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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.

Thomas [

The use of computational fluid dynamics (CFD) has recently increased within the area of fluid-rotor interactions. It was introduced by Dietzen and Nordmann [

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 [

All the computations are made for the Hölleforsen Kaplan turbine model runner, shown in Figure

The computational domain.

The inlet boundary condition was obtained by taking the circumferential average of a separate guide vane calculation, yielding an axisymmetric inlet flow [

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 for

For the convection divergence terms in the turbulence equations the Gamma discretization scheme by Jasak et al. [

To describe how the eigenfrequencies and damping of a torsional dynamic system change due to the flow, the model illustrated in Figure

The mechanical model of a torsional dynamic system.

Inserting (

where

The eigenfrequency of (

In Figure

The torque as a function of time for one of the simulated cases (rotational speed is

Identification of the coefficients of (

Additional polar inertia as a function of perturbing frequency and operating condition.

Additional damping as a function of perturbing frequency and operating condition.

The later coefficients are added to the mechanical system, that is, (

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.

Additional damping due to the flow through the turbine (the damping of the mechanical system is zero).

Both added polar inertia and damping have a significant effect on the eigenfrequency of the mechanical system. The added polar inertia decreases the eigenfrequency

Iso-surfaces are here used to illustrate the difference between the different operating conditions. Figure

Iso-surface of turbulent kinetic energy, (a)

Smearlines and velocity vectors for

Smearlines and velocity vectors for

Smearlines and velocity vectors for

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 Figure

Recent research of added mass of a cylinder by Wang et al. [

The added polar inertia and damping due to the hydraulic system significantly affect the mechanical system. This results in a reduced eigenfrequency of

Angular displacement (

Damped natural frequency (

Damping ratio (

Prescribed frequency (

Rotational speed (

Normal vector at one face (

Pressure one face (

Time (

Area of one face (

Damping (

Added damping (

Force on one face (

Polar moment of inertia (

Added Polar moment of inertia (

Stiffness (

Added stiffness (

External moment (

Moment at one face (

Total torsional torque due to flow (

Sine and cosine components of the torque (

Amplitude of the oscillating part of the torque (

Constant part of the torque (

The research presented in this paper has been carried out with funding by Elforsk AB and the Swedish Energy Agency through their joint Elektra programme and as a part of the Swedish Hydropower Centre (SVC) (