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ENGINEERING
DIPLOMA THESIS
by
Emeka G. Chijioke
Modelling of the Laminar to Turbulent Flow
Transition in a Compressor Cascade Modelowanie przejścia laminarno-turbulentnego w przepływie
przez wieniec sprężarki
Index Number: 209323
Course : Mechanical Engineering (Mechanika i Budowa Maszyn)
Thesis Advisor : Dr inż. Sławomir Kubacki
Warsaw, January 2012.
WARSAW UNIVERSITY OF TECHNOLOGY
FACULTY OF POWER AND AERONAUTICAL
ENGINEERING
DEPARTMENT OF AERODYNAMICS
2
Undersigned, Emeka Chijioke, author of this Thesis.
Aware of criminal liability for making untrue statements I declare that the
following thesis was written personally by myself and that I did not use
any sources but the ones mentioned in the dissertation itself.
I also certify that this version of the work is identical with the attached
electronic version.
--------------------- ------------------
date signature
ACKNOWLEDGEMENTS
I would like to thank Dr. Sławomir Kubacki for his infinite patience, guidance
and support without which this project would never have been completed. Also,
my father and mother and the rest of my family who have always supported me
in everything I have chosen to partake.
3
ABSTRACT
The purpose of this project is to numerically calculate and understand the flow
behaviour over a NACA 65 Series airfoil using Fluent Solver and to correlate the
data obtained with the provided experimental data from Zaki paper (Zaki T.A,
Wissink J.G, Rodi W and Durbin P.A., Direct simulations of transition in a
compressor cascade: the influence of free-stream turbulence). The flow through
a compressor passage with incoming free-stream grid turbulence is simulated.
At moderate Reynolds number, laminar-to-turbulence transition can take place
on both sides of the airfoil, but proceeds in distinctly different manners (Zaki
and Wissink 2009).
The model for the airfoil 65 series was created, drawn and meshed in Gambit
using geometry data that were sent by Zaki. The model was read into FLUENT
where flow boundary conditions were applied and the discretized Navier-
Stokes equations were solved numerically. The airfoil section pressure
coefficient from the numeric simulation was compared with DNS data from
paper by Zaki and the results agree within 20% for angle of attack 42°.
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TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS………..…………………………………………....... 2
ABSTRACT.............................................................................................................. 3
1. INTRODUCTION.............................................................................................. 6
1.1 Motivation………………………………………………………………….. 6
1.2 Laminar versus turbulent flow.…………………………………………....... 7
1.3 Laminar flow……………………………………………………………….. 7
1.4 Turbulent flow………………………………………………………….…... 8
1.5 Viscosity…………………………………………………………............…. 8
1.6 Boundary layer…………………………………………………………....... 8
1.7 Transitional Modes………………………………………………………..... 9
2. THE GOVERNING EQUATIONS…………………..................................... 11
2.1 Time-averaged Navier-Stokes equations………………………….............. 12
2.2 Transitional model by Walters……………………………………………. 14
2.3 Turbulence model (SST k-ω)……………………………………………... 23
3. OBJECTIVES OF THE THESIS..………………………..………............... 24
4. COMPUTAIONAL DETAILS.……………...……………..…..…................ 24
4.1 Computational mesh……………………………………………………… 24
4.2 Boundary Conditions…………………………………………..………….. 29
5
4.3 Convergence Criteria…...…………………………………………......…... 32
5. RESULTS AND DISCUSSIONS...………….………………….………….... 34
5.1 Verification of the inlet conditions………………………...……………… 34
5.2 Analysis of the mean velocity and pressure fields……….....….................. 38
5.3 Contour plots of the turbulent to molecular viscosity ratio……………..... 45
5.4 Transition onset …………………………………………….………..….... 49
5.5 Pressure coefficient …………………………………..………………...… 51
5.6 The Skin friction coefficient…...……………...……..……………...……. 55
6. CONCLUSIONS...……………………………....…………………................ 55
7. REFERENCES………………………………………………….….…....…... 56
6
INTRODUCTION
1.1 Motivation
The ability to accurately predict transitional fluid flow behaviour is important to
the design of engineering systems in a wide variety of applications including
aerospace, automotive, biomedical, heating and cooling, power generation,
marine systems, and chemical processing. Methods for addressing boundary
layer transition in computational fluid dynamics (CFD) simulations can range
from highly empirical approaches based on “engineering insight” for example,
selecting an appropriate transition location and applying a turbulence model
only downstream of this location to direct numerical simulation (DNS).
Almost every flow in nature and in practical engineering applications is
turbulent. After years of research in turbulence, there still does not exist a
precise definition of turbulence. However, some of the characteristics of
turbulent flows can be listed: irregularity, diffusivity, large Reynolds numbers,
three-dimensional vorticity fluctuations, and dissipation (Tennekes and Lumley,
1972). Inspite of all the uncertainties associated with turbulent flows, it has
been encouraging that engineering calculations have been possible with well-
formulated turbulence models.
In 1937, Taylor and von Kármán proposed the following definition of
turbulence:
“Turbulence is an irregular motion which in general makes its appearance in
fluids, gaseous or liquid, when they flow past solid surfaces or even when
neighbouring streams of the same fluid flow past or over one another” (Wilcox,
1994). Turbulence is usually characterized by the presence of a wide range of
length and time scales (Wilcox, 1994).
7
1.2 Laminar versus turbulent flow
Some flows are smooth and orderly , while others are rather chaotic. The flows
characterized by smooth layers of fluid is called Laminar. Example, the flow of
highly viscous fluid such as oils at low velocities is typically laminar.
The highly disordered fluid flows at high velocities and also characterized
by velocity fluctuations is called Turbulent . The flow of low-velocity fluids such
as air at high velocities is typically turbulent. (Yunus A. Cengel ; John M.
Cimbala).
1.3 Laminar Flow
Laminar Flow is the smooth, uninterrupted flow of air over the contour of the
wings, fuselage, or other parts of an aircraft in flight. Laminar flow is most often
found at the front of a streamlined body and is an important factor in flight. If
the smooth flow of air is interrupted over a wing section, turbulence is created
which results in a loss of lift and a high degree of drag. An airfoil designed for
minimum drag and uninterrupted flow of the boundary layer is called a laminar
airfoil.
The theory in using an airfoil of a particular design was to maintain the
adhesion of the boundary layers of airflow which are present in flight as far aft
of the leading edge as possible. On normal airfoils, the boundary layer would
be interrupted at high speeds and the resultant break would cause a turbulent
flow over the remainder of the foil. This turbulence would be realized as drag
up to the point of maximum speed, at which time the control surfaces and
aircraft flying characteristics would be affected. The formation of the boundary
layer is a process of layers of air formed one next to the other, i.e. the term
laminar is derived from the lamination principle involved.
8
1.4 Turbulent Flow
Most flows encountered in engineering practice are turbulent. Turbulent
flow is a complex mechanism dominated by fluctuations, characterized by
random and rapid fluctuations of swirling regions of fluid, called eddies,
throughout the flow. In turbulent flow, the swirling eddies transports mass,
momentum, and energy to other regions of flow much more rapidly than
molecular diffusion , greatly enhancing mass, momentum, and heat transfer.
1.5 Viscosity
Viscosity is the measure of internal stickiness of a fluid. Viscosity is caused by
cohesive forces between the molecules in liquid and by molecular collision in
gases . There is no fluid with zero viscosity, so to some degree , all fluid flows
involve viscous effects (Yunus A. Cengel ; John M. Cimbala).
1.6 Boundary layer
As an object moves through a fluid, or as a fluid moves past an object, the
molecules of the fluid near the object are disturbed and move around the
object. Aerodynamic forces are generated between the fluid and the object.
The magnitude of these forces depend on the shape of the object, the speed of
the object, the mass of the fluid passing by the object and on two other
important properties of the fluid; the viscosity, or stickiness, and the
compressibility, or springiness, of the fluid.
Aerodynamic forces depend in a complex way on the viscosity of the fluid.
As the fluid moves past the object, the molecules right next to the surface stick
to the surface. The molecules just above the surface are slowed down in their
collisions with the molecules sticking to the surface. These molecules in turn
slow down the flow just above them. The farther one moves away from the
surface, the fewer the collisions affected by the object surface. This creates a
thin layer of fluid near the surface in which the velocity changes from zero at
the surface to the free stream value away from the surface. This layer is called
the boundary layer because it occurs on the boundary of the fluid.
9
1.7 Transitional Modes
The important flow characteristic of fluid flow is transition to turbulence.
Transition is the process by which a laminar flow changes to a turbulent flow.
It is known that, typically, the boundary layer flow is laminar over the surface
of the body before it transitions to turbulent flow due to flow instabilities.
Instability of a laminar flow does not immediately lead to turbulence, which is a
severely nonlinear and chaotic stage characterized by macroscopic “mixing” of
fluid particles. Some of the transition modes which lead to turbulence are
natural transition, bypass transition, or separated flow transition. Below are
brief discussion on these different transition modes in summary of what
appears in Mayle (1991).
Natural transition: during this process, initial breakdown of laminar
flow occurs because of amplification of small disturbances, after the initial
breakdown the flow goes through a complex sequence of changes finally
resulting in the chaotic state known as turbulence. Natural transition occurs
when the laminar boundary layer is affected by small disturbances, which grow
into an instability. This instability amplifies within the layer to a point where it
grows and develops into loop vortices with large fluctuations. These highly
fluctuating loop vortices inside the laminar boundary layer develop into
turbulent spots, which then are convected downstream, and eventually, with
time, grow and come together to form a fully developed turbulent boundary
layer.
Bypass transition: this usually occurs at high free-stream turbulence
levels. In this mode of transition, free-stream disturbances influence the
development of turbulent spots that are directly produced within the boundary
layer.
Separated-flow transition: this occurs in the laminar separation bubble.
The flow transitions into turbulent flow over the separated bubble and
reattaches to the surface forming a turbulent shear layer. This usually occurs in
an adverse pressure gradient region that contributes to the separation of the
laminar boundary layer. Separated flow transition is usually found on the
suction surface, near a compressor airfoil’s leading edge, or near the point of
10
minimum pressure. Turbine blades are likely to have separation along the
suction surface in the trailing edge region. High levels of free-stream
turbulence can cause early transition compared to lower turbulence levels.
In gas turbine engines, the flow is periodically unsteady, so is transition,
and this is called periodic-unsteady transition. In “wake-induced” transition,
the periodic passing of wakes from the upstream blades or obstructions causes
unsteadiness in the flow field and affects transition on the downstream blades.
There also exists something called reverse transition, i.e., transition from
turbulent to laminar flow, which is referred to as “relaminarization.” This is
usually expected to occur at low turbulence levels if the acceleration
parameter, ( )( ) , is greater than 3 x 10-6
. In this equation, U
refers to the velocity in the streamwise direction and x refers to the surface
coordinate in the streamwise direction.
Predicting transition becomes very important for improving the efficiencies of
gas turbine engines. Considering transition will lead to improved designs of
turbomachinery airfoils. A significant amount of research effort has been
devoted to determine the transition regime inside the boundary layer. Since
Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) are more
computationally expensive using present computing hardware, the Reynolds-
Averaged Navier-Stokes (RANS) equations continue to be better suited for
engineering calculations with the incorporation of appropriate turbulence and
transition models.
11
2 The Governing Equations
2.1 Time- Average Navier- Stokes Equation (RANS) The Reynolds-averaged Navier–Stokes equations (or RANS equations) are time-averaged equations of motion for fluid flow and are primarily used to describe turbulent flows.
Statistical methods are used to average the fluctuating properties of flow in turbulent case. To obtain the mean values, there are different averaging techniques such as time, spatial and ensemble averaging. For homogenous turbulence with uniform turbulent flow in all directions, spatial averaging is used. For stationary turbulence (i.e. a turbulent flow that, on average, does not vary with time), time averaging is used. Ensemble averaging is the most suitable averaging for flows decaying with time. For the flows that engineers mostly deal with, time averaging is used.
For this thesis, I used the time-averaging technique to couple the models
for both transitional and turbulent cases.
According to Walters (bulletin 80), the Reynolds averaging is also
relevant in the laminar flow region. This means that in principle, the time-
averaging technique can be used basically for the description of the laminar to
turbulent transition.
Time averaging yields an average and a fluctuating part for a certain
variable. These parts could be represented as the part of the instantaneous
parameter, for example, velocity.
( ) ( ) ( ) (1)
where ( ) is expressed as the instantaneous velocity with, ( ); average
and ( ) fluctuating part, see figure 1. below.
12
Fig.1, Time average velocity.
The starting point for the computation of a steady, incompressible and
viscous flow are the conservation equations for continuity and momentum relating the laws of mechanics to the fluid:
(2)
(3)
Where and are the velocity components, t is time, p is the pressure, is
density rho and is the viscous stress tensor defined by
(4)
Where is the molecular viscosity and is the strain-rate tensor,
(
) (5)
The convective term in Equation (3) is written in conservation form to
simplify the time-averaging process, i.e.,
( )
( ) (6)
𝑈𝑖(𝑥 𝑡)
𝑈𝑖(𝑥)
t
13
Taking advantage of Equation (2),
tends to zero.
Then combining Equations (3) through (6) the Navier-Stokes equations can be
written in conservation form.
( )
( ) (7)
Time averaging Equations (2) and (7) yields the Reynolds equations of
motion in conservation form as follows,
(8)
(
)
( ) (9)
The time-average conservation of mass, Equation (8) is identical to
instantaneous Equation (2) with the mean velocity replacing the instantaneous
velocity. Subtracting Equation (8) from Equation (2) shows that the fluctuating
velocity, , also has zero divergence. The appearance the correlation
is
the only difference between the time-averaged and instantaneous momentum
equations.
Formula is needed for computing
in order to compute all mean-flow
properties of the turbulent flow under consideration.
Equation (9) can be written using Equation (6) and this yields the
Reynolds average Navier-Stokes equation (RANS)
(
) (10)
The quantity
is known as the Reynolds-stress tensor denoted by .
Thus,
(11)
14
The Reynolds-stress tensor is a symmetric tensor ( ), and thus has six
independent components that has to be modelled. Therefor to solve Eq. (10), it
is necessary to find enough equations to close the system.
The tensor is modelled using the Boussinesq approximation (Wilcox
book), thus the Reynolds stress tensor is given by
(12)
Where is the turbulent or eddy viscosity, k is the specific kinetic
energy, is the density, the operator is the Kronecker delta.
2.2 Transitional model by Walters
This model described by Walters, is used to predict boundary layer
development and calculate transition onset. The model can be used to
effectively address the transition of the boundary layer from a laminar to a
turbulent regime.
Model Equations. The Governing equations for the new model is
considered to be a three-equation eddy-viscosity type, where three additional
model transport equations are solved for turbulent kinetic energy ( ), the
laminar kinetic energy , and the scale-determining variable ( ), defined here
as ( ⁄ ), where is the isotropic dissipation.
The transport equations in the conservative form provided in the paper
by (Walters and Cokljat) are
( )
(
)
[(
)
] (13)
15
( )
(
)
[
] (14)
( )
[
(
)
( )
√
]
[(
)
] (15)
The various terms in the equations represent production, destruction, and transport mechanisms. The inverse turbulent time-scale ( ) is used here for the new model rather than the dissipation rate ( ).
In the equation, the fully turbulent production, destruction, and gradient transport terms (first, third, and fifth terms on the right-hand side of Equation (15)) are comparable to the similar terms in the and equations, (13 and 14). The transition production term (second term on right-hand side) in Eq. (15) is intended to produce a reduction in turbulence length scale during the transition breakdown process. The fourth term on the right-hand side was included in order to decrease the length scale in the outer region of the turbulent boundary layer, which is necessary to ensure correct prediction of the boundary layer wake region.
The production of turbulent and laminar kinetic energy by mean strain is modeled as:
(16)
(17)
where, is the large-scale eddy-viscosity and is the small-scale eddy-
viscosity. The “small-scale” eddy-viscosity concept is defined as
√ (18)
16
where is the effective small-scale turbulence.
(19)
The kinematic wall effect is included through an effective (wall-limited) turbulence length scale and damping function .
(
) (20)
( ) (21)
The turbulent length scale ( ) is sufficiently large when it is greater than
is the distance from the wall, is a constant and is the turbulent length
scale, defined as
√
(22)
and in Eq.(19) is the shear-sheltering effect, which refers to the damping of
turbulence dynamics that occurs in thin regions of high vorticity. Its effect is to
prevents transfer of energy from streamwise to wall-normal (and spanwise)
components.
[ (
)
] (23)
The viscous wall effect is formed through the viscous damping function, which is computed in terms of the effective turbulence Reynolds number.
( √
) (24)
(25)
The turbulent viscosity coefficient is defined to satisfy the realizability
constraint following Shih et al.
(
) (26)
17
A damping function describing the turbulent production due to intermittency is given by
(
) (27)
The production of laminar kinetic energy is assumed to be governed by the large-scale near-wall turbulent fluctuations based on the correlation of pretransitional fluctuation growth with free-stream low-frequency wall-normal turbulent fluctuations. The large-scale turbulence contribution is given by
(28)
where the small-scale contribution is given in equation (19) above. The production term is
(29)
Where is the large- scale turbulent viscosity, and it is modelled as
{ (
)√
( )
} (30)
where
(31)
( ( )
) (32)
18
[
] (33)
In the new model, shear-sheltering is incorporated through a production damping term, while transition initiation is included through transfer terms in the and equations. The transition process which is the transfer of energy from the laminar kinetic energy to the turbulent kinetic energy assumes the variable to represent the magnitude of fluctuations that display the characteristics of fully turbulent flow, such as strong three dimensionality, multiple length and time-scales, energy cascading, and significant viscous dissipation. The initiation of the transition process in this model is based on local (single-point) flow conditions and adopts an approach for transition initiation based on the concept of shear-sheltering and consideration of relevant time-scales for nonlinear disturbance amplification and dissipation.
The anisotropic (near wall) dissipation terms for and is of the form.
√
√
(34)
√
√
(35)
The turbulent transport terms (diffusion terms) in the and equations
include an effective diffusivity defined as
√ (36)
To properly reproduce the behaviour of the boundary layer wake region (boundary layer production term), a kinematic damping function is defined
[ (
)
] (37)
19
Introducing here the Reynolds stress transport equation for incompressible flow provided in (Walters bulletin), Eq. (38).
The area of interest here is on the production term and the pressure strain term.
( )
(
)
[ ( )
] (38)
Uppercase denotes mean quantities, lowercase denotes fluctuating quantities, and the overbar denotes Reynolds averaging.
The production term, ( ), is responsible for the
transfer energy from the mean to the fluctuating flow, and the pressure strain
term, (
), is responsible for the redistribution of energy among the
normal Reynolds stress components and modify the shear stress components. The effect of the pressure strain term is typically modeled as a "return to
isotropy" in which high energy components transfer energy to lower energy components. Moreover, this term is expressed as the sum of a rapid part, which incorporates interactions between turbulent eddies and the mean velocity field, and a slow part that incorporates inter-eddy interactions.
The three main summarized physical mechanisms involved in the RANS-based description of Reynolds stress dynamics for the transitional boundary layer (Walter bulletin book) include:
1) production of one-dimensional streamwise fluctuation energy in the
pretransitional region by entrained freeestream turbulence interacting with the mean strainrate;
2) no generation of three-dimensional (normal and spanwise) fluctuations in the pretransitional region, due to suppression of the pressure strain mechanism found in turbulent flow;
3) transition initiation due to an increase in magnitude of the pressure strain term, which can be viewed as a transfer of energy from the one-dimensional streamwise fluctuations to the three-dimensional fluctuations more indicative of fully turbulent flow.
20
The transfer of energy from one to another is appropriately interpreted
as energy redistribution (via the pressure-strain mechanism) rather than production (due to interaction with the mean flow) or dissipation (due to viscous mechanisms), Since the total fluctuation energy in the model is
comprised of the sum of and . It is proposed that the suppression of the
pressure strain effect is due to the presence of short molecular diffusion time scales within the pretransitional region, relative to the characteristic time scale of the rapid pressure strain terms. Potential fluctuations associated with energy redistribution are therefore quickly dissipated. shear sheltering here reduces the effective wall-normal length scale of the entrained wall-normal stress component, reducing the diffusion time scale and stabilizing the mechanisms responsible for intercomponent energy transfer.
In the pretransitional region, entrained turbulent fluctuations do not couple strongly with the freestream velocity gradient, it instead act as perturbations on the mean (approximately laminar) velocity profile. Thereafter, giving the turbulent streamwise fluctuations scaled as:
(39)
Where is a fluctuation length scale in the wall-normal direction given by
√
(40)
Where is the local mean vorticity magnitude. An estimate for the molecular diffusion time scale is constructed as:
(41)
The time scale associated with the rapid pressure strain mechanisms is
proportional to the inverse of the vorticity magnitude:
(42)
The ratio of molecular diffusion to rapid pressure strain time scales is
therefore expressed as:
(43)
21
Equation (43) as proposed is the relevant local dimensionless quantity that is governing shear sheltering and transition initiation. When the ratio in Eq. (43) is small, pressure strain is suppressed and one-component fluctuations (i.e. laminar kinetic energy) are generated. When the ratio reaches a critical value, the rapid pressure strain term quickly increases in magnitude to generate three- dimensional turbulent fluctuations, and transition begins.
In Equation (13) – (15), The initiation of bypass transition is included through the term , this term is a function of the timescale ratio expressed in Eq.
(43), with transition initiating at a critical value
⁄ (44)
Where is the threshold function which controls the bypass transition
process:
(
) (45)
[(
) ] (46)
The breakdown to turbulence caused by instabilities is considered to be a natural transition production term, given by
(47)
is used for the modelling of the natural laminar to turbulent flow
transition (for low background turbulence intensity Tu<1). is not
responsible for the activation of the transition onset in my case, this is due to
high background of turbulence intensities.
(
) (48)
[( ) ] (49)
22
( √
) (50)
The sum of the small-scale and large-scale contributions
(51)
which are defined in equations (18) and (30) above, is the turbulent viscosity used in the time-averaged Navier-Stokes equations (9)
The Model Constants
The summary of the model constants for the k-kl-w transition are listed in table 1 below.
Table. 1 Summary of the new model constants.
23
2.3 Turbulence model (SST K-ω)
Transport equation for the Turbulence SST Model
The two transport equation is given by the following:
[( )
] (52)
[( )
] (53)
and the model is supplemented by the definition of turbulent viscosity:
(54)
The model constants are given in (Menter book) Reference 8.
24
3 Objectives of the thesis The overall objective of this research is to understand the transitional behavior for the flow over a NACA65 compressor blade cross-section. As predicting the fluid flow behavior from laminar to transition stages is important in many technical applications including turbines, compressors, airfoils (technique which is based on laminar boundary layer) .
To validate the flow model developed by Walters and comparing the numerical results obtained to the DNS results obtained by Zaki (Zaki T.A,
Wissink J.G, Rodi W and Durbin P.A., Direct simulations of transition in a
compressor cascade: the influence of free-stream turbulence). In particular, the computational studies was done for transitional and
turbulent flow on NACA65 airfoil using Fluent software. The turbulent flow use
the K-w STT- two equation model and the transitional flow predictions use the
K-Kl-w three equation model.
4 Computational Details
For computing transition, the approach used in this work involves: 1) obtaining a fully turbulent solution on the whole computational domain, and 2) restarting the fully turbulent solution with the transition model activated.
4.1 Computational Mesh
The mesh generation software, Gambit has been used to generate meshes.
Both structured and unstructured meshes are used throughout this research so
that accurate CFD calculations can be performed over the airfoil sections which
have reasonably complex geometry. Gambit is a modeling software that is
capable of creating meshed geometries that can be read into FLUENT and other
analysis software. Gambit is essential in the process of doing the CFD analysis:
it creates the working environment where the object is simulated. An
important part is creating the mesh surrounding the airfoil, this needs to be
done in all directions on the boundary wall to have a good resolved flow
around the airfoil after calculations.
25
The mesh and edges must also be grouped in order to set the necessary
boundary conditions effectively. An outline for the Gambit geometry creation is
shown in figure 2.
Fig. 2, Gambit diagram work process
Create Coordinates
Lunch Gambit
Import Coordinates
Define Mesh Boundaries
Create Mesh
Define Boundaries
Export mesh
26
Boundary layer mesh specification on the airfoil
Table 2, shows the parameters specified to obtain the grid on the airfoil
boundary layer,it was used Gambit for these specifications.
Table 2, airfoil boundary layer mesh specification
Final Mesh
Unstructured grid was applied in some parts and block-structured grid around
the blade and inside the channel see figure 3. Detailed views at the leading and
trailing edge are shown in figure 4.
Fig. 3, Final resultant mesh of the geometry.
Algorithm First row
(a)
Growth
Factor (b/a)
Rows Transition
pattern
Uniform 1.2 24 1.1
27
Fig. 4, Detail of the grid topology in the leading and trailing edge region.
28
Refined mesh
Refining the mesh was necessary so as to make sure that the solution is grid
independent. Density of the mesh is higher where the velocity gradients are
higher, that means in the proximities of the leading and trailing edges.
Transitional calculations were done using the refined grid of 350,000 cells.
Detail structure of the refined grid at the leading edge is presented in figure 5.
Structure of the refined grid
Fig. 5, Detail view of refined grid for transitional flow at the leading edge.
29
4.2 Boundary Conditions
A schematic of the computational domain with the boundary conditions is
shown in Figure 6. The blade geometry is designated V103 as used in the
experiments of Hilgenfeld and Pfitzner 2004. The separation of the top and
bottom computational boundaries is one blade pitch P = 0.59L. The streamwise
extent of the domain is 1.9L. The nean Velocity of the object relative to the
fluid ( ) based on the the axial chord L and the simulation Reynolds number,
Re = 138,500, was calculated given by:
(55)
where : Mean Velocity of the object relative to the fluid (m/s).
L – is the characteristic length, L(m), (axial cord length in this case is used).
is the fluid dynamic viscosity kg/(m·s).
is the fluid density (kg/m³).
At the inflow plane, x/L=−0.4, a mean velocity (Uo cos(α);Uo sin(α); 0) is
prescribed, where the flow angle α = 42o.
Note: The X- and Y-component of flow direction are set because of the
angle of attack: ( ) and ( ) .
Pressure outlet is applied at the flow outlet, this allows the static pressure
to be directly specified and is assumed to be equal to the freestream pressure.
Upper profile and lower profile are set to wall type conditions, and periodic
boundary conditions are applied in the y-direction. Table 4 summarizes the
boundary conditions and the numerical method implemented for all the
simulations both turbulent and transitional flow.
30
Fig. 6, Cross section through the computational domain at midspan.
A summary of the boundary conditions and the numerical method implemented for all the simulations both turbulent and transitional flow.
Model Steady, two dimensional, incompressible
Viscous model k-w and k-kl-w
Equations Reynolds-average navier-Stokes equations
Density
Viscosity
Turbulent case Transitional case
Pressure Standard Standard
Pressure- velocity coupling Coupled coupled
Momentum First order upwind Second order upwind
Turbulent kinetic energy First order upwind Second order upwind
Pressure Outlet
Velocity Inlet
Materials
Discretization
31
Laminar kinetic energy First order upwind Second order upwind
Specific dissipation rate First order upwind Second order upwind
Inlet
Velocity components, (method-
magnitude and direction)
The velocity components are established via the inlet
flow angle α =42o, with a velocity magnitude 10m/s
Turbulent case Transitional case
Turbulence intensities
Turbulent length scale Lt=0.008m
kinetic energy
[
]constant
Outlet
Outlet pressure(methods-from
neighbouring cell)
Backflow turbulent intensity
Backflow turbulent length scales Lt=0.008m
Stationary wall
Shear condition No slip
Periodic
Periodic type Translational
Table 3, Numerical method specifications.
Boundary Conditions
The turbulent intensities provided for test case are Tu = {3.25,6.50,10}%
static (gauge) pressure. The value of the specified
static pressure is used only while the flow is subsonic.
if the flow becomes supersonic the specified pressure
will no longer be used.
test case Tu = {3.25,6.50,10}%
32
4.3 Convergence Criteria
After inputting the mesh and the conditions of the flow, FLUENT iterated until
the solution converged. The scaled residuals show how close the flow is to
converging, the lower the residuals, the closer it is to convergence.
The governing equations were solved sequentially with the pressure-
correction SIMPLE method and momentum interpolation was used for the
pressure-velocity coupling. For the momentum and transport equations, the
scaled residual at each iteration step was obtained as a sum of residuals in all
cells of the computational domain divided by the left hand side of the
discretized equation summed over all cells. For the pressure-correction, the
non-scaled residual was first computed as the sum of the mass imbalance in all
cells. Then, scaling was performed by dividing the non-scaled residual by the
maximum continuity equation residual (normalization factor) reported during
the first five iterations. The multigrid method was used to solve the system of
equations at each iteration step while the Gauss-Seidel method was applied to
eliminate the oscillatory modes of the error at each multigrid level.
Since a numerical approach can only approximate, but never give an exact
solution, a limit for convergence has to be provided. Figure.7, shows the
convergence history for the turbulent flow, and Figure.8, shows the
convergence history for the transitional flow.
The residual criterion was set to 10-6 resulting in 2000 iterations for turbulent
case and 3000 iterations for the transitional case.
33
Convergence quality
Fig. 7, Convergence history for Turbulent K-omega model.
Fig. 8, Convergence history for transitional flow, K-Kl-w model.
34
5 Results and Discussions
The meshed geometry was exported from Gambit and was read into the Fluent
solver software. Calculations and observations was made. Computations was
done both on Coarser and Fine Grid, The mesh density is higher where the
velocity gradients are higher, that means in the proximities of the leading and
trailing edges. first the K-omega two-equation (Shear Stress Transport (SST) k-
ω Based Model) was calculated for turbulent prediction and then was applied
three equation turbulence Model K-Kl-omega.
5.1 Verification of the inlet conditions
A test case was done to be sure that the conditions for the incoming flows
are correct, to do this I had to compute for the corresponding turbulent length
scales (Lt) that would correspond to the turbulent intensities (Tu) provided in
the DNS results in Zaki paper.
Figures. 9 and 10 shows the turbulent intensity plots for the turbulent
length scales that were tested. The cases were, Lt = { 0.012, 0.024, 0.006,
0.008}, the results are divided in to two parts :
First part is figure 9, which shows the results for the different
turbulent length scales, Lt = { 0.012, 0.024, 0.006} tested to give the
turbulent intensity (Tu) that would correspond to the turbulent
intensity (Tu) provided in the DNS in Zaki paper. From the results
obtained, I therefore say that the turbulent intensities are increasing
monotonically as the length scales are increased.
Second part is figure 10, which shows the plot of turbulent intensity
for test case Lt= 0.008 compared with the turbulent intensity closer at
the edge of the boundary layers as provided in the DNS in Zaki paper,
and it was the nearest value that matched the DNS result.
35
From the results, I conclude that the comparable and nearest turbulent
length scale that produced the turbulent intensity provided in the DNS Zaki
paper is the length scale Lt = 0.008. This value was used throughout the rest of
the calculations.
Fig. 9, Calculated turbulent intensities (Tu) compared with the DNS for test
cases Lt = ( 0.012, 0.024, 0.006). Turbulent intensities are increasing
monotonically as the length scales are increased.
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1 1.2
Turb
ule
nce
inte
nsi
ty, T
u(%
)
Downstream distance, x/L
Turbulence intensity (Tu) for different Turbulent length Scales(Lt)
DNS
TestCase_Lt_0.012
TestCase_Lt_0.024
TestCase_Lt_0.006
36
Fig. 10, Calculated turbulent intensity (Tu) compared with the DNS for test case
Lt=0.008.The value Lt=0.008 was used for rest of the simulations.
Wall Yplus
In terms of boundary layer theory, y+ is simply a local thickness Reynolds
number. In terms of CFD y+ is a nondimensional distance from the wall to the
first grid point. In a practical sense, we don't resolve the solutions of turbulent
flow by direct numercial simulation. This would require a very fine mesh near
the wall in order to resolve the turbulent eddies in the boundary layer. Also,
turbulence is time varying and random, so CFD models would need to be run as
transient, even if the mean flow is steady state. Modern computers can't
handle this except for the simplest flow, maybe in the future it will be possible
to solve the turbulent Navier Stokes equations by direct numercial simulation
right down to the wall.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.2 0.4 0.6 0.8 1 1.2
Turb
ule
nce
inte
nsi
ty, T
u(%
) Turbulence intensity (Tu) for different
Turbulent length Scales(Lt)
DNS
TestCase_Lt_0.008
Downstream distance,
37
Turbulence models deal with the flow in the boundary layer. The boundary
layer is divided into an inner and outer region, and the inner region can be
further subdivided into a laminar (viscous) sublayer and a fully turbulent
region.
Reynolds number, surface roughness, adverse pressure gradient, etc will
change the value of y+ where the velocity profile swicthes from the viscous
sublayer to the log law of the wall. Resolving the viscous sublayer or laminar
boundary layer (say small y+ < 1) is crucial to get a correct shear stress in the
near wall region. The Upper limits on Y+, however depends on the Reynolds
number and becomes larger as Re increases.
The values of y+ are dependent on these two parameters: the resolution
of the mesh and the Reynolds number of the flow, and are defined only in wall-
adjacent cells. The maximal which cannot be exceeded is .
The value of y+ in the wall-adjacent cells dictates how wall shear stress is
calculated.
The equation for y+ is
√ (56)
Where y is the distance from the wall to the cell center, μ is the molecular viscosity, is the density of the air, and is the wall shear stress. Figure 11. indicates range between 0 .005 and 0.07 for the transitional computation. Therefore, I can conclude that the near-wall mesh resolution for this project is very good.
38
Fig. 11. Wall Y+ range from 0.005 – 0.07 at the lower and upper surface.
5.2 Analysis of the mean velocity and pressure fields
The contour plots of the mean velocity magnitude round the airfoil for the
turbulent and transitional case is shown in figures 12 and 13. The Flow
accelerates or increases significantly on the suction side up to x/C= 0.8 due to
displacement effect of the airfoil and it decelerates starting from x/C= 0.5
towards the trailing edge. It means that the boundary layer becomes prone to
separation on the suction side (adverse pressure gradient).
Flow decelerates gradually along the pressure side up to 80% of the
blade cord. Then slightly accelerates from x/C= 0.6. However there is no big
difference between the turbulent and transitional flow simulation in the bulk of
the flow. Some differences are visible in the separated shear layer close to the
trailing edge.
The velocity is much higher at the upper surface close to the leading
edge of the blade than that at the lower surface. This indicates that there is
lower pressure at the upper surface than the pressure at the lower surface as
shown in the contour plots of the static pressures , figures 19 and 20.
0
0.01
0.02
0.03
0.04
0.05
0.06
0 0.2 0.4 0.6 0.8 1 1.2
Wal
l Yp
lus
Downstream distance x/L
Wall Yplus for transitional case
Upperprofile
Lowerprofile
39
Fig.12, Mean velocity distribution and contour of velocity magnitude around
airfoil for turbulent case.
Fig. 13, Mean velocity distribution and contour of velocity magnitude around
airfoil for the transitional case.
40
Figures, 14 and 15 presents detailed views of the velocity vectors at the leading and trailing edge respectively. Looking closer to Fig. 14 at the area around the airfoil, the velocity approaches zero at the wall of the airfoil due to viscous effects.
Fig.14, Velocity vector detail at the leading edge region for turbulent flow.
Attached flow at the wall blade upper and lower surface.
41
Fig. 15, Velocity vector detail at the trailing edge region for turbulent flow.
The flow on the upper surface decelerates and converges with the flow on the lower surface.
Detail views for the velocity fields for transitional flow are presented below,
figure 16 presents flow details at the trialing edge, at mid-section figure 17 and
at the trailing edge figure 18.
42
Fig.16, Detail of velocity vectors at the leading edge region for transitional flow. The boundary layer is strongly accelerating close to the leading edge.
Fig.17, Detail of velocity vectors at mid-section of the suction and pressure surface, Transitional case. The velocity approaches zero at the wall of the airfoil due to viscous effects. The suction side prone to mild separation downstream of the blade.
43
Fig.18, Detail of the velocity vectors at the trailing edge region for transitional
flow. There is low velocity at the trailing edge and at the wake region. The flow on the upper surface decelerates and converges with the flow on the lower surface.
It is seen that the pressure distribution is negative over most of the surface,
except over small regions near the nose and the tail. The pressure distribution
at the upper surface is negative, see figures 19 and 20 as the velocity on the
upper surface is higher than the reference velocity. Whenever there is high
velocity vectors, we have low pressures and vise versa. The phenomenon
complies with the Bernoulli equation.
The static pressure distribution fig. 19 and 20, shows that there is low pressure
shifted towards the leading edge of the suction surface of the blade from x/C
=0.8 (dark blue zone) and it is reaching a high pressure value at the stagnation
region.
44
At the pressure surface the pressure increases gradually from x/C=0.9 (light
blue zone) towards the trailing edge and finally to the outlet flow plane (light
yellow zone).
From the inlet plane inside the domain, the blue area is approaching to the steady state and the yellow area is approaching to the unsteady state.
The pressure at the trailing edge is related to the airfoil thickness and
shape near the trailing edge. For thick airfoils the pressure here is slightly
positive (the velocity is a bit less than the freestream velocity). For infinitely
thin sections Cp = 0 at the trailing edge. Large positive values of Cp at the trailing
edge imply more severe adverse pressure gradients.
Fig. 19, Contours of static pressure for turbulent flow. The pressure increases
from its minimum value to the value at the trailing edge at the lower surface.
There is no big difference between the turbulent and transitional flow
simulation in the bulk of the flow. Some differences are visible in the separated
shear layer close to the trailing edge.
45
Fig. 20, Contours of static pressure for transitional flow. Pressure over most of
the upper surface are smaller than those over the bottom surface.
5.3 Contour plots of the turbulent to molecular viscosity ratio
The turbulent viscosity ratio
, is the ratio between the turbulent
viscosity, , and the molecular dynamic viscosity, . The ratio directly says something about how strong the influence of the turbulent viscosity is compared to the molecular viscosity.
Damping of the turbulent viscosity in the laminar zone in the simulations with the transition model by Walters is necessary for the expression Klebanoff modes development. The Klebanoff modes are streamwise-oriented fluctuations which are formed as a result of selective filtering of the freestream fluctuations. If the turbulent viscosity will be kept large in laminar region there will not be a possibility to model the pretransitional boundary layer. The streamwise-oriented structures will be damped by the turbulent fluctuations.
46
Figures 21 and 22 depicts the contours of turbulent to molecular viscosity ratio for the turbulent and transitional case respectively. Turbulent flow is still laminar, when transitional case at about x/C= 0.8 starts the turbulent flow.
Closer look at the near wall region on suction side (top blade) of the
profile fig (21), the boundary layer is still laminar (dark blue zone). In the
transitional case at about x/C= 0.8, the boundary layer is already turbulent at
this area. In the turbulent simulation the ratio of turbulent to molecular
viscosity is zero (dark blue colour), when the ratio of turbulent to molecular
viscosity for the transitional simulation at x/C=0.8 is 0.96 (thick yellow colour)
Additionally turbulent viscosity is also rather low in the turbulent zone (rear
part of the blade). This is not fully correct. It means that there is delayed onset
to transition, due to long boundary layer laminar flow prediction.
47
Fig. 21, Contours of Turbulent to molecular viscosity ratio for turbulent flow. Zoomed at x/C=0.8.
48
Fig. 22, Contours of Turbulent to molecular viscosity ratio for transitional flow. Zoomed at distance x/C= 0.8.
49
5.4 Transition Onset The ratio of molecular diffusion to rapid pressure strain time scales as described in (Walters bulletin 80 paper) is the important local dimensionless quantity governing shear sheltering and transition initiation. When this ratio is small, pressure strain is suppressed and one-component fluctuations (i.e. laminar kinetic energy) are generated. When the ratio reaches a critical value, the rapid pressure strain term quickly increases in magnitude to generate three-dimensional turbulent fluctuations, and transition begins. See chapter 2.2 above (transitional model by Walters) for full description of this process.
Figure. 23, shows the contour plot of the term as defined in equation (46) above, is a function of the time scale ratio expressed in Eq. (43) with transition initiating at a critical value .
Normal lines are created on the wall surfaces at distances x/L = {0.2, 0.3, and 0.4} respectively to verify the transition onset see figures 23 and magnified view of the contour plot of at the pressure surface at distance x/L = 0.2, the pressure is continuously increasing up to x/C=0.9. This summarizes the fact that transition onset is predicted too late. The too late transition is due to missing model sensitivity to the strong adverse pressure gradient.
In the DNS (Zaki paper), the transition to turbulent started at location x/L 0.2 on the pressure surface at turbulent intensity Tu=3.25% when in computation is at x/L= 0.25, see the skin friction plot figure 29.
According to Mayle the adverse pressure gradient shifts the transition onset
more upstream. This phenomenon is not well reproduced by the Walters
model.
50
Fig. 23, Contour plot of for transitional flow. Normal distances
created at x/L= 0.2, 0.3, and 0.4. Shows the transition onset at x/C=0.8.
X/L =0.2
51
5.5 Pressure coefficient This section shows the variations in the pressure coefficient around the airfoil, for turbulent flow, transitional flow and experimental data.
The pressure coefficient is prerequisite for reliable simulation of the flow
over the airfoils/blades. It is a dimensionless parameter which describes the relative pressures throughout a flow field in fluid dynamics defined by the equation
( )
(57)
Where is the static pressure, is the reference pressure and
is the reference dynamic pressure given by
(58)
Where is the reference velocity of the fluid, is the density of air.
The reference pressure is simply a device used to avoid the
numerical round-off errors in the numerical calculations involving pressure.
This can occur when the dynamic pressure change in a fluid, which is what
drives the flow, are small compared to the absolute pressure level.
The reference pressure was computed by subtracting off a suitable
constant reference pressure, to obtain the working pressure that is less prone
to round-off errors.
Generally is plotted against x/C. The blade chord length x/C varies
from 0 at the leading edge to 1.0 at the trailing edge. is plotted "upside-
down" with negative values (suction), higher on the plots presented.
The shape of the pressure distribution is directly related to the airfoil
performance. The features that affects airfoil performance are:
Favourable pressure gradient that leads to laminar flow close the leading
edge of the blade.
52
Minimum Cp value which determines the maximum speed at upstream of
the blade that can be useful to indicate shock formation and also
determine the extent of pressure recovery.
Adverse pressure gradients that leads to transition and separation. It is
especially important to avoid strong adverse gradients as we approach
the leading edge.
From the computation the flow behaviour along the blade shows an
initial acceleration on the suction surface up to 20% of the chord from the
leading edge. Thereafter a constant deceleration of the flow towards the
trailing edge is noticeable an on the pressure surface the pressure distribution
remains almost constant along the blade for case Tu=3.25% and 6.50% in both
transitional and turbulent case, see figures 24 and 25, below.
The flow agreement are not perfect at the if comparing with the DNS,
NOTE
Figures 24-27, the top curves corresponds to the pressure surface and the
lower curve corresponds to the suction surface. At the suction surfaces some
parts has negative values at the leading edge as the pressure here is lower than
the reference pressure.
53
Fig. 24, pressure distribution along the blade surface for turbulent
case compared with the transitional case, for case Tu = 6.50%.
Fig. 25, Pressure distribution along the blade surface for the transition
case. Comparison between the increasing turbulent intensities, Tu 3.25%
and 6.50% which was calculated.
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8 1 1.2
pre
ssu
re c
oef
fici
ent 𝐶𝑝
Distance ×/L
Pressure Coefficient Vs. Dwonstream distance for transitional flow
CFD_Transitionalcase_Tu_6.50%
CFD_Turbulentcase_Tu=6.50%
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8 1 1.2pre
ssu
re c
oef
fici
ent 𝐶𝑝
Distance ×/L
Pressure Coefficient Vs. Dwonstream distance for transitional flow
CFD_Tu_3.25%
CFD_Tu=6.50%
54
Fig. 26, pressure distribution along the blade surface for transitional
flow compared with the DNS result by Zaki paper, for the case Tu =
3.25%.
Figure. 27, pressure distribution along the blade surface for transitional flow compared with the DNS for case Tu = 6.50%.
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8 1 1.2
pre
ssu
re c
oef
fici
ent 𝐶𝑝
Distance ×/L
Pressure Coefficient Vs. Dwonstream distance for transitional flow
DNS_Tu_3.25%
CFD_Tu_3.25%
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8 1 1.2
pre
ssu
re c
oef
fici
ent 𝐶𝑝
Distance ×/L
Pressure Coefficient Vs. Dwonstream distance for transitional flow
DNS_Tu=6.50%
CFD_Tu=6.50%
55
5.6 The skin friction coefficient
The skin friction coefficient is defined by
(59)
Where is the local wall shear stress , is the fluid density and is the free-stream velocity (usually taken outside of the boundary layer or at the inlet).
Computations shows that the Walters model is not performing well. Zaki
paper shows that flow with the presence of free-stream perturbations, even at
the lowest intensity , the boundary layer transitions to turbulence.
Transition onset, defined as the point of minimum skin friction, takes place at
x/L=0.2 figure 29, which is upstream of the laminar separation point. The
transitional boundary layer circumvents separation, and transition is complete
at x/L=0.46 for the DNS. There is underprediction for the computational case.
Transition onset is located at x/L=0.24 and transition is completed at x/L= 0.48,
this underprediction compared to the DNS, see the skin friction coefficient
figure 29 for Tu=3.25%.
Figure 28, shows the skin friction for turbulent case along the pressure surface compared with transitional case and DNS, for Tu=6.50%, it shows that the turbulent flow simulation are not able to reproduce the laminar boundary layer. The flow is fully turbulent already at the leading edge.
Comparing the skin friction coefficient Cf figure (28) for the transitional case to that of turbulent case, the turbulent part is predicted but deviated upstream as from the distances x/L=0.04 to x/L=0.9 indicating the transitional flow. There underprediction compared with DNS data.
Figure 29, shows the skin friction along the pressure surface for the turbulent intensities Tu = {3.25, 6.50, 10}%, for transition case (CFD) compared with DNS data. The Transitional location is observed to move upstream monotonically as the turbulent intensity is increased. The earlier transition onset with increasing Tu is physical, this gives that the numerical results are therefore qualitatively correct.
56
Fig. 28, Skin Friction along the pressure surface for turbulent case compared
with transitional case and DNS data, for the case Tu=3.25%.
Fig. 29, Skin Friction Cf , along the pressure surface for transitional case
compared with the DNS data . For cases, Tu – 3.25%, 6.50% and 10%.
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 0.2 0.4 0.6 0.8 1 1.2
DNS
Transitionalcase
Skin
Fri
ctio
n
Pressure surface skin friction
Downstream distance ×/L
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 0.2 0.4 0.6 0.8 1 1.2
Skin
Fri
ctio
n 𝐶𝑓
Downstream distance ×/L
Pressure surface skin friction coefficient
DNS_Tu=10%
DNS_Tu=6.50%
DNS_Tu=3.25%
CFD_Tu=10%
CFD_Tu=6.50%
CFD_Tu=3.25%
57
6 Conclusions
Navier–Stokes equation solver was coupled with a transition model and a Reynolds-averaged three equation model, with the Reynolds number 138,500, at flow angle 42o and in velocity 10m/s with various levels of free- stream turbulent intensities, to understand the flow behavior around a NACA65 compressor blade cross-section. The following conclusions are made based on the observations and results analysis.
The transition onset is predicted too late. The too late transition is due to missing model sensitivity to the strong adverse pressure gradient. According to Mayle the adverse pressure gradient phenomenon that shifts the transition onset more upstream is not well reproduced by the Walters model.
The Transitional location is observed to move upstream monotonically as
the turbulent intensity is increased. The earlier transition onset with
increasing Tu is physical, this shows that the numerical results are therefore
qualitatively correct.
Contours of static pressure, pressure coefficient on the entire blade
surface, were compared for both the turbulent and transitional flow.
Because the top surface of the blade is curved, the air travels a greater
distance at a faster rate over the top of the blade than over the bottom of
the blade. This causes a difference in pressure between the top and bottom
of the blade. There is higher pressure on the bottom of the blade than on
the top. At the leading edge the flow is laminar, and at the trailing edge, the
flow is turbulent.
58
7 REFERENCES
1 Zaki T.A, Wissink J.G, Rodi W and Durbin P.A., Direct simulations of
transition in a compressor cascade: the influence of free-stream
tuebulence.
2 Mayle, R.E., 1991, “ The Role of Laminar-Turbulent Transition in Gas
Turbine Engines,” ASME Journal of Turbomachinery, Vol. 113, pp. 509-
537.
3 Wilcox, D.C., (1994), Turbulence modelling for CFD, DCW Industries, la
Canada, Ca.
4 Walters, D.K and Cokljat D., 2008, “ A Three –Equation Eddy –Viscosity
Model for Reynolds- Averaged Navier-Stokes Simulations of transitional
Flow.
5 Menter, F. R., "Two-Equation Eddy-Viscosity Turbulence Models for
Engineering Applications," AIAA Journal, Vol. 32, No. 8, August 1994, pp.
1598-1605.
6 ECROFTAC Bulletin 80, September 2009, ‘Transition Modelling’.
7 "ANSYS FLUENT 12.1 and FLUENT 6.3.26 ,Flow Modeling Software."
8 Çengel, Yunus A. Fluid mechanics : fundamentals and applications /
Yunus A. Çengel, John M. Cimbala.McGraw-Hill Series in Mechanical
Engineering : McGraw-Hill/Higher Education,cop. 2006.
