chapter 6 differential analysis of fluid flow student
TRANSCRIPT
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Offered by: Hang-Suin Yang
Chapter 6
Chapter 6 Differential Analysis of Fluid Flow
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Outline
6.1 Fluid Element Kinematics
6.2 Conservation of Mass
6.3 The Linear Momentum Equation
6.4 Inviscid Flow
6.5 Some Basic, Plane Potential Flows
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Outline
6.6 Superposition of Basic, Plane Potential Flows
6.7 Other Aspects of Potential Flow Analysis
6.8 Viscous Flow
6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
6.10 Other Aspects of Differential Analysis
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6.1 Fluid Element Kinematics
We can break the element's complex motion into four components: translation, rotation, linear deformation, and angular deformation.
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6.1 Fluid Element Kinematics
6.1.1 Velocity and Acceleration Fields Revisited
i j kV u v w
This method of describing the fluid motion is calledthe Eulerian method. It is also convenient to expressthe velocity in terms of three rectangular componentsso that
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D
D
i j
k
i j kx y z
V V V V Va u v w
t t x y z
u u u u v v v vu v w u v w
t x y z t x y z
w w w wu v w
t x y z
a a a
6.1 Fluid Element Kinematics
The acceleration of a fluid particle can be expressed as
material derivative
or substantial derivative
D( )
Du v w V
t t x y z t
where
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6.1 Fluid Element Kinematics
6.1.2 Linear Motion and Deformation
V x y
6.1.2 Linear Motion and Deformation
u tu
u x tx ( V)
u ud u x t y u t y x y t
x x
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6.1 Fluid Element Kinematics
( V) Vu u
d x y t tx x
the rate at which the volume is changing per unit volume due to the gradient ∂u/∂x is
V
0
1 ( V) lim
V t
d u t u
dt x t x
1-D
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6.1 Fluid Element Kinematics
If velocity gradients ∂v/∂y and ∂w/∂z are also present, then
1 ( V)
V
d u v wV
dt x y z
Volumetric dilatation rate
3-D
For an incompressible flow
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6.1 Fluid Element Kinematics6.1.3 Angular Motion and Deformation
Rotation
Deformation
Rotation + Deformation
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6.1 Fluid Element Kinematics
u t
v t vv x t
x
uu y t
y
uy t
y
vx t
x
x
y
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6.1 Fluid Element Kinematics
uy t
y
vx t
x
x
y
O A
B
OA t t tv
x
( )v
x t xx
dl rd
Similarly
OB
u
t y
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6.1 Fluid Element Kinematics
1
2
1
2
v uvx yx
u v uy x y
z
Similarly
1 1 1 , ,
2 2 2x y z
w v u w v u
y z z x x y
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6.1 Fluid Element Kinematics
1i j k i j k
2
1
2
x y z
w v u w v u
y z z x x y
V
The three components, ωx, ωy, and ωz can be combined to give the rotation vector, ω, in the form
The vorticity, ζ, is defined as a vector that is twice the rotation vector; that is,
2 V
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6.2 Conservation of Mass
xm
y ym
x xm
ym
zm
z zm z
x
y
y
x
z
( )
( )
CVin out
x y z
x x y y z z
mm m
tm m m
m m m
Conservation of mass
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6.2 Conservation of Mass
,
,
,
xx x x x x
yy y y y y
zz z z z z
m um u y z m m x m x y z
x xm v
m v x z m m y m x y zy y
m wm w x y m m z m x y z
z z
( )CVmx y z x y z
t t t
where
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6.2 Conservation of Mass
0 0
0
( ) 0
CVin out
mm m
t
u v wx y z x y z
t x y z
u v wV
t x y z t
u v wu v w
t x y z x y z
DV
Dt
Continuity Equation
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6.2 Conservation of Mass6.2.2 Cylindrical Polar Coordinates
Div in cylindrical coordinates
( )( ) ( )1
( )( ) ( )1 1
r z
r z
vr v r vV
r r z
vr v v
r r r z
( , , )p r z
z
r
x
y
z
reθe
ze
Continuity equation
( )( ) ( )1 10r zvr v v
Vt t r r r z
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6.2 Conservation of Mass
6.2.3 The Stream Function
Steady, incompressible, two-dimensional flow field can be expressed in terms of a stream function.
( ) 0
0
DV
Dt
u v wu v w
t x y z x y z
Continuity Equation
0u v
x y
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6.2 Conservation of Mass
Velocity components in a two-dimensional flow field can be expressed in terms of a stream function.
0 0u v
x y x y y x
u v
ψ(x, y) called the stream function
, u vy x
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6.2 Conservation of Mass
0 constantd dx dy vdx udyx y
dy v
dx u
dy v
dx u
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6.2 Conservation of Mass
1( , )x y c
2c
3c
4c
1c
2c
3c 4c
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6.2 Conservation of Mass
The change in the value of the stream function is related to the volume rate of flow.
d dx dy vdx udy dqx y
the volume rate of flow, q, betweentwo streamlines such as ψ1 and ψ2 canbe determined by
2 1q d
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6.2 Conservation of Mass
In cylindrical coordinates the continuity equation for incompressible, plane, two-dimensional flow reduces to
( )( ) ( )1 10r zvr v v
Vt t r r r z
( )1 1 0r vrv
r r r
( )1 1 1 1 r vrv
r r r r r r r
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6.2 Conservation of Mass
( )1 1 1 1 0r vrv
r r r r r r r
1
r rrv v
r
v vr r
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6.3 The Linear Momentum EquationTo develop the differential momentum equations we can start with the linear momentum equation (Newton's Second Law)
CV n
sys
CS
DPF
Dt
VdV V V dAt
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6.3 The Linear Momentum Equation
Conservation of momentum
ii i i
( )CV
inlet outletports ports
mvmv mv F
t
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6.3 The Linear Momentum Equation
xm
y ym
x xm
ym
zm
z zm z
x
y
y
x
z
ii i
i
( )
CV
inlet outletports ports
mvmv mv
t
F
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6.3 The Linear Momentum EquationIn x-direction
y
z
x
x
u y z u u y z uu y z u x
x
v x z u
, u
v x z uv x z u y
y
u y z u u y z uu y z u x
x
w x y u
w x y uw x y u z
z
, u
i( )( )CVmv
x y z ut t
uu x y z
t t
Newton`s second law of motion
Momentum carried by flow
2i
i
i
inlet at x
inlet at y
inlet at z
mv u y z u u y z
mv v x z u uv x z
mv w x y u uw x y
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6.3 The Linear Momentum EquationIn x-direction
y
z
x
x
u y z u u y z uu y z u x
x
v x z u
, u
v x z uv x z u y
y
u y z u u y z uu y z u x
x
w x y u
w x y uw x y u z
z
, u
i i i
2 2
i i i
i i i
( )
( )
(
inlet at x x inlet at x inlet at x
inlet at y y inlet at y inlet at y
inlet at z z inlet at z inlet at z
mv mv mv xx
u y z u x y zx
mv mv mv yy
uv x z uv x y zy
mv mv mv zz
uw x y uwz
) x y z
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6.3 The Linear Momentum EquationIn x-direction
2i i ( ) ( ) ( )
inlet outletports ports
mv mv u x y z uv x y z uw x y zx y z
u v wu x y z
x y z
u u v w x y zx y z
u u uu v w x y z
x y z
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6.3 The Linear Momentum Equation
ii i i
i
i
i
( )
D( )
D
D
D
CV
inlet outletports ports
mvmv mv F
t
u u v wu u u u v w
t t x y z x y z
u u uu v w F
x y z
u Du V F
t Dt
uF
t
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6.3 The Linear Momentum Equation6.3.1 Description of Forces Acting on the Differential Element
In x-direction Body force: (Ex. gravity)
,body x x xF x y z g g x y z
xx
y
z
x
xz
xy
Surface force: (Ex. pressure, viscous force)
,xx xy xz
Direction offorce
Direction ofsurface
Normal stress Shear stress
yy
yz
yx
zz
zxzy
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6.3 The Linear Momentum EquationIn x-direction
Surface forces:
,
,
,
,, ,
,, ,
,, ,
sur x xx
sur y yx
sur z zx
sur x xxsur x x sur x xx
sur y yxsur y y sur y yx
sur z zxsur z z sur z zx
F y z
F x z
F x y
FF F x y z x y z
x xF
F F y x z x y zy y
FF F z x y x y z
z z
y
z
x
x
xx y z xx
xx x y zx
yx x z
yxyx y x z
y
xx y z xx
xx x y zx
zx x y
zxzx z x y
z
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6.3 The Linear Momentum EquationIn x-direction
,i , , , , , ,sur sur x sur x x sur y sur y y sur z sur z z
yxxx zx
F F F F F F F
x y zx y z
y z
x x
xx y z xx
xx x y zx
yx x z
yxyx y x z
y
xx y z xx
xx x y zx
zx x y
zxzx z x y
z
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6.3 The Linear Momentum Equation
D
Dyxxx zx
x
u u u u uu v w g
t t x y z x y z
In x-direction
In y-direction
In z-direction
D
Dxy yy zy
y
v v v v vu v w g
t t x y z x y z
D
Dyzxz zz
z
w w w w wu v w g
t t x y z x y z
6.3.2 Equations of Motion
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6.4 Inviscid Flow
The shearing stresses are assumed to be negligible are said to be inviscid, nonviscous, or frictionless.
The negative sign is used so that a compressive normal stress (which is what we expect in a fluid)
will give a positive value for p.
xx yy zz p
0yx zx xy zy xz yz
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6.4 Inviscid Flow6.4.1 Euler's Equations of Motion
D
D x
u u u u u pu v w g
t t x y z x
In x-direction
In y-direction
In z-direction
D
D y
v v v v v pu v w g
t t x y z y
D
D z
w w w w w pu v w g
t t x y z z
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6.4 Inviscid Flow
D
D
D
D
D
D
x
y
z
u u u u u pu v w g
t t x y z x
v v v v v pu v w g
t t x y z y
w w w w w pu v w g
t t x y z z
( )V
V V g pt
Euler's equations of motion
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6.4 Inviscid Flow6.4.2 The Bernoulli Equation
We will restrict our attention to steady flow so Euler's equation in vector form becomes
( )V
V V g pt
Body force
kg g g z
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6.4 Inviscid Flow
( )V V g p
Euler's equations of motion
1( ) ( ) ( )
2V V V V V V
use
then
1( ) ( )
2
pV V g z V V
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6.4 Inviscid Flow
1( ) ( ) 0
2
pV V g z ds V V ds
Take the dot product of each term with adifferential length ds along a streamline.Thus,
ds
( )V V ds
where i j kds dx dy dz
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6.4 Inviscid Flow1
( ) 02
pV V g z ds
where1
i j k ( i j k)
1
p p p pds dx dy dz
x y z
p p p dpdx dy dz
x y z
similarly
2 21 1 1( ) ( )
2 2 2V V ds V ds dV
g z ds gdz
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6.4 Inviscid Flow
2 2
2
1 ( ) 0
2
1 1 0 0
2 2
1 constant along a streamline
2
pV V g z ds
dp pdV gdz d V gz
pV gz
Bernoulli equation
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6.4 Inviscid Flow
6.4.3 Irrotational Flow
The vorticity is zero in an irrotational flow field.
If we make one additional assumption—that the flow is irrotational—the analysis of inviscid flow problems is further simplified.
2 0V
, , w v u w v u
y z z x x y
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6.4 Inviscid Flow
flow in the entrance toa pipe may be uniform(if the entrance isstreamlined) and thuswill be irrotational
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6.4 Inviscid Flow6.4.4 The Bernoulli Equation for Irrotational Flow
The Bernoulli equation can be applied between any two points in an irrotational flow field.
1( ) ( ) 0
2
pV V g z ds V V ds
21 constant is not limited along a streamline
2
pV gz
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6.4 Inviscid Flow6.4.5 The Velocity Potential
For an irrotational flow 0V
We know that ( ) 0
So we can let
i j k i j kV u v wx y z
ϕ is called the velocity potential
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6.4 Inviscid Flow
w v
y z y z z y
u w
z x z x x zv u
x y x y y x
, , u v wx y z
check
The velocity components
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6.4 Inviscid Flow
For inviscid, incompressible,irrotational flow fields it followsthat
2
0
0 0
V
V
Laplace's equation
This type of flow is commonly called a potential flow.
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6.5 Some Basic, Plane Potential Flows
2 22 2
2 2
i j k
0 0
0
Vx y z x y
y x
Laplace's equation
Similarly, for the streamlines in an irrotational flow
0 i jV Vy x
then
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6.4 Inviscid Flow
Laplace's equation in cylindrical coordinates
1r zv v v
r r z
r θ z r θ ze e e e e e
1 , , r zv v v
r r z
2 22
2 2 2
1 10r
r r r r z
Velocity components in cylindrical coordinates
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6.5 Some Basic, Plane Potential Flows
Cartesian coordinates
u ux y
orv v
y x
Cylindrical coordinates
1
1
r rv vr ror
v vr r
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6.5 Some Basic, Plane Potential Flows
=conatant
0 along
dy ud dx dy udx vdy
x y dx v
=conatant
0 along
dy vd dx dy vdx udy
x y dx u
Along = constant (called equipotential lines)
Along = constant (called streamlines)
=conatant =conatant
1along along
dy dy u v
dx dx v u
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6.5 Some Basic, Plane Potential Flows
For any potential flow field a “flow net” can be drawn thatconsists of a family of streamlines and equipotential lines.
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6.5 Some Basic, Plane Potential Flows6.5.1 Uniform Flow
1
2
0
0
u Ux
Ux Cv
y
u Uy
Uy C
vx
C is an arbitrary constant, which can be set equal to zero
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6.5 Some Basic, Plane Potential Flows
cos
( cos sin )sin
cos
( cos sin )
sin
u Ux
U x yv U
y
u Uy
U y x
v Ux
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6.5 Some Basic, Plane Potential Flows6.5.2 Source and Sink
Source
Let m be the volume rate of flow
2 2r r
mm rv v
r
1
2 2 1
0 0
r r
m mv v
r r r ror
v vr r
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6.5 Some Basic, Plane Potential Flows
2 ln1 2
0
1
2 2
0
r
r
mv
mr r r
vr
mv
mr r
vr
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6.5 Some Basic, Plane Potential Flows
Sources and sinks do not really exist in real flowfields, and the line representing the source or sink is amathematical singularity in the flow field.
02r r
mv v as r
r
If m is positive, the flow is a source flow.
If m is negative, the flow is a sink flow.
The flowrate, m, is the strength of the source or sink.
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6.5 Some Basic, Plane Potential Flows6.5.3 Vortex
0
1
10
ln
r
r
vr K
Kv
r r
vr K r
Kv
r r
where K is a constantK
vr
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6.5 Some Basic, Plane Potential FlowsIrrotational flow Rotational flow
Both sticks are rotating, theaverage angular velocity of thetwo sticks is zero and the flow isirrotational
The rotational vortex iscommonly called a forced vortex,whereas the irrotational vortex isusually called a free vortex.
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6.5 Some Basic, Plane Potential Flows
A mathematical concept commonly associated with vortex motion is that of circulation.
V ds
0 0 0V ds ds d
For an irrotational flow
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6.5 Some Basic, Plane Potential Flows
2K
V ds v rd rd Kr
2
K
2
ln2
r
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6.5 Some Basic, Plane Potential Flows6.5.4 Doublet
A doublet is formed by an appropriate source-sink pair.
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6.5 Some Basic, Plane Potential Flows
1 12
m
2 22
m
2 0
Laplace's equation
is linear2 2
1 2
2 21 2
( )
0
Superposition of is allowed
1 2 1 2 1 2( )2 2 2
m m m
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6.5 Some Basic, Plane Potential Flows
1 21 2
1 2
tan tan2tan tan( )
1 tan tanm
where
1 2
sin sintan , tan
cos cos
r r
r a r a
then
12 2 2 2
2 2 sin 2 sintan tan
2
ar m ar
m r a r a
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6.5 Some Basic, Plane Potential Flows
The so-called doublet is formed byletting the source and sink approachone another (a → 0) while increasingthe strength m (m → ∞) so that theproduct ma/π remains constant.
sin where
K maK
r
12 2 2 20 0
2 sin 2 sin sinlim tan lim
2 2a a
m ar m ar ma
r a r a r
cosor
K
r
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6.6 Superposition of Basic, Plane Potential Flows6.6.1 Source in a Uniform Stream—Half-Body
sin2 2
ln cos ln2 2
uniform flow source
uniform flow source
m mUy Ur
m mUx r Ur r
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6.6 Superposition of Basic, Plane Potential Flows
0 , rv at r b
1 1sin
2
cos2
r
mv Ur
r r
mU
r
0 cos 2 2
m mU b
b U
sin sin2 2 2stagnation
m m mUr Ub bU
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6.6 Superposition of Basic, Plane Potential Flows
2
mb
U
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6.6 Superposition of Basic, Plane Potential Flows
sin2
sin
stagnation
mbU Ur
Ur bU
( )
sin
br
sin ( )y r b
sin sin2
mv Ur U
r r
and
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6.6 Superposition of Basic, Plane Potential FlowsApplying the Bernoulli equation between a point farfrom the body, where the pressure is p0 and the velocityis U, and some arbitrary point with pressure p andvelocity v, it follows that
22 2 2 2 2
0
22
2
1 1 1( ) cos sin
2 2 2
1 1 2 cos
2
r
bUp U p v v p U U
r
b bp U
r r
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6.6 Superposition of Basic, Plane Potential Flows
An important point to be noted is that the velocity tangent to the surface of the body is not zero; that is, the fluid “slips” by the boundary
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6.6 Superposition of Basic, Plane Potential Flows6.6.2 Rankine Ovals
sink
12 2
12 2 2
2 sin tan
2
2 tan
2
uniform flow source
m arUy
r a
m ayUy
x y a
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6.6 Superposition of Basic, Plane Potential Flows
12 2 2
2 2 2
2 2 2 2 2 2 4
12 2 2
2 2 2 2 2 2
2tan
2
2 ( )
2 ( ) 2 ( )
2tan
2
4
2 ( ) 4
m ayu Uy
y y x y a
m a x y aU
x y a x y a
m ayv Uy
x x x y a
m axy
x y a a y
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6.6 Superposition of Basic, Plane Potential Flows
0 , 0u v at x l y
2 2 2
2 2 2 2 2 2 4 2 2
2 ( 0 )0
2 ( 0 ) 2 ( 0 ) ( )
m a l a m au U U
l a l a l a
1/2
1l m
a U
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6.6 Superposition of Basic, Plane Potential Flows
12 2 2
2tan 0 , 0
2tagnation
m ayUy at x l y
x y a
0 0 , tagnation at x y h
12 2 2
2 2
2 tan 0
2 0
2 tan
2
m ahUh
h a
h a Uhh
a m
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6.6 Superposition of Basic, Plane Potential Flows
1.A large variety of body shapes with differentlength to width ratios can be obtained by usingdifferent values of Ua/m
2.The potential solution for the Rankine ovalswill give a reasonable approximation of thevelocity outside the thin, viscous boundarylayer and the pressure distribution on the frontpart of the body only.
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6.6 Superposition of Basic, Plane Potential Flows
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6.6 Superposition of Basic, Plane Potential Flows6.6.3 Flow around a Circular Cylinder
sin sin sin
cos cos cos
uniform flow doublet
uniform flow doublet
K KUy Ur
r r
K KUx Ur
r r
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6.6 Superposition of Basic, Plane Potential Flows
The stream function for flow around a circular cylinder
0 , 0, rv at r a
2
1 1 sinsin
cos
r
Kv Ur
r r r
KU
r
22
cos 0 K
U K a Ua
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6.6 Superposition of Basic, Plane Potential Flows
2 2
2 21 sin and 1+ cos
a aUr Ur
r r
Then
2
2
2
2
11 cos
1+ sin
r
av U
r r
av U
r r
0 rv at r a
2 sin sv U at r a
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6.6 Superposition of Basic, Plane Potential Flows
The pressure distribution on the cylinder surface is obtained from the Bernoulli equation.
2 2 20
2 2 2
1 1( )
2 21
2 sin2
s r
s s
p U p v v
p v p U
2 20
1 (1 4sin )
2sp p U
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6.6 Superposition of Basic, Plane Potential Flows
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6.6 Superposition of Basic, Plane Potential FlowsThe resultant force (per unit length) developed on the cylinder can be determined by integrating the pressure over the surface.
2
0
2
0
cos 0
sin 0
x s
y s
F p a d
F p a d
升力
阻力
Potential theory incorrectly predicts that the drag on a cylinder is zero.Why?
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6.6 Superposition of Basic, Plane Potential Flows
An additional, interestingpotential flow can bedeveloped by adding afree vortex to the streamfunction or velocitypotential for the flowaround a cylinder. In thiscase
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6.6 Superposition of Basic, Plane Potential Flows
2 2
2 21 sin ln or 1 cos
2 2
a aUr r Ur
r r
2
21+ sin 2 sin
2 2s
av U v U
r r r a
The tangential velocity, vθ, on the surface of the cylinder (r = a) now becomes
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6.6 Superposition of Basic, Plane Potential FlowsAt stagnation point, vθs = 0
2 sin 0 sin2 4s stag stagv U
a Ua
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6.6 Superposition of Basic, Plane Potential Flows
The pressure distribution on the cylinder surface is obtained from the Bernoulli equation.
22 2
0
1 1 12 sin
2 2 2 2s sp U p v p Ua
22 2
0 2 2
1 2 sin 1 4sin
2 4sp p UaU a
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6.6 Superposition of Basic, Plane Potential Flows
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6.6 Superposition of Basic, Plane Potential Flows
2
0
2
0
cos 0
sin
x s
y s
F p a d
F p a d U
x
y
The development of this lift on rotating bodies is called the Magnus effect.
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6.7 Other Aspects of Potential Flow Analysis
1.The method of superposition of basic potentials andstream functions has been used to obtain detaileddescriptions of irrotational flow around certain bodyshapes immersed in a uniform stream.
2.It is possible to extend the idea of superposition byconsidering a distribution of sources and sinks, vortexes,or doublets, which when combined with a uniform flowcan describe the flow around bodies of arbitrary shape.
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6.7 Other Aspects of Potential Flow Analysis
3.Potential flow solutions are always approximatebecause the fluid is assumed to be frictionless.
4.An important point to remember is that regardless ofthe particular technique used to obtain a solution to apotential flow problem, the solution remainsapproximate because of the fundamental assumption ofa frictionless fluid.
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6.7 Other Aspects of Potential Flow Analysis
Outer flowNeglect viscosity Vorticity = 0 Inner flow
Viscosity is importantVorticity generated
Wake regionViscosity is not importantVorticity ≠ 0
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6.8 Viscous Flow
For incompressible Newtonian fluids it is known that the stressesare linearly related to the rates of deformation and can beexpressed in Cartesian coordinates as (for normal stresses)
xx
y
z
x
xz
xy
yy
yz
yx
zz
zxzy 2
2
2
xx
yy
zz
u u up p
x x x
v v vp p
y y y
w w wp p
z z z
stresses are linearly related to the rate of strain
6.8.1 Stress-Deformation Relationships
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xx
y
z
x
xz
xy
yy
yz
yx
zz
zxzy
(for shearing stresses)
6.8 Viscous Flow
xy yx
yz zy
zx xz
v u
x y
w v
y z
u w
z x
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6.8 Viscous Flow
In cylindrical polar coordinates the stresses for incompressible Newtonian fluids are expressed as
2
12
2
rrr
r
zzz
vp
rv v
pr r
vp
z
1
1
rr r
zz z
r zzr rz
v vr
r r r
v v
z r
v v
z r
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6.8 Viscous Flow
In x-direction
In y-direction
In z-direction
D
Dyxxx zx
x
u u u u uu v w g
t t x y z x y z
D
Dxy yy zy
y
v v v v vu v w g
t t x y z x y z
D
Dyzxz zz
z
w w w w wu v w g
t t x y z x y z
6.8.2 The Navier-Stokes Equations
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6.8 Viscous FlowIn x-direction
In y-direction
In z-direction
The Navier-Stokes equations are the basic differential equationsdescribing the flow of Newtonian fluids.
2 2 2
2 2 2
D
D x
u u u u u p u u uu v w g
t t x y z x x y z
2 2 2
2 2 2
D
D y
v v v v v p v v vu v w g
t t x y z y x y z
2 2 2
2 2 2
D
D z
w w w w w p w w wu v w g
t t x y z z x y z
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6.8 Viscous Flow
Navier-Stokes equation in cylindrical coordinates
2 2 2
2 2 2 2
( )1 1 2r r r r r r rr z r
v v vv v v v rv v vpv v g
t r r r z r r r r r r z
r-direction:
θ-direction:
z-direction:
2 2
2 2 2 2
( )1 1 1 2
rr z
r
v v v v v v vv v
t r r r z
rv v vvpg
r r r r r r z
2 2
2 2 2
1 1z z z z z z zr z z
vv v v v v v vpv v g r
t r r z z r r r r z
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
1. A principal difficulty in solving the Navier-Stokes equations is their nonlinearity arising from the convective acceleration terms.
2. The Navier-Stokes equations apply to both laminar and turbulent flow, but for turbulent flow each velocity component fluctuates in an apparently random fashion, with a very short time scale, and this added complication makes an analytical solution intractable.
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
6.9.1 Steady, Laminar Flow between Fixed Parallel Plates
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
D
D
D
D
D
D
x
y
z
u p u u ug
t x x y z
v p v v vg
t y x y z
w p w w wg
t z x y z
2
20
0
0
p u
x y
pg
y
p
z
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
1
22
1 1 22
0 ( , )
( )
1 1
2
pp p x y
zp
g p gy f xy
p u u p py c u y c y c
x y y x x
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
21 2 1
22 2
1 2
B.C. 0 at
10 0
2 1
10 2
2
u y h
ph c h c c
xp
c hph c h c x
x
2 21 ( )
2
pu y h
x
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
The volume rate of flow, q, passing between the plates (for a unit width in the z direction) is obtained from the relationship
32 21 2
( )2 3
h h
h h
p h pq udy y h dy
x x
The pressure gradient ∂p/∂x is negative, since thepressure decreases in the direction of flow.
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
If we let Δp represent the pressure drop between twopoints a distance ℓ apart, then
3 32 2
3 3
h p h pq
x l
The mean velocity V=q/2h, then
2
3
h pV
l
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
The maximum velocity, umax, occurs at y = 0
22 2
max
1 3( )
2 2 2
p h pu y h u V
x x
The pressure gradient in the x direction is constant, then
1 0 0( ) ( 0)p p
f x p x p xx x
0 at 0p p x y where is reference pressure
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
then
1 0( )p
p gy f x gy p xx
3
0 3
2 3
3 2
h p qq p p gy x
x h
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
6.9.2 Couette Flow (g = 0)0 at 0
B.C. at
u y
u U y b
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
D
D
D
D
D
D
x
y
z
u p u u ug
t x x y z
v p v v vg
t y x y z
w p w w wg
t z x y z
2
20
0
0
p u
x y
p
y
p
z
( )p p x
0 at 0B.C.
at
u y
u U y b
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
22
1 22
10
2
p u pu y c y c
x y x
20 at 0 1B.C. ( )
at 2
u y y pu U y by
u U y b b x
In dimensionless form2
12
u y b p y y
U b U x b b
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
The simplest type of Couette flow is one for which thepressure gradient is zero; that is, the fluid motion iscaused by the fluid being dragged along by themoving boundary.
2
12
u y b p y y y
U b U x b b b
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
This situation would be approximated bythe flow between closely spacedconcentric cylinders in which onecylinder is fixed and the other cylinderrotates with a constant angularvelocity, ω.
The flow in an unloaded journal bearing might beapproximated by this simple Couette flow if the gapwidth is very small.
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
6.9.3 Steady, Laminar Flow in Circular Tubes
sin
cosrg g
g g
g
rg
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
Continuity equation
( )D 1 1( ) 0 0
Dr zvrv v
V Vt r r r z
The flow is parallel to the walls, then
0 0 ( , ) ( )zr z z z
vv v v v r v r
z
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
Navier-Stokes equations
2 2
2 2 2 2
2 2
2 2 2 2
2 2
2 2 2
D ( )1 1 2
D
D ( )1 1 1 2
D
D 1 1
D
r r r rr
r
z z z zz
vv rv v vpg
t r r r r r r z
v rv v vvpg
t r r r r r r z
v v v vpg r
t z r r r r z
0
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
0 sin
1 0 cos
10 z
pg
rp
gr
vpr
z r r r
Navier-Stokes equations reduced to
sin ( , )p gr f z
sin ( , )p gr h r z
1( )f zy
1( ) constant
f zp
z z
Suppose that the pressure gradient is constant
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
21
21 2
1 10
2
1 ln
4
z z
z
v vp pr r r c
z r r r r z
pv r c r c
z
This type of flow is commonlycalled Hagen-Poiseuille flow, orsimply Poiseuille flow.
1
22
0finite at 0
B.C. 10 at
4
z
z
cv r
pc Rv r R
z
2 21 ( )
4z
pv r R
z
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
The volume rate of flow, Q, passing through the tube and the pressure gradient.
2
0 0
42 2
0
1 2 ( )
4 8
R
z
R
Q v rdrd
p R pr R rdr
z z
or4
where 8
R p p pQ
l z l
Poiseuille's law relates pressure drop and flowrate for steady, laminar flow in circular tubes
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
The mean velocity
4 22
8 8
R p R pR V Q V
l l
The maximum velocity, vmax occurs at r = 0
2 22 2
max
1(0 ) 2
4 4 4
p R p R pv R V
z z l
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
In dimensionless form2
2 2
max
1( ) 1
4z
z
vp rv r R
z v R
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
6.9.4 Steady, Axial, Laminar Flow in an Annulus
10 zvp
rz r r r
0 at B.C.
0 at z i
z o
v r r
v r r
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
21
21 2
1 10
2
1 ln
4
z z
z
v vp pr r r c
z r r r r z
pv r c r c
z
0 at B.C.
0 at z i
z o
v r r
v r r
2 22 21
ln4 ln( / )
i oz o
o i o
r rp rv r r
z r r r
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
The volume rate of flow Q is2 2 2
2 4 4
0 0
2 2 24 4
( )
8 ln( / )
( ) where
8 ln( / )
Ro i
z o io i
o io i
o i
r rpQ v rdrd r r
z r r
r rp p pr r
l r r z l
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
The maximum velocity, vmax occurs at r = rm that
0 at zm
vr r
r
One can obtain
1/22 2
2ln( / )o i
mo i
r rr
r r
These results for flow through an annulus are valid only if the flow is laminar.
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
For tube cross sections other than simple circular tubes it is common practice to use an “effective” diameter, termed the hydraulic diameter, Dh, which is defined as
4 cross-sectional area
wetted perimeterhD
For circular tube with diameter D24 ( /4)
h
DD D
D
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6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows
For an annulus2 24 cross-sectional area 4 ( )
2( )wetted perimeter 2 2
o ih o i
o i
r rD r r
r r
In terms of the hydraulic diameter, the Reynolds number is
Re hD V
mean velocity
Re < 2100, the flow will be laminar.
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6.10 Other Aspects of Differential Analysis
( ) 0D
VDt
Continuity equation
Navier-Stokes equation
2DVp g V
Dt
Very few practical fluid flow problems can be solved using an exact analytical approach.
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6.10 Other Aspects of Differential Analysis
6.10.1 Numerical Methods
2
( ) 0D
VDt
DVp g V
Dt
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Thanks for your attention