2 fluid mechanics review
TRANSCRIPT
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8/9/2019 2 Fluid Mechanics Review
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HCMC University of Technology,
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Fluid Mechanics Review
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15/09/200957:020 Fluid Mechanics
Fundamental Concepts
Physic Properties: , , , , , ,
Forces on Fluid:
Internal Forces
External ForcesSurface
Volume
Ideal Fluid/Real Fluid
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15/09/200957:020 Fluid Mechanics
Fluid Statics
Basic Differential Equation:
),,(
0
zyxpp
pgradF
=
=
Potential Force:
0:. =+
=
pgradgradEqDiff
gradF
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15/09/200957:020 Fluid Mechanics
Fluid Statics
Gravity Force:
gzgF ==
constzp =+
Hydrostatics:
=2
1
12
z
z
gdzpp Aerostatics:
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15/09/200957:020 Fluid Mechanics
Fluid Statics
Archimedes Buoyancy:
WgPz =
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15/09/200957:020 Fluid Mechanics
Fluid Kinematics
Motion classification:
Viscous Friction Ideal Fluid
Viscous Fluid Laminar
Turbulence Time Stable
Unstable
Dimension 1D, 2D, 3D
Steady/Unsteady
Incompressible/Compressible
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15/09/200957:020 Fluid Mechanics
Fluid Kinematics
Two methods for the description of fluid motion
Lagrangian Method
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15/09/200957:020 Fluid Mechanics
Eulerian Method
Fluid Kinematics
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Fluid Kinematics
)( ugradut
u
dt
uda
+
==
Acceleration of a Fluid Element
z
z
y
z
x
zzz
z
z
y
y
y
x
yyy
y
zx
yx
xxxx
x
uz
u
uy
u
ux
u
t
u
dt
du
a
uz
uu
y
uu
x
u
t
u
dt
dua
uz
uu
y
uu
x
u
t
u
dt
dua
+
+
+
==
+
+
+
==
+
+
+
==
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HCMC University of Technology,
15/09/200957:020 Fluid Mechanics
Fluid Kinematics
Relation of System Derivatives to theControl Volume Formulation
+
=SCVsystem
dAnut
X
dt
dX
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15/09/200957:020 Fluid Mechanics
Fluid Kinematics
0)( =+
udiv
t
01
)(1
0
=
+
+
=
+
+
z
uu
rru
rr
z
u
y
u
x
u
zr
zyx
Continuity Equation
Incompressible ( = const):
Irrotational/Rotational Flow urot
2
1=
1D, steady flow: Q1 = Q2
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15/09/200957:020 Fluid Mechanics
Fluid Dynamics
Cu
t
gradu
=+++
=
2
2
Cu
=++2
2
Irrotational, potential flow
Steady flow (integration along a streamline )
Steady flow (integration along a vortex)
Cu
=++2
2
Steady flow (integration along a normal to the streamline)
( )R
u
n
2
=+
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15/09/200957:020 Fluid Mechanics
Fluid Dynamics
Cup
gzt
=+++
2
2
Cup
gz =++2
2
Irrotational, potential flow
Steady flow (integration along a streamline )
Steady flow (integration along a vortex)
Steady flow (integration along a normal to the streamline)
R
upgz
n
2
=
+
Cup
gz =++2
2
(Bernoulli Equation)
Gravity, ideal, incompressible fluid:
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Fluid Dynamics
dtududivgradupgradF
=++
)(3
1 2
0=udiv
Equations of Motion for Viscous Fluid(Navier-Stokes Equations)
Incompressible Flows:
2
2
2
2
2
22
zyx
+
+
=
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15/09/200957:020 Fluid Mechanics
Fluid Dynamics
dt
QdPP
dt
dE
sm
~
++=
Energy Equation
Int. Formulation
Diff. Formulation
+=
+ dAnqdAndwuFdwu
edt
d
WW
2
2
Gravity, incompressible, steady flow:
fhg
Vpz
g
Vpz +++=++
22
2
22
22
2
11
11
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15/09/200957:020 Fluid Mechanics
Fluid Dynamics
Momentum Equation
( )
[ ] =
+
=
outoutoutoutoutout
W
VQVQF
dAnuudwt
u
F
)(
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15/09/200957:020 Fluid Mechanics
Dimensional Analysis
[ ] tml
TMLa =
)...,,,(121
=
n
aaafa
( ) 1,...,,, 121 == nsf s
Dimensional/Non-dimensional Quantities
Fundamental Dimensions/ Derivative Dimensions
Buckingham Theorem
k fundamental dimensions
Dimensionless Parameterk
k
iki
aaa
a
...21 21
+=
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15/09/200957:020 Fluid Mechanics
Dimensional Analysis
VLVL
ForceViscous
ForceInertia===Re
2
2
1V
ppCa v
=
gL
V
ForceGravity
ForceInertiaFr ==
Significant Dimensionless Group in Fluid Dynamics
Reynolds number:
Cavitation number:
Froude number:
Weber number:
LV
ForceTensionSurface
ForceInertiaWe
2
==
Mach number:c
V
ilityCompressibtodueForce
ForceInertiaM ==
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Similarity
Geometric Similarity
Kinematically Similarity
Dynamically Similarity
Each dimensionless group has the same value
in the model and in the prototype