Prinsip dan Persamaan Dasar

8/12/2019 Prinsip dan Persamaan Dasar

1/40

FUNDAMENTAL PRINCIPLES AND

EQUATIONS

The principle is most important, not the detail.Theodore von Karman, 1954


2/40

Fundamental Principles and EquationsVector

kAjAiAA zyx

cartesian

eAeAeAA

zrr

cylindrical

zkyjxir (Position vector)

zzx

y

yxr

arctan

22

Cartesiancylindrical transformation

eAeAeAA rr

spherical

22

222

222

arccos

arccosarccos

yx

x

zyx

z

r

z

zyxr

Cartesianspherical transformation

...........(2.6)

...........(2.8)


3/40

Fundamental Principles and EquationsScalar and Vector

trTtzrTtzyxTT

trtzrtzyx

trptzrptzyxpp

,,,,,,,,,

,,,,,,,,,

,,,,,,,,,

321

321

321

Scalar field

kVjViVV zyx

Vector field

tzyxVV

tzyxVV

tzyxVV

zz

yy

xx

,,,

,,,

,,,

xyyxzxxzyzzyzyx

zyx

zzyyxx

zyx

zyx

BABAkBABAjBABAi

BBB

AAA

kji

BA

BABABABA

kBjBiBB

kAjAiAA

Scalar and vector products

cartesian

zr

zr

zr

zzrr

zzrr

zzrr

BBBAAA

eee

BA

BABABABA

eBeBeBB

eAeAeAA

cylindrical

BBB

AAA

eee

BA

BABABABA

eBeBeBB

eAeAeAA

r

r

r

rr

rr

rr

spherical

...........(2.9)

...........(2.10)

...........(2.11)

...........(2.12)

...........(2.13)

...........(2.14)


4/40

Fundamental Principles and EquationsGradient of Scalar Field

npds

dp.

kz

pj

y

pi

x

pp

zyxpp

,,

Cartesian

,,,,,, 321 rpzrpzyxpp

Scalar field

p

yx, .3 constp

.2 constp

.1 constp

Isolines of

Pressure

321 ppp

y

x

Direction of the

maximum change

inpat the point (x,y)

The gradient of p, at a given point in space is defined

as a vector such that:

1. Its magnitude is the maximum rate of change ofpper

unit length of the coordinate space at the given point.

2. Its direction is that of the maximum rate of change of p

at the given point.

p

(directional derivativein s direction)

y

x

n

yx,

p

s

...........(2.15)

...........(2.16)


5/40

Fundamental Principles and EquationsGradient of Scalar Field

zr ez

p

e

p

rer

p

p

zrpp

1

,,

Cylindrical

ep

re

p

re

r

pp

rpp

r

sin

11

,,

Spherical

...........(2.17)

...........(2.18)


6/40

Fundamental Principles and EquationsDivergence of a Vector Field

z

V

y

V

x

VV

kVjViVzyxVV

zyx

zyx

,,

Cartesian

z

VV

rrV

rrV

eVeVeVzrVV

zr

zzrr

11

,,

cylindrical

V

rV

rVr

rrV

eVeVeVrVV

r

rr

sin

1sin

sin

11

,,

2

2

spherical

Vector field

,,,,,, rVzrVzyxVV

The divergence of a vector is ascalar quantity.

...........(2.19)

...........(2.20)

...........(2.21)


7/40

Fundamental Principles and EquationsCurl of a Vector Field

y

V

x

Vk

x

V

z

Vj

z

V

y

Vi

VVV

zyx

kji

V

kVjViVV

xyzxyz

zyx

zyx

,,,,,, rVzrVzyxVV

Cartesian

zr

zr

zzrr

VrVVzr

eree

rV

eVeVeVV

1

VrrVVr

erree

rV

eVeVeVV

r

r

rr

sin

sin

sin

12

Cylindrical Spherical

The curl of V is a vector quantity.

...........(2.22)

...........(2.23) ...........(2.24)


8/40

Fundamental Principles and EquationsLine Integrals

Vector field

A curve C in space connecting point aand b, dsis elemental length of the curve.

n is unit vector tangent to the curve.

,,,,,, rAzrAzyxAA

ds

n

A

C

a

b

ds

C

A

Defined the vector ds=nds. Line integralofA along curve C from point ato bis

b

aA

If the curve C is closed, the line integral is given by

ds

CA ds

Counterclockwise direction around Cis considered positive


9/40

Fundamental Principles and EquationsSurface Integrals

S dS

pn

C

Closed surface S

Volume V

dS

nThe three-dimensional surface area S

is bounded by the closed curve CVolume Venclosed by the closed surface S

Define a vector elemental area dS=ndS. In term of dS, the surface

Integral over the surface Scan be difined in three ways

s

p dS

s

A dS

s A dS

= surface integral of a scalarpover the open surface S

(the result is a vector)

= surface integral of a vectorAover the open surface S

(the result is a scalar)

= surface integral of a vectorAover the open surface S(the result is a vector)

Closed surface:

S

p dS

S

A dS

S A dS


10/40

Fundamental Principles and EquationsVolume integral

V

dV

V AdV

is a scalar field in space, volume integral over the volume V of the quantity is

= volume integral of a scalar over the

volume V (the result is a scalar)

= volume integral of a vector Aover thevolume V (the result is a vector)

A is a vector field in space, volume integral over the volume V of the quantity A is

Relation between line, surface and volume integral

dSds

SCAA A : vector filed(Stokes theorem)

VS

dVAA dS (divergence theorem)

VS

pdVp dS (gradient theorem) p: scalar field

...........(2.26)

...........(2.27)

...........(2.25)


11/40

Fundamental Principles and EquationsModel of the fluid

VControl

volume V

Control surface S S

Volume dVVolume dV

Finite control volume fixed in space

with the fluid moving throught it

Finite control volume moving with the fluid

such that the same fluid particles are always in

the same control volume

Finite control volume approach

Infinitesimal fluid element approach

Infinitesimal fluid element fixed in space

with the fluid moving throught it

Infinitesimal fluid element moving along a

stream line with the velocity V equal to the

local flow velocity at each point.


12/40

Fundamental Principles and EquationsPhysical Meaning of The Divergence Of Velocity

n

V

tVS

dS

V

Moving control volume, an infinitesimalelement of the surface dSmoving at the

local velocity V. The change in the volume of

the control volume V, due to just the

movement of dSover a time increament t is

equal to the volume of the long, thin cylinder

with base area dS and altitude ( ).ntV

dSVnV tdStV

Total change in volume of the whole volume: dSV S

tDV (surface integral)

Divided by t : SdSVdSV1

S

ttDt

DV

Divergence theorem dVDt

DV

V

VdSVS

...........(2.28)

...........(2.29)

...........(2.30)


13/40

Fundamental Principles and EquationsPhysical Meaning of The Divergence Of Velocity

V dVDt

DV

V

V

dVDt

VD

V

V

is small enough such that divergence of V essentially the same value throught

V

Dt

VD

V

Dt

VD

V

1V

Divergence of Vis physically the time rate of change of the volume of movingfluid element per unit volume.

V ...........(2.31)

V V

...........(2.32)


14/40

Fundamental Principles and EquationsContinuity Equation

V

V dt

Vndt

A(edge view)

Consider the fluid element with velocity V

that pass through area A

AdtVnvolume

AdtVnmass

AVdt

AdtVm nn Mass flow: nVA

m

fluxMass

Mass flow

...........(2.33)

...........(2.34) ...........(2.35)


15/40


Physical principle: Mass can be neither created nor destroyed

dS

V

S

dS

VdV

Net mass flow out of control

Volume through surface S

Time rate of decrease of mass

inside control volume V=

CB

Elemental mass flow across thew area dS is

dSV dSVn + outflow and - inflow

Net mass flow out of the entire control surface Sis S

B dSV

The mass contained within the elemental volume dVis dV

Total mass inside control volume is V

dV

The time rate of decrease of mass inside control volume is

V

dV

t

C

...........(2.36)

...........(2.37)

...........(2.38)


16/40


S

B dSV

V

dVt

C

VS

dVt

dSV

B=C

SV

dVt

0dSVThe continuity equation

in integral form

Control volume is fixed in space, the limit integration are also fixed, so

0dSVdSV

S SVV

dVt

dVt

...........(2.40)


17/40

dVS V

VdSV Applying the divergence theorem

Substitusi

0V dVdVt

VV

0V

dVt

V

0V

t

The continuity equation

in partial differential form


For steady flow

0V

and

0dSV

S

...........(2.42)

...........(2.41)

...........(2.43)

...........(2.44)

...........(2.45)


18/40

Fundamental Principles and EquationsMomentum Equation

VF:lawsecondsNewton' mdtd

Physical principle Force = time rate of change of momentum

Force:

1. Body force: gravity, electromagnetic, acting on the fluid volume.

2. Surface force: pressure, shear stress, acting on the control surface.

f net body force per unit mass, the body force on the elemental volume dV, dVf

V

dVfforceBody

dSpressuretodueforcesurfaceelemental p

S

pdSforcePressure

SV

pdV viscousFdSfF

...........(2.47)

...........(2.48)

...........(2.49)

...........(2.50)


19/40


Time rate of change of momentum = G + H

G: net flow of momentum out of control volume across surface S

H: time rate of change of momentum due to unsteady fluctuations of flow properties

inside volume V

dVt

V

S

VH

VdSVG

dV

tm

dt

d

VS

VVdSVHGV

viscous

FfdS-VdSVV

dVpdVt

VSSV

FV mdt

d

Momentum equation in integral form

...........(2.54)

...........(2.55)


20/40


Gradient theorem

VS

pdVpdS

viscousFf-VdSVV dVpdVdVtVVSV

substitusi

(vector eq.)

kjiV wvu

viscousxx

Ff-dSV

componentthe

dVdVx

pudVtu

x

VVSV

(scalar eq.)

dVuuu VSS VdSVdSV

Divergence theorem

...........(2.56)

...........(2.57)

...........(2.58)

...........(2.59)


21/40


0FfV

substitusi

viscousx

dVx

pu

t

u

V

x

0FfV

viscousx

xx

pu

t

u

viscousxFfV x

x

pu

t

u

viscousyFfV y

y

pv

t

v

viscouszFfV z

z

pw

t

w

Momentum eq. in

differential form

Navier-Stokes Eq.

...........(2.60)

...........(2.61a)

...........(2.61b)

...........(2.61c)


22/40


For steady, inviscid flow and no body forces, equation become

SS

pdS-VdSV

xpu

V

y

pv

V

zpw V

Euler Equation

...........(2.62)

...........(2.63a)

...........(2.63b)

...........(2.63c)


23/40

Fundamental Principles and EquationsEnergy Equation

Physical principle: Energy can be neither created nor destroyed, it can only change in form

ewg

B1= rate of heat added to fluid inside control volume from surroundings

B2= rate of work done on fluid inside control volumeB3=rate of change of energy of fluid as it flows through control volume

(First law of thermodynamics)

321 BBB

V

QdVqBviscous1

VS

WdVpBviscous2

VfdSV

2

VdSV

2

V 22

3 edVe

tB

SV

V

QdVqviscous

VS

WdVpviscous

VfdSV

SV

edVet

dSV2

V

2

V 22

...........(2.76)

...........(2.77)

...........(2.86)


24/40

Fundamental Principles and EquationsEnergy Equation

''viscousviscous

22

VfVV2

V

2

VWQpqee

t

Partial differential form

For steady, inviscid flow, adiabatic andno body forces, equation become

VV2

V2pe

S

e dSV2

V2

S

p dSV

...........(2.87)

...........(2.88)

...........(2.89)


25/40

Fundamental Principles and EquationsSubstantial Derivative

Fluid element

at time t=t1

1

2

i

j

k

y

x

z

Same fluid

element

at time t=t2

V1

V2

tzyx

tzyxww

tzyxvv

tzyxuu

wvu

,,,

,,,

,,,

,,,

field,Density

wherek,jiVfield,Velocity

22222

11111

isdensity2,point

isdensity1,point

tzyx

tzyx

,,,

,,,

sorder term-higherseries,Taylor12

112

112

1

121

12

tt

tzz

zyy

yxx

x

Dividing by t2-t1, and ignoring the higher-order terms,

112

12

112

12

112

12

112

12

ttt

zz

ztt

yy

ytt

xx

xtt

Dt

D

tttt

12

12

12

limit Instantaneous time rate

Of change of density

derivativelSubstantia

Dt

D

.........(2.92)


26/40

Fundamental Principles and EquationsSubstantial Derivative

wttzz

vtt

yy

uttxx

tt

tt

tt

12

12

12

12

12

12

12

12

12

limit

limit

limit

tzw

yv

xu

Dt

D

........(2.93)

z

w

y

v

x

u

tDt

D

Substantial derivative in

cartesian coordinates

zyx

kji

VtDt

D

Local derivative Convective derivative

z

Tw

y

Tv

x

Tu

t

TT

t

T

Dt

DT

V

example

........(2.94)

........(2.95)

........(2.96)


27/40

Fundamental Principles and EquationsFundamental Equation in Term of the Substantial Derivative

VVV

:identityVector

0V

t

........(2.97)

0VV

t

........(2.43)

Continuity eq.

.....(2.98)

0V

Dt

D.....(2.99)

Substantial derivative of continuity eq.

viscousx FfV xxpu

tu

...........(2.61a)

tu

t

u

t

u

uu

uu

VV

VV

...........(2.100)

...........(2.101)

Substituting into (2.61a)

viscousx

Ff

VV

xx

p

uu

t

u

t

u

viscousx

FfVV xx

pu

tu

t

u

=0 (Continuity eq.)

viscousx FfV xxp

ut

u

...........(2.102)

X componen of momentum eq.


28/40

viscousx

FfV xx

pu

t

u

Fundamental Principles and EquationsFundamental Equation in Term of the Substantial Derivative

viscousx

FfV xx

pu

t

u

Dt

Du

viscousx

Ff xx

p

Dt

Du

...........(2.103)

...........(2.104a)

In similar manner, equations (2.61b) and (2.61c)

viscousy

Ff yy

p

Dt

Dv

viscousz

Ff zz

p

Dt

Dw

...........(2.104b)

...........(2.104c)

Energy equation

''

viscousviscous

2

VfVDt

2V

WQpq

eD

...........(2.105)


29/40

Fundamental Principles and EquationsPathlines and streamlines of a flow

Element A

1

Pathline for element A

Element A at some

later time

Element B at some

later time

Element B1

V

V

Pathlines for two different fluid elements

passing through the same point in space;

unsteady flow

Velocity vector

Streamlines

Streamlines

V

V

1

2

ds

F(x,y,z)=0

Streamlines

Pathlines

Sme pathline for

all fluid element

Going through point 1

x

y

z

For stedy flow, streamlines and pathlines are the same


30/40


Streamline: f(x,y,z)=0, Streamline equation ?

kjiV

kjids

scoordinateCartesian

0Vds

ds.toparallelisV

,streamlinetheofelementdirectedabeds

wvu

dzdydx

wvu

dzdydx

kji

Vds

0k

ji

udyvdx

wdxudzvdzwdy

..........(2.106)

..........(2.107)

0

0

0

udyvdx

wdxudz

vdzwdy ..........(2.108a)

..........(2.108b)

..........(2.108c)

Differential equation for

the streamline


31/40


xfy

u

vV

u

v

dx

dy

Streamline in two dimensional

cartesian space

Streamltube in three-dimensional

cartesian space

u

v

dx

dy ..........(2.109)

0udyvdx


32/40

Fundamental Principles and EquationsKinematic of fluid motion

Consider a two-dimensional fluid element, a square ABCDfor simplicity. when the fluidflows this element is subject to various forces and as a result undergoes a complex motionand a possible deformation as indicated in the figure, and assumes a shape like A`B`C`D`.It appears that the complex deformation of the element can be split into four basicconstituents :1. Translation2. Linear Deformation3. Rotation

4. Angular Deformation


33/40


dy

dyy

uu

dxx

vv

v

u

dx

A

B

C

Fluid element at time t

dy

dx

tdxx

v

tdyy

u

1

B

A

C2

K

Fluid element at time tt

x

y

tv tincreamenttimeduringmovesAthatdirectionyinDistance

tdxx

vv

tincreamenttimeduringmovesCthatdirectionyinDistance

tdxx

v

tvtdxx

v

v

AtorelativeCofdirectionyinntdisplacemeNet


34/40


tx

v

dx

tdxxv

2tan

222angle,smallaissince tan

............(2.110)

tx

v

2 ............(2.111)

Consider lineAB. The x cmponent of the

velocity at pointAat time tis u.

dyyuutB

at timepointofvelocity

tdyyutiA

B

increamentover timetorelative

ofdirectionxin thentdisplacemenet

............(2.112)

ty

u

dy

tdyxu

1tan

anglesmallais-since1

ty

u

1 ............(2.113)

y

u

tdt

d

t

1

0

1 limit

............(2.114)

x

v

tdt

d

t

2

0

2 limit

............(2.115)


35/40


definedisplane,inlocityangular ve xy

y

u

x

v

dt

d

dt

d

z

z

2

1

2

1 21

............(2.116)

............(2.117)

Angular velocity of the fluid element in 3-D space

kji2

1

kji

y

u

x

v

x

w

z

u

z

v

y

w

zyx

kji

2vorticity,

y

u

x

v

x

w

z

u

z

v

y

w

............(2.118)

............(2.119)

V ............(2.120)

Curl of thr velocity

vorticity

x

z

y

i

j

k


36/40

locity.angular vefiniteahaveelementsfluidthat theimpliesthis

,rotationalcalledisflowtheflow,ainpointeveryat0V

locity.angular venohaveelementsfluidthat theimpliesthisal,irrotationcalledisflowtheflow,ainpointeveryat0V


37/40


38/40

kk

(2.119)eqfromthenplane),y-(xldimensiona-twoisflowtheif

y

u

x

v,

0

0al,irrotationisflowtheif

y

u

x

v

............(2.121)

............(2.122)

Is condition of irrotational for two-dimensional flow


39/40

y

u

x

v

dt

d

dt

d

dt

d

xy

xy

(2.115)and(2.114)ngsubstituti

strainofratetime-Strain

inchangetheisplanexyin theseenas

elementfluidtheofstrainthe,definitionBy

2.28fig.from

12

12

12

x

w

z

u

z

v

y

w

xz

yz

planexzandyzIn the

............(2.123)

............(2.124)

............(2.125)

............(2.126a)

............(2.126b)

............(2.126c)

z

w

y

w

x

w

z

v

y

v

x

vzu

yu

xu


40/40

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