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