TY - JOUR

T1 - Rotational test spaces for a fully-implicit FVM and FEM for the DNS of fluid-particle interaction

AU - Hollbacher, Susanne

AU - Wittum, Gabriel

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The authors are grateful to D. Logaschenko for the useful discussions and thank to S. Reiter and A. Vogel for the support with the implementation in UG4.

PY - 2019/5/23

Y1 - 2019/5/23

N2 - The paper presents a fully-implicit and stable finite element and finite volume scheme for the simulation of freely moving particles in a fluid. The developed method is based on the Petrov-Galerkin formulation of a vertex-centered finite volume method (PG-FVM) on unstructured grids. Appropriate extension of the ansatz and test spaces lead to a formulation comparable to a fictitious domain formulation. The purpose of this work is to introduce a new concept of numerical modeling reducing the mathematical overhead which many other methods require. It exploits the identification of the PG-FVM with a corresponding finite element bilinear form. The surface integrals of the finite volume scheme enable a natural incorporation of the interface forces purely based on the original bilinear operator for the fluid. As a result, there is no need to expand the system of equations to a saddle-point problem. Like for fictitious domain methods the extended scheme treats the particles as rigid parts of the fluid. The distinguishing feature compared to most existing fictitious domain methods is that there is no need for an additional Lagrange multiplier or other artificial external forces for the fluid-solid coupling. Consequently, only one single solve for the derived linear system for the fluid together with the particles is necessary and the proposed method does not require any fractional time stepping scheme to balance the interaction forces between fluid and particles. For the linear Stokes problem we will prove the stability of both schemes. Moreover, for the stationary case the conservation of mass and momentum is not violated by the extended scheme, i.e. conservativity is accomplished within the range of the underlying, unconstrained discretization scheme. The scheme is applicable for problems in two and three dimensions.

AB - The paper presents a fully-implicit and stable finite element and finite volume scheme for the simulation of freely moving particles in a fluid. The developed method is based on the Petrov-Galerkin formulation of a vertex-centered finite volume method (PG-FVM) on unstructured grids. Appropriate extension of the ansatz and test spaces lead to a formulation comparable to a fictitious domain formulation. The purpose of this work is to introduce a new concept of numerical modeling reducing the mathematical overhead which many other methods require. It exploits the identification of the PG-FVM with a corresponding finite element bilinear form. The surface integrals of the finite volume scheme enable a natural incorporation of the interface forces purely based on the original bilinear operator for the fluid. As a result, there is no need to expand the system of equations to a saddle-point problem. Like for fictitious domain methods the extended scheme treats the particles as rigid parts of the fluid. The distinguishing feature compared to most existing fictitious domain methods is that there is no need for an additional Lagrange multiplier or other artificial external forces for the fluid-solid coupling. Consequently, only one single solve for the derived linear system for the fluid together with the particles is necessary and the proposed method does not require any fractional time stepping scheme to balance the interaction forces between fluid and particles. For the linear Stokes problem we will prove the stability of both schemes. Moreover, for the stationary case the conservation of mass and momentum is not violated by the extended scheme, i.e. conservativity is accomplished within the range of the underlying, unconstrained discretization scheme. The scheme is applicable for problems in two and three dimensions.

UR - http://hdl.handle.net/10754/656472

UR - https://linkinghub.elsevier.com/retrieve/pii/S0021999119303298

UR - http://www.scopus.com/inward/record.url?scp=85066096415&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2019.05.004

DO - 10.1016/j.jcp.2019.05.004

M3 - Article

SN - 0021-9991

VL - 393

SP - 186

EP - 213

JO - Journal of Computational Physics

JF - Journal of Computational Physics

ER -