What Equation Models Boundary Layer Thickness in Early Stage Pipe Flow?

[brambram]

Hi

I cannot find an equation for a boundary layer in a pipe flow (laminar). I am looking for an equivalent of the equation δ(x)=4.91x/(√Re) that works for a flow between plates (x is the distance downstream). The thing is- I am looking for BL thickness for still undeveloped flow. I would be very grateful for any help

Best regards

 

[Chestermiller]

Hi

I cannot find an equation for a boundary layer in a pipe flow (laminar). I am looking for an equivalent of the equation δ(x)=4.91x/(√Re) that works for a flow between plates (x is the distance downstream). The thing is- I am looking for BL thickness for still undeveloped flow. I would be very grateful for any help

Best regards

The BL thickness for a pipe should be the same as for parallel plates (for the same free stream velocity) because the BL thickness is small compared to the pipe radius, so, to the BL, the pipe wall looks like a flat plane.

 

[brambram]

Dear Chestermiller

Thank you, but I am afraid that my problem tackles pipe with quite small radius. The thing is I need analytical check for CFD simulation, that cannot rely on such a coarse assumption.

Also the function is far from being linear- for the entrance of the pipe that would vitally change the result

 

[brambram]

Chestermiller

I actually made a mistake saying that the equation concerns flow between plates- it is flow over a plate. I don't know what is the one for the parallel plates- do you know it? do you think it could be used here? The radius of the pipe is about 10x of the BL thickness.

Thanks again

 

[Chestermiller]

Chestermiller

I actually made a mistake saying that the equation concerns flow between plates- it is flow over a plate. I don't know what is the one for the parallel plates- do you know it? do you think it could be used here? The radius of the pipe is about 10x of the BL thickness.

Thanks again

Let me see if I understand you correctly. You solved the fluid flow problem for the hydrodynamic entrance region of a pipe using CFD, and now you are trying to compare the results to the momentum integral approximate solution to the same problem in order to roughly validate your results.

The solution to the slit (parallel plate) problem using the momentum integral method is presented in Transport Phenomena by Bird, Stewart, and Lightfoot, as a problem at the end of one of the early chapters. In the same chapter, they also present the solution for flow over a plate, both the exact Blasius analystical solution as well as the solution obtained using the momentum integral approximation. The solution to the flat plane case is going to be the same as that for flow in the entry region of a slit and for the entry region of a circular pipe for the very beginning portion of the entry length. However, further downstream within the entry region, the solution will deviate from flat plane case. You can apply the very same methodology that Bird et al used for the slit problem to solve the pipe problem. They pretty much lead you through how to do it in the homework problem.

Because, in the problem you are solving, the pipe radius is about 10x the BL thickness (for the total length of pipe you are considering, which is a small fraction of the hydrodynamic entry length), in the CFD solution, you need to have used a non-umiform mesh and need to have packed lots of nodal points near the pipe wall. The situation you are describing is also telling me that the flow-over-a-plate momentum integral solution will be adequate for your purposes, since the boundary layer thickness will not even begin to approach the pipe radius. You need to be thinking in terms of the dimensionless variables, rather than dimensional variables. I can see you have started to do this when you compared the BL thickness to the pipe radius.

Chet

 

[brambram]

Thank you! I found this example (slit), but I was not sure if it is a right approximation. But I guess it is the best that I can get- I went on to research on actual velocity profile in the entry stages of flow in a pipe (so 3D), and it turns out that it cannot be described by a single function..

When it comes to the problem outlined in Transport Phenomena, the equation has a ln term, I hope it will be fine to skip it (as δ/radius is low).

Thanks again for the help

 

[Chestermiller]

Thank you! I found this example (slit), but I was not sure if it is a right approximation. But I guess it is the best that I can get- I went on to research on actual velocity profile in the entry stages of flow in a pipe (so 3D), and it turns out that it cannot be described by a single function..

When it comes to the problem outlined in Transport Phenomena, the equation has a ln term, I hope it will be fine to skip it (as δ/radius is low).

Thanks again for the help

No. The ln term is a vital part of the solution. Try expanding the solution in powers of Δ = δ/B and see what you get. (Truncate the expansion at Δ²).