IN DEGREE PROJECT TECHNOLOGY,FIRST CYCLE, 15 CREDITS

, STOCKHOLM SWEDEN 2018

Evaluation of OpenFOAM performance for RANS simulations of flow around a NACA 4412 airfoil

HAMPUS TOBER

ERIK HÄNNINGER

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

INOM EXAMENSARBETE TEKNIK,GRUNDNIVÅ, 15 HP

, STOCKHOLM SVERIGE 2018

Utvärdering av OpenFOAM prestanda för RANS simuleringar av flöde runt en NACA 4412 vingprofil

HAMPUS TOBER

ERIK HÄNNINGER

KTHSKOLAN FÖR TEKNIKVETENSKAP

Abstract

The purpose of this study is to evaluate how well the RANS models implemented in theopen-source software OpenFOAM performs when simulating flow around a NACA 4412 airfoilwith a turbulent boundary layer. After assessing the different models and performing severaltests, the Spalart-Allmaras model was chosen to be the best candidate for this airfoil.

The model was validated against several different references, including a flying-hot-wire study byColes and Wadcock [5] for a chord based Reynolds number of 1.500.000, and a Direct NumericalSimulation (DNS) performed by Vinuesa et al. [18] at the Mechanics department at KTH for aReynolds number of 400.000.

Sammanfattning

Syftet med denna studie var att utvärdera prestandan hos de redan implementeradeRANS-modellerna i programvaran OpenFOAM, vars källkod är öppen, i simuleringar avturbulent flöde runt ett NACA 4412 vingprofil. Efter en studie med en rad tester av de olikamodellerna, valdes Spalart-Allmaras modellen som den bästa modellen för denna vingprofil.

Denna modell kontrollerades mot en rad olika referenser, bland annat en så kallad flying-hot-wirestudie av Coles och Wadcock [5] för ett kordbaserat Reynoldstal av 1.500.000. Validering gjordesäven för ett Reynoldstal 400.000 mot en Direct Numerical Simulation (DNS) utförd av Vinuesa etal. [18] på institutionen för Mekanik vid KTH.

Acknowledgements

We would like to express great appreciation to Marco Atzori and Alvaro Tanarro for their patientguidance and many hours spent supervising. We would also like to thank Ricardo Vinuesa andPhilipp Schlatter for all the good feedback and comments throughout the project.

Contents1 Introduction 1

1.1 Fluid mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fundamental equations of fluid mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Wall-bounded turbulent flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Separation of scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Modelling 42.1 DNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 LES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 RANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Positives and Negatives of RANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 k-� model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.3 Realizable k-� model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.4 k-ω model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.5 k-ω-SST model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.6 Spalart-Allmaras model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Numerical Setup 93.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Grid Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Boundary & Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.7 Post processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Results 134.1 Mesh study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Computational cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4 Case 1 - Re = 400k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.4.1 U+ v. y+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4.2 Cp & Cf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.5 Case 2 - Re = 1.5m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5.1 U+ v. y+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5.2 Cp & Cf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.6 CL, CD and aerodynamic efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Conclusion 255.1 Re = 400.000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Re = 1.500.000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Limitations of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Appendices 266.1 Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1 Introduction

1.1 Fluid mechanicsFluid mechanics is the physics and study of fluids and their behaviour. It is a field of study that has its rootsin ancient Greece when Archimedes formulated his buoyancy principle. It has later been expanded upon bymany known physicists, including Da Vinci, Newton and Bernouilli. There are still many unknowns in the field,especially concerning the nature of turbulence and the equations that govern it.

In more recent years the field has become more reliant on computers to solve complex simulations in order toaid in the design of automobiles, aircraft and other vehicles, the study of flow in pipes (e.g. blood in bloodvessel) and the turbines used in power plants.

1.2 Fundamental equations of fluid mechanicsThe most fundamental pillars of modern fluid mechanics are the Navier–Stokes (NS) equations. The equationsdescribe the motion of viscous fluids. The Navier—Stokes equations were first formulated by Claude-LouisNavier, a French physicist and engineer, and George Gabriel Stoke, an Irish physicist and mathematician, in1822.

The NS equations are derived through assuming continuum mechanics and applying Newton’s second law to ainfinitesimal fluid particle and using the principles of continuity of mass, momentum and energy [11]:

F =∂(mv)

∂t(1)

The Navier–Stokes equations can be written as follows, beginning with the conservation of momentum:

∂ui∂t

+∂

∂xj

[uiuj +

p

ρ− ν ∂ui

∂xi

]= 0 , i = 1, 2, 3 (2)

These equation describes the equilibrium of momentum for a fluid particle in three directions. The next equationis the conservation of mass or the "continuity equation". This equation describes the transfer of mass in anincompressible fluid due to fluctuations in the flow.

∂uj∂xj

= 0. (3)

And lastly, conservation of energy which describes the balance of energy for a fluid particle:

∂e0∂t

+∂

∂xj

[uje0 +

ujp

ρ+qjρ− 2νuiSij

]= 0. (4)

where qj is the heat-flux and Sij is the Strain rate tensor, defined as

Sij ≡1

2

(∂ui∂xj

+∂uj∂xi

)− 1

3

∂uk∂xk

δij . (5)

1

1.3 TurbulenceIn laminar flow the the so called streamlines are all parallel. The diffusion of momentum between particles isin general high, while the dissipation of momentum is very low, i.e. no momentum transport in the directionnormal to the streamlines. The kinetic energy of the flow is dampened by the viscosity of the fluid.

When the viscosity of the fluid no longer can dampen all the kinetic energy, eddies (vorticies) form and theflow turn turbulent. A turbulent flow has a few important characteristics: [11]

• In a turbulent flow, the dependent quantities fluctuate relative to the mean in what appears to be achaotic fashion.

• The vorticity of the flow field also fluctuates, which gives rise to spinning structures called eddies. Aturbulent flow contains a range of different eddy sizes and this range increases with Reynolds number.The largest eddies are the size of the turbulent region and it is these eddies that contain the majority ofthe energy.

• The largest eddies collapse into smaller and smaller eddies until the eddies are small enough for viscousforces to dissipate the energy. This phenomena is know as the energy cascade and discussed further insection (1.3.2). This means that the dissipation of energy is high and thus a persistent energy supply isnecessary to maintain a turbulent flow.

• Due to the presence of eddies, the mixing rate of momentum and energy (heat) is high compared to alaminar flow.

Basically all flows we encounter on a macroscopic scale are in the so called turbulent region, everything fromflow through pipes and ducts, to winds and currents in the atmosphere. The characteristics previouslydescribed can be useful, e.g. a high diffusion is good when mixing is desired like in fuel systems, but they canalso pose great problems. The high diffusion of momentum in turbulent flows is an important factor whendesigning any kind of vehicle, since high diffusion of momentum means large wall shear stresses and thusincreased drag compared to a laminar flow. Increased drag means higher fuel consumption which is bothexpensive and increases the environmental impact.

2

1.3.1 Wall-bounded turbulent flows

A vast majority of turbulent flows are so called wall bounded flows which as the name suggests are boundedby one or more solid walls. These flows can be either internal flows, like flows in channels or ducts, or externalflows like the flow over a wing or around a car.

For the case studied in this report, the external flow around an airfoil, the near wall region is represented by aboundary layer. In this layer the velocity tangential to the wall takes on a distinct profile. Due to viscousforces the velocity at the wall is zero and then increases up to the free-stream velocity. Depending on theReynolds-number and the geometry of the airfoil this layer is more or less turbulent. It is usually laminar nearthe leading edge and then transitions into turbulence as it approaches the trailing edge. The thickness of theboundary layer δ(x) also grows with increasing x. When the boundary layer transitions to become turbulentthe mean velocity profile is no longer constant, which leads to a changing pressure according to

− dp0dx

= ρU0dU0dx

. (6)

This pressure gradient is said to be favourable (FPG) if it is negative, that is if the flow is accelerating(dU0

dx> 0), and adverse (APG) if it is positive and the flow is decelerating.

Figure 1: Boundary layer with APG (Figure extracted from: [7])

In figure 1 the effects of an adverse pressure gradient on a boundary layer is shown. The flow is constantlydecelerated and this effect is most prominent close to the wall where the momentum of the fluid is lower.When the near-wall flow decelerates to a standstill (at (c) in 1) a tipping point is reached and the near-wallflow then switches direction. This is what is known as flow separation and it can have major effects inaeronautical applications, i.e for an airfoil, flow separation leads to a massive decrease in lift. [13]

3

1.3.2 Separation of scales

Turbulent flows consist of fluctuations and eddies (vortices) in a range of sizes. This can be explained throughthe energy cascade, first described by Lewis F. Richardson in the 1920s. [14] Richardson stated that thecharacteristic length of the largest eddies, l0, is comparable to the scale of the flow L and that their velocityscale is similar to the free-stream velocity U∞. This means that the Reynolds number of the largest eddies isclose to the high Reynolds number of the free flow and thus viscous effects are negligible. Richardson thenstated that these large eddies are very unstable and break up to give rise to smaller eddies which in turn breakup into even smaller eddies. This process goes on until stable eddies are formed, with a Reynolds numbersmall enough for viscous shear forces to be prominent. The kinetic energy of these eddies is then finallydissipated by the molecular viscosity into heat. [13]

In order to better understand the scale of the smallest eddies Andrey Kolmogorov developed what is todayknow as the Kolmogorov microscales. They are scales of length, velocity and time based on � the dissipation ofturbulent kinetic energy, and ν the kinetic viscosity of the fluid. They are defined as

η ≡(ν 3�

)1/4Kolmogorov length scale

uη ≡ (�ν)1/4 Kolmogorov velocity scale

τeta ≡(ν�

)1/2Kolmogorov time scale

(7)

Kolmogorov based the definition of these scales on three hypotheses:

1: The hypothesis of local isotropy states that for sufficiently high Reynolds-number flows, the small scaleturublence is statistically isotropic.

2: The first similarity hypothesis states that for every turbulent flow of high enough Reynolds number, thestatistics of the small scale turbulence (i.e statistically isotropic turbulence) have a universal form, determinedby � and ν.

3: The second similarity hypothesis states that for every turbulent flow of high enough Reynolds number, thestatistics of the turbulence of scale l in the range l0 � l� η have a universal form, determined by � andindependent of ν. [10]

Through these hypotheses Kolmogorov also came to the conclusion that the characteristic velocity scale u(l)and time scale τ(l), both decrease as l decreases.

2 ModellingAs in any other field of science, the most favourable approach would be to derive a theory or model that isapplicable to any problem one can imagine. In the case of turbulence this would be to develop an analyticsolving-method for solving the Navier-Stokes equations with turbulence, although this has proven to beimmensely difficult. Physicists and mathematicians are yet to even prove that smooth solutions always existsfor the Navier-Stokes equations, and this has been stated as one of the Millennium Prize Problems by the ClayMath Institute. [6]

One workaround for this problem, which has proven to be successful is Computational Fluid Dynamics (CFD).Through numerical solving and modelling of turbulence we have managed to overcome a number ofengineering challenges and learn a great deal about turbulence. CFD may, from a simplified perspective, bedivided into three categories: Direct Numerical Simulation (DNS), Reynolds-Averaged Navier–Stokes (RANS)and Large-Eddy Simulation (LES). These are further discussed in the following sections.

4

2.1 DNSIt is very much possible to solve the Navier-Stokes equations with turbulence instantaneously, althoughpredicting the complete time-dependence of the flow is by today’s means impossible. A simulation in whichthe Navier-Stokes equations are solved down to a numerical precision, with discrete time steps is called aDirect Numerical Simulation (DNS). Since DNS’s requires the whole range of spatial and temporal scales to beresolved, they often have extremely high computational costs. For this reason the DNS approach was more orless impossible before the 1970s. The sheer number of operations required renders DNS’s too computationallyexpensive even for some of todays most powerful super computers. For this reason DNS’s are mostly used inresearch. [13] Some recent examples of DNS’s of the flow around airfoils are given in Refs. [8] [18]

2.2 LESA Large Eddy Simulation (LES) can be seen as a compromise between a DNS and a RANS. Much like a DNSit solves numerically for the large-scale turbulence although when small enough turbulence scales are reached itinstead models the turbulence. The portion of the energy spectra which is modelled can vary between differentLES’s. In figure (2) an example of what such a division of the energy spectra could look like, is displayed.

As a consequence of this division, an LES can be expected to have significantly higher accuracy for flowsdominated by large-scale turbulence, in comparison to RANS models. Meanwhile the computational cost of aLES simulation is only a fraction of that of a full DNS, since most of the computational power of DNS’s isneeded to resolve the small-scale turbulence. The isotropic behaviour of this small scale turbulence, which wasdiscussed in section 1.3.2, makes it easy to model. [13] A recent example of a well-resolved LES of the flowaround an airfoil at a high Reyonolds number is given in Ref. [19]

Figure 2: Turbulent energy spectrum with RANS, LES and DNS spectra indicated

5

2.3 RANSAs stated earlier, a DNS has to resolve the entire spectra of spatial and temporal scales which in many cases isnot necessary. In most engineering applications the time scale of the overall flow is so large compared to thetiny time scales of the turbulent fluctuations, that it makes much more sense to look at the time-averagedquantities. This is done through the decomposition of the instantaneous quantity into a time-averaged partand a fluctuating part. Such a decomposition was first proposed by Osbourne Reynolds and is thus called aReynolds Decomposition. Using this decomposition the instantaneous quantities can be written as

ũi = Ui + uip̃ = P + pτ̃ij = Tij + τij

(8)

where Ui , P and T are the time-averages and ui , p and τij are the fluctuations with zero mean. Substituting(8) into the Navier-Stokes momentum equations (2) and then time-averaging yields the Reynolds AveragedNavier-Stokes equations, or in short the Reynolds equations. For a complete derivation we refer to FluidMechanics by P. K. Kundu. [11]

∂ui∂t

+∂

∂xj

[uiuj + P −

1

Re

∂ui∂ui

+ 〈uiuj〉]

= 0 , i = 1, 2, 3 (9)

These equations are the basis of the RANS-models (Reynolds-Averaged Navier-Stokes). The goal of suchmodels is to solve the Reynolds equations in order to obtain the mean velocity field. The equations are verysimilar to the instantaneous Navier-Stokes equations (2) presented earlier, with one exception, namely theappearance of a new term: 〈uiuj〉. This new term is a tensor with the dimension of a stress and is thususually called the Reynolds-stress, or the Reynolds-stress tensor.

The introduction of the Reynolds stress tensor does though pose a major problem since no new equation isintroduced with it. This means that there are now more unknowns than equations. This problem is known asthe closure problem. There are mainly two different approaches through which the closure problem can besolved: using turbulent viscosity models or using Reynolds-stress models. In Reynolds-stress models,individual transport equations are solved for each Reynolds-stress component which makes them morecomputationally expensive than turbulent viscosity models.

This report is focused on the use of turbulence viscosity models. In these models the Reynolds stresses aredetermined using the turbulent-viscosity hypothesis (Boussinesq hypothesis), which states that

〈uiuj〉 =2

3kδij − 2νT 〈Sij〉 , (10)

where νT is the so called turbulence (or eddy) viscosity. The challenge here is to determine an expression forthis eddy viscosity. [13] The expression generally consists of the product of a velocity u∗(xi, t) and a lengthl∗(xi, t):

νT = u∗ · l∗ . (11)

The boussinesq hypothesis does though limit turbulent viscosity models since it is not entirely accurate andshould only be used for simple shear flows.

6

2.3.1 Positives and Negatives of RANS

As the results will show later the average required computational time for the final mesh, using 4 cores, was 5minutes which puts the computational cost of the RANS at roughly 0.33 core-hours. The DNS by the KTHdepartment of mechanics, used as reference in this report, had a computational cost of 35.000.000 core-hours.

As stated earlier, for most cases the small fluctuations are negligible in comparison to the average flowalthough there are some exceptions. In acoustics for instance, the fluctuations is the only interesting part ofthe flow which renders RANS-simulations insufficient.

2.3.2 k-� model

The first model which is covered in this report is the k-� model. The model was first described in 1972 in areport by B.E Launder and W.P Jones. [9] Since then the model has developed and improved over time. Themodel was one of the first in a family of models known as the Two-equation models. In Two-equation modelsthe eddy viscosity is expressed using two turbulence quantities which are solved for using two model transportequations, hence the name. [13]

The k-� model uses k - the turbulent kinetic energy, and � - the turbulence dissipation rate, as its twoturbulence quantities. The transport equation for k is

Dk

Dt=

∂

∂xi

(νTσk

∂k

∂xi

)+ 〈uiuj〉

∂ui∂xi− � (12)

and for �D�

Dt=

∂

∂xi

(νTσ�

∂�

∂xi

)+ c1

�

k

(〈uiuj〉

∂ui∂xi

)− c2

�2

k. (13)

For high Reynolds-numbers the eddy viscosity is then determined using the following relation

νT =cνρk

2

�, (14)

where cν is a constant which has been empirically determined to be roughly 0.09 and ρ is the density which inour case is constant. [9]

The k-� model does though struggle when it comes to flows with adverse pressure gradients (which includesmost flows around airfoils). For these flows the model predicts too high shear-stresses and thus fails to predictseparation. [12]

2.3.3 Realizable k-� model

The Realizable k-� model was presented in 1994 as an improvement to the original method. The new modelwas designed to better predict the spreading rate for planar and round jets, and to perform better for flowswith rotation and for boundary layers with strong pressure gradients. This is achieved through theintroduction of a new transport equation for �, [15]

D�

Dt=

∂

∂xi

(νTσ�

∂�

∂xi

)+ c1�

√2SijSij − c2

�2

k +√νT �

. (15)

7

2.3.4 k-ω model

The k-ω model was published by David C. Wilcox in 1998 [20], and is similar to the k-� model, but uses ωinstead of � as the second modelled turbulence property. Wilcox states that ω can be interpreted as "... theratio of the turbulent dissipation rate � to the turbulent mixing energy k." and he defines it as

ω =�

β · k, (16)

where β is a so called closure coefficient equal to 340. The transport equation for k is the same as in the k-�

model and the transport equation for ω is

ρDω

Dt=γω

k〈uiuj〉

∂ui∂xj− βρω2 + ∂

∂xj

[(ν + σνT )

∂ω

∂xj

](17)

Compared to the k-� model, this model has been shown to be very accurate at describing boundary layerswith adverse pressure gradients. There is though one major setback to the model: It requires a non-zeroboundary condition for ω and is very sensitive to this specified value. According to F. R. Menter [12] themagnitude of the eddy viscosity has been shown to change by more than 100% by just changed the freestreamvalue of ω. The model also under-predicts the spreading rate for free shear layers.

2.3.5 k-ω-SST model

The k-ω-SST model (Shear Stress Transport) was published by Florian R. Menter in 1994 and combines thebest parts of the k-� and k-ω models. Put simply it uses the standard k-ω model close to walls (within 50% ofthe boundary layer) and then gradually "switches" to a behaviour more like the k-� model as it approaches theboundary layer edge. This is done through the introduction of a blending function, which takes on the value 0at the wall and then gradually switches to 1 using the wall-normal distance y as switching variable. Themodel is thus able to avoid the strong dependency on free-stream values that the original k-ω model exhibits.

2.3.6 Spalart-Allmaras model

Unlike previous model this is a one-equation model which was developed in the early 1990s by Spalart &Allmaras [16]. Their article states that the motivation for the model was: "The aerodynamics community feelsthe need for, and is ready to invest in, a new generation of turbulence models, more onerous than the algebraicmodels but with a wider envelope in terms of flow and grid complexity". The model was in other wordsdeveloped to give good results in aerodynamics applications, and specially for wall-bounded flows. It was alsomeant to be a good middle-ground between the simpler algebraic models and the more complex two-equationmodels. It performs better than algebraic models, and also most other one-equation models in the sense thatit does not suffer from incompleteness (e.g. The mixing length model requires a specification of the mixinglength). [13]

In the Spalart-Allmaras model the Reynolds stresses are given by

〈uiuj〉 = 2νTSij . (18)

The Spalart-Allmaras model being a one-equation model only needs one transport equation to be solved,namely the transport equation for νT . This equation is on the form

D̄νTD̄t

= ∇ ·(νTσν∇νT

)+ Sν , (19)

where σν is a constant and Sν is a source term which is dependent on viscosities, mean vorticity, the gradientof νT and the distance to the nearest wall. [16]

There are though many limitations to the Spalart-Allmaras model, including its inability to account for thedecay of νT and its over-prediction of the spreading of free jets. [13]

8

3 Numerical Setup

3.1 GeometryThe wing profile studied in this report is of the type NACA 4412. The NACA family of airfoils were developedduring the 1930 by the National Advisory Committee for Aeronautics (NACA) the predecessors of NASA.More specifically our airfoil is part of the NACA four-digit series, where the name of each airfoil is a four-digitnumber derived from its most critical geometric parameters. There exists also a five-digit series.

The first digit represents the maximum camber and the second digit tells where this camber is located intenths of the chord length. For the 4412 profile this means that the maximum camber is 4% at 40% of thechord length. The last two digits represents the maximum thickness of the airfoil as a percentage of the chordlength, so the maximum thickness of the 4412 is 12% of the chord length. [2]

3.2 Grid DependenceAny simulation, no matter how refined a grid is used, will introduce numerical errors. These errors will mostoften depend on how fine a mesh is used in the simulation, i.e. the size of ∆x and ∆y but also on the size ofthe time step, ∆t. There can also be a lot of deviations between similar physical problems that are solved ongrids that are structured differently.

When setting up a numerical simulation it is therefore important to be aware that the results may be affectedby the composition of the grid. A grid study was therefore conducted and a final grid developed over severaliterations. This grid is structured for the regions close to the wall but unstructured in all other regions withthe size of the elements gradually increasing as one approaches the free stream.

In order to avoid the results being interfered with by the boundary, the area being simulated was expanded sothat free stream conditions were imposed at 50 chord lengths in each direction.

3.3 ModelIn this study the open source program "blueCFD-Core" was used to perform the simulations. BlueCFD isbased on the open source software OpenFOAM, which is a free but versatile CFD software capable of runningmany different CFD simulations using a wide variety of numerical methods and models. [4]

OpenFOAM uses a method known as the SIMPLE algorithm (Semi-Implicit Method for Pressure LinkedEquations) in order to reach a steady state solution to the NS-equations. SIMPLE uses "finite volumeschemes" (FVM) in order to solve for the desired quantities. The numerical schemes used in the simulations inthis report were linear Gaussian upwind schemes for most quantities. A more detailed explanation of thismethod can be found in the documentation of OpenFOAM. [3]

9

3.4 MeshIn order to facilitate the simulations and create a easy-to-use method of performing simulations an automaticmesher was used to create all the meshes used in this study. The software used to create the meshes is calledGmsh in which the user can create a mesh from a CAD-model using parameters. This facilitates the process ofcreating a mesh and means that one can create a full mesh within a few hours. [1]

After several iterations and studying which meshes worked well a final mesh was chosen. There is always roomfor improvement, especially since the mesh is automatically generated, creating regions where the resolution ofelements change quite rapidly. However, one must keep in mind that as a mesh becomes more complex and asone increases the total amount of elements, the computational cost also increases.

As such, the final mesh is a compromise between resolution and computational cost. This mesh has 65714 gridpoints and 54586 cells. The y+ of the mesh varies between 0.05 and 6.5 with and average of about 2.5.

Figure 3: Trailing edge of final mesh

This mesh is 50 chord lengths in size in both the x and y-direction from the airfoil in order to reduce the effectof the boundary conditions.

Figure 4: Closeup of airfoil in final mesh. Mesh extends 50 chord lengths in both cardinal directions

10

3.5 Boundary & Initial ConditionsThe Reynolds number is a dimensionless parameter which is generally defined as

Re =Ul

ν(20)

and can be thought of as a ratio between the inertial and viscous forces in a given flow. In order to achievedimensionlessness the free stream velocity U∞ was set equal to 1. The length l was in this case replaced withthe chord length C and was also set equal to 1. Finally the kinematic viscosity was set equal to ν = 1

Re.

The initial free stream condition for νT is ideally zero for the Spalart-Allmaras model, although this can beproblematic for many solvers because of round-off errors, or because the convergence test may require divisionwith νT . The reason why a low initial value for νT is preferred is that νT should be sufficiently small incomparison to ν in order for the laminar boundary layers to remain undisturbed. As recommended in thereport by Spalart & Allmaras, a initial value of νT =

ν

10was used. [16]

Along the wall of the airfoil a no-slip boundary condition is imposed, meaning that the velocity, U , is set tozero along all the points at the wall.

The values of νT along the wall were determined using one of OpenFOAMs built in wallfunction callednutUSpaldingWallfunction. The function assumes that the Law of the wall can be used, i.e. that there aredistinct viscous, buffer and log layers, and then imposes the expected value of νT based on the given velocities.[17]

3.6 SamplingIn the tutorial cases uniform sampling was used for simplicity, although this is far from ideal when samplingvelocities in the boundary layer. Close to the wing, the viscous forces are strong and distort the surroundingflow which leads to a distinct velocity profile. A more ideal sampling tool would use a bias with many pointsclose to the wall, and then fewer and fewer points as the wall-normal distance increases. For this reason aMATLAB-program was constructed which distributes points with exponentially increasing distance (SeeAppendix 1 ). The program gives a set of points which are then fed into OpenFOAM’s sampling utility usingthe "cloud" sampling type.

11

3.7 Post processingAfter the simulation has reached a steady state, it is possible to reconstruct pressure, velocity, shear-stresses aswell as other fields in the domain. This can be called the "raw" data of the case and while useful, it ispreferable to process this data further in order to calculate standardised quantities that are useful to comparethe results between cases and experiments.

Among these quantities are Cp, the pressure coefficient along the airfoil, Cf , the skin friction coefficient alongthe airfoil.

Cp =p− pref

1

2ρU 2∞

(21)

Cf =τw

1

2ρU 2∞

(22)

Where pref is chosen so that Cp is 1 at the stagnation point. Cp and Cf can be used to study how the flowevolves along the wing as well as to assure that the results are in accordance with reality. They are also usedto calculate L, the lift force, D, the drag force, as well as the lift and drag coefficients CL and CD which aredefined as;

CL =L

1

2ρU 2∞S

(23)

CD =D

1

2ρU 2∞S

(24)

Where S is the chord length of the airfoil, 1, since the simulation is done in 2D. When both CL and CD arecalculated it is also possible to calculate the aerodynamic efficiency Aeff as;

Aeff =CLCD

(25)

Another important area of study is the boundary layer and how it develops. In order to be able to drawmeaningful conclusions it is important to non-dimensionalise the velocity and length scale U and y into U+

and y+.

U+ =U

uτ(26)

y+ =yuτν

(27)

Where uτ is defined as uτ =√τw/ρ.

12

4 Results

4.1 Mesh study

Figure 5: Comparison between different meshes of U+ v. y+ at x/c = 0.4

From the mesh study, Mesh 2 performs best compared to the reference DNS data [8]. The plot shows verygood agreement in the viscous and buffer layers, and still good agreement in the log-layer. The mesh seems tobe sufficiently fine to accurately capture the physics of the flow around a NACA 4412 airfoil at aReynolds-number of 400.000.

Number of cells Mean y+ Min/Max y+

Mesh 0 42012 10.4 0.5/21.4Mesh 1 52525 5.2 0.1/12Mesh 2 54586 2.5 0.05/6.5

Table 1: Mesh data

13

4.2 Convergence

Figure 6: Residuals of various quantities for case 2, Re = 1.5m, at angle of attack = 7.5◦

This is a sample of one of the convergence tests. The plot shows that all the flow quantities have convergedand that the residuals are acceptably low with respect to the value of the quantity. The result was the sameand the residuals were of the same order of magnitude for all the different cases and angles of attack.

4.3 Computational costFor the final cases the domain was divided into four parts and then each part was assigned to its ownCPU-core. This allowed the four domain parts to be solved simultaneously in parallel. One case took roughly 5minutes to run which puts the computational cost at: 4 cores · 5 minutes = 20 core-minutes = 0.3 core-hours.As reference, the computational cost of the DNS by Hosseini et al [8] was roghly 35.000.000 core-hours.

14

4.4 Case 1 - Re = 400kFor the first case with a Reynolds number of 400.000 the reference used is a DNS by Hosseini et al, which isdescribed in further detail in [8] and [18]

4.4.1 U+ v. y+

Figure 7: U+ v. y+ - plot at x/c = 0.4 for varying angles of attack

Figure 8: U+ v. y+ - plot at x/c = 0.6 for varying angles of attack

15

Figure 9: U+ v. y+ - plot at x/c = 0.8 for varying angles of attack

From the U+ vs y+ plots (7, 8, 9) the performance of the Spalart-Allmaras model of OpenFOAM seems to bequite good in comparison to the DNS [8], although the RANS-simulation for AoA = 5◦ reaches a slightlyhigher maximum velocity than the DNS for the same AoA.

As expected, the plots show that a higher AoA leads to a higher acceleration of the flow towards the trailingedge, and the maximal velocity seems to grow almost exponentially with increased AoA. The log-layer is quitedistinct in the plots and begins at roughly y+ = 10.

The overall shapes of all of the plots show signs of an adverse pressure gradient, which makes sense for anairfoil with a positive angle of attack.

16

4.4.2 Cp & Cf

Figure 10: −Cp on suction side for varying angles of attack

Figure 11: Cp on pressure side for varying angles of attack

17

Figure 12: Cf on suction side for varying angles of attack

Figure 13: Cf on pressure side for varying angles of attack

18

The Cp plots show great agreement with the DNS, although this was expected since it is fairly easy to get agood estimate of the pressure distribution.

The Cf curves from the DNS differs a lot from the RANS data near the leading edge, although this is alsoexpected since the DNS utilises tripping in order to get a turbulent boundary layer. The tripping occurs at thedistinct peak in plots (12) and (13), near 10% of the chord length. From around 50% of the chord til thetrailing edge the agreement with the DNS is very good.

In figure (12) Cf becomes negative close to the trailing edge (see the magnified box in figure (12)). Thisindicates flow separation and begins at around 0.85 of the chord length with AoA = 10◦, at 0.92 of the chordlength for AoA = 7.5◦ and at 0.97 of the chord length for AoA = 5◦

4.5 Case 2 - Re = 1.5mFor the case with a Reynolds number of 1.500.000 the reference used is a study by Donald Coles and Alan J.Wadcock from the 1970s. [5] The study consisted of hot-wire measurements of the boundary layer of a NACA4412 airfoil in a wind-tunnel.

4.5.1 U+ v. y+

Figure 14: U+ v. y+ - plot at x/c = 0.4 for varying angles of attack

19

Figure 15: U+ v. y+ - plot at x/c = 0.6 for varying angles of attack

Figure 16: U+ v. y+ - plot at x/c = 0.8 for varying angles of attack

From the U+ vs y+ plots (14, 15, 16) for the 1.500.000 Reynolds case there was no data in the referencereport by Coles and Wadcock and thus not much can be said about the Spalart-Allmaras models performance.In figure (14) the transition from buffer-layer to log-layer can be seen clearly at y+ ≈ 0.5. One can alsoconclude that the effect of the AoA on U+ is decreased as the Reynolds numer increases.

20

4.5.2 Cp & Cf

Figure 17: −Cp on suction side for varying angles of attack

Figure 18: Cp on pressure side for varying angles of attack

21

Figure 19: Cf on suction side for varying angles of attack

Figure 20: Cf on pressure side for varying angles of attack

22

In the Cp plots (17 & 18), quite good agreement with the reference can be seen for the AoA = 14◦ case.Although the RANS deviates slightly from the experiment towards the trailing edge.

For the skin friction coefficient there was no data in the reference. The plots does show similar behaviour tothe 400.000 Reynolds case with the biggest impact of the AoA at the leading edge. There are signs of flowseparation (negative Cf ) towards the trailing edge (see the magnified box in figure (19)) beginning at around0.9 of the chord length with AoA = 10◦ and at 0.96 of the chord length for AoA = 7.5◦. Compared to the400.000 Reynolds case, flow separation seems to occur later which implies that the higher Reynolds numbercounteracts separation to some degree.

4.6 CL, CD and aerodynamic efficiency

Figure 21: CL comparison for different angles of attacks between the cases

23

Figure 22: CD comparison for different angles of attacks between the cases

Figure 23: Comparison of the aerodynamic efficiency, Aeff , for different angles of attacks between the cases

24

5 ConclusionIn general it seems that the Spalart-Allmaras model implemented in OpenFOAM works well with the finalmesh for a Reynolds number of 400.000, although for higher Reynolds numbers a finer mesh is most likelyneeded. The results show that the model, paired with a proper mesh, can accurately approximate the flowaround a NACA 4412 airfoil.

The fact that the results of the Spalart-Allmaras simulation shows such good agreement with the DNS byHosseini et al, even though the computational cost is such an infinitesimal fraction of the computational costof the DNS, shows what a powerful tool the model can be.

5.1 Re = 400.000As stated earlier, the simulation with a Reynolds number of 400.000 shows very good agreement in the viscouslayer and buffer layer, and still good agreement in the log layer. The model accurately captures the behaviourof the flow and predicts an adverse pressure gradient and even flow separation close to the trailing edge forsome angles of attack.

5.2 Re = 1.500.000The simulation with a Reynolds number of 1.500.000 showed good agreement with the Cp curve from thestudy by Coles and Wadcock [5] although it deviates slightly towards the trailing edge. This could be aproblem in the experiment, due to the disturbances introduced by the rotating hot-wire rig in the wake of theairfoil, as discussed in the original report [5].

In the skin friction coefficient curves (19 & 20) oscillations of a significant amplitude can be seen close to theleading edge. This could be an indication that the mesh was not sufficiently fine to capture the physics of theflow at this Reynolds number. Since the mesh study could only be performed at a Reynolds number of400.000, due to insufficient reference data, not much can be said about the performance of mesh 2 at higherReynolds numbers.

5.3 Limitations of the studySince RANS models per definition only calculate the mean of the flow quantities, the results from this studycannot be used in applications where it is necessary to consider time-varying fluctuations, such as incalculating stresses that lead to fatigue, or in aeroacoustics applications, etc.

Another limitation of the models used in this study is the fact that they are all based on the turbulenceviscosity hypothesis 10 and this is only valid for simple shear flows. As stated earlier, there are other RANSmodels known as Reynolds stress models, which solves directly for the Reynolds stresses. It would beinteresting to try different Reynolds stress models on the same cases with the same mesh in order to see howmuch the turbulence viscosity hypothesis actually limits the turbulence viscosity models.

5.4 Future workThe focus of this study was to evaluate the RANS models already implemented in OpenFOAM and thus allthe meshes were produced using an automatic mesher. This was partially due to time restrictions, andpartially in the spirit of keeping computational costs and time consumption down. A good starting point forfuture work would be to simply use the same mesher to produce a much finer mesh for the 1.500.000 Re caseto see if the oscillations disappear.

For both the 400.000 Re case and the 1.500.000 Re case, the Spalart–Allmaras model predicted flow separationtowards the trailing edge for angles of attack of 7.5 and upwards. Another interesting study would thus be tovalidate how accurately the Spalart–Allmaras actually predicts flow separation. This could be done byperforming LES’s and/or windtunnel experiments with the same setup to use as references.

25

6 Appendices

6.1 Appendix 1

1 %% openFoam sampleDict point generator − v: 22 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−3 %% NACA Generator4 clc5 clear6

7 points = 10000; %input(’Enter number of points: ’);8

9 sidePoints = points/2;10 xc = linspace(0,1, sidePoints) ;11

12 m = 4/100;13 p = 4/10;14 t = 12/100;15

16 yt = 5 ∗ t ∗ (0.2969 ∗ sqrt(xc) − 0.1260∗xc − 0.3516∗xc.^2 + 0.2843∗xc.^3 − 0.1036∗xc.^4);17

18 yc = zeros(1,sidePoints) ;19 dyc = zeros(1,sidePoints) ;20

21 for n = 1:sidePoints22

23 if xc(n) < p24 yc(n) = m/p^2 ∗ (2∗p∗xc(n) − xc(n)^2);25 dyc(n) = 2∗m/p^2 ∗ (p − xc(n));26 else27 yc(n) = m/(1−p)^2 ∗ ((1−2∗p) + 2∗p∗xc(n) − xc(n)^2);28 dyc(n) = 2∗m/(1−p)^2 ∗ (p − xc(n));29 end30 end31

32 theta = atan(dyc);33

34 xU = xc − yt.∗sin(theta);35 yU = yc + yt.∗cos(theta);36

37 xL = xc + yt.∗sin(theta);38 yL = yc − yt.∗cos(theta);39

40

41 %% Sample line calc42

43 %Log−spacing calc44 % −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−45 a = 0;46

47 b = 0.2; % length of sampling line48 scale = 10^6; % Scaling factor49 nP = 1000; % Number of points50

51 aTemp = log10(a + 1);52 bTemp = log10(b ∗ scale + 1);53

54 % Spaced points55 sPoints = (logspace(aTemp,bTemp,nP) − 1) ./ (scale);56

57 % Angle calc58 % −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−59 cIndex04 = 1;60 cIndex06 = 1;61 cIndex08 = 1;62

63 while 164 if xU(cIndex04) < 0.465 cIndex04 = cIndex04 + 1;

26

66 end67

68 if xU(cIndex06) < 0.669 cIndex06 = cIndex06 + 1;70 end71

72 if xU(cIndex08) < 0.873 cIndex08 = cIndex08 + 1;74 else75 break76 end77 end78

79 k04 = (yU(cIndex04+1) − yU(cIndex04−1)) / (xU(cIndex04+1) − xU(cIndex04−1));80 k06 = (yU(cIndex06+1) − yU(cIndex06−1)) / (xU(cIndex06+1) − xU(cIndex06−1));81 k08 = (yU(cIndex08+1) − yU(cIndex08−1)) / (xU(cIndex08+1) − xU(cIndex08−1));82

83 alpha04 = atand(k04) + 90;84 alpha06 = atand(k06) + 90;85 alpha08 = atand(k08) + 90;86

87 xS04 = sPoints ∗ cosd(alpha04) + xU(cIndex04);88 yS04 = sPoints ∗ sind(alpha04) + yU(cIndex04);89

90 xS06 = sPoints ∗ cosd(alpha06) + xU(cIndex06);91 yS06 = sPoints ∗ sind(alpha06) + yU(cIndex06);92

93 xS08 = sPoints ∗ cosd(alpha08) + xU(cIndex08);94 yS08 = sPoints ∗ sind(alpha08) + yU(cIndex08);95

96 %% Sample line calc for beta97 a = 0;98 b = 0.4;99 nP = 200;

100

101 sPoints = linspace(a,b,nP);102 cIndex = (1:100)∗50−2;103

104 k = (yU(cIndex+1) − yU(cIndex−1)) ./ (xU(cIndex+1) − xU(cIndex−1));105 alpha = atand(k) + 90;106

107 xS = zeros(length(sPoints),length(cIndex));108 yS = zeros(length(sPoints),length(cIndex));109

110 for n = 1:length(cIndex)111 xS(:,n) = sPoints ∗ cosd(alpha(n)) + xU(cIndex(n));112 yS(:,n) = sPoints ∗ sind(alpha(n)) + yU(cIndex(n));113 end114

115 xS0 = xS(1,:);116 yS0 = yS(1,:);117

118 %% PLotter119

120 load af .mat % Imported NACA data121 figure (1) , axis equal, hold on, grid on122

123 % Upper side124 plot(xU,yU,’−k’)125

126 % Lower side127 plot(xL,yL,’−k’)128

129 % Sample lines130 plot(xS04,yS04,’.r ’ )131 plot(xS06,yS06,’.r ’ )132 plot(xS08,yS08,’.r ’ )133

134 % Reference

27

135 plot(af .xU,af.zU,’b.−’)136 plot(af .xL,af.zL,’b.−’)137

138 figure (2) , axis equal, hold on, grid on139

140 % Upper side141 plot(xU,yU,’−k’)142

143 % Lower side144 plot(xL,yL,’−k’)145

146 % Sample lines147 plot(xS,yS,’ . r ’ )148

149 %% Data exporter 2.0150

151 z = 0.225 ∗ ones(length(xS04),1);152

153 p04 = [xS04’ yS04’ z ];154 p06 = [xS06’ yS06’ z ];155 p08 = [xS08’ yS08’ z ];156 p0 = [xS0’ yS0’ ones(100,1) ∗0.225’];157

158 formatSpec = ’(%.8f %.8f %.3f) ’ ;159

160 file04 = fopen(’points04.txt’ , ’w’);161 file06 = fopen(’points06.txt’ , ’w’);162 file08 = fopen(’points08.txt’ , ’w’);163 file0 = fopen(’points0.txt ’ , ’w’);164

165 for n = 1:length(p04)166 fprintf ( file04 ,formatSpec,p04(n,1),p04(n,2),p04(n,3));167 fprintf ( file06 ,formatSpec,p06(n,1),p06(n,2),p06(n,3));168 fprintf ( file08 ,formatSpec,p08(n,1),p08(n,2),p08(n,3));169 end170

171 for n = 1:length(p0)172 fprintf ( file0 ,formatSpec,p0(n,1),p0(n,2),p0(n,3)) ;173 end174

175 fclose ( file04 ) ;176 fclose ( file06 ) ;177 fclose ( file08 ) ;178 fclose ( file0 ) ;179

180 %% Data exporter 3.0 (Beta)181

182 formatSpec1 = ’(%.8f %.8f %.3f) ’ ;183 formatSpec2 = ’);\n }\n \n \n line_%i\n {\n type cloud;\n axis xyz;\n points ( ’ ;184

185 file = fopen(’preSample.txt’,’w’);186

187 for n = 1:length(cIndex)188

189 fprintf ( file ,formatSpec2,n);190

191 for m = 1:length(sPoints)192

193 fprintf ( file ,formatSpec1,xS(m,n),yS(m,n),0.225);194

195 end196 end197

198 fclose ( file ) ;

28

References[1] Gmsh - A three-dimensional finite element mesh generator with built-in pre- and post-processing

facilities. http://gmsh.info/.

[2] A history of the NACA and NASA, 1915-1990. https://history.nasa.gov/SP-4406/contents.html.

[3] OpenFOAM - The open source CFD toolbox). https://www.openfoam.com/.

[4] c© FSD blueCAPE Lda 2016-2018. The blueCFD-Core project.http://bluecfd.github.io/Core/About/.

[5] Donald Coles and Alan J. Wadcock. Flying-Hot-Wire Study of Flow Past an NACA 4412 Airfoil atMaximum Lift. AIAA Journal, 17(4), 1979.

[6] Charles L. Fefferman. Existence and smoothness of the Navier—Stokes equation.http://www.claymath.org/millennium-problems/.

[7] Tom Fletcher, John Altringham, Jeffrey Peakall, Paul Wignall, and Robert Dorrell. Hydrodynamics offossil fishes. Proceedings. Biological sciences / The Royal Society, 281, 08 2014.

[8] S.M. Hosseini, R. Vinuesa, A. Hanifia P. Schlatter, and D.S. Henningson. Direct numerical simulation ofthe flow around a wing section at moderate reynolds number. International Journal of Heat and FluidFlow, 61:117–128, 2016.

[9] W. P. Jones and B. E. Launder. The prediction of laminarization with a two-equation model ofturbulence. International Journal of Heat and Mass Transfer, 15:301–314, 1972.

[10] Andrey N. Kolmogorov. The Local Structure of Turbulence in Incompressible Viscous Fluid for VeryLarge Reynolds’ Numbers. Akademiia Nauk SSSR Doklady, 30:301–305, 1941.

[11] Pijush K. Kundu, Ira Cohen, and David Dowling. Fluid Mechanics. Academic Press, 2011.

[12] F. R. Menter, M. Kuntz, and R. Langtry. Ten years of industrial experience with the sst turbulencemodel. Turbulence, Heat and Mass Transfer, 4, 2003.

[13] Stephen B. Pope. Turbulent Flows. Cambridge University Press, 2011.

[14] Lewis F. Richardson. Weather prediction by numerical process. Cambridge, University Press, 1922.

[15] T.-H. Shih, W. W. Liou, A. Shabbir, and J. Zhu Z. Yang. A new k − � eddy viscosity model for highreynolds number turbulent flows - model development and validation. NASA Technical Memorandum106721, 1994.

[16] P. R. Spalart and S. R. Allmaras. A one-equation turbulence model for aerodynamic flows. Originallypresented at the AIAA 30th Aerospace Sciences Meeting and Exhibit, 1992.

[17] D. B. Spalding. A single formula for the “law of the wall”. Journal of Applied Mechanics, 28(3), 1961.

[18] R. Vinuesa, S. M. Hosseini, A. Hanifi, D. S. Henningson, and P. Schlatter. Pressure-gradient turbulentboundary layers developing around a wing section. Flow Turbulence Combust, 99:613–641, 2017.

[19] R. Vinuesa, P. S. Negi, M. Atzori, A. Hanifi, D. S. Henningson, and P. Schlatter. Turbulent boundarylayers around wing sections up to rec = 1,000,000. International Journal of Heat and Fluid Flow, 2018.

[20] D. C. Wilcox. Reassessment of the scale-determining equation for advanced turbulence models. AIAAJournal, 26(11), 1998.

29

http://gmsh.info/https://history.nasa.gov/SP-4406/contents.htmlhttps://www.openfoam.com/http://bluecfd.github.io/Core/About/http://www.claymath.org/millennium-problems/

www.kth.se

IntroductionFluid mechanicsFundamental equations of fluid mechanicsTurbulenceWall-bounded turbulent flowsSeparation of scales

ModellingDNSLESRANSPositives and Negatives of RANSk- modelRealizable k- modelk- modelk–SST modelSpalart-Allmaras model

Numerical SetupGeometryGrid DependenceModelMeshBoundary & Initial ConditionsSamplingPost processing

ResultsMesh studyConvergenceComputational costCase 1 - Re = 400 kU+ v. y+Cp & Cf

Case 2 - Re = 1.5 mU+ v. y+Cp & Cf

CL, CD and aerodynamic efficiency

ConclusionRe = 400.000Re = 1.500.000Limitations of the studyFuture work

AppendicesAppendix 1