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MECH 221 FLUID MECHANICS(Fall 06/07)
REVIEW
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MECH 221 – Review
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What Have You Learnt? 1. Fluid Statics
2. Fluids in Motions
3. Kinematics of Fluid Motion
4. Integral and Differential Forms of Equations of Motion
5. Dimensional Analysis
6. Inviscid Flows
7. Boundary Layer Flows
8. Flows in Pipes
9. Open Channel Flows
On coming week lectures
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Fluid Statics
It is to calculate the fluid pressure when thefluid is no moving
Shear stress is due to relative motion of fluid,so no shear stress and only normal stress(Pressure) acting on the fluid
The fluid pressure is only due to body force,Gravitational Force
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Fluid Statics
Fluid pressure will increase when the positionof the fluid become deeper, we have followingequation:
g dz
dp
z
y
x
g
0
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Fluid Statics
Total force acting on the surface become:
hdA g A p pdA F atm
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Fluid In Motion (Inviscid Flow)
2 sets equations for solving fluid motion problems
Conservation of Mass
Conservation of Momentum
dV pd d dV )( t S )t ( V S )t ( V
g s s
vvv
0d dV t S )t ( V
s v
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Fluid In Motion (Inviscid Flow)
By invoking the continuity equation, themomentum equation becomes Euler’sequation of motion
Bernoulli equation is a special form of theEuler’s equation along a streamline
constantz
2
g 2
v p
Along streamline incompressible flow
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Fluid In Motion (Inviscid Flow)
A conical plug is used to regulate the air flowfrom the pipe. The air leaves the edge of thecone with a uniform thickness of 0.02m. If
viscous effects are negligible and the flowrateis 0.05m3 /s, determine the pressure within thepipe.
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Fluid In Motion (Inviscid Flow)
Procedure:
Choose the reference point
From the Bernoulli equation
P, V, Z all are unknowns
For same horizontal level, Z1=Z2
Flowrate conservation
Q=AV
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Fluid In Motion (Inviscid Flow)
From the Bernoulli equation,
)(2
2
2
zzlevel,lhorizontiasameat the
Since,
z
2
z
2
2
1
2
221
2
22
2
11
21
2
2
221
2
11
vv p p
g v
g p
g v
g p
g
v
g
p
g
v
g
p
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Fluid In Motion (Inviscid Flow)
From flowrate conservation,
smv
smv
mrt A
m D
A
mr mt m D sm
Q
v Av AQ
894.190251.0/5.0
034.120415.0/5.0
Therefore,
0251.0)02.0)(2.0(22
0415.04
23.0
4
2.0,02.0,23.0,5.0Given
2
1
2
2
222
1
3
2211
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Fluid In Motion (Inviscid Flow)
21
22
1
22
3
21
2
1
2
221
565.148
)034.12894.19(2
184.10
0 p point,reference becomes pSet
184.1C,25atm,air@1standardFor
894.19,034.12
)(2
m N p
p
mkg
smv
smv
vv p p
Sub. into the Bernoulli equation,
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Fluid In Motion (Viscous Flow)
In the mentioned fluid motion is inviscidflows, only pressure forces act on the fluidsince the viscous forces (stress) wereneglected
With the viscous stress, the total stress onthe fluid is the sum of pressure stress ( )
and viscous stress ( ) given by:τ
pσ
τ pσ σ
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Dimensional Analysis
The objective of dimensional analysis is to obtainthe key non-dimensional parameters that governthe physical phenomena of flows
After the dimensional analysis or normalization ofthe complicated Navier-Stokes equations (steadyflow), the non-dimensional parameters areidentified
The equations are reduced to simple equationand solvable analytically under certain conditions
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Dimensional Analysis
By using proper scales, the variables, velocity(u), pressure (p) and length (L) arenormalized to obtain the non-dimensionalvariables, which are order one
*
U
gL
UL p
U
P *
g2
2
2 ivvv
***
L P p p U //vv
directionnalgravitatioinrunit vecto*
g i
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Dimensional Analysis
For simplicity consider the case where thegravitational force has no consequence to thedynamic of the flow, the Navier-Stokes
equations becomes
UL
Re ,
Re
1 2*
2
**vvv
*
p
U
P
,
Re
1 2* **vvv
*
p
When Re >> 1
scalepressureas2U P
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Inviscid Flow Vs. Boundary Layer Flow
where is the viscous diffusion length in anadvection time interval of .
Here, measures the time required forfluid travel a distance L.
2
22
v
L
U / L
LUL Re
forceviscous
forceinertia
U / L
U / Laτ
aU / L τ
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Inviscid Flow Vs. Boundary Layer Flow
When , inertia force is much greater thanviscous force, i.e., the viscous diffusion distance ismuch less than the length L.
Viscous force is unimportant in the flow region of
, but can become very important in the region of
near the solid boundary.
This flow region near the solid boundary is called anboundary layer as first illustrated by Prandtl.
1 Re
)( O ) L( O
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Inviscid Flow Vs. Boundary Layer Flow
Flow in the region outside the boundarylayer where viscous force is negligible isinviscid. The inviscid flow is also called thepotential flow.
U
Boundary layer flowPotential flow
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Inviscid Flow
Inviscid flow implies that the viscous effect isnegligible. The governing equations areContinuity equation and Euler equation.
We introduce a potential function, which isautomatically satisfy the continuity equation
v
02
2
2
2
2
22
z y x
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Inviscid Flow
The continuity equation becomes Laplaceequation. The flow is described by Laplaceequation is called potential flow
For 2D potential flows, a stream function (x,y) can also be defined together with (x,y)
x y y x
and
C i
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Inviscid Flow
If 1 and 2 are two potential flows, the sum =( 1+ 2 ) also constitutes a potential flow
We can combine certain basic solutions toobtain more complicated solution
+ =
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MECH 221 R i
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Boundary Layer Flow
The thin layer adjacent to a solid boundary iscalled the boundary layer and the flow insidethe layer is called the boundary layer flow
Inside the thin layer the velocity of the fluidincreases from zero at the wall (no slip) to thefull value of corresponding potential flow.
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Boundary Layer Flow
There exists a leading edge for all externalflows. The boundary layer flow developing fromleading edge is laminar
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Boundary Layer Flow
When we normalize the governing equations withRe underneath the viscous term and resolve thevariables of y and v inside the boundary flow, thenon-dimensional normalized variables are selected:
V
vv
U
uu
y y
L
x x
L
,,,
V be the scale of v in the boundary layer
L is viscous diffusion layer near the wall (boundary layer)
MECH 221 Review
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Boundary Layer Flow
These results in the boundary layer equations thatin dimensional form are given by:
0
y x
vu
2
2
y
u
x
p
y
uv
x
uu
y
p
0
Continuity:
X-momentum:
Y-momentum:
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Boundary Layer Flow
A boundary layer flow is similar and its velocityprofile as normalized by U depends only on thenormalized distance from the wall:
i.e.,
y xU y
x
2/1
g U
u
(*)
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Boundary Layer Flow
By introduce a stream function
The boundary layer equation in term of thesimilarity variables becomes:
)('
f U y
u
f xvU 21
f f f ' ' asandat 100
02 ' ' ' ' ' ff f
(**)
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After we solve this ordinary equation, we obtain asolution of
We first find the value of by Equ. (*) based on
coordinate of x and y, then find out the value ofby checking the solution table in the reference. Finallythe u at x and y is calculated by Equ. (**)
Therefore, we obtain following results:
Boundary Layer Flow
5
U
vx
)(' f
)(' f
x
w
U
Re
332.0 2
x
f C Re
664.0
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Boundary Layer Flow
Laminar boundary layer flow can becomeunstable and evolve to turbulent boundarylayer flow at down stream. This process iscalled transition
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Boundary Layer Flow
Under typical flow conditions, transition usuallyoccurs at a Reynolds number of 5 x 105
Velocity profile of turbulent boundary layer flows is
unsteady
A good approximation to the mean velocity profilefor turbulent boundary layer is the empirical 1/7
power-law profile given by
71
y
U
u
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Boundary Layer Flow
51Re
37.0
x x
512
Re
0577.0
2/ x
w f
U C
41
200225.0
U
vU w
For turbulent boundary layer, empirically wehave
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If i is the unit vector in the body motion direction,
then magnitude of drag F D becomes:
For two-dimensional flows, we can denotes j as
the unit vector normal to the flow direction, F L is
the magnitude of lift and is determined by:
Boundary Layer Flow
)( ss
..itni dAdA p F w
sb
D F
jtn j )( ss
..
dAdA p F w
sb
L F
Pressure Drag
Friction Drag
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The drag coefficient defined as
For uniform flow passing a flat plate and nopressure gradient is zero and no flow separation, :
Boundary Layer Flow
22
/ AU
F C D
D
Re
072.0
51
L
DC
v
ULC
L L
D ReRe
328.1 whereLaminar Friction Drag
Turbulent Friction Drag
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Boundary Layer Flow
The pressure drag is usually associated withflow separation which provide the pressuredifference between the front and rear faces ofthe body
For low velocity flows passing a sphere ofdiameter D, the drag coefficient then is expressedas:
D
D D
AU
F C
Re
24
2/2
directionflowtheinspheretheof areaprojectedtheis where 42 / πD A