🌬️ Prandtl-Glauert Compressibility Correction Calculator
Apply Prandtl-Glauert, Kármán-Tsien, and Laitone corrections to pressure and lift coefficients for subsonic compressible flow. For supersonic flow, Ackeret linearized theory is used. Reference: §1.11, §7.10.
🌬️ Compressibility Effect Schematic
📝 Configuration
Key Equations:
Prandtl-Glauert:
$C_p(M)=\dfrac{C_{p,0}}{\sqrt{1-M_\infty^2}}$
Kármán-Tsien:
$C_p(M)=\dfrac{C_{p,0}}{\sqrt{1-M^2+\frac{M^4(C_{p,0}+1)^2}{4}}}$
Laitone:
$C_p(M)=\dfrac{C_{p,0}}{\beta+\frac{M^2 C_{p,0}/2}{1+\beta}}$
Ackeret (supersonic):
$C_L=\dfrac{4\alpha}{\sqrt{M^2-1}},\;C_{d,w}=\dfrac{4\alpha^2}{\sqrt{M^2-1}}$
Prandtl-Glauert:
$C_p(M)=\dfrac{C_{p,0}}{\sqrt{1-M_\infty^2}}$
Kármán-Tsien:
$C_p(M)=\dfrac{C_{p,0}}{\sqrt{1-M^2+\frac{M^4(C_{p,0}+1)^2}{4}}}$
Laitone:
$C_p(M)=\dfrac{C_{p,0}}{\beta+\frac{M^2 C_{p,0}/2}{1+\beta}}$
Ackeret (supersonic):
$C_L=\dfrac{4\alpha}{\sqrt{M^2-1}},\;C_{d,w}=\dfrac{4\alpha^2}{\sqrt{M^2-1}}$
📊 Results & Visualization
Configure inputs and click Calculate.
ℹ️ About Compressibility Corrections
At subsonic speeds, increasing Mach number amplifies pressure coefficients. Three classical corrections exist:
- Prandtl-Glauert (simplest, linear): Cp(M) = Cp₀/√(1−M²)
- Kármán-Tsien (more accurate near M=1): includes nonlinear correction
- Laitone (most accurate for higher subsonic M): second-order correction
For supersonic flow, Ackeret linearized theory gives Cp = ±2α/√(M²−1).
At subsonic speeds, increasing Mach number amplifies pressure coefficients. Three classical corrections exist:
- Prandtl-Glauert (simplest, linear): Cp(M) = Cp₀/√(1−M²)
- Kármán-Tsien (more accurate near M=1): includes nonlinear correction
- Laitone (most accurate for higher subsonic M): second-order correction
For supersonic flow, Ackeret linearized theory gives Cp = ±2α/√(M²−1).
📘 Calculation Methodology
Theory
For thin airfoils in subsonic compressible flow, the linearized potential equation yields three correction rules of increasing accuracy:
$$\text{Prandtl-Glauert: } C_p = \frac{C_{p,0}}{\sqrt{1-M_\infty^2}}$$
$$\text{Kármán-Tsien: } C_p = \frac{C_{p,0}}{\sqrt{1-M^2+\frac{M^4(C_{p,0}+1)^2}{4}}}$$
$$\text{Laitone: } C_p = \frac{C_{p,0}}{\beta + \frac{M^2 C_{p,0}/2}{1+\beta}}$$
$$\text{Ackeret (supersonic): } C_L = \frac{4\alpha}{\sqrt{M^2-1}},\; C_{d,w} = \frac{4\alpha^2}{\sqrt{M^2-1}}$$
The lift-curve slope for a thin airfoil is corrected as dCL/dα = 2π/β (subsonic) or 4/β (supersonic, Ackeret).
Worked Example
Problem: Cp₀ = −1.0, CL₀ = 0.5 at M∞ = 0.6 (air, γ = 1.4).
1. β = √(1 − 0.36) = 0.8000
2. Prandtl-Glauert:
Cp = −1.0 / 0.8 = −1.2500
3. Kármán-Tsien:
denom = √(0.64 + 0.1296×0/4) = 0.8
Cp ≈ −1.2500 (identical when Cp₀+1=0)
4. Laitone:
denom = 0.8 + 0.36×(−0.5)/1.8 = 0.7
Cp = −1.0/0.7 = −1.4286
5. CL = 0.5/0.8 = 0.6250
6. dCL/dα = 2π/0.8 = 7.854 /rad
1. β = √(1 − 0.36) = 0.8000
2. Prandtl-Glauert:
Cp = −1.0 / 0.8 = −1.2500
3. Kármán-Tsien:
denom = √(0.64 + 0.1296×0/4) = 0.8
Cp ≈ −1.2500 (identical when Cp₀+1=0)
4. Laitone:
denom = 0.8 + 0.36×(−0.5)/1.8 = 0.7
Cp = −1.0/0.7 = −1.4286
5. CL = 0.5/0.8 = 0.6250
6. dCL/dα = 2π/0.8 = 7.854 /rad
Assumptions & References
Assumptions: Steady, irrotational, isentropic. Small perturbations from freestream. Thin airfoil. 2D flow. Calorically perfect gas. Corrections become inaccurate as M → 1 (transonic regime).
References:
- Anderson, J. D. Fundamentals of Aerodynamics, McGraw-Hill — Ch. 11 (Compressibility corrections).
- Gupta, S. C. Applied Computational Fluid Dynamics — §1.11, §7.10.
- Shapiro, A. H. The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. 2, Wiley.
- Ackeret, J. Über Luftkräfte bei sehr großen Geschwindigkeiten, Helvetica Physica Acta, 1928.