Prinsip dan Persamaan Dasar
-
Upload
revandifitro -
Category
Documents
-
view
223 -
download
1
Transcript of 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
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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
-
8/12/2019 Prinsip dan Persamaan Dasar
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
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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.
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
15/40
Fundamental Principles and EquationsContinuity Equation
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
16/40
Fundamental Principles and EquationsContinuity Equation
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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
Fundamental Principles and EquationsContinuity Equation
For steady flow
0V
and
0dSV
S
...........(2.42)
...........(2.41)
...........(2.43)
...........(2.44)
...........(2.45)
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
19/40
Fundamental Principles and EquationsMomentum Equation
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
20/40
Fundamental Principles and EquationsMomentum Equation
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
21/40
Fundamental Principles and EquationsMomentum Equation
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
22/40
Fundamental Principles and EquationsMomentum Equation
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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.
-
8/12/2019 Prinsip dan Persamaan Dasar
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
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
-
8/12/2019 Prinsip dan Persamaan Dasar
30/40
Fundamental Principles and EquationsPathlines and streamlines of a flow
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
-
8/12/2019 Prinsip dan Persamaan Dasar
31/40
Fundamental Principles and EquationsPathlines and streamlines of a flow
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
-
8/12/2019 Prinsip dan Persamaan Dasar
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
-
8/12/2019 Prinsip dan Persamaan Dasar
33/40
Fundamental Principles and EquationsKinematic of fluid motion
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
-
8/12/2019 Prinsip dan Persamaan Dasar
34/40
Fundamental Principles and EquationsKinematic of fluid motion
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)
-
8/12/2019 Prinsip dan Persamaan Dasar
35/40
Fundamental Principles and EquationsKinematic of fluid motion
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
-
8/12/2019 Prinsip dan Persamaan Dasar
36/40
locity.angular vefiniteahaveelementsfluidthat theimpliesthis
,rotationalcalledisflowtheflow,ainpointeveryat0V
locity.angular venohaveelementsfluidthat theimpliesthisal,irrotationcalledisflowtheflow,ainpointeveryat0V
-
8/12/2019 Prinsip dan Persamaan Dasar
37/40
-
8/12/2019 Prinsip dan Persamaan Dasar
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
-
8/12/2019 Prinsip dan Persamaan Dasar
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
-
8/12/2019 Prinsip dan Persamaan Dasar
40/40