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VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF … · VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF AN AXIAL HYDRAULIC TURBINE A. F. Tiago Mechanical Engineering Department Instituto

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Page 1: VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF … · VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF AN AXIAL HYDRAULIC TURBINE A. F. Tiago Mechanical Engineering Department Instituto

VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF AN AXIAL HYDRAULIC TURBINE

A. F. Tiago

Mechanical Engineering Department Instituto Superior Técnico

Universidade Técnica de Lisboa Av. Rovisco Pais, 1, 1049-001 Lisboa

e-mail: [email protected] Abstract

The paper reports a numerical study of the flow through a 0.5 m diameter axial hydraulic turbine

rotor. The two-dimensional flow around the blade cascade was modelled using a panel method.

Pressure distribution, lift and drag coefficients results are presented for incidence angle values

close to the design condition, assuming inviscid and viscous flow. The latter, includes the

boundary-layer calculation according to a weak viscous-inviscid interaction formulation. Results

are presented for several rotor-blade sections located at different radial positions.

The three-dimensional viscous flow was computed using the FLUENT code. The Spalart-

Allmaras, the standard k-ε and the k-ω (proposed by Wilcox) turbulence models were used. Non-

structured and structured meshes were tested, considering a maximum of about 2x106 elements.

A mesh dimension dependence study was performed. Pressure, velocity and angular momentum

distributions at the inlet and outlet sections and on the blade surface, obtained with the different

turbulence models, are presented and compared with the results from three-dimensional inviscid

simulations. Pressure distributions on the blade surface were also compared with two-

dimensional inviscid (design condition) and viscous results.

Keywords: rotor, axial turbine, FLUENT, panel method

Nomenclature A area c chord

DC drag coefficient

LC lift coefficient

PC pressure coefficient

0C total pressure coefficient

D outer diameter DH hydraulic diameter Din inner diameter H static head I turbulence intensity k angular momentum

gn number of meshes

P static pressure

relP0 relative total pressure Q mass flow volume r radius Re Reynolds number U∞ external velocity ( )φU uncertainty

aV axial velocity

rV radial velocity

refV reference velocity

Vθ tangential velocity

θW relative tangential

velocity υ kinematic viscosity ρ density μ dynamic viscosity

MΔ estimated error φ numerical solution

exactφ exact solution Γ swirl number β radial component angle η efficiency

aζ loss coefficient α angle of attack

Page 2: VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF … · VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF AN AXIAL HYDRAULIC TURBINE A. F. Tiago Mechanical Engineering Department Instituto

Introduction The paper reports the numerical study performed to validate the tubular-type hydraulic turbine

rotor design method described in [1], considering as test case a turbine designed to be used in a

small hydroelectric power plant.

Two methods were used: a two-dimensional cascade panel method coupled with a weak

viscous-inviscid boundary-layer calculation[2]; and a three-dimensional incompressible viscous

flow calculation, solving the Reynolds averaged Navier-Stokes (RANS) equations, using the

FLUENT code (6.2.16 version) [3], with meshes generated with the GAMBIT 2.2 code [4].

Design Method and Geometry Description The turbine blades are contained between two approximately cylindrical surfaces with a

common axis, which coincides with the turbine’s rotation axis. The outer diameter is D = 0.5 m

and the rotor diameter ratio is Din/D = 0.428.

The rotor blades were designed using the streamline curvature throughflow method for the

axis-symmetric calculation and a first-order panel method for the calculation of the flow between

the blades [5].

The cylindrical section radius was assumed to be the meridian flow streamlines average radius

at the rotor inlet and outlet sections. The rotor’s inlet section tangential velocity component is Vθ =

K/r, in which K is the inlet section angular momentum, constant along the radius. In the outlet

section, a zero tangential velocity component was assumed and, consequently, the angular

momentum is zero.

For the design condition, a nominal flow rate Q=3.362 m3, a static head H=75.6 m and a

rotation speed of 2500 r.p.m were imposed [5]. Air at ambient pressure and temperatures was

assumed in the performed simulations. The Reynolds number is approximately 4.6x106.

Two-dimensional Method (Panel Method) In the two-dimensional study, the panel method was used with a boundary-layer computation

based on the weak viscous-inviscid interaction theory. This inviscid flow solution is based on the

Laplace equation using sources and vortices superficial distributions. The airfoil is discretized

using flat panels with constant superficial sources distributions in each panel [6]. The normal

velocity component on the panel is equal to zero for the purely potential flow, in a first calculation.

However, considering the boundary-layer effect, this is equal to the wall-transpiration velocity [2].

As for the viscous flow solution, the influence of the wake is not taken into account and, as a

consequence, only the laminar region and the turbulent boundary-layer are considered [2].

The inviscid and viscous solutions compatibility is obtained by an iterative calculation using the

“weak interaction” theory.

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The twelve profiles used in the rotor blade geometry definition were discretized using 121

panels on the pressure and suction sides, obeying a cosine function distribution. The Reynolds

number υcU∞=Re , the chord-to-pitch ratio, the cascade stagger angle and the attack angle are

the code entry variables. Except for the attack angle, only design values were simulated.

Three-dimensional Method In this study, the flow solution was obtained using the FLUENT 6.2 code [3]. This code solves

the Navier-Stokes equations, for the mean time-dependent values (Reynolds equations).

The turbulence is modelled by different turbulence viscosity models: the standard k-ε model [7]

(used for high Reynolds numbers), the k-ω model proposed by Wilcox [8] and the

Spalart-Allmaras model [9].

The segregated solver was used to solve the flow equations (continuity and momentum). This

solver uses smaller computational resources, and is more suited, as referred in [3], for

simulations using a rotating coordinates system.

Several algorithms can be used to solve the velocity-pressure coupling. The SIMPLEC [10]

algorithm was selected because it is more consistent than the SIMPLE algorithm [3].

The FLUENT code solves the linearized form of the discretized transport equations, using a

Gauss-Siedel point-by-point solution algorithm, together with an algebraic multiple mesh method.

The equations’ diffusive terms are discretized using the second-order central-differences scheme

[3]. The convective terms are interpolated by the third-order QUICK [11] scheme. Among the

different schemes that can be used to obtain the pressure on the control volume faces, the

second-order interpolation scheme was selected [3]. Simple precision variables were used in the

performed simulations.

Relative pressures were used in the computations to minimize the round off error effect. The

reference pressure was defined at the inlet section, with a standard value of 101325 Pa. The air

density and viscosity were assumed constant, and their values at ambient temperature (300K),

are: ρ = 1.225 kg/m3 e μ = 1.789 x 10-5 Pa.s.

In the simulations, five different types of boundary conditions were used: fluid, velocity inlet,

outflow, periodic surfaces and solid surfaces.

Due to the hub rotation, it is necessary to define the rotating coordinates system. To do this, in

the fluid’s boundary condition, a 2500 r.p.m rotation velocity is imposed. At the inlet boundary the

velocity components for the absolute cylindrical referential system are imposed. This boundary

condition is identified in the code FLUENT as velocity inlet. At the inlet the radial velocity

component is null, the axial velocity component is constant, 96.20=AQ m/s and the tangential

velocity component is rV 491.2=θ m/s. The tangential velocity and the axial velocity components

Page 4: VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF … · VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF AN AXIAL HYDRAULIC TURBINE A. F. Tiago Mechanical Engineering Department Instituto

are obtained from the angular momentum and the design flow rate, respectively. The inlet section

turbulence values are obtained from the turbulence intensity, I , and the hydraulic diameter,

286.0=−== inH DDPAD m. For the turbulence intensity, a value of 4% was used, as

suggested in [3].

On the outlet section, the free flow condition was used. This condition, identified as outflow in

the code, corresponds to a variables zero degree extrapolating, from the interior of the domain to

the boundary, followed by a mass flow rate correction.

On the solid surfaces the impermeability condition was imposed for all the simulations.

However, tanking into account the rotation of the domain, it was necessary to define the rotation

of the solid boundaries. On the outer boundary, on the hub inlet and outlet region a zero rotational

velocity was defined. On the central region of the hub and on the pressure and suction faces of

the blade a 2500 r.p.m. velocity was imposed. On the periodic boundaries, the rotational

periodicity was imposed.

Mesh Generation In order to do the three-dimensional simulation, structured hexahedral and non-structured

tetrahedral meshes were built.

The computational domain for the flow between blades was obtained from the boundary lines of

the meridian flow. The radial coordinate ranges from 0.250 to 0.107 m and the axial coordinate

ranges from 0.670 to 1.670 m. The inlet boundary is 1.5 chords upstream of the leading edge and

the outlet boundary is 3 chords downstream of the trailing-edge. The blades shape was defined

from the twelve project sections divided into suction and pressure sides.

For the three turbulence models tested, different mesh thicknesses near the walls were used,

due to different the near-wall treatments. For the k-ε standard model the wall functions are

applied, so the first element height must be included in the logarithmic region, i.e. 30030 << +y .

For the k-ω and Spalart-Allmaras models, the meshes should be thin enough to solve laminar

sub-layer, 5<+y .

To facilitate the mesh generation, the tip clearance was ignored, modifying the load on this

blade region.

For the k-ε standard model four geometrically similar structured hexahedral meshes were

generated, with 129168, 319158, 831138 and 2152516 elements, respectively. Also for this

model, four non-structured tetrahedral meshes were generated, with 148318, 334002, 797531

and 1936342 elements. For the Spalart-Allmaras and k-ω models four geometrically similar

structured hexahedral meshes were generated, with 127188, 299782, 765824 and 2033040

elements, respectively.

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Numerical Error Estimation and Solution Convergence

There are three types of error that influences the numerical solution of a computational fluid

dynamics problem: round-off error, iterative error and discretization error [13, 14]. If the round-off

and the iterative errors are negligible compared to the discretization error the numerical

uncertainty is only due to this last term.

The numerical uncertainty estimation allows us to determine an uncertainty band that includes

the exact solution:

( ) ( )φφφφφ UU exact +≤≤− . (1)

The method used in this study, proposed by Eça [14, 15], estimates the error, MΔ , as the

maximum difference between the available results:

( ) gijM nji ≤≤−=Δ ,1max φφ . (2)

This method is appropriate to complex flows due to the almost inevitability of scattered results.

When monotonic convergence is not observed the uncertainty is estimated by [14]:

( ) MU Δ= 3φ . (3)

The numerical errors for the three turbulence models were compared for some computed

integral parameters. The considered variables were: the swirl number, Γ , and the radial

component angle average values, β , at the inlet and outlet; the efficiency, η ; and the loss

coefficient, aζ .

For the k-ω model four structured meshes were used. However, a convergence solution was

only obtained with two of them, the 127188 and 765824 element meshes, respectively. In those

cases the error estimation is quite limited due to the use of only two mesh solutions.

Table 1 presents the swirl number, Γ , numerical uncertainty for the simulations made. The

uncertainties for the other parameters are presented in [12].

Γinlet Γoutlet

Models Ø1 U1 |U1/Ø1| Ø1 U1 |U1/Ø1|

k-ε Structured 0,63196 0,000334 0,000528 0,0551952 0,05073 0,91908 k-ε Non-Structured 0,63221 0,000925 0,001464 0,0517542 0,03280 0,63366 Spalart-Allmaras 0,63083 0,0015323 0,002429 0,039447 0,00524 0,13271

Project 0,63199 0

Tab. 1 – Discretization error numerical uncertainty for the three models used.

The designt value, for this variable is within the uncertainty band of the obtained solutions

11 U±φ , except for the outlet swirl number. For the Spalart-Allmaras model, the uncertainty

assumes relatively lower values in relation to the k-ε model.

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Results For the two-dimensional method, the drag coefficient is obtained by the Squire & Young

approximations [2], cUcU

LCL∞∞

Γ−=−=

2

21 2ρ

.

Figure 1 shows the friction coefficient evolution as function of the drag coefficient for the

sections where separation doesn’t occur. In most cases, the design attack angle is on the low

range of drag coefficient, named laminar bucket, however, for the sections with radius equal to

0.1945 m and 0.2065 m, nothing could be concluded because the computation couldn’t continue

for the respectively design attack angles.

CD

CL

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-0.2

0

0.2

0.4

0.6

0.8

1

R=0.148 F. RealR=0.165 F. RealR=0.1805 F. Real

CD

CL

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-0.2-0.1

00.10.20.30.40.50.60.70.80.9

1

R=0.1945 F. RealR=0.2065 F. RealR=0.219 F. Real

CD

CL

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2-0.2-0.1

00.10.20.30.40.50.60.70.80.9

1

R=0.230 F. RealR=0.2405 F. RealR=0.250 F. Real

(a) (b) (c)

Fig. 1 – Evolution of the friction coefficient as function of the drag coefficient on the sections where separation doesn’t

occur.

α

CL

-4 -3 -2 -1 0-0.5

0

0.5

1

1.5

2

2.5

R=0.107 F. PerfeitoR=0.118 F. PerfeitoR=0.1295 F. Perfeito

α

CL

-4 -3 -2 -1 0-0.5

0

0.5

1

1.5

2

2.5

R=0.1805 F. PerfeitoR=0.1805 F. Real

α

CL

-4 -3 -2 -1 0-0.5

0

0.5

1

1.5

2

2.5

R=0.250 F. PerfeitoR=0.250 F. Real

(a) (b) (c)

Fig. 2 – Evolution of lift coefficient on the range of α ’s used in some sections

Figure 2 presents the evolution of the lift coefficient in the range of α used for some sections,

for ideal and real fluids cases without flow separation. The evolution of the lift coefficient for the

other sections is presented in [12].

The evolution of the lift coefficient with α is approximately linear in the attack angles range

considered. A linear evolution of the lift coefficient with α is also well verified on real flow for the

range of low attack angles, but with an lower rate than the ideal fluid model.

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The drag coefficients become higher in the proximity of the hub. This increase is caused by the

decrease of the chord on those sections and the constant angular momentum imposed along the

radius.

For the three-dimensional method, the presented results were computed with the finest meshes

for which the obtained results were considered enough accurate: The non-structured mesh with

1936342 elements and structured mesh with 2152516 elements for the k-ε model; the structured

mesh with 2033040 elements for the Spalart-Allmaras model; and the structured mesh with

765824 elements for the k-ω model. This last mesh presents a lower refinement when compared

with the others; it corresponds to the finest mesh for which the desired convergence was

obtained.

D*

Va*

0 0.25 0.5 0.75 10.990

0.995

1.000

1.005

1.010

K-Epsilon Não EstruturadaK-Epsilon EstruturadaK-OmegaSpalart-Allmaras

D*

Va*

0 0.25 0.5 0.75 10.90

0.92

0.94

0.96

0.98

1.00

1.02

1.04

K-Epsilon Não EstruturadaK-Epsilon EstruturadaK-OmegaSpalart-Allmaras

(a) (b)

Fig. 3 – Axial velocity axis-symmetric averages: (a) inlet section; (b) outlet section.

The axial velocity component axis-symmetric averages, *aV , on the inlet section are present in

figure 3a). The values show an increase close to the solid boundaries (hub and casing). This

increase is more evident on the k-ω and Spalart-Allmaras models than on k-ε model, because in

the first ones the laminar sub-layer is solved and in the last one the wall functions are applied.

This velocity increase is caused by the boundary condition imposed at the inlet – uniform axial

velocity – to guarantee the desired mass flow.

The axial velocity axis-symmetric averages profiles at the outlet section (figure 3b)) present

shape closer to the flow inside a duct with the viscous effects near the walls visible.

The angular momentum on the outlet section shows small differences between the turbulence

models. Near the hub, its value is zero or slightly negative and near the casing it presents a small

region with positive values; this means that not all the energy was extracted from the flow.

Figure 5 presents the pressure coefficient distribution on the twelve project sections. The

pressure coefficient is calculated by 20

21

ref

relp

V

PPC

ρ

−= , where

222

222 ⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎞⎜⎜⎝

⎛ ++⎟⎟

⎞⎜⎜⎝

⎛ += rOrIOIaOaI

refVVWWVV

V θθ and each referred variable represents the

circumferential average for the respective radius coordinate.

Page 8: VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF … · VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF AN AXIAL HYDRAULIC TURBINE A. F. Tiago Mechanical Engineering Department Instituto

(a) (b)

(c) (d)

Fig. 4 – Angular momentum distribution on the outlet section: (a) non-structured k-ε model; (b) structured k-ε model; (c) k-ω model; (d) Spalart-Almaras model.

The distribution on the hub is the one that differs most for the three turbulence models used,

the largest difference occurs between the leading edge and the suction peak. In the intermediate

sections a small difference is verified only in the leading edge area [12]. For the casing section,

the distributions are significantly different and vary according to the turbulence models near wall

treatment.

Between R = 0.107 m and R = 0.219 m the pressure coefficient presents a load exchange

close to the leading edge and in those cases the stagnation point is positioned on the suction side

of the blade [12]. The peak suction is predicted between 50% and 75% of the chord, and moves

towards the trailing edge direction as the radius diminishes.

X/C

Cp

0 0.25 0.5 0.75 1-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5R=0.107 Não Estruturada k-epsilonR=0.107 Estruturada k-epsilonR=0.107 k-omegaR=0.107 Spalart-Allmaras

X/C

Cp

0 0.25 0.5 0.75 1-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5R=0.1805 Não Estruturada k-epsilonR=0.180.5 Estruturada k-epsilonR=0.1805 k-omegaR=0.1805 Spalart-Allmaras

X/C

Cp

0 0.25 0.5 0.75 1-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5 R=0.250 Não Estruturada k-epsilonR=0.250 Estruturada k-epsilonR=0.250 k-omegaR=0.250 Spalart-Allmaras

(a) (b) (c)

Fig. 5 – Pressure coefficient distribution in three sections for the used models.

For the three turbulence models used, the k-ω model was the one that presented the biggest

difficulty to obtain convergence. The residual instability required more computation time and, in

some cases, it was not possible to get convergence. Compared to the k-ω model, the k-ε model

does not need such thin meshes near the solid boundaries, so it requires less calculation time.

However, for details on the walls, this model may not be as efficient. Compared to the last two

Page 9: VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF … · VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF AN AXIAL HYDRAULIC TURBINE A. F. Tiago Mechanical Engineering Department Instituto

models, less computational resources are required for the Spalart-Allmaras model and, at the

same time, this model can solve the laminar sub-layer.

For the tested cases, the Spalart-Allmaras model was the one that presented more stability on

the calculation, allowing us to conclude that it has better performance for these types of studies.

So, considering the flow characteristics, it is reasonable to consider that the Spalart-Allmaras

model produces good quality results, and so, only this model results will be compared with the

two-dimensional method ones.

The total pressure coefficient, 2

00

21

refVPP

∞−= , for the Spalart-Allmaras model at the inlet and

outlet sections, respectively, are present in figure 6. At the inlet section, the effect of the boundary

layer is visible. The non-uniformity of the pressure fields at the outlet section central region

suggests that the distance of this section to the trailing edge should be increased.

(a) (b)

Fig. 6 –Total pressure coefficient distribution on Spalart-Allmaras model: (a) inlet section; (b) outlet section.

Two-dimensional and Three-dimensional Methods Comparison For the comparison of the two models used, the pressure coefficient distribution on the same

blade profiles shown on the previous section is presented in figure 7.

The project values were determined by the panel method with ideal fluid [5], as expected they

coincide with the results obtained by the two-dimensional study for ideal fluid which uses the

same method.

X/C

Cp

0 0.25 0.5 0.75 1-1

-0.5

0

0.5

1

1.5

2

2.5 R=0.107 Spalart-AllmarasR=0.107 F. PerfeitoR=0.107 ProjectoR=0.107 Invíscido

X/C

Cp

0 0.25 0.5 0.75 1-1

-0.5

0

0.5

1

1.5

2

2.5R=0.1805 Spalart-AllmarasR=0.1805 F. PerfeitoR=0.1805 F. RealR=0.1805 ProjectoR=0.1805 Invíscido

X/C

Cp

0 0.25 0.5 0.75 1-1

-0.5

0

0.5

1

1.5

2

2.5R=0.250 Spalart-AllmarasR=0.250 F. PerfeitoR=0.250 F. RealR=0.250 ProjectoR=0.250 Invíscido

(a) (b) (c)

Fig. 7 – Pressure coefficient distribution for some profiles using the Spalart-Allmaras and the bi-dimensional models.

The differences between the inviscid [5] and the viscous simulations are not significant, except

for the sections coincident with the hub and the casing. These differences are due to the viscous

Page 10: VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF … · VISCOUS FLOW CALCULATION THROUGH THE ROTOR OF AN AXIAL HYDRAULIC TURBINE A. F. Tiago Mechanical Engineering Department Instituto

effects at those solid boundaries. The load exchange on the leading edge is more visible for the

viscous simulation.

There is a good agreement between three-dimensional and two-dimensional flow simulations in

the central zone of the blade. These results diverge in the hub and in the casing adjacent zones.

Conclusions From the three turbulence models used in the three-dimensional method, k-ω model was the

one which presented more difficulties in converging. The residuals’ instability required more

calculation time. For this model, in some cases, the solution convergence was not possible to

achieve.

The k-ε model using the standard wall function, comparatively to k-ω model, does not need

very refined meshes near the walls, so less computation time is required. However, the quality of

the solution near the walls may be less accurate.

Compared to the last two models, less computational resources are required for the

Spalart-Allmaras model. This model, as the k-ω model, can also solve the laminar sub-layer. For

the cases tested, the Spalart-Allmaras model was the one that presented more stability on the

calculation, allowing us to conclude that it has better performance for these types of studies.

The differences between the inviscid [5] and viscous simulations are not significant, except for

the sections coincident with the hub and the casing.

The design values were determined by the panel method considering ideal fluid [5], and as

expected they coincide with the two-dimensional study results obtained for ideal fluid which uses

the same method.

For a significant part of the flow in the middle of the blade span, the three-dimensional method

may be approached to the two-dimensional, which on a first stage of the design may be a very

useful tool, since it has a much lower demanding level.

References [1] L.M.C. Ferro, Numerical and Experimental Analysis of the Flow Through an Axial

Hydraulic Turbine, in portuguese, Ph.D. Thesis, Instituto Superior Técnico, Lisboa, to be submitted in 2008.

[2] L.R.C. Eça and J.A.C. Falcão de Campos, Analysis of Two-Dimensional Foils Using a Viscous-Inviscid Interaction Method, Int. Shipbuild. Progr., 40, no. 422 (1993) pp. 137-163.

[3] FLUENT 6.2 User's Guide, Fluent Incorporated, (2005). [4] GAMBIT 2.2 Modelling Guide, Fluent Incorporated, (2004). [5] L.M.C. Ferro, A.F. Tiago, L.M.C. Gato e J.M. Paixão Conde, Inviscid Flow Calculation on

a Rotor of a Axial Turbine, in portuguese, 8º Congresso Ibero-americano de Engenharia Mecânica, Cusco, 2007.

[6] J.C.C. Henriques, Analysis and Optimization on Airfoils and Blades Cascade, in portuguese, Final Project, Mechanical Engineering Department, Instituto Superior Técnico, Universidade Técnica de Lisboa, 1993.

[7] B.E. Launder and D.B. Spalding, Lectures in mathematical models of turbulence, Academic Press, (1972).

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[8] D.C. Wilcox, Turbulence Modelling for CFD, second edition, DCW industries. [9] P. Spalart and S. Allmaras. A one-equation turbulence model for aerodynamic flows.

Technical Report AIAA-92-0439, American Institute of Aeronautics and Astronautics, 1992.

[10] J.P. Vandoormaal and G.D. Raithby. Enhancements of the SIMPLE method for predicting incompressible fluid flows, Numer. Heat Transfer Vol. 7, pp. 147 - 163, (1984).

[11] B.P. Leonard and S. Mokhtari. ULTRA-SHARP Non-oscillatory Convection Schemes for High-Speed Steady Multidimensional Flow. NASA TM 1-2568 (ICOMP-90-12), NASA Lewis Research Center, 1990

[12] A.F. Tiago, Viscous Flow Calculation On The Rotor Of An Axial Turbine, in portuguese, Master Thesis Mechanical Engineering Department, Instituto Superior Técnico, Universidade Técnica de Lisboa, Outubro 2007.

[13] P.J. Roache, Verification and Validation in Computational Science and Engineering, Hermosa Publishers, (1998).

[14] L. Eça, Codes and Calculus verification in Computation Fluid Dynamics, in portuguese. I Conferência Nacional em Mecânica dos Fluidos e Termodinâmica, FCT-UNL, Portugal, 2006.

[15] L. Eça and M. Hoekstra, Discretization Uncertainty Estimation based on a Least Squares version of the Grid Convergence Index, Proceedings of the 2nd Workshop on CFD Uncertainty Analysis, Instituto Superior Técnico, Lisboa, 2006.