Analytic Approximate Solution for a Flow of a Second-Grade Viscoelastic Fluid in a Converging Porous Channel

The problem of a two-dimensional steady flow of a second-grade fluid in a converging porous channel is considered. It is assumed that the fluid is injected into the channel through one wall and sucked from the channel through the other wall at the same velocity, which is inversely proportional to the distance along the wall from the channel origin. The equations governing the flow are reduced to ordinary differential equations. The boundary-value problem described by the latter equations is solved by the homotopy perturbation method. The effects of the Reynolds and crossflow Reynolds number on the flow characteristics are examined.     

A preconditioned semi‐staggered dilation‐free finite volume method for the incompressible Navier–Stokes equations on all‐hexahedral elements

A new semi‐staggered finite volume method is presented for the solution of the incompressible Navier–Stokes equations on all‐quadrilateral (2D)/hexahedral (3D) meshes. The velocity components are defined at element node points while the pressure term is defined at element centroids. The continuity equation is satisfied exactly within each elements. The checkerboard pressure oscillations are prevented using a special filtering matrix as a preconditioner for the saddle‐point problem resulting from second‐order discretization of the incompressible Navier–Stokes equations. The preconditioned saddle‐point problem is solved using block preconditioners with GMRES solver. In order to achieve higher performance FORTRAN source code is based on highly efficient PETSc and HYPRE libraries. As test cases the 2D/3D lid‐driven cavity flow problem and the 3D flow past array of circular cylinders are solved in order to verify the accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical solution of the flow of thin viscous sheets under gravity and the inverse windscreen sagging problem

The slumping of a thin sheet of very viscous liquid glass is used in the manufacture of windscreens in the automotive industry. The governing equations for an asymptotically thin sheet with variable viscosity are derived in which the vertical coordinate forms the centre‐line of the sheet. The time‐dependant equations have been solved numerically using the backward Euler method to give results in both two and three dimensions. The flow of an initially flat sheet falls freely under gravity until it becomes curved and the flow becomes very slow in the ‘slumped’ phase. Finally the sheet freefalls as the thickness becomes small at the boundaries. The inverse problem in which the viscosity profile is to be determined for a given shape can be solved as an embedding problem in which a search is made amongst the forward solutions. Possible shapes in the two‐dimensional problem are very restrictive and are shown to be related to the sheet thickness. In three dimensions the range of shapes is much greater. Copyright © 2002 John Wiley & Sons, Ltd.     

配置点谱方法-人工压缩法(SCM-ACM)求解同心圆筒内流体流动

发展了配置点谱方法SCM(Spectral collocation method)和人工压缩法ACM(Artificial compressibility method)相结合的SCM-ACM数值方法,计算了柱坐标系下稳态不可压缩流动N-S方程组。选取典型的同心圆筒间旋转流动Taylor-Couette流作为测试对象,首先,采用人工压缩法获得人工压缩格式的非稳态可压缩流动控制方程;再将控制方程中的空间偏微分项用配置点谱方法进行离散,得到矩阵形式的代数方程;编写了SCM-ACM求解不可压缩流动问题的程序;最后,通过与公开发表的Taylor-Couette流的计算结果对比,验证了求解程序的有效性。结果证明,本文发展的SCM-ACM数值方法能够用于求解圆筒内不可压缩流体流动问题,该方法既保留了谱方法指数收敛的特性,也具有ACM形式简单和易于实施的特点。本文发展的SCM-ACM数值方法为求解柱坐标下不可压缩流体流动问题提供了一种新的选择。     

The Optimal Shape of Riblets in the Viscous Sublayer

Our aim is to find the optimal shape of periodically distributed microstructures on surfaces of swimming bodies in order to reduce their drag. The model describes the flow in the viscous sublayer of the boundary layer of a turbulent flow. The microscopic optimization problem is reduced applying homogenization. In the reduced so-called macroscopic optimization problem we minimize the Navier constant subject to the boundary layer equations which are solved in a very small part of the original domain. Under the assumptions that the microstructures can be represented as smooth functions the sensitivity can be determined analytically. The optimization problem is then solved by a sensitivity based method (steepest descent with optimal step size) and the state equations are solved in each iteration with an external software. Our reduced model is validated by comparing the results from the homogenized model with those obtained by simulating the whole rough channel. An improved shape is found and a drag reduction up to 10% can be shown.     

Computational study of the axial instability of rimming flow using Arnoldi method

Axial instability of rimming flow has been investigated by solving a linear generalized eigenvalue problem that governs the evolution of perturbations of two‐dimensional base flow. Using the Galerkin finite element method, full Navier–Stokes equations were solved to calculate base flow and this base flow was perturbed with three‐dimensional disturbances. The generalized eigenproblem formulated from these equations was solved by the implicitly restarted Arnoldi method using shift‐invert technique. This study presents instability curves to identify the critical wavelength of the neutral mode and the critical β, which measures the importance of gravity relative to viscosity. The axial instability of rimming flow is examined and three‐dimensional flow was reconstructed by using eigenvector and growth rate at a critical wave number. The critical β value in the axial instability analysis was observed to be comparable to the onset β value of the transition between the bump and the homogeneous film state in 2‐D base flow calculations. Copyright © 2006 John Wiley & Sons, Ltd.     

Transient motion of a floating body having a rectangular shape

The twodimensional transient problem of a floating body having a rectangular shape in a fluid layer of finite depth is considered. Vertical displacements of the body are specified. The problem is studied within the framework of the linear theory of potential ideal incompressible flow. The fluid flow equations reduce to an infinite system of Volterra integral equations of the second kind by the method of decomposition of the flow region. The system obtained is studied and solved numerically by the reduction method. A method of solving the problem for the flow velocity potential is proposed. The distribution of the hydrodynamic pressure and force acting on the body is determined.     

裂隙岩体非稳态渗流数值模型及其应用

为解决裂隙岩体非稳态渗流问题, 发展了一种新的数值模型. 对于单裂隙渗 流求解, 其控制方程是基于一定假设的简化Navier-Stokes方程, 数值方法采用有限差分法 和流体体积法. 在裂隙网络中, 交界处渗流可以由专门的控制方程求解. 计算结果表明, 该 数值模型既可以大幅提高非稳态渗流的计算效率, 还可以避免孤立裂隙所带来的影响. 最后, 通过两个工程算例验证该数值模型的适用性.     

弹塑性接触问题的非光滑非线性方程组方法

将求解三维弹性摩擦接触问题的非光滑非线性方程组方法推广到弹塑性(Mises材料)情形，提出了两种应用方法：一种是将非光滑非线性方程组方法和求解弹塑性问题常用的Newton—Raphson迭代方法结合起来；另一种是将问题写成统一的非光滑非线性方程组，直接求解。数值算例验证了两种方法的有效性，并进行了结果比较。     

A finite element analysis of a free surface drainage problem of two immiscible fluids

A sharp interface problem arising in the flow of two immiscible fluids, slag and molten metal in a blast furnace, is formulated using a two-dimensional model and solved numerically. This problem is a transient two-phase free or moving boundary problem, the slag surface and the slag–metal interface being the free boundaries. At each time step the hydraulic potential of each fluid satisfies the Laplace equation which is solved by the finite element method. The ordinary differential equations determining the motion of the free boundaries are treated using an implicit time-stepping scheme. The systems of linear equations obtained by discretization of the Laplace equations and the equations of motion of the free boundaries are incorporated into a large system of linear equations. At each time step the hydraulic potential in the interior domain and its derivatives on the free boundaries are obtained simultaneously by solving this linear system of equations. In addition, this solution directly gives the shape of the free boundaries at the next time step. The implicit scheme mentioned above enables us to get the solution without handling normal derivatives, which results in a good numerical solution of the present problem. A numerical example that simulates the flow in a blast furnace is given.     

Level set方法在自适应Cartesian网格上的应用

采用基于自适应Cartesian网格的level set方法对多介质流动问题进行数值模拟。采用基于四叉树的方法来生成自适应Cartesian网格。采用有限体积法求解Euler方程,控制面通量的计算采用HLLC（Hartern, Lax, van Leer, Contact）近似黎曼解方法。level set方程也采用有限体积法求解,采用Lax-Friedchs方法计算通量,通过窄带方法来减少计算量,界面的处理采用ghost fluid方法。Runge-Kutta显式时间推进,时间、空间都是二阶精度。对两种不同比热比介质激波管问题进行数值模拟,其结果和精确解吻合;对空气/氦气泡相互作用等问题进行模拟,取得令人满意的结果。     

Molecular models and flow calculations: II. Simulation of steady planar flow

A multibead-rod model is used to replace the constitutive equation of continuum mechanics in solving flow problems of steady-state planar flows of rigid-rodlike molecular suspensions. The governing equations then constitute a set of differential equations of the elliptic type, which is more amenable to numerical treatment than those of the mixed type. The conservation equations of the flow fields are solved by the boundary element method with linear boundary elements in physical space and the diffusion equation of the distribution function is solved separately by the Galerkin method in phase space. The solution to the flow problem is obtained when the convergence of the iteration procedure between the two spaces has been reached. Several numerical examples are shown and the interesting features of the present method are discussed in this paper. The project supported by the National Nature Science Fundation of China.     

Numerical solution of the problem of the flow of a supersonic gas stream over the upper surface of a delta wing in the expansion region

A numerical method is described for the calculation of supersonic flow over the arbitrary upper surface of a delta wing in the expansion region. The shock wave must be attached everywhere to the leading edge of this wing from the side of the lower surface. The stream flowing over the wing is assumed to be nonviscous. A problem with initial conditions at some plane and with boundary conditions at the wing surface and the characteristic surface is set up for the nonlinear system of equations of gas dynamics. The difference system of equations, which approximates the original system of differential equations on a grid, has a second order of accuracy and is solved by the iteration system proposed in [1]. The initial conditions are determined by the method of establishment of self-similar flow. A number of examples are considered. Comparison is made with the solutions of other authors and with experiment.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 76–81, November–December, 1973.The author thanks A. S. II'ina who conducted the calculations and V. S. Tatarenchik for advice.     

Transient natural convection flow of a particulate suspension through a vertical channel

Continuum equations governing transient, laminar, fully-developed natural convection flow of a particulate suspension through an infinitely long vertical channel are developed. The equations account for particulate viscous effects which are absent from the original dusty-gas model. The walls of the channel are maintained at constant but different temperatures. No-slip boundary conditions are employed for the particle phase at the channel walls. The general transient problem is solved analytically using trigonometric Fourier series and the Laplace transform method. A parametric study of some physical parameters involved in the problem is performed to illustrate the influence of these parameters on the flow and thermal aspects of the problem.     

Accurate 3D viscous incompressible flow calculations with the FEM

A second-order-accurate (in both time and space) formulation is developed and implemented for solution of the three-dimensional incompressible Navier–Stokes equations involving high-Reynolds-number flows past complex configurations. For stabilization, only a fourth-order-accurate artificial dissipation term in the momentum equations is used. The finite element method (FEM) with an explicit time-marching scheme based on two-fractional-step integration is used for solution of the momentum equations. The element-by-element (EBE) technique is employed for solution of the auxiliary potential function equation in order to ease the memory requirements for matrix. The cubic cavity problem, the laminar flow past a sphere at various Reynolds numbers and the flow around the fuselage of a helicopter are successfully solved. Comparison of the results with the low-order solutions indicates that the flow details are depicted clearly even with coarse grids. © 1997 John Wiley & Sons, Ltd.     

Unsteady aerodynamic interaction of two annular blade rows of thin lightly loaded blades rotating one relative to the other in a subsonic flow

The problem of subsonic unsteady ideal-gas flow over two annular blade rows of thin lightly loaded blades rotating one relative to the other is solved within the framework of linear small perturbation theory. As in the case of the interaction of two-dimensional cascades [1], the problem reduces to an infinite system of singular integral equations for the harmonic components of the oscillations in the distribution of the unknown aerodynamic load on one blade of each row. The system of integral equations for a finite number of harmonics is solved numerically by the collocation method. The kernels of the integral equations are regularized on the basis of the method proposed in [2].Translated from Izvestiya Akademii Nauk.SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 165–174, May–June, 1991.The author is grateful to A. A. Osipov and K. K. Butenko for their considerable assistance in the preparation of this paper.     

Poiseuille Flow of a Third Grade Fluid in a Porous Medium

Effects of porous medium have been investigated on the steady flow of a third grade fluid between two stationary porous plates. The continuity and momentum equations along with modified Darcy??s law are used for the development of mathematical problem. The governing nonlinear problem is solved by a homotopy analysis method. The dimensionless velocity and shear stresses at the plates are analyzed.     

Aiding and Opposing Mixed Convection Flow in Melting From a Vertical Flat Plate Embedded in a Porous Medium

The problem of melting from a vertical flat plate embedded in a porous medium is studied. The main focus is to determine the effect of mixed convection flow in the liquid phase on the melting phenomenon. Both aiding and opposing flows are considered. The conservation equations that govern the problem are reduced to a system of nonlinear ordinary differential equations. The governing equations are solved numerically. Numerical results are obtained for the temperature and flow fields in the melting region. The melting phenomenon decreases the local Nusselt number at the solid–liquid interface.     

Cavitation streamline flow of lamina in transverse gravitational field

The problem of cavitation streamline flow located on the linear base of a lamina in a gravity solution current is solved by the systems of Ryabushinskii and Zhukovskii-Roshko. The method of fragment-continuum approximation of the boundary condition at the free boundary was used, in which this condition is exactly satisfied at a finite number of points. In this way the original problem comes down to a solution of a system of nonlinear equations whose solvability can be shown by the method of V. N. Monakhov [1]. The main consideration in the present work was given to a numerical solution of this system of equations on a computer. The problem is similar to the type for large Froude numbers, when the effect of weight on the flow is small, studied in [2-5]. In [6, 7] the flow problems were solved by the method of finite differences. The approximations of the boundary condition at the free boundary used earlier are based on the use of the smallness of these or other characteristics of flow. Thus, for example, the linearization of Levi-Chivit [8] is rightly used in the assumption of smallness of the change in the modulus and angle of inclination of the velocity at the free flow line; a stronger linearization is based on the requirement of smallness of additional velocities caused by an obstacle in comparison with the velocity of the undisturbed current [9]. In the given work the problems studied lead to a range of cavitation and Froude numbers when the gravitational force exerts a considerable effect on the main characteristics of the flow. As an example of one of the possible applications of the calculation, the solution of the problem of choice of the form of a body of zero buoyancy with a zone of constant pressure is given.Translated from Zhurnal Prikladnoi Mekhanik i Tekhnicheskoi Fiziki, No. 5, pp. 132–136, September–October, 1971.     

Two-phase flow over a surface with the formation of a liquid film by particle deposition

A consistent asymptotic theory of wall flow with film formation is constructed with reference to subsonic two-phase flow over a blunt body. The external flow problem and the film equations are solved simultaneously. This formulation of the problem supplements the investigation carried out in [4] in which particles deposited on the surface were assumed to disappear from the flow. It is shown that depending on the values of the governing parameters the flow in the film should be described either by the boundary layer equations or by the equations of creeping flow in a layer of unknown thickness. At the outer edge of the film the mass, momentum and energy fluxes found from the numerical solution of the flow problem are given. The case of isothermal film flow on the front of a sphere is investigated. The thickness of the film and the friction and heat transfer coefficients near the axis of symmetry are found for nonisothermal flows. The conditions under which the presence of a film significantly reduces the heat flow to the wall are determined. A similar formulation of the problem (but with another type of mass, momentum and energy sources at the outer edge) is encountered in problems of film condensation on a cold surface [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 85–92, July–August, 1989.