Twist & Shout: Maximal EnstrophyGeneration in 3-D Navier-Stokes

LU LU, Ph.D. (2006)Wachovia Investments

CHARLES R. DOERINGUniversity of Michigan

Outline

I. Enstrophy & regularity and uniqueness of solutionsII. Analytic estimate on rate of enstrophy generationIII. Variational formulation of maximal productionIV. Computational results & a reality checkV. Conclusions, remarks & laments

Navier-Stokes equations:

• Periodic box:

• Initial condition:

Open ($1M) question:

Does a smooth solution exist for all t > 0?

!

r ˙ u +

r u "

r #

r u +

r # p = $%

r u

0 =r # "

r u

!

r u (

r x ,0) =

r u

0(r x )

WLOG r u

0(r x )d

3x = 0"( )

!

r x " [0,L]

3

Some definitions & some things we know:

• Kinetic energy:

• Vorticity:

• Enstrophy:

• If solution is smooth enough, dK/dt = –νE.

!

K(t) " 1

2

r u (

r x ,t)

2d3x# = 1

2

r u (

r x ,t)

2

2

!

r " =# $

r u %

r ˙ " +

r u &

r #

r " = '(

r " +

r " &

r #

r u

!

E(t) "r # (

r x ,t)

2

2=

r $

r u (

r x ,t)

2

2

% 8& 2

L2 K(t)

Global (in time) weak solutions exist:

• If K0<∞, there are weak solutions with finite energy,

• … and with finite integrated enstrophy,

• … but only known to satisfy an energy inequality,

• … and there is no assurance that they are unique.

!

K(t) " K0 #t $ 0

!

E(t)dt < "

ta

tb

# $0 % ta% t

b

!

K(tb) " K(t

a) #$ E(t)dt for a.e. t

a> 0

ta

tb

%

Local (in time) strong solutions exist:

• For (8π2/L2 )K0 ≤ E0 < ∞,

• Fact:

• And strong solutions are unique.

!

" T K0,E0,#( ) > 0 $ E(t) <% for 0 & t < T.

!

E(t) <" for ta# t # t

b

cr u ($,t)% C

"([0,L]

3) for t

a< t # t

b.

As long as the enstrophy is finite …

⇑Enstrophy generation rate G{u} = production – dissipation

!

dK

dt= "#E

dE

dt= "2#

r $

r %

2

2

+ 2r % &

r $ '

r u &

r % d3

x

!

= "2# $r u

2

2

+ 2r u %

r & '

r u % $

r u d

3x

1 2 4 4 4 4 4 3 4 4 4 4 4

Vortex stretching & enstrophy production:

Vorticity can be amplified; enstrophy can be produced.

Does this nonlinear process get out of control?

Can G{u} be estimated in terms of K and E?

Hölder & Cauchy-Schwarz⇓

Agmon-Sobolev-Gagliardo-Nirenberg (in 3D)

⇓

!

G{r u } = "2# $

r u

2

2+ 2

r u %

r & '

r u % $

r u d

3x

!

r u

"# ˜ c

r $

r u

2

1/ 2

%r u

2

1/ 2

= ˜ c E1

4 %

r u

2

1/ 2

!

" #2$ %r u

2

2

+ 2r u

&

r '

r u

2

%r u

2

= #2$ %r u

2

2

+ 2r u

& E

12

%r u

2

Hölder-Young (ab ≤ ap/p + bq/q when 1/p + 1/q=1)⇓

Integration by parts & Cauchy-Schwarz⇓

!

E =

r "

r u

2

2

= #r u $% &

r u d

3x '

r u

2 &

r u

2 = 2K &

r u

2

!

G{r u } " #2$ %

r u

2

2+ 2 ˜ c E

34

%r u

2

3 / 2

!

2˜ c E3

4 "

r u

2

3 / 2

#

c

$ 3E

3+ $ "

r u

2

2

System of differential inequations:

!

dK

dt= "#E

dE

dt$ "

#

2

E2

K+c

# 3E3

!

" E # E0

K

K0

$

% &

'

( )

1

2

1+2cK0E0

*4

K

K0

$

% &

'

( )

3

2

+1

,

-

.

.

.

/

0

1 1 1

$

%

& & &

'

(

) ) )

+1

... as long as RHS 2 0.

!

dK

dt= "#E

dE

dt$ "

#

2

E2

K+c

# 3E

3

!

dK

dt= "#E

dE

dt$ "

#

2

E2

K+c

# 3E

3

!

dK

dt= "#E

dE

dt$ "

#

2

E2

K+c

# 3E

3

Enstrophy decreases if E0K0 ≤ ν4/2c.

!

dK

dt= "#E

dE

dt$ "

#

2

E2

K+c

# 3E

3

Global existence and uniqueness if E0K0 ≤ 3ν4/2c.

!

dK

dt= "#E

dE

dt$ "

#

2

E2

K+c

# 3E

3

But does not prevent finite-time singularity if E0K0 > 3ν4/2c.

!

dK

dt= "#E

dE

dt$ "

#

2

E2

K+c

# 3E

3

!

dK

dt= "#E

dE

dt$ "

#

2

E2

K+c

# 3E

3

But does not prevent finite-time singularity if E0K0 ≥ 3ν4/2c.

Question:

How big can G{u} really get in terms of K and E?

• Analytic estimates don’t account for div u = 0 …

• or total competition between production & dissipation.

• Would like to solve the variational problem for max rate:

!

M(K,E;",L) = supr # $

r u = 0

G{r u } |

1

2

r u

2

2= K and

r #

r u

2

2

= E{ }

Settle for slightly less:

• We know that ℜ ≤ cE3/ν3 … but that ≠ $1M.

• “Critical” behavior is ℜ ~ E2 as E → ∞.

• Solve the Euler-Lagrange equations:

Computationally … via gradient ascent method.

!

"(E;#,L) = supr $ %

r u = 0

G{r u } |

r $

r u

2

2

= E{ }

so dE

dt&"(E)

!

0 ="

"r u

G{r u } + p

r # $

r u d

3x% + &

r #

r u

2

d3x%{ }

Starting from exact solution as E → 0 …

• Large E behavior is ℜ ~ E 1.78 (= 7/4?) … subcritial!

What do the maximizers look like?

periodic array of “vortex stretchers”

But this is a non-convex variational problem

• Euler-Lagrange solutions are local extrema …• So must see if there are other, global, maxima.

… another branch emerges at high E:

• Large E behavior is ℜ ~ E 2.997 (= 3?) … as estimated.

What do these maximizers look like?

colliding vortex rings

Another view …

Vorticity in a plane slice

Reality check:

!

Gr u { } = "2#

r $

r %

2

2

+ 2 %&2'

ur

r d

3x

z

r

For velocity fields with cylindrical symmetry …

!

" Gr u { } ~

E3

# 3

Then maximize over a …… max occurs at a ~ ν2/E

!

Circulation "

#

r $ ~

"

a2

%

& '

(

) *

E ~"

a2

%

& '

(

) *

2

a3

="

a

2

Reality check (continued):

!

Gr u { } = "#

r $

r %

2

2

+ 2 %&2'

ur

r d

3x

~ "#1

a

(

a2

)

* +

,

- .

2

a3 +

(

a2

)

* +

,

- .

2

1

a

(

a

)

* +

,

- . a

3

!

~ "#E

a2

+E3 / 2

a3 / 2

Conclusions, remarks & laments:

• Conclusion #1: The analytic asymptotic high-E estimateℜ(E;ν,L) ≤ cE3/ν3 can be saturated by divergence-free fields.

• Lament #1: no $1M to be found down this road!

• Conclusion #2: This “most dangerous” velocity field will

not produce a singularity in N-S.

• Lament #2: no $1M to be found down that road!

• Conclusion #3: K ~ 1/E for the optimizer, so we’re not sureif knowing the full upper limit M(K,E;ν,L) will help …

• Lament #3: so $1M not clearly down that road, either!

Maybe Lu will find $1M in Manhattan …

Just for fun … what do colliding vortex rings do?

(from website of Dr. T.B. Nickels <http://www2.eng.cam.ac.uk/~tbn22/Mov.html>)

The End

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