High-resolution numerical schemes for compressible flows and\ud compressible two-phase flows

Several high-resolution numerical schemes based on the Constrained Interpolation Profile Conservative Semi-Lagrangian (CIP-CSL), Essentially Non-Oscillatory (ENO), Weighted ENO (WENO), Boundary Variation Diminishing (BVD), and Tangent of Hyperbola for INterface Capturing (THINC) schemes have been proposed for compressible flows and compressible two-phase flows. In the first part of the thesis, three high-resolution CIP-CSL schemes are proposed. (i) A fully conservative and less oscillatory multi-moment scheme (CIP-CSL3-ENO) is proposed based on two CIP-CSL3 schemes and the ENO scheme. An ENO indicator is designed to intentionally select non-smooth stencil but can efficiently minimise numerical oscillations. (ii) Motivated by the observation that combining two different types of reconstruction functions can effectively reduce numerical diffusion and oscillations, a better-suited scheme CIP-CSL-ENO5 is proposed based on hybrid-type CIP-CSL reconstruction functions and a newly designed ENO indicator. (iii) To further reduce the numerical diffusion in vicinity of discontinuities, the BVD and THINC schemes are implemented in the CIP-CSL framework. The resulting scheme accurately capture both smooth and discontinuous solutions simultaneously by selecting an appropriate reconstruction function. In the second part of the thesis, the TWENO (Target WENO) scheme is proposed to improve the accuracy of the fifth-order WENO scheme. Unlike conventional WENO schemes, the TWENO scheme is designed to restore the highest possible order interAbstract iv polation when three sub-stencils or two adjacent sub-stencils are smooth. To further minimise the numerical diffusion across discontinuities, the TWENO scheme is implemented with the THINC scheme and the Total Boundary Variation Diminishing (TBVD) algorithm. The resulting scheme TBVD-TWENO-THINC is also applied to solve the five-equation model for compressible two-phase flows. Verified through a wide range of benchmark tests, the proposed numerical schemes are able to obtain accurate and high-resolution numerical solutions for compressible flows and compressible two-phase flows

Predictor-Corrector LU-SGS Discontinuous Galerkin Finite Element Method for Conservation Laws

Efficient implicit predictor-corrector LU-SGS discontinuous Galerkin (DG) approach for compressible Euler equations on unstructured grids is investigated by adding the error compensation of high-order term. The original LU-SGS and GMRES schemes for DG method are discussed. Van Albada limiter is employed to make the scheme monotone. The numerical experiments performed for the transonic inviscid flows around NACA0012 airfoil, RAE2822 airfoil, and ONERA M6 wing indicate that the present algorithm has the advantages of low storage requirements and high convergence acceleration. The computational efficiency is close to that of GMRES scheme, nearly 2.1 times greater than that of LU-SGS scheme on unstructured grids for 2D cases, and almost 5.5 times greater than that of RK4 on unstructured grids for 3D cases

Development and applications of the Finite Point Method to compressible aerodynamics problems

This work deals with the development and application of the Finite Point Method (FPM) to compressible aerodynamics problems. The research focuses mainly on investigating the capabilities of the meshless technique to address practical problems, one of the most outstanding issues in meshless methods. The FPM spatial approximation is studied firstly, with emphasis on aspects of the methodology that can be improved to increase its robustness and accuracy. Suitable ranges for setting the relevant approximation parameters and the performance likely to be attained in practice are determined. An automatic procedure to adjust the approximation parameters is also proposed to simplify the application of the method, reducing problem- and user-dependence without affecting the flexibility of the meshless technique. The discretization of the flow equations is carried out following wellestablished approaches, but drawing on the meshless character of the methodology. In order to meet the requirements of practical applications, the procedures are designed and implemented placing emphasis on robustness and efficiency (a simplification of the basic FPM technique is proposed to this end). The flow solver is based on an upwind spatial discretization of the convective fluxes (using the approximate Riemann solver of Roe) and an explicit time integration scheme. Two additional artificial diffusion schemes are also proposed to suit those cases of study in which computational cost is a major concern. The performance of the flow solver is evaluated in order to determine the potential of the meshless approach. The accuracy, computational cost and parallel scalability of the method are studied in comparison with a conventional FEM-based technique. Finally, practical applications and extensions of the flow solution scheme are presented. The examples provided are intended not only to show the capabilities of the FPM, but also to exploit meshless advantages. Automatic hadaptive procedures, moving domain and fluid-structure interaction problems, as well as a preliminary approach to solve high-Reynolds viscous flows, are a sample of the topics explored. All in all, the results obtained are satisfactorily accurate and competitive in terms of computational cost (if compared with a similar mesh-based implementation). This indicates that meshless advantages can be exploited with efficiency and constitutes a good starting point towards more challenging applications.En este trabajo se aborda el desarrollo del Método de Puntos Finitos (MPF) y su aplicación a problemas de aerodinámica de flujos compresibles. El objetivo principal es investigar el potencial de la técnica sin malla para la solución de problemas prácticos, lo cual constituye una de las limitaciones más importantes de los métodos sin malla. En primer lugar se estudia la aproximación espacial en el MPF, haciendo hincapié en aquéllos aspectos que pueden ser mejorados para incrementar la robustez y exactitud de la metodología. Se determinan rangos adecuados para el ajuste de los parámetros de la aproximación y su comportamiento en situaciones prácticas. Se propone además un procedimiento de ajuste automático de estos parámetros a fin de simplificar la aplicación del método y reducir la dependencia de factores como el tipo de problema y la intervención del usuario, sin afectar la flexibilidad de la técnica sin malla. A continuación se aborda el esquema de solución de las ecuaciones del flujo. La discretización de las mismas se lleva a cabo siguiendo métodos estándar, pero aprovechando las características de la técnica sin malla. Con el objetivo de abordar problemas prácticos, se pone énfasis en la robustez y eficiencia de la implementación numérica (se propone además una simplificación del procedimiento de solución). El comportamiento del esquema se estudia en detalle para evaluar su potencial y se analiza su exactitud, coste computacional y escalabilidad, todo ello en comparación con un método convencional basado en Elementos Finitos. Finalmente se presentan distintas aplicaciones y extensiones de la metodología desarrollada. Los ejemplos numéricos pretenden demostrar las capacidades del método y también aprovechar las ventajas de la metodología sin malla en áreas en que la misma puede ser de especial interés. Los problemas tratados incluyen, entre otras características, el refinamiento automático de la discretización, la presencia de fronteras móviles e interacción fluido-estructura, como así también una aplicación preliminar a flujos compresibles de alto número de Reynolds. Los resultados obtenidos muestran una exactitud satisfactoria. Además, en comparación con una técnica similar basada en Elementos Finitos, demuestran ser competitivos en términos del coste computacional. Esto indica que las ventajas de la metodología sin malla pueden ser explotadas con eficiencia, lo cual constituye un buen punto de partida para el desarrollo de ulteriores aplicaciones.Postprint (published version