hydraulic design and analysis of the saxo-type vertical ......the criteria used in the hydraulic...

HYDRAULIC DESIGN AND ANALYSIS OF THE SAXO-TYPE VERTICAL AXIALTURBINE

Edvard Hofler1, Janez Gale2, Anton Bergant2

1Litostroj Power, d.o.o, retired

2Litostroj Power, d.o.o, Ljubljana, Slovenia

E-mail: [email protected]; [email protected]; [email protected]

Received July 2010, Accepted February 2011No. 10-CSME-58, E.I.C. Accession 3221

ABSTRACT

The paper presents a procedure for hydraulic design and analysis of the blade geometry of ahigh specific speed runner of the Saxo-type double-regulated vertical axial turbine. Themeridional through-flow in the passage from the conical guide vane apparatus to the draft-tubeelbow is designed by a streamline curvature method (SCM). To validate the design method andpredictions and to investigate the design duty point and a number of off-design operatingregimes, an extensive CFD simulation inside the entire turbine water-passage is performed. Theflow patterns downstream the guide vane apparatus and the runner exit flow are analyzed. Thefocus of the analysis is on distribution of the angular momentum alongside the turbine, as wellas on its impact on the flow around the runner blades. The SCM design procedure presented inthe paper proves to be a robust and accurate tool for the runner blade row design.

Keywords: hydraulic turbine; streamlines curvature method; runner design; CFD.

CONCEPTION ET ANALYSE HYDRAULIQUES D’UNE TURBINE AXIALEVERTICALE DE TYPE SAXO

RESUME

Cet article presente un procede de conception et d’analyse hydrauliques applicable a lapalissade d’une roue a vitesse specifique elevee dans le cas d’une turbine axiale verticale adouble reglage de type Saxo. Selon ce procede, d’abord, l’ecoulement meridien entre ledistributeur conique et le coude d’aspirateur est concu par la methode de courbure des lignes decourant (MCLC). Ensuite, pour confirmer les resultats de cette methode et les predictions etpour examiner les points de fonctionnement de la turbine en charges nominale et partielles, nousfaisons une simulation MFN approfondie de l’interieur de l’ensemble du passage d’eau; nousetudions la configuration des ecoulements en aval du distributeur et a la sortie de la roue,notamment la distribution du moment angulaire le long de la turbine et l’influence de cemoment sur l’ecoulement autour des pales. La MLCL presentee dans l’article se revela un outilfiable et precis de conception des pales de roue.

Mots-cles : turbine hydraulique; methode de courbure des lignes de courant; conception deroue; MFN.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 35, No. 1, 2011 119

NOMENCLATURE

A area (m2)a0 dimensionless opening of the

guide vanesA0 opening of the guide vanes (m)B width (m)c absolute velocity (m s21)cd dissipation coefficientCp pressure coefficientd thickness (m)D diameter (m)Dx measuring sphere diameter of

the guide apparatus (m)E total specific energy (m2 s22)Fd dissipation force (N kg21)g acceleration due to gravity

(m s22)G function (m21)H turbine net head (m)I rothalpy (m2 s22)i streamline labelj QO-line labelkx mapping constantK function (m s21)L functionLs length of the profile camber

(m)m distance along the meridional

direction (m)m* dimensionless distance along

the meridional directionn speed of rotation (min21)nq, ns specific speed (min21)N function (s21)p static pressure (Pa)P power (W)q distance along the quasi-

orthogonal line (m)qm mass flowrate (kg s21)Q volume flowrate (m3 s21)r radius (m)Rc radius of the streamline curva-

ture (m)Rcone radius of the cone describing

the guide apparatus (m)

s course on the streamsurface(m)

t blade or vane spacing (m), orpitch (m)

u, U blade speed (m s21)w relative velocity (m s21)x coordinate of the mapping

planey coordinate of the mapping

planez axial coordinate, axis of rota-

tion (m)Z number of blades, or vanesGreek symbols

b relative flow angle, measuredfrom the circumferential direc-tion (rad)

d relative flow deviation angle(rad)

e blade lean angle (rad)w stream surface pitch angle

(rad)W function, or quantityg efficiencyi incidence angle (rad)Q blade-wrapping angle (rad), or

discharge numberq swirl angle (rad)ns specific speedh circumferential coordinate

(rad)p Ludolf ’s numberr fluid density (kg m23)V angular speed (rad s21)y angle between the meridional

direction and QO (rad), orenergy number

Subscripts

2 runner inlet3 runner outletav averaged dissipationi index of the streamlinej index of the QO-linegv guide vane


1. INTRODUCTION

The paper investigates a special type of the double-regulated vertical axial turbine equippedwith the Kaplan runner. In North America, the turbine is known as the Saxo-type turbine (S axialturbine) because of its similarities with the S-type turbine. Figure 1 shows a typical turbine layoutwith two alternative intake structures. A compact elbow with stay vanes located downstream theintake provides a uniform flow field at the entrance of the conical guide vane apparatus designedto generate a sufficient swirl at the entrance to the runner. The conical guide vane apparatus istypical for the bulb-type turbines, whereas the runner with the elbowed-draft tube is typical forthe Kaplan turbines. Due to their compact design, the Saxo-type turbines are suitable for netheads from some metres up to more than 30 metres covering the range of both the bulb-and theKaplan turbines. This type of the turbine has been accepted as an alternative to the small classicalKaplan turbines in a number of industrial projects (Hofler [1], Gale et al. [2]).

Early development in the Saxo-type turbines revealed that the hydraulic design of the turbineshould treat the conical distributor, runner and draft tube with an elbow as a single domain [3].

Fig. 1. Typical layout of the Saxo-type turbine with an alternative intake with a penstock.

h hubloss lossm meridional componentp pressure sider radial componentrb runner bladeref reference values suction side

u, h circumfential - tangential com-ponent

AbbreviationsLE leading edgePS pressure sideQO quasi-orthogonal lineSS suction sideTE trailing edge


The same principle applies also to model tests [4] because only a homologous model of thecomplete water passage provides accurate data on the turbine performance characteristics.

The objective of the paper is to present development and design of a high-specific speed four-blade Saxo turbine on the basis of the layout of the existing medium-specific speed five-bladeturbine. The hydraulic shape of the intake structure, conical guide vane apparatus and drafttube with an elbow of the new high-specific speed turbine are assumed to be the same as theones of the existing medium-specific speed turbine, thus making the new element to bedeveloped as a four-blade runner. The turbine was designed to have its best efficiency at theduty point with the characteristic discharge number Q 5 0.210 and energy number y 5 0.270with the characteristic dimensionless numbers are defined as follows:

Q~4Q

pD2Uy~

2E

U2ð1Þ

where U 5 V.D/2 is the runner tip speed, V is the angular speed and the corresponding specificspeed is ns 5 1.224 (nq 5 193 min21 or ns 5 705 min21). Note that there are several definitionsfor the specific speed:

ns~Q1=2

y3=4nq~n

Q1=2

H3=4ns~n

P1=2

H5=4ð2Þ

where n is the number of the runner rotations per minute, H 5 E/g is the net head in metres, E isthe specific energy, and P is the hydraulic power in horsepower units.

The runner blade design was performed for a turbine in the model scale with the followingcharacteristic parameters:

- turbine rotational speed: n 5 1200 min21

- runner reference diameter: D 5 350 mm

- hub diameter: Dh 5 133 mm

- number of guide vanes: Zgv 5 16

- number of runner blades: Zrb 5 4

Initially, four different blade shapes were designed by using the streamline curvature method(SCM). They were then thoroughly analysed with the aid of commercial computational fluiddynamics (CFD) tools. The criteria used in the hydraulic design were the turbine efficiency andcavitational characteristics.

The paper starts by presenting concepts of SCM used in the basic design of the SAXO-typeturbine runner blade row. To follow is a brief outline on the use of the CFD code in the viscousflow analysis of the SAXO-type turbine. The main focus of the paper is on investigation of theflow characteristics within the SAXO turbine blade row and on a comparison of the SCM andCFD results. Design guidelines are given for hydraulically favourable blade shapes. The resultsof the blade with the best turbine performance characteristics are presented in the paper.

2. BASIC HYDRAULIC DESIGN USING THE STREAMLINE CURVATUREMETHOD

The streamline curvature method (SCM) is a traditional method for the prediction of a flowfield in turbomachinery. The method is widely used in development of different types of the


turbomachinery and is regarded as an appropriate tool in designing the basic fluid passageshapes of the turbomachine components. The final selection of the best aero-hydraulics is basedon numerical simulations using a viscous fluid flow solver (CFD) and model testing (forexample [5]). The SCM is suitable for blade design in the best efficiency point (design dutypoint), flow analysis at off-design conditions and analysis of predicted aero-hydraulic shapes.

The SCM presented in this paper is based on a generalized equation of the radial equilibriumand a procedure known as »throughflow method analysis«. The theoretical background of themethod and the method of solution were developed by Smith [6], followed by Novak [7], Novakand Hearsey [8] and Wilkinson [9] who systematized the computing approach. Wennerstrom[10] made an accurate account of the blade and dissipation forces on the meridional flow field.His approach was then used by Bohn and Kim [11], Casey and Roth [12] and Casey andRobinson [13]. A more recent overview on the structure of SCM was given by Cumpsty [14] andSchobeiri [15]. The latest paper of Templalexis et al. [16] (dealing with axial compressors) showsthat the developments in this area are still in progress. Despite the widespread prevalence of theSCM, in fact all published studies are related to the compressible fluid turbines andcompressors. Rare are readings concerning turbopumps for incompressible fluid. In particular,published applications of the SCM on the hydraulic turbines are indeed very rare and this paperis aiming to fill the gap.

Bajd [17] developed the SCM for design of Francis turbines. Hothersall and Huntsman [18]briefly introduced the SCM as a preliminary design tool for hydraulic turbines. Hofler et al. [3]compared results obtained with a viscous fluid flow solver and SCM; the Wennerstrom’sapproach [10] was used in the SCM. A similar and upgraded method and code for the analysisof meridional flow field in the guide vane cascade and the design of new runner blades of thevertical tubular hydraulic turbine is presented in this paper.

The SCM assumes stationary axisymmetrical flow at the meridional flow surface between thehub and the shroud of the turbine. In the vaneless domain, the flow properties are averaged atthe circumference. At the blade-to-blade surface, the flow properties are treated at thestreamline dividing the flow field between two blades. The approach is the same for the fixedand the rotating blade rows.

2.1 Equation of the Radial EquilibriumThe SCM method is based on a differential equation for the radial equilibrium describing

variations in the meridional component of absolute velocity cm along curved coordinate q.Coordinate q (quasi-orthogonal QO) is arbitrary and almost perpendicular to the meridionalstreamlines. The selected coordinate system (r-h-z) is fixed in the space as shown in Fig. 2. Theequation of radial equilibrium for the incompressible fluid is given as [10]:

cm

dcm

dq~

dE

dq{

cu

r

d rcuð Þdq

zc2

m

Rc

sinyzcm

Lcm

Lmcosyz

cm

r

L rcuð ÞLm

tan ezFd ð3Þ

where E is the total specific energy, cu is the circumferential component of the absolute velocity,y is the angle between meridional direction and QO-line, e is the blade lean angle, Rc is theradius of the streamline curvature and Fd is the dissipation force. The geometric parameters aredepicted in Fig. 2.

The key unknown is meridional velocity cm along the QO-line from the hub A to the shroud B(Fig. 2). The radial equilibrium equation is solved along the QO at distinct nodes defined by


crossing the QO-lines and meridional streamlines whose location is adjusted to preserve partialdischarge balance until a converged solution is obtained.

Herein, the system of equations to be solved includes the following equations: (i) thecontinuity equation, (ii) the energy equation (Euler’s turbomachinery equation), and (iii) themomentum law for fluid on mean flow surface between the blades, Eq. (3), that is valid alongthe QO-line. The mean stream surface in the blade row approximately follows the blade cambersurface.

2.2. Solving the Equation of the Radial EquilibriumBajd [17] and Hofler et al. [19] solved the equation of the radial equilibrium, Eq. (3), in an

analytical manner. The equation is integrated along each QO-curve (j 5 const) introducingspecial solution functions. Equation (3) is rearranged as follows:

dcm

dqzG qð Þcm~N q,cu,cmð Þ ð4Þ

where function G involves gradients of meridional velocities and terms describing geometry ofcoordinates q and m,

G qð Þ~{siny

Rc

{cosy

cm

Lcm

Lmð5Þ

while function N incorporates terms describing the shape of streamlines and energy conversion:

N q,cu,cmð Þ~ 1

cm

dE

dq{

cu

cm

d rcuð Þrdq

z1

r

L rcuð ÞLm

tan ez1

cm

Fd ð6Þ

Equation (4) is a quasi-linear ordinary differential equation with the following solution ateach iteration step along the line j 5 const:

Fig. 2. Coordinate system and definition of geometry.


cm qð Þ~K qð ÞzL qð Þ:cm 0ð Þ ð7Þ

with cm(0) defined at the hub (q 5 0) and function L(q) being the solution of the homogeneousequation:

L qð Þ~exp {

ðq

0

G qð Þ:dq

24

35 ð8Þ

Particular solution K(q) of non-homogeneous equation is:

K qð Þ~L qð Þ:ðq

0

N q,cmð Þ:exp

ðq

0

G qð Þ:dq

24

35

8<:

9=;:dq ð9Þ

Equation (4) is solved simultaneously with the continuity equation to account for the partialdischarge balance along each QO-line:

Q~2p

ðB

A

r:cm:siny: 1{drb=tð Þ:dq ð10Þ

Term (1 2 drb /t) , 1 represents the flow blockage, caused by the actual blade thickness onthe flow section; drb is the blade thickness in the circumferential direction and t is the bladespacing, both accounted for the calculating node. The proposed method for calculation of themeridional velocities along each QO-line proved to be numerically efficient and stable. To closethe system of equations, two additional equations are needed to describe the relation betweentangential velocity cu and meridional velocity cm, and static pressure p and velocity c. Bothclosure relations depend on the stator or rotor domain and position (i, j) inside the calculatingdomain.

2.3. Flow-field Downstream the Guide VanesA particularity of the conical guide vane apparatus in comparison to the cylindrical one,

which is traditionally used for the Francis and Kaplan turbines, is a combined swirl-flow fieldincluding free and forced vortex components. The streamlines close to the shroud gain a largerangular momentum than the streamlines closer to the turbine hub. Figure 3 depicts thegeometry of the conical guide vane apparatus. The dimensionless guide vane opening is givenwith the below conventional equation:

a0~A0:Zgv

Dx

ð11Þ

where Dx represents the characteristic diameter of the guide vane sphere at which opening A0

(line C-D in Fig. 3) has to be measured and Zgv is the number of the guide vanes. An important


quantity to be considered is swirl angle q. The swirl angle is directly given with distributordimensionless opening a0 for a particular guide vane layout; moreover, it is practically the samealong the full span of the guide vane, and the predicted function q 5 q(a0) holds for allstreamlines under consideration (for further information see [3]).

The change in the angular momentum through the runner blade row can be defined with theEuler’s turbine equation:

g23:E~V rcuð Þ2{ rcuð Þ3� �

ð12Þ

where g23 is the total turbine efficiency between sections 2 and 3 (Fig. 4), E is the total specificenergy of the fluid flow at the design point, and rcuð Þ2 and rcuð Þ3 are the flow averaged angularmomenta over sections 2 and 3, respectively.

2.4 Geometry Description in the Meridional Cross SectionTo evaluate the meridional velocities in the computational domain, projection of the

streamlines in the meridional cross-section at each calculating node and at each iteration needsto be known. The computational grid in the meridional cross section is composed of meridionalstreamlines m and QO-lines in Fig. 4. The QO-lines may be arbitrary curves. However, it isrecommended that the QO-lines at the blade inlet and at the blade outlet coincide with themeridional projections of the actual blade inlet and outlet edges. The same principle applies tothe stay vanes and guide vanes.

The initial m-QO grid is approximated. In the iteration procedure, the shape of thestreamlines is modified along the QO-lines to preserve mass conservation. The QO-lines areapproximated as parabolas of the third order. The streamlines, hub and shroud contours areapproximated with segments as quadratic parabolas [12, 19].

The main parameters describing the runner blade camber surface are depicted in Fig. 4.Coordinates of calculating nodes rij, hij and zij are defined in the cylindrical coordinate system.An important parameter defining the length of the blade in the direction of a relative flow iswrapping angle Q. The wrapping angle may be changed during calculation according to thelocal load conditions at each iteration. The leading edge is defined with relative displacementangle DQi.

Fig. 3. Conical guide vane apparatus: its geometry and swirl generation.


2.5 Rotational Stream SurfaceThe next step in the blade design is transformation of the m2h grid laying on the rotational

surface to the x–y plane. The transformation is known as a conformal mapping and is givenwith the following expressions:

x~1

kx

ðq

2

dm

r, y~

1

kx

ðq

2

dh, and kx~

ð3

2

dm

rð13Þ

Fig. 4. Meridional cross-section of the turbine and a grid of streamlines m and QO-lines.


Figure 5 shows a streamline and a blade profile mapped into a plane. The geometry of theblade is governed by the change in angular momentum @(r.cu)/@m along the meridionalstreamline. The shape of the relative streamline from the inlet to the outlet can be approximatedas a power series of the fourth order:

y~X4

i~1

aixii ð14Þ

In this manner, the components of absolute velocity c and the angular momentum:

rcuð Þ~r rV{cm

tanb

� �ð15Þ

can be predicted at each calculating station.

The next step is to define the profile camber line which is different from the relativestreamline due to local flow deviations accounted for with incidence angle i and outflowdeviation angle d in Fig. 5. The camber surface that is finally defined with the r2h2z grid inFig. 4 is thickened in order to obtain an actual hydrodynamic profile along the streamline (j 5

const). At each calculating node, the normal to the stream surface is defined with an addedblade thickness. In our case, the blade profile at the hub follows the NACA 63A standardprofile with a thicker outlet edge. At the shroud, the blade profile is more uniform (British C4)thus increasing the blade stiffness.

Velocity and pressure at the rotational stream surface

The velocity and pressure distribution along the blade are very important for the bladedesign. The change in flow velocity w in circumferential direction h (blade-to-blade surfacem2h, Fig. 6) may be approximated as a vortex-free or potential flow at the rotational streamsurface under consideration.

Fig. 5. Streamline and blade profile mapped into a plane; point q is a calculating node and p and rare adjacent nodes.


The final equation of velocity deviation Dw in the pitch direction of the blade passage is:

Dw~{p

Zrb

1{Zrbdb

2pr

� �sin b

L rcuð ÞLm

ð16Þ

where Dw 5 (ws – wp)/2 is the difference in the average relative velocity, Zrb is the number ofblades, db is the blade thickness in the circumferential direction, b is the relative flow angle andL(r.cu)/Lm is the change in the angular momentum along the meridional streamline.

The pressure difference Dp 5 pp – ps , known as the blade pressure jump, at an assumedconstant total pressure in circumferential direction h at a constant radius r 5 const is:

Dp~{r2p

Zrb

1{Zrbdb

2pr

� �wm

L rcuð ÞLm

ð17Þ

where r is the fluid density and wm is the meridional velocity.

2.6 Energy lossesEnergy losses (profile losses) DEd in the blade channels are estimated according to the models

developed by Denton [20] and his below equation in which for hydraulically smooth runnerblade friction losses are accounted for:

DEd~1

DQcdBLs

ð1

0

w3d s=Lsð Þ ð18Þ

where DQ is the partial turbine discharge, cd is the dissipation coefficient (in this study it isassumed constant cd 5 0.002); B is the wetted width, Ls is the wetted profile length and w is therelative velocity. The same principle is used in predicting the losses in turbine stationary parts.Losses DEd are then used to predict dissipation force Fd appearing in Eqs. (3) and (6). Casey and

Fig. 6. Flow in the blade-to-blade stream surface and the corresponding velocity triangle.


Roth [12] developed an improved equation for the dissipation force including effects of theactual flow angles:

Fd~ cosy sin2bztane sinb cosb� � dEd

dmð19Þ

Besides the friction forces, there are also the mixing losses in the wake downstream thetrailing edge (Denton [20]) considered in the analysis; an example of the mixing losses in aconical guide vane apparatus is given by Hofler et al. [3].

2.7 Summary of the Design ProcedureFigure 7 shows a flow diagram of the runner blading design procedure. The numerical

algorithm is based on a solution of equations including continuity Eq. (10), energy Eq. (12)(Euler’s turbine equation) and Eq. (3) for the radial equilibrium valid along the QO-line asexplained above.

3. CFD VISCOUS-FLOW ANALYSIS IN THE ENTIRE SAXO-TYPE TURBINE

To support and verify the SCM-based blade design an extensive Computational FluidDynamics (CFD) simulation was performed inside the entire turbine water-passage systemusing the commercial computer code ANSYS CFX 11.0 [21]. Keck and Sick [22] performed anexcellent review of the main steps in the application of the CFD in the design of hydraulicturbines and the breakthroughs that were made in the last decades. The CFD computer codecalculates flow field inside the computational domain by using the numerical solution of theNavier-Stokes system of equations. The governing equations in conservative form arenumerically solved for each discrete control volume within the computational domain usingfinite volume method. More details on numerical models and solvers behind the CFD codes canbe found in references [23, 24].

The flow analysis in the present study was undertaken in two steps: (1) CFD code validationusing an existing five-blade SAXO turbine and (2) CFD analysis of a new four-blade SAXOturbine. In the first step, the existing five-blade SAXO turbine was modelled and the resultswere compared to the available results of the extensive model testing. The whole operatingrange was under observation, which means that propeller diagrams at constant runner bladeopening were constructed for runner blade angles ranging from 4u to 28u. Then the hill chartwas made and compared to the hill chart of the model testing. The calculation settings andinitial and boundary conditions had been adjusted in a way that overall agreement betweenCFD and model hill chart was less than 0.5 % in proximity of the best efficiency point and lessthan 1 % over the whole remaining operating range. In the second step, the validatedcomputational model was applied in the development of the four-blade SAXO turbine byreplacing the five-blade runner with the newly developed four-blade runner. The hydraulicshape of all other parts of the turbine flow-passage system (intake, elbow, guide vanes and drafttube) remained the same. The results of the second step are presented and discussed in thispaper.

Figure 8 shows the computational domain of the entire SAXO turbine water passage appliedin the second step, which included the intake, the elbow with guide plates, the vertical shaft, thestay vanes and conical guide vane apparatus, the rotating four-blade runner, and the draft tube.The size of the computational domain was scaled to the model turbine size with a runner


Fig. 7. Flow diagram of the runner blading design procedure.


reference diameter of D 5 350 mm. The computational domain was assembled from severalsubdomains, which were meshed using the structured HEXA type mesh. However, the elbowwith guide plates and the runner used the unstructured TETRA type mesh. The general gridinterface (GGI) was applied at subdomain interfaces and Frozen rotor option for rotatingsubdomains. The grid refinement study was performed and mesh with about 3.106

computational volumes for entire SAXO turbine water passage was selected as optimal(CPU time, RAM memory, convergence, accuracy of the results). All main mesh qualityparameters were checked (Determinant 26262, Angle, Warpage etc.) and the y+ parameter inthe main water-passage sections spanned from 15 to 160.

The computational domain was filled with single-phase water, the reference pressure at thetop of the outlet cross-section was pref 5 0 Pa. The flow was not buoyant; the k-e based Shear-Stress-Transport model (SST) with automatic wall functions was applied. The mass flow rateboundary condition was defined at the intake and the static pressure for entrainment at theoutlet. Walls of the water passage were treated as walls without slip. The steady state analysiswas performed and the convergence criterion targeted was RMS , 1025. The CFD results wereused to study the characteristic flow behaviour in the core part of the turbine and to compareand verify hydraulic design of the turbine based on the SCM method. Fig. 9 shows turbine hillchart as a function of dimensionless energy number and dimensionless discharge number. Notethat the name B14-A1.8-37 in the legend stands for the case with a runner blade angle of bb 5

140, with a dimensionless guide vane opening of a0 5 1.8, and discharge of Q 5 0.37 m3 s21.

4. ANALYSIS OF OBTAINED NUMERICAL RESULTS

The flow characteristics within the SAXO turbine blade row and the SCM and CFDcomparison results were investigated. Local quantities W(r, h, z) change over the consideredcurve, cross-section or plane. The mass-averaged value of the related quantity W on the curve orcross-section was obtained by integrating over the curve or cross-section, respectively:

Fig. 8. Computational domain of the SAXO-type turbine model.


~WW~1

qm

ðr,h,z

W(r,h,z)dqm ð20Þ

4.1 Flow Conditions Downstream the Guide Vanes - Surface A in Fig. 4The flow field downstream the conical guide vane row (surface A in Fig. 4) depends on

dimensionless guide vane opening a0 and discharge Q. Fig. 10 shows analysis results.Meridional component cm is divided with average discharge velocity cav 5 Q/AA and theangular momentum is divided with reference momentum (R.U)ref, where R 5 0.175 m is therunner radius and U is the peripheral speed at this radius. Each value is circular and mass-averaged. The CFD-simulated case that corresponds to the SCM design point is the caselabelled with B14-A1.8-37. The shape of the meridional velocity and normalized angularmomentum curves for the simulated and design operating point is identical except in the areaclose to the wall affected by wall friction and persistent forces.

4.2 Conditions Downstream the Runner 2 Plane C in Fig. 4Figure 11 shows that the velocity relation and the normalized angular momentum are

strongly related to the considered operating point. Although the simulated operating points layclose to the turbine BEP, the guide vane opening strongly affects intensity and direction of the

Fig. 9. Position of simulation points in the discharge – energy hill chart.


departure swirl (cu/cm). The SCM design conditions in plane C were confirmed with the CFDsimulation at the B14-A1.8-37 operating point.

4.3 Distribution of the Angular MomentumAs already mentioned above, the guide vanes generate a swirl giving rise to a flow field with

non-uniform total energy E at a span of the passage. This can also be seen from Fig. 10(b)showing distribution of normalized angular momentum (r.cu). Actually, what we are searchingis distribution of the angular momentum along the entire investigated turbine water passage.Figures 12(a–b) show a normalized angular momentum and a derivative of the angularmomentum along meridional streamline at three fractions of the passage height (span 0.1, 0.5and 0.9).

Figure 12(a) shows distribution of a circumferentially averaged angular momentum alongsidethe water passage in the meridional direction. The angular momentum and the meridionallength are normalized (the runner blade leading edge: mLE* 5 0, the runner blade trailing edge:mTE* 5 1). The drop in the angular momentum through the runner directly shows the hydraulicenergy transferring from the water to the runner blading, and the derivative of the angularmomentum Fig. 12(b) shows the intensity of the momentum change along the streamline. Thederivative at the span 5 0.9 (close to the shroud) is interesting for showing that depreciation ofthe swirl starts before it reaches the leading edge of the blades and it gets the lowest valuescompared to other span sections.

4.4 Flow Around the Runner BladeFigure 13 presents the flow around the runner blade at three typical sections for the B14-

A1.8-37 case. The velocity vectors show a hydraulically perfect flow into the blade and alongthe blade for more than 80 % of the passage span. The last section near the shroud (span 5 0.9)is hydraulically less ideal, particularly at higher velocities on the suction side just after theleading edge. However, this is negligible as it is evident that there is no aware streamlinestagnation or flow separation.

Another interesting analysis of the flow around the runner blade was made by defining thelayer around the blade at an offset distance of 0.4 mm. At this layer, we were looking forcomponents of a relative velocity in the circumferential h and axial z direction. The radial

Fig. 10. Flow and energy conditions at the surface downstream the guide vane apparatus.


component of velocity wr indicates exchanges in the water passage channel (contracting orspreading) or local to the changes in the blade-surface geometry. Each larger discontinuity ofthe radial velocity directly implies a flow-field defect. Figure 14 shows distribution of the radialvelocity component on the layer. At the pressure side, the discontinuity is evident near theleading edge close to the hub, but a much stronger radial flow is arising on the last third of theblade inlet part close to the shroud. In both cases, the layer flow is directed toward the shroud.In the first case, the blade surface around the hub is most oriented toward the shroud thus theflow away from the hub. The other parts of the pressure side surface generally exhibit a circularflow. At the suction side, one can see the velocity vectors pointed inward flow. Thisphenomenon cannot be interpreted as a separated layer, because it would be advisable tocentrifuge a blocking layer.

The influence of the observed phenomenon was further investigated. Figure 15 shows theflow field of one runner blade - the situation for other three blades is identical. The firstimpression suggests that the relative flow is generally rotational and without the local stops,pointing to the trace of the vortices, which could be sourcing upstreams in the blade channel.Obviously, the radial component of relative flow in the boundary layer along the blade leadingedge, which is shown in Fig. 14, provides very low intensity that quickly dissipates and it has nodetectable effect on the flow leaving the runner.

4.5 Runner Blade LoadingThe static pressure coefficient is defined with the following equation:

Cp~p{pref

� �1=2r:w2

ref

ð21Þ

where pref and wref are the pressure and relative velocity at the reference point located in themiddle of the exiting part of the blade-to-blade channel at the radius r 5 125.5 mm (see Fig. 4).The CFD analysis yields the flow field for the whole runner. To eliminate possible

Fig. 11. Flow and energy conditions at plane C downstream the runner.


non-homogeneity of the flow, the results presented herein (i.e. values pref and wref) for one bladewere averaged over all the four blades. The results are presented in Fig. 16.

One of the basic objectives of our investigation was to determine the pressure load on therunner blade. Figure 17 shows the pressure distribution over the pressure and suction side of theblade at typical cross-sections. The highest pressure appears at the inlet part of the profile withthe local and relative stagnation pressure being equal. The highest under pressure appears at theoutside of the blade and close to the leading edge. These results confirm other results obtainedwith our analysis of the angular momentum gradient Fig. 12(b) and Eq. (17) which relates the

Fig. 12. Distribution of circumferentially averaged angular momentum alongside the turbinepassage.

Fig. 13. Flow at the blade-to-blade stream surface; velocity vectors (case B14-A1.8-37). Span 5 0.1.Span 5 0.5. Span 5 0.9.


Fig. 14. Radial component of the relative velocity on a layer 0.4 mm off the blade surface (case B14-A1.8-37). (a) Blade pressure side. (b) Blade suction side.

Fig. 15. Control plane close to the runner blade trailing edge; the relative velocity vectors projectedinto the axial plane (case B14-A1.8-37).


angular momentum derivative and the pressure jump. The SCM predicted pressures are in goodagreement with the CFD simulated pressures over 75 % of the blade surfaces, while on the restof the surfaces, discrepancies are larger due to a slightly misguided flow-field just before therunner blades leading edge.

4.6 Losses in the RunnerIn our consideration of the flow-field through the rotor, the concept of total specific energy,

or the constant rothalpy, in the rotating coordinate system was applied [25]. The rothalpy forthe steady-state flow of non-viscous and non-compressible fluid is given as:

I~p

rz

w2

2{

u2

2~const ð22Þ

The rothalpy along the streamline in an ideal fluid through the rotor is constant. The wallfriction losses due to wall friction and turbulent mixing in viscous flow appear as the loss of thestatic pressure p. Eq. (22) for the viscous fluid is therefore modified by using energy loss termDEloss to account for losses:

IzDEloss~const ð23Þ

Fig. 16. Decrease in the circumferentially averaged static pressure coefficient through the runnerpassage.


Figure 18 shows losses due to a process taking place in the viscous fluid. The two curvesindicate the location of the leading and trailing edge of the runner blade. As the pressure lossesare not uniform in the pitch direction (coordinate h), depending on the velocity and thestreamline location on the blade-to-blade channel, one must calculate a mass-averaged rothalpyin the circumferential direction (see Fig. 18). In the section before the blade row, the rothalpyevenly reduces from the hub to the shroud. If the guide vanes would generate potential swirl anda constant meridional velocity, then the rothalpy before the runner in the radial direction wouldbe almost constant. The first larger rothalpy drop appears at the leading edge. The iso-lines witha low rothalpy are then gradually moved away from the shroud. The second larger rothalpydrop appears due to the flow mixing inside the blade wake. The pressure loss process near thehub is similar but less intense. Generally speaking, the largest source of the pressure loss is in theregion close to the shroud as also confirmed by the rothalpy contours, shown in section at span0.9 in Fig. 19. The core with the lowest energy appears just after the leading edge on the suctionside of the runner blade. However, this core is quickly dissolved in a surrounding of a slowlydecaying wider zone.

5. CONCLUSIONS

By applying the SCM method as a design tool for the runner blade row, the physical shape ofthe blade can be obtained in a fast and transparent way, necessitating no further adjustments inthe viscous analysis when using the CFD tools. To allow for viscous fluid-flow modelling, theSCM method was improved with certain empirical supplements. The calculation of the flowfield in an axially symmetrical part of the Saxo-type turbine passage was presented and the

Fig. 17. Loading of the runner blade sections – expressed with a static pressure coefficient.


optimal shape of the runner blade for the design duty point was calculated. Because of itsimpact on the flow kinematics in front of the runner, the guide vane apparatus was included inthe SCM computations. To have the runner blade designed optimally, runner velocity patternneeds to be known exactly.

In our determination of the preliminary shape of the blade and its profiles, the followingphysical guidelines were considered:

(1) Redirection of the relative velocity in the amount of incidence angle is caused by theinduced velocity close to the blade inlet section. The higher inlet blade region loading resultsin higher negative value of the optimal incidence angle.

(2) Intentional layout of the angular momentum along the stator and rotor region is of a vitalimportance of an efficient design of turbomachinery blading, as well as hydro turbine.Derivation of the angular momentum through the blading - inverse design platform – isthe basic criterion when selecting the correct number of blades and the proper profileshape.

(3) For the elbowed draft tube to operate optimally, the velocity field with a certaindistribution in the cone inlet and independent from the discharge and the turbine headshould be assured (see Fig. 11). Conditions achieved in cases B14-37-A1.8-A1.9 and B20-46are almost optimal for the used draft tube.

Fig. 18. Distribution of a circumferentially- and mass-averaged rothalpy on the meridional surfacealong the runner domain (case B14-A1.8-37).


Drawn up from the extensive CFD analysis, here are shown only the results related to theSCM computing domain and runner blading design results. With a view to correct comparison,the CFD simulations were made with no gap between the casing and the blade tip, whileconsidered the CFD results were used as reference. In principle, a correct prediction of the flowpattern overlaps with the proper shape of the blades. Predictions of the static pressure obtainedby using the SCM method are less accurate, but still giving a satisfactory information.Analysing the flow around the developed blading in the design-duty point shows no separationor origin of the secondary flow. Observing the runner exit flow, it may be concluded that if thehydro turbine blades are generally low hydraulically loaded (in order to avoid cavitation), thesecondary flow is not apparent.

Estimation of energy losses in individual turbine passages by integrating the flow energy inthe relevant control sections is a standard procedure. More important then the knowledge of thelosses themselves is the knowledge of its origin location. By analysing the rothalpy distribution,the place of the low-energy fluid accumulation can be identified. When analysing only theabsolute total specific energy in rotating region, it is difficult to differentiate between thespecific energy drop resulting from changes in angular momentum, and the share of the increasein the energy losses.

Fig. 19. Rothalpy contour at the blade-to-blade surface - Span 5 0.9 (case B14-A1.8-37).


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