stress field.pdf

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

stress field.pdf

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