Numerical simulation of unsteady viscous flow around a circular cylinder

A finite difference solution is obtained for the time-dependent viscous incompressible 2-dimensional flow past a circular cylinder by direct integration of the Navier-Stokes equations expressed in a general curvilinear coordinate system. The solution describes the development of the vortex street developed behind the cylinder. Evolution of flow configuration is studied by means of streamlines, pressure contours, and vorticity contours for different Reynolds numbers. The time-dependent lift and drag coefficients are also obtained.     

圆柱绕流的流场特性及涡脱落规律研究

采用粒子图像测速技术对630、800及950三种雷诺数条件下的圆柱绕流场进行了实验,给出了圆柱下游沿流动方向4倍圆柱直径和垂直方向3倍圆柱直径区域内的速度场、涡量场以及涡脱落现象的时空演化规律.结果表明:圆柱尾流区域位于垂直方向约1.5~2.5倍圆柱直径范围内,随着雷诺数增大,这一范围呈现缩小趋势,而主流对涡的拉伸和输运能力有所增强;涡脱落频率随雷诺数增大而增大,小雷诺数时能够较为完整地捕捉到涡生成、脱落、发展和耗散过程,由于PIV采集频率的限制,大雷诺数条件下涡脱落整个过程不易被完整捕捉到.     

Vortex motion in the early stages of unsteady flow around a circular cylinder

In this paper, processes in the early stages of vortex motion and the development of flow structure behind an impulsively-started circular cylinder at high Reynolds number are investigated by combining the discrete vortex model with boundary layer theory, considering the separation of incoming flow boundary layer and rear shear layer in the recirculating flow region. The development of flow structure and vortex motion, particularly the formation and development of secondary vortex and a pair of secondary vortices and their effect on the flow field are calculated. The results clearly show that the flow structure and vortices motion went through a series of complicated processes before the symmetric main vortices change into asymmetric: development of main vortices induces secondary vortices; growth of the secondary vortices causes the main vortex sheets to break off and causes the symmetric main vortices to become “free” vortices, while a pair of secondary vortices is formed; then the vortex sheets, after breaking off, gradually extend downstream and the structure of a pair of secondary vortices becomes relaxed. These features of vortex motion look very much like the observed features in some available flow field visualizations. The action of the secondary vortices causes the main vortex sheets to break off and converts the main vortices into free vortices. This should be the immediate cause leading to the instability of the motion of the symmetric main vortices. The flow field structure such as the separation position of boundary layer and rear shear layer, the unsteady pressure distributions and the drag coefficient are calculated. Comparison with other results or experiments is also made. This work was presented at the First Asian Congress of Fluid Mechanics, Bangalore in December 1980.     

A new element‐by‐element method for trajectory calculations with tetrahedral finite element meshes

A new method to track massless particles in a three‐dimensional flow field is presented. The method is based on an element‐by‐element approach coupled with a predictor–corrector shooting scheme and does not use any time step. By analogy with time‐dependent schemes, the number of shootings is related to an equivalent number of time steps. The method has been implemented in a finite element framework using unstructured tetrahedral finite element meshes. However, it is general enough so that it can be implemented in finite difference and finite volume frameworks as well. It has been tested on a variety of flow systems namely: the Poiseuille flow in an empty circular pipe, the rotating flow in a stirred tank, the shear flow in a square tank and the flow through a static mixer. Accuracy has been found to depend on the accuracy of the velocity computation, the number of points per element and the level of mesh refinement. Copyright © 2006 John Wiley & Sons, Ltd.     

A coupled BEM and arbitrary Lagrangian–Eulerian FEM model for the solution of two‐dimensional laminar flows in external flow fields

This paper describes a new computational model developed to solve two‐dimensional incompressible viscous flow problems in external flow fields. The model based on the Navier–Stokes equations in primitive variables is able to solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the pressure projection method. The external flow field is simulated using the boundary element method by solving a pressure Poisson equation that assumes the pressure as zero at the infinite boundary. The momentum equation of the flow motion is solved using the three‐step finite element method. The arbitrary Lagrangian–Eulerian method is incorporated into the model, to solve the moving boundary problems. The present model is applied to simulate various external flow problems like flow across circular cylinder, acceleration and deceleration of the circular cylinder moving in a still fluid and vibration of the circular cylinder induced by the vortex shedding. The simulation results are found to be very reasonable and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical simulation of flapping wings using a panel method and a high‐order Navier–Stokes solver

The design of efficient flapping wings for human engineered micro aerial vehicles (MAVs) has long been an elusive goal, in part because of the large size of the design space. One strategy for overcoming this difficulty is to use a multifidelity simulation strategy that appropriately balances computation time and accuracy. We compare two models with different geometric and physical fidelity. The low‐fidelity model is an inviscid doublet lattice method with infinitely thin lifting surfaces. The high‐fidelity model is a high‐order accurate discontinuous Galerkin Navier–Stokes solver, which uses an accurate representation of the flapping wing geometry. To compare the performance of the two methods, we consider a model flapping wing with an elliptical planform and an analytically prescribed spanwise wing twist, at size scales relevant to MAVs. Our results show that in many cases, including those with mild separation, low‐fidelity simulations can accurately predict integrated forces, provide insight into the flow structure, indicate regions of likely separation, and shed light on design–relevant quantities. But for problems with significant levels of separation, higher‐fidelity methods are required to capture the details of the flow field. Inevitably high‐fidelity simulations are needed to establish the limits of validity of the lower fidelity simulations.Copyright © 2012 John Wiley & Sons, Ltd.     

Nodal velocity derivatives of finite element solutions: The FiND method for incompressible Navier–Stokes equations

This paper presents the development and application of the finite node displacement (FiND) method to the incompressible Navier–Stokes equations. The method computes high‐accuracy nodal derivatives of the finite element solutions. The approach imposes a small displacement to individual mesh nodes and solves a very small problem on the patch of elements surrounding the node. The only unknown is the value of the solution ( u , p) at the displaced node. A finite difference between the original and the perturbed values provides the directional derivative. Verification by grid refinement studies is shown for two‐dimensional problems possessing a closed‐form solution: a Poiseuille flow and a flow mimicking a boundary layer. For internal nodes, the method yields accuracy slightly superior to that of the superconvergent patch recovery (SPR) technique of Zienkiewicz and Zhu (ZZ). We also present a variant of the method to treat boundary nodes. The local discretization is enriched by inserting an additional mesh point very close to the boundary node of interest. Computations show that the resulting nodal derivatives are much more accurate than those obtained by the ZZ SPR technique. Copyright © 2008 John Wiley & Sons, Ltd.     

Computation of limit and shakedown loads using a node‐based smoothed finite element method

This paper presents a novel numerical procedure for computing limit and shakedown loads of structures using a node‐based smoothed FEM in combination with a primal–dual algorithm. An associated primal–dual form based on the von Mises yield criterion is adopted. The primal‐dual algorithm together with a Newton‐like iteration are then used to solve this associated primal–dual form to determine simultaneously both approximate upper and quasi‐lower bounds of the plastic collapse limit and the shakedown limit. The present formulation uses only linear approximations and its implementation into finite element programs is quite simple. Several numerical examples are given to show the reliability, accuracy, and generality of the present formulation compared with other available methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Evaluating J‐integral and Q parameter in high‐density polyethylene using a combined experimental finite element method

This work introduces a combined experimental finite element method (FEM) to calculate J‐integral and Q parameter for centre‐crack tension of high‐density polyethylene specimens. In the majority of the studies to date, mostly a pure FEM has been used. However, the several simplified assumptions that are included in such models could result in imperfect predictions. This study aims to overcome this issue by suggesting a novel method that privileges from the displacement fields on specimens' surface that is provided for our finite element model. The method introduced in this work has a merit in presenting Q results for a softening material. The results of our proposed method were in a satisfactory agreement with the pure FEM results of similar models, and thereby validating our approach. Using this model, the effects of parameters like crack length and thickness were also investigated. This method could be utilized in health monitoring of structures.     

Quasi‐dual reciprocity boundary‐element method for incompressible flow: Application to the diffusive–advective equation

This work presents a new boundary‐element method formulation called quasi‐dual reciprocity formulation for heat transfer problems, considering diffusive and advective terms. The present approach has some characteristics similar to those of the so‐called dual‐reciprocity formulation; however, the mathematical developments of the quasi‐dual reciprocity approach reduces approximation errors due to global domain interpolation. Some one‐ and two‐dimensional examples are presented, the results being compared against those obtained from analytical and dual‐reciprocity formulations. The method convergence is evaluated through analyses where the mesh is successively refined for various Peclet numbers, in order to assess the effect of the advective term. Copyright © 2003 John Wiley & Sons, Ltd.     

An enhanced cell‐based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh

In this work, an enhanced cell‐based smoothed finite element method (FEM) is presented for the Reissner–Mindlin plate bending analysis. The smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed FEM is reformulated, and the relationship between the original approach and the present method in curvature smoothing is established. To improve the accuracy of shear strain in a distorted mesh, we span the shear strain space over the adjacent element. This is performed by employing an edge‐based smoothing technique through a simple area‐weighted smoothing procedure on MITC4 assumed shear strain field. A three‐field variational principle is utilized to develop the mixed formulation. The resultant element formulation is further reduced to a displacement‐based formulation via an assumed strain method defined by the edge‐smoothing technique. As the result, a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement/rotation fields and smoothing operators can be shown to improve the solution accuracy in cell‐based smoothed FEM for Reissner–Mindlin plate bending analysis. Several numerical examples are presented to demonstrate the accuracy of the proposed formulation.Copyright © 2013 John Wiley & Sons, Ltd.     

Coupling of finite element and boundary integral methods for a capsule in a Stokes flow

We introduce a new numerical method to model the fluid–structure interaction between a microcapsule and an external flow. An explicit finite element method is used to model the large deformation of the capsule wall, which is treated as a bidimensional hyperelastic membrane. It is coupled with a boundary integral method to solve for the internal and external Stokes flows. Our results are compared with previous studies in two classical test cases: a capsule in a simple shear flow and in a planar hyperbolic flow. The method is found to be numerically stable, even when the membrane undergoes in‐plane compression, which had been shown to be a destabilizing factor for other methods. The results are in very good agreement with the literature. When the viscous forces are increased with respect to the membrane elastic forces, three regimes are found for both flow cases. Our method allows a precise characterization of the critical parameters governing the transitions. Copyright © 2010 John Wiley & Sons, Ltd.     

Hybrid displacement function element method: a simple hybrid‐Trefftz stress element method for analysis of Mindlin–Reissner plate

In order to develop robust finite element models for analysis of thin and moderately thick plates, a simple hybrid displacement function element method is presented. First, the variational functional of complementary energy for Mindlin–Reissner plates is modified to be expressed by a displacement function F, which can be used to derive displacement components satisfying all governing equations. Second, the assumed element resultant force fields, which can satisfy all related governing equations, are derived from the fundamental analytical solutions of F. Third, the displacements and shear strains along each element boundary are determined by the locking‐free formulae based on the Timoshenko's beam theory. Finally, by applying the principle of minimum complementary energy, the element stiffness matrix related to the conventional nodal displacement DOFs is obtained. Because the trial functions of the domain stress approximations a priori satisfy governing equations, this method is consistent with the hybrid‐Trefftz stress element method. As an example, a 4‐node, 12‐DOF quadrilateral plate bending element, HDF‐P4‐11 β, is formulated. Numerical benchmark examples have proved that the new model possesses excellent precision. It is also a shape‐free element that performs very well even when a severely distorted mesh containing concave quadrilateral and degenerated triangular elements is employed. Copyright © 2014 John Wiley & Sons, Ltd.     

The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves

Particle Methods are those in which the problem is represented by a discrete number of particles. Each particle moves accordingly with its own mass and the external/internal forces applied to it. Particle Methods may be used for both, discrete and continuous problems. In this paper, a Particle Method is used to solve the continuous fluid mechanics equations. To evaluate the external applied forces on each particle, the incompressible Navier–Stokes equations using a Lagrangian formulation are solved at each time step. The interpolation functions are those used in the Meshless Finite Element Method and the time integration is introduced by an implicit fractional‐step method. In this manner classical stabilization terms used in the momentum equations are unnecessary due to lack of convective terms in the Lagrangian formulation. Once the forces are evaluated, the particles move independently of the mesh. All the information is transmitted by the particles. Fluid–structure interaction problems including free‐fluid‐surfaces, breaking waves and fluid particle separation may be easily solved with this methodology. Copyright © 2004 John Wiley & Sons, Ltd.     

An adaptive non‐conforming finite‐element method for Reissner–Mindlin plates

Adaptive algorithms are important tools for efficient finite‐element mesh design. In this paper, an error controlled adaptive mesh‐refining algorithm is proposed for a non‐conforming low‐order finite‐element method for the Reissner–Mindlin plate model. The algorithm is controlled by a reliable and efficient residual‐based a posteriori error estimate, which is robust with respect to the plate's thickness. Numerical evidence for this and the efficiency of the new algorithm is provided in the sense that non‐optimal convergence rates are optimally improved in our numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

Use of higher‐order shape functions in the scaled boundary finite element method

The scaled boundary finite element method is a novel semi‐analytical technique, whose versatility, accuracy and efficiency are not only equal to, but potentially better than the finite element method and the boundary element method for certain problems. This paper investigates the possibility of using higher‐order polynomial functions for the shape functions. Two techniques for generating the higher‐order shape functions are investigated. In the first, the spectral element approach is used with Lagrange interpolation functions. In the second, hierarchical polynomial shape functions are employed to add new degrees of freedom into the domain without changing the existing ones, as in the p‐version of the finite element method. To check the accuracy of the proposed procedures, a plane strain problem for which an exact solution is available is employed. A more complex example involving three scaled boundary subdomains is also addressed. The rates of convergence of these examples under p‐refinement are compared with the corresponding rates of convergence achieved when uniform h‐refinement is used, allowing direct comparison of the computational cost of the two approaches. The results show that it is advantageous to use higher‐order elements, and that higher rates of convergence can be obtained using p‐refinement instead of h‐refinement. Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient method of solving the Navier–Stokes equations for flow control

A new method of solving the Navier–Stokes equations efficiently by reducing their number of modes is proposed in the present paper. It is based on the Karhunen–Loève decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system. Employing these empirical eigenfunctions as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear subspace that is sufficient to describe the observed phenomena, and consequently reduce the Navier–Stokes equation defined on a complicated geometry to a set of ordinary differential equations with a minimum degree of freedom. The present algorithm is well suited for the problems of flow control or optimization, where one has to compute the flow field repeatedly using the Navier–Stokes equation but one can also estimate the approximate solution space of the flow field based on the range of control variables. The low-dimensional dynamic model of viscous fluid flow derived by the present method is shown to produce accurate flow fields at a drastically reduced computational cost when compared with the finite difference solution of the Navier–Stokes equation. © 1998 John Wiley & Sons, Ltd.     

High‐order predictor–corrector algorithms

New predictor–corrector algorithms are presented for the computation of solution paths of non‐linear partial differential equations. The predictors and the correctors are based on perturbation techniques and Padé approximants. This extends the Asymptotic Numerical Method (ANM), which is an efficient high‐order continuation technique without corrector. The efficiency and the reliability of the new technique are assessed by several examples within thin shell theory and Navier–Stokes equations. Many variants have been tested to establish an optimal algorithm. Copyright © 2002 John Wiley & Sons, Ltd.     

Stokes flow around an asymmetric channel divider; a computational approach using MATLAB

While computational methods for solving Stokes-flow problems have existed for some time, these have depended on specialized codes developed specifically, for this type of problem. This work shows how to combine traditional applied mathematics and a modern over-the-counter software package Matlab to solve and study Stokes flow in a channel with a splitter plate. Specifically exact unidirectional flow solutions are used as a basis for choosing boundary conditions for Matlab to anticipate the boundary conditions of a Stokes flow. A method for selecting zeroth and first-order approximate boundary conditions is presented, along with a suggestion for finding a second-order approximation. It is also shown that small errors made in choosing the approximate boundary conditions do not grow as one moves away from the boundary into the interior of the flow. Finally several computational examples using this approach are presented.