Lift and Circulation Flashcards
What is the equation for a rotating cylinder in the flow?
Ф = vr(1+R^2/r^2)*cosθ - k/2π *θ
How does the rotating cylinder change for the velocity field on surface of cylinder?
v(r) remains 0, but v(θ) becomes = -k/2πR -2vsinθ
What is Bernoullis equation for the cylinder?
P0 + 1/2ρv^2 = P(s) + 1/2ρv(θ)^2
How do we find the net force on the rotating cylinder?
F = - integral of P(s)(θ) ds, where ds = (cosθ, sinθ)R dθ
What do we get for F(y) on the rotating cylinder?
F(y) = -integral from 0 to 2π of P(s)(θ) sinθR dθ = ρvK
What is Bernoullis equation for a thin aerofoil?
P(bottom)(x) - P(top)(x) = 1/2ρ(v(top)(x)^2 - v(bottom)(x)^2)
How can we rearrange Bernoullis equation for the thin aerofoil?
P(b) - P(t) = ρv0(v(t)-v(b)), where v0 = 1/2*(v(t)-v(b))
What is the equation for the total lift of the thin aerofoil?
L = integral from 0 to d of (P(b)(x) - P(t)(x)) dx
What is the final equation for the total lift of the aerofoil?
L = ρv0k, where k = -k z(hat) (into page) and is the circulation
What is the total lift on a thin aerofoil an example of?
The magnus effect: the sideways deflection.
What is another example of the magnus effect?
Swerve on a ball when it spins backwards and swerves left of initial direction.
What does the flow around a vortex line look like?
Clockwise circle (A) to line (AB) along vortex to anticlockwise circle (B) to line (BA) back along vortex
What is the equation for integral of v.dl for the vortex line?
= 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
What do we do with the integral for the vortex line?
Set equal to zero as path does not enclose vortex, so integral over A of v.dl = -integral over B of v.dl
What happens when there are 2 interacting vortex lines?
Vortex 2 generates a flow field v2 and this leads to a force on vortex 1.
What is the equation for the Force on vortex 1 from vortex 2?
F12 = ρv2k1
What can we say for equal, opposite vortex lines?
|v1|=|v2| = k/2πd, and |F12| = |F21| (if |k1| =/ |k2)
What happens if vortex 1 moves to the right?
Equivalent to k in flow to left, so magnus effect force up. Reaches equilibrium when upward drift v(drift) cancels ρv2k1 force
What happens when v(drift) cancels out the F12 force?
Antiparallel vortex lines move perpendicular to their axis with a drift speed v(drift) = k/2πd
How can we use fluids to explain smoke rings?
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.
What happens with 2 parallel vortex lines?
Vortex’s still propel each other with speed k/2πd, but they rotate about mid-point with angular speed k/2πd^2
What does the theoretical diagram of potential flow over an aerofoil look like?
Flow over and flow under comes back on itself then continues: a stagnation point at leading edge and on uper surface.
What does k equal for the aerofoil?
k = integral of v.dl over surface
What does the lift equal for an aerofoil?
L = ρv0k = 0, no lift.
