A New Artificial Staggered-Grid Central Difference Solution for Checkerboard Problem in Incompressible, Steady, Inviscid, and Quasi-One-Dimensional Flow through Convergent Nozzle

Abstract

In this paper, a novel artificial staggered grid points and under-relaxation free solution for a checkerboard pattern problem in a quasi-one-dimensional, incompressible, steady, and inviscid flow is introduced. The purpose of this numerical development is to obtain a new numerical solution, which is under-relaxation factor free scheme, more accurate, and easier to implement than a conventional staggered grid scheme. The proposed numerical solution can be described as the non-staggered grid/collocated grid central difference scheme which is free of pressure checkerboard pattern or spurious oscillation. The accuracy and convergence speed of the proposed numerical scheme is benchmarked against a conventional SIMPLE-based finite volume scheme and the exact solution for the flow problem in a convergent nozzle. The numerical analysis shows that the proposed numerical scheme outperforms the SIMPLE-based finite volume scheme in terms of accuracy, computational resource, and convergence speed. Also, the proposed numerical scheme has consistent numbers of iteration over the different grid sizes in contrast to the SIMPLE-based scheme which is iteration-grid size dependent. The proposed numerical scheme can be implemented with both uniform and non-uniform grid points and shows good agreement with the exact solution for every grid size. However, the uniform grid approach produces significantly more accurate results than the non-uniform grid approach. Hence, the choice of grid distribution is still an important factor affecting the accuracy of the proposed numerical solution. The proposed numerical technique can be further extended to solve incompressible flow problem in the complex 2D-3D domain with unstructural grids.