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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.

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For an ideal inviscid fluid, the velocity profile is perfectly flat, and the factor a = 1 applies. For a viscous fluid, the velocity profile in laminar pipe flow is parabolic, and the factor of a = 2 applies. But, for a viscous fluid in turbulent flow in a pipe, the velocity profile is very close to being flat, and the factor of a = 1 is very nearly correct. Furthermore, for typical low viscosity fluids like water and air in pipe flow, the Reynolds number is greater than the critical value for transition to turbulent flow of 2100, so the factor a = 1 applies.

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  • $\begingroup$ 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. $\endgroup$ Mar 30 '20 at 23:27
  • $\begingroup$ In these cases, one needs to use the version of the Bernoulli equation that includes viscous friction loss. $\endgroup$ Mar 30 '20 at 23:55
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I can't give a comprehensive account, but I will try to stop your confusion.

  • The ideal fluid equation is only valid in idealized circumstances.
  • The factor $\alpha=2$ only applies to the special case of a cylindrical pipe (and maybe similar configurations)
  • The general (viscous) case is pretty hard, so one tries to approximate (locally) by the ideal case.
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