I am wondering what exactly is the ALE method (or formulation). I am involving in CFD calculation. I often encounter this method. Does anybody kindly give me some hint on it in general or in detail, and also any useful reference would be good.

Thanks a lot

By the way, here ALE means Arbitrary Lagrangian Eulerian.

Behind the impressive name is actually something very simple. There may be many ways to describe and implement it. Let me give you one from the perspective of finite volume methods.

You may know this already: From a strictly Eulerian point of view we consider a control volume as stationary. We describe the flow not by tracking the particles, but by defining flow variables locally (in space and time). With a Lagrangian method, your control volume would actually be following a certain particle. In structure dynamics we usually apply Lagrangian methods, as the structure is clearly confined within its moving surface. ALE: In some CFD applications you may want to apply some kind of hybrid Eulerian-Lagrangian method. For example in aeroelasticity you have to deal with deforming structural boundaries. A body-fitted flow grid will have to be deformed to match the boundary at each time. This is very easily implemented in a finite-volume code. All you need to do is to take the motion and deformation of cell faces into account, when calculating the fluxes. The fluxes are usually formulated as the transport of some quantity (e.g. density) through the cell face by some velocity (flow velocity). Now as the grid is moving, the velocity at which the quantity is transported becomes the flow velocity relative (!) to the grid (flow velocity minus grid velocity). You now basically have a quasi-Eulerian method (ALE) which you can actually still regard as Eulerian but which accounts for Arbitrary motion and deformation of your control volumes. You could now theoretically assign your grid velocities such that you are exactly following a fluid particle, in which case you would end up with a Lagrangian description. (there is more to it, such as a geometric conservation law that should be satisfied by the numerical method, but that's little more than a detail) Mani

hi, Mani

I'm going to impliment GCL with Roe's flux difference splitting scheme. Are there any difference from orginal Roe at fixed coordinate? if it does, can you point out any reference about deducing the modified scheme ?

zyunf

The differences between the ALE and the regular Eulerian description are on the level of the governing equations and in principle should be independent of the applied numerical scheme. Now, with Godunov and Roe schemes, and other schemes which are based on the solution of a local wave-propagation problem, there may be special things to consider. I don't have enough experience with those schemes to give you advice. Go through the derivation of your scheme and consider where you may have to replace the flow velocity by the relative flow velocity (relative to the grid). If you know how to use the Roe scheme on a moving frame of reference, then that's going to tell you how to get to the ALE. I am sure it's been done before.

Thanks, Mani. I have find an article by J.Y.Trepanier, he described how to modify Roe's Riemann solver with GCL.

Go to Journal of Computational Physics U will get host of papers using ALE take any one and read ..

Thanks a lot for all of you!