On accurate boundary conditions for a shape sensitivity equation method

This paper studies the application of the continuous sensitivity equation method (CSEM) for the Navier–Stokes equations in the particular case of shape parameters. Boundary conditions for shape parameters involve flow derivatives at the boundary. Thus, accurate flow gradients are critical to the success of the CSEM. A new approach is presented to extract accurate flow derivatives at the boundary. High order Taylor series expansions are used on layered patches in conjunction with a constrained least‐squares procedure to evaluate accurate first and second derivatives of the flow variables at the boundary, required for Dirichlet and Neumann sensitivity boundary conditions. The flow and sensitivity fields are solved using an adaptive finite‐element method. The proposed methodology is first verified on a problem with a closed form solution obtained by the Method of Manufactured Solutions. The ability of the proposed method to provide accurate sensitivity fields for realistic problems is then demonstrated. The flow and sensitivity fields for a NACA 0012 airfoil are used for fast evaluation of the nearby flow over an airfoil of different thickness (NACA 0015). Copyright © 2005 John Wiley & Sons, Ltd.     

Evaluation of the drag force by integrating the energy dissipation rate in stokes flow for 2D domains using the FEM

The total drag force on the surface of a body, which is the sum of the form drag and the skin friction drag in a 2D domain, is numerically evaluated by integrating the energy dissipation rate in the whole domain for an incompressible Stokes fluid. The finite element method is used to calculate both the energy dissipation rate in the whole domain as well as the drag on the boundary of the body. The evaluation of the drag and the energy dissipation rate are post-processing operations which are carried out after the velocity field and the pressure field for the flow over a particular profile have been obtained. The results obtained for the flow over three different but constant area profiles—a circle, an ellipse and a cross-section of a prolate spheroid—with uniform inlet velocity are presented and it is shown that the total drag force times the velocity is equal to the total energy dissipation rate in the entire finite flow domain. Hence, by calculating the energy dissipation rate in the domain with unit velocity specified at the far-field boundary enclosing the domain, the drag force on the boundary of the body can be obtained.     

Coupled lubrication and Stokes flow finite elements

A method is developed for performing a local reduction of the governing physics for fluid problems with domains that contain a combination of narrow and non‐narrow regions, and the computational accuracy and performance of the method are measured. In the narrow regions of the domain, where the fluid is assumed to have no inertia and the domain height and curvature are assumed small, lubrication, or Reynolds, theory is used locally to reduce the two‐dimensional Navier–Stokes equations to the one‐dimensional Reynolds equation while retaining a high degree of accuracy in the overall solution. The Reynolds equation is coupled to the governing momentum and mass equations of the non‐narrow region with boundary conditions on the mass and momentum flux. The localized reduction technique, termed ‘stitching,’ is demonstrated on Stokes flow for various geometries of the hydrodynamic journal bearing—a non‐trivial test problem for which a known analytical solution is available. The computational advantage of the coupled Stokes–Reynolds method is illustrated on an industrially applicable fully‐flooded deformable‐roll coating example. The examples in this paper are limited to two‐dimensional Stokes flow, but extension to three‐dimensional and Navier–Stokes flow is possible. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical simulation on an interaction of a vortex street with an elliptical leading edge using an unstructured grid

An algorithm for a time accurate incompressible Navier–Stokes solver on an unstructured grid is presented. The algorithm uses a second order, three‐point, backward difference formula for the physical time marching. For each time step, a divergence free flow field is obtained based on an artificial compressibility method. An implicit method with a local time step is used to accelerate the convergence for the pseudotime iteration. To validate the code, an unsteady laminar flow over a circular cylinder at a Reynolds number of 200 is calculated. The results are compared with available experimental and numerical data and good agreements are achieved. Using the developed unsteady code, an interaction of a Karman vortex street with an elliptical leading edge is simulated. The incident Karman vortex street is generated by a circular cylinder located upstream. A clustering to the path of the vortices is achieved easily due to flexibility of an unstructured grid. Details of the interaction mechanism are analysed by investigating evolutions of vortices. Characteristics of the interactions are compared for large‐ and small‐scale vortex streets. Different patterns of the interaction are observed for those two vortex streets and the observation is in agreement with experiment. Copyright © 2004 John Wiley & Sons, Ltd.     

Simulation of single‐ and two‐phase flows on sliding unstructured meshes using finite volume method

The paper presents an efficient finite volume method for unstructured grids with rotating sliding parts composed of arbitrary polyhedral elements for both single‐ and two‐phase flows. Mathematical model used in computations is based on the ensemble averaged conservation equations. These equations are solved for each phase and in case of single‐phase flow reduce to the transient Reynolds‐averaged Navier–Stokes (TRANS) equations. Transient flow induced by rotating impellers is thus resolved in time. The use of unstructured grids allows an easy and flexible meshing for the entire flow domain. Polyhedral cell volumes are created on the arbitrary mesh interface placed between rotating and static parts. Cells within the rotating parts move each time step and the new faces are created on the arbitrary interfaces only, while the rest of the domain remain ‘topologically’ unchanged. Implicit discretization scheme allows a wide range of time‐step sizes, which further reduce the computational effort. Special attention is given to the interpolation practices used for the reconstruction of the face quantities. Mass fluxes are recalculated at the beginning of each time step by using an interpolation scheme, which enhances the coupling between the pressure and velocity fields. The model has been implemented into the commercially available CFD code AVL SWIFT (AVL AST, SWIFT Manual 3.1, AVL List GmbH, Graz, Austria, 2002). Single‐phase flow in a mixing vessel stirred by a six‐bladed Rushton‐type turbine and two‐phase flow in aerated stirred vessel with the four‐blade Rushton impeller are simulated. The results are compared with the available experimental data, and good agreement is observed. The proposed algorithm is proved to be both stable and accurate for single‐phase as well as for the two‐phase flows calculations. Copyright 2004 John Wiley & Sons, Ltd.     

Calculation of impulsively started incompressible viscous flows

The paper's focus is the calculation of unsteady incompressible 2D flows past airfoils. In the framework of the primitive variable Navier–Stokes equations, the initial and boundary conditions must be assigned so as to be compatible, to assure the correct prediction of the flow evolution. This requirement, typical of all incompressible flows, viscous or inviscid, is often violated when modelling the flow past immersed bodies impulsively started from rest. Its fulfillment can however be restored by means of a procedure enforcing compatibility, consisting in a pre‐processing of the initial velocity field, here described in detail. Numerical solutions for an impulsively started multiple airfoil have been obtained using a finite element incremental projection method. The spatial discretization chosen for the velocity and pressure are of different order to satisfy the inf–sup condition and obtain a smooth pressure field. Results are provided to illustrate the effect of employing or not the compatibility procedure, and are found in good agreement with those obtained with a non‐primitive variable solver. In addition, we introduce a post‐processing procedure to evaluate an alternative pressure field which is found to be more accurate than the one resulting from the projection method. This is achieved by considering an appropriate ‘unsplit’ version of the momentum equation, where the velocity solution of the projection method is substituted. Copyright © 2004 John Wiley & Sons, Ltd.     

散射体系A+BC的平-振能量传递

用Alhassid与Levine所提出的动力学李代数的方法(简称A-L理论),研究了含有二级非简谐的共线散射体系A+BC的平-振能量传递问题,计算了散射过程含有主要动力学参量的跃迁矩阵和跃迁几率的解析表示式.     

A fourth‐order compact finite difference scheme for the steady stream function–vorticity formulation of the Navier–Stokes/Boussinesq equations

A fourth‐order compact finite difference scheme on the nine‐point 2D stencil is formulated for solving the steady‐state Navier–Stokes/Boussinesq equations for two‐dimensional, incompressible fluid flow and heat transfer using the stream function–vorticity formulation. The main feature of the new fourth‐order compact scheme is that it allows point‐successive overrelaxation (SOR) or point‐successive underrelaxation iteration for all Rayleigh numbers Ra of physical interest and all Prandtl numbers Pr attempted. Numerical solutions are obtained for the model problem of natural convection in a square cavity with benchmark solutions and compared with some of the accurate results available in the literature. Copyright © 2003 John Wiley & Sons, Ltd.     

Long time behaviour of a flow in infinite pipe conforming to slip boundary conditions

The paper analyses long time behaviour of solutions of the Navier–Stokes equations in a two‐dimensional pipe‐like domain. The system is studied with perfect slip boundary conditions with arbitrary inflow conditions at infinity. The main results show the existence of global in time solutions and of an attractor for the dynamical system generated by the model. The paper also establishes an upper bound for the Hausdorff dimension of the attractor. Copyright © 2005 John Wiley & Sons, Ltd.     

The Ring of Invariants of Three (3 × 3)-Matrices over a Field of Prime Characteristic

We find a minimal system of generators and a homogeneous system of parameters for the algebra of invariants of three matrices of third order over a field of an arbitrary characteristic.