I'd say that you have several regimes that are well defined:
- The behavior of the fluid as it exits in inlet jets and enters the bulk without interference from the cubes. [Length scale set by the exit aperture?]
- Flow of the fluid around isolated cubes when far from the edges of the tank (far being several times the characteristic size of the cube). [Length scale set by the side of the cube.]
- Flow of the fluid toward, along and away from the sides of the tank away from the jets and without interference from the cubes. [Length scale set by the boundary behavior?]
which is the good news, unfortunately you also have all the cases that mix and match the various length scales:
- case with cubes interacting with the jet near the aperture
- case with cubes in motion near the walls
- case with cubes in close proximity to one another
You can probably find existing treatments for all the former cases, but the latter ones are going to be tricky, and you'll note that they feature at least two length scales.
Yuck.
This must be part of why they say CFD is hard.
Actually all of them have to be considered. The flow depends on $\rho,\mu,D,U,\rho_{bubble},\mu_{bubble}$. Any quantity of interest would be a function of dimensionless parameters that are formed from these variables. You may make use of Buckingham's Pi theorem to determine what they are. In this case mere inspection tells you that there are three dimensionless numbers:
\begin{align}
Re\equiv \frac{\rho U d}{\mu},\quad \frac{\rho_{bubble}}{\rho}, \quad \frac{\mu_{bubble}}{\mu}
\end{align}
However dependence on three parameters is complicated, so we seek simplification. Usually if we may adopt the approximation that one of the parameters tends to $0$ or $\infty$, then we may drop that parameter from the list. Another case in which you may drop parameters is if, in the particular case you are studying, those parameters are held constant.
For example if you are studying air bubble in liquid, as an approximation you may analyse governing equations in the limit, $\frac{\rho_{bubble}}{\rho}\to 0, \frac{\mu_{bubble}}{\mu}\to 0$, in which case solutions obtained will be a function of $Re$ alone. If you are an experimenter studying motion of air bubbles of different size $d$ in water, then $\frac{\rho_{bubble}}{\rho}, \frac{\mu_{bubble}}{\mu}$ are constant (for this experiment) and results depend on $Re$ alone. But if you change your experiment to studying motion of bubbles in water of identical size but varying gas inside the bubble, then $Re$ remains constant and results depend on $\frac{\rho_{bubble}}{\rho}$ and $\frac{\mu_{bubble}}{\mu}$. In doing experiments, you exploit the freedom to form dimensionless numbers any way you want in such a way that, as far as possible, resulting dimensionless numbers can be varied independently of each other in an experiment. My point is that there is nothing sacred about adopting $Re$ as defined above, but the choice depends on context and the simplification that results.
Suppose instead of the definition above for dimensionless parameters, you had the following altered definitions
\begin{align}
Re'\equiv \frac{\rho_{bubble} U d}{\mu_{bubble}},\quad \frac{\rho_{bubble}}{\rho}, \quad \frac{\mu_{bubble}}{\mu}
\end{align}
In principle there is nothing wrong with this new set of definitions. Now suppose you conduct the very last experiment of changing gas inside the bubble. Then whenever you change the gas inside the bubble, all three parameters above would change, while with the first set of definitions only two parameters would change. So if you wanted to plot some dependent quantity, say drag on the bubble (suitably non-dimensionalised), then with the first choice you have two independent variables, but with second choice you have three independent variables. However both graphs are equivalent in the sense that they are interconvertible, i.e. they contain the same information. So the first choice in this case would make your life easy.
Best Answer
The difficulty with Reynolds number is that the length scale (and often times the velocity scale) are both completely subjective, as you have identified. In standard nomenclature, an airfoil (or in your case, an ellipse) would use the freestream velocity as the velocity scale and the chord (semi-major axis) as the length scale. This is the assumed standard for aerodynamic analysis.
Your example of a vertical ellipse (ie. the semi-minor axis is aligned with the flow) changes the type of problem slightly because aerodynamicists don't generally consider things like that. However, a problem like that is similar to flow over a "fence", or flow over a "post", etc.. And in these problems, the height of the fence/post is the length scale of choice. For your ellipse, that would mean you still use the semi-major axis.
In general, you need to pick the length scale that is appropriate for the problem. For a wing, it makes sense to choose the chord of the wing and not the length of the conveyor belt that built the wing. For your ellipse in the configurations you listed, the semi-major axis is essentially always the length scale that controls the features of the flow. Other problems may not be so easy to determine and in these cases, you pick one (or several) and give Reynolds numbers based on all of them -- this is typical in turbulence where you might see a Reynolds number based on the integral scale, or the dissipation scales, or the Taylor microscale. All of these are valid and useful provided you state what system you use.