I have trouble understand how Vorticity occurs in a Rotational and not in a Irrotational vortex when I consider it in terms of shear forces.
A Rotational vortex has a velocity profile where
$v∝r$
whereas a Irrotational vortex has a velocity profile
$v = \frac{1}{r}$
So between two different radii $r+$ $r-$ with a clockwise velocity and flow –
Rotational – the velocity is proportional to radius so between a larger $r+$ and smaller radius$r-$ there will be no velocity profile so no shear forces
.
Irrotational – the velocity is inversely related to radius so between a larger $r+$ and smaller radius $r-$ there will be a velocity profile and this will cause shear forces ,resulting in rotation towards the larger radius $r+$
So if we only considering shear forces to this point, then only the Irrotational has any rotation- vorticity.
Other forces ?
In the Rotational example
We have a centripetal force caused by changing the bulk velocity vector around a solid curve
This centripetal reaction force has no counter-clockwise action of the shear to oppose it. So as its acted on by the centripetal force the parcel is free to rotate with the flow. This means that the rotational parcel changes vector as it goes around the bend.
In the Irrotational example
We have a Bernoulli pressure gradient which causes acceleration of the fluid radially this changes the vector clockwise but this is counteracted by the counterclockwise shear forces – so the combination of these two actions (shear and pressure gradient) somehow results in no net rotation
( I guess because both shear and pressure are both proportional to the difference in velocity
Allowing the Irrotational parcel to maintain the same vector as it goes around the bend and the rotational to change heading.
Rotional ……………………………… Irrotational
My question Isn't it more appropriate (considering all forces shear, pressure gradients and resultant forces-centripetal) to say that an Irrotational flow has no net vorticity and a Rotational flow has a net vorticity ?
Best Answer
Your calculation of vorticity is wrong. Vorticity is not zero if fluid element rotates, even it does not deform, which is the case in the rotational case. Just use your finger pointing from one side to another side of a fluid element and follow this element for a while will convince you it is rotating. On the other hand, in the irrotational case if you follow two perpendicular sides of an element, you may find they rotate in opposite directions and the sum of the angular velocity is zero.
You are recommended to watch video #6 of the fluid mechanics films. Just google national committee fluid mechanics films.