Introduction to Vorticity and Vortex Dynamics
CH4 Basic Vortex flow
2D Vortex • Cyli Cylind ndri rica call co coor ordi dina nate te (r, θ, z ), ), and (u, v, w ) 2 u u u v u 1 P 2 • GE: u w ( u 2 ) r 0 r t r z r
• CE:
v v v v uv 2 u w ( v 2 ) r t r z r 2 1 P w w w u w g w 0 z t r z (ru ) (rw) 0 r z
where
2
∇ =
1 ∂
r ∂r
( r
∂
∂
2
)+ 2 ∂r ∂ z
0 symmetry
Potential Vortex • Consid ide er stea ead dy, w=u=0, and v=v(r).
v2 r
2v
1 dp
dr v 2
r
0,
2
v 2
d r
0,
1 dv
• Where Where means me ans the bal balanc ance e bet betwee ween n the dr r dr pressure gradient force and centrifugal force physically. d 2v dr
2
2
1 dv
r dr
v 2
r
0.
Potential Vortex • For For solvi solving ng abov above e ODE, ODE, we we faci facilit litat ate e the the circulation
Ñ c u dx 2 rv, • Then eq(4.1.3) becomes r
d 2 dr
2
d dr
0.
• Solution
• BC:
1 2
Ar B. 2
r , is bounded A=0 r 0, is bounded B =0 ?
Velocity distribution
v
cons constt 2 r ln( z z0 ) w i ,
2 i
Potential Vortex • We can can ob obta tain in th the e sol solut utio ion n v
2 r
,
cons constt
Besides r=0, the flow is irrotational i rrotational and at r=0, v and w are infinite.
• Pressure distribution v 2 1 dp from 0, we can obtain r dr 2 2 2 p p 8 r
r , p p
It’s interesting to estimate the critical value when the p 0 . (evaporation) on) for water 14 m/s (ev m/s (s (supersonic) for air 406 m/
Velocity distribution
Potential Vortex • Discussion: 2 2 • For u , and 0 u 0
visc viscous ous terms terms disa disappe pperr in inco incompr mpres essib sible le N S eq • At la larg rge e Re, Re, th the e vor vorte tex x is is sim simil ilar ar to po pote tent ntia iall vortex outside r=0 when the radius of vortex core is as thinner as possible. • But, at r=0, it’s far from realized. • It It’s ’s unr unrea eali lize zed d that that the the cal calcu cula late ted d K.E. K.E. is inf infin init ite. e.
Rankine Vortex • Ra Ranki nkine ne (18 (1882) 82):: A simp simplif lify y vort vortex ex mode modell Vorticity distribution is uniform inside the vortex core.
2 rv r 2 r a, v ~ r (match at r a ) when r a, v ~ 1 r r r a, 2 v 2 a 2 r r a. Incompressible flow outside the vortex core. Where a is the radius of vortex vore.
A 2 B 0
1 2
A 0 B cons.t
Ar2 B
Rankine Vortex • Pr Pres essu sure re di dist stri ribu buti tio on for r a, in z-dir
dp dr
dp dz
v
2
r
2
4
- g p
r p
2 r 2 8
2 2
8
gz
r c ( z ), c'
• Usi sing ng th the e sam same e met metho hod d for r
a
2 a 4 gz , p c '' 2 8r
• The con contin tinuit uity y cond conditi ition on mat matche ches s at at r=a
2 2 gz 8 r - gz p 2 2 a a 2 (2 - 2 )- gz 8 r
r a, r a.
Rankine Vortex • Th The e sol solut utio ion n of of fre free e sur surfa face ce
r , z 0, P a P when 4 r 0, v 0, P P 2 2 2 a r r a, (2 ) 8 g z 2 4 a r a. 8 g r 2 It’s interesting in r 0, v 0, but p p 2
2
a
and thus, forms a hollow vortex which is similar to the meso-scale typhoon.
Oseen Vortex Potential vortex inviscid unrealistic Rankine vortex steady, in • Osee Oseen(L n(Lam amb) b) (1912 (1912): ): poten potentia tiall vortex vortex will will obey obey the viscous fluid dynamic at t=0, and we can make sure the motion of vortex at t>0. r
BC&
. 2 t r r r
t 0, 0, r 0 IC 0 , r , t 0, , r , t 0. 0
Oseen Vortex • Us Usin ing g the the si simi mila larr var varia iabl bles es f 1 rt , t t 2 f 1 , r r t 1 f . r t 3
r
t
, and f ( )
2
2
2
1 ) f 0 2
f (
f( )
A 2
2
e
B,
0, r 0 B A / 2 , r, t 0 B 0 0 , r , t 0 B 0 0
Oseen Vortex • We can obtain
0 (1 e • As t 0, v
0
r2
4 t
)v
0 or r 2 ?
2 r
v
1 r
0 2 r
(1 e
r 2 4 t
).
4 t ,
(potential vortex)
• For small r
0 0 r 2 v 1 ( 1 h . o . t ) r, v t 2 r 2 t
r (so lid body rotation )
• The unsteady transitional zone at r0 ~ 4 t ,
Where r 0 means the dimension of vortex core. It will incerase with time and causes the decay of vortex.
Oseen Vortex • Th The e dist distri ribu buti tion on of vo vort rtic icit ity y
1 (rv) r
r
0 4 t
r 2
e 4 t ,
1. The above eq. shows the vorticity is i s infinity at r=0 when t=0.
2. The total vorticity is invariable.
2 rdr 0
0
The vorticity distribution of Oseen vortex
Taylor Vortex • Oseen vortex is the simplest one of the solutions of N-S eq., and G. I. Taylor (1918) find another one. H
v M
r 2
4 t
e
r 2 4 t
,
(H
0 2 r rvdr H ,
H 2 t
2
(1
2
r
)e 4 t
const.)
(angular momentum is finite)
r 2
4 t
.
1. for t 0 0 at r=0 , r 3 2. vmax =H / 2 2 et at r0 2 t , 3. v t 3 / 2 from v decays to v 0 o max t t0 (23 / 2 1)t0 0.296r02
2 , it needs
Oseen Vortex & Taylor Vortex • To take take time time der deriva ivatio tion n with with the the vort vortici icity ty of of Oseen vortex can obtain the same result with the vorticity of Taylor vortex. • When t=o for Taylor vortex
for Oseen vortex
0. energy total total angular momentum total energy disppation 0 . energy total total angular momentum total energy disppation
are finite. are infinite.
• Tayl Taylor or vort vortex ex is is more more real realisti istic c than than Osee Oseen n vortex vortex..
The General Solution of 2D AxialSymmetry, Inviscid Vortex • Vo Vort rtic icit ity y eq: eq: (l (lin inea ear) r) r
1 2 , t r r r
• Similarity la law:
T (t ) f ( ), Ar at b , where A, a, b are undeteminited consts.
• Sep epar ara ate vari varia abl ble e: choice A 1/ 4 , a 2, b 1
f ( 1 1) f p f 0 , f( ) T T p / t , d p e . vorticity : ct p e p d
e
d
p e p d
T
ct p ,
Laguerre eq
The General Solution of 2D AxialSymmetry, Inviscid Vortex • The The exa exact ct so solu luti tion on of 2D 2D vort vortex ex is is real realis isti tic c due due to the significant axial flow. b. Burgers vortex c. Rott vortex d. Sullivan vortex e. Long vortex
Burgers Vortex • Burge gerrs (1948) ( , )
A
2
e
2 4
,
4 t (i) without deforming a 0, A 1,
( r , t )
e
t, r
2
r 4
(Oseen vort ex),
4 t (ii) with deforming a( t ) consts, w( z) 2 az ( a 0), u A et , reat , e 2 at 1 / 2a
( r , t )
a 2 (1 (1 e
2 at
let t (r )
)
e
l 2
ar
ar 2 2 (1 e2 at )
e
r 2 2
l
(steady state), where l 2
(2 / a)1/ 2
Burgers Vortex • Velocity a(t ) const . (a 0) v( r )
(1 e
ar 2 2
)
2 r w( z ) 2az u ar
• pressure p p( r, z) p0 p(0,0) u u u v 2 1 P u 2 u w ( u 2 ) t r z r 0 r r w w w 1 P 2 u w g w t r z 0 z 2
p( r, z) p0
2
2
2
(4 a z
a
2 2
r
r
2 gz) 0
v
r
dr.
p(0, z) p0 ( 2 a2 z2 gz) (Negative PGF causes axial
Burgers Vortex • Discussion v( r )
(1 e
ar 2 2
)
2 r w( z ) 2az, u ar
(r )
l 2
e
r 2 l 2
1. r , z u , (unrealistic) 2. r 0, z 0 u v w 0 3. Rott(1958) Rott(1958) : unsteady Burger Burgers' s' solutio solution n
1 e v( r , t ) 2 r
2
ar
1
2 1 e 2 t
,
( : integral const)
Sullivan Vortex • Sul ulli liva van n (195 959) 9):: w=zf(r) ar 2 / 2
], w 2az[1 be (ru ) (rw) from 0, we can obtain u( r) r z
u ar ar 2b [1 e r v
2 r
2
H(
ar
2
ar 2 / 2
u(0) 0
],
) / H (), where H ( x )
x
e 0
( t b
e
t 1
0
d )
dt .
Sullivan Vortex • Tw Twoo-ce cell lled ed str truc uctu ture re b 1, when u 0, we can obtain
r r0 r r 0
u 0, u 0,
ar 02
1 e
2b r r0 w 0, r 0 w 0,
ar 02 / 2
,
w 2az (b 1),
Long Vortex • Lo Long ng (195 (1958) 8) cons conside iderr the sim simil ilar ar solu soluti tion. on. vr
2
k const.
(far from the symmetry axial)
x kr 2 z
u f ( x) k f ( x), 2 z k r v ( x), r w k f ( x), 2r 4 k p gz 2 2 s( x) z
Long Vortex • In Inst stit itut utin ing g u, v, w, p into into th the e NN-S S eq eqs, s, 2 2 x3 s 2 ( f 2 ff x x2 f 8 x4 s xf 4 x5 s) 4 (2 x4 f ) f x f ( f 1) 4 x3 s 2 ( x3 f ) , x ( f 1) 2 (2 x3 4 x2) , wheree , / k , mea wher eans ns th thee re reci cipr proc ocal al of Re no no., ., Long assumes = 1 u / w = 1
2 2 x3 s 0, f x f ( f 1) 4 x3 s 0, x ( f 1) 0.
Long Vortex • As the the same same wit with h Sulli Sullivan van vort vortex, ex, Long Long vor vortex tex can use extensively in the meteorology. • Bec Becaus ause e Long Long doesn doesn’t ’t cons consider ider the surf surface ace effe effect, ct, the results only simulate the flow field far from the surface.
(a) Axial velocity
(b) Azimuthal velocity
