Combination of WENO and Explicit Runge–Kutta Methods for Wind Transport in the Meso-NH Model

This paper investigates the use of the weighted essentially nonoscillatory (WENO) space discretization methods of third and fifth order for momentum transport in the Meso-NH meteorological model, and their association with explicit Runge–Kutta (ERK) methods, with the specific purpose of finding an optimal combination in terms of wall-clock time to solution. A linear stability analysis using von Neumann theory is first conducted that considers six different ERK time integration methods. A new graphical representation of linear stability is proposed, which allows a first discrimination between the ERK methods. The theoretical analysis is then completed by tests on numerical problems of increasing complexity (linear advection of high wind gradient, orographic waves, density current, large eddy simulation of fog, and windstorm simulation), using a fourth-order-centered scheme as a reference basis. The five-stage third-order and fourth-order ERK combinations appear as the time integration methods of choice for coupling with WENO schemes in terms of stability. An explicit time-splitting method added to the ERK temporal scheme for WENO improves the stability properties slightly more. When the spatial discretizations are compared, WENO schemes present the main advantage of maintaining stable, nonoscillatory transitions with sharp discontinuities, but WENO third order is excessively damping, while WENO fifth order provides better accuracy. Finally, WENO fifth order combined with the ERK method makes the whole physics of the model 3 times faster compared to the classical fourth-order centered scheme associated with the leapfrog temporal scheme

Mesoscale modeling and direct simulation of explosively dispersed granular materials

Explosively dispersed granular materials frequently exhibit macroscale coherent particle clustering and jetting structures. The underlying mechanism is of significant interest to study instability and mixing in high-speed gas-solid flows but remains unclear, primarily attributed to the complex mesoscale multiphase interactions involved in the dispersal process. In order to advance the understanding of particle clustering and jetting instabilities, this thesis establishes a numerical framework for solving interface-resolved gas-solid flows with non-deforming bodies that are able to move, contact, and collide. The developed framework is implemented to create a computational solver and then verified using a variety of gas-solid flow problems at different geometric scales. Employing the developed framework and solver, this thesis further studies the particle clustering and jetting instabilities in explosively dispersed granular materials. A Cartesian, 3D, high-resolution, parallelized, gas-solid flow solver is created with the capability of tackling shocked flow conditions, irregular and moving geometries, and multibody collisions. The underlying numerical framework integrates operator splitting for partitioned fluid-solid interaction in the time domain, 2nd/3rd order strong stability-preserving Runge--Kutta methods and 3rd/5th order weighted essentially nonoscillatory schemes for high-resolution tempo-spatial discretization, the front-tracking method for evolving phase interfaces, a new field function developed for facilitating the solution of complex and dynamic fluid-solid systems on Cartesian grids, a new collision model developed for deterministic multibody contact and collision with parameterized coefficients of restitution and friction, and a new immersed boundary method developed for treating arbitrarily irregular and moving boundaries. The developed framework and solver are able to accurately, efficiently, and robustly solve coupled fluid-fluid, fluid-solid, and solid-solid interactions with flow conditions ranging from subsonic to hypersonic states. Employing the developed framework and solver, direct simulations that capture interface-resolved multiphase interactions and deterministic mesoscale granular dynamics are conducted to investigate particle clustering and jetting instabilities. A random sampling algorithm is employed to generate stochastic payload morphologies with randomly distributed particle positions and sizes. Through solving and analyzing cases that cover a set of stochastic payloads, burster states, and coefficients of restitution, a valid statistical dissipative property of the framework in solving explosively dispersed granular materials with respect to Gurney velocity is demonstrated. The predicted surface expansion velocities can extend the time range of the velocity scaling law with regard to Gurney energy in the Gurney theory from the steady-state termination phase to the unsteady evolution phase. When considering the mean surface expansion velocities, the maximum error of the unsteady velocity scaling law is about $0.792\%$ among the investigated Gurney energies. In addition, a dissipation analysis of the current discrete modeling of granular payloads suggests that incorporating the effects of porosity can enhance the prediction of Gurney velocity for explosively dispersed granular payloads. On the basis of direct simulations, an explanation for particle clustering and jetting instabilities is proposed to increase the understanding of established experimental observations in the literature. Results suggest that the development of internal sliding and colliding lines in the shock-compacted granular payload can be critical to the subsequent fracture pattern of the payload. Particle clusters manifested through payload fracture are then maintained by local pressure gradient between surrounding and interstitial flows as well as by dissipative inter-grain collisions. The existence of stable clusters introduce a more non-equilibrium momentum distribution in the overall payload, exhibiting as a form of clustering instability. Under the current assumptions of non-deformable grains, the mesoscale granular dynamics largely depends on the payload morphology as a result of packing methods. Different payload morphologies can develop varied sliding and colliding lines, which lead to a corresponding pattern for payload fracturing and particle clustering. With the rapid development of high-performance computing technology, future direct simulations on stochastic payloads with significantly increased domain sizes, number of particles, and solution times are expected to lead to a better understanding of the flow instability in explosively dispersed granular payloads. It is suggested that statistics collected from a large number of mesoscale computations based on random payload morphologies can potentially evolve into a macroscopic theory of multiphase flow instability for particle clustering and jetting phenomena widely observed in many areas involving dense gas-solid flows

ECHO: an Eulerian Conservative High Order scheme for general relativistic magnetohydrodynamics and magnetodynamics

We present a new numerical code, ECHO, based on an Eulerian Conservative High Order scheme for time dependent three-dimensional general relativistic magnetohydrodynamics (GRMHD) and magnetodynamics (GRMD). ECHO is aimed at providing a shock-capturing conservative method able to work at an arbitrary level of formal accuracy (for smooth flows), where the other existing GRMHD and GRMD schemes yield an overall second order at most. Moreover, our goal is to present a general framework, based on the 3+1 Eulerian formalism, allowing for different sets of equations, different algorithms, and working in a generic space-time metric, so that ECHO may be easily coupled to any solver for Einstein's equations. Various high order reconstruction methods are implemented and a two-wave approximate Riemann solver is used. The induction equation is treated by adopting the Upwind Constrained Transport (UCT) procedures, appropriate to preserve the divergence-free condition of the magnetic field in shock-capturing methods. The limiting case of magnetodynamics (also known as force-free degenerate electrodynamics) is implemented by simply replacing the fluid velocity with the electromagnetic drift velocity and by neglecting the matter contribution to the stress tensor. ECHO is particularly accurate, efficient, versatile, and robust. It has been tested against several astrophysical applications, including a novel test on the propagation of large amplitude circularly polarized Alfven waves. In particular, we show that reconstruction based on a Monotonicity Preserving filter applied to a fixed 5-point stencil gives highly accurate results for smooth solutions, both in flat and curved metric (up to the nominal fifth order), while at the same time providing sharp profiles in tests involving discontinuities.Comment: 20 pages, revised version submitted to A&

Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling. Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms

Large eddy simulation of deflagration to detonation transition

Deflagration to detonation transition (DDT) is a very important research project for both national defense and energy industry. It is the process where a subsonic deflagration transits into a supersonic detonation, which generates shock waves. In the past years, the simulations of DDT were limited in a small domain, usually sev- eral cubic centimeters. If we want to simulate it in a larger space without improving the numerical method, we need to use the more powerful computer. When the computing resources are limited, we must improve the numerical method to achieve the big-domain simulating. There are two technical paths, one is the adaptive mesh refinement and the other is the large eddy simulation. Both of them are difficult to realize. In this project, we focus on the usage of the LES method for simulating DDT. The main challenge in this work is to develop a reliable model. In this research, a new approach for LES modelling was developed. It is a fully compressible variant of the artificial thickened flame model, which adopts the opt- ing functions on the reference flame thickness. This method ensures that the flame is not over-thickened in deflagration or detonation. To control the options on the flame thickness, a detonation sensor is utilized during the computing.Open Acces

Large eddy simulation of primary liquid-sheet breakup

This research project aims at providing the aeronautical industry with a modelling capability to simulate the fuel injection in gas turbine combustion chambers. The path to this objective started with the review of state-of-the-art numerical techniques to model the primary breakup of liquid fuel into droplets. Based on this and keeping in mind the requirements of the industry, our modelling strategy led to the generation of a mass-conservative method for efficient atomisation modelling on unstructured meshes. This goal has been reached with the creation of high-order numerical schemes for unstructured grids, the development of an efficient numerical method that transports the liquid-vapour interface accurately while conserving mass and the implementation of an algorithm that outputs the droplet boundary conditions to separate combustion codes. Both high-order linear and WENO schemes have been created for general polyhedral meshes. The notorious complexity of high-order schemes on 3D mixed-element meshes has been handled by the creation of a series of algorithms. These include the tetrahedralisation of the mesh, which allows generality of the approach while remaining efficient and affordable, together with a novel approach to stencil generation and a faster interpolation of the solution. The performance of the scheme has been demonstrated on typical two-dimensional and three-dimensional test cases for both linear and non-linear hyperbolic partial differential equations. The conservative level set method has been extended to unstructured meshes and its performance has been improved in terms of robustness and accuracy. This was achieved by solving the equations for the transport of the liquid volume fraction with our novel WENO scheme for polyhedral meshes and by adding a flux-limiter algorithm. The resulting method, named robust conservative level set, conserves mass to machine accuracy and its ability to capture the physics of the atomisation is demonstrated in this thesis. To be readily applicable to the simulation of atomisation, the novel interfacecapturing technique has been embedded in a framework — within the open source CFD code OpenFOAM — that solves the velocity and pressure fields, outputs droplet characteristics and runs in parallel. In particular, the production of droplet boundary conditions involves a set of routines handling the selection of drops in the level set field, the calculation of relevant droplet characteristics and their storage into data files. An n-halo parallelisation method has been implemented in OpenFOAM to perform the computations at the expected order of accuracy. Finally, the modelling capability has been demonstrated on the simulation of primary liquid-sheet breakup with relevance to fuel injection in aero-engine combustors. The computation has demonstrated the ability of the code to capture the physics accurately and further illustrates the potential of the numerical approach

Moment Approximations and Model Cascades for Shallow Flow

Shallow flow models are used for a large number of applications including weather forecasting, open channel hydraulics and simulation-based natural hazard assessment. In these applications the shallowness of the process motivates depth-averaging. While the shallow flow formulation is advantageous in terms of computational efficiency, it also comes at the price of losing vertical information such as the flow's velocity profile. This gives rise to a model error, which limits the shallow flow model's predictive power and is often not explicitly quantifiable. We propose the use of vertical moments to overcome this problem. The shallow moment approximation preserves information on the vertical flow structure while still making use of the simplifying framework of depth-averaging. In this article, we derive a generic shallow flow moment system of arbitrary order starting from a set of balance laws, which has been reduced by scaling arguments. The derivation is based on a fully vertically resolved reference model with the vertical coordinate mapped onto the unit interval. We specify the shallow flow moment hierarchy for kinematic and Newtonian flow conditions and present 1D numerical results for shallow moment systems up to third order. Finally, we assess their performance with respect to both the standard shallow flow equations as well as with respect to the vertically resolved reference model. Our results show that depending on the parameter regime, e.g. friction and slip, shallow moment approximations significantly reduce the model error in shallow flow regimes and have a lot of potential to increase the predictive power of shallow flow models, while keeping them computationally cost efficient