Three-dimensional centrifugal instability and the Rayleigh critertion

troymcfont · May 9, 2016, 09:42

Lord Rayliegh postulated that for a two-dimensional symmetric swirling flow, a three-dimensional instability may occur if the angular momentum magnitude decreases in the radial direction with respect to the centre of rotation. This is known as the Rayleigh criterion. This happens because the balance (for a fluid volume) between the centrifugal force and the radial pressure force is disrupted and the fluid volume is further displaced from its original position if perturbed.
My question is: why when this instability happens, the flow creates three-dimensional structures instead of just breaking the steady swirling flow in a two-dimensional manner. Why is there fluid being ejected into the normal direction of the plane?
Thanks.

davidwilcox · May 9, 2016, 23:28

A quick answer to that would be to look at the Biot-Savart law.

troymcfont · May 10, 2016, 03:35

Thank you for your answer. Could you be a but more specific?
The Biot-Savart law just recovers de velocity field from a vortex line, but I can't see how for a vortex line of a 2D velocity field (straight line) a 3D velocity field can be recovered.

FMDenaro · May 10, 2016, 04:24

I think that a useful way is to look also to the vorticity vector field and its equation. In general, instability would onset the stretching and the vorticity vector does not remain normal to the velocity plane

May 9, 2016, 09:42	Three-dimensional centrifugal instability and the Rayleigh critertion	#1
troymcfont New Member B. F Join Date: Feb 2014 Posts: 5 Rep Power: 12	Lord Rayliegh postulated that for a two-dimensional symmetric swirling flow, a three-dimensional instability may occur if the angular momentum magnitude decreases in the radial direction with respect to the centre of rotation. This is known as the Rayleigh criterion. This happens because the balance (for a fluid volume) between the centrifugal force and the radial pressure force is disrupted and the fluid volume is further displaced from its original position if perturbed. My question is: why when this instability happens, the flow creates three-dimensional structures instead of just breaking the steady swirling flow in a two-dimensional manner. Why is there fluid being ejected into the normal direction of the plane? Thanks.

May 9, 2016, 23:28		#2
davidwilcox Senior Member david Join Date: Oct 2012 Posts: 142 Rep Power: 13	A quick answer to that would be to look at the Biot-Savart law.

May 10, 2016, 03:35		#3
troymcfont New Member B. F Join Date: Feb 2014 Posts: 5 Rep Power: 12	Thank you for your answer. Could you be a but more specific? The Biot-Savart law just recovers de velocity field from a vortex line, but I can't see how for a vortex line of a 2D velocity field (straight line) a 3D velocity field can be recovered.

May 10, 2016, 04:24		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	I think that a useful way is to look also to the vorticity vector field and its equation. In general, instability would onset the stretching and the vorticity vector does not remain normal to the velocity plane