Resumen de High-order finite volume methods based on moving least squares for computational fluid dynamics: application to all-speed and incompressible flows on unstructured grids

Luís Ramírez

• The development of high-order methods for unstructured grids remains a very active research field in Computational Fluid Dynamics (CFD). In the engineering field, most of the problems are associated with complex geometries. In these problems the use of structured meshes can lead to distorted elements that could affect the accuracy of the method. The great geometrical flexibility offered by unstructured grids makes them highly effective for dealing with complex geometries. In this context, the development of very accurate numerical methods to work on unstructured grids is very desirable.

  During the past two decades, the interest in high-order methods has grown especially in certain applications where the complex flow structure and small length scales need to be adequately resolved. For example, in the simulation of turbulent flows or the propagation of acoustic waves, high-order methods are more suitable than low-order methods. However, with high-order methods there is a need to obtain high-order reconstructions of the variables.

  Despite the progress made in high-order methods for CFD, common industrial simulations on unstructured meshes are usually based on second-order discretizations. These methods have been typically considered as the right choice due to their simplicity, robustness, and their effectiveness in providing a reasonably accurate solution with comparably low computational cost. However, classical second-order algorithms can be insufficient to accurately predict the flow in complicated geometries and complex physics.

  This thesis presents the development of high-order numerical methods for the numerical simulation of all-speed and incompressible flows on unstructured grids. One possible application of the formulation developed in this thesis is the simulation of turbomachinery flow. The operational flow regimes associated to turbomachinery range from very low Mach numbers, which leads to nearly incompressible flows, to supersonic Mach numbers. In order to simulate the flow with a wide range of Mach numbers, a high-order finite volume density-based formulation for all-speed flows is proposed in this thesis. The high-order reconstruction of the variables is obtained by means of a Taylor series expansion. The gradients and high-order derivatives are obtained with the Moving Least Squares approximants. It is known that density-based solvers present the so-called accuracy problem at low Mach regimes, due to excessive wrong numerical diffusion.

  In this thesis, it is shown that the accuracy problem is alleviated when the order of the method is increased. However, a grid dependency still remains. In order to to circumvent this dependency, several fixes have been proposed in the literature. To the author knowledge, all these fixes have been applied, at most to second-order methods. In this thesis, the use of low-Mach fixes has been extended to high-order numerical methods.

  However, when all the flow in the domain presents a low Mach regime, the resolution of the compressible Navier-Stokes with an all-speed scheme may not be practical. This is motivated by the small time step required due to the large disparity between the acoustic and the flow speed. This regime is common in hydrodynamics and low speed aerodynamics, such as the flow around a tidal turbine. For these cases, the incompressibility assumption can be adopted. In this thesis, a novel high-order pressure-based formulation is proposed for the numerical resolution of the incompressible Navier-Stokes. The Semi Implicit Method for Pressure Linked Equations (SIMPLE) algorithm is used to impose the incompressibility condition on a collocated grid arrangement. The formulation is based on the use of MLS to obtain the high-order approximations of the variables. In order to avoid the checkerboard oscillations and preserve the accuracy of the scheme, MLS is employed to obtain a new Momentum Interpolation Method (MIM).

  The proposed methods have been analyzed on several steady and unsteady numerical test cases with structured and unstructured mesh discretizations. The formal order of convergence is recovered and very accurate results have been obtained.

  In addition, new high-order sliding mesh methods are proposed in order to simulate accurately the flow around rotating geometries. In a high-order framework, to preserve the accuracy of the numerical scheme, the simulation of rotating geometries needs to be of, at least, the same order than the numerical scheme. This is a crucial point in the development of high-order methods for the simulations of turbomachines. In this thesis, the MLS approximants are used to develop a new sliding mesh methodology, which preserves the accuracy of the numerical scheme. The accuracy and robustness of the new methodology has been investigated for several structured and unstructured mesh discretizations. The numerical results have shown that the novel numerical methodologies preserve the formal order of accuracy.