A velocity-decomposition formulation for the incompressible Navier–Stokes equations

The principle of velocity decomposition is used to combine field discretization and boundary-element techniques to solve for steady, viscous, external flows around bodies. The decomposition modifies the Navier–Stokes boundary-value problem and produces a Laplace problem for a viscous potential, and a new Navier–Stokes sub-problem that can be solved on the portion of the domain where the total velocity has rotation. The key development in the decomposition is the formulation for the boundary condition on the viscous potential that couples the two components of velocity. An iterative numerical scheme is described to solve the decomposed problem. Results are shown for the steady laminar flow over a sectional airfoil, a circular cylinder with separation, and the turbulent flow around a slender body-of-revolution. The results show the viscous potential is obtainable even for massively separated flows, and the field discretization must only encompass the vortical region of the total velocity.     

Primitive variable dual reciprocity boundary element method solution of incompressible Navier–Stokes equations

This paper describes the solution to transient incompressible two-dimensional Navier–Stokes equations in primitive variables by the dual reciprocity boundary element method. The coupled set of mass and momentum equations is structured by the fundamental solution of the Laplace equation. The dual reciprocity method is based on the augmented thin plate splines. All derivatives involved are calculated through integral representation formulas. Numerical example include convergence studies with different mesh size for the classical lid-driven cavity problem at Re=100 and comparison with the results obtained through calculation of the derivatives from global interpolation formulas. The accuracy of the solution is assessed by comparison with the Ghia–Ghia–Shin finite difference solution as a reference.     

A dual reciprocity boundary element formulation using the fractional step method for the incompressible Navier–Stokes equations

This paper presents a dual reciprocity boundary element solution method for the unsteady Navier–Stokes equations in two-dimensional incompressible flow, where a fractional step algorithm is utilized for the time advancement. A fully explicit, second-order, Adams–Bashforth scheme is used for the nonlinear convective terms. We performed numerical tests for two examples: the Taylor–Green vortex and the lid-driven square cavity flow for Reynolds numbers up to 400. The results in the former case are compared to the analytical solution, and in the latter to numerical results available in the literature. Overall the agreement is excellent demonstrating the applicability and accuracy of the fractional step, dual reciprocity boundary element solution formulations to the Navier–Stokes equations for incompressible flows.     

A three-step Oseen correction method for the steady Navier–Stokes equations

We present and analyze a two-grid scheme based on mixed finite element approximations for the steady incompressible Navier–Stokes equations. This numerical scheme aims at the simulations of high Reynolds number flows and consists of three steps: in the first step, we solve a finite element variational multiscale-stabilized nonlinear Navier–Stokes system on a coarse mesh, and then, in the second and third steps, we solve Oseen-linearized and -stabilized problems which have the same stiffness matrices with only different right-hand sides on a fine mesh. We provide error bounds for the approximate solutions, derive algorithmic parameter scalings from the analysis, and present some numerical results to verify the theoretical predictions and demonstrate the effectiveness of the proposed method.     

The method of fundamental solutions for solving incompressible Navier–Stokes problems

A novel meshless numerical procedure based on the method of fundamental solutions (MFS) is proposed to solve the primitive variables formulation of the Navier–Stokes equations. The MFS is a meshless method since it is free from the mesh generation and numerical integration. We will transform the Navier–Stokes equations into simple advection–diffusion and Poisson differential operators via the operator-splitting scheme or the so-called projection method, instead of directly using the more complicated fundamental solutions (Stokeslets) of the unsteady Stokes equations. The resultant velocity advection–diffusion equations and the pressure Poisson equation are then calculated by using the MFS together with the Eulerian–Lagrangian method (ELM) and the method of particular solutions (MPS). The proposed meshless numerical scheme is a first attempt to apply the MFS for solving the Navier–Stokes equations in the moderate-Reynolds-number flow regimes. The lid-driven cavity flows at the Reynolds numbers up to 3200 for two-dimensional (2D) and 1000 for three-dimensional (3D) are chosen to validate the present algorithm. Through further simulating the flows in the 2D circular cavity with an eccentric rotating cylinder and in the 3D cube with a fixed sphere inside, we are able to demonstrate the advantages and flexibility of the proposed meshless method in the irregular geometry and multi-dimensional flows, even though very coarse node points are used in this study as compared with other mesh-dependent numerical schemes.     

A matrix-form GSM–CFD solver for incompressible fluids and its application to hemodynamics

A GSM–CFD solver for incompressible flows is developed based on the gradient smoothing method (GSM). A matrix-form algorithm and corresponding data structure for GSM are devised to efficiently approximate the spatial gradients of field variables using the gradient smoothing operation. The calculated gradient values on various test fields show that the proposed GSM is capable of exactly reproducing linear field and of second order accuracy on all kinds of meshes. It is found that the GSM is much more robust to mesh deformation and therefore more suitable for problems with complicated geometries. Integrated with the artificial compressibility approach, the GSM is extended to solve the incompressible flows. As an example, the flow simulation of carotid bifurcation is carried out to show the effectiveness of the proposed GSM–CFD solver. The blood is modeled as incompressible Newtonian fluid and the vessel is treated as rigid wall in this paper.     

Development of a scalable finite element solution to the Navier–Stokes equations

A scalable numerical model to solve the unsteady incompressible Navier–Stokes equations is developed using the Galerkin finite element method. The coupled equations are decoupled by the fractional-step method and the systems of equations are inverted by the Krylov subspace iterations. The data structure makes use of a domain decomposition of which each processor stores the parameters in its subdomain, while the linear equations solvers and matrices constructions are parallelized by a data parallel approach. The accuracy of the model is tested by modeling laminar flow inside a two-dimensional square lid-driven cavity for Reynolds numbers at 1,000 as well as three-dimensional turbulent plane and wavy Couette flow and heat transfer at high Reynolds numbers. The parallel performance of the code is assessed by measuring the CPU time taken on an IBM SP2 supercomputer. The speed up factor and parallel efficiency show a satisfactory computational performance.The authors wish to acknowledge Mr. W. K. Kwan of The University of Hong Kong for his help in using the IBM SP2 supercomputer.     

Convergence acceleration of iterative algorithms. Applications to thin shell analysis and Navier–Stokes equations

This work deals with the convergence acceleration of iterative nonlinear methods. Two convergence accelerating techniques are evaluated: the Modified Mininal Polynomial Extrapolation Method (MMPE) and the Padé approximants. The algorithms studied in this work are iterative correctors: Newton’s modified method, a high-order iterative corrector presented in Damil et al. (Commun Numer Methods Eng 15:701–708, 1999) and an original algorithm for vibration of viscoelastic structures. We first describe the iterative algorithms for the considered nonlinear problems. Secondly, the two accelerating techniques are presented. Finally, through several numerical tests from the thin shell theory, Navier–Stokes equations and vibration of viscoelastic shells, the advantages and drawbacks of each accelerating technique is discussed.     

The localized differential quadrature method for two-dimensional stream function formulation of Navier–Stokes equations

Localized differential quadrature (LDQ) method is employed to solve two-dimensional stream function formulation of incompressible Navier–Stokes equations. Being developed by introducing the localization concept to the general differential quadrature (GDQ) method, the employment of LDQ method becomes efficient and flexible, especially for the simulations of large scale computations. By introducing the Lagrange stream function to vorticity transport equation, the governing equation—the fourth-order partial differential equation (PDE)—is derived. To stably obtain the solutions of the fourth-order PDE, a fictitious point method is included to treat the boundary conditions. To examine the present scheme, two different types of classic benchmark fluid flow problems are proposed, including driven cavity flow problems and backward-facing step flow problems. The good agreement of solutions demonstrate the robustness and feasibility of the proposed scheme. Conclusively, the LDQ method is sufficient and appropriate enough to simulate the solutions of stream function formulation of Navier–Stokes equations with various Reynolds numbers.     

Conservative interpolation for the boundary integral solution of the Navier–Stokes equations

The Dual Reciprocity Method (DRM) is a technique to transform the domain integrals that appear in the boundary element method into equivalent boundary integrals. In this approach the non-linear terms are usually approximated by mathematical interpolation applied to the convective terms of the form of the Navier–Stokes equations. In this paper we introduce a conservative interpolation scheme that satisfies the continuity equation and performs better than pure mathematical interpolation. The new scheme together with a subdomain variation of the dual reciprocity method allows better approximation of the non-linear terms in the Navier–Stokes equations for moderate Reynolds number. Received: 21 January 2000     

A combined BEM–FEM model for the velocity–vorticity formulation of the Navier–Stokes equations in three dimensions

This paper describes a combined boundary element and finite element model for the solution of velocity–vorticity formulation of the Navier–Stokes equations in three dimensions. In the velocity–vorticity formulation of the Navier–Stokes equations, the Poisson type velocity equations are solved using the boundary element method (BEM) and the vorticity transport equations are solved using the finite element method (FEM) and both are combined to form an iterative scheme. The vorticity boundary conditions for the solution of vorticity transport equations are exactly obtained directly from the BEM solution of the velocity Poisson equations. Here the results of medium Reynolds number of up to 1000, in a typical cubic cavity flow are presented and compared with other numerical models. The combined BEM–FEM model are generally in fairly close agreement with the results of other numerical models, even for a coarse mesh.     

Mixed interval–fuzzy two-stage integer programming and its application to flood-diversion planning

Innovative prevention, adaptation, and mitigation approaches as well as policies for sustainable flood management continue to be challenges faced by decision-makers. In this study, a mixed interval–fuzzy two-stage integer programming (IFTIP) method is developed for flood-diversion planning under uncertainty. This method improves upon the existing interval, fuzzy, and two-stage programming approaches by allowing uncertainties expressed as probability distributions, fuzzy sets, and discrete intervals to be directly incorporated within the optimization framework. In its modelling formulation, economic penalties as corrective measures against any infeasibilities arising because of a particular realization of the uncertainties are taken into account. The method can also be used for analysing a variety of policy scenarios that are associated with different levels of economic penalties. A management problem in terms of flood control is studied to illustrate the applicability of the proposed approach. The results indicate that reasonable solutions have been generated. They can provide desired flood-diversion alternatives and capacity-expansion schemes with a minimized system cost and a maximized safety level. The developed IFTIP is also applicable to other management problems that involve uncertainties presented in multiple formats as well as complexities in policy dynamics.     

Photoconductive properties of TiO2 films prepared by the sol–gel method and its application

The Photoconductive characteristics of TiO2 films prepared by the sol-gel method, and the photovoltaic characteristics fabricated with the resulting TiO2 film and phthalocyanine nickel (NiPc) are investigated. For a TiO2 film with hydroxypropyl cellulose (Hpc) heat treated at 500°C for 10 min, the relative sensitivity is about ten times higher than that without Hpc. A space-charge-limited current is observed in the dark current–voltage characteristic of the TiO2 film with Hpc. It is found that the TiO2 film with Hpc has a photosensitizing effect. The photovoltaic characteristics of TiO2(Hpc)/NiPc are as follows: the short-circuit current density, Jsc is 5.6×10-7 A cm-2, the open-circuit voltage, Voc is 0.24 V, the fill factor (F.F) is 0.64 and the power conversion efficiency, is 0.73. Furthermore, the carrier transport mechanisms of the TiO2(Hpc)/NiPc photovoltaic cell are discussed.     

A constitutive multiphysics modeling for nearly incompressible dissipative materials: application to thermo–chemo-mechanical aging of rubbers

In this paper we investigate the modeling of chemo-physical evolution due to thermo-mechanical loadings at finite strain in soft materials. In particular we discuss the question of a proper and consistent thermodynamical formulation in the case of nearly incompressible materials. The objective of this phenomenological modeling is to represent the thermo–chemo-mechanical aging that occurs in filled rubbers during high-cycle fatigue for some specific loading conditions.