1. VORTICITY AND VORTICITY EQUATION
Vorticity is the tendency of fluid elements to rotate

2. Vorticity vector:
A property of vorticity:

3. Just as streamlines were defined to be lines that are everywhere parallel to the velocity vector
vortex lines are defined to be everywhere parallel to the vorticity vector.

4. CIRCULATION
Another measure of the tendency of fluid elements to rotate
Stokes’ theorem: normal flux of vorticity through a cross section of a vortex tube (of area A) = line integral of tangential velocity around it.
For an inviscid fluid, Kelvin’s theorem states that

5. vortex lines are in the y direction and equidistant
For Couette Flow  z H
vortex lines are in the y direction and equidistant

6. For Poseuille Flow

7. Vortex Tubes off Flagler Beach

8. Obtained by taking the curl of:
THE VORTICITY EQUATION
Obtained by taking the curl of:
LHS only:
Can use the vector identity
(Lecture 7):
Because the curl of a gradient = 0

9. = 0 for incompressible flows
RHS only:

10. Stretching and tilting of vortex lines
Isobars and isopycnals not parallel -- baroclinicity
diffusion
Including Earth’s rotation effects:

11. Stretching and tilting of vortex lines
Spatial gradient in the direction of vorticity
If velocity is not constant in the direction of vorticity, the vortex lines will be
distorted, i.e., strained -- increased vorticity with vortex stretching
This term is absent in 2-D flows where vorticity is perpendicular to the plane of the flow, e.g., Couette & Poiseuille flows.
Water Spout
(Lake Buchanan, Texas)
Whirlpool
(la Rance, France)

12. Blue Holes
Bahamas

13. Baroclinic Generation
isobars
isopycnals
increasing depth and density
geopotential
surface x z

14. Turbulence is characterized by fluctuating vorticity in structures -- EDDIES.