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Compressibility & high-speed flow

Our solver assumes air has constant density — true below about Mach 0.3. Fly faster and the air noticeably compresses, changing the pressures and, eventually, the whole character of the flow. This article sketches the high-speed regime the demo deliberately leaves out.

The speed of sound and the Mach number

Pressure disturbances travel at the speed of sound, which for an ideal gas depends only on temperature:

a = \sqrt{\gamma R T}

(1)

$\gamma$: ratio of specific heats (1.4 for air) [—]
$R$: specific gas constant for air [J/(kg·K)]
$T$: absolute temperature [K]

At sea level ( $T = 288\,\text{K}$ ), $a = \sqrt{1.4 \times 287 \times 288} \approx 340\,\text{m/s}$ . The Mach number $M = V/a$ is then the key parameter: how fast you fly relative to how fast the air can “tell” the flow ahead that you are coming.

Subsonic compressibility: the Prandtl–Glauert rule

For attached subsonic flow (up to $M \approx 0.7$ ), compressibility amplifies the pressure coefficients predicted by an incompressible calculation by a simple factor:

C_p = \frac{C_{p,0}}{\sqrt{1 - M_\infty^2}}

(2)

where $C_{p,0}$ is the incompressible (low-speed) value. The lift coefficient scales the same way. So everything our solver computes at low speed can be approximately corrected to higher subsonic speeds just by dividing by $\sqrt{1 - M_\infty^2}$ .

Freestream Mach

M_\infty

0.50

\frac{1}{\sqrt{1 - 0.50^2}}

\times\, 1.15\ \text{amplification}

At sea level ( $a \approx 340\ \text{m/s}$ ) that is $\approx 170\ \text{m/s}$ . Pressures (and lift) are 15% larger than the incompressible estimate. The factor blows up as $M \to 1$ — the theory breaks down before then (drag divergence near $M \approx 0.7\text{–}0.8$ ).

The critical Mach number and drag divergence

Even when the aircraft is subsonic, the flow accelerating over the top of the wing can reach $M = 1$ locally. The freestream Mach at which this first happens is the critical Mach number $M_{cr}$ . Push past it and a region of supersonic flow forms, usually terminated by a shock wave. The shock thickens the boundary layer and can trigger separation, causing a sudden jump in drag — the drag-divergence Mach number, typically $M \approx 0.7\text{–}0.85$ . This is precisely where Prandtl–Glauert (which assumes no shocks) breaks down.