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Prandtl-Glauert Compressibility Correction
Corrects incompressible pressure coefficients for subsonic compressibility effects as Mach number increases.
Interpretation
Prandtl‑Glauert correction: C_p = C_p0 / √(1 − M²). It adjusts pressure coefficients for compressibility at subsonic speeds. Example: C_p0=0.5, M=0.7 → C_p = 0.5/√(1−0.49)=0.5/0.714=0.700.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| C_p | Compressible pressure coefficient | |
| C_p0 | Incompressible pressure coefficient | |
| M | Freestream Mach number |
What it means
The Prandtl‑Glauert correction is a linearised compressibility correction for subsonic flow (M < ~0.7). It relates the pressure coefficient C_p in compressible flow to that in incompressible flow. This correction is important for predicting pressure distributions on airfoils and wings at higher subsonic speeds. It is used in aerodynamic design to estimate lift and drag coefficients with compressibility effects. However, it becomes invalid near transonic speeds (M > 0.7) due to non‑linearities. The correction also appears in the calculation of critical Mach number. Understanding this correction is essential for early design stages and for evaluating the onset of compressibility effects.
Worked example
Prandtl‑Glauert Correction – Two Examples
Real‑World| Parameter | Value |
|---|---|
| C_p0 | -0.3 |
| M | 0.3 |
| Parameter | Value |
|---|---|
| C_p0 | -0.5 |
| M | 0.6 |
Common mistakes
- Prandtl‑Glauert compressibility correction: C_p = C_p0 / √(1 − M²).
- Applies to subsonic (M<1) compressible flow.
- C_p0: Incompressible pressure coefficient.
- Diverges as M→1 – limitation of linear theory.
- Valid only for small perturbations (thin airfoils).
Applications
The Prandtl‑Glauert compressibility correction, C_p = C_p₀ / √(1 − M²), adjusts incompressible pressure coefficients for subsonic compressible flow. It is used in early airfoil design and in predicting the onset of compressibility effects (drag rise). Engineers apply this correction to estimate the pressure distribution at high subsonic speeds, which influences lift, drag, and pitching moments. While limited to M < ~0.7, it provides a quick approximation. By using this correction, aerospace engineers can assess the performance of wings and airfoils at transonic speeds and identify regions of potential shock formation.
- Preliminary design of high‑subsonic airfoils and wings
- Prediction of drag divergence Mach number
- Estimation of pressure distributions for structural loads
- Comparison with CFD results for validation
- Educational tool for compressibility effects
Frequently Asked Questions
It corrects incompressible pressure coefficients for subsonic compressibility effects as Mach number increases. It is a classical linearised theory correction for small perturbations.
Cp = compressible pressure coefficient
Cp0 = incompressible pressure coefficient (at M=0)
M = Mach number
It allows the use of incompressible flow results to estimate subsonic compressible flow behaviour, which is essential for preliminary aircraft design.
It is valid only for subsonic flows with small perturbations (thin airfoils, small angles of attack) and fails near M=1 where the singularity occurs.
- Applying the correction near or above M=1, where the singularity makes the formula invalid.
- Using the correction for thick airfoils or high angles of attack where nonlinear effects dominate.
- Forgetting that the correction is for pressure coefficients, not for lift or drag directly.
At M = 0.5, an airfoil has incompressible Cp0 = −0.4. Then Cp = −0.4 / √(1 − 0.25) = −0.4 / √0.75 = −0.4 / 0.866 = −0.462.
Lift coefficient increases by the same factor (1/√(1−M²)), and drag (wave drag) is not predicted by this correction; it requires more advanced methods.
Göthert’s rule accounts for the effect of the body’s volume on the compressibility correction, providing a more accurate result for non‑slender bodies.
Lift coefficient: CL = CL0/√(1−M²). Moment coefficient: CM = CM0/√(1−M²).
The correction becomes singular at M=1, which is never reached; the critical Mach number is where local sonic flow first appears, and the correction is not valid beyond that.