SYE 3801 Aerodynamics Spring 2015 - KSU | Faculty...

SYE 3801

Aerodynamics

Spring 2015

Dr. Thomas Fallon

Email: [email protected]

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


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


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


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


P


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”



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?


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


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”


Perfect Gas

• Equation of state for a perfect gas

Where:

R = Specific gas constant

= 287 for air

= 1716 for air


P RT

J kg K

( ) ( )ft lb slug R o

Specific Volume: ( ) is the

inverse density

1P RT RT

v


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.


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


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


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


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


“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


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”


, ,P T V



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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