I Help Regarding Application of Bernoulli in a Boundary Layer

[jdgotts]

TL;DR Summary

Professor asked to explain why Bernoulli works in a boundary layer, and I don't believe it can. Any explanations that agree with his reasoning out there?

Hey all,

I recently took an aerodynamics exam that included the question "Please Explain how the Bernoulli Equation can be Applied Inside a Boundary Layer". Now, it is my belief that the Bernoulli equation, defined by my textbook as P+0.5ρV²=ℂ, requires inviscid flow to be properly applied. Because a boundary layer is formed through skin friction and friction in the flow, a boundary layer can therefore not exist in an inviscid flow, i.e. the premise of the question is flawed.

Where Bernoulli works, no boundary layer, where there is a boundary layer, no Bernoulli (as far as I understand it).

My professor has since responded to the widespread criticism of this question with the following explanation:
"Bernoulli's principle can always be applied along a streamline, the latter of which can exist in rotational flow. Bernoulli therefore can be applied in rotational flow, hence in the boundary layer. Bernoulli exists for compressible, time variant, and friction flow. Recall how we used Bernoulli's principle to equate the loss of pressure to drag force in the airfoil inside wind tunnel problem? We were able to do this with the Bernoulli equation despite the existence of viscous flow (if it were inviscid, there would be no drag)."

If anyone could explain either why his explanation makes sense, or explain why or where his explanation is flawed, I'd greatly appreciate it. I feel like the solution that Bernoulli works along a streamline, and streamlines exist in rotational flow, therefore Bernoulli can be applied in viscous flows and boundary layers, is oversimplified. A staggering amount of the internet seems to agree with me based on some Googling, but I can't really disprove what he wrote as his solution with what I know.

Thanks for any help you're able to give!

 

[Arjan82]

In the derivation of the Bernoulli equation that you've quoted (there are more variants) you need to assume steady, inviscid barotropic (i.e. no strong shocks) flow with a conservative external force field (like gravity). In that case you can derive that the following equation must hold along a streamline:

$$ \frac{1}{2} | \vec{u} |^2 + \int \frac{dp}{\rho} + F = constant $$
(Edit: for some reason I cannot get Latex to work anymore...)

So, what you compute here, the 'constant', is in fact the total pressure. If you use this equation in a viscous flow then this means the equation is not constant anymore. So, the way you can use the Bernoulli equation in a flow which includes losses (e.g. viscous flow) is to quantify the losses. In the end, Bernoulli is an energy conservation statement.

You can compute the loss of total pressure in a boundary layer if you have a measurement of the pressure and velocity. For an airfoil that is actually a common way to compute drag. You measure (with a pitot tube for example) the flow behind an airfoil. You see the dip in total pressure (pressure head) due to the boundary layer and you can use that, together with Bernoulli to compute the viscous drag. I think that is what your professor is referring to.