MSc Thesis - Luis Felipe Paulinyi - Separation Prediction Using State of the Art Turbulence Models

University of SouthamptonFaculty of Engineering and the EnvironmentMSc. Race Car Aerodynamics

Separation Prediction Using State of theArt Turbulence Models

Luis Felipe de Aguilar Paulinyi

Supervisor: Richard D SandbergSecond Assessor: John S Shrimpton

SouthamptonSeptember 2013

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To my wife.

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Acknowledgments

This thesis is the last building block of an amazing journey that I had throughoutthis year and I would like to acknowledge and thank from the bottom of my heart thecontribution of the following individuals:

My wife, son and daughter, for understanding and for being there for me;My mother, for the psychological and financial support;My brother, a great motivator;

Professor Richard Sandberg, for sharing his knowledge and for his guidance during thisproject;

My colleagues and friends: Richard Prichler, Jack Whetheritt, Jesus Pozo, Manuel DiazBrito, David Williams and Paulo Gustavo Cervantes for all the help in this project.

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Abstract

A computational fluid dynamics (CFD) examination of the flow over the two-dimensionalT106 turbine cascade blade was conducted using the Reynolds Averaged Navier-Stokes(RANS) equations with different turbulence models using OpenFOAM as the solver. Bybeing an open source solver, OpenFOAM gives an unparalleled flexibility in defining theproblem setup when compared with other commercial software, this flexibility allows theprogramming of new modules such as different turbulent models from the ones provided,which allowed the investigation of two state of the art Explicit Algebraic Stress Models(EASM), a baseline EASM and the ϕ-α-EASM and compare them with classic turbulencemodels such as Spallart-Almaras and k-ω-SST. Classical turbulence models based in theBoussinesq hypothesis, by not being able to describe stress anisotropy, usually fail toproduce accurate results in flows with high streamline curvature such as the T106 blade.Two different Reynolds number conditions were tested 60,000 and 150,000. At the lowerReynolds number the use of the ϕ-α-EASM model have shown a better agreement withexperimental and direct numerical simulation (DNS) results for most of the regions ofattached flow and favorable pressure gradient, however, it failed to predict flow separationdue to the reduced Reynolds number, where laminar flow simulations presented betterresults.

Contents

Acknowledgments v

Abstract vii

Summary x

List of Figures xii

List of Tables xiii

1 Introduction 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Theoretical Background 72.1 Viscous Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 The Navier Stokes Equations . . . . . . . . . . . . . . . . . . . . . 102.2 Turbulence Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Turbulence Closure Strategies . . . . . . . . . . . . . . . . . . . . . 142.3.2 The Turbulence Closure Problem . . . . . . . . . . . . . . . . . . . 162.3.3 Reynolds-Averaged Navier-Stokes Turbulence Models . . . . . . . . 18

3 The Problem 233.1 Geometry and Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Setting up the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 The Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . 313.2.2 Typical OpenFOAM Simulation Setup . . . . . . . . . . . . . . . . 333.2.3 Determination of flow parameters . . . . . . . . . . . . . . . . . . . 34

3.3 Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Results 374.1 Preliminary Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.1 Steady Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.2 Unsteady Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Assessment of Turbulence Models on the T106A Blade . . . . . . . . . . . 484.2.1 Flow at Reynolds Number of 60,000 . . . . . . . . . . . . . . . . . . 494.2.2 Flow at Reynolds Number of 150,000 . . . . . . . . . . . . . . . . . 60

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x CONTENTS

5 Conclusions 65

A Description of Turbulence Models 73A.1 Algebraic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.2 One Equation and Two Equation Models . . . . . . . . . . . . . . . . . . . 76

A.2.1 Turbulence Kinetic Energy Equation . . . . . . . . . . . . . . . . . 76A.2.2 One Equation Turbulence Models . . . . . . . . . . . . . . . . . . . 76A.2.3 Spalart-Allmaras Model . . . . . . . . . . . . . . . . . . . . . . . . 77A.2.4 Two Equation Models . . . . . . . . . . . . . . . . . . . . . . . . . 78A.2.5 k-ω-SST Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

A.3 Reynolds Stress Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 79A.3.1 Expansion of the Boussinesq Hypotesis . . . . . . . . . . . . . . . . 79A.3.2 Algebraic Stress Models . . . . . . . . . . . . . . . . . . . . . . . . 80

A Boundary Conditions of Numerical Simulations 83

List of Figures

2.1 Flow close to a solid surface under the influence of an adverse pressure gra-dient, the deceleration of the flow leads to a reversed flow and a consequentseparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Results obtained numerically for the velocity distribution on the boundarylayers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Measurement of the instantaneous velocity with a probe in a turbulent flow[26] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Combined O-type/H-type mesh from Reference [16] used in the current work. 25

3.2 Details of the leading and trailing edges of the mesh of the T106 blade. . . 26

3.3 Details of the mesh on the connection points of five different blocks. . . . . 26

3.4 Details of the original mesh and the coarser mesh over the suction side ofthe T106 blade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Detail of the technique adopted do eliminate points to generate a coarsermesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6 Details of the connection of blocks of the coarse mesh of the T106 blade. . 29

3.7 Details of the leading and trailing edge on the coarse mesh of the T106 blade. 30

3.8 Name convention for the boundaries of the mesh used in the current work . 30

3.9 Standard convention for one-dimensional mesh in the finite volume method. 32

4.1 Convergence history for cases P1 and P2. . . . . . . . . . . . . . . . . . . . 39

4.2 Pressure and velocity distribution on the upper and lower boundaries ofthe domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 The history of convergence for the simulation P3. . . . . . . . . . . . . . . 41

4.4 Pressure coefficient distribution on the T106 blade for Cases P2 and P3,steady simulation, Reynolds 60,000. . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Wake loss at 40% chord for the T106 blade, Cases P2 and P3, steadysimulation, Reynolds 60,000. . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.6 Friction coefficient on the suction side of the T106 blade for Cases P2 andP3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.7 Velocity magnitude field for Cases P2 and P3, steady simulation, Reynolds60,000, velocity in m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.8 The history of convergence for the simulation P4, steady simulation, coarsemesh, Reynolds 60,000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.9 Pressure coefficient and wake loss for Case P4, steady simulation, coarsemesh, Reynolds 60,000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.10 Friction coefficient on the suction side of the T106 blade for Case P4, steadysimulation, coarse mesh, Reynolds 60,000. . . . . . . . . . . . . . . . . . . 47

4.11 The convergence history for the Cases P5 and P6. . . . . . . . . . . . . . . 48

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xii LIST OF FIGURES

4.12 Convergence history of the unsteady simulation with the Spalart-Allmarasturbulence model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.13 Numerical instability developed while running the baseline-EASM turbu-lence model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.14 Differences on the average of the wake in subsequent time intervals, laminarsimulation, coarse mesh, Reynolds 60,000. . . . . . . . . . . . . . . . . . . 51

4.15 Pressure coefficient distribution, Reynolds number 60,000. . . . . . . . . . 524.16 Zoom-in Pressure coefficient distribution, Reynolds 60,000. . . . . . . . . . 534.17 Friction coefficient on the suction side of the T106 blade, Reynolds number

60,000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.18 Velocity profiles for different turbulence models at four different points of

the suction side, refined mesh, Reynolds 60,000. . . . . . . . . . . . . . . . 564.19 Turbulent kinetic energy profiles for different turbulence models at four

different points of the suction side, refined mesh, Reynolds 60,000. . . . . . 574.20 Wake losses 40% chord downstream of trailing edge, Reynolds 60,000. . . . 584.21 Sequence of snapshots of the flow vorticity magnitude, at an interval of

0.002 seconds of simulated time. Sequence presented from top to bottom,coarse mesh, Reynolds 60,000. . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.22 Pressure coefficient distribution, Reynolds number 150,000. . . . . . . . . . 614.23 Instantaneous velocity magnitude field, velocity in m/s, Reynolds number

150,000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.24 Friction coefficient on the suction side of the T106 blade, Reynolds number

150,000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.25 Wake losses 40% chord downstream of trailing edge, Reynolds number

150,000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.26 Vorticity at the trailing edge of the profile, laminar simulation at Reynolds

number 150,000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

A.1 Diagram with the main RANS closure strategies. . . . . . . . . . . . . . . 74

List of Tables

3.1 Number of elements in each of the two meshes used in the study . . . . . . 28

4.1 Preliminary steady simulations . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Boundary conditions for simulations P1 and P2. . . . . . . . . . . . . . . . 394.3 Description of the cases simulated with Reynolds number and total time,

Reynolds 60,000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4 Description of the cases simulated with Reynolds number and total time,

Reynolds number 150,000. . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.5 Separation points calculated by the different turbulent models. . . . . . . . 64

A.1 Boundary conditions for laminar simulation. . . . . . . . . . . . . . . . . . 83A.2 Boundary conditions for Spalart-Allmaras turbulence model. . . . . . . . . 84A.3 Boundary conditions for k-ω turbulence model. . . . . . . . . . . . . . . . 84A.4 Boundary conditions for baseline-EASM turbulence model. . . . . . . . . . 84A.5 Boundary conditions for ϕ-α-EASM turbulence model. . . . . . . . . . . . 84

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xiv LIST OF TABLES

Nomenclature

αp The under-relaxation factor applied to the pressure

∆t The time step in an unsteady simulation

δ Boundary layer thickness

δij Kroeneker delta

Γ An Interface diffusion coefficient

N The Navier-Stokes operator

µ Molecular viscosity

µt Turbulent eddy viscosity

νt Kinematic turbulent eddy viscosity

νTo Clauser turbulent viscosity in the outer layer

Ω(y) The wake loss at a determined the non-dimensional height y at 40% chord down-stream

Ωij The transport of Reynolds stresses due to rotation

φ A general vector or scalar variable

Πij The transport of Reynolds stresses due to turbulent pressure-strain iterations

τ Shear stress

V Volume

εij The rate of dissipation of Reynolds stresses

C The Courant number

Cp The pressure coefficient

Cijk Turbulent transport tensor

Cij The transport of Reynolds Stresses by convection

D Diameter

Dij The transport of Reynolds Stresses by diffusion

FE Ensemble average function

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xvi LIST OF TABLES

FT Time average function

FV Volume average function

k Turbulence kinetic energy

lmix Mixing length

N number of separate experiments

P Mean static pressure

p Instantaneous static pressure

p′ Correction for pressure field in the solution of pressure-velocity coupling algorithms

p∗ Guessed pressure field in the solution of pressure-velocity coupling algorithms

Pij The rate of production of Reynolds Stresses

Ps The static pressure

Ps(x) The static pressure at a determined coordinate x on the blade

Pt The total pressure, the sum of static pressure and dynamic pressure

Rij The kinematic Reynolds stresses

Re The Reynolds number

Sφ The source term for the variable φ

sij Instantaneous strain-rate tensor

Sji Mean strain-rate tensor

T Characteristic time scale

t Time

tji Instantaneous viscous stress tensor

u′i, u′j Fluctuating velocity in tensor notation

U, V Mean velocity component in x and y directions

Ui, Uj Mean velocity in tensor notation

ui, uj Instantaneous velocity in tensor notation

vmix Mixing velocity

xi, xj Position vector in tensor notation

A Area

CV Control Volume

l A characteristic turbulence length

q A characteristic turbulence velocity

SC Control surface

Chapter 1

Introduction

1.1 Overview

The science of fluid mechanics has accompanied mankind since antiquity; the knowl-edge of fluid behavior and successful management of fluid forces led to the survival ofearly civilizations. Even if the concept of the aqueduct were developed independently indifferent parts of the planet and by different civilizations, the Roman aqueduct is prob-ably one of the best examples of the use of systematic engineering knowledge in fluidmechanics applied to the construction of large-scale civil works to sustain the expansionof the population in the roman cities.

Knowledge of fluid behavior and the correct estimation of the forces involved in certainphenomena also led to competitive advantages as pointed out majestically by Anderson[1]:the early aircraft developed in the beginning of the XX century employed very thin airfoilsin their wings, which resulted in poor high-lift performance due to separation at low anglesof attack, however, the research developed by Ludwig Prandtl on 1917 have shown thatthe use of a thicker airfoil would be advantageous and the concept was implemented byAntony Fokker in the Fokker Dr-1 and later on the Fokker D-VII which outperformed itsopponents during the First World War.

In competitive industries such as the aircraft, energy generation or race car industries,the speed of reaching a solution that will meet the needs of the consumer or give a betterperformance in an ever changing regulations environment could be vital to the survivalof a company or a team. The need of efficient ways to reach an improved solution is evenmore important in the current scenario of an economic recession.

One good example of the need to obtain a solution with improved performance in thefastest time and the smallest effort possible in the race car industry is Formula 1, where theInternational Automobile Federation (FIA) has imposed restrictive rules regarding windtunnel, track tests [2] and CFD simulations, to avoid the escalating expenses of the teams.In the specific case of CFD studies there is a limit on the number of Teraflops allowedfor each team [3]. In other industries it is not much different from racing cars; insteadof crossing the finish line first it is necessary to have the fastest time to market. In theaircraft industry, the enforcement of noise regulation in various parts of the world createdthe demand of an aircraft with reduced noise, which created a tremendous technical andscientific effort in the area.

One of the most important problems in practical aerodynamics is the determination ofthe position of the boundary layer separation. The presence of adverse pressure gradientswill result in a reduction of the velocities on the boundary layer up to a point where thevelocity gradient on the surface is zero. The knowledge of this position is important since

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2 CHAPTER 1. INTRODUCTION

the flow after the separation is dominantly rotational, causing changes on the pressuredistribution over the body and an increase on the drag, therefore, the point of the flowseparation is a very important consideration in any aerodynamic design.

This problem becomes particularly relevant in the design of Low Pressure Turbines(LPT), where the overall efficiency of a gas turbine is highly dependent of the efficiencyon the LPT stage. An increase in 1% of the polytropic efficiency of the LPT can lead to areduction in up to 0.5% in the fuel consumption of the turbine, therefore a lot of researchhas been directed in obtaining a more efficient design for the aerodynamic profile of thatstage and an increase of 13% in its efficiency has been obtained in the past 50 years [4]. Insuch a long timespan the methodology used in designing these components has changedconsiderably especially when considering the role played by the numerical simulations.

Having dependable techniques to estimate the fluid forces is a fundamental need in theaeronautical industry, and the evolution of the computing power associated with the newnumerical methods developed caused a shift of paradigm on aerodynamic calculations.The first step was the implementation of the potential methods for solving airfoils. Toestimate flow properties using these methods could be a quite difficult task since mostof the software had to be developed by the research group itself, as stated by Liebeck inhis 1978 paper [5] that the lack of a multi-element inverse method to solve the pressuredistribution over an airfoil had impaired the obtention of better results. Also the needof a considerably good computer even in geometries with a few hundred of panels canbe realized in the reference document of the XFoil Software [6], developed in 1986 whichrecommends the use of the software in a good workstation, otherwise the solution couldtake too long.

In 1989, Cebeci et al. [7] said that the limitations on the computer power demandedthat most of the development of an aircraft was made by wind tunnel testing with flowcalculation methods being responsible for a small contribution on simple geometries. Theavailable numerical approaches at that time were the use of coupling techniques betweeninviscid and viscous flows solving the boundary layer equations close to the solid surfacesand the solution of the Reynolds Averaged Navier Stokes Equations (RANS) in mesheswith 5×105 cells, which can be very stringent to the type and size of geometry that isgoing to be simulated. The author also states that the results obtained with the firstmethod were reasonably accurate at low angles of attack and only took a fraction of thethin-layer Navier-Stokes equations calculations.

Nowadays the use of some form of simulation using RANS models is almost an indus-trial standard. The evolution from sparse in-house codes to commercial packages with auser-friendly interface combining mesh-generation and post-processing allowed a massiveincrease in the number of users of this tool. In the past ten years the use of CFD byFormula 1 teams jumped from a couple of full models run a year to hundreds of jobs in aweek [3], and the evolution in the solutions obtained is not restricted to a faster turnover,the way in which turbulence is modelled is also constantly evolving.

According to the purpose of simulation that is being performed the use of an averagesolution will not capture the desired details of the flow field under study, whether be-cause the phenomenon under study is intrinsically unsteady or because knowledge of theinstantaneous flow features is needed. The ratio of how much turbulent kinetic energy ismodelled compared with how much is calculated by the solver is normally used to classifythe type of turbulence modelings. By modeling most of the turbulent spectrum RANSmodels can provide an averaged flow field, and be less computationally demanding. LargeEddy Simulation (LES) models compute most of the energy-containing scales of turbu-lence and model most of the dissipative scales, with increased computational cost. Direct

1.1. OVERVIEW 3

Numerical Simulation (DNS) simulates all of the turbulence spectrum, being the mostdemanding method in terms of computing power.

To simulate all turbulence spectrum, DNS simulations must have a mesh refined enoughto capture the turbulence micro scales. As an example on how demanding a DNS cal-culation can be when compared to RANS, the simulation of the flow over an automobilecan be used: in the RANS simulation of a 1/3 size model race car at 30 m/s Giovanettiet al. [8] have used a 12.5 million cells mesh, the most refined cells were in the boundarylayers with approximately 1 mm thick using wall functions. Wilcox [9] estimates the sizeof the turbulent micro scales near the driver’s windows in an automobile at 29 m/s asapproximately 4.6×10−3. Since in DNS the mesh must be sufficiently refined to calculatethe micro scales a simplified analysis would estimate a mesh 2193 ≈ 1.5× 106 times morerefined. These results are discouraging when considering DNS simulations as an indus-trial standard. By comparing the ratio between the large and small scales it is possibleto calculate the computational cost for a DNS method as proportional to Re9/4, whichrestricts its applications to specific problems at reduced Reynolds numbers.

Another comparison is given by Frohlich and von Terzi [10] that states that an LESsimulation is 10 to 100 times more costly than RANS computations, however Gadebusch[11], is optimistic by stating that the technological advances in supercomputing are allow-ing the LES simulations to become a useful engineering tool to predict turbulent flows.While becoming more accessible, LES simulations also suffer from the computational ef-fort scaling with the Reynolds number with a smaller constant than DNS [12], especiallywhen solving wall-bounded flows, since close to the wall the structures that carry mostof the energy become very small. Therefore it seems straightforward to think that somecomputation power could be saved by solving a RANS model in the near wall region and aLES model to solve the outer region of the flow. This and several other ways of using dif-ferent turbulence modeling to different parts of the flow are grouped as DNS/LES/RANShybrid models.

The accessibility of the LES simulation or the hybrid models is in fact getting closerto industrial application, commercial packages such as STAR-CCM+ already offer thepossibility of modeling the turbulence using either LES or a Detached Eddy Simulation(DES) which is a hybrid LES/RANS simulation that uses the RANS modeling for theshear layers and a subgrid scale model where the mesh is enough refined [13] .

The use of a commercial package however presents some limitations: the user isbounded by the implemented solvers and turbulence models. The viable alternative isthe use of Open Field Operation Manipulation CFD Toolbox (OpenFOAM), which is anopen source CFD package developed by ESI-OpenCFD, a company based in Bracknell,UK, established in 2004. OpenFOAM has several modules implemented that allow theuser to solve problems ranging from electromagnetics and solid dynamics to turbulentflows. By being an open source software it allows the possibility of customizing its mod-ules. The software also allows the user to select the level of turbulence modeling; it canperform RANS, LES, DNS and also the hybrid DES.

As shown previously, advances in computational power are gradually allowing the pos-sibility of performing an even more complex range of flow simulations in industrial applica-tions, yet the most refined modeling, such as LES and DNS, still present some limitationsof practical order, therefore, a tool for accurately predicting the flow separation with areduced computational cost is highly desirable. Designers can take advantage of the fastresponse of one of the novel RANS models that can accurately predict the turbulent flowbehavior and save precious CPU time to analyze new and untested geometries.


1.2 Motivation

The use of the RANS models to predict a complex phenomenon such as the flowseparation in aerodynamic surfaces demands a good experimental database or reliableDNS results as benchmark data. The University of Southampton has a long tradition ofresearch in race car aerodynamics. The University’s R. J. Mitchel wind tunnel was used bydifferent Formula 1 and IndyCar teams in the past and many of the University’s formerstudents are currently working in Formula 1 teams. Within this partnership betweenthe University and Formula 1 teams, the original subject of the present thesis was theprediction of separation of the boundary layer on a race car related geometry, however,due to the difficulty of obtaining a suitable geometry with experimental data to developthe studies, a change of path was made and the study of a LPT profile was undertaken.

The study of LPT is also a very active topic of research at the University of Southamp-ton, with several recent studies being developed, especially using DNS to solve the flow,which gives a richness of data, such as meshes and numerical results, that can be used todevelop the current study and to compare the results.

Gas turbines are used in a wide range of applications, in the propulsion of differenttypes of aerial vehicles from big airliners to cruise missiles and in the generation of en-ergy. They are attractive in some applications due to their higher power-to-weight ratio,higher efficiency and smaller size when compared to reciprocating engines of the samepower. These benefits come with an additional cost, gas turbines also operate in a highertemperature and velocity environment which demands a far more complex design andmanufacture.

Turbines are the last stage in a gas turbine; their main function is to transform theenergy from the high pressure gas that comes from the combustion chamber in shaft workoutput. In order to extract most of the energy from the flow, turbines are designed withmulti-stages, and through the stages the flow pressure reduces hence the later stages of agas turbine are nominated the low pressure turbine. The overall efficiency of the turbineis strongly dependent on the LPT efficiency, in order to reduce the fuel consumption injet gas turbines two possible methods can be used: the first is increasing the efficiency ofthe turbine blades; the second way is to reduce the number of blades, with the associatedreduction in the mass of the engine and consequent reduction in consumption. This lastapproach leads to higher loading and pressure gradients at which the individual bladeswill be subjected, which demands the use of more resistant materials.

Since the present thesis is the concluding work of a MSc. in Race Car Aerodynamics,a comparison between similar characteristics of the flow in a LPT and a race car seemsto be appropriate. In aircraft LPT, according to Stieger [4], the Reynolds numbers basedon the blade chord can range from approximately 4 × 104 to 5 × 105 depending on thesize of the engine and the altitude; when analyzing the flow on the rear wing of a racecar the Reynolds numbers can range from zero to 1 ×106 depending on the size of thewing chord and velocity of the car. By being the last stage in a gas turbine, the flow thatreaches the turbine blades is highly disturbed due to interactions with previous stages;similarly the flow that reaches some components of a race car such as the rear wing arehighly disturbed and rotational by the influence of other aerodynamic structures of thecar or by the effect of a car in the front. The aerodynamic profiles in LPT are normallyhighly cambered in order to extract the most of the energy from the flow, the same is trueat the rear wing of race cars, by being the last component of the car, a highly camberedwing is used in order to obtain a great amount of downforce before the flow leaves thecar, sometimes even Gurney flaps are attached to the trailing edge to extract an extra

1.3. OBJECTIVE 5

bit of aerodynamic downforce from the flow. In order to use such high cambered profiles,gas turbines take advantage of the interaction between the successive profiles in a cascadeof blades existent in the blade ring of the LPT, a similar effect is obtained by the useof multi element wings in race cars that allow the flow to pass through a much highercambered wing.

As in the race car industry, the turbo-machinery industry can also profit from accurateRANS models predictions at the conceptual design level, according to Weinmann, M. [14],despite the constant evolution in computing power, the use of the RANS approach willcontinue to have a central role on flow simulations for industrial engineering applications inthe near future. Wissink [15] points out that DNS simulations of the low Reynolds Numberflow at the LPT, despite being feasible are still too costly for engineering applications,therefore, DNS is being used to get a better insight on the physics of the flow and toprovide reference data for the development of turbulence models.

The attempts to develop mathematical models to describe the turbulent stresses startedwith the work of Reynolds more than one century ago by the process of time averagingthe Navier-Stokes equations. From that starting point, two main approaches have beendeveloped, depending whether or not they use the Boussinesq hypothesis, it assumes thatthe Reynolds Stresses are analogous to the shear stresses and therefore calculated basedon the velocity gradients, which simplifies the calculations and the physical phenomena.On the other hand, by obviating the Boussinesq hypothesis, the computations suffer fromincreased cost by modeling the transport of each of the Reynolds stresses.

The Bousinessq hypothesis has been used successfully in several engineering applica-tions, however, due to its linear relationship between the Reynolds Stresses and the meanstrain-rate tensor, it is inaccurate in describing the actual response of turbulence to com-plex mean-flow perturbations and to anisotropy in the Reynolds stresses such as: flowsover curved surfaces, flows in ducts, flows in rotating fluids, secondary flows, tridimen-sional flow, separations and obstacles. In a real flow, the Reynolds stresses adjust to thesechanges in an unrelated form to mean flow processes.

1.3 Objective

The current research will focus on one of these flows that are allegedly poorly pre-dicted by linear turbulence models: the flow in a highly cambered T106 LPT cascadeprofile and compare their performance with two modern non-linear turbulence models:a baseline Explicit Algebraic Stress Model (EASM) and the ϕ-α-EASM. IncompressibleRANS simulations will be performed in OpenFOAM, since these modern RANS mod-els are not available in commercial packages and were programmed into OpenFOAM inprevious research projects developed at the University of Southampton [14]. Simulationswill be compared with experimental and numerical results from compressible and incom-pressible DNS. The performance will also be assessed by testing the models in coarsermeshes.

To reach the proposed objectives several steps had to be completed:

• learning to work in the OpenFOAM framework: the author had no previous experi-ence with the software therefore some time had to be spent in the learning process;

• obtain a suitable mesh to execute the simulations: the development of meshes fornon conventional geometries is rather difficult in OpenFOAM, hence the mesh usedin a DNS study at the University of Southampton was employed [16];


• develop a program to coarsen the mesh: to test the quality of the numerical predic-tions in coarser meshes;

• run a series of preliminary simulations: to get confident with the results generatedby OpenFOAM;

• get the state-of-the-art turbulence models working with the available mesh: themodels were developed for an early version and recently upgraded, therefore theirfunctionality should be tested;

• run the proper simulations: to accomplish the ultimate objective of the thesis;

1.4 Organization of the Thesis

The present work is organized in 5 chapters. Turbulence modeling is a vast subjecthence in Chapter 1 the idea was to present a brief introduction on the subject and themain objectives of the work in order to situate the project in the current practices on thefield. Chapter 2 presents a brief revision of the concepts and theories that were used todevelop the project. The idea was to write a concise chapter that could touch on majorissues, therefore, only the most important equations in their derived form are presented.Chapter 3 presents the specific geometry that was studied and the mesh used to calculatethe numeric solutions, the chapter also discusses the finite volumes approach and how theproblem was set in OpenFOAM environment. Chapter 4 presents the results obtained, inthe first part of the chapter it is shown the preliminary results generated in order to learnhow to use the CFD solver and in the second part the results targeted on the objectivesare presented. Chapter 5 is the conclusion of the work and suggestions to further workare presented.

Chapter 2

Theoretical Background

Since the objective of the present work is to determine the point of separation onaerodynamic profiles, a few considerations about viscous flows were made by defining itand explaining the mechanism of separation, also a short incursion into the vast fieldof the fundamentals of turbulent flows is presented along with the basics of turbulencemodelling. For a more comprehensive discussion on these subjects, the reader is suggestedto consult the following references: Anderson [1], Wilcox [9], Houghton and Carpenter[17], Schlichting [18], Pope [19], Davidson [20], and Mathieu and Scoot [21].

2.1 Viscous Flow

Viscosity is an inherent property of any real fluid. When a fluid is submitted to ashear stress it will sustain continuous deformation and viscosity is the property thatrelates the deformation of the fluid with the amount of shear stress exerted. It is observedby experiment that the intermolecular interaction between solid surfaces and the fluidensure that the velocity of the flow at the body surface is zero, this is called the non-slipcondition, hence if a stream of fluid is passing by a solid surface, it is reasonable to admitthat the velocity close to the wall will vary from zero to the stream velocity. This is theeffect of the viscosity: at the solid boundary the flow velocity is zero, this layer of fluidwill act on the following layer of fluid by generating a frictional force that will reduce itsvelocity and each adjacent layer of fluid will have a decelerating effect on the followinglayer up to a point where the effect of viscosity is so small that the flow velocity reaches thestream velocity. By the action of Newton’s Third Law, the effect of viscosity on the fluidis its deceleration by the formation of a distribution of velocities from zero to the velocityof the stream, and the effect on the solid boundary is the appearance of a tangentialforce in the direction of the flow. For newtonian fluids, the viscosity is the constant ofproportionality between the shear stress and the gradient of velocity as follows:

τ = µdu

dy(2.1)

The variation of fluid velocity from zero to the free stream velocity in a region close toa solid surface is called boundary layer and its discovery was one of the most significantbreakthroughs in the science of the fluid mechanics. For a long time, the connectionbetween the empirical science of hydraulics and the theoretical fluid mechanics was lackinga theory that could unify both fields of knowledge. Several practical problems could besolved without considering the viscous effects on the fluid, since the velocity gradients

7

8 CHAPTER 2. THEORETICAL BACKGROUND

are negligibly small throughout most of the fluid, however, some problems could not besolved properly because the velocity gradients are considerable in the area immediatelyadjacent to the solid boundary and consequently generates high shear stress. Prandtldeveloped the concept in 1904, stating that a variation on the flow velocity from zero tothe free stream velocity is expected in a very thin region close to any solid boundary andthat difference on velocity could lead to a high shearing stress. The drag force1 felt inbodies immersed in a flow is mainly from the shearing stresses on the surface of the body,therefore the knowledge of the behavior of the boundary layer and consequent estimationof these stresses is fundamental to the accurate prediction of drag. Boundary layers canstart on a sharp edge, e.g., the leading edge of a flat plate or at the stagnation point ofthe leading edge of a bluff body and it will grow from zero to a finite thickness, as thefluid flows downstream and the shear stress causes a deceleration of the layers of fluidadjacent to the wall, the size of the area affected by the shear stress will increase, thereforeboundary layers growth in the stream wise direction.

When passing close to a solid boundary, the flow can be subjected to pressure gradientsthat will affect the flow within the boundary layer. Considering the situation where thepressure decreases in the direction of the flow, it can be expected that the pressure forceswill act against the viscous forces. An element inside the boundary layer would have ahigher velocity when compared with a case with no pressure gradient, this is said to bea favorable pressure gradient. The flow is not decelerated as intensely close to the solidsurface, a fuller velocity profile is developed and the boundary layer grows more slowly.In the case of the pressure increasing in the direction of the flow, the pressure force willadd to the effect of the viscous forces and a lower velocity when compared with the caseof zero pressure gradient is expected, this is said to be an adverse pressure gradient andthe boundary layer will grow faster. The deceleration of the fluid particles can become sointense that the velocity can reach zero or even become negative. This effect is illustratedat Figure 2.1a, it can be seen that while moving downstream the velocity profile closeto the surface becomes less inclined up to a point where the inclination is zero, furtherincrease of the pressure causes a reversed flow. The consequence of this reversed flow isthe separation of the boundary layer from the solid surface and the formation of a wakeof recirculating flow downstream that can be seen in Figure 2.1b. The point where theinclination of the velocity profile is zero is defined as the point of separation. In thatpoint the shear stress is zero. The separation of the boundary layer causes an alterationon the flow field and the pressure distribution over the body, in the aerodynamic jargon itis said that the flow does not “see” the body shape as it is but “sees” an altered effectivebody, thicker than the original body due to separation. As it could be expected a changein the pressure distribution will cause an increase of the aerodynamic drag and it is calledpressure drag.

Changes on the pressure distribution over the body due to separation can lead toundesirable results on aerodynamic profiles. A sudden increase of pressure on an airfoil cancompromise most of the lift generated reducing its efficiency or even rendering it useless,hence, the careful management of the airfoil geometry in order to obtain a suitable pressuredistribution is the key to design efficient aerodynamic devices, as mentioned in Section1.2, the efficiency of the LPT stage is fundamental to the design of more economical gasturbines, so the efficient design of the LPT profile is a crucial task in gas turbine design.

As the flow interacts with a solid wall, two different flow regimes are possible onboundary layers, the laminar and turbulent. In the laminar regime, the flow is the smoothmovement between laminae (layers) that are decelerated by the viscous action of the fluid

1The drag force caused by the viscous stresses is also called skin friction.

2.1. VISCOUS FLOW 9

(a) Vector Field

(b) Line integral convolution

Figure 2.1: Flow close to a solid surface under the influence of an adverse pressure gradient,the deceleration of the flow leads to a reversed flow and a consequent separation

between the layers resulting in a well behaved velocity profile, in the turbulent boundarylayers there are fluctuations of velocity in the direction of the flow and perpendicular toit, the perpendicular fluctuations transport mass between adjacent layers and makes thevelocity profile change in time. Since the velocity profile varies in time, a time averagedvelocity profile can be obtained, the movement between the layers of fluid bringing highmomentum flow from higher layers of the boundary layer to lower layers makes the averagevelocity profile fuller than the laminar profile. Figure 2.2 shows the laminar profile andthe turbulent averaged profile obtained from numerical simulations, in the figure, the scaleof the vectors is different in each case, nevertheless it is possible to see the difference onthe shape of the profiles.

In general, when the flow starts its interaction with a solid boundary it is laminar, asit proceeds further downstream, internal instabilities starts to be formed and amplified,this process continues up to a point in which the flow can no longer sustain its smoothand laminar movement, and it suffers a transition to a turbulent flow. According toEquation 2.1, the fuller turbulent profile indicates that the shear stress on the wall willbe higher than the laminar case, therefore, an increase on the drag of a body immersedon the flow can be expected. Nevertheless, turbulent profiles are more energetic and lessprone to separation than laminar profiles, consequently the determination of the regimein which an aerodynamic profile will work is another important design decision, since atrade-off can be obtained by working in the turbulent regime with a higher skin frictionand without any flow separation instead of in a laminar regime with lower skin frictionwith a greater tendency to separation.


(a) Laminar boundary layer (b) Turbulent boundary Layer

Figure 2.2: Results obtained numerically for the velocity distribution on the boundarylayers.

2.1.1 The Navier Stokes Equations

The equations that describe the behavior of nearly all fluids are called the Navier-Stokes equations due to the work of the French engineer Claude-Louis Navier and theIrish mathematician George Gabriel Stokes. They are a set of nonlinear partial differen-tial equations that represent three fundamental physical principles: conservation of mass,Newton’s second law and the conservation of energy. According to Davidson [20], theNavier Stokes equations are “deceptively simple”, as they don’t look more complex thana wave equation or a diffusion equation, which in the author words, “leads to simple solu-tions”. For him, The Navier-Stokes equation “embodies such rich and complex phenomenaas instabilities and turbulence” as a consequence of a “seemingly innocent non-linearity”of the dependent variable in quadratic form.

For the incompressible flow of a Newtonian and continuum fluid, the equations of theconservation of mass and momentum are:

∂ui∂xi

= 0 (2.2)

ρ∂ui∂t

+ ρuj∂ui∂xj

= − ∂p

∂xi+∂tji∂xj

(2.3)

the equations are presented using the subscript notation, with the subscripts takingthe values of the x, y and z components and with the summation convention employed,where a single term containing one or more repeated subscripts represents an implied sumover all three values of each repeated subscript. The viscous stress tensor, tij, and theinstantaneous stress tensor sij are defined as:

tij = 2µsij (2.4)

sij =1

2

(∂ui∂xj

+∂ui∂xj

)(2.5)

2.2. TURBULENCE FUNDAMENTALS 11

2.2 Turbulence Fundamentals

A precise definition for turbulence is often tried by different authors, but often a concisedefinition cannot encompass all its complexity. Some of the greatest minds from Leonardoda Vinci to Richard Feynmann have been puzzled by its characteristics and some evenused poetry to try to express their comprehension over the subject, like the famous verseof the British mathematician Lewis Richardson2.Despite being a phenomenon that is partof people daily lives, it has been a challenge and authors rather try to introduce turbulenceby its general properties.

As it was mentioned in section 2.1, turbulence arises from instabilities of laminar flow.The earliest contributions on the subject are owed to the pioneering work of OsborneReynolds over the behavior of the flow in pipes and how it could change based on pertur-bations on the inlet and a parameter, later named as the Reynolds number, as follows:

Re =ρ UD

µ(2.6)

The Reynolds number gives a measure of the ratio between the inertial forces and theviscous forces in the flow, an increase in its value, corresponds to an increase in the relativeimportance of the non-linear convective term on the right-hand side over the viscous termon the left-hand side of Equation 2.3. The viscous term tends to damp the instabilitieswithin the flow and with not enough damping an increase of the flow internal instabilitieswill occur.

Considering the flow on a pipe, in the laminar regime the velocity profile is parallel tothe axis of the pipe and has a parabolic distribution, which is a possible solution for theNavier-Stokes equation. It was observed that the viscosity tends to damp the perturba-tions on the flow up to a certain Reynolds number, when it is increased, the tendency toinstability increases and the flow starts to suffer a transition by having sporadic burstsof turbulence. As the Reynolds number is further increased a fully turbulent regime isattained, with the parallel velocity profile being substituted by rotational flow structures.It is important to notice that the instabilities by themselves are not an indication of aturbulent regime, one of the best examples of instabilities developing in a laminar flow isthe formation of the Karman vortex street on a cylinder, the vortex wake is formed by anunsteady separation of the boundary layers on the top and bottom sides of the cylinder,as the flow Reynolds number is increased the transition to turbulent flow begins to formon the far wake of the vortex street. Despite the fact that there are some conditions fora turbulent flow to develop and that viscosity also contributes to damp instabilities onthe flow that can even lead to a relaminarization, the laminar flow is more an exceptionthan a rule, as it said by Wilcox [9]: “Virtually all flow of practical engineering interestare turbulent”, mentioning several applications in which turbulent flow is present fromthe flow past vehicles to the mixing of the cream in a cup of coffee. Still quoting theauthor, “Turbulent flows occurs when the Reynolds number is large”, and “large” most ofthe times “correspont to anything stronger than a tiny swirl, a small breeze or a puff ofwind.” The equations 2.2 and 2.3 also describes the turbulent flow. In order to obtain asolution, these equations must be supplied by the appropriate initial and boundary condi-tions, which for a particular set of these conditions is unique, nevertheless when executingexperiments one might never be able to reproduce the same flow due to the sensitivity to

2“Big whorls have little whorls, which feed on their velocity,little whorls have lesser whorls, and so on to viscosity.”


changes in the initial and boundary conditions that cannot be controlled experimentallywith infinite precision, therefore, the theoretical study of turbulence is based on the flowstatistics, which are assumed to be reproducible by sampling a large number of differ-ent realizations. The main turbulence properties pointed out by the above mentionedreferences are:

• Turbulence is a random process: when observing the graph of the measured in-stantaneous velocity obtained by a probe in a turbulent flow, as in Figure 2.3, onecan observe random fluctuations as a function of time. Turbulent flow is time andspace dependent and highly sensitive to initial conditions which makes its instanta-neous properties very difficult to predict, therefore, a statistical approach is used todescribe the flow since the averaged flow properties are reproducible.

• Turbulence has a wide range of different scales: looking again at Figure 2.3, it is alsopossible to see that there are large oscillations on the measured value of velocity andwithin these oscillations smaller oscillations exist, this reflects the movement of largestructures passing by the probe and while they are passing smaller structures thatare living inside the large ones cause smaller fluctuation in the measured velocity.The large scales of the flow are typically defined by the geometry, for example, in thea jet flow they are of the order of magnitude of the width of the jet. It is possible todetermine the smaller scales in turbulence by magnifying the time interval of Figure2.3, eventually it is possible to reach a timescale in which the oscillations in velocityare smooth, this is due to the action of viscosity, therefore, the smallest scales on theflow depend on the viscosity. Flow instabilities continuously generates turbulenceat high Reynolds number, producing large scale eddies which are also unstable andform smaller ones, that also form yet smaller eddies in a continuous energy cascadeup to a point where viscosity becomes important. There is a continuum of spatialscales generated by this energy cascade process, and the spectra get wider as theReynolds number increases since the dissipative smaller scales become smaller atlarger Reynolds numbers.

• Turbulence dissipates energy: as a result of the cascade process viscous flows rapidlydissipate energy as the viscous stresses tend to have it major contribution at thesmaller scales. Turbulent flows require a continuous supply of energy, which is givenby the large scales on the flow.

• Turbulence is a continuum phenomenon: the smallest scales in a turbulent flow aremany orders of magnitude larger than the molecular free path, therefore, turbulentflows can be described within the same continuous approximation used for derivingthe Navier-Stokes equations.

• Turbulence is intrinsically tridimensional: turbulence has a rotational nature, vortexlines form inside of a turbulent flow and they tend to evolve by the action of the strainrate produced by the velocity gradients in a phenomenon called “vortex stretching”.This is one of the fundamental processes in a turbulent flow and it does not occurin two-dimensional flows.

• Turbulence mixing and diffusivity: by the existence of large structures moving in aturbulent flow, large masses of fluid migrate across the flow. These large structurescarry small disturbances within them, this movement greatly increases the mixingand diffusion in a turbulent flow. Another observable phenomenon is the interaction

2.3. TURBULENCE MODELING 13

Figure 2.3: Measurement of the instantaneous velocity with a probe in a turbulent flow[26]

with neighboring regions of laminar flow, in which fluid from the surroundings isbrought into the turbulent region and as an effect a spreading of the turbulent flowoccurs in the flow direction, as is seen in a wall boundary layer.

2.3 Turbulence Modeling

According to Pope [19], the objective of the study of turbulent flows is to obtain aquantitative theory or model that can be used to obtain results to calculate flows ofpractical interest. If, in one hand, experience shows that there are no simple analytictheories that could be applied to solve turbulent flows, on the other hand, the evolution incomputing power of digital computers allows to solve flows with increasing complexity anddetail. Nevertheless turbulent flows present several challenges that must be addressed, forinstance its tridimensionality, time dependency and randomness are characteristics thatmake it difficult to develop an accurate model, moreover the large scales are intrinsicallydependent of the geometry of the flow which makes each different problem unique.

It is a fact that even before the extensive use of computers to solve turbulent flows, thenecessity to predict the behavior of such flows demanded the development of analyticaland experimental methods. One good example of analytical methods was the integralmomentum equation, derived by Von Karman, that allowed practical solutions for someengineering problems, such as the determination of momentum thickness in a turbulentboundary layer over a body. Other analytical methods using simplified equations, experi-mental results and actual performance data from prototypes were also applied in industryin the past and are still in use today. A compendium that collects several methods foraircraft design is Roskam Airplane Design Collection [22]. As an alternative, performingexperiments in different geometries at different flow regimes produced a large numberof charts, tables and practical handbooks such as the classic Hoerner’s Fluid-DynamicDrag and Fluid Dynamic Lift [23], [24], that were used in early stages of development ofaerospace products [25].

There is no doubt that the use of these methods allowed the execution of great engi-neering feats; however, the simplicity of setting up a numerical simulation in graphicalinterfaces, its increasingly accuracy due to the implementation of improved numericalmethods and turbulence models, and the easier access to solvers, makes the use of numer-ical modeling every day more attractive. A short review of the main issues in numericalmodeling of turbulence will be presented in the following subsections.


2.3.1 Turbulence Closure Strategies

Using the numerical simulation approach three different methods can be used to solveturbulence:

• Reynolds-Averaged Navier-Stokes (RANS) models: The Navier-Stokes equa-tions are time averaged in order to obtain a mean velocity field. In the process ofaveraging the extra terms that appear are modeled. The current work was devel-oped using RANS models therefore a more complete explanation will be performedin subsequent sections.

• Large Eddy Simulation (LES): The idea behind LES is to solve the large scalesof turbulence while scales below a certain size are modeled. It is important to noticethat all turbulent scales are dynamically significant on turbulent flows, hence thesmaller scales must be accounted for in the model, this is made by subgrid terms thatare added to the equations of motion. This approximate approach is used in order toreduce computational cost and allows the use of a coarser mesh. The Navier-Stokesequations are solved to determine an instantaneous realization of the flow instead ofan average field. Since some of the smaller scales are not being calculated, LES canonly produce statistical results independent of these scales, such as mean velocityfield and second order velocity moments. According to Matieu and Scott [21], “Theart of LES lies in the appropriate choice of subgrid terms, matched to the particularflow and numerical scheme used”, because in a real flow the turbulent energy cascadeis responsible for transferring the energy from the large scales to viscous dissipationin a LES the size of the grid makes the energy cascade incomplete (it works upto the scales described by the grid resolution) hence the choice of the numericalscheme can include an artificial numerical dissipation and the subgrid turbulencemodel must be selected in order to represent the correct energy transfer otherwisethe flow can become under-dissipative or over-dissipative.

• Direct Numeric Simulation: In this approach the unsteady and tridimensionalNavier-Stokes equations are also solved for an instantaneous realization of the flowand for all turbulent scales. Since none of the scales will be modeled, the meshmust be sufficiently refined to capture the smallest spatial scales, the time marchingmust have time steps short enough to capture the period of the fastest fluctuations,therefore, the method is costly and as it was mentioned earlier the cost increaseswith the increase of Reynolds number. Due to the fact that no approximations arebeing employed, except the ones related to discretization, the solution of the flowfields yields detailed information of instantaneous and statistical properties, whichrises a concern related to storage and treatment of obtained data. In the early stagesof development of DNS techniques, the researchers were concerned in showing thatit was possible to simulate a flow accurately, nowadays the level of confidence hasincreased to a level that it can be also called a “Virtual Wind Tunnel” and it allowsthe execution of numerical experiments that are sometimes impossible to performin a laboratory such as choosing geometries or boundary conditions that cannotbe realized in practice or artificially modifying the governing equations. Alongsidewith experiments, results from DNS can also provide benchmarks against whichother simulation methods can be evaluated and parametrized.

As described in the previous section, turbulent flows exist in a huge variety of appli-cations from the flow on a cup of tea to the atmospheric flows, for such vast applications


a large number of different models have been proposed, Pope [19] presents some criteriathat can be taken into account when evaluating different models:

• Level of description: the level of description can range from the mean flow prop-erties to instantaneous characteristics of the flow, the use of a higher level of de-scription leads to a deeper characterization of turbulence and a wider applicability,its use depends on results needed for a determined application since it is more com-putational costly, for most of the industrial flows low levels of description such asmean-flow closures are sufficient.

• Completeness: the completeness of a model refers to its constituent equations. Ina complete model only fluid properties and boundary conditions have to be specified,they are more costly and has wider applicability, whereas incomplete models needthe specification of other properties normally related to a specific type of flow.

• Cost and ease of use: the general trend observed when considering the evolutionof the computing power over the last decades is that the speed (number of flops) hasincreased by a factor of 30 per decade, which gives and increase of approximately106 in the last forty years. This means that in a short period of time simulationsthat were regarded as research material are currently accessible to a daily CFD user.The cost of computing a turbulent flow can be linked to several different causes: itcan vary as a result of the increase of the complexity of the flow under study or theconsideration of a more complete physical description of the phenomena; it can alsobe affected by type of closure strategy selected, some models are highly sensitiveto the increase of Reynolds number, whereas in some models, the increase of costis insignificant of non existent; it is also important to take into account the timeemployed in obtaining or developing the software to solve a particular flow and thedifficulties in operating such software.

• Range of applicability: Pope summarizes the applicability by stating “A model isapplicable to a flow if the model equations are well posed and can be solved”, there isno point trying to obtain a shock-wave with an incompressible model nor trying tosolve a high Reynolds number flow using DNS, in the first case, the model equationsare not well posed and in the second case they cannot be solved (at least in a timelymanner).

• Accuracy: the accuracy of a certain model can be assessed by comparing its resultswith experimental measurements. The discrepancies between the results can beoriginated from different sources:

– Inaccuracies of the model: discrepancies can arise when the modeling equa-tions do not correspond to the complete phenomenon under study, for example,by using a turbulence model not suitable for a particular application.

– Numerical error: errors can arise from not using a refined enough time orspace steps or by performing calculations with insufficient numerical accuracy.

– Measurement error: results obtained from experiments have their own er-rors.

– Discrepancies in the boundary conditions: this type of error arises fromdifferences between the simulated and actual boundary conditions of a prob-lem, in some cases it is impossible to reproduce the same boundary conditions


of a determined experiment, sometimes the boundaries are approximated orunknown.

When performing an evaluation of the accuracy of the model it is important tomaintain the last three sources of error in a minimum level, so they do not interferein the conclusions of the advantages and shortcomings of the implemented model.Sandham [27] also lists two other sources of errors: iteration errors, which is notallowing the calculations to run far enough to reach a steady state and code errorswhich are an incorrect implementation of the numerical method for the equations,he also recommends that the results from CFD calculations should be verified tosee if the equations are being solved correctly by comparing the results with knownanalytic solutions, and he adds that recently the codes can also be validated bycomparing their results against direct numerical simulations databases.

2.3.2 The Turbulence Closure Problem

As it was mentioned in the previous section, turbulence is a random process with itsaverage properties reproducible; hence, a statistical approach seems to be appropriate.The averaging concept was introduced by Reynolds in 1895, and it consists in perform-ing the averaging of the terms of the Navier-Stokes equations by decomposing the flowquantities into the sum of a mean and a fluctuating part and substituting them into theequations. The process of averaging the equations have the advantage of avoiding theneed of resolving all scales of turbulence, unfortunately during the process of averagingnew terms arise and they have to be modelled in order to solve the averaged equationsnumerically.

In turbulent flows, different types of averages of turbulent quantities can be defined,the most common forms are time averaging, spatial averaging and ensemble averaging.

Time averaging FT (x) of an instantaneous flow variable f(x, t) is well suited to sta-tionary turbulence, since the majority of flows of engineering interest are stationary, thisis the most used form of Reynolds averaging. It is given by:

FT (x) = limT→∞

1

T

∫ t+T

t

f(x, t) dt (2.7)

in practice T cannot be infinity, therefore, it is taken to be a period of time long enoughto capture the largest scales of turbulence that are associated with the slowest variationsof a determined flow variable. The time averaging can also be used in cases of unsteadyflows, as long as there is a separation between the period of the unsteadiness of the flowand the time scale of the turbulence fluctuations, which is known as spectral gap. Inthis case, T must be larger than the turbulence scales, but smaller than the period of theunsteadiness.

Spatial averaging is recommended to be used in cases of homogeneous turbulence whichmeans that the statistics of the turbulence are independent of direction. A volume integralis taken over a volume V in a region where turbulence is uniform in all directions:

FV (x) = limV→∞

1

V

∫∫∫V

f(x, t) dV (2.8)


Ensemble averaging, can be used for flows that can be repeated numerous times asdifferent individual experiments and is defined as:

FE(x, t) = limN→∞

1

N

N∑n=1

fn(x, t) dt (2.9)

where N is the number of separate experiments using the same setup. This type ofdefinition is very robust since it can be applied to almost every type of turbulence problem,however it poses a difficulty in obtaining a statistical convergence since a high number ofrealizations is necessary. Other types of averages can also be defined depending on theparticular case under study.

The averaged Navier-Stokes Equations

Considering the time averaging of a stationary turbulent flow and performing theReynolds decomposition of the velocity it is possible to write:

ui(x, t) = Ui(x) + u′i(x, t) (2.10)

Each of the variables of the Navier-Stokes equations is substituted by variables decom-posed as shown in Equation 2.10, and the whole equation is averaged. It is importantto recall some properties of averaging such as Ui(x) = Ui(x), u′i(x) = 0 and u′iu

′j 6= 0.

For the full derivation of the Reynolds-Averaged Navier-Stokes Equations the reader isrecommended to refer to Versteeg and Malalasekera [28], the equations are presented inthe derived form below:

∂Ui∂xi

= 0 (2.11)

ρ∂Ui∂t

+ ρUj∂Ui∂xj

= −∂P∂xi

+∂

∂xj(2µSji − ρu′ju′i) (2.12)

by comparing them with the Equations 2.2 and 2.3, it can be seen that the term −ρu′ju′iarises. This term is known as the Reynolds-stress tensor, it is a symmetric tensor, hence,in three dimensions it represents six new unknowns to the set of equations and since nonew equations have been derived, the system of equations is not closed (more unknownsthan equations) and this is known as the turbulence closure problem.

It is necessary to develop models that predict the behavior of the Reynolds stresses inorder to be able to compute flows using the RANS approach.

Employing a pure statistical analysis of the component quantities of the Reynolds stresstensor, they can be described as the variance of the velocity fluctuations and they can

give important information on the structure of the flow. The quantities u′i2 are always

non-zero because they contain squared velocity fluctuations, and the quantities u′ju′i are

normally non-zero and they indicate a correlation between the velocity fluctuations indifferent directions, which is expected for the vortical flow structures that compose aturbulent flow. These quantities represent momentum fluxes that are closely linked withthe additional shear stresses present in a turbulent flow.

Their effect can be understood by imagining a moving control volume within a turbulentboundary layer. In a flow moving in the x direction forming a boundary layer, there is amean velocity distribution in the direction normal to the wall (y) which is responsible for


a mean shear stress, the eddying motion through the boundaries of the control volumecontinuously let in parcels of fluid with a higher or lower x-momentum, the interaction ofthese parcels with different momentum with the fluid inside the control volume generatesan additional turbulent stress within the control volume known as Reynolds stresses.

2.3.3 Reynolds-Averaged Navier-Stokes Turbulence Models

The earliest attempts to model the turbulent stresses started in the 1920’s by the workof Ludwig Prandtl when he introduced the mixing length concept which is based on theBoussinesq hypothesis. Most of the work developed in the area in the subsequent yearswas led by Prantdl, von Karman and Kolmogorov and was also based on the Boussinesqhypothesis, the most important developments where the development of models that tookin consideration the kinetic energy of turbulent fluctuations and later on a model that alsoconsidered the dissipation of energy. It is important to mention that the development ofthe models was hampered by the limitations imposed by the unavailability of computersin that time. In the late 1940’s, Chou and Rotta started to work with a different approachby proposing models that did not used the Boussineq approximation. This approach ismore accurate in the physical description of the phenomenon, however, it is more complexin terms of modeling and it leads to the modeling of all the components of the Reynoldsstress tensor.

The most common didactical division to classify the turbulence models is whether theyare based or not on the Boussinesq hypothesis.

Turbulent Viscosity Models

Early experiments on turbulent theory were developed on thin shear layers like jets,mixing layers and wakes where a causal relation between the existence of shear stressesand the development of turbulence was postulated, it is known that the turbulence alsoincreases the viscous dissipation in a flow, hence an analogy was proposed by Boussinesqthat the turbulent stresses could be proportional to the mean rates of deformation justas the viscous stresses in the Navier-Stokes equations are modeled by a viscosity timesthe rate of deformation of the fluid element, as shown on Equations 2.4 and 2.5, theBoussinesq hypothesis can be written as:

τij = −ρu′iu′j = µt

(∂Ui∂xj

+∂Uj∂xi

)− 2

3ρkδij (2.13)

µt is the eddy viscosity, it is also possible to write νt = µt/ρ, being the kinematic eddyviscosity, and k is the turbulence kinetic energy defined as:

k =1

2

(u′iu′i

)(2.14)

Equation 2.13 establishes a linear relationship between the Reynolds stresses and meanstrain rate of the flow, hence it is common to describe the turbulence models based onthe Boussinesq hypothesis as linear eddy viscosity models.

By inspecting the dimensions of the kinematic turbulent viscosity (L2T−1), it is possibleto say that when prescribing a specific turbulent viscosity model, it has been implicitlyprescribed a characteristic turbulent length (l) and a characteristic turbulent velocity (q)in a way that νt = Cql, with C being a non-dimensional coefficient. Different turbulentmodels have different ways of prescribing q and l and defining C as a constant or a fieldvariable.


Within the turbulent viscosity models, another didactical division is made by meansof the number of equations used to calculate νt:

• Algebraic or zero-equation models: these are the simplest turbulent modelsavailable; the turbulent velocity and length scales are calculated through algebraicrelations. These models are also classified as incomplete models since a length scalehas to be provided a priori, called the mixing length. For a series of flows theselength scales are tabulated as a function of a meaningful length of the flow. Thevelocity scale must also be prescribed. The fundamentals of the algebraic models arebased on the mixing-length concept developed by Prandtl in 1925, however, somenew developments to correct some of the shortcomings of the models were made byVan Driest in 1956, Cebeci and Smith in 1974 and Baldwin Lomax in 1978.

Despite its simplicity, these models are well established, extensively validated andthe mixing length concept proved to bear accurate results in simple two-dimensionalflows such as thin shear layers (jets, mixing layers, wakes and boundary layers) withslow changes of direction, this is mostly because in such flows, there is a balancein the production and dissipation and the turbulence properties are proportional tothe mixing length, which is described by algebraic formulae.

These models are cheap in terms of computing resources and easy to implement, inan already existent laminar code, they need no more than a few extra lines to taketurbulent viscosities into account, these features make them attractive to combinewith more sophisticated turbulence models to describe wall behavior.

• One-equation models: An additional partial differential equation has to be solved.In order to improve the prediction of the turbulent properties, Prandtl proposed in1945 a model that related the eddy viscosity with the turbulent kinetic energy(equation of k) and hence the concern in giving more depth to the modelling byadding historical considerations, since the turbulent kinetic energy is affected bywhere the flow has been. These models are also incomplete since a characteristiclenght must be provided. Later on in the 1970’s and the 1990’s new one-equationmodels have been proposed in which a transport Partial Differential Equation (PDE)for νt has to be solved.

The most successful modern one-equation model is the Spalart-Almaras 1992 model,in which a transport equation for an eddy viscosity parameter ν is solved, thelength scale is specified and it determines the rate of dissipation of the transportedturbulence quantity. Due to the fact that it has only one extra PDE, it provides lessexpensive calculations for boundary layers on external flows. According to reference[28], the model constants were tuned for external aerodynamics flows and hencethey provides accurate results for boundary layers at adverse pressure gradients,showing good prediction on stalled flows. The suitability of the model to airfoilapplications have also sparked the interest in the turbo machinery community. Inthe other hand, the model proved to be unsuitable for complex geometries since itis difficult to define an appropriate length scale, and it seems to lack sensitivity totransport processes in rapidly changing flows.

A detailed description of the model equations is presented on the Appendix, sincethis was one of the models used in the current work.

• Two-equation models: The first two-equation model was developed by Kol-mogorov in 1942. To improve the prediction of turbulent properties he devised


a model that apart from the equation of k also included the calculation of anotherPDE for the rate of dissipation of energy. By calculating two turbulent properties,these models does not demand the user to input specific characteristics of the flowother than the boundary conditions. These models are also classified as completemodels. Historically, they had to wait the development of faster computers in orderto be tested and be further developed, this explains the great number of new twoequation models that have arisen after the 1970’s.

Two-equation models in general use the transport equation for the turbulent kineticenergy k to determine the velocity scale, since q = k1/2, different turbulent modelsuse different methods to obtain the length scale with the second transport equationfor the other dependent variable. These two equations allow the model to accountfor flow conditions where convection and diffusion impact in the production anddissipation of turbulence such as in recirculating flows. One of the most used andextensively tested turbulence models is the k-ε, its second equation is the equationof the viscous dissipation, named as ε, therefore the length scale is l = k3/2/ε, inthis model the eddy viscosity is given by:

νt = Cql = Cµk2

ε(2.15)

A detailed presentation of some of the two-equation models is also shown in theAppendix. Close to the wall and at high Reynolds numbers, the standard k-ε modelhas equations to account for the effects close to the wall (wall functions), basedon the universality of the log-law and on the fact that measurements show that theproduction of turbulent kinetic energy is balanced with dissipation. At low Reynoldsnumbers some modifications had to be included in the model to account for the nearwall effects.

According to reference [28], the k-ε model presented good agreement in severalindustrial relevant flows, such as confined flows without the necessity of adjustingits constants. Results with external flows, weak shear layers, axisymmetric jets instagnant surroundings and rotating flows are a little less encouraging. The modelalso have some deficiencies that do not allow it to predict secondary flows in noncircular ducts. To address to some of the shortcomings of the standard k-ε model,modifications have been proposed such as: the two-layer k-ε to deal with the lowReynolds issue; the RNG k-ε model to deal with issues related to large rates ofdeformation on the flow; the Wilcox k-ω and the Menter k-ω Shear Stress Transport(SST) model to provide more accurate aerodynamic calculations, the last one wasalso used in the present work.

Differently from molecular viscosity which is a property of the fluid, the turbulenteddy viscosity is related to several aspects of the flow such as its dimensions, its geometryand its history. The use of the Bussinesq hypothesis in the eddy viscosity models, whilegiving accurate results to a range of flows, can also lead to wrong predictions even whentaking into account sophistications like the history of the flow, in fact the assumption isa simplification and does not reflects what really happens in the flow.

These models assume that exists isotropy of the normal Reynolds stress, which is asimplification of the real fluid behavior and it is not very accurate even in simple two-dimensional flows such as the flow on a flat plate. In this type of modelling, the Reynoldsstress is proportional to the mean rate of strain Sij which is true in the cases where there is


a balance on ratio of production and dissipation of turbulent kinetic energy, so it is difficultto duplicate the actual response of turbulence to complex mean-flow perturbations withthis approach.

Non-eddy viscosity models

In view of the limitations presented in the eddy viscosity models concerning Reynoldsstress anisotropy, two main alternatives have been proposed, the Reynolds Stress EquationModels and the Algebraic Stress Equation Models. The first originates from the work ofLaunder et al. [29] in which they propose a model where each of the Reynolds Stresses isdetermined from the solution of transport equations plus the solution of the equation forturbulence energy dissipation. The second approach uses an algebraic modeling of someof the transport terms of the Reynolds Stress transport equation reducing it to a set ofalgebraic equations.

• Stress transport models: these models are also called as second-order or second-moment closure, according to Wilcox [9] they have the “conceptual advantage”of modeling the stress transport in a natural manner that incorporates non-localand history effects, therefore only the initial and boundary conditions have to besupplied without further adjustments for particular cases. The Reynolds stressesRij are modeled as follows:

∂Rij

∂t+ Cij = Pij +Dij − εij + Πij + Ωij (2.16)

where:

– Rij is the kinematic Reynolds stresses

– Cij is the transport of Reynolds Stresses by convection

– Pij is the rate of production of Reynolds Stresses

– Dij is the transport of Reynolds Stresses by diffusion

– εij is the rate of dissipation of Reynolds stresses

– Πij is the transport of Reynolds stresses due to turbulent pressure-strain iter-ations

– Ωij is the transport of Reynolds stresses due to rotation

this equation represents the transport for each one of the six individual Reynoldsstresses, they are solved along with the equation of turbulent energy dissipationε, hence by solving seven different partial differential equations in a tridimensionalcase, this model has a higher computational cost than the models discussed previ-ously.

The terms for convection (Cij) production (Pij) and rotation (Ωij) are used in theirexact form and the remaining terms are modelled by means of assumptions. Thediffusion term (Dij) is modeled by considering that the rate of transport of Reynoldsstresses by diffusion is proportional to gradients of Reynolds stresses; the modelingof the dissipation rate (εij) assumes isotropy of the small dissipative eddies; thepressure-strain term (Πij) models two different processes; a process that reducesanisotropy that is considered to be proportional to the degree of anisotropy of theReynolds stresses and a process that opposes the production of anisotropic vortices


that is taken as proportional to the production process that generate anisotropy;the pressure-strain term contributes to the reduction of the Reynolds shear stressesand a redistribution of energy among the normal Reynolds stresses.

According to reference [28] these models are complex, but they are “the simplest typeof model capable of describing the mean flow properties and Reynolds stresses withouta case-by-case adjustment”, they have shown to be very accurate in determiningmean flow properties and the Reynolds stresses in many flows including wall jets,channel flows and curved flows, nevertheless they haven’t been as validated as otherturbulence models such as k-ε, they have a higher computational cost and theyare reported to perform as poorly as the k-ε in axisymetric jets and unconfinedrecirculating flows due to problems with the modeling of the ε equation.

• Algebraic Stress Models (ASM): due to the high computational cost of solvingthe Reynolds Stress models it was proposed that some of the terms of the ReynoldsStress Transport Equation were modelled by algebraic expressions. This way a re-duction of the computational effort would be achieved while still taking into accountthe anisotropy of the Reynolds stresses. Gradients of the Reynolds stresses appearin the convective Cij and diffusive Dij terms of Equation 2.16, some authors havetested even neglecting the terms with success in some applications, but in generalthe sum of these terms is replaced by the sum of the convection and diffusion termsof the turbulent kinetic energy equation. The algebraic stress model equation isimplicit, with the Reynolds Stresses appearing in both sides of the equation, theresulting problem was simplified from six transport equations to six algebraic equa-tions and the solution of k and ε transport equations.

• Explicit Algebraic Stress Models (EASM): By being an implicit method, theASM was reported to exhibit numerical issues such as multiple solutions, singular-ities and convergence to non-physical conditions [30]. An alternate approach is toexpand the Reynolds stresses in a series with the Boussinesq approximation as theleading term, this will result in an explicit algebraic model, which makes them morerobust with improved predictive capabilities. Two different EASM models will betested within the current project, a baseline EASM and the ϕ-α-EASM, which is amodel that incorporates improved capabilities for modelling regions close to walls.

Chapter 3

The Problem

The problem that is being studied on the current thesis is the determination of the pointof separation of the flow through a linear LPT cascade with T106 profile sections withRANS/URANS simulations using different turbulence models. The RANS calculationswill be performed in OpenFoam and the results will be compared with experimental andnumerical results.

Stadtmuller [32] performed experimental measurements of the pressure distribution andthe wake losses of the T106 blade in a low pressure linear turbine test rig with seven bladesand aspect ratio of 1.76, which was considered enough to assume a two-dimensional flow inthe middle of the blade. Due to some experimental uncertainties on the inlet conditions,the inlet angle was estimated to be 45.5 with a Reynolds number of 59,634 and a Machnumber of 0.405. The experimental setup also had a possibility to add moving transversalbars on the inlet that allowed him to perform experiments with incoming wakes.

Stieger [4] does an experimental study using a cascade composed by five blades withmoving bars in the inlet, to investigate the wake induced transition in separating boundarylayers. A latter study from Stieger et al. [33] investigates the fluctuation of the surfacepressure in the region of a separation bubble due to the effect of incoming wakes, and astudy from Stieger and Hodson [34] does a deep investigation on the transition mechanismof a boundary layer in a turbine blade subjected to the effect of incoming wakes.

Due to the range of Reynolds numbers on the LPT cascades of gas turbines, DNSstudies are becoming more frequent, however, these studies are still rather expensive andare still not accessible to a daily industrial user. The first incompressible DNS on LPTcascades was performed by Wu and Durbin [35] where they describe the formation of twotypes of longitudinal vortices caused by the passage of wakes through the LPT, they alsocompare the results of DNS with LES over same configurations obtaining good agreementfor well resolved LES.

Wissink [15] performed a three-dimensional incompressible DNS over the T106 profilewith both an undisturbed inlet and a periodically disturbed inlet by incoming wakes, toprovide data for the development of turbulence models and to investigate the effect ofthe incoming wakes on the boundary layer. For the undisturbed inlet case the authorexecutes a simulation with an inlet angle of attack of 45.5 and Reynolds number of51, 831, the pitch between the blades in his study is 0.9306. The flow obtained in thisDNS simulation has a good agreement with experimental results, the author describesthe formation of a separation bubble at the leading edge of the suction side. Due to theaction of the favorable pressure gradient the disturbances originated from the unstableleading edge separation are damped. Beyond the chord position of x/L = 0.6 the pressuregradient becomes adverse, in the undisturbed inlet case the author reports the formation

23

24 CHAPTER 3. THE PROBLEM

of constantly present separation bubble near the trailing edge (x/L ≈ 0.93), in the casewith incoming wakes the formation of an intermittent and less pronounced separation isobserved at x/L ≈ 0.87, it is a very unstable shear layer due to the effect of the incomingdisturbances.

Sandberg et al. [16] uses an in-house compressible multi-block structure curvilinearNavier-Stokes solver to compare the flow over the T106A turbine cascade with experi-mental data and investigate the influence of the inflow turbulence level on the transitionbehavior and profile losses. They find that the laminar boundary layer separation isstrongly dependent to the level of inlet turbulence, moving downstream with increasingturbulence level. They also observe that the turbulence level reduces the peak amplitudeof the wake loss and shifts the peak pitchwise towards the pressure side.

The current chapter will present how the problem was set up in the OpenFOAMenvironment. In the first part it is shown how the mesh for solving the problem wasobtained and modified in order to be used in OpenFOAM and the second part shows howthe cases were defined and run in OpenFOAM.

3.1 Geometry and Mesh

As it could be seen from the previous section, the T106 blade is a well known testcase. The particular profile in the present study is the T106A which have a pitch of 0.799chord lengths between blades, the specification can variate from T106A to T106D withincreasing pitch between blades. The airfoil is highly cambered and has rounded leadingand trailing edges. A turbine cascade is an aerodynamic device composed by a numberof blades placed at a radial distance from one another, and they are connected on a hublike a fan. Since these blades have a high aspect ratio, they are normally simulated astwo dimensional blades positioned on top of one another. Experimental measurementsare normally carried on the middle blade of setups with five or seven blades. In numericalsimulations it is usual to see meshes with one blade in the middle of the domain andperiodical boundary conditions on top and bottom of the domain like in Reference [16]and also domains with the pressure side of the blade on the top of the domain and thesuction side of the blade on the bottom of the domain like in Reference [15], the firstapproach was also used in the current work.

The mesh employed to solve the problem was generated by a program developed withinthe work done by Sandberg et al. [16], it is a high quality mesh designed for a finitedifference DNS solver and it was adapted in the current project to be solved withina finite-volume methodology in OpenFOAM. The mesh was constructed based in themethodology developed by Gross and Fasel [36] for turbine cascades by solving a Poissonequation. It was conceived as a hybrid O-type mesh around the profile and an H-typegrid away from the profile. Due to the rounded leading and trailing edges of the blade,the O-type grid will allow a good resolution in these regions and will ensure orthogonalityon the near wall cells around the profile. The H-type grid employed away from the profilewill allow the implementation of the periodic boundary conditions. The grid is composedby nine blocks, and it is shown in Figure 3.1.

The O-type section is composed of the blocks 3, 4, 5 and 7 and the H-type section iscomposed from the remaining blocks. Detail on the leading and trailling edges of the meshare shown in Figure 3.2. There are four points within the mesh where an intersection offive domains occurs: the connection of blocks 1, 3, 5, 2, 6 and 5, 7, 8, 6, 9, shown inFigure 3.3 and two of them due to the periodic boundary conditions on the top/bottomconnection, elements 1, 3, 4, 2, 6 and 4, 7, 8, 6, 9. According to Sandberg et al. [16] at

3.1. GEOMETRY AND MESH 25

Figure 3.1: Combined O-type/H-type mesh from Reference [16] used in the current work.

these points, the cells cannot be orthogonal and a maximum angle of 72 was enforcedin order to improve the quality of the grid, these points were also moved away from theblade to a region where a high accuracy is not essential.

For the DNS study the authors states that the grid refinement close to the wall hadto be sufficient to resolve the wall structures and that the mesh could be coarser awayfrom the wall, however, one of the objectives of their study was to investigate the effectsof the incoming turbulence on the boundary layer of the blade, then a fine mesh was setupstream of the blade profile and in the passage between the blades in order to resolve theincoming turbulence. To confirm that the mesh was adequate for their study they shownthat the ∆y+ on the suction side is bellow 1.4 and the flow is laminar in regions where∆y+ > 1 with at least 30 points across the boundary layer. On the pressure side, mostof the surface is below ∆y+ = 1 except for the first point and close to the trailing edgewhere the flow is also laminar. On their study, ∆x+ < 10 and ∆z+ < 11 were employedwhich was deemed to be adequate.

The mesh for the current work was obtained from Sandberg et al. [16] as an ASCIIfile with the coordinates of the points for each one of the blocks of the two-dimensionalmesh, initially it was expected that the OpenFOAM utility plot3dToFoam would be ableto read the mesh and extrude it in order to be used by the solver. The extrusion processis necessary because, in OpenFOAM, all simulations have to be performed with tridimen-sional meshes, in two-dimensional simulations the mesh must have one cell thickness andthe two new faces generated by the third dimension must be set with an empty boundarycondition.

A simple Fortran program was written to read the points from the nine ASCII files andcreate a mesh in the Plot 3D format. The utility was executed, and the extruded meshwas created perfectly, however, only the boundary conditions for the blade were set, anda different path had to be undertaken.

Pointwise was the software chosen by Holohan [37] to generate meshes in his thesisusing OpenFOAM as a solver. The program has several grid generation utilities and alsois capable of generating meshes to specific solvers such as OpenFOAM. In the presentcase, what was needed was a program that could extrude the mesh and set the boundaryconditions for the solver, which was accomplished with Pointwise.

The procedure executed with Pointwise was to import the mesh in Plot 3D format,then execute an extrusion, which was made with the command Create/Extrude/Translateand a translation of one unit of length is performed in the z direction. The following stepwas the execution of the command CAE/Set Boundary Conditions to set the boundary


(a) Leading edge

(b) Trailing edge

Figure 3.2: Details of the leading and trailing edges of the mesh of the T106 blade.

(a) Point of connection of blocks 1, 3, 5, 2, 6 (b) Point of connection of blocks 5, 7, 8, 6, 9

Figure 3.3: Details of the mesh on the connection points of five different blocks.


conditions to the mesh: as it was mentioned earlier, the lateral boundaries are definedas empty boundary condition; the profile was defined as wall boundary condition andthe other boundaries were defined as patch boundary condition1. The mesh was thenexported to the OpenFOAM format with the command File/Export/CAE, and the fivefiles that compose the mesh in an OpenFOAM case were created.

OpenFoam meshes are defined by a hierarchical set of files that organizes the celldistribution and it is composed by at least five different files:

• the points file is a list with the coordinates of all points in the mesh;

• the faces file is a list of all the mesh faces composed by the points of the previousfile;

• the owner file is a list of the number of the volumes that own the faces defined inthe previous file;

• the neighbor file is a list of the neighbor faces and

• the boundary file defines the boundary conditions for the mesh, this is the file wherethe periodic boundary condition has to be defined.

The final step to get a working mesh in OpenFOAM environment was to set theperiodic boundary conditions on the top and bottom parts of the mesh, for that operationOpenFOAM’s utility createPatch had to be executed, it uses the information from thedictionary file createPatchDict that determines which boundaries are periodic and whichare the neighbors of the periodic boundaries. It is very important to make sure that thepoint coordinates of the mesh on one of the periodic boundaries matches the correspondentcoordinates of the points on the other.

The original mesh was developed to DNS calculations, therefore, it is was expected tobe more refined than necessary for RANS simulations. The number of points in each of itscomponent blocks, number of cells and number of faces is presented in Table 3.1. Duringthe initial test simulations with the mesh in a two-dimensional configuration, the timeof execution of the iterations was considered short enough to carry on with the studies,with some of the simulations reaching convergence in less than five hours. However it wasthought that a comparison of the results with a less refined mesh could also bring newinformation about simulation times and the quality of the results obtained.

A less refined mesh was created based on the original mesh, since the original meshwas obtained in ASCII format with the cartesian points of the mesh elements, the idea ofeliminating some of the intermediate points of the mesh came to mind. A simple programin Visual Basic within Microsoft Excel environment was made in order to select the xand y coordinates correspondent to a specific point and erase it. The refinement of themesh in the direction normal to the wall was kept the same as the original mesh, and thepoints in the direction tangential to the wall were reduced in its half as shown in detailon Figure 3.4.

The number of points in each direction of all meshes was even, then, to keep the size ofeach of the nine blocks the same, the first and the last column of elements were maintainedand the intermediate columns were eliminated alternately until the last column that wasmaintained to keep the size of the mesh. The neighbor of the last column was eliminated,as shown in Figure 3.5, this process created a column with a wider thickness, in order to

1In OpenFOAM is a list of boundary faces is called a patch, in the current work each patch will beassociated to a distinct boundary


(a) Original Mesh (b) Coarse Mesh

Figure 3.4: Details of the original mesh and the coarser mesh over the suction side of theT106 blade.

Table 3.1: Number of elements in each of the two meshes used in the study

Element Original Mesh Coarse MeshBlock 1 288 x 192 144 x 96Block 2 288 x 48 144 x 48Block 3 144 x 192 144 x 96Block 4 144 x 240 144 x 120Block 5 144 x 240 144 x 120Block 6 48 x 240 48 x 120Block 7 144 x 192 144 x 96Block 8 288 x 192 144 x 96Block 9 288 x 48 144 x 48

Number of points 555,612 216,340Number of cells 270,825 107,409Number of faces 1,084,718 430,397

avoid the problem in critical regions of the mesh, whenever possible these wider elementswere positioned close to the external boundaries of the mesh.

Special care was taken in the mesh intersections, since a previous mesh presentednumerical instabilities on these points, nevertheless, in the central blocks 4, 5 and 6 itwas impossible to avoid a wider element in the connection of the blocks, the detail of theintersection points of the mesh is shown on Figure 3.6, in the left branch it is possible tosee the right-hand side blocks with the wider element in the connection of mesh blocks.

In the H-type blocks 1 and 8 the reduction of the number of cells had to be madeboth in the horizontal direction and in vertical direction, because the number of verticalelements on these blocks defines the number of elements on the wall direction in blocks 3and 7. A detail of the leading and trailing edges on blocks 3 and 7 are shown in Figure3.7. The total number of points in each block and the total number of cells and faces forthe modified and coarser mesh are also presented in Table 3.1, the total number of cellswas reduced in almost 60% when comparing with the original mesh.

A tri-dimensional view of the mesh is shown in Figure 3.8, the lateral faces were hiddento allow a better visualization of the boundaries (patches) and their name convention, italso shows the coordinate convention used when solving the numerical problem.


(a) Original Mesh (b) Coarse Mesh

Figure 3.5: Detail of the technique adopted do eliminate points to generate a coarsermesh.

(a) Connection of blocks 1,3,5, 2 and 6 (b) Connection of blocks 5, 7, 8, 6 and 9

Figure 3.6: Details of the connection of blocks of the coarse mesh of the T106 blade.


(a) Leading edge

(b) Trailing edge

Figure 3.7: Details of the leading and trailing edge on the coarse mesh of the T106 blade.

Figure 3.8: Name convention for the boundaries of the mesh used in the current work

3.2. SETTING UP THE PROBLEM 31

3.2 Setting up the Problem

In order to convert the real problem of solving the flow over the T106 blade in a nu-merical problem, a suitable numerical method should be employed. OpenFOAM uses thefinite volume method to perform its calculations. In general, the solution of a numericalproblem with the finite volume method requires three steps: the first step is the gridgeneration which was covered in the last section; the second step is the discretization ofthe equations in a form that the computer can solve within the available grid and thethird is the solution of the equations; the second and third steps will be covered brieflyin the following subsections, since some of the options given by OpenFOAM when settingup a case demands the knowledge of these concepts. A small description of a typicalsimulation setup is also presented.

3.2.1 The Finite Volume Method

The flow equations in its conservative form can be written it terms of a general variableφ as follows:

∂ρφ

∂t+∇ · (ρφU) = ∇ · (Γ∇φ) + Sφ (3.1)

if φ assumes a value of 1 the continuity equation is obtained, if it assumes u, v or w,the momentum equations are obtained2. Equation 3.1 is known as the transport equationof the variable φ that can also represent any other scalar or vector variable such astemperature or turbulent kinetic energy. It describes the balance between the transportprocesses.

On the left-hand side of the equation the first term represents the rate of change of φof the fluid element, the second term represents the net rate of flow due to the convectiveprocess; on the left hand side, the first term describes the diffusive rate of change in φand the second term represents the variation of φ due to source terms.

The Equation 3.1 can be integrated over a control volume and can be written as follows:∫CV

∂ρφ

∂tdV +

∫CV

∇ · (ρφU) dV =

∫CV

∇ · (Γ∇φ) dV +

∫CV

Sφ dV (3.2)

The Diffusion Term

The first term on the left-hand side of Equation 3.2 is the diffusion term or in Open-FOAM nomenclature Laplacian term, by applying the Gauss divergence theorem it canbe written as: ∫

CV

∇ · (Γ∇φ) dV =

∫SC

(Γ∇φ) · dA (3.3)

using the standard one-dimensional grid convention, with P being the nodal point ofthe control volume in study, “W” and “E” for the left-hand side and right-hand side nodal

2The terms not shared by different equations are all expressed by the source term Sφ


Figure 3.9: Standard convention for one-dimensional mesh in the finite volume method.

points respectively and “w” and “e” for the left-hand side and right-hand side boundaries,as shown in Figure 3.9 the discretization of the diffusion equation can be written as:∫

SC

(Γ∇φ) · dA =

(ΓA

dφ

dx

)e

−(

ΓAdφ

dx

)w

(3.4)

in OpenFOAM, the discretization of the diffusion term follows this equation, called inthe program the Gaussian discretization, in order to allow the computer to perform thecalculations, a suitable form for Γ and dφ/dx has to be derived. The program gives a choiceof different interpolation schemes for Γ and for evaluating the normal gradient at the cellface (dφ/dx). In all calculations performed in the current work, the interpolation schemeselected was linear, and the normal gradient scheme was the corrected, this selection givesan unbounded, second-order and conservative numerical behavior.

The Convection Term

The second term in the right-hand side of Equation 3.2 is the convection term, inOpenFOAM is is named the divergence term, the Gauss divergence theorem can be appliedand the term becomes: ∫

CV

∇ · (ρφU) dV =

∫SC

ρφU · dA (3.5)

using the same conventions as the last sub-section the equation can be discretized asfollows: ∫

SC

ρφU · dA = (ρUAφ)e − (ρUAφ)w (3.6)

which is the Gaussian discretization scheme in OpenFOAM, the program requires theuser to select the interpolation scheme for the independent field φ, several schemes areoffered by the program including the second order central difference scheme and the firstorder upwind scheme.

Time Dependent Problems

In time dependent problems it is also necessary to integrate the transport equationwith respect to time, which gives the following form:

∫∆t

∂

∂t

(∫CV

∂ρφ

∂tdV

)dt+

∫∆t

∫CV

∇·(ρφU) dVdt =

∫∆t

∫CV

∇·(Γ∇φ) dVdt+

∫∆t

∫CV

Sφ dV

(3.7)

3.2. SETTING UP THE PROBLEM 33

in OpenFOAM, the user can determine which discretization scheme for the first timederivative ∂/∂t will be used. In the current work, the only discretization used was theEuler discretization, a first order bounded and implicit scheme.

Solution of the equations

After the discretized equations and appropriate interpolations for the different φ vari-ables are set up for each of the nodal points on the mesh, the result is a system of linearalgebraic equations that has to be solved to obtain the distribution of the variables overthe numerical domain. For each of the discretized equations, OpenFOAM gives a rangeof different linear system solvers for the user to chose from. It is also possible to definethe linear solver parameters such as the tolerance, the smoother and the under-relaxationfactor among others options.

3.2.2 Typical OpenFOAM Simulation Setup

OpenFOAM is a very customizable software, it gives a great deal of liberty for theuser to define different aspects of the simulation that are not available in most of thecommercial packages, this liberty comes with a price of having a steeper learning curvesince the wrong definition of one of its numerous parameters can lead to errors whenrunning the simulation. A typical OpenFOAM case is defined in a specific case folder,within the case folder, three different folders have to be created as explained bellow:

• “constant” folder: in the constant folder, the main constants of the problem, suchas kinematic viscosity and turbulence model has to be defined. It is also the folderwhere the mesh files are stored.

• “system” folder: in the system folder, the parameters associated with the solu-tion procedure are stored, the controlDict file is composed by the controls for thesimulation such as the number of iterations, the type of solver that will be used theoutput interval and so on; the fvSchemes file stores the discretization schemes thatwill be used in the simulation and the fvSolution file stores the equation solvers,tolerances and other algorithm controls used in the simulation.

• “time” folder: the time folders are named with the specific time step in which theuser commanded the program to generate an output, in these folders the instanta-neous and averaged fields are saved by the solver while calculating the numericalsolution. In the folder “0”, the user has to save the files setting the initial fieldconditions for the problem.

each of the folders have determined files that have to be saved on them in order to run asimulation.

OpenFOAM offers a wide range of solvers for different types of fluid flow problems, forthe single-phase incompressible simulations performed in the current work, it was selected,the solvers simpleFoam for steady simulations and pisoFoam, for unsteady simulations.

The simpleFoam solver is the OpenFOAM implementation of the SIMPLE algorithm,developed by Patankar in 1972, the need for this algorithm arises from the fact that whencalculating a flow, the pressure distribution is not known beforehand i.e., the pressuredistribution is also one of the variables the user is trying to obtain. From the incompress-ible Navier-Stokes equations it is possible to see that there is a coupling between pressure


and velocity, if the correct pressure field is applied in the momentum equations the cal-culated velocity field will satisfy the continuity equation. An iterative process is startedby guessing an initial pressure field and obtaining guessed velocity components, these arethen used in the continuity equation to obtain a pressure correction field that updates thevelocity field. The process continues until the velocity and pressure fields converges. Theguessed pressure field is normally denoted by p∗ and the correction applied is denoted byp′, hence it is possible to write:

p = p∗ + p′ (3.8)

The pressure correction can sometimes diverge, specially in the beginning of the itera-tive process when the guessed field is far from the final solution, therefore, under-relaxationfactors are employed to avoid a substantial correction that could lead to numerical insta-bilities. The under-relaxation factor (αp) is a numerical value between 0 and 1 that ismultiplied to p′ as follows:

p = p∗ + αpp′ (3.9)

the lower the value of the under-relaxation factor the smaller the correction applied, whichcan lead to a more stable calculation while allowing evolution of the iterative process. InOpenFOAM, under-relaxation factors can also be applied to other calculated fields suchas velocities and turbulence properties.

The pisoFoam solver is the implementation of the PISO algorithm to transient flows,it is a non-iterative procedure that uses more than one correction of the pressure field3,therefore, a more accurate pressure field is obtained. The algorithm also has a higherorder temporal accuracy due to the splitting technique, and when using a small time stepthe velocity and pressure fields are considered sufficiently accurate to advance to the nexttime step. Normally the time step is defined by maintaining the Courant number smallerthan 1, it is defined as:

C =u∆t

∆x(3.10)

the use of large time steps can lead to numerical instabilities, under-relaxation factorscan also be employed when using the PISO solver.

Another functionality of OpenFOAM is the possibility of running simulations in par-allel on distributed processors. The software does a domain decomposition where thegeometry and the variable fields are divided and allocated to separate processors or com-puters in a network. The number of sub-domains is defined by the user in the dictionaryfile decomposeParDict. The software uses openMPI, an implementation of the MessagePassing Interface (MPI), which is a standardized message-passing system, developed forparallel computing.

3.2.3 Determination of flow parameters

The numerical results obtained in the present work will be compared with the exper-iments performed by Stadtmuller [32], where the reference Mach and Reynolds numbersare calculated for the outlet velocity, based on the isentropic expansion between the inletand the exit. In incompressible simulations these calculations lead to incorrect results dueto the fact that the density is constant, therefore the outlet Reynolds number (Equation2.6) will be calculated based on the outlet velocity calculated from Bernoulli equation as

3The number of corrections performed by the solver is defined by the user in the fvSolution file.

3.3. POST-PROCESSING 35

follows:

Uoutlet =

√2 · Ptinlet − Psoutlet

ρ(3.11)

in practice, to match the experimental and numerical Reynolds, it is necessary to runthe simulation, take an average of the inlet pressure and calculate the outlet velocityto obtain the outlet Reynolds number and adjust the inlet velocity accordingly. Thisprocedure is time consuming since it takes a couple of iterations to obtain the desiredReynolds number. It is suggested that a difference in the Reynolds number within 5% isdeemed acceptable.

The pressure coefficient and the wake loss are defined as follows:

Cp =Ps(x)− PsoutletPtinlet − Psoutlet

(3.12)

Ω(y) =Ptinlet − Pt(y)

Ptinlet − Ptoutlet(3.13)

with Ps(x), the static pressure at a determined blade position x, and Pt(y) the totalpressure along a non dimensional pitchwise taken at 40% chord downstream.

3.3 Post-Processing

OpenFOAM does not provide a complete environment for post processing of the resultsas does some commercial packages available. Instead, it provides a number of utilities thatcan be used to generate raw data to be processed by other software. The output files arein the text format, composed as a list of values (one column for scalar variables andthree columns for vector variables) for the centroid of the mesh cells, and they have to beprocessed in conjunction with the mesh files.

The frequency of the output is set by the user, a normal output is normally composedby: the pressure field, the velocity field and any other transport variable that is beingcalculated. The user can then run a series of OpenFOAM utilities that will be calculatedusing the data from those output files.

The execution of a utility is quite simple, and it is made by typing a command lineon the Unix shell screen. Examples of utilities used in the present work were vorticityto calculate the vorticity field based on the velocity field and wallGradU that calculatedthe gradient of the velocity at the wall. The program also gives the user the possibilityto sample data from a probe or a surface and generate averaged results; there are alsolibraries supplied for the calculation of fluid forces over a selected surface.

To perform graphical visualizations of the mesh, surface plots and vector plots ofthe flow variables, the software ParaView was chosen, it comes with the installation ofOpenFOAM and it is an open source application developed for flow visualization. Itautomatically reads the output files generated by OpenFOAM simulations, therefore thetwo softwares create an integrated work environment that allows the user to check theevolution of the solution while the simulation is running.

The main tasks carried out with ParaView were mesh inspections and surface plots.ParaView also helped in generating data in a suitable format for Microsoft Excel, since itallows the visualization of the mesh patches such as the inlet and the blade, and it allowsthe user to generate a spreadsheet with the fluid properties on those selected points.


Gnuplot was another software used on post-processing, it is an openSource softwareused to plot two and tridimensional graphs. It is a command-line program and withsimple scripts it was used to plot the simulation residuals in real time.

Chapter 4

Results

The current chapter will present the main results obtained in the current project. It willbe divided into two main sections, the initial section, Preliminary Studies, describes theinitial cases simulated with different setups used in order to acquire sufficient knowledgeabout the program capabilities and options to setup a proper simulation to achieve theobjectives proposed in Section 1.3; the final section describes the investigation of thebehavior of the different turbulent models on the chosen geometry, for a Reynolds numberof 60,000 and the effect of coarsen the mesh on the predictions. It also shows a comparisonbetween turbulence models and the laminar simulation in the coarse mesh for a Reynoldsnumber of 150,000.

4.1 Preliminary Studies

The initial cases were simulated using openFOAM’s incompressible steady solver sim-pleFoam with the Spalart-Almaras turbulence model, the use of this particular turbulencemodel is due to the fact that one of the tutorials supplied with OpenFOAM is the solutionof an incompressible flow over an airfoil using it, this way it was easier to implement anew case having the tutorial files as a baseline. In that initial moment, the objective wasto see if the results calculated by the simulations were following the expected trends andoutputting meaningful results. The following subsections will present some of the pre-liminary simulations performed, and the lessons learned from them. In the initial part,the results obtained from steady simulations will be presented. It was observed that thesteady simulations were not able to capture some of the flow instabilities and the unsteadybehavior of a separated flow with vortex shedding, then the later part of the chapter willshow the use of an unsteady solver that was more appropriate to simulate this type ofinstabilities.

4.1.1 Steady Simulations

The results presented in the following subsections are a selection of results obtainedfrom a number of preliminary tests performed with OpenFOAM with the steady solversimpleFoam. A total of six different cases will be presented, all of them using the Spalart-Allmaras turbulence model. The main characteristics that were investigated were theboundary conditions, the order of accuracy of the convective term and properties thatwould affect the simulation time such as the use of a different number of processors, adifferent mesh refinement and the use of different under-relaxation factors.

37

38 CHAPTER 4. RESULTS

Table 4.1: Preliminary steady simulations

Name Mesh Order of accuracy† CPU’s Under-relaxation factor Time‡

P1 Refined 1st 4 0.3 10610P2 Refined 1st 4 0.3 n/aP3 Refined 2st 2, 4, 8 0.3 12267, 10449, 7061P4 Coarse 2st 4 0.3 4334P5 Coarse 2st 4 0.7 n/aP6 Coarse 2st 4 0.5 2575

† of the convection term‡ time to perform 10,000 iterations in seconds

The cases are presented in the Table 4.1, they were named from P1 to P6 (P as anacronym for Preliminary) and the results are presented in the following subsections.

Testing Different Boundary Conditions

One of the initial issues as a novice user was to verify how the different boundaryconditions on OpenFOAM could modify the simulations and if the periodic boundaryconditions were implemented correctly. The boundary conditions defined for simulationsP1 and P2 are described in Table 4.2. The difference between the cases was in thedefinition of the type of the boundary conditions for the inlet and for the outlet for thevariable ν, in case P1, it was set as fixedValue, and in case P2 the boundary conditionwas set as freeStream.

FreeStream is hybrid boundary condition that acts locally at each boundary elementverifying the direction of the mass flow, if the mass is flowing into the numerical domainit sets a fixedValue boundary condition and if the flow is moving out it behaves as azeroGradient boundary condition.

The calculated Reynolds number for the simulation was 60,127; the under-relaxationfactors were set as 0.3 for pressure and 0.7 for velocity and ν. The selected numericalschemes for the simulations were:

• gradSchemes : Gauss linear (second order);

• laplacianSchemes Gauss linear corrected (second order)

• divSchemes bounded Gauss upwind (first order),

Figure 4.1 show the evolution of the residuals of the two cases, it can be seen that forthe freeStream boundary condition the residuals kept reducing1 while for the fixedValueboundaries there is an initial reduction of the values and by the 800 iteration a suddenincrease on ν residual with a posterior stabilization leads to a crash on the simulation bythe 2085 iteration.

Since case P1 did not reached convergence, the following analysis was performed onCase P2. The results were also checked for the behavior of the periodic boundaries (topand bottom of the numerical domain) and empty boundaries (lateral boundaries of thedomain). ParaView capability of plotting a specific patch and generate a spreadsheet with

1It is possible to notice a spike on the graph at the iteration 3000, this was caused by the stop andrestart of the solution on that particular iteration.

4.1. PRELIMINARY STUDIES 39

Table 4.2: Boundary conditions for simulations P1 and P2.

Boundary Velocity† Pressure ν νInlet (0.389 0.396) zeroGradient 3.0× 10−6 ‡ calculated

Outlet zeroGradient 0.0 3.0× 10−6 ‡ calculatedTop cyclic cyclic cyclic cyclic

Bottom cyclic cyclic cyclic cyclicProfile (0.0 0.0) zeroGradient 0.0 0.0Lateral empty empty empty empty

† The velocity is expressed in its cartesian coordinates‡ The type of boundary conditions were different for each case

(a) Case P1, fixedValue boundary condition

(b) Case P2, freeStream boundary condition

Figure 4.1: Convergence history for cases P1 and P2.


(a) Pressure distribution (b) Velocity distribution

Figure 4.2: Pressure and velocity distribution on the upper and lower boundaries of thedomain.

the values of the field variables on that patch was explored to generate a spreadsheet withthe values of the field variables on lateral boundaries and the top/bottom boundaries.

As mentioned before, even for two-dimensional simulations, OpenFOAM uses a tridi-mensional mesh. One of the doubts as a beginner user was if the solution calculated by thesolver was the same throughout the z axis which is one cell thick or, if, by symmetry, onlythe middle plane would have the correct solution. By checking the spreadsheet generatedby ParaView it was noticed that the values calculated at the lateral extremities of themesh were the same, therefore, the mesh kept the same values of the variables throughthe z axis, and that meant that the inspection of the graphical results with ParaViewcould be performed in any plane perpendicular to the z axis.

Concerning the periodic boundary conditions (top and bottom), a graph with thedistribution of pressure and velocity magnitude on these boundaries is plot on Figure 4.2,it is possible to see that the values do not match exactly, as it could be expected, sincethey do not represent the same cell, but in fact they are at a distance of one cell fromeach other. The pressure curve shows an increase of the pressure as the x coordinate getscloser to the stagnation region generated by the leading edge of the blade and it reduceswhen moving downstream until it reaches the reference pressure at the outlet it is alsoobserved a small oscillation at the first wake region. The velocity curve shows a decreaseuntil it passes the stagnation region when it starts increasing until the wake region whereit is possible to see four different valleys on the curve representing the low velocity on thewake region, since the domain is periodic and the wake is pointed in the direction of thetrailing edge (pointed to the bottom boundary) it crosses the domain four times, but infact they represent the wakes from upper blades in the real case.

Testing Different Numerical Schemes

A further step in the learning process had to be taken, the use of a first order numericalscheme for the divergent term of the discretized equations for stability purposes wasimportant in the beginning of the process, to get a simulation running, nevertheless, thenecessity of higher accuracy results led to the next test, which was to verify the influence


Figure 4.3: The history of convergence for the simulation P3.

of a second order numerical scheme in the simulation parameters. A simulation with thesame boundary and initial conditions as P2 was created changing the divSchemes boundedGauss upwind (first order) to Gauss linear (second order) and it was named P3.

The residuals for simulation P3 are shown in Figure 4.3, and by comparing with theresults obtained in P2 (Figure 4.1b), it is possible to see that, after 3000 iterations theparameters converged and started to oscillate around a constant value while in P2 pa-rameters kept decreasing, the values obtained for P3 were much higher than expected toconsider the simulation was rigorously converged, it was postulated that the higher accu-racy allowed a better description of the flow on the trailing edge than the one obtainedwith the lower accuracy which was damping the instabilities on the flow near the trailingedge, and the solution now was beyond the steady state simulation capabilities.

The results for pressure distribution and wake loss for Cases P2 and P3 were comparedwith results obtained experimentally by Stadtmuller [32] and numerically using compress-ible DNS by Sandberg et al. [16], and incompressible DNS by Wissink [15]. The resultsfor the Cp distribution are shown in Figure 4.4, and for the wake losses in Figure 4.5. Inthe Cp plot, a good agreement is found on the pressure side for all results, but a cleardifference is observed on the suction side when comparing the incompressible simulationswith both the compressible simulation and experimental results, as expected, the com-pressible results predict a higher pressure on that region, since part of the flow kineticenergy is converted into fluid compression.

On the suction side, in the favorable pressure region, cases P2 and P3 followed the samebehavior as incompressible DNS, with P2 showing a better agreement than P3 by almostmatching the incompressible DNS results up to almost 60% of the chord. In the adversepressure region cases P2 and P3 also presented a reduction of pressure comparable withthe DNS result up to x/L ≈ 0.85, again with a better agreement of P2 even if not as goodas in the favorable pressure region. After x/L = 0.85 a divergence of the RANS and DNSresults is observed, the pressure of the DNS simulation shows an increase, Reference [15]states that a jump on the Cp curve is observed at x/L ≈ 0.93 that corresponds to theformation of a separation bubble that is “permanently present near the trailing edge”, thatincrease is not observed in cases P2 and P3, the Cp on case P2 shows a continuous decrease,while P3 shows a waggly behavior that varies as the solver calculates new iterations. Thisbehavior can be related to instabilities caused by the adverse pressure gradient close tothe trailing edge, these instabilities were not captured by Case P2 that uses a lower ordernumerical scheme that is more dissipative.


(a) Case P2

(b) Case P3

Figure 4.4: Pressure coefficient distribution on the T106 blade for Cases P2 and P3,steady simulation, Reynolds 60,000.


(a) Case P2 (b) Case P3

Figure 4.5: Wake loss at 40% chord for the T106 blade, Cases P2 and P3, steady simula-tion, Reynolds 60,000.

Figure 4.6: Friction coefficient on the suction side of the T106 blade for Cases P2 and P3.

The skin friction is plotted in Figure 4.6, and it shows for simulations P2 and P3that a separation occurs after x/L = 0.8. In Case P2, the position of separation can bedetermined (x = 0.837) since no reattachment is observed and in P3 the same wagglingseen on Cp plot is observed indicating recirculation on the region and the separation mustbe determined with statistical methods, P3 also shows a separation bubble on the leadingedge. The divergence of results from DNS to RANS can be explained from the fact thatthat the flow on the DNS simulation is laminar in most of the blade, and a separationbubble is formed near the trailing edge when it separates a transition to turbulent flowtakes place and the boundary layer reattaches, and in the RANS simulation, by simulatinga turbulent flow all around the blade it would not capture the formation of the separationbubble on the trailing edge, when the flow separates no reattachment is observed.

When comparing the wake loss calculated in Case P2, it shows a good agreementin shape with compressible DNS and experiment, although a slight shift in the pitchwisedirection that could be justified by the differences on the wake formed at the trailing edge.The wake loss for Case P3 shows narrower defect region with changes in the direction


within the curve, indicating an unsteady behavior, and with a higher peak in a higherpitchwise position than the compared results. In DNS and experimental results, theformation of a separation bubble can contribute to the formation of a thicker wake. InCase P2, the use of a lower order numerical scheme can cause additional dissipation thatcan lead to a wider wake when compared with Case P3.

This effect can also be observed in the velocity field shown in Figure 4.7; Case P3 showsa higher contrast on the wake. On Case P2, the region immediately after the trailing edgehas a low velocity indicated by a blue region. As one keeps moving through the wake itis possible to see that the velocity increases, but the core of the wake still maintains alower velocity than the surrounding fluid, the difference of velocities keeps reducing bythe diffusion of the vorticity on the wake as one keeps moving through the wake, thiscan be noticed by the fact that the green colored core of the wake is slowly fading andbecoming more orange. It is possible to distinguish four wakes crossing the domain, thelast one is very difficult to see since the wake had traveled a long distance. On Case P3,the four wakes are easier to distinguish it is also possible to see that, in P3, the wake iswavy due to the instabilities that can be seen forming close to the trailing edge on thesuction side of the blade, where it is possible to see four different structures forming. Thetrailing edge shows the formation of a low velocity region that is not straight as seen onthe figure of Case P2.

The instabilities observed in Case P3 indicated the existence of a non steady behavioron the flow in study, therefore, the execution of an unsteady solver would be indicatedto better capture this behavior. Nevertheless a few extra tests were performed withsimulation P3 as the baseline, as a better knowledge of OpenFOAM setups was needed.

Testing CPU-Time Parameters

The time to perform a numerical simulation is crucial in CFD and it can be affectedby a different number of factors, from the type of algorithm used to solve a problemto equipment characteristics. Three different aspects were tested in these preliminarystudies: a different number of processors, the solution of the same problem with a coarsermesh and the use of a different under-relaxation factor.

In OpenFoam, it is possible to parallelize the computations by decomposing the meshin parts that will be solved by different CPUs. A test was performed by running the CaseP3 with 2, 4 and 8 processors, using the simple geometric decomposition. The resultsare presented in Table 4.1 they show the execution time of the simulations with differentnumber of CPUs until reaching 10,000 iterations. As expected, it shows the reduction ofthe execution time with the increase of the number of processors, however, the reductionis non linear and depends on how the meshes were divided and on the processes beingrun by the operating system.

From the results obtained it would be expected that running two simulations in parallel,with 4 CPUs each, would be faster than two serial simulations with 8 CPUs, however,it was observed that when using more than one CPU, the system continuously switchesthe CPUs in use. This process tends to stop when there are CPUs available, but whenall CPU’s are taken it keeps changing the CPUs and this is probably what make theexecution of simulations in parallel slower than in serial. It was also observed that usingOpenFOAM’s simple decomposition, the number of cells per processor was not optimizedwith some divisions having a considerable greater amount of cells than others.

It was also found from all these test cases that in OpenFOAM there is a relation betweenthe time per iteration and the residuals of that simulation, in a sense that solutions closer


(a) Case P2

(b) Case P3

Figure 4.7: Velocity magnitude field for Cases P2 and P3, steady simulation, Reynolds60,000, velocity in m/s.


Figure 4.8: The history of convergence for the simulation P4, steady simulation, coarsemesh, Reynolds 60,000.

to convergence presented a shorter time per iteration, then when the problem started torun from an initial condition far from solution, such as a constant field.

A study to compare the execution time of the simulation in meshes with differentrefinements was also performed. The same parameters of the simulation P3 were setusing the coarse mesh described in Section 3.1 and it was named Case P4. The history ofconvergence of Case P4 is shown in Figure 4.8. It can be seen that the solution evolved tolower residuals than Case P3 in the converged solution. With all the quantities reachingthe convergence after 4,500 iterations. Comparing the time to run 10,000 iterations thecoarser mesh took 41% of the time spent to solve the same problem with the refined mesh,as shown in Table 4.1.

The Cp distribution and wake loss for case P4 are shown in Figure 4.9. Comparing theCp distribution with the DNS and experimental results it is seen that a behavior similarto case P2 and P3 was obtained on the pressure side and the favorable pressure gradientregion but with a worse agreement than both cases. On the adverse pressure region itpresents a constant increase in pressure similar to Case P2. The coarser grid used inP4 also have a tendency to diffuse out the transient features that were captured in caseP3 with a finer mesh. The diffusive behavior of the coarser mesh can also be seen bycomparing the wake loss curves for the three cases. Cases P2 and P4 have a lower andwider peak whereas in P3 a sharper peak is obtained.

Figure 4.10 shows the friction coefficient for Case P4. It shows a separation on thetrailing edge at the position x = 0.833; it reproduces the separation bubble at the leadingedge also observed in P3.

As it was discussed before, the definition of the under-relaxation factors plays an impor-tant role in the stability of the Simple algorithm, it can also lead to a faster convergenceof the solution and reduce the total time spent in solving a problem. A test with differentunder-relaxation factors for the pressure was performed, using Case P4 as the benchmark,the simulations named P5 and P6 were created only changing the under-relaxation factorto 0.7 in Case P5 to 0.5 in Case P6, the convergence history is presented on Figure 4.11.

A convergence behavior in the initial iterations of Case P5 can be seen, with a completechange in the tendency around 1500 iterations, from that moment on the values predictedfor the pressure on the numeric field started to increase to non physical values. Case 6have shown a convergence very similar to Case P4. It is important to note that eachproblem has a specific set of ideal under-relaxation factors an the test was performed only


(a) Cp distribution (b) Wake loss at 40% chord downstream

Figure 4.9: Pressure coefficient and wake loss for Case P4, steady simulation, coarse mesh,Reynolds 60,000.

Figure 4.10: Friction coefficient on the suction side of the T106 blade for Case P4, steadysimulation, coarse mesh, Reynolds 60,000.


(a) Case P5 (b) Case P6

Figure 4.11: The convergence history for the Cases P5 and P6.

to have an idea of the numerical behavior shown when a wrong relaxation factor is chosen.

4.1.2 Unsteady Simulations

The execution of steady simulations provided a good starting point for the work andprovided the skills necessary to set simulations in OpenFOAM, however, the poor conver-gence obtained using a steady solver in the refined mesh, demanded a solver that coulddeal with the unsteadiness of the flow.

A simulation using pisoFoam was set taking the solution field of the steady simulationas the initial condition. The time derivative was set as an Euler scheme, and the timestep was set as 0.0005 seconds, which gave a maximum Courant number of 0.854. A timeaverage of the flow variables was taken on all unsteady simulations.

The convergence history for the simulation is shown at Figure 4.12, and it is possible tosee that a proper convergence was achieved, with much smaller residuals when comparingwith Figure 4.3. The curve for the pressure residuals is longer because it shows the initialand the corrected pressure in the same curve. A survey on the numerical values of theflow variables was made, comparing the averaged flow variables with the instantaneousflow variables, and they presented the same values as the instantaneous variables meaningthat the unsteady simulation was showing a steady flow behavior.

From the results obtained with the unsteady simulation, it was decided that the un-steady solver would be used for the remaining cases of the thesis, since an unsteady flowbehavior on the trailing edge and the wake could be expected for this flow from what isdescribed in the References [15] [16] [32], moreover OpenFOAM’s steady solver has notbeen able to deal with the instabilities using the Spalart-Allmaras turbulence model andwith the use of an unsteady solver a time average could be used in unsteady cases tocalculate the parameters in an averaged flow.

4.2 Assessment of Turbulence Models on the T106A

Blade

After the execution of the preliminary studies, with an improved understanding ofthe CFD solver, it was possible to start addressing the main objectives proposed to the

4.2. ASSESSMENT OF TURBULENCE MODELS ON THE T106A BLADE 49

Figure 4.12: Convergence history of the unsteady simulation with the Spalart-Allmarasturbulence model.

present thesis. Four different turbulent models were tested and compared against DNSand experimental results. Since the flow Reynolds number is relatively low, laminar sim-ulations were also performed. For the Reynolds number of 60,000 a total of five differentsimulations were performed for each of the two mesh refinements. For the Reynolds num-ber of 150,000 only three turbulence models and the laminar flow were tested in a coarsemesh, due to time restrictions.

4.2.1 Flow at Reynolds Number of 60,000

Cases Setup

For each of the different turbulence models tested, different setups had to be prepared,because different turbulence parameters are calculated in each model, nevertheless, severalquantities were maintained. For example, to set the initial conditions of a case, a resolvedfield from a steady simulation was taken, that way the solver would start the unsteadysolution with a better guess of the pressure and velocity fields, all cases used the sameinitial condition for the velocity and pressure fields; for the turbulence variables, uniformfields were used as initial conditions, and they were different for each case. A table withthe different boundary conditions for each case is presented on the Appendix 2.

With the exception of the baseline-EASM simulation in the refined mesh, all other caseshave run without any issues. When solving the baseline-EASM turbulence model case, inapproximately 4 seconds of simulation time, a region of low pressure started to developat the wake region close to the outlet, as shown in Figure 4.13, that led the simulationto instabilities and an eventual crash. Since the initial condition for the simulation wasa previous solution of a steady simulation, it was postulated that some unsteadiness thatdeveloped from the difference of the solution calculated and the initial solution, like astarting vortex, was formed and it was moving out of the domain, but as the wake isdirected to the bottom boundary, it had to cross the domain four times before leaving it,the disturbances in the wake must have amplified that vortex and a nonphysical solutionwas developed. The problem was circumnavigated by starting the solution with a first-order scheme, and after approximately 6 seconds of simulation time it was switched tosecond order. From then, no further instabilities were encountered.


Figure 4.13: Numerical instability developed while running the baseline-EASM turbulencemodel.

The Reynolds number for each case is different, as it was explained in Section 3.2.3;it is based in the average inlet pressure, and that value varies slightly from case to case,the values for each simulation are shown in Table 4.3, the total time spent to run eachsimulation is also shown in the same table, this value is only for reference since cases wererun using different number of processors depending on the availability of the computerresources.

All simulations were run using the unsteady solver pisoFoam, and, after an initialtransient that varied for each simulation, the results were time averaged. All simulationswith a turbulence models have evolved to a stable solution and the results shown are basedin averages presented a converged behavior. In the laminar case, due to the instabilitiesdeveloped in the flow, the simulations demanded the execution of longer simulationsin order to reach a converged average. On Table 4.3 it can be seen that the laminarsimulations took the larger times of execution.

It was observed that averaged quantities measured on the blade reached a convergence,however, quantities measured on the wake were still showing small variations in subsequentresults. Figure 4.14 shows the average wake loss measured for the laminar case in a coarsemesh for six different time intervals, representing a simulated time of 6 seconds and 12,000iterations apart. Eventually, due to time constraints the laminar simulations had to beinterrupted and the results are presented with the average obtained in the last time step.

Pressure Coefficient Distribution

Figure 4.15 shows the pressure distribution for the coarse and refined meshes respec-tively, for all five different simulations. A zoom-in of the trailing edge region is shownin Figure 4.16. Regarding the mesh refinement, from the figures it is possible to saythat both meshes have presented very similar results. The most remarkable differencewas between the laminar simulations on the trailing edge region, where the coarser meshpresented a lower pressure peak on the separation region.

On the pressure side of the blade, a good agreement was observed for all simulationsfrom the leading edge to the trailing edge, on the trailing edge region, however, a closeragreement with the incompressible DNS results is seen. The ϕ-α-EASM model presentedan abnormal stagnation pressure, reaching the value of 1.022 for the refined mesh and


Figure 4.14: Differences on the average of the wake in subsequent time intervals, laminarsimulation, coarse mesh, Reynolds 60,000.

Table 4.3: Description of the cases simulated with Reynolds number and total time,Reynolds 60,000.

Simulation Mesh Turbulence Model Reynolds number Time [s]1 Coarse none 61,208 216,3952 Coarse Spalart-Allmaras 59,969 12,7753 Coarse k-ω 59,714 10,4154 Coarse baseline-EASM 59,818 76,4695 Coarse ϕ-alpha-EASM 60,328 92,7136 Refined none 61,009 183,1127 Refined Spalart-Allmaras 60,034 48,9688 Refined k-ω 59,699 92,2849 Refined baseline-EASM 59,854 175,13110 Refined ϕ-alpha-EASM 60,263 159,775

Total 1,068,037


(a) Cp distribution coarse mesh

(b) Cp distribution refined mesh

Figure 4.15: Pressure coefficient distribution, Reynolds number 60,000.


(a) Cp distribution coarse mesh (b) Cp distribution refined mesh

Figure 4.16: Zoom-in Pressure coefficient distribution, Reynolds 60,000.

1.019 for the coarse mesh.

On the suction side, the simulation curves have correctly followed the trends of theDNS and experimental results, showing a closer agreement on the favorable pressuregradient region with the incompressible DNS results. The ϕ-α-EASM model matchesthe DNS results up to x/L ≈ 0.58; all the other turbulence models slightly underpredictthe DNS Cp value, and the laminar simulation slightly over predicts it. On the adversepressure gradient region, all turbulence models show a pressure recovery up to the trailingedge; they are not capable of capturing the intermittent separation bubble in that region,as described by Wissink, it is characterized by a sudden decrease on the Cp seen onincompressible DNS and experimental data after x/L ≈ 0.93, however, when simulatingthe flow with incoming disturbances, the author found a pressure recovery that increasesuntil it reaches the trailing edge. The laminar solution captures a recirculating regionclose to the trailing edge, with a periodic vortex shedding, these results present a goodresemblance to the compressible DNS and experimental results.

The observed Cp overprediction of the laminar simulation and the underprediction ofturbulent models simulation at the suction side, is connected to the fact that a turbulentboundary layer is thicker than a laminar boundary layer, therefore, the displaced bodyseen by the flow is thicker on the turbulent flow, leading to higher velocities and a lesserpressure. The fact that the DNS results occur in the middle of the laminar and turbulentmodels simulations can be related to the fact that a separation bubble is found on theleading edge causing a disturbed flow but it is still not turbulent, and probably it has athicker boundary layer than the pure laminar simulation and thinner than the turbulentmodels simulations.

Friction Coefficient Distribution and Boundary Layer

In Figure 4.17, the friction coefficient is shown on the suction side of the blade forthe coarse and refined mesh respectively, for the five different simulations, where theyare compared with the compressible DNS results. A separation bubble on the leadingedge is predicted by the laminar simulation and the turbulence models k-ω and Spalart-Allmaras. For the k-ω model, a larger separation bubble is predicted with the refinedmesh. It is also noteworthy the waggly behavior on the ϕ-α-EASM model on the region0.05 < x/L < 0.10, seen only on the refined mesh.


(a) Coarse mesh

(b) Refined mesh

Figure 4.17: Friction coefficient on the suction side of the T106 blade, Reynolds number60,000.


On the favorable pressure gradient region, none of the simulations was able to matchthe increase observed on DNS results, they all show a wavy but rather constant value upto x/L ≈ 0.57. A very close agreement was found between the laminar simulation andthe Spalart-Almaras turbulence model.

On the adverse pressure gradient region, a better agreement with DNS results is seenby the laminar simulation and the Spalart-Allmaras turbulence model, both showing atrailing edge separation. The other turbulence models fail to predict separation at thetrailing edge. The separation points for each simulation is shown on table 4.5. TheSpalart-Allmaras model overpredicts the separation point of the DNS simulation. Thelaminar simulation in the coarse mesh slightly underpredicts the DNS separation pointand in the refined mesh it almost matches the DNS result.

From these results, four different points were chosen on the suction side to plot thevelocity and turbulent kinetic energy profiles, they were extracted from the refined meshes.The first position was close to the leading edge where the friction coefficient plot haveshown the formation of a separation bubble for some of the simulations (x/L = 0.017), thesecond point is in a region of favorable pressure gradient (x/L = 0.352), the third pointis already in the separation region predicted by laminar simulation and Spalart-Allmarasmodel and the fourth position is close to the trailing edge (x/L = 0.970). The results areshown in Figure 4.18 and 4.19.

At the first position (Figure 4.18a), the laminar simulation and the turbulence modelsk-ω and Spalart-Allmaras results are very close to each other and show a separated profile.The profile generated in the simulation using the baseline-EASM and the ϕ-α-EASM areresisting to the formation of the separation bubble at the leading edge, they also showa much fuller profile. In the second position (Figure 4.18b), it is possible to see thatthe turbulence models exhibit a fuller profile then the laminar simulation. The Spalart-Allmaras model almost matches the laminar profile. The ϕ-α-EASM model shows thefuller velocity profile. In the third and fourth position (Figures 4.18c and 4.18d), it ispossible to see the separated profiles of the laminar simulation and the Spalart-Allmarasmodel, and the effect of the adverse pressure gradient leading to a less fuller profile onthe other simulations.

The turbulent kinetic energy profiles shown the expected behavior, they increase withthe shear stress up to a maximum point and then all go to zero at the wall, as a resultof the damping effect on the turbulence fluctuations. The ϕ-α-EASM in all stations hasshown the peak on a lower height than the other turbulence models, which agrees withthe fuller velocity profile presented by the model.

Analysis of the Wake

Figure 4.20 shows the total pressure losses measured 40% downstream of the trailingedge, for the coarse and refined mesh. The variable y∗ is a non-dimensional length inthe pitchwise direction, from the suction (zero) to the pressure side (one). The resultshave shown that different meshes have a different effect on the wake. The k-ω model wasthe one that maintained a very similar wake for both meshes, with a peak slightly higherand pitchwise down on the refined mesh, the ϕ-α-EASM model also presented a largerpeak. The other models have presented a smaller peak in the refined mesh. A strongeroscillation on the values outside of the defect region was observed for the coarser mesh.

In general, it can be seen that the turbulence models predicted the location of the peakvalues of the wake loss lower than the DNS and experimental results, also the defect regionis thinner with a larger peak for the RANS simulations, which can be translated into aless disturbed wake. The result is in agreement with what was found in Reference [16]


(a) x/L = 0.017

(b) x/L = 0.352

(c) x/L = 0.856

(d) x/L = 0.970

Figure 4.18: Velocity profiles for different turbulence models at four different points ofthe suction side, refined mesh, Reynolds 60,000.


(a) x/L = 0.017

(b) x/L = 0.352

(c) x/L = 0.856

(d) x/L = 0.970

Figure 4.19: Turbulent kinetic energy profiles for different turbulence models at fourdifferent points of the suction side, refined mesh, Reynolds 60,000.


(a) Coarse mesh (b) Refined mesh

Figure 4.20: Wake losses 40% chord downstream of trailing edge, Reynolds 60,000.

where the cases with higher inlet turbulence also caused a movement of the peak to thepressure side. The formation of a separation bubble on the trailing edge is also reported,this contributes to a higher perturbation on the wake thus a thicker defect region. TheSpalart-Allmaras model, was the only turbulence model to predict separation, however,the wake formed in the solution is quite stable showing a steady behavior.

Since the flow is at a low Reynolds number, it was expected that turbulent models,by simulating a turbulent flow in all over the blade, would not present a good agreementwith DNS results specially in the wake region. As described by Wissink [15], the flow isinitially disturbed by the leading edge separation bubble, but the instabilities are dampeddue to the favorable pressure gradient, then when subjected the adverse pressure gradientthe formation of an intermittent bubble triggers the transition to turbulence. An unstableshear layer is formed and the bubble expands both upstream and downstream formingtwo rolls they are convected downstream and disappear into the wake.

A similar process would be expected in the laminar simulation, a sequence of 12 snap-shots of the vorticity magnitude at the trailing edge is presented on Figure 4.21. There isa simulated time gap of 0.002 seconds between each figure. In the first column of frames,it is possible to observe that the flow formed close to the trailing edge of the laminarsimulation also shows the formation of a pair of rolls of recirculating flow, but in a posi-tion downstream to the one seen on Reference [15]. The rolls are convected downstreamand interact with a rotating structure that is being developed at the trailing edge fromthe pressure side of the blade, as it rolls to the opposite side it weakens the vortices thatare shed to the wake. In a latter moment, the structure developed in the pressure side islarge enough to contribute to push a structure on the suction side upstream, forming abig roll that is shed to the wake. In the last column a third different mechanism can beseen, as a roll from the suction side moves past a still “weak” roll from the pressure sideand strengthen it, resulting in a twin vortex being shed to the wake.

The previous description does not try to address to the complex problem of describingthe multitude of mechanisms that happen on that particular flow condition. It just aimedto show how an intermittent flow on that region can generate a highly disturbed wake,and that explains why the wake loss for the laminar flow has a thicker defect region.


Figure 4.21: Sequence of snapshots of the flow vorticity magnitude, at an interval of0.002 seconds of simulated time. Sequence presented from top to bottom, coarse mesh,Reynolds 60,000.


Table 4.4: Description of the cases simulated with Reynolds number and total time,Reynolds number 150,000.

Simulation Mesh Turbulence Model Reynolds number Time [s]1 Coarse none 152,980 121,1382 Coarse Spalart-Allmaras 148,766 16,5253 Coarse k-ω 148,528 14,2484 Coarse ϕ-alpha-EASM 149,240 32,477

Total 184,388

4.2.2 Flow at Reynolds Number of 150,000

Cases Setup

Once again a different setup had to be adjusted to the simulations at a differentReynolds number. The inlet velocity was increased maintaining the flow with the sameangle of incidence of 45. To reach the desired Reynolds number, the velocity was set to1.38m/s. To initialize the velocity and pressure fields for all unsteady simulations withdifferent turbulence models, a steady simulation with the Spalart-Allmaras model was ranuntil convergence and the fields where used as initial conditions. Again, all simulationswere run using pisoFoam, with the results averaged after the initial transient. The otherinitial conditions are described in the Appendix 2 and Table 4.4 shows all the cases withthe exact Reynolds numbers and time of the simulations performed.

Pressure Coefficient Distribution

Figure 4.22 shows the Cp distribution for the tested turbulence models and the laminarsimulation. All the results have shown a good agreement on the pressure side of theblade. The laminar simulation presented an oscillation of the values in the region of0.5 > x/L > 0.7, when looking at the instantaneous velocity field (Figure 4.23), at thepressure side the region of low velocity formed just after the stagnation point presents awavy pattern different from the continuous pattern seen before e.g., Figure 4.7, these wavesare convected and impinge on the blade at approximatelly the region of 0.5 > x/L > 0.7.

On the suction side, the initial region up to x/L = 0.05; a diversified behavior is ob-served in the different simulations with the laminar simulation reaching a higher pressurein a short space. The Spalart-Allmaras model is the one that takes longer to reach thepressure peak in that region.

On the favorable pressure gradient region, all turbulence models present very close re-sults, and the laminar simulation shows the same behavior with a higher pressure. Movingto the adverse pressure gradient, the turbulence models have presented an even betteragreement among themselves and the laminar simulation maintained a higher pressure,but with a reduced gap. The pressure recovery on the laminar simulation was not asuniform as the other simulations and does not achieve the same level at the trailing edge.An intense oscillation on the curve is seen after x/L > 0.7. An interesting separationoccurs on this particular case, the adverse pressure is acting on the laminar boundarylayer and eventually it separates forming a region of recirculating flow that is convecteddownstream. Differently from the flow that separates in an airfoil in the free stream, theexistence of a cascade of blades the flow coming from the pressure side of the upper blademaintains the separated flow attached to the blade and lead to the formation of rolls


Figure 4.22: Pressure coefficient distribution, Reynolds number 150,000.

(a) Velocity field

(b) Detail on pressure side

Figure 4.23: Instantaneous velocity magnitude field, velocity in m/s, Reynolds number150,000.


Figure 4.24: Friction coefficient on the suction side of the T106 blade, Reynolds number150,000.

on the boundary layer region that are convected to the wake. This kind of “attached-separated” boundary layer leads to an unsteady average as shown in the graph of the Cpdistribution.

Very close to the trailing edge there is a region of pressure drop, in that region acombined effect of the vortex shedding mechanism from the convected rolls on the suctionside with the shedding associated with the shear layer separating from the pressure side,contributes to a highly disturbed flow and lead to the formation of that region of lowpressure.

Friction Coefficient Distribution

Figure 4.24 shows the friction coefficient for the performed simulations on the suctionside at Reynolds number 150,000. On the leading edge the ϕ-α model was the onlyone that did not predict a leading edge separation, showing the same behavior seen forlower Reynolds number. The laminar simulation and the k-ω models predicted separationbubbles at approximately the same size, which was slightly smaller and upstream whencompared with the lower Reynolds case. The Spalart-Allmaras model predicted a bubbletwice the size the ones obtained in the other simulations.

On the favorable pressure gradient region, a higher skin friction was predicted by the k-ω model, showing a different trend from the lower Reynolds number where both turbulencemodels predicted approximately the same skin friction. Once again the laminar simulationhave shown close agreement with the Spalart-Allmaras turbulence model. On the adversepressure region all turbulence models predict a reduction on the skin friction showinggood agreement after x/L > 0.9. It is possible to see that from the curves no trailing edgeseparation was predicted by the turbulent models. The laminar simulation presented theirregular behavior associated to the separation after x/L = 0.67, it is interesting to notethat even if a separation occurs, the average skin friction never gets bellow zero, exceptfor a spike at x/l = 0.959.

Analysis of the wake

Figure 4.25 shows the total pressure losses measured 40% downstream of the trailingedge, for the flow at Reynolds number 150,000. In general, the curves obtained are very


Figure 4.25: Wake losses 40% chord downstream of trailing edge, Reynolds number150,000.

Figure 4.26: Vorticity at the trailing edge of the profile, laminar simulation at Reynoldsnumber 150,000.

similar to the ones obtained for lower Reynolds number, the overall result show a reductionon the peak and a movement of the defect region pitchwise in the direction of the pressureside. The laminar simulation curve presents a narrow defect region, similar to the othersimulations, however, instead of stabilizing at zero outside of the defect region it maintainsa higher value which agrees with the highly disturbed wake seen on Figure 4.23.

Figure 4.26 shows the vorticity at the trailing edge of the profile, it is possible to see theformation of three different recirculating structures on the suction side, they are convecteddownstream and shed to the wake, the formation of a recirculating region at the trailingedge from the pressure side can also be seen. The shedding happens in a higher frequencywhen comparing with the case of lower Reynolds number, more vortical structures witha higher intensity can be seen on the wake.


Table 4.5: Separation points calculated by the different turbulent models.

Simulation Mesh Turbulence Model Reynolds Separation SeparationNumber L. E. [x/L] T.E. [x/L]

1 Coarse none 61,208 0.014 - 0.032 0.7712 Coarse Spalart-Allmaras 59,969 0.014 - 0.031 0.8343 Coarse k-ω 59.714 0.012 - 0.033 n/a4 Coarse baseline-EASM 59,818 n/a n/a5 Coarse ϕ-alpha-EASM 60,328 n/a n/a6 Refined none 61,009 0.013 - 0.032 0.7847 Refined Spalart-Allmaras 60,034 0.013 - 0.032 0.8338 Refined k-ω 59,699 0.015 - 0.028 n/a9 Refined baseline-EASM 59,854 n/a n/a10 Refined ϕ-alpha-EASM 60,263 n/a n/a11 Refined laminar 152.980 0.011 - 0.026 0.67012 Refined Spalart-Allmaras 148,766 0.007 - 0.053 n/a11 Refined k-ω 148,528 0.009 - 0.030 n/a11 Refined ϕ-alpha-EASM 149,240 n/a n/a

Chapter 5

Conclusions

Based on the results presented in the previous chapter it is possible to say that theproject has accomplished its proposed objectives and draw some conclusions regardingthe methodology employed, the results obtained and suggest further work to verify someof the questions raised by the present research.

Conclusions on the Methodology

The development of the project using a CFD tool that was not previously known bythe author, presented an interesting challenge. By being Unix based and open source,it takes the user away from the convenient world of visual tools and Windows based all-in-one packages, to a script based program and command line typing approach. Despitethe learning curve being steep, it was a rewarding experience. The software allows theuser to have a better control of the simulations being executed, since there is unrestrictedaccess to the source code, and modifications can be implemented and compiled to suitthe demands of the user.

Regarding the whole CFD problem cycle in the current project, from the mesh gener-ation to post-processing, it can be said that OpenFOAM, by not having a well developedmesh generating utility with a graphic interface, cause that particular step to have alonger time and demand more effort that was expected and required the use of externalsoftware to generate an useful mesh. The difficulties found in this crucial moment couldhave posed a risk to the entire project, since, without a mesh, it is impossible to obtainresults.

Moving to the actual solving of the problem, several difficulties were encountered inthe process due to the reduced amount of information made available by OpenFOAMdevelopers, fortunately good feedback and valuable information on particular problemswas found on online discussion forums and third party online courses [38].

The upgrade of the baseline-EASM and ϕ-α-EASM turbulence models to OpenFOAMversion 2.2.0, was fundamental to the project, since that is the current version available fordownload and that particular work, performed by Jack Weatheritt, must be acknowledged.Without that upgrade, no simulations with these models could have been performed.

The post-processing in OpenFoam has also demanded the learning of its particularities,the built in utilities to calculate derivate fluid properties had proven to be very useful.The bundle with ParaView has enhanced the post-processing capabilities, it has provento be a very efficient software that was able to execute most of the 3D data visualization.

Overall the timeframe to develop the proposed project, approximately four months,was adequate though the problems had to be overcome quickly not to compromise theresults. The support and experience of the supervisor were fundamental to avoid the waste

65

66 CHAPTER 5. CONCLUSIONS

of precious time in unfruitful pathways, nevertheless, a longer time would be interestingto allow a better investigation of the results obtained, it would also allow the executionof additional test cases.

Conclusion on the Results

Numerical simulations and turbulence modeling are two complex subjects with manyparticularities that can be addressed. The current project was about setting up a simula-tion in a CFD solver and test different turbulent models based on two different modelingstrategies, the classical Boussinesq approximation and the modern Reynolds stresses mod-eling.

The preliminary results obtained in the first part of the work were valuable to show thereliability of the solver, it also gave the experience to deal with eventual troubleshootingon the second part of the project.

At the lower Reynolds number setup, ϕ-α-EASM presented a very good agreementin the region of favorable pressure gradient, where it almost matched the benchmarkdata on the Cp curve. The other turbulence models did not presented the same goodagreement with experimental and DNS results. Regarding separation prediction, theSpallart-Allmaras model, was the only turbulence model to predict it, a better descriptionof the physical phenomenon of separation was obtained when solving the flow with alaminar solver, a remarkably close prediction of the leading edge separation bubble andthe trailing edge separation point was obtained. The fact that turbulent models assumea turbulent flow throughout the blade does not correspond to what happens in practice.

In general, the difference on the refinement of the mesh did not presented a considerabledifference in the results of pressure distribution and friction coefficient, mostly because therefinement on the direction normal to the wall was kept constant, hence a good descriptionof the boundary layer was maintained. A diffusive behavior was also found on the coarsemesh when simulating the flow with a steady solver. The coarse mesh has also givendifferent results on the wake loss curves.

At the higher Reynolds number, despite the inexistence of a benchmark result to com-pare, a better agreement between the turbulence models has been found both in the Cpdistribution as in the wake loss curves. In the prediction of the friction coefficient, a veryclose agreement was found between the laminar simulations and the Spalart-Allmarasturbulence model in both Reynolds numbers.

An interesting behavior was found with the laminar simulation at a higher Reynoldsnumber, where the separated flow remained attached on the blade due to the cascadeeffect, resulting in an intermittent Cp profile that in time average presented approximatelythe same results as simulations with turbulent models.

Further Work

Despite being a well known test case, the study of the flow on a T106 blade is verycomplex, and a further investigation of the flow could bring interesting results. A logicalfirst step would be the investigation of the flow with incoming disturbances that to add alayer of reality on the study, since real LPTs are normally subjected to a highly disturbedflow. In this type of simulation, there is a good chance that the results with turbulencemodels would give a better agreement with DNS and experimental results. The simulationof the flow using compressible simulations would also give additional results to comparewith experimental and compressible DNS results.

67

The possibility of testing a composite model to simulate the flow as laminar up tothe separation point and than switch to a turbulence model simulation could also beattempted to try to increase the accuracy of the RANS modeling, also the use of a hybridLES/RANS simulation could lead to a substantial decrease in simulation time of the DNSsimulations.

Another further step that could be attempted would be the simulation in a tridimen-sional mesh, where the modern turbulence models might probably present better resultsthan the linear turbulence models.

68 CHAPTER 5. CONCLUSIONS

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[5] Liebeck, R. H., “Design of Subsonic Airfoils for High Lift”, J. Aircraft, Vol. 15, No.9, 1978, pp. 547-561

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[9] Wilcox, D.C, “Turbulence Modeling for CFD”, 3rd Edition.

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http://www.formula1.com/inside_f1/rules_and_regulations/sporting_regulations/8713/

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[14] Weinmann, M., “Simulation Strategies for Complex Turbulent Flows”, PhD Thesis,University of Southampton, 2011.

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[16] Sandberg, R. D., Pichler, R., Chen, L., “Assessing the Sensitivity of Turbine CascadeFlow to Inflow Disturbances Using Direct Numerical Simulation”. 13th InternationalSymposium for Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity in Turbo-machinery (ISUAAAT), Tokyo, JP, 11 - 14 Sep 2012.

[17] Houghton, E. L., Carpenter, P.W., “Aerodynamics for Engineering Students”

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[24] Hoerner, S. F., Borst, H. V., “Fluid-Dynamic Lift”, Hoerner Fluid Dynamics, Brick-town New Jersey, 1975.

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[28] Versteeg, H. K., Malalasekera, W., “An Introduction to Computational Fluid Dynam-ics, The Finite Volume Method”, 2nd Edition, Pearson Education Limited, Harlow,UK.

[29] Launder, B. E., Reece, G. J., Rodi, W., “Progress in the development of a Reynolds-stress turbulence closure”, Journal of Fluid Mechanics, vol. 68, part 3, pp. 537 - 566,1975.

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[31] Thakkar, M., “Turbulence Modeling using Hybrid RANS/LES techniques applied toa Tandem Cylinder geometry”, MSc. Thesis, University of Southampton, 2012.

[32] Stadtmuller, P. 2001 “Investigation of Wake- Induced Transition on the LP TurbineCascade T106 A-EIZ”, DFG-Verbundprojekt Fo 136/11, Version 1.0, University ofthe Armed Forces Munich, Germany.

[33] Stieger, R. D., Holis, D., Hodson, H.P., “Unsteady Surface Pressures due toWake Induced Transition in a Laminar Separation Bubble on a LP Turbine Cas-cade”.Proceedings of ASME Turbo Expo 2003 Power for Land, Sea and Air. Atlanta,Georgia, USA. June 16-19, 2003.

[34] Stieger, R. D., Hodson, H.P.,“The Transition Mechanism of Highly Loaded LP Tur-bine Blades”. Proceedings of ASME Turbo Expo 2003 Power for Land, Sea and Air.Atlanta, Georgia, USA. June 16-19, 2003.

[35] Wu, X. and Durbin, P. A., “Evidence of longitudinal vortices evolved from distortedwakes in a turbine passage. Journal of Fluid Mechanics, 446, pp 199-228, 2001.

[36] Gross, A., Fasel, H. F., “Multi-block Poisson Grid Generator for Cascade Simula-tions”, Math and Computers in Simulation 79 (3),416-428.

[37] Holohan, N., “RANS Study of the Sensitivity to Inlet Conditions of Separated Flowin 3D Asymmetric Diffusers”, MSc. Thesis, University of Southampton, 2011.

[38] Guerrero, J.,“2013 UNIGE Introductory OpenFOAM Course”, Dipartimento di In-gegneria dell Costruzioni, dell’Ambiente e del Territorio della Universita di Genoa,[online], available at: www.dicat.unige.it/guerrero/OpenFOAM_course2012.htmlAcessed on 08/08/2013.

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72 BIBLIOGRAPHY

Appendix A

Description of Turbulence Models

This appendix was devised as a reference section explaining briefly the main RANSturbulence modeling strategies and presenting the most important constitutive equationsfor modeling. It was written strongly based on references [9], [14] and [31]. An overviewof the closure problem and the solutions for modeling are presented on the diagram onFigure A.1.

The diagram tries to show the closure problem that arises when averaging the Navier-Stokes equations, and the possible ways of modeling the Reynolds stresses. One is bymodeling the Reynolds stresses by a linear relation, the Boussinesq hypothesis, the otheris by modeling the Reynolds stress equation. In either way one fundamental equation isthe differential equation for the Reynolds stress transport, it is obtained by multiplying thethe Navier-Stokes equation by the fluctuating velocities and time-averaging the product,using the “Navier-Stokes operator”:

u′iN (uj) + u′jN (ui) = 0 (A.1)

N (ui) = ρ∂ui∂t

+ ρuk∂ui∂xk

+∂p

∂xi− µ ∂2ui

∂xk∂xk= 0 (A.2)

the derived equation for the Reynolds stress can be written as follows:

∂τij∂t

+ Uk∂τij∂xk

= −τik∂Uj∂xk

τjk∂Ui∂xk

+ εij − Πij +∂

∂xk

[ν∂τij∂xk

+ Cijk

](A.3)

with,

Πij =p′

ρ

(∂u′i∂xj

+∂u′j∂xi

)(A.4)

εij = 2ν∂u′i∂xk

∂u′j∂xk

(A.5)

ρCijk = ρu′iu′ju′k + p′u′iδjk + p′u′jδik (A.6)

by taking moments of the Navier-Stokes equations, new unknowns are generated. Atthis point, the processes of deriving the Equation A.3, generated 6 new equations but also22 unknowns, therefore the problem is still not closed.

73

74 APPENDIX A. DESCRIPTION OF TURBULENCE MODELS

Figure A.1: Diagram with the main RANS closure strategies.

A.1. ALGEBRAIC MODELS 75

A.1 Algebraic Models

Algebraic models are normally calculated based on the mixing length hypothesis, whichis a theoretical model for turbulent motion, according to this model, masses of fluid withthe same x-directed momentum are aggregated and move together through the fluid. In aturbulent flow, this aggregates move in directions other than the x-direction; the mixinglength (lmix), and maintain their momentum for some time. The mixing length hypothesiswas conceived based on an analogy with the molecular viscosity definition and how it isdefined by the mean free path of the molecules. From the analogy:

τxy = νTdU

dy=

1

2vmixlmix

dU

dy(A.7)

When computing the turbulence viscosity using algebraic models, in general the char-acteristic length is the mixing length and a characteristic velocity is specified as:

vmix = C · lmix∣∣∣∣dUdy

∣∣∣∣ (A.8)

hence:

νT = l2mix

∣∣∣∣dUdy∣∣∣∣ (A.9)

these equations model νT as a function of the mixing length, that has to be determinedfor each particular flow. In wall flows it is a value proportional to the distance of thewall; for jets, wakes and mixing layers it is proportional to the width of the layer. For theboundary layer flow, the basic mixing length model presents good agreement in the regionof the Log layer and fails to provide a close agreement with measured skin friction, andin the defect layer region. A few corrections have been proposed to improve the modelpredictions in these regions of the boundary layer. For the near wall region, the mixinglength should be multiplied by the Van Driest damping function:

lmix = κy[1− e−y+/A

+0

](A.10)

in the defect layer the correction proposed by Clauser is the turbulent viscosity in theouter layer (νTo):

νTo = αUeδ∗ (A.11)

Escudier suggested that the mixing length value was limited by a constant multiplied bythe boundary layer thickness (δ):

(lmix)max = 0.09δ (A.12)

The Klebanoff intermittence factor improves the prediction on the upper layers of theboundary layer, that region is not always turbulent, but rather intermittent, it is calcu-lated as a function of the distance to the wall divided by the boundary layer thickness:

FKleb(y; δ) =

[1 + 5.5

(yδ

)6]−1

(A.13)


A.2 One Equation and Two Equation Models

A.2.1 Turbulence Kinetic Energy Equation

An evolution, from the Equation A.8, in the determination of the velocity scale of theeddy viscosity, to incorporate flow history effects on the calculation of the eddy viscosity,was proposed by Prandtl in 1945, using the turbulent kinetic energy (k) as parameter,which is defined as:

k =1

2u′iu′i (A.14)

the eddy viscosity can be calculated using a length scale (l) as:

νT = C · k1/2l (A.15)

The turbulent kinetic energy is the trace of the Reynolds stress tensor, an equation forthe transport of k can be obtained by taking the trace of the Reynolds stress equation,Equation A.3, the resulting equation:

∂k

∂t+ Uj

∂k

∂xj= τik

∂Ui∂xj− ε+

∂

∂xj

[ν∂k

∂xj− 1

2u′iu′ju′k −

1

ρp′u′i

](A.16)

still have unknowns that have to be modelled. The first and second terms on the left-hand side are the unsteady term and the diffusion term. On the right-hand side: thefirst term is the production term; the second term in the dissipation term; the terms inbrackets are the molecular diffusion, the turbulent transport and the pressure diffusionterms respectively.

In order to have a closed form of the Equation A.16, appropriate modeling for τij,dissipation term, turbulent transport term and pressure diffusion term must be carriedout. The first quantity is approximated by the Bousinessq hypothesis from Equation2.13. Considering that the dissipation term is a property of turbulence, the dimensionalanalysis shows that:

ε ∼ k3/2/l (A.17)

The last two terms are modelled by an analogy with molecular transport process. Themodelled version of the turbulence kinetic energy equation can be written as:

∂k

∂t+ Uj

∂k

∂xj= τij

∂Ui∂xj− ε+

∂

∂xj

[(ν +

νTσk

)∂k

∂xj

](A.18)

A.2.2 One Equation Turbulence Models

The early one equation turbulence models used the kinetic energy equation in conjunc-tion of the dissipation given by Equation A.17 transformed in an equality by means of aconstant:

ε = Ck3/2/l (A.19)

the turbulent viscosity can be written as a relation a characteristic length that has to bespecified and the turbulent kinetic energy:

νT = k1/2l = Ck2/ε (A.20)

A.2. ONE EQUATION AND TWO EQUATION MODELS 77

l is a mean flow scale and k is a turbulence quantity, and the proportionality given by theprevious equation is only true for equilibrium flows with the balance of the productionand dissipation.

Other one-equation models have been proposed based on parameters different from theturbulence kinetic energy. To avoid the shortcomings of the original model Spalart andAlmaras proposed modeling the equation for the eddy viscosity and it is presented asfollows.

A.2.3 Spalart-Allmaras Model

The model that will be presented is the standard Spalart-Allmaras model, which isthe one that is implemented in OpenFOAM, this model has eight closure coefficientsand three closure functions. Alternate versions of the model to correct for specific flowcharacteristics can be found in the NASA Turbulence Modeling Resource [39].

Kinematic Eddy Viscosity:

νT = νfv1 (A.21)

Eddy Viscosity Equation:

∂ν

∂t+ Uj

∂ν

∂xj= cb1Sν − cw1fw

(ν

d

)2

+1

σ

∂

∂xk

[(ν + ν)

∂ν

∂xk

]+cb2σ

∂ν

∂xk

∂ν

∂xk(A.22)

Closure Coefficients and Auxiliary Relations:

cb1 = 0.1355, cb2 = 0.622, cv1 = 7.1, σ = 2/3 (A.23)

cw1 =cb1κ2

+(1 + cb2)

σ, cw2 = 0.3, cw3 = 2, κ = 0.41 (A.24)

fv1 =χ3

χ3 + c3v1

, fv2 = 1− χ

1 + χfv1

, fw = g

[1 + cw36

g6 + cw36

]1/6

(A.25)

χ =ν

ν, g = r + cw2

(r6 − r

), r =

ν

Sκ2d2(A.26)

S = S +ν

κ2d2fv2, S =

√2ΩijΩij, Ωij =

1

2

(∂Ui∂xj− ∂Ui∂xj

)(A.27)

The source terms depend on the gradient of ν and d which is the distance to the closestsurface, hence the model predicts no decay of the eddy viscosity in an uniform stream.


A.2.4 Two Equation Models

Two equation models normally calculate the characteristic velocity by means of theturbulent kinetic energy equation and differ in the way they determine the remainingmodeling quantity, the characteristic length. It can be modeled by the transport equationof the specific dissipation rate ω, the transport equation of the dissipation per unit massε, the transport equation for the turbulence length scale or the transport equation for theturbulence dissipation time:

νT ∼ k/ω (A.28)

νT ∼ k2/ε (A.29)

νT ∼ k(1/2)/ω (A.30)

νT ∼ kτ (A.31)

each of the possible ways to determine νT are dimensionally correct. To model turbulenceproperly, it is important to have a physical insight of the phenomenon. There is no indica-tion that the turbulence viscosity depends on these parameters, hence it cannot be stateda priori that these models are capable to predict turbulence properties more accuratelythan one equation models. Kolmogorov developed the first two equation model, selectingthe specific dissipation rate to determine the length scale and postulated an equationfor the transport of ω, later other researchers worked on the model initial formulationand presented improved versions. One of the most used implementations the Wilcox k-ωadded a cross diffusion term in order to reduce the sensibility of the original model tofree stream conditions. Menter improved the model ability to predict boundary layers inadverse pressure gradients and it also has a shear stress transport (SST) mechanism tobound the over prediction of νT and is presented as follows.

A.2.5 k-ω-SST Model

Kinematic Eddy Viscosity

νT =a1k

max (a1ω,ΩF2)(A.32)

Turbulence Kinetic Energy

∂k

∂t+ Uj

∂k

∂xj= τij

∂Ui∂xj− β∗ωk +

∂

∂xj

[(ν + σkνT )

∂k

∂xj

](A.33)

Specific Dissipation Rate

∂ω

∂t+ Uj

∂ω

∂xj=

γ

νTτij∂Ui∂xj− βω2 +

∂

∂xj

[(ν + σωνT )

∂ω

∂xj

]+ 2(1− F )

σω2

ω

∂k

∂xj

∂ω

∂xj(A.34)

A.3. REYNOLDS STRESS MODELLING 79

Closure Coefficients and Auxiliary Relations:

F1 = tanh(arg41), arg1 = min

[max

( √k

β∗ωd,500ν

d2ω

),

4σω2k

CDkωd2

](A.35)

CDkω = max

(2σω2

1

ω

∂k

∂xj

∂ω

∂xj, 10−20

)(A.36)

F2 = tanh(arg22), arg2 = max

(2

√k

β∗ωd,500ν

d2ω

)(A.37)

The constants are blended via the following blending function:

φ = F1φ1 + (1− F1)φ2 (A.38)

with φ1 represent the inner constant and φ2 represent the outer constant. The constantsare:

γ1 =β1

β∗− σω1κ

2

√β∗

, γ2 =β2

β∗− σω2κ

2

√β∗

(A.39)

σk1 = 0.85, σk2 = 1.0, σω1 = 0.5, σω2 = 0.856, β1 = 0.075, β2 = 0.0828 (A.40)

β∗ = 0.09, κ = 0.41, α1 = 0.31 (A.41)

A.3 Reynolds Stress Modelling

Turbulent models based on the linear relation between the Reynolds stress tensor andthe mean strain rate tensor, limits the prediction of the Reynolds stresses to mean flowproperties, however this approach simplifies the physical description of νT , actually itdepends on flow characteristics such as: the shape and nature of solid boundaries, freestream turbulence intensity, and history effects.

A.3.1 Expansion of the Boussinesq Hypotesis

In order to improve the description of the Reynolds stresses tensor it was postulatedthat the Boussinesq hypothesis is the leading term in a series expansion of functionals asfollows:

τij = −2

3kδij + 2νTSij −B

k

ω2SmnSmnδij − C

k

ω2SikSkj

−D k

ω2(SikΩkj + SjkΩki)− F

k

ω2ΩmnΩmnδij −G

k

ω2ΩikΩki

(A.42)

the closure coefficients B,C,D,F and G are determined by applying the conditions: thetrace of τij = −2k and τxx ≈ 1/2(τyy + τzz). It is simplified to:

τij = −2

3kδij + 2νTSij −D

k

ω2(SikΩkj + SjkΩki) (A.43)

this can lead to the development of turbulence models with a non linear relation betweenReynolds stress tensor and the mean strain rate tensor.


A.3.2 Algebraic Stress Models

By working with the full Reynolds stress equation (Equation A.3), Rodi proposed anapproximation of the convective and turbulent transport term as proportional to theReynolds-stress, as follows:

∂τij∂t

+ Uk∂τij∂xk− ∂

∂xk

(ν∂τij∂xk

+ Cijk

)≈ τij

k

[∂k

∂t+ Uk

∂k

∂xk− ∂

∂xk

(ν∂k

∂xk+

1

2Cjjk

)] (A.44)

Performing approximations of the other terms of the Equation A.3, a non linear al-gebraic relation of the Reynolds stress tensor is obtained. Traditional Algebraic StressModels (ASM) provide implicit equations for the Reynolds stresses and they present diffi-culties to be solved numerically. It has also been proposed that the Reynolds stress tensorwas written as an expansion similar to the one in Equation A.42, leading to an explicitformulation and thus being names Explicit Algebraic Stress Models (EASM).

The ϕ-α-EASM Model

The ϕ-α-EASM was used in the current work. It present modifications to account forthe wall proximity. The parameter ϕ is a wall normal velocity scale, that has a behaviorsimilar to the normal Reynolds stress component close to the wall. The parameter α is ablending parameter. A transport equation is solved for these parameters:

Dϕ

Dt= (1− αp) fwall+αpfhom− ϕ

kPk+

2

k(σkνT )

∂k

∂xj

∂ϕ

∂xj+

∂

∂xj

[(ν + σϕνT )

∂ϕ

∂xj

](A.45)

l2T∇2α− α = −1 (A.46)

the term fhom and fwall are given by:

fhom = − 1

tT

(C1 − 1 + C2

Pkε

)(ϕ− 2

3

), fwall = −ϕε

k(A.47)

in the near wall region the turbulence length (lT ) and time (tT ) scales are limited byKolmogorov length and time scales:

lT = CLmax

[min

(k3/2

ε,

k1/2

√6Cν

µϕ|Sij|

), Cν

(ν3

ε

)1/4]

(A.48)

tT = max

[min

(k

ε,

0.6√6Cν

µϕ|Sij|

), Ctν

√ν

ε

](A.49)

The eddy viscosity is given by the relation:

νT = CνµϕktT (A.50)

The closure coefficients:

σk = σϕ = 1, C1 = 1.7, C2 = 1.2, CL = 0.161, Cν = 90, Ctν = 6, Cνµ = 0.22, p = 3

(A.51)

A.3. REYNOLDS STRESS MODELLING 81

The Baseline EASM

This model is based on the procedure of expanding each of the terms of the Reynoldsstress tensor with the relation of Equation A.42, leading to a more accurate prediction ofthe Reynolds stresses. The equation for the k and ω are also solved as follows:

Dk

Dt= Pk − ε+

∂

∂xj

[(ν + σkνT )

∂k

∂xj

](A.52)

Dω

Dt= γ

ω

k− βω2 +

∂

∂xj

[(ν + σωνT )

∂ω

∂xj

]+σdωmax (CDkω, 0) (A.53)

with:

CDkω =∂k

∂xj

∂ω

∂xj(A.54)


Appendix A

Boundary Conditions of NumericalSimulations

Table A.1: Boundary conditions for laminar simulation.

Boundary Velocity† PressureInlet (0.387 0.394) zeroGradient

Outlet zeroGradient 0.0Top cyclic cyclic

Bottom cyclic cyclicProfile (0.0 0.0) zeroGradientLateral empty empty

† The velocity is expressed in its cartesian coordinates

83

84 APPENDIX A. BOUNDARY CONDITIONS OF NUMERICAL SIMULATIONS

Table A.2: Boundary conditions for Spalart-Allmaras turbulence model.

Boundary Velocity† Pressure ν νInlet (0.389 0.396) zeroGradient 3.0× 10−6 ‡ calculated

Outlet zeroGradient 0.0 3.0× 10−6 ‡ calculatedTop cyclic cyclic cyclic cyclic

Bottom cyclic cyclic cyclic cyclicProfile (0.0 0.0) zeroGradient 0.0 0.0Lateral empty empty empty empty

† The velocity is expressed in its cartesian coordinates‡ With freeStream boundary condition type selected

Table A.3: Boundary conditions for k-ω turbulence model.

Boundary Velocity† Pressure k ωInlet (0.389 0.396) zeroGradient 1.5× 10−4 0.148

Outlet zeroGradient 0.0 zeroGradient zeroGradientTop cyclic cyclic cyclic cyclic

Bottom cyclic cyclic cyclic cyclicProfile (0.0 0.0) zeroGradient 0.0 1.1× 105

Lateral empty empty empty empty


Table A.4: Boundary conditions for baseline-EASM turbulence model.

Boundary Velocity† Pressure k ω f v2Inlet (0.389 0.396) zeroGrad. 1.5× 10−4 0.148 1.0 2.0

Outlet zeroGrad. 0.0 zeroGrad. zeroGradient zeroGrad. zeroGrad.Top cyclic cyclic cyclic cyclic cyclic cyclic

Bottom cyclic cyclic cyclic cyclic cyclic cyclicProfile (0.0 0.0) zeroGradient 1.0× 10−16 1.1× 105 1.0× 10−16 1.0× 10−16

Lateral empty empty empty empty empty empty


Table A.5: Boundary conditions for ϕ-α-EASM turbulence model.

Boundary Velocity† Pressure k ω f v2Inlet (0.389 0.396) zeroGrad. 1.5× 10−4 0.148 1.0 2.0

Outlet zeroGrad. 0.0 zeroGrad. zeroGradient zeroGrad. zeroGrad.Top cyclic cyclic cyclic cyclic cyclic cyclic

Bottom cyclic cyclic cyclic cyclic cyclic cyclicProfile (0.0 0.0) zeroGradient 1.0× 10−16 1.1× 105 1.0× 10−16 1.0× 10−16

Lateral empty empty empty empty empty empty


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MSc Thesis - Luis Felipe Paulinyi - Separation Prediction Using State of the Art Turbulence Models