ComputationalMechanics(1989)4, 11-30
Computational Mechanics © Springer-Verlag1989
Unsteady mixed flows with in-passage shocks K.K. Puff Universityof Maine, Dept. of Mathematics,Orono, ME 04469, USA Abstract. The problem of blade flutter in an infinite cascade of thin, sinusoidalblades is explored in the frameworkof a linearized theory. The model admits normal shocks in the blade-passages.Closedform analytical solutionsare formed in the regionsof both the supersonicand subsonicflows.
1 Introduction
The phenomenon of dynamic aeroelastic instability, known as flutter, is one of the most serious problems encountered by a design engineer in the development of gas-turbine-engines. Its accurate prediction entails the computation of unsteady aerodynamic loads on the blades in an incremental annulus. The focus in the following discussion is on understanding the flutter phenomenon by computing the forces and the moments on the blades in an infinite, rectilinear, two-dimensional cascade which, as is customary, is regarded here as an adequate representation for the blade row in an annulus (Fig. 1 for the definition sketch). Modern fans and compressors operate with supersonic velocities relative to the blades. The axial velocities entering the blade row, however, are usually subsonic. As a result the leading-edge Mach waves extend all the way to infinity upstream, leaving no region ahead of the cascade undisturbed. Moreover, as is evidenced by the experimental investigations of Miller and Bailey (1971) and Strazisar and Chima (1980) these flow conditions exhibit an in-passage shock structure. Over most of the range of operating conditions, these shocks are nearly normal and are discernible in the tip region or in the trailing edge region or even in both. This leads to the requirement that a viable model must contain a strong interaction regime characterized by the presence of in-passage shocks. With a supersonic flow ahead and, therefore, a subsonic flow behind the shock a realistic model should be a mixed flow model relative to the blades. Over the past few years, various numerical and analytical models have been developed to study the phenomenon, both for an isolated airfoil as well as for a cascade. The numerical treatments have led to finite difference schemes that deal with the complete set of Euler's equations (Magnus and Yoshihara 1975), the full potential equations (Isogi 1977) and the transonic small disturbance equation (Ballhaus and Goorinjan 1977). In the flutter analysis, however, one is generally interested in blades and wings executing small oscillations within the framework of a linearized theory. Several authors have attempted this but the assumed basic flow about which the perturbations are constructed, is entirely supersonic and does not show the mixed feature discussed above. The finite difference scheme adopted by Weatherhill, Ehlers and Sebastian (1975) allows for the presence of the shocks but they do not properly account for their displacement in the ensuing motion. Eckhaus (1959) approached the problem analytically for a single two-dimensional airfoil. He replaced the steady disturbance by a single normal shock discontinuity separating two uniform regimes. He obtained the correct jump conditions for the first harmonic disturbances at the undisturbed shock as well as the induced shock displacement and the pressure at the shock. Williams (1979) considered the corresponding problem for a thin three-dimensional planar wing oscillating in a
12
Computational Mechanics 4 (1989)
/ I //Moch lines
111111,,I'
n=2
/1~11~+ ~Ill /// I
/)=1 UI>GI
0:0 I
/7=-7 /7=Z
I
I
ii / 1
+
1-s+
~'2 [71
I
I
___~__ 2
..... ~_.
U2
(I 2
Figs. 1 and 2. 1 Definition sketch. 2 Basic flow configuration
transonic flow with embedded shocks. However, as discussed above in reference to modern fans and compressors, the flow ahead of a cascade is completely disturbed which is not the case of an isolated airfoil. The two flow fields are different and so the problem has to be addressed afresh for the case of a cascade. This was done in a brilliant attempt by Goldstein, Braun and Adamczyk (1977). Considering only the normal shocks in the tip region, they contented that an appropriate basic flow about which the perturbations should be constructed, is correctly represented by a step function (depicted in Fig. 2), wherein the horizontal lines are the so-called skip lines and the vertical lines represent the norma! shocks. They solved the Euler's equations, linearized about the above basic flow. As the steady and unsteady aerodynamics decouple in the linearized theory and in view of the fact that the blade thickness, camber and the mean angle of attack affect only the steady flow, they addressed themselves to the unsteady flow in a cascade where the blades were replaced by zero thickness flat plates at zero angle of incidence. The assumed fluid is an inviscid, non-heat conducting ideal gas with constant specific heats both upstream and downstream. The flow ahead of the shock is assumed irrotational and isentropic the presence of vorticity is admitted in the downstream side of the shock. The viewpoint adopted in this paper is that the plates could assume a form with an arbitrary profile. However such a profile can be expanded into a Fourier Series of oscillatory functions. The objective here, therefore, is to carry out the analysis for a sinusoidal plate geometry. Invoking the superposition principle for linear problems, one can easily generalize the above results for the case of an arbitrary plate profile. The rest of the premises of their paper are admitted in this paper. Obviously some of their results will be applicable in the present case. They will be directly quoted with reference "GBA" to their paper. I wish to express my heartiest thanks to Dr. M. Goldstein, Chief Scientist at Lewis Research Laboratory, Cleveland for suggesting this investigation as well as the benefit of several discussions I had with him and with Drs. W. Braun and J. Adamczyk.
2 Formulation Let the steady basic flow defined above be characterized by the densities 0i, local speeds of sound ai and the flow velocities relative to the blades, Oi where i = 1 or 2 according as the subscript 'i' refers to the quantities related to the supersonic flow upstream or to the subsonic flow downstream of the shocks. All variables appearing in the equations governing the motion have been nondimensionalized with reference to the chord length c, the density Q1 and the velocity 01. Furthermore, the linearization of the equations describing the motion in the region R i is based on the quantities Oi, a;, O; appropriate to the flow in R~. To study the unsteady flow, we allow the blades to execute a harmonic oscillation whose amplitude is small enough to be consistent with the linearized problem. As such the resulting motion of the fluid also has harmonic time dependence. Upstream of the shock, the dimensionless velocity potential 41 (X, y, t) = ~b1(x, y) exp ( - icol t)
must satisfy the equation,
(2.1)
13
K.K. Puri: Unsteady mixed flows with in-passage shocks (8 2
82
~
+ 2iMl
)
kl + 4k1 el(x,y)=0
(2.2)
(.OC. where co1 = ~ is twice the dimensionless reduced frequency, co is the frequency of the motion, U1 M1-
91 > 1, is the free stream Mach number of the supersonic flow, fil = ~ al
1 and
kl - c°l M1. Once 41 is determined, the velocity vector ql = (Ul, Vl) is given by
ql = Vq~l = V!~l exp ( - ico 1t) = (U1, V1) exp (-- ico1t)
(2.3)
and using the m o m e n t u m equation in x direction, we can write the pressure fluctuation, as Pl = P1 exp ( - icolt) =
(2.4)
(°)
ic01 - ~xx 41 exp ( - icoit)
(2.5)
The equation for the conservation of mass across the steady shock is given by
Q1 ~-~1= Q2 ~-~2
(2.6)
Using it, the equations of motion for the subsonic flow downstream of the shock are reduced to
~tt + ~x q2 = -- Vp2
(2.7)
and ~ t -+-
(2.8)
P 2 = - - g " q2
Here M2 = __U2< 1 is the mean steady flow Mach number downstream of the shock and is related a2 to Mt by
M2=I(M2+#221)/(#221M2+l)I
(2.9)
where # = Cp/C~ is the ratio of the specific heats. As mentioned above, the motion of the shock can generate vorticity and entropy gradients in R2. Thus the velocity q2 = (u> v2) can be decomposed into an irrotational field characterized by the velocity potential (2.10)
q52 (x, y, t) = q~2(x, y) exp ( - ico2 t) and a solenoidal field characterized by the stream function,
(2.11)
~P2(x, y, t) = gt2 (x, y) exp ( - ico2 t) so that q2 = Vq~2+ V x (0,0,~2)
(2.12)
as established by Goldstein (1976). This yields the velocity q2 = (u2, v2) = (U2, 112)exp ( - i~2 t) and pressure perturbation P2 = P2 exp ( - leo2 t), with
u2= \ x-x + ay/
v2 = \ ay
ax /
14
Computational Mechanics 4 (1989)
The functions 42, g~2 are governed by,
( 82~y2-[-fl22~2--2iM2f12k28x
~-b~x f14k2)42=0
(2.13)
(ico2 - ~x) }It2= 0
(2.14)
coc where, //2 = 1 - M 2, co2 = ~ - , is the reduced frequency based on the mean velocity Uz of the U2 (-o2M2 subsonic regime and k 2 - - -
The above equations are to be solved subject to certain boundary conditions on the blades which are assumed to be infinitesimally thin. As explained in the introduction, we assume that the blade displacement is, W0 (x; a) = H 0 (a) exp (iax)
(2.15)
and eventually we shall deduce the results for the periodic displacement 1 y = B 0 sin ~ (x -- do) = ~ (W0 (x, 7c) - W0 (x, - ~c))
(2.16)
with, B0 = H0 exp (iado), ao, do are real constants. In order to simplify the calculations we also require that the amplitude H0 ~ the amplitude of the fluid motion so that the boundary condition may be applied at the mean positions of the blades, viz.,
y = ns,
( n - 1)s + < x < ( n - 1)s + + 1,
n = 0, _+ 1, _+ 2, ...
where s is the gap distance measured normal to the chord and s + is the stagger distance measured parallel to the chord. Also, the unsteady wakes, to the order of linearization, are replaced by vortex sheets emanating from the trailing edges of the blades and lying along the lines,
y=ns,
x>s+(n-1)+l,
n=0,_
1,_2,...
The mean shock wave positions are along the line segments,
x = n s +,
ns
n=0,_l,_2,...
Finally, following Lane (1956) we require that all blades oscillate harmonically with the same amplitude and a constant but arbitrary interblade phase angle o-. Thus
Vi(x, y)
Vi(x Ac-ns +, y + ns) = e i~rn
(2.17)
for - s + < x < l - s + , y = 0 + , n = 0 , _ 1 , _+ 2, ..., where y = 0 -+ denote the limit a s y ~ 0 f r o m above or below respectively. The above condition determines the upwash on the n th blade in terms of that on the zero th blade. The upwash on the latter is determined from the kinematic conditions,
V2(x, y) = -
U2 ico2_
Ux
Wo (x; c~) for
- s+ < x <
1-s+,y
= 0-
(2.18)
and
Vl(x,y)=-
icol-
W0(x;~)
for
y=O+,-s+
(2.19)
The pressure and the upwash must be continuous across the wake. The discontinuities in the axial velocities are admitted in order to satisfy the Kutta condition, which, when applied at the trailing edges will render the solution unique.
K.K. Puri: Unsteadymixedflowswith in-passageshocks
15
The flow in the supersonic region is connected to that in the subsonic region by the j u m p conditions across the shock waves. These are calculated in GBA and are reproduced therefrom, as, 1 + M~ P2 + UI + 2M~ 2
V2--
1 + M21 - - M2
P1
( M 2 + ~--1 #2~el - (# + 1) (M1M2~2 (~2~J2 \ fllfl2 // \ - ~ # + 1 ,-1] V
eP2 ~yy
forx=ns+,sn
(2.20) °~2 g*2)
(2.21)
+_1, 4 - 2 , . . .
Finally we require that there be only outward propogating disturbances at large distances from the cascade. As a result no diturbance will propogate upstream of the shock-relative to the blade fixed coordinates.
3 Solution of the problem in the supersonic regime As is frequently done, (Noble 1958), in boundary value problems posed in a frictionless medium, we allow a small damping in the above formulation by regarding k 1, k2 and a as complex quantities with positive imaginary parts ~1, e2, ~3 respectively. They are set equal to zero in the final solution. This device helps avoiding singular integrals, and allows one to replace the radiation condition to be satisfied at oo by a simple boundedness condition. Also the condition (2.17) is assumed to hold for all the physical unknowns. It will be a simple matter to verify that the final solution satisfies all the boundary conditions. Since the disturbance can't propogate upstream in a supersonic flow, the motion of the fluid in region R2 does not affect that in R1. As such, following GBA, we may solve for the supersonic part of the flow by solving the problem of oscillating cascade of semi-infinite blades extending to + oo. We write
~,= ~ ~')
(3.1)
Y / ~ - - O0
where P1n) is the disturbanc_e field generated by the n -th blade and its spatial evolution is determined by the Eq. (2.2) above. Using the technique of separation of variables on it and invoking the linearity of the problem, we obtain q~tn) _ sgn (y~) co+ ia, 2rt S F(nl)(c~) e x p { - i(o~ - M l k l ) S
+}
-oo+ial
x exp{ -- i[(c~ - M l k l ) X ,
-
fll~l ly.I] } dc~
(3.2)
where x , = x - n s + and yn = y - ns, 71 =- ~ k2 which is a single valued function of e with the branch cut shown in the Fig. 3. The contour of integration is taken to be Im~ -- 51 = M1 a~. It will be clear from below that this crucial choice allows us to sum up the series in (3.1) with ~1~) given by the Eq. (3.2). Sgn (y,) -- + 1, - 1 according as the blade y, is approached from above or below. It ensures a pressure discontinuity across each blade and the continuity of the upwash velocity Im~
Integration poth
d>~'
1
p
Branch cut for ~
R-ec~
Fig. 3. Branchcut for 71= ~
k~
16
C o m p u t a t i o n a l M e c h a n i c s 4 (1989)
everywhere. The latter will be verified shortly and the former is guaranteed with the demonstration of a jump discontinuity in the velocity potential. The same across then rtth blade is [~1 (x)]~ = lim [~bI (x, ns + e) - ~1 (x, rIs - -
~)] = [ ~ t n) (X)]n
e-+0 1 oo+i&
= -
F(nl)(~)exp - {i(Mlkl - ~)}x,_ldc~ # 0.
S
(3.3)
7Cl -- oo + i3~
Now the assumed periodicity of various unknowns entails, F (1)(cx) = ein~'F~1)(~)
(3.4)
which together with (3.1) and (3.2) yields, 1 oo+i61 ~1 = ~ ~ F~l)(OOAl(~,y)exp- {i(cz-mlkl)Xl}dO~
(3.5)
where
-~1 = x + s + a n d
Al(e,y)=
~
n=--O0
(sgn y,)exp{i[n(a + s+c~-s+ M l k l ) + fllhlY, l)]}
(3.6)
Then, on the contour Im(c~-Mlkl)--O, in view of ImT1 > O, the common ratio of the above geometric series is less than one in absolute value. " A I ( ~ ' Y)= ~Ll~exp{i(Af+fl171Y)}~lnA11--+ exp{i(A{--[ImTlY)}]sinAi ~
for
O<_y
(3.7)
where 1
A +--
(3.8)
( a - M l k l s+±fllT1 s)
With this, the upwash velocity, Oq31
Vl(X, y) -
1
oo+ielM1
~y - 2~
S --
oo
~A 1
F~I)(~) 0 y e x p { - i(a - Mlkl))~l}dC~
(3.9)
+ ielM1
can be easily seen to be continuous everywhere. The required continuity of the rest of the physical quantities along the lines
Ln:y=ns,-oc
+,
n=0,_+l,_+2,...
follows from the continuity of the velocity potential. Observing that the latter quantity is periodic and so, as a consequence, its continuity on L 0 implies that on Ln for all n, this requirement is satisfied from the Eqs. (3.3) and (3.4) provided, oo +i31
FI1) (~) e x p { - i(c~ - Mlkl))?l} d0~ = 0
for
)71 < 0
(3.10)
-oo+i~l
Finally, to satisfy the kinematic condition on the upwash velocity, we have from the Eq. (2.19) 1 ~+ia
2--x
~
F~l)(00Xl(0q0)exp{ - i ( e - M i k l ) J ? l } d e
-- ao +i~1
=
(
°)
- icol + -~x H°exp(iax) 5/I 1
where xl (~, Y) = - ~y
for
xl > 0
(3.11)
K.K. Puri: Unsteady mixed flows with in-passage shocks
17
The Eqs. (3.10) and (3.11) determine the function F~1) (e). The above equations are the analogues of the Eqs. (3.11), (3.12) in GBA; the main difference being the different expression on the right hand side of our Eq. (3.11) above. Following their implementation of the Wiener-Hopf technique, we solve the integral Eqs. (3.10), (3.11) to obtain, F~I)(c0 =
L1 zi- ( M l k 1 - a) (a + e - m l kl) z{ (c0
(3.12)
where L1 = H0(col - a)exp ( - ias +) ~1(~,0)
(3.13)
zi ~ (c~) _ ~ A
(3.14)
y=0
The expressions for ~+ (e), obtained in GBA, are:
)~-((~)=ez(a)fllYlsin(fllYlS)
v~-v~n=--oO I-I
~nn)
V+V~
11[ ( )1-1 rl
oO
1-~-~2
(3.15)
n4~O
~F (~) = eZ(a)
1-
(3.16)
o~
rT= - - ~ \
1 x ( e ) = - - i a ( s + - ills) 2
(3.17)
d ~ - = (s~- -
(3.18)
fl2s2)1/2
v+ = F(n1) -S-+ + S i l 1 ([r())] 2 -- k2) 1/2, d~- - d ?
(3.19)
F/ = 0, + 1, + 2 , . . . -
(3.20)
F(~1) = ( 2 n n + M l k l s+ - a ) / d {
The insertion of the expression (3.12) for F~1)(e) in the integral in Eq. (3.5) yields ~ ~+i& S z1(Mlk1 - a)Al(~,y)exp{~bl_ 2L1
--oo+i31
i(o~- M l k l ) 4 l } d e
(3.21)
(a + ~ - Mlkl) z~ (~)
Our primary interest is to compute pressures P1 (x, 0) +_ on the blade surfaces as well as the axial velocity U1 (0, y) and the pressure P1 (0, y) on the shock. The latter two quantities are used in the Eqs. (2.20) and (2.21) to find the boundary conditions on the shock for the subsonic regime. Substituting for ~b1(x, 0 + ) from (3.21) in the Eq. (2.4) we get
(°)
PI(x,0 + ) = i co1 - ~xx ~b1(x'0+) i L ~+ia A1 ( ~ , 0 ) ~
( M l k l - a) (col + ~ - M l k l ) e x p { - i(~ - Mlka)~?l}
-
+
a -
(3.22)
Mlkl)
The singularities of the integrand are poles which are the zeros of the denominator. These are given by c~= M1 kl - a and by the zeros of the functions el (e, 0) and zi- (~). The zeros of the latter two A1 (c~,0) functions are investigated in GBA. They are, 20)=
kZ+\flls]
for
n = 0, _+ l, _ 2, ...
(3.23)
18
Computational Mechanics 4 (1989)
and e = v+ defined in the Eqs. (3.19) and (3.20). The zeros 2(1), v7 and ~ = Mlkl line Im ~ < fil and therefore, are in the lower half plane. Those denoted by v+ lie plane. Examining the integrand for large Ic¢1, it is easy to see that, the c o u n t o u r the upper or lower half plane according as xe ( - s +, - Sill ) or xe ( - sill, 0). Using we can invoke the residue theorem to write P I ( X , 0 -t-) =
- a lie below the in the upper half m a y be closed in this information,
filiS1 -~- fi2is2, i = 1,2, ...
(3.24)
where the fi's are the Kronecker's deltas and
iLi
S1
oO S~-~ ~¢i- ( M l k l nzz~= = 0 [zF (v+)] '
a)
V+ -- kl/M~ v++a-Mlkl
exp { - i(v + -Mlkl)21}
_ S+ < X <
d{-r(~l)-v+s +
S~1
(3.25)
The ' on the first term in the denominator denotes derivative with respect to e and oO
s2 = exp ( i M 1k I x) n=~0{ T + exp ( - i 2 (n1)x) + T 2 exp (i 2 (n0 x) } +
L1 (a - coi) (51 sins(51
exp (ia x) { exp (ia) - cos (51s exp (is + a)}
(3.26)
where (51 = [2~1(e) 1 - a) 2 - a2]1/2 ~ (_+2(~ t) - kl/M1)_ _~ T~- = -- L( -'[- ~(n1) -- M1 kl + a ) / Q +
Q+= ~¢F(Mlkl - a)
-s/
l
[exp{i(a - nrc} - e x p { i s + ( M l k 1 + 2(1)}] S("[- )~(1)) (1 +
241(_ t_ )~(1))
6n,0)[1~
These series converge as 0 ( ! ) . To accelerate the convergence we find the asymptotic approximation of the terms of the above series for large n. These terms are added and subtracted from the above series so that
S1
_LliS~ ~ ~F(Mlkl-a)
kn=o
[~i- (v+)] '
v+-klM1
exp{-i(v+-Mlkl)21}
v+ - k 1M 1 + a
di~ F(~0 - v+ s +
i(1 - fin, O) ~F (Mlkl - a) exp [ - i ~ df-F(~l)
2rcnsfll
zr(oo)
(S~Z-~l
a'e pC
°}
Mlkl
(x + s+) 1 x+s + ']]_s + < x-sill
s+- lS)l
(3.27)
and
s2 = exp(iMlklx)
T.+ exp ( - i2(~l)x) + T~-(i2(~l)x)
~i-(Mlkl).(-l) n . . (nr~x))] 2i (1 - fin,0) exp(t~r)sm -- ~fll ~--) ~t"l n i ~i- ( M l k l - a) L1 x exp{ i(o- + MlklX)}
(oc)
+
Z 1 (a -- (-Ol)
(51 sins(51
s
exp (ix a) (exp (ia) - c o s (51S exp (is + a))
-- s]? I < x < 0
(3.28)
19
K.K. Puff: Unsteady mixed flows with in-passage shocks
In view of Eqs. (2.3) and (2.4), the determination of U I (0,y) and P1 (0,y) requires the evaluation of 41 (0, y) and ~
(0, y). The latter are determined from the integral on the right of Eq. (3.21). Setting
c~= i6 + R exp(i0) in the integrand we see that it ~ 0 as R ~ ec in the lower half plane. We can thus close the contour in the lower half plane. The integrand being an even function of ?l has no branch point singularity. The only singularities are the poles which are at,
and
~=Mlkl-a
n2 2 ]1/2
c~n=+2(~)=+
k~+~s2
for
n = O, _+ l, +_ 2 . . . .
Calculating the residues at these points and invoking the residue theorem again, we obtain ~bl(0'Y) = i + L1
Q+ 20)-Mlkl+a COS(/~1 y
+ _ 2(I) Q 7 -Mlkl+a
cos--
s
exp (io-) - cost51 (y -- s) exp(ias +) eS1sinsc51
(3.29)
Substituting the above expression on the right of the Eq. (2.4), we have,
(k,) an+ t'~(1)- MI ) P,(O,y) = -
+
+ L l ( a - COl). = __LoR.COS
an
n~zy
t
n
M
,)1 cos
7
cose51 y exp (ia) - coso51 ( y -- s) exp(ias +) 051sine51 s - L1 (o 1 - a )
ei°cos (51y -- e ias+cos ~1 (Y, s) o51sin ~1 S
(3.30)
Similarly the Eq. (2.3) yields U1 (0, y) = £ Qn cos _n~y n=0 S
_
L1 a
C O S 03 1 y
exp (ia) - cos e51 (y s) exp (ias +) (D1sin s 051
(3.31)
We are now in a position to construct the solution in the subsonic regime.
4 Solution of the problem in the subsonic regime
We need to solve the Eqs. (2.13) and (2.14), subject to the following conditions (Fig. 4). (i) Jump conditions on the shocks as given by the Eqs. (2.20), (2.21), (3.29) and (3.30). (ii) The kinematics condition (2.18) to be satisfied on the blades. (iii) The solution must satisfy the periodicity conditions. (iv) The pressures and the upwash velocity is to be continuous across the wakes which are approximated, to the leading order, by vortex sheets emanating from the trailing edges. (v) The Kutta condition is to be satisfied at the trailing edges. (vi) Only the outward propagating disturbances exist far down stream. It is clear from the earlier discussion, the solution consists of two parts, namely, the vortical solution ~/2(X, y) of the Eq. (2.14) and the acoustic solution ~2(x,y) of the Eq. (2.13). It can be easily seen that, 7t2(x,y) = g?(y) exp(ie)2x )
for
ns < y < (n + 1)s,
n = 0, _ 1, _ 2 . . . .
where the periodicity requirement on it entails, f2(y + ns) = e x p { i n ( a - co2s+)}f2(y)
for
0 < y < s,
n = 0, + 1, ___2 . . . .
(4.1)
20
Computational Mechanics 4 (1989)
We shall modify this solution so that its total contribution to the boundary conditions is zero. Of course, it must have the required outgoing wave behavior at downstream infinity. Consider the reduced potential function = (2(s(n + 1)) coshc~z(y - ns) - (2(ns) coshco2[s(n + 1) - y ] exp (i~o2x)
isinh~o2s ns < y < (n + 1)s,
n = 0, _ 1, 4- 2, ...
(4.2)
This satisfies the potential Eq. (2.13), has an outward going wave behavior at the downstream infinity and it satisfies the periodicity condition. The stream function having the effect on the motion equivalent to that of the velocity potential (4.2) is, (2(s(n + 1)) sinhco2(y - ns) + (2(ns)sinh~oz[s(n + 1) -y]exp(ic#2x )
=
sinh c~2s n=0,_
ns
1,+2,...
(4.3)
We write ~P2 = ~2 - q3. Then it can be easily seen, that, the function ~P2satisfies the vortical Eq. (2.14), has zero upwash velocity on the blades, has no jump in pressure, or upwash velocity across the wakes, and has the required behavior at + oo. On the other hand the effect of ~ is the same as that of q~in the Eq. (4.2) and hence can be regarded as incorporated in the determination of the acoustical component 4 2 of the solution. We therefore regard that the vortical solution is given by ~t2 = (2 (y) exp (/co2x) - v~,
(4.4)
(the 'bar' on ~2 has been removed for convenience) and note that it has no contribution to the boundary conditions so that the solution q~2 must satisfy all the boundary conditions, listed above. In order to solve for the latter, we first allow the plates to be extended upstream to oo as, (Fig. 5), and assume that the mean flow is uniform at the downstream Mach number -4/2. The construction then consists of two parts, namely, q~l), ~b~2). The part, q~l), satisfies the kinematic boundary condition (2.18) on the blades, the Kutta condition at the trailing edges and has the correct wake jump conditions but has no help to satisfy the conditions (2.20), (2.21) at the shocks. The solution q~22)has the right behavior at infinity as well as at the trailing edges and at the wakes but otherwise has zero contribution to the boundary conditions on the blades. On the other hand it has an arbitrarily specified downstream-propogating wave field far upstream of the trailing edge. The composite solution ~u2 + ~b(21)+ ~b(22) then satisfies all the boundary conditions and the arbitrariness in ~2 and q~(22)together is just sufficient to satisfy the conditions at the shock. The solution q~2) is computed in Mani and Harvey (1970). In terms of the terminology of this paper, we have °° ~ exp(hl+X) nny ~b~2) = - i~=0 B, cos s
x
exp (iq+) { } 4hi exp(in~ - io') - e x p ( - i~/+ s +)
oo+i~2 x~_(_2(2)) A 2 ( c ~ , y ) e x p { _ i ( ~ + M 2 k 2 ) 2 2 }
f
~;-~2)~;~
x2(~)
]
dc~j for O
(4.5)
- - oo 4 - i 5 2
Vodicity wove -,-~- Jump conditions ~eq. (2.20)j ( 2.21] 4
Incident I I
Woke..... ..... 5
I Acoustic fiord ~ ~ /J2
Woke
n=1 ...... x Figs. 4 and 5. 4 Boundary conditions for subsonic flow. 5 Extended field for subsonic flow
K.K. Puri: Unsteady mixed flows with in-passage shocks
21
where x~i
=x+s + -1
1 { e x p ( - flz72Y + iA2+) exp(fz72Y + iA2) ] A2(c~'Y) = ~ sinAi+ A~-~-sin
for
0
1
A?
(4.7)
= ~ (~ + M2k2s + + as +) + sfl2y2/2i
(4.8)
-
M 2 k 2 + 2 ( , 2)
(4.9)
~-i
\Sfl2/]
(4.10) - e2 < 62 < ~2 - Imk2 and ~2 -= (e2 _ k2)1/2 with the branches of the root chosen as in Fig. 6. The required arbitrariness is furnished by the unknown constants Bn which are the amplitudes of the incident-infinite duct waves [exp (itl+ x)cos U I .
The second term in the summand results from
the reflection of the above-mentioned incident wave from its impingement on the open end of the cascade. The function x~- (~), is analytic in the lower half plane, arises in the factorization xS(~z) Z~
~ A2(cgy ) ~yy y=O
c~+
for
Im~=6 2
-oo
(4.11)
which is required for the implementation of Wiener-Hopf technique to obtain the solution. Since the details are given in GBA as well as in the original paper, they are omitted here. The second part of the solution, ~b~t), follows from the use of Mani-Harvey procedure in conjunction with the technique adopted for the supersonic case, yielding,
dP~I)
L2 ~ia2 A2(a,y)x2+ ( - M2k2 - a) exp{+ i(~ + M2k2) ;?2}dc~ - 2rc - ~ + i & (c~ + k2/M2)ky (~)(~ + M2k2 + a)
(4.12)
where L2 = (-01((-02 -- a) exp{ia(1 - s+)} 0)2
(4.13)
To evaluate the above integral we observe that the integrand -, 0 as e ---,0 along an infinitely large circle in the upper half plane. Hence we can close the contour in the upper half plane. The singularities
Im(z
Integration path
IRek2 2i Re~
1
Fig. 6. Branch cuts for 72 = ] ~ - - k22
22
Computational Mechanics 4 (1989)
of the integrand are poles, given by, c~= - a the residues on these poles, we obtain GO
H nexp (iG-
-
m2k 2
and the poles of the function A2 (~'Y). Using z 2 (~)
22) COS t'/gy S
q~l) = _ i}-" n=0
+ iL2 exp (ia (2 2 - s+)) I exp (io-)
cos 052Y 0)2-eXPsin(ias+)052 s cos 052(Y - s)]
(4.14)
where L2[exp{i(a + nrc - q y s + ) } - 1]z2+ ( - a - M2k2)
2(~2) +
sfl2~(2)(1 ÷ (~n,0) z~- (~(2)) ()~(2) ÷ a + M2k2)
Hn =
(4.15)
(In the above a is replaced by R e a - ib, b > 0). This expression allows us to c o m p u t e the axial velocity and the pressure on the blade. Thus Hn (,~(2)÷ m 2 k 2 ) e x p (i~/2 22) cos nrcy S
1) _
-
a L 2 exp
-
(ia (2 2 - s +))
[exp (i a) cos 052Y - exp (ics +) c o s 052 (Y + s)] 052 sin 052 S
,
0 < y < s
(4.16)
and
( Pf) =
8) i0)2 -- ~x ~b~l)=
+ L2
~=oHnexp(irln22)(o92÷d~(2)÷M2k2) (2(n2) + k2/M2)
nrc y
COS - -
S
exp { i a 022 -- s + ) } (0) 2 -- a) [exp (i o') co s 052 Y -- exp (i a s +) co s 052 (y - s)] 052sin052s
(4.17)
Proceeding likewise with the Eq. (4.5), we establish
P~2)(x'Y)
=
if°2 - ~ x
_-
.B,
0
q~2)(x,y)
exp(i11+x)
1 - - s +,
(0)2_t/+)+
oo exp____(it/____~; ~=oBmKm,n 2(2)+ 22) (0-)2 _ t/n)
cos
s
(4.18)
0
So that P2 = p~l) + p~2) = n~0 { Bn exp (i~l+ x) + (H, + =
+ Lzexp(ia(x - 1)) (a 0 < Xz < 1 - s +,
-- (.02)
0
m=0
B mkin,n) exp (it/n 22)} cos n roy S
exp (i o-) cos 052Y - exp (i a s + ) co s 052 (Y - s) 052sin052 s
(4.19)
K.K. Puri: Unsteady mixed flows with in-passage shocks
23
Here exp (i t/n+_ 1) gm~// --
[exp {i(mzc - @} - e x p ( - it/m+ S+)]
- 2 (1 + a.,0)
242-( -- ,~(n2))(/, (n2)"q-M~22) x [exp {i(a - t/~-s + - n~z)} - 1] ~-(2(2)) (,~(n2)-~- ~ (m2))Sf122)~(n2)
(4.20)
We now use the arbitrariness implied in the solution for ~2 represented by the Eq. (4.1) to satisfy the shock conditions. '." gt2 vanishes at y = 0, s,/2 (y) can be expanded into a Fourier sine series so that we have g t 2 ( y ) = ,2.~ - o , , s"m |t/n~zY'~ --iexp(ioo2x)
~=i
(4.21)
O<_y
for
ks/
Also from the Eqs. (4.4), (4.21), (4.14), and (4.16), we have
O~(21) Sqb(22) ~gt2 & Bnt/+ exp (it/+ x) (20 ) t/n exp (it/y yc2) ~x + ~xx + ~y - .L-_o ( T i / ~ / ~ - ~ p ) + Bmk~,. (.~(~)+k~/M~)
v~-
_ (2(2) + Mzk2)H~ exp (it/~- x2) cos nrcy + aLzexp{ia(~2 _ s+ )
),(2) +
k2/M2)
S
exp(ia)cosc52y -- exp(ias+)cosch2(y -- s) x
a52 sin a52s
& nrc
nrcy
+ ~=~17 b~ cos
s
(4.22)
The result (4.22) together with (4.4), (4.19), (3.29) and (3.30) when substituted in the conditions (2.20), (2.21) on the shocks leads us to a Fourier series in terms of cos - - , sin
S
nT~
F r o m the fact that cos - - ,
sin nTt form a complete set, we deduce after eliminating the constants
b+s,
s
a + Bn + a2
e x p { - it~2 (s + - 1)} -
s 2
F.
s 1 + b~,o
where
(2---6.,o) L 8 4 \ 7 -
r. -
+ M1M2 ~ - fl~ \ ~ +
+ o,22 Q ~ - a ; ~ e x p { i t / ; ( s + - l ) } + c02M2r 1 22
Rn
( 1)}
{
c(.2)L2exp(- ia) (o~2- a) a2 - 4aM2k~ - k~M~ 3 + MlYM~
+ Ll (C°l - a)c(~l) { (2M2 M22 - f12) (n2rc2 84
\ s 2 +(°22M2r,
where a + = (2(2)) 2 + 2M2k2 2(2) + ( k 2 ~ 2 -
\M1J
Cn~)= exp(ia) - exp{i(o- + nrc)}
n 2 7C2/$2 -M;~; 2 2
.
,
rl = m 2 + 1
)
2M~a n2~ 2 } +
(4.23)
24
Computational Mechanics4 (1989)
The infinite set of Eqs. (4.23) are to be solved for the infinitely many unknowns. The solutions then allow us to determine the pressures on the subsonic portion of the upper surface of the blade n=0. In order to determine the forces on the blades, we also need to calculate the P2 (x, 0 - ) , i.e., the pressure on the lower side of the blade. That cannot be directly obtained from the Eq. (4.19) because of its restrictive validity on 0 < y < s. We therefore have to invoke the periodicity condition on the pressure, namely,
Pz(X q- s +, y + s) = exp(&r)P2(x,y) whence, P2(x, 0 - ) = e x p ( - &r)P2(x + s +, s + 0 - )
(4.24)
Using the Eq. (4.19), we have oo
oo
P2(x, 0 - ) = e-i~rn~oBnexp{itl+ (x + s +) (-- I) n + (H, + m~__oBmkm,n)exp{i~ln (2 2 + s + ) ( - 1)n } + L2 exp { ia (x + s + - 1) } (a - 0)2) (exp (ia) cos o52s - exp (ias+)) c52sin e52s
(4.25)
The above solution is valid on 0 < x + s + < 1 - s +, that is, on - s + < x < 1 - 2s +. To complete it, we now have to examine the pressure on 1 - 2s + < x < 1 - s +, y = 0 - . We write
P2 (x, 0 - ) = e-icr[P~2)(x + s +, s + 0 - ) + P~I)(x + s +, s + 0 - )
(4.26)
p~2~(x+s+,s+O_)_ iL22~o~+i&~A2(e's)e~(-a-M2k2)exp{-i(e+M2k2)(Yc2+s+)}de - ~ +ia2 z~- (e)(e + M2k 2 + a)
(4.27)
for 1 - 2 s + < x < 1 - s +. Using the representation (4.11) for z2 (e) and studying the behavior of the integrand in (4.27) we can split the above integral into two closing one each in the upper and the lower half plane. In view of the Eq. (4.26) then we have, P~2)(x,0 - ) = e x p ( - &r)P~(x + s +, s + 0 - ) = i L 2 e x p ( - ia) (11 + / 2 ) 2re where
(11= 1 f 2,
a - M2 kz) e x p { - i(e + M2k2)(x2 + s +)} ( e + k2 ] exp { i(a + M2k2 s+ + as +} coshfl272s) \ M2J
(e+ M2k2+a)~ ~ ( e ) ( @ y2) y=o sinA2- sinA~12= -
l_~(-a-M2k2)exp{-i(e+M2k2)(x~ +s +) 2~ ~2+ (e) (e + m 2 k 2 -1- a) sinai- sinA1+ de
It is now a simple matter to apply the residue theorem to the integrals I1, 12 to evaluate them. Doing so and collecting the results, we get,
P~2)(x, 0
Hnex p (it/~-)~2) x? ~--0 [exp £i(cr + nrc - ~n s+)} - 1]
L2ex p (ia22) (-- a + 0)2) cotsa32 e52
+ L 2 e x p ( - ia) ~ K + (M2k2 + a ) e x p { - i ( e ; + M2k2) (x{ + s+)} 1 - 2s + < x < 1 - s + n=0
where,
(4.28)
K.K. Puri: Unsteady mixed flows with in-passage shocks
25
~ (-- o:) (df F(n2) -- an s +)
(4.29)
K+ (~) = zy (e;)d~-(s+F(n2) - c¢n d~-) sin(e2 s + - d~-F~)) (a + cA-) 452 = [M~ (a + co2)2 - a2] 1/2
(4.30)
d~ = ~/(s+) ~+ fi~s 2, F~2)d~ = 2nrc - a - M2k2 s+
(4.31)
d~- ~+ = s + F(~2) + s f l / -
(F~2))2 + k~ = s + F(~2) _ isfi ~ -
k 2 + (F(~2))2
(4.32)
The solution p~l) (x, 0 - ) does not depend u p o n the profile of the blade n = 0 and so must be same as obtained in GBA. We can write it, as, P~l)(x'0-) =
+ B~
exp{-
i t ° 2 - ~xx ~
q@(x'O-)=~=okexp{i(rr-Oys+=+ n r 0 } - lm~=OBmKm'n
exp{i(q + exp(it/+ X) i(@-s-J+ ---~-Sr- nrc)} - 1 + 2
Xm=_oo ~ K;n(2!2))exp{-i(Ccm+M2k2)(x-l+
t7)}
2s+))}l
(exp { - i(o- + nrc)} - e x p ( - it/+ s+))
1-2s+
(4.33)
The Eqs. (4.28), (4.33) give complete calculations for the subsonic pressure on the lower side of n = 0 blade.
5 D i s c u s s i o n o f the results
The expressions for the pressure obtained above are rather complicated and are made more so, as a result of the presence of several parameters in them. A general discussion relating the pressure distributions to various M a c h numbers and the interblade angles is given in G B A . Here the concern will be to discuss the differences between the solutions obtained by them in the case of a flat-plate cascade and those obtained here for a sinusoidal cascade. To facilitate the discussion, b o t h the functions Re (pe i°~'t) = Re P and Im (pe ;~°'t) = Im P, representing the in-phase and out-of-phase component of the pressure amplitude respectively, are depicted in graphs, Figs. 7-15, for the reduced frequence (D1 -= 0.5, the stagger angle of 60 ° and the solidity = 1.3 It can be easily seen that the above functions which are practically constant for o~t = 0.5 in the supersonic region in the G B A study are not quite so in the present case. Both these c o m p o n e n t s show an oscillatory tendency. This fact, in a flat-plate cascade shows up only when the dimensionless reduced frequency is of the order of 2.
2 0 -2 3 ~.
-0
1
-8
&
-1 -0.6
I -s +
r -0.4
i -0.3
i -0.2
i -0.1 ~' ,--.-~,.-
I
J
r
0.1
0.2
1-s +
-10 0.4
-12
Fig. 7 a and b. Non-dimensional pressure amplitude: upper
surface, a = 0, --MI--- 1.4,-.-M 1= 1.6,-"-MI = 1.8
b
-14 -0.6
J
J [ 1 - s + -0.4
[ -0.3
I -0.2
I -0.1
0.1
0.2
1-s +
0.4
26
C o m p u t a t i o n a l M e c h a n i c s 4 (1989)
16
\
14-12--
"\
\
10--
\
i
5
'\
q_
\
\
I
'\
8 0 -12
4--
-14 I
2 ~-
-15
i i
/
-18
_2LI
J
-13.6
~
-s + -O.Z~ -0.3 -0.2
&
-0.1 ,r.-----~
f 0.1
_20 I
f r\ 0.2 l - s +
-22 -
Fig. 8 a and b. N o n - d i m e n s i o n a l p r e s s u r e a m p l i t u d e : u p p e r surface, M1 = 1.4, - - a = 0, - - - a = n/4, - . . - a = ~c/2
\.
. -0.3
- s + -0.4
I -0.2
I 0.1
I -0.1 X----~
r 0.2
\.,
/ -0.2
I -0.1 X
a
&4
\,
\
\
J
8
".\
J
.J .J
\ I I -13.6 - s + -0.4 -0.3
J 1-s +
f
-10 I
/
-2
\
I
/
I
I
l-
0.1
0.2
1-s +
/
-12 0.4
.J"
"
-14 -
Fig. 9 a and b. N o n - d i m e n s i o n a l p r e s s u r e a m p l i t u d e : u p p e r surface, M l = 1.6, - - ~ 7 = O, - . - a = n/4, - . . - a = ~ / 2
-1~. 6
fI I -s + -0.4
r -0.3
b
I -0.2
f 0.1 X "-----~
J ~ 0.1
I r O.Z 1-s +
O.Z~
2 0 -2
-6 J
-8 I
I
6 -s +
L
I
-0.4 -0.3
I
-0.2
J
B
-0.1
OJ
J
0.2 1-s +
-"
j"
-10
J 0.4
/
-12
Fig. 1 0 a a n d b. N o n - d i m e n s i o n a l p r e s s u r e a m p l i t u d e : u p p e r surface, M l = 1.8, - - t 7 = 0, - . - a = n/4, - . . - a = ~ / 2
-14 -0.6 b
J
.J"
I
I
- s + -0.4
i -0.3
[ i -0.2 -0.1 X ~
.J I 0.1
I 0.2
I l-s +
0.4
K . K . Puri: U n s t e a d y m i x e d flows with i n - p a s s a g e s h o c k s
27
/.j"J"J"~. ././, ,/z" ~
-2
-
~
8
I +
--0.5
-s
I -0,4
I
I
-0.3
I
-0.2
-0.1
I
I
I
0,1
0,2
1-S +
10 0.4
a
-12
Fig. 11 a and b. N o n - d i m e n s i o n a l p r e s s u r e amplitude: lower surface, cr = 0, - - M 1 = 1.4, - ' - M 1 = 1.6, - - . - M 1 = 1.8
-I/*
I
I
I
-0.0 -s + -0.4
I
I
q
-0.3 -0.2
I
-0,I
0,1
Z,----4~
j..
6
k
I
0.2 1-s +
O.Z~
..
°
4
j'-
2 0
/.. "/// / /
-2 -z,
I
I
-6
c~
-
8
/'
-10 - 12
\----'/
,/
-16
&
_/
J
-18
- s + -0,4
-0.3
i
/
/
-lZ~
-20 -0.6
/
/
-0.2
~ _ _ L _ _ _ L ~ -0.1 0.1 0.2 1-s + f'------~
I 0,4
].6
I
- s + -0.¢
b
Fig. 1 2 a and b. N o n - d i m e n s i o n a l p r e s s u r e amplitude: lower surface, M 1 = 1.4, - - a
I -0.0
I -0.2
I -0.1 f'-"--'-
F
I
I
0.1
0.2
1-S +
0.4
= 0, - . - ~ = n/4, - . . - ~ r = g/2
The above graphs also show that the magnitudes of the jump discontinuities for a = 0 is a decreasing function of the Mach number. Indeed the jump at M 1 = 1.4 (Fig. 7 a and b) is significantly greater than that for the higher speeds. On the other hand, these discontinuities for the in-phase components are increasing functions of the interblade phase angle for each of three Mach numbers, M1 = 1.4, 1.6, 1.8 considered here. Whereas the pressures take on finite value on the surface of the shock and exhibit expected discontinuities across it, their distributions on the lower surfaces are continuous and finite in values. The latter facts stands out in contrast to the case of a purely subsonic cascade.
28
ComputationalMechanics4 (1989) f
J -
f
f . • f
0
..J
I
J
Z-
2 J
-4
/-ij.
%
¢:z
8
-10
10
-12
[2 J I
-0.~
I
[
- s + -0.4
I
f
-0.3 -0.2
a
[
I 8.1
-0.1 X" - - - ~
j
[ 0.2 1-sE + O.Z~
/
/
/,
[
4-06
-s +
-0.4
I
• j " ~ - ' J
I-4 / -8
I
{
r
;
-0.2 -0.1
I
I
I
-0.2 -0.1 X----~
= 0, - . - a
[
0.1
= n/4, - . . - a
I
0.2 1-S +
0.4
= 7c/2
-2
t? ~ " / J
I
0
.~.°./7
-S + -0.4 -0.3
-0.3
J
.J
-0.6
./" J
b
0
- 1~
,
/
Fig. 1 3 a a n d b. N o n - d i m e n s i o n a l p r e s s u r e a m p l i t u d e : l o w e r surface, M1 = 1.6, - - a
i-2
.......
0
8
-14
, f
j- J •
J
/
.....
I
0.1
I
_,J /
I r f -10 -0.6 - s + -0.4 -0.3
I
0.2 1-s +
_...1(,/ /'J
0.4
i
-0.2
[
-0.1
I
I
I
0.1
0.2
1-s +
0.4
15
14
14 and 15. N o n - d i m e n s i o n a l p r e s s u r e a m p l i t u d e : l o w e r surface, M 1 = 1.4, - - a = 0, ---a = ~/4, -..-a = ~c/4 15 M l = 1.8, --a
Figs.
= 0, - . - a
= ~z/4, - . . - a
= ~z/2;
The theory developed here as well as in GBA paper is based on a leading order linear analysis. Some of the graphs in the two papers, show 0 (1) variations which are rather questionable in such a theory. These questions should be addressed in the context of a non-linear analysis to which, currently, we are devoting our efforts.
6 Lifts and moments
The total fluctuating lift and moment on the surface of a blade consists of the contribution to these due to the unsteady surface pressures together with the direct contributions due to the motion of the shock. It has been established in GBA that the amplitude of the shock-induced lift fluctuation, non-dimensionalized by cQ1 U~ is given by LS --
i 2~,ll_#+lpl(o,o+)+a/t? # UI(O,O+) 2klM 1 #+ 1 #+ 1
P2(O,O+)
(5.1)
29
K.K. Puri: Unsteady mixed flows with in-passage shocks
Assuming that the equilibrium position of the shock is slightly ahead of the leading edge of the upper blade, the contribution to the fluctuating lift from the motion of its footprint is zero and, so, the dimensionless moment coefficient about a point x = do is, 1 -s
M=
+
~ ( x - do)[elodx + doLs
(5.2)
--S"
where doLs is the moment of the shock-induced lift and [P]0 is the pressure jump across the n = 0 blade. For torsional motion, the work per cycle done on the flow by the blades is equal to A001 O~ Im M(Fung 1965). When this quantity is positive, the blades receive energy from the flow and develop instability. In terms of the dimensionaless moment coefficient, ~
where A0 is the
instantaneous angle of attack, it follows that the cascade will flutter when Im ( _ ~ 0 ) <0. In order to compute the quantity on the left, we first calculate the pressure jump. lira
P(Y)
o
y-+0+
n=0
1-s +
5 Pl(x,O+)dx+
=
~ P2(x,O+)dx
-s +
0
-sill
0
= ~ P,(x,O+)dx+ I Pl(x'O+)dx+ -s +
-sfl
1-s +
I P2(x,O+)dx
(5.3)
0
Similarly 1 --2s +
1--s +
!+ P2(x,O-)dx + ~ Pz(x, 0 - ) d x
P(x,0 - ) = -
1-2s
(5.4)
+
and 1- s +
M=
(x-do[&dx+4L --S +
1--S +
I
x[P]o dx + doL + doLs
mS+
where 1--S +
L=-
_[+~[P]0dx
Using the various expressions for the pressure from Eqs. (3.24), (3.27), (3.28), (4.19), (4.25), (4.26), and (4.28) to (4.33) in the Eqs. (5.3) and (5.4) we can easily determine L, Ls and M by terms-wise integrations of the involved series.
References Ballhaus, W. F.; Goorinjan, R. M. (1977): Implicit finite difference computations of unsteady transonic flows about airfoils. AIAA J. 15, 1728-1735 Eckhaus, W. (1959): Two-dimensional transonic unsteady flow with shock waves. Office Sci. Res. Tech. Note 59, 459-491 Fung, Y.C. (1965): An introduction to the theory of aeroelasticity, pp. 166-168. New York: Wiley Goldstein, M.E. (1976): Aeroacoustic, pp. 220. New York: McGraw-Hill Goldstein, M.E.; Braun, W.; Adamczyk, J. (1977): Unsteady flow in supersonic cascades with strong in-passage shocks. J. Fluid Mech. 83, 569-604 Isogi, K. (1977): Calculations of unsteady transonic flow over oscillating airfoils using the full potential equations. AIAA pap. 77-448 Lane, F. (1956): System mode shapes in the flutter of compressor blade rows. J. Aeronaut. Sci. 23, 54-66
30
Computational Mechanics 4 (1989)
Miller, G.R.; Bailey, E.E. (1971): Static pressure contours in the blade passage at the tip of several high Math number ratios. NASA Tech. Memo No. X-2170 Mani, R.; Harvey, G. (1970): Sound transmission through blade rows. J. Sound Vib. 12, 59-83 Magnus, B.; Yoshihara, H. (1975): Unsteady transonic flows over an airfoil. AIAA J. 13, 1622-1628 Noble, B. (1958): Methods based on the Wiener-Hopf technique. Los Angeles: Pergamon Strazisar, A.J.; Chima, R.V. (1980): Comparison between optical measurements and a numerical solution of the flow field within a transonic axial-flow compressor rotor, pp. 1-9. Proc. of the AIAA/ASME/SAE 16th Joint Prop. Conf., Hartford/CT Weatherhill, W.H.; Ehlers, F.E.; Sebastian, J.D. (1975): On the computation of transonic perturbation flow fields around two and three dimensional oscillating wings. NASA CR-2599 Williams, M. H. (1979): Linearization of unsteady transonic flows containing shocks. AIAA J. 17, 394-397
Communicates by S.N. Atluri, May 1, 1987