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A gravitation vortex type water turbine, which mainly comprises a runner and a tank, generates electricity by introducing a flow of water into the tank and using the gravitation vortex generated when the water drains from the bottom of the tank. This water turbine is capable of generating electricity using a low head and a low flow rate with relatively simple structure. However, because its flow field has a free surface, this water turbine is extremely complicated, and thus its relevance to performance for the generation of electricity has not been clarified. This study aims to clarify the performance and flow field of a gravitation vortex type water turbine. We conducted experiments and numerical analysis, taking the free surface into consideration. As a result, the experimental and computational values of the torque, turbine output, turbine efficiency, and effective head agreed with one another. The performance of this water turbine can be predicted by this analysis. It has been shown that when the rotational speed increases at the runner inlet, the forward flow area expands. However, when the air area decreases, the backward flow area also expands.

Many large-scale conventional hydraulic power generations mainly use medium- or high-heads and water turbines [

Therefore, we focused on a water turbine used in the Gravitation Water Vortex Power Plant (GWVPP) [

In light of this background, this study aims to clarify the performance of a gravitation vortex type water turbine and elucidate its flow field. We performed numerical analysis by considering the free surface, conducted a performance test and a visualization experiment, and verified the validity of our analysis. Furthermore, we examined the flow field around the runner at the center of the blade width in detail using a numerical analysis.

An overview of the gravitation vortex type water turbine is shown in Figure _{1} = 140 mm, blade outlet diameter (inner diameter) is_{2} = 90 mm, blade inlet width is_{1} = 91 mm, blade outlet width is_{2} = 91 mm, and number of blades is

Specifications of runner.

Outer diameter: | 0.14 m |

Inner diameter: | 0.09 m |

Inlet width: | 0.091 m |

Outlet width: | 0.091 m |

Inlet angle: | 71.9° |

Outlet angle: | 19.0° |

Tip clearance: | 0.5 mm |

Number of blades: | 20 |

Test water turbine.

Runner.

An overview of the experimental apparatus is shown in Figure ^{3}/s. The load to the runner was controlled by a motor and an inverter, and the rotational speed was arbitrarily set. The rotational speed

Experimental apparatus.

Here the torque

Definition of performance evaluation.

Here the upstream water depth_{3} was measured at the tank inlet in the vicinity of the wall surface on the +_{4} was measured by a point gauge (Kenek Corporation; PH-102) at five points from the vicinity of the wall surface on the +_{1} downstream from the downstream atmospheric opening from which the average water depth was obtained. The upstream velocity_{3} and downstream velocity_{4} were calculated by

Here_{3} and_{4} are the waterway widths of the tank inlet and downstream, respectively. In addition, the turbine efficiency

A digital camera (Casio Computer Co., Ltd.; EXILIM EX-F1) was used to visualize the flow field at a frame rate of 30 frames per second (fps).

In this study, three-dimensional unsteady flow analysis was performed by considering the free surface. The general-purpose thermal fluid analysis software, ANSYS CFX15.0 (ANSYS, Inc.), was used for the calculations. Moreover, the volume of fluid (VOF) method [

The entire area of calculation is shown in Figure _{1} in length from the atmospheric opening to the outlet boundary. The reference position of the upstream side was set to the tank inlet, which was the same as that in the experiment. The reference position of the downstream side was set to 6_{1} downstream from the atmospheric opening of the downstream waterway. At these reference positions, the distribution of each water depth was obtained in the width direction, assuming that the water surface is equivalent to that of a water volume fraction of 0.5. The upstream water depth_{3} and downstream water depth_{4} are the averages obtained from the distribution of water depth in the width direction. As an example, the grids used in the runner and tank calculations are illustrated in Figures

Calculating area.

Computational grids.

Runner

Tank

A comparison between the experimental and calculation results for this water turbine in relation to its performance is shown in Figures _{3} and that the resistance of the runner increases with the increase of rotational speed^{−1}. The maximum experimentally determined efficiency is approximately 0.354 at a specific speed ^{−1}, kW, m].

Turbine performance.

Torque and turbine output

Turbine efficiency and effective head

The experimental results for the free surface flow field of this water turbine are shown in Figures

Flow field by experiment.

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Flow field by calculation.

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First, in order to identify the water/air interface, the circumferential distribution of water volume fraction VF_{1} obtained numerically for a runner inlet at the center of the blade width is shown in Figure _{1} is the time average value during one rotation of the runner. Air is represented by 0 ≤ VF < 0.5, water is represented by 0.5 < VF ≤ 1, and the interface between them is represented by VF = 0.5. These notations are the same as those of a runner outlet, which will be described subsequently. According to Figure ^{−1}, the water area of VF > 0.5 is at

Volume fraction of water at runner inlet (Cal.).

Here velocity triangles of the water turbine are illustrated in Figure _{r} is defined in the radially inward direction,_{u} in the rotation direction, and_{a} in the +_{r1}, axial component_{a1}, and circumferential component_{u1} of the absolute velocity of a runner inlet at the center of the blade width in the numerical analysis. Here each component is the time average value during one rotation of the runner and displays a water area identified only from the water volume fraction. These notations are the same as those of a runner outlet, which will be described subsequently. The theoretical_{u1} value obtained from the following expression that is based on the assumption that the tank has a free vortex type flow is also shown in Figure _{1} is the outer radius of the runner inlet and_{3} is the representative radius of the tank inlet. The radius (70.5 m) at the measuring point on the periphery of the runner inlet and the radius (245 mm) in the tank were used as_{1} and_{3}, respectively. The value obtained from (_{3}.

Velocity triangles.

Absolute velocity at runner inlet (Cal.).

Radial component

Axial component

Circumferential component

From Figure _{r1} is not uniform in the circumferential direction at any rotational speed. For ^{−1}, the forward flow area in which_{r1} is positive appears at _{a1} is negatively large, as is shown in Figure _{r1} changes from positive to negative shifts to the large _{u1} values are relatively similar at _{u1} value decreases as the rotational speed increases, the computational_{u1} value increases as the rotational speed increases, and it becomes large at both ends of the air area at any rotational speed. Because the water area for low values of _{1}, at the runner inlet,_{1} = 0.594 m/s at^{−1},_{1} = 0.894 m/s at^{−1}, and_{1} = 1.188 m/s at^{−1}. At^{−1},_{u1} at both ends of the air area is nearly the same as_{1}. Therefore, a flow in the tank of this water turbine is not a perfect free vortex, and it is greatly influenced by the rotation of the runner near the runner inlet. Because this water turbine does not have guide vane upstream of the runner, a uniform and strong circumferential spiral flow can be produced by designing the tank shape that improves the turbine output.

The numerically determined circumferential distribution of the relative flow angle _{1} of a runner inlet at the center of the blade width is shown in Figure ^{−1}, the relative flow angle _{1} for _{b1} = 71.9°. However, at^{−1}, the relative flow angle _{1} dissociates greatly from the blade inlet angle, and the shock loss appears to increase. Therefore, it is necessary to control the flow in the tank and homogenize the relative flow angle _{1} in the circumferential direction in order to decrease the shock loss at the blade inlet.

Relative flow angle at runner inlet (Cal.).

The numerically determined circumferential distribution of water volume fraction VF_{2} of a runner outlet at the center of the blade width is shown in Figure ^{−1}, the largest water area is for

Volume fraction of water at runner outlet (Cal.).

Figures _{r2}, axial component_{a2} and circumferential component_{u2} of the absolute velocity of a runner outlet at the center of the blade width. According to Figure _{r2} distribution at each rotational speed is almost the same and does not show a backward flow at any rotational speed. From Figure _{a2} component of the water area at the large ^{−1},_{u2} is the largest around ^{−1} and a positive rotation remains at^{−1}.

Absolute velocity at runner outlet (Cal.).

Radial component

Axial component

Circumferential component

The flow rate and the angular momentum per unit time that flow in and out at the runner inlet and outlet relates to the torque of a water turbine studied.

The flow rates

_{r1} and_{r2} during one rotation of the runner at each measuring point.

Here

Therefore, the angular momentums,_{1} and_{2}, per unit blade width (1 mm) and unit time at the runner inlet and outlet can be expressed as the following equations:

_{r1}_{u1} and_{r2}_{u2} during one rotation of the runner at each measuring point.

The relationship between the rotational speed,_{1} and_{2}, per unit blade width are shown in Figure _{1} and_{2}, per unit blade width and unit time are shown in Figure

Flow rate per unit blade width.

Angular momentum per unit blade width and unit time.

In Figure _{1} is rather small at^{−1}, it is nearly constant when the rotational speed changes. Therefore, when the rotational speed increases, the forward flow area expands. However, as previously stated, the flow rate that flows in from the center of blade width barely changes. Conversely,_{2} is relatively similar to_{1} at^{−1}. When the rotational speed increases,_{2} at^{−1} is approximately 20.6% lower than_{1}. Since_{a2} is a negative value, as stated above, the flow through the runner comes close to the tip side (bottom of the tank). Therefore, in order to design a high-performance runner, it is necessary to study the three-dimensional flow, including the direction of the blade width.

In Figure _{1} is rather large at^{−1}, it is nearly constant when the rotational speed changes. However,_{2} has a large negative value at^{−1} and a large positive value at^{−1}. When only the center of the blade width is considered, because the difference between_{1} and_{2} is the theoretical torque of the water turbine, change in torque when the rotational speed changes shows that difference of the angular momentum that remains at the runner outlet has a large influence. Since both the positive and negative angular momentums at the runner outlet cause an increase in the loss of waste, they are considered to be one of the factors related to the decrease in efficiency at low or high rotational speeds.

Figures ^{−1}, the relative water flow is relatively smooth along the blade, but, at^{−1}, it flows in at small angles.

Relative velocity vectors and volume fraction of water (Cal.).

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The following matters were determined by our research of the performance of a gravitation vortex type water turbine and the flow field at the center of blade width through experiments and free surface flow analysis:

The experimental and computational values of the torque, turbine output, turbine efficiency, and effective head agree well with one another. Thus, the performance of this water turbine can be predicted by this analysis.

With increase in the rotational speed at a runner inlet, the forward flow area increases, as does the backward flow area because of the reduction in the air area. However, the flow rate that flows in from the center of the blade width barely changes.

The flow in the tank of this water turbine is not a perfect free vortex, and it is greatly influenced by the rotation of the runner near the runner inlet.

The water area of a runner outlet is considerably smaller than that of a runner inlet and does not change with the rotational speed. In addition, backward flow does not occur at a runner outlet.

When the rotational speed changes, the angular momentum per unit time that flows from the runner inlet is nearly constant. The angular momentum per unit time that flows from the runner outlet shows a large negative value at low-speed rotations and a large positive value at high-speed rotations. It also has a large influence on the torque when the rotational speed changes.

Blade width m

Waterway width m

Runner diameter m

Gravitational acceleration m/s^{2}

Water depth m

The difference in height between the bottom surface of tank and the bottom surface of downstream waterway m

Effective head m

Angular momentum per unit blade width and unit time N·m/mm·s

Rotational speed min^{−1}

Specific speed min^{−1}, kW, m

Turbine output W

Flow rate per unit blade width m^{3}/mm·s

Flow rate m^{3}/s

Torque N·m

Circumferential velocity m/s

Absolute velocity m/s

Volume fraction of water

Relative velocity m/s

Relative flow angle °

Blade angle °

Turbine efficiency

Circumferential angle °

Density of water kg/m^{3}

Water area

Runner inlet

Runner outlet

Upstream

Downstream

Axial component

Hub

Radial component

Tip

Circumferential component.

The authors declare that they have no conflicts of interest regarding the publication of this paper.

The authors acknowledge the support of Shinoda Co., Ltd. in the design and production of the experimental apparatus. They are also grateful to Tomoaki Tanemura and Kentaro Hatano, graduate students of Ibaraki University at the time, who supported us with the experiments and numerical analysis. Here, we express our sincere gratitude for their cooperation.