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abj 1
3.1: Introduction to Motion and Velocity Field: Pathlines, Streamlines, and Streaklines
Geometry of Motion
Pathline
Streamline
No flow across a streamline
Local Relative Velocity of Fluid with respect to A Surface : A Preliminary Glimpse at
Flux
Stream Surface and Stream Tube
Streakline
abj 2
1. Pathline
2. Streamline
3. Flux through a surface:
Only the normal component of the local relative velocity of fluid with
respect to the surface ( ) can transport fluid across the surface.
dtw
dzdy
u
dxor
wdtdz
dtdy
udtdx
v,v),( txV
dt
xd
dw
dzdy
u
dx
v),(// txVxd
scalar vanishing-non aiswhere
),(
k
txVkxd
nsfV /
Very Brief Summary of Important Points and Equations
abj 3
A pathline is the path, or trajectory, traced out by an identified fluid particle.
Note that here is the displacement along the pathline of an identified particle.
dtw
dzdy
u
dxor
wdtdz
dtdy
udtdx
v,v
Pathline
Motion and path of an identified particle (A)
Scene of VARYING TIME / VIDEO
x
y
z
xd
dt
xdV A
A
)(txx A
)( dttxx A )( dttVA
)(tVA
),( txVdt
xd
xd
abj 4
A streamline is defined as the line that is everywhere tangent to the
local velocity vector.
xd
Streamline
Scene of FIXED TIME / STILL IMAGE
x
y
z
s (streamline coordinate)
AB
Cxd
abj 5
A streamline is defined as the line that is everywhere tangent to the local velocity vector.
Note that here is the displacement along the streamline.
It is customary to denote the coordinate along streamline as s.
xd
Scene of FIXED TIME / STILL IMAGE
x
y
z
x
xd
),( txdxV
),( txV
xdx
s (streamline coordinate)
dw
dzdy
u
dx
v
0),(
txVxd
),(// txVxd
scalar vanishing-non aiswhere
),(
k
txVkxd
abj 6
• By its definition, it follows that .
• Hence, there can be no flow across a streamline.
• Note that only the normal component of velocity can transport fluid from one side of the curve to
the other.
0
nV
Some Properties of Streamline: No flow across streamline
No flow across streamline
s (streamline coordinate)
tnt VVVV
Not allowed
)( ˆor ˆ enttangunitst ee
)(ˆ normalunitne
ttt eVVV ˆ
0ˆ
nnn eVV
abj 7
Let be the local fluid velocity
be the local surface velocity
Then, the local relative velocity of fluid with respect to the surface is given by
For short, we simply write
• Only the normal component of the local relative velocity of fluid with respect to the surface (
) can transport fluid across the surface.
sfsf VVV
/
fV
sV
Local Relative Velocity of Fluid with respect to A Surface :A Preliminary Glimpse at Flux
nedAAd ˆ ne
ntsf VVVV
/
ttt eVV ˆ
nnn eVV ˆ
te
nsfV /
sfVV /
fV
sV
abj 8
Stream Tube
Arbitrary closed curve C
Stream Surface
Arbitrary open curve C
Stream Surface and Stream Tube
Stream Surface
• Starting from an arbitrary open curve C.
• If we trace out streamlines that start from points on this curve, we have a stream surface that contains C.
Stream Tube
• On the other hand, if we choose a closed curve, we have a stream tube.
From the definition of streamline, no flow can cross a stream tube.
Therefore, a stream tube acts like an imaginary pipe/channel.
Due to this property, stream tube is a useful tool for analysis.
abj 9
A streakline is the line joining fluid particles that once passed through the same fixed point in space.
(It is helpful to think of a dye streak.)
Note that here is the displacement along the streakline.
One way to think of a streakline that passes through a point P is to think of a still image of a trace of dye from an
injection port at P.
xd
Streakline
Still image of a trace of dye from an injection port at P.
Scene of FIXED TIME / STILL IMAGE
)(txC
)(txB
)(txD
)(txA
xdDye injection
P
)( AA dttx
)( BB dttx
)( CC dttx
)( DD dttx
• Use the current time t as a reference time,
• particle A passed through point P at earlier time of t-dtA
• particle B, at t-dtB
• particle C, at t-dtC
• particle D, at t-dtD
abj 10
Coincidence of Pathlines, Streamlines, Streaklines
Unsteady flows:
Pathlines, streamlines, and streaklines are not the same.
Steady flows:
Pathlines, streamlines, and streaklines are identical.
abj 11
Flow past an airfoil, visualized by dye in water tunnel.From Van Dyke, M., 1982, An Album of Fluid Motion, Parabolic Press.
Some Images
abj 12
Flow past a block showing horseshoe vortex (top-right and bottom),
visualized by smoke-wire.
abj 13
Flow past a damper, visualized by smoke-wire.
abj 14
Questions
1. What is the dimension of A?
2. Is the velocity field steady?
3. Is the velocity field uniform along any line parallel to the x axis?
4. Is the flow 1-, 2-,or 3-D?
For below, let the velocity be given in m/s and A = 0.3 s-1.
5. Find the pathline of a particle that is located at point at time
6. Find the streamline that passes through the point at time
7. Can we find the streamline in (6) without having to solve for them in (6)? If so, how?
8. Sketch a vector plot.
9. Sketch a few streamlines.
Example 1: Pathlines and Streamlines
Given the velocity field
jAyiAxvuVVtxV yxˆ)(ˆ)(),(),(),(
)8,2(),( oo yx sto 0
)8,2(),( oo yx sto 0
abj 15
Questions
1. Find the pathline of a particle that is located at point at time
2. Find the streamline that passes through the point at time
3. Can we find the streamline in (2) without having to solve for them in (2)? If so, how?
4. Find the position and the velocity of the particle that is initially (at time ) located at
at time
5. Sketch a vector plot.
6. Sketch a few streamlines.
Example 2: Pathlines and Streamlines
Given the velocity field
11112 6,2,/ˆ)(ˆ)(),( smbsmasmjbxyiaxtxV
)2/1,2(),( oo yx sto 0
)2/1,2(),( oo yx sto 0
)2/1,2(),( oo yx
sto 0sto 1.0
abj 16
Questions
1. Find the pathline of a particle that is located at point at time
2. Find the streamline that passes through the point at time
3. Can we find the streamline in (2) without having to solve for them in (2)? If so, how?
4. Sketch a vector plot.
5. Sketch a few streamlines.
Example 3: Pathlines and Streamlines
Given the velocity field
12 4,1,/ˆ)(ˆ)(),( sbsasmjbxiayttxV
),( oo yx ot
),( oo yx 21 and,, ttto