## This content will become publicly available on August 28, 2020

# Piston driven converging shock waves in a stiffened gas

## Abstract

We present that the problem of a one-dimensional (1D) cylindrically or spherically symmetric shock wave converging into an inviscid, ideal gas was first investigated by Guderley[Starke kugelige und zylinrische verdichtungsstosse in der nahe des kugelmitterpunktes bzw. Der zylinderachse,” Luftfahrtforschung 19, 302 (1942)]. In the time since, many authors have discussed the practical notion of how Guderley-like flows might be generated. One candidate is a constant velocity, converging “cylindrical or spherical piston,” giving rise to a converging shock wave in the spirit of its classical, planar counterpart. A limitation of pre-existing analyses along these lines is the restriction to flows in materials described by an ideal gas equation of state (EOS) constitutive law. This choice is of course necessary for the direct comparison with the classical Guderley solution, which also features an ideal gas EOS. However, the ideal gas EOS is limited in its utility in describing a wide variety of physical phenomena and, in particular, the shock compression of solid materials. This work is thus intended to provide an extension of previous work to a nonideal EOS. The stiff gas EOS is chosen as a logical starting point due to not only its close resemblance to the ideal gas lawmore »

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1570638

- Report Number(s):
- LA-UR-18-30293

Journal ID: ISSN 1070-6631

- Grant/Contract Number:
- 89233218CNA000001

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physics of Fluids

- Additional Journal Information:
- Journal Volume: 31; Journal Issue: 8; Journal ID: ISSN 1070-6631

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Ramsey, Scott D., and Baty, Roy S. Piston driven converging shock waves in a stiffened gas. United States: N. p., 2019.
Web. doi:10.1063/1.5109097.
```

```
Ramsey, Scott D., & Baty, Roy S. Piston driven converging shock waves in a stiffened gas. United States. doi:10.1063/1.5109097.
```

```
Ramsey, Scott D., and Baty, Roy S. Wed .
"Piston driven converging shock waves in a stiffened gas". United States. doi:10.1063/1.5109097.
```

```
@article{osti_1570638,
```

title = {Piston driven converging shock waves in a stiffened gas},

author = {Ramsey, Scott D. and Baty, Roy S.},

abstractNote = {We present that the problem of a one-dimensional (1D) cylindrically or spherically symmetric shock wave converging into an inviscid, ideal gas was first investigated by Guderley[Starke kugelige und zylinrische verdichtungsstosse in der nahe des kugelmitterpunktes bzw. Der zylinderachse,” Luftfahrtforschung 19, 302 (1942)]. In the time since, many authors have discussed the practical notion of how Guderley-like flows might be generated. One candidate is a constant velocity, converging “cylindrical or spherical piston,” giving rise to a converging shock wave in the spirit of its classical, planar counterpart. A limitation of pre-existing analyses along these lines is the restriction to flows in materials described by an ideal gas equation of state (EOS) constitutive law. This choice is of course necessary for the direct comparison with the classical Guderley solution, which also features an ideal gas EOS. However, the ideal gas EOS is limited in its utility in describing a wide variety of physical phenomena and, in particular, the shock compression of solid materials. This work is thus intended to provide an extension of previous work to a nonideal EOS. The stiff gas EOS is chosen as a logical starting point due to not only its close resemblance to the ideal gas law but also its relevance to the shock compression of various liquid and solid materials. Using this choice of EOS, the solution of a 1D planar piston problem is constructed and subsequently used as the lowest order term in a quasi-self-similar series expansion intended to capture both curvilinear and nonideal EOS effects. The solution associated with this procedure provides correction terms to the 1D planar solution so that the expected accelerating shock trajectory and nontrivially varying state variable profiles can be obtained. This solution is further examined in the limit as the converging shock wave approaches the 1D curvilinear origin. Lastly, given the stiff gas EOS is not otherwise expected to admit a Guderley-like solution when coupled to the inviscid Euler equations, this work thus provides the semianalytical limiting behavior of a flow that cannot be otherwise captured using self-similar analysis.},

doi = {10.1063/1.5109097},

journal = {Physics of Fluids},

number = 8,

volume = 31,

place = {United States},

year = {2019},

month = {8}

}

Works referenced in this record:

##
Self-Similar Solutions for Converging Shocks and Collapsing Cavities

journal, April 1981

- Lazarus, Roger B.
- SIAM Journal on Numerical Analysis, Vol. 18, Issue 2

##
An analytic description of converging shock waves

journal, January 1998

- Chisnell, R. F.
- Journal of Fluid Mechanics, Vol. 354

##
A self-similar isochoric implosion for fast ignition

journal, August 2007

- Clark, D. S.; Tabak, M.
- Nuclear Fusion, Vol. 47, Issue 9

##
Double-detonation supernovae of sub-Chandrasekhar mass white dwarfs

journal, October 2007

- Fink, M.; Hillebrandt, W.; Röpke, F. K.
- Astronomy & Astrophysics, Vol. 476, Issue 3

##
The Guderley problem revisited

journal, February 2012

- Ramsey, Scott D.; Kamm, James R.; Bolstad, John H.
- International Journal of Computational Fluid Dynamics, Vol. 26, Issue 2

##
Second‐type self‐similar solutions to the strong explosion problem

journal, April 1993

- Waxman, Eli; Shvarts, Dov
- Physics of Fluids A: Fluid Dynamics, Vol. 5, Issue 4

##
On the question of universality of imploding shock waves

journal, August 2008

- Hornung, H. G.; Pullin, D. I.; Ponchaut, N. F.
- Acta Mechanica, Vol. 201, Issue 1-4

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

journal, September 2017

- Ramsey, Scott D.; Lilieholm, Jennifer F.
- Journal of Verification, Validation and Uncertainty Quantification, Vol. 2, Issue 3

##
On the Existence of Self-Similar Converging Shocks in Non-Ideal Materials

journal, August 2017

- Boyd, Z. M.; Ramsey, S. D.; Baty, R. S.
- The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 70, Issue 4

##
Converging shock flows for a Mie-Grüneisen equation of state

journal, April 2018

- Ramsey, Scott D.; Schmidt, Emma M.; Boyd, Zachary M.
- Physics of Fluids, Vol. 30, Issue 4

##
The Riemann problem for fluid flow of real materials

journal, January 1989

- Menikoff, Ralph; Plohr, Bradley J.
- Reviews of Modern Physics, Vol. 61, Issue 1

##
Initial Behavior of a Spherical Blast

journal, November 1952

- McFadden, J. A.
- Journal of Applied Physics, Vol. 23, Issue 11

##
On the Propagation and Structure of the Blast Wave, I

journal, September 1953

- Sakurai, Akira
- Journal of the Physical Society of Japan, Vol. 8, Issue 5

##
On the Propagation and Structure of a Blast Wave, II

journal, March 1954

- Sakurai, Akira
- Journal of the Physical Society of Japan, Vol. 9, Issue 2

##
A simplified analysis of spherical and cylindrical blast waves

journal, August 1961

- Friedman, Manfred P.
- Journal of Fluid Mechanics, Vol. 11, Issue 1

##
Convergence of Strong Shock in a Van der Waals Gas

journal, January 2006

- Arora, Rajan; Sharma, V. D.
- SIAM Journal on Applied Mathematics, Vol. 66, Issue 5

##
Solutions of the Noh problem for various equations of state using LIE groups

journal, January 2000

- Axford, Roy A.
- Laser and Particle Beams, Vol. 18, Issue 1

##
Symmetries of the gas dynamics equations using the differential form method

journal, November 2017

- Ramsey, Scott D.; Baty, Roy S.
- Journal of Mathematical Physics, Vol. 58, Issue 11

##
Analysis of self-similar problems of imploding shock waves by the method of characteristics

journal, January 1983

- Nakamura, Y.
- Physics of Fluids, Vol. 26, Issue 5

##
On the motion of piston in a polytropic gas

journal, January 1977

- Kozmanov, M. Iu.
- Journal of Applied Mathematics and Mechanics, Vol. 41, Issue 6