M I N I S T R Y O F T E C H N O L O G Y

R. & M. No. 3547

A E R O N A U T I C A L R E S E A R C H C O U N C I L

R E P O R T S A N D M E M O R A N D A

The Effect of Change in Axial Velocity on the Potential Flow in Cascades

By M. R. A. Shaalan and J. H. Horlock

~ '2. i! .": 7: ,-

L O N D O N : H E R M A J E S T Y ' S S T A T I O N E R Y O F F I C E

1968

PNXCE 10s. 6d. NET

The Effect of Change in Axial Velocity on the Potential Flow in Cascades

By M. R. A. Shaa l an and J. H. H o r l o c k

Reports and Memoranda No. 3547*

September, 1966

Summary. An analysis is given for the potential flow through a cascade in which a change in axial velocity occurs.

An approximate solution of the derived potential equation is obtained and applied first to a flat plate cascade and then to a cascade of blades with camber and thickness. In the former application Weinig's exact solution is used as a first approximation while in the latter application Schlichting's analysis is used as a basic solution and then modified to account for the change in axial velocity.

Calculations demonstrate the effect of change in axial velocity on the cascade performance.

Section.

1. Introduction

CONTENTS

2. Analysis

2.1. The equation for the potential

2.2. Solution of the potential equation

3. Applications

3.1. The cascade of flat plates

3.2. The cascade of cambered thick aerofoils

4. Calculations

4.1. The fiat plate cascade

4.2. The cambered aerofoil with thickness

5. Conclusions

6. Acknowledgements

List of Symbols

References

Appendices A and B

Illustrations--Figs. 1 to 12

Detachable Abstract Cards

*Replaces A.R.C. 28 611.

1. Introduction.

Experimental cascade data is widely used in the design of axial flow compressors. Many attempts have been made to predict the performance of compressor cascades with varying degrees of success. Most potential-flow theories assume purely two-dimensional flow through the cascade, which is a reasonable assumption when the cascade is operating well away from stall. However experiments (Rhodenl, Pollard2, Shaalan 3) have shown a remarkable flow contraction across the cascade when the side wall boundary layer is not removed. The increase in axial velocity, which is a measure of the contraction, may be some 30 to 40 per cent of the inlet axial velocity near the stalling point. This enormous contraction appears to be due to the boundary-layer separation initiated in the corner between the blade suction surface and the side wall of the wind tunnel. Such an acceleration severely limits the static-pressure rise that can be achieved by the cascade but it may at the same time delay stalling because of the reduction in the adverse pressure gradient through the centre of the cascade.

In order to predict the stalling incidence, the pressure distribution round the blade profile must be calculated as accurately as possible. In a two-dimensional flow, an inviscid-flow solution, with equal velocities on the upper and lower surfaces close to the trailing edge, is shown to be a good approximation to the real flow (Gostelow, Lewkowicz and Shaalan4). But for accurate determination of the pressure distribution, the effects of the three-dimensional contraction of the flow must be included.

Some attempts have been made to solve the potential flow including an axial velocity change. Mont- gomery 5 has created a reduction in axial velocity through a compressor cascade by placing an obstacle in the downstream flow and has also approached the problem theoretically, replacing the obstacle by a series of doublets. Bollard and Horlock 6. have modified the two-dimensional analysis of Schlichting 7, including strip sources and sinks to account for the change in axial velocity. The strip singularities were of constant strength in both directions, normal to and along the cascade, producing a linear change in axial velocity within the cascade. Recently Norbury a has approached the problem by regarding each aerofoil as an element of an annular cascade. The radial flow, which corresponds to axial velocity in the conventional case, was obtained by locating an infinite line source or sink along the axis of the annulus. This he superimposed on a uniform axial flow and solved the problem for a single conical or near conical aerofoil. Two cases were considered, aerofoils of zero camber and thickness and aerofoils of zero thickness but with camber. Results for the circulation over a range of incidence and stagger are reported in Reference 8 for radially inward and outward flows.

In the present Report the problem is approached differently. The continuity equation is manipulated to arrive at a Poisson partial differential equation for the velocity potential. The solution of this equation follows that suggested by Price 9 for the case of compressible two-dimensional flow where a similar equation arises.

The method of solution is simplified and applied first to a flat-plate cascade then to a cascade of blades with camber and thickness. The latter application may be regarded as an improvement on Pollard and Horlock's modification of the Schlichting analysis.

2. Analysis.

2.1. The Equation for the Potential. Figure 1 shows a diffusing flow through a cascade. An x, y, z co-ordinate system is used. The continuity equation in three-dimensional incompressible flow is :

OCx OC r OCz Ox +O-~+Oz = 0 (1)

*Kubota, S., J.S.M.E. Bulletin, Vol. 5, No. 19 (1962), has also suggested that a change in axial velocity may be introduced by including sources whose strengths vary arbitrarily in the x-direction only. However, for a cascade of arbitrary aerofoils, he restricted the problem to the case of sources uniformly distributed in both the x and y directions.

where Cx is the axial velocity, Cy is the tangenital velocity and C~ is the spanwise velocity. Since the flow is irrotational

8Cx OCt = O,

Oy Ox

and defining a potential function q~ so that:

a¢ a¢ a¢ c~= a~,c,= a-s,c~= N

it follows that

Oz¢ . 02¢ OC~ Ox z ~--~y2 = az " (2)

The approximations made here involve the term on the right hand side of equation (2),

ac~ 8z¢ 8z 8z z "

It is assumed that near the x-axis (the centreline of the cascade) the intersections of the stream surfaces with any (x,z) plane are lines passing through a common point 0 on the x-axis, which is taken as the origin.

It then follows approximately

and

N / Cx + Cz C~ Cx 2 z Z X X 2 "-[- Z 2

8C~ Cx z 8Cx 8z x x" Oz"

For points near the x-axis we may ignore the second term so that

8C~ Cx 8z x

(3)

Hence, equation (2) may be written, for the flow close to the centreline

az¢ .a~¢ 1(a¢) ax ~ . 7 = - x ~ "

(4)

The equation may be solved as it stands if the location of the origin 0 is known. It may be assumed that the mean axial velocity across a blade pitch Cx,. obeys a simple 'source' relation near the axis z = 0

1 Cxm x = constant = g . (5)

Further if the axial velocity changes from Cx,,, at the leading edge (x = xl) to Cx,,~ at the trailing edge (x = x2 = Xl +a , where a is the axial chord) and if the axial velocity ratio is 2 = Cx,,ffC~,,,, then

3

1 -~ = Cxm, x l = C . . . . x2 = 2C~m~ ( x l + a ) . (6)

~ a

For 2 < 1, x 1 = ~ and the location of the origin is determined for a diffusing axial velocity. A B

similar simple source analysis gives the location of 0 downstream of the cascade, for increasing axial velocity across the cascade.

There are several alternative forms of the potential equation (4), V2~b = F(x,y). The function on the right hand side may be written

F = 1 a4) X ~ X ~

cx dc~m a ~4) F - C~m dx - dx (loge C~,,) d--~'

F = - KCx, . dx l - a ) c~m a,k

Cxm, d x '

F - Cx dh • d (log e h ) _ . h dx dx dx (7)

where h is the height of the mean streamline as it passes through the cascade. It is probably a better approximation to assume that the product of axial velocity and channel width

obeys the "source' relation. In this case C~m x = ~ , where d is the blade spacing and Yt half

the thickness. The function F is then increased locally by a factor

d

d - 2yt

2.2. Solution o f the Potential Equation.

Equation (4) may be written as

OzcP ' aEcP F(x ,y) . t x 2 t -~x2 =

Price 9 has given a method for solving numerically this type of Poisson equation, which also arises in two-dimensional compressible flow through cascades. A similar method of solution is followed here, but several simplifications are made.

The steps in the solution are as follows:

2 2 0o. ~ 4~o 1. Solve ~-~T-e 0 ~ - = 0 for 0o, the incompressible two-dimensional solution, as a first approxima-

tion. 2. Evaluate the R.H.S. of (4), F(x,y) from this first approximation. 3. Locate plane sources in the flow field, the strength of a source at any point x,y being equal to the

R.H.S. of (6). 4. Obtain the induced velocities up, VF at the profile boundaries (parallel and normal to the profile)

due to these plane sources F(x,y). (See Appendix A).

5. Locate a line of singularities at the profile to cancel the normal induced velocities. 6. Obtain the total induced velocities in the flow field due to the line singularities and to the plane

sources. 7. Add the total perturbation velocities on to the basic flow and re-evaluate the R.H.S. of (4). 8. Iterate from 4 until convergence has been obtained. 9. Calculate the total induced velocities parallel to the profile at the blade surface and calculate the

circulation. 10. Calculate the outflow angle and lift coefficient from this corrected circulation. The application of this procedure to solve a compressible flow is very complicated. In particular,

numerical calculation of the induced velocities uv, vv (Step 4 above) at the profile is difficult and because of this a simplification to the problem was sought. If is assumed in Step 4 (Appendix A) that the distribu- tion of axial velocity (and hence the source strength) from the suction surface of one blade to the pressure surface of the next blade is linear with y at any x. This enables the area integral which gives the induced velocities to be evaluated analytically in the y direction before numerical integration in the x direction.

3. Applications. Two applications of the analysis are given here, one to a cascade of flat plates and one to a cascade of

cambered aerofoils with thickness. In the first application Weinig's exact solution to the incompressible flow through a cascade of flat plates l° is developed to give the potential solution ~b o required in the first approximation. In the second application, Schlichting's analysis v is used both for the basic incom- pressible flow and in the subsequent analysis.

3.1. The Cascade of Flat Plates. The incompressible two-dimensional velocity distribution (Cxo, Cyo) obtained from Weinig's incom-

pressible two-dimensional solution is given in.Appendix B. It is shown in Appendix A that the induced velocity on a plate or chord line, may be obtained approxi-

mately, equation (A3), if the source strength (-KCxm (x) ~f-~) on each of the two surfaces is known.

Using the two appendices the induced velocities associated with the first approximation, parallel and normal to the surface (x,y) may be obtained.

Ur(X, Y)/C1 -

X2 xZI 2d . m sin fl I 1 + m cos fl I 2 sinh -~- (x - X)

x l

+ b sin fl 13 -k- b cos fl 14 sinh --d- ( x - X) Ax (8)

and

X2 _IZ[ VF(X,Y)/C1 = ~ m c o s f l l l - m s i n f l I 2 s i n h - ~ ( x - X ) x l

q + b cos fl I3 - b sin f114 sinh --d ( x - X) ] A x

where b = F l ( x ) - F . (x), the difference in source strength from suction to pressure surface,

(9)

x[ ] m = F . ( x ) d t an f l F l ( x ) - F . ( x )

and 11, lz, 13 and 14 are integrals given in Appendix A. The limits of integration xa and x 2 are taken as 0-025 I cos fl and 0.990 1 cos ft.

It is next required to distribute line vortices along the plate to cancel out the velocity vv, so that the plate lies along a streamline in the flow.

After Schlichting v it is assumed that the distribution of vorticity along the chord is

_ ® y(x') AoCot~+A~sin®+A2sin2®+ .+A,_~ sin ( n - 1) ® 2C1 " '

(lo)

where A o, At, A 2 . . . . A,_ 1 are Fourier coefficients to be determined, and ® = cos-1 (2x'/l-1). Using Schlichting's analysis and following his notation, it follows that the induced velocities due to

~(x') are

i=n--j

U~

i = 0

Al g~,, (1 la)

i = n - 1

V~, =. ~ , C~ A, f ,, (l lb)

i = 0

where g~, andf~, are factors tabulated by Schlichting. If the L.H.S. of equation (llb) is put equal to - VF / C j from equation (9), then the coefficients A1

may be obtained and the distribution of vorticity 7(x') can be calculated. 1 0¢o

The basic ¢o flow, the flow due to the sources - - - - and the flow due to the vorticity y(x') may now x Ox

be added together to giye the resultant velocity on either plate surface.

Cs = C~ o + uF + u~ _ 7/2 (+ for suction side and - for pressure side) (12)

where C,o is the basic surface velocity. (C, may then be used for a second approximation. In the calculation described here, a second iteration has not been carried out because of the computation time involved).

With the vorticity distribution 7(x') obtained, other cascade data follows. The change in tangenital f' velocity associated with the change in circulation 7(x') dx' is

0

ACt = ~ ~ (Ao +A1/2) C1.

The modified inlet and outlet angles are then (see Fig. 2).

tan ~1 = C1 sin al +½ ACy

_ 2 1 t a n ~ 1 4 C~cos-----~I '

t a n 8 2 = C 2 s i n a 2 - ½ A C y = 2 + l ( ½ACt ) Cx: 22 tan a2 C1 cos cq

where 2 = C~/C,,. It may be shown that the lift coefficient is

CL = 2 (d/l) cos 2 ~i cos fl (tan ~1 - 22 tan a2) + (d/l) cos 2 ~1 sin fi

{(1 + t an 2 ~x)-22 (1 + tan 2 82)+2 (22_ 1)}.

The percentage change in circulation is

(zr(I/d)(Ao+A1/2) sinai ) AF ~ = sin ~l - tan a2 cos al × (sin al +~-ACr /Ci ) x 100

and the modified pressure coefficient is

Cv=T-?-~=~pCl 1-- = 1-- Cl"

It is worth noting that the pressure coefficient (and other quantities) obtained must be compared with that of two-dimensional flow with the inlet angle ~l, not ~ .

3.2. The Cascade of Cambered Thick Aerofoils. The analysis for cambered aerofoils follows that of the fiat plates, but Schlichting's analysis is used

from the start for the two-dimensional flow. The aerofoil is replaced by a source-sink distribution qo(x') and a vorticity distribution V0(x') along the chord line. The distributions are expressed as Fourier series and by matching the thickness slope and the camber-line slope at say n points, n Fourier coefficients in each series are determined.

The velocity distributions on the suction and pressure sides are :

Cm~, + Uqo + Uro + ~o/2,

Crux' + Uqo + Uro - 70/2

respectively where Uqo, Uro are the induced velocities due to qo(x'), 7o(X') respectively. The velocities on the profile surfaces are :

(dy. h 2 )

( )/( : .v)' C,ol = Cm,, + Uqo + Uro - 70/2 1 + \ dx' J

where y,,, Yl are the ordinates of the suction and pressure surfaces. From these velocities the plane source distribution F(x,y) is calculated along both sides of the chord

line and the induced components of velocity (up, re) due to F(x,y) at the chord line are obtained as in application 3.1 and Appendix A.

Next additional line vortices 7(x') and line sources and sinks q(x') are located on the chord line to produce induced velocities ur, v : uq, vq respectively.

These velocities are calculated using the Schlichting equations :

i = n = 1 i=n

2 2" (Uq+U~) _ Aig~l+Bog~o + Bigq,, C1

i = 0 i = 2

(13a)

and

i = n = l i = n

C ~ - A i f ; ' + B ° f q° + Bifq~ i = 0 i = 2

(13b)

where BI are Fourier coefficients defined by the series :

i =n

o Z q(x') _ B® (Cot-~- - 2 sin ®) + Bi sin i ® 2C1

i = 2

(14)

® and Ai are defined as in equation (10). Again g~, g~, f~ , fq are factors tabulated by Schlichting. The slope of the camber line is now

dye _ C, , , + (Vqo + G'o) + (vq + v~) + vv dx' Crux' + (Uqo + U~o) + (uq + u~) + ur "

and the slope of the thickness distribution is

dyt

dx'

{" OUr ½ qo(x')+½ q(x ' ) -y t ~, ~; -x , -F(x ) J

C., , + (u~+ ur) + (Uqo + Uro) + uF

These equations can be solved for n points to yield the unknown Fourier coefficients in the distributions 7(x'), q(x'). The corresponding velocities (uq + u~) and (Vq + v~) are then obtained and the modified velocities on either side of the chord line are then known, and the velocity on the blade surface is obtained as in the normal two-dimensional solution.

The steps in calculating ACy, AF, ~1 and ~2 are all similar to those taken in solving the flat-plate problem.

4. Calculations.

4.1. The Flat-Plate Cascade. Fig. 3 shows the pressure distribution obtained from Weinig's solution for a flat-plate cascade of

space-chord ratio 0.875, set at a stagger of 36 ° and an incidence of 20 ° for which a2 = 36"75°- Fig. 4 shows the distributions of induced velocities due to plane sources calculated from the first

approximation and Fig. 5 shows the perturbation vorticity distribution along the chord. The percentage change in circulation is A F ~ = -7.50. Fig. 6 shows the distribution of chordwise components of velocity due to line vortices on chord.

It follows that

~t = 57"30o, ~2 = 36"30°.

Fig. 7 shows a comparison between the pressure distributions when there is an increase in axial velocity of 15 per cent of a decrease of 20 per cent for the same inlet angle.

4.2. The Cambered Aerofoil with Thickness. The flow past a cascade of aerofoils of profile 10C430C50, set at a stagger fl of 36 ° and a space-chord

ratio of 0-875, was calculated. The inlet angle el was assumed to be 52.00 in the preliminary two- dimensional Schlichting calculation.

Fig. 8 shows the distribution of induced velocities at the chord line due to plane sources F(x,y) calculated from the first approximation.

The modified pressure distribution Cp is shown in Fig. 9 fork = 1-10 and 2 = 0-90. The corresponding air angles are

~1 = 52"800, ~2 = 28"500 (4 = 1.10).

For a two-dimensional flow with the same inlet angle el = 81 -- 52"80°, the outlet angle c~ 2 = 29 ° and the pressure distribution is as shown in Fig. 9.

5. Conclusions. A method is given for calculating the pressure distribution and fluid deflection in a cascade across

with a change in axial velocity takes place. It is shown that there are substantial changes in these quan- tities with axial velocity ratio (4 -- Cx2/C,I). In particular the diffusion on the suction surface of the blade is altered considerably.

It is particularly important to realise that the value of 2 increases towards the stall point because of end wall effects in the cascade, and any two-dimensional calculations of boundary-layer development and separation are not valid unless the effects of axial velocity ratio are included.

The method described is approximate in two ways (a) a linear distribution of equivalent source strength from blade to blade is assumed. (b) an approximate result is used for one of the integrals in the analysis. However it is submitted that

the analysis is sufficiently accurate for practical purposes. A comparison is given in Fig. 10 between calculations based on the present analysis and the analysis

of Ref. 6. The difference between the two pressure distributions occurs mainly on the front part of the suction side due to the inclusion of a non-uniform source distribution in the flow field.

It should be noted that a simple extension of the analysis enables a solution to the compressible flow in cascades to be obtained (an approximate solution of the problem as posed by Price9). Details of this work form the subject of another paper.

6. Acknowledgements. The authors wish to thank Dr. J. W. Cleaver for assistance in obtaining the integrals given in Appendix

A, Dr. R. I. Lewis and Dr. D. Pollard of the English Electric Company for use of a computer program of the Schlichting analysis, and Mr. D. Wilkinson of the English Electric Company for assistance in deriving the potential equation.

Analysis.

x,y,z

C x, C r, C~

r

0

2

h

a

F

K

Applications and x', y'

H~ 1)

(X, Y)

d/l

Yt

Y~

Yu

Y~

0o Cxo, Cy o

C1

C2

G C $ou

C so 1

C rux'

C t n y "

Uqo~ l)qo

H?~ V? t Uq~ Vq

LIST OF SYMBOLS

Reference axes (axial, tangenital and spanwise directions respectively)

Velocity components in the directions of the reference axes

Radial distance in the source flow

Angle included between the x-axis and the source streamline

( outlet axial velocity "~ Axial velocity ratio \ ~ a x - ~ a l v ~ J

Height of mean streamline

Axial chord length

Plane source strength

Potential function

Constant defined in text

Appendices.

Cartesian Co-ordinates with the actual chord as x'-axis and leading edge as origin

Velocity components in the directions x', y'

A point at the plate

Space chord ratio

Blade thickness

Blade camber

Upper surface co-ordinate

Lower surface co-ordinate

Basic velocity potential

Basic velocity components at plate

Inflow velocity

Outflow velocity

Inflow modified velocity (2nd approximation)

Velocity on the upper surface

Velocity on the lower surface

Basic mean flow velocity in the direction of x'

Basic mean flow velocity in the direction of y'

Induced velocity components at chord due to basic singularities

Induced velocity components at chord due to perturbation singularities

t0

UF~ VF

Cxi

Cx2

C~

ACy

0(2

~2 ®

S ,C

S~ C

CL

r

AF

P

Pl

P

Cp

qo(x)

q(x')

n

b, m

13

I1, 12, I3, I4

A~, B i

a,,,f;, }

Subscripts. 1

2

s

Induced velocity components at chord due to plane sources

Inlet axial velocity

Outlet axial velocity

Resultant modified velocity at plate

Basic flow velocity at plate

Increment change in the tangenital velocity due to perturbation singularities

Inlet flow angle (1st approximation)

Outlet flow angle (lst approximation)

Modified outlet flow angle

Modified outlet flow angle

Profile parameter (defined by Equation 10)

Hyperbolic functions defined in Appendix A

Trigonometric functions defined by Appendix A

Stagger angle

Lift coefficient - L (lift force)

s

Circulation

Increment change in circulation

Surface local static pressure

Inlet static pressure

Fluid density

Static-pressure coefficient = (P/Pl)/½ P C~.

Basic singularity distribution in Schlichting's analysis

Perturbation singularities to account for the change in axial velocity

Number of stations on chord to define the profile

Constants used in the approximation

Angle defined in Appendix A

Integrals associated with the approximation

Fourier coefficients

Factors defined in Schlichting's analysis

Inlet condition (or L.E. condition)

Outlet condition (or T.E. condition)

Refers to surface

11

U

l

F

0

q

n

i

Refers to upper surface (suction)

Refers to lower surface (pressure)

Associated with plane source distribution

Basic value

Associated with vorticity distribution

Associated with line source distribution

Associated with n th coefficients

Associated with i th coefficient

No. Author(s)

1 H . G . Rhoden ..

2 D. Pollard . . . .

3 M. R .A . Shaalan . .

4 J .P. Gostelow . . . . . A. K. Lewkowicz and M. R. A. Shaalan

5 S.R. Montgomery . . . .

6 D. Pollard and J. H. Horlock

7 H. Schlichting . . . . . .

8 J .F. Norbury . . . . . .

9 D. Price . . . . . .

10 F. Weinig . . . . . .

REFERENCES

Title, etc.

Effects of Reynolds number on the flow of air through a cascade of compressor blades.

A.R.C.R. & M. 2919, July 1952.

Low speed performance of two-dimensional cascades of aerofoils. Ph.D. Thesis, University of Liverpool. 1964.

The stalling performance of cascades of different aspect ratios (unpublished). 1966.

Viscosity effects on the two-dimensional flow in cascades. A.R.C., C.P. 872. October 1965.

Spanwise variations of lift in compressor cascades. Jl. Mech. Eng. Sci. Vol. 1. 1959.

A theoretical investigation of the effect of changes in axial velocity on the potential flow through a cascade of aerofoils.

A.R.C., C.P. 619. June 1962.

Berechnug der reibungslosen inkompressiblen stromug fur ein vorgegevenes ebenes schaufelgitter.

V.D.I. Forschungshaft 447. 1955.

The circulation about a thin annular aerofoil in an axial stream with axial source or sink.

ULME/B.4. February 1964.

Two dimensional compressible potential flow around profiles in cascade.

G.T.C.C. Aero. Sub. Committee Rep. 547.

Die stromung um die schaufeln von turbomachinen. Joh. Ambr. Barth. Leipzig. 1935.

12

APPENDIX A

The Induced Velocity on the Chord Line due to the Distributed Sources.

Fig. 11 shows the induced velocities ue and vv parallel and normal to the chord line of the blades due to distributed sources of strength F(x,y). Price 9 has shown that the induced velocity on the surface at (X, Y) due to source patterns repeating to infinity is

2= 2n ) uv(X,Y) = - i f F(x'y) sinflsin'-f(y-Y)+cOsflsinh-~-(x-X) dxdy

(A1)

2r~ 2n ) i+f°°F(x,y) cosflsin-7(y-Y)-sinflsinh--~(x-X)

- V i o - ~ 2d cosh-~- ( x - X ) - cos -~- ( y - Y)

It is next assumed that F(x,y) varies linearly from F,(x) on the suction surface to Ft(x) on the pressure surface at a given x

i.e.F(x,y) = (Fl(x)-F"(x)

so by substitutions

x2 y u + d

uv(X,y)=_f f m+b(f) sinfl's+c°sfl'Sdydx 2d C-c '

Xl Yu

x2 y u + d

vF(X'Y)=-f f m+bO0 c°sf l ' s -s inf l 'xdydx2d C-c Xl Yu

(A2)

where

2n 2n s = sin ~ ( y - Y), S = sinh ~ ( x - X),

c = cos ~ ( y - Y), C = cosh ( x - X),

rn= F~(X)-d ( F,(x)-F=(x) ) , b = Fl(x)-F~(x),

t y

y~ - tanfl , f = 3

13

The integration with respect to y may be obtained analytically, and ihe subsequent integration with respect to x is carried out numerically.

Equation (A2) can be written"

where

uF(X,Y) -

X2

~-d 2 (m sin [l l x + m cos [t S I 2 + b sin [J I 3 + b cos fl S I,d Ax ,

x l

VF(X,Y) -

x2

'2 2d (m c°s fl l l - m sin fl S I 2 + b c°s fl I 3 - b sin fl S I 4) A x

XI

11

2~ Y"+/ sin -~ ( y - Y)

J 2n dy , yu C - cos ~- ( y - Y)

(A3)

12

y u + d

f 2n dy , y, C - cos -d- (y - Y)

13

27~ Y" +/(y/d) sin -d- (y - Y)

% !~, 2n dy , C - cos -~- ( y - Y)

yu+d

14 = f (y/d)2n dy. (A4) yu C - cos -d ( y - Y)

It may be shown that these integrals are

I1 = 0 , I 2 = ~ 7 - C - T ~ , I 3 = In C - c o s e f (C) '

C ~ 2 - 1 [ 1 ( x / C 2 - 1 s i n e ) d ] I4 = ~ t a n - 1 + 1 - C cos e

(A5)

where

f (C) "-" C ~ . ( C - 1) ~ . (C+ 1) ~ ,

2n e = -~ (Yu- Y),

I1, 12 and 14 are exact. No analytical solution for I 3 has been obtained, but the approximate result given is obtained by graphical plotting and use of the other integrals.

14

APPENDIX B

W e i n i f s Two-Dimensional Solution for the Flat Plate Cascade.

Weinig 1° gave an early exact solution for the flow past a cascade of flat plates. He used the method of conformal transformation, from the cascade plane to the region outside a circle.

Using the notation of Fig. 12 it may be shown by developing Weinig's analysis that the velocities Cxo, Cro at the point (x,y) are given by:

Cxo 1 0q~o d~ I C m' = C'--mm 0% = [wO COS 6 + (W 1-1- W,~) sin 6] x ~ x

(x,y) cos fl, (B1)

Cro - 1 O~% [WoCOSa+(w~+w2)sin6]x & x C m C m Oy (x,r}

sin fl (B2)

where

4R d Wo = [(R 2 - 1 ) sin fl cos 0 - ( R 2 + 1) cos fl sin 0] x-a--, 1 )2 -4R 2 cos 2 0]

-4R wl = [ ( R 2 + l ) 2 _ 4 R 2 c o s 2 0 ] [ ( R 2 + l ) s i n O s i n f l + ( R 2 - 1 ) c o s O c o s f l ] x ,

4R d w2 = [(R2 + 1) 2 _ 4R 2 cos20] [(R2 + 1) sin 0st sin fl+ (R 2 - 1) cos 0s, cos 1/] x 2-~n'

[ ] G = t a n -~ t a n ~ R 2 + l } ,

where

R 1 X t = c o s 2 0 - ~ [ l + c o s 2 f i ( R 2 - c o s 2 0 ) + s i n 2 f l s i n 2 0 ] ,

1 [R 2 _ cos 20 sin 2 f l - cos 2 fl sin 2 0 ] , Y1 = R sin 2 0 - - ~

X 2 = (R2 +~2) cos 2 0 - 1 +sin 2 2 0 - c o s 2 2 0 and

Y2 = ( R2 +~-~2 - 2 cos 20) sin 2 0 .

0 is varied from Ost to 0n + 2n to obtain the suction and pressure sides successively.

15

Z

0

,/ X

~ . / ~ C~ _...~ ¸

X, C x

FIG. 1. Diffusing flow model (2 < 1).

'x

YlY

/ //I/I tl / /~.. i

, ' / i t .

~ ' X CX i C X I "

Cxz-

1/2ACy f

,/~txcy Id:

I ~. , g ,o, I

A FIG. 2. Co-ordinates and velocity diagrams.

16

1.01

O.'BI

0"60

0"40

P - P l CP~ ! 2

.~pc,

O.ZO

\

I0 20

..~... ~ - - ' - - -

40 50 80 90 I00 60 70

-0 .20

-0"40

-0 .60

- 0 ' 8 0

/ /

/3 = 36 °

d//, = 0"875 = , = 56 °

v F c t

FIG. 4.

- I .00

FIG. 3.

x Io -2

5"0

4 " 0

3 '0

2 .0 ~ -

1'0

0

- I ' 0

-2"0

-3"0 .~"

-4"0 . . ~ " "

-5"0 0 I0 20

Pressure distribution (Weinig's solution).

'a F ,,,..,..."

.4"

S

r '

. s

30 40 50 60 70 80

x' I ¢

x io z

5.0

.~" 4 - 0

3-0

2-0

I '0

o-~ C!

-I.0

-2-0

~ -3-0

-4"0

-5.0 90 I00 °1o

Normal and chordwise components of velocity induced at the flat plate due to plane sources (~l = 560, 2 = 1.15).

17

¢1

xlO 2

2"0

I'O

- I ' 0

- 2 " 0

-3"0 I

-s.o ~ - - - - - ~

-6"0 0 I0 20 30 40

/

50 60 70 80 90 I00 %

x'l Z

FIG. 5. Perturbation vorticity distribution along the chord for flat plate cascade (al = 56°, 2 = 1.15).

xz(~ z

4"0

(u.), +.),/z) 3.0

C I ' 2 . 0 ( /Z) - - - - - - - - i "

u.),- 7 ~-0 '- - " -

0

- 1.0

-2 "0

-3.0/" ~ ' ~

J f

/

-4 .0

0 I0 20 30 40 50 60 70 80 90

x ' l~

\

J

I00 °/o

FIG. 6. Induced velocities at the plate due to perturbation vorticity (:q = 56 °, 2 = 1.15).

18

I '0

0 '80

0"6(

0"41

P - Pl C:p=

0-20

-0.20

0 '40

0 '6

0"0(

0.04

u F v F

c5'~ o-oz

-0"02

- 0 . 0 4

-0"06

I0 20 , . . f ~

/ / / ' / " ? - 40 so 60 x'lt o/,7o 80 ~ - - J

d H . = 0 875 - - . ~ = 0 - 8 0

,8 = 36 ° ~. == 1 ,00

~l = 57"200 X == I ' l S

FIG. 7. Effect of axial velocity change on pressure distribution (fiat plate).

I0

J

20 30

I I 40 60

I

J

!

x'//, 70 80 90 I00°/o

FIG. 8. I.nduced velocities on chord due to plane sources. (10C4-30C50 blade).

19

0"80

/, . . , .p

/ 0"60 ,\ ~\ . I ~ ~ ' - -

0-40 I " .... ~ ' " ~ " . . . .

0.20

p - p Cp = i~_-2

pc, O0 I0 20 30 40 50

. /

-0"2q

-0.4

FIG. 9.

!

-0.6C

/jr-

0"60

l /7 / . ~

/ " 80 90

/3 = 36 ° dll = 0.875

~0 = 52"80o

100

Effect of axial velocity change on pressure distribution. (Cambered aerofoil with thickness).

0"60

-0-20

0"40 P-PI

Cp=

O'ZO

-0-40

-0-60

-0 .80

FIG. 10. Pressure distribution. Comparison between the present analysis and that of Ref. 6.

20

p..

,%

O

©

o "

>

"-0

r.~

I

Y~Y

"o

FIG. 11.

% --~x +d)

l F(x,y) J *

/ <~¥)

~ X

x I x 2

X X

Velocities on plate (or chord) induced by plane sources.

Circle ~Ione ( f -p lane)

,#w

k,~ormol flow

FIG. 12. Conformal transformation of the flat plate cascade as given by Weinig (Ref. 10).

Ro & M. No. 354

© Crown copyright 1968

Published by HER MAJESTY'S STATIONERY OFFICE

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R. & 1VL No. 354,

S,O. Code No. 23 354,