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High-order numerical methods have proven to be an essential tool to improve the accuracy of simulations involving turbulent flows through the solution of conservation laws. Such flows appear in a wide variety of industrial applications and its correct prediction is crucial to reduce the power consumption and improve the efficiency of these processes. The present study implements and analyzes different types of high-order spatial discretization schemes for unstructured grids to assess and quantify their accuracy in simulations of turbulent flows. In particular, high-order Finite Volume methods (FVM) based on least squares and fully constrained deconvolution operators are considered and their accuracy is evaluated in a variety of linear and non-linear test cases and throughanalytical analysis. Special emphasis is placed on the comparison of formally second-order and high-order FVM, showing that the former can over-perform the latter in terms of accuracy and computational performance in under-resolved configurations. High-order Spectral Element methods (SEM), including Spectral Difference (SD) and Flux Reconstruction (FR), are compared in different linear and non-linear configurations. Furthermore, a SD GPU-based solver (based on the open-source PyFR solver) is developed and its performance with respect to other state of the art CPU-based solvers will be discussed, showing that the developed GPU-based solver outperforms other state of the art CPU-based solvers in terms of performance-per-euro and performance-per-watt. The accuracy and behavior of SEM under aliasing are assessed in linear test cases using analytical tools. The use of grids with high-order cells, which allow to better describe the surfaces of interests of a given simulation, in combination with SEM is also analyzed. The latter analysis demonstrates that special care must be taken to ensure appropriate numerical accuracy when utilizing meshes with such elements. This document also presents the development and the analysis of the Spectral Difference Raviart-Thomas (SDRT) method for two and three-dimensional tensor product and simplex elements. This method is equivalent to the SD formulation for tensor product elements and it can be considered as a natural extension of the SD formulation for simplex elements. Additionally, a new family of FR methods, which is equivalent to the SDRT method under certain circumstances, is described. All these developments were implemented in the open-source PyFR solver and are compatible with CPU and GPU architectures. In the context of high-order simulations of turbulent flows found in rotor-stator interaction test cases, a sliding mesh method (which involves non-conformal grids and mesh motion) specifically tailored for massivelyparallel simulations is implemented within a CPU-based solver. The developed method is compatible with second-order and high-order FVM and SEM. Grid movement, needed to simulate rotor-stator test cases due to the relative movement of each domain zone, is treated using the Arbitrary-Lagrangian-Eulerian (ALE) formulation. The analysis of such formulation depicts its important influence on the numerical accuracy and stability of numerical simulations with mesh motion. Moreover, specific non-conformal discretization methodscompatible with second-order and high-order FVM and SEM are developed and their accuracy is assessed on different non-linear test cases. The parallel scalability of the method is assessed with up to 11000 cores, proving appropriate computational efficiency. The accuracy of the implementation is assessed through a set of linear and non-linear test cases. Preliminary results of the turbulent flow around a DGEN 380 fan stage in an under-resolved configuration are shown and compared to available experimental data.
Understanding the motion of fluids is crucial for the development and analysis of new designsand processes in science and engineering. Unstructured meshes are used in this contextsince they allow the analysis of the behaviour of complicated geometries and configurationsthat characterise the designs of engineering structures today. The existing numerical methodsdeveloped for unstructured meshes suffer from poor computational efficiency, and their applicabilityis not universal for any type of unstructured meshes. High-resolution high-orderaccurate numerical methods are required for obtaining a reasonable guarantee of physicallymeaningful results and to be able to accurately resolve complicated flow phenomena thatoccur in a number of processes, such as resolving turbulent flows, for direct numerical simulationof Navier-Stokes equations, acoustics etc. The aim of this research project is to establish and implement universal, high-resolution, veryhigh-order, non-oscillatory finite-volume methods for 3D unstructured meshes. A new classof linear and WENO schemes of very high-order of accuracy (5th) has been developed. Thekey element of this approach is a high-order reconstruction process that can be applied to anytype of meshes. The linear schemes which are suited for problems with smooth solutions, employ a single reconstruction polynomial obtained from a close spatial proximity. In theWENO schemes the reconstruction polynomials, arising from different topological regions, are non-linearly combined to provide high-order of accuracy and shock capturing features. The performance of the developed schemes in terms of accuracy, non-oscillatory behaviourand flexibility to handle any type of 3D unstructured meshes has been assessed in a series oftest problems. The linear and WENO schemes presented achieve very high-order of accuracy(5th). This is the first class of WENO schemes in the finite volume context that possess highorderof accuracy and robust non-oscillatory behaviour for any type of unstructured meshes. The schemes have been employed in a newly developed 3D unstructured solver (UCNS3D). UCNS3D utilises unstructured grids consisted of tetrahedrals, pyramids, prisms and hexahedralelements and has been parallelised using the MPI framework. The high parallel efficiencyachieved enables the large scale computations required for the analysis of new designs andprocesses in science and engineering.
Mathematical models of various natural processes are described by differential equations, systems of partial differential equations and integral equations. In most cases, the exact solution to such problems cannot be determined; therefore, one has to use grid methods to calculate an approximate solution using high-performance computing systems. These methods include the finite element method, the finite difference method, the finite volume method and combined methods. In this Special Issue, we bring to your attention works on theoretical studies of grid methods for approximation, stability and convergence, as well as the results of numerical experiments confirming the effectiveness of the developed methods. Of particular interest are new methods for solving boundary value problems with singularities, the complex geometry of the domain boundary and nonlinear equations. A part of the articles is devoted to the analysis of numerical methods developed for calculating mathematical models in various fields of applied science and engineering applications. As a rule, the ideas of symmetry are present in the design schemes and make the process harmonious and efficient.
Understanding the motion of fluids is crucial for the development and analysis of new designsand processes in science and engineering. Unstructured meshes are used in this contextsince they allow the analysis of the behaviour of complicated geometries and configurationsthat characterise the designs of engineering structures today. The existing numerical methodsdeveloped for unstructured meshes suffer from poor computational efficiency, and their applicabilityis not universal for any type of unstructured meshes. High-resolution high-orderaccurate numerical methods are required for obtaining a reasonable guarantee of physicallymeaningful results and to be able to accurately resolve complicated flow phenomena thatoccur in a number of processes, such as resolving turbulent flows, for direct numerical simulationof Navier-Stokes equations, acoustics etc. The aim of this research project is to establish and implement universal, high-resolution, veryhigh-order, non-oscillatory finite-volume methods for 3D unstructured meshes. A new classof linear and WENO schemes of very high-order of accuracy (5th) has been developed. Thekey element of this approach is a high-order reconstruction process that can be applied to anytype of meshes. The linear schemes which are suited for problems with smooth solutions, employ a single reconstruction polynomial obtained from a close spatial proximity. In theWENO schemes the reconstruction polynomials, arising from different topological regions, are non-linearly combined to provide high-order of accuracy and shock capturing features. The performance of the developed schemes in terms of accuracy, non-oscillatory behaviourand flexibility to handle any type of 3D unstructured meshes has been assessed in a series oftest problems. The linear and WENO schemes presented achieve very high-order of accuracy(5th). This is the first class of WENO schemes in the finite volume context that possess highorderof accuracy and robust non-oscillatory behaviour for any type of unstructured meshes. The schemes have been employed in a newly developed 3D unstructured solver (UCNS3D). UCNS3D utilises unstructured grids consisted of tetrahedrals, pyramids, prisms and hexahedralelements and has been parallelised using the MPI framework. The high parallel efficiencyachieved enables the large scale computations required for the analysis of new designs andprocesses in science and engineering.
Handbook of Grid Generation addresses the use of grids (meshes) in the numerical solutions of partial differential equations by finite elements, finite volume, finite differences, and boundary elements. Four parts divide the chapters: structured grids, unstructured girds, surface definition, and adaption/quality. An introduction to each section provides a roadmap through the material. This handbook covers: Fundamental concepts and approaches Grid generation process Essential mathematical elements from tensor analysis and differential geometry, particularly relevant to curves and surfaces Cells of any shape - Cartesian, structured curvilinear coordinates, unstructured tetrahedra, unstructured hexahedra, or various combinations Separate grids overlaid on one another, communicating data through interpolation Moving boundaries and internal interfaces in the field Resolving gradients and controlling solution error Grid generation codes, both commercial and freeware, as well as representative and illustrative grid configurations Handbook of Grid Generation contains 37 chapters as well as contributions from more than 100 experts from around the world, comprehensively evaluating this expanding field and providing a fundamental orientation for practitioners.
A new implicit and compact optimization-based method is presented for high order derivative calculation for finite-volume numerical method on unstructured meshes. Highorder approaches to gradient calculation are often based on variants of the Least-Squares (L-S) method, an explicit method that requires a stencil large enough to accommodate the necessary variable information to calculate the derivatives. The new scheme proposed here is applicable for an arbitrary order of accuracy (demonstrated here up to 3rd order), and uses just the first level of face neighbors to compute all derivatives, thus reducing stencil size and avoiding stiffness in the calculation matrix. Preliminary results for a static variable field example and solution of a simple scalar transport (advection) equation show that the proposed method is able to deliver numerical accuracy equivalent to (or better than) the nominal order of accuracy for both 2nd and 3rd order schemes in the presence of a smoothly distributed variable field (i.e., in the absence of discontinuities). This new Optimization-based Gradient REconstruction (herein denoted OGRE) scheme produces, for the simple scalar transport test case, lower error and demands less computational time (for a given level of required precision) for a 3rd order scheme when compared to an equivalent L-S approach on a two-dimensional framework. For three-dimensional simulations, where the L-S scheme fails to obtain convergence without the help of limiters, the new scheme obtains stable convergence and also produces lower error solution when compared to a third order MUSCL scheme. Furthermore, spectral analysis of results from the advection equation shows that the new scheme is better able to accurately resolve high wave number modes, which demonstrates its potential to better solve problems presenting a wide spectrum of wavelengths, for example unsteady turbulent flow simulations.
This book presents the select proceedings of the 48th National Conference on Fluid Mechanics and Fluid Power (FMFP 2021) held at BITS Pilani in December 2021. It covers the topics such as fluid mechanics, measurement techniques in fluid flows, computational fluid dynamics, instability, transition and turbulence, fluid‐structure interaction, multiphase flows, micro- and nanoscale transport, bio-fluid mechanics, aerodynamics, turbomachinery, propulsion and power. The book will be useful for researchers and professionals interested in the broad field of mechanics.
The development of high-order solution methods remain a very active field of research in Computational Fluid Dynamics (CFD). These types of schemes have the potential to reduce the computational cost necessary to compute solutions to a desired level of accuracy. The goal of this thesis has been to develop a high-order Central Essentially Non Oscillatory (CENO) finite volume scheme for multi-block unstructured meshes. In particular, solutions to the compressible, inviscid Euler equations are considered. The CENO method achieves a high-order spatial reconstruction based on the k-exact method, combined with hybrid switching to limited piecewise linear reconstruction in non-smooth regions to maintain monotonicity. Additionally, fourth-order Runge-Kutta time marching is applied. The solver described has been validated through a combination of high-order function reconstructions, and solutions to the Euler equations. Cases have been selected to demonstrate high-orders of convergence, the application of the hybrid switching method, and the multi-block techniques which has been implemented.