M. Vandenboomgaerde* and C. AymardCEA, DAM, DIF

IWPCTM12, Moscow, 12-17 July 2010 01/14[1] Submitted to Phys. Fluids * marc.vandenboomgaerde@cea.fr

Analytical theory for planar shock wave focusing

through perfect gas lens [1]

Spherical shock waves (s.w.) and hydrodynamics instabilities are involved

in various phenomena :

Lithotripsy Astrophysics Inertial confinement fusion

(ICF)

There is a strong need for convergent shock wave experiments

A few shock tubes are fully convergent : AWE, Hosseini Most shock tubes have straight test section Some experiments have been done by adding convergent test section

IWPCTM12, Moscow, 12-17 July 2010 02/14[2] Holder et al. Las. Part. Beams 21 p. 403 (2003) [3] Mariani et al. PRL 100, 254503 (2008) [4] Bond et al. J. Fluid Mech. 641 p. 297 (2009)

AWE shock tube [2] IUSTI shock tube [3] GALCIT shock tube[4]

Convergent shock waves

[5] Phys. Fluids 22, 041701 (2010) [6] Phys. Fluids 18, 031705 (2006) IWPCTM12, Moscow, 12-17 July 2010 03/14

Zhigang Zhai et al. [5]

Shape the shock tube to make the incident s.w. convergent

The curvature of the tube depends on the initial conditions (~one shock tube / Mach

number) Theory, experiments and simulations are 2D

Dimotakis and Samtaney [6]

Gas lens technique : the transmitted s.w. becomes convergent

The shape of the lens depends on the initial conditions (~one interface / Mach number) The shape is derived iteratively and seems to be an ellipse Derivation for a s.w. going from light to heavy gas only Theory and simulations are 2D

IMAGE Zhai

IMAGE Dimotakis

Efforts have been made to morph a planar shock wave into a cylindrical one

Present work : a generalized gas lens theory

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The gas lens technique theory is revisited and simplified

Exact derivations for 2D-cylindrical and 3D-spherical geometries Light-to-heavy and heavy-to-light configurations

Validation of the theory

Comparisons with Hesione code simulations

Applications

Stability of a perturbed convergent shock wave Convergent Richtmyer-Meshkov instabilities

Conclusion and future works

Bounds of the theory

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Theoretical assumptions

Perfect and inviscid gases Regular waves

Dimensionality All derivations can be done in the symmetry plane (Oxy)

2D- cylindrical geometry 3D- spherical geometry

The polar coordinate system with the pole O will be used

Boundary conditions As the flow is radial, boundaries are streamlines

Derivation using hydrodynamics equations (1/3)

The transmitted shock wave must be circular in (Oxy) and its center is O

The pressure behind the shock must be uniform

Eqs (1) and (2) must be valid regardless of

IWPCTM12, Moscow, 12-17 July 2010 06/14

The transmitted shock wave must be circular in (Oxy) and its center is O

The pressure behind the shock must be uniform

Eqs (1) and (2) must be valid regardless of

Equation of a conic with eccentricity and pole O in polar coordinates IWPCTM12, Moscow, 12-17 July 2010 06/14

Derivation using hydrodynamics equations (2/3)

As we now know that C is a conic, it can read as :

All points of the circular shock front must have the same radius at the same time

Eqs. (4) and (5) show that the eccentricity of the conic equals

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Light-t

o-heavy

Heavy-to-lig

ht

Derivation using hydrodynamics equations (3/3)

It has been demonstrated that :

The same shape C generates 2D or 3D lenses C is a conic The eccentricity is equal to Wt/Wi => C is an ellipse in the light-to-heavy (fast-slow) configuration and an hyperbola, otherwise. The center of focusing is one of the foci of the conic Limits are imposed by the regularity of the waves => cr => cr

Derivation through an analogy with geometrical optics

Equation (3) can be rearranged as :

This is the refraction law (Fresnel’s law) with shock velocity as index

Optical lenses are conics !

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IMAGE Principles of Optics

To summarize … and another derivation

Hesione code

ALE package Multi-material cells The pressure jump through the incident shock wave is resolved by 20 cells Mass cell matching at the interface

Initial conditions of the simulations

First gas is Air Mi = 1.15 2nd gas is SF6 or He => e = 0.42 or e = 2.75 Height of the shock tube = 80 mm w = 30o

Rugby hohlraum is a natural way to increase P2

Numerical simulations have been performed with Hesione code

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Morphing of the incident shock wave

Focusing and rebound of the transmitted shock wave (t.s.w.)

Validation in the light-to-heavy (fast-slow) case

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The t.s.w. is circular in 2D as in 3DThe t.s.w. stay circular while focusing Spherical s.w. is faster than cylindrical s.w. P = 41 atm is reached in 3D near focusing P = 9.6 atm is reached in 2D near focusing Shock waves stay circular after rebound

Wedge

Cone

Wedge

Cone

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Validation in the heavy-to-light (slow-fast) case

Morphing of the incident shock wave

Focusing and rebound of the transmitted shock wave (t.s.w.)

Wedge

Cone

Wedge

Cone

The t.s.w. is circular in 2D as in 3DThe t.s.w. stay circular while focusing Spherical s.w. is faster than cylindrical s.w. P = 6.9 atm is reached in 3D near focusing P = 2.9 atm is reached in 2D near focusing Shock waves stay circular after rebound

The stability of a pertubed shock wave has been probed in convergent geometry

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We perturb the shape of the lens in order to generate a perturbed t.s.w.

with a0 = 2.871 10-3m and m = 9

Focusing and rebound of the perturbed t.s.w.

The t.s.w. is perturbed in 2D and in 3D

The t.s.w. stabilizes while focusing

Near the collapse, the s.w. becomes circular

These results are consistent with theory [7]

The acoustic waves do not perturb s.w.

Shock waves stay circular and stable after the rebound

[7] J. Fusion Energy 14 (4), 389 (1995)

Richtmyer-Meshkov instability in 2D cylindrical geometry

We add a perturbed inner interface : Air/SF6/Air configuration

with a0 = 1.665 10-3m and m = 12

Richtmyer-Meshkov instability due to shock and reshock

A RM instability occurs at the 1rst passage of the shock through the perturbed interface

The reshock impacts a non-linear interface

Even if the interface is stopped, the instability keeps on growing

High non-linear regime is reached (mushroom structures)

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We have established an exact derivation of the gas lens tehnique The shape of the lens is a conic Its eccentricity is Wt/Wi

The conic is an ellipse in the light-to-heavy case, and hyperbola otherwise The focus of the convergent transmitted shock wave is one of the foci of the

conic

The same shape generates 2D and 3D gas lens

These results have been validated by comparisons with Hesione numerical simulations

The transmitted shock wave is cylindrical or spherical The acoustic waves do not perturb the shock wave The shock wave remains circular after its focusing

This technique allows to study hydrodynamics instabilities in convergent geometries

Numerical simulations show that the RM non-linear regime can be reached Implementation in the IUSTI conventional shock tube is under consideration : a

new test section and new stereolithographed grids [8] for the interface are needed

Inertial Confinement Fusion applications ? e=Wt/Wi stays finite in ICF targets Doped plastic can prevent the radiation wave to perturb the hydrodynamic shock

wave

Conclusion and future works

[8] Mariani et al. P.R.L. 100, 254503 (2008) IWPCTM12, Moscow, 12-17 July 2010 14/14