biology 427 biomechanics lecture 17. drag and the reynolds...
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Biology 427 BiomechanicsLecture 17. Drag and the Reynolds number.•Recap conservation of energy and mass•D’Alembert’s Paradox and the missing energy•A wake is a separate issue•The Reynolds number measures the relevance of viscous and inertial stresses
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Conservation of mass (continuity) and energy (Bernoulli)
12u1,h1,P1
u2,h2,P2
(P2 - P1) /ρ + (u22- u1
2 )/2 = 0 Σ uin Ain = Σ uout Aout
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Conservation of Energy: what is the relationship between fluid motion and pressure? (P2 - P1) /ρ + (u2
2- u12 )/2 = 0
Is the direction drawn here correct?
http://www.asknature.org/strategy/e27b89ebcdec8c9b5b2cd9ac84b8f8a0
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“what quality of air surround birds in flight? The air
surrounding the bird is above thinner than the usual thinness of the other air, as below it is
thicker than the same... in proportion to the velocity of the
bird in its motion forward ...”
Sul volo degli Uccelli (On the flight of birds)
Leonardo da Vinci ~1500
~1500Leonardo da Vinci
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D’Alembert’s Paradox
(P2 - P1) /ρ = (u12- u2
2 )/2
stagnation pointsu = 0, P = highu: high P:low
No net force?
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D’Alembert’s Paradox
(P2 - P1) /ρ = (u12- u2
2 )/2
stagnation pointsu = 0, P = highu: high P:low
No net force?
assumptionssteadyincompressibleinviscidPE + KE + W = const.
no slip condition
viscosity robs fluid of its momentum.There is a shear stress exerted on the sphere and energy is dissipated by viscosity
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(P2 - P1) /ρ = (u12- u2
2 )/2
new stagnation point where the flow separates
Wake
because of viscosity, velocity cannot increase as much as in the inviscid case.
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(P2 - P1) /ρ = (u12- u2
2 )/2There are two mechanisms leading to force:shear stress : friction dragthe fore-aft asymmetry in pressure : pressure drag
The total drag is a composite of these and depends on size and shape of the body
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MOVING
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The total drag is a composite of pressure and shear stresses
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(P2 - P1) /ρ = (u12- u2
2 )/2Pressure stress: P ~ ρ u2
Shear stress: τ ~ µdu/drµu/L =
ρ u LµRe =
The total drag is a composite of pressure and shear stresses
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Bacterium ~10−5
Spermatozoa ~10−4
Ciliate ~10−1
Smallest Fish ~1Drosphila flight~ 1 × 102
Manduca flight~ 1 × 103
Birds flying 1 × 104
Typical pitch in Major League Baseball ~ 2 × 105
Person swimming ~ 4 × 106
Fastest Fish ~106
Blue Whale ~ 3 × 108
A large ship (RMS Queen Elizabeth 2) ~ 5 × 109
Re = ρULµ
=ULν
What is your Reynolds number ?
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Re = ρULµ
=ULν
What is your Reynolds number ?
What would the viscosity of the medium have to be for you to feel the flows like a bacterium does?
How far would a bacterium glide if it turned off its flagellum?
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ρu LµRe =
The Reynolds number measures the relative importance of intertial and viscous stresses in determining the flow. Conservation of Re implies identical flow patterns
Re << 1
Re ~ 1
Re >> 1