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Hugues Egly,B. D. and R.
Sentis
The ablative Hele-Shaw model for ICF flows
Hugues Egly, B. D. and R. Sentis
April 2009
The ablative Hele-Shaw model for ICF flows p. 1 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
A schematic ablation front
Fluid velocityFront velocity
Cold region Hot region
Γ
Γ
Γ
Γb
r
t
lΓ (t)f
J. Sanz proposed to model this problem with harmonicpotentials ∆Θ = 0 solved by integral equations.
The ablative Hele-Shaw model for ICF flows p. 2 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
Starting point
The starting point of our analysis is the compressible Eulermodel with non linear heat flux
∂tρ +∇.(ρu) = 0∂tρu +∇.(ρu⊗ u) +∇p = 0∂t(ρe) +∇(ρue + pu− κnT n∇T ) = 0.
The Spitzer non linear coefficient is
n ∈ [5
2,7
2].
The heat flux boundary condition on the exterior boundary Γr
is non linearκnT n∂nT|Γe
= b given.
The ablative Hele-Shaw model for ICF flows p. 3 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
1D cut of an ablation front
The isobar regime is
p = (γ − 1)ρCvT ≈ C .
Fluid velocityAblation front velocity
x=0 x=x (t)f
xr
Tc
ρc
ρ
Th
h
left=cold right=hot
Two other important ingredients are
ε =
(Tc
Th
)n
, and u(t = 0, x) = uc(t), x ∈ the cold region.
The ablative Hele-Shaw model for ICF flows p. 4 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
Quasi-isobar model
Let us define the acceleration g = u′c(t) and reset the velocityu← u− uc . After rescaling one gets
Dt?ρ? + ρ?∇?.u? = 0
ρ?Dt?u? +1
M2∇?p? =
1
Frρ?g?
1
γ − 1Dt?p? +
γ
γ − 1p?∇ · u? −∇ · (HnT n
?∇T?) = 0.
where
The Mach number M2 = ρ?|u?|2p? .
The Froude number Fr = |u?|2g?l? = |u?|t?
g? is a measure of theacceleration of the particles coming from the left boundaryΓl in the cold region.
H = κn(T?)n+1
p?|u?| = κn(T?)n+1
ρ?|u?|3 measures the velocity of the
particles in the hot region.
The ablative Hele-Shaw model for ICF flows p. 5 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
Slow Mach number expansion
We perform an asymptotic expansion of all variables withrespect to the square of the Mach number,
ρ? = ρ(0)? + M2ρ
(2)? + · · · , . . .
One gets the quasi-isobar model∂tρ +∇.(ρu) = 0∂t(ρu) +∇.(ρu⊗ u) +∇p = ρg∇ · (u− nT n∇T ) = 0,ρT = 1.
or also ∂tT + uvort.∇T − T 2∆T n = 0,∂tu + u.∇u + T∇p = g,u = nT n∇T + uvort = utherm + uvort,∇.uvort = 0.
The ablative Hele-Shaw model for ICF flows p. 6 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
Hele-Shaw model : uvort = 0
Conjecture : For well prepared data, the solution of
∂tT − T 2∆T n = 0 is approximated by max(ε, θ)1n where
θ ≈ T n is a solution of the
Hele-Shaw equation :
∆θ = 0, x ∈ Ω(t),∂nθ = b, x ∈ Γh,θ = 0, x ∈ Γf (t),
∂tΓf (t) = −∇θ|γ(t), x ∈ Γf (t).
A numerical example is
The ablative Hele-Shaw model for ICF flows p. 7 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
Main idea of the 1D proof
Set Θ = T n. Then ∂t
(1
Θ1n
)+ ∂xxΘ = 0. Progressive waves are
defined by Θ = Θ(x + vt). The generating Kull’s function isthe progressive wave with v = 1 and Θ(−∞) = 1
K ′(x) = 1− K− 1n (x), normalisation K (0) =
(n + 2
n + 1
)n
≈ e.
Set T =(εK
(x−x0+vt
ε
)) 1n . Then (T n)(−∞) = ε
1n ,
(T n)′(+∞) = v and ε1n ∂tT − T 2∂xxT
n = 0.
n(T )’=v
y
T
ε1/n
v
The ablative Hele-Shaw model for ICF flows p. 8 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
The classical Hele-Shaw problem (1898)
∆p = 0, x ∈ Ωin(t) = blue region,p = 0, x ∈ ∂Ωin(t),∂nx = −∇p, x ∈ ∂Ωin(t).
The ablative Hele-Shaw model for ICF flows p. 9 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
Numerical result in 2D
We use a Finite Element Method for the Poisson equation, andmarkers for the front. The markers move accordingly to theHele-Shaw model.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
t=0
t=0.06
Solution numeriqueSolution analytique
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0 0.01 0.02 0.03 0.04 0.05 0.06
R(t
)
temps
solution numeriquesolution analytique
Grid 65*65 points. 65 markers are used to discretizeΓint(t) = Γf (t). On the left Γint(t). On the right t 7→ r(t).
The ablative Hele-Shaw model for ICF flows p. 10 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
Stability
Here we see the effect of an initial sinusoidal perturbation onthe evolution of the front. The smoothing effect of theHele-Shaw equation for convergent front is visible.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
In this regime, ablation fronts are stable.
The ablative Hele-Shaw model for ICF flows p. 11 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
The ablative Hele-Shaw model : uvort 6= 0
The model writes∣∣∣∣∣∣∣∣∣∣∣∣∣∣
−∆Θ = 0, x ∈ Ω− Ωin,∂nΘ = v , x ∈ ∂Ω = Γext,Θ = 0, x ∈ ∂Ωin = Γin(t),
x ′(t) = ∇Θ + uvort, x(t) ∈ Γin(t),
∂tV +∇. (uthermV) = S1, x ∈ Ω− Ωin,∆ϕ = TV, x ∈ Ω− Ωin,uvort = ∇∧ ϕ, x ∈ Ω− Ωin.
∣∣∣∣∣∣∣∣∣∣∣∣∣∣The vorticity source is S1.
The thermic velocity if utherm = nn+1∇T n+1 = Θ
1n∇Θ.
The “vorticity” is V = ωT .
The ablative Hele-Shaw model for ICF flows p. 12 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
Modeling S1
In 2D the vorticity is ω = ∇∧ u = ∂x1u2 − ∂x2u1 ∈ R. One has
∇∧(
1
T(∂tu + u.∇u)
)= ∇∧ g
T,
that is
∂tV +∇. (uthermV) = S1 + S2 + S3, V =ω
T.
The sources are
S1 = ∇∧ g
T≈ − 1
T 2g.∇⊥T = g.∇⊥ρ,
S2 =
(∇∧ 1
T
)(∂tu)− ω∂t
1
T, and S3 = · · ·
For FCI flows, it can be assumed that S1 is the maincontribution. At least the effect of g is only in S1.This is why we simplify further the model using S2 ≈ S3 ≈ 0.
The ablative Hele-Shaw model for ICF flows p. 13 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
Degrees of freedom
∣∣∣∣∣∣−∆Θ = 0, Θ = T n, Finite Element Method,
x ′(t) = ∇Θ + uvort, Markers,
∂tV +∇. (uthermV) = S1, V = ∇∧uT , Finite Volume.
∣∣∣∣∣∣
Coupling with the Fat Boundary of Maury and Ismail.
The ablative Hele-Shaw model for ICF flows p. 14 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
Passive vorticity : S1 6= 0 but uvort = 0
The initial data is a front Γint discretize with 100 markers andwith a mode 9. We plot the vorticity at three different times.
The ablative Hele-Shaw model for ICF flows p. 15 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
Active vorticity : S1 6= 0 and uvort 6= 0
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
10
80
140
0.5
0.6
0.7
0.8
0.9
1
0.5 0.6 0.7 0.8 0.9 1
10
80
140
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
140
180
220
0.5
0.6
0.7
0.8
0.9
1
0.5 0.6 0.7 0.8 0.9 1
140
180
220
In this regime, ablation fronts may be unstable.
The ablative Hele-Shaw model for ICF flows p. 16 / 17
Plan
Introduction
Quasi-isobarmodel
Hele-Shawmodel :uvort = 0
The ablativeHele-Shawmodel
Conclusion
Conclusion
• We have derived the ablative Hele-Shaw model for ICF flows∣∣∣∣∣∣∣∣∣∣∣∣∣∣
−∆Θ = 0, x ∈ Ωext(t) = Ω− Ωin(t),∂nΘ = v , x ∈ ∂Ω = Γext,Θ = 0, x ∈ ∂Ωin(t) = Γin(t),
x ′(t) = ∇Θ + uvort, x(t) ∈ Γin(t).
∂tV +∇. (uthermV) = ∇∧ gT , x ∈ Ω− Ωin,
∆ϕ = TV, x ∈ Ω− Ωin,uvort = ∇∧ ϕ, x ∈ Ω− Ωin.
∣∣∣∣∣∣∣∣∣∣∣∣∣∣• Plenty of open problems.• See H. Egly PhD thesis for more numerical results.• Cf the Web page of Howison (Ociam, Oxford) for somehistorical references about Hele-Shaw (and also with freshscience) : “Mr Hele-Shaw (inv. of the variable pitch propeller)worked on the propeller of the cruisers of her gracious majesty”.
The ablative Hele-Shaw model for ICF flows p. 17 / 17