Given the assumptions made for deriving the Bernoulli's Equation, is it possible to use this equation to understand the pressure-velocity relation in the core of a 3D vortex ?
No, one of the assumptions is the absence of dissipation effects. Conversely, a vortex structure in a real flow has a core region where the effect of viscosity becomes relevant.
Vortex flow is simply called whirling. It has two types:- Free vortex flow and forced vortex flow.
Forced vortex flow
In forced vortex flow external supply is required to maintain the constant flow (constant angular momentum).
Free vortex flow
In free vortex flow the velocity is remain same in the flow i.e. no external torque is required. The free vortex flow is irrotational, and therefore, also known as the irrotational vortex. In free vortex flow, Bernoulli’s equation can be applied. Here, I attached a file which is derived equation of Bernoulli using free vortex condition.
inverse square relation as per Bernoulli's theorem IS a fundamental characteristic of vortex pairs
problem is that pairs of vortices are not always what you empirically find to be at work in nature
as there is the continuous expenditure of energy, Bernoulli's equation is not valid for forced vortexes
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