Why naturalness is an issue in particle physics? It is not in fluid dynamics I have often heard that naturalness is a strong argument in favor of Supersymmetry. They say that there must  be energy scales not too far away the EW-energy so that the quantum corrections of the heavy particles (Higgs, top quark, etc) do not diverge. 
The argument goes like this: the Plank Energy scale is so huge that if nothing else exists in between the EW scale and the Plank scale, renormalization would pull all "natural numbers" from the EW to the Plank scale.
However, there are some interesting problems in physics where two extremely large numbers almost exactly cancel: in a 2D turbulent flow of an imcompressible fluid, the integrated vorticity is almost zero, even though positive and negative vorticity numbers are huge.
 A: I think that this is a misguided example. In isotropic turbulence the mean vorticity is zero (by rotational invariance), but the mean square vorticity is not. This means that there is indeed a symmetry that ensures that the mean vorticity is much smaller than the root mean square vorticity. This is precisely what naturalness says: If a number is small, there should be a symmetry reason.
A: The modelling with mathematics of data from particle physics experiments is an evolving in time process. The success of quantum mechanical solutions in fitting the spectra of the atoms, describing the photoelectric effect and black body radiation, led to quantum field theories, which attempt to solve the many body problems of particle scattering and interactions. It is  always  the data which drive the mathematical models, validate them or not. Once the models are in place one can examine the mathematical consequences, but should always remember that it is data which validates or falsifies mathematical conclusions.
There were blind roads before settling on SU(3)xSU(2)xU(1) as the standard model lagrangian in fitting practically all data accumulated since the 1960s: Regge poles and the parton model dominated as mathematical models of particle data. They were overthrown by the experimental verification of the quark model. 
Effective field theories are a mathematical tool in all this process. The desire to get a theory of everything which would describe all of nature in a unified manner is a goal at present, formulated because of the successes up to now: Maxwell's equations , which very successfully describe the data at classical dimensions, unified electricity an magnetism to do so. Electroweak unification led to the standard model. So the impetus is going towards unifying in a GUT model the strong with the electroweak, and finally a theory of everything , all four known forces.
It is encouraging that supersymmetries help in the mathematics, and that data point towards a unified coupling constants scheme. The discovery of the Higgs reinforced this aspiration.
So it is not a matter just of cancelations of large numbers, which is useful, it is a matter of going towards a theory of everything for particle data, and thus for nature.
Arguments like "naturalness", reinforce the accepted expectation of unification. A unified theory should embed all the lower energy ones naturally. 
It might be though a non attainable goal, but it is the present goal.
