AERODYNAMICS 1
Aerodynamics – The science relating to the effects produced by air or other gases in motion. The study of the properties of
moving air, and especially of the interaction between the air and solid bodies moving through it.
Fundamental Physical Quantities of a Flowing Gas:
1.
Pressure - the normal force exerted per unit area due to time rate of change of momentum of gas molecules impacting
on the surface. Common units of pressure N/m2, dyn/cm2, lb/ft2, and atm
2.
Density - Defined as the mass per unit volume of a substance. Common units of density kg/m3, g/cm3, slugs/ft3, and
lbm/ft3
3.
Temperature - The measure of the average kinetic energy of the particles in a gas. The measure of the hotness or
coldness of the body. Common units of temperature are: Kelvin (K), degree Celsius (C), degree Rankine (R), and
degree Fahrenheit (F).
4.
Velocity - The distance travelled by some object per unit time. Velocity connotes direction as well as speed.(Vector
quantity). For a flowing gas, we must further recognize that each region of the gas does not necessarily have the
same velocity; that is, the speed and direction of the gas may vary from point to point in the flow. Hence flow velocity
is a point property.
Streamlines of the flow - the path taken by a moving fluid element.
The Two Main Sources of all Aerodynamic Forces:
1. Pressure Distribution on the surface
2. Shear Stress (friction on the surface)
* Air behaves like a perfect gas. A perfect gas is defined as one in which intermolecular forces are negligible.
Equation of State for a Perfect Gas
P  RT
Where:
P = pressure in Pa or psf
ρ = density in kg/m3 or slugs/ft3
T = temperature in Kelvin or Rankine
R = specific gas constant
R for normal air:
287.08 J/kg . K
or 1716 ft .lb/slug.R or 53.342 ft.lbf/lbm.
I. THE ATMOSPHERE
 The atmosphere is defined as the whole mass of air extending at a specified height
 The atmosphere is the mechanical mixture of gases surrounding the earth
Atmospheric Constituents
Nitrogen -----------------Oxygen ------------------Aragon --------------------Carbon Oxide ------------Hydrogen -----------------Helium --------------------Neon -----------------------
AERODYNAMICS 1
78.03%
20.99%
0.94%
0.03%
0.01%
0.004%
0.0012% and a small amount of water vapor and other gases
Layers of the Atmosphere:
1. Troposphere - The lowest and most turbulent region where clouds, moisture, and weather are formed.
This layer contains about 75% of the total mass of the atmosphere. This is where all plants and animals live
and breathe. Characterized by decreasing temperature with increasing height. The lowest altitude from
0 – 11 km
2. Stratosphere - The calm region of the atmosphere. Ozone in this layer stops many of the sun's harmful ultraviolet rays
from reaching the earth. This layer plus the troposphere make up 99% of the total mass of the atmosphere. The
temperature in this region is constant at 216.7 or 216.5 ⁰ K or 390 or 390.15 ⁰ R or -56.5 ⁰ C
3. Mesosphere - The third layer is the mesosphere. The temperature there is around -90° C (-130° F). This is where we
see "falling stars," meteors that fall to the earth and burn up in the atmosphere. At certain times of the year, we can see
many of these "falling stars" when the earth goes through the pieces of a broken comet. The top of the mesosphere is the
coldest part of Earth's atmosphere.
4. Thermosphere - Solar activity strongly influences temperature in the thermosphere. Although the thermosphere is
considered part of Earth's atmosphere, the air density is so low in this layer that most of the thermosphere is what we
normally think of as outer space. It is also where the space shuttle orbits. The aurora (the Southern and Northern Lights)
primarily occur in the thermosphere.
5. Ionosphere - The portion of the atmosphere which is ionized and contains plasma. The part of the atmosphere that is
ionized by solar radiation. It has practical importance because among other functions, it influences radio propagation to
distant places on the Earth.
6. Exosphere - The highest layer of the atmosphere. Very high up, the Earth's atmosphere becomes very thin. The region
where atoms and molecules escape into space. It extends to 40,000 miles above the earth's surface. The thermosphere
and the exosphere together make up the upper atmosphere.
The International Standard Atmosphere (ISA) – hypothetical model of the properties of atmosphere such as pressure,
temperature, density, etc.
Purposes of ISA:
1. To provide basis for comparing the performance characteristics of airplanes.
2. To allow for the calibration of altimeters, it is desirable to have standard properties of the atmosphere which represent
“average” conditions.
NOTE: Standard properties have been established by the ICAO. The ISA is generally used by the airplane and engine
manufacturer around the world.
Basic Properties of Air:
1. Pressure (P) - The values of standard air pressures at sea level (Po) are:
Po = 14.7 lb/in2 = 2116.8 or 2116.2 lb/ft2 = 29.92 in Hg
= 76 cmHg = 760 mmHg = 101325 Pa = 1 atm
NOTE: Atmospheric pressure decreases with altitude
2. Temperature (T) - The values of standard air temperature at sea level (To) are:
To = 15 C = 59 F
= 288.2 K = 519 or 518.7 R
NOTE: Atmospheric temperature decreases with height in the troposphere region.
3. Density (ρ) - The values of standard air densities at sea level are:
ρo = 0.002378 or 0.002377 slugs/ft3 = 1.225 kg/m3
AERODYNAMICS 1
NOTE: Atmospheric density decreases with height
4. Viscosity (μ) - The ability of the fluid to resist shearing stresses. It is the sticky or adhesive characteristics of a fluid. The
values of standard air viscosities at sea level are:
μu = 3.7372 x 10-7 slug/ft.sec
= 1.7894 x 10-5 kg/m-sec
Kinematic Viscosity - The kinematic viscosity is the dynamic viscosity μ divided by the density of the fluid ρ. It is usually
denoted by the Greek letter nu (ν).



Humidity - Amount of water vapor in air (condition of moisture or dampness). Temperature influence the maximum amount
of water vapor that the air can hold . Higher air temperature  absorb more water vapour. Density of air varies with
humidity. Density on damp day (hot day) is less than density on dry day (cold day).
The Lapse Rate for the Gradient Layers of the Atmosphere:
a
dT
dh
Where: a = temperature lapse rate
dT = change in temperature
dh = change in height
Temperature Variation with Altitude Formula
T  To  ah
Where: T = temperature at any altitude up to 11 km (troposphere)
To = 288.2 K or 519 R
h = height from sea level up to 11 km
a = lapse rate
Standard Values of Temperature Lapse Rate:
a = -0.0065 or -0.00651 K/m
= -6.5 or -6.51 K/km
= -0.003566 R/ft
General Equation for Pressure Variation with Altitude (Troposphere, 0 – 11 km)
P T 
  
P1  T1 
 g o /( aR )
g
 9.81m / s 2

aR  0.0065K / m( 287.08 N .m / kg .K )
 5.26
For Air:
P T 

Po  To 
5.26
Where:
P = pressure at any altitude up to 11 km
Po = standard pressure at sea level
T = temperature at any altitude up to 11 km
To = standard temperature at sea level
General Equation for Density Variation with Altitude (Troposphere, 0 - 11 km)
 T 
 
1  T1 
AERODYNAMICS 1
 ([ g o /( aR )] 1 )


 9.81m / s 2
 g 

1

  0.0065K / m( 287.08 N .m / kg .K )   1
 aR 


 4.26
For Air:
 T 
 
 o To 
Where:
4.26
ρ = density at any altitude up to 11 km
ρo = standard density at sea level
T = temperature at any altitude up to 11 km
To = standard temperature at sea level
Pressure Variation with Altitude (Stratosphere, 11-25 km)
 g 
P
  RT  ( h  h )
e  
P1
1
Where:
P = pressure at any altitude above 11 km
P1 = pressure at 11 km
g = gravitational constant, (9,81 m/s2, 32.2 ft/s2)
R = gas constant, for air
(287.08 J/kg.K, 53.342 ft.lbf/lbm.R)
T = constant temperature at stratosphere
216.5 K, 390.15 R
h = the given altitude above 11,000 m
h1 = 11,000 m
Density Variation with Altitude (Stratosphere, 11-25 km)
 g 

  RT  ( h  h )
e  
1
1
Where:
ρ = density at any altitude above 11 km
ρ1 = density at 11 km
g = gravitational constant, (9.81 m/s2, 32.2 ft/s2)
R = gas constant, for air
(287.08 J/kg.K, 53.342 ft.lbf/lbm.R)
T = constant temperature at stratosphere
216.5 K, 390.15 R
h = the given altitude above 11,000 m
h1 = 11,000 m
Temperature at Stratosphere (11-25 km) has no variation, constant at...
T  390.15 R
T  216.5 K
Problems:
1. Determine the temperature at the following altitudes:
a.) 2,000 ft
- Answer: 447.58 R
b.) 3,500 m
- Answer: 265.45 K
2. Determine the pressure at 15,000 ft altitude – Answer: P = 1194.555 psf
3. Determine the pressure at 4500 m altitude – Answer: P = 57, 656.065 Pa
4. Determine the density at 15,000 ft altitude – Answer: ρ = 1.496 x10-3 slugs/ft3
5. Determine the density at 4500 m altitude – Answer: ρ = 0.776 kg/m3
6. Find the Pressure and Density at 14 km height – Answer: P = 1.41 x 104 Pa ρ = 0.226 kg/m3
Altimeter - a pressure gauge which indicates an altitude in the standard atmosphere corresponding to the measured
pressure. A pressure gauge which translates the measured pressure into an altitude reading which corresponds to that
predicted by the standard atmosphere.
AERODYNAMICS 1
Types of Altitude:
1. Absolute Altitude - of particular importance on space travel because, gravitational constant varies inversely as the
square of the distance from the center of the earth.
ha = hG + r
2. Geometric altitude - the geometric height above sea level.
3. Pressure altitude, hp - is the altitude given by altimeter set to 29.92”Hg. The altitude corresponding to a given pressure
in the standard atmosphere.
4. Density altitude, hd - is the altitude corresponding to a given density in the standard atmosphere. Density altitude is
used for computing the performance of an aircraft and its engines.
5. Temperature altitude, hT - is the altitude corresponding to a given temperature in the standard atmosphere.
NOTE: In an atmosphere with standard conditions, all three altitudes are the same on the other hand; in non standard
atmosphere they will be different.
Problems:
1. If an airplane is flying at an altitude where the actual pressure and temperature are 4.72 x 10 4 N/m2 and 255.7 K,
respectively, what are the pressure, temperature, and density altitudes?
Answers: hp = 5984.485 m ht = 4992.320 m, hd = 6216.129 m.
2. The flight test data for a given airplane refer to a level-flight maximum-velocity run made at an altitude that simultaneously
corresponded to a pressure altitude of 30,000 ft and density altitude of 28,500 ft. Calculate the temperature of the air at the
altitude at which the airplane was flying for the test.
Answers: T = 392.807 R
3. A standard altimeter indicates 15,000 ft when the ambient temperature is 35 deg. F. Calculate the density altitude and the
temperature altitude.
II. BASIC AERODYNAMIC PRINCIPLES
The Continuity Equation
Physical Principle: Mass can neither be created nor destroyed
m1  m2
1 A1V1   2 A2V2
Incompressible and Compressible Flow
Where:
m1 – mass flow rate at section 1
m2 –mass flow rate at section 2
ρ1 – density of fluid at section 1
ρ2 – density of fluid at section 2
A1 – cross sectional area at section 1
A2 – cross sectional area at section 2
Compressible Flow - flow in which the density of the fluid elements can change from point to point. Indeed, all real-life
flows, strictly speaking, are compressible. However, there are some circumstances in which the density changes only
slightly. These circumstances lead to the second definition, as follows.
Incompressible Flow - flow in which the density of the fluid elements is always constant. For incompressible flow,
1   2
A1V1  A2V2
AERODYNAMICS 1
Incompressible flow is a myth. It can never actually occur in nature. However, for those flows in which the actual variation
of p is negligibly small, it is convenient to make the assumption that p is constant, to simplify our analysis. The low
speed flow of air , where V<100 m/s or (V<225 mi/h) can also be assumed to be incompressible.
Problems:
1. Consider a convergent duct with an inlet area A1 = 3 ft2 and an exit area A2 = 2.57 ft2. Air enters this duct with a velocity
VI = 700 ft/s and a density ρI = 0.002 slug/ft3, and air leaves with an exit velocity V2 = 1070 ft/s. Calculate the density of the
air ρ2 at the exit.
Answer: ρ2 = 0.00153 slugs/ft3
2. Consider the incompressible flow of water through a divergent duct. The inlet velocity and area are 2 m/s and 1 m2
respectively, if the exit area is 3 times the inlet area, calculate the water flow velocity at the exit.
Answer: V2 = 0.667 mps
3. A pipe is gradually tapering in size by 0.01 m2/m, what is the change in velocity per meter where the pipe is 0.4 m2 in
cross section, if the velocity there is 30 m/s. Is the velocity increasing or decreasing?
4. A circular pipe, 100 ft long gradually tapers from 3 ft in diameter at one end to 2 ft in diameter at the other. Fluid is flowing
from the bigger toward the smaller end. What is the rate of increase in velocity at the entrance if the velocity there is 80 fps?
The Momentum Equation - states the relation between pressure and velocity in a flow which is not considered in the
continuity equation. The Momentum Equation is the foundational equation for the Bernoulli’s Equation.
dp   VdV
The Bernoulli’s Equation for Incompressible Flow
1
1
2
2
P1  V1  P2  V2
2
2
1
P  V 2  CONSTANT
2
NOTE:
Where:
P1 – pressure at point 1
P2 – pressure at point 2
ρ – density of the fluid
V1 – velocity of the fluid at point 1
V2 – velocity of the fluid at point 2
1. This equations hold only for inviscid (frictionless), incompressible flow.
2. This equations relate properties between different points along a streamline.
Problems:
1. Consider an airfoil (the cross section of the wing, as shown in the figure below) in a flow of air, where far ahead,
(upstream) of the airfoil, the pressure, velocity, and density are 2116 psf, 100 mph, and 0.002377 slugs/ft3, respectively. At
a given point A on the airfoil, the pressure is 2070 psf. What is the velocity at point A?
A
AERODYNAMICS 1
Answer: VA = 245.4 fps
2. Consider a convergent duct. Air enters this duct with a velocity V1 = 10 m/s and leaves the duct exit with a velocity V2 =
30 m/s. If the air pressure and temperature at the inlet are P1 = 1.2 x 105 Pa and T1 = 330 K, respectively, calculate the
pressure at the exit.
Answer: P2 = 1.195 x 105 Pa
3. Consider an airplane flying with a velocity of 50 m/s at a standard altitude of 2.5 km. At a point on the wing, the airflow
velocity is 65 m/s. Calculate the pressure at this point.
The Isentropic Flow
- Reversible and Adiabatic Process
A reversible process is one in which no frictional or other dissipative effects occur.
An adiabatic process is one in which no heat is added or taken away, ∆Q = 0
Isentropic Flow Equations:
P2   2 
 
P1  1 


 1
NOTE: Isentropic Flow Equations are relevant to
Compressible flows only

Where: P1 and P2 – pressures at points 1 and 2
ρ1 and ρ2 – densities at points 1 and 2
T1 and T2 – temperatures at points 1 and 2
P2  T2 
 
P1  T1 

P2   2   T2   1
   
P1   1   T1 
Problem:
An airplane is flying at standard sea level conditions. The temperature at a point on the wing is 250 K. What is the pressure
at this point? Answer: P2 = 6.14 x 104 Pa
The Energy Equation (Applicable for Frictionless Adiabatic Flow):
2
2
V
V
h1  1  h2  2
2
2
2
V
h
 const
2
1 2
1 2
C pT1  V1  C pT2  V2
2
2
1
C pT  V 2  const
2
Since h = cpT, hence:
Where: h1 and h2 – enthalpies
V1 and V2 – velocities
T1 and T2 – temperatures
Cp – constant pressure specific heat
Problem:
Consider an airfoil in a flow of air, where far ahead of the airfoil (the freestream), the pressure, velocity, and density are
2116 psf, 500 mph, and 0.002377 slugs/ft3, respectively. At a given point A on the airfoil, the pressure is 1497 psf. What is
the velocity at point A? Assume isentropic flow. For air Cp = 6006 ft.lb/slug R
∞
A
Answer: VA = 1061 fps
The Speed of Sound
a 
P

AERODYNAMICS 1
Where:
a – speed of sound
γ – 1.4 (for air)
P – pressure
ρ - density
For Metric System
For English System
a  RT
a  gRT
a  20.05 T
a  49.02 T
Where: a = speed of sound in m/s
γ = 1.4
R = 287.08 J/Kg. K
T = temperature in degree Kelvin
Where: a = speed of sound in ft/s
γ = 1.4
R = 53.342 ft.lbf/lbm R
T = temperature in degree Rankine
g = 32.174 lbm.ft/lbf.s2
Principle: The speed of sound in a perfect gas depends only on the temperature of the gas.
Speed of sound is a function of the temperature.
Note: The standard speed of sound at sea level is approximately 340.9 m/s or 1117 ft/s
The Mach Number – the flow velocity divided by the speed of sound.
M
V
a
Where:
M – mach number (unitless)
V – flow velocity
a – speed of sound
The Regimes of Aerodynamic Flows:
1.If M < 1, the flow is subsonic
2. If M = 1, the flow is sonic
3. If M > 1, the flow is supersonic
4. 0.8 < M < 1.2 , the flow is transonic
5. M > 5, the flow is hypersonic
Problems:
1. A jet transport is flying at a standard altitude of 30,000 ft with a velocity of 550 mph. What is its Mach number?
Answer: M = 0.811
2. A Boeing 747 is cruising at a velocity of 250 m/s at a standard altitude of 31,500 ft. What is its Mach number?
3. Compute for the Mach number of the jet airplane flying at a speed of 650 knots at an altitude of 40,000 ft in standard
atmosphere.
The Bernoulli’s Equation for Compressible Flow:
  P 1 2

  V  cons tan t
   1  2
(  1 ) / 
2
2
2

V1  Vo
ao  P1 

 1  0
 
2
  1  Po 

Where:
γ – 1.4 (for air)
P – pressure of the flow
ρ – density of the flow
V – velocity of the flow
a – speed of sound
NOTE: The flows in which M < 0.3 can be treated as essentially incompressible and, conversely, flows where M ≥ 0.3
should be treated as compressible.
AERODYNAMICS 1
Other Compressible Flow Equations (Summary)
Continuity
Energy
1 A1V1   2 A2V2
1 2
1 2
CpT1  V1  CpT2  V2
2
2
Isentropic Relations


P1  1   T1  (  1 )
   
P2   2   T2 
Equation of State
1  1 RT1
 2   2 RT2
Problem:
1. In an undisturbed airstream, the pressure is 14.7 psi, the density is 0.002378 slugs per cu. ft, and the velocity is 500 fps.
What is the velocity where the pressure is 13.5 psi?
Answer: V1 = 632. 281 fps
2. Consider an airplane flying at a standard altitude of 5 km with a velocity of 270 m/s. At a point on the wing of the airplane,
the velocity is 330 m/s. Calculate the pressure at this point.
3. A supersonic transport is flying at a velocity of 1500 mi/h at a standard altitude of 50,000 ft. The temperature at a point in
the flow over the wing is 793.32°R. Calculate the flow velocity at that point.
III. WIND TUNNELS
Wind Tunnel - is a device for testing aircraft and its components in a controlled airstream under laboratory conditions.
Ground-based experimental facilities designed to produce flows of air (or sometimes other gases), which simulate natural
flows occurring outside the laboratory.
Some Parameters that can be measured in a basic wind tunnel:
a. Lift
b. Drag
c. Side Force
d. Pitching Moment
e. Yawing Moment
f. Rolling Moment
Type of Wind Tunnels:
1. Low Speed Wind Tunnel
2. Transonic Wind Tunnel
3. Supersonic Wind Tunnel
4. Hypersonic Wind Tunnel
Low Speed Wind Tunnels - are used for operations at very low Mach number, with speeds in the test section up to 400
Km/h (M=0.3). They are both of open-return type or return-flow. The air is moved with a propulsion system made of a large
axial fan that increases the dynamic pressure to overcome the viscous losses.
AERODYNAMICS 1
Type of Low Speed Wind Tunnels:
1. Open Circuit - Also called (Eiffel or NPL) tunnel – the air flowing follows an essentially straight path from the entrance
through a contraction to the test section, followed by a diffuser, a fan section, and an exhaust of the air.
2. Closed Circuit Also called (Prandtl, Gottingen, or Return Flow) – the air flowing recirculates continuously with little or
no exchange of air with the exterior.
Advantages of an Open Circuit Wind Tunnel:
1. Construction cost is typically much less
2. If one intends to run internal combustion engines or do extensive flow visualizations via smoke, there is no purging
problem provided both inlet and exhaust are open to the atmosphere.
Disadvantages of an Open Circuit Wind Tunnel:
1. May require extensive screening at the inlet to get high quality flow.
2. Can be affected by weather during operation
3. For a given size and speed the tunnel will require more energy to run.
4. In general, open circuit tunnels tend to be noisy.
Advantages of a Closed Circuit Wind Tunnel:
1. Through the use of corner turning vanes and screens, the quality of the flow can be well controlled and most
important will be independent of other activities in the building and weather conditions.
2. Less energy is required for a given test section size and velocity.
3. There is less environmental noise when operating.
Disadvantages of a Closed Circuit Wind Tunnel:
1. The initial cost is higher due to return ducts and corner vanes.
2. If used extensively for a smoke flow visualization experiments or running of internal combustion engines, there
must be a way to purge tunnel.
3. If tunnel has high utilization, it may have to have an air exchanger or some other method of cooling.
Transonic Wind Tunnel - are able to achieve speeds close to the speed of sound. The highest speed is reached in the test
section. Testing at transonic speeds presents additional problems, mainly due to the reflection of the shock waves from the
walls of the test section.
AERODYNAMICS 1
Supersonic Wind Tunnel - The first problem with a supersonic wind tunnel is to produce supersonic speeds (Mach
numbers up to 5). This can be achieved with an appropriate design of a convergent-divergent nozzle. When the sonic
speed is reached in the test section, the flow accelerates in a nozzle supersonically.
Hypersonic Wind Tunnel – is designed to generate a hypersonic flow field in the working section. The speeds of these
tunnels vary from Mach 5 to 15. As with supersonic wind tunnels, these types of tunnels must run intermittently with very
high pressure ratios when initializing.
Low Speed Wind Tunnel Analysis
Test Section Velocity
V2 
A1
V1
A2
Diverging Duct Velocity
V3 
Where:
V2 – test section velocity
V1 – reservoir velocity
A1 – reservoir cross sectional area
A2 – test section cross sectional area
Where:
V3 – diffuser section velocity
V2 – test section velocity
A2 – test section cross sectional area
A3 – diffuser cross sectional area
A2
V2
A3
Pressure at Various Locations in Wind Tunnel
1
1
1
2
2
2
P1  V1  P2  V2  P3  V3
2
2
2
Where:
P1 – reservoir pressure
ρ – density of the fluid inside the wind tunnel
V1 – reservoir velocity
P2 – test section pressure
P3 – diffuser section pressure
V3 – diffuser section velocity
V2 – test section velocity
Test Section Velocity
V2 
2( P1  P2 )
  A 2 
 1   2  
  A1  
Where:
V2 – Test Section Velocity
P1 – P2 – pressure differences
ρ – density
A2/A1 – Area Ratio
Manometer - in subsonic wind tunnels, a convenient method of measuring the pressure difference P1 - P2, hence of
measuring V2 is by means of a manometer.
p1  p2  wh
AERODYNAMICS 1
Where:
P1 – P2 – pressure differences
W – specific weight of the fluid in the manometer
∆h – change in height
Problems:
1. In a low speed subsonic wind tunnel, one side of a mercury manometer is connected to the settling chamber (reservoir)
and the other side is connected to the test section. The contraction ratio of the nozzle A 2/A1 equals 1/15. The reservoir
pressure and temperature are P1 = 1.1 atm and T1 = 300 K, respectively. When the tunnel is running, the height difference
between the two columns of mercury is 10 cm. The density of liquid mercury is 1.36 x 104 kg/m3. Calculate the airflow
velocity in the test section V2.
Answer: V2 = 144 mps
2. Referring to the figure below, consider a low speed subsonic wind tunnel designed with a reservoir cross section area A 1
= 2 m2 and a test section cross section area A2 = 0.5 m2. The pressure in the test section is p2 = 1 atm. Assume constant
density equal to standard sea level density. a.) Calculate the pressure required in the reservoir, p 1, necessary to achieve a
flow velocity V2 = 40 m/s in the test section. b.) Calculate the mass flow through the wind tunnel.
Answers: a.) P1 = 102, 243. 75 Pa
b.) m = 24.5 kg/s
3. Consider a low-speed subsonic wind tunnel with a nozzle contraction ratio of 1 : 20. One side of a mercury manometer is
connected to the settling chamber, and the other side to the test section. The pressure and temperature in the test section
are 1 atm and 300 K, respectively. What is the height difference between the two columns of mercury when the test section
velocity is 80 m/s? Density of liquid mercury is 1.36 x 104kg/m3
IV. MEASUREMENT OF AIRSPEED
Static Pressure (P) - at a given point is the pressure we would feel if we were moving along with the flow at that point. The
static pressure is due simply to the random motion of the molecules.
Total Pressure/Stagnation Pressure (Po or PT) - Total pressure at a given point in a flow is the pressure that would exist
if the flow were slowed down isentropically to zero velocity. At a stagnation point the fluid velocity is zero and all kinetic
energy has been converted into pressure energy (isentropically). Stagnation pressure is equal to the sum of the freestream dynamic pressure and free-stream static pressure.
For the special case of a gas that is not moving, that is, the fluid element has no velocity in the first place, then static and
total pressures are synonymous:
P = Po. This is the case in common situations such as the stagnant air in the room and gas confined in a cylinder.
Pitot Tube - a pressure measurement instrument used to measure fluid flow velocity. An aerodynamic instrument that
actually measures the total pressure at a point in the flow. It consists of a tube placed parallel to the flow and open to the
flow at one end.
Stagnation Point - Any point of a flow where V = 0
Static Ports – where the static pressure is obtained. A flush-mounted hole on the fuselage of an aircraft, and is located
where it can access the air flow in a relatively undisturbed area. In situations where an aircraft has more than one static
port, there is usually one located on each side of the fuselage. With this positioning, an average pressure can be taken,
which allows for more accurate readings in specific flight situations. Static ports are usually located on the fuselage
somewhere between the nose and the wing.
Pitot Static Probe - An instrument that both measures both static pressure and total pressure.
AERODYNAMICS 1
Pitot Static System - a system of pressure-sensitive instruments that is most often used in aviation to determine an
aircraft's airspeed, Mach number, altitude, and altitude trend. It generally consists of a pitot tube, a static port, and the pitotstatic instruments.
A. Subsonic Incompressible Flow Airspeed Measurement
1
2
P  V1  PT
2
P  q  PT
2( PT  P )
V1 

True Airspeed
VTRUE 
Where:
P – static pressure
ρ – density of the flow
V – velocity of the flow
PT – total pressure
q – dynamic pressure (1/2ρV2)
Where:
2( PT  P )
VTRUE – true airspeed
PT – total pressure
P – static pressure
ρ – actual density
( actual )
Equivalent Airspeed
Ve 
2( PT  P )
Where:
Ve – equivalent airspeed
PT – total pressure
P – static pressure
ρ – density at sea level
 ( sealevel )
B. Subsonic Compressible Flow Airspeed Measurement (Fundamental Relations)

(  1 )
PT  1
2
 1  (   1 )M 1 
P1  2

1
(  1 )
T  1
2
 1  (   1 )M 1 
1  2

2  PT
M1 

(   1 )  P1
2



(  1 ) / 

 1

Where:
PT – total pressure
ρT – total density
P1 – pressure at point 1
ρ1 – density at point 1
M – Mach Number
γ – 1.4 (for air)
True Airspeed for Subsonic Compressible Flow
2 a1  PT
V1 

(   1 )  P1
2
2



(  1 ) / 

 1

(  1 ) / 
2


2 a1  PT  P1
V1 
 1
 1

(   1 )  P1


2
Calibrated Airspeed for Subsonic Compressible Flow
VCAL
2
(  1 ) / 
2


2aS  PT  P1

 1
 1

(   1 )  PS


AERODYNAMICS 1
Where:
VCAL – calibrated airspeed
as – speed of sound at sea level
PT – total pressure
P1 – pressure 1
Ps – standard static pressure at sea level
C. Supersonic Compressible Flow Airspeed Measurement
- In supersonic flow a shock wave will form ahead of the Pitot tube which will change the properties of the
flow.
Specific Changes in the Flow through a Shockwave:
1. The Mach number decreases.
2. The static pressure increases.
3. The static temperature increases.
4. The flow velocity decreases.
5. The total pressure PT decreases.
6. The total temperature TT stays the same for a perfect gas.
The Rayleigh Pitot Tube Formula
PT 2  (   1 ) M 1 

2
P1  4M 1  2(   1 ) 
2
2

(  1 )
1    2M 1
 1
2
Where:
PT2 – total pressure behind the shockwave
P1 – free stream static pressure
M1 – free stream supersonic Mach Number
γ – 1.4 (for air)
Problems:
1. The altimeter on a low-speed Cessna 150 private aircraft reads 5000 ft. By an independent measurement, the outside air
temperature is 505°R. If a Pitot tube mounted on the wing tip measures a pressure of 1818 lb/ft 2. What is the true velocity of
the airplane? What is the equivalent airspeed?
Answers: Vequiv = 218. 470 fps, Vtrue = 236. 339 fps
2. A low speed airspeed indicator reads 200 mph. When the airplane is flying at an altitude at which the altimeter reads
6000 ft. While the ambient temperature is found to be 30 deg. F. Calculate the true airspeed.
Answer: Vtrue = 318.27 fps
3. A high-speed subsonic McDonnell-Douglas DC-10 airliner is flying at a pressure altitude of 10 km. A Pitot tube on the
wing tip measures a pressure of 4.24 x 104 N/m2. Calculate the Mach number at which the airplane is flying. If the ambient
air temperature is 230 K. Calculate the true airspeed and the calibrated airspeed.
Answers: M1 = 0.853, V1/Vtrue = 259. 359 mps, Vcal = 157.465 mps
4. Consider an airplane flying at a standard altitude of 25,000 ft at a velocity of 800 ft/sec. To experience the same dynamic
pressure at sea level, how fast must the airplane be flying?
5. An experimental rocket-powered aircraft is flying at a velocity of 3000 mi/h at an altitude where the ambient pressure and
temperature are 151 lb/ft2 and 390°R, respectively. A Pitot tube is mounted in the nose of the aircraft. What is the pressure
measured by the Pitot tube?
Answer: PT2 = 4077. 604 psf
6. The altimeter on a low-speed airplane reads 2 km. The airspeed indicator reads 50 m/s. If the outside air temperature is
280 K, what is the true velocity of the airplane?
7. Pitot tube is mounted in the test section of a high-speed subsonic wind tunnel. The pressure and temperature of the
airflow are 1 atm and 270 K, respectively. If the flow velocity is 250 m/s, what is the pressure measured by the Pitot tube?
8. A high-speed subsonic Boeing 777 airliner is flying at a pressure altitude of 12 km. A Pitot tube on the vertical tail
measures a pressure of 2.96 x 104 N/m2. At what Mach number is the airplane flying?
AERODYNAMICS 1
Airspeed Terminologies:
Airspeed - is the speed of an aircraft relative to the air. Among the common conventions for qualifying airspeed are:
indicated airspeed ("IAS"), calibrated airspeed ("CAS"), true airspeed ("TAS"), equivalent airspeed ("EAS") and density
airspeed.
Indicated Airspeed (IAS) - is the airspeed read directly from the airspeed indicator on an aircraft, driven by the pitot-static
system uncorrected for instrument and position errors.
Calibrated Airspeed (CAS) - is the result of correcting indicated airspeed for errors in the measurement of static pressure.
Calibrated airspeed (CAS) is the IAS corrected for instrument and position errors. Most civilian EFIS displays also show
CAS.
Calibrated Airspeed Formula:
Vc  VI  Vp
Where:
VC – calibrated airspeed
VI – indicated airspeed
∆Vp – calibration correction (read from the correction chart)
Equivalent Airspeed (EAS) - Equivalent airspeed (EAS) is the calibrated airspeed which is adjusted for the correct static
pressure at a given pressure altitude. Equivalent airspeed (EAS) is the CAS corrected for non standard pressure the
reading on airspeed indicator corrected for compressibility effects.
Equivalent Airspeed Formula:
Ve  Vc  Vc
Where:
Ve – equivalent airspeed
Vc – calibrated airspeed
∆Vc – compressibility correction (read from the correction chart)
True Airspeed - The true airspeed (TAS; also KTAS, for knots true airspeed) of an aircraft is the speed of the aircraft
relative to the air mass in which it is flying. True Airspeed (TAS) is the equivalent airspeed corrected for non standard
density (actual value of air density)
V- Speeds
In aviation, V-speeds or Velocity-speeds are standard terms used to define airspeeds important or useful to the
operation of aircraft, such as airplanes, gliders, autogiros, helicopters, blimps, and dirigibles. These speeds are derived from
data obtained by aircraft designers and manufacturers during flight testing and verified in most countries by government
flight inspectors during aircraft type-certification testing. Using them is considered a best practice to maximize aviation
safety, aircraft performance or both.
Some Common V Speeds:
VA - design maneuvering speed (stalling speed at the maximum legal G-force, and hence the maximum speed at which
abrupt, full deflection, elevator control input will not cause the aircraft to exceed its G-force limit).
VFE - maximum flap extended speed
VLE - maximum landing gear extended speed. The maximum speed at which the aircraft may be flown with the landing gear
extended.
VLO - maximum landing gear operating speed. The maximum speed at which the aircraft may be flying while raising or
lowering the gear.
VMC or VMCA - minimum control speed with the critical engine inoperative.
VNE - the VNE, or never exceed speed, is the V speed which refers to the velocity that should never be exceeded because
of the risk of structural failure, due for example to wing or tail deformation, or aeroelastic flutter.
AERODYNAMICS 1
VR - rotation speed. The speed of an aircraft at which the pilot initiates rotation to obtain the scheduled takeoff performance.
VS - the stalling speed or the minimum steady flight speed at which the aircraft is controllable.
VS0 - the stalling speed or the minimum steady flight speed in the landing configuration.
VS1 - the stalling speed or the minimum steady flight speed obtained in a specific configuration (usually a "clean"
configuration without flaps, landing gear and other sources of drag).
VX - speed for best angle of climb. This provides the best altitude gain per unit of horizontal distance, and is usually used
for clearing obstacles during takeoff.
VY - speed for best rate of climb. This provides the best altitude gain per unit of time.
V. VISCOUS FLOWS
Viscosity - The property of fluid to resist shearing stress. The sticky or adhesive characteristic of a fluid.
Boundary Layer - the region of viscous flow which has been retarded owing to friction at the surface.
NOTE: The boundary layer thickness d grows as the flow moves over the body; that is, more and more of the flow is
affected by friction as the distance along the surface increases. In addition, the presence of friction creates a shear stress at
the surface τw. This shear stress has dimensions of force/area and acts in a direction tangential to the surface. τw gives rise
to a drag force called skin friction drag.
Value of Absolute Coefficient of Viscosity of Air at Standard Sea Level Temperature:
μ = 1.7894 x 10-5 kg/(m)(s)
μ = 3.7373 x 10-7 slug/(ft)(s)
NOTE: For liquids, μ decreases as T increases (we all know that oil gets "thinner“when the temperature is increased). But
for gases, μ increases as T increases (air gets "thicker" when temperature is increased).
Two Basic Types of Viscous Flows:
1. Laminar Flow – flow in which the streamlines are smooth and regular and the fluid element moves smoothly along the
streamline.
2. Turbulent Flow – flow in which the streamlines break up and a fluid element moves in a random, irregular, and tortuous
fashion.
Factors that Affect the Type of Flow in the Boundary Layer:
1. Smoothness of the flow approaching the body
2. the shape of the body
3. the surface roughness
4. the pressure gradient
5. Reynold’s Number
6. Heating of the fluid by the surface
Reynold’s Number – Invented by Osborne Reynolds this is a measure of the ratio of inertia forces to viscous forces. A
similarity parameter that helps determines whether the flows in a body and its scaled version are aerodynamically similar.
Reynolds Number can be applied in determining whether all or portion of the boundary layer is laminar or turbulent.
Rex 
AERODYNAMICS 1
 V x

Where:
Re – Reynold’s Number (local) (dimensionless)
ρ∞ – free stream density in slugs/ft3 or kgm/m3
V∞ - free stream velocity in fps or mps
x – Local chord length in ft or m
μ∞ - free stream absolute coefficient of viscosity
Problems:
1. Find the Reynolds number for a model wing of 3 in chord, tests run at 100 mph with standard air.
Answer: Re = 234, 000
2. Find the velocity at which tests should be run in a wind tunnel on a model wing of 4 in chord in order that the RN shall be
the same a for a wing with a 4 ft chord at 100 mph. Air standard conditions in both cases.
Answer: V = 1200 mph
Laminar Flow Analysis
A. Laminar Boundary Layer Thickness

5.2 x
Rex
Where:
δ – boundary layer thickness (ft, m, in, cm)
x – local chord length (ft, m, in, cm)
Rex – local Reynold’s Number
B. Laminar Local Skin Friction Drag Coefficient (For a local value of x or chord length)
0.664
Cf x 
Rex
Cf x 
W
1
 V 2
2

W
q
C. Laminar Total Skin Friction Drag Coefficient (For the whole surface)
Cf 
1.328
ReL
D. Total Skin Friction Drag
D f  C f q S
Where:
Df – Total skin friction drag (lbf or N)
Cf – coefficient of skin friction drag
q∞ - free stream dynamic pressure (psf, psi, Pa)
S – area of the surface (ft2, m2)
Problems:
1. Consider the flow of air over a small flat plate that is 5 cm long in the flow direction and 1 m wide. The freestream
conditions correspond to standard sea level, and the flow velocity is 120 mps. Assuming laminar flow, calculate
a.) The boundary layer thickness at the downstream edge (the trailing edge)
b.) The drag force on the plate
Answers: a.) δ = 4.06 x 10-4 m
b.) Df = 1.826 N
2. For the flat plate in sample problem 1, calculate and compare the local shear stress at the locations 1 and 5 cm from the
front edge (the leading edge) of the plate, measured in the flow direction.
Answer: at x = 1cm, Ʈw = 20.43 Pa, at x = 5cm, Ʈw = 9.135 Pa
Turbulent Flow Analysis
NOTE: Under the same flow conditions, a turbulent boundary layer will be thicker than a laminar boundary layer. Turbulent
boundary layers grow faster than laminar boundary layers.
A. Turbulent Boundary Layer Thickness

0.37 x
0 .2
Rex
AERODYNAMICS 1
B. Turbulent Local Skin Friction Drag Coefficient (For a local value of x or chord length)
Cf x 
0.0592
(Rex )0.2
C. Turbulent Total Skin Friction Drag Coefficient (For the whole surface)
Cf 
0.074
0 .2
ReL
D. Total Skin Friction Drag
D f  C f q S
Problems:
1. Consider the flow of air over a small flat plate that is 5 cm long in the flow direction and 1 m wide. The free stream
conditions correspond to standard sea level, and the flow velocity is 120 mps. Assuming turbulent flow, calculate the
boundary layer thickness at the trailing edge and the drag force on the plate.
Answers: δ = 1.39 x 10-3 m, Df = 4.92 N
2. For the flat plate in sample problem 1 of laminar flow, calculate and compare the local shear stress at the locations 1 and
5 cm from the front edge (the leading edge) of the plate, measured in the flow direction. Assume now the flow is completely
turbulent
Answers: at x = 1cm, Ʈw = 54.33 Pa, at x = 5cm, Ʈw = 39.34 Pa
NOTE: Skin Friction Drag is reduced by maintaining a Laminar Boundary Layer over a surface. Laminar flow is obtained in
the design of the shape of an airfoil. This special type of airfoil designed to encourage laminar flow is called Laminar Flow
airfoil. Standard shaped airfoils are more prone to have turbulent boundary layers and hence greater skin friction drag.
Laminar Flow Airfoil – a special type of airfoil designed to produce laminar flow all throughout the surface and hence
benefiting from reduced skin friction drag. Laminar Flow Airfoils have a maximum thickness near the middle of the airfoil
.Standard Shaped airfoils have a maximum thickness near the leading edge.
Advantages of a Laminar Flow Airfoil:
1. Low Skin friction drag
2. have excellent high-speed properties, postponing to a higher flight Mach number the large drag rise due to shock
waves and flow separation encountered near Mach 1
Disadvantage of a Laminar Flow Airfoil:
 it readily gets unstable and tries to change to turbulent flow.
 For example, the slightest roughness of the airfoil surface caused by such real-life effects as protruding rivets,
imperfections in machining, and bug spots can cause a premature transition to turbulent flow in advance of the
design condition.
Transition Region/Point
 In reality, the flow always starts out from the leading edge as laminar.
 Then at some point downstream of the leading edge, the laminar boundary layer become unstable and small
"bursts" of turbulent flow begin to grow in the flow.
 Finally, over a certain region called the transition region, the boundary layer becomes completely turbulent.
AERODYNAMICS 1
Critical Reynold’s Number for Transition
Rexcr 
 V xcr

Where:
Rexcr – Critical Reynold’s Number
Xcr – Transition Point/Region
Problems:
1. Assume that you have an airfoil of given surface roughness in a flow at a free-stream velocity of 150 m/s. How far from
the leading edge the transition will take place. Rexcr =5 x 105
Answer: Xcr = 0.047 m
2. If now you double the free stream velocity to 300 m/s, the transition point is still governed by the critical Reynolds number
Rexcr = 5 x 105.Thus,
Answer: Xcr = 0.0235 m
3. The wing of the Fairchild Republic A-1 OA twin-jet close-support airplane is approximately rectangular with a wingspan
(the length perpendicular to the flow direction) of 17.5 m and a chord (the length parallel to the flow direction) of 3 m. The
airplane is flying at standard sea level with a velocity of 200 m/s. If the flow is considered to be completely laminar, calculate
the boundary layer thickness at the trailing edge and the total skin friction drag. Assume the wing is approximated by a flat
plate. Assume incompressible flow.
Flow Separation - Friction also causes another phenomenon, called flow separation, which, in turn, creates another
source of aerodynamic drag, called pressure drag due to separation.
 Laminar boundary layers separate more easily than turbulent boundary layers. Hence, to help prevent flow field
separation, we want a turbulent boundary layer.
Adverse Pressure Gradient – the region of increasing pressure distribution on airfoil which is the primary reason for flow
separation on the upper surface of an airfoil at extreme high angles of attack.
Two Major Consequences of Flow Separation on an Airfoil:
1. A drastic loss of Lift (Stalling)
2. A major increase in drag, caused by pressure drag due to separation
Stall - is a condition in aerodynamics and aviation wherein the angle of attack increases beyond a certain point such that
the lift begins to decrease. This occurs when the critical angle of attack of the foil is exceeded. The critical angle of
attack is typically about 15 degrees, but it may vary significantly depending on the fluid, and Reynolds number.
Ways to Avoid or Delay Flow Separation of airflow on Aircraft Wings:
1. By the use of suction or high pressure air source in the boundary layer.
2. By the Use Variable Geometry airfoil - Laminar separations such as those that occur at the sharp leading edge of a
thin profile can frequently be avoided by a change in geometry that alters the pressure field, such as the deflection of a
nose flap.
AERODYNAMICS 1
3. Vortex Generators - is an aerodynamic device, consisting of a small vane usually attached to a lifting surface, such as
an aircraft wing or a propeller blade of a wind turbine, or attached to some part of an aerodynamic vehicle such as an
aircraft fuselage or a car. When the aerofoil or the body is in motion, the VG creates a vortex, which, by removing some
part of the slow-moving boundary layer in contact with the aerofoil surface, delays local flow separation and aerodynamic
stalling, thereby improving the effectiveness of wings and control surfaces, such as flaps, elevators, ailerons,
and rudders.
Two Types of Drag Caused by Viscous Effects (Presence of Friction on the Surface):
1. Skin Friction Drag, Df
2. Pressure Drag due to flow separation, Dp (Form Drag)
Summary of Viscous Effects on Drag:
 skin friction drag is reduced by maintaining a laminar boundary layer over a surface.
 However, we also pointed out that turbulent boundary layers inhibit flow separation; hence, pressure drag due to
separation is reduced by establishing a turbulent boundary layer on the surface. Therefore, we have the following
compromise:
Profile Drag (D) – the other term for the total drag due to viscous effects.
Laminar Vs Turbulent Flow (Summary)
Laminar
Turbulent
Less skin friction drag; pressure drag predominates
Greater skin friction drag
Unstable, tends to favor flow separation
More stable; disorder is favored by nature
Transition region is closer to leading edge
Transition region is farther aft
“Thinner” velocity profile
Fuller velocity profile
Thinner boundary layer
Thicker boundary layer
VI. VENTURI TUBE
Venturi Tube - a convergent-divergent tube with a short cylindrical throat or constricted section. Venturi’s are used in
carburettors and in many types of fluid control devices to produce a pressure drop proportional to the speed of the fluid
passing them. Venturi tubes on airplanes are also used to provide airflow (vacuum) for air driven gyroscopic instruments
Volume Flow Rate Formula for Venturi Tube
Q  A2
P1  P2

A  
1   2  
2   A1  
2
Where: Q – flow rate in cubic feet/second
A – area of the venturi tube in square feet
P - pressure in pounds per square foot
ρ - density in slugs per cubic foot
Problems:
1. The diameter at A, is 12 in.; the diameter at B is 6 in. What is the flow rate of water if the pressure difference between B
and A is 5 in.Hg?
Answer: Q = 3.867 ft3/s
AERODYNAMICS 1
2. A Venturi tube narrows down from 4 in. in diameter to 2 in. in diameter. What is the rate of flow of water if the pressure at
the throat is 2 Ib per sq in. less than at the larger section?
Answer: Q = 0.388 ft3/s
VII. FLAT, CURVED, AND INCLINED FLAT PLATES
A. Flat Plates Normal to the Direction of Flow
F  cf

2
AV 2
Where:
F = Force on a flat plate normal to air stream in lbf or N.
 = free stream density in slug/ft3 or kg/m3
A = cross-sectional area of the plate in ft2 or m2
V = free stream velocity in fps or mps.
cf = force coefficient (1.28 for flat plates normal to wind)
B. Curved Deflecting Surfaces
Fh( D )  AV 2 ( 1  cos  )
Fv( L )  AV 2 sin 
F 
Fh 2  Fv 2
F  AV 2 2( 1  cos  )
Where: FH = horizontal component of force F in lb or N
FV = vertical component of force F in lb or N
F = resultant force in lb or N
V = free stream velocity
 = angle of deflection in deg
A = cross-sectional area of air stream in ft2 or m2
C. Inclined Flat Plates
L  CL
D  CD

2

2
AV 2
AV 2
Where:
L – Lift in lbf or N
D – Drag in lbf or N
CL – coefficient of lift (unit less)
CD – coefficient of drag (unit less)
ρ – Density in slugs/ft3 or kgm/m3
A – Area of the inclined flat plate in ft2 or m2
V – Velocity of the air that strikes the plate in fps or mps
AERODYNAMICS 1
Problems:
1. A 40 mph wind tunnel is blowing against a signboard 8 ft by 10 ft in size. Atmosphere is normal density. What is the force
acting against the signboard?
Answer: F = 419 lbf
2. A stream of air 50 ft wide and 10 ft high is moving horizontally at a speed of 60 mph. What is the magnitude of the force
required to deflect its movement 4 deg. downward without loss of speed?
Answers: F = 642.682 lbf, Fh = 22.429 lbf, Fv = 642. 291 lbf
VIII. AERODYNAMICS OF CYLINDERS AND SPHERES
Laminar Flow and Turbulent Flow Over a Sphere
NOTE: We need turbulent boundary layer on
a sphere to reduce the pressure drag due to
separation
Karman Vortex Street - named after the great aerodynamicist Theodor Von Kármán is a repeating pattern of
swirling vortices caused by the unsteady separation of flow of a fluid around blunt bodies.
Flow about a Circular Cylinder:
Tangential Velocity of a flow in cylinder:
V = 2Vsin
Where:
V = tangential velocity in ft/s or m/s
V = freestream velocity in ft/s or m/s
Ө = angle through the point on the surface of the cylinder with the main direction of the airflow in deg.
Pressure at any point on the surface of a circular cylinder:
P  P 
Where:
 V  2
2
1  4 sin  
2
P = Pressure at any point on the surface of a circular cylinder in psf or Pa
P = Freestream pressure in psf or Pa
 = Freestream density in slug/ft3 or kg/m3
AERODYNAMICS 1
Problem:
A uniform current of air with a speed of 100ft per sec. flows around a circular cylinder. At a distance from the cylinder the
pressure is atmospheric. What is the pressure at a point on the surface of the cylinder so located that a radial line through
the point makes an angle of 15o with the direction of airflow?
Answer: P = 2080.5 psf
Drag on Common Aerodynamic Shapes:
 Blunt Body such as sphere – pressure drag due to separation is greater than skin friction drag
 Streamlined Body such as airfoil – skin friction drag is greater than pressure drag due to separation
IX. AIRFOILS WINGS AND OTHER AERODYNAMIC SHAPES
Airfoil - is a streamlined body which when set at a suitable angle of attack, produces more lift than drag while also
producing a manageable pitching moment. Solid body designed to move through gaseous medium and obtain useful force
reaction other than drag. Examples: wing, control surface, fin, turbine blade, sail, windmill blade.
Two Types of an Airfoil:
1. Symmetrical – the upper and lower cambers are equal in shape
2. Unsymmetrical – the upper and lower cambers are not equal in shape. The upper camber has a greater curvature
than the lower camber.
Airfoil Nomenclature:
 Chord line - straight line connecting the leading and trailing edges of an airfoil .Is the line joining the end points of
the mean camber lines.
 Mean Camber line – locus of all points equidistant from top and bottom of airfoil
 Camber – maximum distance between chord line and the mean camber line
 Thickness – maximum distance between top and bottom surfaces of wing
 Thickness Ratio - is the maximum thickness to chord ratio, t/c.
 Leading Edge – the most forward points of the mean camber line
 Trailing Edge – the most rearward points of the mean camber line
 Leading Edge Radius - is the radius of a circle, tangent to the upper and lower surfaces, with its center located on
a tangent to the mean camber line drawn through the leading edge of this line.
 Wingspan (b) – the distance from wing tip to wing tip, inclusive of ailerons
 Wing Area (S) – the area of the projection of the actual outline on the plane of the chord. Ailerons and flaps are
counted as part of the wing area.
 Aspect Ratio – is the ratio of the span to the chord (AR = b/c). This is for rectangular wings. For wings that are not
rectangular in shape as viewed from above, the aspect ratio is the ratio of the square of the span to the area (AR =
b2/S)
 Angle of Attack - angle between the relative wind and the chord line
 Drag - defined as the component of the aerodynamic force parallel to the relative wind.
 Lift - defined as the component of the aerodynamic force perpendicular to the relative wind.
 Relative Wind – the air far upstream of the airfoil
 Center of Pressure – the point on the airfoil on which the aerodynamic force (resultant force) is considered to be
concentrated. For an unsymmetrical airfoil the center of pressure moves forward as the angle of attack increases
and vice versa. For a symmetrical airfoil, the center of pressure does not move as the angle of attack varies.
NACA (National Advisory Committee for Aeronautics) - Contains all the pertinent information of all the airfoils
developed. The National Aeronautics and Space Act of 1958 created NASA from NACA.
AERODYNAMICS 1
NACA Airfoil Designation System:
1.
4-digit airfoils : Example NACA 4412
4 = camber 0.04c
4 = position of the camber at 0.4c from L.E.
12 = maximum thickness of 0.12c
The first digit specifies the maximum camber in percentage of the chord (airfoil length), the second indicates the
position of the maximum camber in tenths of chord, and the last two numbers provide the maximum thickness of the
airfoil in percentage of chord. For example, the NACA 2415 airfoil has a maximum thickness of 15% with a camber
of 2% located 40% back from the airfoil leading edge (or 0.4c).
2.
5-digit airfoils: Example NACA 23012
2 = camber 0.02c
30 = position of the camber at (0.30/2) = 0.15c from the leading edge (L.E.)
12 = maximum thickness: 0.12c
3. 6-series laminar flow airfoils: Example NACA 65-218
6 = first digit simply identifies the series
5 = the second gives the location of minimum pressure in tenths of chord from the leading edge (for the
basic symmetric thickness distribution at zero lift). In this case it is 0.5c
2 = the third digit is the design lift coefficient in tenths. In this case, it is 0.2
18 = the last two digits give the maximum thickness in hundredths of chord. In this case, it is 0.18 c
Aerodynamic Forces on an Airfoil:
1. Lift
2. Drag
Factors affecting Aerodynamic Forces:
1. Velocity of air, V
2. Air density, 
3. Characteristic area or size, S
4. Coefficient of dynamic viscosity, 
5. Speed of Sound, Va
6. Angle of attack,
Aerodynamic Center - a certain point on the airfoil about which moments essentially do not vary with angle of attack. For
low-speed subsonic airfoils, the aerodynamic center is generally very close to the quarter-chord point (1/4 of chord from the
leading edge)
Lift Force - the component of the aerodynamic force perpendicular to the relative wind.
L  cl
1
SV 2
2
L  q Scl
Where:
L – Lift force in lbf or N
Cl – coefficient of lift (unit less)
q∞ - free stream dynamic pressure in psf or Pa
ρ – density in slugs/ft3 or kgm/m3
S – wing area in ft2 or m2
V – velocity in fps or mps
Drag Force - the component of the aerodynamic force parallel to the relative wind
D  q Scd
Where:
D = Drag Force in N or lbf
CD = coefficient of drag (unitless)
ρ = density in kg/m3 or slugs/ft3
V = velocity in fps of mps
S = wing area in ft2 or m2
AERODYNAMICS 1
Pitching Moment - the force which tends to rotate the wing caused by lift and drag.
M  q Sccm
Where:
M = Moment Force in N-m or lbf -ft
Cm = coefficient of moment (unitless)
ρ = density in kg/m3 or slugs/ft3
V = velocity in fps of mps
S = wing area in ft2 or m2
c = chord of the wing in m or ft
Problems:
1. Compute for the lift force acting on a wing with an area of 56 ft 2, and the air velocity is 75 ft/s at an altitude of 5000 ft. Cl
of the wing is 0.75.
2. Calculate the drag of a certain wing with an area of 20 m2 and the velocity of the relative wind is 75 m/s at an altitude of
2000 m. Cd of the wing is 0.05
3. Compute for the drag of certain rectangular wing with a chord of 4.5 ft and a span of 16 ft if the dynamic pressure is 80
psi. Coefficient of drag of the wing is 0.030.
Airfoil Data
Zero Lift Angle of Attack αL=O
- The value of α when lift is zero
Stalling Velocity:
VSTALL 
2W
  SCL max
Problem:
Consider the Lockheed F-104 . With a full load of fuel, the airplane weighs 10,258 kgf . Its empty weight (no fuel) is 6071 kgf
.The wing area is 18.21 m2. The maximum lift coefficient at subsonic speeds is only 1.15. Calculate the stalling speed at
standard sea level when the airplane has (a) a full fuel tank and (b) an empty fuel tank. Compare the results.
Answers: a.) Vstall = 88.3 mps/197.6 mph, b.) Vstall = 68 mps/152 mph
AERODYNAMICS 1
Low Speed Pressure Coefficient:
Cp 
P  P
P  P

1
2
q
 V
2
V 
C p  1   
 V 
2
Prandtl Glauert Rule/Correction for Compressibility on Pressure Coefficient:
Cp 
C P ,O
1 M
2
Where: Cp,o - the low-speed pressure coefficient
NOTE: This formulas is reasonably accurate only for 0.3 < M∞ < 0.7.
Problems:
1. The pressure at a point on the wing of an airplane is 7.58 x 104 Pa. The airplane is flying with a velocity of 70 mps at
conditions associated with a standard altitude of 2000 m. Calculate the pressure coefficient at this point on the wing.
Answer: Cp = -1.5
2. Consider an airfoil in a flow with a freestream velocity of 150 fps. The velocity at a given point on the airfoil is 225 fps.
Calculate the pressure coefficient at this point.
Answer: Cp = -1.25
3. Consider an airfoil mounted in a low speed subsonic wind tunnel. The flow velocity in the test section is 100 fps, and the
conditions are standard sea level. If the pressure at a point on the airfoil is 2102 psf, what is the pressure coefficient?
Answer: Cp = -1.18
4. Consider sample problem no. 3, if the flow velocity is increased such that the freestream Mach number is 0.6, what is the
pressure coefficient at the same point on the airfoil?
Answer: Cp = -1.48
5. An airplane is flying at a velocity of 100m/sat a standard altitude of 3 km. The pressure coefficient at a point on the
fuselage is -2.2. What is the pressure at this point?
6. Consider a wing mounted in the test section of a subsonic wind tunnel. The test section air temperature is 510 R and the
velocity of the airflow is 160 ft/s. If the velocity at a point on the wing is 195 ft/s, what is the pressure coefficient at this point?
If the velocity is increased to 700 fps, what is the new pressure coefficient at the same point?
Compressibility Correction for Lift Coefficient:
Cl 
Cl ,O
1 M
Where: Cl, O – low speed value of lift coefficient
2
Critical Mach Number - The free-stream Mach number at which sonic flow is first obtained somewhere on the airfoil
surface.
Critical Pressure Coefficient - The specific value of Cp that corresponds to sonic flow.
Local Coefficient of Pressure at any point in the flow field:

 1

(  1 )
2 
 1  (   1 )M  

2 

2
CP 

1

2 

M   1  1 (   1 )M 2 


2


AERODYNAMICS 1
Critical Pressure Coefficient
 At a particular point on the surface where M = 1. Then, by definition, Cp = C p.cr· Putting M = 1 into the equation of
Local Cp
C P ,cr



2 (  1 )
2  2  (   1 )M  

 1

2 

 1
M  



Problem:
What is the critical value of the pressure coefficient Cp, cr for an airplane flying at 500 knots in air at 25 degree Fahrenheit?
Answer: Cpcr = -0.488
Drag Divergence Mach Number - The free stream Mach number at which cd begins to increase rapidly.
Rule of Thumb for Drag Divergence Mach number:
M cr  M dragdivergence  1.0
Theories of Lift
1. The Standard Theory – Bernoulli’s Principle
2. Newton’s Third Law of Motion
3. Circulation Theory – Kutta Joukowsky theorem
Kutta Joukowsky Theorem for Spinning Cylinders:
  2rVt
LT  Vl
Circulation
Total Lift
Where:
r - radius of cylinder in ft or m
Vt - 2πrN
LT - total lift in Lbf or N
l - length of cylinder in ft or m
ρ – density in slugs/ft3 or kgm/m3
N - rotational speed of the cylinder in rps
Problem:
1. A cylinder 3 ft in diameter and 8 ft long is rotating at 150 rpm in an airstream of 50 mph. What is the total lift?
Answer: LT = 309.660 lbf
2. The lift on a spinning circular cylinder in a freestream with a velocity of 30 mps and at standard sea level conditions is 6
N/m of span. Calculate the circulation around the cylinder.
Answer: Circulation = 0.163 m2/sec
Infinite Wings – two dimensional wings where the model wings are spanned the test section of the wind tunnel from one
sidewall to the other. In this fashion, the flow sees essentially a wing with no wing tips; that is, the wing in principle could be
stretching from plus infinity to minus infinity in the spanwise direction.
Finite Wings - three-dimensional wings. There is flow in the wing tips.
Finite Wing Properties:
1.) Geometric Wingspan, b - the distance between tip to tip of the of the wings inclusive of ailerons
AERODYNAMICS 1
2.) Wing Area, S – the area of the projection of the actual outline on the plane of the chord. Ailerons and flaps are counted
as part of the wing area.
3.) Chord Length, c – the length of the projection of the airfoil section on its chord.
4.) Aspect Ratio, AR – the ratio of the square of the span to the wing area.
AR 
b
Ca
AR 
Where:
b - Wing span
S – Wing area
S
Ca 2
Ca 
Ct  Cr
2
Ca – average chord Cr – root chord
Ct – tip chord
Aspect Ratio for Non Rectangular Wings:
b2
AR 
S
5.) Taper Ratio, TR - the ratio of the tip chord Ct to the root chord Cr
TR 
Ct
Cr
6.) Mean Aerodynamic Chord, MAC - the chord of an imaginary simple rectangular wing which, throughout the flight
range, will have the same force vectors as the actual wing. Mean Geometric Chord MGC is sometimes synonymous to
MAC. For a rectangular monoplane wing, the MGC is identical with chord of the wing section.
2  (Ct  Cr ) 2  CtCr 
MAC  

3
Ct  Cr

7.) Sweepback or Sweep Angle – the angle between a line perpendicular to the plane of symmetry of the airplane and the
horizontal projection of a reference line on a wing. If the wing is rectangular in shape, this reference line is the leading edge
of the wing. If the wing is tapered, the reference line is the line passing through the point in each airfoil section that is ¼ of
the chord length back from the leading edge of the section. If the angle of sweepback is 45 degrees or greater, the wing is
termed arrowhead wing.
8.) Dihedral - the sloping upwards of the wing toward the tips as viewed in front of the airplane. The angle between a line
perpendicular to the axis of symmetry of the airplane and the projection of the wing axis on a plane perpendicular to the
longitudinal axis of the airplane.
Drag - drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction) refers
to forces acting opposite to the relative motion of any object moving with respect to a surrounding fluid.
Drag forces always decrease fluid velocity relative to the solid object in the fluid's path.
Two General Types of Drag:
1. Pressure Drag – due to a net imbalance of surface pressure acting in the drag direction
2. Friction Drag - due to the net effect of shear stress acting in the drag direction.
Types of Subsonic Drag:
1. Skin Friction Drag – drag due to frictional shear stress integrated over the surface.
2. Form Drag – drag due to pressure imbalance in the drag direction caused by separated flow.
3. Profile Drag – the sum of skin friction and form drag. Also called section drag. Usually used in conjunction with two
dimensional airfoils.
4. Interference Drag – an additional pressure drag caused by the mutual interaction of the flow fields around each
component of the airplane.
5. Parasite Drag – the term used for the profile drag of the complete airplane. It includes interference drag.
AERODYNAMICS 1
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Induced Drag – A pressure drag due to pressure imbalance in the drag direction caused by the induced flow
(downwash) associated with the vortices created at the tips of the finite wings.
Zero Lift Drag – (Usually used in conjunction with complete airplane configuration.) The parasite drag that exists
when the airplane is at zero lift angle of attack, that is when the lift of the airplane is zero.
Drag due to Lift – (Usually used in conjunction with a complete airplane.) The portion of the total airplane drag
measured above the zero lift drag. It consist of the change in parasite drag when the airplane is at an angle of
attack different from the zero lift angle, plus the induced drag from the wings and other lifting components of the
airplane.
External Store Drag – An increase in parasite drag due to external fuel tanks, bombs, rockets, etc. carried as
payload by the airplane, but mounted externally from the airframe.
Landing Gear Drag – An increase in parasite drag when the landing gear is deployed.
Protuberance Drag – An increase in parasite drag due to “aerodynamic blemishes” on the external surface, such
as antennas, lights, protruding rivets, and rough or misaligned skin panels.
Leakage Drag – An increase in parasite drag due to air leaking into and out of holes and gaps in the surface.
Engine Cooling Drag – An increase in parasite drag due to airflow through the internal cooling passages for
reciprocating engines.
Flap Drag – An increase in both parasite drag and induced drag due to the deflection of flaps for high lift purposes.
Trim Drag – The induced drag of the tail caused by the tail lift necessary to balance the pitching moments about
the airplane’s center of gravity.
Induced Drag – drag due to lift. Caused by wing tip vortices
Coefficient of Induced Drag
2
C D ,i
Where:
C
 L
A Re
CD,i - coefficient of Induced Drag
CL – coefficient of lift
AR – wing aspect ratio
e – span efficiency factor
Induced Drag Formula
Di  CD ,i

2
SV 2
1 W 
Di   
q  b 
2
Total Drag Coefficient for a Finite Wing at Subsonic Speeds
2
C
C D  Cd  L
eAR
Where:
CD – total drag
Cd – profile drag (skin friction drag + pressure drag due to separation)
CD,i – induced drag
Power Required to Overcome Induced Drag
hpDi 
DiV
550
Where:
hpDi – power required in (hp)
Di – induced drag in lbf
V – Velocity in fps
Problems:
1. A Northrop F-5 fighter airplane which has a wing area of 170 ft2 has a velocity of 250 mph at standard sea level
conditions with a lift generated of 18,000 lbs. If the wing span is 25.25 ft, calculate the induced drag coefficient and the
induced drag itself. Assume e = 0.8,
Answers: Cdi = 0.0466, Di = 1266 lbf
2. Consider a flying wing (such as the Northrop YB-49 of the early 1950s) with a wing area of 206m2 , an aspect ratio of 10,
a span effectiveness factor of 0.95, and an NACA 4412 airfoil. The weight of the airplane is 7.5 x 10 5 N. If the density
altitude is 3 km and the flight velocity is 100 m/s, calculate the total drag on the aircraft. Assume profile drag coefficient Cd =
0.006.
Answers: Cdi = 0.021, CD = 0.027, D = 2.53 x 104 N
AERODYNAMICS 1
3. A monoplane weighing 2000 lb has a span of 38 ft. What is the induced drag at 10,000 ft altitude if the airspeed is 80
mph? What horsepower is required to overcome induced drag?
Answer: hpdi = 16.6 hp
4. The Northtrop N-3PB weighs 9200 lb, and its span is 48 ft 11 in. What horsepower is required to overcome the induced
drag when it is flying at sea level at 210 mph?
Answer: 55.9 hp
Aspect Ratio Correction
1   2 
CL  1
1 

  AR1 AR2 
 1
1 


 AR1 AR2 
 1   2  18.24CL 
C  1
1 
 L 

  AR1 AR2 
Angle of Attack for the wing of different AR in radians
Angle of Attack for the wing of different AR in degrees
2
CD1  CD 2
CD for the wing of different AR
Problem:
1. Two identical wings are flying level at 120 mph at sea level conditions. Identical in the sense that they both have NACA
2421 as their airfoil. Where,
– W1 = W2 = 2700 lbs
– b1 = b2 = 40 ft
– S1 = 220 ft2
– S2 = 190 ft2
– CL = 1.62
Calculate the angle of attack at which wing 2 must fly if wing 1 is flying at 8.2 degrees. Solve for the difference in induced
drag
Answers: AR1 = 7.27, AR2 = 8.42, Angle of Attack of wing 2 = 7.65 deg.,
Difference in induced drag = 240.12 lbf
2. An airfoil with aspect ratio of 6, at an angle of attack of 3 degrees has a C L = 0.381 and CD = 0.0170. Find for the same
airfoil section, the angle of attack and the CD that will correspond with the CL of 0.381 if the aspect ratio is 4.
Answer: Angle of attack 2 = 3.579 deg., Cd = 0.0208
3. A glider having has an aspect ratio of 14. For the airfoil with an Aspect ratio of 6, at an angle of attack of 6 degrees, CL =
0.725 and CD = 0.039. Find the angle of attack and CD for the glider wing when CL = 0.725.
X. AIRCRAFT HIGH LIFT DEVICES
Aircraft High Lift Devices - moving surfaces or stationary components intended to increase lift during certain flight
conditions.
Common Aircraft High Lift Devices:
1. Flaps
2. Slats/Slots
Flaps - are hinged surfaces mounted on the trailing edges of the wings of a fixed-wing aircraft to reduce the speed at which
an aircraft can be safely flown and to increase the angle of descent for landing. They shorten takeoff and landing distances.
Flaps do this by lowering the stall speed and increasing the drag. Extending flaps increases the camber or curvature of the
wing, raising the maximum lift coefficient - or the lift a wing can generate. This allows the aircraft to generate as much lift but
at a lower speed, reducing the stalling speed of the aircraft, or the minimum speed at which the aircraft will maintain flight.
Extending flaps increases drag which can be beneficial during approach and landing because it slows the aircraft.
AERODYNAMICS 1
Types of Flaps:
 Plain flap: the rear portion of airfoil rotates downwards on a simple hinge mounted at the front of the flap.
 Split flap: the rear portion of the lower surface of the airfoil hinges downwards from the leading edge of the flap,
while the upper surface stays immobile. Like the plain flap, this can cause large changes in longitudinal trim,
pitching the nose either down or up, and tends to produce more drag than lift. At full deflection, a split flaps acts
much like a spoiler, producing lots of drag and little or no lift.
 Slotted flap: a gap between the flap and the wing forces high pressure air from below the wing over the flap
helping the airflow remain attached to the flap, increasing lift compared to a split flap.
 Fowler flap: split flap that slides backwards flat, before hinging downwards, thereby increasing first chord, then
camber. The flap may form part of the upper surface of the wing, like a plain flap, or it may not, like a split flap but it
must slide rearward before lowering.
Slats and Slots - Another common high-lift device is the slat, a small aerofoil shaped device attached just in front of the
wing leading edge. The slat re-directs the airflow at the front of the wing, allowing it to flow more smoothly over the upper
surface while at a high angle of attack. This allows the wing to be operated effectively at the higher angles required to
produce more lift.
A slot is the gap between the slat and the wing. The slat may be fixed in position, or it may be retractable. If it is fixed, then
it may appear as a normal part of the leading edge of a wing which has slot. The slat or slot may be either full span, or may
occur on only part of the wing (usually outboard), depending on how the lift characteristics need to be modified for good low
speed control. Often it is desirable for part of the wing where there are no controls to stall first, allowing aileron control well
into the stall.
AERODYNAMICS 1