Fractional Transport Equation Research Articles

Numerical and geometric investigation of pool boiling heat transfer in cavities with isothermal rectangular corrugations

This numerical work presents a geometrical investigation of a corrugated isothermal surface placed in a two-dimensional cavity subjected to unsteady, turbulent pool boiling flows. The main purposes are maximizing the heat transfer rate between the isothermal surface and the surrounding water flow, and the volume of vapor generated into the cavity. The geometric investigation followed the constructal design method, being the ratio Hi/Li (i = 1, 2 or 3) of the corrugations varied for three different numbers of corrugations: N = 1, 2, and 3, keeping constant the corrugations area. The volume of fluid (VOF) and Lee's evaporation-condensation models are used to estimate the volume fractions of water vapor/liquid and interfacial mass transfer. The unsteady Reynolds Averaged Navier Stokes (URANS) continuity, momentum and conservation of energy equations, and volume fraction transport equation, are solved using the finite volume method (FVM) available in software Ansys FLUENT. For closure of turbulence, the k - ɛ model is adopted. For validation of the model, the heat flux and convection heat transfer coefficient obtained for a pool boiling bared surface case are compared with Rohsenow's correlations, and differences lower than 7.0 % are reached. Results indicated a strong influence of the ratio Hi/Li and number of corrugations (N) in the heat transfer rate per unit depth (qs) and dimensionless volume of vapor (Vdim) generated into the cavity. The highest intrusion of the corrugations led to the generation of few large and many small scales, benefiting the thermal performance, regardless of the performance indicator employed. The optimal configuration, N = 3 and H3/L3 = 2.0 improved 49 % and 188 % the Vdim and qs compared with the worst corrugated case, showing the importance of the geometry of the corrugation in this problem.

A new error estimate of a finite difference scheme for a fractional transport-advection equation with zero order term

In this work, we propose a finite difference scheme for Caputo’s fractional derivative transport equation in time and space, with a zero-order term. A new error estimation of the approximate solution has been demonstrated. By introducing an approximation of the Caputo derivative, we proved that the convergence is of order 2-α in time, and 2-β in space, for 0<α;β<1. Results of conditional stability and convergence of the numerical method are discussed. Finally a numerical implementation was presented to show the conformity between the theoretical and numerical approach of the finite difference scheme.

Modelling and simulation on compressible multi-component gas-liquid flows with chemical reaction and phase transition effects

In this paper, mathematical models and numerical algorithm are conducted for simulating compressible two-phase multi-component reactive flows, coupling with the finite-rate-based chemistry and the instantaneous-equilibrium phase transition model. Distinctive from previous multi-component two-phase governing equations, all volume fraction transport equations, with non-conservative characteristics, of gaseous components are substituted with a thermal equilibrium assumption inside the gaseous mixture for closure. Moreover, based on the temperature polynomial (NASA-7) fitted gas properties and the stiffened gas equation of state (SG-EOS) described liquid properties, the instantaneous-equilibrium phase transition model is re-derived, and the solving strategy of the transient phase transition model is detailly provided using an algebraical approach rather than the iterative process. Having included these elements, the proposed model is validated on three canonical tests, first for pure reactive flows, second for pure two-phase flows, and third for complex compressible multi-component two-phase reactive flows with phase transition effects: hydrogen-oxygen detonation interacting with a water droplet. The cases have well demonstrated the proposed model capability and the expected fluid physics in both the multiphase and reactive flows are reproduced.

An adaptive-correction algorithm for suppressing interface smearing in incompressible multiphase flows with complex interfacial behavior

In this paper, a numerical framework for modelling multiphase flows with interface correction is proposed. In original Volume-of-Fluid method, the interface smearing may appear since the interface profile deviates from its equilibrium state, especially in the simulations of complex interface deformation. To solve this problem, a novel correction step is introduced into the procedure after solving the volume fraction transport equation, and this step just works on the region of phase interface. The difference between the current method and the previous work is that the correction coefficient can be adjusted adaptively with the gradient of normal velocity at the interface, which is strongly correlated to the intensity of interface smearing and changes with the computational time and interface position. The validation computations are performed for bubble deformation (initial shapes of two-dimensional four-lobed-star and three-dimensional (3D) box with holes), bubble rising in a channel and Rayleigh–Taylor instability problems (Reynolds number of 100 and 1000). The obtained results show that the unphysical phenomenon of interface smearing is suppressed effectively, and the interface sharpness is improved greatly by the present method. In addition, the mass of bubble deformation by the original method will decrease by 13.3%. While these results obtained by the present method are in good agreement with the analytical solutions or published data.

Approximate Maximum Likelihood Estimation in Fractional Stochastic Transport Equation

We estimate the drift of the fractional stochastic transport equation the by maximum likelihood and the minimum contrast methods. We show consistency and asymptotic normality of the estimators. We consider both continuous and discrete time observations.

Analysis of anomalous transport with temporal fractional transport equations in a bounded domain

Anomalous transport in magnetically confined plasmas is investigated using temporal fractional transport equations. The use of temporal fractional transport equations means that the order of the partial derivative with respect to time is a fraction. In this case, the Caputo fractional derivative relative to time is utilized, because it preserves the form of the initial conditions. A numerical calculation reveals that the fractional order of the temporal derivative α (α ∈ (0,1), sub-diffusive regime) controls the diffusion rate. The temporal fractional derivative is related to the fact that the evolution of a physical quantity is affected by its past history, depending on what are termed memory effects. The magnitude of α is a measure of such memory effects. When α decreases, so does the rate of particle diffusion due to memory effects. As a result, if a system initially has a density profile without a source, then the smaller the α is, the more slowly the density profile approaches zero. When a source is added, due to the balance of the diffusion and fueling processes, the system reaches a steady state and the density profile does not evolve. As α decreases, the time required for the system to reach a steady state increases. In magnetically confined plasmas, the temporal fractional transport model can be applied to off-axis heating processes. Moreover, it is found that the memory effects reduce the rate of energy conduction and hollow temperature profiles can be sustained for a longer time in sub-diffusion processes than in ordinary diffusion processes.

Fractional Model of Multiple Trapping with Charge Leakage: Transient Photoconductivity and Transit–Time Dispersion

The model of multiple trapping into energy-distributed states is a successful tool to describe the transport of nonequilibrium charge carriers in amorphous semiconductors. Under certain conditions, the model leads to anomalous diffusion equations that contain time fractional derivatives. From this perspective, the multiple-trapping model can be used to interpret fractional transport equations, formulate initial and boundary conditions for them, and to construct numerical methods for solving fractional kinetic equations. Here, we shortly review the application of fractional multiple-trapping equations to problems of transient photoconductivity relaxation and transit–time dispersion in the time-of-flight experiment and discuss the connection of the multiple-trapping model with generalized fractional kinetic equations. Different types of charge leakage are discussed. The tempered fractional relaxation is obtained for recombination via localized states and distributed order equations arise for the non-exponential density of states presented as a weighted mixture of exponential functions. Analytical solutions for photocurrent decay in transient photoconductivity and time-of-flight experiments are provided for several simplified situations.

Parallel and Momentum Superdiffusion of Energetic Particles Interacting with Small-scale Magnetic Flux Ropes in the Large-scale Solar Wind

A recently developed time-dependent fractional Parker transport equation is solved to investigate the parallel and momentum superdiffusion of energetic charged particles in an inner heliospheric region containing dynamic small-scale flux ropes (SMFRs). Both types of superdiffusive transport are investigated with fractional transport terms containing a fractional time integral combined with normal spatial or momentum derivatives. Just as for normal diffusion, accelerated particles form spatial peaks with a maximum amplification factor that increases with particle energy. Instead of growth of the spatial peaks until a steady state is reached as for normal diffusion, parallel superdiffusion causes the peaks to dissipate into plateaus followed by a rollover at late times. The peaks dissipate at a faster rate when parallel transport is more superdiffusive. Furthermore, the accelerated particle spectral distribution function inevitably becomes an f 0 ∝ p −3 spectrum at late times in the test particle limit near the particle source despite the potential for spectral steepening from other transport terms. All this is a product of the growing domination of parallel spatial and especially momentum superdiffusion over other transport terms with time. Such extreme late time effects can be avoided by a transition to a normal diffusive state. Finally, fitting spatial peaks observed during SMFR acceleration events with the solution of the fractional Parker transport equation can potentially be used as a diagnostic for estimating the level of spatial and momentum superdiffusion in these events and how the levels of superdiffusion vary with distance from the Sun.

Vapor mixing in turbulent vaporizing flows

Scalar mixing and evaporation processes are analyzed in a compressible two-phase homogeneous and isotropic turbulence configuration. To this end, the fully compressible Navier–Stokes equations are solved using a pressure-based method and a mass-conservative interface capturing method (CLSVOF). Temperature and vapor mass fraction transport equations are coupled with the momentum equation via the evaporation rate: i.e. heat and mass transfer has an effect on the flow dynamic through the generation of a Stefan flow at the interface. In this formalism, a pressure equation is developed to include the acoustic, thermal dilatation and compressibility effects. Moreover, the method can consider several gas structures in the same domain, each with independent temperature, density, and thermodynamic pressure. The latter is important to simulate atomization, where interaction among the liquid structures (such as breakup or coalescence) can generate multiple gas inclusions. This work is one of the first to describe with high fidelity the vaporization and turbulent mixing occurring in dense two-phase flows.A statistical analysis of the vapor mass fraction field is performed. The obtained results reveal the influence of the non-homogeneous scalar source at the interface, which depend directly on the local interface temperature, on the PDF shape. Under these conditions, the vapor mass fraction PDF deviates from the Gaussian and beta PDF shapes, frequently used for turbulent combustion modeling. Furthermore, access to the local surface averaged vapor mass fraction and evaporation rate at the interface allows a statistical analysis of these variables. Results show a large dispersion of both quantities, especially at the early times of the simulations, which contradict the usual assumption of a constant vapor mass fraction (or vaporization rate) at the interface used in the literature in standard vaporization models. Finally, the influence of the mean interface curvature on the evaporation rate is quantified and discussed. Large evaporation rate are present when the interface is convex (positive mean curvature). On the contrary, local saturation zones are present near concave interface (negative mean curvature), reducing the evaporation rate magnitude.

Assessment of diffuse-interface methods for compressible multiphase fluid flows and elastic-plastic deformation in solids

This work describes three diffuse-interface methods for the simulation of immiscible, compressible multiphase fluid flows and elastic-plastic deformation in solids. The first method is the localized-artificial-diffusivity approach of Cook [1], Subramaniam et al. [2], and Adler and Lele [3], in which artificial diffusion terms are added to the individual phase mass fraction transport equations and are coupled with the other conservation equations. The second method is the gradient-form approach that is based on the quasi-conservative method of Shukla et al. [4], in which the diffusion and sharpening terms (together called regularization terms) are added to the individual phase volume fraction transport equations and are coupled with the other conservation equations [5]. The third approach is the divergence-form approach that is based on the fully conservative method of Jain et al. [6], in which the regularization terms are added to the individual phase volume fraction transport equations and are coupled with the other conservation equations. In the present study, all three diffuse-interface methods are used in conjunction with a four-equation, multicomponent mixture model, in which pressure and temperature equilibria are assumed among the various phases.The primary objective of this work is to compare these three methods in terms of their ability to: maintain constant interface thickness throughout the simulation; conserve mass, momentum, and energy; and maintain accurate interface shape for long-time integration. The second objective of this work is to consistently extend these methods to model interfaces between solid materials with strength. To assess and compare the methods, they are used to simulate a wide variety of problems, including (1) advection of an air bubble in water, (2) shock interaction with a helium bubble in air, (3) shock interaction and the collapse of an air bubble in water, and (4) Richtmyer–Meshkov instability of a copper–aluminum interface. The current work focuses on comparing these methods in the limit of relatively coarse grid resolution, which illustrates the true performance of these methods. This is because it is rarely practical to use hundreds of grid points to resolve a single bubble or drop in large-scale simulations of engineering interest.

Numerical model of an ultrasonically induced cavitation reactor and application to heavy oil processing

This study describes a numerical approach to model ultrasonically induced cavitation (UIC) reactors. UIC forms vapour-filled cavities in a liquid medium due to an applied acoustic field and their eventual collapse. UIC reactors are characterized by the presence of a vibrating probe that generates pressure waves by high-frequency oscillations (>20 kHz), which control the formation, dynamics, and eventual collapse of the vapour cavities. Those vapour cavities eventually enhance mixing and favour the occurrence of gas-liquid reactions. The zones of high mixing and reactivity coincide with the presence of the bubble cloud, which depends on the shape of the vessel and sonotrode. The development of advanced computational fluid dynamics (CFD) models is crucial to optimizing UIC processes’ geometry and operation parameters. A new algorithm for modelling UIC has been implemented within the OpenFOAM framework in the present study. The volume-of-fluid (VoF) method employs a diffuse interface approach for the volume fraction transport equation. The bubble dynamics are solved with sub-grid models, and the coupling between the main flow field and the sub-grid scales is performed through source terms in the transport equations. The source terms are de-coupled from convective and diffusive components of the volume fraction equation. The history of the bubbles is considered to consist of nucleation, oscillations, and collapse. The oscillations are resolved via the Rayleigh–Plesset equation. The concluding part of the work demonstrates the application of the algorithm to simulate the operation of an UIC reactor, which was designed to desulfurize fuels using the oxidative (ODS) process.

Analysis of anomalous transport based on radial fractional diffusion equation

Anomalous transport in magnetically confined plasmas is investigated by radial fractional transport equations. It is shown that for fractional transport models, hollow density profiles are formed and uphill transports can be observed regardless of whether the fractional diffusion coefficients (FDCs) are radially dependent or not. When a radially dependent FDC is imposed, compared with the case under it is observed that the position of the peak of the density profile is closer to the core. Further, it is found that when FDCs at the positions of source injections increase, the peak values of density profiles decrease. The non-local effect becomes significant as the order of fractional derivative and causes the uphill transport. However, as the fractional diffusion model returns to the standard model governed by Fick’s law.

Generalized Skewed Model for Spatial-Fractional Advective–Dispersive Phenomena

The conventional mathematical model expressed by the advection–dispersion equation has been widely used to describe contaminant transport in porous media. However, studies have shown that it fails to simulate early arrival of contaminant, long tailing breakthrough curves and presents a physical scale-dependency of the dispersion coefficient. Recently, advances in fractional calculus allowed the introduction of fractional order derivatives to model several engineering and physical phenomena, including the anomalous dispersion of solute particles. This approach gives birth to the fractional advection–dispersion equation. This work presents new solutions to the fractional transport equation that satisfies the initial condition of constant solute injection in a semi-infinite medium. The new solution is derived based on a similarity approach. Moreover, laboratory column tests were performed in a Brazilian lateritic soil to validate the new solution with experimental data and compare its accuracy with the conventional model and other fractional solutions. The new solution outperforms the existing ones and reveals an interesting fractal-like scaling rule for the diffusivity coefficients.

Energetic Particle Superdiffusion in Solar System Plasmas: Which Fractional Transport Equation?

Superdiffusive transport of energetic particles in the solar system and in other plasma environments is often inferred; while this can be described in terms of Lévy walks, a corresponding transport differential equation still calls for investigation. Here, we propose that superdiffusive transport can be described by means of a transport equation for pitch-angle scattering where the time derivative is fractional rather than integer. We show that this simply leads to superdiffusion in the direction parallel to the magnetic field, and we discuss some advantages with respect to approaches based on transport equations with symmetric spatial fractional derivates.

Selection of [formula omitted]-Caputo derivatives’ functional parameters in generalized water transport equation by genetic programming technique

The paper considers the usage of genetic programming technique to select an analytic form of functional parameter of the ψ-Caputo fractional derivative. We study one-dimensional space–time fractional water transport equation with such derivatives with respect to both time and space variables that generalizes the classical Richards equation. Having water head values measured by Watermark sensors as inputs, the statement of parameters identification problem is performed. The forms of functional parameters are represented as trees and found using a genetic programming algorithm. We compare the accuracy of field data description by the model with fixed and variable forms of derivatives’ functional parameters and obtained up to 30% increase in accuracy for the training dataset and up to 15% increase for the testing dataset when the considered method was used to select parameters’ forms.

On the maximum principle for the multi-term fractional transport equation

In this paper, we prove a maximum principle for the general multi-term space-time-fractional transport equation and apply it for establishing uniqueness of solution to an initial-boundary-value problem for this equation. We also derive some comparison principles for solutions to the initial-boundary-value problems with different problem data. Finally, we present a maximum principle for the Cauchy problem for a time-fractional transport equation on an unbounded domain.

Heat transfer study of a new hybrid photovoltaic/thermal direct absorption parabolic solar collector by two-phase Buongiorno model.

Solar energy is one of the cleanest sources of renewable energy that is easily accessible in vast geographical areas. As the efficiency of the common solar collectors is very low and is limited by the absorption properties of working fluid, enhancing the thermal performance of these collectors is one of the major challenges of developing parabolic solar thermal power plants. In recent decades, researches revealed that utilizing nanofluid as a novel working fluid has a dramatic effect on the thermophysical and optical properties of the fluid. In this study, the flow and temperature fields of water/magnetite and water/aluminum nanofluids are evaluated by solving the steady form of governing equations including conservation laws of mass, volume fraction transport equation, momentum equation, energy equation, and radiation transfer equation. Moreover, the two-phase Buongiorno model is utilized and Brownian motion, thermophoresis effects, and magnetophoresis movement are taken into account in the nanofluid simulation. The numerical results demonstrate that increasing nanofluid volume fraction and flow rate can increase the thermal performance of the collector tube. It is found that the thermal efficiencies reach maximum values of 151.03% and 158.58% for water/aluminum and water/magnetite nanofluids, respectively. Furthermore, increasing the volume fraction from 0 to 0.3% leads to rise of 24.41% and 21.36% in the maximum temperature of the collector. The effects of different parameters such as nanoparticle volume fraction, flow rate, and nanoparticle kind on the collector thermal and electrical efficiencies, thermal distribution, and entropy production have been studied.

A Focused Transport-based Kinetic Fractional Diffusion-advection Equation for Energetic Particle Trapping and Reconnection-related Acceleration by Small-scale Magnetic Flux Ropes in the Solar Wind

Abstract Analysis of energetic particle inner heliospheric spacecraft data increasingly suggests the existence of anomalous diffusion phenomena that should be addressed to achieve a better understanding of energetic particle transport and acceleration in the expanding solar wind medium. Related to this is fast-growing observational evidence supporting the long-standing prediction from magnetohydrodynamic (MHD) theory and simulations of the presence of an inner heliospheric, dominant quasi-two-dimensional MHD turbulence component that contains coherent contracting and merging (reconnecting) small-scale magnetic flux rope (SMFR) structures. This suggests that energetic particle trapping in SMFRs should play a role in anomalous diffusion in the solar wind that warrants further investigation. However, progress in studying such anomalous energetic particle transport phenomena in the solar wind is hampered by the lack of a fundamental derivation of a general fractional kinetic transport equation linking macroscopic energetic particle fractional transport to the microscopic physics of energetic particle interaction with SMFR structures. Here, we outline details of how one can derive a closed ensemble-averaged focused transport equation in the form of a general kinetic fractional diffusion-advection equation from first principles following the nonlinear Eulerian correlation function closure approach of Sanchez et al. With this equation one can model the anomalous diffusion of energetic particles in ordinary, momentum, and pitch-angle space in response to particle trapping in numerous SMFRs advected with the solar wind flow.

Numerical Approximation for Fractional Neutron Transport Equation

Fractional neutron transport equation reflects the anomalous transport processes in nuclear reactor. In this paper, we will construct the fully discrete methods for this type of fractional equation with Riesz derivative, where the generalized WENO5 scheme is used in spatial direction and Runge–Kutta schemes are adopted in temporal direction. The linear stabilities of the generalized WENO5 schemes with different stages and different order ERK are discussed detailed. Numerical examples show the combinations of forward Euler/two-stage, second-order ERK and WENO5 are unstable and the three-stage, third-order ERK method with generalized WENO5 is stable and can maintain sharp transitions for discontinuous problem, and its convergence reaches fifth order for smooth boundary condition.

Fractional Stochastic Active Scalar Equations Generalizing the Multi-Dimensional Quasi-Geostrophic &amp; 2D-Navier–Stokes Equations: The General Case

We introduce and study the well posedness: global existence, uniqueness and regularity of the solutions, of a class of d-dimensional fractional stochastic active scalar equations defined either on the torus $$ {\mathbb {T}}^d$$ or $$ {\mathbb {R}}^d$$ or $$ O\subsetneq {\mathbb {R}}^d$$ bounded, $$ d\ge 1$$ . This class includes, among others the stochastic; dD-quasi-geostrophic equation, fractional Burgers equation, fractional nonlocal transport equation and the 2D-fractional vorticity Navier–Stokes equation. We consider a locally Lipschitz diffusion term and a cylindrical Q-Wiener process with finite trace covariance Q. In particular, for $$ O={\mathbb {T}}^d$$ or $$ O\subsetneq {\mathbb {R}}^d$$ bounded, we prove the existence and the uniqueness of a global mild solution for the free divergence mode in the subcritical regime ( $$\alpha >\alpha _0(d)\ge 1$$ ), martingale solutions in the general regime ( $$\alpha \in (0, 2)$$ , supercritical, critical and subcritical) and free divergence mode and a local mild solution for the general mode and subcritical regime. Different kinds of regularity are also established for these solutions. For $$ O={\mathbb {R}}^d$$ , we studied the subcritical regime and we proved the existence of a global martingale solution for the free divergence mode and a local mild solution for the general mode.