Pharos UniversityME 253 Fluid Mechanics II
Flow over bodies;Lift and Drag
Bodies in motion, experience fluid forces and moments.
Examples include: aircraft, automobiles, buildings, ships, submarines, turbo machines.
Fuel economy, speed, acceleration, stability, and control are related to the forces and moments.
External External Flows
Airplane in level steady flight:
drag = thrust & lift = weight.
Flow over immersed bodies
flow classification:
2D, axisymmetric, 3D
bodies:
streamlined and blunt
Airplane
Upper surface
(upper side of wing):
low pressure
Lower surface (underside of wing): high pressure
Lift and Dragshear stress and pressure integrated over body surfacedrag: force component in the direction of upstream velocitylift: force normal to upstream velocity
212
212
cos sin
sin cos
x w D
y w L
dF p dA dA CU A
dF p dA dA CU A
DD
LL
AIRFOIL NOMENCLATURE
Mean Chamber Line: Points halfway between upper and lower surfaces
Leading Edge: Forward point of mean chamber lineTrailing Edge: Most reward point of mean chamber lineChord Line: Straight line connecting the leading and trailing edgesChord, c: Distance along the chord line from leading to trailing edgeChamber: Maximum distance between mean chamber line
and chord line
AERODYNAMIC FORCERelative Wind: Direction of V∞
We used subscript ∞ to indicate far upstream conditionsAngle of Attack, Angle between relative wind (V∞) and chord line
Total aerodynamic force, R, can be resolved into two force componentsLift, L: Component of aerodynamic force perpendicular to relative windDrag, D: Component of aerodynamic force parallel to relative wind
Pressure Forces acting on the Airfoil
High PressureLow velocity
High PressureLow velocity
Low PressureHigh velocity
Low PressureHigh velocity
Bernoulli’s equation says where pressure is high, velocity will below and vice versa.
Relationship between L´ and p
EdgeTrailing
EdgeLeading
sideupper sidelower
EdgeTrailing
EdgeLeading
sideupper
EdgeTrailing
EdgeLeading
sidelower
pp
pp
sideupper on Force-sidelower on the acting Forces
direction wind the tonormal Force
dx
dxdx
L
V
Relationship between L´ and p(Continued)
EdgeTrailing
EdgeLeading
sideupper sidelower
EdgeTrailing
EdgeLeading
sideupper sidelower
pp
pp
dxpp
dxL
Divide left and right sides by cV 2
2
1
EdgeTrailing
EdgeLeading
upperlower
c
xd
V
pp
V
pp
cV
L
222
21
21
21
We get:
Pressure Coefficient CpFrom the previous slide,
EdgeTrailing
EdgeLeading
upperlower
c
xd
V
pp
V
pp
cV
L
222
21
21
21
The left side was previously defined as the sectional liftcoefficient Cl.
The pressure coefficient is defined as:
2
21
V
ppC p
Thus, edgeTrailing
edgeLeading
upperplowerpl c
xdCCC ,,
Fluid dynamic forces are due to pressure and viscous forces.
Drag: component parallel to flow direction.
Lift: component normal to flow direction.
Drag and LiftLift and drag forces can be found by integrating pressure and wall-shear stress.
Drag and LiftLift FL and drag FD forces fn ( , A,V )
Dimensional analysis: lift and drag coefficients.
Area A can be frontal area (drag applications), plan form area (wing aerodynamics).
Example: Automobile Drag bile Drag
CD = 1.0, A = 2.5 m2, CDA = 2.5m2 CD = 0.28, A = 1 m2, CDA = 0.28m2
• Drag force FD=1/2V2(CDA) will be ~ 10 times larger for Scion XB
• Source is large CD and large projected area
• Power consumption P = FDV =1/2V3(CDA) for both scales with V3!
Drag and Lift
If CL and CD fn of span location x.
A local CL,x and CD,x are introduced.
The total lift and drag is determined by integration over the span L
Friction and Pressure Drag
Fluid dynamic forces: pressure and friction effects.FD = FD,friction + FD,pressure
CD = CD,friction + CD,pressure
Friction drag
Pressure drag
Friction & pressure drag
Flow Around Objects
Streamlining
Streamlining reduces drag by reducing FD,pressure,
Eliminate flow separation and minimize total drag FD
Streamlining
CD of Common GeometriesFor many shapes, total drag CD is constant for Re > 104
CD of Common Geometries
CD of Common Geometries
Flat Plate Drag
Drag on flat plate is due to friction created by laminar, transitional, and turbulent boundary layers.
Flat Plate Drag
Local friction coefficient
Laminar:
Turbulent:
Average friction coefficient
Laminar:
Turbulent:
Cylinder and Sphere Drag
Cylinder and Sphere DragFlow is strong function of Re.
Wake narrows for turbulent flow since turbulent boundary layer is more resistant to separation.
sep, lam ≈ 80º
sep,Tur ≈ 140º
Lift
Lift is the net force (due to pressure and viscous forces) perpendicular to flow direction.Lift coefficient
A=bc is the planform area
Characteristics of Cl vs.
Angle of Attack, in degrees or radians
Cl
Slope= 2 if is in radians.
= 0
Angle ofzero lift
Stall
30
EXAMPLE: AIRFOIL STALLLi
ft
Angle of Attack,
Effect of Angle of Attack
CL≈2 for < stall
Lift increases linearly with Objective:Maximum CL/CD
CL/CD increases until stall.
Effect of Foil ShapeThickness and camber affects pressure distribution andlocation of flow separation.
End Effects of Wing TipsTip vortex created by flow from high-pressure side to low-pressure side of wing.
Tip vortices from heavy aircraft far downstream and pose danger to light aircraft.
Lift Generated by Spinning
Superposition of Uniform stream + Doublet + Vortex
Drag Coefficient: CD
Supercritical flowturbulent B.L.
Stokes’ Flow, Re<1
Relatively constant CD
Drag
Drag Coefficient
with
or
DRAG FORCE
Friction has two effects:Skin friction due to shear stress at wallPressure drag due to flow separation
pressurefriction DDD Total drag due toviscous effectsCalled Profile Drag
Drag due toskin friction
Drag due toseparation= +
Less for laminarMore for turbulent
More for laminarLess for turbulent
38
COMPARISON OF DRAG FORCES
d
d
Same total drag as airfoil
AOA = 2°
AOA = 3°
AOA = 6°
AOA = 9°
AOA = 12°
AOA = 20°
AOA = 60°
AOA = 90°
Drag Coefficient of Blunt and Streamlined Bodies
Drag dominated by viscous drag, the body is __________.Drag dominated by pressure drag, the body is _______.
streamlined
bluffFlat plate
AU
dd
2
F2C
Drag
Pure Friction Drag: Flat Plate Parallel to the FlowPure Pressure Drag: Flat Plate Perpendicular to the FlowFriction and Pressure Drag: Flow over a Sphere and CylinderStreamlining
Drag
Flow over a Flat Plate Parallel to the Flow: Friction Drag
Boundary Layer can be 100% laminar, partly laminar and partly turbulent, or essentially 100% turbulent; hence several different drag coefficients are available
Drag
Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag
Drag coefficients are usually obtained empirically
Flow past an object
Character of the steady, viscous flow past a circular cylinder: (a) low Reynolds number flow, (b) moderate Reynolds number flow, (c) large Reynolds number flow.
DragFlow over a Sphere and Cylinder: Friction and Pressure Drag (Continued)
StreamliningUsed to Reduce Wake and hence Pressure Drag
Lift
Mostly applies to Airfoils
Note: Based on planform area Ap
Lift
Induced Drag
Experiments for Airfoil Lift & Drag
Examine the surface pressure distribution and wake velocity profile on airfoil 2-D
Compute the lift and drag forces acting on the airfoilPressure coefficient
Lift coefficient
Test Facility:• Wind tunnel.• Airfoil• Temp. sensor• Pitot tubes• Pressure sensors • Data acquisition
Test Design
Airfoil in a wind tunnel with free- stream velocity of 15 m/s.This airfoil has:Forces normal to free stream = LiftForces parallel to free stream = Drag Top of Airfoil:- The velocity of the flow is greater than the free-stream.- The pressure is negativeUnderside of Airfoil:- Velocity of the flow is less than the free-stream. - The pressure is positive
This pressure distribution contribute to the lift & Drag
Pressure taps positions
The lift force, L on the Airfoil will be find by integration of the measured pressure distribution over the Airfoil’s surface.
Data reduction
Calculation of lift forceThe lift force L= Integration of the
measured pressure over the airfoil’s surface.
Pressure coefficient Cp where, pi = surface pressure measured, = P pressure in the free-stream
U∞ = free-stream velocity, ϱ = air density pstagnation = stagnation pressure by pitot tube, L = Lift force, b = airfoil span, c = airfoil chord
cU
dspp
C sL
2
21
sin
2
21
U
ppC i
p
ppU stagnation2
bcU
LCL 2
2
dsppLs
sin
Drag Force The drag force, D on the Airfoil = Integration of the momentum
loss using the axial velocity profile in the wake of the Airfoil.
Data reduction
Calculation of drag forceThe drag force D = integration
of the momentum lossThe velocity profile u(y) is
measured ui at predefined locations
U∞ = free-stream velocity, ϱ = air density
pstagnation = Stagnation pressure by Pitot tube, D = Drag force, b = airfoil span, c = airfoil chord dyuUu
cUC i
y
y
iD
U
L
2
2
pypyu stagnation )(2)(
bcU
DCD 2
2
dyyuUyuDU
L
y
y
)()(
Velocity and Drag: Spheres
C ,Re, , ,d f shape orientationD
M
2
2FC d
d U A
2
2FC Red
d fU A
2C
F2
dd
U A
Spheres only have one shape and orientation!
General relationship for submerged objects
Where Cd is a function of Re
Sphere Terminal Fall Velocity
maF
2
2FC d
d U A
0 WFF bd
gW pp
2
2t
d d P w
VF C A
3
3
4rp 2rAp
W
dF
bF
gF wpb
velocity terminalparticle
tcoefficien drag
gravity todueon accelerati
densitywater
density particle
area sectional cross particle
volumeparticle
t
D
w
p
p
p
V
C
g
ρ
ρ
A
Sphere Terminal Fall Velocity (continued)
bd FWF
2
( )2t
d P w p p w
VC A g
2 2 ( ) p p w
td P w
gV
C A
dAp
p
3
2
2 4
3p w
td w
gdV
C
4
3
p w
td w
gdV
C
General equation for falling objects
Relationship valid for spheres
Drag Coefficient on a Sphere
0.1
1
10
100
1000
0.1 1 10 102 103 104 105 106 107
Reynolds Number
Dra
g Co
effici
ent
Stokes Law
24
RedC Re=500000
Turbulent Boundary Layer
Drag Coefficient for a Sphere:Terminal Velocity Equations
Laminar flow R < 1
Transitional flow 1 < R < 104
Fully turbulent flow R > 104
24
RedC
Re tV d
18
2wp
t
gdV
0.3p w
tw
gdV
0.4dC
4
3p w
td w
gdV
C
Valid for laminar and turbulent
Example Calculation of Terminal Velocity
Determine the terminal settling velocity of a cryptosporidium oocyst having a diameter of 4 m and a density of 1.04 g/cm3 in water at 15°C.
ms
kg1.14x1018
kg/m 999kg/m 1040m/s 189.m 4x10
3
33226
tV
18
2wp
t
gdV
ms
kg1.14x10
m 4x10
m/s 189.
kg/m 999
kg/m 1040
3
6
2
3
3
d
g
ρ
ρ
w
p
m/s1014.3 7 xVt
cm/day 7.2 tVReynolds