An Improved Time Marching Method for Turbomachinery Flow ...

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An Improved Time MarchingMethod for Turbomachinery FlowCalculation

J. D. Denton Time marching solutions of the Euler equations are now very widely used for thecalculation of flow through turbomachinery blade rows. All methods suffer from

Cambridge Universiityy EngineeringWr t Laboratory, the disadvantages of shock smearing, lack of entropy conservation, and corn-

Department, U.K. paratively long run times. A new method is described which reduces all theseproblems. The method is based on the author's opposed difference scheme, but thisis applied to a new type of grid consisting of quadrilateral elements which do notoverlap and have nodes only at their corners. The use of a nonoverlapping gridreduces finite differencing errors and gives complete freedom to vary the size of theelements. Both these factors help to improve entropy conservation. Considerablesavings in run time (by a factor of about 3) are obtained by using a simple multigridmethod whereby the solution is advanced simultaneously on a course and on a finegrid. The resulting method is simpler, faster, and more accurate than itspredecessor.

NOM BCLATURE

A Area vector of face of element

C Sonic velocity

CFP Correction factor on pressure

CFRO Correction factor on density

C Specific heat capacity at constant pressureP

C Specific heat capacity at constant volumev

E Specific internal energy

f Downwinding factor for pressure or density

H Specific stagnation enthalpy

I Streamwise grid point number

AR Length of element controlling stability

P Static pressure

R Gas Constant = C - Cp v

RF Relaxation factor

T Static temperature

At Time step

V Velocity vector

Contributed by the Gas Turbine Division of the ASME.

AV Volume of element

a Distribution function for density

p Static density

SUBSCRIPTS

x

y In coordinate directions

z

o Stagnation conditions

* At sonic condition

INTRODUCTION

Time dependent solutions of the Euler equationsare now widely used for the analysis of the flowthrough turbomachine blade rows. Their mainattraction is the ability to compute mixed subsonic-supersonic flows with automatic capturing of shockwaves. Solutions of the potential flow equation havealso recently been extended to compute transonicshocked flow (e.g. Farrell and Adamczyk (1)).Although these can be computationally much moreefficient than solutions of the Euler equations thelimitation to potential flow rules them out forapplications where strong shock waves can occur.Solving the Euler equations is also the most commonway of computing fully 3D flow in turbomachinery, evenfor subsonic flow, since it is not generally possibleto assume irrotational flow.

The equations may be solved in either finitedifference or finite volume form. In the former(e.g. Veuillot (2), Gliebe (3)) it is usual totransform the computational domain into a uniformrectangular grid and to express the derivatives of theflow variables in terms of values at the nodes of this

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grid. Specialised numerical techniques (e.g.McCormack or Lax-Wendroff schemes) are needed toensure stability of the integration of the equationsthrough time until a steady state is reached. In thefinite volume form of the method (e.g. McDonald (4),Denton (5)) the equations are regarded as equationsfor the conservation of mass, energy and momentumapplied to a set of inter-locking control volumesformed by a grid in the physical plane. Whensolved in this way it is easier to ensure conservationof mass and momentum than in the differentialapproach but similar numerical schemes are necessaryto ensure stability.

The argument as to whether finite difference orfinite volume schemes are preferable is not resolvedand both types are still used. The author's (notunbiased) view is that the finite volume approach issuperior because of its simplicity and its automaticconservation of mass and momentum and also because ofthe better physical understanding of the flowdevelopment which is obtained from working in aphysical grid. The latter is an importantconsideration for design engineers who are not usuallyspecialists in numerical analysis.

The author's opposed-difference scheme for solvingthe Euler equations in finite volume form has beenwidely used since its publication in 1975 (5). Thebasic philosophy of this method is to take a verysimple and fast first order scheme and progressivelyadd on a second or higher order correction as thecalculation converges. The resulting method appearsto have advantages of speed and simplicity overalternative second order schemes and it is alsoextremely 'robust'. Because of these factors it isthe only method which has been widely used for 3Dsolutions of the Euler equations through blade rows(e.g. Denton and Singh (6), Kopper (7), Sarathy (8)Barber (9), Singh (19)). The same basic algorithmhas been used for flow in the meridional plane bySpurr (10), for unsteady flow by Mitchell (11) andfor wet steam flow by Bakhtar et al (12). As aresult a great deal of user experience has beenaccumulated (e.g. Bryce and Litchfield (13)). Thegeneral experience is that satisfactory accuracy canbe obtained for most turbine blades although Singh (lit)shows that in some cases the inviscid solutions isimproved by the iterative addition of a boundary layercalculation. For compressor blades, however,Calvert and Herbert (15) show that satisfactoryaccuracy cannot usually be achieved without theaddition of boundary layer displacement to the inviscidcalculation.

As a result of this experience several defects ofthe basic scheme have come to light. They arisemainly from the use of high order correction factorswhich, although ideal for smoothly varying flows, cancause problems at points of discontinuity such asstagnation points and shock waves. The method alsobecomes unstable when the streamwise component ofvelocity is negative (i.e. for backflow) and so isunable to deal accurately with the leading edge flowon blades where the stagnation point lies on thepressure surface.

It is well know that for steady inviscid flowalong a streamline the streamwise momentum equation,the energy conservation equation and the entropyconservation equation are not independent. Any twoof these equations together imply the third equation.Hence if two of the equations are satisfied in finitedifference form the third equation will also only besatisfied in finite difference form with inevitablysome numerical error. When solving the Euler

equations it is usual to solve the momentumequations and the energy equation and hence entropywill not be conserved exactly. Similarly if themomentum equation is solved and isentropic flow assumed(e.g. McDonald ()4)) then the stagnation enthalpy willnot be conserved. Since both stagnation enthalpyand entropy are associated with a particular elementof mass, errors in either will be convected downstreamfrom their source and will influence the whole of thedownstream flow. Finite differencing errors areparticularly likely to occur in the very rapid changesin flow around a leading edge and these will theninfluence the flow on the whole blade surface. Henceit is essential that accurate differencing schemes andsufficient grid points are used around the leadingedge. At highly loaded leading edges it was foundthat the author's scheme could produce changes ofentropy (stagnation pressure) which had an adverseeffect on the blade surface velocities particularlyon the pressure surface. This is thought to be a(unpublicised) characteristic of all time marchingmethods.

The method described in this paper has beendeveloped to try to minimise these difficulties whilstretaining the advantages of speed and simplicity ofthe original scheme.

EQUATIONS

The 2D Euler equations may be written asconservation equations for a control volume AT overa time step At to give

Continuity Ap = In (pV.dA) At/AV (1)

x Momentum A(oV )x _ (P dAx + pV V.dA)

n x—At/AV (2)

y Momentum A(pVY

) _ (P dA + pVY—V.dA)

n Y —At/AV (3)

Energy A(pE) _ ^n (pH V.dA) At/AV (4)

where dA is a vector representing the area of the faceof the element in the direction of the inwards normalto the face and the summations are over the n facesof the element. These equations must be solved inconjunction with the perfect gas relationships.

H = CpT + '-2 V2

E = CvT + 2 V2 (5)

P = pRT

In 2D flow it is usual to replace the energyequation by the assumption H = constant. Thisassumption is not correct for an unsteady flow so thetrue time dependence of the solution is lost. However,in a steady adiabatic flow with H constant at inlet His everywhere constant and so a correct steady statesolution can be obtained without the need to solvethe energy equation. A similar condition ofconstant rothalpy (H - QrV e ) can be used in quasi-3Dflow through rotating blade rows but in fully 3D flowit is generally necessary to solve the energy equation.

GRID

The finite volume elements used for the newscheme are formed by the same pitchwise lines and quasi-streamlines as were used for the original method(Fig. 1.) However, instead of having a node at the

2


centre of each element nodes are now located ateach of the 4 corners. The fluxes of mass

Grid used in

author's original

method.

element

id block

Fig. 1. Grid Systems.

momentum and energy through each face are then foundusing averages of the flow properties stored at theends of that face. These fluxes may then be usedin the RHS of equations 1 - 3 to obtain the changesin p, pV, pVy for the element in time At.x

The question now arises as to how these changesshould be distributed between the 4 corners of theelement. It is important to realise that thisdistribution only affects the stability and timedependence of the method but not the steady solution.As long as a steady solution is obtained the sum ofthe fluxes of each conserved variable over the facesof each element will be zero, and hence theconservation equations satisfied, irrespective of howthe changes were distributed. The manner ofdistribution must therefore be chosen to satisfystability considerations rather than accuracy. Thelatter is only limited by the accuracy with which theflux across a face can be estimated from an averageof the flow properties at its ends.

DIFFERENCING SCHEMES

A variety of stable distribution schemes havebeen discovered for this grid. The exact analogyof the opposed-difference scheme is to send thechanges of all flow quantities to the two downstreamcorners of the element and then to let the averagepressure calculated on the downstream face act on theupstream face when solving the axial momentum equation(Eqn. 2). Like the basic opposed-difference scheme

this method is only of 1st order accuracy unless it iscorrected, using a lagged correction factor to correctthe downwinded pressure to a value close to the trueone, i.e. the pressure, P A I , acting on face I (Fig.l)is taken as

PA s I = PI+l+ CFP1 (6)

where after every time step (or every few steps)

CFPI = (1 - RF) CFPI + RF(PI -P1+1 )s

NEW OLD (7)

RF is a relaxation factor whose value is typically0.05. In the steady state equations 6 and 7 become

PA,I - PI (8)

This scheme will be referred to as scheme A, ithas exactly the same stability mechanism as theoriginal opposed difference scheme (5), but is simplerbecause correction factors are only needed for pressureand because the correction factors are not based uponinterpolation but on the difference between pressuresstored at two nodes.

A second scheme (scheme B) was discovered as theresult of trying to eliminate the use of correctionfactors completely. In scheme A, as in the opposeddifference scheme, only pressure moves upwind, however,at low Mach numbers pressure is very closely tied todensity so a scheme whereby the changes of densitywere sent to the upstream corners of the element wastried, the changes in pV and pV still being sent tothe downstream corners. x This cheme proved stable,without any correction factors or damping, at low Machnumbers but instability was found to develop at Machnumbers around unity and above. Despite thislimitation scheme B was found to have one importantadvantage, it remained stable for negative values ofstreamwise velocity and permitted solutions with astagnation point on the pressure surface and reverseflow around the leading edge. No theoreticalexplanation of this tolerance of reverse flow has beenfound but a physical explanation must be related tothe fact that mass can now be transported upstream bythe upwinding of density. By using an upwindeddensity to obtain velocity from pV it was possible tostabilise this scheme for all cases where the axialMach number was subsonic (hence covering mostturbomachinery applications) but this loss of generalityremoves one of the main attractions of time marchingmethods.

A third scheme, (scheme C) was discovered whilsttrying to combine the advantages of schemes A and B.As in scheme A changes in all variables are sent tothe downstream corners of the element but the pressureat any point is now calculated from the density at thenext downstream point plus a correction factor i.e.

PI = ( p1+l + CFRO I ) R T I (9)

where after every time step (or few steps)

CFRO I = (1-RF) . CFEO I + RF (pI - p l+l l

NEW OLD

so that in the steady state

P1 = p IR T1 (10)

3


This scheme was found to have good shock capturingproperties. It was stable for all Mach numbers(with an appropriate time step) but would not permitreversed flow.

Further attempts to develop a single schemewhich would permit both reverse flow and supersonicflow were not successful so a linear combination ofschemes B and C was adopted for the final program.The density change obtained for each element isdistributed between upstream and downstream cornersof the element according to

Apu = a Ap

AP D = (1 - a) Ap

(11)

where a is a function of Mach number such that a - 1as M -+ 0 and a -> 0 as M - . The precise form ofa(M) is not critical as regards either the stabilityor the steady state solution. A variation whichis discontinuous at M = 1 i.e.

a = 1 for M < 1

a= 0 for M 3 1

gives very good shock fitting in 1D flow but thediscontinuity caused problems in 2D flow. Becausetemperature is a variable already calcualted in theprogram a is more conveniently expressed as afunction of T than of M and a linear variation ofthe form

T*

0

a 0 (12)

was chosen.Having chosen the distribution of density

change in this way the pressure is calculated as anaverage, weighted with respect to a, of the pressuresfrom schemes B and C

i.e. P1 = RTI (ap I + (1 - a)(P I+l + CFROI )) (13)

The distribution function a does not affect the steadystate solution in any way and is merely a device toensure stability of the scheme at both high and lowMach numbers and with reverse flow. As such it issimilar but not exactly analogous to the rotateddifference schemes used for potential flowcalculations. The difference arises because thelatter introduce numerical damping which thedistribution function does not.

With the pressure calculated in this way allfluxes and pressure foces on the faces of the elementsare, in the steady state, obtained from centraldifference formulae using values stored at thecorners of the elements. As a result the solutionis second order in space for smoothly varying flowsbut does not have enough numerical damping to captureshocks without overshoots and undershoots beingproduced. It is known that numerical schemes needan artificial viscosity term proportional to thesquare of the first derivative of the flow propertiesin order to model the natural dissipative processesin a shock wave. Hence such a term was explicitlyintroduced into the equations in the form of anartificial pressure proportional to the square of thedensity gradient in the streamwise direction.Equation 13 is thereby modified to become

P1 = RT I (ap I + (1 - a)(P I+l + CFRO

(P I+l - PI )( PI+l - pI-1)/PI) (l^+)

The last term is negligible for smoothly varying flows.The scheme has been described so far as if it

were applied to a 1D flow. In practice a grid suchas that shown in Fig. 2 is used for a blade-bladecalculation. The flow variables at each node arethen updated by half the change calculated for theelement below and half that from the one above thenode, with the distribution function a being evaluatedas an average value for each element. As shown inFig. 2 the scheme allows complete flexibility inchosing the spacing of the elements in both thepitchwise and axial directions and as such is animprovement on the original method where non-uniform

Fig. 2. Computational Mesh.

spacing, although possible, caused significantcomplication. It is also apparent that when a steadystate is reached the conservation equations aresatisfied for each individual element. In theoriginal scheme the elements overlapped in thepitchwise direction and so, for the same number ofgrid points, were of twice the pitchwise extent.Hence finite differencing in the pitchwise directionis considerably more accurate on the new grid.

Cusps, which were used at the leading andtrailing edges of a blade in the original method,are not strictly necessary with the new grid.

4


However, a cusped trailing edge is felt to be a betterapproximation to the real viscous flow than is ablunt trailing edge and so is usually used. At aleading edge a large number of grid points arenecessary to resolve the flow accurately and a cuspis a useful means of minimising the number usedwhen details of the leading edge flow are notrequired.

STABILITY

As with all explicit time marching methods thetheoretical maximum stable time step is determinedby the CFL condition i.e.

At < A+Vwhere At is usually the streamwise distance betweenthe upstream and downstream faces of an element.

In practice the stability is found to dependmore on the axial Mach number than on the absoluteone and the less restrictive condition

4xAt C+Vx

can usually be used. For grids which are veryclosely spaced in the pitchwise direction thespacing perpendicular to the streamwise lines maybecome the limit on stability.

As pointed out in reference (6) it is notnecessary to take the same physical time step foreach element or even for each equation to obtain thecorrect steady state solution. Equations 1 - 4show that as long as the conservation equations aresatisfied for every element the solution isindependent of the magnitude of At. Hence themaximum stable time step can be chosen for eachindividual element to obtain the fastest convergenceto the steady state. This spatial variation oftime step permits typically about 30% reduction incomputer time but means that the transients of thecalculation have no physical significance.

This ability to use variable time steps leadsto a very powerful but simple means of controllingstability. If the local time step is madeinversely proportional to the local rate of change ofthe property then incipient instability willimmediately (i.e. for the current property in thecurrent time loop) reduce the time step, and hencethe actual change produced in the element concerned.This technique is analogous to negative feedback andmeans that local instabilities do not grow andcause failure of the calculation. As a result thetimestep does not need to be reduced below the valuewhich is stable for the converged solution in orderto cope with large amplitude initial transients.Typically a 25% reduction in the number of timesteps to convergence can be obtained in this way.

BOUNDARY CONDITIONS

The conditions applied at the inlet and outletflow boundaries are the same as in most other timemarching methods. At the outflow boundary thestatic presure is specified and held constant whichis physically correct as long as the axial Machnumber is subsonic. At the inflow boundary therelative stagnation temperature and stagnationpressure are specified together with either therelative flow direction or the relative whirl velocity(V ). The latter condition must be used if there'ative inflow Mach number is supersonic and thecalculation will then automatically satisfy the unique

incidence condition.The periodicity condition on the bounding

streamwise lines upstream and downstream of the bladerow is easily satisfied by first treating pointson these lines as interior points and then equatingvalues at corresponding points on the two boundaries.This periodicity applied immediately downstream ofthe trailing edge is found to be sufficient to veryclosely satisfy the Kutta condition at the trailingedge and no explicit Kutta condition need be applied.

The boundary conditions on the blade surfacesare the most difficult to satisfy accurately andusually a 1st order boundary condition (e.g. assumingthe second derivative in the y direction to be zeroon the surface) has to be applied. At first sightit would appear that with the new grid no specialcondition, other than zero flow through the surface,is needed. The boundary nodes can be updated bythe changes calculated for the elements adjacent tothe boundary and the only difference between themand internal points is that the latter receive changesfrom two adjacent elements and the former from onlyone. However, this treatment ignores the fact thatif there are n streamwise lines within the blade rowthen there are only n - 1 elements. Hence the numberof equations is less than the number of unknowns andthis leads to a condition whereby the sum of thechanges between the initial guess and final solutionon even numbered nodes is related to the sum of thechanges on odd numbered nodes. The final solutionis therefore not independent of the initial guess.As a simple illustration of this effect considera case with only 3 nodes across the pitch. For eachvariable the change of the centre node must equal theaverage of the changes on the two boundary nodes.Hence if the initial guess was a linear variationacross the pitch the final solution must also belinear. With more quasi-streamlines the dependenceon the initial guess becomes much weaker but remainsan undesirable constraint upon the solution. Thesolution may be made unique by applying a boundarycondition on one surface, e.g. by obtaining all flowproperties on the pressure surface by extrapolation(not necessarily linear) from the interior points.However, a better method is to remove the dependenceon the initial guess by a slight smoothing of the flowproperties across the pitch. By providing a linkbetween changes on odd and even points the smoothingrelaxes the condition that changes on them must berelated and replaces it by a condition that thepitchwise variation must be in some sense smooth.The mathematical implications of the smoothing havenot yet been understood but the treatment works welland is very simple to apply. In practice thesmoothing factor is dropped to near zero as convergenceis approached.

A further refinement is possible for points on theblade surfaces when the upstream flow can be assumedisentropic. As mentioned previously the streamwisemomentum equation can then be replaced by theequations for conservation of enthalpy and entropy.Hence if the flow direction is known, as it is on theblade surfaces, and if the flow from the inlet tosome point on the blade surface can be assumed reversibleand adiabatic (i.e. H. and S constant) the densitycomputed at that point can be used to obtain the flow

velocity via

T = To (P/P 0 )Y-1

V = ✓2 c To - T) (15)

5


Hence the surface velocity can be found withoutusing the streamwise momentum equation and so withoutits inevitable finite differencing errors. Theassumptions used on the blade surface then becomeexactly the same as those used in potential flowmethods, i.e. only the continuity equation is solvedin finite volume form and constant enthalpy andentropy are assumed. This treatment is particularlyvaluable around a heavily loaded but shock freeleading edge where finite differencing errors arelarge and the associated entropy changes can influencethe whole of the downstream flow. It may be appliedover the whole surface if the blade is known to beshock free or to have only weak shocks.

MULTIGRID ANALYSISThe method described so far is marginally faster

than the original method (1.2 x 10 4 sec/point/timestepcf 1.3 x 10 +secs) because of the simplerboundary conditions and fewer correctionfactors and the rate of convergence is similar. Verylarge savings in computer time have been obtainedfor potential flow methods using the so-called multi-grid method (e.g. Jameson (16)). The basic philosophyof this approach is to allow information regardingthe overall flow pattern to propagate rapidly on acoarse grid whilst fine detail is resolved on oneor more finer grids. Changes in potential onthe course grid are interpolated into the finer grid.This philosophy is applicable in principle tosolutions of the Euler equations but to the author'sknowledge has not been previously used.

A coarse grid may be considered as being formedby combining a group of elements into a block (Fig.l)of say 3 x 3 elements. This block may then be

is not necessary to interpolate the block changes ontothe fine grid as it is when dealing with the velocitypotential which has to be subsequently differentiated.It is also apparent that the stability mechanism ofthe fine grid (i.e. opposed differencing) carries overunchanged to the course grid so no special techniquesare needed to ensure convergence.

The multigrid method described above is very easyto program and involves little extra computer workper time step. With two levels of grid,blocksizes of 3 x 3 elements appear about optimal andhence permit timesteps which are effectively 3 timesas large as for the fine grid alone. Figure 3compares the rate of convergence of the multigrid andsingle grid methods for the test case of Figure 6.Mass balance is a suitable measure of convergence sinceit is found that the momentum balances converge at aboutthe same rate. The convergence of the multigridmethod is typically about 4 times faster than asingle grid and uses about one-third the amount ofcomputer time.

No difficulties have been encountered in usingthe multigrid method on a large number of test cases.In general the maximum stable timestep is slightlyless than for a single grid and it is preferable tohave an integral number of blocks across the pitch(e.g. 9 elements can be formed into three three-sidedblocks). The method is equally applicable to theoriginal grid and has been implemented on this in three

10 x 62 GRIDTEST CASE OF FIG. 6

EXACT M z

EXACT SHOCK POSITIONS

10 20 30 40 x 50x STANDARD 2D PROGRAMo NEW MULTIGRID PROGRAM

Fig. 4. Shocks in a one-dimensional nozzle.

TIME STEP NUMBER

100 200 300 400 500 600 700 800 900 1000

Fig. 3. Convergence

treated as a single large element and theconservation equations applied to it exactly as forbasic elements. However, the CFL condition appliedto the block shows that much larger stable timestepscan be taken for the block, e.g. for a 3 x 3 block3 times the timestep can be used. The changes offlow properties for the block can be found eitherfrom summing the fluxes around its faces or, moreeasily, by summing the changes already calculatedfor the elements within it. Each element withinthe block is then updated by its own individualchange plus the(much larger) change for the block.Since the changes involve velocity and density it

dimensional calculations. It should be possible toobtain further savings in computer time by using morethan two levels of grid but preliminary attempts touse three levels have so far given no improvement.In principle the multigrid approach should beapplicable to other numerical schemes for solving theEuler equations the main requirement being that thestability mechanism for the fine grid carries overinto the coarse grid.

EXAMPLES OF APPLICATION

The method has been programmed for two dimensionaland quasi-three dimensional blade to blade flow, thelatter permitting changes of stream tube thicknessand radius. Because of the lack of exact solutionsand of experimental data for Q3D flow all the examples

6


given here are for 2D flow.Figure 4 shows computed Mach number distributions

in a 1D nozzle designed to produce a linear variationof Mach number with distance for isentropic flow.Solutions are shown for 3 different back pressuresall of which produce shocked flow. In all casesthe shock position and exit Mach number (hencestagnation pressure loss) are predicted almostexactly. The shock is smeared over 3 - k gridpoints in all cases with negligible overshoot orundershoot. The ability to capture a wide rangeof shock strengths with equal smearing indicatesthat the correct form of artificial viscosity(Eqn. 14) has been used.

To illustrate the shock capturing ability of themethod for oblique shocks Figure 5 shows the solutionfor a cascade of wedges with inlet Mach number 2.0and completely supersonic flow. The leading edgeshock should be exactly cancelled at the upstream

Static pressure contours

prediction of two oblique shock waves, typical ofthose formed at the trailing edge of a turbine bladewith supersonic exit flow. A 13 x 64 point grid wasused for this example and the CPU time was 50 secson an IBM 370-165.

Figure 6 shows a comparison with experimentalmeasurements on a transonic turbine blade tested bySieverding and McDonald at VKI, results from whichare given in reference (17). The results arecompared at exit Mach numbers of 1.05 and 1.42. Withthis type of blade the trailing edge shock from oneblade reflects off the suction surface of theadjacent blade and accurate results cannot be obtainedunless this shock system is modelled correctly. Despiteusing a comparatively coarse grid a trailing edgeshock system was discernible in the solution although,as shown by Fig. 6, this was smeared by the time itreached the suction surface. Better results haverecently been obtained by modelling the base flowregion behind the trailing edge. Figure 7 comparesthe computed and measured exit angles from this bladeover a range of exit Mach numbers. The trend of thecomputed results is correct and the discrepancy oforder 2 is in the direction expected from viscouseffects. This solution took 48 secs CPU time fora 10 x 62 point grid.

Figure 8 compares the computed and exact designsolutions for the Bauer, Garabedian and Korn(reference 18) shock free supercritical compressorblade (Fig. 2). This blade has a highly loadedleading edge which initially caused significant lossof stagnation pressure on the surface streamlines.This was prevented by use of the isentropic surfaceboundary condition which lead to the good agreementshown in the Figure. For this case 65 secs CPU timewere needed on a 13 x 60 point grid.

M2.27a

2 I2

? ear

1.30P

j.?3R

1.s^a

1.4V

.3"u

Surface Mach numbers exact solution

Fig. 5. Supersonic flow through a cascade ofwedges.

corner giving a uniform flow between the two parallelsurfaces and an expansion off the downstream corner.The computed results show a sharp leading edgeshock which is more highly smeared on reflection andprobably as a result of this does not meet theadjacent blade exactly at the corner. Consequentlycancellation is not complete and a weak expansionpenetrates into what should be the region of uniformflow. The periodicity condition is applied downstreamof the trailing edge and an interesting result is the

M

1.8 =

1.6 x x xj1 X

1.4MZ =1.42

^.x M2=1.05 X xr

0. 8 Xx x x xx7`

0.6 _•_ EXPERIMENT >1

0.4 x CALCULATION

0.2 x x ^ xx

0 0 0.2 0.4 0.6/ 0.80.8 1.0

Fig. 6. VKI rotor blade-surfaceMach numbers.

7


Fig. 8. Supercritical compressor cascade.

DISCUSSION AND CONCLUSIONS

EXIT

(DEG.)

75'

coSH(o/P)COMPUTED

EXPERIMENT700

EXIT MACH No.

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Fig. 7. Exit flow angle from VKI rotor blade.

Perhaps the most important practical applicationof time marching methods is the calculation of flowthrough supersonic fan blades where the strong shockspresent make all other methods inapplicable. The flowthrough such blades is strongly influenced by viscouseffects, particularly downstream of the shock, socomparison of inviscid calculations with test data isnot realistic. Figure 9 illustrates the abilityof the method to compute such flows for a typical fanblade with an inlet Mach number of 1.4 and a strongshock attached at the leading edge. The shock ismuch cleaner than would be obtained from the originalmethod despite the use of a fiarly coarse 10 x 50point grid. CPU time for this solution was 29 secs.

Many more test cases than those presented herehave been calculated with the new method and there isno doubt that the method is faster and more accuratethan its predecessor. The improved accuracy comesabout mainly because interpolation is no longer neededto evaluate the correction factors. Suchinterpolations can increase accuracy for a smooth flowbut experience is that they cause problems at shockwaves and at leading edges. The isentropic surfaceboundary condition is an important aid to improvingaccuracy at highly loaded leading edges. The onlynon-second order term in the new method is thenumerical damping which is under the direct controlof the user and which has negligible effect exceptat shock waves. Hence it seems likely that accuracycan only be further improved by using more complexhigher order schemes or by using more grid points.

For 2D calculations the number of grid points isunlikely to be limited by computer storage and themain limitation is likely to be the consequent

Mach number contours

P 10 2A. 3" a9.r.5P ?3. 30 90 10P.

Surface static pressure distribution.

Fig. 9 Transonic fan blade.

increase in CPU time. Use of the multigrid methodallows more grid points to be used for the same CPUtime (about twice as many) and so permits substantiallygreater accuracy for the same cost. With the presentmethod a large modern computer could obtain solutionsfor a grid of 5000 or more points in few minutesCPU time. Such a large number of points shouldpermit very detailed resolution of the flows aroundhighly loaded leading edges and around supersonictrailing edges which are the main problem areas withprevious calculations. It is considered that themulti-grid method has potential to provide furtherincreases in speed by the use of more levels of gridand this deserves further investigation.

23;W ;e

Iice PPa

1PPPPs PP7

99PPa Pad

9TPPr PPP

^3PNP FPP

SPaPa >Pi^

4Pa?P P2?

3?sPP.'rPP

zaPPP paa

8


The new grid geometry can be easily extended to3D flows and work on this is proceeding. However,difficulty has been found in solving the energyequation (as is necessary in 3D) using schemes B andC and so it may be necessary to use scheme A, whichdoes not permit reverse flow, for 3D calculations.

ACKNOWLEDGMENTS

13. Bryce, J.D., and Litchfield, M., "Experienceof the Denton Blade-Blade Time Marching Programs",NGTE Note 1050, October 1976.

14. Singh, U.K., "Computation of Transonic Flowin Cascade with Shock and Boundary Layer Interaction",Proceedings 1st Int. Conf. Num. Meth. in Laminarand Turbulent Flow, Swansea, July 1978.

The original idea of using a grid with nodesonly at the corners of the elements was obtained fromDr. R. Ni of Pratt and Whitney Aircraft.

The wedge case test of Fig. 5 was provided byBrown Boveri & Co., Baden, Switzerland.

15. Calvert, W.J., andViscous Interaction MethodBlade Performance of Axial80012, February, 1980.

Herbert, M.V., "An Inviscid-to Predict the Blade-Compressors", NGTE Memo

REFERENCES

1. Farrell, C., and Adamczyk, J., "FullPotential Solution of Transonic Quasi-3D Flow Througha Cascade. Using Artificial Compressibility",ASME paper 81-GT-70, 1981.

2. Gopalakrishnan, S., Bozzola, R. "A NumericalTechnique for Calculation of Transonic Flows inTurbomachinery", ASME paper 71-GT-42, 1971.

3. Veuillot, J.P.,"Calculation of Quasi-3D flowin a Turbomachine Blade Row", ASME paper 76-GT-56,1976.

4. McDonald, P.W., "The Computation of TransonicFlow Through Two-Dimensional Gas Turbine Cascades",ASME paper 71-GT-89, 1971.

5. Denton, J.D., "A Time Marching Method forTwo and Three Dimensional Blade to Blade Flow",ARC R. & M. 3775, 1975.

6. Denton, J.D., and Singh, U.K., "Time MarchingMethods for Turbomachinery Flow Calculation", VKILecture Series on Transonic Flows for Turbomachinery,1979.

7. Kopper, F., Milano, R., and Vanco., M.,"An Experimental Investigation of Endwall Profilingin a Turbine Vane Cascade", AIAA Journal, 19, 8, 1981.

16. Jameson, A., "Acceleration of TransonicPotential Flow Calculations on Arbitrary Meshes bythe Multiple Grid Method", AIAA paper 79 -1458, 1979.

17. Sieverding, C., "Base Pressure in SupersonicFlow", VKI Lecture Series on Transonic Flows inTurbomachinery, 1976.

18. Bauer, F., Garabedian, P., Korn, D., andJameson, A. "Supercritical Wing Sections II",Lecture Notes in Economics and Mathematical Systems,Vol. 108, Springer Verlag, 1975.

19. Singh, U.K., "A computation and comparisonwith Measurements of Transonic Flow in an AxialCompressor Stage with Shock and Boundary LayerInteraction", ASME paper 81-Gr/GT-5, 1981

8. Sarathy, K.P., "Computation of Three-DimensionalFlow Fields Through Rotating Blade Rows and Comparisonwith Experiment", ASME paper 81-GT-121, 1981.

9. Barber, T.J. "Analysis of Shearing InternalFlows", AIAA Paper 81-0005, 1981.

10. Spurr, A., "The Prediction of 3D TransonicFlow in Turbomachinery Using a Combined Throughflowand Blade-to-Blade Time Marching Method".,International Journal Heat and Fluid Flow, December,1980.

11. Mitchell, N., "A Time Marching Method forUnsteady 2D Flow in a Blade Passage", InternationalJournal Heat and Fluid Flow, 2.4, 1980.

12. Bakhtar, F., and Tochai, M., "An Investigationof 2D Flows of Nucleating and Wet Steam by theTime-Marching Method", International Journal Heatand Fluid Flow, 2.1, 1980.

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