MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS
Final Exam Review and Closing Comments
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
OVERVIEW OF ACCOMPLISHMENT
• “This book is designed for a 1-year course in aerodynamics. Chapters 1 to 6
constitute a solid semester [bold, italics added for emphasis] emphasizing inviscid,
incompressible flow. Chapters 7 to 14 occupy a second semester dealing with
inviscid, compressible flow.” – John D. Anderson, Jr.
• What we did:
– Chapters 1-5
• Why not Chapter 6? → 3-D incompressible flow (sources, doublet, etc.)
– Chapters 7-9, 11 and 12
• Why not Chapter 10? → Fluids II material (nozzles, diffusers, etc.)
– Multiple examples of applications to flight and projectile mechanics
• What would we do if we had more time:
– Viscous flow
– Laminar and turbulent boundary layer models for drag prediction
– Exact solutions, Faulkner-Skan equations and Thwaites method
2
OUTLINE
1. Basic Ideas
– Can you convey basic ideas in aerodynamics in simple terms: lift, stall,
streamline, Kutta-condition, camber, lifting line, separation, etc.
– Explain in words or pictures what complicated equations are trying to say
2. Stream and Potential Functions: Inviscid, Incompressible Flow
– What is the point? What is the utility? What is weakness?
– How do you set-up and use these simple models?
3. Flow Over Airfoils
– Incompressible flow: Theory vs. experiment
– Compressible flow (why so complicated?): Theory vs. experiment
– Supersonic flow: Why does shape of airfoil want to be so different?
4. Flow Over Wings
– Impact of wing tips? How do you model, how do you proceed?
– What are implications for design?
5. Flight Mechanics
– What do (1)-(4) imply about aerodynamic design and performance impacts?
3
KEY CONCEPTS: CHAPTERS 1 and 2
•
•
•
•
Aerodynamic forces and moments (center of pressure)
– Where do they come from, why do we care?
Mach and Reynolds number matching guarantee flow similarity
Types of flows
– Inviscid vs. Viscous
– Incompressible vs. Compressible
– Mach number regimes
•
Fundamentals Principles
1. Conservation of Mass (integral and control volume form)
2. Conservation of Momentum (integral form)
3. Conservation of Energy (algebraic form)
Angular velocity, vorticity and circulation (why do we care about these concepts?)
•
Stream Function and Velocity Potential (how are these related?)
4
KET CONCEPTS: CHAPTER 3
•
•
•
•
Elementary Flows (Building Blocks, why such a name?)
1. Uniform Flow
2. Source / Sink Flow
3. Doublet Flow
4. Vortex Flow
What is the purpose? → Simulate real shapes in a simple manner
– Combine (1) + (2) → flow over half-body or oval
– Combine (1) + (3) → flow over a cylinder
– Combine with (4) → flow over a lifting cylinder
– Kutta-Joukowski Theorem
– Combinations of sources, vortex, uniform flow, tornados, ground effect, etc.
– Why can we combine so easily (simply add)?
Know how to set up y and f for all cases and combined flows (no time to solve)
Know how to get velocity components u and v
How would you model some basic shapes using these tools?
•
Homework #4 has many practice problems (nothing more difficult than these)
•
5
KEY CONCEPTS: CHAPTER 4, 11 and 12
• Model an airfoil as a vortex sheet
– What does this mean, why can we do this, why would we want to do this?
• Thin airfoil theory: Mean camber line is a streamline of the flow
• Symmetric vs. Cambered Airfoils
– S+C: Lift coefficient: 2pa
– S+C: Lift slope: 2p
– S: Moment Coefficient, c/4 = 0
– C: Moment Coefficient, c/4 = p/4(A2 - A1)
•
•
•
•
•
Role of airfoil thickness (incompressible, subsonic, supersonic)
What are added complexities (physics and math) associated with compressibility?
How can we correct for compressibility (what are strengths and weaknesses)?
Also see key concepts/comments for Chapters 7, 8, and 9
Chapter 12: §12.1- §12.3
6
KEY CONCEPTS: CHAPTER 5
• Airfoils vs. Wings
– What is different about these situations
– Why should we care? When is it important to care?
• How do we model a wing? Is it accurate?
• What is lifting line theory
• Key results
– Elliptical Wings
– Other Wings
• Why do we taper a wing?
• Why do we sweep wing?
• Why do we vary AR (or span) as designers
• Why do modern commercial airplane wings (A320, B757, etc.) look way they do?
• Why do modern fighter wings not look like this?
7
KEY CONCEPTS: CHAPTER 7, 8, and 9
• What are isentropic relations?
– When can we use them?
– Why would we use them? (replace energy equation, simple, algebraic)
– When do they break down?
• If flow speeds are greater than Mach 1, shock waves are present in the flow (why?)
• How do flow properties across normal and oblique shock waves change?
– Is it important to capture these effects?
• Expansion processes
• Make use of Appendix A, B, and C as well as q-b-M diagram
– Don’t waste time calculating, but know where these appendicies and figures
come from (what are equations that generate them)
8
BASIC CONCEPTS
CHAPTERS 1-2
KEY CONCEPTS
•
•
•
•
Aerodynamic forces and moments (center of pressure)
– Where do they come from, why do we care?
Mach and Reynolds number matching guarantee flow similarity
Types of flows
– Inviscid vs. Viscous
– Incompressible vs. Compressible
– Mach number regimes
•
Fundamentals Principles
1. Conservation of Mass (integral and control volume form)
2. Conservation of Momentum (integral form)
3. Conservation of Energy (algebraic form)
Angular velocity, vorticity and circulation (why do we care about these concepts?)
•
Stream Function and Velocity Potential (how are these related?)
10
WHAT DOES EULER’S EQUATION TELL US?
dp   VdV
• Euler’s Equation (Differential Equation)
– Relates changes in momentum to changes in force (momentum equation)
– Relates a change in pressure (dp) to a chance in velocity (dV)
• Assumptions:
– Steady flow and no friction (inviscid flow), body forces, and external forces
• dp and dV are of opposite sign
– IF dp increases dV goes down → flow slows down
– IF dp decreases dV goes up → flow speeds up
• Incompressible and Compressible flows, Irrotational and Rotational flows
11
BERNOULLI’S EQUATION
2
2
2
1
V
V
p2  
 p1  
2
2
2
V
p
 Constant along a streamline
2
• If flow is irrotational p+1/2V2 = constant everywhere
• Remember:
– Bernoulli’s equation holds only for inviscid (frictionless) and
incompressible (=constant) flows
– Relates properties between different points along a streamline or entire
flow field if irrotational
– For a compressible flow Euler’s equation must be used ( is a variable)
– Both Euler’s and Bernoulli’s equations are expressions of F=ma
expressed in a useful form for fluid flows and aerodynamics
12
WHAT CREATES AERODYNAMIC FORCES?
• Aerodynamic forces exerted by airflow comes from only two sources
• Pressure, p, distribution on surface
– Acts normal to surface
• 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”
13
SOME DEFINITIONS
• Relative Wind: Direction of V∞
– We used 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
• Center of Pressure: It is that point on an airfoil (or body) about which the
aerodynamic moment is zero
• Aerodynamic Center: It is that point on an airfoil (or body) about which the
aerodynamically generated moment is independent of angle of attack
14
SAMPLE DATA TRENDS
• 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
cl
Cambered airfoil has
lift at a=0
At negative a airfoil
will have zero lift
15
AIRFOIL DATA (APPENDIX D)
NACA 23012 WING SECTION
Re dependence
at high a
Dependent on Re
cl vs. a
Independent of Re
cd
cl
cd vs. cl
cm,a.c.
cm,c/4
cm,a.c. vs. cl very flat
a
cl
16
HOW DOES AN AIRFOIL GENERATE LIFT?
1. Flow velocity over the 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, velocity must increase
Streamtube A is squashed
most in nose region
(ahead of maximum thickness)
A
B
17
HOW DOES AN AIRFOIL GENERATE LIFT?
2. As velocity increases pressure decreases
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)
18
WHY DOES AN AIRFOIL STALL?
• Key to understanding
– Friction causes flow separation within boundary layer
– Separation then creates another form of drag called pressure drag due to
separation
19
STALL CHARACTER: NACA 4412 VERSUS NACA 4421
•
•
•
•
•
Both NACA 4412 and NACA 4421
have same shape of mean camber
line
Thin airfoil theory predict that linear
lift slope and aL=0 should be the
same for both
Leading edge stall shows rapid drop
of lift curve near maximum lift
Trailing edge stall shows gradual
bending-over of lift curve at
maximum lift, “soft stall”
High cl,max for airfoils with leading
edge stall
•
Flat plate stall exhibits poorest
behavior, early stalling
•
Thickness has major effect on cl,max
20
INVISCID, INCOMPRESSIBLE FLOW
CHAPTER 3
KET CONCEPTS
•
•
•
•
Elementary Flows (Building Blocks, why such a name?)
1. Uniform Flow
2. Source / Sink Flow
3. Doublet Flow
4. Vortex Flow
What is the purpose? → Simulate real shapes in a simple manner
– Combine (1) + (2) → flow over half-body or oval
– Combine (1) + (3) → flow over a cylinder
– Combine with (4) → flow over a lifting cylinder
– Kutta-Joukowski Theorem
– Combinations of sources, vortex, uniform flow, tornados, ground effect, etc.
– Why can we combine so easily (simply add)?
Know how to set up y and f for all cases and combined flows (no time to solve)
Know how to get velocity components u and v
How would you model some basic shapes using these tools?
•
Homework #4 has many practice problems (nothing more difficult than these)
•
22
SUMMARY OF STREAM AND POTENTIAL FUNCTIONS
TABLE 3.1
23
LIFTING FLOW OVER A CYLINDER
 R   r
y  V r sin q 1  2  
ln  
 r  2p  R 
L  V  Kutta-Joukowski Theorem
2
24
FLOW OVER AIRFOILS
INCOMPRESSIBLE: CHAPTER 4
COMPRESSIBLE: CHAPTER 11
KEY CONCEPTS
• Model an airfoil as a vortex sheet
– What does this mean, why can we do this, why would we want to do this?
• Thin airfoil theory: Mean camber line is a streamline of the flow
1   d
dz 

 V a  

2p 0 x  
dx 

c
• Symmetric vs. Cambered Airfoils
– S+C: Lift coefficient: 2pa
– S+C: Lift slope: 2p
– S: Moment Coefficient, c/4 = 0
– C: Moment Coefficient, c/4 = p/4(A2 - A1)
• Role of thickness
26
CENTER OF PRESSURE AND AERODYNAMIC CENTER
• Center of Pressure: It is that point on an airfoil (or body) about which the
aerodynamic moment is zero
– Thin Airfoil Theory:
c
xcp 
• Symmetric Airfoil:
4
• Cambered Airfoil:

c p
xcp  1   A1  A2 
4  cl

• Aerodynamic Center: It is that point on an airfoil (or body) about which the
aerodynamically generated moment is independent of angle of attack
– Thin Airfoil Theory:
c
x

A.C .
• Symmetric Airfoil:
4
• Cambered Airfoil:
c
x A.C . 
4
27
PREVIEW: COMPRESSIBILITY CORRECTION
EFFECT OF M∞ ON CP
Cp 
C p ,0
1  M 2

Sound
Barrier ?
0.5
1  M 2
For M∞ < 0.3,  ~ const
Cp = Cp,0 = 0.5 = const
Effect of compressibility
(M∞ > 0.3) is to increase
absolute magnitude of Cp
and M∞ increases
Called: Prandtl-Glauert Rule
Prandtl-Glauert rule applies for 0.3 < M∞ < 0.7
(Why not M∞ = 0.99?)
28
RESULT
Velocity Potential Equation: Nonlinear Equation
Compressible, Steady, Inviscid and Irrotational Flows
Note: This is one equation, with one unknown, f
a0 (as well as T0, P0, 0, h0) are known constants of the flow
 1  f  2   2f  1  f  2   2f 2  f  f   2f
0
1  2    2  1  2    2  2   
a  x  y  xy
 a  x   x  a  y   y
Velocity Potential Equation: Linear Equation
Incompressible, Steady, Inviscid and Irrotational Flows
 2f  0
29
RESULT
• After order of magnitude analysis, we have
following results
1  M 
2

1  M 
2

uˆ vˆ

0
x y
2 ˆ
ˆ
f f
 2 0
2
x
y
2
• May also be written in terms of perturbation
velocity potential
• Equation is a linear PDE and is rather easy to
solve (see slides 19-22 for technique)
• Recall:
– Equation is no longer exact
– Valid for small perturbations
• Slender bodies
• Small angles of attack
– Subsonic and Supersonic Mach numbers
– Keeping in mind these assumptions
equation is good approximation
30
CRITICAL MACH NUMBER, MCR
• 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
31
DESIGN OPTIONS: SWEEP, AERA RULE, SUPERCRITICAL AIRFOILS
•
Sweep:
– Makes airfoil ‘thinner’ → increases
critical Mach number
– Sweeping wing usually reduces lift for
subsonic flight
•
Area Rule: Drag created related to change in
cross-sectional area of vehicle from nose to tail
•
Supercritical Airfoils: Designed to delay and
reduce transonic drag rise, due to both strong
normal shock and shock-induced boundary
layer separation
32
FLOW OVER WINGS
CHAPTER 5
KEY CONCEPTS
• Airfoils vs. Wings
– What is different about these situations
– Why should we care? When is it important to care?
• How do we model a wing? Is it accurate?
• What is lifting line theory
• Key results
– Elliptical Wings
– Other Wings
• Why do we taper a wing?
• Why do we vary AR (or span) as designers
• Why do modern commercial airplane wings (A320, B757, etc.) look the way they
do?
• Why do modern fighter wings not look like this?
34
PHYSICAL INTERPRETATION
a: Geometric Angle of Attack
ai: Induced Angle of Attack
aeff: Effective Angle of Attack
a geometric  a effective  a induced
Di  L sin a i
Di  La i
•
Finite Wing Consequences:
1. Tilted lift vector contributes a drag component, called induced drag (drag due
to lift) → CL < cl and CD > cd
2. Lift slope is reduced relative to infinite wing (a < a0)
35
PRANDTL’S LIFTING LINE EQUATION
  y0 
1
a  y0  
 a L 0 
pV c y0 
4pV
b
2

b
2
 d 
 
 dy  dy
y0  y
• Fundamental Equation of Prandtl’s Lifting Line Theory
– In Words: Geometric angle of attack is equal to sum of effective angle of attack
plus induced angle of attack
– Mathematically: a = aeff + ai
• Only unknown is (y)
– V∞, c, a, aL=0 are known for a finite wing of given design at a given a
– Solution gives (y0), where –b/2 ≤ y0 ≤ b/2 along span
36
KEY RESULT
•
•
•
•
•
True for all finite wings in general
Define a span efficiency factor, e (also called span efficiency factor)
Elliptical planforms, e = 1
For all other planforms, e < 1
Arbitrary Finite Wing
0.60 < e < 0.99
2
L
For Elliptical Planforms
D ,i
C
ai 
CL
pAR
CL
C L2
Di  La i  L
 q S
pAR
pAR
Di
C L2

q S pAR
C D ,i
C L2

pAR
C

peAR
Span Efficiency Factor
Key Points:
Goes with square of CL
Inversely related to AR
Also called drag due to lift
37
SUMMARY: TOTAL DRAG ON SUBSONIC WING
Profile Drag
Profile Drag coefficient
relatively constant with
M∞ at subsonic speeds
Also called drag due to lift
D  D friction  D pressure  Dinduced
D  D profile  Dinduced
2
Di
CL
CD  cd , profile 
 cd , profile 
q S
peAR
Look up
(Infinite Wing)
May be calculated from
Inviscid theory:
Lifting line theory
38
IMPORTANT STATEMENTS
Fundamental Equation of Thin Airfoil Theory
“The camber line is a streamline of the flow”
1   d
dz 

 V  a  

2p 0 x  
dx 

c
Fundamental Equation of Prandtl’s Lifting-Line Theory
“The geometric angle of attack is equal to the sum of the effective
angle of attack plus the induced angle of attack”
  y0 
1




a y0 
 a L  0 y0 
pV c y0 
4pV
b
2

b
2
 d 
 
 dy  dy
y0  y
39
GENERAL LIFT DISTRIBUTION (2/4)
Substitute expression for (q) and d/dy into fundamental equation of Prandtl’s
lifting line theory
N
a q 0  
N
2b
1


A
sin
n
q

a
q

 n
0
L 0
0
pcq 0  n 1
p
p
 nA
n
cos nq
 cos q  cos q
n 1
0
dq
0
Last term on the right (integral term) is a standard form and may be simplified as:
N
sin nq 0
2b N


a q 0  
A
sin
n
q

a
q

nA
 n

0
L 0
0
n
pcq 0  n 1
sin q 0
n 1
Equation is evaluated at a given spanwise location (q0), just as fundamental equation
of Prandtl’s lifting line theory is evaluated at a given spanwise location (y0)
Only unknowns in equation are An’s
Written at q0 equation is 1 algebraic equation with N unknowns
Write equation at N spanwise locations to obtain a system of N independent algebraic
equations with N unknowns
40
SUPERSONIC AIRFOILS AND WINGS
REVIEW: CHAPTER 7
SHOCK WAVES / EXPANSIONS:
CHAPTERS 8 AND 9
KEY CONCEPTS
• What are isentropic relations?
– When can we use them?
– Why would we use them? (replace energy equation, simple, algebraic)
– When do they break down?
• If flow speeds are greater than Mach 1, shock waves are present in the flow (why?)
• How do flow properties across normal and oblique shock waves change?
– Is it important to capture these effects?
• Expansion processes
• Make use of Appendix A, B, and C as well as q-b-M diagram
– Don’t waste time calculating, but know where these appendicies and figures
come from (what are equations that generate them)
42
SUMMARY OF NORMAL SHOCK RELATIONS
0.9
0.8
M2, P02/P01
20
Downstream Mach Number, M2
Total Pressure Ratio, P02/P01
Density Ratio, Rho1/Rho2
Static Pressure Ratio, P2/P1
Static Temperature Ratio T2/T1
18
16
0.7
14
0.6
12
0.5
10
0.4
8
0.3
6
0.2
4
0.1
2
0
0
1
2
3
4
5
6
7
8
9
 2/ 1, p2/p1, T2/T1
1
10
Upstream Mach Number, M1
43
MEASUREMENT OF AIRSPEED:
SUPERSONIC FLOW (M > 1)
p02  p02  p2 
 
 
p1  p2  p1 
p02    1 2 
 1 
M2 
p2 
2

 1 2
1
M1
2
M 22 
 1
M 12 
2

 1
p2
2
 1
M 12  1
p1
 1


2
p02    1 M 12 


p1  4M 12  2  1
  1
1    2M 12
 1
Rayleigh Pitot Tube Formula
44
SUMMARY OF SHOCK RELATIONS
M n ,1  M 1 sin b
Normal Shocks
M 22 
Oblique Shocks

  1 2
M
1
1
2
M
2
1
  1

M
2
n,2

2
1
  1 M 2
2
M n2,1 
n ,1
  1
2

  1M 12
2

1 2    1M 12
  1M n,1
2

1 2    1M n2,1
2
p2
M 12  1
 1
 1
p1
p2
2
 1
M n2,1  1
p1
 1

2


M2 

M n,2
sin b  q 
45
q-b-M RELATION
Strong
Shock Wave Angle, b
M2 < 1
Weak
M2 > 1
M 12 sin 2 b  1
tan q  2 cot b 2
M 1   cos 2b   2
Deflection Angle, q
46
SWEPT WINGS: SUPERSONIC FLIGHT
 1 

  sin 
 M 
1
•
•
•
If leading edge of swept wing is outside Mach cone, component of Mach number normal
to leading edge is supersonic → Large Wave Drag
If leading edge of swept wing is inside Mach cone, component of Mach number normal
to leading edge is subsonic → Reduced Wave Drag
For supersonic flight, swept wings reduce wave drag
47
EXAMPLE OF SUPERSONIC AIRFOILS
http://odin.prohosting.com/~evgenik1/wing.htm
48
FLIGHT MECHANICS
WING LOADING (W/S), SPAN LOADING (W/b) AND ASPECT RATIO (b2/S)
W W  b
 
b  S  AR
Span loading (W/b), wing loading (W/S)
and AR (b2/S) are related
D0  q SC D , 0
Zero-lift drag, D0 is proportional to wing area
1
Di 
peq
W 
 
b
2
Induced drag, Di, is proportional to square
of span loading
2
Di  1  W  
1

  
D0  peq  b   q SC D , 0
2
2
Re-write W2/(b2S) in terms of AR and substitute into
drag ratio Di/D0
2
1: For specified W/S (set by take-off or landing
requirements) and CD,0 (airfoil choice), increasing AR
will decrease drag due to lift relative to zero-lift drag
2: AR predominately controls ratio of induced drag to
zero lift drag, whereas span loading controls actual value
of induced drag
   
W
2
W
W
S 
S

b2
b2S
AR
S
Di
1

D0 peq2 C D , 0
 S
W
AR
Take ratio of these drags, Di/D0
50
FURTHER IMPLICATIONS FOR DESIGN: VMAX
• Maximum velocity at a given altitude is important specification for new airplane
• To design airplane for given Vmax, what are most important design parameters?

C L2 

D  T  q SC D  q S  C D , 0 
peAR 

W
CL 
q S
Steady, level flight: T = D
Steady, level flight: L = W


W2
W2
  q SC D , 0 
T  q S  C D , 0  2 2
q S peAR 
q SpeAR

q2 SC D , 0  qT 
Turn this equation into a quadratic
equation (by multiplying by q∞)
and rearranging
2
W
0
SpeAR
 T   W   W   T 
4C
  A        A   D ,0
  W  max  S   S   W  max peAR

 CD ,0



2
Vmax
Substitute into drag equation







1
2
Solve quadratic equation and set
thrust, T, to maximum available
thrust, TA,max
51
FURTHER IMPLICATIONS FOR DESIGN: VMAX
 T   W   W   T 
4C
  A        A   D,0
  W  max  S   S   W  max peAR

CD,0



2
Vmax







1
2
• TA,max does not appear alone, but only in ratio (TA/W)max
• S does not appear alone, but only in ratio (W/S)
• Vmax does not depend on thrust alone or weight alone, but rather on ratios
– (TA/W)max: maximum thrust-to-weight ratio
– W/S: wing loading
• Vmax also depends on density (altitude), CD,0, peAR
• We can increase Vmax by
– Increase maximum thrust-to-weight ratio, (TA/W)max
– Increasing wing loading, (W/S)
– Decreasing zero-lift drag coefficient, CD,0
52
THRUST REQUIRED VS. FLIGHT VELOCITY
TR  D  q SC D  q S C D , 0  C D ,i 
C L2
TR  q SC D , 0  q S
peAR
Zero-Lift TR
(Parasitic Drag)
Lift-Induced TR
(Induced Drag)
Zero-Lift TR ~ V2
(Parasitic Drag)
Lift-Induced TR ~ 1/V2
(Induced Drag)
53
THRUST REQUIRED VS. FLIGHT VELOCITY
W2
TR  q SC D , 0 
q SpeAR
dTR  dTR  dV 


 
dq  dV  dq 
dTR
W2
 SC D , 0  2
0
dq
q SpeAR
CD,0
At point of minimum TR, dTR/dV∞=0
(or dTR/dq∞=0)
C L2

 C D ,i
peAR
Zero-Lift Drag = Induced Drag
At minimum TR and maximum L/D
54
POWER REQUIRED
PR  TRV  DV  q SC DV  q S C D , 0  C D ,i V
C L2
PR  q SC D , 0V  q SV
peAR
Zero-Lift PR
Lift-Induced PR
Zero-Lift PR ~ V3
Lift-Induced PR ~ 1/V
55
POWER REQUIRED
1
W2
3
PR   V SC D ,0 
1
2
 V SpeAR
2
dPR 3
1

2 
  V S  C D ,0  C D ,i   0
dV 2
3


At point of minimum PR, PTR/dV∞=0
CD ,0
1
 C D ,i
3
C D , 0  C D ,i
56