Free convection of a dusty‐gas flow along a semi‐infinite vertical cylinder

An analysis is performed to study the free convection of a dusty‐gas flow along a semi‐infinite isothermal vertical cylinder. The governing equations of the flow problem are transformed into non‐dimensional form and the resulting nonlinear, coupled parabolic partial differential equations have been solved numerically using an implicit finite difference scheme of Crank–Nicholson type. The flow variables such as gas–velocity, dust‐particle velocity and temperature, shearing stress and heat transfer coefficients are calculated numerically for various parameters occurring in the problem. It is observed that due to the presence of dust particles, the gas velocity is found to decrease. Copyright © 2009 John Wiley & Sons, Ltd.     

Potential flow around obstacles using the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. The method works by weakening the governing differential equations in one co‐ordinate direction through the introduction of shape functions, then solving the weakened equations analytically in the other (radial) co‐ordinate direction. These co‐ordinate directions are defined by the geometry of the domain and a scaling centre. The method can be employed for both bounded and unbounded domains. This paper applies the method to problems of potential flow around streamlined and bluff obstacles in an infinite domain. The method is derived using a weighted residual approach and extended to include the necessary velocity boundary conditions at infinity. The ability of the method to model unbounded problems is demonstrated, together with its ability to model singular points in the near field in the case of bluff obstacles. Flow fields around circular and square cylinders are computed, graphically illustrating the accuracy of the technique, and two further practical examples are also presented. Comparisons are made with boundary element and finite difference solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

BEM solution to magnetohydrodynamic flow in a semi‐infinite duct

We consider the magnetohydrodynamic flow that is laminar and steady of a viscous, incompressible, and electrically conducting fluid in a semi‐infinite duct under an externally applied magnetic field. The flow is driven by the current produced by a pressure gradient. The applied magnetic field is perpendicular to the semi‐infinite walls that are kept at the same magnetic field value in magnitude but opposite in sign. The wall that connects the two semi‐infinite walls is partly non‐conducting and partly conducting (in the middle). A BEM solution was obtained using a fundamental solution that enables to treat the magnetohydrodynamic equations in coupled form with general wall conductivities. The inhomogeneity in the equations due to the pressure gradient was tackled, obtaining a particular solution, and the BEM was applied with a fundamental solution of coupled homogeneous convection–diffusion type partial differential equations. Constant elements were used for the discretization of the boundaries (y = 0, ?a ? x ? a) and semi‐infinite walls at x = ±a, by keeping them as finite since the boundary integral equations are restricted to these boundaries due to the regularity conditions as y → ∞ . The solution is presented in terms of equivelocity and induced magnetic field contours for several values of Hartmann number (M), conducting length (l), and non‐conducting wall conditions (k). The effect of the parameters on the solution is studied. Flow rates are also calculated for these values of parameters. Copyright © 2011 John Wiley & Sons, Ltd.     

An implicit pseudo‐spectral multi‐domain method for the simulation of incompressible flows

A pseudo‐spectral method for the solution of incompressible flow problems based on an iterative solver involving an implicit treatment of linearized convective terms is presented. The method allows the treatment of moderately complex geometries by means of a multi‐domain approach and it is able to cope with non‐constant fluid properties and non‐orthogonal problem domains. In addition, the fully implicit scheme yields improved stability properties as opposed to semi‐implicit schemes commonly employed. Key components of the method are a Chebyshev collocation discretization, a special pressure–correction scheme, and a restarted GMRES method with a preconditioner derived from a fast direct solver. The performance of the proposed method is investigated by considering several numerical examples of different complexity, and also includes comparisons to alternative solution approaches based on finite‐volume discretizations. Copyright © 2003 John Wiley & Sons, Ltd.     

On the efficiency of semi‐implicit and semi‐Lagrangian spectral methods for the calculation of incompressible flows

Classical semi‐implicit backward Euler/Adams–Bashforth time discretizations of the Navier–Stokes equations induce, for high‐Reynolds number flows, severe restrictions on the time step. Such restrictions can be relaxed by using semi‐Lagrangian schemes essentially based on splitting the full problem into an explicit transport step and an implicit diffusion step. In comparison with the standard characteristics method, the semi‐Lagrangian method has the advantage of being much less CPU time consuming where spectral methods are concerned. This paper is devoted to the comparison of the ‘semi‐implicit’ and ‘semi‐Lagrangian’ approaches, in terms of stability, accuracy and computational efficiency. Numerical results on the advection equation, Burger's equation and finally two‐ and three‐dimensional Navier–Stokes equations, using spectral elements or a collocation method, are provided. Copyright © 2001 John Wiley & Sons, Ltd.     

Slip effects and heat transfer analysis in a viscous fluid over an oscillatory stretching surface

In this study, we investigate the heat transfer problem in a viscous fluid over an oscillatory infinite sheet with slip condition. The sheet is moved back and forth in its own plane. The derived problem involves a dimensionless parameter indicating the relative magnitude of frequency to sheet stretching rate. A system of non‐linear partial differential equations is solved numerically using the finite‐difference scheme, in which a coordinate transformation is employed to transform the semi‐infinite physical space to a bounded computational domain. The physical features of interesting parameters on the velocity and temperature distributions are shown graphically and discussed. The values of the skin‐friction coefficient and the local Nusselt number are given in tabular form. Copyright © 2008 John Wiley & Sons, Ltd.     

A Sinc‐collocation method with boundary treatment for the Stokes equations

In Stokes equations the velocity u and the pressure p are coupled together by the imcompressibility condition div u =0 which makes the equations difficult to solve numerically. In this paper, a method named Sinc‐collocation method with boundary treatment (SCMBT) is applied to the Stokes equations. The numerical results show that our method is of high accuracy, of good convergence with little computational effort. Copyright © 2006 John Wiley & Sons, Ltd.     

A semi‐implicit finite difference method for non‐hydrostatic,free‐surface flows

In this paper a semi‐implicit finite difference model for non‐hydrostatic, free‐surface flows is analyzed and discussed. It is shown that the present algorithm is generally more accurate than recently developed models for quasi‐hydrostatic flows. The governing equations are the free‐surface Navier–Stokes equations defined on a general, irregular domain of arbitrary scale. The momentum equations, the incompressibility condition and the equation for the free‐surface are integrated by a semi‐implicit algorithm in such a fashion that the resulting numerical solution is mass conservative and unconditionally stable with respect to the gravity wave speed, wind stress, vertical viscosity and bottom friction. Copyright © 1999 John Wiley & Sons, Ltd.     

A high‐order mass‐lumping procedure for B‐spline collocation method with application to incompressible flow simulations

This paper presents new developments of the staggered spline collocation method for cost‐effective solution to the incompressible Navier–Stokes equations. Maximal decoupling of the velocity and the pressure is obtained by using the fractional step method of Gresho and Chan, allowing the solution to sparse elliptic problems only. In order to preserve the high‐accuracy of the B‐spline method, this fractional step scheme is used in association with a sparse approximation to the inverse of the consistent mass matrix. Such an approximation is constructed from local spline interpolation method, and represents a high‐order generalization of the mass‐lumping technique of the finite‐element method. A numerical investigation of the accuracy and the computational efficiency of the resulting semi‐consistent spline collocation schemes is presented. These schemes generate a stable and accurate unsteady Navier–Stokes solver, as assessed by benchmark computations. Copyright © 2003 John Wiley & Sons, Ltd.     

Collocated discrete least squares meshless (CDLSM) method for the solution of transient and steady‐state hyperbolic problems

A collocated discrete least squares meshless method for the solution of the transient and steady‐state hyperbolic problems is presented in this paper. The method is based on minimizing the sum of the squared residuals of the governing differential equation at some points chosen in the problem domain as collocation points. The collocation points are generally different from nodal points, which are used to discretize the problem domain. A moving least squares method is employed to construct the shape functions at nodal points. The coefficient matrix is symmetric and positive definite even for non‐symmetric hyperbolic differential equations and can be solved efficiently with iterative methods. The proposed method is a truly meshless method and does not require numerical integration. Advantages of the collocation points are shown to be threefold: First, the collocation points are shown to be responsible for stabilizing the method in particular when problems with shocked solution are attempted. Second, the collocation points are also shown to improve the accuracy of the solution even for problems with smooth solutions. Third, the collocation points are shown to contribute to the efficiency of the method when solving steady‐state problems via faster convergence of the resulting algorithm. The ability of the method and in particular the effect of collocation points are tested against a series of one‐dimensional transient and steady‐state benchmark examples from the literature and the results are presented. A sensitivity analysis is also carried out to investigate the effect of the base polynomials on the accuracy and convergence characteristics of the method in solving steady‐state problems. The results show the ability of the proposed method to accurately solve difficult hyperbolic problems considered. The method is also shown to be particularly stable for problems with shocked solution due to the inherent stabilizing mechanism of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Wind‐driven currents in a sea with a variable eddy viscosity calculated via a Sinc–Galerkin technique

The Sinc–Galerkin method is presented as a new and potentially useful extension of the spectral method in numerical oceanography. To describe and illustrate the technique, a Sinc–Galerkin procedure is used to infer the sensitivity of wind‐driven sub‐surface currents in coastal regions and semi‐enclosed seas when the vertical eddy viscosity coefficient is represented as a continuously differentiable function of depth. Problems with exact solutions are used to explore the accuracy and exponential convergence of expansions using composite translated sinc functions as a basis set. To illustrate the essential idea, we describe applications of the Sinc–Galerkin technique to modifications of the Ekman wind‐drift current problem. Copyright © 2000 John Wiley & Sons, Ltd.     

Viscous dissipative fluid flow past a semi‐infinite vertical plate with variable surface temperature

An analysis of the effect of viscous dissipative heat on two‐dimensional viscous incompressible fluid flow past a semi‐infinite vertical plate with variable surface temperature is carried out. The dimensionless governing equations are unsteady, two‐dimensional, coupled, and non‐linear governing equations. A most accurate, unconditionally stable and fast converging implicit finite‐difference scheme is used to solve the non‐dimensional governing equations. Velocity and temperature of the flow have been presented graphically for various parameters occurring in the problem. The local and average skin friction and Nusselt number are also shown graphically. It is observed that greater viscous dissipative heat causes a rise in the temperature. Copyright © 2007 John Wiley & Sons, Ltd.     

MHD Falkner-Skan flow of Maxwell fluid by rational Chebyshev collocation method

The magnetohydrodynamics （MHD） Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.     

Numerical simulation of high‐Reynolds number flow around circular cylinders by a three‐step FEM–BEM model

An innovative computational model, developed to simulate high‐Reynolds number flow past circular cylinders in two‐dimensional incompressible viscous flows in external flow fields is described in this paper. The model, based on transient Navier–Stokes equations, can solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the projection method. The pressure is assumed to be zero at infinite boundary and the external flow field is simulated using a direct boundary element method (BEM) by solving a pressure Poisson equation. A three‐step finite element method (FEM) is used to solve the momentum equations of the flow. The present model is applied to simulate high‐Reynolds number flow past a single circular cylinder and flow past two cylinders in which one acts as a control cylinder. The simulation results are compared with experimental data and other numerical models and are found to be feasible and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

High‐order boundary conditions for linearized shallow water equations with stratification,dispersion and advection

The two‐dimensional linearized shallow water equations are considered in unbounded domains with density stratification. Wave dispersion and advection effects are also taken into account. The infinite domain is truncated via a rectangular artificial boundary ??, and a high‐order open boundary condition (OBC) is imposed on ??. Then the problem is solved numerically in the finite domain bounded by ??. A recently developed boundary scheme is employed, which is based on a reformulation of the sequence of OBCs originally proposed by Higdon. The OBCs can easily be used up to any desired order. They are incorporated here in a finite difference scheme. Numerical examples are used to demonstrate the performance and advantages of the computational method, with an emphasis is on the effect of stratification. Published in 2004 by John Wiley & Sons, Ltd.     

A semi‐Lagrangian level set method for incompressible Navier–Stokes equations with free surface

In this paper, we formulate a level set method in the framework of finite elements‐semi‐Lagrangian methods to compute the solution of the incompressible Navier–Stokes equations with free surface. In our formulation, we use a quasi‐monotone semi‐Lagrangian scheme, which is both unconditionally stable and essentially non oscillatory, to compute the advective terms in the Navier–Stokes equations, the transport equation and the equation of the reinitialization stage for the level set function. The method we propose is quite robust and flexible with regard to the mesh and the geometry of the domain, as well as the magnitude of the Reynolds number. We illustrate the performance of the method in several examples, which range from a benchmark problem to test the volume conservation property of the method to the flow past a NACA0012 foil at high Reynolds number. Copyright © 2005 John Wiley & Sons, Ltd.     

Simulation of the Navier–Stokes equations in three dimensions with a spectral collocation method

This paper describes a nonlinear, three‐dimensional spectral collocation method for the simulation of the incompressible Navier–Stokes equations under the Boussinesq approximation, motivated by geophysical and environmental flows. These flows are driven by the interaction of stratified fluid with topography, which this model accurately accounts for by using a mapped coordinate system. The spectral collocation method is implemented with both a Fourier trigonometric expansion and the Chebyshev polynomials, as appropriate for the domain boundary conditions. The coordinate mapping prohibits the use of existing, fast solution methods that rely on the separation of variables, so a preconditioner based on the approximate solution of a corresponding finite‐difference problem with geometric multigrid is used. The model is parallelized with the Message Passing Interface library, and it runs effectively on shared and distributed‐memory systems. Copyright © 2013 John Wiley & Sons, Ltd.     

物理信息依赖核函数配点法的研究进展

在过去几十年里, 尽管有限元等传统计算方法已被成功用于众多科学与工程领域, 但是其在数值模拟无限域波传播、大尺寸比结构、工程反演和移动边界问题时仍面临计算量大、计算效率低、网格生成困难等计算难题. 本文介绍一类基于物理信息依赖核函数的无网格配点法及其在上述难点问题中的应用. 物理信息依赖核函数配点法的关键在于构建能反映问题微分控制方程物理信息的基函数. 基于这些物理信息依赖核函数, 该方法无需/仅需少量配点对所求微分控制方程进行离散, 即可有效提高计算效率. 本文首先介绍满足常见齐次微分方程的基本解、调和函数、径向Trefftz函数以及T完备函数等典型物理信息依赖核函数. 接着依次介绍非齐次、非均质、非稳态以及隐式微分方程构造物理信息依赖核函数的方法. 随后, 根据所求问题特点, 选用全域配点或局部配点技术, 建立相应的物理信息依赖核函数配点法. 最后, 通过几个典型算例验证所提物理信息依赖核函数配点法的有效性.      

A computational solution of hydromagnetic‐free convective flow past a vertical plate in a rotating heat‐generating fluid with Hall and ion‐slip currents

Taking Hall and ion‐slip current into account, the unsteady magnetohydrodynamic heat‐generating free convective flow of a partially ionized gas past an infinite vertical plate in a rotating frame of reference is investigated theoretically. A computer program using finite elements is employed to solve the coupled non‐linear differential equations for velocity and temperature fields. The effects of Hall and ion‐slip currents as well as the other parameters entering into the problem are discussed extensively and shown graphically. Copyright © 2006 John Wiley & Sons, Ltd.