Unsteady Flow Analysis in Hydraulic Turbomachinery

Albert Ruprecht

Institute of Fluid Mechanics and Hydraulic MachineryUniversity of Stuttgart, Germany

ABSTRACTIn the field of hydraulic machinery Computational Fluid Dynamics (CFD) is routinely used today in re-search and development as well as in design. At that nearly always steady state simulations are ap-plied. In this paper, however, unsteady simulations are shown for different examples. The presentedexamples contain applications with self excited unsteadiness, e. g. vortex shedding or vortex rope inthe draft tube, as well as applications with externally forced unsteadiness by changing or movinggeometries, e. g. rotor-stator interactions. For these examples the requirements, potential and limita-tions of unsteady flow analysis assessed. Particularly the demands on the turbulence models and thenecessary computational efforts are discussed.

INTRODUCTION

For more than a decade Computational Fluid Dynamics (CFD) is used in the field ofhydraulic machinery in research and development as well as in the daily design busi-ness. Early successful demonstrations are given e. g. in the GAMM workshop [1].The applications are steadily increasing. This is expressed in fig. 1, where the per-centage of papers dealing with CFD is shown, which were presented at the IAHRSymposium on Hydraulic Machinery and Cavitation. Starting with Q3D-Euler and 3D-Euler today usually the Rey-nolds averaged Navier-Stokesequations together with a robustmodel of turbulence (usuallythe k-ε model) is used. It iscommon practice to applysteady state simulations, theunsteadiness in consequence ofthe rotor-stator interactions isaddressed by averaging proce-dures. By this method accurateresults are obtained for manyquestions in the design of com-ponents.

However, different problems inturbomachinery arise from un-steady flow phenomena. In or-der to get information on this phenomena or solutions to the problems an unsteadyflow analysis is necessary. This requires a much higher computational effort, roughlya factor 5-10 compared to steady state, depending of the problem and of the degreeof modeling assumptions. With today’s computers and software, however, unsteadyproblems can be solved.

Fig. 1: Percentage of papers at the IAHR-Symposium dealing with CFD

Two major groups of unsteady problems can be distinguished. The first group areflows with an externally forced unsteadiness. This can be caused by unsteadyboundary conditions or by changing of the geometry with time. Examples are the clo-sure of a valve, the change of the flow domain in a piston pump, or the rotor-statorinteractions. The second group are flows with self excited unsteadiness, which are e.g. turbulent motion, vortex shedding (Karman vortex street) or unsteady vortex be-havior (e. g. vortex rope in a draft tube). Here the unsteadiness is obtained withoutany change of the boundary conditions or of the geometry. There can also occur acombination of both groups (e. g. flow induced vibrations, change of geometrycaused by vortex shedding). All these phenomena can take place in a turbine orpump and require different solution procedures.

BASIC EQUATIONS AND NUMERICAL PROCEDURES

In hydraulic turbomachinery today usually the Reynolds averaged Navier-Stokesequations for an incompressible flow are applied. Compared to the steady state themomentum equations contain an additional term prescribing the unsteady change:

0'x

U

xU

xxP1

xU

Ut

Uij

i

j

j

i

jij

ij

i =

τ−

∂∂

+∂∂

ν∂∂−

∂∂

ρ+

∂∂

+∂

∂ (1)

τ ij’ are the Reynolds stresses, which are calculated from the turbulence model. The

continuity equation for incompressible flow reads

0xU

i

i =∂∂

(2)

and does not contain a time depending term. It hasto be emphasized that the equations (1) and (2)behaves different in time and in space. In spacethey show elliptic behavior, therefore they requireboundary conditions on all surfaces. In time, how-ever, they are of parabolic nature, which mean thatthere is no feed-back from the future to the pres-ent or past. Because of that no boundary condi-tions are required in the future. This is schemati-cally shown in fig. 2. This is the reason, why thetime discretization is generally carried out in a dif-ferent way than the spatial discretization. For spa-tial discretization usually a Finite Volume or a FiniteElement approximation is applied. For time discre-tization, however, mostly the Finite Differencemethod is used. A few of the most popular finitedifference approximations are shown in fig. 3. Inaddition explicit multi-point schemes of Runge-Kutta-type or predictor-corrector schemes are often applied.

Fig. 2: Boundary and initialconditions

Fig. 3: Time discretization schemes

It has to be mentioned that the explicit methods require a restriction of the time stepaccording to stability criteria (CFL-criteria), which depend on the local velocities andthe local grid size. The implicit methods, in contrary, are always stable, there is norestriction of the time step. It can be chosen only according to the physical require-ments. In order to obtain accurate solutions the time discretization should be at leastof 2nd order, similar to the spatial discretization. Otherwise extremely small time stepswould be required.

The above description of the flow in the Eulerian coordinates can be applied for un-steady boundary condition problems as well as for self excited unsteadiness. How-ever, to express problems with moving geometries in Eulerian coordinates is moredifficult. At the moving boundary a Lagrangian description can be applied very easilysince the fluid particles can be traced by this method. Combining these two methodsan Arbitrary Lagrangian Eulerian (ALE) method can be utilized. This method is suit-able for the solution of problems with moving boundaries. In the ALE method the ref-erence coordinates can be chosen arbitrary. In this referential coordinate system thematerial derivative can be described as

( ) ( ) ( ) ( )j

Ei

jj

Ri

Li

xt,xf

wut

t,xft

t,xf∂

∂−+∂

∂=∂

∂(3)

with the coordinatesscooddinateEulerian...x;scooddinatelreferentia...x;scooddinateLagrangian...x E

iRi

Li

and Wi ... reference velocity.

The momentum equations in the ALE formulation can be written as follows

( ) 0~x

U

xU

xxP1

xU

WUt

Uij

i

j

j

i

jij

ijj

i =

τ−

∂∂

+∂∂

ν∂∂−

∂∂

ρ+

∂∂

−+∂

∂(4)

The moving of the reference system Wi can be chosen arbitrary. If Wi is equal to zeroone gets the Eulerian description, on the other hand, if wi is equal to the velocity ofthe fluid particle the Lagrangian formulation is obtained. The convective term in thetransport equations for scalar quantities changes in the same way than in the mo-mentum equations. This applies also to the k- and ε-equations.

The numerical realization of moving or changing grids can either be obtained by de-formation of an existing mesh in each time step. For large deformations this requiresan automatic grid smoothing algorithm or even an automatic remeshing after a fewtime steps. An other method is the use of different embedded grids, which can move

against each other. In this case a sliding interface between the non-matching grids isrequired. This procedure is schematically shown in fig. 4 for two different problems,namely rotor-stator interaction and vibration of a cylinder in a fluid.

In FENFLOSS, the computer code developed at our institute at University of Stutt-gart, the second approach is applied. The interface between the grids is realized bymeans of dynamic boundary conditions, where downstream the node values (veloci-ties and turbulence quantities) are prescribed and upstream pressure and fluxes areintroduced as surfaceconditions. A brief over-view on the numericalprocedures is given in [2],for more details thereader is referred to [3,4].

One point has to be em-phasized. Since the un-steady simulations re-quire a severe increaseof computational effortcompared to steady statesolutions, parallel proce-dures are necessary. Inthis case the ALE formulation with moving grids leads to a dynamic change of com-munication because the location of exchange boundaries varies with time and cantherefore change the computational domain of the processors, see [2].

In FENFLOSS an implicit solution algorithm is applied. As already mentioned this hasthe advantage that there is no stability limitation for the time step. The overall solutionprocedure including the fluid-structure interaction is shown in fig. 5. If the movementof the grid does not depend on the flow situation the fluid-structure loop vanishes.

Fig. 5: Flow chart of FENFLOSS including fluid-structure interaction

Fig. 4: Moving grid examples

APPLICATIONS

In the following selected applications are shown and the specific problems for thisexamples are discussed. Firstly some cases with self excited unsteadiness are pre-sented.

Vortex shedding at the inlet of a power plant

Problem description: The first example shows the flow behavior at the inlet of a low-head power plant. It is an existing plant with two identical bulb turbines. During op-eration the inner turbine showed severe bearing problems whereas the outer turbineoperates smoothly. The reason was expected to be vortex shedding at the inlet. Bynumerical analysis the problem was investigated and it was tried to find a solution tothe problem. In fig. 6 the geometry is shown. The calculation has been carried out in2D as well as in 3D. Firstly it was tried tocarry out a steady state simulation, how-ever, no converged solution could beobtained. Therefore an unsteady simula-tion was undertaken. The results indicatea strong unsteady motion. In fig. 7 thevelocity distribution at a certain time stepis presented. Clearly visible are the vor-tices, shedding from the inlet and movingdownstream into the inner turbine. This isthe reason of thedestruction of thebearings. In or-der to improvethe flow behaviora modified ge-ometry was sug-gested. This ge-ometry, shown infig. 8, has beenbuilt in themeantime. Thereare no longerproblems withvortex shedding. Further detailsabout this application can be foundin [5,6].

Discussion: The physical unsteadi-ness of the flow has been indicatedby the inability to achieve a con-verged steady state solution. This isvery often the case with flowsshowing vortex shedding in reality.

Fig. 6: Geometry of power plant inlet

Fig. 7: Instantaneous velocity vectors, vortex shedding at theinlet pier

Fig. 8: Modified geometry

A necessary condition for that is, that the numerical scheme does not contain seri-ous artificial diffusion, which would suppress the unsteady motion. The same appliesto the used turbulence model. The standard k-ε model usually produces a too higheddy viscosity, especially in swirling flows, and therefore it very often suppresses theunsteady motion. This will be discussed again in other applications. For many casesat least a streamline curvature correction or even a non-linear eddy viscosity formu-lation is necessary in order to avoid a too high turbulence production.

Another point in turbulence modeling is the treatment of the near wall flow. It is wellknown that the use of wall functions usually tends to predict a flow separation toolate. In case of vortex shedding this can cause a severe reduction of the vortex sizesor even a complete suppression of the vortices. More accurate results can be ob-tained by solving the flow up to the wall (if possible) by a low-Reynolds- or a two-layer model. The results shown above are achieved by an algebraic turbulencemodel (Baldwin-Lomax-type) where the flow is resolved up to the wall.

Vortex rope in a draft tube

Problem description: As an other self excited unsteady flow example the simulationof a vortex rope in a draft tube is shown. Here a straight axisymmetrical diffuser isconsidered. The inflow conditions to the diffuser are chosen according to the partload operation of a Francis turbine. This means that the flow shows a strong swirlcomponent. The inlet velocity distribution and the geometry are presented in fig. 9.The instantaneous flow for a certain time step is given in fig. 10, where an iso-pressure surface as well as the secondary velocity vectors in three cross-sectionsare plotted. Clearly the cork-screw type flow with an unsymmetrical form is visible,although the geometry and the boundary conditions are completely axisymmetrical.

Fig. 9: Geometry and inlet conditions

Fig. 10: Iso-pressure and secondary flow of a vortex rope

In fig. 11 the secondary velocity and the low pressure region, which represents thevortex center, is shown in the cross-section S, indicated in fig. 9, for certain timesteps. Clearly the revolution of the vortex center can be observed. This, of course,causes pressure fluctuations and therefore dynamical forces on the draft tube sur-face.

Fig. 11: Secondary motion and low pressure region for different time steps

Discussion: Concerning the numerical scheme and the turbulence models the dis-cussion above also applies here, e. g. application of the standard k-ε model leads toa steady state, symmetrical solution. This is also reported in [7]. The results shownabove are achieved by applying the multi-scale k-ε model of Kim [8] together with astreamline curvature correction. This model shows a much lower eddy viscosity thanthe standard model, especially in swirling flows. The application of wall functionsdoes not give any problems here, since the flow instability has its origin in the centerand is not affected by the prediction of the near-wall region.

Vortex instability in a pipe trifurcation

Problem description: In the following anotherproblem caused by a vortex instability isshown. It is a pipe trifurcation, which is es-tablished in a power plant in Nepal. The tri-furcation distributes the water from the pen-stock to the three turbine units. The prob-lem in this plant arises from severe fluctua-tions of the power output of the both outerturbines. By field measurements the trifur-cation was discovered as the reason for thefluctuations. By means of CFD and bymodel tests, carried out at ASTROE in Graz,the flow behavior should be analyzed and acure of the problem should be found. Thegeometry of the trifurcation is shown in fig. 12. It hasa spherical shape.

The fluctuation in the trifurcation is caused by astrong vortex, which tends to be unstable. It skipsbetween the two situations, sketched in fig.13. In themodel tests the secondary velocity of the vortex couldbe found to be 30 times higher than the transportvelocity. The reason is that at the top of the spherethere is enough space for a huge vortex to form. Thisvortex concentrates in the side branches and there-fore increases the swirl intensity. Because of thisstrong secondary motion there are strong losses atthe inlet of the branch, which reduces the head of the turbine and therefore causesthe reduction of power output.

During the project it was tried to obtain the unsteady behavior by a k-ε simulation onrelatively coarse grids (200-300.000 nodes). However, these calculations did notshow the vortex instability. Merely a vortex forms which extends from one sidebranch to the other. The swirl intensity was underpredicted by more than a factor five.Because of the low swirl rate the vortex is completely stable and has no tendency ofskipping between different stations. Even by a dynamical excitation caused bychanges of the outlet boundary condition of one branch the predicted vortex did notchange its position.

Only when applying finer grids and another turbulence model the predicted swirl in-tensity could be increased. Here an algebraic turbulence model with a limitation ofthe eddy viscosity is applied. The used grids consists of about 500 000 nodes. As aconsequence this leads to an instability of the vortex. In the prediction the vortexskips between the two structures shown in fig. 14. One of these structures corre-sponds quite well with the structure observed in the model tests. In the second situa-tion the vortex expends from one side branch to the other. This complies with theabove mentioned stable results. The calculated swirl intensity is still more than twotimes lower compared to the results of the model tests. Therefore further investiga-

Fig. 12: Geometry of the trifurcation

Fig. 13: Vortex structure

tions with other turbulence models and with finer grids are necessary and will be car-ried out in future.

Fig. 14: Predicted vortex structures

For completeness the solution to the problem isshown. It consists of the installation of two platesin the upper and lower part of the sphere. This isshown in fig. 15. Hence no free space is available,where the vortex can form. Consequently the in-tensity of the vortex is dramatically reduced andthe vortex is completely stable. In the meantimethe reconstruction was carried out and the fluctua-tion of the power output vanished. As a by-productthe losses in the trifurcation are severely reduced,which results in an increase of power output of ap-proximately 5%. Further details of this problem canbe found in [9,10].

Discussion: As already mentioned the calculations using the k-ε model were notsuccessful. It is well known that this model is not able to predict highly swirling flowsaccurately. The unsteady motion of the vortices (especially of very slim vortices),however, very much depends on the swirl intensity. In order to prescribe such typesof flow with sufficient accuracy it is necessary to have highly sophisticated turbulencemodels and very fine grids, maybe the only way to achieve it is the application oflarge eddy simulation.

Rotor-stator interaction in an axial tubine

The following ex-ample belongs tothe second group,the unsteadinessis forced by mov-ing geometries.The problem inquestion is the

Fig. 15: Modified geometry

Fig. 16: Geometry of the investigated axial turbine

flow in an axial turbine. The speciality of this turbine is its relatively low specificspeed. It has been designed for pressure recuperation in piping systems. The ad-vantage is that the discharge is nearly independent of the speed, because of that theturbine cannot introduce waterhammers in the system. The geometry of the turbine isshown in fig. 16. It consists of the inlet confuser, 12 fixed guide vanes, 15 runnerblades and the draft tube. The stator and rotor part is shown in more detail in fig. 17.For the simulation the complete turbine is considered including all flow channels inthe guide vanes and in the runner, although a symmetry condition of 120° could beused. The reason is, that also a variant with unsymmetrical outlet has been investi-gated.

The computational mesh consists of more than 2 million grid nodes, part of the gridis shown in fig. 18. These are roughly 60000 nodes per flow channel. It is a rathercoarse grid, considering that the clearance between runner blades and casing has tobe included in the model, which is necessary since the clearance flow very much af-fects the channel flow because of the short runner blades. The calculations are car-ried out using the standard k-ε model.

In the following someresults of the calcu-lation will be shown.In fig. 19 the instan-taneous flow in therunner is presented.The figure shows thepressure distributionof the runner surfaceas well as stream-lines started at dif-ferent locations.Looking at the pres-sure one clearly seesthe stagnation pointat the leading edge.The location of the

Draft tube

guide vanes

runner

Fig. 18: Part of the computational meshFig. 17: Geometry of rotor and stator

Fig. 19: Instantaneous flow in the runner

stagnation point varies slightly with the runner position. Generally the inlet flow angleseems to be slightly too flat. Therefore the stagnation point is shifted towards thesuction side. Considering the flow in the tip clearance one can observe that at theinlet the shear forces dominate. The flow tends to go from the suction to the pressureside. In the second half of the blade the pressure forces dominate. The flow in theclearance goes from the pressure to the suction side. It can already be seen by thisresults that the design of the runner is not optimal. This is a first version, in themeantime a much better runner has been designed. However this geometry is nu-merically investigated since extensive measurements have been carried out for thisconfiguration and the numerical results can be validated.

In fig. 20 again the instantaneous pressure for a certain time step is shown. One canobserve the low pressure region on the suction side at the top of the runner blades.Clearly visible is the variation of the pressure with the position. The low pressure re-gion corresponds quite well with the cavitation observation at the test rig, see fig. 21.There one also can observe the variation of the cavitation bubbles according to therunner position.

As a quantitative compari-son the pressure at twolocations is shown in fig.22. Position 1 is located infront of the guide vanesand the second positionlies between the guidevanes and the runner. Atboth locations the meas-ured and the calculatedpressure correspondsquite well. One can seethat even in front of theguide vanes pressurefluctuations can be ob-served. Between the stator

Fig. 20: Calculated pressure distribution fora certain runner position

Fig. 21: Cavitation observation in therunner

Fig. 22: Pressure distribution at two spot points

and the rotor fluctuations of nearly 25% of the head of the turbine can be seen. This,of course, leads to dynamical forces on the blades. In fig. 23 the torque on one run-ner blade as well as the torque of the complete runner is shown. The calculatedtorque fluctuation on a single blade arenearly 30% of the averaged torque. Thisis a dynamical force on the blading. Thetotal torque, however, is nearly constantdue to the great number of blades anddue to different phases of the fluctua-tions. Further details concerning the tur-bine and the measurements are pub-lished in [11], details on the calculationsare given in [12].

Discussion: Since the unsteadiness of theflow is forced by the changing of the ge-ometry this problem is easier to attackthan the examples shown above. Heresufficient results are obtained applyingthe same models than in steady statesimulations and similar criteria apply foraccuracy than for the steady state simulations. As seen in the comparison with themeasurements the prediction of pressure (but also of velocities which are not shownhere, see [12]) is quite accurate. Therefore this kind of calculation is suitable to pre-dict dynamical forces.

POTENTIAL, LIMITATIONS, REQUIRED RESOURCES

The applications show that many unsteady problems can be investigated by CFD andmany phenomena can be studied, even rather complicated ones. Here we will dis-cuss again the potential and the limitations as well as the required resources, whichare necessary for an unsteady simulation.

Firstly the rotor-stator problem is discussed. It can be said, that for this type of flowthe unsteady computations behave similar to steady state. The accuracy principallydepends on the grid size and on the turbulence model used as in steady state. Thereasons for inaccuracy also correspond very close to that of steady state simulations,e. g. wake flow, swirling flow etc.

It has to be pointed out, that the requirements of computational effort for unsteadyflows is much higher than for steady state. Looking at a single component an un-steady simulation needs at least 3-5 times more computing time. But due to the ab-sence of any periodicity the complete turbine including all stator and rotor channelshave to be considered, in opposition to steady state, where periodicity can be appliedby circumferential averaging. Depending on the type of machine and on the numberof guide vanes and runner blades the necessary grid nodes can be increased by afactor of 20-30, to achieve a similar accuracy.

Another problem for unsteady computations of a complete turbine or pump is, thatthe flow contains a large range of frequencies. In order to resolve the high frequen-

Fig. 23: Torque on a runner blades

cies sufficiently accurate a small computational step has to be chosen. This, how-ever, results in very long computational times when also low frequency phenomenahave to be resolved, see [2].

Concerning the self excited vortex flow qualitative predictions can be obtained. Thephenomenon of vortex shedding can be calculated quite accurate. Even if the de-tailed flow behavior may not be kept completely correct the frequencies and ampli-tudes of integral quantities (e. g. forces) can be predicted with sufficient accuracy formost of the problems. Flow instabilities and correct vortex movements, however, de-pend very much on the detailed flow situation. Even small changes of velocity canhave a great response in the flow structure. As an example the flow in the sphericaltrifurcation is mentioned. There the vortex instability strongly depends on the swirlingrate. This means it is essential to predict the highly swirling flow very accurate, butthat is a severe problem for all the turbulence models used in practice today. At leastit is necessary to apply non-linear models or Reynolds-stress models, since swirlingflows are dominated by anisotropic effects. Maybe sufficient results can only beachieved by Large Eddy Simulations.

Because the vortices are often very concentrated and consequently show very steepgradients their prediction require very fine computational grids. If the vortices movewith time a self adaptive mesh refinement would be desirable. This, however, israther complicated. Since the required computational effort is very high, parallelcomputing must be applied in order to obtain reasonable response times. An adap-tive grid generation then leads to a dynamical load distribution. Both of the two ap-proaches, either using extremely fine grids in total geometry or using adaptivemeshes with dynamical load balancing, leads to a computational effort, which is atleast 20 times higher than a similar steady state solution.

CONCLUSIONS

Unsteady simulations for the different applications have been shown, among themare applications dominated by vortex shedding or vortex instabilities as well as appli-cations with forced unsteadiness which is e. g. rotor-stator interactions. All simula-tions have in common a quite large requirement of computational resources. Espe-cially for rotor-stator interactions the complete turbine has to be considered and allflow channels in the stator as well as in the rotor have to be included. This leads tomany grid nodes and an enormous computational effort.

On the other hand the flow with vortex instability, e. g. vortex rope in a draft tube, rep-resents a great challenge for an unsteady simulation. There the swirl intensity verymuch affects the overall movement of the vortex. But just strong swirling flows areextremely difficult to be calculated and the turbulence models applied today cannotcapture this behavior sufficiently accurate. For this type of flow it is necessary to useimproved models of turbulence.

Even if there are problems with the computational accuracy for certain applicationsgenerally many information can be obtained from unsteady simulations. For examplethe dynamical forces in the axial turbine could be predicted with sufficient accuracy inorder to allow an assessment of the durability. On the other side flow phenomena canbe predicted and potential of influencing them can be assessed.

References

[1] GAMM-Workshop on 3D Computation of incompressible internal flows, Lausanne, 1989, NNFM,Vieweg, 1993.

[2] Ruprecht, A., Heitele, M., Helmrich, T., Moser, W., Aschenbrenner, T., „Numerical Simulation of aComplete Francis Turbine including unsteady rotor/stator interactions“, 20th IAHR Symposium onHydraulic Machinery and Cavitation, Charlotte, 2000.

[3] Ruprecht, A., Bauer, C., Gentner, C., Lein, G., „Parallel Computation of Stator-Rotor Interaction inan Axial Turbine", ASME PVP Conference, CFD Symposium, Boston, 1999.

[4] Heitele, M., Helmrich, T., Maihöfer, M., Ruprecht, A., "New Insight into an Old Product by HighPerformance Computing", 5th European SGI/CRAY MPP Workshop, Bologna, 1999.

[5] Ruprecht, A., Maihöfer, M., Göde, E., „Flow analysis for the intake of low-head hydro powerplants“, 18th IAHR Symposium on Hydraulic Machinery and Cavitation, Valencia, 1996.

[6] Ruprecht, A., Maihöfer, M., Göde, E., „Untersuchung der Strömung in Kraftwerkseinläufen“, 9. Int.Seminar „Wasserkraftanlagen“, Wien, 1996.

[7] Skotak, A., “Draft tube swirl flow modelling”, IAHR WG "The Behaviour of Hydraulic Machineryunder Steady Oscillatory Conditions", Brno, 1999

[8] Kim, S.-W., Chen, C.-P., “A multiple-time-scale turbulence model based on variable partitioning ofthe turbulent kinetic energy spectrum”, Numerical Heat Transfer 16(B), 1989.

[9] Ruprecht, A., Bauer, C., Göde, E., Janetzky, B., Llosa, P., Maihöfer, M., Welzel, B., „Strömung-seffekte in einem Dreifach-Abzweiger (Trifurkation) - Numerische Analyse und deren Grenzen“,10. Internationales Seminar Wasserkraftanlagen, Wien, 1998.

[10] Hoffman, H., Egger, A., Riener, J., "Rectification of the Marsyangdi trifurcation", Hydropower intothe Next Century, Gmunden, Austria, 1999.

[11] Gentner, Ch., "Experimentelle und numerische Untersuchung der instationären Strömung in einerAxialturbine", Doktorarbeit, Universität Stuttgart, 2000.

[12] Bauer, C., "Instationäre Berechnung einer hydraulischen Axialturbine unter Berücksichtigung derInteraktion zwischen Leit- und Laufrad", Doktorarbeit, Universität Stuttgart, 2000.