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Kybernetes, vol. 41 no. 7/8
Type: Research Article
ISSN: 0368-492X

Article
Publication date: 31 May 2011

S. Askari, M.H. Shojaeefard and K. Goudarzi

The purpose of this paper is to carry out a comprehensive study of compressible flow over double wedge and biconvex airfoils using computational fluid dynamics (CFD) and

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Abstract

Purpose

The purpose of this paper is to carry out a comprehensive study of compressible flow over double wedge and biconvex airfoils using computational fluid dynamics (CFD) and three analytical models including shock and expansion wave theory, Busemann's second‐order linearized approximation and characteristic method (CHM).

Design/methodology/approach

Flow over double‐wedge and biconvex airfoils was investigated by the CFD technique using the Spalart‐Allmaras turbulence model for computation of the Reynolds stresses. Flow was considered compressible, two dimensional and steady. The no slip condition was applied at walls and the Sutherland law was used to calculate molecular viscosity as a function of static temperature. First‐order upwind discretization scheme was used for the convection terms. Finite‐volume method was used for the entire solution domain meshed by quadratic computational cells. Busemann's theory, shock and expansion wave technique and CHM were the analytical methods used in this work.

Findings

Static pressure, static temperature and aerodynamic coefficients of the airfoils were calculated at various angles of attack. In addition, aerodynamic coefficients of the double‐wedge airfoil were obtained at various free stream Mach numbers and thickness ratios of the airfoil. Static pressure and aerodynamic coefficients obtained from the analytical and numerical methods were in excellent agreement with average error of 1.62 percent. Variation of the static pressure normal to the walls was negligible in the numerical simulation as well as the analytical solutions. Analytical static temperature far from the walls was consistent with the numerical values with average error of 3.40 percent. However, it was not comparable to the numerical temperature at the solid walls. Therefore, analytical solutions give accurate prediction of the static pressure and the aerodynamic coefficients, however, for the static temperature; they are only reliable far from the solid surfaces. Accuracy of the analytical aerodynamic coefficients is because of accurate prediction of the static pressure which is not considerably influenced by the boundary layer. Discrepancies between analytical and numerical temperatures near the walls are because of dependency of temperature on the boundary layer and viscous heating. Low‐speed flow near walls causes transformation of the kinetic energy of the free stream into enthalpy that leads to high temperature on the solid walls; which is neglected in the analytical solutions.

Originality/value

This paper is useful for researchers in the area of external compressible flows. This work is original.

Details

Engineering Computations, vol. 28 no. 4
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 30 November 2021

Chongbin Zhao, B.E. Hobbs and Alison Ord

The objective of this paper is to develop a semi-analytical finite element method for solving chemical dissolution-front instability problems in fluid-saturated porous media.

Abstract

Purpose

The objective of this paper is to develop a semi-analytical finite element method for solving chemical dissolution-front instability problems in fluid-saturated porous media.

Design/methodology/approach

The porosity, horizontal and vertical components of the pore-fluid velocity and solute concentration are selected as four fundamental unknown variables for describing chemical dissolution-front instability problems in fluid-saturated porous media. To avoid the use of numerical integration, analytical solutions for the property matrices of a rectangular element are precisely derived in a purely mathematical manner. This means that the proposed finite element method is a kind of semi-analytical method. The column pivot element solver is used to solve the resulting finite element equations of the chemical dissolution-front instability problem.

Findings

The direct use of horizontal and vertical components of the pore-fluid velocity as fundamental unknown variables can improve the accuracy of the related numerical solution. The column pivot element solver is useful for solving the finite element equations of a chemical dissolution-front instability problem. The proposed semi-analytical finite element method can produce highly accurate numerical solutions for simulating chemical dissolution-front instability problems in fluid-saturated porous media.

Originality/value

Analytical solutions for the property matrices of a rectangular element are precisely derived for solving chemical dissolution-front instability problems in fluid-saturated porous media. The proposed semi-analytical finite element method provides a useful way for understanding the underlying dynamic mechanisms of the washing land method involved in the contaminated land remediation.

Details

Engineering Computations, vol. 39 no. 5
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 19 May 2022

Jiří Malík and Ondřej Souček

This paper aims to propose a semi-analytical benchmarking framework for enthalpy-based methods used in problems involving phase change with latent heat. The benchmark is…

Abstract

Purpose

This paper aims to propose a semi-analytical benchmarking framework for enthalpy-based methods used in problems involving phase change with latent heat. The benchmark is based on a class of semi-analytical solutions of spatially symmetric Stefan problems in an arbitrary spatial dimension. Via a public repository this study provides a finite element numerical code based on the FEniCS computational platform, which can be used to test and compare any method of choice with the (semi-)analytical solutions. As a particular demonstration, this paper uses the benchmark to test several standard temperature-based implementations of the enthalpy method and assesses their accuracy and stability with respect to the discretization parameters.

Design/methodology/approach

The class of spatially symmetric semi-analytical self-similar solutions to the Stefan problem is found for an arbitrary spatial dimension, connecting some of the known results in a unified manner, while providing the solutions’ existence and uniqueness. For two chosen standard semi-implicit temperature-based enthalpy methods, the numerical error assessment of the implementations is carried out in the finite element formulation of the problem. This paper compares the numerical approximations to the semi-analytical solutions and analyzes the influence of discretization parameters, as well as their interdependence. This study also compares accuracy of these methods with other traditional approach based on time-explicit treatment of the effective heat capacity with and without iterative correction.

Findings

This study shows that the quantitative comparison between the semi-analytical and numerical solutions of the symmetric Stefan problems can serve as a robust tool for identifying the optimal values of discretization parameters, both in terms of accuracy and stability. Moreover, this study concludes that, from the performance point of view, both of the semi-implicit implementations studied are equivalent, for optimal choice of discretization parameters, they outperform the effective heat capacity method with iterative correction in terms of accuracy, but, by contrast, they lose stability for subcritical thickness of the mushy region.

Practical implications

The proposed benchmark provides a versatile, accessible test bed for computational methods approximating multidimensional phase change problems. The supplemented numerical code can be directly used to test any method of choice against the semi-analytical solutions.

Originality/value

While the solutions of the symmetric Stefan problems for individual spatial dimensions can be found scattered across the literature, the unifying perspective on their derivation presented here has, to the best of the authors’ knowledge, been missing. The unified formulation in a general dimension can be used for the systematic construction of well-posed, reliable and genuinely multidimensional benchmark experiments.

Details

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

Keywords

Article
Publication date: 1 March 2005

Chongbin Zhao, B.E. Hobbs, A. Ord, Ge Lin and H.B. Mühlhaus

In many scientific and engineering fields, large‐scale heat transfer problems with temperature‐dependent pore‐fluid densities are commonly encountered. For example, heat…

Abstract

Purpose

In many scientific and engineering fields, large‐scale heat transfer problems with temperature‐dependent pore‐fluid densities are commonly encountered. For example, heat transfer from the mantle into the upper crust of the Earth is a typical problem of them. The main purpose of this paper is to develop and present a new combined methodology to solve large‐scale heat transfer problems with temperature‐dependent pore‐fluid densities in the lithosphere and crust scales.

Design/methodology/approach

The theoretical approach is used to determine the thickness and the related thermal boundary conditions of the continental crust on the lithospheric scale, so that some important information can be provided accurately for establishing a numerical model of the crustal scale. The numerical approach is then used to simulate the detailed structures and complicated geometries of the continental crust on the crustal scale. The main advantage in using the proposed combination method of the theoretical and numerical approaches is that if the thermal distribution in the crust is of the primary interest, the use of a reasonable numerical model on the crustal scale can result in a significant reduction in computer efforts.

Findings

From the ore body formation and mineralization points of view, the present analytical and numerical solutions have demonstrated that the conductive‐and‐advective lithosphere with variable pore‐fluid density is the most favorite lithosphere because it may result in the thinnest lithosphere so that the temperature at the near surface of the crust can be hot enough to generate the shallow ore deposits there. The upward throughflow (i.e. mantle mass flux) can have a significant effect on the thermal structure within the lithosphere. In addition, the emplacement of hot materials from the mantle may further reduce the thickness of the lithosphere.

Originality/value

The present analytical solutions can be used to: validate numerical methods for solving large‐scale heat transfer problems; provide correct thermal boundary conditions for numerically solving ore body formation and mineralization problems on the crustal scale; and investigate the fundamental issues related to thermal distributions within the lithosphere. The proposed finite element analysis can be effectively used to consider the geometrical and material complexities of large‐scale heat transfer problems with temperature‐dependent fluid densities.

Details

Engineering Computations, vol. 22 no. 2
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 20 August 2021

Salam Adel Al-Bayati and Luiz C. Wrobel

The purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed…

Abstract

Purpose

The purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed for one- and two-dimensional steady-state problems, to analyse transient convection–diffusion problems associated with first-order chemical reaction.

Design/methodology/approach

The mathematical modelling has used a dual reciprocity approximation to transform the domain integrals arising in the transient equation into equivalent boundary integrals. The integral representation formula for the corresponding problem is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. The finite difference method is used to simulate the time evolution procedure for solving the resulting system of equations. Three different radial basis functions have been successfully implemented to increase the accuracy of the solution and improving the rate of convergence.

Findings

The numerical results obtained demonstrate the excellent agreement with the analytical solutions to establish the validity of the proposed approach and to confirm its efficiency.

Originality/value

Finally, the proposed BEM and DRBEM numerical solutions have not displayed any artificial diffusion, oscillatory behaviour or damping of the wave front, as appears in other different numerical methods.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 February 2002

W. Song and B.Q. Li

This paper describes the finite element solution of conjugate heat transfer problems with and without the use of gap elements. Direct and iterative methods to incorporate…

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Abstract

This paper describes the finite element solution of conjugate heat transfer problems with and without the use of gap elements. Direct and iterative methods to incorporate gap elements into a general finite element program are presented, along with their advantages and disadvantages of the two gap element treatments in the framework of finite elements. The numerical performance of the iterative gap element treatment is discussed in detail in comparison with analytical solutions for both 2‐ and 3‐D gap conductance problems. Numerical tests show that the number of iterations depends on the non‐dimensional number Bi = hL/k, and it increases approximately linearly with Bi for Bi≥0.6. Here, for gap heat transfer problems, h is taken to be the inverse of the contact resistance. This conclusion holds true for both 2‐ and 3‐D problems, for both linear and quadratic elements and for both transient and steady state calculations. Further numerical results for conjugate heat transfer problems encountered in heat exchanger and micro chemical reactors are computed using the gap element approach, the direct numerical simulations and analytical solutions whenever solvable. The results reveal that for the standard heat exchanger designs, an accurate prediction of temperature distribution in the moving streams must take into consideration the radial temperature distribution and the accuracy of the calculations depends on the non‐dimensional number Bi = hR/2k. From gap element calculations, it is found that classical analytical solutions are valid for a heat transfer analysis of an exchanger system, only when Bi<0.1. This important point so far has been neglected in virtually all the textbooks on heat transfer and must be included to complete the heat transfer theory for heat exchanger designs. Results also suggest that for thermal fluids systems with chemical reactions such as micro fuel cells, the gap element approach yields accurate results only when the heat transfer coefficient that accounts for the chemical reactions is used. However, when these heat transfer coefficients are not available, direct numerical simulations should be used for an accurate prediction of the thermal performance of these systems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 12 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 10 December 2018

A.A. Avramenko, N.P. Dmitrenko, I.V. Shevchuk, A.I. Tyrinov and V.I. Shevchuk

The paper aims to consider heat transfer in incompressible flow in a rotating flat microchannel with allowance for boundary slip conditions of the first and second order…

Abstract

Purpose

The paper aims to consider heat transfer in incompressible flow in a rotating flat microchannel with allowance for boundary slip conditions of the first and second order. The novelty of the paper encompasses analytical and numerical solutions of the problem, with the latter based on the lattice Boltzmann method (LBM). The analytical solution of the problem includes relations for the velocity and temperature profiles and for the Nusselt number depending on the rotation rate of the microchannel and slip velocity. It was demonstrated that the velocity profiles at high rotation rates transform from parabolic to M-shaped with a minimum at the channel axis. The temperature profiles tend to become uniform (i.e. almost constant). An increase in the channel rotation rate contributes to the increase in the Nusselt number. An increase in the Prandtl number causes a similar effect. The trend caused by the effect of the second-order slip boundary conditions depends on the closure hypothesis. It is shown that heat transfer in a flat microchannel can be successfully modeled using the LBM methodology, which takes into account the second-order boundary conditions.

Design/methodology/approach

The paper is based on the comparisons of an analytical solution and a numerical solution, which employs the lattice Boltzmann method. Both mathematical approaches used the first-order and second-order slip boundary conditions. The results obtained using both methods agree well with each other.

Findings

The analytical solution of the problem includes relations for the velocity and temperature profiles and for the Nusselt number depending on the rotation rate of the microchannel and slip velocity. It was demonstrated that the velocity profiles at high rotation rates transform from parabolic to M-shaped with a minimum at the channel axis. The temperature profiles tend to become uniform (i.e. almost constant). The increase in the channel rotation rate contributes to the increase in the Nusselt number. An increase in the Prandtl number causes the similar effect. The trend caused by the effect of the second-order slip boundary conditions depends on the closure hypothesis. It is shown that heat transfer in a flat microchannel can be successfully modeled using the LBM methodology, which considers the second-order boundary conditions.

Originality/value

The novelty of the paper encompasses analytical and numerical solutions of the problem, whereas the latter are based on the LBM.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 June 2000

A. Savini

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic…

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Abstract

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 19 no. 2
Type: Research Article
ISSN: 0332-1649

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

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