Eulerian vs. Lagrangian

When I first studied fluid dynamics, one of the concepts I struggled with was that of Eulerian and Lagrangian reference frames. Essentially, these are just two different perspectives you can view the fluid from.  Physics is the same in both, but mathematically, you approach them differently. In the Eulerian perspective one sits at a location and watches the flow pass, like an observer watching a river go by. It’s demonstrated in the top animation, where turbulent flow sweeps past in a pipe. This is the usual perspective experimentally – you put an instrument at a certain point in the flow and you gather information as the fluid streams past in time.

In the Lagrangian perspective, on the other hand, one follows a particular bit of fluid around and observes its changes over time. This means that one has to follow along at the mean speed of the flow in order to keep up with the fluid parcel one is observing. It would be like running alongside a river so that you can always be watching the same water as it flows downstream. The Lagrangian view of the same turbulent pipe flow is shown in the bottom animation. Notice how moving alongside the pipe makes it easier to see how the turbulence morphs as it goes along. Experimentally, this can be harder to achieve (at least in a flow with non-zero mean speed), but it’s a useful method of studying unsteadiness. (Image credit: J. Kühnen et al., source)

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