Flux Conservation Research Articles

An implementation of mimetic finite difference method for fractured reservoirs using a fully implicit approach and discrete fracture models

In this paper, we present a fully implicit mimetic finite difference method (MFD) for general fractured reservoir simulation. The MFD is a novel numerical discretization scheme that has been successfully applied to many fields and it is characterized by local conservation properties and applicability to complex grids. In our work, we extend this method to the numerical simulation of fractured reservoirs using discrete fracture models. The MFD scheme supports general polyhedral meshes and full tensor properties which improves the modeling and simulation of subsurface reservoirs. Furthermore, we describe in detail the principle of our MFD approach and the corresponding numerical formulations of the discrete fracture model. In our tests, we use a fully implicit scheme that assures flux conservation and simulation efficiency. Several case studies are conducted to show the accuracy and the robustness of the proposed numerical scheme.

Construction of Conservative Numerical Fluxes for the Entropy Split Method

Construction of Conservative Numerical Fluxes for the Entropy Split Method

Predicting large-scale pool fire dynamics using an unsteady flamelet- and large-eddy simulation-based model suite

A low-Mach, unstructured, large-eddy-simulation-based, unsteady flamelet approach with a generalized heat loss combustion methodology (including soot generation and consumption mechanisms) is deployed to support a large-scale, quiescent, 5-m JP-8 pool fire validation study. The quiescent pool fire validation study deploys solution sensitivity procedures, i.e., the effect of mesh and time step refinement on capturing key fire dynamics such as fingering and puffing, as mesh resolutions approach O(1) cm. A novel design-order, discrete-ordinate-method discretization methodology is established by use of an analytical thermal/participating media radiation solution on both low-order hexahedral and tetrahedral mesh topologies in addition to quadratic hexahedral elements. The coupling between heat losses and the flamelet thermochemical state is achieved by augmenting the unsteady flamelet equation set with a heat loss source term. Soot and radiation source terms are determined using flamelet approaches for the full range of heat losses experienced in fire applications including radiative extinction. The proposed modeling and simulation paradigm are validated using pool surface radiative heat flux, maximum centerline temperature location, and puffing frequency data, all of which are predicted within 10% accuracy. Simulations demonstrate that under-resolved meshes predict an overly conservative radiative heat flux magnitude with improved comparisons as compared to a previously deployed hybrid Reynolds-averaged Navier–Stokes/eddy dissipation concept-based methodology.

Entropy stable, robust and high-order DGSEM for the compressible multicomponent Euler equations

This work concerns the numerical approximation of a multicomponent compressible Euler system for a fluid mixture in multiple space dimensions on unstructured meshes with a high-order discontinuous Galerkin spectral element method (DGSEM). We first derive an entropy stable (ES) and robust (i.e., that preserves the positivity of the partial densities and internal energy) three-point finite volume scheme using relaxation-based approximate Riemann solvers from Bouchut (2004) [9] and Coquel and Perthame (1998) [22]. Then, we consider the DGSEM based on collocation of quadrature and interpolation points which relies on the framework introduced by Fisher and Carpenter (2013) [28] and Gassner (2013) [31]. We replace the physical fluxes in the integrals over discretization elements by entropy conservative numerical fluxes [Tadmor (1987) [71]], while ES numerical fluxes are used at element interfaces. We thus derive a two-point numerical flux satisfying the Tadmor's entropy conservation condition and use the numerical flux from the three-point scheme as ES flux. Time discretization is performed with a strong-stability preserving Runge-Kutta scheme. We then derive conditions on the numerical parameters to guaranty a semi-discrete entropy inequality as well as positivity of the cell average of the partial densities and internal energy of the fully discrete DGSEM at any approximation order. The later results allow to use existing limiters in order to restore positivity of nodal values within elements. The scheme also resolves exactly stationary material interfaces. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and high resolution of the scheme.

A fully implicit coupled pore-network/free-flow model for the pore-scale simulation of drying processes

We present a fully coupled pore-network/free-flow model providing pore-scale insight into drying processes. We solve the Navier–Stokes equations with component transport in the free-flow region, coupled to a dynamic two-phase, two-component pore-network model (PNM) in the porous domain. The dynamic multi-physics model allows to temporally resolve drying processes in-between capillary equilibrium states. All simulations are non-isothermal and use pressure- and temperature-depended fluid properties. Carefully chosen coupling conditions and a monolithic solver ensure local conservation of mass, momentum, and energy fluxes, in particular at the interface between both model domains. We solve for wetting and non-wetting fluid pressure fields and consider advective gas transport in the network. Numerical examples demonstrate that the coupled model is able to cover a wide range of physical processes relevant for drying and show the mutual interaction of the two subregions. The model is implemented in the modular open-source framework DuMu such that extensions are straight-forward.

Flux conservation, radial scalings, Mach numbers, and critical distances in the solar wind: magnetohydrodynamics and Ulysses observations

ABSTRACT One of the key challenges in solar and heliospheric physics is to understand the acceleration of the solar wind. As a super-sonic, super-Alfvénic plasma flow, the solar wind carries mass, momentum, energy, and angular momentum from the Sun into interplanetary space. We present a framework based on two-fluid magnetohydrodynamics to estimate the flux of these quantities based on spacecraft data independent of the heliocentric distance of the location of measurement. Applying this method to the Ulysses data set allows us to study the dependence of these fluxes on heliolatitude and solar cycle. The use of scaling laws provides us with the heliolatitudinal dependence and the solar-cycle dependence of the scaled Alfvénic and sonic Mach numbers as well as the Alfvén and sonic critical radii. Moreover, we estimate the distance at which the local thermal pressure and the local energy density in the magnetic field balance. These results serve as predictions for observations with Parker Solar Probe, which currently explores the very inner heliosphere, and Solar Orbiter, which will measure the solar wind outside the plane of the ecliptic in the inner heliosphere during the course of the mission.

Exponential stability of density-velocity systems with boundary conditions and source term for the H2 norm

In this paper, we address the problem of the exponential stability of density-velocity systems with boundary conditions. Density-velocity systems are typical hyperbolic systems that are omnipresent in physics as they encompass all systems that consist in a flux conservation and a momentum equation. In this paper we show that any such system can be stabilized exponentially quickly in the H2 norm using simple local boundary feedbacks, provided a condition on the source term is valid. This condition holds for most physical systems, even when the source term is not dissipative. Besides, the feedback laws obtained only depend on the target values at the boundaries, which implies that they do not depend on the expression of the source term or the force applied on the system. This makes them both very easy to implement in practice and robust to model errors. For instance, for a river modeled by Saint-Venant equations this means that the feedback law does not require any information on the friction model, the slope or the shape of the channel considered. This feat is obtained by showing the existence of a basic H2 Lyapunov function. We apply it to several systems: the general Saint-Venant equations, the isentropic Euler equations, the motion of water in rigid-pipe, the osmosis phenomenon, the traffic flow, etc.

Finite element analysis predicts Ca2+ microdomains within tubular-sarcoplasmic reticular junctions of amphibian skeletal muscle

A finite element analysis modelled diffusional generation of steady-state Ca2+ microdomains within skeletal muscle transverse (T)-tubular-sarcoplasmic reticular (SR) junctions, sites of ryanodine receptor (RyR)-mediated SR Ca2+ release. It used established quantifications of sarcomere and T-SR anatomy (radial diameter d=220 , mathrm{n}mathrm{m}; axial distance w=12 , mathrm{n}mathrm{m}). Its boundary SR Ca2+ influx densities,{J}_{mathrm{influx}}, reflected step impositions of influxes, it {Phi }_{mathrm{influx}}={J}_{mathrm{influx}}left(frac{pi {d}^{2}}{4}right), deduced from previously measured Ca2+ signals following muscle fibre depolarization. Predicted steady-state T-SR junctional edge [Ca2+], [Ca2+]edge, matched reported corresponding experimental cytosolic [Ca2+] elevations given diffusional boundary effluxit it {Phi }_{mathrm{efflux}}=frac{D [ {{{mathrm{Ca}}^{2+}}}]_{mathrm{edge}}}{lambda } (pi dw), established cytosolic Ca2+ diffusion coefficients (D = 4 times {10}^{7} mathrm{nm}^{2}/mathrm{s}) and exit length lambda = 9.2 , mathrm{n}mathrm{m}. Dependences of predicted [Ca2+]edge upon {J}_{mathrm{influx}} then matched those of experimental [Ca2+] upon Ca2+ release through their entire test voltage range. The resulting model consistently predicted elevated steady-state T-SR junctional ~ µM-[Ca2+] elevations radially declining from maxima at the T-SR junction centre along the entire axial T-SR distance. These [Ca2+] heterogeneities persisted through 104- and fivefold, variations in D and w around, and fivefold reductions in d below, control values, and through reported resting muscle cytosolic [Ca2+] values, whilst preserving the flux conservation (it it {Phi }_{mathrm{influx}}={Phi }_{mathrm{efflux}}) condition, {left[mathrm{C}{mathrm{a}}^{2+}right]}_{mathrm{edge}}=frac{lambda {dJ}_{mathrm{influx}}}{4Dw}. Skeletal muscle thus potentially forms physiologically significant ~ µM-[Ca2+] T-SR microdomains that could regulate cytosolic and membrane signalling molecules including calmodulin and RyR, These findings directly fulfil recent experimental predictions invoking such Ca2+ microdomains in observed regulatory effects upon Na+ channel function, in a mechanism potentially occurring in similar restricted intracellular spaces in other cell types.

Efficient simulation of groundwater solute transport using the multipoint flux approximation method with arbitrary polygon grids

Groundwater solute transport models are critical for simulating subsurface contaminant processes, guiding policy making for groundwater management, and pollution remediation. In this study, we develop an improved groundwater solute transport model which uses arbitrary-shaped polygons for spatial discretization. This allows for the efficient simulation of solutes in complex boundary aquifer systems. The governing equation is discretized in space using the finite volume method to ensure flux conservation both locally and globally. The dispersion term is discretized by the multipoint flux approximation (MPFA) method which is a finite volume method that is suitable on arbitrary polygon grids. A total variation diminishing (TVD) scheme is used to control numerical oscillation in the advection term. The functionality and performance of the new model are demonstrated by comparing the simulated results between our model and MT3DMS model in five case studies. The comparisons show that the new model can well represent the solute transport processes. In a real-world watershed with complex-shaped boundaries, the new model outperforms the conventional groundwater solute transport model with smaller computed root-mean-square errors. These modeling results suggest that our model can provide robust modeling solutions for simulating groundwater solute transport processes, especially in aquifer systems with complex shaped boundaries. Furthermore, our model can provide a more flexible discretization solution for coupling surface water models and groundwater solute transport models.

Acoustic Su-Schrieffer-Heeger lattice: Direct mapping of acoustic waveguides to the Su-Schrieffer-Heeger model

Topological physics strongly relies on prototypical lattice model with particular symmetries. We report here on a theoretical and experimental work on acoustic waveguides that is directly mapped to the one-dimensional Su-Schrieffer-Heeger chiral model. Starting from the continuous two dimensional wave equation we use a combination of monomadal approximation and the condition of equal length tube segments to arrive at the wanted discrete equations. It is shown that open or closed boundary conditions topological leads automatically to the existence of edge modes. We illustrate by graphical construction how the edge modes appear naturally owing to a quarter-wavelength condition and the conservation of flux. Furthermore, the transparent chirality of our system, which is ensured by the geometrical constraints allows us to study chiral disorder numerically and experimentally. Our experimental results in the audible regime demonstrate the predicted robustness of the topological edge modes.

Parallel rotor/stator interaction methods and steady/unsteady flow simulations of multi-row axial compressors

Rotor/stator interaction plays a crucial role in predicting the aerodynamic performance of high-speed multi-row axial compressor. Although mixing plane method has been widely used in steady simulations of multi-row single passage flows, the exchange of spanwise non-matched profiles of flow variable between adjacent blade rows is still unknown. Besides, quite little work was devoted to sliding mesh method for unsteady simulations of multi-row full-annulus compressor flows. In this work, mixing plane method and sliding mesh method were developed and presented, respectively, in the environment of parallel computer platforms. Particular attentions were paid to the accurate exchange of spanwise non-matched profiles of fluxes in mixing plane method, and the fast search algorithms of spanwise and circumferential non-matched meshes in sliding mesh method. The developed parallel rotor/stator interaction methods were validated with two multi-row high-speed axial compressors, i.e. NASA Stage 35 and first three stages of NASA 74A, both of which were calculated with steady flow simulation of one blade passage per row and unsteady flow simulation of full-annulus passages per row. Results highlighted that both the developed mixing plane method and sliding mesh method were able to predict the aerodynamic performance curve accurately and satisfy the interface flux conservation tightly. The flow wake and shock wave were observed to transfer continuously across rotor/stator interface with sliding mesh method in the unsteady full-annulus simulation. This work is valuable for the further development of advanced aerodynamic design and accurate performance prediction of multi-row axial compressors.

On the molecular electronic flux: Role of nonadiabaticity and violation of conservation

Analysis of electron flux within and in between molecules is crucial in the study of real-time dynamics of molecular electron wavepacket evolution such as those in attosecond laser chemistry and ultrafast chemical reaction dynamics. We here address two mutually correlated issues on the conservation law of molecular electronic flux, which serves as a key consistency condition for electron dynamics. The first one is about a close relation between "weak" nonadiabaticity and the electron dynamics in low-energy chemical reactions. We show that the electronic flux in adiabatic reactions can be consistently reproduced by taking account of nonadiabaticity. Such nonadiabaticity is usually weak in the sense that it does not have a major effect on nuclear dynamics, whereas it plays an important role in electronic dynamics. Our discussion is based on a nonadiabatic extension of the electronic wavefunction similar in idea to the complete adiabatic formalism developed by Nafie [J. Chem. Phys. 79, 4950 (1983)], which has also recently been reformulated by Patchkovskii [J. Chem. Phys. 137, 084109 (2012)]. We give straightforward proof of the theoretical assertion presented by Nafie using a time-dependent mixed quantum-classical framework and a standard perturbation expansion. Explicitly taking account of the flux conservation, we show that the nonadiabatically induced flux realizes the adiabatic time evolution of the electronic density. In other words, the divergence of the nonadiabatic flux equals the time derivative of the electronic density along an adiabatic time evolution of the target molecule. The second issue is about the accurate computationability of the flux. The calculation of flux needs an accurate representation of the (relative) quantum phase, in addition to the amplitude factor, of a total wavefunction and demands special attention for practical calculations. This paper is the first one to approach this issue directly and show how the difficulties arise explicitly. In doing so, we reveal that a number of widely accepted truncation techniques for static property calculations are potential sources of numerical flux non-conservation. We also theoretically propose alternative strategies to realize better flux conservation.

Continuous streamline trajectories on complex grids

Streamlines have been used for reservoir modeling and flow visualization in the petroleum industry and in computational fluid dynamics. When applied to the calculation of volumetric sweep and the identification of by-passed hydrocarbons for improved oil recovery, it is important that the velocity models that are used to trace trajectories across the cells of a grid are flux conservative. As such, the requirements on their tracing may be more stringent than in other disciplines. Flux conservation is also important at faults, at locally refined or coarsened embedded grid boundaries, and within unstructured grids, where the modeling of flow within a cell may not be consistent with the connection fluxes from adjacent cells. In such cases, additional degrees of freedom must be introduced to satisfy flux conservation. In this study, we introduce a flux conservative conforming cell face local boundary layer construction to resolve these inconsistencies. In contrast, solutions that rely upon spatial continuity of streamlines between elements are shown to not be flux conservative when these inconsistencies are present. The use of flux conservative conforming elements also allows the solution to be developed in local isoparametric coordinates, without explicit reference to cell or connection geometry. The solution has been implemented for both 3D corner point and for 2.5D PEBI grids. In all cases we utilize the lowest order Raviart–Thomas zeroth order velocity model, for which the trajectories and transit times may be obtained analytically. The results are demonstrated on a sequence of increasingly complex type, sector and full-field model applications.

A two-dimensional geometric multigrid model for Poisson equation with interface on structured adaptive mesh refinement grid

In this study, a two-dimensional Geometric Multigrid (GMG) model for Poisson equation with interface on Structured Adaptive Mesh Refinement (SAMR) grid is developed. The model is designed to be combined with Navier-Stokes solvers with staggered arrangements of variables, although solvers with collocated arrangement are also applicable. Based on a general GMG method, special attention for flux conservation is taken on the coarse-fine interface. As large density ratios commonly exist in interface flows such as free surface flows, Galerkin Coarse grid Approximation (GCA) method is adopted to generate coefficients on coarse grids to enhance the robustness and efficiency of the model. Benchmark case is carried out for the validation of the model, and it is demonstrated to obtain 2nd-order accuracy and acceptable efficiency for even the density ratio reaches 104. Point iteration and line relaxation iteration of Gauss-Seidel methods are compared to obtain a better performance. Furthermore, the model is implemented in a developed Navier-Stokes solver to validate its performance in simulating interface problems. The numerical results are compared with theoretical solution or experimental data from reliable sources.

Locally Conservative Immersed Finite Element Method for Elliptic Interface Problems

In this paper, we introduce a locally conservative enriched immersed finite element method (EIFEM) to tackle the elliptic problem with interface. The immersed finite element is useful for handling interface with mesh unfit with the interface. However, all the currently available method under IFEM framework may not be designed to consider the conservative flux conservation. We provide an efficient and effective remedy for this issue by introducing a local piecewise constant enrichment, which provides the locally conservative flux. We have also constructed and analyzed an auxiliary space preconditioner for the resulting system based on the application of algebraic multigrid method. The new observation in this work is that by imposing strong Dirichlet boundary condition for the standard IFEM part of EIFEM, we are able to remove the zero eigen-mode of the EIFEM system while still imposing the Dirichlet boundary condition weakly assigned to the piecewise constant enrichment part of EIFEM. A couple of issues relevant to the piecewise constant enrichment given for the mesh unfit to the interface has been discussed and clarified as well. Numerical tests are provided to confirm the theoretical development.

Observations of a plectonemic configuration in a stable magnetized plasma jet

Astrophysical jets are collimated high-speed outflows emerging from spinning and accreting matter around celestial objects and may spontaneously result from self-organized processes. Magnetic self-organization is commonly observed in laboratory plasma physics experiments; however, they require close-fitting flux conservers to constrain and stabilize the toroidal or cylindrical structures. Here we report the first observations of a long, stable, free-boundary plasma jet far from chamber walls, embedding a double-helix magnetic structure resembling a force-free plectonemic Taylor state. The jets arise from an experimental setup that mimics an accretion disk and has no close-fitting solid flux conserver. The results support the hypothesis that self-organization could be a universal, intrinsic explanation for jet formation, collimation, and stability and may help explain double-helix features in celestial observations.

Transient particle acceleration by a dawn–dusk electric field in a current sheet

The influence of a dawn–dusk electric field Ey on transient particles in a 1D current sheet (CS), characterized by the normal (Bz) and tangential (Bx) components of the magnetic field, is studied. The motion and energization of particles injected at the edges of a CS are investigated within the framework of the trajectory method. The analytical treatment reveals that in the case of uniform Bz and Ey, the dynamics of transient particles are described by magnetic flux conservation on specific segments of the trajectory, which allows prediction of some specific properties of the velocity space inside the CS. Verification of the analytical treatment by means of test-particle numerical modeling demonstrates good agreement. In particular, it is shown that the CS can play the role of a converging lens that focuses particles to pitch-angle values close to θ∼π. At the same time, the analysis reveals that the particle energy gain stays within the range of ΔW∈2m[(Ey/Bz)2,(Ey/Bz)(v0+Ey/Bz)], where m is the particle mass and v0 is the initial particle speed (i.e., v0=v·v). The limits of the range depend only weakly on the CS half-thickness. The analysis reveals that for the typical parameters of Ey and Bz in the stationary terrestrial magnetotail, protons with v0≃450 km/s (before CS crossing) can be accelerated along the CS up to vx≃1800 km/s.

Analysis of Thermoacoustic Modes in Can-Annular Combustors Using Effective Bloch-Type Boundary Conditions

Abstract Heavy-duty gas turbines are commonly designed with can-annular combustors, in which all flames are physically separated. Acoustically, however, the cans communicate via the upstream located compressor plenum or at the downstream gaps found at the transition to the turbine inlet. In this study, a coupling condition that is based on a Rayleigh conductivity and acoustic flux conservation is derived. It enables acoustic communication between adjacent cans, in which one-dimensional acoustic waves propagate. In addition, because can-annular systems commonly feature a discrete rotational symmetry, the acoustic field can be expressed as a Bloch-periodic wave in the azimuthal direction. We demonstrate how the coupling conditions resulting in a combustion system with N cans can be expressed as an effective impedance for a single can. By means of this Bloch-type boundary condition, the thermoacoustics of a can-annular system can be analyzed considering only one can, thus reducing the size of the problem by a factor of N. Using this method, we investigate in frequency domain the effect of the coupling strength of a generic can-annular combustor consisting of 12 identical cans, which are connected at the downstream end. We describe generic features of can-annular systems that can be efficiently addressed with this framework and derive results on the frequency response of the cans at various Bloch numbers in the low-frequency and high-frequency limits. Furthermore, the formation of eigenvalue clusters with eigenvalues of close frequency and growth rate, but very different mode shapes is discussed.

On the Transmittance of Metallic Superlattices in the Optical Regime and the True Refraction Angle

Transmission of electromagnetic fields through (dielectric/metallic)n superlattices, for frequencies below the plasma frequency ωp, is a subtle and important topic that is reviewed and further developed here. Recently, an approach for metallic superlattices based on the theory of finite periodic systems was published. Unlike most, if not all, of the published approaches that are valid in the n→∞ limit, the finite periodic systems approach is valid for any value of n, allows one to determine analytical expressions for scattering amplitudes and dispersion relations. It was shown that, for frequencies below ωp, large metallic-layer thickness, and electromagnetic fields moving along the so-called “true” angle, anomalous results with an apparent parity effect appear. We show here that these results are related to the lack of unitarity and the underlying phenomena of absorption and loss of energy. To solve this problem we present two compatible approaches, both based on the theory of finite periodic systems, which is not only more accurate, but has also the ability to reveal and predict the intra-subband resonances. In the first approach we show that by keeping complex angles, above and below ωp, the principle of flux conservation is fully satisfied. The results above ωp remain the same as in Pereyra (2020). This approach, free of assumptions, where all the information of the scattering process is preserved, gives us insight to improve the formalism where the assumption of electromagnetic fields moving along the real angles is made. In fact, we show that by taking into account the induced currents and the requirement of flux conservation, we end up with an improved approach, with new Fresnel and transmission coefficients, fully compatible with those of the complex-angle approach. The improved approach also allows one to evaluate the magnitude of the induced currents and the absorbed energy, as functions of the frequency and the superlattice parameters. We show that the resonant frequencies of intra-subband plasmons, which may be of interest for applications, in particular for biosensors, can be accurately determined. We also apply the approach for the transmission of electromagnetic wave packets, defined in the optical domain, and show that the predicted space-time positions agree extremely well with the actual positions of the wave packet centroids.

A stable corridor for toroidal plasma compression

A toroidal plasma compressed by a collapsing flux conserver is analyzed to reveal 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, and comparing with nonlinear simulations. 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. The extension of a length of shaft on axis is also found to be critical at high compression, as the resulting good curvature region in magnetic field stabilizes pressure driven modes that would otherwise be unstable. This work extends from previous studies, which initially showed the existence of a stable scenario, to include findings of more extensive stable zones, detailed effects of geometry, and nonlinear simulations of the instabilities. The nonlinear simulations of the compression are consistent with the linear analyses, confirming both the conservation and stability properties.