# Numerical Estimation of Torsional Dynamic Coefficients of a Hydraulic Turbine

Authors: Martin Karlsson; Håkan Nilsson; Jan-Olov Aidanpää
Journal: International Journal of Rotating Machinery
 (2009)

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## Abstract

The rotordynamic behavior of a hydraulic turbine is influenced by fluid-rotor interactions at the
turbine runner. In this paper computational fluid dynamics (CFDs) are used to numerically predict
the torsional dynamic coefficients due to added polar inertia, damping, and stiffness of a Kaplan
turbine runner. The simulations are carried out for three operating conditions, one at about 35%
load, one at about 60% load (near best efficiency), and one at about 70% load.
The runner rotational speed is perturbed with a sinusoidal function with different frequencies in
order to estimate the coefficients of added polar inertia and damping. It is shown that the added
coefficients are dependent of the load and the oscillation frequency of the runner. This affect the
system's eigenfrequencies and damping. The eigenfrequency is reduced with up to 65% compared
to the eigenfrequency of the mechanical system without the fluid interaction. The contribution to
the damping ratio varies between 30–80% depending on the load. Hence, it is important to consider
these added coefficients while carrying out dynamic analysis of the mechanical system.


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## Body

## 1. Introduction

Thomas [1] initiated the research on fluid-rotor interactions on turbines in 1958. He suggested an analytical model of destabilising forces due to nonsymmetric clearance in steam turbines. Alford [2] developed a similar  model for compressors, where the forces are obtained as a function of the change in efficiency due to increased eccentricity. Ulrich [3] carried out the first research in a test rig and suggested corrections to Thomas and Alford's models.  At the same time Iversen et al. [4], Agostinelli et al. [5], and Csanady [6] introduced models of hydraulic unbalance forces due to asymmetry of the flow channel geometry in centrifugal pumps. Hergt and Krieger [7] studied the influence of radial forces during off-design operating conditions. Colding-Jorgensen [8] used potential flow theory to determine damping and stiffness coefficients. Adkins [9] were the first to introduce an analytical model of both mass, damping and stiffness coefficients and harmonic forces. Adkins and Brennen [10], and Bolleter [11, 12] used test rigs to continue the development of models for fluid-rotor interactions of pump impellers. Childs [13] used bulk flow theory to determine rotordynamical coefficients at the pump-impeller-shroud surface.The use of computational fluid dynamics (CFD) has recently increased within the area of fluid-rotor interactions. It was introduced by Dietzen and Nordmann [14] in 1987, but has due to the computational cost not been widely used in the past. The first applications of CFD within rotordynamics have been in the area of hydrodynamic bearings and seals. Recently, CFD has entered into the research of fluid-rotor interactions in centrifugal pumps [15]. CFD has been more common in research and development of hydraulic machinery. Ruprecht et al. [16, 17] used CFD to calculate forces and pressure pulsations on axial and Francis turbines. However, the results were not used in rotordynamical analysis. Liang et al. [18] carried out finite-element fluid-structure interactions of a turbine runner in still water and showed a reduction of the nonrotating eigenfrequencies compared to a runner in vacuum. The result had good agreement with the experimental results presented by Rodriguez et al. [19]. Karlsson et al. [20] analyzed the influence of different inlet boundary conditions on the resulting rotordynamic forces and moments for a hydraulic turbine runner. The benefits of using CFD to calculate rotordynamical forces and coefficients of hydraulic turbines have not yet been fully explored. In the present work CFD is used for the determination of the torsional dynamic coefficients due to the flow through the turbine.


## 2. Modelling and Simulation

### 2.1. Fluid-Dynamical Model

#### 2.1.1. The OpenFOAM CFD Tool

In the present work the OpenFOAM (www.openfoam.org) open source CFD tool is used for the simulations of the fluid flow through the Hölleforsen water turbine runner. The simpleFoam OpenFOAM application is used as a base, which is a steady-state solver for incompressible and turbulent flow. It is a finite volume solver using the SIMPLE algorithm for pressure-velocity coupling. It has been validated for the flow in the Hölleforsen turbine by Nilsson [21]. New versions of the simpleFoam application have been developed in the present work, including Coriolis and centrifugal terms and unsteady RANS. All the computations use wall-function grids and turbulence is modelled using the standard k-ϵ turbulence model. The computations have been run in parallel on 12 CPUs on a Linux cluster, using the automatic decomposition methods in OpenFOAM. The version number used for the present computations is OpenFOAM 1.4.


#### 2.1.2. Operating Conditions

All the computations are made for the Hölleforsen Kaplan turbine model runner, shown in Figure1. The computational grid is obtained from earlier calculations by Nilsson [21]. The operating conditions used for the present investigations are for runner rotational speeds of 52rad/s, 62rad/s, and 72rad/s, which correspond to loads of about 70%, 60%, and 35%, respectively. The boundary conditions are kept the same for all operating conditions (in the inertial frame of reference). The change in the load due to the rotational speed is explained by the fact that the pressure drop (or head of the system) needed to drive the same flow through the turbine will change with different rotational speed. The runner rotational speed is finally perturbed with a sinusoidal function in order to identify added coefficients for the torsional dynamic system. This is described below.Figure 1
The computational domain.


#### 2.1.3. Boundary Conditions and Computational Grid

The inlet boundary condition was obtained by taking the circumferential average of a separate guide vane calculation, yielding an axisymmetric inlet flow [22]. This corresponds to a perfect distribution from the spiral casing and without any disturbance from the guide vane wakes.Wall-functions and rotating wall velocities were used at the walls, and at the outlet the homogeneous Neumann boundary condition was used for all quantities. Recirculating flow was thus allowed at the outlet, and did occur. The turbulence quantities of the recirculating flow at the outlet are unknown, but to set a relevant turbulence level for the present case the back-flow values fork and ϵ were assumed to be similar to the average of those quantities at the inlet. The background of this assumption is that the turbulence level is high already at the inlet due to the wakes of the stay vanes and the guide vanes. It is thus assumed that the increase in turbulence level is small compared with that at the inlet. It is further believed that the chosen values are of minor importance for the overall flow. For the pressure the homogeneous Neumann boundary condition is used at all boundaries. The computations are made for a complete runner with five blades. The computational domain is shown by  Figure 1. A block-structured hexahedral  wall-function grid was used, consisting of approximately 2 200 000 grid points.


#### 2.1.4. Discretization Schemes

For the convection divergence terms in the turbulence equations the Gamma discretization scheme by Jasak et al. [23] was used. For the convection divergence terms in the velocity equations the GammaV scheme was used, which is an improved version of the Gamma scheme formulated to take into account the direction of the flow field. The Gamma scheme is a smooth and bounded blend between the second-order central differencing (CD) scheme and the first-order upwind differencing (UD) scheme. CD is used wherever it satisfies the boundedness requirements, and wherever CD is unbounded UD is used. For numerical stability reasons, however, a smooth and continous blending between CD and UD is used as CD approaches unboundedness. The smooth transition between the CD and UD schemes is controlled by a blending coefficient βm, which is chosen by the user. This coefficient should have a value in the range 0.2≤βm≤1, the smaller value the sharper switch and the larger value the smoother switch between the schemes. For good resolution, this value should theoretically be kept as low as possible, while higher values are more numerically stable. Studies of different βm values have been made, and the results are however more or less unaffected by the choice of βm. In the present work a value of βm=1.0 has been used. The time derivative is discretized using the Euler implicit method.


### 2.2. Identification of Dynamic Coefficients

To describe how the eigenfrequencies and damping of a torsional dynamic system change due to the flow, the model illustrated in Figure2 is used. In the model, the generator is assumed to be stiff due to the connection to a rigid electric grid, and hence only the torsional motion of the turbine runner is considered. The equation of motion for this system is given by
(1)JPθ̈+Cθ̇+Kθ=M(t),
where JP is the polar inertia, C is the damping, K is the stiffness, M(t) an external moment, t is the time, θ is the angular displacement, θ̇ is the angular velocity, and θ̈ is the angular acceleration. It is further assumed that the flow through a turbine will give additional inertia, damping, and stiffness to the system. With these additional coefficients the equation of motion becomes
(2)(JP+JP,Fluid)θ̈+(C+CFluid)θ̇+(K+KFluid)θ=M(t),
where JP,Fluid is the added polar inertia, CFluid is the added damping, and KFluid is the added stiffness. External moments are negligible (M(t)=0) in the present work. CFD is used to identify the added coefficients from the torque of the turbine runner. Rewriting the moments due to the flow to
(3)T'(t)=-JP,Fluidθ̈-CFluidθ̇-KFluidθ,
where T'(t) is the total torsional moment due to the flow, and inserting this into (2) yields
(4)JPθ̈+Cθ̇+Kθ=T'(t).
To solve T'(t), the forces and moments from the CFD-simulations are calculated at each time step. The force on a control volume boundary face is given by
(5)F⃗face,i=pface,iAface,in⃗face,i,
where pface,i is the pressure of the face, Aface,i is the area of the face, and n⃗face,i is the normal vector of the face. The moment of the centre of gravity of the runner at a face is
(6)M⃗face,i=F⃗face,irface,i,
where rface is the radius from the centre of gravity to the face. The total moment is calculated as
(7)M⃗=∑i=1nM⃗face,i,
where n is the number of faces. The torque is obtained as a scalar product of the moment and the direction vector of the shaft
(8)T(t)=M⃗n⃗y.
During steady conditions the torque is constant in order to provide a constant power to the generator. In case of unsteady conditions, the torque can be written as
(9)T(t)=Tmean+T'(t),
where Tmean is the constant part of the torque. In the present work the rotational speed of the turbine runner is prescribed in order to determine the dynamical coefficients of the turbine runner due to the flow. The angular displacement of the runner is given by
(10)θ=Ωt+acos(ϑt)=Ωt+θ',
where Ω is the constant angular velocity, t is the time, a is an amplitude, ϑ is a frequency of the prescribed runner oscillation, and θ' is the oscillating part of θ. Below, we are only interested in the oscillating part, where
(11)θ'=acos(ϑt),
gives the velocityFigure 2
The mechanical model of a torsional dynamic system.(12)θ̇'=-aϑsin(ϑt),
and the acceleration
(13)θ̈'=-aϑ2sin(ϑt).Inserting (11), (12), and (13) into (3) results in an equation for the fluctuation of the torque
(14)T'(t)=aϑ2JP,Fluidcos(ϑt)+aϑCFluidsin(ϑt)-aKFluidcos(ϑt).
This can be written as
(15)T'(t)=TAmpcos(ϑt-ϕ)=T1cos(ϑt)+T2sin(ϑt),whereTAmp is the amplitude of the torque, ϕ is the phase angle, and T1 and T2 are the cosine and sine components of the amplitude. Then the additional damping due to the fluid can be identified as(16)CFluid=T2aϑ,
and the additional stiffness and polar inertia due to the fluid can be identified by solving
(17)aϑ2JP,Fluid-aKFluid=T1
for two simulations with different values of ϑ.The eigenfrequency of (2) can now be solved as(18)ωD=K+KFluidJP+JP,Fluid-(C+CFluid)24(JP+JP,Fluid)2,
and the corresponding damping ratio is(19)ζ=C+CFluid2(JP+JP,Fluid)(K+KFluid)/(JP+JP,Fluid).


## 2.1. Fluid-Dynamical Model

### 2.1.1. The OpenFOAM CFD Tool

In the present work the OpenFOAM (www.openfoam.org) open source CFD tool is used for the simulations of the fluid flow through the Hölleforsen water turbine runner. The simpleFoam OpenFOAM application is used as a base, which is a steady-state solver for incompressible and turbulent flow. It is a finite volume solver using the SIMPLE algorithm for pressure-velocity coupling. It has been validated for the flow in the Hölleforsen turbine by Nilsson [21]. New versions of the simpleFoam application have been developed in the present work, including Coriolis and centrifugal terms and unsteady RANS. All the computations use wall-function grids and turbulence is modelled using the standard k-ϵ turbulence model. The computations have been run in parallel on 12 CPUs on a Linux cluster, using the automatic decomposition methods in OpenFOAM. The version number used for the present computations is OpenFOAM 1.4.


### 2.1.2. Operating Conditions

All the computations are made for the Hölleforsen Kaplan turbine model runner, shown in Figure1. The computational grid is obtained from earlier calculations by Nilsson [21]. The operating conditions used for the present investigations are for runner rotational speeds of 52rad/s, 62rad/s, and 72rad/s, which correspond to loads of about 70%, 60%, and 35%, respectively. The boundary conditions are kept the same for all operating conditions (in the inertial frame of reference). The change in the load due to the rotational speed is explained by the fact that the pressure drop (or head of the system) needed to drive the same flow through the turbine will change with different rotational speed. The runner rotational speed is finally perturbed with a sinusoidal function in order to identify added coefficients for the torsional dynamic system. This is described below.Figure 1
The computational domain.


### 2.1.3. Boundary Conditions and Computational Grid

The inlet boundary condition was obtained by taking the circumferential average of a separate guide vane calculation, yielding an axisymmetric inlet flow [22]. This corresponds to a perfect distribution from the spiral casing and without any disturbance from the guide vane wakes.Wall-functions and rotating wall velocities were used at the walls, and at the outlet the homogeneous Neumann boundary condition was used for all quantities. Recirculating flow was thus allowed at the outlet, and did occur. The turbulence quantities of the recirculating flow at the outlet are unknown, but to set a relevant turbulence level for the present case the back-flow values fork and ϵ were assumed to be similar to the average of those quantities at the inlet. The background of this assumption is that the turbulence level is high already at the inlet due to the wakes of the stay vanes and the guide vanes. It is thus assumed that the increase in turbulence level is small compared with that at the inlet. It is further believed that the chosen values are of minor importance for the overall flow. For the pressure the homogeneous Neumann boundary condition is used at all boundaries. The computations are made for a complete runner with five blades. The computational domain is shown by  Figure 1. A block-structured hexahedral  wall-function grid was used, consisting of approximately 2 200 000 grid points.


### 2.1.4. Discretization Schemes

For the convection divergence terms in the turbulence equations the Gamma discretization scheme by Jasak et al. [23] was used. For the convection divergence terms in the velocity equations the GammaV scheme was used, which is an improved version of the Gamma scheme formulated to take into account the direction of the flow field. The Gamma scheme is a smooth and bounded blend between the second-order central differencing (CD) scheme and the first-order upwind differencing (UD) scheme. CD is used wherever it satisfies the boundedness requirements, and wherever CD is unbounded UD is used. For numerical stability reasons, however, a smooth and continous blending between CD and UD is used as CD approaches unboundedness. The smooth transition between the CD and UD schemes is controlled by a blending coefficient βm, which is chosen by the user. This coefficient should have a value in the range 0.2≤βm≤1, the smaller value the sharper switch and the larger value the smoother switch between the schemes. For good resolution, this value should theoretically be kept as low as possible, while higher values are more numerically stable. Studies of different βm values have been made, and the results are however more or less unaffected by the choice of βm. In the present work a value of βm=1.0 has been used. The time derivative is discretized using the Euler implicit method.


## 2.1.1. The OpenFOAM CFD Tool

In the present work the OpenFOAM (www.openfoam.org) open source CFD tool is used for the simulations of the fluid flow through the Hölleforsen water turbine runner. The simpleFoam OpenFOAM application is used as a base, which is a steady-state solver for incompressible and turbulent flow. It is a finite volume solver using the SIMPLE algorithm for pressure-velocity coupling. It has been validated for the flow in the Hölleforsen turbine by Nilsson [21]. New versions of the simpleFoam application have been developed in the present work, including Coriolis and centrifugal terms and unsteady RANS. All the computations use wall-function grids and turbulence is modelled using the standard k-ϵ turbulence model. The computations have been run in parallel on 12 CPUs on a Linux cluster, using the automatic decomposition methods in OpenFOAM. The version number used for the present computations is OpenFOAM 1.4.


## 2.1.2. Operating Conditions

All the computations are made for the Hölleforsen Kaplan turbine model runner, shown in Figure1. The computational grid is obtained from earlier calculations by Nilsson [21]. The operating conditions used for the present investigations are for runner rotational speeds of 52rad/s, 62rad/s, and 72rad/s, which correspond to loads of about 70%, 60%, and 35%, respectively. The boundary conditions are kept the same for all operating conditions (in the inertial frame of reference). The change in the load due to the rotational speed is explained by the fact that the pressure drop (or head of the system) needed to drive the same flow through the turbine will change with different rotational speed. The runner rotational speed is finally perturbed with a sinusoidal function in order to identify added coefficients for the torsional dynamic system. This is described below.Figure 1
The computational domain.


## 2.1.3. Boundary Conditions and Computational Grid

The inlet boundary condition was obtained by taking the circumferential average of a separate guide vane calculation, yielding an axisymmetric inlet flow [22]. This corresponds to a perfect distribution from the spiral casing and without any disturbance from the guide vane wakes.Wall-functions and rotating wall velocities were used at the walls, and at the outlet the homogeneous Neumann boundary condition was used for all quantities. Recirculating flow was thus allowed at the outlet, and did occur. The turbulence quantities of the recirculating flow at the outlet are unknown, but to set a relevant turbulence level for the present case the back-flow values fork and ϵ were assumed to be similar to the average of those quantities at the inlet. The background of this assumption is that the turbulence level is high already at the inlet due to the wakes of the stay vanes and the guide vanes. It is thus assumed that the increase in turbulence level is small compared with that at the inlet. It is further believed that the chosen values are of minor importance for the overall flow. For the pressure the homogeneous Neumann boundary condition is used at all boundaries. The computations are made for a complete runner with five blades. The computational domain is shown by  Figure 1. A block-structured hexahedral  wall-function grid was used, consisting of approximately 2 200 000 grid points.


## 2.1.4. Discretization Schemes

For the convection divergence terms in the turbulence equations the Gamma discretization scheme by Jasak et al. [23] was used. For the convection divergence terms in the velocity equations the GammaV scheme was used, which is an improved version of the Gamma scheme formulated to take into account the direction of the flow field. The Gamma scheme is a smooth and bounded blend between the second-order central differencing (CD) scheme and the first-order upwind differencing (UD) scheme. CD is used wherever it satisfies the boundedness requirements, and wherever CD is unbounded UD is used. For numerical stability reasons, however, a smooth and continous blending between CD and UD is used as CD approaches unboundedness. The smooth transition between the CD and UD schemes is controlled by a blending coefficient βm, which is chosen by the user. This coefficient should have a value in the range 0.2≤βm≤1, the smaller value the sharper switch and the larger value the smoother switch between the schemes. For good resolution, this value should theoretically be kept as low as possible, while higher values are more numerically stable. Studies of different βm values have been made, and the results are however more or less unaffected by the choice of βm. In the present work a value of βm=1.0 has been used. The time derivative is discretized using the Euler implicit method.


## 2.2. Identification of Dynamic Coefficients

To describe how the eigenfrequencies and damping of a torsional dynamic system change due to the flow, the model illustrated in Figure2 is used. In the model, the generator is assumed to be stiff due to the connection to a rigid electric grid, and hence only the torsional motion of the turbine runner is considered. The equation of motion for this system is given by
(1)JPθ̈+Cθ̇+Kθ=M(t),
where JP is the polar inertia, C is the damping, K is the stiffness, M(t) an external moment, t is the time, θ is the angular displacement, θ̇ is the angular velocity, and θ̈ is the angular acceleration. It is further assumed that the flow through a turbine will give additional inertia, damping, and stiffness to the system. With these additional coefficients the equation of motion becomes
(2)(JP+JP,Fluid)θ̈+(C+CFluid)θ̇+(K+KFluid)θ=M(t),
where JP,Fluid is the added polar inertia, CFluid is the added damping, and KFluid is the added stiffness. External moments are negligible (M(t)=0) in the present work. CFD is used to identify the added coefficients from the torque of the turbine runner. Rewriting the moments due to the flow to
(3)T'(t)=-JP,Fluidθ̈-CFluidθ̇-KFluidθ,
where T'(t) is the total torsional moment due to the flow, and inserting this into (2) yields
(4)JPθ̈+Cθ̇+Kθ=T'(t).
To solve T'(t), the forces and moments from the CFD-simulations are calculated at each time step. The force on a control volume boundary face is given by
(5)F⃗face,i=pface,iAface,in⃗face,i,
where pface,i is the pressure of the face, Aface,i is the area of the face, and n⃗face,i is the normal vector of the face. The moment of the centre of gravity of the runner at a face is
(6)M⃗face,i=F⃗face,irface,i,
where rface is the radius from the centre of gravity to the face. The total moment is calculated as
(7)M⃗=∑i=1nM⃗face,i,
where n is the number of faces. The torque is obtained as a scalar product of the moment and the direction vector of the shaft
(8)T(t)=M⃗n⃗y.
During steady conditions the torque is constant in order to provide a constant power to the generator. In case of unsteady conditions, the torque can be written as
(9)T(t)=Tmean+T'(t),
where Tmean is the constant part of the torque. In the present work the rotational speed of the turbine runner is prescribed in order to determine the dynamical coefficients of the turbine runner due to the flow. The angular displacement of the runner is given by
(10)θ=Ωt+acos(ϑt)=Ωt+θ',
where Ω is the constant angular velocity, t is the time, a is an amplitude, ϑ is a frequency of the prescribed runner oscillation, and θ' is the oscillating part of θ. Below, we are only interested in the oscillating part, where
(11)θ'=acos(ϑt),
gives the velocityFigure 2
The mechanical model of a torsional dynamic system.(12)θ̇'=-aϑsin(ϑt),
and the acceleration
(13)θ̈'=-aϑ2sin(ϑt).Inserting (11), (12), and (13) into (3) results in an equation for the fluctuation of the torque
(14)T'(t)=aϑ2JP,Fluidcos(ϑt)+aϑCFluidsin(ϑt)-aKFluidcos(ϑt).
This can be written as
(15)T'(t)=TAmpcos(ϑt-ϕ)=T1cos(ϑt)+T2sin(ϑt),whereTAmp is the amplitude of the torque, ϕ is the phase angle, and T1 and T2 are the cosine and sine components of the amplitude. Then the additional damping due to the fluid can be identified as(16)CFluid=T2aϑ,
and the additional stiffness and polar inertia due to the fluid can be identified by solving
(17)aϑ2JP,Fluid-aKFluid=T1
for two simulations with different values of ϑ.The eigenfrequency of (2) can now be solved as(18)ωD=K+KFluidJP+JP,Fluid-(C+CFluid)24(JP+JP,Fluid)2,
and the corresponding damping ratio is(19)ζ=C+CFluid2(JP+JP,Fluid)(K+KFluid)/(JP+JP,Fluid).


## 3. Results

In Figure3 the torque is shown as a function of time for one of the simulated cases. The amplitude of T1/a in (17) is presented as a function of perturbation frequency in Figure 4. The perturbation amplitude is a=4.0×10-6rad for all simulations and is selected in the area where torque/angular velocity is linear and the value is selected in order to separate the response from numerical noise. One can see that it is difficult to identify the coefficients as stated in (17). There are two possible explanations to this: the coefficients depend on frequency and the stiffness is probably small due to the incompressible fluid. The stiffness is therefore assumed to be negligible (KFluid=0 in (17) in the analysis below. The added polar inertia is presented in Figure 5 and the added damping in Figure 6.Figure 3
The torque as a function of time for one of the simulated cases (rotational speed is72rad/s and the oscillating frequency is 1809rad/s).Figure 4
Identification of the coefficients of (17), together with a curve-fitted function (rotational speed is 52rad/s).Figure 5
Additional polar inertia as a function of perturbing frequency and operating condition.Figure 6
Additional damping as a function of perturbing frequency and operating condition.The later coefficients are added to the mechanical system, that is, (2). The polar inertia of the mechanical system is JP=1.57Nms2, the damping is C=0Nms, and the stiffness is K=49000Nm. In Figure 7 the reduced eigenfrequencies (18) and in Figure 8 the damping ratio (19) due to the flow for such a fluid-mechanical system are presented and the influence of the different coefficients is illustrated.Figure 7
Reduction of the eigenfrequency (the eigenfrequency of the mechanical system is 1) due to the flow through the turbine. The “undamped” markers represent the effect of an added polar inertia alone.Figure 8
Additional damping due to the flow through the turbine (the damping of the mechanical system is zero).


## 4. Discussion

Both added polar inertia and damping have a significant effect on the eigenfrequency of the mechanical system. The added polar inertia decreases the eigenfrequency3–5% for all cases (see Figure 7). Concerning the damping, an additionally decrease of the eigenfrequency of 5–60% is observed (see Figure 7). One can see that both damping and polar inertia increases for off-nominal speed and with frequency. Recent research by Liang et al. [18], and Rodriguez et al. [19] has shown that the eigenfrequencies are reduced by 10–39% for a nonrotating Francis runner in still water. The effect of added inertia in these papers are significantly higher than the case of nominal operating condition in the present work and the authors observe no strong effect of damping. An explanation to the difference between the present study and the earlier work is the dependency of frequency for both added inertia and damping and that the present work includes the turbine flow.Iso-surfaces are here used to illustrate the difference between the different operating conditions. Figure9 shows iso-surfaces of regions where the turbulent kinetic energy is high. In Figures 10, 11, and 12 smearlines at the blades are presented in order to see the details of the flow.Iso-surface of turbulent kinetic energy, (a)52rad/s, (b) 62rad/s, and (c) 72rad/s.
(a)(b)(c)Figure 10
Smearlines and velocity vectors for52rad/s.Figure 11
Smearlines and velocity vectors for62rad/s.Figure 12
Smearlines and velocity vectors for72rad/s.The difference in the rotating speed results in different flow conditions for the different operating conditions. The guide vane angle is equal for all cases. Hence, the angle of attack at the leading edge of the runner blades is changed when changing the rotational speed. The tipclearance flow from the pressure side to the suction side is increased when the rotational speed is reduced. For high rotational speeds there is also a tip vortex at the runner blade pressure side due to the unfavorable angle of attack close to the tip. The tip vortex flow is the reason to the high turbulent kinetic energy near the tipclearence, which is shown in Figure9. Figure 9 also shows high turbulence kinetic energy in the flow stagnation at the leading edges of the runner blades, and in separation regions. A major difference in the level of turbulence kinetic energy can be found below the runner cone in the recirculation region. The significant differences of the flow field for the different cases are also illustrated by the smearlines in Figures 10, 11, and 12. Figures 10 and 12 show a large non-axisymmetric recirculation area below the cone. The wakes below the runner vanes are also shown on the cone as well as the tipvortex flow. Figure 11 shows a small axisymmetric recirculation area below the cone.Recent research of added mass of a cylinder by Wang et al. [24] has shown that the added mass is dependent on the velocity around a cylinder. The same effect is suspected in the present study, where the flow velocity differs between the cases.


## 5. Conclusions

The added polar inertia and damping due to the hydraulic system significantly affect the mechanical system. This results in a reduced eigenfrequency of5–65% and an increase in the damping of 30–80%. It is further concluded that the added coefficients are dependent on the turbine load and oscillating frequency. A change in the system properties of the mechanical system is important to consider in design and operation. Future studies should include experimental verification of the results in the present work.



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*Source: 349397-2009-06-22.xml*