vorticity and vorticity equation
DESCRIPTION
VORTICITY AND VORTICITY EQUATION. Vorticity is the tendency of fluid elements to rotate. http://mathicse.epfl.ch/~foureste/. A property of vorticity:. Vorticity vector:. Just as streamlines were defined to be lines that are everywhere parallel to the velocity vector - PowerPoint PPT PresentationTRANSCRIPT
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.