3.1: Introduction to Motion and Velocity Field: Pathlines, Streamlines, and Streaklines

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

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.

3. y x z Pathline 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. Motion and path of an identified particle (A) Scene of VARYING TIME / VIDEO

4. y x z Streamline A streamline is defined as the line that is everywhere tangent to the local velocity vector. s (streamline coordinate) C Scene of FIXED TIME / STILL IMAGE B A

5. y x z 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. s (streamline coordinate) Scene of FIXED TIME / STILL IMAGE

6. s (streamline coordinate) Not allowed Some Properties of Streamline: 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. No flow across streamline

7. 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.

8. 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.

9. Dye injection P Streakline 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. Still image of a trace of dye from an injection port at P. Scene of FIXED TIME / STILL IMAGE • 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

10. Coincidence of Pathlines, Streamlines, Streaklines • Unsteady flows: Pathlines, streamlines, and streaklines are not the same. • Steady flows: Pathlines, streamlines, and streaklines are identical.

11. Some Images Flow past an airfoil, visualized by dye in water tunnel. From Van Dyke, M., 1982, An Album of Fluid Motion, Parabolic Press.

12. Flow past a block showing horseshoe vortex (top-right and bottom), visualized by smoke-wire.

13. Flow past a damper, visualized by smoke-wire.

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.

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.

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.