aoe 2104--aerospace and ocean engineering fall 2009

AOE 2104--Aerospace and Ocean Engineering Fall 2009

Virginia Tech 1 September 2009Lecture 2

AOE 2104

Introduction to Aerospace Engineering

Lecture 2Basic Aerodynamics

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Reminder: The first homework assignment (paper copy) is due AT THE BEGINNING OF

NEXT CLASS!!

Also I would appreciate any feedback on the class that you have. You are welcome to see me after class, tell me during class, or send me an

email.


1 September 2009Lecture 2

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3 steps to determine p, r, and T at any altitude ?

2 equations used to construct the standard atmosphere model ?

Name and define the different types of altitudes.

2 types of regions found in the temperature variations with altitude and their characteristics ?

Any questions ?

Standard Atmosphere



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Basic Aerodynamics



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Basic Aero – Why? How? What do we have so far?

Why are we looking into aerodynamics?To determine the forces acting on a vehicle in flightRemember aerodynamic forces arise from two natural phenomena

How are we going to proceed ? Using Laws of Physics to quantify the interaction between the vehicle and the environment it is evolving in.

What do we have so far ?



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Our Aerodynamic Tool Box

Four aerodynamic quantities that define a flow field

Steady vs unsteady flow

Streamlines

Sources of aerodynamic forces

Equation of state for perfect gases

Hydrostatic Equation

Standard Atmosphere Model

6 different altitudes



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Aerodynamic Tools Needed: Governing Laws

We are going to need the 3 following physical principles to describe the interaction between the vehicle and its associated flow field:

Conservation of MassContinuity Equation (§§ 4.1-4.2)

Newton’s 2nd Law (and Conservation of Momentum)

Euler’s and Bernoulli’s Equations (§§ 4.3-4.4)

Conservation of EnergyEnergy Equation (§§ 4.5-4.7)



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Conservation of Mass – The Continuity Equation

Physical Principle:Mass can neither be created nor destroyed (in other words, input = output).

Eq.(4.2)



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Streamline

A streamline is a line that is tangent to the local velocity vector.

If the flow is steady, the streamline is the path that a particle follows.

v v

v

v

v



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Remarks on Continuity

The equation we just derived assumes that both velocities and densities are uniform across areas 1 and 2.

In reality, both velocities and densities will vary across the area

Continuity Equation is extensively used in the design and operation of wind tunnels and rocket nozzles (we will see how later).A stream tube is delimited by 2 streamlines and does not have to be bounded by a solid wall.




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Compressible Versus Incompressible Flows



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Continuity for Incompressible Flows

• All fluids are compressible in reality.

• However, many flows are “incompressible enough” so that the incompressibility assumption holds.

• Incompressibility is an excellent model for Flows of liquids (e.g. water and oil)Air at low speed (V < 100 m/s or 225 mi/h)

• Equation of Continuity for Incompressible Flows reduces to

• So that if A2 < A1 then V2 > V1.

1

2

2

1

VV

AA



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Continuity – Sample Problem 1

A convergent duct was found in the basement of Randolph. The inlet and exit areas are measured to be Ai = 5m2 and Ae = 2m2. Assuming we use this duct with an inlet velocity of Vi = 9 mi/h, find the exit velocity.

First, we need to be consistent with the unit system. Let’s work in SI units.Vi = 9 mi/h = 9x1609/3600 m/s Vi = 4 m/s.Vi << 100 m/s so the flow is considered incompressible.

From Incompressible Continuity, Therefore, the exit velocity will be 10 m/s.

10m/sV425V

AAV ei

e

ie



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Continuity – Sample Problem 2



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Momentum Equation

Continuity is a great addition to our toolbox, however it says nothing about pressure.

Why is pressure important? Let’s look at Newton’s 2nd Law:Sum of the forces = Time rate of change of momentum

F = d(mv)/dtF = m dV/dt assuming m = const.F = m a

The pressure is going to translate into force, which by Newton’s 2nd Law results in change of momentum. Assuming incompressibility (m = const), this will result in change of velocity (thus impacting performance for example).

To find momentum, simply apply F = ma to an infinitesimally small fluid element moving along a streamline.



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Assume fluid element is moving in the x-direction.3 types of force act on the element:

• Pressure force (normal to the surface) p• Shear stress (friction, parallel to the surface) tw

• Gravity r dxdydz gIgnore gravity (smaller than other forces) and assume inviscid flow (non-viscous i.e. no friction), balance of the forces on x.

vO

Streamline

p F = ma

dxdz

dydx

dxdpp

Momentum Equation – Free Body Diagram



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Momentum Equation – Force Balance



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Momentum for Incompressible Flows – Bernoulli’s Equation

• For incompressible flows, r = const.

• Integrating Euler’s equation between 2 points along a streamline gives:

• This equation is known as Bernoulli’s Equation.

streamline a along

wordsother in

as rewritten be can which

02

211

222

21

2212

pconstρV21p

ρV21pρV

21p

VV21ρpp



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Description of Bernoulli’s Equation

02 pρV

21p

Static Pressure• Pressure felt by an object or person suspended in the fluid and moving with it.• Can be thought of as internal energy.

Dynamic Pressure• Pressure due to the fluid motion.• Can be thought of as kinetic energy.

Total (stagnation) Pressure• Pressure that would be felt if the fluid was brought isentropically to a stop.• Can be thought of as total energy.



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3 New Tools – Continuity, Euler, and Bernoulli’s Equations

• Continuity Equationr A V = constAssumptions: steady flow.

• Euler’s Equationdp = - r V dVAssumptions: steady, inviscid flow.

• Bernoulli’s Equation

Assumptions: steady, inviscid, incompressible flow along a streamline.

Euler and Bernoulli’s equations are essentially applications of Newton’s 2nd Law to fluid dynamics.

02 pρV

21p



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Momentum Equations - Sample Problem 1



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Momentum Equations - Sample Problem 2



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Practical Applications

By combining Continuity, Euler, and Bernoulli’s equation, one can obtain the velocity at any point on an aircraft assuming surrounding conditions are known (either through measurements or using Standard Atmosphere).

Two major applications for this:Low-Speed Subsonic Wind Tunnel testing/designingFlight measurements of velocity



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Low-Speed Subsonic Wind Tunnels (§4.10)



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Wind Tunnel Calculations

• From Bernoulli, between points 1 and 2:

• Using Continuity:

• Combining the two, we get:

• Since the ratio of throat to reservoir area (A2/A1) is fixed for wind tunnel and r is constant for low-speed (incompressible) flows, the quantity driving the tunnel is p1-p2.

• But how can we determine p1-p2 ???

2

12

212

AA1ρ

pp2V

21

21 V

AAV

2121

22 Vpp

ρ2V



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Manometer

fluid. ofheight the toalproportiondirectly is difference pressure that themeans thisfluid, reference for theconstant is since

fluid. reference theof e)unit volumper (weight weight specific theis where,

whwpp

gρwhwAApAp

21

f21



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Wind Tunnels – Sample Problem 1



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Wind Tunnels – Sample Problem 1 Solution

2

12

212

AA1ρ

pp2V

31

5

1

11 1.29kg/mρ

300287)10(1.011.1

RTpρ

• Height of liquid: h = 10cm = 0.1m• Specific weight of liquid mercury: w = (1.36x104)x9.8 = 1.33x105 N/m2

• Actual pressure difference: p1-p2 = w h = 1.33x104 N/m2.• To find V2 from Bernoulli, use

• We computed p1-p2, A1/A2 = 15 is given, so we need to find r.• Since we are in a low-speed wind tunnel, flow is incompressible, so r = const, which

means we can compute it at any point in the tunnel. Since p1 and T1 are given, use Equation of State to find r = r 1:

• Combining all the results we get V2 = 144 m/s (slightly over the incompressible velocity limit, which means compressibility effects should be taken into account).

2

12

212

AA1ρ

pp2V

31

5

1

11 1.29kg/mρ

300287)10(1.011.1

RTpρ



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Measurement of Airspeed (§4.11)

Bernoulli’s equation provides an easy method for determining the velocity of any fluid

Therefore, we need to know p and p0

ρ

pp2V 0

RTpρ



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Total (stagnation) Pressure (p0 ) Measurement

• The total pressure is easy to measure if the flow direction is known. An opened-

end tube aligned with the flow direction is enough. This type of tube is called

"Pitot probe”(named after Henri Pitot who invented it in 1732; see §4.3 for historical

background)



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Static Pressure (P) Measurement

The static pressure is also easy to measure using a tube with a close end and

pressure taps around its circumference.

“Static probe”



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Dynamic Pressure Measurement

Finally, it is possible to measure directly the difference between stagnation and static pressure by combining the Pitot and static probes into a Pitot-static probe (!).

“Pitot-Static probe”



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Airspeed Indicator

r

)0(2 ppv

SLind

ppv

r

)0(2

If the only known density is at sea level,

“Indicated or Equivalent Airspeed”



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True Airspeed

alttrue

ppv

r

)0(2

Therefore, the relationship between true and indicated airspeed is:

and SL

ind

ppv

r

)0(2

21

alt

SLindtrue vv

rr



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www.aeromech.usyd.edu.au/ aero/instruments/ http://www.tech.purdue.edu/at/courses/aeml/airframeimages/pitottube.jpg



http://www.aeromech.usyd.edu.au/aero/instruments/


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www.aeromech.usyd.edu.au/ aero/instruments/ http://home4.highway.ne.jp/t-park/tp/image/seventh/s-port.jpg





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Measurement of Airspeed – Sample Problem



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From Standard Atmosphere (App. B), at 5000ft, p = 1761 lb/ft2.Pitot tube measures stagnation pressure so p0 = 1818 lb/ft2.

Density is found from measured temperature and tabulated pressurer = p/(RT) = 1761/(1716*505) r = 2.03x10-3 slug/ft3.

sftVpp

V truealt

true /2371003.2

)17611818(2)0(23

r

sftVpp

V indsea

ind /21910377.2

)17611818(2)0(23

r

7.6% difference

Measurement of Airspeed – Sample Problem Solution



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For Next Class: Review Chapter 4 and let me know what questions you have

Thursday: HW 1 due. Stay Tuned for HW 2.



aoe 2104--aerospace and ocean engineering fall 2009

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