A note on the calculation of strain histories in orthogonal streamline coordinate systems

An extension of a previous work concerning the calculation of strain histories along streamlines is made to get more complete and useful expressions of Finger's strain tensor in a cylindrical (or Cartesian) coordinate system as well as in an orthogonal streamline coordinate system. One of the results shows that Winter's tracking model is correct.Relations among the recent three results of Winter, Adachi and Crochet et al. are presented clearly. Moreover useful applications of Frenet-Serret's formula to the study of the deformation and flow kinematics along streamlines are shown in comparison with the ordinary tensor approach.     

Predicting transition in two- and three-dimensional separated flows

This paper is concerned with the numerical prediction of two- and three-dimensional transitional separated flows of turbomachinery interest. The recently proposed single-point transition model based on the use of a laminar kinetic energy transport equation is considered, insofar as it does not require to evaluate any integral parameter, such as boundary-layer thickness, and is thus directly applicable to three-dimensional flows. A well established model, combining a transition-onset correlation with an intermittency transport equation, is also used for comparison. Both models are implemented within a Reynolds-averaged Navier–Stokes solver employing a low-Reynolds-number k–ω turbulence model. The performance of the transition models have been evaluated and tested versus well-documented incompressible flows past a flat plate with semi-circular leading edge, namely: tests T3L2, T3L3, T3L5, and T3LA1 of ERCOFTAC, with different Reynolds numbers and free-stream conditions, the last one being characterized by a non-zero pressure gradient. In all computations, the first model has proven as adequate as or superior to the second one and has been then applied with success to two more complex test cases, for which detailed experimental data are available in the literature, namely: the two- and three-dimensional flows through the T106 linear turbine cascade.     

Calculation of stagnation streamline quantities in hypersonic blunt body flows

An approximate method for the efficient calculation of stagnation-streamline quantities in hypersonic flows about spheres or cylinders is suggested. Based on the local similarity of the flow field the two-dimensional Navier-Stokes equations are simplified to a one-dimensional approximation for the stagnation streamline. These equations are solved with an implicit finite-volume scheme. Comparisons with fully two–dimensional Euler and Navier–Stokes calculations for flows about spheres are presented, that include perfect gas flows and flows in chemical non-equilibrium. Comparisons with a number of experiments conclude this report. Received 8 May 1996 / Accepted 31 October 1996     

Simulation of two- and three-dimensional internal subsonic flows using a finite element method

In this paper a finite element method is presented to predict internal subsonic flows. Using a low-Mach-number approximation, the pressure is decomposed into a mean thermodynamic contribution and a dynamic fluctuation to deal with the complex role of the pressure in internal aerodynamics. A semi-implicit time integration and a finite element method with a moving mesh are described to take into account complex geometries and moving boundaries. An Uzawa algorithm accelerated by a preconditioned residual method is introduced to solve the coupled non-symmetric linear system for the velocity components and the pressure. An efficient conjugate gradient method combined with an incomplete LU preconditioning is used to solve the non-symmetric linear systems arising from the discretization. The implementation of the numerical scheme on parallel supercomputers is also discussed. Efficient algorithms for the finite element assembly phase and for the solution of linear systems are described which take advantage of the parallel architecture of the new generation of supercomputers. With this technique a global speed-up of 10 is achieved on a supercomputer with eight processors. To illustrate the capabilities of the numerical method, 2D and 3D simulations of flows in the combustion chamber of a reciprocating engine and around the combustor dome of a gas turbine engine are presented.     

Finite element analysis of two- and three-dimensional flows around square columns in tandem arrangement

Two- (2D) and three-dimensional (3D) finite element analyses for flow around two square columns in tandem arrangement were performed with various column spacings and Reynolds numbers. The computed values were compared with the wind-tunnel results in terms of the aerodynamic characteristics of the leeward column. In most 2D computations, strong vortices were formed behind the windward column, irrespective of widely changed Reynolds numbers. This was different from the experimental phenomena of equivalent spacing, so that the computed time-averaged pressure coefficients were not identical to the experimental values except when the distance between the two columns was adequately wide or narrow. On the other hand, in 3D computation, distinct differences in flow structures behind the column were observed between Reynolds numbers of 10³ and 10⁴ and the pressure coefficient in the 3D analysis with Re=10⁴ agreed well with the experimental value. Thus, the effectiveness of 3D computations and Reynolds number effects on the flow around two square columns have been confirmed. © 1998 John Wiley & Sons, Ltd.     

Numerical study of flows of complex fluids between eccentric cylinders using transformation functions

In this paper, we investigate fluid flows between eccentric cylinders by means of two stream‐tube analyses. The first method considers a one‐to‐one global transformation function that allows the physical domain to be transformed into a mapped domain, used as computational domain, that involves concentric streamlines. The second approach uses local transformations and domain decomposition techniques to deal with mixed flow regimes. Both formulations are particularly adapted for handling time‐dependent constitutive equations, since particle‐tracking problems are avoided. Mass conservation is verified in both formulations and the relevant numerical procedure can be carried out using simple meshes built on the mapped streamlines. Fluids obeying anelastic and viscoelastic constitutive equations are considered in the calculations. The numerical results are consistent with those in the literature for the flow rates tested. Application of the method to the K‐BKZ memory‐integral constitutive equation highlights significant differences between the model predictions and those provided by more simple rheological models. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical simulation of complex flows of non-Newtonian fluids using the stream tube method and memory integral constitutive equations

In this paper a memory integral viscoelastic equation is considered for simulating complex flows of non-Newtonian fluids by stream tube analysis. A formalism is developed to take into account co-deformational memory equations in a mapped computational domain where the transformed streamlines are parallel and straight. The particle-tracking problem is avoided. Evolution in time and related kinematic quantities involved with a K-BKZ integral constitutive model are easily taken into account in evaluating the stresses. Successive subdomains, the stream tubes, may be considered for computing the main flow in abrupt axisymmetric contractions from the wall to the central flow region. The ‘peripheral stream tube’ close to the duct wall is determined by developing a non-conventional modified Hermite element. A mixed formulation is adopted and the relevant non-linear equations are solved numerically by the Levenberg-Marquardt algorithm. Although the singularity at the section of contraction is not involved explicitly, the results obtained for the peripheral stream tube clearly show the singularity effects and the extent of the recirculating zone near the salient corner. The algorithm is stable even at high flow rates and provides satisfactory solutions when compared with similar calculations in the literature.     

The study of quadruple integral equations involving inverse Mellin transforms and its application to a contact problem for a wedge shaped elastic solid in antiplane shear distribution

Closed form solution of quadruple integral equations involving inverse Mellin transforms has been obtained. The solution of quadruple integral equations is used in solving a two dimensional four-part mixed boundary value contact problem for an elastic wedge-shaped region as an application. Closed form expression for shear stress has been obtained. Finally, numerical results for shear stress are obtained and shown graphically.     

Analysis of multiple interfacial cracks in three-dimensional bimaterials using hypersingular integro-differential equation method

By using the concept of finite-part integral, a set of hypersingular integro-differential equations for multiple interracial cracks in a three-dimensional infinite bimaterial subjected to arbitrary loads is derived. In the numerical analysis, unknown displacement discontinuities are approximated with the products of the fundamental density functions and power series. The fundamental functions are chosen to express a two-dimensional interface crack rigorously. As illustrative examples, the stress intensity factors for two rectangular interface cracks are calculated for various spacing, crack shape and elastic constants. It is shown that the stress intensity factors decrease with the crack spacing.     

用CE/SE法对弯曲与分叉河道的溃坝洪水波的数值研究

对作者的二维溃坝洪水波的数学模型进一步推广,得到了一般形式的基于任意四边形网格的时空守恒元和解元方法(简称CE/SE法)的新的格式.CE/SE法从守恒积分型浅水方程出发,设立守恒元和解元,严格保证其物理意义上的守恒律,并且构造思想简单,格式通用性好.首先采用CE/SE法计算等宽矩形河道的溃坝洪水波,并与Stoker解析解进行比较,在此基础上,数值模拟了180度强弯曲河道、45度三支分叉河道的二维溃坝洪水波的演进过程,揭示了溃坝洪水波在弯曲河道中内外两岸速度与水位的变化,在分叉河道中自动进行流量与动量的再分配,在分叉点处形成旋涡,水位变化剧烈等复杂的运动特征,算例结果表明基于任意四边形网格的CE/SE法精度高,稳定性好,该格式对各种不规则几何区域内的溃坝问题具有较强的适应性,对溃坝洪水波的间断具有较高的分辨率.     

Evaluation of strongly singular domain integrals for internal stresses in functionally graded materials analyses using RIBEM

An accurate evaluation of strongly singular domain integral appearing in the stress representation formula is a crucial problem in the stress analysis of functionally graded materials using boundary element method.To solve this problem,a singularity separation technique is presented in the paper to split the singular integral into regular and singular parts by subtracting and adding a singular term.The singular domain integral is transformed into a boundary integral using the radial integration method.Analytical expressions of the radial integrals are obtained for two commonly used shear moduli varying with spatial coordinates.The regular domain integral,after expressing the displacements in terms of the radial basis functions,is also transformed to the boundary using the radial integration method.Finally,a boundary element method without internal cells is established for computing the stresses at internal nodes of the functionally graded materials with varying shear modulus.     

Simulation of three-dimensional incompressible flows in generalized curvilinear coordinates using a high-order compact finite-difference lattice Boltzmann method

In the present study, a high-order compact finite-difference lattice Boltzmann method is applied for accurately computing 3-D incompressible flows in the generalized curvilinear coordinates to handle practical and realistic geometries with curved boundaries and nonuniform grids. The incompressible form of the 3-D nineteen discrete velocity lattice Boltzmann method is transformed into the generalized curvilinear coordinates. Herein, a fourth-order compact finite-difference scheme and a fourth-order Runge-Kutta scheme are used for the discretization of the spatial derivatives and the temporal term, respectively, in the resulting 3-D nineteen discrete velocity lattice Boltzmann equation to provide an accurate 3-D incompressible flow solver. A high-order spectral-type low-pass compact filtering technique is applied to have a stable solution. All boundary conditions are implemented based on the solution of the governing equations in the 3-D generalized curvilinear coordinates. Numerical solutions of different 3-D benchmark and practical incompressible flow problems are performed to demonstrate the accuracy and performance of the solution methodology presented. Herein, the 2-D cylindrical Couette flow, the decay of a 3-D double shear wave, the cubic lid-driven cavity flow with nonuniform grids, the flow through a square duct with 90° bend and the flow past a sphere at different flow conditions are considered for validating the present computations. Numerical results obtained show the accuracy and robustness of the present solution methodology based on the implementation of the high-order compact finite-difference lattice Boltzman method in the generalized curvilinear coordinates for solving 3-D incompressible flows over practical and realistic geometries.     

极坐标平面问题的四阶差分及应用

推导出了极坐标系下双调和方程的差分公式，用逐次超松弛迭代法求出圆板平面应力问题的差分解，并和解析解作比较，验证了差分公式的正确性，为解决圆域及其类似区域的平面问题提供了新方法．     

Explicit formulations for evaluation of velocity gradients using boundary‐domain integral equations in 2D and 3D viscous flows

In this paper, explicit boundary‐domain integral equations for evaluating velocity gradients are derived from the basic velocity integral equations. A free term is produced in the new strongly singular integral equation, which is not included in recent formulations using the complex variable differentiation method (CVDM) to compute velocity gradients (Int. J. Numer. Meth. Fluids 2004; 45 :463–484; Int. J. Numer. Meth. Fluids 2005; 47 :19–43). The strongly singular domain integrals involved in the new integral equations are accurately evaluated using the radial integration method (RIM). Considerable computational time for evaluating integrals of velocity gradients can be saved by using present formulation than using CVDM. The formulation derived in this paper together with those presented in reference (Int. J. Numer. Meth. Fluids 2004; 45 :463–484) for 2D and in (Int. J. Numer. Meth. Fluids 2005; 47 :19–43) for 3D problems constitutes a complete boundary‐domain integral equation system for solving full Navier–Stokes equations using primitive variables. Three numerical examples for steady incompressible viscous flow are given to validate the derived formulations. Copyright © 2007 John Wiley & Sons, Ltd.     

极坐标系下Fourier-Legendre谱元方法与有限差分法数值扩散的比较

提出一种Fourier-Legendre谱元方法用于求解极坐标系下的Navier-Stokes方程,其中极点所在单元的径向采用Gauss-Radau积分点,避免了r=0处的1/r坐标奇异性。时间离散采用时间分裂法,引入数值同位素模型跟踪同位素的输运过程验证数值模拟的精度,分别利用谱元法和有限差分法的迎风差分格式求解匀速和加速坩埚旋转流动中的同位素方程。计算结果表明,有限差分法中的一阶迎风差分格式存在严重的数值假扩散,二阶迎风差分格式的数值结果较精确,增加节点可以有效地缓解数值扩散。然而,谱元法具有以较少节点得到高精度解的优势。     

Three‐dimensional transient Navier–Stokes solvers in cylindrical coordinate system based on a spectral collocation method using explicit treatment of the pressure

A spectral collocation method is developed for solving the three‐dimensional transient Navier–Stokes equations in cylindrical coordinate system. The Chebyshev–Fourier spectral collocation method is used for spatial approximation. A second‐order semi‐implicit scheme with explicit treatment of the pressure and implicit treatment of the viscous term is used for the time discretization. The pressure Poisson equation enforces the incompressibility constraint for the velocity field, and the pressure is solved through the pressure Poisson equation with a Neumann boundary condition. We demonstrate by numerical results that this scheme is stable under the standard Courant–Friedrichs–Lewy (CFL) condition, and is second‐order accurate in time for the velocity, pressure, and divergence. Further, we develop three accurate, stable, and efficient solvers based on this algorithm by selecting different collocation points in r‐, ? ‐, and z‐directions. Additionally, we compare two sets of collocation points used to avoid the axis, and the numerical results indicate that using the Chebyshev Gauss–Radau points in radial direction to avoid the axis is more practical for solving our problem, and its main advantage is to save the CPU time compared with using the Chebyshev Gauss–Lobatto points in radial direction to avoid the axis. Copyright © 2010 John Wiley & Sons, Ltd.     

Simulation of interface and free surface flows in a viscous fluid using adapting quadtree grids

A new adaptive quadtree method for simulating laminar viscous fluid problems with free surfaces and interfaces is presented in this paper. The Navier–Stokes equations are solved with a SIMPLE‐type scheme coupled with the Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) (Numerical prediction of two fluid systems with sharp interfaces, Ph.D. Thesis, Imperial College of Science, Technology and Medicine, London, 1997) volume of fluid (VoF) method and PLIC reconstruction of the volume fraction field during refinement and derefinement processes. The method is demonstrated for interface advection cases in translating and shearing flow fields and found to provide high interface resolution at low computational cost. The new method is also applied to simulation of the collapse of a water column and the results are in excellent agreement with other published data. The quadtree grids adapt to follow the movement of the free surface, whilst maintaining a band of the smallest cells surrounding the surface. The calculation is made on uniform and adapting quadtree grids and the accuracy of the quadtree calculation is shown to be the same as that made on the equivalent uniform grid. Copyright © 2004 John Wiley & Sons, Ltd.     

An unstructured finite‐volume method for coupled models of suspended sediment and bed load transport in shallow‐water flows

The aim of this work is to develop a well‐balanced finite‐volume method for the accurate numerical solution of the equations governing suspended sediment and bed load transport in two‐dimensional shallow‐water flows. The modelling system consists of three coupled model components: (i) the shallow‐water equations for the hydrodynamical model; (ii) a transport equation for the dispersion of suspended sediments; and (iii) an Exner equation for the morphodynamics. These coupled models form a hyperbolic system of conservation laws with source terms. The proposed finite‐volume method consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The gradient fluxes are discretized using a modified Roe's scheme using the sign of the Jacobian matrix in the coupled system. A well‐balanced discretization is used for the treatment of source terms. In this paper, we also employ an adaptive procedure in the finite‐volume method by monitoring the concentration of suspended sediments in the computational domain during its transport process. The method uses unstructured meshes and incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep sediment concentrations and bed load gradients that may form in the approximate solutions. Details are given on the implementation of the method, and numerical results are presented for two idealized test cases, which demonstrate the accuracy and robustness of the method and its applicability in predicting dam‐break flows over erodible sediment beds. The method is also applied to a sediment transport problem in the Nador lagoon.Copyright © 2013 John Wiley & Sons, Ltd.     

Computation of unsteady viscous incompressible flows in generalized non‐inertial co‐ordinate system using Godunov‐projection method and overlapping meshes

Time‐dependent incompressible Navier–Stokes equations are formulated in generalized non‐inertial co‐ordinate system and numerically solved by using a modified second‐order Godunov‐projection method on a system of overlapped body‐fitted structured grids. The projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The second‐order Godunov method is applied for numerically approximating the non‐linear convection terms in order to provide a robust discretization for simulating flows at high Reynolds number. In order to obtain the pressure field, the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain so that the moving‐boundary problem can be solved economically. Numerical results are then presented to demonstrate the performance of this projection method for a variety of unsteady two‐ and three‐dimensional flow problems formulated in the non‐inertial co‐ordinate systems. Copyright © 2002 John Wiley & Sons, Ltd.