# Definition of CFL number in Arbitrary Lagrangian-Eulerian framework

In an Eulerian frame of reference, the CFL number is defined as $$\sigma=\frac{u \Delta t}{\Delta x}$$ with $u$ the magnitude of the fluid velocity. A restriction such as $\sigma<1$ for time integration then roughly means that the initial information travels at most over one cell-width during one time-step.

In an Arbitrary Lagrangian-Eulerian (ALE) framework the grid also moves and the convective velocity becomes $u-u_\mathrm{grid}$. Should the definition of the CFL number now be based on $u-u_\mathrm{grid}$ rather than $u$?

• The Courant number measures the ratio between the physical velocity and the mesh velocity in a frame in which the mesh is at rest. Of course mesh velocity should be greater that physical velocity in order to capture the physics, and therefore to avoid instabilities. According to this definition of the Courant number you are right. – HBR Oct 13 '17 at 7:37
• @chris Yes, it should be based on velocity relative to the mesh. The precise form has to come from a stability analysis. – cfdlab Oct 13 '17 at 11:36