๐ Critical Pressure Coefficient
Calculate critical pressure coefficient (Cp*) for airfoils in compressible flow systems.
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โ๏ธ Airfoil Cp Distribution Schematic
๐ Configuration
Key Equations:
Critical Cp*:
$C_p^* = \dfrac{2}{\gamma M_\infty^2}\left[\left(\dfrac{2}{\gamma+1}\left(1+\dfrac{\gamma-1}{2}M_\infty^2\right)\right)^{\!\gamma/(\gamma-1)}\!\!-1\right]$
Bounds:
$C_{p,\text{vac}} = \dfrac{-2}{\gamma M_\infty^2}, \quad C_{p,\text{stag}} = \dfrac{2}{\gamma M_\infty^2}\left[\left(1+\dfrac{\gamma-1}{2}M_\infty^2\right)^{\!\gamma/(\gamma-1)}\!\!-1\right]$
Critical Mach (Prandtl-Glauert):
$C_p^*(M_{cr}) = \dfrac{C_{p,\min,\text{inc}}}{\sqrt{1-M_{cr}^2}}$
Critical Cp*:
$C_p^* = \dfrac{2}{\gamma M_\infty^2}\left[\left(\dfrac{2}{\gamma+1}\left(1+\dfrac{\gamma-1}{2}M_\infty^2\right)\right)^{\!\gamma/(\gamma-1)}\!\!-1\right]$
Bounds:
$C_{p,\text{vac}} = \dfrac{-2}{\gamma M_\infty^2}, \quad C_{p,\text{stag}} = \dfrac{2}{\gamma M_\infty^2}\left[\left(1+\dfrac{\gamma-1}{2}M_\infty^2\right)^{\!\gamma/(\gamma-1)}\!\!-1\right]$
Critical Mach (Prandtl-Glauert):
$C_p^*(M_{cr}) = \dfrac{C_{p,\min,\text{inc}}}{\sqrt{1-M_{cr}^2}}$
๐ Results & Visualization
Configure inputs and click Calculate to view results.
โน๏ธ About the Critical Pressure Coefficient Cpโ
The critical pressure coefficient Cp* is the value of the pressure coefficient at which the local flow reaches sonic speed (Mlocal = 1).
- If $C_p$ on the airfoil surface exceeds Cp* (more negative), the flow is locally supersonic.
- The critical Mach number Mcr is the freestream Mach at which the minimum Cp on the airfoil first reaches Cp*. Beyond Mcr, a supersonic pocket forms on the surface.
- Cp is bounded: Cpvacuum โค Cp โค Cpstagnation.
The critical pressure coefficient Cp* is the value of the pressure coefficient at which the local flow reaches sonic speed (Mlocal = 1).
- If $C_p$ on the airfoil surface exceeds Cp* (more negative), the flow is locally supersonic.
- The critical Mach number Mcr is the freestream Mach at which the minimum Cp on the airfoil first reaches Cp*. Beyond Mcr, a supersonic pocket forms on the surface.
- Cp is bounded: Cpvacuum โค Cp โค Cpstagnation.
๐ Calculation Methodology
Mathematical Model & Theory
Cp* measures the onset of locally sonic flow on a body in a compressible freestream. Using isentropic relations between freestream and a surface point where Mlocal=1:
$$C_p^* = \frac{2}{\gamma M_\infty^2}\!\left[\left(\frac{2}{\gamma+1}\!\left(1+\frac{\gamma-1}{2}M_\infty^2\right)\!\right)^{\!\gamma/(\gamma-1)}\!\!-1\right]$$
$$C_{p,\text{vac}} = \frac{-2}{\gamma M_\infty^2}, \quad C_{p,\text{stag}} = \frac{2}{\gamma M_\infty^2}\!\left[\!\left(1+\frac{\gamma-1}{2}M_\infty^2\right)^{\!\gamma/(\gamma-1)}\!\!-1\right]$$
$$M_{cr}:\; C_p^*(M_{cr}) = \frac{C_{p,\min,\text{inc}}}{\sqrt{1-M_{cr}^2}}$$
Worked Engineering Example
Problem:
An airfoil has Cp,min,inc = โ1.2 in air (ฮณ=1.4). At Mโ=0.7, is the flow locally supersonic?
Solution:
1. Compute Cp* at Mโ=0.7:
$\text{factor} = 1+0.2(0.49) = 1.098$
$p^*/p_\infty = (2/2.4 \times 1.098)^{3.5} = 0.8929$
$C_p^* = \frac{2}{1.4\times0.49}(0.8929-1) = -0.3120$
2. Cp_min at M=0.7 (Prandtl-Glauert):
$C_{p,\min} = -1.2/\sqrt{1-0.49} = -1.680$
3. Since โ1.680 < โ0.312 (more negative), flow is locally supersonic at Mโ=0.7.
An airfoil has Cp,min,inc = โ1.2 in air (ฮณ=1.4). At Mโ=0.7, is the flow locally supersonic?
Solution:
1. Compute Cp* at Mโ=0.7:
$\text{factor} = 1+0.2(0.49) = 1.098$
$p^*/p_\infty = (2/2.4 \times 1.098)^{3.5} = 0.8929$
$C_p^* = \frac{2}{1.4\times0.49}(0.8929-1) = -0.3120$
2. Cp_min at M=0.7 (Prandtl-Glauert):
$C_{p,\min} = -1.2/\sqrt{1-0.49} = -1.680$
3. Since โ1.680 < โ0.312 (more negative), flow is locally supersonic at Mโ=0.7.
Assumptions & References
Assumptions: Steady, isentropic flow from freestream to the surface point. Calorically perfect gas. Prandtl-Glauert compressibility correction for Cpmin(M). Thin airfoil / small perturbation for Mcr estimation.
References:
- Anderson, J. D. Fundamentals of Aerodynamics, McGraw-Hill โ Ch. 11 (Subsonic compressible flow, critical Mach number).
- Gupta, S. C. Applied Computational Fluid Dynamics โ ยง7.9 (Critical pressure coefficient).
- Abbott, I. H. & Von Doenhoff, A. E. Theory of Wing Sections, Dover.