The Immersed Interface Method for Navier-Stokes Equations with Interfaces in Cylindrical Coordinates

doi:10.1007/s42967-024-00445-3

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (3): 1074-1097.doi: 10.1007/s42967-024-00445-3

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The Immersed Interface Method for Navier-Stokes Equations with Interfaces in Cylindrical Coordinates

Juan Ruiz-álvarez¹, Baiying Dong^2,3, Zhilin Li⁴

1 Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain;
2 School of Civil and Hydraulic Engineering, Ningxia University, Ningxia 750021, China;
3 School of Mathematics and Computer Science, Ningxia Normal University, NingXia 756099, China;
4 Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA

Received:2023-09-28 Revised:2024-06-28 Accepted:2024-06-30 Online:2025-09-20 Published:2025-05-23
Supported by:
J. Ruiz is supported by Fundación Sáneca Grant 21728/EE/22 (Este trabajo es resultado de la estancia financiada por la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia con cargo al Programa Regional de Movilidad, Colaboración Internacional e Intercambio de Conocimiento “Jiménez de la Espada”). B. Dong is partially supported by the National Natural Science Foundation of China Grant No. 12261070 and Ningxia Key Research and Development Project of China Grant No. 2022BSB03048. Z. Li is partially supported by Simons Grant 633724 and by Fundación Séneca Grant 21760/IV/22 (Este trabajo es resultado de la estancia financiada por la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia con cargo al Programa Regional de Movilidad, Colaboración Internacional Intercambio de Conocimiento “Jiménez de la Espada”).

Abstract

Abstract: Many three-dimensional physical applications can be better analyzed and solved using the cylindrical coordinates. In this paper, the immersed interface method (IIM) tailored for Navier-Stokes equations involving interfaces under the cylindrical coordinates has been developed. Note that, while the IIM has been developed for Stokes equations in the cylindrical coordinates assuming the axis-symmetry in the literature, there is a gap in dealing with Navier-Stokes equations, where the non-linear term includes an additional component involving the coordinate ?, even if the geometry and force term are axis-symmetric. Solving the Navier-Stokes equations in cylindrical coordinates becomes challenging when dealing with interfaces that feature a discontinuous pressure and a non-smooth velocity, in addition to the pole singularity at r = 0. In the newly developed algorithm, we have derived the jump conditions under the cylindrical coordinates. The numerical algorithm is based on a finite difference discretization on a uniform and staggered grid in the cylindrical coordinates. The finite difference scheme is standard away from the interface but is modified at grid points near and on the interface. As expected, the method is shown to be second-order accurate for the velocity. The developed new IIM is applied to the solution of some related fluid dynamic problems with interfaces.

Key words: Immersed interface method (IIM), Navier-Stokes equations, Axis-symmetric interface problem, Staggered grid, Pole singularity, Finite difference method

CLC Number:

Juan Ruiz-álvarez, Baiying Dong, Zhilin Li. The Immersed Interface Method for Navier-Stokes Equations with Interfaces in Cylindrical Coordinates[J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 1074-1097.

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The Immersed Interface Method for Navier-Stokes Equations with Interfaces in Cylindrical Coordinates

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