Transcript Compressible and Isentropic Flow - Florida Institute of Technology

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
READING AND HOMEWORK ASSIGNMENTS
• Reading: Introduction to Flight, by John D. Anderson, Jr.
– For this week’s lecture: Chapter 4, Sections 4.10 - 4.21, 4.27
– For next week’s lecture: Chapter 5, Sections 5.1 - 5.13
• 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
ANSWERS TO LECTURE HOMEWORK
• 5.2: L = 23.9 lb, D = 0.25 lb, Mc/4 = -2.68 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 = - 0.925 N m
• 5.4: a = 2°
• 5.6: (L/D)max ~ 112
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
CAD DESIGN CONTEST
CAD DESIGN CONTEST
If you do the PRO|E challenge…
Do not let it consume you!
BERNOULLI’S EQUATION
2
2
2
1
V
V
p2  
 p1  
2
2
2
V
p
 Constant along a streamline
2
• One of most fundamental and useful equations in aerospace engineering!
• Remember:
– Bernoulli’s equation holds only for inviscid (frictionless) and
incompressible ( = constant) flows
– Bernoulli’s equation relates properties between different points along a
streamline
– For a compressible flow Euler’s equation must be used ( is variable)
– Both Euler’s and Bernoulli’s equations are expressions of F = ma
expressed in a useful form for fluid flows and aerodynamics
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
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
MEASUREMENT OF AIRSPEED:
INCOMPRESSIBLE FLOW
1
2
p  V1  p0
2
Static
pressure
V1 
Dynamic
pressure
Total
pressure
2 p 0  p 

Incompressible Flow
Total and Static Ports
16
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
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
p
• Instrument is called a Pitotstatic probe
p0
18
MEASUREMENT OF AIRSPEED:
INCOMPRESSIBLE FLOW
1
2
p  V1  p0
2
Static
pressure
V1 
Dynamic
pressure
p
1
V12  p0
2
Total
pressure
2 p 0  p 

Incompressible Flow
19
TRUE VS. EQUIVALENT AIRSPEED
• What is value of ?
• If  is measured in actual air
around the airplane
• Measurement is difficult to do
• Practically easier to use value at
standard seal-level conditions, s
• This gives an expression called
equivalent airspeed
Vtrue 
Ve 
2 p 0  p 

2 p0  p 
s
20
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
21
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
HOW DOES AN AIRFOIL GENERATE LIFT?
1. 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 AV=constant: IF A↓ THEN V↑
Streamtube A is squashed
most in nose region
(ahead of maximum thickness)
A
B
HOW DOES AN AIRFOIL GENERATE LIFT?
2. As V ↑ p↓
1
p

V 2  constant
– Incompressible: Bernoulli’s Equation
2
– Compressible: Euler’s Equation
dp   VdV
– Called Bernoulli Effect
3. 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?
Incorrect Lift Theory
• http://www.grc.nasa.gov/WWW/k-12/airplane/wrong1.html
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  are constants throughout flow
A1V1  A2V2
continuity
1
1
2
p1  V1  p2  V22
2
2
What if flow is high speed, M > 0.3?
What if there are temperature effects?
How does density change?
Bernoulli
1st LAW OF THERMODYNAMICS (4.5)
Boundary
System
Surroundings
e (J/kg)
• 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):
1. Heat, dq, added to (or removed from) system
2. Work, dw, is done on (or by) system
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?
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?
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):
1. Heat, dq, added to (or removed from) system
2. Work, dw, is done on (or by) system
de  dq  dw
1st LAW IN MORE USEFUL FORM (4.5)
• 1st Law: de = dq + dw
– Find more useful expression for dw, in
terms of p and  (or v = 1/)
ΔW  force distance 
ΔW   pdAs 
dw   psdA  p  sdA
A
dw   pdv
• When volume varies → work is done
• Work done on balloon, volume ↓
• Work done by balloon, volume ↑
de  dq  dw
de  dq  pdv
A
Change in
Volume (-)
ENTHALPY: A USEFUL QUANTITY (4.5)
Define a new quantity
called enthalpy, h:
(recall ideal gas law: pv = RT)
h  e  pv  e  RT
Differentiate
dh  de  pdv  vdp
Substitute into 1st law
(from previous slide)
dq  de  pdv
dq  de  dh  de  vdp
Another version of 1st law
that uses enthalpy, h:
dq  dh  vdp
HEAT ADDITION AND SPECIFIC HEAT (4.5)
• Addition of dq will cause a small change in temperature dT of system
dq
dT
dq
J
c

kg K 
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…
SPECIFIC HEAT: CONSTANT PRESSURE
• Addition of dq will cause a small change in temperature dT of system
• System pressure remains constant
dq
dT
dq
J
c

kg K 
dT
 dq 
cp  

 dT  constant pressure
dq  c p dT
dh  c p dT
h  c pT
Extra Credit #1:
Show this step
SPECIFIC HEAT: CONSTANT VOLUME
• Addition of dq will cause a small change in temperature dT of system
• System volume remains constant
dq
dT
dq
J
c

kg K 
dT
 dq 
cv  

 dT  constant volume
dq  cv dT
Extra Credit #2:
de  cv dT
e  cvT
Show this step
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
dq
J
c

kg K 
dT
• However, for a fixed dq, resulting dT depends on type of process:
Constant Pressure
Constant Volume
dq 
 dq 

cp  

cv  

 dT  constant pressure
 dT  constant volume
dq  c p dT
dq  cv dT
dh  c p dT
de  cv dT
h  c pT
e  cvT
Specific heat ratio
For air,  = 1.4

cp
cv
ISENTROPIC FLOW (4.6)
• 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

p2   2   T2 
     
p1  1   T1 

 1
 = ratio of specific heats
 = cp/cv
air=1.4
DERIVATION: ENERGY EQUATION (4.7)
Energy can neither be created nor destroyed
Start with 1st law
de  dq  dw
Adiabatic, dq=0
1st law in terms of enthalpy
dq  0
dq  dh  vdp  0
Recall Euler’s equation
dp   VdV
Combine
dh  vVdV  0
dh  VdV  0
h2
V2
Integrate

Result: frictionless + adiabatic flow
V22 V12
h2  h1 

0
2
2
h1
dh   VdV  0
V1
ENERGY EQUATION SUMMARY (4.7)
• Energy can neither be created nor destroyed; can only change physical form
– Same idea as 1st law of thermodynamics
2
1
2
2
V
V
h1 
 h2 
2
2
2
V
h
 constant
2
V12
V22
c pT1 
 c pT2 
2
2
2
V
c pT 
 constant
2
Energy equation for frictionless,
adiabatic flow (isentropic)
h = enthalpy = e+p/ = 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
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  are constants throughout flow
A1V1  A2V2
continuity
1
1
2
p1  V1  p2  V22
2
2
Bernoulli
• 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
1 A1V1   2 A2V2

isentropic
energy
equation of state
at any point

p1  1   T1   1
     
p2   2   T2 
1 2
1 2
c pT1  V1  c pT2  V2
2
2
p1  1 RT1
p2   2 RT2
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:
dp
a 
d
2
a 
p

 RT
• 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
V
M 
a
KEY TERMS: CAN YOU DEFINE THEM?
• Streamline
• Stream tube
• Constant pressure process
• Constant volume process
• Steady flow
• Unsteady flow
• Adiabatic
• Reversible
• Viscid flow
• Inviscid flow
• Compressible flow
• Incompressible flow
• Laminar flow
• Turbulent flow
• Isentropic
• Enthalpy
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
1 2
c pT1  V1  c pT0
2
2
T0
V1
 1
T1
2c pT1
T0
 1 2
 1
M1
T1
2
p0    1 2 
 1 
M1 
p1 
2

0    1 2 
 1 
M1 
1 
2


 1
1
 1
cp: specific heat at constant pressure
M1=V1/a1
air=1.4
MEASUREMENT OF AIRSPEED:
SUBSONIC COMRESSIBLE FLOW
• So, how do we use these results to measure airspeed
 1 




p
2
 0 
M 12 
 1
  1  p1 


 1 




p
2
a
 0 
V12 
 1
  1  p1 


2
1
 1 




p

p
2
a
1
 0
V12 
 1
 1
  1  p1



2
1
2
cal
V
 1 


2a  p0  p1 


 1
 1
  1  ps



2
s
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)
EXAMPLE: TOTAL TEMPERATURE
Total temperature
T0
 1 2
 1
M1
T1
2
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!
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)
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
MEASUREMENT OF AIRSPEED:
SUPERSONIC FLOW

p02    1 M


2
p1  4M 1  2  1
2
2
1
  1
1    2M 12
 1
Notice how different this expression is from previous expressions
You will learn a lot more about shock wave in compressible flow course
SUMMARY OF AIR SPEED MEASUREMENT
Ve 
2
Vcal
2 p0  p 
s
• Subsonic,
incompressible
 1 

 • Subsonic,


2a
p0  p1


 1
 1
compressible
  1  ps



2
s
2
p02    1 M 12 


p1  4M 12  2  1
  1
1    2M 12
• Supersonic
 1
HOW ARE ROCKET NOZZLES SHAPPED?
MORE ON SUPERSONIC FLOWS (4.13)
Isentropic flow in a streamtube
Differentiate
Euler’s Equation
Since flow is isentropic
a2=dp/d
Area-Velocity Relation
AV  constant
ln   lnA  lnV  ln constant 
d dA dV


0

A V
dp   VdV
dVdV dA dV



0
dp
A V
VdV dA dV
 2 

0
a
A V
dA
dV
2
 M 1
A
V


CONSEQUENCES OF AREA-VELOCITY RELATION


dA
dV
2
 M 1
A
V
• 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)
TRENDS: CONTRACTION
1: INLET
2: OUTLET
M1 < 1
V2 > V1
M1 > 1
V2 < V1
TRENDS: EXPANSION
1: INLET
2: OUTLET
M1 < 1
V2 < V1
M1 > 1
V2 > V1
PUT IT TOGETHER: C-D NOZZLE
1: INLET
2: OUTLET
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
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  are constants throughout flow
A1V1  A2V2
continuity
1
1
2
p1  V1  p2  V22
2
2
Bernoulli
• 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
1 A1V1   2 A2V2

isentropic
energy
equation of state
at any point

p1  1   T1   1
     
p2   2   T2 
1 2
1 2
c pT1  V1  c pT2  V2
2
2
p1  1 RT1
p2   2 RT2