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Research Papers

The Sedov Blast Wave as a Radial Piston Verification Test

[+] Author and Article Information
Clark Pederson, Bart Brown

XCP-Computational Physics Division,
Los Alamos National Laboratory,
Los Alamos, NM 87545

Nathaniel Morgan

XCP-Computational Physics Division,
Los Alamos National Laboratory,
Los Alamos, NM 87545
e-mail: nmorgan@lanl.gov

1Corresponding author.

Manuscript received September 23, 2015; final manuscript received April 20, 2016; published online June 22, 2016. Editor: Ashley F. Emery.

J. Verif. Valid. Uncert 1(3), 031001 (Jun 22, 2016) (9 pages) Paper No: VVUQ-15-1042; doi: 10.1115/1.4033652 History: Received September 23, 2015; Revised April 20, 2016

The Sedov blast wave is of great utility as a verification problem for hydrodynamic methods. The typical implementation uses an energized cell of finite dimensions to represent the energy point source. This approximation can be avoided by directly finding the effects of the energy source as a boundary condition (BC). The proposed method transforms the Sedov problem into an outward moving radial piston problem with a time-varying velocity. A portion of the mesh adjacent to the origin is removed and the boundaries of this hole are forced with the velocities from the Sedov solution. This verification test is implemented on two types of meshes, and convergence is shown. The results from the typical initial condition (IC) method and the new BC method are compared.

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References

Figures

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Fig. 1

The axisymmetric (RZ) meshes used with the new verification test problem are shown: (a) box mesh and (b) radial mesh. Cell(s) are removed near the origin, and the velocity BCs are applied to the nodes exposed on the inner surface. For both meshes, the horizontal axis is R, and the vertical axis is Z.

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Fig. 2

A comparison of material velocities for particles at three different initial positions as a function of time. The velocity jump for a position of r = 0.02 cm is about 2 orders of magnitude larger than the jump for r = 0.2 cm.

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Fig. 3

Convergence rates of the four ODE solvers are presented. The forward Euler with logarithmic spacing, fourth-order Runge–Kutta with logarithmic spacing, fourth-order Runge–Kutta with quadratic spacing, and the BS23 adaptive time-stepping method were studied.

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Fig. 4

Comparison of density and velocity profiles generated by the BS23 solver with the analytic solution

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Fig. 5

The density and mesh are shown for the (a) IC and (b) BC approaches using a box mesh topology

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Fig. 6

The plots of density, pressure, velocity, and internal energy for various mesh resolutions are provided using a radial mesh. These results were generated using the new BC verification approach. The calculations are approaching the analytic solution as the mesh resolution increases. Convergence plots are provided in Figs. 7 and 8.

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Fig. 7

Convergence plots are shown for the (a) density and (b) pressure fields for the BC approach using a box mesh topology. The initial edge length is dx(t = 0). The holes are squares with lengths = 0.02 cm, 0.1 cm, and 0.2 cm, respectively. For the density, the orders of convergence for the different hole sizes were 0.77, 0.80, and 0.85. For pressure, the orders of convergence were 0.79, 0.82, and 0.83.

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Fig. 8

Convergence plots are shown for the (a) density and (b) pressure fields for the BC approach using a radial mesh topology. The initial edge length is dr(t = 0). The radii of the holes are 0.02 cm, 0.1 cm, and 0.2 cm, respectively. For the density, the orders of convergence for the different radii were 0.93, 1.10, and 1.08. For pressure, the orders of convergence were 1.00, 1.18, and 1.06.

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Fig. 9

Absolute value of the errors as a function of radius and angle, using the box mesh topology, a grid spacing of dx = 0.02 cm, and an initial hole size of 0.02 cm

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Fig. 10

Comparison of the IC and BC verification approaches using box and radial mesh topologies. The errors represented are (a) L1 density error and (b) L1 pressure error. The initial edge length for the box mesh is dx and for the radial mesh is dr. The hole size is 0.2 cm for the BC calculations. The order of convergence and correlation coefficients are provided in Tables3 and 4.

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Fig. 11

Absolute values of the errors are compared for the IC method and the BC method using the box mesh. For the BC method, results for hole sizes of 0.02 cm (red points) and 0.2 cm (blue points) are shown. The initial edge length of the mesh (dx) is 0.02 cm for all calculations shown.

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Fig. 12

Absolute values of the errors are compared for the IC method and the BC method using the radial mesh. For the BC method, results for hole sizes of 0.02 cm (red points) and 0.2 cm (blue points) are shown. The initial edge length of the mesh (dr) is 0.02 cm for all calculations shown.

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