Schematic diagram of axial section of open outlet channel.

3.3.1. Streamline and Axial Velocity Distribution of Outlet Channel 3.3.1. Streamline and Axial Velocity Distribution of Outlet Channel

3.3.1. Streamline and Axial Velocity Distribution of Outlet Channel The streamline and axial flow velocity distribution clouds of the open outlet channel under different flow conditions in the section were drawn, as shown in Figure 10. The streamline and axial flow velocity distribution clouds of the open outlet channel under different flow conditions in the section were drawn, as shown in Figure 10. The streamline and axial flow velocity distribution clouds of the open outlet channel under different flow conditions in the section were drawn, as shown in Figure 10.

(**g**) *Q* = 14.5 L/s (1.26 *Qd*)

**Figure 10.** Streamline and axial velocity distribution of the outlet channel under different flow conditions. **Figure 10.** Streamline and axial velocity distribution of the outlet channel under different flow conditions. **Figure 10.** Streamline and axial velocity distribution of the outlet channel under different flow conditions.

It can be found in Figure 10 that the outflow channel is turbulent near the pump gate, and a return vortex parallel to the flow direction is formed at the outlet of the pump gate guide vane, and the size of the vortex increases as the flow rate increases. When the flow rate is 8.5~10.5 L/s (0.74~0.91 *Qd* ), the axial flow velocity at the outlet of the pump gate is about 3.3 m/s; when the flow rate is 11.5~14.5 L/s (*Qd*~1.26 *Qd* ), the axial flow velocity at the outlet is about 4.1 m/s, and then there is a step transition along the outlet direction, due to the existence of vortex, the axial flow velocity above the pump gate is lower. The axial flow velocity above the outlet is low. The existence of the vortex is mainly due to the open outlet upper part of the stagnant water area, where the flow velocity is low due to

the rotation of the impeller, the lower and middle water flow velocity is fast, making the formation parallel to the direction of the water flow back vortex. the rotation of the impeller, the lower and middle water flow velocity is fast, making the formation parallel to the direction of the water flow back vortex.

It can be found in Figure 10 that the outflow channel is turbulent near the pump gate, and a return vortex parallel to the flow direction is formed at the outlet of the pump gate guide vane, and the size of the vortex increases as the flow rate increases. When the flow rate is 8.5~10.5 L/s (0.74~0.91 *Qd*), the axial flow velocity at the outlet of the pump gate is about 3.3 m/s; when the flow rate is 11.5~14.5 L/s (*Qd*~1.26 *Qd*), the axial flow velocity at the outlet is about 4.1 m/s, and then there is a step transition along the outlet direction, due to the existence of vortex, the axial flow velocity above the pump gate is lower. The axial flow velocity above the outlet is low. The existence of the vortex is mainly due to the open outlet upper part of the stagnant water area, where the flow velocity is low due to

The local flow field at the outlet of the pump gate is taken out, and the streamline and pressure distribution clouds at the outlet of the pump gate of the open outflow channel under different flow conditions in the section are drawn, as shown in Figure 11. The local flow field at the outlet of the pump gate is taken out, and the streamline and pressure distribution clouds at the outlet of the pump gate of the open outflow channel under different flow conditions in the section are drawn, as shown in Figure 11.

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(**g**) *Q* = 14.5 L/s (1.26 *Qd*)

**Figure 11.** Streamline and pressure distribution at the pump gate outlet under different flow conditions.

As can be seen from Figure 11, at different flow rates, there is a backflow vortex above the pump gate outlet. With the increase of flow rate, the size of the vortex area also increases, and the center of the backflow vortex also shifted to the outlet of the outflow channel. Under the action of gravity, the fluid flows as a whole to the bottom of the pool to do offset flow, and due to the existence of vortex, the pressure at the outlet is low, and with the increase of vortex, the low pressure area also increases; pump gate outlet is 5~9 *D* at the bottom of the pool, and there is a semicircular piece of high pressure area, and due to the slope of the outflow channel, the pressure returns to normal level, and the streamline also began to level off. due to the slope of the outflow channel, the pressure returns to normal level, and the streamline also began to level off. due to the slope of the outflow channel, the pressure returns to normal level, and the streamline also began to level off.

**Figure 11.** Streamline and pressure distribution at the pump gate outlet under different flow condi-

**Figure 11.** Streamline and pressure distribution at the pump gate outlet under different flow condi-

As can be seen from Figure 11, at different flow rates, there is a backflow vortex above the pump gate outlet. With the increase of flow rate, the size of the vortex area also increases, and the center of the backflow vortex also shifted to the outlet of the outflow channel. Under the action of gravity, the fluid flows as a whole to the bottom of the pool to do offset flow, and due to the existence of vortex, the pressure at the outlet is low, and with the increase of vortex, the low pressure area also increases; pump gate outlet is 5~9 *D* at the bottom of the pool, and there is a semicircular piece of high pressure area, and

As can be seen from Figure 11, at different flow rates, there is a backflow vortex above the pump gate outlet. With the increase of flow rate, the size of the vortex area also increases, and the center of the backflow vortex also shifted to the outlet of the outflow channel. Under the action of gravity, the fluid flows as a whole to the bottom of the pool to do offset flow, and due to the existence of vortex, the pressure at the outlet is low, and with the increase of vortex, the low pressure area also increases; pump gate outlet is 5~9 *D* at the bottom of the pool, and there is a semicircular piece of high pressure area, and

#### 3.3.2. Hydraulic Loss of Outlet Channel 3.3.2. Hydraulic Loss of Outlet Channel 3.3.2. Hydraulic Loss of Outlet Channel

tions.

tions.

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*Water* **2022**, *14*, x FOR PEER REVIEW 11 of 22

According to the Formula (3) to calculate the hydraulic loss of the outflow channel, draw the hydraulic loss curve of the outflow channel under different flow conditions, as shown in Figure 12. According to the Formula (3) to calculate the hydraulic loss of the outflow channel, draw the hydraulic loss curve of the outflow channel under different flow conditions, as shown in Figure 12. According to the Formula (3) to calculate the hydraulic loss of the outflow channel, draw the hydraulic loss curve of the outflow channel under different flow conditions, as shown in Figure 12.

As can be seen from Figure 12, the hydraulic loss of the open outlet channel under different flow conditions is approximately parabolic with an open upward distribution, reaching a minimum value of 1.121 m at an inlet flow rate of 9.5 L/s (0.83 *Qd*). With the increase of the inlet flow rate, the hydraulic loss also gradually increases, reaching a maximum value of 1.933 m at an inlet flow rate of 14.5 L/s (1.26 *Qd*). As can be seen from Figure 12, the hydraulic loss of the open outlet channel under different flow conditions is approximately parabolic with an open upward distribution, reaching a minimum value of 1.121 m at an inlet flow rate of 9.5 L/s (0.83 *Qd* ). With the increase of the inlet flow rate, the hydraulic loss also gradually increases, reaching a maximum value of 1.933 m at an inlet flow rate of 14.5 L/s (1.26 *Qd* ). As can be seen from Figure 12, the hydraulic loss of the open outlet channel under different flow conditions is approximately parabolic with an open upward distribution, reaching a minimum value of 1.121 m at an inlet flow rate of 9.5 L/s (0.83 *Qd*). With the increase of the inlet flow rate, the hydraulic loss also gradually increases, reaching a maximum value of 1.933 m at an inlet flow rate of 14.5 L/s (1.26 *Qd*).

#### *3.4. Three-Dimensional Flow Regime Analysis 3.4. Three-Dimensional Flow Regime Analysis 3.4. Three-Dimensional Flow Regime Analysis*

3.4.1. Integrated Pump Gate Three-Dimensional Streamline and Characteristic 3.4.1. Integrated Pump Gate Three-Dimensional Streamline and Characteristic Cross-Sectional Flow Rate 3.4.1. Integrated Pump Gate Three-Dimensional Streamline and Characteristic Cross-Sectional Flow Rate

Cross-Sectional Flow Rate The characteristic section is shown schematically in Figure 13, the specific location is The characteristic section is shown schematically in Figure 13, the specific location is shown in Table 3, and the pump gate impeller diameter *D* = 60 mm. The characteristic section is shown schematically in Figure 13, the specific location is shown in Table 3, and the pump gate impeller diameter *D* = 60 mm.

**Figure 13.** Schematic diagram of the characteristic cross section. **Figure 13.** Schematic diagram of the characteristic cross section.

**Figure 13.** Schematic diagram of the characteristic cross section. Setting characteristic sections A-1, A-2, A-3 in order at 10 *D* intervals from the impeller inlet. Setting characteristic sections A-4, A-5, A-6 in order at 10 *D* intervals from the Setting characteristic sections A-1, A-2, A-3 in order at 10 *D* intervals from the impeller inlet. Setting characteristic sections A-4, A-5, A-6 in order at 10 *D* intervals from the Setting characteristic sections A-1, A-2, A-3 in order at 10 *D* intervals from the impeller inlet. Setting characteristic sections A-4, A-5, A-6 in order at 10 *D* intervals from the guide vane outlet. Setting characteristic sections B-1, B-2, B-3 in order at distances 5 *D*, 3 *D* and *D* from the impeller inlet. Setting characteristic sections B-4, B-5, B-6 in order at distances *D*, 3 *D* and 5 *D* from the guide vane outlet.

The position of 5 *D*, 3 *D*, *D* from the impeller inlet and *D*, 3 *D*, 5 *D* from the guide vane outlet are selected to make cross sections, and the sections are shown in the characteristic sections B-1~B-6 in Figure 13, and the three-dimensional streamlines and axial flow velocity distribution of the integrated pump gate in the characteristic sections are shown in Figures 14–20.

guide vane outlet. Setting characteristic sections B-1, B-2, B-3 in order at distances 5 *D*, 3 *D* and *D* from the impeller inlet. Setting characteristic sections B-4, B-5, B-6 in order at

> **Distance from Impeller Inlet L1**

A-4 10 *D*

**Distance from Guide Vane Outlet L2**

**Table 3.** Summary of characteristic selections. A-5 20 *D*

**Table 3.** Summary of characteristic selections.

**Characteristic Section Number**

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distances *D*, 3 *D* and 5 *D* from the guide vane outlet.

A-1 30 *D* A-2 20 *D* A-3 10 *D*

**Figure 15.** *Q* = 9.5 L/s (0.83 *Qd*). **Figure 15.** *Q* = 9.5 L/s (0.83 *Qd* ).

**Figure 16.** *Q* = 10.5 L/s (0.91 *Qd*).

**Figure 17.** *Q* = 11.5 L/s (*Qd*).

(**a**) Before the pump gate (**b**) After the pump gate

(**a**) Before the pump gate (**b**) After the pump gate

(**a**) Before the pump gate (**b**) After the pump gate

(**a**) Before the pump gate (**b**) After the pump gate

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**Figure 15.** *Q* = 9.5 L/s (0.83 *Qd*).

**Figure 15.** *Q* = 9.5 L/s (0.83 *Qd*).

**Figure 18.** *Q* = 12.5 L/s (1.09 *Qd*). **Figure 18.** *Q* = 12.5 L/s (1.09 *Qd* ).

**Figure 19.** *Q* = 13.5 L/s (1.17 *Qd*).

**Figure 20.** *Q* = 14.5 L/s (1.26 *Qd*).

111

 (**a**) Before the pump gate (**b**) After the pump gate

(**a**) Before the pump gate (**b**) After the pump gate

By comparing Figures 14 and 20, it can be seen that before the impeller inlet of the pump gate, with the increase of the inlet flow rate, there is no adverse flow pattern in the inlet channel, and the streamlines are uniformly contracted toward the impeller inlet. Comparing the axial velocity distribution clouds in the sections at different flow rates, it

(**a**) Before the pump gate (**b**) After the pump gate

(**a**) Before the pump gate (**b**) After the pump gate

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**Figure 18.** *Q* = 12.5 L/s (1.09 *Qd*).

**Figure 18.** *Q* = 12.5 L/s (1.09 *Qd*).

**Figure 20.** *Q* = 14.5 L/s (1.26 *Qd*). **Figure 20.** *Q* = 14.5 L/s (1.26 *Qd* ).

**Figure 20.** *Q* = 14.5 L/s (1.26 *Qd*). By comparing Figures 14 and 20, it can be seen that before the impeller inlet of the pump gate, with the increase of the inlet flow rate, there is no adverse flow pattern in the inlet channel, and the streamlines are uniformly contracted toward the impeller inlet. Comparing the axial velocity distribution clouds in the sections at different flow rates, it By comparing Figures 14 and 20, it can be seen that before the impeller inlet of the pump gate, with the increase of the inlet flow rate, there is no adverse flow pattern in the inlet channel, and the streamlines are uniformly contracted toward the impeller inlet. Comparing the axial velocity distribution clouds in the sections at different flow rates, it By comparing Figures 14 and 20, it can be seen that before the impeller inlet of the pump gate, with the increase of the inlet flow rate, there is no adverse flow pattern in the inlet channel, and the streamlines are uniformly contracted toward the impeller inlet. Comparing the axial velocity distribution clouds in the sections at different flow rates, it can be seen that the axial velocity in the same sections gradually increases with the increase of the inlet flow rate. At 5 *D* from the impeller inlet, the axial velocity distribution in the section is more uniform, and the velocity at the side wall is smaller due to the boundary layer effect. At 3 *D* from the impeller inlet, the velocity in the section gradually increases from top to bottom, which is mainly due to the slope that causes the water to collect at the bottom of the pool. At the impeller inlet *D*, the axial flow velocity is distributed outward in a circular pattern, and the high velocity area is located in the lower and middle part of the slice at the level of the impeller, while the flow velocity in the upper and middle part of the section is lower.

From the distribution of streamlines in the outlet section of Figures 14 and 20, it can be concluded that between 3 *D* and 5 *D* from the guide vane outlet, there is separation and stratification of the fluid caused by vortices, which is due to the opposite axial flow velocity of the streamlines. The region of high flow velocity within each section is located in the middle and lower part, and the axial flow velocity of the water in the middle and upper part is opposite to the flow velocity at the bottom of the flow channel due to the presence of vortices. At *D* and 3 *D* from the guide vane outlet, the axial flow velocity in the sections is distributed in a circular band, decreasing from the center line of the guide vane to the top and bottom of the sections. At 5 *D* from the guide vane outlet, when the flow