Introduction to Basic Aerodynamics: Lecture Notes and Homework, Slides of Aerospace Engineering

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AEROSPACE PRACTICUM

Lecture 3: Introduction to Basic Aerodynamics 2

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READING AND HOMEWORK ASSIGNMENTS

• Reading: Introduction to Flight, by John D. Anderson, Jr.
  • For this week’s lecture: Chapter 4, Sections 4.1 - 4.
  • For next week’s lecture: Chapter 4, Sections 4.10 - 4.21, 4.
• Lecture-Based Homework Assignment:
  • Problems: 4.1, 4.2, 4.4, 4.5, 4.6, 4.8, 4.11, 4.15, 4.
    • DUE: Friday, February 8, 2013 by 11:00 am
    • Turn in hard copy of homework
  • Also be sure to review and be familiar with textbook examples in Chapter 4
• Lab this week:
  • Machine shop (remember to dress appropriately, no ‘open-toe’ shoes)
  • Team Challenge #

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3 FUNDAMENTAL PRINCIPLES

1. Mass is neither created nor destroyed (mass is conserved)
   • Conservation of Mass
   • Often called Continuity
2. Force = Mass x Acceleration ( F = m a )
   • Newton’s Second Law
   • Momentum Equation
   • Bernoulli’s Equation , Euler Equation, Navier-Stokes Equation
3. Energy Is Conserved
   • Energy neither created nor destroyed; can only change physical form
   • Energy Equation (1 st^ 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 2 2

2 1 1

1 1 2 2

p V p V

AV AV

( )

2 2 2

1 1 1

2 2 2

2 1 1

1

2

1 2

1 2

1

1 1 1 2 2 2

p RT

p RT

c T V c T V

T

T

p

p

AV AV

p p

γ γ γ

−

• Incompressible flow of fluid along a streamline or in a stream tube of varying area
• Most important variables: p and V
• T and ρ are constants throughout flow
• Compressible, isentropic (adiabatic and frictionless) flow along a streamline or in a stream tube of varying area
• T, p, ρ, and V are all variables

continuity

Bernoulli

continuity

isentropic

energy

equation of state

at any point

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CONSERVATION OF MASS (4.1)

Stream tube

• Consider all fluid elements in plane A (^1)
• During time dt, elements have moved V 1 dt and swept out volume A 1 V 1 dt
• Mass of fluid swept through A 1 during dt: dm=ρ 1 (A 1 V 1 dt)

A 1 : cross-sectional area of stream tube at 1

V 1 : flow velocity normal (perpendicular) to A (^1)

1 2

2 2 2 2

1 1 1 1 s

kg Mass Flow

m m

m AV

m AV dt

dm

 





=

=

 

  

 = = =

ρ

ρ

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SIMPLE EXAMPLE

p 1 = 1.2x10 5 N/m 2 T 1 = 330 K V 1 = 10 m/s A 1 = 5 m 2

p 2 =? T 2 =? V 2 =? m/s A 2 = 1.67 m 2

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

s

m A

A V V

AV AV

m m AV AV

30

1. 67

5 10 2

1 2 1

1 1 2 2

1 2 1 1 1 2 2 2

^ = 

  

 = =

=

Given air flow through converging nozzle, what is exit velocity, V 2?

Conservation of mass could also give velocity, A 2 , if V 2 was known

Conservation of mass tells us nothing about p 2 , T 2 , etc. Docsity.com

APPLYING NEWTON’S SECOND LAW FOR FLOWS

dx

dz

dy

x

y

z

Consider a small fluid element moving along a streamline Element is moving in x-direction

V

What forces act 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 for now) 3. Gravity acting on all mass inside element (neglect for now)

Note on pressure: Always acts inward and varies from point to point in a flow Docsity.com

APPLYING NEWTON’S SECOND LAW FOR FLOWS

dx

dz

dy

p

(N/m 2 )

Area of left face: dydz

Force on left face: p(dydz)

Note that P(dydz) = N/m 2 (m^2 )=N

Forces is in positive x-direction

x

y

z

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APPLYING NEWTON’S SECOND LAW FOR FLOWS

dx

dz

dy

p

(N/m 2 )

p+(dp/dx)dx

(N/m 2 )

Net Force is sum of left and right sides

Net Force on element due to pressure (^) ( dxdydz )

dx

dp

F

dx dydz

dx

dp

F pdydz p

x

y

z

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APPLYING NEWTON’S SECOND LAW FOR FLOWS

V

dx

dV

dt

dx

dx

dV

dx

dx

dt

dV

a

dt

dx

V

dt

dV

a

Now put this into F =m a

First, identify mass of element

Next, write acceleration, a , as

(to get rid of time variable)

( )

mass (^ dxdydz )

volume dxdydz

volume

mass

ρ

ρ

=

=

=

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WHAT DOES EULER’S EQUATION TELL US?

• Notice that dp and dV are of opposite sign: dp = -ρVdV
• IF dp ↑
  • Increased pressure on right side of element relative to left side
  • dV ↓

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WHAT DOES EULER’S EQUATION TELL US?

• Notice that dp and dV are of opposite sign: dp = -ρVdV
• IF dp ↑
  • Increased pressure on right side of element relative to left side
  • dV ↓ (flow slows down)
• IF dp ↓
  • Decreased pressure on right side of element relative to left side
  • dV ↑ (flow speeds up)
• Euler’s Equation is true for Incompressible and Compressible flows

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BERNOULLI’S EQUATION

• =

• = +

2

2 2 2

2 1 1

2 2 2

V p

V p

V p

ρ

ρ ρ

• One of most fundamental and useful equations in aerospace engineering!
• Remember:
  • Bernoulli’s equation holds only for inviscid (frictionless) and incompressible (ρ = constant) flows
  • Bernoulli’s equation relates properties between different points along a streamline
  • For a compressible flow Euler’s equation must be used (ρ is variable)
  • Both Euler’s and Bernoulli’s equations are expressions of F = m a expressed in a useful form for fluid flows and aerodynamics

Constant along a streamline

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WHEN AND WHEN NOT TO APPLY BERNOULLI

YES NO

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