1
Two-Dimensional Inviscid Incompressible
Fluid Flow
SOLO HERMELIN
Updated: 2.03.07 10.05.13
2
2-D Inviscid Incompressible Flow
Laplace’s Homogeneous Differential Equation
SOLO
TABLE OF CONTENT
3
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 ,,,,,,,,
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 ,,,,,,,,,,,
Path Line: The curve described in space by a moving fluid element is known as its trajectory or path line.
tt
tt
t
tt tt 2
t
tt tt 2
Path Line (steady flow)
t
tt
tt 2
tt
Path Line (unsteady flow)
tt 2
tt
4
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
0urd
0
1
1
1111
zdyudxv
ydxwdzu
xdzvdyw
wvu
dzdydx
zyx
w
zd
v
yd
u
xd
Cartesian
t
u
r
rd
5
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
022
11
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
u
0tr
rd
0t
11 cr
22 cr
Streamline Those are two surfaces, the
intersection of which is the streamline.
6
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
u
0tr
rd
0t
11 cr
22 cr
Streamline 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
where μ is a factor that must satisfy the following constraint.
0,, 0201 trtrVr
7
FLUID DYNAMICS
2 .BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
-Fluid mean velocity u r t, sec/m
-Body Forces Acceleration-) gravitation, electromagnetic(..,
G
-Surface Stress 2/ mNT
nnpnT ˆ~ˆˆ~
mV(t)
G
q
T n ~
d E
d t
Q
t
uu
d s n ds -Internal Energy of Fluid molecules
) vibration, rotation, translation( per volume
e
3/ mJ
-Rate of Heat transferred to the Control Volume) chemical, external sources of heat( 3/ mW
Q
t
- Rate of Work change done on fluid by the surrounding (rotating shaft, others)) positive for a compressor, negative for a turbine(td
Ed
3/ mW
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(
q 2/ mW
-Pressure (force per unit surface) of the surrounding on the control surface 2/ mNp
-Shear stress tensor (force per unit surface) of the surrounding on the control surface 2/ mN~
-Stress tensor (force per unit surface) of the surrounding on the control surface 2/ mN
~
8
FLUID DYNAMICS
0,,,,,,
,,,,
vdut
vduut
uvdvdut
vdtD
Dvd
tD
Dvd
tD
D
tD
mD
OOOOOO
OOOO
0,,,,,,
OOOOOO uut
ut
~~2
1
,,,
,
2
,,
.).(
III
II
I
I
I
DM
I
pGG
uuut
uuu
t
u
tD
uD
)2.1 (CONSERVATION OF MASS (C.M.)
vF (t)
m
SF (t)
O
x
y
z
r u,O
tr ,
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 /
)2.2 (CONSERVATION OF LINEAR MOMENTUM (C.L.M.)
mv(t)
G
T n ~ds n ds
I
xy
z
Iu,
Iu,
9
FLUID DYNAMICS
mv(t)
Q
t
uq
u
S(t)
td
Wd
dsnsd ˆ
nT ˆ~
dm
G
q
t
QuG
uupue
tD
D
1~
2
1 2
SOLO
)2.3 (CONSERVATION OF ENERGY (C.E.) –THE FIRST LAW OF THERMODYNAMICS
CHANGE OF INTERNAL ENERGY + KINETIC ENERGY= CHANGE DUE TO HEAT + WORK + ENERGY SUPPLIED BY SUROUNDING
)2.4 (THE SECOND LAW OF THERMODYNAMICS AND ENTROPY PRODUCTION
Enthalpyp
eh :
d e q w T ds p dv pd
hdd
peddvpedsdTv
2
/1
For a Reversible Process
2
d
ppd
edhd
hsTp
Gp
pGuuu
t
uII
III
II
I
,,
,,,
,
2
,
~~
2
1
p
hsT
drpdp
drhdh
drsds
(C.L.M.)
GIBBS EQUATION:Josiah Willard Gibbs
1839-1903
10
FLUID DYNAMICS
2 .BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
)2.5.1.4 (ENTROPY AND VORTICITY
FROM (C.L.M.)
OR
GIBBS EQUATION: T d s d hd p
tld
pd
tdt
pldp
hd
tdt
hldh
sd
tdt
sldsT &
1
SINCE THIS IS TRUE FOR AND d l t
&
T s hp
Ts
t
h
t
p
t
&1
SOLO
hsTGp
Guuut
uII
III
II
I
,,
,,,
,
2
,
~~
2
1
p
hsT
dlpdp
dlhdh
dlsds
11
Luigi Crocco 1909-1986
FLUID DYNAMICS
2 .BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
)2.5.1.4 (ENTROPY AND VORTICITY (CONTINUE)
Define
LET TAKE THE CURL OF THIS EQUATION
Vorticityu
If , then from (C.L.M.) we get:
G
CRROCO’s EQUATION (1937)
~1
0
2
2
uhsTuu
t
SOLO
~
2
1 ,2
,,
I
II
I
uhsTut
u
hsTGuuut
uII
I
II
I
,,
,
,
2
,
~
2
1
From
“Eine neue Stromfunktion fur die Erforshung der Bewegung der Gase mit Rotation,”
Z. Agnew. Math. Mech. Vol. 17,1937, pp.1-7
12
FLUID DYNAMICS
2 .BASIC LAWS IN FLUID DYNAMICS (CONTINUE))2.5.1.4 (ENTROPY AND VORTICITY (CONTINUE)
u u u u u u
0
0
T s T s
~
0
1~1~1
Therefore
~1
sTuuu
t
SOLO
~1
sTuu
tD
D
or
13
FLUID DYNAMICS
2 .BASIC LAWS IN FLUID DYNAMICS (CONTINUE)
)2.5 (CONSTITUTIVE RELATIONS
)2.5.1 (NAVIER–STOKES EQUATIONS (CONTINUE)
)2.5.1.4 (ENTROPY AND VORTICITY (CONTINUE)
~1
sTuu
tD
D
FLUID WITHOUT VORTICITY WILL REMAIN FOREVER WITHOUTVORTICITY IN ABSENSE OF ENTROPY GRADIENTS OR VISCOUSFORCES
-FOR AN INVISCID FLUID 0 0~ ~
sTuutD
DINVISCID
0~~
-FOR AN HOMENTROPIC FLUID INITIALLY AT REST
s const everywhere i e ss
t. ; . . &
0 0
0 0
D
Dts
0 0 0 0 0~ ~, ,
SOLO
14
3-D Inviscid Incompressible Flow
Circulation
SOLO
Circulation Definition:
tV
tVV
S
Sn 1
V
tr
ttr
tC
ttC
C
rdV
:
Material Derivative of the Circulation
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:
C
rdtD
VD
tD
D
integral of an exact differential on a closed curve.
C – a closed curve
15
3-D Inviscid Incompressible FlowSOLO
tV
tVV
S
Sn 1
V
tr
ttr
tC
ttC
S
tC
rdV
:
Material Derivative of the Circulation (second derivation)
Subtract those equations:
tVrdSn t
1
ttC
rdVV
:
S
TheoremsStoke
CC
SnVrdVVrdVttt
1'
S is the surface bounded by the curves Ct and C t+Δ t
tVVrdtVrdVSnVS
t
S
t
S
1
td
d
ttd
rd
tV
ttD
D rdd
Computation of:
tC
rdt
V
t
Computation of:td
d
16
3-D Inviscid Incompressible FlowSOLO
tV
tVV
S
Sn 1
V
tr
ttr
tC
ttC
Material Derivative of the Circulation (second derivation)
tVVrdS
t
When Δ t → 0 the surface S shrinks to the curve C=Ct and the surface integral transforms to a curvilinear integral:
C
t
CC
t
C
t
C
t VVrdV
dVVrdV
rdVVrdtd
d
0
22
22
Computation of: (continue)td
d
Finally we obtain:
tt CC
t
C
rdtD
VDVVrdrd
t
V
td
d
ttD
D
17
3-D Inviscid Incompressible FlowSOLO
tV
tVV
S
Sn 1
V
tr
ttr
tC
ttC
Material Derivative of the Circulation
We obtained:
tC
rdtD
VD
tD
D
Use C.L.M.: hsTp
VVt
V
tD
VDII
I
G
II
II
,,
,
,,
~
0
,
,,
,
,
~~
tttt CC
I
I
C
I
C
I
I
I
hddrdp
sTrdhrdp
sTtD
D
to obtain:
tC
I
I
I
rdp
sTtD
D ~,
,or:
Kelvin’s Theorem (1869)
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~~
G
18
3-D Inviscid Incompressible Flow
Circulation
SOLO
Circulation Definition: C
rdV
:
C – a closed curve
Biot-Savart Formula
1820
Jean-Baptiste Biot1774 - 1862
VorticityV
Space
dVsr
A
4
1
lddSnsr
Ad
4
1
The contribution of a length dl of the Vortex Filament to isA
SS
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
A2 Poisson Equation Solution
Space
dvsr
rA
4
1
Félix Savart1791 - 1841
19
3-D Inviscid Incompressible Flow
Circulation
SOLO
Circulation Definition: C
rdV
:
C – a closed curve
Biot-Savart Formula (continue - 1)
1820
Jean-Baptiste Biot1774 - 1862
VorticityV
lddSnsr
Ad
4
1We found
SS
Stokes
C
SdnSdnVrdV
:
also we have dlld
ldsr
dSnlddSnsr
AdrV r
S
dlld
v
rr
1
4
1
4
1
34 sr
srldrV
Biot-Savart Formula
Félix Savart1791 - 1841
20
3-D Inviscid Incompressible Flow
Circulation
SOLO
Circulation Definition: C
rdV
:
C – a closed curve
Biot-Savart Formula (continue - 2)
1820
Jean-Baptiste Biot1774 - 1862
34 sr
srldrV
Biot-Savart Formula General 3D Vortex
Félix Savart1791 - 1841
21
3-D Inviscid Incompressible Flow
Circulation
SOLO
Circulation Definition: C
rdV
:
C – a closed curve
Biot-Savart Formula (continue - 3)
1820
Jean-Baptiste Biot1774 - 1862
Félix Savart1791 - 1841
34 sr
srldrV
Biot-Savart Formula General 3D Vortex
For a 2 D Vortex:
d
hsr
dl
sr
srld sinˆˆsin23
dh
dlhl2sin
cot
sin/hsr
ˆ
2sinˆ
4 0 hd
hV
Biot-Savart Formula General 2D Vortex
22
3-D Inviscid Incompressible Flow
Helmholtz Vortex Theorems
SOLO
Helmholtz (1858): “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. He 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).
23
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
v
r
1111 vruyvxuV r
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
0
111
0
111
rr vu
zr
zr
vu
zyx
zyx
V
v
u
v
u r
cossin
sincos
i
r eviuviu
i
r eviuviu
24
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
v
r
1111 vruyvxuV r
v
u
v
u r
cossin
sincos
i
r eviuviu
i
r eviuviu
Continuity: 00 uutD
D
rv
ruz
rr
r
xv
yuzy
yx
xzzu
r
111
11
111
11 22
Incompressible: 0tD
D
Irrotational:
rv
ru
yv
xu
u
r
12
0 u
rrv
rru
xyv
yxu
r
11
25
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
v
r
1111 vruyvxuV r
v
u
v
u r
cossin
sincos
i
r eviuviu
i
r eviuviu
00 222 uu
2-D Incompressible:
2-D Irrotational:
222
0
222
222
1110
110
zzz
zzuu
02
2
2
2
Complex Potential in 2-D Incompressible-Irrotational Flow:
yixz
yxiyxzw
,,:
zd
zwdx
ix
yyi
0x
0y
i
r
i
r eviueviuVviu
zd
wdviu
i
r ezd
wdviu
xyyx
Cauchy-Riemann Equations
We found:
26
2-D Inviscid Incompressible Flow
Examples:
SOLO
Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow
sincos 00 UiUV Uniform Stream:
xyUv
yxUu
sin
cos
0
0
yUxU
yUxU
cossin
sincos
00
00
zU
zUzUiw
0
00 sincos
0U
27
2-D Inviscid Incompressible Flow
Examples:
SOLO
Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow
rrv
rrr
mu r
10:
1
2:
x
ymm
yxm
rm
1
22
tan22
ln2
ln2
zm
rem
irm
iw i ln2
ln2
ln2
Definition:
Source , Sink : 0m 0m
Sink 0m
Source 0m
The equation of a streamline is: constm
2
28
2-D Inviscid Incompressible Flow
Examples:
SOLO
Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow
r
Kvvr
rzuvr
rVu rr
0010: 2
22
1
ln2
ln2
tan22
yxr
x
y
zi
rei
riiw i ln2
ln2
ln2
Definition:
Infinite Line Vortex :
rrrv
rru r
1
2:
10:
ddrrdr
rdrV
2111
2Circulation
streamlines:
/222
22ln2
eyx
yx
29
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
y
x
.lim0
constm
z
zm
z
zm
zm
zm
zw
/1
/1ln
2ln
2
ln2
ln2
.lim0
constm
when
zz
m
zO
z
m
zO
zz
m
z
zmzw
m
22
21ln2
11ln2/1
/1ln
2
2
2
2
2
30
2-D Inviscid Incompressible Flow
Examples:
SOLO
Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational Flow
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
Definition:
Doublet at the Origin with Axis Along x Axis (continue):
2
1
2
1
2 z
m
z
m
zd
d
zd
wdviuV
The equation of a streamline is: .22
constyx
y
22
2
22
yx
31
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 cosrU sinrUcosrU yixUzU
Source
ire
kz
kln
2ln
2 r
kln
2
2
k
Doubletier
B
z
B cos
r
B sinr
B
Vortex(with clockwise
Circulation)
ire
iz
iln
2ln
2
2
rln2
90◦ Corner Flow 22
22yix
Az
A yxA 22
2yx
A
32
SOLO 2-D Inviscid Incompressible Flow
Paul Richard Heinrich Blasius(1883 – 1970)
Blasius was a PhD Student of Prandtl at Götingen University
x
y
xy
sd
M
C
C
zdzd
wdzM
zdzd
wdiiYX
2
2
2
2
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
33
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
Re
1911Blasius Theorem
Proof of Blasius Theorem
Consider the Small Element δs on the Boundary C
sysx cos,sin
xpspY
ypspX
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
U∞ = Air Velocity far from Body
x
y
xy
sd
M
X
Y
34
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
Re
1911Blasius Theorem
Proof of Blasius Theorem (continue – 1)
C
ydixdUconstiYiX 2
2
1
but 00 CCC
ydixdconstydxd
yduivuxduivvdyixdviu
dyuixdvdyixdvu
dyvuidyixdvudyixdvudyixdU
22
22
2
2
2222
2222222
viuU and
x
y
xy
sd
M
X
Y
35
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
Re
1911Blasius Theorem
Proof of Blasius Theorem (continue – 2)
CC
zdzd
wdiydixdU
iYiX
22
22
zdzd
wddyixdviudyixdU
2
22
00 xdvyduviuydixdUsd
Since the Flow around C is on a Streamline defined by
therefore yduivuxduivv 22
yixz
yxiyxzw
,,:
and
xyv
yxu
,where
Completes the Proof for the Force
viuzd
wd
x
y
xy
sd
M
X
Y
36
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
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
since 2
2 UconstpBernoulli
and 0C
ydyxdxconst
hence xdyydxvuydyxdxvuzd
zd
wdz
222
2
Re
x
y
xy
sd
M
X
Y
37
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
Re
1911Blasius Theorem
Proof of Blasius Theorem (continue – 4)
CCC
zdzd
wdzydyxdxvuydyxdxpM
2
22
22
Re
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
x
y
xy
sd
M
X
Y
38
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
22
2
C
ii
C
zdz
i
z
eaUeU
izd
zd
wdiYiX
2
2
22
222
ii
C
i
eUiz
eUResiduezd
z
eUiiYiX
22
02
zenclosesCif
z
AResidueAizd
z
A
C
where we used:
X
YL
U
x
y
i
ii ez
i
ez
aUezUzw
ln2
2
39
SOLO 2-D Inviscid Incompressible FlowBlasius Theorem Example
Circular Cylinder with Circulation (continue – 1)
C
ii
C
zdz
i
z
eaUeUzzd
zd
wdzM
2
2
22
222
ReRe
Since the Closed Loop Integral is nonzero only for 1/z component, we have
0'10
012
zenclosendoesCornif
zenclosesCandnifz
AResidueAi
zdz
A
Cn
we used:
04
2224
2
2 2
222
2
222
aUizdzz
aUM
C
ReRe
ieUiYiX
UL
DUieYiXiLD i
0
:
X
YL
U
x
y
Zero Moment around the Origin.
40
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
22
2
aUi
a
ieeUeeUe
zd
Wdeviuviv iiiiii
r
2sin2
2
Stagnation Points are the Points on the Cylinder for which vθ = 0:
02
sin2
aUv
Uastagnation
4sin 1
41
2-D Inviscid Incompressible Flow
42
The Flow Pattern Around a Spinning Cylinderwith Different Circulations Γ Strengths
2-D Inviscid Incompressible Flow
43
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
21
U
aU
U
vv
U
ppC rSurface
Surfacep
Using Bernoulli’s Law:
22
2
1
2
1 UpUp SurfaceSurface
UaUaC
Surfacep
4sin8
44sin41
2
2
44
2-D Inviscid Incompressible Flow
45
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
46
SOLO
Velocity Field
http://www.diam.unige.it/~irro/cilindro_e.html
University of Genua, Faculty of Engineering,
2-D Inviscid Incompressible Flow
47
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
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
48
SOLO 2-D Inviscid Incompressible Flow
49
50
51
52
53
54
55
56
57
58
59
60
61
Flow over a Slender Body of Revolution Modeled by Source Distribution
62
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)
63
Effect of Circulation on the Flow around an Airfoil at an Angle of Attack
64
Kutta-Joukovsky Theorem
Martin Wilhelm Kutta (1867 – 1944)
Nikolay Yegorovich Joukovsky (1847-1921
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:
cos: dlVldV
Circulation
2-D Inviscid Incompressible Flow
UL Kutta–Joukowsky Theorem:
LCUL 2
2
1 Lift:
Kutta in 1902 and Joukowsky in 1906, independently, arrived to this result.
65
SOLO 2-D Inviscid Incompressible Flow
General Proof of Kutta-Joukovsky TheoremUsing the Corollary to Blasius Theorem
Suppose that we with 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
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
LiftiDragUiUi
ii
z
UiResidue
i
zdz
Uiizd
z
i
z
aUU
izd
zd
wdizd
zd
wdiiYX
CCCC
22
1
2
1
2
1
2222 ''
2
2
2
'
22
Therefore 0& DragULLift q.e.d.
02
zenclosesCif
z
AResidueAizd
z
A
C
where we used:
C
'C
UL
D
66
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
2-D Inviscid Incompressible Flow
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.
xcyc
U
R
x
y
0,a
The transform isz
az
2
sincosˆ RiRacicc yx For α=0 we have
czi
cz
RczUzw ˆln
2ˆˆ
2
For any α we have
cezi
cez
RcezUzw i
ii ˆln
2ˆˆ
2
67
Kutta-Joukovsky
Nikolay Yegorovich Joukovsky (1847-1921
2-D Inviscid Incompressible Flow
cezi
cez
RcezUzw i
ii ˆln
2ˆˆ
2
viucez
i
cez
RUe
zd
wdii
i
ˆ1
2ˆ1 2
2
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
sin:,cos:1
21
ˆ1
2ˆ10
2
2
2
2
222
22222222222
22
2
BA
BAAURBAiBABBARBAU
e i
68
2-D Inviscid Incompressible Flow
we have
222
22222222222
22
20
BA
BAAURBAiBABBARBAU
ezd
wd i
az
sinsinsin:,coscoscos: RacaBRaacaA yx
222
2222
coscos2cos12
sinsincoscos
RRaRa
RaaRaBA
2
20 222 BAAURBA sinsin444
22
2
RaUUBUBBA
R
0
22
22222222222
2222222222222222
RBABAUBARBAU
URBBARBAUBABBARBAU
Let check
For this value of Γ, we have
This value of Γ satisfies the Kutta Condition
0az
zd
wd
Joukovsky Airfoils
69
Joukovsky Airfoils Design
1. Move the Circle to ĉ and choose Radius R so that the Circle passes through z = a.
Nikolay Yegorovich Joukovsky (1847-1921
2-D Inviscid Incompressible Flow
xcyc
U
R
x
y
0,a
for Center at z = 0. zi
z
RzUzW ln
2
2
2. Change z-ĉ → z
czi
cz
RczUzW ˆln
2ˆˆ
2
3. Change z → z e-iα
cezi
cez
RcezUzW i
ii ˆln
2ˆˆ
2
4. Compute Γ from Kutta Condition
azazd
Wd
d
Wd
2
0
sin4ˆ
RUac
70
Joukovsky Airfoils Design (continue – 1)2-D Inviscid Incompressible Flow
5. Use the Transformation and computez
az
2
22 /1
//
za
zdWd
zd
d
zd
Wd
d
Wd
6. To Compute Lift use either:
sin4 2RUUL6.1 Kutta-Joukovsky
6.2 Blasius
d
d
WdieFiFeLi i
yxi
2
2''
6.3 Bernoulli
2
2/1
2/
U
zdWd
U
ppC p
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
sin2sin82/ 42
cR
acL c
R
Uc
LC
'yF
'xF 'xF
U 'x
L
plane
'y
71
Joukovsky Airfoils Design (continue – 2)2-D Inviscid Incompressible Flow
7. To compute Pitching Moment about Origin use either:
7.2 Blasius
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
'yF
'xF 'xF
U 'x
L
plane
'y
0pM
2sin4
222
0aUM p
22
20
a
R
a
L
Mx p
p
sin4 2RUL
72
Joukovsky Airfoils Design (continue – 3)2-D Inviscid Incompressible Flow
8. To Pitching Moment about Any Point x0 is given by:
Lmpp C
c
xCcULxMM
x
0220 000 2
'yF
'xF 'xF
U 'x
L
plane
'y
0pM
0x
2sin4 22
0aCc m
sin2LC
a
xaU
c
x
c
acUM
ac
px
0221
4
02
222
882
sin22sin420
a
x
a
xaUM
acpx
00221
418
20
73
2-D Inviscid Incompressible Flow
Theodersen Airfoil Design Method
Theodore Theodersen 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 1
,sinsinh2
,coscosh2
a
a 3
2
,/sinh2
,/sin2
222
222
app
app 5
4
Theodore Theodorsen (1897 – 1978)
planez''y
'x
ea
x
yplanez
0
0aeR
plane
a2a2
Given ξ, η find ψ, θ using
22
221:
aap
where
T. Theodersen, “Theory of Wing Sections with Arbitrary Shapes”, NACA Rept. 411, 1931 T. Theodersen, I.E. Garrick, “General Potential Theory of Arbitrary Wing Sections”, NACA Rept. 452, 1933
74
2-D Inviscid Incompressible Flow
Theodore Theodorsen (1897 – 1978)
plane
a2a2
planez''y
'x
ea
Theodersen Airfoil Design Method (continue – 1)
3. Transform from z’ =a eψ eiθ to z = (a eψ0) eiф) through
10 expexp'
nn
nn
z
BiAzizz
Equaling Real and Imaginary Parts:
1 00
sincosn
nn
nn n
R
An
R
B 8
where An, Bn can be found by the following Iterative Procedure:
2
0
0
2
00
2
00
2
1
sin1
cos1
d
dnR
B
dnR
A
nn
nn 9
10
11
x
yplanez
0
0aeR
Start with
7
1 00
0 sincosn
nn
nn n
R
Bn
R
A
75
2-D Inviscid Incompressible Flow
Theodore Theodorsen (1897 – 1978)
plane
a2a2
planez''y
'x
ea
x
yplanez
0
0aeR
Theodersen Airfoil Design Method (continue – 2)
4. Given Airfoil, Compute An, Bn, Cp, Γ
222
22200
2
2
/1sinsinh
/1sinsin1
12/
0
dd
edd
U
q
U
ppC
T
p
Procedure:
ii 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
4.3 Compute Pressure Distribution
4.1 Assume ε small and take . Compute An, Bn, ψ0 and using (9), (10), (11) and after that using (8).
0
where α0 is the Angle of Attack, and εT is the ε of the Trailing Edge
76
2-D Inviscid Incompressible Flow
Theodore Theodorsen (1897 – 1978)
plane
a2a2
planez''y
'x
ea
x
yplanez
0
0aeR
Theodersen Airfoil Design Method (continue – 3)
4. Given Airfoil, Compute An, Bn, Cp, Γ
TUea 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
UL
5. Given we can compute for the Airfoil 0,
5.1 From Compute An, Bn
1 00
sincosn
nn
nn n
R
An
R
B
5.2 Compute
5.4 Compute ξ and η using (2) and (3).
5.3 Compute
1 00
0 sincosn
nn
nn n
R
Bn
R
A
77
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.
AERODYNAMICS
78
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
zxx
MzxM
zxzUxUzx ,sincos, 0,2 zxBoundary Conditions: The Normal Velocity Component on the Airfoil Surface is Zero
n̂xd
zd
V
Velocity
zz
UxUzz
Uxx
U
zwUxuUzxV
Thin
ˆsinˆcos
sinˆcos,1
1:
Normal to Airfoil Surface zxd
zdxn
Surface
ˆˆˆ
0ˆ
SurfaceSurface
zU
xd
zdUnV
xd
zdU
zw
xd
zdU
zw
Lower
Upper
Lower
Upper
Upper
Upper
79
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 zx
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
..
80
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 xdx
1
111 2
,xx
xdxxxwd
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
c
xdx0
11
xSurface
cx
x xd
zdU
xx
xdxxxwd
1
111
0 2,
1
1
81
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities – Solution (continue – 2)
Skeleton (Camber Profile)
x
z
c0
1x
11 xdx
xSurface xd
zdU
xx
xdx
1
11
2
Perform a transformation of variables
cxxdcxdcx 111111111 &002/sin2/cos1
2/cos1 cx
x
xd
zdUd
0
11
11
coscos
sin
2
1
Solution for a Flat Plate dz/dx = 0
Ud0
11
11
coscos
sin
2
1
The Solution, that must also satisfy the Kutta Condition γ (π) = 0, is
1
11 sin
cos12
U
x
z
c0 1x
11 xdx
82
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 xdx
sin
sin
coscos
cos
0
11
1 nd
n
1
00
n
n
0
11
1
0
11
11
coscos
cos1
coscos
sin
2
1d
Ud
UdU
d0
11
1
0
11
11
coscos
cos1
coscos
sin
2
1
1
11 sin
cos12
U
83
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities – Solution (continue – 4)
Skeleton (Camber Profile)
x
z
c0
1x
11 xdx
Solution for a Given Camber Profile z = z (x)
x
xd
zdUd
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 sin
sin
cos12
nn
PlateFlat
nAAU
To find the parameters An let substitute γ (θ1) in the Integral above
xn
n xd
zdUd
nA
Ud
AU
1 0
11
11
0
11
10
coscos
sinsin
coscos
cos1
84
2-D Inviscid Incompressible Flow
Skeleton (Camber Profile)
x
z
c0
1x
11 xdx
Solution for a Given Camber Profile z = z (x)
xd
zdUnAUAU
nn
10 cos
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
Let compute
1 00
0
0
coscoscoscosn
n dmnAdmAdmxd
zd
nm
nmdmn
2/
0coscos
0
Profile Theory by the Method of Singularities – Solution (continue – 5)
85
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 6)
Skeleton (Camber Profile)
x
z
c0
1x
11 xdx
Solution for a Given Camber Profile z = z (x)
0
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
0
11cos2
dnxd
zdAn
86
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 7)
Skeleton (Camber Profile)
x
z
c0
1x
11 xdx
Solution for a Given Camber Profile z = z (x)
Lift Computation
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
87
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 8)
Skeleton (Camber Profile)
x
z
c0
1x
11 xdx
Solution for a Given Camber Profile z = z (x)
Lift Computation (continue) 1022
AAUc
10
2
22
AAcU
UL
102
2
21
: AAcU
LCL
0
111 cos2
dxd
zdA
2d
Cd L
The Angle of Attack α0 for which Lift is Zero is given by:
0
11
0
1010 cos22
220 dxd
zdd
xd
zdAA
0
110 cos12
dxd
zd
0
10
1d
xd
zdA
88
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 9)
Skeleton (Camber Profile)
x
z
c0
1x
11 xdx
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
Therefore 111 xVxVx LowererUpper
Also because the zero thickness of the Camber Surface 112 xVxVU LowererUpper
We have 12
12
12 xVxVUx LowererUpper
Use Bernoulli’s Equality 12
112
1 2
1
2
1xVxpxVxp LowererLowerUpperUpper
We get
112
12
11 2
1xUxVxVxpxp LowererUpperUpperLower
89
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 10)
Skeleton (Camber Profile)
x
z
c0
1x
11 xdx
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
We can recover the Lift Equation using
UdnAAcU
xdxpxpLdL
nn
x
c
UpperLower
c
0
111
11
10
2cos1
2
1
0
111
0
sinsinsin
cos12
11
10
2
22
AAcU
UL
90
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 11)
Skeleton (Camber Profile)
x
z
c0
1x
11 xdx
Solution for a Given Camber Profile z = z (x)
Pitching Moment
Let MLE be the Pitching Moment about the Leading Edge
1111111 xdxxUxdxpxpxMd UpperLowerLE
The Pitching Moment Coefficient: 2// 22cUMC LEmLE
0
1111
11
10
22
cos12
1
sin2
1cos1
2
1sin
sin
cos12
21
11
dccnAAUcU
UC
nn
x
mLE
0
11
111
1112
0 2sinsin2
1sinsincos1 dnAnAA
nn
nn
210210 224422
AAAAAA
102 AACL 21210 44
122
4AACAAAC LmLE
91
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 21210 44
122
4AACAAAC LmLE
2110
21
10
210
142
14
2/
2/
4
AAC
c
AA
AAc
AA
AAAc
C
Cc
L
Mx
L
L
mLECP
LE
LEML
cx
xM
For any point at a distance x from the Leading Edge we have
10210 2224
AAc
xAAAC
c
xCC Lmm LEx
For x = c/4 we have: 1244
14/
AACCC Lmm LEc
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.
92
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)
93
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 14)Teardrop (Symmetric Airfoil)
xtz xdxc0 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
11 xd
zdUx t
2xd
d
94
2-D Inviscid Incompressible Flow
Profile Theory by the Method of Singularities - Solution (continue – 15)Teardrop (Symmetric Airfoil) (continue – 1)
xtz xdxc0 P
Q
The Algebraic Sum of Sources and Sinks is Zero.
We have 020
0
11
cx
xt
c
xzUxdx
xd
zdUx t
2
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:
1
1 sin2
1
nn nBcfxf
where Bn is given by
0
1 sin4
dnfc
Bn
95
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
Perform coordinate transformation
c0 P
x1x
Pu 11 xdx
c
P xx
xdxdfd
Uxu
0 1
11
1
1111
111
1
1 cos2
1
nn dnBncd
d
fdxd
xd
xfd coscos
2 11 c
xx
0
11
11
coscos
cosd
nBnU
xu nn
Pto obtain
96
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 xdxWe get
Use Glauert Integral
1
2/cos1
sin
sin
n
nx
P
nBnUxu
0
1 sin4
dnfc
Bn
sin
sin
coscos
cos
0
11
1 nd
n
0
11
11
coscos
cosd
nBnU
xu nn
P
to obtain
97
Flow over a Slender Body of Revolution Modeled by Source Distribution
AERODYNAMICS
Profile Theory by the Method of Singularities
98
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
AERODYNAMICS
99
Airfoil DesignAERODYNAMICS
100
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
AERODYNAMICS
101
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
102
NACA Airfoils AERODYNAMICS
103
NACA Airfoils
Geometry of the most important NACA Profiles(a)Four-Digit Profiles(b)Five-Digit Profiles(c)6-Series Profiles
AERODYNAMICS
104
NACA Airfoils
12.04.002.0
2142
c
t
c
xh
c
h
NACA
Lower Surface
Upper Surface
AERODYNAMICS
105
Effects of the Reynolds Number (Viscosity)
cVRe :
Effects of the Reynolds Number on the Lift and Drag characteristics of NACA 4412
AERODYNAMICS
106
SOLO
References
2-D Inviscid Incompressible Flow
AA200A – “Applied Aerodynamics” Stanford University,which I attended in Winter 1984, given by Prof. Holt Ashley
Holt Ashley )1923 – 2006(
107
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”, Dover, 1984
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
L.J. Clancy“Aerodynamics”,
John Wiley & Sons, 1975
J.J. Bertin, M.L. Smith“Aerodynamics for Engineers”,
Prentice-Hall, 1979
SOLOReferences
2-D Inviscid Incompressible Flow
April 21, 2023 108
SOLO
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 –2013
Stanford University1983 – 1986 PhD AA
109
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.
110
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
111
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
112
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.
113
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 BernhardRiemann
1826 - 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.
114
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
115
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
116
SOLO
Glauert Integral Formula (1926) Proof
sin
sin
coscos
cos
0
11
1 nd
n
Consider the Integral
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
But
111
111
sinsin2
1
2
1cos
2
1sin
sinsin2
1
2
1sin
2
1cos
Therefore
1
1
1
1
1
21
sin
21
cos
21
sin
21
cos
2
1sin
coscos
1
Hermann Glauert(1892-1934)
117
SOLO
Glauert Integral Formula (1926) Proof (continue – 1)
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 dxdx
xdx
xnxn
xdx
xnxn
xdnnxx
x
I
2sin
2cossin
2
sin
2sin
2coscos
2
coscos
2sin
2cos
2
1
xdx
xnx
Yn
2sin
2coscos
:
xdx
xnx
Zn
2sin
2cossin
:
Compute
01sinsin
2cos
2sin2
2sin
2cos1coscos
00
1
xdxnxdxnxdxx
xnxdx
xxnxn
YY nn
01coscos
2cos
2
1cos2
2sin
2cos1sinsin
00
1
xdxnxdxnxdx
xnxdx
xxnxn
ZZ nn
118
SOLO
Glauert Integral Formula (1926) Proof (continue – 2)
nn Zn
Yn
dn
I2
sin
2
cos
coscos
cossin:
0
11
1
Therefore
02
sin
2sin
2sin1
2sin
2coscos 2
11
xd
x
x
xdx
xx
YYY nn
2cos12
cos2
2sin
2cossin
211
xdxxd
xxd
x
xx
ZZZ nn
and
ndn
I sincoscos
cossin:
0
11
1
sin
sin
coscos
cos
0
11
1 nd
n
q.e.d.
119
SOLO Linearized Flow Equations
Preasure Field
Stream Lines Streak Lines (α = 0º) Streak Lines (α = 15º)
Streak Lines (α = 30º) Forces in the Body
120
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
121
3-D Inviscid Incompressible Flow
Circulation
SOLO
Circulation Definition: C
rdV
:
C – a closed curve
Biot-Savart Formula (continue - 2)
1820
Jean-Baptiste Biot1774 - 1862
34 sr
srldrV
Biot-Savart Formula General 3D Vortex
Félix Savart1791 - 1841
Lifting-Line Theory