Three-dimensional Transport Equation Research Articles

Numerical model of Earth ionosphere F region based on three-dimensional transport and ambipolar diffusion equations

Abstract The paper provides a detailed description of the numerical implementation of the transport scheme in the Earth’s ionosphere three-dimensional dynamical model of the Institute of Numerical Mathematics (INMIM). The presented version of INM-IM model takes into account the global dynamical processes of ion transport and ambipolar diffusion in the altitude range between 100 and 500 km (corresponding to F region). Based upon the splitting method the model solves the equations of ambipolar diffusion on the first split step and incorporates specifically designed advective transport scheme on the second step. The accuracy of transport scheme implementation in the model has been investigated through analytical solutions. The stability of the numerical algorithm has been demonstrated even for cases when transport velocities approaching extreme values.

Three-Dimensional Numerical Transfer Model Using a Monotonized Z-Scheme

The aim of this work is to investigate a three-dimensional second-order monotone finite-difference scheme for the transport equation. The investigation is conducted for model three-dimensional transport equations of an incompressible medium. The properties of the three-dimensional extension of the Z-scheme with nonlinear correction are studied in this work. This study is an extension of the author’s previous works, where a nonlinear correction of one-dimensional transport equations was constructed. The proposed scheme uses “skew differences” containing values from different time layers instead of fluxes for the correction. The monotonicity of the obtained nonlinear scheme is numerically studied for a family of limiter functions for both smooth and non-smooth continuous solutions. The constructed scheme is absolutely stable but loses the monotonicity property when the Courant step is exceeded. The distinctive feature of the proposed finite-difference scheme is the minimalism of its template. The constructed numerical scheme is designed for models of plasma instabilities of various scales in the low-latitude ionospheric plasma of the Earth. One of the real problems that arise in solving such equations is the numerical modeling of strongly non-stationary medium- and small-scale processes in the low-latitude ionosphere of the Earth under conditions of the occurrence of the Rayleigh–Taylor instability and other types of instabilities, leading to the phenomenon of F‑scattering. Due to the fact that transport processes in the ionospheric plasma are controlled by the magnetic field, it is assumed that the plasma is incompressible in the direction perpendicular to the magnetic field.

Averaging equation of three-dimensional tracer transport by a dual approach

The conventional method for calculating averaged tracer concentration in three-dimensional tracer transport equation relies on classical averaging, which averages tracer concentration data obtained from the tracer transport equation in turbulent flow. However, the averaging tracer concentration quantity calculated by this method is not precisely the same as the one obtained by the dual approach. To address this issue, the author proposes a more intricate dual approach for averaging the three-dimensional tracer transport equation in turbulent flow. This paper describes how the author performs two integrations: one from time t to t + r (where r < T, T being the repeated period of fluctuating concentration), and another from time t to time t + T. The fluctuating concentration quantity in turbulent flow is modeled using the trigonometric Fourier series. The resulting three-dimensional mathematical model of tracer transport obtained by this dual approach is more comprehensive than the one obtained by the classical method.

Laminar drag reduction in surfactant-contaminated superhydrophobic channels

Although superhydrophobic surfaces (SHSs) show promise for drag reduction applications, their performance can be compromised by traces of surfactant, which generate Marangoni stresses that increase drag. This question is addressed for soluble surfactant in a three-dimensional laminar channel flow, with periodic SHSs made of long finite-length longitudinal grooves located on both walls. We assume that bulk diffusion is sufficiently strong for cross-channel concentration gradients to be small. Exploiting long-wave theory and accounting for the difference between the rapid transverse and slower longitudinal Marangoni flows, we derive a one-dimensional model for surfactant transport from the full three-dimensional transport equations. Our one-dimensional model allows us to predict the drag reduction and surfactant distribution across the parameter space. The system exhibits multiple regimes, involving competition between Marangoni effects, bulk and interfacial diffusion, bulk and interfacial advection, shear dispersion and surfactant exchange between the bulk and the interface. We map out asymptotic regions in the high-dimensional parameter space, and derive explicit closed-form approximations of the drag reduction, without any fitting or empirical parameters. The physics underpinning the drag reduction effect and the negative effect of surfactant is discussed through analysis of the velocity field and surfactant concentrations, which show both uniform and non-uniform stress distributions. Our theoretical predictions of the drag reduction compare well with results from the literature solving numerically the full three-dimensional transport problem. Our atlas of maps provides a comprehensive analytical guide for designing surfactant-contaminated channels with SHSs, to maximise the drag reduction in applications.

Turbulent Flame Propagation in Hydrogen-Air and Methane-Air Mixtures in the Field of Synthetic Turbulence: Direct Numerical Simulation

A technique alternative to the direct numerical simulation of turbulent combustion of gas mixtures is proposed. It is based on the solution of the three-dimensional transport equations for species concentrations and the energy conservation equation in the “synthetic” field of constant-pressure homogeneous, isotropic and statistically stationary (forced) turbulence using the detailed reaction mechanism. The synthetic turbulence with given spatial and temporal correlation functions is generated using the Monte Carlo method, assuming that the components of the vector of fluctuation velocity obey the normal Gaussian distribution. The technique is applied to the problem of turbulent combustion of fuel-lean and stoichiometric mixtures of hydrogen and methane with air at a turbulence intensity up to 10 m/s. The calculated turbulent flame propagation velocities agree satisfactorily with the values measured in the fan-stirred bomb. The predicted volume fractions of active reaction centers H, O, and OH in a turbulent flame are shown to be less than in a laminar flame up to an order of magnitude, which also agrees with the experiment. In general, calculations indicate that the “wrinkled flame” model is applicable to fuel-lean and stoichiometric mixtures of hydrogen and methane with air at turbulence intensities up to 10 m/s

CALCULATION OF ATMOSPHERIC AIR POLLUTION IN CASE OF CHEMICALLY DANGEROUS SUBSTANCE SPILL

Problem statement. The task of forecasting for chemical pollution areas at the industrial site during atmospheric inversion and chemically hazardous substance evaporation from the spill zone is considered. The purpose of the work is to develop a method for calculating the intensity and shape of pollution areas in the atmosphere that are formed under inversion conditions. Methodology. The three-dimensional impurity transport equation is used to calculate the atmospheric spread of chemically hazardous substances under inversion conditions, together with the M. Berland’s approach, to determine the vertical diffusion coefficient for this adverse meteorological condition. Numerical integration of the impurity transfer equation is realised using two finite-difference schemes. Both schemes are based on the splitting method. Results. An effective calculating method of atmospheric pollution areas formed during the evaporation of a chemically hazardous substance from the spill zone is proposed. A computer code to allow quick determination of these atmospheric pollution zones under inversion conditions is developed. The results of the computational experiment are presented. Scientific novelty. A method is developed to assess the size and intensity of chemical pollution zones formed in the atmosphere during inversion and evaporation of the substance from the spill area. Practical value. Based on the developed method, a code for a computational experiment to assess atmospheric pollution zones at industrial sites during emergency spills of chemically hazardous substances is generated.

Comprehensive modulation potential for the solar modulation of Galactic cosmic rays

The concept of the modulation potential being utilized to describe the solar modulation of galactic cosmic rays is calculated based on the full three-dimensional Parker transport equation. This is in sharp contrast to the force-field approach, which is based on a series of approximations of this equation and where traditionally all the physical processes are condensed into a single parameter without knowing their relative contribution. In our comprehensive approach to the modulation potential, the contribution of all the different physical processes is given explicitly as derived directly from the parameters of Parker's transport equation. We use the stochastic differential equations approach to study the effects of each of these different physical modulation processes thoroughly and then also determine the rigidity and mass-to-charge-ratio ($A/Z$) of these processes for the first time in the context of a modulation potential. It is found that: (1) This comprehensive modulation potential for galactic cosmic particles with a given rigidity at the Earth is a random variable, which can be fit by an inverse Gaussian distribution; (2) the rigidity and $A/Z$ dependence of this modulation potential can be divided into three categories based on the index value of power law diffusion; (3) the modulation caused by convection becomes weaker with increased rigidity and eventually dissipates, leading to a larger modulation potential for particles with a larger $A/Z$; (4) particle drift significantly reduces this modulation potential and its variation with changes in the tilt angle of the heliospheric current sheet has a distinct pattern for each of the two solar magnetic field polarities.

A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids

A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids

Quantum kinetic equation for fluids of spin-1/2 fermions

Fluid of spin-1/2 fermions is represented by a complex scalar field and a four-vector field coupled both to the scalar and the Dirac fields. We present the underlying action and show that the resulting equations of motion are identical to the (hydrodynamic) Euler equations in the presence of Coriolis force. As a consequence of the gauge invariances of this action we established the quantum kinetic equation which takes account of noninertial properties of the fluid in the presence of electromagnetic fields. The equations of the field components of Wigner function in Clifford algebra basis are employed to construct new semiclassical covariant kinetic equations of the vector and axial-vector field components for massless as well as massive fermions. Nonrelativistic limit of the chiral kinetic equation is studied and shown that it generates a novel three-dimensional transport theory which does not depend on spatial variables explicitly and possesses a Coriolis force term. We demonstrated that the three-dimensional chiral transport equations are consistent with the chiral anomaly. For massive fermions the three-dimensional kinetic transport theory generated by the new covariant kinetic equations is established in small mass limit. It possesses the Coriolis force and the massless limit can be obtained directly.

ASSESSMENT OF GEOECOLOGICAL CONSEQUENCES OF SUBDUCTION PROCESSES

t. The aim of the work is to develop a numerical-analytical method for assessing the geoecological consequences of volcanic activity accompanying the subduction process. The boundary problem was formulated, including the three-dimensional transport and diffusion equation and boundary conditions at the bottom and surface of the water. For research, a block structure with quasi-homogeneous layers is introduced. In each block, a differential factorization method is implemented and integral representations of solutions are obtained. Calculations for a model problem are carried out, conclusions are formulated about the features of the behavior of the heavy and light fractions of volcanic ejections depending on the speed of currents, intensity and duration of the eruption.

Central finite-difference of numerical solution for three-dimensional atmospheric transport equation

The energy transport equation is fundamental for the meteorological analysis; in this work, we analyze this equation in three dimensions using the methods of central finite differences; the analysis of convergence, consistency, and stability of the scheme shows a strong dependence of space and temporal variables. In conclusion, with the central finite differences was possible to predict the three-dimensional dynamics of the temperature.

Developing 3D neutron transport kernel for heterogeneous structures in an improved method of characteristic (MOC) framework

Given the importance and complexity of the three-dimensional (3D) neutron transport equation solution, in the current research, a new Modular Ray Tracing (MRT) Algorithm and 3D characteristic kernel for heterogeneous structures are presented. Improvement of memory management and cache coherency are achieved to some acceptable level. Also, parallel implementation of transport algorithm utilizing OpenMP, cause significant reduction in runtime.To validate our Algorithm, first, a self-constituted pin cell and a lattice arrangement are modeled and results are compared with Monte-Carlo simulation. Second, the well-known 3D benchmark, Takeda model one and two, are investigated and results compared with the well-known MPACT code. Also, as a reliable trial, the obtained multiplication factors from the forward and adjoint form of MOC kernel are investigated. Meanwhile, for the multiplication factor, we achieved 3.5800E-04 and 3.3924E-04 |Δkeffkeff| the difference with Takeda reference models one and two respectively. Also, the maximum speedup that our parallel algorithm gained for the Takeda benchmark is 5.13.

A Three-Dimensional Monotonicity-Preserving Modified Method of Characteristics on Unstructured Tetrahedral Meshes

Slope limiters have been widely used to eliminate nonphysical oscillations near discontinuities generated by finite volume methods for hyperbolic systems of conservation laws. In this study, we investigate the performance of these limiters as applied to three-dimensional modified method of characteristics on unstructured tetrahedral meshes. The focus is on the construction of monotonicity-preserving modified method of characteristics for three-dimensional transport problems with discontinuities and steep gradients in their solutions. The proposed method is based on combining the modified method of characteristics with a finite element discretization of the convection equations using unstructured grids. Slope limiters are incorporated in the method to reconstruct a monotone and essentially nonoscillatory solver for three-dimensional problems at minor additional cost. The main idea consists in combining linear and quadratic interpolation procedures using nodes of the element where departure points are localized. We examine the performance of the proposed method for a class of three-dimensional transport equations with known analytical solutions. We also present numerical results for a transport problem in three-dimensional pipeline flows. In considered test problems, the proposed method demonstrates its ability to accurately capture the three-dimensional transport features without nonphysical oscillations.

Two Numerical Methods for the 3D Anisotropic Propagation of Galactic Cosmic Rays

Abstract Conventional cosmic-ray propagation models usually assume an isotropic diffusion coefficient to account for the random deflection of cosmic rays by the turbulent interstellar magnetic field. Such models very successfully explain many observational phenomena related to the propagation of Galactic cosmic rays, such as broken power-law energy spectra, secondary-to-primary ratios, etc. However, the isotropic diffusion presupposition is facing severe challenges from recent observations. In particular, such observations on the large-scale anisotropy of TeV cosmic rays show that the dipole direction differs from the prediction of the conventional model. One possible reason is that the large-scale regular magnetic field, which leads to an anisotropic diffusion of cosmic rays, has not been included in the model provided by the public numerical packages. In this work, we propose two numerical schemes to solve the three-dimensional anisotropic transport equation: the pseudo-source method and Hundsdorfer–Verwer scheme. Both methods are verified by reproducing the measured B/C and proton spectrum and the radial variation of spectral index expected by former 2D simulation. As a demonstration of the prediction capability, dipole anisotropy is also calculated by a toy simulation with a rough magnetic field.

Central finite-difference of numerical solution for three-dimensional atmospheric transport equation

The energy transport equation is fundamental for the meteorological analysis; in this work, we analyze this equation in three dimensions using the methods of central finite differences; the analysis of convergence, consistency, and stability of the scheme shows a strong dependence of space and temporal variables. In conclusion, with the central finite differences was possible to predict the three-dimensional dynamics of the temperature.

Parallel implementation of a numerical method for solving a three-dimensional transport equation for a mesoscale meteorological model

Abstract A parallel algorithm for numerical solution of a generalized unsteady 3-dimensional advection-diffusion equation for a mesoscale meteorological model is considered. Efficiency of parallel implementations of the solver with Message Passing Interface (MPI), Open Multi-Processing (OpenMP), and NVidia Compute Unified Device Architecture (CUDA) technologies was compared. It was shown that on the mesh of 25625632 nodes the speedup of the program reaches 18 for OpenMP and MPI implementations and 38 for CUDA.

Comparison of the performance of heterogeneous variational nodal method and 2D/1D-MOC on pin-resolved neutron transport calculations of PWR

Heterogeneous Variational Nodal Method (HVNM) and the 2D/1D Method of Characteristics (2D/1D-MOC) are two promising methods for high-fidelity neutronics calculation. HVNM starts from even-parity form of neutron transport equation and employs iso-parameteric finite elements, orthogonal polynomials and piece-wise constants for spatial discretization. Additionally, spherical harmonics are applied for the angular expansion. In terms of 2D/1D-MOC, it starts from first-order transport equation and splits the three-dimensional transport equation into one 2D and one 1D equations which are coupled by the radial and axial leakage terms. Then space and angle are discretized by characteristic lines in radial plane and 1D Sn calculation is employed in axial direction. Both HVNM and 2D/1D-MOC have the capability of dealing with pin-resolved geometry and strong transport effect in PWR core problem. Thus, we developed two codes PANX and NECP-X which based on the methods of HVNM and MOC respectively. In this paper, to compare their performance on pin-resolved problems in PWR, the KAIST problem, the NuScale problem and the Beavrs problem are calculated. Both computational accuracy and efficiency are evaluated. The numerical results shown that both PANX and 2D/1D-MOC can obtain accurate eigenvalues with the error less than 100 pcm. However, PANX obtains more accurate pin power distribution and achieves higher computational efficiency than 2D/1D-MOC.

Effect of Pad and Liner Material Properties on the Static Load Performance of a Tilting Pad Thrust Bearing

AbstractTilting pad thrust bearings (TPTBs) control rotor axial placement in rotating machinery, and their main advantages include low drag power loss, simple installation, and low-cost maintenance. The paper details a novel thermo-elasto-hydrodynamic (TEHD) analysis predictive tool for TPTBs that considers a three-dimensional (3D) thermal energy transport equation in the fluid film, coupled with heat conduction equations in the pads, and a generalized Reynolds equation with cross-film viscosity variation. The predicted pressure field and temperature rise are employed in a finite element (FE) structural model to produce 3D elastic deformation fields in the bearing pads. Solutions of the governing equations delivers the operating film thickness, required flowrate, and shear drag power loss, and the pad and lubricant temperature rises as a function of an applied load and shaft speed. To verify the model, predictions of pad subsurface temperature are benchmarked against published test data for a centrally pivoted eight-pad TPTB with 267 mm in outer diameter (OD) operating at 4–13 krpm (maximum surface speed = 175 m/s) and under a specific load ranging from 0.69 to 3.44 MPa. The current TEHD temperature predictions match well the test data with a maximum difference of 4 °C and 11 °C (&lt;10%) at laminar and turbulent flow conditions, receptively. Next, the TEHD predictive tool is used to study the influence of both pad and liner material properties on the performance of a TPTB. The analysis takes a whole steel pad (without a liner or babbitt), a steel pad with a 2-mm-thick babbitt layer (common usage), a steel pad with a 2-mm-thick hard-polymer (polyether ether ketone, e.g., PEEK®) liner, and a pad entirely made of hard-polymer material, whose elastic modulus is just 12.5 GPa, only 6% that of steel. The bare steel pad reveals the poorest performance among all the pads as it produces the smallest fluid film thickness and consumes the largest drag power loss. For laminar flow operations (Reynolds number Re &lt; 580), the babbitted-steel pad operates with the thickest fluid film and the lowest film temperature rise. For turbulent flow conditions Re &gt; 800, the solid hard-polymer pad, however, shows a 23% thicker film than that in the babbitted pad and produces up to 25% lesser drag power loss. In general, the solid hard-polymer TPTB is found to be a good fit for operation at a turbulent flow condition as it shows a lower drag power loss and a larger film thickness; however, its demand for a too large supply flowrate is significant. Predictions for steel pads with various hard-polymer liner and babbitt thicknesses demonstrate that using a hard-polymer liner, instead of white metal, isolates the pad from the fluid film and results in an up to 30 °C (50%) lower temperature rise in the pads than that for a babbitted-steel pad. For operations under a heavy specific load (&gt;3.0 MPa), however, a thick hard-polymer liner extensively deforms and results in a small film thickness.

Numerical analysis of the groove effect on the tip leakage vortex cavitating flow

In this paper, the groove effect on the tip leakage vortex cavitating flow characteristics of a simplified NACA0009 hydrofoil with tip gap is studied. Considering local rotation characteristics and curvature effects of the tip leakage vortex flow, the rotation-curvature corrected shear-stress-transport turbulence model is applied to simulate the time-averaged turbulent flow. The Zwart–Gerber–Belamri cavitation model is used to simulate the cavitating flow. The results show that the groove could affect the tip leakage vortex cavitating flow. The groove enhances the interaction between the tip leakage flow and main flow, and then it affects the cavitation of the tip leakage vortex. Compared with the non-groove case, for groove cases of αgre ≤75°, the tip leakage vortex cavitating flow is suppressed, the flow pattern in the gap is improved, and the mean leakage velocity Vlk &lt; 0.8. The region of high leakage velocity is eliminated and the distribution of the pressure is more uniform. The tip leakage vortex cavitation area is reduced, and the maximum decrease is 72.90%. While for groove cases of αgre≥90°, neither the tip leakage vortex cavitating flow nor flow pattern in the tip gap is ameliorated, the mean leakage velocity Vlk lies the range from 0.90 to 0.96. The region of high leakage velocity still exists and even the tip leakage vortex cavitation area is increased. Based on three-dimensional streamlines and vorticity transport equation, the interaction between the tip leakage flow and main flow leads to the variation of the tip leakage vortex cavitating flow. This paper aims for a useful reference to mitigate the tip leakage vortex cavitation and control the influence of the tip leakage vortex cavitating flow for the hydraulic machinery.

Some features of semiclassical chiral transport in rotating frames

Semiclassical chiral kinetic theories in the presence of electromagnetic fields as well as vorticity can be constructed by means of some different relativistic or nonrelativistic approaches. To cover the noninertial features of rotating frames one can start from the modified quantum kinetic equation of Wigner function in Minkowski spacetime. It provides a relativistic chiral transport equation whose nonrelativistic limit yields a consistent three-dimensional kinetic theory which does not depend explicitly on spatial coordinates. Recently a chiral transport equation in curved spacetime has been proposed and its nonrelativistic limit in rotating coordinates was considered in the absence of electromagnetic fields. We show that the modified theory can be extended to curved spacetime. The related particle current density and chiral transport equation for an inertial observer in the rotating frame are derived. A novel three-dimensional chiral kinetic transport equation is established by inspecting the nonrelativistic limit of the curved spacetime approach in the rotating frame for a comoving observer in the presence of electromagnetic fields. It explicitly depends on spatial coordinates. We prove that it is consistent with the chiral anomaly, chiral magnetic and vortical effects.