Perturbation solution to 3-D nonlinear supercavitating flow problems

A 3-D nonlinear problem of supercavitating flow past an axisymmetric body at a small angle of attack is investigated by means of the perturbation method and Fourier-cosine-expansion method. The first three order perturbation equations are derived in detail and solved numerically using the boundary integral equation method and iterative techniques. Computational results of the hydrodynamic characteristics and cavity shapes of each order are presented for nonaxisymmetric supercavitating flow past cones with various apex-angles at different cavitation numbers. The numerical results are found in good agreement with experimental data. The project supported by the National Natural Science Foundation of China     

A numerical investigation of primary surface rounded cross wavy ducts

 A three-dimensional numerical study was conducted to assess the hydraulic and heat transfer performance of primary surface type heat exchanger surfaces, called cross wavy (CW) ducts aimed for recuperators. The governing equations, i.e., the mass conservation equation, Navier–Stokes equations and the energy equation, are solved numerically by a finite volume method for boundary fitted coordinates. Periodic boundary conditions are imposed in the main flow direction. In this particular case laminar convective flow and heat transfer prevail. Details of the recuperator ducts and the numerical method as well as relevant results are presented. Received on 4 January 2001 / Published online: 29 November 2001     

Hydromagnetic flow in a viscoelastic fluid due to the oscillatory stretching surface

An analysis is carried out to study the unsteady magnetohydrodynamic (MHD) two-dimensional boundary layer flow of a second grade viscoelastic fluid over an oscillatory stretching surface. The flow is induced due to an infinite elastic sheet which is stretched back and forth in its own plane. For the investigated problem, the governing equations are reduced to a non-linear partial differential equation by means of similarity transformations. This equation is solved both by a newly developed analytic technique, namely homotopy analysis method (HAM) and by a numerical method employing the finite difference scheme, in which a coordinate transformation is employed to transform the semi-infinite physical space to a bounded computational domain. The results obtained by means of both methods are then compared and show an excellent agreement. The effects of various parameters like visco-elastic parameter, the Hartman number and the relative frequency amplitude of the oscillatory sheet to the stretching rate on the velocity field are graphically illustrated and analysed. The values of wall shear stress for these parameters are also tabulated and discussed.     

Stability of membrane in low subsonic flow

Stability of an isolated membrane lying in a uniform two-dimensional low subsonic flow is studied theoretically and experimentally. The problem is formulated in a form of a boundary integral equation and differential equations. The boundary integral equation is solved by the boundary element method and the finite difference method is used to solve the differential equations. An effect of a membrane wake is used in the analysis. The theoretical critical divergence velocity is compared with the experimental value.     

Solution of the incompressible Navier–Stokes equations by the method of lines

The numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two‐dimensional unsteady Navier–Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier–Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier–Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas. The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright © 2015 John Wiley & Sons, Ltd.     

不可压粘流N－S方程的边界积分解法

对原变量的Ｎ－Ｓ方程进行一阶时间离散，采用共轭梯度法解除压强－速度的耦合．对所得的一系列Ｌａｐｌａｃｅ方程、Ｐｏｓｓｉｏｎ方程和Ｈｅｌｍｈｏｔｚ方程均进行边界积分法求解，首次得到了粘性Ｎ－Ｓ方程的边界积分表示式．圆柱的定常、非定常尾迹计算结果表明了本文方法的有效性．     

Numerical simulation of axisymmetric unsteady incompressible flow by a vorticity-velocity method

A new numerical method for solving the axisymmetric unsteady incompressible Navier-Stokes equations using vorticity-velocity variables and a staggered grid is presented. The solution is advanced in time with an explicit two-stage Runge-Kutta method. At each stage a vector Poisson equation for velocity is solved. Some important aspects of staggering of the variable location, divergence-free correction to the velocity field by means of a suitably chosen scalar potential and numerical treatment of the vorticity boundary condition are examined. The axisymmetric spherical Couette flow between two concentric differentially rotating spheres is computed as an initial value problem. Comparison of the computational results using a staggered grid with those using a non-staggered grid shows that the staggered grid is superior to the non-staggered grid. The computed scenario of the transition from zero-vortex to two-vortex flow at moderate Reynolds number agrees with that simulated using a pseudospectral method, thus validating the temporal accuracy of our method.     

Rarefied gas flow in a triangular duct based on a boundary fitted lattice

The rarefied fully developed flow of a gas through a duct of a triangular cross section is solved in the whole range of the Knudsen number. The flow is modelled by the BGK kinetic equation, subject to Maxwell diffuse boundary conditions. The numerical solution is based on the discrete velocity method, which is applied for first time on a triangular lattice in the physical space. The boundaries of the flow and computational domains are identical deducing accurate results with modest computational effort. Results on the velocity profiles and on the flow rates for ducts of various triangular cross sections are reported and they are valid in the whole range of gas rarefaction. Their accuracy is validated in several ways, including the recovery of the analytical solutions at the free molecular and hydrodynamic limits. The successful implementation of the triangular grid elements is promising for generalizing kinetic type solutions to rarefied flows in domains with complex boundaries using adaptive and unstructured grids.     

Analysis of velocity equation of steady flow of a viscous incompressible fluid in channel with porous walls

Steady flow of a viscous incompressible fluid in a channel, driven by suction or injection of the fluid through the channel walls, is investigated. The velocity equation of this problem is reduced to nonlinear ordinary differential equation with two boundary conditions by appropriate transformation and convert the two‐point boundary‐value problem for the similarity function into an initial‐value problem in which the position of the upper channel. Then obtained differential equation is solved analytically using differential transformation method and compare with He's variational iteration method and numerical solution. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright © 2009 John Wiley & Sons, Ltd.     

A control-volume based finite element method for three-dimensional incompressible turbulent fluid flow,heat transfer,and related phenomena

A control-volume based finite element method of equal-order type for three-dimensional incompressible turbulent fluid flow, heat transfer, and related phenomena is presented. The discretization equations are based mainly on the physics of the phenomena under consideration, more than on mathematical arguments. Special emphasis is devoted to the discretization of the convective terms and the continuity equation, and to the treatment of the boundary conditions imposed by the use of a high Reynolds k-?, type turbulence model. The pressure-velocity coupling in the fluid flow calculation is made from a derivative of the original SIMPLER method, without pressure correction. The discretized equations are solved in a sequential, rather than a coupled, form with significant advantage in the required computer time and storage. The method is an extension of a former version proposed by us for two-dimensional, laminar problems, and is here successfully applied to the following situations: three-dimensional deflected turbulent jet, and flows in 90° and 45° junctions of ducts with rectangular cross sections. The calculated results are in very good agreement with the experimental and numerical (obtained with the well established finite difference method) data available in the literature.     

不可压粘流N－S方程的边界积分解法

对原变量的Ｎ－Ｓ方程进行一阶时间离散，采用共轭梯度法解除压强－速度的耦合．对所得的一系列Ｌａｐｌａｃｅ方程、Ｐｏｓｓｉｏｎ方程和Ｈｅｌｍｈｏｔｚ方程均进行边界积分法求解，首次得到了粘性Ｎ－Ｓ方程的边界积分表示式．圆柱的定常、非定常尾迹计算结果表明了本文方法的有效性．     

Theoretical analysis on hydraulic transient resulted by sudden increase of inlet pressure for laminar pipeline flow

Hydraulic transient, which is resulted from sudden increase of inlet pressure for laminar pipeline flow, is studied. The partial differential equation, initial and boundary conditions for transient pressure were constructed, and the theoretical solution was obtained by variable-separation method. The partial differential equation, initial and boundary conditions for flow rate were obtained in accordance with the constraint correlation between flow rate and pressure while the transient flow rate distribution was also solved by variable-separation method. The theoretical solution conforms to numerical solution obtained by method of characteristics (MOC) very well.     

用Level Set方法求解具有自由面的流动问题

为采用Ｌｅｖｅｌ Ｓｅｔ方法来计算有自由的流动问题提出了一种方案,把自由水面视为水和空气的交界面,两种介质用统一的Ｎ－Ｓ方法求解,在自由面两侧分别采用各自的密度和粘性,并在自由面上给以适当的光滑;采用边界元法求解双调和方程来确定距离函数;Ｎ－Ｓ方程用投影法求解,文中给出了二维水池水面振荡和瞬时溃坝问题的算例,可以看出用ＬｅｖｅｌＳｅｔ方法求解有自由面流动问题是有效的。     

A Newton's method scheme for solving free-surface flow problems

A Newton's method scheme is described for solving the system of non-linear algebraic equations arising when finite difference approximations are applied to the Navier–Stokes equations and their associated boundary conditions. The problem studied here is the steady, buoyancy-driven motion of a deformable bubble, assumed to consist of an inviscid, incompressible gas. The linear Newton system is solved using both direct and iterative equation solvers. The numerical results are in excellent agreement with previous work, and the method achieves quadratic convergence.     

Boundary element solution of unsteady magnetohydrodynamic duct flow with differential quadrature time integration scheme

A numerical scheme which is a combination of the dual reciprocity boundary element method (DRBEM) and the differential quadrature method (DQM), is proposed for the solution of unsteady magnetohydrodynamic (MHD) flow problem in a rectangular duct with insulating walls. The coupled MHD equations in velocity and induced magnetic field are transformed first into the decoupled time‐dependent convection–diffusion‐type equations. These equations are solved by using DRBEM which treats the time and the space derivatives as nonhomogeneity and then by using DQM for the resulting system of initial value problems. The resulting linear system of equations is overdetermined due to the imposition of both boundary and initial conditions. Employing the least square method to this system the solution is obtained directly at any time level without the need of step‐by‐step computation with respect to time. Computations have been carried out for moderate values of Hartmann number (M?50) at transient and the steady‐state levels. As M increases boundary layers are formed for both the velocity and the induced magnetic field and the velocity becomes uniform at the centre of the duct. Also, the higher the value of M is the smaller the value of time for reaching steady‐state solution. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.     

Space conservation law in finite volume calculations of fluid flow

In the numerical solutions of fluid flow problems in moving co-ordinates, an additional conservation equation, namely the space conservation law, has to be solved simultaneously with the mass, momentum and energy conservation equations. In this paper a method of incorporating the space conservation law into a finite volume procedure is proposed and applied to a number of test cases. The results show that the method is efficient and produces accurate results for all grid velocities and time steps for which temporal accuracy suffices. It is also demonstrated, by analysis and test calculations, that not satisfying the space conservation law in a numerical solution procedure introduces errors in the form of artificial mass sources. These errors can be made negligible only by choosing a sufficiently small time step, which sometimes may be smaller than required by the temporal discretization accuracy.     

Turbulent flow in converging nozzles,part one: boundary layer solution

The boundary layer integral method is used to investigate the development of the turbulent swirling flow at the entrance region of a conical nozzle. The governing equations in the spherical coordinate system are simplified with the boundary layer assumptions and integrated through the boundary layer. The resulting sets of differential equations are then solved by the fourth-order Adams predictor-corrector method. The free vortex and uniform velocity profiles are applied for the tangential and axial velocities at the inlet region, respectively. Due to the lack of experimental data for swirling flows in converging nozzles, the developed model is validated against the numerical simulations. The results of numerical simulations demonstrate the capability of the analytical model in predicting boundary layer parameters such as the boundary layer growth, the shear rate, the boundary layer thickness, and the swirl intensity decay rate for different cone angles. The proposed method introduces a simple and robust procedure to investigate the boundary layer parameters inside the converging geometries.     

Three-dimensional coupled numerical model of creeping flow of viscous fluid

A three-dimensional coupled numerical model is developed to describe creeping flow in a computational domain that consists of a thick viscous layer overlaid with a thin multilayered viscous sheet. The density of the sheet is assumed to be lower than that of the layer. The model couples the Stokes equations describing the flow in the layer and the Reynolds equations describing the flow in the sheet. We investigate the long-time behavior of the flow in the sheet by using an asymptotic method and derive an ordinary differential equation for the sheet boundary displacements and the velocities at the interface between the sheet and the layer. The Stokes and Reynolds equations are coupled by applying the resulting equation as an internal boundary condition. Numerical implementation is based on a modified finite element method combined with the projection gradient method. The computational domain is discretized into rectangular hexahedra. Piecewise square basis functions are used. The model proposed enables different-type hydrodynamic equations to be coupled without any iterative improvements. As a result, the computational costs are reduced significantly in comparison with available coupled models. Numerical experiments confirm that the three-dimensional coupled model developed is of good accuracy.     

The DRBEM solution of incompressible MHD flow equations

This paper presents a dual reciprocity boundary element method (DRBEM) formulation coupled with an implicit backward difference time integration scheme for the solution of the incompressible magnetohydrodynamic (MHD) flow equations. The governing equations are the coupled system of Navier‐Stokes equations and Maxwell's equations of electromagnetics through Ohm's law. We are concerned with a stream function‐vorticity‐magnetic induction‐current density formulation of the full MHD equations in 2D. The stream function and magnetic induction equations which are poisson‐type, are solved by using DRBEM with the fundamental solution of Laplace equation. In the DRBEM solution of the time‐dependent vorticity and current density equations all the terms apart from the Laplace term are treated as nonhomogeneities. The time derivatives are approximated by an implicit backward difference whereas the convective terms are approximated by radial basis functions. The applications are given for the MHD flow, in a square cavity and in a backward‐facing step. The numerical results for the square cavity problem in the presence of a magnetic field are visualized for several values of Reynolds, Hartmann and magnetic Reynolds numbers. The effect of each parameter is analyzed with the graphs presented in terms of stream function, vorticity, current density and magnetic induction contours. Then, we provide the solution of the step flow problem in terms of velocity field, vorticity, current density and magnetic field for increasing values of Hartmann number. Copyright © 2010 John Wiley & Sons, Ltd.