### Abstract

The equations of motion for spherically-symmetric, unsteady, inviscid, compressible flow are expressed in a form that enables accurate numerical solutions to be obtained using a one-dimensional formulation based on the method of characteristics. Unlike the corresponding equations for uniaxial unsteady flows, the spherically-symmetric equations necessarily include terms involving the reciprocal of the radius and, close to the radial origin, numerical integration is unreliable. Nevertheless, good accuracy is obtained over a wide range of radii, including regions inside the range where the pressure and acceleration are approximately in phase. The range of validity of the method is assessed by comparison with an analytical solution for a single pulse and the method is then used to predict the radiation of acoustic waves from the exit of a duct in which a pulse is propagating internally. A method of obtaining efficient solutions of flows containing both uniaxial and spherically-symmetric domains is then obtained. Interfaces between the domains are solved in a manner that ensures continuity of pressure and flowrate. The use of spherically-symmetric assumptions limits the range of three-dimensional domains that can be approximated, but the combination of the two forms of one-dimensional analysis makes highly efficient use of resources. Furthermore, the two-way coupling is a significant advantage over two-step methods that use independent solutions of the internal, uniaxial domain to provide prescribed boundary conditions for solutions of the external domain.

Original language | English |
---|---|

Pages (from-to) | 810-828 |

Number of pages | 19 |

Journal | Applied Mathematical Modelling |

Volume | 77 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2020 |

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### Keywords

- Method of characteristics
- Spherically-symmetric flow
- Vibrating sphere acoustics
- Wave radiation
- Wave reflection

### Cite this

*Applied Mathematical Modelling*,

*77*(1), 810-828. https://doi.org/10.1016/j.apm.2019.07.037

}

*Applied Mathematical Modelling*, vol. 77, no. 1, pp. 810-828. https://doi.org/10.1016/j.apm.2019.07.037

**Method of characteristics for transient, spherical flows.** / Vardy, Alan E.; Tijsseling, Arris S. (Corresponding author).

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Method of characteristics for transient, spherical flows

AU - Vardy, Alan E.

AU - Tijsseling, Arris S.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - The equations of motion for spherically-symmetric, unsteady, inviscid, compressible flow are expressed in a form that enables accurate numerical solutions to be obtained using a one-dimensional formulation based on the method of characteristics. Unlike the corresponding equations for uniaxial unsteady flows, the spherically-symmetric equations necessarily include terms involving the reciprocal of the radius and, close to the radial origin, numerical integration is unreliable. Nevertheless, good accuracy is obtained over a wide range of radii, including regions inside the range where the pressure and acceleration are approximately in phase. The range of validity of the method is assessed by comparison with an analytical solution for a single pulse and the method is then used to predict the radiation of acoustic waves from the exit of a duct in which a pulse is propagating internally. A method of obtaining efficient solutions of flows containing both uniaxial and spherically-symmetric domains is then obtained. Interfaces between the domains are solved in a manner that ensures continuity of pressure and flowrate. The use of spherically-symmetric assumptions limits the range of three-dimensional domains that can be approximated, but the combination of the two forms of one-dimensional analysis makes highly efficient use of resources. Furthermore, the two-way coupling is a significant advantage over two-step methods that use independent solutions of the internal, uniaxial domain to provide prescribed boundary conditions for solutions of the external domain.

AB - The equations of motion for spherically-symmetric, unsteady, inviscid, compressible flow are expressed in a form that enables accurate numerical solutions to be obtained using a one-dimensional formulation based on the method of characteristics. Unlike the corresponding equations for uniaxial unsteady flows, the spherically-symmetric equations necessarily include terms involving the reciprocal of the radius and, close to the radial origin, numerical integration is unreliable. Nevertheless, good accuracy is obtained over a wide range of radii, including regions inside the range where the pressure and acceleration are approximately in phase. The range of validity of the method is assessed by comparison with an analytical solution for a single pulse and the method is then used to predict the radiation of acoustic waves from the exit of a duct in which a pulse is propagating internally. A method of obtaining efficient solutions of flows containing both uniaxial and spherically-symmetric domains is then obtained. Interfaces between the domains are solved in a manner that ensures continuity of pressure and flowrate. The use of spherically-symmetric assumptions limits the range of three-dimensional domains that can be approximated, but the combination of the two forms of one-dimensional analysis makes highly efficient use of resources. Furthermore, the two-way coupling is a significant advantage over two-step methods that use independent solutions of the internal, uniaxial domain to provide prescribed boundary conditions for solutions of the external domain.

KW - Method of characteristics

KW - Spherically-symmetric flow

KW - Vibrating sphere acoustics

KW - Wave radiation

KW - Wave reflection

UR - http://www.scopus.com/inward/record.url?scp=85070994027&partnerID=8YFLogxK

U2 - 10.1016/j.apm.2019.07.037

DO - 10.1016/j.apm.2019.07.037

M3 - Article

AN - SCOPUS:85070994027

VL - 77

SP - 810

EP - 828

JO - Applied Mathematical Modelling

JF - Applied Mathematical Modelling

SN - 0307-904X

IS - 1

ER -