Potential flows and Helmholtz decomposition

AI Thread Summary
The discussion centers on potential flows and the Helmholtz decomposition in fluid dynamics. The user understands that irrotational flows can be represented as a gradient of a scalar potential, but is confused about how this relates to the Helmholtz theorem, which allows for the decomposition of any vector field into scalar and vector potentials. They struggle to see how imposing the curl of the velocity to zero simplifies the Helmholtz decomposition to a potential form. The user seeks guidance on resolving this confusion and is looking for references or resources that explain the topic more clearly. The conversation highlights the complexity of fluid dynamics and the nuances of potential flow theory.
pigna
Messages
12
Reaction score
1
Hi. I'm studying fluid dynamics and in particular potential flows. I know that for an irrotational flow the velocity field is a conservative field and it can be rapresented by the gradient of a scalar field v=-∇Φ. In this case the explicit form of Φ is something like a line integral between a reference point where Φ=0 and a generic point of the domain.

This can be obtained using the stokes theorem and the domain has to be simply connected. Moreover i know that a generic vectorial field ( without any assuption about the fact it is irrotational or not, solenoidal or not) can be decomposed using the helmholtz theorem in the form v=- ∇Φ +∇×Ψ where Φ is a scalar potential while ψ is a vectorial potential. In this case the explicit form of the two potential require a non local integration over the volume and over the boundaries and greens functions are used to find out this results ( l have found the explicit formulations reporter on wikipedia as on other sources and I have also find them out by myself).

I'm a little confused because I thought that imposing the curl of velocity equal to zero in the explicit formulation of the helmholtz decomposition it should reduce, in some ways, to a potential form as the one obtained previously considering directly the flow as irrotational and using the stokes theorem. I have struggled a lot with this issue, but I haven't obtained any results. Can someone give me a tip or a reference or tell me where I'm wrong...
Thanks...
 
Engineering news on Phys.org
The Helmholtz decomposition is not unique. In general there will be several different combinations of scalar and vector potentials that will give you the same field. However, in the case of an irrotational flow, it is possible to put the vector potential to zero.
 
It is exactly what I'm not able to do. I have tried to do that or to rewrite the curl term as a gradient but I haven't reached any result... do you know some book or internet content that adress this issue in a comprehesible way?
 
Could you show your work?
 
20171023_223425.jpg
20171023_223435.jpg
20171023_223459.jpg
20171023_223514.jpg
20171023_223556.jpg
20171023_223425.jpg
20171023_223435.jpg
20171023_223459.jpg
20171023_223514.jpg
20171023_223556.jpg
 

Attachments

  • 20171023_223425.jpg
    20171023_223425.jpg
    32.2 KB · Views: 566
  • 20171023_223435.jpg
    20171023_223435.jpg
    37.2 KB · Views: 616
  • 20171023_223459.jpg
    20171023_223459.jpg
    33.3 KB · Views: 585
  • 20171023_223514.jpg
    20171023_223514.jpg
    39 KB · Views: 528
  • 20171023_223556.jpg
    20171023_223556.jpg
    27 KB · Views: 508
Ok l have posted two time the same images... sorry... as you can see confusion and no conclusions. Probably when i say that I ' m not sure if an integral goes or not to zero it actually goes because in different case I don't even use the hypothesis of irrotational flows. And as you can see I'm not able to set ∇×Ψ=0. There are a lot of calculations and so I send pictures, I Hope they are understandable... thanks you very much if you take a look...
 
Posted June 2024 - 15 years after starting this class. I have learned a whole lot. To get to the short course on making your stock car, late model, hobby stock E-mod handle, look at the index below. Read all posts on Roll Center, Jacking effect and Why does car drive straight to the wall when I gas it? Also read You really have two race cars. This will cover 90% of problems you have. Simply put, the car pushes going in and is loose coming out. You do not have enuff downforce on the right...
I'm trying to decide what size and type of galvanized steel I need for 2 cantilever extensions. The cantilever is 5 ft. The space between the two cantilever arms is a 17 ft Gap the center 7 ft of the 17 ft Gap we'll need to Bear approximately 17,000 lb spread evenly from the front of the cantilever to the back of the cantilever over 5 ft. I will put support beams across these cantilever arms to support the load evenly
Thread 'What's the most likely cause for this carbon seal crack?'
We have a molded carbon graphite seal that is used in an inline axial piston, variable displacement hydraulic pump. One of our customers reported that, when using the “A” parts in the past, they only needed to replace them due to normal wear. However, after switching to our parts, the replacement cycle seems to be much shorter due to “broken” or “cracked” failures. This issue was identified after hydraulic fluid leakage was observed. According to their records, the same problem has occurred...

Similar threads

Back
Top