number of swirl formula
 

I have a problem concerning the number of swirl.I know that swirl number is equal to the ratio between the axial flux of angular momentum and the flux of axial momentum

S = Gtheta / (R * Gz) with R radius of duct

I suppose that:

angular coordinate : Er, Etheta, and Ez

U -> axial velocity. V -> circunferencial velocity. r -> radial coordinate. theta -> angular coordinate R -> the pipe radius.

I don't understand the formula. where does the axial veloxity term come from in the angular momentum and axial momentum

for me angular momentum is equal (in angular coordinate) to (r.Er)^(density.v.Etheta) =r.v.density.Ez and the flux is equal to the surface integral between 0 to R and 0 to 2Pi of (density.r.v.Ez).(r.dr.dtheta.Ez) = density.r².v.dr.dtheta.

In the doc, we find density.r².v.u.dr.dtheta. idem for axial momentum
see the link below
http://pdf.aiaa.org/jaPreview/JPP/2009/PVJA35723.pdf

Thanks for your help

If the axial momentum is rho*U, and the axial velocity component is the component normal to the constant Z face, then wouldn't the axial flux of axial momentum simply be rho*U*U?