zoology 427 biomechanics lecture 18. shape and drag •a comment about poster...
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Zoology 427 BiomechanicsLecture 18. Shape and drag
•A comment about poster projects•Recap drag and the Reynolds number•Drag and its coefficient•How size, shape, and the Reynolds number determine drag
•Streamlined bodies and swimming energetics•When does drag matter?
(P2 - P1) /ρ = (u12- u2
2 )/2PE + KE + W + Ediss = const
new stagnation point where the flow separates
Wake
Because of viscosity, velocity cannot increase as much as in the inviscid case.
Pressure stress: P ~ ρ u2
Shear stress: τ ~ µ du/drρ u L
µRe =
Drag force arises from pressure and frictional stresses D ~ ρ u2/2
D = Cd ρ S u2/2drag coefficient projected surface area
sizeshapeReynolds number
!" =$
12'()*
!
Shape and Drag
1.25 1.14 0.5 0.3 0.045
!" =$
12'()*
!
Flow
Log Reynolds number
Cd
-1 0 1 2 3 4 5 6 7 8
100
10
1
0.1
!" =$
12'()*
!
streamlined body: same surface area
roughness
!" =24&' +
61 + &' + 0.4!
von Karmanvortex street
Streamlining – how much?
LD
Fineness ratio = L/D
manage your wake and the adverse pressure gradient
LD
Fineness ratio = L/D
Challenge: you are given a fixed volume of fish and must arrange it to minimize the total drag. The volume can be distributed with any fineness ratio. Generate a hypothetical plot for the drag as a function of fineness ratio.
Streamlining – how much? manage your wake and the adverse pressure gradient
Streamlining – how much?D
rag
Fineness ratio = L/D
“You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, a horse splashes”. JBS Haldane 1926. On Being the Right Size.
D = 0.5 Cd ρ S u2
m g
m g = 0.5 Cd ρ S u2
! = #$% = $#$ 43 ()*! ! = #$%&!
D = 0.5 Cd ρ S u2
m g
m g = 0.5 Cd ρ S u2
! = #$% = $#$ 43 ()*! ! = #$%&!
u
r
Assume ρ, g,Cd : const.
Drag is a propulsive force
Drag is a propulsive forceDrag on oar generates thrust
Thrust = DragThrust = Drag
Drag * Velocity = Power output
Thru
st
0 Piangle
When does drag matter?
Energy What is the average power per strideif we consider drag?
(1 − $)& '(&)
8+ +'+-
2) + 12/012+-!
D =
0.5
Cd ρ
S u
2
m g!
=#$%
=$#$4 3(
)*!
!=#$%
& !
u
r
Assu
me ρ,
g,C
d : c
onst
.
DragFo
r a fi
xed
volu
me