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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^‘)kThe 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»

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)


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