the calculation of three dimensional viscous flow through

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The Calculation of Three Dimensional Viscous FlowThrough Multistage Turbomachines

J. D. DENTONWhittle LaboratoryCambridge, U.K.

SUMMARY

The extension of a well established three dimensional flowcalculation method to calculate the flow through multipleturbomachinery blade rows is described in this paper. To avoidcalculating the unsteady flow, which is inherent in any machinecontaining both rotating and stationary blade rows, a mixing processis modelled at a calculating station between adjacent blade rows. Theeffects of this mixing on the flow within the blade rows may beminimised by using extrapolated boundary conditions at the mixingplane.

Inviscid calculations are not realistic for multistage machines and sothe method includes a range of options for the inclusion of viscouseffects. At the simplest level such effects may be included byprescribing the spanwise variation of polytropic efficiency for eachblade row. At the most sophisticated level viscous effects andmachine performance can be predicted by using a thin shear layerapproximation to the Navier Stokes equations and an eddy viscosityturbulence model. For high pressure ratio compressors there is astrong tendency for the calculation to surge during the transientpart of the flow. This is overcome by the use of a new techniquewhich enables the calculation to be run to a prescribed mass flow.Use of the method is illustrated by applying it to to a multistageturbine of simple geometry, a 2 stage low speed experimentalturbine and to two multistage axial compressors.

NOTATION

F body force, fluxj grid index in meridional flow directionM mass flow rateP static pressureS entropyT static temperatureV absolute velocityW relative velocity

Tw wall shear stressp fluid densitylip polytropic efficiency

Subscripts

x in axial directionr in radial direcitont in tangential directionm in streamwise direction

INTRODUCTION

Fully 3D calculations for single blade rows are now wellestablished, the most common method being time dependent solutionsof the Euler or Navier Stokes (N-S) equations. Before applyingsuch methods, however, the boundary conditions for the blade rowmust usually be established by means of a separate calculation forthe whole machine. The latter will usually be performed using anaxisymmetric throughflow method with empirical input to allow forviscous losses and for blade row deviations. If the exit flowpredicted by the 3D calculation is not compatible with the input tothe throughflow calculation, eg different blade exit angles, then itmay be necessary to repeat the whole process until compatibleresults are obtained from the two calculations. This can be atedious process involving a great deal of human intervention.

To help to circumvent this process , about 10 years ago, theauthor introduced a method of calculating the 3D inviscid flowthrough a single stage turbomachine, ie two blade rows , (Denton &Singh 1979). Since the flow through two blade rows in relativerotation is inevitably unsteady ( unless they are very widelyspaced) this type of calculation must involve some modelling of thereal flow to remove the effects of unsteadiness. This was achieved bycircumferentially averaging the flow at an axial stationapproximately mid way between the two rows so that the upstreamrow saw a circumferentially uniform downstream boundarycondition and the downstream row saw a circumferentially uniformupstream flow approaching it. This averaging is not a physicallyrealistic process unless the blade rows are widely spaced and ifperformed too close to the leading or trailing edge can lead tounrealistic predictions of the flow in those regions of the blades.However, it should be realised that this assumption of axisymmetricboundary conditions is no different in principle from that which isalmost universally applied in both 2D, Q3D and full 3D blade toblade calculations. The only difference in practice is that thecondition may have to be applied closer to the blade row than wouldbe usual in such methods.

*Presented at the Gas Turbine and Aeroengine Congress and Exposition—June 11-14, 1990—Brussels, BelgiumThis 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

Copyright © 1990 by ASME

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It is also important to realize that the pitchwise averaging does notaffect the spanwise variation in flow. The spanwise variation ofpressure, velocity, flow angle, etc, at stations between the bladerows is obtained from the 3D calculation. In particular this meansthat the calculation will predict the spanwise variation of stagereaction in the same way as would an axisymmetric throughflowcalculation. Since the blade exit angles are automatically compatiblewith the blade to blade calculation there is now no need to iteratebetween throughflow calculations and blade to blade calculations toobtain the flow conditions between the two blade rows and thecomplete flow through the stage will be predicted by a singlecalculation.

This single stage 3D calculation has proved especially valuable forthe last stages of large steam turbines. These have very low hub totip ratio, high aspect ratio, high exit Mach numbers and largemeridional pitch angles ( up to 60 0 ) and so the flow is highly threedimensional as well as transonic. As a result conventionalthroughflow calculations and coupled blade to blade calculations arenot sufficiently accurate and the fully 3D stage calculation hasproved both a simpler and a more accurate tool. Examples of suchcalculations are given by Denton,1983 and Grant &Borthwick,1 987

Although there is no difficulty in principle in extending such 3Dcalculations to more than a single stage it was not thought to beworthwhile to extend the original method because of the neglect ofviscous effects. An inviscid calculation through a multistage machinewould inevitably give too high a mass flow for a prescribed pressureratio and hence the velocities and flow angles could be in significanterror. The result would be similar to that obtained by running athroughflow calculation without the inclusion of any losses.

Inclusion of viscous effects in 3D calculations is a comparativelyrecent development that is still fairly expensive in terms ofcomputer time. It is the author's opinion that in the complexturbomachine environment the accuracy of such viscous calculationsis severely limited by the limitations of turbulence modelling sothat at present only qualitatively accurate results can be obtainedfrom even the most sophisticated turbulence models. In suchcircumstances there seems little justification for using any but thesimpler turbulence models and combining them with semi empiricalcorrections to enable them to predict the correct magnitude of loss.As a result mixing length eddy viscosity models are by far the mostcommonly used method, the Baldwin Lomax model being especiallypopular. Typical of the methods available for viscous flowcalculation through a single blade row is that of Dawes ( Dawes1988).

In the spirit of modelling the viscous effects in the simplest possibleway the author showed how it was possible to modify a conventionalEuler solver to include approximations to the viscous effects byinclusion of a body force term in the momentum equations. Thisforce term can be calculated within a separate subroutine of theprogram so that the method used to estimate it is completelydecoupled from the main flow solver. As originally proposed,(Denton 1986 ), this force term was estimated by a semiempirical approach but it was pointed out that the model could inprinciple be used with any level of approximation to the full N-Sequations. Over the past few years the author has developed a total ofseven different subroutines for estimating the viscous force. Theserange in complexity from the original specification of an empiricalfrictional pressure loss for the blade row to a thin shear layerapproximation to the full N-S equations. The most recentsubroutine was developed specifically for this multistage calculationand enables the spanwise variation of polytropic efficiency for eachblade row to be input to the calculation.

This approach to the inclusion of viscous effects has the advantagethat different levels of approximation may be used within the sameprogram. For example a calculation may be started with a verysimple completely empirical model of viscous effects and after asolution has been obtained the calculation may be restarted with a

more sophisticated model. The approach is especially attractive forincluding viscous effects in multistage calculations wherelimitations on the number of grid points and the neglect of unsteadyeffects means that more complex methods are even less justifiablethan they are for a single blade row calculation and empirical inputis likely to be essential if realistic results are to be obtained.

Three dimensional calculations for multiple blade rows have alsobeen developed by Adamczyk (1985, 1989) , Arts (1984) and byNi(1 989). Adamcyzk uses what he calls a 'passage average'approach in which 3D calculations are performed separately foreach blade row with the effects of the other blade rows beingmodelled by additional force terms in the equations. Once a 3Dsolution has been obtained for one blade rows it is circumferentiallyaveraged to obtain the force terms needed for the 3D calculations inother blade rows. This approach is much more rigourous than that ofthe author but is inevitable more time consuming because several3D calculations have to be done for every blade row with the forceterms being adjusted iteratively. Although the method makes someattempt to include unsteady effects, only the effect of unsteadiness onthe time average flow is captured. Viscous effects are included in thelatest version of Adamczyk's method. Arts (1984) effectivelycouples two single blade row calculations by a common boundarycondition, again effectively performining a circumferentialaveraging at the interface. Ni (1989) gives very few details of hismethod but shows results for a 2 stage turbine. It is not clearwhether or not viscous effects are included, but good agreementwith experimental data is claimed and it is unlikely that this couldbe achieved without some modelling of viscous effects.

3. CALCULATION METHOD

3.1 EULER SOLVERThe basic Euler solver used in this method is well known and isdescribed by Denton 1982 and Denton 1985 . The method is anexplicit finite volume method with spatially varied time steps andthree levels of multigrid. The multigrid block sizes are not thoseconventionally used, with two elements per block side, but insteadany number of elements can be used along the sides of each block.Usually blocks of 3 elements per side are found to be near optimumand so blocks of 3x3x3, 9x9x9 and 27x27x27 elements would beused for the 3 levels. However, for the present application it wasnot considered advisable to use the third level of multigrid becausethe blocks are likely to extend over more than one blade row. Aspointed out by the author, (Denton 1986) , this level of multigridenables high levels of grid refinement to be used without largedeterioration in the rate of convergence. Typically 70x28x28 meshpoints will be used for a single blade row viscous calculation withthe mesh spacing varied by a factor of 10-15 between pointsadjacent to solid surfaces and points near the centre of the passage.With this number of points convergence to engineering accuracywill usually be achieved in the order of 1000 to 2000 steps.

The grid used by the method is a simple H mesh with all thevariables stored at the cell corners. This may not be the optimaltype of mesh for any one problem but the method is designed to beapplicable to all types of turbomachines, ranging from axial flow toradial flow, and to include such compicating features as splitterblades, part span shrouds and free blade tips ( ie propellers). Suchfeatures are easily accomodated by the H mesh but it it unlikely thatany other type of mesh could be made to include many of them. Theauthor's experience is that the errors incurred by using a highlysheared H mesh in regions such as that around a thick leading edgecan be made acceptable by using sufficient mesh points. The cellcorner storage of the variables is felt to be preferable to cell centrestorage since the formal accuracy does not decrease for rapidchanges of mesh spacing. The mesh used for a calculation on a 4blade row turbine is shown in Fig. 1.

3.2 VISCOUS FORCE TERM

The idea behind the use of a force term in the momentumequations to model viscous effects is describe in Denton,1986. Theonly approximation inherent in this model is the neglect of viscous

('a


effects in the energy equation and it is shown that this is a goodapproximation for the adiabatic flow of fluids with Prandtl numberclose to unity, especially at low speeds. The viscous force acting oneach finite volume must be found by summing the viscous shearstresses and normal stresses around the faces of the element. Anylevel of approximation to these stresses can be used and there is noinherent reason why the stresses should not be obtained from thefull N-S equations. In practice it was found that remarkablyrealistic results could be obtained using very simpleapproximations to the viscous force term. That described in Denton,1986 involved an input value of skin friction coefficient and analgebraic distribution of the shear stress away from solidboundaries. It was therefore completely empirical since theeffective turbulent viscosity did not depend at all upon the flow.

The method most commonly used in the latest version of the programobtains the viscous force from a thin shear layer approximation inwhich all normal components of viscous stress are neglected andonly the shear stresses on the streamwise and bladewise faces of thecuboid shaped elements are considered, viscous stresses on the quasiorthogonal faces (ie faces which are tangential to the thetadirection) of the elements are neglected. In order to avoid the use ofan extremely fine mesh near to solid boundaries, with grid points inthe laminar sublayer, it is thought preferable to use a slip model offlow on the boundaries and to calculate the shear stress on theboundary using wall functions.

The shear stress on the solid boundaries is obtained by assumingthat the first grid point away from the solid surfaces ( ie not thepoint lying on the surface) lies either in the laminar sublayer, or,more likely, in the logarithmic region of a turbulent boundarylayer. In the latter case the well known expression for the log lawboundary layer profile

V+ = 2.5 In Y+ + 5.2 (la)

Is very closely approximated by an explicit relationship betweenReg (= p V2 y2/µ) and Cf (= 2'rW/pV2 2 ). This relationship is

Cf = -0.001767 + ^n(Re 2 ) + (I (Re2))2 (1 b)

Its use avoids the need for iteration to calculate the skin friction

The shear stresses at points off the wall are obtained from a simplemixing length model. No attempt is made to vary the model betweeninner and outer regions of the boundary layer, as is done in the wellknown Baldwin-Lomax and Cebeci-Smith models. Instead theturbulent viscosity is calculated in the conventional way from amixing length which is taken as I = 0.41y, where y is theperpendicular distance from the nearest wall. The mixing lengthcalculated in this way is 'cut off' by an algebraic filter at aprescribed distance from the surface. Since the calculated stressesusually decay to near zero outside the boundary layer the preciselocation of this cut off is not very important provided that it isoutside the boundary layer. It is usually applied at a point which is10% of the blade pitch from the surface as measured in thecircumferential direction. The mixing length behind the trailingedge is held constant at a value which is input to the calculation. Thisvalue is usually taken as 2% of the blade pitch.

Having obtained the viscous shear stresses in this way the viscousforce term is obtained by summing the stresses around thebladewise and streamwise faces of every element. The viscous forceacting on the elements does no work in a stationary coordinatesystem but does do work in a rotating system and this work mustthen appear as a source term in the energy equation. However, asexplained by Denton 1986 the viscous shear work and the heatconduction may be assumed to almost cancel one another and so arenot included at all in the energy equation.

3.3 THE PRESCRIBED POLYTROPIC EFFICIENCY MODEL

For multistage calculations it was thought desireable to be able toinput empirical blade row losses in a similar way to that used insome throughflow calculations. In order to do this the body forcemust be made to depend on the required efficiency. As shown in theAppendix this can be achieved by applying a body force, Fv per unitvolume, where

Fv = (1-li p ) dP for a turbine

dPand Fv = ( 1 -1) for a compressorP

acting along the streamlines in the opposite direction to the flow. Inpractice, for simplicity, the streamlines are assumed to coincide

FIG 1. MESH FOR INVISCID CALCULATION ON A TWOSTAGE TURBINE

3


with the quasi streamlines of the grid and the magnitude of the bodyforce acting on an element is given by

Fz = Vol . Fv

where z is the distance along the quasi streamline. This force mustthen be resolved into its three components before inclusion in themomentum equations. In the present code the value of polytropicefficiency is input as a function of spanwise position for every bladerow.

This loss model will, unfortunately, not always produce a flow withexactly the prescribed value of polytropic efficiency becauseadditional entropy increases occur due to shock waves and tonumerical errors. For subsonic machines the difference between theprescribed and achieved values of polytropic efficiency will besmall unless too coarse a grid, giving rise to numerical entropyproduction, is being used . For transonic machines, especiallytransonic compressors, the additional shock loss can causesignificant differences between the prescribed and achievedpolytropic efficiency.

3.4 THE INTER-ROW MIXING MODEL

The simple averaging of flow quantities at a plane between the bladerows as used previously ( Denton 1983) is equivalent to applying amixing process between the non uniform flow at the calculatingplane upstream of the mixing plane and the mixing plane. Thisprocess conserves mass, energy and momentum, but as with anyreal mixing process entropy will be created by the mixing.However, experience with inviscid flows has been that in practicethe magnitude of entropy creation is negligible. The real problemwith the model is that a circumferentially uniform flow may beforced to exist too close to the leading edge of the downstream bladerow. This does not allow the flow to adjust circumferentially to thepresence of the blade as it would in reality. As a result the leadingedge loading on the blade row may be wrong and may even bephysically unrealistic. The magnitude of this problem depends on theleading edge loading and thickness and on how close the leading edgeis to the mixing plane. For a moderately loaded leading edge the planecan be located as close as 1/4 blade pitch upstream without apparentproblems but this would not be acceptable for highly loaded leadingedges. A similar problem occurs at the trailing edge of the blade rowupstream of the mixing plane but for subsonic exit flows the Kuttacondition ensures zero loading at the trailing edge and so theproblem is not so serious. This simple averaging process isincluded as an option in the present method.

Flux variation at JMIX-1

^ ^ I

Flux arlatlon at upstream

JMIX seenby upstream elele ments

Flux variation at JMIX as seenby downstream elements

variation atJMIX+1

JMIX-1

JMIX averagevalue

FLUX JMIX Fl-OW

Theta direction JMlx+1

Merldlonal flow direction

FIG.2 TREATMENT OF FLUXES AT THE MIXING PLANE

A simple but effective means of relieving this problem is alsoincluded in the method. The method may be briefly described asobtaining the circumferential variation of fluxes at the mixing planeby extrapolation from the upstream and downstream planes whilstadjusting the level of the fluxes to satisfy overall conservation.Thus the fluxes 'seen' by the blade rows are circumferentially non-uniform at the mixing plane with different circumferentialvariations, but the same average value, being 'seen' by theupstream and downstream rows.

In more detail. The circumferential averaging process is firstapplied as before so that the averaged flow at the mixing planeconserves mass, momentum and energy with the circumferentialaverage of the non-uniform fluxes crossing the adjacent calculatingstations. This process may be thought of as treating all the elements(at the same spanwise location) immediately upsteam of the mixingplane as a single large element and conserving fluxes between thenon-uniform flow entering its upstream face and the uniform flowleaving its downstream face which lies on the mixing plane.However, when applying the conservation equations to the individualcells the fluxes crossing the mixing plane are not taken to becircumferentially uniform. Instead, for the elements immediatelyupstream of the mixing plane, the flux crossing the face on themixing plane is obtained by multiplying the flux through theupstream face by the ratio of the circumferentially averaged flux atthe mixing plane to the circumferentially averaged flux at theupstream plane. ie

F jmixF jmix = F jmix-1 ( — ) 4a

jmix-1

The idea is illustrated schematically in Fig 2. This extrapolation ofthe flux variation is done at every spanwise position and isequivalent to obtaining the circumferential variation in flux fromthe upstream face of the element but adjusting its magnitude tosatisfy overall conservation.

The same treatment is applied to the elements immediatelydownstream of the mixing plane . At every spanwise position theflux entering an element through the mixing plane is obtained bymultiplying the flux through its downstream face by the ratio of thecircumferentially averaged flux at the mixing plane to thecircumferentially averaged flux at the location of the downstreamface.

F jmixF jmix = F jmix+1 ( F ) 4b

jmix+1

The treatment described is only approximate because it iseffectively assumed that the derivative of the flux terms along thequasi streamlines of the grid at the mixing plane is only due tochanges in the circumferentially averaged flux. However, it doesallow the flows entering and leaving the mixing plane to varycircumferentially and so is a great improvement on the previousmethod when the mixing plane is close to the leading or trailing edge.

3.5 MASS FLOW FORCING

Traditional Euler solvers always work with boundary conditions ofspecified pressure ratio and allow the mass flow to be predicted bythe solution. This is a well conditioned boundary condition for mostapplications and is essential for choked blade rows. However, forhigh pressure ratio compressor blades, when the blade is operatingnear stall, the calculation may become unstable due to 'numericalsurge'. In this the computed flow separates or generates a high loss,possibly during the transient part of the calculation, the resultingblockage reduces the mass flow which increases the incidence on theblade which makes the separation or loss larger,-----etc. Thusthe calculation may fail as a result of the transient induced by theinitial guess rather than because of a genuinely unstable operatingpoint. This tendency to instability becomes more pronounced as the

4


compressor pressure ratio increases and so it becomes almostessential to be able to ensure that each blade row is operatingreasonably close to design at all times during the transient part ofthe calculation.

There is no known way of removing the boundary condition ofprescribed pressure ratio from a time dependent Euler or N-Smethod and so the alternative is to specify both mass flow andpressure ratio and to make the two compatible by generating thecorrect amount of loss. This is the basis of the method developed forthe present code. In principle the loss is generated by a body forcein the same way as the viscous terms are included, however, thedetails are very different.

The mass flow, M(j), crossing every quasi-orthogonal plane ( onwhich the index j is constant) is calculated and if this differs fromthe prescribed mass flow, Mi n , all the velocities at that plane areadjusted according to

A = RF * (Min -M(j))/Min * (P Vm) mid

A(PVx)= (Vx/Vm)*AA (P Vr) _ (V r/ Vm ) ' 4

A (P Wt) = (Wt/ Vm) *A

Where RF is a relaxation factor.

This adjustment makes a constant change to pV m at all grid points onthe quasi orthogonal surface without changing the relative flowdirection. When a steady state is reached the increment in (density* velocity ) produced by eqns 5 must be exactly cancelled by anopposite change produced by the time stepping procedure. Thismeans that the momentum fluxes are not in balance and an effectivebody force, Fm , is acting on the elements in the direction of therelative velocity. The force can be positive or negative and itsmagnitude may be adjusted by changing the relaxation factor ; it willproduce 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 broughtto a value close to the required value ,Min , and loss is produced at arate just sufficient to make the imposed pressure ratio consistentwith the imposed mass flow. Since A will be zero when the mass flowis equal to that specified the flow rate obtained must always beslightly different from Mi n unless the pressure ratio and mass floware already consistent. The small difference between the specifiedand calculated mass flow will be directly proportional to RF whichis in effect a scaling factor on the imposed force. In practice valuesof RF=0.25 are found to produce negligible difference between theprescribed and the calculated mass flow.

This mass flow forcing process can be applied during the initialstages of a calculation until the initial transients have decayed and anear converged solution is obtained. This solution will have spuriousloss generated by the mass flow forcing and since this loss can beproduced everywhere in the flow it may be physically unrealistic.The value of RF may then be gradually reduced and the calculationcontinued with the mass flow being allowed to find its own value asdetermined by the pressure ratio. This will not prevent 'numericalsurge' in cases where the flow is genuinely unstable, ie if thepressure ratio of the compressor is too high, but it will enable amuch closer approach to the point of instability to be reached beforethe numerical instability prevents a solution from being obtained.It is doubtful whether solutions for multistage high pressure ratioaxial compressors or for single stage high pressure ratiocentrifugal impellers, could be obtained without this procedure.

?0.

—2.

16

Uiw>0

C —3 12

UE

w

o0

LU

w

O >cnL

w 8D

Jo co

0 4J

m

4.

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 flowfor the machine but always to force the local mass flow towards theaverage value for the whole machine, as obtained by averaging thecomputed flow at every calculating station. ie

JMMin = IM(j)/ JM 7

1is used in eqn 5 . This procedure gives almost the same stabilisingeffect as specifying the mass flow but it does not affect the finalsteady solution. In fact this use of the method often enables solutionsto be obtained in significantly fewer time steps even when there isno problem with 'numerical surge'.

4. APPLICATIONS OF THE METHOD

The method has been applied to several multistage turbines andcompressors. There is no limit in principle to the number of bladerows that can be calculated in a single run, the limit in practicebeing set by the available computer power and storage. For realisticviscous solutions a minimum of about 50 axial stations and 19pitchwise and spanwise stations per blade row are needed , makingabout 18000 points per row. Hence 5 blade rows would requireabout 90000 mesh points and about 15 Mbytes of computer storage.A typical run time for this number of points would be 15 hours on asingle processor of an Alliant FX80. If the option to specifypolytropic efficiency without trying to predict detailed viscouseffects is used then the grid need only be sufficient to support aninviscid calculation and about 40 x 13 x 13 grid points per bladerow would be adequate. At this level the program can be run for upto 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 machineswas found to be surprisingly fast as is illustrated in Fig. 3. Onemight expect the number of time steps required to be proportional

150

150

140N

E

NEL

130

120

198 395 594 792 990


J

I

FIG 4. SOLUTION WITH WIDE AXIAL SPACING(0.8 x CHORD)

to the number of axial calculating stations and on this basis theconvergence should be significantly slower. The reason for thisfaster than expected convergence is thought to be that for a highpressure ratio turbine pressure waves are rapidly damped by thework extraction. For example a change in turbine exit pressure willproduce a proportionally much smaller change in inlet pressure tothe last blade row and this in turn will produce an even smallerchange 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 withrepeating stage geometry for two different values of blade rowspacing. The grid used is shown in Fig.1. Figs 4 & 5 show thecalculated static pressure contours for a inter stage gap of 0.8 of theaxial chord and with the gap at a more realistic level , 0.25 of theaxial chord. It can be seen that there is very little differencebetween the two solutions except in the immediate vicinity of themixing plane where the contouring routine is confused by thediscontinuity in grid coordinates at the mixing plane. Figs 4 & 5were obtained with a prescribed polytropic efficiency, by contrastFig 6 shows results for the same turbine with a finer mesh(19x180x19) and with viscous effects calculated from the thinshear layer model. The predicted polytropic efficiency in this casewas 87.8 % which is a very reasonable value for a turbine withpoor blade surface velocity distribution ( the blade sections weredrawn not designed ) and no tip clearance.

To check for any problems that might arise from more stages andhigher pressure ratios the same blade geometry was run with 4stages and a pressure ratio of 5:1. The annulus height was increasedto accomodate the increasing volume flow. No difficulties wereencountered, convergence was naturally rather slower than for 2stages but nevertheless , as shown in Fig.3 , took only about 1000steps. Fig 7 shows the Mach number contours from this solution, thelast blade row is choked with a shock wave which is highly smeareddue to the small number of grid points per blade row.

Fig. 8 shows computed velocity contours from a viscous solution atthe mid span of a two stage low speed experimental turbine.Convergence was very slow for this case due to the low Machnumbers, peak value = 0.19 . Note the very thin boundary layersassociated with the mainly accelerating flow in this well designedturbine. The predicted isentropic efficiency was 88.2% which israther lower than expected for this machine, a possible explanationfor this is the assumption of fully turbulent boundary layers. Theexperimental 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 polytropicefficiency and with calculation of viscous effects. Results cannot yetbe published but experience was that the solutions werecomparatively difficult to obtain because of instabilities at the

FIG 5. SOLUTION WITH REDUCED AXIAL SPACING(0.25 CHORD)

6


FIG 6. VISCOUS SOLUTION FOR TWO STAGE AXIAL TURBINEMACH NUMBER CONTOURS.

mixing plane when using the improved treatment aescrioea aoave.The calculated efficiency was also significantly lower than theprescribed 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 1.6was obtained with no difficulty. Fig 9 illustrates the blade staticpressure distribution at mid span for this machine. An interestingfeature of this solution is the growth of the annulus wall boundarylayers. A knowledge of the blockage due to these boundary layers isan important requirement for conventional throughflow calculationsand its prediction is currently heavily dependent on empiricalcorrelations. Fig.10 illustrates the computed boundary layerdevelopment through the compressor, showng the initial thinboundary layer growing to fill almost the whole annulus. Noexperimental data are available but it is known that a repeatingvelocity profile is usually obtained after a few stages, hence theresults appear very plausible. Ability to predict this annulusboundary layer blockage would be an important advance incompressor design. The predicted isentropic efficiency of thismachine was 82.2 % which is somewhat lower than might beexpected for such a compressor without any tip leakage.

The method has also been used as part of the design process for alarge 2 stage , 6 blade row, axial flow compressor with highsubsonic blade surface Mach numbers. Since a conventionalthroughflow solution was available for this machine it isinformative to compare its predictions with those from the 3Dviscous calculation. To make the comparison meaningfull the 3D

calculation was run without any attempt to model the annulus wallboundary layers. The througflow calculation uses empirical data andcorrelations to predict blade row loss and deviation. Fig. 11compares the two predictions of relative flow angle at exit from bothrotors and stators ( the angles from the 3D calculation are those atthe mixing planes) and shows excellent agreement between the two.Ability to predict blade row deviations without any empiricismrepresents a significant advance over current throughflowcalculations. The predicted isentropic efficiency of this machine was85.5%, that predicted by the throughflow calculation was 88.3%.

5. CONCLUSIONS

The method described is still at an early stage of development but itis hoped that the results presented in this paper illustrate itspotential. The main attraction of the method is its ability to reducethe amount of human intervention needed to obtain solutions formultistage turbomachines by eliminating the need to iterate betweenthroughflow solutions and blade to blade solutions. At the same timeit removes most of the limitations implicit in the quasi threedimensional approach , especially the neglect of stream surfacetwist and the need to assume a distribution of stream surfacethickness within the blade rows.

The inclusion of viscous effects is necessarily approximate and itcannot be claimed that the method will yet provide accurate a-priori predictions of machine efficiency. However, the resultsquoted above, which were obtained without any empirical tuning ofthe method and assuming fully turbulent boundary layers, suggest

FIG 7. MACH NUMBER CONTOURS FOR A 4 STAGE TURBINE

CONTOUR INTERVAL 0.05

rA


0 0 0 5 AXIAL DISTANCE 1 0

FIG 9. BLADE SURFACE PRESSURE DISTRIBUTION FOR A4 STAGE COMPRESSOR WITH IGV.

•

•

••

240000 0

200000 0

w160000 0

NNuJ

a

U

120000 0

U,

80000 0

40000 0

/;"1

FIG 8. VELOCITY CONTOURS FOR A LOW SPEED 2 STAGE TURBINE

that the accuracy of efficiency prediction is already comparable tothat of other methods. Inclusion of tip leakage flows and losses isclearly needed before accurate modelling of the loss mechanisms canbe claimed. This is very easily included in the calculation as it isalready available in the single blade row version of the method. Atpresent it is not felt that sufficient grid points can be used to givemeaningfull predictions of tip leakage flows in the multistageprogram.

In a turbine accurate prediction of viscous effects is often notessential to obtaining a good prediction of the overall flow patternand undesireable flow features may easily be identified using thepresent loss models. In a high pressure ratio compressor the overallflow pattern is often determined by viscous effects and semiempirical modelling of these will be necessary for a long time tocome. Hence, further comparison with experimental data andtuning of the loss models is needed before the method can be usedwith confidence to predict the overall flow pattern in high pressureratio multistage compressors.

The status of the present method is therefore similar to that ofaxisymmetric 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 typewill dominate our design procedures in the way that throughflowcalculations do today.

ACKNOWLEDGEMENT

The data for the calculation on the 2 stage turbine of Fig. 8 wasprepared by Dr H.P.Hodson of the Whittle Laboratory. That for the 4stage compressor of Figs. 9 & 10 was made available by SulzerEscher Wyss Ltd. Preparation of data for such multistagecalculations represents a considerable amount of effort and theauthor is grateful) to all who have helped him with this task.

APPENDIX CONVERSION OF POLYTROPIC EFFICIENCY TO A BODYFORCE

For changes in state taking place with polytropic efficienct 71p smallchanges in pressure and change in temperature are related by

dT Y 1 ` dP= ^ TIp for a turbineT P

and T = (y- 1 1 p

1 dP for a compressor

8


150 00

100 00

100 00

too 00

too 00

100 00

100 00

100 00

0

100 00

100 Do

100 00

50.00

0 00

•N • • • • • • • • • • • N•

• •

IGV inlet

• IGV outlet •

• rotor 1 outlet •••

• • •••• -

;•

stator 1 outlet S'

• rotor 2 outlet •

stator 2 outlet

•f^ rotor 3 outlet •

•

• stator 3 outlet • -

••• rotor 4 outlet S -.

• 0•• stator 4 outlet ' -

The entropy gradient along streamlines is related to the body force, Fm per unit mass, acting along the streamline in a directionopposing the flow by

T dS/dz = Fm

where z is the distance along the streamline.

Hence Fv = IdP/dzl ( Ti p -1) for a turbine

and Fv = IdP/dzl ( g1 -1) for a compressor

where Fv (= Fm p ) is the body force per unit volume acting alongthe streamline in a direction opposing the flow. The force acting ona fluid element is simply obtained by multiplying Fv by thevolume of the element.

120 0

80 0

o % 0

°XR

X° rotor I outlet0X

xo0

tF ° x° °x ° 70 ° X 0 X °Stator I outlet

0 X o 6o 0 A ° X rotor 2 outlet

° ° Xx

40 0

c

00z

c

40 0

00

0200. 0.240 0.280 0.320

RADIAL DISTANCE

FIG. 10. DEVELOPMENT OF THE AXIAL VELOCITY PROFILEIN A 4 STAGE AXIAL COMPRESSOR.

° Xo 0X o 11 0 x ° X O

-40 0 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

-80 0

1 600 2000.

RADIAL DISTANCE m.

FIG 11. COMPARISON OF AXISYMMETRICTHROUGHFLOW PREDICTIONS AND 3D MULTISTAGEPREDICTIONS FOR THE RELATIVE EXIT FLOW ANGLESIN A 2 STAGE COMPRESSOR.

9


REFERENCES

Acamczyk, J.J. Model equation for simulating flows in multistageturbomachinery. ASME paper 85-GT-226, 1985.

Adamczyk, J.J. Celestina, M.L. Beach, T.A. & Barnett, M.Simulation of three dimensional viscous flow within a multistageturbine. ASME paper 89-GT-152, 1989.

Arts, T. Calculation of the 3D, steady, inviscid flow in transonicaxial turbine stages, ASME paper, 84-GT-76, 1984.

Dawes, W.N. Development of a 3D Navier Stokes Solver forapplication to all types of turbomachinery. ASME paper, 88-GT-70, 1988.

Denton, J.D. A time marching method for two and three dimensionalblade to blade flow. Aero Res. Co., R&M 3775, 1975.

Denton, J.D. & Singh, U.K. Time marching methods forturbomachinery flow calculation. VKI lecture Series, 1979-7,1979.

Denton, J.D. An improved time marching method forturbomachinery flows. ASME paper 82-GT-239, 1982.

Denton, J.D. 3D flow calculations on a hypothetical steam turbinelast stage. In: Aerothermodynamics of low pressure steamturbines and condensers. Editors Moore,M.J & Sieverding,C.H.Hemisphere. 1985.

Denton, J.D. Calculation of fully three dimensional flow throughany type of turbomachine blade row. AGARD LS 140, 1985.

Denton, J.D. The Use of a distributed body force to simulate viscouseffects in 3D flow calculations. ASME paper, 86-GT-144 ,1986.

Grant, J. & Borthwick, D. Fully 3D inviscid flow calculations forthe final stage of a large low pressure steam turbine. I. Mech. E.paper C281/87, 1987.

Ni, R.H. Prediction of 3D multi stage turbine flow field using amultiple grid Euler solver. AIAA paper 89-0203. 1989.

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the calculation of three dimensional viscous flow through

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