Multiple-correction hybrid k -exact schemes for high-order compressible RANS-LES simulations on fully unstructured grids

A Godunov's type unstructured finite volume method suitable for highly compressible turbulent scale-resolving simulations around complex geometries is constructed by using a successive correction technique. First, a family of k-exact Godunov schemes is developed by recursively correcting the truncation error of the piecewise polynomial representation of the primitive variables. The keystone of the proposed approach is a quasi-Green gradient operator which ensures consistency on general meshes. In addition, a high-order single-point quadrature formula, based on high-order approximations of the successive derivatives of the solution, is developed for flux integration along cell faces. The proposed family of schemes is compact in the algorithmic sense, since it only involves communications between direct neighbors of the mesh cells. The numerical properties of the schemes up to fifth-order are investigated, with focus on their resolvability in terms of number of mesh points required to resolve a given wavelength accurately. Afterwards, in the aim of achieving the best possible trade-off between accuracy, computational cost and robustness in view of industrial flow computations, we focus more specifically on the third-order accurate scheme of the family, and modify locally its numerical flux in order to reduce the amount of numerical dissipation in vortex-dominated regions. This is achieved by switching from the upwind scheme, mostly applied in highly compressible regions, to a fourth-order centered one in vortex-dominated regions. An analytical switch function based on the local grid Reynolds number is adopted in order to warrant numerical stability of the recentering process. Numerical applications demonstrate the accuracy and robustness of the proposed methodology for compressible scale-resolving computations. In particular, supersonic RANS/LES computations of the flow over a cavity are presented to show the capability of the scheme to predict flows with shocks, vortical structures and complex geometries.A Godunov's type unstructured finite volume method suitable for highly compressible turbulent scale-resolving simulations around complex geometries is constructed by using a successive correction technique. First, a family of k-exact Godunov schemes is developed by recursively correcting the truncation error of the piecewise polynomial representation of the primitive variables. The keystone of the proposed approach is a quasi-Green gradient operator which ensures consistency on general meshes. In addition, a high-order single-point quadrature formula, based on high-order approximations of the successive derivatives of the solution, is developed for flux integration along cell faces. The proposed family of schemes is compact in the algorithmic sense, since it only involves communications between direct neighbors of the mesh cells. The numerical properties of the schemes up to fifth-order are investigated, with focus on their resolvability in terms of number of mesh points required to resolve a given wavelength accurately. Afterwards, in the aim of achieving the best possible trade-off between accuracy, computational cost and robustness in view of industrial flow computations, we focus more specifically on the third-order accurate scheme of the family, and modify locally its numerical flux in order to reduce the amount of numerical dissipation in vortex-dominated regions. This is achieved by switching from the upwind scheme, mostly applied in highly compressible regions, to a fourth-order centered one in vortex-dominated regions. An analytical switch function based on the local grid Reynolds number is adopted in order to warrant numerical stability of the recentering process. Numerical applications demonstrate the accuracy and robustness of the proposed methodology for compressible scale-resolving computations. In particular, supersonic RANS/LES computations of the flow over a cavity are presented to show the capability of the scheme to predict flows with shocks, vortical structures and complex geometries

Multi-moment advection schemes for Cartesian grids and cut cells

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

DisPar Methods and Their Implementation on a Heterogeneous PC Cluster

Esta dissertação avalia duas áreas cruciais da simulação de advecção- difusão. A primeira parte é dedicada a estudos numéricos. Foi comprovado que existe uma relação directa entre os momentos de deslocamento de uma partícula de poluente e os erros de truncatura. Esta relação criou os fundamentos teóricos para criar uma nova família de métodos numéricos, DisPar. Foram introduzidos e avaliados três métodos. O primeiro é um método semi-Lagrangeano 2D baseado nos momentos de deslocamento de uma partícula para malhas regulares, DisPar-k. Com este método é possível controlar explicitamente o erro de truncatura desejado. O segundo método também se baseia nos momentos de deslocamento de uma partícula, sendo, contudo, desenvolvido para malhas uniformes não regulares, DisParV. Este método também apresentou uma forte robustez numérica. Ao contrário dos métodos DisPar-K e DisParV, o terceiro segue uma aproximação Eulereana com três regiões de destino da partícula. O método foi desenvolvido de forma a manter um perfil de concentração inicial homogéneo independentemente dos parâmetros usados. A comparação com o método DisPar-k em situações não lineares realçou as fortes limitações associadas aos métodos de advecção-difusão em cenários reais. A segunda parte da tese é dedicada à implementação destes métodos num Cluster de PCs heterogéneo. Para o fazer, foi desenvolvido um novo esquema de partição, AORDA. A aplicação, Scalable DisPar, foi implementada com a plataforma da Microsoft .Net, tendo sido totalmente escrita em C#. A aplicação foi testada no estuário do Tejo que se localiza perto de Lisboa, Portugal. Para superar os problemas de balanceamento de cargas provocados pelas marés, foram implementados diversos esquemas de partição: “Scatter Partitioning”, balanceamento dinâmico de cargas e uma mistura de ambos. Pelos testes elaborados, foi possível verificar que o número de máquinas vizinhas se apresentou como o mais limitativo em termos de escalabilidade, mesmo utilizando comunicações assíncronas. As ferramentas utilizadas para as comunicações foram a principal causa deste fenómeno. Aparentemente, o Microsoft .Net remoting 1.0 não funciona de forma apropriada nos ambientes de concorrência criados pelas comunicações assíncronas. Este facto não permitiu a obtenção de conclusões acerca dos níveis relativos de escalabilidade das diferentes estratégias de partição utilizadas. No entanto, é fortemente sugerido que a melhor estratégia irá ser “Scatter Partitioning” associada a balanceamento dinâmico de cargas e a comunicações assíncronas. A técnica de “Scatter Partitioning” mitiga os problemas de desbalanceamentos de cargas provocados pelas marés. Por outro lado, o balanceamento dinâmico será essencialmente activado no inicio da simulação para corrigir possíveis problemas nas previsões dos poderes de cada processador.This thesis assesses two main areas of the advection-diffusion simulation. The first part is dedicated to the numerical studies. It has been proved that there is a direct relation between pollutant particle displacement moments and truncation errors. This relation raised the theoretical foundations to create a new family of numerical methods, DisPar. Three methods have been introduced and appraised. The first is a 2D semi- Lagrangian method based on particle displacement moments for regular grids, DisPar-k. With this method one can explicitly control the desired truncation error. The second method is also based on particle displacement moments but it is targeted to regular/non-uniform grids, DisParV. The method has also shown a strong numerical capacity. Unlike DisPar-k and DisParV, the third method is a Eulerian approximation for three particle destination units. The method was developed so that an initial concentration profile will be kept homogeneous independently of the used parameters. The comparison with DisPar-k in non-linear situations has emphasized the strong shortcomings associated with numerical methods for advection-diffusion in real scenarios. The second part of the dissertation is dedicated to the implementation of these methods in a heterogeneous PC Cluster. To do so, a new partitioning method has been developed, AORDA. The application, Scalable DisPar, was implemented with the Microsoft .Net framework and was totally written in C#. The application was tested on the Tagus Estuary, near Lisbon (Portugal). To overcome the load imbalances caused by tides scatter partitioning was implemented, dynamic load balancing and a mix of both. By the tests made, it was possible to verify that the number of neighboring machines was the main factor affecting the application scalability, even with asynchronous communications. The tools used for communications mainly caused this. Microsoft .Net remoting 1.0 does not seem to properly work in environments with concurrency associated with the asynchronous communications. This did not allow taking conclusions about the relative efficiency between the partitioning strategies used. However, it is strongly suggested that the best approach will be to scatter partitioning with dynamic load balancing and with asynchronous communications. Scatter partitioning mitigates load imbalances caused by tides and dynamic load balancing is basically trigged at the begging of the simulation to correct possible problems in processor power predictions

Geometry considerations for high-order finite-volume methods on structured grids with adaptive mesh refinement

2022 Summer.Includes bibliographical references.Computational fluid dynamics (CFD) is an invaluable tool for engineering design. Meshing complex geometries with accuracy and efficiency is vital to a CFD simulation. In particular, using structured grids with adaptive mesh refinement (AMR) will be invaluable to engineering optimization where automation is critical. For high-order (fourth-order and above) finite volume methods (FVMs), discrete representation of complex geometries adds extra challenges. High-order methods are not trivially extended to complex geometries of engineering interest. To accommodate geometric complexity with structured AMR in the context of high-order FVMs, this work aims to develop three new methods. First, a robust method is developed for bounding high-order interpolations between grid levels when using AMR. High-order interpolation is prone to numerical oscillations which can result in unphysical solutions. To overcome this, localized interpolation bounds are enforced while maintaining solution conservation. This method provides great flexibility in how refinement may be used in engineering applications. Second, a mapped multi-block technique is developed, capable of representing moderately complex geometries with structured grids. This method works with high-order FVMs while still enabling AMR and retaining strict solution conservation. This method interfaces with well-established engineering work flows for grid generation and interpolates generalized curvilinear coordinate transformations for each block. Solutions between blocks are then communicated by a generalized interpolation strategy while maintaining a single-valued flux. Finally, an embedded-boundary technique is developed for high-order FVMs. This method is particularly attractive since it automates mesh generation of any complex geometry. However, the algorithms on the resulting meshes require extra attention to achieve both stable and accurate results near boundaries. This is achieved by performing solution reconstructions using a weighted form of high-order interpolation that accounts for boundary geometry. These methods are verified, validated, and tested by complex configurations such as reacting flows in a bluff-body combustor and Stokes flows with complicated geometries. Results demonstrate the new algorithms are effective for solving complex geometries at high-order accuracy with AMR. This study contributes to advance the geometric capability in CFD for efficient and effective engineering applications