Aerodynamics part ii
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Transcript of Aerodynamics part ii
2
Table of Content
AERODYNAMICS
Earth AtmosphereMathematical 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
AERODYNAMICS�PART�I
3
Table of Content (continue – 1)
AERODYNAMICSSOLO
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
AERODYNAMICS�PART�I
4
Table of Content (continue – 2)
AERODYNAMICSSOLO
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
AERODYNAMICS�PART�I
5
Table of Content (continue – 3)
AERODYNAMICSSOLO
Shock & Expansion Waves
Shock Wave Definition
Normal Shock Wave
Oblique Shock Wave
Prandtl-Meyer Expansion WavesMovement 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
Table of Content (continue – 4)
AERODYNAMICSSOLO
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
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 Method2. Singularity-Distribution MethodTheoretical 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
AERODYNAMICS� PART� III
8
Table of Content (continue – 6)
AERODYNAMICSSOLO
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
AERODYNAMICS� PART� III
10
SOLO
- when the source moves at subsonic velocity V < a, it will stay inside the family of spherical sound waves.
a
VM
M=
= − &1
sin 1µ
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
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
SHOCK WAVESSOLO
A shock wave occurs when a supersonic flow decelerates in response to a sharpincrease in pressure (supersonic compression) or when a supersonic flow encountersa 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 isan 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
Normal Shock Wave Over a Blunt Body
Normal S�hock Wave
S�HOCK WAVES�SOLO
Oblique S�hock Wave
Oblique Shock Wave
Return to Table of Content
15
NORMAL S�HOCK WAVES�SOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
Conservation of Mass (C.M.) ρ ρ1 1 2 2u u= η ρρ
= =2
1
1
2
u
u
Conservation of Linear Momentum (C.L.M.) 22221
211 pupu +=+ ρρ ( )p
p
up
2
1
12
1
1
1 1= + −ρ
η
H H h u h u1 2 1 12
2 221
2
1
2= → + = + h
h
u
h2
1
12
121
21
1= + −
η
Conservation of Energy (C.E.)
Field Equations
Constitutive Relations
p R T�= ρIdeal Gas
( )( )
( )e e T� C T�v= =1 2(1) Thermally Perfect Gas
(2) Calorically Perfect Gas
ργγ
ρρρ
γρ pp
CC
CC
p
R
CT�C
peh
v
p
vp CC
v
p
v
pCCR
pT�Rp
p 11 −=
−===+=
≡−==
u
p
ρT
e
u
p
ρT
e
τ11
q
1
1
1
1
1
2
2
2
2
2
1 2
16
NORMAL S�HOCK WAVES�SOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
First Way
h
h
p
pp
p
p
p
u
h
up
2
1
2
2
1
1
2
1
1
2
2
1
12
12
12
1
1
2
1
1
112
11
12
1
11=
−
−
= = = + −
= +
−
−
γγ ρ
γγ ρ
ρρ η η γ
γ ρη
or
( )p
p
up
up
C L M2
1
12
1
1
12
1
1
2
11 1
112
1
11
ηρ
ηη γ
γ ρη
= + −
= +
−
−
( . . .)
after further development we obtain
1 21
11
11
1
201
2
1
1
212
1
1
12
1
1
−−
− +
+ + −
=γγ
ρη
ρη
γγ
ρ
up
up
up
Solving for 1/η , we obtain
1
1 1 21
11
2
11
2
2
1
12
1
1
12
1
1
2
12
1
1
12
1
1
ηρρ
ρ ρ
γγ
ρ
γγ
ργ
γ
= = =
+
− +
− + + −
+u
u
up
up
up
up
u
p
ρT
e
u
p
ρT
e
τ11
q
1
1
1
1
1
2
2
2
2
2
1 2
17
NORMAL S�HOCK WAVES�SOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
We obtain an other relation in the following way:
( )
p
p
up
p
p
up
p
pp
p
p
p
p
p
p
p
p
pp
p
2
1
12
1
1
2
2
1
12
1
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
11
1
21
1
1 1
11
1
1
211
1 1
2
1
21
1
21
1
2
1
21
2
1
2
ηγ
γρ
η
ρη
η γγ η
ηγ
γγ
γγ
γ
η
γγ
γγ
γγ
γγ
− = − −
− = −
⇒−
−= − +
⇓
− − − −
= + − −
⇓
=
+ − −
− + +
η ρρ
γγ
γγ
= = =
+−
−
+ +−
=2
1
1
2
2
1
2
1
2
1
1
2
1
11
1
1
u
u
p
pp
p
p
p
T�
T�
or
Rankine-Hugoniot Equation
Rankine-Hugoniot Equation (1)
William John MacquornRankine
(1820-1872)
u
p
ρT
e
u
p
ρT
e
τ11
q
1
1
1
1
1
2
2
2
2
2
1 2
Pierre-Henri Hugoniot(1851 – 1887)
18
NORMAL S�HOCK WAVES�SOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
η ρρ
γγ
γγ
= = =
+−
−
+ +−
=2
1
1
2
2
1
2
1
2
1
1
2
1
11
1
1
u
u
p
pp
p
p
p
T�
T� Rankine-Hugoniot Equation
Rankine-Hugoniot Equation (2)
p
p2
1
2
1
2
1
1
11
1
1
=
+−
−
+−
−
γγ
ρρ
γγ
ρρ
T�
T�
p
p
p
p
p
pp
p
p
p
pp
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
11
11
1
11
1
1
1
11
1
1
1
1
1
1
1
= =+ +
−+−
−=
+ +−
+−
−
=
+−
−
+−
−=
+−
−
+−
−
ρρ
γγ
γγ
γγ
γγ
γγ
ρρ
γγ
ρρ
ρρ
γγ ρ
ργγ
ρρ
p 2p 1
ρ 2ρ 1
Normal Shock WaveRankine-Hugoniot
Isentropicγp 2
p 1
ρ 2ρ 1
( )=
u
p
ρT
e
u
p
ρT
e
τ11
q
1
1
1
1
1
2
2
2
2
2
1 2
20
NORMAL S�HOCK WAVES�SOLO
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 H2
1
2
1
1
2
2
1
2
1
1
1
1
1→ ∞ ⇒ = → +
−→ −
+− −ρ
ργγ
γγ
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
up
1
2 =∆∆ρ
∆∆
h u
ρ ρ= 1
2
1
u u u u u u21
2
11
1
1
1
1 1
1
1
1
1= =
+=
+≅ −ρ
ρρ
ρ ρ ρρ
ρρ∆ ∆
∆(C.M.)
( ) ( )ρ ρ ρ ρρ1 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 21
2
1
21
1
211
2
11
11222
211
∆+∆−+∆+=
∆−+∆+=+=+ρρ
ρρ
ρρ(C.E.)
u
p
ρT
e
u
p
ρT
e
τ11
q
1
1
1
1
1
2
2
2
2
2
1 2
21
NORMAL S�HOCK WAVES�SOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
Second Wayh h u h u0 1 1
22 2
21
2
1
2≡ + = +Define
−−−=→+−
=
−−−=→+−
=
210
1
121
1
10
220
2
222
2
20
11
2
1
1
11
2
1
1
uhp
up
h
uhp
up
h
γγ
γγ
ρργγ
γγ
γγ
ρργγ
u u h1 2 021
1= −
+γγ Prandtl’s Relation
( )u hu
u
u
p
p
up2 0
1
2 11
2
2
1
1
2
1
1
21
1
11 1= −
+→ = = → = + −γ
γρ ρ ρη
ρηFrom this relation, we obtain:
Prandtl’s Relation
Ludwig Prandtl(1875-1953)
u
p
ρT
e
u
p
ρT
e
τ11
q
1
1
1
1
1
2
2
2
2
2
1 2
(C.M.)(C.L.M.)
ργγ p
h1−
=and use
1222
2
11
1
2211
22221
211 11
uuu
p
u
p
uu
pupu−=−→
=+=+
ρρρρρρ
122121
0 2
1
2
1111uuuu
uuh −=−+−−
−−
γγ
γγ
γγ
( )
−−−=−−γ
γγ
γ2
11
112
21
120 uu
uu
uuh
22
NORMAL S�HOCK WAVES�SOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
(C.M.)
Hugoniot Equation
ρ ρ ρρ1 1 2 2 2 11
2
u u u u= → =
( )ρ ρ ρ ρρ
ρ ρρ
ρρ
ρ ρ
ρ ρρρ ρ ρ
ρρ
ρρ
ρρ
1 12
1 2 22
2 21
2
2
12
2 2 1 12
11
2
2
12 1
2
2 1
1
2 2 1
2
22
2 2 1
2
1
2
2 11
2
2 11
2
u p u p u p p p u u
up p
up p
u u
u u
+ = + =
+ → − = −
= − →
→ = −−
→ = −
−
=
=
(C.L.M.)
( ) ( )
h u h u ep p p
ep p p
e ep p p p p p p p
e ep p
h ep
1 1
2
2 2
2
11
1
2 1
2 1
22
2
2
2 1
2 1
1
2
2 12 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 ρρρ ρ ρ ρ
ρρ
ρ ρρρ
ρρ ρ ρ ρ ρ
ρ ρρ ρ
ρ ρρρ
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 ep p p p
ρ ρρ ρ
ρ ρ ρ ρ ρ ρρ ρ
ρ ρρ ρ
(C.E.)
e ep p
2 11 2
2 12
1 1− = + −
ρ ρHugoniot Equation
u
p
ρT
e
u
p
ρT
e
τ11
q
1
1
1
1
1
2
2
2
2
2
1 2
Pierre-Henri Hugoniot(1851 – 1887)
23
NORMAL S�HOCK WAVES�SOLO
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 ++=+ 22221
2112 ρρ
( )3 1
2
1
21 1
2
2 2
2C T� u C T� u h C T�p p p+ = + =
( )4 1 1 1 2 2 2p R T� p R T�= =ρ ρ
( )5 2 12
1
2
1
s s CT�
T�R
p
pp− = −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 CT�
T�R
T�
T�
h C T�
h C T�
p
p
p
p
p
p
p
2
1
42
1
2
1
2
1
11
2
30 1
0 2
2
1
2
1
0 1
0 2
2 12
1
2
1
0 1
0 2
5
=
= =−
−
→ =−
−→
− = −−
−
( )
( ) ( )
ln ln
ρρ
ρρ
Assume that all the conditionsof the model are satisfied except the moment equation (2)(a flow with friction)
Using , we obtainh C T�p=
ss
1s
2s max
h1
h2
h2
1s s C
h
hR
h
h
h h
h hp2 1
2
1
2
1
0 1
0 2
− = −−−
ln ln
Fanno’s Line for a Perfect Gas
T�his is the Adiabatic, Constant Area Flow.
u
p
ρT
e
u
p
ρT
e
τ11
q
1
1
1
1
1
2
2
2
2
2
1 2
Gino Girolamo Fanno(1888 – 1962)
24
NORMAL S�HOCK WAVES�SOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
Fanno’s Line for a Perfect Gas (2)
ss
1s
2s max
h1
h2
h2
1
We have a point of maximum entropy. Let see the significance of this point
ρρdp
dhdp
dhdsT� =→=−= 0max
Gibbs
u
duddudu −=→=+
ρρρρ 0(C.M.)
duudhu
hd −=→=
+ 02
2(C.E.)
T�herefore)4..(
0
.).(
000
EC
ds
MC
dsdsds u
du
d
dpd
d
dpdpdh =
−
=
==
==== ρρ
ρρρ
0
0
=
=
=
ds
ds d
dpu
ρor
ds CdT�
T�R
dp
p
ds CdT�
T�R
d
C
C
dp
p
d
dp
d p
dp
d
pR T�
p
v
p
v
ds
ds
ds ds
p R T�= − =
= − =
→ ≡ = = → = ==
=
= =
=
max
max
0
0
0
0
0 0ρρ
γρρ
ρρ
ργ
ργ
ρWe have:
udp
dR T� a speed of soundds
ds
=
=
=
= = =0
0ρ
γ
u
p
ρT
e
u
p
ρT
e
τ11
q
1
1
1
1
1
2
2
2
2
2
1 2
25
Ideal Gas
NORMAL S�HOCK WAVES�SOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
Rayleigh’s Line for a Perfect Gas (1)( )
A
muu
== 22111 ρρ
( )2 1 1
2
1 2 2
2
2ρ ρu p u p+ = +
( ) QhuT�CuT�C pp ++=+ 222
211 2
1
2
13
( )4 1 1 1 2 2 2p R T� p R T�= =ρ ρ
( )5 2 12
1
2
1
s s CT�
T�R
p
pp− = −ln ln
(C.M.)
(C.L.M.)
(C.E.)
Assume that all the conditionsof the model are satisfied except the energy equation (3)(a flow with heating and cooling)
Let substitute in (5) , to obtainh C T�p=
Rayleigh’s Line for a Perfect GasT�his is the Frictionless, Constant Area Flow, with Cooling and Heating.
s max
s
s1
s2
h1
h2
h
M>1
M<1Rayleigh2
1
Heating
Heating
Cooling
m
A
R T�
pp
m
A
R T�
pp
x
p1
11
2
22
1
+ = +
( )
21
121
11
212
111
212
&12
1
lnln5
p
R
A
mc
p
T�R
A
mb
hC
abbR
h
hCss
pp
=
+=
−+−=−
We want to find xp
p≡ 2
1
. Let multiply the result byx
p1
xm
A
R T�
p
b
xm
A
R
pc
T�2 1
12
1
12
1
21
2
0− +
+ =
or
xp
pb b a T�= = + −2
1
1 12
1 2T�he solution is:
John William Strutt
Lord Rayleigh
(1842-1919)
u
p
ρT
e
u
p
ρT
e
τ11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2
26
NORMAL S�HOCK WAVES�SOLO
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
duddudu −=→=+
ρρρρ 0(C.M.)
(C.L.M.)
A Normal Shock Wave must be on both Fanno and Rayleigh Lines, thereforethe end points of a Normal Shock Wave must be on the intersection of Fanno and Rayleigh Lines
udp
dR T� a speed of soundds
ds
=
=
=
= = =0
0ρ
γ
d p udp
duu+
= → = −1
202ρ ρ
( )→ = = − −
=dp
d
dp
du
du
du
uu
ρ ρρ
ρ2
ss
1s
2
h1
h2
h
M>1
M<1
Rayleigh
Fanno
2
1
SHOCK
According to the Second Law of Thermodynamicsthe Entropy must increase. T�herefore a Normal S�hockWave from state (1) to state (2) must be such thats2 > s1. (from supersonic to subsonic flow only)
u
p
ρT
e
u
p
ρT
e
τ11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2
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
uu u
C Ea
h
ua
h
ua a u
a a u
ap
. .
. . .
. .
ρ ρρ ρ ρ ρ
γ γ
γ γ
γ γ
γρ
1 1 2 2
1 12
1 2 22
2
1
1 1
2
2 22 1
12
1
12 2
2
2
22
12 2
12
22 2
22
41
1
2 1
1
2
1
2
1
21
2
1
2
=
+ = +
→ − = − →
−+ =
−+ →
=+
−−
=+
−−
=
∗
∗
− = −a
u
a
uu u1
2
1
22
22 1γ γ
Field Equations:
( )
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
+−
−−
++
−= −
↓
+ −+
−− = − →
+= −
−=
+
↓
∗ ∗
∗∗
1
2
1
2
1
2
1
2
1
2
1
2
1
21
1
2
1
2
2
11
2
22 2 1
2 1
1 2
22 1 2 1
2
1 2
a
uu
a
uu u u
u u
u ua u u u u
a
u u
u u a1 22= ∗
u
a
u
aM M1 21 21 1∗ ∗∗ ∗= → =
Prandtl’s Relation
u
p
ρT
e
u
p
ρT
e
τ11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2
Ludwig Prandtl(1875-1953)
28
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
Mach Number Relations (2)
( ) ( ) ( ) ( )
( ) ( )( ) ( )
( )[ ]( ) ( ) ( )
M
MM
M
M
M
M
MM
22
22
1
12
12
12
12
12
21
1
2
1 1
2
11
1 21
2 1 2
1 1 1 1 1
12
=+ − −
=+ − −
=+
+− +
− −=
− ++ / + − / / + − / + − −
∗
=
∗
∗∗
γ γ γ γ
γγ
γγ
γγ γ γ γ γ
or
( )M
M
M
M
MH H
A A
2
12
12
12
121 2
1 21
1
21
2
2
1
11
2
12
11
=+ −
− − =+
+ −
++
−=
=γ
γ γ γ
γ
γγ
( )( )
ρρ
γγ
2
1
1
2
12
1 2
12
2 12 1
2
12
1 2 1
1 2= = = = =
+− +
=
∗∗
A A u
u
u
u u
u
aM
M
M
u
p
ρT
e
u
p
ρT
e
τ11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2
29
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
Mach Number Relations (3)
( )( )
( ) ( )( )
p
p
up
u
u
u
a
MM
MM
M M
M
2
1
12
1
1
2
1
12
12
1
2
12 1
2
12 1
2 12
12
12
1 1 1 1
1 11 2
11
1 1 2
1
= + −
= + −
= + −− +
+
= + / + − / − −
+
ργ ρ
ρ
γγ
γγ
γ γγ
or
(C.L.M.)
( )p
pM2
1121
2
11= +
+−γ
γ
( ) ( )( )
h
h
T�
T�
p
pM
M
M
a
a
h C T� p R T�p2
1
2
1
2
1
1
212 1
2
12
2
1
12
11
1 2
1= = = +
+−
− ++
== =ρ ρ
ργ
γγ
γ
( ) ( )( )
s s
R
T�
T�
p
pM
M
M2 1 2
1
12
1
1
12
1
112
12
1
12
11
1 2
1
−=
= +
+−
− ++
−−
− −
ln ln
γγ
γγ
γγγ
γγ
γ
( ) ( ) ( ) ( )s s
RM M
M2 1
1 1
2 12 3
2
2 12 41
2 2
3 11
2
11
−≈
+− −
+− +
− << γγ
γγ
K Shapiro p.125
u
p
ρT
e
u
p
ρT
e
τ11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2
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
MM02
01
02
2
1
01
2
1
22
12
1
12
11
2
11
2
12
11= =
+ −
+ −
++
−
−γ
γγ
γ
γγ
( )
( )
11
21
1
2
11
21
2
1
2
1
2
1
21
21
2
11
1
2
12
11
22
12
12
12
2
12
12
12
12
+ − = + − + −
− − =− − + − + −
+ ++
−
=
+
++
−
γ γγ
γ γ
γ γ γ γ
γ γγ
γ
γγ
MM
M
M M
M
M
M
( )( )p
p
M
MM02
01
12
12
1
12
1
1
1
2
12
11
12
11=
+
++
−
++
−
−−
−γ
γγ
γγ
γγ
γ
u
p
ρT
e
u
p
ρT
e
τ11
q
Q
1
1
1
1
1
2
2
2
2
2
1 2
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
MM
M
T� T�2 1 02
01
102
01
1
02
01
12
12
12
02 01
1
11
2
11
1
1
2
11
2
−=
= −
=−
++
−
−
−
+
+ −
−−
=ln ln
ln ln
γγ
γγ
γγ
γ
γ
γ
s
s1
s2
T
M>1
M<1Rayleigh
Fanno
2
1
SHOCK
T2
T1
T02
T01=
T 2T 1=* *
p2
p1
p01
p02
Mollier’s Diagram
u
p
ρT
e
u
p
ρT
e
τ11
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(
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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
22
22
211
21
21
1 22u
wuhu
wuh ρρ
++=
++
Continuity Eq.: 2211 uu ρρ =
Moment Eq.:21 ww =
2222
2111 upup ρρ +=+
Energy Eq.:22
22
2
21
1
uh
uh +=+
Summary
Calorically Perfect Gas:T�ch
T�Rp
p== ρ
6 Equations with 6 Unknowns
222222 ,,,,, hwuT�pρ
33
OBLIQUE SHOCK & EXPANSION WAVESSOLO
For a calorically Perfect Gas
( )( )
( )( )[ ]
( )[ ]
2
1
1
2
1
2
21
212
2
21
1
2
21
21
1
2
11/2
1/2
11
21
21
1
ρρ
γγγ
γγ
γγ
ρρ
p
p
T�
T�
M
MM
Mp
p
M
M
n
nn
n
n
n
=
−−−+=
−+
+=
+−+=
βsin11 MM n =
( )θβ −=sin
22
nMM
Now we can compute
( )( ) ( )
( )( )
( )
⋅+−=−
+−+===−
⇒
=
=−
=
θββθβ
βθβ
βγβγ
ρρ
βθβ
θβ
β
tantan1tan
tantan
tan
tan
sin1
sin12
tan
tan
tan
tan
221
221
2
1
1
2
12
2
2
1
1
M
M
u
u
ww
w
u
w
u
34
OBLIQUE S�HOCK & EXPANS�ION WAVES�SOLO
( )
++
−=22cos
1sincot2tan 2
1
221
βγββθ
M
M
M,, βθ relation
12 <M
12 >M
.5max =Mforθ
β θ1M 2M
Strong Shock
Weak Shock
θ
β
We can see that θ = 0 for1.β = 90° (Normal Shock)2.sin β = 1/ M1
35
OBLIQUE S�HOCK & EXPANS�ION WAVES�SOLO
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
OBLIQUE S�HOCK & EXPANS�ION WAVES�SOLO
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
S�T�RONGβ
( )
++
−=22cos
1sincot2tan 2
1
221
βγββθ
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( )
++
−=22cos
1sincot2tan
2
1
22
1
βγββθ
M
M
M,, βθ relation
SOLO OBLIQUE S�HOCK & EXPANS�ION WAVES�
θβ
4.1=γ
θ
maxθ
θ
38
( )[ ]( )[ ]
( )θβ
γγγ
β
−=
−−−+=
=
sin
11/2
1/2
sin
22
21
212
2
11
n
n
nn
n
MM
M
MM
MM
SOLO
θ
maxθ
OBLIQUE S�HOCK & EXPANS�ION WAVES�
Mach Number in Back of Oblique Shock M2 as a Function of the Mach Numberin Front of the Shock M1, for Different Values of Deflection Angle θ (γ=1.4)
39
( )11
21
sin
21
1
2
11
−+
+=
=
n
n
Mp
p
MM
γγ
β
SOLO
θ
θ
OBLIQUE S�HOCK & EXPANS�ION 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
SOLO
θ
θ
OBLIQUE S�HOCK & EXPANS�ION WAVES�
Stagnation Pressure Ratio P20/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)
42
Hodograph Shock Polar
SOLO
-For every deflection angle θ the Hodograph gives two solutions, a strong shock (B outside the sonic circle – M2>1) and a weak shock (D inside the sonic circle – M1<1)
- 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 S�HOCK & EXPANS�ION WAVES�
43
SOLO
Family of Hodograph Shock Polars ( γ= 1.4)
θ
1***1
2
1**
***21
2
1
212
21
2
2
+−
+
−
−=
cV
cV
cV
cV
cV
c
V
c
V
c
V
x
x
xy
γ
A. H. S�hapiro “T�he Dynamics and T�hermodynamics of Compressible Flow Fluid”,pg.543
45.2
OBLIQUE S�HOCK & EXPANS�ION WAVES�
50
SOLO OBLIQUE SHOCK & EXPANSION WAVES
Prandtl-Meyer Expansion Waves
Ludwig Prandtl(1875 – 1953)
Theodor Meyer (1882 – 1972)
The Expansion Fan depicted in Figure wasFirst analysed by Prandtl in 1907 and hisstudent Meyer in 1908.
Let start with an Infinitesimal Change across aMach Wave
Mac
h Wav
e
θd
µ µπ −2
θµπd−−
2
V
VdV +
( )( ) θµθµ
µθµπ
µπdddV
VdV
sinsincoscos
cos
2/sin
2/sin
−=
−−+=+
µθµθ
µθ tan
/tan1
tan1
11
VVddd
dV
Vd =⇒+≈−
≈+
1
1tan
1sin
2
1
−=⇒
= −
MMµµ
V
VdMd 12 −=θ
1907 - 1908
51
SOLO OBLIQUE SHOCK & EXPANSION WAVES
Prandtl-Meyer Expansion Waves (continue-1)
Mac
h Wav
e
θd
µ µπ −2
θµπd−−
2
V
VdV +
V
VdMd 12 −=θ
Integrating this equation gives
∫ −=2
1
12M
M V
VdMθ
Using the definition of Mach Number: V = M.a
a
ad
M
Md
V
Vd +=
For a Calorically Perfect Gas
20
2
0
2
11 M
T
T
a
a −+==
γ
MdMMa
ad1
2
2
11
2
1−
−+−−= γγ
M
Md
MV
Vd
2
21
1
1−+
= γ ∫ −+
−=2
12
2
21
1
1M
M M
Md
M
Mγθ
52
SOLO OBLIQUE SHOCK & EXPANSION WAVES
Prandtl-Meyer Expansion Waves (continue-2)
The integral
∫ −+
−=2
12
2
21
1
1M
M M
Md
M
Mγθ
( ) ∫ −+
−=M
Md
M
MM
2
2
21
1
1γν
is called the Prandtl-Meyer Function and isgiven the symbol ν. Performing the integration we obtain
( ) ( ) ( )1tan11
1tan
1
1 2121 −−−+−
−+= −− MMM
γγ
γγν
Deflection Angle ν and Mach Angle μ as functions of Mach Number
= −
M
1sin 1µ
Finally
( ) ( )12 MM ννθ −=
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53
Movement of Shocks with Increasing Mach Number
Drag rises due to pressureIncrease 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
Movement of Shocks with Increasing Mach Number
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )87654321 ∞∞∞∞∞∞∞∞ <<<<<<< MMMMMMMM
SOLO AERODYNAMICS
55
UpperSurface
LowerSurface
UpperSurface
LowerSurface
UpperSurface
LowerSurface
UpperSurface
LowerSurface
UpperSurface
LowerSurface
( c) Shock on upper surface
UpperSurface
LowerSurface
(d ) Shocks on both surfaces
Shock
Movement of Shocks with Increasing Mach NumberSOLO AERODYNAMICS
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
11
2
11
−
∞
∞
−+
−+=
γγ
γ
γ
A
A
M
M
p
p
p∞, M∞ - Pressure and Mach Number upstream the AirfoilpA, MA- Pressure and Mach Number at a point A on the Airfoil
Critical Mach Number
The Pressure Coefficient Cp is computed using
( )
−
−+
−+=
−=
−
∞
∞∞∞
1
2
11
21
121
2
1/
2
2γγ
γ
γ
γγA
ApA
M
M
Mp
p
MC
Definition of Critical Mach Number.Point A is the location of minimum pressure on the top surface of the Airfoil.
SOLO AERODYNAMICS
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
11
21
121
2
1/
2
2γγ
γ
γ
γγA
ApA
M
M
Mp
p
MC
( )
−
−+
−+=
−
1
2
11
21
12
1/2
γγ
γ
γ
γcr
crp
M
MC
cr
2
0
1 ∞−=
M
CC p
p
( )
−
−+
−+=
−
1
21
1
21
12
1/2
γγ
γ
γ
γcr
crp
M
MC
cr
2
0
1 ∞−=
M
CC 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 orLaiton’s Rule
SOLO AERODYNAMICS
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 ofthe 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
Movement of Shocks with Increasing Mach Number
Drag Divergence Mach Number The Drag at small Mach number, due toProfile Drag with Induced Drag =0 (αi = 0)is constant (points a, b, and c) untilM∞ = 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 ofsupersonic 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 separateFrom 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 bottom surface is supersonic, both terminating with Strong Wave Shocks.
SOLO AERODYNAMICS
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 AirfoilDf – Skin Friction Drag Dp – Pressure Drag due to Flow SeparationDw – 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
61Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2
SOLO
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AERODYNAMICS
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 forWing = 0.7
Airfoil Sectionwith Mcr = 0.7
Airfoil Sectionwith 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
MM
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
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 areindependent of the component of the flight velocity in thedirection 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 AttackΛ – Swept Angle
The Flow Mach components are:
forcesairaffectingnotELtoparallelM
forcesairaffectingPlaneWingtheinELtonormalM
PlaneWingtonormalM
..sincos
..coscos
sin
Λ
Λ
∞
∞
∞
ααα
We have:
( ) ( )[ ] ( )
Λ=
Λ=
Λ=
Λ=
Λ
=
Λ−=Λ+=
−
∞
∞−
∞∞∞∞
coscos:
cos:
cos
tantan
coscos
sintan:
cossin1coscossin:
11
2/1222/122
ττ
αα
αα
ααα
c
t
cc
M
M
MMMM
e
e
e
e Section A-A
Section B-B
65
SOLO Wings in Compressible Flow
65
Swept Wings
Section A-A
Section B-B
( ) bcM
LCL 22/ ∞∞
=ργThe Total Lift is:
( ) ( )[ ] ( )
Λ=
Λ=
Λ=
Λ=
Λ
=
Λ−=Λ+=
−
∞
∞−
∞∞∞∞
coscos:
cos:
cos
tantan
coscos
sintan:
cossin1coscossin:
11
2/1222/122
ττ
αα
αα
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
LC
eeeeeL 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/
cos22 bcM
DC
bcM
DC
eDD ργργ
or: ( ) ( )α222 cossin1cos/ Λ−Λ== ∞∞ eDeeDD CMMCC
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 byR.T. Jones.
Robert Thomas Jones(1910–1999)
Oblique Wing Flight Demonstration by the AD-1.
68Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2
AERODYNAMICS
Swept Wings Drag Variation
SOLO
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
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
73
SOLO
Return to Table of Content
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
77
Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLO
Return to Table of Content
79Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
SOLOReturn to Table of Content
80Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane
Richard T. Whitcomb (1921 – 2009)
SOLO
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
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
89Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2
AERODYNAMICSSOLO
Return to Table of Content
91Nguyen X. Vinh, “Flight Mechanics of High Performance Aircraft”, Cambridge University,1993
AERODYNAMICSSOLO
92
Examples of airfoils in nature and within various vehicles
Lift and Drag curves for a typical airfoil
SOLO
96Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 2
AERODYNAMICSSOLO
Return to Table of Content
100
))B)- Flow field in wing-tail plane, influence of B)- Flow field in wing-tail plane, influence of control deflection control deflection δδ for pitch for pitch
SOLO
103
))C)- Flow field in wing-tail plane, influence of C)- Flow field in wing-tail plane, influence of control deflection control deflection ξξ for roll for roll
SOLO
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
105Density Profile Mach 1.2, Color Contours Modified to see Detail on Shock Waves
More Fun With CFD – RM-10SOLO
106
Density Profiles, Mach 2.41, simulated altitude of 11,000 ft )Re=76.4x106)
More Fun With CFD – RM-10SOLO
107Density Profiles, Mach 2.41 – color contours modified to see detail in shock waves
More Fun With CFD – RM-10SOLO
108Density Profiles, Mach 1.62 – rotated, with plot to show distribution around fins
More Fun With CFD – RM-10SOLO
109The Effect of Leading Edge Slat, Flap, and Trailing Edge FlapUpon Angle of Attack of Basic Wing
Darrol Stinton “ The Design of the Aircraft” SOLO
112
Three-Element AirfoilPressure Coefficient and Streamlines at Maximum Lift M=0.2 )Re=4.1x106)SALSA Computation
AERODYNAMICSSOLO
113Inviscid Transonic Flow Solution Over a 2-D Airfoil at M=0.75 )Re=1000)
AERODYNAMICSSOLO
114Inviscid Supersonic Flow Solution Over a 2-D Airfoil at M=1.50 )Re=1000)
AERODYNAMICSSOLO
116
Linearized Flow Equations
1. Irrotational Flow
SOLO
Assumptions
2. Homentropic
3. Thin bodies
( )0 =×∇ u
=
∂∂
=∇ 0&0..;.t
sseieverywhereconsts
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
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
( ) Gpuuut
uuu
t
u
tD
uD
ρ∂∂ρ
∂∂ρρ +−∇=
×∇×−
∇+=
∇⋅+= 2
2
1
or
( ) Gp
uuut
u
+∇−=×∇×−
∇+
∂∂
ρ2
2
1 Euler’s Equation
0& =∂Ψ∂Ψ−∇=t
G
- Body forces as a function of position( )zyxG ,,
Leonhard Euler1707-1783
Linearized Flow Equations
118
SOLO
Let integrate the Euler’s Equation between two points )1) and )2)
( ) ( ) ( ) ∫∫∫∫∫∫ ⋅Ψ∇+⋅∇+×∇⋅×−⋅
∇+⋅
∂∂=⋅
Ψ∇+∇+×∇×−
∇+
∂∂=
2
1
2
1
2
1
2
1
22
1
2
1
2
2
1
2
10 rd
rdpuurdrdurdu
trd
puuuu
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: ( ) ( ) 02
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:
( ) rdFtrFdconstt
⋅∇==
,
( )2
1
22
1
2
2
1
2
10
Ψ+++
∂Φ∂=
Ψ∇++
+Φ
∂∂= ∫∫
∞
p
p
pdu
t
pdudd
t ρρ
Bernoulli’s Equationfor Irrotationaland Inviscid Flow
Daniel Bernoulli1700-1782
Linearized Flow Equations
119
SOLO
For an isentropic ideal gas we have
2
2
11 a
ad
T
Tdd
p
pd
−=
−==
γγ
γγ
ρργ
where
ργγ
ρρp
TRd
pdpa
s
===∂∂=2 is the square of the speed of sound
In this case
22
2
1
1
1 2ad
a
adppdRTa
RTp
−=
−=
=
=
γργγ
ρ γ
ρ
and[ ]222
1
1
1
12
2
∞−−
=−
= ∫∫∞∞
aaadpd a
a
p
p γγρ
Using the Bernoulli’s Equation we obtain
( ) ( ) ( ) ( )
Ψ−Ψ+−+
∂Φ∂−−=−=− ∞∞∞ ∫
∞
2222
2
111 Uu
t
dpaa
p
p
γρ
γ
( )2
1
22
1
2
2
1
2
10
Ψ+++
∂Φ∂=
Ψ∇++
+Φ
∂∂= ∫∫
∞
p
p
pdu
t
pdudd
t ρρ
Bernoulli’s Equationfor Irrotationaland Inviscid Flow
Linearized Flow Equations
120
SOLO
Let use the conservation of mass )C.M.) equation
)C.M.) 0=⋅∇+ utD
D ρρ
ortD
Du
ρρ1−=⋅∇
Let go back to Bernoulli’s Equation ( ) ( )
Ψ−Ψ+−+
∂Φ∂−= ∞∞∫
∞
22
2
1Uu
t
pdp
p ρ
and use the Leibnitz rule of differentiation: ( ) ( )uxFdxuxFxd
d x
x
,,0
=∫to obtain
ρρ1=∫
∞
p
p
pd
pd
d
Now we can computetD
Da
tD
D
d
pd
tD
pD
tD
pDpd
pd
dpd
tD
Dp
p
p
p
ρρ
ρρρρρρ
211 ===
= ∫∫
∞∞
Therefore ( ) ( )
Ψ−Ψ+−+
∂Φ∂=−=−=⋅∇ ∞∞∫
∞
2222 2
1111Uu
ttD
D
a
pd
tD
D
atD
Du
p
p ρρ
ρ
Since ( )[ ] 0=Ψ−Ψ= ∞∞ tD
Du
tD
D
we have
∇⋅+
∂∂⋅+
∂Φ∂=
∇⋅+
∂Φ∂∇⋅+
∂∂⋅+
∂Φ∂=
=
+
∂Φ∂
∇⋅+
∂∂=
+
∂Φ∂=⋅∇
Φ∇=
22
1
2
1
2
11
2
11
2
2
2
2
2
2
2
2
22
22
uu
t
uu
ta
uu
tu
t
uu
ta
ut
uta
uttD
D
au
u
GOTTFRIED WILHELMvon LEIBNIZ
1646-1716
Linearized Flow Equations
121
SOLO
∇⋅+
∂∂⋅+
∂Φ∂=
∇⋅+
∂Φ∂∇⋅+
∂∂⋅+
∂Φ∂=
=
+
∂Φ∂
∇⋅+
∂∂=
+
∂Φ∂=⋅∇
Φ∇=
22
1
2
1
2
11
2
11
2
2
2
2
2
2
2
2
22
22
uu
t
uu
ta
uu
tu
t
uu
ta
ut
uta
uttD
D
au
u
Let substitute Φ∇=u
Φ∇⋅Φ∇∇⋅Φ∇+Φ∇
∂∂⋅Φ∇+
∂Φ∂=Φ∇⋅∇
2
12
12
2
2 tta
( ) ( ) ( )
Ψ−Ψ+−Φ∇⋅Φ∇+
∂Φ∂−−= ∞∞∞
222
2
11 U
taa γ
Special cases
0≈Φ∇⋅∇ Laplace’s equation
∞∞ >>Ua )subsonic flow) we can approximate the first equation by
1
2 ( ) ( )2
2
tuu
tuuu
∂Φ∂<⋅
∂∂+⋅∇⋅ we can approximate
the first equation by
01
2
2
2=
∂Φ∂−Φ∇⋅∇ta
Wave equation
Pierre-Simon Laplace
1749-1827
Linearized Flow Equations
122
SOLO
Note
The equation
+
∂Φ∂
∇⋅+
∂∂=⋅∇ 2
2 2
11u
tu
tau
can be written as
Φ=
Φ∇⋅+
∂Φ∂
∇⋅+
∂∂=
+
∂Φ∂
∇⋅+
∂∂=Φ∇
2
2
222
22 11
2
11
tD
D
au
tu
tau
tu
tac
c
where the subscript c on and on is intended to indicate that the velocity istreated 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
SOLO
Let compute the local pressure coefficient: 2
2
1:
∞∞
∞−=U
ppC p
ρ
We have:
−
=
−
=
−
=
−=
−
∞∞
=−
∞
∞
∞
=
−
∞∞
∞
=
=
∞∞
∞
∞
∞∞∞
−
∞∞
∞∞∞
12
12
112
12
1
2
2
2
/1
2
2
2
2
1
22
2
1
γγ
γγ
γ
γγ
ρ
γγ
ργγ
a
a
Ma
a
a
U
T
T
UTR
p
p
Up
C
aUMTRa
T
T
p
p
TRp
p
Let use the equation
( ) ( ) ( )
Ψ−Ψ+−Φ∇⋅Φ∇+
∂Φ∂−−= ∞∞∞
222
2
11 U
taa γ
to compute
( ) ( ) ( )
Ψ−Ψ+−Φ∇⋅Φ∇+
∂Φ∂−−= ∞∞
∞∞
2
22
2
2
111 U
taa
a γ
Finally we obtain:
( ) ( ) ( )
−
Ψ−Ψ+−Φ∇⋅Φ∇+
∂Φ∂−−=
−
∞∞∞∞
12
111
2 12
22
γγ
γγ
UtaM
C p
Linearized Flow Equations
124
SOLO
Assuming a stationary flow and neglecting the body forces :
=
∂∂
0t
( )0=Ψ
Φ∇⋅Φ∇∇⋅Φ∇=Φ∇⋅∇2
112a
( ) ( )222
2
1∞∞ −Φ∇⋅Φ∇−−= Uaa
γ
( ) ( )
−
−Φ∇⋅Φ∇−−=−
∞∞∞
12
11
2 12
22
γγ
γγ
UaM
C p
Φ∇=u
Linearized Flow Equations
125
SOLO
1
0
332211
323121
=⋅=⋅=⋅
=⋅=⋅=⋅→→→→→→
→→→→→→
eeeeee
eeeeee
General Coordinates ( )321 ,, uuu
→→→
∂Φ∂+
∂Φ∂+
∂Φ∂=Φ∇ 3
332
221
11
111e
uhe
uhe
uh
( ) ( ) ( )
∂∂+
∂∂+
∂∂=
++⋅∇=⋅∇
→→→
3213
2132
1321321
332211
1Ahh
uAhh
uAhh
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
SOLO
zzyyxx Φ+Φ+Φ=Φ∇=Φ∇⋅∇ 2
Φ+Φ+Φ∇⋅
Φ+Φ+Φ=
Φ∇⋅Φ∇∇⋅Φ∇
→→→222
2
1
2
1
2
1111
2
1zyxzyx zyx
( ) ( )( ) =ΦΦ+ΦΦ+ΦΦΦ+
ΦΦ+ΦΦ+ΦΦΦ+ΦΦ+ΦΦ+ΦΦΦ=
zzzyzyxzxz
yzzyyyxyxyxzzxyyxxxx
yzzyxzzxxyyxzzzyyyxxx ΦΦΦ+ΦΦΦ+ΦΦΦ+ΦΦ+ΦΦ+ΦΦ= 22222
Φ∇⋅Φ∇∇⋅Φ∇=Φ∇⋅∇2
112a
( ) ( )222
2
1∞∞ −Φ∇⋅Φ∇−−= Uaa
γ
( ) 012
2
22111
222
222
2
2
2
2
2
=Φ−ΦΦ+ΦΦ+ΦΦ−ΦΦΦ
−
ΦΦΦ
−ΦΦΦ
−Φ
Φ−+Φ
Φ−+Φ
Φ−
ttztzytyxtxyzzy
xzzx
xyyx
zzz
yyy
xxx
aaa
aaaaa
( ) ( ) ( )
Ψ−Ψ+−Φ+Φ+Φ+
∂Φ∂−−= ∞∞∞
222222
2
11 U
taa zyxγ
We finally obtain
Cartesian Coordinates ( )zuyuxu === 321 ,,
Linearized Flow Equations
Return to Table of Content
127
SOLO
Cylindrical Coordinates ( )θ=== 321 ,, uruxu
→→→→→→++=++= zryrxxzzyyxxR 1sin1cos1111 θθ
→→→→→+−=
∂∂+=
∂∂=
∂∂
zryrR
zyr
Rx
x
R1cos1sin&1sin1cos&1 θθ
θθθ
rR
hr
Rh
x
Rh =
∂∂==
∂∂==
∂∂=
θ
:&1:&1: 321
→→→→
→→→→→→
=+−=
∂∂∂∂
=
=+=
∂∂∂∂
==
∂∂∂∂
=
θθθ
θ
θ
θθ
11cos1sin:
&11sin1cos:&1:
2
21
zyR
R
e
rzy
r
R
r
R
ex
x
R
x
R
e
1
0
332211
323121
=⋅=⋅=⋅
=⋅=⋅=⋅→→→→→→
→→→→→→
eeeeee
eeeeeeWe have
Linearized Flow Equations
128
SOLO
Cylindrical Coordinates )continue – 1) ( )θ=== 321 ,, uruxu
→→→→→→Φ+Φ+Φ=
∂Φ∂+
∂Φ∂+
∂Φ∂=Φ∇ 321321
11e
reee
re
re
x rx θθ
2
2
22 1θΦ+Φ+Φ=Φ∇⋅Φ∇
rrx
→
→→
ΦΦ+ΦΦ+ΦΦ+
Φ−ΦΦ+ΦΦ+ΦΦ+
ΦΦ+ΦΦ+ΦΦ=
Φ+Φ+Φ∇=
Φ∇⋅Φ∇∇
322
22
3212
2
2
22
11
111
1
2
1
2
1
err
err
er
r
rrxx
rrrrxrxxrxrxxx
rx
θθθθθ
θθθθθ
θ
θθθ
θθ
Φ+Φ+Φ+Φ=∂
Φ∂+∂Φ∂+
∂Φ∂+
∂Φ∂=
∂Φ∂
∂∂+
∂Φ∂
∂∂+
∂Φ∂
∂∂=
=Φ∇⋅∇=Φ∇
22
2
22
2
2
2
2
1111
11
rrrrrrx
rrr
rxr
xr
rrrxx
Linearized Flow Equations
129
SOLO
Cylindrical Coordinates )continue – 2) ( )θ=== 321 ,, uruxu
Then equation
Φ∇⋅Φ∇∇⋅Φ∇+Φ∇
∂∂⋅Φ∇+
∂Φ∂=Φ∇⋅∇
2
12
12
2
2 ttabecomes
( ){
ΦΦ+ΦΦ+ΦΦ+
Φ−ΦΦ+ΦΦ+ΦΦ+
ΦΦ+ΦΦ+ΦΦ
Φ+Φ+Φ+
ΦΦ+ΦΦ+ΦΦ+Φ=Φ+Φ+Φ+Φ
→
→
→→→→
322
22
32
12321
22
11
11
11
2111
err
err
er
er
ee
arr
rrxx
rrrrxrx
xrxrxxxrx
ztzytyxtxttrrrxx
θθθθθ
θθθ
θθθ
θθ
or
( ) 02
112
/1
1/1
111
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−
ztzytyxtxtt
rrxxrxrx
rrrr
xxx
aa
rra
a
r
ra
r
raa
θθθθ
θθθ
θ
Linearized Flow Equations
130
SOLO
Cylindrical Coordinates )continue – 3) ( )θ=== 321 ,, uruxu
becomes
( ) ( ) ( )
Ψ−Ψ+−Φ∇⋅Φ∇+
∂Φ∂−−= ∞∞∞
222
2
11 u
taa γ
In cylindrical coordinates, equation
( ) ( )
Ψ−Ψ+
−Φ+Φ+Φ+Φ−−= ∞∞∞
22
2
2222 1
2
11 U
raa rxt θγ
Assuming a stationary flow and neglecting body forces
=
∂∂
0t
( )0=Ψ
0112
/1
1/1
111
222
2
22
2
22
22
2
2
2
=
ΦΦΦ+ΦΦΦ+ΦΦΦ−
Φ+Φ+Φ
Φ−+Φ
Φ−+Φ
Φ−
rrxxrxrx
rrrr
xxx
rra
a
r
ra
r
raa
θθθθ
θθθ
θ
( )
−Φ+Φ+Φ−−= ∞∞
22
2
2222 1
2
1U
raa rx θ
γ
Linearized Flow Equations
Return to Table of Content
131
Linearized Flow Equations SOLO
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 =⋅
Sun
n
- Normal at the Wing Surface
22
1/111
∂∂+
∂∂+
+
∂∂−
∂∂−=
y
z
x
zzy
y
zx
x
zn UUUU
U
( ) ( ) ( ) ( ) zwUyvxuUzwUyvxuUu 1'1'1'1'sin1'1'cos ++++≅++++= ∞∞∞∞ ααα
( ) ( ) 0,,''' =++∂∂−
∂∂+− ∞∞ U
UU zyxwUx
zv
x
zuU α
For Upper Surface
( ) ( )
−
∂∂≅
∂∂+
∂∂+= ∞∞ α
x
zU
x
zv
x
zuUzyxw U
onPerturbatiSmall
UUU '',,'
Therefore( )
( )( ) Sonyxallfor
x
zUzyxw
x
zUzyxw
LL
UU
,
,,'
,,'
−
∂∂≅
−
∂∂≅
∞
∞
α
α
Section AA (enlarged)
Wake region
132
Linearized Flow Equations SOLO
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 theAirfoil in the Camber Distribution zC )x,y) and the Thickness Distribution zt )x,y) by:
( )
( )( ) Sonyxallfor
x
zUyxw
x
zUyxw
CC
tt
,
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 theSolutions for the following Boundary Conditions
Superposition of• Angle of Attack•Camber Distribution•Thickness Distribution
Section AA (enlarged)
Wake region
( ) ( ) ( )
( ) ( ) ( )( ) Sonyxallfor
x
z
x
zUyxwyxwyxw
x
z
x
zUyxwyxwyxw
tCtCL
tCtCU
,
0,,'0,,'0,,'
0,,'0,,'0,,'
∂∂−−
∂∂=−+=±
∂∂+−
∂∂=++=±
∞
∞
α
α
133
Linearized Flow Equations SOLO
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 LeadingEdge Flow
Subsonic TrailingEdge Flow
Supersonic LeadingEdge Flow
Supersonic TrailingEdge 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 SupersonicFlow since the shape and location of theTrailing Edge exert no influence on the flow ahead.
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
211
2222
11
ρρρ
φ
+=+=
+≈+=
+=Φ+=
++=+=
+=
∞
∞
∞∞∞
∞
∞∞
∞
ppp
aaaaaa
xU
uu
uuUUuuu
uUu
O
Small Perturbation Assumptions:
∇⋅+
∂∂⋅+
∂Φ∂=⋅∇
22
1 2
2
2
2
uu
t
uu
tau
(C.M.) +(C.L.M)
(C.M.) +(C.L.M)
12
1
12
1 22
22
−+=
−++
∂∂ ∞
∞ γγφ a
Ua
ut
Bernoulli
121 −
∞
−
∞∞∞
=
=
=
γγ
γγγ
ρρ
a
a
T
T
p
pIsentropic Chain
Development of the Flow Equations:
Flow Equations:
( ) '' 21 φφ ∇=+∇⋅∇=⋅∇ ∞ xUu
( )1
12
2
1
1212
2
2
'''
1
2
1
x
u
a
U
x
uuU
a
uu
a ∂∂≅+
∂∂+≅
∇⋅
∞
∞∞
∞
( )t
uUuUU
tt
u
t
uu
∂∂=+
∂∂≅
∂∂=
∂∂⋅ ∞∞∞
'2'22 1
12
2
( )
∞
∞
∞∞
∞∞
∞∞∞∞ ++
∂∂=⇒
−+=
−++++
∂∂
ρ
γφ
γγφ
p
apuU
t
aU
aaauUU
t
2
1
22
2
12 ''
'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 γγ
γγ
ρργ
γγ
γγ
ρργ Isentropic Chain
Bernoulli
Linearized Flow Equations
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) 52.1&8.00''
2'1
'2
21
1
12
22 ≤≤≤≤
∂∂+
∂∂+
∂∂=∇ ∞∞
∞
MMtt
uU
x
uU
a
φφ
( )''
,,,'' 321
φφφ∇=
=u
xxxt
Bernoulli
+
∂∂−= ∞∞ '
'' 1uU
tp
φρ
∞∞∞∞ −=
−==
a
a
T
T
p
p '
1
2'
1
''
γγ
γγ
ρργIsentropic Chain
Linearized Flow Equations
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
Upxd
ud θ=
L
Lowxd
ud θ−=
∞U
x
z
( ) 0'''
12
2
2
2
2
22 =
∂∂+
∂∂+
∂∂− ∞ zyx
Mφφφ
(1)
( )zyx ,,'φ(2)
zw
yv
xu
∂∂=
∂∂=
∂∂= '
','
','
'φφφ
(3)
α−=≅+ ∞∞ S
xd
zd
U
w
uU
w '
'
'(4)
xUuUp
∂∂−=−= ∞∞∞∞
'''
φρρ(5)
'
21
1
''
1
2'
1
''2
MM
M
U
uM
a
a
T
T
p
p
∞
∞
∞∞
∞∞∞∞−+
−=−=−
=−
== γγγ
γγ
γγ
ρργ(6)
∂∂+
∂∂∂+
∂∂=∇
∞∞∞2
2
2
2
2
2
22 '1'2'1
'tUxtUxM
φφφφ
( )''
,,,''
φφφ∇=
=u
zyxt
+
∂∂−= ∞∞ '
'' uU
tp
φρ
Steady Three Dimensional Flow Small Perturbation Flow Equations: 0
'2
2
=∂∂=
∂∂
tt
52.1
8.00
≤≤≤≤
∞
∞
M
M
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
22 =
∂∂+
∂∂+
∂∂
zyx
φφφβ(1)
Steady Three Dimensional Flow
Subsonic Flow M∞ < 1
01: 22 >−= ∞Mβ
( )
( )
( )
( )α
ξαφ
αξ
αφ
−=−=∂∂=
−=−=∂∂=
∞∞
∞∞
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
( ) ( )
===
=−= ∞
ςηξφφςη
ξβξ
,,,,'
1 2
zyx
z
y
Mx
∂∂=
∂∂⇒
∂∂=
∂∂
∂∂=
∂∂⇒
∂∂=
∂∂
∂∂=
∂∂⇒
∂∂=
∂∂
2
2
2
2
2
2
2
2
2
2
22
2
''
''
1'1'
ςφφ
ςφφ
ηφφ
ηφφ
ξφ
βφ
ξφ
βφ
zz
yy
xx
( ) ( ) SMdcMydycSbb ∞∞ −=−== ∫∫ 2020
11 ηη( ) ( )ηcMyc 21 ∞−=
∞∞ −=
−==
22
22
11 M
AR
SM
b
S
bAR
22 1
2
1
12
∞∞∞∞ −=
∂∂
−−=
∂∂−=
M
C
UMxUC p
p ξφφ
Section AA (enlarged)
Wake region
so 02
2
2
2
2
2
=∂∂+
∂∂+
∂∂
ςφ
ηφ
ξφ
Laplace’s Equation like in Incompressible Flow
Similarity Rules
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)
incpCM 21
1
∞−
incLCM 21
1
∞−
22 1
2
1
1
∞∞ −=
− Md
Cd
M inc
L αα
incMCM 21
1
∞−
inc0α
4
1=
inc
N
c
x
incMCM
021
1
∞−
incLsCM 21
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 FlowBased on Subsonic Similarity Rules
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
Upxd
ud θ=
L
Lowxd
ud θ−=
∞U
x
y
( ) 0''
12
2
2
22 =
∂∂+
∂∂− ∞ yx
Mφφ(1)
( )yx,'φ(2)
yv
xu
∂∂=
∂∂= '
','
'φφ(3)
α==≅+ ∞∞ S
xd
yd
U
v
vU
v '
'
'(4)
xUuUp
∂∂−=−= ∞∞∞∞
'''
φρρ(5)
'
21
1
''
1
2'
1
''2
MM
M
U
uM
a
a
T
T
p
p
∞
∞
∞∞
∞∞∞∞−+
−=−=−
=−
== γγγ
γγ
γγ
ρργ(6)
∂∂+
∂∂+
∂∂=∇ ∞∞
∞2
21
1
12
22 ''
2'1
'tt
uU
x
uU
a
φφ
( )''
,,,'' 321
φφφ∇=
=u
xxxt
+
∂∂−= ∞∞ '
'' uU
tp
φρ
Steady Two Dimensional Flow Small Perturbation Flow Equations: 0
'2
2
=∂∂=
∂∂
tt
52.1
8.00
≤≤≤≤
M
M
Linearized Flow Equations
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
22 =
∂∂+
∂∂
yx
φφβ(1)
Steady Two Dimensional Flow
Subsonic Flow M∞ < 1
01: 22 >−= ∞Mβ
( )
( )
( )
( )αφ
αφ
−=∂∂=
−=∂∂=
∞∞
∞∞
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
,', φβηξφβη
ξ
∂∂=
∂∂
∂∂=
∂∂
∂∂=
∂∂
∂∂+
∂∂
∂∂=
∂∂=
∂∂
∂∂=
∂∂
∂∂+
∂∂
∂∂=
∂∂=
∂∂
2
2
2
2
2
2
2
2 ',
1'
11'
111'
ηφφ
ξφ
βφ
ηφη
ηφξ
ξφ
βφ
βφ
ξφ
βη
ηφξ
ξφ
βφ
βφ
yx
yyyy
xxxx
so 02
2
2
2
=∂∂+
∂∂
ηφ
ξφ
Laplace’s Equation like in Incompressible Flow
Linearized Flow Equations
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 theCompressible Flow over an Airfoil in (x,y) Space to the Incompressible Flowin (ξ,η) over the same Airfoil.
αηφφ −=
∂∂=
∂∂=
∞∞∞ Uper
Upper
xd
yd
UyUU
v 1'1'α
ηφφ −=
∂∂=
∂∂=
∞∞∞ Lower
Lower
xd
yd
UyUU
v 1'1'
( )yx,ρρ =
x
y η
ξ
∞= ρρ
Compressible Flow Incompressible Flow
αηφ −=
∂∂=
∞∞ Uper
Upper
xd
fd
UU
v 1'
αηφ −=
∂∂=
∞∞ Lower
Lower
xd
fd
UU
v 1'
Linearized Flow Equations
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
φβ
φ(1)
( ) ( ) ( )( ) ( ) ( ) yxGyxGyx
yxFyxFyx
Lower
Upper
βννβφβηηβφ
+==+=
−==−=
:,'
:,'(7)(8)
Steady Two Dimensional Flow
Supersonic Flow M∞ > 1
01: 22 >−= ∞Mβα
U
Upxd
yd θ=
L
Lowxd
yd θ−=
∞U
x
y1
12 −
=∞Mxd
yd
1
12 −
−=∞Mxd
yd
Flow
Flow
( )
( )
( )
ηβα
d
Fd
Uxd
yd
U
v
Uper
Upper
∞∞
−=−=1
7
4'
( ) ( )
ηφ
d
Fd
xd
du Upper
73 '' ==
−
−−=
∞
∞ αUpper
Upper xd
yd
M
Uu
1'
2
( )
( )
( )
νβα
d
Gd
Uxd
yd
U
v
Lower
Lower
∞∞
=−=3
8
4'
( ) ( )
νφ
d
Gd
xd
du Lower
83 '' ==
−
−=
∞
∞ αLower
Lower xd
yd
M
Uu
1'
2
−
−=−=
∞
∞∞∞∞ αρρ
Upper
UpperUpper xd
yd
M
UuUp
1''
2
2
−
−−=−=
∞
∞∞∞∞ αρρ
Lower
LowerLower xd
yd
M
UuUp
1''
2
2
Linearized Flow Equations
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
sdxd
ydppD αsin
np
α−Upper
xd
yd
∞U
Upperxd
yd
∞p∞p α( )∫
−−−= ∞
S S
sdxd
ydppL αcos
( )∫
−−≅ ∞
S S
sdxd
ydppD α
( )
Γ
∞∞∞ ∫∫ =
−−−≅
SS S
sduUsdxd
ydppL 'ρα
1<<−αUper
xd
yd
1<<−αUper
xd
yd
Kutta-Joukovsky
Define: 2
21
:
∞∞
∞−=U
ppC p
ρ
( )
( )∫∫
∫∫
−−=
−−−≅
−=
−−≅
∞∞
∞∞
∞∞∞
∞∞
∞∞
∞∞∞
S S
p
S S
S S
p
S S
sdxd
ydCUsd
xd
yd
U
ppUL
sdxd
ydCUsd
xd
yd
U
ppUD
αραρ
ρ
αραρ
ρ
2
2
2
2
2
2
2
1
2
12
1
2
1
2
12
1
Linearized Flow Equations
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
Upperxd
yd
∞p∞p α
We found:
α−=∞ xd
fd
U
v' αξ
−=∞ d
gd
U
v
( ) ( )
−=
−=
=
∞
∞
yxM
yM
x
,'1,
1
2
2
φηξφ
η
ξ
( ) 0''
12
2
2
22 =
∂∂+
∂∂− ∞ yx
Mφφ 0
2
2
2
2
=∂∂+
∂∂
ηφ
ξφ
yv
xu
∂∂=
∂∂= '
','
'φφ
ηφ
ξφ
∂∂=
∂∂= vu ,vv
M
uu =
−=
∞
',1
'2
'' uUp ∞∞−= ρ uUp ∞∞−= ρ
xUU
u
U
ppC p ∂
∂−=−=−=∞∞
∞∞
∞ '2'2
21
':
2
φ
ρ ξφ
ρ ∂∂−=−=−=
∞∞∞∞
∞
UU
u
U
ppC p
22
21
:2
0
21'
∞−=
M
pp
210
∞−=
M
CC p
p
Compressible: Incompressible:
Linearized Flow Equations
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
Upperxd
yd
∞p∞p α
The Relation:
∫∫
∫∫
−−=
−−≅
−
−=
−≅
∞
∞∞
∞∞
∞
∞∞
∞∞
S
p
S S
p
S S
p
S S
p
c
sdC
M
U
c
sd
xd
ydCUL
c
sd
xd
ydC
M
U
c
sd
xd
ydCUD
0
0
2
2
2
2
2
2
12
1
2
1
121
2
1
ραρ
αρ
αρ
210
∞−=
M
CC p
pPrandtl-Glauert
Compressibility Correction
As earlier in 1922, Prandtl is quoted as stating that the LiftCoefficient increased according to (1-M∞
2)-1/2; he mentionedthis at a Lecture at Göttingen, but without a proof. This result wasmentioned 6 years later by Jacob Ackeret, again without proof.The result was finally established by H. Glauert in 1928 based onLinear Small Perturbation.
Ludwig Prandtl(1875 – 1953)
Hermann Glauert(1892-1934)
Linearized Flow Equations Return to
Critical Mach Number
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:
( )[ ] 2/11/10
0
222p
pp
CMMM
CC
∞∞∞ −++−= Karman-Tsien
Rule
Linearized Flow Equations
( )0
0
2222 12/2
111 p
pp
CMMMM
CC
−
−++−
=
∞∞∞∞γ
Laitone’sRule
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
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
φβ
φ(1)
( ) ( ) ( )( ) ( ) ( ) zxFzxGzx
zxFzxFzx
Lower
Upper
βννβφβηηβφ
+==+=
−==−=
:,'
:,'(7)(8)
Steady Two Dimensional Flow
Supersonic Flow M∞ > 1
01: 22 >−= ∞Mβ
αU
Upxd
zd θ=
L
Lowxd
zd θ−=
∞U
x
z1
12 −
=∞Mxd
zd
1
12 −
−=∞Mxd
zd
Flow
Flow
( )
( )
( )
ηβα
d
Fd
Uxd
zd
U
w
Upper
Upper
∞∞
−=−=3
7
4'
( ) ( )
ηφ
d
Fd
xd
du Upper
73 '' ==
( )
( )
( )
νβα
d
Gd
Uxd
zd
U
w
Lower
Lower
∞∞
==−=3
8
4'
( ) ( )
νφ
d
Gd
xd
du Lower
83 '' ==
−
−−=
∞
∞ αUpper
Upper xd
zd
M
Uw
1'
2
−
−=
∞
∞ αLower
Lower xd
zd
M
Uw
1'
2
−
−=−==−
∞
∞∞∞∞∞ αρρ
Upper
UpperUpperUpper xd
zd
M
UwUppp
1''
2
2
−
−−=−==−
∞
∞∞∞∞∞ αρρ
Lower
LowerLowerLower xd
zd
M
UwUppp
1''
2
2
zw
xu
∂∂=
∂∂= '
','
'φφ
(3)
α−=≅+ ∞∞ S
xd
zd
U
w
uU
w '
'
'(4)
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
Upxd
zd θ=
L
Lowxd
zd θ−=
∞U
x
z1
12 −
=∞Mxd
zd
1
12 −
−=∞Mxd
zd
Flow
Flow
Pressure Distribution and Lift Coefficient
−+
−=
−=
∞∞∞
αρ
21
2
2/
''22
LowerUpper
LowerUpperp xd
zd
xd
zd
MU
ppC
1
42 −
=∞M
cL
α
( ) ( ) ( ) ( )
−+−
−−
−=
+
−
−
−=
+
−=
∞∞
∞∫∫∫∫
00
22
1
0
1
02
1
0
1
0
001
2
1
4
21
2
LowerLowerUpperUpper
LowerUpper
ppL
zczzczMM
c
xd
xd
zd
c
xd
xd
zd
Mc
xdC
c
xdCc
LowerUpper
α
α
−
−=
∞
αUpper
p xd
zd
MC
Upper
1
22
−
−−=
∞
αLower
p xd
zd
MC
Lower
1
22
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
Upxd
zd θ=
L
Lowxd
zd θ−=
∞U
x
z1
12 −
=∞Mxd
zd
1
12 −
−=∞Mxd
zd
Flow
Flow
Wave Drag Coefficient
−+
−
−=
−−
−= ∫∫∫∫
∞
1
0
2
1
0
2
2
1
0
1
0 1
2
c
xd
xd
zd
c
xd
xd
zd
Mc
xd
xd
zdC
c
xd
xd
zdCc
UpperUpperLower
p
Upper
pD LowerUpperWαααα
−
−=
∞
αUpper
p xd
zd
MC
Upper
1
22
−
−−=
∞
αLower
p xd
zd
MC
Lower
1
22
( ) ( ) ( ) ( )
+
−+
+
−
−= ∫∫∫∫
=−=−
∞
1
0
2
00
1
0
21
0
2
00
1
0
2
222
1
2
c
xd
xd
zd
c
xd
xd
zd
c
xd
xd
zd
c
xd
xd
zd
M Lower
zcz
LowerUpper
zcz
Upper
LowerLowerUpperUpper
αααα
( )22
22
2
1
2
1
4LowerUpperD
MMC
Wεεα +
−+
−=
∞∞
∫
∫
=
=
1
0
2
2
1
0
2
2
:
:
c
xd
xd
zd
c
xd
xd
zd
Lower
Lower
Upper
Upper
ε
ε
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
LowerUpperxd
zd
xd
zd
Double Wedge Airfoil
1
42
2
−=
∞MC
WD
α
022 == LowerUpper εε
( ) ( ) ( )kkc
tck
c
t
kck
c
t
kcLowerUpper −=
−−
+==14
11
14
1
4
112
2
2
2
22
2
222 εε
( ) ( )
<<−
<<−=
<<−
−
<<=
cxckck
t
ckxck
t
xd
zd
cxckck
t
ckxck
t
xd
zd
LowerUpper
12
02
12
02
( )( )kk
ct
MMC
WD −−+
−=
∞∞1
/
1
1
1
4 2
22
2α
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 twoCircular Arcs of radius R. c – the chordt – maximum thickness at x = c/2
( ) ( ) ( )tcttcRtc
4/4/ 22222 >>
≈+=
θθθθ −≈−=≈= tan,tanLowerUpper
xd
zd
xd
zd
22
2/2
/2
321
0
2
1
0
2
2
3
2
34
11: Lower
ct
ctUpperUpper
Upper c
t
t
cdR
cxd
xd
zd
cc
xd
xd
zd εθθθεδ
δ==≈≈
=
=
+
−
+
−∫∫∫
c
t
R
c
xd
zd
MaxUpper
22/
, ≈≈≈
δδ
( )2
2
22
222
22
2
3
16
1
1
1
4
1
2
1
4
c
t
MMMMC LowerUpperDW −
+−
=+−
+−
=∞∞∞∞
αεεα
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
12( )kk −1
1
( )( ) xckck
xcxtz
212 22 −+−±=
( )
<<−
±
<<±=
cxckxck
t
ckxxck
t
z
12
02
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
12
154
SOLO Wings in Compressible Flow
Double Wedge
Modified Double Wedge
Biconvex
τ2
1
2
1221
2'
2==
=c
t
c
tc
A
τ3
2
3
23321
2'
2==
+
=c
t
c
tc
tc
A
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 CoefficientThe Pitching Moment Coefficient about theLeading Edge for any Thin Airfoil is given by
xdxxd
zd
xd
zd
Mcc
xd
c
xC
c
xd
c
xCc
c
LowerUpper
ppM LowerUpperLE ∫∫∫
−+
−
−−=
+
−=−
∞022
1
0
1
0 1
2 αα
Thus
+−
+−
−= ∫∫∞∞
xdzxdzMcM
cc
Lower
c
UpperM LE 00222 1
2
1
2α
( ) ( ) ( ) ( )[ ] xdzxdzczczcxdzzxxdzzxxdxxd
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
22 −
−=∞M
cM
α
The distance of the Airfoil Center of Pressure aft of the Leading Edge is given by
ccM
Mc
c
c
c
x
L
MN
2
1
1/4
1/22
2
=⋅−
−=⋅−=
∞
∞
αα
αL
∞U
x
Return to Table of Content
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
xd
xd
yd
xd
yd
M
U
c
sdCUL
c
xd
xd
yd
xd
yd
M
U
c
sd
xd
ydCUD
c
LowerUpperS
p
c
LowerUpperS S
p
02
2
2
0
22
2
2
2
21
2
1
2
1
21
21
2
1
ααρ
ρ
ααρ
αρ
αU
Upxd
yd θ=
L
Lowxd
yd θ−=
∞U
x
y1
12 −
=∞Mxd
yd
1
12 −
−=∞Mxd
yd
Flow
Flow
−
−==−
∞
∞∞∞ αρ
Upper
UpperUpper xd
yd
M
Uppp
1'
2
2
−
−−==−
∞
∞∞∞ αρ
Lower
LowerLower xd
yd
M
Uppp
1'
2
2
1
2
1
2
2
2
−
−
−=
−
−
=
∞
∞
M
xdyd
C
M
xdyd
C
Lowerp
Upper
p
Lower
Upper
α
αWe 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 inZeitschhrift fur Flugtechnik und Motorluftschiffahrt, vol. 16, 1925, p.72
Jakob Ackeret (1898–1981)
Linearized Flow Equations
157
AERODYNAMICSSmall 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
−
−−=−
+−=
−
+=−
−=
∞∞
∞∞
MC
MC
MC
MC
pp
pp
αδαδ
αδαδ
( ) ( )
1
4
2
1
1
4
2
1
1
4222
1
2/1
2/1
0 3412
−=
−+
−=
−+
−=
=
∞∞∞
∫∫∫
MMM
c
xdCC
c
xdCC
c
sdCC pppp
S
pX
ααα
( ) ( )
( ) ( )1
4
1
4
22
22 2
2/
2
0
2/
2/
0
3412
3412
−=
−×=−+−=
−+
−=
=
∞
=
∞
−∫∫∫
MMc
tCC
c
tCC
c
t
c
ydCC
c
ydCC
c
ydCC
ct
pppp
ct pp
ct
pp
S
pX
δδ δ
XYXYD
XYXYL
CCCCC
CCCCC
+≈+=
−≈−=<<
<<
ααα
αααα
α
1
1
cossin
sincos
1
4
1
4
1
4
1
4
2
2
2
21
2
2
2
1
−+
−≈
−−
−≈
∞∞
<<
∞∞
<<
MMC
MMC
D
L
δα
αδα
α
α
158
AERODYNAMICSSmall 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
pc−
cx /
0.1
pc−
cx /
0.1
pc−
cx /
0.1
αδα<
δ∞M
δα >
∞M
∞M
δα =α
∞M
Upper Surface
Lover Surface
Expansion
ShockShock
Expansion
ExpansionShock
Expansion
Shock
Shock
Expansion
Expansion
Shock
Shock
Shock
Shock
∞M
∞M
( )1
22 −
−=∞M
c p
αδ
( )1
22 −
+=∞M
c p
αδ
( )1
22 −
−=∞M
c p
αδ
( )1
22 −
+=∞M
c p
αδ
1
42 −
=∞M
c p
α
1
42 −
−=∞M
c p
α
( )1
22 −
+−=∞M
c p
αδ
( )1
22 −
−−=∞M
c p
αδ
( )1
22 −
+−=∞M
c p
αδ
( )1
22 −
−−=∞M
c p
αδ
Supersonic Flow past a Symmetric Biconvex Aerfoil
AERODYNAMICSSmall 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)
22
2
2
22
2
4
1
1
316
316
1
4
LD
L
CM
M
ct
C
ctD
L
Md
Cd
−+
−
=
+
=
−=
∞
∞
∞
α
α
α
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 FlowBased on the Linear Theory Supersonic Rules
−
−=
∞Xd
Zd
Mα
1
12
1
42 −
=∞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
4XdZ
M
S
Wave DragL
D
Cd
Cd 14
1 2 −−= ∞M
( ) ( )
∫
+
−−=
∞
1
0
22
2 1
4Xd
Xd
Zd
Xd
Zd
M
tS
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.
The magnitude of Lift is given by Ackeret Theory
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.
164
AERODYNAMICSSOLO
Return to Table of Content
Continue to Aerodynamics – Part III
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 toTheoretical 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
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 theirHistorical Development”,
Prentice-Hall, 1979
L.J. Clancy“Aerodynamics”,
John Wiley & Sons, 1975
References (continue – 1)
AERODYNAMICSSOLO
167
Frank G. Moore“Approximate Methods
for Missile Aerodynamics”, AIAA, 2000
Thomas J. Mueller, Ed.“Fixed and Flapping WingAerodynamics for Micro Air
Vehicle Applications”, AIAA, 2002
Richard S. Shevell“Fundamentals of Flight”, Prentice Hall, 2nd Ed., 1988 Ascher H. Shapiro
“The Dynamics and Thermodynamicsof 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
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
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February 10, 2015 169
SOLO
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 –2013
Stanford University1983 – 1986 PhD AA
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
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
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
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)