singularities in interfacial fluid dynamics michael siegel dept. of mathematical sciences njit...
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Singularities in interfacial fluid dynamics
Michael SiegelDept. of Mathematical SciencesNJIT
Supported by National Science Foundation
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Outline
•Singularities on interfaces
-Motivation and examples -Methods for analyzing singularities -Kelvin-Helmholtz -Rayleigh-Taylor -Hele-Shaw
•Singularity formation in 3D Euler flow
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Example 1: Breakup of a viscous drop
Shi, Brenner, Nagel ‘94
Similarity solution
Eggers ’93Stone, Lister ’98 (modifications due to exterior fluid)
01/ 2
0
( , ) ;
location of pinch off
time to pinch off
z zr z t tR
t
z
t
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Kelvin-Helmholtz instability
Krasny (1986)
-uu
•Evolution of interface at different precision
7 digits
16 digits
29 digits
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Kelvin-Helmholtz (cont’d)
•Irregular point vortex motion at later times Krasny ‘86
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Importance of singularity
•Mathematical theory (existence of solutions, continuous dependence on data)
• Numerical computation
•Physical importance depends on particular problem
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Roll-up of vortex sheet at edge of circular tubeRegularized vortex sheet
calculation
Krasny 1986Didden 1979
decreasing
0 for gives 'singular' structurect t
Singularity removed by regularization
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Methods for analyzing singularities
-Jet pinch-off: Brenner, Eggers, Lister, Papageorgiou-3D Euler: Childress-Vortex Sheets: Pullin
•Complex Variables
•Numerical
•Similarity solutions
-Bardos, Hou, Frisch, Sinai, Caflisch, Tanveer, S.
•Unfolding
-Caflisch
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0, 0 t x t xu iu u iu
Complex Variables:Canonical example 1
Laplace equation
( )( ) 0tt xx t x t xu u i i u
•Initial value problem is ill-posed
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Complex Variables: Example 2
0 0
0
0
-Implicit solution
( , ) ( )
where is the initial position of
the straight line characteristic through
x,t
u x t u x
x x tu
x
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1/ 30 0
30
3
1/ 3 1/ 31/ 2 1/ 22 3 2 3
Example
From Ca
- Initial data
- x ( )
- ( )
1 1( ,
rdano's formula
Complex singularities collide, formi
)2 4 27 2 4
n
27
u x
u u
x u u tu
x t x tu x t x x
0
30 0
( , )
is example of an unf
g shoc
olding
k
u x t u
x u tu
Burger’s equation example, cont’d
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Numerical analysis of complex singularities Sulem, Sulem, Frisch (1983)Vortex sheets: S., Krasny, Shelley, Baker, Caflisch Taylor-Green: Brachet and collaborators2D Euler: Frisch and collaborators3D Euler: Siegel and Caflisch
* z i
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1-D example 1( )
1 .9 Singularity power 1= 0.5Distance to singularity in lower halfplane is
0.105
ixf x
e
1
kk
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k
k
1
Singularity fit for Burger’s equation
•Initial value problem solved using pseudo-spectral method, u=sin(x) data
ˆln | |ku
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Singularity fit for Burger near shock
1
k
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3
2 2
,
( , ) , =1
ˆFourier coefficients k l
f x y x iAy
f
k
Fit to Fourier coefficients in 2D
Malakuti, Maung, Vi, Caflisch, Siegel 2007
1l 2l
3l 4l
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Outline
•Singularities on interfaces
-Motivation and examples -Methods for analyzing singularities -Kelvin-Helmholtz -Rayleigh-Taylor -Hele-Shaw
•Singularity formation in 3D Euler flow
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Kelvin-Helmholtz instability
•Birkhoff-Rott equation: ( , ) ( , )z x t iy t
| |
2
Linear stability of flat sheet
Argument for singularitie
( ) | | / 2
( 0)
skt ik
kk
k pk
k k
z A e
A e k p
1
2velocity due to point
vortex strength at .
u ivi z z
z
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Moore’s analysis (1979)
20 2
1 51 2 2
0
Moore analyzed singularity formation
through asymptotics
Curvature singular
( )
( ) ( ) ( ) ... ( 0)
( ) (2 ) (1 ) exp (1 ln
ity
4
i
)
2
ikk
k
k kk k k
k
z A t e
A t A t A t k
t tA t t i k k
n complex plane,
reaches real line
1 ln 0
2 4
at with
c c
ct
t
t
t
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Kelvin-Holmholtz (cont’d)
•Rigorous construction of singular solution, demonstration of ill-posedness (Duchon & Robert 1988, Caflisch & Orellana 1989)
•Regularized evolution: vortex blobs (Krasny ’86)
* * *
2 2
1 ( , ) ( , )
2 | ( , ) ( , ) |
z z t z tPV d
t i z t z t
( 0 solution) ( 0 solution) for ct t
•Surface tension regularization: Hou, Lowengrub, Shelley (1994), Baker, Nachbin (1997), S. (1995),Ambrose (2004)
•Numerical validation: Krasny(1986), Shelley(1992), Baker Cowley,Tanveer (1999)
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Vortex sheet singularity for Rayleigh-Taylor
(from Ceniceros and Hou)
Baker, S. , Caflisch 1993( , ) ( , )
is a Lagrangian parameter
Linear stability: (k) | |
z x t iy t
k
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Moore’s construction (Baker, Caflisch, S. ’93 interpretation)
(+ eqn. for )
*
Birkhoff-Rott equation
Look for ,
Ignore interactions between
1 ( , )( , ) ( , )
2 ( , ) ( , )
- Upper analytic (pos. wavenumber) , lower analytic
,
t
tz t B z PV d
i z t z t
z z z
z z
z z
*
*t t*
2
t + *
- Evaluate ( , ) for upper analytic functions , , etc. by
contour integration
1 1
etc. (Moore'
, 2 1 2 1
2
s approx
(1 )
imatio )
(1 )
n
B z z z
z zz z
AiAg
z z*
Traveling wave solutions (complex wavespeed) with singu
( )
larities
z z
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5 5
9 5
( )
( )
i
i
C t e
C t e
2 2 3 3( ) ( )i iA t e B t e
7 7 2 2( ) ( )i iA t e B t e
( , ) ( , ) ( , ) ( , , , )
product of , functions
B z B z B z E z z
E
•Evaluate PV integral by contour integration
Equivalence to Moore’s approximation
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•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
‘Moore’s’ equations for Rayleigh-Taylor (cont’d)
1 11
1
e.g.,
ˆ ˆ ˆ
ˆ, arbitrary
t x
ikx tk
k
u uu iu
u u e u iu
i u
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Singularity formation: comparison of asymptotics and numerics
H L
H L
A
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•Two-phase Hele-Shaw, or porous media, flow
Boundary
Conditions:
1 1,u p
2, 2u p
nV
Vj
Hele-Shaw flow: Problem formulation
1,
2,
2 /(12 )i ik h
as
i Vj yu
Water
Oil
i i iu k p
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Colored water injected into glycerinNJIT Capstone Lab (Kondic)
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Hele-Shaw flow: one phase problem
Forward problems, in which fluid region expands, are stable
Backward problems (fluid region contracts) are unstable
Exact solutions which develop cusps in finite time
(k)
-Sho
| |
w
k
s ill-posedness in nonlinear evolution
•Exact solutions derived using conformal map
(Howison)
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1
2( , ) ln ( , )
Exact solution: ( , ) ln 1( )
n
jj j
z t i f t
f t At
0z
Singularity
Hele-Shaw: Conformal map
plane
planez
Problem is well posed in | |>1 (e.g., Baker, S., Tanveer ’95)
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Zero surface tension limit
1
21 2 3
0 1
( , ) ( , ) 2 ( , ) ( , ) , |
Analytic extension of conformal map
Perturba
|>1
tion theory in | | 1
Expansion is regular except near poles and zero o
s f
tz q t z q t B q t z r t
z z Bz
0
3 3
2 21 0 0 2
(0)1
Near a zero ( )
z ( ) ( ) ( ) ( )
where
Motion of daughter singularity differs from that of z
( )
e
(
r
,
o
)
d
d d
z A t
A t t A t t
t q t
z
t
(Tanveer ’93, S., Tanveer, Dai ’96)
0
( , ) analyticiq t
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1/3
d
0
0
Analysis of inner region suggests localized (O(B ))
cluster of 4 / 3 singularities near
In inner region, differs by O(1) from , even
when is smooth
z z
z
Channel problem
Siegel, Tanveer, Dai ‘96
0z
B=0
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Hele-Shaw: Radial geometry
Comparison of B=0.00025, B=0 evolution
B=0.00025 evolution for overLong time
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•Detailed numerical studies suggest cusps can form (Ceniceros, Hou, Si 1999)
Singularities in two-phase Hele-Shaw (Muskat) problem
•Much less is known about two phase Hele-Shaw flow
•There are no know singular exact solutions
•Originally proposed as a model for displacement of oil by water in a porous medium
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Ceniceros, Hou and Si 1999
Numerical solutions
2/3y x
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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
•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
•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
Howison (2000)
(see also Ambrose (2004), Cordoba et al (2007)
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Strategy to show existence (stable case) and construct singular solutions
•Derive preliminary existence result involving class of solutions of the form
•Remainder terms are estimated using abstract Cauchy-Kowalewski theorem (Caflisch 1990)
•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
( , ), ( , )ss t w t are singular at t=0, e.g.,
Exact decaying solution of linearized system
0 < p < 1
1 1( , ) ( , , , )n n n n sL r w N r w s w ss
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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
•This results in an existence theorem for what appears to be a restricted setof data
0 ( ,0)r r depends on0 ( ,0)s s
•Introduction of a reparameterization converts to existence for any initial data
•First (?) global existence result that relies on stable decay rate k to show that solutions become analytic after initial time
t
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Euler singularity problem is an outstanding open problem in mathematics & physics
• Euler singularity connects Navier-Stokes dynamics to Kolmogorov scaling
2 kinematic viscosity
0 c
u
•Can a solution to the incompressible Euler equations become singular in finite time, starting from smooth (analytic) initial data?
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max
Singularity formation at time T
( , ) T
t dt
xx
Theoretical Results
•Beale-Kato-Majda (1984)
•Constantin-Fefferman-Majda (1996), Deng-Hou-You (2005)
No blow up if direction of vorticity = is smoothy directed
1minimum growth ( )T t (modulo log terms)
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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
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Hou and Li (2006) reconsidered Kerr’s (1993,2005) calculation
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Growth of maximum vorticity from Hou and Li (2006)
•Rapid growth of vorticity
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Growth of vorticity is bounded by double exponential
•No conclusive numerical evidence for singularities
e.g., Kerr’s (1993) numerics suggest singularity formation, buthigher resolution calculations for same initial data by Hou & Li (2006) show double exponential growth of vorticity
From Hou& Li (2006)
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Complex singularities for axisymmetric flow with swirl
•Annular geometry
1 2 , 0 2r r r z •Steady background flow
(0, , )( )zu u ru
chosen to give instabilitywith an unstable eigenmode
•Caflisch (1993), Caflisch & Siegel (2004)
1ˆ ( ) iz tr eu
swirl
1
2
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Traveling wave solution
Construct complex, upper-analytic traveling wave solution
Traveling wave with speed in Im(z) direction
( )
1
( ) ( , , )
in which
ˆ ( ) ik z i tk
k
r r z t
r e
u u u
u u
Baker, Caflisch & Siegel (1993)Caflisch(1993), Caflisch & Siegel (2004)
Traveling wave speed is thus determined from linear
eigenvalue problem and is independent of the amplitude
•Exact solution of Euler
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Motivation for traveling wave form
Construction of solution is greatly simplified
One way coupling among wavenumbers so
mode depends only o
-Degrees of freedom reduced
-Computational errors minimized since no truncation
n
k k k
ˆEquation f
or aliasing errors in r
or has form
is second order ODE operato
estriction to finite number of
Fourier components
ˆ ˆ ˆ( , , , )
rk
k
L
L k
k k 1 k 1u F u u u
u
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Motivation (cont’d)
Singularities at travel with speed
in Im z direction, reach real z line in finite time (for 0)
Singularities detected through asymptotics of
ˆ Fourier coefficients
(Sulem, Sulem & Fris
u
r i
i
z z i z
z
Provide information on generic form of singu
ch 19
lari
83)
ties
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Numerical method forswirl and axial background velocity
Pseudospectral in , 4th order discretization
(in r) for
Numerical method is accurate but unstable
-Instability controlled using high-precision
arithemetic (1
0
k
z
L
-100 )
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Caflisch & Siegel (2004)
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Re
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Perturbation construction of real singular solution
* *Consider where ( )
, , are exact solutions of Euler equations
satisfies system of equations in which forcing
terms are quadratic, i.e.,
z
u u u u u u u
u u u u u
u
u u u
2
We want , ( ) ( )
Full construction requires analysis showing
that singularity of is same or weaker than
that of , (Main tool is Cauchy-Kowalewski Thm)
O O
u
u u u
u
u u
Real remainder
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max
Singularity formation at time T
( , ) T
t dt
xx
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
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3D Traveling wave solution
Construct upper analytic traveling wave, periodic in (x,y,z)
Traveling wave with speed in Im(x) direction
Pos. wavenumbermodes
Const. FourierCoeffs.
•Exact solution of Euler but for complex velocity
0
ˆ( )= exp ( )i i t
kk<N
F x F k x σ
ˆ exp ( )
( , , ), (1,0,0)
( , , )
i x i t
k l m
x y z
kk>0
u u k σ
k σ
x
•Simplify construction -Base flow , instability driven by forcing termu 0
•No observed numerical instability!
Siegel, Caflisch 2007
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Euler equations
ˆ ˆ ˆ ˆ ( , , , )
, 1, ,
Small amplitude singularity by choice
of forcing
Introduce into forcing; when =0, solution
is entire.
For small , sing
k
j
L
j n
1 2 nk k k k ku G u u u
k k
u
ularity amplitude is O( )
3D traveling wave (cont’d)
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Numerical method
kˆ Nonlinear terms N evaluated by pseudospectral method
No truncation error in restriction to finite
Since N is quadratic, padding with zeroes eliminates
aliasing error from pseudospectral part
k
of calculation
Extreme numerical instability eliminated
We compute traveling wave , is real u u u
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k
1
ˆ1D fit: ( , )
log( | | ( , ))
ikxk
k
u u y z e
u c x i t i y z
Fit of singularity parameters (1,0,0), 1
c
•BKM satisfied
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k
Fit of singularity parameters 0.1
c
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Fit of singularity parameters 0.01
k
c
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Singularity amplitude
max u
c
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y
z
(1,0,0)σIm ( , )x y z
Im x
Singular surface
•Geometry of singular surface is useful for analysis
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k
Other singularity types
•Also find square root and cube root singularities
Fit for cube root
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k
N=100 '+'
N=140 ' '
N=160 ' '
1
Square root singularity
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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,