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Proceedings of ASME Turbo Expo 2012GT2012

June 11-15, 2012, Copenhagen, Denmark

GT2012-68708

AN OBJECT-ORIENTED CFD CODE FOR OPTIMIZATION OF HIGH PRESSURERATIO COMPRESSORS

Elmar Gr oschel ∗, Benjamin RemboldFluid Mechanics

ABB Turbo Systems AG Schweiz Bruggerstrasse 71a, 5401Baden, Switzerland

[email protected]

Luca Mangani †, Ernesto CasartelliLucerne University of Applied Sciences and Arts

Technik & ArchitekturTechnikumstrasse 21, 6048

Horw, [email protected]

ABSTRACTThe flow fields and performances of different transonic ra-

dial compressors of varying geometries and conceptual designshave been studied numerically. All the simulations were per-formed with a modified in-house 3D RANS solver based on anobject-oriented open-source library. The solver uses an All-Mach algorithm with a special treatment for the pressure cor-rector equation to deal with highly compressible flows. The 3Dflow field structures, the characteristics and integral quantitieshave been compared to the results of established, state-of-the-artcommercial solvers as well as to measurements whenever possi-ble. This paper demonstrates for various configurations that themain flow features and the flow characteristics have been cap-tured by the new solver. Furthermore, the new solver is also ca-pable of computing the delta variations of similar designs.Thisis an essential step for the broad application of the new solverfor optimization design cycles.

NOMENCLATUREF1 Blending Function for SST Model[−]k Turbulent kinetic energy[m2 ·s−2]Ma Mach[−]MVE Inlet mass flow rate[kg·s−1]ndesign Rotational speed at design point[rad·s−1]p Pressure[Pa]

∗Address all correspondence to this author.†Address all correspondence to this author.

Pk Turbulence production term

= µt

(

∂Ui∂x j

+∂U j∂xi

− 23δi j

∂Uk∂xk

)

∂Ui∂x j

− 23ρk

∂U j∂x j

, [kg·m−1 ·s−3]

PVE∗ Inlet total pressure[Pa]PV2 Pressure impeller outlet[Pa]Re Reynolds number[−]s Blade to blade surface at constant span[−]

S Strain rate tensor= 0.5(

∂Ui∂x j

+∂U j∂xi

)

, [s−1]

T Temperature[K]Tu Turbulence intensity[−]U Velocity vector[m ·s−1]uτ Friction Velocity[m ·s−1]x Cartesian coordinates[m]y+ Non dimensional normal wall distance[−]

GreeksαΩ,αε Constitutive constanst forω equation in SST modelΓ Blending function for SST Modelη∗

is Total Isentropic Efficiency[−]µ Molecular viscosity[kg·m−1 ·s−1]µt Eddy viscosity[kg·m−1 ·s−1]π∗ Total Pressure Ratio[−]ρ Density[kg·m−3]ω Turbulence frequency[s−1]

Subscriptsawt Automatic Wall Treatmentk−ω SST turbulence modelHR High Reynolds number

1 Copyright c© 2012 by ASME

LR Low Reynolds numberp Center of first cell from wallref Reference condition at maximum efficiency0 Total conditions1 Inlet conditions2 Impeller Outlet3 Diffuser OutletAcronymsCC1 Commercial Code 1: density-based solverCC2 Commercial Code 2: pressure-based solverCFD Computational Fluid DynamicsDLR German Aerospace CenterPS Pressure sideRANS Reynolds Averaged Navier-StokesSRV4 Name of DLR radial transonic compressorSS Suction sideSST k−ω Shear Stress Transport Low-Re turbulence modelTM Turbulence model

Symbols< . > Arithmetically averaged quantity

INTRODUCTIONCFD is nowadays a common tool for the development of

turbomachinery components in industry, e.g. for the predictionof turbine and compressor aerodynamics. Besides the integralquantities of a stage, such as total pressure ratio or efficiency,it also provides detailed information of the flow field structureswithout any additional costs as in experiments. From the analy-sis of these information the experienced engineer is able toim-prove the stage performance by e.g. geometrical modificationsuntil the desired characteristics of the stage are achieved. Nev-ertheless, the final design of a compressor stage is determinedby an experimental study of different best-of CFD variants.Thereason for this is the limited reliability of the simulations due tonumerical issues but also to constraints related to the uncertaintyof the unknown geometry such as the tip clearance as outlinedby Denton [1]. The use of CFD in the design process is there-fore very powerful on a comparative basis and not for absoluteperformance predictions.

CFD for radial compressor stages is performed on differentlevels in the design process. There exist numerous simplified butquick commercial and open source 1D as well as 2D through-flow simulation tools for the preliminary design of the compres-sor stage. From this step onwards, usually only a limited num-ber of full 3D Reynolds Averaged Navier-Stokes simulationsarecomputed for the final design variants. In recent years, the use ofoptimization techniques is steadily progressing as can be seen bythe increasing number of published papers related to optimiza-tion and it is becoming part of the design cycle. One of the most

often used optimization methods is a multi-objective optimiza-tion based on evolutionary algorithms such as genetic algorithmsor differential evolution [2]. The evolutionary strategy requiresa sound data base which comes from 3D Navier-Stokes simula-tions. One of the critical points in this process is the time con-sumption of the 3D CFD simulations to compute various pointsin the compressor map. Besides an optimization of the designpoint itself also the optimization of the whole compressor map(surge and choke limit, various speed lines etc.) is important.

The bottleneck of the CFD optimization in industry is of-ten not the simulation time itself, since that can be overcomeby use of massively parallel computations on local computationclusters but it is the limited number of commercial softwareli-cences available. Often the number of computer processors islarger than the number of licences and the computer cluster isonly part-time loaded. To keep the commercial solver costs ona moderate level, the use of open source sofware becomes veryinteresting to industry.

The purpose of the present paper is to show the developmentof a new solver based on the OpenFOAMR© code which can beused for a wide range of turbomachinery problems. Since thestandard OpenFOAMR© library does not contain a solver prop-erly working for compressible flow fields, a new solver has beendeveloped by Mangani [3]. This solver has been already pub-lished in earlier publications such as [4] but has been perma-nently improved since then. This paper compares the resultsofthe new solver to existing, state-of-the-art commercial solversas well as experimental data when available for various radialcompressor cases. That is, first, a radial transonic compressorcalled SRV4 [5] with high pressure ratio and moderate massflow coefficient and, secondly, two ABB Turbo Systems radialtransonic compressors with different designs. These compressorvariants have been part of the design cycle of the best-of CFDperformance results and were experimentally investigated. Thepaper demonstrates, that the results with the new solver matchvery well the simulation results of the commercial solvers.Thenew solver proves its powerfulness on a comparative basis and isready for use in the optimization design cycle.

CASE DESCRIPTIONThree different radial compressors (Case1, Case2a, and

Case2b) have been chosen for this study:

1. The first radial compressor (Case1) is a high pressure ratio,high specific speed and high mass flow coefficient compres-sor developed and designed within the FVV project ”Homo-gene Lauf- und Leitradstromung” and built at DLR Koln.The compressor stage is composed of a centrifugal impellerwith thirteen main and splitter blades and a vaneless diffuser.Further details can be found in [5]. Instead of a backwardsswept vaneless diffusor channel contour a radial contour has


been used for the present simulation.2. The second (Case2a) and third (Case2b) compressors are 2

design variants. Both compressors were design variants forthe same compressor stage. That is, the new solver has toprove its capability to predict the compressor maps at sameaccuracy as the commercial solvers.

For the solver comparisons the same meshes and boundary con-ditions have been used in all cases. An overview of the meshquality is shown in the following section.

COMPUTATIONAL DETAILSCFD Solver

The new solver is compared to two state-of-the-art commer-cial solvers used in the design cycle process at ABB Turbo Sys-tems. In the following, the commercial solvers are called CC1and CC2. The solver CC1 is a density based solver while CC2is a pressure based solver. It is important to notice that thenu-merical scheme of the new solver as well as the implementedturbulence model are close to that of solver CC2. Therefore,incase of differences in results between CC1 and CC2 the samedeviations are expected between the new solver and CC1.

A pressure-based finite volume solver using a colocatedvariable approach suitable for calculating steady-state flows at allspeeds was developed based on the OpenFOAMR© framework.The C++ library, named OpenFOAMR© (Open-source Field Op-eration And Manipulation), offers specific class and polyhedralfinite volume operators suitable for continuum mechanics. Tomake it robust, fast and reliable for 3D RANS simulations in tur-bomachinery, it was necessary to implement additional submod-ules. The package coded by the authors within the environmentincludes a suitable algorithm for compressible steady-state anal-ysis. An algorithm similar to a SIMPLE-C was specifically de-veloped to extend the application fields to a wider range of Machnumbers especially for transonic and supersonic conditions.

Full second order upwind scheme for convection discretiza-tion together with several Low-Reynolds number eddy-viscosityturbulence models, chosen among the best performing in wallbounded flows, were developed and implemented: for further de-tails, see Mangani [3]. The code has recently been used also forheat transfer analysis [6] [7] [8] [9] and to conjugate simulations[10] [11] for gas turbine applications.

Turbulence Modeling & Automatic Near Wall TreatmentLow Reynolds models require a fine grid at the walls (y+ ∼

1) but for cases of a certain complexity this becomes quite astrict constraint. Unfortunately one of the most common issues inthe preparation of accurate calculation meshes for wall boundedflows of complex geometries is the assurance of a propery+

distribution, particularly when the flow field is characterizedby a wide range of flow Mach numbers. Therefore, in order

to increase grid independence a mixed approach between wall-function and Low Reynolds was added to thek−ω SST model.The idea, see [12], is to blend the two approaches via a blend-ing functionΓ calculated algebraically from the non-dimensionalwall distance. Both turbulent production and turbulent specificdissipation are imposed on the first node mixing the effects ofthe Low and the High Reynolds contributions. Starting from thetransport equation forω

∂(ρω)

∂t+

∂(ρωU j)

∂x j−

∂∂x j

[

(µ+µtαΩ)∂ω∂x j

]

=

ρC1Pk

µt−ρC2ω2 +

2ραε(1−F1)

ω∂k∂x j

∂ω∂x j

, (1)

both contributions are formulated as follows,

Pawt,p = PLR,pe−Γ +PHR,pe−1Γ , (2)

ωawt,p = ωLR,pe−Γ +ωHR,pe−1Γ , (3)

whereΓ is calculated with an algebraic expression fory+ definedwith uτ = max(uτ,LR,uτ,HR) . The same blending is applied tothermal quantities following the Kader universal law [13].Forfurther details concerning Automatic Near Wall Treatment thereader is referred to [12].

In the present work the automatic wall treatment versionof the Low Reynolds formulationk−ω Shear Stress Transport(SST) is used in the revised form of Menter [14] with Dirichlettype boundary condition forω at the wall.

Computational Domain & Mesh

The computational domain of all 3 radial compressors con-sists of a flow passage model with cyclic periodic 1:1 match-ing connections (except for Case2a with cyclic periodic non-matching connections). The grid topology is indicated by thesolid wireframe in Fig. 1(a). Between the inlet boundary condi-tion and the impeller a nozzle geometry contour has been placedto accelerate the flow to avoid detached flow upstream of the im-peller. All compressors have a vaneless diffuser and are com-puted in the rotating frame of reference. At the diffuser outlet anozzle has been attached to avoid backward flow. All grids have afine mesh resolution close to the walls to fulfill the requirementsof the low Reynolds number turbulence model of solver CC1.Solver CC2 and the new solver use the adaptive wall treatmentformulation. An example of the mesh resolution and quality isshown in Fig. 1(b) and Fig. 1(c) for Case2b.


(a) Computational setup. Solid lines indicate topology of the flowpassage model.

(b) Mesh details: mesh lines atleading edge main blade.

(c) Mesh details: mesh lines attrailing edge main blade.

Figure 1. Computational domain and mesh.

RESULTSGeneral Results and Validation

In this section the computational results of all 3 solvers arepresented and compared to experimental results. It should be no-ticed that all results are non-dimensionalized by the experimentalefficiency at best operating point, that isη∗

is,max. All other quanti-ties, such as mass flow rate etc., are non-dimensionalized bythisreference point.

In Case1, the comparison is only made between the commer-cial solvers CC2 and the new solver. This case is meant to be avalidation of the new solver against the commercial solver CC2.A similar study has been performed by Mangani et al. [4]. Sinceboth solvers are comparable concerning the numerical schemeand the SST-k-omega turbulence model with adaptive wall func-tions, similar results can be expected.

Case1Fig. 2 displays the total isentropic efficiency ratioη∗

is/η∗is,re f

and the total pressure ratioπ∗/π∗re f over the mass flow rate ratio

MVE/MVEre f . The results between both solvers match almost

Mesh quality of all cases

Case1 Case2a Case2b

No. grid points 3,100,000 1,500,000 2,200,000

y+max 56.0 3.1 17.2

< y+ > 0.6 1.1 4.9

anglemin 17.9 27.2 20.9

expansion ratiomax 16.0 57.0 11.0

aspect ratiomax 12,900.0 16,200.0 1600.0

Table 1. Mesh quality

perfectly. There is a very good agreement over the whole massflow range, indicating that the main flow features such as the 3dimensional shock patterns and separation zones are representedby the new solver in the same way as with the commercial solverCC2.

Case2aThe second case is the first design variant of the compres-

sor stage Case2a. The results of total pressure ratio and totalisentropic efficiency ratio are displayed in Fig. 3 and Fig. 4.Three speed lines (78.50%, 100.00%, and 114%ndesign) havebeen computed by all 3 solvers. Again, the numerical resultsof all 3 solvers are very similar to each other over the wholemass flow rate range. All solvers meet the same choke limit foreach speed line very well. The total pressure ratio is predictedalmost the same for all 3 solvers but differs from the experimen-tal results by a varying offset depending on the speed line andmass flow rate. The experimental results agree well with the nu-merical results close to the choke limit but devitate increasinglytowards lower mass flow rates. The reason for this behaviourcan be explained by different measurement positions in the ex-periment and computation. The measurement positions in theexperiment is downstream of the volute while the numerical re-sults are taken from the vaneless diffuser outlet. The volute isnot simulated in the present numerical studies. The experimentshave been performed at the worst case scenario with regard tothe matching between the compressor and the volute, that is withthe highest compressor trim and a small area-ratio volute. In thatcase, a considerable amount of radial kinetic energy is dissipatedat the diffuser outlet. It is known from various experimental andnumerical studies at ABB including the volute, that the observeddiscrepancy between the compressor characteristics in total pres-sure ratio as well as efficiency are in the same range as thoseobserved in the present study.

To substantiate this statement, a comparison between the nu-


(a) Total isentropic efficiency ratioη∗is/η∗

is,re f over mass flow rateratio MVE/MVEre f . The reference state is at best operating pointwith η∗

is,max.

MVE/MVEref

π* /π* re

f

0.8 0.9 1 1.10.75

0.8

0.85

0.9

0.95

1

1.05

1.1

CC2New Solver

(b) Total pressure ratioπ∗/π∗re f over mass flow rate ratio

MVE/MVEre f . The reference state is at best operating point withη∗

is,max.

Figure 2. Compressor map Case1.

merical and experimental results for static pressure is shown. Inthe experiments seven equidistantly distributed pressureprobesPV2 were taken at the compressor impeller outlet (shroud side)as can be seen in Fig. 5(a). Fig. 5(b) shows the pressure ra-tio PV2/PVE∗ over the mass flow rate for 3 speed lines. Thebars indicate the minmax values of the experimental resultsincircumferential direction and the numerical results as averagedquantities in the same direction. The numerical results arewithinthe range of the minmax values or close to the experimental re-sults for all 3 speed lines, confirming the quality of the numericalresults. The increasing minmax pressure variation of the experi-mental results on the speed line towards the surge point impliesthe strong impact of the volute on the compressor characteris-tics. Remember, that all simulations use constant outlet pressureboundary conditions, whereas the experimental results indicate apressure variation in circumferential direction caused bythe vo-lute. This variation actually increases in radial direction as shown

78.5%ndesign

MVE/MVEref

π*/π

* ref

0.8 1 1.2 1.4

1

1.2

1.4

1.6

1.8

ExperimentCC1CC2New Solver

114%ndesign

100%ndesign

Figure 3. Compressor map of Case2a. Total pressure ratio π∗/π∗re f

over mass flow rate ratio MVE/MVEre f for 3 speed lines. The reference

state is the experimental best operating point with η∗is,max.

in Fig. 6 by the static pressure distribution at the diffuseroutlet.The pressure varies by more than 10% over the diffuser outlet. Aqualitative comparison between solver CC2 and the new solver isshown in Fig. 7 by the relative Mach number distribution in threeblade to blade views from s=0.05 (close to hub) to s=0.95 (closeto shroud). The flow patterns of relative Mach number distribu-tion agree well confirming the comparitative quality of the newsolver solution with solver CC2.

In the following, some discrepancies between the predictionof the solvers CC2 and the new solver against the solver CC1 areinvestigated in more detail. As can be seen from Fig. 3, thereis akink in the total pressure ratio distribution according to the pre-diction of solver CC2 and the new solver which is missing in theCC1 solver prediction. The compressor is driven on this speedline in off-design at 114.00% ndesign. With increasing pressureratio the curve is subject to a sudden jump in the mass flow rate.The flow fields related to this kink in the compressor map areshown in Fig. 8 by the Mach number contours of the blade toblade views at s=0.95 (close to shroud). The flow field is verycomplex with multiple shock patterns. In the present case, theshock system is caused by the main blade curvature as well as bythe low blade number count. Remember, that this shock systemonly appears since the compressor is driven in the off-design at


(a) Total isentropic efficiency ratioη∗is/η∗

is,re f over mass flow rate ratioMVE/MVEre f for speed line 78.50%ndesign.

(b) Total isentropic efficiency ratioη∗is/η∗


(c) Total isentropic efficiency ratioη∗is/η∗


Figure 4. Compressor map of Case2a.

114.00% ndesign. In fact, modern transonic compressors have amuch higher solidity and up to the throat the main blade is prac-tically straight, so that only one shock pattern can develop[15].From P1 to P6 the flow field changes dramatically from a threeshock system in Fig 8(b) to a one shock pattern in Fig. 8(d). Thekink appears when the shock pattern on the main blade suctionside begins to change. At the same time the shock on the mainblade pressure side is influenced by the shock from the mainblade suction side. From P1 to P3 the shock on the main blade

(a) 7 equidistant pressure measurement pointsPV2 at outer radiustrailing edge.

MVE/MVEref

PV

2/P

VE

*

0.8 1 1.2 1.4 1.6

78.50% ndesign , CC278.50% ndesign , Exp.100.0% ndesign , CC2100.0% ndesign , Exp.114.0% ndesign , CC2114.0% ndesign , Exp.

0.2

(b) Averaged pressure ratio< PV2 > /PVE∗ over mass flow rateratioMVE/MVEre f for 3 speed lines. Bars indicate the static pres-sure asymmetry in circumferential direction.

Figure 5. Case2a: Pressure distrubution at compressor outlet.

pressure side almost disappears and suddenly reappears from P3to P4. It becomes obvious that these developing, complex shockpatterns are very sensitive to the solver in particular whenusingdifferent turbulence models, even if the computations are per-formed on the same grid.

Case2bIn order to apply the new solver for an optimization design

cycle, it must accurately predict the delta variation in theinte-gral quantities of different compressor designs. That means, a


Figure 6. Case2a: Measured static pressure distribution in circumferen-

tial direction (from 0 to 360) at diffuser outlet for speed line 100.00%

ndesign. Pressure is nondimensionalized by its averaged value.

(a) Blade2Blade views of relative Mach number at s=0.05.

(b) Blade2Blade views of relative Mach number at s=0.50.

(c) Blade2Blade views of relative Mach number at s=0.95.

Figure 7. Case2a: Comparison of relative Mach number distribution be-

tween solver CC2 (left) and the new solver (right). Blade2Blade views of

relative Mach number at s=0.05, s=0.5 and s=0.95 at the operating best

point on 100.00% ndesign. For all subfigures the same contour scales

have been used.

small geometrical variation resulting in e.g. a change in effi-ciency, must also be accurately predicted by the solver. There-fore, a third compressor, Case2b, has been computed at similarflow conditions and thermodynamic characteristics to Case2a. Aqualitative comparison of the blade geometries is shown in Fig. 9.

In the experiments the same volute as for Case2a has been

(a)

(b) Point P1 (left) and P2 (right).

(c) Point P3 (left) and P4 (right).

(d) Point P5 (left) and P6 (right).

Figure 8. Case2a: Blade2Blade views of relative Mach number at

s=0.95 for Points P1 to P6 from the compressor map in Fig. 8(a) at

114.00% ndesign.

used. It can be seen from Fig. 10 that the numerical results oftotal pressure ratio and total isentropic efficiency between solverCC2 and the new solver agree almost exactly indicating that thequality of the new solver meets the standards of solver CC2.However, there is now a discrepancy between CC2 and the new


Figure 9. Geometric blade comparison between Case2a (red) and

Case2b (blue).

solver versus solver CC1, which predicts the same total pres-sure ratio but a lower isentropic efficiency than the other solvers.From the available experimental results it is impossible toquan-tify which numerical solution is more accurate than the others.But it can be confirmed, that if the commercial solver CC2 isused for an optimzation task it can be substituted without loss ofaccuracy by the new solver.

Similar to Case2b, the numerical results of all three solversdiffer from the experimental results by a deviation which de-pends on the compressor map point. These losses in total pres-sure ratio and total isentropic efficiency are mostly due to themissing volute in the present simulations. The comparison ofstatic pressure near the impeller outlet (shroud side) between ex-periment and simulation is shown in Fig. . The numerical pres-sure results are lower than those of the mean values of the ex-perimental results but match well with the minmax values of theexperiment. Again, the volute has a strong impact on the com-pressor stage and that statement holds true in particular for stageswith vaneless diffusers. This effect is not studied in more detailhere, since the purpose of the paper is to prove the capability ofthe new solver to become part of the design optimization processfor turbomachinery components.

CONCLUSIONSAutomatic optimization is becoming a very important part

of the design cycle of today’s turbomachinery industry. 3D flowfield computation is nowadays a standard procedure for a widerange of cases such as compressor and turbine aerodynamics aswell as for inlet and outlet housings. The engineer is often in-terested in the delta variations of characteristic integral-valueswhich can be predicted by commercial flow solvers very well.However, optimization depends in general on a good-qualitydatabase, which means a huge number of detailed 3D CFD sim-

(a)

MVE/MVErefπ* /π

* ref

0.8 0.9 1 1.1 1.2 1.30.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

Exp.CC1CC2New Solver

(b)

Figure 10. Compressor map Case2b for speed line 100.00% ndesign.

ulations and which makes massive parallelization on computerclusters necessary. The bottleneck is often not the number ofcomputer processors but the number of available commercialsolver licences. Open source software can be used in such a waybut needs to prove its capability to predict the numerical resultswith the same accuracy level and at comparable speed to com-mercial software packages.

An inhouse 3D Navier-Stokes RANS solver based on anobject-oriented open source library has been developed. Inthepresent paper, the new solver fulfills the aforementioned require-ments very well. It is tested against different commercial flowsolvers and experimental results for 3 transonic radial compres-sors. Two of these compressors are industrially relevant and notgeneric geometries. The chosen cases show complex flow phe-nomena such as multiple shock systems. The new solver matchesthe results of the commercial flow solver CC2 very well, whichis very close to the new solver concerning the numerics and theturbulence model. Likewise the comparison in terms of the in-tegral characteristics such as total pressure ratio and total isen-tropic efficiency with the second commercial solver CC1 and tothe experimental results is satisfactory. The differencesin the


MVE/MVEref

PV

2/P

VE

*

CC2Exp.

0.2

Figure 11. Case2a: Pressure distribution at compressor outlet for speed

line 100.00% ndesign. Averaged pressure ratio < PV2 > /PVE∗ over

mass flow rate MVE. Bars indicate the static pressure asymmetry in

circumferential direction.

numerical results to the experimental values can be explained bythe missing volute in the simulations which has a strong impacton compressor stage performance. Local pressure data near theimpeller trailing edge confirmed this statement and the quality ofthe presented results. The new solver proved its reliability and isready for use in the optimizing design cycle.

ACKNOWLEDGMENTThanks go to Dr. Daniel Rusch and Dr. Gerd Mundinger for

the support given to this work.

REFERENCES[1] Denton, J. D., 2010. “Some limitations of turbomachinery

CFD”. ASME Paper(GT2010-22540).[2] M. Joly, T. Verstraete, G. P., 2010. “Attenuation of vanedis-

tortion in a transsonic turbine using optimization strategies,parti - methodology”.ASME Paper(GT2010-22370).

[3] Mangani, L., 2008. “Development and validation of an ob-ject oriented CFD solver for heat transfer and combustionmodeling in turbomachinery application”. PhD thesis, Uni-versita degli Studi di Firenze, Dipartimento di Energetica.

[4] Mangani, L., Casartelli, E., and Mauri, S., 2011. “Asses-ment of various turbulence models in a high pressure ratiocentrifugal compressor with an object oriented CFD code”.ASME Paper(GT2011-46829).

[5] Eisenlohr, G., Krain, H., Richter, F. A., and Tiede, V., 2002.“Investigations of the flow through a high pressure ratiocentrifugal impeller”.ASME Paper(GT2002-30394).

[6] Mangani, L., Bianchini, C., Andreini, A., and Facchini,B.,2007. “Development and validation of a C++ object ori-ented CFD code for heat transfer analysis”.ASME SummerHeat Transfer(AJ-1266).

[7] Di Carmine, E., Facchini, B., Mangani, L., Abba, L., Ar-cangeli, L., and Traverso, S., 2007. “Different manufactur-ing solution of fan-shaped cooling holes. Part ii: Numericalanalysis”.IGTC 2007(ABS-54).

[8] Di Carmine, E., Facchini, B., and Mangani, L., 2008. “In-vestigation of innovative trailing edge cooling configura-tions with enlarged pedestals and square or circular ribs.Part ii - numerical results”.ASME Paper(GT2008/51047).

[9] Mangani, L., and Andreini, A., 2008. “Application of anobject-oriented CFD code to heat transfer analysis”.ASMEPaper(GT2008-5118).

[10] Mangani, L., Facchini, B., and Bianchini, C., 2009. “Con-jugate heat transfer analysis of an internally cooled turbineblades with an object oriented CFD code”.European Tur-bomachinery Congress.

[11] Mangani, L., and Maritano, M., 2010. “Conjugate heattransfer analysis of NASA C3X film cooled vane with anobject-oriented CFD code”.ASME Paper(GT2010-23458).

[12] Menter, F. R., 1993. “Zonal two equationk−ω turbulencemodels for aerodynamic flows”.AIAA Paper(93-2906).

[13] Kader, B. A., 1982. “Temperature and concentration pro-files in fully turbulent boundary layers”.International Jour-nal of Heat and Mass Transfer,24(2), pp. 1541–1544.

[14] Menter, F., Thomas, E., and Wolfgang, V., 16-20/03/2003.“Heat transfer predictions based on two-equation turbu-lence models”. ASME-JSME 2003 Thermal EngineeringJoint Conference.

[15] Eisenlohr, G., Dalbert, P., Krain, H., Proll, H., Richter,F. A., and Rohne, K. H., 1998. “Analysis of the tran-sonic flow at the inlet of a high pressure ratio centrifugalimpeller”. ASME Paper(GT1998-GT-24).


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