Introduction to Vorticity Introduction to Vorticity and Vortex and Vortex

DynamicsDynamics

CH4 Basic Vortex flowCH4 Basic Vortex flow

2D Vortex2D Vortex• Cylindrical coordinate (r, θ, z), and (u, v, w)• GE:

• CE:

22

20

22

2

0

1( )

( )

1

u u u v P uu w u

t r z r r r

v v v uv vu w v

t r z r rw w w Pu w g w

t r z z

0)()(

z

rw

r

ru

2

22 )(

1

zrr

rr

where symmetry 0

Potential VortexPotential Vortex• Consider steady, w=u=0, and v=v(r).

• Where means the balance between the pressure gradient force and centrifugal force physically.

2

22

10,

0,

v dp

r dr

vvr

2

2

2

1d r dvv

dr r dr

2

2 2

10.

d v dv v

dr r dr r

Potential VortexPotential Vortex• For solving above ODE, we facilitate the

circulation

• Then eq(4.1.3) becomes

• Solution

• BC: , is bounded A=0

0, is bounded =0 ?

r

r B

, const2

vr

2 ,cu dx rv

2

20.

d drdr dr

21.

2Ar B

0ln( )2

w i z zi

Velocity distribution

Potential VortexPotential Vortex• We can obtain the solution

• Pressure distribution

, const2

vr

Besides r=0, the flow is irrotational and at r=0, v and w are infinite.

2

2 2 2

1from 0, we can obtain

8

v dp

r dr

p p r

, r p p

It’s interesting to estimate the critical value when the .0 p

for water 14 m/s (evaporation)

for air 406 m/s (supersonic)

Velocity distribution

Potential VortexPotential Vortex• Discussion:• For

• At large Re, the vortex is similar to potential 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’s unrealized that the calculated K.E. is infinite.

2 2, and 0 0

u u

viscous terms disapper in incompressible N S eq

Rankine VortexRankine Vortex• Rankine (1882): A simplify vortex model Vorticity distribution is uniform inside the vortex core.

, ~ (match at )

, ~ 1

r a v rwhen r a

r a v r

Incompressible flow outside the vortex core.

Where a is the radius of vortex vore.

20

AB

0.

AB const

22 rv r

2

,2

.2

v

r r a

ar a

r

21

2Ar B

Rankine VortexRankine Vortex• Pressure distribution

• Using the same method

• The continuity condition matches at r=a

2 2 2 2

2 2

for , ( ),4 8

in z-dir - '8

dp v rr a r p c z

dr r

dp rg p gz c

dz

2 4

2for '' ,

8

ar a p c gz

r

22

2 22

2

-

(2 - )-

,8

.8

pa

r

r gz r a

a gz r a

Rankine VortexRankine Vortex• The solution of free surface

2 2, 0, 4

0, 0,

when ar z P a P

r v P P

2

2 4

2

2 2

(2 ) ,8

.8

za

r

a r r ag

r ag

It’s interesting in and thus, forms a hollow vortex which is similar to the meso-scale typhoon.

0, 0, r v but p p

Oseen VortexOseen Vortex

• Oseen(Lamb) (1912): potential vortex will obey the viscous fluid dynamic at t=0, and we can make sure the motion of vortex at t>0.

steady, inviscid unrealisticPotential vortex

Rankine vortex

2.r

t r r r

0

0

& , ,

, ,

0, 0 0, 0,

0.

BC IC r

r

r tt

t

Oseen VortexOseen Vortex• Using the similar variables , and ( )

rf

t

3

2

2

2

1

2

1

1

,

,

.

1( ) 02

rtf

t t

fr r t

fr t

f f

2

0 0

0 0

0, 0 / 2

( ) , , , 0 2

, , 0

r B AA

f e B r t B

r t B

Oseen VortexOseen Vortex• We can obtain

• As

• For small r

• The unsteady transitional zone at

2 2

04 40 (1 ) (1 ).

2

r r

t te v er

20, 0 or 4 ,t r t

20 01 (1 . . ) , (solid body rotation)

2 2

rv h o t r v r

r t t

0

2v

r

1 (potential vortex)vr

0 ~ 4 ,r t

0Where means the dimension of vortex core.It will incerase with time and causes the decay of vortex.

r

Oseen VortexOseen Vortex• The distribution of vorticity

2

0 41 ( ),

4

r

trve

r r t

1. The above eq. shows the vorticity is infinity at r=0 when t=0.2. The total vorticity is invariable. 00

2 rdr

The vorticity distribution of Oseen vortex

Taylor VortexTaylor Vortex• Oseen vortex is the simplest one of the solutions of N-S e

q., and G. I. Taylor (1918) find another one.2

2

42

02

42

, ( .)4

2 , (angular momentum is finite)

(1 ) .2 4

r

t

r

t

H rv e H const

t

M r rvdr H

H re

t t

3

max 0

3/ 2max 0

3/ 2 20 0 0

r=0r

1. for 0 0 at ,

2. = / 2 2 at 2 ,

3. from decays to 2, it needs

(2 1) 0.296

o

t

v H et r t

v t v v

t t t r

Oseen Vortex & Taylor VortexOseen Vortex & Taylor Vortex• To take time derivation with the vorticity of Oseen vor

tex can obtain the same result with the vorticity of Taylor vortex.

• When t=o

• Taylor vortex is more realistic than Oseen vortex.

0.total energy for Taylor vortex total angular momentum are finite.total energy disppation

0.total energy for Oseen vortex total angular momentum are infinite.total energy disppation

The General Solution of 2D Axial-SymThe General Solution of 2D Axial-Symmetry, Inviscid Vortexmetry, Inviscid Vortex

• Vorticity eq: (linear)

• Similarity law:

• Separate variable:

2

1,r

t r r r

( ) ( ), , where , , are undeteminited consts.a bT t f Ar t A a b

choice 1/ 4 , 2, 1A a b

1 ( ) Laguerre eq( 1) 0,

/ , ,

vorticity : .

pp

p

p pp

dp f e ef f fd

T T p t T ctd

ct e ed

The General Solution of 2D Axial-SymThe General Solution of 2D Axial-Symmetry, Inviscid Vortexmetry, Inviscid Vortex

• The exact solution of 2D vortex is realistic due to the significant axial flow.

a. Burgers vortexb. Rott vortexc. Sullivan vortexd. Long vortex

Burgers VortexBurgers Vortex• Burgers (1948)

224( , ) ,

4

Aet

2

2

4 ( , ) (Oseen vort

(i) without deforming 0, 1, ,

(ii) with deforming ( ) , ( ) 2 ( 0),

e

,

x),4

, / 2

(

1t at a

r

t

a A t r

a t consts w z az a u arA e re

r t

a

t

e

e

2

22 (1 )2

, )2 (1 )

at

ar

eat

ar t e

e

2

2 2 1/ 22

let ( ) (steady state), where (2 / )r

lt r e l al

Burgers VortexBurgers Vortex• Velocity

• pressure

( ) . ( 0)a t const a 2

2( ) (1 )2

( ) 2

ar

v r er

w z azu ar

22

20

2

0

1( )

1

u u u v P uu w u

t r z r r r

w w w Pu w g w

t r z z

( , )p p r z

22 2 2 2

0 0( , ) (4 2 ) .

2

r vp r z p a z a r gz dr

r

0 (0,0)p p

2 20(0, ) (2 )p z p a z gz (Negative PGF causes axial

flow)

Burgers VortexBurgers Vortex• Discussion

2

2

2

2

2

( ) (1 )2

( ) 2 ,

( )

ar

r

l

v r er

w z az u ar

r el

2

2

1

2 1 ( , ) 1 ( : integral const)2

1. , , (unrealistic)2. 0, 0 0

3. Rott(1958) : unsteady Burgers' solution

, t

ar

ev r t er

r z ur z u v w

Sullivan VortexSullivan Vortex• Sullivan (1959): w=zf(r)

2 / 22 [1 ],arw az be

( ) ( )from 0, we can obtain ( ) (0) 0

ru rwu r u

r z

2 / 22 [1 ],aru ar b e

r

0

12 ( )

0( ) / ( ), where ( ) .

2 2

t et b dxar

v H H H x e dtr

Sullivan VortexSullivan Vortex• Two-celled structure 1,b

20

2/ 20when 0, we can obtain 1 ,

2arar

u eb

0

0

0, 0,

r r ur r u

0 0,

0 0,r r wr w 2 ( 1),w az b

Long VortexLong Vortex• Long (1958) consider the similar solution.

. (far from the symmetry axial)2

vr k const

2x kr z

4

2 2

( ) ( ),2

v ( ),

( ),2

( )

ku f x f x

r zk

xrk

w f xr

kp gz s xz

Long VortexLong Vortex• Instituting into the N-S eqs,

2 3 2 2 2 4 5 4 4

3 2 3

2 3 2

2 ( 8 4 ) (2 )( 1) 4 ( ),( 1)

where , / , means the reciprocal of Re no.,Long assu

(2 4 )

me

,

s 1 / 1

x s f ff x x f x s xf x s x ff x f f x s x fx f x x

ku w

2 3

3

2 0,( 1) 4 0,( 1) 0.

x sf x f f x sx f

, , ,u v w p

Long VortexLong Vortex• As the same with Sullivan vortex, Long vortex can

use extensively in the meteorology.• Because Long doesn’t consider the surface effect,

the results only simulate the flow field far from the surface.

(a) Axial velocity (b) Azimuthal velocity