Lecture F06 Mud:
1. Which came ﬁrst, Biot-Savart for E&M or for Fluids? (2 students)
Not sure. Biot-Savart was ﬁrst used for Fluids in the mid 1800’s by Helmholtz.
2. Is there some sort of Gauss’s Law for vortices to avoid doing the Biot-Savart
integral? (1 student)
Nope. Gotta deal with it as is.
3. How is V� = �/2πr for the straight 3-D vortex in the ﬁrst example? (1 student)
I’m not sure what you mean. Carrying out the Biot-Savart integral on the straight
3-D vortex produced V� = �/2πh, where h was the perpendicular distance from the
vortex. This is the same as the 2-D result, except that r was replaced with h (following
4. If the circulation is greatest at the tips, is the lift greatest at the tips also?
I think you’re confusing the circulation � with the wake vortex strength γ = −d�/dy.
The circulation � and hence L� = ρV�� both go to zero at the tips. In contrast, γ is
typically maximum near the tips.
5. Confused about sign conventions for � and γ? (4 students)
� on the wing is deﬁned positive about the y-axis by righthand rule.
γ in the wake is deﬁned positive about the x-axis by righthand rule.
6. What would γ(y) for a wing with a winglet look like? (1 student)
First of all, the vortex sheet in this case is not planar, but is upturned on the edges,
following the winglets. So γ has to be treated as a function of arc length s along the
span – ﬁrst along y, and then along z following the winglet. Qualitatively, γ(s) looks
similar to that of a ﬂat wing. The largest γ typically occurs near the winglet’s tip.
There may also be a “spike” in γ at the wing/winglet junction, depending how the
wing/winglet system is designed.
7. In the PRS, if � is constant, why isn’t d�/dy = 0? (1 student)
� is not completely constant – it sharply drops to zero at the tip over a very small
distance. So d�/dy = 0 over most of the span, but it’s large where � is sharply dropping
8. Are we ﬁnding the downwash at one point yo, or over the whole span? (1
When doing the integral to compute w, we hold yo ﬁxed. But this yo is left as “yo
rather than a speciﬁc number, so the result is an expression in terms of yo. We can
then plug in a range of yo values into the expression to create points for the w versus
9. How did you decide that dw = γ dy/4π(yo − y)? (1 student)
This is an application of the Biot-Savart law to the vortex ﬁlament consisting of a
dy-wide strip of the wake, with circulation γ dy.
10. What does the induced angle distribution tell me? (1 student)
It will be used to compute the loading on the wing, and hence the lift and induced
11. Didn’t understand PRS? (3 students)
I’ll go over it in recitation.
12. No mud (9 students)