Lecture 12 Flashcards
(12 cards)
What is the mathematical definition of circulation and how is it related to vorticity via Stokes’ theorem?
Circulation:
Γ=∮from C 𝑢⋅𝑑𝑙=∬from 𝐴 (∇×𝑢)⋅𝑛𝑑𝐴=∬from𝐴 𝜔⋅𝑛𝑑𝐴
By Stokes’ theorem, circulation around a closed curve equals the flux of vorticity through the enclosed area.
What physical implication does ∇⋅𝜔=0 have in vortex flows?
It means there are no “sinks” or “sources” of vorticity. Vorticity lines cannot begin or end within the fluid—they must form closed loops or terminate on boundaries (Helmholtz’s 2nd theorem).
In incompressible flow, what happens to the velocity magnitude along a streamtube if the cross-sectional area shrinks?
Since ∇⋅𝑢=0, the volume flux is constant. If area decreases, velocity magnitude must increase to maintain 𝑄=const.
What is Helmholtz’s 1st theorem and what does it say about circulation along a vortex tube?
It states that circulation is conserved along a vortex tube in an inviscid fluid. If the cross-sectional area 𝐴 decreases, ∣𝜔∣ must increase to keep Γ=𝜔𝐴=const.
How are streamlines and vortex lines defined, and how do they differ?
Streamlines: tangent to velocity vector 𝑢 everywhere.
Vortex lines: tangent to vorticity vector 𝜔 They generally differ unless 𝜔 is aligned with 𝑢
Define a vortex sheet and explain its physical interpretation.
A vortex sheet is a surface across which tangential velocity is discontinuous. It represents an infinitesimal layer of vorticity filaments. Circulation per unit length is:
𝛾=lim from 𝜖→0 1/𝜖 ∮𝑢⋅d𝑙
What causes the Kelvin-Helmholtz instability in a vortex sheet?
A small kink creates a local pressure difference due to streamline curvature. The pressure difference amplifies the kink, making it grow—this is unstable and leads to vortex roll-up.
Derive the inviscid vorticity equation starting from Euler’s equation.
Start with Euler:
∂𝑢/∂𝑡+(𝑢⋅∇)𝑢=−1/𝜌 ∇𝑝 +𝑔
Take curl:
∂𝜔/∂𝑡=∇×(𝑢×𝜔)−∇(1/𝜌)×∇𝑝
What is the physical meaning of the baroclinic term
∇(1/𝜌)×∇𝑝?
It represents vorticity generation in non-barotropic fluids (where 𝑝 and 𝜌 gradients are not aligned). This term is zero in barotropic flow (i.e.,𝑝=𝑓(𝜌)).
What happens to vorticity in a blob of fluid where pressure and density gradients are not aligned?
The misalignment leads to a net torque, changing the blob’s rotation. This generates vorticity—this is the physical mechanism behind baroclinic vorticity generation.
Define “specific vorticity” and under what conditions it is conserved.
Specific vorticity is 𝜔/𝜌. In an inviscid, barotropic fluid, it satisfies:
𝐷/𝐷𝑡 (𝜔𝜌)=0
It is conserved along fluid element trajectories in such flows.
State and interpret two key consequences for vorticity in barotropic inviscid flows.
If there’s no initial vorticity, vorticity remains zero.
In 2D: 𝜔/𝜌=const. along pathlines ⇒ if 𝜌 decreases downstream, 𝜔 must decrease. Together, these imply that vorticity cannot spontaneously arise or amplify unless introduced via boundary or initial conditions.
