Research Article Numerical Simulation of Water...

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2013 Article ID 473842 8 pageshttpdxdoiorg1011552013473842

Research ArticleNumerical Simulation of Water Flow through a Nano-HydraulicTurbine of Waterfall-Type by Particle Method

Tomomi Uchiyama1 Haruki Fukuhara2 Shouichiro Iio3 and Toshihiko Ikeda3

1 EcoTopia Science Institute Nagoya University Furo-cho Chikusa Nagoya 464-8603 Japan2Hitachi Mitsubishi Hydro CorporationTamachi Nikko Bldg 29-14 Shiba 5-chome Minato-ku Tokyo 108-0014 Japan3 Faculty of Engineering Shinshu University 4-17-1 Wakasato Nagano 380-8553 Japan

Correspondence should be addressed to Tomomi Uchiyama uchiyamaisnagoya-uacjp

Received 4 September 2013 Accepted 10 October 2013

Academic Editor Ningsheng Feng

Copyright copy 2013 Tomomi Uchiyama et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This study simulates the flow through an impulse-type small-scale hydraulic turbine utilizing a waterfall of extra-low headThe two-dimensional Moving Particle Semi-implicit (MPS) method is employed for the simulationThe fluid is discretized by particles andthe flow is computed by the Lagrangian calculation for the particle motion When the distance between the particles discretizingthe waterfall of a width B 119897

0 is set at 119897

0119861 le 0084 the flow can be simulated with the sufficiently high spatial resolution and the

rotor performance can also be favorably predicted The present simulation also successfully analyzes the effect of the rotationalspeed of rotor on the flow and the turbine performance

1 Introduction

Hydropower is one of the promising renewable energyresources It is converted to electric energy through hydraulicturbines In Japan in 2010 approximately 10 of all supplyingelectric energy is provided by hydraulic power generationsThe percentage is expected to increase steadily on the basis ofa government policy promoting the development of renew-able energy As large-scale hydroelectric plants require hugedams and long conduits the places for the construction arehardly remainedThus expectations for the development of asmall-scale hydropower of which output is less than 1000 kWhave been increasingWhennatural disaster occurs the large-scale centralized hydraulic power plants may lose the powersupplying ability due to the collapse of the power grid Sincethe small-scale hydropower existing in small-scale riversirrigation canals and industrial drainages distributes widelyin Japan it realizes the small-scale distributed power gener-ation The small-scale hydropower can contribute the localproduction for local consumption of electric power whichis more resistant to disaster Consequently the developmentis also of great worth from the viewpoint of constructing adisaster-resistant society

To exploit effectively the small-scale hydropower varioustypes of hydraulic turbine have been presented [1ndash5] Ikeda etal [6] developed an impulse-type hydraulic turbine utilizingwaterfalls of extra-lowheadwhich is 2mor less in small riversand agricultural canals and so forth The nano-hydraulicturbine of which output power is less than several kW canbe easily carried to the places where they are necessaryand produces electric power easily without damaging theenvironment Ikeda et al [6] investigated the power charac-teristic of the turbine through a laboratory experiment andmade clear the flow inside the rotor by the experimentalvisualization

This study proposes the simulation method for the flowin the nano-hydraulic turbine of Ikeda et al [6] whichcan be favorably employed for the prediction of the turbineperformance The turbine performance is generally affectedby the geometric conditions such as the blade shape theblade installation angle and the number of blades It is alsoinfluenced by the position of turbine relative to the waterflow The numerical simulation can examine the effect ofeach condition individually and promises to yield designguidelines for high-performance turbineThenano-hydraulic

2 International Journal of Rotating Machinery

turbine is driven by a waterfall The falling water collideswith the rotating blades and it is scattered around therotor Therefore the flow through the turbine includes freesurfaces and it is very complicated The simulation of thisstudy is based on the Moving Particle Semi-implicit (MPS)method [7ndash9] which is one of the particle methods forfree-surface flows The fluid is discretized with particlesand the flow is simulated by the Lagrangian computationof the particle motion There are few numerical simulationsof the flow through small-scale hydraulic turbines utilizingsmall-scale hydropower except for a MPS simulation ofan impulsive turbine driven by a nozzle jet [5] But theparameters for theMPSmethod such as the distance betweenthe particles were not examined and accordingly the suffi-cient knowledge on the simulation of small-scale hydraulicturbines was not obtained First this study elucidates theappropriate distance between the particles for the flow andperformance simulations of the hydraulic turbine Secondlyit is demonstrated that the present simulation can analyzethe effect of the rotational speed of rotor on the turbineperformance

2 Basic Equations and Simulation Method

21 Conservation Equations for Flow If the flow through thehydraulic turbine is incompressible it is governed by themassand momentum conservation equations

D120588D119905

= 0 (1)

DuD119905

= minus1

120588nabla119901 + ]nabla2u + F (2)

where 120588 is the density 119905 is the time u is the velocity 119901 is thepressure ] is the kinematic viscosity and F is the externalforces such as the gravitational force and the surface tension

This study analyzes (1) and (2) by the Moving ParticleSemi-implicit (MPS) method [7ndash9] which is one of theparticle methods In the MPS method the fluid is discretizedby particles and the particle motion is computed by theLagrangian method Equations (1) and (2) are discretizedthrough the interactions between the particles

22 Particle Interaction Model The interactions between theparticles aremodeled with a weight function119908 defined by thefollowing equation

119908 (119903) =

119903119890

119903minus 1 119903 le 119903

119890

0 119903 gt 119903119890

(3)

where 119903 is the distance between two particles and 119903119890stands for

a kernel sizeThe particle number density at the position of the 119894th

particle ⟨119899⟩119894 is defined as

⟨119899⟩119894 = sum

119895 = 119894

119908(10038161003816100381610038161003816r119895minus r119894

10038161003816100381610038161003816) (4)

where r119894is the position vector of the 119894th particle

As the fluid density remains unaltered in the incom-pressible flow the particle number density is required tobe constant The incompressible flow condition in the MPSmethod is satisfied bymaintaining ⟨119899⟩

119894at a constant value 1198990

The Laplacian operator expressing the viscous term onthe right-hand side of (2) is modeled with the weightfunction The Laplacian operator at the position of the 119894thparticle is given as

⟨nabla2120601⟩119894=2119889

1198990Λsum

119895 = 119894

[(120601119895minus 120601119894)119908 (

10038161003816100381610038161003816r119895minus r119894

10038161003816100381610038161003816)] (5)

where 120601 is a physical quantity A parameterΛ is introduced sothat the variance increase is equal to the analytical solution

Λ =int119881119908 (119903) 119903

2dVint119881119908 (119903) dV

(6)

The gradient operator expressing the pressure gradientterm on the right-hand side of (2) is also modeled Thegradient operator at the position of the 119894th particle ismodeledby setting the interparticle force at the repulsion to ensurethe numerical stability [7ndash9] Thus the interacting pressureforces between two particles are not antisymmetric and themomentum is not always conserved To resolve this problemthis simulation employs the following model presented byKhayyer and Gotoh [10]

⟨nabla120601⟩119894=119889

1198990sum

119895 = 119894

[

[

(120601119894+120601119895)minus(120601119894+ 120601119895)

10038161003816100381610038161003816r119895minusr119894

10038161003816100381610038161003816

2(r119895minusr119894)119908 (

10038161003816100381610038161003816r119895minusr119894

10038161003816100381610038161003816)]

]

(7)

where 119889 is the number of space dimensions and 120601119894is defined

as

120601119894= min119895isin119869

(120601119894 120601119895) 119869 = 119895 119908 (

10038161003816100381610038161003816r119895minus r119894

10038161003816100381610038161003816) = 0 (8)

23 Simulation Method Equations (1) and (2) are solved bya semi-implicit method which is used in the SMAC method[11] If the particle velocity u119896

119894and position r119896

119894at time 119905 = 119896Δ119905

are known the flow at time 119905 = (119896 + 1)Δ119905 is simulated by thefollowing two steps

In the first step the temporal velocity and position forthe particle ulowast

119894and rlowast119894 respectively are calculated from (2)

without considering the pressure gradient term Then thetemporal particle number density ⟨119899lowast⟩

119894is computed by using

(4)In the second step the following Poisson equation is

solved for the pressure 119901119896+1 so that the mass conservation issatisfied or ⟨119899lowast⟩

119894is made to coincide with 1198990

⟨nabla2119901119896+1⟩119894= minus

120588

Δ1199052

⟨119899lowast⟩119894minus 1198990

1198990 (9)

International Journal of Rotating Machinery 3

D=200 120∘

40

Blade250

R23

Figure 1 Rotor and blade specifications

Then the temporal velocity and position for the particleare corrected by the obtained pressure gradient

u119896+1119894

= ulowast119894+ u1015840119894

(10)

r119896+1119894

= rlowast119894+ u1015840119894Δ119905 (11)

where

u1015840119894= minus

Δ119905

120588⟨nabla119901119896+1⟩119894 (12)

The viscous term in (2) and the left-hand side of (9) arecomputed by the Laplacian operator (5) The right-hand sideof (12) is calculated by the gradient operator (7)

3 Simulation Conditions

The flow through an impulse-type nano-hydraulic turbinedeveloped by Ikeda et al [6] is simulated The powercharacteristic of the turbine was investigated by a labora-tory experiment and the flow field was made clear by theexperimental visualization [6] Figure 1 shows the rotor andblade specifications The rotor diameter 119863 is 200mm theaxis diameter is 15mm and the number of blades is 12The blade has a circular-arc shape The curvature radiusis 23mm the thickness is 3mm and the chord length is40mm To visualize the flow inside the rotor the blades aresandwiched between two circular plates made of transparentvinyl chloride

In the experiment the rotor is placed at the bottom of awaterfall as shown in Figure 2The origin of coordinates is setat the initial point of the waterfall The 119909-axis is horizontaland the 119911-axis is vertical The flow through the rotor issimulated by a two-dimensionalMPSmethodThewater flowrate119876 is 00035m3s and the head of waterfall119867

119865is 570mm

The nondimensional horizontal distance between the initialpoint of the waterfall and the colliding point of the water withthe blade 119871

119865 119871119865119863 is 133

The square region of 500mm times 500mm around the rotoraxis is chosen as the computational domain as shown inFigure 3 where the coordinates of the axis are supposed tobe (119909119886 119911119886) The simulation is performed at the conditions of

056 le 120582 le 07 where 120582(= 119881119905119880119865) is defined by the rotor tip

speed 119881119905and the impact velocity of waterfall with the blade

119880119865The waterfall flows into the computational domain from

the location A on the upper boundary as indicated inFigure 3 The particles discretizing the falling water are

Rotor

Waterfall

xy

z

LF

HF

Rotation

Gravity

Figure 2 Positional relation between waterfall and rotor

Rotor

Waterfall Rotation

Gravity

A

minus125

125

0

0

125minus125

minus(z

minusz a)D

B

(x minus xa)D

Figure 3 Computational domain

released from the location A into the domain The releasingvelocity and angle are determined from the condition of thefree fall It is required that the released particles are arrangedat a uniform interval in the horizontal and vertical directionsto ensure the computational accuracy The particles arearranged at an interval 119897

0in the horizontal direction and

released into the domain at a constant time interval Δ119905119901 so

that the width of the waterfall 119861 at the location A correspondsto the measured value 84mm The Δ119905

119901value is prescribed

so that the vertical distance between the particles coincideswith 119897

0 The distance 119897

0corresponds to the grid width for

grid-based simulations such as a finite difference methodTherefore the simulated results are considered to depend on1198970 This study investigates the effect of 119897

0on the flow and the

turbine performance to search for the appropriate value of 1198970


A

B

C

(a) 119905 = 1199051[119904]

A

B

C

(b) 119905 = 1199051 + 0005

A

B

C

(c) 119905 = 1199051 + 001

A

B

C

(d) 119905 = 1199051 + 0015

A

B

C

(e) 119905 = 1199051 + 002

Figure 4 Time variation for particle distribution when 120582 = 056

The rotor axis and the blades which are the solidwalls arediscretized by the particles having the same angular velocityas the rotor The distance between these particles is also setat 1198970 The Neumann boundary condition for the pressure

gradient is imposed on the particles contacting with thefluid

The pressure is set at zero on the free surface Theposition of the free surface is detected according to the value


minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

Figure 5 Distribution of superimposed particles when 120582 = 056

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

0 1|u|UF

Figure 6 Time-averaged velocity distribution when 120582 = 056

RotationWaterfall

Rotor

Figure 7 Experimentally visualized flow pattern when 120582 = 056

006 008 01 012 0140

10000

20000

30000

l0B

N

Figure 8 Change in number of particles119873 due to distance betweenparticles 119897

0when 120582 = 056

Table 1 Simulation conditions

Rotor diameter119863 02mNumber of blades 12Water flow rate 119876 00035m3sVertical velocity of waterfallcolliding with blade 119880

119865

334ms

Head of waterfall119867119865

057mHorizontal distance between bladeand initial point of waterfall 133119863

Distance between particlesdiscretizing waterfall 119897

0

0069119861ndash012119861

Maximum of time increment Δ119905 0001 s

of the particle number density When the particle numberdensity ⟨119899lowast⟩

119894obtained by the first-step calculation of each

time step satisfies the following relation the 119894th particle isdecided to be on the free surface

⟨119899lowast⟩119894lt 1205731198990 (13)

where 120573 is a parameter of 120573 lt 1The time incrementΔ119905 is determined from the maximum

particle velocity at each computational time step In thissimulation the initial value is set at 0001 s resulting in themaximum value of 0001 s The value of 119903

119890in (3) is generally

chosen at 2 le 1199031198901198970le 4 [7 8]The value is 21119897

0for the particle

number density and the gradient operator while it is 41198970for

the Laplacian operator [8] The parameter 120573 in (13) is set at097 It is reported that the simulation of a fragmentation offluid scarcely depends on the 120573 value in the case of 08 le 120573 le099 [7]

The simulation conditions are listed in Table 1

4 Results and Discussion

41 Results at Tip Speed Ratio of 120582 = 056 Figure 4 showsthe time variation of the particle distribution for the fully


minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

(a) In case of 1198970119861 = 012

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

(b) In case of 1198970119861 = 0069

Figure 9 Effect of distance between particles 1198970on particle distribution when 120582 = 056

developed flow where the tip speed ratio 120582 is 056The distri-butions at five time points during a period for the interactionbetween the waterfall and a blade are presentedThe distancebetween the particles discretizing waterfall 119897

0is set at 119897

0119861 =

0084When the time is as in Figure 4(a) thewaterfall collidesdirectly with the tip of the concave surface for the blade AAt the subsequent time points of Figures 4(b) 4(c) and 4(d)the collision point moves toward the center of the concavesurface for the blade A as the rotor rotates When the timeis as in Figure 4(e) a part of the waterfall gets in contactwith the tip of the subsequent blade B But the collisionbetween the waterfall and the concave surface for the blade Ais still maintained The collision point is closest to the rotoraxis during the one period for the interaction between thewaterfall and a blade On the concave surface of the blade Athe number of particles or the mass of water increases withthe passage of time during the one periodWhen the time is asin Figure 4(d) the water on the concave surface of the bladeA flows toward the subsequent blade B and it collides withthe convex surface of the blade B at the time of Figure 4(e)It should be noted that the blade C corresponds to a bladewhich has just finished colliding with the waterfallThe wateron the concave surface flows toward the inside and outside ofthe rotor The water flowing toward the inside collides withthe concave surface of the subsequent blade A and then itenters into the rotor The water directing toward the outsidedisperses markedly in the radial direction as the rotor rotatesThe flow rate toward the outside is much larger The particlesalways exist on the convex surface of the blades which areon the opposite side of the waterfall This demonstrates thestagnation of water inside the rotor The blades give theangular momentum to the stagnant water and flick the wateraway from the rotor causing the deterioration of the rotorperformance

Figure 5 shows the superposition for the particle distri-butions at every time interval 3Δ119905 during 30Δ119905 time periodwhere 120582 = 056 and 119897

0119861 = 0084 One can grasp the

water dispersion around the rotor and the water stagnationin the rotor The water scatters mainly toward the lower leftdirection and the waterfall direction

The time-averaged water velocity for 120582 = 056 is shown inFigure 6Thewater dispersion toward the outside of the rotorand the flow into the rotor are reconfirmed

The flow pattern inside and around the rotor experimen-tally visualized by Ikeda et al [6] at 120582 = 056 is presentedin Figure 7 It was acquired by using a CCD camera anda strobe light sheet shaped through a 2mm wide slit Theimage visualizes vividly the water flow along the concavesurface of the blades the water dispersion and flick towardthe outside of the rotor and the water stagnation inside therotorThe simulated flow shown in Figure 4 agrees well withthe experimental visualization demonstrating the validity ofthe present simulation

The distance between the particles discretizing the water-fall 1198970or the initial distance between the particles corresponds

to the grid width in grid-based simulation methods suchas a finite difference method The simulation with smaller 119897

0

has superior space resolution The time-averaged number ofparticles119873 for the fully developed flow is plotted against 119897

0in

Figure 8 For the abovementioned simulation of 1198970119861 = 0084

119873 is 14437 The119873 value increases greatly with the decrementof 1198970 The improvement of the spatial resolution increases the

number of particles and therefore it causes the increment ofthe computational time In the simulation of 119897

0119861 = 0084

348 and 299 particles are used to discretize the axis andthe single blade respectively These particles are included inFigure 8 For the flow simulation during the one revolutionof the rotor about 64 hours are required on a worksta-tion (Processor Intel Xeon X5660 28GHz times 6 Memory12GB)

The superimposed particles at 1198970119861 = 0069 and 012

distribute as shown in Figure 9 where 120582 = 056 In the caseof 1198970119861 = 012 the number of particles is low and the water

dispersion outside the rotor and the stagnation inside the


0

02

04

06

08

1

Cp

006 008 01 012 014l0B

Analysis of Ikeda et al (2010)

Experiment of Ikeda et al (2010)

Figure 10 Change in power coefficient 119862119901due to distance between

particles 1198970when 120582 = 056

rotor are not fully resolved The simulation of 1198970119861 = 0069

composed of more particles yields almost the same resultat 1198970119861 = 0084 shown in Figure 5 It is discovered that the

simulation of 1198970119861 le 0084 can ensure a sufficiently high

spatial resolutionCalculating the rotor power 119875 by estimating the

water kinematic energy at the rotor inlet and outlet thenondimensional value 119862

119901changes as the function of 119897

0119861

as plotted in Figure 10 When the spatial resolution ishigh enough (119897

0119861 le 0084) the 119862

119901value remains almost

unaltered It is slightly larger than the experimental resultof 119862119901

= 066 But it is considered to be valid in dueconsideration of the two-dimensional simulation Ikeda et al[6] calculated the119862

119901value at 120582 = 056 by estimating the force

on a blade from the visualized flow pattern It is also plottedin Figure 10 being in good agreement with the presentsimulation Consequently it is found that the simulation of1198970119861 le 0084 can accurately predict the 119862

119901value

Ikeda et al [6] made it clear by their experiment that thehorizontal distance between the initial point of the waterfalland the blade 119871

119865affects the rotor performance 119862

119901 The

distance119871119865varies with thewater flow rate119876 For the practical

use of the hydraulic turbine it is desirable that 119871119865is always set

at the optimal value irrespective of119876 Ikeda et al [6] proposedamethod to control the119871

119865value by installing a flat plate along

the waterfall The current simulation is effectively employedto search for the applicability of the method

42 Results at Tip Speed Ratios of 120582 = 06 and 07 Theflows at 120582 = 06 and 07 are simulated for the condition of1198970119861 = 0084 Figure 11 depicts the relation between 120582 and119862119901 The 119862

119901value decreases with the increment of 120582 This

change agrees with the experimental result of Ikeda et al [6]indicating that the effect of 120582 on 119862

119901is successfully analyzed

by the simulation The simulated 119862119901value is slightly larger

This is because the current simulation does not sufficientlyresolve the turbulent flow and therefore it ignores the lossescaused by the turbulence on the blade surface as well asinside the rotor It may be also because the current simulation

04 06 080

02

04

06

08

1

Cp

Experiment of Ikeda et al (2010)Analysis of Ikeda et al (2010)

Simulation

120582

Figure 11 Relation between power coefficient119862119901and tip speed ratio

120582

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D


employs the two-dimensional MPS method Using the three-dimensional MPS method the flow and the 119862

119901value would

be simulated more accurately But the number of particlesincreases and a longer computational time is required

The flow fields at 120582 = 07 are shown in Figures 12 and13 The waterfall disperses due to the collision with the rotorWhen compared with the results at 120582 = 056 (Figures 5 and6) the divergence angle of the dispersed water lessens Onecan grasp the decrement of the rotor angular momentumobtained from the waterfall The amount of water inside therotor is less than that at 120582 = 056

5 Conclusions

The flow through an impulse-type small-scale hydraulicturbine utilizing a waterfall of extra-low head is simulated bya two-dimensional MPS method The rotor performance is


0 1|u|UF

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D


also analyzed by using the simulated flow field The resultsare summarized as follows

(1) When the distance between the particles discretizingthe waterfall of a width 119861 119897

0 is set at 119897

0119861 le 0084

the flow simulated at the tip speed ratio 120582 = 056

is confirmed to agree well with the experimentallyvisualized one Thus the simulation of 119897

0119861 le 0084

has a sufficiently high spatial resolution

(2) For the simulation of 1198970119861 le 0084 at 120582 = 056 it is

also confirmed that the simulated power coefficient119862119901agrees nearly with the experiment Thus the

simulation of 1198970119861 le 0084 can also favorably predict

the rotor performance

(3) The 119862119901values simulated at 120582 = 06 and 07 agree

almost with the experimental results Therefore thepresent simulation can successfully analyze the effectof 120582 on the flow and the rotor performance

Nomenclatures

119861 Thickness of waterfall119862119901 Power coefficient = 119875120588119892119876119867

119865

119889 Number of space dimensions119863 Diameter of rotorF External force119867119865 Head of waterfall

1198970 Distance between particles discretizing

waterfall119871119865 Horizontal distance between blade andinitial point of waterfall

119899 Particle number density119873 Time-averaged number of particles119901 Pressure119875 Power output from rotor119876 Water flow rater Position vector of particle119905 Timeu Velocity119880119865 Impactvelocity of waterfall with blade

= (2119892119867119865)12

119881119905 Rotor tip speed = 1205961198632

119908 Weight function119909 119910 119911 Spatial coordinatesΔ119905 Time increment120582 Tip speed ratio = 119881

119905119880119865

] Kinematic viscosity120588 Density120596 Angular velocity of rotor

References

[1] Y Takamatsu A Furukawa K Okuma and K TakenouchildquoExperimental studies on a preferable blade profile for highefficiency and the blade characteristics of Darrieus-type cross-flow water turbinesrdquo JSME International Journal vol 34 no 2pp 149ndash156 1991

[2] M Nakajima S Iio and T Ikeda ldquoPerformance of double-stepSavonius rotor for environmentally friendly hydraulic turbinerdquoJournal of Fluid Science andTechnology vol 3 no 3 pp 410ndash4192008

[3] M Nakajima S Iio and T Ikeda ldquoPerformance of Savoniusrotor for environnotmentally friendly hydraulic turbinerdquo Journalof Fluid Science and Technology vol 3 pp 420ndash429 2008

[4] S Derakhshan and A Nourbakhsh ldquoExperimental study ofcharacteristic curves of centrifugal pumps working as turbinesin different specific speedsrdquo Experimental Thermal and FluidScience vol 32 no 3 pp 800ndash807 2008

[5] Y Nakanishi S Iio Y Takahashi A Kato and T Ikeda ldquoDevel-opment of a simple impulse turbine for nano hydropowerrdquoJournal of Fluid Science and Technology vol 4 pp 567ndash5772009

[6] T Ikeda S Iio andK Tatsuno ldquoPerformance of nano-hydraulicturbine utilizing waterfallsrdquo Renewable Energy vol 35 no 1 pp293ndash300 2010

[7] S Koshizuka and Y Oka ldquoMoving-particle semi-implicitmethod for fragmentation of incompressible fluidrdquo NuclearScience and Engineering vol 123 no 3 pp 421ndash434 1996

[8] S Koshizuka A Nobe and Y Oka ldquoNumerical analysisof breaking waves using the moving particle semi-implicitmethodrdquo International Journal for Numerical Methods in Fluidsvol 26 no 7 pp 751ndash769 1998

[9] K Shibata S Koshizuka M Sakai and K TanizawaldquoLagrangian simulations of ship-wave interactions in roughseasrdquo Ocean Engineering vol 42 pp 13ndash25 2012

[10] A Khayyer and H Gotoh ldquoDevelopment of CMPS methodfor accurate water-surface tracking in breaking wavesrdquo CoastalEngineering Journal vol 50 no 2 pp 179ndash207 2008

[11] A A Amsden and F H Harlow ldquoThe SMAC method anumerical technique for calculating incompressible fluid flowsrdquoLos Alamos Scientific Laboratory Report LA-4370 1970

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turbine is driven by a waterfall The falling water collideswith the rotating blades and it is scattered around therotor Therefore the flow through the turbine includes freesurfaces and it is very complicated The simulation of thisstudy is based on the Moving Particle Semi-implicit (MPS)method [7ndash9] which is one of the particle methods forfree-surface flows The fluid is discretized with particlesand the flow is simulated by the Lagrangian computationof the particle motion There are few numerical simulationsof the flow through small-scale hydraulic turbines utilizingsmall-scale hydropower except for a MPS simulation ofan impulsive turbine driven by a nozzle jet [5] But theparameters for theMPSmethod such as the distance betweenthe particles were not examined and accordingly the suffi-cient knowledge on the simulation of small-scale hydraulicturbines was not obtained First this study elucidates theappropriate distance between the particles for the flow andperformance simulations of the hydraulic turbine Secondlyit is demonstrated that the present simulation can analyzethe effect of the rotational speed of rotor on the turbineperformance

2 Basic Equations and Simulation Method

21 Conservation Equations for Flow If the flow through thehydraulic turbine is incompressible it is governed by themassand momentum conservation equations

D120588D119905

= 0 (1)

DuD119905

= minus1

120588nabla119901 + ]nabla2u + F (2)

where 120588 is the density 119905 is the time u is the velocity 119901 is thepressure ] is the kinematic viscosity and F is the externalforces such as the gravitational force and the surface tension

This study analyzes (1) and (2) by the Moving ParticleSemi-implicit (MPS) method [7ndash9] which is one of theparticle methods In the MPS method the fluid is discretizedby particles and the particle motion is computed by theLagrangian method Equations (1) and (2) are discretizedthrough the interactions between the particles

22 Particle Interaction Model The interactions between theparticles aremodeled with a weight function119908 defined by thefollowing equation

119908 (119903) =

119903119890

119903minus 1 119903 le 119903

119890

0 119903 gt 119903119890

(3)

where 119903 is the distance between two particles and 119903119890stands for

a kernel sizeThe particle number density at the position of the 119894th

particle ⟨119899⟩119894 is defined as

⟨119899⟩119894 = sum

119895 = 119894

119908(10038161003816100381610038161003816r119895minus r119894

10038161003816100381610038161003816) (4)

where r119894is the position vector of the 119894th particle

As the fluid density remains unaltered in the incom-pressible flow the particle number density is required tobe constant The incompressible flow condition in the MPSmethod is satisfied bymaintaining ⟨119899⟩

119894at a constant value 1198990

The Laplacian operator expressing the viscous term onthe right-hand side of (2) is modeled with the weightfunction The Laplacian operator at the position of the 119894thparticle is given as

⟨nabla2120601⟩119894=2119889

1198990Λsum

119895 = 119894

[(120601119895minus 120601119894)119908 (

10038161003816100381610038161003816r119895minus r119894

10038161003816100381610038161003816)] (5)

where 120601 is a physical quantity A parameterΛ is introduced sothat the variance increase is equal to the analytical solution

Λ =int119881119908 (119903) 119903

2dVint119881119908 (119903) dV

(6)

The gradient operator expressing the pressure gradientterm on the right-hand side of (2) is also modeled Thegradient operator at the position of the 119894th particle ismodeledby setting the interparticle force at the repulsion to ensurethe numerical stability [7ndash9] Thus the interacting pressureforces between two particles are not antisymmetric and themomentum is not always conserved To resolve this problemthis simulation employs the following model presented byKhayyer and Gotoh [10]

⟨nabla120601⟩119894=119889

1198990sum

119895 = 119894

[

[

(120601119894+120601119895)minus(120601119894+ 120601119895)

10038161003816100381610038161003816r119895minusr119894

10038161003816100381610038161003816

2(r119895minusr119894)119908 (

10038161003816100381610038161003816r119895minusr119894

10038161003816100381610038161003816)]

]

(7)

where 119889 is the number of space dimensions and 120601119894is defined

as

120601119894= min119895isin119869

(120601119894 120601119895) 119869 = 119895 119908 (

10038161003816100381610038161003816r119895minus r119894

10038161003816100381610038161003816) = 0 (8)

23 Simulation Method Equations (1) and (2) are solved bya semi-implicit method which is used in the SMAC method[11] If the particle velocity u119896

119894and position r119896

119894at time 119905 = 119896Δ119905

are known the flow at time 119905 = (119896 + 1)Δ119905 is simulated by thefollowing two steps

In the first step the temporal velocity and position forthe particle ulowast

119894and rlowast119894 respectively are calculated from (2)

without considering the pressure gradient term Then thetemporal particle number density ⟨119899lowast⟩

119894is computed by using

(4)In the second step the following Poisson equation is

solved for the pressure 119901119896+1 so that the mass conservation issatisfied or ⟨119899lowast⟩

119894is made to coincide with 1198990

⟨nabla2119901119896+1⟩119894= minus

120588

Δ1199052

⟨119899lowast⟩119894minus 1198990

1198990 (9)


D=200 120∘

40

Blade250

R23



u119896+1119894

= ulowast119894+ u1015840119894

(10)

r119896+1119894

= rlowast119894+ u1015840119894Δ119905 (11)

where

u1015840119894= minus

Δ119905

120588⟨nabla119901119896+1⟩119894 (12)





119865is 570mm


119865 119871119865119863 is 133






Rotor

Waterfall

xy

z

LF

HF

Rotation

Gravity


Rotor

Waterfall Rotation

Gravity

A

minus125

125

0

0

125minus125

minus(z

minusz a)D

B

(x minus xa)D














A

B

C

(a) 119905 = 1199051[119904]

A

B

C

(b) 119905 = 1199051 + 0005

A

B

C

(c) 119905 = 1199051 + 001

A

B

C

(d) 119905 = 1199051 + 0015

A

B

C

(e) 119905 = 1199051 + 002






minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D


minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

0 1|u|UF


RotationWaterfall

Rotor


006 008 01 012 0140

10000

20000

30000

l0B

N


0when 120582 = 056



119865

334ms




0

0069119861ndash012119861





⟨119899lowast⟩119894lt 1205731198990 (13)





0for the particle







minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

(a) In case of 1198970119861 = 012

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

(b) In case of 1198970119861 = 0069



0is set at 119897

0119861 =









0


0in




0119861 = 0084






0

02

04

06

08

1

Cp

006 008 01 012 014l0B




particles 1198970when 120582 = 056







0119861


0119861 le 0084) the 119862






119901value



119901 The












04 06 080

02

04

06

08

1

Cp


Simulation

120582


120582

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D



119901value would



5 Conclusions



0 1|u|UF

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D




0 is set at 119897

0119861 le 0084



0119861 le 0084








Nomenclatures


119865





= (2119892119867119865)12

119881119905 Rotor tip speed = 1205961198632


119905119880119865


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










D=200 120∘

40

Blade250

R23



u119896+1119894

= ulowast119894+ u1015840119894

(10)

r119896+1119894

= rlowast119894+ u1015840119894Δ119905 (11)

where

u1015840119894= minus

Δ119905

120588⟨nabla119901119896+1⟩119894 (12)





119865is 570mm


119865 119871119865119863 is 133






Rotor

Waterfall

xy

z

LF

HF

Rotation

Gravity


Rotor

Waterfall Rotation

Gravity

A

minus125

125

0

0

125minus125

minus(z

minusz a)D

B

(x minus xa)D














A

B

C

(a) 119905 = 1199051[119904]

A

B

C

(b) 119905 = 1199051 + 0005

A

B

C

(c) 119905 = 1199051 + 001

A

B

C

(d) 119905 = 1199051 + 0015

A

B

C

(e) 119905 = 1199051 + 002






minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D


minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

0 1|u|UF


RotationWaterfall

Rotor


006 008 01 012 0140

10000

20000

30000

l0B

N


0when 120582 = 056



119865

334ms




0

0069119861ndash012119861





⟨119899lowast⟩119894lt 1205731198990 (13)





0for the particle







minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

(a) In case of 1198970119861 = 012

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

(b) In case of 1198970119861 = 0069



0is set at 119897

0119861 =









0


0in




0119861 = 0084






0

02

04

06

08

1

Cp

006 008 01 012 014l0B




particles 1198970when 120582 = 056







0119861


0119861 le 0084) the 119862






119901value



119901 The












04 06 080

02

04

06

08

1

Cp


Simulation

120582


120582

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D



119901value would



5 Conclusions



0 1|u|UF

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D




0 is set at 119897

0119861 le 0084



0119861 le 0084








Nomenclatures


119865





= (2119892119867119865)12

119881119905 Rotor tip speed = 1205961198632


119905119880119865


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










A

B

C

(a) 119905 = 1199051[119904]

A

B

C

(b) 119905 = 1199051 + 0005

A

B

C

(c) 119905 = 1199051 + 001

A

B

C

(d) 119905 = 1199051 + 0015

A

B

C

(e) 119905 = 1199051 + 002






minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D


minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

0 1|u|UF


RotationWaterfall

Rotor


006 008 01 012 0140

10000

20000

30000

l0B

N


0when 120582 = 056



119865

334ms




0

0069119861ndash012119861





⟨119899lowast⟩119894lt 1205731198990 (13)





0for the particle







minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

(a) In case of 1198970119861 = 012

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

(b) In case of 1198970119861 = 0069



0is set at 119897

0119861 =









0


0in




0119861 = 0084






0

02

04

06

08

1

Cp

006 008 01 012 014l0B




particles 1198970when 120582 = 056







0119861


0119861 le 0084) the 119862






119901value



119901 The












04 06 080

02

04

06

08

1

Cp


Simulation

120582


120582

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D



119901value would



5 Conclusions



0 1|u|UF

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D




0 is set at 119897

0119861 le 0084



0119861 le 0084








Nomenclatures


119865





= (2119892119867119865)12

119881119905 Rotor tip speed = 1205961198632


119905119880119865


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D


minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

0 1|u|UF


RotationWaterfall

Rotor


006 008 01 012 0140

10000

20000

30000

l0B

N


0when 120582 = 056



119865

334ms




0

0069119861ndash012119861





⟨119899lowast⟩119894lt 1205731198990 (13)





0for the particle







minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

(a) In case of 1198970119861 = 012

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

(b) In case of 1198970119861 = 0069



0is set at 119897

0119861 =









0


0in




0119861 = 0084






0

02

04

06

08

1

Cp

006 008 01 012 014l0B




particles 1198970when 120582 = 056







0119861


0119861 le 0084) the 119862






119901value



119901 The












04 06 080

02

04

06

08

1

Cp


Simulation

120582


120582

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D



119901value would



5 Conclusions



0 1|u|UF

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D




0 is set at 119897

0119861 le 0084



0119861 le 0084








Nomenclatures


119865





= (2119892119867119865)12

119881119905 Rotor tip speed = 1205961198632


119905119880119865


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

(a) In case of 1198970119861 = 012

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D

(b) In case of 1198970119861 = 0069



0is set at 119897

0119861 =









0


0in




0119861 = 0084






0

02

04

06

08

1

Cp

006 008 01 012 014l0B




particles 1198970when 120582 = 056







0119861


0119861 le 0084) the 119862






119901value



119901 The












04 06 080

02

04

06

08

1

Cp


Simulation

120582


120582

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D



119901value would



5 Conclusions



0 1|u|UF

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D




0 is set at 119897

0119861 le 0084



0119861 le 0084








Nomenclatures


119865





= (2119892119867119865)12

119881119905 Rotor tip speed = 1205961198632


119905119880119865


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

02

04

06

08

1

Cp

006 008 01 012 014l0B




particles 1198970when 120582 = 056







0119861


0119861 le 0084) the 119862






119901value



119901 The












04 06 080

02

04

06

08

1

Cp


Simulation

120582


120582

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D



119901value would



5 Conclusions



0 1|u|UF

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D




0 is set at 119897

0119861 le 0084



0119861 le 0084








Nomenclatures


119865





= (2119892119867119865)12

119881119905 Rotor tip speed = 1205961198632


119905119880119865


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1|u|UF

minus125

125

0

0

125minus125

minus(z

minusz a)D

(x minus xa)D




0 is set at 119897

0119861 le 0084



0119861 le 0084








Nomenclatures


119865





= (2119892119867119865)12

119881119905 Rotor tip speed = 1205961198632


119905119880119865


References














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Numerical Simulation of Water...

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