Lecture 13

Question 1

What is the vorticity transport equation for a viscous, incompressible fluid with constant ρ and μ?

Answer 1

Dω/Dt=(ω⋅∇)u+ν∇^2ω
This accounts for tilting/stretching of vorticity and viscous diffusion.

Question 2

In 2D inviscid flow, how does vorticity behave along a fluid particle?

Answer 2

Vorticity is conserved:
𝐷𝜔/𝐷𝑡=0
A fluid particle maintains its vorticity over time.

Question 3

What does the third Helmholtz theorem state about vorticity in 3D inviscid flow?

Answer 3

If a fluid particle starts with zero vorticity, it will have zero vorticity for all time:
𝐷𝜔/𝐷𝑡=∇𝑢⋅𝜔⇒If𝜔=0 initially,then𝜔=0always.

Question 4

What does Kelvin’s Circulation Theorem state?

Answer 4

For an inviscid, barotropic flow with conservative body forces:
𝐷Γ/𝐷𝑡=0
Circulation around a material loop (moving with the fluid) is conserved.

Question 5

How can you physically interpret the tilting and stretching terms of vorticity?

Answer 5

Tilting: Vortex tubes rotate or realign with the flow direction.

Stretching: As parts of the vortex tube move at different speeds, the tube stretches and its cross-section shrinks, increasing vorticity magnitude.

Question 6

Why is there no “real” turbulence in 2D flows?

Answer 6

In 2D, the stretching/tilting term (𝜔⋅∇)𝑢 vanishes. Thus, no amplification or reorientation of vorticity occurs—essential for turbulence.

Question 7

In a flat plate boundary layer, how is circulation generated and changed?

Answer 7

Vorticity is generated at the wall due to the no-slip condition.

Distribution of vorticity changes due to diffusion:
Γ=∫𝑢⋅𝑑𝑙 (remainsconstantalongtheplate)

Question 8

Why doesn’t the circulation around a fixed contour decrease when vorticity diffuses out of it?

Answer 8

That applies to inviscid flows. In viscous flows, circulation around a fixed contour can change due to diffusion of vorticity across the boundary.

Question 9

How is circulation conserved in the airfoil-starting vortex example?

Answer 9

When the airfoil starts moving, it generates vorticity. To conserve circulation, it sheds a vortex of opposite sign. When it stops, another vortex of opposite sign is shed to maintain net zero circulation

Question 10

What is the circulation of an isolated vortex tube? What happens for contours that don’t enclose the tube?

Answer 10

Circulation Γ is constant along the tube.

Any contour not enclosing the tube has zero circulation.

Biot–Savart Law relates the velocity field:
∣𝑣_𝜃∣=Γ/2𝜋𝑟