It depends on what you are analysing and what the kind of information you would like to extract from the result.
Some of the non-dimensional numbers you mentioned mostly consist of thermo -physical values and heat capacity. These are valid regradless of your shape or profile and you can apply them to whatever you like.
Others carry the charactristic length (CL), of your geometry which has to be defined for each problem unless given by the standard shapes and profiles. The question of good choice of CL has been discussed in other questions.
As for the Reynolds no, it containes the hydraulic diamter (dh). This value must be defined according to your problem also.
E.g. :Depending on if you are investigating the effect of a complex geometry in a channel to improve heat transfer (Nusselt No, correlation derivation) you might want to choose a dh that is valid for packed columns, containing the porosity of your channel set-up. Here the geometry would not have such an impact, more would the "free-flow" space.
Thus, it strongly depends on your experimental set-up. Or are you working only numerically?
But most importantly, aks yourself what the results should convey?
For complex geometries you will not be able to use one of Reynolds numbers most helpfull features- to compare dynamic similitude and detect different flow regimes (lamiar, transient, turbulent)- as this only works for the given simple geometries unless you investigate the flow regime for your Reynolds numbers with your hydraulic diameters and satisfactory accuracy is still hard to achieve.
So if you know that you can not use the values to compare with other literature (have you checked for similar problems?) it might be whise to choose a more simple dh for each shape and use heat transfer over varying flow velocity as an evaluation model. Given, of course that that is a goal in your research.
No, these numbers are applied for all profiles/shapes. For Reynolds number etc depend properties of fluid which are also depending on shapes or profiles. If you search google for reynolds number formula images you may find formulas for different shapes.
It depends on what you are analysing and what the kind of information you would like to extract from the result.
Some of the non-dimensional numbers you mentioned mostly consist of thermo -physical values and heat capacity. These are valid regradless of your shape or profile and you can apply them to whatever you like.
Others carry the charactristic length (CL), of your geometry which has to be defined for each problem unless given by the standard shapes and profiles. The question of good choice of CL has been discussed in other questions.
As for the Reynolds no, it containes the hydraulic diamter (dh). This value must be defined according to your problem also.
E.g. :Depending on if you are investigating the effect of a complex geometry in a channel to improve heat transfer (Nusselt No, correlation derivation) you might want to choose a dh that is valid for packed columns, containing the porosity of your channel set-up. Here the geometry would not have such an impact, more would the "free-flow" space.
Thus, it strongly depends on your experimental set-up. Or are you working only numerically?
But most importantly, aks yourself what the results should convey?
For complex geometries you will not be able to use one of Reynolds numbers most helpfull features- to compare dynamic similitude and detect different flow regimes (lamiar, transient, turbulent)- as this only works for the given simple geometries unless you investigate the flow regime for your Reynolds numbers with your hydraulic diameters and satisfactory accuracy is still hard to achieve.
So if you know that you can not use the values to compare with other literature (have you checked for similar problems?) it might be whise to choose a more simple dh for each shape and use heat transfer over varying flow velocity as an evaluation model. Given, of course that that is a goal in your research.
In whatever cases those numbers apply they should be fine, but most are for idealized geometries. Most of Rebecca's comments are appropriate for more complex geometries. In addition, you need to consider when the flow might transition from continuum to particle. When that happens you will need to adjust or scale the calculated nondimensional values, as for example with slip-flow.
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