Question

In which equation is total velocity and it double derivative substituted to obtain the perturbation velocity potential equation?

(a) Momentum equation

(b) Velocity potential equation

(c) Perturbation equation

(d) Enthalpy equation

This question was posed to me in an online interview.

This question is from Linearized Velocity Potential Equation topic in section Linearized and Conical Flows of Aerodynamics

Answer 1

Correct answer is (b) Velocity potential equation

Best explanation: The velocity potential equation is given by:

(1 – \(\frac {Φ_x^2}{a^2}\)) Φxx + (1 – \(\frac {Φ_y^2}{a^2}\)) Φyy + (1 – \(\frac {Φ_z^2}{a^2}\)) Φzz – (\(\frac {2Φ_x Φ_y}{a^2}\)) Φxy – (\(\frac {2Φ_x Φ_z}{a^2}\)) Φxz – (\(\frac {2Φ_y Φ_z}{a^2}\)) Φyz

The total velocity potential is related to the perturbation velocity potential by:

Φx = V∞ + Φx, Φy = ϕy, Φz = ϕz

And its double derivative is given by

Φxx = ϕxx, Φyy = ϕyy, Φzz = ϕzz

Substituting these values in the velocity potential equation and multiplying it with a^2 we get

(a^2 – (V∞ + ϕx)^2)ϕxx + (a^2 – ϕy^2) ϕyy + (a^2 – ϕz^2) ϕzz – (2(V∞ + ϕx)ϕy)ϕxy – (2(V∞ + ϕx)ϕz)ϕxz – (2ϕyϕz) ϕyz

The above equation is known as the perturbation velocity potential equation which is a non linear equation.