1. M. Vandenboomgaerde* and C. Aymard CEA, 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]

2. 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

3. [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

4. Present work : a generalized gas lens theory IWPCTM12, Moscow, 12-17 July 2010 04/14 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

5. Bounds of the theory IWPCTM12, Moscow, 12-17 July 2010 05/14 Theoretical assumptions Perfect and inviscid gases Regular waves Dimensionality All derivations can be done in the symmetry plane ( O xy) 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

6. Derivation using hydrodynamics equations (1/3) The transmitted shock wave must be circular in ( O xy) 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

7. The transmitted shock wave must be circular in ( O xy) 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)

8. 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 IWPCTM12, Moscow, 12-17 July 2010 07/14 Light-to-heavy Heavy-to-light Derivation using hydrodynamics equations (3/3)

9. It has been demonstrated that : The same shape C generates 2D or 3D lenses C is a conic The eccentricity is equal to W t /W i => 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 ! IWPCTM12, Moscow, 12-17 July 2010 08/14 IMAGE Principles of Optics To summarize … and another derivation

10. 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 M i = 1.15 2 nd gas is SF6 or He => e = 0.42 or e = 2.75 Height of the shock tube = 80 mm  w = 30 o Rugby hohlraum is a natural way to increase P 2 Numerical simulations have been performed with Hesione code IWPCTM12, Moscow, 12-17 July 2010 9/14

11. 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 IWPCTM12, Moscow, 12-17 July 2010 10/14 The t.s.w. is circular in 2D as in 3D The 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

12. IWPCTM12, Moscow, 12-17 July 2010 11/14 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 3D The 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

13. The stability of a pertubed shock wave has been probed in convergent geometry IWPCTM12, Moscow, 12-17 July 2010 12/14 We perturb the shape of the lens in order to generate a perturbed t.s.w. with a 0 = 2.871 10 -3 m 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)

14. Richtmyer-Meshkov instability in 2D cylindrical geometry We add a perturbed inner interface : Air/SF 6 /Air configuration with a 0 = 1.665 10 -3 m 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) IWPCTM12, Moscow, 12-17 July 2010 13/14

15. We have established an exact derivation of the gas lens tehnique The shape of the lens is a conic Its eccentricity is W t /W i 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=W t /W i 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