POLITECNICO DI MILANO

School of Industrial and Information Engineering

Master of Science in Mechanical Engineering

RANS and LES simulations of a turbulent plane jet.

Supervisor at Politecnico di Milano: Prof. Paolo GAETANI

Supervisor at MTU Aero Engines: Eng. Robert KLUXEN

Master thesis of:

Stefano SALERNO

852551

Academic Year 2017 - 2018

Stefano Salerno: RANS and LES simulations of a turbulent plane jet. | MasterThesis in Mechanical Engineering, Politecnico di Milano.c© Copyright April 2018.

Politecnico di Milano:www.polimi.it

School of Industrial and Information Engineering:www.ingindinf.polimi.it

http://www.polimi.ithttp://www.ingindinf.polimi.it

Acknowledgments

First of all I would like to thank my supervisor at MTU Aero Engines eng.Robert Kluxen, that followed me from the very beginning of this project, offeringme a precious support and being for me example and reference.

I would also like to thank my Professor at Politecnico di Milano prof. PaoloGaetani, for being always ready to answer to my questions and for its supervisingof my work.

I am also thankful for the help that I received from many colleges of MTUAero Engines, in particular from the other students of my office and from theOpenFOAM meeting crew.

Finally, I would like to thank my parents, my family and relatives, my girlfriendand my friends. Without their support and encouragement through all these monthsthis accomplishment would have not been possible.

Milano, April 2018 Stefano Salerno

iii

Contents

Introduction 1

1 Theory of jets 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Structure of the Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Equations of Fluid Dynamics 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Momentum equation . . . . . . . . . . . . . . . . . . . . . . 8

2.2.3 Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Finite volume method 11

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Discretization of a transport equation . . . . . . . . . . . . . . . . . 11

3.3 Solution of Navier Stokes equations . . . . . . . . . . . . . . . . . . 16

3.3.1 Pressure equation . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.2 PISO algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.3 SIMPLE algorithm . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.4 PIMPLE algorithm . . . . . . . . . . . . . . . . . . . . . . . 18

4 Turbulence Models 19

4.1 Introduction to turbulence . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 RANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.1 k-ε model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2.2 k-ω model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.3 k-ω SST model . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 LES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3.1 LES filtered Navier Stokes equations . . . . . . . . . . . . . 24

4.3.2 The Smagorinsky model . . . . . . . . . . . . . . . . . . . . 26

4.3.3 Dynamic models . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3.4 WALE model . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3.5 k-Equation model . . . . . . . . . . . . . . . . . . . . . . . . 33

v

vi CONTENTS

5 Numerical Results - RANS 355.1 First studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3.1 The blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3.2 Mesh independence test . . . . . . . . . . . . . . . . . . . . 41

5.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4.1 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . 445.4.2 Inlet boundary conditions . . . . . . . . . . . . . . . . . . . 44

5.5 Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.6 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.7 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.8 Comparison with experiments . . . . . . . . . . . . . . . . . . . . . 47

6 Numerical Results - LES 516.1 First Attempts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2 Crashing problems: BCs and schemes . . . . . . . . . . . . . . . . . 52

6.2.1 New temperature BCs . . . . . . . . . . . . . . . . . . . . . 526.2.2 Diverging backflow . . . . . . . . . . . . . . . . . . . . . . . 53

6.3 A new mesh - snappyHexMesh . . . . . . . . . . . . . . . . . . . . . 546.4 The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.4.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 576.4.2 fvSchemes and fvSolution . . . . . . . . . . . . . . . . . . . 58

6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.5.1 Convergence in time . . . . . . . . . . . . . . . . . . . . . . 596.5.2 Mesh independence test . . . . . . . . . . . . . . . . . . . . 616.5.3 Axial development . . . . . . . . . . . . . . . . . . . . . . . 636.5.4 SGS models comparison . . . . . . . . . . . . . . . . . . . . 65

6.6 Blending schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Conclusions 75

A Setup 77

List of Figures

1.1 The jet shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4.1 Second filter in dynamic models (McDonough, 2007 [8]) . . . . . . . 29

5.1 The final domain - RANS, L=length=60d W=width=10d. . . . . . 365.2 Width independence test vs experiments, tranverse profiles at x =

40d (exp:[14] in (a,d) and [7] in (c)) . . . . . . . . . . . . . . . . . . 375.3 Length independence test vs experiments, tranverse profiles at x =

40d (exp:[14] in (a,d) and [7] in (c)) . . . . . . . . . . . . . . . . . . 385.4 Mesh 40.000 cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.5 Blocks definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.6 Mesh independence test vs experiments, tranverse profiles at x = 40d

(exp:[14] in (a,d) and [7] in (c)) . . . . . . . . . . . . . . . . . . . . 425.7 Axial evolution of transverse normalized profiles . . . . . . . . . . 495.8 Axial profiles: RANS vs experiments . . . . . . . . . . . . . . . . . 50

6.1 SnappyHexMesh utility [13] . . . . . . . . . . . . . . . . . . . . . . 556.2 SnappyHexMesh - refinement regions . . . . . . . . . . . . . . . . . 556.3 Final mesh - fine case . . . . . . . . . . . . . . . . . . . . . . . . . . 576.4 Transverse profiles of the mean axial velocity at the nozzle exit for

the cases of smooth contraction nozzle and long pipe in a round jet[10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.5 Convergence in time of mesh medium, tranverse profiles at x = 20d(exp:[14]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.6 Points referring to table 6.2 . . . . . . . . . . . . . . . . . . . . . . 636.7 Time to simulate 1 second against mesh size . . . . . . . . . . . . . 646.8 Mesh Independence test: 3D mesh with inlet patch . . . . . . . . . 656.9 Mesh independence test vs experiments: transverse profiles at x=20d

(exp:[14] in (a,d,e) and [7] in (c)) . . . . . . . . . . . . . . . . . . . 666.10 Velocity and temperature half widths compared to [3] in a and to [7]

in b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.11 Mesh independence test vs experiments: transverse profiles at x = 6d 686.12 Axial profiles for the different meshes: comparison with RANS results

and experiments of [3] in plot a,c and of [7] in b,d. . . . . . . . . . . 696.13 Axial development of transverse profiles (mesh medium) (exp:[14] in

(a,d,e) and [7] in (c,f)) . . . . . . . . . . . . . . . . . . . . . . . . . 706.14 SGS models comparison: transverse profiles at x = 20d (exp:[14] in

(a,d,e) and [7] in (c)) . . . . . . . . . . . . . . . . . . . . . . . . . . 71

vii

viii LIST OF FIGURES

6.15 Velocity magnitude (a,b,c) and temperature averaged fields (d,e,f),with mesh coarse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.16 Instantaneous contour plots of temperature for different sub-gridscale models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.17 Axial profiles for the case with blending schemes, experiments of [3]in plot a, c and of [7] in plot d . . . . . . . . . . . . . . . . . . . . . 73

List of Tables

5.1 Mesh refinement: number of cells . . . . . . . . . . . . . . . . . . . 435.2 Mesh quality parameters - RANS . . . . . . . . . . . . . . . . . . . 43

6.1 mesh properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.2 cell sizes [mm] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

ix

Nomenclature

Acronyms

BC Boundary Conditions

CD Central Differencing

CFD Computational Fluid Dynamics

DNS Direct Numerical Simulation

GAMG generalised Geometric-Algebraic Multi-Grid

LES Large Eddy Simulation

OpenFOAM Open Source Field Operation and Manipulation

PISO Pressure-Implicit-Split-Operator

RANS Reynolds Averaged Navier-Stokes equations

SIMPLE Semi-Implicit Method for Pressure Linked Equations

UD Upwind Differencing

WALE Wall-Adapting Local Eddy-viscosity

Greek letters

∆ filer length

ε turbulent dissipation rate

Γ diffusion coefficient

µ dynamic viscosity

µT turbulent viscosity

ν kinematic viscosity

ω turbulent frequency

τ viscous stress tensor

ψ compressibility

xi

xii LIST OF TABLES

ρ density

τSGS sub-grid scale stress tensor

Latin letters

a deviatoric part of Reynolds stresses

D strain rate tensor

d fluctuation strain rate tensor

I identity tensor

r Reynolds stress tensor

F surface forces

f volume forces

g acceleration of gravity

S surface vector

U velocity vector

xF face center position vector

xP cell center position vector

d transverse nozzle dimension

eT total energy per unit volume

F flux

fx weight for Central Differencing interpolation

G kernel function for LES filtering

k turbulent kinetic energy

Ku inclination of axial velocity decay line

lI integral length scale

lT turbulent length scale

P pressure

Re Reynolds number

ReI turbulent Reynolds number

Sϕ source term

LIST OF TABLES xiii

t time

u′i velocity oscillations

Uc core velocity

Un nozzle exit velocity

uT turbulent velocity

V control volume

x axial coordinate

x01 virtual origin of axial velocity decay line

y transverse coordinate

y0.5 velocity half-width length

z third direction

Sommario

In questa tesi di laurea magistrale è stata sviluppata un’analisi numerica diun getto turbolento e planare in aria libera attraverso i metodi RANS (Reynoldsaveraged Navier-Stokes equations) e LES (large-eddy simulation). Il numero diReynolds ed il numero di Mach sono stati scelti di 16,000 e 0.07 rispettivamente.L’aria uscente dall’ugello è stata impostata come leggermente riscaldata rispettoalla temperatura ambiente. In questo modo la temperatura è stata considerata unoscalare passivo, cioè che non influenza la soluzione del campo di moto. Sono stateadoperate entrambe le versioni del sofware open-source OpenFOAM, versione 4.1.0e versione 3.0.1, la prima per le simulazioni RANS e la seconda per le simulazioniLES. Sono state utilizzate mesh differenti, ma sempre a predominanza esaedriche.L’ugello non è stato incluso nel dominio di simulazione, quindi le condizioni di uscitadel getto sono state fissate dall’autore. L’obiettivo di questo progetto è quello divalutare le differenze nei metodi RANS e LES nel caso di un getto d’aria turbolento,per il quale sono presenti molti dati sperimentali in letteratura. I risultati dellesimulazioni LES sono stati mediati nel tempo, mostrando una convergenza solo finoalla distanza assiale di x = 20d. Per questo motivo le comparazioni dei profili dopoquesta distanza sono difficili da considerarsi attenibili. I risultati con il metodo LEShanno mostrato un comportamento più conforme ai dati sperimentali rispetto aquelli con il metodo RANS, in particolare per l’evoluzione assiale della velocità. E’stato rilevato che il decadimento assiale della temperatura è maggiormente concordecon gli esperimenti per il caso LES rispetto al caso RANS, ma la half-width dellatemperatura è stata sovrastimata in modo non fisico dal primo e sottostimata(ma ragionevolmente) dal secondo. Per quanto riguarda l’approccio LES, con unmaggiore tempo a disposizione e ulteriori mezzi computazionali, il lavoro potrebbeessere esteso, introducendo l’ugello all’interno del dominio, rifinendo la mesh nellazona vicino all’ugello e mediando su tempi più lunghi.

Parole chiave: CFD, OpenFOAM, RANS, LES, getto planare.

xv

Abstract

In this master thesis a numerical analysis of a turbulent plane free jet wascarried out by means of both Reynolds averaged Navier-Stokes equations (RANS)and large-eddy simulation (LES) approaches. The Reynolds and the Mach numberschosen for this studies were 16,000 and 0.07 respectively. A small temperaturegradient was enforced, so that the jet nozzle exit temperature was slightly higherthan the ambient one. This allowed to consider the temperature as a passivescalar not influencing the velocity field. Both versions of the open-source softwareOpenFOAM, OF 4.1.0 and OF 3.0.1 were used, the former for the RANS simulationsand the latter for the LES ones. Different meshes were exploited, in any case with apredominant hexaedral cell shape. The nozzle was not included inside the simulationdomain, so that the exit nozzle conditions were fixed by the author. The objectiveof this project was to test the differences between the RANS and LES methods fora simple test case such as the one of the turbulent free jet, in order to have differentpossibilities of comparison with experimental data available literature. Averagingin time of the LES simulations brought to converging results only until an axialdistance of x = 20d, so that data after this axial coordinate were difficult to compare.LES results showed a better behaviour in the near field region, in particular forthe velocity axial decay. Temperature axial decay was found to match better theLES results then the RANS ones, but the temperature half width was found tobe non-physically overestimated by the former, while consistently underestimatedby the latter. For what concerns the LES simulations, with an higher amount oftime and more computational resources the work could be extended, introducing anozzle in the domain, building a finer mesh for the near field region, and averagingon longer times.

Keywords: CFD, OpenFOAM, RANS, LES, plane jet.

xvi

Introduction

In this study an investigation on accuracy of numerical methods such as theRANS equations and the LES method was carried out. In particular, these methodswere adopted in order to study the development of a turbulent plane free jet issuingfrom a nozzle into ambient quiescent air. The flow studied was subsonic and athigh Reynolds numbers, so that it was considered fully turbulent. The results werecompared with some experimental ones available in literature.

Turbulent jets are widely studied in the field of engineering, they have bothmany direct applications and offer a good test case to compare numerical methods.This master thesis was developed inside the company MTU Aero Engines. The aimwas to investigate the accuracy of LES against traditional RANS methods used forcooling systems of turbomachines. For this reason the test case of a turbulent planejet was chosen, thanks to its simple geometry and to the data available in literaturefor comparisons. The turbulent jet is a phenomenon dominated by turbulence, sothat its development and its shape are very dependent on the turbulence modellingtechnique applied. As the flow exits the nozzle, the difference in velocity betweenthe two fluids (jet flow and quiescent ambient air) gives birth to a shear layer, thatbecomes soon unstable. This instability is the cause of the transition from laminarto turbulent state and it is highly depending on the Reynolds number. If thisnumber is above a certain threshold the jet is fully turbulent and the instabilitiesdevelop right after the nozzle exit.

The plane jet was preferred to the axisymmetric one because of its two-dimensional characteristics. This allowed the RANS study to be done withoutsolving the third direction. Also for the 3D mesh accomplished for the LES case inthis way the number of cells was easier to control.

The approach chosen for this study was to begin with the investigations ofthe plane jet with the RANS equations, to fix the domain and the cell size afterthe correspondent independence studies, and then compare these results with theexperiments. After this first solution, the LES method was adopted to model theflow turbulence. In this phase many problems were encountered to stabilize thesolution. The presence of eddies in correspondence of the boundaries brought thesolution to diverge and the simulations to crash. This problem was worked aroundthrough the adoption of very large domains with larger cells close to the boundaries.In a second moment the problem was overcome with the adoption of sponge layersclose to the boundaries, so that not only the cell size was increased close to theoutlet, but also the discretization schemes were of lower order in the critical regions.

In the first chapter an overview on the main characteristics on the turbulentplane free jet is given, in the second one the conservation equations of fluid dynamics

1

2 Introduction

are shown, and in the following chapter their discretization through the finite volumemethod is explained. In chapter four the different ways of modelling turbulenceare shown, with a particular attention to the ones used in this study: k-ω SST forRANS and WALE for LES. Results of both RANS and LES simulations can befound in chapter five and six, as well as the description of the domain and meshgeneration, the set up and the boundary conditions used. Finally the conclusionsof this work are reported.

Chapter 1

Theory of jets

1.1 Introduction

A free jet is the result of a high pressure fluid mass flow undergoing an expansionthrough a nozzle and being injected into an infinite large environment of quiescentair at ambient pressure. This process at high Reynolds numbers is very muchinfluenced by turbulence events, that is why this test case results so interestingin turbulence models testing. In particular, in this thesis the development of theflow field properties of a plane jet is considered. Other two cases largely studied inliterature are the axisymmetric and the square jet 1. Once that the flow is ejected,the boundary layer of the nozzle develops into a shear layer. This shear layeris due to the difference in velocity between the incoming high-speed air and thesurrounding quiescent air. Because of what is known as Kelvin Helmoltz instability,this shear layer is highly unstable, meaning that vortical structures are started anda process of entrainment of ambient air inside the jet is activated. This unstablevortical structures develop in time and collapse together 2. This collapsing bringsthe flow to a full developed state where the mean flow has self-similar characteristics.

1.2 Structure of the Jet

The jet structure can be described as divided into three different main zones:

1. Potential core region: the potential core of the jet is the region that conservesthe nozzle outlet velocity Un. In this work the potential core length is definedas the axial distance at which the core velocity is equal to the 98% of Un andis called xP . Given x the axial coordinate as in figure 1.1, d the transversedimension of the nozzle and Re the Reynolds number as in equation 1.1, thisvalue can vary from xP = 3d to xP = 6d in the case of a subsonic isothermalplane jet issuing from a contraction nozzle with Reynolds number betweenRe = 1500 Re = 16500 (Deo et al., 2008 [3]).

1The jet is issuing from a square-shaped nozzle2In plane jets there are two shear layers, one in the upper part and one in the lower part of

the jet

3

4 Chapter 1. Theory of jets

2. Transition region: here the center line axial velocity is decreasing and theroll-ups of the upper and lower shear layer collapse together. Already inthis region some mean flow quantities show a self-similar behavior. In fact,Mi et al. (2000 [10]) showed that for a round jet the mean temperatureradial profiles are self-similar after x = 10d, while in the same case turbulentquantities become self-similar only after x = 40d.

3. Self-similar region: in this region non-dimensional transverse profiles of meanfields at different axial distances are independent from the axial distanceitself. There are different possibilities to normalize these profiles, as it will bediscussed later in this chapter. This property results very useful to comparedifferent experimental results in literature.

Plane jets are theoretically defined as jets in which the nozzle has the thirddirection which is much higher then the other two, so that the time-averaged quan-tities in the jet region are non not depending on it. In the case of an experimentaltest, this hypothesis is not so easy to represent. In order to study the case of aplane jet, all the three experimental studies taken as reference for this work used ahigh aspect ratio-shaped nozzle and side walls. The latter were used to confine thejet and to preserve it from a development in the third direction.

A turbulent plane jet can be described by several non-dimensional parame-ters that can be divided into two groups: initial conditions parameters and jetdevelopment parameters.

Figure 1.1: The jet shape

Initial conditions are: the velocity and turbulence intensity lateral profiles atthe nozzle exit, the nozzle geometry (e.g. smooth contraction nozzle or pipe), thedensity ratio between the jet exit and ambient fluid, and finally the Reynols number.(Deo et al., 2008 [3])

Re =Un · dν

(1.1)

In this equation Un is the nozzle exit velocity, d is the transverse dimension of thenozzle and ν is the kinemtaic viscosity.

1.2. Structure of the Jet 5

Among the parameters describing the nozzle shape, the most intuitive one is thepotential core length. Other two parameters used to define the jet axial behaviorare the Ku constant and the virtual origin x01. These are parameters describingthe self-similar region behavior of the axial velocity decay as in the equation below(Deo et al., 2008 [3]): (

UcUn

)−2= Ku

(x+ x01

d

)(1.2)

In the equation above Uc is the core axial velocity, Ku is a constant measuring theslope of the line described by (Uc/Un)

−2 while x01 is called virtual origin and isrelated to the intercept of this same line. In the studies of Deo et al (2008 [3]) theresults showed a convergence of the parameter Ku only after Re = 25000. Thisconvergence value is dependent on the aspect ratio and on the dimensions of thenozzle, and it is found to be around 0.15 - 0.17.

A second way of describing the jet is by its non-dimensional shape. Anotherconstant Ky is introduced, as well as a second virtual origin, which represents thevirtual origin of the half-width axial development line. The curve defining theseparameters is given in Deo et al. (2008, [3]):

y05d

= Ky(x+ x02

d) (1.3)

In this equation the term y05 is the half-width distance, defined as the transversedistance at which the axial velocity assumes the half of the core value. Thanksto this quantity the shape of the jet cone is described. The half-width is alsoused to normalize the transverse profiles of the averaged quantities. In this way,normalized lateral profiles of normalized quantities result to be collapsing togetherin the self-preserving region.

Considering now the flow evolution in time, it is known that a turbulent planejet presents coherent structures that can occur in a symmetric mode or in ananti-symmetric mode. Deo et al. (2008, [3]) assess that the near field region canpresent both modes, depending on the velocity exit profiles and on initial conditions.In the far field region instead the mode is only anti-symmetric. These quasi-periodicpassage of the developed coherent structures that are found in the far field regionof plane jets is what can be found in literature as flapping of the jet.

Chapter 2

Equations of Fluid Dynamics

2.1 Introduction

The set of equations describing the flow of a fluid in space and time areconservation equations that describe the conservation of certain physical propertiesof the flow in time and space. The set of equations that has been solved numericallyin this study is known as the Navier-Stokes equations. This is a system of threeequations: mass, momentum, and energy equations. These are conservation lawsapplied to Newtonian fluids in an Eulerian reference approach (the reference systemand the volume on which the equation terms are integrated are both fixed intime and space: this volume is known as the control volume). The Navier Stokesequations contain 6 unknowns: the fields of velocity (three scalars, one for eachdirection), pressure, temperature and density. Since the momentum equations arewritten for each direction, the overall number of equations is 5, so the system canbe closed through the boundary conditions definition and the equation of state.

In this project the open source code OpenFOAM was used, solving numericallythe Navier-Stokes equations through the Finite Volume Method. This methodsolves the integral form of the equations for each cell of the discretized controlvolume: in Computational Fluid Dynamics the control volume takes the name ofdomain, while its discretization in space (many adjacent cells) is called mesh.

2.2 Navier-Stokes equations

2.2.1 Continuity equation

The first equation of the Navier Stokes system is the continuity equation. Itexpresses the conservation of mass in a control volume: it can be written both in itsintegral and punctual form. The two forms are theoretically equivalent and they canbe re-conducted to each other be means of the Gauss theorem. The conservation ofmass theorem postulates that the material (or total) derivative of mass is constantand equal to zero:

dM

dt= 0 (2.1)

7

8 Chapter 2. Equations of Fluid Dynamics

Considering the total mass as the integral of density over the volume and using thetransport theorem, this can be rewritten as:∫

V

(dρ

dt+ ρ∇ ·U

)dV = 0 (2.2)

where V is an arbitrary volume and ρ is the density [kg/m3]. Since this is truefor any volume, the function inside the integral becomes true for every point. Thepunctual formulation of mass conservation can be so written as:

dρ

dt+ ρ∇ ·U = 0 (2.3)

Since the total derivative of a generic property ϕ is defined as its partial derivativein time plus the product of the velocity vector and the gradient of the propertyitself, as stated in equation 2.4, another punctual formulation of the continuityequation can then be written as in 2.5.

dϕ

dt=∂ϕ

∂t+ U · ∇ϕ (2.4)

∂ρ

∂t+∇ · (ρU) = 0 (2.5)

2.2.2 Momentum equation

The second principle of dynamics can be re-written in the form of conservationof momentum. In this approach, the total derivative of the momentum integratedover a generic volume V results to be equal to the integral of the volume forces fplus the integral of the surface forces F applied on its contour, as shown in thefollowing equation.

d

dt

∫V

ρUdV =

∫V

ρfdV +

∫S

FdS (2.6)

Using the transport theorem and the continuity equation, this can be also writtenas: ∫

V

ρdU

dtdV =

∫V

∂(ρU)

∂tdV +

∫S

(ρU)U · ndS =∫V

ρfdV +

∫S

FdS (2.7)

Assuming gravity the only volume force and pressure and viscous stresses the surfaceones, through the Gauss theorem the previous equation becomes the momentumequation in its punctual formulation:

∂(ρU)

∂t+∇ · (ρUU) = ρg −∇P +∇ · τ (2.8)

In the equation above g is the acceleration of gravity, P is the pressure and τ isthe viscous stress tensor, defined in the case of a Newtonian fluid by the Newton’sviscous constitutive law as:

τ = 2µ

(D − 1

3(∇ ·U)I

)(2.9)

2.2. Navier-Stokes equations 9

Where D is the strain rate tensor:

Dij =1

2

(∂Ui∂xj

+∂Uj∂xi

)(2.10)

With this definition the divergence of the viscous stresses can be computed as:

∇ · τ = µ∇2U + 13µ∇ (∇ ·U) (2.11)

This leads to the final formulation of the momentum equation:

∂(ρU)

∂t+∇ · (ρUU) = ρg −∇P + µ∇2U + 1

3µ∇ (∇ ·U) (2.12)

2.2.3 Energy equation

The conservation of energy is stated in the first principle of thermodynamics: thevariation of the total energy of the system, which in the case of a fluid correspondsto the sum of internal and kinetic energy, is equal to the work of the forces that acton the fluid plus the contribution of heat energy. This results in:∫

V

∂(ρet)

tdV +

∫S

ρetU · ndS =∫V

ρf ·UdV +∫S

F ·UdS −∫S

q · ndS (2.13)

in which the last term captures the heat transferred to the system.The Gauss theorem helps re-writing the equation above in its punctual formula-

tion:

∂(ρet)

∂t+∇ · (ρetU) = ρg ·U +∇ · (−PU) +∇ · (τU)−∇ · q (2.14)

The work of the viscous term can be written as:

∇ · (τU) = U∇ · τ + τ : ∇U (2.15)

where the operator ’:’ indicates an element to element multiplication. The lastterm of this equation is called ε, the dissipation term. It is the one responsible ofthe conversion of the energy from its kinetic form to the internal one.

Thanks to Fourier model the heat term can be written as follows:

−∇ · q = k∇2T (2.16)

Putting together the last three equations, a final formulation can be derived:

∂(ρet)

∂t+∇ · (ρetU) = ρg ·U +∇ · (−PU) + U∇ · τ + ε+ k∇2T (2.17)

Chapter 3

Finite volume method

3.1 Introduction

The finite volume method is a way of discretizing the equations of fluid dynamicspresented in the previous chapter. This procedure allows to solve numerically theNavier Stokes equations system. There are two levels of discretization, one for thedomain and one for the equations. The domain is divided into many different sub-domains, each one of them representing a control volume. These sub-domains arecalled cells, which in the case of an unstructured mesh can be of a generic polyhedralshape. In the case used in this work the dependent variables (pressure, temperatureetc.) are all defined for each one of these cells. In particular they correspond to thevalue in the center of the cells (cell centroid). This approach is called a co-locatedor non staggered mesh approach. Once that the domain and the equations arediscretized, a system of linear algebraic equations for each fluid dynamic equationis built. The number of equations of this system corresponds to the number ofcells of the domain. This linear system is then solved through an iterative method,which stops looking for a better solution after that a given tolerance is reached.In OpenFOAM the Navier Stokes equations system is treated in a segregated way,meaning that the equations are solved one at a time. This means for example thatthe momentum equation is solved in each direction separately and consequently.

In this chapter the main passages of the finite volume method are describedreferring to the work of Jasak (1996, [6]).

3.2 Discretization of a transport equation

All the Navier Stokes equations are transport equations, or conservation equa-tions. This means that they can be written in the form of a generic transportequation, like the following (punctual formulation):

∂ρϕ

∂t+∇ · (ρUϕ)−∇ · (ρΓϕ∇ϕ) = Sϕ(ϕ) (3.1)

In the equation above the first term is the temporal derivative, the second onethe convection term. Together they represent the total or material derivative ofthe quantity ρϕ. The third term is the diffusion term, which is depending on the

11

12 Chapter 3. Finite volume method

gradient of the quantity ϕ and on the diffusion coefficient Γϕ. Finally, on the righthand side of the equation there is a source term. Another more general form ofthis equation could be re-written considering the density included in the field ϕ. Inorder to remain similar to the formulations of the momentum and energy equations,the density is here explicited. This generic formulation could be written in itsintegral form as:∫ t+∆t

t

[∂

∂t

∫V

ρϕdV +

∫S

ρϕU · dS−∫S

ρΓϕ∇ϕ · dS]dt =∫ t+∆t

t

[∫V

Sϕ(ϕ)dV

]dt

(3.2)

In this formulation the flux terms are written as surface integrals, thanks to theGauss theorem. From the equation above it is maybe more intuitive to understandthe meaning of the transport equation itself. Given V the control volume and S itscontour, this conservation equation states that the quantity ρϕ can change its valuein time only if there is a source term or if it is exchanged with the outside of thedomain through its boundaries. The control volume communicates with the restof the space through its contour, over which the fluxes of the quantity are defined.These fluxes can be convective or diffusive in nature. The convective flux is thetransport of the quantity through velocity, while the diffusion one is the transportdue to the gradient of the quantity itself (think to Fourier equation of heat). Thefinite volume method, as already expained before, solves the integral form of thetransport equations. The discretization of these terms is a matter of reconstructingthe integral in a numerical way. A numerical error is so introduced, linked to thenumerical approximation of the operation of integration.

Integration is approximated based on the assumption that the field is linearlydistributed inside the control volume and on its surfaces. This assumption is anapproximation of the second order: this can be shown from the Taylor expansion ofϕ. Cells center xP is defined as:∫

V

(x− xP )dV = 0 (3.3)

Face center xF is defined as: ∫S

(x− xF )dS = 0 (3.4)

Taylor expansions result in:

ϕ(x) = ϕP + (x− xP ) · (∇ϕ)P + o((x− xP )2

)(3.5)

ϕ(x) = ϕF + (x− xF ) · (∇ϕ)F + o((x− xF )2

)(3.6)

In this way, a volume integral becomes:∫V

ϕ(x)dV =

∫VP

[ϕP + (x− xP ) · (∇ϕ)P ] dV

= ϕP

∫VP

dV +

[∫VP

(x− xP )dV]· (∇ϕ)P

= ϕPVP

(3.7)

3.2. Discretization of a transport equation 13

Convective term

The convective term can be transformed from a volume to a surface integralthrough the Gauss therem, resulting then in:∫

VP

∇ · (ρUϕ)dV =∑f

S · (ρUϕ)f

=∑f

S · (ρU)fϕf

=∑f

Fϕf

(3.8)

Where F is the flux, which is a scalar defined on the face center as resulting fromthe product of the face vector and the ρU term evaluated again on the face center.The first passage of the previous equations is derived thanks both to Gauss theoremand to the approximation of the surface integral as the product of the face centervalue and the face vector. The term ϕf indicates the value of the quantity ϕ onthe face center. This value is not directly given, so it has to be interpolated fromthe cell centers values. This interpolation can be done in several different waysand following different criteria. It must be noted that this interpolation procedureintroduces another numerical error. It is then a consistent choice the one thatkeeps the order of the numerical error constant and equal to the other ones alreadyintroduced. This interpolation methods are crucial in particular for the momentumequation, in which the convective term represents the convection of velocity itself,so analytically a second order term. The main interpolation scheme is the CentralDifferencing one (linear scheme in OpenFOAM). This method interpolates linearlythe face center value of ϕ from the cell center value of the cell itself ϕP and fromits neighbour one ϕN .

ϕf = ϕPfx + (1− fx)ϕN (3.9)where

fx =Pf

PN

In the equations above P is the centroid of the considered cell, N is the centroid ofthe generic neighbour cell and f is the face center of the face that the two cells aresharing. Since this method can generate non-physical oscillations of the solution,another important scheme was introduced: the Upwind Differencing scheme (UD).This is a first order interpolation scheme that assumes the value on the face centerto be equal to the one of the cell center. In order to determine which one of thetwo cells sharing the considered face is the selected one, direction of the flow, andso the sign of the flux F, is considered:

ϕf =

{ϕP ifF ≥ 0ϕN ifF < 0

(3.10)

This scheme guarantees boundedness of the solution, but it introduces a higheramount of numerical diffusion because of its lower order. A combination of these

14 Chapter 3. Finite volume method

two techniques is represented by the Blending Schemes, which define the facecenter value as an interpolation between the value resulting from a CD and a UDinterpolation method. This interpolation is governed by the blending factor γ:

ϕf = (1− γ)(ϕf )UD + γ(ϕf )CD (3.11)

Diffusion term

As for the case of the convective term, a linear variation of the quantity ϕ isassumed and the resulting formulation can then be written:∫

VP

∇ · (ρΓϕ∇ϕ)dV =∑f

S · (ρΓϕ∇ϕ)f

=∑f

(ρΓϕ)fS · (∇ϕ)f(3.12)

In the case of an orthogonal mesh the following expression is used:

S · (∇ϕ)f = |S|ϕN − ϕP|d|

(3.13)

Another way of computing the gradient on the face center could be the one ofcomputing the gradient in each cell center using the interpolated quantities of theface centers and then linearly interpolate the two resulting values on the face centerlike this:

(∇ϕ)f = fx(∇ϕ)P + (1− fx)(∇ϕ)N (3.14)

Since this method uses a larger computational cell, the first one would be preferred.The problem is that for non-orthogonal cells expression 3.13 loses in accuracy,leading to the necessity, for high non-orthogonal meshes, of introducing a non-orthogonal correction. This is done decomposing the face vector in two components,∆ is parallel to the line passing through the two cell centres, k is normal to it. Inthis way a new formulation can be written as:

S · (∇ϕ)f = |∆|ϕN − ϕP|d|

+ k · (∇ϕ)f (3.15)

In the equation above the non-orthogonal correction gradient term is computedwith the formulation shown in equation 3.14. The model without the correction isbounded, while the correction introduces un-boundedness. In case of non-orthogonalmeshes the choice of setting the correction on the diffusion term depends from atrade off between boundedness and accuracy.

Source terms

All those terms that are not considered convective, diffusive, or temporalderivative terms are considered as sources. The source term Sϕ(ϕ) is a genericfunction of ϕ but in order to be discretized it has to be written in a linerized form:

Sϕ(ϕ) = Su+ Spϕ (3.16)

3.2. Discretization of a transport equation 15

where Su and Sp may also be depending from ϕ. It follows that:∫VP

Sϕ(ϕ)dV = SuVP + SpVPϕP (3.17)

Time discretization

The integral formulation shown already in equation 3.2 is here again reported.∫ t+∆tt

[∂

∂t

∫V

ρϕdV +

∫S

ρϕu · dS−∫S

ρΓϕ∇ϕ · dS]dt =∫ t+∆t

t

[∫V

Sϕ(ϕ)dV

]dt

Using the spatial discretization methods presented above this can be written as:∫ t+∆tt

[(∂ρϕ

∂t

)P

VP +∑f

Fϕf −∑f

(ρΓϕ)fS · (∇ϕ)f

]dt

=

∫ t+∆tt

(SuVP + SPVPϕP ) dt

(3.18)

The main time discretization schemes are the Euler explicit, Euler implicit, CrankNi-colson and Backward ones. In the case of Euler explicit discretization the facecenter values are computed from the old-time field. In this way:

ϕf = fxϕoP + (1− fx)ϕoN (3.19)

and

S · (∇ϕ)f = |∆|ϕoN − ϕoP|d|

+ k · (∇ϕ)of (3.20)

Once the system is discretized also in time, the discratization is completed, and alinear system of algebraic equations is found:

aPϕnP +

∑N

aNϕnN = RP (3.21)

which in matrix form results:[A] [ϕ] = [R] (3.22)

For the Euler explicit case this equation can be written as:

ϕnP = ϕoP +

∆t

ρPVP

[∑f

Fϕf −∑f

(ρΓϕ)fS · (∇ϕ)f + SuVp + SpVPϕoP

](3.23)

Thanks to this approach all the right hand side terms are defined and dependingonly on the old time step fields. The drawbacks of this method are that it is firstorder accurate and that the Courant number (3.24) has to be smaller than unity inorder to ensure stability.

16 Chapter 3. Finite volume method

Co =Uf · d

∆t(3.24)

In the equation above uf is the velocity on the face, d is the distance vector betweenthe two cell centers and ∆t is the time step.

In order to overcome this issue a Euler implicit method can be used:

ϕf = fxϕnP + (1− fx)ϕnN (3.25)

and

S · (∇ϕ)f = |∆|ϕnN − ϕnP|d|

+ k · (∇ϕ)nf (3.26)

Second order schemes for time discretization are instead the CrankNicolson andBackard Diferinceng ones.

For Crank Nicolson: (∂ρϕ

∂t

)P

=ρnPϕ

nP − ρoPϕoP

∆t(3.27)

For Backward Differencing:(∂ϕ

∂t

)P

=32ϕnP − 2ϕoP + 12ϕ

ooP

∆t(3.28)

The Crank-Nicolson method is unconditionally stable but the solution may beunbounded: its accuracy is of the second order. Also for the Backward Differencingthere is still a second order accuracy and boundedness of the solution is notguaranteed. (Jasak, 1996 [6])

3.3 Solution of Navier Stokes equations

Considering the simplified case of an incompressible flow, Navier Stokes equationssystem is decoupled, so that the energy equation can be left out of the system. Inthis way, continuity and momentum equations reduce to:{

∇ ·U = 0∂U∂t

+∇ · (UU)−∇ · (ν∇U) = −∇p(3.29)

The method used in OpenFOAM also for compressible flows is similar to theincompressbile one, so that it built as an extension of it. These methods are basedon pressure-velocity coupling, a practice that was developed first for incompressibleflows and then extended to the compressible ones. This is a way of solving theissue of having no pressure terms inside the continuity equation. A new equation isbuilt from applying the continuity equation (divergence-free velocity field) onto themomentum one. The Poisson equation, or pressure equation, is so defined.

3.3. Solution of Navier Stokes equations 17

3.3.1 Pressure equation

A semi-discretized form of momentum equation can be written as:

apUP = H(U)∇P (3.30)

where the term H is:

H(U) = −∑N

aNUN +Uo

∆t(3.31)

It must be noted that the coefficients aN are depending on the fluxes on the facecenters. This form of the momentum equation can be written as:

UP =H(U)

aP− 1aP∇P (3.32)

The Poisson or pressure equation can be now built applying the continuityequation onto this momentum one:

∇ ·(

1

aP∇p)

= ∇ ·(

H(U)

aP

)=∑f

S ·(

H(U)

aP

)f

(3.33)

The Navier Stokes system will then result in: apUP = H(U)−∑

f S(P )f∑f S ·

[(1aP

)f

(∇P )f]

=∑

f S ·(

H(U)aP

)f

(3.34)

3.3.2 PISO algorithm

The Pressure-Implicit-Split-Operator algorithm is an algorithm for unsteadyflows. It works through the following steps:

1. Momentum equation is solved using the pressure field of the previous timestep. This passage is called momentum predictor.

2. The pressure equation is solved, using as H(U) operator the one updatedwith the new velocity field calculated at step 1.

3. Velocity is re-computed in an explicit manner, so that the pressure field isupdated with the results of step 2 but the H(U) operator is depending onthe velocity field of step 1, and not on the new one.

4. Repeat until convergence.

This procedure corrects the velocity field depending only on the new pressure field,neglecting the correction term introduced by the update of the H(U) operator.This assumption brings to the limitation of the time step to respect the maximumCourant number of one to ensure stability. Another important point is that alsothe fluxes determining the coefficients of H(U) are not updated after the pressureequation solution. This is also descending from the hypothesis the the main velocitydependence is the one from the pressure term.

18 Chapter 3. Finite volume method

3.3.3 SIMPLE algorithm

The Semi-Implicit-Method-for-Pressure-Linked-Equations is an algorithm usedfor steady state solutions. This method takes out the time derivative term fromthe equations and it solves on subsequent iterations the pressure and momentumequations. Its steps are:

1. The momentum equation is solved using the pressure field of the previousiteration. This results is then under-relaxed in order to enhance convergence.

2. The pressure equation is solved based on the new velocity field (H(U) iscomputed with the velocity field of point 1). Also in this case the resultingvalues are under-relaxed.

3. New face fluxes are computed and the H(U) coefficients are updated.

4. Repeat until convergence.

3.3.4 PIMPLE algorithm

The PIMPLE algorithm is a combination of the two solution methods presentedabove. This algorithm aims at increasing the Courant number limitation andensuring anyhow stability. This is done adding to the PISO loop an outer loop thatupdates the mass fluxes and so the coefficients of the H(U) operator inside themomentum predictor. If the number of this outer correction loops is set to zero,the PIMPLE algorithm is equivalent to the PISO one. A more intuitive descriptioncould be the one of thinking the PIMPLE algorithm as a method of looking for asteady state solution in each time step. After that convergence is reached, the timeis advanced.

Chapter 4

Turbulence Models

4.1 Introduction to turbulence

The Navier Stokes equations are non-linear partial differential equations inwhich the main non-linearity is provided by the convection term. This term is theone responsible of turbulence. A non-turbulent flow (laminar) presents a stable,parallel and well ordered velocity field with a low momentum convection. If theflow has a sufficiently high Reynolds number, small perturbations may introduce atransition to the turbulent state. In this state the flow properties are characterizedby a high unsteadiness. They oscillate around a mean value with an apparentlychaotic motion, so that the best approach to deal with them is the statistical one.This phenomenon is characterizes by vorticity, so that eddies are to be found. Theresult is that, since momentum is transferred from larger to smaller scales andthen dissipated into heat in the smallest ones, the diffusion of turbulent flows issignificantly higher.

Turbulent flows are characterized by eddies of different sizes: regions occupiedby bigger eddies may include also smaller ones. The largest ones are characterizedby the integral length scale lI , which is of the same order of magnitude of thegeometrical length scales. These larger eddies can be identified by a turbulentReynolds number like:

ReI =

√klIν

(4.1)

with k representing the turbulent kinetic energy:

k =1

2

√u′x

2 + u′y2 + u′z

2 (4.2)

In the equation above the u′i velocities are the oscillations of velocity with respect tothe mean value. This are the velocity fluctuations, following from a decompositionof the instantaneous velocity field as the summation of a mean and an oscillatingcomponent:

Ui = ui + u′i (4.3)

These larger eddies are dominated by turbulence, so that they are unstable andbreak up into smaller ones. This generates a turbulent cascade that ends up to thesmallest scales, where the Reynolds number is small and viscous effects become more

19

20 Chapter 4. Turbulence Models

significant. These are known as Kolmogorov length scales (η) and are responsibleof dissipation.

In numerical methods solving the Navier-Stokes equations, the problem ofturbulence is fundamental. Turbulence could be chosen to be solved up to theKolmogorov scales, with a DNS1 approach. This has severe implications on the cellsize, so that the number of cells results in being very high. Another approach is theLES2 one, where the equations are spatially filtered in the inertial sub-range (wheredissipation is not present yet) and solved only for the larger eddies. The approachwith the less number of cells needed is the RANS3 one, where the equations areaveraged in time, so that turbulence is modelled and only the largest scales arecomputed.

4.2 RANS

Each flow quantity is considered as sum of a mean and a fluctuating part:

ϕ(x, t) = ϕ(x) + ϕ′(x, t) (4.4)

Where the overline indicates the result of a time-averaging procedure:

ϕ(x) =1

∆t

∫ t+∆t/2t−∆t/2

ϕ(x, t)dt (4.5)

When the time interval is large enough, since the field properties are assumed tovary in a random-like manner, the following properties of this averaging procedurebecome true:

ϕ′ = ϕϕ′ = ϕϕ′ = 0 (4.6)

Please note though that the averaging of the product of the fluctuating terms oftwo different quantities remain instead different from zero:

ϕ′ψ′ 6= 0 (4.7)

Given the properties of this averaging procedure, its application on the continuityequation is straightforward. Only linear terms are involved, so that the resultingequation is formally equal to the original one, only with the averaged fields insteadof the instantaneous ones. More issues arise for the other two transport equations.Starting from the momentum equation, the second order term is the convetive one.This leads to the following expression:

∂(ρU)

∂t+∇ · (ρU U) = ρg −∇P + µ∇2U + 1

3µ∇(∇ ·U)−∇ · (ρu′u′) (4.8)

The equation is formally equal to the non-averaged form if not for the last term,which is not going to zero. This is known as the Reynolds stress tensor:

r = −ρ

u′x2 u′xu′y u′xu′zu′yu′x u′y2 u′yu′zu′zu

′x u

′zu′y u

′z

2

1 Direct numerical simulation2Large eddy simulation3Reynolds Averaged Navier-Stokes equations

4.2. RANS 21

This tensor introduces an unknown, so that a closure problem arises. This meansthat this term has to be modelled in order to be able to solve the RANS equationssystem. In the case of the energy equation, if the deformation work terms areneglected, the only non-linear term is the convective one, that in analogy withthe momentum equation gives birth to the so called turbulent heat flux term(∇ · ρcT ′u′).

The Reynolds stress is a symmetric tensor which trace equals: tr(r) = −2ρk.In this way, the tensor can be written as sum of an isotropic and a deviatoric part:

r = −ρ23k + a (4.9)

The Boussinesq hypothesis is then used to model the deviatoric part, in analogy tothe viscous stress tensor. In this way:

a = −2µtD (4.10)

In this way the Reynolds stresses are proportional to the mean strain rate, so thatits deviatoric part acts together with the viscous stresses. Formally the result isan increase of the viscosity thanks to the turbulent viscosity µt that is added to it.The resulting summation is known as the effective viscosity. In a similar way theturbulent heat flux is modelled through the eddy diffusivity kT :

− ρcT ′u′ = kT∇T (4.11)

In order to link the two quantities a turbulent Prandtl number is used, so that,once PrT = c

µTkT

is chosen, the modelling effort is concentrated only on the eddyviscosity.

4.2.1 k-ε model

From dimensional analysis the following proportionality can be written:

µT ∝ lT · uT (4.12)

where

uT ∝√k (4.13)

and

lT ∝√kk

ε(4.14)

So that the turbulent viscosity can be written as:

µT = Cµρk2

ε(4.15)

Where Cµ is a model constant and ε is the dissipation rate of the turbulent kineticenergy. The k− ε model is a two equation turbulence model, meaning that it solvesa transport equation both for k and ε.

22 Chapter 4. Turbulence Models

The transport equation for k is a rigorous one, that can be derived from theRANS equations:

ρ∂k

∂t+ ρU · ∇k = r : ∇U− 2µ(d : d) +∇ · (−P ′u′ + µ∇k − 1

2u′2u′) (4.16)

In the equation above the term d stands for the fluctuation strain rate tensor.The last terms under the divergence operator are the transport terms of k due topressure, molecular diffusion and turbulent transport. The term r : ∇U is theproduction of k from the mean flow while 2µ(d : d) is its dissipation ε.

The second equation that closes the problem is the dissipation rate equation,which is instead fully modelled:

∂ε

∂t+ U · ∇ε = Cε1

ε

kr : ∇U− Cε2

ε2

k+∇ ·

((µ

ρ+µTρσε

)∇ε)

(4.17)

with

Cε1 = 1.44; Cε2 = 1.92; σε = 1.3; Cµ = 0.09;

These coefficients are not fixed, they can be tuned based on experiments or othersimulations (DNS) for the specific problem. The main problem of this approach isthe near wall region, where the quantity k goes to zero and so the term ε

2

kbecomes

singular. For this reason functions describing the wall behaviour in this region mustbe used in order to overcome the problem (wall functions).

4.2.2 k-ω model

In this model the turbulent kinetic energy dissipation rate ε transport equationis substituted from a turbulent frequency one. The turbulence frequency is definedas

ω =1

tT(4.18)

where tT is the time scale, given by tT =kε. Together with the definitions seen

before in equations 4.12, 4.13, 4.14 the resulting formulation is derived:

µT = Cµρk

ω(4.19)

The turbulent frequency transport equation is also modelled:

∂ω

∂t+ U · ∇ω = Cω1

ω

kr : ∇U− Cω2ω2 +∇ ·

((µ

ρ+

µTρσω

)∇ω)

(4.20)

The three terms on the right hand side are the production, dissipation and transportterms. The main advantage of this model is that the near wall region can be nowsolved up to the wall, without the need of wall functions. A disadvantage is insteadthat the free stream turbulence is less accurate and more problematic.

4.2. RANS 23

4.2.3 k-ω SST model

In order to take the advantages of the other two approaches, the k-ω SST modeladopts a combination of both of them: k-ω near the wall and k-ε in the free streamregion. A blending factor is so introduced inside the transport equation in order toregulate this switching.

24 Chapter 4. Turbulence Models

4.3 LES

Large Eddy Simulation is a turbulence model in which the turbulent scalesunder a certain filter length scale are modelled (like in the RANS approach), whilefor the larger ones Navier Stokes equations are resolved (like in DNS approach).According to Ferzinger and Perić (1999, [4]) the large scale motions contain moreenergy than the small scale ones: for this reason they are the most relevant termsin the transport of the conserved quantities. LES approach takes advantage of thisprinciple and solves only these larger eddies, in order to gain in the description ofthe flow with respect to the RANS approach, and to save computational effort withrespect to DNS. The latter is difficult to employ because the Reynolds number isrestricted by the computational resources. This means that LES must be used incases with Reynolds number above a certain threshold.

In order to separate the large scale motions from the small scale ones it isnecessary to filter the Navier-Stokes equations with a filter kernel G(x,x′). Thisoperation consists of eliminating from the equations all wave-numbers of the velocityspectrum above a threshold one. This filtering operation allows to have equationsfree of aliasing problems that arise when the mesh is not fine enough to resolve allthe scales (up to the Kolmogorov one, as done in DNS). The filtered velocity willbe then defined (as in normal filtering processes) as the convolution between thevelocity and the kernel function G (often a Gaussian) (Ferzinger and Perić 1999,[4]):

ũi(x) =

∫G(x,x′)ui(x

′)dx′ (4.21)

In the equation above, x′ is the variable moving in the points surrounding x, that isinstead the vector coordinate of the position. The velocity will be now decomposedinto two functions (McDonough, 2007 [8]) :

Ui(x) = ũi(x) + u′i(x) (4.22)

Every filter is associated to a correspondent length scale ∆ which determines thethreshold under which the equations are simulated. This ∆ is typically definedas the dimension of the cell, so that what is modelled are actually the turbulentscales that have a length scale lower then the grid length scale itself. Even ifthe formalism is similar to the one of the RANS decomposition, there is a bigdifference in the two approaches, demonstrated by the fact that with RANS acertain amount of information is definitely lost, while with LES it depends on thesize of the filter length scale ∆. In fact, as ∆ tends to Kolmogorov length scale theLES converges to DNS, so that the filtered Navier-Stokes equations converge to thenormal Navier-Stokes equations themselves.4 (McDonough, 2007 [8])

4.3.1 LES filtered Navier Stokes equations

In this section continuity and momentum equations for incompressible flowsare presented. Thanks to their lower level of complexity they allow the reader to

4This is true also because the turbulent stress of the sub-grid scale models is typicallyconstructed so that it converges to zero as the mesh size is refined.

4.3. LES 25

concentrate on the main challenges linked to the LES filtering procedure.The continuity equation for incompressible fluids is ∇ ·U = 0, which leads to:

∂Ux∂x

+∂Uy∂y

+∂Uz∂z

= 0 (4.23)

Applying the filter it will result in:

∂̃Ux∂x

+∂̃Uy∂y

+∂̃Uz∂z

= 0 (4.24)

For typical kernel functions (but not for all of them in general), the commutativitybetween the spatial filter and the spatial differentiation holds. (McDonough, 2007[8]) This means that

∂̃Ui∂xi

=∂ũi∂xi

and∂ũx∂x

+∂ũy∂y

+∂ũz∂z

= 0 (4.25)

Since equation 4.22 holds, it can be easily seen that also ∇ · u′ = 0, meaningthat divergence-free condition is true for both the large and the small scales.(McDonough, 2007 [8])

The only term that is not straightforward in the filtering operations is theconvection term in the momentum equation:

∂ũ

∂x+∇ ·

(ŨU

)= −∇p̃+ ν∇2ũ (4.26)

Looking more into detail at the second term:

∇ ·(ŨU

)= ∇ · ˜((ũ + u′)(ũ + u′))

∇ · ˜((ũi + ui′)(ũj + uj ′)) = ˜̃uiũj + ˜̃uiu′j + ũ′iũj + ũ′iu′jThe first of these terms ˜̃uiũj is part of the Leonard stress (see 4.27 ), and it can stillbe computed directly. The second and the third terms ˜̃uiu′j + ũ′iũj are together thecross stress, which in the RANS decomposition is equal to zero, and which instead

in LES has to be modelled. The final term ũ′iu′j is equivalent to the Reynolds stress

of RANS, so the name is conserved also in this approach.

Lij ≡ ˜̃uiũj − ũiũj Leonard stressCij ≡ ˜̃uiu′j + ũ′iũj Cross stressRij ≡ ũ′iu′j Reynolds stress

(4.27)

In the past each of these terms used to be modelled independently, but it was laterrealized how modeling all them together under one term resulted to be actuallymore convenient, since certain numerical properties were conserved (the sub-gridscale stress is invariant to Galilean transformation, which means that it remanisconstant even if the frame of reference changes its velocity (Meneveau, 2010 [9])).

26 Chapter 4. Turbulence Models

Looking at equation 4.26 it can be seen that the term ∇ ·(ŨU

)has to be

written in the form ∇ · (ũũ), otherwise the system cannot be linearized and solved.In order to do this, a correction term has to be introduced to keep the equationvalid, which is the difference between the two mentioned terms and takes the nameof sub-grid scale stress 5:

∇ · τSGS = ∇ · (ŨU)−∇ · (ũũ) (4.28)

where the sub-grid stress is defined as the sum of Leonard, Cross and Reynoldsstresses:

τSGSij = Lij + Cij +Rij (4.29)

It should now be clear why the second term of the Leonard stress was added,

remembering that Lij ≡ ˜̃uiũj − ũiũj. The final resulting filtered momentumequation will then be:

∂ũ

∂x+∇ · (ũũ) = −∇p̃+ ν∇2ũ−∇ · τSGS (4.30)

where the sub-grid stress term has to be properly modelled (the name stress is onlylinked to how it is treated, not to its physical meaning).

4.3.2 The Smagorinsky model

The Smagorinsky model is an eddy viscosity model based on the principle thatthe sub-grid scale stress affects the equations mainly by increasing transport anddissipation. Since this two phenomena are caused by viscosity in the laminar flowcase it makes sense to exploit the Boussinesq hypothesis.

τij =ŨiUj − ũiũj

=1

3τkkδij +

(τij −

1

3τkkδij

)(4.31)

In the equation above, τ is decomposed in its isotropic and anisotropic part, withτkk indicating the diagonal components of the stress tensor. In a matrix approachthis results in the trace of the tensor, where for a generic matrix A:

tr(A) =n∑i=1

aii = a11 + a22 + ...+ ann

Just like in the RANS approach, the Boussinesq hypothesis can be introduced inorder to model the deviatoric part of the sub-grid stress tensor. In this way theturbulent stresses are believed to behave formally as the viscous stresses.

1

3τkkδij +

(τij −

1

3τkkδij

)' 1

3τkkδij − 2νsgsdev(D̃)ij

(4.32)

5In this chapter the tensors are not represented with a double over-line for the sake of simplicity

4.3. LES 27

In the equation above νsgs is the sub-grid eddy viscosity and D̃ij is the resolved-scale strain rate tensor:

D̃ij =1

2

(∂ũi∂xj

+∂ũj∂xi

)(4.33)

Where dev() is the operator that takes the deviatoric part of a tensor: if applied toa generic tensor A this gives:

dev(A) = A− 13tr(A)I (4.34)

The viscosity term has to be modelled here as well, and it is done through aformulation that is in principle similar to the one of the mixing length concept usedin RANS:

νSGS = (CS∆)2|D̃| (4.35)

where ∆ is, as seen before, the filter width, while Cs is the Smagorinsky constantand

| D̃ |=√

2D̃ : D̃ (4.36)

As it can be assessed from equation 4.35 the Smagorinsky model is completelydissipative, that is why it is not appropriate for wall bounded flows. In fact, inthese cases as much as one third of the turbulent energy that reaches the sub-gridscales is then returned to the larger ones (back-scatter phenomenon).

Smagorinsky in OpenFOAM

In this section the implementation of the Smagorinsy model in OpenFOAMaccording to Nozaki’s CFD blog [12] is shown.6

In analogy with the RANS approach, where the turbulent kinetic energy isdefined as

k = − 12ρtr(r) (4.37)

(with r=Reynolds stress) the sub-grid scale kinetic energy is defined as:

ksgs =1

2τkk =

1

2

(ŨkUk − ũkũk

)(4.38)

Substituting this definition inside equation 4.32 the following result is obtained:

1

3τkkδij − 2νsgsdev(D̃)ij

=2

3ksgsδij − 2νsgsdev(D̃)ij

(4.39)

The ksgs is computed from a local equilibrium hypothesis, so that the productionand the dissipation of it are balanced:

D̃ : τ + Cεksgs

1.5

∆= 0 (4.40)

6τSGS is here referred as only τ for the sake of simplicity.

28 Chapter 4. Turbulence Models

The turbulent viscosity is then computed from the following equation, that followsthe principle already adopted in RANS where the viscosity scales with a lengthscale and velocity

νsgs = Ck∆√ksgs (4.41)

where Ck=0.094 as default value. Given these definitions equation 4.40 can bereformulated as follows:

D̃ :

(2

3ksgsI − 2νsgsdev(D̃)

)+ Cε

ksgs1.5

∆= 0

⇒D̃ :(

2

3ksgsI − 2Ck∆

√ksgsdev(D̃)

)+ Cε

ksgs1.5

∆= 0

⇒√ksgs

(Cε∆ksgs +

2

3tr(D̃)

√ksgs − 2Ck∆

(dev(D̃) : D̃

))= 0

⇒aksgs + b√ksgs − c = 0

⇒ksgs =

(−b+

√b2 + 4ac

2a

)2(4.42)

where: a = Cε

∆

b = 23tr(D̃)

c = 2Ck∆(dev(D̃) : D̃

) (4.43)In order to return to the formulation of 4.35 a simplification is introduced, consid-ering the flow incompressible. In this way the following equations can be written:

a = Cε∆

b = 23tr(D̃) = 0

c = 2Ck∆(dev(D̃) : D̃

)= Ck∆|D̃|2

(4.44)

where

|D̃| =√

2D̃ : D̃ (4.45)

by substituting in equation 4.42 the following relation is found:

ksgs =c

a=Ck∆

2|D̃|2

Cε(4.46)

Introducing now this result into equation 4.41 a formulation for sub-grid scale eddyviscosity of incompressible flows can be written as:

νsgs = Ck

√CkCε

∆2|D̃| (4.47)

Comparing with equation 4.35 a final formulation for the Smagorinsky constant Csis now available:

C2s = Ck

√CkCε

(4.48)

4.3. LES 29

Figure 4.1: Second filter in dynamic models (McDonough, 2007 [8])

4.3.3 Dynamic models

Dynamic models are those models proposed at first by Germano et al. in 1991.These models conserve the Smagorinsky structure, the Smagorinsky constant isnow computed as function of time and space. The idea that stands behind it is thatthe resolved scales close to the filter will have characteristics similar to the onesof the scales just underneath the filter (scale similarity model). This is exploitedthrough another filtering operation: it is typical for the filter width of the new filterto be twice as big as the first one, as shown in figure 4.1. (McDonough, 2007 [8])

Since these models were not part of this study, the detailed explanation of theirfunction is here omitted.

4.3.4 WALE model

The Wall-Adapting Local Eddy-viscosity sub-grid scale model was developed byNicoud and Ducros. The authors presented this new model in [11], published inApril 1999. Here follows a summary of their work.

All the eddy viscosity-based SGS models can be rewritten in the form of 4.32,with νsgs defined as follows:

νsgs = Cm∆2ÕP (x, t) (4.49)

In the equation above OP is an operator in space and time and defined fromthe resolved fields: in the previously described Smagorinsky model this operatorcorresponds for example to |D̃|. Nicoud and Ducros proposed a new operator in orderto upgrade the Smagorinsky model and overcome some of its main disadvantages.The main problem of classical models are:

30 Chapter 4. Turbulence Models

1. Energy is concentrated in the streams7 and energy dissipation in eddies andconvergence zones8: Smagorinsky model does not take into account the former,where the vorticity dominates over the irrotational strain. (Nicoud and Ducros1999, [11])

2. Near wall behavior: with Smagorinsky model a non-zero value of the sub-grid scale viscosity is present near the wall, even though physically all theturbulence is damped and the νsgs should be null.

Several different models have been proposed in order to deal with these two mainissues. Resulting solutions are several, the most diffused of which are probablythe dynamic models, described in the previous subsection, where the Smagorinskyconstant becomes a function of both time and space. However, defining a testfilter in complex geometries as it is required for dynamic models may lead to issues(Nicoud and Ducros 1999, [11]).

As a result of these considerations, the WALE model was proposed in order toguarantee better performances and also an easy implementation on more complicatedmeshes. The WALE model presents the operator OP discussed above with thefollowing properties:

1. It is invariant to any coordinate translation o rotation;

2. It is easy to determine on any kind of computational grid;

3. It is function of both strain and rotation rates;

4. No-slip condition at wall is reproduced without the need of neither dynamicmethods or damping function (e.g. Van Driest).

In the WALE model, the ÕP is defined as follows:

ÕP =

(DdijD

dij

)3/2(D̃ijD̃ij

)5/2+(DdijD

dij

)5/4 (4.50)resulting in a formulation of the sub-grid scale viscosity as:

νsgs = (Cw∆)2

(DdijD

dij

)3/2(D̃ijD̃ij

)5/2+(DdijD

dij

)5/4 (4.51)In the equations above D̃ij corresponds to the deformation tensor of the resolvedfield, as defined in equation 4.33, while Ddij is defined as:

Ddij =1

2(g̃2ij + g̃

2ji)−

1

3δij g̃

2kk (4.52)

7Stream regions are regions where the flow is relatively fast, the flow lines are not much curvedand not significantly converging (Hunt et a., 1988 [5]).

8Convergence zones are characterized by irrotational straining motion and convergence of thestream lines, like in stagnation points (Hunt et a., 1988 [5])

4.3. LES 31

with g̃ij = ∂ũi/∂xj the velocity gradient tensor. Finally Cw is a constant of themodel and ∆ is the filter length scale (in practice usually corresponding to themesh size).

The reasoning of how such an operator was chosen from the authors of theWALE method is now introduced. Considering the near-wall behaviour of the flowin a case of a flat plate, placed at y = 0, with a homogeneous incompressible flow,the following conclusion can be drawn:

1. ∇ · u = 0

2. u(y = 0) = 0

3. Because of point 1, the first invariant of both the velocity gradient tensor g̃and the strain rate one D̃ are zero. Please note that the first invariant of atensor is nothing else but its trace.

4. The second and third invariant of D̃ are not zero. In particular the secondinvariant, since the first one is zero, is reduced to −1

2D̃ijD̃ij. This term is

strongly related to the Smagorinsky operator ÕP =

√2D̃ijD̃ij

The idea behind the WALE method is to change the reference tensor D̃ toextrapolate an invariant, since as explained in point 4 both the second and thirdinvariant remain consistently different from zero approaching the wall. In order todo this , an additional tensor has to be introduced:

Ω̃ij =1

2

(∂ũi∂xj− ∂ũj∂xi

)(4.53)

This is equivalent to half of (g̃ − g̃T ), also known as the rotation rate of the flow.In fact its six non-zero components (the three on the diagonal are equal to zero)represent the three components of the rotor of the velocity field and their opposites.

Once that this tensor is defined, the equation 4.52 can be rewritten as :

Ddij = D̃ikD̃kj + Ω̃ikΩ̃kj −1

3δij

[D̃mnD̃mn − Ω̃mnΩ̃mn

](4.54)

Since this tensor is defined as the deviatoric part of the square of the velocitygradient tensor, by definition its first invariant (or trace) is null. Its secondinvariant instead is finite and found to be proportional to DdijD

dij. Assuming the

case of an incompressible flow, this quantity can be written as :

DdijDdij =

1

6

(D2D2 + Ω2Ω2 +

2

3D2Ω2 + 2IVSΩ

)(4.55)

whereD2 = D̃ijD̃ij Ω

2 = Ω̃ijΩ̃ij IVAΩ = D̃ikD̃kjΩ̃jlΩ̃li

In this way it can be well recognized how both the influences of the strain ratetensor and of the rotation rate tensor are both represented. In the case of pureshear, if for example g12 is the only non-zero component of the velocity gradient

32 Chapter 4. Turbulence Models

tensor, then D2 = Ω2 = 4D̃12 and IVDΩ = −12D2D2, resulting in an invariant equal

to zero. This in accordance with what stated before regarding the stream zones:dissipation is concentrated in the eddies and in the convergence zones, while itis way less relevant in the stream ones. In case of wall-bounded laminar flow noeddy-viscosity is produced. This would permit to describe the laminar to turbulenttransition, thanks to the fact that small instabilities will not be dissipated by themodel, as in the case of Smagorinsky model.

In order to get to the final operator OP that characterizes ultimately the WALEmethod, two further steps are still needed. First, in order to reproduce the law ofthe wall with a third order function (∼ y3), the tensor DdijDdij has to be elevated tothe 3

2power. In fact, without this operation the behaviour of DdijD

dij would be only

of the second order (∼ y2). The second step is to bring the operator to the neededdimensions, dividing it by: (

D̃ijD̃ij

)5/2+(DdijD

dij

)5/4This choice looks quite more complicated than a simple division for

(D̃ijD̃ij

)5/2,

but it is necessary in order to avoid the determinant to have eventually a zero-value.Finally, in order to evaluate the constant Cw, a similitude with the classical

Smagorinsky model is assessed imposing the ensemble-average 〈...〉 sub-grid kineticenergy dissipation to be equal. As a consequence:

C2w = C2s

〈√

2(D̃ijD̃ij

)3/2〉

〈D̃ijD̃ijÕP 〉(4.56)

WALE in OpenFOAM

As for the case of Smagorinsky model, also for the the WALE one OpenFOAMdoes not compute directly the νsgs: instead it first derives the value of the sub-gridscale turbulent kinetic energy ksgs. Once this quantity is known, equation 4.41leads then to νsgs. It must be noted that it might be interesting to compare thismodelled kinetic energy with the resolved one, in order to check how much of thetotal kinetic energy is solved and how much is hidden under the grid scale, and somodelled. In order to do this, the standard OpenFOAM solver has to be modified,otherwise the code computes ksgs without giving it as output.

As explained by Nozaki’s CFD blog [12], OpenFOAM computes the sub-gridscale turbulent kinetic energy as follows:

ksgs =

(C2w∆

Ck

)2 (DdijDdij)3((DijDij

)5/2+(DdijD

dij

)5/4)2 (4.57)Substituing this formulation in νsgs = Ck∆

√ksgs it can be noted that this corre-

sponds exactly to the definition given earlier in this chapter (see equation 4.51).

νsgs = (Cw∆)2

(DdijS

dij

)3/2(D̃ijD̃ij

)5/2+(DdijD

dij

)5/4

4.3. LES 33

4.3.5 k-Equation model

The k-equation model, ore one-equation model, is a sub-grid scale model thatalso applies the formalism of Boussinesq hypothesis, so that up to the formulationof 4.32 it corresponds to all the previously described sub-grid scale models. Thedifference is in how the ksgs is computed. Instead of setting the hypothesis thatk is in equilibrium between the smaller and resolved scales, a complete transportequation is resolved:

∂(ρksgs)

∂t+∂(ρũjksgs)

∂xj− ∂∂xj

[ρ (ν + νsgs)

∂ksgs∂xj

](4.58)

= −ρτij : D̃ij − Cερk

3/2sgs

∆(4.59)

In the equation above the terms are representing, from left to right: temporalderivative, convective, diffusion, production and dissipation of the sub-grid scaleturbulent kinetic energy. The last two terms are the only one considered in thebalance for the Smagorinsky model, as seen in 4.40. (Nozaki’s CFD blog [12])

Once that ksgs is computed, νsgs = Ck∆√ksgs leads to the computation of the

sub-grid scale eddy viscosity. In OpenFOAM, since a whole additional new equationfor the new field has to be solved, like in the RANS approach, boundary conditionsfor k are to be set. The solver will compute and write k as output automatically.

Chapter 5

Numerical Results - RANS

The approach chosen for the first part of the study is based on the Reynoldsaveraged Navier-Stokes equations. Thanks to the high level of modelling of thismethod, a very much lower number of cells inside the discretization domain canbe used, resulting in a drastic reduction of computational time with respect toboth LES and DNS. Through the RANS simulations, a first understanding of theproblem was achieved, as well as some important information abut the domain sizeand the boundary conditions that have been subsequently exploited for the LESstudy.

5.1 First studies

During the first phases of the project several different issues arose and a largeamount of time was required in order to deal with them. The most relevant problemwas the generation of unexpected oscillations of the jet. Since the study was startingfrom scratch, the domain size as well as the nozzle outlet conditions were not given,resulting in a high number of free parameters to be set. The final set up is theresult of many different trials of setting these parameters as correct as possible.The whole project was based on comparisons with results found in literature, inwhich the domain dimensions as well as the nozzle exit flow characteristics werefound to be several different ones.

5.2 Domain

A big advantage of the RANS method is that it allows to solve a planar problemwithout the need of solving the third direction (z in this project).1 This kind ofapproach is allowed by the fact that the problem to be solved is the one of a planarjet, which means that the problem is constant along the third direction (this is nottrue anymore for the LES approach). Even if the problem studied is the one of aplanar jet, the nozzle will be described in this work by a parameter d (diameter)

1This is achieved in OpenFOAM through the definition of the patches in z-direction as empty.This option tells the solver to solve the momentum equation only in the x and y directions (x isthe axial direction, y is the transverse direction).

35

36 Chapter 5. Numerical Results - RANS

Figure 5.1: The final domain - RANS, L=length=60d W=width=10d.

which is not referring to a real diameter, but to the nozzle transverse dimension(y-direction).

The obvious result of a 2D domain is that the number of cells in the thirddirection is reduced to only one, so in this case the total number of cells is reducedby one or two orders of magnitude. In picture 5.1 the 2D domain chosen is shown.The final choice of the domain dimensions is the result of an independence testboth in x and y direction. This means that different similar domains were testedin order to check the influence on the results as the length and the width of thedomain were varied.

As reference to evaluate the independence of the results on the domain, the flowfields were sampled on a transverse line at x = 40d. In this region self similarityof the fields should be reached, allowing comparisons among different experimentscarried out in different conditions. (Terashima et al., 2015 [14])

It must be stressed that the dimensions of the domain are to be chosen asthe result of a trade off between region of interest, influence of boundaries on thesolution and computational time. The problem of computational time is of coursewidely increased as the turbulence model is switched to LES since the 3D domainand the need of cell size to be inside the inertial sub-range of the energy cascademakes the number of cells rise steeply.

The domain independence analysis was developed with a mesh that resultedto be coarser than the final one, allowing the study to be less expansive from acomputational time point of view. This procedure is based on the hypothesis thatthe error introduced by the coarser mesh does not influence the impact of thedomain width and length on the solution. The final results were found to be in

5.2. Domain 37

(a) normalized axial velocity (b) normalized transverse velocity

(c) normalized Temperature (d) normalized velocity fluctuations

Figure 5.2: Width independence test vs experiments, tranverse profiles at x = 40d(exp:[14] in (a,d) and [7] in (c))

38 Chapter 5. Numerical Results - RANS

(a) normalized axial velocity (b) normalized transverse velocity

(c) normalized Temperature (d) normalized velocity fluctuations

Figure 5.3: Length independence test vs experiments, tranverse profiles at x = 40d(exp:[14] in (a,d) and [7] in (c))

5.2. Domain 39

agreement with this hypothesis.The domain was chosen to be inclined for two reasons:

1. In this way the cells were automatically shaped so that those closer to thenozzle had a smaller size (in y direction) with respect to those closer to theoutlet.

2. Since the inclination of the lateral boundaries α was chosen so that theyremained parallel to the jet boundaries, all the domains generated with thedifferent lengths resulted to be similar, meaning that the distance betweenthe jet and the lateral boundaries remained constant.

The results of the width and length domain independence analysis are shown infigures 5.2 and 5.3 respectively: the final choice selected the couple of parameters(W = 10d, L = 60d). It should be noted that as the width was varied the lengthwas kept constant to L = 60d, while as the length was varied the width was keptconstant to W = 10d. In these diagrams four different plots for each independencestudy are presented. All of them are transverse profiles sampled at x = 40d, plottedover the normalized tranverse distance: the first one shows the normalized profileof axial velocity, where the axial velocity is divided by the core axial velocity (aty = 0). The lateral coordinate is divided by the half-width distance. In plot b thenormalized profile of the transverse velocity is shown, where the transverse velocityis divided by the core axial velocity. Plot c represents the normalized temperatureprofile over a different normilized lateral coordinate, where instead of the velocityhalf-width distance the temperature half-width distance y0.5,T is used. This quantityis defined as the lateral distance at which the normalized temperature profile equalsthe value of 0.5. In plot d the normalized velocity oscillations are shown, this timenormalized with respect to the velocity oscillations in the core. This quantity iscalculated from the turbulent kinetic energy as in equation 5.1. The hypothesisthat hides behind this definition is that the turbulence is isotropic, meaning thatvelocity fluctuations, once averaged over time, result to be equal in every direction.Equation 5.1 can be then derived from the definition of turbulent kinetic energy(5.2) simply by setting u′x = u

′y = u

′z. In fact, comparing the experimental results

from Terashima et al. (2015 [14]) with a so-computed velocity fluctuation involvesan error in itself, by definition, since the experiments are reported considering thefluctuations only in axial direction. A better comparison could be for examplecomparing directly the turbulent kinetic energies, which are representative of thefluctuations in all the three directions.

u′ =

√2

3k (5.1)

k =1

2

(u′x

2+ u′y

2+ u′z

2)

(5.2)

Looking at the axial velocity profiles, differences are very hard to capture, so thata much higher zoom would be needed to show a real convergence. With convergenceit is here meant a trend of the profiles that shows a higher independence from thedomain size, as the domain in enlarged. Since the scalar field is transported by

40 Chapter 5. Numerical Results - RANS

(a) whole domain

(b) zoom on inlet

Figure 5.4: Mesh 40.000 cells

all the components of the velocity field, a higher difference between the differentcases can be seen for the temperature field, in particular in the length independencetest. In fact, since the temperature is subjected to a very low gradient, it can beconsidered as a passive scalar transported by the flow. In the case of the lengthindependence test on the transverse velocity profile, convergence can be seen alsofrom a larger scale plot: here it is clear how a domain of L = 40d or L = 45d isnot sufficient to describe the flow field at x = 40d. This makes sense since in theoutlet region velocities are still considerably high and the influence of the boundaryconditions on the fields is particularly strong. The final choice of a domain withW = 10d and L = 60d was then believed to be sufficiently correct for the type ofstudy that this work was meant to carry out.

5.3. Mesh 41

5.3 Mesh

The mesh was generated through the OpenFOAM utility BlockMesh. Thisutility allows to build a Hexahedra cell-shape mesh defining one or more blocks,as well as the number of cells along their edges. The distribution of cell-size iscontrolled by the simpleGrading option (see appendix A). A constraint that thisutility imposes is the correspondence one to one between different blocks, meaningthat if two different blocks share an edge, the number of cells as well as theirdistribution will have to be equal for both blocks.

In order to be able to create different domains with different sizes but with similarmeshes, an Excel file was developed and used: in this way some of the characteristicsof the mesh were computed and monitored directly even before running BlockMesh.For exampl