Transcript Document 7177007

MAE 1202: AEROSPACE PRACTICUM
Lecture 7: Compressible Flow Review and Overview of Airfoils
March 11, 2013
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
UDPATES
• Mid-term grades
• Team project: Introduced in Laboratory this week
• Mid-Term Exam: Monday, March 18, 2013 in class
– Covers Chapter 4 and Chapter 5.1 – 5.7
– Open book / open notes… but no time to study during the exam
– No computers, cell phones, etc.
– Sample Mid-Term with Solution on line
– Review Session: Thursday, March 14, 2013, Crawford
Science Tower, Room 112, 8 – 10 pm
• AIAA Meeting/Fund Raiser Friday, March 15, 2013, 7:00 pm –
11:00 pm, Buffalo Wild Wings (Palm Bay Road)
READING AND HOMEWORK ASSIGNMENTS
• Reading: Introduction to Flight, by John D. Anderson, Jr.
– For March 25, 2013 lecture: Chapter 5, Sections 5.13-5.19
– Read these sections carefully, most interesting portions of Ch. 5
• Lecture-Based Homework Assignment:
– Problems: 5.7, 5.11, 5.13, 5.15, 5.17, 5.19
• DUE: Friday, March 29, 2013 by 5pm
• Turn in hard copy of homework
– Also be sure to review and be familiar with textbook examples in
Chapter 5
ANSWERS TO LECTURE HOMEWORK
•
•
•
•
•
•
5.7: Cp = -3.91
5.11: Cp = -0.183
– Be careful here, if you check the Mach number it is around 0.71, so the flow is
compressible and the formula for Cp based on Bernoulli’s equation is not valid. To
calculate the pressure coefficient, first calculate r∞ from the equation of state and find
the temperature from the energy equation. Finally make use of the isentropic relations
and the definition of Cp given in Equation 5.27
5.13: cl = 0.97
– Make use of Prandtl-Glauert rule
5.15: Mcr = 0.62
– Use graphical technique of Section 5.9
– Verify using Excel or Matlab
5.17: m = 30°
5.19: D = 366 lb
– Remember that in steady, level flight the airplane’s lift must balance its weight
– You may also assume that all lift is derived from the wings (this is not really true
because the fuselage and horizontal tail also contribute to the airplane lift). Also assume
that the wings can be approximated by a thin flat plate
– Remember that Equation 5.50 gives a in radians
HOMEWORK EXAMPLES
• Can you read this?
• Who would you hire?
MODERN ART?
Compressible Flow Applications
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
A1V1  A2V2
continuity
1
1
2
p1  rV1  p2  rV22
2
2
Bernoulli
• 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
continuity
r1 A1V1  r 2 A2V2

isentropic
energy
equation of state
at any point

p1  r1   T1   1
     
p2  r 2   T2 
1 2
1 2
c pT1  V1  c pT2  V2
2
2
p1  r1 RT1
p2  r 2 RT2
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

r0    1 2 
 1 
M1 
r1 
2


 1
1
 1
cp: specific heat at constant pressure
M1=V1/a1
air=1.4
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:
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)
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 
rs
• 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/dr
Area-Velocity Relation
rAV  constant
ln r  lnA  lnV  ln constant 
dr dA dV


0
r
A V
dp   rVdV
drVdV 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
Chapter 5 Overview
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 = PA
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 rAV=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

rV 2  constant
– Incompressible: Bernoulli’s Equation
2
– Compressible: Euler’s Equation
dp   rVdV
– 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?
AIRFOILS VERSUS WINGS
Why do airfoils have such a shape?
How are lift and drag produced?
NACA airfoil performance data
How do we design?
What is limit of behavior?
AIRFOIL THICKNESS: WWI AIRPLANES
English Sopwith Camel
Thin wing, lower maximum CL
Bracing wires required – high drag
German Fokker Dr-1
Higher maximum CL
Internal wing structure
Higher rates of climb
Improved maneuverability
AIRFOIL NOMENCLATURE
• Mean Chamber Line: Set of points halfway between upper and lower surfaces
– Measured perpendicular to mean chamber line itself
• Leading Edge: Most forward point of mean chamber line
• Trailing Edge: Most reward point of mean chamber line
• Chord Line: Straight line connecting the leading and trailing edges
• Chord, c: Distance along the chord line from leading to trailing edge
• Chamber: Maximum distance between mean chamber line and chord line
– Measured perpendicular to chord line
NACA FOUR-DIGIT SERIES
• First digit specifies maximum camber in percentage of chord
• Second digit indicates position of maximum camber in tenths of chord
• Last two digits provide maximum thickness of airfoil in percentage of chord
NACA 2415
Example: NACA
2415
• Airfoil has maximum thickness of 15%
of chord (0.15c)
• Camber of 2% (0.02c) located 40%
back from airfoil leading edge (0.4c)
WHAT CREATES AERODYNAMIC FORCES? (2.2)
•
Aerodynamic forces exerted by airflow comes from only two sources:
1. Pressure, p, distribution on surface
• Acts normal to surface
2. Shear stress, tw, (friction) on surface
• Acts tangentially to surface
•
•
Pressure and shear are in units of force per unit area (N/m2)
Net unbalance creates an aerodynamic force
“No matter how complex the flow field, and no matter how complex the shape of
the body, the only way nature has of communicating an aerodynamic force to a
solid object or surface is through the pressure and shear stress distributions that
exist on the surface.”
“The pressure and shear stress distributions are the two hands of nature that
reach out and grab the body, exerting a force on the body – the aerodynamic
force”
RESOLVING THE AERODYNAMIC FORCE
• Relative Wind: Direction of V∞
– We use subscript ∞ to indicate far upstream conditions
• Angle of Attack, a: Angle between relative wind (V∞) and chord line
• Total aerodynamic force, R, can be resolved into two force components
– Lift, L: Component of aerodynamic force perpendicular to relative wind
– Drag, D: Component of aerodynamic force parallel to relative wind
MORE DEFINITIONS
• Total aerodynamic force on airfoil is summation of F1 and F2
• Lift is obtained when F2 > F1
• Misalignment of F1 and F2 creates Moments, M, which tend to rotate airfoil/wing
– A moment (torque) is a force times a distance
• Value of induced moment depends on point about which moments are taken
– Moments about leading edge, MLE, or quarter-chord point, c/4, Mc/4
– In general MLE ≠ Mc/4
F1
F2
VARIATION OF L, D, AND M WITH a
• Lift, Drag, and Moments on a airfoil or wing will change as a changes
• Variations of these quantities are some of most important information that an
airplane designer needs to know
• Aerodynamic Center
– Point about which moments essentially do not vary with a
– Mac=constant (independent of a)
– For low speed airfoils aerodynamic center is near quarter-chord point, c/4
AOA = 2°
AOA = 3°
AOA = 6°
AOA = 9°
AOA = 12°
AOA = 20°
AOA = 60°
AOA = 90°
Lift (for now)
SAMPLE DATA: SYMMETRIC AIRFOIL
Angle of Attack, a
A symmetric airfoil generates zero lift at zero a
Lift (for now)
SAMPLE DATA: CAMBERED AIRFOIL
Angle of Attack, a
A cambered airfoil generates positive lift at zero a
SAMPLE DATA
Lift (for now)
• Lift coefficient (or lift) linear
variation with angle of attack, a
– Cambered airfoils have
positive lift when a = 0
– Symmetric airfoils have
zero lift when a = 0
• At high enough angle of attack,
the performance of the airfoil
rapidly degrades → stall
Cambered airfoil has
lift at a=0
At negative a airfoil
will have zero lift
Lift (for now)
SAMPLE DATA: STALL BEHAVIOR
What is really going on here
What is stall?
Can we predict it?
Can we design for it?
REAL EFFECTS: VISCOSITY (m)
• To understand drag and actual airfoil/wing behavior we need an understanding of
viscous flows (all real flows have friction)
• Inviscid (frictionless) flow around a body will result in zero drag!
– This is called d’Alembert’s paradox
– Must include friction (viscosity, m) in theory
• Flow adheres to surface because of friction between gas and solid boundary
– At surface flow velocity is zero, called ‘No-Slip Condition’
– Thin region of retarded flow in vicinity of surface, called a ‘Boundary Layer’
• At outer edge of B.L., V∞
• At solid boundary, V=0
“The presence of friction in the flow causes a shear stress at the surface of a body,
which, in turn contributes to the aerodynamic drag of the body: skin friction drag”
p.219, Section 4.20
TYPES OF FLOWS: FRICTION VS. 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
COMMENTS ON VISCOUS FLOWS (4.15)
THE REYNOLDS NUMBER, Re
• One of most important dimensionless numbers in fluid mechanics/ aerodynamics
• Reynolds number is ratio of two forces:
– Inertial Forces

– Viscous Forces
– c is length scale (chord)
rV c
Re 
m
• Reynolds number tells you when viscous forces are important and when viscosity
may be neglected
Outside B.L. flow
Inviscid (high Re)
Within B.L. flow
highly viscous
(low Re)
LAMINAR VS. 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
LAMINAR VS. TURBULENT FLOW
All B.L.’s transition from
laminar to turbulent
Turbulent velocity
profiles are ‘fuller’
cf,turb > cf,lam
FLOW SEPARATION
• Key to understanding: Friction causes flow separation within boundary layer
• Separation then creates another form of drag called pressure drag due to separation
REVIEW: AIRFOIL STALL (4.20, 5.4)
• Key to understanding: Friction causes flow separation within boundary layer
1. B.L. either laminar or turbulent
2. All laminar B.L. → turbulent B.L.
3. Turbulent B.L. ‘fuller’ 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
SUMMARY OF VISCOUS EFFECTS ON DRAG (4.21)
•
Friction has two effects:
1. Skin friction due to shear stress at wall
2. Pressure drag due to flow separation
D  D friction  D pressure
Total drag due to
Drag due to
=
viscous effects
skin friction
Called Profile Drag
+
Less for laminar
More for turbulent
Drag due to
separation
More for laminar
Less for turbulent
So how do you design?
Depends on case by case basis, no definitive answer!
COMPARISON OF DRAG FORCES
d
d
Same total drag as airfoil
TRUCK SPOILER EXAMPLE
•
•
•
•
•
•
Note ‘messy’ or
turbulent flow pattern
High drag
Lower fuel efficiency
Spoiler angle
increased by + 5°
Flow behavior more
closely resembles a
laminar flow
Tremendous savings
(< $10,000/yr) on
Miami-NYC route
LIFT, DRAG, AND MOMENT COEFFICIENTS (5.3)
• Behavior of L, D, and M depend on a, but also on velocity and altitude
– V∞, r ∞, Wing Area (S), Wing Shape, m ∞, compressibility
• Characterize behavior of L, D, M with coefficients (cl, cd, cm)
1
L  rV2 Scl
2
Matching Mach and Reynolds
(called similarity parameters)
M∞, Re
L
L
cl 

1
rV2 S q S
2
cl  f a , M  , Re 
M∞, Re
cl, cd, cm identical
LIFT, DRAG, AND MOMENT COEFFICIENTS (5.3)
• Behavior of L, D, and M depend on a, but also on velocity and altitude
– V∞, r ∞, Wing Area (S), Wing Shape, m ∞, compressibility
• Characterize behavior of L, D, M with coefficients (cl, cd, cm)
1
L  rV2 Scl
2
L
L
cl 

1
rV2 S q S
2
cl  f1 a , M  , Re 
1
D  rV2 Scd
2
D
D
cd 

1
rV2 S q S
2
cd  f 2 a , M  , Re 
1
rV2 Sccm
2
M
L
cm 

1
rV2 Sc q Sc
2
cm  f 3 a , M  , Re 
M
Note on Notation:
We use lower case, cl, cd, and cm for infinite wings (airfoils)
We use upper case, CL, CD, and CM for finite wings
SAMPLE DATA: NACA 23012 AIRFOIL
Flow separation
Stall
Lift Coefficient
cl
Moment Coefficient
cm, c/4
a
AIRFOIL DATA (5.4 AND APPENDIX D)
NACA 23012 WING SECTION
Re dependence at high a
Separation and Stall
Dependent on Re
cl vs. a
Independent of Re
cd
cl
cd vs. a
R=Re
cm,a.c.
cm,c/4
cm,a.c. vs. cl very flat
a
cl
EXAMPLE: BOEING 727 FLAPS/SLATS
EXAMPLE: SLATS AND FLAPS
AIRFOIL DATA (5.4 AND APPENDIX D)
NACA 1408 WING SECTION
• Flaps shift lift curve
• Effective increase in camber of airfoil
Flap extended
Flap retracted
PRESSURE DISTRIBUTION AND LIFT
• Lift comes from pressure distribution
over top (suction surface) and bottom
(pressure surface)
• Lift coefficient also result of pressure
distribution
PRESSURE COEFFICIENT, CP (5.6)
• Use non-dimensional description, instead of plotting actual values of pressure
• Pressure distribution in aerodynamic literature often given as Cp
• So why do we care?
– Distribution of Cp leads to value of cl
– Easy to get pressure data in wind tunnels
– Shows effect of M∞ on cl
p  p
p  p
Cp 

1
q
r V2
2
EXAMPLE: CP CALCULATION
COMPRESSIBILITY CORRECTION:
EFFECT OF M∞ ON CP
C p ,0
p  p

1
2
r V
2
For M∞ < 0.3, r ~ const
Cp = Cp,0 = 0.5 = const
M∞
COMPRESSIBILITY CORRECTION:
EFFECT OF M∞ ON CP
Cp 
C p,0
1  M 2

0.5
1  M 2
For M∞ < 0.3, r ~ const
Cp = Cp,0 = 0.5 = const
Effect of compressibility
(M∞ > 0.3) is to increase
absolute magnitude of Cp as
M∞ increases
Called: Prandtl-Glauert Rule
M∞
Prandtl-Glauert rule applies for 0.3 < M∞ < 0.7
OBTAINING LIFT COEFFICIENT FROM CP (5.7)
1
cl   C p ,lower  C p ,upper dx
c0
c
cl 
cl , 0
1  M 2
COMPRESSIBILITY CORRECTION SUMMARY
• If M0 > 0.3, use a compressibility correction for Cp, and cl
• Compressibility corrections gets poor above M0 ~ 0.7
– This is because shock waves may start to form over parts of airfoil
• Many proposed correction methods, but a very good on is: Prandtl-Glauert Rule
• Cp,0 and cl,0 are the low-speed (uncorrected) pressure and lift coefficients
– This is lift coefficient from Appendix D in Anderson
• Cp and cl are the actual pressure and lift coefficients at M∞
Cp 
C p,0
1 M
2

cl 
cl , 0
1 M
2

CRITICAL MACH NUMBER, MCR (5.9)
• As air expands around top surface near leading edge, velocity and M will increase
• Local M > M∞
Flow over airfoil may have
sonic regions even though
freestream M∞ < 1
INCREASED DRAG!
CRITICAL FLOW AND SHOCK WAVES
MCR
CRITICAL FLOW AND SHOCK WAVES
‘bubble’ of supersonic flow
AIRFOIL THICKNESS SUMMARY
• Which creates most lift?
– Thicker airfoil
• Which has higher critical Mach number?
– Thinner airfoil
• Which is better?
– Application dependent!
Note: thickness is relative
to chord in all cases
Ex. NACA 0012 → 12 %
AIRFOIL THICKNESS: WWI AIRPLANES
English Sopwith Camel
Thin wing, lower maximum CL
Bracing wires required – high drag
German Fokker Dr-1
Higher maximum CL
Internal wing structure
Higher rates of climb
Improved maneuverability
THICKNESS-TO-CHORD RATIO TRENDS
A-10
Root: NACA 6716
TIP: NACA 6713
F-15
Root: NACA 64A(.055)5.9
TIP: NACA 64A203
MODERN AIRFOIL SHAPES
Boeing 737
Root
Mid-Span
Tip
http://www.nasg.com/afdb/list-airfoil-e.phtml
SUMMARY OF AIRFOIL DRAG (5.12)
D  D friction  D pressure  Dwave
cd  cd , f  cd , p  cd , w
Profile Drag
Profile Drag coefficient
relatively constant with
M∞ at subsonic speeds
Only at transonic and
supersonic speeds
Dwave=0 for subsonic speeds
below Mdrag-divergence