International Journal for Numerical Methods in Fluids

Details are given of the computational method used to obtain an accurate solution of the equations describing two‐dimensional natural convection in a square cavity with differentially heated side walls. Second‐order, central difference approximations were used. Mesh refnement and extrapolation led to solutions for 10³⩽Ra⩽10 ⁶ which are believed to be accurate to better than 1 per cent at the highest Rayleigh number and down to one‐tenth of that at the lowest value.

A new continuous reproducing kernel interpolation function which explores the attractive features of the flexible time‐frequency and space‐wave number localization of a window function is developed. This method is motivated by the theory of wavelets and also has the desirable attributes of the recently proposed smooth particle hydrodynamics (SPH) methods, moving least squares methods (MLSM), diffuse element methods (DEM) and element‐free Galerkin methods (EFGM). The proposed method maintains the advantages of the free Lagrange or SPH methods; however, because of the addition of a correction function, it gives much more accurate results. Therefore it is called the reproducing kernel particle method (RKPM). In computer implementation RKPM is shown to be more efficient than DEM and EFGM. Moreover, if the window function is C^∞, the solution and its derivatives are also C^∞ in the entire domain. Theoretical analysis and numerical experiments on the 1D diffusion equation reveal the stability conditions and the effect of the dilation parameter on the unusually high convergence rates of the proposed method. Two‐dimensional examples of advection‐diffusion equations and compressible Euler equations are also presented together with 2D multiple‐scale decompositions.

After several years of planning, the 1st International Workshop on High‐Order CFD Methods was successfully held in Nashville, Tennessee, on January 7–8, 2012, just before the 50th Aerospace Sciences Meeting. The American Institute of Aeronautics and Astronautics, the Air Force Office of Scientific Research, and the German Aerospace Center provided much needed support, financial and moral. Over 70 participants from all over the world across the research spectrum of academia, government labs, and private industry attended the workshop. Many exciting results were presented. In this review article, the main motivation and major findings from the workshop are described. Pacing items requiring further effort are presented. Copyright © 2013 John Wiley & Sons, Ltd.

A global method of generalized differential quadrature is applied to solve the two‐dimensional incompressible Navier‐Stokes equations in the vorticity‐stream‐function formulation. Numerical results for the flow past a circular cylinder were obtained using just a few grid points. A good agreement is found with the experimental data.

The paper describes a new approach to approximating the convection term found in typical steady‐state transport equations. A polynomial‐based discretization scheme is constructed around a technique called ‘curvature compensation’; the resultant curvature‐compensated convective transport approximation is essentially third‐order accurate in regions of the solution domain where the concept of order is meaningful. In addition, in linear scalar transport problems it preserves the boundedness of solutions. Sharp changes in gradient in the dependent variable are handled particularly well. But above all, the scheme, when used in conjunction with an ADI pentadiagonal solver, is easy to implement with relatively low computational cost, representing an effective algorithm for the simulation of multi‐dimensional fluid flows. Two linear test problems, for the case of transport by pure convection, are employed in order to assess the merit of the method.

A number of contributed solutions to the problem of laminar natural convection in a square cavity have been compared with what is regarded as a solution of high accuracy. The purposes of this exercise have been to confirm the accuracy of the bench mark solution and to provide a basis for the assessment of the various methods and computer codes used to obtain the contributed solutions.

In this paper, a thermal lattice BGK model is developed for the Boussinesq incompressible fluids. The basic idea is to solve the velocity field and the temperature field using two independent lattice BGK equations, respectively, and then combine them into one coupled model for the whole system. The porous plate problem and the two‐dimensional natural convection flow in a square cavity with Pr=0.71 and various of Rayleigh numbers are simulated using the model. The numerical results are found to be in good agreement with the analytical solutions or those of previous studies. Copyright © 2002 John Wiley & Sons, Ltd.

A finite volume multigrid procedure for the prediction of laminar natural convection flows is presented, enabling efficient and accurate calculations on very fine grids. The method is fully conservative and uses second‐order central differencing for convection and diffusion fluxes. The calculations start on a coarse (typically 10 × 10 control volumes) grid and proceed to finer grids until the desired accuracy or maximum affordable storage is reached. The computing times increase thereby linearly with the number of control volumes.

Solutions are presented for the flow in a closed cavity with side walls at different temperatures and insulated top and bottom walls. Rayleigh numbers of 10⁴, 10⁵ and 10⁶ are considered. Grids as fine as 640 × 640 control volumes are used and the results are believed to be accurate to within 0–01%. Second‐order monotonic convergence to grid‐independent values is observed for all predicted quantities.

A numerical solution for steady incompressible flow over a two‐dimensional backward‐facing step is developed using a Galerkin‐based finite element method. The Reynolds number for the simulations is 800. Computations are performed on an extended channel length to minimize the effect of the outflow boundary on the upstream recirculation zones. A thorough mesh refinement study is performed to validate the results. Extensive profile data at several channel locations are provided to allow future testing and evaluation of outflow boundary conditions.

The paper outlines the formulation of a novel algorithm which can be used for the solution of both compressible and incompressible Navier‐Stokes or Euler equations. Full incompressibility can be dealt with if the algorithm is used in its semi‐explicit form and its structure permits arbitrary interpolation functions to be used avoiding the Babuška‐Brezzi restriction. In a fully explicit version it introduces a rational form of balancing dissipation avoiding the use of arbitrary parameters and forms for this.