Solution Of Advection Research Articles

A Galerkin finite element technique with Iweobodo-Mamadu-Njoseh wavelet (IMNW) basis function for the solution of time-fractional advection–diffusion problems

In this paper, the authors used wavelet-based Galerkin finite element technique constructed with Iweobodo-Mamadu-Njoseh wavelet as the basis function, for the numerical solution of time-fractional advection–diffusion equations. To achieve this, the authors used the Iweobodo-Mamadu-Njoseh wavelet as well as fractional calculus, wavelet and wavelet transform, and the Galerkin finite element technique. Also, time and space discretization in relation to the finite element technique were considered, followed by the steps in implementing numerical solutions to TFADE with the new technique. The new technique was considered in seeking numerical solutions of some Caputo type TFADE test problems, and the resulting numerical evidence displayed the effectiveness and accuracy of the method as the results obtained with the new method converged at a good pace to the exact solutions. The results obtained at different fractional order were also compared and the resulting evidence showed that at certain fractional value the convergence behavior displayed slight differences. Every numerical computation was done with the use of MAPLE 18 software.

Pore-scale modeling of wettability alteration coupled two-phase flow in carbonate porous media

The Middle East marine carbonate reservoirs are the primary targets for global crude oil exploration and production. In contrast to the clastic rock reservoirs, the marine carbonate reservoirs have undergone more complicated deposition, diagenesis and transformation, and the rock surface tends to be oil-wet or mixed-wet. One of the most effective methods for enhancing oil recovery in marine carbonate reservoirs is the use of an ion matched (IM)-surfactant (S) system to alter the wettability of rock surfaces. In this paper, we propose a microscale mathematical model of wettability alteration-coupled fluid flow by considering oil–water two-phase flow, wettability alteration, solute advection- diffusion and surfactant adsorption. The proposed microscale mathematical model is solved using the multi-component lattice Boltzmann model, and its accuracy is further verified. Finally, the effects of displacing fluid, ion-matched water concentration and surfactant concentration on oil mobilization at pore-scale are studied. Results show that, ion-matched water has a positive effect on wettability alteration, and typically shifts the cross-point of oil–water relative permeability curve to the right. The concentration of ion-matched water primarily affects fluid morphology of the wetting-phase and has little effect on phase permeability. After adding surfactant to the ion-matched water, the Euler coefficient of oil-phase gradually increases as the displacement continues, a large amount of remaining oil is gradually dispersed into scattered oil droplets and the oil-phase relative permeability is further improved.

Quantifying a common inconsistency in RANS-VoF modeling of water and oil core annular flow

Current trends in modeling CAF of oil and water combine the interface capturing methodology of VoF with the computational savings of a RANS approach. As demonstrated mathematically, this results in an inconsistency in the overall RANS-VoF treatment and incurs a number of omissions in the solution of oil fraction advection and momentum. To quantify these inconsistencies, five CAF cases with increasing Rew are considered and solved via DNS. Symptoms of the inconsistencies include qualitative errors in the prediction of the flow behavior in the entrance and fully-developed regions, as well as in the transition between these two regions. From a momentum perspective, the more troubling issue is the absence of unclosed terms resulting from fluctuating viscous and surface tension forces in the RANS equations. For momentum advection, terms associated with fluctuating density can be safely ignored due to the similar magnitude between oil and water density.

New Switchable Calculation Method for Chromatogram and Breakthrough-Curve Considering Both Solute Advection and Distribution Between Mobile Phase and Stationary Phase

New Switchable Calculation Method for Chromatogram and Breakthrough-Curve Considering Both Solute Advection and Distribution Between Mobile Phase and Stationary Phase

Two quantum algorithms for solving the one-dimensional advection–diffusion equation

Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection–diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other. Specifically, we solve the linear partial differential equation with a Quantum Linear Systems Algorithm (QLSA) based on the Harrow–Hassidim–Lloyd method and a Variational Quantum Algorithm (VQA), for resolutions that can be encoded using up to 6 qubits, which corresponds to N=64 grid points on the unit interval. Both algorithms are hybrid in nature, i.e., they involve a combination of classical and quantum computing building blocks. The QLSA and VQA are solved as ideal statevector simulations using the in-house solver QFlowS and open-access Qiskit software, respectively. We discuss several aspects of both algorithms which are crucial for a successful performance in both cases. These are the accurate eigenvalue estimation with the quantum phase estimation for the QLSA and the choice of the algorithm of the minimization of the cost function for the VQA. The latter algorithm is also implemented in the noisy Qiskit framework including measurement noise. We reflect on the current limitations and suggest some possible routes of future research for the numerical simulation of classical fluid flows on a quantum computer.

Analytical Solution for Contaminant Transport through a Soil–Bentonite (SB)/Geosynthetic Clay Liner (GCL)/Soil–Bentonite (SB) Composite Cutoff Wall and an Aquifer

This study develops a one-dimensional analytical solution for contaminant advection, diffusion and adsorption through a soil–bentonite (SB)/geosynthetic clay liner (GCL)/SB–aquifer composite cutoff wall (CCW) system. The solution agrees well with an existing double-layer model. Adopting toluene as a representative contaminant, using the present solution, the analysis systematically investigates the impact of hydraulic gradient (i) and the hydraulic conductivities of GCL (kgcl) and SB (ksb). The results show the following: (1) Increasing i from 0.1 to 1 reduces the concentration breakthrough time (tcb) from 20 to 11 years and mass flux breakthrough time (tfb) from infinite to 11 years, indicating lower i significantly extend both tcb and tfb, which is crucial for optimizing CCW barrier performance; (2) lowering kgcl from 5.0 × 10−11 m/s to 1 × 10−12 m/s and reducing ksb from 1.0 × 10−9 m/s to 1.0 × 10−11 m/s, would increase the tcb by 36% and 100%, respectively. It demonstrates that reducing kgcl and ksb could enhance barrier performance. (3) To achieve equivalent barrier performance, soil–bentonite cutoff wall (SBCW) requires greater thickness compared to SB/GCL/SB CCW, indicating that GCL reduces the required amount of bentonite; and (4) CCWs can use SB with lower adsorption capacity to achieve equivalent performance, further reducing bentonite requirements. The present solution can aid in the design and optimization of GCL-enhanced CCWs.

Weak and parabolic solutions of advection–diffusion equations with rough velocity field

We study the Cauchy problem for the advection–diffusion equation ∂tu+div(ub)=Δu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t u + {{\\,\\mathrm{\ extrm{div}}\\,}}(u\\varvec{b}) = \\Delta u$$\\end{document} associated with a merely integrable divergence-free vector field b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{b}$$\\end{document} defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes.

Unconditionally stable higher order semi-implicit level set method for advection equations

We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of advection by an external velocity and by a speed in the normal direction that are applicable in level set methods. The recommended numerical scheme is third order accurate for the linear advection in the 2D case with a space dependent velocity. Using a combination of analytical and numerical tools in the von Neumann stability analysis, the third order scheme is claimed to be unconditionally stable. We also present a simple high-resolution scheme that gives a TVD (Total Variation Diminishing) approximation of the spatial derivative for the advected level set function in the 1D case. In the case of nonlinear advection, a semi-implicit discretization is proposed to linearize the problem. The compact implicit part of the stencil of numerical schemes contains unknowns only in the upwind direction. Consequently, algebraic solvers like the fast sweeping method can be applied efficiently to solve the resulting algebraic systems. Numerical tests to evolve a smooth and non-smooth interface and an example with a large variation of the velocity confirm the good accuracy of the third order scheme even in the case of very large Courant numbers. The advantage of the high-resolution scheme is documented for examples where the advected level set functions contain large jumps in the gradient.

Flux exchange between fracture and matrix dictates late-time tracer tailing

Understanding and quantifying the solute tailing process in fractured rocks is critical for various subsurface applications but remains challenging. Particularly, the effects of flux exchange between fracture and matrix on the tailing process are considered to be important but have not been explicitly investigated. To fill this knowledge gap, we estimate the power-law scaling of late-time tracer tailing, n, based on three dimensional direct numerical simulations that consider both solute diffusion and advection across fracture-matrix interface. The dependence of solute decay rate n on fracture-matrix interaction is elucidated by simulating cases over a wide range of matrix permeability (km). In the case with matrix diffusion but without advection between fracture and matrix, we find that matrix diffusion significantly enhances tracer tailing. Then, the consideration of advection does not show significant effect on n up to the matrix permeability km of 10−12 m2. When k is greater than 10−12 m2, n declines rapidly down to 1.0 at k = 10−9 m2. We finally demonstrate that the upscaled model of continuous time random work can capture the late-time tailing behavior. This study provides a mechanistic explanation for the heavy tailing process commonly observed in fractured rocks.

Numerical Solution of Advection–Diffusion Equation of Fractional Order Using Chebyshev Collocation Method

This work presents a highly accurate method for the numerical solution of the advection–diffusion equation of fractional order. In our proposed method, we apply the Laplace transform to handle the time-fractional derivative and utilize the Chebyshev spectral collocation method for spatial discretization. The primary motivation for using the Laplace transform is its ability to avoid the classical time-stepping scheme and overcome the adverse effects of time steps on numerical accuracy and stability. Our method comprises three primary steps: (i) reducing the time-dependent equation to a time-independent equation via the Laplace transform, (ii) employing the Chebyshev spectral collocation method to approximate the solution of the transformed equation, and (iii) numerically inverting the Laplace transform. We discuss the convergence and stability of the method and assess its accuracy and efficiency by solving various problems in two dimensions.

Evaluation of advective solute infiltration into porous media by pulsed focused ultrasound-induced acoustic streaming effects.

Acoustic streaming induced by applying transcranial focused ultrasound (FUS) promotes localized advective solute transport in the brain and has recently garnered research interest for drug delivery and enhancement of brain waste clearance. The acoustic streaming behavior in brain tissue is difficult to model numerically and thus warrants an in vitro examination of the effects of using different sonication parameters, in terms of frequency, intensity, and pulse duration (PD). Melamine and polyvinyl alcohol (PVA) foams were used to mimic the porous brain tissue, which contains leptomeningeal fenestrations and perivascular space, while agar hydrogel was used to emulate denser neuropil. FUS was delivered to these media, which were immersed in a phosphate-buffered saline containing toluidine blue O dye, across various frequencies (400, 500, and 600 kHz; applicable to transcranial delivery) in a pulsed mode at two different spatialpeak pulse-average intensities (3 and 4 W/cm2). Image analysis showed that the use of 400 kHz yielded the greatest dye infiltration in melamine foam, while sonication had no impact on infiltration in the agar hydrogel due to the dominance of diffusional transport. Using a fixed spatial-peak temporal-average intensity of 0.4 W/cm2 at 400 kHz, a PD of 75 ms resulted in the greatest infiltration depth in both melamine and PVA foams among the tested range (50-150 ms). These findings suggest the existence of a specific frequency and PD that induce greater enhancement of solute/fluid movement, which may contribute to eventual in vivo applications in promoting waste clearance from the brain.

Internal layer intersecting the boundary of a domain in a singular advection–diffusion equation

We perform an asymptotic analysis with respect to the parameter ε > 0 of the solution of the scalar advection–diffusion equation y t ε + M ( x , t ) y x ε − ε y x x ε = 0, ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T ), supplemented with Dirichlet boundary conditions. For small values of ε, the solution y ε exhibits a boundary layer of size O ( ε ) in the neighborhood of x = 1 (assuming M > 0) and an internal layer of size O ( ε 1 / 2 ) in the neighborhood of the characteristic starting from the point ( 0 , 0 ). Assuming that these layers interact each other after a finite time T > 0 and using the method of matched asymptotic expansions, we construct an explicit approximation P ε satisfying ‖ y ε − P ε ‖ L ∞ ( 0 , T ; L 2 ( 0 , 1 ) ) = O ( ε 1 / 2 ). We emphasize the additional difficulties with respect to the case M constant considered recently by the authors.

A Bottom-up Transient Theory for Concentration Polarization with Applicability to over-Limiting Current Conditions

Renewable power sources have drawn interest due to scarcity of fossil fuels and associated pollution during their combustion-based power production. Renewables are projected to produce approximately 60% of US electricity generation by 2050, with an expected contribution of 2000 TW-h from wind power alone. To overcome the mismatch between electricity demand and intermittent supply, such power plants require large-scale energy-storage device integrated with them. In contrast to other technologies, redox flow batteries (RFBs) offer independently scalable power and energy capacity that is demanded for such applications. Despite significant achievement in their designs through experimentation and modeling, limited understanding has been established to date concerning their transient response while operating under irregular charge/discharge scenarios in renewable-powered grids and micro-grids. Under such conditions pore-scale mass transport physics and different mechanisms of concentration polarization can occur, but the conventional film law of mass transfer (FLoMT) using a constant Sherwood number cannot capture time-varying diffusion layer thickness near electrode surfaces. Further, the mean solute advection rate through porous RFB electrodes can deviate from the value expected from product of superficial velocity and average concentration inside the electrode if pore-scale concentration polarization is substantial. Consideration of these effects in the present bottom-up theory makes it applicable to over-limiting conditions, while predicting concentration polarization in agreement with experimental observations. We accomplish this by creating theory with similarity to the theory of Ralph White and co-workers for solid-state, particle-scale diffusion in Li-ion batteries [J. Power Sources, 2012, Vols. 214 and 218], while our theory is instead applied to an arbitrary representative volume element of flowing liquid electrolyte inside electrode pores.Here, we present a theory to incorporate these transient effects into a bottom-up transient model (B-UTM) by using frequency-dependent transfer functions (TFs) that are derived from the Fourier transformed pore-scale mass conservation equations (MCEs). These TFs preserve pore-scale mass transfer physics during their embedding into an up-scaled model. One of these TFs is the spectral Sherwood number that extends the film law of mass transfer to transient conditions and that correlates reactive flux with concentration polarization through an output-input relationship in the frequency domain. Another TF named the advective-flux transfer function captures the acceleration/suppression of solute advection rate that arises from the inhomogeneity in pore-scale concentration distributions. We first present validation of this theory using transient RFB experiments where the introduced B-UTM predicts the time evolution of concentration polarization in good agreement. Meanwhile the conventional FLoMT model that uses time-independent Sherwood number and neglects the solute acceleration/suppression effect shows systematically larger polarization (up to 400%), which restricts such models from simulating charging/discharging with applied current near and above the limiting current.The utility of the introduced bottom-up theory is demonstrated by incorporating it into a multi-scale RFB model in which redox-active electrolyte flows through a porous electrode comprised of an array of circular cylinders in crossflow that mimics commonly used carbon felt. The frequency dependence of the introduced TFs obtained from numerical solution of pore-scale MCEs are analyzed for different electrode porosity and Péclet number, where both the TFs reach a non-zero finite value in the limit of vanishing frequency. The non-zero value of the spectral Sherwood number in this limit is identical to the Sherwood number obtained in the pseudo-steady limit (PSL) where transient effects are negligible. With increasing frequency both the TFs show a transition from lagless response to semi-infinite Warburg response with orders higher values compared to that of the PSL. Finally, these TFs are embedded into an up-scaled model to obtain the time-domain response of RFBs operating under a step impulse applied current with different step durations. Numerically obtained concentration polarization and the average reactant concentration inside electrodes is used to construct non-dimensional regime maps showing a regime of over-limiting current. The results show that fast current fluctuations are sustained even when current exceeds the limiting current, which provides a means to increase the utilization of charge capacity of existing flow-based electrochemical devices that are operated under transient conditions. The theory introduced here can find applicability in other flow-based electrochemical devices including electrochemical separations and CO2 capture. Figure 1

Determination of diffusivities in binary fluid mixtures with various Lewis numbers as well as positive and negative Soret coefficients by the shadowgraph method.

The present work demonstrates the accessibility of the Fick diffusion coefficient D11 and/or the thermal diffusivity a of the binary mixtures dicyclohexylmethane/diphenylmethane, n-hexane/carbon dioxide, 1-hexanol/carbon dioxide, and methane/propane by the analysis of the dynamics of non-equilibrium fluctuations using the shadowgraph method. It is evidenced that D11 and a can be simultaneously determined for binary mixtures with Lewis numbers Le = a/D11 ranging over two orders of magnitude down to Le ≈ 5 or in the presence of minor advection for binary mixtures possessing a negative Soret coefficient in the investigated temperature and pressure ranges from (298.15 to 473.15) K and from about (0.5 to 25) MPa. The determined diffusivities are compared with those measured by heterodyne dynamic light scattering or obtained from the literature, with a focus on achievable uncertainties. By this comparison, it is shown that the determination of a by the shadowgraph method was hindered by mode-coupling effects for Le ≈ 5, whereas a determination of D11 was always possible for mixtures with Le ≥ 5. Furthermore, it is demonstrated that even in the presence of solutal advection, the description of the purely diffusive behavior of non-equilibrium fluctuations in concentration remains valid.

Convection-Induced Compositional Patterning at Grain Boundaries in Irradiated Alloys.

We consider the stability of precipitates formed at grain boundaries (GBs) by radiation-induced segregation in dilute alloys subjected to irradiation. The effects of grain size and misorientation of symmetric-tilt GBs are quantified using phase field modeling. A novel regime is identified where, at long times, GBs are decorated by precipitate patterns that resist coarsening. Maps of the chemical Péclet number indicate that arrested coarsening takes place when solute advection dominates over thermal diffusion right up to the precipitate-matrix interface, preventing interfacial local equilibrium and overriding capillary effects. This contrasts with liquid-solid mixtures where convection always accelerates coarsening.

A stable and mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations

The exact numerical simulation of plasma turbulence is one of the assets and challenges in fusion research. For grid-based solvers, sufficiently fine resolutions are often unattainable due to the curse of dimensionality. The sparse grid combination technique provides the means to alleviate the curse of dimensionality for kinetic simulations. However, the hierarchical representation for the combination step with the state-of-the-art hat functions suffers from poor conservation properties and numerical instability.The present work introduces two new variants of hierarchical multiscale basis functions for use with the combination technique: the biorthogonal and full weighting bases. The new basis functions conserve the total mass and are shown to significantly increase accuracy for a finite-volume solution of constant advection. Numerical analysis of the new basis functions reveals that their higher dual regularity does not only lead to conservation, but also yields an L2-stable basis for the combination technique. Accordingly, further numerical experiments applying the combination technique to a semi-Lagrangian Vlasov–Poisson solver in six dimensions show a stabilizing effect of the biorthogonal and full weighting bases on the simulations.

Numerical solution of Advection–Diffusion Equation using Graph theoretic polynomial collocation method

Water is one of the main constituents on earth for a living. The Advection Diffusion Equation (ADE) serves as an essential water standard model in environmental engineering since water pollution seriously threatens all life. Hence, the study of ADE has become an increasing concern for researchers in recent years. In the present work, we propose a new numerical method named the Hosoya polynomial collocation method (HPCM) for solving one-dimensional linear ADE. In this work, we developed an operational matrix of Integration of the ortho-normal Hosoya polynomials of path graphs (OHPG) to compute the solute concentration numerically by converting ADE into an algebraic system of equations and solving this system of equations using Newton’s method. With the appropriate collocation approach, we developed a graph theoretic polynomial-based solution to the considered ADE. We presented the convergence analysis, comparisons of HPCM findings with ADE’s analytical solutions, comparisons of error norms, and graphical representations of the numerical conclusions to illustrate the productivity of the HPCM. Comparing numerical results with the outcomes of other current numerical methods ensures the validity of the suggested method’s accuracy and efficacy.

Numerical simulations of the effect on twisted spacer filaments on biofouling and scaling in the feed channel of reverse osmosis membrane modules

Overcoming biofouling and scaling in membrane-based separation processes is significant as it affects the permeate quality. In the present study, the effect of biofouling (on membrane only, spacer only, membrane as well as spacer) and scaling on the velocity field, pressure variation, solute residence time distribution and water flux in the reverse osmosis feed channel consisting of twisted spacer filaments is presented using numerical simulations. The twisted spacer filaments caused fluid mixing, which enhanced membrane performance. The biofouling on spacer filaments and/or membrane surfaces and scaling resulted in a narrower effective area for fluid flow, thus causing fluid acceleration and higher pressure drop. The study of solute residence time distribution for biofouling revealed that the breakthrough time decreased with residence time due to fluid acceleration. The observations of solute tracer movement revealed that the solute advection pattern broadened when the feed time for the solute tracer was increased. An irregular and heterogeneous pattern of solute tracer movement was observed in the reverse osmosis feed channel with biofouling which enhanced the flow dispersion and acceleration. The effect of feed temperature on the biofouling pattern was also observed. Biofouling reduced with temperature because the increased feed temperature improved biomass mobility. The water flux in the computational domain containing spacer with scaling was higher compared to the computational domain containing spacer with biofouling because biofouling reduced the membrane porosity, diffusion coefficient and increased flow resistance. The present study concludes that the twisted spacer can effectively reduce biofouling and scaling with improved water flux in reverse osmosis membrane modules.

A bottom-up, multi-scale theory for transient mass transport of redox-active species through porous electrodes beyond the pseudo-steady limit

New theory is presented for the dynamic response of redox-active electrolyte flowing through porous electrodes under time-dependent applied current. This is done by introducing certain frequency-dependent transfer functions (TFs) that incorporate pore-scale transport physics. One TF – dubbed the spectral Sherwood number – extends the film law of mass transfer (FLoMT) to transient conditions. Another TF captures the acceleration/suppression of solute advection that results from pore-scale velocity/concentration gradients. Numerical results are shown for the frequency-dependent TFs of solid cylinders in crossflow to represent porous electrodes commonly used in flow batteries (FBs). Spectral regions are observed where a transition from lagless response to semi-infinite Warburg response occurs with increasing frequency. The embedding of these TFs into an up-scaled model is also formulated to obtain the time-domain response of FBs. Without adjustable parameters this model predicts polarization in agreement with transient FB experiments, despite systematic overprediction by the conventional FLoMT model. Analysis of concentration polarization and reactant concentration is also used to construct non-dimensional maps of operational space. These predictions show that fast current fluctuations are sustained even when current exceeds the limiting current expected from the time-invariant FLoMT without solute advection suppression/acceleration, suggesting implications for electrochemical conversion and separations devices in addition.

Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach

Abstract In this work, we propose the Ritz approximation approach with a satisfier function to solve fractal-fractional advection–diffusion–reaction equations. The approach reduces fractal-fractional advection–diffusion–reaction equations to a system of algebraic equations; hence, the system can be solved easily to obtain the numerical solution for fractal-fractional advection–diffusion–reaction equations. With only a few terms of two variables-shifted Legendre polynomials, this method is capable of providing high-accuracy solution for fractal-fractional advection–diffusion–reaction equations. Numerical examples show that this approach is comparable with the existing numerical method. The proposed approach can reduce the number of terms of polynomials needed for numerical simulation to obtain the solution for fractal-fractional advection–diffusion–reaction equations.