Lift / drag • When an object is submerged in a flowing fluid, or the

object moves in a stationary fluid the fluid is forced to flow around the object.

• As a result, the object is subjected to forces perpendicular and parallel to free stream velocity

• Drag:

– forces parallel to free stream velocity

• Lift:

– forces perpendicular to free stream velocity

The resultant force exerted by the fluid on the object has two

components : parallel to the incoming velocity DRAG

perpendicular to the incoming velocity LIFT

Drag on a surface – 2 types

• Pressure stress/ distribution > form drag

• Shear stress > skin friction drag

BLACKBOARD

Shortcuts for total drag • For less precise design and/or well-known / well-studied

(simple) objects, we rely on dimensional analysis and

experimental studies for an average coefficient of drag

• Here, A is a reference area,

sometimes “frontal area”

2AV

FC

2

DD

2AVCF 2

DD

from tables

if Re independent

Shortcuts for total drag

• For less precise design

and/or well-known / well-

studied (simple) objects, we

rely on charts for an average

coefficient of drag

• E.g., cylinders & spheres 2AV

FC

2

DD

2AVCF 2

DD

There are two flow regions and regimes that are important : the wake (laminar only

in case 1) and the BL developing on the leading edge (laminar 1 - - 4);

those defines the location where flow start separating (vortex generation)

Re increasing 15

flow separation controls the wake region characterized by low pressure

a change of regime (laminar > turbulent) in the boundary layer of the cylinder

front surface retard the separation : the flow in the wake is more mixed, the

pressure is not as low, as the velocity increases

as the upwind – downwind pressure decreases, Cd decreases considerably

laminar regime in the boundary layer

developing on the leading edge

turbulent regime in the boundary layer

developing on the leading edge

note

analogy with

the flat plate

BL:

Rex ~xV/D

x

Lift

why airplanes change the shape of the airfoil while landing ?

Circulation

given a closed area, circulation measures the average rate of rotation

of fluid particle situated in that area (e.g. flow through a cavity)

dLVL

path integral: goes all around

the curve until the latter is

closed; of course it does not

depend on the initial point VL is the velocity component

tangent to the path

circulation can be applied also on

virtual areas chosen to

comprehend particular regions of

the flow

dAdLVL vorticity

V = C / r

free vortex

V decreases with r

example: flow field rotating with circular streamline

CCdd

Cdrdr

CdLVd L

2

angle r radius l arc :remember

VL

note: this flow can be achieved by

rotating a cylindrical rod in water

(no slip condition applies)

However, using V=C/r the flow

domain remains irrotational (the

rotation is only imposed as a

boundary conditions (e.g. V=R)

in the shaded area like a

Couette flow B.Layer

clock wise positive

counter clock wise negative

C 2

what happens if we impose a uniform

velocity on this flow generated by a

rotating cylinder ?

constant!

does not

depend on r

free irrotational vortex + Uniform flow

=

stagnation

U0

ω𝑅 + U0

−ω𝑅 + U0

The rotation creates a pressure difference associated with the velocity

difference between top an bottom (Bernoulli). The resultant of the pressure

distribution over the cylinder area is the Lift Force (definition)

https://www.youtube.com/watch?v=3ECoR__tJNQ

ω𝑅 + U0

−ω𝑅 + U0

Lift force

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FL

ω𝑅 + U0

−ω𝑅 + U0

note that the uniform flow has the opposite direction

as compared to the case of a tennis ball with a top spin effect

the ball is moving >

this is equivalent to

the flow moving <

as in the drawing

ideal flow inviscid

rotating cylinder

rotating sphere

different object induce a different lift, drag due to the different pressure distribution

(?)dragntranslatio

liftrotation

V

r

Let us note that inviscid flow calculation are different from experiment (Drag is not 0)

~ inverse of

a Rossby number

used in geophys

flows to scale Coriolis force

Total lift

• Similar to our calculations of total drag, we

rely on charts for an average coefficient of lift

• A is a reference area, sometimes “planform

area”

22 AV

FC L

L

22 AVCF LL

Airfoils: profiles designed to induce a circulation (as a rotating cylinder)

Lift force

1) potential flow calculation misplaces the trailing edge stagnation point, as no separation

can be accounted for

2) in viscous fluid the stagnation is at the trailing edge

3) Kutta conditions: defines the circulation (or the vorticity) necessary to move the

stagnation point obtained by irrotational flow calculation to the correct trailing edge point.

(see: http://www.diam.unige.it/~irro/). Note that “irrotational flow” is ok for lift not for drag

angle of attack

Kutta condition : A body with a sharp trailing edge which is moving through a fluid will create about itself a circulation of sufficient strength to

hold the rear stagnation point at the trailing edge.

Kutta-Jukowski theorem:The value of circulation of the flow around the airfoil must be that value which would cause the Kutta

condition to exist

22/ρU)(

l α Uc πρCthen

l cA f

2/ρUC

α Uc πρ

α Uc π

2

0

2

0L

2

0

L

2

00

0

cl

i

A

F

Ul

F

L

L

theoretical circulation for the Kutta condition (c = cord length)

symmetric airfoil, small angle of attack

definition of lift per unit blade length l

airfoil lift coefficient

reference (plan) area based on the chord length

valid for an infinitely long airfoil at small angle of attack and ideal flow

“real” viscous effect and flow separation creates drag and lower lift

Airfoil in a real fluid

note: zero angle of attack

on a symmetric airfoil

generates NO lift

stall conditions

(separation occurs

on top of the airfoil

U L D

worst conditions !

2

linear!

Finite length blades and realistic planes

wing aspect ratio

b/c=wing span/chord length

for low aspect ratio

drag increases and lift decrease

long wings are better

coming back to the initial question: the plane move the flaps to increase

circulation This allows to decrease the velocity (landing) while maintaining

the lift (no crash landing)

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FL