Initial Solutions

Initial Solutions

It can be very beneficial to have a solution that is ‘close’ to the desired solution that can be used as a starting point for the analysis, particularly for cases that are very non-linear, or very sensitive to the flow conditions. There are a number of ways in which this can be achieved.

Software Defaults

Good convergence will most easily be achieved by starting with a solution which is close to the final solution. The default values used are very dependent upon which solver is being used, so the User Documentation should be consulted, to see the recommendations for that particular solver. The default solution for all flow variables being calculated may be zero. However, if most of the flow is likely, for example, to have an x-component of velocity of a certain magnitude, then it may be wise to start with this as a solution. Similarly, if temperatures in the final solution are going to vary between 200 and 300 degrees C, it may be better to start with initial temperatures of 250 degrees C than with the default value of 0.

Simpler Physics

The example given above using a solution close to the final solution to provide a starting condition demonstrates an important technique for obtaining converged solutions of complex problems - use simpler physics. A velocity field for Newtonian flow behaviour can be a good starting point for a non-Newtonian problem and an isothermal analysis could be used as a starting condition for a non-isothermal simulation. For very complex flows it may be appropriate to use a number of intermediate solutions adding further physical phenomena at each step.

For example, a Stokes (linear) solution may show good convergence and give a reasonable initial velocity profile for a non-linear analysis.

Incremental Approach

An alternative step-by-step technique is to use an incremental approach e.g. Reynolds Number progression. This is performed by starting with a higher viscosity (low Reynolds Number) and using this solution as the start guess for a solution at a lower viscosity (higher Reynolds Number). The viscosity can then be gradually decreased to the required value.

Other ways of achieving a similar goal could be:

  • Use an initial solution at the correct parameter values but with a ‘smoother’ solution method. For example an initial solution may be obtained using first order upwind differences, since this will often prove easier to converge than ones with a higher order difference approximation. This first-order solution can then be used as an initial guess for a solution with the higher order approximation. Another example may be to use the results from a simpler turbulence model, eg use a k -  turbulence model result as a starting guess for a Reynolds Stress Model.

  • Use a solution with a coarser grid and interpolate this to the desired grid. Adaptive gridding may achieve this automatically.

  • Use a physical time-stepping approach to reach a converged ‘steady-state’ solution.