Two-Dimensional Compact-Finite-Difference Schemes for Solving the bi-Laplacian Operator with Homogeneous Wall-Normal Derivatives

Jesús Amo-Navarro, Ricardo Vinuesa, J. Alberto Conejero and Sergio Hoyas
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Jesús Amo-Navarro: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 València, Spain
Ricardo Vinuesa: FLOW, Engineering Mechanics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
J. Alberto Conejero: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 València, Spain
Sergio Hoyas: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 València, Spain

Mathematics, 2021, vol. 9, issue 19, 1-13

Abstract: In fluid mechanics, the bi-Laplacian operator with Neumann homogeneous boundary conditions emerges when transforming the Navier–Stokes equations to the vorticity–velocity formulation. In the case of problems with a periodic direction, the problem can be transformed into multiple, independent, two-dimensional fourth-order elliptic problems. An efficient method to solve these two-dimensional bi-Laplacian operators with Neumann homogeneus boundary conditions was designed and validated using 2D compact finite difference schemes. The solution is formulated as a linear combination of auxiliary solutions, as many as the number of points on the boundary, a method that was prohibitive some years ago due to the large memory requirements to store all these auxiliary functions. The validation has been made for different field configurations, grid sizes, and stencils of the numerical scheme, showing its potential to tackle high gradient fields as those that can be found in turbulent flows.

Keywords: DNS; CFD; turbulence; bi-Laplacian; fourth-order elliptic (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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