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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
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2. Very Brief Summary of Important Points and Equations
Pathline
Streamline
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.
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3. Pathline y x z Scene of VARYING TIME / VIDEO
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Pathline
Motion and path of an identified particle (A)
Scene of VARYING TIME / VIDEO x y z
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.
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Streamline x y z
s (streamline coordinate) C
Scene of FIXED TIME / STILL IMAGE B A
A streamline is defined as the line that is everywhere tangent to the local velocity vector.
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5. y x z Scene of FIXED TIME / STILL IMAGE s (streamline coordinate)
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s (streamline coordinate)
Scene of FIXED TIME / STILL IMAGE
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.
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6. Some Properties of Streamline: No flow across streamline
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Some Properties of Streamline: No flow across streamline
s (streamline coordinate)
Not allowed
No flow across streamline
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.
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Local Relative Velocity of Fluid with respect to A Surface : A Preliminary Glimpse at Flux
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.
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8. Stream Surface and Stream Tube
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Stream Surface and Stream Tube
Arbitrary open curve C
Arbitrary closed curve C
Stream Surface
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.
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9. Streakline Scene of FIXED TIME / STILL IMAGE Dye injection P
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Streakline
Still image of a trace of dye from an injection port at P.
Scene of FIXED TIME / STILL IMAGE
Dye injection P
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
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.
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10. Coincidence of Pathlines, Streamlines, Streaklines
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Coincidence of Pathlines, Streamlines, Streaklines
Unsteady flows:
Pathlines, streamlines, and streaklines are not the same.
Steady flows:
Pathlines, streamlines, and streaklines are identical.
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11. Some Images Flow past an airfoil, visualized by dye in water tunnel.
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Some Images
Flow past an airfoil, visualized by dye in water tunnel.
From Van Dyke, M., 1982, An Album of Fluid Motion, Parabolic Press.
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12. Flow past a block showing horseshoe vortex (top-right and bottom),
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Flow past a block showing horseshoe vortex (top-right and bottom),
visualized by smoke-wire.
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13. Flow past a damper, visualized by smoke-wire.
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Flow past a damper, visualized by smoke-wire.
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14. Example 1: Pathlines and Streamlines
Given the velocity field
Questions
What is the dimension of A?
Is the velocity field steady?
Is the velocity field uniform along any line parallel to the x axis?
Is the flow 1-, 2-,or 3-D?
For below, let the velocity be given in m/s and A = 0.3 s-1.
Find the pathline of a particle that is located at point at time
Find the streamline that passes through the point at time
Can we find the streamline in (6) without having to solve for them in (6)? If so, how?
Sketch a vector plot.
Sketch a few streamlines.
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15. Example 2: Pathlines and Streamlines
Given the velocity field
Questions
Find the pathline of a particle that is located at point at time
Find the streamline that passes through the point at time
Can we find the streamline in (2) without having to solve for them in (2)? If so, how?
Find the position and the velocity of the particle that is initially (at time ) located at at time
Sketch a vector plot.
Sketch a few streamlines.
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16. Example 3: Pathlines and Streamlines
Given the velocity field
Questions
Find the pathline of a particle that is located at point at time
Find the streamline that passes through the point at time
Can we find the streamline in (2) without having to solve for them in (2)? If so, how?
Sketch a vector plot.
Sketch a few streamlines.
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