Numerical study of singular behavior incompressible flows

Pierre Gremaud

Department of MathematicsNorth Carolina State University

gremaud@ncsu.edu

Pierre Gremaud Numerical study of singular behavior in compressible flows

Collaborators

I Kristen DeVault, NCSUI Kris Jenssen, PennStateI Joel Smoller, U. Michigan

Pierre Gremaud Numerical study of singular behavior in compressible flows

Main question:

Can vacuum appear in a compressible Navier-Stokes fluid?

or

Does there exist a weak solution to Navier-Stokes where thedensity ρ reaches zero in finite time, assuming ρ(·,0) boundedaway from zero?

Pierre Gremaud Numerical study of singular behavior in compressible flows

Why should you care?

I open problemI need to clarify notion of solutionI inviscid case is understood (Euler)I cool numerical problem

Present work: numerical study of this theoretical question.

Disclaimer: No attempt is made at modeling interstellar matterand/or low density fluids.

Pierre Gremaud Numerical study of singular behavior in compressible flows

Simplifications and notation

Symmetric flows, no swirl:

I ρ(x , t) = ρ(r , t): densityI ~u(x , t) = x

r u(r , t): velocity

x point in space, r = |x |, t is time

This talk: barotropic flows: pressure depends solely on ρ

Pierre Gremaud Numerical study of singular behavior in compressible flows

Navier-Stokes

ρt + (ρu)ξ = 0 mass

ρ(ut + uur ) +1

γM2 (ργ)r =1

Reuξr momentum

whereI M Mach numberI Re Reynolds numberI γ adiabatic coefficientI ∂ξ = ∂r + n−1

rI n spatial dimension (n = 1,2,3)

Pierre Gremaud Numerical study of singular behavior in compressible flows

Euler

Inviscid fluid: Re →∞

ρt + (ρu)ξ = 0 mass

ρ(ut + uur ) +1

γM2 (ργ)r = 0 momentum

Riemann data (r > 0){ρ(r ,0) = 1,u(r ,0) = 1,

1D : u(r ,0) =

{−1 if r < 0,

1 if r > 0.

“strength of the pull" is measured by M

Pierre Gremaud Numerical study of singular behavior in compressible flows

Riemman solution M > 2γ−1

ˆ ρu

˜(r, t) =

8>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

»1−1

–if r

t < −1 − 1M ,

2664“

2γ+1 − M γ−1

γ+1 (1 + rt )

”2/(γ−1)

1M(γ+1)

“2 + (1 − γ)M + 2M r

t

”3775 if −1 − 1

M < rt < −1 + 2

γ−11M ,

»0∅

–if −1 + 2

γ−11M < r

t < 1 − 2γ−1

1M ,

24“2

γ+1 + M γ−1γ+1 (−1 + r

t )”2/(γ−1)

1M(γ+1)

“−2 + (−1 + γ)M + 2M r

t

”35 if 1 − 2

γ−11M < r

t < 1 + 1M ,

»11

–if 1 + 1

M < rt .

Pierre Gremaud Numerical study of singular behavior in compressible flows

Riemman solution 0 < M < 2γ−1

ˆ ρu

˜(r, t) =

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

»1−1

–if r

t < −1 − 1M ,

2664“

2γ+1 − M γ−1

γ+1 (1 + rt )

”2/(γ−1)

1M(γ+1)

“2 + (1 − γ)M + 2M r

t

”3775 if −1 − 1

M < rt < − 1

M + γ−12 ,

24“1 − M

2 (γ − 1)” 2

γ−1

0

35 if − 1M + γ−1

2 < rt < 1

M − γ−12 ,

24“2

γ+1 + M γ−1γ+1 (−1 + r

t )”2/(γ−1)

1M(γ+1)

“−2 + (−1 + γ)M + 2M r

t

”35 if 1

M − γ−12 < r/t < 1 + 1

M ,

»11

–if 1 + 1

M < rt .

Pierre Gremaud Numerical study of singular behavior in compressible flows

Known Euler results: 1DExplicit Riemann solution: vacuum ⇔ M > 2

γ−1

!2 !1.5 !1 !0.5 0 0.5 1 1.5 2!1

!0.8

!0.6

!0.4

!0.2

0

0.2

0.4

0.6

0.8

1M = 2

!u

!2 !1.5 !1 !0.5 0 0.5 1 1.5 2!1

!0.8

!0.6

!0.4

!0.2

0

0.2

0.4

0.6

0.8

1M = 10

!u

Pierre Gremaud Numerical study of singular behavior in compressible flows

2, 3 D “Riemann problem"

ρ = ρ(s), u = u(s), s =tr

⇒

ρs = (n − 1)ρu(1− su)

s2c2 − (1− su)2

us = (n − 1)sc2u

s2c2 − (1− su)2

where ρ(0) = 1, u(0) = 1, c = 1M ρ

γ−12

Phase space analysis (Zheng, 2001) shows existence of criticalMach number M?

Pierre Gremaud Numerical study of singular behavior in compressible flows

Known Euler results: 2, 3 DZheng (2001): vacuum ⇔ M > M?

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1M=2

!u

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1M=10

!u

Pierre Gremaud Numerical study of singular behavior in compressible flows

Euler: phase diagram

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I = su

K =

s !("

!1)

/2/M

Q = (1/", ("!1)/(21/2"))

(s = t/r )

Pierre Gremaud Numerical study of singular behavior in compressible flows

Euler: critical Mach number

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60

0.5

1

1.5

2

2.5

3

!

criti

cal M

ach

num

ber M

*

2!D3!D

Pierre Gremaud Numerical study of singular behavior in compressible flows

Known Navier-Stokes results

I theory far from complet, Danchin (2005), Feireisl (2004),Hoff (1997), P.L. Lions (1998)

I unique uniqueness result for discont. sol., Hoff (2006)I Hoff & Smoller (2001): no vacuum formation for 1D NSI Xin & Yuan (2006): 2, 3D sufficient conditions to rule out

vacuumI results below are consistent with the above

Pierre Gremaud Numerical study of singular behavior in compressible flows

Numerics

I equations are split

(ρn,un)Euler−−−→ (ρ?,u?)

ρ∗ut=1

Re uξr−−−−−−−→ (ρn+1,un+1)

I diffusive step solved by Chebyshev-Gauss-Radaucollocation (avoid coord. singularity at 0)

I diffusive step advanced in time by BDF (can manage“infinite stiffness" when ρ = 0, i.e., index 1 DAE)

I Euler step advanced at each collocation node “à la Zheng"(ODE in s = t/r )

Pierre Gremaud Numerical study of singular behavior in compressible flows

Digression on collocation

Basic collocation principlesI Work on a finite gridI Find p such that p(xj) = uj , ∀xj ∈ gridI approximate derivative is p′(xj).

Non periodic problemsI algebraic polynomials on non-uniform gridsI Chebyshev TN optimalityI TN(x) = cos(Nθ) with θ = arccos x inherits fast

convergence from periodic caseCoordinate singularity at r = 0

I Chebyhsev-Gauss-Radau(Spatial) discretization

I uN(r , t) =∑N−1

i=0 Ui(t)ψi(r); ψi Lagrange interpolation pol.

Pierre Gremaud Numerical study of singular behavior in compressible flows

The mesh

rrN−1 1 0

n

n+1

ss

0

1sn+1

1

s 0

n

n+1

n

r r

t

tt

Pierre Gremaud Numerical study of singular behavior in compressible flows

Euler vs NS, 3D, M = 1.2/2.7, Re = 106

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

I

K

No Vacuum

Vacuum

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

I

K

No Vacuum

Vacuum

Pierre Gremaud Numerical study of singular behavior in compressible flows

result: 3D

105 106 1070

0.5

1

1.5

2

2.5

3

3.5

4

Reynolds number Re

Mac

h nu

mbe

r M

inconclusive

no vacuum

vacuum

criterion: ρN(rN−1, t) < tol = 10−14, for some t , 0 < t < .005

Pierre Gremaud Numerical study of singular behavior in compressible flows

So...

numerics ⇒ possible vacuum formation for multi-D NS flows

Pierre Gremaud Numerical study of singular behavior in compressible flows

Another example: relativistic Euler (2D)

∂t ρ̂+ ∂x(ρ̃v1) + ∂y (ρ̃v2) = 0,

∂t(ρ̃v1) + ∂x(ρ̃v21 +

1γM2 ρ

γ) + ∂y (ρ̃v1v2) = 0,

∂t(ρ̃v2) + ∂x(ρ̃v1v2) + ∂y (ρ̃v22 +

1γM2 ρ

γ) = 0,

where

I ρ̃ =ρ+ 1

γβ2

M2 ργ

1−β2|v |2 , ρ̂ = ρ̃− β2

γM2 ργ ,

I β = v̄c ,

I β → 0 ⇒ classical Euler

Pierre Gremaud Numerical study of singular behavior in compressible flows

Singularity formation

I blow up of smooth compactly supported perturbations ofconstant states Pan & Smoller (2006)

I type of singularity is unknownI shock formationI violation of subluminal conditionI mass concentration

I numerical difficulty: relationship between conserved andphysical variables is non trivial

Pierre Gremaud Numerical study of singular behavior in compressible flows

Preliminary results

shock formation; more to follow...

Pierre Gremaud Numerical study of singular behavior in compressible flows

Conclusions

I analyzed two phenomena of singularity formation incompressible fluids

I discussed corresponding numerical challengesI provided “numerical answers" to two open questions

Pierre Gremaud Numerical study of singular behavior in compressible flows