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09-01841

The Structure of Radiative Shocks

Robert B. Lowrie

Presentation at the Computational Kinetic Transport andHybrid Methods Workshop, Institute for Pure and AppliedMathematics, UCLA, Los Angeles, CA, March 30 - April 3,2009

The Structure of Radiative Shocks

Robert B. Lowrie

Los Alamos National LaboratoryComputational Physics and Methods Group (CCS-2)

[email protected]

CollaboratorsJ. D. Edwards (Texas A&M)

R. M. Rauenzahn (T-3 LANL)

[email protected] LA-UR 09-01841 IPAM 2009 1 / 52

Motivation

Non-trivial, analytic solutions do not exist for radiationhydrodynamics.

I Compromise: Generate semi-analytic solutions; solutions fromnumerically integrating nonlinear ODEs.

I Seek traveling wave solutions⇒ radiative shocks.In this talk, use solutions to

I verify code correctnessI test AMR (rad-shocks are multiscale problems)

Gives physics insight that would be very difficult to obtain by aseries of runs from a computational physics code.


Outline

1 Overview of radiative shocks

2 Equations of radiation hydrodynamicsGrey nonequilibrium diffusionEuler coupled with grey nonequilibrium diffusionEquilibrium diffusion limit

3 Overview of Semi-analytic Approach

4 Sample Solutions

5 Code Comparison


Outline




4 Sample Solutions

5 Code Comparison


Physics Regime

As one example, regimes in astrophysics governed byInviscid hydrodynamics (Euler equations; non-relativistic)Thermal radiation (X-ray) transportHigh-energy density:

I Material temperatures O(1 keV = 11.6× 106 Kelvin)I Radiation pressure affects hydrodynamics


Past Work on Radiative Shocks

Most theory on “thick – thick” shocks: equilibrates on either side ofshock.Overview of Theory:

I Y. B. ZEL’DOVICH and Y. P. RAIZER, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena,1966 (Dover 2002).

I D. MIHALAS and B. W. MIHALAS, Foundations of Radiation Hydrodynamics, 1984 (Dover 1999).

I R.P. DRAKE, High-Energy-Density Physics, Springer, 2006.

Grey VEF/AMR, ion/electron, calculations:I M. W. SINCELL, M. GEHMEYR, and D. MIHALAS, two articles in Shock Waves, 1999.

Theory and Solutions:I Equilibrium: S. BOUQUET, R. TEYSSIER, and J. P. CHIEZE, Astrophysical Journal Supplement Series, 2000.I Equilibrium: R. B. LOWRIE and R. M. RAUENZAHN, Shock Waves, 2007.

I Nonequilibrium: R. B. LOWRIE and J. D. EDWARDS, Shock Waves, 2008.

Extended thermodynamics:I W. Weiss, “Structure of Shock Waves,” in Rational Extended Thermodynamics, Müller & Ruggeri (eds.), Springer,

1998.


Inviscid Hydrodynamic Shocks

Shock Jump

T1

T0

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1x

0

1

2

3

4

5

T

Temperature


Subcritical Radiative Shocks (Tp < T1)

Tp

Ts Relaxation Region

Precursor

Overall ShockJump

Material and Radiation Temperatures

T1

T0

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01x

1

1.2

1.4

1.6

1.8

2

2.2

T,Tr

TTr

Hydro Shock


Supercritical Radiative Shocks (Tp = T1)

-0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015x

1

2

3

4

5

6

7

8

9

10

11

12

T,Tr

TTr

Tp=T1

Ts

{Hydro Shock(Zel’dovich Spike)

Precursor

T1

T0


Zel’dovich Spike

-5e-05 0 5e-05 0.0001 0.00015 0.0002x

8

9

10

11

T,Tr

TTr

Tp=T1

Ts

Hydro Shock(Zel’dovich Spike)

T1

Relaxation Region


Outline




4 Sample Solutions

5 Code Comparison


Grey nonequilibrium diffusion in 1 slide1-D planar (slab) geometry

First two moments of radiative transfer equation:

∂tE + ∂xF = cσa[B(T )− E ]− σt

c

(F − 4

3vE)

v

1c2∂tF + ∂xP = −σt

c

(F − 4

3vE)

Nonequilibrium diffusion sets P = E/3 and drops ∂tF to yield

∂tE +43∂x(vE)− ∂x

(c

3σt∂xE

)= cσa[B(T )− E ] +

13

v∂xE

Underlined terms often dropped in “low-energy density” approximation.


Euler Coupled with Grey Nonequilibrium DiffusionNondimensional form, 1-D planar (slab) geometry

Most of this talk will be about solutions of

∂tρ+ ∂x(ρv) = 0,

∂t(ρv) + ∂x

(ρv2 + p +

13P0Tr

4)

= 0,

∂t(ρE) + ∂x [v(ρE + p)] = P0σa(Tr4 − T 4)− 1

3P0v∂xTr

4,

∂t(ρE + P0Tr4) + ∂x

[v(ρE + p +

43P0Tr

4)]

= P0∂x(κ∂xTr4),

where E = e + 12v2 and for a γ-law EOS

p =ρTγ, e =

Tγ(γ − 1)

, P0 =aRT 4

0γp0

≈ rad. pressuremat. pressure

.

Embedded discontinuities are hydrodynamic [email protected] LA-UR 09-01841 IPAM 2009 13 / 52

Equilibrium Diffusion Limit

We’ll also discuss “1T” solutions. Optically thick limit, Tr → T , and oursystem reduces to

∂tρ+ ∂x(ρv) = 0,

∂t(ρv) + ∂x

(ρv2 + p +

13P0T 4

)= 0,

∂t(ρE + P0T 4) + ∂x

[v(ρE + p +

43P0T 4

)]= P0∂x(κ∂xT 4).

Euler equations with nonlinear heat conduction and a modifiedequation-of-state.For small-P0, heat conduction may dominate.Embedded discontinuities are isothermal shocks.


Problem statement

Assume that far from shock, Tr = T(“thick–thick”).Galilean invariant, so use steady frame.

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01x

1

1.2

1.4

1.6

1.8

2

2.2

T,Tr

TTr

Given: The value γ andI Pre-shock state (x → −∞): ρ0, T0, Tr 0 = T0.I Non-dimensional constants: P0 andM0 (≡ v0/a0, Mach number).I Functions σa(ρ,T ) and κ(ρ,T ) (κ = c/3σt ).

Calculate: ρ(x), v(x), T (x), and Tr (x).Optional: Transform back to a frame where shock is moving.


Outline




4 Sample Solutions

5 Code Comparison


Euler Coupled with Grey Nonequilibrium DiffusionNondimensional form, 1-D planar (slab) geometry

Seek traveling-wave solutions of

∂tρ+ ∂x(ρv) = 0,

∂t(ρv) + ∂x

(ρv2 + p +

13P0Tr

4)

= 0,

∂t(ρE) + ∂x [v(ρE + p)] = P0σa(Tr4 − T 4)− 1

3P0v∂xTr

4,

∂t(ρE + P0Tr4) + ∂x

[v(ρE + p +

43P0Tr

4)]

= P0∂x(κ∂xTr4),


Overall Jump Relation

Integrate conservation equations from −∞ < x <∞ to give: ρvρv2 + p∗

(ρE∗ + p∗)v

0

=

ρvρv2 + p∗

(ρE∗ + p∗)v

1

,

where

p∗ = p +13P0T 4, e∗ = e +

1ρP0T 4, E∗ = e∗ +

12

v2.

Get ninth-order polynomial in T1; see Bouquet et al (2000).Same procedure used for any radiation model.


Hydro Shock Relations

At a discontinuity separating state-p andstate-s:

ρvρv2 + p

(ρE + p)v−κ∂xTr

4 + 43vTr

4

p

=

ρv

ρv2 + p(ρE + p)v

−κ∂xTr4 + 4

3vTr4

s

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01x

1

1.2

1.4

1.6

1.8

2

2.2

T,θ

Tθ

First 3 equations: Standard hydro jump conditions. Holds for anyradiation model.Last equation: Continuity of Eulerian frame radiation flux.


Reduced Equations

4 PDEs reduce to 2 ODEs. We use Mach number (M) as theindependent variable:

dxdM

=3M0(M2 − 1)ρβ

P0,

dTdM

= (γ − 1)β[4M0Tr

3Tr′ + (γM2 − 1)r

],

where β(T ,M) and r(T ,M) are known functions, and

ρ(T ,M) =M0

M√

T, Tr (T ,M) =

1γP0

[Km − 3γ

M20

ρ(T ,M)− 3Tρ(T ,M)

]

If Tr ≈ const., thenM = 1/√γ corresponds to maximum T

(isothermal sonic point – ISP).


Overview of Solution Procedure

1 Compute post-shock state (x →∞): Find root of a ninth-orderpolynomial in T1.

2 End states are typically saddle points. Solve two IVPs:I Find precursor region: Integrate ODEs fromM =M0 toM = 1.I Find relaxation region: Integrate ODEs fromM =M1 toM = 1.

3 If the two ODE solutions do not match atM = 1, then shift thesolutions such that they are connected with a hydro shock.

See Lowrie & Edwards (Shock Waves, 2008) for the details.

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01x

1

1.2

1.4

1.6

1.8

2

2.2

T,Tr

TTr


Outline




4 Sample Solutions

5 Code Comparison


Solution Regimes

Radiative shock solutions may be characterized by the following:Is there an embedded hydro shock?

I If no, all variables continuous.I If yes,

F Subcritical (Tp < T1).F Supercritical (Tp ≈ T1).

Is there an isothermal sonic point (ISP;M = 1/√γ)?

I If yes, then there is a temperature (Zel’dovich) spike =⇒ Tmax > T1.I Lowrie & Edwards (2008) derive an algebraic expression for Tmax

that is very accurate.

May have a hydro shock without a spike.May have a spike without a hydro shock.


Isothermal Shock (Equilib.) ⇔ Spike (Non-equilib.)γ = 5/3. See Lowrie & Rauenzahn (2007).

10-4 10-3 10-2 10-1 100 101

P0 ~ (radiation pressure / material pressure)0

0

10

20

30

40

50

M0

Isothermal Shock / Spike Region


M0 = 1.05 solutionP0 = 10−4, σa = 106, κ = 1, γ = 5/3; subcritical; no embedded shock or isothermalsonic point (ISP).

-0.015 -0.01 -0.005 0 0.005 0.01 0.015x

1

1.01

1.02

1.03

1.04

1.05

TrTT (E-D)

Temperature

-0.015 -0.01 -0.005 0 0.005 0.01 0.015x

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

ρρ (E-D)

Density


M0 = 1.2 solutionSubcritical; shock, but no ISP.

-0.01 -0.005 0 0.005 0.01x

1

1.05

1.1

1.15

1.2

TrTT (E-D)

Temperature

-0.01 -0.005 0 0.005 0.01x

1

1.05

1.1

1.15

1.2

1.25

1.3

ρρ (E-D)

Density


M0 = 2 solutionSubcritical; ISP coincident with shock.

-0.01 -0.005 0 0.005 0.01x

1

1.25

1.5

1.75

2

2.25

TrTT (E-D)ISP

Temperature

-0.01 -0.005 0 0.005 0.01x

1

1.25

1.5

1.75

2

2.25

2.5

ρρ (E-D)ISP

Density


M0 = 3 solutionSubcritical; ISP downstream of shock.

-0.01 -0.005 0 0.005x

1

1.5

2

2.5

3

3.5

4

4.5

TrTT (E-D)ISP

Temperature

-0.01 -0.005 0 0.005x

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

3.25

ρρ (E-D)ISP

Density


M0 = 3 solutionSpike region.

-0.0005 0 0.0005 0.001 0.0015 0.002x

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

TrTT (E-D)ISP

Temperature

-0.0005 0 0.0005 0.001 0.0015 0.002x

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

3.25

ρρ (E-D)ISP

Density


M0 = 5 solutionSupercritical; ISP downstream of shock.

-0.04 -0.03 -0.02 -0.01 0x

0

2

4

6

8

10

TrTT (E-D)ISP

Temperature

-0.04 -0.03 -0.02 -0.01 0x

1

1.5

2

2.5

3

3.5

4

ρρ (E-D)ISP

Density


M0 = 5 solution (continued)Zel’dovich spike region

-1 0 1 2x * 104

8

8.5

9

9.5

10

10.5

11

TrTT (E-D)ISP

Temperature

-1 0 1 2x * 104

1

1.5

2

2.5

3

3.5

4

ρρ (E-D)ISP

Density


Isothermal Shock (Equilib.) ⇔ Spike (Non-equilib.)γ = 5/3. See Lowrie & Rauenzahn (2007).

10-4 10-3 10-2 10-1 100 101

P0 ~ (radiation pressure / material pressure)0

0

10

20

30

40

50

M0

Isothermal Shock / Spike Region


M0 = 27 solutionSupercritical; ISP downstream of shock. NOTE OVER COMPRESSION.

-0.25 -0.2 -0.15 -0.1 -0.05 0x

0

10

20

30

40

50

60

70

TrTT (E-D)ISP

Temperature

-0.25 -0.2 -0.15 -0.1 -0.05 0x

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

ρρ (E-D)ISP

Density


M0 = 27 solution (continued)Zel’dovich spike region

-1 0 1 2 3 4 5 6x * 106

58

59

60

61

62

63

64

TrTT (E-D)ISP

Temperature

-1 0 1 2 3 4 5 6x * 106

3

3.5

4

4.5

5

5.5

6

ρρ (E-D)ISP

Density


M0 = 30 solutionNo shock, but still an ISP!

-0.25 -0.2 -0.15 -0.1 -0.05 0x

0

10

20

30

40

50

60

70

TrTT (E-D)ISP

Temperature

-0.25 -0.2 -0.15 -0.1 -0.05 0x

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

ρρ (E-D)ISP

Density


M0 = 30 solution (continued)Spike region

1.425 1.43 1.435x * 103

62

63

64

65

66

67

TrTT (E-D)ISP

Temperature

1.425 1.43 1.435x * 103

3.5

4

4.5

5

5.5

6

6.5

7

ρρ (E-D)ISP

Density


M0 = 50 solutionNo shock or ISP

-0.25 -0.2 -0.15 -0.1 -0.05 0x

0

10

20

30

40

50

60

70

80

90

TrTT (E-D)

Temperature

-0.25 -0.2 -0.15 -0.1 -0.05 0x

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

ρρ (E-D)

Density


Outline




4 Sample Solutions

5 Code Comparison


Code Comparison Test Cases

Hydrogen gas, γ = 5/3Bremsstrahlung absorption model (Zel’dovich & Razier; assumefully ionized):

σa(ρ,T ) = σa,0ρ2T−7/2

Thomson scattering:σs(ρ) = σs,0ρ

ρ0 = 1 g/ccCases:

1 T0 = 10 eV,M0 = 10, subcritical2 T0 = 100 eV,M0 = 5, supercritical3 T0 = 100 eV,M0 = 45, no embedded hydro shock

Comparison of relaxation rates shows that 3T effects should besmall.Compare with a finite-volume, Godunov-based AMR code(RAGE).


Two Initialization Methods

1 Initialize with exact shock profile.I How well can the code propagate the shock and maintain the

profile?I Pros: Boundary conditions not an issue.I Cons: Requires code initialize from exact solution.I This method was used for all error-norm calculations.

2 Piston problem.I Black-body piston moving at v1, radiating at Tr = T1.I Given enough time, the resulting shock should match the

semi-analytic solution.I Pros: Easy setup; better test.I Cons: Requires “large” solution domain; boundary conditions

become an issue.

With either method, you should propagate the shock long enough toestablish a self-similar profile.


T0 = 10 eV,M0 = 10 Sample Results∆x = 50 µm, propagated 1 cm, T1 ≈ 321 eV, Tmax ≈ 419 eV

0.1 0.2 0.3 0.4 0.5x (cm)

0

5

10

15

20

25

30

35

40

45(T

, Tr) /

T0

T (Exact)Tr (Exact)T (RAGE)Tr (RAGE)


T0 = 10 eV,M0 = 10 Error ConvergenceExact solution computed without flux limiter.

0.0001 0.001 0.01∆x (cm)

0.01

0.1

1L 1(T

-Tex

act)

Flux Limiter OffFlux Limiter OnSlope 1


T0 = 10 eV,M0 = 10, Piston Problem InitializationLegend indicates distance exact profile has moved. Solutions shifted to align.∆x = 50 µ m.

0.1 0.2 0.3 0.4 0.5x (cm)

0

5

10

15

20

25

30

35

40

45T

/ T0

Exact0 cm1 cm2 cm5 cm10 cm


Profile vs. Piston Initialization

0.1 0.2 0.3 0.4 0.5x (cm)

0

5

10

15

20

25

30

35

40

45

T / T

0

T (Exact)T (Profile)T (Piston)


T0 = 100 eV,M0 = 5 Sample ResultsAMR (8 levels): ∆xmax = 5 cm, ∆xmin ≈ 391 µm, propagated 230 cm, T1 ≈ 857 eV,Tmax ≈ 1.08 keV

24 26 28 30 32 34 36 38 40 42 44 46 48 50 52x (cm)

0

1

2

3

4

5

6

7

8

9

10

11

12(T

, Tr) /

T0



T0 = 100 eV,M0 = 5 Spike Region

49.6 49.8 50 50.2 50.4 50.6 50.8 51x (cm)

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8

11(T

, Tr) /

T0

T (Exact)Tr (Exact)T (10 levels)Tr (10 levels)T (8 levels)Tr (8 levels)T (6 levels)Tr (6 levels)


T0 = 100 eV,M0 = 5 Refinement Distribution8 levels =⇒ 814 cells; 10 levels =⇒ 528 cells (10-level mesh derefined in precursor).

20 25 30 35 40 45 50 55x

0

2

4

6

8

10

12

T/T 0

4

5

6

7

8

9

10

11R

efin

emen

t Lev

elMax 10 levelsMax 8 levels


Refinement and Error DistributionEven though 10-level mesh derefined in 27 / x / 49, it is more accurate.

20 25 30 35 40 45 50 55x

0.0001

0.001

0.01

0.1

|T -

T exac

t|

4

5

6

7

8

9

10

11R

efin

emen

t Lev

elMax 10 levelsMax 8 levels


T0 = 100 eV,M0 = 45 Sample Results∆x = 1.25 cm, propagated 20 m, T1 ≈ 8.36 keV, Tmax ≈ 8.48 keV

50 100 150 200 250 300x (cm)

0

10

20

30

40

50

60

70

80

90(T

, Tr) /

T0



T0 = 100 eV,M0 = 45 Spike RegionRAGE results with ∆x = 1.25 cm/2N , with N = 0, 1, 2, 3

280 290 300 310 320x (cm)

82

83

84

85(T

, Tr) /

T0



T0 = 100 eV,M0 = 45 ConvergenceA hard-coded tolerance was limiting the convergence rate.

0.1 1∆x (cm)

0.001

0.01

0.1L 1(T

- T ex

act)

Small ToleranceDefault ToleranceSlope 1


Summary

The radiative shock solutions test several issues:I Fully-coupled radiation hydrodynamicsI Embedded shock problems =⇒ ability to capture hydro shocksI Smooth problems =⇒ ability to attain theoretical convergence rateI Refinement criteria for AMR (solutions are multiscale)I Other radiation models

Future workI Separate ion/electron temperaturesI More advanced radiation models


the structure of radiative shockshelper.ipam.ucla.edu/publications/ktws1/ktws1_8270.pdfeuler...

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