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Aerodynamics

solid pillar 1 · Physical modeling and domain theory

In one minute

  • Lift comes from circulation: \(L' = \rho U \Gamma\). The Kutta condition is what selects \(\Gamma\) for a given shape and incidence.
  • Thin airfoil theory gives \(c_\ell = 2\pi\alpha\) and the quarter-chord aerodynamic centre — good to within a few percent below stall, useless above it.
  • Induced drag is the price of finite span: \(C_{D_i} = C_L^2/(\pi e \mathrm{AR})\). It falls with speed while profile drag rises, which sets minimum-drag speed.
  • Prandtl–Glauert corrects for compressibility until it does not — it is singular at \(M=1\) and unusable transonically.
  • Potential-flow methods cannot predict separation or maximum lift. Those need viscous physics.

Key results

Kutta–Joukowski, and thin airfoil theory for a cambered section:

\[ L' = \rho U_\infty \Gamma, \qquad c_\ell = 2\pi(\alpha - \alpha_{L=0}), \qquad x_{ac} = \frac{c}{4} \]

The lift-curve slope \(2\pi\) per radian (\(\approx 0.11\) per degree) is the reference every real airfoil is measured against; real sections reach 90–95% of it.

Finite wing — Prandtl lifting line:

\[ C_L = \frac{C_{L_\alpha,\,2\mathrm{D}}}{1 + \frac{C_{L_\alpha,\,2\mathrm{D}}}{\pi e \mathrm{AR}}}\,\alpha, \qquad C_{D_i} = \frac{C_L^2}{\pi e\, \mathrm{AR}}, \qquad \alpha_i = \frac{C_L}{\pi e\,\mathrm{AR}} \]

Elliptical loading gives \(e = 1\) and minimum induced drag; typical wings reach \(e \approx 0.7\text{–}0.85\).

Drag decomposition:

Component Scales as Dominant when
Induced \(C_L^2/\mathrm{AR}\), so \(\propto 1/V^2\) slow, heavy, high \(C_L\)
Profile (skin friction + form) \(\propto V^2\) fast, low \(C_L\)
Wave onset above \(M_{crit}\) transonic and above

Minimum drag occurs where induced equals profile drag — which is also where \(L/D\) is maximum.

Drag polar:

\[ C_D = C_{D_0} + \frac{C_L^2}{\pi e \mathrm{AR}}, \qquad \left(\frac{L}{D}\right)_{\max} = \frac{1}{2}\sqrt{\frac{\pi e \mathrm{AR}}{C_{D_0}}} \]

Compressibility. Prandtl–Glauert for subsonic flow:

\[ c_p = \frac{c_{p,0}}{\sqrt{1-M_\infty^2}} \]

Critical Mach \(M_{crit}\) is where local flow first reaches sonic; drag divergence follows shortly after. Sweep delays both by acting on the normal component: \(M_{\text{eff}} \approx M_\infty\cos\Lambda\).

Reynolds effects. \(C_{L,\max}\) and stall behaviour depend strongly on \(\mathrm{Re}\) — transition location changes, and a laminar separation bubble at low \(\mathrm{Re}\) can halve maximum lift. Wind-tunnel data at \(\mathrm{Re}=10^6\) does not transfer to flight at \(10^7\).

Mental model

A wing deflects air downwards; the reaction is lift. The downwash it generates at its own location tilts the local flow, so the effective incidence is less than the geometric one, and the lift vector tilts backwards by that angle — that backward component is induced drag. Nothing is lost to friction; the energy goes into the trailing vortex system.

That picture explains why span matters more than area for induced drag, why gliders have enormous aspect ratio, and why formation flight saves fuel.

Numerics / practice

  • Panel methods (vortex/source/doublet) solve potential flow cheaply and are excellent for attached, thin-boundary-layer cases. Coupled to an integral boundary-layer method (as XFOIL does) they predict separation onset reasonably.
  • Report \(\mathrm{Re}\) and \(M\) with any aerodynamic coefficient. Without both, the number is not interpretable.
  • Check \(e\) against the planform rather than assuming 1. Taper, twist and fuselage interference all reduce it.
  • For CFD, see fluids › CFD; the failure modes there apply directly.
Failure modes

Thin airfoil theory near stall. \(c_\ell = 2\pi\alpha\) has no maximum — extrapolate it past about 10–12° and it predicts lift a real wing cannot produce. It is a linear theory with no viscous physics.

Prandtl–Glauert approaching \(M=1\). The \(1/\sqrt{1-M^2}\) factor diverges. Above roughly \(M=0.7\) it is already poor and transonically it is meaningless. Symptom: pressure coefficients growing without bound as the design Mach rises.

Lifting line at low aspect ratio. The theory assumes high \(\mathrm{AR}\) and small sweep. Below \(\mathrm{AR}\approx4\), or on strongly swept or delta wings, it is wrong in kind — those wings generate lift partly through leading-edge vortices, a mechanism the theory does not contain.

2D section data applied directly to a 3D wing. Ignores downwash, so it overpredicts lift-curve slope and omits induced drag entirely. The correction is not small: at \(\mathrm{AR}=8\) the 3D slope is about 80% of 2D.

Reynolds number mismatch. Sub-scale tunnel data at \(\mathrm{Re}\sim10^5\)–\(10^6\) can show laminar separation bubbles absent at flight \(\mathrm{Re}\). \(C_{L,\max}\) is the most affected quantity and the one most often transferred uncorrected.

Potential-flow methods asked to predict separation. They will return an answer — a smooth, attached, plausible one — at 25° incidence. There is no mechanism in the formulation for the flow to leave the surface.

Ignoring trim drag. The tail download to trim reduces effective lift and adds drag. Cruise polars built from wing-alone data are optimistic.

Worked example

Where minimum drag sits, and why it is also best \(L/D\):

import numpy as np

CD0, e, AR = 0.022, 0.80, 9.0
k  = 1.0 / (np.pi * e * AR)
CL = np.sqrt(CD0 / k)                    # induced drag == profile drag
LD = CL / (CD0 + k * CL**2)
print(f"k            = {k:.5f}")
print(f"CL at min drag = {CL:.3f}")
print(f"(L/D)max       = {LD:.2f}")
print(f"check formula  = {0.5*np.sqrt(np.pi*e*AR/CD0):.2f}")

k            = 0.04421
CL at min drag = 0.705
(L/D)max       = 16.03
check formula  = 16.03

Best \(L/D\) occurs at exactly the \(C_L\) where the two drag contributions are equal — which is why a clean, high-aspect-ratio wing flies its best-range speed slower than intuition suggests.

Connections

Sources

  • Anderson, Fundamentals of Aerodynamics.
  • Katz & Plotkin, Low-Speed Aerodynamics — panel methods.
  • Archive: see the course archive for the Aerospace folders.