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 Lule˚a University of Technology (LTU) during the summer of 2016. The problem was sett so I could work with numerical calculation in MatLab and simulations using ANSYS Workbench. The first part of this thesis was the mathematics that couples movement from a solid to a fluid which was completely 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 a result of that.
I would like to thank my supervisor Michel Cervantes for the chance to work with this subject and also my co-supervisor Arash Soltani Dehkharqani for all the help with the simulations and theoretical aspects.
Lule September 30, 2016
Abstract
A preliminary investigation of the added properties called added mass, added damping and 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 the solid. The classification is done to bring clarity in which of the added properties are of importance for the system.
The diameter of the runner and the hub have been calculated using the power output and the head for a Kaplan turbine. These dimensions have been used to determine the magnitude of the dimensionless numbers along with the velocity of the fluid. It turned out that all added properties affect the turbine, however, the magnitude of them are quite different. The magnitude of the added mass and the added damping are greater than the added stiffness, which often is neglected.
The added mass can be determined if the natural frequencies of the structure in air and in water are known. The difference in natural frequencies can be used to determine the added mass factor and thereby the added mass of the system. The added damping can be determined by the change in damping ratio for different surrounding fluids. This was done using the simulation software ANSYS Workbench v.17.1, where two different types 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 disc and 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 different types of simulation have been compared and it turned out that the added mass could be determined using ”acoustic coupled simulation” and ”two way coupled simulation”, but the added damping could only be determined using the ”two way coupled simulation”.
Sammanfattning
En prelimin¨ar unders¨okning av de adderade egenskaperna kallade, adderad massa, adderad d¨ampning och adderad styvhet har utf¨orts f¨or en Kaplan turbin. Magnituden av dimen- sionsl¨osa tal har anv¨ants f¨or att klassificera interaktionen av fluiden och soliden. Klassi- ficeringen g¨ors f¨or att bringa klarhet i vilka av de adderade egenskaperna ¨ar av betydelse f¨or systemet.
Diametrarna f¨or l¨ophjulet och navet har ber¨aknats utifr˚an effekt och fallh¨ojd f¨or en Ka- plan turbin. Dessa l¨angder har anv¨ants f¨or att best¨amma magnituden av de dimensionl¨osa talen tillsammans med fluidens hastighet. Det visade sig att alla adderade egenskaper p˚averkar turbinen, men omfattningen av dem ¨ar helt annorlunda. Magnituden av den adderade massan och den adderade d¨ampningen ¨ar st¨orre ¨an den adderade styvheten, som ofta f¨orsummas.
Den adderade massan kan best¨ammas om de naturliga frekvenserna av strukturen i luft och vatten ¨ar k¨anda. Skillnaden i egenfrekvenser kan anv¨andas f¨or att best¨amma faktorn av den adderade massan och d¨arigeniom den adderade massan. Den adderade d¨ampnin- gen kan best¨ammas genom ¨andringen i d¨ampningsf¨orh˚allande f¨or olika omgivande fluider.
Detta gjordes med hj¨alp av simuleringsprogrammet ANSYS Workbench v.17.1, d¨ar tv˚a olika typer av simulering anv¨andes, ”acoustic coupled simulation” och ”two way cou- pled simulation”. Komplexiteten i geometrin f¨or en Kaplan turbin f¨orenklades till en skiva och en axel. Resultatet f¨or den adderade massan validerades med resultat fr˚an ett experiment [1]. Den adderade d¨ampningen kunde best¨ammas, men inte valideras. De olika typerna av simulering har j¨amf¨orts och det visade sig att den adderade massan kan best¨ammas med hj¨alp av b˚ade ”acoustic coupled simulation” och ”two way coupled simulation”, men den adderade d¨ampningen kunde endast best¨ammas med hj¨alp av ”two way coupled simulation”.
Contents
1 Introduction 9
1.1 Background . . . 9
1.2 Fluid-Solid Interaction . . . 10
1.3 Thesis aim . . . 12
2 Theory 13 2.1 Design of Kaplan turbine . . . 13
2.2 Dimensionless numbers . . . 15
2.2.1 Velocity of the solid structure . . . 17
2.3 Vibration and frequency . . . 18
2.4 Numerical simulation . . . 21
2.4.1 Acoustic structural coupled simulation . . . 22
2.5 Two way coupled simulation . . . 23
3 Method 25 3.1 Numerical calculation . . . 25
3.1.1 Design of Kaplan turbine . . . 25
3.1.2 Dimensionless numbers . . . 25
3.2 Numerical simulation . . . 26
3.2.1 Experimental setup . . . 26
3.2.2 Geometry . . . 26
3.2.3 Acoustic structural coupled simulation . . . 28
3.2.4 Two way coupled simulation . . . 29
4 Result 33 4.1 Numerical calculations . . . 33
4.1.1 Kaplan turbine . . . 33
4.1.2 Dimensionless numbers . . . 34
4.2 Simulation . . . 35
4.2.1 Acoustic structural coupled simulation . . . 35
4.2.2 Two way coupled simulation . . . 37
5 Conclusion and discussion 43
6 Future work 45
Chapter 1
Introduction
This master thesis report is a part if the final course E7015T Maters thesis of the master program in sustainable energy engineering, specializing in wind and hydropower.
1.1 Background
Hydropower is an environmentally friendly and efficient energy source. The probability to build new hydropower plants is slim in Sweden which makes the maintenance work the main focus in order to keep the energy distribution on a constant level [2]. Instability on the grid, for instance due to the increase in wind power, generates more transient operating conditions for the hydropower plants. The use of numerical tools have made the design 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 concentration in the turbines due to upgrades increases the excitation forces on the runner that can lead to fatigue and damages on the blades [3]. It is therefore important to avoid the excitation frequencies. In oder to do that, the knowledge of the dynamics of a immersed structure, called fluid-solid interaction (FSI), must be studied. If the natural frequency of the turbine submerged in water is known in advance, the excitation frequencies can be taken 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 three and eight runner blades that are attached to a hub. The inflow is optimized by controlling the pitch of the runner blades and the turning of the wicket gates. The water enters the turbine in a radial direction and exit with an axial direction, see Figure (1.1). The flowing water transfers its angular and axial momentum to the blades, which produces torque and rotation. The forces acting on the turbine and also the change in operating conditions due to the operations mentioned earlier creates vibrations. This can lead to negative 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 to cracks and fatigue. If a piece of the structure was separated from the structure, it could lead to major damages and inefficiency.
Figure 1.1: Water flow in a Kaplan turbine. [4]
When studying the interaction of the solid and the fluid, different terms called, added mass, 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 conditions can 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. Vibrations in nuclear reactor due to earthquakes, vibrating hydropower turbines or ship propellers are 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 coupling of movement of the solid structure and movement of the fluid domain. It is important not to neglect the affect of the surrounding fluid on the dynamics of the solid structure, since it is known that the added mass can reduce the natural frequencies of the structure and that 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 damages of the structure and are therefore important aspects to be considered when designing a turbine.
The added properties that arise when a solid structure and a fluid interact have been investigated both analytically and through experiments and later with numerical simu- lations. The dynamics of the interaction have been analytically investigated by E. de Langre [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. He stated that the forces and velocities acting on the interface are equal in both domain and can 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 are affecting a specific cases. This type of investigation was partially performed by Gauthier in 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
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 perform time depending CFD simulation with a given displacement amplitude. The fluid force could be calculated and the stiffness derived. The added stiffness due to the water was determined to a value of about 2% of the structural stiffness. The added stiffness can be neglected when investigating the dynamics of a structure submerged by still water according to Rodriguez et al. [7]. The added damping in [6] was determined by the use of energy dissipation in the system. The results were obtained using CFX with the use of interpolated mode shapes. However the results could not be fully validated, but they were found to be sufficient enough for the needs of engineering design.
A similar investigation of the added mass has been performed by M¨uller [8] on a Francis turbine runner using both experimental data and numerical simulations. The natural frequencies of the structure were excited by a impact force in both air and in water. The results from the experiments were compared with simulations using acoustic-structural coupled simulations in ANSYS and the added mass could be determined experimen- tally and numerically. This type of numerical simulation was also validated by Graf and Chen [9] where a comparison of experimental data and numerical simulations was done for a Francis runner. This type of investigation have also been performed for different geometries by Hengstler [10], Hubner [11] and Egusquiza et al. [12]. The added mass can according to Hubner et al. [11] and Hengstler [10] be determined with an acoustic struc- tural coupled simulation using modal analysis. Hengstler used the acoustic structural coupled simulation to numerically calculate the different frequencies of a disc and shaft, which represented a simplified hydro turbine. The distance from a submerged structure to a rigid wall has an effect on the decrease in natural frequency. This was shown by Rodriguez et al. [13] when the natural frequencies of a cantilever plate submerged in water was investigated for different distances to the rigid wall of a tank. The results from the acoustic structural coupled simulation were compared with experimental data and confirmed that the effect of the nearby rigid wall needs to be taken into account in 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-solid interaction 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 shapes but 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 from these types of simulations is therefore a bit higher than the actual added mass obtained with 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 more time consuming, but more accurate information can be obtained.
1.3 Thesis aim
The aim of this master thesis is to investigate the different proprieties; added mass, added stiffness, and added damping of Kaplan turbine as a function of the specific speed. The main focus of this thesis is to evaluate the added properties using dimensionless numbers for a Kaplan turbine. Acoustic structural coupled simulation and two way simulations are performed in order to determine the magnitude of the added mass and added damping for a simplified turbine runner represented by a disc and shaft.
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 the magnitude of dimensionless numbers.
2.1 Design of Kaplan turbine
To investigate the added properties, the velocity of the working fluid and the diameters of the runner and the hub of the turbine must be determined. The variables used to determined these parameters are listen in Table (2.1). The equations following the table are used to calculate the diameters mentioned and the velocity of the fluid.
Variable Symbol Value
Power output P 42 [M W ]
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 Kaplan turbine was examined by M. N¨asselqvist [16]. The displacement of the shaft will be used for the dimensionless numbers in the next section of this chapter. The design calculations where preformed according to [17].
Q = P
Hηhρwg (2.1)
The net head, Hn is calculated using equation (2.2) below, which is used to calculate the specific speed, Ns. The specific speed was then used to calculate the speed of the runner, N and the diameter of the runner, Dr and the hub, Dh. These steps are shown in equation (2.3)-(2.6) and are made according to [17].
Hn = Hηh (2.2)
Ns= 2.294
Hn0.486 (2.3)
N = Ns
(Hng)3/4
Q1/2 (2.4)
Dr = 84.5(0.79 + 1.602Ns)Hn0.5
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 runner and 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 the area 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 the interaction of the fluid and the runner blade is investigated. To determine the relative velocity, 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, Vtof the fluid can be determined using equation (2.9).
P = ρwωQ(rrVt− rhV1,t) (2.9) The tangential velocity of the fluid at the inlet can be determined by assuming that there is no swirl at the outlet, V1,t = 0 giving equation (2.10).
Vt= P
ρwωQrr (2.10)
where rr is the radius of the runner, rr = Dr/2 and ω is the rotational velocity of the runner, ω = 2πN . The tangential velocity of the runner, U is calculated using equation (2.11).
U = ωrr (2.11)
The relative velocity of the fluid is calculated using equation (2.12) W =
q
Vn2 + (U − Vt)2 (2.12)
Excitation of the runner can occur when the rotor-static interaction coincide with the natural frequencies of the runner. The frequency depends on the pressure pulsations occurring when the runner blade passes a guide vane and is therefore a function of the numbers of guide vanes, Zgand 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 of the shaft have been found to be a factor of 2.4 times the rotational speed of the runner according to M. N¨asselqvist [16], see equation (2.14).
fs,s = 2.4N (2.14)
The result from these calculations will be used for the dimensionless numbers, which are mentioned later in this chapter.
2.2 Dimensionless numbers
The interaction of a fluid and solid can according to de Langre [5] be classified using dimensionless numbers. The most useful dimensionless numbers are the reduced velocity, URand the displacement number, D. The mass number, M and Reynolds number, RE are also useful to determine the importance of the added mass. Moreover, the inertia in the flowing fluid and the decrease of the natural frequencies are governed by the magnitude of M . The inertia in the fluid increases with the value of RE . The mass number is the ratio of the densities of the fluid and the solid, see equation (2.15) and the Reynolds number is presented in equation (2.16).
M = ρf
ρs (2.15)
RE = ρfUFL
µ (2.16)
The displacement number is the ratio between the displacement of the structure, x0 and the characteristic length, L, see equation (2.17). The greater the value of D is, the larger is the deformation of the structure.
D = x0
L (2.17)
In this thesis the diameter of the shaft and diameter of the runner will be used for the characteristic length.
The reduced velocity is the ratio of the time scale of the solid domain, TS and the time scale of the fluid domain, TF, see equation (2.18).
UR = TS
TF (2.18)
The time scales depend on the characteristic length, L and the characteristic velocities of 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 not affect 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 in Figure (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 URand D for this case would be UR<< D, where both D << 1 and UR<< 1 and can be used to determine the added mass of the system. The added 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 the solid could be neglected since the change in the time scale would be slow for the solid and fast for the fluid. This state is presented in Figure (2.3).
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 velocity is UR = O(1) a state called pseudo-static condition occur and the relation of UR2 >> D must be obtained. The time scales are shown in Figure (2.4). The coupling is strong when 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 determine which 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 of the structure is of interest. The vibration of the structure can be a good choice when the forced induced vibrations of the structure is investigated. If the mode of the structure is of interest, the wave propagation could be used for the solid velocity. The wave propagation velocity depends on the molecules’ ability to move in the material, a high density and high tension give a lower velocity. The equation for the wave propagation velocity if
shown in equation (2.21)
US,w = s
E
ρs, (2.21)
where E is the Young’s modulus and ρs is the density of the material used for the solid structure. 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 length and as mentioned earlier this length was chosen to be the diameter for either the runner or the shaft.
2.3 Vibration and frequency
A system undergoing deformations in a periodic manor can be referred to as an oscillating or vibrating system. The system is then storing kinetic and potential energy, as inertia and 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 of oscillation and the magnitude of the displacement will decrease until it finally stops. This means 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 it can be said to be a free vibrating system. However if the structure would to be affected by a repeating force so that the movement of the structure is affected multiple times, the system can be said to be a forced vibrating system [18]. If the external force coincide with the natural frequencies of the structure a state called resonance can occur [18]. This can generate vibrations with large amplitudes in the structure which can lead to cracks and fatigue. This state is crucial to avoid for all kinds of machinery, such as hydropower turbines, 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 stiffness force is shown in equation (2.23). The mass of the structure can gain and lose kinetic energy when the structure accelerates or decelerate. The mass force of the structure is equal to the acceleration times the mass of the structure. This is derived from the second Newton law and the work done on the structure is stored as the kinetic energy of the mass [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 mass of the structure and ¨x is the acceleration of the structure.
If a system is undamped the amplitude of the oscillations will remain the same or even increase, which means that the vibration will continue to infinity [18]. This is a theoretic state 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 potential energy in the system is stored in the spring and remains constant for a free vibrating system 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 consideration since there is no or very little energy dissipation to the surrounding or in the material, see equation (2.25).
msx + k¨ sx = 0 (2.25)
The natural frequency of the system can be determined using equation (2.26), if the mass and the stiffness is known.
fn= 1 2π
r ks
ms (2.26)
Where n denotes the number of degrees of freedom and thereby the number of natural frequencies. The number of natural frequencies of a system is governed by the number of 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 or generation of heat. The damping force can be described used equation (2.27), where cs is the structural damping and ˙x is the velocity of the vibrations of the structure.
FC = cs˙x (2.27)
By adding the mass force, stiffness force and the damping force the equation of motion for the vibrating structure can be expressed, see equation (2.28), where Fs(t) is the force applied on the structure. If Fs(t) = 0 the system can be said to be a free vibrating system.
msx + c¨ s˙x + ksx = Fs(t) (2.28) The equation of motion can also be expressed with the characteristic motion equation, see equation (2.29).
¨
x + 2ξωn˙x + ω2nx = Fs(t)
ms . (2.29)
Where ξ is the damping ratio and ωn, is the angular velocity of the oscillation and is equal to pks/ms. The damping ratio is the ratio of the critical damping constant and the actual damping constant of the oscillation system [19].
ξ = cs
cc = cs 2√
ksms (2.30)
The oscillation is critically damped when ξ = 1, undamped when ξ = 0 and under damped if ξ < 1 [18]. A way to determine the damping is to use the logarithmic decrement where the 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. If the rate of decay increases, amplitudes decrease more rapidly [19]. The rate of decay can also be expressed as equation (2.32) [19].
δ = 2πξ
p1 − ξ2 ⇔ ξ = 1 q
1 + (2πδ )2
(2.32)
When the damping ratio ξ is known the damping constant of the oscillation, cs can be determined.
When an object is surrounded by a fluid, an added fluid force occurs and the equation of motion for the vibrating structure can be describes as equation (2.33).
m¨x + c ˙x + kx = Fs(t) + Ff(t) (2.33) The fluid force would affect the system even if it was a free vibrating system, since it is a medium that surrounds the structure. The fluid force can be separated into three different forces, the force of added mass, the force of added damping and the force of added 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 added stiffness force.
The added mass force, Fa can be considered as part of the inertia in the fluid that affects the solid structure when it moves. When the solid object moves the surrounding fluid also does, since the fluid and the solid can not take up the same space at the same time. The fluid that the solid must push away can be described as the added mass. The added mass always occur when a structure is submerged in a fluid. The added mass force is 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 a pressure 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)
From the solid point of view, the condition of added stiffness is like being connected to a fluid 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 effect of added damping and the added damping force can be expressed as equation (2.37)
Fc= −ca˙x (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 the damping of the oscillations. The natural frequency of the submerged structure can be expressed as equation (2.39)
fa = 1 2π
r ks+ ka
ms+ ma, (2.39)
and the change in damping ratio can be expressed as equation (2.40) ξa = cs+ ca
2p(ks+ ka)(ms+ ma). (2.40) The changes in natural frequencies and in damping ratio can be used to determine the magnitude of the added mass and added damping. The changes can be determined though experiments or simulations using different surrounding fluids. The added mass factor can be calculated using equation (2.41) and the added damping can be calculated using equation (2.42).
α = fwater fair
(2.41)
β = ξwater
ξair (2.42)
The magnitude of the added mass and the added damping can now be determined using the natural frequencies and damping ratio in different fluids.
2.4 Numerical simulation
In this thesis two types of simulation are compared. The acoustic coupled simulation where the pressure wave in the fluid is used to determine the behavior of the structure in the fluid, and the second simulation type is the two way coupled simulation where the displacement of the mesh of the interface is used to determine the fluid force and the displacement of the structure.
2.4.1 Acoustic structural coupled simulation
A numerical simulation can be used when investigating the affect of surrounding water of a runner. To couple the behavior of the structure to the fluid, an acoustic structural coupling 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 in equation (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 and Ks denotes the structural stiffness matrix. When a structure is vibrating freely the load vector Fs is the fluid load due to the pressure distribution over the interface [13]. The pressure distribution is determined with the use of the acoustic wave equation, shown in equation (2.44). The fluid is considered to be non-flowing, irrotational, inviscid and slightly compressible.
∇2P = 1 c2
∂2P
∂t2 (2.44)
The Laplace operator is represented by ∇2 and P is the pressure in the fluid. The speed of sound in the fluid is represented by c in the equation. Equation (2.44) can be written as 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. This makes it possible to express the following equation for the momentum equation, where the relation between the normal acceleration of the structure and the normal pressure gradient 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 as equation (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]. The Equation (2.47) and equation (2.43) can now be written together as equation (2.48).
Fs(t) 0
= Ms 0 Mf s Mf
¨u
¨ p
+Cs 0 0 Cf
˙u
˙ p
+Ks Kf s 0 Kf
u p
(2.48) Where Mf s is the coupling mass matrix and [Kf s] is the coupling stiffness matrix at the interface and occur when a structure is immersed in a fluid. The coupling matrixes are
dependent on the shape function of the pressure and the displacement. The structural elements are all the variables denoted with s and the fluid elements are all the variables subscribed with f including the coupling variables. The natural frequencies are calculated using the Lanczos algorithm. The mechanical solver that was used in the simulations was the Modal analysis. This mechanical solver can only solve the natural frequency of a non-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 coupled using ANSYS Workbench v.17.1. In this thesis the structure is solved using T ransient Structural, which is a mechanical solver and the fluid is solved using F luid F low (CF X).
The fluid solver calculates the changes in mass, mass flux and the mesh deformation in the domain [20]. These three terms are shown in equation (2.49).
d dt
Z
V
ρ dV + Z
ρUjdnj + Z
S
ρWjdnj = 0 (2.49)
The mechanical domain is analysed using the finite element approach shown in equa- tion (2.50).
M ¨u + C ˙u + Ku = F (2.50)
The fluid dynamics and solid dynamics are solved separately, but at the interface the dynamics of the two physically different systems are coupled. The solution from CF X and 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 from the mechanical solver to the fluid solver. The fluid force acting on the structure is also transferred from CF X to the mechanical solver at the same interface. These steps are iterating for each time step for the simulation until a convergence is reached [20]. The transferred data is interpolated between the mesh of the structure and the mesh of the fluid. When using a two way coupled simulation, the added mass and the added damping in 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 sampling frequency and time steps. The Nyquist theorem states that the sampling frequency must be at least twice the size of the highest frequency of interest, see equation (2.51).
fs> 2fc (2.51)
The frequency fcrepresents the highest interesting frequency and fsrepresents the sample frequency that should be used for the two way coupled simulation. The time step can then 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 of the mesh remains high during deformation due to the increase in stiffness at nodes close
to a boundary [20]. The natural frequencies of a structure can be obtained by applying an impact force making the structure vibrate and monitoring the deformation over time.
A Fast Fourier Transformation analysis, FFT-analysis can then be performed using the results from the simulations to determine the natural frequencies of the structure in air and in water. The amplitude of the displacement is used to determine the magnitude of the damping.
Chapter 3
Method
3.1 Numerical calculation
In order to evaluate the magnitude of the dimensionless numbers used for the classification of the interaction, the characteristic length of the runner and shaft must be calculated along with the velocity of the flowing fluid. These numerical calculations have been performed 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 are presented in Table (2.1). The values were chosen so that the displacement of the shaft from [16] could be used for the dimensionless numbers. The equations from section 2.1 were used to calculate the diameters and the fluid velocity. The values of the relative fluid 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 same as the diameter of the hub, Dh and will therefore be used for the investigation of the dimensionless numbers, UR and D.
3.1.2 Dimensionless numbers
The dimensionless numbers are determined using the equations mentioned in chapter 2 for the different velocities of the solid structure using the diameters and velocities calculated for the Kaplan turbine. The relation of the dimensionless number will determine which classification the interaction has and which added properties are affecting the behavior of the structure. The numerical values for the different variables used to calculate the dimensionless numbers will be used to determine the order of magnitude of the reduced velocity, UR. The other dimensionless numbers will also be determined, however the numerical value of, D, M and, RE are interesting. The displacement of the runner blades due to the mode shape, x0,rb is assumed larger than the displacement of the shaft and have a magnitude of about 10−3 [m].
3.2 Numerical simulation
The effect of the flowing water in a turbine is difficult to determine due to the complexity of the system and would take incredible amount of time to solve manually. A tool often used in engineering is numerical simulations were the complex solutions of a moving system is solved by computer power. This is a useful tool when the fluid-solid interaction is investigated in a system. With the right boundary condition the results from numerical simulation represent the reality well. The runner was simplified to a disc and a shaft submerged in a cylindrical fluid domain. The simulations are compared to experimental data 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 has been performed by J. Hengstler and J. Dual [1]. The results from the experiment will be used to validate the results from the acoustic coupled simulation. A disc was excited in both air and in water by a coil interacting with a magnet attached to the disc and the vibrations was measured with a laser scanning vibrometer. In Figure (3.1) the schematics of 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 way coupled simulation. The only difference on the geometry is the mesh. The mesh for the acoustic coupled simulation must have matching nodes at the interface and the mesh of the
two way coupled simulation does not [21]. The two way coupled simulation interpolates between the nodes and the nodes must therefore be separate [20]. The parameters used for the geometry of the solid body and the fluid body are presented in Table (3.1) and are 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 shaft is fixed. For the acoustic coupled simulation the boundary condition called Displacement support is used. The walls are therefore not deformed. This makes all degrees of freedom except the pressure equal to zero. The pressure must be a degree of freedom since it is used to couple the two domains [22]. The displacement in X-, Y-, and Z-axis was set to zero so that the upper surface of the shaft was fixed in space. When using the two way coupled simulation the boundary condition called F ixed support is used, where all degrees 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).
Figure 3.3: Geometry of the fluid body.
The fluid-solid interface is highlighted in Figure (3.4) and the boundary condition is applied 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 obtain the 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 density of the fluid must be specified for this boundary condition. The air surrounding the disc and shaft has a very small effect on the natural frequency so the fluid can be suppressed,
hence, the simulation is performed in vacuum. The speed of sound, c = 1430 [m/s] and density 1000 [kg/m3] are used for the simulation in water.
The Displacement support boundary condition is applied on the walls of the fluid body in Figure (3.3) in order to illustrate the walls of a container [23]. As mentioned earlier the nodes must match when using this type of simulation. This is easily done by merging the bodies in the geometry builder to one part with multiple bodies. This will make the nodes 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. The mechanical solver calculates the displacement of the solid body, which is transferred to the fluid solver via the fluid-solid interface. The fluid solver calculates the fluid force due to the displacement, which is transferred from the mechanical solver. The data is interpolate 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 way simulation. The mechanical solver used for this simulation was T ransient Structural and the F luid F low (CF X) was used for the fluid. In Figure (3.5) the project setup is shown.
Figure 3.5: The coupling link for the simulations.
The line between the geometry cells shows that the geometry is shared between the two 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 correctly assigned and ready for simulation. The solid body and fluid body are meshed separately in 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 fluid will be still since the results from the acoustic coupled simulation and the results form the two way simulation will be compared.
The time step from the simulation was set so that the highest interesting frequency could 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 time step for the two way coupled simulation. The displacement of the mesh at the interface was used in a FFT-analysis to determine the natural frequencies. The FFT-analysis was performed in CF X − post where the results from the simulation can be examined. The
displacement of the interface over time can be used to determine the change in damping when the fluid is changed from air to water. This can not be done in the acoustic coupled simulation [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 the entire upper surface of the disc and the second force was a point force applied close to the edge of the upper surface, see position of M P 1 − 4 in Figure (3.6). The time steps and the magnitude for the forces used in the simulations is presented in Table (3.2).
Time [s] Force [N]
0.00 0
0.0102 100 0.0153 100
0.051 0
1.02 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 be close 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 be wrong. The interface between the solid and the fluid was defined by the F luid − Solid Interf ace boundary condition and data transfer will occur at this surfaces. The setup for the T ransient structural mechanical part is now done and can be coupled to the setup of F luid F low (CF X). It is important that the setup cell in T ransient is updated and have a green check mark in order to share the correct dat.ds file, which is the data transfer file, before the setup for the fluid starts.
The geometry of the structure is suppressed in F luid F low (CF X) so that only the fluid body is present and meshed. The analysis settings for CF X starts with selecting the AN SY S M ultiF ield coupling analysis type and apply the simulation time and the time steps, which should be the same as or smaller than the time step for the mechanical solver. The T ransient analysis type was also chosen. The mesh deformation was acti- vated to make it possible to couple the two domains.
The W all boundary condition was applied on the outer surfaces of the fluid body along with the surfaces which are connected to the F luid-Solid Interf ace of the solid struc- ture. In order to transfer data from the fluid to the structure the mesh motion must be set to AN SY S M ultiF ield for the surfaces of the interaction.
A number of monitor point was applied on the interface of the disc to be able to measure the displacement in order to determine the natural frequencies of the structure. The T otal M esh Displacement should be used, where the measured displacement is always compared with the original mesh for each time step. A total of 32 monitor points were
placed on eight different nodes on the disc to monitor the displacement in X-, Y- and Z-direction and also the total displacement. The position of the points are presented in Figure (3.6).
Figure 3.6: Position of the monitor points.
Four monitor points are used for each node and they have different directions. Each node 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 the simulation is applied in the same node as the monitor point, M P 1−4. All monitor points and the directions are listed in Table 3.3.
Direction Monitor point X 1, 5, 9, 13, 17, 21, 25, 29 Y 2, 6, 10, 14, 18, 22, 26, 30 Z 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 will be presented in chapter 4. The acoustic coupled simulation will be compared with the two way coupled simulation.
Chapter 4
Result
4.1 Numerical calculations
The values from the calculations for the Kaplan runner and the dimensionless numbers are presented 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 runner are presented in Table (4.1).
Flow rate Q 122.05 [m3/s]
Net head Hη 35.15 [m]
Specific speed N s 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 using equation (2.13) and equation (2.14) is presented in Table (4.2). The velocity of the vibrations calculated using equation (2.22) for the shaft and the runner blades and the diameters, Dr and Ds are also presented in the table below. The value of the velocity of the wave propagation in the solid material using equation (2.21) is also presented in same table.
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 the dimensionless numbers used to classify the state of fluid-solid interaction of the Kaplan turbine.
4.1.2 Dimensionless numbers
The order of magnitude of the variables calculated for the Kaplan turbine in previous section are used for the displacement number, D and the reduced velocity are presented in Table (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 is related to the magnetic pull due to the rotation of the generator and the runner vibrations are 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 and the 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, UR2 >> D is fulfilled so 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 stiffness is related to the force acting on the structure due to the velocity of the fluid, the added damping is related to the viscosity of the surrounding fluid, and the added mass is an added term that occur when a solid structure and a fluid interact. It can therefore be said that all the added properties can more or less affect the system. In earlier work by Gauthier the added stiffness was found to have a value of 2 % of the stiffness of the structure for a Francis. The aim of this thesis is not the determine the value of the added properties, 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 low change in the Francis turbine. The magnitude of the added mass and added damping are therefore more important for the investigation of interaction in Kaplan turbine.
The order of magnitude for the reduced velocity of the wave propagation was determined to 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. This relation has not been investigated by de Langre [5]. Gauthier states that this relation is not realistic, but no more is mentioned. However it might be interesting to use this solid velocity 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 in the fluid. This indicates that the added mass and added damping affect the solid body since 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 the experimental investigation by J. Henstler and J. Dual [1]. The added mass factor, added damping 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 are determined. In Table (4.5), the values obtained from the simulations are presented. The mode shapes of interest are the first, second, third and fourth nodal diameter, N D1, N D2, N D3 and N D4. From the experimental results, the nodal diameter for N D2, N D3 and N D4 are used for the validation.
Experimental results
Nodal diameter Frequency in air [Hz] Frequency in water [Hz]
N D1 − −
N D2 72.97 32.02
N D3 169.4 81.64
N D4 296.3 152.97
Simulation results
Nodal diameter Frequency in air [Hz] Frequency in water [Hz]
N D1 41.15 15.01
N D2 73.62 31.61
N D3 166.04 80.21
N D3 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 coupled simulation can be fully compared with the results from the acoustic coupled simulations.
The frequency for N D1 could help to identify the mode shapes in the FFT-analysis since the mode shapes cannot be shown visually. The second nodal diameter, N D2 is shown in Figure (4.1a) and the third and forth nodal diameter, N D3 respectively N D4 are shown in Figure (4.1b) and Figure (4.1c).
(a) N D2 (b) N D3 (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 for the mode shapes are small. This could be due to the quality of the mesh however the difference 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 the experimental values from [1] have been calculated using equation (2.41) from chapter 2 and the values are presented in Table (4.6). Where sim denotes the added mass factor from the simulation and exp denotes the added mass factor from the experiment.
Nodal diameter αsim [−] αexp [−] Difference [%]
N D1 0.36 − −
N D2 0.43 0.44 2.27
N D3 0.48 0.48 0.00
N D4 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 can indicate 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 M P 17 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 monitor point, M P 17 in air is presented in Figure (4.2) and the FFT-analysis in water is presented in Figure (4.3). The dashed vertical lines show the frequencies from the acoustic coupled simulation and the solid vertical lines shows the peak for the frequencies from the two way 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 be a multiple of a frequency, which makes the frequency repeat itself at higher values. This is the case for the prominent peaks at 85 [Hz] in Figure( 4.2). The small peak slightly to the left is therefore chosen.
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 peaks are 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 in Table 4.7. The difference is greater for the frequencies of the disk in water. The peaks in Figure (4.3) are more difficult to find and the mode shape could be changed compared to the mode shape in air.
Frequency [Hz]
Air Water
f1,a 43 f1,w 17 f2,a 73 f2,w 36 f3,a 169 f3,w 87 f4,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 simulation in air are compared in Table (4.8) and the results from the simulations in water are compared in Table (4.9).
Frequency Acoustic Two way Difference [%]
f1,a 41.15 43 4.49
f2,a 73.62 73 0.84
f3,a 166.04 169 1.78
f4,a 290.88 289 0.65
Table 4.8: A comparison of frequencies from acoustic simulation and two way simulation in air
The difference in percentage for the frequencies in air for both types of simulations are small and can therefore be validated.
Frequency Acoustic Two way Difference [%]
f1,w 15.01 17 13.26
f2,w 31.61 36 13.89
f3,w 80.21 87 8.47
f4,w 152.31 156 2.42
Table 4.9: A comparison of frequencies from acoustic simulation and two way simulation in water.
The difference in frequencies using water as the surrounding medium is greater than the results form the simulations using air. The mode shapes for the simulations in the two way coupled simulation is more difficult to investigate since the change in the solid structure is not visible.
The displacement for the disc is of importance when investigating the added damping.
The displacement of M P 17 in air is presented in Figure (4.4) and the displacement of the same monitor point in water is presented in Figure (4.5).
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 for the disc in air and in water. The energy dissipation is clearly greater for the disc in water, which is due to the increase in viscosity of the fluid. The damping ratio have been determined using equation (2.31) and equation (2.32) in chapter 2 and the values for the damping ratio in air and in water are presented in Table (4.10)
Damping rate Air 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 is clear that the surrounding water increases the damping significantly. This shows that the added damping is of great importance for the fluid-solid interaction investigation of a structure surrounded by a fluid medium.
Chapter 5
Conclusion and discussion
The dimensionless numbers calculated using the parameters for a Kaplan turbine with a power output of 42 [M W ] 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 minimum state for the added damping to occur. The wave propagation velocity was found to be difficult to use, since the relation between UR and D did not fulfill any of the conditions de Langre has suggested. The wave propagation velocity could be of interest when the modes of the structure are investigated. The relation of the dimensionless numbers, Ur and D, could be explained by saying that the wave propagation is not affected by the velocity of the fluid or the displacement of the structure.The solid velocity due to forced vibrations fulfill the relation stated for the pseudo-static condition and it can be said that the added mass, stiffness and, damping occur in the system. The added stiffness is 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 of added stiffness with the magnitude of the added damping and the added mass, it is clear that 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 results showed that the different simulations types have similar results. The acoustic simulation can be used to determine the magnitude of the added mass, but not the damping or the stiffness. However, the two way coupled simulation can be used to determine the added damping and added mass. A comparison of the two different simulation setups showed that the acoustic coupled simulation is easy to implement and takes less time to solve, which is good for a time efficient investigation of the interaction for the fluid and the solid. The two way coupled simulation is far more complex and has a greater solving time. The acoustic coupled simulation can be used to get a reference value for the natural frequencies and the mode shapes, and the result from this simulation can be used 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, which can not be considered as time efficient. In order to perform this type of simulation it is important to use a powerful computer. The setup for this type of simulation is more
