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Unsteady Rotor Dynamics in CascadeYU-TAI LEE, THOMAS W. BEIN

David Taylor Research CenterBethesda, Maryland

andJIN ZHANG FENG, CHARLES L. MERKLE

The Pennsylvania State UniversityUniversity Park, Pennsylvania

ABSTRACT

A time-accurate potential-flow calculation method hasbeen developed for unsteady incompressible flows through two-dimensional multi-blade-row linear cascades. The methodrepresents the boundary surfaces by distributing piecewiselinear-vortex and constant-source singularities on discretepanels. A local coordinate is assigned to each independentlymoving object. Blade-shed vorticity is traced at each time step.The unsteady Kutta condition applied is nonlinear and requireszero blade trailing-edge loading at each time. Its influence onthe solutions depends on the blade trailing-edge shapes.

Steady biplane and cascade solutions are presented andcompared to exact solutions and experimental data. Unsteadysolutions are validated with the Wagner function for an airfoilmoving impulsively from rest and the Theodorsen function foran oscillating airfoil. The shed vortex motion and itsinteraction with blades are calculated and compared to ananalytic solution. For multi-blade-row cascade, the potentialeffect between blade rows is predicted using steady and quasiunsteady calculations. The accuracy of the predictions isdemonstrated using experimental results for a one-stage turbinestator-rotor.

NOMENCLATURE

CL = lift coefficientCP= pressure coefficientc = chord of the airfoilCDU = velocity potential due to uniformly distributed vortexCDL = velocity potential due to linearly distributed vortexCD1 = coefficient defined in Eq. (12)CD2 = coefficient defined in Eq. (12)CS = coefficient defined in Eq. (12)ds = small surface elementE = numerical errorF = functions defined in Eq. (10)G = Green's functionH = cascade spacing or pitch

h = spacing between two flat platesN = total number of panelh = surface normalp = field point or local static pressureq = running index over surface integration

= distance vector in (x,y) coordinater = distance vector in ( ,ii) coordinateS = panel lengtht = timeU = reference velocityV = total velocityVn = surface moving velocityV. = onset flow velocity

= disturbance velocity= incoming flow velocity

W = vortex-sheet surface in the wakex,y = global ground-fixed coordinatea = angles defined in Eq. (10) or angle of attackI' = circulation around an airfoily = vortex strengthO = blade stagger angleK = ratio of panel lengthsA = boundary surfaces,l = a constant defined in Eq. (10)E,rl = local panel coordinatep = fluid densitya = source strength(D = total scalar velocity potential

= disturbance velocity potential= inflow velocity potential

ca = angular rotational speed

Sub- or Superscriptse = exact solutionm = cascade modifications = tangent to the surfacesa = single airfoilu = upper surface

Presented at the International Gas Turbine and Aeroengine Congress and ExpositionOrlando, FL June 3-6, 1991

This paper has been accepted for publication in the Transactions of the ASMEDiscussion of it will be accepted at ASME Headquarters until September 30, 1991

Copyright © 1991 by ASME

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= vortex= lower surface

INTRODUCTION

The flow field can be represented by a total scalar velocitypotential 1 as follows,

V = v(D = v4m+v4) = v+v , (1)

Unsteady flow analyses for turbomachinery can becategorized as linear or nonlinear. The linear method useseither the linear potential approach (Verdon and Caspar, 1984)or the linear Euler approach (Hall and Crawley, 1987). Thenonlinear method includes both the nonlinear Euler and theReynolds averaged Navier-Stokes (Rai, 1987 and Giles, 1988)methods. Linear methods use the isentropic and irrotationalassumptions. Quandt (1989) has shown that the solutionobtained from the unsteady linear potential-flow method givescorrect fluid enthalpy, pressure, and velocity changes obtainedfrom the traditional nonlinear Euler turbomachine energyequation. The disadvantage of using the linear potentialanalysis is that the method does not allow for incomingvorticity. Engineering experience (Lee et al., 1990) shows thatlinear potential methods are the simplest models which giveaccurate predictions of the very large changes in lift andmoment for shock-free subsonic flows. Computational effortsincrease sharply when Reynolds averaged Navier-Stokesmethods are used. However, more complete physics of fluidflow, e.g. flow separation, nonlinear vortex interaction andturbulence, can be predicted. Due to the greater completenessof the nonlinear methods, these methods can be used tosupplement the simplified linear approaches during a designprocess when experimental data are not available.

In this paper, a time-accurate 2-D linear potential-flowcalculation method is developed. Viscosity effects are partiallyaccounted for in the analysis by using a nonlinear Kuttacondition at the airfoil trailing edge. In applying the Kuttacondition, the wake of the airfoil is represented by discretevortices (Kim and Mook, 1986 and Yao et al., 1989). Themethod developed is capable of providing flow predictions forvarious combinations of steady and unsteady body motions.The emphasis of the present paper is on cascade flows,particularly unsteady cascade flows. Although the numericalmethod developed is general, cascade flows require specialanalyses. When calculating unsteady flows between two bladerows, the vortex/blade interaction requires detailed treatment.The unsteady cascade predictions are possible only when thebuilding blocks of steady cascade flow calculations and somebasic unsteady single airfoil calculations are predicted correctly.This paper compares calculated results for single airfoils andcascades under similar conditions. It concludes with a quasiunsteady solution for a two-blade-row cascade (Dring et al.,1982).

Mathematical Model

Consider a 2-D system configuration consisting ofmultiple airfoils and/or nonlifting bodies, or a linear multi-blade-row cascade which execute arbitrary relative motionsbetween airfoils/bodies or between cascades in anincompressible potential flow. Let the surface of the solidboundaries be denoted by A(fl,t), where A is a distance vectorin a global ground-fixed coordinate system (x,y) and t is time.The shear layer next to the surface A and the shed vorticity inthe wake are assumed to consist of thin layers modelled asvortex sheets. The vortex sheets in the wake are symbolized byW(fl,t). They are allowed to move with the local fluid particles.

where 4_ and v. represent the inflow velocity potential andvelocity at infinity, including any incoming onset flow andinduced flow due to a near-field disturbance. 4) and "v representthe disturbance velocity potential and velocity due toairfoils/bodies and the wake vorticity. Both the total anddisturbance velocity potentials satisfy Laplace's equation

72 = V24) = O . (2)

The problem posed is to find the velocity potential 4) and thefree wake vortex structure. To ensure a unique solution to theproblem, the disturbance velocity potential is subjected to: (1)a kinematical condition applied at the surface A where fluidparticles maintain the same normal velocity as the movingsurface; (2) the Kutta-Joukowski condition of equal pressurebetween the upper and lower airfoil surfaces applied at thetrailing edge; (3) total circulation is conserved at any time, i.e.Kelvin's theorem; and (4) a dynamical condition of continuouspressure applied at free wake vortex sheets.

Using the classical Green's function approach andMorino's formulation (Morino and Tseng, 1978) in the globalcoordinate system, Eqs. (1) and (2) are transformed into anintegral equation

4)(P) = w4 ) a 4

+ 1 f (3)8G(p,4)2a J A[^(4) an G(P,4) an l ds4 .

Here p represents the field point on A while q is the runningindex over all panels. The symbol 4 w represents the velocitypotential due to shed wake vortices and -A = n x 7 + n. I and-Aw are the unit outward surface normal vectors to A and theupper side of W. The Green's function is given by G=¢n(1/r),where r is the distance between p and q. This formulationallows I V(p) l to be equal to the vortex sheet strength y (p) onA. a4)/an is given from the normal boundary condition

0(D -n = VB-n (4a)

or

(4b)an VB n an

where Vn is the velocity, including translational and rotationalspeeds, of the surface A. For a finite number of airfoils/bodies,a4— /an is prescribed as an onset flow V.. Equation (3) is aFredholm integral equation of the second kind for the unknown

The calculation of unsteady force and moment on eachmoving body/airfoil is reduced to an integration using theunsteady Bernoulli's equation in the global coordinate systemas


.7

q 5,/2

a1D V + p = Pm + V. (5)8t 2 p p 2

Equation (5) can also be nondimensionalized by a referencevelocity U. If one uses the original symbols for nondimensionalvariables, the following formula in a body-fixed coordinate isobtained

CP= "' = -2 A-V{2VB+I^+Vo . (6)p 0/2 at

Here the velocity potential of the onset flow is assumed to besteady.

Numerical Method

Singularity DistributionsThe surface A is discretized into small line segments.

The control point for each segment is located at the center ofthe segment. The normal to the surface is approximated by theperpendicular to the straight segment. The distributions of thevortices and the sources on each segment are assumed to belinear and constant, respectively. The velocity potential at afield point p due to a line segment q of length S with a constantunit-strength source, i.e. a (E ) = 1, in a local line-segmentcoordinate (E ,r^) as shown in Fig. 1 is

2n(%(P,q) = f 5 o(E) In dE 72 ( )

S-ri pF1 -t,F3 - SF4

77

—s/2 q S/2

Fig. 1 Schematics for a line segment

The velocity potential at the point p due to a constant unit-strength vortex, i.e. y (E) =1, is

2a41 0(p,q) = f ZS y(&) arctan Pd^2 T1p (8)

_ P Fl -FZ - I F, .

The velocity potential corresponding to a linearly distributedvortex sheet, i.e. y (E) = 2Z /S, is

2ncYL(P,q) = S y(^) arctan Pd^2 rlp

2_ liP +1(^2 - 11 )Fl-IE*IF3,2 2S P P 4 2S P P

In Eqs. (7)-(9) we have

F1 = a l - a2

F2 = 7C-a l -a 2 +2n.1F3 = ln(R 1/R1)F4 = ln(R1R2)Rl = [(S + 12 +^P)112

R2 = [(2 92 +fpp] 1/2

a 1 =P + S/2

arctanlip

_P- S/2a 2 = arctan

lip

The function F2 is multiple valued when E P > 0 and Irk P Iapproaches zero, i.e.). =1 when it p crosses the positive real axisfrom positive r^ p, 1. =-1 when r^ p crosses the positive real axisfrom negative q P, and ), =0 for all other cases. Thus thevelocity potential at an ith segment due to a jth segment which

Fig. 2 Linearly distributed vortex on a line segment

contains a uniformly and a linearly distributed vortices, i.e.

Y(E)=(yj+yj+i)/2 +(yi+iyi)E/25^ as shown in Fig. 2, and aconstant source is given by

4,,j = , (11)

where

CD1,j = 1 (CDU,7 -CDL)(12)

CD2i, = I (CDU,^+CDL^^) ,

and CDU, J and CDLIJ represent the velocity potentials due tothe uniformly and the linearly distributed vortices from Eqs. (8)

(9)

(10)

3


and (9), and CS is due to the uniformly distributed source fromEq. (7). The velocity potential at p due to a discrete shedvortex in the wake is

2a4w(p,9) = Yw(4) arctanyP -y9 . (13)p X9

is worth noting that this condition does not restrain the trailingedge from having a variation of pressure in the time domain.Using Eq. (6), one obtains

(18)-(V+VBMV+VB) J = 0

Matrix Equation for Bound Vortex YFor a unique solution outside of the surface A described

by Eq. (3), the flow inside the surface A can be assumed at rest.If each body/airfoil surface A is divided into N segments, theinterior flow-quiescent condition requires

V4 = 41i+1 - ^l = 0 i=1,2...,N-1 . (14)

The no-penetration boundary condition, Eq. (4), is imposed atthe control point of each line segment. By substituting Eqs. (3),(4), (Ii) and (12) into Eq. (14), a set of linear algebraicequations for the bound vortex y is formed

(CD1i 1 -CD1 i_1.1)Y1 + CD2i,,-CD2 i_1,N)YN+1N-1

+ (CD1.. -CDI. +CD2 -CD2. )Y .r^+l r-1j+1 ij r—lj 1+1

N i=' riz-1 (15)_ CSii CSi_ljYwi^wi

i=1 ( ) an,

i=1

i =1,2....,N-1 .

Here ITS represents the total time steps. Equation (15) formsN-1 equations for N+1 unknown values of y. Hence two extraequations are required for obtaining a unique solution.

Extra Conditions for Nonlifting BodyAn unsteady potential flow past a nonlifting body

generally does not shed vorticity in the wake. Hence an extracondition can be obtained by requiring

Y1 = YN.i (16)

for a closed body, where y 1 and YN+1 are the first and the lastvortex strengths as defined in Eq. (15). Since no lift will begenerated, a zero net circulation around the body can be used,i.e.

N

E S (Yi+yi+l) = 0 . (17)i=1

Equations (15), (16) and (17) form a set of linear equations fordetermining y on a non-lifting configuration.

Nonlinear Kutta Condition for Lifting BodyThe Kutta-Joukowski condition was originally applied to

two-dimensional steady flow in order to obtain a finite velocityat the trailing edge, and as a consequence, the flow is uniquelydetermined. This hypothesis implicitly accounts for viscouseffects otherwise neglected in the potential-flow theory. Insteadof requiring a velocity condition at the trailing edge, a pressurecondition is used in the present numerical model. Since thetrailing edge does not account for any loading, the pressuresthere on the upper surface (sub- or superscript u) and the lowersurface (sub- or superscript Q) are required to be the same. It

Since the impermeable boundary condition is applied at A, theflow on the upper and lower surfaces is along the tangents (sub-or superscript s) to both surfaces. Use Eq. (1) and thefollowing relations

N 1r = E — (Yi+Yi+1)S

i=1 2(19)

v 1 vsu-vse u+ su e + r 2 ( m v vs is ia)

Eq. (18) is cast as

S1 N 1 S +Si+l SN

( 1+ Y1+E Yi+1+(1+YN+12 j=1 2FT 2 (20)

-F(t -et) s4 SPVT

where VT = 2(0 t)v r and 1 is the circulation of the airfoil anddefined as positive in a clockwise direction. Equation (20) islinear in y if VT is considered to be a constant. In the presentstudy, Eq. (20) is solved iteratively to account for thenonlinearity of VT. Both Yao et al. (1989) and Kim and Mook(1986) applied a more restricted Kutta condition at the trailingedge by requiring y,=Y h+1= 0 and placing a wake vortex ofunknown strength there. This implies that their algebraicequations are linear and require an optimization technique toproduce a deterministic system.

For the present scheme, according to Eq. (15) one moreequation is needed in order to obtain a unique y-distributionfor a lifting body. This condition is provided by requiring thevelocity gradients along the tangents at the trailing edge fromboth the upper and lower surfaces to be the same, i.e.

(ay )u = (ay ) Q • (21)as as

Allowing the equality of velocity gradients from the upper andthe lower surfaces at the trailing edge further ensures thesmooth merge of two jet flows. Experience also indicates thatthis condition offers stable and accurate predictions. When asecond-order backward differencing scheme is used, Eq. (21) istransformed to

KLK:(2+xa)Y—(1 +x 1)2Y2 + Y3] +(1 +xu)2YN —YN-1

K(2 +K)

where K u =SN_ 1 /SN, K 1 =S 2/S 1 and K=SN_ 1(l+x 5 )/S 2 (l+K 1 ).Hence Eqs. (15), (20) and (22) form a determinate system ofnonlinear equations for determining y for a liftingconfiguration. They are solved using an iterative scheme.

1N+1 (22)

■


Determination of Shed VorticityKelvin's theorem states that the total circulation of the

fluid at any instant is conserved. This condition provides amechanism to shed vorticity into the wake. At each time step,a uniformly distributed vortex segment with strength y w isgenerated in the wake adjacent to the airfoil trailing edge. Thelength Sw of the vortex segment is the distance the trailing edgemoves between t-et and t. At a subsequent time step thisuniform line vortex is replaced by a discrete concentratedvorticity of equivalent strength located at the center of thesegment. The generated trailing-edge vortex segment relates tothe airfoil circulation r as follows

P(t) -P(t-At) + S(t)y(t) = U (23)At At

The model depicted in Eq. (23) yields stable solutions which areindependent of the time-step used. However, for the presentcalculations the solutions were found to be dependent on thetime step used if a concentrated vorticity model was adopted formodeling the trailing-edge vortex generation. The latter modelwas used in Kim and Mook's model (1986). Since the time stepused by Kim and Mook was extremely small, they may not beaware of the time dependency of the discrete vortex generatingmechanism at the trailing edge.

The wake vortices are convected downstream anddevelop a vorticity field. In the present numerical scheme,these vortices are tracked through the flow field usingLagrangian techniques. The convection of these discretevortices is modelled by a predictor-corrector scheme as

t+ At) = r(t)+va,[vw(t),t]At (24a)

r)(t+At) = r)(t+At)

+2 ( p,[rwt+At),t] -vw[r,(t),t] )At (24b)

ra,(t+At) = (t+At) . (24c)

The superscripts in parentheses represent the iteration numberwithin each time step. The corrector in Eq. (24b) is particularlyimportant for a body oscillating at low frequency. For such aflow, au/at is relatively small in Eqs. (18) and (19). Bothequations imply that the nonlinear effect in the Kutta conditionis strong and the velocity prediction within each time step playsa dominant role in obtaining a converged solution. Giesing(1969) also used a predictor-corrector scheme to convect thevortices in the wake, but treated the intermediate iteration stepas a complete separate time step". The present calculationonly uses the corrector to locate the new vortex positionswithout performing the rest of the calculations at the next timestep. Hence the calculation time only increases slightly. Kimand Mook (1986) used only Eq. (24a) for vortex convection, buttheir time step is rather small.

Special Considerations for a CascadeFor a cascade of blades with pitch H, each blade

generates a circulation. If the cascade runs along the y-axis,there exists an upwash at upstream infinity of the cascade anda downwash at downstream infinity. In conjunction with aspecified inflow onset condition as shown in Eq. (4), an extraterm is needed to ensure a unique upstream inflow condition,

i.e.

aim ^u r= n E (25)

an mj=1 2Hmj

for a multi-blade-row cascade flow, where MJ is the totalnumber of blade rows, Hmj is the mj-th cascade spacing or pitch,and P m is the blade circulation from the mj-th cascade.

In a cascade configuration, the velocity potentials at afield point p due to a point source and a point vortex on ablade surface q can be written as

40(p,q) = - 2i ln[sinh2H(xp -xq) cosZH(yp -yq)

+ cosh2H(xp -xq) sin2H(yp-yq)]1n (26a)

(P,4) = 2n arctan sinH(yp-y9)

(26b)sinhH(xp -x4)

where FI=n/H. Equation (11) is therefore modified to

4,i = Lr (aj^oij +YA,ij)

J ^ oij j yii , oij iij iij(27)

sa m4+

where superscripts sa and m represent the corresponding singleairfoil and its cascade modification. 4's are defined as

47. 2.^c x?+y. (28a)

= 1 azctany` , (28b)2a xi

and the cascade modification is

sQ 1 sinh2Hxi +sin2Hy i= --ln( -.n (29a)2a (Hxi)2+(Hyt)z

1,,Q1 xsinHyi-ysinhhxi (29b)= —arctan

2n x1sinhHxi+ysinHyi

A complete integration of the velocity potentials for the sourceand the vortex distributions on each segment was performednumerically for the cascade modification term 4 in Eq. (27).

APPLICATIONS

The present calculation method was first validated bycomparison to known steady flows. Steady-flow comparisonsare presented for two cases. The first case is for flow past abiplane, which has an exact conformal mapping solution(Robinson and Laurmann, 1956). The second steady flowcalculation is for flows past a NACA65-1210 cascade for whichthe solutions compared to measured values (Herrig et al., 1957).For unsteady flows, four cases calculated using the presentmethod are presented. They include the following three basicblade motions: an airfoil moving impulsively from rest, anoscillating airfoil, and a vortex interacting with an airfoil.


a a

0.6 -0.4 0.2 0.0 0.2 0.6 0.6

./c

(3b) Varying plate spacing

a1 I ^^^I IX

h/c

50.00

h/c - 0.220

i a - 65.00Ix

o;^0.2 0.0 0.2 0.4 0.6 9.6 0.I 0.3 0.0 0.2 0.1 0.6

(3c) Varying inflow angle

(3d) a =90°

Fig. 3 Pressure distributions on a biplane

Calculations were made for both a single airfoil and a cascade Plate Conformal Presentunder each of the specified blade motions. The last case Mapping Calculationpresented is a quasi-unsteady calculation for a two-blade-row Upper

Elstator-rotor configuration. Lower

Steady FlowsThe main purpose of investigating flows past a biplane,

i.e. two flat plates, is threefold: to examine the accuracy of theprediction when compared to an exact solution; to numericallyexplore the limits of the prediction scheme; and to understandthe prediction capability of the nonlinear Kutta condition.

Figure 3a shows a schematic of the biplane geometry andinflow description. The comparisons between the presentcalculated results and the exact conformal mapping solutions inFig. 3b correspond to a constant inflow angle a =20, andvarious spacings between two plates. When the spacing is large,pressure distributions are identical on the two plates. As thespacing is reduced, a stronger interaction between the twoplates is observed. The pressure of the lower surface of theupper plate approaches that of the upper surface of the lowerplate. Figures 3c and 3d show the effect of varying the inflowangle under the condition of strong plate interaction. As avaries from 20 to 90, the pressure on the lower surface of theupper plate stays nearly the same as that on the upper surfaceof the lower plate. However, the lift generated by the lowerplate is reduced as a increases. The zero-lift a for the lowerplate at h/c=0.228 is between 50° and 65°. For a larger thanthe zero-lift value, the lower plate actually begins to generatenegative lift. When a approaches 90 as shown in Fig. 3d, thepressure curve for the upper surface of the upper plate matchesthat for the lower surface of the lower plate, and similarly forthe other surfaces of both plates. The total lift of the two-platesystem is zero. When the spacing between the two platesincreases at a =90°, the pressures on the lower surface of theupper plate and on the upper surface of the lower plateapproach those of the other surfaces. At h/c =9.708 as shownin Fig. 3d, the pressures on all the surfaces coincide and theinteraction between the two plates is minimum. Although inreality flow separates at high values of a, the present numericalcalculations serve the purpose of validating the implementednumerical procedures under extremely severe flow conditions bycomparing with the conformal-mapping solutions. For all thecases predicted, the present results agree well with the exactsolutions and show extremely "clean" predictions at both theleading and the trailing edges.

Steady flows through a cascade of NACA65-1210 bladeswere also examined. Figure 4a shows the calculated bladepressure distributions compared to measurements (Herrig et al.,1957) at two different inlet flow angles (a 1 =12.1° and 16.1°) andblade stagger angles (0 =-32.9° and -28.9°). Flow is at thedesign condition for the first case. The predicted blade loading

c ---1

II(3a) Geometry schematic


-- PRESENT CALC.1110 0 MEASURED

—zs_a _oo 0 o v. '

U a 1104] -.

•n e _ -12.9°"° - 12.1 0

vo o-ao 0.40 (ISO 080 1.00

FRACTION OF CHORD

II HO

a, u.n0LII • __ o o p O

s on

00 0 10 0 ^-U

aD

8 = -28.9°.8o a «1 = 16.10

- 1211 I-+ t^

0.00 0.20 0.80 0.50 0.80 000

FRACTION OF CHORD

(4a) Predicted pressure distributions

c l)/ Cascade /

a SINGLE NACA0010 a=5deg- e TWO NACA0010,h/c=0.328,a=5deg

a TWO NACA0010,h/c=0.328,a=20deg

20J _

Panel Number

(5b) For one and two NACA 0010 airfoils

Fig. 5 Numerical convergence versus panel numbers

1.20 WAGNER FUNCTIONJ IIW 1'00 — _

0.80

Oa

t 0 .80

0.40zzW

Z 0.20

o!r

— EXACT SINGLE FOIL---- CALCULATED SINGLE FOIL-------- CASCADE (H=1.5c— — — CASCADE (H=1.Oc

DISTANCE TRAVELED, IN SEMICHORDS

(6a) Instantaneous lift

(6b) Wake pattern for a single airfoil

(4b) Blade-to-blade pressure contours 02

Fig. 4 Calculation for NACA65-1210 cascade 1 ^^°°^ SINGLE FOILJ •• CASCADE I (H=1.5c)


agrees well with the measurements. The computed blade-to-blade pressure contours for flows past a single airfoil and acascade are depicted in Figure 4b. The results indicate that thepressure gradients for the cascade due to blockage is largerthan that of the single foil, particularly in the leading andtrailing edge areas. The computed flow turning angles throughthe cascade for both cases are 21.98° and 26.13° versus themeasured values of 19.6° and 23.3°. If an estimated boundary-layer displacement thickness of 0.4% of the chord, obtainedbased on a flat-plate boundary layer, at the trailing edge isadded to the airfoil ordinates, the calculated turning anglesbecome 19.99° and 23.8°. This modification in the calculationprocedure indicates that the cascade exit flow angle dependsalso on the viscous effect in the trailing-edge area.

The numerical error E of the present calculations wasevaluated based on

N

E = i_[ (C -CP )2]'"2 (30)

N ,_ 1 P ^

where CP, represents the exact pressure distribution. There aretwo cases in Fig. 5 to show the numerical convergence versusthe panel numbers used. The first case shown in Fig. 5a is fornonlifting flows past a circular cylinder. The panel numbersused range from 20 to 600. The second case shown in Fig. 5bis for lifting flows past either a single NACA 0010 airfoil or twoNACA 0010 airfoils. The exact pressure distributions used forcalculating errors are the numerical solutions of 1000 panels forthe single airfoil and 500 panels for the bifoil. The numericalconvergence rate shown in Fig. 5b is independent of thenumber of the airfoil, but it decreases as the angle of attackincreases. Since the coarse grid points do not usually matchwith the finest grid points used as the exact solution, theconvergence dependency on the angle of attack relates to theinaccurate interpolated exact C r-distribution in the leading-edgearea when the slope becomes steep for large angles of attack.Nevertheless, based on this analysis using 100 panels for aclosed body indicates that the error of the solution is under 10 4 .

Unsteady FlowsFigure 6a shows the calculated transient lift coefficients

as a function of the airfoil traveling distance for an impulsivelymoving NACA 0012 airfoil of 4% thickness at an angle ofattack of 5° and two cascades with different spacings (H) ascompared to the exact Wagner function (Fung, 1969) for asingle airfoil. The wake pattern of the calculated single airfoilcase is depicted in Fig. 6b. Figure 6c shows the calculatedvortex locations for the three cases. The results indicate thatthe period of the transient phenomenon becomes shorter forthe cascade flow.

Figure 7 shows a similar comparison for oscillatingNACA 0012 blades. The exact solution plotted is theTheodorsen function (Fung, 1969) at a reduced frequency a> c/Uof 17. The time step chosen is to cover one period of theoscillating motion with minimum 25 points. The amplitude ofthe calculated lift coefficients using the present method,referring to (Cr).f versus T(=w t) in Fig. 7a, is smaller for thecascade data. The shed vortex patterns, shown in Fig. 7c, of thesingle foil and the cascades are compared to the flowvisualization of Bratt (1950), shown in Fig. 7b, for a singleNACA 0015 airfoil. The width of the cascade vortex wakes inFig. 7c is predicted to be slightly smaller than that of the single

foil. This result demonstrates that the cascade effect is moreimportant than the wake vortex structure for predicting theblade loading.

Figure 8c shows the time history of the shed vortices fora NACA 0012 section at zero angle of attack interacting withan incoming vortex initially located upstream of the airfoil aty„/c=-0.13. The calculated instantaneous lifts versus vortexlocations (the airfoil is located between 0 and 1) for the singlefoil are shown in Fig. 8a as compared with Lee and Smith's(1987) solutions . The lift decreases first to a negative value,recovers quickly to a positive value, and drops slowly to nearzero when the vortex passes through the airfoil. Two curves fordifferent vertical separation distances between the airfoil andthe vortex are depicted. When the vortex is placed closer tothe airfoil, the effect of producing a fluctuating lift is morepronounced. Figure 8b shows the calculated lifts for both singlefoils and cascades. For the cascade results, the fluctuation inlifts when the vortex passing through the leading edge and thetraining edge of the airfoil is suppressed significantly. This maysuggest that vortical flows through turbomachinery bladepassages are much more orderly distributed as one can conceivebased on a single airfoil motion.

Figure 9 presents calculated 2D quasi unsteady two-blade-row cascade results and corresponding 3D measurements(Dring et al., 1982). The turbine consists of a 22-blade statorwith pitch H = 0.854c and a 28-blade rotor with pitch H = 0.813c.Both the stator and the rotor have round trailing edges. Thegap between the two blade rows is 15% of the stator chord.The calculated steady solutions under uniform flows of zeroangles of attack for each separate blade row are shown inFig. 9a. Due to the unknown inlet flow angle for the rotorblade row was specified incorrectly as an input parameter, thepressure prediction in the rotor leading edge area does notagree with the measured data at all. The pressure predictionin the rotor trailing edge area is, however, independent of theits inlet flow angle and relates to the viscous effect and theapplication of the Kutta condition. Since the rotor operates ata reduced passing frequency of 2.8 (based on the turbinediameter), both the stator trailing edge and the rotor leadingedge sense a variation of pressure. Figure 9b shows the "time-averaged" maximum and minimum pressure distributions of thesteady calculations for both stator-rotor blade rows at 10different relative blade locations, which represent a full cycle ofthe stator-rotor blade interaction. The measured values shownin the Fig. 9b are also the time-averaged pressures and therange of the unsteady fluctuating pressures (represented by theuncertainty symbols centered at the time-averaged value). Thecalculated quasi unsteady pressure ranges are larger at thestator trailing edge and the rotor leading edge than themeasured ranges. They become smaller elsewhere. Thepredicted large fluctuations near the stator and the rotortrailing edges are associated with the round shapes when theKutta condition is enforced at the centers of the trailing-edgecircular arcs. Since a small trailing-edge separation bubbles areexpected for both the rotor and the stator, a more realisticlocation for applying the Kutta condition would be some pointoutside the circular arcs. However, this influence will be verylocalized. These results clearly demonstrate a strong potentialeffect of the stator flow on the rotor flow. It also shows thecapability and accuracy of the 2D potential-flow calculations.


THEODORSEN FUNCTION0.8EXACT SINGLE FOIL

- - — CACULATED SINGLE FOIL------- CASCADE (H=1.5c)

0.6 - - - CASCADE (H=1.0c)

!"7 A A

0.2

ci 0.0

-0.2

-0.4

-0,65 10 15 20

tbcr

(7a) Instantaneous lift

(7b) Flow visualization of Bratt for a NACA 0015 airfoil

0.8

SINGLE FOIL

^^I,

'4ap4 lv Q7^;f s i

0.4 CASCADE (H=1.5c)

0.2

CASCADE (H=1.Oc)^° 4 R p^ yT

0.0 = °"'$np• 7d•°^1 ^ dey E

°

-0.20.0 0.2 0.4 0.6 0.8 1

x/c

(7c) Wake vortex locations

Fig. 7 Predictions for oscillating NACA 0012 airfoils ata reduced frequency of 17

PRESENT SINGLE FOIL (y,=-0.13c)PRESENT SINGLE FOIL (v - -0.52c)

00000 LEE & SMITH (y,---0.13 c)eAAA- LEE & SMITH (y, =-0.52c)

ia

X/C

(8a) Instantaneous lift for single airfoil

— SINGLE FOIL (Y,=-0.13c)SINGLE FOIL (y --0.52c)

o.a - - CASCADE (y,-=-0.13c,H=1.Oc)--------- CASCADE (y, =-0.52c,H=1.0c)

W 0.0

U /

W-0.2

lJi

IM

Fig. 8 Incoming vortex interacting with airfoil

CONCLUSIONS

A time-dependent potential-flow method is described tocalculate the vortex shedding and blade-vortex interaction forcascades and for single airfoils. Without a separation, thesteady potential-flow calculation has the capability of accuratelypredicting the cascade blade loading. A simple inclusion of theboundary-layer displacement effect enables the presentcalculation to predict the cascade exit flow conditions reliably.The unsteady interaction phenomena between the incomingvortex field or the wake shedding vorticity field and the airfoilare found to be similar for the single airfoil and cascade flows.The predicted periodic blade lifts have smaller amplitude forthe cascade flows. For the transient calculations, i.e. theWagner function prediction, the cascade flow approaches steadystate much faster than the single airfoil flow. The presentcalculation method has been shown to be effective in predictingsteady and unsteady flows through a multi-blade-row cascade.

ACKNOWLEDGEMENTS

This work is supported by the Independent ExploratoryDevelopment Program administrated at the David TaylorResearch Center and the Submarine Auxiliary SystemsExploratory Development Project, Program Element 62323N,

9


IfLRL STATOR- -STRC ROTOR

—

c

mowatea xo)°°°^ wa ; : 4^ tion

— ^mcuucl.e ( (2D'

^ ,:m^m°11:c.una (PSuat'a^ni'

^♦Ylnl ' nSTAN(.r rlNc-O

,.a n AXIAL DISTANCE (INCI

(9a) Considering stator and rotor separately under uniformflows at zero angle of attack

UTRC STATORUTRC ROTOR

//Calculated Average

E. Measured Pressure)°e MeasuredSuction)0 „^/////-Calculated Mox and Mln

-

Calculated Aver

Measured (Presqu e ) 1

-- - C.Iceuote (Suction)- C I laled Max and Mln I

I

j

01^I

00 1.5 J0 6.5 610AXIAL DISTANCE (INCH) .0 75 9.0 fO5 tLu 1.5

AXIAL DISTANCE (INCH)

(9b) Considering stator and rotor as one single unit

Fig. 9 Quasi unsteady blade-loading predictions forUTRC stator and rotor

Block ND3A, Project RB23P11. The work is also sponsoredby the Office of Naval Research under Grant N14-89-J-1363.The computing time is provided by the Numerical AerodynamicSimulation Program of the NASA Ames Research Center.

REFERENCE

Bratt, J.B., 1950, "Patterns in the Wake of an OscillatingAerofoil," RoyalAeronautical Establishment, R&M 2773, pp.269-295.

Dring, R.P., Joslyn, H.D., Hardin, L.W., and Wagner, J.H.,1982, "Turbine Rotor-Stator Interaction," Journal of Engineeringfor Power, Vol. 104, pp.729-742.

Fung, Y.C., 1950, An Introduction to the Theory ofAeroelasticity, Dover, New York, pp.207 & 401-407.

Giesing, J.P., 1969, "Vorticity and Kutta Condition forUnsteady Multi-Energy Flows," Journal of Applied Mechanics,Vol.36, Series E, pp.608-613.

Giles, M.B., 1988, "Calculation of Unsteady Wake RotorInteraction," AIAA J. of Propulsion and Power, Vol. 4.

Hall, K.C., and Crawley, E.F., 1987, "Calculation of UnsteadyFlows in Turbomachinery Using the Linearized EulerEquations," Proc. of 4th Sym. on Unsteady Aerodynamics andAeroelasticity of Turbomachines and Propellers.

Herrig, L.J., Emery, J.C., and Erwin, J.R., 1957, "SystematicTwo-Dimensional Cascade Tests of NACA 65-SeriesCompressor Blades at Low Speeds," NACA Technical Note3916.

Kim, M.J., and Mook, D.T., 1986, "Application of ContinuousVorticity Panels to General Unsteady Incompressible Two-Dimensional Lifting Flows," Journal of Aircraft, Vol.23, No.6,pp.464-471.

Lee, D.J., and Smith, C.A., 1987, "Distortion of the VortexCore During Blade/Vortex Interaction," AIAA Paper 87-1243.

Lee, Y.T., Jiang, C.W., and Bein, T.W., 1990, "Rotor/StatorFlow Coupling in Turbomachines," Proc. of 3rd Intern. Sym. onTransport Phenomena and Dynamics of Rotating Machinery, J.H.Kim and W.-J. Yang, ed., Honolulu, Hawaii.

Morino, L., and Tseng, K., 1978, "Time-domain Green'sFunction Method for Three-Dimensional Nonlinear SubsonicFlows," AIAA 11th Fluid and Plasma Dynamics Conference,Seattle, Washington, Paper 78-1204,.

Quandt, E., 1989, "The Acoustic Radiation from Unsteady 3-DPotential Flow Vortex System Produced by TurbomachineBlade Motions," ASME Paper No. 89-WA/NCA-10.

Rai, M.M., 1987, "Unsteady Three-Dimensional Navier-StokesSimulations of Turbine Rotor-Stator Interaction," AIAA Paper87-2058.

Robinson, A., and Laurmann, J.A., 1956, Wing Theory,Cambridge at the University Press, pp.137-142.

Verdon, J.M. and Caspar, J.R., 1984, "A Linearized UnsteadyAerodynamic Analysis for Transonic Cascades," Journal of

Fluids Mechanics, Vol. 149, pp.403-429.Yao, Z.X., Garcia-Fogeda, P., Liu, D.D., and Shen, G., 1989,

"Vortex/Wake Flow Studies For Airfoils in Unsteady Motions,"AIAA Paper 89-2225.

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