Aerodynamics part ii

1

AERODYNAMICS Part II

SOLO HERMELIN

http://www.solohermelin.com

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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

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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



4


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



5


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


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


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

9

Continue from AERODYNAMICS – Part I

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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

11

SOLO SHOCK & EXPANSION WAVES

M < 1

M = 1

M > 1

Mach 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


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



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



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



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


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

19


u

p

ρT

e

u

p

ρT

e

τ11

q

1

1

1

1

1

2

2

2

2

2

1 2

SOLO

20



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



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



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


23



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



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



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



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


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



G Q= =0 0,


( ) ( ) ( ) ( )

( ) ( )( ) ( )

( )[ ]( ) ( ) ( )

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



G Q= =0 0,


( )( )

( ) ( )( )

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



G Q= =0 0,


( )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



G Q= =0 0,


( )

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(


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


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


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


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


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)

41

Hodograph Shock PolarSOLO OBLIQUE S�HOCK & EXPANS�ION WAVES�

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.


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


44

SOLO

θmaxθ

maxθ

maxθ

γ


45

SOLO OBLIQUE SHOCK & EXPANSION WAVES

46


47


48


49



50


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

smb://upload.wikimedia.org/wikipedia/commons/e/ea/Prandtl_portrait.jpg

http://en.wikipedia.org/wiki/File:Theodor-Meyer-at-Munsterlager.jpg

51


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


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 −=


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


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )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


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



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



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


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


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


AERODYNAMICS

http://en.wikipedia.org/wiki/File:Transonic_flow_patterns.svg

62

AERODYNAMICS

Drag Variation with Mach Number

SOLO


63

AERODYNAMICS


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

http://en.wikipedia.org/wiki/File:Adolf_Busemann_at_Langley.jpg

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


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


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.

67

AERODYNAMICS


SOLO


AERODYNAMICS


SOLO

69

AERODYNAMICS


SOLO

70

AERODYNAMICS


Comparison of the Transonic Drag Polar for an Unswept Wing with that for a Swept Wing (data from Schlichting)

SOLO

71


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

AERODYNAMICS


SOLO

73

SOLO


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

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

SOLO


SOLO


SOLO

77

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

SOLO


78

SOLO


Ray Whitford, “Design for Air Combat”


SOLOReturn to Table of Content


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.


Adolph Busemann (1901 – 1986)also NACA & Colorado U.

Origin of Transonic Area Rule

http://history.nasa.gov/SP-4219/Chapter5.html

SOLO




82



Origin of Transonic Area Rule


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

http://www.youtube.com/watch?v=xZWBVgL8I54

http://www.youtube.com/watch?v=Cn0lSoreB1g



SOLO


SOLO


SOLO


SOLO


SOLO

88

SOLO


AERODYNAMICSSOLO


90

AERODYNAMICSSOLO

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

http://en.wikipedia.org/wiki/File:Wing_profile_nomenclature.svg

http://en.wikipedia.org/wiki/File:Examples_of_Airfoils.svg

http://en.wikipedia.org/wiki/File:Lift_drag_graph.JPG

93

SOLO

94

AERODYNAMICSSOLO


AERODYNAMICSSOLO


AERODYNAMICSSOLO


97

Flow regimes for a slender bodySOLO

98

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

99

Slender wing-body combinationSOLO

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

101

Missile controlMissile controlSOLO

102

CHARACTERISTICSCHARACTERISTICS

Summary of aerodynamic designSummary of aerodynamic designSOLO

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


105Density Profile Mach 1.2, Color Contours Modified to see Detail on Shock Waves

More Fun With CFD – RM-10SOLO

f7so, 09/05/2004

Reference used by "More Fun with CFD"[1] "The Zero-Lift Drag of a Slender Body of Revolution (NACA RM-10 Research Model) As Determined From Tests in Several Wind Tunels And In Flight At Supersonic Speeds" Evans, Albert J., NACA, NACA TN 2944, 1953

106

Density Profiles, Mach 2.41, simulated altitude of 11,000 ft )Re=76.4x106)


107Density Profiles, Mach 2.41 – color contours modified to see detail in shock waves


108Density Profiles, Mach 1.62 – rotated, with plot to show distribution around fins


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

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

SOLO

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

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

f7so, 09/05/2004

"Unstructured Navier-Stokes/Euler Solver"Laith Zori & Ganish Rajagopalan

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

AERODYNAMICSSOLO

f7so, 09/05/2004

"Unstructured Navier-Stokes/Euler Solver"Laith Zori & Ganish Rajagopalan

115

AERODYNAMICSSOLO

116


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


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


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


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


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


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


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


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


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


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 ,,



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


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


129

SOLO


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


130

SOLO


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



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


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


Wake region

( ) ( ) ( )

( ) ( ) ( )( ) Sonyxallfor

x

z

x

zUyxwyxwyxw

x

z

x

zUyxwyxwyxw

tCtCL

tCtCU

,

0,,'0,,'0,,'

0,,'0,,'0,,'

∂∂−−

∂∂=−+=±

∂∂+−

∂∂=++=±

∞

∞

α

α

133


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.

http://www.bugman123.com/FluidMotion/KuttaCondition.m1v

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


135

SOLO


'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


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


=×∇ 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


Wake region

so 02

2

2

2

2

2

=∂∂+

∂∂+

∂∂

ςφ

ηφ

ξφ

Laplace’s Equation like in Incompressible Flow

Similarity Rules

138


=×∇ u


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



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


140

SOLO



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


141

SOLO




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'


142

SOLO



0'1'

2

2

22

2

=∂∂−

∂∂

yx

φβ

φ(1)

( ) ( ) ( )( ) ( ) ( ) yxGyxGyx

yxFyxFyx

Lower

Upper

βννβφβηηβφ

+==+=

−==−=

:,'

:,'(7)(8)


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


143

SOLO



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


144

SOLO



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:


145

SOLO



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.


Hermann Glauert(1892-1934)

Linearized Flow Equations Return to


smb://upload.wikimedia.org/wikipedia/commons/e/ea/Prandtl_portrait.jpg

SOLO



Several improved formulas where developed:

( )[ ] 2/11/10

0

222p

pp

CMMM

CC

∞∞∞ −++−= Karman-Tsien

Rule


( )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◦.


147


=×∇ u


0'1'

2

2

22

2

=∂∂−

∂∂

zx

φβ

φ(1)

( ) ( ) ( )( ) ( ) ( ) zxFzxGzx

zxFzxFzx

Lower

Upper

βννβφβηηβφ

+==+=

−==−=

:,'

:,'(7)(8)



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


=×∇ u



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


=×∇ u



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


=×∇ u





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


=×∇ u





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


=×∇ u





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


=×∇ u





Wave Drag at Supersonic Incident Flow versus Relative Thickness Position

for Double Wedge and Parabolic Profiles

k

( )kk −1

1

( )kk −13

12

154


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


=×∇ u


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


156

SOLO



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)


157

AERODYNAMICSSmall Perturbation Flow (Homentropic: , Isentropic ) ( )0~,0,0~,0 ==== τQqsd ( )0

=×∇ u



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


=×∇ u



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


=×∇ u



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


=×∇ u


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.

162

AERODYNAMICS



SOLO AERODYNAMICS

164

AERODYNAMICSSOLO


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


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


AERODYNAMICSSOLO


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

http://upload.wikimedia.org/wikipedia/en/a/ae/RAFAEL_logo.png

170


University of Göttingen

Max Michael Munk (1890—1986)[

also NACA

Theodor Meyer (1882 - 1972


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)


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)


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)

174

Science

Aerodynamics part ii