VORTICITY AND VORTICITY EQUATION

http://mathicse.epfl.ch/~foureste/

Vorticity is the tendency of fluid elements to rotate

zyx ,,Vorticity vector:

zv

yw

x

xw

zu

y

yu

xv

z

u

0 A property of vorticity:

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.

CIRCULATION

Another measure of the tendency of fluid elements to rotate

A

dAn

dlu

For an inviscid fluid, Kelvin’s theorem states that 0DtD

Stokes’ theorem: normal flux of vorticity through a cross section of a vortex tube (of area A) = line integral of tangential velocity around it.

zH

For Couette Flow

zu

zv

yw

x

xw

zu

y

yu

xv

z

y

vortex lines are in the y direction and equidistant

For Poseuille Flow

zHzxpu

2

21

xw

zu

y

Vortex Tubes

off Flagler Beach

http://www.srh.noaa.gov/jax/?n=waterspouts

http://www.floridahurricanelawfirm.com/labels/Hurricane%20Property%20Damage.html

THE VORTICITY EQUATION

upgDt

uD

2

Obtained by taking the curl of:

upg

DtuD

2

uuutDt

uD

uuuuu

2

21Can use the vector identity

(Lecture 7):

u

tDtuD

Because the curl of a gradient = 0

LHS only:

uuuuu

= 0 for incompressible flows

uu

tDtuD u

DtD

uppupg 2

22 11

RHS only:

uuu2

22 u

22

2 1

pupg

22

1 pu

DtD

22

1 pu

DtD

22

12 puDtDIncluding Earth’s

rotation effects:

Stretching and tilting of vortex lines

Isobars and isopycnals not parallel -- baroclinicity

diffusion

uDtD

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 bedistorted, 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)

http://synaesthetical.blogspot.com/2010/07/blue-holes-of-bahamas-wes-c-skiles.html

Blue HolesBahamas

Baroclinic Generation

pDtD

2

1

x

z

1

isobars

isopycnals

increasing depth and density

geopotentialsurface

2

http://mathicse.epfl.ch/~foureste/

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