Flow Simulation over a Joukowski Airfoil

Project Overview: Flow Simulation over a Joukowski Airfoil

This project involves the aerodynamic analysis and potential flow simulation of a Joukowski airfoil. Using the Joukowski transformation in MATLAB, the study explores the relationship between airfoil geometry (thickness and camber) and its aerodynamic performance, approximating a NACA 3408 series profile.

Aerodynamic Specifications and Conditions

Airfoil Profile: Maximum thickness-to-chord ratio 8.5%, maximum camber-to-chord ratio 3.5%.
Chord Length: (c = 1.25\,\mathrm{m})
Angle of Attack: Primary (\alpha = 6^\circ), comparative range (-5^\circ) to (10^\circ)
Free-stream Velocity: (V_\infty = 125\,\mathrm{m/s})

Key Aerodynamic Findings

Flow Visualization: Streamlines show steady, inviscid potential flow with smooth trailing edge separation.
Velocity and Pressure: Upper surface experiences higher velocity and lower pressure; lower surface has lower velocity and higher pressure, generating lift.
Coefficient Variations: Lift coefficient ((C_L)) increases linearly with (\alpha); moment coefficient ((C_M)) at quarter-chord varies inversely with lift.

Velocity Distribution over Airfoil

Technical Implementation

MATLAB was used to parametrize airfoil surface coordinates, create a Cartesian grid, define the stream function, and compute local surface velocities using Bernoulli’s equation.

Core Mathematical Framework

1. Airfoil Surface Coordinates

The Joukowski airfoil geometry is generated parametrically:

\[r = b \left[ 1 + e(1 - \cos\theta) + \beta \sin\theta \right]\] \[x = r \cos\theta, \quad y = r \sin\theta\]

Where (b), (e), and (\beta) are constants derived from chord, thickness, and camber ratios.

2. Stream Function for Uniform Flow

The streamlines of inviscid potential flow at angle of attack (\alpha):

\[\psi = V_\infty (y \cos\alpha - x \sin\alpha)\]

3. Pressure Coefficient

The pressure coefficient (C_p) is computed from the local velocity (V) via Bernoulli’s principle:

\[C_p = 1 - \left( \frac{V}{V_\infty} \right)^2\]

4. Lift and Moment Coefficients

Using thin airfoil theory:

Lift Coefficient: \(C_L = 2\pi \alpha_\mathrm{rad}\)
Moment Coefficient (quarter-chord): \(C_M = -0.25 C_L\)

📄 Full Project Documentation

You can access the full documentation here.