Segregated Flow Guidelines

• The default pressure and velocity under-relaxation factors are conservative. They lead to a converged solution in most cases, including when the grid is poor or the equations are stiff.
• Optimal values of under-relaxation parameters are problem-dependent. The grid aspect also affects the optimum values. In a batch job or any other computation where the progress cannot be monitored, it is safer to use the default values.
• When performing similar calculations, check how the simulation reacts to the variation of under-relaxation factors. Since convergence behavior is similar on different grid densities, it is worthwhile to perform a test on a coarse grid to determine the optimum values for the case. These optimum values can then be applied to a calculation on a denser grid.
• It is an accepted rule of thumb that optimum under-relaxation factors for velocity and pressure add up to 1 for steady-state simulations. If the grid is highly non-orthogonal, keep the under-relaxation factor for pressure lower than this rule suggests. (0.1 is a typical value.)
• If the residuals are converging well, it is acceptable to try increasing the under-relaxation factor for velocity (up to 0.9), and decreasing the pressure under-relaxation factor (to 0.1).
• For compressible flows, you must sometimes decrease the under-relaxation factor for velocity to 0.5 and possibly the pressure under-relaxation factor to 0.1.
• For porous media, you must sometimes to decrease the under-relaxation factor for velocity to 0.5 and possibly the pressure under-relaxation factor to 0.1.
• When solving steady-state problems on fine meshes, start with a much coarser mesh, and then successively refine the mesh by reducing the base size. For example:
  • Design the desired mesh for the highest cell count affordable.
  • Increase the base size by a factor of 8 and start the computation with this coarse mesh.

    You sometimes need lower under-relaxation values for the coarsest mesh, but iterations are fast and this requirement does not increase the computing effort much.

  • After the convergence criteria are satisfied or maximum specified number of iterations is reached, halve the mesh base size, remesh, and restart the simulation. Repeat until the final mesh size is reached.

  In this way, you obtain faster convergence on finer grids by providing a good initial solution. Usually, one needs only a third or a quarter of the number of iterations that would be needed if starting with an initial guess, like constant or zero values.

  In addition, you obtain solutions on a series of grids which are systematically refined (same design, only base sizes reduced), which allows for an estimate of discretization errors using Richardson extrapolation. For example, if you reduce base size by a factor of 2 and use second-order discretization, discretization errors on the finest mesh are equal to one third of the difference between solutions on the finest and next coarser grid. For base size reduction by a factor of 1.5, the errors amount to about 80% of the difference in the two solutions.

• For transient simulations, provided a reasonably well-resolved time-step is chosen, you can often increase the under-relaxation factor for velocity as high as 1.0 and for pressure up to 0.9 (for example, for LES-type simulations).
• Under-relaxation factors of 0.9 (velocities and scalars) and 0.5 (pressure) work well in most cases.
• These values are higher than in steady-state cases since the contribution of the transient term to the discretized equations has the same effect as under-relaxation.

For periodic flow, an order of 50–100 time steps per period is appropriate.

The convective Courant number is a helpful indication for selecting the time step size: for time-accurate simulations, the convective Courant number should be 1.0 on average in the zone of the interest. This value implies that the fluid moves by about one cell per time step.

For flows with free surfaces, if a second-order scheme for time integration is used, the Courant number must be less than 0.5 in all cells. If a free surface moves more than half a cell per time step, the High-Resolution Interface-Capturing scheme can lead to overshoots or undershoots, and eventually to divergence.

When computing wave propagation, use more than twice as many time steps per wave period than cells per wavelength.

Smaller physical time-steps generally mean that the solution is changing less from one time-step to the next, so that fewer inner iterations are required. There is clearly an optimal balance of time-step size, under-relaxation factors, and the number of inner iterations for a given problem and desired transient accuracy.

If you are performing a time-accurate simulation with adequate time step and under-relaxation parameters, 5 iterations per time step are usually sufficient. More iterations per time step (an order of 10) and lower under-relaxation factors (around 0.8) are sometimes required for tough coupled problems like flow with free surface and motion of a floating body. VOF simulations with melting and solidification also require many inner iterations.

If the convergence of iterations is slow, it is a sign that the time step is too large and will result in significant temporal discretization errors. It is better to reduce the time step than to do many iterations within a large time step. For example, halving the time step allows the number of inner iterations to be reduced, so that the computing effort is not much higher while the accuracy of solution is increased.