Computational Fluid Dynamics: A Visual Approach

Computational Fluid Dynamics (CFD) is a powerful tool for simulating and analyzing fluid flow problems. In this post, we'll explore visual approaches to understanding CFD results.

The Navier-Stokes Equations

The fundamental equations governing fluid flow are the Navier-Stokes equations:

$\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}$

$\nabla \cdot \mathbf{u} = 0$

where $\mathbf{u}$ is the velocity field, $p$ is pressure, $\rho$ is density, and $\mu$ is viscosity.

Visualizing Flow Fields

A common approach to visualizing flow fields is using streamlines or particle tracing. Here's a simple example of generating streamlines in Python:

import numpy as np
import matplotlib.pyplot as plt

# Create a grid
x = np.linspace(-2, 2, 20)
y = np.linspace(-2, 2, 20)
X, Y = np.meshgrid(x, y)

# Define velocity components (example: rotating flow)
U = -Y
V = X

# Plot streamlines
plt.figure(figsize=(8, 8))
plt.streamplot(X, Y, U, V, density=1, linewidth=1, arrowsize=1, color='blue')
plt.axis('equal')
plt.title('Streamlines of a Rotating Flow')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.show()