Flow Visualization

Flow visualization is a method to examine the complex flow field feature. It is not just physical experiments, it is also widely used in numerical simulation, computational fluid dynamics (CFD) to visualize the numerical result in virtual environment.

Streamlines

A streamline is a curve by joining everywhere tangent to the instantaneous local velocity vector or motion vector of a flow field with same arbitrary constant. Since the flow field is a function of time, the streamline pattern cannot be easily observed by experiment directly except in a steady flow field.

Consider a point of a streamline in a velocity field,

the local instantaneous velocity vector , must be parallel to the infinitesimal are length along the streamline. By geometric similar, the components of is proportional to those of  . let be the magnitude of and be the magnitude of . Imply .And the equation of streamlines can then be obtained by integration.

For a two dimensional steady flow , imply  and a family of streamlines are found by substituting an arbitrary constant of the integration.

However, velocity cannot be determined directly from the streamlines alone since the streamline pattern from the streamline equation provide the tangential information of the instantaneous local velocity only.

Streamtube

A streamtube is a closed volume formed by collecting a buddle of streamlines. Since fluid flow everywhere parallel to the instantaneous local velocity along the streamline at any instant in time, the streamtube pattern will also change with time subject to the velocity field of the flow except the flow is steady. However, at any instant, the mass flow rate passing through any cross-sectional area remains constant along the streamtube. 

Pathlines

A pathline is the locus of a fluid particle after traveling a period of time as described by Lagrangian concept and is equal to the position vector of the fluid particle . The curve of fluid particle path is just like a moving car lights photograph taken by long time exposure shot.

Mathematically, the location of fluid particle at time can be obtained by integration over time from the starting time at location to time , i.e. . The pathline of a fluid particle can then be obtained by computing the location of the fluid particle between the tracing period of time and plotting the position vector over the tracing period.

If the flow is steady, the velocity field is also steady and the fluid particles will follow the streamlines. The pathlines will be same as the streamlines.

Streaklines

A streakline is a curve by sequentially joining the location of all particles that passing through a prescribed point in a flow after a period of time. In other words, a streakline is a snapshot of a time-integrated flow pattern through a common starting point. This is the most common flow visualization method used in physical experiment.

Similar to pathline, the location of a tracer particle can also be determined mathematically by integration over time from the tracking start time of each injection at location to the end of tracking, i.e. . The streakline can then be obtained by computing the location of each injected tracer particle between the tracing period of time and joining all position vectors of tracer particles at the end of the tracing period sequentially.

If the flow is steady, the velocity field is also steady and all tracer particles will follow the streamlines also. The streaklines will be same as the streamlines and pathlines also.

Timeline

A timeline is a set of marked fluid particles in a straight line across the flow at the same instant in time. Timelines can be used to visualize the change of flow profile along the flow at different instant of time.

Differences of Streamline, Pathline & Streakline

Streamlines is a snapshot of an instantaneous flow pattern at a given time instant. While both pathline and streakline are flow pattern related to a time period.

Mathematically, the pathline is formed from particles with the same starting location and tracing start time, and the end of tracing time is difference. While the streakline is formed from particles with the same starting location and the tracing start time is difference, but the end of tracing time is the same.

If the flow is unsteady, the velocity field will be a function of time also. Since the starting time of tracer particles is difference in streakline, the surrounding velocity field at a prescribed location will be difference from the previous instant also and therefore the streakline can not resemble neither a streamline nor pathline except the flow is steady.

And conceptually, a pathline is an long time exposure shot flow path of a single particle over a period of tracing time. while a streakline is an instantaneous snapshot of time-integrated flow pattern from a collection of tracers.