Two-Dimensional Inviscid Incompressible Fluid Flow

121
1 Two-Dimensional Inviscid Incompressible Fluid Flow SOLO HERMELIN Updated: 2.03.07 10.05.13

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Two-Dimensional Inviscid Incompressible Fluid Flow. SOLO HERMELIN. Updated: 2.03.07 10.05.13. SOLO. 2-D Inviscid Incompressible Flow. TABLE OF CONTENT. Laplace’s Homogeneous Differential Equation. SOLO. 3-D Flow. Flow Description. - PowerPoint PPT Presentation

Transcript of Two-Dimensional Inviscid Incompressible Fluid Flow

Page 1: Two-Dimensional  Inviscid Incompressible Fluid Flow

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Two-Dimensional Inviscid Incompressible

Fluid Flow

SOLO HERMELIN

Updated: 2.03.07 10.05.13

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2-D Inviscid Incompressible Flow

Laplace’s Homogeneous Differential Equation

SOLO

TABLE OF CONTENT

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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

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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

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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.

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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

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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

~

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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,

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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).

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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

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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

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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:

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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

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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

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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

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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

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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

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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

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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

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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

Page 34: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 35: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 36: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 37: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 38: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 39: Two-Dimensional  Inviscid Incompressible Fluid Flow

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.

Page 40: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 41: Two-Dimensional  Inviscid Incompressible Fluid Flow

41

2-D Inviscid Incompressible Flow

Page 42: Two-Dimensional  Inviscid Incompressible Fluid Flow

42

The Flow Pattern Around a Spinning Cylinderwith Different Circulations Γ Strengths

2-D Inviscid Incompressible Flow

Page 43: Two-Dimensional  Inviscid Incompressible Fluid 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

Page 44: Two-Dimensional  Inviscid Incompressible Fluid Flow

44

2-D Inviscid Incompressible Flow

Page 45: Two-Dimensional  Inviscid Incompressible Fluid 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

Page 46: Two-Dimensional  Inviscid Incompressible Fluid 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

Page 47: Two-Dimensional  Inviscid Incompressible Fluid 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

Page 48: Two-Dimensional  Inviscid Incompressible Fluid Flow

48

SOLO 2-D Inviscid Incompressible Flow

Page 49: Two-Dimensional  Inviscid Incompressible Fluid Flow

49

Page 50: Two-Dimensional  Inviscid Incompressible Fluid Flow

50

Page 51: Two-Dimensional  Inviscid Incompressible Fluid Flow

51

Page 52: Two-Dimensional  Inviscid Incompressible Fluid Flow

52

Page 53: Two-Dimensional  Inviscid Incompressible Fluid Flow

53

Page 54: Two-Dimensional  Inviscid Incompressible Fluid Flow

54

Page 55: Two-Dimensional  Inviscid Incompressible Fluid Flow

55

Page 56: Two-Dimensional  Inviscid Incompressible Fluid Flow

56

Page 57: Two-Dimensional  Inviscid Incompressible Fluid Flow

57

Page 58: Two-Dimensional  Inviscid Incompressible Fluid Flow

58

Page 59: Two-Dimensional  Inviscid Incompressible Fluid Flow

59

Page 60: Two-Dimensional  Inviscid Incompressible Fluid Flow

60

Page 61: Two-Dimensional  Inviscid Incompressible Fluid Flow

61

Flow over a Slender Body of Revolution Modeled by Source Distribution

Page 62: Two-Dimensional  Inviscid Incompressible Fluid Flow

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)

Page 63: Two-Dimensional  Inviscid Incompressible Fluid Flow

63

Effect of Circulation on the Flow around an Airfoil at an Angle of Attack

Page 64: Two-Dimensional  Inviscid Incompressible Fluid Flow

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.

Page 65: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 66: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 67: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 68: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 69: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 70: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 71: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 72: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 73: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 74: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 75: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 76: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 77: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 78: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

SurfaceSurface

zU

xd

zdUnV

xd

zdU

zw

xd

zdU

zw

Lower

Upper

Lower

Upper

Upper

Upper

Page 79: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

..

Page 80: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 81: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 82: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 83: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 84: Two-Dimensional  Inviscid Incompressible Fluid Flow

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)

Page 85: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 86: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 87: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 88: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 89: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 90: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 91: Two-Dimensional  Inviscid Incompressible Fluid Flow

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.

Page 92: Two-Dimensional  Inviscid Incompressible Fluid Flow

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)

Page 93: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 94: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 95: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 96: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 97: Two-Dimensional  Inviscid Incompressible Fluid Flow

97

Flow over a Slender Body of Revolution Modeled by Source Distribution

AERODYNAMICS

Profile Theory by the Method of Singularities

Page 98: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 99: Two-Dimensional  Inviscid Incompressible Fluid Flow

99

Airfoil DesignAERODYNAMICS

Page 100: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 101: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 102: Two-Dimensional  Inviscid Incompressible Fluid Flow

102

NACA Airfoils AERODYNAMICS

Page 103: Two-Dimensional  Inviscid Incompressible Fluid Flow

103

NACA Airfoils

Geometry of the most important NACA Profiles(a)Four-Digit Profiles(b)Five-Digit Profiles(c)6-Series Profiles

AERODYNAMICS

Page 104: Two-Dimensional  Inviscid Incompressible Fluid Flow

104

NACA Airfoils

12.04.002.0

2142

c

t

c

xh

c

h

NACA

Lower Surface

Upper Surface

AERODYNAMICS

Page 105: Two-Dimensional  Inviscid Incompressible Fluid Flow

105

Effects of the Reynolds Number (Viscosity)

cVRe :

Effects of the Reynolds Number on the Lift and Drag characteristics of NACA 4412

AERODYNAMICS

Page 106: Two-Dimensional  Inviscid Incompressible Fluid Flow

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(

Page 107: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 108: Two-Dimensional  Inviscid Incompressible Fluid 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

Page 109: Two-Dimensional  Inviscid Incompressible Fluid Flow

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.

Page 110: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 111: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 112: Two-Dimensional  Inviscid Incompressible Fluid Flow

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.

Page 113: Two-Dimensional  Inviscid Incompressible Fluid Flow

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.

Page 114: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 115: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 116: Two-Dimensional  Inviscid Incompressible Fluid Flow

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)

Page 117: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 118: Two-Dimensional  Inviscid Incompressible Fluid Flow

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.

Page 119: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 120: Two-Dimensional  Inviscid Incompressible Fluid Flow

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

Page 121: Two-Dimensional  Inviscid Incompressible Fluid Flow

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