Second-order Boundary Condition Research Articles

Analysis of energy dissipation in hyperbolic problems influenced by internal and boundary control mechanisms

ABSTRACT This article primarily focuses on the rational stabilisation of the wave equation, supplied with a second-order dynamical boundary condition of hyperbolic type, while considering an additional internal damping mechanism within the specified ring. To achieve rational decay rates of the associated energy, it is imperative to exponentially stabilise a portion of the domain using a global Ingham's-type estimate. This paper will subsequently illustrate how this partially localised exponential stabilisation, combined with a Bessel analysis, leads to a rational decrease in the overall energy of the system considered.

Analysis of Fluid Flows in Bounded Domain with Particular Shape of a Cavity using Lattice Boltzmann Method

Background: The present work numerically models the incompressible, continuous phase, viscous flow of Newtonian fluid flow in a bounded domain of two-dimensional cavity that is driven by walls and contains grooves in the shape of squares on the lower wall. With the help of the mesoscopic lattice Boltzmann method (LBM) and D2Q9 square lattice model, simulation results are found stable and reliable. The flow physics of the problem by varying Reynolds number, the height and quantity of lower wall grooves, and other fluid flow characteristics within the bounded domain are studied in detail. It is seen that the effects of the groove heights and wavelengths on the fluid flow are structured within the bounded domain. The study is performed from low Re = 100 to high Re = 3200, with minimum two and maximum four-wavelength grooves evaluated on the bottom surface, each having a height of low 0.25 and high 0.75. Additionally, a thorough discussion of complicated vortex dynamics is provided regarding the input parameters and geometry. Objective: The current study aims to use mesoscopic LBM to analyze incompressible viscous fluid flows on complex geometries other than rectangular shapes. Methods: Mesoscopic approach of kinetic theory-based Lattice Boltzmann method (LBM) is implemented in the current work. The popular Single Relaxation Time Lattice Boltzmann method with D2Q9 square lattice model and second-order accurate boundary condition is adopted for the current study. Results: The numerical approach of LBM is used to simulate fluid flows in a 2D bounded domain with grooved bottom surfaces. The influence of different factors, such as the height of bottom-wall surface grooves, flow Reynolds number, and wavelength of these grooves on flow patterns, is then investigated. Conclusion: The numerical study of the bounded domain is considered, and the Reynolds number is varied from 100 to 3200, with two and four-wavelength grooves evaluated on the bottom surface, each having a height of 0.25 and 0.75. The impacts on the flow pattern both within and slightly above the grooves of the computational findings for different Reynolds numbers, groove heights, and groove wavelengths are evaluated. As the Reynolds number rises, the mixing phenomenon of fluid is shown to flow more quickly in the wall-driven enclosures.

A second-order constitutive theory for polyatomic gases: theory and applications

In the classical irreversible thermodynamics (CIT) framework, the Navier–Stokes–Fourier constitutive equations are obtained so as to satisfy the entropy inequality, by and large assuming that the entropy flux is equal to the heat flux over the temperature. This article is focused on the derivation of second-order constitutive equations for polyatomic gases; it takes the basis of CIT, but most importantly, allows up to quadratic nonlinearities in the entropy flux. Mathematical similarities between the proposed model and the classic Stokes–Laplace equations are exploited so as to construct analytic/semi-analytic solutions for the slow rarefied gas flow over different shapes. A set of second-order boundary conditions are formulated such that the model's prediction for the drag force is in excellent agreement with the experimental data over the whole range of Knudsen numbers. We have also computed the normal shock structure in nitrogen for Mach ${Ma} \lesssim 4$ . A very good agreement was observed with the kinetic theory, as well as with the experimental data.

A General Single-Node Second-Order Dirichlet Boundary Condition for the Convection–Diffusion Equation Based on the Lattice Boltzmann Method

A general single-node second-order Dirichlet boundary condition for curved boundaries for the convection–diffusion equation based on the lattice Boltzmann method has been developed. The boundary condition simply utilizes the bounce back rule for the temperature distribution and linear interpolation. The developed boundary condition is quite general and stable. The asymptotic analysis indicates that the proposed boundary scheme is of second-order accuracy. The comparison in terms of L2-norm between the simulation results and the analytical solutions confirms that it is indeed second-order accurate for 2D and 3D symmetric flows comprising straight and curved boundaries. Moreover, the Nusselt numbers for a spherical particle in uniform flow obtained by current simulations agree very well with the predictions of empirical correlations and the data of previous direct numerical simulations with a maximal relative deviation within 8%.

Evaluation of Nusselt Number for a Flow in a Parallel Plates Using Magnetohydrodynamics Second-Order Slip Model

Abstract This study is the examination of magnetohydrodinamics (MHD) flow and heat transfer between microparallel plates with constant wall heat flux using two different boundary condition coefficients, widely known in the literature, in a second-order slip flow and temperature jump boundary condition model. The stream of incompressible viscous liquid between rigid microtwin plates with electrical conductivity was assumed to be steady, laminar, hydrodynamically and thermally fully improved. The effects of dilution, viscous dissipation, and axial transmission, which are significant at the microscale, are also contained in the analysis. Momentum and energy equations are solved analytically under second-order slip velocity and temperature jump boundary conditions. Closed type answers for the temperature field and the fully developed Nusselt number are derived as a function of the magnetic parameter (MHD), Knudsen number and modified Brinkman number. Particular attention has been paid to the effects of the magnetic field on the second-order slip flow conditions. The use of the second-order boundary condition coefficients proposed in the study varies according to the proposed boundary conditions and tends to increase and decrease the heat transfer. Otherwise, MHD increases the heat transfer for both boundary conditions.

Chemical entropy generation and second-order slip condition on hydrodynamic Casson nanofluid flow embedded in a porous medium: a fast convergent method

The chemical entropy generation analysis is an approach to optimize the performance of different thermal systems by investigating the related irreversibility of the system. The influences of second-order slip with the chemical reaction on the boundary layer flow and heat transfer of a non-Newtonian nanofluid in a non-Darcian porous medium have been investigated numerically. Simultaneous solutions are presented for first and second-order velocity slips. The second-order boundary conditions serve as a closure of a system of the continuity, transport, and energy differential equations. The current work differs from the previous studies in the application of a new second-order slip velocity model. The Casson fluid model is applied to characterize the non-Newtonian fluid behavior. The effect of the second slip parameter on the present physical parameters was discussed through graphs and it was found that this type of slip is a very important one to predict the investigated physical model. The present study provides two fast convergent methods on the semi-infinite interval, namely Chebyshev collocation method and optimal homotopy analysis method are used to analyze the fluid flow, heat, and mass transport. Compared with available analytical and numerical solutions, current methods are effective, quickly converging, and with great accuracy. It was shown that the account for the second-order terms in the boundary conditions noticeably affects the fluid flow characteristics and does not influence on the heat transfer characteristics.

A Comparative Study of 3d Cumulant and Central Moments Lattice Boltzmann Schemes with Interpolated Boundary Conditions for the Simulation of Thermal flows in High Prandtl Number Regime

Thermal flows characterized by high Prandtl number are numerically challenging as the transfer of momentum and heat occurs at different time scales. To account for very low thermal conductivity and obey the Courant-Friedrichs-Lewy condition, the numerical diffusion of the scheme has to be reduced. As a consequence, the numerical artefacts are dominated by dispersion errors commonly known as wiggles. In this study, we explore possible remedies in the framework of the lattice Boltzmann method by applying novel collision kernels, lattices with a large number of discrete velocities, namely D3Q27, and second-order boundary conditions. For the first time, the cumulant-based collision operator is utilised to simulate both the hydrodynamic and the thermal field. Alternatively, the advected field is computed using the central moments' collision operator. Different relaxation strategies have been examined to account for additional degrees of freedom introduced by a higher-order lattice. To validate the proposed kernels for a pure advection-diffusion problem, the numerical simulations are compared against an analytical solution of a Gaussian hill. The structure of the numerical dispersion is shown by simulating advection and diffusion of a square indicator function. Finally, a study of steady forced heat convection from a confined cylinder is performed and compared against a FEM solution. It has been found, that the relaxation scheme of the advected field must be adjusted to profit from lattice with a larger number of discrete velocities. Obtained results show clearly that it is not sufficient to assume that only the first-order central moments/cumulants contribute to solving the macroscopic advection-diffusion equation. In the case of central moments, the beneficial effect of the two relaxation time approach is presented.

Second-order accurate Dirichlet boundary conditions for linear nonlocal diffusion problems

Second-order accurate Dirichlet boundary conditions for linear nonlocal diffusion problems

Gaussian process regression constrained by boundary value problems

We develop a framework for Gaussian processes regression constrained by boundary value problems. The framework may be applied to infer the solution of a well-posed boundary value problem with a known second-order differential operator and boundary conditions, but for which only scattered observations of the source term are available. Scattered observations of the solution may also be used in the regression. The framework combines co-kriging with the linear transformation of a Gaussian process together with the use of kernels given by spectral expansions in eigenfunctions of the boundary value problem. Thus, it benefits from a reduced-rank property of covariance matrices. We demonstrate that the resulting framework yields more accurate and stable solution inference as compared to physics-informed Gaussian process regression without boundary condition constraints.

Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems

We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which is based on a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. The proposed method allows to use large time steps, while avoiding the solution of large linear systems, which would be required by implicit time discretization techniques. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. A novel extrapolation technique is proposed to enforce second-order accurate Dirichlet boundary conditions. We include a theoretical analysis of the scheme, along with numerical experiments which demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations.

On a Class of Isoperimetric Constrained Controlled Optimization Problems

In this paper, we investigate the Lagrange dynamics generated by a class of isoperimetric constrained controlled optimization problems involving second-order partial derivatives and boundary conditions. More precisely, we derive necessary optimality conditions for the considered class of variational control problems governed by path-independent curvilinear integral functionals. Moreover, the theoretical results presented in the paper are accompanied by an illustrative example. Furthermore, an algorithm is proposed to emphasize the steps to be followed to solve a control problem such as the one studied in this paper.

A general single-node second-order boundary condition for the lattice Boltzmann method

In this work, we propose a general single-node nonslip hydrodynamic boundary condition for the lattice Boltzmann method. The construction of the boundary scheme is the combination of the bounce back rule for the nonequilibrium part of the density distribution and linear interpolation. The proposed boundary condition is very simple, universal, stable, and accurate. The asymptotic analysis of the newly proposed boundary condition confirms that is of second-order accuracy. The numerical experiments demonstrate that the boundary condition is indeed second-order accurate for both straight and curved boundaries.

Numerical study on the second-order hydrodynamic force and response of an elastic body – In bichromatic waves

The second-order hydrodynamic force and response of an elastic body in bichromatic waves are studied by using higher-order boundary element method (HOBEM). To solve the boundary value problem, the free-surface wave Green function is adopted in the boundary integral equation and the discretization is conducted by quadratic shape function. In the boundary condition, the normal vector variation on the body surface is re-defined to consider both rigid and elastic body motions. It is invoked especially for the derivation of second-order body boundary condition and several generalized forces in zero forward speed condition. In the second-order quantities, the second-order velocity potential force is obtained by using indirect formulation in the generalized mode and the contribution of each component is investigated. The validation for the second-order forces is first conducted by comparing with published data. The excitation force on the rigid body motion of a hemisphere is calculated and showed a good agreement with other semi-analytic and numerical results. Using simplified structural model, several vertical bending modes of an elastic hemisphere are considered as a numerical test. Second-order hydrodynamic force and response for the two-node vertical bending are obtained and properties of second-order quantities are confirmed.

The Neumann problem in plane deformations of a micropolar elastic solid with micropolar surface effects

We consider the Neumann problem in a theory of plane micropolar elasticity incorporating micropolar surface effects. The incorporation of surface elasticity utilizes the Eremeyev–Lebedev–Altenbach shell model, leading to a set of second-order boundary conditions describing the separate micropolar elasticity of the surface. The Neumann problem is of particular interest, since the question of solvability is complicated by the fact that the corresponding systems of homogeneous singular integral equations admit nontrivial solutions that affect the solvability of both the interior and exterior Neumann boundary value problems. We overcome this difficulty by constructing integral representations of the solutions based on specifically constructed auxiliary matrix functions leading to uniqueness and existence theorems in appropriate classes of smooth matrix functions.

A lattice Boltzmann study of rarefied gaseous flow with convective heat transfer in backward facing micro-step

Rarefied pressure-driven gaseous flow with heat transfer in a microchannel with a backward facing micro-step is investigated in this paper using the lattice Boltzmann method (LBM) in slip and transition flow regimes. In a novel approach, a two-relaxation-time LB equation is used to solve the flow velocity and the single-relaxation-time to handle the heat transfer. The asymmetric relaxation time is determined by equating the analytical second-order slip velocity boundary condition and the slip velocity obtained from applying the implemented bounce back specular boundary condition in the LBM. A second-order implicit temperature jump boundary condition is also implemented to capture the rarefaction effect on the fluid temperature at walls. Velocity slip, temperature jump, centerline temperature, and Nusselt number variations are evaluated for channels with and without the micro-step for a wide range of the Knudsen number. Effects of the micro-step on the rarefied gaseous flow and convective heat transfer are evaluated and discussed. The numerical model is verified by comparing with direct simulation Mont Carlo results.

The hydrodynamic FORCE of fluid–structure interaction interface in lattice Boltzmann simulations

The hydrodynamic force (HF) evaluation plays a critical role in the numerical simulation of fluid–structure interaction (FSI). By directly using the distribution functions of lattice Boltzmann equation (LBE) to evaluate the HF, the momentum exchange algorithm (MEA) has excellent features. Particularly, it is independent of boundary geometry and avoids integration on the complex boundary. In this work, the HF of lattice Boltzmann simulation (LBS) is evaluated by using the MEA. We conduct a comparative study to evaluate two lattice Boltzmann models for constructing the flow solvers, including the LBE with single-relaxation-time (SRT) and multiple-relaxation-time (MRT) collision operators. The second-order boundary condition schemes are used to address the curve boundary. The test case of flow past a cylinder asymmetrically placed in a channel is simulated. Comparing the numerical solutions of Lattice Boltzmann method (LBM) with those of Navier–Stokes equations in the literature, the influence of collision relaxation model, boundary conditions and lattice resolution is investigated. The results demonstrate that the MRT-LB improves the numerical stability of the LBM and the accuracy of HF.

Heat transfer and hydrodynamics of slip confusor flow under second-order boundary conditions

The paper focused on an analytical analysis of the main features of heat transfer in incompressible steady-state flow in a microconfusor with account for the second-order slip boundary conditions. The second-order boundary conditions serve as a closure of a system of the continuity, transport, and energy differential equations. As a result, novel solutions were obtained for the velocity and temperature profiles, as well as for the friction coefficient and the Nusselt number. These solutions demonstrated that an increase in the Knudsen number leads to a decrease in the Nusselt number. It was shown that the account for the second-order terms in the boundary conditions noticeably affects the fluid flow characteristics and does not influence on the heat transfer characteristics. It was also revealed that flow slippage effects on heat transfer weaken with an increase in the Prandtl number.

A Second-Order Boundary Condition Capturing Method for Solving the Elliptic Interface Problems on Irregular Domains

A second-order boundary condition capturing method is presented for the elliptic interface problem with jump conditions in the solution and its normal derivative. The proposed method is an extension of the work in Liu et al. (J Comput Phys 160(1):151–178, 2000) to a higher order. The motivation of proposed method is that the approximated value at the interface can be reconstructed by proper interpolation based on the level set representation from Gibou et al. (J Comput Phys 176(1):205–227, 2002). A second-order accurate method is constructed, both in the solution and its gradient, using second-order finite difference approximation. Several numerical results demonstrate that the proposed method is indeed second-order accurate in the solution and its gradient in the $$L^{2}$$ and $$L^{\infty }$$ norms.

Electrical characteristics of n-ladder network with external load

This paper addresses theoretically and methodologically the electrical characteristics (including node voltage and equivalent resistance) of a class of n-ladder resistor networks with external load. The authors establish, with node voltage as a variable, the second-order difference equation model and boundary condition equation model based on Kirchhoff’s law. They propose the general and specific methods of solution for different equations and at once obtain the voltage formula of any node in the n-ladder network with external load and the equivalent resistance formula between any two nodes as well. Special cases in the conclusion section are analyzed and discussed. In particular, they find the oscillation and resonance characteristics of the LC network under different conditions by carrying out an analysis and research into the equivalent complex impedance characteristics of the LC network.

Heat Transfer in Porous Microchannels with Second-Order Slipping Boundary Conditions

The paper presents results of a study of forced convection in a vertical flat and circular microchannels incorporating porous medium under slip boundary conditions of the first and second orders. The problem was solved analytically and compared with numerical simulations based on the lattice Boltzmann method. Effects of porosity and slip velocity on velocity and temperature profiles were investigated. Behavior of the normalized Nusselt number as a function of the Knudsen and Prandtl numbers, as well as the parameter M characterizing porosity of the medium in the microchannel, was also elucidated. Computations indicated that a decrease in porosity (an increase in the parameter M) causes a decrease in the velocity and temperature jumps on the wall, which contributes to the increase in the Nusselt number. Considering the effect of the second-order slip boundary conditions leads to the decrease in the velocity jump on the wall, when the coefficient A2 (this coefficient considers the second-order slip boundary conditions) changes from negative to positive values. The heat transfer rate at high Prandtl numbers increases with the increasing Knudsen number, because of the improved thermal interaction of the flow with the channel wall. Given the second-order boundary conditions, the effect of parameter A2 was not observed at small Prandtl numbers (Pr ≤ 1). For A2 0 the normalized Nusselt number decreases in comparison with the case of A2 = 0 (the first-order boundary conditions). The results by the analytical solution and the lattice Boltzmann method are in a good agreement with each other for A2 ≥ 0, with the differences ≤ 1%. For A2 < 0, the differences between two models are more noticeable. For A2 ≥ 0, predictions by the analytical model lie higher than the numerical results, whereas for A2 < 0, numerical simulations exceed the analytical solution.