Transcript Slide 1

Singularities in interfacial fluid dynamics
Michael Siegel
Dept. of Mathematical Sciences
NJIT
Supported by National Science Foundation
Outline
•Singularities on interfaces
-Motivation and examples
-Methods for analyzing singularities
-Kelvin-Helmholtz
-Rayleigh-Taylor
-Hele-Shaw
•Singularity formation in 3D Euler flow
Example 1: Breakup of a viscous drop
Shi, Brenner, Nagel ‘94
Similarity solution
 z  z0
r ( z , t )  tR  1 / 2
 t

;

z 0  lo ca tio n o f p in ch o ff
t  tim e to p in ch o ff
Eggers ’93
Stone, Lister ’98 (modifications
due to exterior fluid)
Kelvin-Helmholtz instability
Krasny (1986)
•Evolution of interface at different precision
7 digits
-u
u
16 digits
29 digits
Kelvin-Helmholtz (cont’d)
•Irregular point vortex motion at later times
Krasny ‘86
Importance of singularity
•Mathematical theory (existence of solutions,
continuous dependence on data)
• Numerical computation
•Physical importance depends on particular problem
Singularity removed by regularization
Roll-up of vortex sheet
at edge of circular tube
Didden 1979
Regularized vortex sheet
calculation
Krasny 1986
 d e cre a sin g
  0 for t  t c gives 'singular' structure
Methods for analyzing singularities
•Numerical
•Similarity solutions
-Jet
pinch-off: Brenner, Eggers, Lister, Papageorgiou
-3D Euler: Childress
-Vortex Sheets: Pullin
•Complex Variables
-Bardos,
•Unfolding
-Caflisch
Hou, Frisch, Sinai, Caflisch, Tanveer, S.
Complex Variables:
Canonical example 1
Laplace equation
u tt  u xx  (  t  i  x )(  t  i  x )u  0
u t  iu x  0, u t  iu x  0
•Initial value problem is ill-posed
Complex Variables: Example 2
-Im plicit solution
u ( x , t )  u 0 ( x0 )
x  x 0  tu
w here x 0 is the initial position of
the straight line characteristic through
x,t
Burger’s equation
example, cont’d
E xa m p le
- In itia l d a ta u 0   x 0
1/ 3
- x 0 (u )  u
3
- x ( u )   u  tu
3
F ro m C a rd a n o 's fo rm u la
 1
x
t 
u ( x,t )    x  


 2
4
2
7



2
3
1/ 2




1/ 3
 1
x
t 
  x  


 2
4
2
7



2
C o m p le x sin g u la ritie s co llid e , fo rm i n g sh o c k
u ( x,t )  u0
x   u 0  tu 0 is e xa m p le o f a n u n f o ld in g
3
3
1/ 2




1/ 3
Numerical analysis of complex singularities
Sulem, Sulem, Frisch (1983)
Vortex sheets: S., Krasny, Shelley, Baker, Caflisch
Taylor-Green: Brachet and collaborators
2D Euler: Frisch and collaborators
3D Euler: Siegel and Caflisch
z *    i
1-D example
1
1  .9 e ix
S ingularity pow er   1=  0.5
D istance to singularity in low er halfpla ne is
 0.105
f ( x) 

 1
k
k
Singularity fit for Burger’s equation
•Initial value problem solved using pseudo-spectral method,
u=sin(x) data
 1
k
ln | uˆ k |
k
Singularity fit for Burger near shock
 1
k
Fit to Fourier coefficients in 2D
Malakuti, Maung, Vi, Caflisch, Siegel 2007
f ( x, y )

x  iA y
2


3
2
,  =1
F o u rie r co e fficie n ts fˆk ,l

l 1
l  2
l  3
l  4
k
Outline
•Singularities on interfaces
-Motivation and examples
-Methods for analyzing singularities
-Kelvin-Helmholtz
-Rayleigh-Taylor
-Hele-Shaw
•Singularity formation in 3D Euler flow
Kelvin-Helmholtz instability
z  x (  , t )  iy (  , t )
•Birkhoff-Rott equation:
L in e a r sta b ility o f fla t sh e e t
 (k )   | k | / 2
A rg u m e n t fo r sin g u la ritie s

z  

|k |
Ak e
2
t  ik 
k  
Ak  e
k
k
p
(p  0)
u  iv 
1

2 i z  z 
v e lo city d u e to p o in t
v o rte x stre n g th  a t z .
Moore’s analysis (1979)
M o o re a n a lyze d s in g u la rity fo rm a tio n
th ro u g h a s ym p to tic s

z  

A k ( t )e
ik 
k  
Ak (t )   Ak 0 (t )  
k
1
A k 0 ( t )  t (2  )

1
2
(1  i ) k
k 2

5
2
A k 2 ( t )  ...
(k  0 )
t
t 

e xp  k (1   ln
)
2
4 

C u rv a tu re s in g u la r ity i n c o m p le x p la n e ,
re a c h e s re a l lin e a t t  t c w ith
1
tc
2
 ln
 tc
4
0
Kelvin-Holmholtz (cont’d)
•Numerical validation: Krasny(1986), Shelley(1992), Baker
Cowley,Tanveer (1999)
•Rigorous construction of singular solution, demonstration
of ill-posedness
(Duchon & Robert 1988, Caflisch & Orellana 1989)
•Regularized evolution: vortex blobs
(Krasny ’86)
z
t
*

1
2 i
z (  , t )  z (  , t )
*
PV
*
 | z (  , t )  z (  , t ) |
2

2
d 
(  0 solution)  (   0 solution) for t  t c
•Surface tension regularization: Hou, Lowengrub,
Shelley (1994), Baker, Nachbin (1997), S. (1995),
Ambrose (2004)
Vortex sheet singularity for Rayleigh-Taylor
z  x ( , t )  iy ( , t )
 is a L a g ra n g ia n p a ra m e te r
L in e a r sta b ility:  (k )   | k |
(from Ceniceros and Hou)
Baker, S. , Caflisch 1993
Moore’s construction (Baker, Caflisch, S. ’93 interpretation)
B irkhoff-R ott equation
 t z ( , t )  B ( z ,  ) 
1
*
2 i
PV
 (  , t )
 z ( , t )  z (   , t ) d  
(+ e q n . fo r  )
Look for z    z   z  ,       
- U pper analytic (pos. w avenum ber) z  , low er analytic z 
Ignore interactions betw een z  , z  etc. (M oore's approx im atio n )

- E valuate B ( z ,  ) for upper analytic fu nctions z , z  , etc. by
*
contour integration
 t z 
 t + 

1
2 1   z
A
2
*

, tz  
*


1
2 1   z

2

 iA g   ( z   z  )
*
(1    z  ) (1    z  )
*
T raveling w ave solutions (com plex w avesp eed) w ith singu larities
Equivalence to Moore’s approximation
 A ( t )e
i 2
  B ( t )e
i 3
 C ( t )e
i 5
 A ( t )e
i 7
  B ( t )e
 i 2
 C ( t )e
i 5
2
7
3
2
5
9
B ( z,  )  B ( z  ,   )  B ( z  ,   )  E ( z  ,   , z  ,   )
E  product of  ,  functions
•Evaluate PV integral by contour integration
‘Moore’s’ equations for Rayleigh-Taylor (cont’d)
•The system of `Moore’s’ equations admit traveling wave
solutions (complex wavespeed) with 3/2 singularities
•The speed of the nonlinear traveling wave is independent
of the amplitude and identical to the speed given by a
linear analysis
•This is a general property of upper analytic systems of
PDE’s, as long as system of ODE’s resulting from substitution
of the traveling wave variable is autonomous
e .g ., u t  u u x  iu

u 

ikx   t
uˆ k e
  uˆ 1  iuˆ 1
k 1
  i , uˆ 1 a rb itra ry
gularity formation: comparison of asymptotics and numerics
A 
H  L
H  L
Hele-Shaw flow: Problem formulation
•Two-phase Hele-Shaw, or porous media, flow
Vj
u1 , p1 ,  1
Vn
u2, p2 ,  2
Oil
Water
ui   k i pi
Boundary
Conditions:
u i  V j as y  
k i  h /(12  i )
2
Colored water injected into glycerin
NJIT Capstone Lab
(Kondic)
Hele-Shaw flow: one phase problem
F o rw a rd p ro b le m s, in w h ich flu id re g io n e xp a n d s, a re sta b le
B a ck w a rd p ro b le m s (flu id re g io n co n tra cts) a re u n sta b le
 (k )  | k |
E xa ct so lu tio n s w h ich d e v e lo p cu sp s in fin ite tim e (Howison)
-S h o w s ill-p o se d n e ss in n o n lin e a r e v o lu tio n
•Exact solutions derived using conformal map
Hele-Shaw: Conformal map
z  0
z ( , t )  
2

ln   i  f ( , t )
n
E xact solution: f (  , t ) 

j 1
 plane

 
A j ln  1 




(
t
)
j


Singularity
z plane
P ro b le m is w e ll p o se d in |  |>1
(e.g., Baker, S., Tanveer ’95)
Zero surface tension limit
A n a lytic e xte n sio n o f co n fo rm a l m a p
(Tanveer ’93,
S., Tanveer, Dai ’96)


 1 
z t  q 1 (  , t ) z   q 2 (  , t )  B  2 q 3 (  , t )  z  2   r (  , t )  , |  |> 1







P e rtu rb a tio n th e o ry in |  | 1
q i ( , t ) analytic
z  z 0  B z1 
E xp a n sio n is re g u la r e xce p t n e a r p o le s a n d ze ro s o f z 
N e a r a ze ro z 
A    0 (t ) 
z 1  A0 ( t )     0 ( t ) 
w h e re  d ( t )   q 1
(0 )

3
2
 A2 ( t )     d ( t ) 

3
2
  d ( t ), t    0
M o tio n o f d a u g h te r sin g u la rity d iffe rs fro m th a t o f z e r o
A n a lysis o f in n e r re g io n su g g e sts lo ca lize d (O (B
1 /3
))
clu ste r o f  4 / 3 sin g u la ritie s n e a r  d
0
In in n e r re g io n , z  d iffe rs b y O (1 ) fro m z  , e v e n
0
w h e n z  is sm o o th
z  0
Channel problem
Siegel, Tanveer, Dai ‘96
B=0
Hele-Shaw: Radial geometry
Comparison of B=0.00025, B=0
evolution
B=0.00025 evolution for over
Long time
Singularities in two-phase Hele-Shaw
(Muskat) problem
•Originally proposed as a model for displacement of oil by
water in a porous medium
•Much less is known about two phase Hele-Shaw flow
•There are no know singular exact solutions
•Detailed numerical studies suggest cusps can form
(Ceniceros, Hou, Si 1999)
Numerical solutions
y
x
2/3
Ceniceros, Hou and Si 1999
Construction of singular solution (S., Caflisch,
Howison, CPAM 2004)
•S., Caflisch, Howison prove a global existence theorem for forward problem
with small data. The initial data is allowed to have a curvature (or weaker)
singularity, but the solution is analytic for subsequent times
(see also Ambrose (2004), Cordoba et al (2007)
•Time reversibility implies there are solutions to the backward problem
that start smooth but develop a curvature singularity
-Not a foregone conclusion: bounded finger velocity
in the two fluid case; negative interfacial pressure weakens
“runaway” that leads to cusp formation
Howison (2000)
•In view of waiting time behavior (King, Lacey, Vasquez ’95),
different techniques will be required to show cusp or corner formation
•Shows backward Muskat problem is ill-posed (on non-linear theory)
•Apply ideas to 3D Euler
Strategy to show existence (stable case) and construct singular solutions
•Approach is similar to that for Kelvin-Helmholtz problem (Duchon, Robert 1988,
Caflisch, Orellana 1989)
1. Extend equations to complex 
2. Put singularity in initial data
3. Construct solution within class of analytic functions
•Derive preliminary existence result involving class of solutions of the form
s ( , t ), w ( , t )
s
are singular at t=0, e.g.,
Exact decaying solution
of linearized system
0<p<1
•Remainder terms are estimated using abstract Cauchy-Kowalewski
n 1
n 1
n
n
s
theorem (Caflisch 1990)
L(r ,w
)  N (r ,w , s,w ) ss

New Challenges presented by Muskat problem
•Nonlinear term is considerably more
complicated
•Presence of a nonphysical ``reparameterization’’ mode (neutrally stable mode)
-Analysis is modified to accommodate this mode by prescribing its data
at  , i.e., by requiring it to go to 0 as t  
•This results in an existence theorem for what appears to be a restricted set
of data
r  r ( , 0) depends on s 0  s ( , 0)
0
•Introduction of a reparameterization converts to existence for any initial data
•First (?) global existence result that relies on stable decay rate 
to show that solutions become analytic after initial time
k
Euler singularity problem is an outstanding open
problem in mathematics & physics
•Can a solution to the incompressible Euler equations become
singular in finite time, starting from smooth (analytic)
initial data?
• Euler singularity connects Navier-Stokes dynamics to
Kolmogorov scaling
 
u
0c 
2
  k in e m a tic v isco sity
Theoretical Results
•Beale-Kato-Majda (1984)
S in g u la rity fo rm a tio n a t tim e T 

T
m ax x
 ( x, t ) d t  
m inim um grow th 
(T  t )
1
(modulo log terms)
•Constantin-Fefferman-Majda (1996),
Deng-Hou-You (2005)
N o b lo w u p if d ire ctio n o f v o rticity  =


is sm o o th y d ire cte d
Numerical Studies
•Axisymmetric flow with swirl and 2D Boussinesq convection
-Grauer & Sideris (1991, 1995), Pumir & Siggia (1992)
Meiron & Shelley (1992), E & Shu (1994)
Grauer et al (1998), Yin & Tang (2006)
• High symmetry flows
-Kida-Pelz flow: Kida (1985), Pelz & coworkers (1994,1997)
-Taylor-Green flow: Brachet & coworkers (1983,2005)
• Antiparallel vortex tubes
-Kerr (1993, 2005)
-Hou & Li (2006)
•Pauls, Frisch et al(2006).: Study of complex space singularities
for 2D Euler in short time asymptotic regime
Hou and Li (2006) reconsidered Kerr’s (1993,2005) calculation
Growth of maximum vorticity from Hou and Li (2006)
•Rapid growth of vorticity
•No conclusive numerical evidence for singularities
e.g., Kerr’s (1993) numerics suggest singularity formation, but
higher resolution calculations for same initial data by Hou & Li
(2006) show double exponential growth of vorticity
Growth of vorticity is bounded by double exponential
From Hou
& Li (2006)
Complex singularities for axisymmetric flow with swirl
1
•Caflisch (1993), Caflisch & Siegel (2004)
2
•Annular geometry
r1  r  r2 , 0  z  2
•Steady background flow
swirl
u  (0, u , u z )( r )
chosen to give instability
with an unstable eigenmode
iz   t
uˆ 1 ( r ) e
Traveling wave solution
C o n stru ct co m p le x, u p p e r-a n a lytic tra v e lin g w a v e so lu tio n
Baker, Caflisch & Siegel (1993)
Caflisch(1993), Caflisch & Siegel (2004)
u  u ( r )  u  ( r , z, t )
in w h ich

u 
ik ( z  i  t )
ˆ
u
(
r
)
e
 k
k 1
•Exact solution of Euler
T ra v e lin g w a v e w ith sp e e d  in Im (z) d ire ctio n
T ra v e lin g w a v e sp e e d  is th u s d e te rm in e d fro m lin e a r
e ig e n v a lu e p ro b le m a n d is in d e p e n d e n t o f th e a m p litu d e
Motivation for traveling wave form
C o n stru ctio n o f so lu tio n is g re a tly sim p lifie d
-D e g re e s o f fre e d o m re d u ce d
O n e w a y co u p lin g a m o n g w a v e n u m b e rs so
m o d e k  d e p e n d s o n ly o n k  k 
-C o m p u ta tio n a l e rro rs m in im ize d sin ce n o tru n ca tio n
o r a lia sin g e rro rs in r e strictio n to fin ite n u m b e r o f
F o u rie r co m p o n e n ts
E q u a tio n f o r uˆ k h a s fo rm
L k uˆ k  Fk ( u , uˆ 1 ,
, uˆ k  1 )
L k is se co n d o rd e r O D E o p e ra to r
Motivation (cont’d)
S in g u la ritie s a t z  z r  i z i tra v e l w ith sp e e d 
in Im z d ire ctio n , re a ch re a l z lin e in fin ite tim e (fo r z i  0 )
S in g u la ritie s d e te cte d th ro u g h a sym p to tics o f
F o u rie r co e fficie n ts uˆ
(S u le m , S u le m & F ris ch 1 9 8 3 )
P ro v id e in fo rm a tio n o n g e n e ric fo rm o f s in g u la ri tie s
Numerical method for
swirl and axial background velocity
P s e u d o s p e c tra l in z , 4 th o rd e r d is c re tiz a tio n
(in r) fo r L k
N u m e ric a l m e th o d is a c c u ra te b u t u n s ta b le
-In s ta b ility c o n tro lle d u s in g h ig h -p re c is io n
a rith e m e tic (1 0
-1 0 0
)
Caflisch & Siegel (2004)
Re 
Perturbation construction of real singular solution
Real remainder
C o n sid e r u  u  u   u   u w h e re u   u  ( z )
*
*
u , u  u  , u  u  a re e xa ct so lu tio n s o f E u le r e q u a t io n s
u sa tisfie s syste m o f e q u a tio n s in w h ich fo rcin g
te rm s a re q u a d ra tic, i.e .,
u  u  u  u
W e w a n t u  , u   O ( )  u  O ( )
2
F u ll co n stru ctio n re q u ire s a n a lysis sh o w in g
th a t sin g u la rity o f u is sa m e o r w e a k e r t h a n
th a t o f u  , u  (M a in to o l is C a u ch y-K o w a le w sk i T h m )
Difficulties
•Numerical method is highly unstable, resolved using
high precision arithemetic
•Too numerically intensive for 3D
•Square root singularity does not satisfy Beale-Kato-Majda
theorem
S in g u la rity fo rm a tio n a t tim e T 

T
m ax x
 ( x, t ) d t  
3D Traveling wave solution
Siegel, Caflisch 2007
C o n stru ct u p p e r a n a lytic tra v e lin g w a v e , p e rio d ic in (x,y,z)
Pos.
wavenumber
modes
u 
 uˆ
k
e xp i k ( x  i σ t )
k >0
k  ( k , l , m ), σ  (1, 0, 0 )
x  ( x, y , z )
Const.
Fourier
Coeffs.
•Exact solution of Euler but for complex velocity
T ra v e lin g w a v e w ith sp e e d  in Im (x) d ire ctio n
•Simplify construction
-Base flow u  0, instability driven by forcing term
F ( x )=

0  k <N
Fˆk e xp i k  ( x  i σ t )
•No observed numerical instability!
3D traveling wave (cont’d)
E u le r e q u a tio n s
L k uˆ k  G k ( uˆ k , uˆ k ,
1
k j  k , j  1,
2
, uˆ k )
n
,n
S m a ll a m p litu d e s in g u la rity b y c h o ic e
o f fo rc in g
In tro d u c e  in to fo rc in g ; w h e n  = 0 , s o lu tio n u
is e n tire .
F o r s m a ll  , s in g u la rity a m p litu d e is O (  )
Numerical method
N o n lin e a r te rm s Nˆ k e v a lu a te d b y p se u d o s p e ctra l m e th o d
N o tru n ca tio n e rro r in re strictio n to f in ite k
S in ce N is q u a d ra tic, p a d d in g w ith ze ro e s e lim in a te s
a lia sin g e rro r fro m p se u d o sp e ctra l p a r t o f ca lcu la tio n
E xtre m e n u m e rica l in sta b ility e lim in a te d
W e co m p u te tra v e lin g w a v e u    , u     u    is re a l
Fit of singularity parameters
  (1, 0, 0 ),   1
c


1 D fit: u 

ikx
uˆ k ( y , z )e
k 1
u
c lo g ( x  i  | t |  i  ( y , z ))
•BKM satisfied
k

  0 .1
Fit of singularity parameters

c

k
Fit of singularity parameters
  0 .0 1

c

k
Singularity amplitude
m ax
u
c

Singular surface
 Im x   ( y , z )
σ  (1, 0, 0 )
 Im x
z
y
•Geometry of singular surface is useful for analysis
Other singularity types
•Also find square root and cube root singularities
Fit for cube root

k
Square root singularity
N = 1 0 0 '+ '
N=140 '  '
N=160 '  '
 1
k
Conclusions
•Singularity formation is important in mathematical theory
and numerical computation
•Physical significance depends on particular problem
•Singularities are often removed by regularization, but are
relevant in understanding zero regularization limit
•New results presented concerning complex singularities
for 3D Euler,