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Sedov Blast Wave
Verification Problem

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Contact: F.X.Timmes
my one page vitae,
full vitae,
research statement, and
teaching statement.
This software instrument is released to the public under LA-CC-05-101: sedov3.f.zip. Jeremiah Moskal and Jared Workman have ported/refactored this instrument to python.

von Neumann (1941), Taylor (1941), Sedov (1946), and Bethe et al (1947) derived a self-similar description of the evolution of a blast wave arising from a point explosion in a cold, uniform density, ideal polytropic fluid. It is remarkable that this model yields exact, although algebraically complicated, analytical expressions for the fluid quantities.

Kamm, Bolstad (RIP), & Timmes describe the generation of robust numerical solutions for a Sedov blast wave propagating through a density gradient ρ=ρ0r in planar, cylindrical or spherical geometry for the standard, singular and vacuum cases. In the standard, case a nonzero solution extends from the shock to the origin, where the pressure is finite. In the singular case, a nonzero solution extends from the shock to the origin, where the pressure vanishes. In the vacuum case a nonzero solution extends from the shock to a boundary point, where the density vanishes.

The venerable Sedov problem might appear to be an old solved problem. However, there is a paucity of papers that discuss all possible families of real solutions, in all common geometries, and address all the removable singularities. Jim Kamm's opus and David Book's slightly irreverant paper on generating Sedov solutions are worth reading.

The constant density, spherically symmetric Sedov blast wave is a stalwart test case for verification of hydrodynamic codes. However, it is not a particularily difficult test for a modern hydrocode. In this paper we identify several new verification problems involving more challenging Sedov blast waves. Analytic and numerical solutions for verification purposes re discussed in this paper, this paper, and this paper.

Standard, singular and vacuum solutions:
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standard cases
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singular cases
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vacuum cases


Sedov functions
Four Sedov functions describe the spatial variation of density, material speed, and pressure with distance at any point in time. Although the Sedov functions are analytic, they appear singular for various combinations of parameters and lower limits of integration. All six of these singularities are removable; they are a consequence of the way the solution is formulated. Below are a few example plots of f (red; scaled material velocity), g (blue, scaled mass density), and h (green, scaled pressure) versus scaled distance λ; with γ and ω labeled.

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planar cases
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cylindrical cases
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spherical cases
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the energy integral α is not one!


Code verificaton efforts:
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shock down a density gradient
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analytical and numerical
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convergence study
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1D evolution movie


Density uniform 2D mesh:
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240x240 cells
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480x480 cells
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960x960 cells
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error 240x240 cells
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error 480x480 cells
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error 960x960 cells
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asymmetry 240x240
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asymmetry 480x480
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asymmetry 960x960


Density 2D adaptive mesh:
image
240x240 cells
image
480x480 cells
image
960x960 cells
image
error 240x240 cells
image
error 480x480 cells
image
error 960x960 cells
image
asymmetry 240x240
image
asymmetry 480x480
image
asymmetry 960x960