Singularities in interfacial fluid dynamics

Michael SiegelDept. of Mathematical SciencesNJIT

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

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

Kelvin-Helmholtz instability

Krasny (1986)

-uu

•Evolution of interface at different precision

7 digits

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

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

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

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

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

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

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

1-D example 1( )

1 .9 Singularity power 1= 0.5Distance to singularity in lower halfplane is

0.105

ixf x

e

1

kk

k

k

1

Singularity fit for Burger’s equation

•Initial value problem solved using pseudo-spectral method, u=sin(x) data

ˆln | |ku

Singularity fit for Burger near shock

1

k

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

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

•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

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

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)

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

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

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

•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

Singularity formation: comparison of asymptotics and numerics

H L

H L

A

•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

Colored water injected into glycerinNJIT Capstone Lab (Kondic)

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)

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)

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

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

Hele-Shaw: Radial geometry

Comparison of B=0.00025, B=0 evolution

B=0.00025 evolution for overLong time

•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

Ceniceros, Hou and Si 1999

Numerical solutions

2/3y x

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)

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

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

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?

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)

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

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)

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

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

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

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

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 )

Caflisch & Siegel (2004)

Re

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

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

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

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)

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

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

k

Fit of singularity parameters 0.1

c

Fit of singularity parameters 0.01

k

c

Singularity amplitude

max u

c

y

z

(1,0,0)σIm ( , )x y z

Im x

Singular surface

•Geometry of singular surface is useful for analysis

k

Other singularity types

•Also find square root and cube root singularities

Fit for cube root

k

N=100 '+'

N=140 ' '

N=160 ' '

1

Square root singularity

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,