Scheme Of Finite Difference Method Research Articles

Applying a finite difference scheme to the flow field analysis of rotating detonation engines using multiple fuels

The objective of this study is to apply the finite difference method (FDM) scheme to generate profiles of velocity, vorticity, pressure, temperature, and density of the flow field inside the chamber of a rotating detonation engine. In addition, this study discusses the Kuo's algorithm used in the computation of Chapman-Jouguet (CJ) variables. A new computational algorithm is introduced to compute the CJ variables, the shock, and the post-shock conditions. Five different fuels are tested, H2–O2, H2-air, CH4-air, C2H6-air, and C2H2-air. The results show good performance for the FDM scheme, and the new algorithm is adopted in this study to perform all CJ calculations. The new algorithm's results are compared with the experimental data available for the tested fuels and show close results. Another important result is that the highest vorticity forms along the slip line of the combustion chamber, then it gradually shrinks along the path of the slip line.

Computation of Two-Dimensional Poisson Equation Using the Third-Order Discrete Scheme of Finite Difference Method Based on Node Set Vector

A novel third-order discrete scheme of finite difference method based on node set vector for two dimensional Poisson equation is proposed in this paper. Studies on the basic discrete scheme of this method as well as the discrete scheme of the interior node and boundary node are carried out in detail. Computer programs are also developed to emulate a computation sample. The numerical computation results show that this discrete scheme is effective for the numerical computation of two dimensional Poisson equation.

Magnetohydrodynamics (MHD) Induced Slip Flow of a Non-Newtonian Fluid through Circular Microchannels

The present numerical analysis reveals the nature of non-Newtonian fluid flow through circular microchannels under slip boundary conditions. The power law has been used for the simulation of the fluid flow, which considers a steady, laminar, incompressible non-Newtonian fluid acted upon by a constant, externally applied magnetic field. The flow is axisymmetric and slip boundary conditions are applied in the near wall. A constant magnetic flux has been applied on the wall boundary to analyze the effect of magnetic field on Xanthan solution in formic acid, a type of non-Newtonian fluid having electrical conductivity. Using control volume method of finite difference scheme, a set of dimensionless governing differential equations defining the behavior of the fluid flow in the microchannel under an externally applied magnetic field, has been solved using slip boundary conditions to understand the effect of magnetic field on slip induced flow of non-Newtonian fluids. The results have depicted that the magnetic field affects both the centerline velocity and slip velocity but it is more prominent for the centerline velocities. The main objective of this research is to study the flow of non-Newtonian fluid, Xanthan through a circular microchannel and its corresponding behavior when flow boundary conditions are applied to interpret the characteristics under an externally applied magnetic field. The results obtained from this present study will find its application in the area of the flow of ferrofluids and biofluids.
 HIGHLIGHTS
 
 Most of the bio fluids and ferrofluids are non-Newtonian in nature and it is required to control these fluids when they pass through microchannels of modern devices
 Analysis conducted to find out the flow behaviour of non-Newtonian fluids through circular microchannels when they are exposed to externally applied magnetic fields
 Slip flow occurs in the fluid flow and an externally applied magnetic field controls the flow patterns by affecting slip velocity and centerline velocity which introduces extra flow in the microchannel. The results are helpful for better understanding of the flow of ferrofluids and bio fluids through circular microchannels
 
 GRAPHICAL ABSTRACT

Order Out of Chaos in Soil–Water Retention Curves

Water flow in porous media is one of many phenomena in nature that can demonstrate both simple and complex behaviors. A soil–water retention curve (SWRC) is needed to characterize this flow properly. This curve relates the soil water content and the matric potential (or porepressure), being fundamental for simulating unsaturated soil behaviors. This article proposes a new model based on simple assumptions regarding the saturated and unsaturated branches of soil–water retention curves. Despite its simplicity, the modeling capability of the proposed SWRC is shown for two types of soil. This new SWRC is obtained as a logistic function after solving an ordinary differential equation (ODE). This ODE can also be solved numerically using the Finite Difference Method (FDM), which indicates that the discrete version of the SWRC can be represented as the logistic map for specific parameters. On the other hand, this discrete representation is known to encompass chaotic and fractal behaviors. This link is used to investigate the stability and convergence of the FDM scheme, indicating that for values pre-bifurcation, both the FDM and the analytical solution of the ODE represent the new SWRC. This way, the present paper is the first step to better understating how a chaotic framework could be related to SWRCs and geotechnics in general.

Higher Order Implicit Scheme for Nonlinear Time-Dependent Convection-Diffusion- Reaction Equation

A mathematical model comprising of nonlinear reaction, diffusion, and convection mechanisms seen in natural and anthropogenic processes is numerically investigated here. It is proposed that a higher order numerical scheme of finite difference method be used in conjunction with an iterative approach in order to solve the nonlinear one dimensional convection-diffusion-reaction equation. To account for the wide variety of physical characteristics and boundary conditions, an iterative approach is presented that yields a reliable and precise solution every time. We examined the accuracy and operational efficiency of two distinct finite difference approaches. The efficiency of the system is determined by comparing the estimated results to the appropriate analytical solution by adhering to established norms. Coherence and convergence were analyzed for each approach. The simulation results demonstrate the efficacy and accuracy of these methods in solving nonlinear convection- diffusion-reaction equations. Convection-diffusion-reaction equation modeling is critical for employing the offered results in heat and mass transport processes.

Variable-order optimal implicit finite-difference schemes for explicit time-marching solutions to wave equations

The time-space-domain finite-difference method (FDM) is widely used in forward modeling of wave equations. Conventional explicit FDMs (EFDMs) (with high order in space and the second order in time) would result in apparent temporal and spatial dispersion for high frequencies and large time steps. Moreover, the saturation effect of Taylor expansion seriously restricts the improvement of bandwidth coverage and efficiency of EFDMs. We have developed a variable-order optimization scheme of the implicit FDM (IFDM) to improve the computational efficiency and numerical accuracy of forward modeling. Then, we applied time-dispersion transforms (TDTs) to filter out the temporal dispersion generated by the second-order temporal approximations. Our method greatly alleviates the saturation effect of the high-order spatial finite-difference (FD) operators. Dispersion analysis indicates that the optimized coefficients of the IFDM based on the Remez algorithm can achieve the widest bandwidth (close to the Nyquist wavenumber), which corresponds to the shortest length of the spatial FD operator under a given error threshold. Our method has great potential to approach the highest spectral accuracy but with minimal increase in computational cost. Numerical experiments indicate that the combination of the variable-order optimization of IFDM and TDTs can significantly reduce numerical dispersion. Compared with traditional methods, our scheme is more advantageous for numerical simulation of large-scale geologic models because it has the least amount of calculation burden under the same accuracy requirements.

Računska analiza konvektivnog prenosa toplote preko vertikalne cevi

This paper deals with the numerical investigation of the convective mode of heat transfer across a vertical tube. Experiments were carried out using air as a fluid in a closed room by achieving a steady-state condition. Implicit scheme of finite difference method was adopted to numerically simulate the free convection phenomenon across vertical tube using LINUX based UBUNTU package. Numerical data were collected in the form of velocity, temperature profiles, boundary layer thickness, Nusselt number (Nu), Rayleigh's number (Ra), and heat transfer coefficient. The results of the Nusselt number showed a good agreement with the previous studies. Results data of heat transfer coefficient indicate that there were some minor heat losses due to radiation of brass tube and curvature of the tube.

The computational study of heat conduction in laboratory using Crank-Nicolson finite difference method

The main propose of this research is to study the heat conduction of an aluminium metal bar in an undergraduate thermodynamics laboratory by using the Crank-Nicolson scheme of finite difference method. Heat conduction was simulated to find the thermal conductivity and was compared with the data obtained from the experiment and theory. In the performed boundary and initial conditions, it was found that the predicted thermal conductivity from the simulation was in agreement with those from the theory, however, it was slightly different from the experiment. These results can be used to improve method to find the thermal conductivity in the laboratory more accurately in the laboratory. Besides, these computational approaches can be an optional teaching method for studying thermal conduction for undergraduates, especially for a laboratory that has limited instruments.

BOUNDARY LAYER CONTROL OF A RADIATIVE CASSON FLUID FLOW PAST A PERMEABLE RIGA-PLATE WITH UNIMOLECULAR CHEMICAL REACTION

The buoyancy driven, chemically reacting and radiative Casson fluid flow past an impulsively started permeable Riga-plate is investigated through the numerical solution obtained by Crank-Nicholson implicit scheme of finite difference method. The main aim of this study is to control the boundary layer separation. Escalating modified Hartmann number and the distance from leading edge of the plate reduces the viscous drag so that the separation can be controlled. Effects of permeability on the flow configuration are also elucidated. The results are validated by comparing the solutions of the literature which already exists.

Numerical solutions of the second-order dual-phase-lag equation using the explicit and implicit schemes of the finite difference method

PurposeThe purpose of this paper is the application of the finite difference method (FDM) for numerical modeling of the microscale heat transfer processes occurring in the domain of thin metal film subjected to a laser pulse. The problem discussed is described by the different variants of the second-order dual-phase-lag equation (DPLE). The laser action is taken into account by the introduction of internal volumetric heat source to the governing equation. The capacity of the source is dependent on the geometrical co-ordinates and duration of the laser beam. The modified forms of DPLE presented in the paper, resulting from the certain substitutions introduced to the basic equation.Design/methodology/approachAt the stage of numerical computations, the different variants of the FDM are applied. Both the explicit and implicit FDM schemes are used. The formula determining the capacity of the internal heat source suggests the formulation of the task discussed using the cylindrical co-ordinate system. The in-house programs realizing the numerical computations concern the axially-symmetrical tasks. In this paper, the metal films made of the nickel and gold are considered.FindingsThe algorithms presented make possible to analyze the heating/cooling processes occurring in the domain of metal film having a thickness Z for the different laser parameters (laser intensity, characteristic time of laser pulse and laser beam radius) and the different materials (optical penetration depth, reflectivity of irradiated surface, lag times, thermal conductivity and volumetric specific heat).Research limitations/implicationsNot for all metals, one can find information on lag times. In the literature, analytical formulas can be found to calculate these values, but they are strongly approximated. It should be pointed out that there are some limitations concerning the delay times of material considered, which assure the physical correctness of the second-order DPLE.Originality/valueThe FDM algorithm concerns the three-dimensional cylindrical domain while a large majority of the second-order DPLE numerical solutions have been obtained for the one-dimensional tasks. Both the implicit and explicit numerical schemes are proposed and the testing computations confirm the correctness and effectiveness of the algorithms presented.

Alternating direction implicit scheme as Finite-Difference method to solve coupled groundwater flow and contaminant transport model in the coastal aquifer

The simulation of coupled groundwater flow-contaminant transport equation can be conducted numerically. The Finite-Difference Method (FDM) through Alternating Direction Implicit (ADI) Scheme will be applied in this research. The ADI scheme is chosen because the FDM in two-dimensional form. The ADI scheme recognized to solve two-dimensional PDE (Partial Differential Equation) due the Crank-Nicholson has matrix dimension problem and other problem from another scheme of FDM. This research will use ADI scheme to transform the groundwater flow and contaminant transport equation become the discrete form. The model algorithm translated into VBA (Visual Basic Algorithm) code and the physical parameter determined hypothetically. The result of model simulation describes that the ADI scheme can be applied to solve coupled groundwater flow-contaminant transport equation. The result can be stated as stable because it reached the steady condition after 60 days of simulation.

Evaluation of temperature history of a spherical nanosystem irradiated with various short-pulse laser sources.

Spatiotemporal thermal response and characteristics of net entropy production rate of a gold nanosphere (radius: 50-200 nm), subjected to a short-pulse, femtosecond laser is reported. In order to correctly illustrate the temperature history of laser-metal interaction(s) at picoseconds transient with a comprehensive single temperature definition in macroscale and to further understand how the thermophysical response of the single-phase lag (SPL) and dual-phase lag (DPL) frameworks (with various lag-ratios') differs, governing energy equations derived from these benchmark non-Fourier frameworks are numerically solved and thermodynamic assessment under both the classical irreversible thermodynamics (CIT) as well as extended irreversible thermodynamics (EIT) frameworks is subsequently carried out. Under the frameworks of SPL and DPL with small lag ratio, thermophysical anomalies such as temperature overshooting characterized by adverse temperature gradient is observed to violate the local thermodynamic equilibrium (LTE) hypothesis. The EIT framework, however, justifies the compatibility of overshooting of temperature with the second law of thermodynamics under a nonequilibrium paradigm. The DPL framework with higher lag ratio was however observed to remain free from temperature overshooting and finds suitable consistency with LTE hypothesis. In order to solve the dimensional non-Fourier governing energy equation with volumetric laser-irradiation source term(s), the lattice Boltzmann method (LBM) is extended and a three-time level, fully implicit, second order accurate finite difference method (FDM) is illustrated. For all situations under observation, the LBM scheme is featured to be computationally superior to remaining FDM schemes. With detailed prediction of maximum temperature rise and the corresponding peaking time by all the numerical schemes, effects of the change of radius of the gold nanosphere, the magnitude of fluence of laser, and laser irradiation with multiple pulses on thermal energy transport and lagging behavior (if any) are further elucidated at different radial locations of the gold nanosphere. Last, efforts are further made to address the thermophysical characteristics when effective thermal conductivity (with temporal and size effects) is considered instead of the usual bulk thermal conductivity.

A novel 2.5D finite difference scheme for simulations of resistivity logging in anisotropic media

The objective of this study is to develop a method to model 3D resistivity well logging problems in 2D formation with anisotropy, known as 2.5D modeling. The traditional 1D forward modeling extensively used in practice lacks the capability of modeling 2D formation. A 2.5D finite difference method (FDM) solving all the electric and magnetic field components simultaneously is proposed. Compared to other previous 2.5D FDM schemes, this method is more straightforward in modeling fully anisotropic media and easy to be implemented. Fourier transform is essential to this FDM scheme, and by employing Gauss-Legendre (GL) quadrature rule the computational time of this step can be greatly reduced. In the numerical examples, we first demonstrate the validity of the FDM scheme with GL rule by comparing with 1D forward modeling for layered anisotropic problems, and then we model a complicated 2D formation case and find that the proposed 2.5D FD scheme is much more efficient than 3D numerical methods.

Numerical approach to the uniqueness solution of Von-Karman evolution

The purpose of this paper is to give some theoretical and numerical analysis approach, for local generalised and weakly uniqueness solution, to the models with interior dissipation and clamped boundary conditions of Von-Karman evolution, without rotational inertia and nonlinear internal dissipation. To approach the uniqueness solution we use the scheme of finite difference method.

Numerical approach to the uniqueness solution of Von-Karman evolution

The purpose of this paper is to give some theoretical and numerical analysis approach, for local generalised and weakly uniqueness solution, to the models with interior dissipation and clamped boundary conditions of Von-Karman evolution, without rotational inertia and nonlinear internal dissipation. To approach the uniqueness solution we use the scheme of finite difference method.

The Transient Conjugate Heat Transfer Analysis in Multi-zone of an Apartment House by using Crank-Nicolson Finite Difference Method (FDM)

Purpose: This study was conducted for the purpose of accurate energy analysis of apartment house which is composed of multi-zones such as a bedrooms, a livingroom, and a kitchen. Those of multi-zonal spaces have different indoor thermal environment requirements depending on space usage. For the analysis of this study was selected the 74 ㎡ type unit, which has the largest amount of supply in residential needs. The orientation of the apartment house unit was divided into the southeast and southwest directions, and the annual heating energy demand was calculated and compared. Method: A multi-zone heat transfer is done in dynamic and a very complex, such as conduction, convection, radiation and the solar radiation in a various building elements. In this paper, the Crank-Nicolson FDM scheme and the ISO 13790 algorithm were employed to analyze a complex and transient heat transfer in the multi-zone of apartment house. Result: The results of the analysis are as follows; the total amount of annual energy requirement for the apartment house unit which oriented in the southwest is 756 ㎾h. and equivalent to 10.2 ㎾h /㎡. When the required amount of primary energy was converted equivalent to electricity by unit floor area, it was 28.1 ㎾h /㎡. On the other hand, the total amount of annual energy requirement for a unit oriented in the southeast direction is 1,037 ㎾h and 14.0 ㎾h /㎡. The result of conversion (electricity) to primary energy is 38.5 ㎾h /㎡. This result indicates that the energy demand is about 27% higher than the southwest.

Pemodelan Matematika Perpindahan Panas Konveksi Campuran (Mixed Convection) pada Pelat Horizontal

This study examines and analyzes mathematical model of mixed convection in horizontal plate. The heat transfer uses the model of a two dimensional nonlinear partial differential equations system. Then, this equation is derived first into the dimensionless equation form, and then it is changed into system of nonlinear ordinary differential equations form using similarity transformation. This system of nonlinear ordinary differential equations is solved by using the finite-difference scheme method, also with the mathematics program with software Matlab. The results obtained form this program is to determine skin friction coefficient , wall temperature velocity profiles and temperature profiles

A Cell-Based Framework for Numerical Modeling of Electrical Conduction in Cardiac Tissue

In this paper, we study a mathematical model of cardiac tissue based on explicit representation of individual cells. In this EMI model, the extracellular (E) space, the cell membrane (M) and the intracellular (I) space are represented as separate geometrical domains. This representation introduces modelling flexibility needed for detailed representation of the properties of cardiac cells including their membrane. In particular, we will show that the model allows ion channels to be non-uniformly distributed along the membrane of the cell. Such features are difficult to include in classical homogenized models like the monodomain and bidomain models frequently used in computational analyses of cardiac electrophysiology. The EMI model is solved using a finite difference method (FDM) and two variants of the finite element method (FEM). We compare the three schemes numerically, reporting on CPU-efforts and convergence rates. Finally, we illustrate the distinctive capabilities of the EMI model compared to classical models by simulating monolayers of cardiac cells with heterogeneous distributions of ionic channels along the cell membrane. Because of the detailed representation of every cell, the computational problems that result from using the EMI model are much larger than for the classical homogenized models, and thus represent a computational challenge. However, our numerical simulations indicate that the FDM scheme is optimal in the sense that the computational complexity increases proportionally to the number of cardiac cells in the model. Moreover, we present simulations, based on systems of equations involving ~ 117 million unknowns, representing up to ~ 16000 cells. We conclude that collections of cardiac cells can be simulated using the EMI model, and that the EMI model enable greater modeling flexibility than the classical monodomain and bidomain models.

Sensitivity analysis of temperature field in the heated soft tissue with respect to the perturbations of porosity

• Generalized dual-phase lag equation for axisymmetrical domain is considered. • Sensitivity analysis is used to estimate temperature changes due to the perturbation of the porosity. • Numerical algorithm based on the explicit scheme of the finite difference method is proposed. The thermal processes occurring in the heated soft tissues are described by generalized dual-phase lag model supplemented by appropriate boundary and initial conditions. A key parameter in this model is the porosity associated with the diameter of blood vessels and the diameter of the surrounding tissue. Using the direct approach, the sensitivity analysis of the tissue and blood temperatures with respect to these diameters is discussed. The basic problem and additional ones concerning the sensitivity functions are solved by means of the explicit scheme of finite difference method. In the final part the results of computations are presented and the conclusions are formulated.

Implementation of Crank–Nicolson scheme in cell dynamics simulation for diblock copolymers

Many computational techniques have been designed to study block copolymer systems. Cell Dynamics Simulation (CDS) technique is one of those. It is based on solving partial differential equations (PDEs) and is computationally very fast compared to other simulation methods. In this contribution the focus is made on finite difference schemes and their use in computer simulation of PDEs involved in diblock copolymers systems. The Crank–Nicolson (CN) scheme of finite difference method is implemented in CDS method. The CN scheme is unconditionally stable but slower compared to Forward Euler’s method which is fast but not very stable.