THE PREDICTION OF THE TIP CLEARANCE VORTEX CIRCULATION AND ITS INDUCED FLOW FIELD IN AXIAL FLOW MACHINES

Transcription

1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS E. 47th St., New York, N.Y di' 73' The Society shall not be responsible for statements or opinions advanced in papers or discussion at meetings of the Society or of its Divisionary Sections, or printed in its publications. Discussion is printed only if the paper is published in an ASME Jounial. Authorization to,photocopy, material for internal or personal use under circumstance not falling within the fair use provisions of the Copyright Act is granted by ASME to libraries and other users registered with the Copyright Clearance Center (CCC) Transactional Reporting Service provided that the base fee of $0.30 per page Is paid directly to the CCC, 27 Congress Street, Salem MA Requests for special pennission or bulk reproduction shied be ad:... dressed to the ASME Technical Publishing Department. Copyright C 1996 by ASME MI Rights Reserved Printed in U.S.A.. :, THE PREDICTION OF THE TIP CLEARANCE VORTEX CIRCULATION AND ITS INDUCED FLOW FIELD IN AXIAL FLOW MACHINES I.K. Nikolos, 0.1. Douvikas, K.D. Papailiou Laboratory of Thermal Turbomachines National Technical University of Athens Athens, Greece ABSTRACT A model for the prediction of the leakage vortex circulation was developed, based on the assumption that the leakage jet flow enters as a whole the vortex core, increasing its radius and its moment of momentum in the direction of the vortex axis. Using the assumption that the leakage vortex has a solid body rotation, an expression was derived for the vortex circulation, which demonstrates that this circulation is proportional to the square root of the corresponding tip clearance height. This theoretical result is supported by the available.experimental data for both compressors and turbines. A simple model was developed, which demonstrates the ability of the proposed theory to calculate the leakage vortex circulation, provided that the vortex trace is known. A method for predicting the tip clearance effects in the flow field inside and downstream the blade passage, compatible with a meridional flow calculation procedure, has been developed by the authors. The method uses incompressible considerations and accounts for the calculation of the circumferentially mean deficit radial profiles of the various flow quantities. In the calculation procedure the tip clearance flow effects are considered as a modification to the basic flow, existing in the absence of tip clearance. The complete calculation procedure was used in order to calculate the leakage vortex circulation and the induced velocity field in various axial flow cases, with satisfactory results. distance from the vortex centre (figure (2)) i,j,k orthogonal Cartesian coordinates (figure (1)) static pressure radial position vortex core radius distance along the vortex axis surfaces of the vortex control volume (figures (1), (2)) V velocity V m circumferentially mean flow velocity v r Rotational velocity at a radius r v x Axial velocity component w u Relative peripheral velocity Axial direction Greek symbols P' blade camber angle Pv vortex axis angle with respect to the axial direction leakage vortex circulation As finite length along the vortex axis Ax finite length along the axial direction 0 angle of velocity component V sn (figure (2)) density vortex axis angle with respect tol gthe blade camber to angular velocity of the leakage.,yortex flux of moment of momentum O NOMENCLATURE chord c y axial chord discharge coefficient e tip clearance Subscripts i,j,k orthogonal Cartesian coordinates (figure (1)) jet leakage jet normal to the surface direction normal to the blade camber line direction Presented at the International Gas Turbine and Aeroengine Congress & Exhibition Birmingham, UK June 10-13,1996

2 FIG.(1): SCHEMATIC REPRESENTATION OF THE LEAKAGE VORTEX CIRCULATION MODEL, AND THE CORRESPONDING CONTROL VOLUME. FIG.(2): THE CONTROL VOLUME FOR THE CALCULATION OF THE FLUX OF THE JET MOMENT OF MOMENTUM ENTERING THE VORTEX. pr pressure side su suction side direction along the blade camber (figure (4)) 1,2,3 surfaces of the vortex control volume (figures (1), (2)) INTRODUCTION The calculation of the induced kinematic field as well as the estimation of the losses downstream of the gap exit depend on the correct computation of the strength and position of the leakage vortex. The formation and the evolution of this vortex is quite similar to that of the finite wing case and previous researchers insisted on using prediction methods tailored to this last case. Lifting line theory, originally developed to predict lift forces on finite wings, was introduced by Lakshminarayana (1964) in the turbomachinery field, in order to predict the effects of the leakage vortex in the kinematic field and the induced drag due to the leakage vortex presence. The measurements carried out by Lakshminarayana and Horlock (1965) demonstrated that only a fraction of the bound blade circulation was shed out of the blade tip in the form of the leakage vortex and a portion of the lift was assumed to retain across the gap, in order to predict the correct amount of induced drag. An empirical relation for the fraction K of retained lift through the gap was proposed, based on their experimental data. The calculations of Lewis and Young (1977) also showed that, in order to predict the effects of the vortex, the retained lift assumption was required and they also proposed an empirical relation for the estimation of K. Additional experiments exist which confirm that the circulation of the leakage vortex is a fraction of the blade's bound circulation but do not indicate directly the validity of the "retained lift" theory, which stipulates that the shed out voracity is produced by part of the vortex lines of the blade, the rest going towards the endwall and producing the existing pressure difference along the gap. Measurements of Inoue et al. (1985) and Yaras and Sjolander (1990) showed that the vortex strength is a fraction of the bound circulation which tends to unity as the clearance approaches values greater than the typical used in axial machines. However, the total amount of leakage vorticity in terms of circulation measured by Inoue et al., was found substantially larger than predictions using Ukshminarayana's model. Measurements also made by Yaras et al. (1989) show that the pressure difference across the gap is almost constant for typical values of tip clearance gap and equal to the value at the blade tip. Various researchers, such as Yamamoto (1989), Inoue and Kurumaru (1989), Sjolander and Amrud (1986) demonstrated that the strength of the leakage vortex is increasing with increasing tip clearance height. This increase in the amount of vorticity of the leakage vortex is followed by an increase of the vortex diameter. In addition, the data of Inoue and Kurumaru (1989) demonstrate that the shed circulation depends also on the relative wall speed. In the case of a compressor this relative wall motion increases the shed out vorticity, while in the case of a turbine it decreases the amount of vorticity shed into the blade passage. This fact was confirmed by the experimental work of Yaras et al (1991) for a turbine cascade, where the relative wall motion was simulated by a moving belt. The above experimental evidence seems to suggest that the key factor for the determination of the amount of shed out vorticity is the mass flow rate through the tip clearance. A model for the calculation of the leakage vortex circulation was developed by the authors (Nikolos et al. (1993a)), based on

3 3a 3b FIG.(3): ALTERNATIVE RADIAL POSITIONS OF THE VORTEX CENTRE, FOR THE APPLICATION OF THE CIRCULATION MODEL moment of momentum considerations. Additionally a model was presented by the authors (Nikolos et al. (1995)) for the prediction of the influence of relative wall motion in the mass flow rate through the gap, in both compressors and turbines. It was demonstrated, that the correct computation of the reduced mass flow rate in the turbine case, led to the correct calculation of the leakage vortex strength. Some further developments and more details of the vortex circulation model are presented in this work, while comparisons with experimental results give additional support to the validity of the proposed calculation method. THE LEAKAGE VORTEX CIRCULATION The flow field near the endwall is characterised by the interaction of two opposite flows. The secondary flow, having a direction from the pressure to the suction side of the passage, and the leakage jet flow, coming out of the gap at an angle to the main flow. The leakage flow rolls down away from the enchvall, along a separation line, and forms the leakage vortex. In the outer flow, this vortex interacts with the tip-side passage vortex, rotating contrary to the leakage vortex. At the same time, as we march downstream, the additional mass from the tip clearance jet entering the vortex, increases both its radius and its strength. Flow visualization in incompressible cases suggests that the mass leaving the gap exit enters in its quasi-totality inside the leakage vortex. The measurements performed by Lakshminarayana (1970), Inoue et al. (1989), Yaras et al. (1990) etc., indicate that the leakage vortex possesses a near solid body Fla(4): SCHEMATIC REPRESENTATION OF THE MODEL IN A PLANE PARALLEL TO THE BLADE TIP, rotation structure. The two observations made above, were used, in order to develop a model for the prediction of the leakage vortex circulation. Figure (1) presents schematically the adopted model, along with an elementary control volume for the application of the integral equations, the mass conservation equation and the moment of momentum equation. It is assumed that the mass flow coming out of the gap is wrapped around the existing solid body rotating vortex, increasing its radius and the moment of momentum component in the direction of the vortex axis. Additionally, it is also assumed that the vortex mass flow rate enters and exits the control volume, from the normal to its axis sections, uniformly and practically in the direction of the vortex axis. No mass exchange occurs at the conical surface of the control volume, except for the region corresponding to the leakage jet. The moment of momentum equation We consider a local Cartesian coordinate system i,j,k with i in the direction of the leakage vortex axis as in figure (1). Assuming an axisymetric vortex with a solid body rotation, we have for the rotational velocity V r V, car (I) where w is the angular velocity and r the radial distance from the vortex axis. The circulation at the vortex radius R, associated with a solid body rotation is given 3

4 moment of momentum in the direction of its axis. Using the moment of momentum conservation component in the i direction we have equation for the + = 0 1/2 je (6) where D iet is the flux of moment of momentum of the leakage jet, with respect to the vortex axis. Using equations (5) and (6) we may have Ili R12 4, Pt rar: 4 (7) or P, = (r,vr; 411J) 1 V2.4 (8) The radius R of the vortex at the current section can be calculated assuming that all the jet flow enters the vortex core. Then we have from mass conservation FIG.(5): THE LEAKAGE VORTEX INDUCED VELOCITY FIELD IN CALCULATION PLANES INSIDE AND DOWNSTREAM THE BLADE PASSAGE. R:V; = RV As e V3 It (9) r= f P, = 2n co R 2 (2) The leakaqe let moment of momentum flux in the direction of the vortex axis From figure (2) we have for D iet The flux of the moment of momentum in the direction of the vortex axis, passing through a section of the vortex normal to its axis, associated with the rotation of the vortex, is given 2n R 0, = f f V, r V, rdr d. = 2n f lit r 3 dr (3) 00 Assuming a constant velocity in the direction of the vortex axis we have O f = 2n V. co 4 or using equation (2) a t - ViR 2 4 (4) (5) = 1/2. h V; ds = a sf V3 Z sine h R db 5, Ft = R cos0 a, (10) Assuming a uniform distribution for the jet velocity component normal to the vortex axis we have = As R ic 2 sine h db ia 0 The pressure term is not appearing in the moment of momentum equation, as the surface S3 is normal to the radius and the cross product of R and n 3 is zero, where n3 is the normal to 53 unitary vector. As is a finite distance, measured along the vortex axis, denoting the distance between the sections 1 and 2. From figure (2) we have (12) We consider the control volume of figure (2) where the vortex edge is at the endwall. The leakage jet is assumed to enter as a whole the leakage vortex, increasing its radius and the and then equation (11) becomes 4

5 o, = As R2 V3,2f sine cos0 de (13) level. Then we may have AP V 2p (22) Taking into account (from figure (2)) that and equation (18) becomes cos@ = R-e (14) V3) = V3 s cos( Hsin(,) 2pV., (23) equation (13) fmally yields = As R V 2 e[1- la a l 2R The jet flow velocity V 3p4 normal to the blade camber is (15) calculated as a function of pressure difference With a similar procedure, if the vortex edge is at the tip level (figure (3a)) we have V 3, = C D 2AP (24) Die = As R V3,2 e[1. 2R1 (16) where C D is the discharge coefficient. Then equation (23) becomes while, if the vortex edge is at the level of the mid gap height (figure (3b)), we have 1. = As R V 3 2 e (1 Calculation of the normal to the vortex axis let velocity component The normal to the vortex axis component V 33 of the jet velocity can be written (figure (4)) \I 2AP AP D CO MN C sin(4') (25) As we may see V3 j is a function of the pressure difference distribution and the angle of the vortex axis, with respect to the blade camber. Integral form of the expression for I Assuming that the velocity V i along the vortex axis is equal to the mean flow velocity V m, equation (9) becomes V3, a V -V1/2 sin(ck) (18) As e V3 2 R22 Val = &V at + It (26) where N and S denote the normal and longitudinal components of the mean jet velocity and v is the local vortex angle with respect to the blade camber line. The longitudinal velocity is assumed to be conserved through the gap, due to the dominance of the pressure difference across the gap against the pressure gradient along the blade chord, so which provides the evolution of the vortex radius. Additionally, equation (8) becomes = (r ic ilc2 +4Cl1) 1 a 17,2/2; (27) V 3, = Pr (19) or from equation (26) where V pr is the corresponding velocity at the pressure side of the blade. Taking as V velocity at the suction side and using Bernoulli's equation we may have 2AP We assume that ( Vpr) (Vn-V) (20) = (21) where Vm is the circumferentially mean flow velocity at the tip 2 r2 = (r i f 1.12, +4fain) 2 AseV31 4/ThiR,+ n Using equation (17) for the moment of momentum of the jet, equation (28) finally yields 1 = (1`1Icie+4AsR1eV3i2) Ase V3 (29) cari + IT

6 with V3 given by equation (25). 1 Equation (26) can be written in the form A(R 2K.) - As i V 3 s n and if As tends to ds e d(r 2V.) =ids and finally R 2 V, = f V3 ds o (30) where s is measured along the vortex axis. A similar analysis for equation (27), using equation (17), results to the following expression d (rr 2v,,,) = 4 R V3)2 e ds (33) and finally riev. =44 R 1/3/2 els (34) The radius R can be calculated using equation (32) as (35) Using equations (32), (34) and (35) we finally have for the circulation r = 4 f' 1V3i 2 f Isrv3 ds l ds O (36) 113, ds If cbc is an incremental distance along the axis of the machine and j3 is the vortex axis angle with respect to the axial direction, we have for ds ds = dr cos ( f1 3,) (31) (32) (37) and equation (36) becomes for the circulation shed all along the blade chord 4 v 2 v I "f drift coso),tv : a coos) (38) C. v I " dx coo) As we may observe from equation (38), the circulation r is proportional to the square root of the gap height. From equation (25) it can be seen that the normal to the vortex axis component of the leakage jet velocity is a function of the pressure difference AP at the tip region and the angle of the leakage vortex, with respect to the blade camber. Additionally, from equation (38) it may be seen that r is a function of V 33 and the angle pv of the leakage vortex, with respect to the axial direction. Mean flow velocity V ol is a function of the blade geometry, as it can be approximated as V.= I cos( (39) where p' is the blade camber angle. Consequently, the leakage vortex circulation, seems to depend on the blade loading, the blade geometry and the vortex peripheral position, while it varies with the square root of the gap height. However, the peripheral displacement of the vortex depends on its strength, and this fact demonstrates a complicated connection between the position of the vortex and its circulation. CALCULATION METHODS Two levels of approximation were used for the calculation of the leakage vortex circulation. The more simple one, was set up in order to demonstrate the ability of calculating the variation of leakage vortex circulation with clearance, if the vortex trace is known. The second one uses the complete calculation procedure, developed by the authors for the calculation of tip clearance effects. The simple model A simple model was set up for the calculation of the leakage vortex circulation, as a function of the gap height. Equation (38) is used for the variation of the leakage vortex circulation with clearance, while equation (39) provides the distribution of the mean flow velocity. The static pressure distribution at rnidspan is used for the calculation of AP. A modification is adopted for the first thirty percent of chord (Nikolos et al (1993a)), using a linear variation of AP for the corresponding region. A constant value of the discharge coefficient CD is used, which was set equal to 0.8. A constant value along chord, for each different case, was adopted for the vortex axis angle t v with respect to the blade camber. 6

7 =CXD Yams et al. experiment SAW Lokshminarayana et al. exp. Inoue et al. experiment SORT(e) FIG.(6): THE AVAILABLE EXPERIMENTAL DATA SHOW A LINEAR VARIATION OF VORTEX CIRCULATION WITH THE SOARE ROOT OF GAP HEIGHT. CIZaID `fares et of. experiment 0:1203De Cecco et al. experiment SQRT(e) FIG.(7): THE LINEAR VARIATION OF SHED CIRCULATION WITH THE SOARE ROOT OF CLEARANCE, EXTENDS OVER A WIDE RANGE OF CLEARANCE VALUES. The complete calculation procedure The complete calculation procedure will be described with the remark that the tip clearance flow will be introduced as a modification to the basic flow, existing in the absence of tip clearance effects. Consequently, the tip clearance flow is introduced as a third zone in the already existing two-zone model, which is used for the calculation of the secondary flow effects. The implementation of the complete calculation procedure in a secondary flow method and the models used in this procedure, are discussed in detail by Nikolos etal. (1993a), (1993b), (1995). The meridional-secondary flow calculation method used in this work, is discussed in detail by Douvficas et al. (1989). The flow field without tip clearance effects is established through the meridional flow calculation method, including secondary flow effects. Peripherally mean flow quantities are calculated and it is possible to compute, through momentum considerations, the forces acting on the blade surface. Then, the corresponding static pressure difference can be computed, if it is assumed that the shear stresses at the blade surfaces can be neglected. This pressure difference, provided by the secondary flow calculation procedure, does not include the contribution of the leakage vortex. This contribution may be computed, using Newman's (1959) model for the diffusion of a line vortex, in order to model the effect of the leakage vortex on the static pressure field (Nikolos et al. (1993a)). The vortex strength and position is required for this computation, so that an iterative procedure must be set up in order to provide the various flow quantities, which need for their calculation the modified pressure difference at the blade tip. However, only two iterations are used in the present work, as a compromise between time consumption and correct calculation of the static pressure distribution at the suction side of the blade. The formation of the leakage vortex and the calculation of the induced kinematic field is treated in successive planes normal to the machine axis, in order to provide a better information exchange between a meridional flow calculation procedure and the tip clearance method. At the considered station, a first guess for the static pressure difference may be made, from the results of the secondary flow calculation procedure. Then the mean jet velocity is calculated, using a varying discharge coefficient along the blade chord, depending on the local bladethickness-to-clearance ratio. Additionally, the amount of losses inside the gap is computed. The conservation of the longitudinal component of momentum through the gap provides, finally the angle of the jet at the gap exit. The computation of the shed out voracity is performed between the two successive planes, using equation (29). The resulting vortex strength corresponds to a voracity distribution in the second of the two successive planes. The vortex radius is calculated using the mass balance equation (26) and taking into account the diffusion process. Lamb's (1932) model, for the diffusion of a line vortex, is used for the distribution of the vonicity in the current calculation plane. However, the axis of the leakage vortex is not perpendicular to the calculation planes, so that the angle of the vortex trace must be taken into 7

8 atn Yaras et al. experiment Auss Lakshminarayano et al. exp. 032C Inoue et al. experiment e/c FIG.(8): COMPARISONS BETWEEN THE CALCULATION OF SHED CIRCULATION, USING THE SIMPLE MODEL, AND THE CORRESPONDING EXPERIMENTAL DATA. CEITODYaras et al. experiment MAY Lakshminarayano et al. exp. EC O:Olnoue et al. experiment Simple model 'Yams et al.) Simple model Lokshminarayana et al.) Simple model Inoue et al.) step, resulting from the flow velocity at the vortex centre, considered in the absence of tip clearance effects, and the distance between the calculation planes, are used in order to compute the peripheral displacement of the vortex centre. The radial position is estimated by assuming that the vortex core edge lies at mid gap height, between the endwall and the blade tip (figure (3b)). The induced velocity field is calculated using the solution of a Poisson equation in each calculation plane. For calculation planes inside the blade passage, a zero normal to the wall velocity is used as boundary condition, for all the boundary nodes except for those at the tip clearance exit, where the leakage mass flow rate determines the boundary values of the stream function. The zero normal to the wall velocity condition is used at planes downstream the trailing edge, only for the hub and tip boundaries. Periodic conditions are used for the remaining boundaries. Figure (5) presents the calculated velocity field in successive planes inside and downstream the blade passage. RESULTS AND DISCUSSION The theory presented above for the calculation of the leakage vortex circulation, demonstrated that the vortex circulation is proportional to the square root of gap height. This fact is supported by the available experimental data. Figure (6) contains the experimental measurements for the variations of vortex circulation against the square root of gap height, for three different cases. The cases considered are the Inoue et al. (1985) compressor rotor, Lakshminarayana et al. (1965) compressor cascade arid Yaras et al. (1990) turbine cascade. For the second case the vortex strength, as a fraction of the bound circulation, was calculated from the available retained lift coefficient K, given as (40) SQRT(e) 0.12 FIG.(9): THE CALCULATIONS OF SHED CIRCULATION, USING THE SIMPLE MODEL, AS A FUNCTION OF THE SQUARE ROOT OF GAP HEIGHT. account. The vortex trace, the peripheral position of the vortex centre and its angle, with respect to the axial direction, are computed using the self induced velocity field in the previous plane, in order to move the vortex in the peripheral direction. The time It must be pointed out that the Inoue et al. and Yaras et al. cases concern direct measurements of vorticity, while Lakshminarayana et al. data are based on extrapolated liftdistribution curves. However, in all the cases, it is evident that there exists a linear variation of leakage vortex circulation with the square 'root of gap height, at least for the tip clearances which are of higher interest. It can be observed that the best fit straight lines do not pass through the axis origin, although this should be the normal situation, i.e. having zero shed vorticity for zero clearance. The non zero circulation value for the zero clearance case is connected with the flow without clearance and its presence may be attributed to secondary flow vorticity or endwall vorticity, included in the presented measurements. It may be pointed out that, in the case of Lalcshminarayana et al., the measured points fall closer on a line passing through the origin, something that should be expected, as no secondary or endwall effects were present in this case. The distance of the straight line from the origin could give a useful measure for the distinction of the tip clearance shed circulation from the remaining votticity sources. 8

9 C comd Inoue et al. experiment Complete calculation (rot.) Atiesama Complete calculation (no rot.) Simple model Best fit to experimental data sqrt(e) FIG410): CALCULATION OF THE VORTEX CIRCULATION, WITH AND WITHOUT ROTATION EFFECT, USING THE COMPLETE CALCULATION PROCEDURE AND THE SIMPLE MODEL, AND COMPARISON WITH INOUE ET AL. EXPERIMENTAL DATA. =CO Yoras et al. experiment Simple model Present calculation -- BOUND CIRCULATION o.os 0.06 e/c FIG.(11): CALCULATION OF THE VORTEX CIRCULATION WITH THE SIMPLE MODEL AND THE COMPLETE CALCULATION PROCEDURE AND COMPARISON WITH YARAS ET AL. EXPERIMENTAL DATA. Nevertheless, the observed linearity is, by itself, a critical property of the vortex circulation variation, as it allows the determination of the dependance with the minimum number of measurements. It would be interesting to examine whether this linearity extends over the higher values of tip clearance. Figure (7) presents the measurements of Yaras et al. (1990), plus the measurements of De Cecco et al. (1995) for the same case but with particularly large gap heights (0.1 and 0.15 e/c), (Sjolander (1995)). Although the bound circulation varies considerably for these higher values of the tip clearance, the shed circulation continues to vary with the same law, being independent of the variation of the bound circulation. It must be pointed out that the measured bound circulation for the large clearance (0.15 e/c) is equal to the measured leakage vortex circulation. The simple model presented above was initially used, in order to establish the ability of the proposed theory in predicting the variation of the leakage vortex circulation with clearance. Figure (8) contains comparisons between the calculations using the simple model and the corresponding experimental data. The vortex circulation as a fraction of bound circulation is plotted against the clearance gap as a fraction of chord. The same comparisons are contained in figure (9) but the circulation is plotted against the square root of gap height. The simple model predictions pass through the axis origin, while this is not the case for the experimental data, as it was mentioned above. The complete calculation procedure was used, in order to calculate the leakage vortex circulation for the Inoue et al. compressor rotor and the Yaras et al. turbine cascade. Figure (10) contains the comparisons between the current calculation and experimental data for the first case, while figure (11) contains the corresponding comparisons for the turbine case. In the compressor case, both the calculations with and without rotation effect are included, as in the presented experimental data, the effect of relative wall motion of the casing wall was eliminated by subtracting the circulation of the skewed boundary layer on the casing wall. This subtraction may be the reason for the negative extrapolated vortex circulation for the zero clearance case. However, in both cases, the calculation results are quite satisfactory, especially for the turbine case. The complete calculation procedure was also used in order to calculate the velocity field downstream the trailing edge, for two additional cases. The experimental measurements, performed by Goto (1991) in a low speed single-stage axial compressor, were used first. The rotor consists of 51 blades of C4 profile with.111 m chord and a designed flow coefficient of $=O.55. The measurements concern a flow coefficient of 43=0.51. Three tip clearances were tested, 0.7, 2.0 and 3.0 percent of chord. Figure (12) contains the comparisons between the calculated pitchwise averaged relative flow angle, with and without the tip clearance model and the experimental data. The corresponding comparisons for the meridional velocity are presented in figure (13). The calculation results are quite satisfactory, particularly in view of the fact that there exists a flow separation for half the blade span. starting from the hub. Only for the large clearance, an overestimation of the tip clearance influence in the meridional velocity component is observed, which results in art overturning of the flow angle. 9

10 re Meridional + Tip clearance model Cbomo Gatos experiment Meridional 20 ao 60 BO Rel. angle (deg.).755 e/c ; do 80 Rel. angle (deg.) 2.95 e/t ao so so 100 Rel. angle (deg.) 3.55 e/c An isolated axial-flow compressor rotor, with a varying clearance along chord, was studied by Wagner, Dring and Joslyn (1983) in a series of aerodynamic tests at low Mach numbers. The rotor consists of 28 blades with a hub/tip diameter ratio equal to 0.8 and 6.0 in. constant chord. Two different inlet boundary layers (a thin and a thick) and four different flow coefficients were used. In the present work, the thick case with two flow coefficients (0.65 and 0.85) will be considered. The relative peripheral velocity distributions were reconstructed from the measured relative flow angle and axial velocity. Figure (14) contains the comparisons for the two flow coefficients, between the meridional calculation without clearance effects, the calculation with the clearance effects and the corresponding experimental data. Good agreement is obtained between the calculation results and the experimental data for both flow coefficients. CONCLUSIONS A model for the prediction of the leakage vortex circulation has been proposed, based on the assumption that the whole leakage jet flow enters the vortex core, increasing its radius and its moment of momentum in the direction of the vortex axis. Using the moment of momentum equation in the direction of the leakage vortex and assuming a solid body rotating vortex, an expression was derived for the vortex circulation, which demonstrates that this circulation is proportional to the square root of the tip clearance height. This theoretical result is supported by the available experimental data for both compressors and turbines. It was observed that the best fit to the experimental data lines, do not pass through the axis origin, although it should be the normal situation to have zero shed vorticity for zero clearance. The non zero circulation for the zero clearance case is connected with the flow without clearance and its presence was attributed to secondary flow or endwall vorticity, included in the presented measurements. The extrapolated value of circulation for zero clearance could be a useful criterion for the distinction of the leakage vortex circulation from the remaining vorticity sources. A simple model was developed, which demonstrates the ability of the proposed theory to calculate the variation of leakage vortex circulation with clearance, provided that the vortex trace is known. The complete calculation procedure, already presented by the authors, was used in order to calculate the leakage vortex circulation and the induced velocity field in various axial flow machines, with very satisfactory results, giving additional support to the validity of the proposed calculation method. FIG.(12): COMPARISON BETWEEN THE CALCULATED PITCHWISE AVERAGED RELATIVE FLOW ANGLES, WITH AND WITHOUT THE CLEARANCE PRESENCE, AND THE CORRESPONDING EXPERIMENTAL DATA OF GOT. ACKNOWLEDGEMENTS The Authors would like to express their thanks to Prof. S.A. Sjolander, for providing the experimental values of vortex circulation, for the corresponding test case. The Authors would, also, like to express their thanks to SNECMA, Rolls-Royce, Turbomeca and MTU for permitting them to publish this work, as well as for the discussions that 1 0

11 0.76 Meridional + Tip clearance model OZCCX7 Gates experiment Meridional - Meridional + Tip clearance model Experimental data phi=0.65 thick Meridional calculation v) VM (M/s).7% e c Meridional + lip clearance model WOW Experimental data phi=0.85 thick Meridional calculation VM (m/s) 2.% e c C o Ito 20 Wu (m/s) FIG.(14): COMPARISON BETWEEN THE CALCULATED PITCHWISE AVERAGED RELATIVE PERIPHERAL VELOCITY COMPONENT, WITH AND WITHOUT THE CLEARANCE PRESENCE, AND THE CORRESPONDING EXPERIMENTAL DATA OF WAGNER ET AL 0.61 IC VM (m/s) 3.36 e/c FIG.(13): COMPARISON BETWEEN THE CALCULATED PITCHWISE AVERAGED MERIDIONAL VELOCITY COMPONENT, WITH AND WITHOUT THE CLEARANCE PRESENCE, AND THE CORRESPONDING EXPERIMENTAL DATA OF GOTO. they had with the engineers of the above mentioned companies and the colleagues from the Cranfield Institute of Technology participating to this project. The authors also wish to express their thanks to the European Community which funded this project (AER2 CT AC3A) and to Dr W. Borthwick and Dr R. Dunker who acted as Technical Monitors. REFERENCES De Cecco S., Yaras M.I., Sjolander SA., 1995, "Measurements 11

12 of the Tip-Leakage Flow in a Turbine Cascade With Large Clearances", ASME paper 95-GT-77. Douvikas D., Kaldellis J., Papailiou K.D., 1989, "A Secondary Flow Calculation Method for One Stage Centrifugal Compressor, 9th ISABE, Athens. Goto A., "Three-Dimensional Flow and Mixing in an Axial Flow Compressor with Different Rotor Tip Clearances", ASME paper 91-GT-89. Inoue M., Kurumaru M., Fukuhara M., 1985, "Behaviour of Tip Leakage Flow Behind an Axial Compressor Rotor", ASME paper 85-GT-62. Inoue M., Kurumaru M., 1989," Structure of Tip Clearance Flow in an Isolated Axial Compressor Rotor", ASME Journal of Turbomachinery, Vol. 111 pp Lalcshminarayana B., 1964," Effects of a Chordwise Gap in an Aerofoil of Finite Span in a Free Stream", Journal of the Royal Aeronautical Society, Vol. 68, pp Lalcshminariyana B., Horlock J. H., 1965, "Leakage and Secondary Flow in Compressor Cascades", ARC R&M Lalcshminarayana B., 1970, "Methods of Predicting the Tip Clearance Effects in Axial Turbomachinely", ASME Journal of Basic Eng., Ser. D., Vol. 92, No. 3, pp Lamb H., 1932, "Hydrodynamics", 6th Edition, Dover Publications. Newman B.G., 1959, "Flow in a Viscous Trailing Vortex", The Aeronautical Quarterly pp Lewis RI., Yeung E.H.C., 1977, "Vortex Shedding Mechanisms in Relation to Tip Clearance Flows and Losses in Axial Fans", ARC R&M No Newman B.G., 1959, "Flow in a Viscous Trailing Vortex", The Aeronautical Quarterly, pp Nikolos I.K., Douvllcas Papailiou K.D., 1993a, " A Method for the Calculation of the Tip Clearance Flow Effects in Axial Flow Compressors, Part I: Description of Basic Models", ASME paper 93-GT-150. Nikolos I.K., Douvikas DJ., Papailiou K.D., 1993b, " A Method for the Calculation of the Tip Clearance Flow Effects in Axial Flow Compressors, Part II: Calculation Procedure", ASME paper 93-GT-151. Nilcolos IX., Douvikas D.I., Papailiou K.D., 1995, 'Theoretical Modelling of Relative Wall Motion Effects in Tip Leakage Flow", ASME paper 95-GT-88. Sjolander SA., Amrud K.K., 1986, "Effects of Tip Clearance on Blade Loading in a Planar Cascade of Turbine Blades", ASME paper 85-GT-245. Sjolander S.A. 1995, private communication. Wagner J.H., Dring R.P., Joslyn H.D., 1983, "Axial Compressor Middle Stage Secondary Flow Study", NASA Contractor Report Yamamoto A., 1989, "Endwall Flow/Loss Mechanisms in a Linear Turbine Cascade With Blade Tip Clearance", ASME Journal of Turbomachinery, vol. III, pp Yaras M., Yingkang Z., Sjolander SA., 1989, "Flow Field in the Tip Gap of a Planar Cascade of Turbine Blades", ASME Journal of Turbomachinery, vol. 111, pp Yams M., Sjolander SA., 1990, "Development of the Tip- Leakage Flow Downstream of a Planar Cascade of Turbine Blades: Vorticity Field", ASME Journal of Turbomachinery, Vol. 112, pp Yaras MI., Sjolander SA., Kind RJ., 1991, "Effects of Simulated Rotation on Tip Leakage in a Planar Cascade of Turbine Blades, Part II: Downstream Flow Field and Blade Loading', ASME paper 91-GT