Midterm Prep - Ch5

Question 1

How do aerodynamic forces differ between ideal flow and real flow?

Answer 1

Ideal Flow (inviscid, incompressible, irrotational, steady):
o Symmetric pressure distribution around the body.
o No drag, no lift (D’Alembert’s paradox).

Real Flow (viscous, with boundary layers and vorticity):
o Symmetry breaks due to viscosity.
o Boundary layers form, circulation can develop.
o Nonzero drag and lift appear (what we observe in practice).

Question 2

What is D’Alembert’s paradox?

Answer 2

It is the result that in ideal fluid flow (inviscid, incompressible, irrotational, steady), the net aerodynamic force on a rigid body is zero — meaning both drag and lift vanish. The paradox is that this contradicts reality, where viscosity produces nonzero drag and lift.

Question 3

Write Euler’s equations (ignoring gravity) for steady flow.

Answer 3

∇∙[uu+pI/ρ]=0 ; ∇∙u=0

Question 4

What integral relation follows from Gauss’ theorem for momentum flux?

Answer 4

M= ∫(n∙uu+pn/ρ)dA

Question 5

How is the force on a body related to 𝑀∞?

Answer 5

F=-∫▒pndA=-ρM_∞

Question 6

How is velocity at infinity expressed in terms of perturbations?

Answer 6

u=U+u’
where U is uniform velocity and u’ is a small perturbation

Question 7

Expand n∙uu

Answer 7

n∙uu=n∙UU+n∙u’ U+n∙Uu^’+n∙u’u’

Question 8

How is Bernoulli’s equation modified using perturbation velocity in aerodynamics?

Answer 8

For u=U+u’ Bernoulli gives:
p/ρ=p_∞/ρ-U∙u’-u’2/2
Where -U∙u’ is the main pressure disturbance term and -u^’2/2 is a small quadratic correction that vanishes at infinity

Question 9

After simplifications, what does M_∞ reduce to?

Answer 9

M_∞=U×∫▒〖dA(u’×n)〗

Question 10

Why is the force orthogonal to velocity 𝑈?

Answer 10

Because drag vanishes (D’Alembert’s paradox), but lift may still exist due to circulation.

Question 11

In 2D, what velocity fields can give a net force?

Answer 11

Only those decaying as
1/r (like vortices); dipoles (1/r2) do not produce net force.

Question 12

What condition ensures lift exists around an airfoil?

Answer 12

Circulation around the airfoil (vortex with circulation Γ).

Question 13

What is the Joukowski lift formula?

Answer 13

F=ρUe_z Γ
Gives the lift force directly in terms of circulation without integrating pressure over the entire body.