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

Jesús Amo-Navarro,Ricardo Vinuesa,Sergio Hoyas,J Alberto Conejero

doi:10.3390/math9192508

Jesús Amo-Navarro, Ricardo Vinuesa + Show 2 more

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https://doi.org/10.3390/math9192508

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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.

Highlights

Turbulence is most likely the open subject in physics with the greatest number of applications in everyday life
We propose an efficient algorithm to impose Neumann boundary conditions (BC), taking advantage of the capabilities of compact finite difference (CFD) to compute any kind of derivatives
The accuracy of the CFD method is defined by the employed stencil, which results in a specific truncation order of the Taylor expansion

Summary

Introduction

Turbulence is most likely the open subject in physics with the greatest number of applications in everyday life. DNSs are extremely expensive because every turbulent scale in the flow, both temporal and spatial, must be properly resolved. This requires very fine grids and very small temporal steps. Due to the presence of the side walls, turbulent ducts are more challenging to simulate [17,18,19] than channels, pipes, or boundary layers. This method, like many other high-resolution ones, cannot be used in complex geometries. See [20] for details

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