Research Article Study on an Undershot Cross-Flow Water...

Research ArticleStudy on an Undershot Cross-Flow Water Turbine withStraight Blades

Yasuyuki Nishi,1 Terumi Inagaki,1 Yanrong Li,1 and Kentaro Hatano2

1Department of Mechanical Engineering, Ibaraki University, 4-12-1 Nakanarusawa-cho, Hitachi-shi, Ibaraki 316-8511, Japan2Graduate School of Science and Engineering, Ibaraki University, 4-12-1 Nakanarusawa-cho, Hitachi-shi, Ibaraki 316-8511, Japan

Correspondence should be addressed to Yasuyuki Nishi; [email protected]

Received 31 March 2015; Revised 13 June 2015; Accepted 15 June 2015

Academic Editor: Ryoichi Samuel Amano

Copyright © 2015 Yasuyuki Nishi et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Small-scale hydroelectric power generation has recently attracted considerable attention. The authors previously proposed anundershot cross-flow water turbine with a very low head suitable for application to open channels. The water turbine was of across-flow type and could be used in open channels with the undershot method, remarkably simplifying its design by eliminatingguide vanes and the casing.Thewater turbinewas fittedwith curved blades (such as the runners of a typical cross-flowwater turbine)installed in tube channels. However, there was ambiguity as to how the blades’ shape influenced the turbine’s performance and flowfield. To resolve this issue, the present study applies straight blades to an undershot cross-flow water turbine and examines theperformance and flow field via experiments and numerical analyses. Results reveal that the output power and the turbine efficiencyof the Straight Blades runner were greater than those of the Curved Blades runner regardless of the rotational speed. Comparedwith the Curved Blades runner, the output power and the turbine efficiency of the Straight Blades runner were improved by about31.7% and about 67.1%, respectively.

1. Introduction

As environmental and energy problems become critical,small-scale hydroelectric power generation is attracting sig-nificant attention as ameans of utilizing natural energy.Waterturbines used for small-size hydroelectric power generationare largely categorized into the tube channel-type [1–3](including the Francis and propeller water turbines) and openchannel-type [4–6]. Tube channel water turbines requirerelatively large heads, resulting in a decrease in the numberof suitable construction sites, whereas open channel waterturbines require almost no attached facility, enabling theirdirect installation in open channels, thus being environmen-tally friendly and easier tomaintain. Accordingly, open chan-nel water turbines have been applied in small-to-mediumrivers and irrigation canals at numerous sites. However,as these water turbines have been primarily designed formotive power, their design style is not yet well established;for example, they are of large size and have low turbineefficiency.

Considering these circumstances, the authors aimed todevelop a water turbine with a very low head that was suitablefor application to open channels; thus, an undershot cross-flowwater turbine [7, 8] was proposed. Comparedwith cross-flowwater turbines [9, 10] withmedium-to-low heads appliedto tube channels, the proposed water turbine offered a greatlysimplified design thanks to its lack of guide vanes and the cas-ing, and it was applied to open channels with the undershotmethod. It was revealed [7, 8] that the proposedwater turbine,unlike conventional undershot water turbines, did not whirlthe water with its blades and thus did not accumulate airbubbles between them. It also received the flow twice owing tothe through-flow effect, thereby greatly improving the turbineefficiency. Furthermore, the water turbine’s flow field witha free surface was elucidated by particle image velocimetrymeasurement and numerical analyses [11, 12]. However, sincethe water turbine used curved blades such as the runners ofa typical cross-flow water turbine used for tube channels, theeffect of the blade’s shape on the performance and flow fieldwas unclear.

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2015, Article ID 817926, 10 pageshttp://dx.doi.org/10.1155/2015/817926

2 International Journal of Rotating Machinery

X

Y

Monitor points

𝛽b2

𝛽b1

𝜃

D1

D2

(a) Straight Blades (section A-A)

bA

A

X

Y

Monitor points

𝛽b2

𝛽b1

𝜃

D1

D2

𝛿

𝛾 𝛾

(b) Curved Blades (section A-A)

Figure 1: Test runners.

Table 1: Specifications of test runners.

Straight Blades Curved BladesOuter diameter:𝐷

10.18m

Inner diameter:𝐷2

0.12mInlet angle: 𝛽

𝑏190∘ 30∘

Outlet angle: 𝛽𝑏2

90∘

Runner width: 𝑏 0.24mSide clearance: 𝛾 5mmBottom clearance: 𝛿 5mmNumber of blades: 𝑍 24

This study applies straight blades to an undershot cross-flow water turbine and examines the performance and flowfield through experiments and numerical analyses.

2. Experimental Apparatus and Method

Two types of test runners used in this study are illustratedin Figures 1(a) and 1(b), and their specifications are presentedin Table 1. Figure 1(a) shows another prototype runner whoseoriginal curved blades are replaced by radially arrangedstraight blades, hereafter referred to as the Straight Bladesrunner. Figure 1(b) shows a prototype runner based on across-flow runner [10] used for tube channels; hereafter, thisprototype is referred to as the Curved Blades runner.The onlydifference between the two types of runners is the shape of theblades. Both runners have an outer diameter 𝐷

1of 180mm,

runner width 𝑏 of 240mm, side clearance 𝛾 of 5mm, and bot-tom clearance 𝛿 of 5mm. Other specifications for this runnerwere determined in reference to a typical cross-flow runner[10] used in conjunctionwith a penstock.The circumferentialangle 𝜃 defined the negative direction of the 𝑥-axis as 𝜃 = 0∘and the counterclockwise rotation as positive. Two velocity-monitoring points were established: one was set at a point0.5mm radially outward for the outer periphery of the run-ners, while the other was set at a point 0.5mm radially inwardfor the inner periphery of the runners.

A summarized illustration of the experimental apparatusis shown in Figure 2. The equipment used in this experiment

consisted of an open-air circulation water tank that was usedto simulate an open channel. The experiment was conductedwith a constant flow rate (𝑄 = 0.003m3/s), which wasmeasured with an electromagnetic flowmeter. The load onthe water turbine was altered using a load machine, andthe rotational speed 𝑛 and torque 𝑇 were measured using amagnetoelectric-type rotation detector and a torque meter,respectively, from which we obtained the output power 𝑃.Water depth was measured at 2 points, upstream and down-stream from the runner, providing for an upstream waterdepth ℎ

3and a downstream water depth ℎ

4. These depths

were derived throughmeasurements obtained at a distance of2𝐷1from the center of the runner in both the upstream and

downstream directions. Measurements at each location wereobtained from a point on the wall surface and from the centerpoint of the channel having a width 𝐵 = 0.25m. In addition,both the upstream and downstream flow velocities, V

3and

V4, respectively, were obtained using the measured upstream

water depth ℎ3, the measured downstream water depth ℎ

4,

and the flow rate 𝑄 via the following equations:

V3 =𝑄

(𝐵ℎ3),

V4 =𝑄

(𝐵ℎ4).

(1)

We used a 100mW semiconductor-excited laser sheet asa light source to visualize the flow around the runner byilluminating the central cross-section of the runner widthfrom the bottom of the runner. The thickness of the lasersheet was approximately 1mm. In this experiment, we useda digital camera of 1920 × 1080 pixels as the photographicdevice to take pictures at a shooting speed of 60 fps from adirection that is perpendicular to the central cross-section ofthe illuminated runner width.

3. Numerical Analysis Method and Conditions

This study uses the general thermal fluid analysis codeANSYS-CFX13.0 and analyzes a three-dimensional unsteadyflow considering free surfaces. The multiphase flow model is

International Journal of Rotating Machinery 3

Measuring locationTorque meter

Measuring location Test runner

Load machine

360

Pump

360

250Flow

Tachometer

Flow

X

X

Y

Flow meter

Z

Laser sheet generator

Digital camera

h3 h4

Figure 2: Experimental apparatus.

Inlet

Outlet

Runner

Flow

Opening

360

410

1260

360360

1440

250

Figure 3: Computational domain.

the uniformmodel [13] of an Euler-Euler simulation, with theacting fluids being water and air. The basic equations usedin the model are based upon mass, momentum, and volumeconservation [13]. The standard 𝑘-𝜀 model was adopted asthe turbulent flowmodel, and the standard wall function wasused to handle regions near wall surfaces.

The entire computational domain is shown in Figure 3,and the computational grid of each runner is shown in Fig-ures 4(a) and 4(b). The computational domain is composedof the runner and the upstream and downstream domains.The length of the upstream and downstream domains to thecenter of the runner is 9𝐷

1and 10𝐷

1, respectively. Control

surfaces are established at a distance of 2𝐷1in the upstream

and downstream directions from the runner center. Thewater depth is monitored at these control surfaces as theywould be in an experiment. The computational grid of either

runner has about 462 thousand elements and that of boththe upstream and downstream has about 420 thousandelements, so the total elements number of the computationalgrid for either runner is about 882 thousand. To investigategrid dependence, the number of computational grids wasincreased by approximately 2 times, and analysis was con-ducted at 𝑛 = 22min−1. Although the torque decreased byapproximately 3.7% from the result presented in this paper,the effect of computational grid number was comparativelysmall. For boundary conditions, the mass flow rate wasapplied to the inlet boundary, free outflow (with a relative airpressure of 0 Pa) was applied to the outlet boundary, and rota-tional speed was applied to the runner domain. In addition,the top surface of the computational domain was permeableto the atmosphere in order to allow air to move freely in andout of the computational domain, while nonslip conditionswere applied to all the other walls. The boundary of therotational and static domains was joined by the transientrotor-stator method [14]. For initial conditions, the experi-mental value of the upstream flow velocity V

3was used as the

flow velocity in the simulations. The volume fraction VF𝑖ℎof

water was defined in accordance with the following formulausing the step function:

VF𝑖ℎ= step (ℎ − 𝑦) . (2)

Here, 𝑦 is the coordinate of the height direction inthe computational domain, and the water depth ℎ uses theexperimental value of the upstream water depth ℎ

3in the

simulations. Therefore, the position 𝑦 ≤ ℎ3is the domain of

water and the position𝑦 > ℎ3is the domain of air. In addition,

the time step was adjusted to ensure that the runner would


(a) Straight Blades (b) Curved Blades

Figure 4: Computational grid of the runners.

undergo one rotation every 180 steps, and the time step wasrecalibrated until fluctuations in the flow became negligible.

4. Experimental and AnalysisResults and Considerations

4.1. Comparison of the Water Turbine Performances. Figures5(a)–5(c) present a comparison of the performances of theStraight Blades runner andCurvedBlades runner on the basisof the experimental and computational values. For both run-ners, the computed torque 𝑇 reduces as the rotational speedincreases. The computed torque 𝑇 is greater than its experi-mental value throughout the rotational speed range. Accord-ingly, the computed output power𝑃 is greater than the exper-imental value. However, both computed 𝑇 and 𝑃 are greaterin the case of the Straight Blades runner than in the case ofthe Curved Blades runner throughout the rotational speedrange, and this trend is also seen in the experimental values.The torque with no runner installed is simply measured byrotating the shaft by hand, and the resulting value coincideswith the difference between the experimental and computedtorque values. Thus, although the difference between theexperiment and computation could be attributed to themod-els of multiphase flow or turbulence, the difference occursmainly becausemechanical friction loss is not taken into con-sideration in the computation. While the computed turbineefficiency 𝜂 is greater than its experimental value except inthe low rotational speed range, its trend matches the experi-ment. Comparing the experimental values of both runners, itis revealed that the maximum output power and maximumturbine efficiency of the Straight Blades runner are greaterthan those of the Curved Blades runner by about 31.7% andabout 67.1%, respectively.

A comparison of the time-averaged values of theupstream water depth ℎ

3and downstream water depth ℎ

4of

the Straight Blades runner and Curved Blades runner fromthe experiment and computation is shown in Figures 6(a)and 6(b), while a comparison of the time-averaged values ofthe upstream Froude number 𝐹

𝑟3and downstream Froude

number 𝐹𝑟4

is shown in Figures 7(a) and 7(b). Here, theupstream Froude number 𝐹

𝑟3and the downstream Froude

number 𝐹𝑟4

are obtained using the following equations,respectively:

𝐹𝑟3 =

V3√𝑔ℎ3

,

𝐹𝑟4 =

V4√𝑔ℎ4

.

(3)

For both runners, as seen in Figures 6(a) and 6(b), theupstreamwater depths ℎ

3from the experiment and computa-

tion decrease as the rotational speeds increase; contrarily, thedownstream water depths ℎ

4increase. Furthermore, for both

the experiment and computation, the upstream water depthsℎ3of the Straight Blades runner are smaller than those of the

Curved Blades runner, whereas the downstreamwater depthsℎ4are greater for both runners. This is because the resistance

of the runners against the flow decreases as the rotationalspeed increases, implying that the Straight Blades runner hasa smaller resistance than the Curved Blades runner. However,the downstream water depth ℎ

4from the experiment rapidly

increases at around 𝑛 = 16–19min−1 for the straight bladesrunner and 𝑛 = 22–25min−1 for the Curved Blades runner,before maintaining a nearly constant value even when therotational speed further increases.The constant value is about0.03m for both runners. As seen in Figure 7(b), the Froudenumber 𝐹

𝑟4 from the experiment becomes 𝐹𝑟4 < 1 at around

the point where the downstream water depth ℎ4begins to

rapidly increase, indicating that the flow has shifted fromsupercritical to subcritical. Whereas the rotational speeds atwhich the transition from the supercritical flow to subcriticalflow occur differ between the experiment and computation,the water depths and Froude numbers show rather similarvalues for both runners. Note that the upstream Froudenumber 𝐹

𝑟3 from the experiment and computation is always𝐹𝑟3 < 1 for both runners, meaning that the flow upstream of

the runners is subcritical regardless of the runners’ rotationalspeed.


0 5 10 15 20 25 30 35 400.00

0.05

0.10

0.15

0.20

0.25T

(N·m

)

n (min−1)

(a) Torque

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 5 10 15 20 25 30 35 40n (min−1)

P(W

)

(b) Output power

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30 35 40n (min−1)

𝜂

Straight (exp.)Straight (cal.)

Curved (exp.)Curved (cal.)

(c) Turbine efficiency

Figure 5: Water turbine performance.

From the above findings, this study has demonstrated toa certain extent the qualitative characteristics of the perfor-mances and flow fields of water turbines fitted with bladesof different shapes.

4.2. Comparison of Flow Fields. To compare the flow fieldsaround the Straight Blades runner andCurved Blades runner,the flow field in the middle of the runner width is examinedby numerical analyses (described below) at a rotational speedof 𝑛 = 22min−1 at which the computed output power of bothrunners roughly becomes maximal.

First, the water-air interfaces are identified with the time-averaged values of the volume fractions of water VF

1and

VF2, respectively, at the outer and inner circumferences of the

Straight and Curved Blades runners, as shown in Figure 8.Here, VF = 1 is water, VF = 0 is air, and VF = 0.5 is aninterface of water and air. The water domains of the StraightBlades runner andCurved Blades runner are 𝜃= 27∘–126∘ and𝜃 = 24∘–132∘ and 𝜃 = 42∘–114∘ and 𝜃 = 57∘–138∘ at the runners’

outer and inner circumferences, respectively, demonstratingthat the water domains at both the outer and inner circum-ferences are narrower on the Straight Blades runner than onthe Curved Blades runner.

The velocity triangles [11] of the undershot cross-flowwater turbine are shown in Figure 9. Here, the radial com-ponent V

𝑟of the absolute velocity is set positive in the radial

inner direction, whereas the circumferential component V𝑢

is set positive in the rotation direction (counterclockwiserotation).Moreover, the absolute flow angle 𝛼 and the relativeflow angle𝛽 define the rotational direction (counterclockwiserotation) of the runner as positive.

The time-averaged values of the radial component V𝑟1

and circumferential component V𝑢1

of the absolute velocitiesat the outer circumferences of the Straight and CurvedBlades runners are shown in Figures 10(a) and 10(b), whereasthe time-averaged values of the radial component V

𝑟2and

circumferential component V𝑢2

of the absolute velocities atthe inner circumferences of both runners are shown in


0.00

0.02

0.04

0.06

0.08

0.10

0 5 10 15 20 25 30 35 40n (min−1)



h3

(m)

(a) Upstream

0.00

0.01

0.02

0.03

0.04

0.05

0 5 10 15 20 25 30 35 40n (min−1)



h4

(m)

(b) Downstream

Figure 6: Correlation between the rotational speed and both the upstream (ℎ3) and downstream (ℎ

4) water depths from experiment and

numerical analysis.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30 35 40n (min−1)



Fr3

(a) Upstream

0.0

0.5

1.0

1.5

2.0

2.5

0 5 10 15 20 25 30 35 40n (min−1)



Fr4

(b) Downstream

Figure 7: Correlation between the rotational speed and both the upstream (𝐹𝑟3) and downstream (𝐹

𝑟4) Froude numbers from experiment

and numerical analysis.

Figures 11(a) and 11(b), respectively. Here, V𝑟and V

𝑢of

both runners only denote the values in the water domainsidentified on the basis of the volume fraction of water, asshown in Figure 8. Figure 10(a) shows that the V

𝑟1values of

the two runners greatly differ at 𝜃 ≦ 42∘ and that whereas V𝑟1

of the Curved Blades runner decreases as 𝜃 decreases, V𝑟1of

the Straight Blades runner increases. This implies that, in theregions with small 𝜃, the Straight Blades runner has smallershock loss and smaller resistance to the flow compared withthe Curved Blades runner. At 𝜃 > 42∘, V

𝑟1of both runners

decreases as 𝜃 increases and turns negative at around 𝜃 = 90∘.

At the outer circumference of the runner, V𝑟1

is positive inthe first stage inlet region, and it is negative in the secondstage outlet region.These regions are narrower on the StraightBlades runner than on the Curved Blades runner. As seen inFigure 10(b), V

𝑢1of both runners increases as 𝜃 increases in

the first stage inlet region, reaching amaximum at around 𝜃 =87∘–90∘. However, V

𝑢1of both runners greatly differs in the

second stage outlet region; that is, V𝑢1

of the Curved Bladesrunner decreases as 𝜃 increases, turning from positive tonegative at around 𝜃 = 123∘.Thus, it is conceivable that, in thisregion, the flow in the Curved Blades runner in the direction


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 15 30 45 60 75 90 105 120 135 150 165 180

Free surface

VF 1,V

F 2

𝜃 (∘)

VF1 (Straight)VF2 (Straight) VF2 (Curved)

VF1 (Curved)

Figure 8: Volume fraction of water at the runner’s outer (VF1) and

inner (VF2) circumferences (𝑛 = 22min−1, cal.).

opposite to the rotation of the runner hinders the reversalflow and leakage flow at the bottom and turns sharply to beagainst the main flow. Conversely, V

𝑢1of the Straight Blades

runner switches from decreasing to increasing at around 𝜃 =117∘, showing that the flow in the Straight Blades runner isin the same direction as the rotation of the runner and thatalthough there remain swirling flows, the hindrance to thereversal flow and the sharp turn against the main flow aresuppressed.

Meanwhile, V𝑢2

decreases as 𝜃 increases in both Straightblades runner and Curved Blades runner, as shown inFigure 11(a), and turns negative at 𝜃 = 90∘ and 105∘, respec-tively. At the inner circumference of the runner, V

𝑟2is positive

in the first stage outlet region and is negative in the secondstage inlet region. These regions are shifted to lower 𝜃 valuesfor the Straight Blades runner compared with the CurvedBlades runner. Whereas the second stage inlet region isnarrower in the case of the Straight Blades runner relative tothe Curved Blades runner, the first stage outlet region appearsto have almost the same areas for both runners. Figure 11(b)shows that V

𝑢2of the Straight Blades runner is generally

greater than that of the Curved Blades runner, especially inthe second stage inlet region. Figure 10(b) shows that V

𝑢1of

the Straight Blades runner is slightly smaller than that of theCurved Blades runner in the first stage inlet region.Thus, it isconceivable that, in the first stage region, the Straight Bladesrunner has less swirling flow to enter the second stage outletthan the Curved Blades runner and is unable to sufficientlyrecover the swirling flow. Therefore, there is apparently agreater amount of swirling flow from the first stage outletregion in the case of the Straight Blades runner than in thecase of the Curved Blades runner; part of the flow thenlocally enters the second stage inlet region. However, asmentioned earlier, since the Straight Blades runner has part ofthe swirling flow remaining in the second stage outlet region,

the recovery of this swirling flow is an issue for the improve-ment of performance.

The relative flow angles 𝛽1and 𝛽

2at the outer and inner

circumferences, which are obtained from the time-averagedvalues of V

𝑟and V

𝑢for the Straight and Curved Blades

runners, are shown in Figures 12(a) and 12(b), respectively.The 𝛽

1and 𝛽

2values of both runners decrease as 𝜃 increases.

Observing the Curved Blades runner, it is inferred that, in thefirst stage inlet region, 𝛽

1coincides with a blade angle of 30∘

at around 𝜃 = 70∘, whereas it largely diverges from the bladeangle in the low-𝜃 region, causing great shock loss. On theother hand, observing the Straight Blades runner, it is inferredthat 𝛽

1coincides with a blade angle of 90∘ at around 𝜃 = 49∘,

while the flow runs almost along the blades in the region of𝜃 = 27∘–57∘. Owing to the above results, it is believed that theStraight Blades runner has a small shock loss in the low-𝜃region at the first stage inlet region, with V

𝑟1increasing, as

mentioned earlier. However, in the second stage inlet region,it is thought that shock loss is large because𝛽

2greatly diverges

from the blade angle 90∘.For both Straight and Curved Blades runners, the results

of experimental visualization are presented in Figures 13(a)and 13(b), while the absolute velocity vectors from numericalanalyses are shown in Figures 14(a) and 14(b). Comparingthe experimental and computational results, it is confirmedthat the flow fields including the water depth qualitativelycoincide between the two results for both runners. From boththe experimental and computational results, it is reconfirmedthat the upstream water depth is smaller in the case of theStraight Blades runner than in the case of the Curved Bladesrunner. The computed results demonstrate that the CurvedBlades runner is different from the Straight Blades runner inthat the Curved Blades runner has air intruding between theblades and flow turbulences occurring in the low-𝜃 region atthe first stage inlet and that it whirls upwater, accompanied bycounter flows occurring in the low-𝜃 region at the first stageinlet. Furthermore, it is confirmed that the flow sharply turnsagainst the main flow in the second stage outlet region.Theseconditions seem to be the causes of the loss in the CurvedBlades runner.

5. Conclusions

Applying straight blades to an undershot cross-flow waterturbine and examining its performance and flow field com-pared with those of the Curved Blades runner throughexperiments and numerical analyses showed the followingresults:

(1) The output power and the turbine efficiency of theStraight Blades runner (𝛽

𝑏1= 90∘, 𝛽𝑏2= 90∘) were

greater than those of the Curved Blades runner (𝛽𝑏1=

30∘, 𝛽𝑏2= 90∘) throughout the rotational speed range.

Compared with the Curved Blades runner, the outputpower and the turbine efficiency of the Straight Bladesrunner were improved by about 31.7% and about67.1%, respectively.

(2) The upstream water depth of the Straight Bladesrunner was smaller than that of the Curved Blades


1

1

2

2

3

3

44

u

u

u

u

ww

w

w

𝛼

𝛼

𝛼

𝛼 𝛽

𝛽

𝛽

𝛽

1st stage inlet 2nd stage inlet

1st stage outlet 2nd stage outlet

Figure 9: Velocity triangles at the runner inlet (points 1 and 2) and outlet (points 3 and 4).

0.0

0.1

0.2

0.3

15 30 45 60 75 90 105 120 135 150

1st stage inlet

2nd stage outlet

Boundary line

𝜃 (∘)

r1

(m/s

)

−0.1

−0.2

−0.3

−0.4

−0.5

StraightCurved

(a) Radial component

1st stage inlet 2nd stage outlet

Boundary line0.0

0.1

0.2

0.3

0.4

0.5

15 30 45 60 75 90 105 120 135 150𝜃 (∘)

u1

(m/s

)

−0.1

−0.2

−0.3

−0.4

StraightCurved

(b) Circumferential component

Figure 10: Absolute velocity at outer circumference (𝑛 = 22min−1, cal.).

runner, whereas the downstream water depth of theStraight Blades runner was greater than that of theCurved Blades runner.

(3) The water domains at the outer and inner circumfer-ences of the runners were narrower on the StraightBlades runner than on the Curved Blades runner.Therefore, the first stage inlet region and secondstage outlet region of the Straight Blades runner werenarrower than those of the Curved Blades runner.Whereas the second stage inlet region was narroweron the Straight Blades runner than on the CurvedBlades runner, the first stage outlet regionwas approx-imately the same for both runners.

(4) Whereas the circumferential component of the abso-lute velocity at the high circumferential angle inthe second stage outlet region was positive for theStraight Blades runner, it was negative for the CurvedBlades runner. Thus, it is inferred that the flow in the

aforesaid region on the Curved Blades runner is in thedirection opposite to the runner’s rotation, hinderingthe reversal flow (including the leakage flow at thebottom) and sharply turning against the main flow.

(5) The relative flow angle in the first stage inlet regionof the Straight Blades runner coincides with a bladeangle at a circumferential angle of around 49∘, whilethe flow runs generally along the blades in the circum-ferential angle range of 27∘–57∘. On the other hand,the relative flow angle of the Curved Blades runnercoincides with the blade angle at a circumferentialangle of around 70∘, largely diverting from the bladeangle at the low circumferential angle. Consequently,at the low circumferential angle, the radial componentof the absolute velocity on the Curved Blades runnerdecreases, whereas that on the Straight Blades runnerincreases.


0.0

0.1

0.2

0.3

30 45 60 75 90 105 120 135 150

1st stage outlet

2nd stage inlet

Boundary line (straight)

Boundary line (curved)

𝜃 (∘)

−0.1

−0.2

−0.3

−0.4

StraightCurved

r2

(m/s

)

(a) Radial component

0.0

0.1

0.2

0.3

0.4

0.5

30 45 60 75 90 105 120 135 150

1st stage outlet

2nd stage inlet


Boundary line (curved)

𝜃 (∘)

StraightCurved

u2

(m/s

)

(b) Circumferential component

Figure 11: Absolute velocity at inner circumference (𝑛 = 22min−1, cal.).

0306090

120150180

15 30 45 60 75 90 105 120 135 150

1st stage inlet

2nd stage outlet

Boundary line

𝜃 (∘)

StraightCurved

𝛽1(∘)

−30

−60

−90

−120

−150

(a) Outer circumference

2nd stage inlet

1st stage outlet


Boundary line (curved)0

30

60

90

120

150

30 45 60 75 90 105 120 135 150𝜃 (∘)

StraightCurved

𝛽2(∘)

−30

−60

−90

(b) Inner circumference

Figure 12: Relative flow angle (𝑛 = 22min−1, cal.).


Figure 13: Visualization results (𝑛 = 22min−1, exp.).



Figure 14: Absolute velocity vectors (𝑛 = 22min−1, cal.).

Nomenclature

𝐵: Channel width m𝑏: Runner width m𝐷: Runner diameter m𝑔: Gravitational acceleration m/s2𝐻: Effective head m, equal to ℎ

3+ V23/2𝑔−ℎ

4− V24/2𝑔

ℎ: Water depth m𝑛: Rotational speed min−1𝑃: Output power W, equal to 2𝜋𝑛𝑇/60𝑄: Flow rate m3/s𝑇: Torque N⋅m𝑢: Circumferential velocity m/sVF: Volume fraction of waterV: Absolute velocity m/s𝑤: Relative velocity m/s𝑍: Number of blades.

Greek Letters

𝛼: Absolute flow angle ∘𝛽: Relative flow angle ∘𝛽𝑏: Blade angle ∘

𝛾: Clearance between runner and side walls m𝛿: Clearance between runner and floor m𝜂: Turbine efficiency = 𝑃/𝜌𝑔𝑄𝐻𝜃: Circumferential angle ∘𝜌: Fluid density kg/m3.

Subscripts

1: Inner circumference of runner2: Outer circumference of runner3: Upstream4: Downstream𝑟: Radial component𝑢: Circumferential component.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

References

[1] C. Nicolet, A. Zobeiri, P. Maruzewski, and F. Avellan, “Experi-mental investigations on upper part load vortex rope pressure

fluctuations in francis turbine draft tube,” International Journalof Fluid Machinery and Systems, vol. 4, no. 1, pp. 179–190, 2011.

[2] T. Vu, M. Koller, M. Gauthier, and C. Deschenes, “Flow simula-tion and efficiency hill chart prediction for a Propeller turbine,”International Journal of Fluid Machinery and Systems, vol.4, no. 2, pp. 243–254, 2011.

[3] C. Yang, Y. Zheng, and L. Li, “Optimization design and perfor-mance analysis of a pit turbine with ultralow head,” AdvancesinMechanical Engineering, vol. 2014, Article ID 401093, 7 pages,2014.

[4] T. S. Reynolds, Stronger Than a Hundred Men: A History ofthe Vertical Water Wheel, The Johns Hopkins University Press,Baltimore, Md, USA, 1983.

[5] Y. Fujiwara, H. Kado, and Y. Hosokawa, “History and perfor-mance of traditional water wheel in Western Europe,” Transac-tions of the Japan Society of Mechanical Engineers, vol. 90, no.819, pp. 212–218, 1987.

[6] A. Furukawa, S. Watanabe, D. Matsushita, and K. Okuma,“Development of ducted Darrieus turbine for low headhydropower utilization,” Current Applied Physics, vol. 10, sup-plement, no. 2, pp. S128–S132, 2010.

[7] Y. Nishi, T. Inagaki, R. Omiya, T. Tachikawa, M. Kotera, and J.Fukutomi, “Study on a cross-flow water turbine installed in anopen channel,”Turbomachinery, vol. 39, no. 8, pp. 467–474, 2011.

[8] Y. Nishi, T. Inagaki, Y. Li, R. Omiya, and J. Fukutomi, “Studyon an undershot cross-flow water turbine,” Journal of ThermalScience, vol. 23, no. 3, pp. 239–245, 2014.

[9] C. A. Mockmore and F. Merryfield, “The banki water turbine,”Bulletin Series 25, 1949.

[10] J. Fukutomi, “Characteristics and internal flow of cross-flowtubine for micro hydro power,” Turbomachinery, vol. 28, no. 3,pp. 156–163, 2000.

[11] Y. Nishi, T. Inagaki, R. Omiya, and J. Fukutomi, “Performanceand internal flow of an undershot-type cross-flow water tur-bine,” Transactions of the Japan Society of Mechanical Engineers,Series B, vol. 79, no. 800, pp. 521–532, 2013.

[12] Y. Nishi, T. Inagaki, Y. Li, R. Omiya, and K. Hatano, “The flowfield of undershot cross-flow water turbines based on PIVmeasurements and numerical analysis,” International Journal ofFluid Machinery and Systems, vol. 7, no. 4, pp. 174–182, 2014.

[13] ANSYS, ANSYS CFX-Solver Theoretical Guide, 2010.[14] ANSYS, ANSYS CFX-Solver Modeling Guide, ANSYS, 2010.

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