ERTH 430: Fluid Dynamics in Earth Systems Reading Questions #2 (Written responses due in class Wednesday, 9/20 5 pts total ) Dr. Dave Dempsey Dept. of Earth & Climate Sciences SFSU, Fall 2017

Kinematics: Streamlines, Trajectories, and Other Flow Visualization Methods

A. Wikipedia: Streamlines, streaklines, and pathlines
1. The motion of a fluid (fluid flow) is a vector (velocity) field. What is meant by a steady velocity field? [Note: This idea applies to any field variable, not just velocity.]

2. Streamlines, streaklines, and pathlines (trajectories) are lines ("field lines") that help us visualize the pattern of flow, in time and/or space. Under what circumstance do streamlines, streaklines, and pathlines differ? [Note: The Wikipedia article isn't quite complete on this point: the circumstance in question applies to the flow direction but not the flow speed.]

3. What is a streamline? (What does "instantaneously" mean here?)

4. What is a streakline? What is an example of a streakline in a fluid?

5. What is a pathline (trajectory)?

6. What is a timeline?

7. Given that at each point along a streamline and pathline (trajectory), the fluid velocity there is parallel (tangent) to the line, what is the (essential) difference between a streamline and a trajectory? How does this difference help us understand why the two lines aren't the same when the flow direction is unsteady (but are the same when it is steady)? [Hint: Imagine following a parcel of fluid over time. First consider it's initial location. A streamline and the parcel's trajectory both pass through that location and are parallel to each other there (since at the same location and time, streamlines and trajectories are parallel to the fluid velocity at that spot). Now imagine letting a short time go by, during which the parcel moves to a new location. By definition, the parcel will follow its trajectory, and at the new location there will be a streamline that is parallel to the trajectory there. Compare the orientation of this streamline to the one at the same location but at the earlier, initial time, when the flow direction at the new location is (a) unsteady, and (b) steady.]

8. What is the (essential) difference between a streakline and a trajectory?

9. Can different streamlines intersect? Why or why not? Can different streaklines intersect? Why or why not? Can different trajectories intersect? Why or why not?

10. Your answers to Questions (7) and (8) above should be consistent with the statement that "streamlines and timelines provide a snapshot of [a] flow field ..., whereas streaklines and pathlines [trajectories] depend on the full time history of the flow." Are they? Explain why or why not.

B. Flow Visualization (Barry Belmont, National Committee for Fluid Dynamics Films, 1963) [First 13 minutes, 15 seconds only.]

This is a classic video that uses laboratory fluid flows to illustrate various flow visualization concepts and differences between them.

1. What is one thing that can make fluid flow patterns so complicated to understand?

2. According to the video narrator, can we predict or simulate flow patterns on theoretical grounds alone? [Note that this point is less true today because of advances in our theoretical understanding of fluid flow and the development of supercomputers and sophisticated computational models of fluid flow based on theory, though it is still true that we can't model real flows with complete precision and some regimes of fluid flow are still not fully understood.]

3. What is meant by "flow visualization"? What is one application of flow visualization?

4. What are some visualization methods used for visualizing water flow in the laboratory?

5. Why do tiny hydrogen bubbles injected into water provide a good way to visualize the flow of water, but larger bubbles don't?

6. The video narrator defines four flow visualization concepts to help us interpret images recorded of flow visualizations: pathlines (trajectories), streaklines, timelines, and streamlines. How is each defined? [Make sure you can identify their essential differences.] Under what condition does a pathline (trajectory) coincide with a streakline?

7. As a general rule, we can't make streamlines directly visible, but under one condition we can deduce what streamlines look like because they are the same as streaklines and trajectories. What is that condition?

8. In laboratory studies of steady flow past an airfoil, what does flow visualization tell us about the flow past opposite sides of the airfoil when the airfoil is tilted relative to the direction of the flow?

9. What is a technique for visualizing patterns of flow speed (at least, under certain conditions)? When steady flow through a channel or pipe enters a restricted (narrower) part of the channel or pipe, what does flow visualization show happens to the flow speed? How can we tell?

10. In flow past an oscillating plate, are streaklines and pathlines (trajectories) the same? Are streamlines the same as either? How can we tell, experimentally? (In particular, how are streamlines visualized in this case?)

11. What is meant by transient flow?

C. Streamlines vs. Trajectories in a Translating Rankine Vortex (Lucas Harris and Dale Durran, University of Washington, 2006)

These Web pages show output of a numerical model simulation of idealized (mathematically defined, not observed), two-dimensional flow to illustrate the difference between streamlines and trajectories. Several methods are used to visualize the flow field: vectors (where each vector represents the direction and relative speed of the flow at the location of the base (tail end) of the vector), streamlines, and trajectories.

1. A Rankine vortex is an idealized, mathematically defined pattern of fluid flow. In a Rankine vortex, how does the "tangential" component of the flow velocity vary with distance from the center of the vortex? What does the black circle on the plots of the flow field represent? [Note: The tangential velocity at any point is the component of the flow velocity perpendicular to a line (the radius) drawn to that point from the center of the vortex (the radius). of a circle through that point). The other component of velocity is parallel to the radius. These are the components of velocity in a polar coordinate system. In the case of a Rankine vortex, what is the radial component of the velocity? How can you tell?]

2. To complete the idealized flow field, a uniform, left-to-right velocity is added to the Rankine vortex velocity at each location. The speed of this uniform flow equals the maximum speed in the Rankine vortex. In that case, the added uniform velocity will cancel out the Rankine vortex velocity at exactly one location. Where is that location? What are two ways you can tell, at least roughly?

3. In the third plot, streamlines (red) are calculated and included on the plot. Do they satisfy the definition of streamlines? How can you tell?

4. Click the link to "Now let's put this flow into motion." Note that the initial velocities of three particular fluid parcels are indicated using bold-faced vectors. If the flow were steady, what would you predict the trajectories of these three parcels to look like? Why?

5. Run the animation, which shows what happens to the flow field and the locations of these three parcels over time as the Rankine vortex moves to the right at a speed equal to that of the uniform velocity added to it. Are the trajectories and streamlines the same? Are they parallel to each other? Explain.

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