Aerodynamics Part II of 3 describes aerodynamics of bodies in supersonic flight. For comments please contact me at solo.hermelin@gmail.com. For more presentations on different subjects visit my website at http://www.solohermelin.com.

1. 1
AERODYNAMICS
Part II
SOLO HERMELIN
http://www.solohermelin.com

2. 2
Table of Content
AERODYNAMICS
Earth Atmosphere
Mathematical Notations
SOLO
Basic Laws in Fluid Dynamics
Conservation of Mass (C.M.)
Conservation of Linear Momentum (C.L.M.)
Conservation of Moment-of-Momentum (C.M.M.)
The First Law of Thermodynamics
The Second Law of Thermodynamics and Entropy Production
Constitutive Relations for Gases
Newtonian Fluid Definitions – Navier–Stokes Equations
State Equation
Thermally Perfect Gas and Calorically Perfect Gas
Boundary Conditions
Flow Description
Streamlines, Streaklines, and Pathlines
AEROD

3. 3
Table of Content (continue – 1)
AERODYNAMICS
SOLO
Circulation
Biot-Savart Formula
Helmholtz Vortex Theorems
2-D Inviscid Incompressible Flow
Stream Function ψ, Velocity Potential φ in 2-D Incompressible
Irrotational Flow
Aerodynamic Forces and Moments
Blasius Theorem
Kutta Condition
Kutta-Joukovsky Theorem
Joukovsky Airfoils
Theodorsen Airfoil Design Method
Profile Theory by the Method of Singularities
Airfoil Design
AEROD

4. 4
Table of Content (continue – 2)
AERODYNAMICS
SOLO
Lifting-Line Theory
Subsonic Incompressible Flow (ρ∞ = const.) about Wings
of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
Incompressible Potential Flow Using Panel Methods
Dimensionless Equations
Boundary Layer and Reynolds Number
Wing Configurations
Wing Parameters
References
AEROD

5. 5
Table of Content (continue – 3)
AERODYNAMICS
SOLO
Shock & Expansion Waves
Shock Wave Definition
Normal Shock Wave
Oblique Shock Wave
Prandtl-Meyer Expansion Waves
Movement of Shocks with Increasing Mach Number
Drag Variation with Mach Number
Swept Wings Drag Variation
Variation of Aerodynamic Efficiency with Mach Number
Analytic Theory and CFD
Transonic Area Rule

6. 6
Table of Content (continue – 4)
AERODYNAMICS
SOLO
Linearized Flow Equations
Cylindrical Coordinates
Small Perturbation Flow
Applications: Nonsteady One-Dimensional Flow
Applications: Two Dimensional Flow
Applications: Three Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Drag (d) and Lift (l) per Unit Span Computations for Subsonic Flow (M∞ < 1)
Prandtl-Glauert Compressibility Correction
Computations for Supersonic Flow (M∞ >1)
Ackeret Compressibility Correction

7. 7
SOLO
Table of Contents (continue – 5)
Wings of Finite Span at Supersonic Incident Flow
Theoretic Solutions for Pressure Distribution on a
Finite Span Wing in a Supersonic Flow (M∞ > 1)
1. Conical Flow Method
2. Singularity-Distribution Method
Theoretical Solutions for Compressible Supersonic Flow (M∞ >1)
Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing
in a Supersonic Flow and Subsonic Leading Edge (tanΛ > β)
Theoretic Solution for Pressure Distribution on a Semi-Infinite Triangular Wing
in a Supersonic Flow and Supersonic Leading Edge (tanΛ < β)
Theoretic Solution for Pressure Distribution on a Delta Wing in a
Supersonic Flow and Subsonic Leading Edge (tanΛ > β)
Theoretic Solution for Pressure Distribution on a Delta Wing in a Supersonic
Flow and Supersonic Leading Edge (tanΛ < β)
Arrowhead Wings with Double-Wedge Profile at Zero Incidence
Systematic Presentation of Wave Drag of Thin, Nonlifting Wings having
Straight Leading and Trailing Edges and the same dimensionless profile in
all chordwise plane [after Lawrence]
AERODYNAMICS
A
E
R
O
D
Y
N
A
M
I
C
S
P
A
R
T
I
I
I

8. 8
Table of Content (continue – 6)
AERODYNAMICS
SOLO
Aircraft Flight Control
References
CNα – Slope of the Normal Force Coefficient Computations of Swept Wings
Comparison of Experiment and Theory for Lift-Curve Slope of Un-swept Wings
Drag Coefficient
A
E
R
O
D
Y
N
A
M
I
C
S
P
A
R
T
I
I
I

9. 9
Continue from AERODYNAMICS – Part I
AERODYNAMICS

10. 10
SOLO
- when the source moves at subsonic velocity V a, it will stay inside the
family of spherical sound waves.
a
V
M
M
=





= −
1
sin 1
µ
Disturbances in a fluid propagate by molecular collision, at the sped of sound a,
along a spherical surface centered at the disturbances source position.
The source of disturbances moves with the velocity V.
- when the source moves at supersonic velocity V a, it will stay outside the
family of spherical sound waves. These wave fronts form a disturbance
envelope given by two lines tangent to the family of spherical sound waves.
Those lines are called Mach waves, and form an angle μ with the disturbance
source velocity:
SHOCK EXPANSION WAVES

11. 11
SOLO
SHOCK EXPANSION WAVES
M 1
M = 1
M 1
Mach Waves

12. 12
SOLO
When a supersonic flow encounters a boundary the following will happen:
When a flow encounters a boundary it must satisfy the boundary conditions,
meaning that the flow must be parallel to the surface at the boundary.
- when the supersonic flow, in order to remain parallel to the boundary surface,
must “turn into itself” (see the Concave Corner example) a Oblique Shock will
occur. After the shock wave the pressure, temperature and density will increase.
The Mach number of the flow will decrease after the shock wave.
SHOCK EXPANSION WAVES
- when the supersonic flow, in order to remain parallel to the boundary surface,
must “turn away from itself” (see the Convex Corner example) an Expansion
wave will occur. In this case the pressure, temperature and density will decrease.
The Mach number of the flow will increase after the expansion wave.
Return to Table of Content

13. 13
SHOCK WAVESSOLO
A shock wave occurs when a supersonic flow decelerates in response to a sharp
increase in pressure (supersonic compression) or when a supersonic flow encounters
a sudden, compressive change in direction (the presence of an obstacle).
For the flow conditions where the gas is a continuum, the shock wave is a narrow region
(on the order of several molecular mean free paths thick, ~ 6 x 10-6
cm) across which is
an almost instantaneous change in the values of the flow parameters.
Shock Wave Definition (from John J. Bertin/ Michael L. Smith,
“Aerodynamics for Engineers”, Prentice Hall, 1979, pp.254-255)
When the shock wave is normal to the streamlines it is called a Normal Shock Wave,
otherwise it is an Oblique Shock Wave.
The difference between a shock wave and a Mach wave is that:
- A Mach wave represents a surface across which some derivative of the flow variables
(such as the thermodynamic properties of the fluid and the flow velocity) may be
discontinuous while the variables themselves are continuous. For this reason we call
it a weak shock.
- A shock wave represents a surface across which the thermodynamic properties and the
flow velocity are essentially discontinuous. For this reason it is called a strong shock.

14. 14
Normal Shock Wave Over a Blunt Body
Normal Shock
Wave
SHOCK WAVESSOLO
Oblique
Shock
Wave
Oblique Shock Wave
Return to Table of Content

15. 15
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Conservation of Mass (C.M.) ρ ρ1 1 2 2u u= η
ρ
ρ
= =2
1
1
2
u
u
Conservation of Linear Momentum (C.L.M.) 2
2
221
2
11 pupu +=+ ρρ ( )
p
p
u
p
2
1
1
2
1
1
1 1= + −
ρ
η
H H h u h u1 2 1 1
2
2 2
21
2
1
2
= → + = +
h
h
u
h
2
1
1
2
1
2
1
2
1
1
= + −






η
Conservation of Energy (C.E.)
Field Equations
Constitutive Relations
p R T= ρIdeal Gas
( )
( )
( )
e e T C Tv= =
1 2
(1) Thermally Perfect Gas
(2) Calorically Perfect Gas
ργ
γ
ρρρ
γ
ρ
pp
C
C
C
C
p
R
C
TC
p
eh
v
p
vp C
C
v
p
v
p
CCR
p
TRp
p
11 −
=
−
===+=
≡−==
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2

16. 16
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
First Way
h
h
p
p
p
p
p
p
u
h
u
p
2
1
2
2
1
1
2
1
1
2
2
1
1
2
1
2
1
2
1
1
2
1
1
1
1
2
1
1
1
2
1
1
1
=
−
−
= = = + −





 = +
−
−






γ
γ ρ
γ
γ ρ
ρ
ρ η η γ
γ ρ
η
or
( )
p
p
u
p
u
p
C L M
2
1
1
2
1
1
1
2
1
1
2
1
1 1
1
1
2
1
1
1
η
ρ
η
η γ
γ ρ
η
= + −










= +
−
−






( . . .)
after further development we obtain
1 2
1
1
1
1
1
1
2
01
2
1
1
2
1
2
1
1
1
2
1
1
−
−





 − +










+ +
−










=
γ
γ
ρ
η
ρ
η
γ
γ
ρ
u
p
u
p
u
p
Solving for 1/η , we obtain
1
1 1 2
1
1
1
2
1
1
2
2
1
1
2
1
1
1
2
1
1
2
1
2
1
1
1
2
1
1
η
ρ
ρ
ρ ρ
γ
γ
ρ
γ
γ
ρ
γ
γ
= = =
+










− +










−
+
+
−










+
u
u
u
p
u
p
u
p
u
p
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2

17. 17
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
We obtain an other relation in the following way:
( )
p
p
u
p
p
p
u
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
2
1
1
2
1
1
2
2
1
1
2
1
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
1
1
1
2
1
1
1 1
1
1
1
1
2
1
1
1 1
2
1
2
1
1
2
1
1
2
1
2
1
2
1
2
η
γ
γ
ρ
η
ρ
η
η γ
γ η
η
γ
γ
γ
γ
γ
γ
η
γ
γ
γ
γ
γ
γ
γ
γ
− =
−
−






− = −








⇒
−
−
=
−
+






⇓
−
−
−
−




 = +
−
−






⇓
=
+
−
−
−
+
+
η
ρ
ρ
γ
γ
γ
γ
= = =
+
−
−
+
+
−
=2
1
1
2
2
1
2
1
2
1
1
2
1
1
1
1
1
u
u
p
p
p
p
p
p
T
T
or
Rankine-Hugoniot Equation
Rankine-Hugoniot Equation (1)
William John Macquorn
Rankine
(1820-1872)
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2
Pierre-Henri Hugoniot
(1851 – 1887)

18. 18
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
η
ρ
ρ
γ
γ
γ
γ
= = =
+
−
−
+
+
−
=2
1
1
2
2
1
2
1
2
1
1
2
1
1
1
1
1
u
u
p
p
p
p
p
p
T
T Rankine-Hugoniot Equation
Rankine-Hugoniot Equation (2)
p
p
2
1
2
1
2
1
1
1
1
1
1
=
+
−
−
+
−
−
γ
γ
ρ
ρ
γ
γ
ρ
ρ
T
T
p
p
p
p
p
p
p
p
p
p
p
p
2
1
2
1
1
2
2
1
2
1
2
1
2
1
2
1
2
1
2
1
1
2
2
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
= =
+
+
−
+
−
−
=
+
+
−
+
−
−
=
+
−
−
+
−
−
=
+
−
−
+
−
−
ρ
ρ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
ρ
ρ
γ
γ
ρ
ρ
ρ
ρ
γ
γ ρ
ρ
γ
γ
ρ
ρ
p2
p 1
ρ 2
ρ 1
NormalShockWave
Rankine-Hugoniot
Isentropic
γp2
p 1
ρ 2
ρ 1
( )=
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2

19. 19
Rankine-Hugoniot Equation (3)
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2
SOLO

20. 20
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Strong Shock Wave Definition:
p
p
u
u
T
T
p
p
R H R H
2
1
2
1
1
2
2
1
2
1
1
1
1
1
→ ∞ ⇒ = →
+
−
→
−
+
− −ρ
ρ
γ
γ
γ
γ
Weak Shock Wave Definition:
∆ p
p
p p
p1
2 1
1
1=
−
ρ ρ ρ2 1
2 1
2 1
= +
= +
= +
∆
∆
∆
p p p
h h h
For weak shocks
u
p
1
2
=
∆
∆ρ
∆
∆
h u
ρ ρ
= 1
2
1
u u u u u u2
1
2
1
1
1
1
1
1 1
1
1
1
1
= =
+
=
+
≅ −
ρ
ρ
ρ
ρ ρ ρ
ρ
ρ
ρ∆ ∆
∆
(C.M.)
( ) ( )ρ ρ ρ
ρ
ρ
1 1
2
1 1 1 2 2 1 1 1
1
1 1u p u u p u u u p p+ = + = −





 + +
∆
∆(C.L.M.)

ordernd
uuuhhuuhhuhuh
2
4
1
2
1
2
1
2
1
2
1 2
1
2
1
2
1
1
2
11
2
1
1
11
2
22
2
11 




 ∆
+
∆
−+∆+=




 ∆
−+∆+=+=+
ρ
ρ
ρ
ρ
ρ
ρ(C.E.)
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2

21. 21
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Second Way
h h u h u0 1 1
2
2 2
21
2
1
2
≡ + = +Define







−
−
−
=→+
−
=
−
−
−
=→+
−
=
2
10
1
12
1
1
1
0
2
20
2
22
2
2
2
0
11
2
1
1
11
2
1
1
uh
p
u
p
h
uh
p
u
p
h
γ
γ
γ
γ
ρργ
γ
γ
γ
γ
γ
ρργ
γ
u u h1 2 0
2
1
1
=
−
+
γ
γ
Prandtl’s Relation
( )u h
u
u
u
p
p
u
p2 0
1
2 1
1
2
2
1
1
2
1
1
2
1
1
1
1 1=
−
+
→ = = → = + −
γ
γ
ρ ρ ρη
ρ
ηFrom this relation, we obtain:
Prandtl’s Relation
Ludwig Prandtl
(1875-1953)
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2
(C.M.)
(C.L.M.)
ργ
γ p
h
1−
=
and use
12
22
2
11
1
2211
2
2
221
2
11 11
uu
u
p
u
p
uu
pupu
−=−→



=
+=+
ρρρρ
ρρ
1221
21
0
2
1
2
1111
uuuu
uu
h −=
−
+
−
−





−
−
γ
γ
γ
γ
γ
γ
( ) 




 −
−−=
−−
γ
γ
γ
γ
2
1
1
1
12
21
12
0 uu
uu
uu
h

22. 22
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
(C.M.)
Hugoniot Equation
ρ ρ
ρ
ρ
1 1 2 2 2 1
1
2
u u u u= → =
( )ρ ρ ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ ρ
ρ ρ
ρ
ρ ρ ρ
ρ
ρ
ρ
ρ
ρ
ρ
1 1
2
1 2 2
2
2 2
1
2
2
1
2
2 2 1 1
2
1
1
2
2
1
2 1
2
2 1
1
2 2 1
2
2
2
2 2 1
2
1
2
2 1
1
2
2 1
1
2
u p u p u p p p u u
u
p p
u
p p
u u
u u
+ = + =





 + → − = −





 = − →
→ =
−
−





 → =
−
−














=
=
(C.L.M.)
( )( )
h u h u e
p p p
e
p p p
e e
p p p p p p p p
e e
p p
h e
p
1 1
2
2 2
2
1
1
1
2 1
2 1
2
2
2
2
2 1
2 1
1
2
2 1
2 1
2 1
2 1
2
1
1
2
2
2 1
2 1
2
2
1
2
1 2
1 2 2 1
2
2 1
2 1 1 2
1
2
1
2
1
2
1
2
1
2
+ = + → + +
−
−





 = + +
−
−





 →
→ − =
−
−
−





 + − =
−
−
−
+
−
→
→ − =
− +
= +
ρ
ρ ρ ρ
ρ
ρ ρ ρ ρ
ρ
ρ
ρ ρ
ρ
ρ
ρ
ρ ρ ρ ρ ρ
ρ ρ
ρ ρ
ρ ρ
ρρ
ρ ρ
( ) ( )
+ −
=
+ − − + −
→
→ − =
+ − +














2 2
2
2 2
2
2
1 2 2
1 2
2 2 2 1 2 1 1 1 2 2
1 2
2 1
2 1 2 1 1 2
1 2
p p p p p p p p
e e
p p p p
ρ ρ
ρ ρ
ρ ρ ρ ρ ρ ρ
ρ ρ
ρ ρ
ρ ρ
(C.E.)
e e
p p
2 1
1 2
2 1
2
1 1
− =
+
−






ρ ρ
Hugoniot Equation
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2
Pierre-Henri Hugoniot
(1851 – 1887)

23. 23
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Fanno’s Line for a Perfect Gas (1)
( )1 1 1 2 2ρ ρu u
m
A
= =

( ) frictionpupu ++=+ 2
2
221
2
112 ρρ
( )3
1
2
1
2
1 1
2
2 2
2
C T u C T u h C Tp p p+ = + =
( )4 1 1 1 2 2 2
p R T p R T= =ρ ρ
( )5 2 1
2
1
2
1
s s C
T
T
R
p
p
p
− = −ln ln
(C.M.)
(C.L.M.)
(C.E.)
Ideal Gas
( )
p
p
T
T
u
u
h C T
h C T
p
p
T
T
h C T
h C T
s s C
T
T
R
T
T
h C T
h C T
p
p
p
p
p
p
p
2
1
4
2
1
2
1
2
1
1
1
2
3
0 1
0 2
2
1
2
1
0 1
0 2
2 1
2
1
2
1
0 1
0 2
5
=












= =
−
−







→ =
−
−
→
− = −
−
−
( )
( ) ( )
ln ln
ρ
ρ
ρ
ρ
Assume that all the conditions
of the model are satisfied except
the moment equation (2)
(a flow with friction)
Using , we obtainh C Tp=
s
s 1
s 2
s max
h 1
h 2
h
2
1
s s C
h
h
R
h
h
h h
h h
p2 1
2
1
2
1
0 1
0 2
− = −
−
−
ln ln
Fanno’s Line for a Perfect Gas
This is the Adiabatic, Constant Area Flow.
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2
Gino Girolamo Fanno
(1888 – 1962)

24. 24
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Fanno’s Line for a Perfect Gas (2)
s
s 1
s 2
s max
h 1
h 2
h
2
1
We have a point of maximum entropy. Let see the significance of this point
ρρ
dp
dh
dp
dhdsT =→=−= 0
max
Gibbs
u
dud
dudu −=→=+
ρ
ρ
ρρ 0(C.M.)
duudh
u
hd −=→=





+ 0
2
2
(C.E.)
Therefore
)4..(
0
.).(
00
0
EC
ds
MC
dsds
ds
u
du
d
dpd
d
dpdp
dh =





−





=





==
===
=
ρρ
ρ
ρρ
0
0
=
= 





=
ds
ds
d
dp
u
ρ
or
ds C
dT
T
R
dp
p
ds C
dT
T
R
d
C
C
dp
p
d
dp
d p
dp
d
p
R T
p
v
p
v
ds
ds
ds ds
p R T
= − =
= − =







→ ≡ = = → = ==
=
= =
=
max
max
0
0
0
0
0 0
ρ
ρ
γ
ρ
ρ
ρ
ρ
ρ
γ
ρ
γ
ρ
We have:
u
dp
d
R T a speed of soundds
ds
=
=
=





 = = =0
0
ρ
γ
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2

25. 25
Ideal Gas
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Rayleigh’s Line for a Perfect Gas (1)
( )
A
m
uu

== 22111 ρρ
( )2 1 1
2
1 2 2
2
2ρ ρu p u p+ = +
( ) QhuTCuTC pp ++=+ 2
22
2
11
2
1
2
1
3
( )4 1 1 1 2 2 2
p R T p R T= =ρ ρ
( )5 2 1
2
1
2
1
s s C
T
T
R
p
p
p
− = −ln ln
(C.M.)
(C.L.M.)
(C.E.)
Assume that all the conditions
of the model are satisfied except
the energy equation (3)
(a flow with heating and cooling)
Let substitute in (5) , to obtainh C Tp=
Rayleigh’s Line for a Perfect Gas
This is the Frictionless, Constant Area Flow, with Cooling and Heating.
s max
s
s 1
s 2
h 1
h 2
h
M1
M1
Rayleigh2
1
Heating
Heating
Cooling
 m
A
R T
p
p
m
A
R T
p
p
x
p
1
1
1
2
2
2
1
+ = +
( )
2
1
12
1
1
1
2
12
11
1
2
12
1
2
1
lnln5
p
R
A
m
c
p
TR
A
m
b
h
C
a
bbR
h
h
Css
p
p

=







+=








−+−=−
We want to find x
p
p
≡ 2
1
. Let multiply the result by
x
p1
x
m
A
R T
p
b
x
m
A
R
p
c
T2 1
1
2
1
1
2
1
21
2
0− +





 + =
 
   
or
x
p
p
b b a T= = + −2
1
1 1
2
1 2
The solution is:
John William
Strutt
Lord Rayleigh
(1842-1919)
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2

26. 26
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Rayleigh’s Line for a Perfect Gas (2)
We have a point of maximum entropy. Let see the significance of this point
u
dud
dudu −=→=+
ρ
ρ
ρρ 0(C.M.)
(C.L.M.)
A Normal Shock Wave must be on both Fanno and Rayleigh Lines, therefore
the end points of a Normal Shock Wave must be on the intersection of
Fanno and Rayleigh Lines
u
dp
d
R T a speed of soundds
ds
=
=
=





 = = =0
0
ρ
γ
d p u
dp
du
u+





 = → = −
1
2
02
ρ ρ
( )→ = = − −





 =
dp
d
dp
du
du
d
u
u
u
ρ ρ
ρ
ρ
2
s
s 1
s 2
h 1
h 2
h
M1
M1
Rayleigh
Fanno
2
1
SHOCK
According to the Second Law of Thermodynamics
the Entropy must increase. Therefore a Normal Shock
Wave from state (1) to state (2) must be such that
s2 s1. (from supersonic to subsonic flow only)
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2

27. 27
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Mach Number Relations (1)
( )
( )
( )
 
C M u u
C L M u p u p
p
u
p
u
u u
C E
a
h
u
a
h
u
a a u
a a u
a
p
. .
. . .
. .
ρ ρ
ρ ρ ρ ρ
γ γ
γ γ
γ γ
γ
ρ
1 1 2 2
1 1
2
1 2 2
2
2
1
1 1
2
2 2
2 1
1
2
1
1
2 2
2
2
2
2
1
2 2
1
2
2
2 2
2
2
4
1
1
2 1
1
2
1
2
1
2
1
2
1
2
=
+ = +



→ − = − →
−
+ =
−
+ →
=
+
−
−
=
+
−
−














=
∗
∗



− = −
a
u
a
u
u u1
2
1
2
2
2
2 1
γ γ
Field Equations:
( )
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
+
−
−
−
+
+
−
= −
↓
+ −
+
−
− = − →
+
= −
−
=
+
↓
∗ ∗
∗
∗
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
1
2
1
2
2
1
1
2
2
2 2 1
2 1
1 2
2
2 1 2 1
2
1 2
a
u
u
a
u
u u u
u u
u u
a u u u u
a
u u
u u a1 2
2
= ∗
u
a
u
a
M M1 2
1 21 1∗ ∗
∗ ∗
= → =
Prandtl’s Relation
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2
Ludwig Prandtl
(1875-1953)

28. 28
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Mach Number Relations (2)
( ) ( ) ( ) ( )
( )
( )
( )
( )
( )[ ]
( )( ) ( )
M
M
M
M
M
M
M
M
M
2
2
2
2
1
1
2
1
2
1
2
1
2
1
2
2
1
1
2
1 1
2
1
1
1 2
1
2 1 2
1 1 1 1 1
1
2
=
+
− −
=
+ − −
=
+
+
− +
− −
=
− +
+ / + − / / + − / + − −
∗
=
∗
∗
∗
γ
γ γ γ
γ
γ
γ
γ
γ
γ γ γ γ γ
or
( )
M
M
M
M
M
H H
A A
2
1
2
1
2
1
2
1
21 2
1 2
1
1
2
1
2
2
1
1
1
2
1
2
1
1
=
+
−
−
−
=
+
+
−
+
+
−
=
=
γ
γ
γ γ
γ
γ
γ
( )
( )
ρ
ρ
γ
γ
2
1
1
2
1
2
1 2
1
2
2 1
2 1
2
1
2
1 2 1
1 2
= = = = =
+
− +
=
∗
∗
A A u
u
u
u u
u
a
M
M
M
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2

29. 29
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Mach Number Relations (3)
( )
( )
( ) ( )
( )
p
p
u
p
u
u
u
a
M
M
M
M
M M
M
2
1
1
2
1
1
2
1
1
2
1
2
1
2
1
2 1
2
1
2 1
2 1
2
1
2
1
2
1 1 1 1
1 1
1 2
1
1
1 1 2
1
= + −





 = + −






= + −
− +
+





 = +
/ + − / − −
+
ρ
γ
ρ
ρ
γ
γ
γ
γ
γ γ
γ
or
(C.L.M.)
( )
p
p
M2
1
1
2
1
2
1
1= +
+
−
γ
γ
( )
( )
( )
h
h
T
T
p
p
M
M
M
a
a
h C T p RTp
2
1
2
1
2
1
1
2
1
2 1
2
1
2
2
1
1
2
1
1
1 2
1
= = = +
+
−






− +
+
=
= =ρ ρ
ρ
γ
γ
γ
γ
( )
( )
( )
s s
R
T
T
p
p
M
M
M
2 1 2
1
1
2
1
1
1
2
1
1
1
2
1
2
1
1
2
1
1
1 2
1
−
=






















= +
+
−






− +
+
















−
−
− −
ln ln
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
( )
( )
( )
( )
s s
R
M M
M
2 1
1 1
2 1
2 3
2
2 1
2 41
2
2
3 1
1
2
1
1
−
≈
+
− −
+
− +
− γ
γ
γ
γ
K Shapiro p.125
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2

30. 30
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Mach Number Relations (4)
( )
p
p
p
p
p
p
p
p
M
M
M02
01
02
2
1
01
2
1
2
2
1
2
1
1
2
1
1
2
1
1
2
1
2
1
1= =
+
−
+
−










+
+
−






−γ
γ
γ
γ
γ
γ
( )
( )
1
1
2
1
1
2
1
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
1
1
2
1
2
1
1
2
2
1
2
1
2
1
2
2
1
2
1
2
1
2
1
2
+
−
= +
−
+
−
−
−
=
−
−
+
−
+
−





+
+
+
−






=
+
+
+
−
γ γ
γ
γ
γ
γ
γ γ γ
γ γ
γ
γ
γ
γ
M
M
M
M M
M
M
M
( )
( )
p
p
M
M
M02
01
1
2
1
2
1
1
2
1
1
1
2
1
2
1
1
1
2
1
1=
+
+
+
−












+
+
−






−
−
−
γ
γ
γ
γ
γ
γ
γ
γ
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2

31. 31
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
 
G Q= =0 0,
Mach Number Relations (5)
( )
s s
R
T
T
p
p
p
p
M
M
M
T T
2 1 02
01
1
02
01
1
02
01
1
2
1
2
1
2
02 01
1
1
1
2
1
1
1
1
2
1
1
2
−
=






















= −






=
−
+
+
−





 −
−
+
+
−










−
−
=
ln ln
ln ln
γ
γ
γ
γ
γ
γ
γ
γ
γ
s
s
1
s
2
T
M1
M1
Rayleigh
Fanno
2
1
SHOCK
T
2
T
1
T
02
T
01=
T 2
T 1=* *
p
2
p
1
p
01
p
02
Mollier’s Diagram
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2
John William
Strutt
Lord Rayleigh
)1842-1919(
Gino Girolamo Fanno
)1888–1062(
Return to Table of Content

32. 32
OBLIQUE SHOCK EXPANSION WAVESSOLO
→→
→→
+=
+=
twnuV
twnuV
11
11
222
111


Continuity Eq.: 2211 uu ρρ =
( ) ( ) ( )21222111 ppuuuu +−−=+− ρρ
Moment Eq. Tangential Component:
( ) ( ) 0222111 =+− wuwu ρρ
Moment Eq. Normal Component:
Energy Eq.: 22
2
2
2
2
211
2
1
2
1
1
22
u
wu
hu
wu
h ρρ 






 +
+=






 +
+
Continuity Eq.: 2211 uu ρρ =
Moment Eq.:
21 ww =
2
222
2
111 upup ρρ +=+
Energy Eq.:
22
2
2
2
2
1
1
u
h
u
h +=+
Summary
Calorically Perfect Gas:
Tch
TRp
p=
= ρ
6 Equations with 6 Unknowns
222222 ,,,,, hwuTpρ

33. 33
OBLIQUE SHOCK EXPANSION WAVESSOLO
For a calorically Perfect Gas
( )
( )
( )
( )[ ]
( )[ ]
2
1
1
2
1
2
2
1
2
12
2
2
1
1
2
2
1
2
1
1
2
11/2
1/2
1
1
2
1
21
1
ρ
ρ
γγ
γ
γ
γ
γ
γ
ρ
ρ
p
p
T
T
M
M
M
M
p
p
M
M
n
n
n
n
n
n
=
−−
−+
=
−
+
+=
+−
+
=
βsin11 MMn =
( )θβ −
=
sin
2
2
nM
M
Now we can compute
( )
( ) ( )
( )
( )
( )






⋅+
−
=
−
+
−+
===
−
⇒









=
=−
=
θββ
θβ
β
θβ
βγ
βγ
ρ
ρ
β
θβ
θβ
β
tantan1tan
tantan
tan
tan
sin1
sin12
tan
tan
tan
tan
22
1
22
1
2
1
1
2
12
2
2
1
1
M
M
u
u
ww
w
u
w
u

34. 34
OBLIQUE SHOCK EXPANSION WAVESSOLO
( ) 





++
−
=
22cos
1sin
cot2tan 2
1
22
1
βγ
β
βθ
M
M
M,, βθ relation
12 M
12 M
.5max =Mforθ
β θ
1M 2M
Strong Shock
Weak Shock
θ
β
We can see that θ = 0 for
1.β = 90° (Normal Shock)
2.sin β = 1/ M1

35. 35
OBLIQUE SHOCK EXPANSION WAVESSOLO
1. For any given M1 there is a maximum deflection angle θmax
If the physical geometry is such that θ θmax, then no solution
exists for straight oblique shock wave. Instead the shock will be
curved and detached.

36. 36
OBLIQUE SHOCK EXPANSION WAVESSOLO
2.For any given θ θmax, there are two values of β predicted by
the θ-β-M relation for a given Mach number.
WEAKβ
STRONGβ
( ) 





++
−
=
22cos
1sin
cot2tan 2
1
22
1
βγ
β
βθ
M
M
M,, βθ relation
- the large value of β is called the strong shock solution
In nature the weak shock solution usually occurs.
- the small value of β is called the weak shock solution
- in the strong shock solution M2 is subsonic (M2 1)
- in the weak shock M2 solution is supersonic (M2 1)

37. 37( ) 





++
−
=
22cos
1sin
cot2tan 2
1
22
1
βγ
β
βθ
M
M
M,, βθ relation
SOLO
OBLIQUE SHOCK EXPANSION WAVES
θ
β
4.1=γ
θ
maxθ
θ

38. 38
( )[ ]
( )[ ]
( )θβ
γγ
γ
β
−
=
−−
−+
=
=
sin
11/2
1/2
sin
2
2
2
1
2
12
2
11
n
n
n
n
n
M
M
M
M
M
MM
SOLO
θ
maxθ
OBLIQUE SHOCK EXPANSION WAVES
Mach Number in Back of Oblique Shock M2 as a Function of the Mach Number
in Front of the Shock M , for Different Values of Deflection Angle θ (γ=1.4)

39. 39
( )1
1
2
1
sin
2
1
1
2
11
−
+
+=
=
n
n
M
p
p
MM
γ
γ
β
SOLO
θ
θ
OBLIQUE SHOCK EXPANSION WAVES
Static Pressure Ratio P2
/
P1 as a Function of M1 the Mach Number
in Front of an Oblique Shock, for Different Values of Deflection Angle θ (γ=1.4)

40. 40
SOLO
θ
θ
OBLIQUE SHOCK EXPANSION WAVES
Stagnation Pressure Ratio P2
0/
P1
0
as a Function of M1 the Mach Number
in Front of an Oblique Shock, for Different Values of Deflection Angle θ (γ=1.4)

41. 41
Hodograph Shock Polar
SOLO
OBLIQUE SHOCK EXPANSION WAVES

42. 42
Hodograph Shock Polar
SOLO
-For every deflection angle θ the Hodograph
gives two solutions, a strong shock (B outside
the sonic circle – M21) and a weak shock
(D inside the sonic circle – M11)
- The line OC tangent to the Hodograph gives
the maximum deflection angle θmax.
For θ θmax there is no oblique shock wave.
- For point E θ=0 and β=π/2, therefore a normal
shock. Point A corresponds to the Mach value
before the shock M1.
- The Shock Angle β corresponding to a given
angle θ defined by the points B and D can be
found by drawing the line OH normal to line
AB. β = angle HOA.
OBLIQUE SHOCK EXPANSION WAVES

43. 43
SOLO
Family of Hodograph Shock Polars ( γ= 1.4)
θ
1
***1
2
1
**
*** 21
2
1
212
21
2
2
+−





+
−






−=





c
V
c
V
c
V
c
V
c
V
c
V
c
V
c
V
x
x
xy
γ
A. H. Shapiro “The Dynamics and Thermodynamics of Compressible Flow Fluid”,pg.543
45.2
OBLIQUE SHOCK EXPANSION WAVES

44. 44
SOLO
θmaxθ
maxθ
maxθ
γ
OBLIQUE SHOCK EXPANSION WAVES

45. 45
SOLO
OBLIQUE SHOCK EXPANSION WAVES

46. 46
SOLO
OBLIQUE SHOCK EXPANSION WAVES

47. 47
SOLO
OBLIQUE SHOCK EXPANSION WAVES

48. 48
SOLO
OBLIQUE SHOCK EXPANSION WAVES

49. 49
SOLO
OBLIQUE SHOCK EXPANSION WAVES
Return to Table of Content

50. 50
SOLO
OBLIQUE SHOCK EXPANSION WAVES
Prandtl-Meyer Expansion Waves
Ludwig Prandtl
(1875 – 1953)
Theodor Meyer
(1882 – 1972)
The Expansion Fan depicted in Figure was
First analysed by Prandtl in 1907 and his
student Meyer in 1908.
Let start with an Infinitesimal Change across a
Mach Wave
M
ach
W
ave
θd
µ µ
π
−
2
θµ
π
d−−
2
V
VdV +
( )
( ) θµθµ
µ
θµπ
µπ
dddV
VdV
sinsincoscos
cos
2/sin
2/sin
−
=
−−
+
=
+
µ
θµθ
µθ tan
/
tan1
tan1
1
1
VVd
dd
dV
Vd
=⇒+≈
−
≈+
1
1
tan
1
sin
2
1
−
=⇒





= −
MM
µµ
V
Vd
Md 12
−=θ
1907 - 1908

51. 51
SOLO
OBLIQUE SHOCK EXPANSION WAVES
Prandtl-Meyer Expansion Waves (continue-1)
M
ach
W
ave
θd
µ µ
π
−
2
θµ
π
d−−
2
V
VdV +
V
Vd
Md 12
−=θ
Integrating this equation gives
∫ −=
2
1
12
M
M
V
Vd
Mθ
Using the definition of Mach Number: V = M.
a
a
ad
M
Md
V
Vd
+=
For a Calorically Perfect Gas
20
2
0
2
1
1 M
T
T
a
a −
+==




 γ
MdMM
a
ad
1
2
2
1
1
2
1
−





 −
+
−
−=
γγ
M
Md
MV
Vd
2
2
1
1
1
−
+
=
γ ∫ −
+
−
=
2
1
2
2
2
1
1
1
M
M
M
Md
M
M
γ
θ

52. 52
SOLO
OBLIQUE SHOCK EXPANSION WAVES
Prandtl-Meyer Expansion Waves (continue-2)
The integral
∫ −
+
−
=
2
1
2
2
2
1
1
1
M
M
M
Md
M
M
γ
θ
( ) ∫ −
+
−
=
M
Md
M
M
M
2
2
2
1
1
1
γ
ν
is called the Prandtl-Meyer Function and is
given the symbol ν. Performing the integration we obtain
( ) ( ) ( )1tan1
1
1
tan
1
1 2121
−−−
+
−
−
+
= −−
MMM
γ
γ
γ
γ
ν
Deflection Angle ν and Mach Angle μ as functions of Mach Number






= −
M
1
sin 1
µ
Finally
( ) ( )12 MM ννθ −=
Return to Table of Content

53. 53
Movement of Shocks with Increasing Mach Number
Drag rises due to pressure
Increase across a Shock Wave
•Subsonic Flow
- Local airspeed is less than
sonic
•Transonic Flow
- Local airspeed is less than sonic
at some points, greater than
sonic elsewhere
•Supersonic Flow
- Local Airspeed is greater
than sonic everywhere
SOLO AERODYNAMICS

54. 54
Movement of Shocks with Increasing Mach Number
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )87654321 ∞∞∞∞∞∞∞∞ MMMMMMMM
SOLO AERODYNAMICS

55. 55
Upper
Surface
Lower
Surface
Upper
Surface
Lower
Surface
Upper
Surface
Lower
Surface
Upper
Surface
Lower
Surface
Upper
Surface
Lower
Surface
( c) Shock on upper surface
Upper
Surface
Lower
Surface
(d ) Shocks on both surfaces
Shock
Movement of Shocks with Increasing Mach Number
SOLO AERODYNAMICS

56. 56
Movement of Shocks with Increasing Mach Number
The Mach Number at witch M=1 appears
on the Airfoil Upper Surface is called the
Critical Mach Number for this Airfoil.
The Critical Mach Number can be
calculated as follows. Assuming an
isentropic flow through the flow-field we
have
( )1/
2
2
2
1
1
2
1
1
−
∞
∞












−
+
−
+
=
γγ
γ
γ
A
A
M
M
p
p
p∞, M∞ - Pressure and Mach Number upstream the Airfoil
pA, MA- Pressure and Mach Number at a point A on the Airfoil
Critical Mach Number
The Pressure Coefficient Cp is computed using
( )












−












−
+
−
+
=





−=
−
∞
∞∞∞
1
2
1
1
2
1
1
2
1
2
1/
2
2
γγ
γ
γ
γγ
A
A
pA
M
M
Mp
p
M
C
Definition of Critical Mach
Number.
Point A is the location of
minimum pressure on the
top surface of the Airfoil.
SOLO AERODYNAMICS

57. 57
Movement of Shocks with Increasing Mach Number
Critical Mach Number
This relation gives a unique relation between the upstream values of p∞, M∞ and the
respective values pA, MA at a point A on the Airfoil.
Assume that point A is the point of minimum pressure, therefore maximum velocity,
on the Airfoil and that this maximum velocity corresponds to MA = 1. Then by
definition M∞ = Mcr .
( )












−












−
+
−
+
=





−=
−
∞
∞∞∞
1
2
1
1
2
1
1
2
1
2
1/
2
2
γγ
γ
γ
γγ
A
A
pA
M
M
Mp
p
M
C
( )












−












−
+
−
+
=
−
1
2
1
1
2
1
1
2
1/
2
γγ
γ
γ
γ
cr
cr
p
M
M
C cr
2
0
1 ∞−
=
M
C
C
p
p
( )












−












−
+
−
+
=
−
1
2
1
1
2
1
1
2
1/
2
γγ
γ
γ
γ
cr
cr
p
M
M
C cr
2
0
1 ∞−
=
M
C
C
p
p
To find the Mcr we need on other equation describing
Cp at subsonic speeds. We can use the
Prandtl-Glauert Correction
or the Karman-Tsien Rule or
Laiton’s Rule
SOLO AERODYNAMICS

58. 58
Movement of Shocks with Increasing Mach Number
Critical Mach Number
AirfoilThickAirfoilMediumAirfoilThin
AirfoilThickAirfoilMediumAirfoilThin
crcrcr
ppp
MMM
CCC
000
The point of minimum pressure, therefore maximum velocity, does not correspond
to the point of maximum thickness of the Airfoil. This is because the point of
minimum pressure is defined by the specific shape of the Airfoil and not by a local
property.
The Critical Mach Number is a function of
the thickness of the Airfoil.
For the thin Airfoil the Cp0 is smaller in
magnitude and because the disturbance in the
Flow is smaller. Because of this the Critical
Mach Number of the thin Airfoil is greater
SOLO AERODYNAMICS

59. 59
Movement of Shocks with Increasing Mach Number
Drag Divergence Mach Number
The Drag at small Mach number, due to
Profile Drag with Induced Drag =0 (αi = 0)
is constant (points a, b, and c) until
M∞ = Mcr (point c). As the velocity
increase above Mcr (point d), a finite
region of supersonic flow (Weak Shock
boundary)appears on the Airfoil.
The Mach Number in this bubble of
supersonic flow is slightly above Mach 1,
typically 1.02 to 1.05. If M∞ increases more,
We encounter a point, e, at which is a sudden increase in Drag. The Value of M∞ at
which the sudden increase in Drag starts is defined as the Drag-divergence Mach
Number, Mdrag-divergence 1. At this point Shock Waves appear on the Airfoil. The
Shock Waves are dissipative phenomena extracting energy (Drag) from the kinetic
energy of the Airfoil. In addition the sharp increase of the pressure across the
Shock Wave create a strong adverse pressure gradient, causing the Flow to
separate
From the Airfoil Surface creating Drag increase. Beyond the Drag-divergence
Mach Number, the Drag Coefficient becomes very large, increasing by a factor of
10 or more. As M∞ approaches unity (point f) the Flow on both the top and the
SOLO AERODYNAMICS

60. 60
Movement of Shocks with Increasing Mach Number
Summary of Airfoil Drag
The Drag of an Airfoil can be described as the sum of three contributions:
wpf DDDD ++=
where
D – Total Drag of the Airfoil
Df – Skin Friction Drag
Dp – Pressure Drag due to Flow Separation
Dw – Wave Drag (present only at Transonic and Supersonic Speeds; zero for
Subsonic Speeds below the Drag-divergence Mach Number)
In terms of the Drag Coefficients, we can write:
wDpDfDD CCCC ,,, ++=
The Sum:
pDfD CC ,, + Profile Drag Coefficient
SOLO AERODYNAMICS

61. 61
Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2
SOLO
Return to Table of Content
AERODYNAMICS

62. 62
AERODYNAMICS
Drag Variation with Mach Number
SOLO
Return to Table of Content

63. 63
AERODYNAMICS
Swept Wings Drag Variation
Adolf Busemann and Alfred Betz, discovered around 1930 that Drag at Transonic
and Supersonic Speeds could be reduced using Swept Back Wings.
Assume Mcr for
Wing = 0.7
Airfoil Section
with Mcr = 0.7
Airfoil Section
with Mcr = 0.7
Airfoil ³ sees´
only this
component of
velocity
Mcr for swept wing
Adolph
Busemann
(1901 – 1986)
also NACA
Colorado U.
Albert Betz
(1885 – 1968 ),
Λ
=
cos
_
cr
sweptcr
M
M
From the Figure we see that
if Λ is the Swept Angle, than
Supersonic L.E.
Subsonic L.E.
Mach Cone
For Supersonic Flow M∞ 1
•If the Leading Edge of Swept Wing is outside
the Mach Cone, the component of the Mach
Number normal to the Leading Edge is
Supersonic. As a result a Strong Oblique Shock
Wave will be created on the Wing.
•If the Leading Edge of Swept Wing is inside
the Mach Cone, the component of the Mach
Number normal to the Leading Edge is
Subsonic. As a result a Weaker Oblique Shock
Wave will be created on the Wing
and a Lower Drag will result.
SOLO

64. 64
SOLO Wings in Compressible Flow
64
Swept Wings
The Swept Wing Theory was first presented by Adolf Busemann at the Fifth Volta Conference
in Roma 1935. Busemann made use of so called
“Independence Principle”:
“The air forces on a sufficient long, narrow Wing Panel are
independent of the component of the flight velocity in the
direction of the Wing Leading Edge (disregarding friction).
The air forces the depend only on the reduced component
velocity perpendicular to the Wing Leading Edge”
Adolph
Busemann
(1901 – 1986).
The Wing angles relative to Flow Direction are:
α – Angle of Attack
Λ – Swept Angle
The Flow Mach components are:
forcesairaffectingnotELtoparallelM
forcesairaffecting
PlaneWingtheinELtonormalM
PlaneWingtonormalM
..sincos
..coscos
sin
Λ






Λ
∞
∞
∞
α
α
α
We have:
( ) ( )[ ] ( )
Λ
=
Λ
=
Λ=






Λ
=





Λ
=
Λ−=Λ+=
−
∞
∞−
∞∞∞∞
coscos
:
cos:
cos
tan
tan
coscos
sin
tan:
cossin1coscossin:
11
2/1222/122
τ
τ
α
α
α
α
ααα
c
t
cc
M
M
MMMM
e
e
e
e Section A-A
Section B-B

65. 65
SOLO Wings in Compressible Flow
65
Swept Wings
Section A-A
Section B-B
( ) bcM
L
CL 2
2/ ∞∞
=
ργ
The Total Lift is:
( ) ( )[ ] ( )
Λ
=
Λ
=
Λ=






Λ
=





Λ
=
Λ−=Λ+=
−
∞
∞−
∞∞∞∞
coscos
:
cos:
cos
tan
tan
coscos
sin
tan:
cossin1coscossin:
11
2/1222/122
τ
τ
α
α
α
α
ααα
c
t
cc
M
M
MMMM
e
e
e
e
Therefore: ( ) ( )α222
cossin1/ Λ−== ∞∞ eLeeLL CMMCC
( ) ( ) ( )ΛΛ
==
∞∞∞∞ cos/cos2/2/ 22
bcM
L
bcM
L
C
eeee
eL
ργργ
and:
The Friction Drag is ignored the Tangential Component of Velocity does not
contribute to the Drag and the Pressure Drag is normal to the Leading Edge.
If D is the Total Pressure Drag the component in the M∞ direction is only D cosΛ.
( ) ( ) ( ) ( )ΛΛ
=
Λ
=
∞∞∞∞ cos/cos2/
;
2/
cos
22
bcM
D
C
bcM
D
C
e
DD
ργργ
or: ( ) ( )α222
cossin1cos/ Λ−Λ== ∞∞ eDeeDD CMMCC

66. 66
SOLO Wings in Compressible Flow
66
Swept Wings
Oblique Wing aircraft, AD-1 was built and flown by NASA..
Oblique Wing concept was developed in the USA by
R.T. Jones.
Robert Thomas Jones
(1910–1999)
Oblique Wing Flight Demonstration by the AD-1.

67. 67
AERODYNAMICS
Swept Wings Drag Variation
SOLO

68. 68
Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2
AERODYNAMICS
Swept Wings Drag Variation
SOLO

69. 69
AERODYNAMICS
Swept Wings Drag Variation
SOLO

70. 70
AERODYNAMICS
Swept Wings Drag Variation
Comparison of the Transonic Drag Polar for an Unswept Wing with that for a
Swept Wing (data from Schlichting)
SOLO

71. 71
SOLO Wings in Compressible Flow
Profile Drag Coefficients versus Mach Number for an Un-swept and a Swept-back Wing
(φ=45°), t/c=0.12, AR=4
Swept Wings

72. 72
AERODYNAMICS
Swept Wings Drag Variation
SOLO

73. 73
SOLO
Return to Table of Content
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane

74. 74
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLO

75. 75
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLO

76. 76
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLO

77. 77
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLO
Return to Table of Content

78. 78
SOLO
Return to Table of Content
Ray Whitford, “Design for Air Combat”

79. 79
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLO
Return to Table of Content

80. 80
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
Richard T. Whitcomb
(1921 – 2009)
SOLO

81. 81
German aerodynamicist named Dr. Adolf Busemann, who had come to work at
Langley after World War II, gave a technical symposium on transonic airflows. In
a vivid analogy, Busemann described the stream tubes of air flowing over an
aircraft at transonic speeds as pipes, meaning that their diameter remained
constant. At subsonic speeds, by comparison, the stream tubes of air flowing over a
surface would change shape, become narrower as their speed increased. This
phenomenon was the converse, in a sense, of a well-known aerodynamic principle
called Bernoulli's theorem, which stated that as the area of an airflow was made
narrower, the speed of the air would increase. This principle was behind the design
of venturis,9
as well as the configuration of Langley's wind tunnels, which were
necked down in the test sections to generate higher speeds.10
But at the speed of sound, Busemarm explained, Bernoulli's theorem did not
apply. The size of the stream tubes remained constant. In working with this kind of
flow, therefore, the Langley engineers had to look at themselves as pipefitters.
Busemann's pipefitting metaphor caught the attention of Whitcomb, who was in
the symposium audience. Soon after that Whitcomb was, quite literally, sitting with
his feet up on his desk one day, contemplating the unusual shock waves he had
encountered in the transonic wind tunnel. He thought of Busemann's analogy of
pipes flowing over a wing-body shape and suddenly, as he described it later, a light
went on.
Richard T. Whitcomb
(1921 – 2009)
Adolph Busemann
(1901 – 1986)
also NACA
Colorado U.
Origin of Transonic Area Rule
http://history.nasa.gov/SP-4219/Chapter5.html
SOLO

82. 82
Richard T. Whitcomb
(1921 – 2009)
Adolph Busemann
(1901 – 1986)
also NACA
Colorado U.
Origin of Transonic Area Rule
http://history.nasa.gov/SP-4219/Chapter5.html
In practical terms, the area rule concept meant that something had to
be done in order to compensate for the dramatic increase in cross-
sectional area where the wing joined the fuselage. The simplest
solution was to indent the fuselage in that area, creating what
engineers of the time described as a Coke bottle or Marilyn
Monroe shaped design. The indentation would need to be greatest at
the point where the wing was the thickest, and could be gradually
reduced as the wing became thinner toward its trailing edge. If
narrowing the fuselage was impossible, as was the case in several
designs that applied the area rule concept, the fuselage behind or in
front of the wing needed to be expanded to make the change in
crosssectional area from the nose of the aircraft to its tail less
dramatic.
Throughout the first quarter of 1952, Whitcomb conducted a series of
experiments using various area-rule based wing-body configurations in
Langley's 8-Foot High-Speed Tunnel. As he expected, indenting the
fuselage in the area of the wing did, indeed, significantly reduce the
amount of drag at transonic speeds. In fact, Whitcomb found that
indenting the body reduced the drag-rise increments associated with the
unswept and delta wings by approximately 60 percent near the speed of
sound, virtually eliminating the drag rise created by having to put wings
on a smooth, cylindrical shaped body.
http://www.youtube.com/watch?v=xZWBVgL8I54
http://www.youtube.com/watch?v=Cn0lSoreB1g
SOLO

83. 83
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLO

84. 84
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLO

85. 85
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLO

86. 86
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLO

87. 87
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLO

88. 88
SOLO

89. 89
Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2
AERODYNAMICSSOLO
Return to Table of Content

90. 90
AERODYNAMICSSOLO

91. 91
Nguyen X. Vinh, “Flight Mechanics of High Performance Aircraft”, Cambridge University,1993
AERODYNAMICSSOLO

92. 92
Examples of airfoils in nature and within various vehicles
Lift and Drag curves for a typical airfoil
SOLO

93. 93
SOLO

94. 94
AERODYNAMICSSOLO

95. 95
Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2
AERODYNAMICSSOLO

96. 96
Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2
AERODYNAMICSSOLO
Return to Table of Content

97. 97
Flow regimes for a slender bodySOLO

98. 98
)A)- Flow field in wing-tail plane, influence of angle of attackSOLO

99. 99
Slender wing-body combinationSOLO

100. 100
))B)- Flow field in wing-tail plane, influence ofB)- Flow field in wing-tail plane, influence of
control deflectioncontrol deflection δδ for pitchfor pitch
SOLO

101. 101
Missile controlMissile controlSOLO

102. 102
CHARACTERISTICSCHARACTERISTICS
Summary of aerodynamic designSummary of aerodynamic design
SOLO

103. 103
))C)- Flow field in wing-tail plane, influence ofC)- Flow field in wing-tail plane, influence of
control deflectioncontrol deflection ξξ for rollfor roll
SOLO

104. 104
Types of missile roll control skid-to-turn, bank-to-turnTypes of missile roll control skid-to-turn, bank-to-turnSOLO
Return to Table of Content

105. 105
Density Profile Mach 1.2, Color Contours Modified to see Detail on Shock Waves
More Fun With CFD – RM-10SOLO

106. 106
Density Profiles, Mach 2.41, simulated altitude of 11,000 ft )Re=76.4x106
)
More Fun With CFD – RM-10SOLO

107. 107
Density Profiles, Mach 2.41 – color contours modified to see detail in shock waves
More Fun With CFD – RM-10SOLO

108. 108
Density Profiles, Mach 1.62 – rotated, with plot to show distribution around fins
More Fun With CFD – RM-10SOLO

109. 109
The Effect of Leading Edge Slat, Flap, and Trailing Edge Flap
Upon Angle of Attack of Basic Wing
Darrol Stinton “ The Design of the Aircraft”SOLO

110. 110High Angles of Attack Flows
)Development of a High Resolution CFD)
SOLO

111. 111High Angles of Attack Flows
)Development of a High Resolution CFD)
SOLO

112. 112
Three-Element Airfoil
Pressure Coefficient and Streamlines at Maximum Lift M=0.2 )Re=4.1x106
)
SALSA Computation
AERODYNAMICSSOLO

113. 113Inviscid Transonic Flow Solution Over a 2-D Airfoil at M=0.75 )Re=1000)
AERODYNAMICSSOLO

114. 114Inviscid Supersonic Flow Solution Over a 2-D Airfoil at M=1.50 )Re=1000)
AERODYNAMICSSOLO

115. 115
AERODYNAMICSSOLO

116. 116
Linearized Flow Equations
1. Irrotational Flow
SOLO
Assumptions
2. Homentropic
3. Thin bodies
( )0

=×∇ u






=
∂
∂
=∇ 00..;.
t
s
seieverywhereconsts
This implies also inviscid flow ( )~τ = 0
Changes in flow velocities due to body presence are small
were
- flow velocity as a function of position and time
- flow entropy as a function of position and time
( )tzyxu ,,,

( )tzyxs ,,,

117. 117
SOLO
)C.L.M)
For an inviscid flow conservation of linear momentum gives:( )~τ = 0
Assume that body forces are conservative and stationary
were
- flow pressure as a function of position and time( )tzyxp ,,,
- flow density as a function of position and time( )tzyx ,,,ρ
( ) Gpuuu
t
u
uu
t
u
tD
uD 



ρ
∂
∂
ρ
∂
∂
ρρ +−∇=





×∇×−





∇+=





∇⋅+= 2
2
1
or
( ) G
p
uuu
t
u 

+
∇
−=×∇×−





∇+
∂
∂
ρ
2
2
1 Euler’s Equation
0 =
∂
Ψ∂
Ψ−∇=
t
G

- Body forces as a function of position( )zyxG ,,

Leonhard Euler
1707-1783
Linearized Flow Equations

118. 118
SOLO
Let integrate the Euler’s Equation between two points )1) and )2)
( ) ( ) ( ) ∫∫∫∫∫∫ ⋅Ψ∇+
⋅∇
+×∇⋅×−⋅





∇+⋅
∂
∂
=⋅





Ψ∇+
∇
+×∇×−





∇+
∂
∂
=
2
1
2
1
2
1
2
1
2
2
1
2
1
2
2
1
2
1
0 rd
rdp
uurdrdurdu
t
rd
p
uuuu
t



υρ
We can chose the path of integration as follows:
- along a streamline ) and are collinear; i.e.: )rd

u

0

=×urd
- along any path, if the flow is irrotational ( )0

=×∇ u
to obtain:
( ) ( ) 0
2
1
=×∇⋅×∫ uurd

Assuming that the flow is irrotational we can define a potential ,
such that:
( )0

=×∇ u ( )tr ,

Φ
Φ∇=u

Let use the identity
to obtain:
( ) rdFtrFd constt

⋅∇==
,
( )
2
1
2
2
1
2
2
1
2
1
0








Ψ+++
∂
Φ∂
=





Ψ∇++





+Φ
∂
∂
= ∫∫
∞
p
p
pd
u
t
pd
udd
t ρρ
Bernoulli’s Equation
for Irrotational
and Inviscid Flow
Daniel Bernoulli
1700-1782
Linearized Flow Equations

119. 119
SOLO
For an isentropic ideal gas we have
2
2
11 a
ad
T
Tdd
p
pd
−
=
−
==
γ
γ
γ
γ
ρ
ρ
γ
where
ρ
γ
γ
ρρ
p
TR
d
pdp
a
s
===
∂
∂
=2
is the square of the speed of sound
In this case
2
2
2
1
1
1 2
ad
a
adppd
RTa
RTp
−
=
−
=
=
=
γργ
γ
ρ γ
ρ
and
[ ]222
1
1
1
1
2
2
∞−
−
=
−
= ∫∫
∞∞
aaad
pd
a
a
p
p
γγρ
Using the Bernoulli’s Equation we obtain
( ) ( ) ( ) ( )





Ψ−Ψ+−+
∂
Φ∂
−−=−=− ∞∞∞ ∫
∞
2222
2
1
11 Uu
t
dp
aa
p
p
γ
ρ
γ
( )
2
1
2
2
1
2
2
1
2
1
0








Ψ+++
∂
Φ∂
=





Ψ∇++





+Φ
∂
∂
= ∫∫
∞
p
p
pd
u
t
pd
udd
t ρρ
Bernoulli’s Equation
for Irrotational
and Inviscid Flow
Linearized Flow Equations

120. 120
SOLO
Let use the conservation of mass )C.M.) equation
)C.M.) 0=⋅∇+ u
tD
D 
ρ
ρ
or
tD
D
u
ρ
ρ
1
−=⋅∇

Let go back to Bernoulli’s Equation ( ) ( )





Ψ−Ψ+−+
∂
Φ∂
−= ∞∞∫
∞
22
2
1
Uu
t
pd
p
p
ρ
and use the Leibnitz rule of differentiation: ( ) ( )uxFdxuxF
xd
d
x
x
,,
0
=∫
to obtain
ρρ
1
=∫
∞
p
p
pd
pd
d
Now we can compute tD
Da
tD
D
d
pd
tD
pD
tD
pDpd
pd
dpd
tD
D
p
p
p
p
ρ
ρ
ρ
ρρρρρ
2
11
===








= ∫∫
∞∞
Therefore ( ) ( )





Ψ−Ψ+−+
∂
Φ∂
=−=−=⋅∇ ∞∞∫
∞
22
22
2
1111
Uu
ttD
D
a
pd
tD
D
atD
D
u
p
p
ρ
ρ
ρ

Since ( )[ ] 0=Ψ−Ψ= ∞∞
tD
D
u
tD
D
we have












∇⋅+
∂
∂
⋅+
∂
Φ∂
=











∇⋅+
∂
Φ∂
∇⋅+
∂
∂
⋅+
∂
Φ∂
=
=





+
∂
Φ∂






∇⋅+
∂
∂
=





+
∂
Φ∂
=⋅∇
Φ∇=
2
2
1
2
1
2
11
2
11
2
2
2
2
2
2
2
2
2
2
2
2
u
u
t
u
u
ta
u
u
t
u
t
u
u
ta
u
t
u
ta
u
ttD
D
a
u
u 






GOTTFRIED WILHELM
von LEIBNIZ
1646-1716
Linearized Flow Equations

121. 121
SOLO












∇⋅+
∂
∂
⋅+
∂
Φ∂
=











∇⋅+
∂
Φ∂
∇⋅+
∂
∂
⋅+
∂
Φ∂
=
=





+
∂
Φ∂






∇⋅+
∂
∂
=





+
∂
Φ∂
=⋅∇
Φ∇=
2
2
1
2
1
2
11
2
11
2
2
2
2
2
2
2
2
2
2
2
2
u
u
t
u
u
ta
u
u
t
u
t
u
u
ta
u
t
u
ta
u
ttD
D
a
u
u 






Let substitute Φ∇=u













Φ∇⋅Φ∇∇⋅Φ∇+Φ∇
∂
∂
⋅Φ∇+
∂
Φ∂
=Φ∇⋅∇
2
1
2
1
2
2
2
tta
( ) ( ) ( )





Ψ−Ψ+−Φ∇⋅Φ∇+
∂
Φ∂
−−= ∞∞∞
222
2
1
1 U
t
aa γ
Special cases
0≈Φ∇⋅∇ Laplace’s equation
∞∞ Ua )subsonic flow) we can approximate
the first equation by
1
2 ( ) ( ) 2
2
t
uu
t
uuu
∂
Φ∂
⋅
∂
∂
+⋅∇⋅
 we can approximate
the first equation by
0
1
2
2
2
=
∂
Φ∂
−Φ∇⋅∇
ta
Wave equation
Pierre-Simon
Laplace
1749-1827
Linearized Flow Equations

122. 122
SOLO
Note
The equation






+
∂
Φ∂






∇⋅+
∂
∂
=⋅∇ 2
2
2
11
u
t
u
ta
u

can be written as
Φ=





Φ∇⋅+
∂
Φ∂






∇⋅+
∂
∂
=





+
∂
Φ∂






∇⋅+
∂
∂
=Φ∇ 2
2
22
2
2
2 11
2
11
tD
D
a
u
t
u
ta
u
t
u
ta
c
c

where the subscript c on and on is intended to indicate that the velocity is
treated as a constant during the second application of the operators and .
cu

2
2
tD
Dc
t∂∂/ ( )∇⋅u

This equation is similar to a wave equation.
End Note
Linearized Flow Equations

123. 123
SOLO
Let compute the local pressure coefficient: 2
2
1
:
∞∞
∞−
=
U
pp
C p
ρ
We have:










−







=










−







=










−





=





−=
−
∞∞
=−
∞
∞
∞
=
−
∞
∞
∞






=
=
∞
∞
∞
∞
∞∞∞
−
∞∞
∞∞∞
1
2
1
2
1
1
2
1
2
1
2
2
2
/1
2
2
2
2
1
22
2
1
γ
γ
γ
γ
γ
γ
γ
ρ
γ
γ
ρ γ
γ
a
a
Ma
a
a
U
T
T
U
TR
p
p
U
p
C
aUMTRa
T
T
p
p
TRp
p
Let use the equation
( ) ( ) ( )





Ψ−Ψ+−Φ∇⋅Φ∇+
∂
Φ∂
−−= ∞∞∞
222
2
1
1 U
t
aa γ
to compute
( ) ( ) ( )





Ψ−Ψ+−Φ∇⋅Φ∇+
∂
Φ∂−
−= ∞∞
∞∞
2
22
2
2
11
1 U
taa
a γ
Finally we obtain:
( ) ( ) ( )










−












Ψ−Ψ+−Φ∇⋅Φ∇+
∂
Φ∂−
−=
−
∞∞
∞∞
1
2
11
1
2 1
2
22
γ
γ
γ
γ
U
taM
Cp
Linearized Flow Equations

124. 124
SOLO
Assuming a stationary flow and neglecting the body forces :





=
∂
∂
0
t
( )0=Ψ












Φ∇⋅Φ∇∇⋅Φ∇=Φ∇⋅∇
2
11
2
a
( ) ( )222
2
1
∞∞ −Φ∇⋅Φ∇
−
−= Uaa
γ
( ) ( )










−






−Φ∇⋅Φ∇
−
−=
−
∞
∞∞
1
2
1
1
2 1
2
22
γ
γ
γ
γ
U
aM
Cp
Φ∇=u

Linearized Flow Equations

125. 125
SOLO
1
0
332211
323121
=⋅=⋅=⋅
=⋅=⋅=⋅
→→→→→→
→→→→→→
eeeeee
eeeeee
General Coordinates ( )321 ,, uuu
→→→
∂
Φ∂
+
∂
Φ∂
+
∂
Φ∂
=Φ∇ 3
33
2
22
1
11
111
e
uh
e
uh
e
uh
( ) ( ) ( )





∂
∂
+
∂
∂
+
∂
∂
=






++⋅∇=⋅∇
→→→
321
3
213
2
132
1321
332211
1
Ahh
u
Ahh
u
Ahh
uhhh
eAeAeAA

Using we obtainΦ∇=:A













∂
Φ∂
∂
∂
+





∂
Φ∂
∂
∂
+





∂
Φ∂
∂
∂
=
=Φ∇⋅∇=Φ∇
33
21
322
13
211
32
1321
2
1
uh
hh
uuh
hh
uuh
hh
uhhh
where
We have for ( ) ( )321321 ,,,,, uuuAuuu

Φ
Linearized Flow Equations

126. 126
SOLO
zzyyxx Φ+Φ+Φ=Φ∇=Φ∇⋅∇ 2






Φ+Φ+Φ∇⋅





Φ+Φ+Φ=





Φ∇⋅Φ∇∇⋅Φ∇
→→→
222
2
1
2
1
2
1
111
2
1
zyxzyx zyx
( ) ( )
( )=ΦΦ+ΦΦ+ΦΦΦ+
ΦΦ+ΦΦ+ΦΦΦ+ΦΦ+ΦΦ+ΦΦΦ=
zzzyzyxzxz
yzzyyyxyxyxzzxyyxxxx
yzzyxzzxxyyxzzzyyyxxx ΦΦΦ+ΦΦΦ+ΦΦΦ+ΦΦ+ΦΦ+ΦΦ= 222
22












Φ∇⋅Φ∇∇⋅Φ∇=Φ∇⋅∇
2
11
2
a
( ) ( )222
2
1
∞∞ −Φ∇⋅Φ∇
−
−= Uaa
γ
( ) 0
12
2
22111
222
222
2
2
2
2
2
=Φ−ΦΦ+ΦΦ+ΦΦ−Φ
ΦΦ
−
Φ
ΦΦ
−Φ
ΦΦ
−Φ






 Φ
−+Φ







 Φ
−+Φ






 Φ
−
ttztzytyxtxyz
zy
xz
zx
xy
yx
zz
z
yy
y
xx
x
aaa
aaaaa
( ) ( ) ( )





Ψ−Ψ+−Φ+Φ+Φ+
∂
Φ∂
−−= ∞∞∞
222222
2
1
1 U
t
aa zyxγ
We finally obtain
Cartesian Coordinates ( )zuyuxu === 321 ,,
Linearized Flow Equations
Return to Table of Content

127. 127
SOLO
Cylindrical Coordinates ( )θ=== 321 ,, uruxu
→→→→→→
++=++= zryrxxzzyyxxR 1sin1cos1111 θθ

→→→→→
+−=
∂
∂
+=
∂
∂
=
∂
∂
zryr
R
zy
r
R
x
x
R
1cos1sin1sin1cos1 θθ
θ
θθ

r
R
h
r
R
h
x
R
h =
∂
∂
==
∂
∂
==
∂
∂
=
θ

:1:1: 321
→→→→
→→→→→→
=+−=
∂
∂
∂
∂
=
=+=
∂
∂
∂
∂
==
∂
∂
∂
∂
=
θθθ
θ
θ
θθ
11cos1sin:
11sin1cos:1:
2
21
zy
R
R
e
rzy
r
R
r
R
ex
x
R
x
R
e






1
0
332211
323121
=⋅=⋅=⋅
=⋅=⋅=⋅
→→→→→→
→→→→→→
eeeeee
eeeeee
We have
Linearized Flow Equations

128. 128
SOLO
Cylindrical Coordinates )continue – 1) ( )θ=== 321 ,, uruxu
→→→→→→
Φ+Φ+Φ=
∂
Φ∂
+
∂
Φ∂
+
∂
Φ∂
=Φ∇ 321321
11
e
r
eee
r
e
r
e
x
rx θ
θ
2
2
22 1
θΦ+Φ+Φ=Φ∇⋅Φ∇
r
rx
→
→→






ΦΦ+ΦΦ+ΦΦ+






Φ−ΦΦ+ΦΦ+ΦΦ+





ΦΦ+ΦΦ+ΦΦ=






Φ+Φ+Φ∇=





Φ∇⋅Φ∇∇
322
2
2
3212
2
2
22
11
111
1
2
1
2
1
e
rr
e
rr
e
r
r
rrxx
rrrrxrxxrxrxxx
rx
θθθθθ
θθθθθ
θ
θθ
θ
θθ
Φ+Φ+Φ+Φ=
∂
Φ∂
+
∂
Φ∂
+
∂
Φ∂
+
∂
Φ∂
=












∂
Φ∂
∂
∂
+





∂
Φ∂
∂
∂
+





∂
Φ∂
∂
∂
=
=Φ∇⋅∇=Φ∇
22
2
22
2
2
2
2
1111
11
rrrrrrx
rr
r
rx
r
xr
rrrxx
Linearized Flow Equations

129. 129
SOLO
Cylindrical Coordinates )continue – 2) ( )θ=== 321 ,, uruxu
Then equation 











Φ∇⋅Φ∇∇⋅Φ∇+Φ∇
∂
∂
⋅Φ∇+
∂
Φ∂
=Φ∇⋅∇
2
1
2
1
2
2
2
tta
becomes
( ){












ΦΦ+ΦΦ+ΦΦ+






Φ−ΦΦ+ΦΦ+ΦΦ+









ΦΦ+ΦΦ+ΦΦ





Φ+Φ+Φ+
ΦΦ+ΦΦ+ΦΦ+Φ=Φ+Φ+Φ+Φ
→
→
→→→→
322
2
2
32
12321
22
11
11
11
2
111
e
rr
e
rr
e
r
e
r
ee
arr
rrxx
rrrrxrx
xrxrxxxrx
ztzytyxtxttrrrxx
θθθθθ
θθθ
θθθ
θθ
or
( ) 0
2
112
/
1
1/
1
1
11
22
222
2
22
2
22
22
2
2
2
=ΦΦ+ΦΦ+ΦΦ−
Φ
−






ΦΦΦ+ΦΦΦ+ΦΦΦ−







 Φ
+Φ+Φ






 Φ
−+Φ






 Φ
−+Φ






 Φ
−
ztzytyxtx
tt
rrxxrxrx
rrr
r
xx
x
aa
rra
a
r
ra
r
raa
θθθθ
θ
θθ
θ
Linearized Flow Equations

130. 130
SOLO
Cylindrical Coordinates )continue – 3) ( )θ=== 321 ,, uruxu
becomes
( ) ( ) ( )





Ψ−Ψ+−Φ∇⋅Φ∇+
∂
Φ∂
−−= ∞∞∞
222
2
1
1 u
t
aa γ
In cylindrical coordinates, equation
( ) ( )





Ψ−Ψ+





−Φ+Φ+Φ+Φ−−= ∞∞∞
22
2
2222 1
2
1
1 U
r
aa rxt θγ
Assuming a stationary flow and neglecting body forces





=
∂
∂
0
t
( )0=Ψ
0
112
/
1
1/
1
1
11
222
2
22
2
22
22
2
2
2
=





ΦΦΦ+ΦΦΦ+ΦΦΦ−







 Φ
+Φ+Φ






 Φ
−+Φ






 Φ
−+Φ






 Φ
−
rrxxrxrx
rrr
r
xx
x
rra
a
r
ra
r
raa
θθθθ
θ
θθ
θ
( )






−Φ+Φ+Φ
−
−= ∞∞
22
2
2222 1
2
1
U
r
aa rx θ
γ
Linearized Flow Equations
Return to Table of Content

131. 131
Linearized Flow EquationsSOLO
Boundary Conditions
1. Since the Small Perturbations are not
considering the Boundary Layer the
Flow must be parallel at the Wing
Surface.
The Wing Surface S is defined by
zU )x,y) – Upper Surface
zL )x,y) – Lower Surface
0

=⋅ S
un
n

- Normal at the Wing Surface
22
1/111 





∂
∂
+





∂
∂
+





+
∂
∂
−
∂
∂
−=
y
z
x
z
zy
y
z
x
x
z
n UUUU
U

( ) ( ) ( ) ( ) zwUyvxuUzwUyvxuUu 1'1'1'1'sin1'1'cos ++++≅++++= ∞∞∞∞ ααα

( ) ( ) 0,,''' =++
∂
∂
−
∂
∂
+− ∞∞ U
UU
zyxwU
x
z
v
x
z
uU α
For Upper Surface
( ) ( ) 





−
∂
∂
≅
∂
∂
+
∂
∂
+= ∞∞ α
x
z
U
x
z
v
x
z
uUzyxw U
onPerturbati
Small
UU
U '',,'
Therefore
( )
( )
( ) Sonyxallfor
x
z
Uzyxw
x
z
Uzyxw
L
L
U
U
,
,,'
,,'













−
∂
∂
≅






−
∂
∂
≅
∞
∞
α
α
Section AA
(enlarged)
Wake region

132. 132
Linearized Flow EquationsSOLO
Boundary Conditions )continue -1)
1. Flow must be parallel at the Wing Surface.
The Wing Surface S is defined by
zU )x,y) – Upper Surface
zL )x,y) – Lower Surface
Since the Small Perturbation gives
Linear Equation we can divide the
Airfoil in the Camber Distribution zC )x,y)
and the Thickness Distribution zt )x,y) by:
( )
( )
( ) Sonyxallfor
x
z
Uyxw
x
z
Uyxw
C
C
t
t
,
0,,'
0,,'













−
∂
∂
=
∂
∂
±=±
∞
∞
α
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )[ ]
( ) ( ) ( )[ ]


−=
+=
⇔



−=
+=
2/,,,
2/,,,
,,,
,,,
yxzyxzyxz
yxzyxzyxz
yxzyxzyxz
yxzyxzyxz
LUt
LUC
tCL
tCU
Because of the Linearity the complete solution can be obtained by summing the
Solutions for the following Boundary Conditions
Superposition of
• Angle of Attack
•Camber Distribution
•Thickness Distribution
Section AA
(enlarged)
Wake region
( ) ( ) ( )
( ) ( ) ( )
( ) Sonyxallfor
x
z
x
z
Uyxwyxwyxw
x
z
x
z
Uyxwyxwyxw
tC
tCL
tC
tCU
,
0,,'0,,'0,,'
0,,'0,,'0,,'













∂
∂
−−
∂
∂
=−+=±






∂
∂
+−
∂
∂
=++=±
∞
∞
α
α

133. 133
Linearized Flow EquationsSOLO
Boundary Conditions (continue -2)
2. Disturbances Produced by the Motion
must Die Out in all portion of the Field
remote from the Wing and its Wake
Normally this requirement is met by making
ϕ→0 when y→ ±0, z → ±0, x→-∞
Subsonic Leading
Edge Flow
Subsonic Trailing
Edge Flow
Supersonic Leading
Edge Flow
Supersonic Trailing
Edge Flow
3. Kutta Condition at the Trailing Edge of a
Steady Subsonic Flow
There cannot be an infinite change in velocity at the
Trailing Edge. If the Trailing Edge has a non-zero
angle, the flow velocity there must be zero. At a cusped
Trailing Edge, however, the velocity can be non-zero
although it must still be identical above and below the
airfoil. Another formulation is that the pressure must
be continuous at the Trailing Edge.
http://nylander.wordpress.com/category/engineering/
Kutta Condition does not apply to Supersonic
Flow since the shape and location of the
Trailing Edge exert no influence on the flow
ahead.

134. 134
SOLO
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
'2u
∞+Uu '1∞U
( )
( )
'
'
'2'
'
'0
''2'0
'
222
1
33
2
11
22
22
11
ρρρ
φ
+=
+=
+≈+=
+=Φ
+=
++=+=
+=
∞
∞
∞∞∞
∞
∞∞
∞
ppp
aaaaaa
xU
uu
uuUUuuu
uUu
O
Small Perturbation Assumptions:












∇⋅+
∂
∂
⋅+
∂
Φ∂
=⋅∇
2
2
1 2
2
2
2
u
u
t
u
u
ta
u



(C.M.) +(C.L.M)
(C.M.) +(C.L.M)
12
1
12
1
2
2
2
2
−
+=
−
++
∂
∂ ∞
∞
γγ
φ a
U
a
u
t
Bernoulli
121 −
∞
−
∞∞∞






=





=





=
γ
γ
γ
γ
γ
ρ
ρ
a
a
T
T
p
p
Isentropic Chain
Development of the Flow Equations:
Flow Equations:
( ) '' 2
1 φφ ∇=+∇⋅∇=⋅∇ ∞ xUu

( )
1
1
2
2
1
12
12
2
2
''
'
1
2
1
x
u
a
U
x
u
uU
a
u
u
a ∂
∂
≅+
∂
∂
+≅





∇⋅
∞
∞
∞
∞


( ) t
u
UuUU
tt
u
t
u
u
∂
∂
=+
∂
∂
≅
∂
∂
=
∂
∂
⋅ ∞∞∞
'
2'22 1
1
2
2

( )

∞
∞
∞
∞
∞
∞
∞∞
∞∞ ++
∂
∂
=⇒
−
+=
−
+
+++
∂
∂
ρ
γ
φ
γγ
φ
p
a
puU
t
a
U
aaa
uUU
t
2
1
2
2
2
1
2
''
'
0
12
1
1
'2
'2
2
1'
∞∞∞∞∞∞∞∞ −
=
−
==⇒
−
=
−
==
a
a
T
T
p
p
a
ad
T
Tdd
p
pd '
1
2'
1
''
1
2
1 γ
γ
γ
γ
ρ
ρ
γ
γ
γ
γ
γ
ρ
ρ
γ Isentropic Chain
Bernoulli
Linearized Flow Equations

135. 135
SOLO
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
'2u
∞+Uu '1∞U
Small Perturbation Flow Equations:
(C.M.) +(C.L.M) 52.18.00
''
2
'1
' 2
2
1
1
12
2
2
≤≤≤≤





∂
∂
+
∂
∂
+
∂
∂
=∇ ∞∞
∞
MM
tt
u
U
x
u
U
a
φ
φ
( )
''
,,,'' 321
φ
φφ
∇=
=
u
xxxt

Bernoulli 





+
∂
∂
−= ∞∞ '
'
' 1uU
t
p
φ
ρ
∞∞∞∞ −
=
−
==
a
a
T
T
p
p '
1
2'
1
''
γ
γ
γ
γ
ρ
ρ
γIsentropic Chain
Linearized Flow Equations

136. 136
SOLO
Linearized Flow Equations
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Three Dimensional Flow (Thin Wings at Small Angles of Attack)
α
U
Up
xd
ud
θ=
L
Low
xd
ud
θ−=
∞U
x
z
( ) 0
'''
1 2
2
2
2
2
2
2
=
∂
∂
+
∂
∂
+
∂
∂
− ∞
zyx
M
φφφ
(1)
( )zyx ,,'φ(2)
z
w
y
v
x
u
∂
∂
=
∂
∂
=
∂
∂
=
'
',
'
',
'
'
φφφ
(3)
α−=≅
+ ∞∞ S
xd
zd
U
w
uU
w '
'
'
(4)
x
UuUp
∂
∂
−=−= ∞∞∞∞
'
''
φ
ρρ(5)
'
2
1
1
''
1
2'
1
''
2
M
M
M
U
u
M
a
a
T
T
p
p
∞
∞
∞
∞
∞∞∞∞
−
+
−=−=
−
=
−
==
γ
γ
γ
γ
γ
γ
γ
ρ
ρ
γ(6)








∂
∂
+
∂∂
∂
+
∂
∂
=∇
∞∞∞
2
2
2
2
2
2
2
2 '1'2'1
'
tUxtUxM
φφφ
φ
( )
''
,,,''
φ
φφ
∇=
=
u
zyxt







+
∂
∂
−= ∞∞ '
'
' uU
t
p
φ
ρ
Steady Three Dimensional Flow Small Perturbation Flow Equations:
0
'
2
2
=
∂
∂
=
∂
∂
tt
52.1
8.00
≤≤
≤≤
∞
∞
M
M

137. 137
SOLO
Linearized Flow Equations
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Three Dimensional Flow (Thin Wings at Small Angles of Attack)
0
'''
2
2
2
2
2
2
2
=
∂
∂
+
∂
∂
+
∂
∂
zyx
φφφ
β(1)
Steady Three Dimensional Flow
Subsonic Flow M∞ 1
01:
22
−= ∞Mβ
( )
( )
( )
( )
α
ξ
α
φ
α
ξ
α
φ
−=−=
∂
∂
=
−=−=
∂
∂
=
∞∞
∞∞
LowerLower
Lower
UperUper
Upper
d
zd
xd
zd
zUU
w
d
zd
xd
zd
zUU
w
'1'
'1'
3
4
3
4
Transform of Coordinates
( ) ( )






=
=
=
=−= ∞
ςηξφφ
ς
η
ξβξ
,,,,'
1 2
zyx
z
y
Mx










∂
∂
=
∂
∂
⇒
∂
∂
=
∂
∂
∂
∂
=
∂
∂
⇒
∂
∂
=
∂
∂
∂
∂
=
∂
∂
⇒
∂
∂
=
∂
∂
2
2
2
2
2
2
2
2
2
2
22
2
''
''
1'1'
ς
φφ
ς
φφ
η
φφ
η
φφ
ξ
φ
β
φ
ξ
φ
β
φ
zz
yy
xx
( ) ( ) SMdcMydycS
bb
∞∞
−=−== ∫∫ 2
0
2
0
11 ηη
( ) ( )ηcMyc 2
1 ∞−=
∞∞
−
=
−
==
22
22
11 M
AR
SM
b
S
b
AR
22
1
2
1
12
∞∞∞∞ −
=
∂
∂
−
−=
∂
∂
−=
M
C
UMxU
C
p
p
ξ
φφ
Section AA
(enlarged)
Wake region
so 02
2
2
2
2
2
=
∂
∂
+
∂
∂
+
∂
∂
ς
φ
η
φ
ξ
φ
Laplace’s Equation like in Incompressible Flow
Similarity Rules

138. 138
SOLO
Linearized Flow Equations
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Three Dimensional Flow (Thin Airfoil at Small Angles of Attack)
incpC
M 2
1
1
∞−
incLC
M 2
1
1
∞−
22
1
2
1
1
∞∞ −
=





− Md
Cd
M inc
L α
α
incMC
M 2
1
1
∞−
inc0α
4
1
=





inc
N
c
x
incMC
M
02
1
1
∞−
incLsC
M 2
1
1
∞−
incsα
LsC
sα
0MC
c
xN
MC
0α
αd
Cd L
LC
pCPressure Distribution
Lift
Lift Slope
Zero-Lift Angle
Pitching Moment
Neutral-Point Position
Zero Moment
Angle of Smooth
Leading-Edge Flow
Lift Coefficient of Smooth
Leading-Edge Flow
Aerodynamic Coefficients of a Profile in Subsonic Incident Flow
Based on Subsonic Similarity Rules

139. 139
SOLO
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
α
U
Up
xd
ud
θ=
L
Low
xd
ud
θ−=
∞U
x
y
( ) 0
''
1 2
2
2
2
2
=
∂
∂
+
∂
∂
− ∞
yx
M
φφ(1)
( )yx,'φ(2)
y
v
x
u
∂
∂
=
∂
∂
=
'
',
'
'
φφ
(3)
α==≅
+ ∞∞ S
xd
yd
U
v
vU
v '
'
'
(4)
x
UuUp
∂
∂
−=−= ∞∞∞∞
'
''
φ
ρρ(5)
'
2
1
1
''
1
2'
1
''
2
M
M
M
U
u
M
a
a
T
T
p
p
∞
∞
∞
∞
∞∞∞∞
−
+
−=−=
−
=
−
==
γ
γ
γ
γ
γ
γ
γ
ρ
ρ
γ(6)






∂
∂
+
∂
∂
+
∂
∂
=∇ ∞∞
∞
2
2
1
1
12
2
2 ''
2
'1
'
tt
u
U
x
u
U
a
φ
φ
( )
''
,,,'' 321
φ
φφ
∇=
=
u
xxxt







+
∂
∂
−= ∞∞ '
'
' uU
t
p
φ
ρ
Steady Two Dimensional Flow Small Perturbation Flow Equations:
0
'
2
2
=
∂
∂
=
∂
∂
tt
52.1
8.00
≤≤
≤≤
M
M
Linearized Flow Equations

140. 140
SOLO
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
0
''
2
2
2
2
2
=
∂
∂
+
∂
∂
yx
φφ
β(1)
Steady Two Dimensional Flow
Subsonic Flow M∞ 1
01:
22
−= ∞Mβ
( )
( )
( )
( )
α
φ
α
φ
−=
∂
∂
=
−=
∂
∂
=
∞∞
∞∞
Lower
Lower
Uper
Upper
xd
yd
yUU
v
xd
yd
yUU
v
'1'
'1'
3
4
3
4
∞U
α
Transform of Coordinates
( ) ( )




=
=
=
yx
y
x
,', φβηξφ
βη
ξ











∂
∂
=
∂
∂
∂
∂
=
∂
∂
∂
∂
=





∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
=
∂
∂
∂
∂
=





∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
=
∂
∂
2
2
2
2
2
2
2
2
'
,
1'
11'
111'
η
φφ
ξ
φ
β
φ
η
φη
η
φξ
ξ
φ
β
φ
β
φ
ξ
φ
β
η
η
φξ
ξ
φ
β
φ
β
φ
yx
yyyy
xxxx
so 02
2
2
2
=
∂
∂
+
∂
∂
η
φ
ξ
φ
Laplace’s Equation like in Incompressible Flow
Linearized Flow Equations

141. 141
SOLO
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Steady Two Dimensional Flow
Subsonic Flow M∞ 1 (continue)
The Airfoil is defined in (x,y) plane and by (ξ,η)
( ) ( )ξη gxfy AirfoilAirfoil =⇔=
The above Transformation relates the
Compressible Flow over an Airfoil
in (x,y) Space to the Incompressible Flow
in (ξ,η) over the same Airfoil.
α
η
φφ
−=
∂
∂
=
∂
∂
=
∞∞∞ Uper
Upper
xd
yd
UyUU
v 1'1'
α
η
φφ
−=
∂
∂
=
∂
∂
=
∞∞∞ Lower
Lower
xd
yd
UyUU
v 1'1'
( )yx,ρρ =
x
y η
ξ
∞= ρρ
Compressible Flow Incompressible Flow
α
η
φ
−=
∂
∂
=
∞∞ Uper
Upper
xd
fd
UU
v 1'
α
η
φ
−=
∂
∂
=
∞∞ Lower
Lower
xd
fd
UU
v 1'
Linearized Flow Equations

142. 142
SOLO
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
0
'1'
2
2
22
2
=
∂
∂
−
∂
∂
yx
φ
β
φ
(1)
( ) ( ) ( )
( ) ( ) ( ) yxGyxGyx
yxFyxFyx
Lower
Upper
βννβφ
βηηβφ
+==+=
−==−=
:,'
:,'(7)
(8)
Steady Two Dimensional Flow
Supersonic Flow M∞ 1
01:
22
−= ∞Mβ
α
U
Up
xd
yd
θ=
L
Low
xd
yd
θ−=
∞U
x
y
1
1
2
−
=
∞Mxd
yd
1
1
2
−
−=
∞Mxd
yd
Flow
Flow
( )
( )
( )
η
β
α
d
Fd
Uxd
yd
U
v
Uper
Upper
∞∞
−=−=
1
7
4'
( ) ( )
η
φ
d
Fd
xd
d
u Upper
73
'
' ==








−
−
−=
∞
∞
α
Upper
Upper
xd
yd
M
U
u
1
'
2
( )
( )
( )
ν
β
α
d
Gd
Uxd
yd
U
v
Lower
Lower
∞∞
=−=
3
8
4
'
( ) ( )
ν
φ
d
Gd
xd
d
u Lower
83
'
' ==








−
−
=
∞
∞
α
Lower
Lower
xd
yd
M
U
u
1
'
2








−
−
=−=
∞
∞∞
∞∞ α
ρ
ρ
Upper
UpperUpper
xd
yd
M
U
uUp
1
''
2
2








−
−
−=−=
∞
∞∞
∞∞ α
ρ
ρ
Lower
LowerLower
xd
yd
M
U
uUp
1
''
2
2
Linearized Flow Equations

143. 143
SOLO
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Drag (D) and Lift (L) Computations
( )∫ 







−−= ∞
S S
sd
xd
yd
ppD αsin
np
α−
Upper
xd
yd
∞U
Upper
xd
yd
∞p∞p α( )∫ 







−−−= ∞
S S
sd
xd
yd
ppL αcos
( )∫ 







−−≅ ∞
S S
sd
xd
yd
ppD α
( )

Γ
∞∞∞ ∫∫ =








−−−≅
SS S
sduUsd
xd
yd
ppL 'ρα
1−α
Uper
xd
yd
1−α
Uper
xd
yd
Kutta-Joukovsky
Define: 2
2
1
:
∞∞
∞−
=
U
pp
Cp
ρ
( )
( )
∫∫
∫∫








−−=








−
−
−≅








−=








−
−
≅
∞∞
∞∞
∞
∞∞
∞∞
∞∞
∞
∞∞
S S
p
S S
S S
p
S S
sd
xd
yd
CUsd
xd
yd
U
pp
UL
sd
xd
yd
CUsd
xd
yd
U
pp
UD
αρα
ρ
ρ
αρα
ρ
ρ
2
2
2
2
2
2
2
1
2
12
1
2
1
2
12
1
Linearized Flow Equations

144. 144
SOLO
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Drag (D) and Lift (L) Computations for Subsonic Flow (M∞ 1)
np
α−
Upper
xd
yd
∞U
Upper
xd
yd
∞p∞p α
We found:
α−=
∞ xd
fd
U
v'
α
ξ
−=
∞ d
gd
U
v
( ) ( )











−=
−=
=
∞
∞
yxM
yM
x
,'1,
1
2
2
φηξφ
η
ξ
( ) 0
''
1 2
2
2
2
2
=
∂
∂
+
∂
∂
− ∞
yx
M
φφ 02
2
2
2
=
∂
∂
+
∂
∂
η
φ
ξ
φ
y
v
x
u
∂
∂
=
∂
∂
=
'
',
'
'
φφ
η
φ
ξ
φ
∂
∂
=
∂
∂
= vu ,
vv
M
u
u =
−
=
∞
',
1
'
2
'' uUp ∞∞−= ρ uUp ∞∞−= ρ
xUU
u
U
pp
Cp
∂
∂
−=−=
−
=
∞∞
∞∞
∞ '2'2
2
1
'
:
2
φ
ρ ξ
φ
ρ ∂
∂
−=−=
−
=
∞∞
∞∞
∞
UU
u
U
pp
Cp
22
2
1
:
2
0
2
1
'
∞−
=
M
p
p
2
1
0
∞−
=
M
C
C
p
p
Compressible: Incompressible:
Linearized Flow Equations

145. 145
SOLO
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Drag (D) and Lift (L) Computations for Subsonic Flow (M∞ 1)
np
α−
Upper
xd
yd
∞U
Upper
xd
yd
∞p∞p α
The Relation:
∫∫
∫∫






−
−=













−−≅














−
−
=













−≅
∞
∞∞
∞∞
∞
∞∞
∞∞
S
p
S S
p
S S
p
S S
p
c
s
dC
M
U
c
s
d
xd
yd
CUL
c
s
d
xd
yd
C
M
U
c
s
d
xd
yd
CUD
0
0
2
2
2
2
2
2
1
2
1
2
1
1
2
1
2
1
ρ
αρ
α
ρ
αρ
2
1
0
∞−
=
M
C
C
p
p
Prandtl-Glauert
Compressibility Correction
As earlier in 1922, Prandtl is quoted as stating that the Lift
Coefficient increased according to (1-M∞
2
)-1/2
; he mentioned
this at a Lecture at Göttingen, but without a proof. This result was
mentioned 6 years later by Jacob Ackeret, again without proof.
The result was finally established by H. Glauert in 1928 based on
Linear Small Perturbation.
Ludwig Prandtl
(1875 – 1953)
Hermann Glauert
(1892-1934)
Linearized Flow Equations
Return to
Critical Mach Number

146. SOLO
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Several improved formulas where developed:
( )[ ] 2/11/1 0
0
222
p
p
p
CMMM
C
C
∞∞∞ −++−
= Karman-Tsien
Rule
Linearized Flow Equations
( ) 0
0
2222
12/
2
1
11 p
p
p
CMMMM
C
C






−




 −
++−
=
∞∞∞∞
γ
Laitone’s
Rule
Comparison of several compressibility corrections
compared with experimental results for NACA 4412
Airfoil at an angle of attack of α = 1◦
.
Return to Table of Content

147. 147
SOLO
Linearized Flow Equations
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
0
'1'
2
2
22
2
=
∂
∂
−
∂
∂
zx
φ
β
φ
(1)
( ) ( ) ( )
( ) ( ) ( ) zxFzxGzx
zxFzxFzx
Lower
Upper
βννβφ
βηηβφ
+==+=
−==−=
:,'
:,'(7)
(8)
Steady Two Dimensional Flow
Supersonic Flow M∞ 1
01:
22
−= ∞Mβ
α
U
Up
xd
zd
θ=
L
Low
xd
zd
θ−=
∞U
x
z
1
1
2
−
=
∞Mxd
zd
1
1
2
−
−=
∞Mxd
zd
Flow
Flow
( )
( )
( )
η
β
α
d
Fd
Uxd
zd
U
w
Upper
Upper
∞∞
−=−=
3
7
4'
( ) ( )
η
φ
d
Fd
xd
d
u Upper
73
'
' ==
( )
( )
( )
ν
β
α
d
Gd
Uxd
zd
U
w
Lower
Lower
∞∞
==−=
3
8
4
'
( ) ( )
ν
φ
d
Gd
xd
d
u Lower
83
'
' ==








−
−
−=
∞
∞
α
Upper
Upper
xd
zd
M
U
w
1
'
2








−
−
=
∞
∞
α
Lower
Lower
xd
zd
M
U
w
1
'
2








−
−
=−==−
∞
∞∞
∞∞∞ α
ρ
ρ
Upper
UpperUpperUpper
xd
zd
M
U
wUppp
1
''
2
2








−
−
−=−==−
∞
∞∞
∞∞∞ α
ρ
ρ
Lower
LowerLowerLower
xd
zd
M
U
wUppp
1
''
2
2
z
w
x
u
∂
∂
=
∂
∂
=
'
',
'
'
φφ
(3)
α−=≅
+ ∞∞ S
xd
zd
U
w
uU
w '
'
'
(4)

148. 148
SOLO
Linearized Flow Equations
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Steady Two Dimensional Flow
Supersonic Flow M∞ 1
α
U
Up
xd
zd
θ=
L
Low
xd
zd
θ−=
∞U
x
z
1
1
2
−
=
∞Mxd
zd
1
1
2
−
−=
∞Mxd
zd
Flow
Flow
Pressure Distribution and Lift Coefficient








−+
−
=
−
=
∞∞∞
α
ρ
2
1
2
2/
''
22
LowerUpper
LowerUpper
p
xd
zd
xd
zd
MU
pp
C
1
4
2
−
=
∞M
cL
α
( ) ( ) ( ) ( )








−+−
−
−
−
=








+





−





−
−
=





+





−=
∞∞
∞
∫∫∫∫
    
00
22
1
0
1
02
1
0
1
0
00
1
2
1
4
2
1
2
LowerLowerUpperUpper
LowerUpper
ppL
zczzcz
MM
c
x
d
xd
zd
c
x
d
xd
zd
Mc
x
dC
c
x
dCc LowerUpper
α
α








−
−
=
∞
α
Upper
p
xd
zd
M
C Upper
1
2
2 







−
−
−=
∞
α
Lower
p
xd
zd
M
C Lower
1
2
2

149. 149
SOLO
Linearized Flow Equations
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Steady Two Dimensional Flow
Supersonic Flow M∞ 1
α
U
Up
xd
zd
θ=
L
Low
xd
zd
θ−=
∞U
x
z
1
1
2
−
=
∞Mxd
zd
1
1
2
−
−=
∞Mxd
zd
Flow
Flow
Wave Drag Coefficient
























−+













−
−
=













−−













−= ∫∫∫∫
∞
1
0
2
1
0
2
2
1
0
1
0
1
2
c
x
d
xd
zd
c
x
d
xd
zd
Mc
x
d
xd
zd
C
c
x
d
xd
zd
Cc
UpperUpperLower
p
Upper
pD LowerUpperW
αααα








−
−
=
∞
α
Upper
p
xd
zd
M
C Upper
1
2
2 







−
−
−=
∞
α
Lower
p
xd
zd
M
C Lower
1
2
2
( ) ( ) ( ) ( ) 



























+





−+













+





−
−
= ∫∫∫∫
=−=−
∞
1
0
2
00
1
0
2
1
0
2
00
1
0
2
2
22
1
2
c
x
d
xd
zd
c
x
d
xd
zd
c
x
d
xd
zd
c
x
d
xd
zd
M Lower
zcz
LowerUpper
zcz
Upper
LowerLowerUpperUpper
    
αααα
( )22
22
2
1
2
1
4
LowerUpperD
MM
C W
εε
α
+
−
+
−
=
∞∞
∫
∫














=














=
1
0
2
2
1
0
2
2
:
:
c
x
d
xd
zd
c
x
d
xd
zd
Lower
Lower
Upper
Upper
ε
ε

150. 150
SOLO
Linearized Flow Equations
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Steady Two Dimensional Flow
Supersonic Flow M∞ 1
Wave Drag Coefficient
Flat Plate 







== 0
LowerUpper
xd
zd
xd
zd
Double Wedge Airfoil
1
4
2
2
−
=
∞M
C WD
α
022
== LowerUpper εε
( )
( )
( )kkc
t
ck
c
t
k
ck
c
t
kc
LowerUpper
−
=






−
−
+==
14
1
1
14
1
4
11
2
2
2
2
22
2
2
22
εε
( ) ( )






−
−
=







−
−
=
cxck
ck
t
ckx
ck
t
xd
zd
cxck
ck
t
ckx
ck
t
xd
zd
LowerUpper
12
0
2
12
0
2
( )
( )kk
ct
MM
C WD
−−
+
−
=
∞∞
1
/
1
1
1
4
2
22
2
α

151. 151
SOLO
Linearized Flow Equations
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Steady Two Dimensional Flow
Supersonic Flow M∞ 1
Wave Drag Coefficient
Biconvex Airfoil
( ) ( )222
2/2/ ctRR +−=
The Biconvex Airfoil is obtained by intersection of two
Circular Arcs of radius R.
c – the chord
t – maximum thickness at x = c/2
( ) ( ) ( )tcttcR
tc
4/4/ 222
22
≈+=
θθθθ −≈−=≈= tan,tan
LowerUpper
xd
zd
xd
zd
2
2
2/2
/2
3
2
1
0
2
1
0
2
2
3
2
34
11
: Lower
ct
ctUpperUpper
Upper
c
t
t
c
dR
c
xd
xd
zd
cc
x
d
xd
zd
ε
θ
θθε
δ
δ
==≈≈








=













=
+
−
+
−∫∫∫
c
t
R
c
xd
zd
MaxUpper
2
2/
, ≈≈≈








δδ
( ) 2
2
22
2
22
22
2
3
16
1
1
1
4
1
2
1
4
c
t
MMMM
C LowerUpperDW
−
+
−
=+
−
+
−
=
∞∞∞∞
α
εε
α

152. 02/10/15 152
SOLO
Linearized Flow Equations
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Steady Two Dimensional Flow
Supersonic Flow M∞ 1
Wave Drag Coefficient
Parabolic ProfileDesignation Double Wedge Profile
Contour
Side View
Wave Drag
( )kk −13
1
2( )kk −1
1
( )
( ) xckck
xcxt
z
212 22
−+
−
±=
( )






−
±
±
=
cxckx
ck
t
ckxx
ck
t
z
12
0
2

153. 153
SOLO
Linearized Flow Equations
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Steady Two Dimensional Flow
Supersonic Flow M∞ 1
Wave Drag Coefficient
Wave Drag at Supersonic Incident Flow
versus Relative Thickness Position
for Double Wedge and Parabolic Profiles
k
( )kk −1
1
( )kk −13
1
2

154. 154
SOLO Wings in Compressible Flow
Double Wedge
Modified Double Wedge
Biconvex
τ
2
1
2
122
1
2
' 2
==






=
c
t
c
t
c
A
τ
3
2
3
2332
1
2
' 2
==
+





=
c
t
c
t
c
t
c
A

155. 155
SOLO
Linearized Flow Equations
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Steady Two Dimensional Flow
Supersonic Flow M∞ 1
Pitching Moment Coefficient
The Pitching Moment Coefficient about the
Leading Edge for any Thin Airfoil is given by
xdx
xd
zd
xd
zd
Mcc
x
d
c
x
C
c
x
d
c
x
Cc
c
LowerUpper
ppM LowerUpperLE ∫∫∫ 















−+








−
−
−=











+











−=−
∞
022
1
0
1
0
1
2
αα
Thus




 +
−
+
−
−= ∫∫
∞∞
xdzxdz
McM
c
c
Lower
c
UpperM LE 00222
1
2
1
2α
( ) ( ) ( ) ( )[ ] xdzxdzczczcxdzzxxdzzxxdx
xd
zd
xd
zd c
Lower
c
UpperLowerUpper
c
Lower
cx
xLower
c
Upper
cx
xUpper
c
LowerUpper
∫∫∫∫∫ −−−=−+−=








+
=
=
=
= 00
0
00000   
Using integration by parts
Symmetric Airfoil zUpper = -zLower
1
2
2
−
−=
∞M
cM
α
The distance of the Airfoil Center of Pressure aft of the Leading Edge is given by
cc
M
M
c
c
c
c
x
L
MN
2
1
1/4
1/2
2
2
=⋅
−
−
=⋅−=
∞
∞
α
α
α
L
∞U
x
Return to Table of Content

156. 156
SOLO
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ 1)






















−+








−
−
−=





−≅






















−+








−
−
=













−≅
∫∫
∫∫
∞
∞∞
∞∞
∞
∞∞
∞∞
c
x
d
xd
yd
xd
yd
M
U
c
s
dCUL
c
x
d
xd
yd
xd
yd
M
U
c
s
d
xd
yd
CUD
c
LowerUpperS
p
c
LowerUpperS S
p
0
2
2
2
0
22
2
2
2
2
1
2
1
2
1
2
1
2
1
2
1
αα
ρ
ρ
αα
ρ
αρ
α
U
Up
xd
yd
θ=
L
Low
xd
yd
θ−=
∞U
x
y
1
1
2
−
=
∞Mxd
yd
1
1
2
−
−=
∞Mxd
yd
Flow
Flow








−
−
==−
∞
∞∞
∞ α
ρ
Upper
UpperUpper
xd
yd
M
U
ppp
1
'
2
2








−
−
−==−
∞
∞∞
∞ α
ρ
Lower
LowerLower
xd
yd
M
U
ppp
1
'
2
2
1
2
1
2
2
2
−








−
−=
−








−
=
∞
∞
M
xd
yd
C
M
xd
yd
C
Lower
p
Upper
p
Lower
Upper
α
α
We found:
This relation was first derived by Jacob Ackeret in 1925, in a paper
“Luftkrafte auf Flugel, die mit groserer als Schall-geschwingigkeit bewegt werden”
(“Air Forces on Wings Moving at Supersonic Speeds”), that appeared in
Zeitschhrift fur Flugtechnik und Motorluftschiffahrt, vol. 16, 1925, p.72
Jakob Ackeret
(1898–1981)
Linearized Flow Equations

157. 157
AERODYNAMICS
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ 1)
Supersonic Flow past a Symmetric Double-Edged Airfoil
1
2
3
4
SHOCK LINE
SHOCK LINE
SHOCK LINE
SHOCK LINE
EXPANSION
EXPANSION
Using Ackeret Theory we have
( ) ( )
( ) ( )
1
2
,
1
2
1
2
,
1
2
22
22
43
21
−
−
−=
−
+
−=
−
+
=
−
−
=
∞∞
∞∞
M
C
M
C
M
C
M
C
pp
pp
αδαδ
αδαδ
( ) ( )
1
4
2
1
1
4
2
1
1
4
222
1
2/1
2/1
0 3412
−
=
−
+
−
=






−+





−=





=
∞∞∞
∫∫∫
MMM
c
x
dCC
c
x
dCC
c
s
dCC pppp
S
pX
ααα
( ) ( )
( ) ( )
1
4
1
4
2
2
22 2
2/
2
0
2/
2/
0
3412
3412
−
=
−
×=−+−=






−+





−=





=
∞
=
∞
−∫∫∫
MMc
t
CC
c
t
CC
c
t
c
y
dCC
c
y
dCC
c
y
dCC
ct
pppp
ct
pp
ct
pp
S
pX
δδ δ
XYXYD
XYXYL
CCCCC
CCCCC
+≈+=
−≈−=
ααα
ααα
α
α
1
1
cossin
sincos
1
4
1
4
1
4
1
4
2
2
2
21
2
2
2
1
−
+
−
≈
−
−
−
≈
∞∞
∞∞
MM
C
MM
C
D
L
δα
αδα
α
α

158. 158
AERODYNAMICS
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ 1)

159. 159
pc−
cx /
0.1
pc−
cx /
0.1
pc−
cx /
0.1
α
δα
δ
∞M
δα
∞M
∞M
δα =
α
∞M
Upper Surface
Lover Surface
Expansion
Shock
Shock
Expansion
Expansion
Shock
Expansion
Shock
Shock
Expansion
Expansion
Shock
Shock
Shock
Shock
∞M
∞M
( )
1
2
2
−
−
=
∞M
cp
αδ
( )
1
2
2
−
+
=
∞M
cp
αδ
( )
1
2
2
−
−
=
∞M
cp
αδ
( )
1
2
2
−
+
=
∞M
cp
αδ
1
4
2
−
=
∞M
cp
α
1
4
2
−
−
=
∞M
cp
α
( )
1
2
2
−
+
−=
∞M
cp
αδ
( )
1
2
2
−
−
−=
∞M
cp
αδ
( )
1
2
2
−
+
−=
∞M
cp
αδ
( )
1
2
2
−
−
−=
∞M
cp
αδ
Supersonic Flow past a Symmetric Biconvex Aerfoil
AERODYNAMICS
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ 1)
2
2
2
2
2
2
2
4
1
1
3
16
3
16
1
4
LD
L
C
M
M
c
t
C
c
tD
L
Md
Cd
−
+
−






=






+
=
−
=
∞
∞
∞
α
α
α

160. 160
SOLO
Linearized Flow Equations
Small Perturbation Flow (Homentropic: , Isentropic )( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u
Applications: Three Dimensional Flow (Thin Airfoil at Small Angles of Attack)
Aerodynamic Coefficients of a Profile in Supersonic Incident Flow
Based on the Linear Theory Supersonic Rules






−
−
=
∞
Xd
Zd
M
α
1
1
2
1
4
2
−
=
∞M
2
1
=
0DC
0α
0MC
c
xN
αd
Cd L
pCPressure Distribution
Lift Slope
Neutral-Point Position
Zero Moment
Zero-Lift Angle 0=
( )
∫−
−=
∞
1
02
1
4
XdZ
M
S
Wave Drag
L
D
Cd
Cd 1
4
1 2
−−= ∞M
( ) ( )
∫ 













+





−
−=
∞
1
0
22
2
1
4
Xd
Xd
Zd
Xd
Zd
M
tS

161. 161
SOLO
• Up to point A the flow is Subsonic and it follows Prandtl-
Glauert Linear Subsonic Theory.
• At point B (M∞=0.81) the flow on the Upper Surface exceeds
the Sound Velocity and a Shock Wave occurs. On the Lower
Surface the Flow is everywhere Subsonic.
• At point C (M∞=0.89) the Flow velocity exceeds the Speed of
Sound also on the Lower Surface and a Shock Wave occurs.
• At point D (M∞=0.98) the two Shock Waves on the Upper
and Lower Surface (weaker than at point C) are located at
the Trailing Edge. The Lift is larger than at point C.
• At point E (M∞=1.4) pure Supersonic Flow on both
Surfaces.
Transonic Flow past Airfoils
Lift Coefficient of an Airfoil versus Mach Number.
Solid Line – Measurement. Dashed Lines - Theory
AERODYNAMICS
Transonic Flow over an Airfoil at various
Mach Numbers; Angle of Attack α=2°.
The points A,B, C, D,E correspond to the Lift
Coefficients.

162. 162
AERODYNAMICS
Return to Table of Content

163. 163
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLO AERODYNAMICS

164. 164
AERODYNAMICSSOLO
Return to Table of Content
Continue to Aerodynamics – Part III

165. 165
I.H. Abbott, A.E. von Doenhoff
“Theory of Wing Section”, Dover,
1949, 1959
H.W.Liepmann, A. Roshko
“Elements of Gasdynamics”,
John Wiley Sons, 1957
Jack Moran, “An Introduction to
Theoretical and Computational
Aerodynamics”
Barnes W. McComick, Jr.
“Aerodynamics of V/Stol Flight”,
Dover, 1967, 1999
H. Ashley, M. Landhal
“Aerodynamics of Wings
and Bodies”,
1965
Louis Melveille Milne-Thompson
“Theoretical Aerodynamics”,
Dover, 1988
E.L. Houghton, P.W. Carpenter
“Aerodynamics for Engineering
Students”, 5th
Ed.
Butterworth-Heinemann, 2001
William Tyrrell Thomson
“Introduction to Space Dynamics”,
Dover
References
AERODYNAMICSSOLO

166. 166
Holt Ashley
“Engineering Analysis of
Flight Vehicles”,
Addison-Wesley, 1974
J.J. Bertin, M.L. Smith
“Aerodynamics for Engineers”,
Prentice-Hall, 1979
R.L. Blisplinghoff, H. Ashley,
R.L. Halfman
“Aeroelasticity”,
Addison-Wesley, 1955
Barnes W. McCormick, Jr.
“Aerodynamics, Aeronautics,
And Flight Mechanics”,
W.Z. Stepniewski
“Rotary-Wing Aerodynamics”,
Dover, 1984
William F. Hughes
“Schaum’s Outline of
Fluid Dynamics”,
McGraw Hill, 1999
Theodore von Karman
“Aerodynamics: Selected
Topics in the Light of their
Historical Development”,
Prentice-Hall, 1979
L.J. Clancy
“Aerodynamics”,
John Wiley Sons, 1975
References (continue – 1)
AERODYNAMICSSOLO

167. 167
Frank G. Moore
“Approximate Methods
for Missile Aerodynamics”,
AIAA, 2000
Thomas J. Mueller, Ed.
“Fixed and Flapping Wing
Aerodynamics for Micro Air
Vehicle Applications”,
AIAA, 2002
Richard S. Shevell
“Fundamentals of Flight”,
Prentice Hall, 2nd
Ed., 1988 Ascher H. Shapiro
“The Dynamics and Thermodynamics
of Compressible Fluid Flow”,
Wiley, 1953
Bernard Etkin, Lloyd Duff Reid
“Dynamics of Flight:
Stability and Control”,
Wiley 3d Ed., 1995
H. Schlichting, K. Gersten,
E. Kraus, K. Mayes
“Boundary Layer Theory”,
Springer Verlag, 1999
References (continue – 2)
AERODYNAMICSSOLO

168. 168
John D. Anderson
“Computational Fluid Dynamics”,
1995
John D. Anderson
“Fundamentals of Aeodynamics”,
2001
John D. Anderson
“Introduction to Flight”,
McGraw-Hill, 1978, 2004
John D. Anderson
“Introduction to Flight”,
1995
John D. Anderson
“A History of Aerodynamics”,
1995
John D. Anderson
“Modern Compressible Flow:
with Historical erspective”,
McGraw-Hill, 1982
References (continue – 3)
AERODYNAMICSSOLO
Return to Table of Content

169. February 10, 2015 169
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 –2013
Stanford University
1983 – 1986 PhD AA

170. 170
Ludwig Prandtl
(1875 – 1953)
University of Göttingen
Max Michael Munk
(1890—1986)[
also NACA
Theodor Meyer
(1882 - 1972
Adolph Busemann
(1901 – 1986)
also NACA
Colorado U.
Theodore von Kármán
(1881 – 1963)
also USA
Hermann Schlichting
(1907-1982)
Albert Betz
(1885 – 1968 ),
Jakob Ackeret
(1898–1981)
Irmgard Flügge-Lotz
(1903 - 1974)
also Stanford U.
Paul Richard Heinrich Blasius
(1883 – 1970)

171. 171
Hermann Glauert
(1892-1934)
Pierre-Henri Hugoniot
(1851 – 1887)
Gino Girolamo Fanno
(1888 – 1962)
Karl Gustaf Patrik
de Laval
(1845 - 1913)
Aurel Boleslav
Stodola
(1859 -1942)
Eastman Nixon Jacobs
(1902 –1987)
Michael Max Munk
(1890 – 1986)
Sir Geoffrey Ingram
Taylor
(1886 – 1975)
ENRICO PISTOLESI
(1889 - 1968)
Antonio Ferri
(1912 – 1975)
Osborne Reynolds
(1842 –1912)

172. 172
Robert Thomas Jones
(1910–1999)
Gaetano Arturo Crocco
(1877 – 1968)
Luigi Crocco
(1906-1986)
MAURICE MARIE
ALFRED COUETTE
(1858 -1943)
Hans Wolfgang Liepmann
(1914-2009)
Richard Edler
von Mises
(1883 – 1953)
Louis Melville
Milne-Thomson
(1891-1974)
William Frederick
Durand
(1858 – 1959)
Richard T. Whitcomb
(1921 – 2009)
Ascher H. Shapiro
(1916 — 2004)

173. 173
John J. Bertin
(1928 – 2008)
Barnes W. McCormick
(1926 - )
Antonio Filippone John D. Anderson, Jr. Holt Ashley
)1923–2006(
Milton Denman Van
Dyke
(1922 – 2010)

174. 174