Lecture 16 Flashcards
(10 cards)
In corner flow for inviscid, incompressible flow (ω = 0), why is the potential flow form ϕ(r, θ) = r^λ f(θ)?
No natural length scale exists in the corner region, so we seek scale-invariant (self-similar) solutions. The form ϕ = r^λ f(θ) satisfies Laplace’s equation with this symmetry.
What boundary conditions are used for potential flow in corner geometries?
Normal velocity on the walls is zero (∂ϕ/∂n = 0), but tangential velocity can be nonzero. This reflects inviscid flow assumptions—no normal penetration but slip is allowed.
For a 90° corner, what is the smallest nontrivial λ that satisfies Laplace’s equation?
λ = π/α = π/(3π/2) = 2/3. This sets the power-law behavior near the corner in the potential flow solution.
How does pressure vary near the corner in inviscid flow?
From Bernoulli: p ∝ –ρ|u|² ∝ r^(–2/3). Pressure increases with radius → adverse pressure gradient → risk of separation.
What is D’Alembert’s paradox and how is it resolved?
Inviscid steady flow predicts zero drag. But in real flows, unsteady effects or viscosity (e.g. boundary layers) introduce drag.
What is “added mass” in fluid mechanics?
When a body accelerates, it moves surrounding fluid. This adds an inertial load. Force ∝ (mass of displaced fluid) × acceleration.
Write the expression for the force on an accelerating sphere in an inviscid fluid.
F^H=c_AρVU, where 𝑐_𝐴=1/2 for a sphere. V is the sphere’s volume.
For a light gas bubble rising in a denser fluid, what is the effective acceleration?
U= ((ρ−ρb)/(ρ+c_Aρ) g. Buoyancy is opposed by added mass.
In inviscid, incompressible flow, what equation must the potential ϕ satisfy?
∇^2𝜙=0 (Laplace’s equation).
This holds because 𝑢=∇𝜙 and ∇⋅𝑢=0.
What is the added mass force on a sphere accelerating in a quiescent fluid?
F= 2/3 πa^3 ρU.
Derived using unsteady Bernoulli and symmetry in potential flow.
