Ch.3 Fluid Kinematics

filercaliforniaMechanics

Nov 14, 2013 (3 years and 8 months ago)

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Fluid Kinematics:

In kinematics, the force is not concerned, but the motion.

Ch.3 Fluid Kinematics


유체의

운동학

Fluid motion
involves
position
,
velocity
,
acceleration

of fluid.

Fluid Kinematics :
Description of fluid motion

How to describe
fluid motion
?

(A) Lagrangian approach

(B) Eulerian approach

Describes a defined particle (
position
, velocity, acceleration,

concentration, etc.) as a function of
time, i.e.

(A)
Lagrangian approach

~ motion of
moving particles

Describes the flow field (
velocity
, acceleration, concentration, etc.) as
functions of
position
and
time, i.e.

(B)
Eulerian approach

~
motion at
fixed points

C(x,y,z,t)

Classification of types of fluid motion

Three
-
Dimensional Flow
: All three velocity components are important.

Two
-
Dimensional Flow
: One of the velocity components may be small relative to
the other two.

One
-
Dimensional Flow
: In some situations two of the velocity components may
be small relative to the other one

Velocity Field

Continuum Hypothesis
: the flow is made of tightly packed fluid particles that
interact with each other. Each particle consists of numerous molecules, and we
can describe velocity, acceleration, pressure, and density of these particles at a
given time.

Lagrangian Frame:

Eulerian Frame:

Acceleration Field

Lagrangian Frame:

Eulerian Frame: we describe the acceleration in terms of position and time
without following an individual particle. This is analogous to describing the
velocity field in terms of space and time.

time dependence

spatial dependence

We note:

Then, substituting:

Writing out these terms in vector components:

x
-
direction:

y
-
direction:

z
-
direction:

,

k
z
j
y
i
x
ˆ
ˆ
ˆ
()










V

V

V

V

V

V

V

V

V

V

V

V

V

V

V

V



= del operator (gradient)

convective accleration

It represents the fact the flow
property associated with a fluid
particle may vary due to the motion
of the particle from one point in
space to another.

local acceleration

Applied to the Temperature Field in a Flow:

The derivative of temperature :

In steady flow,

V

= 0

In uniform flow,

a =

0

0

0

0

Streamlines, Pathline and

Streaklines


Streamline
: The line that is tangent to the velocity field; a line that is tangent to
the

instantaneous velocity field


Experimentally, flow visualization with dyes can easily produce the streamlines
for a steady flow, but for unsteady flows these types of experiments don’t
necessarily provide information about the streamlines.

-

has the direction of the velocity vector at each point

-

no flow across the streamline

-

steady flow streamlines are fixed in space, unsteady flow streamlines move

-

streamlines are a
Eulerian

concept

Pathline :

A trajectory of a particle path over time

-

pathlines are a
Lagrangian

concept

-

If the flow is steady, the picture will look like streamlines.

-

If the flow is unsteady, the picture will be of the instantaneous


flow field, but will change from frame to frame, “pathlines”.

Streaklines
: a laboratory tool used to obtain instantaneous photographs of all
marked particles that passed through a given flow field at some earlier time.

-

If the flow is steady, the picture will look like streamlines.

-

If the flow is unsteady, the picture will be of the instantaneous flow field, but


will change from frame to frame, “streaklines”.

If
u

= 2
x
+
t
,
v

=
y
-
t
, find

a) the pathline for the fluid particle which is at the point (1
;
1)

at
t
= 1,

b) the streakline through the point (1
;
1)

at
t
= 1,

c) the streamline through the point (1
;
1)

at
t
= 1.

Example ;

u
dx
dt
x
t



2
v
dy
dt
y
t



2
(a) Pathline

x

y

B.C.s : at t=1, x=1

B.C.s : at t=1, y=1

(b) Streakline

u
dx
dt
x
t



2
v
dy
dt
y
t



2
B.C.s : at t=1 , x=1

B.C.s : at t=1 , y=1

x

y

y

x


t

t

A
ai
bj
ck




B
i
j
k








A
B



A
B







A
ai
bj
ck
B
i
j
k
A
B
a
b
c
A
B
i
j
k
a
b
c
b
c
i
c
a
j
a
b
k











































(
)
(
)
(
)








A
B
A
B
A
B
A
B







1
0
/
/







(
)
(
)
(
)
vdz
wdy
i
wdx
udz
j
udy
vdx
k
0
d
r
dxi
dyj
dzk






r
d
r


r
xi
yj
zk



Streamline(
유선
)


V
ui
vj
wk







V
ui
vj
wk
d
r
dxi
dyj
dzk
V
d
r
i
j
k
u
v
w
dx
dy
dz


















vdz
wdy
wdx
udz
udy
vdx


dx
u
dy
v
dz
w


Streamline function





V
d
r
V
d
r
/
/



0
dx
u
dy
v
dz
w


u
dx
dt
x
t



2
v
dy
dt
y
t



2
dx
x
t
dy
y
t
2
2



dx
x
dy
y
2
1
2



(c) Streamline

At t=1,

y
x









2
2
3
1
2
Integrating, and line passing through (1,1) ;

y

x

Control Volume and System Representations

Systems of Fluid
: a specific identifiable quantity of matter that may consist of a
relatively large amount of mass (the earth’s atmosphere) or a single fluid particle.
They are always the same fluid particles which may interact with their surroundings.

Control Volume
: is a volume or space through which the fluid may flow, usually
associated with the geometry.

Fixed Control Volume:

Fixed or Moving Control Volume:

Deforming Control Volume:

Surface of the Pipe

Surface of the Fluid

Inflow

Outflow

Outflow

Deforming Volume

Homework# 3

If
u

= 2
x
+
t
,
v

=
y
-
t
, find

a) the pathline for the fluid particle which is at the point (1
;
1)

at
t
= 1,

b) the streakline through the point (1
;
1)

at
t
= 1,

c) the streamline through the point (1
;
1)

at
t
= 1.

d) Plot pathline, streakline, and streamline.

e) Find accelerations
at the point (1
;
1)

at
t
= 1 with ;


-

Lagrangian approach


-

Eulerian approach


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