Review Flow Characterization Quantities - Mach number - Reynolds number Flow Physics - Continuity - Conservation of momentum - Conservation of energy Standard Atmosphere - Equation of state - Standard atmosphere Aircraft Instruments Parts of an Airplane

1

Incompressible vs. Compressible Function of Mach number

Air is an example of a compressible fluid Its density changes if temperature changes, or if some

external force is applied A flow is said to be incompressible if there are no

changes in density attributable to (or caused by) the velocity or speed of the flow

Theory and observations in wind tunnels suggest that most flows may be treated as incompressible (i.e., constant density) until the Mach number is sufficiently high (>0.4 or so)

M = Velocity / Speed of Sound 2

Speed of Sound From thermodynamics, and compressible flow theory,

sound travels at the following speed:

where, a is Speed of Sound is Ratio of Specific Heats (1.4 for air) R is Gas Constant T is Temperature (in oK or degrees oR)

So, really depends on Temperature!

RTa =

3

Why a function of M? If there is sufficient time for the sound waves to travel

before the body arrives, the fluid particles downstream will hear the message and tend to get out of the way

Otherwise, there will be a crush (compression), or even a large jump in density (shock wave)

4

Viscid vs. Inviscid Flow Function of Reynolds Number (Rn)

Streamlines describe the path the fluid particles will take

Flow velocity is tangential to the streamline

Viscosity alters the shape of streamlines around bluff bodies

Scientists inject smoke particles into streamlines to make them visible to the naked eye

Reynolds number can be thought of as the ratio of pressure and viscous forces of the fluid.

Inviscid (ideal) flow

Viscous flow dia.Cylinder D

VDNumber Reynolds

=

=

where

5

Streamlines over an Airfoil

6

Flow Physics

Continuity Conservation of Momentum Conservation of Energy

7

Continuity Equation

Area A1 Density 1 Velocity V1

Area A2 Density 2 Velocity V2

Rate at which mass enters=1A1V1 Rate at which mass leaves=2A2V2

8

Conservation of Momentum Newtons 2nd Law

Rate of change of momentum is sum of forces, i.e., F = ma

Consider a small slice of stream tube

Rate of change of momentum of the fluid particles within this stream tube slice must be due to forces acting on it Force = time rate of change of Momentum

9

Momentum Equation: Rate of Change of Momentum

Density velocity V Area = A

Density +d velocity V+dV Area = A+dA

Mass Flow Rate in = Mass Flow rate out (Continuity) VA = (+d)(V+dV)(A+dA)

Momentum rate in = Mass flow rate times velocity = V2A

Momentum Rate out = Mass flow rate times velocity = VA (V+dV)

Rate of change of momentum within this element = Momentum rate out - Momentum rate in

= VA (V+dV) - V2A = VA dV 10

Momentum Equation: Forces Acting on Stream Tube

Pressure times

Area = pA (p+dp)(A+dA)

Horizontal Force = Pressure times area of the ring = (p+dp/2)dA

Area of ring = dA

Net force = pA + (p+dp/2)dA-(p+dp)(A+dA) =- Adp - dpdA/2 -Adp

Product of two small numbers

11

Momentum Equation Rate of change of momentum = AVdV Forces acting on the stream tube = -Adp

Note: We have neglected all other forces: viscous, gravity, electrical and magnetic forces

Equating the two factors (and divide by A), we get

Eulers equation

VdV + dp = 0

12

Momentum Equation for Incompressible Flow

For incompressible flows ( constant), Eulers equation can be integrated:

called Bernoullis Equation This total pressure is constant along a streamline,

High Reynolds number, low Mach number

TpConstpV

dpVdV

dpVdV

==+

=+

=+

2

21

0

0

13

Ideal Gas Law or Equation of State

Most gases satisfy the following relationship between density, temperature and pressure:

p = RT p = Pressure (in lb/ft2 or N/m2) = Rho , density (in slugs/ft3 or kg/m3) T = Temperature (in Degrees R or degrees K) R = Gas Constant, varies from one gas to another. Equals 287.1 J/Kg/K or 1715.7 ft lbf/slug/oR for air

14

What is a standard atmosphere?

Weather conditions vary around the globe, from day to day.

Taking all these variations into design is impractical. A standard atmosphere is therefore defined, that

relates flight tests, wind tunnel tests and general airplane design to a common reference.

This common reference is called a standard atmosphere.

15

International Standard Atmosphere Standard Sea Level Conditions

Pressure 101325 Pa 2116.7 lbf/ft2 Density 1.225 Kg/m3 0.002378 slug/ft3 Temperature 15 oC or 288 K 59 oF or 518.4 oR

16

Standard Atmosphere Variation of temperature, density and

pressure with altitude can be computed for a standard atmosphere.

These properties may be tabulated, as shown in Appendix A of the text by Nelson.

Short programs called applets exist on the world wide web for computing atmospheric properties.

17

Aircraft Instruments Airspeed Indicator Altimeter Rate of Climb Indicator Machmeter Angle of Attack Indicator

18

Airspeed Designations Indicated airspeed (IAS): Airspeed indicated by the

airspeed instrument, which is affected by altitude, compressibility, instrument, and position errors.

Calibrated airspeed (CAS): Indicated airspeed corrected for instrument and position errors.

True airspeed (TAS): Actual airspeed. Equivalent airspeed (EAS): Equivalent airspeed at

sea level standard atmosphere corresponding to the true dynamic pressure. 122 = 122

19

Pitot Tube: Measure Airspeed using Bernoullis Equation

Measure local pressure of air: pS, static pressure

Measure pressure after bringing to zero speed relative to instrument (i.e., bring to stagnation): pT, total pressure

From Bernoulli:

Determine velocity of flow with respect to instrument:

Note: really only need difference between pS and pT

TS ppV =+2

21

( )

ST ppV = 2

20

Pitot Tube

(Wikopedia)

Measure pressure difference ( )

ST ppV = 221

Pitot-static system Total pressure probe(s) not

behind propulsion sources, near front of aircraft usually on sides of fuselage near front or front of a wing

Static pressure port(s) in location with minimum disturbance from aircraft itself usually sides of fuselage

Airspeed indicator shows measured airspeed to pilot

2 on each side of this Boeing 757 22

Airspeed Indicator

is air density at sea level in a standard atmosphere

Mechanically uses Bernoullis equation to show airspeed not corrected for density

Aircraft are typically flown via this so-called indicated airspeed (not corrected for density, mechanical errors, etc.) in knots

( )std

2

ST ppV =

23

Using Mach Number in Energy Equations

20

02

02

0

2

21-1 ,

21-1 :get to1)-/(by throughDivide

121get to

1 Use

2

MTTOr

TT

RTVRT

RTVRTRCC

TCVTC

Vp

pp

+=

=+

=+

==

=+

llyadiabatica and reversiblyrest brought to are they if have, willparticles theproerties theare Theselyrespective pressure, stagnation

and density, stagnation pressure, stagnation called are quantities The2

1-1

21-1

:relations isentropic of aid theWith

1200

11

211

00

000 , T, p

Mpp

MTT

+=

=

+=

=

Mach meter (M

Measuring Airspeed, 0.3 < M < 1

=

11-

21

0

ppM

+

=

111-

21

std

0stdcalibrated

pppaV

=

11-

21

0

ppaV

( )

ppV = 02

Only for incompressible flow!

25

For M>1: Must include the effect of shock waves on Pitot tube measurements. Rayleighs formula:

5.3

2

22

1752.01

617

++

=

MMM

ppo

Where po is the measured stagnation pressure and p is flow static pressure. Note: The above formula is not valid for very high Mach numbers or very high altitudes.

26

Parts of an Airplane

27

Aerodynamic Controls Elevators control pitch angle Ailerons control roll angle Rudder controls yaw angle Flaps increase lift and drag Leading edge slats increase lift Drag brakes increase drag Spoilers reduce lift and increase drag

28

Body Axes of an Airplane

xb

Normal or Vertical axis

yb

zb

29

Roll The longitudinal axis extends

lengthwise through the fuselage from the nose to the tail.

Movement of the airplane around the longitudinal axis is known as roll and is controlled by movement of the ailerons. xb

Normal or Vertical axis

yb

zb

30

Pitch

The lateral axis extends crosswise from wingtip to wing tip.

Movement of the airplane around the lateral axis is known as pitch.

Pitch is controlled by movement of the elevators.

xb

Normal or Vertical axis

yb

zb

31

Yaw

The vertical or normal axis is normal to xbyb passing throug the center of gravity.

Movement of the airplane around the vertical axis is yaw.

Yaw is controlled by movement of the rudder.

xb

Normal or Vertical axis

yb

zb

32

Definition of Forces, Moments, and Velocity Components

In body fixed coordinates: Angular rates: p, q, r Relative air velocity components: u. v. w Aerodynamic force components: X, Y, Z Aerodynamic moment components: L, M, N Moments of Inertia: Ix, Iy, Iz Products of Inertia: Ixy, Iyz, Ixz

33

Definition of airspeed, angle of attack and sideslip angle

Airspeed is the magnitude of aircraft velocity relative to air.

Angle of attack is the angle from

the projected air velocity vector in body xz-plane to body x-axis, positive in a clockwise sense when looking from the port side.

Sideslip angle is the angle from

the body xz-plane to relative air velocity vector, positive in a clockwise sense when looking from the top.

222 wvuV ++=

uw1tan=

222

1sinwvu

v++

=

34

ReviewIncompressible vs. CompressibleFunction of Mach numberSpeed of SoundWhy a function of M?Viscid vs. Inviscid Flow Function of Reynolds Number (Rn)Streamlines over an AirfoilFlow PhysicsContinuity EquationConservation of MomentumMomentum Equation: Rate of Change of MomentumMomentum Equation: Forces Acting on Stream TubeMomentum EquationMomentum Equation for Incompressible FlowIdeal Gas Law orEquation of StateWhat is a standard atmosphere?Slide Number 16Standard AtmosphereAircraft InstrumentsAirspeed DesignationsPitot Tube: Measure Airspeed using Bernoullis EquationPitot TubePitot-static systemAirspeed IndicatorUsing Mach Number in Energy EquationsMeasuring Airspeed, 0.3 < M < 1Slide Number 26Parts of an AirplaneAerodynamic ControlsBody Axes of an AirplaneRollPitchYawDefinition of Forces, Moments, and Velocity ComponentsDefinition of airspeed, angle of attack and sideslip angle