A Higher-Order Unstructured Finite Volume Solver for Three-Dimensional Compressible Flows

Abstract

High-order accurate numerical discretization methods are attractive for their potential to significantly reduce the computational costs compared to the traditional second-order methods. Among the various unstructured higher-order discretization schemes, the k-exact reconstruction finite volume method is of interest for its straightforward mathematical formulation, and its compatibility with the current lowerorder industrial solvers. However, current three-dimensional finite volume solvers are limited to the solution of inviscid and laminar viscous flow problems. Since three-dimensional turbulent flows appear in many industrial applications, the current thesis takes the first step towards the development of a threedimensional higher-order finite volume solver for the solution of both inviscid and viscous turbulent steady-state flow problems.The k-exact finite volume formulation of the governing equations is rederived in a dimension-independent manner, where the negative Spalart-Allmaras turbulence model is employed. This one-equation model is reasonably accurate for many flow conditions, and its simplicity makes it a good starting point for the development of numerical algorithms. Then, the three-dimensional mesh preprocessing steps for a finite volume simulation are presented, including higher-order accurate numerical quadrature, and capturing the boundary curvature in highly anisotropic meshes. Also, the issues of k-exact reconstruction in handling highly anisotropic meshes are reviewed and addressed.Since three-dimensional problems can require much more memory than their two-dimensional counterparts, solution methods that work in two dimensions might not be feasible in three dimensions anymore.As an attempt to overcome this issue, a practical and parallel scalable method for the solution of the discretized system of nonlinear equations is presented. Finally, the solution of four three-dimensional test problems are studied: Poisson's equation in a cubic domain, inviscid flow over a sphere, turbulent flow over a flat plate, and turbulent flow over an extruded NACA 0012 airfoil. The solution is verified, and the resource consumption of the flow solver is measured.The results demonstrate the benefit and practicality of using higher-order methods for obtaining a certain level of accuracy.ii Lay SummaryThe finite volume method is a popular numerical scheme for the solution of aerodynamic flow problems.Although conventional finite volume schemes are mostly second-order accurate, there has been a growing interest in higher-order accurate numerical methods, since they can be considerably more efficient in terms of computational resources for achieving a certain level of accuracy.Higher-order finite volume methods have been successfully developed for a wide range of two-dimensional flow problems. Nevertheless, numerical methods that work in two dimensions might not be feasible in three dimensions anymore, since three-dimensional problems can require much more memory than their two-dimensional counterparts. The current th...

Um método robusto de volumes finitos de alta ordem para advecção em malhas esféricas geodésicas

“…Em seu trabalho, Ollivier-Gooch utiliza um estêncil fixo com pesos dependentes da distância dos dados e a solução numérica é obtida pela reconstrução da solução, integração do fluxo e evolução temporal. Esse método tem sido utilizado com sucesso em diversas aplicações em malhas não estruturadas no plano (Hoshyari e Ollivier Gooch, 2018;Jalali et al, 2014;Nejat e Ollivier-Gooch, 2008;Ollivier-Gooch, 1997;Ollivier-Gooch e Van Altena, 2002).…”

“…O objetivo deste trabalho é estender a metodologia baseada no método de volumes finitos de alta ordem desenvolvido por Ollivier-Gooch e colaboradores (Hoshyari e Ollivier Gooch, 2018;Jalali et al, 2014;Nejat e Ollivier-Gooch, 2008;Ollivier-Gooch, 1997;Ollivier-Gooch e Van Altena, 2002;Ollivier-Gooch et al, 2009), que apresenta resultados relevantes em alta ordem, aplicadas em malhas no plano, para a superfície esférica. A transição do método para esse novo contexto será verificada de forma rigorosa, ou seja, serão feitos testes para validar todas as etapas no processo de implementação.…”

“…Esse polinômio, por sua vez, é utilizado para avaliar os fluxos nas faces dos volumes de controle. O método de reconstrução de alta ordem desenvolvido por Ollivier-Gooch e colaboradores tem diversas aplicações em malhas bidimensionais não estruturadas (Hoshyari e Ollivier Gooch, 2018;Jalali et al, 2014;Nejat e Ollivier-Gooch, 2008;Ollivier-Gooch, 1997;Ollivier-Gooch e Van Altena, 2002).…”