negative displacement thickness
 

Hi all,
I'm working on an incompressible axisymmetric boundary layer code for some bodies of revolution. For some of my results in turbulent flow, I'm getting negative values for the displacement and momentum thicknesses. The rest of the flow field looks correct, and the code appears to be functioning properly in other cases and validated well for flat plate flow and laminar axisymmetric flows. I'm not sure if these negative values are physical, and if they are, how to interpret them. I've done a bit of googling and it seems like they might be indicative of strong streamline convergence leading to an excess of mass and momentum flow in some region or somesuch but I'm not 100% on that. Can these values be negative? What is the best way to interpret that result physically if they can? Finally, are there any good references on this? I'm using Schlichting's and White's books, but haven't been able to find any references in there to negative values for these parameters, at least not so far.
Thanks!

If the flow is incompressible, then I believe that the only way for these values to become 'negative' is if the bulk flow direction is reversed.

Thanks for the reply. I've since realised that the displacement and momentum thickness formula's need to be modified for axisymmetric systems and that has resolved my problem.

I know this is an old topic, but I think people could use some actual answer here. As an experimentalist, I've seen momentum thickness (as well as displacement thickness) go negative many times.

Just look at the formulation of displacement thickness. You're integrating:



The term in the parenthesis can go negative if . This happens quite often in boundary layers where there is a highly favorable pressure gradient near the surface (but not in the free stream). For example, cases where I encountered negative usually occurred near a sharp edge of a bluff body. Say for example, the Ahmed body or a 3D backwards step with low aspect ratio. The suction under the separation bubble after the step "propagates" upstream to the boundary layer prior separation, accelerating it. Since the free stream does not see this effect, you get a greater velocity close to the surface (in this strange boundary layer), than in the free stream. Then the integral becomes negative.

I see people in the internet say that the momentum thickness becomes negative when the flow separates. I think that is not correct, and I have never personally encountered this particular case in my experiments. If the flow separates, sure, becomes negative in the region of reverse flow, but usually velocity magnitudes at this region are rather small compared to the free stream, making the ratio too small to overcome the momentum defect in the remainder of the separation bubble. In all separation bubbles I've analyzed I've never encountered a negative displacement thickness. It could be a limitation of my experience, though (mostly bluff body separation).

With regards to interpretation, I think the negative momentum thickness has a fairly reasonable one: If a boundary layer has negative momentum thickness because , it means it's more energized than the free stream. Thus, it is "highly energized" and will be quite resistant to separation. Obviously, a sharp edge will still separate it.

Thanks,
-FZ

Recirculation is possible
1) if u changes its direction then the integral of momentum defect will be negative.
2) or if u> U, the momentum thickness will be negative too.
But the chances of the 2) option are quite low (0.00001%)
Reason: 'u' can only energize if the tendency of boundary separation exists.
And we should not forget the pressure gradient along the length of the body.