Engineering Mathematics Project 2: Joukowski Transformation

Joukowski Airfoil

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

Project Description

Background

The Joukowski Transformation is defined as:

$$ w = z + \frac{1}{z} $$

Where:

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%)

2. Computational Implementation (40%)

Develop a program to implement and visualize the Joukowski transformation. Tasks include:

  1. Input Parameters:
    • Define a circle in the complex plane (radius $r$ and center $z_0$).​
  2. Transformation:
    • Apply the Joukowski formula to map points on the circle into the $w$-plane.
  3. 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%)

4. Mathematical Analysis (10%)

Technical Requirements

Deliverables

  1. Comprehensive report including:
    • Theoretical background
    • Detailed explanation of implementation
    • Source code with annotations
    • Visualizations and advanced explorations
    • Mathematical analysis and insights
  2. Executable program: Clean and annotated code for the transformation and visualization.
  3. Presentation: A clear and engaging explanation of the project, visual results, and key findings.

Evaluation Criteria

Here’s the enriched version of the Bonus Challenges section, incorporating studies of potential flow and related topics:

Bonus Challenges (Optional)

For students seeking deeper insights and a broader understanding, explore the following extensions that provide opportunities to connect complex analysis with fluid dynamics concepts:

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  1. Joukowsky Airfoil page with engaging interactive plots
  2. All reference Textbooks of the course: Section of Conformal Mapping

Submission Guidelines