I find that, vorticity depends on non-inertia frame of reference: the vorticity expression under cylindrical coordinate in stationary frame of reference is different from that under cylindrical coordinate in a rotating frame of reference. For turbomachinery flow field simulation, the N-S under cylindrical coordinate is usually adopted. When rotor is concerned, a rotating frame of reference is used. When implementing Baldwin-Lomax or Spalart-Allmaras turbulence models, vorticity is needed. According to the above, vorticity in stationary frame is different from that in stationary one, then which one is to be chosen?

>vorticity depends on non-inertia frame of reference

Yes.

>vorticity in stationary frame is different from that in stationary one

Huh?

Don't confuse the distinction between Cartesian and cylindrical coordinates with the distinction between an absolute and a relative reference frame. Those two issues are independent of each other. Of course, in cylindrical coordinates the mathematical expression for vorticity looks different than on Cartesian coordinates, but they should be equivalent. What you should pay attention to is the reference frame, i.e. relative (rotating with the blade row in turbomachinery) or absolute. You simply have to be consistent within your reference frame, e.g. if your N-S equations are expressed in the relative frame than you have to do the same with vorticity.

Thanks! vorticity in stationary frame is different from that in stationary one

this is a typing error, it should be

vorticity in stationary frame is different from that in a rotating one

We can find the quivalent vorticity expression in cylindrical coordinate for the vorticity in cartesian coordinate, however it seems impossible to find an equivalent one in a roating reference frame?

You need to properly account for the transformation from an inertial system to a non-inertial coordinate frame.