Flux-corrected Transport Research Articles

Finite-difference modelling in two-dimensional anisotropic media using a flux-corrected transport technique

Finite-difference modelling in two-dimensional anisotropic media using a flux-corrected transport technique

Nonlinear Filtering for Low-Velocity Gaseous Microflows

Gaseous flows in microfluidic devices are often characterized by relatively high Knudsen numbers. For such flows, the continuum approximation is not valid, and direct simulation Monte Carlo (DSMC) may be used to find an appropriate solution. For low-velocity flows, where the fluid velocity is much smaller than the mean molecular velocity, large statistical fluctuations in the solution mean that the features of the flow may be obscured by noise in the solution. The use of a high-order, nonlinear monotone convection algorithm, flux-corrected transport (FCT), as a filter to extract the solution from the noisy DSMC calculation is described. The diffusion, antidiffusion, and flux-limiting properties of FCT are discussed in terms of their filtering properties. The effects of filtering with FCT are demonstrated for a series of test problems, including a square wave with superimposed random noise, and low-and high-velocity and low- and high-Knudsen-number microchannel flows

Comparisons of numerical methods with respect to convectively dominated problems

AbstractA series of numerical schemes: first‐order upstream, Lax–Friedrichs; second‐order upstream, central difference, Lax–Wendroff, Beam–Warming, Fromm; third‐order QUICK, QUICKEST and high resolution flux‐corrected transport and total variation diminishing (TVD) methods are compared for one‐dimensional convection–diffusion problems. Numerical results show that the modified TVD Lax–Friedrichs method is the most competent method for convectively dominated problems with a steep spatial gradient of the variables. Copyright © 2001 John Wiley & Sons, Ltd.

Particle-in-Cell Simulation of Electrical Gas Discharges

A fluid particle-in-cell (PIC) model is proposed for the numerical solution of the continuity equation of electrons and ions in transient electrical gas discharges. The reactions occurring in a gaseous discharge, such as ionization of neutral molecules, electron attachment, and recombination between electrons and ions, are implemented through the variation of the mass of the computational particles used in the simulation. Two different forms of interpolation of the gain/loss rates from the grid to the computational particles are suggested, depending on the reaction type. The PIC model is first applied to the problem of an idealized electron avalanche in a non-attaching gas. This problem possesses an analytical solution where the electron density grows exponentially in time as it propagates, but keeps the square-wave form of the initial electron distribution. This problem is used to validate the optimum interpolation of the gain/loss rate and to analyze the effect of the mass matrix formulation of the PIC model. Then, a more realistic model is applied to simulate the propagation of a Trichel pulse between a sphere and a plate. In this case, the continuity equation for electrons and positive and negative ions, coupled to the Poisson equation, has been solved. This second test has proved the ability of the present numerical method to deal with those discharges dominated by the space charge effect. The results of the PIC simulation are compared with those obtained from the application of a flux-corrected transport method.

A two-dimensional, finite-element, flux-corrected transport algorithmfor the solution of gas discharge problems

An improved finite-element flux-corrected transport (FE-FCT)scheme, which was demonstrated in one dimension by the authors, is nowextended to two dimensions and applied to gas discharge problems. Thelow-order positive ripple-free scheme, required to produce a FCTalgorithm, is obtained by introducing diffusion to the high-orderscheme (two-step Taylor-Galerkin). A self-adjusting variable diffusioncoefficient is introduced, which reduces the high-order scheme to theequivalent of the upwind difference scheme, but without the complexities of anupwind scheme in a finite-element setting. Results are presented which showthat the high-order scheme reduces to the equivalent of upwinding when the newdiffusion coefficient is used. The proposed FCT scheme is shown to givesimilar results in comparison to a finite-difference time-split FCT codedeveloped by Boris and Book. Finally, the new method is applied for the firsttime to a streamer propagation problem in its two-dimensional form.

Simulation of electrophoretic separations by the flux-corrected transport method

Electrophoretic separations at typical experimental electric field strengths have been simulated by applying the flux-corrected transport (FCT) finite difference method to the transient, one-dimensional electrophoresis model. The performance of FCT on simulations of zone electrophoresis (ZE), isotachophoresis (ITP), and isoelectric focusing (IEF) has been evaluated. An FCT algorithm, with a three-point, central spatial discretization, yields numerical solutions without numerical oscillations or spurious peaks, which have plagued previously-published second-order solutions to benchmark ZE and ITP problems. Moreover, the FCT technique captures sharp zone boundaries and IEF peaks more accurately than previously-published, first-order upwind schemes.

The acoustic wave from a shock–vortex interaction: comparison between theory and computation

Experimental, theoretical, and computational research has shown that the interaction of a shock with a vortex produces a quadrupolar acoustic wave. Although the theoretical basis for this has been accepted for some time, there has not been a detailed comparison of the predictions for the pressure variation around and through the acoustic wave to the pressure fields obtained from numerical simulations. In this paper, Ribner's theory is used to predict the acoustic pressure field evolving from a shock interacting with either a Rankine (or infinite) vortex or with a composite (or finite) vortex. A comparison of these theoretical results indicates the importance of vortex geometry on the acoustic wave. Then the theoretical results are compared to the results of detailed computations. The computational results were obtained by solving the two-dimensional conservation equations for mass, momentum, and energy for a compressible, inviscid fluid using the Flux-Corrected Transport algorithm. These results show that there is good qualitative agreement between the theory and the computation.

Simulation of free surface flows using a Runge–Kutta technique

An accurate Runge–Kutta scheme has been coupled to a flux corrected transport (FCT) technique for solving the free surface flow equations. The main attraction of this approach lies in obtaining oscillatory free solution. Numerical results indicate that this formulation is less sensitive to the CFL stability criteria, and the captured shock front is high resolute.

Compact high‐resolution algorithms for time‐dependent advection on unstructured grids

A technique for constructing monotone, high resolution, multi-dimensional upwind fluctuation distribution schemes for the scalar advection equation is presented. The method combines the second-order Lax–Wendroff scheme with the upwind positive streamwise invariant (PSI) scheme via a fluctuation redistribution step, which ensures monotonicity (and which is a generalization of the flux-corrected transport approach for fluctuation distribution schemes). Furthermore, the concept of a distribution point is introduced, which, when related to the equivalent equation for the scheme, leads to a ‘preferred direction’ for the limiting procedure, and hence to a new distribution of the fluctuation, which retains second-order accuracy from the Lax–Wendroff scheme, even when the solution contains turning points. Experimental comparisons show that the new method compares favourably in terms of speed, accuracy and robustness with other, similar, techniques. Copyright © 2000 John Wiley & Sons, Ltd.

An investigation of FEM-FCT method for streamer corona simulation

This paper proposes the Finite Element Method (FEM) combined with Flux-corrected Transport (FCT) algorithm as a method to simulate streamer corona. The interested region is discretized with unstructured grids and the streamer propagation processes of two models are calculated to illustrate the performance of the proposed method. The proposed method requires substantially smaller number of grids than the conventional method-Finite Difference Method (FDM) combined with FCT, and the appearance of streamer propagation is quite similar to that of the papers which used the FDM-FCT method based on the structured grids.

Numerical study of thermoacoustic waves in an enclosure

The behavior of thermoacoustic waves in a nitrogen-filled two-dimensional cavity is numerically studied in order to investigate how these waves may be used as an effective heat removal mechanism. The compressible, unsteady Navier–Stokes equations were solved for a series of initial conditions by combining a flux-corrected transport algorithm for convection with models for temperature-dependent viscosity and thermal conduction. By considering a one-dimensional test problem and comparing the results to existing data, the accuracy of the present numerical method is verified. In the problems considered, the vertical walls of a cavity were heated or cooled to generate the thermoacoustic waves. Both impulsive and gradual changes of the wall temperatures were considered. When the vertical wall was heated impulsively and nonuniformly, the waves induced two-dimensional flows within the enclosure. The observed thermoacoustic waves oscillate and eventually decay due to viscous and heat dissipation.

Simulation of the coronal development in air at radio frequency: the effects of attachment, secondary emission and diffusion

The finite-element flux-corrected transport (FE-FCT) method is used to examine the transient development of corona discharges in a 1 cm hyperboloid (50 /spl mu/m tip radius) point-plane gap in air at a frequency of 40 MHz. The investigation revealed the formation of streamers in the positive part of the voltage cycle during the transient stage. These are suppressed in the steady state due to the formation of negative ions, while a cathode fall region is fully established. The effects of attachment, secondary emission and diffusion are also examined, both in the transient and steady states.

A high-resolution finite element scheme for convection-dominated transport

A finite element methodology for obtaining non-oscillatory and non-diffusive solutions to convection problems is proposed. The presented technique can be traced back to the concept of flux-corrected transport, but it differs from the existing FEM-FCT methods in that the high-order solution is corrected pointwise. This simplifies implementation and provides a greater freedom in the choice of high and low-order methods. Mass conservation is enforced by post-processing. Numerical examples in one and two dimensions confirm the accuracy properties of the new algorithm. Copyright © 2000 John Wiley & Sons, Ltd.

Compact high-resolution algorithms for time-dependent advection on unstructured grids

A technique for constructing monotone, high resolution, multi-dimensional upwind fluctuation distribution schemes for the scalar advection equation is presented. The method combines the second-order Lax–Wendroff scheme with the upwind positive streamwise invariant (PSI) scheme via a fluctuation redistribution step, which ensures monotonicity (and which is a generalization of the flux-corrected transport approach for fluctuation distribution schemes). Furthermore, the concept of a distribution point is introduced, which, when related to the equivalent equation for the scheme, leads to a ‘preferred direction’ for the limiting procedure, and hence to a new distribution of the fluctuation, which retains second-order accuracy from the Lax–Wendroff scheme, even when the solution contains turning points. Experimental comparisons show that the new method compares favourably in terms of speed, accuracy and robustness with other, similar, techniques. Copyright © 2000 John Wiley & Sons, Ltd.

Numerical Simulation of Neutral Dynamics in Stationary Short - Gap Discharge in Air

The numerical simulation of the variation of the neutral molecule density induced by the strong correlation between the dynamics of the electron gas and that of the neutral gas is studied in a 2-D numerical modelisation. The energy injection is simulated by a mathematical function that represents the spatial dependence of the discharge density. The simulated discharge is a positive point to plane discharge in a gap = 6 mm, in air at atmospheric pressure (760 Torr), with cylindrical symmetry. The hydrodynamic set of equations, i.e. equations of transport for mass, momentum and energy, is solved by the flux corrected transport (F.C.T.) method, in employing the procedure of time splitting for the two space variables. The space and time distributions of the fundamental hydrodynamic quantities, density and temperature of the neutral gas are obtained.

Numerical modelling of methanol liquid pool fires

The focus of this paper is on numerical modelling of methanol liquid pool fires. A mathematical model is first developed to describe the evaporation and burning of a two-dimensional or axisymmetric pool containing pure liquid methanol. Then, the complete set of unsteady, compressible Navier-Stokes equations for reactive flows are solved in the gas phase to describe the convection of the fuel gases away from the pool surface, diffusion of the gases into the surrounding air and the oxidation of the fuel into product species. Heat transfer into the liquid pool and the metal container through conduction, convection and radiation are modelled by solving a modified form of the energy equation. Clausius–Clapeyron relationships are invoked to model the evaporation rate of a two-dimensional pool of pure liquid methanol. The governing equations along with appropriate boundary and interface conditions are solved using the flux-corrected transport algorithm. Numerical results exhibit a flame structure that compares well with experimental observations. Temperature profiles and burning rates were found to compare favourably with experimental data from single- and three-compartment laboratory burners. The model predicts a puffing frequency of approximately 12 Hz for a 1 cm diameter methanol pool in the absence of any air co-flow. It is also observed that increasing the air co-flow velocity helps in stabilizing the diffusion flame, by pushing the vortical structures away from the flame region.

Characterization of point-plane corona in air at radio frequency using a FE-FCT method

The development of corona under RF conditions in point-plane configuration has been investigated in this paper, using the recently developed finite-element flux-corrected transport (FE-FCT) method. The study was made in air for a 1 cm gap with 50 µm tip radius needle at 40 MHz and the effects of different parameters, such as the peak applied voltage (in the range 3-4.5 kV) on the coronal development and particularly on the current and light output, are examined. Polarity and hysteresis effects are observed, as well as a plateau in the current waveform, which are confirmed by previously published experimental results. Finally, a Fourier analysis on the steady-state current waveform is performed.

Twodee: The Health and Safety Laboratory's shallow layer model for heavy gas dispersion Part 2: Outline and validation of the computational scheme

Part 1 of this three part paper described the mathematical and physical basis of twodee, the Health and Safety Laboratory's shallow layer model for heavy gas dispersion. In this part, the numerical solution method used to simulate the twodee mathematical model is developed. The boundary conditions for the leading edge, discussed in part 1, make demanding requirements on the computational scheme used. The flux correction scheme of Zalesak [S.T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, Journal of Computational Physics, 31 (1979) 335–362] is used in twodee as this has all the required properties. The twodee code is then tested against a number of theoretical and computational benchmark problems.

Twodee: the Health and Safety Laboratory's shallow layer model for heavy gas dispersion Part 3: Experimental validation (Thorney Island)

Part 1 of this three-part paper described the mathematical and physical basis of twodee, the Health and Safety Laboratory's shallow layer model for heavy gas dispersion. In part 2, the numerical solution method used to simulate the twodee mathematical model was developed; the flux correction scheme of Zalesak [S.T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, Journal of Computational Physics, 31 (1979) 335–362.] was used in twodee. This paper compares results of the twodee model to the experimental results taken at Thorney Island [J. McQuaid, B. Roebuck, The dispersion of heavier-than-air gas from a fenced enclosure. Final report to the U.S. Coast Guard on contract with the Health and Safety Executive. Technical Report RPG 1185, Safety Engineering Laboratory, Research and Laboratory Services Division, Broad Lane, Sheffield S3 7HQ, UK, 1985.]. There is no evidence to suggest that twodee predictions could be improved by changing any of the entrainment parameters from generally accepted values [R.K.S. Hankin, Heavy gas dispersion over complex terrain, PhD thesis, Cambridge University, 1997.]. The twodee model was broadly insensitive to the exact values of the entrainment parameters.

Flux-corrected transport technique for open channel flow

In modeling flow in open channels, the traditional finite difference/finite volume schemes become inefficient and warrant special numerical treatment in the presence of shocks and discontinuities. The numerical oscillations that arise by making use of a second- and higher-order schemes require some additional smoothing mechanism. A characteristic feature of high-resolution schemes lies in smooth capturing of the shock fronts. We provide a general formulation for a flux-corrected transport algorithm to the one-dimensional open channel flow equations. The preliminary results presented show that the present algorithm is an efficient, conservative and robust tool that can be easily coded. To demonstrate the robustness of the present formulation, results are compared with other published numerical results, experimental data and analytical solutions when available. In particular, a comprehensive study on the effect of the source term, dry bed, variable width channel, steep sloping channel and flow with mixed flow conditions (as in a hydraulic jump) has been carried out to test the efficacy of the present algorithm