differential analysis of fluid flow -...
TRANSCRIPT
Chap. 6
Differential Analysis
of Fluid Flow
Although the control volume analysis gave us accurate
solution of flow physics, it deals with only global information
of a certain volume (control volume). Therefore, it can not
provide the detail information of flow at certain point.
To arrive this goal of gaining the detail flow physics, the
differential analysis should be adopted and this is truly the
governing equation of fluid dynamics written as partial
differential equation (hard to solve by hand CFD
essentially!!!).
6.1 Fluid Element Kinematics
Eulerian derivative:
V
t
Dt
D
zw
yv
xu
tDt
D
termConvective change of rate Convective
termunsteady Local derivate time Local
V s,coordinate
spatial torespectivewith
derivative imesVelocity t
t location, fixed aat
change of Rate
fluid a ofproperty of
change of rate Total
Linear deformation-Stretching
-Rate of the change of volume per unit volume
x
uzyx
x
u
zyx
1
t
zyxzy txx
ux
limzyx
1
dt
Vd
V
1
0t
rate
This rate of change of the volume per unit volume is called
the volumetric dilatation rate, or simply dilatation.
The derivatives, , and cause a linear
deformation (stretching) of the element in the sense that the
shape of the element does not change.
V
z
w
y
v
x
u
dt
Vd
V
1
In general,
x/u y/v z/w
Angular Motion and Deformation
Vorticity(渦度):
If the flow is , then the flow is called “irrotational”.
V
0V
6.2 Conservation of Mass
Conservation of mass: 0mmmt
inoutcv
zyxt
zyxt
mt
cv
z x
2
y
y
vvz y
2
x
x
uumout
z x
2
y
y
vvz y
2
x
x
uumin
Continuity equation:
For incompressible flow,
0z yx
y
vz yx
x
uz yx
t
0
z
w
y
v
x
u
t
0Vt
0z
w
y
v
x
u
Dilatation
0V
or
Ex. 6.2
6.2.2 Cylindrical Polar Coordinates
Cylinderical coordinates
Spherical coordinates
0
z
vv
r
1
r
vr
r
1
t
zr
0Vt
0
v
sinr
1sinv
sinr
1
r
vr
r
1
t
r
2
2
6.2.3 The Stream Function
introduce the stream function, , which satisfy the continuity
equation automatically as
;
substituting these relationships into incompressible continuity
equation,
By introducing the stream function, we can simplify the
problem from two unknowns, u and v, to only one unknown, .
Another merit of using the stream function is that lines along
which is constant are streamlines.
yu
xv
0xyyxxyyxy
v
x
u 22
The change in the value of is given by
Along a line of constant ,
i.e.,
which is the defining equation for a streamline.
Thus, if we know the function (x,y) we can plot lines of
constant to provide the family of streamlines.
udyvdxdyy
dxx
d
udyvdx0d
u
v
dx
dy
Ex. 6.3
6.3 Conservation of Linear Momentum
Newton’s second law of x-direction
gravity,xstress ,xxx FFFa m
gravity,xstress ,xxx FFFa m
Newton’s second law of x-direction
D-2in
y
uv
x
uu
t
u zyx
Dt
Du zyxa m x
zy x yx
z x 2
y
yz x
2
y
y
z y 2
x
xz y
2
x
xF
yxxx
yx
yx
yx
yx
xxxx
xxxxstress,x
xgravity,x g z y x F
Thus, the Newton’s second law in x-direction is
Similarly, in y-direction
x
yxxx
Dt
Du
gyx
1
y
uv
x
uu
t
u
y
yyxy
Dt
Dv
gyx
1
y
vv
x
vu
t
v
The general equation of motion in 3-D is
xzxyxxx
Dt
Du
gyyx
1
z
uw
y
uv
x
uu
t
u
zzzyzxz
Dt
Dw
gyyx
1
z
ww
y
wv
x
wu
t
w
y
zyyyxy
Dt
Dv
gyyx
1
z
vw
y
vv
x
vu
t
v
x-dir:
y-dir:
z-dir:
g1
Dt
VD
6.8.1 Stress-Deformation Relationships
For incompressible Newtonian fluids, it is known that the
stresses are linearly proportional to the rates of deformation
as:
x
u2pxx
y
v2pyy
z
w2pzz
x
v
y
uyxxy
y
w
z
vzyyz
z
u
x
wxzzx
; ;
zzyyxx
3
1p
6.8.2 The Navier-Stokes Equations (Incompressible)
y2
2
2
2
2
2
g z
v
y
v
x
v
y
p1
z
vw
y
vv
x
vu
t
v
x2
2
2
2
2
2
g z
u
y
u
x
u
x
p1
z
uw
y
uv
x
uu
t
u
z2
2
2
2
2
2
g z
w
y
w
x
w
z
p1
z
ww
y
wv
x
wu
t
w
x-dir:
y-dir:
z-dir:
x
2 g ux
p1
Dt
Du
y
2 g vy
p1
Dt
Dv
z
2 g wz
p1
Dt
Dw
; ;
g Vp1
VVt
V 2
(for incompressible flows)
These equations are called Navier-Stokes equations (1822),
named in honor of the French mathematician L. M. H. Navier (1758-
1836) and the English mechanician Sir G. G. Stokes (1819-1903).
Unfortunately, because of the complexity of the Navier-Stokes
equations ( i.e., nonlinear, second order partial differential
equations), they are not amenable to exact solutions except in a few
instances.
CFD (Computational Fluid Dynamics) is essential to solve the Navier-
Stokes equations.
g Vp1
VVt
V 2
Sir George Gabriel Stokes, 1st Baronet FRS
(1819–1903, England)
Claude-Louis Navier(1785–1836, France)
Compressible Navier-Stokes Eq.
For compressible flows,
xzxyxxx gyyx
1
z
uw
y
uv
x
uu
t
u
zzzyzxz gyyx
1
z
ww
y
wv
x
wu
t
w
y
zyyyxyg
yyx
1
z
vw
y
vv
x
vu
t
v
x-dir:
y-dir:
z-dir:
g1
Dt
VD
Continuity eq:
0z
w
y
v
x
u
t
Momentum eq.
Stress relationships:
x
u2pxx
y
v2pyy
z
w2pzz
x
v
y
uyxxy
y
w
z
vzyyz
z
u
x
wxzzx
Energy Equation for Compressible Viscous Flow
Energy Equation:
qt
QVp
Dt
De
agency external
fromin-heat
of Rate
q
t
Q
Dt
Dp
Dt
Dh
or,
Where e=internal energy, Tce v
,/peh Tch ph=enthalpy,
Tkq
: Heat transfer by conduction
: Dissipation function
5 unknowns (u, v, w, p, and ) and 5 equations
6.4 Inviscid Flow (Euler equation)
yg y
p1
z
vw
y
vv
x
vu
t
v
xg x
p1
z
uw
y
uv
x
uu
t
u
zg z
p1
z
ww
y
wv
x
wu
t
w
x-dir:
y-dir:
z-dir:
xg x
p1
Dt
Du
yg y
p1
Dt
Dv
zg z
p1
Dt
Dw
; ;
g p1
VVt
V
Euler’s equation
Swiss mathematician
Leonhard Euler (1707-1783)
Leonhard Euler
6.4.3 Irrotational Flow
An irrotational flow field is defined as the flow
for which , or =0. 0V
0z
v
y
wx
0x
w
z
uy
0y
u
x
vz
That is,
6.4.4 The Bernoulli Equation for Irrotational Flow
For incompressible inviscid irrotational, ( ) flow fields,
the constant of Bernoulli equation is same throughout the
entire flow field and the Bernoulli equation can be applied
between any two points.
(valid between any two points)
- steady incompressible inviscid irrotational flow
0V
22
2
21
1
2
1 gzp
2
Vgz
p
2
V
6.9 Some Simple Solutions for Viscous, Incompressible Fluids
y2
2
2
2
2
2
g z
v
y
v
x
v
y
p1
z
vw
y
vv
x
vu
t
v
z2
2
2
2
2
2
g z
w
y
w
x
w
z
p1
z
ww
y
wv
x
wu
t
w
x2
2
2
2
2
2
g z
u
y
u
x
u
x
p1
z
uw
y
uv
x
uu
t
u
x-dir:
y-dir:
z-dir:
0
z
w
y
v
x
u
t
Continuity :
Incompressible Navier-Stokes Equation
Convective term (nonlinear) Viscous dissipation term (linear)
Mom
entu
m E
quatio
n
in vector form,
- Continuity equation :
- Momentum equation :
0Vt
g Vp1
VVt
V 2
Convective term
(nonlinear)
Viscous term
(linear)
These equations of continuity and Navier-Stokes equations
consist of 4 equations having 4 unknowns (u, v, w, and p).
The unique solution exists mathematically.
CFD(Computational Fluid Dynamics) is essential to solve
the Navier-Stokes and Euler equations.
6.9.1 Steady Laminar Flow Between Fixed Parallel Plates
In this internal flow, v = w = 0.
Continuity eq:
x-dir.
momentum :
“Poiseuille Flow”
0
z
w
y
v
x
u
t
0)(w0
0)(v0
steady0
const)( 0x
u
0
x
dir-zin variationno
0
2
2
2
2
2
2
00
steady0
g z
u
y
u
x
u
x
p1
z
uw
y
uv
x
uu
t
u
y-dir. momentum:
z-dir. momentum:
g
y
0)v( 0
2
2
2
2
2
2
0) v( 0
g z
v
y
v
x
v
y
p1
z
vw
y
vv
x
vu
t
v
0
z
0)w( 0
2
2
2
2
2
2
0)w( 0
g z
w
y
w
x
w
z
p1
z
ww
y
wv
x
wu
t
w
That is,
Continuity:
x-momentum:
y-momentum:
z-momentum:
0x
u
2
2
0)xu/(0
2
2
continuityfrom
0
y
u
x
u
x
p1
x
uu
g-y
p1 0
z
p1 0
x
p1
yd
ud2
2
)x(Cgyp
no variation of u in x-direction, u=u(y only)
no variation of p in z-direction, p=p(x,y)
only) x(fx
p
ode in u
The integration of x-component of momentum gives
Note that is treated as constant because it is not a
function of y.
Two boundary conditions are needed to obtain C1 and C2.
That is, u=0 at
1Cy x
p1
dy
du
dy
du
21
2 CyCy x
p
2
1u
x/p
h y
0C1 2
2 h x
p
2
1C
used for
;
Therefore,
Drag:
Volume Flowrate:
22 hy x
p
2
1u
: Parabolic velocity distribution
y x
p
dy
du
: Linear shear stress distribution
hx
phymax
00ymin ;
hL x
p 2dx h
x
p 2dx2D
L
0
L
0hy
(per unit width in z-dir.)
x
p
3
h2dy hy
x
p
2
1dy uQ
3h
h
22h
h
(per unit width in z-dir.)
( is –’ve value for flowing to +’ve x-direction.)x/p
The volume flowrate is proportional to the pressure gradient
and inversely proportional to the viscosity.
Vmean:
i.e.,
By the way,
Hence,
Here, it was assumed that the present flow is laminar. If the
flow is turbulent , then the above analysis is
not valid.
x
p
3
h2h2VQ
3
mean
x
p
3
hV
2
mean
x
p
2
huu
2
0ymax
meanmax V 5.1u
1400
)h2(VRe mean
6.9.2 Couette Flow
Consider the flow developed by fixing one plate and letting
the other plate move with a constant velocity, U. The flow is
initially at rest.
U
b
x
y
In this internal flow, v = w = 0.
Continuity eq:
x-dir.
momentum :
y-dir.momentum:
z-dir.momentum:
g
y
0)v( 0
2
2
2
2
2
2
0) v( 0
g z
v
y
v
x
v
y
p1
z
vw
y
vv
x
vu
t
v
0
z
w
y
v
x
u
t
0)(w0
0)(v0
steady0
const)( 0x
u
0
x
dir-zin variationno
0
2
2
2
2
continuityfrom
0x
u0
2
2
0
00
continuityfrom
0steady
0
g z
u
y
u
x
u
x
p1
z
uw
y
uv
x
uu
t
u
0
z
0)w( 0
2
2
2
2
2
2
0)w( 0
g z
w
y
w
x
w
z
p1
z
ww
y
wv
x
wu
t
w
Resulting equations are the same as those of Poiseuille flow, except p/x=0.
That is,
Continuity:
x-momentum:
y-momentum:
z-momentum:
0x
u
g-y
p1 0
z
p1 0
0 yd
ud2
2
)x(Cgyp
no variation of u in x-direction,
u=u(y only)
no variation of p in z-direction,
p=p(x,y)
ode in u
The integration of x-component of momentum gives
Two boundary conditions are needed to obtain C1 and C2.
That is, u=0 at y=0 and u=U at y=b
Therefore, ; (constant shear stress throughout the flow)
21 CyCu
b
UC1 0C2 ;
b
yUu
b
U
U
b
x
y
U
The flow in an unloaded journal bearing might be
approximated by this Couette flow if the gap is very small
(i.e., ).iio rrr
irU
io
i
rr
r
;io rrb
U
b
x
y
ri =ri
Couette flow with imposition of streamwise pressure gradient, ( ).
Continuity eq:
x-dir.
momentum :
y-dir.
momentum:
z-dir.
momentum:
g
y
0)v( 0
2
2
2
2
2
2
0) v( 0
g z
v
y
v
x
v
y
p1
z
vw
y
vv
x
vu
t
v
0
z
w
y
v
x
u
t
0)(w0
0)(v0
steady0
const)( 0x
u
0
x
dir-zin variationno
0
2
2
2
2
continuityfrom
0x
u0
2
2
0
00
continuityfrom
0steady
0
g z
u
y
u
x
u
x
p1
z
uw
y
uv
x
uu
t
u
0
z
0)w( 0
2
2
2
2
2
2
0)w( 0
g z
w
y
w
x
w
z
p1
z
ww
y
wv
x
wu
t
w
Couette + Poiseuille Flow
0x/p
That is,
Continuity:
x-momentum:
y-momentum:
z-momentum:
0x
u
g-y
p1 0
z
p1 0
)x(Cgyp
no variation of u in x-direction,
u=u(y only)
no variation of p in z-direction,
p=p(x,y)
ode in u
x
p1
yd
ud2
2
Resulting equations are the same as those of Poiseuille flow.
Only difference is boundary condition.
The integration of x-component of momentum gives
Two boundary conditions are needed to obtain C1 and C2.
That is, u=0 at y=0 and u=U at y=b
Therefore,
1Cy x
p1
dy
du
dy
du
21
2 CyCy x
p
2
1u
0C1
used for
;
U
bx
y
bCb x
p
2
1U 1
2
b
yUbyy
x
p
2
1u 2
0]x/p[
Poiseuille
Couette
Couette+Poiseuille
U
b
x
y
U
U
For positive pressure gradient
For negative pressure gradient
Same and Difference
Flow
Pattern
Resulting
Equation
Boundary Condition
Remark
at y=0 at y=b
Poiseuille 0 0
Couette 0 U
Poiseuille
+Couette0 U
x
p1
yd
ud2
2
x
p1
yd
ud2
2
0 yd
ud2
2
U
bx
y
U
U
2002 WORLD CUP 의 Velocity Vector
2002 WORLD CUP 의 Pressure Contour Line
Take a break
Ex. 6.9
0
z
w
y
v
x
u
t
x2
2
2
2
2
2
g z
u
y
u
x
u
x
p1
z
uw
y
uv
x
uu
t
u
y2
2
2
2
2
2
g z
v
y
v
x
v
y
p1
z
vw
y
vv
x
vu
t
v
z2
2
2
2
2
2
g z
w
y
w
x
w
z
p1
z
ww
y
wv
x
wu
t
w
6.9.3 Steady, Laminar Flow in Circular Tubes
The incompressible Navier-Stokes equations in cylindrical
coordinate system was written as
“Hagen-Poiseuille Flow”
z
vv
r
vv
r
v
r
vv
t
v rz
2
rrr
r
r2
r
2
22
r
2
22
rr g z
vv
r
2v
r
1
r
v
r
vr
rr
1
r
p1
Continuity :
r-momentum :
0Vz
Vr
1rV
rr
1
tzr
z
vv
r
vvv
r
v
r
vv
t
vz
r
r
g
z
vv
r
2v
r
1
r
v
r
vr
rr
1p
r
11
2
2
r
22
2
22
z
vv
v
r
v
r
vv
t
v zz
zzr
z
z2
z
2
2
z
2
2
z g z
vv
r
1
r
vr
rr
1
z
p1
-momentum :
z-momentum :
Since vr = v = 0,
0z
v z
sing
r
p10
cosg
p
r
110
r
vr
rr
1
z
p10 z
Continuity :
r-momentum :
-momentum :
z-momentum :
g
y
x
r
The integration of r- and -direction gives
or
This equation indicates that the pressure is hydrostatically
distributed at any cross-section and is not a function of r or .
The z-momentum equation is
and integration gives
)z(Csinrgp
)z(Cgyp
const.
z
z
pr
r
vr
r
1
2
z Cz
p
2
r
r
vr
r
C
z
p
2
r
r
v 1z
Hence,
Since vz should be finite at r=0, or should be zero at
r=0 from the symmetry condition of velocity profile, C1 must
be zero ( ).
And from vz = 0 at r=R,
Thus,
: Parabolic distribution
21
2
z CrlnCz
p
4
rv
r/vz
)0ln(
2
2 R z
p
4
1C
22
z Rrz
p
4
1v
To obtain a relationship between Q and ,
Mean velocity:
Maximum velocity:
z/p
z
p
8
Rdr r2 Rr
z
p
4
1dr r2vQ
4R
0
22R
0z
z
p
8
R
R
QV
2
2mean
z
p
4
Rvv
2
0rzmax
meanmax V2v
Wall shear stress:
Skin friction coefficient: defined as
Then, the substitution of wall gives
where ReD is called Reynolds number, defined as
mean
RrRr
zwall V
R
4
z
p
2
R
z
p
2
r
r
v
2
mean
wall
f
V2
1C
D
Re
meanmean
mean2
mean
fRe
16
V D 16
V R
18V
R
4
V2
1
1C
D
DVRe mean
D
The flow remains laminar for ReD<2100.
Drag:
z
pLRLV8dz R2 V
R
4 dz R2 D 2
mean
L
0mean
L
0wall
Turbulent
In this flow situation the N-S equations are reduced to the
same equations of the previous section of Hagen-Poiseuille
flow.
Since vr = v = 0,
0z
v z
sing
r
p10
cosg
p
r
110
r
vr
rr
1
z
p10 z
Continuity :
r-momentum :
-momentum :
z-momentum :
6.9.4 Steady, Axial, Laminar Flow in an Annulus
As mentioned before, the integration of r- and -momentum
equation yield that the pressure is hydrostatically distributed
at any cross-section and is not a function of r or .
The integration of z-momentum equation gives
Boundary condition: no-slip condition at r=Ri and r=Ro
Hence,
z/p
21
2
z CrlnCz
p
4
rv
(same as Hagen-Poiseuille)
0vz
o
i
o
2
o
2
i2
o
2
zR
rln
R
Rln
RRRr
z
p
4
1v
Volume Flowrate:
The annular flow is assumed laminar if ReDh <2100, where
ReDh means the Reynolds number based on the hydraulic
diameter, Dh.
i
o
22
i
2
o2
i
4
o
R
Rz
R
Rln
RRRR
z
p
8dr r2vQ
o
i
hmean
D
DVRe
h
io
io
2
i
2
oh RR2
RR2
RR4
perimeter wetted
area sectional-cross4D
Ex. 6.10
p
8
R
z
p
8
RQ
44
i
o
22
i
2
o2
i
4
o
R
Rln
RRRR
z
p
8Q