Fluid Mechanics - Chapter 2: Aerodynamics · Deﬁnition Summary 1 Deﬁnition 2 Flow of a perfect...

LITTORAL CÔTE D’OPALE

Fluid MechanicsChapter 2: Aerodynamics

Mathieu Bardoux

IUT du Littoral Côte d’OpaleDépartement Génie Thermique et Énergie

Summary

1 Definition

2 Flow of a perfect fluid

3 Viscous flow

4 Boundary layerDefinitionDescriptionAirflow separation

5 Aircraft aerodynamicsMach regimesLift and stall

6 Momentum in fluid mechanics

Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 2 / 46

Definition

Summary

1 Definition


3 Viscous flow





Definition

Definition

Aerodynamics

"a branch of dynamics that deals with the motion of air and othergaseous fluids and with the forces acting on bodies in motion relativeto such fluids" 1.

1. Merriam-Webster DictionaryMathieu Bardoux (IUTLCO GTE) Fluid Mechanics 4 / 46

Flow of a perfect fluid

Summary

1 Definition


3 Viscous flow







A body of any shape in uniform motion in an incompressible perfectfluid extending to infinity, undergoes no resistance on the part of thefluid.

V8 V8

Local deformation of the fluid velocity field

The resulting forces applied to the object are therefore zero.Watch out : the sum of the moments of force may not be zero !


Viscous flow

Summary

1 Definition


3 Viscous flow





Viscous flow

Body in motion in a viscous fluid :

Fluid-solid contact on surface dS :

V8

dS

pressure

viscosity

I perpendicular component = pressureI parallel component = viscosity


Viscous flow


Resultant force = pressure forces + viscous forces.

V8

Pressuretotal force

Viscosity total force

Resultant

#»

R =

∫S

# »

PrM ·dS +

∫S

# »

ViM ·dS =# »

Pr +#»

Vi


Viscous flow


Resultant force = lift force + drag force.

V8

Lift force

Drag force

Resultant

#»

R =#»

L +#»

D

Drag⇒ parallel to V∞ ; Lift⇒ perpendicular to V∞.Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 9 / 46

Viscous flow

Determination of aerodynamic forces

Drag :# »

FD =12ρv2CD S

CD = dimensionless drag coefficient.A = cross sectional area

Lift :# »

FL =12ρv2CL S

CL = dimensionless lift coefficient.

CD and CL are functions of body’s shape/orientation, Reynolds’number of the flow, etc. . .


Viscous flow

CD as a function of shape

Airplane wing : 0,005

Car (competition) : 0,14

Car (berlin) : 0,3

Cube : 1,05

Usain Bolt : 1,2


Viscous flow

CD as a function of flow pattern

102 104 106103 105 107

0.1

0.5

1.0

1.5

Re

Cd


Boundary layer

Summary

1 Definition


3 Viscous flow





Boundary layer Definition

Boundary layer : definition

Viscous fluid⇒ slowing down next to the body.

Perfect fluid

Boundarylayer

Viscous fluid

I Boundary layer is the area where flow velocity is modified(v < 0,99 · v∞).

I This layer is very thin and widens downstream.


Boundary layer Description

Boundary layer : description

I High speed gradient⇒ viscousity effectsI Outside, the flow is not significally modifiedI Develops from the stagnation point.I Low Reynolds⇒ high viscosity⇒ laminar flowI High Reynolds⇒ first laminar, then turbulent


Boundary layer Airflow separation

Airflow separation

I When the boundary layer separates from the surface.I A reversed flow appears downstream of the separation point.I At separation point, velocity profile is orthogonal to the wall.



Airflow separation

Separationline

Separationpoint

Boundary layer

Reversed flow (wake)



Airflow separation

I Increased pressure drop (diffusers)I Increased dragI Loss of lift (wings)I Yield loss (turbomachines)I Adjustment difficultiesI Vibrations⇒ structural failuresI Kármán vortex street



Airflow separation

Guadalupe Island – Nasa, public domain


Aircraft aerodynamics

Summary

1 Definition


3 Viscous flow





Aircraft aerodynamics Mach regimes

Mach regimes

Dimensionless ratio between flow velocity v , and sound celerity c :

Subsonic :vc< 1⇐⇒Ma < 1

Transsonic :vc≈ 1⇐⇒Ma ≈ 1

Supersonic :vc> 1⇐⇒Ma > 1

Nota bene :I c is a function of T , ρ. . .I Sound barrier : aerodynamic drag increases dramatically nearMa= 1.



Sound propagationStill source

The sound propagates in all directions around the source.

The wave fronts form concentric circles.



Sound propagationSubsonic source

The sound propagates in all directions around the source.

The wave fronts are closer in front of the source than behind it.⇒ Doppler effect.



Sound propagationTransonic source

The wave fronts accumulate in front of the source.

The drag increases sharply : sound barrier.



Sound propagationSupersonic source

The waves form a Mach cone.

The sound reaches an observer with delay and produces acharacteristic deflagration.


Aircraft aerodynamics Lift and stall

How do planes fly?

Y2432

Weight

Lift



How do planes fly?

Airfoil



Angle of attack

Angle of attack α creates the lift.

Airfoil

Angleof attack

I The flow "adheres" to the wing by viscosity.I The higher α, the stronger the lift.



Angle of attackNot to be confused

Attitude : orientation of an aircraft with respect to the horizon

Slope : orientation of the motion vector with respect to thehorizon

Angle of attack : orientation of an aircraft with respect to the motionvector

Attitude = Slope + angle of attack




Attitude : orientation of an aircraft with respect to the horizon

Slope : orientation of the motion vector with respect to thehorizon

Angle of attack : orientation of an aircraft with respect to the motionvector

Attitude = Slope + angle of attack




Y2432

Slope Attitude

Angle ofattack




Y2432Slope

AttitudeAngle ofattack



Angle of attackPressure field

Xfoil simulation


http://web.mit.edu/drela/Public/web/xfoil/


Angle of attackPressure field

Xfoil simulation


http://web.mit.edu/drela/Public/web/xfoil/


Stall

Too high α⇒ the airflow separates from the wing

Airfoil

Angleof attack

I This is stall.I The lift drops sharply.



Wing liftdepending on the angle of attack

2

1.75

1.5

1.25

1

0.75

0.5

0.25

0−10° −5° 0° 5° 10° 15° 20° 25° 30°

Angle of attack

Lift

coe

ffic

ient

SM701 airfoil (public domain)Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 33 / 46


Wing optimization?

Spitfire Mk IIa (Adrian Pingstone, public domain)



Wing optimization?

Airbus A380 (Simon_sees , licence cc-by-2.0)



Wing optimization?

Concorde (Adrian Pingstone, public domain)



Wing optimization?

Boeing X-29 (Image Nasa, public domain)



Wing optimization?

Leadingedge

Trailingedge

c0

c1

Sweepangle

Upper surface

Lower surface

e

Leading edge

Trailingedge

I Wingspan : distance between thewing tips

I Chord (c) : distance between leadingedge and trailing edge

I Tappering : ratio between tip chordand root chord

I Aspect ratio : the span divided bythe mean or average chord

I Thickness (e) : distance betweenupper and lower surface of the wing

I Sweep angle : angle between thewing and the perpendicular to thelongitudinal axis



Wing optimization?

I Swept wingrepels the appearance of compression effectsdecreases drag force in transonic+ regimestabilizes the roll flightdecreases lift force

I Thicknessincreases stall angleincreases structural strengthincreases drag force



Wing optimization?

The ideal wing shape depends on the planned flying speed :I Subsonic :

unsweptthick

I Supersonic :high sweep anglelow thickness


Momentum in fluid mechanics

Summary

1 Definition


3 Viscous flow






Definition

Linear momentum of :

a body of mass m : #»p = m · #»v

a fluid parcel of mass dm , located in point M :d #»p = # »vM ·dm = # »vM · ρ ·dV

a certain volume of fluide : #»p =∫

V# »vM · ρ ·dV

Three-dimensionnal vector quantity, measured in kg ·m · s−1



Conservation law

Newton’s second law of motion

Let’s consider a body in an inertial reference frame :

Σ#»

F =d #»pdt

If m is constant :

d #»pdt

=d(m #»v )

dt= m

d #»vdt

+ #»vdmdt

= md #»vdt

Momentum conservation law

In a closed system, the total momentum is constant.

d #»pdt

=#»

0

This is equivalent to the shift symmetry of space (Emmy Noether,1918).



Momentum theorem

I Newton’s second law, on a fluid parcel :

d(d #»p )

dt= d

#»

F

I Volume integration :

ddt

∫V

# »vM · ρ ·dV =

∫Vd

#»

F =#»

R

#»

R = resultant of external forcesI Momentum theorem :

d #»pdt

= Σ#»

F ext

⇒ Newton’s second law, generalized to fluids



Navier-Stokes equationaka Cauchy momentum equation

dρ #»vdt

+#»∇ · (ρ #»v ⊗ #»v ) = − #»∇p +µ∇2 #»v



Conclusion

In this chapter, we haveI Studied the forces exerted on a solid in a moving fluidI Defined the boundary layer and saw its effectsI Discovered the physical bases of airplane flightI Generalized Newton’s second law to fluids


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Fluid Mechanics - Chapter 2: Aerodynamics · Deﬁnition Summary 1 Deﬁnition 2 Flow of a perfect...

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