Searching for Singularities in Vortical Flow - personal.psu.edu · IMPA March 2014 Searching for Singularities in Vortical Flow Russel Caflisch Mathematics Department and Institute

IMPA March 2014

Searching for Singularitiesin Vortical Flow

Russel Caflisch

Mathematics Department

and

Institute for Pure & Applied Mathematics

UCLA

IMPA March 2014

Acknowledgements

Collaborator: Mike Siegel, NJIT.

Thanks to Tom Hou and Guo Luo (Caltech)for discussion of their preprint

“POTENTIALLY SINGULAR SOLUTIONS OF THE 3D INCOMPRESSIBLE

EULER EQUATIONS”and for permission to use their slides.

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Outline

• Singularities and turbulence• Numerical construction of Euler singularities• Complex singularities: PDE examples• Numerical construction of complex singularities for Euler

• Numerical construction of possible singular solutions by Luo and Hou

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Navier-Stokes and Euler Equations• Navier-Stokes for incompressible, viscous fluid flow

– Euler equations for inviscid flow, in limit Re → ∞

• Velocity u, pressure p

• Vorticity ω

• Energy dissipation

• Energy dissipation is roughly independent of Re in turbulence

1

0

Ret

u

u u u p u−

∇ × =∂ + ×∇ + ∇ = ∆

1Ret u u

u

ω ω ω ωω

−∂ + ×∇ = ×∇ + ∆∆ = ∇ ×

22 1/ 2 Ret u dx u dx−∂ = − ∇∫ ∫

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Singularities in Turbulence?Onsager’s Conjecture(1949):

– Inviscid singularities are the source of turbulent energy dissipation

• Requirement for energy dissipation– singularity set S of codimension κ, singularity order α

– Onsager; Constatin, E & Titi (1994)• Improvements by Constantin, Cheskidov, Friedlander, Pavlovic & Shvydkoy

(2007)

– Eyink (1995); REC, Klapper & Steele (1997)• Shvydkoy (2008)

3 1α κ+ <

3 1α <

S codim = κ

r

xu ≈ r α

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Derivation of Singularity Requirements for Inviscid Energy Dissipation

• For singularity set S of codimension κ, singularity order α

r = dist(S)

• Time derivative of energy

1Sdx r drdx

u r

κ

α

−=≈

2( ) 2 ( )d dt u dx u u u u pdx/ | | = − × ×∇ + ×∇∫ ∫

S

r

x

u ≈ r α

codim = κ

0tu u u p+ ×∇ + ∇ =

IMPA March 2014

Derivation of Singularity Requirements for Inviscid Energy Dissipation

• For singularity set S of codimension κ, singularity order α

r = dist(S)

• Time derivative of energy

1Sdx r drdx

u r

κ

α

−=≈

2( ) 2 ( )d dt u dx u u u u pdx/ | | = − × ×∇ + ×∇∫ ∫

S

r

x

u ≈ r α

codim = κ

0tu u u p+ ×∇ + ∇ =0

IMPA March 2014

Derivation of Singularity Requirements for Inviscid Energy Dissipation

• For singularity set S of codimension κ, singularity order α

r = dist(S)

• Time derivative of energy

1Sdx r drdx

u r

κ

α

−=≈

2( ) 2 ( )d dt u dx u u u dx/ | | = − × ×∇∫ ∫

S

r

x

u ≈ r α

codim = κ

0tu u u p+ ×∇ + ∇ =

IMPA March 2014

Derivation of Singularity Requirements for Inviscid Energy Dissipation

• For singularity set S of codimension κ, singularity order α

r = dist(S)

• Time derivative of energy

• The convective integral is nonzero, only if it isn’t absolutely integrable; i.e.

1Sdx r drdx

u r

κ

α

−=≈

3 1 1 1

3 1α κ

α κ− + − < −

+ <

2

3 1 1

( ) 2 ( )

SS

d dt u dx u u u dx

r r drdxα κ− −

/ | | = − × ×∇

=

∫ ∫∫ ∫

S

r

x

u ≈ r α

codim = κ

0tu u u p+ ×∇ + ∇ =

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Analysis of Euler Singularities

• Analysis – necessary conditions for singularity at time T from smooth initial data (partial list)– Beal, Kato & Majda (BKM) (1984)

– Constantin, Fefferman & Majda (CFM) (1996) • Either |u| → ∞ or | grad(ω/| ω |) | → ∞

– Chae (2004) Refinements of BKM, using approach of CFM

– Deng, Hou & Yu (DHY) (2004) Local criterion, using approach like CFM.

• Criterion is easy to apply

0( )

T

xsup x t dtω| , | = ∞∫

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Computations of Frisch & Orszag

• “Energy catastrophe” proposal for turbulence onset– finite time singularity vs. instability

– initiation of Kolmogorov spectrum

– Brissaud, Frisch, Lesieur, Pouquet, Sulem, Leorat, Mazure & Sadourny (Ann. Geophysique 1973)

• Taylor-Green cell– highly symmetric flow in cube

– indication of real singularity from Pade summation of Taylor series in time

• Morf, Orszag & Frisch, PRL 1980

– More refined Pade computation indicates no singularity• Brachet, Meiron, Orszag, Nickel, Morf & Frisch, JFM 1983

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Kerr’s Computations

• Colliding vortex tubes– Kerr (Phys Fl 1993)

– predicted singularity at t=18.7

• Computations repeated by Hou & Li – J. Nonlin. Sci. 2006

– computational pictures at right

– no singularity up to t=19

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Pelz’s Computations• Highly symmetric flow of Kida

– J Phys Soc Jap, 1985– 64 fold geometric symmetry, 3 fold component

• Viscous computations – large increase in ω– Pelz, JFM 2001

• Vortex tube model & simulation – collapse of vortex tubes, singularities– Boratav & Pelz, Phys. Fl. 1994

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Complex Singularities Example 1: Cauchy-Riemann Equations

• Laplace equation in x,t

• Complex traveling wave solution

• Singularities in initial data move toward the real axis.

0tt xxu u+ =

( , ) ( ) *( )u x t w x it w x it= + + −

xR

xI

w singularity

w* singularity

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Complex Singularities Example 2: Burgers Equations

• Burgers equation– Initial value problem

– Characteristic form

– Invert initial data

– Implicit solution

• Singularities

i.e.

0

0

(0, ) ( )t xu uu

u x u x

+ ==

0 ( )x x u tu= +

0 0 0( ) : ( ( ))x u u x u u=

0x uu x= ∞ ⇔ =

00 ( )ux u t= ∂ +

00t tu on x u u∂ = ∂ = =

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Burgers Equations (Cont)

• Burgers equation singularity condition

• Example– Initial data

– Singularity condition

– Real singularities for t>0

– Complex singularities for t<0

• Complex singularities collide forming shock

20 3u t= − +

0

1/30 0

30

( )

( )

u x x

x u u

= −

= −

/ 3u t= ±

| | / 3u i t= ±

00 ( )ux u t= ∂ +

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Complex Singularities in Fluid Flows

• Shock-free airfoil design– Garabedian (1972)

• Kelvin-Helmholtz problem– Moore (1979)

– Orellana, REC (1989)

• Rayleigh-Taylor– Siegel, Baker, REC (1993)

• Hele-Shaw with surface tension– Siegel, Tanveer (1996)

• Muskat problem (2-sided Hele-Shaw, porous media)– REC, Howison, Siegel (2004)

– Cordoba (2006)

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Complex Euler Singularities:Numerical Construction

• Axisymmetric flow with swirl– REC (1993)

• 2D Euler– Pauls, Matsumoto, Frisch & Bec (2006)

• 3D Euler (Pelz and related initial data)– Siegel & REC

• Singularity detection via asymptotics of fourier components

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Singularity Analysis

• Fit to asymptotic form of fourier components in 1D• Sulem, Sulem & Frisch (1983)

• Apply 3-point fit, to get singularity parameters c,α,z* as function of k– Successful fit has c,α,z* nearly independent of k– extended to 3D by Siegel & REC (2008)

1ˆ ikzk c k eu

α ∗−−≈ 1*( )u c z z α −≈ −

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Axisymmetric flow with swirl REC (1993)

• Solution method– Moore’s approximation:

• u+ upper analytic in z, u- = u+* lower analytic, no interaction between them

– Traveling wave ansatz (Siegel’s thesis 1989 for Rayleigh-Taylor)

– Ultra-high precision, • needed to control amplification of round-off error

• Singularity type– u ≈ x -1/3

– ω ≈ x -4/3

• Real singularity? No– Violates Deng-Hou-Yu (DHY) criterion, restricted directionality

u u u+ −= +

( , , ) ( , )u r z t u r z i tσ+ += −

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Upper analytic solutions in z• Look for upper analytic solution (k≥0)

• One-way wave coupling– Since k>0 and wavenumbers add (Siegel)

u u u+= +(0 )( )

( )( , )

z

r z

u ru u

u u u u r z t

θ

θ+ + + +

= , ,= , , ,

0 1

0

( )ˆ ˆ ˆ

( )ˆ

k kk k

k k

M A …u u u

M M u

σσ

−= , , ,= ,

1

( ) ( )ˆ ikz ktk

k

u r z t r euσ+ +

≥

, , = ∑

z

r

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Complex upper analytic solution:pure swirling flow

• Flow in periodic anulus, – r1 < r < r2 (no normal flow BCs)– 0 < z < 2 π (periodic BCs)

uθ

ωθ

t < tsing t = tsing

z

z

rr

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2D Euler Pauls, Matsumoto, Frisch & Bec (2006)

• Solution method– Small time asymptotics, spectral

computation

– Ultra-high precision, • needed for singularity detection, since

singularities are far from reals

• Singularity type– ω ≈ x -β with 5/6 ≤ β ≤ 1

• Real singularity? No– Vorticity does not grow in 2D → no

singularities

– u=(u,v,0)

– ω=(0,0, ζ)

0t u uω ω ω∂ + ×∇ = ×∇ =

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3D Euler (Pelz initial data)Siegel & REC (2006)

• Solution method– Moore’s approximation:

• u+++ upper analytic in x,y,z, u--- = u+++* lower analytic, • no interaction between them

– Traveling wave ansatz

– No need for ultra-high precision– Highly symmetric (Kida)

• Singularity type– u+++ ≈ ε log x – ω+++ ≈ ε x -1

• Real singularity? ?– Satisfies known singularity criteria– Attempting to construct real singular solution as

u u u+++ −−−= +

( , , , ) ( , , )u x y z t u x y z i tσ+++ +++= −

2cu u u uε+++ −−−= + +

IMPA March 2014

Numerical Method for u+++ (Siegel)• 3D Euler in periodic cube with forcing

• Fully spectral, no “base flow” u0=0

– Aliasing error remove by zero padding– No numerical error, except roundoff

• Solution driven by forcing

– F chosen to allow O(1) choice of leading Fourier components – Singularities cannot be real: do not satisfy CFM or DHY

1 2 3

( )

0, 0, 0

ˆ( ) i t

k k k

u u u e × −+++

> > >

= = ∑ k xσk

0

t

u

u u Fω ω ω∇ × =

+ ×∇ − ×∇ =

1 2 3

( )

0, 0, 0

ˆ ( ) i t

k k k

F F e × −

> > >

= ∑ k xσk

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k

δ

α

σ ρ

∞

=

+++

=

≅ − +

∑1

ˆ1D fit: ( , )

log( ( , ))

ikxk

k

u u y z e

u c x i t y z

Fit of singularity parameters (1,0,0), 1σ ε= =

c

•BKM satisfied

Numerical construction of possible singular solutions

Luo and Hou (2013)

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Conclusions

• Inviscid singularities may play a role in viscous turbulence.

• Complex singular solutions for Euler constructed numerically.

• Computations of Luo & Hou.– Singuarity is at corner of outer bdry r=1 and symmetry plane z=0

– Amplification of vorticity by 108

– Vorticity singularity in time (ts - t) -2.5

– Accuracy and resolution retained

– Vorticity parallel to second mode of the strain matrix

– Near singularity, solution has similarity form ( )( )

0 03

( , )s

s

r r z zt t f

t t

γ − −− ÷ ÷−