University of the Basque Country - Euskal Herriko UnibertsitateaUniversidad del Paı́s Vasco
Faculty of Science and Technology- Zientzia eta Teknologia FakultateaFacultad de Ciencia y Tecnologı́a
Máster en Iniciación a la Investigación en
Matemáticas
Master Thesis
AERODYNAMIC ANALYSIS OF
AXIAL FAN UNSTEADY SIMULATIONS
September 2012
Author
Imanol Garcı́a de Beristain
Advisors
F. Palacios & L. Remaki
Agradecimientos (Acknowledgements)
Me gustarı́a mostrar mi agradecimiento a todo aquel que haya contribuido a la creación de esta
tesis de alguna manera.
To Basque Center for Applied Mathematics for the trust they put on me. Specially to Sergey
Korotov for letting me join up his group. To Francisco Palacios for his help and patience
during my beginnings in the field. At last, thanks also goes to Lakhdar Remaki for helping me
finishing this thesis and introducing me into my next steps.
Agradezco ası́ mismo al Servicio Técnico de Informática Aplicada a la Investigación de la
UPV/EHU, por haberme facilitado el uso de STAR-CCM+ en mi ordenador personal.
Por último, mi más sincero agradecimiento para todos aquellos con los que he compartido el
dı́a a dı́a de este largo año. Gente que todavı́a sigue ahı́ y gente que ya no esta. Por supuesto,
también 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 simulation performance depending on the turbulence model selected. For this porpoise, simulations
with two in industry extensively used turbulence modes have been carried out: k − ε and k − ω
models. Simulations were performed using the commercial software STAR-CCM+. Results
have been compared between them and with analytically obtained simplified solutions. This
will allow to judge real industrial case simulations.
Flow Rate results showed the same mean value downstream the fan for both turbulence models. However, oscillations induced by unsteady condition had different amplitudes. Boundary
layers have been studied as well. It wasn’t found any difference among both turbulence models results, but simplified analytical problems solution predicted smaller boundary layers than
STAR-CCM+ simulations.
We obtained two main conclusions. First, k − ω is overpredicting blade performance compared
to k − ε turbulence model because the second is more dissipative. However, if whole fan is
considered, the extra blade efficiency on k − ω turbulence model is lost because of bigger
recirculation zones. The second conclusion is the need for further understanding on boundary
layer simulation, since deviations from expected results by both turbulence models can not be
accurately explained by the author. However, it is probably related to surface curvature or
blade edge pressure-gradient induced streamline curvature.
Keywords: CFD, aerodynamics, axial fan, boundary layer, turbulence.
Análisis Aerodinámico de Simulaciones No Estacionarias de Ventiladores
Axiales
Resumen: El objetivo de esta tesis es comprender la dependencia de los modelo de
turbulencia en simulaciones no estacionarias de ventiladores axiales mediante CFD. Con
este fin, se han realizado simulaciones empleando dos modelos de turbulencia ampliamente
utilizados en la industria: los modelos k − ε y k − ω. Dichas simulaciones se llevaron a
cabo empleando el software commcercial STAR-CCM+. Los resultados con los diferentes
modelos de turbulencia se han comparado entre si, y con las soluciones analı́ticas de
problemas simplificados. Con esto se pretende ser capaz de valorar las simulaciones de
casos 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 lı́mite son independientes respecto del modelo de turbulencia
esperado. Sin embargo, los modelos analı́ticos simplificados predicen capas lı́mites más
pequeñas que las obtenidas mediante simulaciones.
Se han obtenido dos conclusiones principales. Primero, el empleo del modelo k − ω
resultará en un rendimiento de álabe mayor que el obtenido mediante k − ε, debido a que
el segundo modelo es más disipativo. Por otro lado, si se considera el ventilador en su
conjunto, el empleo de k − ω no supondrı́a una mayor eficacia porque también implica
reflujos mayores. La segunda conclusión es la necesidad de un mejor entendimiento
sobre la simulacion de las capas lı́mite puesto que no se ha logrado explicar de forma
convincente la diferencia entre los resultados esperados mediante modelos simplificados
y las simulaciones. Lo más probable es que se deba a la curvatura de la superficie o a la
curvatura de las lineas de flujo inducidas por gradientes de presión.
Palabras clave: CFD, aerodinámica, ventilador axial, capa lı́mite, turbulencia.
Contents
1 Introduction
1
2 Background
2.1 Turbomachinery and Centrifugal Fans Essentials . . . . . .
2.2 Fan Aerodynamics . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Airfoils . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Basic Characteristics of the Fan Flow Aerodynamics
2.3 CFD Basics . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Flow Physics Modeling . . . . . . . . . . . . . . . . .
2.3.2 Turbulence Closure . . . . . . . . . . . . . . . . . . .
2.3.3 Numerical Solution Techniques . . . . . . . . . . . .
3 Turbulence
3.1 Reynolds Equations . . . . . . . . . . . . . . . . . .
3.1.1 Reynolds Stresses . . . . . . . . . . . . . . . .
3.1.2 Anisotropy and Tensor Properties . . . . . .
3.2 Turbulent Boundary Layer . . . . . . . . . . . . . . .
3.2.1 Momentum Equations and Velocity Profiles .
3.2.2 Kinetic Energy and Reynolds-Stress Balances
3.2.3 Theory Limitations . . . . . . . . . . . . . . .
3.2.4 The Mixing Length Theory . . . . . . . . . .
3.3 Eddy Viscosity Based Turbulence Modelling . . . . .
3.3.1 Eddy Viscosity Hypothesis . . . . . . . . . .
3.3.2 Wall Treatment . . . . . . . . . . . . . . . . .
3.3.3 Realizable Two-Layer k − ε . . . . . . . . . .
3.3.4 Menters SST k − ω . . . . . . . . . . . . . . .
4 Simulations
4.1 Set-up . . . . . . . . . .
4.2 Results . . . . . . . . . .
4.2.1 Mass Flow . . . .
4.2.2 Force . . . . . .
4.2.3 Boundary Layer
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3
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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 or N Ox reduction in aircraft engines are new research-lines driven by new normative constraints. On the other side, stage efficiency is many times imposed by the manufacturer
itself to keep competitiveness in the market. Stage efficiency translates into specific aerodynamic performance requirements. New components will need much greater complexity during
its design, including a higher degree of three dimensionality analysis and flow-path configurations.
Design tools traditionally available in engineering, such as basic relative velocity triangles, have
strong limitations to comply with demanded complex research. The use of advanced aerodynamic tools have to be developed. At this point Computational Fluid Dynamics (CFD) is
being extended in the turbomachinery community to fulfil this task. Its ability to simulate flow
physics at any point in the domain, with better predicting capabilities every year, is extremely
attractive. CFD has already upgraded design of turbomachinery from 2-D inviscid flow models
to 3-D, viscous, turbulent analysis [Earl Logan and Roy, 2003].
In this thesis unsteady simulation of an axial fan flow field is studied. Although aerodynamic
components such as boundary layers or recirculation zones are considered, boundary layers take
most of the weight of the thesis, since blade performance is directly related to phenomena such
as boundary layer thickness or boundary layer detachment. Although boundary layers have
been target for research for many decades, they are still under study. It is the case of aeroengines, 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 focus
this thesis on a basic and yet extremely valuable boundary layers.
Boundary layers modelling is strongly dependant of viscosity an Reynolds Stresses. For this
reason, turbulence modelling is included besides boundary layers theory. In view of the extensive
area of turbulence in bibliography, it was decided to cover the basics of Reynolds Averaged
Navier Stokes (RANS) models, which will be linked to the boundary layer theory during its
introduction.
This thesis is structured in five main Chapters. Chapter two introduces the basics of turbomachinery 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 the
theoretical core of the thesis. Starting from the Reynolds Equations, boundary layer structure is
deduced, to conclude with how turbulence is modelled within STAR-CCM. In Chapter four the
Simulation set up and its results are reported such that it can be reproduced by the interested
reader. 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 include
pumps, 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 shaft
stems from. Fluid is introduced through pipes in the case. Turbomachinery works transferring
energy between fluid and rotor. When energy is extracted from fluid, the machine is called
turbine, 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 the
fluid flowing between two blades to execute a specific force on the blades. It is commonly find
as 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 exit
direction: 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 application or environment.
Tubeaxial fans have a cylindrical housing. Gas exhaustion is the most common application for
this fans. The main negative outcome is the fast increase in the outlet duct friction losses due
to 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 neutralizes
the spinning air, so the unit is usable for blowing and exhausting (exit and inlet ducts). Use of
venturi inlet is possible as in tubeaxial fan.
Two-stage axial-flow fans are two fans connected in series, so pressure increase add up. It is
useful when excessive tip speeds and noise levels are not tolerated. There might be guided vanes
between two rotors rotating in the same direction. If counter-rotating rotors are used, guided
vanes are unnecessary.
3
4
CHAPTER 2. BACKGROUND
L
F
Suction side
of airfoil
LE
D
V
TE
α
Pressure side
of airfoil
V Relative air velocity D
α Angle of attack
L
Drag LE
Lift
TE
Chord
line
Leading edge F
Trainling edge
Resultant Force
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 later
in airfoil aerodynamics (section (2.2.1)). After flow passes the blades, the flow pattern has
helical shape. Flow can be decomposed into two component: axial velocity and tangential or
circumferential velocity. Axial velocity is the desired velocity since it moves air from/to the
desired spaces. Tangential velocity is an energy loss in propeller fan or tubeaxial fans. In
vaneaxial fans tangential velocity can be converted into static pressure. This makes vaneaxial
fans more efficient.
For good efficiency on the airflow of an axial fan it is usually demanded evenly distributed flow
over the working face of the fan wheel. This means axial velocity should be the same from hub to
tip on each blade. However, blade velocity is function of the radial distance. Velocity gradients
are then compensated by blade twisting, which outcomes in a smaller blade angle toward the
tip. Same hub to tip blade angle results in a loss of fan efficiency since air propulsion will
take place mostly on the outer region of the blade. Incorrect blade twist might cause stall in
the interior portion of the blade strongly hindering efficiency when working with higher static
pressures.
2.2
2.2.1
Fan Aerodynamics
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 perturbations, and because of structural strength characteristics. The trailing edge is rather
sharp.
• The chord line is defined as the line that connects the two lowest points of a two dimensional 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 a
maximum 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 chord
from the leading edge. It might happen the lower surface to be flat instead of concave for
some applications.
• The angle of attack, α, is measured as the angle between the relative air velocity and the
base line.
• As the airfoil moves through the gas, it produces a positive pressure on the lower surface
of the airfoil (or pressure side) and a negative pressure gradient in the upper side (or
suction side). Both forcces have approximately the same direction, but suction force is
close to twice the positive gradient force.
The forces described in the last point define F when they are added. This force can be decomposed into lift and drag forces. The first is perpendicular to the relative air velocity whereas
the second is parallel. Lift is the desired component in most engineering applications. Drag
is undesired since it causes power-consumption. However, this two objectives are conflicting
each other, and a trade off hast to be made while designing to obtain high lift forces, but good
lift-drag ratios. As the maximum section height of the airfoil profile increases, lift usually increases but lift-drag ratio tends to worsen. Selection of airfoil shapes is done according to the
desired application. For example, for compressors, wide airfoils are used. When efficiency is
the important parameter to be considered thinner airfoils are employed. Further, this forces
are strongly dependent on the angle of attack, and the overall range of operation has to be
considered when choosing the appropriate airfoil shape.
Aspect ratio is an important feature. It is the ratio between the total blade height and the
chord length. The bigger the aspect ratio, the better lift and lift-drag ratio. This is explained
by the trailing vortex phenomena. Trailing vortexes are generated when the fluid flows from the
pressure side to the suction surface though the outward space of the airfoil tip (the clearance
space in case of turbomachinery with casing). They strongly hint lift force. Big aspect ratio
is 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 common
in 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 to
symmetry considerations.
• As it can be seen in figure (2.3), as the angle of attack is increased, lift coefficient increases
as well. The maximum for this example curve is found at 15◦ . It is the maximum operating
point of the airfoil.
• The maximum lift-drag ratio is fount at 1◦ . For this example, best operating rate is
regarded 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 follow
the contour of the airfoil.
• When the angle is higher than 15◦ , the airfoil stalls. A huge fall in the efficiency occurs
driven by the boundary layer detachment.
The way airfoils are used in axial fans is shown in figure (2.4).
6
CHAPTER 2. BACKGROUND
Figure 2.2: Trailing Vortex generation around
the tip of an airfoil
2.2.2
Figure 2.3: NACA 6512 airfoil infinite aspect
ratio characteristic curve
Basic Characteristics of the Fan Flow Aerodynamics
Many different and diverse flow features are involved in turbomachinery which goes from supersonic velocity to rotating flows.
It is off special interest complex stress and performance losses that result from viscous flow
phenomena, mainly located at the airfoil boundary layers, but also blade to wall transient
boundary layers, near-wall flow migration, tip clearance and trailing vortexes , wakes and mixing. Research is also being done in relative end-wall motion and transition between rotation
and stationary end walls.
Another trend in aerodynamics research is the unsteadiness of the flow for time-varying conditions, vortex shedding from blade trailing edges (and impact in aspect ratio as described
in 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 in
the second half of the impeller flow passages. Consequently, flow separation is encountered
in the suction surface, causing wakes region, shifting the flow toward the pressure side. Flow
separation diminishes diffusion potential for the impeller and distorted jet/wake structures are
found in the discharge. Mixing follows this jet/wake structure and unsteady flow runs into the
diffuser. 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 centrifugal
compressor. He obtained detailed measurements of flow velocities and directions at multiple
locations in the flow field, from the inducer inlet to the impeller discharge. Eckardt observed
2.2. FAN AERODYNAMICS
7
Trailing edge of airfoil blade
Suction side on convex side
Rot
Hub Whell
dia. o.d.
Rot
D
ai efl
r fl ec
ow te
d
Incoming
air flow
Blade
angle
Leading edge of airfoil blade
Pressure side on concave of airfoil blade
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 impeller
blade chord. At the 60 chord % distance, a flow separation originated in the shroud suctionside corner of the passage. The separation rapidly grew to give raise to a wake. This wake is
produced from secondary flows as will be seen later. In short, vortexes near the shroud and
the 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-energy
passes through the tip clearance space causing the wake to increase through the downstream half
of 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 significant jet/wake flow after halve of the chord. Peak meridional velocities are located at the
blade hub-pressure sides, due to potential flow at high Reynolds numbers. Jet/wake structure
causes 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 or
employing Euler flow solvers (for high Reynolds numbers). After meridional flow is obtained,
normal and binormal vorticity can be numerically evaluated and applied to vorticity equations
to calculate streamwise vorticity. The use of potential flow solvers can be employed to solve
passage circulatory secondary flows.
n-Euler equation
The well known Bernoulli or Euler equations describe the mechanical energy terms (pressure
and velocity) along a streamline with constant energy. Whereas the n-Euler equation describes
the forces normal to this streamline. Thus, it is the equation explaining why lift occurs in
airfoils or why secondary flows/vortexes are created. The equation is obtained balancing the
centripetal acceleration of a fluid particle and the net force in the normal direction to the
streamline. Consider the mass of a fluid element to be ρR dθ dn. The particles centripetal
8
CHAPTER 2. BACKGROUND
Figure 2.5: Separated jet-wake structure from impeller [Tuzson, 2000]
acceleration towards the centre of the streamline is proportional to ρv 2 /R. For an inviscid fluid
in equilibrium, in absence of significant body forces the equation obtained is:
∂p
v2
=ρ
∂n
R
(2.1)
One might read this equation as follows. When a steady streamline is curved there is a centrifugal pressure gradient force acting on the streamline fluid particles. Pressure increases with
curvature radius.
As we stated before, equation (2.1) contains the principle behind airfoil lift and secondary
flows. The concave curvature of the streamline in the pressure side of an airfoil means higher
pressure is located below this surface compared to the undisturbed flow in the middle of the
pitch (line equidistant between two consecutive blades). Same reasoning allows to demonstrate
low pressure is encountered at the suction surface of the airfoil compared to the undisturbed
flow. 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 blade
turning. Employing the n-Euler equation, we deduce a pressure gradient has to be created from
the streamline curvature described in the lift force generation. However, further implications
have to be considered in the near wall region (usually hub and case), where the flow velocity
is smaller although a similar pressure-gradient (because of mechanical stability across the span
of the airfoil) exists. From the n-Euler equation, for constant pressure gradient, we deduce
ρv 2 /R = constant. If v is decreased due to viscosity effects, R has to decrease as well. The
streamlines follow different passage trajectories close to the walls as shown in figure 2.8. The
overturning of the fan moves slow velocity fluid close to the wall from the blade pressure surface
to the blade suction surface, and the motion is compensated by a return flow near the centre
of the passage. The combined effect is that two three-dimensional passage vortexes are set up
in the streamwise directions. These vortexes are one of the main sources of secondary flow in
blade passages. To completely understand this vortexes full momentum equations have to be
2.3. CFD BASICS
Figure 2.6: Lift force description by the pressure gradient of n-Euler equation
9
Figure 2.7: n-Euler equation physical representation
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) [Japikse
and Baines, 1997]
employed. This inertia-generated vortexes might be known as circulatory flow. The overall
effect 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 along
the wall, with a subsequent vortex. The strength of the vortex is determined by the starting
conditions, and its evolution to the conservation of its angular momentum. The vortex flows
are principally generated by the meridional flow field, while the centrifugal and Coriolis forces
act 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 complex flow filed analysis. Bear in mind this two objectives may demand diferent computational
10
CHAPTER 2. BACKGROUND
resources according to the results desired. For example, in we find two stages in component
design: preliminary and detailed design. During preliminary stage many variables are involved
and consequently many prototypes are proposed. At this point we expect a tool which allows
a 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 improvements by blade-row spacing analysis or initial blade shape design. Detailed design focuses on
a small number of design parameters based on preliminary design analysis. Examples for turbomachinery are tip clearance flows, blade-end wall interactions, flow separation, wakes, etc.
Detailed analysis simulations require order of magnitudes longer time because flow physics has
to be usually accurately resolved.
2.3.1
Flow Physics Modeling
For industrial applications Navier Stokes Equations are not tractable due to computational limitations. Mathematical treatment such as averaging is typically done to obtain the Reynoldsaveraged Navier-Stokes equations (RANS) in the case of Reynolds averaging. Other mathematical treatments produce other models such as Large Eddy Simulation (LES), which is more
popular in research. The set of equations including mass, momentum and energy conservation
are known as full Navier Stokes Equations. However, use from auxiliary equations is necessary
such as the equation of state (usually perfect gas law in fans), the Stokes hypothesis which
relates the second coefficient of viscosity (or bulk viscosity) to the molecular viscosity, and
Sutherland’s law, which expresses molecular viscosity as function of temperature. Turbulence
model for the Reynolds stress closure is also added to the previous system of equations. Use
of full RANS equations allows simulation of unsteady, 3D, viscous, turbulent flows in rotation
where the primary dependent variables are density, three velocity directions, total energy, pressure, enthalpy, nine components of the turbulent Reynolds stress tensor and three components
of turbulent heat-flux vector.
Further techniques like the thin layer assumption is widely used to reduce the complexity of
the problem. Within this assumption the streamwise diffusion term is neglected. It is used in
viscous layers, but must not be used when recirculation zones or viscous structures producing
streamwise diffusion is present.
Solution of the PDE must be done altogether with appropriate boundary condition. Three
type of spatial boundaries may be identified for turbomachinery: (1) wall boundaries, (2) inlet
and 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 inlet
and pressure conditions at the outlet. To do this, pressure, temperature, tangential velocity
upstream and pressure at downstream is usually specified. Depending on the turbulence model
selected, some turbulence properties are required. The inlet/exit boundaries should be placed
far enough from the blade, so they are not influenced by its presence. Typical distances to this
boundaries are 50 % to 100 % of the blade chord. Distribution of inlet conditions might be
included in the spanwise and tangential directions.
Periodic boundaries upstream and downstream of the blade are used to model one blade passage
to the next, assuming inlet conditions are periodic. Periodicity is forced by setting dependent
flow variables equal at equivalent positions on the periodic boundaries. Straight forward initialization of the problem can converge into the solution with no problem. However, an appropriate
distribution 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 inviscid 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 of
the 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 Full
Reynolds stress models. Turbulence tretment such as Large eddy simulation (LES) and full
Navier-Stokes equations are to expensive at this stage.
n-equation models represent into some extent the real turbulence physics. They use of n partial
differential equations which model transport of selected turbulence variables. For example, in
two 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 to
obtain what is known as the turbulent viscosity.
Full Reynolds stress model is a much more realistic representation of a turbulent flow but is
makes use of approximately twice the number of equations. It Solves transport equations for
all components of the specific Reynolds stress tensor R = –Tt /ρ ≡ v 0 v 0 . These model naturally
account 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 the
computational cost.
LES and full Navier-Stokes equations are useful in some cases during detailed simulation stage
or very sensitive physics simulation. For example, noise creation due to vortex generation in
airfoil 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 by
the 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 used
method for discretization is the Finite Volume Method. Finite Difference Method had more
users in the past, but requirements of structured grids compared to the flexibility of Finite
Volume Method, hinders its selection. Finite Element Method is also very common especially
in multiphysics simulations.
After discretization, the system of algebraic equations have to be numerically solved. According
to 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 previous
time-intervals. It is less computationally expensive than implicit methods. However, stability
is an issue to be considered when setting up the problem: time-step size is dependent on grid
size. On the other hand, implicit methods are numerically unconditionally stable to any timestep 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. Avoidance
of matrix inversion by factorization procedures is usually done to reduce the computational
cost. Hybrid methods might be used as well, which benefits from both implicit and explicit
characteristics. Some examples include the implicit residual smoothing in an explicit RungeKutta technique to relax the stability criterion. Or the two-step explicit one-step implicit Beam
12
CHAPTER 2. BACKGROUND
and Warming algorithm. Use of local time stepping is another approach. It is only useful for
steady-state simulations. In this case, time-step in time marching methods does not have any
physical 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 all
the mesh.
Non of implicit or explicit methods have demonstrated overriding performance against the other
and methods have similar levels of maturity. However, explicit methods are less computationally
expensive 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 one
for pressure) in a segregated, or uncoupled, manner. The linkage between the momentum and
continuity equations is achieved with a predictor-corrector approach. This model has its roots
in constant-density flows.
The Coupled solver on the other side considers the conservation equations for mass and momentum simultaneously when solving using a time- (or pseudo-time-) marching approach. The
preconditioned form of the governing equations used by the Coupled Flow model makes it suitable for solving incompressible and isothermal flows. One advantage of this formulation is its
robustness for solving flows with dominant source terms, such as rotation. Another advantage of the coupled solver is that CPU time scales linearly with cell count; in other words, the
convergence rate does not deteriorate as the mesh is refined. On the negative aspects of the
coupled 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 exceeded
the coupled solver capabilities in the computer employed. Formulation of Governing-Equations
in STAR-CCM is
Z
I
Z
d
ρχ d V +
ρ v − bmvg · d a =
Su d V
(2.2)
dt V
A
Z
I
I
I V
Z
d
ρχv d V +
ρv ⊗ (v − vg )· da = −
pI· d a +
T · a + (fr + fω ) d V
(2.3)
dt V
A
A
A
V
The terms on the left-hand side of equation (2.3) are the transient term and the convective
flux. On the right-hand side are the pressure gradient term, the viscous flux and the body force
terms. T is the viscous stress tensor (or Reynolds stress tensor). The body force terms have
been simplified to represent exclusively the effects of system rotation.
In turbulent flow, the complete stress tensor is given by:
2
T
T = µef f ∇v + ∇v − (∇· v) I
3
where the effective viscosity is µ = µl + µt , the sum of the laminar and turbulent viscosities.
Chapter 3
Turbulence
Turbulence is, of course, an important component to be considered. It must represent the
characteristics of typical turbomachinery flow fields, such as flow-path curvature, rotating flow,
high-pressure gradients, and separated, recirculating flows. The capability to model unsteady
flow and blade-row interaction is also necessary. In order to study turbulence, a simple introduction to the fluid motion equations will be done. They will be applied to Newtonian
incompressible 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
+ ∇ · (ρU ) = 0
∂t
is simplified for incompressible flows to yield the solenoidal or divergence-free equation
∇ · U = 0.
(3.2)
and applying relationship 3.1,
∇ · hU i = 0
∇ · u = 0.
In the case of the momentum equation some nonlinear terms are created after applying Reynolds
averaging.
First we write the substantial derivative conservative form.
DUj
∂Uj
∂
=
+
Ui Uj ,
Dt
∂t
∂xi
and calculate the mean value
DUj
Dt
∂ Uj
∂ =
+
Ui Uj
∂t
∂xi
(3.3)
Nonlinear average of Ui Uj is
Ui Uj =
hUi i + ui
Uj + uj
D
E
= hUi i Uj + ui Uj + uj hUi i + ui uj
= hUi i Uj + ui uj
13
(3.4)
14
CHAPTER 3. TURBULENCE
The velocity covariance ui uj is the so-called Reynolds stress.
employing equations (3.2), (3.3) and (3.4) we rewrite the total mass derivative
DUj
Dt
∂ Uj
∂ =
+
hUi i Uj + ui uj
∂t
∂xi
∂ Uj
∂ ∂ =
Uj +
ui uj
+ hUi i
∂t
∂xi
∂xi
(3.5)
Further, if we define the mean mass derivative,
which represents the rate of change of a point
moving with the local mean velocity U (x, t) as
D̄
∂
≡
+ hU i · ∇
∂t
D̄t
we obtain the relationship with the mass derivative (3.5)
DUj
Dt
=
D̄ ∂ ui uj .
Uj +
∂xi
D̄t
The momentum equation (or Navier Stokes Equation) for an incompressible Newtonian fluid
is
1
DU
= − ∇p + ν∇2 U .
(3.6)
Dt
ρ
And relating with previous deductions, its mean mass derivative or Reynolds equations
∂ ui uj
D̄ Uj
1 ∂ hpi
2
= ν∇ Uj −
−
.
∂xi
ρ ∂xj
D̄t
(3.7)
The structure of the Reynolds and the Navier-Stokes equations might look similar, but in the
Reynolds 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


!
D̄ Uj
∂ 
∂ hUi i ∂ Uj
=
µ
+
− hpi δij − ρ ui uj  .
ρ
∂xi
∂xj
∂xi
D̄t
(3.8)
The first term between brackets comes from the molecular description of the flow, it is called
the Viscous Stress. In Pope [2000] in shown how the Reynolds Stresses are also deducted when
calculating the mean momentum transfer.
As we saw with the full Navier Stokes equations, a three-dimensional flow is completely described
by four equations (assuming incompressible and temperature independent flows): Reynolds
equations and the continuity equation. However, in practice, the statistical introduction by
Reynolds averaging adds new variables in terms of Reynolds Stresses and the system is underdefined. This problem is known as the turbulence closure problem.
3.1. REYNOLDS EQUATIONS
3.1.2
15
Anisotropy and Tensor Properties
In tensor theory the diagonal components u21 = hu1 u1 i , u22 , u23 are called the normal
stresses,
and the off-diagonal components (ui uj , i 6= j) the shear stresses. When
ui uj = ui uj the tensor is called symmetric. Since the stress tensor is coordinate orientation 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 physical reasons must be positive (hu1 i ≥ 0). This way we obtain a positive semidefinite tensor. For
non-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
1
hu· ui = hui ui i .
2
2
which represents the mean fluctuating kinetic energy per unit mass. Then, the isotropic stress
tensor is obtained as 23 kσij . The difference is the anisotropic part
2
aij ≡ ui uj − kδij .
3
The anisotropic term is many times normalized:
ui uj
aij
1
=
− δij
bij =
2k
hul ul i
3
we solve for the Reynolds stress tensor
2
ui uj = kδij + aij
3 1
δij + bij .
= 2k
3
This last equation helps to understand why the anisotropic component is responsible for momentum transportation. The isotropic part only modifies the pressure term, which is irrelevant
for compressible fluids. Application of this ideas is shown in the modified mean pressure equation (3.9).
∂ ui uj
∂aij
∂ hpi
∂
2
ρ
+
=ρ
+
hpi + ρk .
(3.9)
∂xi
∂xj
∂xi
∂xj
3
We observe the isotropic 32 k is absorved in the mean pressure term.
In irrotational flows the Reynolds Stress tensor produces exclusively a modified pressure. Taking
zero mean and fluctuating vorticity, and thus ∂ui /∂xj − ∂uj /∂xi zero, one obtains
*
!+
∂uj
∂ui
∂
1
∂ ui
−
=
hui ui i −
ui uj = 0
∂xj
∂xi
∂xj 2
∂xi
which leads to the Corrsin-Kistler equation
∂ ∂k
ui uj =
.
∂xi
∂xj
We can appreciate in this equation how the stress tensor has the same effect as the isotropic
stress kδij , which can be absorbed into the modified pressure as before.
16
CHAPTER 3. TURBULENCE
3.2
Turbulent Boundary Layer
Boundary layers are the location of relevant flow features and are of primary interest in turbomachinery. Study of simplest boundary layers, formed by uniform-velocity flows over a plane plate
reveals that statistically can be described by two dimensions. This two-dimensional coordinate
system is defined such that the x-coordinate is set in the flow direction, and y-coordinate is
perpendicular to the surface. We define, U, V, W as the velocity in the positive x, y, z directions
respectively.
Boundary layer thickness, δ(x), increases
with x, and is generally (but not only) defined as the
value 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 experimental
measurements reasons. Some examples are Displacement thickness
Z ∞
hU i
∗
1−
δ (x) ≡
dy
U0
0
and momentum thickness
∞
Z
θ(x) ≡
0
hU i
U0
hU i
1−
dy.
U0
(3.10)
Viscous Scales
Viscous scales are some variables deduced in order to accurately describe the boundary layer
region of a fluid. We will start describing the total shear stress
τ = ρν
∂ hU i
− ρ huvi ,
∂y
(3.11)
which can be seen to be made up of a viscous term and the Reynolds Stress tensors. We will
further define wall shear stress
τw ≡ τ (0).
If normalization is applied one arrives to the normalized wall shear stress or skin-friction coefficient. Normalization is made by division with a velocity factor,
cf ≡
τ
.
1/2ρU02
If non-slip conditions are to be satisfied on the walls (U (x, t) = 0), equation (3.11) simplifies
to
d hU i
τ (y = 0) = τw = ρν
.
(3.12)
dy y=0
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 variables
of interest definition.
r
τw
• friction velocity, uτ ≡
ρ
r
ρ
ν
=
• viscous lengthscale, δν ≡ ν
τw
uτ
• friction Reynolds Number, Reτ ≡
uτ δ
δ
=
ν
δν
3.2. TURBULENT BOUNDARY LAYER
• viscous lengths or wall units y + ≡
17
y
uτ y
=
δν
ν
Notice that y + looks like a Reynolds number, and its magnitude expresses the relative weight
of the viscous and turbulent flows. The viscous contribution is, as reasoned before, 100 % at
the 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. Within
this region there is a dominant effect of molecular viscosity. When y + > 50 the outer layer
starts, where turbulence is the main contribution. Inside the viscous wall region, 3 sublayers
can be found: the viscous sublayer for y + < 5, in which the Reynolds shear stress is negligible
compared with the viscous stress. The range 5 < y + < 30 is called the buffer layer. And the
range 30 < y + < 50 known as the log-law region. As Reynolds number increases, the fraction
of the channel dominated by the viscous wall region decreases, since δν /δ varies as Re−1
τ .
Boundary layer formation and evolution is described in many fluid mechanics books. When the
free fluid stream contacts the surface of the plane plate edge (knonw as the leading edge), a
laminar flow region forms at this point and spreads towards the fluid core as the flow extends
in the surface. When the Reynolds number (which length parameter must be appropriately
defined to account for the laminar layer width) rises up to 106 a transition from laminar to
turbulent flow starts in the growing boundary layer. Boundary layer properties strongly change
from laminar to turbulent flow. The sketch in figure (3.1) shows this transition. Although a
turbulent boundary layer refers to the boundary layer that has undergone this transition, we
still find laminar motion in the viscous sublayer.
of velocity u against distance y from surface at point x.pdf
U0
Boundary layer
U=0.99 U0
Transition
region
δ
Turbulent
y
Laminal
Leading
edge
U
Transition point
Viscous
sublayer
x
Figure 3.1: Graph of velocity u against distance y from surface at point x. Source: Adapted by
the 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 compared
to the cross-stream gradients. Considering all velocity terms are zero at the surface the lateral
mean momentum equation simplifies to
1 ∂ hpi ∂ v 2
+
=0
ρ ∂y
∂y
If we integrate this equation between wall and free-stream we obtain
D E
hpi + ρ v 2 = p0 (x).
18
CHAPTER 3. TURBULENCE
1
1.0
<U>/U0
U/U0
0.8
γ
0.6
τ/τw
0.4
τ/τw
0.2
0
0
0.0
y/δ
1
Figure 3.2: Mean velocity, shear stress
and intermittency factor profiles in a zeropressure-gradient boundary layer, Reθ =
8000. Source: Adapted by the author from
Klebanoff (1954).
0
1
2
3
y/δx
4
5
6
7
Figure 3.3: Nomalized velocity and shearstress profiles from Blasius solution for
the zero-pressure-gradient laminar boundary
layer on a flat plate: y is normalized by
1/2
σx ≡ x/Rex = (xvU0 )1/2 . Source: Adapted by the author from Pope [2000]
.
Since v 2 equals zero at the wall, from previous equation the wall pressure pw (x) equals the
free 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
1 dp0
=−
∂y
ρ dx
If we consider a zero presure gradient flow (free-stream pressure doesn’t change along x coordinate), we finally get
!
∂ 2 hU i
1 ∂τ
=ν
= 0.
ρ ∂y y=0
∂y 2
y=0
Which might be integrated from y = 0 to y = ∞. The obtained equation is known as the
Kármán’s integral momentum equation,
τw =
d 2 dθ
ρU0 θ = ρU02 .
dx
dx
Where θ is defined by equation (3.10). This relation allows us to understand the influence of
the wall shear stress in the momentum thickness. Turbulent experimental results were obtained
by 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 inner
layer and one for the outer. In the inner layer we find the viscous sublayer law, the van Driest
damping function and the log-law which describes the velocity profile in the viscous sublayer the
log-law region and the buffer layer respectively. This three laws together creates what is known
as the law of the wall. In the outer layer there is a portion of the log-law starting from the inner
layer (generally starts to be applicable for y + > 30), and although is not always mentioned the
velocity-defect law is employed for certain big y + values.
3.2. TURBULENT BOUNDARY LAYER
19
We will introduce the universal laws, which means this equations are independent of the flow
characteristics such as Reynolds number and, thus, results are limited by the hypothesis imposed. 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 its
non-universality.
Law of the Wall
The law of the wall is the consequence of an adimensional modelling. A flow is completely
specified by ρ, ν, δ and uτ parameters. Only two adimensional parameters can be constructed
from this variables, so one may write
d hU i
uτ
y y
,
,
=
Φ
dy
y
δv δ
where Φ is a universal non-dimensional function. The idea behind choosing this adimensional
parameters resides in the possibility of neglecting the turbulent δ length scale when considering
flows close to the wall (y + < 50) and viscous δv for further regions. As it was mentioned
previously: viscosity drives the flow close to the wall, whereas turbulent viscosity is the main
source in further regions.
In the inner layer (where velocity is determined by viscous scales) function φ(y/δv , y/δ) tends
to φ1 y/δv . So, if y + ≡ y/δv and u+ (y + ) ≡ hU i /uτ and for y/δ << 1 then it is posible to
express
d hU i
uτ
y
=
ΦI
dy
y
δv
as
1
du+
= + ΦI (y + )
dy +
y
which after integration computes
u+ = fw (y + )
with
+
Z
fw (y ) =
0
y+
(3.13)
1
ΦI y 0 dy 0 .
0
y
It has been extensively validated that the function fw is universal for flows with Reynold numbers far from the transition region.
• From equation (3.12) and no-slip condition fw (0) = 0 and fw0 (0) = 1. Then, the viscous
sublayer 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 of
order 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 the
Inner 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 is
du+
1
= +
+
dy
ky
20
CHAPTER 3. TURBULENCE
which integrates to
1
ln y + + B
(3.15)
k
B is a constant, k is known as the von Kármán constant and, within small variations,
typical values are: k = 0.41, B = 5.2.
u+ =
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 region
description is the van Driest damping function giving rise to the law with this name. According
to the mixing-length hypothesis, which will be introduced in section (3.2.4), the total shear
stress is
τ (y)
∂ hU i
∂ hU i
=ν
+ νT
ρ
∂y
∂y
∂ hU i 2
∂ hU i
2
.
+ lm
=ν
∂y
∂y
+ ≡ l /δ
Normalizing this equation by the viscous scales and solving for ∂u+ /∂y + , defining lm
m v
and setting τ /τw unity for the inner layer, the solution yields
u+ = f2 (y + ) =
Z
0
y+
2τ /τw
h
1+ 1+
+ 2
(4τ /τw )(lm
)
i1/2 .
The mixing length hypothesis is not accurate, but gives an approximation of the real Reynolds
+ = ky + in both log-law
Stresses. According to this hypothesis we might write lm = ky or lm
region and the viscous sublayer. However, on the overlapping buffer layer, model consistency
+ is applied.
doesn’t occur unless a damping function to the parameter lm
+
+
+
+
lm = ky 1 − exp −y /A
The term in brackets is the van Driest damping function. A+ = 26 is a standard value. For
large y + the damping function tends to unity and the log-law is recovered. For a given k, the
specification 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 layer
with y/δ > 0.2). In this region flow deviates
from the log-law. A second function is
slightly
then defined, which added to the log-law fw δyν
fits the velocity profile in the mentioned
layer. This is the wake function w yδ .
hU i
y
Π
y
= fw
+ w
.
(3.16)
uτ
δν
k
δ
Π is called the wake strength parameter, and its value is flow dependent. The wake function
is assumed universal and it is defined to satisfy the normalization condition w(0) = 0 and
w(1) = 2.
Equation (3.16) is usually expressed as the velocity-defect law, where fw is substituted by the
log-law and condition hU iy=δ = U0 is imposed to obtain


"
#

U0 − hU i
1
y
y
=
− ln
+Π 2−w
uτ
k
δ
δ 
3.2. TURBULENT BOUNDARY LAYER
3.2.2
21
Kinetic Energy and Reynolds-Stress Balances
The definition of the kinetic energy is
1
E(x, t) ≡ U (x, t)· U (x, t)
2
We can perform a decomposition equivalent to the equation (3.1).
E(x, t) = Ē(x, t) + k(x, t).
Ē(x, t) is the kinetic energy of the mean flow and K the turbulent kinetic energy.
1
hU i · hU i
2
1
1
k(x, t) ≡ hu· ui = hui ui i
2
2
Ē(x, t) ≡
One might obtain the mean kinetic energy equation from the Reynolds Equation (3.7).
D̄Ē
+ ∇· T̄ = −P − ε̄
D̄t
(3.17)
where
∂Ui
P ≡ − ui uj
,
∂xj
T̄i ≡ Uj ui uj + hUi i hpi /ρ − 2ν Uj S̄ij
∂ 2 ui uj
ε̃ ≡ 2ν S̄ij S̄ij − ν
∂xi ∂xj
!
1 ∂ hUi i ∂ Uj
+
S̄ij =
2
∂xj
∂xi
(3.18)
Similarly, the mean turbulent kinetic energy is obtained after subtracting the Reynolds equations
from the Navier Stokes Equation (3.6)
D̄k
+ ∇· T 0 = P − ε
D̄t
(3.19)
where p0 is the fluctuating pressure term
Ti0 ≡
1
ui uj uj + ui p0 /ρ − 2ν uj sij
2
ε ≡ 2ν S̄ij S̄ij
where Sij is defined by equation (3.18). This equation can alternatively be written as
∂ 1
D̄k
+
ui uj uj + ui p0 /ρ = ν∇2 k + P − ε̃.
∂xi 2
D̄t
Where p0 = p − hpi. Which in the boundary-layer approximation, the equation reduces to
∂k
∂2k
∂ 1
1 ∂ 0
∂k
+ hV i
+ P − ε̃ + ν 2 −
νu· u −
νp .
0 = − hU i
∂x
∂y
∂y
∂y 2
ρ ∂y
The different terms are, from left to right, the mean flow convection, production, pseudodissipation, viscous diffusion, turbulent convection, and pressure transport. The profiles of the
terms are ploted in fig (3.4).
22
CHAPTER 3. TURBULENCE
1.0
gain
turbulent
convection
production
0.5
0.20
viscous
diffusion
0.10
viscous
diffusion
pressure transport
0.00
0.0
turbulent convection
pressure transport
-0.5
production
gain
-0.10
mean
convection
dissipation
mean
convection
dissipation
loss
loss
-0.20
-1.0
0.0
0.2
0.4
0.6
0.8
y/δ
0
1.0
10
20
30
y+
40
50
Figure 3.4: Turbulent kinetic energy budget in a turbulent boundary layer at Reθ = 1410. Left
plot 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. In
the log-law region, P and ε modules decrease with y. From y + ≈ 40 to y/δ ≈ 0.4 the balance
is dominated by production and dissipation. At last, in the outer boundary layer, production
becomes 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-stresses
are obtained from the fluctuating velocity in the Navier Stokes Equations u(x, t).
0=−
D̄ ∂ ui uj −
ui uj uk + ν∇2 ui uj + Pij + Πij − εij
∂xk
D̄t
where Pij is the production tensor
∂ hUi i
∂ Uj
− uj uk
,
Pij ≡ − ui uj
∂xk
∂xk
(3.20)
Πij is the velocity-pressure-gradient tensor
1
Πij ≡ −
ρ
*
∂p0
∂p0
ui
+ uj
∂xj
∂xi
+
and εij is the dissipation tensor
εij ≡ 2ν
∂ui ∂uj
∂xk ∂xk
.
Data for different terms of the Reynolds Stresses can be found in Pope [2000]. Main properties
are reported next.
Concerning to the normal-stress balance, since only ∂ hU i /∂y is a significant mean velocity
gradient, normal-stress production (equation (3.20)) can be approximated by
P11 = 2P = −2 huvi
P22 = P33 = 0
∂ hU i
∂y
3.2. TURBULENT BOUNDARY LAYER
23
Over most boundary layer, P11 is the dominant source term of u2 . Although pressure fluctuation does not play a big role in turbulent kinetic energy balance, it plays a central role in
the Reynolds Stress equations. It is responsible for the energy redistribution
2 among the normal
2
turbulent
components
hu
i.
Resulting
in
the
dominant
sink
term
for
u
and
source
for
i
2
2
2 v
and w . Last but not least, dissipation will be responsible for the sink of v and w bulk
energy.
In huvi shear-stress balance, dissipation is negligible. There is an approximate balance between
production, P, and the pressure term, Π. Mention that the dissipation term is isotropic in the
bulk 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 surface curvature and strong pressure gradient conditions [Patel and Sotiropoulos, 1997]. After few
tests 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 turbomachinery applications and consequently several tests were readily performed, specially for airfoil
shaped blades. Even today, phenomena such as boundary layer detachment, or boundary layer
instability are still under study. Rayleigh criterion is known for boundary layer stabilization
under curved surfaces description: in convex curvatures, which has the centre of curvature
inside the wall, the angular momentum increases with curvature and an stabilizing process occurs. 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 to
Pope [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 as
it was explained in equation (2.1). All this mechanisms for turbulence modelling are still under
study.
3.2.4
The Mixing Length Theory
As mentioned in section (2.3.2), the mixing length hypothesis is a simple turbulence closure
model used in the CFD community. Although its accuracy limitation is a concern in real
applications, it gives a rough approximation of the Reynolds Stress values. Lets remember that
the idea behind the turbulent viscosity theory is to obtain a model of the form
− huvi = vT
d hU i
dy
(3.21)
so that the Reynolds Stress in equation (3.8) can be included in the viscosity term. The term
νT is called the turbulent viscosity.
In the mixing length theory the turbulent viscosity is parametrized by a velocity and a length
variables, u∗ and lm respectively.
νt = u∗ lm .
Next step is to relate this parameters with known flow variables. Since the application of the
theory is high Reynolds number flows, and for this flows − huvi ≈ u2τ , from equation (3.21) it
is possible to write
huvi
u2
νt = − dhU i = − dhUτ i .
(3.22)
dy
dy
24
CHAPTER 3. TURBULENCE
Further, if this theory will be used within the log-law region where
d hU i uτ
d y = ky
as it was demonstrated in section (3.2.1), we further write
u2
νt = − dhUτ i = −uτ ky
dy
where we identify
1/2
u∗ = uτ = huvi
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
∗
2 d hU i νt = u lm = lm .
dy And in log-law region lm = ky.
3.3
3.3.1
Eddy Viscosity Based Turbulence Modelling
Eddy Viscosity Hypothesis
According to the Eddy Viscosity Hypothesis, the anisotropy aij ≡ ui uj − 23 kδij is intrinsically
determined by the mean velocity gradients
!
2
∂ hUi i ∂ Uj
ui uj − kδij = −νT
+
3
∂xj
∂xi
or equivalently
aij = −2νT z S̄ij
(3.23)
where S̄ij is the mean rate-of-strain tensor. In simple shear flows it reduces to
huvi = −νT
∂ hU i
∂y
as the mixing length theory predicts. Unfortunately, many flows don’t obey this last equation as
it 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 zero
equation model. Where as in two-equation models two transport equations are solved to obtain
the turbulent viscous term. Transport equation variables typically are k, ε or ω, giving rise
to the k − ε turbulence model or the k − ω turbulence model. It is accepted that the k − ε
model performs reasonably well for two-dimensional thin shear flows where small streamline
curvature and mean pressure gradients are found. As it could be the case of air surrounding an
airfoil. For strong pressure gradients k − ω model gives a more accurate prediction according
to bibliography.
3.3. EDDY VISCOSITY BASED TURBULENCE MODELLING
3.3.2
25
Wall Treatment
When region in the near-wall is considered, computation of the turbulence models faces some
difficulties. Hence, modification of turbulence models is very common. For example, experimental 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 has
been introduced is the code. Different types of wall treatment are available, high, low and ally + wall treatments. The all-wall treatment makes use of both high and low treatments when
necessary. This way, high wall treatment is employed in coarse-near wall cells, and low-wall
treatment in fine cells. When the wall-cell centroid falls in the buffer region of the boundary
layer, the wall treatment is acconditioned so the fluid motion is appropriately resolved.
Even though wall formulations is modified when choosing different turbulence models, the standard formulation is as follows:
(
+
+
u+
+
lam fory ≤ ym
u =
(3.24)
+
+
u+
turb fory > ym
(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
yu∗
ν
hU
i
u+ = ∗
u
y+ =
(3.26)
(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 particular
turbulence 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
E
ln( y + )
k
f
(3.29)
being k = 0.42 and E = 9.0. Roughness function value, f , is unity for smooth walls. Specific
modifications 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-layer
approach. Model coefficients are still the same in all the fluid domain, but enhanced treatment
is done near the wall region.
In order to understand the Realizable k − ε model, we first first explain the general k − ε
modelling. In this model transport equations are solved for turbulent kinetic energy, k, and
turbulence dissipation, ε. Specific formulation of parameters is done: for lengthscale (L =
26
CHAPTER 3. TURBULENCE
k 3/2 /ε), for timescale (τ = k/ε) and a singular quantity of dimensions νT (k 2 /ε) among other
options. With this definitions, lm kind of specifications are not necesary. It is called a complete
model.
k − ε model evolved for many years, but the roots of the model has always relied on the two
transport equations and the algebraic turbulent viscosity model. Two different approaches were
taken during transport equations development. In the case of k, the exact transport equation
was deduced from the Navier Stokes Equations. In the case of ε viscosity for high Reynolds
numbers 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 of
viscosity. It can be said that the equation is rather “empirical”. This empirical development of
the equation will induce some constrains in the applicability of the model.
The equations reported below are the continuous equations used in STAR-CCM for Realizable
k − ε model [CD-ADAPCO].
Basic Transport Equations
d
dt
Z
Z
ρk(v − vg )· d a =
ρk d V +
A
Z Z h
i
µt
µ+
∇k· d a +
Gk + Gb − ρ (ε − ε0 ) + γ + Sk d V
σk
A
V
V
(3.30)
(3.31)
Z A
µt
µ+
σε
d
dt
Z ∇ε· d a +
V
Z
Z
ρε(v − vg )· d a =
ρε d V +
V
A
ε
ε
√ Cε2 ρ (ε − ε0 ) + Sε d V
Cε1 Sε + (Cε1 Cε3 Gb ) −
k
k + νε
(3.32)
where Sk and Sε are the user-specified source terms, and ε0 is the ambient turbulence
value that counteracts turbulence decay. The production Gk is evaluated as:
2
2
Gk = µt S 2 − ρk∇· v − µt (∇· v)2
3√
√3
T
S = |S| = 2S : S = 2S : S
and
S=
1
∇v + ∇v T
2
(3.33)
Two dots operation refers to double dot product of a tensor. The turbulent viscosity is
computed as:
k2
µt = ρCµ
ε
although Cµ is not a constant.
1
A0 + As U (∗) kε
√
= S :S−W :W
Cµ =
U (∗)
3.3. EDDY VISCOSITY BASED TURBULENCE MODELLING
27
W is the rotation rate tensor,
1
T
W =
∇v − ∇v .
2
Only most important equations are writen here. Other terms and constant values can be
found 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 region
and modify Gk and Gε in the near-wall cell. As explained before, the all y + wall treatment
tries to mimic the high and low y + formulation. It also gives reasonable results in the buffer
layer.
For the all-y + formulation, a blending function g is defined in terms of the wall-distance based
Reynolds number
Rey
g = exp −
11
√
ky/v
q
1/2
u∗ = gνu/y + (1 − g) Cµ k
2 +
1
∂u
2
∗ u
Gk = gµt S + (1 − g)
ρu +
µ
u
∂y +
Rey =
ε=
k 3/2
lε
(3.34)
(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 algebraically
blends ε transport equation with a distance from the
wall formulation. More specifically, it
parametrizes 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 full
two-equation model

!
∗
Rey − Rey
1

λ = 1 + tanh
2
A
where Re∗y defines the applicability limit of the two-layer formulation. In STAR-CCM values
for 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 be
solved
∆Rey A=
atanh 0.98
In STAR-CCM+ ∆Rey = 10.
Turbulent viscosity from k − ε model is blended in the two-layer model as follows:
µt
µt = λ µt |k−ε + (1 − λ) µ
.
µ 2layer
28
CHAPTER 3. TURBULENCE
The discretized transport equation for ε is modified and yields
!
X
X
ap
n
n+1 n
n
− εp
∆εp +
an λ∆εn = λ b − ap εp −
an εn + (1 − λ) ap εp ω
2layer
n
n
3.3.4
Menters SST k − ω
k − ω model was developed by Wilcox when he proposed the substitution of the ε equation by
a modified variable: ω ≡ ε/k.
This leads to an equation similar to ε equation in the standard k − ε model, but an additional
term is created (obviously constant values also change). In the differential form of the equation
the additional term is
2νT
∇ω· ∇k.
(3.36)
σω k
If this term is neglected we obtain the standard k − ω model. It was found that k − ω model was
superior in the viscous near-wall region and in the streamwise pressure gradients. Unfortunately
ω treatment in boundaries is problematic: a value for ω has to be given in each boundary, and
it 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 − ω
models. This is known as the SST k − ω model. In this model, the omega equation includes a
weighting function for the term (3.36). When we consider cells far from the walls the weighting
function is set to one, and the k − ε model is employed. When cells are close to the walls, then
k − ω model is obtained by setting the weighting function to zero.
At last, Menter modified the linear constitutive equation and obtained the model which is
employed in STAR-CCM under SST (Menter) model. It is a model widely used in aerospace industry where viscous flows are well resolved and turbulence models in boundary layers extremely
important.
The basic transport equations from SST k − ω are
Basic Transport Equations
d
dt
Z
Z
ρk d V +
ρk v − vg · d a =
V Z
A
Z
(µ + σk µt ) ∇k· da +
γef f Gk − γ 0 ρβ ∗ fβ (ωk − ω0 k0 ) + Sk d V
A
V
d
dt
Z
Z
ρω d V +
ρω v − vg · da =
V
A
Z
Z 2
2
(µ + σω µt ) ∇ω· da +
Gω − ρβfβ ω − ω0 + Dω + Sω d V
A
V
Sk and Sω are the user-specified source terms. k0 and ω0 are the ambient turbulence
values in source terms that counteract turbulence decay. γef f is the effective intermittency provided by the Gamma ReTheta Transition model (it is unity if this model is not
activated. It basically helps to evaluate the momentum thickness Reynolds number in
unstructured messes) and
h
i
γ 0 = min max γef f , 0.1 , 1 .
3.3. EDDY VISCOSITY BASED TURBULENCE MODELLING
29
The production Gω equations is evaluated as
"
#
2
2
2
2
− ω∇· v
Gω = ργ
S −
∇· v
3
3
where γ is a blended coefficient of the model and S is the modulus of the mean strain rate
tensor, equation (3.33).
The cross derivative term Dω ,
1
Dω = 2 (1 − F1 ) ρσω2 ∇k· ∇ω.
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 + wall
treatments. In the k − ω model this strategy is used to obtain a reference velocity u∗ , a Gk
turbulence production and to compute a specific dissipation ω in wall laws and wall cells.
q
∗
u = gνu/y + (1 − g) β ∗ 1/2 k
2 +
∂u
1
∗ u
2
ρu +
Gk = gµt S + (1 − g)
µ
u
∂y +
∗
6ν
u
ω = g 2 + (1 + g) √ ∗
βy
β ky
with the blending function
Rey
g = exp −
11
and Rey =
√
.
ky/ν is the wall-distance Reynolds number introduced before.
30
CHAPTER 3. TURBULENCE
Chapter 4
Simulations
In this chapter simulation setting up and results are written. In sett up section, information
about the geometries, fan meshing, models employed and solver conditions selection is done. In
the 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 which
approximately 0.12 m belong to the blade and 0.03 m to the hub. The fan has an approximately
constant 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 black
line represents the airfoil plane cross section
location
Figure 4.2: Blade middle chord length cross
section.
The simulation domain is defined by a cylinder co-axial to the fan axis. The cylinder radius is
of 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 surface
remeshing was performed, followed by a Prism layer and a Trimmer volume meshing. The
surface remeshing basically consists on a re-triangulation of the mesh to make it suitable for
Finite Volume Methods. A Prism layer consists on a number of orthogonal cell layers next to
walls in order to improve boundary layer resolution. This layers are defined in terms of its
thickness, the number of cell layers, the size distribution of the cells and the function used to
generate them. It must be combined with another core volume mesh generator such as the
Trimmer volume meshing. This is a method employed in STAR-CCM which produces a high
quality 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
Surface growth rate
Fan surface mesh absolute minimum size
Fan surface mesh absolute Target size
Volume Meshing
Property
Maximum cell size
Number of prism layers
Prism layer stretching
Prism layer thickness
Boundary growth rate
Value
1.3
7.5E − 4 m
0.0015 m
Value
0.5
5
1.5
0.333 m
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. Let
remember in this thesis two simulations have been carried out, one based on k − ε turbulence
model, and the other based on k − ω model.
k − ε turbulence model
k − ω turbulence model
Optional Model
Cell Quality Remediation
Cell Quality Remediation
Reynolds-Averaged Navier-Stokes Turbulence
Two-layer All y + Wall Treatment
All y + Wall Treatment
Realizable k − ε two-layer
SST (Menter) k − ω
Energy
Segregated Fluid Temperature
Segregated Fluid Temperature
Material
Ideal Gas
Ideal Gas
Flow
Segregated Flow
Segregated Flow
Time
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 boundary
conditions . 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 mass
flow inlet boundary condition. Velocity normal to the surface is 0.05 m/s. The default
Intensity and viscosity ratio was left.
• The outlet boundary is a flow split outlet. Which basically will specify all the flow exits
through this boundary. It is compatible only with incompressible flows, but for low speed
of the flow it is a reasonable assumption.
• The far field boundary refers to the cylinder, which is far enough from the fan to have
any influence. We set this boundary as a wall type of boundary condition to reduce
reversed outflow related errors. Yet, slippery wall conditions are chosen in order to simplify
computation 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 framesRotating. A rotating
reference frame in STAR-CCM generates a constant flux in de discretized equations. This
allows 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 a
revolution is done each six time steps.
k − ε turbulence model
k − ω turbulence model
Implicit Unsteady
time step=0.033 s
time step = 0.033 s
1st order
1st order
Wall Distance
default
default
Segregated flow
default
default
Segregated energy
fluid und. relax fact. = 0.5
fluid und. relax fact. = 0.5
Turbulence model
Under-Relax. fact=0.7
Under-Relax. fact=0.7
Viscosity 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 iteration
number (set to 60) and the second is the error-tolerance criterion which is set to an academical
standard 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 energy
variables find the minimum around 0.01 in k − ω model, and in the k − model only energy
finds the same difficulty converging asymptotically to 0.005. It has been found easier to work
with the k − method rather than the k − ω since last one had convergence difficulties. In both
models few iterations were performed with low under-relaxation numbers and smaller time steps
since first time steps to overcome convergence errors.
34
CHAPTER 4. SIMULATIONS
4.2
Results
The objective behind this simulations is not to make any fan performance assessment, but to
compare 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 located
downstream the fan such that captures velocity and mass flow rate through it. The centre of the
plane is at Cartesian coordinates [0, 0, −0.4] with the normal vector parallel to the fan axis. The
plane has the same width and height of 0.6 m. Unless explicitly mentioned flow rates (velocity
or mass flow rates) are always space averaged over this constrained plane using formulas shown
below.
Other mean values are obtained after appropriate time averaging over the 30 seconds of flow
simulation computed. By appropriate it is meant ignoring initial start up of the flow, where
non 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 mean
variables. When no mean value is mentioned, it will refer to the surface averaged flux for a
specific time.
• Mass Mass Flow : Mass flow is a type of report system available in STAR-CCM. Mathematically it is measured as follows
P
f ρf φf v· af
P
(4.1)
ρf vf · af f
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
X pressure
+ ffshear · nf
f=
ff
f
where ffpressure and ffshear 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 force
should be computed.
4.2.1
Mass Flow
Study of Mass flow has been done using equation (4.1). As one could expect, oscillations around
a 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 from
one flow rate type is directly applicable to the other.
A time-averaged mean mass flow rate of 2.15 kg/min, or an equivalent 1.77 m3 /min has been
found with k − ε turbulence model. Mean velocity parallel to the fan axis direction is 0.054 m/s.
For the k − ω turbulence model, mass flow rate is 2.19 kg/min, and mean velocity 0.055 m/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. When
k − 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 20
seconds of flow simulation
Figure 4.4: k − ω turbulence model after 20
seconds of flow simulation
Figure 4.5: k − ε turbulence model after 30
seconds of flow simulation
Figure 4.6: k − ω turbulence model after 30
seconds of flow simulation
k − ε turbulence model the maximum oscillation is 44% amplitude. However, both maximum
amplitudes 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 − ω
model has a wider extension at this values. In another words, velocity distribution variance is
bigger for k − ω despite the same maximum / minimum values are found. This might explain
why 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 force
in the axis direction has been studied. This is, only lift power of the fan is considered, drag
force is not included in this thesis.
As with flow rate analysis, oscillations over a mean value are found. For k − ε model the
36
CHAPTER 4. SIMULATIONS
time averaged force is 0.07 N , and the maximum oscillation is about 28% of the mean value.
With k − ω model, average value is the same, 0.07 N , 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 0.1 m far from the hub centre on the pressure side of the blade: close to the trailing
edge and on the chord line centre. This point was chosen because is far from the blade tip, where
trailing vortexes exist, and boundary layer is developed as much as possible. Boundary layer
velocity is shown in figure (4.8), which is perpendicular to the blade surface. Blade surface
normal vector is [0.993, −0.15, 0.0]. We will go through equations in section (3.2.1) doing a
comparison of theory and simulations for k − ε model. k − ω model will be studied at the end
of this subsection.
Data used is summarized below. Wall velocity gradient and the wall stress values have been
obtained from simulations.
ν = 1.58 10−5 kg/ms
ρ = 1.17 kg/m3
τw and
dhU i
dy
τw = 0.07 P a
d hU i
= 3333.3 m/s.
d y y=0
are related by
τw = νρ
d hU i
dy
y=0
and from previous values it is computed τw = 0.061 P a, which is close to 0.07. Bear in mind
that gradient value has been obtained numerically by the writer from graphical representation
and is subject to error. Using τw = 0.070, viscous sublayer ends at y = y + δν = 0.3 mm when
y + = 5, which is under the value reported in the simulation (shown in figure (4.8) with a value
of 0.45 mm). Hence, simulation and theory show a deviation of 30 % error.
In the log-law region similar results are obtained. Solving for hU i in equation (3.15) with
y + = 30,
1
+
hU i = uτ
ln y
+ 5.2 = 3.3.
0.41
Whereas solution in figure (4.8) shows a value of 2.95 (at a distance y = 2 mm or equivalent
y + = 30). The relative error is 13.3 %. Same operations give for y + = 50 a value hU i = 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 predictions
from STAR-CCM. Thus, we have tried doing the same calculations with the Standard wall
treatment provided by STAR-CCM (equation (3.29) and related). We would expect them to
be closer to simulation results. Unexpectedly, results obtained are very similar to mentioned
analytical solutions. hU i = 3.33 for y + = 30 and hU i = 3.63 for y + = 50.
Measurements at the same blade point at the same simulation time employing k − ω turbulence
model gives very close boundary layer values compared to k − ε. For sake of illustration, taking
4.2. RESULTS
Figure 4.7:
Location of
the point used for boundary
layer profile study
37
Figure 4.8: Boundary layer velocity magnitude parallel to the
wall. It can be appreciated a linear velocity progression up to
almost 0.5 mm were the viscous sublayer ends. The right point
in the figure is located at y + = 30
point y + = 30 both turbulence models reports 2.95 m/s speed. Moreover, a viscous sublayer
size of 0.485 mm is measured. The same as k − ε model. We might summarize that in the
boundary layer, there is no difference between using k − ε and k − ω models at the point and
properties considered.
38
CHAPTER 4. SIMULATIONS
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 a
similar time averaged flow rate. However, for each physical time of the flow, different flow rates
are computed. k − ω model seems to overpredict the area extension at which high or low speed
flows occur. On the other side k − ε predicts a smoother velocity distribution, but still with
similar maximum and minimum values. k − ω also overpredicts the recirculation area behind
the fan. An important conclusion from the analysis above is that, if only blade simulation would
be done, k − ω would overpredict the blade aerodynamic efficiency (lift capabilities) compared
to k − ε. However, if complete fan is taken into account, the gas exhausting power of the fan
yields similar results.
Mention, understanding the strong oscillations of the flow rates is of big interest. Weighting of
the numerical instabilities, simulation setting or fan wrong design contribution to oscillation
should be done, despite the physically sensible results obtained in the simulations.
Similar conclusions are obtained from Force subsection. While k − ω has strong oscillations in
mass flow rate analysis, they are smaller than k − ε for force variable. Both conclusions are
not contradictory, they are complementary in fact. As we mentioned already, both turbulence
models show similar axial direction velocities in the near-blade (where we find the maximum
values). But the bigger force oscillations are bigger in k − ε must strongly damp to obtain a
lower flow oscillations. This matches the theory since it is known k − ε is more dissipative, and
thus velocity distributions are smoother in the flow field.
Most important conclusion from boundary layer subsection is that STAR-CCM overestimates
the size of the boundary layer in comparison with the simpler analytical equations reviewed
in section (3.2), or even the STAR-CCM standard wall value model. This has been proof by
wall distance and mean velocity calculations on the viscous and log law region. This conclusion
applies to both turbulence models. Answer to this prediction mismatch is not clear. Differences in simulations and analytical results might be consequence of the different formulations
employed in each turbulence model during computation. We saw in section (3.3.2) the normalization velocity is computed from turbulence related variables. Another possibility is the
boundary layer model, which was developed for mono-dimensional surface parallel flows, not
being representative of our case (we have certain angle of attack altogether with a complex
geometry which causes multi-dimensional boundary layers). Another possibility is, according
to section (3.2.3), effect of pressure gradients and leading and trailing edge curvatures to affect
the streamline curvature and distorts the boundary layer. A pressure plot from the simulations
is shown in figure (5.1). Experimental validation should be done to contrast simulations and
identify analytical values mismatch reasons.
39
40
CHAPTER 5. CONCLUSIONS AND FUTURE WORK
Figure 5.1: Blade pressure side surface pressure distribution. It is appreciated a few Pa overpressure on the leading edge while an under pressure on the trailing edge. Further analysis of
the simulation reveals a fast decay in over/underpressure with distance from the blade
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