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8/14/2019 Stress Field.pdf
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FUNDAMENTAL CONCEPTSFUNDAMENTAL CONCEPTS
Flow field around a car Flow field around buildings
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Flow around an airfoil, = 00
Flow around an airfoil, = 200
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Flow field around an airfoil
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-4
-3
-2
-1
0
1
2
3
4
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
x
y
x
y
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Continuum FluidMechanics
Inviscid= 0
Viscous
Laminar Turbulent
Compressible Incompressible Internal External
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FUNDAMENTAL CONCEPTSFUNDAMENTAL CONCEPTS
FLUID S CONTINUUM : all fluid properties continuousfunction of position and time
Definition of density at a pointFor continuum assumption to be satisfied: (smallest significant characteristic
dimension of the problem) >> (mean free path of the molecules).
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VELOCITY FIELD
The complete representation of the velocity field is given by:
( )t,z,y,xVVrr
=
k
wjvi
uV ++=
r
Any fluid property: is a function of position & time
Any fluid property: = (x, y, z, t) Cartesian coordinate
= (r, , z, t) Cylindrical coordinate
, p, T, h, , ,Vr
ar
, etc.
- Comprises of scalars and vectors
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Specific Gravity (SG) and Specific Weight ()
Density of liquid or solid is expressed in non-dimensional form as:
SG = /(H2O)maxwhere (H2O)max = 1000 kg/m3 or 1.94 slug/ft3.
Specific weight () is defined as: = g
One-, Two-, and Three-Dimensional Flows.
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One-, Two-, and Three-Dimensional Flows
1-D, u = u (r)
Uniform flow at a section
2-D
u = u (x,y)
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One-, Two-, and Three-Dimensional Flows
3-D flow
u = u (x, y, z)
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Visual representation of flow fields
Single/individual
particle
Single/individual
Location/locus
For steady flow, pathlines, streaklines, streamlinesare identical lines (coincide)
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Timelines
A Timeline is a set of
adjacent fluid particlesthat were marked at the
same (earlier) instant in
time. Timelines can be
generated using a
hydrogen bubble wire.
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Timelines
Timelines produced by a hydrogen bubble wire are used to
visualize the boundary layer velocity profile shape.
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EXAMPLE 2.1
FIND: a) Eq. of streamline at (2, 8, 0)
b) Velocity of particle at (2, 8, 0)
c) Position at t = 20 sec. of particle located at (2, 8, 0) at t = 0.
d) Velocity of particle at position found in (c)
e) Eq. of pathline of particle located at (2, 8, 0) at t = 0.
GIVEN: jayiaxV =
r
, where x and y in meters, a = 0.1 sec.-1
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EXAMPLE 2.1(contd)
b) Particle velocity at (2, 8, 0):
( )mji.sec.jayiaxV 8210 1 == r
Or ( )secmj.i.V 8020 =
r
a) Eq. of streamline:
SOLUTION:
x
y
ax
ay
u
v
dx
dy
streamline
=
==
Or x y = C
Since the streamline passing through (2, 8, 0) then: xy = x0y0 = 16 m2
Separating variables leads to: = xdx
y
dyor ln y = -ln x + C
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EXAMPLE 2.1(contd)
c) Particle position at t = 20 sec.
jayiaxV =r
axdt
dxup == and ay
dt
dyvp ==
Separating variables and integrating:
= tx
xadt
x
dx
00and =
ty
yadt
y
dy
00
ln (x / x0) = atln (y / y0) = -atand
Or
x = x0 eat and y = y0 e
-at
Particle position at t = 20 sec where at t = 0 was at (2, 8, 0) is then:
mj.,i. 081814
EXAMPLE 2 1(contd)
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EXAMPLE 2.1(cont d)
d) The particle velocity at mj.,i. 081814 is then:
jayiaxV =r
.sec/mj.i. 10804881 =
x = x0 eat y = y0 e
-at
e) Pathline Eq.
Then:
e at = x / x0 = y0 / y
Or x y = x0y0 = C
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EXAMPLE 2.1(contd)
0
2
4
6
8
10
12
14
16
18
0 2 4 6 8 10 12 14 16 18
x (m )
y
(m)
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0.0
5.0
10.0
15.0
20.0
0.0 5.0 10.0 15.0 20.0
x (m)
y(m)
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Elevation Mean free path ()Sea level 6 x 10-8 m
100 km 0.10 m
160 km 50 m
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FORMAT PENGERJAAN TUGAS:
- DIKERJAKAN PADA KERTAS A-4
- DIKERJAKAAN PADA SATU SISI KERTAS (TIDAK BOLAK-BALIK) ONE SIDED PAGE
- SETIAP GANTI NOMOR, HARUS DIMULAI PADA
HALAMAN BARU
FORMAT FOR HOMEWORK:
- DO ON A-4 PAPER
- WORK ON ONE SIDED PAGE
- EACH NUMBER PUT ON SEPARATED PAPER SHEET
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STRESS FIELDS
STRESS FIELDS
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STRESS FIELDS
The concept of stress in
a continuum
STRESS
Area
Forces(Fr
A
r
Surface Forces
Body ForcesDistributed through
out the CV
Direct contact on CS
STRESS FIELDS
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Force and stress components on a element of area Ax
STRESS FIELDS
x
x
xxx A
F
A
lim
0=
i
j
iij A
F
A
lim
0=
x
z
x
xz
A
F
A
lim
0
=
STRESS FIELDS stress is a tensor orde two
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Notation for stress
n
n
n
nA
F
A
lim
0=
n
t
n
nA
F
A
lim
0=
STRESS FIELDS - stress is a tensor orde two
zzzyzx
yzyyyx
xzxyxx