All About The Convergence Criteria

In CFD, most of the fluid flow problems are non-linear in nature and deal with inherently unstable phenomena such as turbulence. The solution techniques use an iterative process to successively improve a solution until ‘convergence’ is reached. 

What is Convergence in CFD?

Convergence is the limiting behavior observed in fluid dynamics. The exact solution to the iterative problem is unknown, but you want to be sufficiently close to the solution for a particular required level of accuracy. Convergence, therefore, does need to be associated with a requirement for a particular level of accuracy. This requirement depends upon the purpose to which the solution will be applied.

It is also important to keep in mind that a converged solution is not necessarily an accurate one. Many people called it “budgetary convergence,” which means your solution is converged when you can’t afford any more computer time.

Since the point at which the analysis is deemed converged is defined by the judgment of the analyst, users should have a solid understanding of when the analysis has reached its final solution. Typically, when assessing the convergence of a steady-state CFD analysis, the following three criteria need to be monitored as the analysis progresses:

1. Residual Values

The residual is one of the most fundamental measures of an iterative solution’s convergence, as it directly quantifies the error in the solution of the system of equations.

In a CFD analysis, the residual measures the local imbalance of a conserved variable in each control volume. Therefore, every cell in your model will have its own residual value for each of the equations being solved. A typical CFD simulation includes equations for momentum, pressure, and turbulence, and the solver performs iterative solutions of these equations.  The residuals plot shows the difference between successive solutions of these equations. 

The solutions are aggregated and normalized so that each equation gets represented by a single number, with all numbers scaled to a common range.  All residuals get plotted on one graph (with Y-axis as log scale) to allow the CFD engineer to study interactions between the equations.

In an iterative numerical solution, the residual will never be exactly zero. However, the lower the residual value is, the more numerically accurate the solution. Each CFD code will have its own procedure for normalizing the solution residuals. It is best to check your code’s documentation for guidance on an appropriate criterion when judging convergence.

A good residuals plot has several characteristics to identify.  The key element is decreasing lines and downward slopes.  Residuals should always decrease.  The sawtooth pattern in Figure 2 happens in unsteady simulations.  Each spike represents a new timestep.  Successive spikes should show decreasing peaks, or at least show each peak at the same height.

Figure 1: Residuals for Steady-state simulation.

Figure 2: Residuals for Unsteady state simulation.

For CFD, RMS residual levels of 1E-4 are considered to be loosely converged, levels of 1E-5 are considered to be well converged, and levels of 1E-6 are considered to be tightly converged. For complicated problems, however, it's not always possible to achieve residual levels as low as 1E-6 or even 1E-5.

2. Solution Imbalances

As the word conservation suggests, the quantities from conservation equations (mass, momentum & energy) should conserve till the last iteration. 

The solution imbalances will never be exactly zero because we are representing the physical system into an algebraic(numerical) form. However, the imbalances should be sufficiently small before considering the solution converged. As a good practice, we can aim for solution imbalances of below 1 % as a starting point but, highly sensitive analysis can demand tighter convergence.

Figure 3: Solution Imbalance of temperature.

As the figure shows, after the initial start-up period, the solution imbalances gradually decrease as the solution progresses. In general, the small values of solution residuals lead to small solution imbalances. 

There are a few exceptions to the above statement that sometimes small residuals can have large imbalances. For example, in CHT simulation conduction timescales are much larger than flow timescales.

3. By observing the quantity of Interest.

In an iterative solution, the calculation process starts with initial estimations of quantities (pressure, velocity, temperature, etc.). In every iteration, quantities are updated with respect to the previous iteration's results. In a converged steady-state solution, one should expect to see those quantities come to an expected level and don’t change anymore.

Monitoring integrated quantities such as force, drag, or average temperature can help the user judge when his or her analysis has reached this point. There might be instances where one quantity may have reached its final point, yet the other still shows signs of fluctuations. Thus, considering this point as the final solution can be misleading. It is better to iterate to the point where values flatten out, in order to consider the solution converged.

Figure 4: Temperature over the number of iterations.

We can see in the above figure that the temperature after 25 iterations is within just a few percent of its final value. Once the monitor point values have "flattened out", we can safely assume the solution is converged.

So, to sum it all up, convergence means:

• All discrete conservation equations (momentum, energy, etc.) are obeyed in all cells to a specified tolerance OR the solution no longer changes with subsequent iterations.
• Overall mass, momentum, energy, and scalar balances are achieved.
• Ensuring that overall mass/heat/species conservation is satisfied.
• The drag, lift, and other coefficients reached the expected level.
• The net flux imbalance should be less than 1% of the smallest flux through the domain boundary.

Results of Poor Convergence:

• Diverged solution.
• Stuck residuals.
• Increasing imbalances.
• Unstable Solution.

Troubleshooting Residuals:

The residuals are more than just a tool for judging convergence.  They also provide feedback for troubleshooting bad simulations.  The pattern of the residuals provides clues as to why a simulation fails.

• Ensure that the problem is well-posed.
• If the residuals for a single equation go bad, look for trouble spots in the mesh.
• If the residuals are constantly bad, change the order of interpolation.
• If the residuals are oscillating and not showing a strong downward trend, try reducing the under-relaxation factor.
• For the density-based solver, reduce the Courant number.
• Remesh or refine cells that have a large aspect ratio or large skewness.

Or even, convergence can be accelerated by:

• Supplying better initial conditions.
• Starting from a previous solution (using file/interpolation when necessary).
• Gradually increasing under-relaxation factors or Courant number.
• Save case and data files before continuing iterations.
• Controlling MultiGrid solver settings (not generally recommended).
• Default settings provide a robust Multigrid setup and typically do not need to be changed.

If not, we can decrease the numerical error by:

• Using higher-order discretization schemes (second-order upwind, MUSCL).
• Aligning the grid with the flow to minimize the “false diffusion”
• Refine the mesh to resolve salient features of the flow.
• Interpolation errors decrease with decreasing cell size.
• Minimize variations in cell size in non-uniform meshes.
• Truncation error is minimized in a uniform mesh.
• Adapt the mesh based on cell size variation.
• Minimize cell skewness and aspect ratio.
• Avoid aspect ratios higher than 5:1 (but higher ratios are allowed in boundary layers)
• Maintain optimal quad/hex cell angle as 90 degrees and optimal tri/tet cells as 60 degrees.

As CFD engineers, we do not have the luxury of judging a binary system: pass/fail. Convergence is not an exact science, so we should rely on evidence from all the above-mentioned criteria. As discussed, we have three tools to judge convergence which are residuals, monitors, and flow patterns. We need to combine all three tools and build a composite picture to reliably judge convergence in any simulation.