This should be an easy one, since it’s based on a formula named for Courant / Friedrichs / Lewy, but like a lot of things in CAE there are nuances.

This number represents the time it takes for a sound wave to travel across a cell (or element): it depends on both the material properties of the cell and the cell’s dimensions. That’s it for the easy bit.

The Wikipedia starts by describing it as “a necessary condition for convergence while solving certain partial differential equations“. But convergence itself is a term that’s gloriously rich in meaning, so it’s not that straightforward. In the first place, the CFL Number can be important for both steady-state problems (that is, for problems whose solutions don’t change with time) and transient or time-dependent problems. Further, there can be time-dependent problems which converge just fine without your needing to worry about the CFL criterion. Next, when cells can have all sorts of shapes, what cell “dimensions” should we use? Since most grids are non-uniform, is there a single CFL number that characterises the entire grid?

There’s a lot to unpack there, more than can be covered in this post. All we’ll say here is that you should check your solver’s user-manual: if it says the CFL number is important, then it is. The smaller the cell (for example we may refer to the smallest of the cell’s dimensions) the lower its CFL number. And since this number is the upper limit for the time-step that the solver can use, a lower CFL makes the solution slower. Solvers can work around this, by the way. They can use different time steps for different parts of the grid, they can alter the time step as the solution progresses, and so on.

If this explanation doesn’t fill you with questions, you haven’t read it right. If you note that the CFL number is relevant only for explicit iteration methods, you’re probably OK. All other details should come from your solver’s documentation.

The 20 page English translation (available here) is worth reading even if you don’t fancy wrestling with the mathematics. It will give you an excellent idea as to how much fine print is involved in numerical methods in general.

The aspect ratio of a cell (or element, if you’re more used to finite element terminology) affects the accuracy of the numerical method – but remember that any cell-quality measure is nuanced. Tempting as it is to look for a cast-in-stone value that indicates a pass/fail, the reality is that the cell-quality is only one of several things that determine solution accuracy. In general, wherever the state variable(s) has (or have) high gradients, you’d like to have better shaped cells.

With that caveat out of the way, here’s the easy part: the aspect ratio of a rectangle is the length divided by the breadth (or the other way round, if the breadth exceeds the length). The ideal value is 1. The larger the value, the “worse” the aspect ratio.

Now the harder part: cells can be of different types. And it’s often far from clear what “larger” and “smaller” dimensions should be used. Finite Elements are usually either 1-dimensional (beams, trusses) in which case the aspect ratio is irrelevant, or two-dimensional (plates, shells, plane stress, plane strain – the standard shapes are quadrilaterals and triangles) or three-dimensional (tetrahedra, pyramids, wedges, and hexahedra). But Finite Volume / Finite Difference methods that are used in CFD also use polyhedra.

Don’t, therefore, use the aspect ratio (or pretty much any other cell-quality measure, for that matter) as a pass / fail test. Use it, instead, as a broad guideline. A lot of the adverse effect of poor aspect ratios depends both on the physics (the gradients of the state variables) and the solver itself.

A series is sometimes an excellent way to represent a physical phenomenon. Population growth is an example most of us are familiar with. Zeno’s paradoxes are pretty well known too. The sum of the series is, in many cases, what we are really interested in. If the series has infinite terms though, we are (like Zeno’s Achilles) more interested in what the sum will converge to.

In real life we settle for an approximate sum rather than insist on the precise sum. That is, we’ll keep adding terms until the difference between successive iterations is less than whatever we think is acceptable.

This is a convenient representation of convergence, particularly for CFD users who commonly monitor the convergence of residuals of energy and velocity.

Convergence is a very elegant subject full of nuances. This is as good a place to start as any, though you should really refer to Piskunov for the truth, the whole truth and nothing but the truth.