Which cell size has to be selected for calculation of courant number? If I consider the smallest cell height in my hexahedral mesh for achieving courant number 0.8, I end up having my time step of size 1.2668e-5 which is extremely small. Is it a correct approach to use base size instead of smallest cell size?
The Courant number is by definition a local variable therefore, depending on the velocity field and mesh size, you have a field of the cfl variable. The numerical stability constraint must be based on the most critical value. Do not forget that in presence of explicit scheme for both advection and diffusion, you have to consider also the diffusive constraint.
...but on the other hand side, who is today still using explicit schemes with a strict CFL number stability criterion (CFL
Time step is calculated over each cell with the help of CFL criterion. If you intend to simulate time-accurate solution, then you need to use "global time stepping", i.e. find the minimum of all time-steps and use that for time iterations. However, if you are interested in the steady state solution, you can use "local time-stepping", i.e. update each cell with the local time step for the next time iteration. This leads to a much faster convergence towards the steady state.
Hope this helps.
Dear Thomas,
I could reverse the issue, who is today using CFL numbers of order 100-200? That can makes sense only for steady laminar flow or for RANS solutions. At present, not only LES (and of course DNS) but also URANS and DES require to have a small time step to describe the time dependent behaviour of the flow, a fact that constraint the CFL to be of order 1 also for implicit schemes.
Regards
Anant Diwakar : CFL>1 does not automatically mean, that a transient simulation would be not accurate in time. Imagine a structured hexahedral mesh where close to a wall a near-wall refinement to y+~1 has been made. Now in the geometry is a 90 degree corner and the near-wall mesh refinement in both coordinate directions (as for a backward facing step) is crossing the free flow. There in the volume you will encounter rather large CFL numbers, but the flow is by crossing these refined cells rather not subject of strong gradients or rapid flow changes. So, if the solver can tolerate it from a numerical stability point of view, than rather large CFL numbers in these propagating refinement cells can be accepted and the flow prediction will be nevertheless accurate in space and time, also the flow crosses e.g. 10-20 cells in a single timestep. Situation might be different, if the flow would experience large changes in that region, where these small cells are located. So, (also in reply to Filippo Maria Denaro) it depends. The given scenario describes a flow simulation, where rather large CFL numbers occur, but where it is of zero advantage to enforce CFL~1 in that region, where the flow crosses these propagating near-wall refinement cells. Ok, next you can argument - who is today still using structured hexahedral cell meshes? Hmmm. E.g. people who like to obtain a high accurate CFD solution at minimum possible computational costs and where the mesh generation for the studied flow domain is feasibly simple.
Thomas Frank : "CFL>1 does not automatically mean, that a transient simulation would be not accurate in time"
I was making my point with respect to explicit time-stepping, not implicit.
In explicit scheme, people use "global time-stepping" for time accurate solution and "local time-stepping" for steady state solution. In both cases, CFL < 1.0 for a given cell.
We can also use "global time-stepping" for steady state solution, but then its convergence would be very slow compared to "local time-stepping".
If I am not wrong, people generally use implicit scheme with dual time-stepping for CFL >1.0 for solving time-accurate flows (Proposed by Jameson).
Let me know if my statement is incorrect.
Thanks
Thomas Frank the role of the CFL can be see clearly in the modified equation. It alters the original PDE as it increases. Then, let us simply consider the CFL as the rate between the exact slope of the characteristic line and the discrete (and approximated) slope dt/dx. Using high CFL number has implication on the transport along the discrete pathline.
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