fluid dynamics

1

Fluid Dynamics

SOLO HERMELIN

http://www.solohermelin.com

http://www.solohermelin.com/

SOLO FLUID DYNAMICS

Table of Content

Mathematical NotationsBasic Laws in Fluid Dynamics

1. Conservation of Mass (C.M.)

2. Conservation of Linear Momentum (C.L.M.)

3. Conservation of Moment-of- Momentum (C.M.M.)

4. Conservation of Energy (C.E.), The First Law of Thermodynamics

5. The Second Law of Thermodynamics and Entropy Production

6. Constitutive Relations for Gases

Newtonian Fluid Definitions – Navier–Stokes Equations

State Equation

Thermally Perfect Gas and Calorically Perfect Gas

Boundary Conditions

Dimensionless EquationsMach Number – Flow Regimes

Boundary Layer and Reynolds Number

SOLO FLUID DYNAMICS

Table of Content (continue – 1)Steady Quasi One-Dimensional Flow

Shock and Expansion Waves

Normal Shock Waves

Flow Description

Streamlines, Streaklines, and Pathlines

Circulation

Biot-Savart Formula

Helmholtz Vortex Theorems

2-D Inviscid Incompressible Flow

Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irrotational FlowBlasius TheoremKutta Condition

Kutta-Joukovsky Theorem

Joukovsky Airfoils

Shock Wave Definition

Oblique Shock Wave Prandtl-Meyer Expansion Waves

SOLO FLUID DYNAMICS

Table of Content (continue – 2)

Linearized Flow Equations

Cylindrical Coordinates

Small Perturbation Flow

Applications: Three Dimensional Flow (Thin Airfoil at Small Angles of Attack)

Applications: Two Dimensional Flow (Thin Airfoil at Small Angles of Attack)

Drag (d) and Lift (l) per Unit Span Computations for Subsonic Flow (M∞ < 1) Prandtl-Glauert Compressibility Correction

Computations for Supersonic Flow (M∞ >1) Ackeret Compressibility Correction

References

5

FLUID DYNAMICS

1. MATHEMATICAL NOTATIONS

VECTOR NOTATION CARTESIAN TENSOR NOTATION

1.1 VECTOR

1.2 SCALAR PRODUCT

1.3 VECTOR PRODUCT

u kk 1 2 3, , u u e u e u e 1 1 2 2 3 3

u v u v u v u v 1 1 2 2 3 3 u v kk k 1 2 3, ,

u v

u u

u u

u u

v

v

v

0

0

0

3 2

3 1

2 1

1

2

3

ji

permutjiodd

permutjieven

CevittaLevi

vu

ij

jiij

0

.,

.,1

SOLO

6

FLUID DYNAMICS

1. MATHEMATICAL NOTATIONS (CONTINUE)


1.5 ROTOR OF A VECTOR

1.4 DIVERGENCE OF A VECTOR

1.6 GRADIENT OF A SCALAR

u

u

x

u

x

u

x

1

1

2

2

3

3 i

i

x

u

uu

x

u

xe

u

x

u

xe

u

x

u

xe

3

2

2

31

1

3

3

12

1

2

2

13

u u

uu u

2

2

u

x

u

xi

k

k

i

i

kj

k

ii x

uu

x

uu

xe

xe

xe

x x x

11

22

313

1 2 3

xk

SOLO

7

FLUID DYNAMICS

1 .MATHEMATICAL NOTATIONS (CONTINUE)


1.7 GRADIENT OF A VECTOR

u u e u e u e1 1 2 2 3 3

u

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

1

1

1

2

1

3

2

1

2

2

2

3

3

1

3

2

3

3

u

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

Dik

1

2

1

1

1

1

1

2

2

1

1

3

3

1

2

1

1

2

2

2

2

2

2

3

3

1

3

1

1

3

3

2

2

3

3

3

3

3

ik

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

0

0

0

2

1

3

2

2

3

3

1

1

3

1

3

3

2

2

1

1

2

1

3

3

1

1

2

2

1

u

xi

k

u

x

u

x

u

x

u

x

u

xi

k

i

k

k

i

i

k

k

i

1

2

1

2

Du

x

u

xiki

k

k

i

1

2

iki

k

k

i

u

x

u

x

1

2

SOLO

8

FLUID DYNAMICS



1.8 GAUSS’ THEOREMS

d s

A

V

A analytic in V

A C C const vector .

S V

dvsdGAUSS

2 analytic in V S k

k

V

dvs

ds

SOLO

Johann Carl Friederich Gauss 1777-1855

S V

dvAsdAGAUSS

1

S k

k

kk

V

dvx

AdsA

9

FLUID DYNAMICS



1.8 GAUSS’ THEOREMS (CONTINUE)

S V

dvAsdAGAUSS

3

A A dv

,A

analytic in V

A dsA

xdv

Vk k

k

kS

V k

k

kk x

A

xA

B e e e 1 1 2 2 3 3

S V

dvABBAsdABGAUSS

4 B A ds AB

xB

A

xdv

Vi k k k

i

k

ik

kS

A analytic in V

S VdvAAsdGAUSS

5 ds A ds A

A

x

A

xdv

Vi j j i

j

i

i

jS

SOLO

10

FLUID DYNAMICS

d s

A

C

d rS



1.9 STOCKES’ THEOREM

A d r A d s

C S

A analytic on S

A d rA

x

A

xd si i

C

j

i

i

j

k

S

Gauss’ and Stokes’ Theorems are generalizations of theFundamental Theorem Of CALCULUS

A b A a

d A x

d xd x

a

b

( ) ( )

George Stokes 1819-1903

SOLO

11

FLUID DYNAMICS

1. MATHEMATICAL NOTATIONS


MATERIAL DERIVATIVES (M.D.)

1e

2e

3e

r

u

b

rd d F r tF

tdt dr F

,

d

dtF r t

F

t

dr

d tF

,

d

dtF r t

F

tb F

b

,

for any dr d F r t

F

tdt d r

F

xi ki

ki

k

,

d

d tF r t

F

t

d r

d t

F

xi ki k i

k

,

d

d tF r t

F

tb

F

xb

i ki

ki

k

,

vectoranybbtdrd

Joseph-Louis Lagrange

1736-1813 Leonhard Euler

1707-1783

SOLO

FIXED IN SPACE(CONSTANT VOLUME)

EULER

LAGRANGE

MOVING WITH THE FLUID(CONSTANT MASS)

1e

3e

2e

u

12

FLUID DYNAMICS



MATERIAL DERIVATIVES (CONTINUE)

Fut

FF

tD

DtrF

td

d

u

,

k

ik

iki

u x

Fu

t

FF

tD

DtrF

td

d

,velocityfluiduutd

rdIf

Material Derivatives = Derivative Along A Fluid Path (Streamline)

D

D tu

u

tu u

u

t

uu u

2

2

1e

2e

3e

r

u duu

dr

AccelerationOf The Fluid

k

ik

i

jj

ji

i

k

ik

ii

x

uu

x

uu

uxt

u

x

uu

t

uu

tD

D

2

2

1

SOLO

13

FLUID DYNAMICS



1.10 MATERIAL DERIVATIVES (CONTINUE)

d uu

tdt dr u

d uu

tdt d x

u

xii

ki

k

rdrdDtdt

u

xd

xd

xd

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

t

u

t

u

t

u

ud

ud

ud

ikik

3

2

1

3

3

2

3

1

3

3

2

2

2

1

2

3

1

2

1

1

1

3

2

1

3

2

1 d u

u

td t

u

x

u

xd x

u

x

u

xd x

ii

Translation

i

k

k

i

Dilation

k

i

k

k

i

Rotation

k

1

2

1

2

Dilationrduu

rdurdu

urdrdurdu

rdurdurdD

T

u

u

ik

2

12

1

2

12

1

2

12

1

ik dr u dr Rotation

1

2

SOLO

14

REYNOLDS’ TRANSPORT THEOREM

v (t)

S(t)

SflowV ,

sd

OSV ,

OSOflowSflow VVV ,,,

OSr ,

md OSV ,

OflowV ,

Or,

-any system of coordinatesOxyz

- any continuous and differentiable functions in

trtr OO ,,, ,,

tandrO,

trO ,,

- flow density at point

and time tOr,

SOLO

- mass flow through the element .mdsdV S , sd

- any control volume, changing shape, bounded by a closed surface S(t)v (t)

- flow velocity, relative to O, at point and time t trV OOflow ,,,

Or,

- position and velocity, relative to O, of an element of surface, part of the control surface S(t).

OSOS Vr ,, ,

- area of the opening i, in the control surface S(t).iopenS

- gradient operator in O frame.O,

- flow relative to the opening i, in the control surface S(t).OSiOflowSi VVV ,,,

- differential of any vector , in O frame.O

td

d

FLUID DYNAMICS

15

Start with LEIBNIZ THEOREM from CALCULUS:

ChangeBoundariesthetodueChange

tb

ta

tb

ta td

tadttaf

td

tbdttbfdx

t

txfdxtxf

td

dLEIBNITZ

)),(()),((

),(),(::

)(

)(

)(

)(

and generalized it for a 3 dimensional vector space on a volume v(t) bounded by thesurface S(t).

Using LEIBNIZ THEOREM followed by GAUSS THEOREM (GAUSS 4):

tv

OSOOOSGAUSS

OpotolativedsofMovement

thetodueChage

tSOS

tvO

LEIBNITZ

Otv

vdVVt

GAUSSsdVvd

tvd

td

d,,,,)4(

intRe

)(,

This is REYNOLDS’ TRANSPORT THEOREM

OSBORNEREYNOLDS

1842-1912

SOLO

GOTTFRIED WILHELMvon LEIBNIZ

1646-1716

REYNOLDS’ TRANSPORT THEOREM

v (t)

S(t)

SflowV ,

sd

OSV ,


OSr ,

md OSV ,

OflowV ,

Or,

FLUID DYNAMICS


16

FLUID DYNAMICS


1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE)


)(,,,,)4(

,)()()(

tvOSOOOS

OGAUSS

OStStv

O

LEIBNITZ

Otv

vdVVt

GAUSS

sdVvdt

vdtd

d

)(

,

,)4(

,)()()(

tv k

kOS

i

k

i

kOSi

GAUSS

kkOStS

itv

iLEIBNITZ

tvi

vdx

V

xV

t

GAUSS

sdVvdt

vdtd

d

SOLO

v (t)

S(t)

SflowV ,

sd

OSV ,


OSr ,

md OSV ,

OflowV ,

Or,

17

FLUID DYNAMICS




O

OOS td

RduV

,,

CASE 1 (CONTROL VOLUME vF ATTACHED TO THE FLUID)

kkOS uV ,

)(,,,)4(

,)()()(

tvOOO

OGAUSS

OtStv

OOtv

F

FFF

vduut

GAUSS

sduvdt

vdtd

d

)()4(

)()()(

tv k

kI

k

Ik

I

GAUSS

kKtS

Itv

I

tvI

F

FFF

vdx

u

xu

t

GAUSS

sduvdt

vdtd

d

SOLO

vF (t)

SF(t)

sd

OSV ,

0,,, OSOflowS VVV

OSR ,

OR,

md

OSV ,

OflowV ,

18

FLUID DYNAMICS




1&, kkOS uV1&, uV OS

CASE 2 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND )1

)(

,,)(

,)(

)(

tvOO

tSO

tv

F

FFF

vdusduvdtd

d

td

tvd )()()(

)(

tv k

kk

tSk

tv

F

FFF

dvx

udsudv

td

d

td

tvd

td

tvd

tvu F

Ftv

OOF

)(

)(

1lim

0)(,,

td

tvd

tvx

u F

Ftv

k

k

F

)(

)(

1lim

0)(

EULER 1755

SOLO

vF (t)

SF(t)

sd

OSV ,

0,,, OSOflowS VVV

OSR ,

OR,

md

OSV ,

OflowV ,

19

FLUID DYNAMICS




CASE 3 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND )

&, kkOS uV &, uV OS

or, since this is true for any attached volume vF(t)

)(,,

)(,

)( )(

)(0

tvOO

tSO

tv tv

F

FF F

vdut

sduvdt

vdtd

d

td

tmd

)(

)()( )(

)(0

tvk

k

tSkk

tv tv

F

FF F

vduxt

sduvdt

dvtd

d

td

tmd

Because the Control Volume vF is attached to the fluid and they are not sources or sinks, the mass is constant.

OOOOOO uut

ut ,,,,,,0

k

k

k

k

k x

u

xu

tu

xt

0

SOLO

vF (t)

SF(t)

sd

OSV ,

0,,, OSOflowS VVV

OSR ,

OR,

md

OSV ,

OflowV ,

20

FLUID DYNAMICS




CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,

OSV

Define

.... VC

OOVC

vdt

vdtd

d

.... VC

i

VCi vd

tvd

td

d

r t r t r t, , , i k k i kx t x t x t, , ,

)(,

)()(

tSOS

tvOO

tv

sdV

vdtt

vdtd

d

ktS

kOSi

tvi

i

tvi

sdV

vdtt

vdtd

d

FF

)(,

)()(

We have

but

OOOO ut

ut ,,,, 0

k

k

iik

k

uxt

uxt

0

CASE 5 r t r t r t, , ,

SOLO

21

FLUID DYNAMICS




We have

)(

,,)(

4

.

)(,

)(,,,,,,

)(,

)(,,

)(

tSOOS

tvO

MDG

DerMatGAUSS

tSOS

tvOOOOOO

O

tSOS

tvOO

OOtv

sduVvdtD

D

sdV

vduuut

sdV

vdut

vdtd

d

)(

,)(

4

.

)(,

)(

)(,

)()(

tSkkkOSi

tv

iMDG

DerMatGAUSS

tSkkOSi

tv k

ki

k

ik

k

ik

i

tSkkOSi

tv k

ki

i

tvi

sduVvdtD

D

sdV

vdx

u

xu

xu

t

sdV

vdx

u

tvd

td

d


SOLO

v (t)

S(t)

SflowV ,

sd

OSV ,


OSr ,

md OSV ,

OflowV ,

Or,

22

FLUID DYNAMICS




REYNOLDS 1

)(,,

)(

)(

tSOOS

tvO

Otv

sduVvdtD

D

vdtd

d

)(,

)(

)(

tSkkkOSi

tv

i

tvi

sduVvdtD

D

dvtd

d

REYNOLDS 2

)(

)(,,

)(

tvO

tSOSO

Otv

vdtD

D

sdVuvdtd

d

)(

)(,

)(

tv

i

tSkkOSki

tvi

vdtD

D

sdVuvdtd

d


SOLO

v (t)

S(t)

SflowV ,

sd

OSV ,


OSr ,

md OSV ,

OflowV ,

Or,

23

FLUID DYNAMICS




REYNOLDS 3

CASE 1 (CONTROL VOLUME ATTACHED TO THE FLUID vF(t) )

kkOS uV ,

)()( tv

OOtv FF

vdtD

Dvd

td

d

)()( tv

i

tvi

FF

vdtD

Dvd

td

d

SOLO

O

OOS td

RduV

,,

r t r t r t, , ,

vF (t)

SF(t)

sd

OSV ,

0,,, OSOflowS VVV

OSR ,

OR,

md

OSV ,

OflowV ,

CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,

OSV

REYNOLDS 4

..,

..

..

SCO

OVC

VCO

sduvdtd

d

vdtD

D

....

..

SCkki

VCi

VC

i

sduvdtd

d

vdtD

D

Return to Table of Content

24

BASIC LAWS IN FLUID DYNAMICS

THE FLUID DYNAMICS IS DESCRIBED BY THE FOLLOWING FOUR LAWS:

(1) CONSERVATION OF MASS (C.M.)

(2) CONSERVATION OF LINEAR MOMENTUM (C.L.M.)

(4) THE FIRST LAW OF THERMODYNAMICS

(5) THE SECOND LAW OF THERMODYNAMICS

SOLO

(3) CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.)

FLUID DYNAMICS


(6) CONSTITUTIVE RELATIONS

25

Casing

Control Volumes & Surfacesfor a Turbomachine

TurbomachineAxis

FixedControl Volume

C.V.

FixedControl Surface

C.S.

1V

2V

1r

2r

Inlet

Outlet

Rotor

1V

2V

1r

2r

Inlet

Outlet

Rotor r

1 2

x

y

z

r


)1 (CONSERVATION OF MASS (C.M.)

SOLO

The mass in the Fixed Control Volume (C.V.)is given by:

..VC

CV vdm

Since the mass entering the C.V. is equal to massexiting C.V., using Reynolds’ Transport Theoremwith η = 1, we have:

..

,

Re

..

0SC

md

S

ynolds

VCmd

CV sdVvdtd

d

td

md

Assume: - one inlet (1) of area A1 and mean fluid velocity V,S1 (relative to A1 )and density ρ1.

- one outlet (2) of area A2 and mean fluid velocity V,S2 (relative to A2 ) and density ρ2.

we have: 022,21,1,,.. 21

AVAVsdVsdVsdV SnSnA

SA

SSC

- mass flow rate entering the system through the element of C.S.

mdsdV S , sd

or: 21

22,211,1

flowflow Q

Sn

Q

Sn AVAV

where: - mass flow velocity exiting the system relative to the element of C.S. SSflow VV ,,

sd

FLUID DYNAMICS

26

SOLO

1 2 30 4 5 6

SUPERSONICCOMPRESSION

SUBSONICCOMPRESSION

COMBUSTIONFUEL

INJECTION EXPANSION

NOZZLECOMBUSTIONCHAMBER

DIFFUSER

FLAMEHOLDERS

EXHAUSTJET

0V

0A

fm

(1) CONSERVATION OF MASS (C.M.)

2221110000 AuAuAum

6665554443330 AuAuAuAumm f

DiffuserEnteringRateFlowMassAirm 0

RateFlowMassFuelm f 6,5,4,3,2,1,0,,,,,, 6543210 StationsatDensityGas

6,5,4,3,2,1,0,,,,,, 6543210 StationsatVelocityGasuuuuuuu

6,5,4,3,2,1,0,,,,,, 6543210 StationsatAreaAAAAAAA

BASIC LAWS IN FLUID DYNAMICSFLUID DYNAMICS


27

rjV

Pelton Water Wheel(Impulse Turbine)

j

rV j R

R

Control Volume

rV j

r

2V(Velocity of jet

leaving theControl Volume)

tR

r

j

2V



SOLO

..

:VC

CCV vdRRm

Using the Reynolds’ Transport Theorem we obtain

The Centroid of the mass enclosed by C.V. isCR

The Linear Moment of the mass enclosed by C.V. is defined as

....

,:VC

IVC

ICV vdtD

RDvdVP

..,

..,

..,

..,

....

SCS

m

SCSC

V

I

CCV

SCS

I

CCV

SCS

VC

REYNOLDS

VCI

CV

sdVRsdVRtd

RdmsdVRRm

td

d

sdVRvdRtd

dvd

tD

RDP

CV

C

..

,SC

SCCCVCV sdVRRVmP

or

The Linear Momentum, of the differential mass dm = ρdv is defined as

vdVmdVPd II ,,:

FLUID DYNAMICS

28

rjV


j

rV j R

R

Control Volume

rV j

r

2V(Velocity of jet


tR

r

j

2V


(2) CONSERVATION OF LINEAR MOMENTUM (C.L.M.) (CONTINUE - 1)

SOLO

Using Newton’s Second Law, for the mass element dm = ρdv, we obtain:

extfd

- Differential external forces acting on dm

ijfd int

- Differential internal forces acting on dm

I

I td

RdVV

,: - Velocity of the mass element dm relative to I.

- mass flow rate entering the system through the element of C.S.

mdsdV S , sd

vdtD

VDmd

tD

VDfdfd

II

ext int

VVt

V

tD

VDI

II

,

- Material derivative of the Velocity of the mass element dm relative to I.V

- Velocity of mass exiting the system, relative to the element of C.S.

SV,

sd

FLUID DYNAMICS

29

rjV


j

rV j R

R

Control Volume

rV j

r

2V(Velocity of jet


tR

r

j

2V


(2) CONSERVATION OF LINEAR MOMENTUM (C.L.M.) (CONTINUE - 2)

SOLO

Let integrate the equation

vdtD

VDmd

tD

VDfdfd

II

ext int

over the mass enclosed by C.V.

From the 3rd Newton’s Law the internal forces that particle j applies on particle i is of equal magnitude but of opposite direction to the force that particle i applies on particle j, therefore :

..

0

..int

.. VCI

VCVCext vd

tD

VDfdfd

Using Reynolds’ Transport Theorem we obtain

..

,..

,......

,SC

S

I

CV

SCS

IVC

REYNOLDS

VCI

VCextCVext sdVV

td

PdsdVVvdV

td

dvd

tD

VDfdF

FLUID DYNAMICS

30

extj

jSC

sdTsdVCSC

md

S

I

CV

SCmd

S

IVC

REYNOLDS

VCI

FFdstfnpvdgsdVVtd

PdsdVVvdV

td

dvd

tD

VD

�......

,..

,....

11


)2 (CONSERVATION OF LINEAR MOMENTUM (C.L.M.) (CONTINUE - 3)

SOLO

The external forces acting on the system are:

• Gravitation acceleration (E center of Earth).E

E

RR

MGg

3

dstfnpsdTsdnsd��

111

where:

ndsnnsdsd

111 - vector of surface differential 2/ mN p - pressure on (normal to) the surface .

jj

SCsdTsd

VCVCextext FdstfnpvdgfdF

�......

11

f - friction force per (parallel to) unit surface . 2/ mN• Discrete force exerting by the surrounding on the point , and discrete moments .

jjF

jR

kkM

nT�

1 - force per unit surface 2/ mN

Therefore:

• Surface forces acting on the system:

npp

1t

1

n

1V

ds

wx1

wy1

wz1

tf

1

Pressure force

Friction force

WS

FLUID DYNAMICS

31

FLUID DYNAMICS

BASIC LAWS IN FLUID DYNAMICS (CONTINUE)

(2.2) CONSERVATION OF LINEAR MOMENTUM (CONTINUE)

VECTOR NOTATION CARTESIAN TENSOR NOTATIONC.L.M.-2

Since this is true for all volumes vF(t) attached to the fluid we can drop the volume integral.

~~

~~

2

1

,,,

,

2

,

,

.).(

Ip

pGG

uuut

u

uut

u

tD

uD

III

II

I

I

I

DM

I

ikikik

i

ik

ii

i

iki

k

ik

i

jjjj

i

i

k

ik

iDM

i

p

xx

pG

xG

x

uu

x

uuuu

xt

u

x

uu

t

u

tD

uD

2

1

.).(

SOLO

Derivation From Integral Form (Continue)

)(,

)()()(

,

~

~

tvI

tStvtvI

I

F

FFF

vdG

sdvdGvdtD

uD

)(

)()()(

tv i

iki

tS

kik

tv

i

tv

i

F

FFF

vdx

G

sdvdGvdtD

uD

32

rjV


j

rV j R

R

Control Volume

rV j

r

2V(Velocity of jet


tR

r

j

2V


)2 (CONSERVATION OF LINEAR MOMENTUM (C.L.M.)

SOLO

Let compute the C.L.M. in the tangential to the wheel direction, for the Pelton Water Wheel

ttvtv

extjj RfdfdrVrQVQ

0

int

0

coswhere

outin AS

AS sdVsdVQ

,,:

cos1 rVQR jt

Therefore

The average Torque on the water wheel is cos1 rVrQrRTorque jt

The Power developed is cos1 rVrQrRTorquePower jt

The average Tangential Reaction Force on the bucket is

In steady-state the directions and magnitudes of flows are fixed, therefore

0..

I

VCI

CV vdVtd

d

td

Pd

extj

jSCVC

SCmd

S

I

CV

SCmd

S

IVC

FFsdvdg

sdVVtd

PdsdVVvdV

td

d

�

....

..,

..,

..

Example

FLUID DYNAMICS

33

Ramjet

SOLO


WW AA

x sdxsdxpApumApumF ~1100006666

WW A

WA

A

WA AdAdpApuApu sinsin0020066

266

60602006

266 AppAuAu

60 Ap

WA

WA Adpp sin0 WA

WAdp sin0

0

600 sin

AAAdp

WA

W600 sin ApAdp

WA

W

WA

WA Ad sin

1 2 30 4 5 6


SUBSONICCOMPRESSION

COMBUSTIONFUEL

INJECTION EXPANSION


DIFFUSER

FLAMEHOLDERS

EXHAUSTJET

0V

0A

fm

x


FLUID DYNAMICS

34

Ramjet

SOLO

CONSERVATION OF LINEAR MOMENTUM (C.L.M.) (continue – 1)

DRAGFRICTION

A

WA

DRAGPRESURE

A

WA

THRUST

x

WW

AdAdppAppAuAuF sinsin060602006

266

00000666 & mAummAu f Using C.M.

00060602006

266 umummAppAuAuTHRUST ef

or

we obtain

060600 /:1 mmfAppuufmTHRUST fe T

and

DRAGFRICTION

A

WA

DRAGPRESURE

A

WA

WW

AdAdppDRAGD sinsin0

1 2 30 4 5 6


SUBSONICCOMPRESSION

COMBUSTIONFUEL

INJECTION EXPANSION


DIFFUSER

FLAMEHOLDERS

EXHAUSTJET

0V

0A

fm

x


FLUID DYNAMICS


35

PdRRvdVRRHd OOO

,


)3 (CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.)

SOLO

The Absolute Angular Momentum, of the differential mass and Inertial Velocity ,relative to a reference point O is defined as

vdmd V

The Absolute Angular Momentum of the mass enclosed by C.V. is defined as

Centrifugal Pump

Inlet

Outlet2V

1V

C.V.C.S.

x

z

rz

yO

....,

VCO

VCOOCV PdRRvdVRRH

Let differentiate the Absolute Angular Momentum and use Reynolds’ Transport Theorem

..

,....

,

SCmd

SOVC

I

OREYNOLDS

IVC

O

I

OCV sdVVRRvdtD

VRRDvdVRR

td

d

td

Hd

We have

VVtD

VDRRVVV

tD

VDRR

VtD

RD

tD

RD

tD

VDRR

tD

VRRD

O

I

OO

I

O

I

O

II

O

I

O

FLUID DYNAMICS

36

int, : fdRRfdRRvdtD

VDRRMd OextO

I

OO

..

,......

,

SCmd

SO

P

VCO

VCI

O

REYNOLDS

IVC

O

I

OCV sdVVRRvdVVvdtD

VDRRvdVRR

td

d

td

Hd

CV

Centrifugal Pump

Inlet

Outlet2V

1V

C.V.C.S.

x

z

rz

yO



SOLO

The Moment, of the differential mass dm = ρdv, relative to a reference point O is defined as

Therefore

Let integrate this equation over the control volume C.V.

0

..

int

....

, VC

O

VC

extO

VC I

OOCV fdRRfdRRvdtD

VDRRM

Using the differential of Angular Momentum equation we obtain

..

,..

,..

,

SCmd

SO

P

VCOOCV

IVC

O

I

OCV sdVVRRvdVVMvdVRRtd

d

td

Hd

CV

kk

jjOj

SC sdTsd

O

VC

O

VC

extOOtCV MFRRsdtfnpRRvdgRRfdRRM

�......

, 11

Also

j

jOj FRR

- Moment, relative to O, of discrete forces exerting by the surrounding at point

jR

- Discrete Moments exerting by the surrounding.k

kM

FLUID DYNAMICS

37

kk

jjO

SC sdTsd

O

VC

O

P

VC

O

SC md

SO

IVC

O

VC Ir

OOCV MFrsdtfnprvdgrvdVVsdVVrvdVrtd

dvd

tD

VDRRM

CV

O

�

,

..

,

..

,

....

,,

..

,

Reynolds

..

, 11

,


)3 (CONSERVATION OF MOMENT-OF-MOMENTUM (C.M.M.)

SOLO

Let find the equation of moment around the turbomachine axis.

Centrifugal Pump

Inlet

Outlet2V

1V

C.V.C.S.

x

z

rz

yO

We shall use polar coordinates , where z is the turbomachine axis.

zr ,,

zzrrrOˆˆ

,

zVVrVV zrˆˆˆ

zFFrFF zrˆˆˆ

zVrVrVzrVz

VVV

zr

zr

Vr zrz

zr

Oˆˆ0

ˆˆˆ

,

kkz

jj

tvextCVO

SCS

VC

MFrdfrPVsdVVrvdVrtd

d

0

..,

..

The moment of momentum equation around the turbomachine z axis.

Example

FLUID DYNAMICS

38



SOLO

Centrifugal Pump

Inlet

Outlet2V

1V

C.V.C.S.

x

z

rz

yO

systemoutsidefromexertedTorque

M

llz

jj

tvext

AVVrAVVr

SCS

statesteady

VC

zSnSn

MFrdfrsdVVrvdVrtd

d

22,21111,122

..,

0

..

We obtain

zflow MQVrVr 111122

or

zSnSnSn MAVVrVrAVVrAVVr 11,1112211,11122,222

Euler Turbine Equation

ρ1 - mean fluid density one inlet (1) of area A1. where

ρ2 - mean fluid density one outlet (2) of area A2.

(Vθ )1, r1 - mean fluid tangential velocity and radius one inlet (1) of area A1.

(Vθ )2, r2 - mean fluid tangential velocity and radius one outlet (2) of area A2.

(V,Sn )1 - mean fluid normal velocity (relative to A1) one inlet (1) of area A1.

(V,Sn )2 - mean fluid normal velocity (relative to A2) one outlet (2) of area A2.

- mean flow rate one outlet (1) of area A1. 11,1 : AVQ Snflow

Leonhard Euler(1707-1783)

FLUID DYNAMICSReturn to Table of Content

39

FLUID DYNAMICS

2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)

(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (DIFFERENTIAL FORM)

- Fluid mean velocity 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

40

FLUID DYNAMICS


(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE - 1)

mv(t)

Q

t

uq

u

S(t)

td

Wd

dsnsd ˆ

nT ˆ~

dm

G

- The Internal Energy of the molecules of the fluid plus the Kinetic Energy of the mass moving relative to an Inertial System (I)

The FIRST LAW OF THERMODYNAMICS

CHANGE OF INTERNAL ENERGY + KINETIC ENERGY =CHANGE DUE TO HEAT + WORK + ENERGY SUPPLIED BY SUROUNDING

SOLO

The energy of the constant mass m in the volume vF(t) attached to the fluid, bounded by the closed surface SF(t) is

This energy will change due to

- The Work done by the surrounding

- Absorption of Heat

- Other forms of energy supplied to the mass (electromagnetic, chemical,…)

41

FLUID DYNAMICS



VECTOR NOTATION CARTESIAN TENSOR NOTATIONC.E.-1

systementeringtd

Qd

tSv

systemontnmenenvirobydonetd

Wd

shaft

tSv

v

REYNOLDS

KineticInternal

tv

FF

FF

FF

sdqvdt

Q

td

Wd

ForcesSurface

sdTu

ForcesBody

vdGu

vduetD

Dvdue

td

d

)(

)(

2)3(

)(

2

2

1

2

1

systementeringtd

Qd

tSkk

tv

systemontnemnoenvirbydonetd

Wd

shaft

tSkk

tvkk

tv

REYNOLDS

KineticInternal

tv

FF

FF

FF

dsqvdt

Q

td

Wd

ForcesSurface

sdTu

ForcesBody

vdGu

vduetD

Dvdue

td

d

)()(

)()(

)(

2)3(

)(

2

2

1

2

1

SOLO

42

FLUID DYNAMICS




)()(

)()()(

)1(

)()(

)()()(

)(

2

~

~

2

1

tvtv

tvtvtv

GAUSS

td

Qd

tStv

td

Wd

tStStv

tv

FF

FFF

FF

FFF

F

vdqvdt

Q

vduvdupvdGu

sdqvdt

Q

sdusdupvdGu

KineticInternal

vduetD

D

)()(

)()()(

)1(

)()(

)()()(

)(

2

2

1

tV s

s

tV

tV

kk

iki

tV

kk

k

tV

kk

GAUSS

td

Qd

tS

kk

tV

td

Wd

tS

kiki

tS

kk

tV

kk

KineticInternal

tV

vdx

qvd

t

Q

dsx

uds

x

upvdGu

dsqvdt

Q

dsudsupvdGu

vduetD

D

T n p n n ds n ds ~ ~ & 0

td

Wd shaftassume and use

SOLO

43

FLUID DYNAMICS


(2.4) CONSERVATION OF ENERGY (C.E.) – THE FIRST LAW OF THERMODYNAMICS (CONTINUE-4)


Since the last equation is valid for each vF(t) we can drop the integral and obtain:

qt

Q

uGuupuetD

D

~2

1 2

k

k

kk

k

iik

k

k

x

q

t

Q

uGx

u

x

upue

tD

D

2

2

1

Multiply (C.L.M.-2) byu

~ upuuGtD

uDu

k

iki

k

kkki

i xu

x

puuGu

tD

D

tD

uDu

2

Subtract this equation from (C.E.-3)C.E.-4

D e

D tp u u u

Q

tq

~ ~

D e

D tp

u

xu

u

x

Q

t

q

x

k

kik

i

k

k

k

~ ~ u u 0

ik

i

k

u

x0

(Proof of inequality given later)

SOLO

44

FLUID DYNAMICS




Enthalpy

Use this result and (C.E.-4)

C.E.-5

p

eh :

tD

pDup

tD

hDu

p

tD

pD

tD

hD

tD

Dp

tD

pD

tD

hD

tD

pD

tD

hD

tD

eD

2

tD

pD

x

up

tD

hD

x

up

tD

pD

tD

hD

tD

pDp

tD

hD

tD

pD

tD

pD

tD

hD

tD

eD

k

k

k

k

2

t

Qq

tD

pD

tD

hD

t

Q

x

q

tD

pD

tD

hD

k

k

SOLO

~ ~ u u 0

ik

i

k

u

x0

45

FLUID DYNAMICS




Total Enthalpy

Use this result and (C.E.-3)

C.E.-6

22

2

1

2

1: u

peuhH

t

pup

tD

HD

tD

pDup

tD

HD

p

tD

D

tD

HDue

tD

D

2

2

1

t

pup

xtD

HD

tD

pD

x

up

tD

HD

p

tD

D

tD

HDue

tD

D

kk

k

2

2

1

qt

QuGu

t

p

tD

HD

~

k

kkk

k

iik

x

q

t

QuG

x

u

t

p

tD

HD

SOLO

46


(4) THE FIRST LAW OF THERMODYNAMICS

SOLO

Casing

Impeller

Centrifugal Pump

Control Volume(C.V.)

fluidmd

pvdpv

pv

fluidmd

2V

1V

Inlet

Outlet Let apply the First Law of Thermodynamicsto an element of fluid of mass dmfluid

fluidfluidfluid dmondone

WorkExternaldmto

addedHeatdmof

ChangeEnergyTotal

WQEd

EnergyPotential

fluid

EnergyInternal

fluid

EnergyKinetic

fluid

dmofChngeEnergyTotal

mdhgmdumdV

dEd

fluid

2

2

boundaryliquidatDonemdinside

volumeandpressurechangetodoneWork

fluidfluidfluid

FrictionbyLosesSystem

fluid

fluid

Losses

LiquidShaftboundaryatDone

fluid

fluid

shaft

dmondoneWorkExternal

fluid

fluid

mdpd

mdp

mdpmd

md

Wddmd

md

WddW

We obtain

pd

md

Wdd

md

Wdd

md

Qzdgud

Vd

fluid

loss

fluid

shaft

fluid2

2

First Law of Thermodynamics

FLUID DYNAMICS


47

FLUID DYNAMICS

2. BASIC LAWS IN FLUID DYNAMICS (CONTINUE)SOLO

THERMODYNAMIC PROCESSES

1. ADIABATIC PROCESSES

2. REVERSIBLE PROCESSES

3. ISENTROPIC PROCESSES

No Heat is added or taken away from the System

No dissipative phenomena (viscosity, thermal, conductivity, mass diffusion, friction, etc)

Both adiabatic and reversible

(2.5) THE SECOND LAW OF THERMODYNAMICS AND ENTROPY PRODUCTION

48

FLUID DYNAMICS


(2.5) THE SECOND LAW OF THERMODYNAMICS AND ENTROPY PRODUCTION

2nd LAW OF THERMODYNAMICS

Using GAUSS’ THEOREM

0)()(

tStv FF

AdT

qvds

td

d

00)(

)1(

)()(

tv

GAUSS

tStv FFF

vdT

q

tD

sDAd

T

qvd

tD

sD

- Change in Entropy per unit volumed s

- Local TemperatureT K

- Fluid Density 3/ mKg

d e q w T ds p dv d sd e

T

p

Tdv

SOLO

For a Reversible Process

- Rate of Conduction and Radiation of Heat from the System per unit surface

q

2/ mW

Gibbs Relation

Josiah Willard Gibbs (1839-1903)

49

FLUID DYNAMICS


(2.5) THE SECOND LAW OF THERMODYNAMICS (CONTINUE - 1)

d e q w T ds p dv d sd e

T

p

Tdv

uT

p

tD

eD

Tu

T

p

tD

eD

T

tD

D

T

p

tD

eD

TtD

D

T

p

tD

eD

TtD

vD

T

p

tD

eD

TtD

sD

utD

DMC

v

2

.).(

2

1

1

11

The Energy Equation (C.E.-4) is

k

iik x

uoruu

t

Qqup

tD

eD

~~

Tt

Q

TT

qup

tD

eD

TtD

sD

11

or

t

Qq

tD

sDT

SOLO

50

FLUID DYNAMICS



Define

TD s

D tq

Q

t

D s

Dt

q

T

0 Entropy Production Rate per unit volume

Therefore

q

T T

Q

t T

q

Tdv

V t

10

&

SOLO

51

FLUID DYNAMICS



q q qq conduction rate per unit surface

q radiation rate per unit surfacec r

c

r

q K T K FOURIER s Conduction Lawc 0 '

q

T

q

T

q

T Tq q

Tq

TK T q

T

K TT

T qT

KT

Tq

T

r

r r

1 1 1 1

1 1 12

2

KT

T T T

Q

tq

T

K

Tr

2 1 10

0

0

D s

D t

q

TK

T

T T T

Q

tq

Tr

2 1 10

SOLO

JEAN FOURIER1768-1830

52

FLUID DYNAMICS



SOLO

Gibbs Function

Helmholtz Function

sThG :

sTeH :

Josiah Willard Gibbs (1839-1903)

Hermann Ludwig Ferdinandvon Helmholtz(1821 – 1894)

Using the Relations

vdpsdTed

pdvsdTvpdedhd vpep

eh

:

pdvTdssdTTdshdGd

vdpTdsTdssdTedHd

dvT

p

T

edsd

53

FLUID DYNAMICS



SOLO

Maxwell’s Relations

vdpsdTed

pdvsdThd

pdvTdsGd

vdpTdsHd

Ts

pv

v

Fp

v

e

s

hT

s

e

vp

Ts

T

Fs

T

G

p

Gv

p

h

ps

vs

s

v

p

T

s

p

v

T

vT

pT

T

p

v

s

T

v

p

s

James Clerk Maxwell(1831-1879)


54

FLUID DYNAMICS


(2.6) CONSTITUTIVE RELATIONS FOR GASES

(2.6.1) NEWTONIAN FLUID DEFINITION – NAVIER–STOKES EQUATIONS

~~ Ip

Stress

NEWTONIAN FLUID:

ndn

uu

u

dydx

n

u

x

n

The Shear Stress onA Surface Parallelto the Flow =Distance Rate ofChange of Velocity

SOLO

CARTESIAN TENSOR NOTATION

ikikik p

VECTOR NOTATION

- Stress tensor (force per unit surface) of the surrounding on the control surface 2/ mN

~

- Shear stress tensor (force per unit surface) of the surrounding on the control surface 2/ mN

~

55

FLUID DYNAMICS


(2.6) CONSTITUTIVE RELATIONS

(2.6.1) NEWTONIAN FLUID DEFINITION – NAVIER–STOKES EQUATIONS

M. NAVIER 1822INCOMPRESSIBLE FLUIDS

(MOLECULAR MODEL)

G.G. STOKES 1845COMPRESSIBLE FLUIDS

(MACROSCOPIC MODEL)


IuuuIpIp T ~~

ikk

k

i

k

k

iikikikik x

u

x

u

x

upp

3

232~0 utrutrIutruutrtr T

3

20322

i

iik

k

k

i

iii x

u

x

u

x

u

SOLO

STOKES ASSUMPTION 3

20~ trace

μ, λ - Lamé parameters from Elasticity

56

FLUID DYNAMICS



(2.6.1) NAVIER–STOKES EQUATIONS (CONTINUE)

(2.6.1.2) VECTORIAL DERIVATION

I

x

y

zT n ~

ds n ds

r

druu + du unrdtd

t

uurdtd

t

uud

1

rdnurdnuuntdt

uud

RotationnTranslatio

1

2

11

2

11

OR

DEFINITION OF NEWTONIAN FLUID, NAVIER-STOKES EQUATION

nnunuunnpT

nTranslatio

1~11

2

1121

CONSERVATION OF LINEAR MOMENTUM EQUATIONS

SOLO

57

FLUID DYNAMICS




(2.6.1.2)VECTORIAL DERIVATION (CONTINUE) I

x

y

zT n ~

d s n ds

r

druu + du

CONSERVATION OF LINEAR MOMENTUM EQUATIONS

)(

)()()()()(

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

251

2

2

2

112

1121

tV

GAUSS

tStStStStV

tStStVtStVtV

vd

GAUSS

u

GAUSS

u

GAUSS

u

GAUSS

pG

usdusdusdsdpvdG

sdnunuunsdnpvdGdsTvdGvdtD

uD

BUT

2 2 2 u u u

2 2 u u u u

THEN

SOLO

58

FLUID DYNAMICS


I

x

y

zT n ~

d s n ds

r

druu + du

THEREFORE

)()(

2tVtV

vduuupGvdtD

uD

OR

uupGtD

uD 2

SOLO



(2.6.1.2)VECTORIAL DERIVATION (CONTINUE)

59

FLUID DYNAMICS



CONSERVATION OF LINEAR MOMENTUM

~ p u u

2

k

k

ii

k

k

i

iii

ik

x

u

xx

u

x

u

xx

p

x

2

Du

DtG

G p u u

~

2

k

k

ii

k

k

i

iii

i

iki

i

x

u

xx

u

x

u

xx

pG

xG

tD

uD

2

USING STOKES ASSUMPTION tr ~ 02

3

uupG

GtD

uD

3

4

~

k

k

ki

k

k

i

kki

k

iki

i

x

u

xx

u

x

u

xx

pG

xG

tD

uD

3

4

SOLO



60

FLUID DYNAMICS



Euler Equations are obtained by assuming Inviscid Flow

03

20~

pGtD

uD

ii

i

x

pG

tD

uD

SOLO


(2.6.2) EULER EQUATIONS

pGuut

u

i

ik

ik

i

x

pG

x

uu

t

u

or or

Leonhard Euler (1707-1783)

61

FLUID DYNAMICS


(2.6.1.3) COMPUTATION

BUT

iki

kik

i

kki

k

iik

i

k

k

iik ik

u

x

u

x

u

x

u

x

u

xD

ik ki1

2

1

2

ik ik kk ikD D 2

HENCE ik ik ik kk ik ikD D D D2

OR

2 2

2 2

11 11 22 33 11 22 11 22 33 22

33 11 22 33 33 122

212

132

312

232

322

D D D D D D D D D D

D D D D D D D D D D DD Dij ji

2 2 2 2112

222

332

122

132

232

11 22 33

2 D D D D D D D D DOR

SOLO



62

FLUID DYNAMICS


(2.6.1.3) COMPUTATION (CONTINUE)

USING STOKES ASSUMPTION: tr ~ 02

3

2 2 2 2112

222

332

122

132

232

11 22 33

2 D D D D D D D D D

2

3

4

3

4

3

42

3

11 22 33

2

11 22 11 33 22 33 112

222

332

2

122

132

232

11 22 33

2

112

222

332

D D D D D D D D D D D D

D D D D D D

D D D

OR

2

34 011 22

2

11 33

2

22 33

2

122

132

232 D D D D D D D D D

SOLO



63

FLUID DYNAMICS


(2.6.1.4) ENTROPY AND VORTICITY

From (C.L.M.)

or

Du

Dt

u

t

uu u G p u u

2

2

1 1 12

GIBBS EQUATION: T d s d hd p

tld

pd

tdt

pldp

hd

tdt

hldh

sd

tdt

sldsT &

1

Since this is true for d l t

&

T s hp

Ts

t

h

t

p

t

&1

SOLO



Josiah Willard Gibbs(1903 – 1839)

64

FLUID DYNAMICS


(2.6.1.4) 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 all 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

65

Luigi Crocco 1909-1986

FLUID DYNAMICS


(2.6.1.4) 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

66

FLUID DYNAMICS



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

67

FLUID DYNAMICS





~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


68

FLUID DYNAMICS



(2.6.2) STATE EQUATION

p - PRESSURE (FORCE / SURFACE)

V - VOLUME OF GAS

M - MASS OF GAS

R - 8314

- 286.9

T - GAS TEMPERATURE

- GAS DENSITY

m3

kg

J kg mol Ko/ ( )

J kg Ko/ ( )R

kgmol /

oK

kg m/ 3

2/ mN

IDEAL GAS

TRMVp

TMVp R

DEFINE:

M

Vv

V

M&

1

pv TR

p T ROR

SOLO

69

FLUID DYNAMICS


IDEAL GAS TMVp R

SOLO



a

Pictures from:Lee, Sears:"Thermodynamics", 2nd Edition, 1962

70

FLUID DYNAMICS


VAN DER WAALS (1873) EQUATION

ARISE FROM THE EXISTENCEOF INTERNAL FORCES BETWEENGAS MOLECULES

REAL GAS

TRbvv

ap

2

2/ va

IS PROPORTIONAL TO THEVOLUME OCCUPIED BY THEGAS MOLECULES THEMSELVES

b

070.15100

488.01400

686.0920

510.0350

587.0344

427.08.62

372.057.8

2

2

2

2

3

2

6

Hg

OH

CO

O

Air

H

He

molelbm

ft

molelbm

ftatm

baGAS

SOLO



71

FLUID DYNAMICS


VAN DER WAALS (1873) EQUATION

REAL GAS TRbvv

ap

2

SOLO




72

FLUID DYNAMICS



(2.6.3) THERMALLY PERFECT GAS AND CALORICALLLY PERFECT GAS

A THERMALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH THEINTERNAL ENERGY e IS A FUNCTION ONLY OF THE TEMPERATURE T.

h e T p e T RT h T / ( ) THERMALLY PERFECT GAS

DEFINE

C

C

v

V V

p

p p p p

e

T

q

T

h

T

de pdv v d p

d T

de pdv

d T

dq

d T

A CALORICALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH IS CONSTANT

Cv

CALORICALLY PERFECT GAS e C Tv

SOLO

73

FLUID DYNAMICS



(2.6.3) CALORICALLLY PERFECT GAS (CONTINUE)

A CALORICALLY PERFECT GAS IS DEFINED AS A GAS FOR WHICH IS CONSTANT

Cv

CALORICALLY PERFECT GAS e C Tv

FOR A CALORICALLY PERFECT GAS

h C T RT C R T C T C C Rv v p p v

C

CC R C

Rp

v

C C R

p

R C C

v

p v p v

1 1

air 14.

SOLO

74

FLUID DYNAMICS




(2.6.3.1) ENTROPY CALCULATIONS FOR A CALORICALLLY PERFECT GAS

pv TR p T R IDEAL GAS

ds

de pdv

T

de pdv vdp vdp

T

dh vdp

T

ds CdT

TR

dv

vs s C

T

TR

v

vC

T

TRv v v 2 1

2

1

2

1

2

1

2

1

ln ln ln ln

1

2

1

212 lnln

p

pR

T

TCss

p

dpR

T

dTCds pp

s s Cp

pR C

p

pCv v p2 1

2

1

1

2

2

1

2

1

2

1

ln ln ln ln

ENTROPY

SOLO

75

FLUID DYNAMICS




(2.6.3.1) ENTROPY CALCULATIONS FOR A CALORICALLLY PERFECT GAS

p

p

T

Te

T

Te

p

p

T

T

C

R s s

R

s s

R

isentropic

s sp

2

1

2

1

2

1

12

1

2

1

12 1 2 1 2 1

2

1

2

1

2

1

1

12

1

2

1

1

12 1 2 1 2 1

T

Te

T

Te

T

T

C

R s s

R

s s

R

isentropic

s sv

p

pe e

p

p

C

C s s

R

s s

R

isentropic

s sp

v2

1

2

1

2

1

2

1

2

1

2 1 2 1 2 1

T

T

h

h

p

pe

p

pe

T

T

h

h

p

p

s s

C

s s

C

isentropic

s sv p2

1

2

1

2

1

2

1

2

1

1

2

1

12

1

2

1

2

1

1

2

1

12 1 2 12 1

ISENTROPIC CHAIN

SOLO


76

FLUID DYNAMICS

BASIC LAWS IN FLUID DYNAMICS (CONTINUE)

BOUNDARY CONDITIONS

SOLO

Viscous Flow Inviscid Flow 0 0

0

wallu 0ˆ nwallu

n wallu

0

wallu

Upstream Flow uu

(a) Temperature of the wall given wallTwallT

(b) Temperature of the wall not given

k

q

n

T wall

wall

wallq Heat transfer through the surface

k Thermal conductivity

77

EXAMPLE: BASIC LAWS IN FLUID DYNAMICS

pd

dpmd

Wddhdgud

Vd

fluid

shaft

1

2

2


SOLO

Assume an isentropic process (ds = 0)

0Q- no heat added

0

fluid

loss

md

Wdd- no losses

The First Law of Thermodynamics becomes

From Gibbs Law 01

0

dpudsdT

Gibbs

Isentropic

Combine First Law of Thermodynamics with Gibbs Law, to obtain:

hdgpdV

dmd

Wdd

fluid

shaft

2

2

Casing

Impeller

Centrifugal Pump


fluidmd

pvdpv

pv

fluidmd

2V

1V

Inlet

Outlet

Second Law of Thermodynamics for an isentropic process

FLUID DYNAMICS

78


SOLO

Assume an isentropic process (ds = 0)

hdgpdV

dmd

Wdd

fluid

shaft

2

2

1. For an incompressible fluid (ρ = const, dρ = 0)

and integrate this equation

1212

2

1

2

2

2hhg

ppVV

md

Wdltheoretica

ltheoreticafluid

shaft

2. For a perfect gas and an isentropic process

const

pp

1

1

hdgpd

p

pVd

md

Wdd

fluid

shaft

1

/1

1

2

2

12

11

1

11

2

1

/1

1

2

1

2

2

11

1

2hhgpp

pVV

md

Wdltheoretica

ltheoreticafluid

shaft

12

1

1

1

2

1

/1

1

2

1

2

2

12hhgpp

pVV ltheoretica

11

1

11

1

1

/1

1

111TcTR

pp

pp

12

1

1

21

2

1

2

2 12

hhgp

pTc

VV

md

Wdp

ltheoretica

ltheoreticafluid

shaft

Casing

Impeller

Centrifugal Pump


fluidmd

pvdpv

pv

fluidmd

2V

1V

Inlet

Outlet

12

1

1

21

1

1

/1

1

2

1

2

2 112

hhgp

pp

pVV ltheoretica

FLUID DYNAMICS

EXAMPLE: BASIC LAWS IN FLUID DYNAMICS

79

TURBOMACHINERY EXAMPLE: EFFICIENCY OF A PUMP

SOLO

The efficiency is composed of three parts:

• Volumetric efficiency:L

v QQ

Q

Loss of fluid due to leakage in the impeller-casing clearanceLQ

• Hydraulic efficiency:s

f

h h

h1

1. Shock loss due to imperfect match between inlet flow and blade entrance

2. Friction loss

3. Circulation loss due to imperfect match at the exit side of the blade

has three parts:fh

• Mechanic efficiency:T

Pf

m 1

Power loss due to mechanical friction in the bearings, and other contact points in the pump.

fP

Total efficiency is :mhv :

Casing

Impeller

Centrifugal Pump


80

SOLODimensionless Equations

Dimensionless Variables are:

0/~ 0/~

Uuu gGG /~ 2

00/~ Upp

0/~ lUtt

20/

~UCTT p 2

00/~ U 2

0/~

UHH 20/

~Uhh 2

0/~ Uee 200/~ Uqq 2/

~UQQ

0

~l

Field Equations

(C.M.): 00

00U

lu

t

200

0

~

3

4

U

luupGuu

t

u

(C.L.M.):

300

0~U

lTk

t

QuGu

t

pHu

t

H

q

(C.E.):

0

/

/

000

00

0

U

ul

lUt

000

000

0

00

00

000

0

200

020

0

000

000

0

0

3

4

/

/

U

ull

UlU

ull

Ul

U

pl

g

G

U

lg

U

ul

U

u

lUt

Uu

2

0

00

00

0

000

02

000002

0

0

02

00

020000

200000 /

~

// U

CTl

k

kl

C

k

UlU

Q

lUtU

u

g

G

U

gl

U

u

Ul

U

p

lUtU

H

lUtD

D p

p

0/~ 0/~

Uuu gGG /~ 2

00/~ Upp

0/~ lUtt

20/

~UCTT p 2

00/~ U 2

0/~

UHH 20/

~Uhh 2

0/~ Uee 200/~ Uqq 2/

~UQQ

0

~l

0/~

0/~

kkk

Reference Quantities: ρ0(density), U0(velocity), l0 (length), g (gravity), μ0 (viscosity), k0 (Fourier Constant), λ0 (mean free path)

0/~

81


Dimentionless Field Equations

(C.M.): 0~~~~ u

t

uR

uR

pGF

uut

u

eer

~~~~1

3

4~~~~1~~~~1~~~~

~~

2

(C.L.M.):

TkPRt

QuG

Fu

t

pHu

t

H

rer

11

~

~~~~1~~~

~~~~~

~

~~

2

(C.E.):

Reynolds:0

000

lU

Re Prandtl:0

0

k

CP p

r

Froude:

0

0

gl

UFr

0/~ 0/~

Uuu gGG /~ 2

00/~ Upp

0/~ lUtt

20/

~UCTT p 2

00/~ U 2

0/~

UHH 20/

~Uhh 2

0/~ Uee 200/~ Uqq 2/

~UQQ

0

~l

0/~ 0/~

Uuu gGG /~ 2

00/~ Upp

0/~ lUtt

20/

~UCTT p 2

00/~ U 2

0/~

UHH 20/

~Uhh 2

0/~ Uee 200/~ Uqq 2/

~UQQ

0

~l

0/~

0/~

kkk



0/~

Knudsenl

Kn0

0:

82


Constitutive Relations

TRp

2

2

1uTCH p

Tkq

TCh p

200

200

200

1

U

TC

U

TC

C

R

U

p pp

p

20

20 U

TC

U

h p

2

020

20 2

1

U

u

U

TC

U

H p

20

000

0

000

0300 U

TCl

k

k

C

k

UlU

q p

p

33

2~ Iuuu T 3

00

0000

0

00

00

0000

0

00 3

2~I

U

ul

UlU

ul

U

ul

UlU

T

0/~ 0/~

Uuu gGG /~ 2

00/~ Upp

0/~ lUtt

20/

~UCTT p 2

00/~ U 2

0/~

UHH 20/

~Uhh 2

0/~ Uee 200/~ Uqq 2/

~UQQ

0

~l

0/~ 0/~

Uuu gGG /~ 2

00/~ Upp

0/~ lUtt

20/

~UCTT p 2

00/~ U 2

0/~

UHH 20/

~Uhh 2

0/~ Uee 200/~ Uqq 2/

~UQQ

0

~l

0/~

0/~

kkk



0/~

83


Dimensionless Constitutive Relations

2~2

1~~uTH

Tp~~1~

Ideal Gas

3

~~~

3

2~~~~~~~ IuR

uuR e

T

e

Navier-Stokes

Th~~

Calorically Perfect Gas

TkPR

qre

~~~11~

Fourier Law

Reynolds:0

000

lU

Re

Prandtl:0

0

k

CP p

r

0/~ 0/~

Uuu gGG /~ 2

00/~ Upp

0/~ lUtt

20/

~UCTT p 2

00/~ U 2

0/~

UHH 20/

~Uhh 2

0/~ Uee 200/~ Uqq 2/

~UQQ

0

~l

0/~ 0/~

Uuu gGG /~ 2

00/~ Upp

0/~ lUtt

20/

~UCTT p 2

00/~ U 2

0/~

UHH 20/

~Uhh 2

0/~ Uee 200/~ Uqq 2/

~UQQ

0

~l

0/~

0/~

kkk



0/~


84

SOLO

Mach Number

Mach number (M or Ma) / is a dimensionless quantity representing the ratio of speed of an object moving through a fluid and the local speed of sound.

• M is the Mach number,• U0 is the velocity of the source relative to the medium, and

• a0 is the speed of sound

• M is the Mach number,• U0 is the velocity of the source relative to the medium, and

• a0 is the speed of sound

Mach:0

0

a

UM

The Mach number is named after Austrian physicist and philosopher Ernst Mach, a designation proposed by aeronautical engineer Jakob Ackeret.

Ernst Mach (1838–1916)

Jakob Ackeret (1898–1981)

m

Tk

Mo

TRa B

0

• R is the Universal gas constant, (in SI, 8.314 47215 J K−1 mol−1), [M1 L2 T−2 θ−1 'mol'−1]

• γ is the rate of specific heat constants Cp/Cv and is dimensionless γair = 1.4.

• γ is the rate of specific heat constants Cp/Cv and is dimensionless γair = 1.4.• T is the thermodynamic temperature [θ1]• T is the thermodynamic temperature [θ1]

• Mo is the molar mass, [M1 'mol'−1]

• m is the molecular mass, [M1]

AERODYNAMICS

85

SOLO

Different Regimes of Flow

Mach Number – Flow Regimes

AERODYNAMICS


86

whereρ = air densityV = true speedl = characteristic lengthμ = absolute (dynamic) viscosityυ = kinematic viscosity

Reynolds:

lVlVRe

Osborne Reynolds (1842 –1912)

It was observed by Reynolds in 1884 that a Fluid Flow changes from Laminar to Turbulent at approximately the same value of the dimensionless ratio (ρ V l/ μ) where l is the Characteristic Length for the object in the Flow. This ratio is called the Reynolds number, and is the governing parameter for Viscous Flow.

Reynolds Number and Boundary Layer

SOLO 1884AERODYNAMICS

87

Boundary Layer

SOLO 1904AERODYNAMICS

Ludwig Prandtl(1875 – 1953)

In 1904 at the Third Mathematical Congress, held at Heidelberg, Germany, Ludwig Prandtl (29 years old) introduced the concept of Boundary Layer. He theorized that the fluid friction was the cause of the fluid adjacent to surface to stick to surface – no slip condition, zero local velocity, at the surface – and the frictional effects were experienced only in the boundary layer a thin region near the surface. Outside the boundary layer the flow may be considered as inviscid (frictionless) flow. In the Boundary Layer on can calculate the •Boundary Layer width•Dynamic friction coefficient μ•Friction Drag Coefficient CDf

88

LAMINAR

TRANSIENTTURBULENT

LAMINARFLOW

PROFILE

TURBULENTFLOW

PROFILE

FLATPLATE

- LOW THICKNESS- LOW VELOCITY NEXT TO SURFACE- GRADUAL VELOCITY CHANGE- LOW SKIN FRICTION

- GREATER THICKNESS- HIGHER VELOCITY NEXT TO SURFACE- SHARP VELOCITY CHANGE- HIGHER SKIN FRICTION

DEVELOPMENT OF BOUNDARY LAYERON A SMOOTH FLAT PLATE

LAMINARSUB-LAYER

J.D. NICOLAIDES “FREE FLIGHT MISSILE DYNAMICS” pp. 422The flow within the Boundary Layer can be of two types:•The first one is Laminar Flow, consists of layers of flow sliding one over other in a regular fashion without mixing.•The second one is called Turbulent Flow and consists of particles of flow that moves in a random and irregular fashion with no clear individual path, In specifying the velocity profile within a Boundary Layer, one must look at the mean velocity distribution measured over a long period of time.There is usually a transition region between these two types of Boundary-Layer Flow

SOLO AERODYNAMICS

89

LAMINAR

TRANSIENTTURBULENT

LAMINARFLOW

PROFILE

TURBULENTFLOW

PROFILE

FLATPLATE

- LOW THICKNESS- LOW VELOCITY NEXT TO SURFACE- GRADUAL VELOCITY CHANGE- LOW SKIN FRICTION

- GREATER THICKNESS- HIGHER VELOCITY NEXT TO SURFACE- SHARP VELOCITY CHANGE- HIGHER SKIN FRICTION

DEVELOPMENT OF BOUNDARY LAYERON A SMOOTH FLAT PLATE

LAMINARSUB-LAYER

J.D. NICOLAIDES “FREE FLIGHT MISSILE DYNAMICS” pp. 422

Normalized Velocity profiles within a Boundary-Layer, comparison betweenLaminar and Turbulent Flow.

SOLO

Boundary-Layer

AERODYNAMICS

90

4Re

40Re4

150Re40

5103Re300

65 103Re103

Flow Characteristics around a Cylindrical Body as a Function of Reynolds Number (Viscosity)

AERODYNAMICSSOLO

91

Relative Drag Force as a Function of Reynolds Number (Viscosity)

AERODYNAMICS

Drag CD0 due toFlow Separation

SOLO


92

STEADY QUASI ONE-DIMENSIONAL FLOWSOLO

u

p

T

e

1

1

1

1

1

1 2

u

p

e

2

2

2

2

T2

A 2 1x1x- A 1

q

Q

11

A 3

00

0..

dpududp

dhTdsisentropic

ududhdHEC

increaseudecreasep

decreaseuincreasepu

du

dp

d

M

d

u

ad

d

dp

u

dp

uu

du

ds22

2

022

111

0..

2

A

dA

u

dudMC

u

duM

d

u

duM

Mu

duM

A

dA

u

du

u

du

a

da

u

du

M

dM

a

uM

d

u

duM

a

dad

p

dpisentropic

22

2

2

111

2

1

1

2

u

duM

d

u

du

A

dA12

M

dM

M

M

A

dA

p

dp

MA

dA

2

2

2

21

1

1

11

1

Steady , Adiabatic + Inviscid = Reversible, , q Q 0 0, ~ ~ 0

G 0 t

0

93

STEADY QUASI 1-DIMENSIONAL GASESSOLO

M

dM

M

M

p

dp

M

d

Mu

duM

d

u

du

A

dA

2

2

222

21

1

111

1111

u increase

p decrease

p increase

u decrease

p increase

u decrease

u increase

p decrease

0dA0dA

1M

1M

(1) At M=0 decrease in A gives a proportional

increase in velocity u

du

A

dA

(2) For 0 < M < 1 the relation between A and u is the same as for incompressible flow.

FLOW IN CONVERGING/DIVERGING DUCTS

(3) For M > 1 increase in A increases u . Explanation: When M > 1 , ρ increases faster than u, so A must increase to keep

constAum

(4) M = 1 can be attained only at throat.


G 0 t

0

94


STAGNATION CONDITIONS

(C.E.)constuhuh 2

22211 2

1

2

1

The stagnation condition 0 is attained by reaching u = 0

2

/

21202020

2

11

12

12

122

12

MTR

u

Tc

u

T

T

c

uTTuhh

TRa

auM

Rc

pp

Tch pp

Using the Isentropic Chain relation, we obtain:

2

10102000

2

11 M

p

p

a

a

h

h

T

T


G 0 t

0

95


CRITICAL CONDITIONS 1*; 222 Maau

AareacriticalAA *2u

1A

1u

InfiniteReservoir

0

0

0

0

TT

pp

u

*

*

*

*

TT

pp

au

Isentropic FlowExpansion

u increases, p, T, and a decrease

000 ,, sTp remain constant

An ideal gas flows from aninfinite reservoir

000 ,,,0 ppTTu

through a duct with variablearea A. The area A* at whichthe flow reaches the soundvelocity u*=a* is calledcritical area.

2

1

*****

10102000

p

p

a

a

h

h

T

T2

10102000

2

11 M

p

p

a

a

h

h

T

T

1

M

12

1

21

1

22

1

2

0

0

**

0

0

/1 21

21

11*

21

21

1

21

21

11*

***

***

M

MA

MM

MA

a

a

a

a

u

aA

u

uAA

auM

)C.M.( *** AuAum


G 0 t

0

96

STEADY QUASI ONE-DIMENSIONAL FLOW

H hu

C Tu p u a u

constp

C Rtp

2 2 1

0

0

2 2 2 0

2 2 1 2 1 2

a R T

p

Mu

a

Mu

a

H C Tu

C T

p a

p p

2

0

0

0

0

2

2

1 1

(1)Stagnation pointon a path:

The gas is brought) imaginary (by an

adiabatic process to the rest: u = 0

a

a

R T

R T

R

C

u

R T

M

p

0

2

0

2

2

12

11

2

1

22

21

1

2

2

1

21

2

21

1

21

1 M

MMM

M

T

T

a

a

T

TM

1

2

11

22

T

T

a

a

0 0

21 42

10833

.

.

a u a a

a

2 22 2

2

1 2 1 2

1

1 2

MM

M

MM

M

2

2

2

22

2

1

2 1

2

1 1

M* - Characterisic Mach Number

H H1 2

)2(Any two points1 and 2 where

are related by:

(3)The gas is brought)imaginary (by an

adiabatic processto

u* = a*

Alternative Forms of the Quasi One-Dimensional Energy Equation and Definition of Reference Quantitiesu

p

T

e

1

1

1

1

1

1 2

u

p

e

2

2

2

2

T2

A 2 1x1x- A 1

q

Q

11

A 3


G 0

Ideal and Calorically Perfect Gas (1). p R T h C Tp

t

0

SOLO

97

STEADY QUASI ONE-DIMENSIONAL FLOW

(1)Stagnation pointon a path:

The gas is brought) imaginary (by an

adiabatic process to the rest: u = 0

H H1 2

)2(Any two points1 and 2 where (3)The gas is brought

)imaginary (by anadiabatic processto

u* = a*

p

p

T

T

True on same path

1

2

1

2

1

2

1

Isentropic Chain

0.1 1 10

M

TT0

pp0

0

1

s

T

T0

T *

p *

p

p0

Cp

u* 21

2

Cp

u21

2

isentropicline

p

pM0 2

1

11

2

p

p

M

M

2

1

1

2

2

2

1

11

2

11

2

p

pM

1

2

11

22

1

p

p

0

1 1 42

10528

.

.

0 2

1

1

11

2

M

2

1

12

22

1

1

11

2

11

2

M

M

1

2

11

22

1

1

M

0

1

1 1 42

10 6339

.

.

Mollier’s Diagram

u

p

T

e

1

1

1

1

1

1 2

u

p

e

2

2

2

2

T2

A 2 1x1x- A 1

q

Q

11

A 3Steady , Adiabatic + Inviscid = Reversible, , q Q 0 0, ~ ~ 0

G 0

Ideal and Calorically Perfect Gas (2). p R T h C Tp

t

0

Alternative Forms of the Quasi One-Dimensional Energy Equation and Definition of Reference QuantitiesSOLO

are related by:

98


0

,,

0

000

u

Tp Bp

xM

1

12

1

2

2

12

11

1

*

xM

xMA

xA

120

2

11

1

xM

p

p

12

00

2

11

xM

ppB

20

2

11

1

xMT

T

0p

xp

0T

xT

1

1

528.0

8333.0

Throat

ISENTROPIC SUPERSONIC NOZZLE FLOW (1)

Assume that the gasin a large containerat rest

0,,, 0000 uTp

The gas is released trough an diverging/converging duct toa second containerin which the pressureis regulated with apump such that

12

00

21

1

M

ppB

u

p

T

e

1

1

1

1

1

1 2

u

p

e

2

2

2

2

T2

A 2 1x1x- A 1

q

Q

11

A 3

99


0

,,

0

000

u

Tp Bip

xM

1

0p

xp

0T

xT

1

1

528.0

8333.0

Throat

01 / ppB

02 / ppB

03 / ppB

04 / ppB

4BM

2BM 3BM

1BM

3BT

4BT

2BT1BT

ShockShock

ShockWave



0,,, 0000 uTp

To fit the pressure atthe output a shock wave increases thepressure by a jump.the Mach numberjumps fromSupersonic toSubsonic.

12

00

21

1

M

ppp BBi

the pressure in thesecond container.

Bip

u

p

T

e

1

1

1

1

1

1 2

u

p

e

2

2

2

2

T2

A 2 1x1x- A 1

q

Q

11

A 3

100


12

00

2

11

xM

ppB

0p

xp

1

Bp

120

2

11

1

xM

p

pIsentropic Solution

0

,,

0

000

u

Tp Bp


In this case the ductbetween the two containers has nothroat, therefore ashock wave is notpossible.

u

p

T

e

1

1

1

1

1

1 2

u

p

e

2

2

2

2

T2

A 2 1x1x- A 1

q

Q

11

A 3


0,,, 0000 uTp

the pressure in thesecond container.

Bip

No Throat

101

STEADY ONE-DIMENSIONAL FLOW EQUATIONSSOLO

Steady , 1-D Flow ,Adiabatic, , t

0 0Q

G 0

Ideal and Calorically Perfect Gas. p R T h C Tp

0

32 xx

Field Equations:

u

tu

const

u

xG

p

x xM u p P

001

11

11

111

EquMHQuGqu

xx

Hu

t

H

M

1111111 00

0

No. Equations Unknowns Knowns

1 ,u M

Pp 11,1

1 H q E,

1 T

11

17 Eq. 7 Unknowns

Muu

xtu

t

1

0

0

0

(C.M.)

Du

DtG p

~ ~

11

22

33

0 0

0 0

0 0(C.L.M.)

D H

Dt

p

tu G u q Q ~ ~

u

u11

22

33

10 0

0 0

0 0

0

0

(C.E.)

Constitutive RelationsTRp Ideal Gas

H h u C T up 1

2

1

22 2h C TpCalorically Perfect

q KT

x

1

Fourier Conduction Law

11

1

4

3

u

x

22 33

1

2

3

u

x

11 22 33

12 21

13 31

32 23

0

0

0

0

Newtonian Flow

u

p

T

e

u

p

T

e

11

q

1

1

1

1

1

2

2

2

2

2

1 2

102

SOLO

Steady One-Dimensional Flow t

0

x x2 3

0

Flow between two Equilibrium States (1) and (2)

u

p

T

e

u

p

T

e

11

q

Q

1

1

1

1

1

2

2

2

2

2

1 2

iki

k

k

i

k

kik

u

x

u

x

u

x

2

3

Assume Newtonian fluid (Navier-Stokes Eq.) in each state

111

1

22 331

1

1

1 2

1

4

3

2

30

u

x

u

x

q KT

x

x

equilibrium

We obtain

Let integrate the field equations between state (1) and state (2)

u1

20

u p G dx2

1

2

12

1

2

11

2

0

uH q u G u Q dx1

2

11

1

2

11

2

0 0

No. Equations Unknowns Knowns

2 2,u 1 1,u1

1

1

p2p G1 1,

H2 H1

3 4

STEADY ONE-DIMENSIONAL FLOW EQUATIONS

103

SOLO

Steady One-Dimensional Flow t

0

x x2 3

0

Flow between two Equilibrium States (1) and (2)

u

p

T

e

u

p

T

e

11

q

Q

1

1

1

1

1

2

2

2

2

2

1 2

1 1 2 2u u

We need one more equation to solve the algebraic equations

Normal Shock Wave ( Adiabatic)

G Q 0 0,

2

1

1

2

u

u

22221

211 pupu

H H h u h u1 2 1 12

2 221

2

1

2

11

1

1

21

1

2

pu

p

p

h

h

u

h2

1

1

2

1

21

21

1

p2 p

1

h2 h

1

General iterative solution:

p

p

up

h

h

u

h2

1

1

2

1

1

2

1

1

2

1

1 12

,

)1( Choose

)2( Go to Mollier Diagram 2

Compute

2

1

1

2

u

u

)3( Go to

21

21

1

2

1

1

21

1

2 11

21,11

h

u

h

hpu

p

p

h2

p2

2

lg hR

lg sR

pC1

vC1

Mollier Diagram

Since we didn’t use Constitutive Relations this isTrue for all gases

STEADY ONE-DIMENSIONAL FLOW EQUATIONS

Richard Mollier (1863 – 1935)

104

SOLO

11

1

01

1

0

1

0

h

h

T

T

p

p Tch p

pc

uTT

2

21

1

1'

exp''

221

1

2

2

2

2

2

2

pc

ss

p

p

T

T

h

h

s

Torh

1T

2T

2'T

T

pc

u

2

' 22

pc

u

2

21

p

t

c

h

p

t

c

h'

pc

u

2

22

1p

2p

0p

1

2

2'22

22

'

'

'

hh

uu

hh cc

Expansion

12

12

uu

pp

1 2

0

s

Torh

1T

2T

2'T

T

pc

u

2

21

pc

u

2

' 22

p

c

c

h

p

c

c

h'

pc

u

2

22

1p

2p

0p

1

2

2'

22

22

'

'

'

hh

uu

hh tt

Compression

12

12

uu

pp

1 2'

0

Isentropic Process

Adiabatic Process


G 0

COMPARISON OF ISENTROPIC (ds=0) AND ADIABATIC (Q=0,q=0) FLOW PROCESSES

t

0


105

AERODYNAMICS

Fluid flow is characterized by a velocity vector field in three-dimensional space, within the framework of continuum mechanics. Streamlines, Streaklines and Pathlines are field lines resulting from this vector field description of the flow. They differ only when the flow changes with time: that is, when the flow is not steady.

• Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction a fluid element will travel in at any point in time.

• Streaklines are the locus of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streakline

• Pathlines are the trajectories that individual fluid particles follow. These can be thought of as a "recording" of the path a fluid element in the flow takes over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time.

• Timelines are the lines formed by a set of fluid particles that were marked at a previous instant in time, creating a line or a curve that is displaced in time as the particles move.

The red particle moves in a flowing fluid; its pathline is traced in red; the tip of the trail of blue ink released from the origin follows the particle, but unlike the static pathline (which records the earlier motion of the dot), ink released after the red dot departs continues to move up with the flow. (This is a streakline.) The dashed lines represent contours of the velocity field (streamlines), showing the motion of the whole field at the same time. (See high resolution version.

Flow DescriptionSOLO

106

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 trajectoryor 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

107

3-D Flow

Flow Description

SOLO


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

108

3-D Flow

Flow Description

SOLO



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.

109

3-D Flow

Flow Description

SOLO



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


110

AERODYNAMICS

Streamlines, Streaklines, and PathlinesMathematical description

Streamlines

If the components of the velocity are written and those of the streamline aswe deduce

which shows that the curves are parallel to the velocity vector

Pathlines

Streaklines

where, is the velocity of a particle P at location and time t . The parameter , parametrizes the streakline and 0 ≤ τP ≤ t0 , where t0 is a time of interest .

The suffix P indicates that we are following the motion of a fluid particle. Note that at point

the curve is parallel to the flow velocity vector where the velocity vector is evaluated at the position of the particle at that time t .

SOLO

111

V

Airfoil Pressure Field variation with α

AERODYNAMICS

Airfoil Velocity Field variation with αAirfoil Streamline variation with αAirfoil Streakline with α

Streamlines, Streaklines, and PathlinesSOLO

112

AERODYNAMICSStreamlines, Streaklines, and Pathlines

SOLO

113

AERODYNAMICSSOLO

114

AERODYNAMICSSOLO

115

AERODYNAMICSStreamlines, Streaklines, and Pathlines

SOLO


116


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

117

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

118


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

119


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

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

1869


120


Circulation Definition: C

rdV

:


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

Biot-Savart Formula

121



rdV

:


Biot-Savart Formula (continue - 1)

1820


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


Biot-Savart Formula

122


Circulation

SOLO


rdV

:



1820


34 sr

srldrV

Biot-Savart Formula

General 3D Vortex


123



rdV

:



1820



34 sr

srldrV

Biot-Savart Formula

General 3D Vortex

For a 2 D Vortex:

dhsr

dl

sr

srld sinˆˆsin23

dh

dlhl2sin

cot

sin/hsr

ˆ

2sinˆ

4 0 hd

hV

Biot-Savart Formula General 2D Vortex

Biot-Savart Formula

124



rdV

:



1820


34 sr

srldrV

Biot-Savart Formula General 3D Vortex


Lifting-Line Theory

Biot-Savart Formula


125


Helmholtz Vortex Theorems

SOLO

Helmholtz : “Uber the Integrale der hydrodynamischen Gleichungen, welcheDen Wirbelbewegungen entsprechen”, (“On the Integrals of the Hydrodynamical Equations Corresponding to Vortex Motion”), in Journal fur die reine und angewandte, vol. 55, pp. 25-55. , 1858He introduced the potential of velocity φ.

Hermann Ludwig Ferdinandvon Helmholtz

1821 - 1894

Theorem 1: The circulation around a given vortex line (i.e., the strength of the vortex filament) is constant along its length.

Theorem 2: A vortex filament cannot end in a fluid. It mustform a closed path, end at a boundary, or go to infinity.

Theorem 3: No fluid particle can have rotation, if it did not originally rotate.Or, equivalently, in the absence of rotational external forces, a fluid that is initially irrotational remains irrotational. In general we can conclude that thevortex are preserved as time passes. They can disappear only through the action of viscosity (or some other dissipative mechanism).

1858


126


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

127



SOLO


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

128



SOLO


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:

129


Examples:

SOLO


sincos 00 UiUV Uniform Stream:

xyUv

yxUu

sin

cos

0

0

yUxU

yUxU

cossin

sincos

00

00

zU

zUzUiw

0

00 sincos

0U

130


Examples:

SOLO


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

131


Examples:

SOLO

Stream Function ψ, Velocity Potential φ in 2-D Incompressible Irotational Flow

r

Kvvr

rzuvr

rVu

rrr

0010:0

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

Irotational

132


Examples:

SOLO


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

133


Examples:

SOLO


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

134

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 sinrU yixUzU

Source

irek

zk

ln2

ln2

rk

ln2

2

k

Doubletier

B

z

B cos

r

B sinr

B

Vortex(with clockwise

Circulation)

irei

zi

ln2

ln2

2

rln2

90◦ Corner Flow 22

22yix

Az

A yxA 22

2yx

A


135


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

136




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

137




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

138




C

C

zdzd

wdzM

zdzd

wdiiYX

2

2

2

2

Re

1911Blasius Theorem


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 Equation

viuzd

wd

x

y

xy

sd

M

X

Y

139




C

C

zdzd

wdzM

zdzd

wdiiYX

2

2

2

2

Re

1911Blasius Theorem


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

140




C

C

zdzd

wdzM

zdzd

wdiiYX

2

2

2

2

Re

1911Blasius Theorem


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 Equation

x

y

xy

sd

M

X

Y

141

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

142


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.

143



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

144

The Flow Pattern Around a Spinning Cylinderwith Different Circulations Γ Strengths


145



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

146


147

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


148

SOLO

Velocity Field

http://www.diam.unige.it/~irro/cilindro_e.html

University of Genua, Faculty of Engineering,



149


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

150


151


AERODYNAMICSSubsonic Incompressible Flow (ρ∞ = const.) about Wings of Infinite Span (AR = ∞)

SOLO


152

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)


1902

SOLO

Definition: A Stagnation Point is a point in a flow field where the local velocity of the fluid is zero

153

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


Definition: A Stagnation Point is a point in a flow field where the local velocity of the fluid is zero

SF – Forward Stagnation Point SA – Aft Stagnation Point

Kutta Condition:SA on the Trailing Edge


154

Martin Wilhelm Kutta (1867 – 1944)

Nikolay Yegorovich Joukovsky (1847-1921

Kutta-Joukovsky Theorem

The Kutta–Joukowsky Theorem is a Fundamental Theorem of Aerodynamics. The theorem relates the Lift generated by a right cylinder to the speed of the cylinder through the fluid, the density of the fluid, and the Circulation. The Circulation is defined as the line integral, around a closed loop enclosing the cylinder or airfoil, of the component of the velocity of the fluid tangent to the loop. The magnitude and direction of the fluid velocity change along the path.

The force per unit length acting on a right cylinder of any cross section whatsoever is equal to ρ∞V ∞Γ, and is perpendicular to the direction of V ∞.

Kutta–Joukowsky Theorem:


19061902

UL Kutta–Joukowsky Theorem:

LCUL 2

2

1 Lift:

Kutta in 1902 and Joukowsky in 1906, independently, arrived to this result.

Circulation cos: ldVldV

SOLO

155


General Proof of Kutta-Joukovsky TheoremUsing the Corollary to Blasius Theorem

Suppose that we wish to determine theAerodynamic Force on a Body of Any Shape.Use Corollary to Blasius Theorem, integratingRound a Circle Contour with a Large Radius andCenter on the Body

zi

z

aUzUzw ln

2

2

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

156


D’Alembert Paradox

The fact that the Inviscid Flow Theories give Drag = 0 is called D’Alembert Paradox.

In 1768 d’Alembert enunciated his famous paradox in the following words:

“Thus I do not see, I admit, how one can satisfactorily explain by theory the resistance of fluids. On the contrary, it seems to me that the theory, developed in all possible rigor, gives, at least in several cases, a strictly vanishing resistance; a singular paradox which I leave to future geometers for elucidation.”

Jean-Baptiste le Rond d'Alembert

(1717 – 1783)

The resistance (Drag) experienced by a Real Airfoil is do to a combination of Skin-Friction and Pressure Distribution Distortions due to displacements effects of its Boundary Layers, which are not considered in the Inviscid Flow Theories.

157

The Kutta-Joukowsky Theory can be used to design Wings of Infinite Span that flow at Subsonic Speeds (Incompressible Flows). The design methods for such wings are called methods of “Profile Theory”.

AERODYNAMICS

Subsonic Incompressible Flow (ρ∞ = const.) about Wings of Infinite Span (AR = ∞)

Profile (of Airfoil) Theory can be treated in two different ways:

1.Conformal Mapping This Method is limited to 2 – dimensional problems. The Flow about a given body is obtained by using Conformal Mapping to transform it into a known Flow about another body (usually Circular Cylinder)

2.Method of Singularities The body in the Flow Field is substitute by Sources, Sinks, and Vortices, the so called Singularities.

For practical purposes the Method of Singularities is much simpler than Conformal Mapping. But, the Method of Singularities produces, in general, only ApproximateSolution, whereas Conformal Mapping leads to Exact Solutions, although these often require considerable effort.

SOLO


158

Joukovsky Airfoils

Joukovsky transform, named after Nikolai Joukovsky is a conformal map historically used to understand some principles of airfoil design.


Profile Theory Using Conformal Mapping

It is applied on a Circle of Radius R and Center at cx, cy. The radius to the Point (a,0) make an angle β to x axis. Velocity U∞ makes an angle αwith x axis.

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

AERODYNAMICSSOLO

159

Kutta-Joukovsky


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

Profile Theory Using Conformal MappingAERODYNAMICSSOLO


160

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 Condition0

azzd

wd

Joukovsky Airfoils


161

Joukovsky Airfoils Design1. Move the Circle to ĉ and choose Radius R so that the Circle

passes through z = a.


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


162

Joukovsky Airfoils Design (continue – 1)

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


163


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


164


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

0pM0x 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


165Generation of Joukowsky Profiles through Conformal Mapping

Symmetric Joukowsky Profile

Circular Joukowsky Profile

Cambered Joukowsky Profile


166

Profile Theory Using Conformal Mapping

AERODYNAMICSSOLO

167



168


169


170


171



172

SOLO

SubsonicV < a

a t

V t

Soundwaves

- when the source moves at subsonic velocity V < a, it will stay inside the family of spherical sound waves.

a

VM

M

&

1sin 1

SupersonicV > a

a t

V t

M

1sin 1

Soundwaves

Machwaves

Disturbances in a fluid propagate by molecular collision, at the sped of sound a,along a spherical surface centered at the disturbances source position.

The source of disturbances moves with the velocity V.

- when the source moves at supersonic velocity V > a, it will stay outside the family of spherical sound waves. These wave fronts form a disturbance envelope given by two lines tangent to the family of spherical sound waves. Those lines are called Mach waves, and form an angle μ with the disturbance source velocity:

SHOCK & EXPANSION WAVES

173

SOLO SHOCK & EXPANSION WAVES

M < 1

M = 1

M > 1

Mach Waves

174

SOUND WAVESSOLO

Sound Wave Definition: p

p

p p

p1

2 1

1

1

2 1

2 1

2 1

p p p

h h h

For weak shocks

up

1

2

11

11

1

11

11

2

12

1

1uuuuuu

)C.M.(

ppuuupuupu

11

111122111

211

)C.L.M.(

21

au 1

1p

1

1T

1e

112 uuu

112 ppp

112

112 TTT

112 eee

SOUND

WAVE

Since the changes within the sound wave are small, the flow gradients are small.Therefore the dissipative effects of friction and thermal conduction are negligibleand since no heat is added the sound wave is isotropic. Since

au 1

s

pa

2valid for all gases

175

SPEED OF SOUND AND MACH NUMBERSOLO

21

au 1

1p

1

1T

1e

112 uuu

112 ppp

112

112 TTT

112 eee

SOUNDWAVE

Speed of Sound is given by

0

ds

pa

RTp

C

C

T

dT

R

C

pT

dT

R

C

d

dp

dR

T

dTCds

p

dpR

T

dTCds

v

p

v

p

dsv

p

00

0

but for an ideal, calorically perfect gas

pRTa

TChPerfectyCaloricall

RTpIdeal

p

The Mach Number is defined asRT

u

a

uM

1

2

1

1

111

a

a

T

T

p

pThe Isentropic Chain:

a

ad

T

Tdd

p

pdsd

1

2

10

176

SOLO

12

12

11 M

12 MM

12 pp

12 TT

Concave Corner

When a supersonic flow encounters a boundary the following will happen:

When a flow encounters a boundary it must satisfy the boundary conditions,meaning that the flow must be parallel to the surface at the boundary.

- when the supersonic flow, in order to remain parallel to the boundary surface, must “turn into itself” (see the Concave Corner example) a Oblique Shock will occur. After the shock wave the pressure, temperature and density will increase. The Mach number of the flow will decrease after the shock wave.

SHOCK & EXPANSION WAVES

1

2

12

11 M

12 MM

12 pp

12 TT

Convex Corner

- when the supersonic flow, in order to remain parallel to the boundary surface, must “turn away from itself” (see the Convex Corner example) an Expansion wave will occur. In this case the pressure, temperature and density will decrease. The Mach number of the flow will increase after the expansion wave.


177

SHOCK WAVESSOLO

A shock wave occurs when a supersonic flow decelerates in response to a sharpincrease in pressure (supersonic compression) or when a supersonic flow encountersa sudden, compressive change in direction (the presence of an obstacle).

For the flow conditions where the gas is a continuum, the shock wave is a narrow region(on the order of several molecular mean free paths thick, ~ 6 x 10-6 cm) across which isan almost instantaneous change in the values of the flow parameters.

Shock Wave Definition (from John J. Bertin/ Michael L. Smith, “Aerodynamics for Engineers”, Prentice Hall, 1979, pp.254-255)

When the shock wave is normal to the streamlines it is called a Normal Shock Wave,

otherwise it is an Oblique Shock Wave.

The difference between a Shock Wave and a Mach Wave is that:

- A Mach Wave represents a surface across which some derivative of the flow variables (such as the thermodynamic properties of the fluid and the flow velocity) may be discontinuous while the variables themselves are continuous. For this reason we call it a Weak Shock.

- A Shock Wave represents a surface across which the thermodynamic properties and the flow velocity are essentially discontinuous. For this reason it is called a Strong Shock.

178

Normal Shock Wave Over a Blunt Body

Normal Shock Wave

SHOCK WAVESSOLO

Oblique Shock Wave

Oblique Shock Wave Return to Table of Content

179

NORMAL SHOCK WAVESSOLO

Normal Shock Wave ( Adiabatic), Perfect Gas

G Q 0 0,

Conservation of Mass (C.M.) 1 1 2 2u u

2

1

1

2

u

u

Conservation of Linear Momentum (C.L.M.) 22221

211 pupu p

p

up

2

1

12

1

1

1 1

H H h u h u1 2 1 12

2 221

2

1

2 h

h

u

h2

1

12

12

12

11

Conservation of Energy (C.E.)

Field Equations

Constitutive Relations

p R TIdeal Gas

e e T C Tv

1 2(1) Thermally Perfect Gas (2) Calorically Perfect Gas

pp

CC

CC

p

R

CTC

peh

v

p

vp CC

v

p

v

pCCR

pTRp

p 11

u

p

T

e

u

p

T

e

11

q

1

1

1

1

1

2

2

2

2

2

1 2

180



G Q 0 0,

First Way

h

h

p

pp

p

p

p

u

h

up

2

1

2

2

1

1

2

1

1

2

2

1

12

12

12

1

1

2

1

1

11

21

11

21

11

or

p

p

up

up

C L M2

1

12

1

1

12

1

1

2

11 1

11

21

11

( . . .)

after further development we obtain

1 21

11

11

1

201

2

1

1

212

1

1

12

1

1

up

up

up

Solving for 1/η , we obtain

1

1 1 21

11

2

11

2

2

1

12

1

1

12

1

1

2

12

1

1

12

1

1

u

u

up

up

up

up

u

p

T

e

u

p

T

e

11

q

1

1

1

1

1

2

2

2

2

2

1 2

181



G Q 0 0,

We obtain an other relation in the following way:

p

p

up

p

p

up

p

pp

p

p

p

p

p

p

p

p

pp

p

2

1

12

1

1

2

2

1

12

1

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

11

1

21

1

1 1

11

1

1

21

1

1 1

2

1

21

1

21

1

2

1

21

2

1

2

2

1

1

2

2

1

2

1

2

1

1

2

1

11

1

1

u

u

p

pp

p

p

p

T

T

or

Rankine-Hugoniot Equation

Rankine-Hugoniot Equation (1)

William John MacquornRankine

(1820-1872)

u

p

T

e

u

p

T

e

11

q

1

1

1

1

1

2

2

2

2

2

1 2

Pierre-Henri Hugoniot(1851 – 1887)

182



G Q 0 0,

2

1

1

2

2

1

2

1

2

1

1

2

1

11

1

1

u

u

p

pp

p

p

p

T

T Rankine-Hugoniot Equation


p

p2

1

2

1

2

1

1

11

1

1

T

T

p

p

p

p

p

pp

p

p

p

pp

2

1

2

1

1

2

2

1

2

1

2

1

2

1

2

1

2

1

2

1

1

2

2

1

2

1

1

11

11

1

11

1

1

1

11

1

1

1

1

1

1

1

p2p1

21

Normal Shock WaveRankine-Hugoniot

Isentropicp2

p1

21

( )=

u

p

T

e

u

p

T

e

11

q

1

1

1

1

1

2

2

2

2

2

1 2

183


u

p

T

e

u

p

T

e

11

q

1

1

1

1

1

2

2

2

2

2

1 2

SOLO

184



G Q 0 0,

Strong Shock Wave Definition:p

p

u

u

T

T

p

p

R H R H2

1

2

1

1

2

2

1

2

1

1

1

1

1

Weak Shock Wave Definition: p

p

p p

p1

2 1

1

1

2 1

2 1

2 1

p p p

h h h

For weak shocks

up

1

2

h u

1

2

1

u u u u u u21

2

11

1

1

1

1 1

1

1

1

1

(C.M.)

1 1

2

1 1 1 2 2 1 1 1

1

1 1u p u u p u u u p p

(C.L.M.)

ordernd

uuuhhuuhhuhuh

2

4

1

2

1

2

1

2

1

2

1 21

2

1

21

1

211

2

11

11222

211

(C.E.)

u

p

T

e

u

p

T

e

11

q

1

1

1

1

1

2

2

2

2

2

1 2

185



G Q 0 0,

Second Wayh h u h u0 1 1

22 2

21

2

1

2 Define

210

1

121

1

10

220

2

222

2

20

11

2

1

1

11

2

1

1

uhp

up

h

uhp

up

h

u u h1 2 021

1

Prandtl’s Relation

u hu

u

u

p

p

up2 0

1

2 11

2

2

1

1

2

1

1

21

1

11 1

From this relation, we obtain:


Ludwig Prandtl(1875-1953)

u

p

T

e

u

p

T

e

11

q

1

1

1

1

1

2

2

2

2

2

1 2

(C.M.)(C.L.M.)

p

h1

and use

1222

2

11

1

2211

22221

211 11

uuu

p

u

p

uu

pupu

122121

0 2

1

2

1111uuuu

uuh

2

11

112

21

120 uu

uu

uuh

186



G Q 0 0,

(C.M.)

Hugoniot Equation

1 1 2 2 2 1

1

2

u u u u

1 1

2

1 2 2

2

2 21

2

2

1

2

2 2 1 1

2

11

2

2

1

2 1

2

2 1

1

2 2 1

2

22

2 2 1

2

1

2

2 11

2

2 11

2

u p u p u p p p u u

up p

up p

u u

u u

(C.L.M.)

h u h u ep p p

ep p p

e ep p p p p p p p

e ep p

h ep

1 1

2

2 2

2

11

1

2 1

2 1

22

2

2

2 1

2 1

1

2

2 12 1

2 1

2 1

2

1

1

2

2

2 1

2 1

2

2

1

2

1 2

1 2 2 1

2

2 1

2 1 1 2

1

2

1

2

1

2

1

2

1

2

2 2

2

2 2

2

2

1 2 2

1 2

2 2 2 1 2 1 1 1 2 2

1 2

2 1

2 1 2 1 1 2

1 2

p p p p p p p p

e ep p p p

(C.E.)

e ep p

2 11 2

2 12

1 1

Hugoniot Equation

u

p

T

e

u

p

T

e

11

q

1

1

1

1

1

2

2

2

2

2

1 2

187



G Q 0 0,

Fanno’s Line for a Perfect Gas (1) 1 1 1 2 2 u u

m

A

frictionpupu 22221

2112

31

2

1

21 1

2

2 2

2C T u C T u h C Tp p p

4 1 1 1 2 2 2p R T p R T

5 2 12

1

2

1

s s CT

TR

p

pp ln ln

(C.M.)

(C.L.M.)

(C.E.)

Ideal Gas

p

p

T

T

u

u

h C T

h C T

p

p

T

T

h C T

h C T

s s CT

TR

T

T

h C T

h C T

p

p

p

p

p

p

p

2

1

42

1

2

1

2

1

11

2

30 1

0 2

2

1

2

1

0 1

0 2

2 12

1

2

1

0 1

0 2

5

( )

( ) ( )

ln ln

Assume that all the conditionsof the model are satisfied except the moment equation (2)(a flow with friction)

Using , we obtainh C Tp

ss

1s

2smax

h1

h2

h2

1s s C

h

hR

h

h

h h

h hp2 1

2

1

2

1

0 1

0 2

ln ln

Fanno’s Line for a Perfect Gas

This is the Adiabatic, Constant Area Flow.

u

p

T

e

u

p

T

e

11

q

1

1

1

1

1

2

2

2

2

2

1 2

Gino Girolamo Fanno(1888 – 1962)

188



G Q 0 0,

Fanno’s Line for a Perfect Gas (2)

ss

1s

2smax

h1

h2

h2

1

We have a point of maximum entropy. Let see the significance of this point

dp

dhdp

dhdsT 0max

Gibbs

u

duddudu

0(C.M.)

duudhu

hd

0

2

2(C.E.)

Therefore)4..(

0

.).(

000

EC

ds

MC

dsdsds u

du

d

dpd

d

dpdpdh

0

0

ds

ds d

dpu

or

ds CdT

TR

dp

p

ds CdT

TR

d

C

C

dp

p

d

dp

d p

dp

d

pR T

p

v

p

v

ds

ds

ds ds

p R T

max

max

0

0

0

0

0 0

We have:

udp

dR T a speed of soundds

ds

0

0

u

p

T

e

u

p

T

e

11

q

1

1

1

1

1

2

2

2

2

2

1 2

189

Ideal Gas



G Q 0 0,

Rayleigh’s Line for a Perfect Gas (1)

A

muu

22111

2 1 12

1 2 22

2 u p u p

QhuTCuTC pp 222

211 2

1

2

13

4 1 1 1 2 2 2p R T p R T

5 2 12

1

2

1

s s CT

TR

p

pp ln ln

(C.M.)

(C.L.M.)

(C.E.)

Assume that all the conditionsof the model are satisfied except the energy equation (3)(a flow with heating and cooling)

Let substitute in (5) , to obtainh C Tp

Rayleigh’s Line for a Perfect GasThis is the Frictionless, Constant Area Flow, with Cooling and Heating.

smax

s

s1

s2

h1

h2

h

M>1

M<1Rayleigh2

1

Heating

Heating

Cooling

m

A

R T

pp

m

A

R T

pp

x

p1

11

2

22

1

21

121

11

212

111

212

&12

1

lnln5

p

R

A

mc

p

TR

A

mb

hC

abbR

h

hCss

pp

We want to find xp

p 2

1

. Let multiply the result byx

p1

xm

A

R T

p

b

xm

A

R

pc

T2 1

12

1

12

1

21

2

0

or

xp

pb b a T 2

1

1 12

1 2The solution is:

John William Strutt

Lord Rayleigh

(1842-1919)

u

p

T

e

u

p

T

e

11

q

Q

1

1

1

1

1

2

2

2

2

2

1 2

190



G Q 0 0,

Rayleigh’s Line for a Perfect Gas (2)

We have a point of maximum entropy. Let see the significance of this point

u

duddudu

0(C.M.)

(C.L.M.)

A Normal Shock Wave must be on both Fanno and Rayleigh Lines, thereforethe end points of a Normal Shock Wave must be on the intersection of Fanno and Rayleigh Lines

udp

dR T a speed of soundds

ds

0

0

d p udp

duu

1

202

dp

d

dp

du

du

du

uu

2

ss

1s

2

h1

h2

h

M>1

M<1

Rayleigh

Fanno

2

1

SHOCK

According to the Second Law of Thermodynamicsthe Entropy must increase. Therefore a Normal ShockWave from state (1) to state (2) must be such thats2 > s1. (from supersonic to subsonic flow only)

u

p

T

e

u

p

T

e

11

q

Q

1

1

1

1

1

2

2

2

2

2

1 2

191



G Q 0 0,

Mach Number Relations (1)

C M u u

C L M u p u p

p

u

p

uu u

C Ea

h

ua

h

ua a u

a a u

ap

. .

. . .

. .

1 1 2 2

1 12

1 2 22

2

1

1 1

2

2 22 1

12

1

12 2

2

2

22

12 2

12

22 2

22

41

1

2 1

1

2

1

2

1

21

2

1

2

a

u

a

uu u1

2

1

22

22 1

Field Equations:

1

2

1

2

1

2

1

2

1

2

1

2

1

21

1

2

1

2

2

11

2

22 2 1

2 1

1 2

22 1 2 1

2

1 2

a

uu

a

uu u u

u u

u ua u u u u

a

u u

u u a1 22

u

a

u

aM M1 2

1 21 1


u

p

T

e

u

p

T

e

11

q

Q

1

1

1

1

1

2

2

2

2

2

1 2

Ludwig Prandtl(1875-1953)

192



G Q 0 0,


M

MM

M

M

M

M

MM

22

22

1

12

12

12

12

12

21

1

2

1 1

2

11

1 21

2 1 2

1 1 1 1 1

12

or

M

M

M

M

MH H

A A

2

12

12

12

121 2

1 21

1

21

2

2

1

11

2

12

11

2

1

1

2

12

1 2

12

2 12 1

2

12

1 2 1

1 2

A A u

u

u

u u

u

aM

M

M

u

p

T

e

u

p

T

e

11

q

Q

1

1

1

1

1

2

2

2

2

2

1 2

193



G Q 0 0,


p

p

up

u

u

u

a

MM

MM

M M

M

2

1

12

1

1

2

1

12

12

1

2

12 1

2

12 1

2 12

12

12

1 1 1 1

1 11 2

11

1 1 2

1

or

(C.L.M.)

p

pM2

1121

2

11

h

h

T

T

p

pM

M

M

a

a

h C T p R Tp2

1

2

1

2

1

1

212 1

2

12

2

1

12

11

1 2

1

s s

R

T

T

p

pM

M

M2 1 2

1

12

1

1

12

1

112

12

1

12

11

1 2

1

ln ln

s s

RM M

M2 1

1 1

2 12 3

2

2 12 41

2 2

3 11

2

11

Shapiro p.125

u

p

T

e

u

p

T

e

11

q

Q

1

1

1

1

1

2

2

2

2

2

1 2

194



G Q 0 0,


p

p

p

p

p

p

p

p

M

MM02

01

02

2

1

01

2

1

22

12

1

12

11

2

11

2

12

11

11

21

1

2

11

21

2

1

2

1

2

1

2

1

21

2

11

1

2

12

11

22

12

12

12

2

12

12

12

12

MM

M

M M

M

M

M

p

p

M

MM02

01

12

12

1

12

1

1

1

2

12

11

12

11

u

p

T

e

u

p

T

e

11

q

Q

1

1

1

1

1

2

2

2

2

2

1 2

195



G Q 0 0,


s s

R

T

T

p

p

p

p

MM

M

T T2 1 02

01

102

01

1

02

01

12

12

12

02 01

1

11

2

11

1

1

2

11

2

ln ln

ln ln

s

s1

s2

T

M>1

M<1Rayleigh

Fanno

2

1

SHOCK

T2

T1

T02

T01=

T2T1=* *

p2

p1

p01

p02

Mollier’s Diagram

u

p

T

e

u

p

T

e

11

q

Q

1

1

1

1

1

2

2

2

2

2

1 2

John William Strutt

Lord Rayleigh

(1842-1919)

Gino Girolamo Fanno(1888 – 1962)


196

OBLIQUE SHOCK & EXPANSION WAVESSOLO

11, MV

1

2

1V

a bcd

f e

n1

t1

twnuV

twnuV

11

11

222

111

Continuity Eq.: 2211 uu

21222111 ppuuuu

Moment Eq. Tangential Component: 0222111 wuwu

Moment Eq. Normal Component:

Energy Eq.: 22

22

22

211

21

21

1 22u

wuhu

wuh

Continuity Eq.: 2211 uu

Moment Eq.:21 ww

2222

2111 upup

Energy Eq.:22

22

2

21

1

uh

uh

Summary

Calorically Perfect Gas:Tch

TRp

p

6 Equations with 6 Unknowns

222222 ,,,,, hwuTp

197


For a calorically Perfect Gas

2

1

1

2

1

2

21

212

2

21

1

2

21

21

1

2

11/2

1/2

11

21

21

1

p

p

T

T

M

MM

Mp

p

M

M

n

nn

n

n

n

sin11 MM n

sin2

2nM

M

Now we can compute

tantan1tan

tantan

tan

tan

sin1

sin12

tan

tan

tan

tan

221

221

2

1

1

2

12

2

2

1

1

M

M

u

u

ww

w

u

w

u

11, MV

1

2

1V

a bcd

f e

n1

t1

198


11, MV

2,2

MV

1

1, tM

w 1

1 ,n

Mu

2

2, tM

w2

2 ,nM

u

1

2

1V 2V

Oblique S

hocka b

cd

f e

22cos

1sincot2tan 2

1

221

M

M

M,, relation

12 M

12 M

.5max Mfor

1M 2M

Strong Shock

Weak Shock

199


1. For any given M1 there is a maximum deflection angle θmax

If the physical geometry is such that θ > θmax, then no solution exists for straight oblique shock wave. Instead the shock will be curved and detached.

11 M

11 M

11 M

11 M

max

max max

max

Wedge

Corner Flow

200


2. For any given θ < θmax, there are two values of β predicted by the θ-β-M relation for a given Mach number.

WEAK

STRONG

22cos

1sincot2tan 2

1

221

M

M

M,, relation

11 Mmax

Weak and Strong Shocks

- the large value of β is called the strong shock solution

In nature the weak shock solution usually occurs.

- the small value of β is called the weak shock solution

- in the strong shock solution M2 is subsonic (M2 < 1)

- in the weak shock M2 solution is supersonic (M2 > 1)

201

22cos

1sincot2tan

2

1

22

1

M

M

M,, relation

SOLO OBLIQUE SHOCK & EXPANSION WAVES

4.1

max

11 Mmax


202

sin

11/2

1/2

sin

22

21

212

2

11

n

n

nn

n

MM

M

MM

MM

SOLO

max

OBLIQUE SHOCK & EXPANSION WAVES

11 Mmax


Mach Number in Back of Oblique Shock M2 as a Function of the Mach Numberin Front of the Shock M1, for Different Values of Deflection Angle θ (γ=1.4)

203

11

21

sin

21

1

2

11

n

n

Mp

p

MM

SOLO


Static Pressure Ratio P2/P1

as a Function of M1 the Mach Number in Front of an Oblique Shock, for Different Values of Deflection Angle θ (γ=1.4)

204

SOLO


Stagnation Pressure Ratio P20/P1

0 as a Function of M1 the Mach Number in Front of an Oblique Shock, for Different Values of Deflection Angle θ (γ=1.4)

11 Mmax



205


Prandtl-Meyer Expansion Waves

1

2

12

11 M

12 MM

12 pp

12 TT

Convex Corner


Theodor Meyer (1882 – 1972)

The Expansion Fan depicted in Figure wasFirst analysed by Prandtl in 1907 and hisstudent Meyer in 1908.

Let start with an Infinitesimal Change across aMach Wave

Mac

h Wav

e

d

2

d

2

V

VdV

dddV

VdV

sinsincoscos

cos

2/sin

2/sin

tan

/tan1

tan1

11

VVddd

dV

Vd

1

1tan

1sin

2

1

MM

V

VdMd 12

206


Prandtl-Meyer Expansion Waves (continue-1)

Mac

h Wav

e

d

2

d

2

V

VdV

V

VdMd 12

Integrating this equation gives

2

1

12M

M V

VdM

Using the definition of Mach Number: V = M.a

a

ad

M

Md

V

Vd

For a Calorically Perfect Gas

20

2

0

2

11 M

T

T

a

a

MdMMa

ad1

2

2

11

2

1

M

Md

MV

Vd

2

21

1

1

2

12

2

21

1

1M

M M

Md

M

M

1

2

12

11 M

12 MM

12 pp

12 TT

Convex Corner

207


Prandtl-Meyer Expansion Waves (continue-2)

The integral

2

12

2

21

1

1M

M M

Md

M

M

1

2

12

11 M

12 MM

12 pp

12 TT

Convex Corner

M

Md

M

MM

2

2

21

1

1

is called the Prandtl-Meyer Function and isgiven the symbol ν. Performing the integration we obtain

1tan11

1tan

1

1 2121

MMM

Deflection Angle ν and Mach Angle μ as functions of Mach Number

M

1sin 1

Finally

12 MM


208


1. Irrotational Flow

SOLO

Assumptions

2. Homentropic

3. Thin bodies

0

u

0&0..;.t

sseieverywhereconsts

This implies also inviscid flow ~ 0

Changes in flow velocities due to body presence are small

were

- flow velocity as a function of position and time

- flow entropy as a function of position and time

tzyxu ,,,

tzyxs ,,,

209

SOLO

(C.L.M)

For an inviscid flow conservation of linear momentum gives: ~ 0

Assume that body forces are conservative and stationary

were- flow pressure as a function of position and time tzyxp ,,,- flow density as a function of position and time tzyx ,,,

Gpuuut

uuu

t

u

tD

uD

2

2

1

or

Gp

uuut

u

2

2

1 Euler’s Equation

0&

t

G

- Body forces as a function of position zyxG ,,

Leonhard Euler1707-1783


210

SOLO

Let integrate the Euler’s Equation between two points (1) and (2)

2

1

2

1

2

1

2

1

22

1

2

1

2

2

1

2

10 rd

rdpuurdrdurdu

trd

puuuu

t

We can chose the path of integration as follows:

- along a streamline ( and are collinear; i.e.: )rd

u

0

urd

- along any path, if the flow is irrotational 0

u2

1

u

ld

to obtain: 02

1

uurd

Assuming that the flow is irrotational we can define a potential , such that:

0

u tr ,

u

Let use the identity

to obtain:

rdFtrFdconstt

,

2

1

22

1

2

2

1

2

10

p

p

pdu

t

pdudd

t

Bernoulli’s Equationfor Irrotationaland Inviscid Flow

Daniel Bernoulli1700-1782


211

SOLO

For an isentropic ideal gas we have

2

2

11 a

ad

T

Tdd

p

pd

where

p

TRd

pdpa

s

2 is the square of the speed of sound

In this case

22

2

1

1

1 2ad

a

adppdRTa

RTp

and 222

1

1

1

12

2

aaadpd a

a

p

p

Using the Bernoulli’s Equation we obtain

2222

2

111 Uu

t

dpaa

p

p

2

1

22

1

2

2

1

2

10

p

p

pdu

t

pdudd

t

Bernoulli’s Equationfor Irrotationaland Inviscid Flow


212

SOLO

Let use the conservation of mass (C.M.) equation

(C.M.) 0 utD

D

ortD

Du

1

Let go back to Bernoulli’s Equation

22

2

1Uu

t

pdp

p

and use the Leibnitz rule of differentiation: uxFdxuxFxd

d x

x

,,0

to obtain

1

p

p

pd

pd

d

Now we can computetD

Da

tD

D

d

pd

tD

pD

tD

pDpd

pd

dpd

tD

Dp

p

p

p

211

Therefore

2222 2

1111Uu

ttD

D

a

pd

tD

D

atD

Du

p

p

Since 0 tD

Du

tD

D

we have

22

1

2

1

2

11

2

11

2

2

2

2

2

2

2

2

22

22

uu

t

uu

ta

uu

tu

t

uu

ta

ut

uta

uttD

D

au

u

GOTTFRIED WILHELMvon LEIBNIZ

1646-1716


213

SOLO

22

1

2

1

2

11

2

11

2

2

2

2

2

2

2

2

22

22

uu

t

uu

ta

uu

tu

t

uu

ta

ut

uta

uttD

D

au

u

Let substitute u

2

12

12

2

2 tta

222

2

11 U

taa

Special cases

0 Laplace’s equation

Ua (subsonic flow) we can approximate the first equation by

1

2 2

2

tuu

tuuu

we can approximate

the first equation by

01

2

2

2

ta

Wave equation

Pierre-Simon Laplace

1749-1827


214

SOLO

Note

The equation

22 2

11u

tu

tau

can be written as

2

2

222

22 11

2

11

tD

D

au

tu

tau

tu

tac

c

where the subscript c on and on is intended to indicate that the velocity istreated as a constant during the second application of the operators and .

cu

2

2

tD

Dc

t / u

This equation is similar to a wave equation.

End Note


215

SOLO

Let compute the local pressure coefficient: 2

2

1:

U

ppC p

We have:

12

12

11

21

2

1

2

2

2

/1

2

2

2

2

1

22

2

1

a

a

Ma

a

a

U

T

T

UTR

p

p

Up

C

aUMTRa

T

T

p

p

TRp

p

Let use the equation

222

2

11 U

taa

to compute

2

22

2

2

111 U

taa

a

Finally we obtain:

12

111

2 12

22

UtaM

C p


216

SOLO

Assuming a stationary flow and neglecting the body forces :

0t

0

2

112a

222

2

1

Uaa

12

11

2 12

22

UaM

C p

u


217

SOLO

1

0

332211

323121

eeeeee

eeeeee

General Coordinates 321 ,, uuu

333

222

111

111e

uhe

uhe

uh

3213

2132

1321321

332211

1Ahh

uAhh

uAhh

uhhh

eAeAeAA

Using we obtain:A

33

21

322

13

211

32

1321

2

1

uh

hh

uuh

hh

uuh

hh

uhhh

where

We have for 321321 ,,,,, uuuAuuu


218

SOLO

zzyyxx 2

222

2

1

2

1

2

1111

2

1zyxzyx zyx

zzzyzyxzxz

yzzyyyxyxyxzzxyyxxxx

yzzyxzzxxyyxzzzyyyxxx 22222

2

112a

222

2

1

Uaa

012

2

22111

222

222

2

2

2

2

2

ttztzytyxtxyzzy

xzzx

xyyx

zzz

yyy

xxx

aaa

aaaaa

222222

2

11 U

taa zyx

We finally obtain

Cartesian Coordinates zuyuxu 321 ,,



219

SOLO

Cylindrical Coordinates 321 ,, uruxu

zryrxxzzyyxxR 1sin1cos1111

zryrR

zyr

Rx

x

R1cos1sin&1sin1cos&1

rR

hr

Rh

x

Rh

:&1:&1: 321

11cos1sin:

&11sin1cos:&1:

2

21

zyR

R

e

rzy

r

R

r

R

ex

x

R

x

R

e

1

0

332211

323121

eeeeee

eeeeeeWe have


220

SOLO

Cylindrical Coordinates (continue – 1) 321 ,, uruxu

321321

11e

reee

re

re

x rx

2

2

22 1

rrx

322

22

3212

2

2

22

11

111

1

2

1

2

1

err

err

er

r

rrxx

rrrrxrxxrxrxxx

rx

22

2

22

2

2

2

2

1111

11

rrrrrrx

rrr

rxr

xr

rrrxx


221

SOLO


Then equation

2

12

12

2

2 ttabecomes

322

22

32

12321

22

11

11

11

2111

err

err

er

er

ee

arr

rrxx

rrrrxrx

xrxrxxxrx

ztzytyxtxttrrrxx

or

02

112

/1

1/1

111

22

222

2

22

2

22

22

2

2

2

ztzytyxtxtt

rrxxrxrx

rrrr

xxx

aa

rra

a

r

ra

r

raa


222

SOLO


becomes

222

2

11 u

taa

In cylindrical coordinates, equation

22

2

2222 1

2

11 U

raa rxt

Assuming a stationary flow and neglecting body forces

0t

0

0112

/1

1/1

111

222

2

22

2

22

22

2

2

2

rrxxrxrx

rrrr

xxx

rra

a

r

ra

r

raa

22

2

2222 1

2

1U

raa rx



223

Linearized Flow Equations SOLO

Boundary Conditions

1. Since the Small Perturbations are not considering the Boundary Layer the Flow must be parallel at the Wing Surface.

The Wing Surface S is defined by zU (x,y) – Upper Surface zL (x,y) – Lower Surface

0

S

un

n

- Normal at the Wing Surface

22

1/111

y

z

x

zzy

y

zx

x

zn UUUU

U

zwUyvxuUzwUyvxuUu 1'1'1'1'sin1'1'cos

0,,'''

UUU zyxwUx

zv

x

zuU

For Upper Surface

x

zU

x

zv

x

zuUzyxw U

onPerturbatiSmall

UUU '',,'

Therefore

Sonyxallfor

x

zUzyxw

x

zUzyxw

LL

UU

,

,,'

,,'

Section AA (enlarged)

Wake region

224


Boundary Conditions (continue -1)

1. Flow must be parallel at the Wing Surface.

The Wing Surface S is defined by zU (x,y) – Upper Surface zL (x,y) – Lower Surface

Since the Small Perturbation gives Linear Equation we can divide theAirfoil in the Camber Distribution zC (x,y) and the Thickness Distribution zt (x,y) by:

Sonyxallfor

x

zUyxw

x

zUyxw

CC

tt

,

0,,'

0,,'

2/,,,

2/,,,

,,,

,,,

yxzyxzyxz

yxzyxzyxz

yxzyxzyxz

yxzyxzyxz

LUt

LUC

tCL

tCU

Because of the Linearity the complete solution can be obtained by summing theSolutions for the following Boundary Conditions

Superposition of• Angle of Attack•Camber Distribution•Thickness Distribution


Wake region

Sonyxallfor

x

z

x

zUyxwyxwyxw

x

z

x

zUyxwyxwyxw

tCtCL

tCtCU

,

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

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

225


Boundary Conditions (continue -2)

2. Disturbances Produced by the Motion must Die Out in all portion of the Field remote from the Wing and its Wake

Normally this requirement is met by making ϕ→0 when y→ ±0, z → ±0, x→-∞

Subsonic LeadingEdge Flow

Subsonic TrailingEdge Flow

Supersonic LeadingEdge Flow

Supersonic TrailingEdge Flow

3. Kutta Condition at the Trailing Edge of a Steady Subsonic Flow

There cannot be an infinite change in velocity at the Trailing Edge. If the Trailing Edge has a non-zero angle, the flow velocity there must be zero. At a cusped Trailing Edge, however, the velocity can be non-zero although it must still be identical above and below the airfoil. Another formulation is that the pressure must be continuous at the Trailing Edge.

http://nylander.wordpress.com/category/engineering/

Kutta Condition does not apply to SupersonicFlow since the shape and location of theTrailing Edge exert no influence on the flow ahead.

226

SOLO

Small Perturbation Flow (Homentropic: , Isentropic ) 0~,0,0~,0 Qqsd 0

u

'2uUu '1U

'

'

'2'

'

'0

''2'0

'

222

1

33

211

2222

11

ppp

aaaaaa

xU

uu

uuUUuuu

uUu

O

Small Perturbation Assumptions:

2

21 2

2

2

2

uu

t

uu

tau

(C.M.) +(C.L.M)

(C.M.) +(C.L.M)

12

1

12

1 22

22

a

Ua

ut

Bernoulli

121

a

a

T

T

p

pIsentropic Chain

Development of the Flow Equations:

Flow Equations:

'' 21 xUu

1

12

2

1

1212

2

2

'''

1

2

1

x

u

a

U

x

uuU

a

uu

a

t

uUuUU

tt

u

t

uu

'2'22 1

12

2

p

apuU

t

aU

aaauUU

t

2

1

22

2

12 ''

'0

12

1

1

'2'2

2

1'

a

a

T

T

p

p

a

ad

T

Tdd

p

pd '

1

2'

1

''

1

2

1

Isentropic Chain

Bernoulli


227

SOLO


u

'2uUu '1U

Small Perturbation Flow Equations:

(C.M.) +(C.L.M) 52.1&8.00''

2'1

'2

21

1

12

22

MMtt

uU

x

uU

a

''

,,,'' 321

u

xxxt

Bernoulli

''

' 1uUt

p

a

a

T

T

p

p '

1

2'

1

''

Isentropic Chain


228

SOLO Linearized Flow Equations Small Perturbation Flow (Homentropic: , Isentropic ) 0~,0,0~,0 Qqsd 0

u

Applications: Three Dimensional Flow (Thin Wings at Small Angles of Attack)

U

Upxd

ud

L

Lowxd

ud

U

x

z

0'''

12

2

2

2

2

22

zyxM

(1)

zyx ,,'(2)

zw

yv

xu

'

','

','

'

(3)

S

xd

zd

U

w

uU

w '

'

'(4)

xUuUp

'

''(5)

'

21

1

''

1

2'

1

''2

MM

M

U

uM

a

a

T

T

p

p

(6)

2

2

2

2

2

2

22 '1'2'1

'tUxtUxM

''

,,,''

u

zyxt

''

' uUt

p

Steady Three Dimensional Flow Small Perturbation Flow Equations: 0

'2

2

tt

52.1

8.00

M

M

229


u

Applications: Three Dimensional Flow (Thin Wings at Small Angles of Attack)

0'''

2

2

2

2

2

22

zyx

(1)

Steady Three Dimensional Flow

Subsonic Flow M∞ < 1

01: 22 M

LowerLower

Lower

UperUper

Upper

d

zd

xd

zd

zUU

w

d

zd

xd

zd

zUU

w

'1'

'1'

3

4

3

4

Transform of Coordinates

,,,,'

1 2

zyx

z

y

Mx

2

2

2

2

2

2

2

2

2

2

22

2

''

''

1'1'

zz

yy

xx

SMdcMydycSbb 2020

11 cMyc 21

22

22

11 M

AR

SM

b

S

bAR

22 1

2

1

12

M

C

UMxUC p

p


Wake region

so 02

2

2

2

2

2

Laplace’s Equation like in Incompressible Flow

Similarity Rules

230


u


incpCM 21

1

incLCM 21

1

22 1

2

1

1

Md

Cd

M inc

L

incMCM 21

1

inc0

4

1

inc

N

c

x

incMCM

021

1

incLsCM 21

1

incs

LsC

s

0MC

c

xN

MC

0

d

Cd L

LC

pCPressure Distribution

Lift

Lift Slope

Zero-Lift Angle

Pitching Moment

Neutral-Point Position

Zero Moment

Angle of Smooth Leading-Edge Flow

Lift Coefficient of Smooth Leading-Edge Flow

Aerodynamic Coefficients of a Profile in Subsonic Incident FlowBased on Subsonic Similarity Rules

231

SOLO


u


U

Upxd

ud

L

Lowxd

ud

U

x

y

0''

12

2

2

22

yxM

(1)

yx,'(2)

yv

xu

'

','

'(3)

S

xd

yd

U

v

vU

v '

'

'(4)

xUuUp

'

''(5)

'

21

1

''

1

2'

1

''2

MM

M

U

uM

a

a

T

T

p

p

(6)

2

21

1

12

22 ''

2'1

'tt

uU

x

uU

a

''

,,,'' 321

u

xxxt

''

' uUt

p

Steady Two Dimensional Flow Small Perturbation Flow Equations: 0

'2

2

tt

52.1

8.00

M

M


232

SOLO


u


0''

2

2

2

22

yx

(1)

Steady Two Dimensional Flow

Subsonic Flow M∞ < 1

01: 22 M

Lower

Lower

Uper

Upper

xd

yd

yUU

v

xd

yd

yUU

v

'1'

'1'

3

4

3

4

U

Transform of Coordinates

yx

y

x

,',

2

2

2

2

2

2

2

2 ',

1'

11'

111'

yx

yyyy

xxxx

so 02

2

2

2

Laplace’s Equation like in Incompressible Flow


233

SOLO


u



Subsonic Flow M∞ < 1 (continue)

The Airfoil is defined in (x,y) plane and by (ξ,η) gxfy AirfoilAirfoil

The above Transformation relates theCompressible Flow over an Airfoil in (x,y) Space to the Incompressible Flowin (ξ,η) over the same Airfoil.

Uper

Upper

xd

yd

UyUU

v 1'1'

Lower

Lower

xd

yd

UyUU

v 1'1'

yx,

x

y

Compressible Flow Incompressible Flow

Uper

Upper

xd

fd

UU

v 1'

Lower

Lower

xd

fd

UU

v 1'


234

SOLO


u


0'1'

2

2

22

2

yx

(1)

yxGyxGyx

yxFyxFyx

Lower

Upper

:,'

:,'(7)(8)


Supersonic Flow M∞ > 1

01: 22 M

U

Upxd

yd

L

Lowxd

yd

U

x

y1

12

Mxd

yd

1

12

Mxd

yd

Flow

Flow

d

Fd

Uxd

yd

U

v

Uper

Upper

1

7

4'

d

Fd

xd

du Upper

73 ''

Upper

Upper xd

yd

M

Uu

1'

2

d

Gd

Uxd

yd

U

v

Lower

Lower

3

8

4'

d

Gd

xd

du Lower

83 ''

Lower

Lower xd

yd

M

Uu

1'

2

Upper

UpperUpper xd

yd

M

UuUp

1''

2

2

Lower

LowerLower xd

yd

M

UuUp

1''

2

2


235

SOLO


u


Drag (D) and Lift (L) Computations

S S

sdxd

ydppD sin

np

Upper

xd

yd

U

Upperxd

yd

pp

S S

sdxd

ydppL cos

S S

sdxd

ydppD

SS S

sduUsdxd

ydppL '

1 Uper

xd

yd

1 Uper

xd

yd

Kutta-Joukovsky

Define: 2

2

1:

U

ppC p

S S

p

S S

S S

p

S S

sdxd

ydCUsd

xd

yd

U

ppUL

sdxd

ydCUsd

xd

yd

U

ppUD

2

2

2

2

2

2

2

1

2

12

1

2

1

212

1


236

SOLO


u


Drag (D) and Lift (L) Computations for Subsonic Flow (M∞ < 1)

np

Upper

xd

yd

U

Upperxd

yd

pp

We found:

xd

fd

U

v'

d

gd

U

v

yxM

yM

x

,'1,

1

2

2

0''

12

2

2

22

yxM

02

2

2

2

yv

xu

'

','

'

vu ,vvM

uu

',1

'2

'' uUp uUp

xUU

u

U

ppC p

'2'2

21

':

2

UU

u

U

ppC p

22

21

:2

0

21'

M

pp

210

M

CC p

p

Compressible: Incompressible:


237

SOLO


u


Drag (D) and Lift (L) Computations for Subsonic Flow (M∞ < 1)np

Upper

xd

yd

U

Upperxd

yd

pp

The Relation:

S

p

S S

p

S S

p

S S

p

c

sdC

M

U

c

sd

xd

ydCUL

c

sd

xd

ydC

M

U

c

sd

xd

ydCUD

0

0

2

2

2

2

2

2

12

1

2

1

12

1

2

1

210

M

CC p

pPrandtl-Glauert

Compressibility Correction

As earlier in 1922, Prandtl is quoted as stating that the LiftCoefficient increased according to (1-M∞

2)-1/2; he mentionedthis at a Lecture at Göttingen, but without a proof. This result wasmentioned 6 years later by Jacob Ackeret, again without proof.The result was finally established by H. Glauert in 1928 based onLinear Small Perturbation.


Hermann Glauert(1892-1934)

Linearized Flow Equations Return to

Critical Mach Number

SOLO


u


Several improved formulas where developed:

2/11/10

0

222p

pp

CMMM

CC

Karman-Tsien

Rule


0

0

2222 12/2

111 p

pp

CMMMM

CC

Laitone’sRule

Comparison of several compressibility corrections compared with experimental results for NACA 4412 Airfoil at an angle of attack of α = 1◦.


239


u


0'1'

2

2

22

2

zx

(1)

zxFzxGzx

zxFzxFzx

Lower

Upper

:,'

:,'(7)(8)



01: 22 M

U

Upxd

zd

L

Lowxd

zd

U

x

z1

12

Mxd

zd

1

12

Mxd

zd

Flow

Flow

d

Fd

Uxd

zd

U

w

Upper

Upper

3

7

4'

d

Fd

xd

du Upper

73 ''

d

Gd

Uxd

zd

U

w

Lower

Lower

3

8

4'

d

Gd

xd

du Lower

83 ''

Upper

Upper xd

zd

M

Uw

1'

2

Lower

Lower xd

zd

M

Uw

1'

2

Upper

UpperUpperUpper xd

zd

M

UwUppp

1''

2

2

Lower

LowerLowerLower xd

zd

M

UwUppp

1''

2

2

zw

xu

'

','

'

(3)

S

xd

zd

U

w

uU

w '

'

'(4)

240


u




U

Upxd

zd

L

Lowxd

zd

U

x

z1

12

Mxd

zd

1

12

Mxd

zd

Flow

Flow

Pressure Distribution and Lift Coefficient

21

2

2/

''22

LowerUpper

LowerUpperp xd

zd

xd

zd

MU

ppC

1

42

M

cL

00

22

1

0

1

02

1

0

1

0

001

2

1

4

21

2

LowerLowerUpperUpper

LowerUpper

ppL

zczzczMM

c

xd

xd

zd

c

xd

xd

zd

Mc

xdC

c

xdCc

LowerUpper

Upper

p xd

zd

MC

Upper

1

22

Lower

p xd

zd

MC

Lower

1

22

241


u




U

Upxd

zd

L

Lowxd

zd

U

x

z1

12

Mxd

zd

1

12

Mxd

zd

Flow

Flow

Wave Drag Coefficient

1

0

2

1

0

2

2

1

0

1

0 1

2

c

xd

xd

zd

c

xd

xd

zd

Mc

xd

xd

zdC

c

xd

xd

zdCc

UpperUpperLower

p

Upper

pD LowerUpperW

Upper

p xd

zd

MC

Upper

1

22

Lower

p xd

zd

MC

Lower

1

22

1

0

2

00

1

0

21

0

2

00

1

0

2

222

1

2

c

xd

xd

zd

c

xd

xd

zd

c

xd

xd

zd

c

xd

xd

zd

M Lower

zcz

LowerUpper

zcz

Upper

LowerLowerUpperUpper

22

22

2

1

2

1

4LowerUpperD

MMC

W

1

0

2

2

1

0

2

2

:

:

c

xd

xd

zd

c

xd

xd

zd

Lower

Lower

Upper

Upper

242


u





Flat Plate

0

LowerUpperxd

zd

xd

zd

Double Wedge Airfoil

1

42

2

MC

WD

022 LowerUpper

kkc

tck

c

t

kck

c

t

kcLowerUpper

14

11

14

1

4

112

2

2

2

22

2

222

cxckck

t

ckxck

t

xd

zd

cxckck

t

ckxck

t

xd

zd

LowerUpper

12

02

12

02

kk

ct

MMC

WD

1

/

1

1

1

4 2

22

2

243


u





Biconvex Airfoil

222 2/2/ ctRR

The Biconvex Airfoil is obtained by intersection of twoCircular Arcs of radius R. c – the chordt – maximum thickness at x = c/2

tcttcRtc

4/4/ 22222

tan,tanLowerUpper

xd

zd

xd

zd

22

2/2

/2

321

0

2

1

0

2

2

3

2

34

11: Lower

ct

ctUpperUpper

Upper c

t

t

cdR

cxd

xd

zd

cc

xd

xd

zd

c

t

R

c

xd

zd

MaxUpper

22/

,

2

2

22

222

22

2

3

16

1

1

1

4

1

2

1

4

c

t

MMMMC LowerUpperDW

04/13/23 244


u





Parabolic ProfileDesignation Double Wedge Profile

Contour

Side View

Wave Drag kk 13

12 kk 1

1

xckck

xcxtz

212 22

cxckxck

t

ckxxck

t

z

12

02

245


u





Wave Drag at Supersonic Incident Flow versus Relative Thickness Position

for Double Wedge and Parabolic Profiles

k

kk 1

1

kk 13

12

246

SOLO Wings in Compressible Flow

Double Wedge

Modified Double Wedge

Biconvex

2

1

2

1221

2'

2

c

t

c

tc

A

3

2

3

23321

2'

2

c

t

c

tc

tc

A

247


u


Steady Two Dimensional Flow Supersonic Flow M∞ > 1

Pitching Moment CoefficientThe Pitching Moment Coefficient about theLeading Edge for any Thin Airfoil is given by

xdxxd

zd

xd

zd

Mcc

xd

c

xC

c

xd

c

xCc

c

LowerUpper

ppM LowerUpperLE

022

1

0

1

0 1

2

Thus

xdzxdzMcM

cc

Lower

c

UpperM LE 00222 1

2

1

2

xdzxdzczczcxdzzxxdzzxxdxxd

zd

xd

zd c

Lower

c

UpperLowerUpper

c

Lower

cx

xLower

c

Upper

cx

xUpper

c

LowerUpper

00

0

00000

Using integration by parts

Symmetric Airfoil zUpper = -zLower 1

22

M

cM

The distance of the Airfoil Center of Pressure aft of the Leading Edge is given by

ccM

Mc

c

c

c

x

L

MN

2

1

1/4

1/22

2

L

U

x


248

SOLO


u


Drag (D) and Lift (L) Computations for Supersonic Flow (M∞ >1)

c

xd

xd

yd

xd

yd

M

U

c

sdCUL

c

xd

xd

yd

xd

yd

M

U

c

sd

xd

ydCUD

c

LowerUpperS

p

c

LowerUpperS S

p

02

2

2

0

22

2

2

2

21

2

1

2

1

21

2

1

2

1

U

Upxd

yd

L

Lowxd

yd

U

x

y1

12

Mxd

yd

1

12

Mxd

yd

Flow

Flow

Upper

UpperUpper xd

yd

M

Uppp

1'

2

2

Lower

LowerLower xd

yd

M

Uppp

1'

2

2

1

2

1

2

2

2

M

xdyd

C

M

xdyd

C

Lowerp

Upper

p

Lower

Upper

We found:

This relation was first derived by Jacob Ackeret in 1925, in a paper“Luftkrafte auf Flugel, die mit groserer als Schall-geschwingigkeit bewegt werden”(“Air Forces on Wings Moving at Supersonic Speeds”), that appeared inZeitschhrift fur Flugtechnik und Motorluftschiffahrt, vol. 16, 1925, p.72

Jakob Ackeret (1898–1981)


249

AERODYNAMICSSmall Perturbation Flow (Homentropic: , Isentropic ) 0~,0,0~,0 Qqsd 0

u



Supersonic Flow past a Symmetric Double-Edged Airfoil

1

2

3

4

SHOCK LINE

SHOCK LINE

SHOCK LINE

SHOCK LINE

EXPANSION

EXPANSION

Using Ackeret Theory we have

1

2,

1

2

1

2,

1

2

22

22

43

21

MC

MC

MC

MC

pp

pp

1

4

2

1

1

4

2

1

1

4222

1

2/1

2/1

0 3412

MMM

c

xdCC

c

xdCC

c

sdCC pppp

S

pX

1

4

1

4

22

22 2

2/

2

0

2/

2/

0

3412

3412

MMc

tCC

c

tCC

c

t

c

ydCC

c

ydCC

c

ydCC

ct

pppp

ct pp

ct

pp

S

pX

XYXYD

XYXYL

CCCCC

CCCCC

1

1

cossin

sincos

1

4

1

4

1

4

1

4

2

2

2

21

2

2

2

1

MMC

MMC

D

L

250

Expansion

Shock

Shock

1M

1

2

3

4

1

22

1

Mc p

1

222

Mc p

1

224

Mc p

1

223

Mc p

cx5.0

1

pcUpper Surface

Lower Surface

Lower Surface

Upper Surface

1

42

M

CL

1

42

Md

Cd L

1

4

1

42

2

2

2

MMCD

22

D

L

22

2

2

4

1

1

4LD C

M

MC

Supersonic Flow past a Symmetric Double Wedge Aerfoil

1M


u



251

pc

cx /

0.1

pc

cx /

0.1

pc

cx /

0.1

M

M

M

M

Expansion

ShockShock

Expansion

ExpansionShock

Expansion

Shock

Shock

Expansion

Expansion

Shock

Shock

Shock

Shock

M

M

1

22

Mc p

1

22

Mc p

1

22

Mc p

1

22

Mc p

1

42

M

c p

1

42

Mc p

1

22

Mc p

1

22

Mc p

1

22

Mc p

1

22

Mc p

Supersonic Flow past a Symmetric Biconvex Aerfoil


u



22

2

2

22

2

4

1

1

316

316

1

4

LD

L

CM

M

ct

C

ctD

L

Md

Cd

252


u


Aerodynamic Coefficients of a Profile in Supersonic Incident FlowBased on the Linear Theory Supersonic Rules

Xd

Zd

M

1

12

1

42

M

2

1

0DC

0

0MC

c

xN

d

Cd L

pCPressure Distribution

Lift Slope

Neutral-Point Position

Zero Moment

Zero-Lift Angle 0

1

02 1

4XdZ

M

S

Wave DragL

D

Cd

Cd 14

1 2 M

1

0

22

2 1

4Xd

Xd

Zd

Xd

Zd

M

tS

253

SOLO

• Up to point A the flow is Subsonic and it follows Prandtl-Glauert Linear Subsonic Theory.

• At point B (M∞=0.81) the flow on the Upper Surface exceeds the Sound Velocity and a Shock Wave occurs. On the Lower Surface the Flow is everywhere Subsonic.

• At point C (M∞=0.89) the Flow velocity exceeds the Speed of Sound also on the Lower Surface and a Shock Wave occurs.

• At point D (M∞=0.98) the two Shock Waves on the Upper and Lower Surface (weaker than at point C) are located at the Trailing Edge. The Lift is larger than at point C.

• At point E (M∞=1.4) pure Supersonic Flow on both Surfaces.

The magnitude of Lift is given by Ackeret Theory

Transonic Flow past Airfoils

Lift Coefficient of an Airfoil versus Mach Number.Solid Line – Measurement. Dashed Lines - Theory

AERODYNAMICS

Transonic Flow over an Airfoil at various Mach Numbers; Angle of Attack α=2°.The points A,B, C, D,E correspond to the Lift Coefficients.

254

AERODYNAMICS


255Brenda B. Kulfan, “Aerodynamic of Sonic Flight”, Boeing Commercial Airplane

SOLO AERODYNAMICS


256

SOLO

References

Air Breathing Jet Engines

William F. Hughes“Schaum’s Outline of

Fluid Dynamics”, McGraw Hill, 1999

Ascher H. Shapiro“The Dynamics and Thermodynamics

of Compressible Fluid Flow”, Wiley, 1953

John D. Anderson“Modern Compressible Flow:with Historical erspective”,

McGraw-Hill, 1982

John D. Anderson“Computational Fluid Dynamics”,

1995

Irving Herman Shames“Mechanics of Fluids”McGraw-Hill, 4th Ed,,

2003

D.Pnueli, C. Gutfinger“Fluid Mechanics”

Cambridge UniversityPress, 1997

I.H. Abbott, A.E. von Doenhoff“Theory of Wing Section”, Dover,

1949, 1959

Louis Melveille Milne-Thompson“Theoretical Aerodynamics”,

Dover, 1988Return to Table of Content

April 13, 2023 257

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 2013 - Retired

fluid dynamics

Science

u v u u u u u u

i u x u x

d d t u u t u u u t

u u e u e u e1

d u x u x ik

u v u v u v u v

x e x e x e x x x

x d x d x