High-order WENO scheme for Polymerization-type equations

Polymerization of proteins is a biochimical process involved in different diseases. Mathematically, it is generally modeled by aggregation-fragmentation-type equations. In this paper we consider a general polymerization model and propose a high-order numerical scheme to investigate the behavior of the solution. An important property of the equation is the mass conservation. The fifth-order WENO scheme is built to preserve the total mass of proteins along time

Simulation techniques for cosmological simulations

Modern cosmological observations allow us to study in great detail the evolution and history of the large scale structure hierarchy. The fundamental problem of accurate constraints on the cosmological parameters, within a given cosmological model, requires precise modelling of the observed structure. In this paper we briefly review the current most effective techniques of large scale structure simulations, emphasising both their advantages and shortcomings. Starting with basics of the direct N-body simulations appropriate to modelling cold dark matter evolution, we then discuss the direct-sum technique GRAPE, particle-mesh (PM) and hybrid methods, combining the PM and the tree algorithms. Simulations of baryonic matter in the Universe often use hydrodynamic codes based on both particle methods that discretise mass, and grid-based methods. We briefly describe Eulerian grid methods, and also some variants of Lagrangian smoothed particle hydrodynamics (SPH) methods.Comment: 42 pages, 16 figures, accepted for publication in Space Science Reviews, special issue "Clusters of galaxies: beyond the thermal view", Editor J.S. Kaastra, Chapter 12; work done by an international team at the International Space Science Institute (ISSI), Bern, organised by J.S. Kaastra, A.M. Bykov, S. Schindler & J.A.M. Bleeke

Hypersonic flows around complex geometries with adaptive mesh refinement and immersed boundary method

This thesis develops and validates a computational fluid dynamics numerical method for hypersonic flows; and uses it to conduct two novel investigations. The numerical method involves a novel combination of structured adaptive mesh refinement, ghost-point immersed boundary and artificial dissipation shock-stable Euler flux discretisation. The method is high-order, low dissipation and stable up to Mach numbers $M \lesssim 30$ with stationary or moving complex geometries; it is shown to be suitable for direct numerical simulations of laminar and turbulent flows. The method's performance is assessed through various test cases. Firstly, heat transfer to proximal cylinders in hypersonic flow is investigated to improve understanding of destructive atmospheric entries of meteors, satellites and spacecraft components. Binary bodies and clusters with five bodies are considered. With binary proximal bodies, the heat load and peak heat transfer are augmented for either or both proximal bodies by $+20\%$ to $-90\%$ of an isolated body. Whereas with five bodies, the cluster-averaged heat load varied between $+20\%$ to $-60\%$ of an isolated body. Generally, clusters which are thin in the direction perpendicular to free-stream velocity and long in the direction parallel to the free-stream velocity have their heat load reduced. In contrast, clusters which are thick and thin in directions perpendicular and parallel to the free-stream velocity feel an increased heat load. Secondly, hypersonic ablation patterns are investigated. Ablation patterns form on spacecraft thermal protection systems and meteor surfaces, where their development and interactions with the boundary layer are poorly understood. Initially, a simple subliming sphere case without solid conduction in hypersonic laminar flow is used to validate the numerical method. Where the surface recession is artificially sped-up via the wall Damk\"{o}hler number without introducing significant errors in the shape change. Then, a case with transitional inflow over a backward facing step with a subliming boundary is devised. Differential ablation is observed to generate surface roughness and add vorticity to the boundary layer. A maximum surface recession of $\sim 0.8\times$ and a maximum surface fluctuation of $\sim 0.2\times$ the inflow boundary layer thickness were generated over two flow times.Open Acces

A fifth-order high-resolution shock-capturing scheme based on modified weighted essentially non-oscillatory method and boundary variation diminishing framework for compressible flows and compressible two-phase flows

First, a new reconstruction strategy is proposed to improve the accuracy of the fifth-order weighted essentially non-oscillatory (WENO) scheme. It has been noted that conventional WENO schemes still suffer from excessive numerical dissipation near-critical regions. One of the reasons is that they tend to under-use all adjacent smooth substencils thus fail to realize optimal interpolation. Hence in this work, a modified WENO (MWENO) strategy is designed to restore the highest possible order interpolation when three target substencils or two target adjacent substencils are smooth. Since the new detector is formulated under the original smoothness indicators, no obvious complexity and cost are added to the simulation. This idea has been successfully implemented into two classical fifth-order WENO schemes, which improve the accuracy near the critical region but without destroying essentially non-oscillatory properties. Second, the tangent of hyperbola for interface capturing (THINC) scheme is introduced as another reconstruction candidate to better represent the discontinuity. Finally, the MWENO and THINC schemes are implemented with the boundary variation diminishing algorithm to further minimize the numerical dissipation across discontinuities. Numerical verifications show that the proposed scheme accurately captures both smooth and discontinuous flow structures simultaneously with high-resolution quality. Meanwhile, the presented scheme effectively reduces numerical dissipation error and suppresses spurious numerical oscillation in the presence of strong shock or discontinuity for compressible flows and compressible two-phase flows

New computational techniques for finite-difference Weighted Essentially Non-Oscillatory schemes and related problems

High-Resolution Shock-Capturing (HRSC) schemes constitute the state of the art for computing accurate numerical approximations to the solution of many hyperbolic systems of conservation laws, especially in computational fluid dynamics. A drawback of these schemes is that most of them use the spectral decomposition of the Jacobian matrix of the system to compute the numerical approximations by local projections to characteristic fields. The numerical solutions obtained are often excellent in terms of resolution, but the computational effort needed may be too high for some problems, especially those for which the spectral information of the flux Jacobian matrix is not available or is quite difficult to obtain. In order to reduce the computational cost, we can use component-wise finite-difference WENO schemes, based on Shu-Osher's finite-difference schemes, which compute the numerical fluxes at each cell interface by upwind-biased reconstructions of split upwind fluxes, avoiding the use of the characteristic information, but, unfortunately, they tend to yield results that are too diffusive and oscillatory. In this work we develop some techniques to improve the accuracy of the numerical results obtained with finite-difference WENO schemes, but also the efficiency of those schemes. There are a lot of works that analyze the main parts of WENO schemes, as the definition of the weights, the smoothness indicators or the role of some parameters present in the definition of the weights in the loss of accuracy near discontinuities and extrema. We derive new weights for the WENO scheme and get some constraints on those parameters present in their definition to guarantee maximal order for sufficiently smooth solutions with an arbitrary number of vanishing derivatives. The other basic ingredient of WENO finite-difference schemes is the use of the upwinding when computing the numerical flux function. The sophisticated design of the numerical flux function, that incorporates upwinding through characteristic information that needs to be computed at each cell boundary in the computational domain, tends to be fairly expensive. To speed up computing times, we use component-wise schemes that avoid the use of characteristic information when computing the numerical fluxes. We introduce an alternative flux-splitting to the usual Lax-Friedrichs flux-splitting. The use of this flux-splitting leads to more accurate numerical solutions, especially near discontinuities, where the use of this flux-splitting reduces the dissipation of the numerical solutions. In the case of the numerical simulation of shallow water flows it has been studied that to accurately represent discontinuous behavior, known to occur due to the non-linear hyperbolic nature of the shallow water system, and, at the same time, numerically maintain stationary solutions it is necessary the use of well-balanced shock-capturing (WBSC) schemes. In this work we combine the block structured AMR technique with a well-balanced scheme to develop a combined AMR-WBSC scheme. We show that in order for the combined AMR-WBSC scheme to maintain its well-balanced character it is necessary to implement well-balanced interpolatory techniques in the transfer operators involved in the multi-level structure. It is shown that the new AMR-WBSC scheme is more efficient than usual WBSC schemes and that it preserves the "water at rest" stationary solutions as the underlying WBSC does. We make extensive testing to compare the performance of several schemes and support our discussion

High Fidelity Numerical Simulation of 3D Arc Extinction in a High Voltage Circuit Breaker

Résumé Cette recherche vise la simulation numérique 3D de l’arc dans un disjoncteur à haute tension SF6 à courant zéro. Dans le but de capturer et de quantifier les instabilités 3D de l’arc à courant zéro, la méthode des volumes finis est utilisée pour résoudre des équations Euler (modifiées pour les gaz réel). La méthode WENO (Weighted Essentially Non-Oscillatory) est utilisée pour atteindre une précision spatiale du 5e ordre pour une compréhension à haute résolution des phénomènes complexes impliqués dans la simulation d’arc. Le chauffage ohmique est obtenu en utilisant une méthode de différence finie compacte du 4ème ordre et le transfert d’énergie radiative est modélisé via la méthode P1. Une méthode TVD Runge- Kutta de troisième ordre est implémentée pour l’intégration temporelle. Le domaine de calcul est un cuboïde, discrétisé en maillage cartésien, à l’intérieur de la buse du disjoncteur qui comprend l’arc et exclut les parois solides et les deux électrodes. L’arc est allumé via un code interne à l’intérieur de la buse. Les résultats sont ensuite cartographiés sur le cuboïde et le courant est réduit à zéro une fois que l’état stationnaire est atteint. Pour étudier les effets 3D de l’arc, une légère asymétrie est imposée à la configuration du problème en déplaçant le domaine de calcul de sorte que l’axe de l’arc ne coïncide pas avec la ligne de symétrie du cuboïde. L’effet de ce déplacement est mesuré sur la résistance électrique du milieu gazeux à l’arc. En comparant le profil numérique de température radiale avec la mesure de température, il est conclu que le schéma WENO et les résultats du premier ordre tombent dans une plage de précision acceptable et la méthode WENO prédit un profil de température plus précis. On observe également que les résultats sont plus précis pour le noyau d’arc que pour les frontières d’arc, en raison de l’absence des termes visqueux responsables de la diffusion d’énergie et du mélange turbulent. Il est également conclu que les équations d’Euler sont capables de capturer les effets 3D de l’arc. Ces effets doivent être pris en compte dans la conception des disjoncteurs car ils sont directement proportionnels à la résistance du milieu gazeux qui a un fort impact sur l’efficacité du disjoncteur. ----------Abstract This research aims at 3D numerical simulation of arc inside an SF6 high voltage circuit breaker at current zero. With the goal of capturing and quantifying the arc 3D instabilities at current zero, FVM is used to solve real-gas-modified Euler equations. A Weighted Essentially Non- Oscillatory (WENO) method is employed to reach a 5th order spatial accuracy for a better understanding of complex phenomena involved in the arc simulation. The ohmic heating is solved using a 4th order compact finite difference method and the radiative energy transfer is modelled via P1 method. A 3rd order TVD Runge-Kutta integration method is implemented to represent the solver evolution in time. The computational domain is a cuboid, discretized on a Cartesian grid, inside the circuit breaker nozzle which includes the arc and excludes the nozzle walls and the two electrodes. The arc is ignited via an in-house code inside the nozzle. The results are then mapped to the cuboid and the current is ramped down to zero once the steady state is reached. To investigate the arc 3D effects, a slight asymmetry is imposed to the problem configuration by moving the computational domain so the arc axis does not coincide the line of symmetry of the cuboid. The effect of this displacement is measured on the arc gaseous medium resistance. Comparing the numerical radial temperature profile with temperature measurement, it is concluded that although both the WENO scheme and the 1st order results fall into an acceptable range of accuracy, the WENO method predicts a more accurate temperature profile. It is observed that the results are more accurate for the arc core than the arc boundary, due to the absence of the viscous terms who are responsible for energy diffusion and turbulent mixing in the arc boundary. It is concluded that the Euler equations are capable of capturing the arc 3D effects. These effects should be considered in circuit breakers design since they are directly proportional to the medium resistance which has a strong impact on the circuit breaker efficiency

High-order methods for diffuse-interface models in compressible multi-medium flows: a review

The diffuse interface models, part of the family of the front capturing methods, provide an efficient and robust framework for the simulation of multi-species flows. They allow the integration of additional physical phenomena of increasing complexity while ensuring discrete conservation of mass, momentum, and energy. The main drawback brought by the adoption of these models consists of the interface smearing, increasing with the simulation time, therefore, requiring a counteraction through the introduction of sharpening terms and a careful selection of the discretization level. In recent years, the diffuse interface models have been solved using several numerical frameworks including finite volume, discontinuous Galerkin, and hybrid lattice Boltzmann method, in conjunction with shock and contact wave capturing schemes. The present review aims to present the recent advancements of high-order accuracy schemes with the capability of solving discontinuities without the introduction of numerical instabilities and to put them in perspective for the solution of multi-species flows with the diffuse interface method.Engineering and Physical Sciences Research Council (EPSRC): 2497012. Innovate UK: 263261. Airbus U