Bernoulli's principle and deduction of the velocity field

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?

• The equation you show is a scalar representation, which relates to something called equipotential contours. Knowing the parameters plus one actual stream line gives you the rest because the constant factor is just an offset. – honeste_vivere Jul 21 '17 at 13:26
• @honeste_vivere but the constant is different for different contours – JonTrav1 Jul 21 '17 at 13:27
• What's different? – honeste_vivere Jul 21 '17 at 13:27