Search results

1 – 10 of 375
Article
Publication date: 4 May 2012

Stephen Brill and Eric Smith

The purpose of this paper is to present the analytical solution to the Hermite collocation discretization of a quadratically forced steady‐state convection‐diffusion equation in…

Abstract

Purpose

The purpose of this paper is to present the analytical solution to the Hermite collocation discretization of a quadratically forced steady‐state convection‐diffusion equation in one spatial dimension with constant coefficients, defined on a uniform mesh, with Dirichlet boundary conditions. To improve the accuracy of the method “upstream weighting” of the convective term is used in an optimal way. The authors also provide a method to determine where the forcing function should be optimally sampled. Computational examples are given, which support and illustrate the theory of the optimal sampling of the convective and forcing term.

Design/methodology/approach

The authors: extend previously published results (which dealt only with the case of linear forcing) to the case of quadratic forcing; prove the theorem that governs the quadratic case; and then illustrate the results of the theorem using computational examples.

Findings

The algorithm developed for the quadratic case dramatically decreases the error (i.e. the difference between the continuous and numerical solutions).

Research limitations/implications

Because the methodology successfully extends the linear case to the quadratic case, it is hoped that the method can, indeed, be extended further to more general cases. It is true, however, that the level of complexity rose significantly from the linear case to the quadratic case.

Practical implications

Hermite collocation can be used in an optimal way to solve differential equations, especially convection‐diffusion equations.

Originality/value

Since convection‐dominated convection‐diffusion equations are difficult to solve numerically, the results in this paper make a valuable contribution to research in this field.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 June 1994

Michael M. Grigor’ev

The paper gives the description of boundary element method(BEM) with subdomains for the solution ofconvection—diffusion equations with variable coefficients and Burgers’equations

Abstract

The paper gives the description of boundary element method (BEM) with subdomains for the solution of convection—diffusion equations with variable coefficients and Burgers’ equations. At first, the whole domain is discretized into K subdomains, in which linearization of equations by representing convective velocity by the sum of constant and variable parts is carried out. Then using fundamental solutions for convection—diffusion linear equations for each subdomain the boundary integral equation (in which the part of the convective term with the constant convective velocity is not included into the pseudo‐body force) is formulated. Only part of the convective term with the variable velocity, which is, as a rule, more than one order less than convective velocity constant part contribution, is left as the pseudo‐source. On the one hand, this does not disturb the numerical BEM—algorithm stability and, on the other hand, this leads to significant improvement in the accuracy of solution. The global matrix, similar to the case of finite element method, has block band structure whereas its width depends only on the numeration order of nodes and subdomains. It is noted, that in comparison with the direct boundary element method the number of global matrix non‐zero elements is not proportional to the square of the number of nodes, but only to the total number of nodal points. This allows us to use the BEM for the solution of problems with very fine space discretization. The proposed BEM with subdomains technique has been used for the numerical solution of one‐dimensional linear steady‐state convective—diffusion problem with variable coefficients and one‐dimensional non‐linear Burgers’ equation for which exact analytical solutions are available. It made it possible to find out the BEM correctness according to both time and space. High precision of the numerical method is noted. The good point of the BEM is the high iteration convergence, which is disturbed neither by high Reynolds numbers nor by the presence of negative velocity zones.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 4 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 January 1992

R. SERFATY and R.M. COTTA

A hybrid numerical‐analytical approach, based on recent developments in the generalized integral transform technique, is presented for the solution of a class of non‐linear…

Abstract

A hybrid numerical‐analytical approach, based on recent developments in the generalized integral transform technique, is presented for the solution of a class of non‐linear transient convection‐diffusion problems. The original partial differential equation is integral transformed into a denumerable system of coupled non‐linear ordinary differential equations, which is numerically solved for the transformed potentials. The hybrid analysis convergence is illustrated by considering the one‐dimensional non‐linear Burgers equation and numerical results are presented for increasing truncation orders of the infinite ODE system.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 2 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 January 1991

L.C. WROBEL and D.B. DE FIGUEIREDO

This paper presents a boundary element formulation for transient convection‐diffusion problems employing the fundamental solution of the corresponding steady‐state equation with…

Abstract

This paper presents a boundary element formulation for transient convection‐diffusion problems employing the fundamental solution of the corresponding steady‐state equation with constant coefficients and a dual reciprocity approximation. The formulation allows the mathematical problem to be described in terms of boundary values only. Numerical results show that the BEM does not present oscillations or damping of the wave front as appear in other numerical techniques.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 1 no. 1
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: 2 March 2015

Ching-Chang Cho, Cha’o-Kuang Chen and Her-Terng Yau

– The purpose of this paper is to study the mixing performance of the electrokinetically-driven power-law fluids in a zigzag microchannel.

Abstract

Purpose

The purpose of this paper is to study the mixing performance of the electrokinetically-driven power-law fluids in a zigzag microchannel.

Design/methodology/approach

The Poisson-Boltzmann equation, the Laplace equation, the modified Cauchy momentum equation, and the convection-diffusion equation are solved to describe the flow characteristics and mixing performance of power-law fluids in the zigzag microchannel. A body-fitted grid system and a generalized coordinate transformation method are used to model the grid system and transform the governing equations, respectively. The transformed governing equations are solved numerically using the finite-volume method.

Findings

The mixing efficiency of dilatant fluids is higher than that of pseudoplastic fluids. In addition, the mixing efficiency can be improved by increasing the width of the zigzag blocks or extending the total length of the zigzag block region.

Originality/value

The results presented in this study provide a useful insight into potential strategies for enhancing the mixing performance of the power-law fluids in a zigzag microchannel. The results of this study also provide a useful source of reference for the development of efficient and accurate microfluidic systems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 14 June 2011

Adriano Tiribocchi, Antonio Piscitelli, Giuseppe Gonnella and Antonio Lamura

The purpose of this paper is to present numerical results about phase separation of binary fluid mixtures quenched by contact with cold walls.

Abstract

Purpose

The purpose of this paper is to present numerical results about phase separation of binary fluid mixtures quenched by contact with cold walls.

Design/methodology/approach

The thermal phase separation is simulated by using a hybrid lattice Boltzmann method that solves the continuity and the Navier‐Stokes equations. The equations for energy and concentration are solved by using a finite‐difference scheme. This approach provides a complete description of the thermo‐hydrodynamic effects in the mixture.

Findings

A rich variety of domain patterns are found depending on the viscosity and on the heat conductivity of the mixture. Ordered lamellar structures are observed at high viscosity while domains rounded in shape dominate the phase separation at low viscosity, where two scales characterize the growth of domains.

Research limitations/implications

The present approach provides a numerical method that can be extended to other systems such as liquid‐vapor or lamellar systems. Moreover, a three‐dimensional study can give a complete picture of thermo‐hydrodynamic effects.

Originality/value

This paper provides a consistent thermodynamic theoretical framework for a binary fluid mixture and a numerically stable method to simulate them.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 January 1995

A. Pascau, C. Pérez and D. Sánchez

A new discretization scheme named NOTABLE (New Option forthe Treatment of Advection in the Boundary Layer Equations) ispresented. Despite its name, this scheme is intended to be…

Abstract

A new discretization scheme named NOTABLE (New Option for the Treatment of Advection in the Boundary Layer Equations) is presented. Despite its name, this scheme is intended to be used in a general transport equation to discretize the convective term. It is formally third‐order accurate in regions of smooth solution and first‐order accurate at grid points having local maxima. Within the finite‐volume formulation it relates the face values to the nodal values via a non‐linear function. This scheme has been compared with well‐known high‐order schemes like QUICK and it has always given more accurate solutions. After assessing the scheme in several unidimensional test cases for which an exact solution is available, its performance in a complex swirling flow is addressed.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 5 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 4 January 2016

Mehdi Jamei and H Ghafouri

The purpose of this paper is to present a novel sequential implicit discontinuous Galerkin (DG) method for two-phase incompressible flow in porous media. It is based on the…

Abstract

Purpose

The purpose of this paper is to present a novel sequential implicit discontinuous Galerkin (DG) method for two-phase incompressible flow in porous media. It is based on the wetting phase pressure-saturation formulation with Robin boundary condition (Klieber and Riviere, 2006) using H(div) velocity projection.

Design/methodology/approach

The local mass conservation and continuity of normal component of velocity across elements interfaces are enforced by a simple H(div) velocity projection in lowest order Raviart-Thomas (RT0) space. As further improvements, the authors use the weighted averages and the scaled penalties in spatial DG discretization. Moreover, the Chavent-Jaffre slope limiter, as a consistent non-oscillatory limiter, is used for saturation values to avoid the spurious oscillations.

Findings

The proposed model is verified by a pseudo 1D Buckley-Leverett problem in homogeneous media. Two homogeneous and heterogeneous quarter five-spot benchmark problems and a random permeable medium are used to show the accuracy of the method at capturing the sharp front and illustrate the impact of proposed improvements.

Research limitations/implications

The work illustrates incompressible two-phase flow behavior and the capillary pressure heterogeneity between different geological layers is assumed to be negligible.

Practical implications

The proposed model can efficiently be used for modeling of two-phase flow in secondary recovery of petroleum reservoirs and tracing the immiscible contamination in porous media.

Originality/value

The authors present an efficient sequential DG method for immiscible incompressible two-phase flow in porous media with improved performance for detection of sharp frontal interfaces and discontinuities.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 6 July 2015

Di Zhao

The purpose of this paper is to develop Triple Finite Volume Method (tFVM), the author discretizes incompressible Navier-Stokes equation by tFVM, which leads to a special linear…

Abstract

Purpose

The purpose of this paper is to develop Triple Finite Volume Method (tFVM), the author discretizes incompressible Navier-Stokes equation by tFVM, which leads to a special linear system of saddle point problem, and most computational efforts for solving the linear system are invested on the linear solver GMRES.

Design/methodology/approach

In this paper, by recently developed preconditioner Hermitian/Skew-Hermitian Separation (HSS) and the parallel implementation of GMRES, the author develops a quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.

Findings

Computational results show that, the quick solver HSS-pGMRES-tFVM significantly increases the solution speed for saddle point problem from incompressible Navier-Stokes equation than the conventional solvers.

Originality/value

Altogether, the contribution of this paper is that the author developed the quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.

1 – 10 of 375