Exploring the Formation of Vortices: From Kutta Condition to Fluid Dynamics

[TheWonderer1]

I’ve been reading about the Kutta Condition and how a vortex results but the information discussing how vortices form aren’t too detailed. I’m just interested to know more about them.

 

[boneh3ad]

There are a number of sources. How familiar are you with fluid mechanics in the first place?

 

[TheWonderer1]

I’m self-taught so I guess the basic level at first to start out. If you know of any good articles, I would certainly be willing to read them.

 

[boneh3ad]

Well my main question is whether you are familiar with the vorticity equation or not.

Generally, for an incompressible, inviscid fluid, vorticity cannot be created nor destroyed. If a flow is initially rotational, it will remain so, and vice versa. For a compressible, viscous fluid, there can be several sources of vorticity generation, including baroclinicity (nonparallel density and pressure gradients), viscous shear, or rotational body force fields. For the case of an airfoil, the vorticity source is generally going to be viscous shear.

 

[Andy Resnick]

I’ve been reading about the Kutta Condition and how a vortex results but the information discussing how vortices form aren’t too detailed. I’m just interested to know more about them.

My favorite book on the subject is Saffman's "Vortex Dynamics". Chapter 6 is entitled 'Creation of Vorticity'. which is what you are asking about.

The beginning points are the Helmholtz theorem/Helmholtz decomposition of a vector field and Kelvin's circulation theorem ('conservation of circulation').

https://en.wikipedia.org/wiki/Helmholtz_decomposition
https://en.wikipedia.org/wiki/Kelvin's_circulation_theorem

Kutta's condition is one approach to remove (mathematical) problems associated with the generation of vorticity without violating these theorems. That is, is it possible to create vorticity in a perfect barotropic fluid, acted upon by conservative forces with a single-valued potential?

The answer is yes, as first demonstrated by Klein's "Kaffeelöffel" (coffee spoon) experiment; the 'trick' is that the 'circulation contours' discussed in Kelvin's theorem intersect the coffee spoon, changing the topological properties of the flow. Extension of that initial result to flow past bodies with sharp edges results in infinitely many solutions of the Euler equations- there are an infinite number of choices about where flow separation occurs, and the site of flow separation is where vorticity ('circulation') can be generated. Kutta's condition (the velocity is bounded but not necessarily continuous) restores unique solutions to the Euler equations.

In essence, the Kutta condition allows the complex velocity field around a sharp-edged body to be modeled in a much simpler way- the trailing edge is the 'source of circulation'.

Does that help?

 

[TheWonderer1]

Certainly does!