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Article
Publication date: 5 May 2015

Martin Joseph Guillot and Steve C McCool

The purpose of this paper is to investigate the effect of numerical boundary condition implementation on local error and convergence in L2-norm of a finite volume discretization…

Abstract

Purpose

The purpose of this paper is to investigate the effect of numerical boundary condition implementation on local error and convergence in L2-norm of a finite volume discretization of the transient heat conduction equation subject to several boundary conditions, and for cases with volumetric heat generation, using both fully implicit and Crank-Nicolson time discretizations. The goal is to determine which combination of numerical boundary condition implementation and time discretization produces the most accurate solutions with the least computational effort.

Design/methodology/approach

The paper studies several benchmark cases including constant temperature, convective heating, constant heat flux, time-varying heat flux, and volumetric heating, and compares the convergence rates and local to analytical or semi-analytical solutions.

Findings

The Crank-Nicolson method coupled with second-order expression for the boundary derivatives produces the most accurate solutions on the coarsest meshes with the least computation times. The Crank-Nicolson method allows up to 16X larger time step for similar accuracy, with nearly negligible additional computational effort compared with the implicit method.

Practical implications

The findings can be used by researchers writing similar codes for quantitative guidance concerning the effect of various numerical boundary condition approximations for a large class of boundary condition types for two common time discretization methods.

Originality/value

The paper provides a comprehensive study of accuracy and convergence of the finite volume discretization for a wide range of benchmark cases and common time discretization methods.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 February 1953

THE accountant is the latest competitor for management power. The Institute of Cost & Works Accountants—the value of whose Associate qualification we acknowledge—has re‐cast its…

Abstract

THE accountant is the latest competitor for management power. The Institute of Cost & Works Accountants—the value of whose Associate qualification we acknowledge—has re‐cast its requirements for the grade of Fellowship. Cost Accountants' or for that matter many other kinds of accountant, are now invited to sit for the Fellowship examination, the syllabus for which has just been published. This comprises the now familiar: Management—Factory and Distribution, Statistical Method, Advanced Cost Accountancy, Company Law, Management Accountancy and the Economic Aspects of Industry and Commerce. (The Management Section includes Motion & Time Study). Assuming that they are successful in this and that they satisfy a Reviewing Board of the adequacy of their experience, they may then call themselves “Management Accountants”.

Details

Work Study, vol. 2 no. 2
Type: Research Article
ISSN: 0043-8022

Article
Publication date: 14 October 2019

Nagesh Babu Balam and Akhilesh Gupta

Modelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving…

Abstract

Purpose

Modelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving the higher order temporal accuracy. A fourth-order accurate finite difference method in both space and time is proposed to overcome these numerical errors and accurately model the transient behaviour of natural convection flow in enclosures using vorticity–streamfunction formulation.

Design/methodology/approach

Fourth-order wide stencil formula with appropriate one-sided difference extrapolation technique near the boundary is used for spatial discretisation, and classical fourth-order Runge–Kutta scheme is applied for transient term discretisation. The proposed method is applied on two transient case studies, i.e. convection–diffusion of a Gaussian Pulse and Taylor Vortex flow having analytical solution.

Findings

Error magnitude comparison and rate of convergence analysis of the proposed method with these analytical solutions establish fourth-order accuracy and prove the ability of the proposed method to truly capture the transient behaviour of incompressible flow. Also, to test the transient natural convection flow behaviour, the algorithm is tested on differentially heated square cavity at high Rayleigh number in the range of 103-108, followed by studying the transient periodic behaviour in a differentially heated vertical cavity of aspect ratio 8:1. An excellent comparison is obtained with standard benchmark results.

Research limitations/implications

The developed method is applied on 2D enclosures; however, the present methodology can be extended to 3D enclosures using velocity–vorticity formulations which shall be explored in future.

Originality/value

The proposed methodology to achieve fourth-order accurate transient simulation of natural convection flows is novel, to the best of the authors’ knowledge. Stable fourth-order vorticity boundary conditions are derived for boundary and external boundary regions. The selected case studies for comparison demonstrate not only the fourth-order accuracy but also the considerable reduction in error magnitude by increasing the temporal accuracy. Also, this study provides novel benchmark results at five different locations within the differentially heated vertical cavity of aspect ratio 8:1 for future comparison studies.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 17 October 2019

J.I. Ramos

The purpose of this paper is to develop a new transversal method of lines for one-dimensional reactiondiffusion equations that is conservative and provides piecewise–analytical…

Abstract

Purpose

The purpose of this paper is to develop a new transversal method of lines for one-dimensional reactiondiffusion equations that is conservative and provides piecewise–analytical solutions in space, analyze its truncation errors and linear stability, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficients, reaction rate terms and initial conditions on wave propagation and merging.

Design/methodology/approach

A conservative, transversal method of lines based on the discretization of time and piecewise analytical integration of the resulting two-point boundary-value problems subject to the continuity of the dependent variables and their fluxes at the control-volume boundaries, is presented. The method provides three-point finite difference expressions for the nodal values and continuous solutions in space, and its accuracy has been determined first analytically and then assessed in numerical experiments of reaction-diffusion problems, which exhibit interior and/or boundary layers.

Findings

The transversal method of lines presented here results in three-point finite difference equations for the nodal values, treats the diffusion terms implicitly and is unconditionally stable if the reaction terms are treated implicitly. The method is very accurate for problems with the interior and/or boundary layers. For a system of two nonlinearly-coupled, one-dimensional reactiondiffusion equations, the formation, propagation and merging of reactive fronts have been found to be strong function of the diffusion coefficients and reaction rates. For asymmetric ignition, it has been found that, after front merging, the temperature and concentration profiles are almost independent of the ignition conditions.

Originality/value

A new, conservative, transversal method of lines that treats the diffusion terms implicitly and provides piecewise exponential solutions in space without the need for interpolation is presented and applied to someone.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 11
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 18 May 2021

J.I. Ramos

The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of…

Abstract

Purpose

The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of microfluidic droplet ensembles, water flow in unsaturated flows, infiltration, etc., as functions of the powers of the convection and diffusion fluxes and upstream boundary condition; to study numerically the evolution of the wave for two different initial conditions; and to assess the accuracy of several finite difference methods for the solution of the degenerate, nonlinear, advection--diffusion equation that governs the model.

Design/methodology/approach

The theory of ordinary differential equations and several explicit, finite difference methods that use first- and second-order, accurate upwind, central and compact discretizations for the convection terms are used to determine the analytical solution for steadily propagating waves and the evolution of the wave fronts from hyperbolic tangent and piecewise linear initial conditions to steadily propagating waves, respectively. The amplitude and phase errors of the semi-discrete schemes are determined analytically and the accuracy of the discrete methods is assessed.

Findings

For non-zero upstream boundary conditions, it has been found both analytically and numerically that the shock wave is smooth and its steepness increases as the power of the diffusion term is increased and as the upstream boundary value is decreased. For zero upstream boundary conditions, smooth, cusped and sharp shock waves may be encountered depending on the powers of the convection and diffusion terms. For a linear diffusion flux, the shock wave is smooth, whereas, for a quadratic diffusion flux, the wave exhibits a cusped front whose left spatial derivative decreases as the power of the convection term is increased. For higher nonlinear diffusion fluxes, a sharp shock wave is observed. The wave speed decreases as the powers of both the convection and the diffusion terms are increased. The evolution of the solution from hyperbolic tangent and piecewise linear initial conditions shows that the wave back adapts rapidly to its final steady value, whereas the wave front takes much longer, especially for piecewise linear initial conditions, but the steady wave profile and speed are independent of the initial conditions. It is also shown that discretization of the nonlinear diffusion flux plays a more important role in the accuracy of first- and second-order upwind discretizations of the convection term than either a conservative or a non-conservative discretization of the latter. Second-order upwind and compact discretizations of the convection terms are shown to exhibit oscillations at the foot of the wave’s front where the solution is nil but its left spatial derivative is largest. The results obtained with a conservative, centered second--order accurate finite difference method are found to be in good agreement with those of the second-order accurate, central-upwind Kurganov--Tadmor method which is a non-oscillatory high-resolution shock-capturing procedure, but differ greatly from those obtained with a non-conservative, centered, second-order accurate scheme, where the gradients are largest.

Originality/value

A new, one-dimensional model for microfluidic droplet transport, water flow in unsaturated flows, infiltration, etc., that includes high-order convection fluxes and degenerate diffusion, is proposed and studied both analytically and numerically. Its smooth, cusped and sharp shock wave solutions have been determined analytically as functions of the powers of the nonlinear convection and diffusion fluxes and the boundary conditions. These solutions are used to assess the accuracy of several finite difference methods that use different orders of accuracy in space, and different discretizations of the convection and diffusion fluxes, and can be used to assess the accuracy of other numerical procedures for one-dimensional, degenerate, convection--diffusion equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 February 1957

NO one will pretend that work study occupies the departmental status that it is entitled to occupy in the organisation structure. Whereas there is every indication that work…

Abstract

NO one will pretend that work study occupies the departmental status that it is entitled to occupy in the organisation structure. Whereas there is every indication that work study's overall influence is wide‐spread, the fact remains that individual departments are kept well screwed down. This seems strange since increased production and optimum efficiency springs from its functions.

Details

Work Study, vol. 6 no. 2
Type: Research Article
ISSN: 0043-8022

Article
Publication date: 3 May 2016

Masao Shimada, David Tae, Tao Xue, Rohit Deokar and K K Tamma

The purpose of this paper is to present new implementation aspects of unified explicit time integration algorithms, called the explicit GS4-II family of algorithms, of a…

Abstract

Purpose

The purpose of this paper is to present new implementation aspects of unified explicit time integration algorithms, called the explicit GS4-II family of algorithms, of a second-order time accuracy in all the unknowns (e.g. positions, velocities, and accelerations) with particular attention to the moving-particle simulation (MPS) method for solving the incompressible fluids with free surfaces.

Design/methodology/approach

In the present paper, the explicit GS4-II family of algorithms is implemented in the MPS method in the following two different approaches: a direct explicit formulation with the use of the weak incompressibility equation involving the (modified) speed of sound; and a predictor-corrector explicit formulation. The first approach basically follows the concept of the explicit MPS method, presented in the literature, and the latter approach employs a similar concept used in, for example, a fractional-step method in computational fluid dynamics.

Findings

Illustrative numerical examples demonstrate that any scheme within the proposed algorithmic framework captures the physics with the necessary second-order time accuracy and stability.

Originality/value

The new algorithmic framework extended with the GS4-II family encompasses a multitude of pastand new schemes and offers a general purpose and unified implementation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 3/4
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 21 November 2018

Tao Xue, Xiaobing Zhang and K.K. Tamma

A consistent implementation of the general computational framework of unified second-order time accurate integrators via the well-known GSSSS framework in conjunction with the…

Abstract

Purpose

A consistent implementation of the general computational framework of unified second-order time accurate integrators via the well-known GSSSS framework in conjunction with the traditional Finite Difference Method is presented to improve the numerical simulations of reactive two-phase flows.

Design/methodology/approach

In the present paper, the phase interaction evaluation in the present implementation of the reactive two-phase flows has been derived and implemented to preserve the consistency of the correct time level evaluation during the time integration process for solving the two phase flow dynamics with reactions.

Findings

Numerical examples, including the classical Sod shock tube problem and a reactive two-phase flow problem, are exploited to validate the proposed time integration framework and families of algorithms consistently to second order in time accuracy; this is in contrast to the traditional practices which only seem to obtain first-order time accuracy because of the inconsistent time level implementation with respect to the interaction of two phases. The comparisons with the traditional implementation and the advantages of the proposed implementation are given in terms of the improved numerical accuracy in time. The proposed approaches provide a correct numerical simulation implementation to the reactive two-phase flows and can obtain better numerical stability and computational features.

Originality/value

The new algorithmic framework and the consistent time level evaluation extended with the GS4 family encompasses a multitude of past and new schemes and offers a general purpose and unified implementation for fluid dynamics.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 3 March 2023

David Tae and Kumar K. Tamma

The purpose of this study is to further advance the multiple space/time subdomain framework with model reduction. Existing linear multistep (LMS) methods that are second-order time

Abstract

Purpose

The purpose of this study is to further advance the multiple space/time subdomain framework with model reduction. Existing linear multistep (LMS) methods that are second-order time accurate, and useful for practical applications, have a significant limitation. They do not account for separable controllable numerical dissipation of the primary variables. Furthermore, they have little or no significant choices of altogether different algorithms that can be integrated in a single analysis to mitigate numerical oscillations that may occur. In lieu of such limitations, under the generalized single-step single-solve (GS4) umbrella, several of the deficiencies are circumvented.

Design/methodology/approach

The GS4 framework encompasses a wide variety of LMS schemes that are all second-order time accurate and offers controllable numerical dissipation. Unlike existing state-of-art, the present framework permits implicit–implicit and implicit–explicit coupling of algorithms via differential algebraic equations (DAE). As further advancement, this study embeds proper orthogonal decomposition (POD) to further reduce model sizes. This study also uses an iterative convergence check in acquiring sufficient snapshot data to adequately capture the physics to prescribed accuracy requirements. Simple linear/nonlinear transient numerical examples are presented to provide proof of concept.

Findings

The present DAE-GS4-POD framework has the flexibility of using different spatial methods and different time integration algorithms in altogether different subdomains in conjunction with the POD to advance and improve the computational efficiency.

Originality/value

The novelty of this paper is the addition of reduced order modeling features, how it applies to the previous DAE-GS4 framework and the improvement of the computational efficiency. The proposed framework/tool kit provides all the needed flexibility, robustness and adaptability for engineering computations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 33 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 March 2003

M.T. Manzari

A finite element solution procedure is presented for the simulation of transient incompressible fluid flows using triangular meshes. The algorithm is based on the artificial…

1328

Abstract

A finite element solution procedure is presented for the simulation of transient incompressible fluid flows using triangular meshes. The algorithm is based on the artificial compressibility technique in connection with a dual time‐stepping approach. A second‐order discretization is employed to achieve the required accuracy in real‐time while an explicit multistage Runge‐Kutta scheme is used to march in the pseudo‐time domain. A standard Galerkin finite element method, stabilized by using an artificial dissipation technique, is used for the spatial discretization. The performance of the proposed algorithm is demonstrated by solving a set of internal and external problems including flows with purely transient and periodic behavior.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 13 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

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