Hi..

I would like to know the relationship between the courant number and time step size.

I am trying to solve the VOF model that the liquid, surounding air, is spreading out on the surface in the closed control volume.

Please give me a tip about it.

thank you for your attention have a nice day..

For transient problems, it is difficult to determine the value of the time step necessary to obtain a converging solution, as the stability criteria of the complete Navier-Stokes equations cannot be found analytically. The choice of time step depends on the time scales of the important flow features which need to be resolved in the flow. Using too large a time step can often result in resolution of non-physical flow behaviour. Although from a numerical stability point of view, the time step used by implicit solvers does not have to satisfy the Courant-Friedrichs-Lewy (CFL) condition, it is often advisable for relatively small time steps close to the CFL limit to be initially used:

C = c * (dt/dx) <= 1

where C is the Courant Number, c is the speed of propagation of some important flow feature, dt is the time step and dx is the grid spacing in the direction of propagation.

Using this (CFL) criteria, an estimate of the time step needed in transient problems can be made. Based on this condition, a time step of the order of magnitude of the residence time of a fluid particle passing through a control volume is often used. The residence time, is the time it would take a fluid particle to move through a cell from one face to the opposite face. For example, if a fluid particle moves in the x-direction with a velocity u, the residence time and hence the time step necessary to capture this movement would be given by:

dt = dx/u

This calculation is typically carried out on a control volume, which is known to be located at a point with the greatest flow instability, such as the vortex street in bluff body flows. Care must, however, be taken to ensure that the time step used, is not so small that the Reynolds-averaging assumptions are violated, i.e. averaging time <= dt.