Copyright 1984 by ASME THEORETICAL AND EXPERIMENTAL DETERMINATION OF THE TRANSFER FUNCTION OF A COMPRESSOR. by Jacques Paulon

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1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47 St., New York, N.Y C ^r The Society shall not be responsible for statements or opinions advanced in papers or in G J discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printed only if the paper Is published in an ASME Journal. ^[ Released for general publication upon presentation. Full credit should be given to ASME, the Technical Division, and the author(s). Papers are available from ASME for nine months after the meeting. Printed in USA. Copyright 1984 by ASME 84-GT-283 I THEORETICAL AND EXPERIMENTAL DETERMINATION OF THE TRANSFER FUNCTION OF A COMPRESSOR by Jacques Paulon Head of Compressor Research Group Office National d'etudes et de Recherches Aerospatiales (ONERA) Chatillon (FRANCE) NOMENCLATURE a speed of sound m sec -1 A area of inlet duct m 2 F LAPLACE transform axial length of compressor m L axial length of inlet duct m M Mach number static pressure PO P mean static pressure Pa /J non dimensional LAPLACE parameter t time sec V velocity m sec -1 Z abscissa m P flow acceleration m sec - 2 Is non dimensional abscissa (9 non dimensional time P density kg m3 0 non dimensional parameter 6 time lag sec CO frequency sec reduced frequency Subscripts 1 upstream of compressor 2 downstream of compressor SS steady state C compressor Superscripts complex amplitude of perturbation terms / fluctuation I. INTRODUCTION Aerodynamic operation of a compressor during transients and any unsteady regime is controlled by its response to transient pressure and mass flow fluctuations. Knowledge of such a response is necessary for the prediction of the behavior of the compressor when submitted to inlet total pressure maldistributions. This is the case of airplane engines at high incidence angle conditions or in the case of a transverse gust. Few experimental or theoretical results can be found in the open literature on the transient response of a compressor (1) (2). On the other nand many empirical transfer function have been pr^,ocsed (ref. (3) to (7)) but they were not validated or discarded after comparison with test results. That is why an experimental research at low Mach number and near one dimensional conditions has been started at ONERA (8) in order to be able to choose among the various transfer functions proposed and, if necessary, find new forms for transfer functions. The following tests were made (i) the compressor was subjected to a sudden throttle area change, that induced a corresponding step change in mass flow and compressor outlet pressure ; (ii) a throttle area modulator was installed, that induced periodic mass flow and pressure fluctuation. Examples of the corresponding test results and theoretical data analysis will be presented. 2. DESCRIPTION OF THE TEST FACILITY A schematic view of the test compressor is shown on figure 1. The compressor is a rotating annular cascade followed by an annular stator cascade. In order to reduce the 3-D effects, a high hub-to-tip ratio (0.957) has been choosen. It was verified that for moderate back pressures the flow is actually one dimensional (axisymmetric and practically independent

2 of radius, outside of the wall boundary layers that are relatively small). The inlet duct is equipped with a filter, settling chamber and converging duct that insures uniform flow at the inlet of the rotor. A conical outlet duct, with a conical center-body, decreases the outer diameter of the exhaust channel. 1 Knife Throttle Peripheral gap I'I Outlet duct Table I gives the main geometric and aerodynamic parameters of the facility. Table I Geometric and Aerodynamic parameters Inlet ductllength 0.80 m Diameter 0.20 m Settling chamber Volume 0.12 m3 Wall pressure Transducers 11111^^ P ty Inlet Filter ng Settlin Chamber. `^ Motor Cornpressor diameter m Number of rotor blades 46 Height of the blades m Blade relative outlet angle 40 Number of stator blades 50 Axial length of compressor m Speed of rotation 3000 to 6000 rpm Maximum mass flow 2 kg/sec Maximum power 30 kw Maximum axial velocity 120 m/sec t t t t t 1a) Step throttle area variation Peripheral OD^ gap Electric motor Rotating disk Fixed disk Modulator Outlet Length 0.70 m duct { Outlet diameter m Outlet duct In configuration (i) designed for a step variation of the outlet area (fig. 1a), a paper sheet was placed downstream of the compressor outlet section. A sharp knife with multiple blades driven by a fast actuating device ruptured the paper membrane in a time the duration of which was short compared to the duration of the transients. In order to avoid unsteady operation before the rupture of the membrane a peripheral gap was managed between the diffuser outlet and the membrane. Therefore the step operation took place between two steady operating points of the compressor. In configuration (ii) an area modulator was used instead of the punctured membrane (fig. 1b). This modulator was made of two identical disks, one of them fixed, the other, driven by a small variable speed electric motor, rotating at a fixed speed. Both disks have a great number (16-32 or 64) triangular openings, the maximum passage area being close to half of the duct area. A frequency range between 80 and 1200 Hz was covered. To avoid compressor stall in the case of a completely closed valve operation, a peripheral gap was also used in this configuration. In both cases (i) and (ii) the time-wise area variation was small compared to the area of the pheripheral gap. 3. DATA ACQUISITION AND DATA REDUCTION Transient wall pressure transducers (Kulite) were used in both configurations. The following measurements were made : Wall pressor Transducers Pt I,-... Filter - Motor Inlet 1 b) Modulated throttle area Fig. 1 Schematic view of the test compressor. (i) Step variation of outlet area (fig. 1a) Wall static pressure measurements were made in the three following stations (1) upstream of the rotor (2) downstream of the stator (3) at the outlet of the diffuser channel, slightly upstream of its end section. The transient total pressure at the entrance of the inlet duct was also measured by means of a Kulite equipped Pitot tube. (ii) Modulation of the outlet area (fig. 1b). Wall static pressure measurements were made in stations (1) and (2) only. The total pressure at the entrance of the inlet duct was measured in this case also.

3 The analog pressure signals were recorded on magnetic tapes, digitized and reduced to obtain the transfer function of the compressor as will be explained below. In the case of the modulated throttle area tests, direct measurement of the ratio of upstream to downstream pressure peaks were also measured by means of an electronic amplitude meter. A Fourier analyzer was also used in some cases. Typical examples of oscillogram pictures taken by means of a Polaroid camera are shown on figure 2. The top image, figure 2a, corresponds to a step variation of the throttle area. Comparing wall pressure measurements in stations (3), at the end section of the outlet duct, and (1), at the rotor inlet, one notes the short duration of the throttle opening, compared to the transients. It shall also be noted, that at station (2), outlet from the stator, the pressure signal is, at a different magnitude, similar to the imposed pressure step. The practically constant value of the inlet total pressure shall also be noted. Two examples of wall pressure fluctuations induced by the rotating throttle valve are shown on figure 2b for two frequences of the area modulator. The recorded signals are periodic, but higher frequency disturbances are noted in some cases. Pressure Wall static pressure 1 - Inlet to rotor 2 - Outlet from stator 3- End section of outlet duct Pty - Total inlet pressure Wall static pressure 1 Modulation frequency : Inlet to rotor 2 - Outlet from stator Wall static pressure 0 Modulation frequency : Inlet to rotor Outlet from stator 2b) Modulated throttle area Fig. 2 Examples of pressure measurements. 3

4 No correct explanations have been given for these perturbations. 4 DEFINITION OF THE COMPRESSOR TRANSFER FUNCTION The pressure rise through a subsonic compressor is function of inlet conditions. There exists therefore a transfer function that gives one of the pressures (the upstream or the downstream one) when the other is known. In the case of our test compressor, the axial length of which is small, the volume of air enclosed in the blade channels is small. The height of the passage being the same upstream and downstream of the compressor, it is assumed that upstream and downstream axial velocities are identical, although time dependent. The transfer function can thus be expressed either as a relation between upstream and downstream total pressures (the measurement of which is difficult) or between upstream and downstream static pressure, easy to measure by means of wall pressure transducers. Data reduction, however, is different for the two configurations. 4.1 Step variation of the throttle area It is quite easy to show that direct transfer functions such as the relaxation type relation z d(p2 fzo )_ dp1 / 11,o)-f^<-^xo) (3a) dt dp, on the time-laq., -lie ^2 7z^ = ^P a P I/7+ ^`,a^ (3b) at time t-3 do not correctly represent the phenomenon. In these relations time lag may be taken as the time necessary for a particle to travel from rotor inlet to stator outlet and d c!i is the steady state slope of the upstream to downstream pressure characteristic. The corresponding transfer functions (4a) The LAPLACE transformation of the upstream and downstream pressure variations is the easiest way to get to the transfer function. We use the following integrals for the LAPLACE transformation (9) or P1 14/ ( 1^ Oaf e -(^^(t^^ie(to^)it / at/ (4) _ c` e P t 8(t) -y2 (to)) cl t to (1) = dp e (4b) ^`^ dp where M is the Mach number corresponding to the inlet axial velocity, tend both towards zero as4increases, and do not reflect the asymptotic behavior of the experimental results. where to is the time when pressure starts to decay in station (2). An accurate value of to is easily obtained. Non-dimensional parameterdvaries as t -1 and, for any regular function,d- infinity corresponds to time t o andd- 0 to an infinite time, but no other correspondence exists between time t and parameter 4. Figure 3 gives an example of variation of 't(1) and 'i (4) as well as that of N ^A This function is the LAPLACE transform of the transfer function F/e, that according to the well-known integral relations of the LAPLACE transformation (9). A more sophisticated way to search for a transfer function is therefore necessary. Since it was verified that the flow through the compressor is nearly one-dimensional, the momentum and mass flow equations can be written as axf fvz avf + ap _ at az Paz t Vz. all Vz -O pat Pay az az where f/,( is for the body force at abscissa z. If only small perturbations are considered, and if non-dimensional abscissa and time (5) (t) (t0)_ ` dfx/ t a) (4>(A) ^,<t^da (2) gives the value of'tjt) when e1 (t) is known. The great difficulty in using LAPLACE transformations is that once image F) is known it is difficult to obtain its original F^a(t) Therefore the only way to use f/ (-") is to compare it to various empirical transfer functions. are defined, the following linearized system of equations is obtained avz t M avz + ± DL9 3 pa/ a (6) a) t M ap,.^. avz O paa9 Pc` D` a^ 4

5 u where dashes indicate small perturbations. The steady state value of r' is obtained by setting to zero the time derivatives obtained from experimental values of /i.f and /7,2 with the theoretical one deduced from the simple delay law rss _(7 M)^c(P2^(7) dp, P Using LAPLACE transformation for Eqs. (6) one obtains cl. p a cl a C//2 + t a = O /Oa Pads 'c (8) 5n LAPLACE transform of transfer = ^5 (12) at time t-' and the LAPLACE transform of the transfer function P 1/4 becomes x/^ - r f 1 -'J/il +M e j/ ^ /) a VZ I-I F, e /1/- M 2 `I which emphazises two facts (1) - F/.f increases exponentialy asdbecomes infinite, and this is verified by experimental evidence, (2) -F/a is function of the ratio Uz^, i.e. of the part of the test facility that is upstream of the rotor inlet. (9) 0.^ S , i Non d,mcns,onal LAPLACE PARAMETER Fig. 3 Step throttle area variation. Laplace transforms of upstream and downstream pressures and of the 80,1, LAPLACE transform of transfer function 00, / Fen 201,00 transfer function F2 f =p2/pt. Let us assume that the inlet duct and the inlet settling chamber can be assimilated to a single constant section duct, the area of which A is large compared to the inlet area A c of the compressor. The length of this duct will be called L. By means of a one-dimensional model similar to (8) but in which the mean Mach number can be neglected we obtain the following relation between z and 'St I Theoretical value of f2, using a lag type law for body forces Evpenmental values of _, avz (M 4Q l 0 (10) where L is the length of the equivalent inlet duct and a=//a. Thus, s is 2 25 Non dimensional LAPLACEparameter Fig. 4 Step throttle area variation. Comparison of theoretical and experimental transfer functions. for which the LAPLACE transform is t^ 2 I Mt^tccn^ ( L/ / e h (11) /^ /l N)> ^^e (13) t ' J+ + ^ P P ^^i'h 4.2 Periodic throttle area variation ( n+ atanh(4l_/z) 4'/ The periodic upstream and downstream pressure signals can be developed in FOURIER series of which we will analyze the first term only. Therefore using the above non-dimensional parameters we write We tried out various expressions for P and ^12 B/ several of them could fit correctly the shape of the I _ e curve FL/-T (A) of figure 3. As an examp-le we show on } ^^ trzo/ / n e figure 4 the correspondance between Fl/-f (ii) l 5

6 Li where 'z, and '72 are independent of time (but eventually functions of frequency) and ft is the reduced frequency / indicates the module of a complex parameter. The transfer function can be directly derived from equation (11) h+^a tcth (SLL/) l7 +cotah (rzl/8) n^, 4/ e=m 1 It is now easier to discuss the physical meaning of the various assumptions made on the shape of P" If we assume the time lag law o jiation (12), the expression dp p assumes that for a given inlet pressure the body force intensity and time lag is independent of the frequency. On the other hand if a relaxation type relation such as (3b) is used (14) (15) dp^ ^^n9 - ^^ta(n1) (16) the body force and the time lag are both decaying when frequency increases. If a decrease of body force intensity with increasing frequency seems to be a sound assumption, it is more difficult to adopt an assumption for which the time lag decreases as frequency increases. Therefore we prefer relation (15) and figure 5 shows the correspondance between transfer function Fz^ based on equations (14) and (15) and the corresponding test results. Transfer function defined no ratio of downsrreom 15^ to upstream pressure fluctuation amplitude zn. 10/ Non dimensionof pul Lion + 0, =20 t1/n 0 25() An Frequency F (Hz) 5. CONCLUSION Using test results obtained on a subsonic compressor with near one-dimensional flow and transients induced by a time wise variable area it has been shown that during transients inlet and outlet pressures are functions of both inlet axial velocity and inlet static pressure. The relation between these parameters depends on the inlet duct geometry. It was also shown that prediction of the response of the compressor to the transients requires the resolution of the momentum and the mass conservation equations through the compressor, at least under their simplified one-dimensional form that takes into account body forces. No completely satisfactory assumption on the transient body forces has been found up to now. However the assumption that body-forces lag with a constant delay behind the inlet pressure variation that induces them seems to give the best theoretical description of test results REFERENCES (1) Peacock, R.E. and GAS, D.K. - Compressor response to pulsed transients AIAA/SAE/ASME 16th Joint Propulsion Conference 30 June-2 July 1980, Hartford, AIAA n 80/1080. (2) Peacock, R.E. and Eralp, O.C. - Compressor response to spatially repetitive and non repetitive transients, Israel Joint Gas Turbine Congress, Haifa July 9-11, 1979, paper 79 GT-Isr 14. (3) Fabri, J. - Amplifications of distorsions in an axial flow compressor stage. CIMAC-GT Congress, Tokyo (1977) TP ONERA no (4) Ferrand, P., - Etude theorique des instabilites de 1'ecoulement dar_s les compresseurs axiaux. These de Doctorat de specialite Universite d'aix-marseille UER-IMF (1980). (5) Takata, H. and Nagano, S. - Non linear analysis of rotating stall. ASME 72 GT 3. Journal of Engineering for Power (1972). (6) Adamczik, J.J. - Unsteady fluid dynamic response of an isolated rotor with distorted inflow. AIAA paper (1974). (7) Greitzer, E.M. - Surge and rotating stall in axial flow compressors. Trans. ASME Journal of Engineering for Power (1976). (8) Fabri, J. and Paulon, J. - Experimental and theoretical determination of the transfer function of a compressor. Symposium on aeroelasticity in turbomachines (IUTAM) Lausanne, (1980). TP ONERA n (9) McLachlan, N. and Humbert, M. - Formulaire pour le calcul symbolique. Gauthier Villars (1941). Fig. 5 - Comparison of theoretical and experimental transfer function for a compressor with periodic modulated throttle area. 6