๐ Transonic Similarity Calculator โ K Parameter
Compute the transonic similarity parameter K = (1โMยฒ)/ฯ2/3, visualize the Prandtl-Glauert singularity at Mach 1, estimate drag-divergence Mach (Korn equation), and classify the flow regime. Reference: ยง7.10.
๐ Transonic Flow Schematic
๐ Configuration
Key Equations:
Transonic similarity parameter:
$K = \dfrac{1 - M_\infty^2}{\tau^{2/3}}$
Scaled Cp:
$\bar{C}_p = \dfrac{C_p}{\tau^{2/3}}$
Korn equation:
$M_{dd} + \dfrac{C_L}{10} + \dfrac{t}{c} = \kappa$
$\kappa = 0.87$ (conventional), $0.95$ (supercritical)
Transonic similarity parameter:
$K = \dfrac{1 - M_\infty^2}{\tau^{2/3}}$
Scaled Cp:
$\bar{C}_p = \dfrac{C_p}{\tau^{2/3}}$
Korn equation:
$M_{dd} + \dfrac{C_L}{10} + \dfrac{t}{c} = \kappa$
$\kappa = 0.87$ (conventional), $0.95$ (supercritical)
๐ Results & Visualization
Configure inputs and click Calculate.
โน๏ธ About Transonic Similarity
Near Mach 1, the Prandtl-Glauert correction diverges to infinity. The transonic similarity parameter K provides a framework to compare different airfoils in this regime.
- K >> 0: subsonic-like behavior
- K โ 0: transonic regime โ nonlinear effects dominate
- K << 0: supersonic-like behavior
The Korn equation estimates the drag-divergence Mach number Mdd.
Near Mach 1, the Prandtl-Glauert correction diverges to infinity. The transonic similarity parameter K provides a framework to compare different airfoils in this regime.
- K >> 0: subsonic-like behavior
- K โ 0: transonic regime โ nonlinear effects dominate
- K << 0: supersonic-like behavior
The Korn equation estimates the drag-divergence Mach number Mdd.
๐ Calculation Methodology
Theory
The transonic small-disturbance (TSD) equation, after scaling, yields a single governing parameter K:
$$K = \frac{1 - M_\infty^2}{\tau^{2/3}}$$
$$\bar{C}_p = \frac{C_p}{\tau^{2/3}} = f(K) \quad\text{(universal function)}$$
$$\text{TSD: } (1-M^2)\phi_{xx} + \phi_{yy} = M^2(\gamma+1)\phi_x\phi_{xx}$$
$$\text{Korn: } M_{dd} + \frac{C_L}{10} + \frac{t}{c} = \kappa$$
The Korn equation is empirical: ฮบ โ 0.87 for conventional airfoils, โ 0.95 for supercritical designs (NASA SC series).
Worked Example
Problem: Conventional airfoil, ฯ = 0.12, CLโ = 0.5, Cpโ = โ1.0, Mโ = 0.8
1. ฯ2/3 = 0.120.667 = 0.2432
2. K = (1 โ 0.64) / 0.2432 = 1.480 โ subsonic-like
3. ฮฒ = โ(1โ0.64) = 0.6000
4. Cp_PG = โ1.0/0.6 = โ1.6667
5. Korn (conv): Mdd = 0.87 โ 0.05 โ 0.12 = 0.700
6. Korn (SC): Mdd = 0.95 โ 0.05 โ 0.12 = 0.780
Since Mโ = 0.8 > Mdd,conv, this airfoil is past drag divergence.
1. ฯ2/3 = 0.120.667 = 0.2432
2. K = (1 โ 0.64) / 0.2432 = 1.480 โ subsonic-like
3. ฮฒ = โ(1โ0.64) = 0.6000
4. Cp_PG = โ1.0/0.6 = โ1.6667
5. Korn (conv): Mdd = 0.87 โ 0.05 โ 0.12 = 0.700
6. Korn (SC): Mdd = 0.95 โ 0.05 โ 0.12 = 0.780
Since Mโ = 0.8 > Mdd,conv, this airfoil is past drag divergence.
Assumptions & References
Assumptions: Steady, irrotational, isentropic. Small perturbations. Thin airfoil (ฯ << 1). 2D flow. Perfect gas. TSD uses mixed-type PDE โ solved numerically in practice.
- Anderson, J. D. Fundamentals of Aerodynamics โ Ch. 11 ยง11.6.
- Gupta, S. C. Applied CFD โ ยง7.10.
- Korn, D. G. NASA TN D-6643, 1972.
- Cole, J. D. & Cook, L. P. Transonic Aerodynamics, 1986.