Bernoulli's principle and deduction of the velocity field

Question

Bernoulli's principle states that, on a streamline of a incompressible inviscid steady fluid flow, we have that

$$ \frac{1}{2}v^2 + \Phi + \frac{P}{\rho} = const $$ The constant on the right hand side depends on the streamline chosen, and $v,\phi,P$ depend on position on the streamline. My question is, if the constant depends on the streamline, how can one deduce the entire velocity field from just one streamline? For example, one of the famous concepts using Bernoulli's principle, Venturi's tube, one can deduce the velocity of the entire fluid, just by looking at one horizontal streamline and exploiting Bernoulli's principle. Does one streamline contain information on all the velocity field?

Answer 1

The constant is indeed different on different streamlines, and given the value of constant on one streamline you can get flow parameters on that streamline only. Those who apply knowledge on one streamline to all streamlines are assuming uniform conditions across flow cross-section. Also note that if you assume the flow to be irrotational then Bernoulli equation may be applied between any two points not necessarily on the same streamline.