The Measurement and Prediction of the Tip Clearance Flow in Linear Turbine Cascades

Transcription

1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47 St., New York, N.Y GT-214 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. M Discussion is printed only If the paper is published in an ASME Journal. Papers are available ;L from ASME for fifteen months after the meeting. Printed In USA. Copyright 1992 by ASME The Measurement and Prediction of the Tip Clearance Flow in Linear Turbine Cascades F. J. G. HEYES' and H. P. HODSON 2 Whittle Laboratory, Cambridge University Engineering Department Cambridge, CB3 ODY England ABSTRACT This paper describes a simple two-dimensional model for the calculation of the leakage flow over the blade tips of axial turbines. The results obtained from calculations are compared with data obtained from experimental studies of two linear turbine cascades. One of these cascades has been investigated by the authors and previously unpublished experimental data is provided for comparison with the model. In each of the test cases examined, excellent agreement is obtained between the experimental and predicted data. Although ignored in the past, the importance of pressure gradients along the blade chord is highlighted as a major factor influencing the tip leakage flow. NOMENCLATURE c Chord C Skin friction coefficient C, Static pressure coefficient _(P1 -P) (P1 -P3) h Span Al Width of a computational cell Am Mass flow through a computational cell M Proportion of mass flow mixing over tip Po Stagnation pressure P Static Pressure T Tip gap length in flow direction V Velocity Y Stagnation pressure loss coefficient _(P1 -P) (P1 -P3) a Flow angle measure to axial direction? Surface shear stress p Density a Contraction Coefficient ti Tip clearance 1 Currently Senior Design Engineer, European Gas Turbines Ltd., Lincoln 2Member, ASME Subscripts b In tip separation bubble N Component normal to blade pressure side P Pressure side R Radius of Orifice S Suction side T Component tangential to blade pressure side 1 Upstream of cascade 2 At cascade exit plane 3 At downstream mixed-out conditions INTRODUCTION The necessary clearance between the tips of unshrouded turbine blades and the machine casing is known to produce a significant loss of efficiency. Typically, a clearance of one percent of the blade height is associated with a penalty of one to three percent of the stage efficiency (e.g., Booth, 1985). However few, if any, of the traditional correlations (e.g., Ainley and Mathieson, 1951; Meldahl, 1941) are based on the physical processes involved. There are two important aspects of tip leakage flow. Firstly, there is a reduction in the amount of work that the blade extracts. This occurs because the leakage flow is not turned to the same degree as the mainstream flow and the effect manifests itself as a deviation of the flow from the exit blade angle. Of itself this does not constitute a loss because the work theoretically may be extracted in a subsequent blade row (Denton and Cumpsty, 1987). The second aspect involves the generation of entropy: within the tip gap; in the blade passage and downstream of the blade row. Recent detailed measurements of the flow within the tip gap of linear turbine cascades (e.g. Bindon, 1988; Dishart and Moore, 1989; Yaras and Sjolander, 199) have started to reveal the physics governing leakage loss generation. Bindon (1988) and later Dishart and Moore (1989) showed that the loss generated within the tip gap is approximately of equal magnitude to that generated by the mixing of the leakage and mainstream flows within and downstream of the cascade. However, experiments by Yaras and Sjolander (199) on a thin blade at high tip clearance showed that little of the loss was generated within the tip gap. Heyes et al. (1991) showed that loss generation within the tip gap reduced the leakage mass flow and therefore the overall leakage loss. In addition they showed that the angle that the tip leakage flow made with the mainstream flow at the exit of the tip gap would also influence the overall loss generation. Clearly a number of factors influence the mass flow rate and the Presented at the International Gas Turbine and Aeroengine Congress and Exposition Cologne, Germany June 1-4, 1992 This paper has been accepted for publication in the Transactions of the ASME Discussion of it will be accepted at ASME Headquarters until September 3, 1992

2 leakage trajectory and these factors will vary along the chord and from design to design. Since these affect the loss generation process, it is necessary to determine the distribution of these quantities along the chord if accurate predictions are to be made. Booth et al. (1982) argued that the flow over a blade tip was analogous to that through an orifice plate and that the discharge coefficients were of similar magnitude. Bindon (1986) showed that the leakage flow within the tip gap was dominated by the formation of a vena contracta and the subsequent mixing and diffusion of the flow that would ultimately fill the clearance gap if the blade were sufficiently thick. At tip gap exit he observed that for thinner parts of the blade, the leakage flow was characterized by a region of loss-free flow near the endwall and a region of high total pressure loss in the wake of the vena contracta separation bubble. Moore et al. (1989) and Moore and Tilton (1987) observed, on a much thicker blade, that the gap exit flow velocity was virtually constant across the tip gap. Heyes et al. (1991) put forward a model of the tip gap flow, based on a simpler model of Moore and Tilton, which is illustrated in Figure 1. The model is characterized by the contraction coefficient of the vena contracta, a, and the proportion of the leakage volume flow that mixes to form the separation bubble wake, M. The value of M will vary from, for regions of tips with very low values of T/ti, to 1 for much thicker ones. Booth et al. (1982) have also presented calculations of the leakage flow trajectories obtained from a simple model. They assumed that the chordwise pressure gradients were small so that the component of the leakage flow parallel to the blade pressure side is convected through the tip gap unchanged. The velocity normal to the blade pressure side was then calculated from the static pressure drop from mid-span on the pressure side to tip gap exit using a measured discharge coefficient of.8. Reasonable agreement with their experimental data was obtained. EXPERIMENTAL DETAILS The Cascade Figure 2 shows the incompressible flow cascade that was previously investigated by Heyes et al. (1991) It is based on the tip section of a model turbine that is currently under investigation. It has a large chord and so permits a detailed investigation of the flow field. The blades were cantilevered from the hub endwall. The tip clearance was changed by withdrawing the blade into the hub endwall; the distance between the hub and tip endwall therefore remained constant, independent of the tip clearance. The cascade was fitted to an open jet wind tunnel so that the exhaust was at atmospheric pressure. Table 1 gives further details of the cascade. Flow Thermometer Pitot Static Probe Boundary Layer Bleed Movable Perspex ^ Endwall Aluminium Cascade Body Wake (senvopic Jet Pressure Suction Side Side T FIGURE 1 MODEL OF FLOW OVER BLADE TIP AFTER HEYES ET AL. (1991) U M w 1..8 i.6.4 to c).2 Mid-Span Pressure Mid-Span Suction The present paper is specifically concerned with the nature of the leakage flow over the blade tips and its variation along the chord. Detailed measurements of the leakage flow within the blade tip gap and the flow losses and deviation downstream of a cascade of turbine blades are presented. A model of the leakage flow is then described and results obtained from calculations employing the model are compared with the experimental data of the present authors and of other investigators. TABLE 1 CASCADE GEOMETRY AND TEST CONDITIONS No. of Blades 5 Chord 225mm Axial Chord 13mm Pitch/chord ratio.824 Aspect ratio 2.11 Reynolds No. V 2c/v 4.x15 Inlet flow angle, a t 32.5 Blade exit angle, a Max. thickness / chord 14% Zweifel Coefficient Fraction of Axial Chord FIGURE 2 GENERAL FEATURES OF CASCADE Instrumentation and Experimental Techniques Exit Traverses The flow leaving the cascade was measured at a plane 35 percent axial chord downstream of the blade trailing edge. A 5 hole pneumatic probe, which had been calibrated over an angular range of ±4 of yaw and pitch, was used for these measurements. If the yaw angle onto the probe was found to be outside this range, the probe was rotated about an axis coincident with its head until the flow

3 was within range of the calibration. No such difficulties were found with the pitch angle at the axial location of this traverse plane. The traverses were made over an area of 1 pitch by 28 percent of chord in the spanwise direction. Although this did not cover the whole of the cascade exit flow, the regions of influence of the leakage flow are contained within this area. This region was the same in extent as the area measured by Heyes et al. (1991) with a Pitot rake, although at a location further downstream. The diameter of the probe head was 3.29 mm; the cone semiangle was 45 and the four side holes were drilled perpendicular to the side of the cone. This was to ensure minimum sensitivity of the probe to Reynolds number effects (Hodson and Dominy, 1988). Clearance Gan Measurements Measurements of the flow at the suction side of the tip gap were made using a single axis hot wire probe. The length of the probe prongs was greater than the tip gap to prevent the probe stem from causing significant blockage when fully immersed in the flow. Along each traverse there were 19 spanwise stations between blade tip and endwall. At each spanwise location, the probe was rotated about its axis and voltage measurements made over a range of probe angles. From these measurements the flow speed and direction were deduced. Further details are given by Heyes et al. (1991). U. Measurements of the static pressure were made on the blade tip surface using the "micro-tapping" technique of Bindon (1986). Here 17 slots were machined across the tip of the blade in a direction perpendicular to the chord line and the slots covered with Magic Tape. Seventeen measurements were made using each slot by puncturing the tape with a finely ground needle. The minimum spacing of the measurements was.25 mm or.8 percent of the maximum blade thickness. TEST RESULTS Exit Traverses Figure 3 shows the results of the 5 hole probe traverses downstream of the cascade with a clearance of 1. percent chord. Contours of total pressure are shown projected into a plane perpendicular to the blade exit angle. The contour interval is.1 (P1 - P3). Although measurements were made only over 1 pitch, the results are shown repeated for clarity. The loss core generated by the leakage loss of one blade is seen to be interacting with the wake of its neighbour. The maximum value of Y 2 is.926 Secondary flow lines are also shown. These were obtained by time integration of the secondary flow vector field to obtain the paths of imaginary particles. The total time for each line was identical. The secondary flow was defined as being perpendicular to the mixed-out exit flow. (See below.) These reveal the vortical motion in the leakage loss core. The centres of the vortex and the loss core are seen to coincide..9.2 U Fraction of Pitch CONTOURS OF TOTAL PRESSURE.2 F.1 O Clearance / Chord (%) U ti Fraction of Pitch U, Z 2 V on X SECONDARY FLOW LINES Clearance / Chord (%) FIGURE 3 EXIT TRAVERSE RESULTS FOR T/C = 1.% FIGURE 4 CASCADE MIXED-OUT EXIT FLOW CONDITIONS FOR A RANGE OF TIP GAPS 3

4 Similar traverses were performed for a range of clearances from to 2.82 percent chord. Although they are not shown here, readers may refer to Heyes (1992) for further details. The results of all the traverses were reduced to mixed-out total pressure loss coefficients, Y 3, and exit angles a 3 by performing constant area mixing calculations over the measured flow field, i.e., a region of 1 pitch by 28 percent chord in the spanwise driection. These are plotted below in figure 4. Since the cascade exit flow angle was 75.6, there is approximately.9 of deviation at zero clearance for the part of the flow under consideration. Tip Static Pressure Distributions Measurements of the static pressure on the blade tip for a clearance of 1 and 2.82 percent chord are shown contoured in figure 5. The contour interval is.2(p1 -P3 ). Over most of the blade they clearly show a low pressure region near the pressure side that is associated with the vena contracta. The width of this low pressure region is approximately 1.5r in the leakage flow direction, as observed by Moore et al. (1988) As the flow begins to mix out the pressure rises until eventually it reaches the level of the static pressure on the blade suction side. Although for the smaller clearance case this mixing process has been completed by tip gap exit, for the larger clearance there is still evidence of increasing static pressure at tip gap exit over the thinner trailing third of the blade. This shows that the mixing process is incomplete (and that the mixing parameter M will therefore take a value less than 1) at tip gap exit. coefficient should be the contraction coefficient, which takes the value.611 for incompressible flow (Milne-Thomson, 1968). This is close to the values shown for orifices of short length, and thus, by analogy, for thin parts of the blade tip. For orifices of intermediate length, the tip discharge coefficient rises sharply from its initial low value and reaches a peak at lengths near 5.5 radii. This regime is associated with the mixing of the flow to fill the orifice and thus the value of M will be expected to rise from to 1 over that distance of 4.5 radii. For orifices of length greater than 5.5 radii, the discharge coefficient is seen to reduce with orifice length. This is consistent with the action of shear on the orifice walls. If there were no shear, the discharge coefficient would be expected to reach the value of.845 where M= a^ en 't8.7 Q.65.6 Orifice Length / Radius 1. FIGURE 6 VARIATION OF DISCHARGE COEFFICIENT WITH ORIFICE LENGTH Figure 6 also shows the result of calculations based on a model that assumes a linear variation of M from to I over the length of the mixing region (i.e., from 1 to 5.5 radii). Before mixing, the flow is assumed to accelerate isentropically into the vena contracta. Downstream of the crest the mixing takes place. A constant shear stress is assumed to act on the orifice walls, which is given by =Cf 2pV2 (1) where the velocity, V, is the mean value at orifice exit. The value of C f depends on the Reynolds number of the flow based on the orifice radius, Re R, and is given by the relationship for fully developed turbulent flow in a pipe.67 Cf = ReRU (2) Clearance = 1.% Chord Clearance = 2.82% Chord FIGURE 5 STATIC PRESSURE MEASUREMENTS ON THE BLADE TIP FOR TWO TIP GAPS MODEL OF THE TIP LEAKAGE FLOW Mixing within the tipgan By extending the analogy between the tip gap and the orifice, the degree of mixing that occurs within the tip gap may be linked to the mixing that occurs within a circular orifice. The variation of orifice discharge coefficients with orifice length has been documented by Lichtarowicz et al. (1965). Figure 6 is a reproduction of one of their figures. It shows how the discharge coefficient varies with orifice length. If no mixing was to occur, theoretically the discharge These assumptions allow the variation of the tip discharge coefficient to be calculated, solely by using the conservation equations of mass and momentum. Figure 6 shows the results of such a prediction for a representative Reynolds number of 4. The agreement is sufficiently adequate for the same model to be used for the turbine tip clearance flows, that is, a linear variation of M within the mixing region and a constant shear stress on the tip and endwall surfaces may be used to calculate the clearance flow perpendicular to the blade pressure side. The mixing will be assumed to take place immediately downstream of the vena contracta, over a length of 4.5t (i.e., from T=1.5ti to T=6t) since the hydraulic mean diameter of the tip clearance is 2ti. Trajectories of the Leakage Streamlines To predict the trajectories of the leakage streamlines, a simple finite volume time-marching method is used. The scheme relies on knowledge of the pressure distribution at tip gap exit and on the blade pressure side away from the tip clearance. In this case, measured values are used, but the results of a 3-D Navier-Stokes solver, such as that of Dawes (1987), could be used instead.

5 The blade tip gap is modelled as a chordwise row of quadrilateral computational cells each having one side on the pressure side of the blade and another on the suction side, as shown in figure 7. The sides of each cell, AD and BC, follow the local leakage flow path and hence mass enters each control volume only at AB, and leaves only at CD. An estimate of the leakage flow is needed to determine the initial position of each cell. For this purpose a constant leakage flow angle over the whole blade (normal to the mean camber line) was found to be sufficient. As the computation progresses, the leakage flow trajectories change and thus the shapes of the control volumes are altered to ensure no flow through the sides BC and AD. The pressure distribution around each cell must be known before the conservation of momentum equations can be employed to calculate an updated value of the leakage velocity. Along AB, it is assumed that the conditions correspond to those at the vena contracta, and so the value Pb acts on the cell pressure side. The arithmetic mean of the pressure at the points C and D is assumed to act along CD. These values are interpolated from the measured distribution at tip gap exit. Pressure Side Al, B C Suction Side changes in the components of the leakage flow momentum can be found. These changes are assumed to occur at the mid-point of CD. Once the components of momentum are updated and the value of M is determined, a new value of the mass flow rate leaving the control volume is calculated. This is equal to the mass flow rate Am. In addition, the new leakage flow trajectory is taken to be the direction of the exit velocity. Once the new leakage trajectories are known for the whole blade, the cell boundaries are moved so that each boundary is parallel to the mean of the velocities either side of it. The above procedure is then repeated over successive timesteps until changes in the momentum flux and mass flow leaving each cell between successive iterations are reduced to less than.1 percent for a CFL number of approximately.1. This procedure gave convergence to a consistent solution, independent of the starting conditions. VALIDATION OF FLOW MODEL AGAINST EXPERIMENT Data of Dishart and Moore Before comparisons are made with the present data, calculation results for the cascade of Dishart and Moore (1989) are presented. Figure 8 shows the reported pressure distributions at mid-span and at tip gap exit and the leakage flow trajectories measured at gap exit with a 3 hole pneumatic probe. The cascade Reynolds number was 5.4x1 5 and the tip clearance 2.1 percent chord. A Al, Vena Conrracra FIGURE 7 A COMPUTATIONAL CELL FOR THE TRAJECTORY CALCULATION The mass flow, Am, through the control volume, and the contraction coefficient, 6, are used with Bernoulli's equation to find the pressure at the vena contracta, Pb. If the cell occupies a length Alp of the pressure side, then 2 Pb = Pp J Alp) (3) where Pp is measured away from the tip gap. At the vena contracta, the momentum of the fluid parallel to the pressure side is assumed to be that inferred from the pressure measurements made on the pressure side away from the blade tip where the value is undisturbed by the leakage flow. The pressure acting at the vena contracta at A (interpolated from the Pb values for this cell and its nearest neighbour) is assumed to act over a distance 1.5ti along AD, and then the pressure at D is assumed to act over the remainder of AD. A similar distribution is assumed for BC based on the pressure values at B and C. Although this magnitude and location of the resulting pressure forces will not be strictly correct, the approximations are acceptable if the leakage flow trajectories are parallel up to the crest of the vena contracta. The added complication of calculating the true variation of momentum and pressure up to the vena contracts for converging streamlines could not be justified for the present investigation. Once an estimate for the mass flow through each control volume and the leakage trajectories is obtained, the calculation proceeds by the application of the equations for the conservation of momentum. The components of the pressure forces acting on the control volume and of the shear forces calculated from equation (1) and the mean cell exit dynamic head are resolved parallel and normal to AB. By applying these forces over a small increment in time, the IU o Suction 1.2 Mid-Span U 1. Tip.8 Suction Mid-Span.6 T Pressure Fraction of Axial Chord FIGURE 8 CASCADE AND LEAKAGE FLOW DATA OF DISI-IART AND MOORE (1989) Figure 9 shows the results of the present leakage flow calculation for the Moore cascade as the angle of the leakage flow trajectories. The angle is measured from the axial direction. Also shown are the measured data and the results of a calculation of the leakage flow trajectory put forward by Booth et al. (1982) The latter assumes that the chordwise pressure gradients are small so that the component of the leakage flow parallel to the blade pressure side is convected through the tip gap unchanged. The velocity normal to the blade pressure side is calculated from the static pressure drop from Z

6 mid-span on the pressure side to tip gap exit and a discharge coefficient of.8. The comparison shows excellent agreement between the data and the present calculation. The scheme of Booth et al under-predicts the leakage flow angle by nearly 3 at the leading edge and overpredicts the value by 1 towards the trailing edge. The reason for the discrepancy in the models is the fact that the Booth method has no connection between the discharge coefficient and the leakage trajectories. The present calculation employs the momentum and continuity equations so that when leakage streamlines converge, the discharge coefficient reduces, and when they diverge the discharge coefficient increases. The reason for the convergence and divergence of the streamlines is linked to the static pressure gradients along the blade chord. As the static pressure falls on the blade pressure side towards the rear of the blade, the component of velocity parallel to AB increases. At the same time the pressure difference across the blade tip reduces and so the velocity component normal to AB falls, causing streamline divergence. Hot-wire Traverse Position / Sides of Computational Cells Leakage Flow Vector Sides of computations[ Cells _ Leakage Flow Vector / 6. o 4 ' Data of Dishart and Moore / 2. - Model of Booth et al. / I Fraction of Axial Chord Clearance = 1.% Chord Clearance = 2.12% Chord FIGURE 1 CALCULATED LEAKAGE FLOW TRAJECTORIES FOR THE PRESENT CASCADE 6. I Hot-wire Traverse FIGURE 9 COMPARISON OF FLOW MODELS WITH DATA OF Present Prediction DISHART AND MOORE 4. Present Data Figure 1 shows the calculated distribution of leakage flow trajectories. for the present cascade at two clearance values. Near the leading edge the streamlines are converging and over the remaining part of the blade (except actually at the trailing edge) the trajectories diverge. The convergence of the flow lines over the first part of the blade arises because the pressure difference across the tip increases over the leading part of the blade, increasing the leakage velocity normal to the blade pressure side, while the pressure parallel to the pressure side remains almost constant in this region. For the case of the higher clearance, the minimum pressure on the blade suction side is further downstream (see figure 12) so that more of the blade shows converging streamlines Hot wire measurements at tip gap exit are compared with the calculation for the lower tip clearance value in figure 11. Again, agreement is excellent over the whole of the blade. Equation (3) indicates that an easily accessible measure of the mass flow rate over the blade tip is the static pressure measured in the vena contracta. The data shown in figure 5 are therefore used as a comparison in figure 12 with the calculated values of P b for the present cascade at two clearances. The measured data, presented as a pressure coefficient Cp, are derived from the average pressure acting on the blade tip over the first 1 mm of the blade measured from the pressure side along a predicted leakage streamline. Also shown in figure 12 is the static pressure measured at mid-span on the blade U '^ 1 H 2. O c -2. Flat Tip Clearance = 1.% Chord Fraction of Tangential Chord FIGURE 11 PREDICTED AND MEASURED LEAKAGE FLOW TRAJECTORIES FOR THE PRESENT CASCADE pressure side and at tip gap exit on the suction side. These data were used for the prediction. The scheme shows good agreement with the measurements although the pressure coefficient is slightly overpredicted for much of the blade at the lower clearance. Near the leading edge of the blade in both cases, the low value of CP in the vena contracta is associated with the converging streamlines and consequent low leakage velocity. At the trailing part of each blade the scheme under-predicts the value of C. This is probably due to the simplified model of the over-tip mixing.

7 3 Measured in Separation Bubble Predicted in Separation Bubble - Input to Prediction U ^ 2.5 w 1.5 / suction.5 Mld-sue / ^ Fraction of Tangential Chord CLEARANCE =1 PERCENT CHORD 3. Measured in Separation Bubble - 2. Tip 1.5 a 1. I Predicted in Separation Bubble Input to Prediction suction.5 ^ Mid-Spin Pres""` Fraction of Tangential Chord CLEARANCE = 2.82 PERCENT CHORD FIGURE 12 PREDICTED AND MEASURED VENA CONTRACTA PRESSURES FOR THE PRESENT CASCADE DISCUSSION The model of the flow within the tip gap, presented above, has been shown to agree accurately with available measurements. The model provides an important step towards the ability to calculate the total pressure losses and flow angle deviations associated with the tip clearance flows. Before the final objective can be achieved, however, the evolution of the flow within the blade passage must be determined. The mixing of the passage and over-tip leakage flows is governed by the equations of conservation of mass and momentum (Denton, 1991). These also determine the resulting deviation. In practice, the details of this mixing process do not need to be accurately modelled since the equations for conservation of mass and momentum must be satisfied globally. Of course, the pressure field which evolves as part of the iterative solution affects the magnitude and the trajectory of the leakage flow and these in turn are effectively determined by the same pressure field. In addition, distortions of the static pressure field occur in the vicinity of the tip gap. The static pressure distortion on the pressure side is a result of an inviscid process and has been highlighted by Moore and Tilton (1987). The distortion on the suction side is associated with the formation of the leakage vortex and the method by which the leakage flow and mainstream low interact. These distortions will not be accurately predicted using coarse computational grids. However, the most important effect on the pressure field (Denton, 1991) is that which arises because the leakage flow is underturned and yet expands to the same back-pressure as the passage flow. This means that the mass flow between the blades is reduced over the entire span as the tip leakage flow will have a higher axial velocity. Since the passage flow is reduced, the blade force and, therefore, the pressure differences are also reduced. This is a large scale effect and will be correctly modelled by a 3-D flow prediction code. In the case of a compressor cascade, Storer and Cumpsty (199) showed that the leakage losses arise almost entirely at the tip gap exit, where a shear layer forms between the leakage flow and the mainstream flow. As a consequence, they found that a 3-D Navier- Stoke code (i.e., Dawes, 1987) gave very good agreement with experimental observations, both in terms of leakage flow trajectories and losses without needing to predict the detail of the flow in the tip gap. In the case of a turbine cascade, some mixing usually takes place in the tip gap so that the loss generation. However, the loss which results from the mixing within the gap and the resulting flow can be determined using the above model. Thus it should be possible to combine the present model with a three-dimensional flow solver to evaluate total pressure losses and flow angle deviations associated with the tip clearance flows, although such a calculation is beyond the scope of the present paper. CONCLUSIONS Measurements of stagnation pressure loss and flow angles have been presented for an incompressible linear turbine cascade over a range of tip clearances. In addition, details of the flow field in the tip clearance gap have been presented. A simple iterative calculation of the flow in the tip gap has been put forward. It relies on a time-marching scheme and uses an adaptive mesh. The method was shown to predict the leakage flow with accuracy for a range of blade geometries and flow conditions. An important effect, which has been ignored in the past, is found to be caused by the chordwise pressure gradients.when coupled to the calculation of the main flow, such as in a Navier-Stokes calculation, it is suggested that the flow losses and reduction in turning can be calculated with great accuracy. ACKNOWLEDGEMENT This work was carried out with the support of the Procurement Executive of the Ministry of Defence and Rolls-Royce plc. The authors would like to thank Rolls-Royce plc for their kind permission to publish the paper. REFERENCES Ainley D.G. and Mathieson G.C.R., 1951, "A Method of Performance Estimation for Axial-Flow Turbines" ARC R&M No.2974, December. Bindon J.P., 1986, "Pressure and Flow Field Measurements of Axial Turbine Tip Clearance Flow in a Linear Cascade" Report CUED/A-Turbo TR-123, Cambridge University, England. Bindon J.P., 1988, "The Measurement and Formation of Tip Clearance Loss" ASME Paper No. 88-GT-23. Booth T.C., 1985 "Turbine Loss Correlations and Analysis" VKI Lecture Series , Tip Clearance Effects in Turbomachines. Booth T.C., Dodge P.R. and Hepworth H.K., 1982, "Rotor Tip Leakage: Part 1 - Basic Methodology" ASME Jnl. Eng. Power, Vol 14, 1982, pp Dawes, W.N., 1987, "A Numerical Analysis of the Three- Dimensional Viscous Flow in a Transonic Compressor Rotor and Comparison with Experiment," ASME Jnl. of Turbomachinery, Vol. 19, No. 1, pp Denton J.D. and Cumpsty N.A., 1987, "Loss Mechanisms in

8 Turbomachines" Proc. IMechE Turbomachinery - Efficiency and Improvement, Paper No. C26/87. Denton, J.D., 1991, Private Communication Dishart P.D. and Moore J., 1989, "Tip Leakage Losses in a Linear Turbine Cascade" ASME Paper No. 89-GT-56. Heyes F.J.G., 1992, "Tip Leakage Flow and Control in Axial Turbines", PhD Thesis, Cambridge University, to be published. Heyes F.J.G., Hodson H.P. and Dailey G.M., 1991, "The Effect of Blade Tip Geometry on the Tip Leakage Flow in Axial Turbine Cascades", ASME paper No. 91-GT-135. Hodson H.P. and Dominy R.G., 1988, "An Investigation into the Effects of Reynolds Number and Turbulence upon the Calibration of 5-Hole Cone Probes" Presented at 9th Symp. Transonic and supersonic flow in cascades and turbo machines, St. Catherine's College, Oxford, March, Ed. Baines N.C. Lichtarowicz A., Duggins R.K. and Markland E., 1965 "Discharge Coefficients for Incompressible Non-Cavitating Flow through Long Orifices" Jnl. Mech. Eng. Sci., Vol 7, No.2, pp Meldahl A., 1941, "End Losses of Turbine Blades" The Brown Boveri Review, Vol 28, No.11, November. Milne-Thomson L.M., 1968 "Theoretical Hydrodynamics", MacMillan & Co. Ltd., 5th Edition, pp Moore J. and Tilton J.S., 1987, "Tip Leakage Flow in a Linear Turbine Cascade" ASME Paper No. 87-GT-222. Moore J., Moore J.G., Henry G.S. and Chaudry U., 1988, "Flow and Heat Transfer in Turbine Tip Gaps" ASME Paper No. 88- GT-188 Storer, J.A., and Cumpsty, N.A., 199, "Tip Leakage in Axial Compressors", ASME Paper No. 9-GT-127. Yaras M.I. and Sjolander S.A., 199, "Prediction of Tip Leakage Losses in Axial Turbines" ASME Paper No. 9-GT-154.