# Tecplot Release 2021R2

"Ease of use and a big time saving"

Dealing with divide-by-zeros within the Analyze-functions, time-step specific zone-display, large reductions in temporary disk storage, and improved data loaders are only some of the features in Tecplot’s latest release.

# Reynold’s Number

One of many, many non-dimensional numbers

This is such a widely used number in fluid dynamics that it’s easy to be embarrassed to ask questions about it. That’s a shame. As with so many things in engineering, the fine print can be confusing and is often unclear at best, uncertain at worst.

The Reynold’s number indicates the relative importance of different forms of energy in a flow: inertial and viscous, as most references will explain. Forget that for a while, and consider why we calculate it.

It is most often used to determine whether a flow is turbulent or not. This is great for flow in a circular pipe, where the length of the pipe and its diameter are the only two important dimensions. But there are so many flows that engineers wrestle with that are not in pipe – flow around an aircraft, for example. Unlike circular pipes, the relevant dimensions for such flow domains are not unique.

You should certainly know the formula for Re (it’s easy enough to remember, and you should feel embarrassed if you fumble) but the values associated with it (both the dimensions and the value that separates turbulence from laminar flows) can depend on the engineering application you’re working on.

Coda: if you’re not concerned with fluid dynamics, there’s really no reason to learn anything about this.

# CFL Number

"Consider a rectangular array of points"

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.

# Aspect Ratio

One of many grid-quality measures

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.

# Convergence

Useful, but remember monotone and bounded

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.

# Conservation

System? Universe?

Pretty much all of our physics is built on the basis that energy is conserved. It can be “changed” into something else – mass, linear momentum, angular momentum, … – but it can neither be created nor destroyed. This applies to the universe as a whole. Does it apply to a system (which is the word we often use to refer to the part of the universe that we are investigating)?

That can seem to depend on our view of that system. We commonly refer to friction as a non-conservative force. By this we mean that if friction is present energy changes form: from momentum to heat or sound, for example. Aren’t heat and sound are forms of energy too? That depends on your view of the system. For purposes of ease of calculation you may choose not to include all possible forms of energy in your model.

We use conservation principles, together with constitutive equations, to derive Governing Differential Equations that are “solved” by computational methods such as CFD. Most computational methods do not conserve energy precisely: they may do so in an average sense or at selected locations, and even that only to within a convergence limit. This is fine for most engineering designs.

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