MAE 3241: AERODYNAMICS AND
FLIGHT MECHANICS
Review: Bernoulli Equation and Examples
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
LECTURE OUTLINE
• Review of Euler’s Equation
– Euler’s equation for incompressible flow → Bernoulli’s Equation
• Review of Basic Aerodynamics
– How does an airfoil or wing generate lift?
– What are effects of viscosity?
– Why does an airfoil stall?
– Why are golf balls dimpled?
WHAT DOES EULER’S EQUATION TELL US?
dp   VdV
• Euler’s Equation (Differential Equation)
– Relates changes in momentum to changes in force (momentum equation)
– Relates a change in pressure (dp) to a chance in velocity (dV)
• Assumptions we made:
– Steady flow
– Neglected friction (inviscid flow), body forces, and external forces
• dp and dV are of opposite sign
– IF dp increases dV decreases → flow slows down
– IF dp decreases dV increases → flow speeds up
• Valid for Incompressible and Compressible flows
• Valid for Irrotational and Rotational flows
INVISCID FLOW ALONG STREAMLINES
Relate p1 and V1 at point 1 to p2 and V2 at point 2
Integrate Euler’s equation from point 1 to point 2 taking =constant
dp  VdV  0
p2
V2
p1
V1
 dp    VdV  0
 V22 V12 
  0
p2  p1   

2 
 2
BERNOULLI’S EQUATION
2
2
2
1
V
V
p2  
 p1  
2
2
2
V
p
 Constant along a streamline
2
• If flow is irrotational p+½V2 = constant everywhere
• Remember:
– Bernoulli’s equation holds only for inviscid (frictionless) and
incompressible (=constant) flows
– Relates properties between different points along a streamline or entire
flow field if irrotational
– For a compressible flow Euler’s equation must be used ( is a variable)
– Both Euler’s and Bernoulli’s equations are expressions of F=ma
expressed in a useful form for fluid flows and aerodynamics
HOW DOES AN AIRFOIL GENERATE LIFT?
• Lift is mainly due to imbalance of pressure distribution over the top and bottom
surfaces of airfoil
– If pressure is lower than pressure on bottom surface, lift is generated
– Why is pressure lower on top surface?
• We can understand answer from basic physics
– Continuity
– Newton’s 2nd law
HOW DOES AN AIRFOIL GENERATE LIFT?
1. Flow velocity over the top of airfoil is faster than over bottom surface
– Streamtube A senses upper portion of airfoil as an obstruction
– Streamtube A is squashed to smaller cross-sectional area
– Mass continuity AV=constant, velocity must increase
Streamtube A is squashed
most in nose region
(ahead of maximum thickness)
A
B
HOW DOES AN AIRFOIL GENERATE LIFT?
2. As velocity increases pressure decreases
1
p

V 2  constant
– Incompressible: Bernoulli’s Equation
2
– Compressible: Euler’s Equation
dp   VdV
– Called Bernoulli Effect
3. With lower pressure over upper surface and higher pressure over bottom
surface, airfoil feels a net force in upward direction → Lift
Most of lift is produced
in first 20-30% of wing
(just downstream of leading edge)
EVEN A FLAT PLATE WILL GENERATE LIFT
• Curved surface of an airfoil is not necessary to produce lift
– But it significantly helps to reduce drag
A
B
EXAMPLE 2: WIND TUNNELS
• A wind tunnel is a ground-based experimental facility used to produce air
flow to study flight of airplanes, missiles, space vehicles, etc.
• Many different types of wind tunnels
– Subsonic, transonic, supersonic, hypersonic
Excellent Wind Tunnel Site: http://vonkarman.stanford.edu/tsd/pbstuff/tunnel/
OPEN VS. CLOSED CIRCUIT WIND TUNNELS
Open-Circuit Tunnel
Closed-Circuit Tunnel
Excellent Wind Tunnel Site: http://vonkarman.stanford.edu/tsd/pbstuff/tunnel/
EXAMPLE: LOW-SPEED, SUB-SONIC WIND TUNNEL
• Subsonic wind tunnels generally operate at speeds < 300 MPH
1
Diffuser
Contraction
(Nozzle)
Why build all of this?
Fan
Test Section 2
EXAMPLE: LOW-SPEED, SUB-SONIC WIND TUNNEL
2
1
• At speeds M < 0.3 ( or ~ 100 m/s) flow regarded as incompressible
• Analyze using conservation of mass (continuity) and Bernoulii’s Equation
1V1 A1   2V2 A2
A1
V2  V1
A2
1
1
2
2
p1  V1  p2  V2
2
2
V 
2
2
V2 
2

 p1  p2   V12
2 p1  p2 
  A 2 
 1   2  
  A1  
EXAMPLE 3: MEASUREMENT OF AIRSPEED
• How do we measure an airplanes speed in flight?
• Pitot tubes are used on aircraft as speedometers (point measurement)
STATIC VS. TOTAL PRESSURE
• In aerodynamics, 2 types of pressure: Static and Total (Stagnation)
• Static Pressure, p
– Due to random motion of gas molecules
– Pressure we would feel if moving along with the flow
– Pressure in Bernoulli’s equation is static pressure
• Total (Stagnation) Pressure, p0 or pt
– Property associated with flow motion
– Total pressure at a given point in flow is the pressure that would exist if
flow were slowed down isentropically to zero velocity
• p0 > p
MEASUREMENT OF AIRSPEED:
INCOMPRESSIBLE FLOW
1
2
p  V1  p0
2
Static
pressure
V1 
Dynamic
pressure
Total
pressure
2 p 0  p 

Incompressible Flow
SKETCH OF A PITOT TUBE (4.11)
• Measures total pressure
• Open at A, closed at B
• Gas stagnated (not moving) anywhere in tube
• Gas particle moving along streamline C will be
isentropically brought to rest at point A, giving
total pressure
EXAMPLE: MEASUREMENT OF AIRSPEED (4.11)
• Point A: Static Pressure, p
– Surface is parallel to flow,
so only random motion of
gas is measured
• Point B: Total Pressure, p0
– Aligned parallel to flow, so
particles are isentropically
decelerated to zero velocity
• A combination of p0 and p
allows us to measure V1 at a
given point
• Instrument is called a Pitotstatic probe
p
p0
MEASUREMENT OF AIRSPEED:
INCOMPRESSIBLE FLOW
1
2
p  V1  p0
2
Static
pressure
V1 
Dynamic
pressure
Total
pressure
2 p 0  p 

Incompressible Flow
p
1
V12  p0
2
TRUE VS. EQUIVALENT AIRSPEED
• What is value of ?
• If  is measured in actual air
around the airplane
• Measurement is difficult to do
• Practically easier to use value at
standard seal-level conditions, s
• This gives an expression called
the equivalent airspeed
Vtrue 
Ve 
2 p 0  p 

2 p0  p 
s
MEASUREMENT OF AIRSPEED:
SUBSONIC COMRESSIBLE FLOW
• If M > 0.3, flow is compressible (density changes are important)
• Need to introduce energy equation and isentropic relations
1 2
c pT1  V1  c pT0
2
2
T0
V1
 1
T1
2c pT1
T0
 1 2
 1
M1
T1
2
p0    1 2 
 1 
M1 
p1 
2

0    1 2 
 1 
M1 
1 
2


 1
1
 1
cp: specific heat at constant pressure
M1=V1/a1
air=1.4
MEASUREMENT OF AIRSPEED:
SUBSONIC COMRESSIBLE FLOW
• So, how do we use these results to measure airspeed
 1 




p
2
 0 
M 12 
 1
  1  p1 


 1 




p
2
a
 0 
V12 
 1
  1  p1 


2
1
 1 




p

p
2
a
1
 0
V12 
 1
 1
  1  p1



2
1
2
Vcal
 1 


2a  p0  p1 


 1
 1
  1  ps



2
s
p0 and p1 give
Flight Mach number
Mach meter
M1=V1/a1
Actual Flight Speed
Actual Flight Speed
using pressure difference
What is T1 and a1?
Again use sea-level
conditions Ts, as, ps
(a1=340.3 m/s)
REAL EFFECTS: VISCOSITY (m)
• To understand drag and actual airfoil/wing behavior we need an understanding of
viscous flows (all real flows have friction)
• Inviscid (frictionless) flow around a body will result in zero drag!
– Called d’Alembert’s paradox (Must include friction in theory)
We will derive this streamline
pattern in class next week
REAL EFFECTS: VISCOSITY (m)
• Flow adheres to surface because of friction between gas and solid boundary
– At surface flow velocity is zero, called ‘No-Slip Condition’
– Thin region of retarded flow in vicinity of surface, called a ‘Boundary Layer’
• At outer edge of B.L., V∞
• At solid boundary, V=0
“The presence of friction in the flow causes a shear stress at the surface of a body,
which, in turn contributes to the aerodynamic drag of the body: skin friction drag”
THE REYNOLDS NUMBER
• One of most important dimensionless numbers in fluid mechanics/ aerodynamics
• Reynolds number is ratio of two forces
– Inertial Forces

– Viscous Forces
– c is length scale (chord)
• Reynolds number tells you when viscous forces are important and when viscosity
can be neglected
V c
Re 
m
Outside B.L. flow
Inviscid (high Re)
Within B.L. flow
highly viscous
(low Re)
LAMINAR VERSUS TURBULENT FLOW
• Reynolds number also tells you about two types of viscous flows
– Laminar: streamlines are smooth and regular and a fluid element moves
smoothly along a streamline
– Turbulent: streamlines break up and fluid elements move in a random,
irregular, and chaotic fashion
LAMINAR VERSUS TURBULENT FLOW
All B.L.’s transition from
laminar to turbulent
Turbulent velocity
profiles are ‘fuller’
cf,turb > cf,lam
WHY DOES AN AIRFOIL STALL?
• Key to understanding: Friction causes flow separation within boundary layer
• Separation then creates another form of drag called pressure drag due to separation
WHY DOES AN AIRFOIL STALL?
• Key to understanding
– Friction causes flow separation within boundary layer
– Separation then creates another form of drag called pressure drag due to
separation
WHY DOES BOUNDARY LAYER SEPARATE?
• Adverse pressure gradient interacting with velocity profile through B.L.
• High speed flow near upper edge of B.L. has enough speed to keep moving
through adverse pressure gradient
• Lower speed fluid (which has been retarded by friction) is exposed to same
adverse pressure gradient is stopped and direction of flow can be reversed
• This reversal of flow direction causes flow to separate
– Turbulent B.L. more resistance to flow separation than laminar B.L. because of
fuller velocity profile
– To help prevent flow separation we desire a turbulent B.L.
WHY DOES AN AIRFOIL STALL?
• Two major consequences of separated flow over airfoil
– Dramatic loss of lift (stalling)
• Separated flow causes higher pressure on upper surface of airfoil
– Major increase in drag
• Separation causes lower pressure on trailing edge
• Unbalance of pressure force causes pressure drag due to separation
SUMMARY OF VISCOUS EFFECTS ON DRAG
• 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
Drag due to
=
viscous effects
skin friction
Called Profile Drag
+
Less for laminar
More for turbulent
Drag due to
separation
More for laminar
Less for turbulent
So how do you design?
Depends on case by case basis, no definitive answer!
COMPARISON OF DRAG FORCES
GOLF BALL AERODYNAMICS
• Why are modern golf balls dimpled?
• How important is skin friction?
• How important is pressure drag (separation)?
GOLF BALL AERODYNAMICS
Large Wake of Separated Flow,
High Pressure Drag
Laminar B.L. Separation Point
Reduced Size Wake of Separated Flow,
Lower Pressure Drag
Turbulent B.L. Separation Point
GOLF BALL AERODYNAMICS
DRAG
Large Wake of Separated Flow,
High Pressure Drag
Laminar B.L. Separation Point
Reduced Size Wake of Separated Flow,
Lower Pressure Drag
Turbulent B.L. Separation Point
Drag due to viscous
boundary layer skin friction
GOLF BALL AERODYNAMICS
DRAG
Large Wake of Separated Flow,
High Pressure Drag
Laminar B.L. Separation Point
Reduced Size Wake of Separated Flow,
Lower Pressure Drag
Turbulent B.L. Separation Point
Pressure drag
(due to viscous
flow separation, wake)
GOLF BALL AERODYNAMICS
Large Wake of Separated Flow,
High Pressure Drag
Laminar B.L. Separation Point
Reduced Size Wake of Separated Flow,
Lower Pressure Drag
Turbulent B.L. Separation Point
DRAG
Total Drag
GOLF BALL AERODYNAMICS: SUMMARY
•
•
•
•
Pressure drag dominates spheres and cylinders
Dimples encourage formation of turbulent B.L.
Turbulent B.L. less susceptible to separation
Delayed separation → Less total drag
COMPARISON OF DRAG FORCES
d
d
Same total drag as airfoil