CFD Training Manual
University of Connecticut
Mechanical Engineering
Department
Momtchil Petkov
Mario Roman
Advisor: Professor Barber
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Table of Contents
Abstract……………………………………………………….
3
Nomenclature…………………………………….……………
3
I.
Introduction………………………………..……….……
4
II.
CFD Analysis Roadmap…………………..…….………
4
III.
Theory……………………………….….…..…….
7
IV.
Module Example—Laminar Flow Past a Cylindrical Pipe………
18
V.
Summary of Tutorials……………………………….……..
27
VI.
Conclusion ………………………………………..…….
43
VII.
References……………………………………………….
44
VIII. Appendices - Modules
A--Laminar Pipe Flow……………………….……………
B--Turbulent Pipe Flow…………………….………..……
C--Laminar Flow Over Flat Plate (Geometry and Mesh)…
D--Laminar Flow Over Flat Plate…………………………
E--Nozzle Tutorial ………………………………………..
F--Jets: Turbulent Flow……………………………..…….
G—External Turbulent Compressible and Incompressible Flow across an airfoil….
`
H—Turbulent Incompressible Flow across a Periodic Airfoil …………
I—Discrete Phase Modeling: Particle Injection into a Pipe………………
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Abstract
ANSYS FLUENT, computational fluid dynamics (CFD) software is very difficult to use
especially for new users. To help with this, tutorials have been created to be able to guide users
in the use of ANSYS FLUENT. They were created to mimic a classroom-like structure, where
the fundamentals are learned first. The creation of fundamental tutorials will allow users to have
projects where they will utilize the learning modules as references to guide them in more
complicated projects. Through the use of FLUENT and several validation efforts, which are
referenced from scholarly sources, the user will be able to validate the accuracy of their results.
In addition, with the help of the provided learning modules, the user will be able to create a
roadmap to achieve competence to solve more complicated problems.
Nomenclature
A
Area
𝐶𝑓
Coefficient of friction along wall
𝐶𝐿
Coefficient of Lift
𝐶𝐷
Coefficient of Drag
𝐶𝑝
Coefficient of Pressure
D
Diameter
𝑔
Gravity
𝐾 − 𝐸𝑝𝑠𝑖𝑙𝑜𝑛-Turbulence model to simulate and read turbulent flow
M
Mach number
P
Pressure
Re
Reynolds number
𝑅𝑒𝑥 Reynolds number along a position x
𝑈𝑐𝑙
Centerline Velocity
𝑢∞
Fluid velocity
u
Friction velocity
𝑈𝑚𝑎𝑥 Max. Velocity
V
Velocity
𝑉𝑖𝑛𝑙𝑒𝑡 Inlet Velocity
Yp
Distance to the wall from center of pipe
𝑦+
Non-dimensionalized distance of first grid point from wall
𝜌–
𝜏–
𝜇–
Density
Shear Stress
Dynamic viscosity
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I.
Introduction
One of the biggest challenges in the engineering industry is being able to come up with efficient
and optimal designs for new products. One of the strongest tools offered is FLUENT. FLUENT
is a very useful program recently acquired by ANSYS. It has the capability to model fluid flow
past objects with the ability to design, test, and analyze results all under one program. Although
it is a strong tool for engineering, it is also very difficult to use. For this reason, tutorials have
been created to teach FLUENT with the hope that these tutorials will serve as a fundamental tool
in teaching and as references for future senior design projects. The way the tutorials are set up
are by creating and analyzing basic flow fields and then to ensure the accuracy of each test case
it is then validated against a scholarly reference.
The structure of the tutorials is to first reproduce the fundamentals learned in a Fluid Mechanics
and Thermo Dynamics courses. One of the first scenarios learned in fluid mechanics is the flow
through a cylindrical pipe. The tutorials created follow very closely to how a fluid mechanics
course would be taught. For this reason, the first tutorial is the laminar flow of fluid through a
cylindrical pipe. The next tutorial is turbulent flow of fluid through a cylindrical pipe. By doing
the turbulent case, it will allow the user to see the difference between laminar and turbulent
flows and to gain some insight as to why different methods of analyzing structures in FLUENT
are necessary.
The next created learning module is to analyze flow over a flat plate. Analyzing the flow over a
flat plate is very important because it will give the user a more in-depth look as to what happens
when flow passes over an object. In addition, other tutorials such as a turbulent flow past a
nozzle, turbulent jet flow, turbulent compressible and incompressible flow past an airfoil,
turbulent incompressible flow past a periodic airfoil, and a discrete phase modeling tutorials are
created to be able to serve as fundamental tutorials so that the user may then use them as
precursors to analyzing more complicated problems.
In addition, one of the most important parts in creating the tutorials is the need for validation.
Validation is extremely important when analyzing solutions, because it is the only way to ensure
the accuracy of the results obtained in FLUENT. Validation is made by comparing results from
FLUENT to theoretical and experimental data from scholarly sources.
II.
CFD Analysis Roadmap
The importance of the created learning tutorials is to guide users into ANSYS FLUENT and
provide them with a friendly introduction to the CFD software. For this reason fundamental
learning modules have been created which are: laminar and turbulent fluid flow through a
cylindrical pipe, laminar fluid flow over a flat plate, turbulent flow through a nozzle, turbulent jet
flow, turbulent compressible and incompressible flow past an airfoil, turbulent incompressible
flow past a periodic airfoil and discrete phase modeling. So why were these specific modules
chosen and created?

The purpose for these tutorials is to lead a new user through options of increasing difficulty.
The laminar pipe flow tutorial helped to introduce the icons and tools that ANSYS has to
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


offer. By having the user work with this very simple tutorial, they could familiarize
themselves with where certain icons are and where certain tools are located. Once the user
has completed this tutorial then the next tutorial added increased in complexity. The reason
for this method is to be able to instill confidence in the user to make them feel confident in
doing simple cases and build up to more complicated ones.
After each of the simple cases is run, the user has to validate each result. By validating one’s
results the user is ensured they have created an accurate simulation. For instance, if a first
time user has to analyze an airfoil, they should not start by designing an airfoil. Although
they might obtain results, how would they know if the results are accurate? For this reason,
the user would first figure out how flow develops through a pipe.
The user would take the laminar tutorial and figure out how to model fluid flow and then be
able to validate it. Next, since the airfoil is close to flow over an isolated surface, they would
then want to analyze the flat plate flow tutorial. Again the user should then have to validate
these results.
Next, since the airfoil is going to have a specific set of coordinates, the user could then want
to use the Nozzle tutorial, which explains how to import coordinates in order to create an
object. Now the user is ready to create an airfoil and analyze it. The user now knows how to
model flow and initialize a solution (laminar tutorial), they also know how to model flow
over an isolated surface (flat plate tutorial) and they know how to import coordinates into
FLUENT (nozzle tutorial). Since they have all the information needed to create an accurate
airfoil, they user can now apply the previous knowledge to analyze a complicated geometry.
From here the user is a step closer to creating an accurate airfoil and of course like all the
other tutorials the user needs to validate the results. Refer to Fig.1 for a visual representation
of the mentioned roadmap.
Fig.1—Example of a Roadmap

Another example of a roadmap is for analyzing the flow through a guide vane. First the
novice user will want to be able to model fluid flow so they would begin with a laminar
tutorial. Once they have learned the icons and what each tool does, then they would want to
analyze turbulent flow. They would then refer to the turbulent flow through a pipe and figure
out how to apply biasing and what models to use to analyze turbulence (See appendix B).
The user will then want to again create a 2D airfoil which would give them the knowledge to
analyze how to fluid passes an isolated surface. Once they have validated this they can then
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move to a 3D airfoil and again validate it. Then they would want create a 2d cascade,
validate it, and lastly create a 3D cascade and validate it. Now the user has the necessary
information and knowledge to create a guide vane and have the confidence to know that it is
accurate. Refer to Fig.2 for a visual representation of the described roadmap.
Fig.2—Example of a Guide Vane Roadmap
The creation of a roadmap is of crucial importance in order to be able to build the
knowledge on how to create and analyze complicated geometry. Before wanting to analyze any
complicated geometry, the user should make a roadmap of their own so that they can build
confidence in how they will figure out the problem and make sure it will be accurate. Like
mentioned before the user can create any geometry they want and can get results but how will
they know if it is accurate? The only way of knowing this is by simplifying the complicated
object into several steps (the roadmap) and work part by part in order to have accurate analytical
data for their object. Sometimes validation for complex problems is not readily available. If
however a roadmap is correctly followed then the need for validation for the specific complex
problem in question while needed is not as crucial.
III.
Theory
ANSYS FLUENT is Computational Fluid Dynamics (CFD) software that allows users to
simulate flow problems of ranging complexity. It contains broad physical modeling capabilities
needed to model flow, turbulence, heat transfer, and reactions over objects designed by the user.
Thousands of companies around the world benefit from the use of CFD software as a main part
of their design phases in their product development. It uses the finite-volume method to solve the
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governing Navier-Stokes equations for a fluid which are derived from the conservation mass
equation (1), the conservation of momentum (2) and the conservation of energy (3) equations [6].
𝜕𝜌
⃗)=0
+ ∇ ∙ (𝜌𝑉
𝜕𝑡
⃗
𝜕𝑉
⃗ ∙ ∇)𝑉
⃗ = −∇𝜌 + 𝜌𝑔 + ∇ ∙ 𝜏𝑖𝑗
𝜌 𝜕𝑡 + 𝜌(𝑉
𝜕
𝜕𝑡
𝑝
∫ 𝑒 ∗ 𝜌 𝑑∀ + ∫(𝑢̌ + 𝜌 +
𝑉2
2
̌ dA = 𝑄𝑛𝑒𝑡̇ 𝑖𝑛 + 𝑊𝑛𝑒𝑡̇ 𝑖𝑛
+ 𝑔𝑧)𝜌𝑽 ∙ 𝒏
(1)
(2)
(3)
The difficulty arises from the fact that the conservation of mass, momentum and energy are
coupled and non-linear set of differential equations making them practically impossible to solve
analytically for practical engineering problems. Hence CFD software such as FLUENT is
utilized to provide very reasonable approximation upon solving the specified governing
equations [2].
Additionally, FLUENT also allows the users to model a range of flows such as incompressible or
compressible, inviscid or viscous, laminar or turbulent flow. The advanced solver technology
that FLUENT has, provides fast and accurate results through flexible moving and deforming
meshes to be able to create optimal designs. Ultimately, FLUENT allows engineers to design,
create and analyze a configuration all under one program.
In order to model the object that a user wants to work with, its geometry and mesh must be first
created in ANSYS Workbench. Another option is to import the geometry and mesh from
Computer Aided Design (CAD) software packages such as Unigraphics, ProE or others. In
Workbench, the user creates the object he or she wishes to analyze and Workbench guides the
user through very complex metaphysics for fluid flow with drag and drop simplicity. Once the
geometry has been created, the user can take advantage of several meshing options that
Workbench provides. The user can implement the meshing in the specimen to analyze the
structure as they try to analyze fluid flow past/through their object. As seen in Fig.3 below that
is a mesh for a jet.
Fig.3 Mesh for flow through a jet
A few different ways of modeling and analyzing fluid flow are through turbulence modeling, k-,
and Y+. Turbulence modeling is used to model turbulent flow. Turbulent flows are characterized
by large, nearly random fluctuations in velocity and pressure in both space and time. These
fluctuations arise from instabilities that eventually are dissipated (into heat) by the action of
viscosity. Turbulent flows occur in the opposite limit of high Reynolds numbers. The two
approaches to solving the flow equations for turbulent flow flied can be roughly divided into two
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classes, direct numerical simulations and k- [2]. Direct numerical simulation numerically
integrates the Navier-Stokes equations, resolving all of the spatial and temporal fluctuations
without resorting to modeling. k-, models Reynolds stress in two turbulent parameters, the
turbulent kinetic energy (k) and the turbulent energy dissipation rate ∈ defined below by
Equations 4 and 5 respectively.
1
𝜅 ≡ 2 ( ̅̅̅̅
𝑢′ 2 + ̅̅̅̅
𝑣 ′ 2 + ̅̅̅̅̅
𝑤 ′2)
(4)
𝜕𝑢′
2
𝜕𝑢′
2
𝜕𝑢′
2
𝜕𝑣 ′
2
𝜕𝑣 ′
2
𝜕𝑣 ′
2
𝜕𝑤 ′
2
𝜕𝑤 ′
2
𝜕𝑤 ′
2
𝜖 ≡ 𝜐[( 𝜕𝑥 ) + ( 𝜕𝑦 ) + ( 𝜕𝑧 ) + ( 𝜕𝑥 ) + ( 𝜕𝑦 ) + ( 𝜕𝑧 ) + ( 𝜕𝑥 ) + ( 𝜕𝑦 ) + ( 𝜕𝑧 )
(5)
The next type of modeling is known as y+. Y+ is a mesh-dependent dimensionless
distance that quantifies to what degree the wall layer is resolved. Y+ plus is a non-dimensional
parameter defined by Eq. [6] [10].
𝜌𝑢𝑦𝑝
𝑦+ =
(6)
𝜇
𝜏𝑤
where u= √𝜌 which is the friction velocity and Yp is the distance to the wall.
𝑤
Workbench offers several meshing options, one being structured meshing. In structured meshing
the user decides how many user defined shapes they want placed over the object they are
analyzing. An example is seen in Fig.4. Structured meshing consists of tetrahedrons and exhibits
a clearly pronounced pattern.
Fig.4 Structured mesh for a pipe
The mesh interior to a pipe shown is 100 by 5, meaning 100 elements in the horizontal directions
and 5 elements in the vertical direction. Which is an example of structured mesh, however as the
geometries increase in complexity it is necessary to adjust the meshing accordingly. However,
when dealing with other cases such as flow across an airfoil, it is important to use a different
mesh structure. One such structure is a structured “O- grid” around the airfoil. Because of the
existence of the layers around the airfoil, it can be ensured the flow gradients are properly
captured.
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Fig.5 “O grid” around an airfoil
Referring to the airfoil grid in Fig. 5 it should be noted that FLUENT obtains a solution such that
the mass, momentum, energy and other quantities are conserved for each cell. The code of the
CFD software solves directly the values of the flow variables at the cell centers and the values at
other locations are appropriately interpolated [2]. In other cases, where there is no complicated
geometry, but rather there is more flow gradients occurring around a certain area, the user can
apply a bias. Applying a bias means concentrating the mesh around a certain area. For example,
in the turbulent flow past a cylindrical pipe tutorial a bias is applied because as previously
learned when dealing with turbulent cases, there are large gradients near the wall requiring the
mesh generated as seen in Fig 6.
Fig.6 Bias Mesh for a turbulent flow
In this bias mesh the farther away from the wall, the meshes seem to go back into the same
structured mesh seen before. It is important to mention that at the bottom of this mesh it
represents the centerline because since cylinders are radially symmetric we’re only showing the
top part of the radius. It is expected for the flow to be less turbulent near the centerline and for
that reason the mesh is less biased.
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Finally, once the test object has been drawn and meshed in Workbench, FLUENT then allows
the user to analyze it in different flow parameters. Another modeling capability FLUENT is
capable of using is enhanced wall treatment. When the user chooses to use enhanced wall
treatment, they can use this especially for turbulent cases using the k-epsilon model because it
analyzes the object closer near the wall region. The initial and boundary conditions can be
specified in FLUENT and upon initializing the problem; it can be checked for convergence. If
the convergence is not achieved accurate results will not be obtained. Finally, FLUENT provides
a wide variety of parameters that can be plotted and analyzed.
The topic of convergence requires further explanation in order for a better understanding to be
achieved of the underlying steps, undertaken by FLUENT and other CFD packages, necessary to
derive a solution. It has already been mentioned that the FLUENT code utilizes the finite-volume
method to solve the governing differential equations to obtain a solution for a particulate
problem. For simplicity purposes let us consider the finite-difference method which is in 1D. If
the grid has equally-spaced points with ∆𝑥 being the spacing between successive points, the
truncation error is O (∆𝑥). As a result as the number of grid points is increased, and the spacing
between successive points is reduced, the error in the numerical solution would decrease.
Therefore the obtained numerical solution will closely agree to the exact solution [2]. In
FLUENT during the obtainment of convergence the governing equations are solved for a
predetermined by the user number of times (iterations). Specifically the magnitude of the average
of particulate variable is computed as illustrated in Eq. [7] [2].
𝑅=√
2
∑𝑁
𝑖=1(𝑢𝑖 −𝑢𝑔𝑖 )
(7)
𝑁
where R is the residual, N is the number of iterations to be performed, u indicates a particulate
variable to be computed, and the subscript g indicates a guessed value.
Laminar and Turbulent Flow into a Pipe:
While FLUENT is a very powerful tool in obtaining solutions to a wide range of fluid flow
problems, the results obtained should be carefully validated with known theory or empirical data
to make sure they are accurate. For example, in the case of a laminar flow through a pipe, the
obtained results for the velocity profile can be compared with the theoretical data. For a steady
state (fluid properties are not changing with respect to time) laminar flow in circular tubes the
Navier-Stokes upon making the necessary assumptions can be solved to obtain a theoretical
solution to the velocity profile. The incompressible Navier-Stokes equations in Cartesian
coordinates are shown in Eqs. [8-10] [6].
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝜕𝑝
𝜕2 𝑢
𝜕2 𝑢
𝜕2 𝑢
𝜕𝑣
𝜕𝑣
𝜕𝑣
𝜕𝑣
𝜕𝑝
𝜕2 𝑣
𝜕2 𝑣
𝜕2 𝑣
𝜕𝑤
𝜕𝑤
x-direction:
𝜌 ( 𝜕𝑡 + 𝑢 𝜕𝑥 + 𝑣 𝜕𝑦 + 𝑤 𝜕𝑧 ) = − 𝜕𝑥 + 𝜌𝑔𝑥 + 𝜇 (𝜕𝑥 2 + 𝜕𝑦 2 + 𝜕𝑧 2 ) (8)
y-direction:
𝜌 ( 𝜕𝑡 + 𝑢 𝜕𝑥 + 𝑣 𝜕𝑦 + 𝑤 𝜕𝑧 ) = − 𝜕𝑦 + 𝜌𝑔𝑦 + 𝜇 (𝜕𝑥 2 + 𝜕𝑦 2 + 𝜕𝑧 2 ) (9)
z-direction:
𝜕𝑤
𝜌 ( 𝜕𝑡 + 𝑢 𝜕𝑥 + 𝑣 𝜕𝑦 + 𝑤
𝜕𝑤
𝜕𝑝
𝜕2 𝑤
𝜕2 𝑤
) = − 𝜕𝑧 + 𝜌𝑔𝑧 + 𝜇 ( 𝜕𝑥 2 + 𝜕𝑦 2 +
𝜕𝑧
𝜕2 𝑤
𝜕𝑧 2
)
(10)
The parabolic velocity profile for steady laminar flow in a cylindrical pipe is provided in Eq. [
11]. It should be noted the obtained equation is a result of solving the Navier-Stokes equations in
cylindrical coordinates.
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𝑈
𝑢𝑚𝑎𝑥
𝑟
= 1 − ( )2
𝑅
𝑢𝑚𝑎𝑥 = 2 ∗ 𝑉𝑖𝑛𝑙𝑒𝑡
(11)
(12)
In the case of turbulent flow past a cylinder a comparison can be established by using
Nikuradse’s empirical correlation. In the case of turbulence one can rely only on empirical
correlations because of the randomness associated with turbulent flow. Depending on how
accurate the experimental results correlate to the theoretical or empirical ones, one can decide
whether or not the mesh used needs improvement or whether or not the initial conditions or the
boundary ones specified in FLUENT need enhancement.
The reason why there is such a high dependency on empirical data is due to the randomness
associated with turbulence. In turbulence there is no exact equation, all solutions are empirical
data points. An example of this is the equation for skin friction for turbulence which is
𝑢𝑑
1
𝜆 = .3164( )−4 =
𝑣
.3164
𝑅𝑒 .25
(13)
The main reason that it is an empirical equation, it has a few limiting conditions. Some of those
limiting conditions are that it has to be only for smooth pipes and it is only useful for Reynolds
number less than 100,000.
Also, another very important one is Nikuradse’s empirical correlation for turbulent flow which is
given by Eq. [14] [6].
𝑢
𝑦 1
= ( )𝑛
𝑈
𝑅
(14)
where n, the power-law exponent varies with respect to the Reynolds number. Turbulent cases
are very difficult to analyze because there is no exact answer, however the Moody diagram [6;
Pg 412-413 Fundamentals of fluid Mechanics 6th ed.] in which the coefficient of friction with
respect to the Reynolds number is displayed, serves as a reference as to what the solutions should
appear to be. The Moody diagram is based off of
𝜀
𝑓 = 𝜙(𝑅𝑒, 𝐷)
(15)
where the results are obtained from numerous set of experiments plotted on the Moody diagram
11 | P a g e
Fig.7 Moody diagram
Flow over a Flat Plate
Another case which is very important is the flow over a flat plate. This tutorial is very important
because it will help simulate how flow develops over a flat plate. For this tutorial, a Re of
𝑅𝑒 =
𝜌𝑢∞ 𝑥𝑐
𝜇
= 5𝑥105
(16)
is critical because above this value the flow is turbulent and flow under this value is laminar.
Provided a sufficiently long flat plate eventually turbulent flow will be encountered since the
value of the Reynolds number is related to the length of the plate. The length of the flat plate in
the created learning module (Appendix C and D) is of length 1 meter and the viscosity, density
and inlet velocity are chosen such that the maximum Reynolds number reached is 10,000 which
is well within the laminar flow range.
Fig.8 Flow Distribution past a flat plate [13]
In the laminar layer, the fluid flow is highly ordered and is possible to identify streamlines along
which fluid particles move. This fluid continues until it hits a transition zone, which is where a
conversion from laminar to turbulent conditions occurs. After it passes this region, it reaches
turbulent flow which is where random motion is relatively high. Solution for the velocity profile
for a laminar flow over a flat plate has been done by solving the Navier-Stokes equations. The NavierStokes equation can be simplified for boundary layer flow analysis. It can be assumed that the boundary
layer is thin and the fluid flow is primarily parallel to the plate. Hence:
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𝜕𝑢
+
𝜕𝑣
=0
𝜕𝑥
𝜕𝑦
𝜕𝑢
𝜕𝑣
𝑢
𝜕𝑥
+𝑣
𝜕𝑦
=𝑣
(17)
𝜕2 𝑢
(18)
𝜕𝑦 2
H. Blasius, one of Prandtl’s students was able to solve those equations for flat plate parallel to the flow
[8]. By introducing the dimensionless parameter 𝜂 (the similarity variable) the partial differential
equations are reduced to an ordinary differential equation.
𝜂 =𝑦∗√
𝑈
𝜈∗𝑥
(19)
𝜇
where U is the inlet velocity, and 𝜈 is the kinematic viscosity, 𝜈 = 𝜌.
The convenience of validating the boundary layer velocity profile in terms of the similarity variable is that
the boundary layer velocity profiles (which depends both on x and y) at any point along the plate will
overlap one another and can be analyzed versus the empirical Blasius correlation.
Flow Through a Convergent-Divergent Nozzle
Another tutorial created is for flow through a convergent-divergent nozzle. The channel is
supplied with a flow at high pressure and exhausts into lower pressure at the outlet. An example
is seen in Fig. 9.
Fig. 9 Convergent Divergent nozzles
In addition, from the figure, A[x] is the local cross-sectional area, u[x] the local axial velocity
and p[x] the local static pressure. To analyze how the flow passes through a nozzle, the user can
do it by imposing restrictions on the geometry or on the character of the flow. It can then be
compared to quasi-1D theory which assumes that the nozzle is slender. FLUENT can be used to
analyze flow, which is what the nozzle learning module demonstrates.
Quasi -1D flow is one where the properties across each cross section are assumed uniform.
Changes in flow properties in the x-direction are brought about by area change of the duct. In
order to assume this, another assumption that has to be made, the assumption is that the nozzle
geometry is long and thin. Additionally, it neglects viscous effects on the flow field, and
maximum velocity occurs at the minimum area which is called “the throat”. The dependence of
the axial velocity and static pressure is due to the area variation, and it’s a function of whether
the flow is subsonic or supersonic which is given by Eq. [10].
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𝑑𝐴
𝐴
𝑑𝑢
= [𝑀2 − 1]
𝑢
= [1 − 𝑀2 ]
𝑑𝑝
(20)
𝜌𝑢2
For isentropic flows, the following relations Eqs. [11-13] govern the variation of Mach number,
static pressure and static temperature with nozzle area.
𝑇
𝑇0
𝑃
𝑃0
𝐴
𝐴0
=
𝐴
𝐴𝑇
𝛾−1
= [1 + (
2
) 𝑀2 ]−1
𝛾−1
= [1 + (
=
1
𝑀
[(
2
2
(21)
−𝛾
) 𝑀2 ](𝛾−1)
) (1 + {
𝛾−1
𝛾−1
2
(22)
𝛾+1
} 𝑀2 )]2(𝛾−1)
(23)
Fig. 10 Back pressure effects on Mach & Pressure for a nozzle
The static pressure distributions in the Fig 8 illustrate the dependence of the distribution on the
exit or back pressure for a given A[x] nozzle contour.
Flow Over an Isolated Airfoil
Airfoils operate upon theories of lift and drag. Consider a typical symmetric airfoil as illustrated
in Fig.11.
α
c=1m
𝑀∞
Fig.11
14 | P a g e
𝑀∞ represents the free stream Mach number, ∝ is the angle of attack and c represents the chord
length of the airfoil. The airfoil experiences a net force due to the fluid acting on the object. The
Drag Force, D, acts in the direction of the free stream while the Lift Force, L, is normal to the
free stream. Airfoils are designed to generate lift. However for objects such as cars it is desired
to reduce the lift since the lift on a car reduces the contact force between the wheels and the
ground. Typically the lift and drag are given in terms of the Coefficient of Lift and the
Coefficient of Drag which are dimensionless forms of the Lift and Drag forces.
𝐶𝐿 =
𝐶𝐷 =
𝐿
0.5∗𝜌∗𝑈 2 ∗𝐴
𝐷
0.5∗𝜌∗𝑈 2 ∗𝐴
(24)
(25)
Most of the airfoil lift is a result of the surface pressure distribution which is consistent with
Bernoulli’s equation analysis [6].
1
𝑝1 − 𝑝2 = 2 ∗ 𝜌 ∗ (𝑉22 − 𝑉1 2 )
∴ 𝑉2 > 𝑉1 𝑎𝑛𝑑 𝑝2 < 𝑝1
(26)
The effect of stall in airfoils must be avoided because loss of control and stability issues arise
from it. The stall can visually be inspected on the upper surface in Fig.12 where a transition from
high velocity to low velocity can be observed.
Fig.12--Turbulent Compressible Flow Across an Airfoil, M=0.8, Re= 𝟏𝟗. 𝟎𝟒 ∗ 𝟏𝟎𝟔 ,
∝= 10° ;Velocity Contours
The theoretical value for the coefficient of lift depends on the angle of attack and is provided by
Eq. [27]:
(𝐶𝐿 )𝑡ℎ. = 2 ∗ 𝜋 ∗ 𝛼
(27)
where α is in radians.
15 | P a g e
1° = 1° ∗ 𝜋⁄180° =0.01745 rad
The coefficient of pressure is an important dimensionless parameter used for data validation
specifically for the velocity profile [6]. Since due to no-slip condition meaning the velocity on
the solid boundary on the airfoil relative to the boundary is zero, pressure data obtained in
FLUENT is meaningful to analyze and validate in Excel. In Theory of Wing Sections by Abbott
and Doenhoff [11] data is provided for various airfoil models assuming the flow is turbulent
incompressible with zero degree angle of attack.
𝐶𝑃 =
𝑝𝑠 −𝑝∞
.5∗𝜌∞ ∗𝑈∞ 2
∴(
=1−(
𝑈𝑠 2
𝑈∞
)
(28)
𝑈𝑠 2
𝑝𝑠 − 𝑝∞
) =1−
𝑈∞
. 5 ∗ 𝜌∞ ∗ 𝑈∞ 2
where 𝐶𝑝 is the coefficient of pressure; 𝑝∞ is the free-stream pressure equaling 0; 𝜌∞ is the
density being 1.225 𝑘𝑔/𝑚3 and 𝑈∞ is the free-stream velocity.
Discrete Phase Modeling [DPM] of Particle in Pipe Flow
ANSYS FLUENT can output the trajectory of a discrete phase particle and that case is
showcased in Appendix I. Specifically, the force balance on the particle is integrated under
Lagrangian reference frame considerations. Recall flow can be analyzed either by Eulerian or
Lagrangian considerations. In the Eulerian representation fluid motion is given by completely
describing the necessary properties such as pressure, density, velocity and others. Flow
information is obtained at fixed points in space as fluid flows through those points. The
Lagrangian method is characterized by following individual fluid particles as they move [6].
The force balance of the particle for the x-direction is provided in Eq. [29].
𝑑𝑢𝑝
𝑑𝑡
= 𝐹𝐷 (𝑢 − 𝑢𝑝 ) +
𝑔𝑥 (𝜌𝑝 −𝜌)
𝜌𝑝
+ 𝐹𝑥
(29)
where 𝐹𝑥 is an additional acceleration in force/unit particle mass, 𝐹𝐷 (𝑢 − 𝑢𝑝 ) is the drag force
per unit particle mass, u is the fluid phase velocity and 𝑢𝑝 is the particle velocity. Additionally,
𝐹𝐷 is given by Eq. [30].
𝐹𝐷 =
18𝜇
𝜌𝑝 𝑑𝑝 2
∗
𝐶𝐷 ∗𝑅𝑒
24
(30)
where μ is the fluid’s molecular viscosity, 𝜌 is the fluid density, 𝜌𝑝 is the density of the particle,
Re is the relative Reynolds number provided by Eq. [31] and 𝑑𝑝 is the diameter of the particle.
𝑅𝑒 =
𝜌𝑑𝑝 |𝑢𝑝 −𝑢|
𝜇
(31)
The additional acceleration term 𝐹𝑥 in Equation X can be of particulate importance during special
circumstances. When a force is required to accelerate the fluid surrounding the particle, or the
virtual mass force as it is referred, is taken under consideration, the acceleration term takes the
form of Eq. [32].
𝐹𝑥 =
1 𝜌 𝑑
2 𝜌𝑝 𝑑𝑡
(𝑢 − 𝑢𝑝 )
(32)
16 | P a g e
Equation X is of particulate importance when the fluid density is greater than the density of the
particle. Due to the pressure gradient an additional force must be considered and the acceleration
term takes the form of Eq. [33] [12].
𝜌
𝜕𝑢
𝐹𝑥 = ( ) 𝑢𝑝𝑖
𝜌
𝜕𝑥
𝑝
IV.
(33)
𝑖
Module Example—Laminar Flow Past a Cylindrical Pipe
The major steps taken into the creation of the learning module are outlined and explained. While
the focus is on the laminar pipe flow module it needs to be noted the exact same approach is
undertaken for the other learning modules. The goal of the section is for the user to become more
acquainted with the flow of the tutorial and if more details are desired, the user can consult with
the provided Appendixes—Appendix A for the particular laminar pipe flow module. Hence the
actual step-by-step procedures are explained in much more detail in the Appendices.
The Laminar Pipe Flow Learning Module has seven distinct components.
1. Problem Statement
2. Geometry Creation and Mesh Creation
a. Actual geometry creation of the pipe and the corresponding mesh creation for it are
done here
3. Problem Setup
a. The specific values and important selections that need to be made in ANSYS
FLUENT prior to obtaining convergence and getting results
4. Solution
a. The user learns the important steps that need to be taken in order to obtain
convergence for the problem.
5. Results section
a. Contains the obtainment of various plots such as the velocity and pressure profile,
skin friction coefficient and their manipulations
b. Visual representation of the velocity profile along the pipe in the form of velocity
vectors is also included in the section
6. Validation section
a. This section provides the necessary information of how meaningful validation can be
performed in order for the user to make sure the obtained data is meaningful.
b.
Center Line
𝑉𝑖𝑛𝑙𝑒𝑡
D=0.2m
L=8m
Fig. 13 Geometry of Pipe
17 | P a g e
Laminar Pipe flow modeling
(1) Problem Statement:
The user learns the basics of the problem at hand as well as important given information
𝑘𝑔
in the Problem Statement section. The given parameters are-- ρ=1.0 ⁄𝑚3 ; 𝑉𝑖𝑛𝑙𝑒𝑡 = 1.0 𝑚⁄𝑠;
𝑘𝑔
μ=2.0 ∗ 10−3 ⁄𝑚 ∗ 𝑠, where 𝜌 is the density and µ is the dynamic viscosity.
Based on the given information, the Reynolds number can be computed to determine whether or
not the problem as stated is laminar, transitional or turbulent. Recall that for flow in a round pipe,
the flow is laminar if the Reynolds number is less than approximately 2100; the flow is
transitional if the Reynolds number is between 2100 and 4000 and it is fully turbulent if Re is
greater than 4000. Based on the given parameters the flow that is to be analyzed is laminar.
𝑅𝑒 =
𝜌∗𝑉𝑖𝑛𝑙𝑒𝑡 ∗𝐷
𝜇
= 100
(34)
(2) Geometry Creation:
The user is introduced to the basics of ANSYS Workbench--specifically how to create a sketch
of the problem and then how to obtain a surface from the made sketch. Since a 2d model is
created it can be assumed the shape of the pipe is rectangular. Furthermore the problem is
assumed as axisymmetric hence only the upper portion of the pipe’s diameter is to be taken
under consideration. The learnt basics then can be applied in the geometry creation for more
complicated problems such as when turbulent pipe flow, flow past a flat plate and others are
analyzed.
Fig.14—2d pipe sketched and made into a surface in ANSYS Workbench.
(3) Mesh Creation:
The mesh is created in ANSYS Workbench as well. In the Mesh Creation section it is
explained to how to create a structured mesh—the grid exhibits clearly pronounced shape with
tetrahedral shaped elements. Structured mesh is recommended for problems of such basic
geometry. It is specifically explained to how size individual edges of the geometry as well as
how to ensure elements on opposite ends correspond to each other—mapped face meshing. In
Meshing specific zones of the geometry are named. Those zones will later be further specified in
ANSYS FLUENT. The learned techniques are a very important starting point in ensuring the
user will be able to handle more complicated problems.
Inlet
Outlet
Wall
Fig.15—Structured Grid
Axis
(4) Problem Setup:
18 | P a g e
Careful consideration must be exercised upon entering specific information in ANSYS
FLUENT as it relates an axis symmetric pipe flow. It is recommended a top-bottom approach is
utilized in order not to miss anything. First the mesh must be checked for errors and the
dimensions must be verified. The problem must be set as axisymmetric. Then the model must be
specified as laminar. The given initial information as it relates to the density and viscosity must
be entered in the material properties. Next, the Boundary Conditions must be specified. The user
must first make sure the zones named in Workbench are of the proper type. The inlet zone must
be of velocity-inlet type, outlet—pressure-outlet type, wall—wall type and finally axis—axis
type. The next step is to input the relevant given information in the specific boundaries—the inlet
velocity given as 1 m/s should be entered in the velocity-inlet boundary. The reference value
must be specified at the inlet (ref. Fig.3) in order to be able to obtain certain data such as the skin
friction coefficient.
(5) Solution:
The importance of obtaining convergence is stressed in the solution section. If
convergence is not obtained, it will be impossible to obtain any results. However first the
problem must be initialized to take into an account initially specified parameters—usually the
given parameters specified at the inlet. Upon obtaining convergence the governing equations are
solved by FLUENT. They are the conservation of mass, momentum and energy—Eqs.[2-4].
However since temperature effects are of no concern and neither is the fluid flow taken as
compressible in the particular analysis of laminar pipe flow, the energy equation is not solved.
The convergence residuals are a measure of how well the solution obtained satisfies the discrete
form of the governing equations. As can be seen in Fig. 16, convergence is obtained as those
residuals after certain number of iterations.
Fig. 16--Laminar Pipe Flow Convergence History
(6) Results:
In Results section, the user becomes familiar with various ways of creating XY Plots of
the obtained data as well as methods of manipulating plots to make them more useful. The color
and shape of the curves can be changed and scaling of the axis can be changed. Plots of skin
friction, static pressure and velocity profile are created and the user also learns how to save them
19 | P a g e
so they can be later opened in Excel to perform data validation. Figs. 17-18 show how by
manipulating a plot, it can be more visually appealing or easier to interpret.
Fig. 17 —Velocity Profile XY Plot before Manipulation
Fig. 18-- Velocity Profile XY Plot after Manipulation
These graphs are the graphs that ANSYS has the ability to show however, for closer analyzing to
upload the values to excel it is very simple.
Uploading to Excel
1. Select plot, here you figure out what you would like to plot whether velocity pressure etc.
2. In the Plot box on the left side, there is a radio button which says write to file, select that
button
3. Save it as choosename.xls
4. Open this file in Excel and when prompted if Excel should up load the file select yes and
then finish.
20 | P a g e
In the Results section it is also explained how to obtain the velocity vectors to observe the
parabolic velocity profile along the pipe. Techniques of adjusting the scale, significant figures
and location of the color map are introduced. It is introduced how to mirror the plane to get a
complete grasp of the velocity profile (recall the problem is analyzed as axisymmetric)—Fig. 19.
The user learns how to scale the rectangular shape, representing the pipe in 2d in order for the
whole shape to visible on the screen making the velocity vectors easier to distinguish—Fig. 20.
Having learned the mentioned techniques the user can then proceed to obtain different visual
representations.
Fig.19—Plane is mirror for the full parabolic profile to be seen.
Fig.20—Through Scaling the Whole Shape is Visible
Validation:
The data validation is explained in significant detail in order to ensure the user can apply the
learnt knowledge to other fluid flow problems. However validation does not stop there. As
engineers one should always ask himself whether the obtained data is correct. Computational
Fluid Dynamics (CFD) software like ANSYS FLUENT is not a “magic box”. One cannot rely
21 | P a g e
that the data being outputted is correct. The data will be only as meaningful as the parameters
specified and models and options selected. Hence the user must not only be knowledgeable of
how to use the CFD client properly, but depending on the actual flow problem a significant
knowledge of fluid mechanics is required as well.
First in the validation process it is explained how the written to file XY Plots can be opened in
Excel for further manipulation. As part of the post-processing process the initial mesh is
enhanced in Workbench in order to investigate how increasing the number of elements will
impact the accuracy of the solution. Validation in Excel is performed using dimensional analysis,
meaning both the experimental data and the theoretical validation will be manipulated so that one
will not need to rely on dimension to perform the analysis. It must be noted that while for the
laminar pipe flow it can be relied on theoretical validation as it relates to entry-length (5),
velocity profile (6), skin friction (7, 8) and static pressure drop (9) in other cases empirical
correlation (extensive experiments performed by others such as Nikuradse or Blasius empirical
correlations) may be the only means of obtaining validation. Empirical validation is illustrated in
the turbulent pipe flow and laminar flow past a flat plate learning modules.
Skin Friction Coefficient, Cf
0.30
0.25
Fluent 100 x 5
Fluent 100 x 10
Theoretical
0.20
0.15
0.10
0.0
0.1
0.2
0.3
0.4
0.5
Axial Distance
Fig. 21--Skin Friction vs. Pipe length along the wall, dimensionless
Dimensional analysis is convenient and preferred because one need not depend on unit
limitations and validation can be performed for pipes of various dimensions if need be. Figure 21
represents validation performed for the skin friction coefficient. Correlation for the entry length
is displayed in Figs. 22-24, which illustrate the velocity profile validation along the radius of the
pipe. Figure 25 provides validation for the static pressure drop from the inlet to the outlet of the
pipe. Finally, Fig. 26 investigates the case where two different materials are chosen—air and oil.
If the Reynolds number is kept the same but the inlet velocity is manipulated to obtain equal
Reynolds number for the two materials, then the data in dimensional form should exactly match.
The statement is proven for the velocity profile in Fig. 26.
22 | P a g e
1.4
Centerline Velocity, u/umax
1.2
1.0
0.8
0.6
Fluent 100 x 5
Fluent 100 x 10
Empirical
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
Axial Distance, x/L
Fig. 22 --Centerline Velocity in Axial Direction, Dimensionless
1.0
Radial Location, r/R
0.8
0.6
Fluent 100 x 5
Fluent 100 x 10
Poiseuille
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Velocity, u/ucl
Fig. 23 --Dimensionless F.D Velocity Profile in Radial Direction
23 | P a g e
Radial Location, r/R
0.3
Fluent 100 x 5
Fluent 100 x 10
Poiseuille
0.2
0.1
0.0
0.84
0.88
0.92
0.96
1.00
Velocity, u/ucl
Fig. 24 -Dimensionless F.D Velocity profile in Radial Direction
1.0
Fluent 100 x 5
Fluent 100 x 10
Theoretical
Static Pressure, p/p1
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Axial Distance, x/L
Fig. 25 --Static Pressure Drop vs. Axial length, Dimensionless
24 | P a g e
1.0
0.8
Y Data
0.6
Air Fluent 100 x 10
Oil Fluent 100 x 10
Poiseuille
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
X Data
Fig. 26 --Fully Developed Pipe Velocity Profile for Air and Oil, Re=1369
While basic in nature the laminar pipe flow module is crucial in illustrating the basics that first
must be learned in order to be able to create, mesh, setup, solve and most importantly validate
data obtained for more complicated problems. Other learning modules of increasing difficulty,
included in the Appendices, are turbulent pipe flow, laminar flow past a flat plate,
turbulent flow past a nozzle, analysis of turbulent jets, turbulent compressible and
incompressible flow past an airfoil, turbulent incompressible flow across a periodic airfoil
and discrete phase modeling—particle injection into a pipe. Each of those learning modules
teaches the user something new and by incorporating and applying the learned knowledge from
all tutorials, the user should be able to solve complex problems more efficiently.
V.
Summary of Available Learning Modules
C. Turbulent Pipe Flow Case
The turbulent case, similar to the laminar case, is started the same way all the way up to the
meshing. All the previous steps such as centering and building the sketch are done in the same
way. However, when it comes to the mesh it is very different. From Fluid Mechanics, it is known
that turbulence will be most significant near the wall. For that reason biased meshing is then
applied. The reason for this biased meshing is so that there is more mesh (smaller ∆x) where the
flow gradients are large. In addition, 𝑘 − 𝜖 turbulence model is turned on because this is another
feature that FLUENT has to evaluate.
25 | P a g e
Fig.27 Biased Mesh for Turbulent Pipe Flow
After creating the sketch and specifying the bias, both done in Workbench, the mesh size is then
specified; the first one that is tried is 100X30 Grid (100 element divisions in the axial direction
and 30 element divisions in the radial direction). The second grid choice is 100X54 Grid.
Similarly to the laminar pipe flow tutorial, the relevant initial and boundary conditions are
specified in FLUENT. Then the problem is initialized and checked for convergence. One of the
graphs that are analyzed is for the Y+ values with respect to the pipe’s length, where the graph
represents the Y+ values comparison for the k- model for the two created meshes—Fig.28.
Since Y+ is a dimensionless quantity there is no need to manipulate the column any further.
However the x-axis values are computed by dividing each value in the column for the axial pipe
distance by the total length of the pipe—in this specific case, 8 meters. It can be seen that as the
mesh is refined the Y+ values are getting closer to one signaling the 100X54 is the better mesh
choice. Results however in fig.29 show little difference. Recall that the dimensionless parameter
Y+ has been defined in the theory section.
3.0
2.5
EWT 100 x 54
EWT 100 x 30
Y+
2.0
1.5
1.0
0.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Axial Distance, x/L
Fig. 28—Y+ values for Turbulent Pipe Flow with Re=10,000
26 | P a g e
Lastly, like the laminar tutorial, validation is very important. One of the validation methods used
to analyze the obtained experimental results is the Nikuradse fully develop velocity profile for
smooth pipes. In dimensionless form, the Nikuradse equation is plotted and then analyzed—Fig.
29. It is very important to mention however that this is empirical correlation due to the flow
being turbulent thus exhibiting unique and random behavior. The relevant Nikuradse empirical
correlation theory has been explained in the theory section. One reason for the mismatch may be
the simulation pipe length is not long enough to get a fully developed profile.
1.0
Radial Distance, r/R
0.8
0.6
EWT 100 x 54
EWT 100 x 30
Nikuradse, n=6
Nikuradse, n=7
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Axial Velocity, u/ucl
Fig. 29--Normalized Axial Velocity Profile
D. Laminar Flow over a Flat Plate Case
The tutorial modeling laminar flow over a flat plate will express results as a is function of the
Reynolds number which varies with the distance along the plate, 𝑅𝑒𝑥 . The flow remains laminar
until the Reynolds number reaches 500,000. The geometry of the problem is created in the same
manner as with the laminar pipe flow. Like the turbulent pipe flow case, bias is needed in order
to obtain more accurate results in the areas of biggest flow gradients, even though the flow is
laminar. For the flat plate that is near the wall—the bottom horizontal edge (recall in the
turbulent pipe flow case the bias is used to obtain more meshing towards the upper horizontal
edge).
Symmetry
Air
Inlet
Outlet
Width,w
Fig.19
Wall
Length,L
27 | P a g e
The key thing is to define the various surfaces properly in FLUENT. The inlet is Velocity inlet;
the outlet is a Pressure Outlet; the named wall zone in Workbench should be by default specified
as a type Wall in FLUENT. Finally the Symmetry zone should be specified as type symmetry.
After the boundary conditions are specified and convergence is obtained after initialization of the
problem (see Appendix D) the obtained data is validated in order to make sure it makes sense.
The validation comparisons are for the skin friction coefficient and the velocity profile.
The steady incompressible Navier-Stokes equations which are solved to yield the obtained data
can be simplified for boundary layer flow analysis. It can be assumed that the boundary layer is
thin and the fluid flow is primarily parallel to the plate. Hence for 2D incompressible flow
𝜕𝑢
𝜕𝑥
𝜕𝑣
+ 𝜕𝑦 = 0
𝜕𝑢
𝜕𝑣
(35)
𝜕2 𝑢
𝑢 𝜕𝑥 + 𝑣 𝜕𝑦 = 𝑣 𝜕𝑦 2
(36)
H. Blasius, one of Prandtl’s students, was able to solve those equations for flat plate parallel to
the flow. By introducing the dimensionless similarity parameter 𝜂, the partial differential
equations can be reduced to a single ordinary differential equation in terms of the independent
variable .
𝑈
𝜂 = 𝑦 ∗ √𝜈∗𝑥
(37)
𝜇
where U is the inlet velocity and 𝜈 is the kinematic viscosity, 𝜈 = 𝜌
The convenience of validating the boundary layer velocity profile in terms of the similarity
variable is that the boundary layer velocity profile (which depends both on x and y) at any point
along the plate will overlap one another and can be analyzed versus the Blasius solution.
Fluent, Laminar, 50 x 60 Grid
6
Similarity Variable, 
5
Blasius
Fluent, x/L=0.6
Fluent, x/L=0.8
Fluent, x/L=1.0
4
3
2
1
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Velocity, u/U
Fig. 31 --Boundary Layer profile for Flow over Flat Plate
28 | P a g e
The validation performed for the skin friction coefficient (Fanning friction) can be seen to
correlate accurately against the Blasius numerical solution data. The Blasius prediction is
specified by Eq. [38].
0.664
𝐶𝑓 =
√𝑅𝑒𝑥
(38)
Skin Friction Coefficient, Cf x Rex1/2
1.2
Fluent, Re=10,000
Blasius
1.0
0.8
0.6
0.4
0.0
0.2
0.4
0.6
0.8
1.0
Axial Distance, x/L
Fig. 32 Centerline skin friction coefficient
It needs to be emphasized bias towards the wall surface is to be used otherwise the results will
not be accurate. As it can be seen the most accurate results are yielded upon using structured
biased towards the wall tetrahedral mesh—see Fig. 33.
0.30
Vertical Distance, y/
0.25
60 x 50 Unstructured Mesh
60 x 50 Far Field Bias
60 x 200
Blasius
0.20
0.15
0.10
0.05
0.00
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Axial Velocity, u/U
Fig. 33 --Velocity profile comparison for different Mesh Types
29 | P a g e
E. Turbulent Flow Past a Nozzle Case
The nozzle case is another fundamental case for learning FLUENT. Unlike the previous tutorials.
Specifically an already created mesh (done in FlowLab) is imported into FLUENT. An
additional part of this specific learning module is in-depth description of how a similar nozzle
would be created and meshed in Workbench. Specifically the user learns how to import
coordinates, specified in Excel, to form curves. By importing an already created mesh file into
FLUENT, time is saved and more focus can be placed on the problem setup and its validation.
This is shown in more detail in Appendix E.
Fig.34--Nozzle sketch
The user must possess knowledge in Turbo Machinery and Fluid Mechanics, not only to properly
setup the boundary conditions, but also to correctly validate the problem. Specifically the
turbulent flow past a nozzle module is validated against the 1D Quasi model already described in
the theory section.
Fig.35--Mesh for Nozzle
30 | P a g e
In order to validate the FLUENT results, they are validated against data provided by Professor
Barber, a professor of Turbo Machinery at the University of Connecticut. They can be seen in the
tutorial in Appendix and they correlate very well.
Fig. 36--Mach number vs. Position
Inviscid Results Pe/Pi=0.75
1.8
1.6
Maxis
Mwall
Maxis-visc
Mach Number
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
10
X
Fig. 37--Mach number vs. Position from other tests (cited at the reference page)
F. Turbulent Jet Flow
For the turbulent jet flow, this problem is set up like the laminar and turbulent pipe flow cases.
However when the design of this is done, first a pipe is created. The flow exits (jets) from the
31 | P a g e
pipe into a large domain. It is expected that the flow gradients will be the greatest and spreads
out into the area and the flow gradients decrease. When conducting this tutorial, it was very
important to take into account the turbulent pipe flow tutorial because it allowed the user to
know where to make the mesh more biased.
Fig. 38--Mesh for jet
Fig. 39 Boundaries
In addition, it allows the user to be able to implement the pressure outlet boundaries. See Fig. 39
for the boundary conditions. However, unlike the turbulent case using simply the k- model to
analyze this flow will not be enough to achieve correct results. The k- model is used more for
flow analyzing what happens near a wall however, it is needed to analyze what happens near the
centerline and for this reason the amount of reversed flow allowed must be reduced. For that
reason, specified intensity and viscosity ratio is used. By applying this, the flow is then able to
reach the other side and can then be analyzed completely across the centerline as opposed to
analyzing data closer to the wall.
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Fig. 40 Axial Velocity for jet
When validating this, the velocity across the centerline has to increase in the orifice and
then decrease in a linear manner. (To see the validation refer to Appendix F)
G. External Turbulent Compressible and Incompressible Flow across an Airfoil
This learning module, discussed in detail in Appendix G, showcases the flexibility of ANSYS
FLUENT by importing into the CFD software an already created geometry and mesh—Fig. 41.
By doing so, the time saved can be spent in ensuring the problem is setup properly in FLUENT
and that correct results are obtained by validating the obtained data against theoretical
correlation. The used mesh, Fig. 41, defines the area around the airfoil as pressure far field. To
be able to use a boundary condition of pressure far field, the flow must be considered
compressible. The density of the material chosen must be that for ideal gas conditions and the
energy equation (the governing conservation of energy will be solved during the convergence
process in FLUENT) must be turned on as well. The pressure far field boundary condition is fine
for the first case considered in this module—external turbulent compressible flow over a
NACA0012 airfoil. The Mach number for that case is 0.8. However while the second case
considered, external turbulent incompressible flow over the same airfoil, can be setup exactly
like the compressible case only lowering the Mach number to 0.2. More accurate results can be
obtained if instead of using pressure far field boundary condition with the energy equation on,
velocity inlet and pressure outlet conditions with the energy equation off are utilized. One can
recall that in general compressibility effects cannot be ignored for Mach numbers greater than
0.3. This module explains how to separate the pressure far field into two zones to be defined as
velocity inlet and pressure outlet.—Fig. 42. The module also describes how to obtain
convergence for both the compressible and incompressible cases.
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Fig. 41 Boundary condition definitions – compressible airfoil case
Fig. 42 Boundary condition definitions – compressible airfoil case
It is also explained how the user can determine the specific designation of the airfoil model. In
the case considered, a NACA 0012 is analyzed. The velocity magnitude along the airfoil is zero
because of the no-slip condition. Specific focus is placed on the validation aspect of the case.
The coefficient of lift is validated against theoretical data as given by theory of lift—Eq. [39].
𝐶𝐿 = 2 ∗ 𝜋 ∗ 𝛼
(39)
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where 𝐶𝐿 is the lift coefficient, 𝛼 is the angle of attack in radians. The lift coefficient is validated
for various angles of attack for both the turbulent compressible and the turbulent incompressible
cases. Fig.43 shows a comparison of results for the compressible and incompressible cases. The
incompressible simulation strongly agrees with the theoretical model, which makes sense since
the theory of lift correlation is valid for incompressible inviscid flows. For the compressible
case, the agreement is only good for angles of attack less than 4 degrees.
2.5
Theory
M=0.2, Incomp., Turb., Re=4.76+6
M=0.8, Comp., Turb., Re=19.04+6
Lift Coefficient, CL
2.0
Stall
1.5
1.0
0.5
0.0
0
5
10
15
20
25
Angle of Attack, 
Fig. 43 Coefficient of Lift Validation for Turb. Compr. And Incompr. Cases
80
Lift to Drag Ratio, CL/CD
60
40
20
0
-20
Turb. Icompr. Flow, Energy Off
-40
-60
-80
-10
-5
0
5
10
15
20
Angle of Attack, 
Fig. 44 Determination of Most Optimum Angle of Attack
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The module also explains how to determine the optimum angle of attack which corresponds to
the largest Lift to Drag Coefficient ratio—Fig. 44. In Fig. 45, one can see the velocity contours
on the upper surface, a supersonic region [red] is formed and ends in a shock. The shock induces
a region of separation or stall [blue]. The region of separation is less visible in the contours of
static pressure, Fig.G30, where static pressure is approximately constant through a region of
separation. For an incompressible flow, the region of separation / stall does not occur until angles
of attack near 18 degrees.
Fig.45 Velocity Magnitude Contours [m/s], =10 degs., Turbulent Compr. Case
Furthermore, validation is obtained for the velocity profile utilizing the equation for the
Coefficient of Pressure and data for the static pressure from FLUENT. If the specific NACA
model of an airfoil is known, validation of the solution can be performed by considering the
velocity at different locations along the surface. The theoretical surface velocity can be obtained
from Abbott and Von Doenhoff’s “Theory of Wing Sections” [11]. The computational surface
velocity has to be zero from the imposed no-slip condition. However, one can apply Eq. [40] to
extract a surface velocity from the surface static pressure. The infinity subscript indicates ref.
value and the s subscript—static value.
𝑝 −𝑝
𝑈
𝐶𝑃 = .5∗𝜌𝑠 ∗𝑈∞ 2 = 1 − (𝑈 𝑠 )2
(40)
∞ ∞
∞
𝑈𝑠
𝑝𝑠 − 𝑝∞
∴ ( )2 = 1 −
𝑈∞
. 5 ∗ 𝜌∞ ∗ 𝑈∞ 2
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NACA0012, =0
1.6
Surface Velocity, [u/U] 2
1.4
1.2
1.0
0.8
Abbott & Von Doenhoff
Incomp., Turb., No Energy
Incomp., Turb. Energy
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Axial Position, x/c
Fig. 46
H. Turbulent Incompressible Flow across a Planar [2D] Cascade of Airfoils
This (Appendix H) demonstrates how turbulent incompressible flow past a cascade of laterally
periodic NACA 0012 airfoils can be modeled in FLUENT. It is shown in detail how the
geometry and mesh for a periodic case can be created in Workbench. In the periodic case, the
focus is placed only on one of the blades in a rotor or turbine and the boundary surrounding the
airfoil reflects that. New tools in geometry creation in Workbench are introduced such as [1]
Boolean subtraction to subtract the airfoil walls from the surrounding boundary, [2] vertex
blending to make sure the trailing edge of the airfoil is smooth in order to obtain better mesh for
the case and [3] others—Fig. 47. This module explains how to properly mesh the wake region
after the airfoil. Also it is demonstrated how to obtain an O-type grid around the airfoil by
utilizing inflation. Better results are obtained by employing an O-type grid because the flow
gradients will be analyzed better during the solution process in FLUENT.
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Fig. 47
Fig. 48
Fig. 49
The procedure of obtaining convergence for the turbulent incompressible case of Appendix G is
used to obtain results for the periodic case. The difference is that the user must manually define
the periodic boundary condition in FLUENT by typing specific commands in the command
prompt and the module specifically explains the appropriate steps. The velocity contours for an
angle of attack of 4 degrees can be seen in Fig. 50.
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Fig. 50
I. Discrete Phase Modeling—Particle Injection into 3-D Turbulent Pipe Flow
Senior Design Group 18 working for Alstom has provided the needed Workbench project for the
learning module in question. The Workbench project includes not only the geometry and mesh
creation for the case but the FLUENT problem setup, solution and results. The learning module
(Appendix I) demonstrates in depth how the user can create the geometry for the pipe by
utilizing the Sweep Feature in Workbench. Two sketches are created one in the XY plane
outlining the line shape of the pipe. The second sketch is in the YZ Plane and includes the
diameter of the pipe. The circle is then swept along the line to create a 3D pipe.
Fig. 51
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Fig. 52
Fig. 53
To obtain the mesh for the problem inflation is utilized in Workbench. As the geometry the
whole body is selected and as the boundary the three faces making the pipe—the horizontal, the
bend and the vertical portions. When inflation is specified in 3D, the body of the geometry and
the faces of the boundaries are selected. For a 2D inflation, as done for the turbulent
incompressible flow across a periodic airfoil (Appendix H), the face of the geometry and the
edges of the boundaries are specified.
Fig. 54
The real focus of the learning module is to demonstrate how to specify material to be injected
into the pipe. Specifically the Discrete Phase Model interface is manipulated in which injections
can be created starting from a specified zone, inlet in the particulate case. Additionally the
velocity, temperature, diameter, and other parameters of the injection can be entered. Once
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proper convergence is obtain in similar manner as outlined in other learning modules the
particles’ residence time can be tracked—Fig. 55.
Fig. 55a
Fig. 55b
VI.
Conclusion
Conclusively, what is seen constantly over and over regardless of what type of problem or design
the user is working with is a consistency in working with geometry, meshing, and validation. In
geometry phase, it is important to recall that the user can design the geometry in ANSYS
Workbench. However the software is flexible enough that the geometry can be created in
Unigraphics, ProE or other Computer-Aided (CAD) software packages and the exported into
Workbench. Again a rough sketch can first be generated and then the user can go in the editing
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window and put the dimensions he or she chooses. Additionally, the user is not only limited to
that, the user may also create already primitive designs. This means the user can select basic
shapes in Workbench that have already been done like boxes, pipes, etc. These shapes can save
the user a lot of time if they are just analyzing certain flow over feasible objects. However, if the
user needs a different design that was possibly done on another CAD package, the user has the
option of importing the design and then being able to work with it.
The next very important part once the geometry is determined, is meshing. Again recall that the
user can have structured, unstructured, and bias types of meshing. The structured meshing works
for very simple designs and simple flows. An example was the laminar flow past a cylindrical
pipe (Appendix A) because since the flow is laminar no extra meshing is required near the wall
due to its simplicity. However, when dealing with different types of flow or sketches such as
nozzle, or turbulent cases, the meshing needed to be adjusted accordingly. The unstructured
meshing is extremely useful especially for shapes that are not symmetrical such as bends or
curves.
When dealing with complicated flow or complicated geometry; biased meshing changes how a
mesh is refined near a wall. For instance, in the turbulent flow through a pipe case biased
meshing was implemented since at turbulent cases the most flow gradients occur near the wall.
Therefore, to obtain better results the mesh was refined near the wall to be able to get better data.
However this still leaves one questions, how does one choose what mesh to start at?
The answer is that there is no actual starting point, based on experience the user will begin to
determine what meshing sizes he or she should start at. In the laminar pipe flow tutorial the
meshing chosen is based on other tutorials, however, meshing does not have a starting point but
rather serves as a reference value. Therefore, when choosing a meshing size it is always
recommended to start with simple small mesh and increase gradually based on how the accuracy
increases or decreases, which is why validation is very important
Validation is the utmost important part in the design process. When validating results it allows
the user to see the accuracy of their experiment. By doing hand calculations or referencing to
other scholarly sources it allows the user to see what needs to be improved to get better answers.
In addition, validation allows the user to go back and adjust parameters such as the meshing, like
it was done in the laminar pipe flow tutorial.
Lastly, the importance for all these tutorials is to be able to create a roadmap for future complex
problems. It is strongly suggested that before designing and analyzing complicated geometry, the
user break it down into parts and create a roadmap with each step increasing in complexity to be
able to not only build confidence in the values they get but also build knowledge on the uses of
FLUENT into more complex parts.
References
1. Cornell University.FLUENT Learning Modules. 4/10/11
https://confluence.cornell.edu/display/SIMULATION/FLUENT+Learning+Modules
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2. Cornell University. Introduction to CFD Basics. 5/7/11,
https://confluence.cornell.edu/download/attachments/90736159/intro.pdf?version=1&modifi
cationDate=1222889778000
3. Lee, Huei-Huang. Finite Element Simulations with ANSYS Workbench 12. Schroff
development corporation, 2010.
http://www.sdcpublications.com/pdfsample/978-1-58503-604-2-2.pdf
4. Lawrence, Kent. ANSYS Workbench tutorial. Schroff development corporation, 2007.
5. Moran, Howard Shapiro. Fundamental of Eng. Thermodynamics. USA: NJ, 2008.
6. Munson, Young, Okiishi, Huebsch. Fundamentals of fluid mechanics 6th ed. USA: West
Virginia , 2009.
7. N.Rajaratnam. Turbulent Jets. New York: Elsevier scientific, 1976.
8. Schlichting, Hermann. Boundary Layer Theory. New York: McGraw, 1976.
9. Senior Design Project Website. University of Connecticut. 5/7/11
http://www.engr.uconn.edu/~barbertj/
10. Y+ definition, 5/7/11, http://my.fit.edu/itresources/manuals/fluent6.3/
11. Abbott, Doenhoff. Theory of Wing Sections. Dover Publications, New York, 1959.
12. ANSYS FLUENT 12.1 Theory Guide, Discrete Phase Modeling, 5/7/11
http://www3.hi.is/~halldorp/fluent_theory.pdf
13. Flow Velocity Profile, 12/10/10
http://www.tpub.com/content/doe/h1012v3/css/h1012v3_40.htm
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