SIMULATION AND CONTROL OF FLUID FLOWS

AROUND OBJECTS USING COMPUTATIONAL

FLUID DYNAMICS

by

Sagar Kamat

P.R.No.200801395

A Project Report

submitted in partial fulfillment

of the requirements for the degree of

Bachelor of Engineering

in

Mechanical Engineering

Under the guidance of

B.S.Manohar Shankar

Selection Grade Lecturer

Department of Mechanical Engineering

Goa College of Engineering

Goa University

2012

Project Approval Sheet

The project titled

“SIMULATION AND CONTROL OF FLUID FLOWS AROUND OBJECTS

USING COMPUTATIONAL FLUID DYNAMICS”

by

Mr. Sagar Kamat

completed in the year 2011-2012 is approved as a partial fulfillment of the requirements for the

degree of BACHELOR OF ENGINEERING in MECHANICAL ENGINEERING and is a

record of bonafide work carried out successfully under our guidance.

(Project Guide)

B.S.Manohar Shankar

Selection Grade Lecturer

Department of Mechanical Engineering

(Head of Department) (Principal)

Prof. Uday Amonkar Mr. Vivek Kamat

Head of Dept. of Mechanical Engineering Goa College of Engineering

Place: Farmagudi, Goa

Date:

CERTIFICATE

This is to certify that the project titled

“SIMULATION AND CONTROL OF FLUID FLOWS AROUND OBJECTS

USING COMPUTATIONAL FLUID DYNAMICS”

by

Mr. Sagar Kamat

has been satisfactorily completed in the academic year 2011-2012 as a partial fulfillment of the

requirement for the degree course in BACHELOR OF ENGINEERING in MECHANICAL

ENGINEERING, at Goa College of Engineering, Farmagudi-Goa.

(Internal Examiner) (External Examiner)

(Head of Department)

Place: Farmagudi, Goa

Date:

1

ACKNOWLEDGEMENTS

I would like to sincerely thank my Guide, Prof. B. S. Manohar Shankar for the

inspiration, guidance and support extended to me during the course of the project.

Without his constant motivation and support, I would not have been able to take up

Goa University’s first CFD Project, let alone complete it successfully.

I would like to thank Prof. Uday Amonkar, Head of Mechanical Engineering

Department of Goa College of Engineering, for all the support and encouragement.

Heartfelt gratitude to Prof. Mahesh Caisucar and Prof. Chetan Desai for all the

timely inspiration and motivation they provided when I needed it the most.

Last but not the least, a sincere thanks to my family and friends for providing all

the emotional support required for the ambitious project I set out to complete.

2

ABSTRACT

With the increase in complexity of the challenges faced by mankind, a need has been felt

to come up with increasingly elaborate solutions to these problems. Be it in the field of

advanced aerospace technologies or high speed transportation, man has felt the need to

develop and test vehicles with tailor-made aerodynamic properties. However, complex

technologies also bring along complex problems. Conventional testing methods cannot be

always satisfactorily used for such systems, either due to lack of knowledge or the high

costs involved. A third, complementary approach namely, Computational Fluid

Dynamics (CFD) is therefore fast gaining ground. CFD makes use of the computing

powers of modern-day digital computers to simulate and study physical situations and

test new systems under various conditions.

This project was aimed at understanding the fundamental concepts of this cutting-edge

approach to research and development and applying them to simulate and control the

fluid around objects.

The main objectives of the project were:

To understand and assimilate the fundamentals of Computational Fluid Dynamics

To use the techniques of CFD to simulate and study the effect of various

parameters on flows around objects such as cylinders, flat plates and Airfoils

Owing to the complex nature of the subject, the first phase of the project involved

understanding of the concepts of CFD and the fundamentals of Fluid Dynamics.

In the second phase, we made use of the concepts learned to simulate fluid flows around

various objects and extracted tangible, practical results and conclusions.

3

Table of Contents

ACKNOWLEDGEMENTS ................................................................................. 1

ABSTRACT ......................................................................................................... 2

Chapter 1 COMPUTATIONAL FLUID DYNAMICS ...................................... 7

1.1 What is Computational Fluid Dynamics .................................................................................... 7

1.2 Why Computational Fluid Dynamics .......................................................................................... 8

1.3 Some Applications of CFD ......................................................................................................... 9

Chapter 2 THE CFD PROCESS ........................................................................11

2.1 Pre-Processing........................................................................................................................... 11

2.2 Solving ...................................................................................................................................... 12

2.3 Post Processing ......................................................................................................................... 12

Chapter 3 GOVERNING LAWS OF FLUID DYNAMICS .............................14

3.1 Continuity Equation .................................................................................................................. 15

3.2 Momentum Equations ............................................................................................................... 16

3.3 Energy Equation ........................................................................................................................ 16

3.4 Physical Boundary Conditions .................................................................................................. 17

3.5 Discretization ............................................................................................................................ 18

Chapter 4 MESHING .........................................................................................19

4.1 Structured Grid.......................................................................................................................... 20

4.2 Unstructured Grids .................................................................................................................... 20

4.3 Hybrid Grids ............................................................................................................................. 21

Chapter 5 EXTERNAL FLOWS AROUND OBJECTS ..................................22

5.1 Drag and Lift ............................................................................................................................. 22

5.2 Laminar and Turbulent Flows ................................................................................................... 23

4

Chapter 6 AIRFOILS .........................................................................................25

Chapter 7 ANSYS ...............................................................................................28

7.1 CFD IN ANSYS ....................................................................................................................... 28

7.2 ANSYS FLOTRAN .................................................................................................................. 29

7.3 ANSYS FLUENT ..................................................................................................................... 30

Chapter 8 SIMULATION OF FLOW AROUND OBJECTS ..........................32

8.1 Flow over a Flat Plate ............................................................................................................... 32

8.2 Flow around a Cylinder ............................................................................................................. 35

Chapter 9 CONTROL OF FLUID FLOWS ......................................................38

9.1 Variation of Cd for an elliptical cross section with change in chord-to-thickness ratio ............. 38

9.2 Variation of point of Flow separation with change in flow velocity .......................................... 43

9.3 Variation of Lift generated by an Airfoil with Angle of Attack ................................................ 45

Chapter 10 CONCLUSIONS .............................................................................48

10.1 General Conclusions ................................................................................................................. 48

10.2 Future scope of work ................................................................................................................. 48

REFERENCES ...................................................................................................49

APPENDIX .........................................................................................................50

5

List of Figures

Figure 1-1 Study of Fluid Dynamics ........................................................................................................... 7

Figure 1-2 Design of Turbo Machinery ...................................................................................................... 9

Figure 1-3 IC Engine Design ...................................................................................................................... 9

Figure 1-4 Automobile Design ................................................................................................................... 9

Figure 1-5 Aerospace Engineering ............................................................................................................ 10

Figure 1-6 Sports Equipment Design ........................................................................................................ 10

Figure 1-7 Civil Engineering .................................................................................................................... 10

Figure 4-1 2D Mesh Elements- quads and tris .......................................................................................... 19

Figure 4-2 3D Mesh Elements- hexes, tets, pyramids and prisms ............................................................. 19

Figure 4-3 Structured Grids ...................................................................................................................... 20

Figure 4-4 Unstructured Grids .................................................................................................................. 20

Figure 4-5 Hybrid Grids............................................................................................................................ 21

Figure 4-6 A mesh refined towards the bottom surface ............................................................................ 21

Figure 6-1 An airfoil ................................................................................................................................. 25

Figure 6-2 Some airfoils ........................................................................................................................... 26

Figure 6-3 Airfoil Terminology ................................................................................................................ 27

Figure 7-1 ANSYS MECHANICAL FLOTRAN USER INTERFACE .................................................... 29

Figure 7-2 Internal combustion engine modeled using ANSYS Fluent .................................................... 30

Figure 7-3 ANSYS FLUENT USER INTERFACE .................................................................................. 31

Figure 8-1 Velocity Boundary Layer ........................................................................................................ 32

Figure 8-2 Computational space for simulation of flow over flat plate ..................................................... 32

Figure 8-3 Grid Generated for Simulation of Flow over Flat Plate ........................................................... 33

Figure 8-4 Velocity Contours obtained for Fluid flow over a flat plate .................................................... 34

Figure 8-5 Direction Vectors obtained for Fluid Flow over Flat plate ...................................................... 34

Figure 8-6 Computational Space for Simulation of Fluid Flow around a cylinder .................................... 35

Figure 8-7 Mesh Generated for Simulation of Fluid Flow over a cylinder................................................ 36

6

Figure 8-8 Direction vectors obtained for Fluid Flow around a cylinder .................................................. 36

Figure 8-9 Velocity contours obtained for Fluid Flow around cylinder .................................................... 37

Figure 8-10 Graph of Pressure Coefficient v/s Position for a cylinder ...................................................... 37

Figure 9-1 Computational Space for study of effect of streamlining on drag forces ................................. 38

Figure 9-2 Grid Generated for Study of effect of streamlining on drag forces .......................................... 39

Figure 9-3 Variation of Cd with chord-to-thickness ratio ......................................................................... 40

Figure 9-4 Direction vectors of flow ......................................................................................................... 41

Figure 9-5 Velocity Contours ................................................................................................................... 42

Figure 9-6 Graphs of Pressure coefficient vs Position .............................................................................. 43

Figure 9-7 Variation of Point of Flow Separation with velocity ............................................................... 44

Figure 9-8 C-Mesh generated for Study of Airfoils .................................................................................. 45

Figure 9-9 Lift curve obtained for NACA 0012 ........................................................................................ 46

Figure 9-10 Pressure contours at different Angles of attack...................................................................... 47

7

Chapter 1

COMPUTATIONAL FLUID DYNAMICS

1.1 What is Computational Fluid Dynamics

Computational Fluid Dynamics is the analysis of systems involving fluid flow, heat transfer and

associated phenomena such as chemical reactions by means of computer-based simulation.

Figure 1-1 Study of Fluid Dynamics

Traditionally, analysis of such systems involved two different approaches. In the earlier days ,

the analysis was done using pure physical experimentation and observation of the different

phenomenon. With time, theoretical approach started gaining ground. Until a couple of decades

ago, one would have to use either of these two approaches- pure theory or pure experimentation-

to study problems of fluid dynamics. However, both these approaches had their limitations.

While theoretical approach was limited by our knowledge of the flow at high speeds and high

temperatures, experimental approach was limited by the high costs. Ground test facilities, like

wind tunnels, do not exist for all types of conditions, such as in hypersonic flight regimes, which

need simultaneous simulation of high Mach numbers as well as high flow field temperatures.

However, the advent of high speed digital computers along with the development of accurate

numerical algorithms for solving physical problems on these computers has revolutionized the

study and practice of fluid dynamics today. It has resulted into a third approach in fluid

dynamics—the approach of Computational Fluid Dynamics.

Experimental Fluid

Dynamics

Computational Fluid Dynamics

Theoretical Fluid

Dynamics

8

CFD is basically simulating of physical problems using a computer to study and analyze various

phenomenon. By simulating a wide array of physical conditions and system configurations,

various observations can be made which would be otherwise impossible. CFD is a

complementary third method that has to be used in conjunction with the other two methods. It

will never replace either of the two methods. The future of advancement of Fluid Dynamics rests

upon carefully balancing the three approaches, with CFD helping to interpret and understand the

results of theory and experiment and vice versa.

1.2 Why Computational Fluid Dynamics

CFD is by no means a cheap approach. The investment costs of a CFD capability are not small.

Along with the high cost of hardware and software, an organization needs qualified personnel to

model the situations, run the codes and communicate their results. But the total expense is not

normally as great as that of a high quality experimental facility. Moreover, there are several

unique advantages of CFD over experiment-based approaches to fluid systems design, such as:

Substantial reduction of lead times and costs of new designs

Ability to study systems where controlled experiments are difficult or impossible

to perform (e.g. very large systems)

Ability to study systems under hazardous conditions at and beyond their normal performance limits (e.g. safety studies and accident scenarios)

Practically unlimited level of detail of results

The variable cost of an experiment, in terms of facility hire and/or man-hour costs, is

proportional to the number of data points and the number of configurations tested. In

contrast CFD codes can produce extremely large volumes of results at virtually no

added expense and it is very cheap to perform parametric studies, for instance to

optimize equipment performance

Also, as raw computing power is getting cheaper every day, CFD continues to become a

sssmore lucrative approach

CFD results are analogous to wind tunnel experiments, i.e. they both represent sets of data for a

given flow configuration at different Mach numbers, Reynolds number etc. However, unlike a

wind tunnel which is generally a heavy unwieldy device, a computer program is portable.

Moreover, the CFD program gives us concrete numerical data which can be used to infer various

results. Minor tweaks can be made to the program to study physically disparate phenomenon.

CFD data can be also used to interpret results of physical experimentation.

CFD often provide valuable data that can be directly incorporated while designing systems. For

example, in the development of an aircraft, CFD provides the means for calculating the detailed

flow field around a complete airplane configuration. This data can be used by structural

engineers, aerodynamicists etc to take appropriate design decisions.

9

In short, CFD is playing a strong role as a Research and Design tool and has become a powerful

influence on the way fluid dynamists and aerodynamicists do business.

1.3 Some Applications of CFD

Some of the applications in which CFD is making in impact are as below:

Figure 1-2 Design of Turbo Machinery

Figure 1-3 IC Engine Design

Figure 1-4 Automobile Design

10

Figure 1-5 Aerospace Engineering

Figure 1-6 Sports Equipment Design

Figure 1-7 Civil Engineering

11

Chapter 2

THE CFD PROCESS

CFD codes are structured around the numerical algorithms that can tackle fluid flow problems.

In order to provide easy access to their solving power all commercial CFD packages include

sophisticated user interfaces to input problem parameters and to examine the results. Hence all

codes contain three main elements:

a pre-processor

a solver

a post-processor

We briefly examine the function of each of these elements within the context of a CFD code.

2.1 Pre-Processing

Pre-processing consists of the input of a flow problem to a CFD program by means

of an operator-friendly interface and the subsequent transformation of this input into a form

suitable for use by the solver. The user activities at the pre-processing stage involve:

Definition of the geometry of the region of interest: the computational domain

Grid Generation i.e. sub dividing the computational domain into a no. of smaller, non-

overlapping sub-domains: a grid/mesh of cells

Selection of the physical and chemical phenomena that need to be modeled

Definition of fluid properties

Specification of appropriate boundary conditions at cells which coincide with or touch

the boundary

The solution of a flow problem (such as velocity, pressure, temperature ) is defined at nodes

inside each cell. The accuracy of a CFD solution is governed by the density of the cells. Higher

the density, better the accuracy. However, the cost of hardware and time of calculation also

increases with the fineness of the grid. More-over, optimal grids are non-uniform i.e. coarser in

regions with little variation and finer in regions where more point-to-point variation is likely to

occur. The major part of the CFD process is devoted to the Pre-processing stage. Most CFD

packages now come with built-in modeling and Mesh generating tools. They also allow for

selection of host of material and fluid properties.

12

2.2 Solving

The Solver is responsible for

Approximation of the unknown flow variables by means of simple functions

Discretization or substitution of the approximations into the governing flow equations

and subsequent mathematical manipulations

Solution of Algebraic equations

There are three different methods of discretization -finite difference, finite element and finite

volume methods. The main difference between the three separate methods is associated with the

way in which the flow variables are approximated and with the discretization process.

In finite difference method, the unknown ø of the flow problem is described by means of point

samples at the node points of a grid of co-ordinate lines. Truncated Taylor series expansions are

used to generate finite difference approximations of derivatives of ø in terms of point samples

of ø at each grid point and its immediate neighbors. Those derivatives appearing in the governing

equations are replaced by finite differences yielding an algebraic equation for the values of ø at

each grid point.

Finite Element Methods use simple piecewise functions valid on elements to describe the local

variations of unknown flow variables ø . The governing equation is precisely satisfied by the

exact solution ø. If piecewise approximating functions are substituted into the equation, it will

not hold exactly and a residual is defined to measure the errors. Next, the residuals are

minimized. As a result, we obtain a set of algebraic for the unknown coefficients of the

approximating functions.

Finite volume method started out as a special type of finite difference method in which the

governing equations are first written in the integral form and then the integrals are substituted

with appropriate finite differences.

2.3 Post Processing

Post-Processing is the stage at which the results of the calculation are put in a form required by

the user. In addition to the alpha-numeric result, the output maybe presented in the form of

visuals, animation, graph, bar graphs or other suitable forms.

13

Figure 2-1 The CFD Process

PHYSICAL PROBLEM

Identifying situation to be

studied

PRE-PROCESSING

Modeling, material properties, Grid Generation

SOLVING

Discretization, Solution

POST-PROCESSING

Visual data representation

14

Chapter 3

GOVERNING LAWS OF FLUID DYNAMICS

Fluid dynamics is based on three fundamental conservation laws that govern the physical

properties of a flow. They are:

Mass of a fluid is conserved

Rate of change of momentum equals the sum of the forces on a fluid particle

(Newton’s 2nd

law)

Energy is conserved

The equations derived from these three laws are known as the governing equations of fluid

dynamics.

For obtaining the basic equations of fluid motion, the following methodology is followed:

1. Choose the appropriate fundamental physical principle from the three stated above

2. Apply these physical principles to a suitable model of flow

3. From this application, extract the suitable mathematical equation

The model of flow selected affects the nature of the equation obtained. 4 different models of

flow can be considered, as demonstrated below.

Figure 3-1 Models of a flow (a) Finite control volume approach (b) infinitesimal fluid element approach

15

As shown in Figure 3-1, if we consider a finite control volume for our derivation, we arrive at

Integral form of governing equations. An infinitesimal fluid element gives us equations that

involve partial differential terms. Also, considering elements to be fixed in space, with fluid

flowing through it, gives us the Conservation form of respective equations while considering

elements moving with the flow gives us Non-conservation forms of governing equations.

Based on the model selected we get different forms of the equations. These forms are inter-

convertible, i.e. each form can be converted into the other with mathematical manipulation.Each

of this form has its place in CFD. However, certain forms are more suitable than others. For

example, the integral form of the equations allows for the presence of discontinuities inside the

fixed control volumes. However, the differential form of the governing equations assumes the

flow properties to be differentiable, hence continuous. Hence the integral form is more

fundamental than the differential form. This is of particular importance when calculating a flow

with real discontinuities, such as shock waves.

Before stating the final, governing equations, we must describe another important term in order

to better understand and hence use, the governing equation. This term is the Substantial

derivative. The substantial derivative of a quantity T is defined as

Local Derivative Convective Derivative

The local derivative part of the expression arises due to the change in quantity T at a point with

time while the convective derivative part arises due to change in the quantity as it moves from

one point in space to another.

Based on the three governing laws and various models of flow, we arrive at the following

equations which form the basis of entire CFD. These are known as the Navier-Stokes equations.

3.1 Continuity Equation

Non Conservation Form

Conservation Form

Where ρ is the density of the fluid and V is the velocity with components u, v, w in x, y, z

directions respectively.

16

3.2 Momentum Equations

Non Conservation Form

x component :

y component:

z component:

Conservation Form

x component :

y component:

z component:

Where p is the pressure, fx, fy,fz are components of body forces per unit volume in x, y, z

directions respectively and the τ terms are surface forces in directions given by their respective

subscripts.

3.3 Energy Equation

Non Conservation Form

17

Conservation Form

Where e is the internal energy, q is the heat transferred into the element per unit volume and T is

the temperature.

In many of the cases encountered in practical situations, the transport phenomena such as

viscosity, mass diffusion and thermal conductivity can be neglected. The equations arrived at by

equating the corresponding terms in the above equations to zero are called as the Euler

equations. These equations are easier to deal with and give reasonable accuracy in many

situations.

3.4 Physical Boundary Conditions

The equations derived above govern the flow of a fluid irrespective of the conditions in which

the flow is occurring. The difference in flow fields occurs due to the concept of Boundary

Conditions. The Boundary conditions dictate the solution obtained from the governing equations.

Some of the boundary conditions are as given below

For a viscous flow, the fluid on a surface has velocity equal to 0, i.e. u=v=w=0 at the surface.

Also, for a viscous flow, temp of layer at the surface is equal to the wall temp. (T=Tw) and

at the wall

For an inviscid flow, V.n=0 at the surface, where n is normal to the surface

Other than these standard boundary conditions, other conditions arise as a result of the specific

physical conditions being modeled.

18

3.5 Discretization

Discretization is the process by which a closed form mathematical expression, such as a function

or a differential or integral equation involving functions , is approximated by analogous (but

different) expressions which prescribe values at only a finite number of discrete points or

volumes in the domain. As stated in the section on ‘Solver’, there are 3 techniques of

discretization- finite difference, finite element and finite volume methods.

In the finite difference technique, the differential terms are substituted by the finite difference

expressions. These are as follows

Forward difference

Rearward difference

Central difference

When all the partial derivatives in a partial differential equation have been replaced by finite

differences, the resulting equation is known as a difference equation.

The resulting algebraic equations are then solved using various computational techniques to

arrive at solutions at discrete points in the domain to establish the complete flow field.

19

Chapter 4

MESHING

The solutions of the discretized governing equations in the computational domain, happens over

a number of discreet points. The Process of dividing the computational domain by generating a

grid is known as Meshing or Grid generation. Meshing is one of the most important step in CFD

solutions and the type of grid used can make or break a CFD solution.

For a 2D mesh, all mesh nodes lie in a given plane. 2D mesh elements are quadrilaterals (also

known as quads) and triangles (tris), shown below.

Figure 4-1 2D Mesh Elements- quads and tris

3D mesh nodes are not constrained to lie in a single plane. Most popular 3D mesh elements are

hexahedra (also known as hexes or hex elements), tetrahedra (tets), square pyramids (pyramids)

and extruded triangles (wedges or triangular prisms), shown below. All these elements are bound

by 2D shapes described above. Some of the newer solvers also support polyhedral elements,

which can be bounded by any number and types of faces.

Figure 4-2 3D Mesh Elements- hexes, tets, pyramids and prisms

20

Various types of Grids are used in practice. Some of them will be seen below.

4.1 Structured Grid

A structured grid (Figure 4-3) consists of either a planar cells with 4 edges or volumetric cells with 6

edges. Although the cells may be distorted from rectangular, each cell is numbered according to indices

(i,j,k) that do not necessarily correspond to co-ordinates (x,y,z).

Figure 4-3 Structured Grids

4.2 Unstructured Grids

An Unstructured grid (Figure 4-4) consists of cells of various shapes in theory. But practically, the cells

are triangles and quadrilaterals for planar cells and tetrahedrons for volumetric cells. Unlike the structured

grid, one cannot identify a cell using indices (i,j,k). Unstructured grids are generally used for complex

geometries.

Modern CFD codes can handle both the types of meshes equally well. However, structured grids offer

computational convenience and better division of the computational space in many cases.

In either case, it’s the quality of mesh that is of importance, rather than the type.

Figure 4-4 Unstructured Grids

21

4.3 Hybrid Grids

Sometimes, we use a combination of the above 2 types of grids within a single body. Such meshes are

known as Hybrid Grids (Figure 4-3). This maybe done to provide high resolution near a body while

giving a low resolution away from the body. It can also be used to increase computational efficiency and

avoiding excessive skewing of individual cells.

Figure 4-5 Hybrid Grids

In almost all CFD simulations, meshes are refined towards object under consideration. This is

done as the variation in fluid parameters tends to be higher nearer to a body than in a region

faraway. Hence, more number of cells nearer to the body helps us obtain a higher resolution of

flow parameters.

Figure 4-6 A mesh refined towards the bottom surface

22

Chapter 5

EXTERNAL FLOWS AROUND OBJECTS

Whenever a body is immersed in a flowing fluid, the flow over the body is known as an external

flow. In external flows, the viscous effects are confined to a portion of the flow field such as

boundary layers and wakes, which are surrounded by an outer flow region that involves small

velocity and temperature gradients.

When a fluid moves over a body, it exerts pressure force normal to the surface and shear forces

parallel to the surface. A combination of these two forces results in a resultant force acting on the

body. The component of this resultant force normal to the direction of fluid flow is referred to as

the Lift force while the component acting in the direction of the fluid flow is known as the Drag

force.

5.1 Drag and Lift

A body moving through a fluid, especially a liquid, experiences resistance. A fluid may exert

forces and moments on a body in and about various directions. The net resultant of the forces in

the direction of flow is called Drag. Drag, like friction, is usually undesirable and we try our best

to reduce it as much as possible. In applications like Automobiles, Ships, Aircrafts, reduction of

drag helps reduce the power required to propel the vehicles. Reduction in Drag also helps

increase the durability and safety of structures exposed to highly windy conditions. It also helps

reduce vibration and noise. However, at times, Drag is also a useful force and we try to

maximize it. Examples include Parachutes, retardation of bodies etc.

The net resultant of the forces in the direction normal to the direction of flow is called Lift. Lift

is the component that causes Aircrafts to rise. Both these forces are caused due to Pressure as

well as Shear forces.

Consider a small are dA on the surface of a body. The Pressure force acting on this elemental

area is PdA while the Shear force acting is τdA. The Resulting Lift and Drag forces are given by

dFD = -PdA cosθ+ τdA sinθ

dFL = -PdA sinθ- τdA cosθ

where θ is the angle made by the outward normal to the area dA with the positive direction of

flow. The total Drag and Lift forces are determined by integrating the above equations over the

entire surface of the body.

23

For a thin flat plate aligned to the direction of flow, θ=90° and hence, Drag is due to Shear forces

only. In contrast, when the plate is perpendicular to the direction of flow, θ=0° and the entire

drag is due to pressure forces only. For intermediate angles, the drag force is a combination of

the_two_as_given_by_the_equation.

The wings of an airplane are shaped specifically to generate the required lift while generating

minimal drag. This is done by maintaining a suitable Angle of Attack. Angle of attack is the

angle made by the line joining the front tip of a wing to its rear end with the direction of flow.

Both Lift and Drag are strongly dependent on the Angle of Attack.

Drag and Lift forces also depend on the Density of the fluid , the Upstream velocity V and the

size, shape and orientation of the body, amongst other things and it is not always possible to list

them all, in all situation. Instead, it is more convenient to work with dimensionless numbers that

represent the drag and lift characteristics of a body. These numbers are the Drag Coefficient CD

and Lift Coefficient CL . They are defined as

CD=

CL=

Where A is usually the frontal area of the body, that is, the area projected on a plane in front of

the body. In certain cases, like in case of an airfoil, A is taken as the Planform area, that is, areas

as seen from top.

As mentioned earlier, the drag force is a resultant of Skin friction as well as pressure. Hence,

CD = CD, Friction + CD,pressure

5.2 Laminar and Turbulent Flows

Flows are said to be either Laminar or Turbulent. In Laminar flows, adjacent layers of the fluid

flow smoothly without disturbing each other. They are characterized by smooth streamlines and

highly ordered motion. In contrast, Turbulent flows are characterized by Velocity fluctuations

and highly disordered motion. Turbulent flows are unpredictable in nature. The transition from

Laminar to Turbulent flow does not happen in an instant. Rather, it happens over an intermediate

transition region where the flow is a combination of the above two.

The transition region depends on a number of variables such as surface roughness, flow velocity,

surface temperature etc. It is not always possible to characterize a flow by describing all these

24

properties. Instead, the flow is characterized using a dimensionless number called as Reynold’s

Number Re.

Reynold’s number is the ratio of the Inertial forces acting in a fluid to the viscous forces.

Re=

=

Where, Vavg Average velocity of flow

D=characteristic length of the geometry

= / = kinematic viscosity of the fluid

When viscous forces dominate, Reynold’s number becomes small and the resulting flow is

laminar in nature. As the Inertial forces start to dominate, Reynold’s number increases and the

flow becomes more and more turbulent in nature.

In general, the following classification of flow based on Reynold’s number is generally accepted.

Re≤ 2000-----Laminar flow

2000≤Re≤4000----Transitional flow

Re≥4000----Turbulent Flow

25

Chapter 6

AIRFOILS

An airfoil or aerofoil is the shape of a wing or blade (of a propeller, rotor or turbine) or sail as

seen in cross-section. An airfoil-shaped body moved through a fluid produces an aerodynamic

force. The component of this force perpendicular to the direction of motion is called lift. The

component parallel to the direction of motion is called drag. Subsonic flight airfoils have a

characteristic shape with a rounded leading edge, followed by a sharp trailing edge, often with

asymmetric camber. Foils of similar function designed with water as the working fluid are called

hydrofoils.

Figure 6-1 An airfoil

The lift on an airfoil is primarily the result of its angle of attack and shape. When oriented at a

suitable angle, the airfoil deflects the oncoming air, resulting in a force on the airfoil in the

direction opposite to the deflection. This force is known as aerodynamic force and can be

resolved into two components: Lift and Drag. Most foil shapes require a positive angle of attack

to generate lift, but cambered airfoils can generate lift at zero angle of attack. This "turning" of

the air in the vicinity of the airfoil creates curved streamlines which results in lower pressure on

one side and higher pressure on the other. This pressure difference is accompanied by a velocity

difference, via Bernoulli's principle, so the resulting flowfield about the airfoil has a higher

average velocity on the upper surface than on the lower surface. The lift force can be related

directly to the average top/bottom velocity difference without computing the pressure.

A fixed-wing aircraft's wings, horizontal, and vertical stabilizers are built with airfoil-shaped

cross sections, as are helicopter rotor blades. Airfoils are also found in propellers, fans,

compressors and turbines. Sails are also airfoils, and the underwater surfaces of sailboats, such as

the centerboard and keel, are similar in cross-section and operate on the same principles as

airfoils. Swimming and flying creatures and even many plants and sessile organisms employ

airfoils/hydrofoils: common examples being bird wings, the bodies of fish, and the shape of sand

dollars. An airfoil-shaped wing can create downforce on an automobile or other motor vehicle,

improving traction. Any object with an angle of attack in a moving fluid, such as a flat plate, a

building, or the deck of a bridge, will generate an aerodynamic force (called lift) perpendicular to

the flow. Airfoils are more efficient lifting shapes, able to generate more lift (up to a point), and

to generate lift with less drag.

Airfoil design is a major facet of aerodynamics. Various airfoils serve different flight regimes.

Asymmetric airfoils can generate lift at zero angle of attack, while a symmetric airfoil may better

suit frequent inverted flight as in an acrobatic aeroplane. In the region of the ailerons and near a

wingtip a symmetric airfoil can be used to increase the range of angles of attack to avoid spin-

stall. Thus a large range of angles can be used without boundary layer separation. Subsonic

26

airfoils have a round leading edge, which is naturally insensitive to the angle of attack. The cross

section is not strictly circular, however: the radius of curvature is increased before the wing

achieves maximum thickness to minimize the chance of boundary layer separation. This

elongates the wing and moves the point of maximum thickness back from the leading edge.

Supersonic airfoils are much more angular in shape and can have a very sharp leading edge,

which is very sensitive to angle of attack. A supercritical airfoil has its maximum thickness close

to the leading edge to have a lot of length to slowly shock the supersonic flow back to subsonic

speeds. Generally such transonic airfoils and also the supersonic airfoils have a low camber to

reduce drag divergence. Modern aircraft wings may have different airfoil sections along the wing

span, each one optimized for the conditions in each section of the wing.

Movable high-lift devices, flaps and sometimes slats, are fitted to airfoils on almost every

aircraft. A trailing edge flap acts similar to an aileron, with the difference that it can be retracted

partially into the wing if not used.

Figure 6-2 Some airfoils

27

A laminar flow wing has a maximum thickness in the middle camber line. Analyzing the Navier-

Stokes equations in the linear regime shows that a negative pressure gradient along the flow has

the same effect as reducing the speed. So with the maximum camber in the middle, maintaining a

laminar flow over a larger percentage of the wing at a higher cruising speed is possible.

However, with rain or insects on the wing, or for jetliner speeds, this does not work. Since such a

wing stalls more easily, this airfoil is not used on wingtips (spin-stall again).

Schemes have been devised to define airfoils — an example is the NACA system. Various airfoil

generation systems are also used. An example of a general purpose airfoil that finds wide

application, and predates the NACA system, is the Clark-Y. Today, airfoils can be designed for

specific functions using inverse design programs such as PROFOIL, XFOIL and AeroFoil.

XFOIL is an online program created by Mark Drela that will design and analyze subsonic

isolated airfoils.

Figure 6-3 Airfoil Terminology

The various terms related to airfoils are defined below:

The suction surface (a.k.a. upper surface) is generally associated with higher velocity and thus lower static pressure.

The pressure surface (a.k.a. lower surface) has a comparatively higher static pressure than the suction surface. The pressure gradient between these two surfaces contributes to

the lift force generated for a given airfoil.

The leading edge is the point at the front of the airfoil that has maximum curvature.

The trailing edge is defined similarly as the point of maximum curvature at the rear of the

airfoil.

The chord line is a straight line connecting the leading and trailing edges of the airfoil.

The chord length, or simply chord, c, is the length of the chord line and is the characteristic dimension of the airfoil section.

The mean camber line is the locus of points midway between the upper and lower

surfaces. Its exact shape depends on how the thickness is defined.

28

Chapter 7

ANSYS

ANSYS is an engineering simulation software (computer-aided engineering, or CAE software)

that offers a comprehensive range of engineering simulation solution sets providing access to

virtually any field of engineering simulation that a design process requires. Companies in a wide

variety of industries use ANSYS software. The tools put a virtual product through a rigorous

testing procedure (such as crashing a car into a brick wall, or running for several years on a

tarmac road) before it becomes a physical object. ANSYS is used across the Automotive,

Aerospace, Energy, Electronics and Consumer products Industry, amongst others.

7.1 CFD IN ANSYS

ANSYS fluid dynamics is a comprehensive product suite for modeling fluid flow and other

related physical phenomena. It offers unparalleled fluid flow analysis capabilities, providing all

the tools needed to design and optimize new fluids equipment and to troubleshoot already

existing installations. The ANSYS fluid dynamics suite contains both general purpose

computational fluid dynamics software and additional specialized products to address specific

industry applications.

The general purpose fluid analysis tools are the renowned ANSYS CFX and ANSYS FLUENT

products, which are now available together in the ANSYS CFD bundle. ANSYS FLOTRAN was

also used earlier for CFD purposes. With ANSYS CFD, one has an access to an unprecedented

array of fluid flow physics models, allowing one to analyze products with a great deal of

confidence. ANSYS CFD technology is highly-scalable, allowing for efficient parallel

calculations on thousands of processing cores. ANSYS CFD also includes the full-featured

ANSYS CFD-Post fluid flow post-processing tool. This can be used for advanced quantitative

analysis and high-quality visualizations. When ANSYS CFD is used in combination with

ANSYS Mechanical it is eminently suitable to solve complex fluid-structure interaction

problems.

ANSYS fluid dynamics products have a high degree of interoperability. ANSYS CFD solvers are

designed to handle all types of meshes, with moving and deforming mesh capabilities, advanced

multi-grid methods, and solution-based, adaptive remeshing functionality. They deliver highly

accurate results across all flow regimes — from hypersonic to creeping flows, Newtonian to non-

Newtonian. The adjoint solver in ANSYS software provides specific information that is

otherwise difficult to gather. Because adjoint solutions estimate the effect of a change prior to

actually making the change, this exclusive capability adds to the speed of simulation.

The CFD suites offer both speed and accuracy. Facility to use High Performance Computers and

parallel computing provide powerful and scalable options, so you get more geometric detail,

larger systems and more complex physics (for example, an unsteady turbulence rather than a

steady turbulence model). The result is enhanced insight into product performance, insight that

29

can’t be gained any other way. This detailed understanding can yield enormous benefits by

revealing design issues that might lead to product failure or troubleshooting delays.

7.2 ANSYS FLOTRAN

The ANSYS FLOTRAN derived product and the FLOTRAN CFD (Computational Fluid

Dynamics) option to the other ANSYS products offer comprehensive tools for analyzing 2-D

and 3-D fluid flow fields. Using either product and the FLOTRAN CFD elements FLUID141

and FLUID142, one can achieve solutions for the following:

Lift and drag on an airfoil

The flow in supersonic nozzles

Complex, 3-D flow patterns in a pipe bend

Besides these, FLOTRAN can be used to perform various other CFD simulations. However, The

FLOTRAN user interface is not very user friendly. Also, its ability to work with other modeling

software is severely limited. The modern CFX and Fluent Packages are much more user friendly

and provide users with powerful capabilities for a wide range of situations.

Figure 7-1 ANSYS MECHANICAL FLOTRAN USER INTERFACE

30

7.3 ANSYS FLUENT

ANSYS Fluent software contains the broad physical modeling capabilities needed to model

flow, turbulence, heat transfer, and reactions for industrial applications ranging from air flow

over an aircraft wing to combustion in a furnace, from bubble columns to oil platforms, from

blood flow to semiconductor manufacturing, and from clean room design to wastewater

treatment plants. Special models that give the software the ability to model in-cylinder

combustion, aeroacoustics, turbomachinery, and multiphase systems have served to broaden

its reach.

Figure 7-2 Internal combustion engine modeled using ANSYS Fluent

Thousands of companies throughout the world benefit from the use of ANSYS Fluent

software as an integral part of the design and optimization phases of their product

development. Advanced solver technology provides fast, accurate CFD results, flexible

moving and deforming meshes, and superior parallel scalability. User-defined functions

allow the implementation of new user models and the extensive customization of existing

ones. The interactive solver setup, solution and post-processing capabilities of ANSYS

Fluent make it easy to pause a calculation, examine results with integrated post-processing,

change any setting, and then continue the calculation within a single application. Case and

data files can be read into ANSYS CFD-Post for further analysis with advanced post-

processing tools and side-by-side comparison of different cases.

The integration of ANSYS Fluent into ANSYS Workbench provides users with superior bi-

directional connections to all major CAD systems, powerful geometry modification and

creation with ANSYS DesignModeler technology, and advanced meshing technologies in

ANSYS Meshing. The platform also allows data and results to be shared between

applications .The combination of these benefits with the extensive range of physical

modeling capabilities and the fast, accurate CFD results that ANSYS Fluent software has to

offer results in one of the most comprehensive software packages for CFD modeling

available in the world today.

31

Figure 7-3 ANSYS FLUENT USER INTERFACE

32

Chapter 8

SIMULATION OF FLOW AROUND OBJECTS

In this section, we perform simulation of some commonly occurring daily phenomenon using

ANSYS Fluent. We will compare our observation with established facts about these daily

phenomena.

8.1 Flow over a Flat Plate

Consider a fluid flow over a flat plate in a direction parallel to the plate surface. The fluid is

considered to be made up of adjacent layers flowing over each other. Due to the viscosity of the

fluid, the particles in the layer adjacent to the plate surface have zero velocity as they adhere to

the plate surface. This motionless layer slows down the particles of the adjacent layer due to

friction between the layers. This layer slows down the particles of the next layer and so on. Thus,

the presence of the plate is felt for some normal distance δ from the plate, beyond which the free

stream velocity remains virtually unchanged. The x component of the fluid velocity u varies

from 0 at y=0 to nearly V, the free stream velocity at y= δ .The region of the flow above the plate

in which this effect is felt is known as the Velocity Boundary Layer.

The Reynold’s number for the flow is given by Re =

where x is the distance from the leading

edge. Hence, as we move further away from the pate, the flow regime changes from Laminar, to

transitional region to Turbulent. This is illustrated in Figure 8-1

Figure 8-1 Velocity Boundary Layer

To simulate Flow over a Flat Plate, we consider a computational space as shown below

Figure 8-2 Computational space for simulation of flow over flat plate

33

Since the flow is symmetrical on both the sides of the plate, we consider flow on one side only.

Hence, in Figure 8-2, the lower surface indicates the flat plate, the left side the inlet, outlet on the

right side while the top side represents the far field. The far field is chosen at a distance such that

it does not affect the flow on the plate.

The next step in the simulation is meshing. It is evident that the most variation in flow prpperties

will occur closer to the plate while it will be fairly constant away from the plate. Hence, our grid

has to be chosen such that the resolution increases towards the plate. Since the computational

domain is regular in shape, a structured grid will do. Figure 8-3 shows the grid generated.

Figure 8-3 Grid Generated for Simulation of Flow over Flat Plate

Once the grid is generated, the flow properties were defined. The Inlet velocity was defined as

1m/s, the density of the fluid taken as 1kg/m3

and viscosity of 10-4

kg/m-s. Thus, at a distance of

1 m from the leading edge, the Reynold’s number turns out to be 10,000.

The results obtained are shown in the images. Figure 8-4 shows the contours of velocity obtained

for the simulation. The variation of velocity as we move away from the plate is clearly visible.

The part near the leading edge has been zoomed for added clarity. Notice that the layers adjacent

to the plate are stationary or have very low velocity as indicated by the Blue color. Also notice

that Boundary layer is restricted to a very small region near the plate only.

Figure 8-5 shows the direction vectors for the same flow. The length of the vectors indicates the

velocity. The velocity profile is clearly visible here and is as per our expectation.

It was also found that simulation yielded a Cd, pressure of 0. This agrees with our expectation for a

plate aligned with the flow.

34

Figure 8-4 Velocity Contours obtained for Fluid flow over a flat plate

Figure 8-5 Direction Vectors obtained for Fluid Flow over Flat plate

35

8.2 Flow around a Cylinder

For simulating fluid flow around cylinders, we consider computational space as shown in Figure

8-6. Notice that the computational space is circular in nature and concentric around the cylinder.

The entire left side of the space is the inlet while the right side is the outlet. The diameter of the

computational space is much bigger than the diameter of the cylinder.

Figure 8-6 Computational Space for Simulation of Fluid Flow around a cylinder

The Mesh generated for the simulation is radial in nature. Once again, we increase the resolution

of the grid as we move towards the cylinder. The Grid generated is shown in Figure 8-7.

36

Figure 8-7 Mesh Generated for Simulation of Fluid Flow over a cylinder

Next, we define the fluid parameters. The Reynolds number is chosen to be 20. In order to

simplify the computation, the diameter of the pipe is set to 1 m, the x component of the velocity

is set to 1 m/s and the density of the fluid is set to 1 kg/m3. Thus, the dynamic viscosity must be

set to 0.05 kg/m-s in order to obtain the desired Reynolds number.

With the above initial values, we begin the simulation. The results obtained are shown below

Figure 8-8 Direction vectors obtained for Fluid Flow around a cylinder

37

Figure 8-9 Velocity contours obtained for Fluid Flow around cylinder

The point where the flow separates from the cylinder surface can be found by platting a graph of

Pressure coefficient v/s the geometry. The point of inflexion indicates separation of flow.

Figure 8-10 Graph of Pressure Coefficient v/s Position for a cylinder

38

Chapter 9

CONTROL OF FLUID FLOWS

Now that we have successfully simulated flows around objects, we will now study how varying some

parameters affects the flow and the resulting effect on the objects.

9.1 Variation of Cd for an elliptical cross section with change in chord-to-thickness ratio

Streamlining of a body is the process of reducing the frontal, projected area of a body.

Streamlining reduces the pressure drag on a body by delaying flow separation. However, it is

also known to increase frictional drag due to increase in the area over which the flow is in

contact with the body.

In this experiment, we vary the chord-to-thickness ratio a/b of an ellipse and study the

corresponding changes in the values of Drag Coefficient due to pressure and viscous/frictional

forces. ‘a’ and ‘b’ are the lengths of the semi-major and semi-minor axis of the body

respectively.

We consider a fixed area of cross section of the body and vary the ratio of a/b steadily and note

the corresponding changes. The Reynolds number of the flow is fixed at 20.

The computational Space and grid are same as that taken for the study of flow around a cylinder.

Figure 9-1 Computational Space for study of effect of streamlining on drag forces

39

Figure 9-2 Grid Generated for Study of effect of streamlining on drag forces

The results obtained are as follows

Table 9-1 Variation of Cd with chord-to-thickness ratio

a/b cd-p cd-v

1 1.1541283 0.846896

1.2 0.9965778 0.898961

1.4 0.8936725 0.942757

1.6 0.8127458 0.978765

1.8 0.736508 1.012806

2 0.6837875 1.040246

40

Figure 9-3 Variation of Cd with chord-to-thickness ratio

0.6

0.7

0.8

0.9

1

1.1

1.2

0.5 1 1.5 2 2.5

cd-p

cd-v

41

Figure 9-4 Direction vectors of flow

Figure 8-2 shows the variation of direction vectors at all points within the flow with variation in

the chord-to-thickness ratio of the ellipse. Notice the reduction in the wake generated by the

body with increased streamlining. Also notice the delay in separation of flow. This delay in flow

separation can be better observed in the

42

Figure 9-5 Velocity Contours

43

Figure 9-6 Graphs of Pressure coefficient vs Position

This data can be made use of determine the optimum aspect ratio of a body so as to obtain least

possible drag acting on the body.

9.2 Variation of point of Flow separation with change in flow velocity

For a body such as a cylinder or a wing immersed in a fluid flow, if all other conditions are kept

same, then the point of separation of flow from the cylinder surface changes with a change in

fluid velocity.

44

Figure 9-7 Variation of Point of Flow Separation with velocity

45

9.3 Variation of Lift generated by an Airfoil with Angle of Attack

Angle of attack is a major factor that determines the lift of an Airfoil and hence, an aircraft. In

fact, variation in the angle of attack helps us control the ascent or descent of an aircraft. Hence,

we will try and study the variation of lift forces with change in angle of attack.

The computational space and mesh generated for the simulation are as shown in Figure 9-8.

The mesh is a special type of mesh known as the C-mesh and allows to easily change the angle

of inlet velocity while keeping the mesh unchanged.

Figure 9-8 C-Mesh generated for Study of Airfoils

We vary the angle of inlet velocity, which has the same effect as varying the angle of attack of

the Airfoil. The density of the fluid was taken as 1 kg/m3.

The results obtained are tabulated in Table 9-2. We can see that as the Angle of attack increases,

the Coefficient of Lift CL increases upto a point after which it suddenly drops. This angle of

Maximum Lift is known as the Stall Angle.

46

Table 9-2 Variation of Lift and Drag with Angle of attack

Angle of attack

(in degrees)

CD CL

4 0.004291 0.444514

6 0.007663 0.647821

8 0.013373 0.819867

10 0.020904 0.841042

12 0.032596 0.865794

14 0.053302 0.877937

16 0.076246 0.893525

18 0.099873 0.870853

20 0.122763 0.801841

When plotted on a graph, we obtain what is known as a Lift Curve.

Figure 9-9 Lift curve obtained for NACA 0012 airfoil

The pressure contours obtained can be seen in Figure 9-10.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

CL

CD

47

Figure 9-10 Pressure contours at different Angles of attack

48

Chapter 10

CONCLUSIONS

10.1 General Conclusions

Through the course of this project, Fundamentals of CFD were studied and assimilated. Besides

simulating fluid flows around primitive bodies, concepts of CFD and Fluid Dynamics were used

to arrive at important conclusions.

Flow of fluid over a flat plate was simulated. Generation of Boundary Layer was clearly

observed. The Velocity profile obtained was also found to be coherent with theoretical

velocity profile.

Flow of fluid around a cylindrical body was simulated. Corelation between point of

separation and Pressure coefficient was established.

Effect of streamlining of a body on the Drag coefficient was observed.

Effect of increase of fluid velocity on point of separation was observed.

Effect of Angle of Attack on Lift generated by an Airfoil was observed.

In conclusion, the goal of the project, ie. Simulation and control of Fluid Flows around objects

using CFD was achieved. Many important insights into the world of CFD were gained over the

course of the project.

10.2 Future scope of work

CFD is an advanced research tool and is being increasingly used to solve problems of complex

nature. As such, the scope of future work in this field is endless.

Some of the areas where research can be undertaken are

Effect of using sequential airfoils [1]

Fluid flows around Bluff bodies

[2]

Effect of variation of various shape parameters on flight characteristics of Airfoils

Flight characteristics of full systems, such as aircrafts, vehicles etc.

Research can be undertaken in a wide number of areas.

49

REFERENCES

1) Ofer Aharon; Hydrofoil analysis using CFD; MIT 2008: 2.094 Project

2) Ankur Bajoria; Analyzing wind flow around the square plate using ADINA; MIT

2008: 2.094 Project

3) John D. Anderson Jr.;Computational fluid dynamics-The Basics with applications;

McGraw-Hill series in Mechanical Engineering

4) John D. Anderson Jr.;Fundamentals of Aerodynamics; McGraw-Hill series in

Mechanical engineering

5) H. K. Versteeg & W Malasekara; An introduction to Computational Fluid Dynamics;

Longman Scientific & Technical

6) Abdulnaser Sayma; Computational Fluid Dynamics; Ventus publishing

7) Grant Ingram; Basic Concepts in Turbomachinary; Ventus publishing

8) Prof. D.M. Causon, Prof. C. G. Mingham; Introductory Finite Difference Methods for

PDEs; Ventus publishing

9) Klaus Jurgan Bathe, Finite Element Procedures, PHI publishing

10) T. J. Ching, Finite Element Analysis in Fluid Dynamics

11) P. N. Chatterjee, Fluid Mechanics for Engineers

12) Robert Fox and Alan McDonald, Introduction to Fluid Mechanics

13) Anil W. Date, Introduction to CFD

14) Antony Jameson; CFD for Aerodynamic Design and Optimization: Its Evolution over

the Last Three Decades; 16th AIAA CFD Conference, 2003

15) http://www.cfd-online.com/

16) http://confluence.cornell.edu/

17) http://ansys.com/

18) http://www.ae.illinois.edu/m-selig/ads/aircraft.html

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APPENDIX

Some early simulations performed in ANSYS FLOTRAN

51