flow radial turbine stages but can be transposed easily in axial-flow architectures. ... where the square brackets stand for the difference of a quantity between the ...
Generation and transmission of spiral acoustic waves in multi-stage subsonic radial compressors Michel Roger1 Ecole Centrale de Lyon, Ecully, France Stéphane Moreau2 and Aurélien Marsan3 GAUS Université de Sherbrooke, Sherbrooke, QC, J1K2R1, Canada
The paper is about the analytical modeling of sound generation and transmission in blade and vane rows of radial-flow turbomachines. Feasibility tests are made for a model centrifugal compressor with a vaned diffuser but the analysis holds for radial turbines as well, with the restriction that the flow must be subsonic in the parts of the architecture where the formulation is applied. The mathematical background is a two-dimensional theory of spiral modes superimposed on a spiral base flow, coupled with a mode-matching procedure. Typically, for a compressor, the inlet circle of the diffuser is considered as an interface where incident disturbances are scattered as reflected converging waves and transmitted duct modes into the inter-vane channels. The latter are assimilated to an array of purely radial, bifurcated waveguides. The matching procedure takes the tangentialvelocity mismatch of the mean-flow into account. The incident disturbances are either prescribed acoustic waves, for the study of sound transmission, or hydrodynamic modes for the study of sound generation by wake impingement. Only the second case is addressed in detail. The effects of the mean flow and of the geometry on the scattered waves is discussed.
Nomenclature a, b B c0 Cs H(1,2) j k=/c0 km Knj, nj m n Mr, M r r0 s Tjm vr, v V w 1
= = = = = = = = = = = = = = = = = = =
radial and tangential velocity parameters impeller blade number sound speed reflection coefficient Hankel functions of arbitrary order radial mode index acoustic wave number acoustic wave number at the mth BPF order modal projection constants Blade-Passing Frequency (BPF) order index azimuthal mode index (integer) absolute radial and tangential Mach numbers radial coordinate matching radius or reference radius reflection mode index transmission coefficient radial and tangential velocity fluctuations diffuser or distributer vane number generic wake velocity-deficit notation
Professor, LMFA, 36 Av. Guy de Collongue, Ecully 69134 cedex, France, AIAA Member. Professor, Mechanical Engineering Departement, 2500 boulevard de l’université, Sherbrooke, J1K2R1, QC, Canada, AIAA lifetime member. 3 Post-doc, Mechanical Engineering Departement, 2500 boulevard de l’université, Sherbrooke, J1K2R1, QC, Canada. 1 American Institute of Aeronautics and Astronautics 2
2 s 0 R,T
= = = = = =
impeller exit-flow angle orders of Bessel or Hankel functions mean fluid density azimuthal polar coordinate impeller rotational speed reflected, transmitted acoustic velocity potentials
I. Introduction
T
HE present study is dealing with the tonal noise generated in radial compressors or turbines by wake interactions and with its transmission through the blade and/or vane stages. The focus is on the nearly twodimensional parts of the turbomachines where the interactions usually take place. The work is aimed at providing a better physical understanding of the basic underlying mechanisms with pure analytical investigation. In turbomachinery aeroacoustics, cost and feasibility issues make the prediction of rotating-blade noise by numerical approaches a challenging task with regards to the complicated flows and geometries. This is why analytical techniques offer an alternative, at least for engineering purposes at the preliminary stage of aeroacoustic design. Thanks to drastic simplifications done on both the geometry and the main flow features, they indeed provide useful closed form solutions. They also have possibilities of extension which allow partially handling the aforementioned complexity. In turbomachinery stages composed of rotors and stators, the description of blades and vanes is one of the major difficulties to deal with in analytical formulations. A possible methodology relies on the acoustic analogy, which states that the unsteady loads on the blades and the vanes are acting as distributions of equivalent dipole sources. A model of sound generation and propagation requires two steps with this view. In the first step the unsteady loads are modeled from some assumed knowledge of the relative velocity disturbances they originate from; this can be achieved by some simplified theory of unsteady aerodynamics and is often the most challenging part. In a second step the sound field is calculated from the equivalent sources by resorting to the general background of linear acoustics and more especially to the formalism of propagation modes in a duct. Solving the first step analytically requires assimilating the blades and vanes to rigid flat plates of zero thickness and leads to more or less academic generic problems. The compressible linear inviscid theory of the response of a rectangular airfoil to incident sinusoidal gusts, typically addressed by the so-called Amiet-Schwarzschild’s technique1,2, is an example of such a generic problem. In axial-flow machines it can be applied to annular cuts of a blade or vane row of different radii within the scope of a strip-theory approach, which allows including step-by-step varying geometry and flow conditions in the spanwise direction. The main issue is that the theory only addresses an isolated-airfoil behavior. The presence of adjacent blades or vanes on the aerodynamic and acoustic responses is ignored, which makes the method only reliable for rotors (stators) with non-overlapping blades (vanes) or of low solidity. This is not acceptable anymore in centrifugal fan/compressor architectures, for example, since radial impellers and their optional vaned diffusers usually correspond to a large overlap of blades and vanes. In such cases the cascade effect, defined as the effect of adjacent blades or vanes on the overall aeroacoustic response, crucially needs being included in the analysis. The interest of cascade response functions to model axial-flow rotor-stator stages has been recognized for a long time and has motivated the development of ad hoc mathematical formulations in the literature, amongst which the recent works by Glegg3 and Posson et al 4 are key contributions. These formulations are again based on a striptheory approach in which each annulus is unwrapped and described as a linear cascade. They succeed in taking into account the cascade effect but the blades or vanes are still modeled by parallel flat plates in any strip; this generates spurious resonances that may pollute the solution, even though a correction proposed by Posson et al 5 in a broadband-noise context and more recently by De Laborderie et al 6 in a tonal-noise context significantly limit this spurious effect. Moreover this kind of approach cannot be transposed in radial architectures because the strip-theory has no equivalent and because the linear-cascade approximation would make no sense. Finally the relative lack of reliable analytical methods for radial turbomachines motivated the authors for a dedicated investigation. A first attempt is found in a preliminary analysis by Roger7, in which the definition of spiral acoustic modes was combined with the formalism of bifurcated waveguides8. Converging modes produced at the leading edge of the diffuser of a radial compressor by the rotor-stator wake interaction and transmitted into the inter-blade channels of the impeller were considered. The solution was obtained by a mode-matching procedure which involved the spiral modes of the full annular gap between the impeller and the diffuser, on the one hand, and transmitted cosine modes in the channels. The continuity of pressure and radial velocity was assumed in the tested configurations, which is 2 American Institute of Aeronautics and Astronautics
relevant only in the case of purely radial mean-flow equal on both sides of the interface. For this reason the procedure was only applied for illustration to the academic case of pure radial flow, discarding the effect of impeller rotation. A similar investigation dealing with some aspects of sound transmission through the guide vanes of the return channel of a multi-stage compressor was presented more recently by Roger & Moreau9. Now modeling any rotor or stator of a radial-flow architecture as an array of purely radial channels requires including a tangentialvelocity mismatch at the interface, for physical consistency. Indeed the mathematical problem is stated in a reference frame attached to the channels and the relative mean-flow speed has a non-zero tangential component outside the channels. The present work pursues and extends previous investigations. Firstly it is aimed at including the discontinuity of tangential velocity in the matching procedure, by resorting to more general conditions than the continuity of pressure and of radial velocity. This allows considering sound transmission in arbitrary configurations. Secondly the complementary problem of sound generation by the impingement of wakes on the front interface of a radial rotor or stator is formulated with the same approach, coupling hydrodynamic modal excitation to the acoustic response. This automatically accounts for cascade and flow effects in an ideal circular geometry, for modeling both sound generation and sound transmission mechanisms. The model is dedicated to centrifugal compressor or inwardflow radial turbine stages but can be transposed easily in axial-flow architectures. Because the unsteady loads are not considered, this mode-matching technique fundamentally differs from Amiet’s approach. It is one-step in essence instead of two-step in the sense that it directly addresses the scattering of incident vortical disturbances into acoustic waves, in the same way as what is achieved using Glegg’s approach3 by the Wiener-Hopf technique in unwrapped axial blade rows. However in its present form the technique only complies with purely radial vanes or blades. The stagger angle and the curvature of the radial compressor or turbine blades and vanes must be ignored, which is certainly abusive with regards to the true design of radial turbomachines. Yet the generic problem remains close to the configuration of a real machine. The underlying idea of the simplification in terms of radial bifurcated waveguides is that the cascade effect is more important than the exact orientation of the equivalent dipoles of the analogy. The possibilities of extension to more realistic geometries will be the matter for future work. It is worth noting that the mode-matching procedure is usually applied in mathematical physics to formulate wave transmission problems at an interface, which includes acoustic and electromagnetic waves. But because the linearized gas-dynamics equations coincide with the equations of linear acoustics in a potential base flow, it applies as well for a hydrodynamic excitation (so called abusively because of its incompressible behavior). The difference between prescribed acoustic and hydrodynamic incident waves on an interface is that the latter are simply convected, carried at the mean-flow speed, and pressure-free, according to Chu & Kovasznay’s10 analysis. The associated incompressible velocity field must be specified according to the continuity equation.
II. Jump Relationships The main generic problem addressed in this work is the determination of the acoustic waves produced by the impingement of the wakes of a radial compressor impeller on the vanes of a diffusor (or of the wakes of a distributor on the blades of a radial turbine) in a two-dimensional context. As an example for a compressor, such waves are both converging spiral waves that are transmitted upstream into the impeller and outgoing modes that propagate downstream inside the inter-vane channels of the diffuser. The starting point for the general matching procedure to be developed is provided by the conservation laws of gas dynamics in a radial turbomachine stage. The leading-edge front interface of the downstream stage component is considered as a surface of meanflow discontinuity in a domain of fluid, so that jump conditions associated with the conservation of basic quantities are used directly for the matching. The conservation of mass leads to
U 12 N 0 where the square brackets stand for the difference of a quantity between the sides 2 and 1 of the interface, the normal vector N to the interface pointing from side 1 to side 2. Here , U are the density and the fluid-velocity fields in the proper reference frame. Viscosity and heat conduction are assumed negligible. Therefore the conservations of energy and of angular momentum in a general rotating component state that the rothalpy defined as I H W 2 / 2 ( r ) 2 / 2 is conserved and that
I U 12 N 0 , 3 American Institute of Aeronautics and Astronautics
with H the fluid enthalpy, W the relative velocity and the rotational velocity. As the interface is at a constant radius the rothalpy I can be replaced by the relative total enthalpy for a rotor blade, which in turn becomes the total enthalpy for a stator vane. Writing the enthalpy H as a function of the temperature and invoking the isentropic perfect-gas equations first leads to p H C p T and t , 0 C p where C p is the constant-pressure thermal capacity of the gas. The index zero refers to the steady-state variables and the lower-case variables stand for the fluctuations, for instance T T0 t , etc. The general equations are now specified in a 2D Cartesian context, for clarity (see Fig.1). This configuration y actually corresponds to the unwrapped cascade representation of the outlet guide vanes of an axial-flow N (1) (2) rotor-stator stage for which the vanes are parallel to the axis. The infinitely thin annulus of radius r in cylindrical coordinates is unwrapped and described in the coordinates Ux ( x, y r ) . The conservation of mass leads first to the x condition Uy M a p 0 c0 u cst (1) Ux
in which M a is the axial Mach number U x / c0 and u the disturbance velocity in the x (axial) direction. The second condition deduced from the conservation of enthalpy involves the pressure and the two components of the velocity. Using Eq.(1) it is rewritten equivalently as
(S)
Fig.1 - Stator cascade leading-edge interface as a discontinuity. 2D unwrapped representation.
M a M t v (1 M a2 ) u cst
(2)
where M t is the tangential Mach number U y / c0 of the swirling inflow and v the disturbance velocity in the y direction. In this approach all mean-flow variables are assumed constant on both sides of the discontinuity. Only the meanflow velocity vector is discontinuous. Using the jump condition on the momentum would be more questionable because of the stepwise deviation of tangential velocity in the case of Fig.1. The deviation must be associated to some compatible force per unit surface in the y direction defined in the sense of generalized functions and distributed along the interface. In the special case of zero mean-flow deviation, U y 0 , Eqs.(1) and (2) obviously reduce to the continuity of fluctuating pressure and the continuity of axial velocity. These are precisely the conditions that would be written for the transmission of acoustic waves through a channel with axial bifurcations, with or without pure axial mean flow. The same 2D analysis can be transposed in polar coordinates to describe the excitation of the vanes of a radial diffusor by the wakes of a centrifugal compressor impeller or, in the case of a radial-inward flow turbine, the excitation of the impeller by the wakes of a distributor. Indeed the involved parts of radial-flow machines are embedded between annular disks which delimitate regions of negligible axial extent where the aerodynamic and acoustic motions are essentially radial and tangential. This simplification makes sense in the annulus including the outer part of a compressor impeller and its vaned diffusor (or the outer part of a turbine impeller and its distributor). The corresponding conditions read M r p 0 c0 vr cst , (3)
M r M v (1 M r2 ) vr cst ,
4 American Institute of Aeronautics and Astronautics
(4)
where Mr=a/r0 and M=b/r0 now refer to the radial and tangential Mach numbers at the interface radius r0. All velocities are expressed in the reference frame attached to the row in which the flow is entering, which means that all velocities are in the relative reference frame for the turbine case and in the absolute reference frame for the compressor case. a and b are some constants ensuring that the mean flow is piecewise rotational-free and locally incompressible. Of course M=0 inside the inter-vane channels since they are assumed purely radial.
III. Wake Modeling A. Wake Modes The wake model is illustrated here for a centrifugal compressor. In that case, the model addresses the impingement of the wakes of the impeller blades on the radial diffuser vanes. It can easily be adapted to the case of an inward-flow radial turbine by rewriting the equations in the frame relative to the impeller. The wakes are defined by a spinning pattern of velocity deficit w, function of (r , ) along the centerline (Fig.2). Streamline of the flow relative The quantity w must be prescribed in terms of velocity 2 to the diffuser vanes modes. At the mth multiple of the blade passing frequency mB , B being the number of blades of the impeller W+w and its rotational speed, it is expressed as an azimuthal Fourier series: r
wr
w(r , , t )
~ (r , ) e i mB t w mB
m
Streamlines of the flow relative to the impeller
wmB (r ) e i mB ( t )
m
Fig.2 - Velocity triangle at the impeller outlet, showing the mean velocity deficit w in the wake and its radial component wr.
The incompressibility is ensured for each azimuthal mode if both components of the associated velocity fluctuations have the form: f (r ) i mB v r wmB e ; r (5) i i mB v wmB f (r ) e mB where wmB is some constant characterizing the modal amplitude and where the radial function f (r ) has to be specified.
Both components of the velocity deficit are needed in the present simplified analysis, even if the channels are considred as purely radial, since a mean flow velocity mismatch is included. The modal velocity components of the excitation also read, according to the velocity triangle and the spiral streamlines of the flow issuing from the impeller
~ cos ; v w ~ sin vr w mB 2 mB 2 if 2 is the angle of the wake centerline from the radial direction, taken positive in the clockwise direction. This imposes the form of the function f as
f mB (r ) e i mB tan 2 ln r .
(6)
A typical wake mode (m=1) as produced by the model is qualitatively illustrated in Fig.3-top, for a 15bladed impeller with an exit flow angle of 55°. Both instantaneous radial and tangential velocity components are 5 American Institute of Aeronautics and Astronautics
shown. The latter is of larger amplitude than the former in this example. The wakes follow the spiral streamlines of the flow relative to the impeller. In order to produce an equivalent model of distributor wake for an inward-flow turbine impeller the angle 2 can be given a non-zero imaginary part that ensures that the amplitude of the velocity deficit decreases inwards according to Eq.(6); furthermore the number of blades B is replaced by the number of vanes V . The case of Fig.3-bottom corresponds to the complex angle 2 of – (1+0.05 i) 65°, which remains consistent with the aforementioned conditions. It must be noted that the additional attenuation by means of a complex angle definition can be also used for the impeller wakes of the compressor.
Fig.3 - Top: typical instantaneous wake velocity mode exciting the blade passing frequency; case of a centrifugal compressor. Radial (left) and tangential (right) velocity components. Impeller rotation in the anticlockwise direction. Same arbitrary color scale on both plots. Bottom: case of the stationary wake component of the distributor of a radial turbine.
B. Gaussian Wake Model A physically consistent wake model dedicated to the preliminary assessment of wake-interaction noise is of great engineering interest when accurate numerical simulations are not available or not affordable. It can be needed at the early design stage to select blade and vane counts. For analytical tractability a Gaussian velocity deficit is often considered in axial-flow machines. The same approach is declined in this section for radial-flow architectures, and its relevance is assessed in section C by comparison with reference simulations. According to the Gaussian model the relative velocity variation at a point of the interface reads w(t )
w0 e (t nT )
2
/ 2
n
with T 2 /(B ) . = 0.693 is a constant initially proposed by Lakshminarayana (Reynolds et al 11) and is the typical half passage time of a wake, related to the wake half width. The Fourier coefficient is derived as
6 American Institute of Aeronautics and Astronautics
wmB
B w0 2
( mB )2 /(4 ) e
(7)
If splitters are installed in the inter-blade channels two sets of B wakes having different parameters are superimposed and the expression becomes:
wmB
2 2 a1 w1 e ( m a1 ) / 1m a 2 w2 e ( m a2 ) /
(8)
where the indices 1 and 2 of a1,2 B 1,2 /(2 ) on the one hand and of w1,2 on the other hand refer to the main blades and to the splitters, respectively. In fact even if the velocity deficit is reasonably assimilated to a Gaussian profile normal to the wake centerline, it is cut obliquely by the matching interface. The resulting asymmetry questions the values of the higher harmonics of the azimuthal distortion according to Eqs.(6-7). Consistent but more complicated closed-form expressions could be derived assuming two Gaussian half-profiles of different characteristic widths. This has not been considered in the present work focused on the methodology, both for the sake of simplicity and because the effect of the asymmetry is presumably not essential at low BPF orders.
(a)
(b)
Fig.4 - Typical wake pattern for the distributor of a radial turbine, as produced by a steady-state computation. (a): tangential-speed map; (b): first azimuthal Fourier mode.
C. Validation of the Wake Model In order to assess the relevance of model Gaussian wake profiles for a radial distributor, two steady-state numerical simulations of the flow within the radial distributor of an inward-flow radial turbine have been performed. The first simulation only takes into account the distributor (Fig. 4), whereas the second also includes the turbine impeller. In the latter, an azimuthal averaging is applied at the interface between the distributor and the impeller (mixing-plane approach). Even if this technique does not allow transferring the distributor wake into the impeller domain, it stresses the influence of the mean radial pressure gradient introduced by the impeller on the wake behavior in the overlapping region between the two numerical configurations. Fig. 5 shows the absolute velocity spectrum at two different radii located immediately downstream of the distributor trailing edge (Fig.4-a) and 0.5mm downstream (Fig 5-b). At each radius, a gaussian profile has been tuned in order to get as close as possible to the stator-rotor numerical profile. The amplitude of the Fourier coefficients are compared in the upper figures. All velocity values are taken at the middle of the vein in order to avoid the influence of side-walls boundary layers. At both radii, the Gaussian model fits well the velocity deficit in the wake region. In addition, the velocity profiles are similar in the isolated-distributor and in the stator-rotor test cases, which justifies the use of the former to study the behavior of the wakes. At lower radii, only the results for the isolated distributor are available (Fig. 6). Upstream of the impeller, the Gaussian model still fits quite well the numerical velocity profile. But secondary lobes appear progressively, due to the interaction between the adjacent wakes. This results in nodal lines in the map of the first azimuthal Fourier coefficient (Fig.4-b) that are not observed with the modeled wakes. As a consequence, the Gaussian model can not represent adequately anymore the velocity profile at the impeller leading edge radius. The 7 American Institute of Aeronautics and Astronautics
envelope for the Gaussian wake model drops very fast, only the first two harmonics being significant, whereas the secondary dip and hump in the simulated profile of Fig.6-a correspond to a richer harmonic content. This test suggests that assuming an even rough Gaussian wake model is enough to make analytical predictions reliable as long as only the first two coefficients are needed. This would be precisely the case for diffuser noise due to the impingement of compressor impeller wakes if only the first two BPF tones were investigated. In contrast the tonal noise at any BPF tone for the radial turbine impeller impinged by distributor wakes involves all Fourier components of the wake profile; in this case the third Fourier component in Fig.6-b must be considered and the Gaussian model becomes questionable.
(a)
(b)
Fig.5 - Spectrum and profile of the absolute velocity (a) at the distributor trailing edge ; (b) downstream of the distributor.
(a)
(b)
Fig.6 - Spectrum and profile of the absolute velocity (a) upstream of the impeller leading-edge ; (b) at the impeller leading-edge.
8 American Institute of Aeronautics and Astronautics
In such a case, one way to improve the wake model could be to take into account the interaction between the adjacent wakes in Eq. (5) and Eq. (6), which would be initialized with a Gaussian profile at the distributor trailing edge. Fig 7 shows the evolution of the amplitude of the 3rd and the 6th azimuthal Fourier coefficients downstream of the distributor, for the two numerical test cases and the wake model. The real part of the angle of the wake has been taken from the numerical flow field (Fig. 4), whereas the modal amplitude of the wake model and the imaginary part of the angle have been tuned for each order separately in order to match with the stator-rotor test case. The influence of the interaction between the adjacent wakes on the second azimuthal mode is obvious for the isolated-distributor test case (Fig.7-a). It has a more pronounced effect on the 6th azimuthal mode (Fig.7-b), for which it takes place closer to the trailing edge. The wake-decay model becomes more and more questionable as the order or the distance from the trailing edge increases, which has to be considered with regard to the distance between the distributor and the impeller. In the present test case, the wake model already fails for the third azimutal mode. If quantitative estimates are targeted an improvement of the wake model is needed.
(a)
(b)
Fig.7 – Evolution of the amplitude of radial modes downstream of the distributor (a) Third azimuthal mode ; (b) Sixth azimuthal mode
15 w 2/w 1 = 0.36/0.4 a2/a1 = 0.6 w2/w1 = 0
wmB
10
5
0 0
5
10
15
m
Fig.8 - Fourier coefficients of the Gaussian wake profile of a centrifugal impeller.
Another test dealing with the wakes of a centrifugal impeller is reported for qualitative illustration in Fig.8. The Gaussian model Eqs.(7) and (8) are used, both in the case of an impeller with B identical blades and in the case of an impeller with B blades and B splitters. The wakes of the splitters slightly differ from the wakes of the main blades. As a result the velocity deficit still has the rigorous periodicity 2 / B with some trend to the periodicity / B . The modal spectrum of the azimuthal velocity profile according to Eq.(7) has a Gaussian envelope. In contrast the one for the impeller with splitters, according to Eq.(8), mainly involves the even harmonic orders m. Only paying attention to the first two BPF tones emitted by the diffuser, their compared levels will be not only a matter of splitter versus no-splitter design; they also depend on the number of vanes V of the diffuser.
IV. Mode Matching Technique A. Spiral Acoustic Modes in a Spiral Base Flow A simple two-dimensional theory of spiral acoustic modes in a flow has been proposed by Roger 7. The theory was developed in the frequency domain and in polar coordinates, assuming a potential spiral mean-flow extending in some arbitrary annulus, keeping in mind that the vicinity of the origin of coordinates is ignored. The 9 American Institute of Aeronautics and Astronautics
main motivation was the description of some parts of radial-flow turbomachines. Only the key aspects are reminded here. For analytical tractability small potential disturbances are assumed to develop on a slightly compressible potential mean flow only function of the radial coordinate r. The corresponding linearized equation for the disturbance potential reads (9) Δ 2i k M 0 k 2 ik ( 1) M 0 0 ,
where M 0 (r ) is the mean-flow Mach number vector, k / c0 the acoustic wavenumber and the gas constant. A consistent approximate two-dimensional base flow is considered with the radial and tangential Mach numbers Mr
a , r
b M , r
a and b being some constants. This is equivalent to neglect the compressibility and makes the second term in the square brackets of Eq.(9) zero. The approximation is acceptable as long as the radial extent of the domain remains moderate with respect to the mean radius. For instance, applying the model to the outer region of a compressor impeller including the diffuser, the constants b and a are positive for an anticlockwise rotation; the local streamlines are given by the equation r / r0 e a ( 0 ) / b and feature outgoing spirals. Modal solutions of the generalized Helmholtz equation, Eq.(9) are sought in the form (r, ) qn (r ) e i n , which leads to the equation d 2 qn 1 2 ik a d qn 2 n 2 2 nk b k qn 0 . r d r dr2 r2 The latter reduces to the Bessel equation by the further change of variable
qn e i k a ln r g n (r ) . Therefore the velocity potential of the acoustic modes is expressed with Hankel functions of real argument and of either real or imaginary orders depending on mode order n and flow parameters: g n (r ) An H(1) k r Bn H(2) k r ,
An and Bn being other constants and 2 n (n 2 kb) (ka ) 2 . With this convention the spiral modes appear as just extensions of classical cylindrical harmonics in the physics of waves, H(1) k r and H( 2) k r representing diverging and converging waves, respectively, when the convention e i t is used for the description of harmonic waves. It is worth noting that the solution can be equivalently expressed in terms of Bessel functions of the first and second kinds, as g n (r ) An J k r Bn Y k r , for convenience. This will be used in the solving procedure. B. Generated Acoustic Waves - Diffuser Case For the impeller-diffusor wake-interaction noise mechanism, the sound field at the BPF harmonic of order m is forced by the single wake mode of same order. In contrast the acoustic response includes all possible modes selected by Tyler & Sofrin’s rule12 and by the scattering properties of the diffusor considered as an array of bifurcated waveguides (Mittra & Lee8). As stated by Roger7 in the similar case of incident converging acoustic modes at the outer radius of an impeller, the generated inward spiral modes of the present case will have the acoustic potential
R e i km a ln r
Cs ei (mBsV )
s
H( 2) k m r s
with
k m mB / c0 ,
s 2 (mB sV ) (mB sV 2 k m b) (k m a) 2 , 10 American Institute of Aeronautics and Astronautics
from which the pressure and acoustic velocity fields can be deduced using the identities
p R 0 c0 i k m R M R , with
v R R and
a b M , . r r
Z 0 0 c0 is the characteristic impedance of air. Inside a reference inter-vane channel the acoustic potential has the form
T e i km a ln r
Tm cos
0
mV (1) H k m r with 2
2 ( V / 2) 2 (k m a) 2 ,
which is repeated with a phase shift in the neighboring channels.
C. Matching Equations Developing the derivations leads to the first matching equation involving the pressure and the radial velocity as e i mB i ( mB tan k a ) ln r V 2 2 m 0 Tm cos (1 M r2 ) H(1) A H(1)1 e i k m M r 2 a a 0
Cs ei (mBsV )
s
i M 2 s 2 (mB s V ) H( 2) A H( 2)1 i k m M r (1 M r ) s s a r0
1 with A k m M r 2 1 . M r The common argument of all Hankel functions, kmr0, is omitted for simplicity. One sum of the equation is reduced using the projection integrals on the transverse modes of the channels: 2 / V
cos
0 2 / V
j
e i cos
0
j
2
cos
jV d j ; 2 V
jV i 1 (1) j e i 2 / V d 2 2 ( jV / 2) 2
if
V
V
jV 2
; 00
2 V
if
; 0j 0 if
jV ; 2
j 0
For any value of j this generates an equation that relates Tjm to all coefficients Cs in the form:
j Cs Qs mB sV
s
j D mB
V
R j T jm , with:
i M Qs i k m M r2 s (1 M r2 ) (mB s V ) H( 2) A H( 2)1 , s s a r0 j 1 R j i k m M r2 (1 M r2 ) H(1) A H(1)1 , D e i ( mB tan 2 km a ) ln r0 . j j a a The second matching equation involving the radial and tangential velocity components follows as
11 American Institute of Aeronautics and Astronautics
(10)
(1 tan 2 ) e i mB i ( mB tan k a ) ln r (1) (1) 2 m 0 e i k m M r H k m H 1 r0 r0 i C s e i ( mBsV ) i k m M r s (mB s V ) H( 2) k m H( 2)1 , s s r0 r0 s
Tm cos
0
mV 2
from which the reduced form reads
j Cs Ps mB sV
s
j E mB
V
S j T jm , with:
(11)
M M i Ps i k m M r s (mB s V ) H( 2) k m H( 2)1 , r 2 , s s r r 1 Mr 0 0 j (1) 1 tan 2 i ( mB tan 2 km a ) ln r0 (1) S j i k m M r . e H j k m H j 1 , E r r0 0 It is worth noting that the tangential velocity does not contribute to the matching procedure on the channel side of the interface. Next eliminating the coefficients Tjm from Eqs.(9) and (10) generates the infinite linear system of equations
Ps Qs j E D j C ; S j R j mB sV s S j R j mB s
j 0,1, 2...
(12)
Once truncated and solved by standard matrix inversion, this system provides the values of the converging mode amplitude factors Cs. Going back to Eq.(10) or Eq.(12) finally provides the transmitted modal amplitudes Tjm. In the present case the matrix is badly condionned for inversion. Another set of equations can be derived using a projection integral on the azimuthal modal functions of the full annulus, with 2
e i n e i n d 2 n,n ;
0 2
K nj
e i n cos
0
jV i jV 1 (1) n cos ( j V / 2) 2 n (1) n sin( j V / 2) d if 2 2 ( jV / 2) 2 n 2
K nj
if
n
n
jV ; 2
jV ; K 00 2 ; K n0 K 0j 0 if n, j 0 . 2
Applied to the matching equations this yields a complementary system of equations by which every coefficient Cs is related to all coefficients Tjm:
1 2
T jm R j
j 0
Qs
1 2
T jm S j
j 0
Ps
1 T jm R j D j K mB C0 ; 2 j 0 Q0 Q0 T jm S j 1 E j K mB C0 . if s 0 ; 2 P0 j 0 P0
j K mB sV C s if s 0 ;
j K mB sV C s
Eliminating the coefficients Cs leads to the system on the Tjm: 12 American Institute of Aeronautics and Astronautics
(13)
(14)
Rj S j D E j Q P T jm K mB Q P ; 0 0 0 j 0 0 R Sj j Qs Ps T jm K mBj sV 0 ; s 0 . j 0
Instead of performing matrix inversions, Eqs.(13) and/or (14) could be used with Eqs.(10) and/or (11) to define an iterative solving procedure. Setting all values of Cs as zero in Eqs.(9) and/or (10) imposes initial values of Tjm. The latter can be used to directly calculate updated values of Cs from Eqs.(13) and/or (14) that can be used to update the Tjm in return, and so on. Preliminary implementations of the inversion of Eq.(12) by a standard pseudo-inverse algorithm and of the iterative procedure are discussed in section V.
D. Mode Reversal Issues The direct implementation of Eqs.(10) to (12) would reveal unphysical diverging growth for relatively large values of the orders s or m , obviously related to the contribution of the Bessel function of the second kind
Y (k m r ) in the Hankel functions. It must be noted that the Hankel functions H(1) and H( 2) are naturally used to expand the solution because they represent diverging and converging spiral waves, respectively; their new expressions in the presence of flow extend the definition of classical cylindrical harmonics in the physics of waves. Yet the Bessel functions of the first and second kinds could be used as elementary solutions as well. Considered separately they behave like stationary waves. Some choice of best-suited modal functions is needed here, which can be justified as follows. Assume first a full annulus in absence of flow. A spiral mode of order n cannot be transmitted from some radius to a smaller one if its local tangential phase speed is subsonic. This is equivalent to a cut-off condition simply written as kr / n 1 for the wavenumber k, only indicative in a polar coordinate system. If this condition is fulfilled the converging mode n is totally reflected as the diverging mode of same order and same amplitude. The resulting acoustic field is simply expressed by the doubled Bessel function J n (k r ) which goes to zero below the cut-off radius. Now the Bessel function Yn (k r ) is known to go to for small arguments, faster as its order increases. The diverging part of that function coincides with the range in which J n (k r ) is close to zero. Therefore Yn (k r ) itself has no physical meaning anymore. In such a case the Hankel functions have to be replaced by the Bessel functions J n (k r ) in the solving procedure. This means that the radial reversal of converging waves is implicitly included in the analysis if the cut-off radius enters the annular domain of interest. In the presence of swirling mean flow sound convection by the flow combines with the aforementioned properties. As an example the modal function
e i k a ln r e i n J k m r ,
2 n (n 2 k b) (k a) 2 ,
is plotted in Fig.9 for the mode n=6 that would be cut-off in absence of flow and for various flow conditions. The given radial and tangential Mach numbers refer to the outer radius of the domain (for instance the front diffuser interface). The sonic radius for the phase speed in absence of flow is indicated by the dashed circle in Fig.9-a. Introducing contra-rotating swirl without radial flow ( M =-0.2, Fig.9-b) is found to make the mode penetrate towards the inner radius of the domain; this is associated with a reduced order n . In contrast co-rotating swirl ( M =0.2, Fig.9-d) tends to block the propagation by increasing the cut-off; the equivalent order is increased: n . This is understood as follows. The tangential phase speed of the mode is considered in the reference frame of the analysis. With respect to the moving fluid the effective phase speed Mach number is kr/n – b/r, thus smaller than in absence of flow. The sonic radius is larger, which explains the difference between plots (a) and (d). The circle of radius / k is also plotted in Fig.9-d, where it is found very close to the exact one. 13 American Institute of Aeronautics and Astronautics
Finally the diagram of Fig.9-c suggests that adding a radial flow is also beneficial to the inward sound transmission of the mode. In fact this effect is identical for converging and diverging radial flows as long as the radial exponential factor of the modal function is ignored; it is similar to the flow-induced reduction of cut-off frequencies found in axial ducts, identically for downstream and upstream propagating modes. Though the radialflow effect is less crucial than the tangential-flow effect, it is confirmed later on in Fig.11 where radial profiles of the acoustic potential are plotted for the same test cases. The reversal is more efficient when the zero-value range of the Bessel function is larger. It is clear that for the sake of solving the matching equations from the outer radius in the annular domain of the figure, the function Yn (k r ) must be ignored in case (d). It must be retained in cases (b) and (c), sound being transmitted beyond the inner radius; the reversal must be ignored because it would occur at a smaller radius.
Fig.9 - Illustration of radial mode reversal by means of the Bessel function J n (k r ) .
k r0 =8.13. Azimuthal order n=6. (a): no mean flow; (b): contra-rotating swirl; (c): combined contra-rotating swirl and radial flow; (d): co-rotating swirl.
Depending on the parameters and on the extent of the computational domain in which the acoustic field needs being synthesized, an automatic threshold must be defined to switch from the Hankel function to the Bessel function of the first kind. This threshold is defined by the compared values of the order s or and of the argument k m r at the most determinant radius. It is fixed by the condition:
k m r1
s,
1,
r1 being the inner radius of the domain.
E. Imaginary Orders The mathematical solution generates Bessel or Hankel functions of imaginary orders that must be interpreted, especially because the square roots i
2 i for 2 0 lead to different behaviors. The
14 American Institute of Aeronautics and Astronautics
algorithms have been implemented in the present work from general formulas in Abramowitz & Stegun13. 2 is negative whenever n is within the range n M M r2 M 2 , n k b k b 1 a / b2 or kr0 introducing a reference radius r0 . One determination increases exponentially with and the other one decreases. For a diverging flow with a>0, a converging wave is expected to experience wavefront contraction and increase of amplitude with respect to what it would be without flow; in contrast a diverging wave is attenuated with wavefront dilatation. The negative imaginary part of is therefore taken for the function H( 2) to reproduce the increase and the positive imaginary part for the function H(1) to reproduce the attenuation. The opposite choice is made when a
