railly

R. & M. No. 3 0 3 0
(15,491)
A.R.C. TechnicalReport
MINISTRY OF SUPPLY
AERONAUTICAL
RESEARCH C O U N C I L
REPORTS A N D M E M O R A N D A
2
.
- r
L
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
ELEVEN
SHILLINGS
NET
Some Actuator-Disc Theories for the Flow of Air
Through an 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
MINISTRY
OF
(AIR),
SUPPLY
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 W h i t e )
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 Railly6 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
vorticity 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 attenuation 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 t h e various theories the direct problem
of a single compressor stage has been considered, the stage consisting of one rotor and one s t a t o r .
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 unity at the tip radius,
i.e., tan/3 = r/rs = R.
The hub-tip ratio is 0.4 and the ' aspect r a t i o s ' 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 M a c h i n e . 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 04],~
(4e) and (4).
The radial equilibrium condition at the trailing edge of a rotor is then expressed b y :
dC;
- - dH2
1 (rC,,2) d
~
C., d r -
dr . . . . . . . . . . . . .
+ r2=
where
C,,2 ---- U -- C.~2tan/32
and
Ha
----
H1 + U(C,,2
For the particular example of the model stage, tan/~2
dH2
and
dr
- -
'
C,,I).
r/rs =
=
(2)
R
,
d {U(U -- C~2 tan G)}
--
dr
Then equation (2) reduces to"
dC.2
C.2 / 2R ]
dR qI1 T R - b -
2UsR
(1 + R~) '
which gives an exact solution"
c,2
<-i =
U, R=
+ c
(1 + R 2)
. . . . . . . . . .
where P~ is a constant to be determined from the continuity conditions:
C,2R dR =
Rh
C.,R dR
Rh
and is given thus"
Us
P 1 ~---
2
2
2
,~
(3)
Similarly at the trailing edge of the following stator"
dC~4
1
d
dH4
C~ ~ - + ~(rC..) ~(rC..~) -- dr
where
. . . . . . . . . .
(4)
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
(s)
Where Q1 is a constant and is determined by trial and error from the continuity relation.
It 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.) e x p { - - ( k ~ / Z ) }
2
c~ =
c~, +
(cx~-
2
c,,)
exp
{(kx/l)}
~ > 0
x< 0
}
,
..
..
(6)
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 t h 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)
A2
where C,,p, C,,,,+1 are the axial velocities t h a t would exist at upstream and downstream infinity
of the p t h 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 t h 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 t h trailing edge C,,~ is given b y :
Cxp~
=
q=*
~ (C~q+~2- C,q) {exp
q=I
C,~,+I-
+ £
-
--
(kqXp~q/lq)}
--(kqx,,qllq) }
•
,
.
.
.
.
.
(8)
q=p+l \
where Xp~qis the distance between the p t h 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 t h 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 m a y 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
dr,
l [o dO
--
re
,Trl
.
+
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(0)
Across an actuator disc:
Hoe = H o l +
A W,
. .
.
.
.
.
.
.
.
.
(10)
where A W is the work done on the fluid b y the moving blade row and is given b y :
AW
=
C,,01) .
O'(<,oe-
.
.
.
.
.
.
.
.
.
.
.
.
(11)
From equations (9), (10), (11):
l~o~r + rC,,oe4 (rC,,oe)l = l ~o~r+ rC,,o~~d (rc"°~)l @re d{u(c"°e--C'~°l)}'
which may be written :
d (rC,,0e) = v0~ + (C,,o,- U ) ~d (rC,,0,) . . . . . . . . .
(12)
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) m a y be made, that the tangential vorticity is
approximately a function of radius alone.
Thus
'101 - - / h =
--
--
1
T]02
-t,-
"172 7----
1
(ac4
4
dr
;c.
. . . . . . . .
(13)
Further
C,~ot -----C.m tail ~oi = C.~ tan ~ ,
and if the disc is placed at the trailing edge of the blades"
]
[
(14)
9
l
C,,o~ = U -- C.o~ tan/~o2 -----C.~2tan a2 )
Hence equation (12) becomes"
dC~
d
dr + C, o2tan/~o2 ~ { r ( U -- C,o~ tan/3oJ}
-
But
-
dC,~
dr + (U -- C,I tan ~) ~d {rC, t tan al} .
C,o~ = C, o2 -- C,~ + C,~ (Ref. 8), and the further approximation is made t h a t :
2
rC, o~~--
d~
dr "
d
1
dW
d
rC, o~dr'
so that :
dC.2
tan ~o2 d Ir l u _ (C,I + C,~)
_
tiC,1
dr
2(U -- C,1 tan ~) d (rC.~ tan cq)
r(C,~ + C~)
dr
(15)
. . . . . . .
This may be written as an equation of the form :
C,~~ + A~(r)C,2 ~dC,~
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~
dr @
tan ~o, d I (C~3 + C~,)
1
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 velocity 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. It 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
c.
where
1) R 2
(9 + R2)
P2 =
2('-
,
3
(17)
•. . . . .
-- 2
2u,
from the continuity relation. The equation for the following stator reduces to:
dC~ a
dC,3
2C~,
d (RC,8 tan ~3)
dR + C~3 + C,~ tan e~ ~
dR--
C.~ tan ~ = (U
But
(18)
C~1+C~22tan /~02)
where tan/302 = R and C~2 = C=~ is obtained from equation (17).
The resulting equation is solved graphically.
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
C,,2, = U -- C~2,tan P2, '
and
( 19)
. . . . . . . . . . . .
where C.~, is obtained from equation (6).
C.2,
•
C.,
(c,~
C,1)
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,~ -t- C.~ tall /~2e~
d {r(V - - C.~° tan ~,)}
-
-
dC,
dr l + (U -- C,, tan ~) ~d (rC~ tan ~1)
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~ _ 2 IC,2 ( 1 K~)r(C,2-/+ Cxl
I C (x~, ))
dr
_
_
_
dCx,
dr
tan $ , e-d ldr ( U
r
lC*=( 1 K ~ ) +
2(U -- C,~ tan ~,) d (rC, z tan ~1)
r(C,l -4- Cx~) dr
' ".
~
K~ l tan/~"e)l
.
(21)
. . . . . . .
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)
dr + r(C,~ + Cx~) dr
(22)
.
.
.
.
.
.
.
.
.
.
.
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. It is, however, more convenient to consider these unknowns as Lhe n values of axial
velocity t h a t would exist far downstream of each blade row, since a knowledge of the tangential
vorticity after the p t h row dC~p/dr enables C,p to be determined from continuity. It 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 infinity'
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 t h 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 t h disc will
now be specified by (U -- Cx0ptan/~0p) for a rotor, or by Cxoptan ~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. It should be noted that the same
7
approximation as before is used for d~p -- r{(C~**~ + C~,)/2} dr, although Cxo~ is not now equal
to the arithmetic mean of C,p and C,p+ ~. Thus the equation for a rotor, say the p t h row, similar
to equation (15) is:
2C~o~tan 13op d {r(U -- C~o, tan GJ}
(C~p+l + C~p)r dr
dC~,.~
dr
-
-
2(U -- C~o(p_~)tan ~olp-~)) d
r(Cxp,~ + Cxp)
dr (rC~ol~_~) tan ~o~p-~))
dC~p
dr
. . . . . .
(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 stator:
dC~p+~
d~
2C~o~t a n ~op d
+ (C;7~+; T
=
~
?; (~<o, tan ~o,)
dC~,
2(U -- C~o(,_~)tan G(p-~I) d {r(U
dr +
r(C~p+~+ C~p)
dr
- - Cxo(p_l) t a n
G:p-~)}
. . . . .
(24)
The n differential equations thus formed are integrated graphically 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 approximation.
For the example of the model stage there are two unknowns, the axial velocities Cx2 and C,~
t h a t would exist far downstream of rotor and stator. There is in this case no need to perform
the successive approximation suggested above for the general case, since the two differentia)
equations m a y be combined to give a more direct solution.
From equation (7)"
C.o~ = C.o2 -- C~ + C ~ + K~(C. - c . ) ~
2
2 Cxl) I
Kb(Cx2C~o~ = C~o~ -- C~2 + C.~
2
2
. . . . . . . . . .
(25)
where Kv = exp (-- kb/l), in which b is the axial distance between the blades and l is the blade
length.
F r o m equations (23) and (24)"
dC~
2Cxo2tan 13o2 d {r(U -- C~o2 tan 13o2)}
r dr -- ( 0 ~ 2 + C ~ ) dr
". . . . .
"
and
dC,4
dC.2
2(U -- C~o2 tan t3o2) d {r(U -- C~o2 tan 13o2)}
r dr -- r ~ - +
(C~2 + C~4)
d-)
Combining equations (26) and (27)"
r dr -- r - - d r -[-
Substituting for C~o2 f r o m
1
(C~2 + C~a) C~o2tan 13o2
equation
(25) •
(u - Go2tan &2)(G2 + G~)t
(C~2 -4- C~) Cxo2 tan 13o~
<
<j
+ I,-
t
dr
This differential equation m a y be integrated graphically to give a relation b e t w e e n (Cxd/Cxl) and
(C,~/C,~), which together with equations (25) and (26) enables (C~2/C~1) to be calculated. W i t h
(C,,/C,I) and (C~/C,~) known, (C~o,/C~) and (C~o~/C,I) are directly obtained from equation (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 t h e centres of pressure of the blades as Marble suggests,
t h e n the whirl velocities are defined not by the axial velocities at t h e discs themselves b u t b y
t h e axial velocities at the trailing-edge positions (C~1,, C~2~ . . . C ~ , C,~ . . . C .... ) d o w n s t r e a m
of the discs.
These whirl velocities are:
for the p t h row, if a rotor C,~p, ---- U -- C,p, tan $p~ 1 ,
for the p t h row, if a stator C,,:o~ ---- C~p~t a n ~p~
..
..
..
..
(29)
J
where tips, ab~ are the rotor and stator exit air angles at the trailing edges, measured relative to
t h e blade rows.
Since the value of O ---- rC,~is constant along a streamline b e t w e e n the discs, for small streamline
displacements the whirl velocity C,, is constant.
Thus the whirl velocity Cu just d o w n s t r e a m of an actuator disc is defined b y the axial velocity
at t h e following trailing edge.
The m e t h o d of solution of the problem is similar to t h a t of Method 4. The ' d o w n s t r e a m
i n f i n i t y ' 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 subtracting those extending from u p s t r e a m to d o w n s t r e a m infinity.
For the trailing edge d o w n s t r e a m of the p t h row the axial velocity C~p~ at t h a t station is given
by:
q=' (C~+I
=
C~,,){ e x p
+
(-
q=l
q=p+l
where x~q is t h e distance (always positive) between the qth row and the p t h trailing edge a n d
the flow is incompressible.
For. each disc a differential equation is obtained similar to equations (15) or (16) b u t the whirl
velocities just d o w n s t r e a m of the disc are the same ,as those calculated for the trailing edge:
for t h e p t h row a rotor
C 0p ---- C,,p~ ---- U -- C,p~ t a n ~p~
for the p t h row a stator
C,~op = C~p~ = C,p, t a n %~.
Thus for the p t h row if a rotor :
dC~p+l 2C~p~t a n Sp~ d {r(U -- C~petan/~p~))
dr
r(C,p+l + C~p) dr
dC~p 2(U
C,(p_l)~ t a n ~p_~)0) d
-- dr
r(C~p+~ + C~p)
dr (rC~tp_l)~tan ~(p_~)~) ,
-
-
..
(30)
..
(31)
or for a stator :
dC~p+~ 2C~p~t a n %° d
dr + r(C~,+~ + C,,) -dr{r(C,*~ t a n
dC.p
-27+
%e)}
2(U -- C,(p_~)~ t a n fl(p_~)~)d (r(U -- C~(p_~)~tail fl(p_~),)}...
r(C~+l + C~p)
dr
9
The n equations for the n blade rows are then simply solved b y graphical integration and the
resulting values of ' d o w n s t r e a m i n f i n i t y ' axial velocities m a y be used in a second approximation.
For the model stage"
c . ~ = c.~ + K~_~(C~, -- C . ) _ K~(C~ -- C . ) ~
2
2
and
• .
c~,~ =
c~o -
Ko+~(C.
--
C~)
_
Ko(C~
--
2
where
K~_~ = exp [{-- k(b -- a)}/l],
C.)
.
.
.
.
.
.
(32)
"
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.
Then as in Method 4"
dC,,
dC,~ I
(U -- C,~ t a n / ~ ) ( C , , + C~)I
dr -- ~
1+
(C,~ + C~)(C,~ . t a n / ~ )
"
But
C~2~ = C,~ (1
~_
K~C~I
K~-~-~Kb-~)+ ~_~C,~ +_ff
~
~+(1-K~-~)c.~
(33)
whence the m e t h o d of solution is the same as before.
Ultimate Steady Flow.--Stages deeply e m b e d d e d in an axial-flow turbo-machine m a y be
considered as identical pairs of actuator discs and the distribution of t h e tangential vorticity ~
is the same after each rotor row. C, R is defined as t h a t axial velocity t h a t would exist far downstream of the rotor and is related to ~R by ~R = -- (riCeR/dr). Similarly the distribution of
tangential vorticity ~s = -- (dC, s/dr) is the same after each identical stator.
If the rotor, say t h e kth row is considered t h e n along a streamline"
H,o = H,o_I + A W,, . . . . . . . . . . . . .
where
(34)
H~0_I is t h e stagnation ellthalpy after the (k -- 1)th disc
HT0 is the stagnation e n t h a l p y after the kth disc
A W~
is the work done on the fluid b y the kth row.
.
For the (k + 1)th row, a stator row:
A W~+~ = 0 ,
H~ = H~+~ .
.
.
.
.
.
.
.
Then from equations (34) and (35), along a given streamline
Differentiating with respect to ~0,
(dH~+q (dH~_4 d (~w~),
d~ ]=~ d~ ] + ~
10
.
.
H1,+1
.
.
.
.
.
=
H~_I + A W~.
(35)
but using equation (9) for the (k -- 1)th row, a stator"
(d.~_~
1(
~04
~d.~+~
d~ ] = r~ ~sr + Os d~o! = \
for small streamline displacements.
d (~W~)
d~
d~o ]
Hence d(AWk)/dr = 0.
0,
.
.
.
.
.
.
Or approximately"
.
.
.
(36)
.
result originally given by A. R. Howell 7 and used by Railly 6.
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~oRtan $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
2/kb) + (exp
l(ex -
4kb
,
)1
-)I
+ (C'R 2 ;'s) I (exp -- -
-
C~R+ C,s
-)I
2
CxOS
"
. . . . . .
* "
(37)
Thus equation (36a) becomes"
d [U{U -- C~oR(tan/~oR + tan ~os)}] = 0
dr
'
the result derived by Railly b y 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 particular example
(U,/C~ = 1.83)"
C~oR
C~1
-
-
C.os_ 1-83
C.1
0.381
R~
,
.
.
.
.
.
.
.
°
•
. . . .
(3s)
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
(C's -- CxR) Iexp k -[ (b + a ) l + e x p l - -
; Cx R
(C~R -- Cys) Iexp
k
+
k(3b + a)l +
~-
..
"1
~I + exp!- fk(2b + ~)1 +" 1
( b - - a ) l + exp l-- 7k(3b -- ~)I+ .. .]
( 2 b - a) l + exp l-- Fk (4b -- a) l + . . . l ,
(c~R- c,~)
c,~ = c~
2
(C,s
[exp
k(3b+~)t +... ]
k (b-t-a) I + exp l-- 7
-2 c,~) k[exp
+ (Cx~-2
C~s) [exp
L
+
l-
+ °/1
Cx Re
1
-7,
(C,s--2 C_.R) [exp l - - / k - ( 2 b - - a ) } + e x p l - - 7 , k (4b --
+
(39)
a)l + " , "l
(40)
=/=C,~se .
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,
i.e.,
; [U{(U --
C:~R,tan/~R~) --
C~s~ tan ~s~}] = 0 . . . . . . . . .
(36b)
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
dr
- - C,s. t a n ~s.) d
r(C~R + C~s)
~ (rC~se tan ~s~) . . . . . . . . .
2(U
(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) m a y be used together with (36b).
It 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
tan ~s, ---- 0 a n d ~d
since
Hence
C,
tan
= 0
C,R = C~s + constant.
The continuity relation requires t h a 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 T h e o r y ~ A d d i t i o n 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
(b)
l
~=4.2
Kb1=0"436
Kb2=0"446
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 t h a 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 b u 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 b y 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 b y 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 b y 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 m a y 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
(C~ o~)~i~
Method
-
(C~o~)roo,
(Cx 04)tip -- (C:~ o4)root;
(C~4)tip- (C,4)root
C ~ ~_ .
1.
Radial equilibrium
0"47
1"05
1 "05
2.
Widely spaced discs at blade
trailing edges
0-27
0"80
1"07
. Widely spaced discs at blade
centres of pressure
1-01
O"97
.
5.
6,
7.
Discs at trailing edges. Interference theory (Hawthorne)
0.40(
Discs at centres of pressure.
Interference theory (Hawthorne)
Ultimate steady flow
..
I
2.0
I
0"66
1" 04
0"75
1 "09
0"63
0"82
0"82
0"93
2.0
I
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. It is therefore suggested t h a 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.
It 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
Tangential velocity
C,
Axial velocity
0
---- rC,,
Tangential momentum
=
(aC,/ax) -- (aC,/ar)
Tangential vorticity
~0
Streamline function defined b y :
C,,
=
H
r,
r~,
R
=
X2
U =
fi
l
b
a
rCx
=
-
Stagnation e n t h a l p y .
Tip radius
Root radius
r/rt
Non-dimensional radius ratio
Rotor angular velocity
~ r Blade speed
Absolute air angles
Exit rotor air angle measured relative to the moving blade
Blade length in radial direction
Blade width in axial direction
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
Re
Se
t
h
Conditions far upstream of a rotor
Conditions far downstream of a rotor
Conditions far upstream of a stator
Conditions far downstream of a stator
Conditions immediately upstream of a rotating actuator disc
Conditions immediately downstream of a rotating actuator disc
Conditions immediately upstream of a stationary actuator disc
Conditions immediately downstream of a stationary actuator disc
Conditions at the trailing edge of a rotor-blade row
Conditions at tile trailing edge of a stator-blade row
Conditions far downstream of a rotating actuator disc in ultimate
Conditions far downstream of a stationary actuator disc in ultimate
Conditions at a rotating actuator disc in ultimate steady flow
Conditions at a stationary actuator disc in ultimate steady flow
Conditions at the trailing edge of a rotating blade row in ultimate
Conditions at the trailing edge of a stationary blade row in ultimate
Relating to blade tip
Relating to blade root
steady flow
steady flow
steady flow
steady flow
Superscripts"
t
Denotes perturbation to the axial velocities at infinity (upstream or downstream) in the neighbourhood of an actuator disc, e.g., c ~ ,t C ~2t •
16
REFERENCES
No.
Title, etc.
Author
1 H. Cohen and E. M. White
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.
..
2 W. Merchant, J. T. Hansford
and P. M. Harrison.
3 S. L. Bragg and W. R. Hawthorne.
Some exact solutions of the flow through annular cascade actuator discs.
J. Ae. Sci. Vol. 17. No. 4. April, 1950.
4 F . E . Marble
The flow of a perfect fluid through an axial-flow turbo-machine with predescribed blade loading. J. Ae. Sci. Vol. 15. No. 8. August, 1948.
. . . . . .
5 W.R. Hawthorne
(Unpublished.)
....
1949.
6 J.W. Railly
. . . . . .
The flow of an incompressible fluid through an axial turbo-machine with any
number of rows. Aero. Quart. Vol. III. September, 1951.
7 A.R. Howell
. . . . . .
(Unpublished.)
8 F. Ruden
. . . . . .
Investigation of single-stage axial-flow fans.
9 Jahnke and Emde
1942.
Tables of Functions. Dover Publications.
....
APPENDIX
N.A.C.A.T.M. 1062.
1945.
I
Three-Dimemional Flow in an Axial Turbo-Machine
1. The Apibroximate Solution for Flow through a~¢ Actuator D i s c . - - A n a p p r o x i m a t e s o l u t i o n 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 b e 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 r a i l i n g v o r t e x lines lie o n
c y l i n d r i c a l s u r f a c e s c o n c e n t r i c w i t h t h e axis of t h e a n n u l u s . T h e r a d i a l velocities a r e n o t , h o w e v e r ,
n e g l e c t e d , so t h a t t h e c o n d i t i o n for s t r e a m s u r f a c e s a n d v o r t e x - l i n e s u r f a c e s to b e t h e s a m e is
n o l o n g e r satisfied.
rt
Cx I
Cx i
+
Cx t.
Cx 2
+
Cx 2
+ oo
Cx 2
rh
~=--+X
Cr 2
Cr I
FIG. A.1.
W i t h this a s s u m p t i o n it m a y b e s h o w n (Ref. 8) t h a t t h e axial v e l o c i t y 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 velocities a t i n f i n i t y u p s t r e a m a n d d o w n s t r e a m
r e s p e c t i v e l y . W i t h t h e n o t a t i o n of t h e figure 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 axial
velocities in t h e regions x-----0 to x ~ - - o o
a n d x - - - - 0 to x--~ + o o r e s p e c t i v e l y . A t x - - - - 0
1' =
--
1) 12 and
2' =
--
2)/2.
Since it is a s s u m e d t h a t t h e r i n g v o r t i c e s pass d o w n s t r e a m a l o n g c y l i n d r i c a l surfaces t h e v a l u e
of t h e t a n g e n t i a l v o r t i c i t y , v, a t a n y r a d i u s will b e c o n s t a n t a n d is e q u a l t o its v a l u e at infinity.
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
(!)
a (rC,,)
'~ =
~C~
aC.
a-7-
a-7
-- rar
For incompressible, axially symmetric, flow the continuity equation is
ax
Hence
(rC)
+ a
~;
(rc,,)=
~-
o
.
~cr~
~x
.
.
.
.
.
.
.
.
.
.
~ (c.~ + ~c~') ~r
.
.
.
ac~
Or
.
.
(2)
.
. . . . . .
(3)
~ C . ~ _ OC.(
or
~x
~r
A similar equation applies upstream of the actuator disc.
incompressible flow m a y be written"
aC /
1 ~
ax + r ~ ;
The equation of continuity (2) for
(re,,)=
O.
. . . . . . . .
(4)
Differentiating equation (4) partially with respect to r and substituting from equation (3)"
a=C,
1 aC~
ar 2 q r ar
C~
a2C~
r= +
Ox 2 - - 0
.
.
.
.
.
.
.
~oo
A standard procedure for solving this equation is to write Cr =
~
{exp
.
.
(s)
(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){exp(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)
C~ = Z {exp (k~x)} {AJ,(k~r) + B~Y,(k~r)} . . . . . . .
(9)
Hence the radial velocity is given b y :
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
or
AJ~(k~rh) + B,Y~(k,r~) = 0
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) b y Janke and Erode (p. 205).
18
Since the values of ki only d e p e n d on the values of rt and rh, t h e solutions for C, 1 and C~ 2 are
s y m m e t r i c a l about the actuator disc.
Substituting for C~ in equation (4) and integrating to obtain C,' between the limits x = 0
a n d x = -¢- oo[{exp
(kix)}/ki] [fi'(r) + 1rf~( r )]
= C~' . . . . . . . . .
(11)
Now since Cr and C,' = 0 at x = ± oo, ki is negative for x > 0 and positive for x < 0.
Inserting the values for Cx' at x = 0"
1
r
(12)
If only the first root of equation (11) is taken, then"
C~l' : (exp kx) (C~ - C~I)
X2
,_- - { o x p
(13a)
2
(13b)
"
Hence if l is the blade length"
<::
The value of 3.16 is correct for
from 1.2 to 2 . 5 respectively.
/
-
te:,:p
rt/rt, =
3"16x~(C~--C~)
2
(14)
.
.
.
.
.
.
.
1.5 and varies between 3-146 and 3-23 as
r,/rl, varies
2. Blade-Interference Theory.--Between each blade row in t h e compressor, sheets of tangential
smoke-ring vortices exist which are identified b y the value of C~ at d o w n s t r e a m infinity. These
are shown diagrammatically 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 vortex sheets of neighbouring
blades are superimposed:
~
I_____ _
p+l
.o_oooooooooooooo~P2
p+2
FIG. A.3.
19
(71361)
B2
The sum of all these vortex sheets is pictured below where the compressor has n stages:
p-2
p-!
P
p+ I
2Cxp +
Cxp+ I
+
•
Cx n
+
Cxp _ I
2 Cxp+l
+
Cxp +2
+
C~p
+
Cx 2
p+2
. . . . . . . . .
+
Cx 2
Cx n
"
FIG. A.4.
Hence between the (p -- 1)th and the p t h rows the vortex sheets obtained b y superposition are :
q=2
Downstream of the p t h row and ahead of the (p + 1)th the vortex sheets sum to"
q=2
Hence the vorticity pattern m a y 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.
,vVVVvvvvvv
FIG. A.Sc.
Each row taken separately.
vvvvVvVvvV#v~
vVVvVvvvvVVV~
~vvvVvvvvVvVVVVV
Sum of vortices due to each row taken separately.
20
FIG. A.5d. Vortexsheets extending to infinity which require to be subtracted from (c) to give (a).
The induced axial velocity at the p t h 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
-
-
(16)
k~ x,~)1 ) -c~,~
where Xpqis the magnitude (always positive) of the axial distance between p t h and qth rows and
C, le is the average axial velocity at the qth row:
c.,,
=
W
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)
.
.
(19)
g=2
Hence, tile total induced velocity is"
-- 2 (C•q- C,,q),
. .
.
.
.
.
.
.
q=2
noting that the -¢- sign is taken when q < p and the -- sign when q > p.
may also be written :
p+l
~
"
21
The induced velocity
(20)
Hence the axial velocity at the/sth row is :
q=I
7q
q-q2~2l(C';q+~2C,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 (-- ~qX,peq)l
where x m is the distance (always positive)between the plane x ----xp~ and the qth disc.
22
I
2
4
Mekhod
°:7
I.
'
MeLhod 5.
Blad~ in~.erPerenc¢. Discs
a~. cenLrea o£" p r e s s u r e
oP blades.
z _ 2.1,4.2
e. = 2
-~~ 7
Radial equiHbrium ab
krailing
edges oP blades.
2 , 4¢
Ol 02
23
Mekhod
I'
R
2¢
OR 05
O5 04
2.
Mlekhod g.
UlLimab¢ sb~acl9 ~'low.
Discs ag ~pa;l~n9 edges
o~' blad¢~.
(_~= 2.l,4-2 )
R $ R 5R S R 5 I?5-
W[d¢]~j ojpaced ackuakor
d,scs ab bra;l[ng edges oF
blades.
R~ 5 ¢
4-¢
OR 09
b3
c~
M eLhod 3.
_~.o =
R 9 Rs
Wt'd¢ly spaced acLuakor
all'sos aL ccnkr~s oP
pr¢55ur~ o# blad~¢.
2¢
I
02104
/
4
~--
r
4¢
Ol
R 5 R 5 R5
I
Re 5¢
ot p2 04.
= 2"1, 4 . 2
M ¢ k_had e.
RaHIu'~ .me/r.hod. ID~Scs
ab Lra]hng edges o f
blades,
2,
2e 4¢
FIGs. 1 and 2.
_~ =
4.
Mekhod 4.
Blade inberPer'¢nc¢ d;~cs
ab LrarJ]n9 ~dg¢.~ o P
blad¢~
2q 4¢
Mekhod 7.
Ultimate skeadLj Plow.
Discs a t cen~r¢¢ oP
pressure oP bladzs.
Methods used in calculation of flow through model stage.
OI
03
2,.?,
02
Ac~ua&or
D;sc
04
Acbuabor
Disc
FIo. 3. Model-stage velocity triangles.
c ~20
I
I
I
I
1.6~
I
3O0
I-5
290
l-4
2 GO
I-3
24-0
i.:~
2 ZO
~00
•
I'0
Cm02
0-9
0-B
L
o
~-14-0
0-7
120
0-G
-~ I00
0.5
h lq',a.d;~.l ec~u~l[bP'[urm t.heor-~
a.W[dely elo,~ced discs ~./:.
bJ~d e. ~ r-o.~.;Ir~n9 e c]c3es,
~o
-0-4.
-o-3
×
<
40-
-0.2
~0-
-o.I
0
0-4
I
O" 5
I
0 .G
I
0 "7
R~-dius
R = ~
r'ad~u~
I
O' 9
I
0"~
o
I-o
[FIG. 4. Model stage--Axial velocity at rotor trailing edge.
24
I
320 t
280
I "4
280~-
240
1'2
240
200
I'0
-
I
'1
r6
I
1"4
•
I'2
mi
I'O
C~O. 4
8
--c-~ 7
0.8
$60
b~
C~
-g Ir~
--
k
C:~!
120 I-- 2 /
I /
/
~,/
N
O~
80
/
/
I/
t Radial
Equilibrium
q'heo'm
9
-- 0'6
~. Widely spacod o,gcs ~
6lade
Trailing Edges
-&
<
-- 0"4
80
0.l
40
4oI
0"4
~12Q
I
I
f
I
t
O'G
0-6
Oq
0'8
0-9
I'0
L Radial Equilibrium Theoey
2. WidelM $p&ced Disc6 ab
Bla~:~ Tra41tn9 Edles.
~4
f
I
0.6
f
--
0"6
--
0'4
--
O.&
i
o.7
.Radius
R = Rad[qs
Tip Radius
FIG. 5.
Model stage--Axial velocity at stator
trailing edge.
FIG. 6.
Model s t a g e I A x i a l velocity far downstream of stator.
.
o
.
,
200
I
I
i
I
1"4
I
2~01
I
I
t
I
t
i 1"4
I
a40
1"2
-
aoo
~ 1 . 0
1.0
1.2
I'0
"O2O¢
o~
C~4e
Co
L
160
--
~| CaC
0"8
o_
xI
120 _
> eO
I. Radial
3. Widelg
6lade
_
equillbr;um
Lh¢ory.
spaced
acEu~lor d[scs
ccnEr~s
oR pressure.
b
b
(ii) _z
.- 4-a
b
<
~
-
-
0.4-
2
0.2_
I
I
0' 5
0-~
P=
FIG. 7,
I
I
0 "T
Rad;us
~ p rac~;u~
0-8
I, Ra~cli~l
e%uifibr-lum F-heory
d i s c s ~E
~..Wlcle1~j s p ~ c e d
~o
O0 ~
= ~.z
bl~de
.~ =
x
b
40
0-4
m120
~a
..90
>
"
=
0.6
~
0.9
Model stage----&xial velocity at
trailing edge.
o
t.o
40
~'2
I
0"4
I
0'6
I
O'G
R =
rotor
FIG. 8.
I
0"7
R~dlu s
Tip r~ciius
l
O" ~
I
0.~
I0
1,0
Model stage--Axial velocity at stator
trailing edge.
I
I
I
"°l
.~4C
/
L
g
_o 160
bO
o
~ 120
>
,~ SO
,
2SO
I-2
24.0 _
[.0
200
:
0"8
c-,3cl
0-G
i
J
~
,,v ]p4,
I
. ,-- 1,2
"
~
,,~ ~%~{c~-~
1
.
0
..
1¢0
Ln
i)_
~Lzo
4
"~j//
/I
s. wia%l~
y~n~
,~/
sp~.¢ea a~cs
" o~ p ~ q ~ .
0o-~: ~'~
'~
~.~.
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
ef r-~d[~i
veloc~eLj ~:;eld ~,)
bl~.de
D'4
-~
of-
):.2
I
0'4-
0'5
I
I
0"~
0"7
I
0'~
Io
0"9
I'0
0,4-
~0
>
3,2
40
0'4
0"5
9.
Model stage--Axial velocity far downstream of stator.
I
0.7
I
0.8
I
0"9
I,O
I~d;uB
~ad;u
FIG.
I
0-~,
FIG. 10.
Model stage--Axial velocity at rotor
trailing edges.
2~0
I
.
j
I
I
.
.
2~0
i I.,~.
I
.
_
I
I
I
I
I
1 " 4
24-C
I°-Z
z¢o -
~.z
20C
1.0
i~O0
f,O
0-B
Lo I~0
-- O'g
CaT.~
o
.~ 160
L
oJ
[',,D
GO
C:coz
C~
10
'o
t~ 120
I. Ra.dla.I
.'~
.>
o_
bla-d e
~c~uillbrlum
~r~il/ng
~heor'B,
l, Rmd~&l
4. 86xd¢
Discs
,_
g%ufllbr-kam
P.heor~,
inkerFer'ence,
~.L b l ~ d s
kr'~.ih% 9 c d g e s .
edge.
o.
--0,4
~0
--
'¢ 40
0"4
I
0"5
I
o-g
I
o,7
I
o.s
t
o.s
0"~
0
1.0
11.
model stage--Axial velocity at rotor
t r a i l i n g edge.
2
- 0"4
~40
0,2
J
0'4
I
0.5
I
O.G
I
0.7
I
0.~
I
o.e
,
0
vo
~ =Radius
~
~aol;tas
R = Tip r~.diua
FIG.
-- 0'6,
FIG. 12.
Model stage--Axial velocities at stator
trailing edge.
280
L
i
I
I
I
400~-
1"4
1"4
It
24q
1"2
I'O
200
240 ~
I'O
-0200
C~c4
J
o,e
el_
L
~160
--
0'6
-
0-6
-
Q'4
--
0.'~
I
/ / /
t,O
¢43
6. 5 ~ e r e n c e
DIsGs
m J2o
"5
J9
I. Rad{al Equ;librTum "Theory.
5. 8lade In~,erf'erenca. Discs
a~
Blade
Gentres oF Pr~55ur'~.
0'6
3
~ 8o ~'
.
°I
0'4
80
0'2
I
0"4
-
0.5
1
I
I
I
C~6
0.'7
0'8
0,9
o
I'O
0'4
I
0'5
Radius
FIG. 13.
model stage---Axial velocities far downstream of stator.
I
I
o.~
o.~
0
I
o-s
o.s
~o
Ft~ - Re.dtu5
"~p R~Lchus
FIG. 14.
Model stage--Axial velocity at rotor
trailing edge.
~0
1.4
I
I
I
~4C
200
1,0
~ 160
0.8
o
C× 4 ¢
g
280j
l
i
I.
5.
~12o
Rmdlal ¢Cluilibr~um L h : o r g .
Blad:
"~nh:rPcr¢n¢¢. Ibises
at blachz c ~ n h r ~ oP pressure,
o.
2 6o
7,
~ = 2. ~
(1)
m
-6 =
0.6
2
I 1"4
1.2
oc 20Q
o
I'0
(3160
ab
Bl~de
cenLree
of
8o
g'2
40
= 4.a
-6-
C:~4
Cx- ]
pressure.
"~12C
0.4
0"8
I. Radial Equilibrium Th~or~J.
~. Blade }nLer+'erence. Discs
(il)
"-;
I
240
Cy, I
L
I
--
0-6
--
0"4
--
0"2
-3-
u
2
3 4O
I
0"4
0.5
I
I
I
I
0-5
0"7
0"8
0"9
R"
FIG. 15.
0
~'0
I
0'4
f
I
I
I
I
0'5
0'6
0"7
0'8
O-S
R =
Pad;u~
T(p r'adiue.
Model stage--Axial velocity at stator
trailing edges.
FIG. 16.
O
i'O
Radius
-
Tip Radius
Model stage--Axial velocities far downstream of stator.
,.+
I
.ol
I
I
l
I
1.4
~40
3ZO
~o0
I-0
I
I
I
i
I°G
I
1
1"4
250
~_.~R=C~:o_.~s
acI C~
°O160
0.8
I-2
24-0
L
o
o_
0-G
~J I~0
co
~
1"O
~00
I R - L Radi&l e%ullibrtum ~heo~ 9.
I S_J" For" ~;ncjle ~ L ~ g e .
~0
U6F
0"4
C~o 2
Ca=#
"o
c
o
o
I60
g
Sbe.c~e i n ul~..ima.ke.- ~gea-d 9 "Flow.
r-esull~ f r o n n meLHocJ~ 5, G
lder~LiceLI
0"2
0,6
la0
I. Racli6cl
e%uilibr~urn
I;heor~J
o
> 90
I
2. |sola,~ed d i * c ~ b b l a . d e
(i)~'
= I . $ 3 (ii) u~ = 3-0
I
c~ t
I
I
I /
- 0.4
0-2
40
~om
l
cdqe.
<
-I
!
L~ihh~
c= I
0.5
0 posiLion. I
0'7
O.G
R
Fic. 17.
=
I
i
0"B
0"9
I'O
Ol
0-6
I
0"7
I
0.6
I
0.9
Tip n~dius
FIG. 18.
0
I.O
R~ct~ue
Tip r ~ d l u ~
Model stage--Axial velocities in ultimate
steady flow.
I
O,G
Model s t a g e ~ x i a l velocity at rotor
trailing edge.
R.&M.
No. 3
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