Study of the contraction flow using grid generation technique

We study a model procedure to solve the incompressible Navier-Stokes equations on the flow inside contraction geometry. The governing equations are expressed in the primitive variable formulation. A rectangular computational plane is arises by elliptic grid generation technique. The numerical solution is based on a technique of automatic numerical generation of a curvilinear coordinate system. By transformed the governing equation into computational plane. The time dependent momentum equations are solved explicitly for the velocity field using the explicit marching procedure, the continuity equation is applied at each grid point in the solution of pressure equation, while the successive over relaxation (SOR) method is used for the Neumann problem for pressure. We will apply the technique on several irregular-shape.     

A random integral quadrature method for numerical analysis of the second kind of Volterra integral equations

In this paper, a novel meshless technique termed the random integral quadrature (RIQ) method is developed for the numerical solution of the second kind of the Volterra integral equations. The RIQ method is based on the generalized integral quadrature (GIQ) technique, and associated with the Kriging interpolation function, such that it is regarded as an extension of the GIQ technique. In the GIQ method, the regular computational domain is required, in which the field nodes are scattered along straight lines. In the RIQ method however, the field nodes can be distributed either uniformly or randomly. This is achieved by discretizing the governing integral equation with the GIQ method over a set of virtual nodes that lies along straight lines, and then interpolating the function values at the virtual nodes over all the field nodes which are scattered either randomly or uniformly. In such a way, the governing integral equation is converted approximately into a system of linear algebraic equations, which can be easily solved.     

On the practical use of the spectral homotopy analysis method and local linearisation method for unsteady boundary-layer flows caused by an impulsively stretching plate

This paper expands the ideas of the spectral homotopy analysis method to apply them, for the first time, on non-linear partial differential equations. The spectral homotopy analysis method (SHAM) is a numerical version of the homotopy analysis method (HAM) which has only been previously used to solve non-linear ordinary differential equations. In this work, the modified version of the SHAM is used to solve a partial differential equation (PDE) that models the problem of unsteady boundary layer flow caused by an impulsively stretching plate. The robustness of the SHAM approach is demonstrated by its flexibility to allow linear operators that are partial derivatives with variable coefficients. This is seen to significantly improve the convergence and accuracy of the method. To validate accuracy of the the present SHAM results, the governing PDEs are also solved using a novel local linearisation technique coupled with an implicit finite difference approach. The two approaches are compared in terms of accuracy, speed of convergence and computational efficiency.     

The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation

In this article we describe a numerical method to solve a nonhomogeneous diffusion equation with arbitrary geometry by combining the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunction expansion method (EEM). This forms a meshless numerical scheme of the MFS‐MPS‐EEM model to solve nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for any time and any shape. Nonhomogeneous diffusion equation with complex domain can be separated into a Poisson equation and a homogeneous diffusion equation using this model. The Poisson equation is solved by the MFS‐MPS model, in which the compactly supported radial basis functions are adopted for the MPS. On the other hand, utilizing the EEM the diffusion equation is first translated to a Helmholtz equation, which is then solved by the MFS together with the technique of the singular value decomposition (SVD). Since the present meshless method does not need mesh generation, nodal connectivity, or numerical integration, the computational effort and memory storage required are minimal as compared with other numerical schemes. Test results for two 2D diffusion problems show good comparability with the analytical solutions. The proposed algorithm is then extended to solve a problem with irregular domain and the results compare very well with solutions of a finite element scheme. Therefore, the present scheme has been proved to be very promising as a meshfree numerical method to solve nonhomogeneous diffusion equations with time‐independent source terms of any time frame, and for any arbitrary geometry. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Navier Stokes model of solitary wave collision

Wave collision and its interaction characteristics is one of the important challenges in coastal engineering. This article concerns the collision of solitary waves over a horizontal bottom considering unsteady, incompressible viscous flow with free surface. The method solves the two dimensional Naiver–Stokes equations for conservation of momentum, continuity equation, and full nonlinear kinematic free-surface equation for Newtonian fluids, as the governing equations in a vertical plan. A mapping was developed to trace the deformed free surface encountered during wave propagation, transforms and interaction by transferring the governing equations from the physical domain to a computational domain. Also a numerical scheme is developed using finite element modeling technique in order to predict the solitary wave collision. Consequently results compared with other researches and show the inelastic behavior of solitary wave collision.     

The postbuckling analysis of laminated circular plate with elliptic delamination

Based on the Von Karman plate theory, considering the effect of transverse shear deformation, and using the method of the dissociated three regions, the postbuckling governing equations for the axisymmetric laminated circular plates with elliptical delamination are derived. By using the orthogonal point collocation method, the governing equations, boundary conditions and continuity conditions are transformed into a group of nonlinear algebraically equation and the equations are solved with the alternative method. In the numerical examples, the effects of various elliptical in shape, delamination depth and different material properties on buckling and postbuckling of the laminated circular plates are discussed and the numerical results are compared with available data.     

An operator-splitting algorithm for the three-dimensional diffusion equation

A second-order-accurate and unconditionally stable operator-splitting algorithm for the three-dimensional diffusion equation is presented in this article. The governing equation is split into three one-dimensional equations, and the split equations are solved by a finite-element method. The simulation characteristics of the algorithm are demonstrated by numerical experiments. © 1995 John Wiley & Sons, Inc.     

Effect of variable viscosity and viscous dissipation on the Hagen‐Poiseuille flow and entropy generation

In this article, the steady‐state flow of a Hagen‐Poiseuille modelin a circular pipe is considered and entropy generation due tofluid friction and heat transfer is examined. Because of variationin fluid viscosity, the entropy generation in the flow varies. Inhis model, Arrhenius law is applied for temperature equation‐dependent viscosity, and the influence of viscosity parameters on the entropy generation number and distribution of temperature and velocity is investigated. The governing momentum and energy equations, which are coupled due to the dissipative term in the energy equation, were solved by analytical techniques. The solutions of equations via perturbation method and homotopy perturbation method are obtained and then compared with those of numerical solutions. It is found that the fluid viscosity influences considerably the temperature distribution in the fluid close to the pipe wall, and increasing pipe wall temperature enhances the rate of entropy generation. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 529–540, 2011     

Effects of Soret Dufour,chemical reaction and thermal radiation on MHD non-Darcy unsteady mixed convective heat and mass transfer over a stretching sheet

A study has been carried out to analyze the combined effects of Soret (thermal-diffusion) and Dufour (diffusion-thermo) on unsteady MHD non-Darcy mixed convection over a stretching sheet embedded in a saturated porous medium in the presence of thermal radiation, viscous dissipation and first-order chemical reaction. Energy equation takes into account of viscous dissipation, thermal radiation and Soret effects. The governing differential equations are transformed into a set of non-linear coupled ordinary differential equations and solved using similarity analysis with numerical technique using appropriate boundary conditions for various physical parameters. The numerical solution for the governing nonlinear boundary value problem is based on shooting algorithm with Runge–Kutta–Fehlberg integration scheme over the entire range of physical parameters. The effects of various physical parameters on the dimensionless velocity, temperature and concentration profiles are depicted graphically and analyzed in detail. Favorable comparisons with previously published work on various special cases of the problem are obtained. Numerical results for local skin-friction, local Nusselt number, and local Sherwood number are tabulated for different physical parameters.     

Generalized eigenvalue decomposition of the field autocorrelation in correlation diffusion of photons in turbid media

We propose a novel numerical method based on a generalized eigenvalue decomposition for solving the diffusion equation governing the correlation diffusion of photons in turbid media. Medical imaging modalities such as diffuse correlation tomography and ultrasound‐modulated optical tomography have the (elliptic) diffusion equation parameterized by a time variable as the forward model. Hitherto, for the computation of the correlation function, the diffusion equation is solved repeatedly over the time parameter. We show that the use of a certain time‐independent generalized eigenfunction basis results in the decoupling of the spatial and time dependence of the correlation function, thus allowing greater computational efficiency in arriving at the forward solution. Besides presenting the mathematical analysis of the generalized eigenvalue problem on the basis of spectral theory, we put forth the numerical results that compare the proposed numerical method with the standard technique for solving the diffusion equation. Copyright © 2012 John Wiley & Sons, Ltd.     

Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations

We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg–deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained.     

Entropy Generation and Thermal Radiation Effects on Nanofluid over Permeable Stretching Sheet

In the present paper, effects of entropy generation and nonlinear thermal radiation on Jeffery nanofluid over permeable stretching sheet with partial slip effect were analyzed. The suitable similarity transformation is utilized for the reduction of a set of governing equations, which are solved by using Differential Transformation Method (DTM) with the help of symbolic software MATHEMATICA. The accuracy of impact of slip parameter on coefficient of skin friction by using DTM and numerical method (Shooting technique with fourth-order Runge-Kutta) is illustrated and good agreement is found. Further, velocity, temperature, nanoparticle volume fraction and entropy generation profiles are shown graphically and studied in detail for various physical parameters. We notice that, as slip parameter rises the velocity and entropy generation profile rises. Enhancement in the effect of nonlinear thermal radiation reduces the entropy generation.     

用CT-VOF法在数值波试验槽中生成线性和非线性波

为渡水槽中波的模拟和传播提出了二维的数值模型．假设流动的流体为粘性、不可压缩的,并将Navier-Stokes方程和连续性方程作为控制方程．用标准的k-ε模型来模拟紊流流动;用交错网格的有限差分法,离散化Navier-Stokes方程;并用简化的标识和单元(SMAC)方法进行求解．使用活塞型波发生器生成并传播波;数值渡水槽的端部采用敞开式的边界条件．为了证明模型的有效性,进行了一些标准的试验,如顶盖驱动的方腔测试试验、单向的常速度场试验以及干燥河床上的溃坝试验．为了论证方法的性能及其精度,将所生成波的结果与已有波理论的结果进行比较．最后,采用群集技术(CT)生成网格,并提出最佳的网格生成条件．     

Numerical approach for the Hunter Saxton equation arising in liquid crystal model through Cocktail party graphs Clique polynomial

In this study, we extracted the Clique polynomials from the Cocktail party graph (CPG) and generated the generalized operational matrix of integration through the Clique polynomials of CPG. Then developed an effective computational technique called Cocktail party graphs Clique polynomial collocation method (CCCM) to obtain an approximate numerical solution for the nonlinear liquid crystal model called the Hunter-Saxton equation (HSE). The operational matrix of CPG has been used to reduce HSE into an algebraic system of nonlinear equations that makes the solution quite superficial. These nonlinear algebraic equations have been solved by the Newton Raphson method. This projected that CCCM is considerably efficacious on the computational ground for higher accuracy and convergence of numerical solutions. The solution of the HSE is presented through figures and tables for different values of and . The accuracy and efficiency of the proposed technique are analyzed based on absolute errors. We also provided the convergence and error analysis of our method and verified the results through two examples to confirm the accuracy of the theoretical results.     

一维气液两相漂移模型全隐式AUSMV算法研究

气液两相漂移模型显式AUSMV（advection upstream splitting method combined with flux vector splitting method）算法的时间步长受限于CFL（Courant-Friedrichs-Lewy）条件，为了提高计算效率，建立了一种全隐式AUSMV算法求解气液两相漂移模型.采用AUSM格式结合FVS（flux vector splitting）格式构造连续方程和运动方程的对流项数值通量，AUSM格式构造压力项数值通量.离散控制方程是非线性方程组，采用六阶Newton(牛顿)法结合数值Jacobi矩阵求解.计算经典算例Zuber-Findlay激波管问题和复杂漂移关系变质量流动问题，结果分析表明:全隐式AUSMV算法，色散效应小，无数值震荡，计算精度高.在压力波波速高的条件下，可以显著提高计算效率，耗散效应小.     

Horizontal water flow in unsaturated porous media using a fractional integral method with an adaptive time step

Predicting the horizontal groundwater flow in unsaturated porous media is a challenge in many areas of science and engineering. The governing equation associated with this phenomenon is a nonlinear partial differential equation known as the Richards equation. However, the numerical results obtained using this equation can differ substantially from the experimental results. In order to overcome this difficulty, a new version of the Richards equation was proposed recently that considers a time derivative of fractional order. In this study, we present a numerical method for solving this fractional Richards equation. Our method comprises an adaptive time marching scheme that uses Picard iterations to solve the corresponding nonlinear equations. A computational code was implemented for the proposed method using the Scilab programming language. We performed numerical simulations of the anomalous diffusion of water in a white siliceous brick and showed that the numerical results were consistent with the available experimental data.     

Computational fluid dynamics modelling of gas jets impinging onto liquid pools

Gas jets impinging onto a gas–liquid interface of a liquid pool are studied using computational fluid dynamics modelling, which aims to obtain a better understanding of the behaviour of the gas jets used metallurgical engineering industry. The gas and liquid flows are modelled using the volume of fluid technique. The governing equations are formulated using the density and viscosity of the “gas–liquid mixture”, which are described in terms of the phase volume fraction. Reynolds averaging is applied to yield a set of Reynolds-averaged conservation equations for the mass and momentum, and the k–ε turbulence model. The deformation of the gas–liquid interface is modelled by the pressure jump across the interface via the Young–Laplace equation. The governing equations in the axisymmetric cylindrical coordinates are solved using the commercial CFD code, FLUENT. The computed results are compared with experimental and theoretical data reported in the literature. The CFD modelling allows the simultaneous evaluation of the gas flow field, the free liquid surface and the bulk liquid flow, and provides useful insight to the highly complex, and industrially significant flows in the jetting system.     

Application of Haar wavelet method to eigenvalue problems of high order differential equations

In this work, we present a computational method for solving eigenvalue problems of high-order ordinary differential equations which based on the use of Haar wavelets. The variable and their derivatives in the governing equations are represented by Haar function and their integral. The first transform the spectral coefficients into the nodal variable values. The second, solve the obtained system of algebraic equation. The efficiency of the method is demonstrated by four numerical examples.     

Free edge stress prediction in thick laminated cylindrical shell panel subjected to bending moment

A thick composite cylindrical shell panel with general layer stacking is studied to investigate the free edge and 3D stresses in the panel which is subjected to pure bending moment. To this aim, a Galerkin based layerwise formulation is presented to discretize the governing equation of the panel to ordinary differential equations. Employing a reduced displacement field for the cylindrical panel, the governing equations for thick panel are developed in terms of displacements and a set of coupled ordinary differential equations is obtained. The governing equations are solved analytically for free edge boundary conditions and applied pure bending moment. The accuracy of numerical results is examined and the distribution of interlaminar and in-plane stresses is studied. The free edge stresses are studied and the effect of radius to thickness ratio, width to thickness ratio and layer stacking on the distribution of stresses is investigated. The focus of numerical results is on the prediction of boundary layer and free edge stress distribution.