Midterm Prep - Ch3

Question 1

What is the definition of vorticity?

Answer 1

Vorticity is the curl of the velocity field: ω = ∇ × u. (Eq. 3.1)

Question 2

What physical quantity does vorticity represent?

Answer 2

Local rotation (angular velocity) of fluid elements. For a solid-body rotation with angular velocity Ω, the vorticity equals 2Ω (so vorticity is twice the rotation rate). (Eqs. 3.2–3.3)

Question 3

Show the vorticity of a solid-body rotation u = Ω × r.

Answer 3

Use double cross product: ∇×(Ω × r) = Ω∇·r − Ω·∇r. Since ∇·r = 3 and ∇r = I, this gives ∇×u = 2Ω. (Derivation in text; Eq. 3.3).

Question 4

State Euler’s equations for an incompressible fluid used in the lecture.

Answer 4

Momentum: Du/Dt = ∂u/∂t + u·∇u = −∇p/ρ + g.
Continuity (incompressible): ∇·u = 0. (Eqs. 3.4–3.5)

Question 5

What identity is used to rewrite u·∇u in the derivation?

Answer 5

u × (∇ × u) = ∇(u²/2) − u·∇u. (Eq. 3.6). This identity is used to rearrange Euler’s equation into a form that exposes the vorticity term

Question 6

Write the rearranged momentum equation that explicitly shows the vorticity term.

Answer 6

∂u/∂t + ∇(u²/2 + p/ρ + gz) = u × ω, where ω = ∇ × u. (Eq. 3.7). This separates a gradient term (potential-like) from the vorticity contribution.

Question 7

What is the vorticity-transport equation (final simplified form)?

Answer 7

Dω/Dt = ω·∇u. (Eq. 3.9). This is the evolution equation for vorticity in an incompressible inviscid flow.

Question 8

Physical meaning of the term ω·∇u in Dω/Dt = ω·∇u?

Answer 8

It represents vortex stretching/tilting: velocity gradients aligned with vorticity change the magnitude/direction of ω. If ω aligns with a region of extension, its magnitude can increase.

Question 9

What does persistence of irrotationality mean (as stated in the lecture)?

Answer 9

If the fluid far upstream has ω = 0 and the flow is incompressible and inviscid, then Dω/Dt = 0 → ω remains zero along particle paths; therefore the flow stays irrotational. (Discussion after Eq. 3.9). Caveat: text notes the rigorous proof is subtler (analogy to y’ = y).

Question 10

If ω = 0 everywhere, what simplification applies to the velocity field?

Answer 10

The velocity is a gradient of a scalar potential: u = ∇φ. (Eq. 3.10)

Question 11

What equation does the velocity potential φ satisfy for incompressible irrotational flow?

Answer 11

∇²φ = 0 (Laplace’s equation). (Eq. 3.11). Solving this with boundary conditions gives the velocity field.

Question 12

What is the Bernoulli-like relation for a potential (irrotational) flow (unsteady form)?

Answer 12

From (3.12)/(3.13): ∂φ/∂t + u²/2 + p/ρ + gz = f(t), where f(t) depends only on time (not on spatial coordinates). (Eq. 3.13)

Question 13

How does ∂φ/∂t + u²/2 + p/ρ + gz = f(t) (3.13) differ from the usual Bernoulli equation?

Answer 13

Differences: (i) (3.13) requires irrotational flow (ω = 0) but allows unsteady flows (∂φ/∂t may be nonzero). (ii) The constant f(t) is the same across all streamlines (because flow is irrotational), while classical Bernoulli (steady) gives constants that can vary between streamlines in rotational flows.

Question 14

Typical boundary conditions for solving ∇²φ = 0 in aerodynamics

Answer 14

(a) Far away: velocity → constant uniform stream. (b) On solid objects at rest: normal velocity = 0 → n·∇φ = 0 at body surface. Also f(t) fixed by conditions at infinity.

Question 15

How do you get forces on a body once φ (and p) are known?

Answer 15

Compute pressure p from ∂φ/∂t + u²/2 + p/ρ + gz = f(t) (3.13), then integrate over the body surface: F = −∫ p n dA.

Question 16

Under what practical conditions is the inviscid, incompressible, irrotational model a reasonable approximation?

Answer 16

When: (i) Reynolds number is large (viscous effects negligible except in thin boundary layers), (ii) free stream vorticity is zero, and (iii) flow is not separated (attached flow). Good for many full-size aircraft at moderate angles of attack. Not good for very small flyers (low Re) or strongly separated bluff-body flows.

Question 17

Why is the boundary layer important for the irrotational assumption?

Answer 17

Even when the outer flow is nearly irrotational, viscous boundary layers next to solid surfaces are rotational and can generate vorticity; separation creates large vortical regions that violate the irrotational model.

Question 18

Does Bernoulli’s equation apply to unsteady flows, viscous flows, compressible flows?

Answer 18

Steady Bernoulli (u²/2 + p/ρ + gz = const along a streamline) applies only for steady, inviscid flow (and incompressible unless modified). For unsteady flows, a time-dependent form like (3.13) applies for irrotational flows. Viscous flows generally violate Bernoulli; compressible flows require using energy & compressible relations (classical Bernoulli must be modified).

Question 19

Exercise 2: List differences between Bernoulli’s equation and equation (3.13).

Answer 19

Bernoulli (steady): constant along streamlines only, does not require irrotationality. Equation (3.13): needs irrotational flow (ω = 0), but allows unsteady flows and gives a spatially uniform quantity (equal for all streamlines) up to f(t).

Question 20

Exercise 3: How is vorticity related to local angular velocity?

Answer 20

For solid-body rotation with angular velocity Ω, ω = 2Ω. Thus vorticity is twice the local angular velocity vector.

Question 21

Exercise 4: What special properties do inviscid flows with uniform far upstream velocity enjoy?

Answer 21

Far-upstream uniform → ω = 0 initially. For incompressible inviscid flow this leads to persistence of irrotationality, allowing potential-flow formulation (u = ∇φ), Laplace’s equation, and a single Bernoulli-like relation valid across the domain (up to f(t)). Simplifies lift/drag computations.

Question 22

Exercise 5: Is the irrotational model applicable at modest Reynolds number? At high Re when flow separates?

Answer 22

At modest Re (low-to-moderate), viscous effects are more important — the irrotational model is less applicable. At high Re, the model can be good if flow remains attached; if the flow separates (creating vortices), the irrotational model fails

Question 23

Exercise 6: How is f(t) in (3.13) determined?

Answer 23

f(t) is set by boundary conditions (commonly conditions at infinity). The lecture notes state it is “normally determined from boundary conditions given at infinity.” In practice you pick the reference pressure/velocity at far field to fix f(t).