THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47th St., New York, N.Y

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1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47th St., New York, N.Y The Society shall not be responsible for statements or opinions advanced in papers or discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printed only if the paper is published in an ASME Journal. Papers are available from ASME for 15 months after the meeting. Printed in U.S.A. Copyright 1993 by ASME 93-GT-109 NUMERICAL INVESTIGATIONS OF CENTRIFUGAL COMPRESSOR FLOWS WITH TIP LEAKAGE USING A PRESSURE CORRECTION METHOD A. Tourlidakis and R. L. Elder School of Mechanical Engineering Cranfield Institute of Technology Cranfield, Bedford, United Kingdom 1 ABSTRACT In this paper, a three-dimensional computational model for the solution of the time-averaged Navier-Stokes equations, based on a pressure correction method and the k-c turbulence model, is presented and implemented for the viscous flow modelling through a series of centrifugal compressors. Theoretical calculations with the current fully elliptic method are carried out and the results are compared critically with available experimental data and with results from other computational models. A radial and two backswept highspeed subsonic compressors with different geometrical and operating characteristics are analysed at design and off-design conditions. In all cases, a wake flow pattern is evident and strong secondary flows are discerned. The tip clearance effects on the relative flow pattern are found to be important and are appropriately simulated. The predictive capability of the current flow model is judged to be encouraging taking into consideration the limitations of the physical models and the numerical schemes involved in the computations. A, AP C P C 1.1' C 1' C 2 G c 1 P R S T V, V, W V, k m' 2 NOMENCLATURE : Coefficients of the finite difference equations : Specific heat at constant pressure : Constants in the k-e turbulence model : Production rate of turbulence kinetic energy : Rotation and curvature modification term : Rothalpy : Jacobian of the coordinate transformation : Total pressure : Distance from the axis of rotation. : Source term in the general transport equation : Static temperature : Curvilinear velocity components : Relative velocity : Grid metric coefficients : Turbulence kinetic energy : Local mass source : Static pressure : Static pressure correction u, v, w x, y, z Il O 4> tz E iji< 11y Cjk, E Subscripts i,j,k m P T 0 2 a (I) E, rl, Superscripts i,j,k Cartesian velocity components Cartesian coordinates Diffusion coefficient for the general transport equation Cell boundary dimensions in the transformed plane Relative flow angle General scalar quantity Rotational speed of the impeller Dissipation rate of turbulence energy 1 if ijk cyclic, -1 if ijk anti-cyclic, 0 otherwise Viscosity Curvilinear coordinates Density Effective Prandtl numbers for k and e : Coordinate direction indices : Meridional : Control grid point : Turbulent : Upstream reference conditions : Value at impeller exit : General neighbouring grid point : Related to scalar 4> : Partial derivatives in respect to E, ri and C : Coordinate direction indices : Arising from the solution of the momentum equations 3 INTRODUCTION Centrifugal compressors currently find a wide range of applications due to their ability to achieve high pressure ratios per stage resulting in lightweight and compact compression systems. The aerodynamics of the impeller are very complex and involve boundary layer growth and separation on the various surfaces, shock waves, unsteady and turbulence effects, secondary flows due to rotation, passage curvature and tip leakage and eventually the formation of Presented at the International Gas Turbine and Aeroengine Congress and Exposition Cincinnati, Ohio May 24-27, 1993

2 wake patterns in the impeller passages. The process of wake formation is associated with high losses and affects also the operating range of both the rotating impeller and the downstream diffuser. There is an urgent need for improving the aerodynamic performance of centrifugal impellers which is strongly associated with seeking ways to control the flow phenomena involved and to eliminate wake formation whilst maintaining high levels of diffusion within the impeller. To achieve these objectives it is essential to fully understand all the flow phenomena and their interaction, occurring in the rotating passages. The most appropriate way forward appears to firstly involve the acquisition of measurement data of the internal flow field and secondly the theoretical analysis of the flow using three-dimensional, compressible and viscous flow solutions which during the past decade have shown promising developments. A thorough examination of the already existing experimental data reveals that the length of our database is very limited and much more effort is required for the acquisition of "benchmark" data for the evaluation of the computational viscous flow methods and the development of more sophisticated approaches in terms of physical modelling. Elder and Forster (1987) provide a description of laser anemometry methods used to investigate flow in a variety of low speed and high speed impellers. Eckardt (1976), performed detailed measurements, using a Laser-2-Focus velocimeter, in a high speed radial impeller and he observed flow separation from the blade suction side and the rapid development of a wake pattern in the impeller's radial region. The effects of the blade backsweep on the performance and on the local flow phenomena of the same high speed impeller were presented by Eckardt (1980). Krain (1988) presented results of laser measurements in the blade passages of a backswept impeller. A smooth velocity profile was observed at the rotor exit which was quite different from the typical jet/wake flow pattern found in all the previous investigations. The quality and comprehensiveness of the provided experimental data in the above investigations attracted the interest of several flow modellers to use them for "benchmark" comparisons against their theoretical results. Fagan and Fleeter (1990) performed a series of experiments using Laser Doppler Velocimetry in order to investigate the 3-D flow field in the passages of a low speed research mixed-flow centrifugal compressor at its design point. In addition, the new low-speed centrifugal compressor facility at NASA Lewis Research Centre, Hathaway et al (1992), has been built to provide a more detailed understanding of the complex nature of the flow phenomena occurring in the curved rotating passages of centrifugal compressors and to obtain high quality "benchmark" experimental data for the verification of three-dimensional Navier-Stokes solvers. Experimental work at Cranfield has also continued providing some limited three-dimensional flow data in a small high-speed impeller, Ahmed and Elder (1990), using a novel Doppler anemometer, and other data from a series of laser anemometry measurements inside the blade passages of a high speed Rolls-Royce impeller of a current in service engine, Forster (1988). The state of the art in turbomachinery computations is the three-dimensional viscous flow analysis using the Reynolds-averaged Navier-Stokes equations with a turbulence model. One of the common computational approaches for the solution of these equations is the, so called, "pressure correction" method which was implemented with a great deal of success initially for incompressible recirculating flow calculations and later for compressible flow prediction. The second approach is the "time marching" method which became commonly used where shocks were of significance, Denton (1983). The first attempt to simulate the flow in a centrifugal compressor using the "pressure correction" solution of the time-averaged Navier-Stokes equations, was performed by Moore and Moore (1980a) who implemented a partially parabolic procedure for the incompressible flow prediction in a low speed shrouded impeller. Although the turbulence, compressibility and tip leakage effects were neglected, this simulation represented a major step towards the application of Navier-Stokes analysis in centrifugal compressors. Moore and Moore (19806) then extended their method to compute the turbulent, compressible flow in a high speed impeller with tip leakage and a stationary shroud. Rhie et al (1984) used a partially parabolic procedure for the analysis of the three-dimensional flow in a high speed impeller with tip leakage. Hah et al (1988) developed a fully elliptic three-dimensional viscous flow analysis method with a finite volume relaxation procedure and presented comparisons between numerical and experimental data for the detailed flowfields and overall performance of a backswept impeller at design, choke and near-surge operating conditions. This method was used later by Hah and Krain (1990) for the flow analysis in a high efficiency and high pressure ratio modern backswept impeller. All the above methods are based on the pressure correction relaxation approach for solving the Reynolds-averaged Navier-Stokes equations. Dawes (1988) presented the results obtained for the flow inside a backswept impeller using a time marching process for the solution of the fully three-dimensional equations of motion expressed in a finite volume form. For the prediction of the turbulence effects on the main flow a mixing-length closure was employed. The complicated interactions of the flow phenomena render the accurate theoretical representation of the flow field in a centrifugal compressor a significantly difficult task. The losses in stage performance associated with this flow field provide a challenging incentive for viscous flow models to contribute towards the achievement of improved compressors aerodynamic performance through the further physical insight and understanding of the complex flow features. In general, the methods which were discussed above are based on different model assumptions and numerical solution procedures. Despite their successes, they have their own limitations especially insofar as the accurate prediction of the losses, the secondary flows and the separated flow regions are concerned. Beyond the uncertainties of the physical models, there are drawbacks associated with the accuracy and the stability of the numerical schemes and the robustness of the solution methods whit% are currently in development or in use. Hence, work continues on the further development and validation of already existing computational methods and on the derivation of alternative schemes since the most suitable way forward is still not well-defined. Theoretical methods for the prediction of flows in centrifugal compressors based on the solution of the system of the Navier-Stokes equations, have been initiated at Cranfield Institute of Technology with the work of Lapworth (1987) who presented a computational method to handle incompressible flows in shrouded centrifugal impellers. The present work has been aimed towards the development of this method to high speed subsonic flows in centrifugal compressors with or without tip leakage and with stationary shroud walls, Tourlidakis (1992). The theoretical flow analysis is carried out using a threedimensional, finite volume, pressure correction relaxation method with a two equation k-c turbulence model, for the solution of the time-averaged Navier-Stokes equations in a rotating frame of reference. The current method is similar to that suggested by Rhie et al (1984) but it is fully elliptic instead of partially parabolic, it solves the energy equation instead of assuming a constant rothalpy condition and it employs a different pressure correction scheme from that used by Rhie et al (1984). In addition, an upwind scheme is utilized unlike the explicit artificial viscosity damping term which is introduced by Rhie et al (1984) to suppress numerical oscillations arising from the use of central difference approximations. The final objective has been pursued through the validation study and comparison of the predictions obtained with the proposed method against the results of Eckardt's experimental investigation of 2

3 the flow in high speed radial and backswept impellers, at design and off-design conditions, Eckardt (1976 and 1980); and, against the experimental results obtained for the flow within the much smaller passages of the Rolls Royce GEM impeller, Forster (1988). 4 MATHEMATICAL MODELLING The conservation equations for mass, momentum and energy which govern the steady, viscous and compressible flow, expressed in a Cartesian coordinate system in a rotating frame of reference, are as follows: Mass a (puj ) -0 (1) Momentum ax; - a(pui ui) - a f aui i + a [ au i_ ap axj ax ax reff ()xi ] ax [Ileff ax,j Energy -2 pt ijki2j u k -p (C2 ixj)0,+p0 2xi (1=1, 2, 3) a(pu,i) _ a [ ail 3 ax,11eff axj j where p is the density; p.eff is the effective viscosity; 0 is the rotational speed; Co is the alternating tensor; and there is a summation over all repeated indices. The rotation is along the z coordinate axis. The energy equation is written in the form of transport of the rothalpy defined as: Rothalpy (I) = CpT+4 (nxl-z.) (nxk. ) (4) The energy equation is used for the calculation of the static temperature field and the density field is determined through the perfect gas law. The effective viscosity tteff is defined as the summation of the laminar viscosity (b) and the turbulent viscosity (p,r). The turbulent viscosity 1.tr is calculated using a modified twoequation k-c eddy viscosity turbulence model, Launder and Spalding (1974). The turbulent viscosity is related to the turbulence kinetic energy, k, and the rate of dissipation of turbulence energy, 6, by: = Co pt k 2 (5) The empirical constants arising in the k-c model are given the values: C u = 0.09, C 1 = 1.47, C2 = 1.92, ak = 1.0 and a c = 1.3. In order to reduce the computer storage and run time requirements, the dependent variables at the walls are linked to those at the first grid node from the wall by relations consistent with the logarithmic law of the wall and are known as "wall functions", Launder and Spalding (1974). All the preceding equations for mass, momentum, energy and turbulence scalars are elliptic partial differential equations and can be expressed in the form of a general transport equation for an arbitrary dependent variable (I) : a ( pui(d) _ a ax;axj r ax; s(x, y, z) (8) where F is the effective diffusion coefficient and S is the source term. By expressing the equations in the form of the general equation (8), a unique calculation process can be used for the solution of these equations. The mesh which is employed is a general structured grid where curvilinear coordinate lines coincide with the boundaries of the flow domain and consequently a coordinate transformation is necessary. If new independent variables E, fl and are introduced, the form of the general transport equation (8) changes according to the general transformation = n=n(x,y,z) and =Ux,y,z), to the following form: a( p Uitt) _ a p r jk,() (9) 8E jg as k where J is the Jacobian of the transformation, U = (U,V,W) are the contravariant velocity components scaled by J, g" are the metric tensor components and S(E,n,C) is the transformed source term. The faces of each control volume constructed around each grid point, are placed at locations midway between the corresponding neighbouring nodes. The governing generalised equations are integrated over the control volumes and the resulting finite difference equation for the general transport equation can be expressed as relation between the value of (I) at P and its value (1) o. at the neighbouring grid points: AA) =?la d). + E n AC (10) where the coefficients A u involve convection, diffusion and geometrical parameters of the control volume. The convective term is calculated using a first order upwind differencing scheme which introduces implicitly numerical diffusion but avoids numerical instability associated with the centred differencing of the convective term. The diffusive term is evaluated using a central difference scheme. where C I is a constant and k and a satisfy the scalar transport equations: a (pu,k) _ a [II, ak], G _ p G, ax ax; ax3 a (pu,e) _ a [II, ac _f_r,c-c2 pc+gc] (7) axjaxj a, a -7 k where G is the rate of production of turbulence kinetic energy and G c is an additional term which accounts for the influence of rotation and streamline curvature on the turbulence structure, Rhie et at (1984). ( 6) For the solution of the pressure field, a pressure-correction procedure is used whereby the pressure field is gradually established through successive corrections which promote the satisfaction of the global and local continuity condition and decouple the linearised momentum equations. A non-staggered grid arrangement is used in which all the variables are stored and calculated at the grid nodes. The pressure field is evaluated through a three-step procedure. In the first step, a one-dimensional global correction is implemented at each cross-stream plane. The uniform correction to the streamwise velocity component is obtained by applying global continuity and the corresponding one-dimensional correction to the pressure is obtained from the approximate streamwise momentum equation, Lapworth and Elder (1988). This step accelerates the 3

4 establishment of the correct pressure field. In the second step, a two-dimensional elliptic pressure correction is performed to the cross-stream velocities and pressure to satisfy the local continuity. The values u', v and w coming from the solution of the momentum equations are used for the calculation of U', V and W at the grid nodes. Subsequently, U ', V and W are calculated at the control volume faces by applying the interpolation practice suggested by Rhie and Chow (1983), in order to suppress the oscillations in the pressure field which might arise due to the nonstaggered location of the flow variables. In order that the local continuity to be satisfied, corrections are added to U', V and W to yield U, V and W given by: U=LT. +Bi4 V= V. +CA,/W= +Dip ( 11) where p' is the value of the correction to p *. The substitution of the above relations for U, V and W into the continuity equation yields an equation for the pressure correction: A lfp ip = E AaPpi mip a (12) where m ' is the integrated mass imbalance based on the U *, V * and W velocity components. In the third step, a three-dimensional correction to the pressure field alone is executed for the acceleration of the elliptic pressure effects transmission. The combination of the relations (11) with the interpolation relations suggested by Rhie and Chow (1983) provides relations between U, V and W and the pressure gradients in the three local coordinate directions. An equation for the pressure, similar to the pressure correction equation (12), is obtained from the substitution of U, V and W into the continuity equation. The pressure equation directly incorporates the boundary pressure values and is solved three dimensionally. More details about this pressure correction step can he found in Tourlidakis (1992). A space-marching process is implemented at each cross-stream plane in the primary flow direction. The solution of the algebraic system of equations for each (I), at each cross-plane, is carried out by implementing an Alternating-Direction-Implicit (ADI) method with the use of the Tri-Diagonal-Matrix-Algorithm (TDMA) on a line-byline marching, Patankar (1980). The marching from the inlet to the outlet of the flow domain updates the values for all the dependent flow variables and is repeated many times till a sufficiently convergent solution has been accomplished. 5 PREDICTED CENTRIFUGAL COMPRESSOR FLOWS The computational method is implemented for the prediction of the flow in a radial centrifugal compressor operating at its design point and in a hackswept impeller at its design and choking conditions. These two impellers were tested by Eckardt (1976) and (1980). Eventually the flow through the Rolls-Royce GEM impeller is also simulated. Experimental tests for this impeller, operating at a condition near surge, were carried out by Forster (1988). cylindrical sections, and for circular-arc huh and shroud lines. Five of the computational planes are coincident with the five measurement stations for which measured velocity data are available, thus allowing a direct comparison between the computed and the measured results. Figure la shows the geometrical domain and the outline of the grid which is used to cover the impeller passage, the upstream region and the vaneless diffuser. The shroud and the vaneless diffuser downstream of the impeller are stationary and the impeller is assumed to operate with a clearance gap of 1% of the local blade height. The computations start with axial flow upstream of the impeller inlet, proceed through the rotor passage to its radial exit and march from the impeller outlet at radius R2 = 0.2 m to the constant area diffuser up to a radius ratio of R/R 2 = Two different computational grids consisting of 20 x 20 x 50 points and 32 x 32 x 65 points, are employed for the present study. The predictions based on the fine grid present negligible changes compared to those obtained on the coarse one and hence the 20 x 20 x 50 grid can he assumed to provide solutions which are almost grid independent. It is somewhat surprising that the encouraging predictions presented in the next section have been obtained using such a coarse mesh. In the streamwise direction 28 cross-planes are located inside the rotor passage. Three of the grid points in the hub to shroud direction are located in the tip region. The flow is modelled at the compressor's maximum efficiency condition, that is, a rotational speed of rpm and a mass-flow rate of 5.31 kg/s. The impeller tip speed is equal to m/s and the measured values of total-to-total pressure ratio and efficiency are equal to and 0.88 respectively. At the inlet of the flow domain a total pressure equal to bar and a uniform total temperature equal to K are taken. The flow upstream of the impeller is assumed to be axial and in the absence of information on the velocity profile, an inlet turbulent boundary layer thickness of 15% of the huh to shroud distance is defined on the shroud surface and a thickness of 1% on the hub. A uniform static pressure is maintained at the inlet. The turbulence parameters are kept uniform at the inlet based on an inlet turbulence level of 4%. The flow is simulated in the rotating frame of reference where it is assumed to be steady, and the velocity components of the momentum equations are the relative ones. The absolute velocity on the stationary shroud and vaneless diffuser walls are set equal to zero. The relative velocity is set equal to zero on the impeller blades and on the huh. The walls are assumed to be adiabatic. Along the periodic boundaries upstream and downstream of the impeller passage a circumferentially repeating condition is imposed for all flow variables. The periodic boundary condition is also employed for the grid points which are located in the tip gap. The grid which is located in the tip clearance gap presents the structure which is illustrated in Figure lb. The blade is considered to have a sharp edge at the tip which is found to be an appropriate assumption which does not introduce significant errors in the case where the blade thickness is small at the blade tip as is the case in many centrifugal impellers. This assumption simplifies the computer coding because it treats the tip region without any alterations of the general computational procedure. At the exit of the flow domain, all the independent variables are obtained by extrapolating their values from the plane immediately upstream. 5.1 Eckardt's Radial impeller 5.L1 Boundary conditions and computational grid As a first test case, the current method is applied for the prediction of the flow in the high-speed radial impeller for which Eckardt (1976) performed detailed flow investigations. The geometry of the impeller is constructed according to Schuster et al (1980) and is described by equations for elliptic-shaped blade camber lines in Computational results The predicted primary velocity contour plots at the five measurement planes are illustrated in Figure 2h. The results are compared with Eckardt's measurements as they are presented in contour plots by Moore and Moore (1980b) and shown in Figure 2a. The velocities are normalised by the impeller tip speed Cr R 2. At all stations the velocity distribution demonstrates a large region with potential flow characteristics. One of the major features 4

5 Figure 2a : Radial impeller, design condition. Measured contours of meridional velocity Vni/(0*R2). Blade Tip clearance Casing wall Figure 1 : (a) Outline of the computational grid. (b) Computational grid at the tip clearance gap. Figure 2b : Radial impeller, design condition. Predicted contours of meridional velocity V m/(0*r2). of the flow is the boundary layer thickening on the casing wall commencing at station III and its rapid growth at downstream stations. The substantial blockage caused by the growing wake, results in the displacement of the main flow towards the pressure side of the hub wall. This can be observed from the velocity contours. For example, consider the 0.4 contour which follows an anti-clockwise motion in the impeller passage, its position is found near the suction/shroud corner at plane II, moves towards the suction side at plane III, occupies the hub/suction corner at plane IV and eventually moves to the pressure side of the passage to occupy the largest area of the cross-stream plane at plane V. The wake is located near the shroud-suction corner and grows rapidly after the plane III. A jet flow pattern is also captured at the suction side of the huh wall at plane V, indicated by the contour Vm/(0*R2) 0.5. In general, encouraging agreement between experimental and computed results is shown at all the meridional planes. The discrepancies observed could be attributed to errors in the blade thickness distribution, especially around the blade leading edge, and to the assumptions related to the boundary layer thickness at the upstream inlet boundary. The results of a study of tip clearance effect and the boundary condition for the shroud wall (rotating or stationary) are illustrated in Figure 3. For each shroud boundary condition, the tip clearance size is considered to be successively 0%, 1% and 2% of the local blade height. In the case of I% tip leakage, 3 grid points are uniformly distributed in the tip gap whereas in the case of the 2% gap, 4 points are used. In the case of zero clearance and stationary shroud the wake region is located towards the suction/shroud corner. Increasing the tip gap and imposing the stationary wall condition carry the wake away from the suction/shroud corner region along the shroud wall. In addition the area of the low velocity region increases slightly as the tip gap increases. The calculated static pressure distribution on the shroud wall normalised by the inlet total pressure, is compared with the measured values in Figure 4. The contours demonstrate very good agreement and exhibit the development of the blade to blade pressure gradients and the steep pressure rise in the radial part of the impeller to yield a 1.5 pressure ratio at the impeller discharge. The blade-to-blade predicted velocity vectors at mid-span, nearhub and in the middle of the tip-gap surfaces are presented in Figure 5. Velocity vectors in the meridional planes near the pressure, the suction and on the mid-pitch surface are shown in Figure 6. Low energy fluid is transported radially towards the shroud surface close to the blade surfaces and afterwards it moves across near the shroud forming two counter-rotating vortices which move this boundary layer fluid towards the wake region. This motion explains the rapid increase in the wake size between planes III and V. The secondary flow pattern predicted at the impeller outlet plane is shown in Figure 7a. Three discrete secondary flows can be observed; a major anticlockwise vortex moves along the huh, pressure side and casing wall, a weaker second motion dumps low energy fluid into the wake region from the suction surface and finally the tip leakage flow increases the wake size. It appears that air of relatively high energy from the pressure side of the 5

6 P.S P.S. P.S. Figure 3 : Radial impeller, design condition. Contours of meridional velocity V m/(0*r2). (a) Rotating shroud, no tip leakage. (b) Rotating shroud, 1% tip leakage. (c) Rotating shroud, 2% tip leakage. (d) Stationary shroud, no tip leakage. (e) Stationary shroud, 1% tip leakage. (f) Stationary shroud, 2% tip leakage. passage, passes through the clearance space, expands into the area of the suction surface, and moves the low-energy air along the shroud. The same pattern is more clearly observed when a fine grid of 32 x 32 x 65 is used in the computations and is depicted in Figure 7b. Experimental data Prediction Figure 4 : Radial impeller, design condition. Pressure distribution p/po on the shroud wall. Near shroud Figure 5 : Radial impeller, design condition. Velocity vectors on blade-to-blade surfaces. 11 A qualitative approach for the visualisation of the secondary flows in the impeller is performed by defining the trajectories of particles which are released at various locations of the cross-stream computational plane at the entrance of the passage, as shown in Figure 8. The particle trajectory prediction model, Tan (1988), is based on the solution of a set of particle equations of motion in the flow field obtained with the current flow model. It can he discerned that the particles which are located very close to the shroud wall are driven by the tip clearance flow and move along the shroud opposite to the direction of rotation (in the relative frame of reference). The loss analysis at the impeller exit is performed through the definition of the relative total pressure, Eckardt (1976), as a function of the entropy production. The experimental and predicted relative total pressure distributions normalised by the isentropic total pressure, at plane V, are illustrated in Figures 9a and 9h. Both of the plots show the high-loss region in the position where the wake appears in the velocity contour plots, Figure 2. Despite the -±2% uncertainty of the measured values, it is felt that the experimental results agree closely with those computed although the contours representing the measured relative total pressure levels from 0.90 to 0.96 appear to be more condensed than those predicted. In addition, the shape of the contour lines in this region indicates that the secondary flow along the shroud wall which accumulates low energy fluid into the wake region, is underpredicted in the computations. Calculation without tip leakage produces the distribution shown in Figure 9c where the impact of the tip leakage on the loss production in the impeller is illustrated. Acceptable convergence was obtained using around 700 iterations of the marching procedure which required 9 hours of CPU time on a DEC 5000/200 computer system, corresponding to seconds per grid point per iteration. 6

7 5.2 Backswept impeller Figure 6 : Radial impeller, design condition. Velocity vectors on meridional surfaces. O P.S. 100m/s..7_ N \ / \ / N \ \ \\ \I I / I 100m/s Figure 7 : Radial impeller, design condition. Predicted secondary velocity vectors at the rotor exit. (a) 20 x 20 x 50 grid. (b) 32 x 32 x 65 grid. Eckardt Radial Impeller Design condition S.S Operating and boundary conditions and computational grid The second centrifugal impeller for which the present computational method is implemented, is a typical high-speed backswept impeller for which Eckardt (1980) described detailed measurements. The backswept impeller is very similar to the radial unit and its geometry is again constructed according to Schuster et al (1980). Both impellers have the same tip diameters, shroud contours, axial lengths and blade numbers (20 blades). The backward curvature commences at R/R 2 = 0.8 and terminates at a blade exit angle of 60 (30 of backsweep). The compressor operates at a rotational speed of rpm firstly at its maximum efficiency condition at a mass-flow rate of 4.54 kg/s and secondly at its choking condition at a mass-flow rate of 6.75 kg/s. The rest of the inlet conditions and all the boundary conditions are identical to those used for the radial impeller and are described in section The 20 x 20 x 50 computational grid is also similar to that used for the radial impeller. The computer resources used for the backswept impeller are the same as for the radial impeller. A grid independence study was not performed for this configuration because it was assumed that the results from this study would he similar to those obtained for the radial impeller Computational results for Design condition The primary velocity contour plots at the five measurement planes are shown in Figure 10 where they are also compared against the corresponding Eckardt's measurements (1980). The observed velocity distribution is similar to that found in the radial impeller. The velocity distribution exhibits potential flow features at planes 1 and II within the axial inducer. The blade-to-blade loading increases at plane III and the low energy flow area near the shroud wall thickens. The velocity distribution in planes IV and V show the wake development and the jet/wake structure at the impeller discharge. Comparing these velocity distributions with those obtained for the radial impeller, it can be observed that there is a circumferential unloading especially near the huh surface and a smoothing of the steep velocity gradients which occurs towards the exit of the backswept rotor. The backwards channel curvature appears to enhance the mixing process and an extended potential flow area appears at the rotor exit. The above observation agrees with the results of the experimental investigation which show that the blade backwardcurvature weakens the secondary flows and reduces the feeding of the low-energy fluid into the wake. In general, good agreement between the predicted and the measured values is shown. Figure 11 and 12 compare the predicted streamwise velocity profiles at stations IV and V with the experimental distributions and the distributions predicted by Hah et al (1988). As it will be noted, the present model provides a better correlation with the experimental profiles than Hah's distribution which appear to yield a rather smooth wake whose location is relatively shifted towards the middle of the casing wall. The computed static pressure distribution on the shroud wall in terms of isobaric lines of pressure ratio p/p o is illustrated in Figure 13 where a very good agreement exists with the corresponding measured results. The predicted pressure values within the inducer develop similarly to those in the radial rotor. In the downstream radial region, the pressure rise is governed by centrifugal forces yielding a steep rise in the radial direction. The blade-to-blade pressure variation in the radial region of the impeller is less abrupt in the backswept than in the radial impeller. Figure 8 : Radial impeller, design condition. Predicted streamlines. The developed secondary flow patterns are similar to those predicted for the radial discharge impeller. The calculated velocity vectors in three blade to blade and three meridional grid surfaces are 7

8 above 0.9 Experimental data IX tip leakage - ssem /7.,... -7,L.W WZONV, 0,Vel Z: k/// ANg below 0.90 illustrated in Figures 14 and 15 respectively. The calculated particle trajectories in the passage of the hackswept impeller are shown in Figure 16. The trajectories of the particles which are released from the hub boundary layer region roll in a clockwise manner opposite to the direction of the rotation. As they emerge radially from the rotor outlet they cannot maintain their already low radial momentum and are carried away tangentially by the high momentum fluid. Prediction without tip leakage Prediction with IX tip leakage Figure 9 : Radial impeller, design condition. Distribution of Pro/P0 at the rotor exit. (a) Measured. (b) Predicted with 1% tip leakage. (c) Predicted without tip leakage. The predicted relative total pressure distribution normalised by the isentropic total pressure, at plane V, is illustrated in Figure 17, where it is compared with the measured distribution. The position of the predicted wake is shifted towards the suction/shroud corner of the impeller outlet. The discrepancies between the predicted and the experimental data can be attributed to the limitations of the turbulence model which is employed and the numerical diffusion caused by the first order upwind discretization of the convective term which tends to underestimate the secondary flows. Other areas of uncertainty are in the inlet conditions of the impeller, the blade geometry and thickness definition, the crude simulation of the tip clearance flow using only three grid points and possibly in the experimental data. A calculation without tip clearance provides a distribution with higher levels of relative total pressure contours as illustrated in Figure 17c. The clearance jet causes the formed wake to move towards the centre of the shroud wall. 6 &WM.. FIAISTME A rr AVAIWAr Computational results for Off-design condition Figure 18 shows meridional velocity distributions in the five cross-planes at which Eckardt performed laser traverses. The experimental results have been obtained from Eckardt (1980) and Dawes (1987). A strong wake develops on the shroud wall towards the suction surface and grows rapidly between planes III and IV. The rate of growth decreases downstream of the plane IV until the exit of the rotor. The growing of the wake displaces the main flow towards the pressure side of the passage and a jet-wake pattern is clearly observed. The secondary flows in the impeller follow similar patterns to those predicted in the radial impeller. The flow vectors in meridional planes near the pressure, the suction and the mid-pitch of the impeller passage are plotted in Figure 19. Figure 10a : Backswept impeller, design condition. Measured meridional velocity V m/(0*r2). 0 NUS The relative total pressure distribution at plane V is shown in Figure 20. The agreement between the predicted and the measured values is good and the position of the wake away from the suction/shroud corner is accurately predicted. As expected, the level of losses is larger than at design condition, Figure 17, and demonstrate the capability of the present method to respond to the increase in the mass flow. 5 Figure 10b : Backswept impeller, design Condition. Predicted contours of meridional velocity V i.n/(0 R2). O 5 5 The ability of the method to "sense" the change in the mass flow rate is also exhibited in Figure 21 where the predicted adiabatic efficiency figures for the design and near choke conditions are compared with the measured efficiency curve as it is presented in Hah et al (1988). The experimental data for the rotor efficiency values are based on total pressure measurements at R/R 2 = and temperature traverses at R/R2 = The numerical results are calculated by mass-averaging the predicted flow-field values at R/R 2 = A very good agreement appears to be accomplished despite the various assumptions involved in the description of the inlet conditions and of the impeller geometry. Similar agreement has been achieved by Hah et al (1988). 8

9 0 Prediction Exper.data Near Shroud inside the Clearance Design condition Plane IV Figure 11 : Backswept impeller, design condition. Distribution of meridional velocity V m/(0*r2) at plane IV. (a) Measured. (b) Predicted. (c) Predicted by Hah et al (1988). 0 Exper data Prediction.08 CD Hah el al (1988) Figure 14 : Backswept impeller, design condition. Velocity vectors on blade-to-blade surfaces. Design condition Plane V Figure 12 : Backswept impeller, design condition. Distribution of meridional velocity Vm/(0*R2) at plane V. (a) Measured. (b) Predicted. (c) Predicted by Hah et al (1988). Figure 15 : Backswept impeller, design condition. Velocity vectors on meridional surfaces. Eckardt Backswept Impeller Design condition Experimental data Prediction Figure 13 : Backswept impeller, design condition. Pressure distribution p/p o on the shroud wall. Figure 16 : Backswept impeller, design condition. Predicted streamlines. 9

10 Arsperimental data fx tip leakage 65:11wil 4 SI= vim m/wf s..= / NN _ below 0.90 revue HJB &,livis 22 P ccm/e Near Pressure ////1A1/4 Mid Pitch II! Prediction without tip leakage %ROW Prediction with lx tip leakage Figure 17 : Backswept impeller, design condition. Distribution of Prel/P0 at the rotor exit. (a) Measured. (b) Predicted with 1% tip leakage. (c) Predicted without tip leakage. Near Suction Figure 19 : Backswept impeller, choking condition. Velocity vectors on meridional surfaces. Szperimental data IX tip leakage above 0.94 Prediction with IX tip leakage plane II below 0.80 Figure 18a : Backswept impeller, choking condition. Measured meridional velocity Vin/(C)*R2). Mug O Figure 20 : Backswept impeller, choking condition. Distribution of Prei/P0 at the rotor exit. (a) Measured. (b) Predicted with 1% tip leakage BACKSTIEPT IMPELLER Adiabatic Efficiency P3 34P0. OAS Pro.. an rgy r.onta rn/rn, 1.3 1,3 Figure 21 : Backswept impeller, adiabatic efficiency. 31.4ROL/0 Figure 186 : Backswept impeller, choking condition. Predicted contours of meridional velocity V ni(f2*r2). 10

11 5.3 Rolls Royce GEM impeller Operating conditions and computational grid The GEM impeller was tested at Cranfield and is part of the axial + centrifugal compression system of the currently in service GEM engine, Birdi, Forster and Tourlidakis (1991). The impeller has a pressure ratio of 3.5. The unshrouded impeller has 19 blades, an axial inlet with a radius of mm at the huh and mm at the shroud, and discharges the air radially at a radius of mm. The impeller has a backsweep angle of 40 from the radial direction and the current measurements were carried out using a vaneless radial diffuser downstream of the rotor. This impeller is more typical of an aerospace impeller than that studied by Eckardt (1980) with a significantly lower specific speed and tip width at outlet (6 mm instead of 26 mm). Results are presented for three planes inside the rotating passage and for a plane just upstream of the leading edge, Figure 22a, where a fairly comprehensive mapping of the flow patterns has been achieved. The impeller is assumed to operate with a linearly distributed tip clearance between 1.2% of the local blade height at the leading edge and 2.9% at the trailing edge. The flow proceeds from the impeller discharge at radius R2 = mm to the diffuser up to a radius ratio of R/R 2 = 1.2. After a grid refinement study, it was found that a computational grid of 20 x 20 x 70 yields solutions which are sufficiently grid independent. In the streamwise direction 40 crossplanes are located inside the rotor passage. Three of the grid points in the spanwise direction are uniformly distributed in the tip gap region. The compressor is simulated to operate at a rotational speed of rpm at a flow condition near surge. The rest of the inlet conditions and all the boundary conditions are the same as those described for the radial Eckardt impeller in section In the following computations, sufficiently convergent solutions were again obtained after about 720 iterations of the marching procedure and required 13 hours of CPU time on a DEC 5000/200 computer system, which corresponds to seconds per grid point per iteration Computational results At section A, the magnitude of the measured blade relative flow velocity presents its peak value in the shroud/suction corner and decreases across the passage to the hub/pressure corner, Figure 22a. The predicted flow characteristics at this station, are "potential" with the existence of a jet pattern close to the shroud/suction corner. The location of the jet and the actual values of the magnitude of the relative velocity are in good agreement to the measured data, Figure 22b. At station B, the experimental velocity contours show a somewhat complicated pattern as illustrated in Figure 22a. On the hub the velocity decreases from 183 m/s on the suction side to 145 m/s on the pressure side. At mid passage the velocity remains almost constant at approximately 150 m/s. On the casing wall the velocity is generally lower with a local region of much lower velocity near the mid passage (93 m/s). According to the predictions shown in Figure 22h, a jet appears towards the suction side of the blade in the middle of the distance between the hub and the tip unlike the experimentally detected jet location towards the huh wall of the suction side. Also a wake pattern is predicted to emerge on the shroud wall towards its pressure side. The agreement at this station is not as precise as at the other stations. The flow at station C is characterised by the existence of a strong wake pattern in the middle of the shroud wall, Figure 22h. The "depth" of the wake is accurately captured by the computational model, but its location is predicted closer to the suction side. The absence of experimental information in the huh area is likely to hide the true magnitude of the jet pattern velocity which appears to be slightly over-predicted. At station D, close to the impeller exit, the relative velocity contours show a region of low velocity in the middle of the shroud surface and generally increasing velocities towards the huh, Figure 22a. The jet flow pattern which is predicted, as indicated by the contour levels of 160 m/s in the two passage corners of the huh wall, Figure 22b, agrees closely with the measured pattern. The depth of the wake is again accurately represented but its location on the shroud wall is relatively shifted closer to the suction surface. The tip leakage flow contributes to the migration of the wake towards the centre of the shroud wall. This motion is under-predicted in the current computations and it appears probable that the magnitude of the simulated tip gap is less than the real one which was kept relatively high, especially in the radial region of the rotor, due to mechanical constraints. Figure 23a, illustrates the measured blade relative flow angle contours. The region of high "slip" coincides with the region of lowest streamwise velocity. The secondary flow pattern is dominated by a clockwise passage vortex which moves fluid towards the pressure side along the casing and towards the suction side along the huh. The predicted flow angle distribution is shown in Figure 23h. As found in the optical measurements as well, the angle remains almost unchanged and equal to 40 to most of the cross-section and increases rapidly towards the shroud wall. The non-dimensional static pressure distribution on the shroud wall is represented in Figure 24. The main flow characteristics which are predicted at the near surge condition for the GEM impeller are similar to those for the Eckardt's impellers. The jet flow pattern follows an anti-clockwise motion, the wake is located on the shroud wall where the low energy fluid from the huh and blade surface boundary layers, is directed due to the secondary flows and the tip leakage action. The relative flow angle increases from the hub to the casing wall where the "slip" is very high. Although the main trends of the relative flow angle distribution at all the measurement planes are captured by the computational model, there are slight discrepancies in the actual distribution. The predicted turbulence intensity level increases rapidly in the wake region and presents a rather smooth distribution in the rest of the cross-sections. The excessive fluctuations are restricted in the wake region attached on the shroud showing a weak turbulence damping due to the Coriolis effects. 6 CONCLUSIONS The computer code developed is capable of predicting the involved flow phenomena occurring in high speed subsonic centrifugal compressor configurations with both stationary and rotating parts and with or without tip leakage. The method can he applied to different centrifugal compressor designs and at different operating conditions providing a great deal of detailed information about the complex flow field patterns occurring. For the test cases related to Eckardt's impellers, the development of a strong wake on the shroud wall near the suction surface is predicted. The behaviour of this wake is affected by the backward curvature at the radial part of the rotating channel which tends to smooth the circumferential pressure and velocity gradients. The classical jet/wake flow pattern is clearly observed. The influence of the increase in flow rate has been well represented and good predictions of the efficiency were possible. In general, good agreement is achieved with the measured meridional velocity distributions at the five measurement planes and with the measured pressure rise distribution. The code provides substantial insight into the evolution of the secondary flows in the impeller which affect the wake formation and location. The computational method can capture the effects of the tip leakage which decreases the efficiency and shifts the 1 1

12 (c) Section C S.S. P.S (d) Section D Is, P.S. A Figure 22a : GEM impeller, near surge condition. Measured relative velocity V R distribution in m/s. (b) Section B (a) Section A (c) Section C P.S. (d) Section D S.S P.S. Figure 22b : GEM impeller, near surge condition. Predicted relative velocity VR distribution in m/s. wake towards the centre of the casing wall. Nevertheless, it is felt that a more refined tip leakage calculation is required with more grid points located in the tip clearance and more accurate representation of the blade thickness distribution near the blade tip. The computed flow field data in the high-speed Rolls-Royce GEM impeller, indicate a curvature dominated flow with a wake flow pattern observed on the casing wall of its radial discharge and a jet pattern close to the hub wall. The implementation of the computational model for the flow analysis in the GEM impeller was less successful than in Eckardt's impellers with the main reasons for that being the existing uncertainties in the inlet conditions, the running tip clearances and the completeness of the available experimental data for the much smaller blade passages. Nevertheless, good agreement has been accomplished for most of the predicted flow parameters with the available experimental results. However, further work is required for a more accurate representation of the secondary flows and the loss production mechanisms occurring in the rotating passage. REFERENCES Ahmed N.A. and Elder R.L., 1990, "Flow Investigation in a Small High Speed Impeller Passage Using laser Anemometry", ASME paper 90-GT-233. Birdi K., Forster C.P. and Tourlidakis A., 1991, "Comparison of the Experimentally Defined and Computed Flow Field in the Rolls Royce GEM Impeller", 10th ISABE, Nottingham U.K,