Prandtl Glauert⚠ unverified
Aerospace / Aerodynamics · Apply the Prandtl-Glauert compressibility correction
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| CL0 | CL0 | — | 1.0 | Incompressible lift coefficient (dimensionless) |
| M | M | — | 1.0 | Freestream Mach number (dimensionless) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | CL | — | Compressibility-corrected lift coefficient (dimensionless). Returns the uncorrected ``CL0`` when ``M >= 1`` |
The science & history
Understanding the Parameters
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Incompressible coefficient $C_{L0}$ — the value measured or predicted at low speed (from thin-airfoil theory or a low-Mach wind-tunnel test), where density changes are negligible.
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Mach number $M$ — the correction grows as $M\to 1$: at $M = 0.6$ the factor $1/\sqrt{1-0.36} = 1.25$ (25 % boost); at $M = 0.8$ it is $1.67$. This steepening is real up to the point where shocks appear.
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The singularity — at $M = 1$ the formula predicts infinite $C_L$, which is unphysical. It signals the breakdown of linearised subsonic theory as the flow becomes transonic; real wings encounter drag divergence and shock stall well before Mach 1.
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Same $\beta$ elsewhere — the factor $\beta = \sqrt{1-M^2}$ is the same compressibility parameter that enters the finite-wing lift-curve slope (Lift Curve Slope), tying the two together.
Derivation (Approaching a Proof)
For small disturbances in subsonic compressible flow, the linearised velocity-potential equation is
$$(1 - M^2)\,\phi_{xx} + \phi_{yy} = 0.$$
This is a stretched Laplace equation. The Prandtl-Glauert transformation rescales the streamwise (or lateral) coordinate by $\beta = \sqrt{1-M^2}$ to map it onto the incompressible Laplace equation $\phi_{\xi\xi} + \phi_{yy} = 0$. Solving the equivalent incompressible problem and transforming back multiplies the pressure coefficients — and hence the lift and moment coefficients — by $1/\beta$:
$$C_L = \frac{C_{L0}}{\sqrt{1 - M^2}}.$$
The derivation assumes small perturbations (thin bodies, small angles) and subsonic flow, which is exactly why it fails as $M\to1$ (the coefficient of $\phi_{xx}$ vanishes and the equation changes type from elliptic to hyperbolic).
Dimensional check. $C_L = C_{L0}/\sqrt{1-M^2}$ — $C_{L0}$ is dimensionless and $M$ is dimensionless, so $C_L$ is dimensionless, as a coefficient must be.
History and Development
The correction is named for Ludwig Prandtl and Hermann Glauert, who developed the subsonic compressibility scaling in the 1920s. It was the first practical tool for extending low-speed aerodynamic data to the higher speeds of 1930s–40s aircraft, and its singularity at Mach 1 dramatised the "sound barrier" — though the true transonic physics (shock waves, drag divergence) required the later Kármán-Tsien and Laitone refinements and, ultimately, transonic CFD. It remains the standard first-order compressibility estimate.
Related Concepts: Lift Curve Slope, Mach Number, Lift Force, Moment Coefficient, Speed Of Sound, Dynamic Pressure
Notes: Subsonic only, singular at $M=1$ (returns $C_{L0}$ for $M\ge1$). Accurate to $M\lesssim0.7$; transonic needs Kármán-Tsien/Laitone or CFD. $\beta=\sqrt{1-M^2}$ is the same factor in Lift Curve Slope. From the linearised subsonic potential equation.