📐 Local Pressure Coefficient (Cp)
Calculate local pressure coefficient from velocity/pressure differences in incompressible and compressible flows.
⚡ Fortran 90 Engine
Double Precision (IEEE 754)
✓ ISO / ASME Validated
Cfd
📊 Solver Telemetry
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📥 Pressures & Freestream Flow
📖 Mathematical Relations:
$$C_{p,0} = \frac{p - p_\infty}{\frac{1}{2}\rho V_\infty^2}, \quad M_\infty = \frac{V_\infty}{\sqrt{\gamma p_\infty / \rho}}$$
$$C_{p,\text{PG}} = \frac{C_{p,0}}{\sqrt{1 - M_\infty^2}}, \quad C_{p,\text{KT}} = \frac{C_{p,0}}{\sqrt{1-M_\infty^2} + \frac{M_\infty^2}{1+\sqrt{1-M_\infty^2}}\frac{C_{p,0}}{2}}$$
-1.3265
Incompressible $C_{p,0}$
-1.7388
Prandtl-Glauert ($C_{p,\text{PG}}$)
-2.1902
Kármán-Tsien ($C_{p,\text{KT}}$)
M = 0.646
Freestream Mach ($M_\infty$)
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Aerodynamic Flow Diagnostic
At $M_\infty = \mathbf{0.646}$ and dynamic pressure $q_\infty = \mathbf{29.65\ \text{kPa}}$, this surface point has $\Delta p = \mathbf{-39.33\ \text{kPa}$.}$
This corresponds to a suction / expansion region. Due to compressibility, the suction intensity amplifies by 65.1% under Kármán-Tsien correction ($C_p = -2.1902$).
📈 Compressibility Amplification: $C_p$ vs Mach Number $M_\infty$
🟢 Kármán-Tsien | 🔵 Prandtl-Glauert🔍 View Raw GNU Fortran Double-Precision Solver Output
MODE=1 MODE_NAME=Single Point P= 6.20000000E+04 P_INF= 1.01325000E+05 RHO= 1.22500000E+00 V_INF= 2.20000000E+02 GAMMA= 1.40000000E+00 Q_INF= 2.96450000E+04 CP= -1.326531 MACH= 0.646500 INTERPRETATION=Favorable pressure gradient (suction)
📘 Calculation Methodology: Pressure Coefficient & Compressibility Corrections
Mathematical Model & Theory
The pressure coefficient $C_p$ measures local static pressure relative to freestream dynamic pressure. Linear compressibility rules (Prandtl-Glauert, Kármán-Tsien) correct for subsonic density changes:
$$C_p = \frac{p - p_\infty}{\frac{1}{2}\rho_\infty V_\infty^2}, \quad C_{p,PG} = \frac{C_{p,0}}{\sqrt{1 - M_\infty^2}}$$
$$C_{p,KT} = \frac{C_{p,0}}{\sqrt{1 - M_\infty^2} + \frac{M_\infty^2}{1 + \sqrt{1 - M_\infty^2}}\frac{C_{p,0}}{2}}$$
Assumptions
- Inviscid, irrotational potential flow with small perturbations.
- Subcritical Mach number without local shock waves ($M_\infty < M_{cr}$).
Academic References
- Anderson, J. D.: Fundamentals of Aerodynamics, McGraw-Hill, Ch. 11.
- Liepmann, H. W., & Roshko, A.: Elements of Gasdynamics, Dover.
Worked Engineering Example
Problem Statement:
An airfoil has low-speed $C_{p,0} = -1.25$. Calculate the Prandtl-Glauert corrected $C_p$ at $M_\infty = 0.65$.
Step-by-step Solution:
1. $\beta = \sqrt{1 - 0.65^2} = \sqrt{0.5775} \approx 0.7599$.
2. $C_{p,PG} = -1.25 / 0.7599 \approx -1.645$.
Final Result:
Corrected pressure coefficient is $C_p = \mathbf{-1.645}$.
An airfoil has low-speed $C_{p,0} = -1.25$. Calculate the Prandtl-Glauert corrected $C_p$ at $M_\infty = 0.65$.
Step-by-step Solution:
1. $\beta = \sqrt{1 - 0.65^2} = \sqrt{0.5775} \approx 0.7599$.
2. $C_{p,PG} = -1.25 / 0.7599 \approx -1.645$.
Final Result:
Corrected pressure coefficient is $C_p = \mathbf{-1.645}$.