Boundary layer thickness confusion

[MexChemE]

Hi, PF!

Recently, while reading chapter 6 of Incropera's Fundamentals of Heat and Mass Transfer I got into a confusion regarding the velocity boundary layer. The book first states that, as the flow becomes more turbulent, the boundary layer gets thicker, as indicated by both figures attached at the bottom of the post. However, I don't get why this is the case, since we can see in the first figure that the region where the velocity gradient is significant in the turbulent boundary layer is much smaller than in the laminar boundary layer. Then, further down the chapter, the author states that, for a greater Reynolds number (turbulent flow) we should expect a smaller boundary layer thickness. This blatantly contradicts what was said earlier. Maybe it is just a typo, and the author wanted to say we should expect a greater boundary layer thickness instead of smaller. What is actually going on here?

Thanks in advance for any input!

 

[gfd43tg]

I think those eddy currents would cause the larger boundary layer thickness. They are flowing in the vertical direction as well as horizontal, as seen in the figure 6.6

 

[OrangeDog]

I am not an expert on boundary layer flows, maybe @boneh3ad can answer in more detail. Ill give you an introductory summary of what the book is trying to explain. The book might be confusing because you probably don't have the background in fluids. I would strongly recommend taking a course in fluid mechanics as it is crucial for convective heat transfer and some cases with conduction.

What the book is trying to say is that the boundary layer characteristics change as you move along the plate. In the earliest stages you have smooth flow where very little mixing is involved. This is called the laminar region. The height of the laminar region grows approximately as a square root function according to the Blasius solution. After a certain point imperfections in the flow will cause mixing to occur. The boundary layer will go from a laminar to a turbulent state and continue to grow. The laminar region, however, will decrease in thickness. The overall thickness (laminar + turbulent) will decrease with increasing Reynolds number regardless.

 

[boneh3ad]

In a turbulent flow, one of the fundamental characteristics is the transport of various quantities through turbulent mixing. In the case of the velocity boundary layer, this quantity is momentum. When the boundary layer is laminar, there is very little mixing, and the transport of momentum occurs due to what is effectively a diffusion process. This is essentially what viscosity represents: a diffusion coefficient for momentum, and the diffusion is facilitated by the zero momentum enforced by the wall (no slip condition).

In a turbulent flow, in addition to this diffusion mechanism, you have turbulent mixing, so it is going to tend to mix more of the low-momentum fluid from near the wall in with the higher-momentum fluid out near the boundary layer edge. The net result is that the boundary layer gets thicker. At the same time, that mixing process is drawing high-momentum fluid near the edge farther in toward the wall. The result of this is that the boundary-layer profile looks "fuller" (e.g. the extremely high gradient area is smaller even though the velocity doesn't actually reach the free-stream value until much farther from the wall). The net result is a thicker, fuller boundary layer that has a higher velocity gradient near the wall (higher shear stress) and mixes all sorts of quantities much more effectively (momentum, temperature, etc.) much more efficiently than in the laminar case.

 

[MexChemE]

Wow! These were pretty good explanations! Suddenly everything clicked while reading these posts, though I still got a lot more to read on boundary layers. Basically, if we increase Re while keeping the length of the plate constant, we will be reducing the BL thickness, regardless of the flow regime.

And you're right, OrangeDog, the only exposure I've had to fluid mechanics was in my transport phenomena class, last semester. But the only FM we actually did was setting up momentum balances and solving the balances or simplified N-S equations for velocity profiles, we didn't get to see the more technical stuff like boundary layers. I will be taking a fluids class next semester but it takes a more practical approach, like pressure drop calculations, friction factors and so on.

 

[boneh3ad]

Yes, if you increase $Re$ and hold everything else constant (e.g. at a given point, if the flow was laminar before, it still is and the same with turbulence), then an increase in $Re$ leads to a decrease in the boundary-layer thickness, $\delta$. For a flat plate, $\delta$ scales approximately with $Re^{-1/2}$ for laminar flow and with $Re^{-1/5}$ for turbulent flow.

Of course, as you increase $Re$, the flow is, in reality, likely to transition earlier, for for a given point along the plate, if the increase in $Re$ caused the flow to now be turbulent at that location, then the thickness will actually be larger. In that sense it is a very complex problem.

If you are interested in more viscous flow and boundary-layer theory, head to your library and check out one or both of the following:
https://www.amazon.com/dp/0072402318/?tag=pfamazon01-20
Boundary-Layer Theory by Schlichting

 

[MexChemE]

Thanks for the recommendations. I've taken a look at White's Fluid Mechanics before and seems like an excellent book, it reminded me of Çengel's style. So I guess I would enjoy his books.

I'll be checking both books for sure. I definitely need to master the boundary layer concept, I'm using it in both my heat transfer and mass transfer courses.

 

[odcardenasp]

I think there is another important point here, before to define Blasius's boundary layer thickness equation, books defines its value as 0.99V where V is free Velocity, If you make this assumption you will understand that boundary layer decreases as Re is higher.

In reality a mixing process makes that thickness to reach V be greater in a turbulence case, however, it'll reach 0.99V rapidly.

I'm sorry by my english.