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Sedov 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

Jeremiah Moskal, working with Dr. Jared Workman, has ported/refactored the instrument into python, avaliable in this zip file.

von Neumann (1941), Taylor (1941) and Sedov (1946) independently derived a self-similar description of the evolution of the blast wave arising from a powerful explosion in a cold, uniform density background (also see Bethe et al 1947). They treated the explosion as an instantaneous release of energy at a point and assumed that the background material through which the expanding blast sweeps behaves as an 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. In this paper, this paper, and this paper, we also discuss the analytic and numerical solutions for the Sedov problem.

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 to become 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. Expressions and techniques for the removal of the apparent singularities are described.

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. The constant density, spherically symmetric Sedov blast wave stalwart test case in the 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. Jim Kamm's opus and David Book's irreverant paper on generating Sedov solutions are worth reading.

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


Plots of the Sedov functions
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 case
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cylindrical case
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spherical case1
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spherical case2
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spherical case3
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spherical case4
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spherical case5
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spherical case6
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entropies
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alpha 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 mesh:
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density 240x240 cells
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density 480x480 cells
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density 960x960 cells
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density error 240x240 cells
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density error 480x480 cells
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density error 960x960 cells
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density asymmetry 240x240
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density asymmetry 480x480
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density asymmetry 960x960


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