m. vandenboomgaerde* and c. aymard cea, dam, dif iwpctm12, moscow, 12-17 july 2010 01/14 [1]...
TRANSCRIPT
M. Vandenboomgaerde* and C. AymardCEA, DAM, DIF
IWPCTM12, Moscow, 12-17 July 2010 01/14[1] Submitted to Phys. Fluids * [email protected]
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
IWPCTM12, Moscow, 12-17 July 2010 07/14
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