Fixed-grid Method Research Articles

Numerical method research and characteristics analysis on frozen start-up process of high-temperature heat pipe

Future space exploration technology requires a long-life and reliable power source that is not reliant on solar energy. Space micro-reactors are able to meet this need, with Heat Pipe Cooled Reactors (HPR) emerging as a notable type of space micro-reactor that has attracted widespread attention in recent years. The HPR utilizes high-temperature alkali metal heat pipes for heat transfer, which presents certain complexities due to the solid state of the alkali metals working medium at room temperature. This results in a three-phase transition during the high-temperature heat pipes start-up process, which significantly impacts the heat transfer characteristics and dynamic behavior of the HPR start-up process. Consequently, thorough research is necessary in this area. Numerical simulation is a crucial tool that can effectively analyze, predict, and guide experiments. This article utilizes the Finite Volume Method (FVM) to develop a simulation code for high-temperature heat pipe frozen start-up. Various physical models are integrated to describe different components of the heat pipe: the container is represented by a two-dimensional axisymmetric heat conduction equation, the wick region utilizes a Fixed Grid Method (FGM) to depict the melting process of the medium, and the vapor channel is described through a two-dimensional axisymmetric compressible laminar flow. The wick region and vapor channel are coupled through the evaporation and condensation of the medium. For the vapor channel, numerical methods such as SIMPLE and PISO are used for solving. Adaptive time step and OpenMP acceleration are employed in the code to enhance computational efficiency. Finally, by comparing the calculated results with experimental data, the feasibility and accuracy of the code are assessed, highlighting special phenomena during the start-up process. The findings confirm that the developed code accurately predicts parameter changes during start-up, and can serve as a heat pipe analysis module for multi-physics coupling analysis of HPR.

Tackling of hydraulic cavitation in pressurized pipe flow using high- or low-density polyethylene penstock and short-section

Abstract Recent studies have proved that the utilization of polyethylene (PE) short-section or penstock is a promising water hammer control tool. However, the interplay between the magnitude attenuation and the phase offset of pressure-wave oscillations remains challenging. This study aimed at inspecting the capacity of a dual PE penstock/short-section-based control technique, with regard to the aforementioned interplay. In this technique, a PE penstock was lumped to the transient initiating zone of the main pipe and a short-section of the counter extremity of the pipe was replaced with PE. The transient pressure-wave behavior in a gravitational viscoelastic pipe involving cavitation was described by the extended 1D water hammer equations embedding the Vitkovsky and Kelvin–Voigt add-ons. The numerical solution was performed by the fixed grid method of characteristics. The high- (HDPE) and low-density (LDPE) were demonstrated in this study. Analysis revealed that upgrading techniques based on LDPE enabled a desirable tradeoff between the magnitude attenuation and the phase offset of pressure-wave oscillations. Particularly, the dual penstock/short-section specific upgrading technique allowed a more important attenuation magnitude of pressure peak (or crest), and led to a similar expansion of the wave oscillation period. Furthermore, results evidenced that the proposed technique outperformed the renewal of the original piping system.

Simulation of Nonlinear Free Surface Waves using a Fixed Grid Method

The simulation of nonlinear surface waves is of significant importance in safety studies of fluid containers and reservoirs. In this paper, nonlinear free surface flows are simulated using a fixed grid method which employs local exponential basis functions (EBFs). Assuming the flow to be inviscid and irrotational, the velocity potential Laplace’s equation is spatially discretized and solved by considering the nonlinear Bernoulli’s equation for irrotational flow as the boundary condition on the free surface. The nonlinear boundary conditions are imposed through a semi-implicit iterative time marching. The fixed grid feature of the method, based on a Lagrangian description of fluid flow, allows for retaining the portion of the discretization performed in the first time step for the bulk of the fluid. Thus, the portion which pertains to the regions near the moving boundaries is reprocessed during the time marching. The accuracy and efficiency of the existing solution is shown by simulating various problems such as liquid sloshing induced by external excitation of the reservoir or initial deformed shape of liquid, seiche phenomena and solitary wave propagation in a basin with constant depth or with a step, and comparing the results with those which are analytically available or those from available codes such as Abaqus. The proposed method shows far better stability of the results when compared with those of Abaqus which sometimes exhibit divergence after a relatively large number of time steps. For instance, in the propagation of the considered solitary wave in an infinite-like domain problem, the wave height is calculated by the maximum error of 1.6% and 9% using the present method and Abaqus, respectively.

Water Hammer in Steel–Plastic Pipes Connected in Series

This paper experimentally and numerically investigates the water hammer phenomenon in serially connected steel and HDPE pipes with different diameters. The aim of the laboratory tests was to obtain the time history of the pressure head at the downstream end of the pipeline system. Transient tests were conducted on seven different pipeline system configurations. The experimental results show that despite the significantly smaller diameter of the HDPE pipe compared to the steel pipe, introducing an HDPE section makes it possible to suppress the valve-induced pressure surge. By referring to the results of the experimental tests conducted, the comparative numerical calculations were performed using the fixed-grid method of characteristics. To reproduce pressure wave attenuation in a steel pipe, Brunone-Vitkovský instant acceleration-based model of unsteady friction was used. To include the viscoelastic behavior of the HDPE pipe wall, the one-element Kelvin–Voigt model was applied. By calibrating the unsteady friction coefficient and creep parameters, satisfactory agreement between the calculated and observed data was obtained. The calibrated values of parameters for a single experimental test were introduced in a numerical model to simulate the remaining water hammer runs. It was demonstrated that using the same unsteady friction coefficient and creep parameters in slightly different configurations of pipe lengths can be effective. However, this approach fails to reliably reproduce the pressure oscillations in pipeline systems with sections of significantly different lengths.

A second order accurate fixed-grid method for multi-dimensional Stefan problem with moving phase change materials

This paper proposes a front-tracking fixed grid method for solving Stefan problem with moving phase change materials in any arbitrary (irregular) bounded domains. To achieve this objective, the governing partial differential equations are transformed, using a curvilinear coordinate transformation, in to a fixed rectangular domain, at each time level and an alternate direction implicit (ADI) scheme is used to solve the resultant differential system. The proposed scheme is consistent, unconditionally stable and also produces second-order accurate results. Many numerical experiments are conducted, in both one and two phase environments, to validate the proposed method. All these validations confirmed the theoretical accuracy of the produced results which are presented in the form of error graphs and tables in each case. Importance is given to understand the influence of the physical parameters like Peclet number, Stefan number, material velocity, latent heat, and thermal conductivity, etc., on the rate of change of phase and observed a continuous enhancement of the same with in the chosen range of the parameters. Also noticed a faster movement of the interface with the increase in these parameters in the direction of melting or freezing.

Modelling and analysis of initial icing roughness with fixed-grid enthalpy method based on DPM-VOF algorithm

Ice particles could form under the continuous impingement of incoming supercooled droplets in icing conditions, which will change the surface roughness to enhance the further heat and mass transfer during icing process. A fixed-grid porous enthalpy method based on the improved Discrete Phase Model (DPM) and Volume of Fluid (VOF) integrated algorithm is developed to solve the multiphase heat transfer problem to give more detailed demonstration of the formation of initial ice roughness. The algorithms to determine the criterion of transformation from DPM to VOF and the allocation of source items during transformation are improved to the general DPM-VOF algorithm. Two verification cases, namely two glycerine-solution droplets impact and single droplet freeze, are conducted to verify the accuracy and reliability of the enthalpy-DPM-VOF method, where the simulation results match well with experiment phenomena. Ice roughness on a NACA0012 airfoil is precisely captured and the effects on convective heat transfer characteristics are preliminarily revealed. The results illustrate that the enthalpy-DPM-VOF method could successfully capture the characteristics of motion and the phase change process of droplet, as well as balance the calculation accuracy and efficiency.

Assessment of a numeric variational method for the solution of confined multielectron atoms.

The confinement of atoms in impenetrable spherical boxes at the Hartree-Fock level was evaluated using a non-uniform fixed-grid variational method defined by a q-exponential. Applications from He to Ne in the ground state and some of them in excited states demonstrated that the method depends essentially on the boundary conditions to produce results as accurate as those reported in the literature with a relatively low number of integration points. An expression was defined for the virial theorem for atoms confined by spherical boxes and the confinement effect on the virial ratio evaluated. The results showed that the confinement of atoms results in much more significant changes to the kinetic than to the potential energy.

A method for solving heat transfer with phase change in ice or soil that allows for large time steps while guaranteeing energy conservation

Abstract. The accurate simulation of heat transfer with phase change is a central problem in cryosphere studies. This is because the non-linear behaviour of enthalpy as function of temperature can prevent thermal models of snow, ice, and frozen soil from converging to the correct solution. Existing numerical techniques rely on increased temporal resolution in trying to keep corresponding errors within acceptable bounds. Here, we propose an algorithm, originally applied to solve water flow in soils, as a method to solve these integration issues with guaranteed convergence and conservation of energy for any time step size. We review common modelling approaches, focusing on the fixed-grid method and on frozen soil. Based on this, we develop a conservative formulation of the governing equation and outline problems of alternative formulations in discretized form. Then, we apply the nested Newton–Casulli–Zanolli (NCZ) algorithm to a one-dimensional finite-volume discretization of the energy–enthalpy formulation. Model performance is demonstrated against the Neumann and Lunardini analytical solutions and by comparing results from numerical experiments with integration time steps of 1 h, 1 d, and 10 d. Using our formulation and the NCZ algorithm, the convergence of the solver is guaranteed for any time step size. With this approach, the integration time step can be chosen to match the timescale of the processes investigated.

Fast and accurate lacunarity calculation for large 3D micro-CT datasets

Microcomputed-tomography (micro-CT) is a 3D imaging method capable of revealing the complete inner structure of materials. Besides imaging, micro-CT also provides quantitative information about numerous structural features including lacunarity, which describes the heterogeneity of samples quantitatively. Theoretically, lacunarity is easily calculated using the gliding box method. However, when implemented in 3D, the computational costs of this method increase enormously, thus preventing its widespread use for large micro-CT datasets. Here we suggest a faster alternative method, based on the fixed-grid algorithm, which offers a viable alternative and renders 3D lacunarity calculations on micro-CT data feasible. Since a possible shortcoming of this alternative is that its reduced data could result in an inferior description of the real spatial heterogeneity of the structures, the two methods are compared concerning the accuracy, computational time, and applicability in materials science. The calculations are carried out on real 3D micro-CT datasets. Our implementation of the fixed-grid method can approximate gliding box lacunarity values rapidly and accurately, especially for large datasets of homogeneous structures. Therefore, we propose adding the fixed-grid method lacunarity calculation to the routine micro-CT analysis toolbox. Our image acquisition platform-independent software (Lac3D) to carry out this calculation is made freely accessible here.

Improved Bi-Direction Evolutionary Structural Optimization Method and Its Application

In the process of lightweight design of structures, the topology optimization method does not depend on the experience of designers, so it can design the optimal products more flexibly and creatively, which has become a hot issue in structural design. However, the current topological optimization results usually have problems such as difficult processing and high cost, which limit their wide application in engineering design. Therefore, this paper is based on the two-way progressive structure optimization method (BESO), considering the existence of jagged edges in the optimization results. In the finite element analysis of the fixed grid method, the boundary element is judged by the sensitivity of the element node, and the boundary element is localized divide and realize the partial deletion of the unit, so as to achieve a smooth boundary of the optimization result. The sensitivity of element nodes during structural flexibility optimization is deduced in detail, its basic principles and specific steps are explained, and finally the feasibility of this method is verified by an example.

Parameter analysis and fast prediction of the optimum eccentricity for a latent heat thermal energy storage unit with phase change material enhanced by porous medium

The melting performance of porous-medium-enhanced phase change material (PCM) in a horizontal shell-and-tube latent-heat thermal energy storage unit (LHTESU) with eccentric structure is studied through a fixed-grid numerical method. The effects of ratio of outer and inner diameter, thermal conductivity of porous medium, porosity and heating temperature on the recommended eccentricity of LHTESU are discussed. Correlations for fast prediction of the optimum eccentricity are built. Results show that the mixed use of porous medium and eccentric structure can save as much as 43.1% of the total melting time compared with the pure use of porous medium. There exists an optimum eccentricity to get the shortest total melting time. The optimum eccentricity increases with the increase of ratio of outer and inner diameter, porosity and heating temperature, and decreases with the increase of thermal conductivity. For fast prediction of the optimum eccentricity and saved melting time, a modified Rayleigh number is proposed to reflect the effect of porous medium on natural convection, especially the flow resistance caused by porous medium. The effects of the four parameters on the optimum eccentricity can be integrated by the relationship between the optimum eccentricity and the modified Rayleigh number. Correlations of The optimum eccentricity and saved melting time are obtained respectively for convenient calculation and evaluation of the feasibility of eccentricity.

Implementation of an adaptive energy grid routine in NDEX for the processing of thermal neutron scattering cross sections

Thermal neutron scattering laws tabulated in File 7 for various materials the ENDF/B database require processing to generate the cross section and distribution data needed for use in Monte Carlo reactor simulations. Typically, the total scattering cross section is calculated on a fixed energy grid and interpolated on this grid. Nevertheless, this method may result in the loss of physical features in the cross section related to the details of the thermal scattering law, especially the oscillatory behavior of solid hydride moderators (e.g., ZrH, YH2). An adaptive energy grid has been implemented in the NDEX nuclear data code, initialized from the tabulated β grid of the processed evaluation. Cross section libraries for Monte Carlo codes generated using this adaptive grid demonstrate greatly improved prediction of oscillations in solid metal hydride moderators when compared to the fixed grid method.

A novel fixed-grid interface-tracking algorithm for rapid solidification of supercooled liquid metal

In this study, we present a novel fixed-grid interface-tracking method using finite volume method to simulate multidimensional rapid solidification (RS) of under-cooled pure metal. The discretized advection equation for solid fraction function is solved using the THINC/WLIC method, which is a VOF method. The governing equations for fluid flow are solved numerically using pressure-velocity coupling SIMPLE algorithm in a 2-D model with incompressible Newtonian fluid. The energy equation is modeled using an enthalpy-based formulation. The nonequilibrium solidification kinetics, interface tracking, undercooling, nucleation, heat transfer, and movement of liquid are included in the presented RS model.

Numerical Modeling of Flow and Local Scour around Pipeline in Steady Currents Using Moving Mesh with Masked Elements

AbstractThis work reports on a computational fluid dynamics (CFD) model for flow and local scour around a pipeline in steady currents that uses a mixed moving-mesh and fixed-grid method. It perform...

Topological design of 3D phononic crystals for ultra-wide omnidirectional bandgaps

As a unique kind of artificial periodic composite materials, phononic crystals have attracted the wide attention in recent years. Unremitting efforts have been made to investigate the potential existence of phononic bandgaps, which can be used to manipulate the propagation of acoustic/elastic waves. This paper proposes a new topology optimization algorithm to achieve the optimised bandgap structures of 3D phononic crystals based on fixed-grid finite element method (FEM) and evolutionary procedure. The maximum normalised gap ratio obtained is as large as 97% between the 12th and 13th order of bands, with the optimised structure of tungsten carbide spheres embraced in epoxy matrix. Symmetric unit cells for simple cubic (SC) lattice are presented for the specified bandgaps to validate the effectiveness of the proposed optimization algorithm for designing 3D phononic bandgap crystals. Meanwhile, the proposed topology optimization represents structures with clear topologies and smooth boundaries.

Reviewing Theoretical and Numerical Models for PCM-embedded Cementitious Composites

Accumulating solar and/or environmental heat in walls of apartment buildings or houses is a way to level-out daily temperature differences and significantly cut back on energy demands. A possible way to achieve this goal is by developing advanced composites that consist of porous cementitious materials with embedded phase change materials (PCMs) that have the potential to accumulate or liberate heat energy during a chemical phase change from liquid to solid, or vice versa. This paper aims to report the current state of art on numerical and theoretical approaches available in the scientific literature for modelling the thermal behavior and heat accumulation/liberation of PCMs employed in cement-based composites. The work focuses on reviewing numerical tools for modelling phase change problems while emphasizing the so-called Stefan problem, or particularly, on the numerical techniques available for solving it. In this research field, it is the fixed grid method that is the most commonly and practically applied approach. After this, a discussion on the modelling procedures available for schematizing cementitious composites with embedded PCMs is reported.

Numerical Approach to Study the Behavior of an Artificial Ventricle: Fluid-Structure Interaction Followed By Fluid Dynamics With Moving Boundaries.

Heart failure is a progressive and often fatal pathology among the main causes of death in the world. An implantable total artificial heart (TAH) is an alternative to heart transplantation. Blood damage quantification is imperative to assess the behavior of an artificial ventricle and is strictly related to the hemodynamics, which can be investigated through numerical simulations. The aim of this study is to develop a computational model that can accurately reproduce the hemodynamics inside the left pumping chamber of an existing TAH (Carmat-TAH) together with the displacement of the leaflets of the biological aortic and mitral valves and the displacement of the pericardium-made membrane. The proposed modeling workflow combines fluid-structure interaction (FSI) simulations based on a fixed grid method with computational fluid dynamics (CFD). In particular, the kinematics of the valves is accounted for by means of a dynamic mesh technique in the CFD. The comparison between FSI- and CFD-calculated velocity fields confirmed that the presence of the valves in the CFD model is essential for realistically mimicking blood dynamics, with a percentage difference of 2% during systole phase and 13% during the diastole. The percentage of blood volume in the CFD simulation with a shear stress above the threshold of 50Pa is less than 0.001%. In conclusion, the application of this workflow to the Carmat-TAH provided consistent results with previous clinical studies demonstrating its utility in calculating local hemodynamic quantities in the presence of complex moving boundaries.

Eccentricity optimization of a horizontal shell-and-tube latent-heat thermal energy storage unit based on melting and melting-solidifying performance

As for a horizontal single-pass shell-and-tube latent-heat thermal energy storage unit (LTESU), the eccentricity between inner and outer tube is designed to improve the melting and melting-solidifying performances. A fixed-grid numerical method with enthalpy-double-porosities model is proposed to accurately predict the melting or solidifying characteristics of LTESUs with different eccentricities. Some optimal eccentricities are obtained to decrease the total melting or melting-solidifying time. The effects of Rayleigh number on the optimal eccentricities are also discussed. The results show that, when melting process is just concerned, vertically moving down the inner tube from the center of outer tube can obviously decrease the total melting time. However, a greater eccentricity does not always bring a better melting performance. That is to say, there exists an optimal eccentricity for the shortest melting time. It is found that the optimal eccentricity for melting process is linearly dependent on the Rayleigh number. As for a complete heat storage process including charging and discharging process, the eccentric arrangement of inner tube has benefit for decreasing the total melting-solidifying time only when the Rayleigh number ratio of solidifying process to melting process is larger than 2.0. The optimal value of eccentricity for the melting-solidifying process increases sharply with the increase of Rayleigh number ratio when the value of Rayleigh number ratio varies from 2.0 to 3.0, but the growth of optimal eccentricity slows down when Rayleigh number ratio is greater than 3.0.

Extension of local front reconstruction method with controlled coalescence model

The physics of droplet collisions involves a wide range of length scales. This poses a challenge to accurately simulate such flows with standard fixed grid methods due to their inability to resolve all relevant scales with an affordable number of computational grid cells. A solution is to couple a fixed grid method with subgrid models that account for microscale effects. In this paper, we improved and extended the Local Front Reconstruction Method (LFRM) with a film drainage model of Zang and Law [Phys. Fluids 23, 042102 (2011)]. The new framework is first validated by (near) head-on collision of two equal tetradecane droplets using experimental film drainage times. When the experimental film drainage times are used, the LFRM method is better in predicting the droplet collisions, especially at high velocity in comparison with other fixed grid methods (i.e., the front tracking method and the coupled level set and volume of fluid method). When the film drainage model is invoked, the method shows a good qualitative match with experiments, but a quantitative correspondence of the predicted film drainage time with the experimental drainage time is not obtained indicating that further development of film drainage model is required. However, it can be safely concluded that the LFRM coupled with film drainage models is much better in predicting the collision dynamics than the traditional methods.

MHD phase change heat transfer in an inclined enclosure: Effect of a magnetic field and cavity inclination

ABSTRACTThe MHD phase change heat transfer of a phase change substance in the presence of a uniform magnetic field is theoretically studied in a cavity. A fixed grid method associated with the enthalpy–porosity method is utilized. The governing equations are transformed into a non-dimensional form and solved using the finite element method. The impacts of the crucial parameters such as the Hartmann number and the inclination angle on the phase change process are investigated. It is found that any increase in Hartmann number and the inclination angle of the cavity leads to a decrease in the rate of the melting process.