Drag and lift

CHEE 3363Spring 2014Handout 23

�Reading: Fox 9.7--9.8

�1

Learning objectives for lecture

1.

boundary layer theory.�

2. Calculate the lift force on airfoils.

�2

Recall: drag on a sphere

force on sphere

What is the natural way to generalize this to arbitrary-shaped objects?

FD

cross-sectional area of sphere:

�

CD is only a function of Re!�3

τw = µ∂u

∂y

∣

∣

∣

∣

y=0

drag forcewetted area�

wall shear stress

frictional drag:

Turbulent:

V

�4

contribute; only pressure contributes to drag:

V

Calculating pressure drop analytically after separation not possible; use

�5

Drag on a sphere

��

d V ��

For larger Re > 3 × 105 ��

Turbulent boundary can better resist adverse pressure gradient�

�

delayed until Re ~ 4 × 105��

FD = 3πµV d CD =

24

Re

�6

Object CD

Square prismwidth/length = ∞: 2.05width/length = 1: 1.05

Disk 1.17

Ring 1.2

Hemisphere (open end facing flow) 1.42

Hemisphere (open end facing downstream) 0.38

C-section (open side facing flow) 2.3

C-section (open side facing downstream) 1.2

CD =

FD

1

2ρV 2A

�7

ν = 1.30 × 10−6

m2

s

1 T U

L × W �

: drag force

and

At y but we need δ(x�

2

�9

Calculate wall stress:

3

�10

back into force equation:

1

L × WU

T�

�11

2

�12

Solve for stress:

L × W

substitute new value of L into expression above

3

�13

∆α ≈

CL

πar

Lift

(Ap

Key ratio for airfoils:

Reduction related to wing aspect ratio: (b

�14

ar ≡

b2

Ap

∆CD ≈ CL∆α ≈

C2

L

πar

1V through air at standard conditions.

CL CD M. �: effective lift area and required engine thrust and power.

For level speed:

�15

2

�16

the forces:

Power:

�17

M (in C L

distance x V �: spin on ball.

x

L R

Basic equations:

Apply Newton’s second law:

�

M (in C L

distance x V �: spin on ball.

x

L R

Solve for the radius of curvature:

�19

M (in C L

distance x V �: spin on ball.

x

L R