Diffusion Term Research Articles

Convergence analysis of novel discontinuous Galerkin methods for a convection dominated problem

In this paper, we propose and analyze a numerically stable and convergent scheme for a convection-diffusion-reaction equation in the convection-dominated regime. Discontinuous Galerkin (DG) methods are considered since standard finite element methods for the convection-dominated equation cause spurious oscillations. We choose to follow a DG finite element differential calculus framework introduced in Feng et al. (2016) and approximate the infinite-dimensional operators in the equation with the finite-dimensional DG differential operators. Specifically, we construct the numerical method by using the dual-wind discontinuous Galerkin (DWDG) formulation for the diffusive term and the average discrete gradient operator for the convective term along with standard DG stabilization. We prove that the method converges optimally in the convection-dominated regime. Numerical results are provided to support the theoretical findings.

The energetics and colour for linearised models of wall turbulence

By comparing the budget of a data-driven quasi-linear approximation (DQLA) (Holford, Lee & Hwang, J. Fluid Mech., vol. 980, 2024, A12) and direct numerical simulation (DNS) (Lee & Moser, J. Fluid Mech., vol. 860, 2019, pp. 886–938), the energetics of linear models for wall-bounded turbulence are assessed. The DQLA is implemented with the linearised Navier–Stokes equations with a stochastic forcing term and an eddy viscosity diffusion model. The self-consistent nature of the DQLA allows for a global comparison across all wavenumbers to assess the role of the various terms in the linear model in replicating the features present in DNS. Starting from the steady-state second-order statistics of a Fourier mode, a spectral budget equation is derived, connecting Lyapunov-like equations to the transport budget equations obtained from DNS. It is found that the DQLA and DNS are in good qualitative agreement for the streamwise-elongated structures present in DNS, comparing well for production, viscous transport and wall-normal turbulent transport. However, the DQLA does not have an energy-conservative nonlinear term. This results in no dissipation under molecular viscosity, with energy instead being dissipated locally through the eddy viscosity model, which models the energy removal by the nonlinear term at integral length scales. Comparison of the pressure–strain statistics also highlights the absence of the streak instability, with production and forcing mainly being retained in the streamwise and wall-normal components or shifted to the spanwise component. It is demonstrated that the eddy viscosity diffusion term locally enforces a self-similar budget, making the model for the nonlinear term self-consistent with a logarithmic mean profile. Implications and recommendations to improve the current eddy viscosity enhanced linear models are also discussed concerning the comparison with DNS, as well as considerations with regard to pressure statistics to mimic the role of the streak instability through colour of turbulence models.

Global existence of a weak solution for a reaction–diffusion system in a porous medium with membrane conditions and mass control

Abstract In this paper, we prove the global exstence of weak solutions for a porous medium dynamics of m species moving between two domains separated by a zero-thickness membrane. On this membrane, Kedem–Katchalsky conditions are considered, and the study is characterized by natural structural conditions applied to the nonlinear reactive terms. The global existence is established under the assumption that these reactive terms are bounded in $L^1$ . This problem has already been analyzed in the linear diffusion case by Ciavolella and Perthame in Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540). The present work constitutes an extension for nonlinear diffusion, particularly of the porous medium type, in the form $\partial _t v_i - \Delta v_i^{r_i} = R_i$ , for an exponent $r_i < 2$ . The case $r_i \geq 2$ remains an open problem. This paper is an adaptation of the ideas from Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540), with new strategies to overcome the appearance of nonlinearity and degeneracy in the diffusion term.

On the origin of counter-gradient transport in turbulent scalar flux: physics interpretation and adjoint data assimilation

The present study offers a twofold contribution on counter-gradient transport (CGT) of turbulent scalar flux. First, by examining turbulent scalar mixing through synchronized particle image velocimetry and planar laser-induced fluorescence on an inclined jet in cross-flow, we clarify the previously unexplained phenomenon of CGT, revealing key flow structures, their spatial distribution and modelling implications. Statistical analysis identifies two distinct CGT regions: local cross-gradient transport in the windward shear layer and non-local effects near the wall after injection. These behaviours are driven by specific flow structures, namely Kelvin–Helmholtz vortices (local) and wake vortices (non-local), suggesting that scalar flux can be decomposed into a gradient-type term for gradient diffusion and a term for large-eddy stirring. Second, we propose a new approach for reconstruction of turbulent mean flow and scalar fields using continuous adjoint data assimilation (DA). By rectifying model-form errors through anisotropic correction under observational constraints, our DA model minimizes discrepancies between experimental measurements and numerical predictions. As expected, the introduced forcing term effectively identifies regions where traditional models fall short, particularly in the jet centreline and near-wall regions, thereby enhancing the accuracy of the mean scalar field. These enhancements occur not only within the observation region but also in unseen regions, underscoring present DA approach's reliability and practicality for reproducing mean flow behaviours from limited data. These findings lay a solid foundation for adjoint-based model-consistent data-driven methods, offering promising potential for accurately predicting complex flow scenarios like film cooling.

Mean random dynamical systems and random attractors of stochastic Hopfield neural models with locally Lipschitz noise and time delay

In this paper, we investigate the long term dynamical behaviour of stochastic Hopfield neural network models driven by white noise with time delay, where the nonlinear drift and diffusion terms are both locally Lipschitz continuous. We establish the global-in-time solvability of these systems with locally Lipschitz continuous nonlinear drift and diffusion terms by using mean square uniform estimates of solutions. Furthermore, we demonstrate that the mean random dynamical system generated by the solution operators possesses a unique weak pullback mean random attractor in the Hilbert space L 2 ( Ω ~ , F ι ; l 2 ) × L 2 ( Ω ~ , F ι ; L 2 ( ( − ϱ , 0 ) , l 2 ) ) .

Dynamics of the modified Rotation-Camassa-Holm Equation with backward diffusion term

Dynamics of the modified Rotation-Camassa-Holm Equation with backward diffusion term

Langevin dynamics for a heavy particle immersed within a flow of light particles

A micro-hydrodynamics model based on elastic collisions of light point solvent particles with a heavy solute particle is investigated in the setting where the light particles have velocity distribution corresponding to a background flow. Considering a range of stationary background flows and distributions for the solvent particle velocities, the macroscopic Langevin-type description of the behaviour of the heavy particle is derived in the form of a generalized Ornstein–Uhlenbeck process. At leading order, the drift term in this process depends upon both the geometric structure of the background flow and the size of the heavy particle, while both drift and diffusion terms scale with moments of the light particle velocity distribution. Computational methods for simulating the micro-hydrodynamics model are then designed to confirm the theoretical results. To enable long-time calculations, simulations are performed in a frame co-moving with the heavy particle. Efficient methods for sampling the position and velocity distributions of incoming solvent particles at the boundaries of the co-moving frame are derived for a range of distributions of solvent particles. The simulations show good agreement with the theoretical results.

Calibration of the Reynolds stress model for turbulent round free jets based on jet half-width

Reynolds stress model (RSM) turbulence models are expected to yield more accurate numerical results for flows with strong anisotropy, such as round free jets, because they directly solve Reynolds stresses rather than modeling them. However, when computational fluid dynamics (CFD) analyses were performed at moderate jet Reynolds numbers using the isotropization by production (IP) RSM model, it was observed that the calculated jet half-widths, decay constants, and spreading rates differed from experimental results due to uncertainties inherent in the turbulence model. In this study, the closure coefficients of the IP RSM turbulence model were calibrated using a variant of the Multi-Objective Genetic Algorithm based on jet half-width data obtained experimentally in the near-field region of the jet. With the use of appropriate discretization schemes and computational grids, the calibrated coefficient combination for the IP RSM turbulence model showed improved accuracy in modeling jet half-widths at Reynolds numbers of 10 000 and 20 000, reducing the errors of calculated decay constants and spreading rates approximately from 2% to 1% and from 16% to 5%, respectively. A detailed examination of the turbulence budget along the longitudinal axis in the self-similar region revealed that the new model coefficients enhanced the modeling of diffusion term but compromised the advection term. As a result of the altered advection term, increased error margins were observed in turbulence intensity (TI) and velocity distribution along the jet centerline, although dissipation along the axis was improved. Consequently, the modeling error in jet half-width calculations using the CFD method was decreased, enhancing the computational cost-effectiveness of the RSM turbulence model compared to more complex turbulence models.

Thermosensitive double-membrane neurons and their network dynamics

Abstract Cell membrane of biological neurons has distinct geometric structure, and involvement of diffusive term is suitable to estimate the spatial effect of cell membrane on neural activities. The gradient field diversity between two sides of the cell membrane can be approached by using a double-layer membrane model for the neuron. Therefore, two capacitive variables and diffusive terms are used to investigate the neural activities of cell membrane, and the local kinetics is described by a functional circuit composed of two capacitors. The voltages for the two parallel capacitors describe the inner and outer membrane potentials, and the diffusive effect of ions is considered on the membrane surface. The results reveal that neural activities are relative to the capacitance ratio between the inside and outside of the membrane and diffusive coefficient. High-energy periodic external stimulation induces the target waves to spread uniformly, while low-energy chaotic stimulation results in wave fragmentation. Furthermore, when the capacitance ratio exhibits exponential growth under an adaptive control law, the resulting energy gradient within the network induces stable target waves. That is, energy distribution affects the wave propagation and pattern formation in the neuron. The result indicates that the spatial diffusive effect and capacitance diversity between outer and inner cell membranes are important for selection of firing patterns and signal processing during neural activities. This model is more suitable to estimate neural activities than using generic oscillator-like or map neurons without considering the spatial diffusive effect.

Free evolution in the Ginzburg-Landau equation and other complex diffusion equations

Abstract New ordinary differential equations (ODEs) for the evolution of spectral components are derived from the complex Ginzburg-Landau equation (CGLe) for one-dimensional spatial domains without boundaries (free evolution) and with one fixed boundary (semi-free evolution). For such evolution, a complex or imaginary diffusion term creates a tendency for waves to lengthen. This requires a novel ansatz and auxiliary condition that treat wavenumbers as time-varying. The ansatz consists of a discrete spatial Fourier transform modified with a time-dependent wavenumber for the peak spectral component. The wavenumbers of the other components are fixed relative to this wavenumber. The new auxiliary condition is the terminal condition for complex diffusion (after wavenumbers evolve to zero, they remain at zero). The derived free and semi-free ODEs are solved along characteristic lines located symmetrically about a fixed spatial point. Waves lengthen with time away from this point in both directions. Laboratory experiments on the formation of channel sandbars, theoretically described by the CGLe, show two regions whose evolutionary behaviour is qualitatively predicted by the free and semi-free evolution equations. This analysis applies to other time-dependent partial differential equations with complex or imaginary diffusion terms. New freely evolving solutions are derived for the complex heat equation and Schrödinger equation (linear and nonlinear).

Revised and Generalized Results of Averaging Principles for the Fractional Case

The averaging principle involves approximating the original system with a simpler system whose behavior can be analyzed more easily. Recently, numerous scholars have begun exploring averaging principles for fractional stochastic differential equations. However, many previous studies incorrectly defined the standard form of these equations by placing ε in front of the drift term and ε in front of the diffusion term. This mistake results in incorrect estimates of the convergence rate. In this research work, we explain the correct process for determining the standard form for the fractional case, and we also generalize the result of the averaging principle and the existence and uniqueness of solutions to fractional stochastic delay differential equations in two significant ways. First, we establish the result in Lp space, generalizing the case of p=2. Second, we establish the result using the Caputo–Katugampola operator, which generalizes the results of the Caputo and Caputo–Hadamard derivatives.

Comparative study on predicting turbulent kinetic energy budget using high-order upwind scheme and non-dissipative central scheme

The accurate computation of different turbulent statistics poses different requirements on numerical methods. In this paper, we investigate the capabilities of two representative numerical schemes in predicting mean velocities, Reynolds stress and budget of turbulent kinetic energy (TKE) in low Mach number flows. With concerns on numerical order of accuracy, dissipation and dispersion properties, a high-order upwind scheme with relatively good dispersion and dissipation and a second-order non-dissipative central scheme with perfect dissipation but poor dispersion are adopted for this comparative study. By carrying out a series of numerical simulations including Taylor-Green vortex, turbulent channel flow at Reτ=180\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re_{\ au }=180$$\\end{document} and turbulent flow over periodic hill at Reb=10595\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re_{b}=10595$$\\end{document}, it can be obtained that although the high-order upwind scheme lacks perfection of dissipation in high wave number range, it still demonstrates superior predictive capability compared with the second-order non-dissipative central scheme, especially with relatively coarse grids. Finally, by taking the high-order upwind scheme as a suitable selection for turbulence simulation, the turbulent flow over a 30P30N multi-element airfoil is investigated as an application study. After briefly comparing the simulated profiles and spectrum with reference experimental results as validation, the budget of TKE is analyzed to locate the dominant flow structures and regions. It is found that the production and dissipation terms behave in a “monopole” pattern in the locations with strong shears and wakes. Whereas the advection and diffusion terms show an “inward” pattern and an “outward” pattern, which indicate the spatial transport of TKE between the center of the shear layer and nearby locations.

The Effect of Grain Boundary Misorientation on Hydrogen Flux Using a Phase‐Field‐Based Diffusion and Trapping Model

Understanding hydrogen–grain boundary (GB) interactions is critical to the analysis of hydrogen embrittlement in metals. This work presents a mesoscale fully kinetic model to investigate the effect of GB misorientation on hydrogen diffusion and trapping using phase‐field‐based representative volume elements (RVEs). The flux equation consists of three terms: a diffusive term and two terms for high and low angle grain boundary (H/LAGB) trapping. Uptake simulations show that decreasing the grain size results in higher hydrogen content due to increasing the GB density. Permeation simulations show that GBs are high‐flux paths due to their higher enrichment with hydrogen. Since HAGBs have higher enrichment than LAGBs, due to their higher trap‐binding energy, they generally have the highest hydrogen flux. Nevertheless, the flux shows a convoluted behavior as it depends on the local concentration, alignment of GB with external concentration gradient as well as the GB network connectivity. Finally, decreasing the grain size resulted in a larger break‐through time and a larger steady‐state exit flux.

Effective diffusion constant of stochastic processes with spatially periodic noise.

We discuss the effective diffusion constant D_{eff} for stochastic processes with spatially dependent noise. Starting from a stochastic process given by a Langevin equation, different drift-diffusion equationscan be derived depending on the choice of the discretization rule 0≤α≤1. We initially study the case of periodic heterogeneous diffusion without drift, and we determine a general result for the effective diffusion coefficient D_{eff}, which is valid for any value of α. We study the case of periodic sinusoidal diffusion in detail, and we find a relationship with Legendre functions. Then we derive D_{eff} for general α in the case of diffusion with periodic spatial noise and in the presence of a drift term, generalizing the Lifson-Jackson theorem. Our results are illustrated by analytical and numerical calculations on generic periodic choices for drift and diffusion terms.

Relaxation of weakly collisional plasma: Continuous spectra, discrete eigenmodes, and the decay of echoes.

The relaxation of a weakly collisional plasma, which is of fundamental interest to laboratory astrophysical plasmas, can be described by the self-consistent Boltzmann-Poisson equations with the Lenard-Bernstein collision operator. We perform a perturbative (linear and second-order) analysis of the Boltzmann-Poisson equations and obtain exact analytic solutions which resolve some longstanding controversies regarding the impact of weak collisions on the continuous spectra, the discrete Landau eigenmodes, and the decay of plasma echoes. We retain both damping and diffusion terms in the collision operator throughout our treatment. We find that the linear response is a temporal convolution of two types of contribution: a continuum that depends on the continuous velocities of particles (crucial for the plasma echo), and another, consisting of discrete modes that are coherent modes of oscillation of the entire system. The discrete modes are exponentially damped over time due to collective effects or wave-particle interactions (Landau damping), as well as collisional dissipation. The continuum is also damped by collisions but somewhat differently than the discrete modes. Up to a collision time, which is the inverse of the collision frequency ν_{c}, the continuum decay is driven by the diffusion of particle velocities and is cubic exponential, occurring over a timescale ∼ν_{c}^{-1/3}. After a collision time, however, the continuum decay is driven by the collisional damping of particle velocities and diffusion of their positions and occurs exponentially over a timescale ∼ν_{c}. This slow exponential decay causes perturbations to damp the most on scales comparable to the mean free path but very slowly on larger scales. This establishes the local thermal equilibrium, which is the essence of the fluid limit. The long-term decay of the linear response is driven by the discrete modes on scales smaller than the mean free path but, on larger scales, is governed by a combination of the slowly decaying continuum and the least damped discrete mode. This slow exponential decay implies that the echo, which results from the interference of the continuum response to two subsequent pulses, is detectable even on scales comparable to the mean free path, as long as the second pulse is introduced within a few phase-mixing timescales after the first.

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.

Effect of magnetic entropy in the thermoelectric properties of Fe-doped Fe2VAl full-Heusler alloy

The effect of spin entropy on the transport of heat/charge carriers in the Fe-doped full-Heusler alloy Fe2+xVAl1-x with x = 0–0.1 has been studied through low-temperature magnetic and thermoelectric measurements. Magnetization (M) measurements confirm itinerant-electron weak-ferromagnetic behavior. A systematic increase of the magnetic transition temperature TC (from 40 K to 223 K) and of the saturation magnetization (from 0.13 to 0.41μB/Fe) with increasing Fe doping (from x = 0 to 0.1) is observed. Applying a magnetic field causes significant suppression of the Seebeck coefficient (S) and the entropy term (S/T) with a negative magnetoresistance near TC for all weak-ferromagnetic samples, demonstrating a clear effect of spin fluctuations. Analyzing M(T) and S(T), we rule out sizeable magnon drag contributions. A large spin fluctuations-induced enhancement in the thermoelectric power factor PF of about 18 % is achieved for x = 0.1 near TC when compared to measurements in a magnetic field of 7 T. The actual improvement in PF is even much higher, as the S shows a significant enhancement (about 34 %) compared to the estimated diffusion term of S(T) at TC. The number of point defects also increases with Fe doping, causing a significant reduction of the lattice thermal conductivity. This study demonstrates the role of spin fluctuations in enhancing the thermopower/thermoelectric performance of Fe-doped Fe2VAl and opens a vista for the strategy's applicability for various thermoelectric materials.

Optimization of Low-Pressure Turbine Blade by Means of Fine Inspection of Loss Production Mechanisms

Abstract In this study, Proper Orthogonal Decomposition (POD) was applied to Large Eddy Simulations (LES) of two high-loaded low-pressure turbine cascades under unsteady inflow conditions to investigate entropy production within different parts of the blade passage. The terms for Turbulent Kinetic Energy (TKE) production, diffusion, and dissipation from the stagnation pressure transport equation were integrated across the computational domain. The POD-based method enables the decomposition of contributions from various coherent flow dynamics to TKE production and dissipation, such as the migration, bowing, tilting, and reorientation of incoming wake filaments, as well as the breakdown of streaky structures in the blade boundary layer and the formation of Von Karman vortices in the blade wake. This approach helps designers identify the dominant POD modes (i.e., turbulent flow structures) responsible for loss generation, understand their dynamics and locate where they primarily act. Using this optimization strategy, a new low-pressure turbine profile was designed. LES calculations on the optimized geometry showed that it is possible to design a higher-loaded profile with a lower global loss coefficient. The new loading distribution, characterized by stronger acceleration in the early part of the blade passage, results in reduced upstream wake migration losses and lower TKE production in the trailing edge wake zone due to early suction side boundary layer transition.

On weak and strong solutions of time inhomogeneous Itô’s equations with VMO diffusion and Morrey drift

We prove the existence of weak solutions of Itô’s stochastic time dependent equations with irregular diffusion and drift terms of Morrey spaces. Weak uniqueness (generally conditional) and a conjecture pertaining to strong solutions are also discussed. Our results are new even if the drift term vanishes.

Stability and bipartite synchronization of fractional-order coupled reaction–diffusion neural networks under unbalanced graph

This paper delves into the analysis of stability and bipartite synchronization (BS) in fractional-order coupled neural networks with reaction–diffusion terms (FORDNNs) under structurally unbalanced signed graphs. Central to our investigation is the novel observation in structurally unbalanced graphs without cycles, where specific nodes under particular structures are capable of achieving BS, while others maintain stability. Subsequently, for unbalanced graphs with negative rooted cycles, we present criteria that guarantee the stability of FORDNNs. Moreover, under structurally balanced graphs, sufficient conditions for achieving BS are established. A novel approach is introduced, wherein synchronization is attained without the presence of a leader node, marking a significant departure from conventional models. Finally, illustrative examples are provided, demonstrating the effectiveness and relevance of our findings.