Aliran Inkompresible
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Transcript of Aliran Inkompresible
FUNDAMENTALS OF INVISCID, INCOMPRESSIBLE FLOW
Bernoulli’s Equation
Inviscid flow, no body forces, momentum equation given by eq. (2.104a) becomes
x
p
z
uw
y
uv
x
uu
t
ux
p
Dt
Du
.............(3.1)
For steady flow
dxx
pdx
z
uwdx
y
uvdx
x
uu
dx
x
p
z
uw
y
uv
x
uu
1
by multiply
1
,
.............(3.2)
Consider the flow along a streamline in 3-D space. The equation of streamline is given by eq. (2.108a to c)
0
0
udyvdx
wdxudz
.......(3.3)
.............(2.108b)
.............(2.108c)
dxx
pdz
z
udy
y
udx
x
uu
dxx
pdz
z
uudy
y
uudx
x
uu
1
1
(3.3), eq. into ngsubstituti
.............(3.4)
.............(3.5)du
dxx
pud
dxx
pudu
1
2
1
1
2 .............(3.6)
dzz
pwd
dyy
pvd
zy
1
2
1
1
2
1
component and for
2
2 .............(3.7)
.............(3.8)
Bernoulli’s Equation
dzz
pdy
y
pdx
x
pwvud
1
2
1
(3.8) through (3.6) eq. adding
222
.............(3.9)
dpdzz
pdy
y
pdx
x
p
wvu
2222 V
however,
.............(3.10)
.............(3.11)
VV
V2
1
(3.9), into (3.11) and (3.10) eq. ngsubstituti
2
ddp
dpd
.............(3.12)
Euler’s equation
2
22
2
11
2
1
2
212
V2
1V
2
1
2
V
2
V
VV
constant,
2
1
2
1
pp
pp
ddpp
p
V
V
.............(3.13)
Bernoulli’s equation
streamline along V2
1 2 constp
flow t the throughou V2
1 2 constp
Bernoulli’s equation for irrotional flow
.............(3.14)
.............(3.15)
The venturi and Low-Speed Wind Tunnel
xAA
1
1
1
A
V
2
2
2
A
V
1 2
Quasi-one-dimensional flow in a duct
S
V S
dV
0dSV
flowsteady for
0dSVt
eq., continuity theof form Integral
...........(2.39)
.............(3.16)
1 2
0dSVdSVdSV
duct, theapply to
A A wall
.............(3.17)
wall
0dSV
0dSV
wall, thealong
.............(3.18)
1
111VdSV
1station at
A
A .............(3.19)
2
222VdSV
2station at
A
A .............(3.20)
222111
222111
VV
00VV
(3.17) into (3.20) to(3.18) ngSubstituti
AA
AA
.............(3.21)
2211VV
flow, ibleincompress
AA .............(3.22)
The venturi and Low-Speed Wind Tunnel
1
1
1
A
p
V
2
2
2
A
p
V
x
p
Throat
Pressure is aminimum atthe throat
12
V
V2
V
VV
V2
V
2
21
2121
21
2
2
112
21
12
12
2212
21
AA
pp
A
App
A
A
pp
.............(3.23)
.............(3.24)
.............(3.25)
.............(3.26)
The venturi and Low-Speed Wind Tunnel
1
1
1
A
V
22
2
A
V
,3
3
3
A
V
model
Settlingchamber
nozzle Test section diffuser
fan
2121
22
233
222
211
23
23
12
12
V2
V
V2
1V
2
1V
2
1
VV
VV
pp
ppp
A
A
A
A
.............(3.27)
.............(3.28)
.............(3.29)
.............(3.30)
2
12
2122
22
2
1
221
22
1
2V
V2
V
AA
pp
A
App
.............(3.31)
.............(3.32)
Pitot TubeTotal pressure Static pressure
oB
B
pp
V
0
1V
B
A
101
02
11
22
2V
pressure totalpressure dynamic press static
0V2
1
V2
1V
2
1
pp
pp
ppBBAA
.............(3.32)
.............(3.34)
Pressure coefficient
2V2
1
q
where
q
ppC p
.............(3.36)
2
2
2
22
22
22
V
V1
V21
VV21
(3.36), into ngsubstituti
VV2
1
V2
1V
2
1
flow, ibleincompress
p
p
C
q
ppC
pp
pp
.............(3.37)
.............(3.38)
0V
0V0
constant flow, ibleincompress
0Vt
equation, continuity
0V
flow, ibleincompress
.............(3.39)
.............(2.43)
Condition on Velocity For Incompressible Flow
.............(3.39)
Governing Equation For Irrational, Incompressible Flow: Laplace’s Equation
0
0
,irrational and ibleincompress
V
, potential velocity a flow, irrational
0V
flow, ibleincompress
2
.............(3.39)
.............(2.145)
.............(3.40)
Laplace’s Equation
0
s,coordinatecartesian
2
2
2
2
2
22
zyx
zyx
,,
.............(3.41)
011
s,coordinate lcylindrica
2
2
2
2
22
zrrr
rr
zr
,,
.............(3.42)
011
s,coordinate spherical
22
2
sinsinsin
sin
,,
rr
rr
r
.............(3.43)
Governing Equation For Irrational, Incompressible Flow: Laplace’s Equation
0yx
(3.44), into b) and (2.141a ngsubstituti
0V
function, stream flow, ibleincompress D,2
22
xyyxxy
y
v
x
ux
v
yu
0
0yx
(2.122), into b) and (2.141a ngsubstituti
0
irrational flow, ibleincompress D,2
2
2
2
2
yx
yx
y
u
x
v
.............(3.44)
.............(2.141a)
.............(2.141b)
.............(3.45)
.............(2.122)
.............(3.46)
Conclusions:1.Any irrotational, incompressible flow has a velocity potential and stream function
(for 2-D) that both satisfy Laplace’s Equation.2. Conversely, any solution of Laplace’s equation represents the velocity potential or
stream function (2-D) for an irrotational, incompressible flow.
V V
V
V
xfyb
Boundary Conditions
0
V
infinity,at
xyv
yxu
Boundary conditions on velocity at infinity
.............(3.47a)
.............(3.47b)
surface
b
u
v
dx
dy
orS
n
0
or
0
0nnV
wall,at the
.............(3.48a)
.............(3.48b)
.............(3.48c)
.............(3.48e)
Boundary conditions on velocity at the wall
x
y
Uniform Flow:1st Elementary Flow
h
l
C curve
V
cont
cont
rx
y
0
V
vy
and
ux
.............(3.49a)
.............(3.49b)
x
x
then
xgconst
yfx
V
const.V
(3.49b), eq from
V (3.49a), eq from
)
.............(3.52)
........(3.50).......(3.51)
.............(3.53)
Velocity potential for a uniform flow
y
vx
and
uy
V
then
0
V
.............(3.54a)
.............(3.54b)
.............(3.55)
Stream function for an incompressibleuniform flow.
sin
cos
sincos
r
and
r
ryrx
V
V
and s,coordinatepolar in
.............(3.56)
.............(3.57)
Uniform Flow:1st Elementary Flow
0
000dSV
C curve closeconsider
dSV
flow, uniform ain n circulatio
hlVhlVC
C
0
0,V,irrational is flow uniform
dSV
0
00VdSVdSV
V
S
CC
const.
.............(2.127)
.............(3.58)
.............(2.128)
Source/Sink Flow: 2nd Elementary Flow
0
Vr
cVr
r
rr
rlm
v
drllrdm
V2
rate flow volume
VV
flow mass total
2
0
2
0
r
rl
v
r
r
2V
V2
length unit per rate flow volume
...............3.59a)
...............3.59b)
. .....3.60)
.....3.60)
.....3.62)
.....3.61)
Source/Sink Flow: 2nd Elementary Flow
0V1
2V
r
potentialvelocity 2
r
r
c
r
rfcont
fr
respect to gintegratin2
r respect to gIntegratin
ln
rln
2
0V
2V
1
function stream the
r
rr r
rf
2
respect to w/ gintegratin
2
0dSV S
.....3.63)
.....3.64)
.....3.65)
.....3.66)
.....3.67)
.....3.68)
.....3.69)
.....3.70)
fconst
r
respect to w/ gintegratin
.....3.71)
.....3.72)
Combination of a uniform flow w/ a source and sink
const.2
streamline2
function stream
sin
sin
rV
rV .....3.74)
.....3.75)
sin
cos
Vr
V
rV
rVr 2
1
fieldvelocity
.....3.76)
.....3.77)
,
sin
cos
Vr,θ
r
Vr
V
2
at located exists,point stagnation
one that find we, and for solving
0
02
points stagnation
.....3.78)
.....3.79)
3.17 fig.in ABC
curve asshown is streamline This const.2
by described ispoint stagnation the
throughgoes that streamline theHence,
const2
const22
(3.75) eq. intopont stagnation ngsubstituti
sinV
V
21
21
2
22
function stream
sin
sin
rV
rV
const2
bygiven is streamline theofequation The
such that located are points stagnation
21
2
sinrV
V
bbOBOA
.....3.80)
.....3.81)
.....3.82)
022
streamline stagnation
Bpoint at 0
Apoint at
streamline stagnation
21
21
21
sinrV .....3.83)
Doublet flow
strengthdoublet
2const
0
limit
doublet. as definedpattern flow
special aobtain weconstant,remain while0 as22
isfunction stream the
flow in the point any At distance aby separated
-strength ofsink a and strength of source a
21
:
.
d
l
l
ll
Pl
.....3.84)
.....3.85)
l
P
d
a
br
3.85 into 3.86 eq. ngSubstituti
Hence
cossin
cos
sin
lr
l
b
ad
b
ad
lrb
la
.....3.86)r
lrl
l
lr
l
l
l
sin
cossin
cossin
2
or
2const
0
limit
or
2const
0
limit
.....3.87)
Doublet flow
sin
sin
cos
cr
cr
r
2
or
const2
flowdoublet for streamline the2
potential velocity the
.....3.88)
.....3.89)
Nonlifting flow over a circular cilinder
221
or 2
function stream the
rVrV
rrV
sin
sinsin
.....3.91)
2
2
2
1
as written becan 3.91 eq ,V2let
r
RrV
R
sin .....3.92)
0 1
0 1
points stagnation oflocation the
1
1 2
1
1 11
fieldvelocity
2
2
2
2
2
2
2
2
3
2
2
2
2
2
sin
cos
sin
sinsin
cos
cos
Vr
R
Vr
R
Vr
RV
Vr
R
r
RrV
rV
Vr
RV
r
RrV
rrV
r
r
.....3.93)
.....3.94)
.....3.95)
.....3.96)
sin
sin
VV
V
Rr
VR
r
RrV
r
2
0
with 3.94 and 3.93 eq.
by given iscylinder theof theof
surface on theon distributiVelocity
2
01
is points stagnation he through tpasses that streamline ofequation
2
2 .....3.97)
.....3.98)
.....3.99)
.....3.100)
2
2
41
3.38 and 3.100equation combining
1
tcoefficien pressure
sin
p
p
C
V
VC .....3.38)
.....3.101)
dyCC
cc
dxCCc
c
TE
LElpupa
c
uplpn
1
1
become 1.16 and 1.15equation
0
,,
,, .....3.102)
.....3.103)
Vortex flow
2
2
2dSV radius
of streamlinecircular given a aroundn circulatio
C
andr
V
or
rVr
r
constV
C,
.....3.105)
.....3.106)
.....3.104)
dS
πCor
dSπC
dSdS
r
dSπC
S
SS
S
2 V
V2
3.109 and 3.108 eq combining
V V
0Let
VdSV2
flow of plane thelar toperpendicuboth
direction, same in the are dS and VBoth
dSVC2
2.137 and 3.106 eq. combining
origin. at theexcept alirrotation is flowvortex
.....3.107)
.....3.108)
.....3.109)
.....3.110)
V
become 1103eq hence
00
.
,,dSr
π
Γφ
rV
r
Vr r
2
b and 3.111a eq. gintegratin2
1
0
flow for vortex potential velocity the
.....3.111a)
.....3.111b)
.....3.112)
rπ
Γ
rV
r
Vr r
ln2
b and 3.113a eq. gintegratin2
01
flow for vortexfunction stream the
.....3.113a)
.....3.113b)
.....3.114)
Lifting Flow Over A Cylinder
2
2
1 1
radius
cylinder aover flow liftingnon
r
RrV
R
sin .....3.92)
R
r
π
Γ
R
rπ
Γ
ln
ln
ln
2
2constant and,
constant 2
strength w/ for vortexfunction stream the
2
2
.....3.115)
.....3.116)
.....3.117)
R
r
r
RrV lnsin
21
2
2
21
.....3.118)
RVθ
rV
r
RV
rVr
RV
rV
r
RV
rVr
RV
r
r
4
obtain we, for solving and
3.122 eq. intoresult thisngsubstituti R,r 3.121, eq. from
02
1
01
points stagnation
21
1
fieldvelocity
2
2
2
2
2
2
2
2
arcsin
sin
cos
sin
cos .....3.119)
.....3.120)
.....3.121)
.....3.122)
.....3.123)
22
22
2
241
2211
tcoefficien pressure2
2
cylinder, theof surface on thevelocity
RVRVC
orRVV
VC
RVVV
Rr
p
p
sinsin
sin
sin
TE
LE
lp
TE
LE
upd
TE
LE
lpupad
dyCc
dyCc
c
or
dyCCc
cc
1
1
1
,,
,,
0
0
2
1
2
1
2 and
3.127 into 3.128 eq. ngsubstituti
n
coordpolar
to3.127 eq. converting
dC
dCc
Rc
dRdyR siy
lp
upd
cos
cos
cos,
,
,
.....3.125)
.....3.126)
.....3.127) .....3.129)
.....3.128)
0
obtainy immedietel we
0 and
0
0
thatnoting and 3.130 into 3.126 eq. ngsubstituti
2
1
2
1
2
1
2
0
2
0
2
2
0
2
0
2
0
d
pd
ppd
plpup
c
d
d
d
dCc
dCdCc
CCC
cossin
cossin
cos
cos
coscos
, ,,
.....3.130)
.....3.131a)
.....3.131b)
.....3.131c)
.....3.132)
c
up
c
lpnl dxCc
dxCc
cc
00
1
1
,,
2
0
02
2
1
2
1
2
1
3.133 into 3.134 eq. ngsubstituti
coordpolar toconverting
dCc
dCdCc
dRdxR x
pl
uplpl
sin
sinsin
sin,cos
,,
.....3.133)
.....3.134)
.....3.135)
.....3.136)
RVcl
obtainy immedietel we
0 and
0
0
thatnoting and 3.136
into 3.126 eq. ngsubstituti
2
0
2
2
0
3
2
0
d
d
d
sin
sin
sin .....3.137a)
.....3.137b)
.....3.137b)
.....3.138)
VL
RVRVLRS
ScVScqL
L
ll
'
',
'
'
22
1 12
2
1
span perunit lift
2
2 .....3.139)
.....3.140)
The lift per unit span is directly proportional to circulation
Kutta-Joukowski theorem
The Kutta-Joukowski Theorem and The Generation of Lift