influence inter-row value on turbine losses

International Journal of Rotating Machinery2001, Vol. 7, No. 5, pp. 335-349Reprints available directly from the publisherPhotocopying permitted by license only

(C) 200l OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science Publishers imprint,member of the Taylor & Francis Group.

Influence of Inter-row Gap Valueon Turbine Losses

PAUL G. A. CIZMASa’*, CORBETT R. HOENNINGERa, SHUN CHENb and HARRY F. MARTINb

aDepartment OfbAerospace Engineering, Texas A & M University, College Station, Texas 77843-3141, USA,"Siemens Westinghouse Power Corporation, Orlando, Florida 32826-2399, USA

(Received in finalform 23 May 2000)

This paper presents the results of a numerical investigation of the gap influence on theturbine efficiency. The rotor-stator interaction in a (1/2)-stage turbine is simulated bysolving the quasi-three-dimensional unsteady Euler/Navier-Stokes equations using aparallelized numerical algorithm. The reduced turnaround time and cost/MFLOP ofthe parallel code was crucial to complete the numerous run cases presented in thispaper. The inter-row gap effect is evaluated for 4 gaps, 3 radial positions and 3 angu-lar velocities. As expected, the results presented in this paper show that the efficiencyincreases and losses decrease while the gap size increases. The maximum efficiencylocation, however, corresponds to values of the gap size which may be too large forpractical use (approximately inch). Fortunately, a local maximum efficiency andminimum losses location has been found at approximately 0.5 inches gap size. Theefficiency variation near the local optimum is large, in some configurations being as highas 1.4 points for a gap size variation of only 0.076 inches. Data produced by thenumerical simulations can be used to develop a design rule based on the inter-row gapsize.

Keywords: Unsteady flow; Rotor-stator interaction; Turbine flow; Numerical simulation; Parallelcomputation

INTRODUCTION

Numerical simulation of unsteady effects inturbomachinery is a necessary step for advanceddesign and analysis. The requirement to furtherincrease performance and improve reliability inturbomachinery has motivated designers to betterunderstand unsteady effects. One reason to

simulate unsteady flows in turbomachines is to beable to predict dangerous phenomena such as stallflutter or rotating stall. Neither stall flutter norrotating stall can be predicted using steady flowsimulation. These dangerous phenomena mighthappen due to the fact that blade loading andturbine inlet temperature are constantly increased,quite often pushing the engine out of the envelope

*Corresponding author. Tel.: 979 845-5952, Fax: 979 845-6051, e-mail: [email protected]

335

336 P. G. A. CIZMAS et al.

of traditional designs. Another reason to simulateunsteady flows in turbomachinery is to be ableto predict rotor-stator interaction. Experimentalinvestigations and numerical simulations haveshown that efficiency can be improved by optimiz-ing the circumferential relative position of con-

secutive airfoil rows. The purpose of this projectis to numerically investigate another importantparameter of rotor-stator interaction, the inter-row gap size. This paper will present the influenceof gap size on the turbine efficiency.An important part of the unsteady effects in tur-

bomachinery results from the rotor-stator inter-action. The main sources of unsteadiness presentin the rotor-stator interaction are potential flowinteraction and wake interaction. Potential flowinteraction is a purely inviscid interaction due tothe pressure variation caused by the relativemovement of the blades and vanes. Potential flowinteraction mainly affects adjacent airfoil rows.

Wake interaction is the unsteadiness generatedby the vortical and entropic wakes shed by one or

more upstream rows. These wakes interact withthe downstream airfoils and other wakes. Addi-tional sources of unsteadiness present’ in therotor-stator interaction include vortex shedding,hot streak interaction, shock/boundary layerinteraction, and flutter.One way of taking advantage of the rotor-

stator interaction effects is through airfoil "clock-ing" or "indexing". Stator clocking consists ofvarying the circumferential relative position ofconsecutive stator airfoils. Consecutive rotor air-foils can be clocked as well, as shown in a pre-vious paper (Cizmas and Dorney, 1999b). Theeffects of airfoil clocking on turbine performancehave been investigated both experimentally andnumerically. The experimental results of Huberet al. (Huber, Johnson, Sharma, Staubach andGaddis, 1996) showed a 0.8% efficiency variationdue to clocking. For the same turbine, a two-dimensional numerical analysis for the midspangeometry by Griffin et al. (Griffin, Huber andSharma, 1996) correctly predicted the maxi-mum efficiency clocking positions. However, the

predicted efficiency variation was only 0.5%.Clocking effects in a l(1/2)-stage turbine havealso been numerically simulated by Eulitz et al.

(Eulitz, Engel and Gebbing, 1996) and Dorneyand Sharma (Dorney and Sharma, 1996). The ef-fects of airfoil clocking on a six-row turbine havebeen also investigated (Cizmas and Dorney, 1999b).This paper presented for the first time rotor rows

clocking, clocking multiple rows and introducedthe concept of "fully clocking", i.e., clocking allthe rotor and stator rows. In all these analyses,the highest efficiencies occurred when the wake ofthe upstream airfoil impinged on the leading edgeof the downstream airfoil, while the lowest effi-ciencies were observed when the upstream wakewas convected through the middle of the down-stream airfoil passage. For the turbine investi-gated, the clocking of the second-stage producedlarger efficiency variations than the clocking ofthe third-stage. This conclusion was true forboth rotor and stator clocking. The predictedresults also showed that rotor clocking producesan efficiency variation which is approximatelytwice the efficiency variation produced by statorclocking.Another way of taking advantage of the rotor-

stator interaction is through variation of the inter-row gap. If one assumes that the potential flowinteraction is not a function of the inter-row gap,then a linear relation could be obtained betweenthe clocking position and the inter-row gap thatare producing the same efficiency. However, thepotential flow interaction varies with the inter-row

gap. Consequently, the relationship between theclocking position and the inter-row gap is more

complicated. The focus of the present paper is to

investigate the influence of inter-row gap on theturbine efficiency. Once the variation of efficiencywith inter-row gap is obtained, a correlation be-tween the clocking position and inter-row gapis possible.The first part of this paper briefly presents the

flow model and the numerical approach. The next

part of the paper presents the results of the inter-

row gap variation on losses and turbine efficiency.

INTER-ROW GAP VALUE 337

The paper ends with conclusions and suggestionsfor future work.

parallelized using Message Passing Interface li-braries and exhibits good parallel efficiency (Ciz-mas and Subramanya, 1997).

NUMERICAL MODEL

The PaRSI2 code used to simulate the flow inthe turbine is presented in detail in (Cizmas andSubramanya, 1997). The numerical approach usedin the code is based on the work done by Rai (Raiand Chakravarthy, 1986). The PaRSI2 code wasdeveloped as a parallel version of the STAGE-2analysis, which was originally developed at NASAAmes Research Center. The numerical approach isbriefly described here.

Governing Equations

The quasi-three-dimensional, unsteady, compres-sible flow through a multi-stage axial turboma-chine with arbitrary blade counts is modeled byusing the Navier-Stokes and Euler equations. Thecomputational domain associated with each airfoilis divided into an inner region, near the airfoil, andan outer region, away from the airfoil. The thin-layer Navier-Stokes equations are solved in theregions near the airfoil, where viscous effects are

strong. Euler equations are solved in the outerregion, where the viscous effects are weak. Theflow is assumed to be fully turbulent. The eddyviscosity is computed using the Baldwin-Lomaxmodel and the kinematic viscosity is computedusing Sutherland’s law.The Navier-Stokes and Euler equations are

written in the strong conservation form. The fullyimplicit, finite-difference approximation is solvediteratively at each time level, using an approximatefactorization method. Two Newton-Raphsonsub-iterations are used to reduce the linearizationand factorization errors at each time step. Theconvective terms are evaluated using a third-orderaccurate upwind- biased Roe scheme. The vis-cous terms are evaluated using second-orderaccurate central differences and the scheme issecond-order accurate in time. The code has been

Grid Generation

The domain is discretized using two types of grids.O-grids are used to resolve the Navier-Stokesequations near the airfoil, where the viscous effectsare important. The O-grids are body-fitted to thesurfaces of the airfoils and consequently theyprovide a good mesh in the critical zones aroundthe leading and trailing edges. Algebraically gen-erated H-grids are used to discretize the restof the passage, permitting the application ofperiodic boundary conditions without interpola-tion. The H-grids discretize the areas away fromthe airfoil, where the flow is modeled by the Eulerequations. The O-grid/H-grid pairs around theairfoil are overlaid and the H-grids are patchedat the sliding boundary between each stationaryand moving blade row. The flow variables are

communicated between the O- and H-gridsthrough bilinear interpolation.

Boundary Conditions

Since multiple grids are used to discretize theNavier-Stokes and Euler equations, two classesof boundary conditions must be enforced on thegrid boundaries: natural boundary conditions andzonal boundary conditions. The natural bound-aries include inlet, outlet, periodic and the airfoilsurfaces. The zonal boundaries include the patchedand overlaid boundaries.The inlet boundary conditions include the

specification of flow angle, average total pressureand downstream propagating Riemann invariant.The upstream propagating Riemann invariant isextrapolated from the interior of the domain. Atthe outlet, the average static pressure is specified,while the downstream propagating Riemann in-

variant, circumferential velocity, and entropy are

extrapolated from the interior of the domain.

Periodicity is enforced by matching flow


conditions between the lower surface of thelowest H-grid of a row and the upper surface ofthe top most H-grid of the same row. At the airfoilsurface, no-slip, adiabatic wall and zero pressuregradient condition are enforced.For the zonal boundary conditions of the

overlaid boundaries, data is transferred from theH-grid to the O-grid along the O-grid’s outermostgrid line. Data is then transferred back to the H-grid along its inner boundary. At the end of eachiteration, an explicit, corrective, interpolation pro-cedure is performed. The patch boundaries aretreated similarly, using linear interpolation toupdate data between adjoining grids (Rai, 1985).

Parallel Computation

The parallel code uses message-passing interface(MPI) libraries and runs on symmetric multi-processors (Silicon Graphics Challenge) and mas-sively parallel processors (Cray T3D and CrayT3E). The development of the quasi-three-dimen-sional parallel code was done such that a three-dimensional parallel version can be an easyextension. As a result of this requirement, oneprocessor was allocated for each airfoil in thetwo-dimensional simulation. Consequently, thenumber of processors necessary for a typicalthree-dimensional turbomachinery configurationwill not exceed the number of processors avail-able on today’s computers. Further details on theparallel algorithm are presented in (Cizmas andSubramanya, 1997).

RESULTS

Geometry and Flow Conditions

The results focus on the first three rows of a three-stage test turbine. The analysis of the flow in thewhole turbine was presented in (Cizmas andDornery, 1999b; Cizmas and Dornery, 1999a).The first three rows of the test turbine have 58first-stage stators, 46 first-stage rotors and 52

second-stage stators. A dimensionally accuratesimulation of this geometry would require model-ing 29 first-stage stators, 23 first-stage rotors and26 second-stage stators. To reduce the computa-tional effort, it was assumed that there were an

equal number of blades (58) in each turbine row.To maintain the same flow area, the airfoils werere-scaled by the factors shown in Table I.The inlet Mach number is 0.083 and the ratio

of the exit static pressure and inlet stagnationpressure is 0.6636. The ratio of specific heats at thegiven temperature and pressure is 9’ 1.284. Theinlet Reynolds number is 1,969,200 per inch, basedon the axial chord of the first-stage stator. Theinlet flow angle is 0 degrees and is kept constantfor all rotational speeds. The Prandtl number is1.03 and the turbulent Prandtl number is 1.2875.Results are presented at hub, mid-span and tip forrotational speeds of 4800, 3600 and 5200 RPM.The hub, mid-span and tip radii are 11, 12.25 and13.5 inches, respectively. The values of the flowcoefficient, 05, used for the numerical simulationare presented in Table II.To study the variation of losses and efficiency as

a function of the inter-row gap size, the size of theinter-stage gap, i.e., the gap between the first-stagerotor and the second-stage vane was varied. Thegap between the first-stage vane and the first-stagerotor was kept constant. Results corresponding tofour inter-stage gap values are presented in this

paper. The four gap values are 0.375, 0.518, 0.5919and 0.883 inches. The length of the axial chord ofthe first-stage stator is 1.25 inches.The results presented in this paper were

computed using two Newton sub-iterations pertime-step and 3000 time-steps per cycle. Here, acycle is defined as the time required for a rotorto travel a distance equal to the pitch length at

TABLE Airfoil rescaling factors

Airfoil Rescaling factor

First-stage statorFirst-stage rotor 46/58Second-stage stator 52/58

INTER-ROW GAP VALUE

TABLE II Values of flow coefficient

Hub Mid Tip

4800 RPM 0.3688 0.3312 0.30053600 RPM 0.4918 0.4416 0.40075200 RPM 0.3405 0.3057 0.2774

339

midspan. To ensure time-periodicity, each simula-tion was run in excess of 25 cycles.The computations were performed on the

Silicon Graphics Origin2000 computers at NASAAmes and Texas A & M University and on theCray T3E computer at Pittsburgh Supercomput-ing Center. The computation can however bedone on any parallel computer, including an. inex-pensive Beowulf type PC cluster. Five processorswere used for this analysis, as shown in Figure 1.The computation time for this simulation was6.24 x 10 -6secs/grid point/iteration on the CrayT3E and 7.68 x 10-6secs/grid point/iteration onthe Silicon Graphics Origin2000.

Accuracy of Numerical Results

To validate the accuracy of the numerical results itis necessary to show that the results are indepen-dent of the grid which discretizes the computa-tional domain. Three grids were used to asses thegrid independence of the solution. The coarse gridhas 37 grid points normal to the airfoil and 112grid points along the airfoil in the O-grid, and 67

grid points in the axial direction and 45 grid pointsin the circumferential direction in the H-grid,as shown in Table IIi. The number of grid pointsof the medium and fine grids is also presentedin Table III. The medium grid is presented inFigure 2, where for clarity every other grid point ineach direction is shown.The distance between the grid points on the

airfoil and the next layer of grid points aroundthe airfoil is the same for the coarse, mediumand fine grids in order to have the same y+number. The grid was generated such that, forthe given flow conditions, the y+ number is lessthan 1.The flow in the last row includes the influences

of all the upwind rows. As a result, if there aredifferences between the results due to different gridsizes, these differences will be largest in the lastrow. For this reason the last row of the turbinewas used to assess the grid independence of thenumerical results.

Before validating the grid independence of thenumerical results, one has to verify that theunsteady solution is periodic. Solution perio-dicity is assessed by comparing the pressure

PE=I

PE=5

FIGURE Processor allocation.


TABLE III Grid points of the coarse, medium and fine meshes at mid-span

No. ofgrids Coarse Grid points medium Fine

H-grid inlet 75 x 45 100 x 60 100 x 60H-grid airfoil, 3 67 x 45 90 x 60 108 x 72all rowsO-grid airfoil, 3 112 x 37 150 x 50 180 x 60all rowsH-grid outlet 75 x 45 100 x 60 100 x 60Total gridpoints 28,227 50,700 67,728

FIGURE 2 Computational grid (every other point in each direction shown).

0 0.2 0.4 0.6 0.8Axial Distance, X/Chord

FIGURE 3 Maximum, averaged and minimum pressure coefficients at three consecutive cycles.

INTER-ROW GAP VALUE

4500

3500

2500

1500

500

-500

15000 0.2 0.4 0.6 0.8

Axial distance, X/Chord

FIGURE 4 Maximum, averaged and minimum skin friction at three consecutive cycles.

341

coefficient and skin friction results of consecutivecycles, as shown in Figures 3 and 4. Sincethe values of the pressure coefficient, Cp, and skinfriction, -, are almost identical for the three con-secutive cycles, one can conclude that the solutionis periodic.

Solution periodicity is also proven by the factthat the efficiency averaged over one period reach-es an asymptotic value, independent of the cyclenumber. The variation of the total-to-total effici-ency with respect to a fixed baseline efficiency isshown in Figure 5 vs. the number of cycles. Theefficiency variation at mid-span radius is shown forthe last 20 cycles. In this paper, the flow is consi-dered to be periodic if the variation of averagedefficiency from one cycle to the next one is lessthan 0.02 points.To validate the grid independence, three values

of the pressure coefficient and skin friction arecompared: the averaged, minimum and maximumover one period. The comparison of the pressurecoefficients computed using the three grids is pre-sented in Figure 6. Good agreement is obtainedamong the averaged pressure coefficients of the

three grids. Maximum and minimum pressurecoefficients corresponding to the medium and finegrids agree well, except around 60% axial chordon the suction side and 70% axial chord on thepressure side.The comparison of the skin friction computed

using the three grids is shown in Figure 7. Asexpected, the skin friction is more sensitive to thegrid size variation than the pressure coefficient.Consequently, larger differences among the resultscorresponding to the three grids are observed com-pared to the case of the pressure coefficient shownin Figure 6. Even for the averaged values, theskin friction predicted by the coarse grid issignificantly different from the skin friction pre-dicted by the medium and fine grids. As shown inFigure 7, the averaged and minimum values of theskin friction predicted by the medium and finegrids agrees rather well. Differences between themaximum skin friction produced by the mediumand fine grids are observed at mid-chord, on bothsuction and pressure sides.As a result, one can conclude that the medium

grid is the best compromise between accuracy


0,055

0.045

0.0350.065

MID, 5200 RPM

0.055

0.045

0.035

MID,3600 RPM

0 5 10 15 20 25Number of Cycles

FIGURE 5 Efficiency variation at mid-span vs. number ofcycles.

and computational cost. In addition, for themedium and fine grids, the numerical results areindependent of the grid size and consequently thenumerical accuracy is proven.

Efficiency Variation

The variation of the total-to-total efficiency as a

function of the inter-stage gap, radius and angularvelocity is shown in Figure 8. The definition of thetotal-to-total efficiency is (Lakshminarayana,1996):

r/t-t (1 T’ca’ta ")/p, ((7-1)/7)Tl,ca,ta / 192,ca,ta / 1,ca,ta;

where subscript "ca" denotes circumferential-averaged, "ta" denotes time-averaged, and thesuperscript "," denotes total (or stagnation). Toillustrate the efficiency variation as a function ofthe inter-row gap size, it is sufficient to present theefficiency variation, r/var, relative to a fixed baselineefficiency, r/e, such that 7-]vat--r/t-t--r/b. The valueof baseline efficiency is not specified since it is notcritical for the point we are trying to make in thispaper. Note that efficiency variation, as opposedto absolute value of efficiency, is shown in Figures5 and 8.

In Figure 8 the efficiency variation is shown as a

function of the inter-stage gap and angular velo-city, while the radius is kept constant. For allradial positions, the lowest efficiency correspondsto the low angular velocity (3600 RPM) and thehighest efficiency corresponds to the high angularvelocity (5200 RPM), except at hub for a gap sizevalue of 0.592 inches. The efficiency increaseswith the gap value, except for high angular velocity(5200 RPM) at the mid-radius. The efficiencyvaries significantly around a gap value of approxi-mately 0.5 inches. The efficiency variation can beas high as 1.4 points for a gap size variation ofonly 0.076 inches.

Besides the maximum efficiency value corre-sponding to the maximum gap, a local maximumefficiency value exists for intermediate gap values.For tip and hub, the local maximum efficiencycorresponds to a gap value of 0.516 inches, forall angular velocities. At mid-radius location, themaximum efficiency at nominal angular velocityand 5200 RPM corresponds to a gap value of

INTER-ROW GAP VALUE

-7.6

-8.6

-9.60 0.2 0.4 0.6 0.8


FIGURE 6 Maximum, averaged and minimum pressure coefficient variation on the coarse, medium and fine grids.

343

4500

3500

2500

1500

500

." [ Coarse grid_+ 4- Medium grido-- Fine grid

-15000 0.2 0.4 0.6 0.8


FIGURE 7 Maximum, averaged and minimum skin friction variation on the coarse, medium and fine grids.

0.592 inches. The efficiency variation at 3600 RPMhas a minimum value for the 0.516-inches gap size.For all radial positions, the efficiency variation at

nominal angular velocity is less affected by the gapsize than at 3600 RPM and 5200 RPM angularvelocities.


0.06

._o, 0.04

>, 0.03

._otu 0.02

0.01

TIP

0.000.07

0.06

o0.04

, o.03

._o0.02

0.01

MID

o.o6

0.05

o0.04

, 0.03.,0.02

0.01

HUB

0"0%5 o.4 0. 0.65 0. 0.85 0.9Gap [in]

FIGURE 8 Efficiency variation vs. gap at tip, mid-span and hub.

At nominal angular velocity, the largest effi-ciency corresponds to the hub while the lowestefficiency corresponds to the tip. For the 5200 and3600 RPM angular velocities, the efficiency at thehub is higher than at the other radii, except for a

gap size of 0.592 inches where mid-radius efficiencyis the highest. For all angular velocities, the lowestefficiency corresponds to the tip. The lowest tipefficiency corresponds to the 3600 RPM angularvelocity.


Losses Variation

Turbine losses are presented using the stagnationpressure loss coefficient, Y. Similar results canbe obtained using the enthalpy loss coefficient,u. The definitions of these coefficients are

(Dixon, 1998):

y_-P (1)

(2)

0.006.00

HUB, 4800 RPM._._o 5.00

o4.00o

3.00

2.00

1.00

0.006.00

HUB, 5200 RPM

8 3.00

._o 2.00._1.00

oZ

0.35 0.45 0. 5 0. 5 0.75 0.85 0.95

Gap [in]

FIGURE 9 Stagnation pressure loss coefficient Y at hub for 3600, 4800 and 5200 RPM.

346 P.G.A. CIZMAS et al.

where ()N is the velocity coefficient, N--C2/C2ideal. mid-span, gap size 0.518 inches and angularThe total pressures are computed using the velocity of 4800 RPM.absolute velocity in the stator rows and the rela- For low gap values, the rotor Y losses aretive velocity in the rotor row. The losses shown dominant, as shown in Figures 9-11. Twoin Figures 9-11 are non-dimensionalized by exceptions are noticed for the Y losses at tipthe value of losses on the first-stage stator at region for nominal angular velocity (4800 RPM)

6.00

E._.- 5,00

4.00

3,00

2.00

& 1.00

Z

5,00

4.00

3,00

2.00

1.oo

MID, 4800 RPM iiiiiiii

._ 5.00

4.00

3,00

2.00

E? 1.oo

MID, 5200 RPM

0.000.35 0.45 0.55 0.65 0,75 0.85 0.95

Gap [in]

FIGURE 10 Stagnation pressure loss coefficient Y at mid-span for 3600, 4800 and 5200 RPM.

INTER-ROW GAP VALUE

9.00

8,00

7.00

6.00

5.00

4.00

3.00

2.00

1.oo

TIP, 3600 RPM

TIP,4800 RPM ]iiiiiiiiiiiiiii"E5.00

4,00

3.00

8 2.00

& 1.000z

0,006.00

TIP, 52005.00

4.00

3.00

2.00

1,oo

0,000.: 0,45 0.55 0.65 0,75 0.85 0.95

Gap [in]

FIGURE 11 Stagnation pressure loss coefficient Y at tip for 3600, 4800 and 5200 RPM.

347

and at 5200 RPM. In these cases the rotor lossesare exceeded by the second-stage vane losses.The pressure and enthalpy losses most aff-

ected by the gap size are the rotor losses. In allthe cases, the rotor losses decrease while thegap size increases. As expected, the first-stage

vane is least affected by the gap size. The 0.592-inches gap produces the minimum losses at mid-

radius for the nominal angular velocity. Thesame 0.592-inches gap produces the highestefficiency at mid-radius and nominal angularvelocity.


CONCLUSIONS

A quasi-three-dimensional unsteady Euler/Navier-Stokes analysis, based on a parallelizednumerical algorithm, has been used to investigatethe effects of inter-stage gap size in a 1(1/2)-stage turbine. Four gap sizes, three angularvelocities and three radial locations have beeninvestigated.The results presented in this paper show that

the efficiency increases and losses decrease whilethe gap size increases. The maximum efficiencyand the minimum losses location, however, corre-sponds to values of the gap size which may betoo large for practical use (approximately inch).Fortunately, a local maximum efficiency loca-tion has been found at approximately 0.5 inchesgap size. Predicting the optimum gap sizefor local maximum efficiency has a significantimpact on turbine aerodynamics, resulting in ap-proximately 0.5 points improved efficiency and1.5 points reduced losses. Data produced bythe numerical simulations can be used to devel-op a design rule based on the inter-row gap size.

P PressureRadiusTemperatureAbsolute velocityEfficiencyRatio of specific heats of a gasViscosityAngular velocityFlow coefficient,Non-dimensional skin friction,Ov)/(_(u_ /c))

Subscripts

2

t-t

ta

InletExitCircumferential-averagedTotal-to-totalTime-averagedUpstream infinity

Superscripts

Total (or stagnation)

u(Ou/

Acknowledgments

We want to thank Mr. Lynn Layman of West-inghouse Electric Company for making availablethe resources of the Pittsburgh SupercomputingCenter and to Dr. Karen Gundy-Burlet ofNASA Ames Research Center for allowing us toaccess the NASA computers. Last but not theleast, we would like to acknowledge the supportreceived from the Texas A & M SupercomputingCenter.

NOMENCLATURE

Pressure coefficient, Cp (p p*_oo)/[(1/2)p(cor)2]Absolute or relative velocity, or chord

References

Cizmas, P. G. A. and Dorney, D. J. (1999a) The influence ofclocking on the unsteady forces of compressor and turbineblades, In: Waltrup, P. J. (Ed.), Fourteenth InternationalSymposium on Air Breathing Engines, Florence, Italy.

Cizmas, P. G. A. and Dorney, D. J. (1999b) Parallelcomputation of turbine blade clocking, International Journalof Turbo and Jet-Engines, 16(1), 49-60.

Cizmas, P. G. A. and Subramanya, D. J. (1997) Parallel com-putation of rotor-stator interaction, In: Fransson, T. H.(Ed.), The Eight International Symposium on InternationalSymposium on Unsteady Aerodynamics and Aeroelasticity ofTurbomachines, Stockholm, Sweden.

Dixon, S. L. (1998) Fluid Mechanics and Thermodynamics ofTurbomachinery, Fourth edn., Butterworth-Heinemann.

Dorney, D. J. and Sharma, O. P. (1996) A study of turbineperformance increases through airfoil clocking, 32nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Lake BuenaVista, FL, AIAA Paper 96-2816.

Eulitz, F., Engel, K. and Gebbing, H. (1996) Numerical in-vestigation of the clocking effects in a multistage turbine,International Gas Turbine and Aeroengine Congress, Birming-ham, UK, ASME Paper 96-GT-26.


Griffin, L., Huber, F. and Sharma, O. (1996) Performanceimprovement through indexing of turbine airfoils: Part II-numerical simulation, ASME Journal of Turbomachinery,118(3), 636-642.

Huber, F., Johnson, P., Sharma, O., Staubach, J. and Gaddis,S. (1996) Performance improvement through indexing ofturbine airfoils: Part I- experimental investigation, ASMEJournal of Turbomachinery, 118(3), 630-635.

Lakshminarayana, B. (1996) Fluid Dynamics and Heat Transferof Turbomachinery, Wiley, Chapter 2, p. 58.

Rai, M. M. (1985) Navier-Stokes simulation of rotor-statorinteraction using patched and overlaid grids, AIAA 7th Com-putational Fluid Dynamics Conference, Cincinnati, Ohio,AIAA Paper 85-1519.

Rai, M. M. and Chakravarthy, S. (1986) An implicit form forthe Osher upwind scheme, AIAA Journal, 24, 735-743.

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