Reader Questions

Reader Question: Lagrangian Vs. Eulerian


Reader isotropicposts writes:

Hi, I’m taking a fluids class and I’m not sure I understand the whole lagrangian-eulerian measurements of velocity and acceleration. Could you explain this?

This is a really great question because the Eulerian versus Lagrangian distinction is not obvious when you first learn about it. If you think about a fluid flowing, there are two sensible reference frames from which we might observe. The first is the reference frame in which we are still and the fluid rushes by. This is the Eulerian frame. It’s what you get if you stand next to a wind tunnel and watch flow pass. It’s also how many practical measurements are made. The photo above shows a Pitot tube on a stationary mount in a wind tunnel. With the air flow on, the probe measures conditions at a single stationary point while lots of different fluid particles go past.

The other way to observe fluid motion is to follow a particular bit of fluid around and see how it evolves. This is the Lagrangian method. While this is reasonably easy to achieve in calculations and simulations, it can be harder to accomplish experimentally. To make these kinds of measurements, researchers will do things like mount a camera system to a track that runs alongside a wind tunnel at the mean speed of the flow. The resulting video will show the evolution of a specific region of flow as it moves through time and space. The video below has a nice example of this type of measurement in a wave tank. The camera runs alongside the the wave as it travels, making it possible to observe how the wave breaks.

In the end, both reference frames contain the same physics (Einstein would not have it any other way), but sometimes one is more useful than the other in a given situation. For me, it’s easiest to think of the Eulerian frame as a laboratory-fixed frame, whereas the Lagrangian frame is one that rides alongside the fluid. I hope that helps! (Photo credit: N. Sharp; video credit: R. Liu et al.)

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