Download Fluid Flow Lecture 1 - Pharos University in Alexandria

Pharos University
ME 253 Fluid Mechanics II
Flow over bodies;
Lift and Drag
External External Flows
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.
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 Drag
shear stress and pressure
integrated over body surface
drag: force component in the
direction of upstream velocity
lift: force normal to upstream
velocity
D   dFx   p cos  dA    w sin  dA
CD 
D
2
1

U
A
2
L   dFy   p sin  dA    w cos  dA
CL 
L
2
1

U
A
2
AIRFOIL NOMENCLATURE
Mean Chamber Line: Points halfway between upper
and lower surfaces
Leading Edge: Forward point of mean chamber line
Trailing Edge: Most reward point of mean chamber line
Chord Line: Straight line connecting the leading and trailing edges
Chord, c: Distance along the chord line from leading to trailing edge
Chamber: Maximum distance between mean chamber line
and chord line
AERODYNAMIC FORCE
Relative Wind: Direction of V∞
We used subscript ∞ to indicate far upstream conditions
Angle of Attack, a: Angle between relative wind (V∞) and chord line
Total aerodynamic force, R, can be resolved into two force components
Lift, L: Component of aerodynamic force perpendicular to relative wind
Drag, D: Component of aerodynamic force parallel to relative wind
Pressure Forces acting on the Airfoil
Low Pressure
High velocity
High Pressure
Low velocity
Low Pressure
High velocity
Bernoulli’s equation says where pressure is high, velocity will be
low and vice versa.
High Pressure
Low velocity
Relationship between L´ and p
V
L  Force normal to the wind direction
 Forces acting on the lower side - Force on upper side
Trailing
Edge

p
Trailing
Edge
dx 
lower side
Leading
Edge
Trailing
Edge

 p
Leading
Edge
lower side
p
dx
upper side
Leading
Edge
 p upper side dx
Relationship between L´ and p
(Continued)
Trailing
Edge
L 
 p
lower side
 p upper side dx
Leading
Edge
Trailing
Edge

 p
lower side


 p   p upper side  p dx
Leading
Edge
1
V2 c
2
Divide left and right sides by



plower  p pupper  p  x
L
d
  

1
1
1
2
2

 c
V2c Leading

V

V




Edge
2
2
 2

Trailing
Edge
We get:
Pressure Coefficient Cp
From the previous slide,



plower  p pupper  p  x
L
d
  

1
1
1
2
2

 c
V2c Leading

V

V




Edge
2
2
 2

Trailing
Edge
The left side was previously defined as the sectional lift
coefficient Cl.
The pressure coefficient is defined as:
Cp 
Thus,
Trailing
edge
Cl 
 C
Leading
edge
p  p
1
V2
2
p ,lower  C p ,upper d
x
c
Fluid dynamic
forces are due to
pressure and
viscous forces.
Drag: component
parallel to flow
direction.
Lift: component
normal to flow
direction.
Lift
and
drag
forces
can
be
found
by
Drag and Lift
integrating pressure and wall-shear stress.
Drag and Lift
Lift 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=1/2V2(CDA) will be ~ 10 times larger for Scion XB
• Source is large CD and large projected area
• Power consumption P = FDV =1/2V3(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
Friction drag
Fluid dynamic forces:
pressure and friction
effects.
FD = FD,friction + FD,pressure
CD = CD,friction + CD,pressure
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
4
For
many
shapes,
total
drag
C
is
constant
for
Re
>
10
of Common Geometries
D
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 Drag Flow 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. a
Stall
Cl
Slope= 2p if a is in radians.
a = a0
Angle of
zero lift
Angle of Attack, a in degrees
or radians
Lift
EXAMPLE: AIRFOIL STALL
Angle of Attack, a
30
Effect of Angle of Attack
CL≈2pa for a < astall
Lift increases linearly
with a
Objective:Maximum
CL/CD
CL/CD increases until
stall.
Thickness and camber affects pressure distribution and
Effectofof
Foil
Shape
location
flow
separation.
End Effects of Wing Tips
Tip 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
Stokes’ Flow, Re<1
Supercritical flow
turbulent B.L.
Relatively constant CD
Drag
Drag Coefficient
with
or
DRAG FORCE
Friction has two effects:
Skin friction due to shear stress at wall
Pressure drag due to flow separation
D  D friction  D pressure
Total drag due to
viscous effects
Called Profile Drag
=
Drag due to
skin friction
Less for laminar
More for turbulent
+
Drag due to
separation
More for laminar
Less for turbulent
COMPARISON OF DRAG FORCES
d
d
Same total drag as airfoil
38
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,
streamlined
the
body is
__________.
Drag
dominated by
pressure
drag,
bluff
the body is
_______.
Flat plate
Cd 
2Fd
U 2 A
Drag
Pure Friction Drag: Flat Plate Parallel to the
Flow
Pure Pressure Drag: Flat Plate
Perpendicular to the Flow
Friction and Pressure Drag: Flow over a
Sphere and Cylinder
Streamlining
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.
Drag
Flow over a Sphere and Cylinder: Friction
and Pressure Drag (Continued)
Streamlining
Used 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 airfoil
Pressure 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 = Lift
Forces parallel to free stream = Drag
Top of Airfoil:
- The velocity of the flow is greater
than the free-stream.
- The pressure is negative
Underside 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 force
 The 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
pi  p
Cp 
1
U 2
2
2 pstagnation  p 
U 

2L
CL 
U 2 bc
L    p  p sin  ds
s
p

CL 
 p sin  ds
s
1
U 2 c
2
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 force
 The drag force D = integration
of the momentum loss
 The 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
u( y) 
2 pstagnation( y)  p 

yU
D    u ( y )U   u ( y ) dy
yL
2D
CD 
U 2 bc
yU
2
CD  2  ui U   ui dy
U  c yL
Velocity and Drag: Spheres


Cd  f  , Re, M, shape, orientation  General relationship for
D
 submerged objects
Spheres only have one shape and orientation!
2Fd
Cd 
U 2 A
2Fd
 Cd  f  Re 
2
U A
Cd U 2 A
Fd 
2
Where Cd is a function of Re
Sphere Terminal Fall Velocity
 p  particle volume
 F  ma
Fb
Fd  Fb  W  0
ρ p  particle density
Fd
W  ppg
ρw  water density
g  acceleration due to gravity
C D  drag coefficient
Fb   p  w g
Vt 2
Fd  Cd AP  w
2
4
 p  pr
3
3
Ap  pr
Ap  particle cross sectional area
Vt  particle terminal velocity
W
2
2Fd
Cd 
U 2 A
Sphere Terminal Fall Velocity (continued)
Fd  W  Fb
Vt 2
Cd AP  w
  p (  p  w ) g
2
Vt 
2
2 p (  p   w ) g
Cd AP  w
p
2
 d
Ap 3
Relationship valid for spheres
4 gd   p   w 
Vt 
3 Cd
w
2
General equation for falling objects
4 gd   p   w 
Vt 
3 Cd
w
Drag Coefficient on a Sphere
Drag Coefficient
1000
100
Stokes Law
10
1
0.1
0.1
1
24
Cd 
Re
10
102
103
104
Reynolds Number
105
106
Re=500000
Turbulent Boundary Layer
107
Drag Coefficient for a Sphere:
Terminal Velocity Equations
4 gd   p   w 
Vt 
3 Cd
w
Valid for laminar and turbulent
24
Laminar flow R < 1 Cd 
Re
Transitional flow 1 < R < 104
Fully turbulent flow R >
Re 
Vt d 

104
Cd  0.4
Vt 
d 2 g  p   w 
18
gd   p   w 
Vt 
0.3
w
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.
ρ p  1040 kg/m
3
Vt 
ρw  999 kg/m 3
g  9.81 m/s
2
Vt
d  4x10 6 m
  1.14x10 3
kg
sm
4x10

6
d 2 g  p   w 
18
m  9.81 m/s 2 1040 kg/m 3  999 kg/m 3 


3 kg


181.14x10


sm
2
Vt  3.14 x107 m/s
Vt  2.7 cm/day
Reynolds