fulltext.pdf

On the part load vortex in draft tubes of hydroelectric power plants

E. Göde, A. Ruprecht, and F. Lippold

University of Stuttgart, Institute of Fluid Mechanics and Hydraulic Machinery,Pfaffenwaldring 10, D-70550 Stuttgart, Germany [email protected]

Summary. For a given draft tube geometry numerical flow simulations have beencarried out. Equivalent to part load operation of a Turbine with fixed runner bladessuch as Francis- or Propellerturbines a set of different inlet boundary conditions havebeen specified to simulate the draft tube vortex. The intention was to find out somecorrelation between inlet condition and the draft tube vortex structure. The resultscan be essential for design purposes in turbine engineering.

1 Introduction

The draft tube vortex is one of the most fascinating flow phenomena in a hy-draulic turbine, but sometimes with considerable consequences on the oper-ation of the power plant. For turbine runners with fixed runner blades, whichare installed in Francis- as well as in Propellerturbines, the fluid leaves therunner with more or less swirl depending on the operating condition, if therotational speed is constant.

From kinematics it turns out that, the more the operating point is far awayfrom best condition the higher the swirl is. Well known is the cork screwrolled up part load vortex (figure 1), that rotates with a fraction of the runnerspeed. Typically the vortex rotates at a speed between 30 and 50% of the run-ner speed. Accordingly, the pressure field rotates, and since the pressure isneither constant along the circumference nor constant in time there is perma-nently an excitation to vibration. The pressure fluctuations can lead to severeoperational problems at critical operating conditions.

In recent years great progress has been achieved in order to numericallysimulate slender vortices in turbulent flows and the draft tube vortex re-spectively [1, 2, 3]. Contributions necessary for this achievement have beenmade especially in the field of turbulence modelling taking into account theanisotropic character of turbulence in the flow around a slender vortex. In ad-dition, multi scale numerical approaches have been introduced (e. g. VLES:very large eddy simulation).

218 E. Göde, A. Ruprecht, and F. Lippold

Fig. 1. Part load vortex, experiment and simulation (Francisturbine)

Fig. 2. Turbine hill chart and part load operating point (sketched)

Now it seems to be possible to answer questions such as which kinematicparameters are of major influence on the development of the draft tube vor-tex. In fact, it would be a great progress to find out by simulation to what ex-tent and through which measure the vortex and the corresponding unsteadyflow field can be changed.

In this paper the approach to simulate the draft tube flow is as follows:For a given operating point corresponding to a part load turbine operation(figure 2) a set of different boundary conditions at draft tube inlet have beenspecified. Since the operating point of the machine is fixed, the dischargeas well as the swirl at runner outlet are given for a given head. Therefore,the different boundary conditions are in fact different distributions over theradius for the through flow as well as for the swirl.

Part load vortex 219

2 Influence of runner design

For a given operating point of the turbine depending on the actual head aswell as on the discharge the flow at draft tube inlet is specified by the flowfield at runner outlet. Since the runner rotates (normally) with constant speedaccording to the frequency of the electric grid, the flow angles β2 relativeto the runner blades are nearly independent from the actual discharge, seefigure 3.

Fig. 3. Velocity triangles at runner outlet for three different turbine operating condi-tions (runner blade cascade in conformal mapping)

As a consequence, the velocity c2 in the absolute frame is very sensitiveon changes in discharge. Figure 3 shows for increasing discharge the turningof the flow vector c2, when the turbine operation is moved from part load(dark grey) to full load (green). In terms of circumferential component ofthe absolute velocity (cu2), the flow down stream of the runner rotates atpart load in the same direction as the runner. At best condition, the flow hasroughly no swirl (cu2 = 0, black colour), and at full load the flow rotates inthe opposite direction as the runner (light grey colour).

As indicated in figure 3, the relative flow angle down stream of the run-ner is similar to the blade angle at runner trailing edge. However, the two an-gles are not identical but somewhat different, which defines one of the majorproblems in the process of the runner design. In addition, the discharge atrunner outlet is a priori not constant over the outlet area but depends on thelocal turning of the flow inside the runner blade cascade. This is why evenif the power of the turbine is the same, the flow field at runner outlet can be


Fig. 4. Francis runner design, blade profiles at band

different in terms of through flow as well as swirl over the radius from hubto band.

In figure 4 an example is given to demonstrate how complex the shapeof the runner blades in a Francis turbine can be. A great number of designparameters such as number of blades, blade length, profile thickness distri-bution, blade curvature, blade angles at leading and trailing edge and soon have influence on the flow through the runner. It is obvious that for dif-ferent bladings the resulting flow field is different. However, for a requiredturbine power the necessary runner torque is given. This torque must be pro-duced no matter which pressure distribution acts locally on the blade surfaceand no matter which through flow and swirl distribution at runner outlet isachieved.

Finally, the runner wake can roughly be divided into two regions: theinner tail water region down stream of the hub and the outer through flowregion down stream of the blade trailing edges. The shape of the hub hasan influence on the tail water region, and the shape of the blades has aninfluence on the through flow as well as the swirl distribution from hub toband.

To take into account the different strategies to design a turbine runner forthe same operating condition, a set of possible boundary conditions at drafttube inlet have been specified.


3 Computational modelling

3.1 Numerical algorithms

The calculations are carried out using the program FENFLOSS which hasbeen developed at the institute for more than a decade [4, 5].

The partial differential equations are solved by a Galerkin Finite ElementMethod. The spatial discretization of the domain is performed by 8-nodehexahedral elements. For the velocity components and the turbulence quan-tities a tri-linear approximation is applied. The pressure is assumed to be con-stant within the element. For advection dominated flow a Petrov-Galerkinformulation with skewed upwind orientated weighting functions is applied.The time discretization is done by a three-level fully implicit finite differenceapproximation of 2nd order.

For the solution of the momentum and continuity equation a segregatedsolution algorithm is applied. Each momentum equation is solved indepen-dently. The momentum equations are linearized by a Picard iteration. Thelinear systems of equations are solved by the BICGSTAB2 algorithm of vander Vorst [6] with an incomplete LU decomposition (ILU) for precondition-ing. The pressure is treated by a modified Uzawa type pressure correctionscheme [5, 7]. The pressure correction is carried out in a local iteration loopwithout reassembling the system matrices until the continuity error is re-duced by a given order (usually 6-10 iterations needed).

After the solution of the momentum and continuity equations the tur-bulence quantities are calculated and a new turbulence viscosity is obtained.The turbulence equations (e. g. k- and ε-equations) are also linearized by suc-cessive substitution and the linear systems are also solved by the BICGSTAB2algorithm with ILU preconditioning.

The whole procedure is carried out in a global iteration until convergenceis obtained. For unsteady simulations the global iteration has to be carriedout in each time step.

The parallelization of the code is introduced by domain decompositionusing overlapping grids. The linear equation solver BICGSTAB2 is carriedout in parallel and the data exchange between the domains is organized onthe level of the matrix-vector multiplication in the BICGSTAB2 solver. Thepreconditioning is carried out locally on each domain. The data exchangeis organized using MPI (Message Passing Interface) on machines with dis-tributed memory. On shared-memory-computers the code is also parallel byapplying OpenMP.

3.2 Turbulence modelling

The simulation of unsteady vortex motion needs quite sophisticated turbu-lence models. When applying the “wrong” models the vortices are severelydamped and motions are unpredictable. A better approach compared to


the usually applied Reynolds-averaged Navies-Stokes simulations is a VeryLarge Eddy Simulation (VLES).

Large Eddy Simulation (LES) from the turbulence research point of viewrequires an enormous computational effort since all anisotropic turbulencescales have to be resolved in the computation and only the influence of thesmallest isotropic eddies are treated by a turbulence model. Consequentlythis method can not be applied for industrial problems today, it requires amuch to high computational effort.

Fig. 5. Modelling approach for RANS and LES

Today’s calculations of flows of practical relevance (characterized bycomplex geometry and very high Reynolds number) are usually based onthe Reynolds-averaged Navier-Stokes (RANS) equations. This means thatthe influence of the complete turbulence behaviour is expressed by meansof an appropriate turbulence model. To find a turbulence model, which isable to capture a wide range of complex flow effects quite accurate is im-possible. Especially for unsteady flow behaviour this approach often leadsto rather poor results. The RANS and LES approach is schematically shownin figure 5, where a typical turbulent spectrum and its division in resolvedand modeled parts is presented.

The recently new established approach of Very Large Eddy Simulationleads to quite promising results, especially for unsteady vortex motion. Incontrary to unsteady RANS the very large turbulent eddies are captured bythe unsteady simulation, consequently there is a requirement to the appliedturbulence model, that it can distinguish between resolved unsteady motionand not resolved turbulent motion which must be included in the model. It is


Fig. 6. Turbulence treatment in VLES

similar to LES, but only a minor part of the turbulence spectrum is resolved(schematically shown in figure 6), and therefore it is available for industrialflows today. For for details the reader is referred to [8, 9].

For comparison the vortex rope in a straight diffusor is shown for a mod-ified k-ε model (Chen&Kim version [10]) and for VLES, figure 7. It can beobserved that the damping of the vortices is reduced severely by the VLESapproach and the results are in a better agreement with measurements andobservations in the experiment.

Fig. 7. Vortex rope in a straight diffuser, k-ε Kim-Chen model vs. VLES


4 Computational grid, boundary conditions and evaluationmethods

In order to take advantage from experience with previous investigations andsince detailed measurement data were available, the draft tube geometry de-scribed in Ruprecht [11] was used for further examinations.

The new and important work to be done here is the definition of appro-priate boundary conditions. Since the impact of the direction of the velocityvectors at the draft tube inlet has to be examined, the choice of these has tobe made thoughtfully. Furthermore, the operational point fixed by the mea-surements has to be maintained. The final step in the preparation of the ex-amination of the draft tube vortex is the definition of the evaluation methodand the appraisal criteria.

4.1 Geometrical model and computational grid

The geometry examined in the paper is an elbow draft tube. It consists ofa straight intake region with a slightly opening cross section. In the elbowthe flow is redirected and finally distributed into three outlet channels. Forthe numerical analysis the geometry is discretized, which leads to a compu-tational grid consisting of about 190000 grid points and 175000 hexahedralelements, see figure 8. To carry out the computations the grid was distrib-uted onto six processors of a PC-cluster.

4.2 Definition of the boundary condition sets

In order to keep the conditions given by the chosen turbine operating point,there are certain constraints the inlet boundary conditions have to meet. Thefirst one is the conservation of the given discharge, which means that

Q =R∫

0

cz · 2π · r dr (1)

has to be constant. Furthermore, the integral swirl value must not differ fromthe original operational point. Therefore, the second condition the boundaryconditions will satisfy is

m =R∫

0

r cu · cz · 2π · r dr = const. (2)

To judge the quality of a chosen velocity distribution the relative error isused. These values are defined as follows

∆q = (Qnew − Qorig)/

Qorig ∗ 100.0%∆γ = (mnew − morig)

/morig ∗ 100.0%. (3)


Fig. 8. Computational (surface grid), approx. 175000 elements

At the outlet boundaries a constant pressure of p = 0 Pa is prescribed. Thereference inlet boundary conditions was obtained from a numerical flowcomputation in a Francis runner. As it was already shown in [11] a vortexis formed in this point of operation.

Boundary condition set 1 (cucz1)

This first generic boundary condition set models a high transport componentcz on a small radius decreasing linearly with increasing radius. The back-flow region in the centre is equivalent to the original one. Assuming a rigid-body-like swirl distribution at the inner and a constant cu at the middle andouter part yields the r*cu distribution shown in figure 9. The relative errorsare ∆q = 0.001% and ∆γ = 0.047%.

Boundary condition set 2 (cucz1)

In order to obtain comparable results only one value of the boundary con-ditions should be changed at once. So, this set models the same swirl dis-tribution described above (cucz1), but, an increasing transport velocity com-ponent with increasing radius, see figure 10. The relative errors are ∆q =0.01% and ∆γ = −0.023%.


Fig. 9. Boundary condition set 1 (cucz1)

Fig. 10. Boundary condition set 2 (cucz2)

Boundary condition sets 3 & 4

These sets combine the cz-distributions of set 2 and 1 with a cu-curve de-creasing from inner to outer radii. This yields the declining swirl curveshown below in figure 11. Relative errors are again considerably low ∆q =0.01% and ∆γ = −0.049% and ∆q = 0.001% and ∆γ = 0.013%, re-spectively.

4.3 Appraisal factors

Two main issues coming with the draft tube vortex are the pressure fluctu-ations exciting the whole hydraulic system and the pressure amplitudes. Toobtain comparable values for all test cases some characteristic numbers re-flecting the behaviour under the given conditions have to be defined. First,this is the frequency of the pressure pulsations in a certain point. Second, theintermediate pressure amplitude, which means the mean peak to peak value.The curves in figure 12 give an impression of how the amplitude, which isthe distance between the two lines (p_max/min,mean), is obtained. The peak


Fig. 11. Swirl distribution for sets 3 & 4

value curves (p_max/min) are integrated in the time domain and the meanvalue is then obtained from the integral value divided by the total time pe-riod. This is, admittedly, an estimation, but, it will show the right trend be-tween the single cases.

Fig. 12. Determination of the mean pressure amplitude

5 Simulation results

5.1 Geometrical model and computational grid

In order to demonstrate the accuracy of the numerical simulation, a com-parison with measurement data is shown according to [11] for the originalboundary condition. In figure 14 the measured and computed pressure fluc-tuations are given for point no. 3 at the draft tube cone, figure 13. The FFT-analysis carried out for both measured and computed pressure signals veri-fies a quite good coincidence in terms of frequency as well as the amplitude


Fig. 13. Positions of pressure measurement probes

of the signals, figure 14. This is why the same set-up for the flow analysiswas chosen here using the specified sets of swirl distribution.

5.2 Influence of boundary condition

In figure 15 two vortices are visualized by using a constant pressure sur-face. It can be seen that the rope produced with the cucz1-boundary condi-tion (light) is shorter and more slender than the original one (dark). This is incontrast to the cucz2-case where the rope is thicker and longer than the orig-inal one. In the case of high discharge near the hub the rope has less room tobe formed and vice versa, which is the reason for the phenomenon describedabove.

A remarkable discovery made here is the correlation between frequencyand amplitude. The values for point 3 (CH3) given in figure 16 show thatthe pressure fluctuations increase with decreasing frequency and vice versa.Since the rope hits the wall next to point 4 (CH4) there is a higher amplitudethan at the upper point 3. Both frequencies are identical.

Another aspect is that there seems to be no major influence of the swirldistribution on the values analyzed here. The results presented indicate thatfirst of all the transport velocity has an impact on frequency and amplitudeof the pressure pulsations.

Furthermore, it turns out that the time-averaged pressure recovery co-efficient of the draft tube increases with increasing frequency, figure 17 vs.figure 16. Here the amplitude of the pressure fluctuations comes into play.Since, the higher the discharge at the hub yields a longer and more slenderrope the blocking of the cross section will decrease. Hence, the losses forthese cases are less and the recovery coefficient is higher.

6 Further investigations and outlook

Since the effects found above are hard to explain in detail for this complex ex-ample further examinations on a simpler geometry could be useful. Severalimpacts on the flow have to be taken into account, for instance the velocity


-60000

-50000

-40000

-30000

-20000

-10000

0

10000

20000

30000

40000

0 0.5 1 1.5 2 2.5 3

pres

sure

[Pa]

time [s]

ch3 simulationch3 messurement

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

ch3 simulationch3 measurement

rela

tive

pres

sure

am

pli tu

d e

Frequency [Hz]

Fig. 14. Measured and computed pressure signals (above), FFT-analysis (below)

distribution at draft tube inlet, second, the conical shape of the draft tube,third, the elbow itself, fourth the turbulence modelling. To understand thebasic mechanism it would be useful to separate all these effects as far as pos-sible and to start with simplified geometries, and then to increase the com-plexity of the problem successively. Moreover, the consistency of the bound-ary condition sets has to be studied more deeply, this can be accomplishedby a theoretical approach of Resiga [12]. Finally, instead of using artificialboundary conditions, an actual runner design should be used to determinemore realistic boundary conditions for the draft tube inlet.


Fig. 15. Iso pressure surfaces obtained by the simulation of the vortex rope for (above)original (dark) and modified (light) cucz1 boundary condition and original (dark) andmodified (light) cucz2 boundary condition

Fig. 16. Rope frequencies and mean pressure amplitudes


Fig. 17. Pressure recovery coefficients (time averaged) for all bc-sets [ cp = (pin –pout)/(v2

in/2) ]

References

1. Ruprecht A, Helmrich Th, Aschenbrenner Th, Scherer Th (2001) Simulation ofpressure surge in a hydro power plant caused by an elbow draft tube. In: Proc.of the IAHR WG 1 Symposium The Behaviour of Hydraulic Machinery underSteady Oscillatory Conditions, Trondheim.

2. Helmrich Th, Ruprecht A (2001) Simulation of unsteady vortex rope in turbinedraft tubes. In: Proc. of the Hydroturbo 2001, Podbanske, Slovak Republic.

3. Ruprecht A, Helmrich Th, Aschenbrenner T, Scherer T (2002) Simulation of vortexrope in a turbine draft tube. In: Proc. of the 21th IAHR Symposium on HydraulicMachinery and Systems, Lausanne

4. Ruprecht A (1989) Finite Elemente zur Berechnung dreidimensionaler turbu-lenter Strömungen in komplexen Geometrien. Doctorate Thesis, University ofStuttgart.

5. Ruprecht A (2003) Numerische Strömungssimulation am Beispiel hydraulischerStrömungsmaschinen. Habilitationsschrift, Universität Stuttgart

6. Van der Vorst HA (1994): Recent developments in hybrid CG methods. In: Proc.of the High Performance Computing & Networking, München.

7. Zienkiewicz OC, Vilotte JP, Toyoshima S, Nakazawa S (1985) Comp Meth ApplMech Eng 51: 3-29

8. Ruprecht A, Helmrich Th, Buntic I (2003) Very large eddy simulation for the pre-diction of unsteady vortex motion. In: Proc. of the Conference on Modeling FluidFlow, Budapest.

9. Helmrich Th, Buntic I, Ruprecht A (2002) Very Large Eddy Simulation for flowin hydraulic turbo machinery. In: Proc. of the Classics and Fashion in Fluid Me-chanics, Belgrade.

10. Kim SW, Chen CP (1989) Numer Heat Transfer 16(B): 193-22111. Ruprecht A (2002) Unsteady flow simulation in hydraulic machinery. In: IAHR,

Task quarterly 6 No 1., 187-208.12. Romeo S-R (2004) Swirling flow downstream a francis turbine runner. In: Proc.

of the 6th international conference on Hydraulic Machinery and Hydrodynamics,Timisoara, Romania.

fulltext.pdf

Documents

Transcript of fulltext.pdf