Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis

8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis

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Verona 14-17 Sept., 2009

Renzo L. RiccaRenzo L. Ricca

Department of Mathematics & Applications, U.Department of Mathematics & Applications, U. MilanoMilano--BicoccaBicocca, ITALY, ITALY

E-mail:E-mail: [email protected] URL:URL: http://www.ttp://www.matappatapp.unimibnimib.it/~it/~riccaicca

Aims

Theoretical goals:- describe and classify complex morphologies;

- study possible relationships between energy and complexity;

- understand and predict energy localization and transfer;

Applications:- implement new visiometric tools and diagnostics;

- develop real-time energy analysis of dynamical processes.


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Coherent structures

Leonardo da Vinci

(Water studies, 1506)

Werl, ONERA, 1974

(Van Dyke, 1982)


3/23

Kida et al.

(Toki-Kyoto, 2002)

Vorticity localization in classical and quantum fluids

Miyazaki et al.

(Physica D, 2009)


4/23

Geometric approach to vortex filament motion

Solitons and integrable vortex dynamics

Stationary solutions

Topological properties and fluid invariants

Localized induction approximation(LIA) and intrinsic kinematics

Structural complexity analysis of vortex dynamics

Measures of morphological complexity

Energy/complexity relations

Shapefinders and eigenvalue analysis

Dynamical properties in terms of graph analysis

Geometry and topology of fluid flows

T


5/23

Localized induction approximation (LIA)

homogeneous

incompressible

inviscid

fluid in : in

as

u = 0

u = 0

X

u = u X,t( )

= u

Space curve , given by:

Ct

Ct:

X(s,t) := Xt(s) C

s [0,L]R 3

Intrinsic reference on , given by:(Frenet frame)

Ct

t:= X(s,t) = Xs

Vortex line on :

Ct

t

n

b

Ct

asymptotic theory

( )

no self-intersections

X(s,t) X

t X X = cb u

LIA

t, n, b{ },

LIA:

=0t

0, = constant

R / a 1(Da Rios, 1906; Hama & Arms, 1961)

3 3


6/23

Intrinsic equations under LIA and NLSE

Intrinsic description: , curvature, torsion;

under LIA: , , .

u = (ut,u

n,u

b)

uLIA

= cb

ut= u

n= 0 u

b= c

Ct:=

t(C)

c = (c ) c

= c c2c

+ c c

Da Rios, 1906

Betchov, 1965

NLSE via Madelung transform:

c s ,t ( )

s,t( )

s,t( ) = c s,t( ) ei , t( )d

0

s

c

Da Rios-Betchov eqs. from NLSE:

by taking log-derivative of

(Hasimoto, 1972)

e

m

i

+

+ 12

2

=

0NLSE:

s,t( )c = fc (s,t)

= f(s,t)

Ct:=

t(C)

2


7/23

Higher-order LIAs and integrable geometric dynamics

Higher-order effects:

: inhomogeneities(Lakshmanan & Ganesan, 1985)

,

: non-linear stretching(Onuki, 1985)

: axial flow and vorticity(Fukumoto & Miyazaki, 1991)

u = + s( )cb + + s( )t+ 12c2t+ c n + c b( )

Higher-order LIAs:

u0( )= u

LIA

(Langer & Perline, 1991) (Nakayama et al., 1992)

=P unei( ) +Q ubei( )

Integrable geometric dynamics:

uj+1( )

= F=1

j+1

u

0( )

s

= R u

0( )( )

u = ut,u

n,u

b( )

s = fs t( )

c = fc (s,t)

= f(s,t)

Germano, 1983

Ricca, 1991Ct :=t(C)


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Conserved quantities from integrable dynamics

Geometric invariants:

P L( ) = IjLj

j= 0

= const.

(Langer & Perline, 1991; Ricca, 1991)

Marsden &

Weistein, 1983

kinetic energy:

pseudo-helicity:

enstrophy:

linear momentum:

angular momentum:

Fukumoto, 1987

Ricca, 1992

Physical conserved quantities:

= const. Arms &Hama, 1965

;

... ...

A = X X ds

I0 = ds

L = ds

I1= c

2ds

I2= c

2ds

I3=

c4

4

c 2 c 22

ds

H= u u( ) d3X

K=1

2u

2

d3X

E= d3X

P =1

2X( ) d3X

M=1

3X X( ) d3X


9/23

Structural complexity analysis of 3D vortex tangles

domain extraction projected diagramanalysis

experimentor simulation

Dynamical systems analysis:

- topological entropy

- eigenvalue analysis

Measures of structural complexity:Geometric information:

- tropicity directions

- coiling, writhing

- alignment

- signed areaTopological information:- minimal crossing number

- linking numbers

- topological changes

Algebraic information:

- average crossing number


10/23

Energy and helicity of a vortex tangle

Kinetic energy:

If , then the kinetic helicity is given by (Moffatt, 1969; Moffatt & Ricca, 1992):

fromLamb (1932),we have:

;

,

with total length of vortex filaments given by .

E(T) 2

8ti t

j

Xi Xjij dsidsj

ij

L(T) = ti

i

i

dsi

u = 0t

where Lkij = Lk i ,j( )

i

Calugareanu-White invariant

Gauss linking number

i-th vortexcirculation

E(T)

1

2u

T2

d3 X = u X( )

T d3X

H(T) u d3XT

= Lkii2+ 2 Lkijij

i j

i

Lki= Lk

i,

i( ) =Wri + Twi


11/23

Tangle analysis by indented projections

i

Let

i

i=(T

i) be the indented projection of the oriented tangleT

i

component ; assign the value to each apparent crossing in .i i

Ti

writhing:

average crossing number:

linking:

Wr =Wr(T) = rrT

Cij

= C(i

,j

) = rri j

, ;

, ;

;

.

1

1

+1

i

Wri=Wr(

i) =

r

ri

Lkij = Lk(i ,j) =1

2r

ri j

i j

Lktot = LkijrT

ij

,

C=

CijrT

r= 1

estimated values:

W r = rrT

,

C = rrT

T= ii


12/23

A vortex tangle test case (Barenghi, Ricca & Samuels, 2001)

t = 0

t = 0.015

t= 0.050 t= 0.087

ABC-type flow field super-imposedon initial vorticity distribution


13/23

Vortex tangle test case: energy-complexity relation

time

logH

logC

logC

logWr

logLktot

log(L /L0)

mature tangle

O(165 s1

)

O(83 s1)

C(t) E(t)[ ]2

log(E /E0)


14/23

Energy-complexity relation

: since

C=

1

4

Xi Xj( ) ti tjXi Xj

3 ij ds

idsjij

we have

(E/ E0)

s

Xi Xj

, andCs

Xi X

j

2

,

.C (E / E0 )2

C(t) E(t)[ ]2

;

hence

If , , then we have(Ricca, 2008):

and .H(t) 22C(t)

d H(t)

dt 2

2 dC(t)

dt

Lkii = 0 i = i T

Helicity-complexity bounds


15/23

Tropicity interpretation of eigenvalues

Eigenvalue analysis: let .1

2

3

From (Vilanova et al., 2006), we have:

Cf =

1

2

ii

Cp =

2 2

3( )i

i

Cs =

33

ii

, , .

Shapefinders: let

V = d3XD A = d

2XD HG = 12 1R

1

+ 1R

2

d2X

D E = 12 1R1R

2

d2XD, , , .

ThenL =

HG

4E

W =A

HG

T =3V

A

(Sahni et al., 1998), , ;

If is convex and , we can defineL W T > 0D

CF=

L W

L +W + T C

P=

2(W T)

L +W + T C

S=

3T

L +W + T

so that1 L

2W

3T, , .

, , ,


16/23

Interpretation of momenta in terms of projected areas

Linear momentum:

Angular momentum:

Under Euler (and LIA) equations:

Then (Arms & Hama, 1965; Ricca, 1992):

,

.

P

1

2X d3X

T =1

2

i

i

X X ds

i

M1

3X X( ) d3X

T =1

3

i

i

X X X( ) ds

i

dP

dt

= 0

dM

dt

= 0, .

Pxy= Axy Pyz = Ayz Pzx = Azx

Mxy = dzAxy Myz = dxAyz Mzx = dyAzx

, , ;

, , .

p(T) : 3 2Plane graph by standard projection

xy p

zT( )

Axy A xy( )xy

...

yz zx

.. ., , .


17/23

Signed area and weight of projected graphs

+1

t

Rj

j

j =kkLk, j

Lj

Let and :j = p(j) Rj int j ( )

j

j

p Cauchy index : at eachintersection assign

and take

i

= 1

.

I

j= I

j

( )

Signed area:

I

j= I

j( ) = i

j

Aj ( ) = IjAj(Rj)

j

Weighted circulation:

+1

1

12

0

Example

j


18/23

Momenta by signed area interpretation:geometric method(Ricca, 2008)

: , , ;

T= ii Let be a vortex tangle. Then, by considering the associatedprincipal projected graphs, we have

P = Pxy ,Pyz ,Pzx( ) Pxy = jIjAxy Rj( )

j

Pyz = ... Pzx = ...

M = Mxy,Myz,Mzx( ) Mxy = dz jIjAxy Rj( )

j

Myz = ... Mzx = ...: , , .

Examples

(a) (b)


19/23

Vortex analysis by geomertic method

FollowingAref & Zawadzki (Nature, 1991):

before after

:

weighted areas

+1

+1

+2

+1

1

1 +1

0

Zero-area momentum (P= 0) interaction:

+

+


20/23

Head-on collision and breakdown of vortex rings (Lim, Nature, 1992)


21/23

Head-on collision of vortex rings (Lim, Nature, 1992):Re=1071


22/23

Head-on collision of vortex rings (Lim, Nature, 1992):Re=1573


23/23

Selected references

Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis

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