Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

A high-order cell-centered finite volume scheme for simulating three dimensional anisotropic diffusion equations on unstructured grids

We present a finite volume based cell-centered method for solving diffusion equations on three-dimensional unstructured grids with general tensor conduction. Our main motivation concerns the numerical simulation of the coupling between fluid flows and heat transfers. The corresponding numerical scheme is characterized by cell-centered unknowns and a local stencil. Namely, the scheme results in a global sparse diffusion matrix, which couples only the cell-centered unknowns. The space discretization relies on the partition of polyhedral cells into sub-cells and on the partition of cell faces into sub-faces. It is characterized by the introduction of sub-face normal fluxes and sub-face temperatures, which are auxiliary unknowns. A sub-cell-based variational formulation of the constitutive Fourier law allows to construct an explicit approximation of the sub-face normal heat fluxes in terms of the cell-centered temperature and the adjacent sub-face temperatures. The elimination of the sub-face temperatures with respect to the cell-centered temperatures is achieved locally at each node by solving a small and sparse linear system. This system is obtained by enforcing the continuity condition of the normal heat flux accross each sub-cell interface impinging at the node under consideration. The parallel implementation of the numerical algorithm and its efficiency are described and analyzed. The accuracy and the robustness of the proposed finite volume method are assessed by means of various numerical test cases

Face- and Cell-Averaged Nodal-Gradient Approach to Cell-Centered Finite-Volume Method on Mixed Grids

In this paper, the averaged nodal-gradient approach previously developed for triangular grids is extended to mixed triangular-quadrilateral grids. It is shown that the face- averaged approach leads to deteriorated iterative convergence on quadrilateral grids. To develop a convergent solver, we consider cell-averaging instead of face-averaging for quadri- lateral cells. We show that the cell-averaged approach leads to a convergent solver and can be efficiently combined with the face-averaged approach on mixed grids. The method is demonstrated for various inviscid and viscous problems from low to high Mach numbers on two-dimensional mixed grids

A first order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total Lagrangian scheme

This paper builds on recent work developed by the authors for the numerical analysis of large strain solid dynamics, by introducing an upwind cell centred hexahedral Finite Volume framework implemented within the open source code OpenFOAM [http://www.openfoam.com/http://www.openfoam.com/]. In Lee, Gil and Bonet [1], a first order hyperbolic system of conservation laws was introduced in terms of the linear momentum and the deformation gradient tensor of the system, leading to excellent behaviour in two dimensional bending dominated nearly incompressible scenarios. The main aim of this paper is the extension of this algorithm into three dimensions, its tailor-made implementation into OpenFOAM and the enhancement of the formulation with three key novelties. First, the introduction of two different strategies in order to ensure the satisfaction of the underlying involutions of the system, that is, that the deformation gradient tensor must be curl-free throughout the deformation process. Second, the use of a discrete angular momentum projection algorithm and a monolithic Total Variation Diminishing Runge-Kutta time integrator combined in order to guarantee the conservation of angular momentum. Third, and for comparison purposes, an adapted Total Lagrangian version of the Hyperelastic-GLACE nodal scheme of Kluth and Despr´es [2] is presented. A series of challenging numerical examples are examined in order to assess the robustness and accuracy of the proposed algorithm, benchmarking it against an ample spectrum of alternative numerical strategies developed by the authors in recent publications

Adaptive mesh refinement method for CFD applications

The main objective of this thesis is the development of an adaptive mesh refinement (AMR) algorithm for computational fluid dynamics simulations using hexahedral and tetrahedral meshes. This numerical methodology is applied in the context of large-eddy simulations (LES) of turbulent flows and direct numerical simulations (DNS) of interfacial flows, to bring new numerical research and physical insight. For the fluid dynamics simulations, the governing equations, the spatial discretization on unstructured grids and the numerical schemes for solving Navier-Stokes equations are presented. The equations follow a discretization by conservative finite-volume on collocated meshes. For the turbulent flows formulation, the spatial discretization preserves symmetry properties of the continuous differential operators and the time integration follows a self-adaptive strategy, which has been well tested on unstructured grids. Moreover, LES model consisting of a wall adapting local-eddy-viscosity within a variational multi-scale formulation is used for the applications showed in this thesis. For the two-phase flow formulation, a conservative level-set method is applied for capturing the interface between two fluids and is implemented with a variable density projection scheme to simulate incompressible two-phase flows on unstructured meshes. The AMR algorithm developed in this thesis is based on a quad/octree data structure and keeps a relation of 1:2 between levels of refinement. In the case of tetrahedral meshes, a geometrical criterion is followed to keep the quality metric of the mesh on a reasonable basis. The parallelization strategy consists mainly in the creation of mesh elements in each sub-domain and establishes a unique global identification number, to avoid duplicate elements. Load balance is assured at each AMR iteration to keep the parallel performance of the CFD code. Moreover, a mesh multiplication algorithm (MM) is reported to create large meshes, with different kind of mesh elements, but preserving the topology from a coarser original mesh. This thesis focuses on the study of turbulent flows and two-phase flows using an AMR framework. The cases studied for LES of turbulent flows applications are the flow around one and two separated square cylinders, and the flow around a simplified car model. In this context, a physics-based refinement criterion is developed, consisting of the residual velocity calculated from a multi-scale decomposition of the instantaneous velocity. This criteria ensures grid adaptation following the main vortical structures and giving enough mesh resolution on the zones of interest, i.e., flow separation, turbulent wakes, and vortex shedding. The cases studied for the two-phase flows are the DNS of 2D and 3D gravity-driven bubble, with a particular focus on the wobbling regime. A study of rising bubbles in the wobbling regime and the effect of dimensionless numbers on the dynamic behavior of the bubbles are presented. Moreover, the use of tetrahedral AMR is applied for the numerical simulation of gravity-driven bubbles in complex domains. On this topic, the methodology is validated on bubbles rising in cylindrical channels with different topology, where the study of these cases contributed to having new numerical research and physical insight in the development of a rising bubble with wall effects.El objetivo principal de esta tesis es el desarrollo de un algoritmo adaptativo de refinamiento de malla (AMR) para simulaciones de dinámica de fluidos computacional utilizando mallas hexaédricas y tetraédricas. Esta metodología numérica se aplica en el contexto de simulaciones Large-eddie (LES) de flujos turbulentos y simulaciones numéricas directas (DNS) de flujos interfaciales, para traer nuevas investigaciones numéricas y entendimiento físicas. Para las simulaciones de dinámica de fluidos, se presentan las ecuaciones governantes, la discretización espacial en mallas no estructuradas y los esquemas numéricos para resolver las ecuaciones de Navier-Stokes. Las ecuaciones siguen una discretización conservativa por volumenes finitos en mallas colocadas. Para la formulación de flujos turbulentos, la discretización espacial preserva las propiedades de simetría de los operadores diferenciales continuos y la integración de tiempo sigue una estrategia autoadaptativa, que ha sido bien probada en mallas no estructuradas. Además, para las aplicaciones que se muestran en esta tesis, se utiliza el modelo LES que consiste en una viscosidad local que se adapta a la pared dentro de una formulación multiescala variable. Para la formulación de flujo de dos fases, se aplica un método de conjunto de niveles conservador para capturar la interfaz entre dos fluidos y se implementa con un esquema de proyección de densidad variable para simular flujos de dos fases incompresibles en mallas no estructuradas. El algoritmo AMR desarrollado en esta tesis se basa en una estructura de datos de quad / octree y mantiene una relación de 1: 2 entre los niveles de refinamiento. En el caso de las mallas tetraédricas, se sigue un criterio geométrico para mantener la calidad de la malla en una base razonable. La estrategia de paralelización consiste principalmente en la creación de elementos de malla en cada subdominio y establece un número de identificación global único, para evitar elementos duplicados. El equilibrio de carga está asegurado en cada iteración de AMR para mantener el rendimiento paralelo del código CFD. Además, se ha desarrollado un algoritmo de multiplicación de malla (MM) para crear mallas grandes, con diferentes tipos de elementos de malla, pero preservando la topología de una malla original más pequeña. Esta tesis se centra en el estudio de flujos turbulentos y flujos de dos fases utilizando un marco AMR. Los casos estudiados para aplicaciones de LES de flujos turbulentos son el flujo alrededor de uno y dos cilindros separados de sección cuadrada, y el flujo alrededor de un modelo de automóvil simplificado. En este contexto, se desarrolla un criterio de refinamiento basado en la física, que consiste en la velocidad residual calculada a partir de una descomposición de escala múltiple de la velocidad instantánea. Este criterio garantiza la adaptación de la malla siguiendo las estructuras vorticales principales y proporcionando una resolución de malla suficiente en las zonas de interés, es decir, separación de flujo, estelas turbulentas y desprendimiento de vórtices. Los casos estudiados para los flujos de dos fases son el DNS de la burbuja impulsada por la gravedad en 2D y 3D, con un enfoque particular en el régimen de oscilación. Además, el uso de AMR tetraédrico se aplica para la simulación numérica de burbujas impulsadas por la gravedad en dominios complejos. En este tema, la metodología se valida en burbujas que ascienden en canales cilíndricos con topología diferente, donde el estudio de estos casos contribuyó a tener una nueva investigación numérica y una visión física en el desarrollo de una burbuja con efectos de pared

Efficient and Robust Weighted Least-Squares Cell-Average Gradient Construction Methods for the Simulation of Scramjet Flows

The ability to solve the equations governing the hypersonic turbulent flow of a real gas on unstructured grids using a spatially-elliptic, 2nd-order accurate, cell-centered, finite-volume method has been recently implemented in the VULCAN-CFD code. The construction of cell-average gradients using a weighted linear least-squares method and the use of these gradients in the construction of the inviscid fluxes is the focus of this paper. A comparison of least-squares stencil construction methodologies is presented and approaches designed to minimize the number of cells used to augment/stabilize the least-squares stencil while preserving accuracy are explored. Due to our interest in hypersonic flow, a robust multidimensional cell-average gradient limiter procedure that is consistent with the stencil used to construct the cellaverage gradients is described. Canonical problems are computed to illustrate the challenges and investigate the accuracy, robustness and convergence behavior of the cell-average gradient methods on unstructured cell-centered finite-volume grids. Finally, thermally perfect, chemically frozen, Mach 7.8 turbulent flow of air through a scramjet engine flowpath is computed and compared with experimental data to demonstrate the robustness, accuracy and convergence behavior of the preferred gradient method for a realistic 3-D geometry on a non-hex-dominant grid

Three-dimensional preliminary results of the MOOD method: A Very High-Order Finite Volume method for Conservation Laws.

The Multi-dimensional Optimal Order Detection (MOOD) method has been designed by authors in [5] and extended in [7] to reach Very-High-Order of accuracy for systems of Conservation Laws in a Finite Volume (FV) framework on 2D unstructured meshes. In this paper we focus on the extension of this method to 3D unstructured meshes. We present preliminary results for the three-dimensional advection equation which confirm the good behaviour of the MOOD method. More precisely, we show that the scheme yields up to sixth-order accuracy on smooth solutions while preventing oscillations from appearing on discontinuous profiles