Advection Terms Research Articles

Extinction of multiple shocks in the modular Burgers’ equation

We consider multiple shock waves in the Burgers’ equation with a modular advection term. It was previously shown that the modular Burgers’ equation admits a traveling viscous shock with a single interface, which is stable against smooth and exponentially localized perturbations. In contrast, we suggest in the present work with the help of energy estimates and numerical simulations that the evolution of shock waves with multiple interfaces leads to finite-time coalescence of two consecutive interfaces. We formulate a precise scaling law of the finite-time extinction supported by the interface equations and by numerical simulations.

Representation of the Mozambique channel trough and its link to southern African rainfall in CMIP6 models

The topography of Madagascar and the strength of the Mozambique Channel Trough (MCT) modulate summer rainfall over southern Africa. A strong MCT hinders the penetration of moisture bearing easterlies from the South Indian Ocean into the mainland, thus reducing rainfall there and vice versa for weak MCT summers. Given the link between the MCT and rainfall, it is important to analyse how climate models represent the trough. Here, output from 20 models within the CMIP6 ensemble of Coupled General Circulation Models (CGCMs) are analyzed to investigate how state-of-the-art CGCMs represent the MCT and its link to southern African rainfall. Overall, the ensemble mean insignificantly underestimates the observed MCT. There is a large spread among the models, with the strength of the MCT significantly correlated with the Froude number based on the mountain height over Madagascar. In models, the vorticity tendency in the MCT area is dominated by the stretching and friction terms, whereas the vertical advection, tilting and residual terms dominate in the ERA5 reanalysis. The link between MCT and rainfall in the southern African subcontinent is missing in the models. Large rainfall biases are depicted over mainland even in models with a very strong MCT. It is found that the impacts of the MCT in the models could be masked by a complex mix of processes such as the strength of the Angola low, moisture fluxes from the Indian and South Atlantic Oceans as well as overestimated convection in the Mozambique Channel area.

The q-homotopy analysis method for a solution of the Cahn–Hilliard equation in the presence of advection and reaction terms

In this paper, we provide a solution to the Cahn–Hilliard equation using the q-homotopy analysis method (q-HAM). The q-HAM is a more general, simple and widely used method for solving stiff nonlinear partial differential equations. The Cahn–Hilliard equation is a classical model in material sciences that describe spinodal decomposition and phase separation in two-phase flows. Using the q-HAM, the effect of various parameters of physical interest such as diffusive parameter, thickness parameter, advection and reaction terms on concentration is studied. The comparison of the computed solution with the exact solution is presented for some fixed parameter values to validate the solution obtained using the q-HAM.

Effect of rigid body motion in phase-field models of solid-state sintering

In the last two decades, many phase-field models for solid-state sintering have been published. Two groups of models have emerged, with and without the contribution of rigid body motion. This paper first describes the previously published phase-field model with an advection term driven by rigid body motion. The model is then used to investigate the differences between models with and without rigid body motion in new benchmark geometries exhibiting markedly different behavior. Sensitivity studies concerning the parameters of the rigid-body motion model are conducted and their effects on equilibrium and kinetic properties explored. In particular, it is shown by simulations that a shrinkage rate independent of system size requires the inclusion of an advection term. Finally, the reason behind this behavior is explored and implications for diffusion-only models are drawn.

Effects of wave-current interactions on sediment dynamics in Hangzhou Bay during Typhoon Mitag

The hydrodynamics and sediment characteristics of muddy estuaries and coasts during typhoons are closely related to the geomorphic evolution, ecological environment, and economic development of coastal zones. Taking the macro-tidal turbid Hangzhou Bay (HZB) as an example, the sediment characteristics and effects of wave-current interactions on sediment dynamics during Typhoon Mitag were studied using a fully-calibrated numerical model. The model considered tide-wave sediment interactions and the reconstructed typhoon wind field. Net sediment fluxes were controlled by residual currents and suspended sediment concentration (SSC). The combined interactions of currents and waves led to a high SSC during the typhoon. Under calm weather conditions, the impact of wave-current interactions (wave dissipation, form drag, wave radiation stress, mean current advection and refraction) was small, except for the combined bottom stresses. The combined bottom stress was the primary wave-current interaction that changed sediment resuspension and increased SSC, particularly in shallow waters or during storms. The advection term, which played an essential role in reducing SSC in HZB, mainly affected SSC by increasing the velocity. The wave dissipation term enhanced vertical mixing, which involved the vertical exchange of suspended sediment and currents. In the shallow waters of the southern bay, the wave dissipation term mostly led to decreased bottom stresses, increased currents, decreased SSC, and increased SSC in deep waters. The effects of form drag, wave radiation stress, and refraction terms on the suspended sediment dynamics were relatively small. These findings provide a theoretical foundation for the study of dynamic geomorphology in macro-tidal estuaries.

SUPG-YZβ computation of chemically reactive convection-dominated nonlinear models

In this computational study, we deal with the stabilized finite element solutions of convection-dominated models having nonlinear reaction mechanisms. The existence of advection terms in these models causes the numerical solutions obtained by standard discretization methods to exhibit nonphysical oscillations. The Galerkin finite element method is stabilized using the Streamline-Upwind/Petrov–Galerkin formulation to avoid such spurious oscillations. The stabilized formulation is also complemented with the YZβ shock-capturing technique to resolve steep gradients and discontinuities accurately. The nonlinear equation systems arising from the spatial discretizations are solved with the Newton–Raphson method supplemented with the ILU-preconditioned GMRES search technique. The proposed methods are tested on a comprehensive set of nonlinear reactive models. Numerical experiments show that the presented methods and techniques eliminate spurious oscillations significantly and resolve strong gradients accurately. All computations are performed in the FEniCS environment.

Convergence of a multi-point flux approximation finite volume scheme for a sharp–diffuse interfaces model for seawater intrusion

This paper deals with development and analysis of a finite volume (FV) method for the coupled system describing seawater intrusion in coastal aquifers. The problem is modeled by the recently sharp–diffuse interfaces approach. This process is formulated by a coupled system of two nonlinear parabolic of cross-diffusion type equations describing two immiscible phase seawater/freshwater flow tacking into account the width of transition zones. A fully coupled, fully implicit approach is developed to discretize the coupled system. The method combines advantages of the MPFA method to accurately solve fluxes and diffusive terms and upstream for advective terms with implicit Euler’s time discretization. The non-negativity of the discrete solution is proved and an existence result is shown using a fixed point theorem. Based on a priori estimates and compactness arguments, we prove the convergence of the numerical approximation to the weak solution. We have developed and implemented this scheme in a new module in the context of the open source platform DuMuX. Two numerical experiments are presented to demonstrate the efficiency of this scheme, one of which is related to flows in a fractured porous aquifer.

A robust scheme for numerical simulation of heat transfer in two-fluid flows with high volumetric heat capacity contrasts

PurposeThis paper aims to propose a new method for robust simulations of passive heat transfer in two-fluid flows with high volumetric heat capacity contrasts.Design/methodology/approachThis paper implements a prediction–correction scheme to evolve the volumetric heat capacity. In the prediction substep, the volumetric heat capacity is evolved together with the temperature. The bounded downwind version of compressive interface capturing scheme for arbitrary meshes and central difference scheme are used for the spatial discretization of the advection and diffusion terms of the heat transfer equation, respectively. In the correction substep, the volumetric heat capacity is updated in accordance with the interface captured by using a coupled level-set and volume-of-fluid method to capture the interface dynamics precisely.FindingsThe proposed method is verified by simulating the advection of a hot droplet with high volumetric heat capacity, a stationary air–water tank with temperature variation between top and bottom walls and heat transfer during wave plunging at . The test results show that the proposed method is practical and accurate for simulating two-fluid heat transfer problems, especially for those feature high volumetric heat capacity contrasts.Originality/valueTo ensure the numerical stability, this paper solves an additional conservative form of volumetric heat capacity equation along with the conservative form of temperature equation by using consistent spatial-discretization and temporal-integration schemes.

On Stable Solutions to a Weighted Degenerate Elliptic Equation with Advection Terms

On Stable Solutions to a Weighted Degenerate Elliptic Equation with Advection Terms

Interscale transfer of turbulent energy in grid-generated turbulence with low Reynolds numbers

Direct numerical simulations are performed for grid-generated turbulence with low Reynolds numbers to study the interscale transfer of turbulent energy. The Reynolds numbers based on the uniform velocity and grid mesh size, M, are set to ReM=5000, 9000, and 15000. The results show that, for ReM=9000 and ReM=15000, the turbulent dissipation constant increases in the upstream region but it becomes nearly constant in the region of x/M>15. In contrast, for ReM=5000, it continues to increase and do not level off even in the downstream region. Scale-by-scale analysis using the Karman–Howarth–Monin–Hill equation indicates that, the non-linear transfer term balances with the dissipation term for all the cases. Instead, it is found that, in the downstream region of the ReM=5000 case, where the turbulent Reynolds number becomes Reλ<25, the contribution of the advection term varies depending on the streamwise positions. Further analysis for the one-point statistics indicates that, in such conditions, the coherences among the advection term, dissipation term, and diffusion term become small at small scales, while they are generally large in the other cases including the upstream region for ReM=5000. These indicate that energy cascade mechanism in the downstream region for ReM=5000 is different from the cases of 30<Reλ<60, even though both of them are generally considered as small Reλ, and it is caused by the increase of the dissipation scale range appearing in the transition period to the final decay.

Impact of Typhoon on Coastal Upwelling Off The Eastern Hainan Island: A Case Study of Typhoon Rammasun (2014)

The Hainan Island coastal upwelling (HIU hereafter) off the eastern Hainan Island (EHI) is an important phenomenon in the northern South China Sea. Though this phenomenon has been extensively studied, the fast responses of its inner oceanic processes to typhoon passage remain unclear. Based on satellite data and a validated three-dimensional numerical model, the HIU response to the passage of Typhoon Rammasun (2014) are explored in this study. Impacted by Rammasun, the climbing of deep water off the EHI was depressed and the bottom cold water moved seaward for at least 10 km. Though the HIU was weakened, a cooling event and Chl-a bloom in the HIU were formed after the passage of Rammasun. Diagnostic analyses revealed that both advection and diffusion terms contributed to controlling the local rate of temperate change during the typhoon passage and the advection terms showed significant internal wave signals along the shelf break off southeastern Hainan Island. At the same time, strong near-inertial waves (NIWs) with 0.705 cpd were triggered in the HIU. The NIWs lasted for about two weeks and caused strong vertical mixing in the HIU after Rammasun. Overall, typhoon wind and NIWs combined to dominate temperature variation in the HIU during the typhoon period.

Critical bait casting threshold of cage culture in open advective environments

Cage culture, as a kind of aquaculture, is increasingly becoming the main way to provide aquatic products due to its merits of high output and easy management. However, the main issue for the fishermen to be concerned about is the determination of the reasonable amount of casting bait since excess feeding bait will not only deteriorate water quality but also increase farming costs while too little bait casting cannot meet the demand for capture. To this end, in this letter, we formulate a diffusive bait-fish model with advection term, aiming to determine the bait casting threshold that meets the demand for the breeding and causes less pollution to the water environment. We first prove the well-posedness of the proposed model. Then we define the net reproductive rate R0 determining the survival of the model and reveal the relationship between R0 and the bait casting rate. Besides, we show that the proposed model undergoes a forward supercritical transcritical bifurcation when the bait casting rate is equal to the critical value. Finally, we resort to a numerical example to support our theoretical results.

Uncertainty quantification of turbulent systems via physically consistent and data-informed reduced-order models

This work presents a data-driven, energy-conserving closure method for the coarse-scale evolution of the mean and covariance of turbulent systems. Spatiotemporally non-local neural networks are employed for calculating the impact of non-Gaussian effects to the low-order statistics of dynamical systems with an energy-preserving quadratic nonlinearity. This property, which characterizes the advection term of turbulent flows, is encoded via an appropriate physical constraint in the training process of the data-informed closure. This condition is essential for the stability and accuracy of the simulations as it appropriately captures the energy transfers between unstable and stable modes of the system. The numerical scheme is implemented for a variety of turbulent systems, with prominent forward and inverse energy cascades. These problems include prototypical models such as an unstable triad system and the Lorentz-96 system, as well as more complex models: The two-layer quasi-geostrophic flows and incompressible, anisotropic jets where passive inertial tracers are being advected on. Training data are obtained through high-fidelity direct numerical simulations. In all cases, the hybrid scheme displays its ability to accurately capture the energy spectrum and high-order statistics of the systems under discussion. The generalizability properties of the trained closure models in all the test cases are explored, using out-of-sample realizations of the systems. The presented method is compared with existing first-order closure schemes, where only the mean equation is evolved. This comparison showcases that correctly evolving the covariance of the system outperforms first-order schemes in accuracy, at the expense of increased computational cost.

Traveling Band Solutions in a System Modeling Hunting Cooperation

A classical Lotka–Volterra model with the logistical growth of prey-and-hunting cooperation in the functional response of predators to prey was extended by introducing advection terms, which included the velocities of animals. The effect of velocity on the kinetics of the problem was analyzed. In order to examine the band behavior of species over time, traveling wave solutions were introduced, and conditions for the coexistence of both populations and/or extinction were found. Numerical simulations illustrating the obtained results were performed.

Dispersion tensor in stratified porous media.

Dispersion in porous media is of great importance in many areas of science and engineering. While dispersion in porous media has been generally well discussed in the literature, little work has been done regarding a generalization of Taylor dispersion in stratified media. In this work, we generalized the Taylor dispersion theory and Stokes flow in porous media to derive a reduced-order model for tracer dispersion in stratified porous media. Our findings revealed that for a simple case of two-layer porous media, the hydrodynamic coupling between the two layers leads to the tensorial nature of dispersion and advection. The results showed that the obtained dispersion tensor and advection are not symmetric unless both porous layers have similar thickness, porosity, and molecular diffusion. We found that the main elements of the coefficient of the dispersion tensor remain positive while the off-diagonal elements can take negative values. On the contrary, all elements of the advection matrix may take negative values. On the basis of these observations, we report the manifestation of the dispersion barrier, uphill dispersion and advection, and osmotic dispersion during tracer transport in stratified porous media. In particular, the identified uphill advection reveals that the injected tracer in one layer could be transported countercurrent to the adjacent layer. Furthermore, we have shown that in the limiting case of Darcy flow, the Taylor dispersion is absent, and the tracer mixing between the two layers is restricted to the cross-diffusive flux between them. The results revealed that the field scale mixing may not necessarily originate from the Taylor dispersion and could be due to the modified advection terms and the cross-diffusive flux between the two layers.

Analysis of a population model with advection and an autocatalytic-type growth

This paper analyzes a population model with time-dependent advection and an autocatalytic-type growth. As opposed to a logistic growth where the rate of growth of the population decreases from the onset, an autocatalytic growth has a point of inflection where the rate of growth switches from an increasing trend to a decreasing trend. Employing the idea of Painlevé property, we show that a variety of exact traveling wave solutions can be obtained for this model depending on the choice of the advection term. In particular, due to situations in resource distribution or environmental variations, if the advection is represented as a decaying function in time or an oscillating function in time, we are able to find exact solutions with interesting behavior. We also carry out a computational study of the model using an exponentially upwinded numerical scheme and illustrate the movement of the solutions and their characteristics pictorially.

Stable timestep formulas for high-order advection–diffusion and Navier–Stokes solvers

We derive three-dimensional stable timestep formulas for high-order explicit time integration of the advection–diffusion equation. The proposed formulas cover explicit first, second, third, and fourth-order Runge–Kutta integrators in time as well as upwind, central, second-order, high-order upwind (k-schemes), and flux-limiters for the advection term along with central discretization on the diffusion term. The stability rules are then compared to numerical computations of stable timesteps of the advection–diffusion equation and show excellent agreement. Finally, the formulas are implemented in a finite volume Navier–Stokes code and used to guide stable timestep selection to assess their suitability for Navier–Stokes solvers. All cases that were tested showed stable non-diverging solutions and significant improvement over common CFL-type stability specification. Despite its basis in the linear advection–diffusion equation, this work could be useful in providing a starting point for stable timestep selection for nonlinear evolutionary-type PDEs such as the Navier–Stokes equations.

Nanomaterial efficacy on freezing of PCM with involvement of numerical simulation

To reduce the freezing time for a unit with curved walls which filled with water, nanoparticles has been dispersed. The shapes of CuO nano-powder can affect the conductivity of mixture and this factor efficacy has been scrutinized in current article. Freezing process is mostly affected by condition and due to negligible magnitude of velocity, only the energy equation with neglecting advection term should be considered as main equation which has been solved via FEM. The proposition of utilizing homogeneous mixture is applicable for condition of low concentration of additives and for this study, this is practical assumption. Conduction mode is significant for freezing and dispersing nanoparticle especially with greater shape factor can expedite the process. When shape factor of copper oxide nanoparticles augments with fraction of 0.04, the freezing time decline about 10.71%. The required time declines about 31.39% with raising the fraction of particles from zero to 0.04.

Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations

This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincaré type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Peclét numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.

Laboratorial and analytical study for prediction of porosity changes in carbonaceous shale coupling reactive flow and dissolution

Molecular diffusion of acid into a porous carbonate medium is usually referred to as acid imbibition. Accurate prediction of acid penetration into a formation and reaction with rock is not possible without a comprehensive understanding of the acid diffusion process and associated parameters.This work aimed to derive an analytical solution for mineral-dissolution reactive diffusion when acid imbibes into a rock surface and where there is no flow condition under laboratory conditions. For the experimental part of the study, different disk samples of Eagle Ford shale were selected, and the effect of acid diffusion on the rock surface was fully investigated. Digital microscopic images confirm that with differing concentrations of consumed acid and saturation front movement, varying degrees of reactivity are conceivable, including surface reaction, surface dissolution, matrix acidizing, and induced fracturing.For the mathematical modeling section, a continuity equation without an advection term and only a diffusion term was considered. An analytical solution was found for the dissolution of minerals as a result of reactive imbibition in a porous medium with porosity variations. The final form of the equation and the analytical solution is proposed here for the first time. Additional experiments were performed on Indiana limestone samples as control tests, with all experimental results matching perfectly with the developed analytical solution. Data from the literature relating to different temperatures also matched appropriately with the mathematical solution, effectively affirming the validity of the suggested solution. The proposed approach can be used as a predictive tool to verify the reactive models leading to porosity creation in the field scale.