Project Overview
This project explores the elegant mathematical concept of the Joukowski Transformation—a conformal mapping used to transform simple geometric shapes into airfoil-like structures in the complex plane. By leveraging your understanding of complex numbers, this project bridges theoretical mathematics with visually intriguing outcomes.
Learning Objectives
- Understand the concept of conformal mapping in the complex plane.
- Apply complex number transformations to practical geometries.
- Develop computational skills to visualize and analyze mapped structures.
- Gain a stronger foundation in complex analysis and its applications.
Project Description
Background
The Joukowski Transformation is defined as:
$$ w = z + \frac{1}{z} $$
Where:
- $z = x + iy$ is a point in the complex plane.
- $w$ is the transformed point in a new plane.
This transformation maps circles in the $z$-plane into airfoil-like shapes or flattened ovals in the $w$-plane. It serves as a fundamental concept in aerodynamics and complex analysis.
Project Stages
1. Theoretical Foundation (30%)
- Study the concept of conformal mappings and transformations.
- Derive and explain the effect of the Joukowski transformation on:
- A unit circle $|z| = 1$.
- A shifted circle $|z - z_0| = r$ with parameters $z_0$ and $r$.
- Discuss the geometric meaning of singularities and mapping behavior.
2. Computational Implementation (40%)
Develop a program to implement and visualize the Joukowski transformation. Tasks include:
- Input Parameters:
- Define a circle in the complex plane (radius $r$ and center $z_0$).
- Transformation:
- Apply the Joukowski formula to map points on the circle into the $w$-plane.
- Visualization
- Plot the original circle and the transformed shape.
- Allow adjustments to the circle parameters and observe how the resulting shape changes.
3. Geometric Exploration (20%)
- Explore how changing the circle’s radius or center influences the transformed shape.
- Investigate special cases where the Joukowski transformation produces:
- Symmetric shapes.
- Sharp edges resembling airfoil trailing edges.
4. Mathematical Analysis (10%)
- Analyze and describe why the transformation generates flattened or stretched shapes
- Discuss the geometric properties preserved by the transformation.
- Identify limitations, such as singularities in the transformation.
Technical Requirements
- Implement in a programming language of choice (Python, MATLAB, or FORTRAN recommended)
- Ensure proper parameter control for circle position and radius.
- Present clear and well-annotated plots showing transformations.
Deliverables
- Comprehensive report including:
- Theoretical background
- Detailed explanation of implementation
- Source code with annotations
- Visualizations and advanced explorations
- Mathematical analysis and insights
- Executable program: Clean and annotated code for the transformation and visualization.
- Presentation: A clear and engaging explanation of the project, visual results, and key findings.
Evaluation Criteria
- Report Quality (20%): Clarity, depth, and presentation of theoretical and computational details
- Code Efficiency and Quality (30%): Correctness, modularity, and readability
- Presentation (50%): Delivery, visual appeal, and depth of understanding
Here’s the enriched version of the Bonus Challenges section, incorporating studies of potential flow and related topics:
Bonus Challenges (Optional)
- Explore the mapping of ellipses or other shapes instead of circles.
- Add an interactive feature to dynamically adjust parameters and observe the transformation.
- Investigate and explain how singularities appear during the transformation.
For students seeking deeper insights and a broader understanding, explore the following extensions that provide opportunities to connect complex analysis with fluid dynamics concepts:
Potential Flow Around the Joukowski Airfoil
- Study the potential flow (ideal, inviscid, and irrotational flow) around the Joukowski airfoil.
- Analyze how the Joukowski transformation simplifies solving the Laplace equation for potential flow.
- Visualize streamlines and equipotential lines around the airfoil shape.
Lift Calculation Using the Kutta-Joukowski Theorem
- Introduce the Kutta-Joukowski theorem for lift generation: $$ L = \rho V \Gamma $$ where $L$ is lift, $\rho$ is fluid density, $V$ is freestream velocity, and $\Gamma$ is the circulation.
- Apply the theorem to estimate lift for the Joukowski airfoil, assuming ideal flow conditions.
- Relate lift generation to the geometry of the transformed shape.
Effect of Circle Parameters on Airfoil Geometry
- Systematically study how varying the parameters $z_0$ (circle center) and $r$ (circle radius) influences the resulting airfoil geometry.
- Document relationships between input parameters and airfoil features, such as thickness, camber, and chord length.
Comparison with Real Airfoil Shapes
- Compare the Joukowski airfoil with real-world airfoils used in aerodynamics (e.g., NACA airfoils).
- Discuss the limitations of the Joukowski transformation in modeling practical airfoil shapes.
Numerical Simulation of Flow
- Use numerical tools (e.g., CFD software like OpenFOAM or COMSOL) to simulate flow over the Joukowski airfoil.
- Compare idealized potential flow predictions with numerical results to identify discrepancies caused by viscosity and other real-world factors.
Explore Related Transformations
- Study alternative conformal mappings (e.g., Kármán-Trefftz transformation) and compare their ability to generate airfoil shapes.
- Discuss how modifications to the Joukowski transformation can produce different aerodynamic features.
Recommended Reading
- Joukowsky Airfoil page with engaging interactive plots
- All reference Textbooks of the course: Section of Conformal Mapping
Submission Guidelines
- Submission deadline: TBA
- Format: PDF report, source code, and presentation slides
- Groups of up to three members are allowed, provided each member’s contribution is documented clearly.
- For every project, only one group with the best performance will be selected to present.