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
(
주의
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