Exercises 3

Question 1

The fluid dynamic equivalent of the principle of energy conservation from mechanics is Bernouilli’s
equation. But does Bernoulli’s equation apply to unsteady problems? How about viscous problems, or compressible flows?

Answer 1

No Bernoulli’s equation assumes steady inviscid and incompressible flow alongthe streamline.

Question 2

What are the differences between Bernoulli’s equation and equation (3.13)?

Answer 2

For steady problems dphi/dt =0 for f(t).
- For f(t) the constant in the irrotational problem ω=0, is the same for all streamlines. For Bernoulli that constant could vary along streamlines.
- Bernoulli applies only for steady flows and along streamlines and does not require irrotational flow. F(t) is less general in that ω must be 0 but more general in allowing for unsteady situations.

Question 3

How is the vorticity of a fluid related to the local angular velocity?

Answer 3

Vorticity is defined as ω=∇xu. The velcoity field u can be defined as u=Ωxr, where Ω is the angular velcoity and r is the position vector. Such that: ω=∇x(Ωxr). By using the double cross product property the aforementioned equation can be rewritten as: ω=∇xu=2Ω. This concludes that the local velcoity is double the local rotation rate.

Question 4

What special properties are enjoyed by inviscid flows in which the velocity field far upstream is uniform?

Answer 4

When assuming that the fluid far way from the object is moving at uniform velocity, the assumption is that also the fluid upstream is not rotating.

Question 5

Is the model of an irrotational flow applicable at modest values of the Reynolds number? Does it apply
at high Re for the flight of an aerodynamic object through a quiescent fluid when the flow separates from
the object?

Answer 5

No, the irrotational flow model is not applicable at modest RE#. That is because unlike the high Re# assumption, the flow cannot be considered inviscid and the inertial and viscous forces lead to the development of rotational phenomena. Also, the moment the flow separates that implies rotational flow.

Question 6

How is the function f(t) appearing in the right hand side of equation (3.13) determined?

Answer 6

From ω=0 at all times and places –> u=∇Φ, which after some math it reduces to ∇^2Φ=0 and ∇(dΦ/dt + u^2 /2 + p/ρ + gz) which then gives 3.13, which would notmally be determiend from boundary conditions at infinity.