1. MAE 1202: AEROSPACE PRACTICUM
Lecture 5: Compressible and Isentropic Flow 1
February 11, 2013
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. READING AND HOMEWORK ASSIGNMENTS
Reading: Introduction to Flight, by John D. Anderson, Jr.
For this week’s lecture: Chapter 4, Sections , 4.27
For next week’s lecture: Chapter 5, Sections
Lecture-Based Homework Assignment:
Problems: 4.7, 4.11, 4.18, 4.19, 4.20, 4.23, 4.27
DUE: Friday, February 22, 2013 by 5 PM
Problems: 5.2, 5.3, 5.4, 5.6
DUE: Friday, March 1, 2013 by 5 PM
Turn in hard copy of homework
Also be sure to review and be familiar with textbook examples in Chapter 5

3. ANSWERS TO LECTURE HOMEWORK
5.2: L = 23.9 lb, D = 0.25 lb, Mc/4 = lb ft
Note 1: Two sets of lift and moment coefficient data are given for the NACA 1412 airfoil, with and without flap deflection. Make sure to read axis and legend properly, and use only flap retracted data.
Note 2: The scale for cm,c/4 is different than that for cl, so be careful when reading the data
5.3: L = 308 N, D = 2.77 N, Mc/4 = N m
5.4: a = 2°
5.6: (L/D)max ~ 112

4. CREO DESIGN CONTEST
Create most elaborate, complex, stunning Aerospace Related project in Creo
Criteria: Assembly and/or exploded view
First place
Either increase your grade by an entire letter (C → B), or
Buy your most expensive textbook next semester
Second place: +10 points on final exam
Third place: +10 points on final exam

5. CAD DESIGN CONTEST

6. CAD DESIGN CONTEST

11. If you do the PRO|E challenge… Do not let it consume you!

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

13. EXAMPLE: MEASUREMENT OF AIRSPEED (4.11)
How do we measure an airplanes speed in flight?
Pitot tubes are used on aircraft as speedometers (point measurement) 13

14. STATIC VS. TOTAL PRESSURE
In aerodynamics, 2 types of pressure: Static and Total (Stagnation)
Static Pressure, p
Due to random motion of gas molecules
Pressure we would feel if moving along with flow
Strong function of altitude
Total (or Stagnation) Pressure, p0 or pt
Property associated with flow motion
Total pressure at a given point in flow is the pressure that would exist if flow were slowed down isentropically to zero velocity
p0 ≥ p 14

15. MEASUREMENT OF AIRSPEED: INCOMPRESSIBLE FLOW
Static
pressure
Dynamic
pressure
Total
pressure
Incompressible Flow

16. Total and Static Ports

17. TOTAL PRESSURE MEASUREMENT (4.11)
Measures total pressure
Open at A, closed at B
Gas stagnated (not moving) anywhere in tube
Gas particle moving along streamline C will be isentropically brought to rest at point A, giving total pressure 17

18. EXAMPLE: MEASUREMENT OF AIRSPEED (4.11)
Point A: Static Pressure, p
Only random motion of gas is measured
Point B: Total Pressure, p0
Flow is isentropically decelerated to zero velocity
A combination of p0 and p allows us to measure V1 at a given point
Instrument is called a Pitot-static probe p p0 18

19. MEASUREMENT OF AIRSPEED: INCOMPRESSIBLE FLOW
Static
pressure
Dynamic
pressure
Total
pressure
Incompressible Flow 19

20. TRUE VS. EQUIVALENT AIRSPEED
What is value of r?
If r is measured in actual air around the airplane
Measurement is difficult to do
Practically easier to use value at standard seal-level conditions, rs
This gives an expression called equivalent airspeed 20

21. TRAGIC EXAMPLE: Air France Crash
Aircraft crashed following an aerodynamic stall caused by inconsistent airspeed sensor readings, disengagement of autopilot, and pilot making nose-up inputs despite stall warnings
Reason for faulty readings is unknown, but it is assumed by accident investigators to have been caused by formation of ice inside pitot tubes, depriving airspeed sensors of forward-facing air pressure.
Pitot tube blockage has contributed to airliner crashes in the past

22. HOW DOES AN AIRFOIL GENERATE LIFT?
Lift due to imbalance of pressure distribution over top and bottom surfaces of airfoil (or wing)
If pressure on top is lower than pressure on bottom surface, lift is generated
Why is pressure lower on top surface?
We can understand answer from basic physics:
Continuity (Mass Conservation)
Newton’s 2nd law (Euler or Bernoulli Equation)
Lift Force = SPA

23. HOW DOES AN AIRFOIL GENERATE LIFT?
Flow velocity over top of airfoil is faster than over bottom surface
Streamtube A senses upper portion of airfoil as an obstruction
Streamtube A is squashed to smaller cross-sectional area
Mass continuity rAV=constant: IF A↓ THEN V↑
Streamtube A is squashed
most in nose region
(ahead of maximum thickness) A B

24. HOW DOES AN AIRFOIL GENERATE LIFT?
As V ↑ p↓
Incompressible: Bernoulli’s Equation
Compressible: Euler’s Equation
Called Bernoulli Effect
With lower pressure over upper surface and higher pressure over bottom surface, airfoil feels a net force in upward direction → Lift
Most of lift is produced
in first 20-30% of wing
(just downstream of leading edge)
Can you express these ideas in your own words?

25. Incorrect Lift Theory

26. SUMMARY OF GOVERNING EQUATIONS (4.8)
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 r are constants throughout flow
continuity
Bernoulli
What if flow is high speed, M > 0.3?
What if there are temperature effects?
How does density change?

27. 1st LAW OF THERMODYNAMICS (4.5)
Boundary
System
e (J/kg)
Surroundings
System (gas) composed of molecules moving in random motion
Energy of molecular motion is internal energy per unit mass, e, of system
Only two ways e can be increased (or decreased):
Heat, dq, added to (or removed from) system
Work, dw, is done on (or by) system

28. THOUGHT EXPERIMENT #1
Do not allow size of balloon to change (hold volume constant)
Turn on a heat lamp
Heat (or q) is added to the system
How does e (internal energy per unit mass) inside the balloon change?

29. THOUGHT EXPERIMENT #2
*You* take balloon and squeeze it down to a small size
When volume varies work is done
Who did the work on the balloon?
How does e (internal energy per unit mass) inside the balloon change?
Where did this increased energy come from?

30. 1st LAW OF THERMODYNAMICS (4.5)
Boundary
e (J/kg)
SYSTEM
(unit mass of gas)
SURROUNDINGS dq
System (gas) composed of molecules moving in random motion
Energy of all molecular motion is called internal energy per unit mass, e, of system
Only two ways e can be increased (or decreased):
Heat, dq, added to (or removed from) system
Work, dw, is done on (or by) system

31. 1st LAW IN MORE USEFUL FORM (4.5)
1st Law: de = dq + dw
Find more useful expression for dw, in terms of p and r (or v = 1/r)
When volume varies → work is done
Work done on balloon, volume ↓
Work done by balloon, volume ↑
Change in
Volume (-)

32. ENTHALPY: A USEFUL QUANTITY (4.5)
Define a new quantity
called enthalpy, h:
(recall ideal gas law: pv = RT)
Differentiate
Substitute into 1st law
(from previous slide)
Another version of 1st law
that uses enthalpy, h:

33. HEAT ADDITION AND SPECIFIC HEAT (4.5)
Addition of dq will cause a small change in temperature dT of system dq dT
Specific heat is heat added per unit change in temperature of system
Different materials have different specific heats
Balloon filled with He, N2, Ar, water, lead, uranium, etc…
ALSO, for a fixed dq, resulting dT depends on type of process…

34. SPECIFIC HEAT: CONSTANT PRESSURE
Addition of dq will cause a small change in temperature dT of system
System pressure remains constant dq dT
Extra Credit #1:
Show this step

35. SPECIFIC HEAT: CONSTANT VOLUME
Addition of dq will cause a small change in temperature dT of system
System volume remains constant dq dT
Extra Credit #2:
Show this step

36. HEAT ADDITION AND SPECIFIC HEAT (4.5)
Addition of dq will cause a small change in temperature dT of system
Specific heat is heat added per unit change in temperature of system
However, for a fixed dq, resulting dT depends on type of process:
Constant Pressure
Constant Volume
Specific heat ratio
For air, g = 1.4

37. ISENTROPIC FLOW (4.6) g = ratio of specific heats g = cp/cv gair=1.4
Goal: Relate Thermodynamics to Compressible Flow
Adiabatic Process: No heat is added or removed from system
dq = 0
Note: Temperature can still change because of changing density
Reversible Process: No friction (or other dissipative effects)
Isentropic Process: (1) Adiabatic + (2) Reversible
(1) No heat exchange + (2) no frictional losses
Relevant for compressible flows only
Provides important relationships among thermodynamic variables at two different points along a streamline
g = ratio of specific heats
g = cp/cv
gair=1.4

38. DERIVATION: ENERGY EQUATION (4.7)
Energy can neither be created nor destroyed
Start with 1st law
Adiabatic, dq=0
1st law in terms of enthalpy
Recall Euler’s equation
Combine
Integrate
Result: frictionless + adiabatic flow

39. ENERGY EQUATION SUMMARY (4.7)
Energy can neither be created nor destroyed; can only change physical form
Same idea as 1st law of thermodynamics
Energy equation for frictionless,
adiabatic flow (isentropic)
h = enthalpy = e+p/r = e+RT
h = cpT for an ideal gas
Also energy equation for
frictionless, adiabatic flow
Relates T and V at two different points along a streamline

40. SUMMARY OF GOVERNING EQUATIONS (4.8) STEADY AND INVISCID FLOW
Incompressible flow of fluid along a streamline or in a stream tube of varying area
Most important variables: p and V
T and r are constants throughout flow
continuity
Bernoulli
continuity
Compressible, isentropic (adiabatic and frictionless) flow along a streamline or in a stream tube of varying area
T, p, r, and V are all variables
isentropic
energy
equation of state
at any point

41. EXAMPLE: SPEED OF SOUND (4.9)
Sound waves travel through air at a finite speed
Sound speed (information speed) has an important role in aerodynamics
Combine conservation of mass, Euler’s equation and isentropic relations:
Speed of sound, a, in a perfect gas depends only on temperature of gas
Mach number = flow velocity normalizes by speed of sound
If M < 1 flow is subsonic
If M = 1 flow is sonic
If M > flow is supersonic
If M < 0.3 flow may be considered incompressible

42. 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
Constant pressure process
Constant volume process
Adiabatic
Reversible
Isentropic
Enthalpy

43. MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW
If M > 0.3, flow is compressible (density changes are important)
Need to introduce energy equation and isentropic relations
cp: specific heat at constant pressure
M1=V1/a1
gair=1.4

44. MEASUREMENT OF AIRSPEED: SUBSONIC COMRESSIBLE FLOW
So, how do we use these results to measure airspeed
p0 and p1 give
Flight Mach number
Mach meter
M1=V1/a1
Actual Flight Speed
Actual Flight Speed
using pressure difference
What is T1 and a1?
Again use sea-level conditions Ts, as, ps (a1=340.3 m/s)

45. EXAMPLE: TOTAL TEMPERATURE
Static temperature
Vehicle flight
Mach number
A rocket is flying at Mach 6 through a portion of the atmosphere where the static temperature is 200 K
What temperature does the nose of the rocket ‘feel’?
T0 = 200(1+ 0.2(36)) = 1,640 K!

46. MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW
What can happen in supersonic flows?
Supersonic flows (M > 1) are qualitatively and quantitatively different from subsonic flows (M < 1)

47. HOW AND WHY DOES A SHOCK WAVE FORM?
Think of a as ‘information speed’ and M=V/a as ratio of flow speed to information speed
If M < 1 information available throughout flow field
If M > 1 information confined to some region of flow field

48. MEASUREMENT OF AIRSPEED: SUPERSONIC FLOW
Notice how different this expression is from previous expressions
You will learn a lot more about shock wave in compressible flow course

49. SUMMARY OF AIR SPEED MEASUREMENT
Subsonic, incompressible
Subsonic, compressible
Supersonic

50. HOW ARE ROCKET NOZZLES SHAPPED?

51. MORE ON SUPERSONIC FLOWS (4.13)
Isentropic flow in a streamtube
Differentiate
Euler’s Equation
Since flow is isentropic
a2=dp/dr
Area-Velocity Relation

52. CONSEQUENCES OF AREA-VELOCITY RELATION
IF Flow is Subsonic (M < 1)
For V to increase (dV positive) area must decrease (dA negative)
Note that this is consistent with Euler’s equation for dV and dp
IF Flow is Supersonic (M > 1)
For V to increase (dV positive) area must increase (dA positive)
IF Flow is Sonic (M = 1)
M = 1 occurs at a minimum area of cross-section
Minimum area is called a throat (dA/A = 0)

53. 2: OUTLET 1: INLET TRENDS: CONTRACTION M1 < 1 M1 > 1 V2 > V1

54. 2: OUTLET 1: INLET TRENDS: EXPANSION M1 < 1 M1 > 1 V2 < V1

55. PUT IT TOGETHER: C-D NOZZLE
1: INLET
2: OUTLET

56. MORE ON SUPERSONIC FLOWS (4.13)
A converging-diverging, with a minimum area throat, is necessary to produce a supersonic flow from rest
Supersonic wind tunnel section
Rocket nozzle

57. SUMMARY OF GOVERNING EQUATIONS (4.8) STEADY AND INVISCID FLOW
Incompressible flow of fluid along a streamline or in a stream tube of varying area
Most important variables: p and V
T and r are constants throughout flow
continuity
Bernoulli
continuity
Compressible, isentropic (adiabatic and frictionless) flow along a streamline or in a stream tube of varying area
T, p, r, and V are all variables
isentropic
energy
equation of state
at any point