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
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 3 / 46
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
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 5 / 46
Flow of a perfect fluid
Flow of a perfect fluid
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 !
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 6 / 46
Viscous flow
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 7 / 46
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
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 8 / 46
Viscous flow
Body in motion in a viscous fluid :
Resultant force = pressure forces + viscous forces.
V8
Pressuretotal force
Viscosity total force
Resultant
#»
R =
∫S
# »
PrM ·dS +
∫S
# »
ViM ·dS =# »
Pr +#»
Vi
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 9 / 46
Viscous flow
Body in motion in a viscous fluid :
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. . .
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 10 / 46
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
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 11 / 46
Viscous flow
CD as a function of flow pattern
102 104 106103 105 107
0.1
0.5
1.0
1.5
Re
Cd
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 12 / 46
Boundary layer
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 13 / 46
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.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 14 / 46
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
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 15 / 46
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.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 16 / 46
Boundary layer Airflow separation
Airflow separation
Separationline
Separationpoint
Boundary layer
Reversed flow (wake)
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 17 / 46
Boundary layer Airflow separation
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
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 18 / 46
Boundary layer Airflow separation
Airflow separation
Guadalupe Island – Nasa, public domain
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 19 / 46
Aircraft aerodynamics
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 20 / 46
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.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 21 / 46
Aircraft aerodynamics Mach regimes
Sound propagationStill source
The sound propagates in all directions around the source.
The wave fronts form concentric circles.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 22 / 46
Aircraft aerodynamics Mach regimes
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.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 23 / 46
Aircraft aerodynamics Mach regimes
Sound propagationTransonic source
The wave fronts accumulate in front of the source.
The drag increases sharply : sound barrier.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 24 / 46
Aircraft aerodynamics Mach regimes
Sound propagationSupersonic source
The waves form a Mach cone.
The sound reaches an observer with delay and produces acharacteristic deflagration.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 25 / 46
Aircraft aerodynamics Lift and stall
How do planes fly?
Y2432
Weight
Lift
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 26 / 46
Aircraft aerodynamics Lift and stall
How do planes fly?
Airfoil
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 27 / 46
Aircraft aerodynamics Lift and stall
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.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 28 / 46
Aircraft aerodynamics Lift and stall
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
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 29 / 46
Aircraft aerodynamics Lift and stall
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
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 29 / 46
Aircraft aerodynamics Lift and stall
Angle of attackNot to be confused
Y2432
Slope Attitude
Angle ofattack
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 30 / 46
Aircraft aerodynamics Lift and stall
Angle of attackNot to be confused
Y2432Slope
AttitudeAngle ofattack
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 30 / 46
Aircraft aerodynamics Lift and stall
Angle of attackPressure field
Xfoil simulation
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 31 / 46
Aircraft aerodynamics Lift and stall
Angle of attackPressure field
Xfoil simulation
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 31 / 46
Aircraft aerodynamics Lift and stall
Stall
Too high α⇒ the airflow separates from the wing
Airfoil
Angleof attack
I This is stall.I The lift drops sharply.
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 32 / 46
Aircraft aerodynamics Lift and stall
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
Aircraft aerodynamics Lift and stall
Wing optimization?
Spitfire Mk IIa (Adrian Pingstone, public domain)
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 34 / 46
Aircraft aerodynamics Lift and stall
Wing optimization?
Airbus A380 (Simon_sees , licence cc-by-2.0)
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 35 / 46
Aircraft aerodynamics Lift and stall
Wing optimization?
Concorde (Adrian Pingstone, public domain)
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 36 / 46
Aircraft aerodynamics Lift and stall
Wing optimization?
Boeing X-29 (Image Nasa, public domain)
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 37 / 46
Aircraft aerodynamics Lift and stall
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
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 38 / 46
Aircraft aerodynamics Lift and stall
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
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 39 / 46
Aircraft aerodynamics Lift and stall
Wing optimization?
The ideal wing shape depends on the planned flying speed :I Subsonic :
unsweptthick
I Supersonic :high sweep anglelow thickness
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 40 / 46
Momentum in fluid mechanics
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 41 / 46
Momentum in fluid mechanics
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
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 42 / 46
Momentum in fluid mechanics
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).
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 43 / 46
Momentum in fluid mechanics
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
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 44 / 46
Momentum in fluid mechanics
Navier-Stokes equationaka Cauchy momentum equation
dρ #»vdt
+#»∇ · (ρ #»v ⊗ #»v ) = − #»∇p +µ∇2 #»v
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 45 / 46
Momentum in fluid mechanics
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
Mathieu Bardoux (IUTLCO GTE) Fluid Mechanics 46 / 46
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