Hydrodynamics Lecture 1

7/21/2019 Hydrodynamics Lecture 1

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Lecture One

CENG 6601- Hydrodynamics

Department of Civil EngineeringEi-!" !e#elle $niversity

1


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Lecture 1Content%ntroduction

&luid !ec'anics (pplications

)roperties of &luid!at'ematical )reliminaries and ensor

(nalysis*ector (nalysis

Dot productCross product

+inematics

,
http://var/www/apps/conversion/tmp/scratch_5/Hydrodyamics%20Course%20outline.docxhttp://var/www/apps/conversion/tmp/scratch_5/Hydrodyamics%20Course%20outline.docx


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%ntroduction( &luid is a sustance .'ic' deforms continuously

under t'e action of a s'earing stress/( olid is a sustance t'at resist a s'ear stress y

static deformation

&luids- li2uid and gas

3


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'e study of t'e motion of

4uids t'at are practicallyincompressile 5suc' as li2uids"

especially .ater" and gases atlo. speeds is usually referredto as hydrodynamics.

( sucategory of'ydrodynamics is hydraulics,.'ic' deals .it' li2uid 4o.s in

7


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&luids in )ure ciences

ag o a c rcu a on ong-range.eat'er prediction8 analysis ofclimate c'ange 5gloal.arming

mesoscale .eat'er patterns8

s'ort-range .eat'er prediction8tornado and 'urricane.arnings8 pollutant transport

,/Oceanograp'yaocean circulation patterns

causes of El Nino" e9ects ofocean currents on .eat'er andclimate

e9ects of pollution on livingorganisms

3/Geop'ysicsaconvection 5t'ermally-driven4uid motion in t'e Eart':smantle understanding of platetectonics" eart'2ua#es"volcanoes

convection in Eart':s molten


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&luids in )ure ciences7/ (strop'ysicsagalactic structure and

clusteringstellar evolution=fromformation ygravitational collapse

to deat' as asupernovae" from.'ic' t'e asicelements are

distriuted t'roug'outt'e universe" all via4uid motion

iological sciences

acirculatory and respiratorysystems in animals6


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&luids in tec'nology1/ %nternal comustionengines=all types of

transportation systems,/uro?et" scram?et" roc#et

engines aerospacepropulsion systems

3/ @aste disposalac'emical treatment

incineration

cse.age transport and treatment

7/ )ollution dispersal in t'e

atmosp'ere 5smog8 inrivers and oceans


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Contd/5a crude oil and natural gas

transferral

5 irrigation facilities

5c oBce uilding and'ouse'old pluming

A/ &luidstructureinteraction5a design of tall uildings

5 continental s'elf oil-drillingrigs

5c dams" ridges" etc/

5d aircraft and launc' ve'icleairframes and controlsystems

/ Heating" ventilatingand air conditioning5H*(C systems

/ Cooling systems for'ig'-densityelectronic devices

digital computers


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Contd/

10/olar 'eat and geot'ermal 'eatutiliFation

11/(rti;cial 'earts" #idney dialysismac'ines" insulin pumps

1,/!anufacturing processes5a spray painting automoiles" truc#s" etc/

5 ;lling of containers" e/g/" cans of soup" cartonsof mil#" plastic ottles of soda

5c operation of various 'ydraulic devices5d c'emical vapor deposition" dra.ing of synt'etic

;ers" .ires" rods" etc/


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Continuum (ssumption

!icroscopic approac' (nalyFe molecularstructure and associated collisions 5e/g/pressure is due to t'e net ec'ange ofmomentum at a solid surface

!acroscopic 5continuum approac' (nalyFeul# e'avior of 4uid 5e/g/ pressure is forceeerted y 4uid per unit area of solid surface

@'ile a ody of 4uid is comprised of molecules"

most c'aracteristics of 4uids are due toaverage molecular e'avior/

'at is" a 4uid often e'aves as if it .erecomprised of continuous matter t'at is in;nitely

divisile into smaller and smaller parts/10


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11


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&luid )roperties

&luid properties a9ect 'o. a 4uid reacts toapplied forces" t'us t'e 4uid:s motion

%ntensive and etensive properties%ntensive properties are independent of siFe or

volume

e/g/" density" pressure" temperatureEtensive properties are dependent of siFe or

volume and are additive for susystems/

e/g/" mass" volume" area" force%ntensive properties are otained from t'e ratio

of etensive properties/

e/g/" densitymassvolume" pressure1,


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Density 5mass%t is a ratio of t'e mass of 4uid element to

its volume

Density of a 4uid a9ects its 4o. in t.o.ays%t determines t'e inertia of a unit volume

of 4uid t'us its

acceleration .'en su?ected to a givenforcefor t'e same force" lo. density 4uids

accelerate more readily t'an 'ig' density

4uids

13


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>ul# !odulus E CoeBcient of compressiility

CoeBcient of 'ermal Epansion >

&or .ater E ,/1 10E

Nm3

> 1/


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)ressure&luids do not support s'ear stresses)ressure is 5compression stress at a point

in a static 4uid/

Net to velocity " it is t'e most dynamicvariale in 4uid dynamics

Di9erence in pressure causes a 4uid to 4o.

(tmosp'eric pressure at m/s/l 101/3 #pa"

.ill e set to Fero for convenience

16


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emperature

emperature is t'e measure oft'e internal energy of a 4uid/

Generally" temperaturedi9erences cause 'eat transfer/

%n t'is course" .e treat

isot'ermal situation/

1A


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*iscosity*iscosity 5also called dynamic viscosity" or

asolute viscosity is a measure of a 4uidsresistance to deformation under s'earstress/

&or eample" crude oil 'as a 'ig'erresistance to s'ear t'an does .ater/

'e symol used to represent viscosity is

5mu and its unit is 5#gm/s/%t is given y t'e ratio of t'e s'earing stress

to rate of deformation/

Ne.tonian and Non Ne.tonian &luids1


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Shear strain angle will grow as f(t)

For fluids such as water, oil, air

stress strain rate

Viscosity = Resistance to shear

1

y

u tt

u

x

t


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However,

As , , 0

But

here dyna!ic viscosity" #his is a constitutive

relation, which relates forces to !aterial $fluid% &ro&erties"

For fluids'

(Stress is &ro&ortional to

strain rate("

For solids'(Stress is &ro&ortional to

strain( $=)%

,0

y

tu

=tan

dy

du

dt

d=

t y

dt

d

t

dydu=


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Notes on shear stress

$i% Any shear stress, however small, &roduces relative!otion"

$ii% *f =0, du+dy=0, ut -0"$iii% Velocity &rofile cannot e tangent to a solid

oundary . #his re/uires an infinite shear stress"

(o.sli&( condition' u=0 at solid oundary"

y

10,1


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1

Bingha! 2lastic

Real 2lastic

Shear.#hinning Fluid

Newtonian

Shear.#hic3ening Fluid

Types of fluids

ewtonian fluid' Stress is linearly &ro&ortional to strain rate"

Shear.thinning' 4etchu&, whi&&ed crea!

Shear.thic3ening' 5orn starch in water

,,

dydu +

=

dy

du


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Units

6yna!ic Viscosity

e"g" S*'

,3

dydU +

=

7+87878 dydU =

99978

LT

M

LT

ML

Area

Force==

=

dy

dU

TLT

L ::==

T

ML

LT

M

TLT

M

=== 978s2as+!9


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e"g" S*'

4ine!atic Viscosity

,7

7+87878 =v

T

L

M

L

LT

Mv

9;

78 ==

LT

M=78

;78

L

M=

Sto3es:+:0 9< = sm

=v


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athe!atical &reli!inaries and tensor analysis

,6


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Di9erentiation%f a function is di9erentiale at a point "

t'e derivative value at t'at point is"

One sided limit

,A


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Vector AnalysisRight-handed, Cartesian coordinate system

1nit vectors

2osition vector

here are co!&onents of

,

;xz

:xx

9xy;a

:a

9

a

>x

ka

ja

ia

?%:,0,0$

?%0,:,0$

?%0,0,:$

;

9

:

=

=

=

;

;

9

9

:

: xaxaxax ++=

x%,,$ ;9: xxx


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Vector: 4undu. @Any /uantity whose co!&onents

change li3e the co!&onents of a &osition

vector under the rotation of the coord"

syste!"

Scalar: Any /uantity that does # change withrotation or translation of the coord" syste!

e"g" density $% or te!&erature $#%

,

: 9 ;>

$ %x x x x


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Vector AnalysisConsider vectors and

Dot )roduct

Cross )roduct

30

AB

A

B B


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'e 5PDelQ" PGradientQ" or PGradQOperator

(/'e Gradient 5Grad and DirectionalDerivatives

'e rate of c'ange of a scalar function in an aritrary direction ise2ual to t'e scalar product of t'e gradient .it' t'e unit vector" " int'at direction/ince

@$ % is &er&endicular to

lines and gives !agnitude

and direction of !ai!u! s&atial

rate of change of

31

zk

yj

xi

+

+

=

???

zk

yj

xi

++=???

:C=

;C=

9C=

C=


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'e 5PDelQ" PGradientQ" or PGradQOperator

>/ Divergence and Divergence'eorem (pplication of to vector 2uantity

Consider vector ;eld

'e dot product of .it' is called t'edivergence

%n volume terms" t'e net volume 4u out oft'e volume d* e2uals t'e product of and d*/

3,

kfjfiff zyx ??? ++=

z

f

y

f

x

f

f zyx

+

+

=


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Divergence 'eorem 5Gauss'eorem

Given an aritrary volume "enclosed y t'e surface " .it'out.ard unit normal vector at allpoints on " t'e divergence t'eorem

states

$outward unit nor!al vector

to surface ele!ent%

$infinitesi!al surface area%

$infinitesi!al volu!e%

33

dSnfdVfSV =

>n

dV

dA

AAreaSurface

VVoume


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*f is a scalar, vector, or any order tensor

S&ecifically, if is a vector

or

6ivergence #heore!' *ntegral over volu!e of divergence

of flu = integral over surface of the flu itself37

V A >i

dV n dAC C

=

%$C x

%$C x !

CC

=

i

i iV A

i

dV n dAx

> = R RV A!dV ! n dA


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)a!&les@ $a%

$%

$c% *f a scalar is $advectively% trans&orted

y the velocity>

U

or

6ivergence of flu within volu!e et flu across

3/ dV / n dA =

V A ># dV # n dA =

%$>>

ii

advadvUFUF ==

=

A advVi

i

advdAnFdV

x

F

>>

= AV adv dAnFdVF >>>


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C/ Curl and to#es: 'eorem'e cross product of .it' t'e

vector is called t'e curl

'e integral of around t'e contourenclosing e2uals t'e component of int'e direction normal to multiplied y

36

ky

f

x

fj

x

f

z

fi

z

f

y

f

fffzyx

kji

f xyzxyz

zyx

???

???

+

+

=

=


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5 $ounding curve%

A $o&en surface%

to#es: 'eorem

Let e a t.o-dimensional surfaceenclosed y C" t'en

@Surface integral of the curl of a vector,>

U , e/uals

the line integral of>

U along the ounding curve3A

>

n

ds

dA

= CS dxfndSf

> > >$ % = A Cu n dA u ds


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'e LaplacianDivergence of a gradient for scalar

Descries t'e net 4u of t'e scalar2uantity into volume

3


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ensors and %nde Notation

ome .ays of multiplying scalars or vectorsresult in tensor

calar is a tensor of ran# 5order Fero

*ector is a tensor of order one

%f order is not speci;ed" a tensor implies oforder t.o

Order of tensor speci;es t'e numer of

indices to descrie it

3


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Tensor:Assigns a vector to each direction in s&ace $ 9nd

order%

e"g"

Rows 5olu!ns

$a% *sotro&ic D 5o!&onents are unchanged y a rotation of

fra!e of reference $i"e" inde&endent of direction . e"g"4ronec3er 6elta iE%

$% Sy!!etric ' AiE = AEi $in general AiE = A#Ei%

$c% Anti.sy!!etric' AiE = .AEi

$d% 1seful result' AiE = :+9 $AiEAiE%:+9 $AEi.AEi%

= :+9 $AiEAEi%:+9 $AiE.AEi%=

70

11 12 13

21 22 23

31 32 33

A A A

A A A A

A A A

RRijA


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!instein summation convention

A% *f an inde occurs twice in a ter! a su!!ation over the

re&eated inde is i!&lied

B% Higher.order tensors can e for!ed y !ulti&lying

lower order tensors'

a% *f 1iand VEare :st

.order tensors then their &roduct1i VE= iE is a 9

nd.order tensor" Also 3nown as

vector outer &roduct $ %"

% *f AiEand B3lare 9nd.order tensors then their &roduct

AiE B3l = 2iE3l is a

= U

0

9

=

=

ji

k

ijkj

k

ijki xx

U

x

U

x

> > > > > >$ % $ 9% = a a a a a a

>> &aLet =

( )

( )

mjkmijk

mkm

kk

kjijki

x

aaaa

x

aa&And

&a&a

=

==

=

>>

>

>>


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: if : if : if : if

7

( )

mjjim

mjjmi

x

aa

x

aaaa

= >>

i= mj= mi= j=

>>

>>

>

9

%$%9+$

aaaa

aaaax

x

aa

x

aa

jjmm

i

j

ij

i

mm

=

=

=

ja%$>


48/127

*elocity Gradient ensor

Laplacian *elocity


49/127

*f is a scalar, vector, or any order tensor

S&ecifically, if is a vector

or

6ivergence #heore!' *ntegral over volu!e of divergence

of flu = integral over surface of the flu itselfi

dV n dAC C

=

%$C x

%$C x !

CC

=

i

i iV A

i

dV n dAx

> = R RV A

!dV ! n dA


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C'ain Sule 5di9erentiation


51/127

C'ain Sule


52/127

Curvilinear Coordinates


53/127

Cylindrical


54/127

p'erical coordinate


55/127

p'erical


56/127

tudies 4uid motion .it'out concernfor t'e force causing t'e 4o.

+inematics


57/127

De;nition


58/127

*elocity &ield

&ield p'ysical 2uantity associated .it' every point in inspace-time continuously de;ned in t'e region/

'e velocity at point C is otained y ta#ing t'e averagevelocity of t'e molecules in *T

'e velocity ;eld at any point can e de;ned li#e .ise

leading to *elocity &ield $/$ is continuous function of space and time" i/e/ $5"y"F

$ is a vector function/ (t any point 5"y"F at any instanttime 5t it 'as t'ree component" u" v" ./

u5"y"F"t- along -direction

v5"y"F"t- along y-direction

@5"y"F"t- along F direction

60


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&lo. *isualiFations%f t'e 4o. ;eld 5velocity is #no.n eit'er y

solving t'e motion e2uations or measuring itla;eld" t'e 4o. ;eld can e displayed usingpat'lines" strea#lines" streamlines/

Pathlines

( pat' t'at a 4uid particle traces over time/

'e position of a 4uid particle is given y t'reenumers 5" y" F/ 'e position is related to t'e

instantaneous velocity y Given t'e velocity ;eld" t'e successive positions 51"

y1" F1 of a 4uid particle from its initial position 50" y0"

F0 can e determined8 and

at'lines are generated eperimentally61


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Streaklines

6,

( line t'at connects at some instant of time

all particles t'at 'ave gone t'roug' a ;edposition or point/

trea#lines are generated eperimentally/ (dye is in?ected continously at a c'osen

point

St li


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Streamline%t is a line in a 4o. ;eld t'at is every.'ere

tangent to t'e velocity vector $ at eac'point along t'e streamline for any instant oftime t/

treamlines display a snaps'ot of t'e entire

4o. ;eld at a single instant in time"63

s

ds

c

o flu By definition' $i%

*elocity is tangent to ds'ence parallel

%,,$>

'vuU=

%,,$> dzdydxds=

0>

= dsU

;

>

9

>

:

>

;

>

9

>

:

>

000%$

%$%$

aaaaudyvdx

a'dxudzavdz'dy

++=++

udyvdx'dxudzvdz'dy === KK

'

dz

v

dy

u

dx==


62/127

treamtue

treamlines never intersect eac' ot'erecause" at any point" t'ere can e onlyone direction of t'e velocity/

( streamtubeis a surface in t'e 4o.

formed from streamlines and closed uponitself to form a tue of variale cross-section/

67

)article pat's and streamlines are not in


63/127

)article pat's and streamlines are not ingeneral coincident/ >ut"teady &lo. streamlines" pat'lines and

strea#lines coincide)at'line 5

(rc lengt' along t'e pat'line is

Di9erential e2uation for pat' lines and streamlines

are t'e same

$nsteady 4o. .'ic' direction doesnotc'ange .it' time

6x

>>xx +

>x

>u

>>duu+

L

>x

%,$ 0>>

txu

%,N$O 0>>>>

txxudu ++

#herefore after ti!e t


88/127

#herefore, after ti!e

Relative !otion of two &oints de&ends on the velocity

gradient, , a 9nd.order tensor"

to first order

I

'

2I'

#aylor series e&ansion of

.. $A%

$% !eans order of =

&ro&ortional to

0

t

ttxuxtx(tuxttxuxxx %,$%PP$PP%,$%$ 0>>>

9

>>>0

>>>>

L

> ++++=

t

x

uxtuxxxse)arationinC*an+e

j

ij

===

>>>

L

>

tuxxx >>>

L

>+=

j

i

x

u

ttxux %,$ 0>>>

+

tx(uxtxuxx

ttxxuxx

%NPP$PP%,$O%$

%,$%$

9

>>>0

>>>>

0>>>>>

++++=+++

%,$ 0>>>

txxu +

$9% 6eco!&osition of otion


89/127

$ % &

@Any tensor can e re&resented as the su! of a sy!!etric

&art and an anti.sy!!etric &art@

=O rate of strain tensorN Orate of rotation tensorN

ote'

$i% Sy!!etry aout

diagonal

$ii% Q uni/ue ter!s

inear angular straining1

+

+

=

i

j

j

i

i

j

j

i

j

i

x

u

x

u

x

u

x

u

x

u

9

:

9

:

iE iEe r= +

+

+

+

+

+

+

=

;

;

;

9

9

;

;

:

:

;

9

;

;

9

9

9

9

:

:

9

:

;

;

:

:

9

9

:

:

:

9

9

9

9

:

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

eij


90/127

ote'

$i% Anti.sy!!etry aout

diagonal

$ii% ; uni/ue ter!s $r:9, r:;, r9;%

Rotation

#er!s in

,

=

0

0

0

9

:

;

9

9

;

;

:

:

;

9

;

;

9

9

:

:

9

:

;

;

:

:

9

9

:

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

rij

: :iE 9 9

>$ % = ijk k ijk k r u

; 9

; :

9 :

0

: 09

0

=

ijr

: ::9 :9; ; ;9 9

: : :;9 ;9: : : :9 9 9

" "

$ %

= =

= = =

e + r

r

I h 3 hi i


91/127

etIs chec3 this assertion aout rij

#he reci&e'$a% ! = i and l =E

$% l = i and ! = E

gives

3

9

=

=

=

=

=

mijk k ijk km

mjik km

mim j i jm

ji

j i

ij

u

x

u

x

ux

uu

x x

r

*nter&retation tu

xxx ij

=L


92/127

*nter&retation

Relative velocity due

to defor!ation of

fluid ele!ent

Relative velocity due

to rotation of

ele!ent at rate :+9

7

>

$ %

:

9

:9

:$ %

9

= +

=

= +

= + R

i j ij ij

ij j ijk k j

ij j ikj k j

i ij j i

u x e r

e x x

e x x

u e x x

tx

xxxj

j >>L

> >

= =

ii j

j

x x uu x

t x

$;% #y&es of !otion and defor!ation "


93/127

$i% 2ure #ranslation

M9

M:

t:

t0

$ii% inear 6efor!ation . 6ilatation

M9

M:

t:

t0

a

>

=V

dVm

'e La. of Conservation of!


103/127

!ass

@it'in some prolem domain de;ned y acontrol volume" t'e net mass of 4uidpassing from outside to inside t'roug' t'econtrol surfaces e2uals t'e net increase of

mass in t'e control volume/%n its most general form t'e la. stat t'at

P mass is neit'er created not destroyed ina closed systemQ

10

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Hydrodynamics Lecture 1