calculation of complex singular solutions to the 3d ...-100 1 1 singularities detected through...
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Calculation of complex singular solutionsto the 3D incompressible Euler equations
Michael SiegelDepartment of Mathematical SciencesNew Jersey Institute of Technology
Russel CaflischDepartment of MathematicsUCLA
Supported by NSF
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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 et al(2006).: Study of complex space singularities for 2D Euler in short time asymptotic regime
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Axisymmetric flow with swirl
•Annular geometry
1 2 , 0 2r r r z π< < < <•Steady background flow
(0, , )( )zu u rθ=uchosen to satisfy Rayleigh’s criterion for instabilityand an unstable eigenmode
•Caflisch (1993), Caflisch & Siegel (2004)
1ˆ ( ) σ+iz tr eu
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Background flow
0Background flow is smoothed vortex sheet at (motivated by Caflisch, Li, Shelley 1991)
ri
1 2
Pure swirling flow is unstable if (Rayleigh criterion)Γ > Γ
i
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Traveling wave solution
iConstruct complex, upper-analytic traveling wave solution
σiTraveling wave with speed in Im(z) direction
σ
+
∞−
+=
= +
= ∑ ( )
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)
1ˆ is linearly unstable eigenmode with eigenvalue Traveling wave speed is thus determined from linear
eigenvalue problem and is independent of the amplitude
σσ
uii
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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 truncationn
k k k′ ′<
i
i
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
LL −=
k
k k 1 k 1u F u u uu
i…
i
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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 zz
σ= +
≤
i
i
Provide information on generic form of singuch 19
lari83)
tiesi
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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 uu u u u uu
u u u
iii
2
2sin
We want , ( ) ( )
Full construction requires analysis showingthat singularity
( ), ( ) ( )
of is same or weaker than that of ,
reg gu O T u O T O
O O
ε
ε
ε
ε+
+ −
+ −
⋅∇
= ⇒ =
+
uu u u
uu u
i∼
i∼
Real remainder
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•Similar approach used in studies of singularityformation on vortex sheets-Caflisch & Orellana (1989), Duchon & Robert (1988)-Siegel, Caflisch, Howison (2004)-Cordoba(2006)
•For vortex sheets, singularity formation is associatedwith ill-posedness
•For Euler equations, traveling wave solution comes from balance between instability and nonlinearity
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2
Numerical construction in Caflisch (1993) was for 0Singularity position depends on , i.e., Im z= (r) Result: ( ) where 1/ 3 Amplitude of (i.e., ) is (1)
r
r
ru c z i t i r
u c O
α
ρσ α+
+
=
−
− − = −
zui
ii ∼i
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γ
γ
= = ⇒
i
i
i
- vortex sheet strength A -
Vortex sheets in Boussinesq approximation Siegel (1992), (1995)
Pure Boussinesq ( 1, 0) traveling waves of O(1) amplitud
density difference
e Pure vortex sh
A
γ
γ εε
= = ⇒
<< =→ →
i
eet ( 0, 1) no traveling waves due to conservation of vorticity on sheet For 1, 1 small amplitude traveling waves
0 as A 0.
A
A
Vortex sheet analogue
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Numerical method
0 0 sin , cos2 2
0 1
-Instability co
Pseudospectral in , 4th order discretization (in r) for Background velocities
Numerical method is accurate but unstablent
kz L
γ γθ θ
γ
θ π θ π
θ
⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠< <
z zu u u u
ii
i
-100
1
1
Singularities detected through asymptotics of Fourier components (Sulem, Sulem, Frisch 1983)
rolled using high-precision arithemetic (10 )
( ( ))ˆ exp( (
))k
u c z i
u c k k i
α
α
µ δ
δ µ
−+
−
≈ − −
≈ − +
i
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Caflisch & Siegel (2004)
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0
0
0
Shift in time by mult. of
Fourier coeff. by shift in imag.component of sing. position by
kt
t kth
et
σ
σ
≡
≡
1ˆAdjustable parameters: u , γθ
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•Square root singularity does not satisfyBeale, Kato, Majda theorem
sup
Singularity formation at time T
( , ) T
t dtω
⇔
= ∞∫ xx
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3D traveling wave
Control numerical instability Look for traveling wave solution, periodic in ( , , )
ˆ exp ( )
( , , ), ( , , )
Simplify construction -Base flow -Instability driven by f
k
x y z
x y zi i t
k l m σ σ σ>
= ⋅ −
= =
=
∑k 0
u u k x σ
k σ
u 0
ii
i
orcing termˆ ( )= exp ( )
Euler equations ˆ ˆ ˆ ˆ ( , , , )
, 1, ,k
j
i i t
L
j n
⋅ −
=
< =
∑
1 2 n
kk<N
k k k k k
F x F k x σ
u G u u uk k
i…
…
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{ }( )
( ){ } ( )
2 2 ( ) ( ) ( )1
( ) 2 2 ( ) ( )
1 ( ) ( ) 1
ˆ ˆ ˆˆ( )( )
ˆ ˆ ˆ ˆ
ˆ ˆˆ ˆ w ( )
where 0
Small amplitude singularity by choice of
, 0ˆ ˆ ˆ
x y z
x y z
z x
l m M lkM kmMuv lkM m k M lmM
k kM mM mk u
k
−
− −
⎛ ⎞+ − −⎛ ⎞ ⎜ ⎟= ⋅ ⋅⎜ ⎟ ⎜ ⎟− + + −⎝ ⎠ ⎝ ⎠
= ⋅ − +
⋅ ≠ ≠
= +
k k kk
k k k k
k k k k
k k k
σ k k k
σ k
σ kM F Ni
forcing Introduce into forcing; when =0, solution
is entire. For small , singularity amplitude is O( )
ε ε
ε ε
ui
i
( )℘ − ⋅∇k u u
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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
kiii
of calculation Extreme numerical instability eliminated; very mild
instability controlled by spectral filtering i
We compute traveling wave , is real+++ +++ −−−+u u ui
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k
δ
α
1
ˆ1D fit: ( , )
log( ( , ))
ikxk
ku u y z e
u c x i t y zσ ρ
∞
=
+++
=
− +
∑∼
Fit of singularity parameters (1,0,0), 1σ ε= =
c
•BKM satisfied
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k
δ
α
Fit of singularity parameters 0.1ε =
c
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δ
α
Fit of singularity parameters 0.01ε =
k
c
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Singularity amplitude
max u+++
ε
c
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yz
(1,0,0)=σIm ( , )x y zρ− =
Im x−
Singular surface
•Geometry of singular surface is useful for analysis
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Conclusion
•Introduced new method to compute singular solutions to3D Euler equations with complex velocity
•Eliminated numerical instability observed in earlier calculations;introduced techniques to achieve small amplitude singularity
•Results suggest a traveling wave singularity to 3D complex Euler equations in which the velocity blows up;satisifies Beale, Kato, Majda theorem, smooth singular surface
• Easily generalized to other problems,e.g., 2D and 3D MHD, quasi-geostrophic equation, etc.