An actuator disc analysis of unsteady supersonic cascade flow

Journal of Sound and Vibration (1983) 88(2), 197-206

AN ACTUATOR

DISC ANALYSIS

SUPERSONIC D. S. WHITEHEAD

OF UNSTEADY

CASCADE AND

M.

R.

FLOW D.

DAVIES

Whittle Laboratory, Cambridge University Engineering Department, Cambridge, England (Received 3 June 1982, and in revised form 9 August 1982)

A rather simple analytical result is derived for the aerodynamic forces and moments acting on a cascade of unloaded flat plates vibrating in a supersonic flow. The principal assumptions are that the axial velocity is subsonic, that the blades are sufficiently closely spaced so that a Mach wave cannot propagate upstream through the cascade, and that the frequency parameter and inter-blade phase angle are both small. The unique incidence condition is used. Results show that bending vibration is always damped, but flutter in pure torsion is always predicted.

1. INTRODUCTION

disc theory was applied by Whitehead [l] to analyze unsteady incompressible flow in cascades. The limitation to small frequency parameters was eased by Tanida and Okazaki [2] who introduced the semi-actuator disc model. The method was extended to subsonic compressible flow by Kaji and Okazaki [3]. The method has further been used for the prediction of stall flutter in supersonic blades by Adamczyk [4] and by Kaji [5]. The present paper is concerned with unloaded cascades of vibrating flat plates in two-dimensional supersonic flow, and therefore small perturbations of a uniform flow are considered. In applying actuator disc theory to this problem there are two fundamental assumptions. The first of these is that the time taken for the fluid to flow through the cascade must be small compared with the time for one oscillation. This is equivalent to assuming that the frequency parameter, A, must be small. The case is considered in which all blades vibrate with the same amplitude, and the phase angle for each blade is p radians ahead of the blade below it. The second fundamental assumption is that p must also be small. Practical interest in flutter prediction now centres on numerical analyses such as those by Verdon and McCune [6] and by Nagashima and Whitehead [7] in which it is not necessary to assume that A and p are small. Nevertheless the limit in which A -+ 0 and p + 0 is of at least academic interest, and it can also be used for checking numerical programs, since results depend on the ratio h/P, and a rather simple analytical result can be obtained. The actuator disc assumptions enable the flow to be considered from two viewpoints. First there is a picture (Figure 1) drawn to a scale comparable with the wavelength of the disturbance, in which the blades are very small and the cascade is equivalent to an actuator disc. The wavelength in the tangential direction is 21&/p, and the wavelength in the flow direction is 27&/A, so that p and A are of the same order of magnitude. Terms of order /3 or A will be said to be of first order. A second picture (Figure 2) is drawn to a scale comparable to the blade chord, and just shows a few of the blades near the origin of Figure 1. Because the frequency Actuator

197 0022460X/83/100197+

10 $03.00/O

@ 1983 Academic Press Inc. (London) Limited,

198

D. S. WHITEHEAD

Figure

Second waves

AND

M. R. D. DAVIES

1. Actuator

Second

order

sheets, edges

shed

disc plane.

vortex from

trolling

/ /

\

/

order pressure from leading edges

First from

Figure

2. Cascade

order trailing

pressure

waves

edges

plane.

parameter, A, based on the blade chord is small, in this second picture the flow may be regarded as quasi-steady. A cascade of flat plates in steady supersonic flow can operate only at zero incidence, as first shown by Kantrowitz [8]. So, just upstream of the blades in Figure 2, it will be assumed that instantaneously the flow comes into the blades with zero incidence. In subsonic flow a Kutta condition would be applied at the trailing edges, but in supersonic flow a wave pattern emanating from the trailing edges can make the flow exit angle different from the blade angle. Quasi-steady waves of this kind are shown in Figure 2. It will be assumed that the axial component of the velocity is always subsonic and that the blades are not so widely spaced that the wave from the trailing edge of one blade passes ahead of the leading edge of the blade above it. These conditions are always

UNSTEADY

SUPERSONIC

CASCADE

199

FLOW

satisfied in practical supersonic blading. Therefore the upward going quasi-steady wave is reflected once from the next blade above, and leaves the cascade parallel to the wave of opposite kind originating at the same trailing edge. Since the waves are of equal magnitude but of opposite kind the average effect is to deflect the flow. Since the deflection in the actuator disc plane is of first order in A or p, the strength of the waves is also of first order. There are in addition some smaller unsteady effects. The circulation round each blade is of first order in A or /?. The rate of change of circulation is therefore additionally proportional to the frequency, and this gives rise to vortex sheets shed from the trailing edges which are of second order in A. Because of these vortex sheets the magnitudes of the two waves shed from each trailing edge are in fact not quite equal, but differ by a quantity of second order in A. This sets up the large scale pressure field downstream of the actuator disc in Figure 1. The large scale pressure field upstream of the actuator disc is set up similarly by second order waves emanating from the leading edges. These are also shown in Figure 2. In the actuator disc plane it is assumed that the vortex sheets are mixed out to give a smooth variation of downstream vorticity, and the pressure waves emanating from the leading and trailing edges are spread out to give a smooth variation of pressure. It is then not necessary to consider the details of the wave patterns. It may perhaps help the reader to visualize the way in which the pressure waves spread out from something looking like steady Mach waves close to the cascade, to a smooth variation far away, if it is remembered that the sources are in fact slightly unsteady and have the classical Bessel function solution showing a discontinuity at the Mach angle which decays with distance from the source, and a perturbation over the whole of the region behind the discontinuities. Kurosaka [9] showed that the unique incidence condition holds in the limit A + 0 when 0 is zero. The actuator disc concept and the unique incidence condition have been used in an analogous way for steady three-dimensional flow by Horlock and Grainger [lo]. 2. ACTUATOR

DISC CONSIDERATIONS

Start by considering axes Ox’y’ which move with the mean velocity of the fluid (U, 6) with Ox’ normal to the cascade. Small pressure waves propagating at an angle 0 at a speed d have a pressure variation given by iexpiw’(t-x’cose/Z-y’sin0/6),

(1)

where w’ is the intrinsic frequency, the frequency observed by an observer moving with the fluid. w’ will be defined in such a way that it may be negative. Note that 8 gives the inclination of the wavefronts, and is not the direction of energy propagation. (A list of notation is given in the Appendix.) In fixed axes Oxy, x = x’+ tit and y = y ’+ i3, the pressure variation is governed by wave numbers k and I in the x and y directions, and an absolute frequency w, so it is p’exp i(wt + kx + ly ) = p’exp i{(o + kz7 + lC)t + kx’ + ly’}.

(2)

Comparing this with equation (1) gives w’=w+kii+lv,

k = -w’ cos $/ii,

I= -w'

sin e/n.

13-5)

Eliminating 8 between equations (4) and (5) gives a2(k2+

12) = wf2 = (w + kc + f6)2.

(6)

200 The

D. S. WHITEHEAD

AND

M. R. D. DAVIES

wave number 1 is related to the interblade phase angle by I = p/s.

(7)

Solving the quadratic equation (6) for k then gives k ={u(w+1~)*67}/(6’-ii2),

T2 = (w + fq2 - Z2(62-

ii2).

(899)

Consider first the range of f for which i2 is positive. This is 2(it b;ing assumed that U 0) f o/{-r? -da -U }. Equation (8) then gives two real roots for k, corresponding to two propagating pressure waves. It will be shown that one of these waves carries energy upstream and the ather wave carries energy downstream. Upstream of the cascade, only the wave carrying energy upstream must be retained, and conversely downstream of the cascade only the wave carrying energy downstream must be retained. One can write for the upstream region kl = {zz(w + lfi) +n7}/(n2

- n2),

(10)

k2 = {fi(w + Ifi) - ~T}/(~’ - E2),

(11)

and for the downstream region

and it is now necessary to choose the correct sign for r to give the correct direction of propagation. In the upstream region, the group velocity in the axial direction, which is also the rate of axial transfer of wave energy, must be negative. This is c,, = G + d cos 8r = c - d2kI/W ‘1 from equation (4), = -&/w;, from equations (3) and (8). If 1 w/{-V - &???} then w i > 0, so the positive value of r is required. In the downstream region cX2= +dr/w h and since w i has the same sign as W; this shows that energy is propagated downstream, away from the cascade as is required. If 1 is in the range for which r2 is negative, then propagating waves cannot exist, and the analysis leading to equations (8) and (9) is not valid. However, by starting from the continuity and momentum equations, it may be shown (see for instance reference [4]) that equations (8) and (9) are still valid, but T is imaginary and k is complex. The solution then represents wave-like disturbances which grow or decay exponentially in the axial direction. The solution which shows decay in the direction away from the cascade is the one which is required. Hence for w/{-r? + e} < I< w/{-V -e} 7 = -i{-(0

+ 16)2+ Z2(a2- zZ~)}~‘~.

(12)

These values of r are tabulated in Table 1. The points where r = 0 are the limiting points for the existence of propagating waves, and are known as the cut-off points. At these points there is no propagation of wave energy in the axial direction, but the energy can propagate in the tangential direction and in subsonic flow this gives rise to a resonance effect. For these pressure waves, the corresponding velocity and density perturbations are given by u”exp i(wt + kx + ly), ~7exp i(wt + kx + ly) and p’exp i(ot + kx + ly), where u”= -(k/pa’)@,

27= -(l/&0$7,

p’= (l/LT2)@.

(13-15)

There is in addition in the region downstream of the cascade a vorticity field. This consists of velocity perturbations which are convected downstream with the mean flow,

201

UNSTEADY SUPERSONICCASCADE FLOW TABLE 1

The expressions for T and T in the three parameter ranges ; it is assumed that 1~ M < 1 /c and s>O

w/i-fi+Jn”-a’)
i
JI -MICE}

~/{-B-J~~-E~}
JI -M%*} c F


Decaying waves

Propagating waves -{(w + IBy - 12(ci2 - ii2)}1’2 T -{(1+2~~+~~)-~*/MZ)1/2

Propagating waves

-i{ - (w + 16)‘+ 12(6’- z?‘)}“~ -i{-(1+2~~+~~)+~~/M~}~‘*

7

with no pressure and density perturbations. fir =p””=o,

+{(w+lfi)2-/Z($-Q*)}‘;Z +{(l+21Ls+K2)-K2/M2}11Z

For these perturbations, o+k,u+lc=O.

k,,& + 117~= 0,

denoted by &Fix v, (16-18)

3. CASCADE CONSIDERATIONS Near the origin of the cascade plane, in which the flow may be regarded as quasi-steady, the fluid velocities are given by u = ii + u”exp iot and v = U + v’exp iwt. The cascade consists of flat plates at an average stagger angle of Cuand initially only torsional oscillation with an amplitude ac’will be considered, so the instantaneous stagger angle is IY= G +a7 exp iwt. The zero incidence condition for supersonic inlet flow gives ul/ul = tan LY. This gives for the steady component (19)

V/c = tan 5, and for the unsteady component,

to first order,

(cl/a) -(cl/c)

=&/sin Cycos 5.

The continuity equation for the cascade is plul = p2u2. The unsteady component this to first order is (p’zld

+

GZI~)

=

(P’llP)

+

$1/U).

(20) of (21)

The energy equation requires conservation across the cascade of stagnation enthalpy, which is h + $(u’ + u*). For small perturbations, one has conservation of dh+udu+vdv=(dp/p)+Tds+udu+vdv. Since the flow is isentropic, ds = 0 and the unsteady component to first order gives (p’2/~)+~~*+b~*=(p”l/p)+~u”~+vu”~.

i22)

The axial force per unit span acting on the blade at the origin of the cascade plane is -p~u$. Taking the first order component of this, and simplifying with use of equations (21) and (22) gives

F,/S =p~ +p~u:--p~

r’, =Spa(U”Z-t?&

(23)

The tangential force per unit span is Fy = -S&7(& - 271).

(24)

202

D. S. WHITEHEAD

AND

M. R. D. DAVIES

Equations (23) and (24) show that there is no drag force parallel to the mean direction of the blade chord. This condition might have been used instead of the energy equation. The lift force normal to the mean direction of the blade chord is a=Jm,

E = -psw (62 - &),

(25,26)

being the absolute fluid velocity. This lift force acts at a point midway between the trailing edge and the point where a Mach wave from the trailing edge of the next blade below hits the blade at the origin. This is illustrated in Figure 2. This point of action of the lift force is a distance ${sin Cr- &?? cos (Y}upstream from the trailing edge. 4. ELIMINATION Using equations obtained directly:

(13), (14), (15) and (20) enables the inlet flow perturbations cl= 12,

z?l= kZ,

2 = -&w*/(k,v

&= -pw ;z,

- lfi).

to be (27-30)

Downstream of the cascade there are both pressure waves and vorticity waves, so the complete velocity perturbations are zZ*= z& + l&,,

l?*= l& + I&,,

(31932)

where the suffix 2p indicates the downstream pressure waves. Equations (13), (14), (15), (17), (20), (21), (31) and (32) may then be used to solve for the unknowns &,, &,, ii*, r&,, &,, &, p”*and b2. The results are &, = kZYJ L&, =koZ(kl-k&i2-ii2)/Y2,

2i& = IZYJ Y*,

(33934)

62, =-k,wZ(kl-k,)(n2--*)/Y2,

(35936)

Y2,

62 = -pw;zYJ Y1=ii21”+(kld2-~;~)(I~-k,~),

Y2,

(37)

Y2 = ii21w + (kzd2-o$i)(la

- k,u).

(3% 39)

Equation (25) then gives for the lift force on the blade at the origin E = -~S~Z(k~-k&i2-n2)(Z2~2+2Zijw It is convenient

to express this non-dimensionally. I CF, =

F

m=

p = pKJ/ws = G/w,

+w2)/(fiY2).

(40)

With T = T/W,

(41)

2MT(p * + ~s/..L + l)( 1 -M2c2) [M2c(s+~)-~c++sT][c~

-(p

+s)MT]’

(42)

where s = sin 6 and c = cos E. Expressions for T are given in Table 1. 5. DISCUSSION OF RESULTS

Some examples of numerical results are shown in Figures 3(a) and (b), in which C,,, the force coefficient due to torsion, is plotted against p, which is a non-dimensional ratio of phase angle to frequency parameter. C,, is always real over the range of (u for which propagating waves occur, since all quantities in equation (42) are then real. C is zero at the cut-off points where T = 0. This occurs where p = {-&%*M(l -M*c~)~‘~}/(M*- 1). At these points the upstream and downstream pressure waves are identical, and they carry energy in a purely tangential direction. The blades

UNSTEADY

-2.5

SUPERSONIC

CASCADE

3. (a) Real and (b) imaginary

parts of C,,

FLOW

1I

Figure

us. p for & = 4.5”.

move to match the wave, like small corks on the surface of a water wave, and have no force on them. The circulation is therefore zero and there is no downstream vorticity wave. C,, is also zero when (1 -M2c2) = 0, corresponding to sonic axial velocity, but this is at the limit of applicability of the theory. The case Cu= 45”, A4 = 1.4 in Figures 3(a) and (b) is close to this condition. There is no point where CF, becomes infinite. Equations (30) and (40) show that this would occur if (k10 - G) or Y2 become zero, but this does not happen. There are points given by cc.= MS/(-M f l), where (k26 - lzT) and Y1 become zero, but the only thing that is special about these points is that there is then no downstream pressure wave, as shown by equations (33), (34) and (37). Comparison may be made with the quasi-steady results given by Kurosaka [9] and by Nagashima and Whitehead [7]. In these results the limit o + 0 is taken. If the further

204

D. S. WHITEHEAD

limit 0 + 0 is subsequently

AND

M. R. D. DAVIES

taken, the result is valid when p/w or CLis large, and is

c,, = -(2/B)(s-cB)+(2/~.B3)(s-CB)*+0(1/EL2), where B2 = M* - 1. This agrees with the limit of equation (42) as CC+ cc to terms of order (l/p). The two types of theory are thus completely compatible over the range where both are valid. However, this range does not include values of p for which decaying waves occur, Comparison has also been made with the program LINSUP described in the paper by Nagashima and Whitehead [7]. This program fails close to the cut-off points, and also close to three other points at which mathematical singularities having no physical significance occur. These regions of failure are normally very narrow, but if the frequency parameter and phase angle are both small they occupy large sections of the range of values of ,x. The smallest value of A for which the program covers a reasonable range of p was found to be 0.05, and the results for this case are compared with the actuator disc solution in Figures 4(a) and (b). Agreement for the real part of the force coefficient shown in Figure 4(a) is good. The imaginary part shown in Figure 4(b) shows less good agreement, but nevertheless this agreement is considered to be entirely satisfactory after considering the not very small value of A used in the numerical solution. 0

(

i ‘/ -I-

”

-2-

1 1

I

z 0)

[r

-

_

.3-

-4-

i

-5I --b

--3

I

(b)

3

-‘I

I

I

--5

--c

-1

I

I

I

”

I

I

I ^

I _

L

3

II

1

00 B

2c I I-t 2 E -

LO n

d &odO

CIA

0

0

rJo

01

1

-1 1

-2 1 -31

' -6

,M=I.2 -5

-4

, -3

I -I

-2

0

I I

I 2

I 3

I 4

P

Figure 4. (a) Real and (b) imaginary parts of C,, us. p for E = 60“, comparison with points from LINSUP, forA=0.0.5:U,M=1.2;A,kf=1.4.

UNSTEADY

SUPERSONIC

CASCADE

FLOW

205

The moment (anticlockwise positive) about a torsional axis distant n from the trailing edge is given by {n -$?(s-cl3)}g The result may also be applied to bending vibration of the blades, since if they have an upward velocity due to their vibration given by 4, this is equivalent to a torsional displacement given by q”/W,which gives the same upwash velocity normal to the surface of the blade. This is true since @/w is large compared with the blade chord. Damping of pure torsional vibration is governed by the imaginary part of the moment coefficient. Over the regimes where propagating waves exist, the force and moment coefficients are purely real, so there is no aerodynamic damping. Over the regime where decaying waves exist, Figures 3(b) and 4(b) show that the imaginary part of the force and moment coefficients takes both positive and negative values, so wherever the torsional axis may be it is always possible to find a phase angle for which the damping is negative. Pure torsional flutter is therefore always predicted in the absence of mechanical damping. But if the torsional axis can be kept near the centre of pressure towards the back of the blade the exciting moment will be small. Damping of pure bending vibration is governed by the real part of the force coefficient, and a positive value corresponds to excitation. Figures 3(a) and 4(a) show that the real part of the coefficient is always negative, so that bending vibration is always damped. Because of the limitation to small frequency parameters and small phase angles, actuator disc theory is not of much use for the prediction of the vibration performance of real blades. The value of the theory is thought to lie in the physical insight it gives into the way in which the various waves are generated and propagate, and into the use of the unique incidence condition. It is also of value for checking the operation of numerical programs of more general applicability, in what for them is a rather tricky limiting

case.

6. CONCLUSIONS A rather simple analytical result for C,, has been derived. C,, is zero at the cut-off points, and where the axial velocity becomes sonic, but never infinite. The result is compatible with previous quasi-steady theories, which apply in the limit F + 03. The result is also compatible with a numerical solution. The results show that bending vibration is always damped, but that flutter in pure torsion is always predicted. ACKNOWLEDGMENT This work was done under a contract from Rolls-Royce, and grateful acknowledgment is made to Rolls-Royce for permission to publish this paper. REFERENCES 1. D. S. WHITEHEAD 1959 Proceedings of the Institution of Mechanical Engineers 173, 555-574. Vibration of cascade blades treated by actuator disc methods. 2. Y. TANIDA and T. OKAZAKI 1963 Bulletin of the Japan Society of Mechanical Engineers 6, 744-7.57. Stall flutter in cascade. 3. S. KAJI and T. OKAZAKI 1972 Transactions of the Japan Society of Mechanical Engineers 38, 1023-1033. Cascade flutter in compressible flow. 4. J. J. ADAMCZYK 1978 NASA TP 1345. Analysis of supersonic stall bending flutter in axial-flow compressor by actuator disc theory. 5. S. KAJI 1980 Proceedings of the Institution of Mechanical Engineers C 282180, 209-214. Stall flutter of linear cascade in compressible flow.

206

D. S.WHITEHEAD

AND M. R. D. DAVIES

6. J. M. VERDON and J.E. MCCUNE 1975 American Institute of Aeronautics and Astronautics Journal 13, 193-201. Unsteady supersonic cascade in subsonic axial flow. 7. T. NAGASHIMA and D. S.WHITEHEAD 1977 R. & M. 3811. Linearized supersonic unsteady flow in cascades. 8. A. KANTROWITZ 1946 NACA Report 974. The supersonic axial-flow compressor. 9. M. KUROSAKA 1973 Transactions of the American Society of Mechanical Engineers A96, 13-31. On the unsteady supersonic cascade with a subsonic leading edge. An exact first order theory. Parts I and II. 10. J. H. HORLOCK and C. F: GRAINGER 1980 Transactions of the American Society of Mechanical Engineers, Journal of Fluids Engineering 102, 330-337. Linearized solutions for the supersonic flow through turbomachinery blade rows using actuator disc theory. APPENDIX: a

speed of sound cos cr axial group velocity enthalpy

C CX

h

i k, 1 p 9 S t 4

V

W x9

Y

B c C Fa F M s T Y Z ; 7 e A CL P 7 w _

,

Jw

axial and tangential wave numbers pressure velocity due to bending vibration sin 5 time axial and tangential velocities absolute velocity axial and tangential axes

J(M*-1)

blade chord force coefficient due to torsion, F/$%*6 force on blade Mach number, +/a blade spacing r/o defined by equations (38) and (39) defined by equation (30) stagger angle of blades phase angle between blades distance of torsional axis from trailing edge wave angle frequency parameter, WC/W pG/,S density defined in Table 1 radian frequency indicates mean value indicates perturbation indicates relative to moving axes

Subscripts

1 2 P V

indicates indicates indicates indicates

value upstream of cascade value downstream of cascade pressure wave vorticity wave

NOTATION