Computational aerodynamics — panel method (intro), CFD overview

1. WHY do we need computational methods?

WHY Laplace? Incompressible $\Rightarrow \nabla\cdot\vec V=0$. Irrotational $\Rightarrow \vec V=\nabla\phi$. Combine: $\nabla\cdot(\nabla\phi)=\nabla^2\phi=0$.

WHAT changes for real flow? Add viscosity ($\mu\nabla^2\vec V$ term) and compressibility ($\rho$ varies) and you get the full Navier–Stokes equations, which are nonlinear — no superposition, must use CFD.

2. Elementary solutions (the LEGO bricks)

Because Laplace is linear, any sum of these is also a solution:

3. Panel method — DERIVATION from scratch

Step 1 — The physical condition we must enforce

The body is solid: no flow goes through the wall. So the velocity component normal to the surface is zero: $\vec V \cdot \hat n = 0 \quad \text{on the body.}$ Why this step? It is the only boundary condition that defines the shape's effect — everything is built to satisfy it.

Step 2 — Discretise the surface

Replace the smooth contour with $N$ flat panels. Put a source sheet of unknown strength $\lambda_j$ (constant) on each panel $j$. Why this step? We can't find a continuous $\lambda(s)$ by hand, but $N$ unknown constants give a solvable linear algebra problem.

Step 3 — Total potential

$\phi(P) = V_\infty(x\cos\alpha+y\sin\alpha) + \sum_{j=1}^{N}\frac{\lambda_j}{2\pi}\int_{\text{panel }j}\ln r\,ds_j$ Why this step? Superposition: freestream + every panel's contribution. The integral spreads $\lambda_j$ over the panel.

Step 4 — Apply boundary condition at each control point

At the midpoint ("control point") of panel $i$, demand zero normal velocity: $V_\infty\cos\beta_i + \sum_{j=1}^{N} \frac{\lambda_j}{2\pi}\,I_{ij} = 0, \qquad i=1\dots N$ where $\beta_i$ is the angle between the freestream and the panel's outward normal, and $I_{ij}=\dfrac{\partial}{\partial n_i}\int_j \ln r\,ds_j$ is the influence coefficient (normal velocity induced at panel $i$ by a unit source on panel $j$).

Why this step? One equation per panel → $N$ equations for $N$ unknowns $\lambda_j$.

Step 5 — Solve the linear system

$\boxed{\;A\boldsymbol{\lambda} = \mathbf{b}\;}\qquad A_{ij}=\frac{I_{ij}}{2\pi},\quad b_i=-V_\infty\cos\beta_i$ Why this step? It's just a matrix inversion — the computer's job.

Step 6 — Post-process

Once $\lambda_j$ known, compute the tangential velocity at each panel, then pressure via Bernoulli: $C_{p,i} = 1 - \left(\frac{V_{t,i}}{V_\infty}\right)^2$ Integrate $C_p$ around the surface to get lift (need vortices/Kutta condition for lift) and moment.

4. CFD overview — when panels are not enough

5. Common mistakes (Steel-man + fix)

6. Active recall

Connections

• Laplace's Equation & Potential Flow
• Sources, Sinks, Doublets, Vortices
• Kutta–Joukowski Theorem & Circulation
• Bernoulli's Equation
• Navier–Stokes Equations
• Boundary Layer Theory & Skin Friction Drag
• Turbulence Modelling — RANS, LES, DNS
• Finite Volume Method
• d'Alembert's Paradox

Concept Map

Hinglish (regional understanding)