simultaneous prediction of external flow-field and...

Copyright © 2000 by ASME1

ASME paper 2000-GT-0253, ASME IGTI Turbo-Expo-2000, Munich, Germany, May 8-11, 2000.

SIMULTANEOUS PREDICTION OF EXTERNAL FLOW-FIELD AND TEMPERATURE

IN INTERNALLY COOLED 3-D TURBINE BLADE MATERIAL

Zhen-Xue Han1

Department of Mechanical andAerospace Engineering

The University of Texas at ArlingtonBox 19018

Arlington, TX 76019, [email protected]

Brian H. Dennis 2

Department of Aerospace Engineering233 Hammond

The Pennsylvania State UniversityUniversity Park. PA 16802, U.S.A.

[email protected]

George S. Dulikravich 3

Department of Mechanical andAerospace Engineering

Box 19018University of Texas at Arlington

Arlington, TX 76019, [email protected]

ABSTRACTA two-dimensional (2-D) and a three-dimensional (3-D)

conjugate heat transfer (convection-conduction) prediction codeswere developed where the compressible turbulent flow Navier-Stokes equations are solved simultaneously in the flow-field andin the solid material of the structure thus automatically predictingcorrect magnitudes and distribution of surface temperatures andheat fluxes. The only thermal boundary conditions are theconvection heat transfer coefficients specified on the surfaces ofthe internal coolant flow passages and the coolant bulktemperature of internally cooled gas turbine blade. Thisapproach eliminates the need to specify hot surface temperatureor heat flux distribution. The conjugate codes use hybridunstructured triangular/quadrilateral grids in 2-D andunstructured prismatic grids in 3-D throughout the flow-field andin the surrounding structure. The codes are capable of conjugateheat transfer prediction in arbitrarily shaped internally cooledconfigurations. The computer codes have been successfullytested on internally cooled turbine airfoil cascades and 3-Dturbine blades by the conjugate solution of the flow-field and thetemperature field inside the structure.

1 Visiting Research Associate. Associate Professor, BUAA, P. R.

China. Member ASME.2 Graduate Research Assistant. Student member ASME.3 Professor and Director of MAIDO Laboratory. Fellow ASME.

INTRODUCTIONDuring thermal analysis of any structure exposed to a

moving fluid of a different temperature, it is essential to specifythermal boundary conditions (temperature, heat flux, or acombination of these) on the solid surface in contact with themoving fluid. In the case of internally cooled gas turbine blades,this task still requires a large degree of experience on the part ofthe thermal analyst. Such thermal boundary conditions arespecified on both the “hot” outside surface of the blade (so thatthe hot gas flow-field governing equations could be integrated)and on the “cold” surfaces of the coolant passages inside theblade. By specifying all thermal boundary conditions we areforcing both the hot gas flow-field and the coolant flow-field toperform the tasks that they physically might not be able toachieve with the given geometry of the blade and the globalflow-field parameters. Thus, this is an inconsistent procedure.


The problem is usually remedied by a tedious and costly iterationthat involves sequentially performing numerical predictions ofthe temperature field in the moving hot gas and temperature fieldinside the blade material.

A less risky procedure is the sequential conjugate heattransfer analysis (Martin et al. 1999). That is, a numericalprediction procedure that iteratively combines two analysiscodes: the prediction of the entire hot gas flow-field, and aprediction of the heat conduction inside the blade material. Thisapproach can be unstable since both temperatures and heat fluxeson a solid surface exposed to the moving fluid will have to beupdated iteratively by both codes each of which can use differentnumerical method, have significantly different convergenceproperties, and provide different numerical accuracies.

The full conjugate (concurrent) analysis is where a singlecomputational fluid dynamics (CFD) analysis code is used tointegrate the conservation laws simultaneously in the entiredomain comprised of both the hot gas flow-field and the solidblade material. The only changes needed in the CFD code arethat in the solid blade material it should: a) not solve mass andmomentum balance equations, b) explicitly use zero velocities, c)use solid blade material specific heat coefficient and thermalconductivity coefficient. Such a fully conjugate code thereforerequires thermal boundary conditions to be prescribed on thecoolant passage walls only. Thus, this type of the conjugate heattransfer analysis automatically predicts the correct hot surfacetemperature and heat flux distributions that are compatible withthe predicted hot gas flow-field.

Pelletier et al. (1995) numerically studied the hydrodynamicand thermal conjugated heat transfer of forced convection aroundcylinders in a channel and free convection around a pin-fin. Anadaptive finite element method was used. The numerical resultsagreed well with the experimental data. Recently, Heidmann etal. (1999) numerically simulated the three-dimensional coupledinternal/external flow-fields of a realistic film-cooled turbinevane. However, the blade wall was treated as an isothermalsurface. This was remedied by a code named CHT-Flow whichis a conjugate fluid flow and heat transfer solver developed byBohn et al. (1999). They studied a convectively cooled axialturbine blade with a ribbed serpentine-shaped cooling passageand cooling gas ejection at the blade tip and the trailing edge.Both NASA code (Heidmann et al., 1999) and CHT-Flow codeuse multi-block structured grids. Due to the complex geometryof the ribbed coolant flow passages, the multi-block gridgeneration in the coolant flow regions and the solid materialregions of a realistic hot gas turbine require significant time togenerate.

This paper presents the numerical results of 2-D and 3-Dfluid flow and heat transfer conjugate simulations that utilizehybrid structured/unstructured grids. The main advantage of thisformulation is a higher degree of flexibility in treating realisticgeometric configurations. In addition, the generation of non-structured grids and especially the hybrid prismatic gridsrequires less time than the block-structured grid generation forcomplex realistic internally cooled turbine blades. These are

important aspects of any future attempts to use conjugateanalysis codes in optimization of 3-D cooled blade shapes.

NUMERICAL PROCEDUREThe integral form of Favre averaged Navier-Stokes

equations and ω−k turbulence model equations with transitionmodel (Chima, 1994, Larsson, 1997) were solved numerically.The spatial domain was divided into a finite number of trianglesand/or quadrilaterals (2-D) or tetrahedrons and/or hexahedronsand/or prisms and/or pyramids (3-D). The governing equationswere discretized with finite volume method and integrated intime using an explicit multi-stage Runge-Kutta scheme with localtime stepping. The vertex-centered scheme was used. In thisscheme the volume-averaged state variables were solved at eachgrid node---the centroid of each ‘dual mesh’ which serves as acontrol volume. To accelerate the convergence to steady statesolution, local time-stepping and pseudo-Laplacian residualsmoothing techniques were used (Jameson and Mavriplis, 1986).A preconditioning method (Weiss and Smith, 1995) was used toaccelerate the convergence to a steady state in the low-Mach-number domains of the flow-field.

The finite volume approach requires computation of thefluxes across each cell face. The convective fluxes werecomputed using the Roe flux-difference splitting approach(Roe, 1986) with upwind-biased interpolations. The centraldifference scheme was used to calculate the viscous terms in thegoverning equations. The Cartesian derivatives were calculatedin the stencil presented by Frink (1994) and Mitchell (1994) orusing least-squares approach. In this work, the flow-fieldgoverning equations and the turbulence model equations weresolved separately at each Runge-Kutta stage.

BOUNDARY CONDITIONSFour different types of flow-field boundary conditions

were involved with this numerical simulation: the inletboundary, the outlet boundary, the solid wall boundary, and theperiodic boundary. All the boundary conditions wereimplemented with a half-boundary cell stencil. At the inlet andoutlet boundaries, one-dimensional Riemann invariants wereused to specify the incoming and outgoing boundary conditions.At the periodic boundaries, the flow-field variables wereduplicated for the periodic cell pairs or vertex pairs. At thesolid walls, no-slip boundary condition was imposed forvelocity, and pressure was extrapolated from the nearest flow-field grid point. The density on the wall was calculated fromthe thermal boundary condition and the equation of state.

When the flow-field around the blade and the heatconduction inside the blade material are solved simultaneously,there is a heat transfer across the solid wall. Continuities oftemperature and heat flux must be enforced at this fluid/solidinterface. That is,


solidfluid TT = (1)

solidsolid

fluidfluid n

TnT

∂∂κ=

∂∂κ (2)

From these equalities, we can see that fluidn

T∂∂

is not equal to

solidnT

∂∂

because fluidκ is different from solidκ . In other

words, temperature is continuous, but normal temperaturederivative is discontinuous across the fluid/solid interface.During the numerical procedure, specifying the temperatureboundary condition of the flow-field with the wall temperaturecalculated from the solid side satisfies the temperaturecontinuity. Specifying the heat flux boundary condition of thesolid temperature field with the value calculated from the flow-field side satisfies the heat flux continuity condition. In thisway, equation (2) is iteratively enforced at the fluid/solidinterface during flow-field and heat conduction conjugatecalculations.

TWO-DIMENSIONAL VALIDATIONTo demonstrate the numerical methods mentioned above for

the combined flow-field/heat conduction problems, several caseswere studied. Firstly, the 2-D code was validated for a transonicturbine airfoil cascade flow against the experimental dataprovided by Giel et al. (1998; 1999). This was to test thecapability and accuracy of the code to predict the complextransonic turbulent flow-fields. The airfoil wall was assumed tobe adiabatic. The 2-D hybrid unstructured grid consisted ofeleven structured clustered O-grid layers (quadrilaterals) aroundthe airfoil with the rest of the flow-field discretized with triangles(Fig. 1). The predicted pressure (non-dimensionalized by theinlet total pressure) distribution along the airfoil contour wascompared with the experimental data (Fig. 2) showing goodagreement.

The CFD code was also validated against aero-thermal datafor a highly loaded VKI transonic turbine guide vane withisothermal walls (Arts et al. 1990). Experimental data areavailable for this cascade at different flow conditions includingincoming free stream turbulence intensity. The test caseMUR222 was chosen to test the ability of our code to simulatethe transonic flows with high free stream turbulence intensity andheat transfer across the wall. The inlet total pressure was 0.822bar, inlet total temperature was 409.2 K, inlet turbulenceintensity was 6%, and the exit average pressure was 0.369 bar.Figure 3 shows the computational grids with enlarged leadingedge and trailing edge regions. The first O-grid line distance

from the blade surface was 610− m. There were 12376 gridpoints, 11718 triangles and 18363 quadrilaterals. The computedMach number contours are shown in Fig. 4. The computed inletMach number was 0.15. This was the same values as in the

experimental results. The computed exit Mach number was alsoaround experimental level of 1.135. The trailing edge shockwave was observed from both Mach number contours andtemperature contours (Fig. 5). A comparison of computedconvective heat transfer coefficient, h, distribution along theairfoil surface against the experimental data is shown in Fig. 6.The predicted pattern of heat transfer matches the experimentaldata reasonably well, but the actual predicted values in theleading edge region and on the pressure side are lower than inthe experimental data. This shift is likely caused by theturbulence and transition models used. They appear to needfurther research in order to make them suitable for highturbulence heat transfer problems as already noted in the VKIreport (Arts et al. 1990, page 13). Another likely cause of thisshift is the non-smoothness of the geometric data provided in theVKI report. Consequently, these geometric data need to besmoothed with higher order splines (Larsson, 1997).

The next test case was the conjugate heat transfer predictionfor a 2-D transonic turbulent flow around an internally cooledaxial turbine airfoil cascade with four internal coolant passages(Fig. 7). This configuration was generated by our own geometrygenerator for internally cooled axial turbine blades since we werenot able to find any publicly available geometrical andexperimental data for such configurations. The hybridtriangular/quadrilateral grid in the fluid region (Fig. 8) wascomposed of seven structured O-grid layers (2208 quadrilaterals)which were used to resolve the boundary layer flow in thevicinity of the airfoil surface and 6496 triangles (3493 points) inthe outer flow region. Inside the solid material the grid consistedof 3242 triangular elements. Figure 9 shows the combined gridsin the leading edge region. The cascade operated under thefollowing conditions: inlet total temperature of 1651.0 K, inlettotal pressure of 597312.1 Pa, and average exit static pressure of225740.0 Pa.

As stated originally, the objective of this conjugate heattransfer prediction algorithm is to automatically predict bothtemperature and heat flux distributions on the hot surface of theairfoil. This still leaves the user with a difficult task of having tospecify thermal boundary conditions on the coolant passagewalls since these boundary conditions strongly depend on thecoolant flow-field conditions in the passages. In the presentwork, we used an approximate method to specify thermalboundary conditions there. Given the coolant core flow bulktemperature, cT , and the coolant wall average convective heat

transfer coefficient, h, the local heat flux wq on the coolant

passage wall is

)/T(T hq wcw −−= (3)

Here, is thermal conductivity of the blade material, and wT is

the coolant passage wall non-uniform temperature which wasiteratively updated during the conjugate heat transfercomputations. Because the coolant bulk temperature, cT , and


the convective heat transfer coefficient, h, may be roughlypredicted by empirical methods, it is realistic to specify heatflux on the coolant walls with equation (3) without solving thecoolant flow-field. In our test case, we used cT = 900.0 K, h =

2000 W m-2 K-1 and = 25.0 W m-1 K-1.The computed iso-Mach number contours are presented in

Fig. 10. The maximum Mach number in the flow-field is 1.653and there is a strong shock-boundary layer interaction on thesuction surface causing flow separation and an asymmetric pairof vortices attached to the trailing edge (Fig. 11).

The computed isotherms were non-dimensionalized by theinlet total temperature. These isotherms are depicted in theentire computational region (Fig. 12), the near leading edgeregion (Fig. 13), and in the trailing edge region, (Fig. 14). Atthe leading edge, predicted material temperature was about 82%of the inlet total temperature. At the trailing edge, it was 76%of the inlet total temperature. This difference is due to the largeamount of heat removed from the flow-filed. The predictedtemperature distributions on the blade surface and coolantpassages (Fig. 15) are fully compatible with both the hot gasflow-field, the temperature field in the blade material, and thespecified coolant bulk temperature and coolant passage wallheat transfer coefficients.

THREE-DIMENSIONAL VALIDATIONThe 3-D heat transfer simulation code was first validated

for the case of a pure heat conduction in a hollow sphere. Theunstructured grid discretizing this domain had 104,544tetrahedrons. Isothermal boundary conditions were specified onboth inner and outer spherical surfaces as 1.0 and 0.25respectively. A comparison of analytical and computedtemperature distributions in the radial direction (Fig. 16)confirms the accuracy of the 3-D heat conduction analysis code.

Due to the lack of availability of hybrid unstructured gridgeneration tools for 3-D arbitrarily shaped turbine blades withinternal coolant flow passages, our 2-D triangular gridgeneration code and structured quadrilateral grid generationcode were used to generate hybrid unstructured 3-D grid aroundeach airfoil and inside the airfoil with coolant holes. The 2-Dtriangular/quadrilateral hybrid grids outside an airfoil aredepicted in Fig. 8. The 2-D hybrid grids outside and inside theairfoil were combined (as shown in Fig. 9) and stacked inspanwise direction with seven sections to form a 3-D aero-thermo conjugate computational prismatic grid of the turbineblade (Fig. 17). This prismatic/hexahedral mixed grid isapplicable only to straight planar turbine blades.

The airfoil of the straight internally cooled turbine bladeand its flow conditions were the same as in the case of the 2-Dinternally cooled cascade. The objective was to test theaccuracy of the 3-D conjugate heat transfer analysis code thatutilizes a hybrid non-structured grid. The 3-D blade geometryused slip boundary condition specified on the flat hub and flatshroud surfaces thus creating flow-field conditions that are

identical to those in a 2-D cascade. The thermal boundaryconditions on the coolant passage walls were specified in thesame way as in the 2-D case, that is, heat flux wq was given by

equation (3). The values for h, , and cT were set to be the

same in all regions and equal to the values used in the 2-D case.Figures 18 – 21 show the results of the aero/thermo

conjugate computation on the mid-span surface of the straightplanar turbine blade. Figure 18 shows computed iso-Machnumber contours around the blade. Comparison with the 2-Dresult (Fig. 10) indicates that, because of the slip boundaryconditions on the flat shroud and hub surfaces, the 3-D and the2-D computations result in an effectively identical flow-field.The predicted isotherms in the leading edge (Fig. 19) and thetrailing edge (Fig. 20) regions compare very well with thecorresponding 2-D calculations (Fig. 13 and Fig. 14). Theminimum non-dimensional temperature predicted inside theblade is about 0.68 (in the middle of the second strut) and themaximum is 0.822 (in the leading edge), while the coolant coretemperature is 0.545. The non-dimensional temperaturepredicted in the trailing edge region inside the blade is around0.77. In the supersonic expansion region aft of the throat of thisblade cascade the maximum Mach number is about 1.56 and thenon-dimensional temperature is about 0.681.

The computed temperature distributions on the blade hotsurface and the coolant passage surfaces are shown in Fig. 21.The temperature on the blade pressure side is higher than on itssuction side. In a region before trailing edge temperature on thesuction side is higher than on the pressure side (also see Figs.19 and 20) and both of them are increasing possibly due to flowseparation and transition. The temperatures on the suctionsurface and on the pressure surface become closer at the strutlocations and form necks there. This phenomenon is expected,because of the high heat conductivity of the struts, whichconnect the pressure side with suction side of the blade.

CONCLUSIONSWe have developed both 2-D and 3-D hybrid unstructured

grid CFD analysis codes that can simultaneously predict the hotgas flow-field and the temperature field inside internally cooledconfigurations such as axial gas turbine blades. With thisapproach, temperatures and heat fluxes on the solid surfaces incontact with the moving fluid do not need to be specified sincethey are iteratively captured by the conjugate analysis code.These fully conjugate heat transfer prediction codes have beentested against analytical solutions for temperature fields due tosteady heat conduction for simple geometries and experimentalheat transfer results for uncooled airfoil cascades. The codeshave also been successfully applied to realistically shaped 2-Dinternally cooled turbine airfoil cascades and 3-D internallycooled turbine blades. This general approach can be easilyextended to include 3-D coolant flow-field computations thusaltogether eliminating the need for specification of thermalboundary conditions on external and internal blade surfaces.


ACKNOWLEDGMENTSThe authors would like to express their gratitude for the

National Science Foundation Grant DMI-9522854 monitored byDr. George A. Hazelrigg, the NASA Glenn Research CenterGrant NAG3-1995 facilitated by Dr. John K. Lytle andmonitored by Dr. Kestutis Civinskas, the Lockheed MartinSkunk Works grant monitored by Mr. Thomas Oatway, and thePenn State - Applied Research Laboratory grant monitored byProf. Steven Garrett. The authors are also grateful for theexperimental data provided by Dr. Paul Giel and Prof. J.Larsson.

REFERENCESArts, T., Lambert de Rouvroit, M, and Rutherford, A. W.,

September 1990, “Aero-Thermal Investigation of a HighlyLoaded Transonic Linear Turbine Guide Vane Cascade”, VKITechnical Note 174.

Pelletier, D., Ignat, L. and Ilinca, F., 1995, “An AdaptiveFinite Element Method for Conjugate Heat Transfer”, AIAApaper 95-0637.

Heidmann, J., Rigby, D. and Ameri, A., 1999, “A Three-Dimensional Coupled Internal/External Simulation of a Film-Cooled Turbine Vane,” ASME paper 99-GT-186.

Bohn, D. E., Becker, V. J., Kusterer, K. A., Otsuki, Y.,Sugimoto, T. and Tanaka, R., 1999, “3-D Internal Flow andConjugate Calculations of Convective Cooled Turbine Bladewith Serpentine-Shaped and Ribbed Channels,” ASME paper 99-GT-220.

Chima, R. V., 1996, “A k-ω Turbulence Model for Quasi-Three Dimensional Turbomachinery Flows,” AIAA Paper 96-0248 and NASA TM-107051.

Frink, N. T., 1994, “Recent Progress Toward a Three-Dimensional Unstructured Navier-Stokes Flow Solver,” AIAApaper 94-0061.

Giel, P. W., Thurman, D. R., Van Fossen, G. J.,Hippensteele, S. A., and Boyle, R. J., April 1998, "Endwall HeatTransfer Measurements in a Transonic Turbine Cascade", ASMEJournal of Turbomachinery, Vol. 120, pp. 305-313.

Giel, P. W., Van Fossen, G. J., Boyle, R. J., Thurman, D. R.,and Civinskas, K. C., 1999, "Blade Heat TransferMeasurements and Predictions in a Transonic TurbineCascade", ASME paper 99-GT-125, ASME International GasTurbine Conference, Indianapolis, IN.

Han, Z.-X., Fang, R. and Liu, Z.-J., 1998, "2-D FlowfieldsCalculation with Multi-Unstructured Grids", Journal ofAerospace Power, No. 3, 1998 (in Chinese).

Jameson, A., and Mavriplis, D., 1986, “Finite VolumeSolution of the Two-Dimensional Euler Equations on a RegularTriangular Mesh,” AIAA Journal, Vol. 24, No. 4, pp. 611-618.

Larsson, J., 1997, “ Two-equation Turbulence Models forTurbine Blade Heat Transfer Simulations,” ISABE paper 97-7163.

Martin, T. J., Dulikravich, G. S., Han, Z.-X. and Dennis, B.H., 1999, “Minimizing Coolant Mass Flow Rate in InternallyCooled Gas Turbine Blades”, ASME paper 99-GT-146, ASMETurbo Expo, Indianapolis, IN, June 7-10, 1999.

Mitchell, C. R., 1994, “Improved Reconstruction Schemefor the Navier-Stokes Equations on Unstructured Meshes”,AIAA paper 94-0642.

Roe, P. L., 1986, “Characteristic Based Schemes for theEuler Equations”, Annual Review of Fluid Mechanics, Vol. 18,pp.337-365.

Weiss, J. M. and Smith, W. A., November 1995,“Preconditioning Applied to Variable and Constant DensityFlows”, AIAA Journal, Vol. 33, No. 11, pp. 2050-2057.

Fig. 1 Hybrid unstructured grid (leading edge region)

0 0.25 0.5 0.75 1X/Cax

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p/p 0i

n

Numerical result

Exp. data

Fig. 2 Nondimensionalized pressure on the blade surface(experimental data provided by P. Giel (1999)).


Fig. 3 Hybrid grids for CFD computations.

1.260

1.140

1.020

1.110

1.140

0.150

1.020

0.330

0.240

0.630

0.180

0.600

1.050

Fig. 4 Computed iso-Mach number contours for an isothermalwall uncooled VKI airfoil cascade.

1.0000.992

0.992

0.907

0.848

0.789

0.75 5

0.941

0.8560.823

0.789

0.789

0.797

Fig. 5 Computed isotherms for an isothermal wall uncooledVKI airfoil cascade.

-60 -40 -20 0 20 40 60 80s

-250

-200

-150

-100

-50

0

50

100

150

200

250

300

350

400

450

500

550

600

h

Numerical Results

Exp. Data, Tu= 6 %

Pressure Side Suction Side

Fig. 6 Heat transfer coefficient vs. isothermal wall uncooledVKI airfoil cascade surface contour-following coordinate.


Fig. 7 Cascade airfoil with four internal coolant passages.

Fig. 8 Triangle-quadrilateral hybrid grid for the flow-field.

Fig. 9 Conjugate grids in the leading edge region.

0.19

0.23

1.46

0.16

1.42

1.23

1.26

1.46

0.42

1.04

0.261.46

0.52

1.00

Fig. 10 Computed iso-Mach contours for a 2-D VKI cascade.


Fig. 11 Computed stream lines in the trailing edge regionof a 2-D internally cooled cascade airfoil.

0.99

0.69

0.85

0.70

0.71

0.99

0.71

0.98

0.93

0.7

0

Fig. 12 The entire field of conjugate isotherms computedfor a 2-D internally cooled cascade.

0.99

0.980.95

0.84

0.71

0.820.68

0.67

0.75

0.73

0.68

0.73

0.72

Fig. 13 Conjugate isotherms computed in the leading edgeregion of a 2-D internally cooled cascade airfoil.

0.69

0.85

0.71

0.98

0.93

0.69

0.73

0.76

0.96

0.70

0.69

0.76

0.74

Fig. 14 Conjugate isotherms computed in the trailing edgeregion of a 2-D internally cooled cascade airfoil.


X/Cax

Tem

eper

atu

re

0 0.25 0.5 0.75 10.65

0.7

0.75

0.8

0.85

Fig. 15 Temperatures computed on the external and internalsurfaces of a 2-D internally cooled cascade airfoil.

R

Tem

pera

ture

0 2 4 6 8 10 12 140.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Numerical Solution

Analytical Solution

Fig. 16 Radial temperature distribution in a hollow sphere.

Fig. 17 Flow-field/heat transfer conjugate 3-Dnon-structured hybrid grid (z-clip).

0.233

0.200

0.233

1.267

0.867

0.533

1. 467

1.233

1.00

0

1.467

0.900

1.200

0.433

Fig. 18 Iso-Mach number contours computed by the 3-Dconjugate analysis.


1.000

0.979

0.957

0.908

0.836

0.79

4

0.801

0.723

0.737

0.687

0.716

0.680

0.680

0.758

0.680

0.993

Fig. 19 Conjugate isotherms computed in the leading edgeregion of a 3-D cooled blade.

Fig. 20 Conjugate isotherms computed in the trailing edgeregion of a 3-D internally cooled blade.

0 0.25 0.5 0.75 1X/Cax

0.65

0.7

0.75

0.8

0.85

Tem

pera

ture

outer wall

holes

Fig. 21 Temperatures computed on the external and internalsurfaces of a 3-D internally cooled blade.

0.69

0.71

0.70

0.690.97

0.96

0.93

0.83

0.74

0.76

0.74

0.860.76

0.94

0.76

0.710.99

0.84

0.810.88

simultaneous prediction of external flow-field and...

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