Diffuse conditions for efficient solution of multislab radiation transport problems

In this article, we make use of recently developed spectral nodal methods for anisotropically scattering media and we derive mathematical conditions for the diffuse reflection and transmission of radiation in the discrete ordinates formulation of particle transport theory for plane-parallel applications. The conditions arise from a suitable reformulation of spatially discretized equations defined on the boundary layers of a multislab domain. As a result, the boundary layers can be removed from the radiation transport calculations and replaced with exact and numerically stable equivalent conditions. In order to illustrate the applicability and computational merit of our discrete ordinates conditions for diffuse reflection and transmission in radiation transport calculations, we perform numerical experiments with atmospheric radiative transfer and nuclear reactor core models.     

Stretched Eulerian coordinate model of coastal sediment transport

A one-dimensional vertical model of sediment and dissolved material transport in a coupled, vertically resolved benthic-pelagic system has been developed. An apparently unique feature of the model is that the governing equations are first formulated in a time-varying Eulerian frame and then solved numerically, which provides a consistent approach for incorporating resuspension/deposition, swelling/consolidation, and biodiffusion processes within a single modelling framework. Numerical experiments were carried out with varying grids to test diffusive properties of the model. A stretched numerical grid that maintains high vertical resolution of the top sediments and constrains numerical diffusion in deep layers is an integral part of the model. The developed model provides a test bed for testing scientific hypothesis pertaining to coastal sediment transport. The model can also be used to support biogeochemical studies in a coupled benthic-pelagic system.     

Stable numeric scheme for diffusion equation with a stiff transport

It is known that the anomalous transport in fusion devices is governed by gradient-driven instabilities and is characterised by an offset linear dependence of the heat and particle fluxes on the corresponding gradients. The dependence is very strong so that a small change in gradients causes a huge variation of fluxes thus giving rise to the so-called stiff transport. This feature makes the standard numeric schemes for a parabolic equation strongly unstable so that plasma simulations with transport codes require very small time steps. In this paper, a modification of the standard finite difference scheme is suggested that eliminates this kind of numerical instability. It is shown that the implementation of the technique allows the time step for stiff transport models to be increased by several orders of magnitude. Generalisation to more advanced numeric schemes and to a system of parabolic equations is straightforward.     

Numerical predictions of backward-facing step flows in microchannels using extended Navier–Stokes equations

Flows in microchannels were successfully predicted, in the past, both analytically and numerically, employing the extended Navier–Stokes equations (ENSE). In ENSE, the self-diffusion transport of mass, together with the resulting momentum and heat transport, is taken into account properly and the same is omitted in the classical Navier–Stokes equations. The ENSE have been employed here to numerically predict backward-facing step flows in microchannels, and the predictions are summarized in this paper. The results obtained by employing ENSE are compared with the available literature data computed by both direct simulation Monte Carlo and slip-velocity-based simulations. The good agreement of the present results with those given in the literature evidently points out that the ENSE can be applied to gas flows through complex microchannel geometries.     

Lie group analysis of stagnation-point flow of a nanofluid

This article concerns with a steady two-dimensional boundary layer flow of an electrically conducting incompressible nanofluid over a stretching sheet in a porous medium with internal heat generation/absorption. The transport model includes the effect of Brownian motion with thermophoresis in the presence of chemical reaction and magnetic field. Lie group analysis is applied to the governing equations. The transformed self similar non-linear ordinary differential equations along with the boundary conditions are solved numerically. The influences of various relevant parameters on the flow field, temperature and nanoparticle volume fraction as well as wall heat flux and wall mass flux are elucidated through graphs and tables.     

Sediment transport models in Shallow Water equations and numerical approach by high order finite volume methods

This paper is concerned with the numerical approximation of bedload sediment transport due to water evolution. For the hydrodynamical component we consider Shallow Water equations. The morphodynamical component is defined by a continuity equation, which is defined in function of the solid transport discharge. We present several deterministic models, such as Meyer-Peter & Müller, Van Rijn or Grass model. We also present an unified definition for the solid transport discharge, and we compare with Grass model. Both components define a coupled system of equations that can be rewrite as a non-conservative hyperbolic system. To discretize it, we consider finite volume methods with or without flux limiters and high order state reconstructions. Finally we present several tests, where we observe numerically the order of the numerical schemes. Comparisons with analytical solutions and experimental data are also presented.     

Effect of heat generation on mixed convection in porous cavity with sinusoidal heated moving lid and uniformly heated or cooled bottom walls

Effect of heat generation and absorption on mixed convection flows in a sinusoidal heated lid-driven square cavity filled with a porous medium is investigated numerically. Both the vertical walls of the enclosure are insulated while the bottom wall is uniformly heated or cooled. The top wall is moving at a constant speed and is heated sinusoidally. The governing equations and boundary conditions are non-dimensionalized and solved numerically by using finite volume method approach along with SIMPLE algorithm together with non-uniform grid system. The effect of Darcy and heat generation parameters are investigated in terms of the flow, heat transfer, and Nusselt number. The results for stream function and isotherm are plotted and it is found that there have significant influence with the presence of heat generation and porous medium.

     

A one-dimensional reactive multi-component landfill leachate transport model

A one-dimensional reactive multi-component landfill leachate transport model coupled to three modules (geochemical equilibrium, kinetic biodegradation, and kinetic precipitation–dissolution) is presented to simulate the migration of contaminants in soils under landfills. A two-step sequential operator splitting method is applied to solve the coupled transport equations and the biogeochemical reaction equations. The geochemical module is based on the equilibrium speciation model (MINTEQA2), which uses ion-association equilibrium–constant approach to represent the various geochemical reactions. The biodegradation module describes the sequential biological degradation of organic compounds by multiple functional bacterial populations. Analytical equations based on macroscopic approach are used to model changes in porosity and permeability caused by biomass accumulation and mineral precipitation in soils. The model has been evaluated by comparing the model results with the widely used one-dimensional mixing-cell model PHREEQM for acidic mine tailings discharge in a carbonate aquifer. The composite leachate transport model is applied to a hypothetical landfill to simulate the effect of biological degradation of organic matter on the local inorganic geochemistry and also to demonstrate the effect of microbial activity on the evolution of porosity reduction of soils under the landfill.     

F.E. in environmental engineering: Coupled thermo-hydro-mechanical processes in porous media including pollutant transport

Summary This paper presents a general model for the analysis of coupled thermo-hydro-mechanical problems in porous media with possible pollutant transport. The governing equations are described and discrete solution techniques using the finite element method in space and finite differences in time are shown. Emphasis is put on a direct solution procedure, where the coupled system of equations is solved without use of matrix partitioning. Both the Newton-Raphson method and fixed point method are employed. Application examples involving pollutant transport, heat and mass transfer in partially saturated geomaterials, dynamic strain localization and durability of concrete show the range of applicability of this model in the field of evironmental engineering.     

Inverse reactive transport simulator (INVERTS): an inverse model for contaminant transport with nonlinear adsorption and source terms

A numerically based simulator was developed to assist in the interpretation of complex laboratory experiments examining transport processes of chemical and biological contaminants subject to nonlinear adsorption and/or source terms. The inversion is performed with any of three nonlinear regression methods, Marquardt–Levenberg, conjugate gradient, or quasi-Newton. The governing equations for the problem are solved by the method of finite-differences including any combination of three boundary conditions: 1) Dirichlet, 2) Neumann, and 3) Cauchy. The dispersive terms in the transport equations were solved using the second-order accurate in time and space Crank–Nicolson scheme, while the advective terms were handled using a third-order in time and space, total variation diminishing (TVD) scheme that damps spurious oscillations around sharp concentration fronts. The numerical algorithms were implemented in the computer code INVERTS, which runs on any standard personal computer. Apart from a comprehensive set of test problems, INVERTS was also used to model the elution of a nonradioactive tracer, ¹⁸⁵Re, in a pressurized unsaturated flow (PUF) experiment with a simulated waste glass for low-activity waste immobilization. Interpretation of the elution profile was best described with a nonlinear kinetic model for adsorption.     

Electron transport through cubic InGaN/AlGaN resonant tunneling diodes

We theoretically study the electron transport through a resonant tunneling diode (RTD) based on strained Al_xGa_1−xN/In_0.1Ga_0.9N/Al_xGa_1−xN quantum wells embedded in relaxed n- Al_0.15Ga_0.85N/strained In_0.1Ga_0.9N emitter and collector. The aluminum composition in both injector and collector contacts is taken relatively weak; this does not preclude achieving a wide band offset at the border of the pre-confinement wells. The epilayers are assumed with a cubic crystal structure to reduce spontaneous and piezoelectric polarization effects. The resonant tunneling and the thermally activated transfer through the barriers are the two mechanisms of transport taken into account in the calculations based on the Schrödinger, Poisson and kinetic equations resolved self-consistently. Using the transfer matrix formalism, we have analyzed the influence of the double barrier height on the resonant current. With an Al composition in the barriers varying between 30% and 50%, we have found that resonant tunneling dominates over the transport mediated by the thermally activated charge transfer for low applied voltages. It is also found that the designed n-type InGaN/AlGaN RTD with 30% of Al composition in the barriers is a potential candidate for achieving a resonant tunneling diode.     

Study of the effect of electric field and electroneutrality on transport of biomolecules in microreactors

In this article, the effect of electrophoresis on the transport of a sample (like biomolecules) in active microreactors is numerically investigated. Navier–Stokes equations are solved along with the equations of electrostatics, species mass transport in the buffer, and chemical reaction kinetics at reactive surfaces. Unlike previous studies, in which the effect of the charge of the sample molecules on the electric field has been neglected (i.e., the assumption of electroneutrality), here the space charge density is assumed to be nonzero and a function of biomolecule concentration. As a result, the governing equations become fully coupled. The validity of the assumption is examined against experimental results. Then, the effect of electroneutrality on the efficiency of the microreactor device is analyzed for the parallel plate open channel geometry, commonly used in biomolecule separation. It is shown that the electroneutrality assumption can drastically influence the final adsorbed concentration depending on the device configuration. Average adsorbed surface concentration and capture efficiency are compared as measures of the performance of the device for a wide range of physiochemical parameters. The sensitivity of the simulation with respect to the ionic concentration of the buffer is investigated. It is also discussed how the electric field and nonzero space charge density alter the bulk concentration profile and the velocity field inside the microreactor.     

A Level Set Framework for Capturing Multi-Valued Solutions of Nonlinear First-Order Equations

We introduce a level set method for the computation of multi-valued solutions of a general class of nonlinear first-order equations in arbitrary space dimensions. The idea is to realize the solution as well as its gradient as the common zero level set of several level set functions in the jet space. A very generic level set equation for the underlying PDEs is thus derived. Specific forms of the level set equation for both first-order transport equations and first-order Hamilton-Jacobi equations are presented. Using a local level set approach, the multi-valued solutions can be realized numerically as the projection of single-valued solutions of a linear equation in the augmented phase space. The level set approach we use automatically handles these solutions as they appear     

Large eddy simulation of turbulent heat transport in the Strait of Gibraltar

We develop a numerical model for large eddy simulation of turbulent heat transport in the Strait of Gibraltar. The flow equations are the incompressible Navier–Stokes equations including Coriolis forces and density variation through the Boussinesq approximation. The turbulence effects are incorporated in the system by considering the Smagorinsky model. As a numerical solver we propose a finite element semi-Lagrangian method. The solution procedure consists of combining a non-oscillatory semi-Lagrangian scheme for time discretization with the finite element method for space discretization. Numerical results illustrate a buoyancy-driven circulations along the Strait of Gibraltar and the sea-surface temperature is flushed out and move to northeast coast. The Ocean discharge and the temperature difference are shown to control the plume structure.     

Large eddy simulation of turbulent heat transport in the Strait of Gibraltar

We develop a numerical model for large eddy simulation of turbulent heat transport in the Strait of Gibraltar. The flow equations are the incompressible Navier–Stokes equations including Coriolis forces and density variation through the Boussinesq approximation. The turbulence effects are incorporated in the system by considering the Smagorinsky model. As a numerical solver we propose a finite element semi-Lagrangian method. The solution procedure consists of combining a non-oscillatory semi-Lagrangian scheme for time discretization with the finite element method for space discretization. Numerical results illustrate a buoyancy-driven circulations along the Strait of Gibraltar and the sea-surface temperature is flushed out and move to northeast coast. The Ocean discharge and the temperature difference are shown to control the plume structure.     

Effect of temperature gradient orientation on the characteristics of mixed convection flow in a lid-driven square cavity

The effect of temperature gradient orientation on the fluid flow and heat transfer in a lid-driven differentially heated square cavity is investigated numerically. The transport equations are solved using the high-order compact scheme. Four cases are considered depending on the direction of temperature gradient imposed. The differentially heated top and bottom walls result in gravitationally stable and unstable temperature gradients. While the differentially heated left and right side walls lead to assisting and opposing buoyancy effects. The governing parameters are Pr = 0.7 and Ri = 0.1, 1, and 10. It is found that both Richardson number and direction of temperature gradient affect the flow patterns, heat transport processes, and heat transfer rates in the cavity. Computed average Nusselt number indicates that the heat transfer rate increases with decreasing Ri regardless the orientation of temperature gradient imposed. And the assisting buoyancy flows have best performance on heat transport over the other three cases.     

Field, force and transport analysis for magnetic particle-based gene delivery

Magnetic particles are used to deliver gene vectors to target cells for uptake in a process known as magnetofection. Magnetic particle-based gene delivery has been successfully demonstrated for all types of nucleic acids and across a broad range of cell lines. It is well suited for multiwell culture plate systems wherein magnetic particles with surface-bound gene vectors are introduced into culture wells, and a magnetic force provided by rare-earth magnets beneath and aligned with the wells attracts the particles to the cells for uptake. In this paper, models are presented for analyzing and optimizing this process. These include closed-form equations for predicting the magnetic field and force and a drift–diffusion equation for predicting the transport and accumulation of particles in a well. The closed-form equations enable rapid parametric analysis of the spatial distribution of the field and force in a well as a function of key parameters including its dimensions, the magnet-to-well spacing, the strength of the magnet, the influence of neighboring magnets and the properties of the particles. The particle transport equation accounts for the field-induced drift of particles as well as fluidic drag and Brownian diffusion. It is solved numerically using the finite volume method. The theory is demonstrated via application to a multiwall plate magnetofection system and the impact of various factors that govern gene delivery is assessed. The models provide insight into gene delivery and are well suited for parametric analysis of particle accumulation in the wells. They enable the rational design of novel magnetofection systems.     

Nonlinear dynamic analysis of the viscoelastic string with a harmonically varying transport speed

Qualitative aspects of parametric excitation due to the non-constant traveling velocity of a viscoelastic string are investigated. The problem considered is an initially stressed viscoelastic string subjected to steady-state and harmonic variation of axially traveling motion. The string material is considered as a Violet element in series with a spring (three-parameter model). The partial differential equation of motion is derived first, and then is reduced to be a set of third-order nonlinear ordinary differential equations by applying Galerkin's method. Finally, the effects of elastic and viscoelastic parameters, constant and non-uniform transport speed, wave propagation speed ratio, and nonlinear terms on the transient amplitudes are investigated numerically.     

The influence of slip conditions, wall properties and heat transfer on MHD peristaltic transport

The effects of both wall slip conditions and heat transfer on peristaltic flow of MHD Newtonian fluid in a porous channel with elastic wall properties have been studied under the assumptions of long-wavelength and low-Reynolds number. The analytical solution has been derived for the stream function and temperature. The results for velocity, temperature, stream function and heat transfer coefficient obtained in the analysis have been evaluated numerically and discussed briefly. The numerical result shows that more trapped bolus appears with increasing Knudsen number.