validation.md

Validation Cases

Phase 1 Validation

Case 1: Impulsively Started Flat Plate

A flat plate at angle of attack $\alpha$ impulsively set into motion. The analytical solution for the initial lift (Wagner function) is known:

$$C_L(t) = 2\pi\alpha \cdot \Phi(s)$$

where $s = 2U_\infty t / c$ is the reduced time and $\Phi(s)$ is the Wagner function:

$$\Phi(s) \approx 1 - 0.165 e^{-0.0455 s} - 0.335 e^{-0.3 s}$$

Compare your solver's $C_L(t)$ against this. At $t \to \infty$, it should approach $C_L = 2\pi\alpha$ (thin airfoil theory).

Parameters: $\alpha = 5°$, $N_p = 100$ panels, $\Delta t^* = 0.01$, run for $s = 20$ chord lengths of travel.

Case 2: NACA 0012 Steady-State Lift

Run until the lift coefficient converges. Compare against:

• Thin airfoil theory: $C_L = 2\pi\alpha$
• Experimental data (Abbott & von Doenhoff): $C_L \approx 0.11$ per degree at low $\alpha$

Parameters: $\alpha = 4°$, $N_p = 100-200$ panels. Expected $C_L \approx 0.44$.

Case 3: Pitching Airfoil (Theodorsen Function)

Sinusoidal pitching: $\alpha(t) = \alpha_0 + \alpha_1 \sin(2k t^*)$ where $k = \omega c / (2U_\infty)$ is the reduced frequency.

The analytical solution for the lift is given by the Theodorsen function $C(k)$:

$$C_L(t) = 2\pi\left[\alpha_{3/4}(t) C(k) + \frac{\dot{h}}{U_\infty} + \frac{c}{4U_\infty}\dot{\alpha}\right] + \pi\frac{c}{2U_\infty}\left[\dot{\alpha} + \frac{c}{4U_\infty}\ddot{\alpha}\right]$$

The Theodorsen function (complex-valued, using Hankel functions) captures the effect of the unsteady wake:

$$C(k) = \frac{H_1^{(2)}(k)}{H_1^{(2)}(k) + iH_0^{(2)}(k)}$$

Compare your solver's amplitude and phase of the unsteady lift against the Theodorsen solution for $k = 0.1, 0.5, 1.0$.

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Phase 2 Validation

Accuracy: For $N = 5000$ random vortices, compute velocity on all particles using both direct summation and treecode. Report max and RMS relative error vs. $\theta$ and expansion order $p$.

Performance: Wall-clock time vs. $N$ for $N = 10^3, 10^4, 10^5, 10^6$. Plot on log-log scale. Should see:

• Direct: slope = 2 (O(N²))
• Treecode: slope ≈ 1.0-1.3 (O(N log N))

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Phase 3 Validation

Tandem Airfoils: Rear airfoil in the wake of the front. At sufficient separation, the rear airfoil should experience reduced effective angle of attack (downwash from front airfoil's wake). Qualitative comparison with potential flow theory and experimental data.

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Phase 4 Validation

Elliptic Wing: A wing with elliptic planform has a known analytical solution:

$$C_{D,i} = \frac{C_L^2}{\pi AR}$$

where $AR$ is the aspect ratio. Compare your computed induced drag against this.

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Performance Targets

Phase	N particles	Expected time per step	Hardware
1 (O(N²), GPU)	10,000	~1 ms	Modern GPU (RTX 3080 class)
1 (O(N²), GPU)	100,000	~100 ms	Modern GPU
2 (Treecode, GPU)	100,000	~5-10 ms	Modern GPU
2 (Treecode, GPU)	1,000,000	~50-100 ms	Modern GPU

These are rough targets. Actual performance depends heavily on implementation quality, memory access patterns, and GPU model.