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Research Papers

Verification Assessment of Piston Boundary Conditions for Lagrangian Simulation of the Guderley Problem

[+] Author and Article Information
Scott D. Ramsey

Mem. ASME
Applied Physics,
Los Alamos National Laboratory,
PO Box 1663, MS T082,
Los Alamos, NM 87545
e-mail: ramsey@lanl.gov

Jennifer F. Lilieholm

Department of Physics,
University of Washington,
3910 15th Ave NE,
Seattle, WA 98195
e-mail: liliej@uw.edu

Manuscript received March 30, 2017; final manuscript received September 5, 2017; published online September 26, 2017. Assoc. Editor: Jeffrey E. Bischoff.The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States government purposes.

J. Verif. Valid. Uncert 2(3), 031001 (Sep 26, 2017) (14 pages) Paper No: VVUQ-17-1015; doi: 10.1115/1.4037888 History: Received March 30, 2017; Revised September 05, 2017

This work is concerned with the use of Guderley's converging shock wave solution of the inviscid compressible flow equations as a verification test problem for compressible flow simulation software. In practice, this effort is complicated by both the semi-analytical nature and infinite spatial/temporal extent of this solution. Methods can be devised with the intention of ameliorating this inconsistency with the finite nature of computational simulation; the exact strategy will depend on the code and problem archetypes under investigation. For example, scale-invariant shock wave propagation can be represented in Lagrangian compressible flow simulations as rigid boundary-driven flow, even if no such “piston” is present in the counterpart mathematical similarity solution. The purpose of this work is to investigate in detail the methodology of representing scale-invariant shock wave propagation as a piston-driven flow in the context of the Guderley problem, which features a semi-analytical solution of infinite spatial/temporal extent. The semi-analytical solution allows for the derivation of a similarly semi-analytical piston boundary condition (BC) for use in Lagrangian compressible flow solvers. The consequences of utilizing this BC (as opposed to directly initializing the Guderley solution in a computational spatial grid at a fixed time) are investigated in terms of common code verification analysis metrics (e.g., shock strength/position errors, global convergence rates). For the examples considered in this work, the piston-driven initialization approach is demonstrated to be a viable alternative to the more traditional, direct initialization approach.

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Figures

Grahic Jump Location
Fig. 1

Guderley density solution (k = 2, γ = 1.4, ρ0 = 1.0) as a function of r, evaluated at t = –0.2. At this time, the shock wave is moving to the left.

Grahic Jump Location
Fig. 2

Guderley density solution (k = 2, γ = 1.4, ρ0 = 1.0) as a function of r, evaluated at t = 0.2. At this time, the shock wave is moving to the right.

Grahic Jump Location
Fig. 3

Direct initialization of a similarity solution onto a computational spatial grid

Grahic Jump Location
Fig. 4

Piston initialization of a similarity solution onto a computational spatial grid

Grahic Jump Location
Fig. 5

Piston velocity versus time for the Guderley problem (k = 2, γ = 1.4)

Grahic Jump Location
Fig. 6

Piston trajectory (solid line) for the Guderley problem (k = 2, γ = 1.4), with initial condition given by Eq. (41). The dashed line denotes the shock trajectory.

Grahic Jump Location
Fig. 7

Guderley problem (k = 2, γ = 1.4) density simulation results for direct (top) and piston (bottom; the right edge of this plot denotes the piston location) initialization methods; t = –0.2 (converging shock)

Grahic Jump Location
Fig. 8

Guderley problem (k = 2, γ = 1.4) density simulation results for direct (top) and piston (bottom; the right edge of this plot denotes the piston location) initialization methods; t = –0.2 (diverging shock)

Grahic Jump Location
Fig. 9

Guderley problem (k = 2, γ = 1.4) density error simulation results for direct (top) and piston (bottom; the right edge of this plot denotes the piston location) initialization methods; t = –0.2 (converging shock)

Grahic Jump Location
Fig. 10

Guderley problem (k = 2, γ = 1.4) density error simulation results for direct (top) and piston (bottom; the right edge of this plot denotes the piston location) initialization methods; t = 0.2 (diverging shock)

Grahic Jump Location
Fig. 11

Guderley problem (k = 2, γ = 1.4) L1 error convergence results for shock position (rs) and magnitude (Ms). Clockwise from top left: rs at t = –0.2 (converging shock), rs at t = 0.2 (diverging shock), Ms at t = 0.2 (diverging shock), Ms at t = –0.2 (converging shock). Fit data are summarized in Tables 5 and 6.

Grahic Jump Location
Fig. 12

Guderley problem (k = 2, γ = 1.4) mass-weighted L1 error norm convergence results for flow variables. Clockwise from top left: direct method at t = –0.2 (converging shock), piston method at t = –0.2 (converging shock), piston method at t = 0.2(diverging shock), and direct method at t = 0.2 (diverging shock). Fit data are summarized in Tables 11 and 12.

Grahic Jump Location
Fig. 13

Piston velocity (top) and trajectory (bottom) for the Caramana–Whalen–Shashkov Guderley problem (k = 2, γ = 5/3). Solid lines result from an exact solution of Eqs. (28)(30), and (39) with 1/λ = 0.6883545, t0 = 0.75, and xs = –0.75; dashed lines result from use of Eq. (A1).

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