This project report summarizes numerical simulations of inviscid and viscous supersonic flow over a diamond head airfoil at zero angle of attack. The simulations were conducted using commercial CFD software. Results for Mach number, drag and lift coefficients are presented and compared between the inviscid and viscous cases. Key differences observed include the development of boundary layers and increased drag in the viscous case due to skin friction effects.

1. MECH 6111 Gas Dynamics
Project Report:
Numerical investigation of inviscid and viscous supersonic
flow over a diamond head airfoil
Submitted to:
Dr. Wahid Ghaly
November 30, 2015
Name Student ID Email address
Jay Adhvaryu 40002804 jayadhvaryu42@gmail.com
Nishant Patel 27853378 nishantpatel9493@gmail.com

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Abstract
In this project we have simulated a steady-state supersonic flow over a diamond
head airfoil for two types of fluids – (i) viscous and (ii) inviscid. The angle of
attack is zero. We have compared the results and explain the reasons for the
differences observed in simulation results.
At first we considered the case of inviscid flow and got the results that includes
coefficient of drag and lift. Then viscous flow is taken into consideration for the
same airfoil. It includes the study of flow behavior, drag characteristics and
variation of velocity along the airfoil. The simulation is carried out on commercial
CFD code. The outcomes of both the viscous and inviscid flow are compared in
the end.

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Table of Contents
1. Introduction………………………………….………………………………………….….6
1.1. Supersonic Airfoils………………………………………………………………....….6
1.2. Airfoil Terminology…………………………………………………………………….6
2. Simulation…………………………………………………….….…………………………7
2.1.Introduction
2.2. Pre-processing…………………………………………………………………………..7
2.2.1. Geometry of Airfoil……………………………………………………………….7
2.2.2. Meshing…………………………………………………………………………..8
2.2.3. Selection of solver………………………………………………………………..9
2.2.4. Boundary Conditions………………………………………….………………….9
2.2.5. Turbulence Model…………………………………………….…………………10
2.3. Post-processing……………………………………………………….………………..10
3. Results and Discussion…………………………………………………………………11
3.1.Mach Number………………………………………………………....………………11
3.1.1. Inviscid Flow…………………………………………..………..………………11
3.1.2. Viscous Flow……………………………………………………...……………13
3.2. Drag and Lift Coefficients……………………………………………………………14
3.2.1. Inviscid Flow…………………………………………………..………..………14
3.2.2. Viscous Flow…………………………………………………….………..……15
4. Conclusion………………………………………………………………………...………16
References…………………………………………………………………………………….18

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List of Figures
2.1 Airfoil Geometry……………………………………………………………………………...7
2.2 Mesh generated on control surface …………………………………………………………8
2.3 Mesh and Airfoil (zoomed) ………………………………………………………………….9
3.1 Mach number variation over diamond head airfoil ……………………………………….11
3.2 Mach number variation along the chord length …………………………………………..12
3.3 Mach number variation over diamond head airfoil …………………………………….…13
3.4 Mach number variation along the chord length………………………………………...…14
4.1 Total pressure variation along the chord length of the airfoil (Inviscid Flow) ……………16
4.2 Total pressure variation along the chord length of the airfoil (Viscous Flow) ……………17

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List of Acronyms
𝑐" - Coefficient of lift
𝑐$ - Coefficient of drag
𝑐%- skin friction coefficient
Pa – Pascal
m – Meter
M – Mach
K – Kelvin

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Chapter 1
Introduction
1.1 Supersonic Airfoils
An airfoil is basically the shape of a wing that creates an aerodynamic force which helps the
plane to get the required lift. The airfoils designed for the exposure to supersonic flows are
called supersonic airfoils. Supersonic airfoils have sharp edge in the front to avoid formation of
detached bow shocks in front of the airfoil as it moves in the air [1] whereas the subsonic
airfoils are generally rounded in the front part. The sharp edge in the supersonic airfoils makes
it more sensitive to the angle of attack.
1.2 Airfoil Terminology
Some of the terms associated with the airfoils and defined as follows:
• Leading Edge: It is the point at the front of the airfoil which has maximum curvature or
minimum radius. [2]
• Trailing Edge: It is defined similarly as leading edge at the rear end of the airfoil.
• Chord Length: The length of the line connecting the leading edge and trailing edge is
known as chord length of the airfoil.
• Mean Camber Line: It is the line midway between the upper and the lower surfaces.
• Angle of Attack: The angle between the flow direction and chord line is known as the
angle of attack.

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Chapter 2
Simulation
2.1 Introduction
In the essence of the technology, a new tool, computational fluid dynamics (CFD) is very useful
to analyze the fluid system. The differential and integral forms of equations are first discretized
so that the computer can understand them and then various schemes are developed (numerical
methods) to solve the problem and output is made available by the software is various ways
such as graphs and animation.
Computational Fluid Dynamics’ simulation process is divided into two parts- (i) Pre-processing
and (ii) Post-processing. Here we have used ICEMCFD 16.2 Academic, Fluent 16.2 Academic and
CFDPost 16.2.
2.2 Pre-processing
Pre-processing is the phase of simulation in which we define the geometry of as object, control
volume or control surface, mesh, etc.
2.2.1 Geometry of Airfoil
We have considered a double-wedge (diamond-head) airfoil as shown in Fig 3.1. It has a
chord length of 20m and thickness of 2m. Consequently, the thickness to chord ratio is
1:10.
Fig 2.1 Airfoil Geometry
The airfoil geometry is made in ICEMCFD 16.2 Academic.
The figure shown above was made in Catia v5r19.

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2.2.2 Meshing
For the software to carry out calculations by numerical methods, we need to define the
small area which will be considered as elemental area for calculation purpose. This task
is accomplished by creating mesh in the control volume or, as in this case, on control
surface. This process is known as Meshing.
Here 2-D mono-block structured mesh is generated using ICEMCFD 16.2 Academic. In
order to get a good mesh quality and hence better flow visualization, H-grid is used.
Fig 2.2 Mesh generated on control surface
The figure above shows the airfoil on the control surface and the mesh generated. The
orthogonal mesh quality attained here is 0.98.

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Fig 2.3 Mesh and Airfoil (zoomed)
This figure shows clearly that the mesh density is higher near the airfoil for better and
precise results.
2.2.3 Selection of solver
There are two types of solver available in Fluent 16.2 Academic, (i) Pressure Based
Solver and (ii) Density Based Solver.
In our simulation, Density Based Solver is chosen due it’s higher accuracy in calculations
of supersonic flow. The Pressure Based Solver on the other hand gives better results for
incompressible and subsonic flows.
2.2.4 Boundary Conditions
The flow over airfoil has been analyzed at 10km altitude where the ambient pressure is
26500Pa and temperature is 223.5K. [3] The airfoil is exposed to the supersonic flow at
M=3.5. Angle of attack is taken to be zero.
For the first part, the flow is considered to be inviscid and in the second part it is viscous
where viscosity is calculated by the Sutherland Law (Three Coefficient Method).

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2.2.5 Turbulence Model
Turbulence model is important to analyze viscous flow field. In this project, we have
used K-ω SST model for the purpose as it is highly accurate for boundary layer
simulation and high pressure gradient.
2.3 Post-processing
The solution data gathered after iterations are converged and represented in graphical form. In
this project, CFDPost is used for this purpose.

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Chapter 3
Result and Discussion
In this section, results from Fluent and CFDPost like Mach number, Coefficient of pressure and
Total pressure are shown for inviscid and viscous flows.
3.1 Mach number
3.1.1 Inviscid Flow
Fig 3.1 shows the variation of mach number as the supersonic flow (M=3.5) flows over
the diamond head airfoil.
Fig 3.1 Mach number variation over diamond head airfoil
The Fig 3.2 shows the variation of mach number along the chord length of the airfoil.
As the angle of attack is zero, the variation pattern is symmetric along the chord line. The
incidence of flow on the airfoil’s leading edge sees a sharp drop in mach number. This is due a
generation of an attached oblique shock wave at the apex of the airfoil. The flow is still
supersonic.

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From the apex of the airfoil to the point of maximum thickness, mach number remains almost
constant. The flow direction is parallel to the surface of the airfoil now. After the point of
maximum thickness, the flow passes over the rear part of the foil where its thickness starts
decreasing. Due this abrupt change in flow direction, expansion of the flow takes place and as
the flow is supersonic, there is a great acceleration which results in a very high mach number in
the flow over the second half of the airfoil and the flow is again parallel to its surface.
At the trailing edge of the airfoil, the flow at high mach number from the upper and lower
portion of the airfoil encounters each other and there is again an oblique shock wave
generated. This neutralizes the raise in mach number and the mach number again goes back to
3.5.
Fig 3.2 Mach number variation along the chord length
Here the black dots indicate the mach number variation along the lower surface and those in
red shows the same along the upper surface of the airfoil.

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3.1.2 Viscous Flow
Fig 3.3 shows the variation of mach number when a viscous supersonic flow (M=3.5)
flows over a diamond head airfoil.
Fig 3.3 Mach number variation over diamond head airfoil
As seen in the figures 3.3 and 3.4, just like in inviscid flow, there is a sharp drop in mach
number when in flows over the apex of the airfoil and there is an increase in it when it
passes over the second half of the airfoil. But it can be clearly seen that when the
viscous flow passes over the increasing thickness and decreasing thickness of the airfoil,
the mach number is not constant but is decreasing all along the path steadily. Even the
raise in mach number when the thickness starts decreasing is not so high as that in the
inviscid flow.
Another remarkable thing observed is a sleeve along the airfoil with very small mach
number. This is because of the viscosity of the fluid. A boundary layer is generated
where the velocity of the first layer of air that comes in contact of the surface reduces to
zero.

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Fig 3.4 Mach number variation along the chord length
3.2 Drag and Lift Coefficients
3.2.1 Inviscid Flow
As the angle of attack is zero, flow is inviscid and the airfoil is symmetric along chord
length, there will be no lift force generated, so the lift coefficient can be expected to be
zero. Below are the results derived from the simulations carried out, which agrees with
the expectation.
𝑐$ = 0.012076
c. = −4.137x1034
≈ 0

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3.2.2 Viscous Flow
Unlike inviscid flow, in viscous flow, there is an another form of drag called skin friction
drag due to viscous effect.
The lift force will still be zero due to symmetry and zero angle of attack. The results of
the simulations are shown below.
Total Coefficient of Drag 𝑐$ = 0.014211
c. = 1.0147x1035
≈ 0
𝑐% = 0.0022281101

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Chapter 4
Conclusion
In the present work, numerical study was carried out over Diamond Head airfoil in viscous and
inviscid medium at supersonic Mach number of 3.5. From the details of the analysis we come to
the following conclusion:
• Due to viscous flow there is a loss in total pressure. It is clearly seen from the total
pressure variation along the chord length of the airfoil (as shown is Fig 4.1), that in
inviscid flow the total pressure is constant along the chord length after the shock
formation and after the expansion occurs. On the other hand, in viscous fluid (as shown
in Fig 4.2), due to viscous dissipation there is a continuous loss in total pressure along
the surface of the airfoil after the shock is formed.
Fig 4.1 Total pressure variation along the chord length of the airfoil (Inviscid Flow)

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Fig 4.2 Total pressure variation along the chord length of the airfoil (Viscous Flow)
• From the observation of drag coefficient, we can say that drag contribution due to
viscous flow is 𝑐% = 0.0022281101, which is not as significant as wave drag. That is why in
most of 2-D supersonic cases viscous effect is neglected.
• There is also remarkable difference in the Mach number variation along the chord length in
inviscid and viscous flows. The reason for a continuous and steady decrement in the mach
number observed in viscous flow along the airfoil surface is the result of viscous dissipation.

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References
[1] Courant & Friedrichs. Supersonic Flow and Shock Waves. Pages 357:366. Vol I.New York: Inter
science Publishers, inc, 1948
[2] Houghton, E. L.; Carpenter, P.W. (2003). Butterworth Heinmann, ed. Aerodynamics for Engineering
Students (5th ed.). ISBN 0-7506-5111-3. p.18
[3] James E. A. John and Theo G. Keith, Gas Dynamics. Page 281 Third Edition. Pearson Education,
Inc., ISBN 0-13-120668-0