calculation of the ﬂow around hydrofoils at moderate ... · calculation of the ﬂow around...

Calculation of the flow around hydrofoils at moderateReynolds numbers

Rui Miguel Alves Lopes

Thesis to obtain the Master of Science Degree in

Aerospace Engineering

Supervisor: Prof. Luís Rego da Cunha de Eça

Examination Committee

Chairperson: Prof. Filipe Szolnoky Ramos Pinto CunhaSupervisor: Prof. Luís Rego da Cunha de Eça

Member of the Committee: Prof. André Calado Marta

December 2015

Acknowledgments

I would like to express my gratitude and appreciation for all of those who, in one way or another, helped

me and contributed for the development of this thesis:

To Professor Luıs Eca, for providing this opportunity and guiding me along the way with great patience.

To my family, for all the support they have given me during all of my life.

To Fabio Silva and Luıs Caldeira for all the friendship throughout the graduation.

To Hugo Abreu, for being calm, optimistic and for all the discussions on the subject.

To Joao Baltazar, for his fantastic hospitality, constant good mood and helpfulness.

To Ana Reis, who has been an amazing friend during these last five years.

To Nuno Lopes and Vanessa Eugenio, the two most fun people I know, for all their support.

iii

Resumo

O efeito da regiao de transicao de escoamento laminar a turbulento, que pode ser responsavel por cara-

teristicas importantes em escoamentos a baixos numeros de Reynolds, nao e capturado pelos modelos

de turbulencia comuns, sendo assim ignorado na maioria dos calculos computacionais. Recentemente

foi proposto um novo modelo de transicao que apresenta resultados promissores - o modelo γ − Reθ.

Este modelo e facilmente implementado em codigos modernos gracas a sua formulacao local, con-

trariamente as alternativas ate agora usadas. O objetivo desta tese e estudar o seu comportamento,

avaliando a sua robustez numerica e comparando a solucao obtida com resultados experimentais e uma

solucao de base recorrendo ao modelo de turbulencia k − ω SST, ja extensivamente validado. Como

calculo preliminar foi analisado o escoamento sobre uma placa plana em gradiente de pressao nulo, de

modo a estudar a influencia das quantidades turbulentas a entrada. Numa segunda fase foi simulado o

escoamento em torno de tres perfis sustentadores diferentes, por forma a avaliar a melhoria nos resul-

tados causada pelo modelo adicional. Os resultados obtidos mostraram que o novo modelo e capaz de

captar bolhas de separacao laminar e transicao natural revelando uma melhor modelacao na maioria

dos casos. No entanto, o comportamento numerico do modelo nao apresenta boas caracterısticas,

devido a uma maior dificuldade na convergencia iterativa e maior influencia do erro de discretizacao.

O modelo sofre ainda de uma sensibilidade elevada as condicoes de entrada da turbulencia, agravada

pelo decaimento exagerado das variaveis do modelo de turbulencia.

Palavras-chave: Transicao, Equacoes de Reynolds, Perfis Sustentadores , Mecanica de

Fluidos Computacional, Modelos de Turbulencia

v

Abstract

The effect of laminar to turbulent flow transition region, which can be responsible for important features

at low Reynolds numbers flows, is not captured by common turbulence models, and thus ignored in

most numerical calculations. Recently a new transition model presenting promising results has been

proposed - the γ − Reθ model. This model is easily implemented in modern codes which resort to non-

structured grids, unlike the alternatives used until now. The goal of this thesis is to study the behaviour of

this model, assessing its numerical robustness and comparing the obtained solution with experimental

results and a baseline solution that uses the already extensively validated k − ω SST turbulence model.

As a preliminary calculation, the flow over a zero pressure gradient flat plate was analysed, to study

the effect of the inlet turbulent quantities. On a second stage the flow over three different airfoils was

simulated, in order to evaluate the improvement of the results obtained due to the additional model.

The results obtained show that the new model is capable of capturing laminar separation bubbles and

natural transition, improving the modelling accuracy in most cases. However, its numerical behaviour

presents poor characteristics due to higher difficulties with iterative convergence and higher influence of

the discretization error. The model also suffers from a high sensitivity to the turbulence inlet conditions,

aggravated by the overestimated decay of the turbulence model variables.

Keywords: Transition, Reynolds Equations, Airfoils, Computational Fluid Dynamics, Turbulence

Models

vii

Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

1 Introduction 1

2 Mathematical Overview 5

2.1 Reynolds Averaged Navier Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Transition Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Chosen Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Flat Plate - Preliminary Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Numerical Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Numerical Setup 19

3.1 ReFRESCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Grids and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Turbulence Decay in Uniform Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Tested Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.1 NACA 0012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.2 NACA 66-018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.3 Eppler 387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Iterative Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.6 Domain Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Results 29

4.1 NACA 0012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 NACA 66-018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Eppler 387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

ix

4.3.1 Re = 3× 105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.2 Re = 1× 105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Conclusions 57

Bibliography 62

A Discretization Schemes 63

x

List of Tables

2.1 Basis for the inlet turbulence variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Distance to each boundary for each domain . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Inlet turbulence conditions tested for the NACA 0012 airfoil . . . . . . . . . . . . . . . . . 23

3.3 Difference (in %) in the lift and drag coefficient components with different stopping criteria

for α = 0o. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1 Transition start chordwise location and uncertainty estimate for the NACA 0012 airfoil. . . 32

4.2 Transition end chordwise location and uncertainty estimate for the NACA 0012 airfoil. . . 32

4.3 Lift force uncertainty estimate for the NACA 0012 airfoil. . . . . . . . . . . . . . . . . . . . 36

4.4 Drag force uncertainty estimate for the NACA 0012 airfoil. . . . . . . . . . . . . . . . . . . 36

4.5 Transition chordwise location and uncertainty estimate for the upper surface of the NACA

66-018 airfoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.6 Uncertainty estimate for the total lift and drag forces for the NACA 66-018 airfoil . . . . . . 42

4.7 Transition chordwise location for the upper surface of the Eppler 387 airfoil at Re = 3× 105. 46

4.8 Transition chordwise location uncertainty estimate for the upper surface of the Eppler 387

airfoil at Re = 3× 105. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.9 Lift and drag forces uncertainty estimate for the Eppler 387 airfoil at Re = 3× 105. . . . . 49

4.10 Transition chordwise location and uncertainty estimate for the upper surface of the Eppler

387 airfoil at Re = 1× 105. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.11 Lift and drag forces uncertainty estimate for the Eppler 387 airfoil at Re = 1× 105. . . . . 55

A.1 Discretization choices for each airfoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

xi

List of Figures

2.1 (a) Airfoil nomenclature, (b) Aerodynamic Forces . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Airfoils studied: (a) NACA 0012, (b) NACA 66-018, (c) Eppler 387 . . . . . . . . . . . . . . 11

2.3 Residual behaviour with varying inlet viscosity ratio and Tu = 0.874%. . . . . . . . . . . . 12

2.4 Local skin friction behaviour in the transition region with varying inlet turbulence intensity. 12

2.5 Local skin friction behaviour with varying inlet viscosity ratio and Tu = 3.3%. . . . . . . . . 13

2.6 Local skin friction behaviour with varying inlet viscosity ratio and Tu = 6.5%. . . . . . . . . 14

2.7 Turbulence kinetic energy along the plate for varying inlet viscosity ratios and Tu = 3.3%.

A different scale is used for case (a) for illustrative purposes. . . . . . . . . . . . . . . . . 15

2.8 Eddy viscosity along the plate for varying inlet viscosity ratios and Tu = 3.3%. . . . . . . . 16

3.1 (a) Blocks on the grid, (b) Coarsest grid for the NACA 0012 airfoil at α = 0o . . . . . . . . 20

3.2 Local skin friction for varying inlet viscosity ratios with frozen decay. . . . . . . . . . . . . 22

3.3 Iterative convergence effect with the SST model on local quantities for α = 0o. US - Upper

surface, LS - Lower Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Iterative convergence effect with the γ − Reθ model on local quantities for α = 0o. US -

Upper surface, LS - Lower Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Iterative convergence effect with the SST model on aerodynamic forces . . . . . . . . . . 25

3.6 Iterative convergence effect with the γ −Reθ model on aerodynamic forces . . . . . . . . 25

3.7 Domain size influence on local quantities. US - Upper surface, LS - Lower Surface . . . . 26

3.8 Domain size influence on lift and drag coefficients . . . . . . . . . . . . . . . . . . . . . . 27

4.1 Pressure coefficient and skin friction distributions for the NACA 0012 airfoil and α = 0o . . 29

4.2 Pressure coefficient and skin friction distributions for the NACA 0012 airfoil and α = 2o,

α = 4o and α = 6o. US - Upper surface, LS - Lower Surface . . . . . . . . . . . . . . . . . 30

4.3 Pressure coefficient and skin friction distributions for the NACA 0012 airfoil and α = 8o.

US - Upper surface, LS - Lower Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4 Transition chordwise location for the NACA 0012 airfoil . . . . . . . . . . . . . . . . . . . . 32

4.5 Friction drag uncertainty estimate for the NACA 0012 airfoil. . . . . . . . . . . . . . . . . 33

4.6 Pressure drag uncertainty estimate for the NACA 0012 airfoil. . . . . . . . . . . . . . . . . 34

4.7 Pressure lift uncertainty estimate for the NACA 0012 airfoil. . . . . . . . . . . . . . . . . . 35

4.8 Total lift and drag forces for the NACA 0012 airfoil. . . . . . . . . . . . . . . . . . . . . . . 36

xiii

4.9 Pressure coefficient distribution for the NACA 66-018 airfoil. US - Upper surface, LS -

Lower Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.10 Skin friction distribution for the NACA 66-018 airfoil. US - Upper surface, LS - Lower

Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.11 Transition chordwise location for the NACA 66-018 airfoil . . . . . . . . . . . . . . . . . . . 40

4.12 Uncertainty estimate for the lift and drag forces for the NACA 66-018 airfoil . . . . . . . . 41

4.13 Pressure coefficient and skin friction distributions for α = −2o and α = −1o for the Eppler

387 airfoil at Re = 3× 105. US - Upper surface, LS - Lower Surface . . . . . . . . . . . . 43

4.14 Pressure coefficient and skin friction distributions for α = 0o and α = 1o for the Eppler 387

airfoil at Re = 3× 105. US - Upper surface, LS - Lower Surface . . . . . . . . . . . . . . . 44

4.15 Pressure coefficient and skin friction distributions for α = 3o and α = 7o for the Eppler 387

airfoil at Re = 3× 105. US - Upper surface, LS - Lower Surface . . . . . . . . . . . . . . . 45

4.16 Transition chordwise location for the Eppler 387 airfoil at Re = 3× 105. . . . . . . . . . . . 46

4.17 Friction drag uncertainty estimate for the Eppler 387 airfoil at Re = 3× 105. . . . . . . . . 47

4.18 Pressure drag uncertainty estimate for the Eppler 387 airfoil at Re = 3× 105. . . . . . . . 47

4.19 Pressure lift uncertainty estimate for the Eppler 387 airfoil at Re = 3× 105. . . . . . . . . 48

4.20 Lift and drag forces for the Eppler 387 airfoil at Re = 3× 105. . . . . . . . . . . . . . . . . 48

4.21 Pressure coefficient and skin friction distributions for α = −1o, α = 0o and α = 1o for the

Eppler 387 airfoil at Re = 1× 105. US - Upper surface, LS - Lower Surface . . . . . . . . 50

4.22 Pressure coefficient and skin friction distributions for α = 2o, α = 4o and α = 6o for the

Eppler 387 airfoil at Re = 1× 105. US - Upper surface, LS - Lower Surface . . . . . . . . 51

4.23 Pressure coefficient and skin friction distributions for α = 7o for the Eppler 387 airfoil at

Re = 1× 105. US - Upper surface, LS - Lower Surface . . . . . . . . . . . . . . . . . . . 52

4.24 Transition chordwise location for the Eppler 387 airfoil at Re = 1× 105. . . . . . . . . . . . 52

4.25 Friction drag uncertainty estimate for the Eppler 387 airfoil at Re = 1× 105. . . . . . . . . 53

4.26 Pressure drag uncertainty estimate for the Eppler 387 airfoil at Re = 1× 105. . . . . . . . 53

4.27 Pressure lift uncertainty estimate for the Eppler 387 airfoil at Re = 1× 105. . . . . . . . . 54

4.28 Lift and drag forces for the Eppler 387 airfoil at Re = 1× 105. . . . . . . . . . . . . . . . . 54

xiv

Nomenclature

List of abbreviations

CFD Computational Fluid Dynamics

DNS Direct Numerical Simulation

LCTM Local Correlation-Based Transition Modelling

LES Large Eddy Simulation

LS Lower Surface

RANS Reynolds-Averaged Navier-Stokes

SST Shear-Stress Transport

US Upper Surface

List of symbols

α Angle of attack.

u Velocity vector.

εφ Discretization error

γ Intermittency.

∞ Free-stream condition.

µ Molecular viscosity.

µt Eddy viscosity.

ρ Density.

θ Momentum thickness.

Reθt Transition momentum thickness Reynolds number.

c Chord.

CD Drag coefficient.

xv

Cf Skin friction coefficient.

CL Lift coefficient.

Cp Pressure coefficient.

D Drag force.

hi Typical cell spacing of grid i

i, j, k Computational indexes.

k Turbulence kinetic energy.

L Lift force.

n Normal component.

p Pressure, observed order of convergence.

Rex Reynolds number.

U Numerical uncertainty.

u, v, w Velocity Cartesian components.

x, y, z Cartesian coordinates.

xvi

Chapter 1

Introduction

The contributions of the field of fluid mechanics are evident in everyday life: from applications in hy-

draulics and meteorology based on concepts from the branch of fluid statics [1], to the discipline of fluid

dynamics, which serves as a stepping stone to the design of the wings of an airplane, the blade profile

of propellers and turbines and the shapes of automobiles and high-speed trains [2].

These applications are studied in more detail in the subjects of aerodynamics and hydrodynamics,

based on the fundamental equations of fluid dynamics which govern fluid flow - the mass conservation

and Navier-Stokes equations. For an incompressible, Newtonian fluid flow, they form a coupled set of

four partial, non-linear, differential equations for four variables: the three components of the velocity

vector and pressure. Due to their complexity, limited analytical solutions are known, and only for very

simple cases [3].

Nonetheless, relevant data can be obtained from the equations when performing appropriate as-

sumptions and simplifications. When considering an external flow around a body, one of the regions

where viscous effects are important, near solid bodies, is the boundary layer. Boundary layer flow can

exhibit two different regimes: the laminar regime, characterized by a steady behaviour and orderly move-

ment of the flow particles, and the turbulent regime in which there are random velocity fluctuations in

both the streamwise and normal directions, which increase the mixing capacity of the flow making it

three-dimensional and unsteady [4].

In the recent years, there has been an increasing focus on numerical simulations: these consist in the

discretization of the governing equations using one of several approaches [5] such as finite differences,

finite volume and finite elements, giving rise to the field of Computational Fluid Dynamics (CFD).

CFD presents itself as an attractive option due to the wealth of data that can be obtained. Additionally,

the constant growth in processing and memory capabilities makes it an engineering tool, as increasingly

accurate and detailed solutions are calculated.

Due to its physics, the turbulent regime has been one of the greatest challenges for CFD. The more

accurate approaches, such as Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) are

very resource demanding, which only makes them suitable for research purposes. A Reynolds Averaged

Navier Stokes (RANS) simulation may not be as precise, due to additional modelling, but is a much more

1

viable alternative for an engineering application.

Nevertheless, nowadays good results are obtained for the turbulent flow region due to the wealth of

existing turbulence models, which have been extensively validated, covering a wide range of applica-

tions.

As such, the next step is to accurately model the transition region from laminar to turbulent flow, as

it can be responsible for important features of the flow. The currently used turbulence models such as

k − ω and k − ε models fail to model transition accurately, which is not surprising since they were not

built for that purpose. Transition modeling is only now getting attention as the new obstacle that has to

be dealt with [6].

Transition is relevant for the flow characteristics in low Reynolds number applications. Although large

transport aircraft usually deal with high Reynolds numbers, over 107, unmanned aerial vehicles (UAVs)

and micro air vehicles (MAVs) operate in the low Reynolds number range: 105 to 106 [7]. These UAVs

can be used for a variety of purposes, such as aircraft tracking, weather monitoring and ocean surveil-

lance [8, 9]. The development of wind energy systems is another growing area where low Reynolds

number phenomena is relevant, due to the increasing attention and investment in this particular renew-

able energy source, which requires efficient solutions: the airfoil section is critical for the determination

of the loads that the structure is subjected to. Small gas turbines’ blades also operate in the mentioned

range [10].

There are methods to achieve good accuracy regarding transition for simple flows, such as DNS

approaches. Another alternative for 2-D geometries is the en method [11, 12] used in the X-FOIL code of

Drela and Giles [13]. However, these options are limited in their usefulness: the use of DNS calculations

is still very resource demanding and the remaining methods are not easily introduced in modern CFD

codes.

Recently, in 2006, a new transition model was published: the γ − Reθ model proposed by Langtry

et al. [14]. This model seems to provide a good basis to incorporate the transition region in RANS

simulations, compatible with modern CFD characteristics: applicable to unstructured grids and parallel

computation due to its local formulation.

The goal of this thesis is to study the influence of transition modeling on the flow over three different

airfoils, by performing a verification and validation exercise, using the flow solver ReFRESCO [15] along

with the γ − Reθ model. Preliminary calculations will be done for a zero pressure gradient flat plate

test, which is a standard test case given its simplicity and the results shall be compared with available

experimental data, as a means to demonstrate the importance of including a transition model. The

numerical properties of said model shall also be studied, regarding iterative and mesh convergence as

well as sensitivity to the inlet turbulence boundary conditions.

The transition model shall then be put to use in a more practical application, by computing the flow for

the NACA 0012, NACA 66-018 and Eppler 387 airfoils. The results regarding drag and lift coefficients, as

well as pressure distributions are then compared to available experimental data, as part of the validation

stage. For each airfoil, and for the flat plate as well, two sets of calculations will be done: one using only

the k−ω Shear Stress Transport (SST) turbulence model and another using the previously mentioned γ -

2

Reθ transition model, which is coupled with the SST model. The two cases are compared to understand

the changes in the solution provided by the transition model, and the potential improvement in modelling

accuracy regarding the lift and drag coefficients.

The following describes the structure of this thesis: chapter 2 discusses briefly the mathematical

models behind the RANS approach, mentioning the closure problem and a general overview of some

turbulence and transition models. The results concerning the flat plate are presented in chapter 2 as well.

Chapter 3 is dedicated to all the numerical aspects concerning the airfoil calculations: the ReFRESCO

flow solver, the grids used, along with domain and iterative convergence criteria aspects and other

relevant details. Finally, the results obtained for each of the airfoils studied are presented and discussed

in chapter 4. The conclusions drawn from this work are presented in chapter 5, as well as considerations

and suggestions for future studies on the subject.

3

Chapter 2

Mathematical Overview

2.1 Reynolds Averaged Navier Stokes Equations

The Navier Stokes equations, together with the mass conservation equation serve as a starting point

for any fluid flow analysis. These equations represent the mass and momentum conservation principles

written for a control volume. Considering the case of an incompressible, Newtonian fluid with constant

properties, the equations, in index notation, are written as [4]:

∂ui∂xi

= 0 (2.1)

∂ui∂t

+ uj∂ui∂xj

= −1

ρ

∂p

∂xi+ ν

∂2ui∂x2j

(2.2)

where ui is the velocity in the xi direction, t is time and p is the pressure and ρ and ν are the

density and the molecular viscosity of the fluid, respectively. The RANS equations are obtained from the

Navier-Stokes equations, by considering that each variable φ is given by the sum of an average, time

independent component Φ, and a fluctuation component φ.

φ = Φ + φ (2.3)

Due to its construction, although the fluctuation component can be a function of time, its mean value

is zero.

The Navier-Stokes equations are then time-averaged and the decomposition above is employed.

Simplifying terms, the following is obtained:

∂Ui∂xi

= 0 (2.4)

∂Ui∂t

+ Uj∂Ui∂xj

= −1

ρ

∂P

∂xi+ ν

∂2Ui∂x2j

+1

ρ

∂

∂xj(−ρuiuj) (2.5)

Comparing the previous equations with the Navier-Stokes equations, a new term appears on the

5

right hand side of the momentum equations, due to the non-linearity of the convective terms. This new

quantity represents the effect of the velocity fluctuations on the average momentum transport, and can

be interpreted as a tensor of turbulent stresses - the Reynolds stress tensor. The appearance of the

Reynolds stresses also give rise to a closure problem: they represent additional unknowns for which

more equations must be found. This issue is in the root of all the effort that has been done to obtain

proper numerical results for the turbulent region and is the reason why so many different turbulence

models exist. The first distinction that can be made between different models is how they tackle the

closure problem.

One possible way is to introduce the transport equations for the Reynolds stresses, modelling them

directly, which results in a larger computational load. Some additional modelling is still required as the

equations contain even more unknowns. Models that fit this category are known as Reynolds stress

transport models [16].

The second, more common alternative, is to introduce the concept of eddy viscosity and make use

of the Boussinesq hypothesis [17]:

− ρuiuj = µt

(∂Ui∂xj

+∂Uj∂xi

)− 2

3ρδijk (2.6)

This relation does not solve the problem immediately, since a new variable appeared - the eddy

viscosity. However, a simplification was already introduced: in the case of a three dimensional problem,

there would be six Reynolds stresses in play. The usage of the Boussinesq hypothesis reduces them all

to only one unknown. This also means that models based on the eddy viscosity consider turbulence to

be isotropic.

Several types of eddy viscosity based models exist, usually classified according to the number of dif-

ferential equations they use: zero-equation or algebraic models, one-equation models and two-equation

models.

Algebraic models, such as Prandtl’s mixing length model use an algebraic equation to relate a char-

acteristic length scale to the eddy viscosity [17].

One and two-equation models include one or two differential equations for turbulent quantities. One

commonly used one-equation model is the Spalart and Allmaras model [18], in which the transported

variable is proportional to the eddy viscosity.

In two-equation models, the first of the transported variables is usually the turbulence kinetic energy,

k, while the second one depends on the model. One of the first formulations derived is to introduce the

turbulence dissipation ε, giving rise to the k − ε turbulence model [19]. It is still commonly used today,

but it performs poorly in adverse pressure gradient situations [20] and cannot be used all the way down

to the wall.

Another popular alternative are k − ω models, in which the second transported variable is the spe-

cific dissipation ω. The model was initially devised by Wilcox [21], and avoids the shortcomings of the

standard k − ε model. However, it suffers from a high sensitivity to the freestream turbulence properties

[22].

6

One popular k− ω formulation is the Shear Stress Transport (SST) model, proposed by Menter [23].

This model acts as a mixture of both models described above: near the wall it uses a k − ω formulation

and switches to a k − ε behaviour in the free-stream. This blend eliminates the disadvantages of each

individual model maintaining their strengths: the model can be used all the way down to the wall, and

the free-stream turbulence influence is eliminated. The two transport equations that form the basis of

the model are as follows:

∂(ρk)

∂t+∂(ρujk)

∂xj= P − β∗ρωk +

∂

∂xj

[(µ+ σkµt)

∂k

∂xj

](2.7)

∂(ρω)

∂t+∂(ρujω)

∂xj=ψ

νtP − βρω2 +

∂

∂xj

[(µ+ σωµt)

∂ω

∂xj

]+ 2(1− F1)

ρσω2ω

∂k

∂xj

∂ω

∂xj(2.8)

In these equations, P is the source term, F1 is a blending function and β, β∗, σω2, σω and σk are

constants. The complete formulation can be consulted in [23]. This will be the turbulence model used in

this thesis, due to its good behaviour in adverse pressure gradients and due to the fact that it is coupled

with the transition model that will be used as well, and referred to in the next section.

All of the previous discussion reveals the existence of many different options when simulating turbu-

lent flows. Depending on the application, some models perform better than others due to the calibration

of the constants, but overall, it is possible to obtain good results if choosing the appropriate model. How-

ever, the development of turbulence models was focused only on the fully turbulent flow region thus the

transition phenomena is not correctly modelled, and whenever its influence is important - for example on

low Reynolds numbers applications - turbulence models fail to capture its effect.

2.2 Transition Modeling

Unlike turbulence, there still is not a wide range of transition models which produce adequate results for

various applications.

The difficulty in modelling transition arises from its non-linearity, wide range of scales at play, and

the fact that it can occur through different mechanisms, depending on the application. Natural transition

occurs when the freestream turbulence level is low (<1%) and Tollmien-Schlichting waves grow and

become unstable, originating turbulent spots which further erupt into a fully turbulent regime [17]. When

the freestream turbulence level is high (>1%), typical of turbomachinery flows, the first stages of natural

transition are bypassed, and turbulent spots are produced due to freestream disturbances [24] - bypass

transition. Separation induced transition [25] occurs when an adverse pressure gradient causes a lam-

inar boundary layer to separate. In the separated shear layer it is possible for transition to develop and

the flow becomes turbulent, occasionally reattaching to the wall, due to the increased mixing capability

of the flow, forming a laminar separation bubble.

Nevertheless, there are several options in order to model this phenomenon [26]. The first approach

is based on stability theory, in which the continuity and momentum equations are linearized, and a form

is assumed for a single disturbance. The idea behind this method is to evaluate when the amplitude of

7

the disturbance is amplified, which reduces the problem to an eigenvalue problem. This is the basis for

the much used en method [11, 12] used in the popular X-FOIL code [13], in which a single disturbance

is followed along streamlines, and transition is considered to occur when the amplification ratio reaches

a certain value - the n factor. Thus this method still relies on some empiricism, in the form of the choice

of the n factor. Despite providing good results, this method is only applicable to simple geometries, and

cannot be easily implemented in modern CFD codes due to the non-local operations involved. Addition-

ally, the principle is based on natural transition, so it should not be used when a different mechanism

causes transition, and it is only able to predict the transition point, not the entire region.

DNS and LES [27] are also valid alternatives which produce good results and a large wealth of

information which is excessive for engineering purposes. Considering the spatial and temporal mesh

requirements, and their impact on the required computational means for an acceptable simulation time,

they are excluded from practical applications.

Another possible alternative are low Reynolds number turbulence models [28]. However the results

exhibited are strongly dependent on the calibration of the models, namely for the viscous sub-layer [29],

and the physics of the phenomenon are not correctly captured.

A new approach to modelling transition is based on the concept of kinetic laminar energy. It is

observed in experiments that in the pre-transitional region, before the increase in skin friction, there are

already some high-amplitude streamwise fluctuations. This led to the development of a laminar kinetic

energy equation [30]. Although the model has not yet been extensively validated, some initial analysis

show promising results [31] despite evidence from LES calculations that some of the assumptions upon

which the model was built are not correct. Other models resorting to the same concept have since then

been proposed as well [32, 33].

One common approach to model transition is to make use of empirical correlations along with a trans-

port equation for intermittency [34–36]. This new variable is closely related to transition and is defined as

the fraction of time that a given point of the flow is turbulent: it takes the value of 0 in regions where the

flow is still laminar, while those immersed in fully turbulent flow have an intermittency value of 1. These

models introduce the effect of the new variable by multiplying it by the eddy viscosity in the equations

of an eddy viscosity based turbulence model. However, the models suffer from the drawback that the

correlations used invoke non-local quantities such as the momentum thickness Reynolds number.

Recently, in 2006, a new alternative has been proposed by Langtry et al. [14], based on the intermit-

tency model, which avoids the usage of non-local quantities - the γ−Reθ model. Two additional transport

equations are used: one for intermittency and another for the transition onset Reynolds number, and it is

coupled with the k−ω SST turbulence model. One marking distinction between this model and other in-

termittency based models is the coupling with the turbulence model used. Unlike the previous case, the

eddy viscosity is not multiplied directly by intermittency: instead, the connection is made by multiplying

the production term of the k equation by the intermittency. When it was first proposed, the model lacked

two main correlations which were deemed proprietary. This caused the first research efforts related to

the model to be aimed at developing possible alternatives for the missing correlations [37, 38]. Finally,

in 2009, the full model was released to the scientific community [6]. The initial formulation lacks the in-

8

fluence of certain aspects such as roughness or crossflow instabilities - the latter may be relevant in 3D

flows - but some steps are already being taken in that direction [39]. Additionally, some efforts concern-

ing the validation of the model and comparison of the results with well established methods such as the

en method have already been published [40–42]. This is the model used in this thesis, chosen due to the

easy implementation on modern CFD codes as well as the results presented so far. This model is also

part of a new concept of transition modelling based on empirical correlations - Local Correlation-Based

Transition Modelling (LCTM). This designation will be used throughout this thesis to refer to the γ −Reθmodel. The two transport equations are as follows:

∂(ργ)

∂t+∂(ρUjγ)

∂xj= Pγ − Eγ +

∂

∂xj

[(µ+

µtσf

)∂γ

∂xj

](2.9)

∂(ρReθt)

∂t+∂(ρUjReθt)

∂xj= Pθt +

∂

∂xj

[σθt(µ+ µt)

∂Reθt∂xj

](2.10)

In these equations, γ is the intermittency, Reθt is the local transition momentum thickness Reynolds

number, Pγ and Eγ are the production and destruction terms of the intermittency equation, while Pθt is

the source term for the transition momentum thickness Reynolds number transport equation. Finally, σf

and σθt are constants. The complete formulation of the model can be found in [6].

2.3 Chosen Airfoils

Figure 2.1 shows the basic nomenclature used for airfoils, which are the cross-sections of lifting bodies

such as the wing of an aircraft. Their shape is designed specifically to provide an aerodynamic force to

counter the weight of the aircraft, by decreasing the pressure on the upper surface of the wing.

(a) (b)

Figure 2.1: (a) Airfoil nomenclature, (b) Aerodynamic Forces

The resulting force is decomposed in two components, one normal to the incoming undisturbed flow

which is the lift force, L, and one in the direction of the incoming flow which is the drag force, D.

Besides the pressure contribution, the aerodynamic force also has a contribution from the friction

force that arises from the no slip condition at the wall. This friction contribution does not affect signifi-

cantly the lift force, but can be relevant for the drag force.

These forces are often referred to in a non-dimensional way, the lift and drag coefficients, Cl and Cd,

respectively:

9

Cl =L

12ρU

2∞c

(2.11)

Cd =D

12ρU

2∞c

(2.12)

The pressure at each point is often considered in its non-dimensional form as well, the pressure

coefficient, Cp:

Cp =p− p∞12ρU

2∞

(2.13)

For potential flow, the pressure coefficient can be expressed as Cp = 1 − ( UU∞

)2. When displaying

the pressure coefficient graphically, its usual to have −Cp on the chart instead of Cp. The definition for

the skin friction, Cf is obtained the same way as for the pressure coefficient:

Cf =τw

12ρU

2∞

(2.14)

In which the shear-stress at the wall, τw, is obtained directly from the no-slip condition:

τw = µ∂U

∂y

∣∣∣y=0

(2.15)

One fundamental parameter when dealing with flows with viscous effects is the Reynolds number,

Re, defined as:

Re =ρU∞L

µ(2.16)

In which L is the characteristic length of the body involved, µ is the dynamic viscosity and U∞ is the

velocity of the incoming, undisturbed flow. The Reynolds number can be interpreted as the ratio of the

inertial forces to the viscous forces.

The chosen airfoils are the NACA 0012, NACA 66-018 and the Eppler 387 airfoil (Figure 2.2). The first

is a conventional, symmetrical 12% thickness airfoil. Its thickness distribution is given by the NACA 4-

digit equation, and it is commonly used as a benchmark airfoil. The NACA 66-018 is a symmetrical, 18%

thickness laminar airfoil, whose geometry involves conformal mapping. Laminar airfoils are designed to

increase the portion of the flow that is in the laminar regime. This results in a lower drag coefficient within

their working range, but worse aerodynamic characteristics than conventional airfoils when outside of the

adequate Reynolds and angle of attack ranges. The Eppler 387 airfoil is also a laminar airfoil, designed

for use on model sailplanes using inverse conformal mapping. It is non-symmetrical, and is widely used

as a common benchmark for low Reynolds number calculations.

The selection of these airfoils was motivated by the available experimental data for low Reynolds

numbers [10, 43–46].

10

(a) (b)

(c)

Figure 2.2: Airfoils studied: (a) NACA 0012, (b) NACA 66-018, (c) Eppler 387

2.4 Flat Plate - Preliminary Calculations

2.4.1 Numerical Settings

A steady-state, bi-dimensional analysis was performed on the flow over a zero pressure gradient flat

plate using the ReFRESCO flow solver [15]. The Reynolds number at the trailing edge of the plate

is 5×106 and the inlet turbulence conditions were based on the available experimental data from the

ERCOFTAC database [47], which is commonly used as a benchmark for transition models, and has

been used to validate the model before [6]. These base conditions are shown in table 2.1.

Designation Turbulence Intensity % Viscosity RatioT3A- 0.874 8.72T3A 3.3 12T3B 6.5 100

Table 2.1: Basis for the inlet turbulence variables.

The computational domain has a total length of 1.5L, where L is the length of the plate. The inlet is

located 0.25L upstream of the plate’s leading edge, and has turbulence kinetic energy and dissipation

prescribed, as well as velocity. The outlet is located 0.25L downstream of the plate’s trailing edge, and

an outflow boundary condition (derivatives with respect to x set to 0) is imposed. The top boundary of

the domain is located at an height of 0.25L, and has a zero pressure condition. The boundary condition

on the plate is a no-slip wall. The remaining regions (downstream and upstream of the plate) have a

symmetry boundary condition.

All sets of boundary conditions were tested on 9 different grids, as part of a grid sensitivity study. The

coarsest grid has 385 grid points in the flow direction and 49 grid points in the normal direction while

the finest grid is about 4 times finer, with 1537 grid points in the flow direction and 193 points in the

normal direction. For all grids the maximum y+ value on the first grid point above the wall was below 1,

as suggested by Langtry [48] for the use of the transition model.

The convergence criterion for the computations without the transition model was set as a maximum

normalized residual of all equations below 10−6. The L∞ residual norm was used, which makes the

normalized residuals equivalent to the variable change in a single Jacobi iteration. When the transition

model was included, this was changed to 10−5. This value allowed for the residuals of the variables

11

from the momentum and pressure equations to be below 10−6, as the highest residual is usually for the

transport equation of either γ or Reθ.

2.4.2 Results

Iteration

Res

idu

al

0 5000 10000 1500010-7

10-6

10-5

10-4

10-3

10-2

10-1t / = 0.01

t / = 0.1

t / = 8.72

Tu = 0.874%

Iteration

Res

idu

al

0 5000 10000 1500010-7

10-6

10-5

10-4

10-3

10-2

10-1LCTM - t / = 1LCTM - t / = 8.72

Tu = 0.874%

(a) SST (b) LCTM

Figure 2.3: Residual behaviour with varying inlet viscosity ratio and Tu = 0.874%.

Figure 2.3 shows the residual for the x component of the velocity vector, obtained when the turbu-

lence intensity is set to 0.87%. Only this residual is shown as it is representative of the behaviour of the

remaining ones. Unlike turbulence intensity, the viscosity ratio has a major influence on the convergence

of the solution. Whenever µt

µ >> 1, the finer grids start exhibiting convergence problems. This trend

happens regardless of the transition model being active or not, which suggests that the cause lies in the

turbulence model.

Rex

Cf

103

20000 30000 40000 50000 60000

4

4.5

5

5.5

SST - Tu = 0.874%SST - Tu = 2%SST - Tu = 3.3%SST - Tu = 6.5%

t / = 1

Rex

Cf

103

2.6E+06 2.8E+06 3E+06 3.2E+06 3.4E+060

2

4

LCTM - Tu = 0.874%LCTM - Tu = 2%LCTM - Tu = 3.3%LCTM - Tu = 6.5%

t / = 1

(a) SST (b) LCTM

Figure 2.4: Local skin friction behaviour in the transition region with varying inlet turbulence intensity.

12

As can be seen from Figure 2.4, the effect of the turbulence intensity goes against what one would

reasonably expect: for lower values of the turbulence intensity at the inlet, transition starts earlier, al-

though the sensitivity shown is small. Note the different scales used: without the γ−Reθ model, transition

starts at a Reynolds of 4 × 104, and is delayed until a Reynolds number of 3 × 106 when the transition

model is used - a difference of 2 orders of magnitude. The start of transition is here considered as the

minimum of the skin friction curve. The large difference in the location of the transition region represents

the reason why transition modelling may be critical for some applications, namely those in which the

laminar region is expected to occupy a significant portion of the flow. In the present case, using only

the SST turbulence model results in less than 1% of the flow over the plate being laminar. When using

the transition model, the fraction increases to 60%. For more practical applications, such as the flow

over an airfoil, this will result in a significant change in the friction component of the lift and drag coeffi-

cients, which may, specially for the drag, have a significant influence over the final results. Furthermore,

changes in the pressure drag are also to be expected due to the change in the location of the transition

region, especially in the cases where laminar separation bubbles occur.

Rex

Cf

103

104 105 1060

2

4

6

8

10

SST - t / = 0.01SST - t / = 0.1SST - t / = 1SST - t / = 12Experimental

Tu = 3.3%

Rex

Cf

103

104 105 1060

2

4

6

8

10

LCTM - t / = 1LCTM - t / = 12LCTM - t / = 20LCTM - t / = 40LCTM - t / = 60LCTM - t / = 80LCTM - t / = 100Experimental

Tu = 3.3%

(a) SST (b) LCTM

Figure 2.5: Local skin friction behaviour with varying inlet viscosity ratio and Tu = 3.3%.

The location of the transition region changes considerably when the viscosity ratio is changed: an

increase in the viscosity ratio causes transition to start earlier. This can be easily seen from Figures

2.5 and 2.6, which show the local skin friction along the plate for the inlet conditions used on [6] as

referred in table 2.1. In both cases, transition is predicted downstream of the location indicated by the

experimental points, even for unrealistically high viscosity ratios. One important factor that explains why

transition takes place so late is the decay of turbulence that occurs from the inlet until the leading edge

of the plate.

The decay is evident in Figures 2.7 and 2.8, which show the turbulence kinetic energy and the eddy

viscosity fields along the plate, for an inlet turbulence intensity of 3.3%, corresponding to an inlet value

for k of 1.634× 10−3, and varying viscosity ratio. Figure 2.7 (a) has a different scale than Figures 2.7 (b)

and (c) to better illustrate the decay. Naturally, different scales are also used for Figures 2.8 (a), (b) and

13

Rex

Cf

103

104 105 1060

2

4

6

8

10 SST - t / = 0.01SST - t / = 1SST - t / = 100Exp - T3BBlasius

Tu = 6.5%

Rex

Cf

103

104 105 1060

2

4

6

8

10 LCTM - t / = 1LCTM - t / = 100Exp - T3BBlasius

Tu = 6.5%

(a) SST (b) LCTM

Figure 2.6: Local skin friction behaviour with varying inlet viscosity ratio and Tu = 6.5%.

(c). Due to the decay, at the leading edge of the plate, for µt

µ = 1, k is already lower than 6× 10−6 - two

orders of magnitude lower than the inlet value. The increase of the inlet eddy viscosity decreases the

decay: for a value of µt

µ = 100, at the leading edge k is around 5 × 10−4. The eddy viscosity itself also

decreases, although the effect is much less pronounced than what happens with the turbulence kinetic

energy. This also explains why increasing k at the inlet produces no significant change in the location

of transition - the decay causes k to decrease so much that the value at the leading edge of the plate is

nearly the same for all cases. This occurrence is troubling since, if such a sharp decrease takes place

for k over such a small distance (0.25L), when computing the flow over an airfoil where the inlet is often

placed at 10 chords or more of the airfoil, the effect will be of even more significance. From what was

shown, increasing the viscosity ratio seems to be a possible solution to the problem. However, care must

be taken since using high enough values - and 100 can be considered as high enough - causes the skin

friction to deviate from the laminar value. This can already be seen in figure 2.6. Additionally, in the

present work, using a high viscosity ratio introduces iterative convergence issues. One other alternative

is to simply change the equations of the turbulence model and prevent the decay from happening [49].

This issue of turbulence decay will be detailed further on in the next chapter.

The flat plate calculations led to the following conclusions:

• A large portion of the plate is in the laminar regime when using the transition model, while when

only the SST model is used, the laminar part is almost negligible.

• The solution fails to converge for high viscosity ratios: µt

µ >> 1. This happens whether the transi-

tion model is active or not.

• The model exhibits a low sensitivity to the inlet turbulence intensity. Instead, the viscosity ratio has

a large impact on the location of the transition region.

• The decay of the turbulence variables in the region upstream of the plate causes the turbulence

14

(a) µt

µ = 1

(b) µt

µ = 12

(c) µt

µ = 100

Figure 2.7: Turbulence kinetic energy along the plate for varying inlet viscosity ratios and Tu = 3.3%. Adifferent scale is used for case (a) for illustrative purposes.

15

(a) µt

µ = 1

(b) µt

µ = 12

(c) µt

µ = 100

Figure 2.8: Eddy viscosity along the plate for varying inlet viscosity ratios and Tu = 3.3%.

16

kinetic energy to be very low at the leading edge and along the plate, which results in delayed

transition. Increasing the inlet eddy viscosity causes a smaller decay of the turbulence model’s

variables.

17

Chapter 3

Numerical Setup

3.1 ReFRESCO

ReFRESCO is a viscous-flow CFD code that solves multiphase (unsteady) incompressible flows us-

ing the Reynolds-Averaged Navier-Stokes equations, complemented with turbulence models, cavitation

models and volume-fraction transport equations for different phases [50]. The equations are discretized

using a finite-volume approach with cell-centered collocated variables, in strong-conservation form, and

a pressure-correction equation based on the SIMPLE algorithm is used to ensure mass conservation

[51]. Time integration is performed implicitly with first or second-order backward schemes. At each

implicit time step, the non-linear system for velocity and pressure is linearised with Picard’s method and

either a segregated or coupled approach is used. A segregated approach is always adopted for the

solution of all other transport equations. The implementation is face-based, which permits grids with

elements consisting of an arbitrary number of faces (hexahedrals, tetrahedrals, prisms, pyramids, etc.)

and if needed h-refinement (hanging nodes). State-of-the-art CFD features such as moving, sliding and

deforming grids, as well as automatic grid refinement are also available. The code is parallelised using

MPI and sub-domain decomposition, and runs on Linux workstations and HPC clusters. ReFRESCO is

currently being developed and verified at MARIN (in the Netherlands), in collaboration with IST (in Portu-

gal), USP-TPN (University of Sao Paulo, Brazil), TUDelft (Technical University of Delft, the Netherlands),

UoS (University of Southampton, UK) and recently UTwente (University of Twente, the Netherlands) and

Chalmers (Chalmers University, Sweden) [15].

3.2 Grids and Boundary Conditions

The quality of the grid used in a calculation has a significant impact on the quality of the solution obtained

after the calculation is performed. For this thesis an existing grid generator [52] was used, resulting in

structured, multiblock grids with 5 blocks: The first block is a C-shaped block around the airfoil. The

second and third blocks are the top and bottom parts of the domain. The fourth block is the area

upstream of the airfoil, while the last block is the area downstream of the airfoil, which captures the

19

wake. Each angle of attack is calculated on a different grid - the effect of the angle of attack is captured

by rotating the C block.

Figure 3.1 (a) shows a schematic of the grid configuration: the generated blocks as well as the

domain boundaries.

(a) (b)

Figure 3.1: (a) Blocks on the grid, (b) Coarsest grid for the NACA 0012 airfoil at α = 0o

• The inlet was located 12 chords away from the leading edge of the airfoil, where the velocity vector

and turbulence quantities were given by a Dirichlet boundary condition.

• The top and bottom boundaries were located each 12 chords away from the airfoil, with a slip

boundary condition.

• The outlet was located 24 chords downstream of the leading edge of the airfoil. An outflow bound-

ary condition was imposed: derivatives with respect to x set to 0.

• The surface of the airfoil is treated as a solid wall: impermeability and no-slip conditions are used.

All calculations were performed on a set of five geometrically similar grids with different refinement

levels, in which the refinement ratio was doubled between the coarsest and finest grid. They are iden-

tified by the number of points along the surface of the airfoil: the coarsest grid has 301 points, while

the finest has 601 grid points. A higher density of grid points occured in both the leading and trailing

edges. All grids obeyed the condition y+ < 1, as recommended by Langtry [48] for the use of the tran-

sition model. Figure 3.1 (b) shows an example of one of the grids used. Performing the calculations

on all 5 grids will allow for an estimation of the discretization error. The procedure is based on Richard-

son extrapolation, which assumes that the discretization is the only source of uncertainty. To fulfil that

assumption, round-off and iterative errors should be minimized accordingly. In the present work, calcu-

lations were performed in double precision, which makes the round-off error negligible. To reduce the

iterative error, the calculations ran until the maximum L∞ residual norm was below 10−6 when using

20

only the turbulence model. When the transition model was used, this was changed to 10−5, similarly

to what was done with the flat plate computations. The choice of these values is justified further on in

section 3.4. To obtain an estimate for the discretization error εφ of a variable φ, the error is assumed to

be of the following form:

εφ = δRE = φi − φo = αhpi (3.1)

Here p is the observed order of convergence, φi is the value of the variable φ on grid i, α is a constant

and hi is the typical cell spacing of grid i. At least three different grids are required to estimate εφ.

However, to avoid the influence of numerical noise on the solution, more than three grids can be used.

When this happens, the error estimation is performed in the least-squares sense. Equation 3.1 applies

only to cases that exhibit monotonic convergence. For the remaining cases, alternative expansions

for the error are used. When presenting the results, the monotonic cases are indicated by the value

obtained for the observed order of convergence, while the non monotonic cases are indicated by NM.

The final objective is to estimate the uncertainty of a given quantity, Uφ, the interval which contains

the exact solution with 95 % coverage,

φi − Uφ ≤ φexact ≤ φi + Uφ (3.2)

The estimation of the uncertainty is done considering the least-squares adjustment, and is comple-

mented by a safety factor based on the reliability of the adjustment. This safety factor possesses high

values for the non-monotonic cases resulting in high, conservative values for the estimated uncertainty.

The complete procedure is described in [53].

3.3 Turbulence Decay in Uniform Flow

As has been previously pointed out on subsection 2.4.2, turbulence decay has a significant influence

over the behaviour of the turbulence quantities, which translates in a large impact when using the γ−Reθmodel. For calculations using only the k− ω SST model, the effect is hidden, as transition usually starts

at very low Reynolds numbers.

This phenomenon has been discussed before [49] , and can be shown mathematically as the k and

ω equations in an uniform incoming flow, assuming a low value for the viscosity ratio (smaller than 1)

simplify to:

U∂k

∂x= −β∗kω ⇒ k = kin(1 + β(x− xinωin))−β

∗/β (3.3)

U∂ω

∂x= −βω2 ⇒ ω = ωin(1 + β(x− xinωin))−1 (3.4)

The decrease in k and ω is much more pronounced than in the eddy viscosity, as the former can drop

over one order of magnitude. Additionally, increasing µt causes the decay of the remaining variables to

be much slower, but too high values of µt compromise the laminar part of the solution. For applications

where the flow can be considered as fully turbulent, this would not be an issue, but the decay proves to

21

be important to control the onset of transition, which implies that the laminar part may not be negligible.

Some solutions have been proposed to deal with this issue [49] such as the introduction of weak

source terms, that become negligible in the boundary layer, and establish what are referred to as ambient

values for the turbulence variables, which behave as a bounded minimum for the uniform flow areas.

One other option, which was used in some of the calculations for this thesis, is to deactivate the

destruction terms in the turbulence model equations, preventing the decay in the regions where the x

coordinate is lower than a certain value, xinlet. This approach suffers from the drawback that it may

not be practical for complex geometries, and leads to one more variable that must be decided upon.

Nevertheless, this was the used approach and was first tested on the flat plate, leading to a much higher

sensitivity to the eddy viscosity than before, as can be seen from figure 3.2. This method is further

referred to as frozen decay.

Rex

Cf

103

104 105 1060

2

4

6

8

10

LCTM - t / = 1LCTM - t / = 6LCTM - t / = 12T3A

Tu = 3.3%

Rex

Cf

103

104 105 1060

2

4

6

8

10

LCTM - t / = 1LCTM - t / = 6LCTM - t / = 12T3A

Tu = 3.3%

(a) xinlet = −0.01 (b) xinlet = −0.001

Figure 3.2: Local skin friction for varying inlet viscosity ratios with frozen decay.

3.4 Tested Conditions

This section pretends to give a global overview on the flow conditions that were used for each airfoil.

Different inlet turbulence conditions and Reynolds numbers were tested for each case. Additionally,

initial calculations were performed on the Eppler 387 airfoil for a range of angles of attack to make a

brief study on the influence of iterative convergence, i.e, stopping criterion, as well as the effect of the

size of the computational domain. The choices for numerical discretization schemes are indicated in

Appendix A.

The iterative convergence study was performed separately for the SST and LCTM models. The

stopping criterion always considered the L∞ normalized residuals, and was varied from 10−2 to 10−8 for

the SST calculations, and from 10−2 to 10−6 for the LCTM calculations.

As for the domain size, three different domains were tested, specified in table 3.1.

22

Domain Inlet distance Outlet Distance Top/BottomSD 6 12 6MD 12 24 12LD 24 48 24

Table 3.1: Distance to each boundary for each domain

The large domain (LD) grids were built as an extension of the medium domain (MD), as to preserve

the grid point density. The grids of the small domain however, had the same number of points as the

medium domain.

3.4.1 NACA 0012

For the NACA 0012 calculations, the Reynolds number is 2.88×106. The inlet turbulence intensity was

set to 1% for all cases, resulting in k = 1.5 × 10−4U2∞. Three different combinations for the viscosity

ratio and the distance until which the turbulence variables remained unchanged were tested. These

conditions are expressed in table 3.2.

Case µt

µ xinletA 1.65 -0.04B 1.65 -1C 0.5 -0.04

Table 3.2: Inlet turbulence conditions tested for the NACA 0012 airfoil

The value for the viscosity ratio for the A case was selected to match the available experimental data

for the drag coefficient at the zero lift angle. Only the LCTM calculations were performed with frozen

decay - free decay occurred for the SST solution. Five different angles of attack were calculated: 0o,

2o, 4o, 6o and 8o. The Reynolds Number and selected angles of attack were chosen based on available

experimental data for the pressure coefficient distributions, as well as lift and drag coefficients. [45, 46].

3.4.2 NACA 66-018

The Reynolds number for this airfoil is 3 × 106. Only one set of turbulence conditions was considered,

which corresponds to case A of the NACA 0012 conditions: turbulence intensity set to 1%, µt

µ = 1.65

and xinlet = −0.04. Both the LCTM and SST calculations had frozen decay. For the SST case, not much

difference in the results is to be expected, but it helped the iterative convergence. As before, the angles

of attack chosen were based on experimental data against which the results could be compared [43] -

0o, 2o, 3o and 6o.

3.4.3 Eppler 387

Two different Reynolds numbers are calculated for the Eppler 387 airfoil: 1×105 and 3×105. Free decay

occurred for both the SST and LCTM cases, and the inlet turbulence conditions were the same for either

Reynolds number: turbulence intensity set to 1%, µt

µ = 0.003. The decay was not frozen for this case,

23

as unlike the previous two airfoils in which transition occurs naturally, the Eppler 387 forms a laminar

separation bubble for the mentioned Reynolds numbers and separation-induced transition takes place.

For the lower Reynolds number, the following angles of attack were calculated: -1o, 0o, 1o, 2o, 4o, 6, and

7o. For Re = 3 × 105 the angles of attack were: -2o, -1o, 0o, 1o, 3 and 7o. Both cases were based on

wind-tunnel data [10, 44, 54–56].

3.5 Iterative Convergence

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

1E-2 - US1E-2 - LS1E-3 - US1E-3 - LS1E-4 - US1E-4 - LS1E-5 - US1E-5 - LS1E-8 - US1E-8 - LS

x/c

Cf ×

103

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

1E-3 - US1E-3 - LS1E-4 - US1E-4 - LS1E-5 - US1E-5 - LS1E-8 - US1E-8 - LS

(a) Pressure coefficient (b) Skin friction

Figure 3.3: Iterative convergence effect with the SST model on local quantities for α = 0o. US - Uppersurface, LS - Lower Surface

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

1E-2 - US1E-2 - LS1E-3 - US1E-3 - LS1E-4 - US1E-4 - LS1E-5 - US1E-5 - LS1E-6 - US1E-6 - LS

x/c

Cf

103

0 0.2 0.4 0.6 0.8 1-5

0

5

10

15

20 1E-3 - US1E-3 - LS1E-4 - US1E-4 - LS1E-5 - US1E-5 - LS1E-6 - US1E-6 - LS


Figure 3.4: Iterative convergence effect with the γ−Reθ model on local quantities for α = 0o. US - Uppersurface, LS - Lower Surface

Figures 3.3 and 3.4 show the effect of the iterative convergence criterion on the pressure and skin

24

friction distributions for the finest grid for the Eppler 387 airfoil at an angle of attack of 0o and a Reynolds

number of 3 × 105 . For the pressure distribution, for both the lower and the upper surfaces it can be

seen that the behaviour exhibited is similar for all cases, although the less demanding criteria show

a quantitative difference. The skin friction distribution obtained for a convergence criterion of 10−2 is

omitted since, similarly to the one obtained for 10−3, it differs significantly from the remaining curves.

For the calculations using the SST model, the results for the stopping criterion of 10−6 and 10−7 are not

shown since they overlap (visually) with those for 10−8. The behaviour exhibited by both the skin friction

and the pressure distribution is the same for the remaining angles of attack, and exhibits the same trend

for both the SST and the γ −Reθ models.

α [º]

Cl

-4 -2 0 2 4 6 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1E-21E-31E-41E-51E-8

Cl

Cd ×

102

0 0.2 0.4 0.6 0.8 1 1.2

1.2

1.4

1.6

1.81E-41E-51E-61E-71E-8

(a) Lift coefficient (b) Drag coefficient

Figure 3.5: Iterative convergence effect with the SST model on aerodynamic forces

[º]

Cl

-2 0 2 4 6

0.2

0.4

0.6

0.8

1

1.21E-21E-31E-41E-5

Cl

Cd

102

0.2 0.4 0.6 0.8 1 1.20.5

1

1.5

2

2.5

3 1E-31E-41E-5


Figure 3.6: Iterative convergence effect with the γ −Reθ model on aerodynamic forces

The Cl versus α curves show the same trend as before: only when using a convergence criterion

25

of 10−5 (or one more restrictive) do the results show a negligible variation. The drag polar is shown in

figure 3.5 (b) and figure 3.6 (b). For the drag coefficient, a sufficiently converged solution only appears

to have been obtained for a limit of 10−6. The curves corresponding to the two most relaxed criteria

are not shown as the difference to the remaining cases is tremendous. This is evidenced in table 3.3,

which shows the error in both lift and drag for each stopping criteria, relative to the most demanding one

(maximum residual lower than 10−8) as a reference value.

Stopping Criteria 10−2 10−3 10−4 10−5 10−6 10−7

Cd − SST 2037.8% 300.5% 1.9% 1.3% 0.2% 0.01%Cl − SST 58.2% 21.4% 4.7% 0.4% 0.05% 0.004%

Cd − LCTM 1985% 158.4% 1.51% 1.17% - -Cl − LCTM 56.4% 0.70% 4.90% 0.10% - -

Table 3.3: Difference (in %) in the lift and drag coefficient components with different stopping criteria forα = 0o.

Taking into account the curves presented, along with the data from the table, using an iterative

convergence stopping criterion of 10−6 seems to yield reliable results, with an error lower than 1%.

As such, this will be the criterion used for the SST calculations. When using the transition model, the

maximum residual will be set to 10−5. Although this results in a loss of accuracy, the convergence

properties exhibited by the model justify this change: very often extremely low relaxations have to be

used for the transition model to achieve convergence, which results in long computation times: a whole

set of grids for only one angle of attack took over 5 days when using a stopping criterion of 10−6. This

resulted in over 600,000 iterations, using 16 processors.

3.6 Domain Influence

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1MD - USMD - LSSD - USSD - LSLD - USLD - LS

x/c

Cf ×

103

0 0.2 0.4 0.6 0.8 10

5

10

15

20MD - USMD - LSSD - USSD - LSLD - USLD - LS


Figure 3.7: Domain size influence on local quantities. US - Upper surface, LS - Lower Surface

Figure 3.7 shows the influence of the domain size on the pressure coefficient and skin friction distri-

26

butions for the Eppler 387 airfoil at a Reynolds number of 3× 105 and α = 0o. The former shows a slight

increase in the pressure curve with the increase of the domain, which is to be expected: the increase

of the domain size causes slightly lower values for the local velocity near the airfoil, which results in an

higher value for the pressure. However, the changes do not appear to be significant. For the skin friction

distribution, transition occurs earlier in the smaller domain, for the same reason: an increase in velocity

can be considered as an increase in the Reynolds number, which triggers transition earlier. Again, the

difference between each case is not very significant.

α [º]

Cl

-4 -2 0 2 4 6 80

0.2

0.4

0.6

0.8

1

1.2

MD SDLD

Cl

Cd ×

102

0 0.2 0.4 0.6 0.8 1 1.21

1.2

1.4

1.6

1.8

2MDSDLD


Figure 3.8: Domain size influence on lift and drag coefficients

The effect of the domain size on the lift coefficient is negligible, as shown by figure 3.8. However,

for the drag coefficient, the behaviour is quite different: for high angles of attack there is a pronounced

difference between each domain - the small and large domains differ by 10 % - while for low angles of

attack the difference is considerably smaller.

Considering the results obtained, the domain used for the remaining calculations will be the medium

domain, as the small difference when compared to the large domain does not justify the increase in size

and grid points which would result in higher computation times.

27

Chapter 4

Results

4.1 NACA 0012

Figure 4.1 shows the pressure coefficient and skin friction distribution for all cases tested for α = 0o.

The SST calculations exhibit transition right after the leading edge of the airfoil, much earlier than the

remaining cases. Regarding the LCTM solutions, conditions A are the first to transition to turbulent

flow, as would be expected. For this angle of attack it would appear that the effect of the xinlet is of

higher importance than the inlet eddy viscosity, since transition occurs later for conditions C. However,

for the remaining angles, as can be seen from figure 4.2, conditions B exhibit transition in the lower

surface earlier than case C, while the order remains the same as before for the upper surface. This

indicates that there is also influence from the pressure gradient. The pressure coefficient curves do not

show significant differences, apart from a slight local increase in pressure in the region where transition

occurs for each case, and all conditions show good agreement with the experimental points. The same

occurs for the remaining angles, plotted in figures 4.2 and 4.3.

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

SSTLCTM - ALCTM - BLCTM - CExperimental

x/c

Cf

103

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5

6

7

SSTLCTM - ALCTM - BLCTM - C


Figure 4.1: Pressure coefficient and skin friction distributions for the NACA 0012 airfoil and α = 0o

29

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1SST - USSST - LSLCTM - US - ALCTM - LS - ALCTM - US - BLCTM - LS - BLCTM - US - CLCTM - LS - CExperimental

x/c

Cf

103

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10SST - USSST - LSLCTM - US - ALCTM - LS - ALCTM - US - BLCTM - LS - BLCTM - US - CLCTM - LS - C

(a) Pressure coefficient, α = 2o (b) Skin friction, α = 2o

x/c

-Cp

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

1

SST - USSST - LSLCTM - US - ALCTM - LS - ALCTM - US - BLCTM - LS - BLCTM - US - CLCTM - LS - CExperimental

x/c

Cf

103

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10 SST - USSST - LSLCTM - US - ALCTM - LS - ALCTM - US - BLCTM - LS - BLCTM - US - CLCTM - LS - C

(c) Pressure coefficient, α = 4o (d) Skin friction, α = 4o

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

1.5

2

2.5SST - USSST - LSLCTM - US - ALCTM - LS - ALCTM - US - BLCTM - LS - BLCTM - US - CLCTM - LS - CExperimental

x/c

Cf

103

0 0.2 0.4 0.6 0.8 1-2

0

2

4

6

8

10

12

14

16SST - USSST - LSLCTM - US - ALCTM - LS - ALCTM - US - BLCTM - LS - BLCTM - US - CLCTM - LS - CExperimental

(e) Pressure coefficient, α = 6o (f) Skin friction, α = 6o

Figure 4.2: Pressure coefficient and skin friction distributions for the NACA 0012 airfoil and α = 2o,α = 4o and α = 6o. US - Upper surface, LS - Lower Surface

30

x/c

-Cp

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

SST - USSST - LSLCTM - US - ALCTM - LS - ALCTM - US - BLCTM - LS - BExperimental

x/c

Cf

103

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10SST - USSST - LSLCTM - US - ALCTM - LS - ALCTM - US - BLCTM - LS - B


Figure 4.3: Pressure coefficient and skin friction distributions for the NACA 0012 airfoil and α = 8o. US- Upper surface, LS - Lower Surface

The increase of the angle of attack causes transition to move downstream in the lower surface and

shift towards the leading edge in the upper surface. However, it becomes evident that conditions B do

not move as much as the others, particularly in the lower surface. In fact for an angle of attack of 6o,

transition in the lower surface occurs first for case B and only later for case A which goes against what

would be expected since the turbulence quantities should remain higher until a smaller distance to the

airfoil. For this same angle, the case with lower eddy viscosity suffers from laminar separation and only

then transition occurs. It is also clear a small distinction between the skin friction curve in the upper

surface between each case, most likely caused by the different transition locations.

Finally for the highest angle, figure 4.3, case C is not exhibited as laminar separation took place

without reattachment. The skin friction distribution again displays the coupling between the pressure

gradient and the effect of xinlet: in the lower surface, transition starts right near the trailing edge for case

A and the flow leaves the airfoil before it is completed, while in case B fully turbulent flow is still obtained

at the trailing edge.

Figure 4.4 shows the numerical and experimental locations of transition for both the upper and

lower surfaces. The numerical location for the start/end of transition was obtained by locating the mini-

mum/maximum of the skin friction curve and fitting a second order polynomial. The minimum/maximum

of the polynomial was then considered as the intended location. This procedure meant to minimize the

error that would occur if the grid point was used, since the different grids do not have the same grid

points. The baseline conditions (A) predict transition earlier than the experimental location for the up-

per surface, but show the best agreement out of the three alternatives for the lower surface, while also

presenting the smallest uncertainty.

Conditions B consistently predict transition further downstream than the experimental location for

the upper surface, while presenting reasonable accuracy on the lower side. The uncertainties for these

conditions are, however, in general much higher than for case A. Case C seems to be the one that has

31

x/c

α [º

]

-0.2 0 0.2 0.4 0.6 0.8-2

0

2

4

6

8

10

ExperimentalA - UpperB - UpperC - Upper

x/c

α [º

]

0.4 0.6 0.8 1 1.2-2

0

2

4

6

8

10

ExperimentalA - LowerB - LowerC - Lower

(a) Upper surface (b) Lower surface

Figure 4.4: Transition chordwise location for the NACA 0012 airfoil

xtrans−begin U(xtrans−begin)α [o] Surface A B C A B C

0 - 0.26 0.44 0.39 0.08 0.05 0.07Upper 0.08 0.29 0.22 0.02 0.06 0.122Lower 0.45 0.54 0.54 0.01 0.05 0.13Upper 0.03 0.15 0.08 0.02 0.02 0.054Lower 0.58 0.62 0.73 0.08 0.14 0.09Upper 0.02 0.06 0.04 0.006 0.07 0.0046Lower 0.74 0.71 0.97 0.007 0.02 0.03Upper 0.01 0.02 - 0.001 0.007 -8Lower 0.97 0.79 - 0.02 0.02 -

Table 4.1: Transition start chordwise location and uncertainty estimate for the NACA 0012 airfoil.

xtrans−end U(xtrans−end)α [o] Surface A B C A B C

0 - 0.41 0.62 0.59 0.08 0.02 0.01Upper 0.13 0.45 0.39 0.02 0.01 0.0072Lower 0.64 0.75 0.76 0.007 0.04 0.05Upper 0.07 0.29 0.18 0.04 0.08 0.194Lower 0.79 0.83 0.91 0.004 0.15 0.01Upper 0.04 0.14 0.10 0.007 0.006 0.026Lower 0.93 0.91 0.995 0.004 0.03 0.001Upper 0.03 0.04 - 0.001 0.007 -8Lower - 0.98 - 0.02 0.04 -

Table 4.2: Transition end chordwise location and uncertainty estimate for the NACA 0012 airfoil.

32

hi/h1

Cd

,f

103

0 0.5 1 1.5 26.5

7

7.5

8

8.5

= 0º - SSTp= 1.61, U= 0.54%

= 2º - SSTp= 1.56, U= 0.62%

= 4º - SSTp= 1.53, U= 0.75%

= 6º - SSTp= 1.49, U= 0.96%

= 8º - SSTp= 1.47, U= 1.3%

hi/h1

Cd

,f

103

0 0.5 1 1.5 24

4.5

5

5.5

6

6.5

7

7.5

8

= 0º - LCTMNM, U= 14.9%

= 2º - LCTMNM, U= 0.22%

= 4º - LCTMNM, U= 0.68%

= 6º - LCTMp= 1.14, U= 1.3%

= 8º - LCTMp= 1.56, U= 1.1%

(a) SST (b) LCTM - A

hi/h1

Cd

,f

103

0 0.5 1 1.5 23

4

5

6

7

= 0º - LCTMNM, U= 3.1%

= 2º - LCTMNM, U= 2.9%

= 4º - LCTMNM, U= 31.0%

= 6º - LCTMNM, U= 3.5%

= 8º - LCTMNM, U= 1.6%

hi/h1

Cd

,f

103

0 0.5 1 1.5 24

4.5

5

5.5

6 = 0º - LCTM

NM, U= 3.9% = 2º - LCTM

p= 0.55, U= 0.62% = 4º - LCTM

NM, U= 11.4% = 6º - LCTM

NM, U= 4.4%

(c) LCTM - B (d) LCTM - C

Figure 4.5: Friction drag uncertainty estimate for the NACA 0012 airfoil.

the most accurate prediction for the upper surface, while transition occurs too late on the lower surface.

It is also the case that presents the highest uncertainty for most angles.

Nevertheless, for all three cases the numerical uncertainty is quite high, as evidenced by Tables 4.1

and 4.2, which present both the location as well as the estimated uncertainty for the start and end of

transition. Although the grid dependence is not exhibited here, most cases presented non-monotonic

behaviour, which justifies the high values for the uncertainty. This occurs because the coarsest grids do

not have enough grid points in the streamwise direction to capture transition correctly.

The uncertainty estimation for the friction drag is presented in figure 4.5. The SST calculations

exhibit the highest values, due to higher extent of turbulent flow, decreasing with the increase of the

angle of attack, due to transition moving slightly upstream in the lower surface. The LCTM calculations

all predict lower values for the friction drag than those of the SST model, which is to be expected since

a considerable part of the flow is laminar. The change of the friction drag with the angle of attack is not

as simple as before, since there are two opposing effects: the contribution from the upper surface tends

33

to increase since transition moves upstream, while the reverse occurs in the lower surface. The change

in total drag depends on which effect is the more relevant, which also depends on the conditions tested.

As an example, for case A the increase of α from 2o to 4o causes a decrease in the friction drag,

while the same change in the angle of attack for conditions B results on higher friction drag. In fact, case

B shows the same trend as the SST which is consistent with what was mentioned before, regarding

transition in the lower surface shifting much less with changes in the angle of attack than the other

conditions.

Overall, the numerical uncertainties are low, although the LCTM calculations present slightly higher

values. This occurs due to the same reason as the high numerical uncertainty on the transition location:

the coarsest grid does not have enough grid points to capture transition accurately, which causes non-

monotonic behaviour, unlike with the SST model.

The pressure drag has similar values for the SST and LCTM calculations, which is to be expected

hi/h1

Cd

,p

103

0 0.5 1 1.5 20

2

4

6

8

10 = 0º - SST

NM, U= 36.1% = 2º - SST

NM, U= 35.2% = 4º - SST

NM, U= 24.3% = 6º - SST

NM, U= 3.1% = 8º - SST

NM, U= 4.7%

hi/h1

Cd

,p

103

0 0.5 1 1.5 2-2

0

2

4

6

8

10

= 0º - LCTMNM, U= 96.2%

= 2º - LCTMNM, U= 150%

= 4º - LCTMNM, U= 3.0%

= 6º - LCTMNM, U= 25.3%

= 8º - LCTMNM, U= 3.8%


hi/h1

Cd

,p

103

0 0.5 1 1.5 20

2

4

6

8

= 0º - LCTMNM, U= 31.0%

= 2º - LCTMNM, U= 59.0%

= 4º - LCTM NM, U= 5.5%

= 6º - LCTMNM, U= 5.1%

= 8º - LCTMNM, U= 5.2%

hi/h1

Cd

,p

103

0 0.5 1 1.5 20

1

2

3

4

5

= 0º - LCTMNM, U= 25.1%

= 2º - LCTMNM, U= 1.8%

= 4º - LCTMp= 1.02, U= 6.0%

= 6º - LCTMNM, U= 21.0%


Figure 4.6: Pressure drag uncertainty estimate for the NACA 0012 airfoil.

34

considering the overlapping pressure distributions. Both models cause the same behaviour - increase in

the pressure drag along with increasing angle of attack, as seen from figure 4.6. The value is particularly

high for α = 8o, in which trailing edge separation is imminent. There is no significant difference between

different turbulence conditions for the LCTM solutions, although case B exhibits slightly lower values

than the other alternatives. The numerical uncertainties are much higher now, and exhibit large values

mostly for the lower angles.

The pressure component of lift is shown in figure 4.7. All cases show good convergence properties.

While no significant difference is found when the transition model is employed, the SST calculations

exhibit slightly higher values for the lift coefficient. This is also evidenced by figure 4.8 (a) and table 4.3.

The values obtained for the lift coefficient for α = 0o are a consequence of the numerical error in the

solution.

The drag polar is shown in figure 4.8 (b), with the estimated numerical uncertainty. The SST model

hi/h1

Cl,p

0 0.5 1 1.5 2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6 = 0º - SST

p= 1.48 = 2º - SST

NM, U= 1.4% = 4º - SST

NM, U= 1.2% = 6º - SST

p= 0.53, U= 0.57% = 8º - SST

p= 1.25, U= 0.26%

hi/h1

Cl,p

0 0.5 1 1.5 2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

= 0º - LCTMNM

= 2º - LCTMNM, U= 3.5%

= 4º - LCTMNM, U= 0.81%

= 6º - LCTMp= 0.55, U= 0.42%

= 8º - LCTMNM, U= 1.6%


hi/h1

Cl,p

0 0.5 1 1.5 2-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

= 0º - LCTMNM

= 2º - LCTMNM, U= 0.34%

= 4º - LCTMNM, U= 0.36%

= 6º - LCTMNM, U= 0.17%

= 8º - LCTMNM, U= 0.17%

hi/h1

Cl,p

0 0.5 1 1.5 2-0.2

0

0.2

0.4

0.6

0.8

1

1.2 = 0º - LCTMNM

= 2º - LCTMNM, U= 0.25%

= 4º - LCTMNM, U= 2.9%

= 6º - LCTMNM, U= 0.71%


Figure 4.7: Pressure lift uncertainty estimate for the NACA 0012 airfoil.

35

[º]

Cl

-2 0 2 4 6 8 10-0.2

0

0.2

0.4

0.6

0.8

1

LCTM - ALCTM - BLCTM - CSSTExperimental - Ladson

Cl

Cd ×

103

-0.2 0 0.2 0.4 0.6 0.8 14

6

8

10

12

14SSTLCTM - ALCTM - BLCTM - CExperimental - GregoryExperimental - Ladson


Figure 4.8: Total lift and drag forces for the NACA 0012 airfoil.

Cl U(Cl) [%]α SST LCTM (A) LCTM (B) LCTM (C) SST LCTM (A) LCTM (B) LCTM (C)0o 6.2×10−7 1.3×10−7 -1.4×10−6 -1.5×10−8 - - - -2o 0.218 0.205 0.216 0.212 1.4 3.5 0.34 0.254o 0.434 0.420 0.432 0.421 1.2 0.79 0.36 2.96o 0.648 0.633 0.644 0.632 0.53 0.42 0.17 0.708o 0.857 0.840 0.847 - 0.25 1.5 0.17 -

Table 4.3: Lift force uncertainty estimate for the NACA 0012 airfoil.

Cd × 103 U(Cd) [%]α SST LCTM (A) LCTM (B) LCTM (C) SST LCTM (A) LCTM (B) LCTM (C)0o 9.19 6.83 4.97 5.27 7.5 27.5 10.8 6.92o 9.39 7.46 5.36 5.58 8.3 31.9 17.2 2.14o 9.97 7.95 6.31 6.65 7.7 1.1 17.8 12.36o 11.0 8.95 7.94 7.92 6.7 12.0 6.9 9.068o 12.6 10.4 10.2 - 7.7 11.0 1.9 -

Table 4.4: Drag force uncertainty estimate for the NACA 0012 airfoil.

presents the highest value, followed by conditions A of the LCTM calculations. It is important to remem-

ber that the accuracy shown by case A is not fortuitous, since the turbulence conditions were chosen in

order to match the drag for α = 0o. As expected, cases B and C exhibit lower values, since transition is

delayed, and the difference to case A is quite significant. All cases present high numerical uncertainties,

with the SST model exhibiting the lowest, which can also be seen from table 4.4.

The main conclusions from the analysis of this airfoil are the following:

• The inclusion of the transition model does not affect the pressure distribution in the laminar and

turbulent regimes. Instead, small changes only occur in the transition region. No conclusion can

be drawn regarding separation bubbles, as they did not occur for this airfoil

• Calibrating the inlet turbulence variables considering the drag for one angle of attack produced

good results when calculating other angles of attack for the same conditions, both for the drag as

36

well as the transition regions.

• It was shown that without this calibration, significant differences in the results could be obtained,

as changing the inlet turbulence conditions resulted in considerably different drag predictions.

• The calculations with the transition model commonly exhibited higher numerical uncertainties. It is

expected that this occurs due a lack of grid resolution, especially in the areas where transition takes

place, for the coarsest grids. This suggests finer meshes are required when transition modelling is

intended, particularly in the transition region.

37

4.2 NACA 66-018

Figure 4.9 shows the pressure coefficient distribution for the 4 tested angles of attack. Overall, the

results obtained are slightly below the available experimental data, but exhibit the same trend. Including

the transition model seems to affect very little the pressure coefficient: the only remarkable difference

occurs around 60 % of the chord for angles of attack higher than 0o, when the adverse pressure gradient

starts and the calculations with the transition model exhibit a localized decrease in the pressure for the

lower surface, owning to laminar separation.

This is supported by figure 4.10 which presents the skin friction distribution for each angle of attack.

Here, the differences between the SST and LCTM calculations are significant: fully turbulent flow occurs

in both the upper and lower surface of the airfoil when using the SST model, as is to be expected. When

the transition model is used, a significant portion of the airfoil is in the laminar regime: transition takes

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1SSTLCTMExperimental

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

SST - USSST - LSLCTM - USLCTM - LSExperimental

(a) α = 0o (b) α = 2o

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

SST - USSST - LSLCTM - USLCTM - LSExperimental

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

1.5

2

2.5SST - USSST - LSLCTM - USLCTM - LS

(c) α = 3o (d) α = 6o

Figure 4.9: Pressure coefficient distribution for the NACA 66-018 airfoil. US - Upper surface, LS - LowerSurface

38

x/c

Cf

103

0 0.2 0.4 0.6 0.8 10

2

4

6

8SSTLCTM

x/c

Cf

103

0 0.2 0.4 0.6 0.8 1-2

0

2

4

6

8

10

SST - USSST - LSLCTM - USLCTM - LS

(a) α = 0o (b) α = 2o

x/c

Cf ×

103

0 0.2 0.4 0.6 0.8 1-2

0

2

4

6

8

10

12


x/c

Cf ×

103

0 0.2 0.4 0.6 0.8 1-5

0

5

10

15SST - USSST - LSLCTM - USLCTM - LS

(c) α = 3o (d) α = 6o

Figure 4.10: Skin friction distribution for the NACA 66-018 airfoil. US - Upper surface, LS - Lower Surface

place at around 60% of the chord for α = 0o and moves upstream with the increase of the angle of

attack in the upper surface. For the lower surface, the increase of the angle of attack causes a laminar

separation bubble to appear, which triggers transition and subsequent reattachment. Increasing α shifts

the bubble slightly downstream: at α = 6o the bubble is located at around 70 % of the chord.

The available experimental data for transition are not in accordance with the obtained results - Figure

4.11. For α = 0o the predicted location of transition matches well with the experimental measurements,

but there is no flow separation. For α = 2o the predicted location of transition on the upper surface

shifts to 10-20% of the chord of the airfoil, while the experimental data show almost no change from the

previous angle of attack. Finally, for α = 6o, transition is located near the leading edge of the airfoil,

although the numerical prediction is still slightly upstream than that of the experimental data. Table 4.5

presents the chordwise location as well as the estimated numerical uncertainties, which exhibit high

values in some cases. The numerical location for the transition points was obtained using the same

procedure as for the NACA 0012 airfoil.

39

x/c

[º]

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-1

0

1

2

3

4

5

6

7

Laminar Separation - ExperimentalTransition - ExperimentalTransition start - LCTMTransition end - LCTM

Figure 4.11: Transition chordwise location for the NACA 66-018 airfoil

α [o] xtrans−begin U(xtrans−begin) xtrans−end U(xtrans−end)0 0.65 0.011 0.70 0.0102 0.12 0.394 0.21 0.2973 0.02 0.008 0.09 0.0526 0.01 0.006 0.03 0.014

Table 4.5: Transition chordwise location and uncertainty estimate for the upper surface of the NACA66-018 airfoil.

Figure 4.12 shows the uncertainty estimation for the pressure and friction components of the drag

force, as well as the pressure contribution for the lift force. The SST calculations provide a higher value

for the friction drag, consequence of the fully turbulent flow condition which decreases slightly with the

increase of the angle of attack, with low uncertainties around 1%. Using the transition model results in

lower friction drag, increasing along with the angle of attack due to the upstream movement of transition

on the upper surface, and significantly higher uncertainties are obtained. The pressure drag exhibits

higher values when the transition model is used. High numerical uncertainty is obtained for the two

models - as high as 78% for the LCTM calculations and 38% for the SST predictions. In both cases, this

results due to the non-monotonic behaviour that is displayed by all angles of attack. Finally, the pressure

contribution for the lift coefficient is nearly the same for both models, but slightly lower when transition

is included. In this case, the uncertainties for both cases are very low, with the exception of the case

where α = 0o, which should result in zero lift since the airfoil is symmetrical.

Focusing now on the complete lift coefficient, calculations with the transition model predict slightly

lower lift, most likely due to the small bubble in the lower surface, which results in lower pressure than

that obtained with the SST model. In general, both models overpredict the lift, although the LCTM results

are closer to the experimental data.

As should be expected, the drag coefficient obtained with the transition model is significantly lower

than that of the SST model, due to the smaller friction component - figure 4.12 (e). However, the shape

of the drag bucket, typical of laminar airfoils, is not correctly captured. Comparing with experimental

40

hi/h1

0 0.5 1 1.5 23

4

5

6

7

8

9

10

= 0º - SSTp= 1.12, U= 1.31%

= 2º - SSTp= 1.11, U= 1.38%

= 3º - SSTp= 1.16, U= 1.35%

= 6º - SSTp= 1.36, U= 1.29%

Cd

,f

103

0 0.5 1 1.5 23

4

5

6

7

8

9

10

= 0º - LCTMNM, U= 5.43%

= 2º - LCTMNM, U= 16.8%

= 3º - LCTMNM, U= 1.0%

= 6º - LCTMp= 1.88, U= 0.86%

hi/h1

0 0.5 1 1.5 20

2

4

6

8

10

= 0º - SSTNM, U= 38.2%

= 2º - SSTNM, U= 37.1%

= 3º - SSTNM, U= 35.8%

= 6º - SSTNM, U= 26.6%

(a) Friction drag (b) Pressure drag

hi/h1

Cl,p

0 0.5 1 1.5 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 0º - SSTp= 1.20, U= 141.8%

= 2º - SSTNM, U= 0.68%

= 3º - SSTNM, U= 0.76%

= 6º - SSTNM, U= 0.15%

0 0.5 1 1.5 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 0º - LCTMNM, U= 253.8%

= 2º - LCTMNM, U= 12.8%

= 3º - LCTMNM, U= 0.71%

= 6º - LCTMNM, U= 0.11%

(c) Pressure lift

[º]

Cl

0 2 4 6

0

0.2

0.4

0.6

ExperimentalSST - Re

Cl

Cd

102

-0.2 0 0.2 0.4 0.6 0.80

0.5

1

1.5

2ExperimentalSSTLCTM

(d) Lift coefficient (e) Drag Polar

Figure 4.12: Uncertainty estimate for the lift and drag forces for the NACA 66-018 airfoil

41

SST LCTMα Cl U(Cl)[%] Cd × 102 U(Cd)[%] Cl U(Cl)[%] Cd × 102 U(Cd)[%]0o 2.10x10−6 - 1.08 13.1 1.18x10−6 - 0.452 21.62o 0.212 0.676 1.10 13.5 0.194 12.7 0.760 5.723o 0.317 0.762 1.13 13.8 0.298 0.691 0.860 7.696o 0.621 0.156 1.30 13.3 0.613 0.115 1.10 9.84

Table 4.6: Uncertainty estimate for the total lift and drag forces for the NACA 66-018 airfoil

results, the drag increases steadily with increasing angle of attack, instead of remaining constant over

the region of the bucket/low angles of attack - the drag is under-predicted for the zero lift condition and

for α = 6o and is over predicted for α = 2o and α = 3o. Table 4.6 shows the estimated numerical

uncertainty for the lift and drag forces as well as the values obtained in the finest grid.

The calculations for this airfoil provided the following conclusions:

• The model failed to capture the experimental trend for both the transition region as well as for the

drag coefficient.

• Once again, calculations with the transition model exhibited higher numerical uncertainties, due to

the non-monotonic convergence obtained when using all of the five grids.

• However, there are some doubts as to whether the simulated flow conditions actually correspond

to the experimental settings, suggested by the differences found in the pressure coefficient.

42

4.3 Eppler 387

4.3.1 Re = 3× 105

Figure 4.13 shows the pressure coefficient and skin friction distribution for the two lowest angles of

attack tested. For α = −2o, the LCTM calculations exhibit a laminar separation bubble on each side

of the airfoil, which is responsible for triggering transition, unlike when using the SST model in which

natural transition always takes place. Differences in the pressure distribution are only found near the

beginning of the adverse pressure gradient and in the separated flow zones. In these regions, the

pressure coefficient remains almost constant and undergoes a sharp increase when the flow reattaches.

This behaviour is only obtained when the transition model is used. The experimental data available

supports the existence of a bubble on the upper surface, although the numerical prediction seems to

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

1.5SST - USSST - LSLCTM - USLCTM - LSExperimental - MuellerExperimental - LTPT

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1 SST - USSST - LSLCTM - USLCTM - LSExperimental - MuellerExperimental - LTPT

(a) Pressure coefficient, α = −2o (b) Pressure coefficient, α = −1o

x/c

Cf

103

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10


x/c

Cf

103

0 0.2 0.4 0.6 0.8 1-5

0

5

10

15

20 SST - USSST - LSLCTM - USLCTM - LS

(c) Skin friction, α = −2o (d) Skin friction, α = −1o

Figure 4.13: Pressure coefficient and skin friction distributions for α = −2o and α = −1o for the Eppler387 airfoil at Re = 3× 105. US - Upper surface, LS - Lower Surface

43

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1SST - USSST - LSLCTM - USLCTM - LSExperimental - MuellerExperimental - LTPT

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

SST - USSST - LSLCTM - USLCTM - LSExperimental - MuellerExperimental - LTPT

(a) Pressure coefficient, α = 0o (b) Pressure coefficient, α = 1o

x/c

Cf

103

0 0.2 0.4 0.6 0.8 1

0

10


x/c

Cf

103

0 0.2 0.4 0.6 0.8 1-5

0

5

10

15


(c) Skin friction, α = 0o (d) Skin friction, α = 1o

Figure 4.14: Pressure coefficient and skin friction distributions for α = 0o and α = 1o for the Eppler 387airfoil at Re = 3× 105. US - Upper surface, LS - Lower Surface

reattach further downstream. Regarding the lower surface, although the LCTM calculations also suggest

separation-induced transition, such is not in accordance with the experimental data. Increasing the angle

of attack to α = −1o, causes the bubble on the lower surface to disappear, resulting in fully laminar flow,

while in the upper surface, the bubble shifts slightly upstream. In the SST calculations, transition moves

upstream in the upper surface and downstream in the lower surface, but most of the flow is still turbulent.

The pressure coefficient presents the same behaviour as before: significant differences are only found

in the bubble region, which is again predicted further downstream than the experimental results.

For α = 0o, α = 1o and α = 3o, the flow characteristics remain the same as for α = −1o: laminar flow

on the lower surface and a bubble on the upper surface which moves upstream with the increase of the

angle of attack for the LCTM calculations. The same movement occurs with the SST model, in which

natural transition takes place on both sides. In all cases, the laminar bubble is predicted downstream of

the experimental location.

44

x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1


x/c

-Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

1.5

2

2.5

3SST - USSST - LSLCTM - USLCTM - LSExperimental - MuellerExperimental - LTPT

(a) Pressure coefficient, α = 3o (b) Pressure coefficient, α = 7o

x/c

Cf

103

0 0.2 0.4 0.6 0.8 1

0

10

20


x/c

Cf

103

0 0.2 0.4 0.6 0.8 1-5

0

5

10


(c) Skin friction, α = 3o (d) Skin friction, α = 7o

Figure 4.15: Pressure coefficient and skin friction distributions for α = 3o and α = 7o for the Eppler 387airfoil at Re = 3× 105. US - Upper surface, LS - Lower Surface

Finally, for the highest angle of attack tested, α = 7o, the experimental data indicate the disappear-

ance of the bubble on the upper side - figure 4.15. However, it remains in the numerical prediction.

In all cases there seems to be a good agreement between the numerical and experimental pressure

coefficient distribution, although one set of experimental data [10] consistently presents slightly higher

pressure in the lower surface. This may be due to blockage effects, since the other set of points do not

exhibit the same behaviour.

Figure 4.16 shows the experimental and numerical location of the points of interest concerning transi-

tion and the separated flow region, with the respective numerical uncertainty. The separation/reattachment

locations were obtained by fitting a second order polynomial in the location where the skin friction be-

came negative/positive, and finding the zero of said polynomial. For the beginning of the transition

region, the same procedure was performed, but for the maximum that occurs before the sharp decrease

in the skin friction. The end of the transition was calculated in the same way, considering the maximum

45

x/c

[º]

0 0.2 0.4 0.6 0.8 1-3

-2

-1

0

1

2

3

4

5

6

7

8

Laminar Separation - ExperimentalTransition - ExperimentalReattachment - ExperimentalLaminar Separation - LCTMTransition start - LCTMTransition end - LCTMReattachment - LCTM

Figure 4.16: Transition chordwise location for the Eppler 387 airfoil at Re = 3× 105.

α [o] xsep xtrans−begin xtrans−end xreattach-2 0.52 0.76 0.90 0.82-1 0.50 0.72 0.87 0.790 0.48 0.69 0.84 0.761 0.46 0.66 0.81 0.733 0.42 0.60 0.75 0.677 0.31 0.47 0.58 0.52

Table 4.7: Transition chordwise location for the upper surface of the Eppler 387 airfoil at Re = 3× 105.

α [o] U(xsep) U(xtrans−begin) U(xtrans−end) U(xreattach)-2 0.002 0.010 0.062 0.007-1 0.003 0.012 0.021 0.0100 0.004 0.009 0.012 0.0111 0.004 0.005 0.013 0.0113 0.002 0.010 0.075 0.0097 0.007 0.017 0.015 0.013

Table 4.8: Transition chordwise location uncertainty estimate for the upper surface of the Eppler 387airfoil at Re = 3× 105.

of the skin friction after the flow had reattached. The values obtained are displayed in Table 4.7

For α = −2o, the position of laminar separation is predicted further upstream than the numerical mea-

surement, while reattachment is predicted quite accurately, resulting in a larger bubble for the numerical

solution. For the intermediate angles of attack, the end result is the same, under different circumstances:

the location of laminar separation for the LCTM calculations coincides with the experimental data, but

reattachment occurs later. Finally, for the largest angle of attack, as mentioned before, separation did

not occur in the experimental testing. The location of laminar separation exhibits the smallest numerical

uncertainty when compared to the other relevant points - Table 4.8. The onset of transition as well as

turbulent reattachment also exhibit a small uncertainty. The end of transition presents the largest error

bar, a consequence of non-monotonic behaviour caused by the coarsest grids.

The friction component of drag is presented in figure 4.17. The predictions obtained when using the

46

hi/h1

Cd

,f

103

0 0.5 1 1.5 28

8.5

9

9.5

10

10.5

11

= 1º - LCTM

1h + 2h2, U= 2.62%

= 3º - LCTMp= 1.91, U= 0.95%

= 7º - LCTMh2, U= 0.71%

0 0.5 1 1.5 28

8.5

9

9.5

10

10.5

11

= -2º - LCTMh2, U= 0.70% = -1º - LCTMh2, U= 0.47% = 0º - LCTM

1h + 2h2, U= 2.67%

0 0.5 1 1.5 28

8.5

9

9.5

10

10.5

11

= -2º - SSTNM, U= 0.59%

= -1º - SSTp= 1.88, U= 0.49%

= 0º - SSTp= 1.86, U= 0.40%

hi/h1

Cd

,f

103

0 0.5 1 1.5 28

8.5

9

9.5

10

10.5

11

= 1º - SSTp= 1.60, U= 0.43%

= 3º - SSTp= 1.55, U= 0.46%

= 7º - SSTNM, U= 0.61%

0 0.5 1 1.5 24

5

6

7

= -2º - SSTNM, U= 0.59%

= -1º - SSTp= 1.88, U= 0.49%

= 0º - SSTp= 1.86, U= 0.40%

hi/h1

Cd

,f

103

0 0.5 1 1.5 24

5

6

7

= 1º - SSTp= 1.60, U= 0.43%

= 3º - SSTp= 1.55, U= 0.46%

= 7º - SSTNM, U= 0.61%

hi/h1

Cd

,f

103

0 0.5 1 1.5 24

5

6

7 = 1º - LCTM

NM, U= 2.62% = 3º - LCTM

p= 1.91, U= 0.95% = 7º - LCTM

NM, U= 0.71%

0 0.5 1 1.5 24

5

6

7

= -2º - LCTMNM, U= 0.70%

= -1º - LCTMNM, U= 0.47%

= 0º - LCTMNM, U= 2.67%

(a) SST (b) LCTM

Figure 4.17: Friction drag uncertainty estimate for the Eppler 387 airfoil at Re = 3× 105.

transition model are always lower than when using the SST model, due to delayed transition and the

appearance of the laminar separation bubbles. Overall, the increase of the angle of attack causes the

friction drag to increase as well, with some exceptions. The first takes place for the SST model and

α = 7o, which possesses the lowest value for this model, as is due to transition moving downstream in

the lower surface. The second and third exceptions occur for the LCTM solutions, in which the highest

value occurs for the lowest angle and vice-versa. The high value for α = −2o is justified by the flow

undergoing transition on the lower surface at around 20 % of the chord, while for the remaining angles

the flow remains laminar. The low value for α = 7o is explained by the lower values for Cf in both

the laminar and turbulent part on the upper surface. The numerical uncertainties are very low for both

models, generally lower than 1%.

hi/h1

Cd

,p

0 0.5 1 1.5 20

2

4

6

8

10

= 1º - LCTMp= 1.96, U= 3.21%

= 3º - LCTMh, U= 24.4% = 7º - LCTM

1h + 2h2, U= 30.8%

0 0.5 1 1.5 20

2

4

6

8

10

= -2º - LCTMh2, U= 0.91% = -1º - LCTMh2, U= 2.17% = 0º - LCTMh2, U= 5.63%

0 0.5 1 1.5 20

2

4

6

8

10

= -2º - SSTNM, U= 7.60%

= -1º - SSTNM, U= 5.14%

= 0º - SSTNM, U= 2.82%

hi/h1

Cd

,p

103

0 0.5 1 1.5 20

2

4

6

8

10

= 1º - SSTNM, U= 1.19%

= 3º - SSTNM, U= 0.89%

= 7º - SSTNM, U= 3.27%

0 0.5 1 1.5 22

4

6

8

10

12

14

16

= -2º - SSTNM, U= 7.60%

= -1º - SSTNM, U= 5.14%

= 0º - SSTNM, U= 2.82%

hi/h1

Cd

,p

103

0 0.5 1 1.5 22

4

6

8

10

12

14

16

= 1º - SSTNM, U= 1.19%

= 3º - SSTNM, U= 0.89%

= 7º - SSTNM, U= 3.27%

0 0.5 1 1.5 22

4

6

8

10

12

14

16

= -2º - LCTMNM, U= 0.91%

= -1º - LCTMNM, U= 2.17%

= 0º - LCTMNM, U= 5.63%

hi/h1

Cd

,p

103

0 0.5 1 1.5 22

4

6

8

10

12

14

16

= 1º - LCTMp= 1.96, U= 3.21%

= 3º - LCTMNM, U= 24.4%

= 7º - LCTMNM, U= 30.8%

(a) SST (b) LCTM

Figure 4.18: Pressure drag uncertainty estimate for the Eppler 387 airfoil at Re = 3× 105.

47

The pressure drag, illustrated on Figure 4.18 exhibits higher values for the LCTM calculations, in-

creasing with the angle of attack - α = 7o presents the highest value for both models by a considerable

margin. The numerical uncertainties follow an erratic behaviour, ranging from 0.9 % to 30%. Like the

previous discussed airfoils, non-monotonic behaviour is displayed for most angles.

0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

= 1º - LCTMNM, U= 0.16%

= 3º - LCTMNM, U= 1.27%

= 7º - LCTMp= 0.89, U= 0.68%

hi/h1

Cl,p

0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

= -2º - LCTMNM, U= 5.42%

= -1º - LCTMNM, U= 1.96%

= 0º - LCTMNM, U= 3.93%

hi/h1

Cl,p

0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

= -2º - SSTp= 0.61, U= 2.41%

= -1º - SSTp= 0.57, U= 1.5%

= 0º - SSTNM, U= 1.94%

0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

= 1º - SSTp= 0.59, U= 0.89%

= 3º - SSTNM, U= 1.02%

= 7º - SSTNM, U= 0.88%

0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

= 1º - SSTp= 0.59, U= 0.89%

= 3º - SST

1h + 2h2, U= 1.02%

= 7º - SST

1h + 2h2, U= 0.88%

hi/h1

Cl,p

0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

= -2º - SSTp= 0.61, U= 2.41%

= -1º - SSTp= 0.57, U= 1.5%

= 0º - SST

1h + 2h2, U= 1.94%

0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

= 1º - LCTMNM, U= 0.16%

= 3º - LCTMNM, U= 1.27%

= 7º - LCTMp= 0.89, U= 0.68%

hi/h1

Cl,p

0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

= -2º - LCTMNM, U= 5.42%

= -1º - LCTMNM, U= 1.96%

= 0º - LCTMNM, U= 3.93%

(a) SST (b) LCTM

Figure 4.19: Pressure lift uncertainty estimate for the Eppler 387 airfoil at Re = 3× 105.

The pressure contribution for the lift is shown in figure 4.19. For a given angle of attack, slightly

higher values are obtained when using the transition model. The estimated numerical uncertainties are

smaller than 3% with only two exceptions, both occurring for the LCTM case.

Finally, figure 4.20 presents the total drag and lift. The prediction for the lift force matches well

with experimental data, although the LCTM values present slightly higher uncertainty and may seem to

overestimate the lift.

[º]

Cl

-4 -2 0 2 4 6 80

0.2

0.4

0.6

0.8

1

1.2MuellerLTPTLow Speed (V1)Low Speed (V3)Low Speed (V5)LCTMSST

Cl

Cd

102

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

Experimental - LTPTLow Speed (V1)Low Speed (V3)Low Speed (V5)LCTMSST

(a) Lift coefficient (b) Drag Polar

Figure 4.20: Lift and drag forces for the Eppler 387 airfoil at Re = 3× 105.

48

SST LCTMα Cl U(Cl)[%] Cd × 102 U(Cd)[%] Cl U(Cl)[%] Cd × 102 U(Cd)[%]

-2o 0.151 2.43 1.23 0.70 0.175 5.41 1.15 0.285-1o 0.260 1.55 1.21 2.07 0.273 1.96 0.854 1.320o 0.369 1.94 1.22 1.00 0.381 3.92 0.911 2.571o 0.478 0.89 1.24 1.51 0.490 0.156 0.973 3.153o 0.691 1.03 1.35 1.54 0.705 1.27 1.11 12.07o 1.09 0.88 1.84 0.99 1.26 0.688 1.43 19.9

Table 4.9: Lift and drag forces uncertainty estimate for the Eppler 387 airfoil at Re = 3× 105.

For the drag, the SST solutions constantly predict higher drag than the experimental data, a conse-

quence of failing to capture the laminar separation bubble. The values estimated by the transition model

are much closer to the experimental data, although they appear to be located on the upper bound of the

data range, and seem to follow the evolution with angle of attack quite accurately. The sharp decrease

in drag from α = −2o to α = −1o, for the numerical solution, has been justified before due to transition

occurring in the lower surface for the lowest angle, resulting in a turbulent flow over a significant portion

of the surface. Given that the same behaviour is presented by the experimental data, and since there

is no other apparent cause - judging by the pressure distribution, no bubble was present in the lower

surface for any situation - it would seem that transition stops occurring for angles higher than α = −2o.

In general, the transition model presents higher uncertainties than calculations performed with the SST

model, for both the lift and drag, namely the two highest angles of attack exhibit values higher than 10%.

The estimated numerical uncertainty as well as the lift and drag force for the finest grid for all angles of

attack can be consulted in table 4.9.

4.3.2 Re = 1× 105

Figures 4.21, 4.22 and 4.23 present the skin friction and pressure distribution for the Eppler 387 airfoil at

a Reynolds number of 1× 105. Calculations performed with the SST turbulence model exhibit transition

in the upper surface, moving upstream with the increase in the angle of attack. The lower surface also

exhibits transition for the lowest angles, but fully laminar flow is obtained for α > 2o. Including the

transition model causes fully laminar flow for the lower surface and a laminar separation bubble in the

upper surface, for all angles of attack. The increase of the angle of attack causes the bubble to move

upstream. The pressure coefficient differs slightly in the upper surface between the two models, both

in the region where the bubble is located, as well as in the region before it. This difference becomes

gradually more evident for the higher angles. For the lower surface, the two pressure distributions

are almost overlapping. Similarly to the previous section, the experimental pressure data show some

differences particularly in the lower surface and again the numerical solution is more in accordance with

the LTPT data [44].

Unlike the other tested Reynolds number, the agreement between the experimental and numerical

position of laminar separation is not as good - figure 4.24. Some scatter can be seen for the lowest an-

gles where the numerical prediction is sometimes earlier and in other cases later than the experimental

position, although in general, both move towards the leading edge with the increase of α. The numerical

49

x/c

-Cp

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5


x/c

Cf

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

30


(a) Pressure coefficient, α = −1o (b) Skin friction, α = −1o

x/c

-Cp

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5


x/c

Cf

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

30



x/c

-Cp

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5


x/c

Cf

0 0.2 0.4 0.6 0.8 1

0

10

20

30


(d) Pressure coefficient, α = 1o (e) Skin friction, α = 1o

Figure 4.21: Pressure coefficient and skin friction distributions for α = −1o, α = 0o and α = 1o for theEppler 387 airfoil at Re = 1× 105. US - Upper surface, LS - Lower Surface

50

x/c

-Cp

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5


x/c

Cf

0 0.2 0.4 0.6 0.8 1

0

10

20

30



x/c

-Cp

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

1


x/c

Cf

0 0.2 0.4 0.6 0.8 1-5

0

5

10

15

20

25



x/c

-Cp

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

1

1.5


x/c

Cf

0 0.2 0.4 0.6 0.8 1

-5

0

5

10

15

20

25

30 SST - USSST - LSLCTM - USLCTM - LS

(e) Pressure coefficient, α = 6o (f) Skin friction, α = 6o

Figure 4.22: Pressure coefficient and skin friction distributions for α = 2o, α = 4o and α = 6o for theEppler 387 airfoil at Re = 1× 105. US - Upper surface, LS - Lower Surface

51

x/c

-Cp

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

1

1.5


x/c

Cf

0 0.2 0.4 0.6 0.8 1

-5

0

5

10

15

20

25



Figure 4.23: Pressure coefficient and skin friction distributions for α = 7o for the Eppler 387 airfoil atRe = 1× 105. US - Upper surface, LS - Lower Surface

x/c

[º]

0 0.2 0.4 0.6 0.8 1

-2

-1

0

1

2

3

4

5

6

7

8

9

Laminar Separation - ExpTransition - ExpReattachment - ExpLaminar Separation - NumTransition start - NumReattach - Num

Figure 4.24: Transition chordwise location for the Eppler 387 airfoil at Re = 1× 105.

prediction for turbulent reattachment is also worse for this Reynolds number: the numerical location is

consistently downstream than that obtained from the experimental measurements and moves very little

when the angle of attack is increased. Additionally, the numerical uncertainty for the reattachment posi-

tion increases considerably when increasing the angle of attack - Table 4.10. One striking consequence

from the combined behaviour of the separation and reattachment locations is in the size of the bubble:

the numerical solution always exhibits a larger separated region than the wind-tunnel data. Particularly,

for the higher angles, the bubble has clearly decreased in size while the numerical solution indicates

almost the same size than that obtained for the lowest angle. This occurs since transition is also taking

place further downstream than the experimental location.

This results in slightly lower friction drag for the higher angles, since the flow on the lower surface

remains laminar, and transition is delayed in the upper surface. The friction drag for a given angle is

52

α [o] xsep U(xsep) xtrans−begin U(xtrans−begin) xreattach U(xreattach)-1 0.49 0.003 0.79 0.012 0.999 0.0020 0.47 0.003 0.75 0.012 0.98 0.0241 0.44 0.004 0.71 0.007 0.95 0.0282 0.42 0.005 0.67 0.017 0.92 0.0354 0.37 0.009 0.60 0.003 0.89 0.0656 0.30 0.012 0.52 0.007 0.93 0.0857 0.24 0.022 0.47 0.039 0.85 0.676

Table 4.10: Transition chordwise location and uncertainty estimate for the upper surface of the Eppler387 airfoil at Re = 1× 105.

h_i/h_1

F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

0.009

0.0095

0.01

0.0105

= -1º - LCTMh, U= 3.56% = 0º - LCTM

1h + 2h2, U= 5.26%

= 1º - LCTM

1h + 2h2, U= 5.11%

= 2º - LCTM

1h + 2h2, U= 9.25%

= 4º - LCTMh2, U= 3.00% = 6º - LCTMh2, U= 3.63% = 7º - LCTM

p= 1.44, U= 2.53%

hi/h1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.28.5

9

9.5

10

10.5

11

11.5

12

= 4º - SSTp= 1.79, U= 0.210%

= 6º - SSTp= 1.81, U= 0.374%

= 7º - SSTp= 0.99, U= 1.88%

hi/h1

Cd

,f

103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.28.5

9

9.5

10

10.5

11

11.5

12

= -1º - SSTNM, U= 0.277%

= 0º - SSTp= 1.78, U= 0.237%

= 1º - SSTp= 1.69, U= 0.191%

= 2º - SSTp= 1.51, U= 0.205%

hi/h1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.28.5

9

9.5

10

10.5

11

11.5

12

= 4º - SSTp= 1.79, U= 0.210%

= 6º - SSTp= 1.81, U= 0.374%

= 7º - SSTp= 0.99, U= 1.88%

hi/h1

Cd

,f

103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.28.5

9

9.5

10

10.5

11

11.5

12

= -1º - SSTNM, U= 0.277%

= 0º - SSTp= 1.78, U= 0.237%

= 1º - SSTp= 1.69, U= 0.191%

= 2º - SSTp= 1.51, U= 0.205%

hi/h1

Cd

,f

103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.24.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5 = 2º - LCTMNM, U= 9.25%

= 4º - LCTMNM, U= 3.00%

= 6º - LCTMNM, U= 3.63%

= 7º - LCTMp= 1.44, U= 2.53%

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.24.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5 = -1º - LCTMNM, U= 3.56%

= 0º - LCTMNM, U= 5.26%

= 1º - LCTMNM, U= 5.11%

(a) SST (b) LCTM

Figure 4.25: Friction drag uncertainty estimate for the Eppler 387 airfoil at Re = 1× 105.

h_i/h_1

F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

0.009

0.0095

0.01

0.0105

= -1º - LCTMh2, U= 5.38% = 0º - LCTM

p= 1.60, U= 4.32% = 1º - LCTM

p= 1.87, U= 4.24% = 2º - LCTMh2, U= 4.17% = 4º - LCTMh2, U= 14.5% = 6º - LCTM

p= 1.54, U= 10.6% = 7º - LCTM

p= 0.86, U= 19.1%

hi/h1

Cd

,p

103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.22

4

6

8

10

12

14

16

18

= -1º - SSTNM, U= 2.69%

= 0º - SSTNM, U= 1.52%

= 1º - SSTNM, U= 8.14%

hi/h1

Cd

,p

103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.22

4

6

8

10

12

14

16

18

= 2º - SSTNM, U= 7.49%

= 4º - SSTNM, U= 6.14%

= 6º - SSTNM, U= 1.66%

= 7º - SSTNM, U= 17.9%

hi/h1

Cd

,p

103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.22

4

6

8

10

12

14

16

18

= -1º - SSTNM, U= 2.69%

= 0º - SSTNM, U= 1.52%

= 1º - SSTNM, U= 8.14%

hi/h1

Cd

,p

103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.22

4

6

8

10

12

14

16

18

= 2º - SSTNM, U= 7.49%

= 4º - SSTNM, U= 6.14%

= 6º - SSTNM, U= 1.66%

= 7º - SSTNM, U= 17.9%

hi/h1

Cd

,p

103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.210

15

20

25

30

35

40

= 1º - LCTMp= 1.87, U= 4.24%

= 2º - LCTMNM, U= 4.17%

= 4º - LCTMNM, U= 14.5%

= 6º - LCTMp= 1.54, U= 10.6%

= 7º - LCTMp= 0.86, U= 19.1%

hi/h1

Cd

,p

103

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.210

15

20

25

30

35

40

= -1º - LCTMNM, U= 5.38%

= 0º - LCTMp= 1.60, U= 4.32%

(a) SST (b) LCTM

Figure 4.26: Pressure drag uncertainty estimate for the Eppler 387 airfoil at Re = 1× 105.

considerably higher for the SST case, due to the higher extent of turbulent flow in the upper surface.

The estimated uncertainty is also lower for the SST case, which is smaller than 2% for all angles, and

as high as 9% for the LCTM calculations.

53

hi/h1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

0.2

0.4

0.6

0.8

1

1.2

1.4

= -1º - SSTp= 1.46, U= 0.22%

= 0º - SSTp= 1.78, U= 0.11%

= 1º - SSTNM, U= 0.18%

hi/h1

Cl,p

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

0.2

0.4

0.6

0.8

1

1.2

1.4

= 2º - SSTNM, U= 0.14%

= 4º - SSTNM, U= 0.077%

= 6º - SSTNM, U= 0.63%

= 7º - SSTNM, U= 5.4%

hi/h1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

0.2

0.4

0.6

0.8

1

1.2

1.4

= -1º - SSTp= 1.46, U= 0.22%

= 0º - SSTp= 1.78, U= 0.11%

= 1º - SSTNM, U= 0.18%

hi/h1

Cl,p

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

0.2

0.4

0.6

0.8

1

1.2

1.4

= 2º - SSTNM, U= 0.14%

= 4º - SSTNM, U= 0.077%

= 6º - SSTNM, U= 0.63%

= 7º - SSTNM, U= 5.4%

hi/h1

Cl,p

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

0.2

0.4

0.6

0.8

1

1.2

1.4

= 2º - LCTMNM, U= 15.6%

= 4º - LCTMNM, U= 10.6%

= 6º - LCTMNM, U= 17.2%

= 7º - LCTMNM, U= 1.0%

hi/h1

Cl,p

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

0.2

0.4

0.6

0.8

1

1.2

1.4

= -1º - LCTMNM, U= 90.8%

= 0º - LCTMNM, U= 19.5%

= 1º - LCTMNM, U= 9.8%

(a) SST (b) LCTM

Figure 4.27: Pressure lift uncertainty estimate for the Eppler 387 airfoil at Re = 1× 105.

[º]

Cl

-4 -2 0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2MuellerLTPTLow Speed (V1)Low Speed (V3)Low Speed (V5)SSTLCTM

Cl

Cd

102

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

3.5

4LTPTLow Speed (V1)Low Speed (V3)Low Speed (V5)SSTLCTM

(a) Lift coefficient (b) Drag Polar

Figure 4.28: Lift and drag forces for the Eppler 387 airfoil at Re = 1× 105.

Despite having lower friction drag, the pressure contribution for the drag force is significantly higher

for the LCTM case, growing with increasing angle of attack as seen from figure 4.26. Again, higher

uncertainties are obtained when using the transition model for both the pressure drag and lift.

The total lift prediction matches well with the experimental data for both models - figure 4.28 (a). The

SST model predicts lower lift than the transition model for the lowest angles, and higher lift for the highest

angles of attack. The total drag obtained from the SST model is much lower than both experimental data

and transition model calculations, as can be seen from figure 4.28 (b). The latter matches well with the

former for the lowest angle, but clearly overpredicts the drag for the highest angle when the effect of

the bubble should be decreasing. Much like before, the numerical uncertainty for both lift and drag is

higher for the LCTM case, and for the drag, a clear growth is seen when increasing the angle of attack,

as shown on Table 4.11.

54

SST LCTMα Cl U(Cl)[%] Cd × 102 U(Cd)[%] Cl U(Cl)[%] Cd × 102 U(Cd)[%]

-1o 0.251 0.22 1.424 2.65 0.263 90.7 1.715 3.690o 0.358 0.11 1.418 2.62 0.374 19.5 1.851 3.371o 0.465 0.18 1.440 2.52 0.479 9.82 2.010 3.162o 0.570 0.14 1.485 2.55 0.578 15.6 2.205 3.234o 0.775 0.08 1.643 2.68 0.765 10.6 2.693 10.06o 0.969 0.63 1.927 4.13 0.932 17.2 3.385 8.987o 1.064 5.39 2.182 12.5 1.023 0.99 3.623 16.3

Table 4.11: Lift and drag forces uncertainty estimate for the Eppler 387 airfoil at Re = 1× 105.

Considering the presented results, the following conclusions were reached:

• The transition model was able to capture the effect of laminar separation bubbles, and good agree-

ment was obtained for the pressure coefficient.

• Substantial improvements for the drag coefficient occurred when using the transition model for the

lower angles of attack, due to the higher pressure drag caused by the laminar separation bubble.

• The location of laminar separation and turbulent reattachment appeared to be correctly captured

by all grids. However, once again transition requires finer grids.

• Additionally, it would appear that reattachment takes place further upstream than the experimental

data due to delayed transition. This might be avoidable by increasing the inlet viscosity ratio and

freezing the decay of the turbulence variables.

55

Chapter 5

Conclusions

In this thesis, the effect of transition modelling on the flow around three different airfoils was analyzed.

The γ − Reθ transition model of Langtry and Menter was used and the results were compared with

baseline calculations using only the SST k − ω turbulence model and available experimental data. The

three airfoils tested were the NACA 0012, which is a conventional symmetrical airfoil, the NACA 66-018,

a thick, laminar symmetrical airfoil and the Eppler 387, which is also a laminar airfoil. Different boundary

conditions were tested: the Eppler 387 airfoil was calculated for two Reynolds numbers, while different

inlet turbulence conditions were used on the NACA 0012 airfoil.

The work performed provided the following conclusions:

• Both the preliminary calculations, as well as the main work, showed that the transition model is not

as numerically robust as the SST turbulence model: iterative convergence becomes much more

difficult.

• The initial analysis of the flow on the flat plate also demonstrated the influence of the inlet boundary

conditions and importance of turbulence decay. The transition model is much more sensitive to the

inlet turbulence settings, namely the eddy viscosity, unlike the SST model. This higher sensitivity

is not a negative aspect however, as it refers to a real effect. Nevertheless, this means that

information regarding turbulence should also be provided in experimental testing, in order to extract

proper conclusions regarding model validation.

• The issue of turbulence decay was also highlighted in the flat plate calculations. The decay of the

turbulence variables is greatly overestimated by common turbulence models, and although it may

be of little importance for applications at high Reynolds numbers, it becomes relevant and must be

dealt with whenever transition modelling is intended. The approach followed in this thesis, which

consisted in deactivating the destruction terms of the turbulence model equations until a given

distance of the airfoil, is not unique and required the decision on one more numerical setting.

• The uncertainty estimation for the pressure and friction components of drag and lift showed that

higher discretization errors are obtained when the γ−Reθ model is used and can reach extremely

57

high values in some cases. Transition can occur at regions where the grid is not sufficiently refined

in the streamwise direction, causing non-monotonic behaviour.

• The calculations on the Eppler 387 airfoil showed that the transition model correctly predicted

laminar separation but causes reattachment to occur too late. Separation induced transition is the

case that shows the best iterative convergence properties, and was the only one that could be

calculated without any modifications for the inlet conditions.

• On the contrary, cases where natural transition is expected, such as in the NACA 0012 airfoil,

are highly dependent on the combination of turbulence boundary conditions and turbulence decay.

This not only affects the iterative convergence of the solution, as it has a significant impact on the

location of the transition region, which translates into different aerodynamic characteristics mostly

observed in the drag coefficient.

• Overall, the calculations with the transition model showed an improvement in accuracy when com-

pared to the SST solutions. The skin friction curves showed clear differences, but a clear com-

parison is not evident since no experimental data was available. The lift coefficient suffered little

change for all airfoils, while the drag showed a clear improvement for most cases, associated with

the higher portion of flow under the laminar regime.

Future work on this topic should include a more detailed study on the effect of the inlet turbulence

variables, as their specification seems to have a major impact on the solution. Although significant

improvements were obtained for each airfoil for most angles, it is possible that an unfortunate yet rea-

sonable choice for the turbulence intensity and eddy viscosity leads to an inaccurate solution, which

may be worse than the SST prediction. This is valid not only for cases where natural transition takes

place, but also for geometries were laminar separation bubbles are formed, since their size is most likely

influenced by the turbulence quantities as well. It would be interesting to compare the γ − Reθ model

with other available alternatives for transition modelling, or even implementations with other turbulence

models besides the k−ω SST model. Finally, the behaviour of the model in three-dimensional problems

is also worthy of some attention, given the higher complexity of the boundary layer in these situations.

58

Bibliography

[1] J. J. Baik, J. J. Kim, and H. J. S. Fernando. A CFD model for simulating urban flow and dispersion.

Journal of Applied Meteorology, 42(11):1636–1648, 2003.

[2] J. Carlton. Marine propellers and propulsion. Butterworth-Heinemann, 2012.

[3] J. D. Anderson. Fundamentals of aerodynamics. 3rd Edition, McGraw-Hill Education, 1985.

[4] V. Brederode. Fundamentos de aerodinamica incompressıvel. Author’s Edition, 1997.

[5] J. H. Ferziger and M. Peric. Computational methods for fluid dynamics. 3rd Ed., Springer, 2012.

[6] R. B. Langtry and F. R. Menter. Correlation-based transition modeling for unstructured parallelized

computational fluid dynamics codes. AIAA journal, 47(12):2894–2906, 2009.

[7] T. J. Mueller and J. D. DeLaurier. Aerodynamics of small vehicles. Annual Review of Fluid Mechan-

ics, 35(1):89–111, 2003.

[8] T. J. Mueller. Low Reynolds number vehicles. Technical report, DTIC Document, 1985.

[9] W. Shyy, Y. Lian, J. Tang, D. Viieru, and H. Liu. Aerodynamics of low Reynolds number flyers,

volume 22. Cambridge University Press, 2007.

[10] G. M. Cole and T. J. Mueller. Experimental measurements of the laminar separation bubble on an

Eppler 387 airfoil at low Reynolds numbers. NASA STI/Recon Technical Report N, 90:15380, 1990.

[11] A. M. O. Smith and N. Gamberoni. Transition, pressure gradient and stability theory. Douglas

Aircraft Company, El Segundo Division, 1956.

[12] J. L. Van Ingen. A suggested semi-empirical method for the calculation of the boundary layer

transition region. Technical report, Delft University of Technology, 1956.

[13] M. Drela and M. B. Giles. Viscous-inviscid analysis of transonic and low Reynolds number airfoils.

AIAA journal, 25(10):1347–1355, 1987.

[14] F. R. Menter, R. B. Langtry, S. R. Likki, Y. B. Suzen, P. G. Huang, and S. Volker. A correlation-based

transition model using local variables—part i: model formulation. Journal of turbomachinery, 128

(3):413–422, 2006.

[15] ReFRESCO, 2015. URL http://www.refresco.org.

59

http://www.refresco.org

[16] B. E. Launder, G. J. Reece, and W. Rodi. Progress in the development of a Reynolds-stress

turbulent closure. Journal of Fluid Mechanics, 68(3):537–566, 1975.

[17] H. Schlichting and K. Gersten. Boundary-layer theory. Springer, 2000.

[18] P. Spalart and S. Allmaras. A one-equation turbulence model for aerodynamic flows. American

Institute of Aeronautics and Astronautics, January 1992.

[19] W. P. Jones and B. E. Launder. The prediction of laminarization with a two-equation model of

turbulence. International journal of heat and mass transfer, 15(2):301–314, 1972.

[20] W. Rodi and G. Scheuerer. Scrutinizing the k-ε turbulence model under adverse pressure gradient

conditions. Journal of fluids engineering, 108(2):174–179, 1986.

[21] D. C. Wilcox. Reassessment of the scale-determining equation for advanced turbulence models.

AIAA journal, 26(11):1299–1310, 1988.

[22] F. R. Menter. Influence of freestream values on k-omega turbulence model predictions. AIAA

journal, 30(6):1657–1659, 1992.

[23] F. R. Menter. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA

Journal, 32(8):1598–1605, August 1994.

[24] M. V. Morkovin. On the many faces of transition. In Viscous drag reduction, pages 1–31. Springer,

1969.

[25] R. E. Mayle. The role of laminar-turbulent transition in gas turbine engines. Journal of Turboma-

chinery, 113(4):509–536, 1991.

[26] D. D. Pasquale, A. Rona, and S. J. Garrett. A selective review of CFD transition models. 39th

AIAA Fluid Dynamics Conference, 2009, San Antonio, Texas, American Institute of Aeronautics

and Astronautics, June 2009.

[27] R. G. Jacobs and P. A. Durbin. Simulations of bypass transition. Journal of Fluid Mechanics, 428:

185–212, 2001.

[28] D. C. Wilcox. Turbulence Modeling for CFD. 2nd Edition, DCW Industries, Inc., 2004.

[29] A. M. Savill. Some recent progress in the turbulence modelling of bypass transition. Near-wall

turbulent flows, pages 829–848, 1993.

[30] R. E. Mayle and A. Schulz. The path to predicting bypass transition. In ASME 1996 International

Gas Turbine and Aeroengine Congress and Exhibition. American Society of Mechanical Engineers,

1996.

[31] S. Lardeau, M. A. Leschziner, and N. Li. Modelling bypass transition with low-Reynolds-number

nonlinear eddy-viscosity closure. Flow, Turbulence and Combustion, 73(1):49–76, 2004.

60

[32] D. K. Walters and J. H. Leylek. A new model for boundary-layer transition using a single-point RANS

approach. In ASME 2002 International Mechanical Engineering Congress and Exposition, pages

67–79. American Society of Mechanical Engineers, 2002.

[33] D. K. Walters and D. Cokljat. A three-equation eddy-viscosity model for Reynolds-averaged Navier–

Stokes simulations of transitional flow. Journal of Fluids Engineering, 130(12):121401, 2008.

[34] J. R. Cho and M. K. Chung. A k—ε—γ equation turbulence model. Journal of Fluid Mechanics,

237:301–322, 1992.

[35] J. Steelant and E. Dick. Modelling of bypass transition with conditioned Navier-Stokes equations

coupled to an intermittency transport equation. International journal for numerical methods in fluids,

23(3):193–220, 1996.

[36] Y. B. Suzen and P. G. Huang. Modeling of flow transition using an intermittency transport equation.

Journal of fluids engineering, 122(2):273–284, 2000.

[37] W. Piotrowski and W. Elsner. Transition prediction on turbine blade profile with intermittency trans-

port equation. Journal of Turbomachinery, 132(1):011020, 2010.

[38] K. Suluksna, P. Dechaumphai, and E. Juntasaro. Correlations for modeling transitional boundary

layers under influences of freestream turbulence and pressure gradient. International Journal of

Heat and Fluid Flow, 30(1):66–75, 2009.

[39] W. Elsner and P. Warzecha. Modeling of rough wall boundary layers with intermittency transport

model. TASK Q, 14(3):271–282, 2010.

[40] J. G. Coder and M. D. Maughmer. Comparisons of theoretical methods for predicting airfoil aero-

dynamic characteristics. Journal of Aircraft, 51(1):183–191, 2014.

[41] P. Khayatzadeh and S. Nadarajah. Laminar-turbulent flow simulation for wind turbine profiles using

the γ-Reθ transition model. Wind Energy, 17(6):901–918, 2014.

[42] C. Seyfert and A. Krumbein. Evaluation of a correlation-based transition model and comparison

with the en method. Journal of Aircraft, 49(6):1765–1773, 2012.

[43] D. E. Gault. An experimental investigation of regions of separated laminar flow. Number 3505.

National Advisory Committee for Aeronautics, 1955.

[44] R. J. Mcghee, B. S. Walker, and B. F. Millard. Experimental results for the Eppler 387 airfoil at low

Reynolds numbers in the Langley low-turbulence pressure tunnel. 1988.

[45] N. Gregory and C. L. OReilly. Low-speed aerodynamic characteristics of NACA 0012 aerofoil sec-

tion, including the effects of upper-surface roughness simulating hoar frost. Technical report, 1970.

[46] C. Ladson. Effects of independent variation of Mach and Reynolds numbers on the low-speed

aerodynamic characteristics of the NACA 0012 airfoil section. Technical report, 1988.

61

[47] J. Coupland. Ercoftac classic database, 1993.

[48] R. B. Langtry. A correlation-based transition model using local variables for unstructured paral-

lelized CFD codes. PhD thesis, Universitat Stuttgart, Holzgartenstr. 16, 70174 Stuttgart, 2006.

URL http://elib.uni-stuttgart.de/opus/volltexte/2006/2801.

[49] P. R. Spalart and C. L. Rumsey. Effective inflow conditions for turbulence models in aerodynamic

calculations. AIAA journal, 45(10):2544–2553, 2007.

[50] G. Vaz, F. Jaouen, and M. Hoekstra. Free-surface viscous flow computations: Validation of URANS

code FreSCo. In ASME 2009 28th International Conference on Ocean, Offshore and Arctic Engi-

neering, pages 425–437. American Society of Mechanical Engineers, 2009.

[51] C. M. Klaij and C. Vuik. SIMPLE-type preconditioners for cell-centered, colocated finite volume

discretization of incompressible Reynolds-averaged Navier–Stokes equations. International Journal

for Numerical Methods in Fluids, 71(7):830–849, 2013.

[52] L. Eca. Grid generation tools for structured grids. Technical Report Report D72-18, Instituto Supe-

rior Tecnico, 2003.

[53] L. Eca and M. Hoekstra. A procedure for the estimation of the numerical uncertainty of CFD calcu-

lations based on grid refinement studies. Journal of Computational Physics, 262:104–130, 2014.

[54] M. S. Selig, J. J. Guglielmo, A. P. Broeren, and Giguere P. Summary of low speed airfoil data,

volume 1. SoarTech, 1995.

[55] M. S. Selig, C. A. Lyon, A. P. Broeren, Giguere P., and Gopalarathnam A. Summary of low speed

airfoil data, volume 3. SoarTech, 1997.

[56] G. A. Williamson, B. D. McGranahan, B. A. Broughton, R. W. Deters, J. B. Brandt, and M. S. Selig.

Summary of low speed airfoil data, volume 5. SoarTech, 2012.

62

http://elib.uni-stuttgart.de/opus/volltexte/2006/2801

Appendix A

Discretization Schemes

Table A.1 presents the discretization schemes chosen for the convective fluxes for each calculation on

each airfoil, as well as whether excentricity corrections were used or not.

NACA 0012 NACA 66-018 Eppler 387Numerical SettingsSST LCTM SST LCTM SST LCTM

Momentum - Discretization Scheme QUICK QUICK QUICK QUICK QUICK QUICKMomentum - Flux Limiters True True True True True True

Momentum - Excentricity Corrections False False True False True TruePressure - Excentricity Corrections False False True False True TrueTurbulence - Discretization Scheme Upwind QUICK QUICK QUICK QUICK QUICK

Turbulence - Flux Limiters - True True True True TrueTurbulence - Excentricity Corrections False False True False True True

Transition - Discretization Scheme - QUICK - QUICK - QUICKTransition - Flux Limiters - True - True - False

Transition - Excentricity Corrections - False - False - True

Table A.1: Discretization choices for each airfoil.

63

calculation of the ﬂow around hydrofoils at moderate ... · calculation of the ﬂow around...

Documents