### Abstract

An extended volume of fluid method is developed for two-phase direct numerical simulations of systems with one viscoelastic and one Newtonian phase. A complete set of governing equations is derived by conditional volume averaging the local instantaneous bulk equations and interface jump conditions. The homogeneous mixture model is applied for the closure of the volume-averaged equations. An additional interfacial stress term arises in this volume-averaged formulation which requires special treatment in the finite-volume discretization on a general unstructured mesh. A novel numerical scheme is proposed for the second-order accurate finite-volume discretization of the interface stress term. We demonstrate that this scheme allows for a consistent treatment of the interface stress and the surface tension force in the pressure equation of the segregated solution approach. Because of the high Weissenberg number problem, an appropriate stabilization approach is applied to the constitutive equation of the viscoelastic phase to increase the robustness of the method at higher fluid elasticity. Direct numerical simulations of the transient motion of a bubble rising in a quiescent viscoelastic fluid are performed for the purpose of experimental code validation. The well-known jump discontinuity in the terminal bubble rise velocity when the bubble volume exceeds a critical value is captured by the method. The formulation of the interfacial stress together with the novel scheme for its discretization is found crucial for the quantitatively correct prediction of the jump discontinuity in the terminal bubble rise velocity.

Language | English |
---|---|

Pages | 326-355 |

Number of pages | 30 |

Journal | Journal of computational physics |

Volume | 387 |

DOIs | |

Status | Published - 2019 |

### Fingerprint

### Keywords

- Extended volume of fluid method
- Negative wake
- Rising bubble
- Velocity jump discontinuity
- Viscoelastic liquid

### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy (miscellaneous)

### Fields of Expertise

- Sonstiges

### Cite this

*Journal of computational physics*,

*387*, 326-355. https://doi.org/10.1016/j.jcp.2019.02.021

**An extended volume of fluid method and its application to single bubbles rising in a viscoelastic liquid.** / Niethammer, Matthias; Brenn, Günter; Marschall, Holger; Bothe, Dieter.

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

*Journal of computational physics*, vol. 387, pp. 326-355. https://doi.org/10.1016/j.jcp.2019.02.021

}

TY - JOUR

T1 - An extended volume of fluid method and its application to single bubbles rising in a viscoelastic liquid

AU - Niethammer, Matthias

AU - Brenn, Günter

AU - Marschall, Holger

AU - Bothe, Dieter

PY - 2019

Y1 - 2019

N2 - An extended volume of fluid method is developed for two-phase direct numerical simulations of systems with one viscoelastic and one Newtonian phase. A complete set of governing equations is derived by conditional volume averaging the local instantaneous bulk equations and interface jump conditions. The homogeneous mixture model is applied for the closure of the volume-averaged equations. An additional interfacial stress term arises in this volume-averaged formulation which requires special treatment in the finite-volume discretization on a general unstructured mesh. A novel numerical scheme is proposed for the second-order accurate finite-volume discretization of the interface stress term. We demonstrate that this scheme allows for a consistent treatment of the interface stress and the surface tension force in the pressure equation of the segregated solution approach. Because of the high Weissenberg number problem, an appropriate stabilization approach is applied to the constitutive equation of the viscoelastic phase to increase the robustness of the method at higher fluid elasticity. Direct numerical simulations of the transient motion of a bubble rising in a quiescent viscoelastic fluid are performed for the purpose of experimental code validation. The well-known jump discontinuity in the terminal bubble rise velocity when the bubble volume exceeds a critical value is captured by the method. The formulation of the interfacial stress together with the novel scheme for its discretization is found crucial for the quantitatively correct prediction of the jump discontinuity in the terminal bubble rise velocity.

AB - An extended volume of fluid method is developed for two-phase direct numerical simulations of systems with one viscoelastic and one Newtonian phase. A complete set of governing equations is derived by conditional volume averaging the local instantaneous bulk equations and interface jump conditions. The homogeneous mixture model is applied for the closure of the volume-averaged equations. An additional interfacial stress term arises in this volume-averaged formulation which requires special treatment in the finite-volume discretization on a general unstructured mesh. A novel numerical scheme is proposed for the second-order accurate finite-volume discretization of the interface stress term. We demonstrate that this scheme allows for a consistent treatment of the interface stress and the surface tension force in the pressure equation of the segregated solution approach. Because of the high Weissenberg number problem, an appropriate stabilization approach is applied to the constitutive equation of the viscoelastic phase to increase the robustness of the method at higher fluid elasticity. Direct numerical simulations of the transient motion of a bubble rising in a quiescent viscoelastic fluid are performed for the purpose of experimental code validation. The well-known jump discontinuity in the terminal bubble rise velocity when the bubble volume exceeds a critical value is captured by the method. The formulation of the interfacial stress together with the novel scheme for its discretization is found crucial for the quantitatively correct prediction of the jump discontinuity in the terminal bubble rise velocity.

KW - Extended volume of fluid method

KW - Negative wake

KW - Rising bubble

KW - Velocity jump discontinuity

KW - Viscoelastic liquid

UR - http://www.scopus.com/inward/record.url?scp=85063061986&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2019.02.021

DO - 10.1016/j.jcp.2019.02.021

M3 - Article

VL - 387

SP - 326

EP - 355

JO - Journal of computational physics

T2 - Journal of computational physics

JF - Journal of computational physics

SN - 0021-9991

ER -