aerodynamics part i
TRANSCRIPT
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
Dimensionless Equations
Boundary Layer and Reynolds Number
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
Flow Description
Streamlines, Streaklines, and Pathlines
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
Wing Configurations
Wing Parameters
References
5
Table of Content (continue – 3)
AERODYNAMICSSOLO
Linearized Flow Equations
Cylindrical Coordinates
Small Perturbation Flow
Applications: Nonsteady One-Dimensional Flow
Applications: Two Dimensional Flow
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
AERODYNAMICSPARTII
6
Table of Content (continue – 4)
AERODYNAMICSSOLO
Analytic Theory and CFD
Transonic Area Rule
Aircraft Flight Control
AERODYNAMICSPARTII
7Wright Brothers First Flight
AERODYNAMICSSOLO
SOLO
Atmosphere
Continuum FlowLow-density and
Free-molecular Flow
Viscous Flow Inviscid Flow
Incompressible Flow
Compressible Flow
Subsonic Flow
Transonic Flow
Supersonic Flow
Hypersonic Flow
AERODYNAMICS
AERODYNAMICS
9
Percent composition of dry atmosphere, by volume
ppmv: parts per million by volume
Gas Volume
Nitrogen (N2) 78.084%
Oxygen (O2) 20.946%
Argon (Ar) 0.9340%
Carbon dioxide (CO2) 365 ppmv
Neon (Ne) 18.18 ppmv
Helium (He) 5.24 ppmv
Methane (CH4) 1.745 ppmv
Krypton (Kr) 1.14 ppmv
Hydrogen (H2) 0.55 ppmv
Not included in above dry atmosphere:
Water vapor (highly variable) typically 1%
Gas Volume
nitrous oxide 0.5 ppmv
xenon 0.09 ppmv
ozone 0.0 to 0.07 ppmv (0.0 to 0.02 ppmv in winter)
nitrogen dioxide 0.02 ppmv
iodine 0.01 ppmv
carbon monoxide trace
ammonia trace
•The mean molecular mass of air is 28.97 g/mol.
Minor components of air not listed above include:
Composition of Earth's atmosphere. The lower pie represents the trace gases which together compose 0.039% of the atmosphere. Values normalized for illustration. The numbers are from a variety of years (mainly 1987, with CO2 and methane from 2009) and do not represent any single source
Earth AtmosphereSOLO
10
Earth AtmosphereSOLO
Earth AtmosphereSOLO
The Earth Atmosphere might be described as a Thermodynamic Medium in a Gravitational Field and in Hydrostatic Equilibrium set by Solar Radiation. Since Solar Radiation and Atmospheric Reradiation varies diurnally and annually and with latitude and longitude, the Standard Atmosphere is only an approximation.
SOLO
12
The purpose of the Standard Atmosphere has been defined by the World Metheorological Organization (WMO). The accepted standards are the COESA (Committee on Extension to the Standard Atmosphere) US Standard Atmosphere 1962, updated by US Standard Atmosphere 1976.
Earth Atmosphere
The basic variables representing the thermodynamics state of the gas are the Density, ρ, Temperature, T and Pressure, p.
SOLO
13
The Density, ρ, is defined as the mass, m, per unit volume, v, and has units of kg/m3.
v
mv ∆
∆=→∆ 0
limρ
The Temperature, T, with units in degrees Kelvin ( K). Is a measure of the average kinetic energy of gas particles.
The Pressure, p, exerted by a gas on a solid surface is defined as the rate of change of normal momentum of the gas particles striking per unit area.
It has units of N/m2. Other pressure units are millibar (mbar), Pascal (Pa), millimeter of mercury height (mHg)
S
fp n
S ∆∆=
→∆ 0lim
kPamNbar 100/101 25 ==
( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 === The Atmospheric Pressure at Sea Level is:
Earth Atmosphere
14
Physical Foundations of Atmospheric Model
The Atmospheric Model contains the values of Density, Temperature and Pressure as function of Altitude.
Atmospheric Equilibrium (Barometric) Equation
In figure we see an atmospheric element under equilibrium under pressure and gravitational forces
( )[ ] 0=⋅+−+⋅⋅⋅− APdPPHdAg gρ
or ( ) gg HdHgPd ⋅⋅=− ρ
In addition, we assume the atmosphere to be a thermodynamic fluid. At altitude bellow 100 km we assume the Equation of an Ideal Gas
where V – is the volume of the gas N – is the number of moles in the volume V m – the mass of gas in the volume VR* - Universal gas constant
TRNVP ⋅⋅=⋅ *
V
m
M
mN == ρ&
MTRP /* ⋅⋅= ρ
Earth AtmosphereSOLO
( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2 ===
Earth AtmosphereSOLO
We must make a distinction between:- Kinetic Temperature (T): measures the molecular kinetic energy and for all practical purposes is identical to thermometer measurements at low altitudes. - Molecular Temperature (TM): assumes (not true) that the Molecular Weight at any altitude (M) remains constant and is given by sea-level value (M0)
SOLO
16
TM
MTM ⋅= 0
To simplify the computation let introduce:- Geopotential Altitude H- Geometric Altitude Hg
Newton Gravitational Law implies: ( )2
0
+
⋅=gE
Eg HR
RgHg
The Barometric Equation is ( ) gg HdHgPd ⋅⋅=− ρ
The Geopotential Equation is defined as HdgPd ⋅⋅=− 0ρ
This means thatg
gE
Eg Hd
HR
RHd
g
gHd ⋅
+
=⋅=2
0
Integrating we obtaing
gE
E HHR
RH ⋅
+
=
Earth Atmosphere
17
Atmospheric Constants
Definition Symbol Value Units
Sea-level pressure P0 1.013250 x 105 N/m2
Sea-level temperature T0 288.15 K
Sea-level density ρ0 1.225 kg/m3
Avogadro’s Number Na 6.0220978 x 1023 /kg-mole
Universal Gas Constant R* 8.31432 x 103 J/kg-mole - K
Gas constant (air) Ra=R*/M0 287.0 J/kg--K
Adiabatic polytropic constant γ 1.405
Sea-level molecular weight M0 28.96643
Sea-level gravity acceleration g0 9.80665 m/s2
Radius of Earth (Equator) Re 6.3781 x 106 m
Thermal Constant β 1.458 x 10-6 Kg/(m-s- K1/2)
Sutherland’s Constant S 110.4 K
Collision diameter σ 3.65 x 10-10 m
Earth AtmosphereSOLO
18
Physical Foundations of Atmospheric Model
Atmospheric Equilibrium Equation
HdgPd ⋅⋅=− 0ρAt altitude bellow 100 km we assume t6he Equation of an Ideal Gas
TRMTRP a
MRR
a
aa
⋅⋅=⋅⋅==
ρρ/
**
/
HdTR
g
P
Pd
a
⋅=− 0
Combining those two equations we obtain
Assume that T = T (H), i.e. function of Geopotential Altitude only. The Standard Model defines the variation of T with altitude based on experimental data. The 1976 Standard Model for altitudes between 0.0 to 86.0 km is divided in 7 layers. In each layer dT/d H = Lapse-rate is constant.
Earth AtmosphereSOLO
19
Layer Index
GeopotentialAltitude Z,
km
GeometricAltitude Z;
km
MolecularTemperature T,
K
0 0.0 0.0 288.150
1 11.0 11.0102 216.650
2 20.0 20.0631 216.650
3 32.0 32.1619 228.650
4 47.0 47.3501 270.650
5 51.0 51.4125 270.650
6 71.0 71.8020 214.650
7 84.8420 86.0 186.946
1976 Standard Atmosphere : Seven-Layer Atmosphere
Lapse RateLh;
K/km
-6.3
0.0
+1.0
+2.8
0.0
-2.8
-2.0
Earth AtmosphereSOLO
20
Physical Foundations of Atmospheric Model
• Troposphere (0.0 km to 11.0 km). We have ρ (6.7 km)/ρ (0) = 1/e=0.3679, meaning that 63% of the atmosphere lies below an altitude of 6.7 km.
( ) HdHLTR
gHd
TR
g
P
Pd
aa
⋅⋅+
=⋅=−0
00
kmKLHLTT /3.60−=⋅+=
Integrating this equation we obtain
( )∫∫ ⋅⋅+
=−H
a
P
P
HdHLTR
g
P
PdS
S 0 0
0 1
0
( )0
00 lnln0
T
HLT
RL
g
P
P
aS
S ⋅+⋅⋅
−=
HenceaRL
g
SS HT
LPP
⋅−
⋅+⋅=
0
0
0
1
and
−
⋅=
⋅
10
0
0g
RL
S
S
a
P
P
L
TH
Earth AtmosphereSOLO
21
Physical Foundations of Atmospheric Model
HdTR
g
P
Pd
Ta
⋅=− *0
Integrating this equation we obtain
( )T
TaS
S HHTR
g
P
P
T
−⋅⋅
−= *0ln
Hence( )T
Ta
T
HHTR
g
SS ePP−⋅
⋅−
⋅=*
0
andS
STTaT P
P
g
TRHH ln
0
*
⋅⋅+=
∫∫ =−H
HTa
P
P T
S
TS
HdTR
g
P
Pd*
0
• Stratosphere Region (HT=11.0 km to 20.0 km). Temperature T = 216.65 K = TT* is constant (isothermal layer), PST=22632 Pa
Earth AtmosphereSOLO
22
Physical Foundations of Atmospheric Model
( )[ ] HdHHLTR
gHd
TR
g
P
Pd
SSTaa
⋅−⋅+⋅
=⋅=− *00
( ) ( ) PaPHPkmKLHHLTT SSSSSST 5474.9,/0.1* ===−⋅−=
Integrating this equation we obtain
( )[ ]∫∫ ⋅−⋅+
=−H
H SSTa
P
P S
S
SS
HdHHLTR
g
P
Pd*
0 1
( )[ ]*
*0 lnln
T
ST
aSSS
S
T
HHLT
RL
g
P
P −⋅+⋅⋅
=
Hence ( ) aRL
g
S
T
SSSS HH
T
LPP
⋅−
−⋅+⋅=
0
*1
and
−
⋅+=
⋅
10
* g
RL
SS
S
S
TS
aS
P
P
L
THH
Stratosphere Region (HS=20.0 km to 32.0 km).
Earth AtmosphereSOLO
23
1962 Standard Atmosphere from 86 km to 700 km
Layer Index GeometricAltitude
km
MolecularYemperature
,K
KineticTemperature
K
MolecularWeight
LapseRateK/km
7 86.0 186.946 186.946 28.9644 +1.6481
8 100.0 210.65 210.02 28.88 +5.0
9 110.0 260.65 257.00 28.56 +10.0
10 120.0 360.65 349.49 28.08 +20.0
11 150.0 960.65 892.79 26.92 +15.0
12 160.0 1110.65 1022.20 26.66 +10.0
13 170.0 1210.65 1103.40 26.49 +7.0
14 190.0 1350.65 1205.40 25.85 +5.0
15 230.0 1550.65 132230 24.70 +4.0
16 300.0 1830.65 1432.10 22.65 +3.3
17 400.0 2160.65 1487.40 19.94 +2.6
18 500.0 2420.65 1506.10 16.84 +1.7
19 600.0 2590.65 1506.10 16.84 +1.1
20 700.0 2700.65 1507.60 16.70
Earth AtmosphereSOLO
24
1976 Standard Atmosphere from 86 km to 1000 km
Geometric Altitude Range: from 86.0 km to 91.0 km (index 7 – 8)
78
/0.0
TT
kmKZd
Td
=
=
Geometric Altitude Range: from 91.0 km to 110.0 km (index 8 – 9)
2/12
8
2
8
2/12
8
1
1
−
−−
−⋅−=
−−⋅+=
a
ZZ
a
ZZ
a
A
Zd
Td
a
ZZATT C
kma
KA
KTC
9429.19
3232.76
1902.263
−=−=
=
Geometric Altitude Range: from 110.0 km to 120.0 km (index 9 – 10)( )
kmKZd
Td
ZZLTT Z
/0.12
99
+=
−⋅+=
Geometric Altitude Range: from 120.0 km to 1000.0 km (index 10 – 11)
( ) ( )
( )
( )
+
+⋅−=
+
+⋅−⋅=
⋅−⋅−−=
∞
∞∞
ZR
ZRZZ
kmKZR
ZRTT
Zd
Td
TTTT
E
E
E
E
1010
1010
10
/
exp
ξ
λ
ξλ
KT
kmR
km
E
1000
10356766.6
/01875.03
=
×=
=
∞
λ
Earth AtmosphereSOLO
25
Sea Level Values
Pressure p0 = 101,325 N/m2
Density ρ0 = 1.225 kg/m3
Temperature = 288.15 K (15 C)Acceleration of gravity g0 = 9.807 m/sec2
Speed of Sound a0 = 340.294 m/sec
Earth AtmosphereSOLO
26
Earth AtmosphereSOLO
27
Winds Winds represents the relative motion of the Atmosphere
Earth Atmosphere
Although in the standard atmosphere the air is motionless with respect to the Earth, it is known that the air mass through which an airplane flies is constantly in a state of motion with respect to the surface of the Earth. Its motion is variable both in time and space and is exceedingly complex. The motion may be divided into two classes: (1) large-scale motions and (2) small-scale motions. Large-scale motions of the atmosphere (or winds) affect the navigation and the performance of an aircraft.
SOLO
Return to Table of Content
28
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS
VECTOR NOTATION CARTESIAN TENSOR NOTATION
1.1 VECTOR
1.2 SCALAR PRODUCT
1.3 VECTOR PRODUCT
u kk = 1 2 3, , u u e u e u e= + +1 1 2 2 3 3
u v u v u v u v⋅ = + +1 1 2 2 3 3 u v kk k = 1 2 3, ,
u v
u u
u u
u u
v
v
v
× =−
−−
0
0
0
3 2
3 1
2 1
1
2
3
=−+
±=−=
ji
permutjiodd
permutjieven
CevittaLevi
vu
ij
jiij
0
.,
.,1
ε
ε
SOLO
29
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
1.5 ROTOR OF A VECTOR
1.4 DIVERGENCE OF A VECTOR
1.6 GRADIENT OF A SCALAR
∇ ⋅ = + +u
u
x
u
x
u
x
∂∂
∂∂
∂∂
1
1
2
2
3
3 i
i
x
u
∂∂
∇× = −
+ −
+ −
uu
x
u
xe
u
x
u
xe
u
x
u
xe
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
3
2
2
31
1
3
3
12
1
2
2
13
u u
uu u×∇× =∇
− ⋅∇
2
2
∂∂
∂∂
u
x
u
xi
k
k
i
−
i
kj
k
ii x
uu
x
uu
∂∂
∂∂
−
∇ = + +
=
φ∂ φ∂
∂ φ∂
∂ φ∂
∂ φ∂
∂ φ∂
∂ φ∂
xe
xe
xe
x x x
11
22
313
1 2 3
∂ φ∂ xk
SOLO
30
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
1.7 GRADIENT OF A VECTOR
∇ = ∇ + ∇ + ∇ u u e u e u e1 1 2 2 3 3
∇ =
u
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
1
1
1
2
1
3
2
1
2
2
2
3
3
1
3
2
3
3
∇ =
+ + +
+ + +
+ + +
u
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
Dik
1
2
1
1
1
1
1
2
2
1
1
3
3
1
2
1
1
2
2
2
2
2
2
3
3
1
3
1
1
3
3
2
2
3
3
3
3
3
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
+
ik
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
Ω
−−
−−
−−
+
0
0
0
2
1
3
2
2
3
3
1
1
3
1
3
3
2
2
1
1
2
1
3
3
1
1
2
2
1
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
u
xi
k
∂∂
∂∂
∂∂
∂∂
∂∂
u
x
u
x
u
x
u
x
u
xi
k
i
k
k
i
i
k
k
i
= +
+ −
1
2
1
2
Du
x
u
xiki
k
k
i
= +
∆ 1
2
∂∂
∂∂
Ω∆
iki
k
k
i
u
x
u
x= −
1
2
∂∂
∂∂
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31
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
1.8 GAUSS’ THEOREMS
d s
A
V
∇⋅A analytic in V
↓ = = A C C const vectorη .
( ) ∫∫ ∫∫∫∇=S V
dvsdGAUSS ηη 2 ∇η analytic in V ∫∫ ∫∫∫=
S k
k
V
dvs
ds∂
η∂η
SOLO
Johann Carl Friederich Gauss 1777-1855
( ) ∫∫ ∫∫∫ ⋅∇=⋅S V
dvAsdAGAUSS
1
∫∫ ∫∫∫=S k
k
kk
V
dvx
AdsA
∂∂
32
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
1.8 GAUSS’ THEOREMS (CONTINUE)
( ) ( ) ( )∫∫ ∫∫∫ ⋅∇=⋅S V
dvAsdAGAUSS
ηη3
( )= ⋅∇ + ∇⋅∫∫∫ A A dvη η
η∇⋅∇ ,A
analytic in V
( )η∂ η
∂A ds
A
xdv
Vk k
k
kS
= ∫∫∫∫∫
∫∫∫
+=
V k
k
kk x
A
xA
∂∂η
∂η∂
↓ = + + B e e eη η η1 1 2 2 3 3
( ) ( ) ( )[ ]∫∫ ∫∫∫ ⋅∇+∇⋅=⋅S V
dvABBAsdABGAUSS
4 B A ds AB
xB
A
xdv
Vi k k k
i
k
ik
kS
= +
∫∫∫∫∫
∂∂
∂∂
∇ ×A analytic in V( ) ∫∫ ∫∫∫ ×∇=×
S VdvAAsdGAUSS
5 ( )ds A ds A
A
x
A
xdv
Vi j j i
j
i
i
jS
− = −
∫∫∫∫∫
∂∂
∂∂
SOLO
33
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
1.9 STOCKES’ THEOREM
A d r A d s
C S
⋅ = ∇ × ⋅∫ ∫∫ ∇ ×A analytic on S
A d rA
x
A
xd si i
C
j
i
i
j
k
S∫ ∫∫= −
∂∂
∂∂
Gauss’ and Stokes’ Theorems are generalizations of theFundamental Theorem Of CALCULUS
( )A b A a
d A x
d xd x
a
b
( ) ( )− = ∫
George Stokes 1819-1903
SOLO
SOLO
Variational Principles of Hydrodynamics
Joseph-Louis Lagrange
1736-1813 Leonhard Euler
1707-1783 FIXED IN SPACE
(CONSTANT VOLUME)
EULER
LAGRANGE
MOVING WITH THE FLUID(CONSTANT MASS)
1e
3e
2e
u
The phenomena considered in Hydrodynamics are macroscopic and the atomic or molecular nature of the fluid is neglected. The fluid is regarded as a continuous medium. Any small volume element is always supposed to be so large that it still contains a large number of molecules.
There are two representations normally employed in the study of Hydrodynamics:
- Euler representation: The fluid passes through a Constant Volume Fixed in Space
- Lagrange representation: The fluid Mass is kept constant during its motion in Space.
Hydrodynamic Field
SOLO
Variational Principles of Hydrodynamics
Material Derivatives (M.D.)
Vector Notation Cartesian Tensor Notation
1e
2e
3e
r
u
b
rd( ) Frddtt
FtrFd
∇⋅+=
∂∂
,
( )d
dtF r t
F
t
dr
d tF
, = + ⋅∇
∂∂
( )d
dtF r t
F
tb F
b
, = + ⋅∇
∂∂
rdanyfor ( )d F r t
F
tdt d r
F
xi ki
ki
k
, = +∂∂
∂∂
( )d
d tF r t
F
t
d r
d t
F
xi ki k i
k
, = +∂∂
∂∂
( )d
d tF r t
F
tb
F
xb
i ki
ki
k
, = +∂∂
∂∂
vectoranybbtdrd
=
( ) Fut
FF
tD
DtrF
td
d
u
∇⋅+=≡
∂∂
,( )
k
ik
iki
u x
Fu
t
FF
tD
DtrF
td
d
∂∂
∂∂
+=≡,velocityfluiduu
td
rdIf
=
uuu
t
u
uut
uu
tD
D
×∇×−
∇+=
∇⋅+=
2
2
∂∂
∂∂
⋅−⋅−
+=
+=
k
ik
i
jj
ji
i
k
ik
ii
x
uu
x
uu
uxt
u
x
uu
t
uu
tD
D
∂∂
∂∂∂∂
∂∂
∂∂
∂∂
2
2
1
Acceleration Of The Fluid
1e
2e
3e
r
u duu +
dr
Material Derivatives = = Derivative Along A Fluid Path (Streamline) tD
D
Hydrodynamic Field
36
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
1.10 MATERIAL DERIVATIVES (CONTINUE)
d uu
tdt dr u
= + ⋅∇∂∂
d uu
tdt d x
u
xii
ki
k
= + ⋅∂∂
∂∂
rdrdDtdt
u
xd
xd
xd
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
x
u
t
u
t
u
t
u
ud
ud
ud
ikik
Ω++=
=
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂∂
∂∂∂
3
2
1
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
3
2
1
3
2
1 d u
u
td t
u
x
u
xd x
u
x
u
xd x
ii
Translation
i
k
k
i
Dilation
k
i
k
k
i
Rotation
k
= +
+ +
+ −
∂∂
∂∂
∂∂
∂∂
∂∂
1
2
1
2
( )
( )
( )
( )[ ] Dilationrduu
rdurdu
urdrdurdu
rdurdurdD
T
u
u
ik
⇒⋅∇+∇=
⋅∇+⋅∇=
∇⋅−⋅∇+⋅∇=
××∇−⋅∇=
2
12
1
2
12
1
2
12
1
( )Ω ik dr u dr Rotation = ∇ × × ⇒
1
2
SOLO
37
REYNOLDS’ TRANSPORT THEOREM
- any system of coordinatesOxyz
- any continuous and differentiable functions in
( ) ( )trtr OO ,,, ,,
ηχ( )tandrO,
( )trO ,,
ρ - flow density at point
and time tOr,
SOLO
- mass flow through the element .mdsdV S
=⋅− ,ρ sd
- any control volume, changing shape, bounded by a closed surface S(t)v (t)
- flow velocity, relative to O, at point and time t( )trV OOflow ,,,
Or,
- position and velocity, relative to O, of an element of surface, part of the control surface S(t).
OSOS Vr ,, ,
- area of the opening i, in the control surface S(t).iopenS
- gradient operator in O frame.O,∇
- flow relative to the opening i, in the control surface S(t).OSiOflowSi VVV ,,,
−=
- differential of any vector , in O frame.O
td
d ζ
ζ
FLUID DYNAMICS
38
Start with LEIBNIZ THEOREM from CALCULUS:( ) ( )
ChangeBoundariesthetodueChange
tb
ta
tb
ta td
tadttaf
td
tbdttbfdx
t
txfdxtxf
td
dLEIBNITZ
−+= ∫∫ )),(()),((
),(),(::
)(
)(
)(
)( ∂∂
and generalized it for a 3 dimensional vector space on a volume v(t) bounded by thesurface S(t).
Using LEIBNIZ THEOREM followed by GAUSS THEOREM (GAUSS 4):
( ) ( )( )
( )∫∫∫∫∫
⋅∇+∇⋅+=⋅+
→
=tv
OSOOOSGAUSS
OpotolativedsofMovement
thetodueChage
tSOS
tvO
LEIBNITZ
Otv
vdVVt
GAUSSsdVvd
tvd
td
d,,,,)4(
intRe
)(,
χχ∂χ∂χ
∂χ∂χ
This is REYNOLDS’ TRANSPORT THEOREM
OSBORNEREYNOLDS
1842-1912
SOLO
GOTTFRIED WILHELMvon LEIBNIZ
1646-1716
REYNOLDS’ TRANSPORT THEOREM
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
39
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
( )
∫∫∫
∫∫∫∫∫∫∫∫
⋅∇+∇⋅+=
⋅+=
)(,,,,)4(
,)()()(
tvOSOOOS
OGAUSS
OStStv
O
LEIBNITZ
Otv
vdVVt
GAUSS
sdVvdt
vdtd
d
χχ∂
χ∂
χ∂
χ∂χ
∫∫∫
∫∫∫∫∫∫∫∫
++=
+=
)(
,
,)4(
,)()()(
tv k
kOS
i
k
i
kOSi
GAUSS
kkOStS
itv
iLEIBNITZ
tvi
vdx
V
xV
t
GAUSS
sdVvdt
vdtd
d
∂∂
χ∂
χ∂∂
χ∂
χ∂
χ∂χ
SOLO
40
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
O
OOS td
RduV
== ,,
CASE 1 (CONTROL VOLUME vF ATTACHED TO THE FLUID)
kkOS uV =,
( )
∫∫∫
∫∫∫∫∫∫∫∫
⋅∇+∇⋅+=
⋅+=
)(,,,)4(
,)()()(
tvOOO
OGAUSS
OtStv
OOtv
F
FFF
vduut
GAUSS
sduvdt
vdtd
d
χχ∂
χ∂
χ∂
χ∂χ
∫∫∫
∫∫∫∫∫∫∫∫
++=
+=
)()4(
)()()(
tv k
kI
k
Ik
I
GAUSS
kKtS
Itv
I
tvI
F
FFF
vdx
u
xu
t
GAUSS
sduvdt
vdtd
d
∂∂χ
∂χ∂
∂χ∂
χ∂χ∂χ
SOLO
41
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
1&, == χkkOS uV1&, == χuV OS
CASE 2 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND )1=χ
∫∫∫∫∫∫∫∫ ⋅∇=⋅==)(
,,)(
,)(
)(
tvOO
tSO
tv
F
FFF
vdusduvdtd
d
td
tvd ∫∫∫∫∫∫∫∫ ===)()()(
)(
tv k
kk
tSk
tv
F
FFF
dvx
udsudv
td
d
td
tvd
∂∂
=⋅∇
→ td
tvd
tvu F
Ftv
OOF
)(
)(
1lim
0)(,,
=
→ td
tvd
tvx
u F
Ftv
k
k
F
)(
)(
1lim
0)(∂∂
EULER 1755
SOLO
42
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
CASE 3 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND )
ρχ == &, kkOS uVρχ == &, uV OS
ρχ =
or, since this is true for any attached volume vF(t)
( )∫∫∫
∫∫∫∫∫ ∫∫∫
⋅∇+=
⋅+===
)(,,
)(,
)( )(
)(0
tvOO
tSO
tv tv
F
FF F
vdut
sduvdt
vdtd
d
td
tmd
ρ∂
ρ∂
ρ∂
ρ∂ρ
( )∫∫∫
∫∫∫∫∫ ∫∫∫
+=
+===
)(
)()( )(
)(0
tvk
k
tSkk
tv tv
F
FF F
vduxt
sduvdt
dvtd
d
td
tmd
ρ∂
∂∂
ρ∂
ρ∂
ρ∂ρ
Because the Control Volume vF is attached to the fluid and they are not sources or sinks, the mass is constant.
( ) OOOOOO uut
ut ,,,,,,0
⋅∇+∇⋅+=⋅∇+= ρρ
∂ρ∂ρ
∂ρ∂ ( )
k
k
k
k
k x
u
xu
tu
xt ∂∂ρ
∂ρ∂
∂ρ∂ρ
∂∂
∂ρ∂ ++=+=
0
SOLO
43
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,
=OSV
Define
∫∫∫∫∫∫ =.... VC
OOVC
vdt
vdtd
d
∂χ∂χ
∫∫∫∫∫∫ =.... VC
i
VCi vd
tvd
td
d
∂χ∂χ
( ) ( ) ( ) χ ρ ηr t r t r t, , ,≡ ( ) ( ) ( )χ ρ ηi k k i kx t x t x t, , ,≡
( )∫∫
∫∫∫∫∫∫
⋅+
+=
)(,
)()(
tSOS
tvOO
tv
sdV
vdtt
vdtd
d
ηρ
∂ρ∂η
∂η∂ρηρ
ktS
kOSi
tvi
i
tvi
sdV
vdtt
vdtd
d
FF
∫∫
∫∫∫∫∫∫
+
+=
)(,
)()(
ηρ∂
ρ∂η∂η∂ρηρ
We have
but
( ) ( )OOOO ut
ut ,,,, 0
ρη∂
ρ∂ηρ∂
ρ∂ ⋅∇−=⇒=⋅∇+ ( ) ( )k
k
iik
k
uxt
uxt
ρ∂
∂η∂
ρ∂ηρ∂
∂∂
ρ∂ −=⇒=+ 0
CASE 5 ( ) ( ) ( ) χ ρ ηr t r t r t, , ,≡
SOLO
44
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
We have
( )
( )
( ) ( )[ ]
( )
( )[ ]∫∫∫∫∫=
∫∫
∫∫∫
∫∫
∫∫∫∫∫∫
⋅−+
⋅+
⋅∇+∇⋅−
∇⋅+=
⋅+
⋅∇−=
+
+)(
,,)(
4
.
)(,
)(,,,,,,
)(,
)(,,
)(
tSOOS
tvO
MDG
DerMatGAUSS
tSOS
tvOOOOOO
O
tSOS
tvOO
OOtv
sduVvdtD
D
sdV
vduuut
sdV
vdut
vdtd
d
ρηρη
ρη
ρηηρη∂
η∂ρ
ρη
ρηρ∂
η∂ρη ( )
( )
( ) ( )
( )
( )[ ]∫∫∫∫∫=
∫∫
∫∫∫
∫∫
∫∫∫∫∫∫
−+
+
+−
+=
+
−=
+
+)(
,)(
4
.
)(,
)(
)(,
)()(
tSkkkOSi
tv
iMDG
DerMatGAUSS
tSkkOSi
tv k
ki
k
ik
k
ik
i
tSkkOSi
tv k
ki
i
tvi
sduVvdtD
D
sdV
vdx
u
xu
xu
t
sdV
vdx
u
tvd
td
d
ρηρη
ρη
∂ρ∂η
∂η∂ρ
∂η∂
∂η∂ρ
ρη
∂ρ∂ηρ
∂η∂ρη
CASE 5 ( ) ( ) ( ) χ ρ ηr t r t r t, , ,≡
SOLO
45
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
REYNOLDS 1
( )[ ]
⋅−+= ∫∫∫∫∫
∫∫∫
)(,,
)(
)(
tSOOS
tvO
Otv
sduVvdtD
D
vdtd
d
ρηρη
ρη
( )[ ]
−+= ∫∫∫∫∫
∫∫∫
)(,
)(
)(
tSkkkOSi
tv
i
tvi
sduVvdtD
D
dvtd
d
ρηρη
ρη
REYNOLDS 2
( )[ ]
=
⋅−+
∫∫∫
∫∫∫∫∫
)(
)(,,
)(
tvO
tSOSO
Otv
vdtD
D
sdVuvdtd
d
ρη
ρηρη
( )[ ]
=
−+
∫∫∫
∫∫∫∫∫
)(
)(,
)(
tv
i
tSkkOSki
tvi
vdtD
D
sdVuvdtd
d
ρη
ρηρη
CASE 5 ( ) ( ) ( ) χ ρ ηr t r t r t, , ,≡
SOLO
46
FLUID DYNAMICS
1. MATHEMATICAL NOTATIONS (CONTINUE)
1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
REYNOLDS 3
CASE 1 (CONTROL VOLUME ATTACHED TO THE FLUID vF(t) )
kkOS uV =,
∫∫∫∫∫∫ =)()( tv
OOtv FF
vdtD
Dvd
td
d ρηρη
∫∫∫∫∫∫ =
)()( tv
i
tvi
FF
vdtD
Dvd
td
d ρηρη
SOLO
O
OOS td
RduV
== ,,
( ) ( ) ( ) χ ρ ηr t r t r t, , ,≡
CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,
=OSV
REYNOLDS 4
( )
⋅+= ∫∫∫∫∫
∫∫∫
..,
..
..
SCO
OVC
VCO
sduvdtd
d
vdtD
D
ρηρη
ρη
( )
+= ∫∫∫∫∫
∫∫∫
....
..
SCkki
VCi
VC
i
sduvdtd
d
vdtD
D
ρηρη
ρη
Return to Table of Content
47
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS
THE FLUID DYNAMICS IS DESCRIBED BY THE FOLLOWING FIVE LAWS:
SOLO
(1) CONSERVATION OF MASS (C.M.)
(2) CONSERVATION OF LINEAR MOMENTUM (C.L.M.)
(3) CONSERVATION OF MOMENT OF MOMENTUM (C.M.M.)
(4) THE FIRST LAW OF THERMODYNAMICS
(5) THE SECOND LAW OF THERMODYNAMICSReturn to Table of Content
48
FLUID DYNAMICS2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.1) CONSERVATION OF MASS (C.M.)
Control Volume attached to the fluid (containing a constant mass m) bounded by
the Control Surface SF (t).
( )tvF
( )tr ,
ρ ( )3/ mkgFlow density
SOLO
Because vF(t) is attached to the fluid and there are no sources or sinks in this volume,the Conservation of Mass requires that:
d m t
d t
( ) = 0
( ) ( )trVtru OfluidO ,, ,,
= Flow Velocity relative to a predefined
Coordinate System O (Inertial orNot-Inertial) ( )sm /
49
FLUID DYNAMICS2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.1) CONSERVATION OF MASS (CONTINUE - 1)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
d m t
d t
( ) = 0
( )∫∫∫=
∫∫∫∫∫ ∫∫∫
⋅∇+
⋅+===
)(,,
1
)(,
)( )(
)(0
tvOO
GAUSS
tSO
tv tv
REYNOLDS
F
FF F
vdut
sduvdt
dvtd
d
td
tmd
ρ∂
ρ∂
ρ∂
ρ∂ρ
( )∫∫∫=
∫∫∫∫∫ ∫∫∫
+
+===
)(
1
)()( )(
)(0
tvk
k
GAUSS
tSkk
tv tv
REYNOLDS
F
FF F
vduxt
sduvdt
dvtd
d
td
tmd
ρ∂
∂∂
ρ∂
ρ∂
ρ∂ρ
The Control Volume mass rate is zero as long as vF(t) is attached to the fluid and therefore contains the same amount of mass.
0),,,()(
=∫∫∫tvF
vdtzyxtd
d ρ is true in any Coordinate System (O) and so is:
( ) ( ) ( ) ( )( ) 0,,,,,,,,,
,,,)(
,,)(
=
⋅∇+= ∫∫∫∫∫∫
tvOO
tv FF
vdtzyxutzyxt
tzyxvdtzyx
td
d ρ
∂ρ∂ρ
SOLO
50
FLUID DYNAMICS2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.1) CONSERVATION OF MASS (CONTINUE - 2)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
For any Control Volume v (t) (not necessarily attached to the fluid)
The following is true for any Coordinate System (for points that are not sources orsinks – mathematically equivalent to analytic ) ( )OO u
t ,,,ρ
∂ρ∂ ⋅∇
( ) OOOOOO uut
ut ,,,,,,0
⋅∇+∇⋅+=⋅∇+= ρρ∂
ρ∂ρ∂
ρ∂ ( )k
k
k
kO
k x
u
xu
tu
xt ∂∂ρ
∂ρ∂
∂ρ∂ρ
∂∂
∂ρ∂ ++=+= ,0
( )
( ) 0)(
,,
4
).(
,
).()(
≠=
⋅∇+
⋅+
∫∫∫=
∫∫∫∫∫=∫∫∫
mvdVt
sdVvdt
vdtd
d
tv
OSO
GAUSS
tS
OS
tv
LEIBNITZ
tv
ρ∂
ρ∂
ρ∂
ρ∂ρ
( ) 0)(
,
4
).(
,
).()(
≠=
+
+
∫∫∫=
∫∫∫∫∫=∫∫∫
mvdVxt
sdVvdt
vdtd
d
tvkOS
k
GAUSS
tS
kkOS
tv
LEIBNITZ
tv
ρ∂
∂∂
ρ∂
ρ∂
ρ∂ρ
The integral above is not zero because the mass in v (t) is not constant.
SOLO
51
FLUID DYNAMICS2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.1) CONSERVATION OF MASS (CONTINUE - 3)
Material Derivative of vdmd ρ=
Let use EULER’s 1755 expression ( ) ( )( ) ( )vd
tD
D
vdtd
tvd
tvu F
Ftv
OOF
11lim
0,, =
=⋅∇
→
and the (C.M.):
to develop the following:
( ) 0,, =⋅∇+ OO ut
ρ∂ρ∂
( ) ( )
( ) 0,,,,,,
,,,,
=
⋅∇+
∂∂=
⋅∇+∇⋅+
∂∂=
⋅∇+
∇⋅+
∂∂=+==
vdut
vduut
uvdvdut
vdtD
Dvd
tD
Dvd
tD
D
tD
mD
OOOOOO
OOOO
ρρρρρ
ρρρρρρ
SOLO
52
FLUID DYNAMICS
∑+=openings
iiopenW SCSCSC ....
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.1) CONSERVATION OF MASS (CONTINUE – 4)
SOLO
Control Volume with fixed shape C.V. and boundary C.S. in O Coordinates ( ) 0,
=OSV
There are no sources or sinks in the volume C.V. The change in the mass of the system is due only to the flow through the surface openings C.Sopen i (i=1,2,…). The surface C.S. can be divided in:
• C.Sw the impermeable wall through which the fluid can not escape .
=−= 0
0
,,,
OSOs VuV
• C.Sopen i the openings (i=1,2,…) through which the fluid enters or exits .( )0>m ( )0<m
∑∑ ∫∫∫∫∫∫∫∫∫ =⋅−⋅−=⋅−==openings
ii
openings
i
m
SCO
SCO
SCO
VC
msdusdusduvdtd
d
td
md
i
iopenw
.,
.0
,..
,..
ρρρρ
Therefore
where is the flow rate entering through the opening Sopen i.∫∫ ⋅−=iopenSC
Oi sdum.
,
ρ
Return to Table of Content
53
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.2) CONSERVATION OF LINEAR MOMENTUM (C.L.M.)
- Fluid density at he point and time t ( )tr ,ρ
r ( )3/ mKg
- Fluid inertial velocity at the point and time t
( )tru I ,,
r
( )sec/m
- Surface Stress ( )2/ mNT
- Pressure (force per unit surface) of the surrounding on the control surface ( )2/ mN
p
- Stress tensor (force per unit surface) of the surrounding on the control surface ( )2/ mN
σ~
- Body forces acceleration-(gravitation, electromagnetic,..)
G ( )2sec/m
nnpnT ˆ~ˆˆ~ ⋅+−=⋅= τσ
Consider a volume vF(t) attached to the fluid, bounded by the closed surface SF(t).
SOLO
- unit vector normal to the surface S(t) and pointing outside the volume v (t)n
vF (t)
m
SF (t)
O
x
y
z
r u,O
np ˆ−
n~ ⋅τ
n~ ⋅σdSn
- Shear stress tensor (force per unit surface) of the surrounding on the control surface ( )2/ mN
τ~
54
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 1)
Derivation From Integral Form
The LINEAR MOMENTUM of the Constant Mass in vF(t) is given by:
∫∫∫=)(
,tv
I
F
vduP
ρ
The External Forces acting on the mass are Body and Surface Forces:
( )
ForcesSurface
tS
ForcesBody
tvexternal
FF
sdTvdGF ∫∫∫∫∫∑ +=)(
ρ
According to NEWTON’s Second Law, for a constant mass in vF(t), we have:
I
external td
=∑
SOLO
55
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 2)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
( )
I
IMomentumLinear
tvI
REYNOLDS
tvI
I
ForcesSurface
tS
ForcesBody
tvexternal
Ptd
dvdu
td
dvd
tD
uD
sdvdGF
FF
FF
===
⋅+=
∫∫∫∫∫∫
∫∫∫∫∫∑
)(,
3
)(
,
)()(
~
ρρ
σρ
itv
i
REYNOLDS
tv
i
tSkik
tviiex
Ptd
dvdu
dt
dvd
tD
uD
dsvdGF
FF
FF
===
+=
∫∫∫∫∫∫
∫∫∫∫∫∑
)(
3
)(
)()(_
ρρ
σρ
C.L.M.-1
T ds n ds dsds n ds
= ⋅ ⋅=
=~ ~σ σ T ds n ds dsi ik k
ds n ds
ik k
k k
==
=σ σ
C.L.M.-2
( )∫∫∫
∫∫∫∫∫∫∫∫
⋅∇+=
⋅+=
)(,
)()()(
,
~
~
tvI
tStvtvI
I
F
FFF
vdG
sdvdGvdtD
uD
σρ
σρρ
∫∫∫
∫∫∫∫∫∫∫∫
+=
+=
)(
)()()(
tv i
iki
tS
kik
tv
i
tv
i
F
FFF
vdx
G
sdvdGvdtD
uD
∂σ∂ρ
σρρ
SOLO
Derivation From Integral Form (Continue)
56
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 3)
VECTOR NOTATION CARTESIAN TENSOR NOTATIONC.L.M.-2
Since this is true for all volumes vF (t) attached to the fluid we can drop the volume integral.
[ ] [ ] [ ]τστρσρ
∂∂ρ
∂∂ρρ
~~
~~
2
1
,,,
,
2
,
,
.).(
+−=⋅∇+∇−=⋅∇+=
×∇×−
∇+=
∇⋅+=
Ip
pGG
uuut
u
uut
u
tD
uD
III
II
I
I
I
DM
I
ikikik
i
ik
ii
i
iki
k
ik
i
jjjj
i
i
k
ik
iDM
i
p
xx
pG
xG
x
uu
x
uuuu
xt
u
x
uu
t
u
tD
uD
τδσ∂τ∂
∂∂ρ
∂σ∂ρ
∂∂
∂∂
∂∂
∂∂ρ
∂∂
∂∂ρρ
+−=
+−=+=
⋅−⋅−
+=
⋅+=
2
1
.).(
SOLO
Derivation From Integral Form (Continue)
( )∫∫∫
∫∫∫∫∫∫∫∫
⋅∇+=
⋅+=
)(,
)()()(
,
~
~
tvI
tStvtvI
I
F
FFF
vdG
sdvdGvdtD
uD
σρ
σρρ
∫∫∫
∫∫∫∫∫∫∫∫
+=
+=
)(
)()()(
tv i
iki
tS
kik
tv
i
tv
i
F
FFF
vdx
G
sdvdGvdtD
uD
∂σ∂ρ
σρρ
57
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE - 4)
Derivation From a Cartesian Differential Volume
VECTOR NOTATION CARTESIAN TENSOR NOTATION
σ∂ σ
∂xxxx
xdx+ 1
2
σ ∂ σ∂xx
xx
xdx− 1
2
τ∂ τ
∂yxyx
ydy+ 1
2
τ∂ τ
∂yzyz
ydy− 1
2
τ∂τ
∂zxzx
zdz+ 1
2
τ∂τ
∂zxzx
zdz− 1
2
τ∂ τ
∂xyxy
xdx+ 1
2
τ∂ τ
∂xyxy
xdx− 1
2σ
∂ σ∂yy
yy
ydy+ 1
2σ
∂ σ∂yy
yy
ydy− 1
2
τ∂τ
∂zyzy
zdz+ 1
2
τ∂τ
∂zy
zy
zdz− 1
2
τ∂ τ
∂xzxz
xdx− 1
2
τ∂ τ
∂yz
yz
ydy+ 1
2
τ∂ τ
∂yx
yx
ydy−
1
2
σ∂ σ
∂zzzz
zdz+ 1
2
σ∂ σ
∂zzzz
zdz− 1
2
z
y
xd y
d x
d z
O
τ∂ τ
∂xzxz
xdx+ 1
2
∂σ∂
∂τ∂
∂τ∂
ρ ρ
∂τ∂
∂σ∂
∂τ∂
ρ ρ
∂τ∂
∂τ∂
∂σ∂
ρ ρ
xx yx zxxB x
xy yy zyyB y
xz yz zzzB z
x y zG a
x y zG a
x y zG a
+ + + =
+ + + =
+ + + =
CAUCHY’s First Law of Motion
ItD
uDa
aG
=
=+⋅∇ ρρσ~
tD
uDa
aGx
ii
iii
ij
=
=+ ρρ∂σ∂
SOLO
AUGUSTIN LOUIS CAUCHY
)1789-1857(
58
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE-5)
Derivation For Any Control Volume v (t) (the velocity of an element of surface is )d s
ISV ,
V (t)
bds
V*(t)
I
T d s= ⋅~σ
G
m
u Use REYNOLDS’ Transport Theorem (REYNOLDS 2)with and O = I, and then the Conservation
of Linear Momentum (C.L.M.)Iu,
=η
VECTOR NOTATION CARTESIAN TENSOR NOTATION
( )[ ]( ) ( ) ( )
∑∫∫∫∫∫
∫∫∫∫∫∫
∫∫∫∫∫
=⋅+=
⋅∇+==
⋅−+
iexternaltStv
tvI
MLC
tvI
REYNOLDS
tSISII
Itv
I
FsdvdG
vdGvdtD
uD
sdVuuvdutd
d
FF
FF
FF
)()(
)(,
...
)(
2
)(,,,
)(,
~
~
σρ
σρρ
ρρ ( )( ) ( )
∑∫∫∫∫∫
∫∫∫∫∫∫
∫∫∫∫∫
=+=
+==
−+
iexternaltS
kiktv
i
tv k
iki
MLC
tv
iREYNOLDS
tSkkISki
tvi
FsdvdG
vdx
GvdtD
uD
sdVuuvdutd
d
)()(
)(
...
)(
2
)(,
)(
σρ
∂σ∂ρρ
ρρ
SOLO
Return to Table of Content
59
( ) ( ) PdRRvdVRRHd OOO
×−=×−= ρ,
2. BASIC LAWS IN FLUID DYNAMICS
(2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.)
SOLO
The Absolute Angular Momentum, of the differential mass and Inertial Velocity ,relative to a reference point O is defined as
vdmd ρ=V
The Absolute Angular Momentum of the mass enclosed by C.V. is defined as
( ) ( )∫∫∫∫∫∫ ×−=×−=....
,VC
OVC
OOCV PdRRvdVRRH
ρ
Let differentiate the Absolute Angular Momentum and use Reynolds’ Transport Theorem
( ) ( ) ( ) ( )∫∫∫∫∫∫∫∫ ⋅−×−+×−
=×−=..
,....
,
SCmd
SOVC
I
OREYNOLDS
IVC
O
I
OCV sdVVRRvdtD
VRRDvdVRR
td
d
td
Hd
ρρρ
We have ( ) ( )
( ) ( ) ( ) VVtD
VDRRVVV
tD
VDRR
VtD
RD
tD
RD
tD
VDRR
tD
VRRD
O
I
OO
I
O
I
O
II
O
I
O
×−×−=×−+×−=
×
−+×−=×−
FLUID DYNAMICS
60
( ) ( ) ( ) int, : fdRRfdRRvdtD
VDRRMd OextO
I
OO
×−+×−=×−= ρ
( ) ( ) ( ) ( )∫∫∫∫∫∫∫∫∫∫∫ ⋅−×−+×−×−=×−=..
,......
,
SCmd
SO
P
VCO
VCI
O
REYNOLDS
IVC
O
I
OCV sdVVRRvdVVvdtD
VDRRvdVRR
td
d
td
Hd
CV
ρρρρ
2. BASIC LAWS IN FLUID DYNAMICS
(2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.)
SOLO
The Moment, of the differential mass dm = ρdv, relative to a reference point O is defined as
Therefore
Let integrate this equation over the control volume C.V.
( ) ( ) ( )
0
..int
...., ∫∫∫∫∫∫∫∫∫∑ ×−+×−=×−=
VCO
VCextO
VCI
OOCV fdRRfdRRvdtD
VDRRM ρ
Using the differential of Angular Momentum equation we obtain
( ) ( ) ( )∫∫∫∫∫∑∫∫∫ ⋅−×−+×−=×−=..
,..
,..
,
SCmd
SO
P
VCOOCV
IVC
O
I
OCV sdVVRRvdVVMvdVRRtd
d
td
Hd
CV
ρρρ
( ) ( ) ( ) ( ) ( ) ∑∑∫∫∫∫∫∫∫∫∑ +×−++−×−+×−=×−==⋅
kk
jjOj
SCsdTsd
OVC
OVC
extOOtCV MFRRsdtfnpRRvdgRRfdRRM
......, 11
σ
ρ
Also
( )∑ ×−j
jOj FRR
- Moment, relative to O, of discrete forces exerting by the surrounding at point jR
- Discrete Moments exerting by the surrounding.∑k
kM
FLUID DYNAMICS
61
( )( ) ( )
∑∑∫∫∫∫∫∫∫∫ +×+×+×=×+⋅−×−×k
kj
jOtv
Otv
extO
P
VCO
SCmd
SO
IVC
O MFrfdrfdrvdVVsdVVrvdVrtd
d
CV
,
0
int,,....
,,..
, ρρρ
2. BASIC LAWS IN FLUID DYNAMICS
(2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.)
SOLO
Let find the equation of moment around the turbomachine axis. We shall use polar coordinates , where z is the turbomachine axis.
zr ,,θ
zzrrrOˆˆ
, +=
zVVrVV zrˆˆˆ ++= θθ
zFFrFF zrˆˆˆ ++= θθ
( ) zVrVrVzrVz
VVV
zr
zr
Vr zrz
zr
Oˆˆ0
ˆˆˆ
, θ
θ
θ+−+−==×
( ) ( )
∑∑∫∫∫∫ ++=×+⋅−−k
kzj
jtv
extCVOSC
SVC
MFrdfrPVsdVVrvdVrtd
dθθθθ ρρ
0
..,
..
The moment of momentum equation around the turbomachine z axis.
Example
FLUID DYNAMICS
62
2. BASIC LAWS IN FLUID DYNAMICS
(2.3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.)
SOLO
( )( ) ( ) ( ) ( )
( )
systemoutsidefromexertedTorque
M
llz
jj
tvext
AVVrAVVr
SCS
statesteady
VC
zSnSn
MFrdfrsdVVrvdVrtd
d ∑∑∫∫∫∫ ++=⋅−−
+−−→
θθ
ρρ
θθ
θθ
ρρ
22,21111,122
..,
0
..
We obtain
( ) ( )[ ] zflow MQVrVr =− 111122 ρθθ
or
( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) zSnSnSn MAVVrVrAVVrAVVr =−=− 11,1112211,11122,222 ρρρ θθθθ
Euler Turbine Equation
ρ1 - mean fluid density one inlet (1) of area A1. where
ρ2 - mean fluid density one outlet (2) of area A2.
(Vθ )1, r1 - mean fluid tangential velocity and radius one inlet (1) of area A1.
(Vθ )2, r2 - mean fluid tangential velocity and radius one outlet (2) of area A2.
(V,Sn )1 - mean fluid normal velocity (relative to A1) one inlet (1) of area A1.
(V,Sn )2 - mean fluid normal velocity (relative to A2) one outlet (2) of area A2.
- mean flow rate one outlet (1) of area A1.( ) 11,1 : AVQ Snflow =
FLUID DYNAMICS
Return to Table of Content
63
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (DIFFERENTIAL FORM)
- Fluid mean velocity [m/sec[( ) u r t,
- Body Forces Acceleration- (gravitation, electromagnetic,..)
G
- Surface Stress [N/m2[T
nnpnT ˆ~ˆˆ~ ⋅+−=⋅= τσ
mV(t)
G
q
T n= ⋅~σ
d E
d t
∂∂
Q
t
uu
d s n ds=- Internal Energy of Fluid molecules (vibration, rotation, translation per
mass [W/kg[
e
- Rate of Heat transferred to the Control Volume (chemical, external sources of heat) [W/m3[
∂∂
Q
t
- Rate of Work change done on fluid by the surrounding (rotating shaft, others) positive for a compressor, negative for a turbine) [W[td
Ed
SOLO
Consider a volume vF(t) attached to the fluid, bounded by the closed surface SF(t).
- Rate of Conduction and Radiation of Heat from the Control Surface (per unit surface) [W/m3[
q
64
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 1)
- The Internal Energy of the molecules of the fluid plus the Kinetic Energy of the mass moving relative to an Inertial System (I)
The FIRST LAW OF THERMODYNAMICS
CHANGE OF INTERNAL ENERGY + KINETIC ENERGY =CHANGE DUE TO HEAT + WORK + ENERGY SUPPLIED BY SUROUNDING
SOLO
The energy of the constant mass m in the volume vF(t) attached to the fluid, bounded by the closed surface SF(t) is
This energy will change due to
- The Work done by the surrounding
- Absorption of Heat
- Other forms of energy supplied to the mass (electromagnetic, chemical,…)
65
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 2)
VECTOR NOTATION CARTESIAN TENSOR NOTATIONC.E.-1
systementeringtd
Qd
tSv
systemontnmenenvirobydonetd
Wd
shaft
tSv
v
REYNOLDS
KineticInternal
tv
FF
FF
FF
sdqvdt
Q
td
Wd
ForcesSurface
sdTu
ForcesBody
vdGu
vduetD
Dvdue
td
d
∫∫∫∫∫
∫∫∫∫∫
∫∫∫∫∫∫
⋅−+
+⋅+⋅=
+=
+
+
)(
)(
2)3(
)(
2
2
1
2
1
∂∂
ρ
ρρ
systementeringtd
Qd
tSkk
tv
systemontnemnoenvirbydonetd
Wd
shaft
tSkk
tvkk
tv
REYNOLDS
KineticInternal
tv
FF
FF
FF
dsqvdt
Q
td
Wd
ForcesSurface
sdTu
ForcesBody
vdGu
vduetD
Dvdue
td
d
∫∫∫∫∫
∫∫∫∫∫
∫∫∫∫∫∫
−+
++=
+=
+
+
)()(
)()(
)(
2)3(
)(
2
2
1
2
1
∂∂
ρ
ρρ
SOLO
66
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 3)
VECTOR NOTATION CARTESIAN TENSOR NOTATIONC.E.-2
( ) ( )
∫∫∫∫∫∫
∫∫∫∫∫∫∫∫∫
∫∫∫∫∫
∫∫∫∫∫∫∫
∫∫∫
⋅∇−+
⋅⋅∇+⋅∇−⋅=
⋅−+
⋅⋅+⋅−⋅=+
+
)()(
)()()(
)1(
)()(
)()()(
)(
2
~
~
2
1
tvtv
tvtvtv
GAUSS
td
Qd
tStv
td
Wd
tStStv
tv
FF
FFF
FF
FFF
F
vdqvdt
Q
vduvdupvdGu
sdqvdt
Q
sdusdupvdGu
KineticInternal
vduetD
D
∂∂
τρ
∂∂
τρ
ρ
( ) ( )
∫∫∫∫∫∫
∫∫∫∫∫∫∫∫∫=
∫∫∫∫∫
∫∫∫∫∫∫∫
∫∫∫
−+
+−
−+
+−=
+
+
)()(
)()()(
)1(
)()(
)()()(
)(
2
2
1
tV s
s
tV
tV
kk
iki
tV
kk
k
tV
kk
GAUSS
td
Qd
tS
kk
tV
td
Wd
tS
kiki
tS
kk
tV
kk
KineticInternal
tV
vdx
qvd
t
Q
dsx
uds
x
upvdGu
dsqvdt
Q
dsudsupvdGu
vduetD
D
∂∂
∂∂
∂τ∂
∂∂ρ
∂∂
τρ
ρ
T n pn n ds n ds= ⋅ = − + ⋅ =~ ~ &σ τ0=
td
Wd shaftassume and use
SOLO
67
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE-4)
VECTOR NOTATION CARTESIAN TENSOR NOTATIONC.E.-3
Since the last equation is valid for each vF(t) we can drop the integral and obtain:
( ) ( )
qt
Q
uGuupuetD
D
⋅∇−+
⋅+⋅⋅∇+⋅−∇=
+
∂∂
ρτρ ~2
1 2 ( ) ( )
k
k
kk
k
iik
k
k
x
q
t
Q
uGx
u
x
upue
tD
D
∂∂
∂∂
ρ∂τ∂
∂∂ρ
−+
++−=
+ 2
2
1
Multiply (C.L.M.-2) byu
τρρ ~⋅∇⋅+∇⋅−⋅=⋅ upuuGtD
uDu
( )
k
iki
k
kkki
i xu
x
puuGu
tD
D
tD
uDu
∂τ∂
∂∂ρρρ +−== 2
Subtract this equation from (C.E.-3)C.E.-4
( )[ ]ρ τ τ
∂∂
D e
D tp u u u
Q
tq
= − ∇⋅ + ∇⋅ ⋅ − ⋅∇⋅
+ −∇⋅
~ ~
Φ
ρ∂∂ τ
∂∂
∂∂
∂∂
D e
D tp
u
xu
u
x
Q
t
q
x
k
kik
i
k
k
k
=− +
+ −
Φ
( )Φ ≡ ∇ ⋅ ⋅ − ⋅ ∇ ⋅ >~ ~τ τ u u 0
Φ ≡ >τ ∂∂ik
i
k
u
x0
(Proof of inequality given later)
SOLO
68
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 5)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
Enthalpy
Use this result and (C.E.-4)
C.E.-5
ρp
eh +=:
( )tD
pDup
tD
hDu
p
tD
pD
tD
hD
tD
Dp
tD
pD
tD
hD
tD
pD
tD
hD
tD
eD
−⋅∇−=⋅∇−+−=
+−=
−=
ρρ
ρρ
ρρρ
ρρρ
ρρρρ
2
tD
pD
x
up
tD
hD
x
up
tD
pD
tD
hD
tD
pDp
tD
hD
tD
pD
tD
pD
tD
hD
tD
eD
k
k
k
k −−=
−+−=
+−=
−=
∂∂ρ
∂∂ρ
ρρ
ρρ
ρρρ
ρρρρ
2
Φ++⋅∇−=t
tD
pD
tD
hD
∂∂ρ
Φ++−=t
Q
x
q
tD
pD
tD
hD
k
k
∂∂
∂∂ρ
SOLO
( )Φ ≡ ∇ ⋅ ⋅ − ⋅ ∇ ⋅ >~ ~τ τ u u 0
Φ ≡ >τ ∂∂ik
i
k
u
x0
69
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 6)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
Total Enthalpy
Use this result and (C.E.-3)
C.E.-6
22
2
1
2
1: u
peuhH ++=+=
ρ
( )t
pup
tD
HD
tD
pDup
tD
HD
p
tD
D
tD
HDue
tD
D
∂∂ρρ
ρρρρ
−⋅∇−=−⋅∇−=
−=
+
2
2
1
( )t
pup
xtD
HD
tD
pD
x
up
tD
HD
p
tD
D
tD
HDue
tD
D
kk
k
∂∂
∂∂ρ
∂∂ρ
ρρρρ
−−=−−=
−=
+
2
2
1
( ) qt
QuGu
t
p
tD
HD ⋅∇−+⋅+⋅⋅∇+=
∂∂ρτ
∂∂ρ ~ ( )
k
kkk
k
iik
x
q
t
QuG
x
u
t
p
tD
HD
∂∂
∂∂ρ
∂τ∂
∂∂ρ −+++=
SOLO
Return to Table of Content
70
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)SOLO
THERMODYNAMIC PROCESSES
1. ADIABATIC PROCESSES
2. REVERSIBLE PROCESSES
3. ISENTROPIC PROCESSES
No Heat is added or taken away from the System
No dissipative phenomena (viscosity, thermal, conductivity, mass diffusion, friction, etc)
Both adiabatic and reversible
(2.5) THE SECOND LAW OF THERMODYNAMICS AND ENTROPY PRODUCTION
71
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.5) THE SECOND LAW OF THERMODYNAMICS AND ENTROPY PRODUCTION
2nd LAW OF THERMODYNAMICS
Using GAUSS’ THEOREM
0)()(
≥+ ∫∫∫∫∫tStv FF
AdT
qvds
td
d
ρ
00)(
)1(
)()(
≥
⋅∇+⇒≥+ ∫∫∫∫∫∫∫∫
tv
GAUSS
tStv FFF
vdT
q
tD
sDAd
T
qvd
tD
sD
ρρ
- Change in Entropy per unit volumed s
- Local TemperatureT [ ]K
- Fluid Densityρ [ ]3/ mKg
d e q w T ds p dv= + = −δ δ d sd e
T
p
Tdv= +
SOLO
For a Reversible Process
- Rate of Conduction and Radiation of Heat from the System per unit surface
q
[ ]2/ mW
72
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 1)
d e q w T ds p dv= + = −δ δ d sd e
T
p
Tdv= +
uT
p
tD
eD
Tu
T
p
tD
eD
T
tD
D
T
p
tD
eD
TtD
D
T
p
tD
eD
TtD
vD
T
p
tD
eD
TtD
sD
utD
DMC
v
⋅∇+=
⋅∇+=
−+=
+=+=
⋅∇−=
=
ρρρ
ρρ
ρρ
ρρρ
ρρρρρ
ρρ
ρ
2
.).(
2
1
1
11
The Energy Equation (C.E.-4) is
( )k
iik x
uoruu
t
Qqup
tD
eD
∂∂τττ
∂∂ρ =Φ⋅∇⋅−⋅⋅∇=ΦΦ++⋅∇−⋅∇−= ~~
Tt
Q
TT
qup
tD
eD
TtD
sD Φ++⋅∇−=
⋅∇+=
∂∂ρ 11
or
Φ++⋅−∇=t
tD
sDT
∂∂ρ
SOLO
73
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 2)
Define
ρ∂∂
TD s
D tq
Q
t= −∇ ⋅ + + Φ
Θ ≡ + ∇ ⋅ ≥ρ
D s
Dt
q
T
0 Entropy Production Rate per unit volume
Therefore
( )Θ
ΦΘ= −
∇ ⋅+ + + ∇ ⋅
≥∫∫∫
q
T T
Q
t T
q
Tdv
V t
10
∂∂
&
SOLO
or
01 ≥Φ++⋅∇⋅−=Θ
nDissipatio
Systemtoadded
Heat
Systemfrom
RadiationHeat
t
QTq
TT
∂∂
74
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 3)
q q qq conduction rate per unit surface
q radiation rate per unit surfacec r
c
r
= +
q K T K FOURIER s Conduction Lawc = − ∇ > 0 '
( )−∇ ⋅
+ ∇ ⋅
= −
∇ ⋅+ ∇ ⋅ + ⋅ ∇
= ⋅ ∇
= − ∇ + ⋅ ∇
= − ∇ ⋅ − ∇
+ ⋅ ∇
=
∇
+ ⋅ ∇
q
T
q
T
q
T Tq q
Tq
TK T q
T
K TT
T qT
KT
Tq
T
r
r r
1 1 1 1
1 1 12
2
ΘΦ
Φ=∇
+ + + ⋅ ∇
>>>
KT
T T T
Q
tq
T
K
Tr
2 1 10
0
0
∂∂
ΘΦ
≡ + ∇⋅
=
∇
+ + + ⋅∇
≥ρ
∂∂
D s
D t
q
TK
T
T T T
Q
tq
Tr
2 1 10
SOLO
JEAN FOURIER1768-1830
75
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 4)
SOLO
Gibbs Function
Helmholtz Function
sThG ⋅−=:
sTeH ⋅−=:
Josiah Willard Gibbs (1839-1903)
Hermann Ludwig Ferdinandvon Helmholtz(1821 – 1894)
Using the Relations
vdpsdTed ⋅−⋅=
( ) pdvsdTvpdedhd ⋅+⋅=⋅+=vpep
eh ⋅+=+=ρ
:
pdvTdssdTTdshdGd ⋅+⋅−=⋅−⋅−=
vdpTdsTdssdTedHd ⋅−⋅−=⋅−⋅−=
dvT
p
T
edsd +=
76
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 5)
SOLO
Maxwell’s Relations
vdpsdTed ⋅−⋅=
pdvsdThd ⋅+⋅=
pdvTdsGd ⋅+⋅−=
vdpTdsHd ⋅−⋅−=
Ts
pv
v
Fp
v
e
s
hT
s
e
∂∂=−=
∂∂
∂∂==
∂∂
vp
Ts
T
Fs
T
G
p
Gv
p
h
∂∂=−=
∂∂
∂∂==
∂∂
ps
vs
s
v
p
T
s
p
v
T
∂∂=
∂∂
∂∂−=
∂∂
vT
pT
T
p
v
s
T
v
p
s
∂∂=
∂∂
∂∂−=
∂∂
James Clerk Maxwell(1831-1879)
Return to Table of Content
77
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6) CONSTITUTIVE RELATIONS FOR GASES
(2.6.1) NEWTONIAN FLUID DEFINITION – NAVIER–STOKES EQUATIONS
[ ] τσ ~~ +−= Ip
Stress
NEWTONIAN FLUID:
The Shear Stress onA Surface ParallelTo the Flow =Distance Rate ofChange of Velocity
SOLO
CARTESIAN TENSOR NOTATION
ikikik p τδσ +−=
VECTOR NOTATION
- Stress tensor (force per unit surface) of the surrounding on the control surface ( )2/ mN
σ~
- Shear stress tensor (force per unit surface) of the surrounding on the control surface ( )2/ mN
τ~
78
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6) CONSTITUTIVE RELATIONS
(2.6.1) NEWTONIAN FLUID DEFINITION – NAVIER–STOKES EQUATIONS
M. NAVIER 1822INCOMPRESSIBLE FLUIDS
(MOLECULAR MODEL)
G.G. STOKES 1845COMPRESSIBLE FLUIDS(MACROSCOPIC MODEL)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
[ ] [ ] ( )[ ] [ ]IuuuIpIp T ∇+∇+∇+−=+−= λµτσ ~~ik
k
k
i
k
k
iikikikik x
u
x
u
x
upp δ
∂∂λ
∂∂
∂∂µδτδσ +
++−=+−=
( )[ ] [ ]( ) ( ) ( ) µλλµλµτ3
232~0 −=⇒∇+∇=∇+∇+∇== utrutrIutruutrtr T ( ) µλ
∂∂λµδ
∂∂λ
∂∂µτ
3
20322 −=⇒=+=+=
i
iik
k
k
i
iii x
u
x
u
x
u
SOLO
STOKES ASSUMPTION µλ3
2−=0~ =τtrace
μ, λ - Lamé parameters from Elasticity
79
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6) CONSTITUTIVE RELATIONS
(2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
(2.6.1.2) VECTORIAL DERIVATION
I
x
y
zT n= ⋅~σ
d s n ds=
r
druu + du( )unrdtd
t
uurdtd
t
uud
∇⋅+=∇⋅+= 1∂∂
∂∂
( ) ( ) ( ) rdnurdnuuntdt
uud
RotationnTranslatio
1
2
11
2
11 ××∇+
××∇−∇⋅+=
∂∂
OR
DEFINITION OF NEWTONIAN FLUID, NAVIER-STOKES EQUATION
( ) ( ) nnunuunnpT
nTranslatio
1~11
2
1121 ⋅=⋅∇+
××∇−∇⋅+−≡ σλµ
CONSERVATION OF LINEAR MOMENTUM EQUATIONS
SOLO
80
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6) CONSTITUTIVE RELATIONS
(2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
(2.6.1.2)VECTORIAL DERIVATION (CONTINUE) I
x
y
zT n= ⋅~σ
d s n ds=
r
druu + du
CONSERVATION OF LINEAR MOMENTUM EQUATIONS
( ) ( )
( ) ( ) ( )
( ) ( )
( )( )
( )( )
( )∫∫∫
∫∫∫∫∫∫∫∫∫∫∫
∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫
⋅∇∇+×∇×∇+∇⋅∇+∇−=
=⋅∇+×∇×+∇⋅+−=
⋅∇+
××∇−∇⋅+⋅−=+=
)(
)()()()()(
)()()()()()(
251
2
2
2
112
1121
tV
GAUSS
tStStStStV
tStStVtStVtV
vd
GAUSS
u
GAUSS
u
GAUSS
u
GAUSS
pG
usdusdusdsdpvdG
sdnunuunsdnpvdGdsTvdGvdtD
uD
λµµρ
λµµρ
λµρρρ
BUT
( ) ( ) ( )∇× ∇× ≡ ∇ ∇⋅ − ∇⋅ ∇2 2 2µ µ µ u u u
( ) ( ) ( ) ( )∇⋅ ∇ + ∇× ∇× = ∇ ∇⋅ − ∇× ∇×2 2µ µ µ µ u u u u
THEN
SOLO
81
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
I
x
y
zT n= ⋅~σ
d s n ds=
r
druu + du
THEREFORE
( ) ( ) ( ) ∫∫∫∫∫∫ ⋅∇∇+×∇×∇−⋅∇∇+∇−=)()(
2tVtV
vduuupGvdtD
uD
λµµρρ
OR
( ) ( )[ ]uupGtD
uD
⋅∇+∇+×∇×∇−∇−= µλµρρ 2
SOLO
(2.6) CONSTITUTIVE RELATIONS
(2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
(2.6.1.2)VECTORIAL DERIVATION (CONTINUE)
82
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
CONSERVATION OF LINEAR MOMENTUM
( ) ( )[ ]∇ ⋅ = − ∇ − ∇ × ∇ × + ∇ + ∇ ⋅~σ µ µ λp u u
2 ( )
++
++−=
k
k
ii
k
k
i
iii
ik
x
u
xx
u
x
u
xx
p
x ∂∂λµ
∂∂
∂∂
∂∂µ
∂∂
∂∂
∂σ∂
2
( ) ( )[ ]ρ ρ σ
ρ µ µ λ
D u
DtG
G p u u
= + ∇ ⋅
= − ∇ − ∇ × ∇ × + ∇ + ∇ ⋅
~
2 ( )
++
++−=
+=
k
k
ii
k
k
i
iii
i
iki
i
x
u
xx
u
x
u
xx
pG
xG
tD
uD
∂∂λµ
∂∂
∂∂
∂∂µ
∂∂
∂∂ρ
∂σ∂ρρ
2
USING STOKES ASSUMPTION tr ~τ λ µ= ⇒ = −02
3
( )
⋅∇∇+×∇×∇−∇−=
⋅∇+=
uupG
GtD
uD
µµρ
σρρ
3
4
~
+
++−=
+=
k
k
ki
k
k
i
iii
i
iki
i
x
ui
xx
u
x
u
xx
pG
xG
tD
uD
∂∂µ
∂∂
∂∂
∂∂µ
∂∂
∂∂ρ
∂σ∂ρρ
3
4
SOLO
(2.6) CONSTITUTIVE RELATIONS
(2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
83
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
VECTOR NOTATION CARTESIAN TENSOR NOTATION
Euler Equations are obtained by assuming Inviscid Flow
03
20~ =−=⇒= µλτ
pGtD
uD ∇−=
ρρi
ii
x
pG
tD
uD
∂∂ρρ −=
SOLO
(2.6) CONSTITUTIVE RELATIONS
(2.6.2) EULER EQUATIONS
Leonhard Euler (1707-1783)
pGuut
u ∇−=
∇⋅+
∂∂
ρρi
ik
ik
i
x
pG
x
uu
t
u
∂∂ρρ −=
∂∂+
∂∂
or or
84
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6.1.3) COMPUTATION
BUT
Φ
Φ = = +
= +
=
=τ ∂
∂τ ∂
∂τ ∂
∂τ ∂
∂∂∂
ττ τ
iki
kik
i
kki
k
iik
i
k
k
iik ik
u
x
u
x
u
x
u
x
u
xD
ik ki1
2
1
2
τ µ λ δik ik kk ikD D= +2
HENCE ( )Φ = = +τ µ λ δik ik ik kk ik ikD D D D2
OR( )[ ] ( )[ ]
( )[ ] ( )Φ = + + + + + + +
+ + + + + + + + + + ⇒=
2 2
2 2
11 11 22 33 11 22 11 22 33 22
33 11 22 33 33 122
212
132
312
232
322
µ λ µ λ
µ λ µ
D D D D D D D D D D
D D D D D D D D D D DD Dij ji
( ) ( )Φ= + + + + + + + +2 2 2 2112
222
332
122
132
232
11 22 33
2µ λD D D D D D D D DOR
SOLO
(2.6) CONSTITUTIVE RELATIONS
(2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
85
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6.1.3) COMPUTATION (CONTINUE)
USING STOKES ASSUMPTION: tr ~τ λ µ= ⇒ = −02
3
Φ
( ) ( )Φ= + + + + + + + +2 2 2 2112
222
332
122
132
232
11 22 33
2µ λD D D D D D D D D
( ) ( ) ( )( )
( )
( )
Φ = + + − + + + + +
+ + + − + +
+ +
2
3
4
3
4
3
42
3
11 22 33
2
11 22 11 33 22 33 112
222
332
2
122
132
232
11 22 33
2
112
222
332
µ µ µ
µµ
λ
µ
D D D D D D D D D D D D
D D D D D D
D D D
OR
( ) ( ) ( )[ ] ( )Φ = − + − + − + + + >2
34 011 22
2
11 33
2
22 33
2
122
132
232µ µD D D D D D D D D
SOLO
(2.6) CONSTITUTIVE RELATIONS
(2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
86
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6.1.4) ENTROPY AND VORTICITY
From (C.L.M.)
or
( ) ( )[ ]D u
D t
u
t
uu u G p u u
= + ∇
− × ∇ × = − ∇ − ∇ × ∇ × + ∇ + ∇ ⋅
∂∂ ρ ρ
µρ
λ µ2
2
1 1 12
GIBBS EQUATION: T d s d hd p
= −ρ
∀
+⋅∇−
+⋅∇=
+⋅∇
→→→→tld
pd
tdt
pldp
hd
tdt
hldh
sd
tdt
sldsT &
1
∂∂
ρ∂∂
∂∂
Since this is true for all d l t→
&
T s hp
Ts
t
h
t
p
t∇ = ∇ −
∇= −
ρ∂∂
∂∂ ρ
∂∂
&1
SOLO
(2.6) CONSTITUTIVE RELATIONS
(2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
Josiah Willard Gibbs(1903 – 1839)
87
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6.1.4) ENTROPY AND VORTICITY
from (C.L.M.)
or
GIBBS EQUATION: T d s d hd p
= −ρ
∀
+⋅∇−
+⋅∇=
+⋅∇
→→→→tld
pd
tdt
pldp
hd
tdt
hldh
sd
tdt
sldsT &
1
∂∂
ρ∂∂
∂∂
Since this is true for all d l t→
&
T s hp
Ts
t
h
t
p
t∇ = ∇ −
∇= −
ρ∂∂
∂∂ ρ
∂∂
&1
SOLO
hsTGp
Guuut
uII
III
II
I
,,
,,,
,
2
,
~~
2
1 ∇−∇+⋅∇
+=⋅∇
+∇
−=
×∇×−
∇+
ρτ
ρτ
ρ∂∂
ρp
hsT
dlpdp
dlhdh
dlsds∇
−∇=∇→
⋅∇=
⋅∇=
⋅∇=
88
Luigi Crocco 1909-1986
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6.1.4) ENTROPY AND VORTICITY (CONTINUE)
Define
Let take the CURL of this equation
Vorticityu
×∇≡Ω
If , then from (C.L.M.) we get:
G = − ∇ Ψ
CRROCO’s EQUATION (1937)
( ) ( )
⋅∇×∇+
Ψ++∇×∇−∇×∇=×Ω×∇+×∇Ω
τρ∂
∂ ~1
0
2
2
uhsTuu
t
SOLO
ρτ
∂∂ ~
2
1 ,2
,,
⋅∇+
Ψ++∇−∇=×Ω+ I
II
I
uhsTut
u
hsTGuuut
uII
I
II
I
,,
,
,
2
,
~
2
1 ∇−∇+⋅∇
+=
×∇×−
∇+
ρτ
∂∂
From
89
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6.1.4) ENTROPY AND VORTICITY (CONTINUE)
( ) ( ) ( ) ( ) ( )∇ × × = ⋅ ∇ − ∇ ⋅ + ∇ ⋅ − ⋅ ∇ ← ∇ ⋅ = ∇ ⋅ ∇ × =
Ω Ω Ω Ω Ω Ωu u u u u u
0
0
( )∇ × ∇ = ∇ × ∇T s T s
τρ
τρ
τρ
~
0
1~1~1 ⋅∇×∇+⋅∇×
∇=
⋅∇×∇
Therefore ( ) ( ) ( ) τρ∂
∂ ~1 ⋅∇×
∇−∇×∇=∇⋅Ω−Ω⋅∇+Ω∇⋅+
ΩsTuuu
t
SOLO
( ) ( ) τρ
~1 ⋅∇×
∇−∇×∇+⋅∇Ω−∇⋅Ω=
ΩsTuu
tD
D
or
90
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6) CONSTITUTIVE RELATIONS
(2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)
(2.6.1.4) ENTROPY AND VORTICITY (CONTINUE)
( ) ( ) τρ
~1 ⋅∇×
∇−∇×∇+⋅∇Ω−∇⋅Ω=
ΩsTuu
tD
D
FLUID WITHOUT VORTICITY WILL REMAIN FOREVER WITHOUTVORTICITY IN ABSENSE OF ENTROPY GRADIENTS OR VISCOUSFORCES
- FOR AN INVISCID FLUID ( )λ µ τ= = → =0 0~ ~
( ) ( ) sTuutD
DINVISCID
∇×∇+⋅∇Ω−∇⋅Ω=Ω =
0~~τ
- FOR AN HOMENTROPIC FLUID INITIALLY AT REST
s const everywhere i e ss
t. ; . . &∇ = =
0 0
∂∂( )( )
Ω0 0=
( )D
D ts
Ω
Ω= = = ∇ =0 0 0 0 0~ ~, ,τ
SOLO
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91
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6) CONSTITUTIVE RELATIONS
(2.6.2) STATE EQUATION
p - PRESSURE (FORCE / SURFACE)
V - VOLUME OF GAS
M - MASS OF GAS
R - 8314
- 286.9
T - GAS TEMPERATURE
- GAS DENSITY
[ ]m3
[ ]kg
[ ]J kg mol Ko/ ( )⋅
[ ]J kg Ko/ ( )⋅R
[ ]kgmol /−η
[ ]oK
[ ]kg m/ 3ρ
[ ]2/ mN
IDEAL GAS
TRMVp η=
TMVp R=
DEFINE: ρρ
= = =∆ ∆M
Vv
V
M&
1
pv T= R
p T= ρ ROR
SOLO
92
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
IDEAL GAS TMVp R=
SOLO
(2.6) CONSTITUTIVE RELATIONS
(2.6.2) STATE EQUATION
Return to Table of Content
93
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6) CONSTITUTIVE RELATIONS
(2.6.3) THERMALLY PERFECT GAS AND CALORICALLLY PERFECT GAS
A THERMALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH THEINTERNAL ENERGY e IS A FUNCTION ONLY OF THE TEMPERATURE T.
( ) ( )h e T p e T RT h T= + = + =/ ( )ρ THERMALLY PERFECT GAS
DEFINE
C
C
v
V V
p
p p p p
e
T
q
T
h
T
de pdv v d p
d T
de pdv
d T
dq
d T
= =
= = = =
+ +
+
∆
∆
∂∂
∂∂
∂∂
A CALORICALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH Cv IS CONSTANT CALORICALLY PERFECT GAS e C Tv=
SOLO
94
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6) CONSTITUTIVE RELATIONS
(2.6.3) CALORICALLLY PERFECT GAS (CONTINUE)
A CALORICALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH Cv IS CONSTANT CALORICALLY PERFECT GAS e C Tv=
FOR A CALORICALLY PERFECT GAS
( )h C T RT C R T C T C C Rv v p p v= + = + = → = +
γγ
γ γ= ⇒ =
−⇒ =
−
= + = −∆ C
CC R C
Rp
v
C C R
p
R C C
v
p v p v
1 1
γ air = 14.
SOLO
95
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6) CONSTITUTIVE RELATIONS
(2.6.3) CALORICALLLY PERFECT GAS (CONTINUE)
(2.6.3.1) ENTROPY CALCULATIONS FOR A CALORICALLLY PERFECT GAS
pv T= R p T= ρ R IDEAL GAS
( )ds
de pdv
T
de pdv vdp vdp
T
dh vdp
T=
+=
+ + −=
−∆
ds CdT
TR
dv
vs s C
T
TR
v
vC
T
TRv v v= + → − = + = −2 1
2
1
2
1
2
1
2
1
ln ln ln lnρρ
1
2
1
212 lnln
p
pR
T
TCss
p
dpR
T
dTCds pp −=−→−=
s s Cp
pR C
p
pCv v p2 1
2
1
1
2
2
1
2
1
2
1
− = ⋅
− = −ln ln ln ln
ρρ
ρρ
ρρ
ENTROPY
SOLO
96
FLUID DYNAMICS
2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
(2.6) CONSTITUTIVE RELATIONS
(2.6.3) CALORICALLLY PERFECT GAS (CONTINUE)
(2.6.3.1) ENTROPY CALCULATIONS FOR A CALORICALLLY PERFECT GAS
p
p
T
Te
T
Te
p
p
T
T
C
R s s
R
s s
R
isentropic
s sp
2
1
2
1
2
1
12
1
2
1
12 1 2 1 2 1
=
=
=
−− − −
− = −
⇒γ
γγ
γ
ρρ
ρρ
γγ γ
2
1
2
1
2
1
1
12
1
2
1
1
12 1 2 1 2 1
=
=
=
−− − −
− = −
⇒T
Te
T
Te
T
T
C
R s s
R
s s
R
isentropic
s sv
p
pe e
p
p
C
C s s
R
s s
R
isentropic
s sp
v2
1
2
1
2
1
2
1
2
1
2 1 2 1 2 1
=
=
=
−−
−− =
⇒ρρ
ρρ
ρρ
γ γ
T
T
h
h
p
pe
p
pe
T
T
h
h
p
p
s s
C
s s
C
isentropic
s sv p2
1
2
1
2
1
2
1
2
1
1
2
1
12
1
2
1
2
1
1
2
1
12 1 2 12 1
= = ⋅ =
=
= =
=
−−
− − −− = −
−
⇒ρρ
ρρ
ρρ
γγ
γγ
γγ
ISENTROPIC CHAIN
SOLO
Return to Table of Content
97
FLUID DYNAMICS
BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
BOUNDARY CONDITIONS
SOLO
Return to Table of Content
98
SOLODimensionless Equations
Dimensionless Variables are:
0/~ ρρρ = 0/~
Uuu = gGG /~
= ( )200/~ Upp ρ=
0/~ lUtt =
20/
~UCTT p=( )2
00/~ Uρττ =2
0/~
UHH = 20/
~Uhh = 2
0/~ Uee = ( )200/~ Uqq ρ= ( )2/
~UQQ =
∇=∇ 0
~l
Field Equations
(C.M.): ( )00
00U
lu
t ρρ
∂ρ∂ =⋅∇+
( )200
0
~
3
4
U
luupGuu
t
u
ρµµρ
∂∂ρ
τ
⋅∇
⋅∇∇+×∇×∇−∇−=
∇⋅+(C.L.M.):
( ) ( )300
0~U
lTk
t
QuGu
t
pHu
t
H
qρ∂
∂ρτ∂∂ρ
∇⋅∇−+⋅+⋅⋅∇+=
∇⋅+
∂∂
(C.E.):
( )( ) ( ) 0
/
/
000
00
0 =
⋅∇+
U
ul
lUt
ρρ
∂ρρ∂
( ) ( ) ( )
( ) ( )
⋅∇∇
+
×∇×∇
−
∇−=
∇⋅+
000
000
0
00
00
000
0
200
020
0
000
000
0
0
3
4
/
/
U
ull
UlU
ull
Ul
U
pl
g
G
U
lg
U
ul
U
u
lUt
Uu
ρµ
µµ
ρµ
ρρρ
∂∂
ρρ
( ) ( ) ( ) ( ) ( ) ( )
∇⋅∇
−+⋅+
⋅⋅∇+
∂
∂=
2
0
00
00
0
000
02
000002
0
0
02
00
020000
200000 /
~
// U
CTl
k
kl
C
k
UlU
Q
lUtU
u
g
G
U
gl
U
u
Ul
U
p
lUtU
H
lUtD
D p
pµρµ
∂∂
ρρ
ρτ
ρρρρ
0/~ ρρρ = 0/~
Uuu = gGG /~
= ( )200/~ Upp ρ=
0/~ lUtt =
20/
~UCTT p=( )2
00/~ Uρττ =2
0/~
UHH = 20/
~Uhh = 2
0/~ Uee = ( )200/~ Uqq ρ= ( )2/
~UQQ =
∇=∇ 0
~l
0/~ µµµ =
0/~
kkk =
Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path)
0/~ λλλ =
99
SOLODimensionless Equations
Dimentionless Field Equations
(C.M.): ( ) 0~~~~
=⋅∇+ ut
ρ∂
ρ∂
( ) ( )uR
uR
pGF
uut
u
eer
~~~~1
3
4~~~~1~~~~1~~~~
~~
2
⋅∇∇+×∇×∇−∇−=
∇⋅+ µµρ
∂∂ρ(C.L.M.):
( ) ( )TkPRt
QuG
Fu
t
pHu
t
H
rer
∇⋅∇−+⋅+⋅⋅∇+=
∇⋅+
∂∂ 11
~
~~~~1~~~
~~~~~
~
~~
2 ∂∂ρτ
∂∂ρ
(C.E.):
Reynolds:0
000
µρ lU
Re = Prandtl:0
0
k
CP p
r
µ= Froude:
0
0
gl
UFr =
0/~ ρρρ = 0/~
Uuu = gGG /~
= ( )200/~ Upp ρ=
0/~ lUtt =
20/
~UCTT p=( )2
00/~ Uρττ =2
0/~
UHH = 20/
~Uhh = 2
0/~ Uee = ( )200/~ Uqq ρ= ( )2/
~UQQ =
∇=∇ 0
~l
0/~ ρρρ = 0/~
Uuu = gGG /~
= ( )200/~ Upp ρ=
0/~ lUtt =
20/
~UCTT p=( )2
00/~ Uρττ =2
0/~
UHH = 20/
~Uhh = 2
0/~ Uee = ( )200/~ Uqq ρ= ( )2/
~UQQ =
∇=∇ 0
~l
0/~ µµµ =
0/~
kkk =
Dimensionless Variables are:
Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path)
0/~ λλλ =
Knudsenl
Kn0
0:λ=
100
SOLODimensionless Equations
Constitutive Relations
TRp ρ=
2
2
1uTCH p +=
Tkq ∇−=
TCh p=
−==200
200
200
1
U
TC
U
TC
C
R
U
p pp
p ρρ
γγ
ρρ
ρ
=
20
20 U
TC
U
h p
2
020
20 2
1
+
=
U
u
U
TC
U
H p
( )
∇
−=
20
000
0
000
0300 U
TCl
k
k
C
k
UlU
q p
pµρµ
ρ
( ) [ ]33
2~ Iuuu T ⋅∇−∇+∇= µµτ [ ]30
00000
0
00
00
0000
0
00 3
2~I
U
ul
UlU
ul
U
ul
UlU
T
⋅∇
−
∇+∇
=
µµ
ρµ
µµ
ρµ
ρτ
0/~ ρρρ = 0/~
Uuu = gGG /~
= ( )200/~ Upp ρ=
0/~ lUtt =
20/
~UCTT p=( )2
00/~ Uρττ =2
0/~
UHH = 20/
~Uhh = 2
0/~ Uee = ( )200/~ Uqq ρ= ( )2/
~UQQ =
∇=∇ 0
~l
0/~ ρρρ = 0/~
Uuu = gGG /~
= ( )200/~ Upp ρ=
0/~ lUtt =
20/
~UCTT p=( )2
00/~ Uρττ =2
0/~
UHH = 20/
~Uhh = 2
0/~ Uee = ( )200/~ Uqq ρ= ( )2/
~UQQ =
∇=∇ 0
~l
0/~ µµµ =
0/~
kkk =
Dimensionless Variables are:
Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path)
0/~ λλλ =
101
SOLODimensionless Equations
Dimensionless Constitutive Relations
2~2
1~~uTH +=
Tp~~1~ ρ
γγ −= Ideal Gas
( ) [ ]3
~~~
3
2~~~~~~~ IuR
uuR e
T
e
⋅∇−∇+∇= µµτ Navier-Stokes
Th~~ = Calorically Perfect Gas
TkPR
qre
~~~11~∇−= Fourier Law
Reynolds:0
000
µρ lU
Re =
Prandtl:0
0
k
CP p
r
µ=
0/~ ρρρ = 0/~
Uuu = gGG /~
= ( )200/~ Upp ρ=
0/~ lUtt =
20/
~UCTT p=( )2
00/~ Uρττ =2
0/~
UHH = 20/
~Uhh = 2
0/~ Uee = ( )200/~ Uqq ρ= ( )2/
~UQQ =
∇=∇ 0
~l
0/~ ρρρ = 0/~
Uuu = gGG /~
= ( )200/~ Upp ρ=
0/~ lUtt =
20/
~UCTT p=( )2
00/~ Uρττ =2
0/~
UHH = 20/
~Uhh = 2
0/~ Uee = ( )200/~ Uqq ρ= ( )2/
~UQQ =
∇=∇ 0
~l
0/~ µµµ =
0/~
kkk =
Dimensionless Variables are:
Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path)
0/~ λλλ =
Return to Table of Content
102
SOLO
Mach Number
Mach number (M or Ma) / is a dimensionless quantity representing the ratio of speed of an object moving through a fluid and the local speed of sound.
• M is the Mach number,• U0 is the velocity of the source relative to the medium, and
• a0 is the speed of sound
Mach:0
0
a
UM =
The Mach number is named after Austrian physicist and philosopher Ernst Mach, a designation proposed by aeronautical engineer Jakob Ackeret.
Ernst Mach (1838–1916)
Jakob Ackeret (1898–1981)
m
Tk
Mo
TRa Bγγ ==0
• R is the Universal gas constant, (in SI, 8.314 47215 J K−1 mol−1), [M1 L2 T−2 θ−1 'mol'−1]
• γ is the rate of specific heat constants Cp/Cv and is dimensionless γair = 1.4.• T is the thermodynamic temperature [θ1]
• Mo is the molar mass, [M1 'mol'−1]
• m is the molecular mass, [M1]
AERODYNAMICS
103
SOLOMach Number – Flow Regimes
Regime Mach mph km/h m/s General plane characteristics
Subsonic <0.8 <610 <980 <270Most often propeller-driven and commercial turbofan aircraft with high aspect-ratio (slender) wings, and rounded features like the nose and leading edges.
Transonic 0.8-1.2 610-915
980-1,470 270-410Transonic aircraft nearly always have swept wings, delaying drag-divergence, and often feature design adhering to the principles of the Whitcomb Area rule.
Supersonic 1.2–5.0915-3,840
1,470–6,150 410–1,710
Aircraft designed to fly at supersonic speeds show large differences in their aerodynamic design because of the radical differences in the behaviour of flows above Mach 1. Sharp edges, thin aerofoil-sections, and all-moving tailplane/canards are common. Modern combat aircraft must compromise in order to maintain low-speed handling; "true" supersonic designs include the F-104 Starfighter, SR-71 Blackbird and BAC/Aérospatiale Concorde.
Hypersonic 5.0–10.03,840–7,680
6,150–12,300
1,710–3,415
Cooled nickel-titanium skin; highly integrated (due to domination of interference effects: non-linear behaviour means that superposition of results for separate components is invalid), small wings, such as those on the X-51A Waverider
High-hypersonic
10.0–25.07,680–16,250
12,300–30,740
3,415–8,465
Thermal control becomes a dominant design consideration. Structure must either be designed to operate hot, or be protected by special silicate tiles or similar. Chemically reacting flow can also cause corrosion of the vehicle's skin, with free-atomic oxygen featuring in very high-speed flows. Hypersonic designs are often forced into blunt configurations because of the aerodynamic heating rising with a reduced radius of curvature.
Re-entry speeds >25.0
>16,250 >30,740 >8,465 Ablative heat shield; small or no wings; blunt shape
104
SOLO
Different Regimes of Flow
Mach Number – Flow Regimes
AERODYNAMICS
105
whereρ = air densityV = true speedl = characteristic lengthμ = absolute (dynamic) viscosityυ = kinematic viscosity
Reynolds:υµ
ρ ρµυ
lVlVRe
=
==
Osborne Reynolds (1842 –1912)
It was observed by Reynolds in 1884 that a Fluid Flow changes from Laminar to Turbulent at approximately the same value of the dimensionless ratio (ρ V l/ μ) where l is the Characteristic Length for the object in the Flow. This ratio is called the Reynolds number, and is the governing parameter for Viscous Flow.
Reynolds Number and Boundary Layer
SOLO 1884AERODYNAMICS
106
Boundary Layer
SOLO 1904AERODYNAMICS
Ludwig Prandtl(1875 – 1953)
In 1904 at the Third Mathematical Congress, held at Heidelberg, Germany, Ludwig Prandtl (29 years old) introduced the concept of Boundary Layer. He theorized that the fluid friction was the cause of the fluid adjacent to surface to stick to surface – no slip condition, zero local velocity, at the surface – and the frictional effects were experienced only in the boundary layer a thin region near the surface. Outside the boundary layer the flow may be considered as inviscid (frictionless) flow. In the Boundary Layer on can calculate the •Boundary Layer width•Dynamic friction coefficient μ•Friction Drag Coefficient CDf
107
The flow within the Boundary Layer can be of two types:•The first one is Laminar Flow, consists of layers of flow sliding one over other in a regular fashion without mixing.•The second one is called Turbulent Flow and consists of particles of flow that moves in a random and irregular fashion with no clear individual path, In specifying the velocity profile within a Boundary Layer, one must look at the mean velocity distribution measured over a long period of time.There is usually a transition region between these two types of Boundary-Layer Flow
SOLO AERODYNAMICS
108
Normalized Velocity profiles within a Boundary-Layer, comparison betweenLaminar and Turbulent Flow.
SOLO
Boundary-Layer
AERODYNAMICS
109
Flow Characteristics around a Cylindrical Body as a Function of Reynolds Number (Viscosity)
AERODYNAMICSSOLO
110
Relative Drag Force as a Function of Reynolds Number (Viscosity)
AERODYNAMICS
Drag CD0 due toFlow Separation
SOLO
111
Relative Drag Force as a Function of Reynolds Number (Viscosity)
AERODYNAMICS
Drag due to Viscosity:1.Skin Friction2.Flow Separation (Drop in pressure behind body)
∫∫
∫∫
⋅+⋅−−=
⋅+⋅−=
∧∧∞
∧∧
W
W
S
S
fpD
dswtV
fwn
V
pp
S
dswtCwnCS
C
xx
xx
11
11
ˆ2/
ˆ2/
1
ˆˆ1
22 ρρ
SOLO
112
Parasite Drag
Pressure Differential,Viscous Shear Stress,and Separation
AERODYNAMICS
Relative Drag Force as a Function of Reynolds Number (Viscosity)
DragFrictionSkin
ET
EL
ll
ET
EL
uu
DragPressure
ET
EL
ll
ET
EL
uu
sdfsdf
sdpsdpD
∫∫
∫∫
++
+−=
..
..
..
..
..
..
..
..
coscos
sinsin
θθ
θθ
SOLO
113
AERODYNAMICS
Relative Drag Force as a Function of Reynolds Number (Viscosity)
• Blunt Body: Most of Drag is Pressure Drag.• Streamlined Body: Most of Drag is Skin Friction Drag.
SOLO
114
AERODYNAMICS
Relative Drag Force as a Function of Reynolds Number (Viscosity)
SOLO
115
AERODYNAMICS
Relative Drag Force as a Function of Reynolds Number (Viscosity)
SOLO
116
AERODYNAMICS
Relative Drag Force as a Function of Reynolds Number (Viscosity)
SOLO
Variation of total skin-friction coefficient with Reynolds number for a smooth, flat plate.[From Dommasch, et al. (1967).]
117
Typical Effect of Reynolds Number on Parasitic Drag
Flow may stay attachedfarther at high Re,reducing the drag
AERODYNAMICSSOLO
Return to Table of Content
118
FluidsSOLO
Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of the body in a fluid. The number is named after Danish physicist Martin Knudsen.
Knudsenl
Kn0
0:λ= Martin Knudsen
(1871–1949).
For a Boltzmann gas, the mean free path may be readily calculated as:
• kB is the Boltzmann constant (1.3806504(24) × 10−23 J/K in SI units), [M1 L2 T−2 θ−1]p
TkB20
2 σπλ =
• T is the thermodynamic temperature [θ1]
λ0 = mean free path [L1]
Knudsen Number
l0 = representative physical length scale [L1].
• σ is the particle hard shell diameter, [L1]
• p is the total pressure, [M1 L−1 T−2].
See “Kinetic Theory of Gases” Presentation
For particle dynamics in the atmosphere and assuming standard atmosphere pressure i.e. 25 °C and 1 atm, we have λ0 ≈ 8x10-8m.
119
FluidsSOLO
Martin Knudsen (1871–1949).
Knudsen Number (continue – 1)
Relationship to Mach and Reynolds numbers
Dynamic viscosity,
Average molecule speed (from Maxwell–Boltzmann distribution),
thus the mean free path,
where
• kB is the Boltzmann constant (1.3806504(24) × 10−23 J/K in SI units), [M1 L2 T−2 θ−1]
• T is the thermodynamic temperature [θ1]
• ĉ is the average molecular speed from the Maxwell–Boltzmann distribution, [L1 T−1]
• μ is the dynamic viscosity, [M1 L−1 T−1]
• m is the molecular mass, [M1]
• ρ is the density, [M1 L−3].
02
1 λρµ c=
m
Tkc B
π8=
Tk
m
B20
πρµλ =
120
FluidsSOLO
Martin Knudsen (1871–1949).
Knudsen Number (continue – 2)
Relationship to Mach and Reynolds numbers (continue – 1)
The dimensionless Reynolds number can be written:
Dividing the Mach number by the Reynolds number,
and by multiplying by
yields the Knudsen number.
The Mach, Reynolds and Knudsen numbers are therefore related by:
Reynolds:Re0
000
µρ lU=
Tk
m
lmTklallU
aUM
BB γρµ
γρµ
ρµ
µρ 00
0
00
0
000
0
0000
00
//
/
Re====
KnTk
m
lTk
m
l BB
==22 00
0
00
0 πρµπγ
γρµ
2Re
πγMKn =
121
FluidsSOLO
Knudsen Number (continue – 3)
Relationship to Mach and Reynolds numbers (continue –2)
According to the Knudsen Number the Gas Flow can be divided in three regions:1.Free Molecular Flow (Kn >> 1): M/Re > 3 molecule-interface interaction negligible between incident and reflected particles2.Transition (from molecular to continuum flow) regime: 3 > M/Re and M/(Re)1/2 > 0.01 (Re >> 1). Both intermolecular and molecule-surface collision are important.3.Continuum Flow (Kn << 1): 0.01 > M/(Re)1/2. Dominated by intermolecular collisions.
2Re
πγMKn =
FluidsSOLO
Knudsen Number (continue – 4)
InviscidLimit Free
MolecularLimitKnudsen Number
Boltzman EquationCollisionless
Boltzman Equation
DiscreteParticlemodel
Euler Equation
Navier-Stokes Equation
Continuummodel
Conservation Equationdo not form a closed set
Validity of conventional mathematical models as a function of localKnudsen Number
Return to Table of Content
123
AERODYNAMICS
Fluid flow is characterized by a velocity vector field in three-dimensional space, within the framework of continuum mechanics. Streamlines, Streaklines and Pathlines are field lines resulting from this vector field description of the flow. They differ only when the flow changes with time: that is, when the flow is not steady.
• Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction a fluid element will travel in at any point in time.
• Streaklines are the locus of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streakline
• Pathlines are the trajectories that individual fluid particles follow. These can be thought of as a "recording" of the path a fluid element in the flow takes over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time.
• Timelines are the lines formed by a set of fluid particles that were marked at a previous instant in time, creating a line or a curve that is displaced in time as the particles move.
The red particle moves in a flowing fluid; its pathline is traced in red; the tip of the trail of blue ink released from the origin follows the particle, but unlike the static pathline (which records the earlier motion of the dot), ink released after the red dot departs continues to move up with the flow. (This is a streakline.) The dashed lines represent contours of the velocity field (streamlines), showing the motion of the whole field at the same time. (See high resolution version.
Flow DescriptionSOLO
124
3-D FlowFlow Description
SOLO
Steady Motion: If at various points of the flow field quantities (velocity, density, pressure)associated with the fluid flow remain unchanged with time, the motion is said to be steady.
( ) ( ) ( )zyxppzyxzyxuu ,,,,,,,, === ρρ
Unsteady Motion: If at various points of the flow field quantities (velocity, density, pressure)associated with the fluid flow change with time, the motion is said to be unsteady.
( ) ( ) ( )tzyxpptzyxtzyxuu ,,,,,,,,,,, === ρρ
Path Line: The curve described in space by a moving fluid element is known as its trajectoryor path line.
tt
tt ∆+
t
tt ∆+tt ∆+ 2
t
tt ∆+tt ∆+ 2
Path Line (steady flow)
t
tt ∆+
t
tt ∆+ 2
tt ∆+
t
Path Line (unsteady flow)
tt ∆+ 2
tt ∆+
t
125
3-D Flow
Flow Description
SOLO
Path Line: The curve described in space by a moving fluid element is known as its trajectoryor path line.
ttt ∆+ tt ∆+ 2
Streamlines: The family of curves such that each curve is tangent at each point to the velocity direction at that point are called streamlines.
Consider the coordinate of a point P and the direction of the streamline passingthrough this point. If is the velocity vector of the flow passing through P at a time t,then and parallel, or:
r
rdu
u
rd
0=×urd
( )( )( )
0
1
1
1111
=
−
−
−
=
zdyudxv
ydxwdzu
xdzvdyw
wvu
dzdydx
zyx
w
zd
v
yd
u
xd==
Cartesian
t
u
r
rd
126
3-D Flow
Flow Description
SOLO
Path Line: The curve described in space by a moving fluid element is known as its trajectoryor path line.
Streamlines: The family of curves such that each curve is tangent at each point to the velocity direction at that point are called streamlines.
( ) ( ) ( )tzyxw
zd
tzyxv
yd
tzyxu
xd
,,,,,,,,,==
t
u
r
rd
Those are two independent differential equations for a streamline. Given a point the streamline is defined from those equations.( )0000 ,,, tzyxr
( ) ( ) ( )( ) ( ) ( )tzyxw
zd
tzyxv
yd
tzyxv
yd
tzyxu
xd
,,,,,,2
,,,,,,1
=
= ( ) ( ) ( )( ) ( ) ( ) 0,,,,,,,,,
0,,,,,,,,,
222
111
=++=++
zdtzyxcydtzyxbxdtzyxa
zdtzyxcydtzyxbxdtzyxa
( ) ( )( ) ( )21
21
22
11
•+••+•
βαβα
022
11 ≠βαβα
Pfaffian Differential Equations
For a given a point the solution of those equations is of the form:( )0000 ,,, tzyxr
( )( ) 2,,,
1,,,
02
01
consttzyx
consttzyx
==
ψψ
u
( )0tr
rd
0t
( ) 11 cr =ψ
( ) 22 cr =
ψStreamline Those are two surfaces, the
intersection of which is the streamline.
127
3-D Flow
Flow Description
SOLO
Path Line: The curve described in space by a moving fluid element is known as its trajectoryor path line.
Streamlines: The family of curves such that each curve is tangent at each point to the velocity direction at that point are called streamlines.
( ) ( ) ( )tzyxw
zd
tzyxv
yd
tzyxu
xd
,,,,,,,,,==
t
u
r
rd
For a given a point the solution of those equations is of the form:( )0000 ,,, tzyxr
( )( ) 2,,,
1,,,
02
01
consttzyx
consttzyx
==
ψψ
u
( )0tr
rd
0t
( ) 11 cr =
ψ
( ) 22 cr =
ψStreamline Those are two surfaces, the
intersection of which is the streamline.
The streamline is perpendicular to the gradients (normals) of those two surfaces.
( ) ( ) ( )0201 ,, trtrVr ψψµ ∇×∇=
where μ is a factor that must satisfy the following constraint.
( )( ) ( ) ( ) 0,, 0201 =∇×∇⋅∇=⋅∇ trtrVr ψψµ
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128
AERODYNAMICS
Streamlines, Streaklines, and PathlinesMathematical description
Streamlines
If the components of the velocity are written and those of the streamline aswe deduce
which shows that the curves are parallel to the velocity vector
Pathlines
Streaklines
where, is the velocity of a particle P at location and time t . The parameter , parametrizes the streakline and 0 ≤ τP ≤ t0 , where t0 is a time of interest .
The suffix P indicates that we are following the motion of a fluid particle. Note that at point
the curve is parallel to the flow velocity vector where the velocity vector is evaluated at the position of the particle at that time t .
SOLO
129
∞V
Airfoil Pressure Field variation with α
AERODYNAMICS
Airfoil Velocity Field variation with αAirfoil Streamline variation with αAirfoil Streakline with α
Streamlines, Streaklines, and PathlinesSOLO
130
AERODYNAMICSStreamlines, Streaklines, and Pathlines
SOLO
131
AERODYNAMICSSOLO
132
AERODYNAMICSSOLO
133
AERODYNAMICSStreamlines, Streaklines, and Pathlines
SOLO
Return to Table of Content
134
3-D Inviscid Incompressible Flow
Circulation
SOLO
Circulation Definition:
tV ∆
( ) tVV ∆∆+
S∆
Sn ∆1
V
×∇
tr
∆
ttr ∆+∆
tC
ttC ∆+
∫ ⋅=ΓC
rdV
:
Material Derivative of the Circulation
( )∫∫∫ ⋅+⋅=
⋅=
Γ
CCC
rdtD
DVrd
tD
VDrdV
tD
D
tD
D
From the Figure we can see that:
( ) tVrtVVr ttt ∆+∆=∆∆++∆ ∆+
( ) VdrdtD
DV
t
rr tttt
=→∆=
∆∆−∆ →∆
∆+0
( ) 02
2
=
=⋅=⋅ ∫∫∫
CCC
VdVdVrd
tD
DV
Therefore:
∫ ⋅=Γ
C
rdtD
VD
tD
D
integral of an exact differential on a closed curve.
C – a closed curve
135
3-D Inviscid Incompressible FlowSOLO
tV ∆
( ) tVV ∆∆+
S∆
Sn ∆1
V
×∇
tr
∆
ttr ∆+∆
tC
ttC ∆+
S
∫ ⋅=ΓtC
rdV
:
Material Derivative of the Circulation (second derivation)
Subtract those equations:
tVrdSn t ∆×=∆
1
( )∫∆+
⋅∆+=Γ∆+ΓttC
rdVV
:
( ) ( )∫∫∫∫ ∆⋅×∇=⋅∆+−⋅=Γ∆−∆+ S
TheoremsStoke
CC
SnVrdVVrdVttt
1'
S is the surface bounded by the curves Ct and C t+Δ t
( ) ( ) ( ) tVVrdtVrdVSnVS
t
S
t
S
∆
×∇×⋅=∆×⋅×∇=∆⋅×∇=Γ∆− ∫∫∫∫∫∫
1
td
d
ttd
rd
tV
ttD
D rdd Γ+
∂Γ∂
=Γ∇⋅+∂
Γ∂=Γ∇⋅+
∂Γ∂
=Γ Γ∇⋅=Γ
Computation of: ∫ ⋅∂∂
=∂
Γ∂
tC
rdt
V
t
Computation of:td
d Γ
136
3-D Inviscid Incompressible FlowSOLO
tV ∆
( ) tVV ∆∆+
S∆
Sn ∆1
V
×∇
tr
∆
ttr ∆+∆
tC
ttC ∆+
Material Derivative of the Circulation (second derivation)
( ) tVVrdS
t ∆
×∇×⋅=Γ∆− ∫∫
When Δ t → 0 the surface S shrinks to the curve C=Ct and the surface integral transforms to a curvilinear integral:
( ) ( ) ( )∫∫∫∫∫ ∇⋅⋅+
−=∇⋅⋅+
∇⋅−=×∇×⋅−=Γ
C
t
CC
t
C
t
C
t VVrdV
dVVrdV
rdVVrdtd
d
0
22
22
Computation of: (continue)td
d Γ
Finally we obtain:
( ) ∫∫∫ ⋅=∇⋅⋅+⋅∂∂
=Γ
+∂
Γ∂=
Γ
tt CC
t
C
rdtD
VDVVrdrd
t
V
td
d
ttD
D
137
3-D Inviscid Incompressible FlowSOLO
tV ∆
( ) tVV ∆∆+
S∆
Sn ∆1
V
×∇
tr
∆
ttr ∆+∆
tC
ttC ∆+
Material Derivative of the Circulation
We obtained: ∫ ⋅=Γ
tC
rdtD
VD
tD
D
Use C.L.M.: hsTp
VVt
V
tD
VDII
I
G
II
II
,,
,
,,
~∇−∇+
⋅∇+Ψ∇=
∇⋅+=
τ∂∂
( ) ( )
0
,
,,
,
,
~~
∫∫∫∫ −Ψ+⋅
⋅∇+∇=⋅∇−Ψ∇+⋅
⋅∇+∇=
Γ
tttt CC
I
I
C
I
C
I
I
I
hddrdp
sTrdhrdp
sTtD
D ττ
to obtain:
∫ ⋅
⋅∇+∇=
Γ
tC
I
I
I
rdp
sTtD
D τ~,
,or:
Kelvin’s Theorem
William Thomson Lord Kelvin(1824-1907)
In an inviscid , isentropic flow d s = 0 with conservative body forces the circulation Γ around a closed fluid line remains constant with respect to time.
0~~ =τ
Ψ∇=G
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1869
138
3-D Inviscid Incompressible FlowSOLO
Circulation Definition: ∫ ⋅=ΓC
rdV
:
C – a closed curve
Biot-Savart Formula
1820
Jean-Baptiste Biot1774 - 1862
VorticityV
×∇≡Ω
∫ −Ω=
Space
dVsr
A
π4
1
( )lddSnsr
Ad
⋅−Ω=
π4
1
The contribution of a length dl of the Vortex Filament to isA
∫∫∫∫∫ ⋅Ω=⋅×∇=⋅=ΓSS
Stokes
C
SdnSdnVrdV
:
If the Flow is Incompressible 0=⋅∇ u
so we can write , where is the Vector Potential. We are free tochoose so we choose it to satisfy .
AV
×∇=A A
0=⋅∇ A
We obtain the Poisson Equation that defines the Vector Potential A
Ω−=∇
A2 Poisson Equation Solution( ) ∫ −Ω=
Space
dvsr
rA
π4
1
Félix Savart1791 - 1841
Biot-Savart Formula
139
3-D Inviscid Incompressible FlowSOLO
Circulation Definition: ∫ ⋅=ΓC
rdV
:
C – a closed curve
Biot-Savart Formula (continue - 1)
1820
Jean-Baptiste Biot1774 - 1862
VorticityV
×∇≡Ω
( )lddSnsr
Ad
⋅−Ω=
π4
1We found
∫∫∫∫∫ ⋅Ω=⋅×∇=⋅=ΓSS
Stokes
C
SdnSdnVrdV
:
also we have dlldΩΩ=
( ) ( ) ∫∫∫∫∫ ×−
∇⋅Ω=⋅−Ω×∇=×∇=
Γ
ΩΩ=
ldsr
dSnlddSnsr
AdrV r
S
dlld
v
rr
1
4
1
4
1
ππ
( ) ( )∫ −
−×Γ= 34 sr
srldrV
π Biot-Savart Formula
Félix Savart1791 - 1841
Biot-Savart Formula
140
3-D Inviscid Incompressible Flow
Circulation
SOLO
Circulation Definition: ∫ ⋅=ΓC
rdV
:
C – a closed curve
Biot-Savart Formula (continue - 2)
1820
Jean-Baptiste Biot1774 - 1862( ) ( )
∫ −−×Γ= 34 sr
srldrV
πBiot-Savart Formula
General 3D Vortex
Félix Savart1791 - 1841
141
3-D Inviscid Incompressible FlowSOLO
Circulation Definition: ∫ ⋅=ΓC
rdV
:
C – a closed curve
Biot-Savart Formula (continue - 3)
1820
Jean-Baptiste Biot1774 - 1862
Félix Savart1791 - 1841
( ) ( )∫ −
−×Γ= 34 sr
srldrV
πBiot-Savart Formula
General 3D Vortex
For a 2 D Vortex:
( ) θθθθθd
hsr
dl
sr
srld sinˆˆsin23 =
−=
−−×
θθ
θ dh
dlhl2sin
cot =⇒=−
θsin/hsr =−
θπ
θθθπ
πˆ
2sinˆ
4 0 hd
hV
Γ=Γ= ∫ Biot-Savart Formula
General 2D Vortex
Biot-Savart Formula
142
3-D Inviscid Incompressible FlowSOLO
Circulation Definition: ∫ ⋅=ΓC
rdV
:
C – a closed curve
Biot-Savart Formula (continue - 4)
1820
Jean-Baptiste Biot1774 - 1862( ) ( )
∫ −−×Γ= 34 sr
srldrV
πBiot-Savart Formula General 3D Vortex
Félix Savart1791 - 1841
Lifting-Line Theory
Biot-Savart Formula
Return to Table of Content
143
3-D Inviscid Incompressible Flow
Helmholtz Vortex Theorems
SOLO
Helmholtz : “Uber the Integrale der hydrodynamischen Gleichungen, welcheDen Wirbelbewegungen entsprechen”, (“On the Integrals of the Hydrodynamical Equations Corresponding to Vortex Motion”), in Journal fur die reine und angewandte, vol. 55, pp. 25-55. , 1858He introduced the potential of velocity φ.
Hermann Ludwig Ferdinandvon Helmholtz
1821 - 1894
Theorem 1: The circulation around a given vortex line (i.e., the strength of the vortex filament) is constant along its length.
Theorem 2: A vortex filament cannot end in a fluid. It mustform a closed path, end at a boundary, or go to infinity.
Theorem 3: No fluid particle can have rotation, if it did not originally rotate.Or, equivalently, in the absence of rotational external forces, a fluid that is initially irrotational remains irrotational. In general we can conclude that thevortex are preserved as time passes. They can disappear only through the action of viscosity (or some other dissipative mechanism).
Return to Table of Content
1858
144
( ) ( ) ( )
MomentFriction
S
C
Momentessure
S
CCA
WW
dstRRfdsnRRppM ∫∫∫∫∑ ×−+×−−= ∞ 11
Pr
/
( )
FrictionSkinorFrictionViscous
S
essureNormal
S
A
WW
dstfdsnppF ∫∫∫∫∑ +−= ∞ 11
Pr
AERODYNAMIC FORCES AND MOMENTS.
SOLO
Aerodynamic Moments Relative to a point can be divided in Pressure Moments and Friction Moments Relative to this point.
CR
Aerodynamic Forces can be divided in Pressure Forces, normal to Body Surface , and Friction Forces, tangent to Body Surface .
dsn
1
dst
1
CR
AERODYNAMICS
145
SOLO
Body Coordinates (B)
The origin of the Body coordinate systemis located at the instantaneous center ofgravity CG of the vehicle, with xB pointedto the front of the Air Vehicle, yB pointedtoward the right wing and zB completingthe right-handed Cartesian reference frame.
ψθ
φBx
Lx
Bz
Ly
LzBy
Rotation Matrix from LLLN to B (Euler Angles):
[ ] [ ] [ ]
−++−
−==
θφψφψθφψφψθφθφψφψθφψφψθφ
θψθψθψθφ
cccssscsscsc
csccssssccss
ssccc
C BL 321
ψ - azimuth (yaw) angle
θ - pitch angle
φ - roll angle
AERODYNAMICS
146
SOLO
Wind Coordinates (W)
The origin of the Wind coordinate systemis located at the instantaneous center ofgravity CG of the vehicle, with xW pointedin the direction of the vehicle velocity vectorrelative to air .AV
[ ] [ ]
−−−=
−
−=−=
ααβαββαβαββα
αα
ααββββ
αβcos0sin
sinsincossincos
cossinsincoscos
cos0sin
010
sin0cos
100
0cossin
0sincos
23WBC
The Wind coordinate frame (W) is defined by the following two angles, relative toBody coordinates (B):
α - angle of attack
β - sideslip angle
AERODYNAMICS
( )BBB
BA zwyvxuV
111 ++=
[ ]
=
⇒
=
βαβ
βα
cossin
sin
coscos
0 V
w
v
u
o
V
C
w
v
uTW
B
147
LowerSurface
UpperSurface
AERODYNAMIC FORCES AND MOMENTS.
SOLO
∞V
Airfoil Pressure Field variation with α
Distribution of Pressure around an Airfoil causes Aerodynamic Forces and Moments
148
SOLO Linearized Flow Equations
Preasure Field
Stream Lines Streak Lines (α = 0º) Streak Lines (α = 15º)
Streak Lines (α = 30º) Forces in the Body
149
SOLO Linearized Flow Equations
Velocity Field
Sum of the elementary Forces on the Body
Lift as the Sum of the elementary Forces on the Body
150
SOLO
Aerodynamic Forces
( )
−−−
=L
C
D
F WA
ForceLiftL
ForceSideC
ForceDragD
−−−
L
C
D
CSVL
CSVC
CSVD
2
2
2
2
12
12
1
ρ
ρ
ρ
=
=
= ( )( )( ) tCoefficienLiftRMC
tCoefficienSideRMC
tCoefficienDragRMC
eL
eC
eD
−−−
βαβαβα
,,,
,,,
,,,
viscositydynamic
lengthsticcharacteril
soundofspeedHa
numberReynoldslVR
BodytoRelativeVelocityFlowV
numberMachaVM
e
−−−
−=−−=
µ
µρ)(
/
/
AERODYNAMICS
( )WWW
WA zLyCxDF
111 −−−=
151
SOLO
Aerodynamic Forces (continue -1)
∫∫
⋅+⋅−=
∫∫
⋅+⋅−=
∫∫
⋅+⋅−=
∧∧
∧∧
∧∧
W
W
W
SfpL
SfpC
SfpD
dswztCwznCS
C
dswytCwynCS
C
dswxtCwxnCS
C
1ˆ1ˆ1
1ˆ1ˆ1
1ˆ1ˆ1
Wf
Wp
SsurfacetheontcoefficienfrictionV
fC
SsurfacetheontcoefficienpressureV
ppC
−=
−−= ∞
2/
2/
2
2
ρ
ρ
ntonormalplanonVofprojectiont
dstonormaln
ˆˆ
ˆ
−
−
AERODYNAMICS
152
SOLO
( )
=
Y
P
RB
CA
M
M
M
M /
MomentYawM
MomentPitchM
MomentRollM
Y
P
R
−−−
YY
PP
RR
ClSVM
ClSVM
ClSVM
2
2
2
2
12
12
1
ρ
ρ
ρ
=
=
= ( )( )( ) tCoefficienMomentYawRMC
tCoefficienMomentPitchRMC
tCoefficienMomentRollRMC
eY
eP
eR
−−−
βαβαβα
,,,
,,,
,,,
viscositydynamic
lengthsticcharacteril
soundofspeedHa
numberReynoldslVR
BodytoRelativeVelocityFlowV
numberMachaVM
e
−−−
−=−−=
µ
µρ)(
/
/
AERODYNAMICS Aerodynamic Moments Relative to CR
CR
( )BYBPBR
BCA zMyMxMM
111/ ++=
Return to Table of Content
( )∫
∫∫
−=
−=
==′
EdgeTrailing
EdgeLeading
sideupper sidelower
EdgeTrailing
EdgeLeading
sideupper
EdgeTrailing
EdgeLeading
sidelower
pp
pp
sideupper on Force-sidelower on the acting Forces
direction wind the tonormal Force
dx
dxdx
L
Relationship between Lift and Pressure on Airfoil
SOLO
( )
[ ] [ ]( )∫
∫
∞∞
=
−−−=
−=′
EdgeTrailing
EdgeLeading
sideupper sidelower
cos
EdgeTrailing
EdgeLeading
sideupper sidelower
pp
cospcosp
dxpp
dsL
sdxd
USLS
θ
θθ
Divide left and right sides of the first equation by cV 2
2
1∞ρ
∫
−
−−=′
∞
∞
∞
∞
∞
EdgeTrailing
EdgeLeading
upperlower
c
xd
V
pp
V
pp
cV
L
222
21
21
21 ρρρ
We get:
Relationship between Lift and Pressure on Airfoil (continue – 1)
LowerSurface
UpperSurface
( )∫ −=−EdgeTrailing
EdgeLeading
sideupper sidelower sinpsinp dsD USLS θθ
Lift – Aerodynamic component normal to VDrag – Aerodynamic component opposite to V
SOLO
From the previous slide,
∫
−
−−=′
∞
∞
∞
∞
∞
EdgeTrailing
EdgeLeading
upperlower
c
xd
V
pp
V
pp
cV
L
222
21
21
21 ρρρ
The left side was previously defined as the sectional lift coefficient C l.
The pressure coefficient is defined as: 2
21
∞
∞−=V
ppC p
ρThus, ( )∫ −=
edgeTrailing
edgeLeading
upperplowerpl c
xdCCC ,,
LowerSurface
UpperSurface
Relationship between Lift and Pressure on Airfoil (continue – 2)
SOLO
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156
2-D Inviscid Incompressible Flow
In 2-D the velocity vector
SOLO
Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow
x
y V
u
vru
θv
rθ
θθ 1111 vruyvxuV r +=+=
θθ
∂∂
++∂∂
=∂∂
+∂∂
=⋅∇v
r
u
r
u
y
v
x
uV rr
zu
r
v
z
ur
z
vz
y
u
x
vy
z
ux
z
vV rr 111111
∂∂
−∂∂
+∂∂
+∂∂
−=
∂∂
−∂∂
+∂∂
+∂∂
−=×∇θ
θ θθ
0
111
0
111
rr vu
zr
zr
vu
zyx
zyx
V∂∂
∂∂
∂∂
=∂∂
∂∂
∂∂
=×∇θ
θ
−
=
v
u
v
u r
θθθθ
θ cossin
sincos ( ) θθ
i
r eviuviu +=+( ) θ
θi
r eviuviu −+=+
157
2-D Inviscid Incompressible Flow
In 2-D the velocity vector
SOLO
Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow
x
y V
u
vru
θv
rθ
θθ 1111 vruyvxuV r +=+=
−
=
v
u
v
u r
θθθθ
θ cossin
sincos ( ) θθ
i
r eviuviu +=+( ) θ
θi
r eviuviu −+=+
Continuity: 00 =⋅∇→=⋅∇+ uutD
D ρρ
( )
∂∂
−=∂∂
=×
∂∂
+∂∂
∂∂
−=∂∂
=×
∂∂
+∂∂
=×∇=×∇=
rv
ruz
rr
r
xv
yuzy
yx
xzzu
r
ψθψθ
θψψ
ψψψψ
ψψ
θ
111
11
111
11 22
Incompressible: 0=tD
D ρ
Irrotational:
∂∂
=∂∂
=
∂∂
=∂∂
==∇=
θφφ
φφ
φθ r
vr
u
yv
xu
u
r
12
0=×∇ u
rrv
rru
xyv
yxu
r ∂∂−=
∂∂=
∂∂=
∂∂=
∂∂−=
∂∂=
∂∂=
∂∂=
ψθφ
θψφ
ψφψφ
θ11
158
2-D Inviscid Incompressible Flow
In 2-D the velocity vector
SOLO
Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow
x
y V
u
vru
θv
rθ
θθ 1111 vruyvxuV r +=+=
−
=
v
u
v
u r
θθθθ
θ cossin
sincos ( ) θθ
i
r eviuviu +=+( ) θ
θi
r eviuviu −+=+
00 222 =∇⋅∇→∇=+=⋅∇ φφuu
2-D Incompressible:
2-D Irrotational:
( )( ) ( ) ( )ψψψ
ψψ
222
0
222
222
1110
110
∇⋅∇−∇∇⋅=×∇×∇=
→×∇=×∇=+=×∇
zzz
zzuu
02
2
2
2 =∇=∇ ψφ
Complex Potential in 2-D Incompressible-Irrotational Flow:( ) ( ) ( )
yixz
yxiyxzw
+=+= ,,: ψφ
( )=
zd
zwdx
ix ∂
∂+
∂∂ ψφ
yyi
∂∂
+∂∂
−ψφ0=x
0=y
( )[ ] ( ) θθ
θθ
i
r
i
r eviueviuVviu −∗∗ −=+==−
zd
wdviu =− θ
θi
r ezd
wdviu =−
xyyx ∂∂
−=∂∂
∂∂
=∂∂ ψφψφ
Cauchy-Riemann Equations
We found:
159
2-D Inviscid Incompressible Flow
Examples:
SOLO
Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow
αα sincos 00 UiUV +=Uniform Stream:
xyUv
yxUu
∂∂
−=∂∂
==
∂∂
=∂∂
==
ψφα
ψφα
sin
cos
0
0
( ) ( )( ) ( ) yUxU
yUxU
ααψααφ
cossin
sincos
00
00
+−=+=
( ) ( )zU
zUzUiw∗=
−=+=
0
00 sincos ααψφ
0U
α
160
2-D Inviscid Incompressible Flow
Examples:
SOLO
Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow
∂∂
−=∂∂
==
∂∂
=∂∂
==
rrv
rrr
mu r
ψθφ
θψφ
π
θ
10:
1
2:
( )
==
+==
−
x
ymm
yxm
rm
1
22
tan22
ln2
ln2
πθ
πψ
ππφ
( ) ( ) zm
rem
irm
iw i ln2
ln2
ln2 ππ
θπ
ψφ θ ==+=+=
Definition:
Source , Sink :( )0>m ( )0<m
Sink 0<m
Source 0>m
The equation of a streamline is: constm == θπ
ψ2
161
2-D Inviscid Incompressible Flow
Examples:
SOLO
Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irotational Flow
( ) ( ) ( )r
Kvvr
rzuvr
rVu
rrr =→=
∂∂→=
∂∂−
∂∂=×∇→=
≠ θθθ θ0010:
02
( )
+Γ=Γ=
Γ−=Γ−= −
22
1
ln2
ln2
tan22
yxr
x
y
ππψ
πθ
πφ
( ) ( ) zi
rei
riiw i ln2
ln2
ln2 ππ
θπ
ψφ θ Γ=Γ=+−Γ=+=
Definition:
Infinite Line Vortex :
∂∂
−=∂∂
−=Γ−=
∂∂
=∂∂
−==
rrrv
rru r
ψθφ
π
θψφ
θ
1
2:
10:
( ) Γ−=Γ−=+⋅
Γ−=⋅ ∫∫∫ θπ
θθθπ
ddrrdrr
drV2
1112
Circulation
streamlines:
( )Λ=+
→+Γ=
/222
22ln2
ψπ
πψ
eyx
yx
Irotational
162
2-D Inviscid Incompressible Flow
Examples:
SOLO
Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow
Definition: Let have a source and a sink of equal strength m = μ/ε situated at x = -εand x = ε such that
Doublet at the Origin with Axis Along x Axis :
m+ m−
ε+ε−
y
x
.lim0
constm ==→
µεε
( ) ( ) ( )
−+=
−+=
−−+=
z
zm
z
zm
zm
zm
zw
/1
/1ln
2ln
2
ln2
ln2
εε
πεε
π
επ
επ
.lim0
constm ==→
µεεwhen
( )
zz
m
zO
z
m
zO
zz
m
z
zmzw
m
πµε
πεε
π
εεεπε
επ
µε =
=≈
++≈
++
+≈
−+=
22
21ln2
11ln2/1
/1ln
2
2
2
2
2
163
2-D Inviscid Incompressible Flow
Examples:
SOLO
Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow
( ) ( )θθπππ
sincos2
1
2ln
2: i
r
m
z
mz
m
zd
d
zd
Wdzw Source
Doublet −==
==
+==
+==
=
=
22
2/
22
2/
sin
cos
yx
y
r
yx
x
r
m
m
πµθ
πµψ
πµθ
πµφ
µ
µ
Definition:
Doublet at the Origin with Axis Along x Axis (continue):
2
1
2
1
2 z
m
z
m
zd
d
zd
wdviuV
ππ−=
==−=∗
The equation of a streamline is: .22
constyx
y=
+=
πµψ
22
2
22
=
++
ψµ
ψµ
yx
164
SOLO 2-D Inviscid Incompressible Flow
Stream Functions (φ), Potential Functions (ψ) for Elementary Flows
Flow W (z=reiθ)=φ+i ψ φ ψ
Uniform Flow θcosrU∞ θsinrU∞( )yixUzU += ∞∞
Source ( )θ
ππire
kz
kln
2ln
2= r
kln
2πθ
π2
k
Doubletθier
B
z
B = θcosr
B θsinr
B−
Vortex(with clockwise
Circulation)
( )θ
ππire
iz
iln
2ln
2
Γ=Γ θπ2
Γ−rln2πΓ
90 Corner Flow ( ) 22
22yix
Az
A += yxA( )22
2yx
A −
Return to Table of Content
165
Wing Types Computations1. Subsonic Incompressible Flow (ρ∞ = constant)
1.1 Infinite Span (2-D, AR = ∞) (Profile Theory)
1.2 Finite Span (3-D, AR ≠ ∞) (Lifting Line Theory)
2. Supersonic Incompressible Flow (ρ∞ = constant)
1.1.1 Kutta-Joukowsky Lift Theorem
1.1.2 Profile Theory by the Method of Conformal Mapping
1.1.3 Profile Theory by the Method of Singularities
1.2.1 Wing Theory by the Method of Vortex Distribution (Prandtl Wing Theory)
1.2.2 Profile Theory by the Method of Conformal Mapping
1.1.3 Profile Theory by the Method of Singularities
SOLO
Return to Table of Content
166
SOLO 2-D Inviscid Incompressible Flow
Paul Richard Heinrich Blasius(1883 – 1970)
Blasius was a PhD Student of Prandtl at Götingen University
x
y
xδyδ
βsd
M
−=
=−
∫
∫
C
C
zdzd
wdzM
zdzd
wdiiYX
2
2
2
2
ρ
ρ
Re
where-w (z) – Complex Potential of a Two-Dimensional Inviscid Flow -X, Y – Force Components in x and y directions of the Force per Unit Span on the Body-M – the anti-clockwise Moment per Unit Span about the point z=0-ρ – Air Density-C – Two Dimensional Body Boundary Curve
1911Blasius Theorem
167
SOLO 2-D Inviscid Incompressible Flow
Paul Richard Heinrich Blasius(1883 – 1970)
Blasius was a PhD Student of Prandtl at Götingen University
−=
=−
∫
∫
C
C
zdzd
wdzM
zdzd
wdiiYX
2
2
2
2
ρ
ρ
Re
1911Blasius Theorem
Proof of Blasius Theorem
Consider the Small Element δs on the Boundary C
sysx δβδδβδ cos,sin =−=
xpspY
ypspX
δδβδδδβδ
⋅=⋅−=⋅−=⋅−=
sin
costhen
p = Normal Pressure to δs
The Total Force on the Body is given by
( ) ( )∫∫ −⋅−=+⋅−=−CC
ydixdpixdiydpYiX
Use Bernoulli’s Theorem .2
1 2constUp =+ ∞ρ
U∞ = Air Velocity far from Body
x
y
xδyδ
βsd
M
X
Y
168
SOLO 2-D Inviscid Incompressible Flow
Paul Richard Heinrich Blasius(1883 – 1970)
Blasius was a PhD Student of Prandtl at Götingen University
−=
=−
∫
∫
C
C
zdzd
wdzM
zdzd
wdiiYX
2
2
2
2
ρ
ρ
Re
1911Blasius Theorem
Proof of Blasius Theorem (continue – 1)
( )∫ −⋅
−−=− ∞
C
ydixdUconstiYiX 2
2
1 ρ
but ( ) 00 =−⋅⇒== ∫∫∫CCC
ydixdconstydxd
( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) yduivuxduivvdyixdviu
dyuixdvdyixdvu
dyvuidyixdvudyixdvudyixdU
+−+++−=
−++−=
+−++=−+=−∞
22
22
2
2
2222
2222222
viuU +=∞and
x
y
xδyδ
βsd
M
X
Y
169
SOLO 2-D Inviscid Incompressible Flow
Paul Richard Heinrich Blasius(1883 – 1970)
Blasius was a PhD Student of Prandtl at Götingen University
−=
=−
∫
∫
C
C
zdzd
wdzM
zdzd
wdiiYX
2
2
2
2
ρ
ρ
Re
1911Blasius Theorem
Proof of Blasius Theorem (continue – 2)
( ) ∫∫ ⋅
=−⋅=− ∞
CC
zdzd
wdiydixdU
iYiX
22
22ρρ
( ) ( ) ( ) zdzd
wddyixdviudyixdU
2
22
=+−=−∞
( ) ( ) 00 =−⇒+×+=×= ∞ xdvyduviuydixdUsd
Since the Flow around C is on a Streamline defined by
therefore ( ) ( ) yduivuxduivv +=+ 22
( ) ( ) ( )yixz
yxiyxzw
+=+= ,,: ψφ
and
xyv
yxu
∂∂−=
∂∂=
∂∂=
∂∂= ψφψφ
,where
Completes the Proof for the Force Equation
viuzd
wd −=
x
y
xδyδ
βsd
M
X
Y
170
SOLO 2-D Inviscid Incompressible Flow
Paul Richard Heinrich Blasius(1883 – 1970)
Blasius was a PhD Student of Prandtl at Götingen University
−=
=−
∫
∫
C
C
zdzd
wdzM
zdzd
wdiiYX
2
2
2
2
ρ
ρ
Re
1911Blasius Theorem
Proof of Blasius Theorem (continue – 3)
( ) ( ) ( ) ( ) ( )ydxixdyiydyxdxvuivudyixdviuyixzdzd
wdz ++−−−=+−+=
2222
2
The Moment around the point z=0 is defined by
( ) ( )∫∫ +⋅−=+⋅= ∞CC
ydyxdxUydyxdxpM2
2
ρ
since 2
2 ∞−= UconstpBernoulli ρ
and ( ) 0=+⋅∫C
ydyxdxconst
hence( ) ( ) ( )xdyydxvuydyxdxvuzd
zd
wdz ++−−=
222
2
Re
x
y
xδyδ
βsd
M
X
Y
171
SOLO 2-D Inviscid Incompressible Flow
Paul Richard Heinrich Blasius(1883 – 1970)
Blasius was a PhD Student of Prandtl at Götingen University
−=
=−
∫
∫
C
C
zdzd
wdzM
zdzd
wdiiYX
2
2
2
2
ρ
ρ
Re
1911Blasius Theorem
Proof of Blasius Theorem (continue – 4)
( ) ( ) ( )
−=+⋅+−=+⋅= ∫∫∫
CCC
zdzd
wdzydyxdxvuydyxdxpM
2
22
22
ρρRe
hence
( ) ( ) ( )xdyydxvuydyxdxvuzdzd
wdz ++−−=
222
2
Re
Since the Flow around C is on a Streamline we found that u dy = v dx
( ) ( ) ( ) ydyuxdxvxdvyuyduxvxdyydxvu 22 22222 +=+=+
( ) ( )ydyxdxvuzdzd
wdz ++=
22
2
2Re
Completes the Proof for the Moment Equation
x
y
xδyδ
βsd
M
X
Y
172
SOLO 2-D Inviscid Incompressible FlowBlasius Theorem Example
Circular Cylinder with Circulation
Let apply Blasius Theorem
Assume a Cylinder of Radius a in a Flow of Velocity U∞ at an Angle of Attack αand Circulation Γ.The Flow is simulated by:-A Uniform Stream of Velocity U∞
-A Doublet of Strength U∞ a2.-A Vortex of Strength Γ at the origin.
Since the Closed Loop Integral is nonzero only for 1/z component, we have
viuz
i
z
eaUeU
zd
wd ii −=Γ−−=
+∞−
∞ π
αα
22
2
∫∫ ⋅
Γ−−=⋅
=−
+∞−
∞C
ii
C
zdz
i
z
eaUeU
izd
zd
wdiYiX
2
2
22
222 πρρ
αα
ααα
ρπ
ρπ
ρ ii
C
i
eUiz
eUResiduezd
z
eUiiYiX −
∞
−∞
−∞ Γ=
Γ=⋅
Γ−=− ∫ 22
02 =
==⋅∫ zenclosesCif
z
AResidueAizd
z
A
C
πwhere we used:
α
αX
Y−L
∞U
x
y
( ) ( )αα
α
πi
ii ez
i
ez
aUezUzw −
−∞−
∞Γ−+= ln2
2
173
SOLO 2-D Inviscid Incompressible FlowBlasius Theorem Example
Circular Cylinder with Circulation (continue – 1)
Γ−−−=
−= ∫∫
+∞−
∞C
ii
C
zdz
i
z
eaUeUzzd
zd
wdzM
2
2
22
222 πρρ α
αReRe
Since the Closed Loop Integral is nonzero only for 1/z component, we have
=≠
==
=
=⋅∫0'10
012
zenclosendoesCornif
zenclosesCandnifz
AResidueAi
zdz
A
Cn
πwe used:
04
2224
2
2 2
222
2
222
=
Γ−−=
Γ−−= ∞∞∫ π
πρπ
ρaUizd
zz
aUM
C
ReRe
αρ ieUiYiX −∞Γ=−
( )
Γ==
⇒Γ=−=+∞
∞−
UL
DUieYiXiLD i
ρρα 0
:
α
αX
Y−L
∞U
x
y
Zero Moment around the Origin.
174
SOLO 2-D Inviscid Incompressible FlowBlasius Theorem Example
Circular Cylinder with Circulation (continue – 2)
On the Cylinder z = a e iθ
We found: viuz
i
z
eaUeU
zd
wd ii −=Γ−−=
+∞−
∞ π
αα
22
2
( )
( )
Γ−−=
=Γ−−==−=−
∞
−+∞
−∞
aUi
a
ieeUeeUe
zd
Wdeviuviv iiiiii
r
παθ
πθαθαθθ
θ
2sin2
2
Stagnation Points are the Points on the Cylinder for which vθ = 0:
( ) 02
sin2 =Γ−−=− ∞ aUv
παθθ
Γ+=∞
−
Uastagnation παθ
4sin 1
175
2-D Inviscid Incompressible FlowSOLO
176
The Flow Pattern Around a Spinning Cylinderwith Different Circulations Γ Strengths
2-D Inviscid Incompressible FlowSOLO
177
SOLO 2-D Inviscid Incompressible FlowBlasius Theorem Example
Circular Cylinder with Circulation (continue – 3)
The Pressure Coefficient on the Cylinder Surface is given by:
( )2
2
2
22
2
2sin2
11
2
1∞
∞
∞∞
∞
Γ−−−=+−=
−=
U
aU
U
vv
U
ppC rSurface
Surfacep
παθ
ρθ
Using Bernoulli’s Law:
22
2
1
2
1∞∞ +=+ UpUp SurfaceSurface ρρ
( ) ( )
Γ−+
Γ−−−=∞∞ UaUa
CSurfacep π
αθπ
αθ4
sin84
4sin412
2
178
2-D Inviscid Incompressible FlowSOLO
179
SOLO
Stream Lines
Flow Around a Cylinder
Streak Lines (α = 0º)
Preasure Field
Streak Lines (α = 5º)
Streak Lines (α = 10º) Forces in the Body
http://www.diam.unige.it/~irro/cilindro_e.html
2-D Inviscid Incompressible Flow
180
SOLO
Velocity Field
http://www.diam.unige.it/~irro/cilindro_e.html
University of Genua, Faculty of Engineering,
2-D Inviscid Incompressible Flow
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181
SOLO 2-D Inviscid Incompressible Flow
C
'C
''C '''C
Corollary to Blasius Theorem
−=
−=
=
=−
∫∫
∫∫
'
22
'
22
22
22
CC
CC
zdzd
wdzzd
zd
wdzM
zdzd
wdizd
zd
wdiiYX
ρρ
ρρ
ReRe
C – Two Dimensional Curve defining Body BoundaryC’ – Any Other Two Dimensional Curve inclosing C such that No Singularity exist between C and C’
Proof of Corollary to Blasius Theorem
Add two Close Paths C” and C”’ , connecting C and C’, in opposite direction, s.t.
∫∫ −=''''' CC
then, since there are No Singularities between C and C’, according to Cauchy:
0'
0
'''''
=−++ ∫∫∫∫CCCC
q.e.d.∫∫ ='CC
therefore
182
SOLO 2-D Inviscid Incompressible Flow
183
2-D Inviscid Incompressible Flow
Louis Melville Milne-Thomson
(1891-1974)
SOLO
184
2-D Inviscid Incompressible FlowSOLO
185
2-D Inviscid Incompressible FlowSOLO
186
2-D Inviscid Incompressible FlowSOLO
187
2-D Inviscid Incompressible Flow
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Infinite Span (AR = ∞)
SOLO
Return to Table of Content
188
Kutta Condition
We want to obtain an analogy between a Flow around an Airfoil and that around a Spinning Cylinder. For the Spinning Cylinder we have seen that when a Vortex isSuperimposed with a Doublet on an Uniform Flow, a Lifting Flow is generated.The Doublet and Uniform Flow don’t generate Lift. The generation of Lift is alwaysassociated with Circulation. Suppose that is possible to use Vortices to generate Circulation, and thereforeLift, for the Flow around an Airfoil. • Figure (a) shows the pure non-circulatory Flow around an Airfoil at an Angle of Attack. We can see the Fore SF and Aft SA Stagnation Points.•Figure (b) shows a Flow with a Small Circulation added. The Aft Stagnation Point Remains on the Upper Surface.•Figure (c) shows a Flow with Higher Circulation, so that the Aft Stagnation Point moves to Lower Surface. The Flow has to pass around the Trailing Edge. For an Inviscid Flow this implies an Infinite Speed at the Trailing Edge.•Figure (d) shows the only possible position for the Aft Stagnation Point, on the Trailing Edge. This is the Kutta Condition, introduced by Wilhelm Kutta in 1902, “Lift Forces in Flowing Fluids” (German), Ill. Aeronaut. Mitt. 6, 133.
Martin Wilhelm Kutta
(1867 – 1944)
2-D Inviscid Incompressible Flow
1902
SOLO
Definition: A Stagnation Point is a point in a flow field where the local velocity of the fluid is zero
189
Effect of Circulation on the Flow around an Airfoil at an Angle of Attack
2-D Inviscid Incompressible FlowSOLO
Definition: A Stagnation Point is a point in a flow field where the local velocity of the fluid is zero
SF – Forward Stagnation Point SA – Aft Stagnation Point
Kutta Condition:SA on the Trailing Edge
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190
Martin Wilhelm Kutta (1867 – 1944)
Nikolay Yegorovich Joukovsky (1847-1921
Kutta-Joukovsky Theorem
The Kutta–Joukowsky Theorem is a Fundamental Theorem of Aerodynamics. The theorem relates the Lift generated by a right cylinder to the speed of the cylinder through the fluid, the density of the fluid, and the Circulation. The Circulation is defined as the line integral, around a closed loop enclosing the cylinder or airfoil, of the component of the velocity of the fluid tangent to the loop. The magnitude and direction of the fluid velocity change along the path.
The force per unit length acting on a right cylinder of any cross section whatsoever is equal to ρ∞V ∞Γ, and is perpendicular to the direction of V ∞.
Kutta–Joukowsky Theorem:
2-D Inviscid Incompressible Flow
19061902
Γ= ∞∞UL ρKutta–Joukowsky Theorem:
LCUL 2
2
1∞∞= ρLift:
Kutta in 1902 and Joukowsky in 1906, independently, arrived to this result.
Circulation ∫∫ =⋅=Γ θcos: ldVldV
SOLO
191
SOLO 2-D Inviscid Incompressible Flow
General Proof of Kutta-Joukovsky TheoremUsing the Corollary to Blasius Theorem
Suppose that we wish to determine theAerodynamic Force on a Body of Any Shape.Use Corollary to Blasius Theorem, integratingRound a Circle Contour with a Large Radius andCenter on the Body
( ) zi
z
aUzUzw ln
2
2
πΓ−+= ∞
∞
The proof is identical to development in the Example ofFlow around a Two Dimensional Cylinder using
According to Corollary to Blasius Theorem we use C’ instead of C for Integration
z
i
z
aUU
zd
wd 1
22
2
πΓ−−= ∞
∞
LiftiDragUiUi
ii
z
UiResidue
i
zdz
Uiizd
z
i
z
aUU
izd
zd
wdizd
zd
wdiiYX
CCCC
+=Γ=
Γ−=
Γ−=
Γ−=
Γ−−=
=
=−
∞∞∞
∞∞∞ ∫∫∫∫
ρπ
πρπ
ρ
πρ
πρρρ
22
1
2
1
2
1
2222 ''
2
2
2
'
22
Therefore 0& =Γ== ∞ DragULLift ρq.e.d.
02 =
==⋅∫ zenclosesCif
z
AResidueAizd
z
A
C
πwhere we used:
C
'C
∞UL
D
192
SOLO 2-D Inviscid Incompressible Flow
D’Alembert Paradox
The fact that the Inviscid Flow Theories give Drag = 0 is called D’Alembert Paradox.
In 1768 d’Alembert enunciated his famous paradox in the following words:
“Thus I do not see, I admit, how one can satisfactorily explain by theory the resistance of fluids. On the contrary, it seems to me that the theory, developed in all possible rigor, gives, at least in several cases, a strictly vanishing resistance; a singular paradox which I leave to future geometers for elucidation.”
Jean-Baptiste le Rond d'Alembert
(1717 – 1783)
The resistance (Drag) experienced by a Real Airfoil is do to a combination of Skin-Friction and Pressure Distribution Distortions due to displacements effects of its Boundary Layers, which are not considered in the Inviscid Flow Theories.
193
The Kutta-Joukowsky Theory can be used to design Wings of Infinite Span that flow at Subsonic Speeds (Incompressible Flows). The design methods for such wings are called methods of “Profile Theory”.
AERODYNAMICS
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Infinite Span (AR = ∞)
Profile (of Airfoil) Theory can be treated in two different ways:
1.Conformal Mapping This Method is limited to 2 – dimensional problems. The Flow about a given body is obtained by using Conformal Mapping to transform it into a known Flow about another body (usually Circular Cylinder)
2.Method of Singularities The body in the Flow Field is substitute by Sources, Sinks, and Vortices, the so called Singularities.
For practical purposes the Method of Singularities is much simpler than Conformal Mapping. But, the Method of Singularities produces, in general, only ApproximateSolution, whereas Conformal Mapping leads to Exact Solutions, although these often require considerable effort.
SOLO
194
Joukovsky Airfoils
Joukovsky transform, named after Nikolai Joukovsky is a conformal map historically used to understand some principles of airfoil design.
Nikolay Yegorovich Joukovsky (1847-1921
Profile Theory Using Conformal Mapping
It is applied on a Circle of Radius R and Center at cx, cy. The radius to the Point (a,0) make an angle β to x axis. Velocity U∞ makes an angle αwith x axis.
β
xcyc
∞U
R
αx
y
( )0,a
The transform isz
az
2
+=ζ
( ) ββ sincosˆ RiRacicc yx +−=+=For α=0 we have
( ) ( ) ( )czi
cz
RczUzw ˆln
2ˆˆ
2
−Γ+
−
+−= ∞ πFor any α we have
( ) ( ) ( )cezi
cez
RcezUzw i
ii ˆln
2ˆˆ
2
−Γ+
−
+−= −−
−∞
αα
α
π
AERODYNAMICSSOLO
195
Kutta-Joukovsky
Nikolay Yegorovich Joukovsky (1847-1921
( ) ( ) ( )cezi
cez
RcezUzw i
ii ˆln
2ˆˆ
2
−Γ+
−
+−= −−
−∞
αα
α
π
( ) viucez
i
cez
RUe
zd
wdii
i −=
−Γ+
−−= −−∞
−
ˆ1
2ˆ1 2
2
ααα
π
we have
Kutta Condition: The Flow Leaves Smoothly from the Trailing Edge.This is an Empirical Observation that results from the tendency ofViscous Boundary Layer to Separate at Trailing Edge.
Martin Wilhelm Kutta (1867 – 1944)
( ) ( )
( ) yxi
ii
i
azaz
caBcaABiA
i
BiA
RUe
cea
i
cea
RUe
zd
wdivu
+=−=
−Γ+
−
−=
−Γ+
−−===−
∞−
−−∞−
==
ααπ
π
α
ααα
sin:,cos:1
21
ˆ1
2ˆ10
2
2
2
2
( ) ( )[ ] ( ) ( )( )
+
Γ++−+
Γ+−−−+=
∞∞−
222
22222222222
22
2
BA
BAAURBAiBABBARBAU
e i ππα
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
196
we have
( ) ( )[ ] ( ) ( )( )
+
Γ++−+
Γ+−−−+==
∞∞−
=222
22222222222
22
20
BA
BAAURBAiBABBARBAU
ezd
wd i
az
ππα
( ) βααβαα sinsinsin:,coscoscos: RacaBRaacaA yx +=+=−−=−=
( ) ( )( ) ( )[ ] 222
2222
coscos2cos12
sinsincoscos
RRaRa
RaaRaBA
≈−−++−=++−+=+
ββααβαβα
( )π2
20 222 Γ++−= ∞ BAAURBA ( )βαπππ sinsin44422
2
RaUUBUBBA
R +=≈+
=Γ ∞∞∞
( ) ( )[ ] ( ) ( ) ( )[ ]( ) ( )[ ] ( ) ( ) 0
22
22222222222
2222222222222222
≈−++=+−+=
−−−+=Γ+−−−+
∞∞
∞∞∞
RBABAUBARBAU
URBBARBAUBABBARBAUπ
Let check
For this value of Γ, we have
This value of Γ satisfies the Kutta Condition0=
=azzd
wd
Joukovsky Airfoils
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
197
Joukovsky Airfoils Design1. Move the Circle to ĉ and choose Radius R so that the Circle
passes through z = a.
Nikolay Yegorovich Joukovsky (1847-1921
β
xcyc
∞U
R
αx
y
( )0,a
for Center at z = 0.( ) zi
z
RzUzW ln
2
2
πΓ+
+= ∞
2. Change z-ĉ → z
( ) ( )czi
cz
RczUzW ˆln
2ˆˆ
2
−Γ+
−
+−= ∞ π3. Change z → z e-iα
( ) ( )cezi
cez
RcezUzW i
ii ˆln
2ˆˆ
2
−Γ+
−
+−= −−
−∞
αα
α
π4. Compute Γ from Kutta Condition
azazd
Wd
d
Wd
==
==2
0ςς ( )βαπ +=Γ ∞
<<
sin4ˆ
RUac
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
198
Joukovsky Airfoils Design (continue – 1)
5. Use the Transformation and computez
az
2
+=ζ
22 /1
//
za
zdWd
zd
d
zd
Wd
d
Wd
−== ς
ς6. To Compute Lift use either:
( )βαρπρ +=Γ= ∞∞ sin4 2RUUL6.1 Kutta-Joukovsky
6.2 Blasius( )
=−=− ∫ ς
ςραα d
d
WdieFiFeLi i
yxi
2
2''
6.3 Bernoulli( )
2
2/1
2/ ∞∞
∞ −=−=U
zdWd
U
ppC p ρ
−=
−= ∫∫∫∫
−−
∞
−−
a
a
p
a
a
p
a
a
Upp
a
a
Low xdCxdCU
xdpxdpLUL
2
2
2
2
22
2
2
2
''cos
2/''
cos
1
αρ
α
( ) ( )βαπβαπρ
+≈+==≈
≈∞
sin2sin82/ 42
cR
acL c
R
Uc
LC
'yF
'xF 'xF
∞U 'x
L
α
plane−ς
'y
α
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
199
Joukovsky Airfoils Design (continue – 2)
7. To compute Pitching Moment about Origin use either:
7.2 Blasius
= ∫ ςς
ςρ
dd
WdiM p
2
20Re
7.1 Bernoulli
+−=
+−=
∫∫
∫∫
−−
∞
−−
a
a
p
a
a
p
a
a
Upp
a
a
Low
SpanUnitper
p
xdxCxdxCU
xdxpxdxpM
UL
2
2
2
2
2
2
2
2
2
''''2
''''0
ρ
'yF
'xF 'xF
∞U 'x
L
α
plane−ς
'y
α
0pM
απρ2sin4
222
0aUM p
= ∞
22
20
a
R
a
L
Mx p
p ≈==
( )βαρπ += ∞ sin4 2RUL
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
200
Joukovsky Airfoils Design (continue – 3)
8. To Pitching Moment about Any Point x0 is given by:
+=+= ∞ Lmpp C
c
xCcULxMM
x
0220 000 2
ρ 'yF
'xF 'xF
∞U 'x
L
α
plane−ς
'y
α
0pM0xαπ 2sin4 22
0aCc m =
( )βαπ += sin2LC
( )
( )
++≈
++=
∞
<<+
≈
∞
βαπαπρ
βαπαπρ
βα
a
xaU
c
x
c
acUM
ac
px
0221
4
02
222
882
sin22sin420
+
+
≈ ∞
<<+
≈βαπρβα
a
x
a
xaUM
acpx
00221
418
20
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
201Generation of Joukowsky Profiles through Conformal Mapping
Symmetric Joukowsky Profile
Circular Joukowsky Profile
Cambered Joukowsky Profile
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
202
Profile Theory Using Conformal Mapping
AERODYNAMICSSOLO
203
Nikolay Yegorovich Joukovsky (1847-1921
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
204
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
205
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
206
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
207
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
Return to Table of Content
208
Theodorsen Airfoil Design MethodTheodore Theodorsen working at NACA applied the Joukovsky inReverse and developed the following Design Method:
1. Given an Airfoil in ζ = ξ+i η Plane, arrange it with the Trailing Edge at ξ = 2a and Leading Edge at ξ=-2a
2. Transform from ζ = ξ+i η to z’ =a eψ eiθ through
''
2
z
az +=ς( )1
( )( )
=←==←=
θψηηθψηθψξξθψξ,sinsinh2
,coscosh2
a
a( )( )32
( ) ( )( ) ( )θψηηηψ
θψξξηθ
,/sinh2
,/sin2
222
222
=←++−=
=←++=
app
app( )( )54
Theodore Theodorsen (1897 – 1978)
planez''y
'xθ
ψea
x
yplanez −
φ0
0ψaeR =
ξplane−ς
η
a2a2−
Given ξ, η find ψ, θ using
22
221:
−
−=
aap
ηξwhere
T. Theodorsen, “Theory of Wing Sections with Arbitrary Shapes”, NACA Rept. 411, 1931 T. Theodorsen, I.E. Garrick, “General Potential Theory of Arbitrary Wing Sections”, NACA Rept. 452, 1933
Profile Theory Using Conformal MappingAERODYNAMICS
1931
SOLO
209
Theodore Theodorsen (1897 – 1978)
ξplane−ς
η
a2a2−
planez''y
'xθ
ψea
Theodorsen Airfoil Design Method (continue – 1)
3. Transform from z’ =a eψ eiθ to z = (a eψ0) eiф) through
( )
+=−−= ∑
∞
=10 expexp'
nn
nn
z
BiAzizz εψψ
Equaling Real and Imaginary Parts:
( ) ( )∑∞
=
+−=−=
1 00
sincosn
nn
nn n
R
An
R
B φφθφε( )8where An, Bn can be found by the following Iterative Procedure:
( ) ( )
( ) ( )
( )∫
∫
∫
=
=
=
π
π
π
φφψπ
ψ
φφφψπ
φφφψπ
2
0
0
2
00
2
00
2
1
sin1
cos1
d
dnR
B
dnR
A
nn
nn( )9
( )10
( )11x
yplanez −
φ0
0ψaeR =
Start with ( ) ( )φψθψ ≅
( )7 ( ) ( )∑∞
=
+=−
1 00
0 sincosn
nn
nn n
R
Bn
R
A φφψψ
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
210
Theodore Theodorsen (1897 – 1978)
ξplane−ς
η
a2a2−
planez''y
'xθ
ψea
x
yplanez −
φ0
0ψaeR =
Theodorsen Airfoil Design Method (continue – 2)
4. Given Airfoil, Compute An, Bn, Cp, Γ
( ) ( )[ ] ( )[ ]( ) ( )[ ]222
22200
2
2
/1sinsinh
/1sinsin1
12/
0
θψθψθεεαφα
ρψ
dd
edd
U
q
U
ppC
T
p
++++++−=
−=−=
∞∞
∞
Procedure:
ii φφ −+1
4.2 Take , compute again An, Bn, ψ0 and εi+1 using (9), (10), (11) and (8) until is less then some predefined value .
ii εθφ +=
4.3 Compute Pressure Distribution
θφε −=4.1 Assume ε small and take . Compute An, Bn, ψ0 and using (9), (10), (11) and after that using (8).
θφ =0
where α0 is the Angle of Attack, and εT is the ε of the Trailing Edge
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
211
Theodore Theodorsen (1897 – 1978)
ξplane−ς
η
a2a2−
planez''y
'xθ
ψea
x
yplanez −
φ0
0ψaeR =
Theodorsen Airfoil Design Method (continue – 3)
4. Given Airfoil, Compute An, Bn, Cp, Γ
( )TUea εαπ ψ +=Γ ∞ 0sin4 0
Procedure (continue):
4.4 Compute Γ
where α0 is the Angle of Attack, and εT is the ε of the Trailing Edge
4.5 Compute Lift
Γ= ∞UL ρ
5. Given we can compute for the Airfoil( ) 0,ψφε
5.1 From Compute An, Bn( ) ( ) ( )∑∞
=
+−=
1 00
sincosn
nn
nn n
R
An
R
B φφφε
5.2 Compute ( ) ( )φεφφθ −=
5.4 Compute ξ and η using (2) and (3).
5.3 Compute ( ) ( )∑∞
=
++=
1 00
0 sincosn
nn
nn n
R
Bn
R
A φφψψ
Profile Theory Using Conformal MappingAERODYNAMICSSOLO
Return to Table of Content
212
Profile Theory by the Method of Singularities
The Profile Theory was Initiated by Max Munk a student of Prandtl, who worked with him at the development of “Lifting Line Theory”, at Götingen University in Germany, between 1918-1919. He moved in 1920 to USA and worked at NACA for six years. At NACA, Munk developed an engineering-oriented method for Theoretical Prediction of Airfoil Lift and Moments, a method still in use today.His Theory applies to Thin Airfoils (t/c < 10%) and Small Angles of Attack. He approximate an Infinitely Thin Airfoil with its Main Camber Line. He published his results in a 1922 report, “General Theory of Thin Wings Sections” NACA Report 142.
Michael Max Munk(1890 – 1986)
Hermann Glauert(1892-1934)
Munk derived his results by using the idea of Conformal Mapping, from the Theory of Complex Variables. One year later, W. Birnbaum, in Germany, derived the same results by replacing the Main Camber Line with a Vortex Sheet (Singularities), given a simpler derivation of the Equations of Thin Airfoils. Finally in 1926 Hermann Glauert, in England, applied the solution of Fourier Series to the Solutions of those Equations. Glauert Hermann, “The Elements of Airfoil and Airscrew Theory”, Cambridge University Press, 1926.It is Glauert’s formulation that is still in use today.
2-D Inviscid Incompressible FlowSOLO
213
2-D Inviscid Incompressible FlowProfile Theory by the Method of Singularities
Assumptions:1.Two Dimensional (x, z)2.Low Velocities (Incompressible)3.Irrotational4.Thin Airfoils5.Small Angles of Attack
Use Small Perturbation Theory:
( ) ( ) 0,0, 20
2
222 =Φ∇⇒=
∂Φ∂−Φ∇
≈zx
xMzx
M
( ) ( ) ( ) ( )zxzUxUzx ,sincos, ϕαα ++=Φ ∞∞ ( ) 0,2 =∇ zxϕBoundary Conditions: The Normal Velocity Component on the Airfoil Surface is Zero
nxd
zd
V
Velocity( ) ( ) ( )
zz
UxUzz
Uxx
U
zwUxuUzxV
Thin
∂∂++≈
∂∂++
∂∂+=
+++=Φ∇=
∞∞
<<
<<∂∂∞∞
∞∞
ϕαϕαϕα
ααα
ˆsinˆcos
sinˆcos,1
1:
Normal to Airfoil Surface zxd
zdxn
Surface
ˆˆˆ +−≅
0ˆ =
∂∂++−=⋅ ∞∞
SurfaceSurfacez
Uxd
zdUnV
ϕα
−=
∂∂=
−=
∂∂=
∞
∞
αϕ
αϕ
xd
zdU
zw
xd
zdU
zw
Lower
Lower
Lower
Upper
Upper
Upper
SOLO
214
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities- Solution:
Solution is a Superposition (Linear Equations) of the Solutions for:• Skeleton (Camber Profile) • Teardrop (Symmetric Airfoil with same Thickness as the Original Airfoil)
Since the Small perturbation Theory leads to a Laplace’s Equationwe may use distribution of solutions (Singularities) to Laplace’s Equation-Sheet of Infinite Line Vortices on the Skeleton (needed for Lift production)-Sheet of Sources, Sinks on the Teardrop
( ) 0,2 =∇ zxϕ
The concept of replacing the Airfoil Surface with a Vortex Sheet is more then justa mathematical device; it also has physical significance. In the real life there is a Thin Boundary Layer on the Surface, due to friction between Flow and Airfoil.
Thickness
tt
Camber
CC
xd
zdU
zxd
zdU
zxd
zdU
zwCB
±=
∂∂+
−=
∂∂=
−=
∂∂= ∞∞∞
ϕαϕαϕ..
SOLO
215
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities – Solution (continue -1)
Skeleton (Camber Profile)
Assume a Infinite Line (in +y direction) Vortex Sheet γ (x1) (to be defined) distributed on the x axis, between 0 ≤ x ≤ c.The total Circulation Γ is given by
x
z
c0
1x
( ) 11 xdxγ
( ) ( )( )1
111 2
,xx
xdxxxwd
−=
πγ
The contribution of the Vortex Sheet γ (x1) distributed between 0 ≤ x ≤ c must satisfy the Boundary Conditions on the Airfoil Surface
Using Biot-Savart Formula for a Two Dimensional Flow the tangent velocity caused by γ (x1) at x is given by
( )∫=Γc
xdx0
11γ
( ) ( )( )
−=
−= ∞
=
=∫∫
xSurface
cx
x xd
zdU
xx
xdxxxwd α
πγ
1
111
0 2,
1
1
SOLO
216
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities – Solution (continue – 2)
Skeleton (Camber Profile)
x
z
c0
1x
( ) 11 xdxγ
( )( )
−=
− ∞∫xSurface xd
zdU
xx
xdx απ
γ1
11
2
Perform a transformation of variables
( ) cxxdcxdcx =→==→==→−= 111111111 &002/sin2/cos1 πθθθθθ( ) 2/cos1 θ−= cx
( )
−=
−⋅
∞∫x
xd
zdUd αθ
θθθγθ
π
π
0
11
11
coscos
sin
2
1
Solution for a Flat Plate dz/dx = 0
( ) αθθθ
θγθπ
π
∞=−⋅
∫ Ud0
11
11
coscos
sin
2
1
The Solution, that must also satisfy the Kutta Condition γ (π) = 0, is
( )1
11 sin
cos12
θθαθγ += ∞U
x
z
c0 1x
( ) 11 xdxγ
SOLO
217
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities – Solution (continue – 3)
Skeleton (Camber Profile)
Solution for a Flat Plate dz/dx = 0
To check the solution let substitute it in the Integral
Use Glauert Integral (1926)
Therefore
x
z
c0 1x
( ) 11 xdxγ
θθπθ
θθθπ
sin
sin
coscos
cos
0
11
1 nd
n =−∫
==
=1
00
n
n
π
( )∫∫ −
+=−⋅ ∞
ππ
θθθ
θπαθ
θθθγθ
π 0
11
1
0
11
11
coscos
cos1
coscos
sin
2
1d
Ud
( ) αθθθ
θπαθ
θθθγθ
π
ππ
∞∞ =
−+=
−⋅
∫∫ UdU
d0
11
1
0
11
11
coscos
cos1
coscos
sin
2
1
( )1
11 sin
cos12
θθαθγ += ∞U
SOLO
218
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities – Solution (continue – 4)
Skeleton (Camber Profile)
x
z
c0
1x
( ) 11 xdxγ
Solution for a Given Camber Profile z = z (x)
( )
−=
−⋅
∞∫x
xd
zdUd αθ
θθθγθ
π
π
0
11
11
coscos
sin
2
1
To determine the Vorticity Distribution we will write γ (θ1) as a Fourier Series(suggested by Glauert) that has to satisfy the Kutta Condition γ (θ1=π) = 0.
( )
++= ∑∞
=∞
11
1
101 sinsin
cos12
nn
PlateFlat
nAAU θθ
θθγ
To find the parameters An let substitute γ (θ1) in the Integral above
−=
−
+−
+∞
∞
=
∞∞ ∑ ∫∫xn
n xd
zdUd
nA
Ud
AU αθθθθθ
πθ
θθθ
π
π
π
π
1 0
11
11
0
11
10
coscos
sinsin
coscos
cos1
SOLO
219
2-D Inviscid Incompressible Flow
Skeleton (Camber Profile)
x
z
c0
1x
( ) 11 xdxγ
Solution for a Given Camber Profile z = z (x)
−=− ∞
∞
=∞∞ ∑ xd
zdUnAUAU
nn αθ
10 cos
( )[ ] ( )[ ]
( )[ ] ( )[ ] θθ
θθθ
θθ
θθθ
θθπ
θθθθθ
π
ππ
nnnn
dnn
dn
IntegralGlauert
cossin
cossin2
2
1
sin
1sin1sin
2
1
coscos
1cos1cos
2
1
coscos
sinsin1
0
11
11
0
11
11
−=−=+−−=
−+−−=
− ∫∫
Therefore
or ∑∞
=
+−=1
0 cosn
n nAAxd
zd θα
Let compute
( ) ∑ ∫∫∫∞
=
+−=1 00
0
0
coscoscoscosn
n dmnAdmAdmxd
zd πππ
θθθθθαθθ
=≠
=∫ nm
nmdmn
2/
0coscos
0 πθθθ
π
Profile Theory by the Method of Singularities – Solution (continue – 5)
SOLO
220
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 6)
Skeleton (Camber Profile)
x
z
c0
1x
( ) 11 xdxγ
Solution for a Given Camber Profile z = z (x)
∫−=π
θπ
α0
10
1d
xd
zdA
For a Symmetric Airfoil the Skeleton has d z/d x =0 (like for a Flat Plate)A0 = α, An = 0 for n=1,2,…
∑∞
=
+−=1
0 cosn
n nAAxd
zd θα
∫=π
θθπ 0
11cos2
dnxd
zdAn
SOLO
221
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 7)
Skeleton (Camber Profile)
x
z
c0
1x
( ) 11 xdxγ
Solution for a Given Camber Profile z = z (x)
Lift Computation
( )( )
( )
( )
++=
++⋅=
==Γ
∫∑∫
∫ ∑
∫∫
∞
=∞
∞
=∞
−=
ππ
π
πθ
θθθθθ
θθθθ
θ
θθθγγ
0
11110
110
0
111
11
10
0
111
2/cos1
0
11
sinsincos1
sinsinsin
cos12
2
1
sin2
111
dnAdAUc
dnAAUc
dcxdx
nn
nn
cxc
( )1022
AAUc +=Γ ∞π
SOLO
222
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 8)
Skeleton (Camber Profile)
x
z
c0
1x
( ) 11 xdxγ
Solution for a Given Camber Profile z = z (x)
Lift Computation (continue) ( )1022
AAUc +=Γ ∞π
( )10
2
22
AAcU
UL +=Γ= ∞∞ πρρ
( )102
2
21
: AAcU
LCL +==
∞
πρ ∫=
π
θθπ 0
111 cos2
dxd
zdA
πα
2=d
Cd L
The Angle of Attack α0 for which Lift is Zero is given by:
∫∫ +−=+=ππ
θθπ
θπ
α0
11
0
1010 cos22
220 dxd
zdd
xd
zdAA ( )∫ −=
π
θθπ
α0
110 cos12
dxd
zd
∫−=π
θπ
α0
10
1d
xd
zdA
SOLO
223
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 9)
Skeleton (Camber Profile)
x
z
c0
1x
( ) 11 xdxγ
Solution for a Given Camber Profile z = z (x)
Chordwise Load Distribution
The Difference between the Upper and Lower Surface Flow Velocities can be computed in the following way:
( ) ( ) ( ) ( ) ( )[ ] 1111111 :& xdxVxVxdxdxxd LowererUpper −=Γ=Γ γ
Therefore ( ) ( ) ( )111 xVxVx LowererUpper −=γ
Also because the zero thickness of the Camber Surface( ) ( )112 xVxVU LowererUpper +≈∞
We have ( ) ( ) ( )121
212 xVxVUx LowererUpper −=∞γ
Use Bernoulli’s Equality ( ) ( ) ( ) ( )1211
21 2
1
2
1xVxpxVxp LowererLowerUpperUpper ρρ +=+
We get
( ) ( ) ( ) ( )[ ] ( )112
12
11 2
1xUxVxVxpxp LowererUpperUpperLower γρρ ∞=−=−
SOLO
224
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 10)
Skeleton (Camber Profile)
x
z
c0
1x
( ) 11 xdxγ
Solution for a Given Camber Profile z = z (x)
Chordwise Load Distribution (continue)
We get
( ) ( ) ( )( )
++==− ∑
∞
=∞
−=
∞1
11
10
2cos1
2
1
111 sinsin
cos12
11
nn
x
UpperLower nAAUxUxpxp θθ
θργρθ
We can recover the Lift Equation using
( ) ( )[ ]( )
Γ=
++=
−==
∞
∞
=∞
−=
∫ ∑
∫∫
UdnAAcU
xdxpxpLdL
nn
x
c
UpperLower
c
ρθθθθ
θρπθ
0
111
11
10
2cos1
2
1
0
111
0
sinsinsin
cos12
11
( )10
2
22
AAcU
UL +=Γ= ∞∞ πρρ
SOLO
∫=π
θθπ 0
111 cos2
dxd
zdA
∫−=π
θπ
α0
10
1d
xd
zdA
225
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 11)
Skeleton (Camber Profile)
x
z
c0
1x
( ) 11 xdxγ
Solution for a Given Camber Profile z = z (x)
Pitching Moment
Let MLE be the Pitching Moment about the Leading Edge
( ) ( )[ ] ( ) 1111111 xdxxUxdxpxpxMd UpperLowerLE γρ ⋅−=−⋅−= ∞
The Pitching Moment Coefficient: ( )2// 22cUMC LEmLE ∞= ρ
( )
( )∫ ∑ −
++−=
∞
=∞
∞
∞−= πθ
θθθθθ
θ
ρ
ρ
0
1111
11
10
22
cos12
1
sin2
1cos1
2
1sin
sin
cos12
21
11
dccnAAUcU
UC
nn
x
mLE
( )∫ ∑∑
−+−−=
∞
=
∞
=
π
θθθθθθ0
11
111
1112
0 2sinsin2
1sinsincos1 dnAnAA
nn
nn
( )210210 224422
AAAAAA −+−=
−+−= ππππ
( )102 AACL += π( ) ( )21210 44
122
4AACAAAC LmLE
−−−=−+−= ππ
SOLO
∫−=π
θπ
α0
10
1d
xd
zdA ∫=
π
θθπ 0
111 cos2
dxd
zdA
226
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 12)
Skeleton (Camber Profile)
Solution for a Given Camber Profile z = z (x)Pitching Moment (continue)
Define the Center of Pressure Position as
( )102 AACL += π( ) ( )21210 44
122
4AACAAAC LmLE
−−−=−+−= ππ
( )
−+=
+
−+=
+−+=−=−=
2110
21
10
210
142
14
2/
2/
4
AAC
c
AA
AAc
AA
AAAc
C
Cc
L
Mx
L
L
mLECP
LE
π
LEML
cx
xM
For any point at a distance x from the Leading Edge we have
( ) ( )10210 2224
AAc
xAAAC
c
xCC Lmm LEx
++−+−=+= ππ
For x = c/4 we have: ( )1244
14/
AACCC Lmm LEc−=+= π
For a Thin Airfoil the Aerodynamic Center of the Section is at the Quarter-Chord Point, x = c/4.
Since A1 and A2 depend on the camber only, the section moment is independent of Angle of Attack. The point about which the section Moment Coefficient is independent of the Angle of Attack is called Aerodynamic Center of the Section.
SOLO
227
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 13)
Skeleton (Camber Profile)
Solution for a Given Camber Profile z = z (x)Pitching Moment (continue)
Comparison of the Aerodynamic Coefficients calculated using Thin Airfoil Theory for two Cambered Airfoils:(a)NACA 2412 (b) NACA 2418 Data from Abbott and von Doenhoff (1949)
Comparison of the theoretical and the experimental Section Moment Coefficient (about the Aerodynamic Center) for two Cambered Airfoils:(a)NACA 2412 (b) NACA 2418 Data from Abbott and von Doenhoff (1949)
SOLO
228
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 14)Teardrop (Symmetric Airfoil)
xtz ( ) xdxσc0 P
Q
The Teardrop (Symmetric Airfoil) Surface is defined by
( ) ( ) ( ) 000 ==≤≤= cffcxxfzt
To find the Velocity Distribution over the Airfoil we use the Teardrop that has the same thickness as the Airfoil. The Flow, at Zero Angle of Attack, is symmetric on the Teardrop, producing Zero Lift (the Lift was computed on the Camber Profile). Therefore we will use a Sheet of Source, Sinks,σ (x1), distributed on x axis, 0 ≤ x1 ≤ c, to compute the Perturbed Velocity Distribution.
We shall make a First Order Approximation, that the Flow Perturbation are small, compared to Free Stream Velocity U∞, and that zt is small. Then the Flux cross any line such as PQ= 2 zt ,located at x, is 2 zt U ∞ . But all the Fluid generated by the Sources between Leading Edge and x must pass the line PQ. Therefore
( ) t
x
zUxdx ∞=∫ 20
11σ ( )xd
zdUx t
∞= 2σxd
d
SOLO
229
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 15)Teardrop (Symmetric Airfoil) (continue – 1)
xtz ( ) xdxσc0 P
Q
The Algebraic Sum of Sources and Sinks is Zero.
We have ( ) ( ) 020
0
11 == =
=∞∫ cx
xt
c
xzUxdxσ
( )xd
zdUx t
∞= 2σ
At the Leading Edge d zt/ dx > 0 (Sources), at the Trailing Edge d zt/ dx < 0 (Sinks).
To find the Perturbed Velocity Distribution let define first x = (1-cos θ)/2, write the function f as a function of θ, and express the function as a Fourier Series:
( ) ( ) ∑∞
===
11 sin
2
1
nn nBcfxf θθ
where Bn is given by
( )∫=π
θθθπ 0
1 sin4
dnfc
Bn
SOLO
230
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 16)Teardrop (Symmetric Airfoil) (continue – 2)
For Sources and Sinks we found that exists only a Radial Velocity Component.In our case the Source/Sink σ (x1) dx1 will produce at a point P (x ) a Velocity Perturbation in x direction d uP given by
The Total Perturbation Velocity, due to all Sources/Sinks, is given by
( ) ( )( ) ( )1
11
1
111 2
2
2,
xx
xdxd
zdU
xx
xdxxxud
t
P −=
−=
∞
ππσ
Perform coordinate transformation
c0 P
x1x
Pu( ) 11 xdxσ
( ) ( )∫ −= ∞
c
P xx
xdxdfd
Uxu
0 1
11
π
( ) ( ) ∑∞
=
==1
1111
111
1
1 cos2
1
nn dnBncd
d
fdxd
xd
xfd θθθθθ ( )θθ coscos
2 11 −=− cxx
( ) ( )∫∑
−=
∞
=∞π
θθθ
θ
π 0
11
11
coscos
cosd
nBnU
xu nn
Pto obtain
SOLO
231
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 17)Teardrop (Symmetric Airfoil) (continue – 3)
c0 P
x1x
Pu( ) 11 xdxσWe get
Use Glauert Integral
( )( )
∑∞
=∞
−==
1
2/cos1
sin
sin
n
nx
P
nBnUxu
θθθ
( )∫=π
θθθπ 0
1 sin4
dnfc
Bn
θθπθ
θθθπ
sin
sin
coscos
cos
0
11
1 nd
n =−∫
( ) ( )∫∑
−=
∞
=∞π
θθθ
θ
π 0
11
11
coscos
cosd
nBnU
xu nn
P
to obtain
SOLO
232
Flow over a Slender Body of Revolution Modeled by Source Distribution
AERODYNAMICS
Profile Theory by the Method of Singularities
2-D Inviscid Incompressible Flow
SOLO
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233
Airfoil DesignThe velocities at the Aviation beginning were Low Subsonic, therefore theAirfoils were designed for Subsonic Velocities. The Design was for HighLift to Drag Ratios.
In 1939 Eastman Jacobs at the NACA Langley, designed and tested the first Laminar Flow Airfoil. He create a Family of Airfoils calledNACA Sections.
Eastman Nixon Jacobs (1902 –1987)
Historical Overview of Subsonic Airfoils Shapes.
Examples of airfoils in nature and within various vehicles
AERODYNAMICSSOLO
234
Airfoil DesignAERODYNAMICSSOLO
235
NACA Airfoils
Airfoil geometry can be characterized by the coordinates of the upper and lower surface. It is often summarized by a few parameters such as: •maximum thickness, •maximum camber, •position of max thickness, •position of max camber, •nose radius.
The Airfoil here is of an Infinite Span, flying in a Incompressible Flow. The Wing Profileis the Cross Section of the Wing.
The NACA 4 digit and 5 digit airfoils were created by superimposing a simple meanline shape with a thickness distribution that was obtained by fitting a couple of popular airfoils of the time:
( ) ( )5325.0 1015.2843.3537.126.2969.2.0/ xxxxxty ⋅−⋅+⋅−⋅−⋅⋅±=The camberline of 4-digit sections was defined as a parabola from the leading edge to the position of maximum camber, then another parabola back to the trailing edge.
NACA 4-Digit Series: 4 4 1 2 max camber position max thickness in % chord of max camber in % of chord in 1/10 of c
AERODYNAMICSSOLO
236
NACA Airfoils
After the 4-digit sections came the 5-digit sections such as the famous NACA 23012. These sections had the same thickness distribution, but used a camberline with more curvature near the nose. A cubic was faired into a straight line for the 5-digit sections. NACA 5-Digit Series: 2 3 0 1 2approx max position max thickness camber of max camber in% of chord in% chord in 2/100 of c
The 6-series of NACA airfoils departed from this simply-defined family. These sections were generated from a more or less prescribed pressure distribution and were meant to achieve some laminar flow.
NACA 6-Digit Series: 6 3 2 - 2 1 2Six- location half width ideal Cl max thickness Series of min Cp of low drag in tenths in% of chord in 1/10 chord bucket in 1/10 of Cl
SOLO
237
AERODYNAMICS
NACA Airfoils
SOLO
238
NACA Airfoils
Geometry of the most important NACA Profiles(a)Four-Digit Profiles(b)Five-Digit Profiles(c)6-Series Profiles
AERODYNAMICSSOLO
239
NACA Airfoils
12.04.002.0
2142
===
↓↓↓↓
c
t
c
xh
c
h
NACA
Lower Surface
Upper Surface
AERODYNAMICSSOLO
240
Effects of the Reynolds Number (Viscosity)µ
ρ cVRe =:
Effects of the Reynolds Number on the Lift and Drag characteristics of NACA 4412
AERODYNAMICS
NACA Airfoils
SOLO
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241
Lifting-Line Theory
The Prandtl Lifting-Line Theory, also called the Lanchester–Prandtl Wing Theory is a mathematical model for predicting the Lift Distribution over a Three-Dimensional Wing based on its geometry.
Frederick William Lanchester
(1868 –1946)
Ludwig Prandtl(1875 – 1953)
The theory was expressed independently by Frederick W. Lanchester in 1907, and by Ludwig Prandtl in 1918–1919 after working with Albert Betz and Max Munk.In this model, the vortex strength reduces along the wingspan, and the loss in vortex strength is shed as a vortex-sheet from the trailing edge, rather than just at the wing-tips.
Albert Betz
(1885 – 1968 ),
Michael Max Munk(1890 – 1986)
1907 1918–1919
The Lifting-Line Theory makes use of the concept of Circulation and of the Kutta–Joukowski Theorem,
so that instead of the lift distribution function, the unknown effectively becomes the distribution of circulation over the span, Γ(y).
The lift distribution over a wing can be modeled with the concept of Circulation
A vortex is shed downstream for every span-wise change in lift
The Upwash and DownwashInduced by the Shed Vortex canBe computed at each NeighborSegment
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
SOLO
242
Frederick William Lanchester
(1868 –1946)
In 1907 Frederick William Lanchester, an English engineer(automotive and aerodynamics) published a two-volume work, Aerodynamics, dealing with the problems of powered flight. In it, he developed a model for the vortices that occur behind wings during flight, which included the first full description of Lift and Drag. His book was not well received in England, but created interest in Germany where the scientist, Ludwig Prandtl mathematically confirmed the correctness of Lanchester’s vortex theory (Lanchester visited Prandtl and vonKarman in Gotingen in 1908). In his second volume, he turned his attention to aircraft stability, aerodonetics, developing Lanchester's Phugoid Theory which contained a description of oscillations and stalls. During this work he outlined the basic layout almost all aircraft have used since then. Lanchester’s contribution to aeronautical science was not recognised until the end of his life.
1907Lifting-Line Theory
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)SOLO
243
Aerodynamic Load Distribution for a Rectangular Wingin Subsonic Airstream(a)Differential Pressure distribution along the chord for several spanwise stations.(b) Spanwise Lift Distribution
AERODYNAMICS
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
Consider a Subsonic IncompressibleFlow passing over a Finite Span Wing.
As a consequence of the tendency of thePressure acting on the Top Surface near the Tip of the Wing to equalize with those on the Bottom Surface, the Lift Force perUnit Span decreases toward the Tips.
The difference between the 3-D Flow andthe 2 – D Flow (over an Infinite Span Wing)is the Spanwise variation of Lift.
Lifting-Line Theory
SOLO
244
Generation of the Trailing Vortices due to the SpanwiseLoad Distribution:(a)View from bottom(b)View from Trailing Edge(c)Formation of the Tip Vortex(d)Smoke Flow Pattern showing Tip Vortex
AERODYNAMICS
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
Where the Flow from Upper Surface and theLower Surface join at the Trailing Edge, the difference in Spanwise Velocity Components will cause the Air to roll up into a number of Streamwise Vortices, distributed along the Span.
Since the Lift Force acting on the Wing SectionAt a given Spanwise location is related to theStrength of the Circulation. Therefor to evaluateThe Spanwise Lift distribution we can use a VortexSystem.
Lifting-Line Theory
SOLO
245
AERODYNAMICS
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
Vortex System of a Wing of Finite Span
Vortex System
Bound Vortex
Starting Vortex
The Vortex System consists of:•Bound Vortex around the Wing•Trailing (free) Vortices•Starting Vortex
The Bound Vortex around the Wing,the Two Free Vortices and theStarting Vortex form a closed VortexLine in agreement with HelmholtzVortex Theorem.
Lifting-Line Theory
SOLO
246
AERODYNAMICS
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
Bound + Trailing Vortex= Horseshoe Vortex
A schematic Vortex System for a Unswept Wing
Finite Unswept Wing
We replace the Spanwise Lift Distribution by a Single Bound Vortex System (the axis of which is normal to the plane of symmetry and passes through the Aerodynamic Center of the Lifting Surface). The single Vortex has a Circulation Γ whose strength varies along the Span Γ = Γ (y).
The Vortex System consists of theBound Vortex System and the related System of Trailing Vortices.The strength of the Trailing Vortex is
yyd
d ∆Γ=Γ∆
Lifting-Line Theory
SOLO
247
Start with Biot-Savart Formula for a Semi-infinite 2D Vortex
θπ
θθθπ
π
π
ˆ4
sinˆ4 2/ h
dh
VΓ=Γ= ∫
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
ydyd
d Γ
1yy
( )ywy1δ
1yy −
Trailing Vortex
θ
2
πθ =
h
θ
Semi-infinite 2D Vortex
The downstream-drifting free vortices produce a downwash velocity w behind and at the wing. The induced Velocity δ wy1 caused by theSemi-infinite Trailing Vortex located at y1,at a point y is given, using Biot-Savart, by
( ) ( )14
11 yy
yyd
dywy −
∆Γ=∆π
Hence the downwash induced velocity is
( ) ( )∫+
− −
Γ
=2/
2/ 11 4
1 b
b
i ydyy
ydd
ywπ
Lifting-Line Theory
SOLO
248
AERODYNAMICS
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
Di – Induced drag
Lift Effective Lift,acts normal to theEffective Flow Direction
Induced Flow
Because of the downwash velocity theFlow is disturbed. We have: α – Angle of Attack relative to Undisturbed Flow αi - Induced Angle of Attack αe - Effective Angle of Attack
( )
( )∫+
− −
Γ
−=
−=−=
2/
2/ 14
1
:
b
b
iie
ydyy
ydd
V
V
wy
πα
αααα
( ) ( )yVyl Γ= ∞ρBased on Kutta-Joukowsky Theorem, the Lift on an elemental Airfoil Section of theWing is
and the Induced Drag
Integrating over the entire span we get
( ) ( ) ( ) ( )yywylyd ii Γ−=−= ∞ραtan
( )∫+
− ∞ Γ=2/
2/
b
bydyVL ρ
( ) ( )∫+
− ∞ Γ−=2/
2/
b
bi dyyywD ρ
Lifting-Line Theory
SOLO
249
AERODYNAMICS
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Infinite Span (AR → ∞)
For Wings of Infinite Span (AR →∞) Γ is constant along the Span, i.e.
( ) ( )yVyl Γ= ∞ρBased on Kutta-Joukowsky Theorem, the Lift on an elemental Airfoil Section of theWing is
and the Induced Drag ( ) ( ) ( ) ( ) 0tan =Γ−=−= ∞ yywylyd ii ρα
Lifting-Line Theory
( ) ( ) 04
1
11
=−
∆Γ=∆yy
yyd
dywy π
0=Γ
∞→ARyd
d
( ) ( ) 04
1
11 =
−
Γ
= ∫∞+
∞−
ydyy
ydd
ywi π
For Wings of Infinite Span (AR →∞) the Induced Drag is zero
and
The Downwash Induced Velocity is zero for Infinite Span.
SOLO
250
AERODYNAMICS
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
Elliptic Circulation Distribution
Elliptic Circulation Distribution
A simple Circulation Distribution, which has also practical applications, is given by the Elliptical relation
( )2
12
0
bs
s
yy =
−Γ=Γ
then
( )2
1
1
20
bs
s
y
ssy
yyd
d =
−
−
Γ=Γ
( ) ( )
( )
( )∫
∫
∫
+
−
+
−
+
−
−−Γ−=
−
−
Γ−=
−
Γ
=
s
s
s
s
s
s
i
ydyyys
y
s
yd
yysy
sy
s
ydyy
ydd
yw
122
0
1
2
0
11
4
14
4
1
π
π
π
Lifting-Line Theory
SOLO
251
AERODYNAMICS
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
Elliptic Circulation Distribution (continue – 1)
Developing the two integrals gives
( )( )
( )( ) ( )
−−+
−−−Γ−=
−−Γ−=
∫∫
∫+
−
+
−
+
−
s
s
s
s
s
s
i
yyys
ydy
yyys
ydyy
s
yyys
ydy
syw
122
1
122
10
122
01
4
4
π
π
πθθ
θθ π
π
π
π
θ==
−=
− ∫∫∫+
−
+
−
⋅=+
−
2/
2/
2/
2/222
sin
22 sin
cosd
ss
d
ys
yd sys
s
( ) ( )Iy
yyys
ydy
yyys
ydy
I
s
s
s
s
1
1221
122
1 =−−
=−− ∫∫
+
−
+
−
( ) [ ]Iys
ywi 10
1 4+Γ−= π
πtherefore
but
( ) ( ) [ ] [ ]( )
044
122
0011 =
−−=⇒+Γ−=−Γ−⇒+==−= ∫
+
−
s
s
iiyyys
ydIsI
ssI
ssywsyw π
ππ
π
and ( )bs
ywbs
i 240
2/0
1
Γ−=Γ−==
Elliptic Circulation Distribution and theResultant Downwash Velocity
Lifting-Line Theory
SOLO
252
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
Elliptic Circulation Distribution (continue – 2)
The Lift Coefficient of the Wing (Area = S)
alsoElliptic Circulation Distribution and the
Resultant Downwash Velocity
( )
∫
∫∫+
−∞
⋅=
+
−∞
+
− ∞
⋅⋅Γ=
−Γ=Γ=
2/
2/0
sin
2
0
coscos
1
π
π
φφφφρ
ρρ
dsV
yds
yVydyVL
sy
s
s
s
s
0
2/
0 42Γ=Γ= ∞
=
∞ VbsVLbs
ρππρ
SV
b
SV
LCL
0
2 221
Γ==∞
π
ρ
( ) ( ) ss
yds
y
sdyyywD
s
s
s
si 241
4
20
2
00 πρρρ Γ=
−ΓΓ=Γ= ∞
+
−∞
+
− ∞ ∫∫
208
Γ= ∞ρπiD
SVSV
DC i
Di 2
20
2 421
Γ== ∞
∞
ρπ
ρ
Lifting-Line Theory
SOLO
253
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
Elliptic Circulation Distribution (continue – 3)
We found
Elliptic Circulation Distribution and theResultant Downwash Velocity
SV
bCL
0
2
Γ= πSV
CiD 2
20
4
Γ= ∞ρπ
b
SVCL
π2
0 =Γ2
22
2
21
4 b
SC
b
SVC
SVC LL
Di ππρπ =
= ∞
Since the Wing Aspect Ratio is
S
bAR
2
=
we have
AR
CC L
Di π
2
=
Lifting-Line Theory
For AR → ∞0lim =
∞→ iDAR
C ( ) 0
2
0 1lim Γ=
−Γ=Γ
∞→ s
yy
s
SOLO
254
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
Elliptic Circulation Distribution (continue – 4)
We found
Elliptic Circulation Distribution and theResultant Downwash Velocity
SV
bCL
0
2
Γ= π
b
SVCL
π2
0 =ΓAR
C
b
SC
b
SVC
VbLLL
i πππα ===
2
2
2
1
Since the Wing Aspect Ratio isS
bAR
2
=
Lifting-Line Theory
SOLO
VbV
w bw
ii
i
20
20
Γ=−=
Γ−=
α
α
π
απ
AR
aa
AR
CL0
0
12
1
2
+=
+=
( )
−=−=
AR
CC L
i
a
L παπααπ 22
0
we have
255
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
( ) ( ) ( )
−
Γ
−=Γ ∫+
−
2/
2/ 1
1
4
1
2
1 b
b
L ydyy
ydd
VCycVy
παα
We have
and
c (y) is the local chord length
Di – Induced drag
Lift Effective Lift,acts normal to theEffective Flow Direction
Induced Flow
( ) ( )yVyl Γ= ∞ρ
( ) ( ) ( ) ( )
( )
( ) ( )
−
Γ
−=
∂∂≈=
∫+
−∞
∞∞
2/
2/ 1
12
22
4
1
2
1
2
1
2
1
b
b
L
e
C
LL
ydyy
ydd
VCycV
yC
ycVyCycVyl
L
παρ
αα
ρρ
α
α
From those two relations we get
Use the Transformation
1111 sin2
cos2
,cos2
φφφ byd
by
by =⇒−=−=
( ) ( )( )
−
Γ
−=Γ ∫+π
α φφφφ
φφπ
αφφ0
11
1
11 sin2coscos
2
sin2
4
1
2
1d
bb
bdd
VCcV L
Techniques for General Spanwise Circulation Distribution
Lifting-Line Theory
SOLO
256
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
We have
Assuming Γ ( =0) = Γ ( =π)=0 let consider theϕ ϕfollowing Fourier development of Γ ( ) ϕ
where the coefficients An have to be determined. Substitute this in the previous equation and use Glauert Integral Formula
Di – Induced drag
Lift Effective Lift,acts normal to theEffective Flow Direction
Induced Flow
( ) ( ) ( )
−
Γ
−=Γ ∫+π
α φφφ
φπ
αφφ0
11
1
coscos2
1
2
1d
dd
VCcV L
Techniques for General Spanwise Circulation Distribution (continue – 1)
1111 sin2
cos2
,cos2
φφφ byd
by
by =⇒−=−=
( ) ∑∞
==Γ
1sin2
n n nAbV φφ
φφπφ
φφφπ
sin
sin
coscos
cos
0
11
1 nd
n =−∫
( ) ( ) ( )
( )
−=
−−==Γ
∑
∫ ∑∑
∞
=
+ ∞
=∞
=
1
0
11
11
sinsin2
1
coscos
cos2
2
1
2
1sin2
nn
L
n nLn n
nnAb
CcV
dnnAbV
VCcVnAbV
φφ
ππ
αφ
φφφ
φπ
αφφφ
α
π
α
Lifting-Line Theory
SOLO
257
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
We have
Techniques for General Spanwise Circulation Distribution (continue – 2)
1111 sin2
cos2
,cos2
φφφ byd
by
by =⇒−=−=
( ) ( ) ( )
( )
−=
−−==Γ
∑
∫ ∑∑
∞
=
+ ∞
=∞
=
1
0
11
11
sinsin2
1
coscos
cos
2
1sin2
nn
L
n nLn n
nnA
bCcV
dnnA
bCcVnAbV
φφ
παφ
φφφφ
αφφφ
α
π
α
Rearranging( ) ( ) φαφφφφ
µ
α
µ
α sin44
sinsin1
Ln Ln C
b
cC
b
cnnA =
+∑∞
=
or
[ ] ( )α
φµφαµµφφ Ln n Cb
cnnA
4:sinsinsin
1==+∑∞
= Monoplane Equation
[ ] NinnA i
N
n iin ,,1sinsinsin1
==+∑ =φαµµφφ
Di – Induced drag
Lift Effective Lift,acts normal to theEffective Flow Direction
Induced Flow
Lifting-Line Theory
SOLO
Assume that we know c (y)=c (-b cos /2), and using Cϕ Lα (2-D) = 2π, and we want to find the first N Fourier coefficients (Ai, i=1,…,N). We can chose N different0 <ϕ i < π (i=1,…,N), and compute the N equations, with N unknowns (Ai, i=1,…,N).
258
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
We have
Techniques for General Spanwise Circulation Distribution (continue – 3)
( ) ( ) ∑∞
=∞∞ =Γ=1
2 sin42
1n n nAbVVl φρφρφ
( ) ( )
( ) ( )[ ] 122
1 0
22
0 10
cos22/
2/
2
11cos1cos
2
1
sinsin22
1sin
2
AbVdnnAbV
dnAbVbVdb
VydyVL
n n
n n
by
b
b
πρφφφρ
φφφρφφφρρ
π
ππφ
∞∞
=∞
∞
=∞∞
−=+
− ∞
=+−−=
=Γ=Γ=
∑ ∫
∫ ∑∫∫
( ) ∑∞
==Γ
1sin2
n n nAbV φφ
Lift Span
Total Lift
11
2
2
21
AARAS
b
SV
LCL ⋅⋅=⋅⋅==
∞
ππρ
122
2
1AbVL πρ∞=
Di – Induced drag
Lift Effective Lift,acts normal to theEffective Flow Direction
Induced Flow
Lifting-Line Theory
SOLO
259
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
We have
Techniques for General Spanwise Circulation Distribution (continue – 4)
( ) ( ) ( ) ( )yywylyd ii Γ−=−= ∞ραtan
( ) ∑∞
==Γ
1sin2
n n nAbV φφ
Induced Velocity
Induced Drag on Span
( ) ( ) ( )
VnnAVdn
AnV
b
db
bnnAVbyd
yy
ydd
yw
in nn
nGlauert
n
yd
yd
dd
d
n n
byb
b
i
αφφφφ
φφπ
φφ
φφ
φφ
ππ
φφπ
π
π
φφ
φ
≈=−
=
−−=
−
Γ
=
∑∑ ∫
∫ ∑∫
∞
=
∞
=
Γ
∞
=
−=+
−
1 11
1
sin
sin
01
0
1
1
cos22/
2/ 11
sinsincoscos
cos
coscos2
sin2
sin2
1cos2
4
1
4
1
1
1
Di – Induced drag
Lift Effective Lift,acts normal to theEffective Flow Direction
Induced Flow
Lifting-Line Theory
SOLO
260
AERODYNAMICS
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
We have
Techniques for General Spanwise Circulation Distribution (continue – 4)
( ) ( ) ( )
( ) ( ) ∑∑ ∑ ∫∫
∑ ∑ ∫
∫ ∑∑∫
∞
=∞∞
=
∞
=
⋅
∞
∞
=
∞
=∞
∞
=
∞
=∞
−=+
− ∞
=
++−=
=
=Γ=
1
222
1 1
0
00
22
1 1 0
22
0 11 1
cos22/
2/
2coscos
2
sinsin22
sin2
sin2sinsin
,
n nn m mn
n m mn
m mn n
by
b
bi
AnbV
dnmdnmAAnbV
dmnAAnbV
db
mAbVnnAV
dyyywD
nm
πρφφφφρ
φφφρ
φφφφφ
ρρ
π
δπ
π
π
πφ
( ) ∑∞
==Γ
1sin2
n n nAbV φφ
Induced Velocity
Induced Drag on Span
VnnAVb
yw in ni αφφ
φ ≈=
−= ∑∞
=1 11
11 sinsin
cos2
Total Drag on Wing
∑∞
=∞=
1
222
2 n ni AnbV
D πρ
Di – Induced drag
Lift Effective Lift,acts normal to theEffective Flow Direction
Induced Flow
( ) ( ) ( ) ( )yywylyd ii Γ−=−= ∞ραtan
Lifting-Line Theory
SOLO
261
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
We have
Techniques for General Spanwise Circulation Distribution (continue – 5)
∑∞
=∞ ⋅⋅=
1
222
2 n ni AnbV
D πρ
∑∑ ∞
=
∞
=∞
⋅⋅=⋅⋅==1
2
1
22
2
2
n nn ni
D AnARAnS
b
SV
DC
iππ
ρ
Di – Induced drag
Lift Effective Lift,acts normal to theEffective Flow Direction
Induced Flow
11
2
2
21
AARAS
b
SV
LCL ⋅⋅=⋅⋅==
∞
ππρ
We found
∑∞
=
++++⋅
⋅=
⋅
⋅=
1 21
27
21
25
21
23
22
1
2 7531
nL
onsDistributilSymmetrica
TermsOddOnly
nLD A
A
A
A
A
A
AR
C
A
An
AR
CC
i
ππ
( ) 0753
:12
1
27
21
25
21
23
2
≥+++=+⋅⋅
= A
A
A
A
A
A
AR
CC L
Diδδ
π
CDi is minimum when δ = 0 : A1≠0, A3=A5=A7=…=0. In this case( ) onDistributiEllipticAbV φφ sin2 1=Γ
Lifting-Line Theory
SOLO
262
AERODYNAMICS
Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
Experimental Drag Polar for a Wing with anAspect Ratio AR = 5 compared with theTheoretical Induced Drag
AR
CC D
Dv π
2
=
Effect of the Aspect Ratio on the Drag Polar for Rectangular Wings (AR from 1 to 7) (a) Measured Drag Polars (b) Drag Polar converted to AR = 5
SOLO
Return to Table of Content
263
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
The Lifting-Line Theory gives good results for Unswept Wings with high AspectRatios. For small Aspect Ratios or Highly Swept Wings or Delta Wings we need an improved method to compute the Lifting Flow Field (for Incompressible Flow and Small Angles of Attack). The Lifting-Surface Theory approximates the continuous distribution of bound vorticity over Wing Surface by a finite number of Horseshoe Vortices.
Sketch of Coordinate System, Elemental Panels, and Horseshoe Vortices for a typical Wing in theVortex Lattice Method
The individual Horseshoe Vortices are placed in Trapezoidal Panels called Finite Elements or Lattices. Hence the procedure of obtaining a Numerical Solution for the Flow is namedVortex Lattice Method (VLM).
The VLM models the lifting surfaces, such as a wing, of an aircraft as an infinitely thin sheet of discrete vortices to compute lift and induced drag. The influence of the thickness, viscosity is neglected.
SOLO
264
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
Boundary Conditions
The Boundary Conditions require that the Flow is Tangent to the Wing Surface(Non-penetrating Flow Condition). For our simplified Lattice Model we require that this condition is satisfied at one point on the Flat Panel . This Point, where theBoundary Conditions are satisfied is called Control Point. We want that the Numerical Coefficients will not be affected by a change (small) in the Angle of Attack. From the Thin Airfoil Theory we found that Aerodynamic Center of the Section is at the Quarter-Chord Point, x = c/4. Therefore we placethe Bound Vortex on the Quarter-Chord Point of the Flat Panel.
BoundVortex
Lattice
TrailingVortex
TrailingVortex
∞Uαsin∞U
α
rV
π2
Γ=Lattice
4
c
BoundVortex
Boundary Conditions: Vr
UU =Γ=≈ ∞∞ παα
2sin
Γ=Γ== ∞∞
−
∞∞∞∞∞ U
rUcUcUl
JoukowskyKutta
AirfoilSymmetric
ρπ
πρπαρ2
22
1 222
cr =
Control Point Position
SOLO
265
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
Boundary Conditions
Sketch of the Distributed Horseshoe Vortices representing the Lifting Flow Field over a Swept Wing. It incudes the position of Bound Vortices (at c/4) and of Control Points (at 3c/4). This is known as the “1/4 – ¾ rule”. This placement works well and has become a rule of tumb. It was discovered by Italian Pistolesi.
ENRICO PISTOLESI(1889 - 1968)
SOLO
266
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
( )34 r
rldrVd n
×Γ=
π( )rVd
The Velocity induced at point C by a Vortex Filament of strength Γn (n is the index of n-Panel) and length dl is given by Biot-Savart Law
From Figure
θsinpr
r =
( )[ ] ( ) ( )( )
( )( ) θ
θθθθ
θθθθ
θθθθθθθθθ
2sinsinsin
sin
sinsin
sincossincoscotcot
dr
d
dr
d
ddrdrld
pp
pp
≈+
=
++−+=+−=
( )21
2121
21
21/ coscos
4sin
4
2
1rr
rr
rrr
rrd
rV
p
n
p
nABC
××−Γ=
××
Γ= ∫ θθπ
θθπ
θ
θ
The Velocity induced at point C by a Vortex Filament A B is given by
Velocity Induced by a General Horseshoe Vortex
SOLO
267
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
CV
( )21
2121/ coscos
4 rr
rr
rV
p
nABC
××−Γ= θθ
π
The Velocity induced at point C by a Vortex Filament A B is given by
Velocity Induced by a General Horseshoe Vortex
Define ABr =:0
We have
20
202
10
101
0
21 cos,cos,rr
rr
rr
rr
r
rrrp
⋅=⋅=
×= θθ
−⋅
××Γ=
2
2
1
102
21
21/ 4 r
r
r
rr
rr
rrV n
ABC
πTherefore
SOLO
268
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
Let write
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) kzzjyyixxr
kzzjyyixxr
kzzjyyixxr
nnn
nnn
nnnnnn
2222
1111
1212120
−+−+−=
−+−+−=
−+−+−=
ABAB
r
r
r
rr
rr
rrV n
ABC
ΩΨ
−⋅
××Γ=
2
2
1
102
21
21/ 4π
( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]
−−−−−+
−−−−−+
−−−−−
−−−−−+
−−−−−+
−−−−−
=
××=Ψ
21221
22112
21221
1221
2112
1221
2
21
21
nnnn
nnnn
nnnn
nnnn
nnnn
nnnn
AB
yyxxyyxx
zzxxzzxx
zzyyzzyy
kyyxxyyxx
jzzxxzzxx
izzyyzzyy
rr
rr
( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( ) 2
22
22
2
212212212
21
21
21
112112112
2
2
1
10
nnn
nnnnnnnnn
nnn
nnnnnnnnn
AB
zzyyxx
zzzzyyyyxxxx
zzyyxx
zzzzyyyyxxxx
r
r
r
rr
−+−+−
−−+−−+−−−
−+−+−
−−+−−+−−=
−⋅=Ω
Contribution of Bounded Vortex
SOLO
269
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
Let write ( )( ) ( ) ( )( ) ( ) ( ) kzzjyyixxr
kzzjyyixxr
ixxADr
nnn
nnn
nn
1112
1131
310
−+−+−=
−+−+−=
−==
ADAD
r
r
r
rr
rr
rrV n
ADC
ΩΨ
−⋅
××Γ=
2
2
1
102
21
21/ 4π
( ) ( )( ) ( )[ ] ( )nnnn
nn
AD
xxyyzz
kyyjzz
rr
rr
132
12
1
11
2
21
21
−−+−−−−=
××=Ψ
( ) ( )( ) ( ) ( )
( )( ) ( ) ( )
−+−+−
−+−+−+−
−−=
−⋅=Ω
21
21
21
1
21
21
23
313
2
2
1
10
nnn
n
nnn
nnn
AD
zzyyxx
xx
zzyyxx
xxxx
r
r
r
rr
Contribution of Trailing Vortex A∞
The Velocity induced at point C by a Vortex Filament A D (D→∞)
( ) ( )( ) ( )[ ]
( )( ) ( ) ( )
−+−+−
−+−+−−−−Γ=∞ 2
12
12
1
12
12
1
11/ 1
4nnn
n
nn
nnnAC
zzyyxx
xx
yyzz
kyyjzzV
π
Taking x3n →∞, we obtain
SOLO
270
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
Finally we obtain, for the nth Panel.
( ) ( )( ) ( )[ ]
( )( ) ( ) ( )
−+−+−
−+−+−−−−Γ=∞ 2
12
12
1
12
12
1
11/ 1
4nnn
n
nn
nnnAC
zzyyxx
xx
yyzz
kyyjzzV
π
( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( )
AB
AB
nnn
nnnnnnnnn
nnn
nnnnnnnnn
nnnn
nnnn
nnnn
nnnn
nnnn
nnnn
nABC
zzyyxx
zzzzyyyyxxxx
zzyyxx
zzzzyyyyxxxx
yyxxyyxx
zzxxzzxx
zzyyzzyy
kyyxxyyxx
jzzxxzzxx
izzyyzzyy
V
Ω
Ψ
−+−+−
−−+−−+−−−
−+−+−
−−+−−+−−
−−−−−+
−−−−−+
−−−−−
−−−−−+
−−−−−+
−−−−−
Γ=
22
22
22
212212212
21
21
21
112112112
21221
22112
21221
1221
2112
1221
/ 4π
( ) ( )( ) ( )[ ]
( )( ) ( ) ( )
−+−+−
−+−+−−−−Γ−=∞ 2
22
22
2
22
22
2
22/ 1
4nnn
n
nn
nnnBC
zzyyxx
xx
yyzz
kyyjzzV
π
∞∞ ++= BCACABCC VVVV ///
SOLO
271
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
We obtained, for the nth Panel.
∞∞ ++= BCACABCC VVVV ///
Let the point (x,y,z) be the Control Point of the mth Panel, which will be designated as (xm, ym, zm).The Velocity induced at the m Control Point by the Vortex representing the nth Panel is designated as .We saw that this can be written as
where the coefficients depend on the geometry of the nth Horseshoe Vortex and its position relative to the mth Control Point.
nmV ,
nnmnm CV Γ= ,,
nmC ,
∑=
Γ=N
nnnmm CV
2
1,
Since the governing equations are Linear, the Velocity induced at mth Panel, by the 2N (Symmetry of the Wing) Vortices is the sum of the influence of all Vortices.
We have 2N equations, one for each Control Point, and 2N unknowns Γn.
SOLO
272
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
SOLO
The total induced velocity at the point m on the surface is due to the 2N vortices (N on each side of the planform) is
∑ =Γ=++= N
n nnmindmindmindmindm CkwjviuV2
1 ,,,,,
The solution requires the satisfaction of Boundary Conditions for the Total Velocity, which is the sum of the induced and free stream velocity. The freestream velocity is introduce the possibility of considering vehicles at combined angle of attack and sideslip
kVjViVV
βαββα cossinsincoscos ∞∞∞∞ +−=
so that the total velocity at point m is:
( ) ( ) ( ) kwVjvViuVVVV indmindmindmindmm
,,,, cossinsincoscos +++−++=+= ∞∞∞∞ βαββα
The values of the unknown circulations Γn, are found by satisfying the non-penetration boundary condition at all the control points simultaneously. For steady flow this is
0=⋅ nV
were the surface is given by
( ) 0,, =zyxF
Therefore 00 =∇⋅⇒=∇∇⋅ FV
F
FV
273
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
SOLO
This can be written as
00 =∇⋅⇒=∇∇⋅ FV
F
FV
( ) ( ) ( )[ ] 0cossinsincoscos ,,, =
∂∂+
∂∂+
∂∂⋅+++−++ ∞∞∞ k
z
Fj
y
Fi
x
FkwVjvViuV indmindmindm
βαββα
( ) ( ) ( )[ ]0
cossinsincoscos2
1 .
2
1 .
2
1 .
=
∂∂+
∂∂+
∂∂
⋅Γ++Γ+−+Γ+ ∑∑∑ =∞=∞=∞
kz
Fj
y
Fi
x
F
kCVjCViCVN
n nnm
N
n nnm
N
n nnm kji
βαββαor
( ) ( ) ( ) 0cossinsincoscos2
1 .
2
1 .
2
1 . =Γ+∂∂+Γ+−
∂∂+Γ+
∂∂ ∑∑∑ =∞=∞=∞
N
n nnm
N
n nnm
N
n nnm kjiCV
z
FCV
y
FCV
x
F βαββα
Performing the dot product we obtain:
Nmz
F
y
F
x
FVC
z
FC
y
FC
x
FN
n nnmnmnm kji2,,1cossinsincoscos
2
1 ... =
∂∂+
∂∂−
∂∂−=Γ
∂∂+
∂∂+
∂∂∑ = ∞ βαββα
Rearranging:
This is the general equation to obtain the circulations coefficients Γn.
274
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
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If the surface is in the x – y plane, and the sideslip is zero (β = 0), we obtain a simpler form.In this case the natural description of the surface is
and
( )yxfz ,=
The gradient of F is
( ) ( ) 0,,, =−= yxfzzyxF
1,, =∂∂
∂∂−=
∂∂
∂∂−=
∂∂
z
F
y
f
y
F
x
f
x
F
Nmx
fVCC
y
fC
x
fN
n nnmnmnm kji2,,1sincos
2
1 ... =
−
∂∂=Γ
+
∂∂−
∂∂−∑ = ∞ αα
This equation provides the solution for the circulations coefficients Γn, for this case.
Using this we obtain
275
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
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Nmx
fVCw
N
n
m
cnnmm k
2,,12
1 . =
−
∂∂=Γ= ∑ = ∞ α
Consider the simple planar surface case, where there is no dihedral. Use the thin airfoil theory were boundary conditions can be applied on the mean surface, and not the actual camber surface. We also use the small angle approximations. Under these assumptions:
We have the following equation which satisfies the boundary conditions and can be used to relate the circulation distribution and the wing camber and angle of attack:
Nmx
f
VC
N
n
m
cnnm k
2,,12
1 . =
−
∂∂=
Γ∑ =∞
α
We have two cases:1.Given camber slopes and α , solve for the circulation strengths, (Γ/V∞) [ a system of 2N simultaneous linear equations].or2.Given (Γ/V∞), which corresponds to a specified surface loading, we want to find the camber and α required to generate this loading (only requires simple algebra, no system of equations must be solved).
276
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
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The Classical Vortex Lattice Method
There are many different vortex lattice schemes. In this section we describe the “classical”implementation. Knowing that vortices can represent lift from our airfoil analysis, and that oneapproach is to place the vortex and then satisfy the boundary condition using the “1/4 - 3/4 rule,”we proceed as follows:1. Divide the planform up into a lattice of quadrilateral panels, and put a horseshoevortex on each panel.
2. Place the bound vortex of the horseshoe vortex on the 1/4 chord element line of each panel.3. Place the control point on the 3/4 chord point of each panel at the midpoint in the spanwise direction (sometimes the lateral panel centroid location is used) .4. Assume a flat wake in the usual classical method.5. Determine the strengths of each Γn required to satisfy the boundary conditions by solving a system of linear equations. The implementation is shown schematically in Figure.
277
AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Finite Span (AR < ∞)
3D Lifting-Surface Theory through Vortex Lattice Method (VLM)
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Return to Table of Content
A comparison between Panel Methods and Vortex Lattice Methods are:
Similar to Panel methods:• singularities are placed on a surface• the non-penetration condition is satisfied at a number of control points• a system of linear algebraic equations is solved to determine singularity strengths
Different from Panel methods:• Oriented toward lifting effects, and classical formulations ignore thickness• Boundary conditions (BCs) are applied on a mean surface, not the actual surface (not an exact solution of Laplace’s equation over a body, but embodies some additional approximations, i.e., together with the first item, we find ∆Cp, not Cpupper and Cplower)• Singularities are not distributed over the entire surface• Oriented toward combinations of thin lifting surfaces (recall Panel methods had no limitations on thickness).
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The Laplace’s Equation is used for Incompressible Flow. One of the key features of Laplace’s Equation is the property that allows the equation to be converted from a 3D problem to a 2D problem for finding the potential on the surface. The solution is then found using by distributing “singularities” of unknown strength over discretized portion of the surface: panels.The flow is found by representing the surface by a number of panels, and solving a linear set of algebraic equations to determine the unknown strengths of the singularities.
Subsonic Flow: Elliptic PDE, each point influences every other point.Supersonic Flow: Hyperbolic PDE, discontinuities exist, “zone of influence”
solution dependency.
Incompressible Potential Flow Using Panel Methods
279
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The flow pattern is uniquely defined by using either:
on ɸ ∑ + κ Dirichlet Problem Designor ∂ /∂n on ɸ ∑ + κ Neumann Problem Analysis
We can have also a mixed boundary condition, a + b ∂ /∂n ɸ ɸ on ∑ + κ.
The Dirichlet Problem corresponds to aerodynamic case where a surface pressure distribution is specified and the surface shape must be found. The Neumann Problem is used when the flow over the surface is defined (usually parallel to the surface.
Incompressible Potential Flow Using Panel Methods
Johann Peter Gustav Lejeune Dirichlet
1805-1859
280
AERODYNAMICS
Wing Configurations
Low wing Mid wingShoulder wing
High wingParasol wing
Monoplane - one wing plane. Since the 1930s most airplanes have been monoplanes. The wing may be mounted at various positions relative to the fuselage.
Biplane - two wing planes of similar size, stacked one above the other. The most common configuration until the 1930s, when the monoplane took over. The Wright Flyer I was a biplane.
Biplane Unequal-span biplane Sesquiplane Inverted sesquiplane
http://en.wikipedia.org/wiki/Wing_configuration
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AERODYNAMICS
Wing ConfigurationsTriplane - three planes stacked one above another. Triplanes such as the Fokker Dr.I (Manfred von Richthofen - Red Baron, WWI As with 80 victories) enjoyed a brief period of popularity during the First World War due to their manoeuvrability, but were soon replaced by improved biplanes
Quadruplane - four planes stacked one above another. A small number of the Armstrong Whitworth F.K.10 were built in the First World War but never saw service.
Multiplane - many planes, sometimes used to mean more than one or more than some arbitrary number. The term is occasionally applied to arrangements stacked in tandem as well as vertically. The 1907 Multiplane of Horatio Frederick Phillips flew successfully with two hundred wing foils, while the nine-wing Caproni Ca.60 flying boat was airborne briefly before crashing.
Triplane Quadruplane Multiplane
Fokker DR1 Triplane2 × 7.92 mm (.312 in)
"Spandau" lMG 08 machine guns
http://en.wikipedia.org/wiki/Wing_configuration
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AERODYNAMICS
Wing ConfigurationsWing Sweep
• Straight - extends at right angles to the line of flight. The most structurally-efficient wing, it is common for low-speed designs, such as the P-80 Shooting Star and sailplanes.
• Swept back, (aka "swept wing") - The wing sweeps rearwards from the root to the tip. In early tailless examples, such as the Dunne aircraft, this allowed the outer wing section to act like a conventional empennage (tail) to provide aerodynamic stability. At transonic speeds swept wings have lower drag, but can handle badly in or near a stall and require high stiffness to avoid aeroelasticity at high speeds. Common on high-subsonic and early supersonic designs e.g. the Hawker Hunter.
• Forward swept - the wing angles forward from the root. Benefits are similar to backwards sweep, also it avoids the stall problems and has reduced tip losses allowing a smaller wing, but requires even greater stiffness to avoid aeroelastic flutter as on the Sukhoi Su-47. The HFB-320 Hansa Jet used forward sweep to prevent the wing spar passing through the cabin. Small shoulder-wing aircraft may use forward sweep to maintain a correct CoG.
Straight Swept back, Forward swept
http://en.wikipedia.org/wiki/Wing_configuration
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AERODYNAMICS
Wing ConfigurationsWing Sweep
Some types of variable geometry vary the wing sweep during flight:
• Swing-wing - also called "variable sweep wing". The left and right hand wings vary their sweep together, usually backwards. Seen in a few types of military aircraft, such as the General Dynamics F-111.
• Oblique wing - a single full-span wing pivots about its midpoint, so that one side sweeps back and the other side sweeps forward. Flown on the NASA AD-1 research aircraft.
Variable sweep Variable-geometry
NASA AD-1
General Dynamics F-111
http://en.wikipedia.org/wiki/Wing_configuration
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AERODYNAMICS
Wing ConfigurationsChord Variation Along Span
The wing chord may be varied along the span of the wing, for both structural and aerodynamic reasons.
• Constant chord - parallel leading & trailing edges. Simplest to make, and common where low cost is important, e.g. in the Piper J-3 Cub but inefficient as the outer section generates little lift. Sometimes known as the Hershey Bar wing in North America due to its similarity in shape to a chocolate bar
• Tapered - wing narrows towards the tip, with straight edges. Structurally and aerodynamically more efficient than a constant chord wing, and easier to make than the elliptical type. It is one of the most common wing planforms, as seen on the F4F Wildcat
• Trapezoidal - a low aspect ratio tapered wing, where the leading edge sweeps back and the trailing edge sweeps forwards as on the Lockheed F-22 Raptor.
• Inverse tapered - wing is widest near the tip. Structurally inefficient, leading to high weight. Flown experimentally on the XF-91 Thunderceptor in an attempt to overcome the stall problems of swept wings.
• Compound tapered - taper reverses towards the root. Typically braced to maintain stiffness. Used on the Westland Lysander army cooperation aircraft to increase visibility for the pilot.
• Constant chord with tapered outer section - common variant seen for example on many Cessna types and the English Electric Canberra.
Constant Chord Tapered Trapezoidal Reverse tapered
Compound tapered
Constant chord,tapered outer
http://en.wikipedia.org/wiki/Wing_configuration
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AERODYNAMICS
Wing ConfigurationsDelta WingsTriangular planform with swept leading edge and straight trailing edge. Offers the advantages of a swept wing, with good structural efficiency and low frontal area. Disadvantages are the low wing loading and high wetted area needed to obtain aerodynamic stability. Variants are
• Tailless Delta - a classic high-speed design, used for example in the widely built Dassault Mirage III series.
• Tailed Delta - adds a conventional tailplane, to improve handling. Popular on Soviet types such as the Mikoyan-Gurevich MiG-21.
• Cropped Delta - tip is cut off. This helps avoid tip drag at high angles of attack. At the extreme, merges into the "tapered swept" configuration.
• Compound Delta or double delta - inner section has a (usually) steeper leading edge sweep e.g. Saab Draken. This improves the lift at high angles of attack and delays or prevents stalling. Seen in tailless form on the Tupolev Tu-144 and the Space Shuttle. The HAL Tejas has an inner section of reduced sweep.
• Ogival Delta - a smoothly blended "wineglass" double-curve encompassing the leading edges and tip of a cropped compound delta. Seen in tailless form on the Concorde supersonic transports.
Tailless Delta Tailed Delta Cropped Delta Compound Delta
Ogival Delta
http://en.wikipedia.org/wiki/Wing_configuration
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AERODYNAMICS
Wing ConfigurationsTailplanes and foreplanes
The classic aerofoil section wing is unstable in pitch, and requires some form of horizontal stabilizing surface. Also it cannot provide any significant pitch control, requiring a separate control surface (elevator) mounted elsewhere.
• Conventional - "tailplane" surface at the rear of the aircraft, forming part of the tail or empennage.• Canard - "foreplane" surface at the front of the aircraft. Common in the pioneer years, but from the
outbreak of World War I no production model appeared until the Saab Viggen appeared in 1967.• Tandem - two main wings, one behind the other. Both provide lift; the aft wing provides pitch stability
(as a usual tailplane) . An example is the Rutan Quickie. To provide longitudinal stability, the wings must differ in aerodynamic characteristics : wing loading and aerofoils must be different between the two wings.
• Three surface - used to describe types having both conventional tail and canard auxiliary surfaces. Modern examples include the Sukhoi Su-33 and Piaggio P.180 Avanti. Pioneer examples included the Voisin-Farman I and Curtiss No. 1.
• Tailless - no separate surface, at front or rear. The lifting and stabilizing surfaces may be combined in a single plane, as on the Short SB.4 Sherpa whose whole wing tip sections acted as elevons. Alternatively the aerofoil profile may be modified to provide inherent stability. Aircraft having a tailplane but no vertical tail fin have also been described as "tailless".
Conventional
http://en.wikipedia.org/wiki/Wing_configuration
Canard Tandem Three Surfaces Tailless
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AERODYNAMICS
Wing ConfigurationsDihedral and Anhedral
http://en.wikipedia.org/wiki/Wing_configuration
Angling the wings up or down spanwise from root to tip can help to resolve various design issues, such as Stability and Control in Flight.
• Dihedral - the tips are higher than the root as on the Boeing 737, giving a shallow 'V' shape when seen from the front. Adds lateral stability.
• Anhedral - the tips are lower than the root, as on the Ilyushin Il-76; the opposite of dihedral. Used to reduce stability where some other feature results in too much stability
Dihedral Biplane with Dihedralon both wings
Biplane with Dihedralon lower wing
Some biplanes have different degrees of dihedral/anhedral on different wings; e.g. the Sopwith Camel had a flat upper wing and dihedral on the lower wing, while the Hanriot HD-1 had dihedral on the upper wing but none on the lower.
Anhedral
Ilyushin Il-76Boeing 737, Sopwith Camel
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AERODYNAMICS
Wing ConfigurationsPolyhedral Wings
http://en.wikipedia.org/wiki/Wing_configuration
In a polyhedral wing the dihedral angle varies along the span.
• Gull wing - sharp dihedral on the wing root section, little or none on the main section, as on the PZL P.11 fighter. Sometimes used to improve visibility forwards and upwards and may be used as the upper wing on a biplane as on the Polikarpov I-153.
• Inverted gull - anhedral on the root section, dihedral on the main section. The opposite of a gull wing. May be used to reduce the length of wing-mounted undercarriage legs or allow a larger propeller. Two well-known examples of the inverted gull wing are World War II's American F4U Corsair , and the German Junkers Ju 87 Stuka dive bomber.
• Cranked - tip section dihedral differs from the main section. The wingtips may crank upwards as on the F-4 Phantom II or downwards as on the Northrop XP-56 Black Bullet. (Note that the term "cranked" varies in usage. Here, it is used to help clarify the relationship between changes of dihedral nearer the wing tip vs. nearer the wing root. See also Cranked arrow planform.)
Gull WingInverted Gull Wing Upward Cranked Tips Downward Cranked Tips
PZL P.11 Junkers Ju 87 Stuka F-4 Phantom
Northrop XP-56 Black Bullet
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AERODYNAMICSWing Configurations
Variable Planform
http://en.wikipedia.org/wiki/Wing_configuration
• Variable-Sweep Wing or Swing-Wing. The left and right hand wings vary their sweep together, usually backwards. The first successful wing sweep in flight was carried out by the Bell X-5 in the early 1950s. In the Beech Starship, only the canard foreplanes have variable sweep.
• Oblique Wing - a single full-span wing pivots about its midpoint, as used on the NASA AD-1 , so that one side sweeps back and the other side sweeps forward.
• Telescoping Wing - the outer section of wing telescopes over or within the inner section of wing, varying span, aspect ratio and wing area, as used on the FS-29 TF glider. The Makhonine Mak-123 was an early example.[22]
• Extending Wing or Expanding Wing - part of the wing retracts into the main aircraft structure to reduce drag and low-altitude buffet for high-speed flight, and is extended only for takeoff, low-speed cruise and landing. The Gérin Varivol biplane, which flew in 1936, extended the leading and trailing edges to increase wing area.[23]
• Bi-directional Wing - a proposed design in which a low-speed wing and a high-speed wing are laid across each other in the form of a cross. The aircraft would take off and land with the low-speed wing facing the airflow, then rotate a quarter-turn so that the high-speed wing faces the airflow for supersonic flight
Variable sweep(swing-wing)
Variable-geometryoblique wing
Telescoping wing Extending wing Bi-directional flying wing
Bell X-5 NASA AD-1 Makhonine Mak-123
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AERODYNAMICSWing ConfigurationsMinor Aerodynamic Surfaces
http://en.wikipedia.org/wiki/Wing_configuration
Aircraft may have additional minor aerodynamic surfaces. Some of these are treated as part of the overall wing configuration:
• Winglet - a small vertical fin at the wingtip, usually turned upwards. Reduces the size of vortices shed by the wingtip, and hence also tip drag.
• Strake - a small surface, typically longer than it is wide and mounted on the fuselage. Strakes may be located at various positions in order to improve aerodynamic behaviour. Leading edge root extensions (LERX) are also sometimes referred to as wing strakes.
• Chine - long, narrow sideways extension to the fuselage, blending into the main wing. As well as improving low speed (high angle of attack) handling, provides extra lift at supersonic speeds for minimal increase in drag. Seen on the Lockheed SR-71 Blackbird.
• Moustache - small high-aspect-ratio canard surface having no movable control surface. Typically is retractable for high speed flight. Deflects air downward onto the wing root, to delay the stall. Seen on the Dassault Milan and Tupolev Tu-144
Moustache, chines, wingletsand nose and ventral strakes
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AERODYNAMICSWing Configurations
Minor Surface Features
http://en.wikipedia.org/wiki/Wing_configuration
Additional minor features may be applied to an existing aerodynamic surface such as the main wing:
• High-lift devices - some of these are visible aerodynamic features: • Slot - a spanwise gap behind the leading edge section, which forms
a small aerofoil or slat extending along the leading edge of the wing. Air flowing through the slot is deflected by the slat to flow over the wing, allowing the aircraft to fly at lower air speeds. Leading edge slats are moveable extensions which open and close the slot.
• Flap - trailing-edge (or leading-edge) wing section which may be angled downwards for low-speed flight, especially when landing. Some types also extend backwards to increase wing area.
• Wing spanwise flow control devices : • Vortex generator - small triangular protrusion on the upper leading wing surface; usually,
several are spaced along the span of the wing. The vortices re-energise the boundary layer and thereby both reduce the stall speed and improve the effectiveness of control surfaces at low speeds.
• Wing fence - a flat plate extending along the wing chord and for a short distance vertically. Used to control spanwise airflow over the wing.
• Vortilon - a flat plate attached to the underside of the wing near its leading edge, roughly parallel to normal airflow, used to increase lift and reduce stalling at low speeds.
• Notched leading edge.[27]
• Dogtooth leading edge
Vortex generators, root fillet, flap,anti-shock body and wing fence
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AERODYNAMICSWing Configurations
Minor Surface Features
http://en.wikipedia.org/wiki/Wing_configuration
Vortex generators, root fillet, flap,anti-shock body and wing fence
• Leading edge extensions of various kinds.• Anti-shock body - a streamlined "pod" shaped body added to the
leading or trailing edge of an aerodynamic surface, to delay the onset of shock stall and reduce transonic wave drag. Examples include the Küchemann carrots on the wing trailing edge of the Handley Page Victor B.2, and the tail fairing on the Hawker Sea Hawk.
• Fillet - a small curved infill at the junction of two surfaces, such as a wing and fuselage, blending them smoothly together to reduce drag.
• Fairings of various kinds, such as blisters, pylons and wingtip pods, containing equipment which cannot fit inside the wing, and whose only aerodynamic purpose is to reduce the drag created by the equipment
Handley Page Victor B.2 Hawker Sea Hawk
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Wing Parameters
Airfoil: The cross-sectional shape obtained by the intersection of the wing with the perpendicular plane
1. Wing Area, S, is the plan surface of the wing.
2. Wing Span, b, is measured tip to tip.
3. Wing average chord, c, is the geometric average. The product of the span andthe average chord is the wing area (b x c = S).
4. Aspect Ratio, AR, is defined as:
( )∫−
=2/
2/
b
b
dyycS
( )b
Sdyyc
bc
b
b
== ∫−
2/
2/
1
S
bAR
2
=
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294
Wing Parameters (Continue)
5. The root chord, , is the chord at the wing centerline, and the tip chord, is the chord at the tip.
6. Taper ratio,
7. Sweep Angle, is the angle between the line of 25 percent chord and the perpendicularto root chord.
8. Mean aerodynamic chord,
rc
Λ
r
t
c
c=λ
tc
λ
( )[ ]∫−
=2/
2/
21~b
b
dyycS
c
c~
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Wing Parameters (Continue)
AERODYNAMICS
Illustration of Wing Geometry
Planform, xy plane
Dihedral (V form), yz plane
Profile, twist xz plane
Geometric Designation of Wings of various planform
Swept-backWing
DeltaWing
EllipticWing
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Wing Design Parameters
•Planform - Aspect Ratio - Sweep - Taper - Shape at Tip - Shape at Root•Chord Section - Airfoils - Twist•Movable Surfaces - Leading and Trailing-Edge Devices - Ailerons - Spoilers•Interfaces - Fuselage - Powerplants - Dihedral Angle
AERODYNAMICSSOLO
297Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 5
AERODYNAMICSSOLO
298Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 5
Airfoil Effects
•Camber increases Zero-α Lift Coefficient•Thickness - Increases α for stall and the stall break - Reduces Subsonic Drag - Increases Transonic Drag - Causes abrupt Pitching Moment variation•Profile Design - Can reduce C.P. (Static Margin) variation with α - Affects Leading-Edge and Trailing-Edge Flow Separation
AERODYNAMICSSOLO
299Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 5
AERODYNAMICSSOLO
300Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 5
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301Stengel, Aircraft Flight Dynamics, Princeton, MAE 331, Lecture 5
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At High Angles of Attack
-Flow Separates
-Wing loses Lift
Flow SeparationProduces Stall
AERODYNAMICS
Stall is a reduction in the lift coefficient generated by a foil as angle of attack increases. This occurs when the critical angle of attack of the foil is exceeded. The critical angle of attack is typically about 15 degrees, but it may vary significantly depending on the fluid, foil, and Reynolds number.
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Continue to Aerodynamics – Part II
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311
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
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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)
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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)
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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 9, 2015 315
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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
316
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)
317
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)
318
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)
319
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)
320
321
SOLO Complex VariablesConformal Mapping
Transformations or Mappings
x
y
u
v
r
xd
yd
r
ud
vdA B
CD
'A
'B
'C'DThe set of equations ( )
( )
==
yxvv
yxuu
,
,
define a general transformation or mapping between (x,y) plane to (u,v) plane.
If for each point in (x,y) plane there corresponds one and only one point in (u,v)plane, we say that the transformation is one to one.
vdv
rud
u
rvdy
v
yx
v
xudy
u
yx
u
x
yvdv
yud
u
yxvd
v
xud
u
xyydxxdrd
u
r
u
r
∂∂+
∂∂=
∂∂+
∂∂+
∂∂+
∂∂=
∂∂+
∂∂+
∂∂+
∂∂=+=
∂∂
∂∂
1111
1111
If is a vector that defines a point A in (x,y) plane, we have: ( ) ( )vuryxr ,,
=r
The area dx dy of a region A,B,C,D, in (x,y) plane is mapped in the area A’,B’,C’,D’, du dv in the (u,v) plane. We have
zvdudu
y
v
x
v
y
u
xvdudy
v
yx
v
xy
u
yx
u
x
vdudv
r
u
rzydxdydxd
y
r
x
rSd
yx
11111
1
11
∂∂
∂∂−
∂∂
∂∂=
∂∂+
∂∂×
∂∂+
∂∂=
∂∂×
∂∂==
∂∂×
∂∂=
If x and y are differentiable
322
SOLO Complex VariablesConformal Mapping
Transformations or Mappings( )( )
==
yxvv
yxuu
,
,
The transformation is one to one if and only if, for distinct points A, B, C, D, in (x,y)we obtain distinct points A’,B’,C’,D’, in (u,v). For this a necessary (but not sufficient)condition:
''''det1det
11
DCBA
ABCD
Sd
v
y
u
y
v
x
u
x
zvdud
v
y
u
y
v
x
u
x
zvdudu
y
v
x
v
y
u
xzydxdSd
∂∂
∂∂
∂∂
∂∂
=
∂∂
∂∂
∂∂
∂∂
=
∂∂
∂∂−
∂∂
∂∂==
Transformation is one to one 00 '''' ≠⇔≠ DCBAABCD SdSd
( )( ) 0det:
,
, ≠
∂∂
∂∂
∂∂
∂∂
=∂∂
v
y
u
y
v
x
u
x
vu
yxJacobian of theTransformation
By symmetry (change x,y to u,v) we obtain:
ABCDDCBA Sd
y
v
x
v
y
u
x
u
Sd
∂∂
∂∂
∂∂
∂∂
=det''''
1detdet =
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
v
y
u
y
v
x
u
x
y
v
x
v
y
u
x
u
one to one
transformation ( )( )
( )( ) 1
,
,
,
, =∂∂
∂∂
vu
yx
yx
vu
x
y
u
v
r
xd
yd
r
ud
vdA B
CD
'A
'B
'C'D
323
SOLO Complex VariablesConformal Mapping
Complex Mapping
In the case that the mapping is done by a complex function, i.e.
( ) ( )yixfzfviuw +==+=
we say that f is a complex mapping.If f (z) is analytic, then according to Cauchy-Riemann equation:
( )( )
( ) 2222
det,
,
zd
zfd
y
ui
x
u
y
u
x
u
x
v
y
u
y
v
x
u
y
v
x
v
y
u
x
u
yx
vu =∂∂+
∂∂=
∂∂+
∂∂=
∂∂
∂∂−
∂∂
∂∂=
∂∂
∂∂
∂∂
∂∂
=∂∂
x
v
y
u
y
v
x
u
∂∂−=
∂∂
∂∂=
∂∂
&
If follows that a complex mapping f (z) is one to one in regions where df/dz ≠ 0.
Points where df/dz = 0 are called critical points.
324
SOLO Complex VariablesConformal Mapping
Complex Mapping – Riemann’s Mapping Theorem
In the case that the mapping is done by a complex function, i.e.( ) ( )yixfzfviuw +==+=
Georg Friedrich BernhardRiemann1826 - 1866
we have:
x
y
u
vC 'C
1
RR' Let C be the boundary of a region R in the z plane,
and C’ a unit circle, centered at the origin of thew plane, enclosing a region R’.
The Riemann Mapping Theorem states that for each pointin R , there exists a function w = f (z) that performs aone to one transformation to each point in R’.
Riemann’s Mapping Theorem demonstrates the existence of theone to one transformation to region R onto R’, but it not providesthis transformation.
325
SOLO Complex VariablesConformal Mapping
Complex Mapping (continue – 1)( )( )
==
yxvv
yxuu
,
,
x
y
u
v
r
2zd
1zd
r
2wd
1wdA
B
C
'A
'B
'C
( ) ( )yixfzfviuw +==+=
Consider a point A in (x,y) plane mapped to pointA’ in (u,v) plane
Consider a small displacement from A to Bdefined as dz1, that is mapped to a displacementfrom A’ to B’ defined as dw1
( ) ( ) ( )
+
===1
1
argarg
11
arg
11
zdzd
zfdi
AA
wdi Aezdzd
zfdzd
zd
zfdewdwd
Consider also a small displacement from A to C defined as dz2, that is mapped to a displacement from A’ to C’ defined as dw2
( ) ( ) ( )
+
===2
2
argarg
22
arg
22
zdzd
zfdi
AA
wdi Aezdzd
zfdzd
zd
zfdewdwd
We can see that dw ≠ 0 if dz ≠ 0, i.e. a one-to-one transformation, if and only if
( )0≠
Azd
zfd
326
SOLO Complex VariablesConformal Mapping
Complex Mapping (continue – 2)( )( )
==
yxvv
yxuu
,
,
x
y
u
v
r
2zd
1zd
r
2wd
1wdA
B
C
'A
'B
'C
( ) ( )yixfzfviuw +==+=
Consider a point A in (x,y) plane mapped to pointA’ in (u,v) plane
( ) ( ) ( )
+
===1
1
argarg
11
arg
11
zdzd
zfdi
AA
wdi Aezdzd
zfdzd
zd
zfdewdwd
( ) ( ) ( )
+
===2
2
argarg
22
arg
22
zdzd
zfdi
AA
wdi Aezdzd
zfdzd
zd
zfdewdwd
We can see that:
( ) ( ) ( ) ( )
12
1212
argarg
argargargargargarg
zdzd
zdzd
zfdzd
zd
zfdwdwd
AA
−=
+−
+=−
Consider two small displacements from A to BAnd from A to C, defined as dz1 and dz2, that are mapped to displacements from A’ to B’ and from A’ to C’, defined as dw1 and dw2
Therefore the angular magnitude and sense between dz1 to dz2 is equal to that between dw1 to dw2. Because of this the transformation or mapping is called aConformal Mapping.Return to Joukovsky Airfoils
327
SOLO
Glauert Integral Formula (1926) Proof
θθπθ
θθθπ
sin
sin
coscos
cos
0
11
1 nd
n =−∫
Consider the Integral
∫ −=
π
θθθθ
θ
0
11
1 sincoscos
cos: d
nI
( ) ( )
( )
( )
( )
( ) ( ) ( ) ( ) ( )11111
1
1
1
111
21
cos21
sin21
sin21
cos
1
21
sin2
21
cos
21
sin2
21
cos
21
sin21
sin2
1
coscos
1
θθθθθθθθθθ
θθ
θθ
θθ
θθθθθθ
−++−+
−
−+
+
+=
−+=
−
But( ) ( ) ( )
( ) ( ) ( )111
111
sinsin2
1
2
1cos
2
1sin
sinsin2
1
2
1sin
2
1cos
θθθθθθ
θθθθθθ
+=−+
−=−+
Therefore( )
( )
( )
( )
−
−+
+
+=
−1
1
1
1
1
21
sin
21
cos
21
sin
21
cos
2
1sin
coscos
1
θθ
θθ
θθ
θθθ
θθ
Hermann Glauert(1892-1934)
328
SOLO
Glauert Integral Formula (1926) Proof (continue – 1)
( )
( )
( )
( )
( )
( )∫ ∫∫
− +
+=
−
−+
+
+=
−=
π π
π
π
θθθθ
θθθθ
θθ
θθ
θθ
θθθθ
θθθ
0
11
1
1
11
1
1
1
1
0
11
1 cos
21
sin
21
cos
2
1cos
21
sin
21
cos
21
sin
21
cos
2
1sin
coscos
cos: dndnd
nI
Change variables
Define
11 θθθ dxdx =⇒+=
( ) ∫∫∫+
−
+
−
+
−
+=−=πθ
πθ
πθ
πθ
πθ
πθ
θθθ xdx
xnxn
xdx
xnxn
xdnnxx
x
I
2sin
2cossin
2
sin
2sin
2coscos
2
coscos
2sin
2cos
2
1
∫+
−
=πθ
πθ
xdx
xnx
Yn
2sin
2coscos
: ∫+
−
=πθ
πθ
xdx
xnx
Zn
2sin
2cossin
:
Compute
( )[ ]( ) 01sinsin
2cos
2sin2
2sin
2cos1coscos
00
1 =−+=
−=
−−=− ∫∫∫∫
+
−
+
−
+
−
+
−−
πθ
πθ
πθ
πθ
πθ
πθ
πθ
πθ
xdxnxdxnxdxx
xnxdx
xxnxn
YY nn
( )[ ]( ) 01coscos
2cos
2
1cos2
2sin
2cos1sinsin
00
1 =−+=
−=
−−=− ∫∫∫∫
+
−
+
−
+
−
+
−−
πθ
πθ
πθ
πθ
πθ
πθ
πθ
πθ
xdxnxdxnxdx
xnxdx
xxnxn
ZZ nn
329
SOLO
Glauert Integral Formula (1926) Proof (continue – 2)
nn Zn
Yn
dn
I2
sin
2
cos
coscos
cossin:
0
11
1 θθθθθ
θθπ
+=−
= ∫
Therefore
02
sin
2sin
2sin1
2sin
2coscos 2
11 =
−
===== ∫∫+
−
+
−−
πθ
πθ
πθ
πθ
xd
x
x
xdx
xx
YYY nn
( ) ππθ
πθ
πθ
πθ
πθ
πθ
2cos12
cos2
2sin
2cossin
211 =+====== ∫ ∫∫
+
−
+
−
+
−− xdxxd
xxd
x
xx
ZZZ nn
and
θπθθθ
θθπ
ndn
I sincoscos
cossin:
0
11
1 =−
= ∫
θθπθ
θθθπ
sin
sin
coscos
cos
0
11
1 nd
n =−∫
q.e.d.
330Ray Whitford, “Design for Air Combat”
331Ray Whitford, “Design for Air Combat”
Flap configurations and (graphs) effect on section lift and drag characteristics of a 25%-chord flap of each type deflected 30°.
332Ray Whitford, “Design for Air Combat”
Fig 75 Northrop F-5E wing flaps
Fig 76 Lift and drag benefits at various flop settings. The angles given ore for leading and trailing-edge flaps respectively.6
Fig 77 F-18 trimmed drag-due-to-lift. The curve indicated by circles is for leading and trailing-edge flap angles of 0° and 0° respectively; triangles ore for 5°/8°; and squares far 10°/12°.25
333Ray Whitford, “Design for Air Combat”
Fig 169 Supersonic area-ruling.
334
Aircraft designed to fly at greater angles of attack use a delta wing: a wing shaped like a triangle when viewed from above. Many delta wings additionally feature tapered leading edges, which are used to combat the increased drag that occurs as the angle of attack increases (during a dog fight, for example). These tapered leading edges affect the airflow over the wing in a way that decreases drag and increases lift.
http://www.concept2.com/oars/how-made-and-tested/vortex-edge
Vortex lift is produced by two vortices which separate along the entire length of the side edges and roll up rapidly into two nearly conical, spiral shaped coils above the leeward surface.Such fast spiraling vortices induce large suction (low pressure) over the leeward surface of the foil generating extra lift
http://seagatesail.com/technology/delta-wing-sail/
335http://defence.pk/threads/design-characteristics-of-canard-non-canard-fighters.178592/
336http://defence.pk/threads/design-characteristics-of-canard-non-canard-fighters.178592/
337http://defence.pk/threads/design-characteristics-of-canard-non-canard-fighters.178592/