Dimensional analysis with space-distributed variables

Question

I read several books about Dimensional Analysis and the "Pi theorem".

It frequently happens that both the "governed" variable and some of the "governing" variables are entities which are "distributed" ones, meaning that their values can physically change with the position. Some examples are (in fluid dynamics): pressure, temperature, density, velocity etc

However, in nearly all examples that I could find on the books, it is assumed that these variables only have one specific "average" value for the problem under investigation. For example: it is shown how to derive the relation which links the average velocity of the fluid and its average density (plus some other variables), which is valid for the specific geometry and conditions under investigation.

I wonder: is it possible to use Dimensional Analysis to derive a relation which takes into account the variation with position of all the "distributed" variables involved? (Avoiding a brutal averaging operation).

A possible solution could be considering the position "x" as an additional governing variable. However, this fixes the problem only if the "governed" variable is the only one which actually depends on the position. In case some of the "governing" variables depend on the position too (as I stated in the example abobe), I still don't know how to approach the problem.

Can anybody give me any insight?

Answer 1

In many cases, trying to take into account spatial variations and locally computing the non-dimensional parameters sort of misses the point to using the non-dimensional parameters. There are many benefits to non-dimensionalization, but the two most common are:

1. Identify which terms in an equation are negligible with respect to others;
2. Allow comparison/similarity between two different bodies or conditions.

In both of those cases, we're interested in the global characteristics of something and so we don't care about local variations. We want to use nominal values that represent the important length, time, and mass scales of our problem. If we have a problem where important scales are drastically different throughout critical regions, then non-dimensionalization isn't the most useful tool for analysis.

That is why you are only seeing "global" examples -- dimensional analysis is most useful when trying to compare global phenomena.

That said, there are instances where we will use local values of non-dimensional parameters. In my field (large-eddy simulation --- LES --- of turbulent combustion), we will often use the Reynolds number, Mach number, Karlovitz number, Prandtl number, and Schmidt number calculated locally at a length scale based on our LES filter size in order to determine how our models should behave. Similar things are done in Reynolds-Averaged Navier Stokes (RANS) for their models.

When we do this, we do it because these numbers give us an idea of how important one term is relative to another -- for example, Reynolds number is the ratio of inertial forces to viscous forces, and so if it is small at the filter scale, we know we don't need to model the viscous terms. This is computed at every location in space and time.