Validity of dimensionless numbers

Question

Dimensionless numbers are often used when studying complex phenomena such as in fluid dynamics or thermodynamics. In my studies we often used them, especially in these two subjects, where they helped quite a lot.

The question

I need help in understanding if dimensionless numbers are always valid in reality. In other words, is it always true that: if a real system I am studying has the same dimensionless numbers as another system, the two are completely similar and all missing data of the studied system can be recovered from the dimensionless numbers?

(In the question I assume the two systems I am comparing are doing the same thing, eg two air flows out of two nozzles, not comparing a nozzle flow with a convective heat flow.)

Some context

I have this doubt for some reasons:

• I think there are some phenomena, like change of state of matter or magnetic saturation, that maybe aren't taken into account by dimensionless numbers, or at least I've never heard of such dimensionless numbers in my studies.
• I once tried using dimensionless numbers in an electrodynamic simulation in Femm software. At first, changing some parameters, like width of the coil or current through the wire, while keeping the dimensionless numbers the same worked and I could calculate the missing parameters, like induced force on an iron piece, and indeed the hand calculations gave the same answer as the simulation. If a parameter became too different from the starting value however I would start getting different values by hand and from the simulation.

EDIT

Reasons to why I think some dimensionless numbers are not always relevant in reality:

Sometimes the effect of the change of one or more dimensionless numbers is negligible on the behaviour of the system due to certain conditions that limit the physical mechanisms represented by such numbers to nearly zero.

I'll show an example:

Let's say we are analyzing a fluid flow in a tube and we have some simplifying conditions like the tube is insulated such that thermal exchanges with the outside are negligible (negligible also in reality, not only from a theoretical point of view) and we can consider the flow as adiabatic. In this case I wouldn't consider to check the Nusselt number because, while it can still be calculated, its value wouldn't influence the system by much, because the tube wall tends to block any heat exchange between the fluid and the ambient outside the tube. In this case I would be mainly interested in Reynolds number and Darcy friction factor, as these are the numbers that matter more in the system.

I would also be interested in Mach number, to be sure that the flow is still able to receive a pressure feedback from the opposite side of the flow and respond to it, and to also check for compressibility effects.
Another number which would be of interest is one relating heat transfer through the tube by conduction and momentum of the fluid, to show that the second is more important than the first.

Answer 1

What matters is the physical laws, not the dimensionless numbers themselves. If you write the Navier Stokes momentum and energy balances in non-dimensional form, (by suitable choices like $u'=u/U_0,\, x'=x/L,\, p'=p/(0.5\rho U_0^2)$ etc., the Reynolds number, Prandtl number and others will appear naturally as part of the equations. You can look this up in any common reference to see what I mean.

Some other dimensionless groups are more empirical, shown to predict behavior by experiment.

To your question on whether dimensionless numbers will predict a real life situation all the time: any physical model is just that, a model, with many assumptions and idealizations. It is meant to get us relatively close to an answer, which is often the best we can hope for with fluid dynamics. Also keep in mind that all dimensionless numbers must match for there to be a true similarity, e.g. Reynolds, Mach, geometric aspect ratios, which can be very difficult to achieve in practice. Often a lab like a wind tunnel testing a model aircraft must make do with an imperfect match, and get what data they can.