Numerical solutions of 2D Navier-Stokes equations based on generalized harmonic polynomial cell method with non-uniform grid

It is essential for a Navier-Stokes equations solver based on a projection method to be able to solve the resulting Poisson equation accurately and efficiently. In this paper, we present numerical solutions of the 2D Navier-Stokes equations using the fourth-order generalized harmonic polynomial cell (GHPC) method as the Poisson equation solver. Particular focus is on the local and global accuracy of the GHPC method on non-uniform grids. Our study reveals that the GHPC method enables use of more stretched grids than the original HPC method. Compared with a second-order central finite difference method (FDM), global accuracy analysis also demonstrates the advantage of applying the GHPC method on stretched non-uniform grids. An immersed boundary method is used to deal with general geometries involving the fluid-structure-interaction problems. The Taylor-Green vortex and flow around a smooth circular cylinder and square are studied for the purpose of verification and validation. Good agreement with reference results in the literature confirms the accuracy and efficiency of the new 2D Navier-Stokes equation solver based on the present immersed-boundary GHPC method utilizing non-uniform grids. The present Navier-Stokes equations solver uses second-order FDM for the discretization of the diffusion and advection terms, which may be replaced by other higher-order schemes to further improve the accuracy.

Transformation Method for Solving System of Boolean Algebraic Equations

: In recent years, various methods and directions for solving a system of Boolean algebraic equations have been invented, and now they are being very actively investigated. One of these directions is the method of transforming a system of Boolean algebraic equations, given over a ring of Boolean polynomials, into systems of equations over a field of real numbers, and various optimization methods can be applied to these systems. In this paper, we propose a new transformation method for Solving Systems of Boolean Algebraic Equations (SBAE). The essence of the proposed method is that firstly, SBAE written with logical operations are transformed (approximated) in a system of harmonic-polynomial equations in the unit n-dimensional cube Kn with the usual operations of addition and multiplication of numbers. Secondly, a transformed (approximated) system in Kn is solved by using the optimization method. We substantiated the correctness and the right to exist of the proposed method with reliable evidence. Based on this work, plans for further research to improve the proposed method are outlined.

A 2D Navier-Stokes Equations Solver Based on Generalized Harmonic Polynomial Cell Method With Non-Uniform Grid

It is essential for a Navier-Stokes equations solver based on a projection method to be able to solve the resulting Poisson equation accurately and efficiently. In this paper, we present a new 2D Navier-Stokes equation solver based on a recently proposed fourth-order method, namely the generalized harmonic polynomial cell (GHPC) method, as the Poisson equation solver. The GHPC method is a generalization of the 2D HPC method originally developed for the Laplace equation. In the recent development of the HPC method, loss of accuracy on highly stretched or distorted grids has been reported when solving the Laplace equation, while the performance of the GHPC method on non-uniform grids is still not explored and discussed in the literature. Therefore, the local accuracy of the GHPC method is investigated in detail in the present study, which reveals that the GHPC method allows for the use of much larger grid aspect ratio than that for the original HPC method. Global accuracy of the GHPC method on stretched non-uniform girds is also thoroughly analyzed by considering cases with analytical solutions. Obvious advantages of using the GHPC method in terms of accuracy are demonstrated by comparing with a second-order central Finite Difference Method (FDM). The present Navier-Stokes equations solver uses second-order FDMs for the discretization of the diffusion and advection terms, which may be replaced by other higher-order schemes to further improve the accuracy. Meanwhile, an immersed boundary method [1] is used to study the fluid-structure-interaction problems. The Taylor-Green vortex and flow around a smooth circular cylinder are studied to confirm the accuracy and efficiency of the new 2D Navier-Stokes equation solver. The predictions show good agreements with the experimental and numerical results in the literature.

A Coupled Harmonic Polynomial Cell and Higher-Order Spectral Method for Nonlinear Wave Propagation

The present work deals with wave generation in fully nonlinear numerical wave tanks (NWT). As an alternative to modelling a moving (physical) wavemaker, a two-dimensional (2D) potential-flow NWT is coupled with an external spectral wave data (SWD) application programming interface (API). The NWT uses the harmonic polynomial cell (HPC) method to solve the governing Laplace equations for the velocity potential and its time derivative, and has previously been extensively validated and verified for numerous nonlinear wave-propagation problems using traditional wave-generation mechanisms. Periodic waves of different steepness generated with a stream-function theory as reference solution in the SWD API are first considered to investigate the method’s numerical accuracy. Thereafter, with a higher-order spectral method (HOSM) as the SWD API solution, irregular waves with different wave heights and water depths relevant for e.g. aquaculture and offshore structures are simulated. Differences between the HPC and HOSM solutions in and near steep crests are investigated. The study aims to demonstrate a robust method to generate and propagate general wave fields for further studies of nonlinear waves and wave-body interaction in both two and three dimensions.