Cell-average WENO with progressive order of accuracy close to discontinuities with applications to signal processing

In this paper we translate to the cell-average setting the algorithm for the point-value discretization presented in Amat el al. (2020). This new strategy tries to improve the results of WENO-(2r−1) algorithm close to the singularities, resulting in an optimal order of accuracy at these zones. The main idea is to modify the optimal weights so that they have a nonlinear expression that depends on the position of the discontinuities. In this paper we study the application of the new algorithm to signal processing using Harten’s multiresolution. Several numerical experiments are performed in order to confirm the theoretical results obtained.This work was funded by project 20928/PI/18 (Proyecto financiado por la Comunidad Autónoma de la Región de Murcia a través de la convocatoria de Ayudas a proyectos para el desarrollo de investigación científica y técnica por grupos competitivos, incluida en el Programa Regional de Fomento de la Investigación Científica y Técnica (Plan de Actuación 2018) de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia) and through the national research project (MINECO/FEDER) PID2019-108336GB-I00. 2 The author has been supported through the NSF grant DMS-2010107 and AFOSR grant FA9550-20-1-0055. 3 The author has been supported through the Spanish MINECO project MTM2017-83942-P

Direct numerical simulation of supersonic flow and acoustics over a compression ramp

We present direct numerical simulations of the shock wave boundary layer interaction (SBLI) at Mach number 2.9 over a 24° ramp. We study both the numerical accuracy and flow physics. Two classes of spatial reconstruction schemes are employed: the monotonic upstream-centered scheme for conservation laws and the Weighted Essentially Non-Oscillatory (WENO) scheme, of accuracy ranging from 2nd- to 11th-order. Using the canonical Taylor–Green vortex test-case, a simple and computationally inexpensive rescaling of the candidate stencil values—within the context of the high-order WENO scheme—is proposed for reducing the numerical dissipation, particularly in under-resolved simulations. For the compression ramp case, higher-order schemes are shown to capture the size of the SBLI separation zone more accurately, a consequence of resolving much finer turbulence structures. For second- and fifth-order schemes, the energy of the unresolved small scale turbulence shifts the cascade of the turbulence kinetic energy (TKE) spectrum, thus resulting in more energetic large scale turbulent structures. Consequently, the λ-shock foot shifts further downstream, leading to a smaller separation bubble size. Nonetheless, other statistical quantities, such as the turbulence anisotropy invariant map and the turbulence kinetic energy budget terms, show little dependence on the type and order of the spatial reconstruction scheme. Finally, using the more accurate ninth-order WENO results, it is reasoned that the interaction of the λ-shock with the post-shock relaxation region drives the low-frequency oscillation of the λ-shock

Investigation of high-order, high-resolution methods for axisymmetric turbulent jet using ILEs

This Philosophiae Doctor thesis presents the motivation, objectives and reasoning behind the undertaken project. This research, study the capability of compressible Implicit Large Eddy Simulation (ILES) in predicting free shear layer flows, under different free stream regimes (Static and Co-flow jets). Unsteady flows or jet flows are non-uniform in structure, temperature, pressure and velocity. Turbulent mixing is of particular importance for the developing of this class of flows. As a shear layer is formed immediately downstream of the jet exhaust, an early linear instability involving exponential growth of small perturbations is introduced at the jet discharge. Beyond this development stage, in the non-linear Kelvin-Helmholtz instability region large scale vortex rings roll up, and their dynamics of formation and merging become the defining feature of the transitional shear flow into fully developed regime. This class of flows is particularly relevant to numerical predictions, as the extreme nature of the flow in question is considered as a benchmark; however, experimental data should be selected carefully as some results are controversial. To qualify the behaviour of unsteady flows, some important criteria have been selected for the analysis of the flow quantities at different regions of the flow field (average velocities, Reynolds stresses and dissipation rates). A good estimation of high-order statistics (Standard Deviation, Skewness and Kurtosis) correspond to mathematical steadiness and convergence of results. From the physical point of view, similarity analysis between jet’s wake sections reveals physical steadiness in results. Spectral analysis of the different regions of the flow field could be used as a sign that the energy cascade is correctly predicted or efficiently enough since this is where the smallest scales are usually present and which in effect require to be modelled by the different numerical schemes. The flow solver has been reviewed and improved. The former, a revised version of the reconstruction numerical schemes (WENO 5th and WENO 9th orders) has been performed and tested, the correspondent results have been compared against analytical data; the latter, correction of the method to compute the Jacobian of the transformation (singularity correction), by changing from the standard algebraic to geometric method, and augmented with transparent boundary condition, giving mathematical and physical meaning to the obtained results. The flow solver improvements and review have been verified and validated through simulations of a compressible Convergent-Divergent Nozzle (CDN), and the standard and a modified version of the Shock tube test cases, where the results are gained with minimal modelling effort. The study of numerical errors associated with the simulations of turbulent flows, for unsteady explicit time step predictions, have been performed and a new formula proposed. Ten different computational methods have been employed in the framework of ILES and computations have been performed for a jet flow configuration for which experimental data and DNS are available. It can be seen that a numerical error bar can be defined that takes into account the errors arising from the different numerical building blocks of the simulation method. The effects of different grids, Riemann solvers and numerical reconstruction schemes have been considered, however, the approach can be extended to take into account the effects of the initial and boundary conditions as well as subgrid scale modelling, if applicable. From the physical analysis several observations were established, revealing that differences in terms of jet’s core size are not an important parameter in terms of quantification and qualification of predictions, in other words, data should be reduced to the jet’s inertial reference system. Moreover, the comparative study has been performed to identify the differences between Riemann solvers (CBS and HLLC), Low Mach number Limiting/ Corrections (LMC), numerical reconstruction schemes (MUSCL and WENO) and spatial order of accuracy (2nd-order LMC, 5th-order LMC and 9th-order schemes) in combination with the most efficient cost/resolution discretization level (Medium mesh). The comparisons between results reveals for the Static and Co-Flow jets that the CBS MUSCL 5th-order LMC and the HLLC MUSCL 5th-order LMC as the most accurate schemes in predicting this class of flows, accordingly. Furthermore, the selected numerical methods show to be in accordance with the empirical (Static) and experimental (Co-flow) results in terms of resonance frequency and/or Strouhal number; also, the expected behaviour in terms of spectral energy decay rate throughout the jet’s central line is observed. To conclude the study of the Static jet case, a possible explanation for the jet’s buoyancy effect is presented.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Numerical solutions of the general relativistic equations for black hole fluid dynamics

Related data - see record for Numerical Solutions of the General Relativistic Equations for Black Hole Fluid Dynamics at http://www.dspace.cam.ac.uk/handle/1810/226117 containing the data from the DVD referred to in the main text.The aims of this thesis are to develop and validate a robust and efficient algorithm for the numerical solution of the equations of General Relativistic Hydrodynamics, to implement the algorithm in a computationally efficient manner, and to apply the resulting computer code to the problem of perturbed Bondi-Hoyle-Lyttleton accretion onto a Kerr black hole. The algorithm will also be designed to evolve the space-time metric, and standardised tests will be applied to this aspect of the algorithm. The algorithm will use up-to-date High-Resolution Shock-Capturing numerical schemes that have been developed for the stable and accurate solution of complex systems of equations. It will be built around the Adaptive Mesh Refinement and overlapping, curvilinear grid methodologies in order to extend these schemes to the efficient solution of two and three-dimensional problems. When implementing the algorithm, we will use previously written code libraries, where appropriate, to avoid excessive software development. We will validate the algorithm against standard test-cases for Special and General Relativistic Hydrodynamics, and for Einstein's equations for the evolution of the space-time metric. The methodologies we use will be tested to ensure that they lead to the stable and accurate numerical solution of these problems. Finally, the implemented algorithm will be applied to the problem of Bondi-Hoyle-Lyttleton flow onto a Kerr black hole in three dimensions. It will be validated against existing exact and numerical solutions of the problem, and then be used to perform an extensive parametric study of the problem, varying the spin of the black hole and the incident wind direction, and allowing for the perturbation of the fluid density upstream of the black hole. We will then analyze the results of the study, and present the complete set of results on a DVD accompanying this thesis.This work was supported by an EPSRC Doctoral Training Grant. Some simulations were performed using the Darwin Supercomputer of the University of Cambridge High Performance Computing Service (http://www.hpc.cam.ac.uk/), provided by Dell Inc. using Strategic Research Infrastructure Funding from the Higher Education Funding Council for England

Advanced numerical approaches in the dynamics of relativistic flows

[no abstract

Conservative finite-volume framework and pressure-based algorithm for flows of incompressible, ideal-gas and real-gas fluids at all speeds

A conservative finite-volume framework, based on a collocated variable arrangement, for the simulation of flows at all speeds, applicable to incompressible, ideal-gas and real-gas fluids is proposed in conjunction with a fully-coupled pressure-based algorithm. The applied conservative discretisation and implementation of the governing conservation laws as well as the definition of the fluxes using a momentum-weighted interpolation are identical for incompressible and compressible fluids, and are suitable for complex geometries represented by unstructured meshes. Incompressible fluids are described by predefined constant fluid properties, while the properties of compressible fluids are described by the Noble-Abel-stiffened-gas model, with the definitions of density and specific static enthalpy of both incompressible and compressible fluids combined in a unified thermodynamic closure model. The discretised governing conservation laws are solved in a single linear system of equations for pressure, velocity and temperature. Together, the conservative finite-volume discretisation, the unified thermodynamic closure model and the pressure-based algorithm yield a conceptually simple, but versatile, numerical framework. The proposed numerical framework is validated thoroughly using a broad variety of test-cases, with Mach numbers ranging from 0 to 239, including viscous flows of incompressible fluids as well as the propagation of acoustic waves and transiently evolving supersonic flows with shock waves in ideal-gas and real-gas fluids. These results demonstrate the accuracy, robustness and the convergence, as well as the conservation of mass and energy, of the numerical framework for flows of incompressible and compressible fluids at all speeds, on structured and unstructured meshes

Implicit large eddy simulation for unsteady multi-component compressible turbulent flows

Numerical methods for the simulation of shock-induced turbulent mixing have been investigated, focussing on Implicit Large Eddy Simulation. Shock-induced turbulent mixing is of particular importance for many astrophysical phenomena, inertial confinement fusion, and mixing in supersonic combustion. These disciplines are particularly reliant on numerical simulation, as the extreme nature of the flow in question makes gathering accurate experimental data difficult or impossible. A detailed quantitative study of homogeneous decaying turbulence demonstrates that existing state of the art methods represent the growth of turbulent structures and the decay of turbulent kinetic energy to a reasonable degree of accuracy. However, a key observation is that the numerical methods are too dissipative at high wavenumbers (short wavelengths relative to the grid spacing). A theoretical analysis of the dissipation of kinetic energy in low Mach number flows shows that the leading order dissipation rate for Godunov-type schemes is proportional to the speed of sound and the velocity jump across the cell interface squared. This shows that the dissipation of Godunov-type schemes becomes large for low Mach flow features, hence impeding the development of fluid instabilities, and causing overly dissipative turbulent kinetic energy spectra. It is shown that this leading order term can be removed by locally modifying the reconstruction of the velocity components. As the modification is local, it allows the accurate simulation of mixed compressible/incompressible flows without changing the formulation of the governing equations. In principle, the modification is applicable to any finite volume compressible method which includes a reconstruction stage. Extensive numerical tests show great improvements in performance at low Mach compared to the standard scheme, significantly improving turbulent kinetic energy spectra, and giving the correct Mach squared scaling of pressure and density variations down to Mach 10−4. The proposed modification does not significantly affect the shock capturing ability of the numerical scheme. The modified numerical method is validated through simulations of compressible, deep, open cavity flow where excellent results are gained with minimal modelling effort. Simulations of single and multimode Richtmyer-Meshkov instability show that the modification gives equivalent results to the standard scheme at twice the grid resolution in each direction. This is equivalent to sixteen times decrease in computational time for a given quality of results. Finally, simulations of a shock-induced turbulent mixing experiment show excellent qualitative agreement with available experimental data

Implicit large eddy simulation of turbulent duct flows

Ducts can be found in ventilation systems, cooling ducts and blade passages of turbines, centrifugal pumps and many other engineering installations. The properties of the flow in ducts can significantly affect the performance and efficiency of these installation areas. The majority of the flows in ducts and engineering applications are turbulent. The work presented in this thesis focuses on the analysis of turbulent flows inside square sectioned ducts and ducts with bends. The accuracy of three different high resolution high order schemes in the context of Implicit Large Eddy Simulation (ILES) is analysed. The influence of a low Mach limiting technique, Low Mach Number Treatment (LMNT) is also studied. The schemes employed are Monotonic Upwind Scheme for Scalar Conservation Laws (MUSCL) with a 2nd order Monotonized Central (MC) and 5th order limiter, and a 9th order Weighted Essential Non-Oscillatory (WENO) limiter. The first case studied is a duct of square cross section . In the absence of experimental data for the duct case, the data from a plain channel flow is used to shed light on the results. The flow analysis points out the generation of secondary motions created by the existence of surrounding walls. All schemes employed lead to a similarly developed turbulent flow that is used to provide the turbulent boundary profile for the following case. LMNT proves to significantly assist MUSCL 2nd and 5th, that use it, in providing a turbulent profile similar to that of WENO 9th that did not employ the technique but is inherently less dissipative. The second case under study is that of a square sectioned duct with a 90o bend. The simulation output is in good agreement both qualitatively and quantitatively with the experimental data available in the literature. The generation of secondary flows inside the bend is observed without flow separation. Although the turbulent flow entering the domain is almost the same for all cases, differences between the schemes are noticed especially after the middle of the bend. LMNT leads to an overprediction of turbulence after that area for both schemes employing it while WENO 9th without LMNT provides the most accurate results compared to those provided by the experiment. The results demonstrate applicability of ILES to strongly confined flows with secondary motions and shed light on cognitive properties of a wide range of state of the art schemes.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Mathematical and numerical modelling of dispersive water waves

Fecha de lectura de Tesis: 4 diciembre 2018.En esta tesis doctoral se expone en primer lugar una visión general del modelado de ondas dispersivas para la simulación de procesos tsunami-génicos. Se deduce un nuevo sistema bicapa con propiedades de dispersión mejoradas y un nuevo sistema hiperbólico. Además se estudian sus respectivas propiedades dispersivas, estructura espectral y ciertas soluciones analíticas. Así mismo, se ha diseñado un nuevo modelo de viscosidad sencillo para la simulación de los fenómenos físicos relacionados con la ruptura de olas en costa. Se establecen los resultados teóricos requeridos para el diseño de esquemas numéricos de tipo volúmenes finitos y Galerkin discontinuo de alto orden bien equilibrados para sistemas hiperbólicos no conservativos en una y dos dimensiones. Más adelante, los esquemas numéricos propuestos para los sistemas de presión no hidrostática introducidos se describen. Se pueden destacar diferentes enfoques y estrategias. Por un lado, se diseñan esquemas de volúmenes finitos implícitos de tipo proyección-corrección en mallas decaladas y no decaladas. Por otro lado, se propone un esquema numérico de tipo Galerkin discontinuo explícito para el nuevo sistema de EDPs hiperbólico propuesto. Para permitir simulaciones en tiempo real, una implementación eficiente en GPU de los métodos es llevado a cabo y algunas directrices sobre su implementación son dados. Los esquemas numéricos antes mencionados se han aplicado a test de referencia académicos y a situaciones físicas más desafiantes como la simulación de tsunamis reales, y la comparación con datos de campo. Finalmente, un último capítulo es dedicado a medir la influencia al considerar efectos dispersivos en la simulación de transporte y arrastre de sedimentos. Para ello, se deduce un nuevo sistema de dos capas de aguas someras, se diseña un esquema numérico y se muestran algunos test académicos y de validación, que ofrecen resultados prometedores