Computational aerodynamics — panel method (intro), CFD overview

3.1.30 · Physics › Compressible Flow & Aerodynamics

1. CFD methods ki ZAROORAT kyun hai?

Laplace kyun? Incompressible $\Rightarrow \nabla \cdot \vec{V}=0$. Irrotational $\Rightarrow \vec{V}=\nabla \varphi$. Combine karo: $\nabla \cdot (\nabla \varphi )={\nabla }^2\varphi =0$.

Real flow ke liye KYA badalta hai? Viscosity ($\mu {\nabla }^2\vec{V}$ term) aur compressibility ($\rho$ varies) add karo aur tumhe full Navier–Stokes equations milti hain, jo nonlinear hain — koi superposition nahi, CFD use karna zaroori.

2. Elementary solutions (LEGO bricks)

Kyunki Laplace linear hai, inme se kisi bhi sum ko bhi solution kehte hain:

3. Panel method — scratch se DERIVATION

Step 1 — Physical condition jo hume enforce karni hai

Body solid hai: koi flow wall ke through nahi jaati. Isliye surface ke normal velocity component zero hai: $\vec{V}\cdot \hat{n}=0\text{on the body.}$ Yeh step kyun? Yeh woh akela boundary condition hai jo shape ka effect define karta hai — sab kuch isi ko satisfy karne ke liye bana hai.

Step 2 — Surface ko discretise karo

Smooth contour ko $N$ flat panels se replace karo. Har panel $j$ pe unknown strength ${\lambda }_j$ (constant) ki ek source sheet daalo. Yeh step kyun? Hum haath se continuous $\lambda (s)$ nahi dhundh sakte, lekin $N$ unknown constants ek solvable linear algebra problem dete hain.

Step 3 — Total potential

$\varphi (P)=V_{\infty }(x\cos \alpha +y\sin \alpha )+{\sum }_{j=1}^N\frac{{\lambda }_j}{2\pi }{\int }_{\text{panel }j}\ln rd{s}_j$ Yeh step kyun? Superposition: freestream + har panel ka contribution. Integral ${\lambda }_j$ ko panel ke upar spread karta hai.

Step 4 — Har control point pe boundary condition apply karo

Panel $i$ ke midpoint ("control point") pe, zero normal velocity demand karo: $V_{\infty }\cos {\beta }_i+{\sum }_{j=1}^N\frac{{\lambda }_j}{2\pi }{I}_{ij}=0,i=1\dots N$ jahan ${\beta }_i$ freestream aur panel ke outward normal ke beech ka angle hai, aur $I_{ij}=\frac{\partial }{\partial n_i}{\int }_j\ln rd{s}_j$ influence coefficient hai (panel $j$ pe unit source se panel $i$ par induced normal velocity).

Yeh step kyun? Har panel ke liye ek equation → $N$ unknowns ${\lambda }_j$ ke liye $N$ equations.

Step 5 — Linear system solve karo

$\boxed{A\mathbf{\lambda }=\mathbf{b}}{A}_{ij}=\frac{I_{ij}}{2\pi },b_i=-V_{\infty }\cos {\beta }_i$ Yeh step kyun? Yeh sirf ek matrix inversion hai — computer ka kaam.

Step 6 — Post-process karo

Jab ${\lambda }_j$ pata chal jaaye, har panel par tangential velocity compute karo, phir Bernoulli se pressure: $C_{p,i}=1-{\left(\frac{V_{t,i}}{V_{\infty }}\right)}^2$ Lift ke liye surface ke around $C_p$ integrate karo (lift ke liye vortices/Kutta condition chahiye) aur moment ke liye bhi.

4. CFD overview — jab panels enough nahi hote

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