Advection-diffusion Transport Equation Research Articles

Acceleration of cosmic rays in presence of magnetohydrodynamic fluctuations at small scales

ABSTRACT This work investigates the evolution of the distribution of charged particles (cosmic rays) due to the mechanism of stochastic turbulent acceleration (STA) in presence of small-scale turbulence with a mean magnetic field. STA is usually modelled as a biased random walk process in the momentum space of the non-thermal particles. This results in an advection-diffusion type transport equation for the non-thermal particle distribution function. Under quasi-linear approximation, and by assuming turbulent spectra with power being available only in the sub-gyroscale range, we find that the Fokker–Planck diffusion coefficients Dγγ and Dμμ scale with the Lorentz factor γ as Dγγ ∝ γ−2/3 and Dμμ ∝ γ−8/3. We consider Alfvèn and fast waves in our calculations, and find a universal trend for the momentum diffusion coefficient irrespective of the properties of the small-scale turbulence. Such universality has already been reported regarding the spatial diffusion of the cosmic rays, and, here too, we observe a universality in the momentum diffusion coefficient. Furthermore, with the calculated transport coefficients, we numerically solve the advection-diffusion-type transport equation for the non-thermal particles. We demonstrate the interplay of various mircophysical processes such as STA, synchrotron loss, and particle escape on the particle distribution by systematically varying the parameters of the problem. We observe that the effect of the small-scale turbulence is more impactful for the high-energy protons as compared to the electrons and such turbulence is capable of sustaining the energy of the protons from catastrophic radiative loss processes. Such a finding is novel and helps us to enhance our understanding about the hadronic emission processes that are typically considered as a competitor for the leptonic emission for certain astrophysical systems.

Coupling a neural network technique with CFD simulations for predicting 2-D atmospheric dispersion analyzing wind and composition effects

The Computational Fluid Dynamics (CFD) tool has a remarkable applicability for the prediction of gas dispersion flows by numerically solving the proper governing equations in realistic scenarios. Depending on the problem complexity, undesirably high computational costs can be incurred, which has encouraged the combined use of Machine Learning (ML) seeking to attenuate the CFD simulations requirement for multiple scenario studies. The present work aims at demonstrating the employment the coupling between CFD and the Artificial Neural Network (ANN) algorithm for representative problems in atmospheric dispersion in a preliminary assessment. A limited set of CFD simulation results was used for training neural networks, whose output is given by flow field interpolators, the potential uses of which include digital twin designing and optimization procedures. One possible strategy is the local approach, which treats the network as a transition rule in the scope of Cellular Automata (CA) modeling, allowing it to learn the dynamic behavior of the addressed physics locally. This method gives rise to simpler neural network architectures with closer computing relatively to the CFD calculation. Assessments have been done by predicting, initially, a scalar field time evolution governed by a 1-D advection-diffusion transport equation to verify the method implementation. Subsequently, species concentration distributions were sought in atmospheric dispersion cases from CFD simulations datasets, comprising four case studies followed in the performed analysis, all considering a bidimensional flow domain and a scenario involving methane leaks. The first one indicated an accurate reproduction of subsequent time steps concentration field referring to the displacement of a methane cloud. The second and third cases concerned a plume formation, in transient and steady-state regimes, respectively; their main outcome was the evidence of the CA-ANN methodology's flexibility to address time-dependent and permanent flow simulations interpolation. The last CFD-based case study comprised an additional complexity feature of gas dispersion problems: the wind influence. By redesigning the investigated data-driven approach in terms of ANN's features and labels choice, promising results followed from the analysis with respect to the simultaneous capturing of two global simulation parameters (wind and leakage speeds boundary conditions) in the species concentration field interpolation.

An a posteriori strategy for adaptive schemes in time for one-dimensional advection-diffusion transport equations

Stability condition is a more restrictive constraint that leads to unnecessary small-time steps with respect to the accuracy and results in computational time wastage. We propose a node by node adaptive time scheme to relax the stability constraint enabling a larger global time step for all the nodes. A nonlinear procedure for optimising both the schemes in time and space is proposed in view of increasing the numerical efficiency and reducing the computational time. The method is based on a four-parameter family of schemes we shall tune in function of the physical data (velocity, diffusion), the characteristic size in time and space, and the local regularity of the function leading to a nonlinear procedure. The a posteriori strategy we adopt consists in, given the solution at time tn, computing a candidate solution with the highest accurate schemes in time and space for all the nodes. Then, for the nodes that present some instabilities, both the schemes in time and space are modified and adapted in order to preserve the stability with a large time step. The updated solution is computed with node-dependent schemes both in time and space. For the sake of simplicity, only convection-diffusion problems are addressed as a prototype with a two-parameters five-points finite difference method for the spatial discretisation together with an explicit time two-parameters four-stages Runge–Kutta method. We prove that we manage to obtain an optimal time-step algorithm that produces accurate numerical approximations exempt of non-physical oscillations.

Solving one-dimensional advection diffusion transport equation by using CDV wavelet basis

Solving one-dimensional advection diffusion transport equation by using CDV wavelet basis

Sixth-Order Combined Compact Finite Difference Scheme for the Numerical Solution of One-Dimensional Advection-Diffusion Equation with Variable Parameters

A high-accuracy numerical method based on a sixth-order combined compact difference scheme and the method of lines approach is proposed for the advection–diffusion transport equation with variable parameters. In this approach, the partial differential equation representing the advection-diffusion equation is converted into many ordinary differential equations. These time-dependent ordinary differential equations are then solved using an explicit fourth order Runge–Kutta method. Three test problems are studied to demonstrate the accuracy of the present methods. Numerical solutions obtained by the proposed method are compared with the analytical solutions and the available numerical solutions given in the literature. In addition to requiring less CPU time, the proposed method produces more accurate and more stable results than the numerical methods given in the literature.

Residual-based variational multiscale 2D simulation of sediment transport with morphological changes

Moving sediments are intrinsic to several engineering and natural phenomena. Consequently, the study of sediment transport induced by fluid flows is critical. The transport of sediments in fluids encompasses the suspended load and the bed-load. In the suspended load turbulence carries the sediment particles. The particles near the bottom are moved by sliding, rolling or by saltation, due to shear stresses in the bed-load region. The bed-load transport implies morphological changes which are also induced by erosion as a consequence of sediment entrainment into suspension. In this work, we implement these phenomena as numerical models for 2D simulations in libMesh, an open finite element library that provides a framework for multiphysics, considering adaptive mesh refinement. Here, the mathematical model results from the incompressible Navier–Stokes equations combined with an advection-diffusion transport equation for suspended sediments using an Arbitrary Lagrangian–Eulerian description. Both equations are treated with the residual-based variational multiscale finite element formulation. Empirical models are used to represent the entrainment and the bed-load transport rate of sediments. Special boundary conditions at the bottom are introduced to take into account sediment entrainment, as well as deposition. After each time step, we update the mesh according to the morphological changes at the bottom due the bed-load transport and erosion, considering smoothing techniques to avoid numerical instabilities. We also introduce a new smoothing technique, called upwind sand-slide, in which we consider the flow direction combined with the slope angle to avoid elevations higher than the sediment angle of repose. We validate the algorithm with test cases where experimental and numerical data are available. Finally, we discuss the results, concluding that our model is very promising.

Numerical solution of 2-d advection-diffusion equation with variable coefficient using du-fort frankel method

The advection-diffusion equation or transport equation is investigated further. A simple numerical approximation such as Du-Fort Frankel methods for advection-diffusion equation with variable coefficient is employed. The obtained results are compared with its analytical solution in a simple unit square domain. The method and results are then being used to solve advection-diffusion in the real-world problems for the more complicate irregular domain.

1D contaminant transport using element free Galerkin method with irregular nodes

The present study deals with the numerical modelling for the one dimensional contaminant transport through saturated homogeneous and stratified porous media using meshfree method. A numerical algorithm based on element free Galerkin method is developed. A one dimensional form of the advectivediffusive transport equation for homogeneous and stratified soil is considered for the analysis using irregular nodes. A Fortran program is developed to obtain numerical solution and the results are validated with the available results in the literature. A detailed parametric study is conducted to examine the effect of certain key parameters. Effect of change of dispersion, velocity, porosity, distribution coefficient and thickness of layer is studied on the concentration of the contaminant.

Two Dimensional Contaminant Transport Through Layered Soil Using Element Free Galerkin Method

In the present study a numerical model of two dimensional contaminant transport through stratified saturated porous media has been analysed using meshfree method called element free Galerkin method. A two dimensional form of advective–diffusive transport equation for the multi-layered soil is considered for the analysis. A FORTRAN program has been developed to obtain numerical solution and the results are validated with the results in the literature. The results are in good agreement and further parametric study has been done. Effect of change of dispersion, velocity, porosity, distribution coefficient and thickness of layer has been observed on the concentration of the contaminant.

Granular segregation in circular tumblers: theoretical model and scaling laws

AbstractWe model bidisperse size segregation of granular material in quasi-two-dimensional circular tumbler flow using the advection–diffusion transport equation with an additional term to account for segregation due to percolation. Segregation depends on three dimensionless parameters: the ratio of segregation to advection, ${\it\Lambda}$; the ratio of advection to diffusion, $\mathit{Pe}$; and the dimensionless flowing layer depth, ${\it\epsilon}$. The degree of segregation in steady state depends only on the ratio of segregation effects to diffusion effects, ${\it\Lambda}\,\mathit{Pe}$, and the degree of segregation increases as ${\it\Lambda}\mathit{Pe}$ increases. The transient time to reach steady-state segregation depends only on advection, which is manifested in ${\it\epsilon}$ and $\mathit{Pe}$ when ${\it\Lambda}\mathit{Pe}$ is constant. This model is also applied to unsteady tumbler flow, where the rotation speed varies with time.

PARTICLE ACCELERATION VIA RECONNECTION PROCESSES IN THE SUPERSONIC SOLAR WIND

An emerging paradigm for the dissipation of magnetic turbulence in the supersonic solar wind is via localized small-scale reconnection processes, essentially between quasi-2D interacting magnetic islands. Charged particles trapped in merging magnetic islands can be accelerated by the electric field generated by magnetic island merging and the contraction of magnetic islands. We derive a gyrophase-averaged transport equation for particles experiencing pitch-angle scattering and energization in a super-Alfvenic flowing plasma experiencing multiple small-scale reconnection events. A simpler advection-diffusion transport equation for a nearly isotropic particle distribution is derived. The dominant charged particle energization processes are (1) the electric field induced by quasi-2D magnetic island merging and (2) magnetic island contraction. The magnetic island topology ensures that charged particles are trapped in regions where they experience repeated interactions with the induced electric field or contracting magnetic islands. Steady-state solutions of the isotropic transport equation with only the induced electric field and a fixed source yield a power-law spectrum for the accelerated particles with index α = –(3 + M{sub A} )/2, where M{sub A} is the Alfven Mach number. Considering only magnetic island contraction yields power-law-like solutions with index –3(1 + τ {sub c}/(8τ{sub diff})), where τ {sub c}/τ{sub diff} is themore » ratio of timescales between magnetic island contraction and charged particle diffusion. The general solution is a power-law-like solution with an index that depends on the Alfven Mach number and the timescale ratio τ{sub diff}/τ {sub c}. Observed power-law distributions of energetic particles observed in the quiet supersonic solar wind at 1 AU may be a consequence of particle acceleration associated with dissipative small-scale reconnection processes in a turbulent plasma, including the widely reported c {sup –5} (c particle speed) spectra observed by Fisk and Gloeckler and Mewaldt et al.« less

Influence of variation in temperature on groundwater flow nets in a geothermal system

Flow nets are one of the effective tools for illustrating groundwater flow rate and direction. Flow nets of a thermal groundwater system under different reference temperatures are examined in this paper. A 3D mathematical model describing water flow and heat transport is established. The stream function and the equivalent hydraulic head are used to describe the flow field in terms of mass flux. The partial differential governing equations are given by the mass conservation equation of the fluid phase and by the advection–diffusion transport equations of heat. They are solved with the standard Galerkin finite element method. Influence of variation in temperature on thermal groundwater flow nets is illustrated with a simplified 2D vertical steady-state mathematical model. The results show that the equipotential lines are not always orthogonal to the streamlines in a geothermal system of a confined aquifer when the heat flow from below is present and when the temperature increases, the equipotential lines are clearly curved and the skew intersection angles between the streamlines and the equipotential lines decrease. For the node of coordinate of (5000, 0) in the aquifer, when the temperature difference between the top boundary and the bottom boundary is 4 and 10 °C, the skew intersection angle is 85.54° and 78.73°, respectively. When the hydraulic gradients are low, the equipotential lines are clearly curved. However, when the hydraulic gradients are high, the bending of the equipotential lines is not obvious. When the hydraulic gradient increases, the skew intersection angles change relatively little. For the same node when the hydraulic head of the left boundary is 600, 700, 800 and 1,000 m, the skew intersection angle is 78.73°, 84.3°, 86.19° and 87.71°, respectively. When the reference temperature increases, the hydraulic heads and the skew intersection angles increase.

Excitation of flow instabilities due to nonlinear scale invariance

A novel route to instabilities and turbulence in fluid and plasma flows is presented in kinetic Vlasov-Maxwell model. New kind of flow instabilities is shown to arise due to the availability of new kinetic energy sources which are absent in conventional treatments. The present approach is based on a scale invariant nonlinear analytic formalism developed to address irregular motions on a chaotic attractor or in turbulence in a more coherent manner. We have studied two specific applications of this turbulence generating mechanism. The warm plasma Langmuir wave dispersion relation is shown to become unstable in the presence of these multifractal measures. In the second application, these multifractal measures are shown to induce naturally non-Gaussian, i.e., a stretched, Gaussian distribution and anomalous transport for tracer particles from the turbulent advection-diffusion transport equation in a Vlasov plasma flow.

Numerical simulation of particle‐laden flows by the residual‐based variational multiscale method

SUMMARYWe present a finite element residual‐based variational multiscale formulation applied to the numerical simulation of particle‐laden flows. We employ a Eulerian–Eulerian framework to describe the flows in which the mathematical model results from the incompressible Navier–Stokes equation combined with an advection–diffusion transport equation. Special boundary conditions at the bottom are introduced to take into account sediments deposition. Computational experiments are organized in two examples. The first example deals with the well‐known gravity current benchmark, the lock‐exchange configuration. The second also employs for the current initiation the lock configuration, but the sediment particles are endowed with a deposition velocity and are allowed to leave the domain in the moment they reach the bottom. This is intended to mimic, partially, as the bed morphology is not allowed to change, the deposition process, in which sediment deposits are no longer carried by the flow. The spatial pattern of the deposition and its correlation with flow structures are the main focus of this analysis. Numerical experiments have shown that the present formulation captures most of the relevant turbulent flow features with reasonable accuracy, when compared with highly resolved numerical simulations and experimental data. Copyright © 2013 John Wiley & Sons, Ltd.

Restricted scaling range models for turbulent velocity and scalar energy transfers in decaying turbulence

The effect of finite Reynolds numbers and/or internal intermittency on the total kinetic energy and scalar energy transfers is examined in detail. For this purpose, two distinct models for velocity and scalar energy transfer are proposed in the specific context of freely decaying isotropic turbulence. The first one extends the already existing dynamical models (hereafter DYM, i.e. based on transport equations originated in Navier–Stokes and advection-diffusion transport equations). The second one relies on the characteristic time of the strain at a specific scale (hereafter SBM). Both models account for the Reynolds number dependence of the scaling exponent of the second-order structure functions, over a range of scales where such exponents may be defined, i.e. a restricted scaling range (RSR). Therefore, the models developed aim at reproducing the energy transfer over the RSR. The predicted energy transfer is very sensible to variations of the scaling exponent, especially at low Reynolds numbers. The approach towards the asymptotic 4/3 law is closely reproduced by the two models. The dynamical model reproduces the experimental results accurately especially in the vicinity of the Taylor microscale, while the SBM agrees almost perfectly with measurements at nearly all scales.

Modeling and Simulations of Laminar Mixed Convection in a Vertical Pipe Conveying Slurries of a Microencapsulated Phase-Change Material in Distilled Water

Steady, laminar, mixed convection in a straight and vertically oriented pipe conveying slurries of a microencapsulated phase-change material (MCPCM) suspended in distilled water (flowing upwards), with essentially uniform heat flux imposed on its outside surface, are considered. A cost-effective homogenous mathematical model is proposed and shown to be applicable to the aforementioned mixed convection phenomena with slurries of a sample MCPCM. Correlations for the effective properties of the sample MCPCM slurries and procedures for their implementation are presented. The energy equation, in which the latent-heat effects are handled using an effective specific heat, is cast in a form akin to that of a general advection-diffusion transport equation. Difficulties with the standard definition of bulk temperature when the specific heat of the fluid changes significantly with temperature are elaborated, and a modified bulk temperature that overcomes these difficulties is proposed. A finite volume method (FVM) was used to solve the mathematical model. The proposed model and FVM were validated by using them to solve problems involving slurries of the sample MCPCM, and comparing the results to those of a complementary experimental investigation. The numerical results compare very well with those of the complementary experimental investigation. They also demonstrate the need for optimizing the various parameters involved, if full benefits of the MCPCM slurries are to be achieved for specific applications.

A finite element model for the ocean circulation driven by wind and atmospheric heat flux

A finite element model for solving ocean circulation forced by winds and atmospheric heat fluxes is presented. It is vertically integrated 1 1/2 layer model which solves the motion and continuity hydrodynamic equations coupled with the advection-diffusion transport equation for temperature. A space-time Petrov-Galerkin formulation is used to minimize the undesirable numerical spurious oscillation effects of unresolved boundary layers solution of classical Galerkin method. The model is employed to simulate the circulation of the eastern South Pacific Ocean represented by a mesh covering the area between the equator and 30oS and between the 70oW to 100oW. Monthly climatological data are used to determine the wind and heat fluxes forcing functions of the model. The model simulates the main features of observed sea surface temperature (SST) pattern during the summer months showing the upwelling along the coastal boundary with currents oriented northwestward and the presence of southward flows and warm water intrusion in offshore side. The calculated SST fields are compared with the mean observed SST showing that the coastal processes and the interaction with the equatorial band are physically better resolved.

Eulerian–Lagrangian multiscale methods for solving scalar equations – Application to incompressible two-phase flows

The present article proposes a new hybrid Eulerian–Lagrangian numerical method, based on a volume particle meshing of the Eulerian grid, for solving transport equations. The approach, called Volume Of Fluid Sub-Mesh method (VOF-SM), has the advantage of being able to deal with interface tracking as well as advection–diffusion transport equations of scalar quantities. The Eulerian evolutions of a scalar field could be obtained on any orthogonal curvilinear grid thanks to the Lagrangian advection and a redistribution of particles on the Eulerian grid. The Eulerian concentrations result from the projection of the volume and scalar informations handled by the particles. The particle velocities are interpolated from the Eulerian velocity field. The VOF-SM method is validated on several scalar interface tracking and transport problems and is compared to existing schemes within the literature. It is finally coupled to a Navier–Stokes solver and applied to the simulation of two free-surface flows, i.e. the two-dimensional buckling of a viscous jet during the filling of a square mold and the three-dimensional dam-break flow in a tank.

Analytical solution of the advection–diffusion transport equation using a change-of-variable and integral transform technique

This paper presents a formal exact solution of the linear advection–diffusion transport equation with constant coefficients for both transient and steady-state regimes. A classical mathematical substitution transforms the original advection–diffusion equation into an exclusively diffusive equation. The new diffusive problem is solved analytically using the classic version of Generalized Integral Transform Technique (GITT), resulting in an explicit formal solution. The new solution is shown to converge faster than a hybrid analytical–numerical solution previously obtained by applying the GITT directly to the advection–diffusion transport equation.

A generalized Richards equation for surface/subsurface flow modelling

Summary An approach for modelling surface/subsurface flow and transport in a fully integrated way is presented. The diffusive wave approximation used to model runoff is formulated as a nonlinear Darcy law and coupled with the Richards equation for variably saturated flow. The water dynamics in the three physical domains, land surface, vadose zone and saturated zone, is described through a single Richards type equation with domain-dependent parameters. This approach naturally provides pressure and fluxes continuity along land surface. An advective–diffusive transport equation is incorporated in the model in order to follow and identify pre-event and event water. This multi-domain Richards equation is solved with a mixed hybrid finite element formulation. The time discretization is implicit and the nonlinear equations are solved within a sequential iterative Picard scheme. This model can describe both infiltration excess or saturation excess runoff. In order to assess this modelling approach, several classical validation, verification and application test cases are presented. For overland flow solely, the model is compared to an analytical solution and to commonly used hydrological models. The integrated model is then validated with a sandbox laboratory experiment. Finally, the dynamic response of a saturated contributing area at the toe of a 3D virtual hillslope is simulated to illustrate the applicability of the model to 3D situations.