Exploring Subsonic, Compressible Flow in Curved Ducts

[komega]

Hi,

I'm new here so I'd like to apologise in advance if this has been posted before. I tried searching first but nothing came up.

I'm doing a project on subsonic, compressible flow in curved ducts and have a question about the physics involved as the flow navigates a bend. Allow me to further simplify the problem by stating that the curvature is mild to avoid separation and that the flow is bounded by two infinite plates (top and bottom). I understand that secondary flows play a significant part in internal flows but for now, I would like to constrain my discussion to the primary or bulk flow per say. As the flow turns, a radial pressure gradient is generated that results in the flow decelerating near the concave side (outer wall) and accelerating (for the initial part) near the convex side (inner wall).

So my question is essentially: How is this radial pressure gradient generated? (I guess one may assume an ideal flow for the purpose of explanation)

I have encountered various ways of explaining it (among them/combination of; Bernoulli's equation normal to the flow direction, mass/momentum continuity, Coanda effect) but have arrived at the "chicken-and-egg" situation. Most references I tried usually highlighted the balance between centrifugal and pressure forces. I would appreciate it if anyone would kindly shed some light on my situation. Apologies if I have not fully clarified the problem.

Thanks,
Dave

 

[RandomGuy88]

Hi Dave, Welcome to PF!

Viscosity forces the the fluid near the wall to follow the curvature of the wall. As the fluid begins to follow this curved path a pressure gradient is developed to balance the centrifugal force. The gradient is such that pressure is lower at the inner part of the bend. In the outer "inviscid" flow the centrifugal force and pressure gradient exactly balance and no secondary flow develops. However, in the boundary layer the centrifugal force is much less because the velocity of the boundary layer flow is much less, but the pressure gradient of the outer flow is still felt by this flow in the boundary layer. So in the boundary layer a secondary flow towards the inner part of the bend develops.

 

[komega]

@ RandomGuy88

Hello there,

Thanks very much for your response.

As the fluid begins to follow this curved path a pressure gradient is developed to balance the centrifugal force. The gradient is such that pressure is lower at the inner part of the bend.
I agree with your statement here. However, am I correct by elaborating as such? - The pressure gradient is generated primarily because the core flow is forced to turn more (i.e. smaller radius of curvature: steeper radial pressure gradient) near the outer wall while turning less (i.e. larger radius of curvature: milder radial pressure gradient) near the inner wall. Hence, in relative terms the fluid near the outer wall is decelerated more which results in an increased static pressure region. And finally by mass/momentum continuity, the flow near the inner wall has to accelerate which results in a low static pressure region.

In the outer "inviscid" flow the centrifugal force and pressure gradient exactly balance and no secondary flow develops.
I agree with you here mostly too but when you say "exactly balanced", does that take into account the fact that the faster (accelerated) core flow near the inner wall is deflected toward the outer wall as it navigates the bend? Apologies if you implied that as well.

However, in the boundary layer the centrifugal force is much less because the velocity of the boundary layer flow is much less, but the pressure gradient of the outer flow is still felt by this flow in the boundary layer. So in the boundary layer a secondary flow towards the inner part of the bend develops.
No problems here.