1202 Lecture 2

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MAE 1202: AEROSPACE PRACTICUM

Lecture 2: Introduction to Basic Aerodynamics 1

January 24, 2011

Mechanical and Aerospace Engineering DepartmentFlorida Institute of Technology

D. R. Kirk

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READING AND HOMEWORK ASSIGNMENTS

Reading: Introduction to Flight, 6th Edition, by John D. Anderson, Jr.

For this weeks lecture: Chapter 4, Sections 4.1 - 4.9 For next weeks lecture: Chapter 4, Sections 4.10 - 4.21, 4.27

Lecture-Based Homework Assignment:

Problems: 4.1, 4.2, 4.4, 4.5, 4.6, 4.8, 4.11, 4.15, 4.16

DUE: Wednesday, February 2, 2011 by 11am

Turn in hard copy of homework

Also be sure to review and be familiar with textbook examples inChapter 4

Laboratory Homework #2 (assigned this week) will be due Friday,

February 4, 2011 by 11 AM

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ANSWERS TO LECTURE HOMEWORK

4.1: V2 = 1.25 ft/s

4.2: p2-p1 = 22.7 lb/ft2

4.4: V1 = 67 ft/s (or 46 MPH)

4.5: V2 = 102.22 m/s

Note: it takes a pressure difference of only 0.02 atm to produce such a high

velocity

4.6: V2 = 216.8 ft/s 4.8: Te = 155 K and e = 2.26 kg/m

3

Note: you can also verify using equation of state

4.11: Ae = 0.0061 ft2 (or 0.88 in2)

4.15: M = 0.847 4.16: V = 2,283 MPH

Notes:

Include a brief comment on your answer, especially if different than above

If you have any questions come to office hours or consult GSAs

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WEEK #3: LABORATORY SESSIONS

Session #1:

Machine Shop Session (2 of 6) Report directly to machine shop for your session

Make sure you are on time to laboratory

Do not wear open-toe shoes or sandals

Avoid wearing loose clothing, jewelry, etc.

Safety glasses are provided

Detailed training guide is online

Session #2:

MATLAB Lecture (2 of 3)

Add a 4th lecture on MATLAB toward end of semester

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REVIEW OF BASIC CONCEPTS

Review: Introduction to Flight by Anderson

Chapter 2: 2.1-2.7

Chapter 3: 3.1-3.5

Be sure that you are familiar with example problems

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REVIEW OF BASIC DEFINITIONS (2.1-2.3)

Streamline (2.1)

Set of points that form a line that is everywhere tangent to local velocity

vector No flow across streamlines

For a steady flow, moving fluid element traces out a fixed path in space

Stream tube

A set of streamlines that intersect a closed loop in space

Steady Flow: A flow that does not fluctuate with time (all flows in MAE 1202)

Unsteady Flow: A flow that varies with time

Equation of State for a Perfect Gas (2.3), applies at a point Ideal Gas Law:p = RT or pv = RT (v = 1/) R universal = 8,314 J/kg mole K

R forair= 8,314 / 28.96 = 287 J/kg K(or1,716 ft lb / slug R)

If you do not remember these concepts review Section 2.1-2.3

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EXAMPLE: STREAMLINES AND STREAM TUBES

IN STEADY FLOW

Streamlines

Stagnation

Point

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HARRIER INSTANTANEOUS STREAMLINES

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WATER STREAMLINES ON F-16 MODEL

http://www.aerolab.com/water.html

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TYPES OF FLOWS: FRICTION VS. NO-FRICTION

Viscous: Flows with friction

All real flows are viscous

Inviscid flow is a useful idealization

By neglecting friction analysis of flow is usually much easier!

Inviscid: Flows with no friction

Flow very close to surface of airfoil is

Influenced by friction and is viscous

(boundary layer flow)

Stall (separation) is a viscous phenomena

Flow away from airfoil is not influenced

by friction and is wholly inviscid

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LAMINAR VERSUS TURBULENT FLOW

Two types of viscous flows

Laminar: streamlines are smooth and regular and

a fluid element moves smoothly along a streamline

Turbulent: streamlines break up and fluid

elements move in a random, irregular, and chaotic

fashion

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FRCTION EXAMPLE: AIRFOIL STALL (4.20, 5.4)

Key to understanding: Friction causes flow separation within boundary layer

1. Boundary layers are eitherlaminar orturbulent

2. All laminar B.L. turbulent B.L.3. Turbulent B.L. fuller or fatter than laminar B.L., more resistant to

separation

Separation creates another form of drag called pressure drag due to separation

Dramatic loss of lift and increase in drag

We will examine these airfoils next lecture in detail

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TYPES OF FLOWS:

COMPRESSIBLE VS. INCOMPRESSIBLE

Compressible: Density of fluid elements may change from point to point

All real flows are compressible

Important for gases (rarely important for liquids)

Most important at high speeds

Incompressible: Density of fluid elements is always constant

General Rule of Thumb:

If flow speed is less than about 100 m/s (or less than 225 MPH) flow

can be considered incompressibleor

If flow is less than Mach 0.3, flow can be considered incompressible

Mach number, M: ratio of local velocity to local speed of sound, V/a

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DENSITY DISCONTINUITY: SHOCK WAVESPhotograph of a T-38 at Mach 1.1,

altitude 13,700 feet, taken at NASA

Wallops in 1993.

Schlieren photography (fromGerman word for "streaks") allows

visualization of density changes, and

therefore shock waves, in fluid flow

Schlieren techniques have been used

for decades in laboratory wind

tunnels to visualize supersonic flowabout model aircraft, but not full

scale aircraft until recently.

Dr. Leonard Weinstein of NASA

Langley Research Center developed

first Schlieren camera, which he calls

SAF (Schlieren for Aircraft inFlight), that can photograph shock

waves of a full sized aircraft in

flight. He successfully took a picture

which clearly shows shock waves

about a T-38 aircraft on December

13, 1993 at Wallops Island, MD.

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KEY TERMS: CAN YOU DEFINE THEM?

Streamline

Stream tube

Steady flow

Unsteady flow

Viscid flow

Inviscid flow

Compressible flow

Incompressible flow

Laminar flow

Turbulent flow

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BASIC AERODYNAMICS

Introduction to Flight by Anderson

Chapter 4: 4.1-4.9

This chapter is going to be a challenge to you. There are

lots of new concepts, ideas, and ways of looking at things.

Expect it to be different, and go at it with enthusiasm.

Be sure that you are familiar with example problems

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WHY STUDY AERODYNAMICS?

Study of aerodynamics is important to determine forces and moments

(torques) acting on flying vehicles

Forces and moments are caused as a result of interaction between a

body (airplane, rocket, etc.) and air surrounding it

Interaction depends on flow conditions (fluid properties, relative

velocity, pressure, temperature, etc.) and body shape (geometry)

GOALS:

Develop foundation of theoretical development (mathematical) Gain insight into physical phenomena taking place

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3 FUNDAMENTAL PRINCIPLES

1. Mass is neither created nor destroyed (mass is conserved)

Conservation of Mass

Often also called: Continuity

1. Sum of Forces = Time Rate Change of Momentum (Newtons 2nd Law)

Often reduces to: Sum of Forces = Mass x Acceleration (F = ma)

Momentum Equation Bernoullis Equation, Euler Equation, Navier-Stokes Equation

1. Energy neither created nor destroyed (energy is conserved)

Can only change physical form Energy Equation (1st Law of Thermodynamics)

How do we express these statements mathematically?

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SUMMARY OF GOVERNING EQUATIONS (4.8)

STEADY AND INVISCID FLOW

2

22

2

11

2211

2

1

2

1VpVp

VAVA

+=+=

( )

222

111

2

22

2

11

1

2

1

2

1

2

1

222111

2

1

2

1

RTp

RTp

VTcVTc

T

T

p

p

VAVA

pp

=

=

+=+

=

=

=

Incompressible flow of fluid along a

streamline or in a stream tube ofvarying area

Most important variables: p and V

T and are constants throughout flow

Compressible, isentropic

(adiabatic and frictionless)

flow along a streamline or in a

stream tube of varying area

T, p, , and V are all variables

continuity

Bernoulli

continuity

isentropic

energy

equation of state

at any point

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EXAMPLES OF WHAT WE WILL BE ABLE TO DO

Wind TunnelsAir Speed Supersonic Flow
http://www.airliners.net/open.file?id=276074&size=L&sok=V0hFUkUgIChNQVRDSCAoYWlyY3JhZnQsYWlybGluZSxwbGFjZSxwaG90b19kYXRlLGNvdW50cnkscmVtYXJrLHBob3RvZ3JhcGhlcixlbWFpbCx5ZWFyLHJlZyxhaXJjcmFmdF9nZW5lcmljLGNuLGNvZGUpIEFHQUlOU1QgKCcrInBpdG90IiArInR1YmUiJyBJTiBCT09MRUFOIE1PREUpKSAgT1JERVIgQlkgcGhvdG9faWQgREVTQw%3D%3D&photo_nr=10http://www.airliners.net/open.file?id=425665&size=L&sok=V0hFUkUgIChNQVRDSCAoYWlyY3JhZnQsYWlybGluZSxwbGFjZSxwaG90b19kYXRlLGNvdW50cnkscmVtYXJrLHBob3RvZ3JhcGhlcixlbWFpbCx5ZWFyLHJlZyxhaXJjcmFmdF9nZW5lcmljLGNuLGNvZGUpIEFHQUlOU1QgKCcrInBpdG90IiArInR1YmUiJyBJTiBCT09MRUFOIE1PREUpKSAgT1JERVIgQlkgcGhvdG9faWQgREVTQw%3D%3D&photo_nr=7

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CONSERVATION OF MASS (4.1)

Physical Principle: Mass can be neither created nor destroyed

Stream tube

As long as flow is steady, mass that flows through cross section at point 1(at entrance) must be same as mass that flows through point 2 (at exit)

Flow cannot enter or leave any other way (definition of a stream tube)

Also applies to solid surfaces, pipe, funnel, wind tunnels, airplane engine

What goes in one side must come out the other side

A1

A2

V1V2

Funnel wall

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CONSERVATION OF MASS (4.1)

Stream tube

Consider all fluid elements in plane A1

During time dt, elements have moved V1dt and swept out volume A1V1dt

Mass of fluid swept through A1 during dt: dm=1(A1V1dt)

A1: cross-sectional area

of stream tube at 1

V1: flow velocity

Normal (perpendicular) to A1

22

2222

2222skgFlowMass

mm

VAm

VAmdtdm

=

=

===

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SIMPLE EXAMPLE

p1=1.2x105 N/m2

T1=330 K

V1=10 m/s

A1= 5m2

p2=?

T2=?

V2=30 m/s

A2=?

IF flow speed < 100 m/s assume flow is incompressible (1=2)

2

2

112

2211

22211121

67.13

5

30

105 m

V

VAA

VAVA

VAVAmm

==

==

=

===

Given air flow through converging nozzle, what is exit area, A2?

Conservation of mass could also give velocity, V2

, if A2

was known

Conservation of mass tells us nothing about p2, T2, etc.

SC O Q A O (4 3)

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INVISCID MOMENTUM EQUATION (4.3)

Relation between pressure and velocity

Differences in pressure from one point to another in a flow create forces

Physical Principle: Newtons Second Law

Notes on pressure:

Always acts inward Pressure varies from point to point in a flow

How to apply F = ma for air flows?

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APPLYING NEWTONS SECOND LAW FOR FLOWS

dx

dz

dy

x

y

z

onsider a small fluid element moving along a streamline

lement is moving in x-direction

V

What are forces on this element?

1. Pressure (force x area) acting in normal direction on all six faces

2. Frictional shear acting tangentially on all six faces (neglect)

3. Gravity acting on mass inside element (neglect)

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dx

dz

dyp

(N/m2)

Area ofleft face: dydz

Force on left face: p(dydz)

Note that P(dydz) = N/m2(m2)=N

Forces is in positive x-direction

x

y

z

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dx

dz

dyp

(N/m2)

Area of left face: dydz

Force on left face: p(dydz)

Forces is in positive x-direction

p+(dp/dx)dx

(N/m2)

Change in pressure per length: dp/dx

Change in pressure along dx is (dp/dx)dx

Force on right face: [p+(dp/dx)dx](dydz)

Forces acts in negative x-direction

x

y

z

ressure varies from point to point in a flow

here is a change in pressure per unit length, dp/dx

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dx

dz

dyp

(N/m2)

p+(dp/dx)dx

(N/m2)

Net Force is sum of left and right sides

Net Force on element due to pressure ( )dxdydz

dx

dpF

dydzdxdx

dpppdydzF

=

+=

x

y

z


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Vdx

dV

dt

dx

dx

dV

dx

dx

dt

dVa

dtdxV

dt

dVa

===

=

=

Now put this into F=ma

First, identify the mass of the element

Next, write acceleration, a, as

(to get rid of time variable)

( )

( )dxdydzmass

dxdydzvolume

volume

mass

=

==

SUMMARY EULERS EQUATION

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SUMMARY: EULERS EQUATION

( ) ( )

VdVdp

dx

dVVdxdydzdxdydz

dx

dp

maF

=

=

=

Eulers Equation

Eulers Equation (Differential Equation)

Relates changes in momentum to changes in force (momentum equation)

Relates a change in pressure (dp) to a chance in velocity (dV)

Assumptions we made:

Neglected friction (inviscid flow)

Neglected gravity

Assumed that flow is steady

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WHAT DOES EULERS EQUATION TELL US?

Notice that dp and dV are of opposite sign: dp = -VdV

IF dp increases

Increased pressure on right side of element relative to left side

dV goes down, flow slows down

IF dp decreases

Decreased pressure on right side of element relative to left side

dV goes up, flow speeds up

Eulers Equation is true for Incompressible and Compressible flows

NAVIER STOKES EQUATIONS

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NAVIER-STOKES EQUATIONS

INVISCID FLOW ALONG STREAMLINES

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INVISCID FLOW ALONG STREAMLINES

0

22

0

0

2

1

2

2

12

2

1

2

1

=

+

=+

=+

VV

pp

VdVdp

VdVdp

V

V

p

p

Relate p1 and V1 at point 1 to p2 and V2 at point 2Integrate Eulers equation from point 1 to point 2 taking = constant

BERNOULLIS EQUATION

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BERNOULLIS EQUATION

=+

+=+

2

222

2

1

1

2

2

2

Vp

Vp

Vp

One of most fundamental and useful equations in aerospace engineering!

Remember:

Bernoullis equation holds only for inviscid (frictionless) and

incompressible ( = constant) flows

Bernoullis equation relates properties between different points along a

streamline

For a compressible flow, Eulers equation must be used ( is variable)

Both Eulers and Bernoullis equations are expressions ofF = ma

expressed in a useful form for fluid flows and aerodynamics

Constant along a streamline

WHEN AND WHEN NOT TO APPLY BERNOULLI

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WHEN AND WHEN NOT TO APPLY BERNOULLI

YES NO

SIMPLE EXAMPLE
http://arjournals.annualreviews.org/na101/home/literatum/ar/journals/production/fluid/2003/35/1/annurev.fluid.35.101101.161128/images/large/fm35_0295_3.jpeg

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SIMPLE EXAMPLE

p1=1.2x105 N/m2

T1=330 K

V1=10 m/s

A1= 5m2

p2=?

T2=?

V2=30 m/s

A2=?

Since flow speed < 100 m/s assume flow is incompressible (1=2)

Given air flow through converging nozzle, what is the exit pressure, p2?

( )( )

( ) ( ) ( )2

52252

2

2

112

3

5

1

1

1

10195.1301027.12

1102.1

2

1

27.1330287

102.1

m

NxxVVpp

m

kgx

RT

p

=+=+=

===

Since the velocity is increasing along the flow, it is an accelerating flow

Notice that even with a 3-fold increase in velocity the pressure decreases

by only about 0.8 %, which is characteristic of low velocity flow

SUMMARY OF GOVERNING EQUATIONS (4 8)

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SUMMARY OF GOVERNING EQUATIONS (4.8)

222

211

2211

21

21 VpVp

VAVA

+=+

= Steady, incompressible flow of an

inviscid (frictionless) fluid along a

streamline or in a stream tube of

varying area

Most important variables: p and V

T and are constants throughout flow

continuity

Bernoulli

What if flow is high speed?

What if there are temperature effects?

How does the density change?

1202 Lecture 2

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