This work details a multigrid-accelerated cut-cell Cartesian mesh methodology for the solution of a single partial differential equation set that describes incompressible as well as compressible flow. The latter includes sub-, trans- and supersonic flows. Cut-cell technology is developed which furnishes body-fitted meshes with an overlapping Cartesian mesh as starting point, and in a manner which is insensitive to surface definition inconsistencies. An edge-based vertex-centred finite volume method is employed for the purpose of spatial discretisation. Further, an alternative dual-mesh construction strategy is developed and the standard discretisation scheme suitably enhanced. Incompressibility is dealt with via a locally preconditioned artificial compressibility algorithm, and stabilisation is in all cases achieved with scalar-valued artificial dissipation. In transonic flows, shocks are captured via pressure switch-activated upwinding. The solution process is accelerated by the use of a full approximation scheme (FAS) multigrid method where coarse meshes are generated automatically via a volume agglomeration methodology. The developed modelling technology is validated by application to the solution of a number of benchmark problems. The standard discretisation as well as the alternative method are found to be equivalent in terms of both accuracy and computational cost. Finally, the multigrid implementation is shown to achieve decreases in CPU time of between a factor two to one order of magnitude. In the context of cut-cell Cartesian meshes, the above work has resulted in the following novel contributions: the development of an alternative vertex-centred discretisation method; the use of volume agglomerated multigrid solution technology and the use of a single equation set for both incompressible and compressible flows. / Dissertation (MEng (Mechanical Engineering))--University of Pretoria, 2007. / Mechanical and Aeronautical Engineering / unrestricted

La méthode LS-STAG est une méthode cartésienne pour le calcul d’écoulements incompressibles en géométries complexes, qui propose une discrétisation précise des équations de Navier-Stokes dans les cut-cells, cellules polyédriques de forme complexe créées par l’intersection du maillage cartésien avec la frontière du solide immergé. Originalement développée pour les géométries 2D, son extension aux géométries 3D se heurte au défi posé par le grand nombre de types de cut-cells (108) à considérer. Récemment, la méthode LS-STAG a été étendue aux géométries complexes 3D dont la frontière est parallèle à l’un des axes du repère cartésien, où sont uniquement présentes les contreparties extrudées des cut-cells 2D. Cette étude a notamment souligné deux points à élucider pour le développement d’une méthode totalement 3D : premièrement, le calcul des flux diffusifs par un simple schéma à deux points s’est révélé insuffisamment précis dans les cut-cells 3D-extrudées du fait de la non orthogonalité. Ensuite, l’implémentation de ces flux à la paroi, qui s’effectue en imposant une discrétisation distincte pour chaque type de cut-cell extrudée, se révèle trop complexe pour être étendue avec succès aux nombreux types supplémentaires de cut-cells 3D, et doit être simplifiée et rationalisée. Dans cette thèse, le premier point est résolu en utilisant l’outil des schémas diamants, d’abord étudié en 2D pour l’équation de la chaleur puis les équations de Navier-Stokes dans l’approximation de Boussinesq, puis étendu en 3D. En outre, les schémas diamants ont permis de revisiter intégralement la discrétisation du tenseur des contraintes des équations de Navier-Stokes, où disparaît le traitement au cas par cas selon la disposition de la frontière solide dans les cut-cells. Cela a permis d’aboutir à une discrétisation systématique, précise et algorithmiquement efficace pour les écoulements en géométries totalement 3D. La validation numérique de la méthode LS-STAG avec schémas diamants est présentée pour une série de cas tests en géométries complexes 2D et 3D. Sa précision est d’abord évaluée par comparaison avec des solutions analytiques en 2D, puis en 3D par la simulation d’un écoulement de Stokes entre deux sphères concentriques. La robustesse de la méthode est notamment mise en évidence par l’étude d’écoulements autour d’une sphère en rotation, dans les régimes laminaires (stationnaire et instationnaire), ainsi que pour un régime faiblement turbulent. / The LS-STAG method is a cartesian method for the computations of incompressible flows in complex geometries, which consists in an accurate discretisation of the Navier-Stokes equations in cut-cells, polyhedral cells with complex shape made by the intersection of cartesian mesh and the immersed boundary. Originally developed for 2D geometries, where only three types of generic cut-cells appear, its extension to 3D geometries has to deal with the large amount of cut-cells types (108). Recently, the LS-STAG method had been extended to 3D complex geometries whose boundary is parallel to an axis of the cartesian coordinate system, where there are only the extruded counterparts of 2D cut-cells. This study highlighted two points to deal with in order to develop a totally 3D method: firstly, the computation of diffusive fluxes by a simple 2-points scheme has shown to be insufficiently accurate in 3D-extruded cut-cells due to the non-orthogonality. In addition to that, implementation of these fluxes on the immersed boundary, which is done with a case by case discretisation according to the type of the cut-cells, appears to be too difficult for its successful extension to the several extra types of 3D cut-cells, and needs to be simplified and rationalized. In this thesis, the first point is solved by using the diamond scheme tool, firstly studied in 2D for the heat equation then for the Navier-Stokes equations in Boussinesq approximation, and finally extended to 3D. Moreover, the diamond schemes have been used to fully revisit the discretisation of shear stresses from Navier-Stokes equations, where the case by case procedure is removed. These modifications have permitted to come up with a systematic discretisation that is accurate and algorithmically efficient for flows in totally 3D geometries. The numerical validation of the LS-STAG method with diamond schemes is presented for a series of test cases in 2D and 3D complex geometries. The precision is firstly assessed by comparison with analytical solutions in 2D, then in 3D by the simulation of Stokes flow between two concentric spheres. The robustess of the method is highlighted by the simulations of flows past a rotating sphere, in laminar modes (steady and unsteady), as well as in a weakly turbulent mode.

Computational fluid dynamics has progressed to the point where it is now possible to simulate flows with large eddy turbulence, free surfaces and other complex features. However, the success of these models often depends on the accuracy of the advection scheme supporting them. Two such schemes are the constrained interpolation profile method (CIP) and the interpolated differential operator method (IDO). They share the same space discretisation but differ in their respectively semi-Lagrangian and Eulerian formulations. They both belong to a family of high-order, compact methods referred to as the multi-moment methods. In the absence of sufficient information in the literature, this thesis begins by taxonomising various multi-moment space discretisations and appraising their linear advective properties. In one dimension it is found that the CIP/IDO with order (2N -1) has an identical spectrum and memory cost to the Nth order discontinuous Galerkin method. Tests confirm that convergence rates are consistent with nominal orders of accuracy, suggesting that CIP/IDO is a better choice for smooth propagation problems. In two dimensions, six Cartesian multi-moment schemes of third order are compared using both spectral analysis and time-domain testing. Three of these schemes economise on the number of moments that need to be stored, with one CIP/IDO variant showing improved isotropy, another failing to maintain its nominal order of accuracy, and one of the conservative variants having eigenvalues with positive real parts: it is stable only in a semi-Lagrangian formulation. These findings should help researchers who are interested in using multi-moment schemes in their solvers but are unsure as to which are suitable. The thesis then addresses the question as to whether a multi-moment method could be implemented on a Cartesian cut cell grid. Such grids are attractive for supporting arbitrary, possibly moving boundaries with minimal grid regeneration. A pair of novel conservative fourth order schemes is proposed. The first scheme, occupying the Cartesian interior, has unprecedented low memory cost and is proven to be conditionally stable. The second, occupying the cut cells, involves a profile reconstruction that is guaranteed to be well-behaved for any shape of cell. However, analysis of the second scheme in a simple grid arrangement reveals positive real parts, so it is not stable in an Eulerian formulation. Stability in a hybrid formulation remains open to question.

Interface tracking methods for segregated flows such as breaking ocean waves are an important tool in marine engineering. With the development in marine renewable devices increasing and a multitude of other marine flow problems that benefit from the possibility of simulation on computer, the need for accurate free surface solvers capable of solving wave simulations has never been greater. An important component of successfully simulating segregated flow of any type is accurately tracking the position of the separating interface between fluids. It is desirable to represent the interface as a sharp, smooth, continuous entity in simulations. Popular Eulerian interface tracking methods appropriate for segregated flows such as the Marker and Cell Method (MAC) and the Volume of Fluid (VOF) were considered. However these methods have drawbacks with smearing of the interface and high computational costs in 3D simulations being among the most prevalent. This PhD project uses a level set method to implicitly represent an interface. The level set method is a signed distance function capable of both sharp and smooth representations of a free surface. It was found, over time, that the level set function ceases to represent a signed distance due to interaction of local velocity fields. This affects the accuracy to which the level set can represent a fluid interface, leading to mass loss. An advection solver, the Cubic Interpolated Polynomial (CIP) method, is presented and tested for its ability to transport a level set interface around a numerical domain in 2D. An advection problem of the level set function demonstrates the mass loss that can befall the method. To combat this, a process known as reinitialisation can be used to re-distance the level set function between time-steps, maintaining better accuracy. The goal of this PhD project is to present a new numerical gradient approximation that allows for the extension of the reinitialisation method to unstructured numerical grids. A particular focus is the Cartesian cut cell grid method. It allows geometric boundaries of arbitrary complexity to be cut from a regular Cartesian grid, allowing for flexible high quality grid generation with low computational cost. A reinitialisation routine using 1st order gradient approximation is implemented and demonstrated with 1D and 2D test problems. An additional area-conserving constraint is introduced to improve accuracy further. From the results, 1st order gradient approximation is shown to be inadequate for improving the accuracy of the level set method. To obtain higher accuracy and the potential for use on unstructured grids a novel gradient approximation based on a slope limited least squares method, suitable for level set reinitialisation, is developed. The new gradient scheme shows a significant improvement in accuracy when compared with level set reinitialisation methods using a lower order gradient approximation on a structured grid. A short study is conducted to find the optimal parameters for running 2D level set interface tracking and the new reinitialisation method. The details of the steps required to implement the current method on a Cartesian cut cell grid are discussed. Finally, suggestions for future work using the methods demonstrated in the thesis are presented.

We present a novel dimensionally split Cartesian cut cell method to compute inviscid, viscous and turbulent flows around rigid geometries. On a cut cell mesh, the existence of arbitrarily small boundary cells severely restricts the stable time step for an explicit numerical scheme. We solve this `small cell problem' when computing solutions for hyperbolic conservation laws by combining wave speed and geometric information to develop a novel stabilised cut cell flux. The convergence and stability of the developed technique are proved for the one-dimensional linear advection equation, while its multi-dimensional numerical performance is investigated through the computation of solutions to a number of test problems for the linear advection and Euler equations. This work was recently published in the Journal of Computational Physics (Gokhale et al., 2018). Subsequently, we develop the method further to be able to compute solutions for the compressible Navier-Stokes equations. The method is globally second order accurate in the L1 norm, fully conservative, and allows the use of time steps determined by the regular grid spacing. We provide a full description of the three-dimensional implementation of the method and evaluate its numerical performance by computing solutions to a wide range of test problems ranging from the nearly incompressible to the highly compressible flow regimes. This work was recently published in the Journal of Computational Physics (Gokhale et al., 2018). It is the first presentation of a dimensionally split cut cell method for the compressible Navier-Stokes equations in the literature. Finally, we also present an extension of the cut cell method to solve high Reynolds number turbulent automotive flows using a wall-modelled Large Eddy Simulation (WMLES) approach. A full description is provided of the coupling between the (implicit) LES solution and an equilibrium wall function on the cut cell mesh. The combined methodology is used to compute results for the turbulent flow over a square cylinder, and for flow over the SAE Notchback and DrivAer reference automotive geometries. We intend to publish the promising results as part of a future publication, which would be the first assessment of a WMLES Cartesian cut cell approach for computing automotive flows to be presented in the literature.

La méthode LS-STAG est une méthode de type frontière immergée/cut-cell pour le calcul d’écoulements visqueux incompressibles qui est basée sur la méthode MAC pour grilles cartésiennes décalées, où la frontière irrégulière est nettement représentée par sa fonction level-set, résultant en un gain significatif en ressources informatiques par rapport aux codes MFN commerciaux utilisant des maillages qui épousent la géométrie. La version 2D est maintenant bien établie et ce manuscrit présente son extension aux géométries 3D avec une symétrie translationnelle dans la direction z (configurations extrudées 3D). Cette étape intermédiaire sera considérée comme la clé de voûte du solveur 3D complet, puisque les problèmes de discrétisation et d’implémentation sur les machines à mémoire distribuée sont abordés à ce stade de développement. La méthode LS-STAG est ensuite appliquée à divers écoulements newtoniens et non-newtoniens dans des géométries extrudées 3D (conduite axisymétrique, cylindre circulaire, conduite cylindrique avec élargissement brusque, etc.) pour lesquels des résultats de références et des données expérimentales sont disponibles. Le but de ces investigations est d’évaluer la précision de la méthode LS-STAG, d’évaluer la polyvalence de la méthode pour les applications d’écoulement dans différents régimes (fluides newtoniens et rhéofluidifiants, écoulement laminaires stationnaires et instationnaires, écoulements granulaires) et de comparer ses performances avec de méthodes numériques bien établies (méthodes non structurées et de frontières immergées) / The LS-STAG method is an immersed boundary/cut-cell method for viscous incompressible flows based on the staggered MAC arrangement for Cartesian grids where the irregular boundary is sharply represented by its level-set function. This approach results in a significant gain in computer resources compared to commercial body-fitted CFD codes. The 2D version of LS-STAG method is now well-established and this manuscript presents its extension to 3D geometries with translational symmetry in the z direction (3D extruded configurations). This intermediate step will be regarded as the milestone for the full 3D solver, since both discretization and implementation issues on distributed memory machines are tackled at this stage of development. The LS-STAG method is then applied to Newtonian and non-Newtonian flows in 3D extruded geometries (axisymmetric pipe, circular cylinder, duct with an abrupt expansion, etc.) for which benchmark results and experimental data are available. The purpose of these investigations is to evaluate the accuracy of LS-STAG method, to assess the versatility of method for flow applications at various regimes (Newtonian and shear-thinning fluids, steady and unsteady laminar to turbulent flows, granular flows) and to compare its performance with well-established numerical methods (body-fitted and immersed boundary methods)

Ce travail est dédié à la simulation d’écoulements multidimensionnels de gaz raréfiés dans un domaine où l’interface avec le solide est mobile. Le comportement du gaz est modélisé par un modèle de type BGK de l’équation de Boltzmann et une méthode déterministe de vitesses discrètes est utilisée pour discrétiser l’espace des vitesses microscopiques.Dans ce document, nous proposons tout d’abord trois discrétisations spatiales du modèle qui permettent la prise en compte du mouvement des parois solides, grâce à un traitement spécifique des conditions aux limites. Ces approches sont implémentées et validées pour plusieurs cas unidimensionnels et à la suite de cette étude, la méthode maille coupée est choisie pour une extension à des écoulements de dimensions plus élevées.La suite du travail présente l’algorithme utilisé pour la simulation d’écoulements 2D et 3D. La précision et la robustesse de l’implémentation sont mises en avant grâce à la simulation de nombreux cas tests, dont les résultats sont comparés à ceux issus de la littérature. La méthode maille coupée a notamment été optimisée par une technique de raffinement de maillage adaptatif. La simulation instationnaire 3D de la rotation des pâles du radiomètre de Crookes illustre pleinement le potentiel de la méthode. / This work is devoted to the multidimentional simulation of rarefied gases in a domain with moving boundary. The governing equation is given by BGKtype model of Boltzmann equation and velocity space is discretized with a standard discrete velocity method.We first propose three space discretizations that take boundary motion into account by specific treatment of the boundary conditions. These approaches are implemented and validated for several 1D flows. Based on this study, the cut cell method is chosen to be extend to multidimentional flows.Then we detail the cut cell algorithm for 2D and 3D flow simulations. Robustness and accuracy of the implementation are investigated through the simulation of numerous test cases. Our results are rigorously compared to the ones coming from the literature and good agreement is shown. The cut cell method has been optimized with an adaptive refinement mesh technique. The 3D unstationary simulation of the Crookes radiometer rotating vanes is a perfect illustration of the method potential.

Despite the high cost of memory and CPU time required to resolve the boundary layer, a viscous unstructured grid solver has many advantages over a structured grid solver such as the convenience in automated grid generation and shock or vortex capturing by solution adaption. Since the geometry and flow phenomenon of a helicopter are very complex, unstructured grid-based methods are well-suited to model properly the rotor-fuselage interaction than the structured grid solver. In present study, an unstructured Cartesian grid solver is developed on the basis of the existing solver, NASCART-GT. Instead of cut-cell approach, immersed boundary approach is applied with ghost cell boundary condition, which increases the accuracy and minimizes unphysical fluctuations of the flow properties. The standard k-epsilon model by Launder and Spalding is employed for the turbulence modeling, and a new wall function approach is devised for the unstructured Cartesian grid solver. It is quite challenging and has never done before to apply wall function approach to immersed Cartesian grid. The difficulty lies in the inability to acquire smooth variation of y+ in the desired range due to the non-body-fitted cells near the solid wall. The wall function boundary condition developed in this work yields stable and reasonable solution within the accuracy of the turbulence model. The grid efficiency is also improved with respect to the conventional method. The turbulence modeling is validated and the efficiency of the developed boundary condition is tested in 2-D flow field around a flat plate, NACA0012 airfoil, axisymmetric hemispheroid, and rotorcraft applications. For rotor modeling, an actuator disk model is chosen, since it is efficient and is widely verified in the study of the rotor-fuselage interaction. This model considers the rotor as an infinitely thin disk, which carries pressure jump across the disk and allows flow to pass through it. The full three dimensional calculations of Euler and RANS equations are performed for the GT rotor model and ROBIN configuration to test implemented actuator disk model along with the developed turbulence modeling. Finally, the characteristics of the rotor-fuselage interaction are investigated by comparing the numerical solutions with the experiments.