Lift and Circulation

Question 1

What is the equation for a rotating cylinder in the flow?

Answer 1

Ф = vr(1+R^2/r^2)*cosθ - k/2π *θ

Question 2

How does the rotating cylinder change for the velocity field on surface of cylinder?

Answer 2

v(r) remains 0, but v(θ) becomes = -k/2πR -2vsinθ

Question 3

What is Bernoullis equation for the cylinder?

Answer 3

P0 + 1/2ρv^2 = P(s) + 1/2ρv(θ)^2

Question 4

How do we find the net force on the rotating cylinder?

Answer 4

F = - integral of P(s)(θ) ds, where ds = (cosθ, sinθ)R dθ

Question 5

What do we get for F(y) on the rotating cylinder?

Answer 5

F(y) = -integral from 0 to 2π of P(s)(θ) sinθR dθ = ρvK

Question 6

What is Bernoullis equation for a thin aerofoil?

Answer 6

P(bottom)(x) - P(top)(x) = 1/2ρ(v(top)(x)^2 - v(bottom)(x)^2)

Question 7

How can we rearrange Bernoullis equation for the thin aerofoil?

Answer 7

P(b) - P(t) = ρv0(v(t)-v(b)), where v0 = 1/2*(v(t)-v(b))

Question 8

What is the equation for the total lift of the thin aerofoil?

Answer 8

L = integral from 0 to d of (P(b)(x) - P(t)(x)) dx

Question 9

What is the final equation for the total lift of the aerofoil?

Answer 9

L = ρv0k, where k = -k z(hat) (into page) and is the circulation

Question 10

What is the total lift on a thin aerofoil an example of?

Answer 10

The magnus effect: the sideways deflection.

Question 11

What is another example of the magnus effect?

Answer 11

Swerve on a ball when it spins backwards and swerves left of initial direction.

Question 12

What does the flow around a vortex line look like?

Answer 12

Clockwise circle (A) to line (AB) along vortex to anticlockwise circle (B) to line (BA) back along vortex

Question 13

What is the equation for integral of v.dl for the vortex line?

Answer 13

= integral over A + integral over AB + integral over B + integral over BA = integral over A of v.dl + integral over B of v.dl, since AB and BA cancel out

Question 14

What do we do with the integral for the vortex line?

Answer 14

Set equal to zero as path does not enclose vortex, so integral over A of v.dl = -integral over B of v.dl

Question 15

What happens when there are 2 interacting vortex lines?

Answer 15

Vortex 2 generates a flow field v2 and this leads to a force on vortex 1.

Question 16

What is the equation for the Force on vortex 1 from vortex 2?

Answer 16

F12 = ρv2k1

Question 17

What can we say for equal, opposite vortex lines?

Answer 17

|v1|=|v2| = k/2πd, and |F12| = |F21| (if |k1| =/ |k2)

Question 18

What happens if vortex 1 moves to the right?

Answer 18

Equivalent to k in flow to left, so magnus effect force up. Reaches equilibrium when upward drift v(drift) cancels ρv2k1 force

Question 19

What happens when v(drift) cancels out the F12 force?

Answer 19

Antiparallel vortex lines move perpendicular to their axis with a drift speed v(drift) = k/2πd

Question 20

How can we use fluids to explain smoke rings?

Answer 20

Opposite element has opposite circulation and ring moves perpendicular to plane: fluid in core moves at v(drift) and any smoke in the core remains there.

Question 21

What happens with 2 parallel vortex lines?

Answer 21

Vortex’s still propel each other with speed k/2πd, but they rotate about mid-point with angular speed k/2πd^2

Question 22

What does the theoretical diagram of potential flow over an aerofoil look like?

Answer 22

Flow over and flow under comes back on itself then continues: a stagnation point at leading edge and on uper surface.

Question 23

What does k equal for the aerofoil?

Answer 23

k = integral of v.dl over surface

Question 24

What does the lift equal for an aerofoil?

Answer 24

L = ρv0k = 0, no lift.

Question 25

What does the real flow pattern on a aerofoil look like?

Answer 25

Just flow above and flow below, with stagnation point on trailing edge, not on top