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SYE 3801
Aerodynamics
Spring 2015
Dr. Thomas Fallon
Email: tfallon@spsu.edu
Aerodynamics
• The science that deal with the flow of any gas
• Heavier than air machine can fly due to flow of air
over its surface
• Applications of aerodynamics;
• Rockets, jet engines, propeller, space vehicle, wind tunnels,
projectiles, motion of global atmosphere, flow of gases
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 2
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 3
Review: Derivative and Differential ElementsRef: Calculus and Analytic Geometry, Thomas/Finney, Addison-Wesley Publishing Co., ISBN: 0-201-07540-7
𝑓′ 𝑥 =𝑑𝑦
𝑑𝑥= lim
Δ𝑥→0
𝑓 𝑥 + Δ𝑥 − 𝑓(𝑥)
Δ𝑥= 𝑚
Algorithm using f(x) = mx + b:
1. f(x + Δx) = m(x + Δx) + bf(x) = mx + b
2. Subtract: f(x + Δx) – f(x) = mΔx
3. Divide by Δx: f(x + Δx)−f(x)
Δx=
mΔ𝑥
Δ𝑥= 𝑚
4. Calculate limΔ𝑥→0
: f ′ x = limΔ𝑥→0
f(x + Δx)−f(x)Δx = 𝑚
12/16/2014SYE 3801 - Aerodynamics - Spring 2015 4
Example: For 𝑓 𝑥 = 𝑥2 +1
𝑥, derive 𝑓′ 𝑥 using the derivative algorithm
1. 𝑓 𝑥 + Δ𝑥 = 𝑥 + Δ𝑥 2 +1
(𝑥+Δ𝑥)= 𝑥2 + 2𝑥Δ𝑥 + (Δ𝑥)2+
1
(𝑥+Δ𝑥)
2. 𝑓 𝑥 = 𝑥2 +1
𝑥
3. Subtract: 𝑓 𝑥 + Δ𝑥 − 𝑓 𝑥 = 2𝑥Δ𝑥 + (Δ𝑥)2+1
(𝑥+Δ𝑥)−
1
𝑥
= 2𝑥Δ𝑥 + (Δ𝑥)2+𝑥−(𝑥+Δ𝑥)
𝑥(𝑥+Δ𝑥)= Δ𝑥 2x + Δ𝑥 −
1
𝑥 𝑥+Δ𝑥
4. Divide by Δ𝑥 : 𝑓 𝑥+Δ𝑥 −𝑓(𝑥)
Δ𝑥= 2x + Δ𝑥 −
1
𝑥 𝑥+Δ𝑥
5. Calculate limΔ𝑥→0
: 𝑓′ 𝑥 = limΔ𝑥→0
f(x + Δx)−f(x)Δx = lim
Δ𝑥→02x + Δ𝑥 −
1
𝑥 𝑥+Δ𝑥=
2𝑥 + 0 −1
𝑥 𝑥+0= 2𝑥 −
1
𝑥2
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 5
Differential Elements
𝑑𝑦 = 𝑓′ 𝑥0 𝑑𝑥,𝑤ℎ𝑒𝑟𝑒 𝑑𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒in the x−direction
𝑓𝑟𝑜𝑚 𝑥0,and dy is a dependent variable of change in the y-direction
to the tangential o𝑓𝑥0.
Example: For x = cos( 𝑡), 𝑦 = sin(𝑡), 𝑓𝑖𝑛𝑑𝑑𝑦
𝑑𝑥
𝑑𝑥 = −sin(𝑡)𝑑𝑡𝑑𝑦 = cos 𝑡 𝑑𝑡
𝑑𝑦
𝑑𝑥=
cos 𝑡 𝑑𝑡
−sin(𝑡)𝑑𝑡=
𝑥
−𝑦
Pressure, ( )• Pressure = Force per unit area
“Pressure is the normal force per unit area exerted on a
surface due to the time rate of change of momentum
of the gas molecules impacting on that surface”
p = mv
dp/dt = m(dv/dt) = ma = F
P = F/A
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 6
P
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 7
Where;
dA = an incremental area around B
dF = Force on one side of dA due to pressure
0limdA
dFP
dA
Units:
2 2 2( ), ( ), ,
N lb dynepascal psi atm
m in cm
We apply pressure at point: points in a flow, hand outside car window
What about a thumbtack?
Density, ( )
• Density = Mass per unit volume
“The density of a substance is the mass of that
substance per unit volume”
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 8
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 9
Where;
dm = Mass of gas inside dV
dV = Elemental volume around point B
0limdV
dm
dV
Units:
*
3 3 3 3, , , mlbkg slug g
m ft cm ft
Temperature, ( )
“The measure of average kinetic energy of the particles
of gas”
KE = 3/2kT, where k = Boltzmann constant (1.38 x 10-23 J/K)
K = 273.13 + C
Question: What does -120ºC equals to in degree Kelvin?
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 10
Units:
( ), ( ), ( ), ( )Kelvin K Celsius C Rankine R Fahrenheit Fo o o
T
Relationship between P, ρ, and T (all scalars)
is referred to the Equation of State
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 11
Perfect Gas
“A perfect gas is one in which intermolecular forces are
negligible”
“Air under normal conditions (e.g. subsonic flight
through the atmosphere) behaves like a perfect gas”
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 12
Perfect Gas
• Equation of state for a perfect gas
Where:
R = Specific gas constant
= 287 for air
= 1716 for air
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 13
P RT
J kg K
( ) ( )ft lb slug R o
Specific Volume: ( ) is the
inverse density
1P RT RT
v
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 14
P RTN/m2 = kg/m3 x J/(kg)(K) x K = J/m3
Are they equal?
True Equation of State is referred to as the Berthelot Equation:
P/ρRT = 1 + aP/T – bP/T3
As P decreases or as T increases the Berthelot Equation approaches the Equation of State
for a perfect gas.
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 15
Compute the temperature of a point on a wing of an
Airbus A320, where the pressure and density are given
as 1.20 x 105 N/m2 and 1.1 kg/m3, respectively.
𝑃 = 𝜌𝑅𝑇 → 𝑇 =𝑃
𝜌𝑅=
1.20𝑥105 𝑁/𝑚2
1.1𝑘𝑔/𝑚3𝑥287𝐽/(𝑘𝑔)(𝑘)= 318°K = 45°C
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 16
P, ρ, and T under standard conditions (sea level):
Ps = 1.01325 x 105 N/m2 = 1atm
ρs = 1.225 kg/m3
Ts = 288.15K
Compute the total air mass in a classroom with dimensions
30m x 20m x 3m at sea level under standard atmospheric
conditions.
ρ =𝑚
𝑉→ 𝑚 = 𝜌𝑉 =
1.225𝑘𝑔
𝑚3𝑥 30𝑚 𝑥 20𝑚 𝑥 3𝑚 = 2205𝑘𝑔
12/16/2015SYE 3801 - Aerodynamics – Spring 2015 17
Compute the density and specific volume of air in a wind
tunnel at P = 0.25atm and -120ºC
𝑃𝑣 = 𝑅𝑇 → 𝑣 =𝑅𝑇
𝑃=
287𝑥153
0.25𝑥1.01𝑥105=
1.74𝑚3
𝑘𝑔
𝜌 =1
𝑣=
0.57𝑘𝑔
𝑚3
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 18
Given: Cabin pressure = 0.9 atm, T = 15°C, V = 1800m3 (of air). Air is recirculated every 20 minutes. Calculate the mass flow rate:
1atm = 1.01325 x 105N/M2
P = 0.9atm = 0.9x1.01325x105 = 0.909x105N/M2
T = 273 + 15 = 288Kρ = P/RT = 0.909x105/(287)(288) = 1.1kg/m3
Total mass m = ρV = (1.1)(1800) = 1980kgMass flow rate = m = 1980/1200 = 1.65kg/s
Speed“Displacement per unit time”
“Speed + Direction”
- Vector quantity
- The velocity of gas may vary from point to point in a flow
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 19
Units:
, ,km
mph machhrVelocity, ( )V
12/16/2015SYE 3801 - Aerodynamics - Spring2015 20
Given: Baseball thrown by a pitcher at 85 mph. Calculate the minimum
and maximum velocity at the surface and their locations:
Theoretical expression of the field velocity:
V = 3/2V∞sinϴ, where V∞ = Free stream velocity
Flow (V∞ = 85 mph)
Vmin = 0 at ϴ = 0°; ie.e when ϴ = 0° and ϴ = 180°Vmax = 3/2(85)(1) = 127.5 mph at ϴ = 90°
ϴ
Streamline
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 21
“As long as the flow is steady (does not fluctuate with time) a moving fluid
element is seen to trace out a fixed path in space”
The path taken by a moving fluid element is called a streamline of a flow.
The velocity at point B is represented by an infinitesimally small element
of fluid as is moves through B; V is tangential to the streamline at point B.
V = ds/dt (m/s)Note the stagnation point where V = 0
Streamline
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 22
Laminar flow = low Reynolds number
Turbulent flow = high Reynolds number Small vortices of turbulence
Wind Tunnel Video: http://tinyurl.com/lq85lba
Streamline
08/20/2012SYE 3801 - Aerodynamics - Fall 2012 23
Source of Aerodynamic Forces
“The knowledge of and at each point of a
flow fully defines the flow field”
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 24
, ,P T V
Source of Aerodynamic Forces
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 25
Source of Aerodynamic Forces
12/16/2015SYE 3801 - Aerodynamics - Spring 2015 26
Pressure: always acts normal to
the surface
Shear stress: ( ) is the force
per unit area acting tangentially
on the surface due to friction.
The velocity decreases down
through the boundary layer as it
goes to 0 at the surface.
w
Which of the two forces contributes most to lift?
τw contributes more to lift at high α
08/20/2012SYE 3801 - Aerodynamics - Fall 2012 27
Standard International (SI) Units
• NACA (National Advisory Committee for Aeronautics) NASA
requires the use of SI units
• AIAA (American Institute of Aeronautics and Astronautics)
encourages the use of SI units
Length Mass Time Temp Pressure Density
SI m kg sec Kelvins N/m2 kg/m3
English foot slug sec Rankine lb/ft2 Slugs/ft3
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