Drag force of a cylinder with it's axis parallel to the flow

[dpk31]

Hello,

I would like to know how to calculate the drag force per unit length of an infinite cylinder with it's axis parallel to the flow field, I had read about a post here previously titled "
Drag force on cylinder in parallel flow" but the slender body approximation is not valid for inifinite cylinders, could you please point me to some references regarding this?

Thanks and regards,
dpk

 

[dpk31]

Hello,

I forgot to add, I am looking for an expression in the stokes regime.

 

[Chestermiller]

Why don't you look at the problem of steady laminar flow in annulus, and look at what happens when the ratio of the outer diameter to the inner diameter becomes unbounded?

Chet

 

[dpk31]

Hello Chester,

Thanks for this, I will look into it!

 

[Chestermiller]

Hello Chester,

Thanks for this, I will look into it!

Make sure you look at it for a situation where the volume flow rate per unit area (average flow velocity) is held fixed.

Chet

 

[boneh3ad]

The main problem I foresee here is that with an annular flow, you can assume a fully-developed profile that is either laminar or turbulent and therefore the concept of drag per unit length makes a little bit of sense. Taking that to infinity, it is now a boundary layer that will grow perpetually (from some undefined start point) and the shear stress will change as a function of downstream distance and laminar-turbulent state of the boundary layer.

In other words, I am not sure that this problem has a useful answer that can then be applied to a real-world situation.

 

[Chestermiller]

The main problem I foresee here is that with an annular flow, you can assume a fully-developed profile that is either laminar or turbulent and therefore the concept of drag per unit length makes a little bit of sense. Taking that to infinity, it is now a boundary layer that will grow perpetually (from some undefined start point) and the shear stress will change as a function of downstream distance and laminar-turbulent state of the boundary layer.

In other words, I am not sure that this problem has a useful answer that can then be applied to a real-world situation.

Hi bh,

I respectfully beg to differ. As I said, I am assuming steady state flow, with a fixed axial velocity. I am not looking at the growth of a boundary layer thickness with downstream distance because the flow is fully developed. I am looking at a comparison of an array of annular systems, each with different outer radius, but with the same inner radius and the same average axial velocity.

I'm pretty sure that when one solves this problem, one finds that the force per unit axial length at the inner radius increases monotonically with increasing outer radius and becomes infinite as the outer radius becomes unbounded (even for laminar flow).

Chet

 

[boneh3ad]

As I said, I am assuming steady state flow, with a fixed axial velocity. I am not looking at the growth of a boundary layer thickness with downstream distance because the flow is fully developed. I am looking at a comparison of an array of annular systems, each with different outer radius, but with the same inner radius and the same average axial velocity.

This is where I suppose I am not following you, then. How can you concoct a fully-developed flow when one of your boundaries is at infinity? There doesn't appear (to me) to be a steady state velocity profile to attain in the first place. So in short, I have a few issues with this approach:

1. Flows with finite boundaries can reach a fully-developed state. I just don't see how this can transfer to a boundary at infinity first of all because the outer boundary conditions are fundamentally different ($u = 0$ at the walls vs. $u/U_{\infty} = 1$ at $\infty$).
2. With a bounded domain, the problem is essentially Poiseuille in nature and is forced by some unknown pressure gradient. and feature a parabolic profile. Moving the outer wall progressively farther away also moves the maximum velocity point progressively farther away from the inner surface. This is not consistent behavior with an unbounded flow such as the original question posed by the OP.
3. A bounded domain like the annular problem necessarily requires a pressure gradient to maintain flow and the concept of $U_{\infty}$ is rather undefined. Indeed, for a constant pressure gradient, if you increase the outer radius to infinity, and just assume the pressure gradient doesn't change, then the velocity blows up in addition to the shear stress. If you then try to just prescribe a set maximum velocity and let the pressure gradient float to match it, then you run into Problem 2 above.

If you drop the assumption that the flow has to be "fully developed" then you get back to the boundary layer problem. If you keep that requirement, the boundary conditions don't make sense as an analog to the OP's question and the steady, fully-developed profile will be substantially different than a boundary layer.

 

[Chestermiller]

Bh.

I see what you are saying now. You make very good points.

Regarding the pressure gradient being held constant, I was thinking more of holding the average velocity in the duct constant. Under these circumstances, as you move the outer boundary further away, the location of maximum velocity also moves further away and the pressure gradient decreases. However...

I thought about the case of steady state flow between parallel plates where the upper plate in placed further away. This is analogous to the annulus problem. Under these circumstances, the location of maximum velocity movers further away, the pressure gradient becomes less, but also, the shear rate and shear stress at the fixed plate decreases. After all, the shear rate at the fixed plate is 6v_av/h, where h is the distance between the plates. So, rather than my initial thought that the drag force per unit length would increase as the boundary was moved further away, the reality would be that the drag force per unit length would decrease and approach zero as the upper plate was moved toward infinity. This is a pretty dis-satisfying result. I'm not sure that taking into account the transition to turbulent flow would make the situation any better. What is your judgement on what the situation would be like with turbulence? Would the same thing happen as occurs assuming laminar flow, or would the shear stress at the wall stop decreasing as the plates were placed further apart and level off?

Chet

 

[boneh3ad]

I think the decrease would remain, whether laminar or turbulent. The drag value would change but the trend wouldn't.

The bigger issue is that this is not an analogous problem. The bounded domain means the profile will be totally different. A better model is Stokes' first problem applied to a cylinder. The problem is that it isn't a steady problem and the viscous layer over the impulsively-started cylinder would continue to grow in perpetuity. The shear stress would also continually decrease in that situation. That's an analytically solvable problem, too.

 

[Chestermiller]

I think the decrease would remain, whether laminar or turbulent. The drag value would change but the trend wouldn't.

The bigger issue is that this is not an analogous problem. The bounded domain means the profile will be totally different. A better model is Stokes' first problem applied to a cylinder. The problem is that it isn't a steady problem and the viscous layer over the impulsively-started cylinder would continue to grow in perpetuity. The shear stress would also continually decrease in that situation. That's an analytically solvable problem, too.

So you are saying that there is no steady solution to the OP's question, right?

Chet

 

[boneh3ad]

So you are saying that there is no steady solution to the OP's question, right?

Chet

For an infinite cylinder, that is correct. In fact, Stokes' first problem would essentially be an exact solution if it is laminar. If the cylinder had finite length, it becomes a boundary-layer problem, which can obviously get pretty hairy and would have an answer for total drag, though it would be nearly impossible to calculate.