Post on 18-Jan-2016
1
Chapter 6
Flow Analysis Using Differential Methods(Differential Analysis of Fluid Flow)
2
• In the previous chapter--• Focused on the use of finite control volume for the
solution of a variety of fluid mechanics problems.• The approach is very practical and useful since it
doesn’t generally require a detailed knowledge of the pressure and velocity variations within the control volume.
• Typically, only conditions on the surface of the control volume entered the problem.
• There are many situations that arise in which the details of the flow are important and the finite control volume approach will not yield the desired information
3
• For example --• We may need to know how the velocity varies over the cross
section of a pipe, or how the pressure and shear stress vary along the surface of an airplane wing.
we need to develop relationship that apply at a point,
or at least in a very small region ( infinitesimal volume)
within a given flow field.
involve infinitesimal control volume (instead of finite
control volume)
differential analysis (the governing equations are
differential equation)
4
• In this chapter—
• (1) We will provide an introduction to the differential equation that describe (in detail) the motion of fluids.
• (2) These equation are rather complicated, partial differential equations, that cannot be solved exactly except in a few cases.
• (3) Although differential analysis has the potential for supplying very detailed information about flow fields, the information is not easily extracted.
• (4) Nevertheless, this approach provides a fundamental basis for the study of fluid mechanics.
• (5) We do not want to be too discouraging at this point, since there are some exact solutions for laminar flow that can be obtained, and these have proved to very useful.
5
• (6) By making some simplifying assumptions, many other analytical solutions can be obtained.
for example , μ small 0 neglected inviscid flow.• (7) For certain types of flows, the flow field can be conceptually
divided into two regions— (a) A very thin region near the boundaries of the system in
which viscous effects are important. (b) A region away from the boundaries in which the flow is
essentially inviscid.• (8) By making certain assumptions about the behavior of the fluid
in the thin layer near the boundaries, and using the assumption of inviscid flow outside this layer, a large
class of problems can be solved using differential analysis .
• the boundary problem is discussed in chapter 9.• Computational fluid dynamics (CFD) to solve differential eq.
6
)(.)((.)(.)(.)(.)(.)(.)
.
.
problem. particular afor t,z, y, on x, dependly specifical components
velocity thesehow determine tois analysis aldifferenti of goals theof One
vtz
wy
vx
utt
D
a
a
z
uw
y
uv
x
uu
t
ua
dt
vd
z
vw
y
vv
x
vu
t
va
z
y
x
7
elementtheofndeformatioangularx
v
y
u
elementtheofndeformatiolinearz
w
y
v
x
uNote
v
ratedilationvolumetricvz
w
y
v
x
u
dt
d
,
,,:
fluid, ibleincompressan for 0
)(1
8
flow alIrrotation0
zero is axis-z thearound Rotation
only when ) (i.e.block undeformed an as axis-zabout Rotation
)(2
1 as such (6.12) Eq.From
2 Define
vor
y
u
x
v
x
v
y
uww
y
u
x
vw
Vcurlvwvorticity
OBOA
z
Vcurlv
wvuzyx
kji
ky
u
x
vj
x
w
z
ui
z
v
y
w
kwjwiw
vectorrotationtheW
zyx
2
1)(
2
1
2
1
})()(){(2
1
9
§ 6.2.1 Differential Form of Continuity Equation
inout
cv
vAvA
dAnv
zyxt
dt
dt
zyxd
)()(
element theof surfaces the through flow mass of rate The)(
)(
0
10
s
yx
xx
xxx
x
uuu
y
x
uuu
)(,)( as such , terms
order high neglecting --expansion seriesTaylor
2
)(|
2
)(|
direction-x theinflow The
2
2
2
zyxz
w
zyxy
v
zyxx
uzy
x
x
uuzy
x
x
uu
)(direction -zin rateNet
)24.6()(
direction -yin rateNet
similarly
)23.6()(
]2
[]2
[
direction-in x outflow mass of rateNet
11
. form aldifferenti in equation continuity The
)27.6(0
outflow mass of rateNet :
0][
0)( Since
z
w
y
v
x
u
t
zyxz
wzyx
y
vzyx
x
uNote
zyxz
wzyx
y
vzyx
x
uzyx
t
dAnvdt cscv
12
)31.6(0
)30.6(0
0
flow ibleincompressFor --
)29.6(0
0)(
fluid lecompressib of flowsteady For --
)28.6(0
form In vector
mechanics fluid of equations lfundamenta theof One --
z
w
y
v
x
uor
v
tconst
z
w
y
v
x
uor
v
vt
13
equation. continuity hesatisfy t torequired , w: Determine
?
flow ibleincompressan For
2 6. Example
222
w
zyzxyv
zyxu
),(2
3n Integratio
3)(2
0)()(
0
continuity ofequation thefrom
:Solution
2
222
yxcz
xzw
zxzxxz
w
z
wzyzxy
yzyx
x
z
w
y
v
x
u
14
§ 6.2.2 Cylindrical Polar Coordinates
15
01)(1
) flowunsteady or steady ( flow ibleincompressFor
0)()(1
)(1
flow lecompressib steady,For
scoordinate lcylindricain
equation y continuall theof form aldifferenti theis This
)33.6(0)()(1)(1
z
vv
rr
rv
r
vz
vr
vrrr
z
vv
rr
vr
rt
zr
zr
zr
16
§ 6.2.3 The Stream Function
)36.6(0
)2(0)(0
flow D-2 & plane, ible,incompress steady, of equation continuity For the
0
equation Continuity
y
v
x
u
flowDz
wcte
twhere
z
w
y
v
x
u
t
0)()(
eq. continuity thesatisfiesit that so ; where
function, stream the),(function a Define
xyyxy
v
x
u
xv
yu
yx
satisfied be willmass of onconservati
unknow one unknows two functionstream using
v
u
17
6.3. Example
)42.6(1
01)(1
0)()(1)(1
flow. D-2 place, , ibleIncompress
for equation y continuall the, scoordinate lcylindricaIn
rv
rv
v
rr
rv
rz
vv
rr
vr
rt
r
rzr
18
Example 6.3 Stream FunctionExample 6.3 Stream Function
• The velocity component in a steady, incompressible, two The velocity component in a steady, incompressible, two dimensional flow field aredimensional flow field are
Determine the corresponding stream function and show on a Determine the corresponding stream function and show on a sketch several streamlines. Indicate the direction of glow sketch several streamlines. Indicate the direction of glow along the streamlines.along the streamlines.
4xv2yu
19
Example 6.3 Example 6.3 SolutionSolution
(y)fx2(x)fy 22
12
Cyx2 22
From the definition of the stream functionFrom the definition of the stream function
x4x
vy2y
u
For simplicity, we set C=0For simplicity, we set C=0
22 yx2 Ψ=0Ψ=0
ΨΨ≠≠0012/
xy 22
20
§ 6.3 Conservation of Linear Momentum
amF
amFordt
vdmFEq
VCsmallFor
FdAnvvdvtdt
vmD
dmvdvP
Pdt
Ddv
dt
D
dt
vmD
sys
cvcv
csvc
cv
sys
syssys
sys
sys
systema for law 2nd Newtons The
)44.6(
..
)44.6()()(
momentumlinear for the t theorem transpor Reynolds theFrom
where
)(
momentumlinear For the
..
21
Figure 6.9 (p. 287)Components of force acting on an arbitrary differential area.
22
Figure 6.10 (p. 287)Double subscript notation for stresses.
23
Figure 6.11 (p. 288)Surface forces in the x direction acting on a fluid element.
24
§ 6.3.2 Equation of Motion
Velocities stresses ----- Unknowns
.rest at or motion in fluid)or (solid continuumany toapplicable also areThey
fluid. afor motion ofequation aldifferenti General
)50.6()(
)50.6()(
)50.6()(
using
cz
ww
y
wv
x
wu
t
w
zyxg
bz
vw
y
vv
x
vu
t
v
zyxg
az
uw
y
uv
x
uu
t
u
zyxg
dzyxm
zzyzxzz
zyyyxyy
zxyxxxx
zszbzzz
ysybyxx
xsxbxxx
maFFmaF
maFFmaF
maFFmaF
25
§ 6.4 Inviscid Flow
stress normal ecompressiv
0&
0 flow invisicidFor
. ssfrictionleor , nonviscous , inviscid be tosaid are
negligible be toassumed are stresses shearing thein which field Flow
zzyyxxP
0&0&
& , water andair assuch , fluidcommon Some
waterair
waterair small
26
§ 6.4.1 Euler’s Equations of Motion
)51.6()(
)51.6()(
)51.6()(
00 with (6.50c) & (6.50b) (6.50a) Eq From
cz
ww
y
wv
x
wu
t
w
z
Pg
bz
vw
y
vv
x
vu
t
v
y
Pg
az
uw
y
uv
x
uu
t
u
x
Pg
P
and
z
y
x
zzyyxx
(6.52) Eq solve to ) (usingSimplify
. solve toDifficulty
)52.6(])([
motion of equations Eulers as toreferredCommonly are equations These
vvt
vPg
27
§ 6.4.2 The Bernoulli Equation
equation Eulers equation Bernoulli section In this
law 2nd Newtons equation Bernoulli 2 3.section In
)(2
1
)()(2
becomes (6.53) Eq
)()(2
1)(
identity vector and ng Usi
)53.6()(
) statesteady ( 0 where
)52.6(])([
equation Eulers Form
2 vvzgvp
vvvvpzg
vvvvvv
zgg
vvpg
t
v
vvt
vpg
28.2
02
1
)]([
./)(0)]([
)()(
)(
Similarly
state.)steady if (
,,,,
)]([)()(2
1
streamline alonglength aldifferenti aLet
)(2
1
22
22
2
2
constgzvdp
gdzdvdp
sdvv
papeyofoutinvvbecausesdvv
dzkdzjdyidxkz
zdsz
dvsdv
dpdzz
pdy
y
pdx
x
p
dzdydxz
p
y
p
x
pdsp
sdvvsdzgsdvsdp
kdzjdyidx
ds
dsvvzgvp
29
streamline a along Flow
flow ibleIncompress
flowSteady
flow Inviscid
2
fluid ibleincompress Inviscid,For
2
constgzvp
30
§ 6.4.3 Irrotational Flow
flow alirrolationan of exampleAn
0
0
.)(
does. flow uniform a However,equations. threehesesatisfy t
not could field flow generalA
Vorticity)( 0Vorticity00)(2
1
flow alIrrotation
w
v
constUu
x
w
z
uz
v
y
wy
u
x
v
VorVorVw
31
§ 6.4.5 The Velocity Potential
mass. ofon conservati of econsequenc a --
function stream The
. field flow theofally irrotation theof econsequenc a --
potential velocityThe :Note
flow D-2 torestricted is
flow D-3 general afor defined becan
potential velocity function scale a is),,( where
,,
0
00)(2
1 flow alirrotationFor --
kwjviuv
zyx
zw
yv
xu
x
w
z
uz
v
y
wy
u
x
v
wvuzyx
kji
v
vorvw
32
flow. potential a calledcommonly is flow of typeThis
. field flow alIrrotation
, ibleincompress , Inviscid
)66.6(00
thatfollowsit ,)( flow alirrotation and
)0( fluid ibleincompressan For
2
2
2
2
2
22
equationLaplace
zyxor
v
v
�
pressures calculate Toequation Bernoulliwith
determined becan
conditionsboundary
withEq.(6.66) from
known is If
vor
w
v
u
33
)71.6(01
)(1
)70.6(;1
;
)69.6( Since
),,(
)68.6(1
)67.6((.)(.)1(.)
(.)
,, s,Coordinate
2
2
2
2
22
zrrr
rr
zv
rv
rv
evevevv
zrwhere
ez
er
er
ez
er
er
zrlCylindricaIn
zr
zzrrr
zr
zr
34
2133
1
22
,/10,30 if
(2)point at pressure (b)
potential velocity(a)
:e Determin
right on re Figu
&/2sin2 : Given
function stream 6.4 Example
zzmkgkpaP
mrsmr
)1()(2cos2)(2cos4
;1
;
2sin42sin2
2cos42)2(cos21
2sin211
)(
:
12
1
2
22
CrCdrrdrvdrv
zv
rv
rv
conditonslcylindricainpotentialVelocity
flowalIrrotation
rrr
v
rrr
rr
v
rv
rv
massofonconservati
conditionslcylindricainfunctionstreama
Solution
rr
zr
rr
35
)(2cos2(2) & (1) Eq
)2(.)(0)(
)(2)2sin(22sin4
)](2cos2[1
2sin4
1 Since
)1()(2cos2
)1(
2
11
122
12
12
AnsCr
constCCorC
Crr
Crr
r
rv
Cr
EqFrom
36
)(36/)416(/102
11030
(3) Eq From
/245.01616
5.0 (2),point At
/41616
1 (1),point At
16)2sin4()2cos4(
sin
)3(2
1
2
1
)(2
1
2
1
as written becan equation Bernoulli the
fluid, ibleincompress ,nonviscous a of flow alirrotationan For (b)
223332
2222
2
122
1
2222
222
22
2112
21222
21
21
1
Anskpasmmkgpap
smvrv
mr
smvrv
mr
rrrv
vvvevevvce
vvpp
zzgzvp
gzvp
rrr
37
§ 6.5 Some Basic, Plane Potential Flows
00
0)()(0
flow alIrrotation , ibleincompressfor (6.66) Eq From
0)()(
(6.74) Eq Usingflow) nal(Irrotatio
(6.72) Eq From
2
2
2
2
2
2
2
2
2
2
2
2
2
2
yxzyx
yyxxy
v
x
u
yxxxyy
x
v
y
u
flowplaneyx
38
§ 6.8 Viscous Flow
es. velociti& stresses hebetween t iprelationsh a establish tonecessary isIt
equations. than unknowns more are There
. fluid afor motion ofequation aldifferenti General
)50.6()(
)50.6()(
)50.6()(
. assuch
Eq.6.50, motion, of equations general derived previously thereturn tomust we
motion, fluid of analysis aldifferenti theinto effects viscouseincorporat To
cz
ww
y
wv
x
wu
t
w
zyxg
bz
vw
y
vv
x
vu
t
v
zyxg
az
uw
y
uv
x
uu
t
u
zyxg
zzyzxzz
zyyyxyy
zxyxxxx
39
§ 6.8.1 Stress - Deformation Relationships
)125.6(2
)125.6(2
)125.6(2
2
2
2
n.deformatio of rate the torelatedlinearly are
stresses t theknown tha isit , fluids Newtonian , ibleincompressFor
cz
wP
by
vP
ax
uP
z
wP
y
vP
x
uP
zz
yy
xx
zz
yy
xx
)125.6()(
)125.6()(
)125.6()(
fx
w
z
u
ey
w
z
v
dx
v
y
u
xzzx
zyyz
yzxy
40
)126.6(][
)126.6(]1
[
)126.6(]1
)([
)126.6(2
)126.6()1
(2
)126.6(2
fluids ibleincompress Newtonian,for stresses The
scoordinatepolar lcylindricaIn
fr
V
z
V
eV
rz
V
dV
rr
V
rr
cz
VP
br
VV
rP
ar
VP
zrrzzr
zzz
rrr
zzz
r
rrr
41
§ 6.8.2 The Navier–Stokes Equations
)127.6()()(
direction-z
)127.6()()(
direction -y
)127.6()()(
direction -x
(6.31), Eq ,continuity of Eq. and
(6.125f) ~ (6.125a) with (6.50c) ~ Eq.(6.50a) From
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
cz
ww
y
wv
x
wu
t
w
z
w
y
w
x
w
z
Pg
bz
vw
y
vv
x
vu
t
v
z
v
y
v
x
v
y
Pg
az
uw
y
uv
x
uu
t
u
z
u
y
u
x
u
x
Pg
z
y
x