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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
* [email protected]
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 tube[4]
[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) IWPCTM12, Moscow, 12-17 July 2010
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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
IMAGE Zhai
 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 Dimotakis
 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
[5] Phys. Fluids 22, 041701 (2010)
[6] Phys. Fluids 18, 031705 (2006)
IWPCTM12, Moscow, 12-17 July 2010
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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
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Validation of the theory
 Comparisons with Hesione code simulations
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Applications
 Stability of a perturbed convergent shock wave
 Convergent Richtmyer-Meshkov instabilities
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Conclusion and future works
IWPCTM12, Moscow, 12-17 July 2010
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Bounds of the theory
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Theoretical assumptions
 Perfect and inviscid gases
 Regular waves
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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
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Boundary conditions
 As the flow is radial, boundaries are streamlines
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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 =>
IWPCTM12, Moscow, 12-17 July 2010
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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 =>
Equation of a conic with eccentricity
and pole O in polar coordinates
IWPCTM12, Moscow, 12-17 July 2010
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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
IWPCTM12, Moscow, 12-17 July 2010
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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 =>
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 => a < acr => q < qcr
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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 !
IMAGE Principles of Optics
IWPCTM12, Moscow, 12-17 July 2010
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Numerical simulations have been performed with Hesione code
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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
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Initial conditions of the simulations
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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
qw = 30o
IWPCTM12, Moscow, 12-17 July 2010
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Validation in the light-to-heavy (fast-slow) case
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Morphing of the incident shock wave
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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
IWPCTM12, Moscow, 12-17 July 2010
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Validation in the heavy-to-light (slow-fast) case
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Morphing of the incident shock wave
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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
IWPCTM12, Moscow, 12-17 July 2010
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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
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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)
IWPCTM12, Moscow, 12-17 July 2010
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Richtmyer-Meshkov instability in 2D cylindrical geometry
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We add a perturbed inner interface : Air/SF6/Air configuration
with a0 = 1.665 10-3m and m = 12
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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
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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
[8] Mariani et al. P.R.L. 100, 254503 (2008)
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