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SOUND PROPAGATION IN AN ANNULAR DUCT WITH MEAN POTENTIAL SWIRLING FLOW
Journal of Sound and Vibration (1996) 198(5), 601–616
SOUND PROPAGATION IN AN ANNULAR DUCT WITH MEAN POTENTIAL SWIRLING FLOW V. V. G H. M. A Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN. 46556, U.S.A. (Received 18 October, and in final form 10 June 1996) The propagation of pressure disturbances is investigated in an annular duct with mean potential axial and swirling flow. The swirling mean motion causes refraction of the acoustic waves as they propagate along the duct. A normal mode analysis leads to a non-linear eigenvalue problem. The numerical solutions are validated with an asymptotic analysis in the limit of large radius. The results show that while for a uniform flow the evanescent modes propagate at the same speed only upstream, for a swirling flow such modes propagate at different speeds both upstream and downstream. Moreover, most non-axisymmetric acoustic modes propagate opposite to the mean flow swirl. 7 1996 Academic Press Limited
1. INTRODUCTION
In many applications such as in turbomachinery, the flow in a duct is not uniform but rather has a swirling motion. The amount of swirl velocity varies and is of the order of 15–45% of the mean axial velocity. The present paper examines the effect of a mean axial and swirling motion on the propagation of acoustic disturbances in an annular duct. In addition to understanding the physical phenomena associated with the propagation of pressure waves, the results of the present analysis can be used in the formulation of inflow/outflow boundary conditions for the simulation of aerodynamic and aeroacoustic problems in aircraft engine duct systems. The effects of a non-uniform mean flow on sound propagation have been studied mainly to account for the effects of shear near the duct wall and of acoustic liners. These studies point out the importance of the mean flow shear and its refraction effects on sound waves. A detailed review of the subject was given by Nayfeh et al. [1]. The analysis is usually simplified by assuming that the mean flow is parallel or near-parallel to the duct wall. In addition, the velocity field is assumed to be only a function of the co-ordinate normal to the duct wall. Although the governing equations have non-constant coefficients, solutions are sought in terms of normal modes subject to the wall boundary conditions. This results in an eigenvalue problem which is generally not self-adjoint, and therefore the normal modes may not form a complete set. In other instances, the acoustic liner imposes boundary conditions which also lead to a non-self-adjoint eigenvalue problem. Tester [2], Zorumski and Mason [3] showed that in this case algebraically growing modes may exist for multiple roots of the eigenvalue equations. The propagation of small disturbances in a duct with swirling flow was first considered by Kerrebrock [4] and more recently by Kousen [5]. Both used the normal mode analysis applied directly to the linearized Euler equations, and showed the presence of nearly-convected shear-like disturbances and pressure waves. Their results show a variety 601 0022–460X/96/500601 + 16 $25.00/0
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of possible solutions. However, in some cases the physical meaning of the unstable and nearly-convected modes remains to be clarified. In this paper we considered the case of potential swirling flows. Such flow can be generated, for example, by adding an axial vortex to a uniform mean axial flow. The study of the propagation of disturbances in potential swirling flows will help elucidate some of the basic physical phenomena associated with sound propagation in such flows, and will pave the way for understanding the wave propagation in vortical flows. The mathematical formulation can be simplified by the use of Goldstein’s [6] splitting of the perturbation velocity field into potential and vortical components. As it is shown below, such an approach gives a convenient method for the determination of the normal modes. Moreover, Goldstein’s partial splitting clearly shows that vortical disturbances are independent of acoustic disturbances, while acoustic disturbances are produced by vortical ones. This approach was used by Wundrow [7] to study the asymptotic behavior of the vortical and the vorticity-induced unsteady pressure disturbances. Golubev and Atassi [8] investigated the evolution of the unsteady pressure and vorticity in response to incident vortical waves. The results show an algebraic decay of the unsteady pressure and an algebraic growth of the vorticity with strong radial variation of their phase. These solutions correspond to the convective eigenmodes of the linearized Euler equations. The present paper examines the propagation of sound waves in an annular duct with potential swirling flow. The normal mode analysis yields a non-linear eigenvalue problem. Numerical solutions are obtained for propagating and evanescent modes. The results are compared with those of uniform mean flow and show the significant effect of the swirl, particularly on the non-axisymmetric modes.
2. MATHEMATICAL FORMULATION
The behavior of small-amplitude unsteady perturbations in an annular duct is considered with hub and tip radii rh and rt , respectively. It is further assumed that the swirling mean flow is composed of an axial flow and a free vortex flow. Thus, U (x ) = U0eˆ x + (G/r)eˆu ,
(1)
where G is the strength of the free vortex, eˆx is the unit vector in the axial direction, and
Figure 1. Geometry.
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eˆu is the unit vector in the circumferential direction (Figure 1). The velocity, pressure and density of the fluid can be written as V (x , t) = U (x ) + u (x , t),
p(x , t) = p0 (x ) + p'(x , t),
r(x , t) = r0 (x ) + r'(x , t),
(2)
where the time-dependent perturbation quantities are assumed to be much smaller than those of the mean flow, and governed by the linearized Euler equations: D0 u /Dt + (u · 9)U = −9(p'/r0 ),
D0 /Dt(p'/c02 r0 ) + (1/r0 )9 · r0 u = 0,
(3)
where D0 /Dt = 1/1t + U · 9; r0 and c0 are the density and the speed of sound of the mean flow. Goldstein’s [6] decomposition of the perturbation velocity in terms of potential and vortical parts is then used: u = u (R) + 9f,
(4)
where the vortical velocity u (R) satisfies the homogeneous equation D0 u (R)/Dt + (u (R) · 9)U = 0,
(5)
and the potential function f is the solution to the inhomogeneous equation Lf = (1/r0 )9 · (r0 u (R)),
(6)
where L is the convective wave operator L 0 D(0 /Dt)(1/c02)(D0 /Dt) − (1/r0 )9 · (r0 9 ).
(7)
The unsteady pressure p' is then given solely in terms of the potential f, p' = −r0 D0 f/Dt.
(8)
This splitting of the velocity field has been used to study the interaction of vortical disturbances imposed on potential mean flows with uniform upstream conditions. A recent review of the subject is given by Atassi [9]. The coupling between the vortical and potential modes in equation (6) shows that, due to the non-uniformity of the mean flow, inlet vortical disturbances will produce an unsteady pressure field. However, equation (5) shows that, in the absence of an incident vortical perturbation of the velocity field, an incident acoustic field will not create vorticity if the mean flow is irrotational. Solutions to equations (5) and (6) depend on the upstream conditions of the unsteady flow. If the upstream imposed perturbation field consists only of acoustic disturbances, then u (R) = 0 and the propagation or decay of acoustic modes in governed by a homogeneous equation (6). On the other hand, if the imposed upstream disturbance is purely convected, then the inhomogeneous form of equation (6) must be used to find the particular solution representing the unsteady ‘‘hydrodynamic’’ pressure induced by the incident vortical waves. When a solid body is located downstream, acoustic waves will be created as a result of the coupling between the vortical and potential solutions along the surface of the body. This problem was examined by Golubev and Atassi [8]. In what follows the propagation of acoustic disturbances in an annulus with swirling mean flow but without the presence of a solid body is considered. 2.1. By considering only acoustic waves, the problem reduces to solving the homogeneous equation Lf = 0.
(9)
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604
In order to determine the spectrum of allowed eigenmodes, the following expansion of the potential function is assumed: f(x, r, u; t) =
g
a
a
s
a
s f mn (r) ei[kmn x + mu − vt] dv,
(10)
−a m = −a n = 1
where m and n are integer modal numbers characterizing the circumferential and radial eigenmodes, respectively. Due to the linear character of the problem, each Fourier component of the potential field can be considered separately. Using equations (8), the pressure associated with each of these components is given by p'mn = ifmn r0 (r)(v − U0 kmn − (G/r 2 )m).
(11)
The present formulation reduces the eigenvalue problem to a single second order ordinary differential equation. This brings about a significant simplification for the determination of the eigenvalues kmn and their eigenfunctions f mn. Substituting a single harmonic of equation (10) in equation (9), one obtains, after reduction, the eigenvalue problem for f mn,
6 0
1 0
d2 1 d(ln r0 ) d L2 m2 2 + + mn 2 − kmn − 2 2+ dr r dr dr c0 r
17
f mn (r) = 0,
(12)
with 1f mn (r)/1r=0
(13)
at r = rh and r = rt . The non-constant eigenvalue of the convective operator D0 /Dt is defined by Lmn = −v + U0 kmn + mUs /r,
(14)
where the swirl component of the mean velocity, Us = G/r. The properties of the eigenfunctions resulting from the eigenvalue problem (12, 13) are now investigated, and then the numerical approach used to calculate the eigenvalues and eigenfunctions discussed. Note that equation (12) can be rewritten in the form L f mn = w(r, kmn )f mn ,
(15)
where L is written in the self-adjoint form L
= (d/dr)[p(r) d/dr] + q(r),
(16)
with p(r) = rr0 ,
q(r) = rr0 [(v/c0 − mMs /r)2 − m 2/r 2 ],
and 2 b02 + 2kmnM0 (v/c0 − mMs /r)], w(r, kmn ) = rr0 [kmn
(17)
where the axial and swirl Mach numbers, M0 = U0/c0 and Ms = Us /c0 are introduced and b02 = 1 − M02 . Since the boundary conditions are homogeneous, for any two arbitrary eigenfunctions f mi and f mj ,
f mj =0, L
f mi f mj −f mi , L
(18)
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where the inner product is defined as f mi , f mj =
g
rt
f mi f mj dr.
(19)
rh
Hence, it follows from equations (15) and (18) that
g
rt
f mi f mj [w(r, kmi ) − w(r, kmj )] dr = 0.
(20)
rh
For a typical Sturm-Liouville problem, equation (20) would represent the orthogonality condition for the eigenmodes, and the function w(r, kmn ) could be factored as f(kmn )w˜ (r). In the present case, however, w(r, kmn ) cannot be factored. As a result, the orthogonality condition does not have a universal form. Therefore, the coefficients of the expansion of an arbitrary function in terms of the eigenfunctions cannot be determined using the orthogonality condition (20). However, these coefficients can be obtained numerically using, for instance, a least squares method as in the case of the sheared mean flows (Shankar [10]). It should also be pointed out that since the boundary value problem (12, 13) is not a Sturm-Liouville problem, it is not possible to conclude that the set of eigenfunctions {f mn } is complete. However, the continuous spectrum of the convected eigenvalues for equation (5), which is not considered in the present work, may complete the set. It is interesting to note that for the two limiting cases, M0 = 0, Ms $ 0 and Ms = 0, M0 $ 0, the eigenvalue problem is of the Sturm-Liouville type. In the first case, a swirl without an axial flow, the orthogonality condition (20) has a universal form with 2 2 w(r, kmn ) = kmn . In the case of a uniform flow in a duct, w(r, kmn ) = kmn b02 + 2kmnvM0 . These limiting cases can be used as the zeroth order approximations in the perturbation analyses of the eigenvalue problem for small M0 or Ms , respectively. 2.2. (rt − rh )/rm :0 The solution of the eigenvalue problem (12, 13) is not straightforward since the eigenvalues enter the problem non-linearly, so that the problem is not of the Sturm-Liouville type. An approximate solution can be found in the limit (rh − rt )/rm :0, where rm is the mean radius of the annulus, rm = (rh + rt )/2. This is the case in many important applications of the present analysis, such as the unsteady swirling flows through the fan and compressor stages. It is further assumed that the gap (rt − rh ) remains finite, so that rm may be very large. By taking e = (rt − rh )/rm 1,
(21)
the eigenvalue problem (12, 13) reduces after the co-ordinate transformation r = rm (1 + er˜ ) to the form
6
0
1
0
17
1 d2 1 d(ln r0 ) 1 d L 2 (r ) m2 2 + + mn 2 m − kmn − 2 2+ e r dr˜ rm dr erm dr˜ c0 rm 2 2 m
f mn (r) = 0.
(22)
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606
Letting 1/rm :0, and taking into account that d(ln r0 )/dr = O(1/rm ) while erm and m/rm are of O(1), one obtains
6
7
d2 2 + gmn f mn (r) = 0, dr 2
(23)
with 1f mn (r)/1r = 0
(24)
at r = rh and r = rt . In equation (23), we defined the eigenvalue g
2 mn
as
2 2 2 = Lmn (rm )/c02 − kmn − m 2/rm2 . gmn
(25)
Solution of the eigenvalue problem (23, 24) gives a set of eigenfunctions f mn = {cos gmn (r − rh )}, where 2 gmn = n 2p 2/(rt − rh )2,
n = 0, 1, 2, . . .
(26)
and n represents the integer radial modal number. The expression for the eigenvalues kmn can now be readily obtained: kmn =
$
6
0
1 n 2p 2 m2 − M0 v˜ 2 v˜ 2 − b02 2 + 2 b0 rm (rt − rh )2
17 % 1/2
,
n = 0, 1, 2, . . . ,
(27)
where v˜ = (v/c0 ) − (mMs M0 /rm ). This approximation is compared with the numerical solution obtained below. 2.3. The eigenvalue problem (12, 13) can be written as a standard eigensystem by introducing the new variable (Press et al. [11], p. 455) cmn (r) = kmn b02 f mn (r).
(28)
Equation (12) can then be rewritten as
0
0 L
1/b02 (2M0 /b )(mMs /r − v/c0 ) 2 0
10 1
0 1
f mn f
= kmn mn . cmn cmn
(29)
The eigensystem (29) now can be solved numerically for a spectrum of eigenvalues kmn and eigenfunctions f mn using a standard technique. The numerical algorithm implemented by the IMSL library reduces the matrix to a real upper Hessenberg matrix, and then the shifted QR algorithm is used to compute the eigenvalues and eigenvectors of the Hessenberg matrix (e.g., Press et al. [11], pp. 478–486). A fourth order finite-difference scheme is used to obtain the numerical approximation for the differential operator, L
. If N is the number of grid points used in the numerical discretization of the domain rh Q r Q rt , then 2N different eigenvalues (km,1 , km,s , . . . , km,2N ) are obtained for the given frequency v and circumferential wavenumber m. The homogeneous boundary conditions (13) can be added and incorporated in the system (29). Since the original eigenvalue problem (12, 13) is not of the Sturm-Liouville type, the eigenvalues of (29) may be complex. As for the problem of acoustic wave propagation in a duct, the calculations show that, in a frame of reference moving with the axial mean flow (u) (d) velocity U0 , there are two families of eigenvalues, kmn and kmn , corresponding to upstream and downstream propagating modes, respectively. For each family of eigenvalues, there are two branches corresponding to positive and negative values of the circumferential
607
wavenumbers m. These branches are related to modes rotating in the direction of the mean swirl, or in the opposite direction. Thus, such modes will propagate with different phase speeds in the axial direction. The eigenvalues are arranged as follows. Those eigenvalues which are real and thus correspond to propagating acoustic waves are considered first. The evanescent and amplifying modes are arranged according to the increasing imaginary parts of the eigenvalues. Note that since the amplifying modes correspond to exponentially growing pressure waves, they should be discarded as non-physical. Although the evanescent modes may not be significant for the study of the propagation of sound, they play an important role in the formulation of inflow/outflow boundary conditions in unsteady aerodynamics and acoustics (Fang and Atassi [12]). 3. RESULTS AND DISCUSSIONS
In this section the spectrum of the eigenmodes resulting from the eigenvalue analysis of equation (29) is considered and examination of the effect of the mean flow on the propagation of the modes in the upstream and downstream directions for various values of the modal numbers m and n undertaken. Note that the values of m can be related, for instance, to the number of blades in a rotor-stator interaction problem. The inflow/outflow boundary conditions in these problems are expressed in terms of the upstream and downstream propagating and decaying eigenmodes. In a swirling flow, the direction of propagation of the eigenmodes depends on m. To illustrate this remarkable effect, the results of the eigenvalue analysis are presented in terms of the double set of eigenmodes (u,d) kmn for m = −30, . . . , 30 and fixed n = 1, 2, 3. The eigenfunctions are shown only for m = 0, 25 and n = 1, 5. The pressure modes are normalized so that the norm >p'> defined by N
>p'> = s (=Re (p'j ) = + =Im (p'j ) =)h = 1,
(30)
j=1
where h is the discretization step. For numerical calculations, the frequency is non-dimensionalized by the mean radius of the annulus and the stagnation speed of sound, v¯ = vrm /c¯0 . The results are presented for two frequencies, v¯ = 2·5 and v¯ = 15 for the cases of no flow, uniform flow with M0 = 0·3, and swirling flows with M0 = 0·3, Ms = 0·5 and M0 = 0·7, Ms = 0·5. The comparison between the approximate solution of Section 2.2, and the exact numerical solution of Section 2.3 is given for the case of the swirling flow. The annulus used in the numerical calculations has tip and hub radii rt = 6 and rh = 4, respectively. The results for the eigenvalues are normalized with respect to the mean radius of the annulus. 3.1. To estimate the accuracy of the numerical method, the results of the calculations are compared with the analytical solution obtained for the case of the uniform flow (M0 = 0·3 and v¯ = 15). Equation (33) (below) gives the analytical solution for the eigenvalues. Figure 2 shows the natural logarithm of the error against the natural logarithm of the number of grid points in the radial direction. The eigenvalues correspond to three downstream propagating modes, with m = 0 and n = 2, 3, 5. More accurate results could be achieved by using spectral or pseudospectral methods for evaluation of eigenmodes and eigenvalues. For the present numerical calculations, a relative error of less than one percent is achieved for N = 100, for low order eigenvalues.
. . . . A
608
Figure 2. Numerical convergence.
3.2. For Ms = M0 = 0, one obtains from equation (12) an eigenvalue problem for the Bessel equation of order m with the eigenfunctions (in terms of Hankel functions) H(1,2) m (mmn r) and eigenvalues mmn , and a simple dispersion relation, (u,d) 2 kmn = 2 zv¯ 2−mmn .
(31)
The spectrum of mmn is determined from the wall condition 1f mn /1r = 0, which reduces to
b
(d/dr)H(1) m (mmn rh ) (d/dr)H(1) m (mmm rt )
b
(d/dr)H(2) m (mmn rh ) = 0. (d/dr)H(2) m (mmn rt )
(u,d) Figure 3. Eigenvalues kmn for M0 = Ms = 0, v¯ = 2·5, m = 0 . . . 30. (a) n = 1, (b) n = 2, (c) n = 3.
(32)
609
(u,d) Figure 4. Eigenvalues kmn for M0 = 0·3, Ms = 0, v¯ = 2·5, m = 0 . . . 30. (a) n = 1, (b) = n = 2, (c) n = 3.
Figure 3 illustrates this result. For v¯ = 2·5, there are 3 acoustic modes for n = 1 allowed to propagate upstream and downstream without decay, and an infinite number of non-propagating modes. For higher n all the modes are non-propagating. However, if M0 $ 0, (u,d) 2 kmn = −v¯ M0/b02 2 {(v¯ M0/b02)2 − (mmn − v¯ 2 )/b02 }1/2,
(33)
there is a shift in Re(kmn ), and thus all the evanescent modes will propagate only upstream. Figures 4 and 5 show the eigenvalues of the pressure modes for M0 = 0·3, for two frequencies, v = 2·5 and v¯ = 15. As the frequency increases, more acoustic modes cut on, but still all decaying modes propagate upstream (Shankar [10]). This effect is due to the Doppler frequency shift. In fact, in the moving frame of reference, these modes still formally exist, but they do not propagate in any direction. However, in the laboratory
(u,d) Figure 5. Eigenvalues kmn for M0 = 0·3, Ms = 0, v¯ = 15, m = 0 . . . 30. (a) n = 1, (b) n = 2, (c) n = 3.
610
. . . . A
Figure 6. Complex eigenfunction: M0 = 0·3, Ms = 0, v¯ = 2·5, m = 0.
Re (pmn )
(solid
line)
and
Im (pmn )
(dashed
line)
for
frame of reference, such modes appear as propagating upstream, and this fact is formally reflected through the dispersion relation (33) with the shifted stationary-frame frequency v. Complex eigenfunctions p'mn are shown in Figures 6, 7 for v¯ = 2·5. The solid and dashed lines represent the real and imaginary parts of the eigenfunctions, respectively. The case of the axisymmetric radial modes with n = 1 and 5 is shown in Figure 6. Note that the pressure eigenfunctions corresponding to the cut-on propagating modes have zero real parts. For higher n, the eigenfunctions exhibit faster variation in the radial direction. For m = 5 (Figure 7), all the modes are decaying.
Figure 7. Complex eigenfunction: M0 = 0·3, Ms = 0, v = 2·5, m = 5.
Re (pmn )
(solid
line)
and
Im (pmn )
(dashed
line)
for
611
T 1 Eigenvalues for upstream-propagating modes in a swirling flow, M0 = 0·7, Ms = 0·5, v¯ = 15, m=5
n 1 2 3 4 5 6 7 8 9 10
Asymptotic solution Full numerical solution ZXXXXXXXXCXXXXXXXXV ZXXXXXXXXCXXXXXXXXV Re (k) Im (k) Re (k) Im (k) −0·44514E + 02 −0·41879E + 02 −0·30866E + 02 −0·19022E + 02 −0·19022E + 02 −0·19022E + 02 −0·19022E + 02 −0·19022E + 02 −0·19022E + 02 −0·19022E + 02
0·00000E + 00 0·00000E + 00 0·00000E + 00 −0·22286E + 02 −0·37261E + 02 −0·50348E + 02 −0·62740E + 02 −0·74782E + 02 −0·86621E + 02 −0·98331E + 02
−0·45035E + 02 −0·40816E + 02 −0·29863E + 02 −0·18915E + 02 −0·18927E + 02 −0·18932E + 02 −0·18935E + 02 −0·18936E + 02 −0·18937E + 02 −0·18938E + 02
0·00000E + 00 0·00000E + 00 0·00000E + 00 −0·22679E + 02 −0·37515E + 02 −0·50562E + 02 −0·62939E + 02 −0·74979E + 02 −0·86821E + 02 −0·98537E + 02
T 2 Eigenvalues for upstream-propagating modes in a swirling flow, M0 = 0.7, Ms = 0.5, v¯ = 15, m = −5
n 1 2 3 4 5 6 7 8 9 10
Asymptotic solution Full numerical solution ZXXXXXXXXCXXXXXXXXV ZXXXXXXXXCXXXXXXXXV Re (k) Im (k)) Re (k) Re (k) −0·63008E + 02 −0·61212E + 02 −0·55157E + 02 −0·39926E + 02 −0·26630E + 02 −0·26630E + 02 −0·26630E + 02 −0·26630E + 02 −0·26630E + 02 −0·26630E + 02
0·00000E + 00 0·00000E + 00 0·00000E + 00 0·00000E + 00 −0·26739E + 02 −0·43145E + 02 −0·57121E + 02 −0·70135E + 02 −0·82643E + 02 −0·94845E + 02
−0·67068E + 02 −0·60237E + 02 −0·55390E + 02 −0·40849E + 02 −0·26901E + 02 −0·26918E + 02 −0·26927E + 02 −0·269313 + 02 −0·26934E + 02 −0·26936E + 02
0·00000E + 02 0·00000E + 00 0·00000E + 00 0·00000E + 00 −0·26378E + 02 −0·42946E + 02 −0·56999E + 02 −0·70066E + 02 −0·82615E + 02 −0·94852E + 02
3.3. Tables 1 and 2 show the comparison of the asymptotic solution for the eigenvalues obtained from equation (27), Section 2.2, and the numerical values obtained using the method of Section 2.3, for M0 = 0.7, Ms = 0.5, v¯ = 15, m = 5, −5 and n = 1, . . . ,10. Only the eigenvalues corresponding to the modes propagating or decaying upstream are listed. Note that the asymptotic solution gives very good results, and thus may be used in the formulation of the outflow conditions for unsteady aerodynamic problems. Tables 1 and 2 show three propagating modes rotating in the direction of the mean swirl (m = 5) and four modes rotating opposite to the mean swirl (m = −5). The rest of the modes are evanescent. Figures 8 and 9 show the eigenvalues of the pressure modes for M0 = 0·3 and Ms = 0·5, for two frequencies v¯ = 2·5 and 15. Comparison with the results for a uniform flow shows that the Doppler effect has a more pronounced influence on the acoustic modes propagating in a swirling mean flow. The figures illustrate how the initially symmetrical pattern of the axial wavenumbers for a uniform flow becomes distorted for Ms $ 0.
612
. . . . A
(u,d) Figure 8. Eigenvalues kmn for M0 = 0·3, Ms = 0·5, v¯ = 2·5, m = −30 . . . 30 (a) n = 1, (b) n = 2, (c) n = 3.
Compared to the case of the uniform flow, the Doppler shift causes an infinite set of evanescent eigenmodes to propagate in both directions. As before, the increase of v¯ (Figure 9) cuts on more acoustic modes. Such modes are clustered very densely for negative circumferential wavenumbers, which corresponds to the waves swirling in the direction opposite to the mean flow rotation. This is justified by the fact that the effective circumferential phase speed decreases, and thus more modes start to propagate along the duct to keep the total phase velocity in the local moving frame of reference equal to the speed of sound in the direction of the mode propagation. We now compare the effect of larger axial velocity component (M0 = 0·7, Ms = 0·5), which is more typical of turbomachinery flows. Figures 10 and 11 show the behavior of the eigenvalues for n = 1, 2 and frequency v¯ = 2·5. Compared to the results of lower M0
(u,d) Figure 9. Eigenvalues kmn for M0 = 0·3, Ms = 0·5, v¯ = 15, m = −30 . . . 30 (a) n = 1, (b) n = 2, (c) n = 3.
613
(u,d) Figure 10. Eigenvalues kmn for M0 = 0·7, Ms = 0·5, v¯ = 2·5, m = −30 . . . 30, n = 1.
(Figure 8), the number of propagating modes for n = 1 increases from 5 to 19 for those with negative m, and from 2 to 3 for those rotating in the direction of the flow. For higher n, there are no acoustic modes. It is also noted that the eigenvalues with positive m = 0, . . . , 30 are distributed over a wider range of Re (kmn ) than those with negative m. Therefore, modes propagating upstream will have their wavelengths clustered more densely compared to those propagating downstream. The complex eigenfunctions p'mn are illustrated for m = 0 in Figure 12. As for uniform flow, the propagating acoustic modes have zero real parts. Figures 13 and 14 show the pressure modes which rotate in the direction of the mean swirl with m = 5, and those in the opposite direction with m = −5, respectively. For m = 5, all modes are decaying, whereas for m = −5, the first radial mode is propagating.
(u,d) Figure 11. Eigenvalues kmn for M0 = 0·7, Ms = 0·5, v¯ = 2·5, m = −30 . . . 30, n = 2.
614
. . . . A
Figure 12. Complex eigenfunction: Re (pmn ) (solid line) and Im (pmn ) (dashed line) for M0 = 0·7, Ms = 0·5, v¯ = 2·5, m = 0.
4. CONCLUSIONS
The behavior of acoustic waves in an annular duct with a potential swirling flow composed of a uniform flow and a free axial vortex rotation was investigated. In the absence of viscosity and thermal conduction, the main effects governing the behavior of the waves are convection and refraction by the moving non-uniform medium. The splitting of the disturbance velocity field into vortical and potential parts was used to show that in the case of a potential swirl, the behavior of the pressure modes is governed solely by a single homogeneous convective wave equation for the velocity potential. The eigenvalue problem for the discrete spectrum of acoustic eigenmodes is not of the Sturm-Liouville type. A special transformation is used to reduce the problem to that of an eigensystem for two coupled functions. The method gives an effective technique to solve the eigenvalue problem. A direct normal mode analysis of the linearized Euler equations would require solving a system of four coupled equations [4, 5].
Figure 13. Complex eigenfunction: M0 = 0·7, Ms = 0·5, v¯ = 2·5, m = 5.
Re (pmn )
(solid
line)
and
Im (pmn )
(dashed
line)
for
615
Figure 14. Complex eigenfunction: Re (pmn ) (solid line) and Im (pmn ) (dashed line) for M0 = 0·7, Ms = 0·5, v v¯ = 2·5, m = −5.
The results of the numerical analysis show that in the case of a uniform flow, there are no evanescent modes in the downstream direction. As the frequency of the incident wave increases, more acoustic (non-decaying) modes cut on. For modes propagating in a swirling flow, the Doppler frequency shift causes an infinite set of evanescent eigenmodes to propagate both upstream and downstream. The acoustic eigenvalues are clustered densely for the negative circumferential wavenumbers, which corresponds to waves propagating in the direction opposite to the mean flow swirl. ACKNOWLEDGMENTS
The research was supported by the Office of Naval Research Grant No. N00014-92-J-1165, monitored by Mr. L. Patrick Purtell, and NASA Lewis Research Center Grant No. NAG3-732, monitored by Mr. Dennis L. Huff. The authors would also like to thank Alexander Balandin for his help in the numerical solution of the eigenvalue problem. Vladimir V. Golubev would like to thank the Center for Applied Mathematics at the University of Notre Dame for its support. REFERENCES 1. A. H. N, J. E. K and D. P. T 1975 AIAA Journal 13, 130–153. Aeroacoustics of aircraft engine-duct systems. 2. B. J. T 1973 Journal of Sound and Vibration 28, 151–203. The propagation and attenuation of sound in lined ducts containing uniform or ‘plug’ flow. 3. W. E. Z and J. P. M 1974 The Journal of the Acoustical Society of America 55, 1158–1165. Multiple eigenvalues of sound absorbing circular and annular ducts. 4. J. L. K 1977 AIAA Journal 15, 794–803. Small disturbances in turbomachine annuli with swirl. 5. K. A. K 1995 Proceedings of the First Joint CEAS/AIAA Aeroacoustics Conference, Munich, Germany, 1085–1094. Eigenmode analysis of ducted flows with radially dependent axial and swirl components. 6. M. E. G 1978 Journal of Fluid Mechanics 89, 433–468. Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. 7. D. W. W 1994 Contractor Report 195406, NASA. Small-amplitude disturbances in turbomachine flows with swirl.
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8. V. V. G and H. M. A 1995 Proceedings of the First Joint CEAS/AIAA Aeroacoustics Conference, Munich, Germany, 167–176. Aerodynamic and acoustic response of a blade row in unsteady swirling flow. 9. H. M. A 1994 Aerodynamics and Aeroacoustics (Ed. K.-Y. F, World Scientific) IV, 121–172. Unsteady aerodynamics of vortical flows: early and recent developments. 10. P. N. S 1972 Journal of Sound and Vibration 22, 233–246. Acoustic refraction and attenuation in cylindrical and annular ducts. 11. W. H. P, S. A. T, W. T. V and B. P. F 1988 Numerical Recipes in FORTRAN: The Art of Scientific Computing, Cambridge University Press. 12. J. F and H. M. A 1991 Unsteady Aerodynamics, Aeroacoustics, and Aeroelasticity of Turbomachines and Propellers (Ed. H. M. A), Berlin: Springer-Verlag. pp. 149–176. Compressible flows with vortical disturbances around a cascade of loaded airfoils. 13. P. M. M and K. U. I 1968. Theoretical Acoustics. New York: McGraw-Hill.
SOUND PROPAGATION IN AN ANNULAR DUCT WITH MEAN POTENTIAL SWIRLING FLOW V. V. G H. M. A Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN. 46556, U.S.A. (Received 18 October, and in final form 10 June 1996) The propagation of pressure disturbances is investigated in an annular duct with mean potential axial and swirling flow. The swirling mean motion causes refraction of the acoustic waves as they propagate along the duct. A normal mode analysis leads to a non-linear eigenvalue problem. The numerical solutions are validated with an asymptotic analysis in the limit of large radius. The results show that while for a uniform flow the evanescent modes propagate at the same speed only upstream, for a swirling flow such modes propagate at different speeds both upstream and downstream. Moreover, most non-axisymmetric acoustic modes propagate opposite to the mean flow swirl. 7 1996 Academic Press Limited
1. INTRODUCTION
In many applications such as in turbomachinery, the flow in a duct is not uniform but rather has a swirling motion. The amount of swirl velocity varies and is of the order of 15–45% of the mean axial velocity. The present paper examines the effect of a mean axial and swirling motion on the propagation of acoustic disturbances in an annular duct. In addition to understanding the physical phenomena associated with the propagation of pressure waves, the results of the present analysis can be used in the formulation of inflow/outflow boundary conditions for the simulation of aerodynamic and aeroacoustic problems in aircraft engine duct systems. The effects of a non-uniform mean flow on sound propagation have been studied mainly to account for the effects of shear near the duct wall and of acoustic liners. These studies point out the importance of the mean flow shear and its refraction effects on sound waves. A detailed review of the subject was given by Nayfeh et al. [1]. The analysis is usually simplified by assuming that the mean flow is parallel or near-parallel to the duct wall. In addition, the velocity field is assumed to be only a function of the co-ordinate normal to the duct wall. Although the governing equations have non-constant coefficients, solutions are sought in terms of normal modes subject to the wall boundary conditions. This results in an eigenvalue problem which is generally not self-adjoint, and therefore the normal modes may not form a complete set. In other instances, the acoustic liner imposes boundary conditions which also lead to a non-self-adjoint eigenvalue problem. Tester [2], Zorumski and Mason [3] showed that in this case algebraically growing modes may exist for multiple roots of the eigenvalue equations. The propagation of small disturbances in a duct with swirling flow was first considered by Kerrebrock [4] and more recently by Kousen [5]. Both used the normal mode analysis applied directly to the linearized Euler equations, and showed the presence of nearly-convected shear-like disturbances and pressure waves. Their results show a variety 601 0022–460X/96/500601 + 16 $25.00/0
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of possible solutions. However, in some cases the physical meaning of the unstable and nearly-convected modes remains to be clarified. In this paper we considered the case of potential swirling flows. Such flow can be generated, for example, by adding an axial vortex to a uniform mean axial flow. The study of the propagation of disturbances in potential swirling flows will help elucidate some of the basic physical phenomena associated with sound propagation in such flows, and will pave the way for understanding the wave propagation in vortical flows. The mathematical formulation can be simplified by the use of Goldstein’s [6] splitting of the perturbation velocity field into potential and vortical components. As it is shown below, such an approach gives a convenient method for the determination of the normal modes. Moreover, Goldstein’s partial splitting clearly shows that vortical disturbances are independent of acoustic disturbances, while acoustic disturbances are produced by vortical ones. This approach was used by Wundrow [7] to study the asymptotic behavior of the vortical and the vorticity-induced unsteady pressure disturbances. Golubev and Atassi [8] investigated the evolution of the unsteady pressure and vorticity in response to incident vortical waves. The results show an algebraic decay of the unsteady pressure and an algebraic growth of the vorticity with strong radial variation of their phase. These solutions correspond to the convective eigenmodes of the linearized Euler equations. The present paper examines the propagation of sound waves in an annular duct with potential swirling flow. The normal mode analysis yields a non-linear eigenvalue problem. Numerical solutions are obtained for propagating and evanescent modes. The results are compared with those of uniform mean flow and show the significant effect of the swirl, particularly on the non-axisymmetric modes.
2. MATHEMATICAL FORMULATION
The behavior of small-amplitude unsteady perturbations in an annular duct is considered with hub and tip radii rh and rt , respectively. It is further assumed that the swirling mean flow is composed of an axial flow and a free vortex flow. Thus, U (x ) = U0eˆ x + (G/r)eˆu ,
(1)
where G is the strength of the free vortex, eˆx is the unit vector in the axial direction, and
Figure 1. Geometry.
603
eˆu is the unit vector in the circumferential direction (Figure 1). The velocity, pressure and density of the fluid can be written as V (x , t) = U (x ) + u (x , t),
p(x , t) = p0 (x ) + p'(x , t),
r(x , t) = r0 (x ) + r'(x , t),
(2)
where the time-dependent perturbation quantities are assumed to be much smaller than those of the mean flow, and governed by the linearized Euler equations: D0 u /Dt + (u · 9)U = −9(p'/r0 ),
D0 /Dt(p'/c02 r0 ) + (1/r0 )9 · r0 u = 0,
(3)
where D0 /Dt = 1/1t + U · 9; r0 and c0 are the density and the speed of sound of the mean flow. Goldstein’s [6] decomposition of the perturbation velocity in terms of potential and vortical parts is then used: u = u (R) + 9f,
(4)
where the vortical velocity u (R) satisfies the homogeneous equation D0 u (R)/Dt + (u (R) · 9)U = 0,
(5)
and the potential function f is the solution to the inhomogeneous equation Lf = (1/r0 )9 · (r0 u (R)),
(6)
where L is the convective wave operator L 0 D(0 /Dt)(1/c02)(D0 /Dt) − (1/r0 )9 · (r0 9 ).
(7)
The unsteady pressure p' is then given solely in terms of the potential f, p' = −r0 D0 f/Dt.
(8)
This splitting of the velocity field has been used to study the interaction of vortical disturbances imposed on potential mean flows with uniform upstream conditions. A recent review of the subject is given by Atassi [9]. The coupling between the vortical and potential modes in equation (6) shows that, due to the non-uniformity of the mean flow, inlet vortical disturbances will produce an unsteady pressure field. However, equation (5) shows that, in the absence of an incident vortical perturbation of the velocity field, an incident acoustic field will not create vorticity if the mean flow is irrotational. Solutions to equations (5) and (6) depend on the upstream conditions of the unsteady flow. If the upstream imposed perturbation field consists only of acoustic disturbances, then u (R) = 0 and the propagation or decay of acoustic modes in governed by a homogeneous equation (6). On the other hand, if the imposed upstream disturbance is purely convected, then the inhomogeneous form of equation (6) must be used to find the particular solution representing the unsteady ‘‘hydrodynamic’’ pressure induced by the incident vortical waves. When a solid body is located downstream, acoustic waves will be created as a result of the coupling between the vortical and potential solutions along the surface of the body. This problem was examined by Golubev and Atassi [8]. In what follows the propagation of acoustic disturbances in an annulus with swirling mean flow but without the presence of a solid body is considered. 2.1. By considering only acoustic waves, the problem reduces to solving the homogeneous equation Lf = 0.
(9)
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604
In order to determine the spectrum of allowed eigenmodes, the following expansion of the potential function is assumed: f(x, r, u; t) =
g
a
a
s
a
s f mn (r) ei[kmn x + mu − vt] dv,
(10)
−a m = −a n = 1
where m and n are integer modal numbers characterizing the circumferential and radial eigenmodes, respectively. Due to the linear character of the problem, each Fourier component of the potential field can be considered separately. Using equations (8), the pressure associated with each of these components is given by p'mn = ifmn r0 (r)(v − U0 kmn − (G/r 2 )m).
(11)
The present formulation reduces the eigenvalue problem to a single second order ordinary differential equation. This brings about a significant simplification for the determination of the eigenvalues kmn and their eigenfunctions f mn. Substituting a single harmonic of equation (10) in equation (9), one obtains, after reduction, the eigenvalue problem for f mn,
6 0
1 0
d2 1 d(ln r0 ) d L2 m2 2 + + mn 2 − kmn − 2 2+ dr r dr dr c0 r
17
f mn (r) = 0,
(12)
with 1f mn (r)/1r=0
(13)
at r = rh and r = rt . The non-constant eigenvalue of the convective operator D0 /Dt is defined by Lmn = −v + U0 kmn + mUs /r,
(14)
where the swirl component of the mean velocity, Us = G/r. The properties of the eigenfunctions resulting from the eigenvalue problem (12, 13) are now investigated, and then the numerical approach used to calculate the eigenvalues and eigenfunctions discussed. Note that equation (12) can be rewritten in the form L f mn = w(r, kmn )f mn ,
(15)
where L is written in the self-adjoint form L
= (d/dr)[p(r) d/dr] + q(r),
(16)
with p(r) = rr0 ,
q(r) = rr0 [(v/c0 − mMs /r)2 − m 2/r 2 ],
and 2 b02 + 2kmnM0 (v/c0 − mMs /r)], w(r, kmn ) = rr0 [kmn
(17)
where the axial and swirl Mach numbers, M0 = U0/c0 and Ms = Us /c0 are introduced and b02 = 1 − M02 . Since the boundary conditions are homogeneous, for any two arbitrary eigenfunctions f mi and f mj ,
f mj =0, L
f mi f mj −f mi , L
(18)
605
where the inner product is defined as f mi , f mj =
g
rt
f mi f mj dr.
(19)
rh
Hence, it follows from equations (15) and (18) that
g
rt
f mi f mj [w(r, kmi ) − w(r, kmj )] dr = 0.
(20)
rh
For a typical Sturm-Liouville problem, equation (20) would represent the orthogonality condition for the eigenmodes, and the function w(r, kmn ) could be factored as f(kmn )w˜ (r). In the present case, however, w(r, kmn ) cannot be factored. As a result, the orthogonality condition does not have a universal form. Therefore, the coefficients of the expansion of an arbitrary function in terms of the eigenfunctions cannot be determined using the orthogonality condition (20). However, these coefficients can be obtained numerically using, for instance, a least squares method as in the case of the sheared mean flows (Shankar [10]). It should also be pointed out that since the boundary value problem (12, 13) is not a Sturm-Liouville problem, it is not possible to conclude that the set of eigenfunctions {f mn } is complete. However, the continuous spectrum of the convected eigenvalues for equation (5), which is not considered in the present work, may complete the set. It is interesting to note that for the two limiting cases, M0 = 0, Ms $ 0 and Ms = 0, M0 $ 0, the eigenvalue problem is of the Sturm-Liouville type. In the first case, a swirl without an axial flow, the orthogonality condition (20) has a universal form with 2 2 w(r, kmn ) = kmn . In the case of a uniform flow in a duct, w(r, kmn ) = kmn b02 + 2kmnvM0 . These limiting cases can be used as the zeroth order approximations in the perturbation analyses of the eigenvalue problem for small M0 or Ms , respectively. 2.2. (rt − rh )/rm :0 The solution of the eigenvalue problem (12, 13) is not straightforward since the eigenvalues enter the problem non-linearly, so that the problem is not of the Sturm-Liouville type. An approximate solution can be found in the limit (rh − rt )/rm :0, where rm is the mean radius of the annulus, rm = (rh + rt )/2. This is the case in many important applications of the present analysis, such as the unsteady swirling flows through the fan and compressor stages. It is further assumed that the gap (rt − rh ) remains finite, so that rm may be very large. By taking e = (rt − rh )/rm 1,
(21)
the eigenvalue problem (12, 13) reduces after the co-ordinate transformation r = rm (1 + er˜ ) to the form
6
0
1
0
17
1 d2 1 d(ln r0 ) 1 d L 2 (r ) m2 2 + + mn 2 m − kmn − 2 2+ e r dr˜ rm dr erm dr˜ c0 rm 2 2 m
f mn (r) = 0.
(22)
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606
Letting 1/rm :0, and taking into account that d(ln r0 )/dr = O(1/rm ) while erm and m/rm are of O(1), one obtains
6
7
d2 2 + gmn f mn (r) = 0, dr 2
(23)
with 1f mn (r)/1r = 0
(24)
at r = rh and r = rt . In equation (23), we defined the eigenvalue g
2 mn
as
2 2 2 = Lmn (rm )/c02 − kmn − m 2/rm2 . gmn
(25)
Solution of the eigenvalue problem (23, 24) gives a set of eigenfunctions f mn = {cos gmn (r − rh )}, where 2 gmn = n 2p 2/(rt − rh )2,
n = 0, 1, 2, . . .
(26)
and n represents the integer radial modal number. The expression for the eigenvalues kmn can now be readily obtained: kmn =
$
6
0
1 n 2p 2 m2 − M0 v˜ 2 v˜ 2 − b02 2 + 2 b0 rm (rt − rh )2
17 % 1/2
,
n = 0, 1, 2, . . . ,
(27)
where v˜ = (v/c0 ) − (mMs M0 /rm ). This approximation is compared with the numerical solution obtained below. 2.3. The eigenvalue problem (12, 13) can be written as a standard eigensystem by introducing the new variable (Press et al. [11], p. 455) cmn (r) = kmn b02 f mn (r).
(28)
Equation (12) can then be rewritten as
0
0 L
1/b02 (2M0 /b )(mMs /r − v/c0 ) 2 0
10 1
0 1
f mn f
= kmn mn . cmn cmn
(29)
The eigensystem (29) now can be solved numerically for a spectrum of eigenvalues kmn and eigenfunctions f mn using a standard technique. The numerical algorithm implemented by the IMSL library reduces the matrix to a real upper Hessenberg matrix, and then the shifted QR algorithm is used to compute the eigenvalues and eigenvectors of the Hessenberg matrix (e.g., Press et al. [11], pp. 478–486). A fourth order finite-difference scheme is used to obtain the numerical approximation for the differential operator, L
. If N is the number of grid points used in the numerical discretization of the domain rh Q r Q rt , then 2N different eigenvalues (km,1 , km,s , . . . , km,2N ) are obtained for the given frequency v and circumferential wavenumber m. The homogeneous boundary conditions (13) can be added and incorporated in the system (29). Since the original eigenvalue problem (12, 13) is not of the Sturm-Liouville type, the eigenvalues of (29) may be complex. As for the problem of acoustic wave propagation in a duct, the calculations show that, in a frame of reference moving with the axial mean flow (u) (d) velocity U0 , there are two families of eigenvalues, kmn and kmn , corresponding to upstream and downstream propagating modes, respectively. For each family of eigenvalues, there are two branches corresponding to positive and negative values of the circumferential
607
wavenumbers m. These branches are related to modes rotating in the direction of the mean swirl, or in the opposite direction. Thus, such modes will propagate with different phase speeds in the axial direction. The eigenvalues are arranged as follows. Those eigenvalues which are real and thus correspond to propagating acoustic waves are considered first. The evanescent and amplifying modes are arranged according to the increasing imaginary parts of the eigenvalues. Note that since the amplifying modes correspond to exponentially growing pressure waves, they should be discarded as non-physical. Although the evanescent modes may not be significant for the study of the propagation of sound, they play an important role in the formulation of inflow/outflow boundary conditions in unsteady aerodynamics and acoustics (Fang and Atassi [12]). 3. RESULTS AND DISCUSSIONS
In this section the spectrum of the eigenmodes resulting from the eigenvalue analysis of equation (29) is considered and examination of the effect of the mean flow on the propagation of the modes in the upstream and downstream directions for various values of the modal numbers m and n undertaken. Note that the values of m can be related, for instance, to the number of blades in a rotor-stator interaction problem. The inflow/outflow boundary conditions in these problems are expressed in terms of the upstream and downstream propagating and decaying eigenmodes. In a swirling flow, the direction of propagation of the eigenmodes depends on m. To illustrate this remarkable effect, the results of the eigenvalue analysis are presented in terms of the double set of eigenmodes (u,d) kmn for m = −30, . . . , 30 and fixed n = 1, 2, 3. The eigenfunctions are shown only for m = 0, 25 and n = 1, 5. The pressure modes are normalized so that the norm >p'> defined by N
>p'> = s (=Re (p'j ) = + =Im (p'j ) =)h = 1,
(30)
j=1
where h is the discretization step. For numerical calculations, the frequency is non-dimensionalized by the mean radius of the annulus and the stagnation speed of sound, v¯ = vrm /c¯0 . The results are presented for two frequencies, v¯ = 2·5 and v¯ = 15 for the cases of no flow, uniform flow with M0 = 0·3, and swirling flows with M0 = 0·3, Ms = 0·5 and M0 = 0·7, Ms = 0·5. The comparison between the approximate solution of Section 2.2, and the exact numerical solution of Section 2.3 is given for the case of the swirling flow. The annulus used in the numerical calculations has tip and hub radii rt = 6 and rh = 4, respectively. The results for the eigenvalues are normalized with respect to the mean radius of the annulus. 3.1. To estimate the accuracy of the numerical method, the results of the calculations are compared with the analytical solution obtained for the case of the uniform flow (M0 = 0·3 and v¯ = 15). Equation (33) (below) gives the analytical solution for the eigenvalues. Figure 2 shows the natural logarithm of the error against the natural logarithm of the number of grid points in the radial direction. The eigenvalues correspond to three downstream propagating modes, with m = 0 and n = 2, 3, 5. More accurate results could be achieved by using spectral or pseudospectral methods for evaluation of eigenmodes and eigenvalues. For the present numerical calculations, a relative error of less than one percent is achieved for N = 100, for low order eigenvalues.
. . . . A
608
Figure 2. Numerical convergence.
3.2. For Ms = M0 = 0, one obtains from equation (12) an eigenvalue problem for the Bessel equation of order m with the eigenfunctions (in terms of Hankel functions) H(1,2) m (mmn r) and eigenvalues mmn , and a simple dispersion relation, (u,d) 2 kmn = 2 zv¯ 2−mmn .
(31)
The spectrum of mmn is determined from the wall condition 1f mn /1r = 0, which reduces to
b
(d/dr)H(1) m (mmn rh ) (d/dr)H(1) m (mmm rt )
b
(d/dr)H(2) m (mmn rh ) = 0. (d/dr)H(2) m (mmn rt )
(u,d) Figure 3. Eigenvalues kmn for M0 = Ms = 0, v¯ = 2·5, m = 0 . . . 30. (a) n = 1, (b) n = 2, (c) n = 3.
(32)
609
(u,d) Figure 4. Eigenvalues kmn for M0 = 0·3, Ms = 0, v¯ = 2·5, m = 0 . . . 30. (a) n = 1, (b) = n = 2, (c) n = 3.
Figure 3 illustrates this result. For v¯ = 2·5, there are 3 acoustic modes for n = 1 allowed to propagate upstream and downstream without decay, and an infinite number of non-propagating modes. For higher n all the modes are non-propagating. However, if M0 $ 0, (u,d) 2 kmn = −v¯ M0/b02 2 {(v¯ M0/b02)2 − (mmn − v¯ 2 )/b02 }1/2,
(33)
there is a shift in Re(kmn ), and thus all the evanescent modes will propagate only upstream. Figures 4 and 5 show the eigenvalues of the pressure modes for M0 = 0·3, for two frequencies, v = 2·5 and v¯ = 15. As the frequency increases, more acoustic modes cut on, but still all decaying modes propagate upstream (Shankar [10]). This effect is due to the Doppler frequency shift. In fact, in the moving frame of reference, these modes still formally exist, but they do not propagate in any direction. However, in the laboratory
(u,d) Figure 5. Eigenvalues kmn for M0 = 0·3, Ms = 0, v¯ = 15, m = 0 . . . 30. (a) n = 1, (b) n = 2, (c) n = 3.
610
. . . . A
Figure 6. Complex eigenfunction: M0 = 0·3, Ms = 0, v¯ = 2·5, m = 0.
Re (pmn )
(solid
line)
and
Im (pmn )
(dashed
line)
for
frame of reference, such modes appear as propagating upstream, and this fact is formally reflected through the dispersion relation (33) with the shifted stationary-frame frequency v. Complex eigenfunctions p'mn are shown in Figures 6, 7 for v¯ = 2·5. The solid and dashed lines represent the real and imaginary parts of the eigenfunctions, respectively. The case of the axisymmetric radial modes with n = 1 and 5 is shown in Figure 6. Note that the pressure eigenfunctions corresponding to the cut-on propagating modes have zero real parts. For higher n, the eigenfunctions exhibit faster variation in the radial direction. For m = 5 (Figure 7), all the modes are decaying.
Figure 7. Complex eigenfunction: M0 = 0·3, Ms = 0, v = 2·5, m = 5.
Re (pmn )
(solid
line)
and
Im (pmn )
(dashed
line)
for
611
T 1 Eigenvalues for upstream-propagating modes in a swirling flow, M0 = 0·7, Ms = 0·5, v¯ = 15, m=5
n 1 2 3 4 5 6 7 8 9 10
Asymptotic solution Full numerical solution ZXXXXXXXXCXXXXXXXXV ZXXXXXXXXCXXXXXXXXV Re (k) Im (k) Re (k) Im (k) −0·44514E + 02 −0·41879E + 02 −0·30866E + 02 −0·19022E + 02 −0·19022E + 02 −0·19022E + 02 −0·19022E + 02 −0·19022E + 02 −0·19022E + 02 −0·19022E + 02
0·00000E + 00 0·00000E + 00 0·00000E + 00 −0·22286E + 02 −0·37261E + 02 −0·50348E + 02 −0·62740E + 02 −0·74782E + 02 −0·86621E + 02 −0·98331E + 02
−0·45035E + 02 −0·40816E + 02 −0·29863E + 02 −0·18915E + 02 −0·18927E + 02 −0·18932E + 02 −0·18935E + 02 −0·18936E + 02 −0·18937E + 02 −0·18938E + 02
0·00000E + 00 0·00000E + 00 0·00000E + 00 −0·22679E + 02 −0·37515E + 02 −0·50562E + 02 −0·62939E + 02 −0·74979E + 02 −0·86821E + 02 −0·98537E + 02
T 2 Eigenvalues for upstream-propagating modes in a swirling flow, M0 = 0.7, Ms = 0.5, v¯ = 15, m = −5
n 1 2 3 4 5 6 7 8 9 10
Asymptotic solution Full numerical solution ZXXXXXXXXCXXXXXXXXV ZXXXXXXXXCXXXXXXXXV Re (k) Im (k)) Re (k) Re (k) −0·63008E + 02 −0·61212E + 02 −0·55157E + 02 −0·39926E + 02 −0·26630E + 02 −0·26630E + 02 −0·26630E + 02 −0·26630E + 02 −0·26630E + 02 −0·26630E + 02
0·00000E + 00 0·00000E + 00 0·00000E + 00 0·00000E + 00 −0·26739E + 02 −0·43145E + 02 −0·57121E + 02 −0·70135E + 02 −0·82643E + 02 −0·94845E + 02
−0·67068E + 02 −0·60237E + 02 −0·55390E + 02 −0·40849E + 02 −0·26901E + 02 −0·26918E + 02 −0·26927E + 02 −0·269313 + 02 −0·26934E + 02 −0·26936E + 02
0·00000E + 02 0·00000E + 00 0·00000E + 00 0·00000E + 00 −0·26378E + 02 −0·42946E + 02 −0·56999E + 02 −0·70066E + 02 −0·82615E + 02 −0·94852E + 02
3.3. Tables 1 and 2 show the comparison of the asymptotic solution for the eigenvalues obtained from equation (27), Section 2.2, and the numerical values obtained using the method of Section 2.3, for M0 = 0.7, Ms = 0.5, v¯ = 15, m = 5, −5 and n = 1, . . . ,10. Only the eigenvalues corresponding to the modes propagating or decaying upstream are listed. Note that the asymptotic solution gives very good results, and thus may be used in the formulation of the outflow conditions for unsteady aerodynamic problems. Tables 1 and 2 show three propagating modes rotating in the direction of the mean swirl (m = 5) and four modes rotating opposite to the mean swirl (m = −5). The rest of the modes are evanescent. Figures 8 and 9 show the eigenvalues of the pressure modes for M0 = 0·3 and Ms = 0·5, for two frequencies v¯ = 2·5 and 15. Comparison with the results for a uniform flow shows that the Doppler effect has a more pronounced influence on the acoustic modes propagating in a swirling mean flow. The figures illustrate how the initially symmetrical pattern of the axial wavenumbers for a uniform flow becomes distorted for Ms $ 0.
612
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(u,d) Figure 8. Eigenvalues kmn for M0 = 0·3, Ms = 0·5, v¯ = 2·5, m = −30 . . . 30 (a) n = 1, (b) n = 2, (c) n = 3.
Compared to the case of the uniform flow, the Doppler shift causes an infinite set of evanescent eigenmodes to propagate in both directions. As before, the increase of v¯ (Figure 9) cuts on more acoustic modes. Such modes are clustered very densely for negative circumferential wavenumbers, which corresponds to the waves swirling in the direction opposite to the mean flow rotation. This is justified by the fact that the effective circumferential phase speed decreases, and thus more modes start to propagate along the duct to keep the total phase velocity in the local moving frame of reference equal to the speed of sound in the direction of the mode propagation. We now compare the effect of larger axial velocity component (M0 = 0·7, Ms = 0·5), which is more typical of turbomachinery flows. Figures 10 and 11 show the behavior of the eigenvalues for n = 1, 2 and frequency v¯ = 2·5. Compared to the results of lower M0
(u,d) Figure 9. Eigenvalues kmn for M0 = 0·3, Ms = 0·5, v¯ = 15, m = −30 . . . 30 (a) n = 1, (b) n = 2, (c) n = 3.
613
(u,d) Figure 10. Eigenvalues kmn for M0 = 0·7, Ms = 0·5, v¯ = 2·5, m = −30 . . . 30, n = 1.
(Figure 8), the number of propagating modes for n = 1 increases from 5 to 19 for those with negative m, and from 2 to 3 for those rotating in the direction of the flow. For higher n, there are no acoustic modes. It is also noted that the eigenvalues with positive m = 0, . . . , 30 are distributed over a wider range of Re (kmn ) than those with negative m. Therefore, modes propagating upstream will have their wavelengths clustered more densely compared to those propagating downstream. The complex eigenfunctions p'mn are illustrated for m = 0 in Figure 12. As for uniform flow, the propagating acoustic modes have zero real parts. Figures 13 and 14 show the pressure modes which rotate in the direction of the mean swirl with m = 5, and those in the opposite direction with m = −5, respectively. For m = 5, all modes are decaying, whereas for m = −5, the first radial mode is propagating.
(u,d) Figure 11. Eigenvalues kmn for M0 = 0·7, Ms = 0·5, v¯ = 2·5, m = −30 . . . 30, n = 2.
614
. . . . A
Figure 12. Complex eigenfunction: Re (pmn ) (solid line) and Im (pmn ) (dashed line) for M0 = 0·7, Ms = 0·5, v¯ = 2·5, m = 0.
4. CONCLUSIONS
The behavior of acoustic waves in an annular duct with a potential swirling flow composed of a uniform flow and a free axial vortex rotation was investigated. In the absence of viscosity and thermal conduction, the main effects governing the behavior of the waves are convection and refraction by the moving non-uniform medium. The splitting of the disturbance velocity field into vortical and potential parts was used to show that in the case of a potential swirl, the behavior of the pressure modes is governed solely by a single homogeneous convective wave equation for the velocity potential. The eigenvalue problem for the discrete spectrum of acoustic eigenmodes is not of the Sturm-Liouville type. A special transformation is used to reduce the problem to that of an eigensystem for two coupled functions. The method gives an effective technique to solve the eigenvalue problem. A direct normal mode analysis of the linearized Euler equations would require solving a system of four coupled equations [4, 5].
Figure 13. Complex eigenfunction: M0 = 0·7, Ms = 0·5, v¯ = 2·5, m = 5.
Re (pmn )
(solid
line)
and
Im (pmn )
(dashed
line)
for
615
Figure 14. Complex eigenfunction: Re (pmn ) (solid line) and Im (pmn ) (dashed line) for M0 = 0·7, Ms = 0·5, v v¯ = 2·5, m = −5.
The results of the numerical analysis show that in the case of a uniform flow, there are no evanescent modes in the downstream direction. As the frequency of the incident wave increases, more acoustic (non-decaying) modes cut on. For modes propagating in a swirling flow, the Doppler frequency shift causes an infinite set of evanescent eigenmodes to propagate both upstream and downstream. The acoustic eigenvalues are clustered densely for the negative circumferential wavenumbers, which corresponds to waves propagating in the direction opposite to the mean flow swirl. ACKNOWLEDGMENTS
The research was supported by the Office of Naval Research Grant No. N00014-92-J-1165, monitored by Mr. L. Patrick Purtell, and NASA Lewis Research Center Grant No. NAG3-732, monitored by Mr. Dennis L. Huff. The authors would also like to thank Alexander Balandin for his help in the numerical solution of the eigenvalue problem. Vladimir V. Golubev would like to thank the Center for Applied Mathematics at the University of Notre Dame for its support. REFERENCES 1. A. H. N, J. E. K and D. P. T 1975 AIAA Journal 13, 130–153. Aeroacoustics of aircraft engine-duct systems. 2. B. J. T 1973 Journal of Sound and Vibration 28, 151–203. The propagation and attenuation of sound in lined ducts containing uniform or ‘plug’ flow. 3. W. E. Z and J. P. M 1974 The Journal of the Acoustical Society of America 55, 1158–1165. Multiple eigenvalues of sound absorbing circular and annular ducts. 4. J. L. K 1977 AIAA Journal 15, 794–803. Small disturbances in turbomachine annuli with swirl. 5. K. A. K 1995 Proceedings of the First Joint CEAS/AIAA Aeroacoustics Conference, Munich, Germany, 1085–1094. Eigenmode analysis of ducted flows with radially dependent axial and swirl components. 6. M. E. G 1978 Journal of Fluid Mechanics 89, 433–468. Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. 7. D. W. W 1994 Contractor Report 195406, NASA. Small-amplitude disturbances in turbomachine flows with swirl.
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