Biot Savart Gauss's Law-Fluid Mechanics-Lecture Notes, Study notes for Fluid Mechanics. National Institute of Industrial Engineering

Fluid Mechanics

Description: This lecture handout was provided by Prof. Anirvan Khan at National Institute of Industrial Engineering for Fluid Mechanics. It includes: Guass, Straight, Wing, Constant, Winglets, Vortex, Law, Fluid, Biot, Savart, Lift
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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
Anderson’s notation).
4. If the circulation is greatest at the tips, is the lift greatest at the tips also?
(1 student)
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
to zero.
y
8. Are we finding the downwash at one point yo, or over the whole span? (1
student)
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
o plot.
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.
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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
drag.
11. Didn’t understand PRS? (3 students)
I’ll go over it in recitation.
12. No mud (9 students)
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