I'm curious why you think there are velocities in the flow different from the bubble's velocity: according to your drawing there shouldn't be such velocities.

The way you've drawn it - I personally haven't encountered such bubbles in my life, but I can assume their existence - I would assume negligible losses.

The way I would've drawn a bubble - convex ends instead of concave ends - I would, first, say that the bubble is "closed", i.e. its entire surface is covered with water, second, assume laminar flow of water in the narrow gap between the bubble and the wall and use friction loss formula for laminar flow as the first approximation.

Also, Bernoulli's equation is valid for non-viscous flows only. If you take losses into account, you're working with the simplified energy balance equation which looks like Bernoulli+losses, but it's not Bernoulli.

P.S. after rereading your question, it seems that you do assume some flow around the bubble. Then, I guess, you need to provide the derivation of your formula for K to see what exactly you assumed to obtain it.

Hi Ivan Nepomnyashchikh

Firstly thank you for answering my question is is much appreciated. Please allow me to answer some of your questions.

So when the bubble is moving I would assume its velocity to be the same as the flowing water around it, but in this case, i am assuming that the velocity of water that is flowing around the bubble isn't enough to dislodge it and hence it is stationary.

The way i have drawn it is due to the contact angle of both materials the bubble is attached to. Both create a contact angle less than 90 degrees and hence the only logical way to achieve this (in my opinion) is to assume the bubble to have a concave shape that is in contact with the flowing water around it. I have attached an image to better display the bubble as from my previous drawing it is not easy to see that the bubble is present in a cylindrical pipe sandwiched by another cylindrical pipe.

I am assuming that the flowing fluid is in fact water, a viscous fluid, and the bubble is composed of air. Should the energy balance equation of Bernoulli be suitable for this application?

A derivation of calculating the value for K an obstruction causes is seen here: https://www.youtube.com/watch?v=JUB_xyw3qCk

During college I have not come across obstructions of flow like this and would much appreciate your input in the validity of this derivation.

Much appreciated,

James

That's an interesting case.

First, you've got to be completely sure the bubble is not moving.

Second, provided the first is true, you can analyze it as a stationary obstruction. What he tells in the video is not fully intelligible to me, though, might be right to some degree. If you go with his suggestion, note, that in your case pipe area A is, in fact, the annulus area (I'm not sure if you accounted for that in your formula). I think that he assumed the form of the wake behind the obstruction. But wake's form can be different at different Re numbers. I'm not completely sure about the contraction-expansion assumption of his.

Third, I would go with classics: using drag coefficient to find drag force on the object. Then convert it to the head loss. Drag force is proportional to the width of the wake anyways. Drag force is 0.5*CD*rho*U2*A. To get pressure drop, divide by A. To get head loss from the pressure drop, divide by rho*g. And you end up having (0.5*CD*U2 )/ g. Here, A is the cross section of your bubble (cross section that faces the flow, not tangential to the flow), U is the upstream velocity (some assumed or calculated velocity before the bubble), rho is the density of water, g is gravitational acceleration and - the most difficult part - CD is the drag coefficient. Drag coefficient depends on the geometrical form of the bubble and Re of the external flow and is tabulated for some forms and Re. For instance, see here (one of the last entries in the table: square flat plate at 90 deg https://www.engineeringtoolbox.com/drag-coefficient-d_627.html): the drag coefficient is 1.17. The form more or less matches yours, but they don't specify Re and, most likely, that's for air. But as a very rough first estimate you can go with it. Try searching the Internet for a better match to your bubble.

With regards to Bernoulli, I'm more than sure that the equation you're using and calling "Bernoulli's equation" is correct. I was just pointing out that sometimes it shouldn't be called "Bernoulli's equation".

Also, pay attention that assessments like this one are crude. The level of crudeness may or may not suffice. That depends on what you want. In my experience, the crude assessments I made for my setup led to overestimates of about 50% of the total pressure drop. That was primarily because my flow doesn't have many analogues and well established formulas, just like your bubble in the annulus. But overestimate is good. Underestimate is bad. In all my experience, crude assessments have led to overestimates only.