The Calculation of Three Dimensional Viscous Flow Through Multistage Turbomachines

Transcription

1 C THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47 St., New York, N.Y 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. s Discussion is printed only if the paper is published in an ASME Journal. Papers are available ^^C from ASME for fifteen months after the meeting. Printed in USA. Copyright 1990 by ASME 90-GT-19 The Calculation of Three Dimensional Viscous Flow Through Multistage Turbomachines J. D. DENTON Whittle Laboratory Cambridge, U.K. SUMMARY The extension of a well established three dimensional flow calculation method to calculate the flow through multiple turbomachinery blade rows is described in this paper. To avoid calculating the unsteady flow, which is inherent in any machine containing both rotating and stationary blade rows, a mixing process is modelled at a calculating station between adjacent blade rows. The effects of this mixing on the flow within the blade rows may be minimised by using extrapolated boundary conditions at the mixing plane. Inviscid calculations are not realistic for multistage machines and so the method includes a range of options for the inclusion of viscous effects. At the simplest level such effects may be included by prescribing the spanwise variation of polytropic efficiency for each blade row. At the most sophisticated level viscous effects and machine performance can be predicted by using a thin shear layer approximation to the Navier Stokes equations and an eddy viscosity turbulence model. For high pressure ratio compressors there is a strong tendency for the calculation to surge during the transient part of the flow. This is overcome by the use of a new technique which enables the calculation to be run to a prescribed mass flow. Use of the method is illustrated by applying it to to a multistage turbine of simple geometry, a 2 stage low speed experimental turbine and to two multistage axial compressors. NOTATION F body force, flux j grid index in meridional flow direction M mass flow rate P static pressure S entropy T static temperature V absolute velocity W relative velocity Tw wall shear stress p fluid density lip polytropic efficiency Subscripts x in axial direction r in radial direciton t in tangential direction m in streamwise direction INTRODUCTION Fully 3D calculations for single blade rows are now well established, the most common method being time dependent solutions of the Euler or Navier Stokes (N-S) equations. Before applying such methods, however, the boundary conditions for the blade row must usually be established by means of a separate calculation for the whole machine. The latter will usually be performed using an axisymmetric throughflow method with empirical input to allow for viscous losses and for blade row deviations. If the exit flow predicted by the 3D calculation is not compatible with the input to the throughflow calculation, eg different blade exit angles, then it may be necessary to repeat the whole process until compatible results are obtained from the two calculations. This can be a tedious process involving a great deal of human intervention. To help to circumvent this process, about 10 years ago, the author introduced a method of calculating the 3D inviscid flow through a single stage turbomachine, ie two blade rows, (Denton & Singh 1979). Since the flow through two blade rows in relative rotation is inevitably unsteady ( unless they are very widely spaced) this type of calculation must involve some modelling of the real flow to remove the effects of unsteadiness. This was achieved by circumferentially averaging the flow at an axial station approximately mid way between the two rows so that the upstream row saw a circumferentially uniform downstream boundary condition and the downstream row saw a circumferentially uniform upstream flow approaching it. This averaging is not a physically realistic process unless the blade rows are widely spaced and if performed too close to the leading or trailing edge can lead to unrealistic predictions of the flow in those regions of the blades. However, it should be realised that this assumption of axisymmetric boundary conditions is no different in principle from that which is almost universally applied in both 2D, Q3D and full 3D blade to blade calculations. The only difference in practice is that the condition may have to be applied closer to the blade row than would be usual in such methods. *Presented at the Gas Turbine and Aeroengine Congress and Exposition June 11-14, 1990 Brussels, Belgium This paper has been accepted for publication in the Transactions of the ASME Discussion of it will be accepted at ASME Headquarters until September 30, 1990

2 It is also important to realize that the pitchwise averaging does not affect the spanwise variation in flow. The spanwise variation of pressure, velocity, flow angle, etc, at stations between the blade rows is obtained from the 3D calculation. In particular this means that the calculation will predict the spanwise variation of stage reaction in the same way as would an axisymmetric throughflow calculation. Since the blade exit angles are automatically compatible with the blade to blade calculation there is now no need to iterate between throughflow calculations and blade to blade calculations to obtain the flow conditions between the two blade rows and the complete flow through the stage will be predicted by a single calculation. This single stage 3D calculation has proved especially valuable for the last stages of large steam turbines. These have very low hub to tip ratio, high aspect ratio, high exit Mach numbers and large meridional pitch angles ( up to 60 0 ) and so the flow is highly three dimensional as well as transonic. As a result conventional throughflow calculations and coupled blade to blade calculations are not sufficiently accurate and the fully 3D stage calculation has proved both a simpler and a more accurate tool. Examples of such calculations are given by Denton,1983 and Grant & Borthwick,1 987 Although there is no difficulty in principle in extending such 3D calculations to more than a single stage it was not thought to be worthwhile to extend the original method because of the neglect of viscous effects. An inviscid calculation through a multistage machine would inevitably give too high a mass flow for a prescribed pressure ratio and hence the velocities and flow angles could be in significant error. The result would be similar to that obtained by running a throughflow calculation without the inclusion of any losses. Inclusion of viscous effects in 3D calculations is a comparatively recent development that is still fairly expensive in terms of computer time. It is the author's opinion that in the complex turbomachine environment the accuracy of such viscous calculations is severely limited by the limitations of turbulence modelling so that at present only qualitatively accurate results can be obtained from even the most sophisticated turbulence models. In such circumstances there seems little justification for using any but the simpler turbulence models and combining them with semi empirical corrections to enable them to predict the correct magnitude of loss. As a result mixing length eddy viscosity models are by far the most commonly used method, the Baldwin Lomax model being especially popular. Typical of the methods available for viscous flow calculation through a single blade row is that of Dawes ( Dawes 1988). In the spirit of modelling the viscous effects in the simplest possible way the author showed how it was possible to modify a conventional Euler solver to include approximations to the viscous effects by inclusion of a body force term in the momentum equations. This force term can be calculated within a separate subroutine of the program so that the method used to estimate it is completely decoupled from the main flow solver. As originally proposed, (Denton 1986 ), this force term was estimated by a semi empirical approach but it was pointed out that the model could in principle be used with any level of approximation to the full N-S equations. Over the past few years the author has developed a total of seven different subroutines for estimating the viscous force. These range in complexity from the original specification of an empirical frictional pressure loss for the blade row to a thin shear layer approximation to the full N-S equations. The most recent subroutine was developed specifically for this multistage calculation and enables the spanwise variation of polytropic efficiency for each blade row to be input to the calculation. This approach to the inclusion of viscous effects has the advantage that different levels of approximation may be used within the same program. For example a calculation may be started with a very simple completely empirical model of viscous effects and after a solution has been obtained the calculation may be restarted with a ('a more sophisticated model. The approach is especially attractive for including viscous effects in multistage calculations where limitations on the number of grid points and the neglect of unsteady effects means that more complex methods are even less justifiable than they are for a single blade row calculation and empirical input is likely to be essential if realistic results are to be obtained. Three dimensional calculations for multiple blade rows have also been developed by Adamczyk (1985, 1989), Arts (1984) and by Ni(1 989). Adamcyzk uses what he calls a 'passage average' approach in which 3D calculations are performed separately for each blade row with the effects of the other blade rows being modelled by additional force terms in the equations. Once a 3D solution has been obtained for one blade rows it is circumferentially averaged to obtain the force terms needed for the 3D calculations in other blade rows. This approach is much more rigourous than that of the author but is inevitable more time consuming because several 3D calculations have to be done for every blade row with the force terms being adjusted iteratively. Although the method makes some attempt to include unsteady effects, only the effect of unsteadiness on the time average flow is captured. Viscous effects are included in the latest version of Adamczyk's method. Arts (1984) effectively couples two single blade row calculations by a common boundary condition, again effectively performining a circumferential averaging at the interface. Ni (1989) gives very few details of his method but shows results for a 2 stage turbine. It is not clear whether or not viscous effects are included, but good agreement with experimental data is claimed and it is unlikely that this could be achieved without some modelling of viscous effects. 3. CALCULATION METHOD 3.1 EULER SOLVER The basic Euler solver used in this method is well known and is described by Denton 1982 and Denton The method is an explicit finite volume method with spatially varied time steps and three levels of multigrid. The multigrid block sizes are not those conventionally used, with two elements per block side, but instead any number of elements can be used along the sides of each block. Usually blocks of 3 elements per side are found to be near optimum and so blocks of 3x3x3, 9x9x9 and 27x27x27 elements would be used for the 3 levels. However, for the present application it was not considered advisable to use the third level of multigrid because the blocks are likely to extend over more than one blade row. As pointed out by the author, (Denton 1986), this level of multigrid enables high levels of grid refinement to be used without large deterioration in the rate of convergence. Typically 70x28x28 mesh points will be used for a single blade row viscous calculation with the mesh spacing varied by a factor of between points adjacent to solid surfaces and points near the centre of the passage. With this number of points convergence to engineering accuracy will usually be achieved in the order of 1000 to 2000 steps. The grid used by the method is a simple H mesh with all the variables stored at the cell corners. This may not be the optimal type of mesh for any one problem but the method is designed to be applicable to all types of turbomachines, ranging from axial flow to radial flow, and to include such compicating features as splitter blades, part span shrouds and free blade tips ( ie propellers). Such features are easily accomodated by the H mesh but it it unlikely that any other type of mesh could be made to include many of them. The author's experience is that the errors incurred by using a highly sheared H mesh in regions such as that around a thick leading edge can be made acceptable by using sufficient mesh points. The cell corner storage of the variables is felt to be preferable to cell centre storage since the formal accuracy does not decrease for rapid changes of mesh spacing. The mesh used for a calculation on a 4 blade row turbine is shown in Fig VISCOUS FORCE TERM The idea behind the use of a force term in the momentum equations to model viscous effects is describe in Denton,1986. The only approximation inherent in this model is the neglect of viscous

3 effects in the energy equation and it is shown that this is a good approximation for the adiabatic flow of fluids with Prandtl number close to unity, especially at low speeds. The viscous force acting on each finite volume must be found by summing the viscous shear stresses and normal stresses around the faces of the element. Any level of approximation to these stresses can be used and there is no inherent reason why the stresses should not be obtained from the full N-S equations. In practice it was found that remarkably realistic results could be obtained using very simple approximations to the viscous force term. That described in Denton,1986 involved an input value of skin friction coefficient and an algebraic distribution of the shear stress away from solid boundaries. It was therefore completely empirical since the effective turbulent viscosity did not depend at all upon the flow. The method most commonly used in the latest version of the program obtains the viscous force from a thin shear layer approximation in which all normal components of viscous stress are neglected and only the shear stresses on the streamwise and bladewise faces of the cuboid shaped elements are considered, viscous stresses on the quasi orthogonal faces (ie faces which are tangential to the theta direction) of the elements are neglected. In order to avoid the use of an extremely fine mesh near to solid boundaries, with grid points in the laminar sublayer, it is thought preferable to use a slip model of flow on the boundaries and to calculate the shear stress on the boundary using wall functions. The shear stress on the solid boundaries is obtained by assuming that the first grid point away from the solid surfaces ( ie not the point lying on the surface) lies either in the laminar sublayer, or, more likely, in the logarithmic region of a turbulent boundary layer. In the latter case the well known expression for the log law boundary layer profile V+ = 2.5 In Y (la) The shear stresses at points off the wall are obtained from a simple mixing length model. No attempt is made to vary the model between inner and outer regions of the boundary layer, as is done in the well known Baldwin-Lomax and Cebeci-Smith models. Instead the turbulent viscosity is calculated in the conventional way from a mixing length which is taken as I = 0.41y, where y is the perpendicular distance from the nearest wall. The mixing length calculated in this way is 'cut off' by an algebraic filter at a prescribed distance from the surface. Since the calculated stresses usually decay to near zero outside the boundary layer the precise location of this cut off is not very important provided that it is outside the boundary layer. It is usually applied at a point which is 10% of the blade pitch from the surface as measured in the circumferential direction. The mixing length behind the trailing edge is held constant at a value which is input to the calculation. This value is usually taken as 2% of the blade pitch. Having obtained the viscous shear stresses in this way the viscous force term is obtained by summing the stresses around the bladewise and streamwise faces of every element. The viscous force acting on the elements does no work in a stationary coordinate system but does do work in a rotating system and this work must then appear as a source term in the energy equation. However, as explained by Denton 1986 the viscous shear work and the heat conduction may be assumed to almost cancel one another and so are not included at all in the energy equation. 3.3 THE PRESCRIBED POLYTROPIC EFFICIENCY MODEL For multistage calculations it was thought desireable to be able to input empirical blade row losses in a similar way to that used in some throughflow calculations. In order to do this the body force must be made to depend on the required efficiency. As shown in the Appendix this can be achieved by applying a body force, Fv per unit volume, where Is very closely approximated by an explicit relationship between Reg (= p V2 y2/µ) and Cf (= 2'r W/pV2 2 ). This relationship is Fv = (1-li p ) dp for a turbine Cf = ^n(re 2) + (I (Re2))2 (1 b) Its use avoids the need for iteration to calculate the skin friction dp and Fv = ( 1-1) for a compressor P acting along the streamlines in the opposite direction to the flow. In practice, for simplicity, the streamlines are assumed to coincide FIG 1. MESH FOR INVISCID CALCULATION ON A TWO STAGE TURBINE 3

4 with the quasi streamlines of the grid and the magnitude of the body force acting on an element is given by Fz = Vol. Fv where z is the distance along the quasi streamline. This force must then be resolved into its three components before inclusion in the momentum equations. In the present code the value of polytropic efficiency is input as a function of spanwise position for every blade row. This loss model will, unfortunately, not always produce a flow with exactly the prescribed value of polytropic efficiency because additional entropy increases occur due to shock waves and to numerical errors. For subsonic machines the difference between the prescribed and achieved values of polytropic efficiency will be small unless too coarse a grid, giving rise to numerical entropy production, is being used. For transonic machines, especially transonic compressors, the additional shock loss can cause significant differences between the prescribed and achieved polytropic efficiency. 3.4 THE INTER-ROW MIXING MODEL The simple averaging of flow quantities at a plane between the blade rows as used previously ( Denton 1983) is equivalent to applying a mixing process between the non uniform flow at the calculating plane upstream of the mixing plane and the mixing plane. This process conserves mass, energy and momentum, but as with any real mixing process entropy will be created by the mixing. However, experience with inviscid flows has been that in practice the magnitude of entropy creation is negligible. The real problem with the model is that a circumferentially uniform flow may be forced to exist too close to the leading edge of the downstream blade row. This does not allow the flow to adjust circumferentially to the presence of the blade as it would in reality. As a result the leading edge loading on the blade row may be wrong and may even be physically unrealistic. The magnitude of this problem depends on the leading edge loading and thickness and on how close the leading edge is to the mixing plane. For a moderately loaded leading edge the plane can be located as close as 1/4 blade pitch upstream without apparent problems but this would not be acceptable for highly loaded leading edges. A similar problem occurs at the trailing edge of the blade row upstream of the mixing plane but for subsonic exit flows the Kutta condition ensures zero loading at the trailing edge and so the problem is not so serious. This simple averaging process is included as an option in the present method. Flux variation at JMIX-1 A simple but effective means of relieving this problem is also included in the method. The method may be briefly described as obtaining the circumferential variation of fluxes at the mixing plane by extrapolation from the upstream and downstream planes whilst adjusting the level of the fluxes to satisfy overall conservation. Thus the fluxes 'seen' by the blade rows are circumferentially nonuniform at the mixing plane with different circumferential variations, but the same average value, being 'seen' by the upstream and downstream rows. In more detail. The circumferential averaging process is first applied as before so that the averaged flow at the mixing plane conserves mass, momentum and energy with the circumferential average of the non-uniform fluxes crossing the adjacent calculating stations. This process may be thought of as treating all the elements (at the same spanwise location) immediately upsteam of the mixing plane as a single large element and conserving fluxes between the non-uniform flow entering its upstream face and the uniform flow leaving its downstream face which lies on the mixing plane. However, when applying the conservation equations to the individual cells the fluxes crossing the mixing plane are not taken to be circumferentially uniform. Instead, for the elements immediately upstream of the mixing plane, the flux crossing the face on the mixing plane is obtained by multiplying the flux through the upstream face by the ratio of the circumferentially averaged flux at the mixing plane to the circumferentially averaged flux at the upstream plane. ie F jmix F jmix = F jmix-1 ( ) 4a jmix-1 The idea is illustrated schematically in Fig 2. This extrapolation of the flux variation is done at every spanwise position and is equivalent to obtaining the circumferential variation in flux from the upstream face of the element but adjusting its magnitude to satisfy overall conservation. The same treatment is applied to the elements immediately downstream of the mixing plane. At every spanwise position the flux entering an element through the mixing plane is obtained by multiplying the flux through its downstream face by the ratio of the circumferentially averaged flux at the mixing plane to the circumferentially averaged flux at the location of the downstream face. F jmix F jmix = F jmix+1 ( F ) 4b jmix+1 ^ ^ I Flux arlatlon at JMIX seen by upstream ele ments Flux variation at JMIX as seen by downstream elements variation at JMIX+1 The treatment described is only approximate because it is effectively assumed that the derivative of the flux terms along the quasi streamlines of the grid at the mixing plane is only due to changes in the circumferentially averaged flux. However, it does allow the flows entering and leaving the mixing plane to vary circumferentially and so is a great improvement on the previous method when the mixing plane is close to the leading or trailing edge. JMIX-1 FLUX JMIX average value JMIX Theta direction JMlx+1 Merldlonal flow direction FIG.2 TREATMENT OF FLUXES AT THE MIXING PLANE Fl-OW MASS FLOW FORCING Traditional Euler solvers always work with boundary conditions of specified pressure ratio and allow the mass flow to be predicted by the solution. This is a well conditioned boundary condition for most applications and is essential for choked blade rows. However, for high pressure ratio compressor blades, when the blade is operating near stall, the calculation may become unstable due to 'numerical surge'. In this the computed flow separates or generates a high loss, possibly during the transient part of the calculation, the resulting blockage reduces the mass flow which increases the incidence on the blade which makes the separation or loss larger,-----etc. Thus the calculation may fail as a result of the transient induced by the initial guess rather than because of a genuinely unstable operating point. This tendency to instability becomes more pronounced as the

5 compressor pressure ratio increases and so it becomes almost essential to be able to ensure that each blade row is operating reasonably close to design at all times during the transient part of the calculation. 2.? There is no known way of removing the boundary condition of prescribed pressure ratio from a time dependent Euler or N-S method and so the alternative is to specify both mass flow and pressure ratio and to make the two compatible by generating the correct amount of loss. This is the basis of the method developed for the present code. In principle the loss is generated by a body force in the same way as the viscous terms are included, however, the details are very different. The mass flow, M(j), crossing every quasi-orthogonal plane ( on which the index j is constant) is calculated and if this differs from the prescribed mass flow, Mi n, all the velocities at that plane are adjusted according to A = RF * (Min -M(j))/Min * (P Vm) mid A(PVx)= (Vx/Vm)*A A (P Vr) _ (V r/ V m ) ' 4 A (P Wt) = ( Wt/ Vm) *A Ui w > 0 16 C 3 12 w U E L U 0 o w O > cn L w 8 D o 0 4 J J co m N E N E L Where RF is a relaxation factor. 4. This adjustment makes a constant change to pv m at all grid points on the quasi orthogonal surface without changing the relative flow direction. When a steady state is reached the increment in (density * velocity ) produced by eqns 5 must be exactly cancelled by an opposite change produced by the time stepping procedure. This means that the momentum fluxes are not in balance and an effective body force, F m, is acting on the elements in the direction of the relative velocity. The force can be positive or negative and its magnitude may be adjusted by changing the relaxation factor ; it will produce an entropy gradient along the streamlines according to F m = T ds/dz z being the distance along a streamline. When this method is applied the mass flow is very rapidly brought to a value close to the required value,mi n, and loss is produced at a rate just sufficient to make the imposed pressure ratio consistent with the imposed mass flow. Since A will be zero when the mass flow is equal to that specified the flow rate obtained must always be slightly different from Mi n unless the pressure ratio and mass flow are already consistent. The small difference between the specified and calculated mass flow will be directly proportional to RF which is in effect a scaling factor on the imposed force. In practice values of RF=0.25 are found to produce negligible difference between the prescribed and the calculated mass flow. This mass flow forcing process can be applied during the initial stages of a calculation until the initial transients have decayed and a near converged solution is obtained. This solution will have spurious loss generated by the mass flow forcing and since this loss can be produced everywhere in the flow it may be physically unrealistic. The value of RF may then be gradually reduced and the calculation continued with the mass flow being allowed to find its own value as determined by the pressure ratio. This will not prevent 'numerical surge' in cases where the flow is genuinely unstable, ie if the pressure ratio of the compressor is too high, but it will enable a much closer approach to the point of instability to be reached before the numerical instability prevents a solution from being obtained. It is doubtful whether solutions for multistage high pressure ratio axial compressors or for single stage high pressure ratio centrifugal impellers, could be obtained without this procedure. NUMBER OF TIME STEPS FIG 3. CONVERGENCE OF A CALCULATION ON A 4 STAGE TURBINE An alternative use of the same idea is not to specify the mass flow for the machine but always to force the local mass flow towards the average value for the whole machine, as obtained by averaging the computed flow at every calculating station. ie JM Min = IM(j)/ JM 7 1 is used in eqn 5. This procedure gives almost the same stabilising effect as specifying the mass flow but it does not affect the final steady solution. In fact this use of the method often enables solutions to be obtained in significantly fewer time steps even when there is no problem with 'numerical surge'. 4. APPLICATIONS OF THE METHOD The method has been applied to several multistage turbines and compressors. There is no limit in principle to the number of blade rows that can be calculated in a single run, the limit in practice being set by the available computer power and storage. For realistic viscous solutions a minimum of about 50 axial stations and 19 pitchwise and spanwise stations per blade row are needed, making about points per row. Hence 5 blade rows would require about mesh points and about 15 Mbytes of computer storage. A typical run time for this number of points would be 15 hours on a single processor of an Alliant FX80. If the option to specify polytropic efficiency without trying to predict detailed viscous effects is used then the grid need only be sufficient to support an inviscid calculation and about 40 x 13 x 13 grid points per blade row would be adequate. At this level the program can be run for up to 4 blade rows on a PC with 4 Mbytes of storage. The program was initially developed for axial turbines where 'numerical surge' is not a problem. Convergence for such machines was found to be surprisingly fast as is illustrated in Fig. 3. One might expect the number of time steps required to be proportional 120

6 J I FIG 4. SOLUTION WITH WIDE AXIAL SPACING (0.8 x CHORD) to the number of axial calculating stations and on this basis the convergence should be significantly slower. The reason for this faster than expected convergence is thought to be that for a high pressure ratio turbine pressure waves are rapidly damped by the work extraction. For example a change in turbine exit pressure will produce a proportionally much smaller change in inlet pressure to the last blade row and this in turn will produce an even smaller change in the inlet pressure to the penultimate blade row, etc. To check the effectiveness of the mixing plane treatment, calculations were performed on a 2 stage axial turbine with repeating stage geometry for two different values of blade row spacing. The grid used is shown in Fig.1. Figs 4 & 5 show the calculated static pressure contours for a inter stage gap of 0.8 of the axial chord and with the gap at a more realistic level, 0.25 of the axial chord. It can be seen that there is very little difference between the two solutions except in the immediate vicinity of the mixing plane where the contouring routine is confused by the discontinuity in grid coordinates at the mixing plane. Figs 4 & 5 were obtained with a prescribed polytropic efficiency, by contrast Fig 6 shows results for the same turbine with a finer mesh (19x180x19) and with viscous effects calculated from the thin shear layer model. The predicted polytropic efficiency in this case was 87.8 % which is a very reasonable value for a turbine with poor blade surface velocity distribution ( the blade sections were drawn not designed ) and no tip clearance. To check for any problems that might arise from more stages and higher pressure ratios the same blade geometry was run with 4 stages and a pressure ratio of 5:1. The annulus height was increased to accomodate the increasing volume flow. No difficulties were encountered, convergence was naturally rather slower than for 2 stages but nevertheless, as shown in Fig.3, took only about 1000 steps. Fig 7 shows the Mach number contours from this solution, the last blade row is choked with a shock wave which is highly smeared due to the small number of grid points per blade row. Fig. 8 shows computed velocity contours from a viscous solution at the mid span of a two stage low speed experimental turbine. Convergence was very slow for this case due to the low Mach numbers, peak value = Note the very thin boundary layers associated with the mainly accelerating flow in this well designed turbine. The predicted isentropic efficiency was 88.2% which is rather lower than expected for this machine, a possible explanation for this is the assumption of fully turbulent boundary layers. The experimental efficiency is not yet available. Solutions have been obtained for a 3 stage high pressure ratio (4.5:1) transonic compressor using both a prescribed polytropic efficiency and with calculation of viscous effects. Results cannot yet be published but experience was that the solutions were comparatively difficult to obtain because of instabilities at the FIG 5. SOLUTION WITH REDUCED AXIAL SPACING (0.25 CHORD) 6

7 FIG 6. VISCOUS SOLUTION FOR TWO STAGE AXIAL TURBINE MACH NUMBER CONTOURS. mixing plane when using the improved treatment aescrioea aoave. The calculated efficiency was also significantly lower than the prescribed polytropic efficiency because of the neglect of shock loss. A viscous solution for a 4 stage, 9 blade row ( IGV + 4 stages), industrial compressor with an overall pressure ratio of about 1.6 was obtained with no difficulty. Fig 9 illustrates the blade static pressure distribution at mid span for this machine. An interesting feature of this solution is the growth of the annulus wall boundary layers. A knowledge of the blockage due to these boundary layers is an important requirement for conventional throughflow calculations and its prediction is currently heavily dependent on empirical correlations. Fig.10 illustrates the computed boundary layer development through the compressor, showng the initial thin boundary layer growing to fill almost the whole annulus. No experimental data are available but it is known that a repeating velocity profile is usually obtained after a few stages, hence the results appear very plausible. Ability to predict this annulus boundary layer blockage would be an important advance in compressor design. The predicted isentropic efficiency of this machine was 82.2 % which is somewhat lower than might be expected for such a compressor without any tip leakage. The method has also been used as part of the design process for a large 2 stage, 6 blade row, axial flow compressor with high subsonic blade surface Mach numbers. Since a conventional throughflow solution was available for this machine it is informative to compare its predictions with those from the 3D viscous calculation. To make the comparison meaningfull the 3D calculation was run without any attempt to model the annulus wall boundary layers. The througflow calculation uses empirical data and correlations to predict blade row loss and deviation. Fig. 11 compares the two predictions of relative flow angle at exit from both rotors and stators ( the angles from the 3D calculation are those at the mixing planes) and shows excellent agreement between the two. Ability to predict blade row deviations without any empiricism represents a significant advance over current throughflow calculations. The predicted isentropic efficiency of this machine was 85.5%, that predicted by the throughflow calculation was 88.3%. 5. CONCLUSIONS The method described is still at an early stage of development but it is hoped that the results presented in this paper illustrate its potential. The main attraction of the method is its ability to reduce the amount of human intervention needed to obtain solutions for multistage turbomachines by eliminating the need to iterate between throughflow solutions and blade to blade solutions. At the same time it removes most of the limitations implicit in the quasi three dimensional approach, especially the neglect of stream surface twist and the need to assume a distribution of stream surface thickness within the blade rows. The inclusion of viscous effects is necessarily approximate and it cannot be claimed that the method will yet provide accurate a- priori predictions of machine efficiency. However, the results quoted above, which were obtained without any empirical tuning of the method and assuming fully turbulent boundary layers, suggest FIG 7. MACH NUMBER CONTOURS FOR A 4 STAGE TURBINE CONTOUR INTERVAL 0.05 ra

8 /;"1 FIG 8. VELOCITY CONTOURS FOR A LOW SPEED 2 STAGE TURBINE that the accuracy of efficiency prediction is already comparable to that of other methods. Inclusion of tip leakage flows and losses is clearly needed before accurate modelling of the loss mechanisms can be claimed. This is very easily included in the calculation as it is already available in the single blade row version of the method. At present it is not felt that sufficient grid points can be used to give meaningfull predictions of tip leakage flows in the multistage program. In a turbine accurate prediction of viscous effects is often not essential to obtaining a good prediction of the overall flow pattern and undesireable flow features may easily be identified using the present loss models. In a high pressure ratio compressor the overall flow pattern is often determined by viscous effects and semi empirical modelling of these will be necessary for a long time to come. Hence, further comparison with experimental data and tuning of the loss models is needed before the method can be used with confidence to predict the overall flow pattern in high pressure ratio multistage compressors. w N N uj a U U, The status of the present method is therefore similar to that of axisymmetric throughflow calculations when, some 20 years ago, they were first developed and applied to multistage turbomachines. It is anticipated that in 10 years time methods of the present type will dominate our design procedures in the way that throughflow calculations do today AXIAL DISTANCE 1 0 FIG 9. BLADE SURFACE PRESSURE DISTRIBUTION FOR A 4 STAGE COMPRESSOR WITH IGV. ACKNOWLEDGEMENT The data for the calculation on the 2 stage turbine of Fig. 8 was prepared by Dr H.P.Hodson of the Whittle Laboratory. That for the 4 stage compressor of Figs. 9 & 10 was made available by Sulzer Escher Wyss Ltd. Preparation of data for such multistage calculations represents a considerable amount of effort and the author is grateful) to all who have helped him with this task. APPENDIX CONVERSION OF POLYTROPIC EFFICIENCY TO A BODY FORCE For changes in state taking place with polytropic efficienct 71p small changes in pressure and change in temperature are related by dt Y 1 `dp T = ^ TIp for a turbine P and T = (y dp for a compressor p 8

9 N N IGV inlet The entropy gradient along streamlines is related to the body force, F m per unit mass, acting along the streamline in a direction opposing the flow by T ds/dz = F m where z is the distance along the streamline. too 00 too 00 IGV outlet rotor 1 outlet ; ' - stator 1 outlet S Hence Fv = IdP/dzl ( Ti p -1) for a turbine and Fv = IdP/dzl ( g1-1) for a compressor where Fv (= F m p ) is the body force per unit volume acting along the streamline in a direction opposing the flow. The force acting on a fluid element is simply obtained by multiplying F v by the volume of the element. rotor 2 outlet stator 2 outlet 0 f^ rotor 3 outlet stator 3 outlet Do rotor 4 outlet S c 0 xo 0X o % 0 R X X rotor I outlet 0 stator 4 outlet ' - z 00 tf x x 70 X 0 X Stator I outlet c RADIAL DISTANCE FIG. 10. DEVELOPMENT OF THE AXIAL VELOCITY PROFILE IN A 4 STAGE AXIAL COMPRESSOR X o 6 o 0 A Xrotor 2 outlet X x Xo 0X o 11 0 x X O stator 2 outlet From the second law T ds = Cp dt - dp/p which for a polytropic expansion gives T ds = ( Tlp -1) dp/p where dp is negative and for a polytropic compression gives TdS = ( gi -1) dp/p where dp is positive. P RADIAL DISTANCE m. FIG 11. COMPARISON OF AXISYMMETRIC THROUGHFLOW PREDICTIONS AND 3D MULTISTAGE PREDICTIONS FOR THE RELATIVE EXIT FLOW ANGLES IN A 2 STAGE COMPRESSOR. 9

10 REFERENCES Acamczyk, J.J. Model equation for simulating flows in multistage turbomachinery. ASME paper 85-GT-226, Adamczyk, J.J. Celestina, M.L. Beach, T.A. & Barnett, M. Simulation of three dimensional viscous flow within a multistage turbine. ASME paper 89-GT-152, Arts, T. Calculation of the 3D, steady, inviscid flow in transonic axial turbine stages, ASME paper, 84-GT-76, Dawes, W.N. Development of a 3D Navier Stokes Solver for application to all types of turbomachinery. ASME paper, 88-GT- 70, Denton, J.D. A time marching method for two and three dimensional blade to blade flow. Aero Res. Co., R&M 3775, Denton, J.D. & Singh, U.K. Time marching methods for turbomachinery flow calculation. VKI lecture Series, , Denton, J.D. An improved time marching method for turbomachinery flows. ASME paper 82-GT-239, Denton, J.D. 3D flow calculations on a hypothetical steam turbine last stage. In: Aerothermodynamics of low pressure steam turbines and condensers. Editors Moore,M.J & Sieverding,C.H. Hemisphere Denton, J.D. Calculation of fully three dimensional flow through any type of turbomachine blade row. AGARD LS 140, Denton, J.D. The Use of a distributed body force to simulate viscous effects in 3D flow calculations. ASME paper, 86-GT-144,1986. Grant, J. & Borthwick, D. Fully 3D inviscid flow calculations for the final stage of a large low pressure steam turbine. I. Mech. E. paper C281/87, Ni, R.H. Prediction of 3D multi stage turbine flow field using a multiple grid Euler solver. AIAA paper