Rotordynamic Forces Due to Turbine Tip Leakage (Part1-Blade Scale Effects)

8/10/2019 Rotordynamic Forces Due to Turbine Tip Leakage (Part1-Blade Scale Effects)

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S. J. SongDepartment ofAerospace Engineering,

Inha University,Namgu, Inchon, Korea

M. Martinez-SanchezDepartment ofAeronautics/

Astronautics,MIT,

Cambridge, MA02139

Rotordynamic Forces Due toTurbine TipLeakage: PartIBlade Scale Effects

An experimental and theoretical investigation hasbeen conducted on rotordynamic

forces due to nonaxisymmetric turbine tip leakage effects. This paper presents anactuator disk model that describes theflow response to afinite clearanceat therotortip. The model simplifies the flow fieldbyassuming that the radially uniform flowsplits into twostreams as itgoes through therotor. Thestream associated withthetip clearance, or the underturned stream, induces radially uniform unloading oftherestofthe flow, called thebladed stream. Thus,ashear layer forms between the twostreams. The fraction ofeach streamand thestrength of shear layer between the twoare found as functions of the turbine loading andflow parameters without resortingto empirical correlations. Theresults show that this model's efficiency predictionscompare favorably withtheexperimental dataandpredictions from various correlations. A companion paper builds on this analysis to yield a model of the three-dimensional disturbances around an offset turbine and topredict the subsequentcross forces.

1 Introduction

Oneofthe major problems thathas always limited thedevelopmentof high-performance turbomachineryis thevibrationofits structure. Ehrich andChilds (1984) have reviewed thevarious types of self-excited vibrations encountered in practice.Self-excited vibrations that occur in rotor systems are referredtoas rotordynamic instability. Themost publicized rotordynamicinstability occurred during the development of the cryogenicturbopumps for the Space Shuttle Main Engine (SSME).Oneof the major causeswasattributed to aphenomenon referred tointhe US as the "Alford force." This phenomenon occursdueto fluid excitation forces causedby anasymmetric tipclearancedistribution, feeding energy into the whirling motion of theshaft.

Thomas (1958) first identified the problem in high powersteam turbines,and Alford (1965) identified thesame problemin jet engines. They independently proposedanidentical mechanism to explain the observed aerodynamic instability. Essentially,thelocal efficiency ishigherin thesmaller tip gap region,and the local torque and the tangential force exerted on theturbine by the fluid increase with the local efficiency. Thisassumption isbased on empirical evidence that the efficiencyof a turbine stage varies linearly with the tip clearance gap(Kofskey and Nusbaum, 1968). When integrated around thecircumference,the netresult is a force acting orthogonalto thedisplacement which adds energy to the forward whirling motion.Figure 1,from Ehrich andChilds, illustrates themechanism.

Although other rotordynamic instabilities (e.g., destabilizingforces in labyrinth seals) have been extensively studied, rela

tively little effort has been expendedonthe tip clearance excitation force. Urlichs (1983) and Vance and Laudadio (1984)established theexistenceandlinearity ofthe Alford force in anunshrouded turbineand anaxial fan, respectively. However,abasic physical understandingofthe influenceofvarious geometricandflow-related parametersin thegeneration andscalingofthe tip clearance excitation force wasstill lacking.

Contributed by theInternationalGasTurbine Instituteandpresentedat the41stInternational Gas TurbineandAeroengine Congressand Exhibition, Birmingham,United Kingdom, June 10-13, 1996. Manuscript receivedatASME Headquarters

February 1996.PaperNo. 96-GT-364. Associate Technical Editor:J. N.Shinn.

Therefore, the objective of this investigation was to gainphysical understandingof the force generation mechanismviaexperimental and theoretical methods. The current investigation's experimental methods and results were previously reported byMartinez-Sanchezet al. (1995). This paper presentsthe first partof a model developed during the theoretical phaseof the investigation.

2 Model Description

The flow responseto aneccentric turbine involves the threescalesof the tip gap,t, the blade span, Hh, and the turbineradius, R. They are typically in ratios of theorder tlHb ~0.01 andHIR ~ 0.1-0.3 (Fig. 2) .Therefore, thegap-scale

effects suchas the leakage flow velocity, which isdrivenbylocal tip conditions, can be decoupled from the other twolarger length scales. The larger length scale effects such asthet ip blade loading become boundary conditions.Thebladescale effects include the radial migration of throughflow toward the tip gap and theunderturning of flow in the outerregion of the blade spandue to the gap leakage flow. Theseeffects are influenced notonlyby the tip gapeffects, whichdetermine the leakage rate, but also by the radius scale effects, which determine the local turbine inlet andoutlet conditions. In reality, theinlet and outlet conditions would varyazimuthally for an eccentric turbine. However, at the bladescale, these boundary conditions can beassumed to be axi-symmetric at the local value. Thus, theblade scale effectslink theeffects of the tip gap and the turbine radius. Finally,

at the turbine radius scale, the turbine eccentricity causesazimuthal variations of flow variables (e.g., theflow coefficient). Theeffects of two smaller scales are seen mainly asconnecting conditions between theupstreamanddownstreamflows.

The differences in the relative importance of unsteady effects arealso pronounced atdifferent scales.A rotor simultaneously spinning at angular frequency, w, and whirling atangular frequency, Q, canhave a radius scale reduced frequency, QR/cx, close to theorder of unity. However, at thetwo smaller scales, the reduced frequencies, U,HJcx and ilt/cx, are orders of magnitude smaller than unity. Therefore,the unsteady effects needto be considered onlyat theradius

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Fig.1 Thomas-Alfo rd destabilizing mechanism

scale, and the flow can be assumed to be quasi-steady at theother two smaller scales.

For the case of a turbine with a whirling rotor, both thestator row and the rotor blade rows are unwrapped and collapsed into a single actuator disk (Fig. 3). In this coordinatesystem,x is the direction of the throughflow, y is the tangen

tial (azimuthal) direction, and z is the radial direction.The analysis proceeds at two levels, as shown in Fig. 3. Thefirst is a quasi-steady blade scale analysis in the meridional (xz)plane. This analysis focuses on the radial flow redistributioneffects due to the existence of rotor tip gap and does not considerthe tangential flow redistribution that occurs upstream anddownstream of the disk. The second is an unsteady radius scaleanalysis in the radial (xy) plane, which focuses on the tangentialflow redistribution effects due to a whirling turbine. This moreglobal, radius scale (xy) analysis, presented in a companionpaper, utilizes the results from the blade scale (xz) analysis asconnecting conditions across the actuator disk. Such a framework can accommodate a flexible, modular approach in whichvarious submodels at different length scales can be separatelyincorporated into the analysis.

t scale flow

Fig.2 Flow features at different length scales in a turbine with an eccentric rotor

This paper exclusively presents the blade scale (xz) model.It is capable of analytically predicting the effects of turbinerotor tip clearance on the losses and the turbine performance.The blade scale model builds on previous efforts by Gauthier(1990) and Martinez-Sanchez and Gauthier (1990).

Nomenc la ture

U

w

BL = total enthalpy of meridionalflow

c = absolute flow velocity; axial

blade chordCi = lift coefficient

D = 2R = turbine mean diameterG = factor defined in Eq. (40a)

H = annulus heightHb =blade span

Q, q = shear layer strengthP, p =pressure

RD = radius of rolled-up vortex

s = blade pitcht = radial tip clearance

= uR = turbine rotational speed atmean radius

w = relative velocity; work defectfactor

, W~ work done per unit mass byunderturned and main flow

ZW = Zweifel coefficienta pitch angle; absolute flow

anglep = -drjld(tlH) = efficiency

loss factor

P = yaw angle; relative flow anglePm =mean rotor blade angle6ia - clearance jet thickness

A = thickness of underturned layerdownstream of the actuator disk

r = circulation of clearance vortex

rj =turbine efficiency

9= azimuthal angle measured in direction of rotation from minimum gaplocation; underturning relative tomean flow angle

\ = nondimensional underturned flow

rate = rolled-up vorticity

TT= turbine pressure ratio; pi

p = density4> = cJU = turbine flow coefficient\\i = meridional stream functionty = (hm - h0?.)/U

2= (turbine stage loadingfactor, or work coefficient

u = angular velocity of rotor shaft rotation

Loy =vorticity of meridional flow

U =angular velocity of rotor shaft whirl

Subscripts

0 = far upstream of actuator disk inblade scale analysis

1 = stator inlet in blade scale analysis2r = rotor inlet in blade scale analysis2s = stator exit in blade scale analysis

3 = rotor exit in blade scale analysis

4 = far downstream of actuator disk inblade scale analysis, axially at thesame location as 0+ of the radiusscale analysis

r = rotors= stator, static; length along clearance

vortex-L = azimuthal component;

perpendicular to rolled-up vortex

Superscripts

= part of flow downstream that hascrossed bladed part of turbine flow

+ = part of flow downstream that wasunderturned due to rotor tip gap

696 / Vol. 119, OCTOBER 1997 Transactions of the ASME

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, . , ~ .

q Q i Q

F i g . 3

Hbscale low

+ % . o + . . '

X

R sca/e flow

Model ing the f low f ie ld a t b lade and radius sca les

The following is assumed in deriving the governing equa-

tions:

1 Incompres sible, invisc id flow.

2 Turbi ne stage collapsed in the axial direction to x = 0.

3 The blades guide the flow perfectly (except for the leak-

age flow) so that the relative flow exit angle is same as

the blade exit angle.

4 The flow is axisymmetric O / O y = 0 ) .

5 Except for the rotor tip gap, the blade geometry is as-

sumed to be radially uniform O / O z = 0) and to be equiv-

alent to that at the mean radius.

6 Flow conditio ns are radially unifo rm at the stator exit.

The blade actuator disk consists of a full-span stator row

and a partial-span rotor row as shown in Fig. 4. Far upstream

the stator row is referred to as 0. The inlet to the stator row

is referred to as Station 1 and the stator exit is called 2s. The

inlet to the rotor is referred to as 2r and rotor exit is referred

to as 3. Far downstre am of the rotor row on the blade scale is

Station 4.

m

X 04

Fig 4 A

b l a d e s c a l e v i e w o f t h e a c t u a t o r d i s k s h o w i n g v a r i o u s a x ia l

s t a t i o n s

I I

U

U

Fig. 5 Turbine velocity tr iangles

y

Figure 5 shows the turbine blading geometry with the velocity

triangles. U is the turbine rotational speed; c is the absolute

flow ve locity; and w is the relative flow velocity. The angles ce

and /3 refer, respec tively, to the absolute and the relative flow

angles.

Tip Scale Analysis

Martinez-Sanchez and Gauthier (1990) showed that, because

of the leakage flow roll-up, an u n d e r t u r n e d l a y e r can be identi-

fied downstream of the rotor, conta ining equal amount s of leak-

age and blade-region flow. The fluid in this region has under-

gone less turning than the main bladed flow, but not zero turn-

ing, and has therefore done a finite amount of work. This

underturned fluid is dealt with in this section.

The blade tip region has been analyzed using a variety of

approaches, The simple model of Rains (1954), which is most

appropriate for thin, lightly loaded blades, uses ideal, pressure-

driven flow concepts to derive the velocity of the gap je t.

Even for the case of the thicker turbine blading, ideal flow is a

fairly good approximation. For example, Rains gave a criterion

for viscous forces to be negligible, and many turbines satisfy

this condition. On the other hand, although the effects of

chordwise pressure gradients on thick blade tip flows, as well

as that of relative wall motion are still potentially significant,

they have not been modeled.

The gap jet is known to interact strongly with the passage

flow and to roll up into a concentrated vortex-like structure.

Rains derived a semi-empirical expression for its trajectory.

Lakshminarayana (1970) also used empirical information on

the tip vortex location and strength to predict details of the

blade pressure distrib ution. In fact, the strength of the vortex

was explicitly related to a partial blade tip loading parameter,

K, varying from 0 to 1, and inferred from extrapolation of

surface pressure measurements near the tip to the end wall.

This section introduces an analytical approach that leads to

simple, but accurate expressions for the location and size of the

leakage vortex. This can then be used in calculating the flow

leaving angle of, and hence the work done by, the leakage flow.

Figure 6 shows schematically the essential features of the

leakage flow. The fluid approaches a blade (here represented

as a flat plate) with a relative velocity, ~'2, which evolves int o

the passage flow velocity, %,s, at locations away from the tip

gap. Under the action of the pressure dif ferential across the

blade, a jet of leakage flow at velocity I,~je escapes under the

blade. This jet penetrates a certain distance into the passage,

but is eventu ally stopped by the ma in flow, which separates the

jet from the wall, turns it backwards, and leads to the formation

of a rolled-up structure containing both leakage and passage

fluid. This col lis ion of the two streams is again shown in

Fig. 7 in plan form, and Fig. 8 shows a schematic of the flow

J o u r n a l o f T u r b o m a c h i n e r y O C T O B E R 1 9 9 7 V o l. 1 1 9 / 6 9 7



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w.

Fig.6 Schematicof thecolliding leakagejetandpassage flows

Fig.7 Planform viewofFig.6

vl*w (Pipndlcularto OP)

Fig.8 Flow patterninplanea-a ofFig.7

structure seen in a cut such asa-a in Fig. 7 wit h leakag e fluid

shown dashed.

Consider thesituation at points along the jet separation line,

such asP inFigs.7 and 8. Ignoring frictional effects, the two

streams that meet there (jet and pass age flows) can both be

traced back, along different paths, to the inle t flow, and hence

have equal total pressures and tempera tures. Si nce they also

have equal static pressures along their contact line (and gener

ally similar static pressures throughout th e region), these two

streams must have equal velocity magnitudes. If the section a-

a isperpendicular to OP, we can think ofpoint P (Fi g. 8) as

the common stagnation point of the two "c oll id ing " flows,

approaching each other with equal velocities, which are each

the component of wia and perpendicular to line OP. It

follows that line OP must bisect th e angle made by wjet andvvpass. This gives a firstand important pieceofinformation about

the location of the rolled-up structure, but,since this struct ure

has afinite andincreasing transvers e dimension, itdoes not yet

locate its center.

Notice that th etransverse momentum balance of a fluid ele

ment near point P requir es that both transve rse collidi ng flows

must bring equal andopposite mo men tum fluxes to the rolled-

up structure. Since the two velocities are equal, we find that

equal mass flows must be entering th erolled-up st ructure from

both fluids. In other words, th eclear and dashed areas in Fig.

8 must occupy equal fractions of thetotal ''vorte x' ' cross sec

tion.Let


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ity r rolled up into the structure increases with downstreamdistance is that the growth of RD gradually overlaps more andmore of the shear vorticity. In this sense, the commonly invokedview of the rolled-up vortex growing by the convection of shedvortices must be used with caution.

Equations (5), (6), and (11) can now be used to calculatethe vortex geometry if the suction and pressure sidecpdistributions are prescribed. A simple approximation can be obtainedusing the theory of lightly loaded thin wing profiles. In thisapproximation, (wp + ws)!2 = vv2, which when used in Eqs.(7) , (8) reduces both (cp)p and {cp)s to functions of c, = {cp)p (c

p)

s alone. Using this in Eq. (11 ) gives finally

d tan (12)

Fig.9 Position and width of rolled-up leakage vortex

where c,is the lift coefficient per unit blade span and is relatedto the Zweifel coefficient, a blade loading parameter. Zweifelcoefficient, ZW, and the lift coefficient, ch are determined asfollows:

w, ,.

PressureSlds

W. - Wput

vorticity

SuctionSid*

Fig. 10 Velocities and angles associated with roll-up

through the gap. It can be verified that the net magnitude wiaof the jet velocity is then equal to ws, as indicated previously.We then obtain (Fig. 10)

tanP=wG (cp)p - (cp)s

1 - (c,,)P(10)

where cp = 2(p p2)l pw\ in each case. Note that (cp)p -

(cp)s is the local lift coefficient, ch referred to the relative turbineinlet velocity. Using the half-angle trigonometric formulae,

tan 0 =V(c;,),, - (c,,),(

Vl - (c)s +Vl - (cp)(11)

Notice that, as shown in Fig. 9, the vorticity vector corresponding to the shear between the jet and the adjacent passageflow is inclined at $ with respect to the blade, i.e., it is parallelto the outer edge OP (Fig. 7) of the rolled-up structure. Thisis also the direction of the mean flow between the two sides ofthe shear layer, which means that the shear vorticity is notconvected at all toward the lineOP.The only reasons the vortic-

ZW = 2* - cos2 /33(tan f32+ tan ft) (13)

wheresic is the blade pitch normalized by the axial blade chord,and

ZW cos2

ftcos2 ft

cos y cos ft,, (14)

where y is the rotor blade stagger angle.

4 Comparison to Vorticity Dynamics Model and toData

Chen's similarity analysis (1991) has provided a means ofcorrelating a variety of rolled-up vortex data. Transverse distances are normalized by gap width, t, and axial distance (ortime-of-flight) are characterized by a parameter

(15)

where x and cx are the axial distance and velocity and AP =pp ps. The data from many experiments (mainly from compressor cascades) correlate well with t*. In addition, a calcula-tional method was developed by Chen to track a series of shedtip vortices from an impulsively started plate, which representsthe situation seen from a convective frame as the flow passesover a blade. The calculated results were shown to also correlatewell with t* and with the data.

We use the correspondences cxlw2 = cos /32,xlxM =cos/3mwhere /32 and /?, represent the inlet and average relative flowangles at the rotor, respectively, to derive

XH = -U w2 co s p2

cD wGcos/3,(16)

where cD = 6jalt is the gap discharge coefficient. Note alsothat w2/wG = 1/vc/.

For an approximate comparison, Rains' (1954) values cD =0.785, c, = 1.35, and cos /3,,,/cos /32= 1.1 are used to relate ;*tox/,i,and then calculate the vortex trajectory using Eqs. (5 ),(6) , (11), and (12). The results are compared in Fig. 11 tothose reported in Chen. The agreement with the data is satisfactory. Additional verification against the theory of Chen is provided by comparing the predictions of both theories regardingthe "center of vorticity" location in a cross-plane similar tothat shown in Fig. 8. In order to be consistent with Chen'scalculations, we have included here both the rolled-up vorticityr = 6.&3wLRDand a vorticity2wj_per unit length (perpendicular

Journal of Turbomachinery OCTOBER 1997, Vol. 119/699



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1

o InoueA Takala

24.0 - m Rains> Johnson

20.0 -6 Smith Analysls\s y'yS

16.0 - .'"/" Theory ofJ0f Chen,1991

12.0 - ,,-obiJ^*

8.0 - ^#Vi V ^

4.0 - AIJSIP

^W\ ' I i I 1 1 ' 1 1 i

0.0 10.0 20.0 30.0 40.0

T' (Non-Dlmanslonal Straamwlse Distance)

50.0

Fig. 11 Trajectory of vortex centroid compared to data and theory ofChen (1991)

to ) of the not-yet-rolled shear layer. In calculating the distancezc between the center of vorticity and the wall, we took thislatter contribution to be at a distance Sia, and that of the rolled-up vortex to be at 0.

From the definition of Bx, with the continuity constraint andthe assumption of spanwise uniform blading,

Bi B,Pi ~ P3

(22)

Vcv = 0 ( 1 7 )

where cL is the meridional velocity defined as cx = icx +kcz. The vorticity equation also reduces to

Vw = 0dz

dc,

dx(18)

Also, the Bernoulli equation reduces to

cL-VB = 0

p 2 p \2 2(19)

Continuity is ensured by introducing the stream function ip(x,z) for the meridional flow where cx = dip/dz, cz = - (dij//dx), so that

- Chen.1991-

Present Analysis- Analysis (Z r=0.46R+h,)- Analysis (Z,-0.46R)

Tlme.t*

Fig. 12 Coordinates of vorticity centroid for tip clearance vortex

Since dBuldtp = 0 by the assumption of irrotationality, Eq.(21) can be rewritten as

Wy 3 =

d(Bu - B3)

difi

_d_( P\ ~ Pi

dip(23)

Thus,the radial distribution of static pressure drop can be used

to determine the downstream vorticity.For the bladed stream,(pt - pi)/p is equivalent to the stagna

tion enthalpy drop, given by Euler's turbine equation, minusthe kinetic energy gain. According to Euler's equation, the stagnation enthalpy drop, -Ah,, is given as

-Ah, = U[cylr (24)

where U is the turbine's rotational speed. The additional subscript rin cy2r denotes the rotor end of the axial space betweenstator and rotor. The stator exit end would, in general, have adifferent tangential velocity, cv2l .' The kinetic energy gain,AK.E., is AK.E. = (l/2)(c; 3 )

2- At the rotor exit (3), the flow

has split into two streams. For the bladed stream,

cy7, = U - cxi tan /33 (25)

Thus,

Pi ~ Pa

PUcx0 tan a2 - - [U

2 - (cx3)2 ta n2 /33] (26)

For the underturned stream, cy2r is the same as that for thebladed stream because the flow is assumed to be radially uniform upstream of the rotor. Thus,

1This interblade gap effectcompanion paper.

modeled and explained in Section 3.4 of the




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Pi - Pi = U[cx0 tana2 - (U - c+3)]

1+ 2(^ >

W + sing

2\Cxi

cospn(27)

One featureof the actuator disk approximation isthat only halfof the total change visiblefardownstreamofthe disk occursatthe disk with theother half occurringin theflow downstreamfrom thedisk (Horlock, 1978). Therefore, fardownstreamonthe blade scale (at 4 orx = 0+ ) , the axial velocities are asfollows:

The last terminEq. (27) isincluded toaccountfor thekineticenergy dissipated when the flow, which leaked through the rotorgap collides with an equal amountof thepassage flow beforerollingupinto vortices.

dldty inEq. (23) can beexpressedas

J- }_dcA d_

cx dzA-o+dcx(28)

Then from Eqs. (23) , (26), (27), (28), theequation for ipbecomes

Upstream (x


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5.2 Model Predictions. We can now compare the calculated losses to those reported in the experimental literature. Werely for this on the compilation by Waterman (1986), whichgives data for ten cases (nine different turbines) over a widerange of parameters. Waterman reports for each case the tipvalues of the work coefficient, 'P, degree of reaction, R, flowcoefficient, and individual blade loading, Zweifel coefficient.

One potential difficulty in application is that only tip parameters are given, whereas from the nature of our theory we suspectthat mean parameters might be more appropriate.

Scanning Table 1, we first notice a large disagreement for

Case 1 (Kofskey turbine). This is an impulse rocket turbopumpstage with an extremely large tip loading (ip = 7.0). Case 4,with very high reaction, is also substantially underpredicted,which may point out an insufficient predicted underturningangle 9 for these conditions. The rest of the cases are jye llpredicted. Excluding Case 1, the mean squared error is e2 =0.1162 and the mean error is f = -0 .1408. If Case 4 is removed,the two quantities would be, respectively, 0.0363 and 0.0498.Perhaps more effort should be devoted to an understanding ofthe leakage and underturning for high reaction rotors.

6 Summary and Conclusions

A model has been developed to illuminate the effects ofspanwise flow redistribution caused by the presence of a small

rotor blade-tip gap. To this end, the blade-to-blade details areignored by using an incomplete actuator disk formulation, whichcollapses both stator and rotor to a plane, across which connecting conditions are imposed.

The model assumes that near the gap region, the underturnedlayer originates partly from the gap flow and partly from thepassage flow, both leaving the passage in the form of rolled-uptip vortex. The trajectory and other details of this vortex arecalculated using a simple model involving the collision of theideal pressure-driven leakage jet with the passage fluid. Thismodel was calibrated against both data and theory of Chen(1991).The new theory allows prediction of the vortex strengthand trajectory.

The model predictions were then compared to a set of datainvolving nine different turbines. With the exception of one

anomalous case, the calculated efficiency loss factors are reasonably close to the data, showing less deviation than the losscorrelations of Ainley, Soderberg, Roelke, Kofskey, and Lak-shminarayana.

Table 1 Efficiency loss and work defect factors (calculated from the blade scale analysis) compared todata. The last line is computed with modified work coefficient chosen for near-exit flow.

Case Author ZW % R t/Hb Pdala Pcalc w0alc Kcalc

1 Kofskey 55 0.79 7.00 0.02 0.05 1.02 2.90 4.096 1.0252A Marshall-

Rogo1.02 0.50 1.48 0.32 0.035 1.51 1.44 2.563 0.346

2B Marshall-Roqo

1.09 0.44 1.25 0.35 0.035 1.23 1.42 2.757 0.290

3 Szanca-Behning-Schum

1.59 0.57 1.46 0.47 0.033 1.90 1.68 3.066 0.369

4 Holeski-Futral

0.35 0.26 0.69 0.69 0.031 2.53 1.66 4.880 0.365

5 Ewen-Huber-Mitchell

0.70 0.25 1.05 0.45 0.02 1.50 1.46 3.914 0.411

6 Lart 0.92 0.51 1.41 0.51 0.02 1.80 1.92 3.500 0.3757 Yamamoto 0.79 0.42 1.52 0.47 0.03 1.63 1.80 3.558 0.452

8 Patel 0.70 0.28 1.15 0.61 0.01 1.81 1.42 4.813 0.6839 Haas-

Kofskey0.80 0.35 1.37 0.47 0.03 1.80 1.64 4.714 0.468

1M Kofskey (if%-2)

0.55 0.79 2.0 0.02 0.05 1.02 0.93 1.614 0.473


5.1 Work Defect and Efficiency Loss Determinat ion.

Application of the Euler equation to both fluids gives thework done per unit mass by each stream:

W+ = U(cx2s tan a2 - c^)

W~ = U(cx2stan a2 c}73)

Then, the total turbine work per unit mass is

W = \W+ + (1 - \)VT

(43)

(44)

(45)

In coefficient form, the power extracted by the turbine, and,hence, the tip loss coefficient can also be calculated easily,

$ =mU

2 {haJa-h,i)pdili+\ (hn - hti)pdtp (46)

The total pressure drop in the mixed-out region is given by

Pio - Pt4 ___ Pa ~ PA 1(47)

where p4 is at far downstream of the disk (on the blade scale),and we have taken advantage of axial and radial velocities beingzero at this location. The static pressure drop between Stations

0 and 4 is given by Eq. (42 ). Then, the efficiency is

*

Pto - Pn

PU2

(48)

where ty is as given by Eq. (46). The efficiency loss factorfollows as

0 =l -*7

tlH(49)

The work defect factor (different from /3) is defined as

\)>o -

* 0

tlH(50)

where


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These results suggest that upstream flow redistributions,which have been largely ignored so far, may be of importancein understanding the basic physics of tip leakage effects.

Acknowledgments

This work was performed under NASA Contract NAS 8-35018. Glenn E. Wilmer, Jr., is the Technical Monitor.

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Gauthier, R. P., 1990, "An Investigation of Flow Field Perturbation Caused

by Constant Blade Tip Clearance in a Turbine," S. M. Thesis, Department ofAeronautics and Astronautics, MIT.

Horlock, J. H 1978,Actuator Disc Theory, McGraw-Hill.

Kofskey, M. G., and Nusbaum, W. J 1968, "Performance Evaluation of aTwo Stage Axial Flow Turbine for Two Values of Tip Clearance," NASA TND4388.

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Journal of Turbomachinery OCTOBER 1997, Vol. 119/703

Rotordynamic Forces Due to Turbine Tip Leakage (Part1-Blade Scale Effects)

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