I'm confused about the kinetic energy correction factor $\alpha$ in the Bernoulli (energy conservation) equation. The uncorrected equation is derived for an ideal fluid, and then reads $$ \frac{1}{2} \rho v^2 + \rho g h + p = const $$ or a variant thereof. Here $v$ is assumed to be the flow everywhere in the pipe, so while this is technically true only for a particular flow-line, ignoring height differences this is true for the average flow through a pipe as well.
Once viscosity is introduced, however, one can show that (assuming laminar and steady flow, and the no-slip condition) a parabolic velocity profile develops in a pipe, with $v=0$ along the edge and $v_{max} = 2 v_{avg}$ in the center. From this profile, one can show by considering the energy flux into the pipe that energy conservation for the average velocity needs to be corrected by a "kinetic energy correction factor" $\alpha = 2$, leading to $$ \frac{\alpha}{2} \rho v^2 + \rho g h + p = \frac{\alpha}{2} \rho v^2 + \rho g h + p + \rho g H_{loss} $$ with $H_{loss}$ due to friction.
Now, my question is - why is the ideal-fluid equation ever valid? It is always parabolic, so it always should have the factor os 2 there.
I would have expected the flow to become ideal-like for low viscosity, but it is $always$ parabolic, irrespective of how low the viscosity is. The velocty never approaches the limit of being an equal value throughout the pipe (except maybe in a boundary layer near the edges). So it seems one should always have this factor of 2 in there. Yet clearly, the ideal-fluid equation must be valid, or else it won't be so heavily quoted and used.
I'm confused.
