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8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
1/23
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.
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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Coherent structures
Leonardo da Vinci
(Water studies, 1506)
Werl, ONERA, 1974
(Van Dyke, 1982)
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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Kida et al.
(Toki-Kyoto, 2002)
Vorticity localization in classical and quantum fluids
Miyazaki et al.
(Physica D, 2009)
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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)
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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)
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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, , .
, , ,
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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
.. ., , .
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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)
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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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:
+
+
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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Head-on collision and breakdown of vortex rings (Lim, Nature, 1992)
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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Head-on collision of vortex rings (Lim, Nature, 1992):Re=1071
8/3/2019 Renzo L. Ricca- Vortex dynamics by geometric and structured complexity analysis
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Head-on collision of vortex rings (Lim, Nature, 1992):Re=1573
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Selected references