A case-study in open-source CFD code verification. Part II: Boundary condition non-orthogonal correction
Abstract This investigation constitutes the follow-up to a previous work in which we applied the manufactured solution procedure to verify the OpenFOAM diffusion operator on different types of meshes. Theoretical convergence orders were observed for Poisson’s equation solver on orthogonal hexahedral grids using several boundary conditions. We noticed that on non-orthogonal grids even slight mesh distortions reduce the theoretical second order convergence rate to first order. Our investigation showed that this loss in convergence order takes place when Dirichlet and/or Neumann boundary conditions are used on non-orthogonal meshes. In this paper, we introduce ways to achieve theoretical second order convergence accuracy, mainly based on applying non-orthogonal correction to Dirichlet and/or Neumann boundary condition schemes. A peculiarity was discovered for the Least-Squares gradient scheme that slightly affects second-order convergence, and which we attribute to the non-orthogonal correction at boundaries in the Least-Squares vector reconstruction library. The verification procedure is performed using the method of manufactured solutions for the Poisson equation and an analytical solution, the SIMPLE solver, for the Navier–Stokes equations. The average and maximum error norms were used to calculate the convergence rate. Comparative results are presented.
A Semi-Automatic Approach Based on the Method of Manufactured Solutions to Assess the Convergence Order in OpenFOAM
Code verification is an intricate and crucial part of code developed. Due to the nature of the partial differential equations being solved, an analytical solution might not exit. To prove that the code is solving the correct equations, the method of manufactured solutions (MMS) is the go to approach. In this method we create a forcing term to drive the solution to a predetermined function. 
By performing a mesh refinement study, we can verify if the code is solving correctly the governing equations. In this work, we present the procedure of MMS as well as a tool developed in an open-source software for symbolic computation of the forcing term, appropriate boundary conditions and error norm computation withing the OpenFOAM® framework.
ANALYSIS OF SAUDI ARABIAN MARITIME TRANSPORTATION INDUSTRY
This paper analyses the past, present and future of the maritime transportation industry in the Kingdom of Saudi Arabia. Saudi Arabian seaborne trade and the shipping capacity are analysed for the past 15 years. Relationships between the declining Saudi shipping capacity and other economic measures are identified. Analysis of the Saudi Arabian maritime transportation is performed with respect to the world carrier fleets and models are developed to determine the theoretical shipping capacity required for transportation of the national seaborne cargo. Forecasts of the future shipping capacity requirement for the Kingdom are also presented. This paper illustrates the general approach for analysing a country's maritime industry and thus could help other researchers in that respect.