Numerical Estimation of Torsional Dynamic Coefficients of a Hydraulic Turbine

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.


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.Urlichs [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 fluidrotor 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 finiteelement 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.

Operating Conditions.
All the computations are made for the Hölleforsen Kaplan turbine model runner, shown in Figure 1.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 52 rad/s, 62 rad/s, and 72 rad/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.

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 for k and 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.

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 β m , which is chosen by the user.This coefficient should have a value in the range 0.2 ≤ β m ≤ 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 β m values have been made, and the results are however more or less unaffected by the choice of β m .In the present work a value of β m = 1.0 has been used.The time derivative is discretized using the Euler implicit method.

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 Figure 2 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 where J P 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, θ is the angular velocity, and θ 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 where J P,Fluid is the added polar inertia, C Fluid is the added damping, and K Fluid 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 where T (t) is the total torsional moment due to the flow, and inserting this into (2) yields To solve T (t), the forces and moments from the CFDsimulations are calculated at each time step.The force on a control volume boundary face is given by where p face,i is the pressure of the face, A face,i is the area of the face, and − → n face,i is the normal vector of the face.The moment of the centre of gravity of the runner at a face is where r face is the radius from the centre of gravity to the face.The total moment is calculated as 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 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 where T mean 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 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 gives the velocity θ = −aϑ sin(ϑt), (12) and the acceleration θ = −aϑ 2 sin(ϑt).( 13) Inserting ( 11), (12), and ( 13) into (3) results in an equation for the fluctuation of the torque This can be written as where T Amp is the amplitude of the torque, φ is the phase angle, and T 1 and T 2 are the cosine and sine components of the amplitude.Then the additional damping due to the fluid can be identified as and the additional stiffness and polar inertia due to the fluid can be identified by solving for two simulations with different values of ϑ.
International Journal of Rotating Machinery  The eigenfrequency of ( 2) can now be solved as and the corresponding damping ratio is

Results
In Figure 3 the torque is shown as a function of time for one of the simulated cases.The amplitude of T 1 /a in ( 17) is presented as a function of perturbation frequency in Figure 4.The perturbation amplitude is a = 4.0 × 10 −6 rad 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 (K Fluid = 0 in (17)) in the analysis below.The added polar inertia is presented in Figure 5 and the added damping in Figure 6.
The later coefficients are added to the mechanical system, that is, (2).The polar inertia of the mechanical system is J P = 1.57Nms 2 , the damping is C = 0 Nms, and the stiffness is K = 49000 Nm.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.

Discussion
Both added polar inertia and damping have a significant effect on the eigenfrequency of the mechanical system.The added polar inertia decreases the eigenfrequency 3-5% for all cases (see Figure 7).Concerning the damping, an additionally decrease of the eigenfrequency of 5-60% is observed (see Figure 7).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.Figure 9 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.
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 9. 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.

Conclusions
The added polar inertia and damping due to the hydraulic system significantly affect the mechanical system.This results in a reduced eigenfrequency of 5-65% 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.

Call for Papers
For safety and reliability reasons, many dynamic engineering systems excited by intensive forces with irregular time histories have to be analyzed and designed by employing statistical concepts.The development of jet and rocket propulsion systems in the 1940's and subsequently the development of commercial jet aeroplanes in the 1950's that introduced a problem of structural vibration are a good example.In the middle of 1950's and early 1960's, the accelerated development of computers and computational techniques, such as the finite-element method, has enabled the designers to provide less conservative designs for more sophisticated and complex structural systems.Since the early 1950's, many national and international conferences as well as symposia have partially or entirely devoted to presenting results in the field of computational stochastic structural dynamics.It is believed that the latter has reached a stage in which a detailed comparison of the various classes of computational techniques and an examination of design issues for largescale structural systems are warranted.
The goal of this Special Issue is, therefore, to solicit and publish the most recent research and development in the field, with particular focus on applications of various classes of computational techniques, and to design issues of existing as well as future structural dynamic engineering systems.
Topics of interest include, while strictly confined to the field of computational stochastic structural dynamics, but are not limited to: • Linear mechanical and structural systems under random excitations • Nonlinear mechanical and structural systems in aerospace, offshore and ocean, earthquake, automotive, bridge, railway and tunnel, and nuclear engineering • Future mechanical and structural systems in random environments Before submission, authors should carefully read over the journal's Author Guidelines, which are located at http://www .hindawi.com/journals/ame/guidelines.html.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking Sys-tem at http://mts.hindawi.com/,according to the following timetable:

Call for Papers
In the lean premixed prevaporized (LPP) combustion, a homogenous lean fuel-air mixture is delivered to the primary zone, and combustion occurs at lower temperatures as the fuel burns at leaner equivalence ratios causing reduction in NOx level.However, dynamic coupling between the acoustics and the flame generates an unstable system with severe pressure oscillations (very important for aircraft engines like LM6000 OLE combustion system) that can lead to flame blowoff causing significant performance loss and also can lead to high noise level and structural damage to components.High pressure and high temperature operations also have paramount impact on the fuel efficiency, combustion stability, heat release rates, burning speeds, and global flame structure for both natural gas and alternate fuel combustion.In a gas turbine or automotive engine, fuel spray injected into a hot gas atmosphere evaporates by heat transfer from its surroundings.Fuel vaporization, fuel-air mixing, ignition delay, and the position of the ignition start affect the spray combustion characteristics and the exhaust emissions significantly.All are functions of the distribution of fuel vapor and liquid within the combustion chamber as well as the surface temperature of the liquid droplets in the fuel spray.Accordingly, improved engine/turbine/combustor design depends on understanding fuel vaporization and mixing.
We invite authors to present original research articles as well as review articles that will stimulate the continuing efforts in the areas of combustion dynamics and heat transfer studies of liquid or gaseous fuels with particular emphasis on alternative fuels in a gas turbine.We are particularly interested in manuscripts that report both fundamental and applied studies of the latest advances in understanding combustion stability, shock/blast combustion, advanced diagnostics for combustion measurements, fuel droplet dynamics, pollutant formation, flame-wall interaction, and new emerging technologies with respect to combustor design, fuel injection methodologies, and cooling schemes for biofuels and bioblends.Potential topics include but are not limited to: • High pressure combustion dynamics and stability studies of alternative biofuels and syngas fuels ZT, can reach 3 or higher.However, it is very challenging to create a perfect TE material with easy electron passage and low phonon flow resistance at the same time.Most recently, nanoscale thermoelectric materials with ZTs as high as 1.4 have been reported, which translates to an efficiency of 14%.Another example is the possibility of using sunlight to breakdown water into hydrogen and oxygen by mimicking photosynthesis.To enable efficient artificial photosynthesis, it is necessary to make artificial multiple electron systems.
A breakthrough has been made to accept multiple electrons and release them from single-walled carbon nanotubes (SWNTs) and Phthalocyanines (PCs).Using a catalyst is yet another way to make hydrogen gas from water.A cheap inorganic catalyst has been developed to effectively split hydrogen from water by using sunlight.Additionally, the advanced micro-and nanoscale surface treatment technologies and structures are also very important to the improvement of renewable energy producing and end use efficiencies.Advances in micro-and nanoscale transports can play a crucial role in exploiting renewable energy and developing energy saving technologies in the future.These technologies hold the possibility of providing groundbreaking increases in renewable energy process yields, efficiencies, and lower overall costs, but there is still much to be done before industrializing these technologies.This special issue focuses on these challenges and solicits papers in relevant areas, including, but not limited to, the following:

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 steadystate solver for incompressible and turbulent flow.It is a finite volume solver using the SIMPLE algorithm for pressurevelocity 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 − 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.

Figure 2 :
Figure 2: The mechanical model of a torsional dynamic system.

Figure 3 :
Figure 3: The torque as a function of time for one of the simulated cases (rotational speed is 72 rad/s and the oscillating frequency is 1809 rad/s).

Figure 7 :
Figure7: 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 :
Figure 8: Additional damping due to the flow through the turbine (the damping of the mechanical system is zero).
Angular displacement (rad) ω D : Damped natural frequency (rad/s) ζ: Damping ratio (−) ϑ: Prescribed frequency (rad/s) Ω: Rotational speed (rad/s) − → n face,i : Normal vector at one face (−) p face,i : Pressure one face (N/m 2 ) t: Time(s) A face,i : Area of one face (Nm) C: Damping (Nms/rad) C Fluid : Added damping (Nms/rad) − → F face,i : Force on one face (N) J P : Polar moment of inertia (kgm 2 ) J P,Fluid : Added Polar moment of inertia (kgm 2 ) K: Stiffness (Nm/rad) K Fluid : Added stiffness (Nm/rad) M(t): External moment (Nm) − → M face,i : Moment at one face (Nm) T (t): Total torsional torque due to flow (Nm) T 1,2 : Sine and cosine components of the torque (Nm) T Amp : Amplitude of the oscillating part of the torque (Nm) T Mean : Constant part of the torque (Nm) Particle transport phenomena in renewable energy conversion processes • Micro/nanoscale engineered materials for energy conversion and storage • Fluid transport in micro/nanoscale structures for energy producing process and end use • Multiphysical transport in microporous media • Systematical optimization of energy efficiency Before submission authors should carefully read over the journal's Author Guidelines, which are located at http://www.hindawi.com/journals/ame/guidelines.html.Authors should follow the Advances in Mechanical Engineering manuscript format described at the journal site http://www .hindawi.com/journals/ame/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: Manuscript Due October 1, 2009 First Round of Reviews January 1, 2010

in Mechanical Engineering Special Issue on Micro/Nanotransport Phenomena in Renewable Energy and Energy Efficiency Call for Papers Energy
has become one of the most important issues of our time as a result of serious concerns about climate change, high oil prices, and peak oil.Renewable energy and energy-saving technologies are potentially crucial parts of the ultimate solutions to both energy sustainability and climate change.Although there are plenty of renewable energy sources available, such as solar, wind, geothermal, they each have some drawbacks: being intermittent, low efficiency and high capital cost, which, at present, limits their applicability.Micro-and nanoscale transport phenomena can play critical role in developing technologies to supply clean energy with both low cost and high efficiency.For example, thermoelectric materials (TE) have application in waste heat recovery and solar energy as well as in the construction of compact coolers or even AC if TE merit, or Swarnendu Sen, Department of Mechanical Engineering, Jadavpur University, India; drssen@mech.jdvu.ac.inAdvances