Present Numerical Method Research Articles

Partially heated lid-driven flow in a hexagonal cavity with inner circular obstacle via FEM

In the present article, numerical analysis is performed over the lid-driven hexagon cavity filled with water. The cavity is partially heated at the upper wall, that is moving with uniform velocity. Moreover, cylindrical obstacle is placed inside of hexagonal cavity and different constraint are defined at the surface. Governing partial diifferential equations are nondimensionlized before being solved numerically using Finite Element Method (FEM). The present numerical method is also applied to the previous reported models to validate the results reported in this article. The simulation is performed in order to determine the heat transfer rate, steam lines and isotherms behavior due to the variation of contained physical and geometrical parameters, such as: Reynolds number (200 ≤ Re ≤ 550), Richards number (10−6 ≤ Ri ≤ 1), Hartman number (0 ≤ Ha ≤ 20), heated length (0.1 ≤ HL ≤ 0.3) and cylindrical obstacle (cold, adiabatic and heated). The circular obstacle played an important role in the formation of isotherms by adjusting the various constraint at the surface. The outcome reveals that the heat length, Richardson number and Reynold number are strongly associated with heat transfer.

Investigation of double-diffusive mixed convection effect on the particles dissolution in the shear flow using coupled SPM–LBM

In the present study, double-diffusive mixed convection related to the heat and mass transfer of the solid particles dissolution in a shear flow was numerically investigated. For this purpose, the study employed the combined two-dimensional thermal-concentration smoothed profile-lattice Boltzmann method scheme. The governing equations of the flow, concentration and temperature fields are solved by applying the LBM. The interplay amongst the fluid–solid particle and the boundary condition of the no-slip at the interface of the fluid–solid particle is treated by using the SPM. Initially, a comparison made amongst the results obtained in the present numerical method and to those that had been obtained in the previous works, showing a well compatibility. Then, the impacts of Reynolds number, thermal and concentration Grashof numbers and buoyancy ratio on the flow characteristics and dissolution process have been addressed. Following that, a comparison is made amongst the impacts of pure forced convection and double-diffusive mixed convection on the solid particle dissolution behavior. It is shown that different competitive mechanisms have impact on the migration of the solid particles under the double-diffusive mixed convection; these are the particle mass force, the buoyancy force resulting from the fluid weight, the thermal convection and finally, the concentration convection. The numerical results also revealed the identification of the critical buoyancy ratio, with the dissolution time of the solid particles being sensitive to it.

Optimization of the airside thermal performance of mini-channel-flat-tube radiators by using composite straight-and-louvered fins

In this study, to optimize the thermal performance on the airside of a flat-tube radiator, four parameters, including fin types, fin pitch, fin thickness, and louver angle, are numerically examined in detail. Existed experimental results are employed to verify the present numerical method with maximum deviations to be less than 1.65%, and 0.27% for heat transfer rate and outlet temperature, respectively. The frontal velocity ranges from 1 m/s to 6 m/s. Copper and aluminum are employed as the tube and fin materials, respectively. For all cases in this study, it is found that the thermal performance is significantly affected by the fin thickness, fin pitch, and fin types. The thermal performance significantly increases with the increase of the louver angle from 0° to 27°; however, it is only slightly increased with a further rise of the louver angle from 27° to 40.5°. A novel composite straight-and-louvered fin structure is proposed, and an optimal fin structure is found. The optimal fin structure could improve the thermal performance up to 17.52%, 18.31%, and 19.54% or 55.76%, 59.27%, and 60.22% compared to that of the SD louvered fin or the baseline at pumping powers of 2 W, 10 W, and 20 W, respectively.

Elasticity Solutions for In-Plane Free Vibration of FG-GPLRC Circular Arches with Various End Conditions

In-plane free vibration of functionally graded graphene platelets reinforced nanocomposites (FG-GPLRCs) circular arches is investigated by using the two-dimensional theory of elasticity. The graphene platelets (GPLs) are dispersed along the thickness direction non-uniformly, and the material properties of the nanocomposites are evaluated by the modified Halpin-Tsai multi-scaled model and the rule of mixtures. A state-space method combined with differential quadrature technique is employed to derive the governing equation for in-plane free vibration of FG-GPLRCs circular arch, the semi-analytical solutions are obtained for various end conditions. An exact solution of FG-GPLRCs circular arch with simply-supported ends is also presented as a benchmark to valid the present numerical method. Numerical examples are performed to study the effects of GPL distribution patterns, weight fraction and dimensions, geometric parameters and boundary conditions of the circular arch on the natural frequency in details.

Simplified unified wave-particle method with quantified model-competition mechanism for numerical calculation of multiscale flows.

A quantified model-competition (QMC) mechanism for multiscale flows is extracted from the integral (analytical) solution of the Boltzmann-BGK model equation. In the QMC mechanism, the weight of the rarefied model and the weight of the continuum (aerodynamic and hydrodynamic) model are quantified. Then, a simplified unified wave-particle method (SUWP) is constructed based on the QMC mechanism. In the SUWP, the stochastic particle method and the continuum Navier-Stokes method are combined together. Their weights are determined by the QMC mechanism quantitatively in every discrete cell of the computational domain. The validity and accuracy of the present numerical method are examined using a series of test cases including the high nonequilibrium shock wave structure case, the unsteady Sod shock-tube case with a wide range of Kn number, the hypersonic flow around the circular cylinder from the free-molecular regime to the near continuum regime, and the viscous boundary layer case. In the construction process of the present method, an antidissipation effect in the continuum mechanism is also discussed.

Asymptotic numerical method for third-order singularly perturbed convection diffusion delay differential equations

In this paper, an asymptotic numerical method based on a fitted finite difference scheme and the fourth-order Runge–Kutta method with piecewise cubic Hermite interpolation on Shishkin mesh is suggested to solve singularly perturbed boundary value problems for third-order ordinary differential equations of convection diffusion type with a delay. An error estimate is derived using the supremum norm and it is of almost first-order convergence. A nonlinear problem is also solved using the Newton’s quasi linearization technique and the present asymptotic numerical method. Numerical results are provided to illustrate the theoretical results.

Crack Growth and Energy Release Rate for an Angled Crack under Mixed Mode Loading

The evaluation of energy release rate with angle is still a challenging task in metal crack propagation analysis, especially for the mixed Mode I-II-III loading situation. In this paper, the energy release rate associated with stress intensity factors at an arbitrary angle under mixed mode loadings has been investigated using both a numerical method and theoretical derivation. A relatively simple and precise numerical method was established through a series of spatial-inclined ellipses in Mode I-II and ellipsoids in Mode I-II-III, with different propagation angles computed from simulation. Meanwhile, a theoretical expression of the energy release rate with angle for a crack tip under a I-II-III mixed mode crack was deduced based on the propagation mechanism of the crack tip under the influence of a stress field. It is confirmed that the theoretical expression deduced could provide results as accurately as the present numerical method. The present results were confirmed to be effective and accurate by comparison with experimental data and other literature.

Failure of single hat-shaped thin-walled tubular composite T-joints under impact loading

Abstract This paper presents an experimental and numerical investigation into the failure characteristics of single hat-shaped thin-walled tubular T-joints made of twill-woven carbon fiber reinforced thermoplastic (CFRTP) under impact loading. One flat panel and one hat-shaped shell were prepared via a high temperature and pressure molding process first, and then were joined together via (a) adhesive bonding and (b) hybrid bonding and riveting, to form the T-shaped thin-walled tubular joints. Drop weight impact tests were conducted for the T-joints with three impact velocities. Load-time/displacement curves were measured and four distinct failure modes were identified through analyzing damage images and micrographs. It is noted that the specific energy absorption (SEA) capability increases with impact velocity and can be affected significantly by the deformation of the hat-shaped shell, but appears to be less sensitive to the joining forms. Nonlinear finite element analyses based on selected progressive damage model were conducted to predict the failure and energy absorption characteristics of the T-joints under an impact loading. There exists a good correlation between the simulation and experimental results, which validates the present numerical modeling method.

Three-temperature nonlinear generalized magneto-thermoelastic stresses in anisotropic cylindrical shells

The main purpose of this paper is to investigate the transient three-temperature nonlinear generalized magneto-thermoelastic stresses in an anisotropic cylindrical shell. The system of governing equations is solved by means of a boundary element method (BEM) and numerical calculations are carried out for temperature and thermal stresses. The numerical results indicate that the effects of three-temperature on the thermal stresses are very pronounced, where we used two-dimensional three temperature nonlinear radiative heat conduction equations coupled with electron, ion and phonon temperatures. Also, these numerical results demonstrate the validity and accuracy of the present numerical method.

Numerical study of wake and aerodynamic forces on a twin-box bridge deck with different gap ratios

Two-dimensional Delayed Detached Eddy Simulation (DDES) was carried out to investigate the uniform flow over a twin-box bridge deck (TBBD) with various gap ratios of L/C=5.1%, 12.8%, 25.6%, 38.5%, 73.3% and 108.2% (L: the gap-width between two girders, C: the chord length of a single girder) at Reynolds number, Re=4x104. The aerodynamic coefficients of the prototype deck with gap ratio of 73.3% obtained from the present simulation were compared with the previous experimental and numerical data for different attack angles to validate the present numerical method. Particular attention is devoted to the fluctuating pressure distribution and forces, shear layer reattachment position, wake velocity and flow pattern in order to understand the effects of gap ratio on dynamic flow interaction with the twin-box bridge deck. The flow structure is sensitive to the gap, thus a change in L/C thus leads to single-side shedding regime at L/C󖼩.6%, and co-shedding regime at L/C󖼳.8% distinguished by drastic changes in flow structure and vortex shedding. The gap-ratio-dependent Strouhal number gradually increases from 0.12 to 0.27, though the domain frequencies of vortices shedding from two girders are identical. The mean and fluctuating pressure distributions is significantly influenced by the flow pattern, and thus the fluctuating lift force on two girders increases or decreases with increasing of L/C in the single-side shedding and co-shedding regime, respectively. In addition, the flow mechanisms for the variation in aerodynamic performance with respect to gap ratios are discussed in detail.

New Analytical and Numerical Solutions of the Particle Breakup Process

Objective: In this work, we obtained the analytical and approximate solutions of the population balance equations (PBEs) involving the breakup process in batch and continuous flow by applying the Adomian decomposition method and piecewise continuous basis functions, respectively. Methods: The key to the advanced numerical method is to represent the number distribution function of the dispersed phase through the orthogonal Chebyshev basis polynomials. It is a straightforward and effective method that has the advantage of simultaneously giving the distribution and the different required moments. Therefore, it does not require the construction of the distribution from moments computations obtained by the transformation of the initial problem and the lost information. Results: The performance of this numerical approach is evaluated by solving breakup equation and comparison against analytical solutions obtained from the Adomian decomposition method, which generally allows the analysis of this approach. Conclusion: The numerical results obtained by the present numerical method were compared with the new analytical solutions of the PBE. It was found that both piecewise continuous basis functions and analytical solutions have comparable results.

Numerical Investigation on Nonlinear Dynamic Responses of a Towed Vessel in Calm Water

In this study, we numerically investigated the nonlinear dynamic responses of an autonomous towing system where a vessel is passively towed by a tug via a towline. A three-degrees-of-freedom maneuvering mathematical model is utilized to describe the nonlinear dynamics of the towed vessel in calm sea. The hydrodynamic force acting on the towed vessel is modelled as a modular-type hull force model, which includes linear and nonlinear (third order) damping forces in sway and yawing directions. The towline force is simply modeled as a linear spring. First, the motion responses of a towing system, showing large sway-yaw coupled motions due to unstable towing characteristics, are studied by applying phase plane analysis. For the validation of the present numerical method, the simulation results are directly compared with the model test data. Then, simulation parameters, such as towing speed, initial positions and hull force coefficients, are changed and their resulting limit cycles are investigated. Finally, the effects of towline and tug motion are discussed based on the simulation results. It is found that the dynamic characteristics of the towed vessel come closer to being chaotic due to the nonlinear stiffness effect of the towline and tug motion effect.

Computational and experimental investigation of a swirl nozzle for viscous fluids

Highly viscous flow in a large-scale pressure-swirl atomizer is studied by (1) 3d scale-resolving large-eddy simulations and volume-of-fluid method, and (2) experiments based on laser-Doppler anemometry, imaging techniques and pressure measurements. Here, a low Reynolds number regime (600 ≤ Re ≤ 910) is investigated by varying the mass flow rate of the water-glycerol mixture. The aim of the study is to perform a comprehensive comparison between the simulations and experiments at a parameter range and nozzle geometry relevant for biomass based fuels. We report the inner-nozzle velocity profiles noting good agreement for mean velocities inside the swirl chamber between the simulations and the experiments. Consistent with the earlier work (Laurila et al., 2019), the simulations indicate the flow mode to be laminar with weak or non-existent gaseous core inside the swirl chamber. As revealed by both approaches, liquid film shapes after the nozzle discharge orifice are qualitatively similar, of hollow cone type, and highly unstable. Both approaches indicate linear scaling of the liquid film velocity with the inlet Reynolds number and discharge coefficients to be in the range 0.57–0.64. The experimentally measured mean opening angles are reported to be 45–62∘, while the numerical counterparts show reasonable correspondence with the experiments. The results demonstrate the predictive ability of the present numerical method in swirl injector analysis.

Numerical study on ship-generated unsteady waves based on a Cartesian-grid method

The added resistance of a ship in waves can be related to ship-generated unsteady waves. In the present study, the unsteady wave-pattern analysis is applied to calculate the added resistance in waves for two modified Wigley models using a Cartesian-grid method. In the present numerical method, a first-order fractional-step method is applied to the velocity-pressure coupling in the fluid domain, and one of volume-of-fluid (VOF) methods is adopted to capture the fluid interface. A ship is embedded in a Cartesian grid, and the volume fraction of the ship inside the grid is calculated by identifying whether each grid is occupied by liquid, gas, and solid body. The sensitivity to the location of measuring position of unsteady waves as well as the number of solution grids is examined. The added resistance computed by direct pressure integration and wave pattern analysis is compared with experimental data. In addition, nonlinear characteristics of the added resistance in waves are investigated by detailed analyses of unsteady flow field and resulting wave pattern.

A Numerical Study on Hydrodynamic Performance of an Inclined OWC Wave Energy Converter with Nonlinear Turbine–Chamber Interaction based on 3D Potential Flow

In this study, a time-domain numerical method based on three-dimensional potential flow was developed to analyze the hydrodynamic characteristics of an inclined oscillating-water-column (OWC) wave energy converter (WEC). A finite element method was applied to solve the potential flow around and inside the OWC chamber. A turbine–chamber interaction was considered to take into account the pressure drop inside the OWC chamber, which is a nonlinear function of airflow speed via turbine operation. The instantaneous pressure drop was updated on the free-surface boundary condition inside the chamber in the time-domain to account for the coupling effect between the turbine and the chamber. The present numerical method was verified by comparing it with the model test results. The hydrodynamic characteristics of an inclined OWC chamber in terms of potential flow, such as the water column motion and the three-dimensional flow distribution around the chamber, were investigated. In terms of hydrodynamic performance, the energy conversion efficiency of the chamber showed a nonlinear response characteristic dependent on the incident wave height. In addition, numerical calculations were carried out to clarify the relationship between the main geometric parameters and the hydrodynamic response of the inclined OWC chamber.

Legendre Collocation Method for the Nonlinear Space–Time Fractional Partial Differential Equations

In this study, nonlinear space–time fractional partial differential equations with variable coefficients are considered. The fractional derivatives are described in the conformable sense and Caputo sense. The numerical approach is based on the shifted Legendre polynomials and the finite difference method. Several illustrative examples are given to observe the validity, effectiveness and accuracy of the present numerical method.

A local mesh free numerical method with automatic parameter optimization

The concern of the paper is a local mesh free numerical method, implemented with automatic optimization of the discretization parameters, to solve problems of two-dimensional linear elasticity.For a nodal discretization, the mesh free local method uses a node-by-node process to generate the global system of equilibrium equations; in the local domain of each node, the respective equilibrium equations are generated with a reduced numerical integration and used as the formulation basis of the numerical method.Application of the local numerical method, to each node of a mesh free discretization, requires the use of two arbitrary parameters which specify the size of, respectively the compact support and the local domain of integration. In this paper, these parameters are automatically defined by means of a multi-objective optimization process, based on genetic algorithms, which is the novelty of the present numerical method. For the sake of efficiency a comparison between genetic algorithms and another robust evolutionary optimization method, the symbiotic organism search algorithm, is carried out.A benchmark problem was analyzed to assess the accuracy and efficiency of these techniques. The results presented in the paper are in perfect agreement with those of analytical solutions and therefore, make reliable and robust this mesh free numerical method, implemented with automatic parameter optimization.

Coupling mechanism of mass transport and electrochemical reaction within patterned anode of solid oxide fuel cell

Patterned electrodes are widely used in the development of novel electrodes of solid oxide fuel cells (SOFCs) because of their well-controlled geometries, distinguishable catalytically active sites and simple transport paths. In the existing studies the patterned electrodes are usually adopted to reveal relevant reaction mechanisms and to investigate the electrochemical characteristics of new materials of SOFCs, however, the effects of electrode geometry are not taken into consideration. In the present paper, a lattice Boltzmann model for simulating the charge transport and electrochemical reaction in an SOFC patterned anode is established, and the key dimensionless parameters governing the above electrode process are deduced. This model is then used to investigate the effects of the key dimensionless parameters on the electrochemical performance of a patterned anode. More importantly, the influences of the patterned anode geometry on the coupling of the charge transport and electrochemical reaction are unraveled. According to the sensitivity of the electrode performance to the dimensionless parameters, a dimensionless phase map, which is divided into maximum area, transition area and minimum area, is built. It is concluded that the transition area, in which the electrode performance varies dramatically with the parameters of design and operation, is regarded as the optimal range for studying the relevant reaction mechanism. Meanwhile, it is found that although the electron transport does not restrict the electrode performance, the moderate decrease of the height-to-width ratio of electronic conductor is capable of enlarging the transition area, which is beneficial to revealing the relevant reaction mechanism. Conversely, the ion transport is the rate-limiting step, however, the transition area remains unchanged under different ionic conductor geometries. The present numerical method and conclusions could offer guidance for rationally designing and operating the patterned electrodes.

Influences of radiation on the prediction of thermal environment in a transient buoyancy-driven natural ventilation classroom

Computational fluid dynamics (CFD) simulations are performed to reveal the significant effects of radiation on predictive accuracy of thermal environment in a transient buoyancy-driven natural ventilation classroom. The present numerical method is verified by the experimental results. The time evolutions of airflow field, radiation heat flux, buoyancy force, and airflow rate are analyzed in detail. In addition, the effects of radiation on thermal comfort are evaluated by three different indices. It is found that radiation plays a critical role in the establishment of the indoor thermal environment. The proportion of radiation heat flux increases with the effective radiation area but decreases with the power of the occupants. Finally, the indoor space can be divided into five different occupied zones depending on how much radiated heat the occupants emit. Radiation can change the indoor heat flux distribution, directly leading to a smaller buoyancy force and ventilation rate. As a result, the thermal comfort except for any draft is linearly affected by both the power and the effective radiation area, and the greater the proportion of radiation heat flux is, the better the indoor thermal comfort is. These conclusions demonstrate that radiation has a noticeable impact on the dynamic evaluations of indoor thermal environment.

A high‐order compact scheme for solving the 2D steady incompressible Navier‐Stokes equations in general curvilinear coordinates

SummaryIn this paper, a high‐order compact finite difference algorithm is established for the stream function‐velocity formulation of the two‐dimensional steady incompressible Navier‐Stokes equations in general curvilinear coordinates. Different from the previous work, not only the stream function and its first‐order partial derivatives but also the second‐order mixed partial derivative is treated as unknown variable in this work. Numerical examples, including a test problem with an analytical solution, three types of lid‐driven cavity flow problems with unusual shapes and steady flow past a circular cylinder as well as an elliptic cylinder with angle of attack, are solved numerically by the newly proposed scheme. For two types of the lid‐driven trapezoidal cavity flow, we provide the detailed data using the fine grid sizes, which can be considered the benchmark solutions. The results obtained prove that the present numerical method has the ability to solve the incompressible flow for complex geometry in engineering applications, especially by using a nonorthogonal coordinate transformation, with high accuracy.