I have a puzzle, in the experiments, we usually set prototype v.s. model with: 1)geometric similitude 2)dynamic similtude

For the dynamic similitude, I can understand, because the Pi theorem stands there. But I don't see rational explanation of geometric similitude, for there is no physical evidence and mathematical proof that one has to do so.

e.g. if we want to know the Drag of an elliptic cylinder, then we may do a circular cylinder exp and interpet it onto elliptic cylinder case by just introducing a coefficient. The instinct is , elliptic and circular are just so much similar to each other, why can't we just do one and explain the other?

In ocean eng, people do experiement, if stay with geometric similitude, the water depth would go near zero in the model. For instance, 100km wide by 20m deep prototype lake with be mapped to 100m by 0.02m experimental basin (but 0.02m depth doesn't make sense for gravity dominated phenomena). If we can breakthrough this geometric similitude requirement, we will be more powerful. Samething will apply to CFD.

Aren't you forgetting boundary conditions? The shape of a surface has a profound effect on the flowfield that develops. Your example of ocean engineering merely points out the limitations that are faced in testing - not the ability to ignore similitude. A similar statement can be made with respect to wind-tunnel testing. Many times Re matching is sacrificed in order to match Mach number, but the data must be carefully weighed in light of Re-scaling. It does not mean that the Re has little or no effect.

i suggest you may like to go thru the book by johnstone and thring (pilot plants, models and scale up methods).

strictly speaking one has to go for nondimensionalisation of the relevant expressions which mathematically describe the physics of the system. boundary conditions are very much included, else how does one define a system. pi theorem is based on mathematics and unfortunately does not take care of the physics in the system. you may like to refer to the book by bird, stewart and lightfoot ('transport phenomena').

for the ocean engineering example which you have cited, a careful analysis is to be made with regard to what one wants to study. there is something called 'distorted model' which is applied to study such phenomena in particular.

by definition (please look at the above books at least), in order to have dynamic similarity, one has to maintain geometric similarity. the scope for defining the system may change with the requirement to be addressed and that is where concept of model distortion comes in.