2D Potential Flow Solver: Vortex & Source Panel Method solver for built-in NACA 4-digit series foils and arbitrary user defined bodies.

Authors: Bora Ugurcan Ekizoglu, Kurt Spiteri
Language: Julia (Pluto.jl)

📌 Overview

This repository contains an interactive, fully parameterized interactive 2D numerical solver built in Julia (using Pluto.jl) to estimate the lift and flow field around fully submerged NACA 4-digit series hydrofoils/airfoils.

The tool utilizes inviscid potential flow theory, allowing users to seamlessly toggle between First-Order Source Panel and First-Order Vortex Panel methodologies to evaluate continuous pressure, velocity magnitudes, and integrated force coefficients as well as the Kutta-Joukowski lift at various geometric angles of attack.

Figure 1: Interactive flow field visualization (Velocity Magnitude) over a parametrised NACA foil.

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✨ Key Features

• Interactive Geometry Generation: Sliders to dynamically manipulate the number of panels ($N$), chord length, thickness, camber, and angle of attack for any NACA 4-digit foil. The solver can also handle user defined arbitrary body functions.
• Dual Singularity Solver: Fully interchangeable between purely Source (thickness modeling) and purely Vortex (lifting modeling) mathematical Green's functions.
• Kutta Condition Enforcement: Explicit matrix augmentation to enforce the trailing edge stagnation point, allowing the mathematical generation of physical circulation and lift. The panel $x$ and $y$ coordinate arrays should start and end at the trailing to be able to enforce the Kutta condition on the correct panels.
• Aerodynamic Post-Processing: Automated integration of the continuous pressure field (via Bernoulli's Equation) to calculate integrated lift and numerical drag, alongside theoretical Kutta-Joukowski checks.
• Flow Visualization: Superposition of the freestream and panel-induced velocity grids to map continuous Velocity Magnitude ($V$), directional vectors ($U_x$, $U_y$), and Pressure Coefficient ($C_p$) contours.

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📐 Mathematical Methodology

The solver discretizes the continuous airfoil curve into $N$ flat line segments (panels). The potential flow velocity on any panel surface is governed by the boundary condition:

$$V_{i} = V_{\infty}\hat{p} + \sum^{n}_{j=1} \lambda_{j} \int_{s_{j}} \frac{\partial G(r)}{\partial s}ds_j$$

Where $V_{\infty}$ is the freestream velocity, $\hat{p}$ is the geometric normal ($\hat{n}$) or tangent ($\vec{t}$) vector, $\lambda_{j}$ is the unknown singularity strength, and $G$ is the Green's function.

Green's Functions

The potential induced at a point $(x,y)$ by a singularity situated at $(x_s, y_s)$ is defined by:

• Point Source: $G = \frac{1}{2}\ln(r)$
• Point Vortex: $G = \arctan\left( \frac{y-y_{s}}{x-x_{s}}\right)$

The Matrix System

By evaluating the influence of every panel on every other panel's collocation point (the midpoint), we assemble an aerodynamic influence matrix $[A]$ and an excitation vector $[B]$ (the freestream boundary condition).

$$[A][\Lambda] = [B]$$

The Kutta Condition

To prevent infinite velocities around the sharp trailing edge, the system is modified to physically lock the trailing-edge stagnation point. We sacrifice the final matrix equation and replace it with the Kutta Condition, enforcing zero net vorticity at the trailing edge:

$$1 \cdot \gamma_{top} + 1 \cdot \gamma_{bottom} = 0$$

Figure 2: Pressure Coefficient ($C_p$) contour mapping the leading-edge suction peak.

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📈 Validation and Grid Convergence

A rigorous validation study (NACA_XXXX_PM_validation_convergence.jl) accompanies the core tool. The matrix assembly and mathematical integration are validated against closed-body sanity checks (circles and ellipses) before being applied to the lifting foil.

As the panel resolution ($N$) increases, the solver demonstrates textbook spatial convergence. The integrated Lift Coefficient ($C_l$) successfully flatlines to its true grid-independent value, perfectly matching the theoretical Kutta-Joukowski theorem. Simultaneously, as expected from D'Alembert's Paradox, the numerical spatial truncation error (Integrated $C_d$) decays exponentially toward zero.

Figure 3: Grid convergence study demonstrating stable singularity distribution and asymptotic lift convergence.

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🏗️ Custom Geometries & Arbitrary Bodies

While this tool includes a built-in NACA 4-digit generator, the core solver is entirely geometry-agnostic. You can compute the flow around any arbitrary body by simply feeding 1D arrays of $x$ and $y$ panel endpoint coordinates into the solve() function.

For example, to simulate flow around a custom circular body, you can define a simple coordinate generator:

# 1. Define a custom geometry function
function circle(N, R=1.0)
    # N panels require N+1 node points to close the loop
    theta = range(-π, π, length=N+1)
    x = R .* cos.(theta)
    y = R .* sin.(theta)
    return x, y
end

# 2. Generate the coordinates for 32 panels
x_circ, y_circ = circle(32, 0.5)

# 3. Feed the coordinate arrays directly into the solver!
U_free = (1.0, 0.0)
q_circ = solve(x_circ, y_circ, U_free; G=source, loopdir=:CCW)

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🚀 How to Use

This project is built as a reactive Pluto.jl notebook.

1. Install Julia.
2. Open the Julia REPL and install Pluto:

   julia> ]
   (@v1.x) pkg> add Pluto
   julia> import Pluto
   julia> Pluto.run()