Convective Terms Research Articles

On a parabolic p-Laplacian system with a convective term

On a parabolic p-Laplacian system with a convective term

Upwind block-centered multistep differences for semiconductor device problem with heat and magnetic influences

In this paper, a mathematical model of semiconductor device with two important factors is discussed. Heat and magnetic influences are considered together. Its numerical approximation is important in information science and other technological innovation fields. The major physical unknowns (the potential, electron concentration, hole concentration, and the heat) are defined by four nonlinear PDEs: an elliptic equation, two convection-diffusion equations and a heat conductor equation. A conservative block-centered method is used to approximate the elliptic equation. The derivatives to time t, ∂∂t(⋅), are discretized by multistep differences for improving the computational accuracy. The diffusions and convection terms are treated by block-centered differences and upwind approximations, respectively. This composite numerical method is suitable for this nonlinear system. Firstly, numerical dispersion and nonphysical oscillations are avoided. Secondly, the unknowns and their adjoint vectors are obtained simultaneously. Thirdly, the law of conservation is preserved. An error estimates is derived. Finally, simple examples are given for showing the efficiency and accuracy.

An asymptotic-preserving and exactly mass-conservative semi-implicit scheme for weakly compressible flows based on compatible finite elements

We present a novel asymptotic-preserving semi-implicit finite element method for weakly compressible and incompressible flows based on compatible finite element spaces. The momentum is sought in an H(div)-conforming space, ensuring exact pointwise mass conservation at the discrete level. We use an explicit discontinuous Galerkin-based discretization for the nonlinear convective terms, while treating the pressure and viscous terms implicitly, so that the CFL condition depends only on the fluid velocity. To handle shocks and damp spurious oscillations in the compressible regime, we incorporate an a posteriori limiter that employs artificial viscosity and is based on a discrete maximum principle. By using hybridization, the final algorithm requires solving only symmetric positive definite linear systems. As the Mach number approaches zero and the density remains constant, the method naturally converges to an H(div)-based discretization of the incompressible Navier-Stokes equations in the vorticity-velocity-pressure formulation. Several numerical tests validate the proposed method.

Heat transfer on MHD oscillatory flow in a vertical double-passage channel

Thermal propagation of MHD oscillatory flow of optically thin fluid in a vertical double-passage with varying temperature was investigated. The thermal distribution of the fluid flow is enhanced through a perfectly thin conductive baffle, which divides the channel into two passages. The equations governing the fluid flow are formulated based on purely oscillatory flow with pressure gradient. The flow velocity and thermal equations governing the flow are analytically solved. The closed-form results gotten were analysed for the influences of Reynolds number, magnetic term, thermal convection term, Peclet number and frequency of oscillation on the flow and heat transfer characteristics with the aid of graphs. It was observed that magnetic parameter reduces the flow rate in the channel.

Gaussian quadrature method with exponential fitting factor for two-parameter singularly perturbed parabolic problem

The parabolic convection-diffusion-reaction problem is examined in this work, where the diffusion and convection terms are multiplied by two small parameters, respectively. The proposed approach is based on a fitted operator finite difference method. The Crank-Nicolson method on uniform mesh is utilized to discretize the time variables in the first step. Two-point Gaussian quadrature rule is used for further discretizing these semi-discrete problems in space, and the second order interpolation of the first derivatives is utilized. The fitting factor’s value, which accounts for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method’s applicability is validated by two examples, which yielded more accurate results than some other methods found in the literatures.

Surface energy balance-based surface urban heat island decomposition at high resolution

Urban heat island (UHI) is among the most pronounced human impacts on Earth. To formulate locally adapted mitigation strategies, a comprehensive understanding of the influencing mechanisms of UHI at high resolution is imperative. Based on surface energy balance, we attributed surface UHI (SUHI) into five biophysical terms (surface radiation, anthropogenic heat, convection, evapotranspiration and heat storage term) using Sentinel-2 and Landsat-8 images in Beijing. The simulated SUHI intensity, derived by combining all five contribution terms, exhibited a good consistency but a higher spatial resolution, than SUHI intensity extracted from Landsat-8 land surface temperature product. SUHI intensity tended to decrease from the old city to outsides, attributed to the decrease of evapotranspiration, solar radiation and anthropogenic heat term. The convection and heat storage term play a positive role in reducing SUHI. Among urban morphological blocks, low-rise and high-density blocks had the strongest SUHI, with the evapotranspiration term contributing the most. The results highlighted the capacity of the urban surface to evaporate water in affecting Beijing SUHI. The proposed method provides one useful tool to analyze the drivers of SUHI from the aspect of heat formation, which can be potentially applied worldwide for large-scale comparisons of how urbanization affects UHI.

Simulation of solidification for saving energy with using nanomaterial involving conduction heat transfer

The current article studies the improvement of the discharging rate in cold storage systems by modifying the tank configuration and incorporating additives. Specifically, the study inspects how varying the diameter (dp) and fraction (ϕ) of nano-powders affects the process duration. The governing equations, derived under the assumption of negligible slip velocity of nanoparticles and convection terms, were solved using the Galerkin method. The computational grid was modified owing to location of the ice front, and unsteady terms were discretized using an unconditionally stable approach. The results indicate that initially, increasing dp decreases the process duration by approximately 20.01 %, but further increases in dp lead to a 49.53 % rise in the duration. As the process time increases, the amount of ice produced also increases, with nanoparticle loading resulting in a significantly higher ice yield. Specifically, the incorporation of nanoparticles enhances the storage rate by approximately 41.37 %.

Phase-space density control based on configuration invariant of spacecraft swarm

This paper investigates a phase-space density control based on configuration invariant for large-scale spacecraft swarms. By introducing Jordan-reduced dynamics and the configuration invariant to characterize relative orbits, the macroscopic swarm dynamics is presented with respect to the temporal evolution of phase space density. The density migration of the spacecraft swarm in phase space is modeled as a specific convection–diffusion process, where the diffusion term promotes swarm spreading and the convection term guides the swarm towards a designated region for uniform distribution. Through local density estimation and defining the convection and diffusion terms, the feedback control’s specific form is proposed and its exponential stability is proved. Particularly, the formulation of the convection term outside the target region and the combined kernel function for local density estimation help in avoiding sporadic isolated spacecraft. Numerical results of spacecraft swarm deployment validate the effectiveness and superiority of phase-space density control.

3D Radiation-hydrodynamical Simulations of Shadows on Transition Disks

Shadows are often observed in transition disks, which can result from obscuring by materials closer to the star, such as a misaligned inner disk. While shadows leave apparent darkened emission as observational signatures, they have significant dynamical impact on the disk. We carry out 3D radiation-hydrodynamical simulations to study shadows in transition disks and find that the temperature drop due to the shadow acts as an asymmetric driving force, leading to spirals in the cavity. These spirals have zero pattern speed following the fixed shadow. The pitch angle is given by tan−1(c s /v ϕ ) (6° if h/r = 0.1). These spirals transport mass through the cavity efficiently, with α ∼ 10−2 in our simulation. Besides spirals, the cavity edge can also form vortices and flocculent streamers. When present, these features could disturb the shadow-induced spirals. By carrying out Monte Carlo radiative transfer simulations, we show that these features resemble those observed in near-infrared scattered light images. In the vertical direction, the vertical gravity is no longer balanced by the pressure gradient alone. Instead, an azimuthal convective acceleration term balances the gravity–pressure difference, leading to azimuthally periodic upward and downward gas motion reaching 10% of the sound speed, which can be probed by Atacama Large Millimeter/submillimeter Array line observations.

Nonlinear twofold saddle point-based mixed finite element methods for a regularized μ(I)-rheology model of granular materials

We propose and analyze new mixed finite element methods for a regularized μ(I)-rheology model of granular flows with an equivalent viscosity depending nonlinearly on the pressure and the Euclidean norm of the symmetric part of the velocity gradient. To this end, and besides the velocity, the pressure and the aforementioned strain rate, we introduce a modified stress tensor that includes the convective term, and the skew-symmetric vorticity, as auxiliary tensor unknowns, thus yielding a mixed variational formulation within a Banach spaces framework. Then, the pressure is obtained through an iterative postprocess suggested by the incompressibility condition of the fluid, which allows us to express this unknown in terms of the aforementioned stress and the velocity. A fixed-point strategy combined with a solvability result for a class of nonlinear twofold saddle point operator equations in Banach spaces, is employed to show, along with the classical Banach fixed-point theorem, the well-posedness of the continuous and discrete formulations. In particular, PEERS and AFW elements of order ℓ⩾0 for the stress, the velocity, and the skew-symmetric vorticity, and piecewise polynomials of degree ⩽ℓ+n (resp. ⩽ℓ+1) for the strain rate with PEERS (resp. with AFW), yield stable Galerkin schemes. Optimal a priori error estimates are derived and associated rates of convergence are established. Finally, numerical results confirming the latter and illustrating the good performance of the methods, are reported.

Implicit-Explicit Schemes for Compressible Cahn–Hilliard–Navier–Stokes Equations

The isentropic compressible Cahn–Hilliard–Navier–Stokes equations are a system of fourth-order partial differential equations that model the evolution of some binary fluids under convection. The purpose of this paper is the design of efficient numerical schemes to approximate the solution of initial-boundary value problems with these equations. The efficiency stems from the implicit treatment of the high-order terms in the equations. Our proposal is a second-order linearly implicit-explicit time stepping scheme applied in a method of lines approach, in which the convective terms are treated explicitly and only linear systems have to be solved. Some experiments are performed to assess the validity and efficiency of this proposal.

A uniformly convergent analysis for multiple scale parabolic singularly perturbed convection-diffusion coupled systems: Optimal accuracy with less computational time

This study addresses time-dependent multiple-scale reaction-convection-diffusion initial boundary value systems characterized by strong coupling in the reaction matrix and weak coupling in the convection terms for a locally optimal accurate solution. The discrete problem, which typically loses its tridiagonal structure, expands its bandwidth to four in such coupled systems, resulting in a substantial computational load. Our objective is to mitigate this computational burden through a splitting approach that transforms the non-tridiagonal matrix into a tridiagonal form while maintaining the consistency, local optimal accuracy in space, and stability of the numerical scheme. We employ equidistributed non-uniform grids, guided by a carefully chosen monitor function, to approximate the continuous space domain. The discretization strategy targets local optimal linear accuracy across space and time on the domain's interior points. In addition, we have also provided the global convergence analysis of the present splitting approach, mathematically. The mathematical evidence is also obtained from the numerical experiments by comparing the splitting approach (either diagonal or triangular forms) of the reaction matrix to its coupled form. The results strongly confirm the effectiveness of this approach in delivering uniform linear accuracy, based on the present problem discretizations while significantly reducing the computational costs.

Model order reduction for the input–output behavior of a geothermal energy storage

In this article, we consider a geothermal energy storage system in which the spatio-temporal temperature distribution is modeled by a heat equation with a time-dependent convection term. Such storage systems are often embedded in residential heating systems. The control and management of such systems requires knowledge of aggregated characteristics of the temperature distribution in the storage. These describe the input–output behavior of the storage, the associated energy flows, and their response to charging and discharging processes. Our aim is to derive an efficient, approximate description of these characteristics by using low-dimensional systems of ordinary differential equations (ODEs). This leads to a model order reduction problem for a large-scale linear system of ODEs resulting from the semidiscretization of the heat equation combined with a linear algebraic output equation. In a first step, we approximated the nonautonomous system of ODEs by a linear time-invariant system. Then, we applied Lyapunov balanced truncation model order reduction to approximate the output by a reduced-order system that has only a few state equations but almost the same input–output behavior. The results of our extensive numerical experiments show the efficiency of the applied model order reduction method. We found that only a few suitably chosen ODEs are sufficient to achieve good approximations of the input–output behavior of the storage.

The limiting behavior of solutions to p-Laplacian problems with convection and exponential terms

We consider, for $a,l\geq1$, $b,s,\alpha> 0$, and $p> q\geq1$, the homogeneous Dirichlet problem for the equation $-\Delta_{p}u=\lambda u^{q-1}+\beta u^{a-1}\left\vert \nabla u\right\vert ^{b}+mu^{l-1}e^{\alpha u^{s}}$ in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$. We prove that under certain setting of the parameters $\lambda$, $\beta$ and $m$ the problem admits at least one positive solution. Using this result we prove that if $\lambda,\beta> 0$ are arbitrarily fixed and $m$ is sufficiently small, then the problem has a positive solution $u_{p}$, for all $p$ sufficiently large. In addition, we show that $u_{p}$ converges uniformly to the distance function to the boundary of $\Omega$, as $p\rightarrow\infty$. This convergence result is new for nonlinearities involving a convection term.

Fluid flow between two parallel active plates

This paper investigates the fluid flow phenomenon arising from the combined action of two parallel plates, which can expand/squeeze, absorb/inject, and stretch/shrink at different rates. These physical mechanisms are incorporated into the governing unsteady Navier–Stokes equations, which are then reduced to a fourth-order nonlinear differential equation with boundary conditions reflecting the imposed wall constraints. By letting the permeable Reynolds number (controlling the nonlinear convective terms) limit to zero, we demonstrate the existence of exact solutions expressed in terms of advanced mathematical functions. Additionally, in the absence of wall expansion/contraction, elementary exponential solutions are obtained under particular relationships between the stretching/shrinking and permeability parameters. A shear-like exact solution with broader applicability across various physical parameters is also identified. For moderate values of the expansion/squeezing parameters and permeable Reynolds numbers, we propose an efficient double-expansion perturbation analysis to approximate the flow behavior. Otherwise, for general physical parameters, a comprehensive mathematical analysis is provided and numerical simulations are employed to extract insights into the complex fluid motion between the parallel plates.

Enhanced framework for solving general energy equations based on metropolis-hasting Markov chain Monte Carlo

Due to the widespread presence of heat and mass transfer phenomena in industrial applications, numerous studies have been devoted to the accurate solution of energy equations, providing a foundation for the analysis of heat and mass transfer processes in practical applications. In this study, a Green's Function Markov Superposition Monte Carlo (GMSMC) for solving general energy equations has been developed based on probability and statistical principles owing to its advantageous features of insensitivity towards dimension and geometric complexity as well as the capability to handle multiple integrals in complex domains. The energy equation is first decomposed, and corresponding probability models are established for each component, considering their interrelationships. Subsequently, a solution framework for solving the general energy equation is constructed by integrating these probability models based on a Markov chain structure. The mathematical principles and formulae of the proposed method are derived in detail. The performance of the proposed method is validated by several heat transfer systems with different combinations of boundary conditions and features, which mainly include the distribution of the internal heat source and whether the convection or transient term is included. Results of the validation show that the temperatures obtained by the proposed method are in good agreement with the FEM based on a fine grid, no matter whether the calculation is for a single point or a distribution.

A fractional time-stepping method for unsteady thermal convection in non-Newtonian fluids

We propose a fractional-step method for the numerical solution of unsteady thermal convection in non-Newtonian fluids with temperature-dependent physical parameters. The proposed method is based on a viscosity-splitting approach, and it consists of four uncoupled steps where the convection and diffusion terms of both velocity and temperature solutions are uncoupled while a viscosity term is kept in the correction step at all times. This fractional-step method maintains the same boundary conditions imposed in the original problem for the corrected velocity solution, and it eliminates all inconsistencies related to boundary conditions for the treatment of the pressure solution. In addition, the method is unconditionally stable, and it allows the temperature to be transported by a non-divergence-free velocity field. In this case, we introduce a methodology to handle the subtle temperature convection term in the error analysis and establish full first-order error estimates for the velocity and the temperature solutions and 1/2-order estimates for the pressure solution in their appropriate norms. Three numerical examples are presented to demonstrate the theoretical results and examine the performance of the proposed method for solving unsteady thermal convection in non-Newtonian fluids. The computational results obtained for the considered examples confirm the convergence, accuracy, and applicability of the proposed time fractional-step method for unsteady thermal convection in non-Newtonian fluids.

Numerical Study on the Impact of the Richardson Number on Convection in an Enclosure, with Moving Sidewalls for Three Aspect Ratios: A = 1, 2 and 0.5

Using mathematical models, this research investigation seeks to explore how varying aspect ratios and the Richardson number impact the mixing effectiveness, fluid flow velocity, and heat transfer characteristics in a rectangle-shaped space where cooling occurs along the sides and warm with a steady temperature exists throughout the 80% of the bottom section's center. Discretizing the governing equations through the finite volume approach, we solved them iteratively using the Tri-diagonal Matrix Algorithm, where pressure-velocity coupling was successfully implemented via the SIMPLER procedure. Innovative approaches for convection and diffusion terms involved employing a power-law difference scheme alongside a central differential scheme. In various simulations with specified parameters (Re = 100, Pr = 0.71), we visualized results in forms like streamlines, isothermal lines, and temperature/velocity profiles, along with local and average Nusselt numbers. We observe five unique structure formations during flow behavior by analyzing how Richardson numbers impact heat transmission within a specific geometric configuration (enclosure size A = 1). A change in aspect ratio leads to improved control over fluid movement while minimizing structural distortion. A subtle rise in heat transmission emerged as the average Nusselt number and Richardson number intersection strengthened force convection's domination while experiencing significant escalation under conditions conductive to natural convection. Decreasing the aspect ratio increased heat transfer efficiency when the Richardson number rose.

Reducing spatial discretization error on coarse CFD simulations using an openFOAM-embedded deep learning framework

AbstractWe propose a method for reducing the spatial discretization error of coarse computational fluid dynamics (CFD) problems by enhancing the quality of low-resolution simulations using deep learning. We feed the model with fine-grid data after projecting it to the coarse-grid discretization. We substitute the default differencing scheme for the convection term by a feed-forward neural network that interpolates velocities from cell centers to face values to produce velocities that approximate the down-sampled fine-grid data well. The deep learning framework incorporates the open-source CFD code OpenFOAM, resulting in an end-to-end differentiable model. We automatically differentiate the CFD physics using a discrete adjoint code version. We present a fast communication method between TensorFlow (Python) and OpenFOAM (c++) that accelerates the training process. We applied the model to the flow past a square cylinder problem, reducing the error from 120% to 25% in the velocity for simulations inside the training distribution compared to the traditional solver using an x8 coarser mesh. For simulations outside the training distribution, the error reduction in the velocities was about 50%. The training is affordable in terms of time and data samples since the architecture exploits the local features of the physics.

Adaptation of the Low Dissipation Low Dispersion Scheme for Reactive Multicomponent Flows on Unstructured Grids Using Density-Based Solvers

Abstract The low dissipation low dispersion (LD2) second-order accurate scheme is effective for scale-resolving simulations in finite volume computational fluid dynamics (CFD) solvers, thanks to its combination of a skew-symmetric split form for convective terms and matrix-valued artificial dissipation fluxes. However, reactive flow simulations face challenges due to steep gradients and varying gas properties. This study replaces the skew-symmetric scheme with the kinetic energy and entropy preserving (KEEP) scheme, utilizing quadratic and cubic split forms for convective terms, enhancing stability. The nonsmooth fluid interfaces in reactive flow simulations necessitate upwind fluxes for reactive scalars to limit total variation (TV), also requiring upwind fluxes for the mixture-dependent internal energy fluxes. Other convective terms use central discretizations from the KEEP scheme, leveraging LD2’s spatial reconstruction to minimize dispersive errors. Numerical assessments show this approach reduces spurious pressure oscillations in single and multicomponent flows. The absolute flux Jacobian for dissipation flux calculation is efficiently computed using an expanded Turkel’s approach for thermally perfect gas mixtures. Partial pressure derivatives are approximated when using the flamelet generated manifolds (FGM) combustion model. The proposed scheme is evaluated through scale-resolving simulations of the Cambridge burner flame SWB1 on an unstructured grid using the density-based solver TRACE, employing both finite rate chemistry (FRC) and FGM combustion models. Comparative analysis with the all-speed SLAU2 scheme shows the superior performance of the proposed scheme in handling turbulent reactive multicomponent flows.