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Analytical theory for planar shock wave
focusing through perfect gas lens 1
M. Vandenboomgaerde and C. Aymard CEA, DAM, DIF
1 Submitted to Phys. Fluids
marc.vandenboomgaerde_at_cea.fr
IWPCTM12, Moscow, 12-17 July 2010
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Convergent shock waves

• 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

AWE shock tube 2
IUSTI shock tube 3
GALCIT shock tube4
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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)
3

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

• 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

IMAGE Zhai
IMAGE Dimotakis
5 Phys. Fluids 22, 041701 (2010) 6
Phys. Fluids 18, 031705 (2006)
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Present work a generalized gas lens theory

• 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

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Bounds of the theory

• 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

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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 q
  gt

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Derivation using hydrodynamics equations (2/3)

• 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 q
  gt

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Derivation using hydrodynamics equations (3/3)

• 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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To summarize and another derivation

• 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 gt
• 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
  gt a lt acr gt q lt qcr
• Derivation through an analogy with geometrical
  optics
• Equation (3) can be rearranged as
• This is the refraction law (Fresnels law)
• with shock velocity as index

IMAGE Principles of Optics
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Numerical simulations have been performed with
Hesione code

• 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 gt e 0.42 or e 2.75
• Height of the shock tube 80 mm
• qw 30o

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

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

• 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
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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.)

• 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

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

• 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)
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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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Conclusion and future works

• 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

8 Mariani et al. P.R.L. 100, 254503 (2008)
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