Question

What is the relation between linearized pressure coefficient for compressible Cp and incompressible flow Cp,0?

(a) Cp = βCp,0

(b) Cp = \(\frac {C_{p.0}}{β}\)

(c) Cp = \(\frac {β}{C_{p.0}}\)

(d) Cp = β^2Cp,0

The question was asked in quiz.

Query is from Prandtl-Glauert Compressibility topic in chapter Subsonic Compressible Flow over Airfoils : Linear Theory, Linearized Supersonic Flow of Aerodynamics

Answer 1

The correct choice is (b) Cp = \(\frac {C_{p.0}}{β}\)

To explain: Linearized pressure coefficient for compressible flow Cp is derived using the Prandtl – Glauert compressibility equation. It is given by:

Cp =  –\(\frac {2\hat {u}}{V_∞}\)

Where, \(\hat {u} = \frac {∂\hat {ϕ}}{∂x}\) is the velocity perturbation

Cp =  –\(\frac {2}{V_∞} \frac {∂\hat {ϕ}}{∂x}\)

\(\frac {∂\hat {ϕ}}{∂x} =  -\frac {1}{β} \frac {∂\bar {ϕ}}{∂ξ}\)

Where, \(\bar {ϕ}\) is the perturbation velocity for incompressible flow

β = \(\sqrt {1 – M_∞^2}\)

Cp =  –\(\frac {2}{V_∞} \frac {1}{β} \frac {∂\bar {ϕ}}{∂ξ} = \frac {C_{p.0}}{β}\)

This relation helps in finding out the coefficient of pressure for a compressible flow using the incompressible pressure coefficient value.