Zoology 427 Biomechanics
Lecture 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?
Because of viscosity,
velocity cannot increase as
much as in the inviscid
case.
PE + KE + W + Ediss = const
Pressure stress: P ~ ρ u2
Shear stress:
τ ~ µ du/dr
(P2 - P1) /ρ = (u12- u22 )/2
Re =
ρuL
µ
new stagnation point
where the flow separates
Wake
Drag force arises from pressure and frictional stresses
D ~ ρ u2/2
D = Cd ρ S u2/2
projected surface area
drag coefficient
shape
Reynolds number
!" =
$
1 )
'( *
2
!
size
Shape and Drag
1.25
1.14
0.5
0.3
Flow
!" =
$
1 )
'( *
2
!
0.045
!" =
$
1 )
'( *
2
!
100
Cd
roughness
von Karman
vortex street
10
1
0.1
-1
0
24
6
!" =
+
+ 0.4!
&' 1 + &'
1
2
3
4
5
6
Log Reynolds number
7
8
streamlined body:
same surface area
Streamlining – how much?manage your wake
and the adverse
pressure gradient
D
L
Fineness ratio = L/D
Streamlining – how much?
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.
manage your wake
and the adverse
pressure gradient
D
L
Fineness ratio = L/D
Drag
Streamlining – how much?
Fineness ratio = L/D
D = 0.5 Cd ρ S u2
“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”.
mg
JBS Haldane 1926. On Being
the Right Size.
m g = 0.5 Cd ρ S u2
4 *
! = #$% = $#$ () !
3
! = #$% & !
Assume ρ, g,Cd : const.
D = 0.5 Cd ρ S u2
u
mg
r
m g = 0.5 Cd ρ S u2
4 *
! = #$% = $#$ () !
3
! = #$% & !
Drag is a propulsive force
Drag is a propulsive force
Drag on oar generates thrust
Drag * Velocity = Power output
Thrust = Drag
Thrust
Thrust = Drag
0
Pi
angle
When does drag matter?
Energy
What is the average power per stride
if we consider drag?
&
'(
)
'+
1
(1 − $)&
+
+ /012 + - !
8+
2)
2
u
Drag
r
Assume ρ, g,Cd : const.
For a fixed volume
m
4
! = #$% = $#$ () *
3
g
! = #$% & !
D = 0.5 Cd ρ S u2