Discrete Transport Equation Research Articles

On the Accuracy of Transport Spatial Discretization Methods for Diffusive Regimes

An infinite-medium analysis is performed for neutron transport spatial discretization methods in planar geometry. Angular flux solutions of the spatially continuous transport equation, which are driven by a linear (or quadratic) source, are shown to vary linearly (or quadratically) in space and angle; these are used to assess whether the discretized transport equations preserve certain cell-averaged and edge quantities. Each of the continuous angular flux solutions has a scalar flux that satisfies the standard diffusion equation; our analysis predicts whether the transport discretizations yield an accurate diffusion coefficient and (diffusion) spatial differencing scheme. The linear moment–based discretization methods under consideration, which are found to preserve certain features of the linear (or quadratic) infinite-medium angular flux solutions, are the familiar linear discontinuous (LD), lumped linear discontinuous (LLD), and linear characteristic (LC) schemes. The step characteristic scheme, which yields an unphysically large diffusion coefficient, is revisited and shown to possess, for diffusive problems, a solution error that would occur if an unphysical anisotropic scattering term had been included in the starting discretized S N transport equations. The numerical results verify the theoretical predictions and demonstrate the accuracy of the LD, LLD, and LC schemes in highly scattering problems that are optically thick. Our numerical results also illustrate the impact of inaccuracies in the diffusion coefficient on the numerical solutions of eigenvalue problems. The analysis in this paper has practical implications in the choice of spatial schemes used to solve realistic eigenvalue problems.

On the momentum resolution limit in solving the discrete Wigner transport equation

The Wigner transport equation can take quantum-mechanical effects into account via the Wigner potential performing the non-local potential correlation. The discretization of the Wigner potential in the phase space makes the maximum correlation length finite and inversely proportional to the grid size (or resolution) in the momentum space (Δk). It has been known that the use of too small Δk leads to unphysical simulation results and there has been a lower bound called the momentum resolution limit for getting physically correct results. Here, we show that these numerical artifacts result from the abrupt regime change (quantum to classical or vice versa) across the contact between the simulated device and its boundary. We also demonstrate that its effect can be mitigated by introducing non-rectangular window functions to the Wigner potential.

Realizability-preserving time-stepping for the differential Reynolds stress turbulence models

In the context of incompressible turbulence modelling, differential Reynolds stress models allow a better representation of turbulence anisotropy than classical linear eddy viscosity models and are less expensive than large eddy simulations in terms of computational cost. They rely on a modelled transport equation for the Reynolds stress tensor, which is by definition a covariance tensor and must remain symmetric positive semi-definite. However, the solution of the modelled and discretized transport equation may not preserve these mathematical properties. In the present article, a methodology to time-split Reynolds stress R__ terms in the transport equation is proposed using a decomposition theorem which proves the realizability of R__ provided realizable initial conditions. Realizability-preserving time-steppings are designed for several differential Reynolds stress models. The numerical schemes are presented in a finite volume context but may be used with others space discretization schemes. The realizability, precision and robustness of the numerical scheme is verified through the anisotropy time evolution in homogeneous shear-flow simulations with various initial conditions.

Hybridized nanofluidic convection in umbrella-shaped porous thermal systems with identical heating and cooling surfaces

PurposeThis study aims to investigate the impact of different heater geometries (flat, rectangular, semi-elliptical and triangular) on hybrid nanofluidic (Cu–Al2O3–H2O) convection in novel umbrella-shaped porous thermal systems. The system is top-cooled, and the identical heater surfaces are provided centrally at the bottom to identify the most enhanced configuration.Design/methodology/approachThe thermal-fluid flow analysis is performed using a finite volume-based indigenous code, solving the nonlinear coupled transport equations with the Darcy number (10–5 ≤ Da ≤ 10–1), modified Rayleigh number (10 ≤ Ram ≤ 104) and Hartmann number (0 ≤ Ha ≤ 70) as the dimensionless operating parameters. The semi-implicit method for pressure linked equations algorithm is used to solve the discretized transport equations over staggered nonuniform meshes.FindingsThe study demonstrates that altering the heater surface geometry improves heat transfer by up to 224% compared with a flat surface configuration. The triangular-shaped heating surface is the most effective in enhancing both heat transfer and flow strength. In general, flow strength and heat transfer increase with rising Ram and decrease with increasing Da and Ha. The study also proposes a mathematical correlation to predict thermal characteristics by integrating all geometric and flow control variables.Research limitations/implicationsThe present concept can be extended to further explore thermal performance with different curvature effects, orientations, boundary conditions, etc., numerically or experimentally.Practical implicationsThe present geometry configurations can be applied in various engineering applications such as heat exchangers, crystallization, micro-electronic devices, energy storage systems, mixing processes, food processing and different biomedical systems (blood flow control, cancer treatment, medical equipment, targeted drug delivery, etc.).Originality/valueThis investigation contributes by exploring the effect of various geometric shapes of the heated bottom on the hydromagnetic convection of Cu–Al2O3–H2O hybrid nanofluid flow in a complex umbrella-shaped porous thermal system involving curved surfaces and multiphysical conditions.

A hybridizable discontinuous Galerkin method for the fully coupled time-dependent Stokes/Darcy-transport problem

We present a high-order hybridized discontinuous Galerkin (HDG) method for the fully coupled time-dependent Stokes–Darcy-transport problem where the fluid viscosity and source/sink terms depend on the concentration and the dispersion/diffusion tensor depends on the fluid velocity. This HDG method is such that the discrete flow equations are compatible with the discrete transport equation. Furthermore, the HDG method guarantees strong mass conservation in the Hdiv sense and naturally treats the interface conditions between the Stokes and Darcy regions via facet variables. We employ a linearizing decoupling strategy where the Stokes/Darcy and the transport equations are solved sequentially by time-lagging the concentration. We prove well-posedness and optimal a priori error estimates for the velocity and the concentration in the energy norm. We present numerical examples that respect compatibility of the flow and transport discretizations and demonstrate that the discrete solution is robust with respect to the problem parameters.

Solution Irregularity Remediation for Spatial Discretization Error Estimation for SN Transport Solutions

The discrete ordinates linear Boltzmann transport equation is typically solved in its spatially discretized form, incurring spatial discretization error. Quantification of this error for purposes such as adaptive mesh refinement or error analysis requires an a posteriori estimator, which utilizes the numerical solution to the spatially discretized equation to compute an estimate. Because the quality of the numerical solution informs the error estimate, irregularities, present in the true solution for any realistic problem configuration, tend to cause the largest deviation in the error estimate vis-a-vis the true error. In this paper, an analytical partial singular characteristic tracking (pSCT) procedure for reducing the estimator’s error is implemented within our novel residual source estimator for a zeroth-order discontinuous Galerkin scheme, at the additional cost of a single inner iteration. A metric-based evaluation of the pSCT scheme versus the standard residual source estimator is performed over the parameter range of a Method of Manufactured Solutions test suite. The pSCT scheme generates near-ideal accuracy in the estimate in problems where the dominant source of the estimator’s error is the solution irregularity, namely, problems where the true solution is discontinuous and problems where the true solution’s first derivative is discontinuous and the scattering ratio is low. In problems where the scattering ratio is high and the true solution is discontinuous in the first derivative, the error in the scattering source, which is not converged by the pSCT scheme, is greater than the error incurred due to the irregularity. Ultimately, a pSCT scheme is judged to be useful for error estimation in problems where the computational cost of the scheme is justified. In the presence of many irregularities, such a scheme may be intractable for general use, but in benchmarks, as an analytical tool, or in problems that have nondissipative discontinuities, the scheme may prove invaluable.

An Adjoint Sensitivity Model for Radionuclide Transport in Heterogeneous Discrete Fractured Porous Formation in Steady‐State Regime

The development of the adjoint state method for the system of partial differential equations (PDEs) describing radionuclide transport in heterogeneous discrete fractured porous media in steady-state regime is presented in this study for both continuous and discrete adjoint approaches. The continuous adjoint approach has its own advantage in the sense that the adjoint system can be derived in terms of PDEs which can be solved analytically for specific performance measures. These PDEs can then be used to obtain a physical interpretation of the adjoint state functions. The discrete adjoint approach permits a simpler mathematical treatment based on the use of the same matrix (or its transpose) of the linear system of the discretized transport equations referring to from hereon as the forward problem. Analytical solutions for both the forward problem and its associated adjoint problem have been derived for radionuclide transport in layered formations including a fracture using an analytical inversion of tridiagonal matrices. A set of 14 formulas of sensitivity coefficients with respect to pertinent system parameters has been derived and presented in integral forms. Analytical results have been compared with numerical results and have shown excellent agreements. The discrete adjoint approach has been coupled with the differential sensitivity analysis method and applied to a real field case involving the performance assessment of a site currently being considered for underground nuclear storage in northern Switzerland, Zürich Nordost. The results show that when a fracture is present in the vicinity of waste repository, the state parameter controlling flow through the fracture, referred to herein as fracture transmissivity, is among the important parameters which should be considered in sensitivity analysis studies.

IMPROVING A POSTERIORI SPATIAL DISCRETIZATION ERROR ESTIMATION FOR SN TRANSPORT BY SINGULAR CHARACTERISTICS TRACKING

Previously, we have developed a novel spatial discretization error estimator, the “residual source” estimator, in which an error transport problem, analogous to the discretized transport equation, is solved to acquire an estimate of the error, with a residual term acting as a fixed source. Like all error estimators, the residual source estimator suffers inaccuracy and imprecision in the proximity of singular characteristics, lines across which the solution is irregular. Estimator performance worsens as the irregularities become more pronounced, especially so if the true solution itself is discontinuous. This work introduces a modification to the residual approximation procedure that seeks to reduce the adverse effects of the singular characteristics on the error estimate. A partial singular characteristic tracking scheme is implemented to reduce the portion of the error in the numerical solution born by irregularities in the true solution. This treated numerical solution informs the residual approximations. The partial singular characteristic tracking scheme greatly enhances the numerical solution for a problem with prominent singular characteristics. The residual approximation and resultant residual source error estimate are likewise improved by the scheme, which only incurs the computational cost of an extra inner iteration.

Dynamic Coarsening and Local Reordered Nonlinear Solvers for Simulating Transport in Porous Media

Summary We present a robust and flexible sequential solution approach in which the flow equation is solved on the original grid, whereas the transport equations are solved with a new dynamic coarsening method that adapts the grid resolution locally to reduce the number of cells as much as possible. The resulting grid is formed by combining precomputed coarse partitions of an underlying fine model. Our approach is flexible and makes very few assumptions on cell geometries and the topology of the grid. To further accelerate the transport step, we combine dynamic coarsening with a local nonlinear solver that permutes the discrete transport equations into an optimal block-triangular form so that these can be solved very efficiently using a nonlinear back-substitution method. Efficiency and utility of the overall approach are assessed through a number of conceptual test cases, including the Olympus field model.

Numerical Research of Movement Characteristic and Deposition Ratio on Cylinder Wall of Secondary Water Droplet between Penultimate Stage and Last Stage

Secondary water droplet between the penultimate and last stage of a 300 MW steam turbine was researched by CFD on the basis of a two-phase flow mathematical model for steam and water droplet, and the Lagrangian methed was adopted to solve the discrete transport equation of water droplet. It indicated that secondary water droplet shows tendency to slop up to and get deposited on cylinder wall. The water deposition ratio on cylinder wall will be greater as the water droplets size increases or the blade height where the water droplets begin to move rises. Local deposition ratio increases as the blade height heightens according to the water deposition on cylinder wall behind penultimate stage and distribution of the secondary water droplet and steam moisture.

Diffusion synthetic acceleration with the fine mesh rebalance of the subcell balance method with tetrahedral meshes for SN transport calculations

Abstract A diffusion synthetic acceleration (DSA) technique for the SN transport equation discretized with the linear discontinuous expansion method with subcell balance (LDEM-SCB) on unstructured tetrahedral meshes is presented. The LDEM-SCB scheme solves the transport equation with the discrete ordinates method by using the subcell balances and linear discontinuous expansion of the flux. Discretized DSA equations are derived by consistently discretizing the continuous diffusion equation with the LDEM-SCB method, however, the discretized diffusion equations are not fully consistent with the discretized transport equations. In addition, a fine mesh rebalance (FMR) method is devised to accelerate the discretized diffusion equation coupled with the preconditioned conjugate gradient (CG) method. The DSA method is applied to various test problems to show its effectiveness in speeding up the iterative convergence of the transport equation. The results show that the DSA method gives small spectral radii for the tetrahedral meshes having various minimum aspect ratios even in highly scattering dominant mediums for the homogeneous test problems. The numerical tests for the homogeneous and heterogeneous problems show that DSA with FMR (with preconditioned CG) gives significantly higher speedups and robustness than the one with the Gauss-Seidel-like iteration.

1D thermal creep channel flow in the BGK approximation by adding and doubling

In the past, the method of adding and doubling has received considerable attention as an effective approach to solving the linear transport equation. Here, we consider doubling for the problem of thermal creep in Rarefied Gas Dynamics. The method, called the Discrete/Discrete Ordinates (D/DO) method is in contrast to the well-known Analytical/Discrete Ordinates (A/DO) method. The D/DO method solves the fully discretized transport equation to demonstrate that straightforward numerical approximation also yields high quality numerical solutions. The method combines the fundamental forward Euler evaluation of the matrix exponential with Richardson extrapolation to generate a high quality benchmark. We then generalize the method using Padé approximants for the matrix exponential and present a 7-place benchmark.

The Neumann numerical boundary condition for transport equations

In this article, we show that prescribing homogeneous Neumann type numerical boundary conditions at an outflow boundary yields a convergent discretization in \begin{document}$ \ell^\infty $\end{document} for transport equations. We show in particular that the Neumann numerical boundary condition is a stable, local, and absorbing numerical boundary condition for discretized transport equations. Our main result is proved for explicit two time level numerical approximations of transport operators with arbitrarily wide stencils. The proof is based on the energy method and bypasses any normal mode analysis.

Accuracy balancing for the simulation of gate-all-around junctionless nanowire transistors using discrete Wigner transport equation

We report the application of the discrete Wigner transport equation to the simulation of gate-all-around junctionless nanowire transistors (JLNWTs). We show that unphysical simulation results are obtained unless we pay attention to the accuracy balancing problem. An empirical criterion based on the subthreshold swing of the JLNWT is proposed for the accuracy balancing. Using this criterion, we investigate the characteristics of JLNWTs with the quantum effect fully considered.

Stability of Finite Difference Schemes for Hyperbolic Initial Boundary Value Problems: Numerical Boundary Layers

AbstractIn this article, we give a unified theory for constructing boundary layer expansions for discretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer expansion. In particular, this expansion yields discrete semigroup estimates that are compatible with the continuous semigroup estimates in the limit where the space and time steps tend to zero. The novelty of our approach is to cover numerical schemes with arbitrarily many time levels.

A novel GEEC (Geometry, Energy, and Entropy Compatible) procedure applied to a staggered direct-ALE scheme for hydrodynamics

The simulation of transient compressible single-fluid flows, modeled by Euler’s classical equation, requires numerical schemes that comply with various stringent physical constraints such as thermodynamic consistency, robustness, and stability when shocks, large volume changes, and transport effects are present. For extreme conditions as encountered in defense applications, the present work introduces a novel generic GEEC (Geometry, Energy, and Entropy Compatible) procedure to derive physics-compatible numerical schemes with the following built-in properties: (i) geometric compatibility whereby volume variations (compressions and expansions) are taken into account consistently with advection relative to the grid; (ii) energetic compatibility whereby total energy is exactly conserved at discrete level for an accurate capture of shock levels and shock velocities, even if solving with primitive variables; and (iii) entropic compatibility whereby compliance with the second law of thermodynamics is ensured (here at least to second-order) – entropy must increase in general and must be conserved for isentropic flows.This novel generic GEEC procedure consists of a three-step mimicking derivation: (i) for a prescribed mass transport equation, a variational least action principle is used to generate the proper pressure forces in the discrete momentum evolution equation – thus ensuring a compatible exchange between kinetic and internal energies; – (ii) corrections are systematically performed on numerical residues in order to force conservation at discrete level up to round-off errors; and (iii) an artificial viscosity term is added as a pressure-like contribution in the evolution equations in order to capture shocks and to stabilize the scheme.As a proof of concept, this procedure is applied in the present work to produce a novel ALE (Arbitrary Lagrangian–Eulerian) compressible hydro-scheme with the following features: (i) a direct ALE formalism where mass, momentum, and energy transport fluxes are directly taken into account without separation in the discrete equations; (ii) continuity with “historic” Lagrangian solvers where velocity fields are space-and-time staggered; (iii) second-order accuracy in the Lagrangian limit; and as a consequence of the least action principle, (iv) a non-standard downwind formulation of the pressure gradient, dual of the upwind transport operator.Results on standard numerical test cases involving shocks and large deformations confirm the expected built-in properties of the scheme. In particular, the entropic compatibility leads to the preservation of entropy to second-order for isentropic flows, regardless of mesh motions and regardless of the first-order accuracy of the discrete mass transport equation. Indifference and versatility with respect to strenuous grid motion strategies are demonstrated in various situations – including supersonic shearing for new variants of Sod’s shock tube and Sedov’s blast wave – highlighting the advantages of the compatibility between velocity fields and volume variations in the presence of strong fluids advection.

Accuracy balancing for the finite-difference-based solution of the discrete Wigner transport equation

The Wigner transport equation based on the Wigner function which is defined on the phase space describes two actions in orthogonal directions of the phase space: movement (diffusion) in position space and transition in momentum space. Here, we show that for the proper analysis of a resonant tunneling diode using the finite-difference-based solution of the Wigner transport equation, the degree of numerical accuracy in the calculation of the movement in position space should be balanced with that in the evaluation of the transition in momentum space.

An efficient numerical scheme for the discrete Wigner transport equation via the momentum domain narrowing

We propose a numerical scheme that narrows down the momentum domain of the Wigner function to enhance numerical efficiency. It enables us to decrease the number of mesh points while maintaining the same mesh spacing in the momentum coordinate. The proposed scheme thus not only requires less memory but can significantly reduce the computation time. To minimize resultant loss of numerical accuracy, we also propose the partial local potential averaging method.

Design of the Catalyst Layers in Pemfcs Using an Adjoint Sensitivity Analysis Approach

The design of the anode and cathode catalyst layers is one of the most important processes in the overall design of low-temperature proton exchange membrane fuel cells (PEMFCs) because of two major reasons: (1) the discharge voltage and the power density of PEMFC strongly depend on the porosity and catalyst distribution in the catalyst layers and (2) the final price of the PEMFC depends significantly on the total mass of the platinum in these layers. Hence, it is particularly important to carefully design the microstructure and platinum distribution in the catalyst layers in order to both increase the power density and reduce the price of the cells. There are many efforts in the literature to optimize the catalyst layers in PEMFCs; a review of the existing can be found in [1]. Most of this work focuses on solving the discretized transport equations on 1-D or 2-D finite element grids and designing the platinum loading, catalyst layer thickness, platinum-to-carbon ratio, or agglomerate composition that result in the largest discharge voltage at a given discharge current. In all the existing works in the literature, the total number of design variables ranges between 3-5 variables that are designed using either gradient-based techniques, evolutionary, or genetic algorithms. In this work, for the first time, we present a large scale optimization method in which we design the optimum platinum distribution in the catalyst layers at each mesh point on the finite element grid. Hence, the total number of design variables is equal to the number of degrees of freedom of the grid, which is of the order of 104 in 2-D meshes and 106 in 3-D meshes. In this way, we are able to predict the precise distribution of the platinum content at each location inside the cell. Our technique is based on evaluating the sensitivity functions of the platinum deposition in the catalyst layer and uses a new gradient-based approach to compute the optimum distribution of the platinum in the catalyst layer. The sensitivity functions are computed in the adjoint space of discretized transport variables by using a new algorithm that requires solving only one sparse system of linear equations instead of a large number of systems to find the Gâtaux derivatives of the discharge current of the cell with respect to the design variables. The technique is relatively general and can also be applied to the optimization of the void fractions (i.e. porosity) and platinum-to-carbon ratios at each mesh point. The technique can be applied to any transport system of equations including traditional diffusion-migration models coupled with Butler-Volmer kinetics at the anode and cathode, as well as to models based on Monte-Carlo techniques or models that require solving the nonlinear Poisson equation. In our presentation we use a simplified but well calibrated model developed by Cetinbas et al [3]. The figure below shows the simulation region, the transport equations and the fluxes of species throughout the cell in the case of a 2-D geometry. Our numerical results show that one can increase the power density of the current state-of-the-art design presented in [2] by at least 15 % by using the same amount of platinum and redistributing it non-uniformly throughout the catalyst layers. More details about the technique, the numerical implementation, and the final optimized structures will be presented at the conference. Figure 1. Simplified schematics of a PEMFC showing the simulation region, the transport equations, and the corresponding fluxes of the chemical species. References. [1] M. Secanell, J. Wishart, and P. Dobson, "Computational design and optimization of fuel cells and fuel cell systems: A review," Journal of Power Sources, vol. 196, pp. 3690-3704, 2011. Figure 1

Fast Simulation of Polymer Injection in Heavy-Oil Reservoirs on the Basis of Topological Sorting and Sequential Splitting

SummaryWe present a set of algorithms for sequential solution of flow and transport that can be used for efficient simulation of polymer injection modeled as a compressible two-phase system. Our formulation gives a set of nonlinear transport equations that can be discretized with standard implicit upwind methods to conserve mass and volume independent of the timestep. In the absence of gravity and capillary forces, the resulting nonlinear system of discrete transport equations can be permuted to lower triangular form with a simple topological-sorting method from graph theory. This gives an optimal nonlinear solver that computes the solution cell by cell with local iteration control. The single-cell systems can be reduced to a nested set of nonlinear scalar equations that can be bracketed and solved with standard gradient or root-bracketing methods. The resulting method gives orders-of-magnitude reduction in run times and increases the feasible timestep sizes. In fact, one can prove that the solver is unconditionally stable and will produce a solution for arbitrarily large timesteps. For cases with gravity, the same method can be applied as part of a nonlinear Gauss-Seidel method. Altogether, our results demonstrate that sequential splitting combined with single-point upwind discretizations can become a viable alternative to streamline methods for speeding up simulation of advection-dominated systems.