Added Properties in Kaplan Turbine

A Preliminary Investigation

Stina Bergström

Sustainable Energy Engineering, masters level

2016

Luleå University of Technology

Department of Engineering Sciences and Mathematics

Preface

This master thesis was written at the division of Fluid and Experimental Mechanics, De-partment of Engineering Sciences and Mathematics, at Lulea University of Technology(LTU) during the summer of 2016. The problem was sett so I could work with numericalcalculation in MatLab and simulations using ANSYS Workbench. The first part of thisthesis was the mathematics that couples movement from a solid to a fluid which wascompletely new for me and I did not have much experience in work with simulations.The urge of learning new things have inspired and motivated me and this thesis is aresult of that.

I would like to thank my supervisor Michel Cervantes for the chance to work with thissubject and also my co-supervisor Arash Soltani Dehkharqani for all the help with thesimulations and theoretical aspects.

Lule September 30, 2016

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Abstract

A preliminary investigation of the added properties called added mass, added dampingand added stiffness have been performed for a Kaplan turbine. The magnitude of dimen-sionless numbers have been used in order to classify the interaction of the fluid and thesolid. The classification is done to bring clarity in which of the added properties are ofimportance for the system.

The diameter of the runner and the hub have been calculated using the power outputand the head for a Kaplan turbine. These dimensions have been used to determine themagnitude of the dimensionless numbers along with the velocity of the fluid. It turnedout that all added properties affect the turbine, however, the magnitude of them are quitedifferent. The magnitude of the added mass and the added damping are greater than theadded stiffness, which often is neglected.

The added mass can be determined if the natural frequencies of the structure in airand in water are known. The difference in natural frequencies can be used to determinethe added mass factor and thereby the added mass of the system. The added dampingcan be determined by the change in damping ratio for different surrounding fluids. Thiswas done using the simulation software ANSYS Workbench v.17.1, where two differenttypes of simulation were used, ”acoustic coupled simulation” and ”two way coupled sim-ulation”. The complexity of the geometry of the Kaplan turbine was simplified to a discand a shaft. The result for the added mass was validated using results from an exper-iment [1]. The added damping could be determined, but not validated. The differenttypes of simulation have been compared and it turned out that the added mass could bedetermined using ”acoustic coupled simulation” and ”two way coupled simulation”, butthe added damping could only be determined using the ”two way coupled simulation”.

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Sammanfattning

En preliminar undersokning av de adderade egenskaperna kallade, adderad massa, adderaddampning och adderad styvhet har utforts for en Kaplan turbin. Magnituden av dimen-sionslosa tal har anvants for att klassificera interaktionen av fluiden och soliden. Klassi-ficeringen gors for att bringa klarhet i vilka av de adderade egenskaperna ar av betydelsefor systemet.

Diametrarna for lophjulet och navet har beraknats utifran effekt och fallhojd for en Ka-plan turbin. Dessa langder har anvants for att bestamma magnituden av de dimensionlosatalen tillsammans med fluidens hastighet. Det visade sig att alla adderade egenskaperpaverkar turbinen, men omfattningen av dem ar helt annorlunda. Magnituden av denadderade massan och den adderade dampningen ar storre an den adderade styvheten,som ofta forsummas.

Den adderade massan kan bestammas om de naturliga frekvenserna av strukturen i luftoch vatten ar kanda. Skillnaden i egenfrekvenser kan anvandas for att bestamma faktornav den adderade massan och darigeniom den adderade massan. Den adderade dampnin-gen kan bestammas genom andringen i dampningsforhallande for olika omgivande fluider.Detta gjordes med hjalp av simuleringsprogrammet ANSYS Workbench v.17.1, dar tvaolika typer av simulering anvandes, ”acoustic coupled simulation” och ”two way cou-pled simulation”. Komplexiteten i geometrin for en Kaplan turbin forenklades till enskiva och en axel. Resultatet for den adderade massan validerades med resultat fran ettexperiment [1]. Den adderade dampningen kunde bestammas, men inte valideras. Deolika typerna av simulering har jamforts och det visade sig att den adderade massankan bestammas med hjalp av bade ”acoustic coupled simulation” och ”two way coupledsimulation”, men den adderade dampningen kunde endast bestammas med hjalp av ”twoway coupled simulation”.

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Contents

1 Introduction 91.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Fluid-Solid Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Thesis aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Theory 132.1 Design of Kaplan turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Dimensionless numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Velocity of the solid structure . . . . . . . . . . . . . . . . . . . . 172.3 Vibration and frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.1 Acoustic structural coupled simulation . . . . . . . . . . . . . . . 222.5 Two way coupled simulation . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Method 253.1 Numerical calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Design of Kaplan turbine . . . . . . . . . . . . . . . . . . . . . . . 253.1.2 Dimensionless numbers . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.3 Acoustic structural coupled simulation . . . . . . . . . . . . . . . 283.2.4 Two way coupled simulation . . . . . . . . . . . . . . . . . . . . . 29

4 Result 334.1 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.1 Kaplan turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 Dimensionless numbers . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.1 Acoustic structural coupled simulation . . . . . . . . . . . . . . . 354.2.2 Two way coupled simulation . . . . . . . . . . . . . . . . . . . . . 37

5 Conclusion and discussion 43

6 Future work 45

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Chapter 1

Introduction

This master thesis report is a part if the final course E7015T Maters thesis of the masterprogram in sustainable energy engineering, specializing in wind and hydropower.

1.1 Background

Hydropower is an environmentally friendly and efficient energy source. The probabilityto build new hydropower plants is slim in Sweden which makes the maintenance workthe main focus in order to keep the energy distribution on a constant level [2]. Instabilityon the grid, for instance due to the increase in wind power, generates more transientoperating conditions for the hydropower plants. The use of numerical tools have made thedesign of turbine more efficient both in use of material and power generation. However,the vibration of the structure can cause major failure. The increase in power concentrationin the turbines due to upgrades increases the excitation forces on the runner that canlead to fatigue and damages on the blades [3]. It is therefore important to avoid theexcitation frequencies. In oder to do that, the knowledge of the dynamics of a immersedstructure, called fluid-solid interaction (FSI), must be studied. If the natural frequencyof the turbine submerged in water is known in advance, the excitation frequencies can betaken into consideration when the renovation of the turbines are planed and design.A Kaplan turbine is a reaction turbine that operates at low head and has between threeand eight runner blades that are attached to a hub. The inflow is optimized by controllingthe pitch of the runner blades and the turning of the wicket gates. The water enters theturbine in a radial direction and exit with an axial direction, see Figure (1.1). Theflowing water transfers its angular and axial momentum to the blades, which producestorque and rotation. The forces acting on the turbine and also the change in operatingconditions due to the operations mentioned earlier creates vibrations. This can lead tonegative effects on the lifespan of the turbine. When the flow rate increases or decreases,the structure experience high mechanical stresses, which in the long term, can lead tocracks and fatigue. If a piece of the structure was separated from the structure, it couldlead to major damages and inefficiency.

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Figure 1.1: Water flow in a Kaplan turbine. [4]

When studying the interaction of the solid and the fluid, different terms called, addedmass, stiffness and damping arise from the interaction of the flowing water and the runner.With a greater understanding of the effects of the added terms, the operating conditionscan be optimized and the lifespan of the turbine can be enhanced.

1.2 Fluid-Solid Interaction

Fluid-solid interaction can be applied in many different engineering problems. Vibrationsin nuclear reactor due to earthquakes, vibrating hydropower turbines or ship propellersare all common engineering problem where fluid-solid interaction plays a part in the dy-namics of the systems. A fluid-solid interaction can in general be described as a couplingof movement of the solid structure and movement of the fluid domain. It is important notto neglect the affect of the surrounding fluid on the dynamics of the solid structure, sinceit is known that the added mass can reduce the natural frequencies of the structure andthat the damping due to the surrounding fluid is higher than the structural damping.Vibrations caused by the pressure pulsation due to the rotor-stator interaction (RSI),vortex induced vibrations and self-excitating vibrations can all lead to major damagesof the structure and are therefore important aspects to be considered when designing aturbine.

The added properties that arise when a solid structure and a fluid interact have beeninvestigated both analytically and through experiments and later with numerical simu-lations. The dynamics of the interaction have been analytically investigated by E. deLangre [5]. E. de Langre proposed that added properties can be expressed by dimen-sionless governing equations and that the dynamics at the interface can be coupled. Hestated that the forces and velocities acting on the interface are equal in both domain andcan therefore be coupled. The magnitude of the dimensionless numbers in the dimen-sionless governing equations can be used to determine which of the added properties areaffecting a specific cases. This type of investigation was partially performed by Gauthierin 2015 [6] on a Francis turbine where the inertia effects were found to be dominant,which are the added mass and the added damping. The added mass was determined by

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comparing the natural frequencies in vacuum and in water with the use of a acoustic cou-pled simulation along with the mode shapes. The mode shape were exported to performtime depending CFD simulation with a given displacement amplitude. The fluid forcecould be calculated and the stiffness derived. The added stiffness due to the water wasdetermined to a value of about 2% of the structural stiffness. The added stiffness canbe neglected when investigating the dynamics of a structure submerged by still wateraccording to Rodriguez et al. [7]. The added damping in [6] was determined by the useof energy dissipation in the system. The results were obtained using CFX with the useof interpolated mode shapes. However the results could not be fully validated, but theywere found to be sufficient enough for the needs of engineering design.

A similar investigation of the added mass has been performed by Muller [8] on a Francisturbine runner using both experimental data and numerical simulations. The naturalfrequencies of the structure were excited by a impact force in both air and in water. Theresults from the experiments were compared with simulations using acoustic-structuralcoupled simulations in ANSYS and the added mass could be determined experimen-tally and numerically. This type of numerical simulation was also validated by Graf andChen [9] where a comparison of experimental data and numerical simulations was donefor a Francis runner. This type of investigation have also been performed for differentgeometries by Hengstler [10], Hubner [11] and Egusquiza et al. [12]. The added mass canaccording to Hubner et al. [11] and Hengstler [10] be determined with an acoustic struc-tural coupled simulation using modal analysis. Hengstler used the acoustic structuralcoupled simulation to numerically calculate the different frequencies of a disc and shaft,which represented a simplified hydro turbine. The distance from a submerged structureto a rigid wall has an effect on the decrease in natural frequency. This was shown byRodriguez et al. [13] when the natural frequencies of a cantilever plate submerged inwater was investigated for different distances to the rigid wall of a tank. The resultsfrom the acoustic structural coupled simulation were compared with experimental dataand confirmed that the effect of the nearby rigid wall needs to be taken into accountin the simulations. This type of simulation can be used for more complex geometries,like turbines and the effect of the clearance can also be considered when the fluid-solidinteraction is investigated. This was also confirmed by Valentin et al. [14] where a vibrat-ing circular disc was investigated through experiments and acoustic coupled simulations.The results indicated that the radial gap have an impact on the transverse mode shapesbut not on the radial mode shapes.

The results form a acoustic coupled simulation can not fully explain the operating con-dition of a turbine since the fluid only can be considered as still. The added mass fromthese types of simulations is therefore a bit higher than the actual added mass obtainedwith rotation of the disc [15]. In order to take into account the effects of the disk rotation,the two way coupled simulation can be used. Although this type of simulation is moretime consuming, but more accurate information can be obtained.

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1.3 Thesis aim

The aim of this master thesis is to investigate the different proprieties; added mass, addedstiffness, and added damping of Kaplan turbine as a function of the specific speed. Themain focus of this thesis is to evaluate the added properties using dimensionless numbersfor a Kaplan turbine. Acoustic structural coupled simulation and two way simulations areperformed in order to determine the magnitude of the added mass and added dampingfor a simplified turbine runner represented by a disc and shaft.

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Chapter 2

Theory

The presence of a fluid in a Kaplan turbine adds properties to the system, as added mass,added stiffness, and added damping. To perform an accurate rotor-dynamic analysis,these added properties need to be known. The interaction can be classified using themagnitude of dimensionless numbers.

2.1 Design of Kaplan turbine

To investigate the added properties, the velocity of the working fluid and the diametersof the runner and the hub of the turbine must be determined. The variables used todetermined these parameters are listen in Table (2.1). The equations following the tableare used to calculate the diameters mentioned and the velocity of the fluid.

Variable Symbol ValuePower output P 42 [MW ]

Head H 37 [m]Density, water ρw 1000 [kg/m3]

Hydraulic efficiency ηh 95[%]Gravity g 9.81[kg/s2]

Table 2.1: Design parameters

This parameters were chosen because the displacement of the shaft of this Kaplanturbine was examined by M. Nasselqvist [16]. The displacement of the shaft will be usedfor the dimensionless numbers in the next section of this chapter. The design calculationswhere preformed according to [17].

Q =P

Hηhρwg(2.1)

The net head, Hn is calculated using equation (2.2) below, which is used to calculatethe specific speed, Ns. The specific speed was then used to calculate the speed of therunner, N and the diameter of the runner, Dr and the hub, Dh. These steps are shownin equation (2.3)-(2.6) and are made according to [17].

Hn = Hηh (2.2)

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Ns =2.294

H0.486n

(2.3)

N = Ns(Hng)3/4

Q1/2(2.4)

Dr =84.5(0.79 + 1.602Ns)H0.5

n

60N(2.5)

Dh = (0.25 +0.0951

Ns

)Dr (2.6)

The area where the water is flowing can be calculated using the diameters of the runnerand hub, see equation (2.7).

A = π((Dr

2)2 − (

Dh

2)2) (2.7)

The axial velocity of the fluid at the inlet can be calculated using the flow rate and thearea where the water flows, see equation (2.8).

Vn =Q

A(2.8)

The relative velocity of the fluid is of interest when the dimensionless number for theinteraction of the fluid and the runner blade is investigated. To determine the relativevelocity, W the velocity triangle at the inlet is used, see Figure (2.1).

Figure 2.1: Velocity triangle. [17]

The tangential velocity component of the fluid, Vt of the fluid can be determined usingequation (2.9).

P = ρwωQ(rrVt − rhV1,t) (2.9)

The tangential velocity of the fluid at the inlet can be determined by assuming that thereis no swirl at the outlet, V1,t = 0 giving equation (2.10).

Vt =P

ρwωQrr(2.10)

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where rr is the radius of the runner, rr = Dr/2 and ω is the rotational velocity ofthe runner, ω = 2πN . The tangential velocity of the runner, U is calculated usingequation (2.11).

U = ωrr (2.11)

The relative velocity of the fluid is calculated using equation (2.12)

W =

√V 2n + (U − Vt)2 (2.12)

Excitation of the runner can occur when the rotor-static interaction coincide with thenatural frequencies of the runner. The frequency depends on the pressure pulsationsoccurring when the runner blade passes a guide vane and is therefore a function of thenumbers of guide vanes, Zg and the rotational speed of the runner, N , see equation (2.13).

fs,r = ZgN (2.13)

The number of guide vanes is assumed to be 28 for this case. The vibrating frequency ofthe shaft have been found to be a factor of 2.4 times the rotational speed of the runneraccording to M. Nasselqvist [16], see equation (2.14).

fs,s = 2.4N (2.14)

The result from these calculations will be used for the dimensionless numbers, which arementioned later in this chapter.

2.2 Dimensionless numbers

The interaction of a fluid and solid can according to de Langre [5] be classified usingdimensionless numbers. The most useful dimensionless numbers are the reduced velocity,UR and the displacement number, D. The mass number, M and Reynolds number, RE arealso useful to determine the importance of the added mass. Moreover, the inertia in theflowing fluid and the decrease of the natural frequencies are governed by the magnitudeof M . The inertia in the fluid increases with the value of RE . The mass number isthe ratio of the densities of the fluid and the solid, see equation (2.15) and the Reynoldsnumber is presented in equation (2.16).

M =ρfρs

(2.15)

RE =ρfUFL

µ(2.16)

The displacement number is the ratio between the displacement of the structure, x0 andthe characteristic length, L, see equation (2.17). The greater the value of D is, the largeris the deformation of the structure.

D =x0L

(2.17)

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In this thesis the diameter of the shaft and diameter of the runner will be used for thecharacteristic length.

The reduced velocity is the ratio of the time scale of the solid domain, TS and thetime scale of the fluid domain, TF , see equation (2.18).

UR =TSTF

(2.18)

The time scales depend on the characteristic length, L and the characteristic velocitiesof the fluid and the solid domain, W respectively US.

TF =L

W(2.19)

TS =L

US(2.20)

Depending on the magnitude of the reduced velocity, different conditions can be expected.

A small value of the reduced velocity indicates that the velocity of the fluid does notaffect the movement of the solid structure, so it can be neglected. Therefore, the sys-tem can be expressed using the velocity of the solid structure. This state is shown inFigure (2.2), where the displacement of the structure and the fluid over time is shown.

Figure 2.2: Fluid and solid time scales for a small UR [6].

The realistic relation between UR and D for this case would be UR << D, where bothD << 1 and UR << 1 and can be used to determine the added mass of the system. Theadded damping and added stiffness would be neglected if this relation was to be obtained.

If the reduced velocity were to be larger than one, UR >> 1 then the velocity of thesolid could be neglected since the change in the time scale would be slow for the solidand fast for the fluid. This state is presented in Figure (2.3).

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Figure 2.3: Fluid and solid time scales for a large UR [6].

The relation between the reduced velocity and the displacement number would be,UR >> D. In this state both added damping and stiffness occur.

If the two time scales are similar and the order of magnitude of the reduced velocityis UR = O(1) a state called pseudo-static condition occur and the relation of U2

R >> Dmust be obtained. The time scales are shown in Figure (2.4). The coupling is strongwhen the time scales are similar and the acceleration at the interface can be neglected,which makes it the minimal state where added damping can be observed [6].

Figure 2.4: Fluid and solid time scales for UR = O(1) [6].

The order of magnitude of the dimensionless numbers are often used to determinewhich state occur at the interface an how they relate to each other.

2.2.1 Velocity of the solid structure

The characteristic velocity of the solid can be chosen depending on which movement ofthe structure is of interest. The vibration of the structure can be a good choice when theforced induced vibrations of the structure is investigated. If the mode of the structure is ofinterest, the wave propagation could be used for the solid velocity. The wave propagationvelocity depends on the molecules’ ability to move in the material, a high density andhigh tension give a lower velocity. The equation for the wave propagation velocity if

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shown in equation (2.21)

US,w =

√E

ρs, (2.21)

where E is the Young’s modulus and ρs is the density of the material used for the solidstructure. The solid velocity due to the forced vibration is shown in equation (2.22).

US = fsL, (2.22)

where fs is the frequency of the vibration occurring and L is the characteristic lengthand as mentioned earlier this length was chosen to be the diameter for either the runneror the shaft.

2.3 Vibration and frequency

A system undergoing deformations in a periodic manor can be referred to as an oscillatingor vibrating system. The system is then storing kinetic and potential energy, as inertiaand stiffness, and losing energy which can be referred to as the damping. When a spring-mass-damper system is set in motion some of the energy will dissipate for each cycle ofoscillation and the magnitude of the displacement will decrease until it finally stops. Thismeans that the surrounding medium offers damping for all vibrating systems [18].

There are different types of vibration, free vibration and forced vibration. If a struc-ture is set in motion by an applied force and vibrate without any other involvement itcan be said to be a free vibrating system. However if the structure would to be affectedby a repeating force so that the movement of the structure is affected multiple times, thesystem can be said to be a forced vibrating system [18]. If the external force coincidewith the natural frequencies of the structure a state called resonance can occur [18]. Thiscan generate vibrations with large amplitudes in the structure which can lead to cracksand fatigue. This state is crucial to avoid for all kinds of machinery, such as hydropowerturbines, to insure that the lifespan is not shorted.

The stiffness of a vibrating system can be described as a spring, which is the deflec-tion of the structure due to an applied force and it has a constant value. The stiffnessforce is shown in equation (2.23). The mass of the structure can gain and lose kineticenergy when the structure accelerates or decelerate. The mass force of the structure isequal to the acceleration times the mass of the structure. This is derived from the secondNewton law and the work done on the structure is stored as the kinetic energy of themass [18]. The mass force is presented in equation (2.24)

FK = ksx (2.23)

FM = msx (2.24)

Where ks is the stiffness constant, x is the displacement of the structure, ms is the massof the structure and x is the acceleration of the structure.

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If a system is undamped the amplitude of the oscillations will remain the same or evenincrease, which means that the vibration will continue to infinity [18]. This is a theoreticstate and will not occur in reality since there always are some form of energy dissipation.However, the energy dissipation in air is often considered equal to vacuum. The potentialenergy in the system is stored in the spring and remains constant for a free vibratingsystem in vacuum since the magnitude of the displacement remains constant. The equa-tion of motion of a free vibrating system only take the mass and stiffness in considerationsince there is no or very little energy dissipation to the surrounding or in the material,see equation (2.25).

msx+ ksx = 0 (2.25)

The natural frequency of the system can be determined using equation (2.26), if the massand the stiffness is known.

fn =1

2π

√ksms

(2.26)

Where n denotes the number of degrees of freedom and thereby the number of naturalfrequencies. The number of natural frequencies of a system is governed by the numberof degrees of freedom, which are the number of possible directions of movement [19].

The damping of a system is the energy dissipation due to gravity, friction forces orgeneration of heat. The damping force can be described used equation (2.27), where csis the structural damping and x is the velocity of the vibrations of the structure.

FC = csx (2.27)

By adding the mass force, stiffness force and the damping force the equation of motionfor the vibrating structure can be expressed, see equation (2.28), where Fs(t) is the forceapplied on the structure. If Fs(t) = 0 the system can be said to be a free vibratingsystem.

msx+ csx+ ksx = Fs(t) (2.28)

The equation of motion can also be expressed with the characteristic motion equation,see equation (2.29).

x+ 2ξωnx+ ω2nx =

Fs(t)

ms

. (2.29)

Where ξ is the damping ratio and ωn, is the angular velocity of the oscillation and isequal to

√ks/ms. The damping ratio is the ratio of the critical damping constant and

the actual damping constant of the oscillation system [19].

ξ =cscc

=cs

2√ksms

(2.30)

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The oscillation is critically damped when ξ = 1, undamped when ξ = 0 and under dampedif ξ < 1 [18]. A way to determine the damping is to use the logarithmic decrement wherethe change in amplitude after a number of time periods is used, see equation (2.31)

δ =1

Nln(

A1

AN) (2.31)

where N is the number of elapsed time periods and A is the amplitude of oscillation. Ifthe rate of decay increases, amplitudes decrease more rapidly [19]. The rate of decay canalso be expressed as equation (2.32) [19].

δ =2πξ√1− ξ2

⇔ ξ =1√

1 + (2πδ

)2(2.32)

When the damping ratio ξ is known the damping constant of the oscillation, cs can bedetermined.

When an object is surrounded by a fluid, an added fluid force occurs and the equation ofmotion for the vibrating structure can be describes as equation (2.33).

mx+ cx+ kx = Fs(t) + Ff (t) (2.33)

The fluid force would affect the system even if it was a free vibrating system, since itis a medium that surrounds the structure. The fluid force can be separated into threedifferent forces, the force of added mass, the force of added damping and the force ofadded stiffness, see equation (2.34).

Ff = Fa + Fc + Fk (2.34)

Where Fa is the added mass force, Fc is the added damping force and Fk is the addedstiffness force.

The added mass force, Fa can be considered as part of the inertia in the fluid thataffects the solid structure when it moves. When the solid object moves the surroundingfluid also does, since the fluid and the solid can not take up the same space at the sametime. The fluid that the solid must push away can be described as the added mass. Theadded mass always occur when a structure is submerged in a fluid. The added mass forceis expressed in equation (2.35)

Fa = −max, (2.35)

where ma is the added mass.

The added stiffness force, Fk occurs when the displacement of a solid object generates apressure change in the surrounding fluid. The stiffness force is expressed in equation (2.36)and ka is the added stiffness constant.

Fk = −kax (2.36)

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From the solid point of view, the condition of added stiffness is like being connected to afluid spring [5]. The difficulty of calculating the added stiffness depends on the geometry.If the geometry is complex then the problem will be difficult to solve.

The added damping force, Fc is proportional to the viscosity in the surrounding fluid.The energy absorption and damping effect in water is much higher than in air. The effectof added damping and the added damping force can be expressed as equation (2.37)

Fc = −cax (2.37)

where ca is the added damping factor.

By combining the added fluid forces from equation (2.35) to equation (2.37) in the equa-tion of motion in equation (2.33) the following equation (2.38) can be expressed.

(ms +ma)x+ (cs + ca)x+ (ks + ka)x = 0. (2.38)

The added terms will lead to changes in the natural frequency of the structure and thedamping of the oscillations. The natural frequency of the submerged structure can beexpressed as equation (2.39)

fa =1

2π

√ks + kams +ma

, (2.39)

and the change in damping ratio can be expressed as equation (2.40)

ξa =cs + ca

2√

(ks + ka)(ms +ma). (2.40)

The changes in natural frequencies and in damping ratio can be used to determine themagnitude of the added mass and added damping. The changes can be determinedthough experiments or simulations using different surrounding fluids. The added massfactor can be calculated using equation (2.41) and the added damping can be calculatedusing equation (2.42).

α =fwaterfair

(2.41)

β =ξwaterξair

(2.42)

The magnitude of the added mass and the added damping can now be determined usingthe natural frequencies and damping ratio in different fluids.

2.4 Numerical simulation

In this thesis two types of simulation are compared. The acoustic coupled simulationwhere the pressure wave in the fluid is used to determine the behavior of the structurein the fluid, and the second simulation type is the two way coupled simulation where thedisplacement of the mesh of the interface is used to determine the fluid force and thedisplacement of the structure.

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2.4.1 Acoustic structural coupled simulation

A numerical simulation can be used when investigating the affect of surrounding waterof a runner. To couple the behavior of the structure to the fluid, an acoustic structuralcoupling simulation using ANSYS Workbench v.17.1. can be performed. The struc-tural dynamic equation in matrix notation used in the mechanical solver is presented inequation (2.43) [3].

[Ms]{u}+ [Cs]{u}+ [Ks]{u} = {Fs} (2.43)

Where u denotes the displacement of the structure and Fs is the load vector. Ms rep-resents the structural mass matrix, Cs, represents the structural damping matrix andKs denotes the structural stiffness matrix. When a structure is vibrating freely the loadvector Fs is the fluid load due to the pressure distribution over the interface [13]. Thepressure distribution is determined with the use of the acoustic wave equation, shownin equation (2.44). The fluid is considered to be non-flowing, irrotational, inviscid andslightly compressible.

∇2P =1

c2∂2P

∂t2(2.44)

The Laplace operator is represented by ∇2 and P is the pressure in the fluid. The speedof sound in the fluid is represented by c in the equation. Equation (2.44) can be writtenas the following equation (2.45).

1

c2∂2P

∂t2− {L}T ({L}P ) = 0 (2.45)

The displacement of the fluid and of the structure at the interface must be equal. Thismakes it possible to express the following equation for the momentum equation, wherethe relation between the normal acceleration of the structure and the normal pressuregradient in the fluid is shown.

{n}{∇P} = −ρf{n}∂2U

∂t2(2.46)

By using the finite element shape function for the displacement, u and the pressure,p and through integration of the pressure the dynamics of the fluid can be written asequation (2.47)

[Mf ]{p}+ [Cf ]{p}+ [Kf ]{p} = {Fsf}. (2.47)

Where the Fsf is the load vector at the interface due to the pressure in the fluid [3]. TheEquation (2.47) and equation (2.43) can now be written together as equation (2.48).{

Fs(t)0

}=

[Ms 0Mfs Mf

]{up

}+

[Cs 00 Cf

]{up

}+

[Ks Kfs

0 Kf

]{up

}(2.48)

Where Mfs is the coupling mass matrix and [Kfs] is the coupling stiffness matrix at theinterface and occur when a structure is immersed in a fluid. The coupling matrixes are

22

dependent on the shape function of the pressure and the displacement. The structuralelements are all the variables denoted with s and the fluid elements are all the variablessubscribed with f including the coupling variables. The natural frequencies are calculatedusing the Lanczos algorithm. The mechanical solver that was used in the simulations wasthe Modal analysis. This mechanical solver can only solve the natural frequency of anon-moving solid structure and acoustic body.

2.5 Two way coupled simulation

In a two way coupled simulation a fluid solver and a mechanical solver are coupledusing ANSYS Workbench v.17.1. In this thesis the structure is solved using TransientStructural, which is a mechanical solver and the fluid is solved using Fluid F low (CFX).The fluid solver calculates the changes in mass, mass flux and the mesh deformation inthe domain [20]. These three terms are shown in equation (2.49).

d

dt

∫V

ρ dV +

∫ρUj dnj +

∫S

ρWj dnj = 0 (2.49)

The mechanical domain is analysed using the finite element approach shown in equa-tion (2.50).

Mu+ Cu+Ku = F (2.50)

The fluid dynamics and solid dynamics are solved separately, but at the interface thedynamics of the two physically different systems are coupled. The solution from CFXand the mechanical solver are shared at the interface by using a boundary called fluid-solid interface. At the interface, the displacement of the structure is transferred fromthe mechanical solver to the fluid solver. The fluid force acting on the structure is alsotransferred from CFX to the mechanical solver at the same interface. These steps areiterating for each time step for the simulation until a convergence is reached [20]. Thetransferred data is interpolated between the mesh of the structure and the mesh of thefluid. When using a two way coupled simulation, the added mass and the added dampingin a system can be investigated, which was not the case for a acoustic coupled simulation.However, the result from the acoustic simulation can be used to determine the samplingfrequency and time steps. The Nyquist theorem states that the sampling frequency mustbe at least twice the size of the highest frequency of interest, see equation (2.51).

fs > 2fc (2.51)

The frequency fc represents the highest interesting frequency and fs represents the samplefrequency that should be used for the two way coupled simulation. The time step canthen be calculated using equation (2.52).

∆t =1

fs(2.52)

The nodes at the interface are connected as springs with varying stiffness. The quality ofthe mesh remains high during deformation due to the increase in stiffness at nodes close

23

to a boundary [20]. The natural frequencies of a structure can be obtained by applyingan impact force making the structure vibrate and monitoring the deformation over time.A Fast Fourier Transformation analysis, FFT-analysis can then be performed using theresults from the simulations to determine the natural frequencies of the structure in airand in water. The amplitude of the displacement is used to determine the magnitude ofthe damping.

24

Chapter 3

Method

3.1 Numerical calculation

In order to evaluate the magnitude of the dimensionless numbers used for the classificationof the interaction, the characteristic length of the runner and shaft must be calculatedalong with the velocity of the flowing fluid. These numerical calculations have beenperformed according to the equations in chapter 2 and are presented in this chapter.

3.1.1 Design of Kaplan turbine

The variables used for the calculation for the diameter of the runner and the hub arepresented in Table (2.1). The values were chosen so that the displacement of the shaftfrom [16] could be used for the dimensionless numbers. The equations from section 2.1were used to calculate the diameters and the fluid velocity. The values of the relativefluid velocity and the diameter on the runner, W and Dr will be used for the dimen-sionless numbers for the runner. The diameter of the shaft is assumed to be the sameas the diameter of the hub, Dh and will therefore be used for the investigation of thedimensionless numbers, UR and D.

3.1.2 Dimensionless numbers

The dimensionless numbers are determined using the equations mentioned in chapter 2 forthe different velocities of the solid structure using the diameters and velocities calculatedfor the Kaplan turbine. The relation of the dimensionless number will determine whichclassification the interaction has and which added properties are affecting the behaviorof the structure. The numerical values for the different variables used to calculate thedimensionless numbers will be used to determine the order of magnitude of the reducedvelocity, UR. The other dimensionless numbers will also be determined, however thenumerical value of, D, M and, RE are interesting. The displacement of the runner bladesdue to the mode shape, x0,rb is assumed larger than the displacement of the shaft andhave a magnitude of about 10−3 [m].

25

3.2 Numerical simulation

The effect of the flowing water in a turbine is difficult to determine due to the complexityof the system and would take incredible amount of time to solve manually. A tool oftenused in engineering is numerical simulations were the complex solutions of a movingsystem is solved by computer power. This is a useful tool when the fluid-solid interactionis investigated in a system. With the right boundary condition the results from numericalsimulation represent the reality well. The runner was simplified to a disc and a shaftsubmerged in a cylindrical fluid domain. The simulations are compared to experimentaldata from [1] in order to determine the accuracy of the simulations.

3.2.1 Experimental setup

An experimental investigation of the natural frequencies of a submerged structure hasbeen performed by J. Hengstler and J. Dual [1]. The results from the experiment will beused to validate the results from the acoustic coupled simulation. A disc was excited inboth air and in water by a coil interacting with a magnet attached to the disc and thevibrations was measured with a laser scanning vibrometer. In Figure (3.1) the schematicsof the setup is shown.

Figure 3.1: Experimental Setup [1].

3.2.2 Geometry

The same geometry is used for both the acoustic coupled simulation and the two waycoupled simulation. The only difference on the geometry is the mesh. The mesh for theacoustic coupled simulation must have matching nodes at the interface and the mesh of the

26

two way coupled simulation does not [21]. The two way coupled simulation interpolatesbetween the nodes and the nodes must therefore be separate [20]. The parameters usedfor the geometry of the solid body and the fluid body are presented in Table (3.1) andare the same as for the experimental setup.

Radius,shaft Rs 15 [mm]Length of shaft hs 200 [mm]

Radius,disc R 190 [mm]Thickness,disc hd 2 [mm]

Radius, fluid domain r 200 [mm]Upper height h1 100[mm]Lower height h2 100[mm]

Table 3.1: Parameters for the geometry

The solid body is presented in Figure (3.2) and the green surface shows where the shaftis fixed. For the acoustic coupled simulation the boundary condition called Displacementsupport is used. The walls are therefore not deformed. This makes all degrees of freedomexcept the pressure equal to zero. The pressure must be a degree of freedom since it isused to couple the two domains [22]. The displacement in X-, Y-, and Z-axis was setto zero so that the upper surface of the shaft was fixed in space. When using the twoway coupled simulation the boundary condition called Fixed support is used, where alldegrees of freedom are set to zero.

Figure 3.2: Geometry of solid body and surface where support boundary is applied.

The fluid body used in the simulations is presented in Figure (3.3).

27

Figure 3.3: Geometry of the fluid body.

The fluid-solid interface is highlighted in Figure (3.4) and the boundary condition isapplied for both simulation types.

Figure 3.4: Fluid-solid interface boundary condition.

3.2.3 Acoustic structural coupled simulation

To be able to use an acoustic structural coupled simulation an extension to ANSYS,called ACT, must be installed and used in the Modal analysis solver in order to obtainthe natural frequencies in air and water. When this is done, the acoustic boundary con-dition can be used for the fluid body and the speed of sound in the fluid and the densityof the fluid must be specified for this boundary condition. The air surrounding the discand shaft has a very small effect on the natural frequency so the fluid can be suppressed,

28

hence, the simulation is performed in vacuum. The speed of sound, c = 1430 [m/s] anddensity 1000 [kg/m3] are used for the simulation in water.

The Displacement support boundary condition is applied on the walls of the fluid bodyin Figure (3.3) in order to illustrate the walls of a container [23]. As mentioned earlierthe nodes must match when using this type of simulation. This is easily done by mergingthe bodies in the geometry builder to one part with multiple bodies. This will make thenodes match when the mesh generates in the Modal analysis.

3.2.4 Two way coupled simulation

In this type of simulation the fluid and structure are solved in separate solvers. Themechanical solver calculates the displacement of the solid body, which is transferred tothe fluid solver via the fluid-solid interface. The fluid solver calculates the fluid forcedue to the displacement, which is transferred from the mechanical solver. The data isinterpolate between the nodes at the interface since the nodes are not exactly matched.The same geometry as used in the acoustic coupled simulation was used in the two waysimulation. The mechanical solver used for this simulation was Transient Structuraland the Fluid F low (CFX) was used for the fluid. In Figure (3.5) the project setup isshown.

Figure 3.5: The coupling link for the simulations.

The line between the geometry cells shows that the geometry is shared between thetwo solvers. The line between the setup cells shows the link where the data transfer oc-curs. the green check marks indicate that the mesh and setup arrangement are correctlyassigned and ready for simulation. The solid body and fluid body are meshed separatelyin this type of simulation. The fluid can have some velocity as well as the structure,which can be useful when simulating a rotating system. However, in this thesis, the fluidwill be still since the results from the acoustic coupled simulation and the results formthe two way simulation will be compared.

The time step from the simulation was set so that the highest interesting frequencycould be captured, which is determined by the use of the Nyquist theorem in chapter 2.The results from the acoustic coupled simulation was used to determine the suitable timestep for the two way coupled simulation. The displacement of the mesh at the interfacewas used in a FFT-analysis to determine the natural frequencies. The FFT-analysis wasperformed in CFX − post where the results from the simulation can be examined. The

29

displacement of the interface over time can be used to determine the change in dampingwhen the fluid is changed from air to water. This can not be done in the acoustic coupledsimulation [10].

In order to excite the disc a force was applied for a number of time steps. Two dif-ferent types of forces were tested in the simulations. First a distributed load on theentire upper surface of the disc and the second force was a point force applied close tothe edge of the upper surface, see position of MP1 − 4 in Figure (3.6). The time stepsand the magnitude for the forces used in the simulations is presented in Table (3.2).

Time [s] Force [N]0.00 0

0.0102 1000.0153 1000.051 01.02 0

Table 3.2: Impact force on the disc

The end time for the simulation was chosen to 1.02 [s] so that the vibrations could beclose to fully damped and the end time must be a factor of the time steps. When per-forming a two way coupled simulation it is important to turn off the Auto time stepping,the data can otherwise be interpolated in a unsuitable manor and the result will bewrong. The interface between the solid and the fluid was defined by the Fluid − SolidInterface boundary condition and data transfer will occur at this surfaces. The setupfor the Transient structural mechanical part is now done and can be coupled to thesetup of Fluid F low (CFX). It is important that the setup cell in Transient is updatedand have a green check mark in order to share the correct dat.ds file, which is the datatransfer file, before the setup for the fluid starts.

The geometry of the structure is suppressed in Fluid F low (CFX) so that only thefluid body is present and meshed. The analysis settings for CFX starts with selectingthe ANSY S MultiF ield coupling analysis type and apply the simulation time and thetime steps, which should be the same as or smaller than the time step for the mechanicalsolver. The Transient analysis type was also chosen. The mesh deformation was acti-vated to make it possible to couple the two domains.

The Wall boundary condition was applied on the outer surfaces of the fluid body alongwith the surfaces which are connected to the Fluid-Solid Interface of the solid struc-ture. In order to transfer data from the fluid to the structure the mesh motion must beset to ANSY S MultiF ield for the surfaces of the interaction.

A number of monitor point was applied on the interface of the disc to be able to measurethe displacement in order to determine the natural frequencies of the structure. TheTotal Mesh Displacement should be used, where the measured displacement is alwayscompared with the original mesh for each time step. A total of 32 monitor points were

30

placed on eight different nodes on the disc to monitor the displacement in X-, Y- andZ-direction and also the total displacement. The position of the points are presented inFigure (3.6).

Figure 3.6: Position of the monitor points.

Four monitor points are used for each node and they have different directions. Eachnode shown in Figure (3.6) have four monitor points, which are measuring the displace-ment in X-, Y- and Z-direction and the total displacement. The point force used in thesimulation is applied in the same node as the monitor point, MP1−4. All monitor pointsand the directions are listed in Table 3.3.

Direction Monitor pointX 1, 5, 9, 13, 17, 21, 25, 29Y 2, 6, 10, 14, 18, 22, 26, 30Z 3, 7, 11, 15, 19, 23, 27, 31

Total 4, 8, 12, 16, 20, 24, 28, 32

Table 3.3: Direction of monitor points.

The results from the simulations and the added mass and added damping factors willbe presented in chapter 4. The acoustic coupled simulation will be compared with thetwo way coupled simulation.

31

32

Chapter 4

Result

4.1 Numerical calculations

The values from the calculations for the Kaplan runner and the dimensionless numbers arepresented in this section. The relation of the dimensionless numbers are also presented.

4.1.1 Kaplan turbine

The results form the calculations using equation (2.1) to (2.12) for the Kaplan runnerare presented in Table (4.1).

Flow rate Q 122.05 [m3/s]Net head Hη 35.15 [m]

Specific speed Ns 0.407 [Hz]Rotational speed N 2.95 [Hz]Diameter, runner Dr 4.09 [m]

Diameter, hub Dh 1.98 [m]Area A 10.04 [m2]

Axial fluid velocity Vn 12.15 [m/s]Tangential fluid velocity Vt 9.12 [m/s]

Velocity of runner U 37.81 [m/s]Relative fluid velocity W 31.16 [m/s]

Table 4.1: Properties of a Kaplan runner.

The values for the frequencies of vibration for the runner blades and the shaft usingequation (2.13) and equation (2.14) is presented in Table (4.2). The velocity of thevibrations calculated using equation (2.22) for the shaft and the runner blades and thediameters, Dr and Ds are also presented in the table below. The value of the velocityof the wave propagation in the solid material using equation (2.21) is also presented insame table.

33

Runner blade fs,r 82.48 [Hz]Shaft fs,s 7.08 [Hz]

Velocity of runner blade vibration Us,rb 337.34 [m/s]Velocity of shaft vibration Us,s 14.02 [m/s]

Velocity of wave propagation Us,w 5047.5 [m/s]

Table 4.2: Frequencies of vibration and velocities of solid structure

The values from the Table (4.2) above are used for the order of magnitude of thedimensionless numbers used to classify the state of fluid-solid interaction of the Kaplanturbine.

4.1.2 Dimensionless numbers

The order of magnitude of the variables calculated for the Kaplan turbine in previoussection are used for the displacement number, D and the reduced velocity are presented inTable (4.3). Equation (2.22) can therefore be used for two different vibrating movements,the vibration for the shaft and the variation of the runner. The vibrations of the shaft isrelated to the magnetic pull due to the rotation of the generator and the runner vibrationsare related to the well known rotor-stator interaction phenomenon.

Relative velocity W O(101)Shaft Runner blades

Displacement x0,s O(10−4) x0,rb O(10−3)Characteristic length Ls O(100) Lr O(100)

Solid velocity US,s O(101) US,rb O(101)Displacement number Ds O(10−4) Dr O(10−3)

Reduced velocity UR,s O(1) UR,rb O(1)

Table 4.3: The order of magnitude for the displacement and frequency for the shaft andthe runner blade for a Kaplan turbine.

The order of magnitude of the reduced velocity is for both the shaft and the runner,UR = O(1) and the relation between the two dimensionless numbers, U2

R >> D is fulfilledso the pseudo-static condition can be said to occur for both the runner and the shaft.This means all the added properties are of significance in the system. The added stiffnessis related to the force acting on the structure due to the velocity of the fluid, the addeddamping is related to the viscosity of the surrounding fluid, and the added mass is anadded term that occur when a solid structure and a fluid interact. It can therefore besaid that all the added properties can more or less affect the system. In earlier workby Gauthier the added stiffness was found to have a value of 2 % of the stiffness of thestructure for a Francis. The aim of this thesis is not the determine the value of the addedproperties, but to investigate which of them can be of importance for Kaplan turbine.It can therefore be assumed that the added stiffness is not of importance due to the lowchange in the Francis turbine. The magnitude of the added mass and added damping aretherefore more important for the investigation of interaction in Kaplan turbine.

34

The order of magnitude for the reduced velocity of the wave propagation was determinedto UR,w = O(10−2) and when comparing with the displacement numbers in Table (4.3)the relation can be described as, UR >> D However, the values for both the displace-ment number and reduced velocity are smaller than one, D << 1 and UR << 1. Thisrelation has not been investigated by de Langre [5]. Gauthier states that this relation isnot realistic, but no more is mentioned. However it might be interesting to use this solidvelocity when the mode shape of the structure is investigated.

The dimensionless number, M , have a fixed value of M = 0.127 for the different ve-locities of the solid. The values for the Reynolds number are presented in Table (4.4).

Hub RE,h 2.4*107

Runner blades RE,rb 3.7*107

Table 4.4: Reynolds number for the flowing water.

The values in Table (4.4) show that the flow is turbulent and that inertia is high inthe fluid. This indicates that the added mass and added damping affect the solid bodysince they are both part of the inertia forces.

4.2 Simulation

The result from the different simulations are presented and compared in this section.The values for the acoustic coupled simulation is also compared with the result from theexperimental investigation by J. Henstler and J. Dual [1]. The added mass factor, addeddamping factor and change in natural frequencies and damping are presented.

4.2.1 Acoustic structural coupled simulation

From the modal analysis, the natural frequencies of the disc in air and in water aredetermined. In Table (4.5), the values obtained from the simulations are presented. Themode shapes of interest are the first, second, third and fourth nodal diameter, ND1,ND2, ND3 and ND4. From the experimental results, the nodal diameter for ND2,ND3 and ND4 are used for the validation.

35

Experimental resultsNodal diameter Frequency in air [Hz] Frequency in water [Hz]

ND1 − −ND2 72.97 32.02ND3 169.4 81.64ND4 296.3 152.97

Simulation resultsNodal diameter Frequency in air [Hz] Frequency in water [Hz]

ND1 41.15 15.01ND2 73.62 31.61ND3 166.04 80.21ND3 290.88 152.31

Table 4.5: The natural frequencies of the disc from the acoustic structural coupled sim-ulation compared with the experimental results from [1].

The first nodal frequency is presented so that the results from the two way coupledsimulation can be fully compared with the results from the acoustic coupled simulations.The frequency for ND1 could help to identify the mode shapes in the FFT-analysis sincethe mode shapes cannot be shown visually. The second nodal diameter, ND2 is shown inFigure (4.1a) and the third and forth nodal diameter, ND3 respectively ND4 are shownin Figure (4.1b) and Figure (4.1c).

(a) ND2 (b) ND3 (c) ND4

Figure 4.1: The nodal diameters from the acoustic coupled simulation.

Comparing the results in Table (4.5), shows that the differences in frequencies forthe mode shapes are small. This could be due to the quality of the mesh however thedifference is so small that the results from the simulations can be considered as validated.

The added mass factor for the results form the simulations and the results from theexperimental values from [1] have been calculated using equation (2.41) from chapter 2and the values are presented in Table (4.6). Where sim denotes the added mass factorfrom the simulation and exp denotes the added mass factor from the experiment.

36

Nodal diameter αsim [−] αexp [−] Difference [%]ND1 0.36 − −ND2 0.43 0.44 2.27ND3 0.48 0.48 0.00ND4 0.52 0.50 4.00

Table 4.6: Added mass factor for simulation and experimental results.

The magnitude of the added mass factors increase for higher frequencies, which canindicate that they are sightly less affected by the surrounding water.

4.2.2 Two way coupled simulation

The natural frequencies from the two way coupled simulation is presented in this section.Monitor point MP17 was chosen to present the result, since peaks close to the frequen-cies from the acoustic coupled simulation could be found. The FFT-analysis for monitorpoint, MP17 in air is presented in Figure (4.2) and the FFT-analysis in water is presentedin Figure (4.3). The dashed vertical lines show the frequencies from the acoustic coupledsimulation and the solid vertical lines shows the peak for the frequencies from the twoway coupled simulation.

Figure 4.2: FFT analysis of the displacement in air.

The analysis of the peaks in an FFT-analysis is not an easy task. Some peaks can bea multiple of a frequency, which makes the frequency repeat itself at higher values. Thisis the case for the prominent peaks at 85 [Hz] in Figure( 4.2). The small peak slightlyto the left is therefore chosen.

37

Figure 4.3: FFT analysis of the displacement in water

The FFT-analysis for the disc submerged in water is not easy to interpret. The peaksare therefore chosen close to the values fro the acoustic coupled simulation.

The values of the solid vertical lines for the disc in air and in water is presented inTable 4.7. The difference is greater for the frequencies of the disk in water. The peaks inFigure (4.3) are more difficult to find and the mode shape could be changed comparedto the mode shape in air.

Frequency [Hz]Air Water

f1,a 43 f1,w 17f2,a 73 f2,w 36f3,a 169 f3,w 87f4,a 289 f4,w 156

Table 4.7: The values for the solid lines for simulations in air and in water.

The results from the acoustic coupled simulation and the two way coupled simulationin air are compared in Table (4.8) and the results from the simulations in water arecompared in Table (4.9).

38

Frequency Acoustic Two way Difference [%]f1,a 41.15 43 4.49f2,a 73.62 73 0.84f3,a 166.04 169 1.78f4,a 290.88 289 0.65

Table 4.8: A comparison of frequencies from acoustic simulation and two way simulationin air

The difference in percentage for the frequencies in air for both types of simulationsare small and can therefore be validated.

Frequency Acoustic Two way Difference [%]f1,w 15.01 17 13.26f2,w 31.61 36 13.89f3,w 80.21 87 8.47f4,w 152.31 156 2.42

Table 4.9: A comparison of frequencies from acoustic simulation and two way simulationin water.

The difference in frequencies using water as the surrounding medium is greater thanthe results form the simulations using air. The mode shapes for the simulations in thetwo way coupled simulation is more difficult to investigate since the change in the solidstructure is not visible.

The displacement for the disc is of importance when investigating the added damping.The displacement of MP17 in air is presented in Figure (4.4) and the displacement ofthe same monitor point in water is presented in Figure (4.5).

39

Figure 4.4: The displacement of MP17 in air.

Figure 4.5: The displacement of MP17 in water.

The added damping can easily be seen by comparing the displacement over time forthe disc in air and in water. The energy dissipation is clearly greater for the disc inwater, which is due to the increase in viscosity of the fluid. The damping ratio have beendetermined using equation (2.31) and equation (2.32) in chapter 2 and the values for thedamping ratio in air and in water are presented in Table (4.10)

40

Damping rateAir ξa 0.223

Water ξw 0.821

Table 4.10: Damping rate in air and in water.

The added damping factor using the values in Table (4.10) is β = 3.68 and it isclear that the surrounding water increases the damping significantly. This shows thatthe added damping is of great importance for the fluid-solid interaction investigation ofa structure surrounded by a fluid medium.

41

42

Chapter 5

Conclusion and discussion

The dimensionless numbers calculated using the parameters for a Kaplan turbine witha power output of 42 [MW ] and a head of 37 [m] show that the pseudo-static assump-tion of the forced vibrations for the runner and shaft is suitable, which is the minimumstate for the added damping to occur. The wave propagation velocity was found to bedifficult to use, since the relation between UR and D did not fulfill any of the conditionsde Langre has suggested. The wave propagation velocity could be of interest when themodes of the structure are investigated. The relation of the dimensionless numbers, Urand D, could be explained by saying that the wave propagation is not affected by thevelocity of the fluid or the displacement of the structure.The solid velocity due to forcedvibrations fulfill the relation stated for the pseudo-static condition and it can be saidthat the added mass, stiffness and, damping occur in the system. The added stiffnessis usually neglected due to it’s low value, which has been determined for a Francis tur-bine to be about 2 % of the structural stiffness [6]. By comparing the magnitude ofadded stiffness with the magnitude of the added damping and the added mass, it is clearthat the added mass and damping affect the system more and are therefore more impor-tant. However the effect of the added stiffness on the mode shapes has not been discussed.

The added mass for a submerged disc using two different numerical simulations, acous-tic coupled simulation and two way coupled simulation was calculated and the resultsshowed that the different simulations types have similar results. The acoustic simulationcan be used to determine the magnitude of the added mass, but not the damping orthe stiffness. However, the two way coupled simulation can be used to determine theadded damping and added mass. A comparison of the two different simulation setupsshowed that the acoustic coupled simulation is easy to implement and takes less timeto solve, which is good for a time efficient investigation of the interaction for the fluidand the solid. The two way coupled simulation is far more complex and has a greatersolving time. The acoustic coupled simulation can be used to get a reference value forthe natural frequencies and the mode shapes, and the result from this simulation can beused to determine the size of the time steps used in the two way coupled simulation.

The two way coupled simulation requires, as mentioned before, a long solving time, whichcan not be considered as time efficient. In order to perform this type of simulation itis important to use a powerful computer. The setup for this type of simulation is more

43

difficult to execute in a proper manor, due to the amount of setup steps compared to theacoustic coupled simulation. The complexity of the setup makes this type of simulationa bit difficult to work with. The error messages in the software could in some cases bedifficult to decipher, which lead to a lot of detective work to be able to correct the errorsand often the ”trial and error” concept was used to figure out the right way of handlingthe errors. The position of the monitor points could not be changed or added when thesimulation was complete, so the simulation needed to be redone if a different position waswanted. The applied force was chanced from a distributed force and a point force, anddifferent shape for the contour plots of the surface of the disc were found. The distributedload gave a circular shape and the point load gave the shape of the first nodal diameter.However the same natural frequencies were found in the FFT-analyses. The solving timeof the simulations combined with the changes of position of the monitor points and thedifferent applied loads have been very time consuming.

The two different types of simulation were compared and the decrease in natural fre-quencies from air to water using the acoustic coupled simulation is 48.0 % - 63.5 % forthe different mode shapes. The decrease of the natural frequencies using the two waycoupled simulation is 46.1 % to 60.5 %. The added mass has a greater impact on thelower frequencies for both simulation types. The values for the acoustic coupled simula-tion is slightly higher than the values from the two way coupled simulation. The cause ofthe difference could be due to the quality of the mesh or the size of the time steps. Theadded damping increases with 268.2 % and was determined using the two way coupledsimulation.

The results from the simulations show that the added mass and the added dampingcan be determined using numerical simulations. A decrease of the natural frequenciesand an increase of the damping was observed, which was expected. The acoustic coupledsimulation should be used when only the added mass is of interest. The two way coupledsimulation should be used when the added damping and added mass for a Kaplan tur-bines are investigated. When the added damping and added mass are known, they canbe taken into consideration when designing the runner and the natural frequencies andhazardous vibrations can be minimized and the lifespan of the turbine can be increased.

44

Chapter 6

Future work

The interaction of the wave propagation could be of interest for the change in modeshapes or the added stiffness due to the flowing water. The added stiffness is dependenton the force from the water on the interface and the displacement of the interface, whichcould be affected by the wave propagation velocity.

A proper mesh quality analysis should be performed for both types of simulation, sothat the difference in results can be improved, along with the size of the time steps. Afiner mesh and smaller time steps could lead to results that are more compatible withthe results from the acoustic coupled simulation. The interpolation between the nodeson the fluid domain and the solid domain could also lead to errors, however the errorfrom the interpolation can be determined when the importance of the mesh quality andthe size of the time steps are known. The difference in results from the two simulationsperformed in this thesis could be explained by these simple investigations. An analysisof the change in mode shapes should be performed in order to see which mode shapes aremostly affected by the surrounding fluid. Since the different loads gave different shapeson the disc a suggestion is to apply both types of load on the disc in the same simulationto see if more mode shapes can be visual in the contour plots. Another suggestion isto evaluate the possibility of interpolating the mode shapes from the acoustic coupledsimulation to the two way coupled simulation so that the mode shapes are more easilyfound. In order to validate the result of the added damping, an experiment should beperformed along with an investigation of the effect of the added damping.

The effect of rotation of the disc and on the fluid should be the next step in this typeof simulations and also to find the mode shapes of the structure in water, with a morethorough modal investigation of the peaks in FFT-analysis. In order to determine theeffect of the added properties on a Kaplan runner, a simulation with the geometry of therunner should be conducted. The simulation should consider the rotation of the runnerand the working fluid in order to determine the added properties as exact as possible,however this might not be possible with the two way coupled simulation since the solvingtime could be extremely long, depending on the standard of the computer. A finer meshcould also increase the solving time and the same can be mentioned for the size of thetime step. The behavior of the increase of the solving time is not known. The solving timecould have a linear increase if the geometry is more complex,have a fine mesh and the

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time steps are small, or it could be exponential. This should be taken into considerationwhen choosing between the two different types of simulations.

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