Flux Conservation Research Articles

Thermodynamic consistency of cell-centered lagrangian schemes

It is well known that gas dynamics equations may develop singularities,i.e., shock waves, at a finite time. The related discontinuous solutions are thus sought under a weak form which corresponds to the integral form of the underlying conservation laws of mass, momentum and total energy. Weak solutions being not uniquely defined, the physically relevant solution is singled out by means of an extra admissibility criterion termed entropy condition. In other words, this means that the solutions of gas dynamics equations have to be consistent with the second law of thermodynamics: for smooth flows entropy has to be conserved, whereas for non smooth flows, such as shock waves, it has to be dissipated ensuring the conversion of kinetic energy into internal energy.Bearing this in mind, the purpose of the present work is to address the thermodynamic consistency of cell-centered Finite Volume discretizations dedicated to the numerical simulation of Lagrangian hydrodynamics. Firstly, we describe explicitly a general procedure to construct affordable entropy conservative numerical fluxes, extending Tadmor’s work to the Lagrangian framework. Secondly, the entropy stability of these fluxes enhanced by proper dissipation operators is investigated. Then, a multi-dimensional extension of this work is explored. Finally, these theoretical studies are assessed by various numerical experiments.

Magnetars from Neutron Star–White Dwarf Mergers: Application to Fast Radio Bursts

Abstract It is widely believed that magnetars could be born in core-collapse supernovae (SNe), binary neutron star (BNS) or binary white dwarf (BWD) mergers, or accretion-induced collapse (AIC) of white dwarfs. In this paper, we investigate whether magnetars could also be produced from neutron star–white dwarf (NSWD) mergers, motivated by FRB 180924-like fast radio bursts (FRBs) possibly from magnetars born in BNS/BWD/AIC channels suggested by Margalit et al. (2019). By a preliminary calculation, we find that NSWD mergers with unstable mass transfer could result in the NS acquiring an ultra-strong magnetic field via the dynamo mechanism due to differential rotation and convection or possibly via the magnetic flux conservation scenario of a fossil field. If NSWD mergers can indeed create magnetars, then such objects could produce at least a subset of FRB 180924-like FRBs within the framework of flaring magnetars, since the ejecta, local environments, and host galaxies of the final remnants from NSWD mergers resemble those of BNS/BWD/AIC channels. This NSWD channel is also able to well explain both the observational properties of FRB 180924-like and FRB 180916.J0158+65-like FRBs within a large range in local environments and host galaxies.

A study of the propagation of a solitary wave along the magnetic field in a cold collision-free plasma

A detailed study has been carried out on the propagation of a solitary wave along the magnetic field in a cold collision-free plasma. With the quasi-neutral approximation and the conservation of momentum flux and energy flux in the wave-frame, the exact analytical solutions of the stationary solitary pulse in terms of particle densities, parallel and transverse velocities, and transverse magnetic fields are presented without performing numerical integration. Graphical representations of the solutions and in particular the 3D structures of the transverse magnetic and velocity field lines are presented.

Scattering of in-plane elastic waves at metamaterial interfaces

In this paper, we consider the problem of the scattering of in-plane waves at an interface between a homogeneous medium and a metamaterial. The relevant eigenmodes in the two regions are calculated by solving a recently described non self-adjoint eigenvalue problem particularly suited to scattering studies. The method efficiently produces all propagating and evanescent modes consistent with the application of Snell’s law and is applicable to very general scattering problems. In a model composite, we elucidate the emergence of a rich spectrum of eigenvalue degeneracies. These degeneracies appear in both the complex and real domains of the wave-vector. However, since this problem is non self-adjoint, these degeneracies generally represent a coalescing of both the eigenvalues and eigenvectors (exceptional points). Through explicit calculations of Poynting vector, we point out an intriguing phenomenon: there always appears to be an abrupt change in the sign of the refraction angle of the wave on two sides of an exceptional point. Furthermore, the presence of these degeneracies, in some cases, hints at fast changes in the scattered field as the incident angle is changed by small amounts. We calculate these scattered fields through a novel application of the Betti–Rayleigh reciprocity theorem. We present several numerical examples showing a rich scattering spectrum. In one particularly intriguing example, we point out wave behavior which may be related to the phenomenon of resonance trapping. We also show that there exists a deep connection between energy flux conservation and the biorthogonality relationship of the non self-adjoint problem. The proof applies to the general class of scattering problems involving elastic waves (under self-adjoint or non self-adjoint operators).

Stable compression of a spherical tokamak plasma

An analysis of toroidal plasma compression by a collapsing flux conserver reveals linearly stable scenarios of operation to high compression ratios. The resistive and ideal MHD stability is calculated in full toroidal geometry using the asymptotic matching method in realistic conditions. A time dependent MHD simulation of the compression is conducted to confirm the conservation principles used to calculate the equilibrium states for the analysis. The near edge current profile, controlled by toroidal field ramping during compression, is shown to be critical to stability due to coupling between poloidal components of the least stable mode. Two cases with slight differences in edge current are examined, one with a stable corridor to high compression and one without.

Study on the influence of dynamic/static interface processing methods on CFD simulation results of the axial-flow blood pump

Computational fluid dynamics is an essential tool for the flow field analysis of the blood pump. The interface processing method between the dynamic/static regions will affect the accuracy of simulation results, but its influence on the simulation results is still unclear. In this study, the axial-flow blood pump was taken as the research object, and the effects of the mixing plane, frozen rotor, and sliding mesh methods on the following results were compared: flux conservation at the interface, hydraulic characteristics, and velocity field distribution. In parallel, the particle image velocimetry experiment was carried out to measure the velocity field of the impeller, the inlet, and the outlet area of the blood pump. The results show that the above methods have significant differences in flux conservation between the impeller and the back vane. The average surface energy flux’s error of frozen rotor and sliding mesh are 0.7% and 0.72%, respectively, while the mixing plane method reaches 3.6%. This nonconservative transfer affects the distribution of the downstream velocity field, and the velocity field predicted by the mixing plane at the outlet is quite different. It is suggested to use the frozen rotor method and the sliding mesh method in the simulation of the blood pump.

Heating of the real polar cap of radio pulsars

ABSTRACT The heating of the real polar cap surface of radio pulsars by the bombardment of ultra-relativistic charges is studied. The real polar cap is a significantly smaller area within or close by the conventional polar cap, which is encircled by the last open field lines of the dipolar field $\vec{B}_\mathrm{ d}$. It is surrounded by those field lines of the small-scale local surface field $\vec{B}_\mathrm{ s}$ that join the last open field lines of $\vec{B}_\mathrm{ d}$ in a height of ∼105 cm above the cap. As the ratio of radii of the conventional and real polar cap Rdip/Rpc ∼ 10, flux conservation requires Bs/Bd ∼ 100. For rotational periods P ∼ 0.5 s, Bs ∼ 1014 G creates a strong electric potential gap that forms the inner accelerating region (IAR) in which charges gain kinetic energies ∼3 × 1014 eV. This sets an upper limit for the energy that backflowing charges can release as heat in the surface layers of the real polar cap. Within the IAR, which is flown through with a dense stream of extremely energetic charges, no stable atmosphere of hydrogen can survive. Therefore, we consider the polar cap as a solidified ‘naked’ surface consisting of fully ionized iron ions. We discuss the physical situation at the real polar cap, calculate its surface temperatures Ts as functions of Bs and P, and compare the results with X-ray observations of radio pulsars.

Flexible and Modular Simultaneous Modeling of Flow and Reactive Transport in Rivers and Hyporheic Zones

AbstractInvestigations of coupled multiphysics processes in rivers and hyporheic zones have extensively used numerical models. Most existing models use a sequential, one‐way coupling between the surface and subsurface domains. Such one‐way coupling potentially introduces error. To overcome this, a fully coupled model, hyporheicFoam, was developed using the open‐source computational platform OpenFOAM. It captures the coupled flow and multicomponent reactive transport processes within both surface and subsurface domains and across their interface. The coupling between two domains is implemented by mapping conservative flux boundary conditions at the interface through an iterative algorithm. Reactive transport is enabled by specifying a reaction network. To start, we have implemented reaction kinetics following the double Monod‐type model with inhibition. The model capability is illustrated through modeling of both conservative and reactive hyporheic flow and transport through dune bedforms. With the novel coupled model, it is now possible to quantify reactions wherein the reactants and products are constantly exchanging between domains and have feedbacks. hyporheicFoam can simulate large, three‐dimensional cases owing to the computational flexibility and power offered by the code structure and parallel design of OpenFOAM.

Entropy-stable schemes for relativistic hydrodynamics equations

In this article, we propose high-order finite difference schemes for the equations of relativistic hydrodynamics, which are entropy stable. The crucial components of these schemes are a computationally efficient entropy conservative flux and suitable high-order entropy dissipative operators. We first design a higher-order entropy conservative flux. For the construction of appropriate entropy dissipative operators, we derive entropy scaled right eigenvectors. This is then used with ENO-based sign-preserving reconstruction of scaled entropy variables, which results in higher-order entropy-stable schemes. Several numerical results are presented up to fourth order to demonstrate entropy stability and performance of these schemes.

Optimized Schwarz and finite element cell-centered method for heterogeneous anisotropic diffusion problems

The paper is concerned with the derivation and analysis of the optimized Schwarz type method for heterogeneous, anisotropic diffusion problems discretized by the finite element cell-centered (FECC) scheme. Differently from the standard finite element method (FEM), the FECC method involves only cell unknowns and satisfies local conservation of fluxes by using a technique of dual mesh and multipoint flux approximations to construct the discrete gradient operator. Consequently, if the domain is decomposed into nonoverlapping subdomains, the transmission conditions (on the interfaces between subdomains) associated with the FECC scheme are different from those of the standard FEM. We derive discrete Robin-type transmission conditions in the framework of FECC discretization, which include both weak and strong forms of the Robin terms due to the construction of the FECC's discrete gradient operator. Convergence of the associated iterative algorithm for a decomposition into strip-shaped subdomains is rigorously proved. Two dimensional numerical results for both isotropic and anisotropic diffusion tensors with large jumps in the coefficients are presented to illustrate the performance of the proposed methods with optimized Robin parameters.

Application of non-equilibrium dendrite growth model considering thermo-kinetic correlation in twin-roll casting

Upon non-equilibrium solidifications, dendrite growth, generally as precursor of as-solidified structures, has severe effects on subsequent phase transformations. Considering synergy of thermodynamics and kinetics controlling interface migration and following conservation of heat flux in solid temperature field, a more flexible modeling for the dendrite growth is herein developed for multi-component alloys, where, two inherent problems, i.e. correlation between thermodynamics and kinetics (i.e. the thermo-kinetic correlation), and theoretical connection between dendrite growth model and practical processing, have been successfully solved. Accordingly, both the thermodynamic driving force ΔG and the effective kinetic energy barrier Qeff have been found to control quantitatively the dendrite growth (i.e. especially the growth velocity, V), as reflected by the thermo-kinetic trade-off. Compared with previous models, it is the thermo-kinetic correlation that guarantees quantitative connection between the practical processing parameters and the current theoretical framework, as well as more reasonable description for kinetic behaviors involved. Applied to the vertical twin-roll casting (VTC), the present model, realizes a good prediction for kissing points, which influences significantly alloy design and processing optimization. This work deduces quantitatively the thermo-kinetic correlation controlling the dendrite growth, and by proposing the parameter-triplets (i.e. ΔG - Qeff - V), further opens a new beginning for connecting solidification theories with industrial applications, such as the VTC.

Numerical simulation of the unsteady combustion of solid rocket propellants at a harmonic pressure change

This study focuses on a numerical investigation of the unsteady burning rate of solid propellants at a harmonic pressure change in the combustion chamber of a solid propellant rocket engine. The physico-mathematical model includes the equations of heat transfer and decomposition of the oxidizer in the solid phase and two phases, the dual velocity, and the two-temperature reaction flow of gasification products. The boundary conditions on the solid fuel surface implement the conservation of energy fluxes and the mass of components. We numerically calculate the unsteady burning rate of metallized solid propellant and nitroglycerin powder under a harmonic pressure change in the combustion chamber of a solid propellant rocket engine and determine the dependence of the burning rate amplitude on the frequency of pressure oscillations. The amplitude of the burning rate depends nonmonotonously on the oscillation frequency. With increasing frequency, the amplitude first rises and then declines.

Influence of the lateral source/building separation on vapour intrusion: A numerical study

Various vapour intrusion (VI) models have been proposed in order to predict indoor concentration of Volatile Organic Compounds (VOCs) in buildings. However, these models tend to be conservative, and overestimate or underestimate vapour flux emissions due to several assumptions. Particularly, most of these VI models only consider an infinite uniform contaminated groundwater as the principal source of VOCs in the soil, and lateral pollution source in the vadose zone are disregarded. It has been shown that ignoring the lateral source position may lead to uncertainties on the estimations. In this paper, a numerical model is developed in order to better understand the relationship between the lateral source position in the soil, including both a source in the vadose zone and a source located at the groundwater level, and the resulting indoor air concentration. Results show that source position plays a significant role on vapour intrusion attenuation. In fact, indoor concentration of VOCs decreases with increasing lateral separation. Finally, it is shown that considering the source position can significantly improve the quality of VI predictions.

Entropy Symmetrization and High-Order Accurate Entropy Stable Numerical Schemes for Relativistic MHD Equations

This paper presents entropy symmetrization and high-order accurate entropy stable schemes for the relativistic magnetohydrodynamic (RMHD) equations. It is shown that the conservative RMHD equations are not symmetrizable and do not admit a thermodynamic entropy pair. To address this issue, a symmetrizable RMHD system, equipped with a convex thermodynamic entropy pair, is proposed by adding a source term into the equations, providing an analogue to the non-relativistic Godunov--Powell system. Arbitrarily high-order accurate entropy stable finite difference schemes are developed on Cartesian meshes based on the symmetrizable RMHD system. The crucial ingredients of these schemes include (i) affordable explicit entropy conservative fluxes which are technically derived through carefully selected parameter variables, (ii) a special high-order discretization of the source term in the symmetrizable RMHD system, and (iii) suitable high-order dissipative operators based on essentially non-oscillatory reconstruction to ensure the entropy stability. Several numerical tests demonstrate the accuracy and robustness of the proposed entropy stable schemes.

Constitutive relations for polar continua based on statistical mechanics and spatial averaging

The purpose of the current work is the formulation of macroscopic constitutive relations, and in particular continuum flux densities, for polar continua from the underlying mass point dynamics. To this end, generic microscopic continuum field and balance relations are derived from phase space transport relations for expectation values of point fields related to additive mass point quantities. Given these, microscopic energy, linear momentum and angular momentum, balance relations are obtained in the context of the split of system forces into non-conservative and conservative parts. In addition, divergence–flux relations are formulated for the conservative part of microscopic supply-rate densities. For the case of angular momentum, two such relations are obtained. One of these is force-based, and the other is torque-based. With the help of physical and material theoretic restrictions (e.g. material frame-indifference), reduced forms of the conservative flux densities are obtained. In the last part of the work, formulation of macroscopic constitutive relations from their microscopic counterparts is investigated in the context of different spatial averaging approaches. In particular, these include (weighted) volume-averaging based on a localization function, surface averaging of normal flux densities based on Cauchy flux theory and volume averaging with respect to centre of mass.

Implementation of streamline simulation based on finite element method in FEniCS

Understanding flow and transport behaviors in subsurface porous rock is extremely important for various applications such as oil/gas reservoirs, groundwater, geothermal energy explorations, and CO2 sequestration. However, the relevant calculation is never an easy task because of the large areal extend and high heterogeneity of subsurface system. Although streamline-based simulations are suitable for solving large, geologically complex, and heterogeneous systems, they still present difficulties while working with the finite element method. This paper introduces a new algorithm to trace streamlines based on the underlying flow field in unstructured triangular mesh systems. The new solution procedure only requires velocities on each element vertex but not on cell facets, and therefore it does not rely on conservative fluxes. The algorithm is based on coupled ODEs derived from shape functions as well as the transformation formula between the actual and master elements. The analytical solution describing the trajectory of a streamline segment in the master element is derived so that the time-of-flight and exit coordinate (leaving the element) can be determined accurately and efficiently. Additionally, the presented algorithm allows the implementation of streamline-based methods in the programming framework, FEniCS, so as to extend the technique to other research areas.

Modelling two-layer nanofluid flow in a micro-channel with electro-osmotic effects by means of Buongiorno’s mode

A fully developed steady immiscible flow of nanofluid in a two-layer microchannel is studied in the presence of electro-kinetic effects. Buongiorno’s model is employed for describing the behavior of nanofluids. Different from the previous studies on two-layer channel flow of a nanofluid, the present paper introduces the flux conservation conditions for the nanoparticle volume fraction field, which makes this work new and unique, and it is in coincidence with practical observations. The governing equations are reduced into a group of ordinary differential equations via appropriate similarity transformations. The highly accurate analytical approximations are obtained. Important physical quantities and total entropy generation are analyzed and discussed. A comparison is made to determine the significance of electrical double layer (EDL) effects in the presence of an external electric field. It is found that the Brownian diffusion, the thermophoresis diffusion, and the viscosity have significant effect on altering the flow behaviors.

Evolutionary equations and constraints: Maxwell equations

By fixing a reference frame in spacetime, it is possible to split the Euler-Lagrange equations associated with a degenerate Lagrangian into purely evolutionary equations and constraints on the allowed Cauchy data with respect to the notion of space and time associated with the given reference frame. In the context of classical electrodynamics, we introduce a “covariantization procedure” that allows us to invert the perspective and to recover a full-fledged covariant formulation of Maxwell’s equations starting only with constraint equations (i.e., Gauss law and the local version of the law of conservation of magnetic flux) as perceived in a sufficient number of (inertial) reference frames on Minkowski spacetime.

The finite volume scheme preserving maximum principle for diffusion equations with discontinuous coefficient

We construct a new scheme whose primary unknowns include both cell-centered unknowns and edge unknowns on discontinuous line to solve diffusion equations with discontinuous coefficient. First, two linear fluxes are given on both sides of the cell edge, respectively. In order to deal with the defect of the existing scheme preserving maximum principle for solving diffusion problem with discontinuous coefficient, in addition to cell-centered unknowns, we also introduce cell-edge unknowns on discontinuous line as basic unknowns. Second, the conservative flux is constructed by using nonlinear weighted combination of these two linear fluxes. For the cell-edge unknowns on discontinuous line, we add an equation by using the continuity of normal flux. Compared to the classical cell-centered nonlinear finite volume scheme, the introduction of cell-edge unknowns on discontinuous edge is the key point for our scheme to solve diffusion equations with discontinuous coefficient. Then we prove that the scheme satisfies the discrete maximum principle. Based on this, the existence of a solution for the scheme is also obtained. Numerical results are presented to show that our scheme obtains almost second order accuracy for solution on random meshes, preserves discrete maximum principle, and is superior to the existing scheme (in Sheng and Yuan (2011)) preserving the maximum principle on dealing with the problems with strong anisotropic discontinuous coefficient on distorted meshes.

Voxel agglomeration for accelerated estimation of permeability from micro-CT images

Direct numerical methods are widely used to solve for flow on micro-computed tomography (micro-CT) images of rocks. Generally, direct numerical methods are computationally demanding, especially on large micro-CT images (10003 voxels and more). We develop a fast, direct numerical approach to solve the elliptic flow equation and estimate the absolute permeability of micro-CT images of rocks. The approach involves voxel agglomeration, which is a common technique in computational fluid dynamics. By allowing specific locations of the pore voxels to be agglomerated, the number of pore voxels or active cells in a sparse matrix can be reduced by approximately 60%–80% depending on the type of rock and its pore size distribution. Nevertheless, the fine details obtained from the micro-CT scan are maintained. The results compared with the Pore-scale Finite Volume Solver (PFVS) are within 1.6% difference on level 1 agglomerated grids and 1.9% for level 2 agglomerated grids without loss of the flux conservation. Moreover, the computation time reductions are at least 59% for the micro-CT images tested in this study. This approach is suitable for rapid estimation of absolute permeability from micro-CT images of rocks, particularly on large systems with high pore voxel counts.