Lecture 15

Question 1

How does vortex shedding from a wing in high-Re flow contribute to the generation of lift and induced drag?

Answer 1

Shed vortices (trailing + starting) form a vortex system that induces a downward velocity near the wing. This tilts the relative flow downward, decreasing the effective angle of attack, reducing lift, and generating induced drag—a 3D effect. Longer wings reduce induced drag.

Question 2

What is the role of the ground in the “ground effect” on wings?

Answer 2

The ground interferes with trailing vortex-induced downwash, reducing the induced downward velocity near the wing. This increases lift and reduces induced drag, making takeoff easier and landing faster due to reduced deceleration forces.

Question 3

Define the condition for irrotational flow and express the velocity field using the velocity potential.

Answer 3

Irrotational flow:
∇×𝑢=0⇒𝑢=∇𝜙
The scalar potential 𝜙 defines the velocity field in 3D space and is valid even for compressible, unsteady flows.

Question 4

What condition must the velocity potential 𝜙 satisfy for incompressible flow?

Answer 4

In incompressible flow,
∇⋅𝑢=0⇒∇^2𝜙=0
Hence, 𝜙 satisfies Laplace’s equation and is harmonic, allowing linear superposition of solutions.

Question 5

Give the potential function 𝜙 for a uniform flow and explain its physical meaning.

Answer 5

Uniform flow:
𝜙=𝑈𝑥
This means fluid flows from low to high potential regions. Streamlines and equipotentials are orthogonal.

Question 6

Derive the potential function 𝜙 for a 2D source of strength 𝑄

Answer 6

Mass flow
𝑄=2𝜋𝑟𝑢_𝑟⇒𝑢_𝑟=𝑄/2𝜋𝑟=∂𝜙/∂𝑟⇒𝜙=𝑄/2𝜋 ln𝑟

This represents radial flow from a point source.

Question 7

What is the velocity potential 𝜙 for a vortex line and what is its implication?

Answer 7

For circulation Γ,
𝜙=Γ/2𝜋 𝜃
Flow is purely tangential; streamlines are concentric circles. No radial flow implies zero net mass flow.

Question 8

What are key properties of the stream function 𝜓 in 2D incompressible flows?

Answer 8

ψ=const defines streamlines.

Difference in 𝜓 gives flow rate between streamlines.

𝜓⊥𝜙; they’re orthogonal in irrotational flows.

If ∇×𝑢=0, then ∇^2𝜓=0, so 𝜓 is also harmonic.

Question 9

What is the effect of adding a source and sink of equal strength 𝑄 at a small distance 2𝜖?

Answer 9

You get a doublet with potential
𝜙=𝑄_𝜖/𝜋 𝑥/(𝑥^2+𝑦^2),
modeling localized disturbances like flow over a slender body.

Question 10

How do you construct potential flow over a circular cylinder?

Answer 10

Superpose uniform flow and a doublet at origin:
𝜙=𝑈𝑥+𝑈𝑎^2𝑥/(𝑥^2+𝑦^2),

or in polar coords:
𝜙=𝑈(𝑟+𝑎^2/𝑟)cos⁡𝜃

From Bernoulli, 𝑝=𝑝_∞ −1/2 𝜌𝑈^2(1−2cos^2𝜃)

Question 11

Describe how adding a vortex to the cylinder flow alters the flow and pressure distribution.

Answer 11

Adding a vortex (e.g. clockwise) to cylinder flow creates circulation Γ, resulting in lift:
𝜙=𝑈(𝑟+𝑎^2/𝑟)cos⁡𝜃+Γ/2𝜋 𝜃ϕ

and stream function
𝜓=𝑈(𝑟−𝑎^2/𝑟)sin𝜃−Γ/2𝜋ln𝑟

This is the classical lift on a rotating cylinder problem.

Question 12

Find the stagnation points on a rotating cylinder with circulation Γ.

Answer 12

At stagnation,
𝑢_𝑟=𝑢_𝜃=0 Solve:
2𝑈sin𝜃+Γ/2𝜋𝑎=0⇒sin𝜃=−Γ/4𝜋𝑈𝑎
Depending on ΓΓ, stagnation points move or merge—related to lift and Kutta condition.