Flux-corrected Transport Method Research Articles

A modified Flux Corrected Transport method coupled with the MPFA-H formulation for the numerical simulation of two-phase flows in petroleum reservoirs using 2D unstructured meshes

A modified Flux Corrected Transport method coupled with the MPFA-H formulation for the numerical simulation of two-phase flows in petroleum reservoirs using 2D unstructured meshes

A cross-diffusion system modeling rivaling gangs: Global existence of bounded solutions and FCT stabilization for numerical simulation

In this paper, we study a gang territorial model consisting of two parabolic and two ordinary differential equations, where a taxis-type mechanism models that the two rivaling gangs are repelled by each other’s graffiti. Our main analytical finding shows the existence of global, bounded classical solutions. By making use of quantitative global estimates, we prove that these solutions converge to homogeneous steady states if the initial data are sufficiently small. Moreover, we perform numerical experiments which show that for different choices of parameters, the system may become diffusion- or convection-dominated, where in the former case the solutions converge toward constant steady states while in the latter case nontrivial asymptotic behavior such as segregation is observed. In order to perform these experiments, we apply a nonlinear finite element flux-corrected transport method (FEM-FCT) which is positivity-preserving. Then we treat the nonlinearities in both the system and the proposed nonlinear scheme simultaneously using fixed-point iteration.

Modeling Two-Phase Water Hammer Flow Using Shock-Capturing Scheme

This article focuses on simulating two-phase water hammer flow. A mathematical formulation has been pointed out to describe the transient one-dimensional homogeneous flow, modelled by gas-liquid mixture flow through an elastic pipe. The governing equations obtained from conservation laws have been carried out. Richtmyer two steps scheme stabilized by the flux corrected transport method have been investigated in the numerical simulations. Despite the fact that the conservative form of the equations gives good results in the simulation, it is useful to present the comparison between the numerical solutions of conservative, semi-conservative and non-conservative forms of equations during the simulation in order to legitimate these approximations often countered and used in such equations. A comparison between obtained results and experimentally reported data indicates that even with non-conservative form, the methods have been in good agreement with experimental observations and are better than the method of characteristics widely used in numerical computation describing water hammer phenomena.

A suitability analysis of transient one-dimensional two-fluid numerical models for simulating two-phase gas-liquid flows based on benchmark problems

Gas-liquid two-phase flows in pipes are present in a variety of engineering applications. These applications require transient one-dimensional mathematical models coupled with robust numerical methods to simulate the flow behavior in time and space. As there exist several possible combinations of models and numerical methods, questions arise as to the suitability of these numerical models to best accommodate the problem features. Aiming to partially answer this question, we present in this work numerical simulations of two well-known gas-liquid benchmarks, the water faucet, and the shock tube problems, carried out with two two-fluid models – 4E1P (four equations and one pressure) and 5E2P (five equations and two pressures) – in association with three numerical methods – FCT (Flux-Corrected Transport method), FORCE (First-Order Centered Scheme) and ModFORCE. The numerical methods used herein are written in a general form and do not require the solution of the associated Riemann problem. The obtained results are sought to serve as a reference to choose the best approaches, among the ones studied, to tackle some transient one-dimensional two-phase flow problems, highlighting the advantages and disadvantages of each model and method combination regarding hyperbolicity, diffusive and dispersive effects. Overall, the combination of the 5E2P model with the ModFORCE numerical method provides the most accurate results compared to the analytical solutions available.

A straightforward strategy for leak localization in two-phase gas pipelines

The leak localization (LL) functionality is an important piece of leak detection systems that continuously monitor the operation of pipelines in real-time. After a leak is detected, it must be located as soon as possible so that mitigating actions can be taken to remedy the undesirable hazardous leak consequences. The most popular methodology used to locate leaks in single-phase pipelines is based on the pressure-gradient along the line. This technique uses the SCADA data polled and collected at the ends of the line segment, along with a fluid dynamics mechanical model capable of describing the flow in steady-state, to infer the leak position. By using the flow data at the pipe ends, the LL methodology seeks the interception of the pressure gradients computed by marching forward and backward along the line to estimate the leak site. This paper presents a detailed study to extend the LL strategy based on the pressure-gradient to two-phase flows in gas pipelines. To achieve this goal, without appealing to field or experimental data, we have built a leak database for typical real-world two-phase gas pipelines under the stratified-pattern flow by means of numerical simulations. By using a transient four-equation single-pressure two-fluid model, with suitable source terms to account for the leak existence, numerical simulations emulating leaks in a pipeline were carried out using the Flux-Corrected Transport method. After being validated by the commercial simulator OLGA, the mathematical model was employed to generate a leak database contemplating leaks of three different magnitudes at three distinct locations along the line for two different scenarios considering low and high holdup contents. The performance of the LL strategy, which has used as mechanical model the steady-state parcel of the transient mathematical model described above along with the simulated data at the pipe inlet and outlet, was evaluated by computing the leak location relative errors. The obtained results have shown that the pressure-gradient methodology has presented a particularly good performance under the circumstances considered herein. It was able to locate leaks of magnitudes between 1% and 10% of the total mass flowrate at the pipe inlet with relative errors ranging from 0.26% to 10.0%.

FCT finite difference forward simulation of the submarine viscoelastic seismic wavefield

The actual seafloor is the typical viscoelastic medium, covered with unconsolidated geological bodies like sand, gravel, sludge, and detritus on the surface. When the seismic wave is propagating therein, viscosity of the seafloor strata will cause seismic wave energy loss, amplitude attenuation and gradual frequency reduction at the wavelet center, and then the accurate subsurface information and high-resolution images cannot be obtained directly from the seismic data due to the reduced signal resolution and SNR. Considering seabed viscoelastic media properties, a model including quality factor, density, and other physical parameters, the flux corrected transport (FCT) algorithm is established to describe the target seafloor and solve the wave equation, in which, the finite difference numerical simulation method of FCT is used to suppress the numerical dispersion generated by differential calculation in coarse grid, improve the seismic wavefield simulation accuracy and computational efficiency. Based on the submarine ancient channel types and the basic characteristics of the real and typical marine geological phenomena, the seismic responses to the three types of submarine ancient channel models, which are straight, curved and braided, are calculated, with good results acquired.

Ozone production and chemical kinetic corona discharge of pure gas O2 species

The work presented in this paper aims to study the chemical kinetics of some species previously chosen by corona discharge at atmospheric pressure P = 1 atm and at room temperature T = 300°K, carried out in wire–cylinder geometry in a pure gas O2. In this perspective, a computer program is designed, efficient in time and space, is used to solve the system of fluid dynamic classical equations based on the resolution of the Boltzmann equation. It participates also, in the chemical kinetics of the gas including the equations of continuity, momentum and energy. For numerical resolution, the flux-corrected transport method is applied successfully to electric discharge, and the generated species of pure gas O2 kinetics are obtained. The reactivity of the gas takes into account nine species that interact according to 23 reactions selected in a way calculated to favor the most dominant. The chosen reduced field values are 100, 120, 140,160, 180 and 200 Td. The results obtained show an evolution closely related to the reduced electric field, and the production of ozone is accompanied by the appearance and disappearance of other species.

The development and validation of a \u2018flux-corrected transport\u2019 based solution methodology for the plasmasphere refilling problem following geomagnetic storms

The refilling of the plasmasphere following geomagnetic storms remains one of the longstanding and interesting problems in ionosphere–magnetosphere coupling research. The objective of this paper is the formulation and development of a one-dimensional (1D) refilling model using the flux-corrected transport method, a numerical method that is well-suited to handling problems with shocks and discontinuities. In this paper, the developed methodology has been validated against exact, analytical benchmarks, and good agreement has been obtained between these analytical benchmarks and numerical results. The objective of this research is the development of a three-dimensional (3D) multi-ion model for ionosphere–magnetosphere coupling problems in open and closed line geometries.

Numerical simulation of pig motion in gas and liquid pipelines using the Flux-Corrected Transport method

The transportation of fluids in long pipelines occurs very often in the oil and gas industries. To keep high efficiency and productivity in pipeline operations, a device called pig is routinely passed throughout the pipeline for the purpose of cleaning, inspection, sealing, corrosion evaluation, among others, despite the uncertainties and risks associated with its passage inside a long duct. For these reasons, it is important to use numerical simulations to follow the motion of a pig inside a pipeline and estimate the pig travelling time. This paper presents a one-dimensional mechanical model, along with a numerical scheme, to obtain approximate solutions for the pig movement and for the transient one-phase flow inside the pipeline, which are physically and mathematically connected. In this model the pig is treated as a moving singular boundary and its mathematical model reduces to an initial-boundary-value problem. This system of equations is coupled to a boundary value problem for the transient fluid flow inside the duct. Since the pig is modeled as a moving surface, the pipeline is divided in two sections, one upstream and one downstream of the pig, and these two separate fluid flows are coupled with the pig motion system. The mathematical domain is discretized within the finite-volume framework and is solved using the Flux-Corrected Transport (FCT) method, which is second-order in space, as our results attest. We present simulations for the pig velocity history and for the distributions of flow pressure and velocity along the pipe in air and water single-phase flows. The results show physically consistent results for no-hole and one-hole pigs, for a rapid transient valve closure problem and in a direct comparison with a commercial software.

Two-dimensional simulation and experiments of helium discharge between parallel-plate electrodes in short gaps at atmospheric pressure

ABSTRACT Helium gas is used as the coolant in high temperature gas-cooled reactor while its insulation property is not good and brings big challenges to insulation design work. In this paper, breakdown voltages of two parallel-plate electrodes in millimeter-scale gaps at atmospheric pressure are calculated numerically with a two-dimensional hydrodynamic model. The Finite-Difference Flux-Corrected Transport method (FD-FCT) is used to avoid spurious oscillation caused by convection and diffusion restrictions. The simulation results are in good agreement with our experimental work. Different discharge characteristics are presented for a breakdown or a non-breakdown voltage. Importantly, when a breakdown is applied, ground-state ionization will make up the most in early stages while later accumulation of metastable atoms and molecules will prompt fractions of penning ionization to increase rapidly. It is predicted that penning ionization will eventually become comparable with ground-state ionization, especially near the anode, as the current density keeps increasing to order of 1 A. Some other simulation results are presented such as time evolutions of the distribution of electrons, ions, field strength and axial potential, etc. Attainment of these breakdown voltages and characteristics of helium discharge is going to help with insulation design work of electric equipment in reactor engineering.

A three-phase VOF solver for the simulation of in-nozzle cavitation effects on liquid atomization

The development of a single-fluid solver supporting phase-change and able to capture the evolution of three fluids, two of which are miscible, into the sharp interface capturing Volume of Fluid (VOF) approximation, is presented. The transport of each phase-fraction is solved independently by a flux-corrected transport method to ensure the boundedness of the void fraction over the domain. The closure of the system of equations is achieved by a cavitation model, that handles the phase change between the liquid and the fuel vapor and it also accounts for the interaction with the non-condensable gases. Boundedness and conservativeness of the solver in the transport of the volume fraction are verified on two numerical benchmarks: a two-dimensional bubble rising in a liquid column and a cavitating/condensing liquid column. Finally, numerical predictions from large-eddy simulations are compared against experimental results available from literature; in particular, validation against high-speed camera visualizations and Laser Doppler Velocimetry (LDV) measurements of cavitating microscopic in-nozzle flows in a fuel injector is reported.

A time-space flux-corrected transport finite element formulation for solving multi-dimensional advection-diffusion-reaction equations

We present a time-space flux-corrected transport (FCT) finite element formulation for the multi-dimensional time-dependent advection-diffusion-reaction equation. Monotonic solutions can be achieved with the presented method while large time steps (with Courant number Cr>1) are used. Numerical verification, as well as a grid convergence analysis, are carried out for 1D and 2D benchmark problems. A 1D Burgers' equation with various Reynolds numbers is solved with the time-space FCT method. In addition, a 2D transport problem with (and without) a nonlinear reaction inside of a cavity is considered. Finally, the newly developed time-space FCT formulation is applied for modeling biofilm growth problems based on a continuum mathematical model. It turns out that the time-space FCT method helps to reduce numerical dissipation and guarantees comparable numerical dispersion of the solution at the same time comparing to numerical solutions obtained by a time-space finite incremental calculus (FIC) method.

Multiple harmonics control of edge pseudomagnetoplasmons in strained graphene

Valley-resolved edge plasmons are relevant to nano-optics at subwavelength scales. However, less attention has been paid to their tunable properties in time domain. In this work we investigate edge pseudomagnetoplasmons in a strained graphene modulated by multiple harmonics with frequency in the THz regime. The edge plasmon is described by a set of nonlinear hydrodynamic equations, which are self-consistently solved by the flux-corrected transport method. Without the applied voltage, there exist two unidirectional-propagating edge-plasmon modes with weak valley polarization P. It is demonstrated that by varying the amplitude of multiple harmonics one can alter both the amplitude and the polarity of the valley polarization in the edge plasmon. One can achieve a full valley polarization P=1 at the instant of half cycle of the multiple harmonics and P=-1 at the instant of one cycle. The edge-plasmon density and the transverse velocity vanish for the frozen valley.

One cause of pulse-like anomalies observed at Guza before the Wenchuan earthquake

Many precursor-like anomaly observations prior to the Wenchuan earthquake were reported and analyzed, especially the abnormal strain pulses observed at the Guza station, but there are few discussions of the causes. Stick–slip motion is the basis for description of a great variety of phenomena characterized by the presence of sliding friction. In this article, perturbed Sine–Gordon (SG) equation is established from Bykov’s unsteady-state slip model. Stable solitary solutions of displacement and strain dimension are obtained and nonlinear pulse propagation is simulated using finite-difference modeling, while numerical stability is obtained by the flux-corrected transport method. Considering the solution of SG equation as initial source, a comparison between the modeling results and actual data at the Guza station gives one possible interpretation for this anomaly. During the seismogenic process of the Wenchuan earthquake, faults may likely occur as stick–slip tectonic movements which might be described by SG equation and would generate solitary wave signal. This kind of pulse experiences a forward tilting distortion due to nonlinear effect of the Earth and is received by the borehole strainmeter. Two kinds of nonlinear effects could lead to these special pulses. One is the nonlinear effects in the wave propagation process, and the other is effects of friction and stress in the process of unsteady-state slip. Because of the convergence effect of pulse, the wave would be collapsed at a certain time due to an excessive increase in the tilt angle. Hence, this kind of pulse cannot propagate for a long distance.

Flux-corrected transport techniques applied to the radiation transport equation discretized with continuous finite elements

The Flux-Corrected Transport (FCT) algorithm is applied to the unsteady and steady-state particle transport equation. The proposed FCT method employs the following: (1) a low-order, positivity-preserving scheme, based on the application of M-matrix properties, (2) a high-order scheme based on the entropy viscosity method introduced by Guermond [1], and (3) local, discrete solution bounds derived from the integral transport equation. The resulting scheme is second-order accurate in space, enforces an entropy inequality, mitigates the formation of spurious oscillations, and guarantees the absence of negativities. Space discretization is achieved using continuous finite elements. Time discretizations for unsteady problems include theta schemes such as explicit and implicit Euler, and strong-stability preserving Runge–Kutta (SSPRK) methods. The developed FCT scheme is shown to be robust with explicit time discretizations but may require damping in the nonlinear iterations for steady-state and implicit time discretizations.

On the effect of different flux limiters on the performance of an engine gas exchange gas-dynamic model

A suitable tool for the design of intake and exhaust systems of internal combustion engines is provided by time-domain non-linear finite volume models. These models, however, are affected by overshoots at discontinuities and numerical dispersion unless some flux limiter is used. In this paper, the effect of the most relevant of such flux limiters on a non-linear staggered-mesh finite-volume model is evaluated. Flux-Corrected-Transport (FCT) and Total Variation Diminishing (TVD) schemes, together with a Momentum Diffusion Term (MDT) are presented for such a model, and the performance of the resulting methods is checked in different problems representative of the influence of engine gas exchange flows on engine performance and intake and exhaust noise. First, two one-dimensional cases are considered: the shock-tube problem, and the propagation of a finite amplitude pressure pulse. Secondly, a simple but representative three-dimensional geometry is studied. From the results obtained, it can be concluded that, even if none of the methods is able to handle properly the three problems considered, the FCT method provides the best overall performance.

Numerical simulation of stratified-pattern two-phase flow in gas pipelines using a two-fluid model

Two-phase flows in pipelines occur in a variety of processes in the nuclear, petroleum and gas industries. Because of the practical importance of accurately predicting steady and unsteady flows along the line, two-fluid models have been extensively used in numerical simulations. These models are usually written as a system of non-linear partial-differential equations. A reliable prediction of such flows is a difficult task to accomplish due to the numerous sources of uncertainties, such as the basic two-phase flow model, the flow-pattern models, the initial and boundary conditions and the numerical method used to solve the initial-boundary-value problem. Several numerical methods, conservative or not, may be used to discretize the problem. In this paper we analyze the global performance of the numerical model, obtained when we couple a classical one-dimensional single-pressure four-equation two-fluid model to a numerical discretization based on the flux-corrected transport (FCT) method, to solve the stratified-pattern two-phase flow that occurs in typical gas-gathering offshore pipelines. The gas phase is considered to be compressible and the liquid phase is incompressible, whereas the flow is assumed to be isothermal. We show that the FCT method is second-order in space and that our numerical kernel produces accurate simulations for steady, as well as unsteady, simulations. We also analyze the hyperbolicity property of the two-fluid model and the adequate prescription of the boundary conditions. The results show that hyperbolicity is lost in part of the area of the stratified-pattern region of a classical flow-pattern diagram. Moreover, the eigenvalues alternate signs at the boundaries, which has implications on how to impose the boundary conditions. Numerical simulations of typical offshore gas pipeline flows show very good agreement when compared to a commercial software.

An improved method to calculate the radio interference of a transmission line based on the flux-corrected transport and upstream finite element method

The radio interference (RI) from HVDC transmission lines can block communication systems. In this paper the RI of a HVDC transmission line has been analyzed by the superposition of the electromagnetic field generated by the corona current based on phase-model transformation, where the corona current can be solved by the flux-corrected transport and finite difference method combining the upstream finite element method for bipolar conductor model instead of corona cage or excitation function. Our calculated values for 0.5 MHz RI from the proposed method compare well with measured values from a 800 kV HVDC test line established in China. With this validation, we find that RI should increase with temperature, increase with altitude, and vary in a complex way with relative humidity. Therefore, the proposed method can be adopted in transmission lines design and electromagnetic environment evaluation.

One-dimension nonlinear and dispersive seismic wave modeling in solid media

The nonlinear theory in Earth Science is very important for solving the problems of the earth. When considering some of the nonlinear properties of the medium, solitary wave (a special wave with a finite amplitude and a single peak or trough) may appear. Previous studies showed that it may be related to the rupture in the earthquake process. Therefore, it would be very helpful to explain some special phenomena in actual observation data if we fully understand the characteristics of nonlinear waves.#br#In this paper, based on the nonlinear acoustic wave equation, we first perform 1-D nonlinear acoustic wave modeling in solid media using a staggered grid finite difference method. To get the stable and accurate results, a flux-corrected transport method is used. Then we analyze several different types of nonlinear acoustic waves by setting different parameters to investigate their nonlinear characteristics in the solid media. Compared with the linear wave propagation, our results show that the nonlinear coefficients have important influences on the propagation of the acoustic waves. When the equations contain only a third-order nonlinear term (consider the case β 1 ≠ 0, β 2=0, α =0), the main lobe of the wave is tilted backward and its amplitude gradually attenuates with the wave spreading, and the amplitude of its front side-lobe attenuates slowly while the back side-lobe attenuates quickly. The whole shape and amplitude of the wave remain unchanged after propagating a certain distance. When the equations contain only a fourth-order nonlinear term (consider the case β 2 ≠ 0, β 1=0, α =0), the main lobe and the two side-lobes of the wave are all slowly damped, but the shape of the whole wave is unchanged with the wave spreading.#br#In addition, for some combinations of nonlinear and dispersive parameters (consider the case β 1 ≠ 0, α ≠ 0, β 2=0), the wave acts like the linear wave, and the nonlinear acoustic wave is equal to solitary wave which is usually obtained by Kortewegde de Vries (KdV) equation. We validate our modeling method by comparing our results with the analytic solitary solutions. Solitary wave propagates with a fixed velocity slightly less than that of the linear compressional wave, which is probably due to the balance between nonlinear and dispersion effects, making the stress-strain constitutive relations show the nature of linear wave.

Numerical simulation of equatorial plasma bubbles over Cachimbo: COPEX campaign

The problem of day-to-day variability in onset of equatorial spread F (ESF) is addressed using data from the 2002 COPEX observational campaign in Brazil and numerical modeling. The observational results show that for values of virtual height of the F layer base less than 355km at around 18:35 LT, and for the prereversal peak enhancement of the vertical plasma drift (Vp) less than 30m/s, the spread-F (ESF) was absent on four nights over Cachimbo (9.5°S, 54.8°W, dip latitude=−2.1°). In this work we analyze the geophysical conditions for the generation of the irregularities by comparing the nights with and without the ESF. In the comparison a numerical code is used to simulate plasma irregularity development in an extended altitude range from the bottom of the equatorial F layer. The code uses the flux corrected transport method with Boris–Book’s flux limiter for the spatial integration and a predictor–corrector method for the direct time integration of the continuity equation for O+ and the SOR (Successive-Over-Relaxation) method for electric potential equation. The code is tested with different evening eastward electric fields (or vertical drifts Vp<30m/s and Vp>30m/s) in order to study the influence of the prereversal enhancement in the zonal electric field on plasma bubble formation and development. The code also takes into account the zonal wind, the vertical electric field and the collision frequency of ions with neutrals and the amplitude of initial perturbation. The simulation shows a good agreement with the observational results of the ESF. The results of the code suggest that the instability can grow at the F layer bottomside by the Rayleigh–Taylor mechanism only when the Vp>30m/s. In the analyzed cases we have considered the competition of other geophysical parameters in the generation of plasma structures.