Aerodynamics Task 7

Anantha Hari Arun Pedapudi

June 2024

Contents

1 Elements of Aircraft Performance 3

1.1 Micro Air Vehicles (MAVs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Quest to Aerodynamic Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Measure of Aerodynamic Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Source of Aerodynamic Drag; Drag Reduction . . . . . . . . . . . . . . . . . . . 5
1.2.3 Example of the Wright Flyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.4 Jones Idea of Streamlining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Types of Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Protuberance Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Cooling Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.3 Induced Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.4 Wave Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
L
1.4 Some Innovative Aircraft Configuration for High D
. . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Truss-Braced Wing Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.2 Flying Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Principles of Stability and Control 8

2.1 Aircraft and the Plane of Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Flight Controls: A review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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 2.4 Definition of Stability and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Dynamic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Moments on the Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.7 Absolute Angle of Attack (AOA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.8 Criterion for Longitudinal Static Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.9 Equation for Longitudinal Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.10 Neutral Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.11 Static Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 REFERENCES 12

List of Figures

1 Micro Air Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Laminar Separation Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Truss-Braced Wing Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 B-2 Spirit an example of Flying Wing Configuration . . . . . . . . . . . . . . . . . . . . 7
5 The xyz-axis system of an aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6 Stability and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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 1 Elements of Aircraft Performance

1.1 Micro Air Vehicles (MAVs)

Unmanned Air Vehicles (UAVs) with a wingspan of approximately 15cm or less and weighing under 0.09kg
are classified as Micro Air Vehicles (MAVs). Typically, these vehicles are employed for the detection of
biological agents, chemical compounds, and nuclear substances within limited areas. They also serve a
crucial role in anti-criminal and anti-terrorism surveillance operations. The aerodynamic performance of
MAVs is contingent upon the Reynolds number, which varies significantly between high and low Reynolds
numbers.

Figure 1: Micro Air Vehicle

At lower Reynolds numbers, the flow exhibits laminar characteristics. Even at a zero angle of attack, flow
separation occurs over the airfoil due to the presence of a laminar separation bubble. This laminar sepa-
ration bubble significantly impacts the performance of most model aircraft, particularly at low Reynolds
numbers. The formation of such a separation bubble is attributed to a pronounced adverse pressure gradi-
ent, leading to the detachment of the laminar boundary layer from the curved airfoil surface.
At low Reynolds numbers, flow separation results in diminished lift and increased drag at higher angles of
attack. This phenomenon translates to reduced lift and heightened drag for Micro Air Vehicles (MAVs).

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The peak lift-to-drag ratio for MAVs typically ranges from 4 to 6. In essence, the aerodynamics of MAVs
are predominantly influenced by low Reynolds numbers and low aspect ratios.

Figure 2: Laminar Separation Bubbles

1.2 Quest to Aerodynamic Efficiency

1.2.1 Measure of Aerodynamic Efficiency

The primary method for assessing the efficiency of an aircraft is the lift-to-drag ratio. A higher ratio results
in a greater rate of climb. In the case of a glider, the ratio is high at small glide angles, which allows for
a longer distance to be covered. The lift-to-drag ratio is pivotal in determining an aircraft’s efficiency in
terms of range and endurance. A higher ratio directly correlates to increased range and enhanced efficiency.
3/2
CL CL
For a propeller-driven plane, range ∝ CD
and endurance ∝ CD
. For a jet-propelled aircraft,
1/2
CL Cl
range ∝ CD
and endurance ∝ CD
.

CL L
R ∝ V∞ = V∞
CD D

V∞ = a∞ M∞ where a∞ is the speed of sound and M∞ is flight MACH number.

At a constant standard altitude,
L
R ∝ M∞
D
L L
If D
increased, it would cost less drag. It is also important to note that increasing lift cannot change D
. It
can be increased by reducing drag. Therefore, the quest to increase aerodynamic efficiency is a quest to
reduce drag.

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1.2.2 Source of Aerodynamic Drag; Drag Reduction

The forces exerted on an object moving in a fluid are:

1. Pressure Distribution
2. Skin Friction (Shear Stress) distribution.
The summation of pressure and shear stress yields the net aerodynamic force exerted on an object moving
in a fluid. This aerodynamic force, which aligns with the relative airflow, is commonly referred to as drag.
Drag is attributed to both pressure and skin friction. Consequently, the diverse drag forces acting on an
aircraft arise from variations in pressure and shear stress distributions.

1.2.3 Example of the Wright Flyer

The Wright Flyer stood as the pioneering functional aircraft in aviation history, featuring a biplane config-
uration supported by struts. Notably, the primary source of drag stemmed from pressure drag induced by
flow separation in the vicinity of the struts. It is essential to recognise that pressure drag is synonymous
with form drag. It is widely acknowledged that form drag mitigation can be achieved through aerodynamic
streamlining, which has the potential to significantly reduce or even eliminate flow separation at the surface.

1.2.4 Jones Idea of Streamlining

The power absorbed by the vortices in the airflow around an aircraft often far exceeds the combined
absorption resulting from skin friction drag and induced drag. In real-world aviation, the total drag of
an aircraft surpasses the sum of induced drag and skin-friction drag, indicating suboptimal streamlining.
Theoretical optimal aircraft design, as envisioned by Jones, eliminates pressure drag caused by flow
separation through perfect streamlining.

1.3 Types of Drag

1.3.1 Protuberance Drag

The flow over protuberances typically leads to separation, resulting in pressure drag, commonly referred to
as protuberance drag. This form of drag can be mitigated through the process of streamlining the protuber-
ance or by reducing the frontal area of the projections. Notably, the implementation of retractable landing
gears has been an innovative approach to significantly diminishing drag. A further aspect contributing

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to drag was the round-headed rivet, which protruded from the aircraft’s metal surface and exacerbated
aerodynamic drag. This issue was effectively rectified through the introduction of flush riveting in the
1930s.

1.3.2 Cooling Drag

The term is commonly employed in the context of utilising a segment of airflow to cool the engine in a
piston-driven aircraft. In the case of air-cooled radial engines, the airflow passes directly over the cylinder,
resulting in form drag. To mitigate this drag, a suitably designed cowling enveloping the cylinder can
effectively channel the airflow, facilitating efficient cooling. The advent of NACA cowling culminated in
a substantial reduction of drag for air-cooled piston engines. For liquid-cooled piston engines, the coolant
circulates through the engine and radiator, thereby cooling the airflow, which in turn leads to an increase
in pressure drag.

1.3.3 Induced Drag

Induced drag, a result of the redistribution of surface pressure over finite wings due to wingtip vortices, is
classified as a form of pressure drag. This phenomenon can be mitigated by augmenting the wing’s aspect
ratio, although it is important to consider the structural limitations associated with this approach. Another
effective strategy for reducing induced drag is the incorporation of winglets at the wingtips, yielding
significant improvements in aerodynamic performance.

1.3.4 Wave Drag

The generation of wave drag can be attributed to the formation of a shockwave at supersonic velocities.
This shockwave results in increased pressure drag acting upon the aircraft. In light of this, it is reasonable
to classify wave drag as a form of pressure drag. Established methods to alleviate this drag include the
implementation of swept wings, the utilisation of wings featuring sharp leading edges, and the meticulous
design of the fuselage.
Note: A turbulent skin friction is more considerable than a laminar friction.

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 L
1.4 Some Innovative Aircraft Configuration for High D

1.4.1 Truss-Braced Wing Configuration

The Truss-Braced Wing Configuration enables the utilisation of extremely high aspect ratio wings. This
design consists of a wing that is supported by an external truss anchored at the base of the fuselage
and connected to the underside of a high-mounted wing. By incorporating this design, additional lift
is generated, accompanied by a marginal contribution to drag, all while upholding a high aspect ratio.
Consequently, this configuration yields a lift-to-drag ratio of 26, marking a 25% increase compared to the
conventional configuration.

Figure 3: Truss-Braced Wing Configuration

1.4.2 Flying Wing

The flying wing integrates a centred body, a thick airfoil shape, and a bullet-shaped nose. This configuration
enables the body to contribute to the lift generated. The spanwise lift distribution closely approximates that

Figure 4: B-2 Spirit an example of Flying Wing Configuration

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of an ideal elliptical wing, resulting in an improved lift-to-drag ratio of 30, which exceeds the conventional
design by 50%.

2 Principles of Stability and Control

Center of Gravity: The centre of gravity can be defined as a point where the weight of the aircraft
effectively acts.

2.1 Aircraft and the Plane of Reference

The aircraft utilizes a fixed XYZ axis system. The x-axis aligns with the fuselage, the y-axis with the
wingspan, and the z-axis is oriented vertically downward, perpendicular to the XY-plane. The aircraft’s
translational motion is delineated by the velocity components U, V, and W along the X, Y, and Z axes,
respectively. At free steam velocity, V∞ is the vector sum of U, V and W.
The description of rotational motion includes the angular components P, Q, and R along the X, Y, and Z
axes, correspondingly. The rotational velocity is determined by the components L’, M, and N around the
X, Y, and Z axes, respectively. Roll is the term for rotational motion about the X-axis, characterised by the
rolling moment (L’) and velocity (P). Similarly, pitch represents motion about the Y-axis, denoted by the
pitching moment (M) and velocity (Q), while yaw corresponds to motion about the Z-axis, indicated by
the yawing moments (N) and velocity (R).

Figure 5: The xyz-axis system of an aircraft

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2.2 Flight Controls: A review

The three primary flight controls on an aircraft include:

1. Ailerons: Ailerons are located at the trailing edge of the wings, at the wingtips.
2. Elevators: Elevators are located on the horizontal stabiliser. In some modern aircraft, the horizontal
stabiliser as a whole acts as the elevator.
3. Rudders: Rudders are present on the trailing edge of the vertical stabiliser.
When a control surface is deflected downward, it increases lift on the wing or tail. Conversely, the upward
deflection of the elevator generates negative lift at the tail, a phenomenon known as pitching. Additionally,
the rightward deflection of the rudder induces a leftward aerodynamic force, referred to as yawing.

Figure 6: Stability and Control

2.3 Stability

Rolling, referred to as lateral motion in aviation, is controlled by the aileron and is therefore termed lateral
control. Pitching, also known as longitudinal stability, is controlled by the elevator and is thus referred
to as longitudinal control. Yawing, also known as directional stability, is controlled by the rudder and is
thereby designated as directional control.

2.4 Definition of Stability and Control

There are two fundamental types of stability: static stability and dynamic stability. When a disturbance
causes forces and moments to act upon a body that results in a return to its original equilibrium position,

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it is said to exhibit positive static stability. Conversely, a body is deemed statically unstable, or to have
negative static stability, when the forces and moments cause it to deviate from its equilibrium position.

2.5 Dynamic Stability

A body naturally exhibits dynamic stability, returning to and maintaining equilibrium over time. However,
in certain scenarios, an aircraft may experience dynamic instability, characterised by increasing amplitude
oscillations without reaching equilibrium. Alternatively, an aircraft may undergo constant amplitude
oscillations, displaying dynamically neutral stability. Notably, while an aircraft may not necessarily be
dynamically stable, it is statically stable.
Control: The analysis of the deflection of control surfaces required for an aircraft to execute the necessary
manoeuvres constitutes an integral component of aircraft control.

2.6 Moments on the Aircraft

The analysis of stability and control centres on the examination of moments acting on the aircraft as well
as those acting on the control surfaces.
Aerodynamic Center: A particular point about which the moments are independent of Angle of Attack.
The force and moment system acting on the wings of an aircraft can be accurately defined by the lift and
drag experienced at the aerodynamic centre. Equilibrium in pitch occurs when the moments around the
aircraft’s centre of gravity (CG) are equal to zero. At Mcg = CM,cg = 0, the aircraft is said to be trimmed.

2.7 Absolute Angle of Attack (AOA)

An AOA where the lift is zero is called zero lift AOA. αL = 0.

Zero-lift line: A line through the trailing edge parallel to the relative wind. It is fixed for a particular
airfoil. A conventional airfoil has a slightly negative zero lift AOA. Therefore, the zero-lift line is slightly
above the chord.
Geometrical AOA: The geometrical AOA is the angle between the free stream relative wind and chord
line.
The angle between the zero-lift line and the relative wing is equal to the sum of α and αL=0 . This angle is
defined as absolute AOA.

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2.8 Criterion for Longitudinal Static Stability

The concept of static stability in aviation pertains to an aircraft’s inherent tendency to return to its state
of equilibrium following a disturbance. A downward pitch due to external forces generates a moment
around the centre of gravity (CG), causing the nose to rise and, conversely, creating longitudinal stability.
Conversely, an upward pitch induced by external forces results in a positive moment of the CG, further
elevating the nose, thereby leading to longitudinal static instability. Therefore, the necessary criteria for
longitudinal balance and static stability are:
1. CM,0 must be positive.
∂CM,cg
2. ∂αa
must be negative.

2.9 Equation for Longitudinal Stability

CM,0 is the value of CM,cg when αa = 0, i.e., when the lift is zero.

at ∂ϵ
∴ CM,0 = (CM,cg )L=0 = CM,acwb + VH (1 − )
a ∂α

w.k.t CM,0 must be positive to balance the aircraft.

w.k.t CM,acwb mist be negative for conventional aircrafts.
∴ VH at (it + ϵo ) must be a large positive value to counterbalance the negative CM,ac . Both VH and at are
positive values, and ϵo is a very small value. This implies that the value of it must be positive.
Consider the slope of the moment coefficient curve:

∂CM,cg at ∂ϵ
= a[h − hacwb − VH (1 − )]
∂αa a ∂α

The equation clearly shows that the location of h of the CG and volume ratio VH have a powerful influence
on determining longitudinal static stability.

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2.10 Neutral Point

In the given scenario, the position of the centre of gravity is permitted to vary while all other factors are
kept constant. Then, there will exist a point such that:

∂CM,cg
=0
∂αa

The value of h at which the above condition holds good is called the neutral point. It is denoted by hn .
When h = hn , the slope of the moment coefficient curve is zero. The location of the neutral point can be
obtained by:
at ∂ϵ
hn = hacwb − VH (1 − )
a ∂α

For a given aircraft, the neutral point is fixed. The neutral point is independent of the actual location of h of
CG. For longitudinal stability, the position of CG must always be ahead of the neutral point. When h = hn ,
CM,cg is independent of AOA. Therefore, we can conclude that the neutral point is the aerodynamic centre
for the whole aircraft.

2.11 Static Margin

The distance between the neutral point and the location of CG is termed a static margin. It can be calculated
using the formula:
∂CM,cg
= −a(h − hn )
∂αa

The static margin serves as a direct measure of longitudinal static stability. A positive static margin is
essential for achieving static stability, with greater margins correlating to heightened aircraft stability.

3 REFERENCES

1. https://www.mh-aerotools.de/airfoils/bubbles.html
2. https://www.mh-aerotools.de/airfoils/images/turbula2.gif
3. https://media.springernature.com/lw685/springer-static/image/prt%3A978-90-481-9751-4
2F10/MediaObjects/978-90-481-9751-4_10_Part_Fig6-256_HTML.gif
4. https://journals.sagepub.com/cms/10.1177/0954410020923060/asset/images/large/

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10.1177_0954410020923060-fig12.jpeg
5. https://www.researchgate.net/figure/The-coordinate-system-of-an-aircraft-8_fig1_
322311856
6. Anderson, J. D., Hunter, L. P. (2021). Introduction to flight. http://ci.nii.ac.jp/ncid/
BA71376688

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