Hi,

Can anyone give me a good definition of helicity ?

And what to used it to ?

Thanks in advance.

Regards

Christian

Yup:

1) it is a dot-product of the vorticity vector and the velocity vector

2) it is a dot product of the vorticity vector and the unit velocity vector

Vorticity vector is the Hodge dual of the vorticity tensor (grad U - (grad U)T), or, if you like, e_ijk d/dx_j U_k

The difference in definition is a detail (scales with velocity magnitude), and simply depends who you ask...

(Fun thing to look at, isn't it!)

Hi,

Helicity is a scalar quantity defined as a inner (dot) product of velocity and vorticity vectors. By the definition, helicity and normailized helicity vanish in two-dimension. Like many other qualtites, you can write a conservation equation for helicity. Helicity has been a subject of active research with regard to its possible connection with fluid turbulence.

The physical meaning of helicity becomes a bit more clearer when we normalized it. The normalized helicity is thus defined as;

v.omega/(|v||omega|)

where v is velocity vector, omega is vorticity vector and |v| and |omega| are their magnitudes. The value of normalized helicity range from -1 to 1.

As you can see from this expression, normalized helicity physically represents the angle between velocity vector and vorticity vector.

As such, normalized helicity can be used as a uselful indicator of how velocity vector field is oriented with respect to vorticity vector-field for a given flow field. For instance, at the center of streamwise vortices observed behind ground-vehicles, ships, aerodynamic bodies at incidence, and in many other flows, the velocity and vorticity vector tend to align themselves parallel to each other. As a result, the normalized helicty value will be very close to 1.0 or -1.0 at the core of the streamwise vortices. This fact is utilized to locate the core of those streamwise vortices.

The explanation given by Sung-Eun Kim is on the target.

I have used normalized helicity very effectively to visualize 3D flow fields. Most people color code velocity vectors with speed - but that is such a waste since the vector lengths already give you the same information (in terms of gross visualization of the flow). However, if you color code the velocity vectors with normalized helicity then on top of the velocity field you have an indication of how the vorticity field behaves - where are the so-called three dimensionality effects, or what parts of the flow are mainly 2D. Since normalized helicity is the cosine of the angle between velocity and vorticity then the color extremes of blue and red would correspond to the extreme cos(angles) of -1 and 1 (where the flow is highly 3D; i.e., vorticity is streamwise) and the center value of 0 (green) means the flow is basically 2D (vorticity is normal to velocity)

Check the following paper for a good intro to the subject:

Y. Levy, A. Seginer and D. Degani, AIAA paper 88A-40769 (1988)

Adrin Gharakhani