aerothermodynamic effects and modeling of the tangential...

Research ArticleAerothermodynamic Effects and Modeling of the TangentialCurvature of Guide Vanes in an Axial Turbine Stage

Tzong-Hann Shieh

Department of Aerospace and Systems Engineering Feng Chia University No 100 Wenhwa Rd Seatwen Taichung 40724 Taiwan

Correspondence should be addressed to Tzong-Hann Shieh thshiehfcuedutw

Received 16 January 2017 Accepted 20 March 2017 Published 23 May 2017

Academic Editor Jechin Han

Copyright copy 2017 Tzong-Hann Shieh This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

By tangential curvature of the stacking line of the profiles guide vanes can be designed which have on both ends an obtuse anglebetween suction side and sidewall This configuration according to literature is capable of reducing secondary loss This type ofvanes develops considerable radial components of the blade force and effects a displacement of the meridional flow towards bothsidewalls In this paper we work with a finite-volume-code for computations of the three-dimensional Reynolds averaged Navier-Stokes equations for an axial turbine stage with radial and two types of tangentially curved guide vanesWith computational resultsmathematical formulations are developed for a new flow model of deflection of such blades that are formally compatible with theassumption of a rotation-symmetrical flow and with the existing throughflow codes in order to predict the deflection angle overthe blade height for the tangential leaned and curved blades

1 Introduction

The classic design method for the blading of axial turbinesapplied for many decades stipulates that the profiles consti-tuting the blade body are stacked in radial direction Suchstacked profiles can be different forms in the case of twistedbladesWhen such blading profiles are arrangedwith the pro-file centers of inertia on a radial line the stacking line bend-ing stress by centrifugal force is avoided in rotor blades Thenormal of the surface elements of blades designed accord-ingly is directed essentially to be tangential-axial so that thepressure forces of the blades have no or quite small radialcomponents these are usually neglected in the aerodynamiccalculation Consequently only the radial pressure gradientand the centrifugal force resulting from the circumferentialcomponent of the flow velocity are acting in radial directionon the fluid particles passing through the turbine stageMinoradditional radial forces may result from streamline curvatureand inclination They enter as important parts into the equa-tion of radial equilibrium which is one of the basic equationsfor the fluid-dynamic design of axial turbomachines

The classic blade design results in difficulties particularlyin the case of long blades as in the LP-part of condensingsteam-turbines (eg [1]) because the large radial pressure

gradient in the gap between stator and rotor causes a consid-erable variation of the degree of reaction over the blade length(eg 0 at the hub and 80 at the tip) which complicatesblade design and reduces efficiency

Consequently the idea was born by Deich et al [2] toproduce additional blade forces in radial inward direction bycircumferential inclination of the stator blades in order tocounteract the centrifugal forces and so to reduce the radialpressure gradient The idea proved successful and causedextensive research on different blade forms with nonradialsurface elements (eg [3ndash6])

It was found in addition that a blade inclination in thesidewall area which produces an obtuse angle between bladesuction side and sidewall can reduce secondary loss whilean acute angle will increase it A pure inclination of thestacking line in circumferential direction (lean)will thereforecreate a positive effect on secondary loss at one end of theblade and a negative effect at the other end Such inclinedblade has been discussed by several authors see for exampleDeich et al [2 7ndash9] Han et al [10 11] Harrison [12] Hour-mouziadis and Hubner [13] Suslov and Filippov [14] Wanget al [15] an experiment research and Deich et al [2 7ndash9] Hourmouziadis and Hubner [13] Pioske and Gallus [16]

HindawiInternational Journal of Rotating MachineryVolume 2017 Article ID 3806356 16 pageshttpsdoiorg10115520173806356

2 International Journal of Rotating Machinery

Smith Jr and Yeh [5] Suslov and Filippov [14] a numericalresearch as well as the references therein

In order to realize the loss reducing obtuse angle betweensuction side and sidewall on both ends of the blade a curva-ture (bow) of the stacking line of the profiles is necessary Forstator blades this can be realized without mechanical troubleThis design appears favourable especially for short bladesas in height pressure (HP) and middle pressure (MP) partof steam-turbines since there secondary loss together withclearance loss amounts to 20ndash40 of the total aerodynamicloss while a reduction of the radial pressure gradient notrealizable with this type of blade is not important Theresearch work of the flow in axial turbine stages with curvedstator blades is therefore an important research topic foractual turbomachinery and aeroengines And some resultsof the curvature blades can be found in Deich et al [7ndash9]Filippov and Wang [17 18] Wanjin et al [11] Harrison [12]Hourmouziadis and Hubner [13] Jiang et al [19] Vogt andZippel [20] Wang and Zheng [21] Wolf and Romanov [22]an experiment research and Filippov et al [18 23] Hour-mouziadis andHubner [13] Jiang et al [19] Pioske andGallus[16] Suslov and Filippov [14]Wang and Zheng [21] a numer-ical research where the curvature form of such researches ishowever mostly a simple curvature (circle) form

Experimental researches and numerical computations areavailable as tools to obtain a detailed knowledge of the 3D-flow field and to understand the effect of geometric modifi-cations Simulation offers the opportunity of a very detailedinsight in the complete flow field which is hardly accessible byexperimental means In addition it is ideally suitable for sys-tematic research of geometry variations as necessary in thepresent case However the quality of simulation results willbe influenced by numerical iteration procedure turbulencemodel mesh-structure wall-flow model and so forth in away that is difficult to control It is therefore highly recom-mendable to validate the results of simulation with the aid ofexperimental data as has been possible in the present case

The object of the present paper is the numerical simu-lation of the flow in an axial turbine stage equipped withseveral variations of tangentially curved stator blades by usinga finite-volume-code for the 3D Reynolds averaged Navier-Stokes equations The research work is centered on

(i) effects of blade curvature on the flow in the stator(ii) effects of the stator blade curvature on the flow in the

rotor (which remained unchanged in geometry)(iii) effects of the blade curvature on stage characteristic(iv) newmodeling of the radial blade force and the deflec-

tion angle over the blade height for the tangentialleaned and curved blades that are formally compatiblewith the assumption of a rotation-symmetrical flowand with the existing throughflow codes

Definition of Curved Blade The research is restricted to statorblades curved of both ends as followsThe stacking line of theprofiles is curved in circumferential direction as for examplein Figure 1

The curvature is defined by the angle 120576 between thestacking line and the sidewall at the blade end and by the

angleAcute

angleAcute

SSDS

minus휀G

+휀N

ℎkG

ℎkN

Figure 1 Definition of guide vanes curved on both ends by itsstacking line in 1198783-plane

extension ℎ119896 of the nonradial part of the stacking line Thesevalues can be different at both ends of the blade

(i) Positive Tangential Blade Curvature If an acute anglebetween the pressure side DS and the sidewall is realized thecurvature is defined as positive

(ii) Negative Tangential Blade Curvature A blade with nega-tive curvature shows acute angles between suction side SS andsidewall

2 Computational Methods

The applied numerical procedure solves the 3D-time-depend-ent Reynolds averagedNavier-Stokes equationswith the finitevolume method and comprises different turbulence modelsSome validations showed a very good quality of the resultsFor turbomachinery as well as aeroengine applications theprogram contains a procedure for circumferential averagingof flowparameters in planes between blade rows for examplebetween stators and rotor in order to make a steady-statecalculation feasible in both blade rows A suitable averagingplane was provided in the grid design The calculation usedthe Baldwin-Lomax turbulence model [24] and a three-levelfull multigrid method for convergence acceleration Oursimulation results showed a difference of mass flow in theinlet and outlet plane of less than about 002 Details onthe computational methods of this code are available in someliteratures for example [25ndash30]

21 Governing Equations The applied numeric procedureregards the so-called Reynolds averaged Navier-Stokes(RANS) equations as the physical basic model Their hyper-bolic system of conservation equations applied for massimpulse and energy depends on velocity pressure andenthalpyThey can be mostly written as a recapitulatory form

120597119880120597119905 + nabla sdot 119865 = 119876 (1)

Thereby 119880 = (120588 120588V 120588119864)119879 denotes the conservative variables119865 the matrices of the fluxes and 119876 the source terms column-vector 119865 is actually combined from the matrices of the

International Journal of Rotating Machinery 3

frictionless flux (Euler) 119865Eu and the dissipative flux (Navier-Stokes) 119865NS It can be rewritten as the following compactform further into the following general conservative form[25 27 29 30]

120597120597119905 [[[120588120588V120588119864]]]+ nabla sdot [[[

[120588V

120588V otimes V + 119901119868 minus 120591120588V119867 minus 120591 sdot V minus 119896nabla119879

]]]]= [[[[

0120588119891

119890119882119891 + 119902119867

]]]] (2)

where 120588 V 119864119867 119868 120591119891119890119882119891 119902119867 denote the density the veloc-

ity vector the total energy the total enthalpy the 3 times 3 unitmatrix tensor the viscous shear stress tensor the externalforce vector the column-vector of characteristic variablesand the source term whereby the velocity vector has compo-nents 119906 V 119908 According to the character of flow we can dif-ferentiate in each case according to the hyperbolic parabolicand elliptical type of equation The time-dependent Navier-Stokes equation possesses a hyperbolic-parabolic charac-ter within the temporal-spatial ranges because the steadyNavier-Stokes equations are a mixed type in the spatialdomain From it we obtain the elliptical-parabolic type whichdescribes subsonic flows and the hyperbolic-parabolic typesupersonic flows For the flows with high Reynolds numbersthe system of the conservative equations causes convectionobtained in most flow domains

22 Method of the Spatial Discretization The spatial fluxesof the applied basic equations of (1) and (2) are actually dis-cretized using the finite volume scheme The finite volumesscheme has been usually spread using numeric technologyin particular for flow problems because the integral form ofthe conservative equations can be interpreted by direct dis-cretization for a volume cell of the grids with the structuredas well as unstructured grids very flexibly and easily [25 2729 30]

Under the condition that the temporal derivatives areexchangeable with the spatial integrations (ie the integrandsare constantly differentiable in 119905) the integral forms can benormally deduced from the above differential expression (1)For this reason the following integral formof the conservativeequations is given by

120597120597119905 int119881119880119889119881 + ∮120597119881119865 sdot 119889119878 = int

119881119876119889119881 (3)

therefore it is formulated for any volume 119881 with the controlsurface 119878 = 120597119881 The discretization in the spatial is realizedwith an approximation of the volume 119881 by elementary cellswhich contain respectively an element Δ119878 of control surface119878Then the above integral equation (3) can be discretized andrewritten as follows

120597120597119905 int119881119880119889119881 + sumfaces119865Δ119878 = int119881119876119889119881 (4)

whereby 119865Δ119878 denotes the flux where they are combinedrespectively from frictionless 119865EuΔ119878 and dissipative 119865NSΔ119878fluxes

221 Numerical Fluxes The frictionless flux 119865EuΔ119878 and thedissipative flux 119865NSΔ119878 are discretized for example in eachcase by the upwind scheme and an application of Gausstheorems The treatment of the two fluxes will be not imple-mented here their detailed description is represented in [2527 29ndash35] The physical fluxes are mainly discretized in theapproximation of the central scheme as follows

[119865 sdot Δ119878]119894+(12) = 12 [119865119894 + 119865119894+1] sdot Δ119878 (5)

This treatment for the numeric flux does not lead actually to astable scheme because it does not give sufficient numeric dis-sipationTherefore one needs an artificial dissipation term inorder to eliminate odd-even oscillations because of the cen-tral discretization (high frequency oscillations) and in orderto capture shocks without oscillation The usual formulationprosecutes the type of the dissipation from Jameson et al[31] it is actually formulated as sum of conservative variablederivation respectively in second order with nonlinearcoefficients for shock capturing and in linear fourth order forthe background oscillations In this way the numeric flux (5)transferred into the form [25 27 29 30]

119865⋆119894+12 sdot Δ119878 = 12 [119865119894 + 119865119894+1] sdot Δ119878 minus 119889119894+12 (6)

where 119889119894+(12) is the artificial dissipation This term fromJameson et al is defined as follows in one-dimensional form[31]

119889119895

119894+12

= 120598(2)119894+12 (119906119894+1 minus 119906119894)minus 120598(4)119894+12 [(119906119894+2 minus 119906119894+1) minus 2 (119906119894+1 minus 119906119894) + (119906119894 minus 119906119894+1)]

(7)

The scalar coefficients 120598 are defined in the following way

120598(2) = 120581(2) 10038161003816100381610038161198861198881003816100381610038161003816 120601120598(4) = max [0 (120581(4) 10038161003816100381610038161198861198881003816100381610038161003816 minus 120598(2))] (8)

and connectedwith the variables of pressure and temperature

120601119894 = max(10038161003816100381610038161003816100381610038161003816119901119894+1 minus 2119901119894 + 119901119894minus1119901119894+1 + 2119901119894 + 119901119894minus1

10038161003816100381610038161003816100381610038161003816 10038161003816100381610038161003816100381610038161003816119879119894+1 minus 2119879119894 + 119879119894minus1119879119894+1 + 2119879119894 + 119879119894minus1

10038161003816100381610038161003816100381610038161003816) (9)

The pressure quantities are only used for liquidsFor the Navier-Stokes equation the friction-affected dis-

sipation is normally not sufficient in the solution and there-fore is the above term of the artificial dissipation necessarilyby the numeric solution of the Navier-Stokes equation Theunexpected difficulty with this approximation hence comesfrom the fact that the solution and the convergence can bestrongly dependent on the coefficients [25 27] This approxi-mation will be used with either explicit or implicitly temporalintegration


57mm

Inlet OutletPosition 0 Position 1 Position 2

1485 mm 385 mm

353mm

101mm

6737mm

Stator Rotorℎm

Axial gap with

65mm

훿axsp = 14mm amp

Figure 2 Meridional plane of the experimental and computational domain for the applied axial turbine

23 Turbulence Model Since there is still no universal andgeneral turbulence model the turbulence model will governthe results simulated [24 25 27] In this study Baldwin-Lomax turbulencemodel which is suitable for the wider classof interferencemodel to achieve enough accurate and reliableresults for a turbomachinery is chosen for the fully turbu-lence condition The added blended wall function imple-mented in the Baldwin-Lomax turbulence model leads to theboundary layer at the solid walls with further refinement by119910+ value with 119910+ asymp 13 Computational Conditions

As a starting point data from an experimental research of asingle stage air turbine with different types of curved guidevanes were made available to us by the department for powernuclear turbine construction and aircraft engines of StateSt Petersburg Technical University (Russia)They comprisedstage characteristics as well as detailed flow measurementswith aerodynamic probes Comparison of these data withthe results of numerical simulation of the same geometriesreported in [36] showed good agreement and served asvalidation of the simulation technique chosen A numberof stator configurations with different types of bow wasderived from the geometry of the experimental turbine andinvestigated numerically

31 Single Stage Experimental Air Turbine Figure 2 shows themeridional section of the turbine the test results of whichhave been used for the validation of the numerical simulationprocedure For themeasuring planes positions 0 1 and 2 dataof probe measurements were available The extension of themeridional section in Figure 2 shows too the field of thenumerical calculation

32 Geometrical Data of the Researched Blades The exper-imental investigations were made for a radial guide vane1198611198861199041198941199042119860 and a curved guide vane 1198611199001199082119861 both combined withthe same rotor Characteristic data of the center-line profilesare given in Table 1 For the numerical research 8 differentstator blade rows have been calculated with a systematicvariation of curvature by tangential displacement of theunchanged profiles of the radial stator 1198611198861199041198941199042119860 Results are

F13g1Bow F32gBow2ABasis

Figure 3 Schematic three-dimensional representation of the radialand the curvature guide vanes

given here for the basis blading and the two most character-istic bow stators Data describing the curvature are shown inTable 2 the parameters describing the curvature are 120576 and therelative extension ℎ119896

bez = ℎ119896ℎ according to Figure 1 Figure 3shows a 3D representation of the three guide vanes

33 Computational Mesh The calculation mesh for thenumerical simulation is identical in form as well as in cellnumber for all investigated blading Table 3 gives the cellnumbers in three coordinate directions (119894 circumferential 119895radial and 119896 axial direction)The entire computation areawasdivided into three zones in order to realize a rather orthogo-nal mesh Near the blade profiles an O-mesh was used forthe inlet channel an H-mesh and for the outlet area of thestator an I-mesh with H-mesh extension in order to adapt thewake of the blade In the outlet channel of the rotor only anH-mesh was applied A total of 973191 nodes resulted for thecomplete calculation area

Figure 4 shows the grid configuration for the appliedguide vanes and rotor blades The interfaces of the twodomains for stator and rotor are the only overlapped regionand there is only one accepted virtual translating directionalong the interfaces that is it is unacceptable to translateacross the interfaces Considering the complex geometry ofthis system for accuracy of the prediction the regions nearinterfaces and boundary layers of the stator and rotor finer


Table 1 Geometrical data of the center line profiles

Notation Unit 1198611198861199041198941199042119860 1198611199001199082119861 RotorBlade height [mm] 656 656 6687Middle diameters [mm] 36398 36398 36424Blade chord [mm] 31159 32101 2026Max camber [mm] 7047 5254 6063Stagger-angle [deg] 4300 3600 5720Inlet angle [deg] 8000 9000 4200Outlet angle [deg] 1537 1395 2133Rel pitch [mdash] 04898 04745 07433Blade number [mdash] 74 74 76120575sp [mm] mdash mdash 05

Table 2 Geometrical data of the investigated guide vanes

Notation Abbreviation 120576119873 120576119866 ℎ119896bez119873 ℎ119896

bez119866Unit [mdash] [deg] [deg] [mdash] [mdash]1 1198611198861199041198941199042119860 00 00 mdash mdash2 119861119900119908119865131198921 +132 minus122 0453 04383 11986111990011990811986532119892 +320 minus290 0409 0374

Table 3 The mesh distributions of the turbine stage

NotationH-mesh O-mesh I + HH-mesh(Inlet) (Blade) (Outlet)

[119894 times 119895 times 119896] [119894 times 119895 times 119896] [119894 times 119895 times 119896]le 33 times 57 times 33 33 times 57 times 217 33 times 57 times 49la 41 times 57 times 17 41 times 57 times 177 41 times 57 times 25structured mesh was employed in the present numericalcomputation on the other side regions far from stator androtor are meshed with coarser structured mesh

This mesh configuration considers the calculation andcomparison of turbulence model and combines it with thenear wall modified function with low Reynolds numbers inorder to effectively offset the interference that occurs in theinternal flow field to improve the near wall computationalgridrsquos deficiencies

Therefore the distance of the first cell layer from theblade surface was 0001mm the added blended wall functionimplemented in the turbulence model leads to the boundarylayer at the solid walls with further refinement by 119910+ valuethis gave a value of the dimensionless wall-distance 119910+ asymp 1

The computation of entire mesh used a three-level fullmultigrid method for convergence acceleration Our simula-tion results for such amesh configuration showed a differenceof mass flow in the inlet and outlet plane of less than about002 This validation showed a very good quality of theapplied mesh configuration

34 Operating Conditions of the Turbine Stage Thenumericalcalculation of the flow in the turbine stage was realized forits optimum operating point deduced from the experimental

data all relevant data for inlet and outlet are summarized inTable 4 The total-static pressure ratio Π equiv 119901aus2119904 119901ein0119905 = 07

The applied 3D calculations of total stage efficiency withaxial gap 65mm were performed for two different ranges ofthe run number ]

(i) Run Number ] in the Range 03 to 06 In this range of therun number the pressure ratio Π = 07 remained constantso the enthalpy change remained also constant But therotational speed was changed for different operating pointsThe applied data are shown in Table 5

(ii) Run Number ] in the Range 06 to 09 In this range ofthe run number the rotational speed was no longer changedbut the reduced rotational speedwasmaintained at a constantvalue 4000 However on the contrary the pressure ratio waschanged by various static pressures on entry This data isshown in Table 6 Therefore the enthalpy or the mass flowis also changed

4 Results and Discuss

41 Validation of Numerical Results For the operating pointsin both Tables 5 and 6 the turbine stage was computed witha real axial gap 65mm the results are shown in Figure 5 Itshows that the stage efficiency is a function of the run number] The run number ] has been divided into two ranges for thecomputations of the stage efficiency In the range of the runnumber between 03 and 06 the stage efficiency of the radialblade Basis2A agrees very well with the experimental dataHowever the computations of this radial blade in the rangebetween 06 and 09 also show a pretty good consistency out-side the range between 065 and 085 since in this range only


(a) (b)

Figure 4 Configuration of the calculating mesh in the middle section of the radial blades (mesh shown in (b) (rotor) and mesh shown in (a)(stator))

04 05 06 07 0803Run number ] (mdash)

Basis2A-exp average

Basis2A-comp ] = 03 06ndashBasis2A-comp ] = 06 09ndash

Tota

l sta

ge effi

cien

cy휂 t

(mdash)

092

09

088

086

084

082

08

078

076

Figure 5 Total stage efficiency as a function of the run number ]for the radial blade Basis2A with real axial gap 65mm

a deviation of about 03 occurs The accuracy of presentednumerical results can therefore be validated by this compari-son result clearly

42 Effect on the Meridional Streamlines The form of merid-ional streamlines is essentially determined by the radial bladeforces acting on the flowmainly the centrifugal force 1198882120593119903 andthe radial pressure gradient 120597119901120597119903 In the familiar stage designaccording to the free vortex law (approximately it is realizedin the 1198611198861199041198941199042119860-stage) these are balanced so that no radialdisplacement of streamlines occurs as in Figure 6

The type of bow applied in two guide vanes presentedhere will create additional radial blade forces acting towards

Table 4 The aerodynamic data in the optimal operation point

Notation Unit Value119901ein0119905 [Pa] 142857119901ein0119904 [Pa] 1408125119901aus2119904 [Pa] 100000119879ein0119905 [K] 3231119879ein0119904 [K] 321772

Tu [] 1 sdot sdot sdot 2ℎ119878Bi [mdash] 21le119888ein119911 [ms] 51657119888ein119903 [ms] 00119888ein120593 [ms] 00 [kgs] 493119899 [rpm] 7190119877 [J(kgK)] 28714120588 [kgm3] 153

the hub and the casing in the sidewall regions The balanceof radial forces and the radial flow displacement will vary toextent in the circumferential direction too and then a 3D-flowfield will be formed really A simplified presentation is shownin Figure 6 a circumferential averaged flow over the pitchis calculated and applied to the calculation of meridionalstreamlines in this figure a similar result would be achievedby a 2D-calculation of the flow field

By the comparison with both curved guide vanes of themeridional flow in Figure 6 it shows that the radial displace-ment of streamlines towards the guide vane ends has the sameeffect of bow The bow-effect for the vane 11986111990011990811986532119892 whoseinclination (bow) angles 120576 have more than double the valueof the vane 119861119900119908119865131198921 is more distinct The radial deflectiontapers off in the gap between stator and rotor and is con-centrated in the outlet part of guide vanes where the radialblade force has its full effect Consequently the deflection of



Figure 6 Circumferentially averaged meridional streamlines for radial and bow stators

Table 5 The operating points of rotating speed as a variable in the range of run number ] between 03 and 06

] [mdash] 0314 0366 0455 0484 0539 0550 0563 0578119899 [rpm] 4113 4790 5955 6326 7046 7190 7366 7567

Table 6 The operating points of total pressure at the inlet as a variable in the range of run number ] between 03 and 06

] [mdash] 0600 0612 0672 0704 0778 0800119901ein0119905 [Pa] 134543 132957 126460 123753 118911 117797

streamlines towards the sidewall will cause a deceleration ofthe axial flow velocity in the middle part of guide vaneswhich can be seen under the increased distance of streamlinescompared with the radial guide vane 1198611198861199041198941199042119860 such aneffect for the velocity distribution over the blade height isimportant

43 Radial Blade Forces Produced by Blade Curvature

431 Treatment andModeling of the Radial Blade Force Blad-ing designed according to standard principles with radialstacking line produces blade forces acting essentially incircumferential and axial directions and the radial com-ponent remains very small In addition the inclined andcurved guide vanes develop the radial blade force which canattain the same magnitude as the circumferential and axialcomponents and therefore enter into the radial equilibriumas important terms which influences the flow field

A schematic representation of three-dimensional bladeforce and an element angle of inclined or curved guide vaneare represented as a definition of balance direction in Figure 7at the blade trailing edge It shows that on the flow thereaction blade forces of guide vane are exerted on the pressureside119891DS

119904and the suction side119891SS

119904 respectivelyThey act on the

blade surface perpendicularly and have different values onboth sides

The local blade force 119891119904is defined by the inclined surface

element of the blade and its axial and radial length elementsΔ119911119911 and Δℎ119903 can be calculated from the local static pressure119901119904 at the blade surface in the following form

119891119904= 119901119904Δ119860119864 = 119901119904

Δ119911119911cos ]

Δℎ119903

cos 120576 (10)

where 120576 is the inclination angle against the radial directionand the deflection angle ] against the axial direction of thesurface element In addition the element length along theblade contour Δ119911119864 and the element height along the inclinedblade height Δℎ119864 can be computed by the forms Δ119911119864 =Δ119911119911cos V and Δℎ119864 = Δℎ119903cos 120576 respectively

The local blade force 119891119904in Figure 7 can be divided into

two components of blade force 1198911199041

in the 1198781-plane (blade toblade) and119891

1199043in the 1198783-planeThe blade force in the 1198781-plane

arises from the blade force 119891119904with the inclined angle 120576119904 in the

following form

1198911199041= 119891

119904cos 120576119904 (11)

thereby the inclined angle 120576119904 is determined over the tangentthrough the radial blade force 119891

119903119904and the blade force in the1198781 plane 1198911199041

120576119904 = arctan(1198911199031199041198911199041

) = arctan( 119891120593119904

tan 120576119891

120593119904 cos ])

= arctan (tan 120576 cos ]) (12)

Both blade forces in the 1198781 and 1198783-planes can be furtherdivided into three coordinate directions 119891

120593119904 119891

119903119904 and 119891

119911119904

describing the blade forces in circumferential radial andaxial directions respectively By the blade force in the 1198781-plane the circumferential blade force can be also indicatedin the form

119891120593119904= plusmn119891

1199041cos ] (13)


Inclined angles

Deflection angleDS SS

fs1

minusfrs

fs3

f휑s fs

fs

minusfzs

minusfs1

fzs

frs

]

]

]

휀

휀

휀s

휑

r

z

minusfs3

minusf휑s

휀s

휀 and 휀s

Figure 7 Schematic representation of 3D-blade forces and angles on the surface element of inclined or curved guide vane as a definition ofthe force direction (DS = pressure side SS = suction side)

with this definition the radial blade force in the 1198783-planearises

119891119903119904= plusmn119891

120593119904tan 120576 = 119891(120576 119891

120593119904) (14)

The local radial blade force 119891119903119904can be therefore obtained

by the following new general form

119891119903119904= [[119901119904

Δ119911119911cos ]

Δℎ119903

cos 120576 cos [arctan (tan 120576 cos ])]]sdot cos ]] tan 120576

(15)

432 Distribution of the Resultant Radial Blade Force Thedistribution of the radial blade force over the entire bladeheight must be described in detail for the effect of the bladecurvature The total radial blade force

119903119904of a profile section

with the elementary height 119889ℎ results from the integrationover all surface elements on this profile

119903119904= minus 119898sum

119894=1

(119891DS119903119904)119894+ 119899sum

119895=1

(119891SS119903119904)119895 (16)

where the indices 119894 119895 are the 119894th and 119895th surface elementof the blade on the pressure (DS) and suction (SS) sidesrespectivelyThe total radial blade force is the sumof the pres-sure side part and the suction side part it is always directedfrom the pressure side to the suction sideThe distribution ofthe total radial blade force per length-element of the bladeis shown in Figure 8 for the radial guide vane (assumingno radial blade force) and both curved guide vanes FromFigure 8 we find the following phenomena

(i) The region for the radial blade force is about equal tothe region of blade zones for bow (ca 40 of bladeheight on both ends)

(ii) The radial blade force is increasing in the inclination(curved) angle 120576 as well as on the same blade indirection of the increasing bow as from blade to bladewith increased bow

minus003 minus001minus005 003 005001Radial blade force (total) (Nm)

0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g

Figure 8 Distributions of the resultant radial blade force for theguide vanes over the blade height

(iii) The radial blade force tapers off in the proximityof sidewall even if the inclination (curved) angleincreases there This is probably due to reduced aero-dynamic loading of profiles there

The distributions and magnitude of the radial bladeforce correspond directly with the deflection of meridionalstreamlines in Figure 6 introducing the radial blade force inthe equation of radial equilibrium their form can be calcu-lated quantitatively

44 Effect of the Blade Curvature on the Static Pressure Distri-bution at the Profiles Theaerodynamic blade forces and theircomponents result from the static pressure on the profile andthe geometric orientation of its surface elementsThepressuredistribution itself is also of great importance for the flowover


Basis2A BowF13g1 BowF32g

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)

02 04 06 08 10Normalized axial chord length (mdash)


095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)


Figure 9 Pressure distributions over the blade profile of the stator at the 0063 (thick line) 0256 05 0744 and 0937 normalized bladeheight

the profile especially the boundary layer and its developmentBoundary layer thickness transition and eventually separa-tion depend to a great deal on the pressure distribution It isof great interest therefore how the inclination and curvatureof a blade influence the pressure distribution

441 Aerodynamic Load of the Blade Profiles If the staticpressure distribution over a profile is presented over the axialchord as in Figure 9 the area circumscribed by the pressureside and the suction side curve is directly proportional tothe circumferential (or working) total force of the bladeprofile the difference between the two curves at an axialstation describes the local aerodynamic loading and the formespecially of the suction side curve is important for theboundary layer development and hence for the profile loss

Figure 9 shows the static pressure distributions over theprofile sections at different height positions for the threeguide vanes treated here Two curves are near the sidewalls(0063 and 0937 normalized blade height) two are in themiddle of the bow zones (0256 and 0744) and one is in themiddle of the blades (05) It is obvious that because of theradial pressure gradient between stator and rotor the outletpressure increases with increasing blade height as visible inFigure 9 for all three stators Noteworthy is the decrease inaerodynamic loading (circumscribed area) towards the side-wall with increasing bow especially in the hub area Finallyremarkable and quite unexpected is the fact that the pressuredistributions of the profiles close to the sidewalls of thecurved vanes are distinctly deformed so that the aerody-namic loading is not only reduced but also displaced towardsthe trailing edge and acceleration on the suction side isdelayed This effect seems too to be proportional to thedegree of curvature

442 Pressure Distribution on the Suction Side The variationof the pressure distribution especially on the suction side bythe blade curvature is shown in Figure 9 for several distinctprofile sections It can also be demonstrated in a continuous

way by a graph of the pressure distribution on the suction sideover the blade height as shown in Figure 10 for the three vanesThe isobaric lines of the radial basis vane extend essentiallyin radial direction over the blade height the peaks near thetrailing edge are a result of a slight overspeed on the profileIn direct proximity of the sidewalls they bow a little down-streamThis somewhat more distinct effect near the casing isprobably due to the conical contour

Blade curvature now effects a clear displacement of theisobars in downstream direction near both sidewalls morepronounced by stronger blade curvature and resulting fromthe downstream movement of the blade loading describedin Section 441 While the local inclination of the bladeproduces for the bulk of the flow an additional force directedtowards the sidewalls as shown before it creates somewhatunexpected near the suction surface of the blade a compo-nent of the pressure gradient which is directed towards themiddle of the blade As is well-known the low energy fluidof the secondary flow accumulates in the corner betweensuction side and sidewall This static pressure componentdirected towards the center of the blade might be able totransport part of the low energy fluid out of the suction cornerand might be assisted by the obtuse angle between sidewalland suction side resulting from the blade inclination Here isa reason for the repeatedly observed reduction of secondaryloss by bowing of blades

45 Effect of the Curvature on the Vane Outlet Angle Sincethe inclination of the blade changes the aerodynamic loadingas well as the circumferential component of the blade forcethe turning of the flow (eg [37]) that is the outlet angle ofthe vane will be influenced too (eg [38]) Figure 11 showsthe distribution of the flow outlet angle averaged over onepitch over the blade height The angle of the radial blade isequal to the value according to the sine-rule (effective angle)over most of the blade heightThe curved blades exhibit clearunderturning over 20 of the blade length at both endsintensified by higher blade inclination which is certainly due


Pressure isolines on surface (Pa)Blade tip Blade tip Blade tip

Blade hubBlade hubBlade hub

TE

LE

TE

LE

TE

LE

1394e + 05

1311e + 05

1232e + 05

1149e + 05


1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1111e + 05

Figure 10 Pressure distributions on the suction side of the blade surface where the unit of the pressure isolines on the blade surface is inPascal Therefore LE and TE denote the leading edge and the trailing edge respectively

to the reduced blade loading in these areas Not so obviousis the explanation for the overturning by 2 to 4 deg in themiddle zone Probably it can be attributed to the decelerationof the axial flow velocity in this area because of the deflectionof the flow towards the sidewalls

46 Effect of the Blade Curvature on the Loss Coefficient ofthe Vanes As explained initially the advantage gained by useof partly or completely inclined vanes is the reduction ofthe radial pressure gradient and hence the gradient of thedegree of reaction especially for long blades The interest inreal bow vanes in turbines accruesmainly from the possibilityof reducing the flow losses Therefore the research of theinfluence of vane curvature on aerodynamic loss is of centralimportance The vane loss coefficient was defined accordingto Traupel [39] as

120577le = 1 minus 11988821 2Δℎle119904 + 11988820 2 (17)

The isentropic enthalpy drop Δℎle119904 of the stator is

Δℎle119904 = 120581120581 minus 11198771198790119904 [1 minus (11990111199041199010119904

)(120581minus1)120581] (18)

The loss coefficient at a sufficient number of stations alongthe streamtubes of constant blade height was determinedfrom the results of the 3D-simulation with circumferentially

Eff angle

10 15 20 25 3050

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Absolute flow angle 훼1 (deg)

Basis2ABowF13g1

BowF32g

Figure 11 Circumference averaged exit flow angle of the stators overthe channel

averaged values of the parameters and is presented for thethree different vane types in Figure 12 whereby 1199010119904 is afunction of the radius in (18) and all parameters are used inthis connection along the streamlines over blade height in


005 01 015 02 0250Loss coefficient (mdash)

0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g

Figure 12 Circumference averaged loss distributions of the statorsover the channel height

each case for stator and rotor The curves are not simple tointerpret but it is obvious that the vane with moderate bowhas lower losses near both sidewalls and a certain increasein the middle zone compared with the radial vane The vanewith high bow exhibits a sharp reduction of the loss coeffi-cient in direct proximity of the sidewalls but then a consid-erable increase around 02 and 08 normalized blade lengthwhile in the middle of the blade the loss is equal to that of theradial blade

The changes of the loss distribution andmagnitudewill bedue to a combination of influences as different developmentof profile boundary layer because of different pressure dis-tribution oblique movement of boundary layer on inclinedparts of the blade different secondary flow and so forthFinally the well-known deficiencies of Navier-Stokes-solversin calculating dissipation correctly must not be forgotten

The integrated loss of the three stators is calculated by thepresented curves of the illustrations according to

120577int = 1ℎbez intℎbez

0120577 (ℎbez) 119889ℎbez (19)

and their differences according to

Δ120577rech = 120577rech minus 120577Basis

2A

rech

120577Basis2Arech

(20)

The values are presented in Table 7 Such results are identicalto Figure 6 for the effects of the radial displacement of themeridional streamlines for radial and curved blades It turnsout that amoderate bow effects a reduction in total loss whilethe high bow vane exhibits higher loss than the radial vane

Table 7 Integrated loss coefficient of all investigated vanes

Notation 120577rech Δ120577rech []1198611198861199041198941199042119860 00767 mdash119861119900119908119865131198921 00744 minus294311986111990011990811986532119892 00782 +2005Table 8 Integrated loss coefficient of the rotor for the calculatedblading

Notation 120577rech Δ120577rech []1198611198861199041198941199042119860 00814 mdash119861119900119908119865131198921 00771 minus534611986111990011990811986532119892 00728 minus10531

5 Effects of the Vane Curvature onthe Rotor Flow

Any modification of the stator of a turbine stage whichchanges the flow field at its outlet will directly influence theflow entering the rotor and therefore its aerodynamic effec-tivity (eg [40ndash45]) In our case the curvature of the statorvanes changes the outlet angle and the axial flow velocitytogether they define in the velocity triangle the relative inletflow angle for the rotorThe difference between aerodynamicinlet angle and the geometric inlet angle of the rotor blade theincidence angle is of central importance for the aerodynamicaction of the rotor blade profiles High positive inlet anglescause an aerodynamic overload of the profile and high lossNegative incidence angle may result in adverse profile pres-sure distribution and increased loss too

Figure 13 showing the rotor incidence angles for the threecases makes it clear that the basis stage which had beenconceived for the experiments at the department for powernuclear turbine construction and aircraft engines of StateSt Petersburg Technical University had already not beendesigned in an optimum way since in the top half of therotor blade a considerable negative incidence prevails Theloss distribution presented in Figure 14 shows however thatthis design seemed not to have a negative consequence Theinfluence of the vane bow on rotor incidence in Figure 13results directly from the variation of the vane outlet angle 1205721Therefore its valuesmove towards negative in the two sidewallareas and towards positive on the middle of the bladeInteresting though not quite easy to explain is the reactionof the rotor loss coefficient on the bow of the stators Theremarkable reduction of the loss coefficient in the lower halfof the blade is not directly attributable to the variation of theincidence Probably it results from a reduction of profile lossas well as secondary loss and would require a detailed studyof the loss characteristics of the rotor profiles in function ofthe incidence Values of the integrated loss coefficient of therotor for the three cases in question are given in Table 8

As could be expected from Figure 14 they show a distinctreduction of loss for the cases with bow vanes Finally therotor outlet angle 1205732 is presented in Figure 15 Since theoutlet angle of turbine blade profiles is within a certain range


Reference

minus20 minus10minus30minus40 10 200Incidence (deg)

0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A

BowF13g1

BowF32g

Figure 13 Circumference averaged incidence of the rotor over thechannel height

004 008 012 016 020Loss coefficient of rotor (mdash)

0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g

Figure 14 Circumference averaged loss coefficient of the rotor overthe channel height

independent of the inflow angle it could be expected thatthe 1205732-distribution is little influenced by the bow of thevanes It shows the familiar underturn near the wall and theoverturn directly at the wall both due to secondary flow ina pronounced way near the hub and less clear near the tipwhere directly at the wall the clearance flow predominatesThe influence of the vane bow is small The absolute flowangle 1205722 behind the rotor is defined not only by 1205732 but via thevelocity triangle by the axial velocity too This is as we haveseen influenced considerably by the curvature of the vanes

Geom angle

20 25 3015Relative flow angle 훽2 (deg)

0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A BowF13g1

BowF32g

Figure 15 Circumference averaged relative exit flow angle 1205732 of therotor in position 2 (MP2)

45 55 65 75 85 95 105 115 125350

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g


Figure 16 Circumference averaged absolute exit flow angle 1205722 overthe channel height in position 2 (MP2)

and this influence is still sensible behind the rotorThereforeas Figure 16 shows the flow angle1205722 can be changed by 10 degand more through the vane bow even if 1205732 differs by one ortwo deg only for the different cases

6 2D-Modeling of the Flow Deflection

In preliminary design and optimization of turbines (eg[46]) the two-dimensional flow calculation (throughflowmethod) assuming rotation-symmetric flow is still widely in


use and of high practical importance since the blades cannotbe introduced directly into the calculation models for flowdeflection and loss representing the chosen blading has to beused For radial blading the familiar sine-rule is quite appro-priate to describe the exit flow angle 1205721 of the cascade in asimple form

1205721 = arcsin(11988611199051 ) (21)

where 1198861 is the exit channel width and 119905 the blade pitch Wedenote the exit angle and so calculated the effective angle

61 Correction Function for the Exit Flow Angle of CurvedBlade Tangential curvature will change the exit flow angleof a blade over the blade height in a waymainly influenced bythe local tangential inclination angle 120576 In a general way a cor-rection function for the effective flow angle 1205721 can be definedwhich renders the exit flow angle 1 of the curved blade

1 (ℎbez) = 119891120572120576 (ℎbez) 1205721 (ℎbez) (22)

This function will depend on the local inclination 120576 and theeffective flow angle 1205721 and varies over the blade height

611 Correction Function Based on Experimental DataExperimental investigations weremade at the department forpower nuclear turbine construction and aircraft engines ofState St Petersburg Technical University for the guide vane1198611199001199082119861 (not treated here in detail) which is inclined in thelower half and curved towards radial near the casing Therelevant data in our definition are 120576119873 = +132∘ 120576119866 = 0∘and ℎ119896

bez119873 = 10 Because of the moderate curvature a rathersimple correction function is proposed

119891120572120576 = 1 + 119896 tan 120576 (23)

From the experimental data a value for 119896 = 0435 is deductedA validation is given in Section 62

612 Correction Function for Vanes withHigh Curvature Thenumerically investigated vanes with high curvature on bothends produce high underturning in the sidewall zones whichcannot be described by a correction approximately propor-tional to 120576 as in (23) A nonlinear version for the correctionparameter 119896 has been devised which makes use of severaladditional parameters

119896 = 119896119896 minus 1120572119896 minus 120572eff119896 (120572119896 minus 1205721)119899 (24)

where 1205721 is again the sine-rule flow angle along the bladeheight 120572eff119896 the effective flow angle at the extreme blade end(hub or tip) and 120572119896 an assumed maximum value of the exitflow angle of the curved blade at its end

The value 120572eff119896 depends on the geometry of the hub profilecascade and is invariable (eg 13∘ for the vane 1198611198861199041198941199042119860 andthe curved vanes deduced from it) The two coefficients 120572119896

and 119896119896 have to be chosen suitably for the different curvatures

Table 9 Values of the empirical coefficients in (25) for the curvatureblading (119886 = general)

Notation 1198611199001199082119861 11986111990011990811986532119892 119870119899 3 3 119896120572eff119896 13 13 1198961205721015840119896 20 1535 1198861198961015840119896 1195 1085 119886

The exponent 119899 will take care of the nonlinear character ofthe angle distribution a value 119899 = 3 seems appropriate Fromthe results of experimental investigations of turbine stageswith curved vanes it seems that an optimal angle of tangentialinclination of approximately 20∘ exists We assume thereforethat the value of 120572119896 can be chosen up to a maximum of20∘ only similarly the coefficient 119896119896 should not be assumedhigher than 12 If (24) is inserted in (23) we receive a newgeneral formulation for the exit flow angle of curved vanes

1 = [1 + ( 119896119896 minus 1120572119896 minus 120572eff119896 (120572119896 minus 1205721)119899) tan 120576] 1205721 (25)

62 Validation of the Correction Function Since the numer-ically investigated vanes have been developed by tangentialdisplacement of the unvaried profiles of the radial basis statortheir effective exit flow angle distribution is identical with thatof the stator 1198611198861199041198941199042119860 The geometry and hence the effectiveflow angle distribution of the stator 1198611199001199082119861 for which exper-imental data are available are quite different The exit flowangle distribution of the two curved stators 1198611199001199082119861 and11986111990011990811986532119892 has been calculated with the correction functiondescribed above The values of the geometrical and empiricalcoefficients that have been used are shown in Table 9Further validations of other blade types from the numericalsimulation are carried out in this investigation respectively(not treated here in detail)

Figure 17 shows the results for the vane 1198611199001199082119861 whichcomprises inclination mainly in the lower half The resultingexit flow angle differs sensibly in this region from the sine-rule value the corrected angle according to (25) and thecoefficients in Table 9 agree well with the experimental andthe values of the 3D-computation Because of the moderatedegree of inclination the correction in the simplified form of(23) with a constant value of 119896 = 0435 would render goodresults

Figure 18 shows results for the vane 11986111990011990811986532119892 with consid-erable curvature on both ends The corrected angles accord-ing to (25) with the coefficients in Table 9 represent the resultsof the 3D-simulation very well in both sidewall regions Sincethe correctionmodel cannot take care of the redistribution ofthe aerodynamic blade load into the middle of the blade thereduction of the flow angle in that region will not be correctlydescribed (see Figure 11) The validation of this correlation istested at present likewise for the throughflow calculation bythe TU St Petersburg as well as industry in St Petersburg

Because the sine-rule is valid for two-dimensional cas-cade flow the exit flow angle distribution will show somedeviations from these values in the sidewall zones of radial


3D-computationSinus-rule

Correction-functionMeasurement

5 10 15 20 250Absolute flow angle 훼1 (grad)

0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Figure 17 Validation of the deflection model of (25) over the bladeheight for the curvature blade 1198611199001199082119861

Sinus-ruleCorrection-function3D-computation

5 10 15 20 25 30 35 400Absolute flow angle 훼1 (grad)

0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)


blades too Secondary flow will cause underturning whenapproaching the sidewall and overturning in its direct prox-imity radial clearance will cause considerable underturningThese deviations from the sine-rule values can be calculatedby specific flow models described for example in [47] andthen introduced in our correction procedure instead of theeffective flow angle 1205721 for improved modeling In Figure 17these influences are visible to some extent A distinct tangen-tial curvature however will render flow angle modificationswhich surpass the secondary flow influence considerably asis clearly visible in Figure 18

7 Conclusions

Results of numerical 3D flow simulations of an axial turbinestagewith radial and twodifferent curved guide vanes are pre-sented The curved guide vanes are developed by tangentialdisplacement of the profiles of the radial vaneThe simulationprocedure was validated with the aid of experimental datanot presented here

The guide vanes with curvature on both ends forman obtuse angle between suction side and sidewall Theygenerate radial pressure forces which displace the fluid flowtowards the sidewalls The axial velocity is increased there aswell as the exit flow angle The pressure distribution of theblade profiles is deformed so that the aerodynamic loadingis shifted backwards This produces a radial component ofthe pressure gradient on the suction side directed towards themiddle of the blade This phenomenon probably promotes adisplacement of the secondary flow and a reduction of thesecondary loss All effects are intensified by increased bladeinclination and extended area of the curvature

The modification of vane exit angle and axial velocitydistribution will cause changes of the incidence to the rotorup to 10∘ and more so that adaption of the rotor geometrywhen applying curved stators seems necessary

In order to render the flow calculation of curved sta-tors feasible by the widespread 2D-methods (through flowmethod) a radial force model and an exit angle model forcurved stators were developed It departs from the famil-iar sine-rule and comprises a correction function whichdepends mainly on the local inclination angle 120576 of the vaneGeneral coefficients permit the adaption to different bladecurvatures Further research by numerical simulation finallyvalidated by experiments which is extended to other profiletypes profile modifications in the inclined zone other typesof curvature and so forth will probably lead to optimumforms of bow vanes with reduced secondary loss andincreased efficiency

Nomenclature

Symbols

119886 Exit channel width119888 Flow velocity in abs system119891 Vector of the forceℎ119904 Specific enthalpyℎ Blade and channel heightΔℎ119904 Change of the isentropic spe enthalpy119894 Incidence119896 Coefficient Mass flow rate119899 Speed in normal direction119901 Pressure119903 Radius119877 Gas constant119878 Blade chord length119879 TemperatureTu Turbulence degree120572 Flow angle in the abs system


120573 Flow angle in the rel systemΔ Difference120575 Curving angles of meridian streamlines120575sp Gap width of tip clearance120576 Curvature angle of blade stacking line120577 Loss coefficient120581 Specific heat ratio] Angle of the meridional stream surface120588 Flow densityΠ Pressure ratio of turbine stage

Indices

0 1 2 Exp measured positionsaus Outlet of the stageax Axialbez NormalizedBi BitangentDS Pressure sideein Inlet of the stage mesheff Effective according to sine-rule119866 At the casingint Integrationis Isentropic119896 Curvature zone and coefficientla Rotorle Stator119898 Meridional comp and middle value119873 At the hub119903 Radial componentrech Computation119904 Staticsp Gap losses or gapSS Suction side119905 Total119911 119911-axial component120593 Circumference component

Conflicts of Interest

The author declares that they have no conflicts of interest

Acknowledgments

This study described here was supported by the Ministryof Science and Technology and the National Science Coun-cil (MOST104-2221-E-035-052 MOST103-2632-E-035-001-MY3 MOST103-2221-E-035-064 NSC102-2221-E-035-029NSC101-2221-E-035-010) and was completed due to the sup-port of Professor W Riess and Professor J Seume of theInstitute for Turbomachinery and Fluid Dynamics of LeibnizUniversity ofHannoverThe author is very grateful for this aidand wants to thank their continuously encouragement andcooperation as well as their helpful discussions to this workfor the writingThe basic blade data and experimental resultswere completed and available due to the support of ProfessorLapschin of the Department for Power Nuclear TurbineConstruction and Aircraft Engines of State St PetersburgTechnical University

References

[1] J A MacDonald ldquoIncreasing Steam Turbine Power GenerationEfficiencyrdquo June 2003 issue of Energy-Tech magazine (2003)

[2] M E Deich A V Gubarev G A Filippov and Z Q Wang ldquoAnew method of profiling the guide vane cascades of stages withsmall ratios of diameter to lengthrdquo Thermal Engineering no 8(in Russian) pp 42ndash46 1962 Associated Electrical IndustriesResearch Laboratory Manchester Translation No 3277

[3] A S Leyzerovich Steam Turbine for Modern Fossil-Fuel PowerPlants Taylor amp Francis UK 2008

[4] M Deckers and E W Pfitzinger ldquoThe exploitation of advancedblading technologies for the design of highly efficient steamturbinesrdquo in Proceedings of the 6th International Charles ParsonsTurbine Conference Dublin Ireland 2003

[5] LH Smith Jr andH Yeh ldquoSweep andDihedral Effects inAxial-Flow Turbomachinery Trans of the ASMErdquo Journal of BasicEngineering vol 85 pp 401ndash406 1963

[6] E Y Watanabe Y Tanaka T Nakano et al ldquoDevelopment ofNew High Efficiency Steam Turbinerdquo Mitsubishi Heavy Indus-tries Technical Review vol 40 no 4 pp 1ndash6 2003

[7] M E Deich and B M Trojanovskij Investigation and Designof Axial Turbine Stages (Untersuchung und Berechnung AxialerTurbinenstufen) VEB Verlag Technik Berlin Germany 1973

[8] M E Deich B M Troyanovskii and G A Filippov ldquoAn effec-tive way of improving the efficiency of turbine stagesrdquoThermalEngineering vol 37 no 10 pp 520ndash523 1990

[9] M E Deich Gas Dynamics of Turbine Blade Rows Edited byGA Filippov Energoatomizdat Moscow 1996

[10] W Han Z Wang and W Xu ldquoAn experimental investigationinto the influence of blade leaning on the losses downstreamof annular cascades with a small diameter-height ratiordquo in Pro-ceedings of the International Gas Turbine and Aeroengine Con-gress and Exposition (GT rsquo88) pp 1ndash9 June 1988 ASME Paper88-GT-19 (1988)

[11] HWanjin W Zhongqi T Chunqing S Hong and Z MochunldquoEffects of leaning and curving of blades with high turningangles on the aerodynamic characteristics of turbine rectangu-lar cascadesrdquo Journal of Turbomachinery vol 116 no 3 pp 417ndash424 1994

[12] S Harrison ldquoInfluence of blade lean on turbine lossesrdquo Journalof Turbomachinery vol 114 no 1 pp 184ndash190 1992

[13] J Hourmouziadis and N Hubner ldquo3-D Design of TurbineAirfoilsrdquo ASME Paper 85-GT-188 1985 pp 1-7

[14] D I Suslov and G A Filippov ldquoAn approximate method forcalculating and profiling cascades of curved vanesrdquo ThermalEngineering vol 42 no 3 pp 259ndash264 1995

[15] Z Wang W Xu W Han and B Jie ldquoAn experimental inves-tigation into the reasons of reducing secondary flow losses byusing leaned blades in rectangular turbine cascades with inci-dence anglerdquo in Proceedings of the ASME 1988 International GasTurbine and Aeroengine Congress and Exposition (GT rsquo88) June1988

[16] C Pioske and H E Gallus ldquoDreidimensionale Turbinenbes-chaufelungrdquo MTZ Motortechnische Zeitschrift pp 358ndash362 1Band 58 Heft 6 (1997)

[17] G A Filippov and Z Q Wang ldquoDie Berechnung der rotation-ssymmetrischen Stromung fur die letzte Stufe der turbomaschi-nenrdquo Publikation des Moskauer Energetisches Institut (MEI) no47 1963


[18] G A Filippov and Z Q Wang ldquoThe effect of flow twisting onthe characteristics of guide rowsrdquo Thermal Engineering (Teplo-energetika) no 5 pp 69ndash73 1964

[19] H D Jiang Y L Lu F Z Zhou and S J Wang ldquoStudy of asubsonic combind-leaning turbine guide vane with tip endwallcontouringrdquo Journal of Aerospace Power vol 8 no 1 pp 41ndash441993

[20] H-F Vogt and M Zippel ldquoSekundarstromungen in turbinen-gittern mit geraden und gekrummten schaufeln visualisierungim ebenen wasserkanalrdquo Forschung im IngenieurwesenEngi-neering Research vol 62 no 9 pp 247ndash253 1996

[21] ZWang and Y Zheng ldquoResearch status and development of thebowed-twisted blade for turbomachinesrdquo Engineering Sciencevol 2 pp 40ndash48 2000

[22] R Wolf and K Romanov Steam Turbines Siemens ReactiveBlading - Designed for Highest Efficiency and Minimal Perfor-mance Degradation 2014 Russia Power 2014 (March 2003) pp1-19

[23] G Filippov V Gribin A Tischenko I Gavrilov and V Tis-chenko ldquoExperimental studies of polydispersedwet steamflowsin a turbine blade cascaderdquo Proceedings of the Institution ofMechanical Engineers Part A Journal of Power and Energy vol228 no 2 pp 168ndash177 2014

[24] B Baldwin and H Lomax ldquoThin layer approximation andalgebraic model for separated turbulent flowrdquo in Proceedings ofthe 16th Aerospace Sciences Meeting vol 78-0257 AIAA 1978

[25] Ch Hirsch Numerical Computational of Internal and Exter-nal Flows Fundamentals of Computational Fluid DynamicsButterworth-Heinemann 2nd edition 2007

[26] Ch Hirsch et al ldquoAn Integrated CFD System for 3D Turboma-chinery Applicationsrdquo ADARD-CP-510 1992

[27] Ch Hirsch CFDMethodology and Validation for Turbomachin-ery Flows AGARD Lecture Series on rsquoTurbomachinery DesignUsing CFDrsquo 1994 pp 41-417 May to June

[28] S Kang and C Hirsch ldquoNumerical simulation of three-dimen-sional viscous flow in a linear compressor cascade with tipclearancerdquo Journal of Turbomachinery vol 118 no 3 pp 492ndash502 1996

[29] D Kroner Numerical Schemes for Conservation Laws WileyTeubner 1997

[30] J W Slooff and W Schmidt ldquoComputational AerodynamicsBased on the Euler Equationsrdquo AGARD-AG-325 1994

[31] A Jameson W Schmidt and E Turkel ldquoNumerical simulationof the euler equations by finite volume methods using runge-kutta time stepping schemesrdquo in Proceedings of the AIAA 5thComputational FluidDynamics Conference AIAAPaper 81-1259(1981)

[32] T-H Shieh and M-R Li ldquoNumeric treatment of contactdiscontinuity with multi-gasesrdquo Journal of Computational andApplied Mathematics vol 230 no 2 pp 656ndash673 2009

[33] T-H Shieh T-M Liou M-R Li C-H Liu and W-J WuldquoAnalysis on numerical results for stage separation with differ-ent exhaust holesrdquo International Communications in Heat andMass Transfer vol 36 no 4 pp 342ndash345 2009

[34] T-H Shieh M-R Li Y-T Li and M-C Chen ldquoA comparativestudy of flux limiters using new numerical methods in unsteadysupersonic flowsrdquo Numerical Heat Transfer Part B Fundamen-tals vol 67 no 2 pp 135ndash160 2015

[35] T-H Shieh andM-C Chen ldquoStudy of high-order-accurate lim-iters for time-dependent contact discontinuity and shock cap-turingrdquoNumerical Heat Transfer Part B Fundamentals vol 70no 1 pp 56ndash79 2016

[36] T H Shieh Untersuchung von Axialturbinen-Beschaufelungenmit dreidimensionalen Gestaltungselementen [PhD disserta-tion] Universitat Hannover 2003

[37] D Wang H Ding and J Zhong ldquoThe influence of tailboard onthe exit flow fields of compressor cascade with curved bladerdquoin Proceedings of the China Engineering Thermal Physics Society(HMArsquo00) pp 344ndash348 2000

[38] H Jujii T Kimura and K Segawa ldquoA high-efficiency steam tur-bine utilizing optimized reaction bladesmdashan application to thekwangyang combined-cycle power plant of K-power of Repub-lic of Koreardquo Hitachi Review vol 56 no 4 pp 104ndash108 2007

[39] W TraupelThermische Turbomaschinen vol 1 Springer-VerlagBerlin Germany 2000

[40] Z-M Feng H-B Gu J-D Zhang and Y Lu ldquoDifferent three-dimensional blades aerodynamic performance research com-parisonrdquo Information Technology Journal vol 12 no 11 pp2219ndash2224 2013

[41] Z M Feng H B Gu and J D Zhang ldquoCurved rotor blade cas-cades aerodynamic performance at off-design incidence in axialsteam turbinerdquo International Journal of Digital Content Technol-ogy and its Applications vol 7 no 7 pp 1086ndash1093 2013

[42] M Murugan A Ghoshal F Xu et al ldquoArticulating turbinerotor blade concept for improved off-design performance of gasturbine enginesrdquo in Proceedings of ASME 2016 Conference onSmartMaterials Adaptive Structures and Intelligent Systems vol1 2016 Paper No SMASIS2016-9045 pp V001T04A004

[43] K Segawa et al ldquoDevelopment of a highly loaded rotor blade forsteam turbines (2nd report performance verification)rdquo JSMEInternational Journal Series B vol 45 no 4 2002

[44] K Segawa et al ldquoA high performance optimized reaction bladefor high pressure steam turbinesrdquo in Proceedings of ASMEPowerConference pp 307ndash314 March 2004

[45] B Mischo Axial Turbine Rotor Aero-thermal Blade Tip Perfor-mance Improvement through Flow Control [PhD dissertation]TU Kaiserslautern 2008

[46] P Lampart and L HirT ldquoComplex multidisciplinary optimiza-tion of turbine blading systemsrdquo Archives of Mechanics vol 64no 2 pp 153ndash175 2012

[47] M Petrovic Berechnung der Meridianstromung in mehrstufigenAxialturbinen bei Nenn- und Teillastbetrieb [PhD dissertation]Universitat Hannover VDI Fortschritt-Berichte Netherlands1995

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Smith Jr and Yeh [5] Suslov and Filippov [14] a numericalresearch as well as the references therein

In order to realize the loss reducing obtuse angle betweensuction side and sidewall on both ends of the blade a curva-ture (bow) of the stacking line of the profiles is necessary Forstator blades this can be realized without mechanical troubleThis design appears favourable especially for short bladesas in height pressure (HP) and middle pressure (MP) partof steam-turbines since there secondary loss together withclearance loss amounts to 20ndash40 of the total aerodynamicloss while a reduction of the radial pressure gradient notrealizable with this type of blade is not important Theresearch work of the flow in axial turbine stages with curvedstator blades is therefore an important research topic foractual turbomachinery and aeroengines And some resultsof the curvature blades can be found in Deich et al [7ndash9]Filippov and Wang [17 18] Wanjin et al [11] Harrison [12]Hourmouziadis and Hubner [13] Jiang et al [19] Vogt andZippel [20] Wang and Zheng [21] Wolf and Romanov [22]an experiment research and Filippov et al [18 23] Hour-mouziadis andHubner [13] Jiang et al [19] Pioske andGallus[16] Suslov and Filippov [14]Wang and Zheng [21] a numer-ical research where the curvature form of such researches ishowever mostly a simple curvature (circle) form

Experimental researches and numerical computations areavailable as tools to obtain a detailed knowledge of the 3D-flow field and to understand the effect of geometric modifi-cations Simulation offers the opportunity of a very detailedinsight in the complete flow field which is hardly accessible byexperimental means In addition it is ideally suitable for sys-tematic research of geometry variations as necessary in thepresent case However the quality of simulation results willbe influenced by numerical iteration procedure turbulencemodel mesh-structure wall-flow model and so forth in away that is difficult to control It is therefore highly recom-mendable to validate the results of simulation with the aid ofexperimental data as has been possible in the present case

The object of the present paper is the numerical simu-lation of the flow in an axial turbine stage equipped withseveral variations of tangentially curved stator blades by usinga finite-volume-code for the 3D Reynolds averaged Navier-Stokes equations The research work is centered on

(i) effects of blade curvature on the flow in the stator(ii) effects of the stator blade curvature on the flow in the

rotor (which remained unchanged in geometry)(iii) effects of the blade curvature on stage characteristic(iv) newmodeling of the radial blade force and the deflec-

tion angle over the blade height for the tangentialleaned and curved blades that are formally compatiblewith the assumption of a rotation-symmetrical flowand with the existing throughflow codes

Definition of Curved Blade The research is restricted to statorblades curved of both ends as followsThe stacking line of theprofiles is curved in circumferential direction as for examplein Figure 1

The curvature is defined by the angle 120576 between thestacking line and the sidewall at the blade end and by the

angleAcute

angleAcute

SSDS

minus휀G

+휀N

ℎkG

ℎkN

Figure 1 Definition of guide vanes curved on both ends by itsstacking line in 1198783-plane

extension ℎ119896 of the nonradial part of the stacking line Thesevalues can be different at both ends of the blade

(i) Positive Tangential Blade Curvature If an acute anglebetween the pressure side DS and the sidewall is realized thecurvature is defined as positive

(ii) Negative Tangential Blade Curvature A blade with nega-tive curvature shows acute angles between suction side SS andsidewall

2 Computational Methods

The applied numerical procedure solves the 3D-time-depend-ent Reynolds averagedNavier-Stokes equationswith the finitevolume method and comprises different turbulence modelsSome validations showed a very good quality of the resultsFor turbomachinery as well as aeroengine applications theprogram contains a procedure for circumferential averagingof flowparameters in planes between blade rows for examplebetween stators and rotor in order to make a steady-statecalculation feasible in both blade rows A suitable averagingplane was provided in the grid design The calculation usedthe Baldwin-Lomax turbulence model [24] and a three-levelfull multigrid method for convergence acceleration Oursimulation results showed a difference of mass flow in theinlet and outlet plane of less than about 002 Details onthe computational methods of this code are available in someliteratures for example [25ndash30]

21 Governing Equations The applied numeric procedureregards the so-called Reynolds averaged Navier-Stokes(RANS) equations as the physical basic model Their hyper-bolic system of conservation equations applied for massimpulse and energy depends on velocity pressure andenthalpyThey can be mostly written as a recapitulatory form

120597119880120597119905 + nabla sdot 119865 = 119876 (1)

Thereby 119880 = (120588 120588V 120588119864)119879 denotes the conservative variables119865 the matrices of the fluxes and 119876 the source terms column-vector 119865 is actually combined from the matrices of the



120597120597119905 [[[120588120588V120588119864]]]+ nabla sdot [[[

[120588V


]]]]= [[[[

0120588119891

119890119882119891 + 119902119867

]]]] (2)





120597120597119905 int119881119880119889119881 + ∮120597119881119865 sdot 119889119878 = int

119881119876119889119881 (3)





[119865 sdot Δ119878]119894+(12) = 12 [119865119894 + 119865119894+1] sdot Δ119878 (5)




119889119895

119894+12


(7)


120598(2) = 120581(2) 10038161003816100381610038161198861198881003816100381610038161003816 120601120598(4) = max [0 (120581(4) 10038161003816100381610038161198861198881003816100381610038161003816 minus 120598(2))] (8)




10038161003816100381610038161003816100381610038161003816) (9)




57mm


1485 mm 385 mm

353mm

101mm

6737mm

Stator Rotorℎm

Axial gap with

65mm

훿axsp = 14mm amp
































(a) (b)


04 05 06 07 0803Run number ] (mdash)

Basis2A-exp average


Tota

l sta

ge effi

cien

cy휂 t

(mdash)

092

09

088

086

084

082

08

078

076














] [mdash] 0314 0366 0455 0484 0539 0550 0563 0578119899 [rpm] 4113 4790 5955 6326 7046 7190 7366 7567


] [mdash] 0600 0612 0672 0704 0778 0800119901ein0119905 [Pa] 134543 132957 126460 123753 118911 117797










119891119904= 119901119904Δ119860119864 = 119901119904

Δ119911119911cos ]

Δℎ119903

cos 120576 (10)







following form

1198911199041= 119891

119904cos 120576119904 (11)



120576119904 = arctan(1198911199031199041198911199041

) = arctan( 119891120593119904

tan 120576119891

120593119904 cos ])

= arctan (tan 120576 cos ]) (12)


120593119904 119891

119903119904 and 119891

119911119904


119891120593119904= plusmn119891

1199041cos ] (13)


Inclined angles


fs1

minusfrs

fs3

f휑s fs

fs

minusfzs

minusfs1

fzs

frs

]

]

]

휀

휀

휀s

휑

r

z

minusfs3

minusf휑s

휀s

휀 and 휀s



119891119903119904= plusmn119891

120593119904tan 120576 = 119891(120576 119891

120593119904) (14)



119891119903119904= [[119901119904

Δ119911119911cos ]

Δℎ119903


(15)




119903119904= minus 119898sum

119894=1

(119891DS119903119904)119894+ 119899sum

119895=1

(119891SS119903119904)119895 (16)





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g







0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)



095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)













TE

LE

TE

LE

TE

LE

1394e + 05

1311e + 05

1232e + 05

1149e + 05


1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1111e + 05




120577le = 1 minus 11988821 2Δℎle119904 + 11988820 2 (17)


Δℎle119904 = 120581120581 minus 11198771198790119904 [1 minus (11990111199041199010119904

)(120581minus1)120581] (18)


Eff angle

10 15 20 25 3050

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)


Basis2ABowF13g1

BowF32g





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g






0120577 (ℎbez) 119889ℎbez (19)



2A

rech

120577Basis2Arech

(20)










Reference


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A

BowF13g1

BowF32g



0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g



Geom angle


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A BowF13g1

BowF32g


45 55 65 75 85 95 105 115 125350

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g








1205721 = arcsin(11988611199051 ) (21)



1 (ℎbez) = 119891120572120576 (ℎbez) 1205721 (ℎbez) (22)




119891120572120576 = 1 + 119896 tan 120576 (23)








Notation 1198611199001199082119861 11986111990011990811986532119892 119870119899 3 3 119896120572eff119896 13 13 1198961205721015840119896 20 1535 1198861198961015840119896 1195 1085 119886











0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)




0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)



7 Conclusions





Nomenclature

Symbols




Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120597120597119905 [[[120588120588V120588119864]]]+ nabla sdot [[[

[120588V


]]]]= [[[[

0120588119891

119890119882119891 + 119902119867

]]]] (2)





120597120597119905 int119881119880119889119881 + ∮120597119881119865 sdot 119889119878 = int

119881119876119889119881 (3)





[119865 sdot Δ119878]119894+(12) = 12 [119865119894 + 119865119894+1] sdot Δ119878 (5)




119889119895

119894+12


(7)


120598(2) = 120581(2) 10038161003816100381610038161198861198881003816100381610038161003816 120601120598(4) = max [0 (120581(4) 10038161003816100381610038161198861198881003816100381610038161003816 minus 120598(2))] (8)




10038161003816100381610038161003816100381610038161003816) (9)




57mm


1485 mm 385 mm

353mm

101mm

6737mm

Stator Rotorℎm

Axial gap with

65mm

훿axsp = 14mm amp
































(a) (b)


04 05 06 07 0803Run number ] (mdash)

Basis2A-exp average


Tota

l sta

ge effi

cien

cy휂 t

(mdash)

092

09

088

086

084

082

08

078

076














] [mdash] 0314 0366 0455 0484 0539 0550 0563 0578119899 [rpm] 4113 4790 5955 6326 7046 7190 7366 7567


] [mdash] 0600 0612 0672 0704 0778 0800119901ein0119905 [Pa] 134543 132957 126460 123753 118911 117797










119891119904= 119901119904Δ119860119864 = 119901119904

Δ119911119911cos ]

Δℎ119903

cos 120576 (10)







following form

1198911199041= 119891

119904cos 120576119904 (11)



120576119904 = arctan(1198911199031199041198911199041

) = arctan( 119891120593119904

tan 120576119891

120593119904 cos ])

= arctan (tan 120576 cos ]) (12)


120593119904 119891

119903119904 and 119891

119911119904


119891120593119904= plusmn119891

1199041cos ] (13)


Inclined angles


fs1

minusfrs

fs3

f휑s fs

fs

minusfzs

minusfs1

fzs

frs

]

]

]

휀

휀

휀s

휑

r

z

minusfs3

minusf휑s

휀s

휀 and 휀s



119891119903119904= plusmn119891

120593119904tan 120576 = 119891(120576 119891

120593119904) (14)



119891119903119904= [[119901119904

Δ119911119911cos ]

Δℎ119903


(15)




119903119904= minus 119898sum

119894=1

(119891DS119903119904)119894+ 119899sum

119895=1

(119891SS119903119904)119895 (16)





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g







0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)



095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)













TE

LE

TE

LE

TE

LE

1394e + 05

1311e + 05

1232e + 05

1149e + 05


1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1111e + 05




120577le = 1 minus 11988821 2Δℎle119904 + 11988820 2 (17)


Δℎle119904 = 120581120581 minus 11198771198790119904 [1 minus (11990111199041199010119904

)(120581minus1)120581] (18)


Eff angle

10 15 20 25 3050

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)


Basis2ABowF13g1

BowF32g





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g






0120577 (ℎbez) 119889ℎbez (19)



2A

rech

120577Basis2Arech

(20)










Reference


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A

BowF13g1

BowF32g



0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g



Geom angle


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A BowF13g1

BowF32g


45 55 65 75 85 95 105 115 125350

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g








1205721 = arcsin(11988611199051 ) (21)



1 (ℎbez) = 119891120572120576 (ℎbez) 1205721 (ℎbez) (22)




119891120572120576 = 1 + 119896 tan 120576 (23)








Notation 1198611199001199082119861 11986111990011990811986532119892 119870119899 3 3 119896120572eff119896 13 13 1198961205721015840119896 20 1535 1198861198961015840119896 1195 1085 119886











0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)




0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)



7 Conclusions





Nomenclature

Symbols




Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










57mm


1485 mm 385 mm

353mm

101mm

6737mm

Stator Rotorℎm

Axial gap with

65mm

훿axsp = 14mm amp
































(a) (b)


04 05 06 07 0803Run number ] (mdash)

Basis2A-exp average


Tota

l sta

ge effi

cien

cy휂 t

(mdash)

092

09

088

086

084

082

08

078

076














] [mdash] 0314 0366 0455 0484 0539 0550 0563 0578119899 [rpm] 4113 4790 5955 6326 7046 7190 7366 7567


] [mdash] 0600 0612 0672 0704 0778 0800119901ein0119905 [Pa] 134543 132957 126460 123753 118911 117797










119891119904= 119901119904Δ119860119864 = 119901119904

Δ119911119911cos ]

Δℎ119903

cos 120576 (10)







following form

1198911199041= 119891

119904cos 120576119904 (11)



120576119904 = arctan(1198911199031199041198911199041

) = arctan( 119891120593119904

tan 120576119891

120593119904 cos ])

= arctan (tan 120576 cos ]) (12)


120593119904 119891

119903119904 and 119891

119911119904


119891120593119904= plusmn119891

1199041cos ] (13)


Inclined angles


fs1

minusfrs

fs3

f휑s fs

fs

minusfzs

minusfs1

fzs

frs

]

]

]

휀

휀

휀s

휑

r

z

minusfs3

minusf휑s

휀s

휀 and 휀s



119891119903119904= plusmn119891

120593119904tan 120576 = 119891(120576 119891

120593119904) (14)



119891119903119904= [[119901119904

Δ119911119911cos ]

Δℎ119903


(15)




119903119904= minus 119898sum

119894=1

(119891DS119903119904)119894+ 119899sum

119895=1

(119891SS119903119904)119895 (16)





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g







0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)



095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)













TE

LE

TE

LE

TE

LE

1394e + 05

1311e + 05

1232e + 05

1149e + 05


1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1111e + 05




120577le = 1 minus 11988821 2Δℎle119904 + 11988820 2 (17)


Δℎle119904 = 120581120581 minus 11198771198790119904 [1 minus (11990111199041199010119904

)(120581minus1)120581] (18)


Eff angle

10 15 20 25 3050

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)


Basis2ABowF13g1

BowF32g





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g






0120577 (ℎbez) 119889ℎbez (19)



2A

rech

120577Basis2Arech

(20)










Reference


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A

BowF13g1

BowF32g



0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g



Geom angle


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A BowF13g1

BowF32g


45 55 65 75 85 95 105 115 125350

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g








1205721 = arcsin(11988611199051 ) (21)



1 (ℎbez) = 119891120572120576 (ℎbez) 1205721 (ℎbez) (22)




119891120572120576 = 1 + 119896 tan 120576 (23)








Notation 1198611199001199082119861 11986111990011990811986532119892 119870119899 3 3 119896120572eff119896 13 13 1198961205721015840119896 20 1535 1198861198961015840119896 1195 1085 119886











0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)




0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)



7 Conclusions





Nomenclature

Symbols




Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





























(a) (b)


04 05 06 07 0803Run number ] (mdash)

Basis2A-exp average


Tota

l sta

ge effi

cien

cy휂 t

(mdash)

092

09

088

086

084

082

08

078

076














] [mdash] 0314 0366 0455 0484 0539 0550 0563 0578119899 [rpm] 4113 4790 5955 6326 7046 7190 7366 7567


] [mdash] 0600 0612 0672 0704 0778 0800119901ein0119905 [Pa] 134543 132957 126460 123753 118911 117797










119891119904= 119901119904Δ119860119864 = 119901119904

Δ119911119911cos ]

Δℎ119903

cos 120576 (10)







following form

1198911199041= 119891

119904cos 120576119904 (11)



120576119904 = arctan(1198911199031199041198911199041

) = arctan( 119891120593119904

tan 120576119891

120593119904 cos ])

= arctan (tan 120576 cos ]) (12)


120593119904 119891

119903119904 and 119891

119911119904


119891120593119904= plusmn119891

1199041cos ] (13)


Inclined angles


fs1

minusfrs

fs3

f휑s fs

fs

minusfzs

minusfs1

fzs

frs

]

]

]

휀

휀

휀s

휑

r

z

minusfs3

minusf휑s

휀s

휀 and 휀s



119891119903119904= plusmn119891

120593119904tan 120576 = 119891(120576 119891

120593119904) (14)



119891119903119904= [[119901119904

Δ119911119911cos ]

Δℎ119903


(15)




119903119904= minus 119898sum

119894=1

(119891DS119903119904)119894+ 119899sum

119895=1

(119891SS119903119904)119895 (16)





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g







0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)



095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)













TE

LE

TE

LE

TE

LE

1394e + 05

1311e + 05

1232e + 05

1149e + 05


1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1111e + 05




120577le = 1 minus 11988821 2Δℎle119904 + 11988820 2 (17)


Δℎle119904 = 120581120581 minus 11198771198790119904 [1 minus (11990111199041199010119904

)(120581minus1)120581] (18)


Eff angle

10 15 20 25 3050

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)


Basis2ABowF13g1

BowF32g





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g






0120577 (ℎbez) 119889ℎbez (19)



2A

rech

120577Basis2Arech

(20)










Reference


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A

BowF13g1

BowF32g



0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g



Geom angle


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A BowF13g1

BowF32g


45 55 65 75 85 95 105 115 125350

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g








1205721 = arcsin(11988611199051 ) (21)



1 (ℎbez) = 119891120572120576 (ℎbez) 1205721 (ℎbez) (22)




119891120572120576 = 1 + 119896 tan 120576 (23)








Notation 1198611199001199082119861 11986111990011990811986532119892 119870119899 3 3 119896120572eff119896 13 13 1198961205721015840119896 20 1535 1198861198961015840119896 1195 1085 119886











0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)




0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)



7 Conclusions





Nomenclature

Symbols




Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













] [mdash] 0314 0366 0455 0484 0539 0550 0563 0578119899 [rpm] 4113 4790 5955 6326 7046 7190 7366 7567


] [mdash] 0600 0612 0672 0704 0778 0800119901ein0119905 [Pa] 134543 132957 126460 123753 118911 117797










119891119904= 119901119904Δ119860119864 = 119901119904

Δ119911119911cos ]

Δℎ119903

cos 120576 (10)







following form

1198911199041= 119891

119904cos 120576119904 (11)



120576119904 = arctan(1198911199031199041198911199041

) = arctan( 119891120593119904

tan 120576119891

120593119904 cos ])

= arctan (tan 120576 cos ]) (12)


120593119904 119891

119903119904 and 119891

119911119904


119891120593119904= plusmn119891

1199041cos ] (13)


Inclined angles


fs1

minusfrs

fs3

f휑s fs

fs

minusfzs

minusfs1

fzs

frs

]

]

]

휀

휀

휀s

휑

r

z

minusfs3

minusf휑s

휀s

휀 and 휀s



119891119903119904= plusmn119891

120593119904tan 120576 = 119891(120576 119891

120593119904) (14)



119891119903119904= [[119901119904

Δ119911119911cos ]

Δℎ119903


(15)




119903119904= minus 119898sum

119894=1

(119891DS119903119904)119894+ 119899sum

119895=1

(119891SS119903119904)119895 (16)





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g







0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)



095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)













TE

LE

TE

LE

TE

LE

1394e + 05

1311e + 05

1232e + 05

1149e + 05


1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1111e + 05




120577le = 1 minus 11988821 2Δℎle119904 + 11988820 2 (17)


Δℎle119904 = 120581120581 minus 11198771198790119904 [1 minus (11990111199041199010119904

)(120581minus1)120581] (18)


Eff angle

10 15 20 25 3050

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)


Basis2ABowF13g1

BowF32g





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g






0120577 (ℎbez) 119889ℎbez (19)



2A

rech

120577Basis2Arech

(20)










Reference


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A

BowF13g1

BowF32g



0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g



Geom angle


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A BowF13g1

BowF32g


45 55 65 75 85 95 105 115 125350

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g








1205721 = arcsin(11988611199051 ) (21)



1 (ℎbez) = 119891120572120576 (ℎbez) 1205721 (ℎbez) (22)




119891120572120576 = 1 + 119896 tan 120576 (23)








Notation 1198611199001199082119861 11986111990011990811986532119892 119870119899 3 3 119896120572eff119896 13 13 1198961205721015840119896 20 1535 1198861198961015840119896 1195 1085 119886











0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)




0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)



7 Conclusions





Nomenclature

Symbols




Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Inclined angles


fs1

minusfrs

fs3

f휑s fs

fs

minusfzs

minusfs1

fzs

frs

]

]

]

휀

휀

휀s

휑

r

z

minusfs3

minusf휑s

휀s

휀 and 휀s



119891119903119904= plusmn119891

120593119904tan 120576 = 119891(120576 119891

120593119904) (14)



119891119903119904= [[119901119904

Δ119911119911cos ]

Δℎ119903


(15)




119903119904= minus 119898sum

119894=1

(119891DS119903119904)119894+ 119899sum

119895=1

(119891SS119903119904)119895 (16)





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g







0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)



095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)













TE

LE

TE

LE

TE

LE

1394e + 05

1311e + 05

1232e + 05

1149e + 05


1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1111e + 05




120577le = 1 minus 11988821 2Δℎle119904 + 11988820 2 (17)


Δℎle119904 = 120581120581 minus 11198771198790119904 [1 minus (11990111199041199010119904

)(120581minus1)120581] (18)


Eff angle

10 15 20 25 3050

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)


Basis2ABowF13g1

BowF32g





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g






0120577 (ℎbez) 119889ℎbez (19)



2A

rech

120577Basis2Arech

(20)










Reference


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A

BowF13g1

BowF32g



0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g



Geom angle


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A BowF13g1

BowF32g


45 55 65 75 85 95 105 115 125350

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g








1205721 = arcsin(11988611199051 ) (21)



1 (ℎbez) = 119891120572120576 (ℎbez) 1205721 (ℎbez) (22)




119891120572120576 = 1 + 119896 tan 120576 (23)








Notation 1198611199001199082119861 11986111990011990811986532119892 119870119899 3 3 119896120572eff119896 13 13 1198961205721015840119896 20 1535 1198861198961015840119896 1195 1085 119886











0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)




0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)



7 Conclusions





Nomenclature

Symbols




Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

0063ℎbez

0256ℎbez

0500ℎbez

0744ℎbez

0937ℎbez

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)



095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)

095

105

115

125

135

145

Stat

ic p

ress

ure (

bar)













TE

LE

TE

LE

TE

LE

1394e + 05

1311e + 05

1232e + 05

1149e + 05


1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1111e + 05




120577le = 1 minus 11988821 2Δℎle119904 + 11988820 2 (17)


Δℎle119904 = 120581120581 minus 11198771198790119904 [1 minus (11990111199041199010119904

)(120581minus1)120581] (18)


Eff angle

10 15 20 25 3050

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)


Basis2ABowF13g1

BowF32g





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g






0120577 (ℎbez) 119889ℎbez (19)



2A

rech

120577Basis2Arech

(20)










Reference


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A

BowF13g1

BowF32g



0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g



Geom angle


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A BowF13g1

BowF32g


45 55 65 75 85 95 105 115 125350

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g








1205721 = arcsin(11988611199051 ) (21)



1 (ℎbez) = 119891120572120576 (ℎbez) 1205721 (ℎbez) (22)




119891120572120576 = 1 + 119896 tan 120576 (23)








Notation 1198611199001199082119861 11986111990011990811986532119892 119870119899 3 3 119896120572eff119896 13 13 1198961205721015840119896 20 1535 1198861198961015840119896 1195 1085 119886











0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)




0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)



7 Conclusions





Nomenclature

Symbols




Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












TE

LE

TE

LE

TE

LE

1394e + 05

1311e + 05

1232e + 05

1149e + 05


1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1394e + 05

1311e + 05

1232e + 05

1149e + 05

1111e + 05

1191e + 05

127e + 05

135e + 05

139e + 05

1111e + 05




120577le = 1 minus 11988821 2Δℎle119904 + 11988820 2 (17)


Δℎle119904 = 120581120581 minus 11198771198790119904 [1 minus (11990111199041199010119904

)(120581minus1)120581] (18)


Eff angle

10 15 20 25 3050

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)


Basis2ABowF13g1

BowF32g





0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g






0120577 (ℎbez) 119889ℎbez (19)



2A

rech

120577Basis2Arech

(20)










Reference


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A

BowF13g1

BowF32g



0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g



Geom angle


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A BowF13g1

BowF32g


45 55 65 75 85 95 105 115 125350

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g








1205721 = arcsin(11988611199051 ) (21)



1 (ℎbez) = 119891120572120576 (ℎbez) 1205721 (ℎbez) (22)




119891120572120576 = 1 + 119896 tan 120576 (23)








Notation 1198611199001199082119861 11986111990011990811986532119892 119870119899 3 3 119896120572eff119896 13 13 1198961205721015840119896 20 1535 1198861198961015840119896 1195 1085 119886











0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)




0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)



7 Conclusions





Nomenclature

Symbols




Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

02

04

06

08

1

Nor

mal

ized

bla

de h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g






0120577 (ℎbez) 119889ℎbez (19)



2A

rech

120577Basis2Arech

(20)










Reference


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A

BowF13g1

BowF32g



0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g



Geom angle


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A BowF13g1

BowF32g


45 55 65 75 85 95 105 115 125350

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g








1205721 = arcsin(11988611199051 ) (21)



1 (ℎbez) = 119891120572120576 (ℎbez) 1205721 (ℎbez) (22)




119891120572120576 = 1 + 119896 tan 120576 (23)








Notation 1198611199001199082119861 11986111990011990811986532119892 119870119899 3 3 119896120572eff119896 13 13 1198961205721015840119896 20 1535 1198861198961015840119896 1195 1085 119886











0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)




0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)



7 Conclusions





Nomenclature

Symbols




Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Reference


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A

BowF13g1

BowF32g



0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g



Geom angle


0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2A BowF13g1

BowF32g


45 55 65 75 85 95 105 115 125350

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)

Basis2ABowF13g1

BowF32g








1205721 = arcsin(11988611199051 ) (21)



1 (ℎbez) = 119891120572120576 (ℎbez) 1205721 (ℎbez) (22)




119891120572120576 = 1 + 119896 tan 120576 (23)








Notation 1198611199001199082119861 11986111990011990811986532119892 119870119899 3 3 119896120572eff119896 13 13 1198961205721015840119896 20 1535 1198861198961015840119896 1195 1085 119886











0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)




0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)



7 Conclusions





Nomenclature

Symbols




Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1205721 = arcsin(11988611199051 ) (21)



1 (ℎbez) = 119891120572120576 (ℎbez) 1205721 (ℎbez) (22)




119891120572120576 = 1 + 119896 tan 120576 (23)








Notation 1198611199001199082119861 11986111990011990811986532119892 119870119899 3 3 119896120572eff119896 13 13 1198961205721015840119896 20 1535 1198861198961015840119896 1195 1085 119886











0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)




0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)



7 Conclusions





Nomenclature

Symbols




Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













0

05

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)




0

02

04

06

08

1

Nor

mal

ized

chan

nel h

eigh

t (mdash

)



7 Conclusions





Nomenclature

Symbols




Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Indices




Acknowledgments


References

















































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation








































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









aerothermodynamic effects and modeling of the tangential...

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