Mechanical Engineering Experiment (II) teaching manual
Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination
M. Z. Jiang*
* Lecturer rank specialist of mechanical department, National Taiwan university.
Introduction
To introduce what is aerodynamics, let’s start with the fluid mechanics which you may have learned
during your sophomore or junior year. Fluid mechanics aims to study either the static or dynamic fluid
behaviors, the “fluid” here represents not only a pure liquid but also a pure gas or a hybrid two-phase flow
(liquid-gas, liquid-solid, or solid-gas). One of the branches of fluid mechanics is the aerodynamics, which focus
on how the air affects a dynamic body behavior; in other words, aerodynamics studies a restricted area of fluid
mechanics which involves only interactions between the air and a dynamic body. The most important difference
between air and a water-based fluid is the compressibility1. Compressibility is defined by the relative volume
change of fluid under a specific pressure change. Fortunately, when the Mach number (the ratio of body moving
speed and local sound speed) of a moving body is below 0.3, which satisfies our experimental environment, we
can treat the air is nearly incompressible (compressibility < 5%).
For an aircraft, four forces are involved to determine whether the aircraft can takeoff: the thrust by its
propeller, the gravity by its weight, the lift by its airfoil, and at last, the drag, which represents a combination
effect of all thrust-reduced parameters. In the first unit, we will introduce what is the drag and how to estimate it
by different methods.
Part 1-1. Definition and classification of drag
All fluid has viscosity1, which can be imagined as the different feelings when your finger touches the
glue or a cup of water. When a dynamic body is moving in the air, the viscous force happens immediately
among air molecules at the air-to-body interface, which is defined as the parasite drag. Three kinds of drag
forces belong to the parasite drag, namely the skin friction drag, the form drag, and the interference drag, while
the former two belong to the profile drag since they result from the aircraft frame design.
Beside the parasite drag caused mainly by the viscous force, the other type of drag is known as the lift
induced drag. The lift induced drag is produced due to the down wash, which locates behind the aircraft when
the airfoil tip “slice” the air and produce a tip vortex. The pressure drop among the tip vortex then becomes a
1
These two parameters will be introduced in the latter unit.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -1
drag. Fig. 1.1 reveals the classification of drag and corresponding dominant parameters. To begin our
introduction on these drags, we can start with the definition of laminar flow and turbulent flow, the essential
differences among these two flow types are responsible for the generation of drag.
Figure 1.1. Classification of drag.
A dimensionless number called the Reynolds number is often applied to determine the laminar flow and
the turbulent flow. The Reynolds number quantify the ratio of inertial force and the viscous force for a certain
flow. The former force is proportional to the flow speed and the latter one is proportional to the liquid viscosity.
When viscous force is dominant, the flow will be steady along its streamline and the energy transferring across
different streamlines happens barely. When the inertial force is dominant, on the other hand, the flow will start
to generate unsteady vortex which involves very complex energy consumption and therefore creates a
dramatically pressure drop. The Reynolds number can be defined as:
𝑅𝑒 =
𝜌𝑢𝑑
𝜇
(1).
In formula (1), 𝜌 denotes the liquid density, u is the flowing speed and 𝜇 is the liquid viscosity. d represents the
characteristic length and should be determined depends on the flowing conditions. For example, d equals to the
sphere diameter when a spherical ball is falling in the liquid. In aerodynamics, we classify the flow belongs to a
laminar flow when 𝑅𝑒 < 104 , and belongs to a pure turbulent flow when 𝑅𝑒 > 106 . The transition stage, which
involves both behaviors, falls in the range 104 < 𝑅𝑒 < 106 . Reader should be noted that this judgment is not a
universal criterion but only suitable for aerodynamics since the Reynolds number alters dramatically in different
flow fields.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -2
Part 1-2. Interference drag
The interference drag is generated by the mixing of streamlines when air passes the interfaces or
connections of different components, such as the airfoil-to-body root or the tail roots. For a certain surface
profile such as the airfoil and the cabin, the passing air flow will respond to the geometrical shape variation and
produces a gradual pressure change along the surface. At the interface of different surfaces, however, two air
flows with different pressures will merge together. To fulfill the continuity, the different air flows must
experience a sharp property variation to reach the same matching state. This sharp variation has a high
possibility to create a turbulent flow, and therefore produces a local low-pressure zone to increase the overall
drag. The best way to reduce the interference drag is to design the aircraft frame properly or install the fairings
and fillets to ensure the interface of different components become much smoother. Fig. 1.2 reveals the
illustration of aircraft fairing and fillet.
Figure 1.2. Left: Fairing of Boeing 747 [5]. Right: Fillet at airfoil root [6].
Part 1-3. Skin drag and form drag
The next two drag belong to the profile drag, namely skin drag and form drag, while the latter one is also
known as the pressure drag. Skin drag is generated by the viscous force between the air and the surface
roughness of aircraft frame. A polishing procedure can reduce the surface roughness effectively to reduce the
skin drag. Skin drag is usually combined with the parasite drag and is very seldom evaluated independently.
Due to the fact that air viscosity varies with different atmosphere conditions such as humidity and temperature,
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -3
besides, it is also difficult to install an instrument on airfoil surface without interfering the passing flow field.
Both reasons lead to the result that skin drag is very hard to determine.
The form drag, or the pressure drag, is produced due to the pressure difference between the front side
and the rare side of the moving body. The form drag can then be estimated by the pressure difference multiply
by the normal-to-flow area. There exists a theoretical estimation for the pressure difference along the streamline,
which is also probably the most famous fluid mechanics theory: the Bernoulli equation.
Unfortunately, due to its simple equation and intuitive physical concept, numerous misuses of this
equation fulfill the internet regardless of its rigorous derivation assumptions. Here, we will derive the Bernoulli
equation step by step again and clarify its limitations on aerodynamics.
Consider a certain fluid with density 𝜌 passes through a specific streamline as shown in Fig. 1.3. With
the inlet area 𝐴1 and the exit area 𝐴2 ; the inlet fluid velocity 𝑢1 and the exit fluid velocity 𝑢2 ; the streamline
possesses a certain height ℎ1 and ℎ2 away from the relative horizontal at the inlet and exit, respectively.
Figure 1.3. A certain fluid passes through a specific streamline.
Now, the most important assumption of the Bernoulli equation is energy conservation along the streamline. This
assumption establishes when the fluid is frictionless and has no viscosity, which implies no energy consumption
during the traveling path. The pressure exerts on the inlet and exit area can be estimated by the local fluid
pressure, therefore, the infinite small work exerts on a sliced fluid with an infinite small thickness 𝑑𝑥1 can be
evaluated by the reaction force exerting on the inlet area so that:
𝑑𝑤𝑖𝑛𝑙𝑒𝑡 = 𝑃1 𝐴1 𝑑𝑥1
,
with the same derivation, the work exerts on the same sliced fluid at exit will be:
𝑑𝑤𝑒𝑥𝑖𝑡 = 𝑃2 𝐴2 𝑑𝑥2
.
For an incompressible fluid, the total volume (dV=Adx) remains the same, the total work exerts on the fluid is:
𝑑𝑤 = 𝑃1 𝐴1 𝑑𝑥1 − 𝑃2 𝐴2 𝑑𝑥2 = (𝑃1 − 𝑃2 )𝑑𝑉
(2).
With no heat flux (dQ = 0), we can infer the work will be the summation of kinetic energy K and the potential
energy U of fluid, therefore:
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -4
𝑑𝑤 + 𝑑𝑄 = 𝑑𝐾 + 𝑑𝑈
,
with the mass of fluid can be described by 𝜌𝑑𝑉, we can obtain:
1
(𝑃1 − 𝑃2 )𝑑𝑉 = 𝜌𝑑𝑉(𝑢2 2 − 𝑢1 2 ) + 𝜌𝑑𝑉𝑔(ℎ2 − ℎ1 )
2
(3).
Equation (3) is the Bernoulli equation for an infinite small volume fluid travels along its streamline. Since we
treat the fluid as incompressible, equation (3) can then be simplified to your familiar Bernoulli equation so that:
∆𝑃 = 0.5𝜌∆(𝑢2 ) + 𝜌𝑔∆ℎ.
One of the misuse reasons of Bernoulli equation is that people just misunderstand its causality. There
must always exists a pressure difference first and then the velocity difference will be produced. The pressure
difference requires an initiating energy no matter it belongs to either mechanical work or heat flux. To conclude,
Bernoulli equation describes: “how the fluid reacts to a given pressure, which comes from a given work or heat
flux”.
Now, back to the form drag. When an aircraft is traveling, its airfoil keeps squeezing the front air
molecules and therefore produces a relative high pressure, this pressure difference then drives the air molecules
move to the relative low-pressure area, which locates at the rare of airfoil. To fulfill the continuity, the
molecules flowing velocity must equals to the aircraft speed, and then we can estimate the streamwise pressure
difference by converting the aircraft speed to the driving pressure difference. This estimation, however, will not
equal to the real pressure difference due to the following reasons: (1) air has viscosity, which leads to additional
energy consumption, and (2) even for a faired airfoil, vortex still happens when flowing speed exceeds a certain
magnitude, vortex involves a complex energy dissipation mechanism and the energy conservation assumption is
no longer valid. Therefore, the Bernoulli estimation is less accurate as the rare vortex increasing, which implies
an increasing form drag.
Part 1-4. Lift induced drag
As aforementioned, the lift induced drag is generated by the down wash behind the aircraft. The down
wash is produced by the airfoil tip vortex and therefore possesses a relative low pressure. The pressure
difference between the down wash region and the air craft nose results in a drag to reduce the thrust. It is worth
noting that the lift induced drag will decrease as the air craft speed increasing. In general, lift induced drag is
important only when the air craft is taking off or landing.
At last, readers may notice that an efficiency estimation of individual drag is still lacking because of the
complex interaction between the aircraft frame, the local atmosphere conditions, and the involved turbulent
flow. To quantify the overall drag for an aircraft, engineers usually adopt the total drag (D) for a certain target
to represent the overall effect from all the aforementioned drags and define the total drag coefficient (𝐶𝑑 ) as
below:
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -5
𝐷
𝐶𝑑 = 0.5𝜌𝑢2 𝐴
(4).
The coefficient 𝜌 and 𝑢 denote the fluid density and the relative flowing speed, respectively. While A represents
the projection area2. For example, in this week experiment, our airfoil projection area represents the area you
observe from the top view of airfoil when its chord line is horizontal, and will not change with different angle of
attack (AOA3). Fig. 1.4 reveals a typical drag characteristic curve under different flying speed and a certain
AOA.
Figure 1.4. Illustration diagram (not real scale) for a typical drag characteristic curve under a specific AOA.
Part 1-5. Theoretical estimation of total drag
In the last section, since we concern about the total drag of our target instead of individual drag. There
exist two methods to obtain the total drag, one is experimental and the other one is theoretical. For the
experimental measurement, to simulate the real aerodynamic behavior, a wind tunnel is usually applied. Wind
tunnel is a certain structure to apply aerodynamic test with different scales, from an experimental (<3 m) scale
to an industrial (~20 m) or a real space shuttle scale. A wind tunnel requires different regulations depend on its
application and is defined by different institutions, some common components and corresponding functions of a
typical wind tunnel is shown in Fig. 1.5. The “windbreak ring” represents the rectification area which is applied
to uniformize the inlet flow field, the inlet nozzle is adopted to speed up the inlet flow, the measuring section
2
This projection area is estimated by setting your airfoil chord line to align with the horizontal. And does not alter with the angle of
attack (AOA). Check Fig. 1.1 of unit 1 if you forget the chord line definition.
3
This parameter will be introduced in the next unit.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -6
contains the test target (the airfoil in the image) and all necessary instruments. Behind the measuring section is
the blower (or propeller) and diffuser. Blower can drive the desire flow field by converting the rotational work
into pressure difference, the diffuser can reduce the exit dynamic pressure to avoid a sharp pressure gradient to
atmosphere. With proper instruments, the overall drag of applied airfoil under different flowing speed can be
measured through a wind tunnel.
Figure 1.5. Illustration diagram for an experimental wind tunnel and real test section image [7].
A theoretical estimation called control volume analysis can also be applied to estimate the overall drag
of the target. The fundamental concept of control volume analysis comes from the Reynolds transport theorem,
which describes how the control mass responds to the shift of control volume. Let’s start with an imaginary
black square (control volume (CV)) at an initial time t as shown in Fig. 1.6. At the instant t, assume we are
observing a segment of a continuum flow (control mass (CM)) within this control volume. At a latter instant 𝑡 +
∆𝑡, our observing region shifts to the blue dashed square, which represents a new control volume. For the new
control volume, we can infer a new control mass segment (the green block) enters its boundary while an
original control mass segment (the red block) leaves. Now assume B is a certain extensive property4 and can be
expressed by multiplying the intensive property3 b with the control mass m so that: B = mb, or we can express it
with a control volume integral so that: 𝐵 = ∫𝐶𝑉 𝑏𝜌𝑑𝑉 , in which V denotes the occupied volume of control
volume and 𝜌 is the density of continuum flow.
4
Extensive property can be influenced by mass while the intensive property is independent to mass. For example, the momentum is an
extensive property and can be described by multiplying the velocity (intensive property) with the body mass.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -7
Figure 1.6. Illustration diagram for Reynolds transport theorem.
For the new control volume, we can infer that:
𝐵𝐶𝑀 (𝑡+∆𝑡) = 𝐵𝐶𝑉 (𝑡+∆𝑡) − 𝐵𝐶𝑀1 + 𝐵𝐶𝑀2
(5).
The time rate change of the extensive property of the control mass can be described as:
∆𝐵𝐶𝑀
∆𝑡
=
𝐵𝐶𝑀 (𝑡+∆𝑡) −𝐵𝐶𝑀 (𝑡)
,
∆𝑡
substitute equation (5) for 𝐵𝐶𝑀 (𝑡+∆𝑡) we can obtain:
∆𝐵𝐶𝑀
∆𝑡
=
𝐵𝐶𝑉 (𝑡+∆𝑡) −𝐵𝐶𝑀1 +𝐵𝐶𝑀2 −𝐵𝐶𝑀 (𝑡)
(6).
∆𝑡
Since control mass at the instant t coincides with the control volume at the same instant (𝐵𝐶𝑀 (𝑡) = 𝐵𝐶𝑉 (𝑡) ),
therefore, equation (6) can be rewrote as:
∆𝐵𝐶𝑀
∆𝑡
=
𝐵𝐶𝑉 (𝑡+∆𝑡) −𝐵𝐶𝑀1 +𝐵𝐶𝑀2 −𝐵𝐶𝑉 (𝑡)
∆𝑡
=
∆𝐵𝐶𝑉
∆𝑡
−
∆𝐵𝐶𝑀1
∆𝑡
+
∆𝐵𝐶𝑀2
∆𝑡
(7).
When the time interval approaches zero (∆𝑡 → 0), the left-hand side of equation (7) becomes the material
derivative, which is applied to describe how the time rate of change of a physical quantity responds to a
macroscopic velocity field and is usually denoted by D/Dt. Therefore, we can obtain:
𝐷𝐵𝐶𝑀
𝐷𝑡
𝑑
= 𝑑𝑡 ∫𝐶𝑉 𝑏𝜌𝑑𝑉 − 𝐵̇𝐶𝑀1 + 𝐵̇𝐶𝑀2
(8).
Recall from the definition of extensive property that B = mb. Assume the flow is steady so that 𝑏̇ = 0, the time
rate change of the extensive property for both leaving and entering control mass can then be treated as 𝐵̇ = 𝑚̇𝑏,
in which the 𝑚̇ represents the mass flow rate and can be estimated by multiplying the perpendicular-to-control
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -8
surface velocity (𝑢
⃑ 𝑖𝑛 ∙ 𝑛⃑𝐶𝑆 )5 to the control surface area (A) and fluid density (𝜌) so that 𝑚̇ = 𝜌(𝑢
⃑ 𝑖𝑛 ∙ 𝑛⃑𝐶𝑆 )𝐴, as
shown in Fig. 1.7. At last, the Reynolds transport theorem can be described as:
𝐷𝐵𝐶𝑀
𝐷𝑡
𝑑
= 𝑑𝑡 ∫𝐶𝑉 𝑏𝜌𝑑𝑉 + ∫𝐶𝑆 𝑏(𝑢
⃑ ∙ 𝑛⃑)𝜌𝑑𝐴
(9).
Figure 1.7. Illustration diagram for the entering velocity component.
To conclude the physical meaning of the Reynolds transport theorem, it states that: “the material derivative for
a certain extensive property (𝐷𝐵𝐶𝑀 ⁄𝐷𝑡) equals to the summation of time rate change of intensive property
𝑑
within a control volume (𝑑𝑡 ∫𝐶𝑉 𝑏𝜌𝑑𝑉 ) and the total flux of intensive property over all control surfaces (∫𝐶𝑆 𝑏(𝑢
⃑ ∙
𝑛⃑)𝜌𝑑𝐴)”
Next, we can relate the Reynolds transport theorem with the drag and reveals a simple example, which
belongs to our experimental flow field. Now we treat the extensive property as the momentum so that B = mu,
and for a fixed control volume, the material derivative is:
𝐷(𝑚𝑢)
𝐷𝑡
=
𝐷𝑚
𝐷𝑡
𝐷𝑢
6
𝑢 + 𝑚 𝐷𝑡
,
since we fix the control volume, the mass conservation leads to the time rate change of mass is zero, the
material derivative of momentum becomes:
𝐷(𝑚𝑢)
𝐷𝑡
=
𝐷𝑚
𝐷𝑡
𝐷𝑢
𝐷𝑢
𝑢 + 𝑚 𝐷𝑡 = 𝑚 𝐷𝑡 = 𝑡𝑜𝑡𝑎𝑙 𝐹𝑜𝑟𝑐𝑒
5
The symbol “ ∙ ” here represents the vector dot product.
6
The material derivative still satisfies with the Leibniz rule, you can try to prove it if you are interested.
(10)7.
7
The term Du/Dt can be separated into the body force and the surface force, which denotes the total external force exerts on the
control volume. Although the concepts are similar, do not confuse with the traditional velocity derivative du/dt.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -9
As for the momentum flux over the control surface, if the flow field is nearly parallel to the relative
horizontal, we can infer that the flow field is perpendicular to both the right and the left surface of control
volume and therefore we can rewrite the term ∫𝐶𝑆 𝑏(𝑢
⃑ ∙ 𝑛⃑)𝜌𝑑𝐴 of formula (9):
⃑ ∙ 𝑛⃑)𝜌𝑑𝐴 = ∫𝑙𝑒𝑓𝑡 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑢𝑜𝑢𝑡 2 𝜌𝑑𝐴 − ∫𝑟𝑖𝑔ℎ𝑡 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑢𝑖𝑛 2 𝜌𝑑𝐴
∫𝐶𝑆 𝑢(𝑢
(11).
Note that we assume the inward flow field locates at the right surface while the outward flow field at the left.
The inward flow direction is inverse to the surface normal vector and therefore produces a negative sign.
Readers should not be confused with the positive sign for CM2 (the entering mass flux) of the Reynolds
transport theorem, since in Reynolds transport theorem we shift the control volume to the right, which equals to
a leftward flow pass through a fixed control volume. Combine equation (9), (10) with (11) under a steady
𝑑
condition (𝑑𝑡 ∫𝐶𝑉 𝑢𝜌𝑑𝑉 = 0), we can obtain:
𝑡𝑜𝑡𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 = ∫𝑙𝑒𝑓𝑡 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑢𝑜𝑢𝑡 2 𝜌𝑑𝐴 − ∫𝑟𝑖𝑔ℎ𝑡 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑢𝑖𝑛 2 𝜌𝑑𝐴
(12).
This result states that for a 2D control volume within a horizontal flow field, the total force exerts on the control
volume can be estimated by the difference between the entering and the leaving momentum flux. Therefore,
with a properly chose of control volume for a body within a flow field, we can then estimate its overall drag by
measure the upstream and downstream velocity field distribution.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -10
Part 2-1. Definition of lift
During the last week, we defined and classified the drag and introduced two methods to obtain the
overall drag. Unlike the drag which exists among all components, the lift, on the other hand, is significant only
for an airfoil.
The principle of lift is very simple, when air pass the airfoil, the upper surface will produce a relative
low-pressure region while the lower surface will produce a higher one. By multiplying the pressure difference
between different regions with the airfoil projection area we can obtain the lift force, an illustration diagram is
shown in Fig. 2.1. Again, readers should be aware of the causality of Bernoulli equation, which is the most
common explanation about how the lift is produced. Though the velocity at different surfaces indeed can be
estimated follow the Bernoulli equation, unfortunately, this is a tragedy when people want to explain why the
lower surface velocity should be slower than the upper surface. For example, a well-known “longer path theory”
states that because the upper surface usually possesses a longer profile, to ensure the air molecules merge
together at the airfoil tail, the air molecules at the upper surface must travels faster than those at the lower
surface.
All these kind explanations just misunderstand the causality of Bernoulli equation in which they believe
“faster velocity is responsible for lower pressure”. The true reason is that when air molecules are squeezed by
the airfoil, the different surface (the upper surface vs. the lower one) contours result in different pressure
distributions. And the overall pressure magnitude exerts on the upper surface is lower than that of the lower
surface. An evidence can prove this explanation, there always exists a zero-lift line of an airfoil by altering its
relative angle between the chord line 8 and the free flow direction. The zero-lift line denotes the pressure
difference between the upper and the lower surface becomes zero if the airfoil moves along this line.
Figure 2.1. Illustration diagram for how the lift is generated.
8
Check Fig. 1.1 of unit 1 if you forget the chord line definition.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -11
Part 2-2. Angle of attack
As we mentioned in the last section, airfoil has a zero-lift line which denotes the airfoil cannot provide
any lift force if the relative-to-flow moving direction is parallel to this line. Define the angle of zero-lift line
then becomes an important issue and therefore many extended characteristic angles are also defined. Consider
the airfoil shown in Fig. 2.2. The angle of attack (AOA, 𝛼) is defined as the angle between the airfoil chord line
and the relative flow direction. If we define the angle between the zero-lift line and the chord line as 𝛼𝐿 , then
the absolute angle of attack 𝛼𝑎 is equal to 𝛼 + 𝛼𝐿 .
Figure 2.2. Definition of angle of attack (AOA).
Next, recall from Fig. 1.1 of the previous unit, there always exists a down wash behind an aircraft. The down
wash is generated by the airfoil tip vortex, tangential velocity of the tip vortex is perpendicular to the chord line
and therefore produces a local relative flow vector as shown in Fig. 2.3. This local relative flow will separate
the original AOA into two different angles. We define the induced angle of attack 𝛼𝑖 as the angle between the
local relative vector and the original relative flow direction. The effective angle of attack 𝛼𝑒 is then defined as
𝛼𝑒 = 𝛼 − 𝛼𝑖 .
Figure 2.3. Definition of effective angle of attack.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -12
The down wash only exists in a 3D airfoil, therefore, when we design the airfoil shape with a 2D diagram, we
should notice that the real AOA will becomes the effective AOA and therefore the real lift is always smaller
than the simulated lift of a 2D airfoil.
As we defined the drag coefficient in the previous section, a similar lift coefficient can also be defined
as below with respect to the overall lift L:
𝐿
𝐶𝐿 = 0.5𝜌𝑢2 𝐴
(13).
All involved parameters in equation (13) have the same definition in equation (4). The reason why AOA is
important is due to its influence on drag and lift. Unfortunately, there still lacks of an overall theory to estimate
the lift, therefore, engineers usually design a measurement module which can measure simultaneously the drag
and lift of an airfoil in the wind tunnel. A typical characteristic D-L curve for an airfoil with respect to different
AOAs under a constant flowing speed is shown in Fig. 2.4. For an airfoil, two characteristic parameters belong
to the main concern. The first one is the lift to drag ratio L/D, the maximum magnitude of L/D implies the best
performance of the airfoil since at this AOA the airfoil possesses a maximum lift with a minimum drag. The
second one is the Stall state, the stall state denotes the maximum lift coefficient 𝐶𝐿,𝑚𝑎𝑥 of the airfoil, and when
AOA exceeds the 𝐶𝐿,𝑚𝑎𝑥 state, the lift will vanish quickly and may cause a serious incident. During this week,
we will measure the characteristic curve of an airfoil under a certain flow speed in an experimental wind tunnel.
Figure 2.4. Characteristic curve for an airfoil with respect to different AOAs [8].
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -13
Reference
* The following reference list contains some free internet resources which may help you to establish a basic or
advanced understanding for the theories revealed in this manual.
[1] Downwash Effects on Lift
[2] Harish, Vejju & Akkinepally, Bhargav. (2016). “Drag Reduction using Suction Slit to the Main Wing.”
10.14741/Ijcet/22774106/6.4.2016.3.
[3] How Interference Drag Affects Your Plane's Performance
[4] Interference Drag in a Transport Airplane Illustration
[5] Aircraft fairing CC BY-SA 4.0
[6] Horizontal stabiliser fillets
[7] Laminar Wind Tunnel | Institute of Aerodynamics and Gas Dynamics | University of Stuttgart
[8] U.S. Dept. of Transportation, FAA, (2016) “Pilot's Handbook of Aeronautical Knowledge, FAA-H-808325B”, Figure 5-5.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -14
Laboratory Manual
Unit 3: Introduction
During the following two consecutive experiments, we will apply an experimental wind tunnel with
each component is shown in Fig. I. In the first week experiment, we install a rectangular obstacle in the wind
tunnel through the test window. After measuring the flow velocity distribution on different heights for both the
upstream and the downstream of the obstacle, we can estimate the overall drag exerting on this obstacle via
control volume analysis.
Figure I. Experimental wind tunnel. The wind tunnel contains a rectification honeycomb to smooth the inlet air,
an inlet nozzle to accelerate the flow, a main test section to apply desired tests, a diffuser to reduce the
discharge pressure gradient, and a driving fan with control panel.
Experimental outlines and procedures
Experimental outlines for unit 3 are concluded in the following list. You can check the detailed procedures,
instrument operation, and important notes in the following contents for each outline.
■ Dimension determination for the obstacle and wind tunnel – Page 16.
■ Put obstacle in the wind tunnel and set the driving fan duty– Page 16~17.
■ Measure the flow velocity distribution at both measurement slots– Page 17~18.
■ Repeat step 3 with different fan duties– Page 18.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -15
Step1: Dimension determination for the obstacle and wind tunnel.
1.1: Measure the obstacle dimension and record them. (Length-width-height)
1.2: Measure the interval of measurement slots as shown in Fig. II.
Figure II. Definition of measurement slots interval.
1.3: Identify the wind tunnel midline for the later alignment.
Step 2: Put obstacle in the wind tunnel and set the driving fan duty.
2.1: Remove the test window, set obstacle in the wind tunnel with its midline align to that of wind tunnel
as shown in Fig. III. You will obtain a better calculation result if your obstacle is closer to the
downstream slot.
Figure III. Installation diagram for obstacle and supporting structure.
Note: do not forget to set the supporting structure behind the obstacle.
2.2: Close the test window and fix it with four corner screws.
2.3: Set fan duty to 15% and activate it, the control panel manipulation is shown in Fig. IV.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -16
Figure IV. Operation guidelines for the wind tunnel control panel.
Note: The button RUN is only required for activation, it is not essential for a dynamic adjustment.
Step 3: Measure the flow velocity distribution at both measurement slots.
3.1: Insert your hot-wire anemometer through the downstream measurement slot until its detection head
reaches the midline of obstacle. The hot-wire anemometer is a simple and convenient device which can
measure the heat loss from its detection head and then converts it to an equivalent flow speed.
3.2: Incline your anemometer slightly to reach the height of 4 cm from wind tunnel bottom. Be aware of
the normal direction of detection head should be perpendicular to the flow field. ** Follow the lecturer
demonstration during the lab.
3.3 Follow the anemometer operation guidelines as shown in Fig. V to measure a ten seconds average
flow speed at this location.
Note: When you operate the anemometer, keep your eye on it to prevent from any collision.
3.4 Repeat step 3.3 but increase your measurement height with a 4 cm interval until it reaches at least
10cm above your obstacle.
3.5 Repeat step 3.1~3.4 but alter your anemometer to the upstream measurement slot.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -17
Figure V. Operation guidelines for the hot wire anemometer.
Step 4: Repeat step 3 with different duties. (Duty = 17% and 20%).
4.1: After finish each duty measurement, remember to record the room temperature.
4.2: After finish step 4, remember to recover the protection hat of anemometer.
End of unit 3 experiment.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -18
Laboratory Manual
Unit 4: Introduction
During the unit 4 experiment, we will apply the same experimental wind tunnel during unit 3. An airfoil
(NACA 23012) is installed to a force measurement module as shown in Fig. I, which contains two normal load
cells to detect the lift force and one tension load cell to detect the overall drag force. After measuring the lift and
drag for the airfoil under different angles of attack (AOA), we can obtain an experimental performance
characteristic curve for this airfoil under a specific wind speed.
Figure I. Force measurement module. The module contains two normal load cells (red square), one tension load
cell (blue square), a supporting cylinder is screwed to the tension cell with its top is fixed to the connection
plate. A clipped rotation plate can be applied to support and adjust AOA of your airfoil, a tighten screw can
adjust the clip strength to maintain the rotation plate.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -19
Experimental outlines and procedures
Experimental outlines for unit 4 are concluded in the following list. You can check the detailed procedures,
instrument operation, and important notes in the following contents for each outline.
■ Install your blade on the sensing module (optional) – Page 20.
■ Adjust the AOA to 0°, return your measurement module to zero– Page 20.
■ Remove your angle meter, close the test window, set the fan duty, and measure the wind speed– Page 21.
■ Record the sensing force from LabVIEW program– Page 22.
■ Cease your wind tunnel fan and repeat step 2~4 for a new AOA– Page 22..
■ Repeat step 5 until you observe the lift force experiences a dramatically descent (stall) – Page 22..
Step1: Install your blade on the sensing module (optional).
1.1: If you already bring your own airfoil, follow the lecturer instruction to install it on the module.
Step2: Adjust the AOA to 0°, return your measurement module to zero.
2.1: A digital angel meter shown in Fig. II (a) is provided. Press its red power button to switch it on,
then align its top side with the bottom of rotation plate. Adjust your rotation plate until the AOA is
smaller than 0.5°.
2.2: We will monitor the sensing force from the module by LabVIEW program. The program interface is
shown in Fig. II (b): for the tension cell and (c): for the normal cells. Press the run button
and an
initial loading due to the contained component weight will reveal. Copy this initial magnitude and paste
it to the corresponding zero-calibration slot.
Figure II. (a) the digital angel meter, and the interface of LabVIEW program: (b) drag, (c) lift.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -20
Step3: Remove your angle meter, close the test window, set the fan duty, and measure the wind speed.
3.1: Remove your angle meter, and close the test window by screwing four door corners.
3.2: Set fan duty to 15% and activate it, the control panel manipulation is shown in Fig. III.
Figure III. Operation guidelines for the wind tunnel control panel.
Note: The button RUN is only required for activation, it is not essential for a dynamic adjustment.
3.3: Follow the anemometer operation guidelines as shown in Fig. IV to measure a ten seconds average
flow speed in front of the airfoil midline.
Note: When you operate the anemometer, keep your eye on it to prevent from any collision.
Figure IV. Operation guidelines for the hot wire anemometer.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -21
Step4: Record the sensing force from LabVIEW program.
4.1: Record the sensing force from the “gage reading” slot of both LabVIEW programs.
Step5: Cease your wind tunnel fan and repeat step 2~4 for a new AOA.
5.1: Press the stop button on the fan control panel to cease your wind tunnel fan.
5.2: Repeat step 2~4 for a new AOA. The interval of AOA is not assigned, just ensure your AOA is
increasing. But an over 1° interval is not recommended.
Note: The tighten screw should be screwed slightly by your hand.
Step6: Repeat step 5 until you observe the lift force experiences a dramatically descent (stall).
6.1: Once the AOA exceeds a certain value, you will observe the lift force drops dramatically. This
AOA is known as the “stall” point. You can finish your experiment now since we should avoid to
exceed the stall point in most conditions.
End of unit 4 experiment.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -22
Preview Questions (Former week unit):
Q1: (Introduction, page 1). Under what conditions should we consider the compressibility of air? ** This is the
so-called “compressible flow”.
Q2: (Part 1-1, page 1~2). What are the differences between the parasite drag and the lift induced drag?
Q3: (Part 1-1, page 2). What is the definition of Reynolds number? Next, assume the air is passing a cylindrical
obstacle, if we are interested in the form drag, how should we define the characteristic length for the current
Reynolds number? Describe your explanation. ** This question aims to help you to understand the importance
of characteristic length, try your best to figure out the potential explanations.
Q4: (Part 1-2, page 3). Consider the following structure design candidates, assume they belong to the same part
of aircraft structure. Which one provides a smaller interference drag? Describe your explanation.
Q5: (Part 1-3, page 3~5). Describe briefly the reason(s) for generating the form drag (or a pressure drag).
Q6: (Part 1-4, page 5~6). Describe briefly the reason(s) for generating the lift induced drag.
Q7: (Part 1-4, page 6). How do we define the total drag coefficient? Describe clearly all the involved
parameter definitions.
Q8: (Part 1-5, page 7~8). At the beginning of the derivation of control volume analysis, please tell me what is
the difference between extensive property and intensive property?
Q9: (Part 1-5, page 9~10). Try your best to get through the control volume analysis derivation. Please describe
clearly how to estimate the total external force exerting on a control volume for a 2D, steady, and horizontal
flow?
End of the former week preview Questions
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -23
Preview Questions (Latter week unit):
Q1: (Part 2-1, page 11). Describe how the lift is generated when air passing an airfoil?
Q2: (Part 2-1, page 11). For the Bernoulli equation, which parameter is dominant? Pressure difference or
velocity difference?
Q3: (Part 2-1, page 11). For the Bernoulli equation, describe clearly under what conditions lead to its invalidity?
** This question aims to help you to review the correct assumptions for the Bernoulli equation, the contents in
page 4~5 may be helpful.
Q4: (Part 2-2, page 12). Describe clearly each definition of the following angle of attack (AOA): (a) Absolute
AOA, (b) induced AOA, and (c) effective AOA.
Q5: (Part 2-3, page 13). How do we define the lift coefficient? Describe clearly all the involved parameter
definitions.
Q6: (Part 2-3, page 13). What is the “stall” state? And why the ratio between the lift and the total drag (L/D) is
important?
End of the latter week preview Questions
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -24
Report Questions:
Q1
(a): Plot both the downstream and upstream velocity distribution (with velocity at x-axis and height at y-axis)
under different fan duties. This question aims to help you to draw a scientific plot and to show you what is a
typical “distribution plot”. **A clear plot requires: (1) axis name and unit, (2) clear data points, and (3) clear
plot legends (color and data marker for each curve) if you decide to draw different curves on the same plot.
(b): By your collected data, estimate the total drag for each duty.
** The equation (12) in page 10 is presented by its original integral form:
𝑡𝑜𝑡𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 = ∫𝑙𝑒𝑓𝑡 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑢𝑜𝑢𝑡 2 𝜌𝑑𝐴 − ∫𝑟𝑖𝑔ℎ𝑡 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑢𝑖𝑛 2 𝜌𝑑𝐴
(12),
which can be simplified by a discrete form:
𝑡𝑜𝑡𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 = 𝜌(∑𝑛𝑖=1 𝑢𝑖,𝑜𝑢𝑡 2 ∆𝐴 − ∑𝑛𝑖=1 𝑢𝑖,𝑖𝑛 2 ∆𝐴)
.
The symbol 𝜌 is the air density at the room temperature, the velocity 𝑢𝑖 denotes your measured ten-seconds
average velocity at each height, the subscript “out” locates at the downstream while the subscript “in” locates at
the upstream. At last, each control surface area ∆𝐴 can be estimated by the measurement interval (4 cm)
multiplied with the box width.
Q2
(a): The following characteristic curves are required, you can decide by yourself how to reveal your results in an
efficient way ** Remember to check how to draw a clear plot in Q1:
1. Lift to AOA.
** Because of the structure design, remember to estimate your true lift (𝐿𝑡𝑟𝑢𝑒 ) by:
𝐿𝑡𝑟𝑢𝑒 = 𝐿𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 − 𝐷𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 × tan 𝐴𝑂𝐴
2. Drag to AOA.
3. Lift coefficient to AOA. ** Remember to apply the true lift.
4. Drag coefficient to AOA.
5. Ratio of L/D to AOA. ** Remember to apply the true lift.
(b): Identify the stall point and L/D maximum point by your experimental result.
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -25
.
Q3－Error prediction. ** This question aims to help you to “predict” the theoretical error source, which
is the premise of “analysis”.
(a): First, one obvious error can be considered. We derive the CV analysis base on a 2D control volume and
assume it to possess a uniform distribution across our wind tunnel width. Answer the following sub questions
with brief explanations.
(a-1): In Q1(b), which symbol is responsible for the “uniform distribution” assumption?
(a-2): When the flow passing our box, does this assumption still hold?
(b): Second, an underlying error can be considered. You may notice that the interval length between the
measurement slots is not used. Consider the following simplified & ideal conditions and answer each sub
question to help you to clarify how an improper control volume affects our estimated result.
(b-1): Consider an empty wind tunnel with an imaginary 2D control volume and a steady flow as shown
below. To simplify your expression, let’s just consider two points attach directly on the imaginary CV
boundary. These points possess an enter/exit velocity and a small control surface area dA. Describe the
total force estimated by equation (12) with these two points? (just like the discrete formula in Q1(b), but
the summation symbol is not required.)
(b-2): Next, check the Bernoulli equation (3) in page 5. Can you relate the Bernoulli equation (∆𝑃 =
0.5𝜌∆(𝑢2 )) with your answer in Q3(b-1)? (you can notice that the CV analysis leads to the similar result:
force = pressure*area).
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -26
(b-3): Put a box which is identical to the imaginary CV, flow velocity at both points in (b-2) become
zero since they are attached to the box surface. So, flow speed will slow down along the stream line to
fulfill the continuity. Consider two new upstream CVs with their length equal to segment 1 and segment
2, respectively. Answer the following sub questions with brief explanations.
(b-3-1): Does your derivation in (b-2) still hold in the following segments? Segment 1, segment 2,
and segment 3.
(b-3-2): Consider segment 1 and 2, which one provides a bigger force exerting on CV measured?
** Recall the reaction force theory, water pressure exerts on a wall equals to the wall pressure
exerts on water.
(c): At last, can you conclude that (1): why the downstream measurement point should be closer to the box?
And (2) As you may observe, our upstream measurement point during your experiment has a distance from our
box, what is its influence on the true drag of the box?
End of the report Questions
機械工程實驗(II) Unit 3 and 4 -Introduction to aerodynamics and airfoil characteristic curve determination -27