Modified Transport Equation Research Articles

Improving numerical accuracy for the viscous-plastic formulation of sea ice

Accurate modeling of sea ice dynamics is critical for predicting environmental variables and is important in applications such as navigating ice breaker ships, and as such there have been numerous investigations on modeling sea ice dynamics. The 1979 viscous-plastic (VP) sea ice model introduced by W.D. Hibler remains the most widely accepted. Due to its highly nonlinear features, the Hibler model is intrinsically challenging for computational solvers. This study therefore focuses on improving the numerical accuracy of the VP sea ice model. Since the poor convergence observed in existing numerical simulations stems from the nonlinear nature of the VP formulation, this investigation proposes using the celebrated weighted essentially non-oscillatory (WENO) scheme – as opposed to the frequently employed centered difference (CD) scheme – for the spatial derivatives in the VP sea ice model. We then proceed to numerically demonstrate that WENO yields higher-order convergence for smooth solutions, and that furthermore it is able to resolve the discontinuities in the sharp features of sea ice covers – something that is not possible using CD methods. Finally, our proposed framework introduces a potential function method that utilizes the phase field method that naturally incorporates the physical restrictions of ice thickness and ice concentration in transport equations, resulting in modified transport equations which include additional forcing terms. Our method does not require post-processing, thereby avoiding the possible introduction of discontinuities and corresponding negative impact on the solution behavior. Numerical experiments are provided to demonstrate the efficacy of our new methodology.

Improving Numerical Accuracy for the Viscous-Plastic Formulation of Sea Ice

Accurate modeling of sea ice dynamics is critical for predicting environmental variables and is important in applications such as navigating ice breaker ships. Research for both modeling and simulating sea ice dynamics is ongoing, with the most widely accepted model based on the viscous-plastic (VP) formulation introduced by Hibler in 1979. Due to its highly nonlinear features, this model is intrinsically challenging for computational solvers. In particular, sea ice simulations often significantly differ from satellite observations. This study therefore focuses on improving the numerical accuracy of the VP sea ice model. Since the poor convergence observed in existing numerical simulations stems from the nonlinear nature of the VP formulation, this investigation proposes using the celebrated weighted essentially non-oscillatory (WENO) scheme -- as opposed to the frequently employed centered difference (CD) scheme -- for the spatial derivatives in the VP sea ice model. We then proceed to numerically demonstrate that WENO yields higher-order convergence for smooth solutions, and that furthermore it is able to resolve the discontinuities in the sharp features of sea ice covers -- something that is not possible using CD methods. Finally, our proposed framework integrates a potential function method that utilizes the phase field method to naturally incorporates the physical restrictions of ice thickness and ice concentration in transport equations, resulting in a modified transport equations which includes additional forcing terms. Our method does not require post-processing, thereby avoiding the possible introduction of discontinuities and corresponding negative impact on the solution behavior. Numerical experiments are provided to demonstrate the efficacy of our new methodology.

Спектры излучения в потоках ионизующихся газов для установки КСПУ-Т с продольным полем

The results of calculations of the spectral and integral characteristics of radiation in flows of ionizing gases in the channel of the quasi-stationary plasma accelerator in the presence of an additional longitudinal magnetic field are presented. The numerical model of the radiative magnetogasdynamics is based on the modified transport equations for the multicomponent medium consisting of atoms, ions, and electrons. The MHD equations for two-dimensional axisymmetric flows are presented in terms of the azimuthal components of magnetic field and vector potential of magnetic field, taking into account the electrical conductivity, thermal conductivity, and radiation transport. The spectral and integral characteristics of radiation were determined within the framework of the 3D model using the method of long characteristics in the multigroup approximation.

Effect of fractional interporosity flow on elastic waves propagation through a fluid-saturated double-porosity interlayer

The reflection and transmission behaviors of elastic waves through a fluid-saturated double-porosity interlayer are studied in present work. In order to simulate the interporosity flow, which cannot be appropriately described by the Darcy’s law, a modified transport equation using fractional order derivative is introduced to the double-porosity, dual-permeability model for the wave propagation in fluid-saturated double-porosity solid. The reflection and transmission coefficients are derived by satisfying the interfacial conditions of tractions and displacements. As a main feature of fluid-saturated double-porosity solid, the influences of interporosity flow on the wave propagation behavior are mainly studied. The numerical results are provided and shown graphically. It is observed that the reflection/transmission and the energy dissipation behavior are affected noticeably by the interporosity flow.

Axially and reflection symmetric systems and structure scalars in [formula omitted] gravity

This manuscript is devoted to a general study on axial and reflection-symmetric self-gravitating sources in the background of f(R,T) gravity, where T represents the trace of stress energy-tensor, while R is the Ricci scalar. For this purpose, all the fundamental equations and notions set to carry out the general study are established. The corresponding field, as well as dynamical equations, are also exhibited. In order to discuss the thermodynamics aspects of the selected system, the modified transport equation is presented. The Weyl-tensor is utilized to evaluate the generalized version of scalar variables termed structure scalars. To produce modified scalar equations, we employ the structure scalars. The applications of our obtained results are examined on the dynamics of the source in f(R,T) gravity. We conclude that the modified structure scalars play an important role in the dynamics of compact objects.

ElectroMicroTransport v2107: Open-source toolbox for paper-based electromigrative separations

electroMicroTransport v2107: Open-source toolbox for paper-based electromigrative separations

A Stable Method for Solving the Simplified Time-Dependent Neutron Transport Equation with Background Motion

The neutron transport equation is usually solved against a stationary background medium. When the background material is moving, the transport equation will need to be modified. Solving the transport equation with moving material is quite complicated, especially for the curved coordinate system because of the double angular redistributions. In this study, the discretization method of the simplified transport equation considering the moving-material effect is implemented in three-dimensional cylindrical geometry. Directly solving this modified transport equation with the standard solution technique is problematic since the advection term introduced by moving material may render the transport solver numerically unstable. The speed ratio λ is defined for stability analysis. A forced-stable method is proposed in this study to achieve good numerical stability for any material speeds and time-step sizes. The accuracy of this new method is verified using manufactured solutions. Steady numerical results demonstrate that the effects introduced by background motion cannot be neglected as the material speed starts to approach one-tenth of the neutron speed. Moreover, transient analysis indicates that the moving background has a considerable impact on the criticality of a system.

Mathematical foundations of the three-dimensional neutron hydrodynamics coupled transport equations

The neutron transport equation is usually solved against stationary background materials. When the background material is moving, the transport equation will need to be modified. Two historical approaches, “effective” cross-sections and “co-moving” frame, have been used in the past decades. The neutron transport community prefers to use the second method. However, the modified transport equation in the “co-moving” frame is extremely complicated. In this study, we discuss these modifications for non-relativistic neutrons. Transport equations considering background motions have been derived under the three-dimensional (3-D) Cartesian, cylindrical, and spherical coordinates, which can be conveniently reduced to lower-dimensional systems. Moreover, methods of transforming the coupled equation between different phase spaces have been studied.

Modified transport equation for the turbulent kinetic energy dissipation of the grid turbulence in the transition period of decay

This paper describes the modified transport equation of grid turbulence at low to moderate Reynolds number ( $$R_{\lambda} =\frac{u^{\prime } \lambda }{\nu }$$ , where $$\lambda $$ is the Taylor micro-scale and $$u^{\prime }$$ is the rms of the turbulent fluctuation) varies from $$R_{\lambda} =0$$ to $$R_{\lambda} =100$$ at $$10 \le U_ot/M \le 530$$ in the transition period of decay. Three different perforated passive grids have been used. It is well established that power law decay is described as $$q^2 \propto x^{-n}$$ . However, this power law can also be described as $$q^2 \propto x^{-(n+m(x))}$$ (here, n is the decay exponent and m is a nonzero integer) in the transition region. Therefore, main aim is to establish a new transport equation in the grid turbulence within the transition region at low $$R_{\lambda} $$ . To achieve results, hot-wire experiments were conducted in the fluid mechanics laboratory at the University of Newcastle, Australia. Higher-order curve fitting is used for obtaining the destruction coefficient (G) from the one-dimensional energy spectrum and curve fitting has helped us to optimize the noise of $$K^4$$ weighted energy spectrum. A modified equation of the destruction coefficient (G) is also shown, and the present results are well agreed with the modified equation of G as well as previous literature’s. The results of the Skewness (S) are observed as a function of $$R_{\lambda} $$ in the transition region, and it is found that behaviour of S is paradoxical. Relationship of skewness (S), destruction coefficient (G) with $$R_{\lambda} $$ has been discussed, and it is noticed that $$(S+2G/ R_{\lambda} ) \propto R_{\lambda} ^{-1}$$ . Obtained results also confirm that local isotropy is satisfied at low $$R_{\lambda} $$ .

Numerical study of cavitating flows in a wide range of water temperatures with special emphasis on two typical cavitation dynamics

Abstracts The objectives of this paper are to pursue accurate numerical methods of unsteady cavitating flows at a reasonable cost and investigate the dynamics evolution of cavitating flows in a wide range of water temperatures. A modified transport equation cavitation model including heat transfer effect, and a modified density corrected turbulence model for varying temperature water are presented in this study. The cavitating flows around a NACA0015 hydrofoil with the angle of attack α 0 = 5° and α 0 = 8° in a wide range of water temperatures (from 298 K to 393 K) are numerically investigated. Good agreements are obtained between the numerical predictions and the experimental measurements, including the pressure and temperature distributions, the cavity structures as well as the quasi-period of cavitating flows. The B -factor temperature drop Δ T B -factor , which is based on a heat balance between two phases, could accurately predict the actual temperature drop inside the cavity. It is found that the inflection-point temperature of two typical cavitation dynamics in varying temperature water is 370 K (±1 K). When the temperature of water is below 370 K, the liquid/vapor density ratio D dominates the change of the cavitation dynamics, and the attached cavity tends to be mushier and longer with the increasing temperature. However, the thermodynamic effects dominate the change of the cavitation dynamics, and significantly decrease the attached cavity when the temperature of water is above 370 K. The thermodynamic effects slightly affect the cavitation dynamics between 343 K and 370 K, but do not change the cavitation dynamics. The scale of shedding cloud cavity increases while the magnitude of turbulence kinetic energy decreases with the increasing temperature at the same free-stream conditions and flow stage. The cavitation dynamics in high temperature water (above 343 K) including pressure and temperature fluctuations, the cavity structures depend on the isothermal free-stream conditions and the thermodynamic effects which could suppress the development of the cavitation behaviors. The thermodynamic effects can be assessed by the thermophysical properties and free-stream conditions. The thermodynamic parameter C -factor (defined as ( ρ v L ev dp v )/(0.5 C l ρ l 2 U ∞ 2 dT )) presented in this paper could quantitatively predict the thermodynamic effects on cavitating flow dynamics.

Studies of oxygen reduction in 1-butyl-1-methylpyrrolidinium bis(trifluoromethylsulfonyl)imide by microdisk voltammetry

The electrochemical behavior of oxygen reduction in the room temperature ionic liquid 1-butyl-1-methylpyrrolidinium bis(trifluoromethylsulfonyl) imide (BMPTFSI) was examined using staircase cyclic voltammetry on a 50μm diameter platinum microdisk electrode. The diffusion coefficients for oxygen and superoxide, and the heterogeneous rate constants were obtained with a fit to experimental voltammograms. A modified transport equation in 1D spherical symmetry and Tsallis’ q-exponential function were utilized in the fit procedure, and its applicability was compared to 2D simulations made with Comsol Multiphysics. In addition, the formal standard potential of oxygen reduction was referenced to the cobaltocene/cobaltocenium (Co(cp)2+/0) redox couple. It was found that oxygen reacts with homogeneously with Co(cp)2+, making the quantitative determination of oxygen reaction kinetics rather difficult. Also, small amounts of water in the liquid cause complex reaction mechanisms. Finally, proton reduction was also measured, and its formal potential resides at ca. 1.45V vs. Co(cp)2+/0 reference, which more than 1V positive to those of water or oxygen reduction.

A new formulation of scale-adaptive simulation approach to predict complex wall-bounded shear flows

A new formulation of scale-adaptive simulation (SAS) approach for complex wall-bounded shear flows is presented. This approach makes use of a unique modelling representation and requires moderate computational costs. Based on the Rotta original transport equation for turbulence integral length scale, the suggested model is able to resolve unsteady turbulent structures with sufficient spatial resolution. Similar to the classical SAS-turbulence model (SST–SAS) proposed by Menter et al. [5] that represents an improvement of the proven turbulence model for unsteady calculations, the new model appears advantageous where classical LES (large eddy simulation) or hybrid LES–RANS models are too expensive. However, in its zonal formulation, the new model provides accurate predictions in (1) so-called stable flows (e.g. channel flow) in which the classical SAS () model will not be able to switch from Reynolds-averaged Navier–Stokes (RANS) to scale-resolving simulations without an explicit introduction of synthetic turbulence, (2) flows with relatively weak instabilities also challenging for the classical SAS-type models (e.g. swirling flow with sudden expansion). Such improvements result from the appropriately treating additional source terms in modified transport equation of dissipation rate. For demonstration, comparisons with experimental data and direct numerical simulation are provided.

MODIFIED TRANSPORT EQUATIONS FOR THE THREE-PHASE FLOW OF IMMISCIBLE, INCOMPRESSIBLE FLUIDS THROUGH WATER-WET POROUS MEDIA

MODIFIED TRANSPORT EQUATIONS FOR THE THREE-PHASE FLOW OF IMMISCIBLE, INCOMPRESSIBLE FLUIDS THROUGH WATER-WET POROUS MEDIA

Axial Dispersion Coefficients in Laminar Flows of Water-Distribution Systems

Because longitudinal dispersion is becoming increasingly germane to the problem of accurately determining water quality, water-resource managers and engineers seeking to represent solute transport in drinking-water systems must be able to compute relevant dispersion coefficients. Accordingly, the present study was undertaken to develop and experimentally verify a modified advection-dispersion-reaction transport equation as well as the formulas used to calculate the axial dispersion coefficient. The analysis assumes laminar flows, constant mean velocities, and short travel times (dimensionless time, T<0.01). With regard to the modified transport equation, the dispersion term was assumed to be direction-dependent. Thus, two distinct dispersion rates (forward and backward) were recognized and quantified as opposed to the single value used in conventional dispersion models. With the dimensionless travel time taken to be the independent variable, the developed dispersion coefficients increased at about one-fou...

Determination of concentration-dependent transport coefficients in nanofiltration: Experimental evaluation of coefficients

To characterize solute transport in nanofiltration (NF) the Spiegler–Kedem equation requires that two coefficients be determined for two-component solutions (a solute in water), solute permeability ω and reflection coefficient σ. For salts both coefficients strongly and in a complex way depend on concentration, which greatly complicates their evaluation from experiments. For this reason, the parameters are usually assumed constant for a given feed and the concentration dependence is assessed from flux–rejection curves for several feeds. This procedure however ignores the fact that the solute concentration and hence the coefficients significantly vary across the membrane. One way to overcome this inconsistency and address concentration dependence is to use physical models explicitly introducing exclusion mechanism(s) and fitting relevant membrane-specific parameters, such as fixed charge or dielectric properties. This procedure often fails to produce unique values of parameters for a given membrane and different salts. In the present study a new phenomenological approach is proposed and critically analyzed, based on the assumption of a similar concentration dependence of ω and 1 − σ, previously shown to be valid under fairly general conditions, thereby the Peclét coefficient A = (1 − σ)/ ω may be assumed to be independent of concentration. The coefficients and their concentration dependence for a given solute may be directly and consistently evaluated by fitting flux–rejection data for several feeds and fluxes to numeric solution of the modified transport equations without the need to invoke specific physical models. The values of transport parameters deduced in this way for representative membranes and salts allow important conclusions regarding the transport mechanism. In particular, the roles of different mechanisms in overall salt exclusion could be addressed directly from the variation of ω or 1 − σ with concentration. On the other hand, the value of the Peclét coefficient, free of the effect of salt partitioning, may be analyzed in terms of hindered transport. Using the proposed method, this value was found to be very small for studied thin-film composite membranes, which may significantly simplify the transport equations.

Wave transport in stellar radiation zone influenced by the Coriolis acceleration

Internal gravity waves constitute an efficient process for angular momentum transport over large distances. They are now seen as an important ingredient in understanding the evolution of rotation, and could explain the Sun's quasi-flat rotation profile. Because the Sun's rotation frequency is of the same order as that of the waves, it is necessary to refine our description of wave propagation and to take into account the action of the Coriolis acceleration in a coherent way. To achieve this aim, we adopt the Traditional Approximation which is verified in stellar radiation zones. We present the modified transport equations and their numerical evaluation in a parameter range that is significant for the Sun. The effectiveness of gravity waves, which become gravito-inertial waves, is reduced while new type of waves, namely the Rossby, the Yanai and the Kelvin waves appear with their associated transport.

Investigations of Interfacial Coupling Phenomena and its Impact on Recovery Factor

Abstract Interfacial coupling phenomena in multiphase flow through porous media and its impact on the recovery factor have been of interest for sometime, especially in the SAGD process and in fractured reservoirs. In this study, a numerical simulator that takes into account inlet end effects was developed to solve the modified transport equations proposed by Bentsen(1, 2), Ayub and Bentsen(3), Ayub(4) and Ayodele(5) to account for interfacial coupling phenomena and hydrodynamic effects. Moreover, the simulator was validated using experimental data. Finally, sensitivity analysis studies were carried out with the simulator to show the effect of interfacial coupling phenomena on the recovery factor. Introduction It is well known that Darcy's equation was developed on the basis of experiments involving single-phase flow through porous media. To account for multiphase flow, Darcy's equation was modified by Muskat et al.(6) into a form that has been widely used in petroleum reservoir engineering. However, as pointed out by several researchers(1, 3, 7–13), the modified equation can not give accurate recovery predictions when used to simulate multiphase flow in petroleum reservoirs. This is because, in multiphase flow, the presence of one fluid affects the flow of the other fluids; that is, interfacial coupling effects (viscous and capillary coupling effects) influence the flow. The viscous coupling effect, first identified by Yuster(14), refers to the coupling that arises due to the viscous drag exerted by one fluid on the other when they flow through the same porous medium, and it is usually associated with the mobility of the fluids(5). The capillary coupling effect, recently postulated by Babchin and Yuan(15) and by Bentsen(16), refers to the coupling that arises due to coupling, through the capillary function(2), of pressure across the interfaces of the fluids. Moreover, counter-current experimental data(17–19) has been used to show that Muskat's extension of Darcy's equation does not correctly describe the physics of multiphase flow through porous media, because the magnitude of the relative permeabilities for a given phase obtained from counter-current flow is always less than that acquired from a co-current experiment conducted in the same porous medium. To closely capture the physics of flow, Bentsen(1, 2), Ayub and Bentsen(3), Ayub(4) and Ayodele(5) developed a set of modified transport equations that incorporate interfacial coupling and hydrodynamic effects. In the following sections, a one-dimensional form of these modified transport equations is solved numerically by developing an Interfacial Coupling Simulator (ICS). Simulation results are compared with immiscible displacement experimental data and some conclusions are drawn as well. Mathematical Formulation To describe the one-dimensional immiscible displacement flow by using the fractional flow concept, two governing equations, the fractional flow equation and the frontal advance equation, are needed. Fractional Flow Equation With the assumption that two immiscible and incompressible fluids flow through a homogenous and isotropic water-wet porous medium, Bentsen(1, 2), Ayub and Bentsen(3), Ayub(4) and Ayodele(5) developed a set of modified transport equations whose one-dimensional form is as follows: V1 = Equation 1 Available In Full Paper V2 = Equation 2 Available In Full Paper The definitions for the relat

Angular Momentum Transport in the Sun’s Radiative Zone by Gravito-Inertial Waves

Internal gravity waves constitute an efficient process for angular momentum transport over large distances. They are now seen as an important ingredient in understanding the evolution of stellar rotation and can explain the Sun’s quasi-flat internal-rotation profile. Because the Sun’s rotation frequency is of the same order as that of the waves, it is now necessary to refine our description of wave propagation and to take into account the action of the Coriolis acceleration in a coherent way. To achieve this goal, we adopt the traditional approximation, which can be applied to stellar radiation zones under conditions that are given. We present the modified transport equations and their numerical evaluation in a parameter range that is significant for the Sun. Consequences for the transport of angular momentum inside solar and stellar radiative regions are discussed.

Pre-asymptotic corrections to fractional diffusion equations

The motion of contaminant particles through complex environments such as fractured rocks or porous sediments is often characterized by anomalous diffusion: the spread of the transported quantity is found to grow sublinearly in time due to the presence of obstacles which hinder particle migration. The asymptotic behavior of these systems is usually well described by fractional diffusion, which provides an elegant and unified framework for modeling anomalous transport. We show that pre-asymptotic corrections to fractional diffusion might become relevant, depending on the microscopic dynamics of the particles. To incorporate these effects, we derive a modified transport equation and validate its effectiveness by a Monte Carlo simulation.

Dynamic structure subgrid‐scale models for large eddy simulation

AbstractLarge eddy simulation (LES) is based on separation of variable of interest into two parts—resolved and subgrid. The resolved part is obtained numerically using modified transport equation while the effect of the subgrid part is modelled using subgrid‐scale (SGS) models. In this paper we present and discuss new one‐equation LES models for SGS scalar flux, SGS scalar dissipation and SGS energy dissipation. The proposed models belong to a new family of SGS models—dynamic structure (DS) models. The DS models borrow the structure of the modelled term from the corresponding Leonard term, and a special scaling factor is then used which does not contain user‐specified constants. The models are evaluated a priori using available DNS data for a non‐reacting mixing layer and decaying isotropic turbulence; the evaluation results compare well with viscosity and similarity models. During the a priori tests, the DS models were found to perform better than dynamic viscosity and similarity models for various test‐to‐base filter size ratios and non‐symmetric filters. For a posteriori evaluation, the models are implemented into a high‐order finite‐difference code and an LES of decaying isotropic turbulence is performed. The results match the data available from the literature and DNS simulations. Copyright © 2005 John Wiley &amp; Sons, Ltd.