Search results

1 – 10 of 245
Article
Publication date: 1 April 2020

Jing-Kui Zhang, Miao Cui, Ben-Wen Li and Ya-Song Sun

The purpose of this paper is to develop a combined method for three-dimensional incompressible flow and heat transfer by the spectral collocation method (SCM) and the artificial…

157

Abstract

Purpose

The purpose of this paper is to develop a combined method for three-dimensional incompressible flow and heat transfer by the spectral collocation method (SCM) and the artificial compressibility method (ACM), and further to study the performance of the combined method SCM-ACM for three-dimensional incompressible flow and heat transfer.

Design/methodology/approach

The partial differentials in space are discretized by the SCM with Chebyshev polynomial and Chebyshev–Gauss–Lobbatto collocation points. The unsteady artificial compressibility equations are solved to obtain the steady results by the ACM. Three-dimensional exact solutions with trigonometric function form and exponential function form are constructed to test the accuracy of the combined method.

Findings

The SCM-ACM is developed successfully for three-dimensional incompressible flow and heat transfer with high accuracy that the minimum value of variance can reach. The accuracy increases exponentially along with time marching steps. The accuracy is also improved exponentially with the increasing of nodes before stable accuracy is achieved, while it keeps stably with the increasing of the time step. The central processing unit time increases exponentially with the increasing of nodes and decreasing of the time step.

Research limitations/implications

It is difficult for the implementation of the implicit scheme by the developed SCM-ACM. The SCM-ACM can be used for solving unsteady impressible fluid flow and heat transfer.

Practical implications

The SCM-ACM is applied for two classic cases of lid-driven cavity flow and natural convection in cubic cavities. The present results show good agreement with the published results with much fewer nodes.

Originality/value

The combined method SCM-ACM is developed, firstly, for solving three-dimensional incompressible fluid flow and heat transfer by the SCM and ACM. The performance of SCM-ACM is investigated. This combined method provides a new choice for solving three-dimensional fluid flow and heat transfer with high accuracy.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 12
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 April 2014

Jean Batina, Serge Blancher and Tarik Kouskou

Mathematical and numerical models are developed to study the melting of a Phase Change Material (PCM) inside a 2D cavity. The bottom of the cell is heated at constant and uniform…

Abstract

Purpose

Mathematical and numerical models are developed to study the melting of a Phase Change Material (PCM) inside a 2D cavity. The bottom of the cell is heated at constant and uniform temperature or heat flux, assuming that the rest of the cavity is completely adiabatic. The paper used suitable numerical methods to follow the interface temporal evolution with a good accuracy. The purpose of this paper is to show how the evolution of the latent energy absorbed to melt the PCM depends on the temperature imposed on the lower wall of the cavity.

Design/methodology/approach

The problem is written with non-homogeneous boundary conditions. Momentum and energy equations are numerically solved in space by a spectral collocation method especially oriented to this situation. A Crank-Nicolson scheme permits the resolution in time.

Findings

The results clearly show the evolution of multicellular regime during the process of fusion and the kinetics of phase change depends on the boundary condition imposed on the bottom cell wall. Thus the charge and discharge processes in energy storage cells can be controlled by varying the temperature in the cell PCM. Substantial modifications of the thermal convective heat and mass transfer are highlighted during the transient regime. This model is particularly suitable to follow with a good accuracy the evolution of the solid/liquid interface in the process of storage/release energy.

Research limitations/implications

The time-dependent physical properties that induce non-linear coupled unsteady terms in Navier-Stokes and energy equations are not taken into account in the present model. The present model is actually extended to these coupled situations. This problem requires smoother geometries. One can try to palliate this disadvantage by constructing smoother approximations of non-smooth geometries. The augmentation of polynomials developments orders increases strongly the computing time. When the external heat flux or temperature imposed at the PCM is much greater than the temperature of the PCM fusion, one must choose carefully some data to assume the algorithms convergence.

Practical implications

Among the areas where this work can be used, are: buildings where the PCM are used in insulation and passive cooling; thermal energy storage, the PCM stores energy by changing phase, solid to liquid (fusion); cooling and transport of foodstuffs or pharmaceutical or medical sensitive products, the PCM is used in the food industry, pharmaceutical and medical, to minimize temperature variations of food, drug or sensitive materials; and the textile industry, PCM materials in the textile industry are used in microcapsules placed inside textile fibres. The PCM intervene to regulate heat transfer between the body and the outside.

Originality/value

The paper's originality is reflected in the precision of its results, due to the use of a high-accuracy numerical approximation based on collocation spectral methods, and the choice of Chebyshev polynomials basis in both axial and radial directions.

Article
Publication date: 1 March 1992

J. FRÖHLICH and R. PEYRET

The low Mach number approximation of the Navier—Stokes equations is of similar nature to the equations for incompressible flow. A major difference, however, is the appearance of a…

Abstract

The low Mach number approximation of the Navier—Stokes equations is of similar nature to the equations for incompressible flow. A major difference, however, is the appearance of a space‐ and time‐varying density that introduces a supplementary non‐linearity. In order to solve these equations with spectral space discretization, an iterative solution method has been constructed and successfully applied in former work to two‐dimensional natural convection and isobaric combustion with one direction of periodicity. For the extension to other geometries efficiency is an important point, and it is therefore desirable to devise a direct method which would have, in the best case, the same stability properties as the iterative method. The present paper discusses in a systematic way different approaches to this aim. It turns out that direct methods avoiding the diffusive time step limit are possible, indeed. Although we focus for discussion and numerical investigation on natural convection flows, the results carry over for other problems such as variable viscosity flows, isobaric combustion, or non‐homogeneous flows.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 2 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 14 November 2023

Mostafa Abbaszadeh, AliReza Bagheri Salec and Shurooq Kamel Abd Al-Khafaji

The space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is…

Abstract

Purpose

The space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.

Design/methodology/approach

The fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.

Findings

A fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.

Originality/value

The spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

Details

Engineering Computations, vol. 40 no. 9/10
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 6 January 2012

Precious Sibanda, Sandile Motsa and Zodwa Makukula

The purpose of this paper is to study the steady laminar flow of a pressure driven third‐grade fluid with heat transfer in a horizontal channel. The study serves two purposes: to…

Abstract

Purpose

The purpose of this paper is to study the steady laminar flow of a pressure driven third‐grade fluid with heat transfer in a horizontal channel. The study serves two purposes: to correct the inaccurate results presented in Siddiqui et al., where the homotopy perturbation method was used, and to demonstrate the computational efficiency and accuracy of the spectral‐homotopy analysis methods (SHAM and MSHAM) in solving problems that arise in fluid mechanics.

Design/methodology/approach

Exact and approximate analytical series solutions of the non‐linear equations that govern the flow of a steady laminar flow of a third grade fluid through a horizontal channel are constructed using the homotopy analysis method and two new modifications of this method. These solutions are compared to the full numerical results. A new method for calculating the optimum value of the embedded auxiliary parameter ∼ is proposed.

Findings

The “standard” HAM and the two modifications of the HAM (the SHAM and the MSHAM) lead to faster convergence when compared to the homotopy perturbation method. The paper shows that when the same initial approximation is used, the HAM and the SHAM give identical results. Nonetheless, the advantage of the SHAM is that it eliminates the restriction of searching for solutions to the nonlinear equations in terms of prescribed solution forms that conform to the rule of solution expression and the rule of coefficient ergodicity. In addition, an alternative and more efficient implementation of the SHAM (referred to as the MSHAM) converges much faster, and for all parameter values.

Research limitations/implications

The spectral modification of the homotopy analysis method is a new procedure that has been shown to work efficiently for fluid flow problems in bounded domains. It however remains to be generalized and verified for more complicated nonlinear problems.

Originality/value

The spectral‐HAM has already been proposed and implemented by the authors in a recent paper. This paper serves the purpose of verifying and demonstrating the utility of the new spectral modification of the HAM in solving problems that arise in fluid mechanics. The MSHAM is a further modification of the SHAM to speed up converge and to allow for convergence for a much wider range of system parameter values. The utility of these methods has not been tested and verified for systems of nonlinear equations. For this reason as much emphasis has been placed on proving the reliability and validity of the solution techniques as on the physics of the problem.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 24 September 2019

Mangwiro Magodora, Hiranmoy Mondal and Precious Sibanda

The purpose of this paper is to focus on the application of Chebyshev spectral collocation methodology with Gauss Lobatto grid points to micropolar fluid over a stretching or…

Abstract

Purpose

The purpose of this paper is to focus on the application of Chebyshev spectral collocation methodology with Gauss Lobatto grid points to micropolar fluid over a stretching or shrinking surface. Radiation, thermophoresis and nanoparticle Brownian motion are considered. The results have attainable scientific and technological applications in systems involving stretchable materials.

Design/methodology/approach

The model equations governing the flow are transformed into non-linear ordinary differential equations which are then reworked into linear form using the Newton-based quasilinearization method (SQLM). Spectral collocation is then used to solve the resulting linearised system of equations.

Findings

The validity of the model is established using error analysis. The velocity, temperature, micro-rotation, skin friction and couple stress parameters are conferred diagrammatically and analysed in detail.

Originality/value

The study obtains numerical explanations for rapidly convergent solutions using the spectral quasilinearization method. Convergence of the numerical solutions was monitored using the residual error analysis. The influence of radiation, heat and mass parameters on the flow are depicted graphically and analysed. The study is an extension on the work by Zheng et al. (2012) and therefore the novelty is that the authors tend to take into account nanoparticles, Brownian motion and thermophoresis in the flow of a micropolar fluid.

Details

Multidiscipline Modeling in Materials and Structures, vol. 16 no. 2
Type: Research Article
ISSN: 1573-6105

Keywords

Article
Publication date: 1 September 2005

M.J. Chern, A.G.L. Borthwick and R. Eatock Taylor

The research is directed at development of an efficient and accurate technique for modelling incompressible free surface flows in which viscous effects may not be neglected. The…

1093

Abstract

Purpose

The research is directed at development of an efficient and accurate technique for modelling incompressible free surface flows in which viscous effects may not be neglected. The paper describes the methodology, and gives illustrative results for simple geometries.

Design/methodology/approach

The pseudospectral matrix element method of discretisation is selected as the basis for the CFD technique adopted, because of its high spectral accuracy. It is implemented as a means of solving the Navier‐Stokes equations coupled with the modified compressibility method.

Findings

The viscous solver has been validated for the benchmark cases of uniform flow past a cylinder at a Reynolds number of 40, and 2D cavity flows. Results for sloshing of a viscous fluid in a tank have been successfully compared with those from a linearised analytical solution. Application of the method is illustrated by the results for the interaction of an impulsive wave with a surface piercing circular cylinder in a cylindrical tank.

Research limitations/implications

The paper demonstrates the viability of the approach adopted. The limitation of small amplitude waves should be tackled in future work.

Practical implications

The results will have particular significance in the context of validating computations from more complex schemes applicable to arbitrary geometries.

Originality/value

The new methodology and results are of interest to the community of those developing numerical models of flow past marine structures.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 15 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 14 September 2023

Huseyin Tunc and Murat Sari

This study aims to derive a novel spatial numerical method based on multidimensional local Taylor series representations for solving high-order advection-diffusion (AD) equations.

Abstract

Purpose

This study aims to derive a novel spatial numerical method based on multidimensional local Taylor series representations for solving high-order advection-diffusion (AD) equations.

Design/methodology/approach

The parabolic AD equations are reduced to the nonhomogeneous elliptic system of partial differential equations by utilizing the Chebyshev spectral collocation method (ChSCM) in the temporal variable. The implicit-explicit local differential transform method (IELDTM) is constructed over two- and three-dimensional meshes using continuity equations of the neighbor representations with either explicit or implicit forms in related directions. The IELDTM yields an overdetermined or underdetermined system of algebraic equations solved in the least square sense.

Findings

The IELDTM has proven to have excellent convergence properties by experimentally illustrating both h-refinement and p-refinement outcomes. A distinctive feature of the IELDTM over the existing numerical techniques is optimizing the local spatial degrees of freedom. It has been proven that the IELDTM provides more accurate results with far fewer degrees of freedom than the finite difference, finite element and spectral methods.

Originality/value

This study shows the derivation, applicability and performance of the IELDTM for solving 2D and 3D advection-diffusion equations. It has been demonstrated that the IELDTM can be a competitive numerical method for addressing high-space dimensional-parabolic partial differential equations (PDEs) arising in various fields of science and engineering. The novel ChSCM-IELDTM hybridization has been proven to have distinct advantages, such as continuous utilization of time integration and optimized formulation of spatial approximations. Furthermore, the novel ChSCM-IELDTM hybridization can be adapted to address various other types of PDEs by modifying the theoretical derivation accordingly.

Details

Engineering Computations, vol. 40 no. 9/10
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 5 September 2016

Abbas Saadatmandi and Zeinab Sanatkar

The purpose of this paper is to develop an efficient method for solving the magneto-hydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid over a porous…

Abstract

Purpose

The purpose of this paper is to develop an efficient method for solving the magneto-hydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid over a porous isothermal stretching sheet.

Design/methodology/approach

The paper applied a collocation approach based on rational Legendre functions for solving the third-order non-linear boundary value problem, describing the MHD boundary layer flow of an UCM fluid over a porous isothermal stretching sheet. This method solves the problem on the semi-infinite domain without transforming domain of the problem to a finite domain.

Findings

This approach reduces the solution of a problem to the solution of a system of algebraic equations. The numerical values of the skin friction coefficient are presented and analyzed for various parameters of interest in the problem. The authors also compare the results of this work with some recent results and show that the new method is efficient and applicable.

Originality/value

The method solves this problem without use of discrete variables and linearization or small perturbation. Also it was confirmed by the theorem and figure of absolute coefficients that this approach has exponentially convergence rate.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 30 September 2014

Asghar Zajkani, Abolfazl Darvizeh and Mansour Darvizeh

The purpose of this paper is to introduce a computational time dependent modeling to investigate propagation of elastic-viscoplastic zones in the shock wave loaded circular…

Abstract

Purpose

The purpose of this paper is to introduce a computational time dependent modeling to investigate propagation of elastic-viscoplastic zones in the shock wave loaded circular plates.

Design/methodology/approach

Constitutive equations are implemented incrementally by the Von-Kármán finite deflection system which is coupled with a mixed strain hardening rule and physical-base viscoplastic models. Time integrations of the equations are done by the return mapping technique through the cutting-plane algorithm. An integrated solution is established by pseudo-spectral collocation methodology. The Chebyshev basis functions are utilized to evaluate the coefficients of displacement fields. Temporal terms are discretized by the Houbolt marching method. Spatial linearizations are accomplished by the quadratic extrapolation technique.

Findings

Results of the center point deflections, effective plastic strain and stress (dynamic flow stress) and temperature rise are compared for three features of the Von-Kármán system. Identifying time history of resultant stresses, propagations of the viscoplastic plastic zones are illustrated for two circumstances; with considering strain rate and hardening effects, and without them. Some of modeling and computation aspects are discussed, carefully. When the results are compared with experimental data of shock wave loadings and finite element simulations, good agreements between them are observed.

Originality/value

This computational approach makes coupling the structural equations with the physical descriptions of the high rate deformation through step-by-step spectral solution of the constitutive equations.

Details

Engineering Computations, vol. 31 no. 7
Type: Research Article
ISSN: 0264-4401

Keywords

1 – 10 of 245