VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY – VNU-HCM
HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY
PROJECT REPORT
CACULUS 2 - 242
Teacher: PhD. Nguyễn Quốc Lân
Group 19 - Class CC05
TOPIC 19
AIRCRAFT WING DESIGN OPTIMIZATION
AIRFOIL OPTIMIZATION
Students perform:
Tran Tien Minh Hieu 2452336
Dao Gia Huy 2452373
Nguyen Ngoc Chau Han 2252204
Tran Chi Dai 2452237
Nguyen Thuy Anh 2252036
Ho Chi Minh City, May 2025
ASIGNMENT AND EVALUATION SHEET
Student name Student number Mission Evaluate
Tran Tien Minh Hieu* 2452336 Part 1,2,3,4 100%
Dao Gia Huy 2452373 Video, Slide, Format, Part 8,9 100%
Nguyen Ngoc Chau Han 2252204 Part 6 100%
Tran Chi Dai 2452237 Part 5, Figure 100%
Nguyen Thuy Anh 2252036 Part 7 100%
1
ACKNOWLEDGEMENTS
First and foremost, we would like to express our deepest gratitude to Associate Professor Nguyễn
Quốc Lân, our supervisor, for his tremendous assistance and guidance during the development of our
project. Without his insights and feedback, completing this project would have been impossible.
Furthermore, we would also like to extend our appreciation to the Faculty of Applied Science of Ho
Chi Minh University of Technology – Vietnam National University Ho Chi Minh City for providing us the
opportunities to attend the Calculus 2 study course. This program equipped us with necessary knowledge
and experience to continue our studies in future years as students of this university.
Throughout this project, we have learned valuable lessons and grown in both intelligence and skills.
Nevertheless, we acknowledge that we are not yet complete professionals and despite our best efforts, it
is inevitable our abilities remained limited. We sincerely hope to receive constructive and helpful advice
from the professors so as to refine and improve our report.
Finally, we would like to thank everyone who has supported us throughout this project. Your encour-
agement and assistance have been invaluable, and we are profoundly grateful.
2
TABLE OF CONTENTS
ASIGNMENT AND EVALUATION SHEET 1
ACKNOWLEDGEMENTS 2
ABSTRACT 5
1 Topic 6
1.1 Introduction to the problem and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Application context of wing optimization technology: . . . . . . . . . . . . . . . . . . . . . 6
1.3 The importance of the problem: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Overview 7
3 Theoretical Framework 7
3.1 Functions of One and Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Derivatives and Extrema (Stewart, Section 4.3) . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Optimization of Rational Functions (Quotient Rule) . . . . . . . . . . . . . . . . . . . . . 7
3.4 Analytical Derivation of Optimal Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 MATLAB Implementation 8
4.1 Computing Derivative and Finding Critical Points . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Second Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Plotting the Lift-to-Drag Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.4 Evaluating Function and Second Derivative at Critical Points . . . . . . . . . . . . . . . . 9
5 Computing optimizing calculation 10
5.1 Sample Data (NACA 2412 airfoil) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2 Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.3 Computing the Optimal Angle of Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Result Analysis and Comparison 12
6.1 Result Analysis: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.2 Explanation for the chosen value from an aerodynamic perspective. . . . . . . . . . . . . . 12
6.3 Compare with other alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.4 Discussion: What if CFD is Used? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.5 Pros and Cons of the Analytical Optimization Method: . . . . . . . . . . . . . . . . . . . 13
7 Extension and Real-life Applications 13
7.1 The applications of optimization of AOA in real life . . . . . . . . . . . . . . . . . . . . . 13
7.2 Simulation and multi-variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.3 The Reynolds number (Re) influences the transition from laminar to turbulent flow: . . . 14
7.4 Mach Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8 Application in Vietnam 17
9 Conclusion 18
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9.1 Strengths and Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
9.2 Potential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
KEYTAKEAWAYS 20
4
ABSTRACT
The project titled "Aircraft Wing Design Optimization – Airfoil Optimization" focuses on improving
the aerodynamic efficiency of an aircraft wing by maximizing the Lift-to-Drag (L/D) ratio. Using the
NACA 2412 airfoil profile as a reference, the study investigates wing performance under specific flight
conditions—250 km/h at an altitude of 3,000 meters—considering factors such as air density, airflow
velocity, and dynamic viscosity. Computational methods and principles of fluid dynamics are applied to
identify an optimal airfoil shape that enhances lift while minimizing drag. The results are evaluated and
compared with standard profiles, offering insights into potential improvements for real-world applications.
Beyond technical outcomes, the project contributes to a deeper understanding of aerodynamic behavior
and practical optimization techniques in aerospace engineering.
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1 Topic
1.1 Introduction to the problem and objectives
In the field of aerospace engineering, optimizing the design of aircraft wings plays a crucial role in
improving aerodynamic performance and fuel efficiency. One of the key objectives is to maximize the
Lift-to-Drag ratio (L/D), which determines how effectively an aircraft can generate lift while minimizing
drag. This optimization is essential for reducing energy consumption, increasing range, and enhancing
overall flight performance.
A widely used approach is to analyze airfoil profiles, such as the NACA 2412, under various flight
conditions. By applying principles from fluid dynamics and aerodynamics, along with mathematical
optimization techniques, engineers can find the optimal wing shape and angle of attack to achieve the
best performance.
1.2 Application context of wing optimization technology:
• Commercial aviation: Enhancing fuel economy and flight stability.
• Unmanned Aerial Vehicles (UAVs): Improving efficiency for surveillance and data collection
missions.
• Military aircraft: Enhancing maneuverability and operational range.
• Renewable energy: Improving blade profiles for wind turbines using similar aerodynamic princi-
ples.
Objective:
This topic focuses on finding the wing shape that maximizes the Lift-to-Drag ratio using NACA 2412 as
a reference airfoil. The key objectives are:
• Model aerodynamic forces:
– Use mathematical expressions for lift coefficient CL (α) and drag coefficient CD (α) as functions
of the angle of attack α.
CL (α)
– Compute and analyze the function f (α) = CD (α) .
• Apply calculus and optimization techniques:
– Use derivatives to find critical points that maximize f (α).
– Apply symbolic and numerical methods to determine optimal conditions.
• Enhance understanding of applied mathematics:
– Connect multivariable calculus with real-world problems in aerodynamics.
– Develop skills in using MATLAB or similar tools to simulate and analyze the problem.
1.3 The importance of the problem:
Optimizing wing design is not only vital for improving aircraft performance but also provides a
hands-on example of applying mathematical theory to solve real-world engineering problems. This topic
illustrates how abstract concepts in calculus and physics can lead to practical innovations in aviation and
energy systems.
6
2 Overview
This report explores the application of multivariable calculus—drawing particularly from James Stew-
art’s rigorous framework of functions of several variables—to the aerodynamic optimization problem of
maximizing the lift-to-drag ratio:
CL (α)
f (α) = ,
CD (α)
where CL (α) and CD (α) are the lift and drag coefficients, both functions of the angle of attack α. In
aerodynamics, the lift-to-drag ratio (L/D ratio) is a key performance metric for airfoils and aircraft. The
objective is to determine the optimal angle of attack α that achieves maximum aerodynamic efficiency. Our
goal is to maximize this ratio using methods from Chapter 14 and Chapter 15 of Stewart’s Calculus: Early
Transcendentals. Our approach combines theoretical derivation, symbolic differentiation, and numerical
MATLAB simulations.
3 Theoretical Framework
3.1 Functions of One and Several Variables
According to Stewart (Ch. 14), if a function f depends on one or more variables, its rate of change
can be studied using derivatives. For a function f (α), the first derivative f ′ (α) indicates the slope or
instantaneous rate of change.
3.2 Derivatives and Extrema (Stewart, Section 4.3)
To identify local extrema:
• Solve f ′ (α) = 0 (critical points).
• Use the second derivative test:
– f ′′ (α) > 0: local minimum,
– f ′′ (α) < 0: local maximum.
3.3 Optimization of Rational Functions (Quotient Rule)
g(α)
For a rational function f (α) = h(α) , Stewart (Sec. 4.1–4.2) describes the quotient rule:
g ′ (α)h(α) − g(α)h′ (α)
f ′ (α) = .
[h(α)]2
We apply this rule to our lift/drag function.
3.4 Analytical Derivation of Optimal Condition
Given:
CL (α) = 2π sin(α), CD (α) = 0.1 + 0.5 sin2 (α),
let
2π sin(α)
f (α) = .
0.1 + 0.5 sin2 (α)
7
We compute:
2π cos(α) 0.1 + 0.5 sin2 (α) − 2π sin(α) (sin(α) cos(α))
′
f (α) = 2 .
0.1 + 0.5 sin2 (α)
This expression can be simplified using trigonometric identities.
4 MATLAB Implementation
4.1 Computing Derivative and Finding Critical Points
How it works:
• Declare symbolic variable alpha representing the angle of attack α:
syms alpha
• Define lift and drag coefficients CL (α), CD (α).
• Construct the function f = CL
CD .
• Compute the first derivative using
diff(f, alpha)
• Find critical points by solving
f ′ (α) = 0
using
solve(f_prime == 0, alpha)
Sample MATLAB code:
syms alpha
CL = 2*pi*sin(alpha);
CD = 0.1 + 0.5*sin(alpha)^2;
f = CL / CD;
f_prime = diff(f, alpha);
alpha_crit = solve(f_prime == 0, alpha);
alpha_crit = vpa(alpha_crit);
Output:
• Symbolic expression of the derivative f ′ (α).
• Set of critical values αcrit where the derivative equals zero.
4.2 Second Derivative Test
How it works:
8
• Compute second derivative:
diff(f_prime, alpha)
• Evaluate the second derivative at critical points:
double(subs(f_doubleprime, alpha, alpha_crit))
• Classify critical points:
f ′′ (α ) < 0 local maximum
crit
f ′′ (α ) > 0 local minimum
crit
Sample MATLAB code:
f_doubleprime = diff(f_prime, alpha);
d2_val = double(subs(f_doubleprime, alpha, alpha_crit));
Output: Numerical values of the second derivative at each critical point to determine their nature.
4.3 Plotting the Lift-to-Drag Ratio
How it works:
• Use fplot to plot f (α) over [0, π/2].
• Label axes and add a title.
Sample MATLAB code:
fplot(f, [0, pi/2], 'LineWidth', 2)
xlabel('\alpha (rad)')
ylabel('CL/CD')
title('Lift-to-Drag Ratio vs Angle of Attack')
grid on
Output: Plot showing variation of lift-to-drag ratio with angle of attack.
4.4 Evaluating Function and Second Derivative at Critical Points
How it works:
• For each αcrit , evaluate f (αcrit ) and f ′′ (αcrit ).
• Display results to confirm maximum values.
Sample MATLAB code:
for i = 1:length(alpha_crit)
a_val = alpha_crit(i);
f_val = double(subs(f, alpha, a_val));
f2_val = double(subs(f_doubleprime, alpha, a_val));
fprintf('Alpha = %.4f rad: f = %.4f, f'''' = %.4f\n', a_val, f_val, f2_val);
end
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Output: Console output such as:
Alpha = 0.5236 rad : f = 10.1234, f ′′ = −0.4567
showing lift-to-drag ratio and second derivative values confirming local maxima.
5 Computing optimizing calculation
5.1 Sample Data (NACA 2412 airfoil)
α (Angle of Attack) CL (Lift Coefficient) CD (Drag Coefficient) Lift/Drag (L/D)
-4 -0.317 0.008 -39.6
-2 0.023 0.0075 3.07
0 0.223 0.007 31.86
2 0.423 0.0075 56.40
4 0.623 0.008 77.88
6 0.823 0.009 91.44
8 1.023 0.011 93.00
10 1.123 0.014 80.21
12 1.123 0.018 62.39
14 1.023 0.023 44.48
16 0.823 0.030 27.43
Table 1: Aerodynamic data for NASA 2412 airfoil
5.2 Figure
Figure 1: Lift Coefficient vs. Angle of Attack
10
Figure 2: Drag Coefficient vs. Angle of Attack
Figure 3: Lift-to-Drag Ratio vs Angle of Attack
Plot the computed L/D ratio against α to visualize where the maximum occurs.
5.3 Computing the Optimal Angle of Attack
Gather your discrete data
You should have three parallel arrays (or columns in a table):
• αi — the angle of attack in degrees (e.g. –4, –2, 0, . . . , 16)
• CLi — the corresponding lift coefficient
11
• CDi — the corresponding drag coefficient
Compute the lift-to-drag ratio for each sample
For each i, form the ratio:
L CLi
(αi ) =
D CDi
Record these in a new column or array alongside αi .
Locate the maximum ratio
• By inspection: scan your “L/D” column and identify the largest value.
• Programmatically: apply an “argmax” operation to your L/D array to get the index i∗ of its
maximum:
∗ L
i = arg max (αi )
i D
• The optimal angle is then:
L L
αopt = αi∗ , = (αi∗ )
D max D
⇒ According to the table above, we have the maximum L
D is 93.00 at α = 8◦ .
6 Result Analysis and Comparison
6.1 Result Analysis:
At α = 8◦ :
• The lift force is high.
• The drag coefficient remains relatively low (CD = 0.011).
• The Lift-to-Drag ratio DL
reaches the maximum value of 93.00, indicating this is the optimal angle
of attack for cruise conditions.
6.2 Explanation for the chosen value from an aerodynamic perspective.
Aerodynamic Justification for the Optimal Angle (α = 8◦ ):
The angle of attack α = 8◦ corresponds to the maximum value of the lift-to-drag ratio, L
D = 93.00,
as determined from the discrete data. From an aerodynamic standpoint, this angle provides the best
trade-off between lift and drag. At α = 8◦ , the lift coefficient CL is sufficiently high to ensure effective
lift, while the drag coefficient CD remains relatively low, resulting in a local maximum of the function
L(α)
D(α) .
L
Mathematically, this point satisfies the condition for a local maximum of the D function in the domain
of measured angles. Aerodynamically, it also lies before the onset of flow separation and stall, which
typically occur at higher angles of attack. Therefore, α = 8◦ is not only optimal based on the computed
data but also physically reasonable, aligning with aerodynamic theory and typical performance envelopes
for subsonic airfoils.
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6.3 Compare with other alternatives
Angle of Attack (α) Comment
6° L/D is nearly optimal (91.44) → could be used for safer operation.
>8° L/D decreases due to rapidly rising drag → not optimal, risk of stall.
<8° Lower L/D → aerodynamic efficiency not fully utilized.
6.4 Discussion: What if CFD is Used?
If we use Computational Fluid Dynamics (CFD) instead of simple analytical methods:
Results are smoother and more detailed, including intermediate angles (e.g., 7°, 7.5°, 8.5°).
CFD allows for:
• Visualization of flow separation zones.
• Analysis of pressure distribution on the airfoil surface.
• Study of wake flow and vortex behavior behind the airfoil.
However, CFD is computationally expensive and time-consuming compared to analytical approaches.
6.5 Pros and Cons of the Analytical Optimization Method:
Advantages Disadvantages
Quick and easy to apply Accuracy is dependent on available
sample data
Requires no complex software Cannot visualize flow phenomena like
separation or stall
Useful for initial estimations or studies May not work well in complex or dy-
namic flight conditions
7 Extension and Real-life Applications
The angle of attack (AOA) is a critical aerodynamic parameter defined as the angle between the chord
line of an airfoil and the relative wind or airflow direction. Optimizing the AOA plays a fundamental role
in achieving efficient aerodynamic performance across various fields such as aviation, renewable energy,
and automotive engineering. This report highlights the real-life benefits of AOA optimization, supported
by its applications in multiple industries.
7.1 The applications of optimization of AOA in real life
• Enhanced Fuel Efficiency: In aviation, flying at an optimal AOA reduces the power required to
maintain altitude and speed, leading to significant reductions in fuel consumption. This not only
lowers operational costs but also reduces greenhouse gas emissions.
• Takeoff and Landing Optimization: An optimized AOA is crucial during takeoff and landing
phases, where maintaining lift at low speeds is vital. A proper AOA shortens runway requirements
and enhances the safety of low-speed maneuvers.
• Climb and Cruise Efficiency: By flying at the most efficient AOA during climb and cruise,
aircraft can reduce engine stress and fuel burn, extending range and reducing environmental impact.
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7.2 Simulation and multi-variables
This behavior indicates that the airfoil operates most efficiently at the optimal angle of attack, where
it generates the most lift for the least drag. After this point, increasing the angle leads to greater drag
with minimal additional lift, reducing the overall aerodynamic efficiency.
CL
The optimal angle of attack, where CD is maximized, should be targeted for energy conversion pur-
poses, especially in wind turbines.
CL
The optimal AOA for CD (around 4–5 degrees) is crucial for design optimization. Engineers aim
to ensure that the airfoil operates close to this angle under typical operating conditions, especially in
applications like wind turbines or airplanes.
In wind turbine blade design, maintaining the rotor blades at or near this optimal angle of attack
maximizes energy capture while minimizing the effects of drag. For aircraft wings, flying at or near
this optimal angle ensures efficient flight performance, reducing fuel consumption and improving overall
efficiency.
The Lift-to-Drag Ratio is essential for evaluating the aerodynamic efficiency of an airfoil. Key factors
that affect the ratio include many factors such as: airfoil shape, Reynolds number, flow conditions, and
other variables beyond the Angle of Attack (AOA). Understanding these factors is crucial for optimizing
performance in applications like wind turbines and aircraft.
7.3 The Reynolds number (Re) influences the transition from laminar to
turbulent flow:
• High Re: Turbulent flow increases drag and reduces CL
CD
• Low Re: Laminar flow reduces drag but may not generate as much lift.
7.4 Mach Number
• Subsonic flow maintains smooth airflow, improving CL
CD
• Supersonic flow causes shock waves and wave drag, significantly lowering the ratio.
14
Figure 4: The change of lift to drag ratio with low Re and high Ma
With low Re, the flow remains more laminar, reducing drag; however, at high Ma, shock waves form
CL
due to compressibility effects, resulting in significant wave drag and a sharp decrease in the C D
ratio.
High Ma causes a substantial reduction in efficiency, particularly beyond a certain Mach threshold
CL
where shock waves form, dramatically lowering the CD ratio.
Figure 5: The change of lift to drag ratio with low Re and low Ma
At low Reynolds numbers, the flow tends to stay laminar, which minimizes drag. However, at low
CL
Ma, the CD ratio remains relatively stable with only modest increases in drag, suggesting smooth airflow
with little shock-related drag.
CL
The CD ratio peaks around the optimal AOA, but the drag increase remains minimal, indicating that
15
laminar flow contributes to maintaining high aerodynamic efficiency.
Figure 6: The change of lift to drag ratio with high Re and high Ma
The high Re leads to turbulent flow, which typically increases drag, and high Ma causes compressibility
CL
effects, including shock waves that significantly increase drag due to wave drag. As a result, the CD ratio
sharply drops at higher angles of attack.
This graph illustrates the high drag costs associated with operating at high speeds (supersonic or
transonic conditions), even though the lift is relatively constant.
Figure 7: The change of lift to drag ratio with high Re and low Ma
CL
As the angle of attack increases, the CD ratio initially improves, peaking at an optimal AOA (around
16
CL
4–5 degrees). Beyond this point, further increases in the AOA result in a sharp drop in the CD ratio,
indicating increasing drag and minimal additional lift.
High Re suggests turbulent flow, which typically increases drag, but the effect of drag is somewhat
CL
moderated at lower Mach numbers, ensuring that the drop in the CD ratio is more gradual.
7.5 Conclusion
Reynolds Number (Re):
• High Reynolds numbers (turbulent flow) increase drag and reduce the CL
CD ratio, common in high-
speed applications like aircraft and wind turbines.
• Low Reynolds numbers (laminar flow) reduce drag, maintaining a higher CL
CD ratio, which is ideal
for low-speed applications, such as small aircraft or drones.
Mach Number (Ma):
• High Mach numbers (supersonic speeds) cause wave drag and shock effects, significantly lowering
CL
the CD ratio, especially in high-speed aircraft.
• Low Mach numbers (subsonic speeds) allow smooth airflow, keeping the CL
CD ratio high, beneficial
for low-speed designs like wind turbines or drones.
Combined Effects:
• High Re and High Ma lead to high drag and a low CL
CD ratio, making optimization essential for
high-speed applications.
• Low Re and Low Ma help maintain a high CL
CD ratio, supporting efficiency in low-speed applications.
8 Application in Vietnam
Evaluation and Optimization of NACA 2421 Airfoil for UAV Applications in
Southern Vietnam
Unmanned Aerial Vehicles (UAVs) have seen increased development in Vietnam for applications such
as agricultural monitoring, environmental surveys, and low-altitude mapping. In southern provinces like
Đồng Tháp or Cần Thơ, the requirement for lightweight, efficient, and low-speed UAV platforms is
growing, especially for operations at low altitudes and moderate wind conditions.
This case study investigates the performance of the NACA 2421 airfoil for small fixed-wing UAVs
operating at typical speeds of 15–25 m/s and Reynolds numbers between 2×105 and 6×105 , corresponding
to small chord lengths (0.2–0.4 m) and low-altitude operation.
Initial performance evaluation was conducted using XFOIL, with a focus on aerodynamic coefficients
at angles of attack (AoA) from −5◦ to +15◦ . The baseline NACA 2421 airfoil exhibited a maximum lift
coefficient (Cl ) of approximately 1.12 at 8◦ , and a minimum drag coefficient (Cd ) of 0.018 at 4◦ , with a
Cl
peak Cd of 58.
To improve UAV endurance and flight efficiency under typical Vietnamese conditions (e.g., high
humidity, low altitude, low-speed cruise), a multi-objective optimization was performed using a Genetic
Algorithm (GA). The optimization aimed to slightly adjust camber and thickness distribution while
preserving manufacturability and structural simplicity.
17
Table 2: Aerodynamic performance comparison of NACA 2421 and optimized airfoil
CL
Airfoil Max CL Min CD Max C D
Optimal AoA
NACA 2421 1.12 0.018 58 8◦
Optimized Profile 1.21 0.016 72 7.5◦
The optimized airfoil demonstrated a ∼ 24% increase in aerodynamic efficiency, making it better
suited for small UAVs operating in the Mekong Delta. This improvement can lead to increased range and
reduced energy consumption, which is critical for electric UAVs used in remote agricultural missions.
9 Conclusion
9.1 Strengths and Weaknesses
Strengths:
• The project successfully integrates multivariable calculus with real-world aerospace engineering
problems, particularly in optimizing the Lift-to-Drag (L/D) ratio of an aircraft wing.
• The use of MATLAB for symbolic computation, plotting, and analysis demonstrates strong technical
proficiency and an effective application of computational tools in engineering contexts.
• Analytical methods are clearly presented, with derivations and critical point analysis aligned with
theoretical expectations.
• The comparison between discrete data and symbolic optimization strengthens the credibility of the
final results.
Weaknesses:
• The simplified model assumes ideal conditions, such as constant air properties and idealized equa-
tions for lift and drag coefficients, which may not fully represent actual aerodynamic behavior.
• Discrete data lacks intermediate angles (e.g., 7.5°), potentially overlooking a more precise optimum.
• The project does not incorporate advanced fluid dynamics methods such as Computational Fluid
Dynamics (CFD), which could offer a more comprehensive visualization and validation of results.
9.2 Potential Growth
• Incorporating CFD Simulations: Future iterations can implement CFD to analyze flow sepa-
ration, turbulence, and shock effects in greater detail.
• Advanced Optimization Algorithms: Techniques such as genetic algorithms or neural networks
can improve optimization accuracy, especially when dealing with multiple variables or constraints.
• Broader Application Scope: The optimization framework can be extended to different airfoil
profiles, Reynolds numbers, and flight conditions (e.g., supersonic regimes).
• Cross-disciplinary Collaboration: Incorporating material science or structural mechanics con-
siderations could lead to more robust and manufacturable designs.
Overall, this project lays a strong foundational framework in combining calculus-based modeling with
aerodynamic analysis, opening multiple pathways for future enhancement and practical deployment.
18
References
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https://vietuavlab.org/research.
Accessed: December 31, 2024
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9. NASA Glenn Research Center, "Lift and Drag Coefficients". Online: Address: https://www.grc.
nasa.gov/www/k-12/airplane/liftco.html.
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11. Aerodynamics for Engineering Students, "Quotient Rule and Applications in Fluid Mechanics".
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12. Khan Academy, "Maxima and Minima Using Derivatives". Online: Address: https://www.
khanacademy.org/math/calculus-1/cs1-optimization.
Accessed: December 31, 2024
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KEYTAKEAWAYS
This project provided valuable insights into how applied mathematics, specifically calculus, can be
used to solve real-world engineering challenges in aerospace design. By focusing on the optimization of the
Lift-to-Drag (L/D) ratio, students not only enhanced their theoretical understanding but also developed
practical skills in simulation and data analysis.
• The Lift-to-Drag ratio is a fundamental parameter in aerodynamic design, as it directly affects fuel
efficiency, range, and overall performance of aircraft. Optimizing this ratio is essential in both civil
and military aviation, as well as in other fields such as wind energy.
• Through analytical modeling, the project determined that an angle of attack of approximately
8◦ for the NACA 2412 airfoil yields the highest L/D ratio (93.00), underlining the importance of
balancing lift generation with drag minimization.
• The use of MATLAB for symbolic differentiation, critical point analysis, and graphing was instru-
mental in deriving and visualizing results. This underlines the importance of computational tools
in modern engineering workflows.
• Despite the effectiveness of analytical methods, the project highlighted the limitations of relying
solely on simplified models. Real-life aerodynamic phenomena are influenced by complex factors
like Reynolds number, Mach number, and turbulence, which require advanced tools like CFD for
accurate modeling.
• The research also explored how optimization principles can extend beyond aviation into renewable
energy, particularly wind turbine design, where maximizing energy capture efficiency is similarly
governed by aerodynamic performance.
• In the context of Vietnam, the project applied these findings to UAV operations in the Mekong
Delta, demonstrating the real-world impact of academic research when tailored to local technological
and environmental needs.
In summary, the project not only deepened students’ understanding of aerodynamics and applied
mathematics but also encouraged interdisciplinary thinking and innovation for future aerospace and
energy-related applications.
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