Question

What is the velocity potential for a slender body in uniform flow with perturbations?

(a) Φ(x, y, z) = V∞ x + ϕ(x, y, z)

(b) Φ(x, y, z) = V∞ z + ϕ(x, y, z)

(c) Φ(x, y, z) = V∞ y + ϕ(x, y, z)

(d) ∇Φ = u^‘i + v^‘j + (V∞ + w^‘)k

The question was asked by my college director while I was bunking the class.

Query is from Linearized Velocity Potential Equation topic in portion Linearized and Conical Flows of Aerodynamics

Answer 1

Right answer is (a) Φ(x, y, z) = V∞ x + ϕ(x, y, z)

Easy explanation: When the body is placed in a uniform flow, the y and z components of the local velocity are zero. Since the velocity potential is given by V = ∇Φ and the local velocity is given by V = (V∞ + u^‘)i + v^‘j + w^‘k, we can use perturbation velocity potential to derive the relation.

Perturbation velocity potential is related to the perturbations in x, y, z components as follows:

\(\frac {∂ϕ}{∂x}\) = u^‘, \(\frac {∂ϕ}{∂y}\) = v^‘, \(\frac {∂ϕ}{∂z}\) = w^‘

Substituting this in the equation V = ∇Φ = (V∞ + u^‘)i + v^‘j + w^‘k we get,

Φ(x, y, z) = V∞ x + ϕ(x, y, z)