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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
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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
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Drag on a surface – 2 types
• Pressure stress/ distribution > form drag
• Shear stress > skin friction drag
BLACKBOARD
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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
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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
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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
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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
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Lift
why airplanes change the shape of the airfoil while landing ?
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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
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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
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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
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ω𝑅 + U0
−ω𝑅 + U0
Lift force
0Ul
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
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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
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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
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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
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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
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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!
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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
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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)
0Ul
FL