AERODYNAMIC ANALYSIS OF AXIAL FAN …conjunto, el empleo de k !no supondr a una mayor e cacia porque tambi en implica re ujos mayores. La segunda conclusi on es la necesidad de un

University of the Basque Country - Euskal Herriko Unibertsitatea-Universidad del Paıs Vasco

Faculty of Science and Technology- Zientzia eta Teknologia Fakultatea-Facultad de Ciencia y Tecnologıa

Master en Iniciacion a la Investigacion en

Matematicas

Master Thesis

AERODYNAMIC ANALYSIS OF

AXIAL FAN UNSTEADY SIMULATIONS

September 2012

Author

Imanol Garcıa de Beristain

Advisors

F. Palacios & L. Remaki

Agradecimientos (Acknowledgements)

Me gustarıa mostrar mi agradecimiento a todo aquel que haya contribuido a la creacion de estatesis de alguna manera.

To Basque Center for Applied Mathematics for the trust they put on me. Specially to SergeyKorotov for letting me join up his group. To Francisco Palacios for his help and patienceduring my beginnings in the field. At last, thanks also goes to Lakhdar Remaki for helping mefinishing this thesis and introducing me into my next steps.

Agradezco ası mismo al Servicio Tecnico de Informatica Aplicada a la Investigacion de laUPV/EHU, por haberme facilitado el uso de STAR-CCM+ en mi ordenador personal.

Por ultimo, mi mas sincero agradecimiento para todos aquellos con los que he compartido eldıa a dıa de este largo ano. Gente que todavıa sigue ahı y gente que ya no esta. Por supuesto,tambien a mis padres por llevar conmigo esta carga.

iii

Aerodynamic Analysis of Axial Fan Unsteady Simulations

Abstract: The objective of this thesis is to understand turbomachinery unsteady CFD simu-lation performance depending on the turbulence model selected. For this porpoise, simulationswith two in industry extensively used turbulence modes have been carried out: k− ε and k−ωmodels. Simulations were performed using the commercial software STAR-CCM+. Resultshave been compared between them and with analytically obtained simplified solutions. Thiswill allow to judge real industrial case simulations.

Flow Rate results showed the same mean value downstream the fan for both turbulence mod-els. However, oscillations induced by unsteady condition had different amplitudes. Boundarylayers have been studied as well. It wasn’t found any difference among both turbulence mod-els results, but simplified analytical problems solution predicted smaller boundary layers thanSTAR-CCM+ simulations.

We obtained two main conclusions. First, k − ω is overpredicting blade performance comparedto k − ε turbulence model because the second is more dissipative. However, if whole fan isconsidered, the extra blade efficiency on k − ω turbulence model is lost because of biggerrecirculation zones. The second conclusion is the need for further understanding on boundarylayer simulation, since deviations from expected results by both turbulence models can not beaccurately explained by the author. However, it is probably related to surface curvature orblade edge pressure-gradient induced streamline curvature.

Keywords: CFD, aerodynamics, axial fan, boundary layer, turbulence.

Analisis Aerodinamico de Simulaciones No Estacionarias de VentiladoresAxiales

Resumen: El objetivo de esta tesis es comprender la dependencia de los modelo deturbulencia en simulaciones no estacionarias de ventiladores axiales mediante CFD. Coneste fin, se han realizado simulaciones empleando dos modelos de turbulencia ampliamenteutilizados en la industria: los modelos k − ε y k − ω. Dichas simulaciones se llevaron acabo empleando el software commcercial STAR-CCM+. Los resultados con los diferentesmodelos de turbulencia se han comparado entre si, y con las soluciones analıticas deproblemas simplificados. Con esto se pretende ser capaz de valorar las simulaciones decasos industriales reales.

Los resultados muestran una media de aire igual para ambos modelos de turbulencia.Sin embargo, las oscilaciones alrededor de esta media son diferentes para ambos modelos.Los resultados de las capas lımite son independientes respecto del modelo de turbulenciaesperado. Sin embargo, los modelos analıticos simplificados predicen capas lımites maspequenas que las obtenidas mediante simulaciones.

Se han obtenido dos conclusiones principales. Primero, el empleo del modelo k − ωresultara en un rendimiento de alabe mayor que el obtenido mediante k− ε, debido a queel segundo modelo es mas disipativo. Por otro lado, si se considera el ventilador en suconjunto, el empleo de k − ω no supondrıa una mayor eficacia porque tambien implicareflujos mayores. La segunda conclusion es la necesidad de un mejor entendimientosobre la simulacion de las capas lımite puesto que no se ha logrado explicar de formaconvincente la diferencia entre los resultados esperados mediante modelos simplificadosy las simulaciones. Lo mas probable es que se deba a la curvatura de la superficie o a lacurvatura de las lineas de flujo inducidas por gradientes de presion.

Palabras clave: CFD, aerodinamica, ventilador axial, capa lımite, turbulencia.

Contents

1 Introduction 1

2 Background 32.1 Turbomachinery and Centrifugal Fans Essentials . . . . . . . . . . . . . . . . . . 32.2 Fan Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Basic Characteristics of the Fan Flow Aerodynamics . . . . . . . . . . . . 6

2.3 CFD Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.1 Flow Physics Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Turbulence Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 Numerical Solution Techniques . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Turbulence 133.1 Reynolds Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Reynolds Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 Anisotropy and Tensor Properties . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Turbulent Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.1 Momentum Equations and Velocity Profiles . . . . . . . . . . . . . . . . . 173.2.2 Kinetic Energy and Reynolds-Stress Balances . . . . . . . . . . . . . . . . 213.2.3 Theory Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.4 The Mixing Length Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Eddy Viscosity Based Turbulence Modelling . . . . . . . . . . . . . . . . . . . . . 243.3.1 Eddy Viscosity Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.2 Wall Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.3 Realizable Two-Layer k − ε . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.4 Menters SST k − ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Simulations 314.1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2.1 Mass Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.2 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2.3 Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Conclusions and future work 39

Bibliography 41

v

Chapter 1

Introduction

Research and investment in technology is often driven by market or legislation requirements.Turbomachinery is subject to the same conditions and fans are not an exception. For example,noise damping orNOx reduction in aircraft engines are new research-lines driven by new normat-ive constraints. On the other side, stage efficiency is many times imposed by the manufactureritself to keep competitiveness in the market. Stage efficiency translates into specific aerody-namic performance requirements. New components will need much greater complexity duringits design, including a higher degree of three dimensionality analysis and flow-path configura-tions.

Design tools traditionally available in engineering, such as basic relative velocity triangles, havestrong limitations to comply with demanded complex research. The use of advanced aerody-namic tools have to be developed. At this point Computational Fluid Dynamics (CFD) isbeing extended in the turbomachinery community to fulfil this task. Its ability to simulate flowphysics at any point in the domain, with better predicting capabilities every year, is extremelyattractive. CFD has already upgraded design of turbomachinery from 2-D inviscid flow modelsto 3-D, viscous, turbulent analysis [Earl Logan and Roy, 2003].

In this thesis unsteady simulation of an axial fan flow field is studied. Although aerodynamiccomponents such as boundary layers or recirculation zones are considered, boundary layers takemost of the weight of the thesis, since blade performance is directly related to phenomena suchas boundary layer thickness or boundary layer detachment. Although boundary layers havebeen target for research for many decades, they are still under study. It is the case of aero-engines, which consider boundary layer induced vortexes because of the impact in aeroelasticity,blade thermal resistance, or noise generation. For this reason it was found appropriate to focusthis thesis on a basic and yet extremely valuable boundary layers.

Boundary layers modelling is strongly dependant of viscosity an Reynolds Stresses. For thisreason, turbulence modelling is included besides boundary layers theory. In view of the extensivearea of turbulence in bibliography, it was decided to cover the basics of Reynolds AveragedNavier Stokes (RANS) models, which will be linked to the boundary layer theory during itsintroduction.

This thesis is structured in five main Chapters. Chapter two introduces the basics of turboma-chinery aerodynamics and CFD. It also gives an overview of the state-of-the-art in the topic.It covers from airfoil lift explanation to latest trends in CFD simulations. Chapter three is thetheoretical core of the thesis. Starting from the Reynolds Equations, boundary layer structure isdeduced, to conclude with how turbulence is modelled within STAR-CCM. In Chapter four theSimulation set up and its results are reported such that it can be reproduced by the interestedreader. At last, the conclusions are explained in Chapter five.

1

2 CHAPTER 1. INTRODUCTION

Chapter 2

Background

2.1 Turbomachinery and Centrifugal Fans Essentials

Turbomachinery is found everywhere in the modern world. This big family of machines includepumps, turbines and fans. The essential components are the rotor, which obviously is rotating;a shaft from which the energy is extracted or added to the rotor and a casing where the shaftstems from. Fluid is introduced through pipes in the case. Turbomachinery works transferringenergy between fluid and rotor. When energy is extracted from fluid, the machine is calledturbine, in the opposite case it will be a pump, fan or compressor.

The rotor is mainly made from blades. This blades have a specific shape in order to make thefluid flowing between two blades to execute a specific force on the blades. It is commonly findas well some fixed components in order to drive the fluid smoothly.

There are other ways of turbomachinery classification. A common way is the rotor flow exitdirection: axial, radial or mixed.

• Axial: Fluid flow is parallel to the axis. Ideally there is no radial component of the flow,only axial and tangential

• Radial: Fluid flow is orthogonal to the axis, there is no axial velocity of the fluid.

• Mixed: The flow has 3 velocity components within the rotor: axial, radial and tangential.

Acording to Bleier [1998], there are 4 types of axial flow fans,

1. Propeller fans (PFs)

2. Tubeaxial fans (TAFs)

3. Vaneaxial fans (VAFs)

4. Two-stage axial-flow fans

Propeller fan, sometimes called panel fan, is the most commonly used fan in any kind of applic-ation or environment.

Tubeaxial fans have a cylindrical housing. Gas exhaustion is the most common application forthis fans. The main negative outcome is the fast increase in the outlet duct friction losses dueto air spin. When venturi inlet is used instead of a duct friction losses are reduced about 10 %of and noise level damped.

Vaneaxial fans has a housing, like tubeaxial fans, but they have guided vanes that neutralizesthe spinning air, so the unit is usable for blowing and exhausting (exit and inlet ducts). Use ofventuri inlet is possible as in tubeaxial fan.

Two-stage axial-flow fans are two fans connected in series, so pressure increase add up. It isuseful when excessive tip speeds and noise levels are not tolerated. There might be guided vanesbetween two rotors rotating in the same direction. If counter-rotating rotors are used, guidedvanes are unnecessary.

3

4 CHAPTER 2. BACKGROUND

LE

D

L F

VTE

Pressure sideof airfoil

Chordline

V Relative air velocityα Angle of attack L Lift

LE Leading edgeD DragTE Trainling edge

F Resultant Force

Suction sideof airfoil

α

Figure 2.1: Shape of a typical NACA airfoil. Source: Adapted by the author from [Bleier, 1998]

The operating principle of axial-flow fans is simply deflection of air as will be described laterin airfoil aerodynamics (section (2.2.1)). After flow passes the blades, the flow pattern hashelical shape. Flow can be decomposed into two component: axial velocity and tangential orcircumferential velocity. Axial velocity is the desired velocity since it moves air from/to thedesired spaces. Tangential velocity is an energy loss in propeller fan or tubeaxial fans. Invaneaxial fans tangential velocity can be converted into static pressure. This makes vaneaxialfans more efficient.

For good efficiency on the airflow of an axial fan it is usually demanded evenly distributed flowover the working face of the fan wheel. This means axial velocity should be the same from hub totip on each blade. However, blade velocity is function of the radial distance. Velocity gradientsare then compensated by blade twisting, which outcomes in a smaller blade angle toward thetip. Same hub to tip blade angle results in a loss of fan efficiency since air propulsion willtake place mostly on the outer region of the blade. Incorrect blade twist might cause stall inthe interior portion of the blade strongly hindering efficiency when working with higher staticpressures.

2.2 Fan Aerodynamics

2.2.1 Airfoils

Airfoils used in fan blades are asymmetric. The best well known airfoils are NACA airfoils,which have been developed by the National Advisory Committee for Aeronautics. A NACA n◦

6512 is shown in figure (2.1). Some features of this airfoils are acording to Bleier [1998] :

• The airfoil has a blunt leading edge which provides robustness to small inlet flow per-turbations, and because of structural strength characteristics. The trailing edge is rathersharp.

• The chord line is defined as the line that connects the two lowest points of a two dimen-sional airfoil section when is laid on a flat surface. The airfoil chord, c, is the length of

2.2. FAN AERODYNAMICS 5

the blade profile orthogonal projection onto the chord line.

• The airfoil has a convex upper surface, with the maximum at 36 % of the chord, and amaximum section height from the chord of 13.3 %.

• A concave lower surface, with maximum distance of 2.5 % of c located at 64 % of the chordfrom the leading edge. It might happen the lower surface to be flat instead of concave forsome applications.

• The angle of attack, α, is measured as the angle between the relative air velocity and thebase line.

• As the airfoil moves through the gas, it produces a positive pressure on the lower surfaceof the airfoil (or pressure side) and a negative pressure gradient in the upper side (orsuction side). Both forcces have approximately the same direction, but suction force isclose to twice the positive gradient force.

The forces described in the last point define F when they are added. This force can be decom-posed into lift and drag forces. The first is perpendicular to the relative air velocity whereasthe second is parallel. Lift is the desired component in most engineering applications. Dragis undesired since it causes power-consumption. However, this two objectives are conflictingeach other, and a trade off hast to be made while designing to obtain high lift forces, but goodlift-drag ratios. As the maximum section height of the airfoil profile increases, lift usually in-creases but lift-drag ratio tends to worsen. Selection of airfoil shapes is done according to thedesired application. For example, for compressors, wide airfoils are used. When efficiency isthe important parameter to be considered thinner airfoils are employed. Further, this forcesare strongly dependent on the angle of attack, and the overall range of operation has to beconsidered when choosing the appropriate airfoil shape.

Aspect ratio is an important feature. It is the ratio between the total blade height and thechord length. The bigger the aspect ratio, the better lift and lift-drag ratio. This is explainedby the trailing vortex phenomena. Trailing vortexes are generated when the fluid flows from thepressure side to the suction surface though the outward space of the airfoil tip (the clearancespace in case of turbomachinery with casing). They strongly hint lift force. Big aspect ratiois favorable because trailing vortexes have influence on a certain distance of the total wing.The longer the wing, the smaller the overall efect of the vortex. The use of hubs is commonin turbomachinery because reduces turbulence and trailing vortexes at turbomachinery tips,increasing lift forces.

Some basic airfoil performance facts are introduced next,

• If the airfoil was symmetric, zero lift force would be found at angle of atack 0◦ due tosymmetry considerations.

• As it can be seen in figure (2.3), as the angle of attack is increased, lift coefficient increasesas well. The maximum for this example curve is found at 15◦. It is the maximum operatingpoint of the airfoil.

• The maximum lift-drag ratio is fount at 1◦. For this example, best operating rate isregarded to be the range [1◦ − 10◦], where ratio is still high and airflow is smooth.

• Angles of attack from 10◦ to about 15◦ are acceptable. Fluid streamline can still followthe contour of the airfoil.

• When the angle is higher than 15◦, the airfoil stalls. A huge fall in the efficiency occursdriven by the boundary layer detachment.

The way airfoils are used in axial fans is shown in figure (2.4).


Figure 2.2: Trailing Vortex generation aroundthe tip of an airfoil

Figure 2.3: NACA 6512 airfoil infinite aspectratio characteristic curve

2.2.2 Basic Characteristics of the Fan Flow Aerodynamics

Many different and diverse flow features are involved in turbomachinery which goes from su-personic velocity to rotating flows.

It is off special interest complex stress and performance losses that result from viscous flowphenomena, mainly located at the airfoil boundary layers, but also blade to wall transientboundary layers, near-wall flow migration, tip clearance and trailing vortexes , wakes and mix-ing. Research is also being done in relative end-wall motion and transition between rotationand stationary end walls.

Another trend in aerodynamics research is the unsteadiness of the flow for time-varying con-ditions, vortex shedding from blade trailing edges (and impact in aspect ratio as describedin section 2.2.1), flow separation, or intersections between rotating and stationary blow rows,which imparts unsteady loading on the blades and thus the life span of the fan.

In centrifugal fans, when high flow is impelled, non-negligible boundary layers are formed inthe second half of the impeller flow passages. Consequently, flow separation is encounteredin the suction surface, causing wakes region, shifting the flow toward the pressure side. Flowseparation diminishes diffusion potential for the impeller and distorted jet/wake structures arefound in the discharge. Mixing follows this jet/wake structure and unsteady flow runs into thediffuser. Overall, this wakes reduces the efficiency of the fan.

A well-known study of this kind is the one performed by Eckardt [1976] in a radial centrifugalcompressor. He obtained detailed measurements of flow velocities and directions at multiplelocations in the flow field, from the inducer inlet to the impeller discharge. Eckardt observed

2.2. FAN AERODYNAMICS 7

Trailing edge of airfoil blade

Deflec

ted

air fl

ow

Incomingair flow

Bladeangle

Rot

RotHubdia.

Whello.d.

Leading edge of airfoil blade

Pressure side on concave of airfoil blade

Suction side on convex side

Figure 2.4: Airfoil shaped blade axial fan [Bleier, 1998]

that flow kept undisturbed within the axial inducer and during the first 60 % of the impellerblade chord. At the 60 chord % distance, a flow separation originated in the shroud suction-side corner of the passage. The separation rapidly grew to give raise to a wake. This wake isproduced from secondary flows as will be seen later. In short, vortexes near the shroud andthe hub/suction-side corners causes detachment in the boundary layers off the channel walls(including blade surfaces) and fed the low-energy fluid into the wake. Additional low-energypasses through the tip clearance space causing the wake to increase through the downstream halfof the impeller flow path. The pattern of high and low-energy fluid (jet and wake respectively)is sustained up until the impeller discharge supported by the system rotation and curvature,that causes non-mixing between jet and wake structures.

Meridional flow in centrifugal compressors is usually highly non-uniform, dominated by sig-nificant jet/wake flow after halve of the chord. Peak meridional velocities are located at theblade hub-pressure sides, due to potential flow at high Reynolds numbers. Jet/wake structurecauses the flow to separate at the hub, and peak velocities locate at the blade tip-pressure side.The prediction of this meridional profiles can be done by simple modelling, empirical models oremploying Euler flow solvers (for high Reynolds numbers). After meridional flow is obtained,normal and binormal vorticity can be numerically evaluated and applied to vorticity equationsto calculate streamwise vorticity. The use of potential flow solvers can be employed to solvepassage circulatory secondary flows.

n-Euler equation

The well known Bernoulli or Euler equations describe the mechanical energy terms (pressureand velocity) along a streamline with constant energy. Whereas the n-Euler equation describesthe forces normal to this streamline. Thus, it is the equation explaining why lift occurs inairfoils or why secondary flows/vortexes are created. The equation is obtained balancing thecentripetal acceleration of a fluid particle and the net force in the normal direction to thestreamline. Consider the mass of a fluid element to be ρR dθ dn. The particles centripetal


Figure 2.5: Separated jet-wake structure from impeller [Tuzson, 2000]

acceleration towards the centre of the streamline is proportional to ρv2/R. For an inviscid fluidin equilibrium, in absence of significant body forces the equation obtained is:

∂p

∂n= ρ

v2

R(2.1)

One might read this equation as follows. When a steady streamline is curved there is a cent-rifugal pressure gradient force acting on the streamline fluid particles. Pressure increases withcurvature radius.

As we stated before, equation (2.1) contains the principle behind airfoil lift and secondaryflows. The concave curvature of the streamline in the pressure side of an airfoil means higherpressure is located below this surface compared to the undisturbed flow in the middle of thepitch (line equidistant between two consecutive blades). Same reasoning allows to demonstratelow pressure is encountered at the suction surface of the airfoil compared to the undisturbedflow. A pressure difference force driven by the airfoil surface is created as sketched in figures(2.6) and (2.7)

Using n-Euler equation we will describe the secondary flow formation. Imagine an axial fan bladeturning. Employing the n-Euler equation, we deduce a pressure gradient has to be created fromthe streamline curvature described in the lift force generation. However, further implicationshave to be considered in the near wall region (usually hub and case), where the flow velocityis smaller although a similar pressure-gradient (because of mechanical stability across the spanof the airfoil) exists. From the n-Euler equation, for constant pressure gradient, we deduceρv2/R = constant. If v is decreased due to viscosity effects, R has to decrease as well. Thestreamlines follow different passage trajectories close to the walls as shown in figure 2.8. Theoverturning of the fan moves slow velocity fluid close to the wall from the blade pressure surfaceto the blade suction surface, and the motion is compensated by a return flow near the centreof the passage. The combined effect is that two three-dimensional passage vortexes are set upin the streamwise directions. These vortexes are one of the main sources of secondary flow inblade passages. To completely understand this vortexes full momentum equations have to be

2.3. CFD BASICS 9

Figure 2.6: Lift force description by the pres-sure gradient of n-Euler equation

Figure 2.7: n-Euler equation physical repres-entation

Figure 2.8: Secondary flow development in turbomachine blade passage. Cross-stream free flow(A) and the end wall boundary layer streamline (B) (left). Passage vortexes (right) [Japikseand Baines, 1997]

employed. This inertia-generated vortexes might be known as circulatory flow. The overalleffect of the vortex is a efficiency loose in the turbomachinery.

In general, secondary flows are always caused by static pressure and kinetic energy imbalance.The most studied vortex generation mechanism is the horseshoe vortex by a stagnation line:an incoming boundary layer meets a stagnation line and causes a motion of the fluid alongthe wall, with a subsequent vortex. The strength of the vortex is determined by the startingconditions, and its evolution to the conservation of its angular momentum. The vortex flowsare principally generated by the meridional flow field, while the centrifugal and Coriolis forcesact on the vortex change of the direction (tilting of the plane) [Brun and Kurz, 2005].

2.3 CFD Basics

This section analyses the state-of-the-art in CFD for both turbomachinery design and com-plex flow filed analysis. Bear in mind this two objectives may demand diferent computational


resources according to the results desired. For example, in we find two stages in componentdesign: preliminary and detailed design. During preliminary stage many variables are involvedand consequently many prototypes are proposed. At this point we expect a tool which allowsa selection of the most suitable options as fast as possible. Accurate solutions are not needed.CFD methods are suited to this requirements. We might be looking for efficiency improve-ments by blade-row spacing analysis or initial blade shape design. Detailed design focuses ona small number of design parameters based on preliminary design analysis. Examples for tur-bomachinery are tip clearance flows, blade-end wall interactions, flow separation, wakes, etc.Detailed analysis simulations require order of magnitudes longer time because flow physics hasto be usually accurately resolved.

2.3.1 Flow Physics Modeling

For industrial applications Navier Stokes Equations are not tractable due to computational lim-itations. Mathematical treatment such as averaging is typically done to obtain the Reynolds-averaged Navier-Stokes equations (RANS) in the case of Reynolds averaging. Other mathem-atical treatments produce other models such as Large Eddy Simulation (LES), which is morepopular in research. The set of equations including mass, momentum and energy conservationare known as full Navier Stokes Equations. However, use from auxiliary equations is necessarysuch as the equation of state (usually perfect gas law in fans), the Stokes hypothesis whichrelates the second coefficient of viscosity (or bulk viscosity) to the molecular viscosity, andSutherland’s law, which expresses molecular viscosity as function of temperature. Turbulencemodel for the Reynolds stress closure is also added to the previous system of equations. Useof full RANS equations allows simulation of unsteady, 3D, viscous, turbulent flows in rotationwhere the primary dependent variables are density, three velocity directions, total energy, pres-sure, enthalpy, nine components of the turbulent Reynolds stress tensor and three componentsof turbulent heat-flux vector.

Further techniques like the thin layer assumption is widely used to reduce the complexity ofthe problem. Within this assumption the streamwise diffusion term is neglected. It is used inviscous layers, but must not be used when recirculation zones or viscous structures producingstreamwise diffusion is present.

Solution of the PDE must be done altogether with appropriate boundary condition. Threetype of spatial boundaries may be identified for turbomachinery: (1) wall boundaries, (2) inletand exit boundaries, and (3) periodic boundaries. Wall boundaries refers to blade surfaces,passage walls, etc. Solid surfaces might be rotating, non-rotating or a combination of them.Zero-relative-velocity (non-slip) conditions should be used.

The most natural form of inlet and exit boundary specifications are mass flow rate for inletand pressure conditions at the outlet. To do this, pressure, temperature, tangential velocityupstream and pressure at downstream is usually specified. Depending on the turbulence modelselected, some turbulence properties are required. The inlet/exit boundaries should be placedfar enough from the blade, so they are not influenced by its presence. Typical distances to thisboundaries are 50 % to 100 % of the blade chord. Distribution of inlet conditions might beincluded in the spanwise and tangential directions.

Periodic boundaries upstream and downstream of the blade are used to model one blade passageto the next, assuming inlet conditions are periodic. Periodicity is forced by setting dependentflow variables equal at equivalent positions on the periodic boundaries. Straight forward initial-ization of the problem can converge into the solution with no problem. However, an appropriatedistribution will fasten the convergence. This might be done by imposing the solution from a

2.3. CFD BASICS 11

preliminary design. For example, unsteady simulation from a previous steady solution, or in-viscid flow solution for an viscid simulation.

2.3.2 Turbulence Closure

As will be seen in next section, there are a handful of turbulence models providing closure ofthe Navier Stokes equations. They range from simple algebraic relationship to a set of PDE.The most commonly used models during preliminary design are two-equation models and FullReynolds stress models. Turbulence tretment such as Large eddy simulation (LES) and fullNavier-Stokes equations are to expensive at this stage.

n-equation models represent into some extent the real turbulence physics. They use of n partialdifferential equations which model transport of selected turbulence variables. For example, intwo equation models kinetic energy and turbulent energy dissipation are most often selected.When solution of transport equations are computed, they are introduced in algebraic models toobtain what is known as the turbulent viscosity.

Full Reynolds stress model is a much more realistic representation of a turbulent flow but ismakes use of approximately twice the number of equations. It Solves transport equations forall components of the specific Reynolds stress tensor R = –Tt/ρ ≡ v′v′. These model naturallyaccount for effects such as anisotropy due to strong swirling motion, streamline curvature,rapid changes in strain rate and secondary flows in ducts. However, this accuracy increases thecomputational cost.

LES and full Navier-Stokes equations are useful in some cases during detailed simulation stageor very sensitive physics simulation. For example, noise creation due to vortex generation inairfoil tips is a good example for use of LES turbulence models.

2.3.3 Numerical Solution Techniques

Solution of fluid dynamic equations is a huge area in CFD which basics should be known bythe interested people in the field. A basic introduction will be done here.

First of all, PDE equations to be solved are discretized. Within fluid simulation, the most usedmethod for discretization is the Finite Volume Method. Finite Difference Method had moreusers in the past, but requirements of structured grids compared to the flexibility of FiniteVolume Method, hinders its selection. Finite Element Method is also very common especiallyin multiphysics simulations.

After discretization, the system of algebraic equations have to be numerically solved. Accordingto Earl Logan and Roy [2003], the most appropriate solution techniques are the time-marching(unsteady) explicit or implicit schemes, although steady Governing-Equations are also possible.Explicit time marching schemes are simpler and use information at grid cells from previoustime-intervals. It is less computationally expensive than implicit methods. However, stabilityis an issue to be considered when setting up the problem: time-step size is dependent on gridsize. On the other hand, implicit methods are numerically unconditionally stable to any time-step chosen. In this method, equations are solved all together in a coupled matrix. Obviously,the number of equations increases in the same rate as number of grid points does. Avoidanceof matrix inversion by factorization procedures is usually done to reduce the computationalcost. Hybrid methods might be used as well, which benefits from both implicit and explicitcharacteristics. Some examples include the implicit residual smoothing in an explicit Runge-Kutta technique to relax the stability criterion. Or the two-step explicit one-step implicit Beam


and Warming algorithm. Use of local time stepping is another approach. It is only useful forsteady-state simulations. In this case, time-step in time marching methods does not have anyphysical relevance, and different time steps might be set to different grid points in the mesh.This allows stability to be satisfied while solution convergence is as fast as possible through allthe mesh.

Non of implicit or explicit methods have demonstrated overriding performance against the otherand methods have similar levels of maturity. However, explicit methods are less computationallyexpensive and are more efficient in parallel-processor and vector computers.

Whether implicit/explicit unsteady/steady has been chosen, two type of solvers are available:Coupled or Segregated solvers.

The Segregated solver considers the flow equations (one for each component of velocity, and onefor pressure) in a segregated, or uncoupled, manner. The linkage between the momentum andcontinuity equations is achieved with a predictor-corrector approach. This model has its rootsin constant-density flows.

The Coupled solver on the other side considers the conservation equations for mass and mo-mentum simultaneously when solving using a time- (or pseudo-time-) marching approach. Thepreconditioned form of the governing equations used by the Coupled Flow model makes it suit-able for solving incompressible and isothermal flows. One advantage of this formulation is itsrobustness for solving flows with dominant source terms, such as rotation. Another advant-age of the coupled solver is that CPU time scales linearly with cell count; in other words, theconvergence rate does not deteriorate as the mesh is refined. On the negative aspects of thecoupled system we find the need of high computational memory resources.

In this thesis segregated flow solver has been employed since the number of cells used exceededthe coupled solver capabilities in the computer employed. Formulation of Governing-Equationsin STAR-CCM is

d

d t

∫VρχdV +

∮Aρ(v − bmvg

)· da =

∫VSu dV (2.2)

d

d t

∫Vρχv dV +

∮Aρv ⊗ (v − vg)· da = −

∮ApI· da +

∮AT ·a +

∫V

(fr + fω) dV (2.3)

The terms on the left-hand side of equation (2.3) are the transient term and the convectiveflux. On the right-hand side are the pressure gradient term, the viscous flux and the body forceterms. T is the viscous stress tensor (or Reynolds stress tensor). The body force terms havebeen simplified to represent exclusively the effects of system rotation.

In turbulent flow, the complete stress tensor is given by:

T = µeff

[∇v +∇vT − 2

3(∇·v) I

]

where the effective viscosity is µ = µl + µt , the sum of the laminar and turbulent viscosit-ies.

Chapter 3

Turbulence

Turbulence is, of course, an important component to be considered. It must represent thecharacteristics of typical turbomachinery flow fields, such as flow-path curvature, rotating flow,high-pressure gradients, and separated, recirculating flows. The capability to model unsteadyflow and blade-row interaction is also necessary. In order to study turbulence, a simple in-troduction to the fluid motion equations will be done. They will be applied to Newtonianincompressible flows.

3.1 Reynolds Equations

Decomposition of the velocity field U(x, t) into the mean flow⟨U(x, t)

⟩and the fluctuation

term u(x, t) is a common procedure.

u(x, t) = U(x, t)−⟨U(x, t)

⟩. (3.1)

This decomposition is known as the Reynolds decomposition.

The well known continuity equation

∂ρ

∂t+∇ · (ρU) = 0

is simplified for incompressible flows to yield the solenoidal or divergence-free equation

∇ ·U = 0. (3.2)

and applying relationship 3.1,

∇ · 〈U〉 = 0

∇ · u = 0.

In the case of the momentum equation some nonlinear terms are created after applying Reynoldsaveraging.

First we write the substantial derivative conservative form.

DUjDt

=∂Uj∂t

+∂

∂xi

(UiUj

),

and calculate the mean value ⟨DUjDt

⟩=∂⟨Uj⟩

∂t+

∂

∂xi

⟨UiUj

⟩(3.3)

Nonlinear average of⟨UiUj

⟩is⟨

UiUj⟩

=

⟨(〈Ui〉+ ui

) (⟨Uj⟩

+ uj

)⟩=⟨〈Ui〉

⟨Uj⟩

+ ui⟨Uj⟩

+ uj 〈Ui〉+ uiuj

⟩= 〈Ui〉

⟨Uj⟩

+⟨uiuj

⟩(3.4)

13

14 CHAPTER 3. TURBULENCE

The velocity covariance⟨uiuj

⟩is the so-called Reynolds stress.

employing equations (3.2), (3.3) and (3.4) we rewrite the total mass derivative⟨DUjDt

⟩=∂⟨Uj⟩

∂t+

∂

∂xi

(〈Ui〉

⟨Uj⟩

+⟨uiuj

⟩)=∂⟨Uj⟩

∂t+ 〈Ui〉

∂

∂xi

⟨Uj⟩

+∂

∂xi

⟨uiuj

⟩(3.5)

Further, if we define the mean mass derivative, which represents the rate of change of a pointmoving with the local mean velocity

⟨U(x, t)

⟩as

D

Dt≡ ∂

∂t+ 〈U〉 ·∇

we obtain the relationship with the mass derivative (3.5)⟨DUjDt

⟩=

D

Dt

⟨Uj⟩

+∂

∂xi

⟨uiuj

⟩.

The momentum equation (or Navier Stokes Equation) for an incompressible Newtonian fluidis

DU

Dt= −1

ρ∇p+ ν∇2U . (3.6)

And relating with previous deductions, its mean mass derivative or Reynolds equations

D⟨Uj⟩

Dt= ν∇2

⟨Uj⟩−∂⟨uiuj

⟩∂xi

− 1

ρ

∂ 〈p〉∂xj

. (3.7)

The structure of the Reynolds and the Navier-Stokes equations might look similar, but in theReynolds equations the Reynolds stresses appear, which give rise to turbulence modelling.

3.1.1 Reynolds Stresses

U(x, t) and⟨U(x, t)

⟩show very different behaviour due to the Reynolds Stresses, which are

one of the big puzzles of the field

This stresses are better understood under the form

ρD⟨Uj⟩

Dt=

∂

∂xi

µ(∂ 〈Ui〉∂xj

+∂⟨Uj⟩

∂xi

)− 〈p〉 δij − ρ

⟨uiuj

⟩ . (3.8)

The first term between brackets comes from the molecular description of the flow, it is calledthe Viscous Stress. In Pope [2000] in shown how the Reynolds Stresses are also deducted whencalculating the mean momentum transfer.

As we saw with the full Navier Stokes equations, a three-dimensional flow is completely describedby four equations (assuming incompressible and temperature independent flows): Reynoldsequations and the continuity equation. However, in practice, the statistical introduction byReynolds averaging adds new variables in terms of Reynolds Stresses and the system is under-defined. This problem is known as the turbulence closure problem.

3.1. REYNOLDS EQUATIONS 15

3.1.2 Anisotropy and Tensor Properties

In tensor theory the diagonal components⟨u21⟩

= 〈u1u1〉 ,⟨u22⟩,⟨u23⟩

are called the nor-mal stresses, and the off-diagonal components (uiuj , i 6= j) the shear stresses. When⟨uiuj

⟩=

⟨uiuj

⟩the tensor is called symmetric. Since the stress tensor is coordinate ori-

entation dependant, the principal directions are defined such that the shear stresses are zero.Then, normal stresses coincide with the eigenvalues of the stress tensor matrix, which for phys-ical reasons must be positive (〈u1〉 ≥ 0). This way we obtain a positive semidefinite tensor. Fornon-principal directions, isotropic and anisotropic stress definitions are useful.

let the turbulent kinetic energy be half of the trace of the Reynolds Stress tensor:

k ≡ 1

2〈u·u〉 =

1

2〈uiui〉 .

which represents the mean fluctuating kinetic energy per unit mass. Then, the isotropic stresstensor is obtained as 2

3kσij . The difference is the anisotropic part

aij ≡⟨uiuj

⟩− 2

3kδij .

The anisotropic term is many times normalized:

bij =aij2k

=

⟨uiuj

⟩〈ulul〉

− 1

3δij

we solve for the Reynolds stress tensor⟨uiuj

⟩=

2

3kδij + aij

= 2k

(1

3δij + bij

).

This last equation helps to understand why the anisotropic component is responsible for mo-mentum transportation. The isotropic part only modifies the pressure term, which is irrelevantfor compressible fluids. Application of this ideas is shown in the modified mean pressure equa-tion (3.9).

ρ∂⟨uiuj

⟩∂xi

+∂ 〈p〉∂xj

= ρ∂aij∂xi

+∂

∂xj

(〈p〉+

2

3ρk

). (3.9)

We observe the isotropic(23k)

is absorved in the mean pressure term.

In irrotational flows the Reynolds Stress tensor produces exclusively a modified pressure. Takingzero mean and fluctuating vorticity, and thus ∂ui/∂xj − ∂uj/∂xi zero, one obtains⟨

ui

(∂ui∂xj− ∂uj∂xi

)⟩=

∂

∂xj

(1

2〈uiui〉

)− ∂

∂xi

⟨uiuj

⟩= 0

which leads to the Corrsin-Kistler equation

∂

∂xi

⟨uiuj

⟩=

∂k

∂xj.

We can appreciate in this equation how the stress tensor has the same effect as the isotropicstress kδij , which can be absorbed into the modified pressure as before.


3.2 Turbulent Boundary Layer

Boundary layers are the location of relevant flow features and are of primary interest in turboma-chinery. Study of simplest boundary layers, formed by uniform-velocity flows over a plane platereveals that statistically can be described by two dimensions. This two-dimensional coordinatesystem is defined such that the x-coordinate is set in the flow direction, and y-coordinate isperpendicular to the surface. We define, U, V,W as the velocity in the positive x, y, z directionsrespectively.

Boundary layer thickness, δ(x), increases with x, and is generally (but not only) defined as thevalue of y at which the mean velocity,

⟨U(x, y)

⟩, equals 99 % of the free-stream velocity, U0(x).

Other definitions are based on integrals, which makes them more reliable for experimentalmeasurements reasons. Some examples are Displacement thickness

δ∗(x) ≡∫ ∞0

(1− 〈U〉

U0

)dy

and momentum thickness

θ(x) ≡∫ ∞0

〈U〉U0

(1− 〈U〉

U0

)dy. (3.10)

Viscous Scales

Viscous scales are some variables deduced in order to accurately describe the boundary layerregion of a fluid. We will start describing the total shear stress

τ = ρν∂ 〈U〉∂y

− ρ 〈uv〉 , (3.11)

which can be seen to be made up of a viscous term and the Reynolds Stress tensors. We willfurther define wall shear stress

τw ≡ τ(0).

If normalization is applied one arrives to the normalized wall shear stress or skin-friction coef-ficient. Normalization is made by division with a velocity factor,

cf ≡τ

1/2ρU20

.

If non-slip conditions are to be satisfied on the walls (U(x, t) = 0), equation (3.11) simplifiesto

τ(y = 0) = τw = ρν

(d 〈U〉dy

)y=0

. (3.12)

For wall shear stress only viscous stress participate. As will be seen through next sections,viscosity plays a central role on near-wall regions and is the reason for viscous scales variablesof interest definition.

• friction velocity, uτ ≡√τwρ

• viscous lengthscale, δν ≡ ν√

ρ

τw=

ν

uτ

• friction Reynolds Number, Reτ ≡uτδ

ν=

δ

δν

3.2. TURBULENT BOUNDARY LAYER 17

• viscous lengths or wall units y+ ≡ y

δν=uτy

ν

Notice that y+ looks like a Reynolds number, and its magnitude expresses the relative weightof the viscous and turbulent flows. The viscous contribution is, as reasoned before, 100 % atthe the wall (y+ = 0), 50 % at y+ ≈ 10 and less than 10 % when y+ = 50.

Several layers are defined near the wall. Viscous wall region is defined as y+ < 50. Withinthis region there is a dominant effect of molecular viscosity. When y+ > 50 the outer layerstarts, where turbulence is the main contribution. Inside the viscous wall region, 3 sublayerscan be found: the viscous sublayer for y+ < 5, in which the Reynolds shear stress is negligiblecompared with the viscous stress. The range 5 < y+ < 30 is called the buffer layer. And therange 30 < y+ < 50 known as the log-law region. As Reynolds number increases, the fractionof the channel dominated by the viscous wall region decreases, since δν/δ varies as Re−1τ .

Boundary layer formation and evolution is described in many fluid mechanics books. When thefree fluid stream contacts the surface of the plane plate edge (knonw as the leading edge), alaminar flow region forms at this point and spreads towards the fluid core as the flow extendsin the surface. When the Reynolds number (which length parameter must be appropriatelydefined to account for the laminar layer width) rises up to 106 a transition from laminar toturbulent flow starts in the growing boundary layer. Boundary layer properties strongly changefrom laminar to turbulent flow. The sketch in figure (3.1) shows this transition. Although aturbulent boundary layer refers to the boundary layer that has undergone this transition, westill find laminar motion in the viscous sublayer.

of velocity u against distance y from surface at point x.pdf

Laminal

Turbulent

Transition region

Boundary layerU=0.99 U0

δ

U

y

Leading edge

U0

Transition pointViscoussublayer

x

Figure 3.1: Graph of velocity u against distance y from surface at point x. Source: Adapted bythe author from Krause et al. [2004]

3.2.1 Momentum Equations and Velocity Profiles

Stress and velocity gradients parallel to the wall in boundary layers are much smaller comparedto the cross-stream gradients. Considering all velocity terms are zero at the surface the lateralmean momentum equation simplifies to

1

ρ

∂ 〈p〉∂y

+∂⟨v2⟩

∂y= 0

If we integrate this equation between wall and free-stream we obtain

〈p〉+ ρ⟨v2⟩

= p0(x).


1

100

y/δ

τ/τw

<U>/U0

γ

Figure 3.2: Mean velocity, shear stressand intermittency factor profiles in a zero-pressure-gradient boundary layer, Reθ =8000. Source: Adapted by the author fromKlebanoff (1954).

0 1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

U/U0

τ/τw

y/δx

Figure 3.3: Nomalized velocity and shear-stress profiles from Blasius solution forthe zero-pressure-gradient laminar boundarylayer on a flat plate: y is normalized by

σx ≡ x/Re1/2x = (xvU0)

1/2. Source: Adap-ted by the author from Pope [2000]

.

Since⟨v2⟩

equals zero at the wall, from previous equation the wall pressure pw(x) equals thefree stream pressure p0(x).

A similar deduction for the mean-axial-momentum equation (parallel to the wall) might be done.After corresponding simplifications of the velocity and axial gradient terms, one obtains

∂τ

∂y= −1

ρ

dp0dx

If we consider a zero presure gradient flow (free-stream pressure doesn’t change along x coordin-ate), we finally get

1

ρ

(∂τ

∂y

)y=0

= ν

(∂2 〈U〉∂y2

)y=0

= 0.

Which might be integrated from y = 0 to y = ∞. The obtained equation is known as theKarman’s integral momentum equation,

τw =d

dx

(ρU2

0 θ)

= ρU20

dθ

dx.

Where θ is defined by equation (3.10). This relation allows us to understand the influence ofthe wall shear stress in the momentum thickness. Turbulent experimental results were obtainedby Klebanoff [1954] and for laminar flow by Blasius [1908]. Results are shown in figures (3.2)and (3.3). It can be seen turbulent profile is much steeper than laminar mean velocity profile.Lets analyse the turbulent velocity profiles closer.

A simple physical description of the flow will consider four velocity laws. Three in the innerlayer and one for the outer. In the inner layer we find the viscous sublayer law, the van Driestdamping function and the log-law which describes the velocity profile in the viscous sublayer thelog-law region and the buffer layer respectively. This three laws together creates what is knownas the law of the wall. In the outer layer there is a portion of the log-law starting from the innerlayer (generally starts to be applicable for y+ > 30), and although is not always mentioned thevelocity-defect law is employed for certain big y+ values.


We will introduce the universal laws, which means this equations are independent of the flowcharacteristics such as Reynolds number and, thus, results are limited by the hypothesis im-posed. The reader must be aware that more complex descriptions are available in the literature.Nevertheless, this laws provide a good qualitative description, which is the aim of this section.Among the universal-laws mentioned the wake region has been more consistently target for itsnon-universality.

Law of the Wall

The law of the wall is the consequence of an adimensional modelling. A flow is completelyspecified by ρ, ν, δ and uτ parameters. Only two adimensional parameters can be constructedfrom this variables, so one may write

d 〈U〉dy

=uτy

Φ

(y

δv,y

δ

),

where Φ is a universal non-dimensional function. The idea behind choosing this adimensionalparameters resides in the possibility of neglecting the turbulent δ length scale when consideringflows close to the wall (y+ < 50) and viscous δv for further regions. As it was mentionedpreviously: viscosity drives the flow close to the wall, whereas turbulent viscosity is the mainsource in further regions.

In the inner layer (where velocity is determined by viscous scales) function φ(y/δv, y/δ) tendsto φ1

(y/δv

). So, if y+ ≡ y/δv and u+(y+) ≡ 〈U〉 /uτ and for y/δ << 1 then it is posible to

expressd 〈U〉

dy=uτy

ΦI

(y

δv

)as

du+

dy+=

1

y+ΦI(y

+)

which after integration computes

u+ = fw(y+) (3.13)

with

fw(y+) =

∫ y+

0

1

y′ΦI

(y′)

dy′.

It has been extensively validated that the function fw is universal for flows with Reynold num-bers far from the transition region.

• From equation (3.12) and no-slip condition fw(0) = 0 and f ′w(0) = 1. Then, the viscoussublayer is constructed after this results by a Taylor-series expansion.

fw(y+) = y+ +O(y+2) (3.14)

An in detail examination of this Taylor expansion concludes that next non-zero term is oforder y+4. So that equation (3.14) is quite accurate.

• As stated before, the log-law region ranges from 30 < y+ < 50. In this section of theInner Layer, equation (3.13) is applicable since the viscous factor y/δv is still dominant.However, ΦI will adopt a constant value, usually expressed as k−1.

the mean velocity gradient isdu+

dy+=

1

ky+


which integrates to

u+ =1

kln y+ +B (3.15)

B is a constant, k is known as the von Karman constant and, within small variations,typical values are: k = 0.41, B = 5.2.

We have obtained the mean velocity behaviour of the viscous sublayer and the log law region.But the buffer layer remains undetermined. A popular approximation to obtain this regiondescription is the van Driest damping function giving rise to the law with this name. Accordingto the mixing-length hypothesis, which will be introduced in section (3.2.4), the total shearstress is

τ(y)

ρ= ν

∂ 〈U〉∂y

+ νT∂ 〈U〉∂y

= ν∂ 〈U〉∂y

+ l2m

(∂ 〈U〉∂y

)2

.

Normalizing this equation by the viscous scales and solving for ∂u+/∂y+, defining l+m ≡ lm/δvand setting τ/τw unity for the inner layer, the solution yields

u+ = f2(y+) =

∫ y+

0

2τ/τw

1 +[1 + (4τ/τw)(l+m)2

]1/2 .The mixing length hypothesis is not accurate, but gives an approximation of the real ReynoldsStresses. According to this hypothesis we might write lm = ky or l+m = ky+ in both log-lawregion and the viscous sublayer. However, on the overlapping buffer layer, model consistencydoesn’t occur unless a damping function to the parameter l+m is applied.

l+m = ky+[1− exp

(−y+/A+

)]The term in brackets is the van Driest damping function. A+ = 26 is a standard value. Forlarge y+ the damping function tends to unity and the log-law is recovered. For a given k, thespecification of A+ determines B. In this case A+ = 26 forces B = 5.3.

Last but no least, we find the velocity-defect law. It is applied in the defect layer (outer layerwith y/δ > 0.2). In this region flow deviates slightly from the log-law. A second function is

then defined, which added to the log-law

(fw

(yδν

))fits the velocity profile in the mentioned

layer. This is the wake function w(yδ

).

〈U〉uτ

= fw

(y

δν

)+

Π

kw

(y

δ

). (3.16)

Π is called the wake strength parameter, and its value is flow dependent. The wake functionis assumed universal and it is defined to satisfy the normalization condition w(0) = 0 andw(1) = 2.

Equation (3.16) is usually expressed as the velocity-defect law, where fw is substituted by thelog-law and condition 〈U〉y=δ = U0 is imposed to obtain

U0 − 〈U〉uτ

=1

k

− ln

(y

δ

)+ Π

[2− w

(y

δ

)]


3.2.2 Kinetic Energy and Reynolds-Stress Balances

The definition of the kinetic energy is

E(x, t) ≡ 1

2U(x, t)·U(x, t)

We can perform a decomposition equivalent to the equation (3.1).⟨E(x, t)

⟩= E(x, t) + k(x, t).

E(x, t) is the kinetic energy of the mean flow and K the turbulent kinetic energy.

E(x, t) ≡ 1

2〈U〉 · 〈U〉

k(x, t) ≡ 1

2〈u·u〉 =

1

2〈uiui〉

One might obtain the mean kinetic energy equation from the Reynolds Equation (3.7).

DE

Dt+∇· T = −P − ε (3.17)

where

P ≡ −⟨uiuj

⟩ ∂Ui∂xj

,

Ti ≡⟨Uj⟩ ⟨uiuj

⟩+ 〈Ui〉〈p〉 /ρ− 2ν

⟨Uj⟩Sij

ε ≡ 2νSijSij − ν∂2⟨uiuj

⟩∂xi∂xj

Sij =1

2

(∂ 〈Ui〉∂xj

+∂⟨Uj⟩

∂xi

)(3.18)

Similarly, the mean turbulent kinetic energy is obtained after subtracting the Reynolds equationsfrom the Navier Stokes Equation (3.6)

Dk

Dt+∇·T ′ = P − ε (3.19)

where p′ is the fluctuating pressure term

T ′i ≡1

2

⟨uiujuj

⟩+⟨uip′⟩ /ρ− 2ν

⟨ujsij

⟩ε ≡ 2νSijSij

where Sij is defined by equation (3.18). This equation can alternatively be written as

Dk

Dt+

∂

∂xi

[1

2

⟨uiujuj

⟩+⟨uip′⟩ /ρ] = ν∇2k + P − ε.

Where p′ = p−〈p〉. Which in the boundary-layer approximation, the equation reduces to

0 = −(〈U〉 ∂k

∂x+ 〈V 〉 ∂k

∂y

)+ P − ε+ ν

∂2k

∂y2− ∂

∂y

⟨1

2νu·u

⟩− 1

ρ

∂

∂y

⟨νp′⟩.

The different terms are, from left to right, the mean flow convection, production, pseudo-dissipation, viscous diffusion, turbulent convection, and pressure transport. The profiles of theterms are ploted in fig (3.4).


gain

loss

y/δ

1.0

0.5

0.0

-0.5

-1.00.0 0.2 0.4 0.6 0.8 1.0

production

dissipation

viscousdiffusion

turbulentconvection

gain

loss

0.20

0.10

0.00

-0.10

-0.20

50403020100

dissipationmean

convectionmean

convection

pressure transport

pressure transport

viscousdiffusion

turbulent convection

production

y+

Figure 3.4: Turbulent kinetic energy budget in a turbulent boundary layer at Reθ = 1410. Leftplot is normalized as function of y. Right plot is normalized by the viscous scales. Source:Adapted by the author from Pope [2000].

According to the figures, the mean-flow convection is negligible in the viscous wall region. Inthe log-law region, P and ε modules decrease with y. From y+ ≈ 40 to y/δ ≈ 0.4 the balanceis dominated by production and dissipation. At last, in the outer boundary layer, productionbecomes small and the balance is between dissipation and the convective transport terms.

In a similar fashion to equation (3.19) and (3.17), balance equations for the Reynolds-stressesare obtained from the fluctuating velocity in the Navier Stokes Equations u(x, t).

0 = − D

Dt

⟨uiuj

⟩− ∂

∂xk

⟨uiujuk

⟩+ ν∇2

⟨uiuj

⟩+ Pij + Πij − εij

where Pij is the production tensor

Pij ≡ −⟨uiuj

⟩ ∂ ⟨Uj⟩∂xk

−⟨ujuk

⟩ ∂ 〈Ui〉∂xk

, (3.20)

Πij is the velocity-pressure-gradient tensor

Πij ≡ −1

ρ

⟨ui∂p′

∂xj+ uj

∂p′

∂xi

⟩and εij is the dissipation tensor

εij ≡ 2ν

⟨∂ui∂xk

∂uj∂xk

⟩.

Data for different terms of the Reynolds Stresses can be found in Pope [2000]. Main propertiesare reported next.

Concerning to the normal-stress balance, since only ∂ 〈U〉 /∂y is a significant mean velocitygradient, normal-stress production (equation (3.20)) can be approximated by

P11 = 2P = −2 〈uv〉 ∂ 〈U〉∂y

P22 = P33 = 0


Over most boundary layer, P11 is the dominant source term of⟨u2⟩. Although pressure fluc-

tuation does not play a big role in turbulent kinetic energy balance, it plays a central role inthe Reynolds Stress equations. It is responsible for the energy redistribution among the normalturbulent components 〈ui〉. Resulting in the dominant sink term for

⟨u2⟩

and source for⟨v2⟩

and⟨w2⟩. Last but not least, dissipation will be responsible for the sink of

⟨v2⟩

and⟨w2⟩

bulkenergy.

In 〈uv〉 shear-stress balance, dissipation is negligible. There is an approximate balance betweenproduction, P, and the pressure term, Π. Mention that the dissipation term is isotropic in thebulk of the fluid, but anisotropic closer to the wall.

3.2.3 Theory Limitations

As early as in the 60s, scientist were interested in the boundary layer development under sur-face curvature and strong pressure gradient conditions [Patel and Sotiropoulos, 1997]. After fewtests it was made clear a strong dependency. Reynold Stresses and boundary layer thickness(δ) are function of curvature radius, Rc. This dependency has deep consequences in turboma-chinery applications and consequently several tests were readily performed, specially for airfoilshaped blades. Even today, phenomena such as boundary layer detachment, or boundary layerinstability are still under study. Rayleigh criterion is known for boundary layer stabilizationunder curved surfaces description: in convex curvatures, which has the centre of curvatureinside the wall, the angular momentum increases with curvature and an stabilizing process oc-curs. Opposite solutions are found for concave surfaces, which curvature centre is in the fluid.Production terms are added to Reynolds stresses to account for curvature, but according toPope [2000], this production terms are an order of magnitude smaller than physical evidences.Curvature of the streamlines produces static pressure gradients through the boundary layer asit was explained in equation (2.1). All this mechanisms for turbulence modelling are still understudy.

3.2.4 The Mixing Length Theory

As mentioned in section (2.3.2), the mixing length hypothesis is a simple turbulence closuremodel used in the CFD community. Although its accuracy limitation is a concern in realapplications, it gives a rough approximation of the Reynolds Stress values. Lets remember thatthe idea behind the turbulent viscosity theory is to obtain a model of the form

− 〈uv〉 = vTd 〈U〉d y

(3.21)

so that the Reynolds Stress in equation (3.8) can be included in the viscosity term. The termνT is called the turbulent viscosity.

In the mixing length theory the turbulent viscosity is parametrized by a velocity and a lengthvariables, u∗ and lm respectively.

νt = u∗lm.

Next step is to relate this parameters with known flow variables. Since the application of thetheory is high Reynolds number flows, and for this flows −〈uv〉 ≈ u2τ , from equation (3.21) itis possible to write

νt = −〈uv〉d〈U〉d y

= − u2τd〈U〉d y

. (3.22)


Further, if this theory will be used within the log-law region where∣∣∣∣d 〈U〉d y

∣∣∣∣ =uτky

as it was demonstrated in section (3.2.1), we further write

νt = − u2τd〈U〉d y

= −uτky

where we identify

u∗ = uτ =∣∣〈uv〉∣∣1/2

lm = ky

Absolute values in the equation above ensures that u∗ is non-negative for all y.

This relation is known as Prandtl’s mixing-lenth hypothesis. To sum up

νt = u∗lm = l2m

∣∣∣∣d 〈U〉d y

∣∣∣∣ .And in log-law region lm = ky.

3.3 Eddy Viscosity Based Turbulence Modelling

3.3.1 Eddy Viscosity Hypothesis

According to the Eddy Viscosity Hypothesis, the anisotropy aij ≡⟨uiuj

⟩− 2

3kδij is intrinsicallydetermined by the mean velocity gradients

⟨uiuj

⟩− 2

3kδij = −νT

(∂ 〈Ui〉∂xj

+∂⟨Uj⟩

∂xi

)or equivalently

aij = −2νT zSij (3.23)

where Sij is the mean rate-of-strain tensor. In simple shear flows it reduces to

〈uv〉 = −νT∂ 〈U〉∂y

as the mixing length theory predicts. Unfortunately, many flows don’t obey this last equation asit has been demonstrated by experimentation in the Axisymmetric contraction test for example.As it was done in section (3.2.4) νT (x, t) is obtained by a velocity, u∗(x, t), and a length, l∗(x, t)parameters product

νT = u∗l∗

Based on how the modelling is done different modelling families are found. Algebraic models(such as mixing-length model), which models νT in a simple algebraic way, is called a zeroequation model. Where as in two-equation models two transport equations are solved to obtainthe turbulent viscous term. Transport equation variables typically are k, ε or ω, giving riseto the k − ε turbulence model or the k − ω turbulence model. It is accepted that the k − εmodel performs reasonably well for two-dimensional thin shear flows where small streamlinecurvature and mean pressure gradients are found. As it could be the case of air surrounding anairfoil. For strong pressure gradients k − ω model gives a more accurate prediction accordingto bibliography.

3.3. EDDY VISCOSITY BASED TURBULENCE MODELLING 25

3.3.2 Wall Treatment

When region in the near-wall is considered, computation of the turbulence models faces somedifficulties. Hence, modification of turbulence models is very common. For example, experi-mental cases have revealed an appropriate Cµ coefficient reduction when y+ < 50.

In STAR-CCM, this approach is followed and the so called two-layer and wall treatment hasbeen introduced is the code. Different types of wall treatment are available, high, low and all-y+ wall treatments. The all-wall treatment makes use of both high and low treatments whennecessary. This way, high wall treatment is employed in coarse-near wall cells, and low-walltreatment in fine cells. When the wall-cell centroid falls in the buffer region of the boundarylayer, the wall treatment is acconditioned so the fluid motion is appropriately resolved.

Even though wall formulations is modified when choosing different turbulence models, the stand-ard formulation is as follows:

u+ =

{u+lam fory+ ≤ y+mu+turb fory+ > y+m

(3.24)

(3.25)

the intersection of the viscous and the fully turbulent regions ym is found by Newtons algorithm.The adimensional values to be employed are

y+ =yu∗

ν(3.26)

u+ =〈U〉u∗

(3.27)

although the reference velocity is often related to the wall shear stress(u∗ = τw/ρ

), in actual

practice the reference velocity is derived from a turbulence quantity specific to the particularturbulence model.

In STAR-CCM, viscous sublayer is modelled as it was done in section (3.2.1).

u+lam = y+ (3.28)

On the other hand, the log-law region velocity distribution is:

u+turb =1

kln(

E

fy+) (3.29)

being k = 0.42 and E = 9.0. Roughness function value, f , is unity for smooth walls. Specificmodifications to this formulation is explained in each of the turbulence models.

3.3.3 Realizable Two-Layer k − ε

The Realizable Two-Layer k− ε model combines the Realizable k− ε model with the two-layerapproach. Model coefficients are still the same in all the fluid domain, but enhanced treatmentis done near the wall region.

In order to understand the Realizable k − ε model, we first first explain the general k − εmodelling. In this model transport equations are solved for turbulent kinetic energy, k, andturbulence dissipation, ε. Specific formulation of parameters is done: for lengthscale (L =


k3/2/ε), for timescale (τ = k/ε) and a singular quantity of dimensions νT (k2/ε) among otheroptions. With this definitions, lm kind of specifications are not necesary. It is called a completemodel.

k − ε model evolved for many years, but the roots of the model has always relied on the twotransport equations and the algebraic turbulent viscosity model. Two different approaches weretaken during transport equations development. In the case of k, the exact transport equationwas deduced from the Navier Stokes Equations. In the case of ε viscosity for high Reynoldsnumbers was neglected. The reason for this last assumption is that the main kinetic energy(flow energy) is coming from the large scale turbulence structures, which are independent ofviscosity. It can be said that the equation is rather “empirical”. This empirical development ofthe equation will induce some constrains in the applicability of the model.

The equations reported below are the continuous equations used in STAR-CCM for Realizablek − ε model [CD-ADAPCO].

Basic Transport Equations

d

d t

∫Vρk dV +

∫Aρk(v − vg)· d a =∫

A

(µ+

µtσk

)∇k· d a+

∫V

[Gk +Gb − ρ

((ε− ε0) + γ

)+ Sk

]dV (3.30)

(3.31)

d

d t

∫VρεdV +

∫Aρε(v − vg)·d a =∫

A

(µ+

µtσε

)∇ε·d a+

∫V

[Cε1Sε+

ε

k(Cε1Cε3Gb)−

ε

k +√νεCε2ρ (ε− ε0) + Sε

]dV

(3.32)

where Sk and Sε are the user-specified source terms, and ε0 is the ambient turbulencevalue that counteracts turbulence decay. The production Gk is evaluated as:

Gk = µtS2 − 2

3ρk∇·v − 2

3µt (∇·v)2

S = |S| =√

2S : ST =√

2S : S

and

S =1

2

(∇v +∇vT

)(3.33)

Two dots operation refers to double dot product of a tensor. The turbulent viscosity iscomputed as:

µt = ρCµk2

ε

although Cµ is not a constant.

Cµ =1

A0 + AsU (∗) kε

U (∗) =√S : S −W : W


W is the rotation rate tensor,

W =1

2

(∇v −∇vT

).

Only most important equations are writen here. Other terms and constant values can befound at CD-ADAPCO.

wall treatment

In k − ε model the wall treatment is employed to set the value u∗ in the law of the wall regionand modify Gk and Gε in the near-wall cell. As explained before, the all y+ wall treatmenttries to mimic the high and low y+ formulation. It also gives reasonable results in the bufferlayer.

For the all-y+ formulation, a blending function g is defined in terms of the wall-distance basedReynolds number

g = exp

(−Rey

11

)Rey =

√ky/v (3.34)

u∗ =

√gνu/y + (1− g)C

1/2µ k

Gk = gµtS2 + (1− g)

1

µ

(ρu∗

u

u+

)2 ∂u+

∂y+

ε =k3/2

lε(3.35)

being Rey the mentioned Reynolds number

two-layer Realizable k − ε

The two-layer Realizable k−ε model always solves the transport equation for k, but algebraicallyblends ε transport equation with a distance from the wall formulation. More specifically, itparametrizes a length scale function lε = f

(y,Rey

)and a turbulent viscosity ratio function,

µt/µ = f(Rey

)such that Rey and ε are defined by equations (3.34) and (3.35).

The following blending function is used to combine the two-layer formulation with the fulltwo-equation model

λ =1

2

1 + tanh

(Rey − Re∗y

A

)where Re∗y defines the applicability limit of the two-layer formulation. In STAR-CCM valuesfor this parameters are, Re∗y = 60. Constant A determines the width of the blending function,such that the value λ will be within 1% of the far-field value of a given ∆Rey. Thus, it can besolved

A =

∣∣∆Rey∣∣

atanh 0.98In STAR-CCM+ ∆Rey = 10.

Turbulent viscosity from k − ε model is blended in the two-layer model as follows:

µt = λ µt|k−ε + (1− λ)µ

(µtµ

)2layer

.


The discretized transport equation for ε is modified and yields

apω

∆εp +∑n

anλ∆εn = λ

(b− apεnp −

∑n

anεnn

)+ (1− λ) ap

(εn+1p

∣∣∣2layer

− εnp)

3.3.4 Menters SST k − ω

k − ω model was developed by Wilcox when he proposed the substitution of the ε equation bya modified variable: ω ≡ ε/k.

This leads to an equation similar to ε equation in the standard k − ε model, but an additionalterm is created (obviously constant values also change). In the differential form of the equationthe additional term is

2νTσωk∇ω· ∇k. (3.36)

If this term is neglected we obtain the standard k−ω model. It was found that k−ω model wassuperior in the viscous near-wall region and in the streamwise pressure gradients. Unfortunatelyω treatment in boundaries is problematic: a value for ω has to be given in each boundary, andit turns out the solution is very sensitive to this value.

Menter proposed a two-equation model which blends the best properties of both k−ε and k−ωmodels. This is known as the SST k − ω model. In this model, the omega equation includes aweighting function for the term (3.36). When we consider cells far from the walls the weightingfunction is set to one, and the k− ε model is employed. When cells are close to the walls, thenk − ω model is obtained by setting the weighting function to zero.

At last, Menter modified the linear constitutive equation and obtained the model which isemployed in STAR-CCM under SST (Menter) model. It is a model widely used in aerospace in-dustry where viscous flows are well resolved and turbulence models in boundary layers extremelyimportant.

The basic transport equations from SST k − ω are

Basic Transport Equations

d

d t

∫Vρk dV +

∫Aρk(v − vg

)· da =∫

A(µ+ σkµt)∇k· da +

∫V

(γeffGk − γ′ρβ∗fβ (ωk − ω0k0) + Sk

)dV

d

d t

∫Vρω dV +

∫Aρω(v − vg

)· da =∫

A(µ+ σωµt)∇ω· da +

∫V

(Gω − ρβfβ

(ω2 − ω2

0

)+Dω + Sω

)dV

Sk and Sω are the user-specified source terms. k0 and ω0 are the ambient turbulencevalues in source terms that counteract turbulence decay. γeff is the effective intermit-tency provided by the Gamma ReTheta Transition model (it is unity if this model is notactivated. It basically helps to evaluate the momentum thickness Reynolds number inunstructured messes) and

γ′ = min[max

(γeff , 0.1

), 1].


The production Gω equations is evaluated as

Gω = ργ

[(S2 − 2

3

(∇·v2

))− 2

3ω∇·v

]

where γ is a blended coefficient of the model and S is the modulus of the mean strain ratetensor, equation (3.33).

The cross derivative term Dω,

Dω = 2 (1− F1) ρσω21

ω∇k· ∇ω.

Turbulent viscosity is computed as:

µt = ρkT

Other constants and values might be found at CD-ADAPCO.

wall treatment

Wall treatment might be employed as in k − ε model. There are high-y+ and low-y+ walltreatments. In the k − ω model this strategy is used to obtain a reference velocity u∗, a Gkturbulence production and to compute a specific dissipation ω in wall laws and wall cells.

u∗ =

√gνu/y + (1− g) β∗ 1/2 k

Gk = gµtS2 + (1− g)

1

µ

(ρu∗

u

u+

)2 ∂u+

∂y+

ω = g6ν

βy2+ (1 + g)

u∗√β∗ky

with the blending function

g = exp

(−Rey

11

).

and Rey =√ky/ν is the wall-distance Reynolds number introduced before.


Chapter 4

Simulations

In this chapter simulation setting up and results are written. In sett up section, informationabout the geometries, fan meshing, models employed and solver conditions selection is done. Inthe results section, results obtained after post-processing and data assimilation is done.

4.1 Set-up

Geometry

Fan surface mesh was available in X B format. The blade has 0.15 m radius, from whichapproximately 0.12 m belong to the blade and 0.03 m to the hub. The fan has an approximatelyconstant 0.054 m long chord length with a straight blade and an angle of attack close to 9.13 ◦.The fan speed was set to 300 rpm.

Figure 4.1: Axial fan geometry. The blackline represents the airfoil plane cross sectionlocation

Figure 4.2: Blade middle chord length crosssection.

The simulation domain is defined by a cylinder co-axial to the fan axis. The cylinder radius isof 3 m and the total height of 6 m, centred at the fan.

Mesh

All simulations have been done under the same meshing conditions. First of all, a surfaceremeshing was performed, followed by a Prism layer and a Trimmer volume meshing. Thesurface remeshing basically consists on a re-triangulation of the mesh to make it suitable forFinite Volume Methods. A Prism layer consists on a number of orthogonal cell layers next towalls in order to improve boundary layer resolution. This layers are defined in terms of itsthickness, the number of cell layers, the size distribution of the cells and the function used togenerate them. It must be combined with another core volume mesh generator such as theTrimmer volume meshing. This is a method employed in STAR-CCM which produces a highquality grid. It combines several mesh attributes,

• It is composed mostly by hexahedral meshes with minimal cell skewness

31

32 CHAPTER 4. SIMULATIONS

Surface Meshing

Property Value

Surface growth rate 1.3Fan surface mesh absolute minimum size 7.5E− 4m

Fan surface mesh absolute Target size 0.0015m

Volume Meshing

Property Value

Maximum cell size 0.5Number of prism layers 5Prism layer stretching 1.5Prism layer thickness 0.333mBoundary growth rate Very slow

Table 4.1: Relevant mesh information in the fan surface and the fluid core

• They have a curvature and proximity refinement based upon surface cell size

• They are surface quality independent

• They can be aligned in any direction based on a user specified cartesian coordinate system.

Relevant setting values are gathered in table (4.1).

Setting

We will expose next the most important modelling settings.

In STAR-CCM most settings concerning modelling is under the Continua tab. In table (4.2)the set of models considered under Physics Continua section in STAR-CCM is reported. Letremember in this thesis two simulations have been carried out, one based on k − ε turbulencemodel, and the other based on k − ω model.

k − ε turbulence model k − ω turbulence model

Optional ModelCell Quality Remediation Cell Quality Remediation

Reynolds-Averaged Navier-Stokes TurbulenceTwo-layer All y + Wall Treatment All y + Wall Treatment

Realizable k − ε two-layer SST (Menter) k − ωEnergy

Segregated Fluid Temperature Segregated Fluid TemperatureMaterial

Ideal Gas Ideal GasFlow

Segregated Flow Segregated FlowTime

Implicit Unsteady Implicit Unsteady

Table 4.2: Summarise of most relevant Continua � Physics section in STAR-CCM simulations

Modelling of Region Boundaries allows the specification of the differential equations boundaryconditions . We have set four regions: the complete fan, the inlet boundary, the outlet boundary,and the far field that connects inlet and outlet boundaries.

4.1. SET-UP 33

• The Inlet boundary has been set to velocity-inlet type. Other common option is massflow inlet boundary condition. Velocity normal to the surface is 0.05m/s. The defaultIntensity and viscosity ratio was left.

• The outlet boundary is a flow split outlet. Which basically will specify all the flow exitsthrough this boundary. It is compatible only with incompressible flows, but for low speedof the flow it is a reasonable assumption.

• The far field boundary refers to the cylinder, which is far enough from the fan to haveany influence. We set this boundary as a wall type of boundary condition to reducereversed outflow related errors. Yet, slippery wall conditions are chosen in order to simplifycomputation effort in this non-relevant area of the field.

• The fan is wall type of boundary as well. But in this case non-slippery condition is set.

All the system is under constant rotation: Tools � Reference frames�Rotating. A rotatingreference frame in STAR-CCM generates a constant flux in de discretized equations. Thisallows solution for steady-state conditions.

Solver and Stopping Criteria

Solver conditions are summarised in table (4.3). Time step value has been chosen such that arevolution is done each six time steps.

k − ε turbulence model k − ω turbulence model

Implicit Unsteadytime step=0.033 s time step = 0.033 s

1st order 1st orderWall Distance

default defaultSegregated flow

default defaultSegregated energy

fluid und. relax fact. = 0.5 fluid und. relax fact. = 0.5Turbulence model

Under-Relax. fact=0.7 Under-Relax. fact=0.7Viscosity under-Relax. fact=0.95 Viscosity under-Relax. fact=0.7

Table 4.3: Most relevant Solver values in STAR-CCM simulations

Each time step iteration might stop under two criteria: One is the maximum inner iterationnumber (set to 60) and the second is the error-tolerance criterion which is set to an academicalstandard value of 10−3. Some of the parameters are found difficult to converge to this minimum,in which case the value to which converge is set as its minimum. Intermittency and energyvariables find the minimum around 0.01 in k − ω model, and in the k − ε model only energyfinds the same difficulty converging asymptotically to 0.005. It has been found easier to workwith the k− ε method rather than the k−ω since last one had convergence difficulties. In bothmodels few iterations were performed with low under-relaxation numbers and smaller time stepssince first time steps to overcome convergence errors.


4.2 Results

The objective behind this simulations is not to make any fan performance assessment, but tocompare simulation differences when using two different turbulence models. In order to do this,two averaging have been used: time and space.

Scalar fluxes are scalar space averaged values over a constrained plane. This plane is locateddownstream the fan such that captures velocity and mass flow rate through it. The centre of theplane is at Cartesian coordinates [0, 0,−0.4] with the normal vector parallel to the fan axis. Theplane has the same width and height of 0.6 m. Unless explicitly mentioned flow rates (velocityor mass flow rates) are always space averaged over this constrained plane using formulas shownbelow.

Other mean values are obtained after appropriate time averaging over the 30 seconds of flowsimulation computed. By appropriate it is meant ignoring initial start up of the flow, wherenon representative values are found. In order to make the report clear, whenever “mean values”are mentioned during this section, it will refer to the time averaged values of the surface meanvariables. When no mean value is mentioned, it will refer to the surface averaged flux for aspecific time.

• Mass Mass Flow : Mass flow is a type of report system available in STAR-CCM. Math-ematically it is measured as follows ∑

f ρfφf∣∣v·af ∣∣∑

f ρf∣∣vf ·af ∣∣ (4.1)

If mass Mass Flow is chosen, scalar variable φ is substituted with ρf

• Velocity Mass Flow : As in the previous case, mass flow averaging (equation 4.1) is used.However, in this case φf is substituted by vf

• Force is another type of report available. It is computed as

f =∑f

(fpressuref + f shearf

)·nf

where fpressuref and f shearf are the pressure and shear force vectors on the surface face f ,and nf is a user-specified direction vector that indicates the direction in which the forceshould be computed.

4.2.1 Mass Flow

Study of Mass flow has been done using equation (4.1). As one could expect, oscillations arounda mean value are obtained. This is logical since unsteady simulation are being done.

Velocity and mass flow rates are proportional by a constant factor. Conclusions obtained fromone flow rate type is directly applicable to the other.

A time-averaged mean mass flow rate of 2.15 kg/min, or an equivalent 1.77m3/min has beenfound with k−ε turbulence model. Mean velocity parallel to the fan axis direction is 0.054m/s.For the k − ω turbulence model, mass flow rate is 2.19 kg/min, and mean velocity 0.055m/s.We could say equal mean results are obtained with two turbulence models.

Oscillations on the other side are different based on which turbulence model is used. Whenk− ω model is used, maximum mass flow rate maximum oscillation is about 50 %. Whereas in

4.2. RESULTS 35

Figure 4.3: k − ε turbulence model after 20seconds of flow simulation

Figure 4.4: k − ω turbulence model after 20seconds of flow simulation

Figure 4.5: k − ε turbulence model after 30seconds of flow simulation

Figure 4.6: k − ω turbulence model after 30seconds of flow simulation

k − ε turbulence model the maximum oscillation is 44% amplitude. However, both maximumamplitudes coincide in the flow time of the oscillation.

Report of the local velocity field for both turbulence models have some differences. In figures(4.3)-(4.6) velocity fields for both turbulence models at two different time intervals are shown.It can be appreciated that for the same maximum and minimum velocity predictions, k − ωmodel has a wider extension at this values. In another words, velocity distribution variance isbigger for k − ω despite the same maximum / minimum values are found. This might explainwhy oscillations are bigger in the first model even if mean values are the same.

4.2.2 Force

As introduced in this section, force is calculated over the whole fan. In our analysis, only forcein the axis direction has been studied. This is, only lift power of the fan is considered, dragforce is not included in this thesis.

As with flow rate analysis, oscillations over a mean value are found. For k − ε model the


time averaged force is 0.07N , and the maximum oscillation is about 28% of the mean value.With k − ω model, average value is the same, 0.07N , but with a maximum oscillation of 21%amplitude.

4.2.3 Boundary Layer

The point of the blade at which boundary layer analysis has been done is shown in figure (4.7).It is located 0.1m far from the hub centre on the pressure side of the blade: close to the trailingedge and on the chord line centre. This point was chosen because is far from the blade tip, wheretrailing vortexes exist, and boundary layer is developed as much as possible. Boundary layervelocity is shown in figure (4.8), which is perpendicular to the blade surface. Blade surfacenormal vector is [0.993,−0.15, 0.0]. We will go through equations in section (3.2.1) doing acomparison of theory and simulations for k − ε model. k − ω model will be studied at the endof this subsection.

Data used is summarized below. Wall velocity gradient and the wall stress values have beenobtained from simulations.

ν = 1.58 10−5 kg/ms

ρ = 1.17 kg/m3

τw = 0.07Pa(d 〈U〉d y

)y=0

= 3333.3m/s.

τw and(d〈U〉d y

)are related by

τw = νρ

(d 〈U〉d y

)y=0

and from previous values it is computed τw = 0.061Pa, which is close to 0.07. Bear in mindthat gradient value has been obtained numerically by the writer from graphical representationand is subject to error. Using τw = 0.070, viscous sublayer ends at y = y+δν = 0.3mm wheny+ = 5, which is under the value reported in the simulation (shown in figure (4.8) with a valueof 0.45 mm). Hence, simulation and theory show a deviation of 30 % error.

In the log-law region similar results are obtained. Solving for 〈U〉 in equation (3.15) withy+ = 30,

〈U〉 = uτ

(1

0.41ln y+

)+ 5.2 = 3.3.

Whereas solution in figure (4.8) shows a value of 2.95 (at a distance y = 2mm or equivalenty+ = 30). The relative error is 13.3 %. Same operations give for y+ = 50 a value 〈U〉 = 3.65,but 3.15 in STAR-CCM. It is 13.7 % error.

It seems analytical boundary layer equations from (3.2) don’t match accurately with predictionsfrom STAR-CCM. Thus, we have tried doing the same calculations with the Standard walltreatment provided by STAR-CCM (equation (3.29) and related). We would expect them tobe closer to simulation results. Unexpectedly, results obtained are very similar to mentionedanalytical solutions. 〈U〉 = 3.33 for y+ = 30 and 〈U〉 = 3.63 for y+ = 50.

Measurements at the same blade point at the same simulation time employing k−ω turbulencemodel gives very close boundary layer values compared to k− ε. For sake of illustration, taking

4.2. RESULTS 37

Figure 4.7: Location ofthe point used for boundarylayer profile study

Figure 4.8: Boundary layer velocity magnitude parallel to thewall. It can be appreciated a linear velocity progression up toalmost 0.5mm were the viscous sublayer ends. The right pointin the figure is located at y+ = 30

point y+ = 30 both turbulence models reports 2.95m/s speed. Moreover, a viscous sublayersize of 0.485 mm is measured. The same as k − ε model. We might summarize that in theboundary layer, there is no difference between using k − ε and k − ω models at the point andproperties considered.


Chapter 5

Conclusions and future work

Conclusions will be reported for the three main subsections on (4.2) independently.

From Mass Flow subsection we conclude that both k− ε and k−ω turbulence models predict asimilar time averaged flow rate. However, for each physical time of the flow, different flow ratesare computed. k−ω model seems to overpredict the area extension at which high or low speedflows occur. On the other side k − ε predicts a smoother velocity distribution, but still withsimilar maximum and minimum values. k − ω also overpredicts the recirculation area behindthe fan. An important conclusion from the analysis above is that, if only blade simulation wouldbe done, k − ω would overpredict the blade aerodynamic efficiency (lift capabilities) comparedto k − ε. However, if complete fan is taken into account, the gas exhausting power of the fanyields similar results.

Mention, understanding the strong oscillations of the flow rates is of big interest. Weighting ofthe numerical instabilities, simulation setting or fan wrong design contribution to oscillationshould be done, despite the physically sensible results obtained in the simulations.

Similar conclusions are obtained from Force subsection. While k − ω has strong oscillations inmass flow rate analysis, they are smaller than k − ε for force variable. Both conclusions arenot contradictory, they are complementary in fact. As we mentioned already, both turbulencemodels show similar axial direction velocities in the near-blade (where we find the maximumvalues). But the bigger force oscillations are bigger in k − ε must strongly damp to obtain alower flow oscillations. This matches the theory since it is known k− ε is more dissipative, andthus velocity distributions are smoother in the flow field.

Most important conclusion from boundary layer subsection is that STAR-CCM overestimatesthe size of the boundary layer in comparison with the simpler analytical equations reviewedin section (3.2), or even the STAR-CCM standard wall value model. This has been proof bywall distance and mean velocity calculations on the viscous and log law region. This conclusionapplies to both turbulence models. Answer to this prediction mismatch is not clear. Differ-ences in simulations and analytical results might be consequence of the different formulationsemployed in each turbulence model during computation. We saw in section (3.3.2) the nor-malization velocity is computed from turbulence related variables. Another possibility is theboundary layer model, which was developed for mono-dimensional surface parallel flows, notbeing representative of our case (we have certain angle of attack altogether with a complexgeometry which causes multi-dimensional boundary layers). Another possibility is, accordingto section (3.2.3), effect of pressure gradients and leading and trailing edge curvatures to affectthe streamline curvature and distorts the boundary layer. A pressure plot from the simulationsis shown in figure (5.1). Experimental validation should be done to contrast simulations andidentify analytical values mismatch reasons.

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40 CHAPTER 5. CONCLUSIONS AND FUTURE WORK

Figure 5.1: Blade pressure side surface pressure distribution. It is appreciated a few Pa over-pressure on the leading edge while an under pressure on the trailing edge. Further analysis ofthe simulation reveals a fast decay in over/underpressure with distance from the blade

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J. Anderson. Computational fluid dynamics: the basics with applications. McGraw-Hill, 1995.

H. Blasius. Grenzschichten in Flussigkeiten mit kleiner Reibung. Annu. Rev. Fluid. Mech, 56:1–37, 1908.

F. Bleier. Fan Handbook: Selection, Application, and Design. McGraw-Hill, 1998.

K. Brun and R. Kurz. Analysis of Secondary Flows in Centrifugal Impellers. InternationalJournal of Rotating Machinery, Issue 1:45–52, 2005.

CD-ADAPCO. STAR-CCM+ USER GUIDE. STAR-CCM+ Version 6.06.017.

J. Earl Logan and R. Roy. Handbook of Turbomachinery. Marcel Dekker, 2003.

D. Eckardt. Detailed Flow Investigations Within a High-Speed Centrifugal Compressor Impeller.Journal of Fluids Engineering, Transactions of the ASME, 98 Ser I(3):390–402, 1976. CitedBy (since 1996): 94.

D. Japikse and N. Baines. Introduction to turbomachinery. Concepts ETI, 1997.

P. Klebanoff and U. S. N. A. C. for Aeronautics. Characteristics of Turbulence in a Bound-ary Layer with Zero Pressure Gradient. Technical note. National Advisory Committee forAeronautics, 1954.

E. Krause, H. Oertel, H. Schlichting, K. Gersten, and C. Mayes. Boundary-Layer Theory.Physics and astronomy online library. Springer, 2004.

E. Logan, J. Sherma, and B. Fried. Handbook of Turbomachinery. Chromatographic ScienceSeries. Marcel Dekker Incorporated, 2003.

V. Patel and F. Sotiropoulos. Longitudinal curvature effects in turbulent boundary layers.Progress in Aerospace Sciences, 33(1–2):1–70, 1997.

S. Pope. Turbulent Flows. Cambridge University Press, 2000.

J. Tuzson. Centrifugal Pump Design. A Wiley-Interscience publication. John Wiley & Sons,2000.

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AERODYNAMIC ANALYSIS OF AXIAL FAN …conjunto, el empleo de k !no supondr a una mayor e cacia porque tambi en implica re ujos mayores. La segunda conclusi on es la necesidad de un