The Three-Dimensional Flow and Blade Wake in an Axial Plane Downstream of an Axial Compressor Rotor

Transcription

1 U Copyright 1978 by ASME $3. PER COPY 61.5 TO ASME MEMBERS 1. at Wembley,V The Society shall not be responsible for statements or opinions advanced in papers or in discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printed only it the paper is published in an ASME journal or Proceedings. Released for general publication upon presentation. Full credit should be given to ASME, the Technical Division, and the author(s). 78-GT-66 The Three-Dimensional Flow and Blade Wake in an Axial Plane Downstream of an Axial Compressor Rotor P. KOOL Research Assistant J. DeRUYCK Research Assistant CH. HIRSCH Professor, Department of Fluid Mechanics Vrije University of Brussels, Brussels, Belgium The three-dimensional flow field has been measured in an axial plane downstream of a low speed axial compressor rotor with a rotated single slanted hot wire. A method is described which allows one to calculate three mutually perpendicular velocity components from hot-wire data, and use is made of the technique of periodic sampling and averaging to extract the pitchwise fluctuating flow from the stationary hot-wire signals. These data contain useful information. The radial distribution of the pitchwise averaged flow variables is compared with classical pneumatic measurements and with the results of a quasi three-dimensional finite-element calculation and a three-dimensional end-wall boundary layer calculation. Finally, the wake characteristics are given and a simple correlation is presented which allows one to determine the wake velocity defect from a single wake shape factor. *Aangesteld Navorser aan het Belgisch Nationaal Fonds voor Welenschappelyk Onderzoek. Contributed by the Gas Turbine Division of The American Society of Mechanical Engineers for presentation at the Gas Turbine Conference & Products Show, London, England, April 9-13, Manuscript received at ASME Headquarters December 2, Copies will be available until January 1, THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS, UNITED ENGINEERING CENTER, 345 EAST 47th STREET, NEW YORK, N.Y. 117

2 The Three-Dimensional Flow and Blade Wake in an Axial Plane Downstream of an Axial Compressor Rotor P. KOOL J. DeRUYCK CH. HIRSCH NOMENCLATURE A end points of prong carrying hot-wire B a = increment of probe angle C = blade chord, constant in King's law CL = blade lift coefficient d = half-wake width D = wake deficit parameter E = anemometer voltage F = wake function G = percent span from hub H = shape factor of blade wake H 12 = ratio of $ 1 to a2 H32 = ratio of 6 3 to 62 H^ 3 = ratio of g 1 to g 3 K' = wake shape factor K = wake shape factor m = massflow deficit in wake N = number of angular positions of probe p = pressure R = test facility S = blade pitch, least-squares criterion U = rotor speed V = absolute velocity W = relative velocity y = tangential coordinate a = angle o, = solidity = yaw angle of hot-wire, blade deflection A = blade stagger angle S = displacement, momentum and energy thickness P = air density 4 = downstream distance from trailing edge T= friction force Subscripts and Abbreviations ax = axial d = defect, drag e = effective, edge of wake g = gaussian i = current index in = at mean radius n = normal to hot-wire sensor o = constant p = pressure side of wake, probe r = radial in hot-wire reference frame R = radial in meridional plane s = suction side of wake t = trailing edge value tot = total 3d = three-dimensional INTRODUCTION In the past, several methods have been used to measure the blade-to-blade flow field in rotating machinery (1-8). 1 In all of them, use is made of three hot-wire readings obtained with a single hot-wire or an array of wires, and three mutually perpendicular velocity components are derived from it. In the first part of this paper, a method is presented which allows one to improve the accuracy with which the velocity components can be derived from the hot-wire data. The method is more reliable, too, because more than three data are used, but it is more timeconsuming than previous methods, or an array with more than three wires is needed. In the present investigation, up to six data were recorded with a single hot-wire turned around into different positions. All data are taken into account as the results are reduced with a least squares technique. The method is described in the case of a steady flow vector. However, it is easily extended to the measurement of the blade-to-blade flow as the technique of synchronous sampling reduces the time-periodic flow to a space-periodic flow. The hot-wire reduction technique is applied to successive points in the latter space. In a second part of this paper, experimental results, obtained with this technique, are compared to an inviscid quasi-three-dimensional finite-element calculation and a three-dimensional 1 Underlined numbers in parentheses designate References at end of paper. 1

3 reference pressure dust filters disk light! phot cell settling chumbei rotor IGV zeroing transdu- - cer lgalvcinomj left- right pressure go lvano m. transdu. total static downstream throttle-valve pitot probe i^ ^ axial pu se counter anemometer oscilloscope Imeanvoltaoel averager Fj x y recorder paper rms puncher meter Fig. 1 Sketch of the blowing facility and setup of the instrumentation end-wall boundary-layer calculation. At the hub, the calculation is less satisfactory due to the jump conditions between the stationary inner wall and the rotating rotor blade row at inlet and exit. Moreover, the need for detailed measurements close to the exit plane of the rotor is stressed. In the third part of this paper, the experimental blade wake profile parameters, such as width, defect, and shape, are given. It will be shown that the wake defect is well predicted by either the shape factor, H 12 (ratio of displacement thickness to momentum thickness), or the shape factor, H 13 (ratio of displacement to energy thickness). One of these parameters is sufficient if the wake shape is known, as is the case in the far wake where the shape is nearly gaussian or sinusoidal. These results follow from the assumption of a defect model for the wake, to connecting the blade-wake parameters to the trailing-edge boundary-layer parameters is possible. EXPERIMENTAL SETUP Fig. 1 is a sketch of the blowing facility, R1, at the Von Karman Institute, Brussels, Belgium. The driving motor is a 75-hp d-c motor, continuously variable in speed. Air enters the settling chamber, goes through a set of dust filters, the inlet guide vane, the rotor, the diffuser, and the throttle valve. The flow rate is adjusted by opening or closing this valve. The air speed is conventionally measured using pitot-static NACA short prism probes. Pressures are measured using Statham strain gage pressure transducers with direct readout on a galvanometer. Hot-wire measurements have been performed with a Disa 55M1 anemometer. The mean anemometer voltage is measured with a Disa 55D31 digital voltmeter. An oscilloscope is used to check waveforms and for the selection of the angular position of the hot-wire probe. The hot-wire signal is further processed by the box-car integrator which is used for synchronous detection. The averaged waveform is recorded on an X-Y recorder and punched on paper tape. The box-car integrator is synchronized with the rotor by a series of pulses generated with a disk on the rotor axis and a photocell. In the disk, there are as many holes as blades on the rotor, and these pulses are also used to measure the rotor speed. The blade characteristics are listed in Table 1. The working point is indicated on the rotor characteristic in Fig. 3. MEASURING TECHNIQUE Hot-Wire Configuration The hot wires used to measure the threedimensional flow are slanted; i.e., the wire sensor makes an angle, a o, with respect to the sensor of a straight wire (Fig. 2). Several values for a o have been used in the past, and the reader is referred to Reference (3) for a discussion. The probe axis is introduced radially through the compressor wall, and if the flow is nearly two-dimensional, the configuration shown in Fig. 2 is obtained. A spherical coordinate system is defined in the wire reference frame in which a velocity vector is defined by its magnitude, V, the probe angle, a p, and the radial angle, a r. The yaw angle,, which primarily determines the effective cooling velocity, is then given by: sin = cosao cos a cos a + sin a sin ar (1) The wire is calibrated for changes in velocity at the position characterized by 4=. and ar. (2) 2

4 Table 1 Blade Characteristics of the Rotor and the IGV Rotor blades Naca 65 - (A1C) 6 Number of blades : 25 C % Tip CL A S(mm) C(rm) a Tip clearance :.75 mm Tip radius : 35 1.mm Hub - tip radius ratio :. 71 Number of blades:39 G % A S (mm) C (mm) Thickness ratio 6 % I.G.V Fig. 2 The hot-wire configuration showing the probe-angle, a. the radial angle, a r (positive to rotor hub), the yaw angle,, of the sensor and the absolute velocity, V LD tot This calibration yields a relationship which will be used to convert measured anemometer voltages at any radial or yaw angle into effective velocities. This may formally be written with King's law: F2 - E2 V= ( C 2 ) ( 3) Additional calibrations are performed to obtain information about the directional sensi-.3 tivity of the. slanted hot wire. Experiments have.3 shown that in the configuration of Fig. 2, the '5Vax use of a simple cosine directional sensitivity law is too simple to be realistic for slanted Utip wires. These additional calibrations are actually Fig. 3 Location of the investigated working performed by selecting a few values of the radial point on the characteristic curve of the rotor angle, a r ' and of the velocity magnitude which cover the operating range. At these parameter equation (3) and the yaw angle is determined from combinations, directional calibrations are per- equation (1). By inspecting the velocity and yaw formed by rotating the hot-wire about its axis, angle data so obtained, an improved directional The probe angle, a, can hereby be deduced from calibration law can be deduced for that particuthe protractor readings. The measured voltages lar wire. are converted into effective velocities with This relation will formally be written as 3

5 ve = f^ (v,) (4) More details about a practical and accurate equation for (4) can be found in References ( 1-3 ). The formulations about the existence of a unique solution and the limitations in the use of slanted wires, including the effect and the choice of a particular wire angle, a, can also be found in Reference ( 3 ). Some authors would call the angle, a. the "yaw" angle of the probe. This name would have been in conflict with the cosine law of the yaw angle of the hotwire, which considers the velocity component normal to the wire as the cooling velocity, or V = V Cos e Description of Measuring Method To obtain the three velocity components of a three-dimensional flow field, hot-wire data must be taken at at least three probe orientations. The use of more than three data allows reducing the data with least squares techniques, thereby decreasing measuring errors. A close inspection of the calibration law, equation (4), will reveal that among the several hot-wire positions which can be realized by rotating the probe around its axis, there is at least one at which the wire is most sensitive to the magnitude of the velocity and that there are several positions where the wire is primarily influenced by changes in the flow direction. This is most important in these applications, as the absolute flow angle is largely fluctuating as has been demonstrated several times in the past (1-5 ). Some angular positions will have to be discarded as the longest prong should be kept behind the smaller one, but sufficient positions are available in a wide range of probe angles covering at least 18 deg. In principle, any three locations can be selected, but good results are obtained with the following choice. A first position is chosen to coincide approximately with the position of maximum sensitivity to the velocity magnitude. As this maximum is quite flat versus the probe turning angle, the true maximum is ill-defined. To select the second and third measuring position, the probe is turned farther while keeping the short prong in front of the longest until the anemometer voltage reaches a minimum. The wire sensor is now aligned with the flow, and this position occurs at approximately 9 deg from the previous one. The second and third measuring position can now be selected to the left and the right of the position where this (5) minimum occurs. Hereby one can take advantage of the fact that there are two symmetrical positions of the probe where the wire is most sensitive to fluctuations in the probe angle, a p, and thus to the fluctuations of the absolute flow angle. These positions are easily revealed by a strong increase of the amplitude of the fluctuations of the anemometer signal, but they can also be derived mathematically from the calibration law in equation (4). If more than three measurements are used, consecutive positions at 1- or 2-deg increments can be realized covering a range of about 18 deg. A comparison of different measuring techniques including those of References ( 4-6, 8 ) is presented in Reference (3). Data Reduction The measured anemometer voltage at the several protractor readings for one velocity are first converted into effective velocities with equation (3); and with equations (1) and (4), these are expressed as functions of the yaw angle and the still unknown velocity. In each angular position of the probe, equation (1) is applied as: sin 4 i = cos a or cos a cos (a P + a i ) sin + a sin ar i = 1, 2,...N In the first measuring position, we have (6) a 1 = (7) The other increments of the protractor reading, a i, are then referenced to the first position. The set of equations (6) then contains V, a p, a as basic unknowns. Actually, these can be computed by different iterative techniques ( 1, 2). To avoid the straight solution of these nonlinear equations, one can use nonlinear optimization techniques. In this method, different combinations of values of the unknowns are selected on a logical basis and are put in equations (1) and (4) to determine an effective cooling velocity, Vi, until the following least squares criterion is satisfied: 1 N E( V(Ei) - vi)2 ` S (8) N i=1 The effective velocity V(E) is derived from the measured voltage, E, and the effective velocity, Vi, is computed from the chosen values of the unknowns. The sum is extended over all data 4

6 L w , 3, d.5.fl absolute velocity '.1 '2.3 '.4 '.5 '.6 '.7 '.8 '.1. relative velocity I A B tangential angle abe G=56.7 s G=56.7 Q.Q z7.q TTT7II 1A:5 2: G=2.l =12,1 27.a IIIII 2l.4Q tangential angle rel O..l LQ'^ o, radial velocity axial velocity Fig. 4 Typical blade-to-blade evolution of the flow variables points and the best choice of parameters is found when the sum of squares is smaller than the se- lected criterion S. In practice, a value of about 1-6 was used. The criterion is made small enough to get accurate results. A too small Fig. 5 Contour plot of axial velocity between rotor hub and rotor tip and covering one blade pitch value increases the number of iterations. Therefore, the value of S is a compromise between accuracy and computing time. Once the velocity components are determined in the reference frame of the wire, these can be transformed to a reference frame fixed to the compressor. Therefore, the coordinates of the wire sensor have to be determined, but this does not usually represent any problem. It should also be noted that the absolute flow quantities are always obtained first. Therefore, the relative flow angles are strongly influenced by the accuracy with which the absolute velocity is obtained, whereas the relative velocity is influenced primarily by the absolute flow angle. RESULTS The foregoing method has been applied to the flow downstream of an axial compressor rotor at,34 chords from the trailing edge. Fig. 4 shows some typical results at different blade height (measured from rotor hub). As the absolute velocity and the relative velocity are approximately orthogonal, the relative velocity profiles and the absolute flow angle profiles in the pitchwide direction are similar except for the sign of their change. The same holds true for the absolute velocity and the relative flow angle. Therefore, a large defect in the blade wake leads to high local skewing in the absolute frame. The blade wake is well defined and the characteristic parameters are discussed later. A review of the 5

7 In/ s ^C co'^d^ ^,cdodo ox xquasi-3d _3BL 1 X HOT W. PITOT Fig. 6 Contour plot of absolute flow angle between rotor hub and rotor tip covering one blade pitch -2 HUB Fig. 7 Contour plot of radial velocity covering one blade span and one blade pitch (positive toward hub of rotor) most important flow variables is shown in Figs. 5, 6, and 7, giving the blade-to-blade flow between rotor hub and rotor tip (1 percent span). As can be seen from Fig. 5, the end-wall boundary layer is quite thick. At.34 chords downstream, the rotor wakes have spread out considerably as can be seen on the same figure and also in Fig. 6, where the high flow skewing in the rotor wake and in the end-wall boundary layer can be observed. Out of the wake, the flow is quite uniform. Fig. 7 shows that this region of uniform 5 G 1% Fig. 8 Comparison of mass-averaged hot-wire axial velocity with pneumatic data and theoretical results flow is characterized by a radial velocity which is nearly zero except near the pressure side of the wake at about 3 percent span, where the largest positive values indicate flow toward the rotor hub. At the same height, the wake is centrifugated. In the end-wall boundary layer, the radial velocity decreases and becomes negative near the wall where the flow is directed toward the end-wall. The mass averages of these data are compared with the results of an inviscid quasithree-dimensional finite-element calculation ( 1 ), and a three-dimensional end-wall boundary layer calculation ( 11 ) in Figs. 8, 9, and 1. As the amount of blockage upstream of the IGV was not determined experimentally, this parameter was adjusted in the end-wall calculation until satisfactory agreement with experiment was obtained. The error bound on the experimental results amounts to 2 deg on the absolute flow angle and to 1 percent in absolute velocity. The high level of the error on the absolute velocity is due to the problem of keeping the hot-wire clean during the tests and, hence, to have a reliable and reproducible calibration curve. As shown in Fig. 8, the agreement is worse at rotor hub. This is evidenced in the absolute flow angle profile in Fig. 9 which shows that the skewing is too high and the boundary layer too thick at the hub. The need to include the end-wall boundary layer calculations is stressed in Fig. 1 in which the mass-averaged hot-wire results are compared with the quasi-three-dimensional calculation. The error bound is 2 deg, and, therefore, the agreement 6

8 a R 7 1 X 5 x X X 3DBL x xxx X 6 XX X X q-3d X) X quasi-3d jx xx 5 X XX 5 hot-w. xxxxxx 5 G 1% 5 G 1 Fig. 9 Comparison of mass-averaged tangential Fig. 1 Comparison of mass-averaged radial meflow angle obtained with hot-wire with pneumatic ridional flow angle obtained with hot-wire with data and theoretical results the results of the quasi-three-dimensional calculation is satisfactory out of the boundary layer, but the increasing values near the end-wall are not predicted. +1 W (Y I ) in = pw d (1 - )dy' (9) Description of the Blade Wake Profiles d e We In the pitchwise relative velocity pro- -1 files, the blade wake profiles are often strongly distorted. In the following, a description of where y represents the dimensionless tangential the relative velocity profile will be given, distance defined as (Fig. 9) Radial velocities contribute to this profile but are not described. Therefore, the profile is,- y (1) treated as the two-dimensional evolution of am- Y d plitude versus pitch, though it is three-dimensional. The profile will be described by the with d the half-wake width and W e the inviscid shape, the width, and the velocity defect of the edge velocity. The gaussian wake is described as wake, and the analysis is equally valid for the axial velocity profile. _ n,2 The classic definition of the wake width W(Y') = We - W e Y (11) as being the extent at a specified velocity level is difficult to apply with distorted profiles, where W d represents the defect velocity. The such as in turbomachines. Therefore, an alterna- equivalent mass defect is set equal to the meative definition is proposed here. As the overall sured defect, and shape is often well described by a gaussian curve, the distorted wake is replaced by a (12) and = p Wd de gaussian-shaped wake having the same velocity defect and the same mass defect. The latter is from which an equivalent half-width can be obgiven by tained and _ (13) dg pwd X TI 7

9 WI a H32 = a2 (19) W V2d then we obtain the following results We-Wd d D F (Y')dy' 1 (2) H 12 F(Y')dY,. y Fig. 11 Blade wake nomenclature Fig. 1 shows the evolution of the equivalent gaussian half-wake width from rotor hub to rotor tip. If a defect-law is used to describe the wake as in equation (11), the correlation between the shape factors, H12 and H 13, is independent of the wake defect. This will be proven and a useful expression for the wake defect is given. The relative velocity profile is written as a defect law: W(Y')=1 - D.F(Y') (14) W e H 32 = 2H 12 ) (1- HL ) + ( 1 - H1 )2K with (21) J F 3dy'. J Fdy' 8 K (22) ( ( F2 dy, ) 2 Table 2 shows that for current shape functions as the gaussian curve and a sine curve and even for less common shapes, the value of this K parameter does not change much. If K is accordingly put equal to unity equation (21) can be easily solved for 1-1/H 12 to yield with D the velocity defect in dimensionless form H 1-1 = ( 1 _F H 12 3 (23) Wd (15) D W e The wake is completely defined by its shape, expressed by the function F, its width -d- and its defect -D-. The functional forms, which come into consideration in describing the wake, are monotonic functions and they satisfy the boundary conditions F()=1 (16) F(1)= (17) If we now define the displacement thickness, the momentum thickness, b 2, and the energy thickness, a 3, with these profiles and determine the dimensionless parameters, H 12 and H32, given by: It is interesting to note that the velocity defect parameter is not present in equation (21). The correlation between the shape factors, H12 and H32, therefore, only depends on the shape of the wake functions. Hence, if the velocity defect model, which was assumed at the beginning of this paragraph, is considered to describe the wake with sufficient accuracy, using different shape functions, F, then the correlation equation (21) may be used. The correspondence between equation (23) and the observed values are shown in Fig. 13(b) and the good agreement shows that even for the very distorted profiles of Fig. 13 (a), the correlation is quite well observed. Equation (2) also shows that the velocity defect parameter, D, should be put in the functional form d H 12 = a1 (18) 2 D = (1- ---)K' (24) 12 where the parameter, K', is only dependent on the 8

10 Table 2 Change of the Wake Shape Parameter, K, with Different Shape Functions Description < / y'/ < 1 F(y') K square cosine sin 7 (y' -. 5) 1.13 Gauss e Try.968 Linear 1 - y' 1 sine 1 - sin (Try' /2) parabolic (1 - y') exponential e yrkm.8)+)+. 1/n - powerlaw 1 - y 1/n 3 (n+3) 4 (n+2) n - powerlaw (1 - fl9 (3n+1)(n+1) ') 8 (2n+1)2 shape of the wake profile. We believe that it should be possible to correlate this K' parameter with the downstream pressure gradient. As the gradient was not known during our experiments, it was not possible to verify this hypothesis. However, the fact that equation (21) is independent of the velocity defect suggests that it should also be possible to apply the model to boundary layers. This is confirmed by experiment ( 6 ), and an application of this principle is found in Reference ( 11 ). From this investigation followed that the parameter, Kr, is closely related to the streamwise pressure gradient, an average value of KI outside the wall boundary layer is given by K i 1.45 (25) one has further 1<K' - 2 A typical profile is given in Fig. 14. With (26) H31 = H32 (27) equations (24), (25), and (23) yield a second equation to predict D D = 2(1-i 5H31) (28) The agreement between equation (28) used with the experimental value for H 31 and the experimental defect is shown in Fig. 15. The very low defect velocities measured in Reference (8), downstream of a rotor, are very well predicted with equations (24) and (25). The shape factors, H 12, were derived from the wake profiles in stations 1, 2, and 3 given in Reference (8). These were found to be 1.54, 1.35, and 1.33, and these low values are in agreement with the rapid decay of the wakes. The higher values of 9

11 d S r)u.5 o1.4.3 o o percent span Fig. 12 Radial evolution of the equivalent gaussian wake width a5.o =7725 % P S Fig. 13(b) Comparison of the theoretical correlation with the experimental values of the shape factors, H 12 and H 13, obtained at various axial locations and different radii and operating conditions K,I Q 1 '^ '^ '4 '^ ' ^ '^ ' ^ `^1 ' re i at i ve ve i oc i tx Fig. 13(a) An example of a distorted relative PERCENT SPANS Fig. 14 Evolution of the blade wake shape factor, KT, from rotor hub ( percent) to rotor tip (1 percent) velocity profile whose shape factor has been included in Fig. 13(b) of the wake are different as is currently observed in turbomachinery wakes, equations (18) to H12 in our experiments are probably due to an (23) should be applied separately to both sides. increased static pressure with downstream dis- However, it is more practical to add the distance, as could be observed with the pitot-static placement and momentum thicknesses of pressure probe. This adverse pressure influences the and suction side to get a single H 12 parameter change in momentum thickness as shown in Refer- and to extend the integrations in equation (2) ence ( 8 ), and thus also the change of H 12 and the over the whole wake. wake defect. In principle, the decay of the shape param- If the pressure side and the suction side eter, H 12, with downstream distance can be 1 1

12 or 6 ^ 8 O x ^^ oo x x x X p_side x x x O O x o s -side 7 )L x EXP. X a o 6 o From H31 xx o x S 5 G 1 Fig. 15 Radial evolution of the blade wake velocity defect and comparison with results derived from the shape factor, H 13, obtained from the relative velocity profiles determined if the downstream pressure gradient is known or if the gradient of the inviscid edge velocity is given. Neglecting the changes in flow angle, one can determine the displacement thickness from the continuity equation W e (Scosa - d l ) = constant (^9) and the momentum thickness from the Von Karman equation +2) dwe d6 (H12 62dC + Wed^ = (3) The parameters of the wake close to the trailing edge of the blades can be linked to the blade boundary layer profile parameters, if the model equation presented in Reference ( 6 ) is used to describe the blade boundary layers. In this model, a velocity defect is defined for turbulent three-dimensional boundary layers and an empirical correlation is given. At the trailing edge, it is logical to approximate the wake defect by the blade boundary layer defect. If the pressure gradient is not taken into account, this defect is related to the friction force at the trailing edge 2T Dt = 1-1 (31) PWe 1, xo X 9 9 x 6 XO x x X X X Q 1 cs2 S2 5 G 1 Fig. 16 Radial evolution of the wake shape factor, H 12, for the pressure side and the suction side of the wake This correlation predicts a large defect when the friction decreases, as it is the case also near separation. In classical correlations, the wake defect is proportional to the square root of the drag coefficient which is in contradiction with equation (31), Fig. 16 shows a typical evolution of the wake shape parameter, H 12, from rotor hub to rotor tip. Near the tip, the low values indicate that the wake has diffused strongly as is evidenced also by the increased wake width in Fig. 12. This is in qualitative agreement with the well-known fact that the three-dimensional skewed wall boundary layer is damped by the action of the wall. This effect modifies the absolute flow angle, and, therefore, measurements must be performed close to the trailing edge of the blades to obtain correct information about the exit flow. As an example, the deviation angles will increase with downstream distance especially near the wall. CONCLUSIONS An improved technique, based on least squares optimization, to reduce hot-wire data in three-dimensional flows has been presented as an extension of a previous method. Reasonable agreement between hot-wire and pneumatic data were obtained. At the end wall, the absolute flow is highly skewed. Downstream of the trailing edge, the absolute flow angle is modified due to the damping of the skewing. Therefore, the flow deviation angles determined at the measuring station are too high and a comparison with blade data is 11

13 inaccurate. Reasonable agreement between the experimental results and finite-element calculations, including a three-dimensional end-wall boundary layer calculation, are obtained. The end-wall boundary layer is highly skewed, and fairly large radial velocities are obtained in the blade wake. The blade wakes are often irregular. Nevertheless, a defect-law model seems to give good results. The predicted correlation between the shape factors, H 12 and H 13, is very well observed by experiment. A wake defect law is proposed, but information is needed about the decay of the shape factor, H 12, and the shape of the wake at different distances from the trailing edge. The shape of the wake can be characterized by a single parameter, and, as in many circumstances, this shape is well described by a sinusoidal or a gaussian curve of this parameter is known. At the trailing edge, the wake shape can be related to the boundary layer shape, but a decay law for the shape parameter should be used especially in the near-wake region where the wake shape changes quickly. ACKNOWLEDGMENT We would like to express our gratitude to the Von Karman Institute (Brussels, Belgium) who made these measurements possible. REFERENCES 1 Kool, P., "Experimental Investigation of the Three-Dimensional Flow Field Downstream of Axial Compressors," Ph.D. thesis, Vrije Universiteit Brussel, Stromingsmechanica, Jan Schmidt, D. P., and Okiishi, T. H., "Multistage Axial-Flow Turbomachine Wake Production," Transport and Interaction," Interim Report, ISU-ERI-Ames-7713, Iowa State University, Nov Hirsch, Ch,, and Kool, P., "Measurement of the Three-Dimensional Flow Field Behind an Axial Compressor Stage," Transactions of the ASME, Vol. 99, Series A, No. 2, April 1977, pp Whitfield, C. E., Kelly, J. C., and Barry, B., "A Three-Dimensional Analysis of Rotor Wakes," Aeronautics Quarterly, Vol. 23, Part 4, Evans, R. L., "Turbulence and Unsteadiness Measurements Downstream of a Moving Blade Row," ASME Paper No. 74-GT Kool, P, "A Model Equation for Three- Dimensional Turbulent Boundary Layers," Report, Vrije Universiteit Brussel, Stromingsmechanica, VTJB-STR-6, Raj, R., and Lakshminarayana, B., "Three-Dimensional Characteristics of Turbulent Wakes Behind Rotors of Axial Flow Turbomachinery," ASME Paper No. 75-GT-4. 8 Raj, R., "On the Investigation of Cascade and Turbomachinery Rotor Wake Characteristics," Ph.D. thesis, Department of Aerospace Engineering, The Pennsylvania State University, Nov Raj, R., and Lakshminarayana, B., "Characteristics of Wakes Behind a Cascade of Airfoils," Journal of Fluid Mechanics," Vol. 61, 1973, p Hirsch, Ch., and Warzee, G., "An Integrated Quasi-3D Finite Element Calculation Program for Turbomachinery Flows," Paper submitted to the 1978 Annual International Gas Turbine Conference, London, England. 11 De Ruyck, J., Hirsch, Ch., and Kool, P., "An Axial Compressor End-Wall Boundary Layer Calculation Method," Paper submitted to the 1978 Annual International Gas Turbine Conference, London, England. 12