Eulerian vs Lagrangian approach in fluid mechanics (wave example)

[Urmi Roy]

Hi All,

Recently we've been working on the distinction between the Eulerian and Lagrangian approaches in Fluid mechanics.
I understand the simpler examples like a running stream of hot water etc. However one example is really tripping me up.

So what's confusing me is that in analyzing water waves in a sea, the Eulerian and Lagrangian descriptions (incidentally) look the same i.e. they both trace a circle.

Where I've been learning about this is this video (link below) by John Lumley from Penn State university :
https://www.youtube.com/watch?v=mdN8OOkx2ko&list=PL0EC6527BE871ABA3"

I thought that Eulerian approach is just looking at something from a fixed reference frame (which might be moving at a constant velocity)...sort of like a control volume analysis. However that doesn't explain this example.

So I'm thinking maybe in Eulerian description instead of fixing a spatial reference frame, its more like fixing a slice of fluid...does this sound right?

 

[olivermsun]

Can you elaborate on what your question is with regard to the water wave?

On the second point, no: the Eulerian is attached to a fixed point in your reference frame. The Lagrangian velocity follows a specific bit of water even if it moves within your reference frame.

 

[Urmi Roy]

Hi olivermsun. I'm trying to figure out why the Lagrangian and Eulerian descriptions have an 'identical appearance' as mentioned at 9 mins 47 seconds on the video.

 

[olivermsun]

Well I guess one way to answer your question is to say that the particle orbits aren't very large, i.e., the Lagrangian particles never move very far from the original (Eulerian) starting points. Since the flow field is continuous, it means the Lagrangian velocities following the particles are never very different from the Eulerian velocity at the (nearby) starting point.

 

[Urmi Roy]

I'm not sure I agree. It says the Eulerian frame of reference is 'unsteady', so its not a fixed reference frame.

 

[olivermsun]

Eulerian velocities are defined at fixed points relative to the reference frame.

What Lumley means by "unsteady" is that the velocity changes with time. When you pick a reference frame moving with the wave speed, the Eulerian velocities become "stationary," that is, unchanging with time.

 

[Urmi Roy]

Okay, I'm going to try explaining this in my own words...could you tell me if I'm right?

So in the Eulerian description (in a stationary reference frame), if we draw velocity vectors at each point, they'll trace a circle...so maybe that's why it looks just like the Lagrangian description?
Also, in this case its unsteady because the velocity vectors are in a circle and therefore there is acceleration.

What I'm still unsure about is when in a stationary reference frame, does the velocity vector just rotate in one spot, or are they just arrows that form a circle?

 

[olivermsun]

I guess I am unsure what you mean by arrows that form a circle.

By convention, the velocity vector is drawn with the base of the vector at the point where the vector is defined. The length of the vector is just drawn according to some arbitrary scale, so the fact that the Eulerian arrow heads seem to trace the same circle as the Lagrangian particle trajectories is just a coincidence.

In the video, the Eulerian vectors show the velocities at the same points in space but as the velocities vary in time. Hence the base of the vectors always originate from the same spots while the arrowheads move around.

By contrast, the bases of the Lagrangian vectors move around following the circular particle trajectories; the arrow heads always point toward where the respective particles are heading at any given instant.

Hope this helps.

 

[AlephZero]

Hi olivermsun. I'm trying to figure out why the Lagrangian and Eulerian descriptions have an 'identical appearance' as mentioned at 9 mins 47 seconds on the video.

I don't think there is any deep reason. It's just a coincidence, for that particular type of flow.

If you follow the motion of one particle of water, it moves round a circle for each period of the wave. That's the Lagrangian description.

If look at one fixed point in the water, you see water flowing past to the right, then down, then left, then up, as each wave crest and trough goes past you. That's the Euleran description. Unless you make some analysis of what you see, you don't know that you see the same particles of water going past the fixed point, for each period of the wave.

I suppose the fact that both happen to be "motion in circles" makes the point that you need to understand what the quantities in your equations represent, before you start interpreting what the math functions mean!

In a first course in solid mechanics (e.g. topics like beam bending, torsion of shafts, etc), you work with a Lagrangian description, which is easy because each particle of the solid never moves far from its original position. But in fluid flow you might be interested in "small" perturbations about a large scale flow, where each individual particle of fluid arrives from "infinity" and departs to "infinity". In that case, looking at a fixed point in space and watching the flow go past the point (Euleran) is often easier.

 

[Urmi Roy]

If look at one fixed point in the water, you see water flowing past to the right, then down, then left, then up, as each wave crest and trough goes past you. That's the Euleran description. Unless you make some analysis of what you see, you don't know that you see the same particles of water going past the fixed point, for period of the wave.

Thanks for your explanation, AlephZero! So from the part in quotes, you're saying that we fix a point in the stationary reference frame and the fixed point sees the fluid that just went through it subsequently move in a circle? And the Eulerian description is not steady because the wave is making the fluid elements go in a circular trajectory?

If my understanding, as explained above is right, then the rotating arrow, with its base fixed at a point (in the video) is the observer at the Eulerian fixed point observing what happens to the fluid as it passes through...but the point under observation stays at one place and doesn't go in a circle? This would further mean that the 'identical' nature of the Lagrangian and Eulerian descriptions in this case is simply because they both see a circular motion.

Please let me know if I'm getting this wrong!

 

[olivermsun]

No, the fluid that is flowing through the Eulerian fixed point at any given moment is going to the right, then down, then left then up. The Eulerian velocity is never the velocity of the fluid that already went through the point.

The "similar" nature of the Lagrangian and Eulerian velocities is because they aren't taken at locations that are very far apart with respect to a wavelength and because the overall motion is circular in nature.

 

[Urmi Roy]

The "similar" nature of the Lagrangian and Eulerian velocities is because they aren't taken at locations that are very far apart with respect to a wavelength and because the overall motion is circular in nature.

Thanks olivermsun. The only part that's still causing me confusion is the part in quotes here. If the Eulerian observation point keeps measuring new fluid, how does it know that motion is circular? Sorry for all the confusion!

 

[olivermsun]

The motion at that point is changing directions all the time, going through a full circle every wave period.

Also remember, in the example there's a flow field around the observation point that obeys the physics associated with the wave motion (in which particles are moving in approximately circular orbits). You are merely observing at that point. That's as far as the Eulerian observation point "knows" anything.

 

[Urmi Roy]

Thanks olivermsun, I think its clear now!

 

[olivermsun]

Great! You're welcome!