Implicit Finite Difference Method Research Articles

A new eighth-order implicit finite difference method to solve the three-dimensional Helmholtz equation

This paper introduces a novel method for deducing high-order finite difference schemes for the three-dimensional Helmholtz equation, Δp±λ2p=f. The new method can reach an eighth-order of accuracy. It is based on an implicit formulation by computing simultaneously the unknown variable and its corresponding derivatives. The scheme is obtained from Taylor series expansion and plane wave theory analysis, and it is constructed from a few modifications to the standard centered finite differences. A classical dispersion analysis and the error between the numerical wavenumber and the exact wavenumber is given. Numerical experiments are presented to verify the feasibility and accuracy of the proposed methods for a broad range of λ-values. Moreover, the numerical method is capable to solve with great precision the monotone and oscillatory Helmholtz problem with variable λ-values. Thus, the proposed formulation results in an attractive method, easy to implement, with high order of accuracy but nearly the same computational cost as those of an explicit formulation.

Variable thickness flow over a rotating disk under the influence of variable magnetic field: An application to parametric continuation method

The present work explores the behavior of three-dimensional incompressible viscous fluid flow and heat transfer over the surface of a non-flat stretchable rotating disk. A variable thickness fluid is subjected under the influence of an external variable magnetic field and heat transfer. Navier–Stokes equation is coupled with Maxwell equations to examine the hydrothermal properties of fluid. The basic governing equations of motion are diminished to a system of nonlinear ordinary differential equations using appropriate similarity framework, which are further treated with numerical scheme known as parametric continuation method. The parametric continuation method has combined interesting characteristics of both shooting and implicit finite difference methods. For validity of the present numerical scheme, a comparison with the published work is performed and it is found that the results are in excellent agreement with each other. Numerical and graphical results for the velocity, temperature, and magnetic strength profiles as well as skin fractions and Nusselt number are presented and discussed in detail for various physical parameters. The heat transfer process is reduced with positive increment of no-flatness parameter [Formula: see text], while Prandtl number increases the heat transfer rate at the surface of the disk.

Numerical Simulation Study on Temperature Field of Dry Hot Rock Gas Drilling Wellbore

In this paper, gas drilling is used for dry hot rock drilling. According to the first law of thermodynamics and the basic principles of heat transfer, a mathematical model for the temperature field of dry hot rock gas drilling wellbore is established. The model is solved by the full implicit finite difference method. Then, relevant sensitivity analysis was carried out. The model can be used for the temperature distribution of drill string, annulus and formation under actual dry hot rock well circulation conditions.

Numerical solution of unsteady two dimensional decelerating flow over a wedge with temperature dependent viscosity

The present work brings into focus the numerical solution of unsteady two dimensional decelerating laminar boundary layer flow and heat transfer of an incompressible fluid above a moving wedge in the existence of variable viscosity. Suitable transformation is used to form a system of coupled nonlinear partial differential equations for governing both the flow and heat transfer. These equations have been solved numerically by using an implicit finite difference method in combination with quasilinearization strategy. The obtained numerical results have been presented graphically in terms of local nusselt number, skin friction, temperature distribution, and velocity distribution for different values of variable viscosity parameter (e) along with Prandtl number (Pr). The boundary layer isolates itself from its surface, past which the dual results exist up to a critical value of an unsteady parameter (λ).

Mixed convection flow past a yawed cylinder

In this study, a new-fangled problem is considered, wherein, for the first time, the effects of mixed convection past a yawed cylinder are analysed. In reality, the concept of mixed convection flow past a yawed cylinder arises in applications pertaining to heat exchangers. To study the diffusion of heat in the mixed convection system, the problem is formulated mathematically in terms of partial differential equations which are nonlinear and coupled in nature. To avoid the mathematical complexity of dimensions, these equations are converted to dimensionless form through non-similar transformations. The linearity in the system of equations is achieved through Quasi-linearization technique. Then, implicit finite difference method is used for the discretization and the linear system of equations is solved by employing Varga's algorithm. The investigation reveals several important results pertaining to yaw angle, mixed convection and non-similarity phenomena. The fluid velocity and friction at the cylinder's surface in all directions increase along with the heat transfer rate due to mixed convection, while the fluid temperature reduces significantly. Yaw angle has prominent effects in increasing the fluid velocity in the chordwise (x-axis) as well as the spanwise (z-axis) directions in case of aiding buoyancy flow.

The impact of spatial and temporal variability of anomalous diffusion flow mechanisms on reservoir performance in structurally complex porous media

The main objective of this paper is understanding the phenomenal anomalous diffusion flow mechanisms in unconventional fractured porous media. The targets are hydraulically fractured unconventional reservoirs characterized by porous media with scomplex structures. These media are consisted of a matrix and naturally induces fractures embedded in the matrix as well as hydraulic fractures.Several analytical models for pressure drop and decline rate as wells productivity index in ultralow permeability reservoirs are presented in this study for the two inner wellbore conditions. Numerical solution is also presented in this study for pressure behavior using a linearized implicit finite difference method. The analytical models are developed from trilinear flow models presented in the literature with a consideration given to the temporal and spatial fractional pressure derivative for the anomalous diffusion flow that could be the dominant flow mechanism in the stimulated reservoir volume between hydraulic fractures. Mittag-Leffler functions are used for solving fractional derivatives of pressure and flow rate considering that temporal and spatial fractional exponents are less than one. Analytically, two solutions are developed in this study for the two inner wellbore conditions. The first represents the transient state condition that controls fluid flow in unconventional reservoirs for a very long production time. The second is the solution of pseudo-steady state condition that might be observed after transient state flow. While numerically, the temporal and spatial domains are discretized into several time steps and block-centered grids respectively.The study has pointed out: 1) Temporal and spatial diffusion flows have a significant impact on pressure drop, flow rate, and productivity index. 2) Wellbore pressure drop for constant Sandface flow rate declines rapidly as the temporal diffusion flow mechanism is the dominant flow pattern in the porous media. 3) Wellbore pressure drop for constant Sandface flow rate slightly increases during transient state flow as the spatial diffusion flow mechanisms increase and rapidly increases during pseudo-steady state flow. 4) Productivity index of diffusion flow is higher than the index of normal diffusion flow during transient and pseudo-steady state conditions. 5) The linear flow regime is most affected by anomalous diffusing flow and can be used to characterize the type of diffusion flow.

Stagnation point flow of Maxwell nanofluid containing gyrotactic micro‐organism impinging obliquely on a convective surface

AbstractA numerical study is performed to discuss the nonaligned stagnation of a rate type fluid over a convective surface. The rheology of the fluid is presented by the constitutive equation of the Maxwell fluid model. Buongiorno's model is used to elaborate on the effects of Brownian motion and thermophoresis and motile microorganisms are introduced for the stability of the nanoparticles. The governing equations were solved by the implicit finite difference method. Graphical illustrations for velocity, temperature, nanoparticle concentration and motile microorganism profiles for various involved parameters are presented for both convective and nonconvective surfaces. It is depicted that the temperature, nanoparticle, and microorganism concentration profiles decease while both axial and tangential velocities increase with the velocity ratio parameter for both Newtonian and Maxwellian fluids. The magnitude of temperature, nanoparticle, and microorganism concentration profiles is large for the nonconvective surface as compared to the convective surface. The Nusselt number, Sherwood number, and motile organism number decrease as we move from Newtonian fluid to non‐Newtonian fluid. Furthermore, the increase in the Brownian motion parameter and thermophoresis parameter decreases the density of the motile organism over the convective as well as nonconvective surface.

Effect of regeneration heat and energy storage on thermal drying performance in a hardwood solar kiln

The use of energy efficiency tools and solar energy in wood drying can help reduce the often heavy energy consumption of industrial dryers. The rational management of energy in terms of drying has attracted a lot of attention and the main objective of this work is to adapt the techniques of renewable energy and energy saving in solar kilns. The present work is therefore a contribution to the improvement of energy efficiency and the modeling of a solar wood kiln with thermal storage and heat regeneration. The drying system consists of five main units, a drying chamber, a multi-pass solar air collector, a cylindrical parabolic solar collector, a thermal storage tank and a Closed Feed Air Heater. The investigation carried out in this work is based on the establishment of mass and energy conservation equations in different components of the drying system. The governing equations of heat and mass transfer are solved using the implicit finite difference method. The discrepancies between the experimental and numerical results do not exceed 5%. The results show that the drying time is shorter in June and longer in December. The drying time decreases as the collector area increases and the boards thickness decreases. By incorporating a Closed Feed Air Heater with an effectiveness of 0.75, in the solar dryer, the collector and drying efficiency values are increased from 0.48 to 0.56 and 0.43 to 0.88, in June, respectively. The integration of the thermal storage unit in the solar kiln has the effect of reducing the drying time up to 40 and 60%, in June and December, respectively. Moreover, the combined use of thermal storage and regeneration heat is efficient in reducing the energy consumption ratio (kWh.m−3) up to 50% and 54%, in June and December, respectively. The proposed solution can significantly improve thermal drying performance and thus overcoming the problem of longer drying time, especially in winter.

Transient development of MHD natural convection flow in vertical concentric annulus

<p>This paper examines the role of magnetic field on fully developed natural convection flow in an annulus due to symmetric of surfaces. The transport equations concerned with the model under consideration are rendered non-dimensional and transformed into the ordinary differential equation using Laplace transform technique. The solution obtained is then transformed to time domain using the Riemann-sum approximation approach. The governing equations are also solved using implicit finite difference method so as to establish the accuracy of the Riemann-sum approximation approach at transient as well as at steady state solution. The solutions obtained are graphically represented and the effects of pertinent parameters on the flow formation are investigated in detail. The Hartmann number (M), is seen to have a retarding effect on the velocity, skin-frictions and the mass flow rate. Also, skin-friction at both surfaces and the mass flow rate within the annulus are found to be directly proportional to the radii ratio (λ).</p>

Improve the calculation accuracy of the optimal insulation thickness in building walls as determined by a dynamic heat transfer model

To improve the calculation accuracy of optimum insulation thickness of building walls, the effects of the initial conditions and the choice of the representative day used as boundary conditions on results have been studied. In this context, the unsteady heat transfer across the multilayer wall was considered as one-dimensional and the problem was solved numerically by an implicit finite difference method. A computational FORTRAN code has been developed and validated to calculate, under real meteorological data (RMD), the cooling transmission loads and the optimum insulation thickness during the summer period in the Sahara of Algeria. For steady periodic conditions, the influence of the initial conditions on the precision of the numerical result has been studied for two cases of representative days (July 15 and 21). By comparing the results with the case of real conditions (RMD), we find that the choice of July 15 as a typical day gives more accurate results than the choice of July 21. In addition, the difference of cooling transmission loads of 76.2% for July 15 and 76.31% for July 21 is observed, between the use of a repeated and another non-repeated typical day.

Exponential finite difference methods for solving Newell\u2013Whitehead\u2013Segel equation

This work presents two different finite difference methods to compute the numerical solutions for Newell–Whitehead–Segel partial differential equation, which are implicit exponential finite difference method and fully implicit exponential finite difference method. Implicit exponential methods lead to nonlinear systems. Newton method is used to solve the resulting systems. Stability and consistency are discussed. To illustrate the accuracy of the proposed numerical methods, some examples are delivered at the end.

Padé schemes with Richardson extrapolation for the sine-Gordon equation

Four novel implicit finite difference methods with (q+s)-th order in space based on (q, s)-Padé approximations have been analyzed and developed for the sine-Gordon equation. Specifically, (4,0)-, (2,2)-, (4,2)-, and (4,4)-Padé methods. All of them share the treatment for the nonlinearity and integration in time, specifically, the one that results in an energy-conserving (2,0)-Padé scheme. The five methods have been developed with and without Richardson extrapolation in time. All the methods are linearly, unconditionally stable. A comparison among them for both the kink–antikink and breather solutions in terms of global error, computational cost and energy conservation is presented. Our results indicate that the (4,0)- and (4,4)-Padé methods without Richardson extrapolation are the most cost-effective ones for small and large global error, respectively; and the (4,4)-Padé methods in all the cases when Richardson extrapolation is used.

Nonlinear mixed convective nanofluid flow along moving vertical rough plate

The objective of the current research paper is to investigate the effects of surface roughness on magnetohydrodynamic nonlinear mixed convection nanofluid flow over vertically moving plate. The highly coupled dimensional nonlinear partial differential equations (NPDE) are converted into dimensionless NPDE along with the boundary conditions with the help of non-similar transformations. The resulting set of dimensionless nonlinear PDE’s are solved by using Quasilinearization technique and implicit finite difference method. Impacts of various dimensionless parameters, namely, Brownian diffusion (Nb), nonlinear mixed convection ( ), nanoparticle buoyancy ratio (Nr), Lewis number (Le), thermophoresis (Nt), frequency (n), magnetic (M) and small parameter ( ) are studied in detail on profiles as well as gradients. The results reveal that increasing values of increase the velocity profile, while increasing values of Nr decrease the same. Further, increasing values of and n exhibit sinusoidal variations on skin-friction coefficient, heat and nanoparticle mass transfer rates. Moreover, the presence of nonlinear mixed convection parameter has significant effects on fluid flow compared to its absence. In addition to this, rate of heat transfer is analyzed in presence and absence of nanoparticles.

Effect of the integration of hemp wool as an insulation material for the construction of the roof and external walls of a typical Moroccan building

This paper deals with the integration of thermal insulation in Moroccan buildings roofs and external walls in order to reduce energy consumption due to the use of air conditioning systems. The insulation material under investigation is the hemp wool, an ecofriendly material whose use allows the storage of the Carbone and help reducing greenhouse gas emissions. In order to assess the dynamic thermal performance of the typical Moroccan insulated building, a life cycle cost analysis is conducted based on annual cooling transmission loads specific to the climate of Meknes, Morocco. Optimum insulation thicknesses, energy savings and payback periods are determined for a building life-span of 20 years. Investigations are carried out based on an implicit finite difference method using a home-made code developed in Matlab.

Mixed Bioconvective Flow Over a Wedge in Porous Media Drenched with a Nanofluid

A mathematical model is accentuated the mixed bioconvective flow on a vertical wedge in a Darcy porous medium filled with a nanofluid containing both nanoparticles and gyrotactic microorganisms. Thermophoresis and Brownian motion impacts are addressed to consolidate energy and concentration equations with passivelycontrolled boundary conditions. A mixed convective parameter for the whole regime of the mixed convective is appointed. The system of governing partial differential equations is converted into a non-similar set, which are then solved by an implicit finite difference method. By taking the impacts of the varying pertinent parameters, namely, the bioconvection nanofluids and wedge angle parameters in the entire mixed convection regime, the numerical results are analyzed graphically for the dimensionless the velocity, temperature, nanoparticle volume fraction and the density motile microorganisms profiles as well as the local Nusselt and motile microorganism numbers.

Radiation and Magnetic Field Impacts on Time-Dependent Mixed Convection Flow and Heat Transmission of Maxwellian Fluid Past A Stretching Sheet

The current study was devoted to explicating the impacts of heat transmission in an unsteady mixed convection flow of an upper convected Maxwell (UCM) fluid passing over a continuously stretching surface under the influence of radiation and magnetic field. Appurtenant similarity transmutations were adopted in order to express the constitutive boundary layer Equations of flow and heat transmission in non-dimensionalized form. The reduced system of partial differential Equations was solved by implementing the implicit finite difference method (IFDM). Our center of attention was to scrutinize the behavior of influential flow parameters on some significant features of flow and heat transmission, which were briefly examined, discussed, and presented in both graphical and tabular formats. Finally, a comparison was established with existing literature in limiting cases to support the present results, and a good agreement was found, corroborating our work. It was predicted that the thermal diffusion rate could be controlled by varying the Prandtl number. Moreover, a rise in radiation and magnetic field parameters reduced the skin friction coefficient and led to enhance the heat transmission rate at the surface. The outcomes of the study might have viable implementations in order to improve the quality of industrial products.

Numerical studies of high-dimensional nonlinear viscous and nonviscous wave equations by using finite difference methods

The numerical solutions of two-dimensional (2D) and three-dimensional (3D) nonlinear viscous and nonviscous wave equations via the unified alternating direction implicit (ADI) finite difference methods (FDMs) are obtained in this paper. By making use of the discrete energy method, it is proven that their numerical solutions converge to exact solutions with an order of two in both time and space with respect to [Formula: see text]-norm. Numerical results confirm that they are relatively accurate and high-resolution, and more successfully simulate the conservation of the energy for nonviscous equations, and the dissipation of the energy for viscous equation.

Impact of glazing area on the thermal performance of buildings

In this paper, the impact of glazing area on the thermal performance of buildings is numerically investigated by considering orientation, wall insulation and type of glazing. The investigation is carried out for all months of the year in the climatic conditions of Elazığ, Turkey. Firstly, the transmission loads through the wall are calculated using an implicit finite difference method. Secondly, the transmission loads through the window glass are also calculated for single and double glass. Then, while the glazing area percentage (GAP) is increased from %0 to %100, the transmission loads are determined according to the type of glazing for uninsulated and insulated walls. It is seen that in single glazing, as GAP increases, the heating and cooling transmission loads increase for all wall orientations. The results show that double glazing provides a significant reduction in especially the heating load for all building orientations when compared to single glazing.

Magnetohydrodynamic natural convection flow of a nanofluid due to sinusoidal surface temperature variations

We examine the natural convection flow that arises in an electrically conducting nanofluid due to the sinusoidal surface temperature variations along a vertical surface. The effect of thermal radiation is also incorporated. The reduced system of governing equations is solved using an efficient implicit finite difference method which is also known as the Keller box method. The results are presented in terms of the shear stress and the rate of heat transfer as well as the velocity and temperature profiles. In general, the amplitude of the undulation of the shear stress gradually decreases and that of the rate of heat transfer increases along the streamwise direction. The significant finding is that, with an increase in the volume fraction of nanoparticles, the shear stress diminishes, whereas the rate of heat transfer considerably increases. Due to the increase in the amplitude of oscillation of the surface temperature, conduction–radiation parameter, and surface temperature, a substantial increase is observed in both the shear stress and the rate of heat transfer. Contrary to this, the shear stress and the rate of heat transfer decrease with the increase in the magnetic field parameter. For any value of the relevant parameters, the use of nanoparticles in a pure fluid reduces the shear stress and enhances the rate of heat transfer. Moreover, when a nanofluid is used instead of pure fluid, the thicknesses of momentum and thermal boundary layers are found to increase, irrespective of the physical parameters.

Optimization of the slotted liners parameters during dual tubing steam injection process

To study the distribution of steam parameters along the wellbore and the influence of slotted liners parameters on the effect of reservoir uniform preheating during the dual tubing steam injection process. Based on the structural characteristics of screen pipe completion and the simultaneous steam injection technology of long and short tubing, a coupled numerical model of fluid flow and heat transfer in reservoir and wellbore was established, and the model was solved by full implicit finite difference method and iterative technique. The results show that when the density, width and length of the slotted liners increase from 350 to 550/m, 0.20 to 0.40 mm and 35 to 75 mm, respectively, the difference of steam quality at the two ends of steam convergence position in the annuli increases by 0.41%, 2.26% and 1.71%, and the reservoir heating uniformity decreases by 1.49%, 2.68% and 2.08%, respectively. Optimizing the slotted liners parameters and reducing the steam quality difference at the convergence position of annuli can improve the uniform heating degree of reservoir along the horizontal wellbore.