Numerical study of singular behavior incompressible flows
Pierre Gremaud
Department of MathematicsNorth Carolina State University
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