In most of the lagrangian co-ordinates formulations of fluid dynamics in 1D, a function h is used as a co-ordinate defined as:
h_x=rho
Integrating this up shows that h is the mass contained in the interval integrated over.
So my question is this:
1) Do I need to change h at each timestep?
2) How do I compute the length of my domain at each timestep?
In fluid mechanics, there are two main approaches to describe fluid flow: Eulerian and Lagrangian. The Eulerian approach focuses on observing the flow at fixed points in space, while the Lagrangian approach follows the motion of individual fluid particles over time.
In the context of 1D fluid dynamics, the Lagrangian coordinates formulation can be particularly useful for certain problems, especially those involving particles or material lines.
Let's discuss how to do fluid mechanics using Lagrangian coordinates in 1D:
It's important to note that while the Lagrangian formulation is advantageous for some problems, it might not be the most practical or efficient approach for all fluid dynamics scenarios. It is commonly used for studying particle trajectories, material deformation, or specific cases where the motion of individual fluid elements is of particular interest. In more general fluid flow problems, the Eulerian formulation is more commonly used, as it describes the flow at fixed points in space and can be more easily applied to complex systems.
Hello Mat,
From what I understood from your document, ℎ needs to be updated at each time step based on the current density distribution ρi(r). So, If ℎ is the Lagrangian coordinate, then the length of the domain is ℎend−ℎstart. In this case, ℎstart will be ∫0 to xstart ρi(r)dr and ℎend would be ∫0 to xend ρi(r)dr, where xstart and xend will be the coordinates of the start and end of the domain in the Eulerian coordinates. The difference between ℎ (end and start) gives the length of the domain in the Lagrangian coordinates at each time step.
Correct me if I am wrong.
Regards
As stated earlier in the first response, the Lagrangian approach involves studying the properties of an individual particle over time. Hence, the determined property is a function of time (timestep). The length of the domain in this approach then will be space interval between two timesteps.
Hi Mohammad Rahjoo
I was thinking that I would have to update the value of h at each timestep. That also means that my discretisation is flawed, there is no one \delta h, it changes at each timestep and also changes from point to point. So I would have to store that information as well. Is this a reasonable thing to do?
Hi Mat Hunt
My answer is yes and this is indeed one of the complexities of this method and that is why they are more computationally expensive. There are some ways to manage your strategy to store δh but at the end it's a matter of trade-off for what you aim to achieve vs what you can ignore.
Hi Mohammad Rahjoo
I see your points, and I realised that I had to redo my complete approach. The numerics seem to be the most complicated when differencing, and I think that I have done it correctly this time. I need to store the values of h for each timestep, and this will be a third variable.
- Badges
- Science topic
- Similar topics
- Mathematics