# Why doesn't Bernoulli's equation contain the kinetic energy correction factor?

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.

• The factor of 2 disappears for turbulent flow, which means it often does, I agree. But then the whole concept of energy conservation collapses due to the loss to friction $H_L$. So Bernoulli's equations remains useless. And the whole idea of developing it does describing laminar effectively-ideal flow also collapses. Mar 30 '20 at 23:27
• The factor $$\alpha=2$$ only applies to the special case of a cylindrical pipe (and maybe similar configurations)