2 . - r

MINISTRY OF SUPPLY

R. & M. No. 3030 (15,491)

A.R.C. Technical Report

L

A E R O N A U T I C A L RESEARCH COUNCIL

REPORTS AND M E M O R A N D A

Some Actuator-Disc Theories for the Flow of Air Through an

Axial Turbo-Machine

j . H. HORLOCK,

of the University of Cambridge, Department of Engineering

© Crown Copyright i958

LONDON : HER MAJESTY'S STATIONERY OFFICE

1958 E L E V E N S H I L L I N G S N E T

Some Actuator-Disc Through an

Theories for the Flow Axial Turbo-Machine

By J. H. HORr~OCK,

of the University of Cambridge, Department of Engineering

COMMUNICATED BY THE ]~)IR~CToR-GENERAL OF SCIENTIFIC RESEARCH (AIR), M I N I S T R Y O F S U P P L Y

of Air

Reports anti Memoranda No. 3 o3 o

December, 19 5 2

Summary.--Using actuator-disc theory, simplified methods are given for the solution of the direct problem of the incompressible flow of air through an axial-flow turbo-machine.

Calculations based on these methods are compared with other approximate solutions to the flow through a model compressor stage.

1. Introduction.--Compressor and turbine design is usually based upon the assumption that radial equilibrium conditions exist behind rotor and stator rows and this approximate theory is fully given by Cohen and Whi te )

Actuator-plane theory, in which a blade row is replaced by an infinitely thin disc which causes a sudden discontinuity in tangential velocity and vorticity has been developed by Merchant 2, Bragg and Hawthorne ~ and Marble. ~

Using one of Marble's results and expressing the radial velocity in the form

Cr = ~ f / r ) exp (k~x) . . . . . . . . . . . . . . . . (1) i ~ 1

Hawthorne 5 and Railly 6 have obtained expressions for the axial velocity at any point away from the disc.

The effect of neighbouring blade rows is calculated (a) by superimposing sheets of tangential vort ici ty due to single isolated rows and subtracting the sum of all the vortex sheets extending from upstream to downstream infinity (Hawthorne); (b) by adding the radial velocity fields due to individual blade rows (Railly).

Differences ill these two methods lie in the determination of a value of the at tenuation constants k~ in equation (1). Hawthorne determines ki from a solution of the equations for an isolated disc, but Railly obtains mean values which are dependent upon the extent of mutual blade interference and are obtained by successive approximation.

General methods for the solution of the direct problem of the incompressible flow of air through a turbo-machine of known blading are given in this paper"

(a) for isolated actuator discs (i.e., for machines with blade spacings such that aerodynamic interference may be neglected)

(b) for actuator discs closely spaced.

The actuator discs may be placed in the plane of the trailing edges of the blades, or at the blade centres of pressure.

To estimate the magnitude of the differences between the various theories the direct problem of a single compressor stage has been considered, the stage consisting of one rotor and one s ta tor .

The velocities at entry to the rotor and at exit from the stator are axial; the tangent of the outlet angle from the rotor varies linearly with radius and is equal to uni ty at the tip radius, i.e., tan/3 = r/rs = R.

The hub-tip ratio is 0.4 and the ' aspect ra t ios ' of the blades 1/b are chosen as (i) 2.1, and (ii) 4.2, assuming negligible axial clearances. When the actuator discs are placed at the centres of pressure of the blades it is assumed that the distance of the centre of pressure from the blade trailing edge is two-thirds of the blade width, i.e., (a/b = 2/3).

Such a stage is similar to an axial-flow-compressor test stage installed in the Cambridge University Engineering Laboratories. The tip diameter of this compressor is 14 in. and the rig is designed for 6,000 r.p.m. If a mean axial velocity of 200 ft/sec is assumed, the flow parameter (U~/Cx~) = 1.83, where Us = blade-tip speed; Cx, = axial velocity at upstream infinity and is assumed constant.

2. Approximate Methods of Calculating the Flow through an Axial Turbo-Compressor Mach ine . - The general methods developed using actuator-disc theory are used to calculate the flow through the model stage and are compared with Railly's theory, and with radial equilibrium solutions. The positions of the actuator planes are illustrated in Figs. 1 and 2 for each method.

Method 1.--Radial Equilibrium Conditions at the Trailing Edges of the Blade Rows For this theory conditions at stations (02), (2e) and (2) become identical, as do stations

(4e) and (4).

The radial equilibrium condition at the trailing edge of a rotor is then expressed by :

dH2 dC; 1 (rC,,2) d - - (2) C., d r - + r 2= ~ dr . . . . . . . . . . . . . '

where C,,2 ---- U -- C.~2 tan/32

and Ha ---- H1 + U(C,,2 - - C,,I).

For the particular example of the model stage, tan/~2 = r / r s = R ,

and dH2 d {U(U -- C~2 tan G)} d r - - d r

Then equation (2) reduces to"

dC.2 C.2 / 2R ] 2UsR dR q- I1 T R - b - (1 + R~) '

which gives an exact solution"

U, R= c,2 + c <-i = (1 + R 2) . . . . . . . . . . (3)

where P~ is a constant to be determined from the continuity conditions:

C,2R dR = C.,R dR Rh Rh

and is given thus" Us 2

P 1 ~--- 2 ,~

04],~

2

Similarly at the trailing edge of the following stator"

dC~4 1 d dH4 C~ ~ - + ~(rC..) ~(rC..~) -- dr

where C,,~ = C~4 tan ~ .

For the particular example, tan ~4 ----- 0 ,

so £hat : Cx~ dC.~ dH~

dr -- dr

d -- d;{U(U -- C.~ tan 3~)},

whence it may be shown"

C~4

. . . . . . . . . . (4)

(s)

Where Q1 is a constant and is determined by trial and error from the continuity relation.

I t iS 0f interest to note that if the outlet air angles /~ and ~ are assumed constant with iflcideiice, then the off-design conditions are easily obtained by using a different value of U,/C~I .

Actuator-Disc General Theory.

By using the result that the axial velocity at an actuator disc stationed at the plane x ---- 0 is approximately (C~1 + Cx~)/2, where Cxl and Cx~ are the axial velocities at upstream and downstream infinity (x = - - ~ , + ~) and also assuming that the radial velocity at any point (r, x) is given by"

$=oo

c~ = ~ {exp (k,.)}f;(r) . . . . . . . . . . . . . (2) i = 1

it may be shown that the axial velocity is given approximately by:

Cx= C. (Cx,~-- C.) } - - 2 e x p { - - ( k ~ / Z ) } ~ > 0

c~ = c~, + ( c x ~ - c , , ) , . . . . (6) 2 exp {(kx/l)} x < 0

where l is the blade height and k is determined uniquely by the hub-tip ratio of the actuator disc.

This theory is developed in Ref. 5, and is given in Appendix I.

Further, the effect of neighbouring blade rows is obtained by superimposing the vortex sheets due to each disc, and then subtracting the vortex sheets at infinity. (The analysis is based Upon the values of k obtained from the isolated actuator-disc theory and the justification for this approximation is investigated in the calculations for the model stage.)

if the actuator discs are placed at the trailing edges of the blade rows the axial velocity at the p th disc C~0p is then given by:

C.op Cx,+~ + C~, __ q=P-I~C -- ) = ~, j . + , _ c ~ { e x p - ( k q ~ , & ) } 2 q-~=l \ 2

3 (71361) A 2

where C,,p, C,,,,+1 are the axial velocities tha t would exist at upstream and downstream infinity of the p th row if that row were isolated, due to the discontinuities in tangential velocity and vorticity at that row.

C,q, C,q+~ are similarly defined and x,~ is the distance (always positive) between the p th and qth rows.

(Equation (7) is also obtained in Ref. 5 as in Appendix I.)

If, however, the actuator discs are placed at the blade centres of pressure then the axial velocity at the p th trailing edge C,,~ is given by :

q=* (C~q+~ - C,q) {exp -- (kqXp~q/lq)} Cxp~ = C , ~ , + I - ~ 2 q = I

, . . . . . + £ - • --(kqx,,qllq) } (8) q = p + l \

where Xp~q is the distance between the p th trailing edge and the qth disc.

Marble has given a solution of the inverse problem in which the distribution of tangential velocity is initially specified, but the direct problem (that of determining the flow conditions if the air angles are specified) results in a series of functional equations, since the change in whirl velocity at the p th row is itself dependent npon C, 0p. Methods of successive approximation to solve this problem are given below in Methods 3 and 4.

Method 2.--Isolated A ctuator Discs placed at the Trailing Edges of the Blades If the rows are sufficiently far apart the effects of blade interference may be neglected and the

following analysis is developed.

Bragg and Hawthorne 3 have given a general equation for the incompressible, axially symmetric flow on either side of an actuator disc:

dH l [o dO ,Trl d r , - - r e + . . . . . . . . . . . . . ( 0 )

Across an actuator disc: Hoe = Hol + A W , . . . . . . . . . . . . (10)

where A W is the work done on the fluid by the moving blade row and is given by:

AW = O ' ( < , o e - C,,01) . . . . . . . . . . . . . (11)

From equations (9), (10), (11):

l l d (rc"°~)l @re d{u(c"°e--C'~°l)}' l~o~r + rC,,oe 4 (rC,,oe) = ~o~r + rC,,o~ ~

which may be written :

d (rC,,0,) (12) d (rC,,0e) = v0~ + (C, ,o,- U ) ~ . . . . . . . . .

But Bragg and Hawthorne have shown that O, H are functions of ~, and for small streamline displacements, will be approximately constant at a given radius on either side of the disc. Further, Ruden's assumption (Appendix I) may be made, that the tangential vorticity is approximately a function of radius alone.

Thus '101 - - / h = -- -- dr 1 1

T ] 0 2 - t , - "172 7---- (ac 4 ; c .

. . . . . . . . (13)

4

Further C,~ot ----- C.m tail ~oi = C.~ tan ~ , ] [ and if the disc is placed at the trailing edge of the blades" 9

C,,o~ = U -- C.o~ tan/~o2 ----- C.~2 tan a2 ) l

(14)

But

Hence equation (12) becomes"

dC~ d dr + C, o2 tan/~o2 ~ { r ( U -- C,o~ tan/3oJ}

dC,~ d {rC, t tan al} - - dr + (U -- C,I tan ~) ~ .

C,o~ = C, o2 -- C,~ + C,~ (Ref. 8), and the further approximation is made tha t : 2

rC, o~ ~-- d~ dr "

d 1 d dW rC, o~ dr'

so that : dC.2 tan ~o2 d Ir l u _ (C,I + C,~ )

_ tiC,1 2(U -- C,1 tan ~) d (rC.~ tan cq) (15) dr r(C,~ + C~) dr . . . . . . .

This may be written as an equation of the form :

dC,~ C,~ ~ + A~(r)C,2 ~7- + B~(r)C.2 - / C~(r) dC,~ -)7- q- D2(r) = 0 . . . . . (lSa)

A similar analysis for a disc representing a stator (AH = O) gives:

dC~ tan ~o, d I (C~3 + C~,) 1 dr @ r dr r 2 tan ~o~

dC,3 2C,3 tan ~3 d = dr + (C.3 + C,~) r dT(rC*" tan ~) . . . . . . . . . (16)

which reduces to a similar differential equation in C, ~:

C,~ 2 + A~(r)C,~ dC.~ ~- + B4(r)C.4 + C~(r) dC.4 - - 5 + D,(r) = 0 . . . . . (16a)

Equations (15) and (16) are general for any actuator discs placed at the blade trailing edges and operating at planes sufficiently far from other discs that the effect of blade interference is negligible. Thus the veloci ty profiles through a turbo.machine may be obtained row by row by assuming the axial velocity upstream of a row is given by that at downstream infinity in the solution to the preceding disc, and the whirl velocity upstream is specified by the conditions immediately downstream of the preceding disc.

Thus A(r), B(r), C(r) and D(r) are all known for the direct problem and equations similar to (15) and (16) may be found. These are usually non-linear differential equations but are easily solved graphically, by successive approximation and integration. I t is of interest to note that equations (15) and (16) take into account a variation in inlet total head and axial velocity.

For the model stage accepting a uniform upstream profile equation (15) reduces to:

dR 1 -[- + C~2R = ( 2 U , - C~,)R,

which gives an exact solution:

{2Ut 1) R 2

c . (9 + R2) , • . . . . .

2( ' - 2u, where P2 = 3 -- 2

from the continuity relation. The equation for the following stator reduces to:

dC~ a

d R - -

But C.~ tan ~ =

where tan/302 = R and C~2 =

dC,3 2C~, d (RC,8 tan ~3) dR + C~3 + C,~ tan e~ ~

(U C~1+C~22 tan /~02)

C=~ is obtained from equation (17).

The resulting equation is solved graphically.

(17)

(18)

Method &--Isolated Actuator Discs placed at the Centres of Pressure of the Blades By equating dH/dw on each side of an actuator disc placed at the blade centres of pressure,

equation (12) is obtained as before. However, the whirl velocity C,02 is no longer specified by the axial velocity at the disc, but by the conditions at the trailing edge.

The whirl velocity is approximately constant downstream of the disc since 0 = rC, is constant along a streamline and radial displacements are small.

Thus C,, o2 ---- C,, ~, ]

J ( 19)

and C,,2, = U -- C~2, tan P2, ' . . . . . . . . . . . .

where C.~, is obtained from equation (6).

( c ,~ C,1) C.2, C., • 2 ' • • . o

where a is the axial distance between the planes of the centre of pressure and the trailing edge, and K~ = {exp (-- karl)}.

Thus

dC,~ d { r (V - - C.~° tan ~ , ) } -t- C.~ tall /~2e~

dC, l d (rC~ tan ~1) - - dr + (U -- C,, tan ~) ~

The same approximation as before is now made for d/d~o at "the disc, i.e.,

d 2 d d~ r(C,~ + C~) dr'

so that

dC~ dr

_ _ _ 2 IC,2 ( 1 K~)r(C,2 -/+ Cxl ( ~ ) I C x , ) tan $ , e d l r ( U - d r lC*=( 1 K ~ ) + ~ K~ l tan/~"e)l

_ dCx, 2(U -- C,~ tan ~,) d (rC, z tan ~1) . (21) dr r(C,l -4- Cx~) dr ' " . . . . . . . .

which may be solved for C,~ if the distribution of C,~, tan a,, tan g2~ with r are known. Similarly for a stator"

dC.~ 2C., tan ~3 d (rC.a tan ~a) (22) - dr + r(C,~ + Cx~) dr . . . . . . . . . . .

Once again a step-by-step solution from a known entry velocity profile is possible through a turbo-machine.

Such equations have been solved for the model stage in which

1/b = 2- 1, 4.2 and a/b = 2/3. k (for r,/rt, =: 2'. 5) - 3.23 (from Jahnke and Emde~.

Blade Interference.--For a turbo-machine of n blade rows, if blade interference is to be considered, the problem may be treated as one involving n unknowns. These unknowns are the n distributions of trailing vorticity/7 with radius that exist behind the n actuator discs replacing each blade row. These values of trailing vorticity are assumed constant with radius between the discs. I t is, however, more convenient to consider these unknowns as Lhe n values of axial velocity tha t would exist far downstream of each blade row, since a knowledge of the tangential vorticity after the p th row dC~p/dr enables C,p to be determined from continuity. I t is assumed that the velocity distribution far upstream of the first row of blades is known.

The method of solution of this problem first involves guessing the 'downstream inf ini ty ' distributions of axial velocity and it is suggested that the values obtained from radial equilibrium theory (Ref. 1) (calculated from the known variations of outlet air angle with radius), form a suitable starting point. From the values of hub-tip ratio the values of hi, k, . . . kp, kq . . . k,, for each actuator disc are established.

Method 4.--Actuator Discs placed at the Trailing Edges of the Bla&s--Blade-Interference Theory The simplest approximation consists of supposing that the actuator discs are placed at tile

trailing edges of the blades. Then the axial velocity at the p th disc (C,0~) is calculated using equation (8) and the guessed values for the ' downstream infinity ' axial velocities.

For each blade row it is now possible to obtain a differential equation similar to equations (15) and (16) for the isolated discs, but the whirl velocities at the trailing edges of the p th disc will now be specified by (U -- Cx0p tan/~0p) for a rotor, or by Cxop tan ~0p for a stator.

The flow through each blade row is ill fact treated as an inverse problem, i.e., an actuator disc across which there is a specified change in whirl velocity. I t should be noted that the same

7

approximat ion as before is used for d~p -- r{(C~**~ + C~,)/2} dr, al though Cxo~ is not now equal to the ar i thmet ic mean of C,p and C,p+ ~. Thus the equation for a rotor, say the p t h row, similar to equation (15) is:

dC~,.~ 2C~o~ tan 13op d {r(U -- C~o, t an GJ} dr (C~p+l + C~p)r dr

dC~p 2(U -- C~o(p_~)tan ~olp-~)) d - - dr r(Cxp,~ + Cxp) dr (rC~ol~_~) tan ~o~p-~)) . . . . . . (23)

where C~op, C~o/p-~l are given from equation (8) and the guessed values of C~2, C~, . . . C~,~.

Or for a s ta tor :

dC~p+~ 2C~o~ t an ~op d d~ + (C;7~+; T ~ ?; (~<o, tan ~o,)

dC~, 2(U -- C~o(,_~)tan G(p-~I) d {r(U - - Cxo(p_l) t a n G:p-~)} (24) = dr + r(C~p+~ + C~p) dr . . . . .

The n differential equations thus formed are in tegra ted graphical ly to give n new values of C~2,...Cxp, C~ e . . . C~,+~, which m a y be used to repeat the procedure for a second approximat ion.

For the example of the model stage there are two unknowns, the axial velocities Cx2 and C,~ t ha t would exist far downst ream of rotor and stator. There is in this case no need to perform the successive approximat ion suggested above for the general case, since the two differentia) equat ions m a y be combined to give a more direct solution.

From equat ion (7)"

C.o~ = C.o2 -- C~ + C~ 2

C~o~ = C~o~ -- C~2 + C.~ 2

+ K~(C. - c . ) ~

2 I Kb(Cx2- Cxl ) 2

. . . . . . . . . . (25)

where Kv = exp (-- kb/l), in which b is the axial distance between the blades and l is the blade length.

F rom equations (23) and (24)"

dC~ 2Cxo2 tan 13o2 d {r(U - - C~o2 t an 13o2)} r dr - - ( 0 ~ 2 + C ~ ) dr " " . . . . .

dC,4 dC.2 2(U -- C~o2 tan t3o2) d {r(U - - C~o2 t an 13o2)} and r dr - - r ~ - + (C~2 + C~4) d-)

Combining equations (26) and (27)"

r dr - - r - - d r -[- (C~2 + C~a) C~o2tan 13o2 dr

Subst i tu t ing for C~o2 from equation (25) •

1 (u - Go2tan &2)(G2 + G~)t (C~2 -4- C~) Cxo2 tan 13o~ t

< < j + I , -

This differential equat ion ma y be integrated graphically to give a relation be tween (Cxd/Cxl) and (C,~/C,~), which together wi th equations (25) and (26) enables (C~2/C~1) to be calculated. Wi th (C,,/C,I) and (C~/C,~) known, (C~o,/C~) and (C~o~/C,I) are directly obta ined from equat ion (25). This has been done for two examples ill which 1/b = 2-1 and 4.2.

Method 5. - -A ctuator Discs placed at the Centres of Pressures of the Blades--Blade-Interference Theory If the actuator discs are placed at the centres of pressure of the blades as Marble suggests,

then the whir l velocities are defined not by the axial velocities at the discs themselves but by the axial velocities at the trailing-edge positions (C~1,, C~2~ . . . C ~ , C,~ . . . C .... ) downst ream of the discs.

These whirl velocities are:

for the p t h row, if a rotor C,~p, ---- U -- C,p, tan $p~ 1 , . . . . .. . . (29)

for the p t h row, if a s tator C,,:o~ ---- C~p~ tan ~p~ J where tips, ab~ are the rotor and stator exit air angles at the trailing edges, measured relative to the blade rows.

Since the value of O ---- rC,~ is constant along a streamline be tween the discs, for small s t reamline displacements the whirl velocity C,, is constant.

Thus the whirl velocity Cu just downst ream of an actuator disc is defined by the axial veloci ty at the following trailing edge.

The me thod of solution of the problem is similar to tha t of Method 4. The ' downs t ream inf in i ty ' values of axial velocity are guessed and the velocities at the trailing-edge stations are calculated using the principle of superimposing the individual trailing vortex sheets due to each blade row, and subtract ing those extending from ups t ream to downst ream infinity.

For the trailing edge downst ream of the p t h row the axial velocity C~p~ at tha t s tat ion is given by :

q=' (C~+I C~,,) = + { e x p ( -

q=l

q=p+l

where x~q is the distance (always positive) be tween the qth row and the p t h trailing edge and the flow is incompressible.

For. each disc a differential equat ion is obta ined similar to equations (15) or (16) but the whirl velocities just downst ream of the disc are the same ,as those calculated for the trailing edge:

for the p t h row a rotor C 0p ---- C,,p~ ---- U -- C,p~ tan ~p~

for the p t h row a stator C,~op = C~p~ = C,p, t an %~.

Thus for the p t h row if a rotor :

dC~p+l 2C~p~ tan Sp~ d {r(U -- C~pe tan/~p~)) dr r(C,p+l + C~p) dr

dC~p 2(U - - C,(p_l)~ tan ~p_~)0) d -- dr r(C~p+~ + C~p) dr (rC~tp_l)~ tan ~(p_~)~) ,

or for a s tator :

dC~p+~ 2C~p~ tan %° d dr + r(C~,+~ + C,,) -dr {r(C,*~ t an %e)}

dC.p - 2 7 +

2(U -- C,(p_~)~ tan fl(p_~)~) d (r(U -- C~(p_~)~ tail fl(p_~),)}... r(C~+l + C~p) dr

9

. . (30)

. . ( 3 1 )

The n equations for the n blade rows are then s imply solved by graphical integrat ion and the result ing values of ' downst ream inf in i ty ' axial velocities may be used in a second approximation. For the model stage"

and

c . ~ = c.~ + K~_~(C~, -- C. ) _ K~(C~ -- C . ) ~ 2 2

c~ ,~ = c~o - K o + ~ ( C . - - C ~ ) _ K o ( C ~ - - C . ) "

2 2

K~_~ = exp [{-- k(b -- a)}/l], where

• . . . . . . . ( 3 2 )

K~ = exp (--hall) , K~+b = exp [{-- k(a + b)}/1],

in which b is the axial distance between the discs (i.e., the blade-row centres of pressure), and a is the axial distance between the actuator-disc stations and the trailing edges of the blades.

But

Then as in Method 4"

dC,, dC,~ I (U -- C,~ t a n / ~ ) ( C , , + C~)I dr -- ~ 1 + (C,~ + C~)(C,~ . t a n / ~ ) "

C~2~ = C,~ (1 K~ K~-~-~Kb-~) + ~_~C,~ +_ff C~I

~_ ~ ~ + ( 1 - K ~ - ~ ) c . ~

whence the me thod of solution is the same as before.

(33)

Ultimate Steady Flow.--Stages deeply embedded in an axial-flow turbo-machine may be considered as identical pairs of actuator discs and the distr ibution of the tangent ia l vort ic i ty ~ is the same after each rotor row. C, R is defined as tha t axial velocity tha t would exist far down- s t ream of the rotor and is related to ~R by ~R = -- (riCeR/dr). Similarly the distr ibution of tangent ia l vort ic i ty ~s = -- (dC, s/dr) is the same after each identical stator.

If the rotor, say the kth row is considered then along a streamline"

H,o = H,o_I + A W,, . . . . . . . . . . . . . (34)

where H~0_I is the s tagnat ion el l thalpy after the (k -- 1)th disc

HT0 is the s tagnat ion en tha lpy after the kth disc

A W~ is the work done on the fluid by the kth row. .

For the (k + 1)th row, a s tator row:

A W~+~ = 0 ,

H~ = H~+~ . . . . . . . . . . . . . . . (35)

Then from equations (34) and (35), along a given streamline H1,+1 = H~_I + A W~.

Differentiating with respect to ~0,

(dH~+q (dH~_ 4 d (~w~), d~ ]=~ d~ ] + ~

10

but using equation (9) for the (k -- 1)th row, a stator"

(d.~_~ 1( ~0 4 ~d.~+~ d~ ] = r ~ ~sr + Os d~o! = \ d~o ]

for small streamline displacements. Hence d(AWk)/dr = 0. Or approximately"

d (~W~) 0 , . . . . . . . . . .

d~

result originally given by A. R. Howell 7 and used by Railly 6.

(36)

Method 6.--Actuator Discs placed at the Trailing Edges of the Blades--Ultimate Steady Flow If the actuator discs are placed at the trailing edges of the blades then"

d dr [U{(U -- C~oR tan $oR) -- C~os tan ~os}] = 0 . . . . . . . . . (36a)

Then if all infinite number of identical rows is considered, using equation (12) the axial velocity at any rotor disc C~oR is given by"

% - I

C~R+ C,s 2

C x O S " * "

Thus equation (36a) becomes"

l(ex

+ (C'R 2 ;'s) I (exp -- -

2/kb) + (exp 4kb , )1

- - ) I

- - ) I

d [U{U -- C~oR(tan/~oR + tan ~os)}] = 0 dr '

the result derived by Railly by superimposition of radial velocity fields.

Thus for the model stage in ultimate steady flow"

d (AWk)= d {U(U--C~oRR)}=O, dr

whence C,0R = f2 + (Constant/R2) .

From the continuity relation the constant is determined and for the (U,/C~ = 1.83)"

C~oR C.os_ 1-83 0.381 - - , . . . . . . . ° •

C~1 C.1 R ~

. . . . . . (37)

particular example

. . . . ( 3 s )

Method 7.--Actuator Discs placed at the Centres of Pressure of the Blades--Ultimate Steady Flow If the actuator discs are placed at the centres of pressure of the blades then the equation (37)

remains unchanged and C~oR -- C~ os, although in general the values of C~ R and C~ s will be different from those used in Method 6.

11

However, the axial velocity at the trailing edge of any rotor is not the same as that at the 'trailing edge of a stator. For if the blade spacing is b, and the distance between the plane of the centre of pressure and the trailing edge is a, then for identical blade rows"

Cx Re ; Cx R

c , ~ = c ~

k (C's -- CxR) Iexp -[

k (C~R -- Cys) Iexp

+

(c~R- c,~) 2 [exp

(C,s - c,~) k [exp 2

(Cx~- C~s) L [exp + 2

+ (C,s--2 C_.R) [exp

Cx Re

k(3b + a)l + "1 (b + a ) l + e x p l - - ~- . .

k(2b + ~)1 +" 1 ~I + exp!- f

k(3b ~)I+ .] ( b - - a ) l + exp l - - 7 -- . .

k (4b a) l + . . . l , ( 2 b - a) l + exp l - - F --

k(3b+~)t + . . . ] k (b-t-a) I + exp l - - 7

+ l- + °/1 1 -7,

k (4b a)l + " "l l - - / k - ( 2 b - - a ) } + e x p l - - 7 , -- ,

=/= C,~se .

(39)

(40)

The criterion d(A Wl,)ldr = 0 is still valid but the work done across a disc is defined by the whirl velocities at the trailing edge positions,

; [U{(U -- C:~R, tan/~R~) -- C~s~ tan ~s~}] = 0 . . . . . . . . . (36b) i.e.,

This equation is not sufficient to determine the flow as in Method 6 since C,R~ ~ C,s,, and a method of successive approximation must be used. C,R and C.~s may be guessed and two differential equations are integrated graphically.

Thus for a rotor:

dC, R 2C, R~ tan /3R, d d r -- r(C,R + C,s) {r(g -- C,R~ tan /3R,)}

__ dC, s 2(U - - C,s. t a n ~s . ) d dr r(C~R + C~s) ~ (rC~se tan ~s~) . . . . . . . . . (41)

And for any stator:

dC, s 2C~s~ tan ~se d (rC, s~ tall ~s~) :-d7 + r(C~R + C~s) dr

dC~R 2(U -- CxR~ tan ~R~) d {r(U -- C, tan/~R~)} dr + r(CxR + C,s) dr R . . . . .

12

(42)

These two equations together with the values of C,R~, C,s~ obtained from equations (39) and (40) and the guessed values of C,R, C,s enable a solution to be obtained, or alternatively either equation (41) or equation (42) may be used together with (36b).

I t is of interest to note that for the model stage the criterion d ( A W J d r ) = 0 gives the distribution of C~ R~ with radius immediately since tan ~s~ = 0, and this gives C, Re as identical to C, oR = C, os calcalated in Method 6. Then referring to equation (41) :

dC, R dC~s dr - - d r '

d d C, tan = 0 since tan ~s, ---- 0 a n d ~

Hence C,R = C~s + constant.

The continuity relation requires tha t this constant should be zero so that C,R = C,s = C, oR = C, os = C,R~ ---- C,s~, i.e., there are no discontinuities in tangential vorticity across the discs.

Method 8.--Actuator-Disc Theory~Addi t ion of Radial Fields ( Railly)

Calculations for the model stage receiving a uniform upstream profile alone using the theory of addition of radial velocity fields have been made for comparison .with Methods 2, 4 and 6 (1/b = 2.1, 4.2).

Values of K~ calculated from Railly's analysis are:

1 (a) ~ = - 2 . 1 K~1---- 0"191 K~ ---- 0.192 cf. K b = 0 . 2 1 4

from the isolated-disc theory l

(b) ~ = 4 . 2 Kb1=0"436 K b 2 = 0 " 4 4 6 cf. 0.462.

The addition of radial velocity fields gives the same result as Methods 6 and 7, for the model stage in ultimate steady flow. Methods 6 and 8 would give identical results for any given distribution of outlet air angles, /~R and as, in ultimate steady flow, but Method 7 in general gives different values for C,R,, C,s~.

Off-design Performance. The exact solutions for Methods 1 and 2 have been used to compute the axial velocity at exit from the model stage rotor for U~/C,I = 3 .0 assuming that the outlet angles ~, remain unaltered.

3. Discussion of Calculations.---The calculation of axial velocity at rotor and stator trailing edges for the eight methods used are shown in Figs. 4 to 18.

The flow parameter UtICa1 has been chosen as 1-83, and for the methods involving the estimation of blade interference values of lib = 2.1, 4.2 and a/b = 2/3 have been used.

In general the radial equilibrium solution (Method 1) has been used as a reference velocity.

In the discussion the effects of blade interference, positive interference from another disc is defined as that causing tile axial velocity at the station considered to become more distorted from the value Cx/C,I ~ 1.0. Negative interference implies tha t an external blade row is inducing a return to the undisturbed upstream velocity distribution. Thus it will be apparent that for the model stage the effect oi the stator upon the rotor is a positive interference effect, but negative interference is induced at the stator disc by the rotor.

13

Figs. 4 to 6 compare the radial equilibrium approximation (Method 1) with a calculation for isolated actuator discs at the blade trailing edges (Method 2). There is a considerable difference in the computed axial velocities at the rotor trailing edges and at the stator-blade root.

Figs. 7 to 9 show the results obtained if the actuator disc is placed at the centre of pressure of the blade (Method 3). For the low aspect ratio blade (l/b = 2.1 a/b = 2/3), Fig. 7 shows this calculation to be near the radial equilibrium solution but the ratio l/b = 4.2 gives all axial velocity C,~ little different from Method 2.

Figs. 8 and 9 suggest only small differences between the calculations for the stator. The calculations for l/b = 2.1 are almost identical with those for lib ---- 4.2 and are not plotted.

Comparisons between Methods 4 and 8 and for blade interference have been made and excellent agreement is obtained. This is illustrated in Fig. 10. The rotor trailing-edge velocity is identical for l/b = 2.1, and the differences for l/b = 4.2 are small. Similar calculations for Cxo~/C,~ and C~4/C~ give close agreement, and the determination of the attenuation constant k~ shows bu t small differences (see above).

The general effects of blade interference are therefore shown using the methods employing the concept of constancy of trailing vorticity (Hawthorne. Ref. 5. Appendix B).

Fig. 11 illustrates the small positive interference effect of the stator upon the rotor for 1/b = 2 .1 and a larger induced distortion from C~/C,~ = 1 for lib = 4.2. Method 4, in which the discs are placed at the trailing edge, is employed.

Negative interference is shown by this method in Fig. 12, but Fig. 13 again suggests that the calculation of C,~ varies little with the method chosen.

Figs. 14 to 16 illustrate Method 5, using blade-interference theory with the discs placed at the blade centres of pressure.

For the axial velocity at the rotor trailing edge, the separate effects of (i) placing the disc at the trailing edge (Fig. 7) and (if) positive interference from the stator (Fig. 11) are combined (Fig. 14) and the result is little different from the radial equilibrium result. The calculations for 1/b = 2- 1 are not shown, as these are almost identical with the Method 1 results.

However, i f the radial equilibrium calculation is fortuitously accurate for the rotor because of positive interference, negative interference at the stator trailing edge illustrates wide differences between Methods 1 and 5.

The axial velocity for the model stage in ultimate steady flow (C,0~ --Cx0s = C~R -~ C~s) is shown in Fig. 17, and suggests a reversal of flow at the root. Once again the comparison is made with the radial equilibrium solutions for the single stage.

Although the differences between the various theories will be greater because of the low hub-tip ratio chosen, the flow parameter U~/Cxl is small judged by modern compressor practice, and this small value tends to minimize these differences. This is illustrated in Fig. 18, which compares Methods 1 and 2 for the rotor, at U~/C,I = 1.83 and 3.0.

The results of these calculations are summarized in Table I which shows the difference in axial velocity from root to tip, as calculated by various methods for positions 02, 04 and 4.

If Method 5 is considered to be that giving the most accurate calculations, then it is seen from the table that while other methods may give agreement at one trailing-edge position, the results for the other trailing edge are not accurate.

14

TABLE 1

Differences in Cx root to tip

Method

1. Radial equilibrium

2.

.

.

5.

Widely spaced discs at blade trailing edges

Widely spaced discs at blade centres of pressure

Discs at trailing edges. Inter- ference theory (Hawthorne)

Discs at centres of pressure. Interference theory (Haw- thorne)

(C~ o~)~i~ - (C~o~)roo,

0"47

0-27

0.40(

(Cx 04)tip - - (C:~ o4)root;

1"05

0"80

0"66

0"75

0"63

0 " 8 2

I 6, Ultimate steady flow .. 2.0 I 2.0 7. I

(C~4)tip- (C,4)root C ~ ~_ .

1 " 0 5

1"07

1-01

O" 97

1" 04

1 "09

0"82

0"93

4. Conclusions.--The comparisons of the actuator-disc theories suggest that the choice of the position of the plane of the actuator disc is important, and that methods in which the discs are placed at the trailing edge of the blades may not be as accurate as the original radial equilibrium theory.

For large aspect ratios (lib > 3) the trailing-edge-disc methods may be more satisfactory, but then the effects of blade interference are considerable unless the axial clearances are large compared with the blade width, a configuration not encountered in modern turbo-machines.

For low aspect ratios (lib < 3) blade interference effects become less important, but the placing of the disc at the trailing edge becomes a larger approximation.

The excellent agreement between the theories for blade interference suggest that the use of the attenuation constants k~, as determined from isolated-disc theory, is justified in a theory taking into account blade interference. I t is therefore suggested tha t Method 5 is of practical use for axial-flow compressor and turbine designers. The disc is placed at the centre of pressure of the blades, but the possibly considerable effects of blade interference are not neglected. The values of k~ are determined directly from the hub-tip ratio.

In turbo-machine design, Method 5 would be used for the first few stages, after which the flow would be expected to approximate to the ultimate steady flow condition soluble by Method 7. Equation (8) may be readily adapted to give the axial velocities at the blade leading edges for determination of the incidences.

A further modification of these methods for the off-design condition may be to allow for variation in outlet air angle and blade losses in the equating of dH/d~ across the disc, using cascade resulfs and the incidences obtained from the first approximation to obtain values of these two new variables.

I t is important to note that Methods 5 and 7, while taking into account any non-uniformity in:entry axial velocity profile give no estimation of the growth of the boundary layer through a compressor.

15

NOTATION

Referring to Figs. 1, 2 and 3, the following notation is used"

Co-ordinates" r, O, x .

C, Radial velocity C,, Tangential velocity C, Axial velocity 0 ---- rC, , Tangential momentum

= ( a C , / a x ) - - ( a C , / a r ) Tangential vorticity ~0 Streamline function defined by :

= r C x = -

H Stagnation entha lpy . r, Tip radius r~, Root radius R = r / r t Non-dimensional radius ratio X2 Rotor angular velocity U = ~ r Blade speed

Absolute air angles fi Exit rotor air angle measured relative to the moving blade l Blade length in radial direction b Blade width in axial direction a Axial distance between planes of blade trailing edge and centre of pressure

Subscripts"

1

2

3

4

01

02

03

06

2e

4e

R

S

OR

0S

R e

S e

t

h

Superscripts" t

Conditions Conditions Conditions Conditions Conditions Conditions Conditions Conditions Conditions Conditions Conditions Conditions Conditions Conditions Conditions

far upstream of a rotor far downstream of a rotor far upstream of a stator far downstream of a stator immediately upstream of a rotating actuator disc immediately downstream of a rotating actuator disc immediately upstream of a stat ionary actuator disc immediately downstream of a stat ionary actuator disc at the trailing edge of a rotor-blade row at tile trailing edge of a stator-blade row far downstream of a rotating actuator disc in ultimate steady flow far downstream of a stat ionary actuator disc in ultimate steady flow at a rotating actuator disc in ultimate steady flow at a s tat ionary actuator disc in ultimate steady flow at the trailing edge of a rotating blade row in ultimate steady flow

Conditions at the trailing edge of a stat ionary blade row in ultimate steady flow Relating to blade tip Relating to blade root

Denotes perturbation to the axial velocities at infinity (upstream or down- t C t stream) in the neighbourhood of an actuator disc, e.g., c ~ , ~2 •

16

No. Author

1 H. Cohen and E. M. White ..

2 W. Merchant, J. T. Hansford and P. M. Harrison.

3 S. L. Bragg and W. R. Haw- thorne.

4 F .E. Marble . . . . . .

5 W.R. Hawthorne . . . .

6 J .W. Railly . . . . . .

7 A.R. Howell . . . . . .

8 F. Ruden . . . . . .

9 Jahnke and Emde . . . .

R E F E R E N C E S

Title, etc.

The theoretical determination of the three-dimensionM flow in an axial compressor with special reference to constant reaction blading. A.R.C. 6842. July, 1943.

The flow of an ideal fluid in an annulus. (Unpublished.) March, 1946.

Some exact solutions of the flow through annular cascade actuator discs. J. Ae. Sci. Vol. 17. No. 4. April, 1950.

The flow of a perfect fluid through an axial-flow turbo-machine with pre- described blade loading. J. Ae. Sci. Vol. 15. No. 8. August, 1948.

(Unpublished.) 1949.

The flow of an incompressible fluid through an axial turbo-machine with any number of rows. Aero. Quart. Vol. III. September, 1951.

(Unpublished.) 1942.

Investigation of single-stage axial-flow fans. N.A.C.A.T.M. 1062.

Tables of Functions. Dover Publications. 1945.

A P P E N D I X I

Three-Dimemional Flow in an Axial Turbo-Machine

1. The Apibroximate Solution for Flow through a~¢ Actuator Di sc . - -An a p p r o x i m a t e so lu t ion of t h e a c t u a t o r - d i s c p r o b l e m m a y be o b t a i n e d b y a s s u m i n g t h a t t h e t ra i l ing v o r t e x lines lie on cy l indr ica l sur faces concen t r i c w i t h t h e axis of t h e annulus . T h e rad ia l veloci t ies a re not , however , neg lec ted , so t h a t t h e cond i t ion for s t r e a m surfaces a n d vo r t ex - l ine sur faces to be t h e s a m e is no longer satisfied.

Cx I

r t + oo

Cx i + Cx t. Cx 2 + Cx 2 Cx 2

rh

Cr 2 Cr I

~=--+X

FIG. A.1.

W i t h th is a s s u m p t i o n it m a y be shown (Ref. 8) t h a t t h e axia l ve loc i ty a t t h e a c t u a t o r disc is (C~1 + C~2)/2 w h e r e C~1 a n d C ~ are t h e axial veloci t ies a t in f in i ty u p s t r e a m a n d d o w n s t r e a m respec t ive ly . W i t h t h e n o t a t i o n of t h e f igure C~I' a n d C~( are t h e p e r t u r b a t i o n s to t h e axia l veloci t ies in t he regions x-----0 to x ~ - - o o a n d x- - - -0 to x--~ + o o respec t ive ly . At x - - - -0

1' = - - 1) 1 2 a n d 2' = - - 2 ) / 2 .

Since it is a s s u m e d t h a t t h e r ing vor t ices pass d o w n s t r e a m a long cy l indr ica l surfaces t he va lue of t h e t a n g e n t i a l vo r t i c i ty , v, a t a n y rad ius will be c o n s t a n t a n d is equa l to its va lue a t inf ini ty.

17 (71361) 13

Vorticity components in a flow with velocities C . C,. C. and co-ordinates r, 0, x, are $, ~, ¢, where for axially symmetric flow:

~x

~C~ aC. '~ = a - 7 - a-7

a (rC,,) -- rar (!)

For incompressible, axially symmetric, flow the continuity equation is

(rC) + a (rc,,)= o ax ~; . . . . . . . . . . . . . . . . .

~cr~ ~ (c.~ + ~c~') - a c ~ Hence ~ - ~x ~r Or . . . . . .

or ~ C . ~ _ OC.( ~x ~r

A similar equation applies upstream of the actuator disc. incompressible flow may be written"

a C / 1 ~ (re,,)= O. ax + r ~ ; . . . . . . . .

Differentiating equation (4) partially with respect to r and substituting from equation (3)"

a=C, 1 aC~ C~ a2C~ ar 2 q r ar r = + Ox 2 - - 0 . . . . . . . . .

(2)

(3)

The equation of continuity (2) for

(4)

(s)

~ o o

A standard procedure for solving this equation is to write Cr = ~ {exp (k~x)}f~(r), where i=I

k~ with i = I, 2, 3, etc., are a number of constants whose values depend on the boundary conditions. Then :

{exp (k~x)}f~ "(r)+ r l ~ { e x p ( k ~ x ) } f ~ ' ( r ) + ~ ( - - ~ l ) { e x p ( k ~ x ) } f ~ ( r ) = O . . . (6i

Hence for all values of i since this equation must be satisfied at all r and x > 0 or x < 0,

The general solution of this Bessel's differential equation is"

f,(r) = A & ( < , ) + B~Vl(<r) . . . . . . . . . . . (8)

Hence the radial velocity is given by :

C~ = Z {exp (k~x)} {AJ,(k~r) + B~Y,(k~r)} . . . . . . . (9) i

Since at the boundaries of the annulus r = r~ and r = r,, C, = 0.

A~J,(k~r,) + B~Y,(k,rt) = 0

and AJ~(k~rh) + B,Y~(k,r~) = 0

or J~(k,r~,) Y~(k~r,) -- J~(k,r,) Y~(k,r~) = 0 . . . . . . . . . (10)

This equation gives an infinite number of values of k~, the first six of which are tabulated for various values of (r,/rh) by Janke and Erode (p. 205).

18

Since the values of ki only depend on the values of rt and rh, t h e solutions for C, 1 and C~ 2 are symmetr ica l about the actuator disc.

Subst i tut ing for C~ in equat ion (4) and integrat ing to obtain C, ' be tween the limits x = 0 a n d x = -¢- oo-

1 r [{exp (kix)}/ki] [fi'(r) + rf~( )] = C~' . . . . . . . . . (11)

Now since Cr and C,' = 0 at x = ± oo, ki is negat ive for x > 0 and positive for x < 0.

Insert ing the values for Cx' at x = 0"

1 r (12)

If only the first root of equat ion (11) is taken, then"

C~l' : (exp kx) (C~ - C~I)

,_- - { o x p X2 2 "

Hence if l is the blade length"

(13a)

(13b)

/ 3"16x~(C~--C~) (14) < : : - te:,:p 2 . . . . . . .

The value of 3 .16 is correct for rt/rt, = 1.5 and varies between 3-146 and 3-23 as r,/rl, varies from 1.2 to 2 .5 respectively.

2. Blade-Interference Theory.--Between each blade row in the compressor, sheets of tangent ia l smoke-ring vortices exist which are identified by the value of C~ at downst ream infinity. These are shown diagrammatical ly in Fig. 2.

FIG. A.2. Diagrammatic representation of sheets of tangential vorticity between blade rows.

If tile p t h row of blades shown in Fig. A.3 is considered and all the vor tex sheets of neighbouring blades are superimposed:

~ I___ __ _

p + l

.o_oooooooooooooo~P- 2 p+2

FIG. A.3.

19 (71361) B 2

The sum of all these vortex sheets is pictured below where the compressor has n stages:

p - 2 p - ! P p + I p + 2

2Cxp +

Cxp+ I + Cxp _ I

+ • . . . . . . . . .

Cx n + Cx 2

2 Cxp+l +

Cxp +2 + C~p

Cx n + Cx 2 "

FIG. A.4.

Hence between the (p -- 1)th and the p th rows the vortex sheets obtained by superposition are :

q=2

Downstream of the p th row and ahead of the (p + 1)th the vortex sheets sum to"

q=2

Hence the vorticity pat tern may be obtained by summing the vorticities of all rows taken separately and subtracting the sum of all the vortex sheets due to each blade stretching from upstream to downstream infinity. This procedure is shown diagrammatically in Fig. A.5 for a single stage :

FIG. A.Sa. Actual vorticity.

A

FIG. A.Sb. Each row taken separately.

, v V V V v v v v v v v v v v V v V v v V # v ~ v V V v V v v v v V V V ~ ~ v v v V v v v v V v V V V V V

FIG. A.Sc. Sum of vortices due to each row taken separately.

20

FIG. A.5d. Vortex sheets extending to infinity which require to be subtracted from (c) to give (a).

The induced axial velocity at the p th row due to the qth row when the vortices of all rows are taken separately is from equation (14),

When q < p"

lex ( _--(C~,q+~2 + C~,q) + (C~q+I2-- C'~q)( 1 - - I exP (--kqlq x~q) }) -- C~q . . . . . (15)

When q > p"

= C~q + {exp ( ---/q 2

= - - k~ x,~)1 ) - c ~ , ~ (16)

where Xpq is the magnitude (always positive) of the axial distance between p th and qth rows and C, le is the average axial velocity at the qth row:

W c . , , = p . , = ( r = _ rZ) . . . . . . . . . . . . . . . . . . . (17)

From the sum of the above induced velocities must be subtracted the effect of all the infinitely long vortices due to each blade row as shown in Fig. A.5. This latter effect is:

( c . - < , , ) . . . . . . . . . . . . . (18 ) g=2

Hence, tile total induced velocity is"

-- 2 (C•q- C,,q), . . . . . . . . . . ( 1 9 ) q=2

noting that the -¢- sign is taken when q < p and the -- sign when q > p. The induced velocity may also be written :

p+l ~ "

21

(20)

Hence the axial velocity at the/sth row is :

q=I 7q

q-q2~2l (C';q+~2 C,q)lexp ( _ ~xpq) I . . . . . . . . (21)

or for incompressible flow:

q=p+l 2 lq . . . . . . . . .

Similarly. it may be shown that the axial velocity at a point Xpe between the/5th and (15 + 1)th rows is given by:

C:;pe : Cxp+l-- '2~=i (Cxq+l; C~;q) lexp (-- ~q X, peq) l

where x m is the distance (always positive) between the plane x ---- xp~ and the qth disc.

22

b3 c~

I 2 4

Ol 02 23 O5 04

R I' 2 ¢ 4-¢

2¢ 4¢

I Ol 02104 4 /

2q 4¢

Mekhod I.

Radial equiHbrium ab krailing edges oP blades.

Mekhod 2.

W[d¢]~j ojpaced ackuakor d,scs ab bra;l[ng edges oF blades.

M eLhod 3.

Wt'd¢ly spaced acLuakor all'sos aL ccnkr~s oP pr¢55ur~ o# blad~¢.

Mekhod 4. Blade inberPer'¢nc¢ d;~cs ab LrarJ]n9 ~dg¢.~ oP blad¢~

2 ,

° : 7 '

2 , 4¢ OR 05

R $ R 5R S R 5 I ? 5 -

R~ 5 ¢

OR 09

_~.o = R 9 Rs R 5 R 5 R5 ~ - -

r Re 5¢

I ot p2 04. 4.

2e 4¢

MeLhod 5.

Blad~ in~.erPerenc¢. Discs a~. cenLrea o£" p ressu re oP blades.

z _ 2.1,4.2 e. = 2 -~- ~ 7

Mlekhod g. UlLimab¢ sb~acl9 ~'low. Discs ag ~pa;l~n 9 edges o~' blad¢~.

(_~= 2.l,4-2 )

Mekhod 7.

Ult imate skeadLj Plow. Discs at cen~r¢¢ oP pressure oP bladzs.

= 2"1, 4 . 2 _~ =

M ¢ k_had e. RaHIu'~ .me/r.hod. ID~Scs ab Lra]hng edges o f blades,

FIGs. 1 and 2. Methods used in calculation of flow through model stage.

OI 02 2,.?, 03 04

Ac~ua&or Acbuabo r D;sc Disc

FIo. 3. Model-stage velocity triangles.

c ~ 2 0

3O0

2 9 0

2 G O

24-0

2 ZO

~00

L o

~-14-0

120

-~ I00

~o

× <

4 0 -

~ 0 -

0 0-4

[FIG. 4.

I I I I I

h lq',a.d;~.l ec~u~l[bP'[urm t.heor-~ a.W[dely elo,~ced d iscs ~./:.

bJ~d e. ~ r-o.~.;Ir~n 9 e c]c3e s,

I I I I I O" 5 0 .G 0 "7 O' 9 0"~

R~-dius R = ~ r'ad~u~

1.6~

I-5

l-4

I-3

i.:~

• I ' 0

Cm02 0-9

0-B

0-7

0-G

0 . 5

- 0 - 4 .

- o - 3

- 0 . 2

- o . I

o I-o

Model stage--Axial velocity at rotor trailing edge.

24

b~ C~

280 I " 4

240 1'2

200 I'0

C~O. 4 8 --c-~ 7

$60 0.8

k 120 I-- 2 / / t Radial Equilibrium q'heo'm 9 -- 0'6

I / / ~. Widely spacod o,gcs ~ ~,/ I/ 6lade Trailing Edges / N 80 -- 0"4

4o I I I f I t

40

0"4 O'G

FIG. 5.

0-6 Oq 0'8 0-9 I'0

R = Rad[qs Tip Radius

Model stage--Axial velocity at stator trailing edge.

320 t

280~-

240

mi

-g Ir~

~12Q

-& <

80

~4

I - I '1 I r6

1"4

f

FIG. 6.

L Radial Equilibrium Theoey 2. WidelM $p&ced Disc6 ab

Bla~:~ Tra41tn 9 Edles.

I f 0.6 o.7

.Radius

0. l

• I'2

I'O

- - O ~

C:~!

- - 0 " 6

- - 0 ' 4

- - O.&

i

Model s tageIAxia l velocity far down- stream of stator.

. o . ,

200 I I i I I 1"4 2~01 I I t t I i 1"4

a40

aoo

Co

1 6 0

o_

120

> eO

<

40

0 - 4

I

- ~ 1 . 0

_ I. Radial e q u i l l b r ; u m L h ¢ o r y .

3. Widelg s p a c e d acEu~lor d[scs ~ 6lade ccnEr~s oR pressure.

_ b b " - 0.4-

(ii) _z .- 4-a ~ = 2 b b

0.2_

I o 0-8 0.9 t . o

FIG. 7,

I I I

0' 5 0-~ 0 "T Rad;us

P= ~ p rac~;u~

Model stage----&xial velocity at rotor trailing edge.

1"2

1.0

- - 0 " 8

x I

- 0 . 6

"O2O¢ o~

L ~| CaC

m120 ~a

..90 >

~o

x

40

I, Ra~cli~l e%uifibr-lum F-heory ~.. Wlcle1~j s p ~ c e d discs ~E bl~de

O0 ~ = ~.z .~ =

1.2

I I I I l I 0"4 0'6 O'G 0"7 O" ~ 0.~

R~dlu s R = Tip r~ciius

I'0

C~4e

FIG. 8. Model stage--Axial velocity at stator trailing edge.

I0 1,0

~'2

b O

"° l

.~4C

L g _o 160

o ~ 120

>

,~ SO

I I I

/ : "~j// s. wia%l~ sp~.¢ea a ~ c s ~.~. bl~.de /I y ~ n ~ " o~ p ~ q ~ . ,~/ 0o-~: ~'~ ' ~ -~

I-2

o f- 0 ' 4 -

[ .0

I I I I Io 0'5 0"~ 0"7 0'~ 0"9 I'0

~ad;u

FIG. 9. Model stage--Axial velocity far down- stream of stator.

0"8

c-,3cl

0-G

D'4

):.2

2SO

24.0

200

1¢0 Ln

i)_

~Lzo

~ 0 >

4 0

, i J ~ ,,v ]p4,

I _ . ,-- 1,2

" ~ 1 . 0

,,~ ~%~{c~-~ ..

4 Bl&de inl:.er"~er-ence (,~-ddJ:.fon 04" vor-k..e× .she..e~.s) S Bl~.de. in~,er4:~renc.e.. O~sc~ ~b ~r~ilin~3 edge.s.

(Addieion e f r-~d[~i veloc~eLj ~: ;eld ~,)

0 ' 4 0 " 5

FIG. 10.

I I I I 0-~, 0 . 7 0 . 8 0 " 9 I,O

I~d;uB

Model stage--Axial velocity at rotor trailing edges.

0,4-

3,2

[',,D GO

2~0 I j I I I

24-C

20C

o .~ 160 L oJ

10 ' o

.'~ ~0

.>

'¢ 40

i I.,~.

. . . . _

I. Ra.dla.I ~c~u i l lb r lum ~heor 'B ,

bla-d e ~r~ i l /ng edge .

I 0"4 0 " 5

FIG. 11.

I I I t o-g o,7 o.s o.s

~aol;tas R = Tip r~.diua

model s tage--Axial velocity at rotor t r a i l i n g edge .

I°-Z

1.0

--0,4

- - 0 " ~

0-B

C:coz C~

0 1.0

2~0 I I I I I 1 " 4

z¢o - ~.z

i~O0 f,O

CaT.~

Lo I~0 -- O'g

o_

t~ 120 l, Rmd~&l g%ufllbr-kam P.heor~, -- 0'6,

4. 86xd¢ i n k e r F e r ' e n c e , ,_ D i s c s ~.L b l ~ d s kr'~.ih% 9 c d g e s .

o. 2 - 0"4

~ 4 0 0,2

J I I I I I , 0 0 . 5 O.G 0 . 7 0 . ~ o .e v o

Radius ~ = ~

Model stage--Axial velocities at stator trailing edge.

0'4

FIG. 12.

t,O ¢43

280

24q

2 0 0

el_

m J2o "5 J9

~ 8o

L i I I I 400~- I t

/ / / 6. 5 ~ e r e n c e DIsGs

~ ' . -

°I 0"4

FIG. 13.

I 0 .5

1"4

1"2

1 I I I o C~6 0.'7 0'8 0,9 I'O

Radius

model stage---Axial velocities far down- stream of stator.

I'O

C~c 4

o , e

0'6

0'4

0'2

240 ~

-0200

J L ~160 -

I

80

0 ' 4 I

0 '5

FIG. 14.

I. Rad{al Equ;librTum "Theory. 5. 8lade In~,erf'erenca. Discs

a~ Blade Gentres oF

3

Pr~55ur'~.

I I I o.~ o.~ o-s o.s ~o

Re.dtu5 Ft~ - - "~p R~Lchus

Model stage--Axial velocity at rotor trailing edge.

1"4

I'O

-- 0'6

- 0 - 6

- Q ' 4

- - 0 . '~

0

~ 0

~4C

2 0 0

~ 160

o g

L ~12o

o.

2 6o u

2

3 4O

I I I

I. Rmdlal ¢Cluilibr~um Lh :o rg . 5. B lad : "~nh:rPcr¢n¢¢. Ibises

at blachz c~nhr~ oP pressure,

7, m 2 (1) ~ = 2. ~ -6 = "-;

I I I I I 0"4 0.5

1.4

1,0

0.8

C× 4 ¢

Cy, I

0 .6

0 . 4

FIG. 15.

0-5 0"7 0"8 0"9

Pad;u~ R"

T(p r'adiue.

Model stage--Axial velocity at stator trailing edges.

g '2

0 ~'0

280j

240

o c 20Q o

(3- 160

"~12C

8o

40

I f 0'4 0'5

FIG. 16.

l i I I I 1"4

1.2

I. Radial Equilibrium Th~or~J. ~. Blade }nLer+'erence. Discs

ab Bl~de cenLree of pressure.

(il) = 4.a -6- -3-

I'0

0"8

C:~ 4 Cx- ]

- - 0-6

-- 0"4

-- 0"2

I I I I O 0'6 0"7 0'8 O-S i'O

Radius R = -

Tip Radius

Model stage--Axial velocities far down- stream of stator.

,.+

co

.ol

~40

~o0

°O 160

L o o_

~J I~0

~ ~0

I I I l I

~_.~R =C~:o_.~s ac I C~

I R - L Radi&l e%ullibrtum ~heo~ 9. I S_J" For" ~;ncjle ~L~ge.

U6F Sbe.c~e in ul~..ima.ke.- ~gea-d 9 "Flow. lder~LiceLI r -esul l~ fronn meLHocJ~ 5, G

I I

I I

I /

- I

~ o m 0 posiLion . I

l 0 . 5 O . G 0'7 !

R = Tip r~d lu~

Fic. 17.

I i 0"B 0 " 9

1.4

I-0

0.8

0-G

0"4

0"2

I'O

Model stage--Axial velocities in ultimate steady flow.

3 Z O

250

24-0

~00

"o c o o

I60

g

la0

o

> 90

<

40

Ol

I I I i I I ° G

1 1"4

I. Racli6cl e%uilibr~urn I;heor~J

2. |sola,~ed d i * c ~ b bla .de ( i ) ~ ' = I .$3 (ii) u~ = 3-0

c ~ t c = I

I I I 0-6 O,G 0"7 0 . 6

R~ct~u e Tip n~dius

FIG. 18.

I-2

1"O

C~o 2 Ca=#

Model s t a g e ~ x i a l velocity at rotor trailing edge.

L~ihh~ cdqe. - 0.4

0-2

I 0 0.9 I.O

0,6

R . & M . No. 3

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