Search Results (34)

Nanoscopic quantum dots exhibit discrete energy spectra and size- and shape-dependent thermal properties that cannot always be adequately described within the conventional Boltzmann–Gibbs statistical framework. In systems with strong confinement, finite size, and reduced symmetry, deviations from extensivity may emerge, affecting the occupation of energy levels and the resulting thermodynamic response. In this context, this work elucidates how GaAs quantum dot geometry, external electric fields, and nonextensive statistical effects jointly influence the thermal response of quantum dots with different geometries—cubic, cylindrical, ellipsoidal, and pyramidal. These energy levels are calculated by solving the Schrödinger equation under the effective mass approximation, employing the finite element method for numerical computation. These energy levels are then incorporated into an iterative numerical procedure to calculate the specific heat for different values of the nonextensivity parameter, thereby enabling exploration of both extensive (Boltzmann–Gibbs) and nonextensive regimes. The results demonstrate that the shape of the quantum dots strongly influences the energy spectrum and, consequently, the thermal properties, producing distinctive features such as Schottky-type anomalies and geometry-dependent shifts under an external electric field. In subextensive regimes, a discrete behavior in the specific heat emerges due to natural cutoffs in the accessible energy states. In contrast, in superextensive regimes, a smooth, saturation-like behavior is observed. These findings highlight the importance of geometry, external-field effects, and nonextensive statistics as complementary tools for tailoring the energy distribution and thermal response in nanoscopic quantum systems. Full article

We report calculations for electron–radon scattering using a complex relativistic optical potential method. The energy range of this study is 0–1000 eV, with results for the elastic (total, momentum-transfer and viscosity-transfer) cross section, summed discrete electronic-state integral excitation cross sections and electron-impact ionization cross sections presented. Here, we obtain our cross sections from a single theoretical relativistic calculation. Since radon is a heavy element, a relativistic treatment is very desirable. The electron transport coefficients are subsequently calculated for reduced electric fields ranging from 0.01 to 1000 Td, using a multi-term solution of Boltzmann’s equation. Full article

The dramatic increase in domestic and industrial waste over recent centuries has significantly polluted water bodies, threatening aquatic life and human activities such as drinking, recreation, and commerce. Understanding pollutant dispersion is essential for designing effective waste management systems, employing both experimental and computational techniques. Among Computational Fluid Dynamics (CFD) techniques, the Lattice Boltzmann Method (LBM) has emerged as a novel approach based on a discretized Boltzmann equation. The versatility and parallelization capability of this method makes it particularly attractive for fluid dynamics simulations using high-performance computing. Motivated by its successful application across various scientific disciplines, this study explores the potential of LBM to model pollutant mixing and dilution from outfalls into surface water bodies, focusing specifically on vertical dense jets in crossflow (JICF), a key scenario for the diffusion of brine from desalination plants. A full-LBM scheme is employed to model both the hydrodynamics and the transport of the saline concentration field, and Large Eddy Simulations (LES) are employed in the framework of LBM to reduce computational costs typically associated with turbulence modeling, together with a recursive regularization procedure for the collision operator to achieve greater stability. Several key aspects of vertical dense JICF are considered. The simulations successfully capture general flow characteristics corresponding to jets with varying crossflow parameter $u_{r}F$ and most of the typical vortical structures associated with JICF. Relevant quantities such as the terminal rise height, the impact distance, the dilution at the terminal rise height, and the dilution at the impact point are compared with experimental results and semi-empirical relations. The results show a systematic underestimation of these quantities, but the key trends are successfully captured, highlighting LBM’s promise as a tool for simulating wastewater dispersion in aquatic environments. Full article

The Fokker–Planck (FP) equation represents the drift and diffusive processes in kinetic models. It can also be regarded as a model for the collision integral of the Boltzmann-type equation to represent thermo-hydrodynamic processes in fluids. The lattice Boltzmann method (LBM) is a drastically simplified discretization of the Boltzmann equation for simulating complex fluid motions and beyond. We construct new two FP-based LBMs, one for recovering the Navier–Stokes equations for fluid dynamics and the other for simulating the energy equation, where, in each case, the effect of collisions is represented as relaxations of different central moments to their respective attractors. Such attractors are obtained by matching the changes in various discrete central moments due to collision with the continuous central moments prescribed by the FP model. As such, the resulting central moment attractors depend on the lower-order moments and the diffusion tensor parameters, and significantly differ from those based on the Maxwell distribution. The diffusion tensor parameters for evolving higher moments in simulating fluid motions at relatively low viscosities are chosen based on a renormalization principle. Moreover, since the number of collision invariants of the FP-based LBMs for fluid motions and energy transport are different, the forms of the respective attractors are quite distinct. The use of such central moment formulations in modeling the collision step offers significant improvements in numerical stability, especially for simulations of thermal convective flows under a wide range of variations in the transport coefficients of the fluid. We develop new FP central moment LBMs for thermo-hydrodynamics in both two and three dimensions, and demonstrate the ability of our approach to simulate various cases involving thermal convective buoyancy-driven flows especially at high Rayleigh numbers with good quantitative accuracy. Moreover, we show significant improvements in the numerical stability of our FP central moment LBMs when compared to other existing central moment LBMs using the Maxwell distribution in achieving high Peclet numbers for mixed convection flows involving shear effects. Full article

The cross-domain reentry unmanned flight vehicle passes through thin atmospheres and dense atmospheres when it comes across atmospheres in the near-space area. For the early transition regime, the classical macroscopic and mesoscopic approaches are either not accurate or computational too expensive. The hybrid macro-/mesoscopic method is proposed for simulating rarefied gas flows past a cross-domain reentry spheroid–cone unmanned flight vehicle in the present study. The R26 moment scheme is applied in the main flow from a macroscopic point of view, and the discrete velocity method (DVM) is used for solving the Boltzmann equation from a mesoscopic point of view. The simulation results show that the hybrid macro-/mesoscopic scheme is well-suited for non-equilibrium rarefied gas flows past a cross-domain reentry unmanned flight vehicle. The results obtained in this study are consistent with benchmark results, with a maximum density error of 9%. The maximum errors of the heat transfer coefficient and pressure coefficient are 2% and 4.6%, respectively. In addition, as the Knudsen number (Kn) becomes larger, the thickness of the shock layer at the head of the flight vehicle becomes thicker, and non-equilibrium effects become more critical for the aircraft. Since the Boltzmann–Shakhov equation has only been solved close to the wall of the spacecraft, the computational cost can be considerably saved. Full article

We present a new formulation of the central moment lattice Boltzmann (LB) method based on a minimal continuous Fokker-Planck (FP) kinetic model, originally proposed for stochastic diffusive-drift processes (e.g., Brownian dynamics), by adapting it as a collision model for the continuous Boltzmann equation (CBE) for fluid dynamics. The FP collision model has several desirable properties, including its ability to preserve the quadratic nonlinearity of the CBE, unlike that based on the common Bhatnagar-Gross-Krook model. Rather than using an equivalent Langevin equation as a proxy, we construct our approach by directly matching the changes in different discrete central moments independently supported by the lattice under collision to those given by the CBE under the FP-guided collision model. This can be interpreted as a new path for the collision process in terms of the relaxation of the various central moments to “equilibria”, which we term as the Markovian central moment attractors that depend on the products of the adjacent lower order moments and a diffusion coefficient tensor, thereby involving of a chain of attractors; effectively, the latter are nonlinear functions of not only the hydrodynamic variables, but also the non-conserved moments; the relaxation rates are based on scaling the drift coefficient by the order of the moment involved. The construction of the method in terms of the relevant central moments rather than via the drift and diffusion of the distribution functions directly in the velocity space facilitates its numerical implementation and analysis. We show its consistency to the Navier-Stokes equations via a Chapman-Enskog analysis and elucidate the choice of the diffusion coefficient based on the second order moments in accurately representing flows at relatively low viscosities or high Reynolds numbers. We will demonstrate the accuracy and robustness of our new central moment FP-LB formulation, termed as the FPC-LBM, using the D3Q27 lattice for simulations of a variety of flows, including wall-bounded turbulent flows. We show that the FPC-LBM is more stable than other existing LB schemes based on central moments, while avoiding numerical hyperviscosity effects in flow simulations at relatively very low physical fluid viscosities through a refinement to a model founded on kinetic theory. Full article

The Boltzmann equation with multiple-relaxation-time (MRT) collision operators has been widely employed in kinetic theory to describe the behavior of gases and liquids at the macro-level. Given the successful development of deep learning and the availability of data analytic tools, it is a feasible idea to try to solve the Boltzmann-MRT equation using a neural network-based method. Based on the canonical polyadic decomposition, a new physics-informed neural network describing the Boltzmann-MRT equation, named the network for MRT collision (NMRT), is proposed in this paper for solving the Boltzmann-MRT equation. The method of tensor decomposition in the Boltzmann-MRT equation is utilized to combine the collision matrix with discrete distribution functions within the moment space. Multiscale modeling is adopted to accelerate the convergence of high frequencies for the equations. The micro–macro decomposition method is applied to improve learning efficiency. The problem-dependent loss function is proposed to balance the weight of the function for different conditions at different velocities. These strategies will greatly improve the accuracy of the network. The numerical experiments are tested, including the advection–diffusion problem and the wave propagation problem. The results of the numerical simulation show that the network-based method can obtain a measure of accuracy at $O\left({10}^{-3}\right)$. Full article

This paper proposes the D2Q5 Lattice Boltzmann method (LBM) method, in two dimensions with five discrete lattice velocities, for simulating linear sound wave propagation in closed rooms. A second-order linear acoustic equation obtained from the LBM method was used as the model equation. Boundary conditions at the domain boundary use the bounce-back scheme. The LBM numerical calculation algorithm in this paper is relatively simpler and easy to implement. Parallelization with the GPU CUDA was implemented to speed up the execution time. The calculation results show that the use of parallel GPU CUDA programming can accelerate the proposed simulation 27.47 times faster than serial CPU programming. The simulation results are validated with analytical solutions for acoustic pulse reflected by the flat and oblique walls, the comparisons show very good concordance, and the D2Q5 LBM has second-order accuracy. In addition, the simulation results in the form of wavefront propagation images in complicated shaped rooms are also compared with experimental photographs, and the comparison also shows excellent concordance. The numerical results of the D2Q5 LBM are promising and also demonstrate the great capability of the D2Q5 LBM for investigating room acoustics in various complexities. Full article

In this work, the explicit boundary-condition-enforced immersed boundary method (EIBM) and the lattice Boltzmann flux solver (LBFS) are integrated into OpenFOAM to efficiently solve incompressible flows with complex geometries and moving boundaries. The EIBM applies the explicit technique to greatly improve the computational efficiency of the original boundary-condition-enforced immersed boundary method. In addition, the improved EIBM inherits the accurate interpretation of the no-slip boundary condition and the simple implementation from the original one. The LBFS uses the finite volume method to discretize the recovered macroscopic governing equations from the lattice Boltzmann equation. It enjoys the explicit relationship between the pressure and density, which avoids solving the pressure Poisson equation and thus saves much computational cost. Another attractive feature of the LBFS lies in its simultaneous evaluation of the inviscid and viscous fluxes. OpenFOAM, as an open-source CFD platform, has drawn increasing attention from the CFD community and has been proven to be a powerful tool for various problems. Thus, implementing the EIBM and LBFS into such a popular platform can advance the practical application of these two methods and may provide an effective alternative for complicated incompressible flow problems. The performance of the integrated solver in OpenFOAM is comprehensively assessed by comparing it with the widely used numerical solver in OpenFOAM, namely, the Pressure-Implicit with Splitting of Operators (PISO) algorithm with the IBM. A series of representative test cases with stationary and moving boundaries are simulated. Numerical results confirm that the present method does not have any streamline penetration and achieves the second-order accuracy in space. Therefore, the present method implemented in the open-source platform OpenFOAM may have good potential and can serve as a powerful tool for practical engineering problems. Full article

How to improve the computational efficiency of flow field simulations around irregular objects in near-continuum and continuum flow regimes has always been a challenge in the aerospace re-entry process. The discrete velocity method (DVM) is a commonly used algorithm for the discretized solutions of the Boltzmann-BGK model equation. However, the discretization of both physical and molecular velocity spaces in DVM can result in significant computational costs. This paper focuses on unlocking the key to accelerate the convergence in DVM calculations, thereby reducing the computational burden. Three versions of DVM are investigated: the semi-implicit DVM (DVM-I), fully implicit DVM (DVM-II), and fully implicit DVM with an inner iteration of the macroscopic governing equation (DVM-III). In order to achieve full implicit discretization of the collision term in the Boltzmann-BGK equation, it is necessary to solve the corresponding macroscopic governing equation in DVM-II and DVM-III. In DVM-III, an inner iterative process of the macroscopic governing equation is employed between two adjacent DVM steps, enabling a more accurate prediction of the equilibrium state for the full implicit discretization of the collision term. Fortunately, the computational cost of solving the macroscopic governing equation is significantly lower than that of the Boltzmann-BGK equation. This is primarily due to the smaller number of conservative variables in the macroscopic governing equation compared to the discrete velocity distribution functions in the Boltzmann-BGK equation. Our findings demonstrate that the fully implicit discretization of the collision term in the Boltzmann-BGK equation can accelerate DVM calculations by one order of magnitude in continuum and near-continuum flow regimes. Furthermore, the introduction of the inner iteration of the macroscopic governing equation provides an additional 1–2 orders of magnitude acceleration. Such advancements hold promise in providing a computational approach for simulating flows around irregular objects in near-space environments. Full article

An electrically controlled rotor (ECR) is a kind of swashplateless rotor that implements the primary control via the trailing-edge flap system instead of a swashplate and demonstrates great potential in vibration reduction and noise alleviation. In this paper, the mesoscopic numerical simulation method known as the lattice Boltzmann method (LBM) is employed to investigate the aerodynamic characteristics of an ECR. In the LBM, the discretized Boltzmann transport equation is solved to simulate the macroscopic motion of the fluid, and the D3Q27 model is applied for this study. The effects of the flap deflection on the ECR aerodynamic characteristics can be accurately included with the appropriate refined wall lattice resolution. On this basis, the adaptive wake-refinement strategy is applied to track the evolution of the wake and adequately capture details of the wake structure in the wake flow field. Based on this method, an aerodynamic analysis model for the ECR can be established on the XFlow simulation platform. The aerodynamic analysis model is validated, and the results indicate that the LBM can accurately capture the details of the rotor flow field and calculate blade aerodynamic load, as well as predict the downwash of the rotor. Therefore, based on this model, the ECR aerodynamic characteristics under hovering and forward flight conditions are analyzed, and the effects of the flap deflection on the wake structure, induced inflow, and disc load can be captured. The results indicate that a relatively large flap deflection required to trim the rotor will cause the additional intense flap wake vortex in the ECR wake flow field, apart from the concentrated vorticity at the blade tip and root demonstrated in the conventional rotor wake flow field, and thus significantly change the distributions of the disc-induced inflow and aerodynamic load. Full article

In the present paper, a novel approach for image denoising based on the numerical solution to the nonlinear diffusion equation is proposed. The Perona–Malik-type equation is solved by employing a hybrid lattice Boltzmann model with five discrete velocities. In this method, the regions with large values of the diffusion coefficient are modeled with the lattice Boltzmann scheme for which hyper-viscous defects are reduced, while other regions are modeled with the conventional lattice Boltzmann model. The new method allows us to solve Perona–Malik-type equations with relatively large time steps and good accuracy. In numerical experiments, the removal of salt and pepper, speckle and Gaussian noise is considered. For salt and pepper noise, the novel scheme yields better peak signal-to-noise ratios in image denoising problems compared to the standard lattice Boltzmann approach. For certain non-small values of time steps, the novel model shows better results for speckle and Gaussian noise on average. Full article

This paper is concerned with the lattice Boltzmann (LB) method for a class of time fractional partial differential equations (FPDEs) in the Caputo sense. By utilizing the properties of the Caputo derivative and discretization in time, FPDEs can be approximately transformed into standard partial differential equations with integer orders. Through incorporating an auxiliary distribution function into the evolution equation, which assists in recovering the macroscopic quantity u, the LB model with spatial second-order accuracy is constructed. The numerical experiments verify that the numerical results are in good agreement with analytical solutions and that the accuracy of the present model is better than the previous solutions. Full article

Physics-Informed Neural Networks (PINNs) improve the efficiency of data utilization by combining physical principles with neural network algorithms and thus ensure that their predictions are consistent and stable with the physical laws. PINNs open up a new approach to address inverse problems in fluid mechanics. Based on the single-relaxation-time lattice Boltzmann method (SRT-LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator, the PINN-SRT-LBM model is proposed in this paper for solving the inverse problem in fluid mechanics. The PINN-SRT-LBM model consists of three components. The first component involves a deep neural network that predicts equilibrium control equations in different discrete velocity directions within the SRT-LBM. The second component employs another deep neural network to predict non-equilibrium control equations, enabling the inference of the fluid’s non-equilibrium characteristics. The third component, a physics-informed function, translates the outputs of the first two networks into physical information. By minimizing the residuals of the physical partial differential equations (PDEs), the physics-informed function infers relevant macroscopic quantities of the flow. The model evolves two sub-models that are applicable to different dimensions, named the PINN-SRT-LBM-I and PINN-SRT-LBM-II models according to the construction of the physics-informed function. The innovation of this work is the introduction of SRT-LBM and discrete velocity models as physical drivers into a neural network through the interpretation function. Therefore, the PINN-SRT-LBM allows a given neural network to handle inverse problems of various dimensions and focus on problem-specific solving. Our experimental results confirm the accurate prediction by this model of flow information at different Reynolds numbers within the computational domain. Relying on the PINN-SRT-LBM models, inverse problems in fluid mechanics can be solved efficiently. Full article

A variety of particle-based methods have been developed for the purpose of computationally modelling processes that involve, for example, complex topological changes of interfaces, significant plastic deformation of materials, fluid flow in conjunction with heat transfer and phase transformation, flow in porous media, granular flow, etc. Being different from the conventional methods that directly solve related governing equations using a computational grid, the particle-based methods firstly discretize the continuous medium into discrete pseudo-particles in mathematics. The methods then mathematically solve the governing equations by considering the local interaction between neighbouring pseudo-particles. Such solutions can reflect the overall flow, deformation, heat transfer and phase transformation processes of the target materials at the mesoscale and macroscale. This paper reviews the fundamental concepts of four different particle-based methods (lattice Boltzmann method—LBM, smoothed particle hydrodynamics—SPH, discrete element method—DEM and particle finite element method—PFEM) and their application in computational modelling research on welding, casting and additive manufacturing. Full article