Solution of Flows Around Airfoils Using RANS with Wall-Functions

Gonçalo Gouveia Velez Bidarra Saraiva

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisor: Prof. Luís Rego da Cunha de Eça

Examination Committee

Chairperson: Prof. Viriato Sérgio de Almeida Semião Supervisor: Prof. Luís Rego da Cunha de Eça

Member of the Committee: João Manuel Melo de Sousa

October 2014

iii

Acknowledgments

I would like to express my deep gratitude to Professor Luís Eça for giving me the opportunity to

work on this subject, for all his guidance, shared knowledge and incredible patience during these last 8

months.

To Dr. João Baltazar, for being welcoming, helpful in every occasion and for the great football

discussions along these last months.

To my parents, brothers, sister and grandmother, for they support, love and care without which

these five years would have been much more difficult. This thesis is dedicated to them.

To Diogo Sampaio, Sérgio Afonso and Tomás Lúcio with whom I had the pleasure to share five

amazing years.

To Ricardo Pires, João Castro, Gonçalo Mendes and André Duarte for their friendship and

unconditional support.

To all my friends and colleagues that in many ways helped me along the course of this degree.

And last, but certainly not least, to Flávio Lopes, for his crucial technical support along the

elaboration of this thesis.

v

Resumo

O cálculo de forças de atrito é crucial em aplicações de hidrodinâmica e off-shore. No entanto, os

elevados gradientes existentes junto a paredes requerem o uso de uma das seguintes abordagens: usar

malhas muito finas junto à parede para calcular a tensão de corte na parede directamente pela sua

definição; ou usar leis da parede para calcular indirectamente a tensão de corte na parede e fornecer

condições de fronteira para as variáveis dos modelos de turbulência. O objectivo desta tese é avaliar a

validade das leis da parede para o cálculo dos coeficientes de atrito e pressão, assim como as

características aerodinâmicas de um perfil convencional e de um perfil laminar. O código ReFRESCO

foi usado para resolver as equações de Reynolds com a versão SST do modelo de turbulência � − �.

As principais conclusões resultantes desta tese foram: as leis da parede podem produzir resultados

razoáveis se o número de Reynolds for suficientemente alto para que a transição ocorra perto do bordo

de ataque; se a parte laminar do escoamento for significativa os resultados produzidos são irrealistas

pois correspondem a escoamento totalmente turbulento; os resultados dos coeficientes de pressão e

sustentação são sempre melhores que os do coeficiente de atrito e de resistência, devido à relação

directa da tensão de corte na parede com estes dois últimos parâmetros; por fim, os resultados das leis

da parede são fortemente dependentes da localização do primeiro nó acima da parede, mesmo em

números de Reynolds elevados.

Palavras-chave: Leis da parede, Equações de Reynolds, Perfis sustentadores, Modelos de

turbulência.

vii

Abstract

The calculation of the friction forces is essential in hydrodynamic and off-shore applications.

However, the high gradients that exist in near-wall regions require the use of one of the following

approaches: grids that are very fine near the wall to calculate the wall shear-stress directly from its

definition; or wall-functions (WF) to calculate indirectly the wall shear-stress and provide boundary

conditions for the variables of the turbulence models. The objective of this thesis is to assess the

validity of WF boundary conditions for the calculation of friction and pressure coefficients, as well as

aerodynamic forces coefficients of a conventional and a laminar airfoil. The ReFRESCO solver was

used to solve the RANS equations with the SST version of the � − � eddy-viscosity turbulence

model. The main conclusions obtained were: WF can yield acceptable results if the Reynolds number

is high enough to promote transition near the leading edge; if the laminar part of the flow is

significant, the results are not realistic because WF lead to a fully turbulent flow; the results for the

pressure and lift coefficient are always better than for friction and drag coefficients due to the direct

connection of the wall shear-stress with the last two; last but not least, the results of the WF approach

are strongly dependent on the location of the first interior grid node, even at high Reynolds number.

Keywords: Wall Functions, RANS, Airfoils, Turbulence Models.

ix

Contents

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

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

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

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Wall Functions in Viscous Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Wall Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Implementation of Wall Functions in ReFRESCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3. Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Choice of Grid Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Generation of Base Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Block I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.2 Blocks II and V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.3 Blocks III and IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Interpolation for Near-Wall Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.1 The W-grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.2 The Standard grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1 Numerical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Flow Over the Conventional Airfoil NACA 0012 at � = 0º . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 W-Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.2 Standard Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.3 Comparison to No-WF solution and Experimental Results . . . . . . . . . . . . . . . . 36

4.3 Flow Over the Conventional Airfoil NACA 0012 at � = 4º . . . . . . . . . . . . . . . . . . . . . 39

4.3.1 W-Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.2 Standard Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.3 Comparison to No-WF solution and Experimental Results . . . . . . . . . . . . . . . . 45

4.4 Flow Over the Laminar Airfoil Eppler 374 at � = 0º . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.4.1 W-Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.4.2 Standard Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4.3 Comparison to No-WF and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 52

5. Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

xi

List of Figures

Figure 2.1 - Airfoil features and designations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Figure 2.2 - Geometry of the NACA 0012 and Eppler 374 airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Figure 3.1 - Computational domain (a) and close-up of interior C-shaped grid (b) . . . . . . . . . . . . . . . 19

Figure 3.2 - Blocks of base grid and its designations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Figure 3.3 - Transformation from computational (left) to physical (right) domain . . . . . . . . . . . . . . . . 21

Figure 3.4 - Close-up of the trailing edge of base and W-grid for � = 121 . . . . . . . . . . . . . . . . . . 24

Figure 3.5 - Close-up of the trailing edge of W-grid for � = 121 . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Figure 3.6 - Close-up of the trailing edge W-grid and Standard grid for � = 241 . . . . . . . . . . . . . 27

Figure 4.1 - Grid convergence of aerodynamic coefficients as function of grid refinement

ratio for W-grid of NACA 0012 airfoil at � = 0º . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 4.2 - Skin friction coefficient distribution of group A96, No-WF and CFL3D

for NACA 0012 at � = 0º . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Figure 4.3 - Skin friction coefficient distribution of W-grid for NACA 0012 at � = 0º . . . . . . . . . . . 34

Figure 4.4 - Pressure coefficient distribution and relative deviation from No-WF for

W-grid of NACA 0012 at � = 0º . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Figure 4.5 - Skin friction coefficient distribution of Standard grid for NACA 0012 at � = 4º . . . . . . 35

Figure 4.6 - Pressure coefficient distribution and relative deviation from No-WF for

Standard grid of NACA 0012 at � = 0º . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Figure 4.7 - Variation of aerodynamic coefficients with ��� for NACA 0012 at � = 0º . . . . . . . . . . . 38

Figure 4.8 - Grid convergence of aerodynamic coefficients as function of grid refinement

ratio for W-grid of NACA 0012 airfoil at � = 4º . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Figure 4.9 - Skin friction coefficient distribution of W-grid for NACA 0012 at � = 4º . . . . . . . . . . . 41

Figure 4.10 - Pressure coefficient distribution of W-grid for NACA 0012 at � = 4º . . . . . . . . . . . . . . 42

Figure 4.11 - Skin friction coefficient distribution of Standard grid for NACA 0012 at � = 4º . . . . . . 43

Figure 4.12 - Pressure coefficient distribution for Standard grid of NACA 0012 at � = 4º . . . . . . . . . 44

Figure 4.13 - Variation of relative deviation of aerodynamic coefficients with ��� for NACA

0012 at � = 4º . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Figure 4.14 - Variation of total aerodynamic coefficients with ��� for NACA 0012 at � = 4º . . . . . . . 47

xii

Figure 4.15 - Grid convergence of aerodynamic coefficients as function of grid refinement

ratio for W-grid of Eppler 374 airfoil at � = 0º . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Figure 4.16 - Skin friction and pressure coefficients distribution of W-grid for Eppler 374 at

� = 0º . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Figure 4.17 - Skin friction coefficient distribution of Standard grid for Eppler 374 at � = 0º . . . . . . . 51

Figure 4.18 - Pressure coefficient distribution of Standard grid for Eppler 374 at � = 0º . . . . . . . . . . . 51

Figure 4.19 - Skin friction and pressure coefficients distribution of W-grid, Standard grid and

No-WF for Eppler 374 at � = 0º . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Figure 4.20 - Variation of total aerodynamic coefficients with ��� for Eppler 374 at � = 0º . . . . . . . . 53

Figure 4.21 - Variation of relative deviation of aerodynamic coefficients with ��� for Eppler

374 at � = 0º . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

xiii

List of Tables

Table 2.1 - Reynolds number used in the calculations of each airfoil . . . . . . . . . . . . . . . . . . . . . . . . . 15

Table 3.1 - � and refinement ratio of each grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Table 3.2 - Grid lines in each block and total number of volumes of coarsest and finest grid . . . . . . 23

Table 3.3 - Near-wall spacing factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Table 3.4 - Number of lines removed in each refinement level and for each w-group . . . . . . . . . . . . 25

Table 3.5 - Number of volumes and relative difference of each w-group to W-grid with no

lines removed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Table 4.1 - ��� value of each group used in NACA 0012 calculations for � = 0º . . . . . . . . . . . . . . . 31

Table 4.2 - Reference values and estimated uncertainties for NACA 0012 at � = 0º . . . . . . . . . . . . 37

Table 4.3 - ��� value of each group used in NACA 0012 calculations for � = 4º . . . . . . . . . . . . . . . 39

Table 4.4 - Reference values and estimated uncertainties for NACA 0012 at � = 4º . . . . . . . . . . . . 45

Table 4.5 - ��� value of each group used in Eppler 374 calculations for � = 0º . . . . . . . . . . . . . . . . 47

Table 4.6 - Reference values and estimated uncertainties for Eppler 374 at � = 0º . . . . . . . . . . . . . 53

xv

List of Acronyms

CFD Computational Fluid Dynamics

DNS Direct Numerical Simulation

LES Large Eddy Simulation

LRN Low-Reynolds Number

LS Lower Surface

NACA National Advisory Committee for Aeronautics

NSE Navier-Stokes Equations

QUICK Quadratic Upstream Interpolation for Convective Kinematics

RANS Reynolds-Averaged Navier-Stokes

RST Reynolds Stress Tensor

SIMPLE Semi-Implicit Method for Pressure Linked Equations

SST Shear-Stress Transport

US Upper Surface

VSL Viscous Sub-Layer

WF Wall Functions

xvii

List of Symbols

�� Velocity component

�� Cartesian direction

� Time variable, maximum airfoil thickness

� Fluid density

� Static pressure, order of grid convergence, location of maximum airfoil camber

�� External body force vector component

��� Viscous stress tensor

� Fluid kinematic viscosity

��� Strain-rate tensor

� Instantaneous variable

�� Mean value

�� Fluctuation component

�� Eddy-viscosity

��� Kronecker’s delta

� Turbulence kinetic energy

� ! Effective viscosity

" Rate of dissipation of turbulence

# Turbulence dissipation

$%, $' Blending functions of the turbulence model

(∗, *�, *", +% Turbulence model constants

, Vorticity magnitude

- Fluid dynamic viscosity

. Velocity component parallel to the wall, estimated uncertainty

/0 Wall shear-stress

�/ Friction velocity

�� Non-dimensional velocity

1� Non-dimensional distance to the wall

2 Von-Kármán constant

xviii

3 Empirical constant in log-law

1'� Non-dimensional distance of the first interior grid node to the wall

�', 1' Velocity and distance to the wall of first interior grid node

4, 54 Lift force and coefficient

6,56 Drag force and coefficient

.7 Freestream velocity

8 Airfoil chord

5� Pressure coefficient

�7 Freestream static pressure

5! Skin friction coefficient

9 Reynolds number

: Typical cell size

;<= Number of points on airfoil surface

>, ? Coordinates in computational domain

1� Thickness distribution

@ Maximum airfoil camber

A Near-wall spacing factor

56,�B Pressure component of drag coefficient

56,CC Friction component of drag coefficient

54,�B Pressure component of lift coefficient

54,CC Friction component of lift coefficient

D Angle of attack

E Variable relative deviation

5�,F Pressure coefficient in incompressible conditions

5�,8G@� Pressure coefficient in compressible conditions

H+ Mach number

1

Chapter 1

Introduction

Lifting surfaces have been around for so long that it is difficult to state precisely when the first ones

appeared. These devices are present in many different applications: aircraft wings, propeller blades in

ships, aerodynamic appendages in sports cars, wind-turbine blades and so on. They are constantly

being developed and evolving in shape, size, type of applications and, most importantly, in efficiency.

Before the existence of computers, the aerodynamic characteristics of such surfaces, i.e. the lift and

drag forces, were determined in experimental facilities such as wind tunnels, towing tanks or

cavitation tunnels. With the development of computers, the subsequent increase in memory capacity

and calculation speed allows for the numerical study of fluid dynamics problems. The method of

studying fluid flow related problems with computer based simulations is called Computational Fluid

Dynamics (CFD). CFD covers a wide range of application areas such as aerodynamics of aircrafts and

vehicles, hydrodynamics of ships, weather prediction, combustion in engine cylinders and flow

through pumps, compressors and turbines, just to give a few examples [1].

In CFD, the Navier-Stokes equations are discretized using different possible approaches, being the

most used finite differences, finite elements and finite volumes. The Finite Volume Method is used in

most commercially available CFD codes. It is also the technique used in the ReFRESCO [2] solver,

which was used in this thesis.

Wilcox [3] states that there are three key elements in CFD: algorithm development, grid generation

and turbulence modelling. This work deals mainly with the second element but also with the last one.

Wilcox [3] also says that turbulence modelling is by far the least developed of those three elements

because the objective is to approximate an extremely complex phenomenon. The complexity of

turbulence arises from its inherent features. Turbulence is described by Ferziger and Peric [4] as

highly unsteady (the flow variables show random fluctuations), three-dimensional, with high

quantities of vorticity, highly diffusive (it increases greatly the rate at which mass, momentum and

energy are transferred), dissipative (the turbulent energy is transferred from the larger eddies to the

smallest ones, where it is eventually lost as heat) and as containing a wide range of time and length

scales. There are several approaches when it comes to numerically solve turbulent flows. Some are

2

more accurate than others because they use less approximations to the governing equations but are

also more demanding in terms of computational resources. Three common approaches are direct

numerical simulation (DNS), large-eddy simulation (LES) and the Reynolds-Averaged Navier-Stokes

(RANS) equations. The modelling increases from DNS to RANS but the computer requirements of

DNS make it only feasible for simple flows at low Reynolds numbers. RANS is the most common

approach in engineering, whereas LES lies somewhere in between, but closer to DNS. Thus, DNS and

LES are mainly research tools. The works [3-5] provide good descriptions of DNS and LES. The focus

of this thesis will be only on the RANS modelling approach.

When solving the RANS equations in wall-bounded (turbulent) flows, such as a flow around an

airfoil, one of the goals is to calculate the wall shear-stress that arises from the no-slip condition on

solid surfaces. It is then necessary to ensure that the near-wall region has a locally fine grid in order to

properly compute the high gradients that exist in those regions. However, in regions close to walls, the

turbulent fluctuations are damped due to the no-slip condition at the wall. This small region adjacent

to a solid wall is called the viscous sub-layer (VSL). Some turbulence models are formulated with the

capability to deal with this thin region, e.g. the Shear Stress Transport (SST) version of the � − �

turbulence model by Menter [6]. However, some turbulence models do not have this kind of near-wall

formulation, thus failing to predict the profiles for the velocity and the turbulence quantities in the

near-wall region if no further treatment is done on these models. Two different paths are usually taken

when it comes to overtaking those difficulties: the use of turbulence models with damping functions

for the turbulence variables (the so-called low-Reynolds numbers (LRN) models) and the use of wall-

functions (WF) boundary conditions. The former represents a more accurate approach even for

complex flows but the ability (due to cells with very high aspect-ratio that may affect convergence)

and the computational costs (because of the high number of cells) of numerically solving the RANS

equations all the way down to the wall can be considerably high, thus justifying the latter approach.

Obviously, turbulence models such as the � − � SST can also be used with WF boundary conditions

in order to avoid solving the RANS equations all the way down to the wall, through the VSL.

Craft et al. [7] state that although the region where viscous effects are dominant occupies no more

than around 1% of the flow, it requires between 3 to 300 times as much computing time as if the mesh

was as dense as in the fully-turbulent part of the flow. Moreover, the small height of the first grid cell,

which is necessary for the direct application of the no-slip condition, is extended to the wake where it

is not necessary to have such fine grids, if one uses flow-aligned and/or single-block structured grids

[8].

As mentioned before, in this thesis the RANS equations were solved with ReFRESCO.

ReFRESCO is a CFD code that solves multi-phase (unsteady) incompressible flows with the RANS

equations, complemented with turbulence models, cavitation models and volume-fraction transport

3

equations for different phases [9]. The equations are discretised using a finite-volume approach with

cell-centered collocated variables. The equations are discretised in strong-conservation form and a

pressure-correction equation based on the SIMPLE algorithm (see for example [4]) is used to ensure

mass conservation. 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 the coupled linear system is solved with a matrix-free Krylov subspace method

using a SIMPLE-type preconditioner [10]. A segregated approach is adopted for the solution of the

turbulence model 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). The code is parallelised using MPI and subdomain

decomposition, and runs on Linux workstations, clusters and super-computers. It is currently being

developed and verified at MARIN (in the Netherlands) [6-12] in collaboration with IST (in Portugal)

[13] and other universities [14-18].

ReFRESCO is mainly used in ship hydrodynamics and off-shore problems where of the main goals

is to calculate the shear forces caused by the viscosity of the fluids on solid surfaces. Moreover, both

applications often deal with lifting surfaces, such as ship propellers or turbine blades, where the

pressure distributions are also relevant. Because the problems where ReFRESCO is used are extremely

complex and with very large computational domains, wall-functions boundary conditions may be used

to deal with near-wall regions. In fact, WF are the default option in the solver for boundary conditions

at solid walls. This leads to the purpose of this work.

The main objectives of this thesis are to study the effects of using WF on the predictions of the

friction and pressure forces on conventional and laminar airfoils. Different Reynolds numbers were

used in order to assess the effect of using WF in flows where the transition from laminar to turbulent

flow occurs close to the leading edge and when it occurs further from it. The aerodynamic

characteristics (lift and drag) of the airfoils (which are a consequence of the friction and pressure

distributions) were also studied. Furthermore, it was intended to examine the differences between the

most usual approach for grid generation in WF applications and a more suitable grid type. For this, a

grid generator was developed that allows for an easy and efficient way to generate both types of grids.

Grid refinement studies were performed to assess the convergence properties of the force coefficients

and to evaluate the numerical uncertainty of the results. It is intended with this work to show the

importance of the location of the wall-adjacent cell center on the predictions of the lift and drag forces

in different flow conditions and to quantify the numerical error inherent to grids with continuously

increasing spacing in the direction normal to the airfoils surface.

The structure of this thesis is the following: chapter 2 presents a description of the mathematical

models used and an explanation of the WF method and its implementation in ReFRESCO. Chapter 3

4

presents the details of the Grid Generation process, the choices made for the type of grid used and the

implementation of a grid generator that allows for an easy choice of the grid configuration and control

of near-wall distances. Chapter 4 shows the results obtained for the two types of grids studied, for a

conventional and a laminar airfoil, the NACA 0012 and the Eppler 374, respectively. Finally, chapter

5 presents the main conclusions drawn from this work as well as some suggestions for future work.

5

Chapter 2

Wall Functions in Viscous Flows

In this chapter, the use of WF boundary conditions in turbulent flows is explained. Section 2.1

presents the main mathematical models used in this thesis, viz. the RANS equations and the turbulence

model used in this work, section 2.2 introduces the concept of WF, section 2.3 presents a short

description of their implementation in the solver ReFRESCO and section 2.4 presents the airfoils that

were studied and explains the different conditions used in each one.

2.1 Mathematical models

RANS is the most widely used approach for computing turbulent flows in engineering applications.

It combines a satisfactory degree of accuracy with viable computation times. This model is directly

derived from the equations that describe the fluid motion: the Navier-Stokes equations (NSE). If

written in differential form, they represent the conservation of mass (continuity) and momentum in an

infinitesimal control volume. Analytical solutions for the NSE are possible to obtain but only for

laminar flows in simple geometries. Due to the random fluctuations that characterize turbulent flows,

one needs to use a statistical approach if a solution of the NSE is desired. Reynolds [19] proposed a

procedure where all quantities are expressed as a sum of mean and fluctuation components. If a certain

flow is statistically steady, meaning that, on average, it does not vary with time, then the continuity

and NSE are time-averaged and one solves the resulting equations for the mean quantities. There are

other ways to average the NSE (ensemble averaging, spatial averaging, phase averaging, Favre

averaging [3]) but those methods are out of the scope of this thesis. Following the procedure

introduced by Reynolds, the RANS equations are derived starting from the NSE. For an

incompressible Newtonian fluid, the continuity (2.1a) and the NSE (2.1b) in tensor notation are the

following:

IJKILK = 0 (2.1a)

IJKIM + JO IJKILO = −1P IQILK +

IMOKILO + RK (2.1b)

6

where JK is the velocity component in direction LK, M is time, P is the density, Q is the pressure, RK is a

vector of external body forces and MKO is the viscous stress tensor. This tensor is defined as

MKO = 2STKO (2.2)

where S is the kinematic viscosity of the fluid and TKO is the strain-rate tensor. This tensor is defined as

TKO = 12UIJKILO +IJOILKV (2.3)

All quantities here presented are instantaneous quantities that, as suggested by Reynolds, are

separated in an average and a fluctuation component:

WXLK, MY = WZXLKY + W�XLK, MY (2.4)

where W is an instantaneous variable, WZ is the mean value and W′ is the fluctuation around the mean

value, which is the turbulence. If this decomposition is applied to the NSE it results in:

IXJZK + JK�YILK = 0 (2.5a)

IXJZK + JK�YIM + \JZO + JO�] IXJZK + JK�YILO = −1P IXQ̅ + Q�YILK + I\MO̅K + MOK� ]ILO + \RZK + RK�] (2.5b)

Time-averaging equations (2.5a) and (2.5b) yields:

IXJZ_ + J_�YIL_ZZZZZZZZZZZZZ = 0 (2.6a)

IXJZ_ + J_�YIMZZZZZZZZZZZZZ + \JZ` + J�̀]ZZZZZZZZZZZZ IXJZ_ + J_�YIL̀

ZZZZZZZZZZZZZ = −1P IXQ̅ + Q�YIL_ZZZZZZZZZZZZ + I\M̅̀ _ + M̀ _� ]IL̀

ZZZZZZZZZZZZZZ + \RZ_ + R_�]ZZZZZZZZZZZZ (2.6b)

The continuity equation is linear so it is satisfied for both the mean and the fluctuation components.

The time derivative of a mean value is, by definition, zero and so are the mean values of the

fluctuations. Using these properties, equation (2.6b) simplifies to the so-called Reynolds-Averaged

Navier-Stokes equations:

7

JZO IJZKILO = −1P IQ̅ILO + IILO a2STK̅O − J_�J�̀ZZZZZZb + RZK (2.7)

Comparing equations (2.7) and (2.1b) it is possible to see that a new term has appeared as

consequence of the averaging process. This new term, −J_�J̀�ZZZZZZ, is commonly known as the Reynolds

stresses. It is a symmetrical tensor so, for a tridimensional flow, 6 new variables exist but no additional

equations to determine the new variables. Transport equations can be derived for each component of

the Reynolds stress tensor (RST) but more unknowns than equations appear. Therefore, the system

needs additional equations in order to be closed. This is known as the closure problem of the RANS

equations. The RST represents the fundamental problem of turbulence in engineering problems. The

computation of the RST requires some kind of approximation or modelling. Some approaches are

more intricate than others but are also more accurate. The simplest way to compute the RST is by

using the so-called Boussinesq assumption (or Boussinesq eddy-viscosity approximation). Boussinesq

[20] proposed an approach to determine the RST. It consists of assuming that the Reynolds stresses are

proportional to the mean velocity gradients, as shown in equation (2.8):

−J_�J�̀ZZZZZZ = Sc UIJZKILO +IJZOILKV − 23�eKO (2.8)

where Sc is the turbulent viscosity (also called eddy-viscosity),eKO is Kronecker’s delta and � is the

turbulence kinetic energy. The latter is defined as

� = 12J_�J_�ZZZZZZ (2.9)

Very often the RANS equations are written with a single term for the viscous and Reynolds

stresses. Both terms are condensed in one using an effective viscosity, Sgh defined as

Sgh = S + Sc (2.10)

Equation (2.7) then becomes

JZO IJZKILO = −1P IQ̅ILO + IILO iSgh UIJKILO +IJOILKVj + RZK (2.11)

8

Several types of models use the Boussinesq assumption: algebraic models, one-equation and two-

equation models. These models differ from each other in the way the eddy-viscosity is computed.

Algebraic models use a single correlation which involves a characteristic length scale (most often the

so-called mixing length) and a velocity gradient. One-equation models solve a partial differential

equation for a turbulent quantity, which can be the turbulence kinetic energy or a modified eddy

viscosity as in the Spalart-Allmaras turbulence model [21], in order to compute the turbulence velocity

scale. However, if a single equation for � is solved, the turbulent length scale has to be related to some

characteristic length. Two-equation models solve, in addition to the turbulence kinetic energy

equation, one more equation for a turbulence length scale (or equivalent) depending on the model. As

stated in [22], two-equation models are based on the assumption that a statistical description of

turbulence requires at least two independent scales (which can be velocity, length and/or time).

Examples of two-equation turbulence models are the widely used � − k model by Launder and

Spalding [23] and the � − � model by Wilcox [24]. In the present work, the turbulence model used

was the Shear Stress Transport (SST) version of the � − � by Menter [6]. This variant of the � − �

model takes advantage of the good behaviour in the near-wall region of the original � − � model by

Wilcox [24] and the small sensitivity of the � − k model [23] to the inlet freestream turbulence

quantities. It is often used in flow around airfoils because it deals well with adverse pressure gradients

and flow separation although it may produce too large turbulence levels in regions close to stagnation

points and strong acceleration [25]. For this reason, a production limiter was introduced [26]. The

transport equations that form the � − � SST turbulence model are the following [6]:

IXP�YIM + IXPJK�YILK = � − l∗P�� + IILK mXn + opncY I�ILKq (2.12a)

IXP�YIM + IXPJK�YILK = rSc � − lP�� + IILK mXn + osncY I�ILKq + 2X1 − tuY Pos�� I�ILK I�ILK (2.12b)

where � = min yzKO {|}{~� , 10l∗��� is the production of turbulence kinetic energy (with limiter), �

is the specific rate of dissipation of turbulence, l∗, op, r, os and os� are constants and tu is a

blending function of �and � defined in [6] that is zero away from the wall (thus resulting in the � − k

model) and switches to one inside the boundary layer (activating the � − � model). The eddy-

viscosity is then computed from:

Sc = �u�maxX�u�,Ωt�Y (2.13)

9

where Ω is the vorticity magnitude, �u is equal to 0.31 and t� is another blending function also

presented in [6]. The eddy-viscosity is then used to calculate the Reynolds stresses with equation (2.8)

in order to close the system of the RANS equations.

2.2 Wall functions

As mentioned previously, the near-wall region requires special treatment. One of the options for

such treatment, the WF method, is the main subject of this thesis. Wall functions are algebraic

formulae that allow for indirectly obtaining the wall-shear stress and provide boundary conditions for

the turbulence quantities at a certain distance away from the solid wall [27]. The main motivations

behind the WF approach is the possibility for considerable savings in computational effort [28] and the

possibility to introduce additional empirical information in special cases, such as wall roughness [23].

The fundamental principle in WF is to match the solution of the first grid node to a semi-empirically

derived law for the velocity in the near-wall region, hence relating flow quantities to the wall shear-

stress. When the velocity, �, and the distance to the wall, �, are expressed in its dimensionless form,

the law-of-the-wall is universal, which means that in a quasi-equilibrium boundary layer (e.g. flow

over a flat plate with zero pressure gradient) the form of the velocity profile between the wall and the

outer edge of the logarithmic layer is invariant when appropriate scaling is used [30]. To derive the

law-of-the-wall it is assumed that the total shear-stress (viscous part plus turbulent part) is constant,

and equal to the wall shear stress, z�. In a two-dimensional boundary layer flow this equates to:

�����J��M������ + ���������M������ = n I�I� − PJ�ZZZZ = z� (2.14)

where n is the dynamic viscosity of the fluid, � is the velocity component parallel to the wall, � is the

coordinate in the direction normal to the wall, P is the density of the fluid and −PJ�ZZZZZ are the Reynolds

stresses. To transform the velocity and length scales into their dimensionless form, the kinematic

viscosity, S, and the so-called friction velocity, J� = �z�/P are used in the following way:

�� = J��S (2.15a)

and

J� = �J� (2.15b)

10

The region where the assumption of constant total shear-stress is valid is divided in three

subregions: the viscous sub-layer (VSL) (sometimes called linear sub-layer), where the Reynolds

stresses are negligible; the log-layer, where the viscous stresses are negligible; and the buffer-layer,

where neither component is negligible. The VSL is valid from the wall up to approximately �� = 5.

The buffer-layer is located on the region 5 < �� < 30 − 50. Finally, the log-layer is valid from

�� = 30 − 50 to the outer edge of the law-of-the-wall region, which depends on the Reynolds

number. The law-of-the-wall is expressed in dimensionless form in the following way:

J� = ������� < 5 (2.16a)

J� = 1� lnX���Y ����� > 30 − 50 (2.16b)

where � is the so-called Von-Kármán constant, which is equal to 0.41 and � is an empirical constant

equal to 8.53 in smooth surfaces.

Menter and Esch [31] proposed a blending of equations (2.16a) and (2.16b) to construct a velocity

profile that is valid in the regions described earlier: the VSL, the buffer-layer and the log-layer. The

new velocity profile is shown in equation 2.17:

J� = Uy 1���� + y �lnX���Y�

�V��.�  (2.17)

This blending is the foundation of the Automatic WF that are implemented in ReFRESCO. This type

of WF are able to automatically switch from the VSL to the log-layer depending on the local ��. The

turbulence quantities can also be computed as a function of the �� in the law-of-the-wall region.

Following the blending proposed by Menter, the (dimensionless) turbulent kinetic energy is computed

from:

�� = �J�� = Uy 20X��Y��� + ¡¢V��.  (2.18)

where ¡¢ = 0.09.

The second turbulent variable � is obtained from:

11

�� = �SJ�� = ¤y 6lX��Y��� + U 1�\�¡¢��]V

�¦�.  (2.19)

Despite the usefulness of the law-of-the-wall, not all flows (or very few, actually) are as simple as

those on which the said law is based. Almost all flows with aerodynamic applications have stagnation

points, regions of laminar or not fully developed turbulent flow, reasonable adverse pressure gradient

and regions of separation and/or reattachment [32]. Flows around airfoils can have some or all of the

mentioned features, thus justifying an extensive study of the influence of the use of WF in the solution

of the RANS equations in such flows.

WF offer a reduction in the number of grid nodes in the direction normal to the wall. For instance,

in the work of Knopp et al. [32] it is stated that for a flow around an airfoil at Reynolds number of

6 × 10¨, LRN models with ��� = 1 (being ��� the dimensionless distance of the center of the wall-

adjacent cell to the wall) require a grid with around 35 nodes in the wall-normal direction to resolve

the boundary layer. A WF grid with ��� = 70 would only need around 17. However, these savings are

not the only advantage in the WF method. The LRN models’ requirement that ��� = 1 (or even

smaller) can be difficult to fulfil in certain flows and always leads to a very large aspect ratio of the

cells near the walls, which is prejudicial for the numerical robustness [33]. Craft [7] showed that the

computation time is reduced by one order of magnitude when changing from a LRN model to WF in a

simulation of an axisymmetric impinging jet. WF can require the first grid node to be on the log-law

region but this is a too severe constraint. WF that do not have this restriction (meaning that the first

grid node away from the wall can either be on the viscous sub-layer, the buffer region or the log-layer)

are called automatic (or hybrid, adaptive or enhanced (e.g. in FLUENT)) wall functions [34]. Several

studies have been made in order to construct simple but effective hybrid WF for several turbulence

models and tested in different types of flows. Craft [7] developed a new WF where a sub-grid is

embedded in the wall-adjacent cell and boundary-layer-type transport equations are solved in that sub-

grid. The results for complex flows such as impinging jets and rotating-disc flows showed excellent

agreement with LRN results. The computing times were slightly larger than those with standard WF

but were still one order of magnitude less than standard LRN models. This represents an excellent

compromise between numerical accuracy and computational effort. Knopp [35] used hybrid WF and a

grid adaptation method to test the use of WF in subsonic flow over the A-airfoil and compared it to the

LRN solution. The � − � SST model showed very good results in the prediction of the skin-friction

coefficient for ��� up to 80.

Kalitzin et al. [34] tested the accuracy of the hybrid WF by eliminating the numerical error inherent

to coarse grids by using a so-called “delta-grid” in which the wall integration grid is shifted upwards

by a distance (delta) to provide the desired ��� at the first cell for the WF. It was shown that the

12

computations of the flow over a flat plate collapsed onto the wall integration profile when using the

“delta-grid” for ��� values ranging from 0.11 up to 111. The same computations using coarse grids

showed relevant deviations from the wall integration profile. Those deviations were dependent on the

��� value and the turbulence model. It is this numerical error that constitutes the second main goal of

this thesis.

Most of the commercial grid generators available allow the user to choose the dimensionless

distance to the wall of the first node (���) but construct the grid so that the grid line spacing increases

progressively as the distance to the wall increases. This is usually done by means of a geometric

progression. For values of ��� greater than 1, where the WF method is justifiable, this way of

generating a grid leads to very few nodes inside the boundary layer thus lacking the resolution needed

to compute the high gradients that exist in that region. This thesis intends to study a more accurate

approach for grid generation by eliminating a number of grid lines adjacent to the wall while

maintaining the high resolution above the first volume. This grid generation method is then applied to

two different airfoils and tested for different angles of attack and Reynolds numbers. The goal was to

obtain the aerodynamic characteristics of the airfoils, viz. the lift and drag coefficients and the skin

friction and pressure distributions over the airfoil upper and lower surfaces.

2.3 Implementation of Wall Functions in ReFRESCO

As mentioned in section 2.2, the WF implemented in ReFRESCO make use of a blending of the

VSL and log-layer behaviours for the velocity and turbulence variables [31]. When WF boundary

conditions are applied the effective viscosity at the wall is modified to obtain the correct wall shear-

stress. This is done using the assumption that the shear-stress is constant in the near-wall region, i.e.

z = z� = PXJ�Y� (2.20)

The shear stress in the first grid node above the wall is defined as

z�P = XS + ScY� J��� = J�� (2.21)

where the subscript 2 indicates values in the first interior grid node. The eddy-viscosity is then tuned at

the wall so that the correct z� is obtained:

13

Sc = J�� ��J� − S (2.22)

However, the friction velocity is not known so it has to be determined. The detailed procedure for

the numerical determination of J� is presented in [8]. It uses equation (2.17) to compute ���. The

friction velocity is calculated from the velocity magnitude, the rate of dissipation of turbulence � is

fixed at the first interior grid node away from the wall and the turbulence kinetic energy � is solved in

the near-wall cell with a corrected production term based on z�.

2.4 Airfoils

Airfoils (or aerofoils) are two-dimensional representations of lifting devices, e.g. the cross-section

of an airplane wing, that, due to its curved shape, impose curvature on the streamlines of a fluid

flowing around the airfoil. An example of a cambered airfoil is shown in Figure 2.1. The airfoil

curvature causes the static pressure to be higher on the lower surface (LS) than on the upper surface

(US) thus creating a net force, called the lift force [36]. The existence of viscosity leads to the

appearance of a force component in the streamwise direction: the drag force, which (in normal

operating conditions of the airfoil) is much lower than the lift component.

Figure 2.1 – Airfoil features and designations

The aerodynamic forces are usually presented in its non-dimensional form, i.e. the lift (¡ª) and

drag (¡«) coefficients:

¡ª = ¬12P�7� � (2.23)

and

L

�

14

¡« = ­12P�7� � (2.24)

where �7 is the free-stream velocity and � is the chord of the airfoil.

As for the friction and pressure distributions, these are local quantities and are defined in the

following way:

¡® = Q − Q712P�7� (2.25)

where Q is the static pressure in the airfoil surface and Q7 is the static pressure in the free-stream flow.

¡h = z�12P�7� (2.26)

with z� the shear-stress at the wall calculated from

z� = n I�I�¯°±� (2.27)

where � is a coordinate perpendicular to the wall and � = 0 represents the wall. One important note is

that ¡h is a direct measure of the wall shear-stress which, in the WF method, is calculated using the

law-of-the-wall as mentioned in section (2.2). Thus, it is natural that ¡h is more sensitive to the

accuracy of the calculation of z� than the remaining parameters.

Two airfoils were studied in this thesis: a conventional airfoil, the NACA 0012, and a laminar

airfoil, the Eppler 374. The NACA 0012 is a symmetrical (without camber) airfoil with a maximum

thickness of 12% of the chord, designed by the National Advisory Committee for Aeronautics

(NACA), now known as NASA. It is one of the most widely used airfoils in verification and validation

exercises and a great quantity of experimental data is available.

The Eppler 374 is a laminar airfoil, meaning that, in comparison to conventional airfoils, the point

of maximum thickness is located further from the leading edge, thus increasing the length of the

region with favourable pressure gradient and, consequently, the length of the laminar part of the flow.

The two airfoils are shown in Figure 2.2.

15

Figure 2.2 – Geometry of the NACA 0012 and Eppler 374 airfoils

These airfoils were studied for two Reynolds numbers, ��, based on the chord, i.e.

�� = �7�S (2.28)

The objective is two assess the influence of WF in two distinct conditions: one where the �� is

sufficiently high so that transition occurs close to the leading edge resulting in a flow that is turbulent

in almost all chord length; and another where the �� is low enough for the transition to occur further

from the leading edge, that is, in the region of adverse pressure gradient. Since the transition is

handled by the turbulence model it is expected that the predicted transition point occurs sooner that in

the real flow [37]. With this in mind, the following Reynolds numbers were selected:

Table 2.1 – Reynolds number used in the calculations of each airfoil

Airfoil Chord-based Reynolds number

NACA 0012 6 × 10¨

Eppler 374 3 × 10 

The choice for these two particular �� resides on the existence of experimental data for each airfoil

with which the numerical results could be compared.

NACA 0012 Eppler 374

17

Chapter 3

Grid Generation

This chapter presents the description of a grid generator developed in this thesis suitable to be used

with wall functions boundary conditions. Section 3.1 enumerates some of the properties desired in

viscous flows calculations and the reasons behind the choice of the type of grid used, i.e. a multi-block

structured grid. Section 3.2 explains the definition of the boundary lines of the computational domain

and the numerical methods for generating the interior grid. Section 3.3 addresses the interpolation

procedure that generates a grid that is refined in the near-wall region and becomes gradually coarser as

the distance to the wall increases. This final grid is then used to generate the two types of grids that

will be the object of study of this thesis: the Standard grid and a new and more accurate approach, the

W-grid.

3.1 Choice of grid configuration

CFD calculations comprise three main steps: pre-processing, solving and post-processing. The first

step is as important as the other two since it consists of defining the computational domain and the

grid on which the discretized equations of mass conservation and momentum (and possibly energy)

balance are solved. A good grid is essential for a good quality of the solution because it has to

guarantee that all the important features of the flow are adequately solved. Some of those features

(mainly if dealing with turbulent flows) exhibit large gradients (e.g. boundary layer), thus the grid has

to be sufficiently dense in such regions so that the numerical approximation is an accurate one but it

cannot be so dense that the solution is impossible (or too costly) to obtain. It should ensure a

satisfactory relation between computational cost and solution quality, meaning that grids that are

unnecessarily fine in the whole domain will result in a computational time that could easily be reduced

without damaging the quality of the solution. Therefore, generating grids for complex flows such as

turbulent ones can be (and it usually is) a very time consuming task.

One of the main goals of this thesis is to present an efficient approach to generate good quality

grids suitable for the application of wall functions boundary conditions. The concept of “good quality

18

grid” can be dependent on the problem but a few properties are desirable in most cases. In viscous

flow calculations, grid line orthogonality is highly desirable and is of major importance near solid

boundaries due to the boundary layer. Grid line distances close to surfaces are also very important

since they should be small where high resolution is needed but should increase smoothly when moving

away from the wall. Curved surfaces can be problematic when controlling grid line distances and

require special attention.

The first decision one has to make when generating a grid is between using a structured or an

unstructured grid. The former consists of families of intersecting lines, one for each space dimension,

with the mesh points located at the intersection of one line of each family, whereas the latter consists

of arbitrary distributions of mesh points, connected by triangles, quadrilaterals or polygons (in a two-

dimensional problem).

Both types of grids have important advantages: from a CFD point of view, structured grids are

often more efficient in terms of accuracy, CPU time and memory requirement [1]. Also, in flows with

predominant gradients in one of the directions, high resolution is necessary in areas where such

gradients occur. It is much more difficult to generate accurate solutions if highly distorted cells (e.g.

triangles) are used in an unstructured grid. Moreover, structured curvilinear grids are easily generated

aligned with the predominant flow direction, which helps the convergence of the CFD solvers. Such

alignment is not possible in unstructured grids. Finally, the application of boundary conditions and

turbulence models benefit from a good definition of the direction normal to flow features such as walls

or wakes, which is easily obtained in structured grids. Regardless of its benefits, generating structured

grids in complex geometries can be extremely time-consuming. Unstructured grids offer the

possibility to develop automatic grid generation tools that significantly reduce the time spent on the

grid generation process.

From all mentioned advantages offered by structured grids, the most relevant for this work is the

control of the resolution in the near-wall region. Control of near-wall distances is crucial to study the

influence of the first cell height used in the WF method on the aerodynamic characteristics of airfoils.

Another important feature of structured grids is the possibility to add and remove lines from one or

both families of lines so that grids with different levels of refinement are geometrically similar and can

therefore be compared. In this thesis, each set of grids contains 9 systematically refined, geometrically

similar grids. Therefore, we will have data from 9 refinement levels which will be used to estimate the

numerical uncertainty using the method presented in [38]. The way geometrical similarity is obtained

is by parameterizing the grid when defining the boundaries, i.e. defining the number of points in all

boundary lines as a fraction of the points that define the airfoil surface, hereafter referred to as �.

The different � and the respective refinement ratio are listed in Table 3.1. The refinement ratio is

defined as

19

X

Y

-12 0 12 24-12

0

12

X

Y

-0.5 0 0.5 1 1.5-0.7

0

0.7

ℎKℎu = ³�J´R����������������Mµ����J´R�����������µ���� (3.1)

And represents the ratio between typical cell sizes (ℎ) of grid � and the finest grid.

Table 3.1 – � and refinement ratio of each grid

;<= 241 301 361 421 481 601 721 841 961 :�/:% 4 3.20 2.67 2.29 2 1.60 1.33 1.14 1

In cases where a structured arrangement is necessary but the geometry is not simple, a common

practice is to divide the domain into regions called “blocks” that are generated separately and whose

boundary points are matched to the boundaries of neighbouring blocks. These types of grids offer

higher flexibility since it is possible to refine the grid only where it is necessary without it propagating

unnecessarily to other parts of the domain.

3.2 Generation of the base grid

The grids used in this thesis can be described as having two distinct regions: an inner C-shaped

region where the shape of the airfoil is defined and rotates according to the angle of attack; and an

outer rectangular region where the external boundary conditions are defined. Figures 3.1a and 3.1b

show the computational domain and the two regions mentioned for � equal to half of the coarsest

grid, i.e. � = 121.

(a) (b)

Figure 3.1 – Computational domain (a) and close-up of interior C-shaped grid (b)

20

The following procedure is used to generate the base grids for each airfoil:

1) Points are distributed on the boundary of an inner C-shaped block (Block I) and along the

surface of the airfoil. The method for the distribution of points on the boundaries of Blocks I to V is

explained afterwards.

2) An orthogonal grid generator based on a system of partial differential equations [39] is used to

obtain a smooth distribution of points along the boundaries of Block I.

3) An elliptical grid generator with control functions based on the Grape approach [40] (which

gives a good compromise between grid line distances and orthogonality as shown in [41]) is used to

obtain the interior grid.

4) Once the grid of Block I is generated, the boundaries of the remaining blocks are defined,

starting from Block II on the top, followed by Block III on the bottom, Block IV on the left and,

finally, Block V on the right, as shown in Figure 3.2. The nodes on the interfaces of different blocks

are matched.

Figure 3.2 – Blocks of base grid and its designations

5) The orthogonal grid generator and the Grape approach mentioned in 2) and 3) are used to

generate the grid of Block II in a similar way to that of Block I.

6) As for the remaining Blocks, only the Grape approach is used since the boundary nodes are

defined by Blocks I and II.

Let us first address step 1) of the procedure listed above.

3.2.1 Block I

Block I is a C-shaped block that one can imagine as a rectangle in the computational domain (¶, ·)

that is transformed into a C on the physical domain (L, �). A simple representation of this

transformation is shown in Figure 3.3:

X

Y

-0.5 0 0.5 1 1.5-1

-0.5

0

0.5

1

V

II

III

I

IV

21

Figure 3.3 – Transformation from computational (left) to physical (right) domain

Block I rotates according to the angle of attack that is being studied. The other blocks are adapted

to this rotation by defining the points on which the different blocks are matched to Block I. This is

addressed afterwards.

The first step to construct Block I is defining the shape of the airfoil, which represents part of the

bottom boundary of the computational domain. In this thesis, this is performed in two distinct ways.

For the NACA 0012, the equations that define the thickness distribution (3.2) and the mean camber

line (3.3a) and (3.3b) for the NACA 4-digit series airfoils [42] are used to create the surface.

�c = M0.20 \0.29690√L − 0.12600L − 0.35160L� + 0.28430Lº − 0.10150L�] (3.2)

where M is the maximum thickness in percentage of the chord (0.12 in this case) and L is the

position along the chord. The thickness is then added perpendicularly to the mean camber line, which

is defined by equations (3.3a) and (3.3b) for points before and after the point of maximum camber,

respectively.

�» = Q́� X2QL − L�Y (3.3a)

and

�» = ´X1 − QY� ¼X1 − 2QY + 2QL − L�½ (3.3b)

where ́ is the maximum camber in percentage of the chord (0 in this case since it is a symmetrical

airfoil) and Q is the location of the maximum camber in tenths of chord (also 0 in this case). A cosine

distribution is used in order to guarantee a finer distribution close to the leading and the trailing edges.

For the Eppler 374, the coordinates of a number of points from [43] are used to perform an

interpolation procedure described in [44] to add points to the surface.

4

1

B3

B1

B4

3

B2

2 4

1

B3

B1B4

3

B2

2L

�

·

¶

22

Once the surface of the airfoil is defined, the next step is to create the other lines of Block I. The

boundaries of Block I and their designations are shown in Figure 3.3. The first line is the region of the

near wake from the trailing edge to the point that defines the right boundary of Block I, which is

located at 1.1 chords from the leading edge. The near-wake contains X� − 1Y/6 + 1 points and

their distribution is defined by a stretching function [45] with the purpose of making a smooth

transition from one region to the other so that there is not a sudden increase or decrease in the size of

two adjacent elements. This stretching function uses a reference element to determine the size of the

first element of the line on which the function is used. In this region, the reference element is the last

element on the trailing edge. The first boundary of Block I is now defined and consists of a total of

4/3 ∗ � − 1/3 points.

The outer boundary of Block I consists of:

B2: vertical line with X� − 1Y/6 + 1 points uniformly distributed.

B3: two horizontal lines at a distance of 0.25 chords above and below the chord, plus a semicircle.

It contains 4/3 ∗ � − 1/3 points. For the horizontal lines, the region directly above and below

boundary B1 has the same point distribution as B1. For the rest of B3, the distribution of points is

tuned automatically so that each point corresponds to one point at the airfoil surface.

B4: The same as B2.

This completes the definition of all the boundaries of Block I, which contains 321x41 points (321

in the ¶-direction and 41 in the ·-direction) in the coarsest grid and 1281x161 in the finest. At this

point, step 1) of the grid generation procedure presented in section 3.2 is complete and the interior grid

for Block I is generated as mentioned in steps 2) and 3). First, the grid is calculated using an

orthogonal grid generator based on a system of partial differential equations [36] to obtain a smooth

distribution of grid nodes along the outer boundaries with good orthogonality on the wall. That

distribution is then fixed and used as boundary condition on an elliptical grid generator with control

functions based on the Grape approach [40] which gives a good compromise between grid line

distances and orthogonality. The other 4 blocks are constructed based on the distribution of points on

the outer boundary of Block I. The procedure for the construction of Blocks II to V is explained next.

3.2.2 Blocks II and V

Block II is located above Block I and is divided in three sub-blocks: II-A, II-B and II-C. The centre

sub-block (II-B) is located directly above Block I so the point distribution on the bottom boundary of

this sub-block coincides with the one on the top boundary of Block I. The remaining boundaries of

Block II are defined using stretching functions based on different reference elements. Block II

contains 309x41 points in the coarsest grid and 1233x161 in the finest. The boundaries on Block II are

23

then completely defined and the exact same procedure is done symmetrically for the block below

Block I (Block V).

3.2.3 Blocks III and IV

The last two blocks (Block III and Block IV) have all their boundaries defined except one: the inlet

boundary in Block III and the outlet boundary in Block IV. In these boundaries, a two-sided stretching

function is used so it takes as reference elements the closest element in Block II and Block V. Block

III has 81x25 grid nodes in the coarsest grid and 321x97 in the finest and Block V contains 81x81 and

321x321 points in the coarsest and the finest grid, respectively. This completes the definition of all the

boundaries in the domain and steps 5) and 6) are executed to create the grid of Blocks II to V. Table

3.4 shows the number of nodes in each block in directions ¶ and · for the coarsest and finest grids as

well as the total number of volumes.

Table 3.2 – Grid lines in each block and total number of volumes of coarsest and finest grid

3.3 Interpolation for near-wall refinement

The base grid mentioned in the previous section was constructed in order to ensure good grid

characteristics, namely orthogonality and grid line distances. However, this grid does not take into

account the need for very small size of the near-wall cells. It is then necessary to modify it in order to

fulfil such requirement. With this purpose, the base grid is used to perform an interpolation routine in

order to create a new grid which is fine close to the wall and becomes gradually coarser as the distance

to the wall increases.

Interpolation is used for near-wall refinement on Block I and to add or remove lines from the

remaining blocks, maintaining the node distribution of the base grid. The C shape chosen for Block I

requires to use the same node clustering in the centreline of Block IV XL = 0Y. For this reason, Blocks

I and IV are put in one single block, hereafter referred to only as Block I.

The interpolation procedure consists of a sweep along the ¶-direction, i.e. lines of ·=constant,

followed by a sweep in the ·-direction. Both sweeps are equivalent and comprise the following steps:

;<= = '¾% ;<= = ¿À%

Block I 441x81 1761x321

Block II 269x41 1073x161

Block III 269x41 1073x161

Block IV 61x25 241x97

# of volumes 58080 929280

24

1) A spline representation of the line in the XL, �Y coordinate system is created for an independent

variable s from 1 to NP, where NP is the number of points of that line in the base grid.

2) The points are then distributed along the spline according to the desired node distribution.

More or less points than the existing on the base grid can be use in this step.

3) Finally, the points in the XL, �Y coordinate system are determined by interpolation.

The second step of this procedure determines the type of grid on which the WF method is used.

Two alternatives exist: the “standard” approach and a new approach, called W-grid that seeks to

minimize the numerical error inherent to the “standard” case.

3.3.1 The W-grid

The W-grid is generated by using a stretching function in the sweep in the ·-direction of Block I

and then removing a number of lines ·=constant adjacent to the wall. The stretching parameter (ratio

between desired first cell height and equidistant spacing) is tuned to produce the desired value of ���.

The near-wall spacing is controlled by multiplying this stretching parameter by a factor Á. Five grid

sets are created with different Á values. Each set and the respective near-wall spacing factor are listed

in Table 3.2.

Table 3.3 – Near-wall spacing factor

Set A Set B Set C Set D Set E A 20 10 50 100 150

The difference between the base grid and the W-grid with no lines removed is shown in Figure 3.4.

(a) Base grid

(b) W-grid with no lines removed

Figure 3.4 – Close-up of the trailing edge of base and W-grid for � = 121

X

Y

0.8 0.85 0.9 0.95 1 1.05 1.1-0.15

-0.1

-0.05

0

0.05

0.1

0.15

X

Y

0.8 0.85 0.9 0.95 1 1.05 1.1-0.15

-0.1

-0.05

0

0.05

0.1

0.15

25

The choice for a multi-block structured grid allows for an easy elimination of grid lines which

represents one of the reasons this arrangement was chosen. By removing more or less lines, one can

choose the location of the first node above the wall where the wall-function boundary condition is

applied. For different refinement levels, a different number of lines are removed in order to achieve

approximately the same value of ��� in each group. Each group of grids with the same (dimensional,

not ��) distance to the wall is represented by w** , where ** represents the number of lines removed

in the finest grid (� = 961) of that group. Obviously, the correspondent ��of each group will be

similar for the different grid refinements, but not exactly the same, since it depends on the calculation

of the friction, as shown in equation (2.15a). The number of lines removed for each refinement level

and each group is shown in Table 3.4. The first column of Table 3.4 represents the different groups

(being w16 the group with fewer lines removed, thus the one with lower ��� and w160 the one with

more lines removed) and the first line represents the different refinement levels (indicated by the �,

which ranges from 241 up to 961, hence covering a refinement ratio of 4).

Table 3.4 – Number of lines removed in each refinement level and for each w-group

241 301 361 421 481 601 721 841 961

w16 4 5 6 7 8 10 12 14 16

w32 8 10 12 14 16 20 24 28 32

w48 12 15 18 21 24 30 36 42 48

w64 16 20 24 28 32 40 48 56 64

w80 20 25 30 35 40 50 60 70 80

w96 24 30 36 42 48 60 72 84 96

w112 28 35 42 49 56 70 84 98 112

w128 32 40 48 56 64 80 96 112 128

w160 40 50 60 70 80 100 120 140 160

The number of lines removed is a function of the � which again shows the importance of using

geometrically similar grids. Each group is calculated with all the 9 refinement levels in order to

estimate the uncertainty of the calculations. The W-grid with no lines removed and with 16 lines

removed (with � = 121) is depicted in Figure 3.5. Two important features are obtained when

constructing the WF grids in this way: possibility to easily choose the value of ��� (simply by choosing

how many lines are removed); and to keep the small size of the cells immediately above the first one

which are necessary to proper resolve the high gradients in the near-wall region. Since one of the goals

of WF is to reduce the number of cells, thus saving time in the computations, Table 3.5 shows the

number of cells in the finest grid (� = 961) of each group (w16 to w160), as well as the relative

difference in total cells number from the grid with no lines removed.

26

(a) W-grid with no lines removed (b) W-grid with 16 lines removed

Figure 3.5 – Close-up of the trailing edge of W-grid for � = 121

Table 3.5 – Number of volumes and relative difference of each w-group to W-grid with no lines removed

w16 w32 w48 w64 w80 w96 w112 w128 w160

# of volumes 901120 872960 844800 816640 788480 760320 732160 704000 647680

Savings (%) 3.0 6.1 9.1 12.1 15.2 18.2 21.2 24.2 30.3

3.3.2 The Standard grid

The second goal of this thesis is to compare the results obtained with the W-grid with those from

the Standard approach. Most commercial grid generators, when dealing with WF, allow the user to

choose the ��� but construct the grid with increasing grid line distances, resulting in too coarse grids.

In this work, this type of grids is generated by using a stretching function with the first element size

being the same as in the equivalent W-grid. The difference between the W-grid and the Standard grids

is illustrated in Figure 3.6.

X

Y

0.95 1 1.05-0.06

0

0.06

X

Y

0.95 1 1.05-0.06

0

0.06

27

(a) W-grid

(b) Standard grid

Figure 3.6 – Close-up of the trailing edge of W-grid and Standard grid for � = 241

It was mentioned previously that it is desired to have grid lines aligned with the flow and it was one

of the reasons a C-shaped configuration for the inner block was chosen. However, when the W-grid or

the Standard grid line distribution is used, this results in a region immediately downstream of the

trailing edge with too large grid line distances. Since in this near-wake region, high gradients are still

present, this gap may produce significant numerical error. This issue could be avoided by using locally

an O-grid [5], but then the grid lines would not be aligned with the flow, so some kind of transition to

a C-shape would be necessary, resulting in additional work that is unnecessary considering the

objectives of this study.

X

Y

0.95 1 1.05-0.06

0

0.06

X

Y

0.95 1 1.05-0.06

0

0.06

29

Chapter 4

Results

In this chapter the results obtained with the wall-functions boundary conditions for the two airfoils

with each grid type are presented and analysed. The calculations were also performed with the direct

application of the no-slip condition to serve as reference for the WF results. First of all, a simple

description of the numerical settings of the calculations is presented in section 4.1. The results for the

NACA 0012 at zero and four degrees of angle of attack are presented in sections 4.2 and 4.3,

respectively. The results for the Eppler 374 are presented in section 4.4.

4.1 Calculations settings

All the calculations of the NACA 0012 and Eppler 374 airfoils were performed in a rectangular

domain of 24 chords in the vertical direction and 36 chords in the horizontal direction. For the

conventional airfoil two angles of attack, �, were studied: zero and four degrees. As for the laminar

airfoil, only a zero degrees angle of attack was calculated. The boundary conditions used were the

following:

- Inlet: all three velocity components are given by a Dirichlet boundary condition (only the

horizontal component has non-zero value), the pressure is extrapolated from the interior of the

domain using a zeroth-order extrapolation (meaning that the boundary value is taken equal to the

nearest cell center) and the freestream values of the turbulence kinetic energy � and the turbulence

dissipation k are prescribed as follows: for the NACA 0012 computations, � = 10��7� and

k = 9 ∗ 10�Ã�7º /¬. For the Eppler 374 calculations, these values were changed to � =1.5 ∗ 10���7� and k = 1.251�7º /¬ because the turbulence kinetic energy was too low.

- Top: the pressure and the velocity component normal to the boundary are extrapolated from

the interior of the domain using zeroth order extrapolation. The variables of the turbulence model

(� and �) are also extrapolated from the interior of the domain.

- Bottom: the same as in the top boundary.

30

- Airfoil surface: impermeability condition (velocity component normal to the wall is set to zero

as Dirichlet boundary condition). Calculations with direct application of the no-slip condition

(“No-WF” calculations) compute the wall shear-stress directly from its definition (from the normal

velocity gradient) and wall functions calculations use the law-of-the-wall to compute the wall-shear

stress and tune the eddy-viscosity at the wall to obtain the correct z�.

- Outlet: zeroth order extrapolation is used for the convective fluxes.

Since all boundaries contain zero normal derivatives for the pressure, it has to be prescribed in at

least one point in the domain. That point was located in the upper left corner of the domain (on the

inlet boundary) in the NACA 0012 calculations and was changed to the upper right corner (on the

outlet boundary) in the Eppler 374 calculations because although the results do not depend on the

location of this point, it helped the iterative convergence properties. Its value is set to zero.

The mass and momentum equations are solved with a segregated approach that solves the

governing equation (e.g. mass, momentum, turbulence models, etc) separately in each outer loop

iteration. The convergence tolerance of the calculations varied from 2.5 ∗ 10� to 5 ∗ 10�¨ (the higher

values were used in almost all simulations of the Eppler airfoil due to iterative convergence difficulties

experienced throughout this work) with the normalisation of the residual performed with the L-infinity

norm (maximum value) of the normalized residual (equivalent to the variable change in a single Jacobi

iteration) of all transport equations.

The under-relaxation parameters were tuned to achieve iterative convergence criteria. For this, the

minimum implicit sub-relaxation varied from 0.5 to 0.7 and the maximum varied from 0.96 from 0.98.

The explicit sub-relaxation varied from 0.1 to 0.3. The discretization schemes used for the convective

fluxes were the QUICK scheme for the momentum equations and the 1st order upwind differencing

scheme for the turbulence model equations. Gradients were computed using Gauss’ theorem.

The first exercise was performed for the NACA 0012 airfoil at angles of attack of 0 and 4 degrees

with the purpose of calculating the lift and drag coefficients and the friction and pressure distributions

along the chord of the airfoil. The lift and drag forces are calculated as a sum of two components:

- One resultant from the integration of the wall shear-stress along the airfoil surface, indicated

by the subscript ��; - And a second component due to the integration of the pressure along the airfoil surface,

indicated by the subscript Q�.

The convergence of each force with the grid refinement is presented for groups w0 (no lines

removed from the grid suitable for direct application of the no-slip condition, but with WF boundary

conditions applied) to w160 of set A. The variation of the forces as a function of the maximum ���

31

(which is simply represented hereafter as ���, for simplicity) is illustrated for the W-grid and the

Standard grid. The results are presented in the form of deviation relative to a reference value taken

from solutions with direct application of the no-slip condition, hereafter called “No-WF” solutions.

4.2 Flow over the conventional airfoil NACA 0012 at � = 0º 4.2.1 W-grid

The calculations for this angle of attack were performed for all the groups of the five sets (A to E).

A total of 26 groups from various sets were then selected in order to accurately define curves for the

variation of the aerodynamic forces with ���. The selected groups are listed in Table 4.1 as well as the

correspondent ���. Each group is named as the letter of the set it belongs to, followed by the group,

e.g. group w64 from set A is listed as A64.

Table 4.1 - 1'� value of each group used in NACA 0012 calculations for � = 0º A0 B16 A16 B32 A32 B48 A48 E16 A64 D32 A80 D48 A96 1'� 0.09 0.97 1.9 2.6 5.4 6.0 11 15 20 24 36 46 62

D64 A112 D80 A128 D96 E96 D112 E112 A160 E128 C160 D160 E160 1'� 79 105 128 173 201 267 306 396 452 581 709 998 1219

Figure 4.1 illustrates the convergence of the pressure and friction components of ¡« and the total

Drag and Lift coefficients with the grid refinement for the NACA 0012 at an angle of attack of zero

degrees. The results show a significant sensitivity of the ¡«,ÄÄ to the non-dimensional distance of the

first grid node to the wall. All groups show monotonic convergence, although 2 groups do not allow

the determination of the observed order of grid convergence (Q). The other groups show values of Q

between 0.86 and 1.73. As for the ¡«,®Å, only 3 groups show monotonic convergence and the observed

order of grid convergence of those groups ranges from 0.73 to 1.49. The asymptotic values (when

ℎK ℎu⁄ → 0) show a small sensitivity to the ��� value with the exception of the group with the highest

��� (A160), which has a considerably larger asymptotic value of ¡«,®Å. The total Drag results show

that monotonic convergence is not always present and the ��� sensitivity is significant. This is not

surprising since this force is the sum of the two previous contributions, thus the convergence

properties are a consequence of the properties of the friction and pressure components. The total Lift

coefficient is presented for illustrative purposes only. Since the iterative error in these calculations was

not reduced to the machine’s accuracy, the results of quantities with values very close to zero (as in

32

the case of the ¡ª of a symmetrical airfoil at an angle of attack of zero degrees) illustrate the level of

the iterative error, resulting in a large noise in the calculation of such quantities as can be seen in

Figure 4.1d.

(a) Friction component of drag coefficient

(b) Pressure component of drag coefficient

(c) Total drag coefficient

(d) Total lift coefficient

Figure 4.1 – Grid convergence of aerodynamic coefficients as function of grid refinement ratio for W-grid of NACA 0012 airfoil at � = 0º

Figure 4.2a shows the distribution of the friction coefficient, ¡h, along the chord and Figure 4.2b

presents a detail of the leading edge region for 0 ≤ L/� ≤ 0.1. One important result is observed in

Fig. 4.2b. The No-WF results show very good agreement with the result from the extensively tested

CFL3D code with the SST turbulence model [46] along the chord, including the prediction of the

transition from laminar to turbulent flow at approximately 1% of the chord. However, the solution

obtained with WF does not exhibit any transition point, instead assuming turbulent flow from the

hi/h1

CD

,ss

0 0.5 1 1.5 2 2.5 3 3.5 40.006

0.0065

0.007

0.0075

0.008

0.0085A0p=1.27A16p=0.87A32p=0.86

hi/h1

CD

,ss

0 0.5 1 1.5 2 2.5 3 3.5 40.006

0.0065

0.007

0.0075

0.008

0.0085A48ah²A64p=1.23A80p=1.73

hi/h1

CD

,ss

0 0.5 1 1.5 2 2.5 3 3.5 40.006

0.0065

0.007

0.0075

0.008

0.0085A96ah²A112p=1.68A128p=1.22A160p=1.27

hi/h1

CD

,pr

0 0.5 1 1.5 2 2.5 3 3.5 40

0.001

0.002

0.003

0.004

0.005A0a1h+a2h²A16a1h+a2h²A32ah²

hi/h1

CD

,pr

0 0.5 1 1.5 2 2.5 3 3.5 40

0.001

0.002

0.003

0.004

0.005A48ah²A64a1h+a2h²A80ah²

hi/h1

CD

,pr

0 0.5 1 1.5 2 2.5 3 3.5 40

0.001

0.002

0.003

0.004

0.005A96ah²A112p=1.49A128p=0.97A160p=0.73

hi/h1

CD

0 0.5 1 1.5 2 2.5 3 3.5 40.007

0.008

0.009

0.01

0.011

0.012 A0a1h+a2h²A16a1h+a2h²A32p=1.69

hi/h1

CD

0 0.5 1 1.5 2 2.5 3 3.5 40.007

0.008

0.009

0.01

0.011

0.012 A48p=0.90A64a1h+a2h²A80a1h+a2h²

hi/h1

CD

0 0.5 1 1.5 2 2.5 3 3.5 40.007

0.008

0.009

0.01

0.011

0.012A96p=1.96A112p=1.34A128p=0.73A160a1h+a2h²

hi/h1

CL

0 0.5 1 1.5 2 2.5 3 3.5 4-1E-06

0

1E-06

2E-06

3E-06

4E-06

5E-06

6E-06

7E-06

8E-06 A0p=1.86A16p=1.69A32p=1.91

hi/h1

CL

0 0.5 1 1.5 2 2.5 3 3.5 4-1E-06

0

1E-06

2E-06

3E-06

4E-06

5E-06

6E-06

7E-06

8E-06 A48p=1.66A64p=1.71A80p=1.97

hi/h1

CL

0 0.5 1 1.5 2 2.5 3 3.5 4-1E-06

0

1E-06

2E-06

3E-06

4E-06

5E-06

6E-06

7E-06

8E-06A96p=1.95A112p=1.97A128ah²A160p=1.73

33

leading edge onwards. For L/� > 0.10, the result from A96 is in good agreement with the remaining

two results.

The friction distribution of groups A16, A48, A96 and A112 is shown in Figure 4.3. These groups

are presented because the respective ��� values are located in the most relevant regions of the law-of-

the-wall: the viscous sub-layer (A16), the buffer-layer (A48), the log-layer (A62) and the outer edge of

the log-layer (A112). The No-WF solution is also shown for comparison. Figure 4.3a and 4.3b show

the absolute ¡h values whereas Fig. 4.3c shows the difference of each group to the No-WF solution.

The highest differences are observed in the first 20% of the chord. None of the WF solutions show a

transition point and group A48 (��� = 11) shows significant deviation from the No-WF solution in the

majority of the chord length.

(a) Friction coefficient for 0 < L/� < 1

(b) Friction coefficient for 0 < L/� < 0.1

Figure 4.2 – Skin friction coefficient distribution of group A96, No-WF and CFL3D for NACA 0012 at � = 0º

The pressure distribution over the surface of the NACA 0012 of the No-WF solution, group A96

and CFL3D [46] is depicted in Figure 4.4a. The difference of the same 4 groups as in the ¡h to the No-

WF solution is illustrated in Figure 4.4b. Group A96 and the No-WF solution show excellent

agreement with the results from [46]. The highest deviations of the four groups with WF boundary

conditions to the No-WF are close to the pressure peak and to the trailing edge. In the first region, the

velocity profile is likely to be very different from the conditions for which the law-of-the-wall is

derived, hence the differences to the solution obtained with the wall shear stress computed directly on

the surface. The second region is poorly resolved because the grid configuration (C-shape) leads to a

“gap” in the wake region when lines are removed for the application of WF boundary conditions.

x/c

Cf(x

10³)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

U.S. A96 (y+=62)U.S. No WF (y+=0.04)U.S. CFL3D [46]

x/c

Cf(x

10³)

0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

U.S. A96 (y+=62)U.S. No WF (y+=0.04)U.S. CFL3D [46]

34

(a) Friction coefficient for 0 < L/� < 1

(b) Friction coefficient for 0 < L/� < 0.1

(c) Relative deviation from No-WF solution

Figure 4.3 – Skin friction coefficient distribution and relative deviation from No-WF solution of W-grid for NACA 0012 at � = 0º

(a) Pressure distribution

(b) Relative deviation from No-WF solution

Figure 4.4 – Pressure coefficient distribution and relative deviation from No-WF for W-grid of NACA 0012 at � = 0º

x/c

Cf(x

10³)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

U.S. A16 (y+=2)U.S. A48 (y+=11)U.S. A96 (y+=62)U.S. A112 (y+=105)U.S. No WF (y+=0.04)

x/c

Cf(x

10³)

0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

U.S. A16 (y+=2)U.S. A48 (y+=11)U.S. A96 (y+=62)U.S. A112 (y+=105)U.S. No WF (y+=0.04)

x/c

∆Cf(x

10³)

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

U.S. A16 (y+=2)U.S. A48 (y+=11)U.S. A96 (y+=62)U.S. A112 (y+=105)

x/c

Cp

0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

U.S. A96 (y+=62)U.S. No WF (y+=0.04)U.S. CFL3D [46]

x/c

∆Cp

0 0.2 0.4 0.6 0.8 1-0.004

-0.002

0

0.002

0.004

U.S. A16 (y+=2)U.S. A48 (y+=11)U.S. A96 (y+=62)U.S. A112 (y+=105)

35

4.2.2 Standard grid

The friction coefficient is depicted in Figure 4.5 as a function of the chord length. Figure 4.5a and

4.5b show group A96n (group 96 from set A with the Standard grid) (��� = 60) compared to the No-

WF and the CFL3D [46] solutions. Figure 4.5c presents the difference of ¡h of 4 selected groups (the

same as in section 4.2.1) to the No-WF solution. Similarly to the W-grid, the flow is treated as fully

turbulent since the leading edge when WF are applied, which translates in considerable deviations

from the No-WF solution, mainly in the first 20% of the chord. Groups A16n and A48n give the

higher deviations whereas group A96n yields the smallest.

(a) Skin friction coefficient for 0 < L/� < 1

(b) Skin friction coefficient for 0 < L/� < 0.1

(c) Relative deviation from No-WF

Figure 4.5 – Skin friction coefficient distribution and relative deviation from No-WF solution of Standard grid for NACA 0012 at � = 0º

x/c

Cf(x

10³)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

U.S. A96n (y+=60)U.S. No WF (y+=0.04)U.S. CFL3D [46]

x/c

Cf(x

10³)

0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

6

7

8

U.S. A96n (y+=60)U.S. No WF (y+=0.04)U.S. CFL3D [46]

x/c

∆Cf(x

10³)

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

U.S. A16n (y+=2)U.S. A48n (y+=11)U.S. A96n (y+=60)U.S. A112n (y+=102)

36

Figure 4.6 presents the pressure coefficient distribution along the chord. Group A96n is compared

in Figure 4.6a to the No-WF and the CFL3D solutions and very good agreement is observed between

these 3 results. Figure 4.6b shows the differences of solutions obtained with WF to the No-WF

solution. Very small differences are observed in the mid-chord region (0.2 ≤ L/� ≤ 0.8) for all

groups. Higher differences are obtained near the leading and trailing edges but are still only 1%

different, approximately.

(a) Pressure coefficent

(b) Relative deviation from No-WF

Figure 4.6 – Pressure coefficient distribution and relative deviation from No-WF for Standard grid of NACA 0012 at � = 0º

4.2.3 Comparison to No-WF solution and experimental results

In this section the results obtained with the Standard and the W-grid are compared to those

obtained without WF, i.e. with the direct application of the no-slip condition. The variation of each

aerodynamic force with the ��� is presented as a relative deviation, using as reference the respective

value obtained without WF (set B with ��� = 0.04), calculated in the following form:

�X∅Y = ∆∅X%Y = ∅ÌÍ − ∅Ågh∅Ågh × 100 (4.1)

where � is the relative deviation, ∅ is the force component that is being presented and ∅ÌÍ and ∅Ågh

designate the solutions obtained with and without WF, respectively. The references values of each

force component and its estimated uncertainty are presented in Table 4.2.

x/c

Cp

0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

U.S. A96n (y+=60)U.S. No WF (y+=0.04)U.S. CFL3D [46]

x/c

∆Cp

0 0.2 0.4 0.6 0.8 1-0.004

-0.002

0

0.002

0.004

U.S. A16n (y+=2)U.S. A48n (y+=11)U.S. A96n (y+=60)U.S. A112n (y+=102)

37

Table 4.2 – Reference values and estimated uncertainties for NACA 0012 at � = 0º

The variation of the friction, pressure and total Drag coefficient with ��� is illustrated in Figure 4.7,

for the Standard and W-grid. The data points of the W-grid results are presented with its uncertainty,

which is estimated using error propagation in the following way:

�X�∅Y = I�∅I∅Ågh �\∅Ågh] + I�∅I∅ÌÍ �X∅ÌÍY (4.2)

where �XÎY is the estimated uncertainty of variable Î.

The ¡«,ÄÄ calculated with the W-grid show good results for ��� values roughly between 25 and 100

and for very small ��� values (in the order of 0.1). Outside these regions, the relative deviation reaches

8% in the region of the buffer layer and diverges from the reference solution for ��� values above 100

due to the small number of grid nodes that are used to resolve the boundary layer. The Standard grid

does not appear to yield satisfactory results in any location of the first grid node above the wall. For

��� between 25 and 100, where the W-grid produces results with less than 1% of relative deviation, the

Standard grid shows deviations from the reference solution greater than 2%.

Concerning the ¡«,®Å, from Figure 4.7b it is possible to observe that the two points with smallest

��� in the W-grid show very small deviation, but for such small values of ��� the WF method is not

necessary. A region of ��� values ranging from 25 to 80 yields very good results. Outside that region

the results again exhibit a peak in the buffer layer with deviations up to 10%. For ��� greater than 100

the ¡«,®Å is highly overpredicted. The Standard grid shows reasonable results for a narrow region of

��� from 35 to 100 with relative deviation between 2.5% and 3.5%. Outside this region the results are

very poor.

As for the total ¡«, for ��� values below 1, the W-grid shows small deviation from the solution

obtained with direct application of the no-slip condition. Good results are also obtained when the first

grid node above the wall is placed at ��� from 25 to 100. For this grid type the region of the buffer

layer shows a relative deviation of 8% at ��� = 11. For ��� greater than 100 the deviation increases

significantly and reaches values up to 50%, which are not shown in Figure 4.7c. The Standard grid

appears to yield reasonable results for ��� greater than 35 but it is only due to an error cancelation

effect, produced by the increase in the ¡«,®Å and the decrease in the ¡«,ÄÄ. It is important to note that

56,CC U(56,CC) 56,�B U(56,�B) 56 U(56)

0.00685 1.6x10-5 0.00135 1.3x10-4 0.00820 1.8x10-4

38

the region where the deviation is the lowest, namely ��� between 25 and 80, also yields the smallest

uncertainty values.

Figure 4.7d shows a comparison between the total Drag coefficient calculated with the Standard

and the W-grid and two reference values: a numerical result [46] and an experimental result [47].

Once more the ¡ª results are not presented due to the influence of the iterative error on the results. It is

confirmed that the location of the first node must be carefully chosen, i.e. one should avoid using a ���

between 5 and 25 (that is, the buffer layer) and greater than 80. Outside these regions, the results for

this airfoil at this angle of attack seem to be in good agreement with the references.

(a) Relative deviation of friction component of drag coefficient

(b) Relative deviation of pressure component of drag coefficient

(c) Relative deviation of total drag coefficient

(d) Total drag coefficient

Figure 4.7 – Variation of aerodynamic coefficients with ��� for NACA 0012 at � = 0º

y+max

∆CD

,ss(%

)

10-1 100 101 102 103-15

-10

-5

0

5

10

W-gridStandard grid

y+max

∆CD

,pr(%

)

10-1 100 101 102 103-15

-10

-5

0

5

10

15

20

25

30

W-gridStandard grid

y+max

∆CD

(%)

10-1 100 101 102 103-10

0

10

20

30

40

50

W-gridStandard grid

y+max

CD

10-1 100 101 102 1030.007

0.008

0.009

0.01W-gridStandard gridNo WFLadson [47]CFL3D [46]

39

4.3 Flow over the conventional airfoil NACA 0012 at � = 4º 4.3.1 W-grid

The calculations for � = 4º were performed only for groups from set A resulting in 10 groups.

These were considered enough to cover an appropriate range of ��� without taking too much

calculation time. The groups used and the respective ��� are listed in Table 4.3.

Table 4.3 - 1'� value of each group used in NACA 0012 calculations for � = 4º A0 A16 A32 A48 A64 A80 A96 A112 A128 A160 1'� 0.12 2.7 8.0 16 29 51 87 143 228 548

Figure 4.8 illustrates the convergence of the pressure, friction and total Drag coefficients, and of

the total Lift coefficient with the grid refinement for the NACA 0012 airfoil at an angle of attack of

four degrees. The convergence properties of the ¡«,ÄÄ show an improvement from the same force

component at zero degrees of angle of attack. All groups show monotonic convergence and the

observed order of grid convergence ranges from 0.61 to 1.99. The asymptotic values still show a

considerable dependence on the location of the first grid node above the wall, particularly in groups

A160 and A48. The former because the correspondent ��� is probably located outside the region of

validity of the log-law and the latter because it is located in the buffer-layer. As for the ¡«,®Å not all

groups show monotonic convergence and the group with the largest ��� shows values two times higher

than the rest of the groups, which show a small sensitivity to the ��� of the first node. The results for

the total Drag coefficient show that monotonic convergence is not present in all groups and the ¡«

calculated with group A160 yields a larger value than the remaining groups, due to the differences

referred in the analysis of the pressure component.

The results of the ¡ª,®Å show small sensitivity to the location of the first grid node above the wall,

the exception being group A160, which produces a lower value than the remaining groups. The

observed order of grid convergence varies from 0.54 to 1.99. The friction component of the Lift

coefficient is not presented because its value is sufficiently close to zero for the iterative error to be a

source of noise in the solution. Due to the very small values of the friction component of the ¡ª, the

results for the total ¡ª are very similar to the ¡ª,®Å, both in terms of sensitivity to the ��� and the

observed order of grid convergence.

40

(a) Friction component of drag coefficient

(b) Pressure component of drag coefficient

(c) Total drag coefficient

(d) Pressure component of lift coefficient

(e) Total lift coefficient

Figure 4.8 – Grid convergence of aerodynamic coefficients as function of grid refinement ratio for W-grid of NACA 0012 airfoil at � = 4º

hi/h1

CD

,ss

0 0.5 1 1.5 2 2.5 3 3.5 40.005

0.0055

0.006

0.0065

0.007

0.0075

0.008

0.0085

0.009A0p=1.33A16p=1.41A32p=0.61

hi/h1

CD

,ss

0 0.5 1 1.5 2 2.5 3 3.5 40.005

0.0055

0.006

0.0065

0.007

0.0075

0.008

0.0085

0.009A48p=1.99A64p=1.44A80p=1.89

hi/h1

CD

,ss

0 0.5 1 1.5 2 2.5 3 3.5 40.005

0.0055

0.006

0.0065

0.007

0.0075

0.008

0.0085

0.009A96p=1.70A112p=1.14A128a1h+a2h²A160a1h+a2h²

hi/h1

CD

,pr

0 0.5 1 1.5 2 2.5 3 3.5 40

0.002

0.004

0.006

0.008

0.01A0a1h+a2h²A16ah²A32ah²

hi/h1

CD

,pr

0 0.5 1 1.5 2 2.5 3 3.5 40

0.002

0.004

0.006

0.008

0.01A48ah²A64ah²A80ah²

hi/h1

CD

,pr

0 0.5 1 1.5 2 2.5 3 3.5 40

0.002

0.004

0.006

0.008

0.01 A96p=1.67A112p=1.10A128p=0.84A160p=0.59

hi/h1

CD

0 0.5 1 1.5 2 2.5 3 3.5 40.008

0.01

0.012

0.014 A0a1h+a2h²A16a1h+a2h²A32ah²A48a1h+a2h²

hi/h1

CD

0 0.5 1 1.5 2 2.5 3 3.5 40.008

0.01

0.012

0.014 A64a1h+a2h²A80ah²A96p=1.66

hi/h1

CD

0 0.5 1 1.5 2 2.5 3 3.5 40.008

0.01

0.012

0.014A112p=1.07A128p=1.26A160p=0.71

hi/h1

CL,

pr

0 0.5 1 1.5 2 2.5 3 3.5 40.41

0.42

0.43

0.44

0.45

0.46

0.47A0p=0.95A16p=1.99A32a1h+a2h²

hi/h1

CL,

pr

0 0.5 1 1.5 2 2.5 3 3.5 40.41

0.42

0.43

0.44

0.45

0.46

0.47A48p=0.71A64ah²A80ah²

hi/h1

CL,

pr

0 0.5 1 1.5 2 2.5 3 3.5 40.41

0.42

0.43

0.44

0.45

0.46

0.47A96p=1.20A112p=0.54A128p=0.96A160p=0.67

hi/h1

CL

0 0.5 1 1.5 2 2.5 3 3.5 40.41

0.42

0.43

0.44

0.45

0.46

0.47A0p=0.99A16ah²A32a1h+a2h²

hi/h1

CL

0 0.5 1 1.5 2 2.5 3 3.5 40.41

0.42

0.43

0.44

0.45

0.46

0.47A48p=0.66A64ah²A80ah²

hi/h1

CL

0 0.5 1 1.5 2 2.5 3 3.5 40.41

0.42

0.43

0.44

0.45

0.46

0.47A96p=1.18A112p=0.54A128p=0.94A160p=0.67

41

Figure 4.9 illustrates the ¡h distribution along the chord for the upper and lower surfaces of the

NACA 0012 airfoil. It is possible to see from Figures 4.9a and 4.9b that the WF method yields

reasonable results, except near the leading edge, probably due to the existence of conditions very

different that those on which the law-of-the-wall is based (flow over a flat plate with no pressure

gradient). Figure 4.9c shows that the greatest differences from the No-WF solution occur for group

A48, which is not surprising since this group has a ��� value in the buffer-layer region. Figure 4.9b is a

detail of the first 10% of the chord length with the purpose to show that, contrary to the No-WF

solution, none of the WF results predict transition.

(a) Skin friction coefficient for 0 < L/� < 1

(b) Skin friction coefficient for 0 < L/� < 0.1

(c) Relative deviation from No-WF

Figure 4.9 – Skin friction coefficient distribution of W-grid for NACA 0012 at � = 4º

The pressure distribution is depicted in Figure 4.10. The results from group A80 are compared in

Figure 4.10a to the ones form the No-WF solution and compressible-flow experimental data from [48],

corrected to incompressible conditions in the following way:

x/c

Cf(x

10³)

0 0.2 0.4 0.6 0.8 1-2

0

2

4

6

8

10

12

14

16 U.S. A16 (y+=3)L.S. A16 (y+=3)U.S. A48 (y+=16)L.S. A48 (y+=16)U.S. A80 (y+=51)L.S. A80 (y+=51)U.S. A112 (y+=143)L.S. A112 (y+=143)U.S. No WF (y+=0.06)L.S. No WF (y+=0.06)

x/c

Cf(x

10³)

0 0.02 0.04 0.06 0.08 0.1-2

0

2

4

6

8

10

12

14

16 U.S. A16 (y+=3)L.S. A16 (y+=3)U.S. A48 (y+=16)L.S. A48 (y+=16)U.S. A80 (y+=51)L.S. A80 (y+=51)U.S. A112 (y+=143)L.S. A112 (y+=143)U.S. No WF (y+=0.06)L.S. No WF (y+=0.06)

x/c

∆Cf(x

10³)

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1 U.S. A16 (y+=3)L.S. A16 (y+=3)U.S. A48 (y+=16)L.S. A48 (y+=16)U.S. A80 (y+=51)L.S. A80 (y+=51)U.S. A112 (y+=143)L.S. A112 (y+=143)

42

¡®,� = ¡®,»ÏЮ × �1 −Ñ�� (4.3)

where ¡®,� and ¡®,»ÏЮ are the pressure coefficients at incompressible conditions and at Mach number

equal to 0.30 (test conditions), respectively, and � is the Mach number. The A80 results are very

similar to the No-WF results but both exhibit some deviation from the experimental data, namely in

the pressure peak region. However it is important to note that the transformation from compressible to

incompressible conditions is based on an inviscid approximation. All results obtained with WF are in

very good agreement with the No-WF solution, as can be seen in Figures 4.10b and 4.10c.

(a) Pressure coefficient distribution of group A80, No-WF and experimental data

(b) Pressure coefficient distribution W-grid and No-WF

(c) Deviation from No-WF

Figure 4.10 – Pressure coefficient distribution of W-grid for NACA 0012 at � = 4º

x/c

Cp

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1U.S. No WF (y+=0.06)L.S. No WF (y+=0.06)U.S. Ladsonet al. [48]L.S. Ladsonet al. [48]U.S. A80 (y+=51)L.S. A80 (y+=51)

x/c

Cp

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1 U.S. A16 (y+=3)L.S. A16 (y+=3)U.S. A48 (y+=16)L.S. A48 (y+=16)U.S. A80 (y+=51)L.S. A80 (y+=51)U.S. A112 (y+=143)L.S. A112 (y+=143)U.S. No WF (y+=0.06)L.S. No WF (y+=0.06)

x/c

∆Cp

0 0.2 0.4 0.6 0.8 1-0.003

-0.002

-0.001

0

0.001

0.002

0.003 U.S. A16 (y+=3)L.S. A16 (y+=3)U.S. A48 (y+=16)L.S. A48 (y+=16)U.S. A80 (y+=51)L.S. A80 (y+=51)U.S. A112 (y+=143)L.S. A112 (y+=143)

43

4.3.2 Standard grid

Figure 4.11 presents the friction coefficient distribution as a function of the chord length. The four

selected groups with WF applied are plotted with the No-WF solution and it is possible to see that the

leading edge area (until 20% of the chord) is where the higher deviations occur. This happens due to

the proximity to the stagnation point, where a highly favourable pressure gradient creates a near-wall

velocity profile that is very far from the assumptions behind the WF formulation, i.e. a local

equilibrium turbulent boundary layer. Obviously, none of the WF groups show a transition point,

similarly to previous cases. Group A80n seems to produce very good results for L/� > 0.10 which is

not surprising since this group is on the location where the WF shows higher accurary, i.e. the log-

layer.

(a) Skin friction coefficient for 0 < L/� < 1

(b) Skin friction coefficient for 0 < L/� < 0.1

(c) Deviation of skin friction coefficient from No-WF

Figure 4.11 – Skin friction coefficient distribution of Standard grid for NACA 0012 at � = 4º

x/c

Cf(x

10³)

0 0.2 0.4 0.6 0.8 1-2

0

2

4

6

8

10

12

14

16 U.S. A16n (y+=3)L.S. A16n (y+=3)U.S. A48n (y+=15)L.S. A48n (y+=15)U.S. A80n (y+=50)L.S. A80n (y+=50)U.S. A112n (y+=141)L.S. A112n (y+=141)U.S. No WF (y+=0.06)L.S. No WF (y+=0.06)

x/c

Cf(x

10³)

0 0.02 0.04 0.06 0.08 0.1-2

0

2

4

6

8

10

12

14

16 U.S. A16n (y+=3)L.S. A16n (y+=3)U.S. A48n (y+=15)L.S. A48n (y+=15)U.S. A80n (y+=50)L.S. A80n (y+=50)U.S. A112n (y+=141)L.S. A112n (y+=141)U.S. No WF (y+=0.06)L.S. No WF (y+=0.06)

x/c

∆Cf(x

10³)

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1 U.S. A16n (y+=3)L.S. A16n (y+=3)U.S. A48n (y+=15)L.S. A48n (y+=15)U.S. A80n (y+=50)L.S. A80n (y+=50)U.S. A112n (y+=141)L.S. A112n (y+=141)

44

The pressure distribution is presented in Figure 4.12. Group A80n is compared in Figure 4.12a to

the same corrected pressure distribution as in section 4.3.1. Very small differences from the WF

solution to the No-WF solution are obtained with the Standard grid as can be seen in Figures 4.12b

and 4.13c. Both grid types are plotted against the experimental data from [48] and the No-WF and it

can be easily seen that the WF solutions are virtually coincident for ��� values of 50 and 51 for the

Standard and the W-grid, respectively. This suggests that Standard-type grids can be used with very

small loss in numerical accuracy if the distance of the first grid node above the wall is carefully

chosen.

(a) Pressure coefficient distribution of group A80n, No-WF and experimental data

(b) Pressure coefficient distribution of Standard grid and No-WF

(c) Deviation of pressure coefficient from No-WF

(d) Pressure coefficient distribution of W-grid, Standard grid, No-WF and experimental data

Figure 4.12 – Pressure coefficient distribution for Standard grid of NACA 0012 at � = 4º

x/c

Cp

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1U.S. No WF (y+=0.06)L.S. No WF (y+=0.06)U.S. Ladsonet al. [48]L.S. Ladsonet al. [48]U.S. A80n (y+=50)L.S. A80n (y+=50)

x/c

Cp

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1 U.S. A16n (y+=3)L.S. A16n (y+=3)U.S. A48n (y+=15)L.S. A48n (y+=15)U.S. A80n (y+=50)L.S. A80n (y+=50)U.S. A112n (y+=141)L.S. A112n (y+=141)U.S. No WF (y+=0.06)L.S. No WF (y+=0.06)

x/c

∆Cp

0 0.2 0.4 0.6 0.8 1-0.01

-0.005

0

0.005

0.01U.S. A16n (y+=3)L.S. A16n (y+=3)U.S. A48n (y+=15)L.S. A48n (y+=15)U.S. A80n (y+=50)L.S. A80n (y+=501)U.S. A112n (y+=141)L.S. A112n (y+=141)

x/c

Cp

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

1.5U.S. No WFL.S. No WFU.S. Ladsonet al. [48]L.S. Ladsonet al. [48]U.S. A80 (y+=51)L.S. A80 (y+=51)U.S. A80n (y+=50)L.S. A80n (y+=50)

45

4.3.3 Comparison to No-WF solution and experimental results

In this section the results obtained with the Standard and the W-grid are compared to those obtained

without WF. The variation of each aerodynamic force with the ��� is presented as a relative deviation,

using as reference the respective value obtained without WF (using set B with ��� = 0.06), calculated

the same way as in section 4.2.3. The references values of each force component and its estimated

uncertainty, are presented in Table 4.4.

Table 4.4 – Reference values and estimated uncertainties for NACA 0012 at � = 4º 56,CC U(56,CC) 56,�B U(56,�B) 56 U(56) 54,�B U(54,�B) 54 U(54)

0.0068 1.7x10-5 0.0021 2.5x10-5 0.0089 1.1x10-4 0.4398 4.6x10-4 0.4397 4.4x10-4

The variation of the friction, pressure and total Drag and Lift coefficients with ��� are depicted in

Figure 4.13 for the Standard and W-grid. The results for the ¡«,ÄÄ show that the calculations performed

with the W-grid produced deviations lower than 1% for ��� values from 50 to 230, whereas those made

with the Standard grid have deviations between 2% and 6% in the same region. Outside this range,

none of the grid types seems to be appropriate for these calculations. Similarly to previous cases, the

W-grid correctly predicts the ¡«,®Å for very small ��� values. In the region of the log-layer the

Standard grid yields smaller deviations from the solution obtained without WF than the W-grid,

although the W-grid shows considerably high uncertainty values. This happens because the numerical

error resultant from the high near-wall spacing in the Standard grids is cancelled by the modelling

error introduced by the WF boundary conditions. For larger ��� both types of grids highly overpredict

de ¡«,®Å with deviations up to 60% (Standard grid) and 160% (W-grid). Regarding the ¡«, the W-grid

shows a narrow region in the log-layer (��� between 50 and 85) where the ¡« is correctly calculated,

with less than 1% of deviation, whereas the Standard grid appears to predict reasonably well the ¡« for

��� between 50 and 230, where the deviation is between 2% and 3%.

As for the ¡ª,®Å, the results from both types of grids are in excellent agreement with the solution

obtained without WF up to values of ��� of 230 for the W-grid and 580 for the Standard grid, including

in the buffer-layer region. Surprisingly, the Standard grid produces very good results even for very

large values of ��� (relative deviation of 1.2% at ��� = 580). The results for the ¡ª are very similar to

those of the ¡ª,®Å due to the very small values of ¡ª,ÄÄ. The conclusions are the same as in the ¡ª,®Å: excellent agreement with the reference values from very small to very high ��� values, including in the

buffer-layer.

46

(a) Friction component of drag coefficient

(b) Pressure component of drag coefficient

(c) Total drag coefficient

(d) Pressure component of lift coefficient

(e) Total lift coefficient

Figure 4.13 – Variation of relative deviation of aerodynamic coefficients with ��� for NACA 0012 at � = 4º

y+max

∆CD

,ss(%

)

10-1 100 101 102 103-25

-20

-15

-10

-5

0

5

10

W-gridStandard grid

y+max

∆CD

,pr(%

)

10-1 100 101 102 103-20

-10

0

10

20

30

40

50

W-gridStandard grid

y+max

∆CD

(%)

10-1 100 101 102 103-10

-5

0

5

10

15

20

25

30

35

40

W-gridStandard grid

y+max

∆CL,

pr(%

)

10-1 100 101 102 103-6

-4

-2

0

2

W-gridStandard grid

y+max

∆CL

(%)

10-1 100 101 102 103-6

-4

-2

0

2

W-gridStandard grid

47

The results obtained in this work are compared to experimental results from [47] in Figure 4.14.

The ¡ª results show very good agreement with the experimental results for ��� up to 230 for the W-

grid and 600 for the Standard grid. The ¡« results show similar trends as the ¡« at zero degrees of

angle of attack: good agreement with the experimental data for small ��� and in the log-law region

(between 50 and 90).

(a) Total drag coefficient

(b) Total lift coefficient

Figure 4.14 – Variation of total aerodynamic coefficients with 1'� for NACA 0012 at � = 4º

4.4 Flow over the laminar airfoil Eppler 374 at � = 0º 4.4.1 W-grid

Calculations for an angle of attack of zero degrees were made using groups w16 to w160 from set

A and group w96 from set E. The ��� of each group is presented in Table 4.5.

Table 4.5 - 1'� value of each group used in Eppler 374 calculations for � = 0º A16 A32 A48 A64 A80 A96 A112 A128 E96 A160 1'� 1.6 3.9 6.5 9.4 13 19 28 41 74 85

Figure 4.15 illustrates the convergence of the Drag and Lift coefficients (pressure, friction

components and total values) with the grid refinement for the Eppler 374.

The results for the pressure component of the Drag show a high dependence on the ��� with the

asymptotic values ranging from 0.0105 to 0.012. Only group A32 does not show monotonic

y+max

CD

10-1 100 101 102 1030.008

0.009

0.01

0.011

W-gridStandard gridNo WFLadson [47]

y+max

CL

10-1 100 101 102 1030.4

0.42

0.44

0.46

W-gridStandard gridNo WFLadson [47]

48

convergence but only for half the groups an observed order of convergence was possible to calculate,

which has values lower than the theoretical value of 2 (between 0.6 and 1.8). Concerning the ¡«,®Å, the

results are of the order of 10�º so the results from some groups may be contaminated with the

iterative error because the convergence criteria was not sufficiently low. Only the 3 groups with lower

��� do not have monotonic convergence and only 4 groups have an observed order of convergence

(which is much lower than the theoretical value, from 0.5 to 1.5). The total ¡« is a consequence of the

¡«,ÄÄ and ¡«,®Å, therefore the same trends are observed.

As for the Lift coefficient, the friction component is of the order of 10�� so the results are

definitely polluted by the iterative error (asymptotic values range from 1.4 × 10�� to 3.4 × 10��) and

the convergence is only shown for illustrative purposes. The ¡ª,®Å results show monotonic

convergence for 8 of the 10 groups but only 3 show an observed order of convergence within the

acceptable values (Q between 0.5 and 2) of the method used for uncertainty estimation. The asymptotic

values (¡ª,®Å when ℎK/ℎu → 0) still show a significant dependence on the distance of the first node to

the wall. Due to the very small values of ¡ª,ÄÄ, the total ¡ª results are very similar to the ¡ª,®Å in terms

of convergence properties and ��� dependence.

Next, the surface distributions of the pressure and friction coefficients (¡® and ¡h, respectively) of

the Eppler 374 airfoil at zero degrees of attack of angle for calculations performed using the W-grid

are presented and studied by comparing the calculations performed with WF to the results obtained

with the RANS equations solved down to the wall. Four groups from set A are used in this

comparison: groups A32, A80, A112 and A160, with ��� values ranging from 13 to 85. These groups

were chosen because they are representative of 4 important regions: the viscous sub-layer, the buffer-

layer, the log-layer and its outer edge.

Figure 4.16a shows the absolute values of ¡h, and the difference of each group to the No-WF

solution is shown in Figure 4.16b. Although the No-WF solution exhibits transition from laminar to

turbulent flow at approximately 25% of the chord on the upper surface and 20% on the lower surface,

it is still sooner than the adverse pressure gradient (that occurs at approximately 40% of the chord)

thus leading to unsatisfactory results for the friction and pressure distributions (mainly in the

distribution of ¡h) and, consequently, for the lift and drag coefficients, showing that when the laminar

part of the flow is relevant, significant deviations from the real flow will arise. None of the groups

from the W-grid show transition, assuming turbulent flow from the leading edge onwards.

49

(a) Friction component of drag coefficient

(b) Pressure component of drag coefficient

(c) Total drag coefficient

(d) Friction component of lift coefficient

(e) Pressure component of lift coefficient

(f) Total lift coefficient

Figure 4.15 – Grid convergence of aerodynamic coefficients as function of grid refinement ratio for W-grid of Eppler 374 airfoil at � = 0º

hi/h1

CD

,ss

0 1 2 3 40.01

0.011

0.012

0.013

0.014 A16ah²A32p=1.78A48p=1.02

hi/h1

CD

,ss

0 1 2 3 40.01

0.011

0.012

0.013

0.014 A112p=0.58A128a1h+a2h²A160a1h+a2h²E96a1h+a2h²

hi/h1

CD

,ss

0 1 2 3 40.01

0.011

0.012

0.013

0.014 A64a1h+a2h²A80p=1.36A96p=0.82

hi/h1

CD

,pr

0 1 2 3 40.002

0.003

0.004

0.005

0.006

0.007

0.008A16a1h+a2h²A32ah²A48p=1.51A64a1h+a2h²

hi/h1

CD

,pr

0 1 2 3 40.002

0.003

0.004

0.005

0.006

0.007

0.008A80p=1.00A96p=0.68A112p=0.52

hi/h1

CD

,pr

0 1 2 3 40.002

0.003

0.004

0.005

0.006

0.007

0.008A128a1h+a2h²A160a1h+a2h²E96a1h+a2h²

hi/h1

CD

0 1 2 3 40.012

0.014

0.016

0.018

0.02

0.022A16a1h+a2h²A32a1h+a2h²A48p=1.21A64a1h+a2h²

hi/h1

CD

0 1 2 3 40.012

0.014

0.016

0.018

0.02

0.022A128a1h+a2h²A160a1h+a2h²E96a1h+a2h²

hi/h1

CD

0 1 2 3 40.012

0.014

0.016

0.018

0.02

0.022A80ah²A96p=1.25A112a1h+a2h²

hi/h1

CL,

ss

0 1 2 3 40.0001

0.0002

0.0003

0.0004

0.0005

0.0006A16p=1.11A32p=1.37A48p=1.05A64p=0.59

hi/h1

CL,

ss

0 1 2 3 40.0001

0.0002

0.0003

0.0004

0.0005

0.0006A80a1h+a2h²A96a1h+a2h²A112a1h+a2h²

hi/h1

CL,

ss

0 1 2 3 40.0001

0.0002

0.0003

0.0004

0.0005

0.0006A128p=1.51A160a1h+a2h²E96a1h+a2h²

hi/h1

CL,

pr

0 1 2 3 40.16

0.17

0.18

0.19

0.2

0.21A16ah²A32p=0.94A48p=0.84

hi/h1

CL,

pr

0 1 2 3 40.16

0.17

0.18

0.19

0.2

0.21A64p=1.53A80a1h+a2h²A96a1h+a2h²

hi/h1

CL,

pr

0 1 2 3 40.16

0.17

0.18

0.19

0.2

0.21A112a1h+a2h²A128a1h+a2h²A160a1h+a2h²E96a1h+a2h²

hi/h1

CL

0 1 2 3 40.16

0.17

0.18

0.19

0.2

0.21A16ah²A32p=0.96A48p=0.83

hi/h1

CL

0 1 2 3 40.16

0.17

0.18

0.19

0.2

0.21A64p=1.56A80a1h+a2h²A96a1h+a2h²

hi/h1

CL

0 1 2 3 40.16

0.17

0.18

0.19

0.2

0.21A112a1h+a2h²A128a1h+a2h²A160a1h+a2h²E96a1h+a2h²

50

(a) Skin friction coefficient

(b) Deviation of skin friction coefficient from No-WF

(c) Pressure coefficient

(d) Deviation of pressure coefficient from No-WF

Figure 4.16 – Skin friction and pressure coefficients distribution of W-grid for Eppler 374 at � = 0º

4.4.2 Standard grid

Figure 4.17 illustrates the distribution along the chord of the skin friction coefficient, ¡h. The 4

groups from the Standard grid are plotted together with the No-WF in Figure 4.17a and the difference

between the WF and the No-WF solutions is depicted in Figure 4.17b. Naturally, the Standard grid

also does not predict transition, but instead computes the flow as turbulent since the leading edge. This

leads to very inaccurate results as can be seen in Figure 4.17b from the very large deviations (up to

50% at L/� = 0.20). Also, this type of grid introduces very high numerical error in the trailing edge

region. This is caused by the chosen grid configuration: the C-shaped grid that is used on this thesis

x/c

Cf(x

10³)

0 0.2 0.4 0.6 0.8 10

5

10

15

20U.S. A32 (y+=4)L.S. A32 (y+=4)U.S. A80 (y+=13)L.S. A80 (y+=13)U.S. A112 (y+=27)L.S. A112 (y+=27)U.S. A160 (y+=85)L.S. A160 (y+=85)U.S. No WF (y+=0.08)L.S. No WF (y+=0.08)

x/c∆C

f(x

10³)

0 0.2 0.4 0.6 0.8 1

0

2

4

6

U.S. A32 (y+=4)L.S. A32 (y+=4)U.S. A80 (y+=13)L.S. A80 (y+=13)U.S. A112 (y+=27)L.S. A112 (y+=27)U.S. A160 (y+=85)L.S. A160 (y+=85)

x/c

Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1U.S. A32 (y+=4)L.S. A32 (y+=4)U.S. A80 (y+=13)L.S. A80 (y+=13)U.S. A112 (y+=27)L.S. A112 (y+=27)U.S. A160 (y+=85)L.S. A160 (y+=85)U.S. No WF (y+=0.08)L.S. No WF (y+=0.08)

x/c

∆Cp

0 0.2 0.4 0.6 0.8 1-0.1

-0.05

0

0.05

0.1U.S. A32 (y+=4)L.S. A32 (y+=4)U.S. A80 (y+=13)L.S. A80 (y+=13)U.S. A112 (y+=27)L.S. A112 (y+=27)U.S. A160 (y+=85)L.S. A160 (y+=85)

51

generates a big “gap” in the region immediately downstream of the trailing edge, leading to unsuitable

grid line distances on the wake. However, the objective of the C-grid was to be easily constructed and

to allow for an easy control of the near-wall distances which is perfectly accomplished with such

configuration. The “gap” in the wake region could be avoided by using locally an O-grid [5], as

mentioned previously.

The pressure coefficient along the chord is presented in Figure 4.18a and the difference of the

Standard grid groups to the No-WF solution is depicted in Figure 4.18b. Very small deviations are

observed even for the highest ��� which indicates that WF can be used to properly compute pressure

distributions in the conditions that were tested in this work.

(a) Skin friction coefficient

(b) Deviation of skin friction coefficient from No-WF

Figure 4.17 – Skin friction coefficient distribution of Standard grid for Eppler 374 at � = 0º

(a) Pressure coefficient

(b) Deviation of pressure coefficient from No-WF

Figure 4.18 – Pressure coefficient distribution of Standard grid for Eppler 374 at � = 0º

x/c

Cf(x

10³)

0 0.2 0.4 0.6 0.8 10

5

10

15

20U.S. A32n (y+=4)L.S. A32n (y+=4)U.S. A80n (y+=13)L.S. A80n (y+=13)U.S. A112n (y+=27)L.S. A112n (y+=27)U.S. A160n (y+=85)L.S. A160n (y+=85)U.S. No WF (y+=0.08)L.S. No WF (y+=0.08)

x/c

∆Cf(x

10³)

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6U.S. A32n (y+=4)L.S. A32n (y+=4)U.S. A80n (y+=13)L.S. A80n (y+=13)U.S. A112n (y+=27)L.S. A112n (y+=27)U.S. A160n (y+=85)L.S. A160n (y+=85)

x/c

Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1U.S. A32n (y+=4)L.S. A32n (y+=4)U.S. A80n (y+=13)L.S. A80n (y+=13)U.S. A112n (y+=27)L.S. A112n (y+=27)U.S. A160n (y+=85)L.S. A160n (y+=85)U.S. No WF (y+=0.08)L.S. No WF (y+=0.08)

x/c

∆Cp

0 0.2 0.4 0.6 0.8 1-0.1

-0.05

0

0.05

0.1U.S. A32n (y+=4)L.S. A32n (y+=4)U.S. A80n (y+=13)L.S. A80n (y+=13)U.S. A112n (y+=27)L.S. A112n (y+=27)U.S. A160n (y+=85)L.S. A160n (y+=85)

52

Figure 4.19a shows a comparison between the No-WF results and group A112 from the W-grid and

the Standard grid (this group yields the smallest relative deviation in the ¡ª predictions as will be

shown in section 4.4.3). An interesting (and surprising) result is observed: the Standard grid seems to

yield better results (i.e. closer to the No-WF solution) than the W-grid in the whole chord length in

both the upper and the lower surfaces. The same comparison is made for the pressure coefficient but

the results in this case are very similar in all three solutions. This highlights two important conclusions

of this thesis: first, wall functions can be useful even in laminar airfoils if the goal is to compute

pressure distributions. However, if the objective is to compute shear-stress, one should be very careful

or simply avoid using WF in this type of airfoils; and second, Standard grids (which are easier to

generate) can also be useful if, once more, the aim is to compute pressure distributions. Obviously,

both conclusions were drawn from results where the ��� was properly chosen (27 may seem a low

value but one must keep in mind that the Reynolds number is relatively low). However, iterative

convergence may be troublesome at such small Reynolds numbers.

(a) Friction coefficient

(b) Pressure coefficient

Figure 4.19 – Skin friction and pressure coefficients distribution of W-grid, Standard grid and No-WF for Eppler 374 at � = 0º

4.4.3 Comparison to No-WF solution and experimental data

In this section the results obtained with the Standard and the W-grid are compared to those

obtained with the direct application of the no-slip condition. Similarly to the NACA 0012 results, the

variation of each aerodynamic force with the ��� is presented as a relative deviation, using as reference

the respective value obtained without WF (set A with ��� = 0.08). The references values of each force

component and its estimated uncertainty are presented in Table 4.6.

x/c

Cf(x

10³)

0 0.2 0.4 0.6 0.8 10

5

10

15

20

U.S. A112n (y+=27)L.S. A112n (y+=27)U.S. No WF (y+=0.08)L.S. No WF (y+=0.08)U.S. A112 (y+=27)L.S. A112 (y+=27)

x/c

Cp

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

U.S. A112n (y+=27)L.S. A112n (y+=27)U.S. No WF (y+=0.08)L.S. No WF (y+=0.08)U.S. A112 (y+=27)L.S. A112 (y+=27)

53

Table 4.6 – Reference values and estimated uncertainties for Eppler 374 at � = 0º 56,CC U(56,CC) 56,�B U(56,�B) 56 U(56) 54,�B U(54,�B) 54 U(54) 0.010 1.5x10-4 0.002 9.3x10-5 0.012 7.5x10-5 0.180 9.0x10-4 0.180 8.8x10-4

Figure 4.20 shows the Drag and Lift coefficients as a function of the X���YÒÓÔ as well as

experimental data from [43]. It is clear that the turbulence model, even without using WF, fails to

predict both coefficients, due to the miscalculation of the transition point. Comparing only the WF

results with the ones obtained with direct application of the no-slip condition, the WF results show

significant deviations, namely in the buffer-layer region. However, small values of ��� seem to

correctly predict the ¡ª and ¡«.

(a) Total drag coefficient

(b) Total lift coefficient

Figure 4.20 – Variation of total aerodynamic coefficients with ��� for Eppler 374 at � = 0º The variation of the aerodynamic forces with ��� is illustrated in Figure 4.21, for the Standard and

W-grid. The ¡ª,ÄÄ is not shown due to the influence of the iterative error on the results, as was

mentioned in section 4.3.1. The data points of the W-grid results are presented with its uncertainty,

which is estimated in the same way that was mentioned in the analysis of the NACA 0012 results.

The results for the friction component of the Drag coefficient show that the Standard grid yields

lower deviations than the W-grid, which is again due to the error cancellation effect discussed in

section 4.3.3. The W-grid shows deviations greater than 4% for small ��� values (below 5) and

superior to 20% in the region of the log-law (��� from 30 to 85), whereas the Standard grid produces

results with smaller deviations from the No-WF solution (below 2%) for ��� below 5 and above 70.

y+max

CD

100 101 1020.008

0.01

0.012

0.014

0.016

0.018

W-gridStandard gridNo WFSeliget al. [43]

y+max

CL

100 101 1020.16

0.17

0.18

0.19

0.2

W-gridStandard gridNo WFSeliget al. [43]

54

Both grid types show a peak in the buffer-layer (��� = 13) but only the Standard grid shows the

decrease in the deviation observed in the NACA 0012 calculations for higher ��� values. As for the

¡«,®Å, the W-grid shows results with deviation always higher than 8% and reaching 90% in ��� = 85.

The Standard grid again shows results with smaller deviations from the No-WF solution than the

W-grid but the deviations are still above 17% for ��� greater than 5. The region in the log-law region

where the NACA 0012 calculations showed good results (��� between 30 and 80 in most cases)

appears to occur for smaller ��� values in the case of the Eppler 374 and seems to be narrower. The

total ¡« results show the same trends as the individual components: Standard grid yields smaller

deviations than W-grid. However, good results are only obtained using small ��� values, i.e. using

grids where the number of volumes is not much smaller than the used on the solution without WF. The

results obtained for the ¡ª,®Å show smaller deviations than the previous components. The Standard

grid yields good results for ��� smaller than 4 and between 25 and 85, whereas the W-grid only shows

good results for small ��� values. Unlike the Standard grid, The W-grid does not show a region above

the buffer-layer where the results tend to a constant value of the deviation. Also, the uncertainty in the

W-grid results for ��� greater than 20 is very significant. As mentioned before, the ¡ª,ÄÄ is very small

compared to the ¡ª,®Å thus the total ¡ª has virtually the same variation with ��� as the ¡ª,®Å. The same

conclusions as the pressure component can be drawn for the total Lift coefficient.

55

(a) Friction component of drag coefficient

(b) Pressure component of drag coefficient

(c) Total drag coefficient

(d) Pressure component of lift coefficient

(e) Total lift coefficient

Figure 4.21 – Variation of relative deviation of aerodynamic coefficients with ��� for Eppler 374 at � = 0º

y+max

∆CD

,ss(%

)

100 101 1020

5

10

15

20

25

30

W-gridStandard grid

y+max

∆CD

,pr(%

)

100 101 102

0

20

40

60

80

100

W-gridStandard grid

y+max

∆CD

(%)

100 101 1020

5

10

15

20

25

30

35

40

45

W-gridStandard grid

y+max

∆CL,

pr(%

)

100 101 102-10

-8

-6

-4

-2

0

2

4

6

8

10

W-gridStandard grid

y+max

∆CL

(%)

100 101 102-10

-8

-6

-4

-2

0

2

4

6

8

10

W-gridStandard grid

57

Chapter 5

Conclusions and Future Work

This thesis presented a study of the influence of the use of wall functions on the aerodynamic

characteristics of two airfoils: a conventional airfoil, the NACA 0012; and a laminar airfoil, the Eppler

374. The effect of using WF in flows in which the laminar part of the flow around the airfoil is

relevant was examined and compared to flows where the flow is turbulent in almost all the chord

length. A grid generation procedure was presented that allows systematic grid refinement for the

assessment of the grid convergence properties of the force coefficients. Two types of grid line

distribution were compared: the Standard grid approach, which has continuously increasing spacing of

grid lines in the direction normal to the airfoil surface; and a W-grid that consists of removing specific

number of lines adjacent to the wall from a grid appropriate for calculations with direct application of

the no-slip condition. The objective was to study the importance of the numerical error inherent to

grids with very few nodes in the boundary layer. This required the development of a grid generator

that would easily and efficiently construct grids with specific near-wall spacing (��) and one of the

two configurations: the Standard and the W-grid. The choice for the type of grid is made simply by

specifying the number of lines to be removed in the near-wall region and the desired grid line

distribution in the wall-normal direction. This grid generator was successfully implemented and was

applied to generate all the grids used in this work.

In order to study the effect of using WF boundary conditions in flows with different extensions of

the laminar part, the two airfoils were studied at different Reynolds numbers: the conventional airfoil

at a high Reynolds number (and two angles of attack) and the laminar airfoil at a low Reynolds

number (only one angle of attack).

The main conclusions from this thesis are the following:

- The calculations performed at high Reynolds number (conventional airfoil), showed that if the

laminar part of the flow is very small (only a few percent of the chord), the differences in ¡h and ¡®

observed between the WF and the No-WF solutions are not significant if the distance of the first

interior grid node to the wall is properly chosen.

58

- At zero degrees of angle of attack, the pressure distributions show very good agreement to the

No-WF solutions almost independently of the ��� value and the higher deviations appear in the leading

and trailing edge regions. The friction distribution shows higher dependence on the value of ��� and

the higher deviations occur close to the leading edge.

- As for the aerodynamic forces (¡ª and ¡«) a clear ���-dependence is observed in the W-grid

solutions. Depending on the ��� value, the force coefficients can be well predicted or completely

miscalculated. Four regions can be identified: very small ��� values (lower than 5) where the solution

is very similar to the No-WF solution; the buffer-layer (between 5 and 20) where the results show

deviations from the No-WF solution as high as 10%, an intermediate region (��� between 30 and 80)

where the results are consistent with the No-WF solution; and a region of high ��� (above 100) where

the results diverge from the No-WF results.

- The Standard grid produces higher deviations in the ¡h and ¡® distributions than the W-grid

and lift and drag force results show that it is more difficult with this grid type to identify regions of

acceptable deviations from the No-WF solutions.

- At four degrees of angle of attack the W-grid shows very small ���-dependence of the pressure

distribution but significant deviations from the No-WF results are observed. The same trends are

present in the Standard grid. The W-grid shows better results in the drag predictions than the Standard

grid, but both grid produce very good results (comparing to the No-WF results and to experimental

data) for the lift coefficient. Since the lift force is mainly due to pressure distribution over the airfoil

surface and the drag force is mainly due to friction (¡«,ÄÄ is more than 3 times higher than ¡«,®Å) it was shown that he usefulness of the WF method is dependent on which parameter is being calculated.

For the same flow conditions, one can obtain very good predictions for the lift coefficient and the

pressure distributions and poor predictions for the drag coefficient and friction distribution.

- The calculations at low Reynolds number (laminar airfoil) showed that reasonable predictions

of the pressure distribution can be obtained with WF but the friction distribution is very poorly

predicted due to the large extent of the laminar part of the flow. The transition is predicted sooner than

the adverse pressure gradient, thus highly over-predicting the drag coefficient (due to higher shear

forces in turbulent flows than in laminar ones). Only for very small ��� values the W-grid and the

Standard grid showed small deviations of ¡« and ¡ª from the No-WF solutions.

Since only one turbulence model was used in this thesis, it would be interesting to test other

turbulence models and see how the results would be affected by such modifications.

As was shown, one of the main weaknesses of the turbulence model used was the prediction of the

transition point, even without WF. Since this strongly affects the results (in the case of the laminar

airfoil), some investigation should be done in order to improve the prediction of transition to turbulent

flow in flows affect by curvature effects, stagnation points and strong pressure gradients.

59

References

[1] Versteeg, H. K. and Malalasekera, W. (2007), An Introduction to Computational Fluid Dynamics –

The Finite Volume Method, 2nd Ed., Pearson Education.

[2] Vaz, G. and Hoekstra, M. (2006), Theoretical and Numerical Formulation of FreSCo code, Tech.

Rep. 18572-WP4-2, Maritime Research Institute of The Netherlands

[3] Wilcox, D. (1994), Turbulence Modelling for CFD, DCW Industries

[4] Ferziger, J. H. and Peric, M., Computational Methods for Fluid Dynamics, 3rd Ed., Springer

[5] Hirsch, C., Numerical Computation of Internal and External Flows – Volume 1, 2nd Ed., Elsevier

[6] Menter, F. R. (1994), “Two-Equation Eddy-Viscosity Turbulence Models for Engineering

Applications”, AIAA Journal, 32 (8), pp. 1598-1605

[7] Craft, T. J., Gant, S. E., Iacovides, H. and Launder B. E. (2004), “A new wall-functions strategy

for complex turbulent flows”, Numerical Heat Transfer, 45 (4), pp 301-318

[8] Eça, L. (2007), “Implementation of Wall Functions in PARNASSOS”, Report VP-1, Instituto

Superior Técnico

[9] Vaz, G., Jaouen, F. and Hoekstra, M. (2009), “Validation of URANS code FRESCO”, In

Proceedings of OMAE2009, Hawaii, USA

[10] Klaij, C. M. and Vuik, C. (2013), “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), pp. 830–849.

[11] Eça, L. and Hoekstra, M. (2012), “Verification and validation for Marine applications of CFD”,

In Proceeding of 29th Symposium on Naval Hydrodynamics (ONR), Gothenburg, Sweden

[12] Toxopeus, S., Simonsen, C., Guilmineau, E., Visonneau, M., Xing, T., and Stern, F. (2013),

“Investigation of Water Depth and Basin Wall Effects on KVLCC2 in Manoeuvring Motion Using

Viscous-flow Calculations”, Journal of Marine Science and Technology, 18(4), pp. 471–496.

60

[13] Pereira, F., Eça, L., and Vaz, G. (2013), “On the Order of Grid Convergence of the Hybrid

Convection Scheme for RANS Codes”, In Proceedings of CMNI2013, Barcelona, Spain.

[14] Rijpkema D., and Vaz G. (2011), “Viscous Flow Computations on Propulsors: Verification,

Validation and Scale Effects”, In Proceedings of RINA 2011, London, UK.

[15] Windt, J. (2013), “Adaptive Mesh Refinement in Viscous Flow Solvers: Refinement in the Near-

wall Region, Implementation and Verification”, In Proceedings of NUTTS2013, Mulheim, Germany.

[16] Kerkvliet, M., Vaz, G., Carette, N. and Gunsing, M. (2014), “Analysis of U-Type Anti-Roll

Tanks using RANS. Sensitivity and Validation”, In Proceedings of OMAE2014, San-Francisco, USA.

[17] Schuiling, B. (2013), “The Design and Numerical Demonstration of a New Energy Saving

Device”, In Proceedings of NUTTS2013, Mulheim, Germany.

[18] Koop, A., Klaij, C. and Vaz, G. (2013), “Viscous-Flow Calculations for Model and Full-Scale

Current Loads on Typical Offshore Structures”, In Proceedings of MARINE 2011, IV International

Conference on Computational Methods in Marine Engineering, Lisboa, Portugal

[19] Reynolds, O. (1895), “On the Dynamical Theory of Incompressible Viscous Fluids and the

Determination of the Criterion”, Philosophical Transactions of the Royal Society of London. A, Vol.

186, pp. 123-164

[20] Boussinesq, J. (1877), “Essai sur la théorie des eaux courantes", Mémoires présentés par divers

savants à l'Académie des Sciences, 23 (1), pp. 1-680

[21] Spalart, P. and Allmaras, S. (1992), “A one-equation turbulence model for aerodynamics flows”,

in Proceedings of AIAA 30th Aerospace Sciences Meeting, Reno, USA

[22] Menter, F. R., Egorov, Y. and Rusch, D. (2006), “Steady and Unsteady Modelling Using the

� − √�¬ Model”, Turbulence, Heat and Mass Transfer 5, Proceedings of the International

Symposium on Turbulence, Heat and Mass Transfer, Dubrovnik, Croatia

[23] Launder, B. E. and Spalding, D. B. (1974), “The Numerical Calculation of Turbulent Flows”,

Computer Methods in Applied Mechanics and Engineering, 3 (2), pp. 269-289.

[24] Wilcox, D. (1998), “Reassessment of the Scale-Determining Equation for Advanced Turbulence

Models”, AIAA Journal, 26 (11), pp. 1299-1310.

[25] CFD Online. Available at: http://www.cfd-online.com/ [Accessed September 2014]

[26] Menter, F. R. (1993), “Zonal Two Equation k-omega Turbulence Models for Aerodynamic

Flows”, AIAA Paper 93-2906

61

[27] Craft, T. J., Gant, S. E., Gerasimov, A. V., Iacovides, H. and Launder, B. E. (2004),

“Development and application of wall-function treatments for turbulent forced and mixed convection

flows”, Fluid Dynamics Research, 38 (2), pp. 127-144

[28] Mohammadi, B. and Puigt, G. (2005), “Wall Functions in Computational Fluid Dynamics”,

Computers and Fluids, 35 (10), pp. 1108-1115

[29] von Kármán, T. (1930), “Mechanical Similitude and Turbulence - NACA Technical

Memorandum No. 611”, National Advisory Committee for Aeronautics

[30] Medic, G., Kalitzin, G., Iaccarino, G. and van der Weide, E. (2006), “Adaptive Wall Functions

With Applications”, AIAA Paper 2006-3744

[31] Menter, F. and Esch, T. (2001), “Elements of Industrial Heat Transfer Predictions”, 16th

Brazilian Congress of Mechanical Engineering (COBEM), pp. 117-127

[32] Knopp, T., Alrutz, T. and Schwamborn, D. (2006), “A grid and flow adaptive wall-function

method for RANS turbulence modelling”, Journal of Computational Physics, 220 (1), pp. 19-40.

[33] Goncalves, E. and Houdeville R. (2000), “Reassessment of the wall functions approach for RANS

computations”, Aerospace Science and Technology, 5 (1), pp. 1-14.

[34] Kalitzin, G., Medic, G., Iaccarino, G. and Durbin, P. (2004), “Near-wall behaviour of RANS

turbulence models and implications for wall functions”, Journal of Computational Physics, 204 (1),

pp. 265-291.

[35] Knopp, T. (2006), “On grid-independence of RANS predictions for aerodynamic flows using

model-consistent universal wall-functions”, European Conference on Computational Fluid Dynamics

ECCOMAS CFD, Delft, The Netherlands

[36] Brederode, V. (1997), Fundamentos de Aerodinâmica Incompressível, Author’s Edition.

[37] Eça, L. and Hoekstra, M. (2008), “The numerical friction line”, Journal of Marine Science and

Technology, 13 (4), pp. 328-345.

[38] Eça, L. and Hoekstra, M. (2014), “A procedure for the estimation of the numerical uncertainty of

CFD calculations based on grid refinement studies”, Journal of Computational Physics, 262, pp. 104-

130.

[39] Thompson, J. F., Sony, B. K. and Weatherill, N. P. (1999), Handbook of Grid Generation, CRC

Press

62

[40] Sorenson, R. L. (1982), “Grid Generation by Elliptical Partial Differential Equations for a Tri-

Element Augmentor-Wing Airfoil”, Applied Mathematics and Computation, Vol. 10-11, pp. 653-665

[41] Eça, L. (2001), “Practical Tools for 2-D, 3-D and Surface Grid Generation”, Report D72-10,

Instituto Superior Técnico.

[42] Jacobs, E. N., Ward, K. E. and Pinkerton, R. M. (1935), The characteristics of 78 related airfoil

sections from tests in the variable-density wind tunnel - Report No. 460, National Aeronautics and

Space Administration

[43] Selig, M. S., Guglielmo, J. J., Broeren, A. P. and Giguère, P. (1995), Summary of Low-Speed

Airfoil Data, SoarTech Publications

[44] Farinha, G., Quental, P., Eça, L. and Chaves, D. M. (2014), “Ground-Effect Simulations based on

an Efficient Boundary Element Method”, V Conferência Nacional de Mecânica dos Fluidos,

Termodinâmica e Energia, MEFTE 2014, Porto, Portugal

[45] Vinokur, M. (1980), “On One-Dimensional Stretching Functions for Finite-Difference

Calculations”, Journal of Computational Physics, 50 (2), pp. 215-234.

[46] 2D NACA 0012 Airfoil Validation Case - SST Model Results. Available at:

http://turbmodels.larc.nasa.gov/naca0012_val_sst.html [Accessed September 2014]

[47] Ladson, C. L. (1988), Effects of Independent Variation of Mach and Reynolds Numbers on the

Low-Speed Aerodynamic Characteristics of the NACA 0012 Airfoil Section – NASA Technical

Memorandum 4074, National Aeronautics and Space Administration

[48] Ladson, C. L., Hill, A. S. and Johnson, W. G. (1987), Pressure Distributions from High Reynolds

Number Transonic Tests of an NACA 0012 Airfoil in the Langley 0.3-Meter Transonic Cryogenic

Tunnel – NASA Technical Memorandum 100526, National Aeronautics and Space Administration