A transformation‐free HOC scheme for steady convection–diffusion on non‐uniform grids

A higher order compact (HOC) finite difference solution procedure has been proposed for the steady two‐dimensional (2D) convection–diffusion equation on non‐uniform orthogonal Cartesian grids involving no transformation from the physical space to the computational space. Effectiveness of the method is seen from the fact that for the first time, an HOC algorithm on non‐uniform grid has been extended to the Navier–Stokes (N–S) equations. Apart from avoiding usual computational complexities associated with conventional transformation techniques, the method produces very accurate solutions for difficult test cases. Besides including the good features of ordinary HOC schemes, the method has the advantage of better scale resolution with smaller number of grid points, with resultant saving of memory and CPU time. Gain in time however may not be proportional to the decrease in the number of grid points as grid non‐uniformity imparts asymmetry to some of the associated matrices which otherwise would have been symmetric. The solution procedure is also highly robust as it computes complex flows such as that in the lid‐driven square cavity at high Reynolds numbers (Re), for which no HOC results have so far been seen. Copyright © 2004 John Wiley & Sons, Ltd.     

The (9,5) HOC formulation for the transient Navier–Stokes equations in primitive variable

We recently proposed an improved (9,5) higher order compact (HOC) scheme for the unsteady two‐dimensional (2‐D) convection–diffusion equations. Because of using only five points at the current time level in the discretization procedure, the scheme was seen to be computationally more efficient than its predecessors. It was also seen to capture very accurately the solution of the unsteady 2‐D Navier–Stokes (N–S) equations for incompressible viscous flows in the stream function–vorticity (ψ – ω) formulation. In this paper, we extend the scope of the scheme for solving the unsteady incompressible N–S equations based on primitive variable formulation on a collocated grid. The parabolic momentum equations are solved for the velocity field by a time‐marching strategy and the pressure is obtained by discretizing the elliptic pressure Poisson equation by the steady‐state form of the (9,5) scheme with the Neumann boundary conditions. In particular, for pressure, we adopt a strategy on the collocated grid in conjunction with ideas borrowed from the staggered grid approach in finite volume. We first apply this extension to a problem having analytical solution and then to the famous lid‐driven square cavity problem. We also apply our formulation to the backward‐facing step problem to see how the method performs for external flow problems. The results are presented and are compared with established numerical results. This new approach is seen to produce excellent comparison in all the cases. Copyright © 2007 John Wiley & Sons, Ltd.     

A class of higher order compact schemes for the unsteady two‐dimensional convection–diffusion equation with variable convection coefficients

A class of higher order compact (HOC) schemes has been developed with weighted time discretization for the two‐dimensional unsteady convection–diffusion equation with variable convection coefficients. The schemes are second or lower order accurate in time depending on the choice of the weighted average parameter μ and fourth order accurate in space. For 0.5?μ?1, the schemes are unconditionally stable. Unlike usual HOC schemes, these schemes are capable of using a grid aspect ratio other than unity. They efficiently capture both transient and steady solutions of linear and nonlinear convection–diffusion equations with Dirichlet as well as Neumann boundary condition. They are applied to one linear convection–diffusion problem and three flows of varying complexities governed by the two‐dimensional incompressible Navier–Stokes equations. Results obtained are in excellent agreement with analytical and established numerical results. Overall the schemes are found to be robust, efficient and accurate. Copyright © 2002 John Wiley & Sons, Ltd.     

A locally DIV‐free projection scheme for incompressible flows based on non‐conforming finite elements

We present a projection scheme whose end‐of‐step velocity is locally pointwise divergence free, using a continuous ?₁ approximation for the velocity in the momentum equation, a first‐order Crouzeix–Raviart approximation at the projection step, and a ?₀ approximation for the pressure in both steps. The analysis of the scheme is done only for grids that guarantee the existence of a divergence free conforming ?₁ interpolant for the velocity. Optimal estimates for the velocity error in L²‐ and H¹‐norms are deduced. The numerical results demonstrate that these estimates should also hold on grids on which the continuous ?₁ approximation for the velocity locks. Since the end‐of‐step velocity is locally solenoidal, the scheme is recommendable for problems requiring good mass conservation. Copyright © 2005 John Wiley & Sons, Ltd.     

Parallel overlapping scheme for viscous incompressible flows

A 3D parallel overlapping scheme for viscous incompressible flow problems is presented that combines the finite element method, which is best suited for analysing flow in any arbitrarily shaped flow geometry, with the finite difference method, which is advantageous in terms of both computing time and computer storage. A modified ABMAC method is used as the solution algorithm, to which a sophisticated time integration scheme proposed by the present authors has been applied. Parallelization is based on the domain decomposition method. The RGB (recursive graph bisection) algorithm is used for the decomposition of the FEM mesh and simple slice decomposition is used for the FDM mesh. Some estimates of the parallel performance of FEM, FDM and overlapping computations are presented. © 1997 John Wiley & Sons, Ltd.     

Genuinely multidimensional characteristic‐based scheme for incompressible flows

This paper presents a novel multidimensional characteristic‐based (MCB) upwind method for the solution of incompressible Navier–Stokes equations. As opposed to the conventional characteristic‐based (CB) schemes, it is genuinely multidimensional in that the local characteristic paths, along which information is propagated, are used. For the first time, the multidimensional characteristic structure of incompressible flows modified by artificial compressibility is extracted and used to construct an inherent multidimensional upwind scheme. The new proposed MCB scheme in conjunction with the finite‐volume discretization is employed to model the convective fluxes. Using this formulation, the steady two‐dimensional incompressible flow in a lid‐driven cavity is solved for a wide range of Reynolds numbers. It was found that the new proposed scheme presents more accurate results than the conventional CB scheme in both their first‐ and second‐order counterparts in the case of cavity flow. Also, results obtained with second‐order MCB scheme in some cases are more accurate than the central scheme that in turn provides exact second‐order discretization in this grid. With this inherent upwinding technique for evaluating convective fluxes at cell interfaces, no artificial viscosity is required even at high Reynolds numbers. Another remarkable advantage of MCB scheme lies in its faster convergence rate with respect to the CB scheme that is found to exhibit substantial delays in convergence reported in the literature. The results obtained using new proposed scheme are in good agreement with the standard benchmark solutions in the literature. Copyright © 2007 John Wiley & Sons, Ltd.     

A short note on the general boundary element method for viscous flows with high Reynolds number

In this paper, the general boundary element method and the parallel computation are employed to solve laminar viscous flows in a driven square cavity, governed by the exact Navier–Stokes equations. Using the solution at Re=0 as the initial approximation, the convergent numerical results for high Reynolds number at Re=7500 are obtained, for the first time, by the boundary element method. This verifies the validity and great potential of the general boundary element method for highly non‐linear problems, which may greatly enlarge application regions of the boundary element method in science and engineering. Copyright © 2003 John Wiley & Sons, Ltd.     

Development of a convection–diffusion‐reaction magnetohydrodynamic solver on non‐staggered grids

This paper presents a convection–diffusion‐reaction (CDR) model for solving magnetic induction equations and incompressible Navier–Stokes equations. For purposes of increasing the prediction accuracy, the general solution to the one‐dimensional constant‐coefficient CDR equation is employed. For purposes of extending this discrete formulation to two‐dimensional analysis, the alternating direction implicit solution algorithm is applied. Numerical tests that are amenable to analytic solutions were performed in order to validate the proposed scheme. Results show good agreement with the analytic solutions and high rate of convergence. Like many magnetohydrodynamic studies, the Hartmann–Poiseuille problem is considered as a benchmark test to validate the code. Copyright © 2004 John Wiley & Sons, Ltd.     

A streamfunction–velocity approach for 2D transient incompressible viscous flows

Electrodeposition is a widely used technique for the fabrication of high aspect ratio microstructures. In recent years, much research has been focused within this area aiming to understand the physics behind the filling of high aspect ratio vias and trenches on substrates and in particular how they can be made without the formation of voids in the deposited material. This paper reports on the fundamental work towards the advancement of numerical algorithms that can predict the electrodeposition process in micron scaled features. Two different numerical approaches have been developed, which capture the motion of the deposition interface and 2‐D simulations are presented for both methods under two deposition regimes: those where surface kinetics is governed by Ohm's law and the Butler–Volmer equation, respectively. In the last part of this paper the modelling of acoustic forces and their subsequent impact on the deposition profile through convection is examined. Copyright © 2009 John Wiley & Sons, Ltd.     

A new stable space–time formulation for two‐dimensional and three‐dimensional incompressible viscous flow

A space–time finite element method for the incompressible Navier–Stokes equations in a bounded domain in ?^d (with d=2 or 3) is presented. The method is based on the time‐discontinuous Galerkin method with the use of simplex‐type meshes together with the requirement that the space–time finite element discretization for the velocity and the pressure satisfy the inf–sup stability condition of Brezzi and Babu?ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two‐dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

A consistent splitting scheme for unsteady incompressible viscous flows I. Dirichlet boundary condition and applications

A well‐recognized approach for handling the incompressibility constraint by operating directly on the discretized Navier–Stokes equations is used to obtain the decoupling of the pressure from the velocity field. By following the current developments by Guermond and Shen, the possibilities of obtaining accurate pressure and reducing boundary‐layer effect for the pressure are analysed. The present study mainly reports the numerical solutions of an unsteady Navier–Stokes problem based on the so‐called consistent splitting scheme (J. Comput. Phys. 2003; 192 :262–276). At the same time the Dirichlet boundary value conditions are considered. The accuracy of the method is carefully examined against the exact solution for an unsteady flow physics problem in a simply connected domain. The effectiveness is illustrated viz. several computations of 2D double lid‐driven cavity problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite volume method for viscous incompressible flows using a consistent flux reconstruction scheme

An incompressible Navier–Stokes solver using curvilinear body‐fitted collocated grid has been developed to solve unconfined flow past arbitrary two‐dimensional body geometries. In this solver, the full Navier–Stokes equations have been solved numerically in the physical plane itself without using any transformation to the computational plane. For the proper coupling of pressure and velocity field on collocated grid, a new scheme, designated ‘consistent flux reconstruction’ (CFR) scheme, has been developed. In this scheme, the cell face centre velocities are obtained explicitly by solving the momentum equations at the centre of the cell faces. The velocities at the cell centres are also updated explicitly by solving the momentum equations at the cell centres. By resorting to such a fully explicit treatment considerable simplification has been achieved compared to earlier approaches. In the present investigation the solver has been applied to unconfined flow past a square cylinder at zero and non‐zero incidence at low and moderate Reynolds numbers and reasonably good agreement has been obtained with results available from literature. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical simulation of flows around two circular cylinders by mesh‐free least square‐based finite difference methods

In this paper, the mesh‐free least square‐based finite difference (MLSFD) method is applied to numerically study the flow field around two circular cylinders arranged in side‐by‐side and tandem configurations. For each configuration, various geometrical arrangements are considered, in order to reveal the different flow regimes characterized by the gap between the two cylinders. In this work, the flow simulations are carried out in the low Reynolds number range, that is, Re=100 and 200. Instantaneous vorticity contours and streamlines around the two cylinders are used as the visualization aids. Some flow parameters such as Strouhal number, drag and lift coefficients calculated from the solution are provided and quantitatively compared with those provided by other researchers. Copyright © 2006 John Wiley & Sons, Ltd.     

Control strategies for timestep selection in finite element simulation of incompressible flows and coupled reaction–convection–diffusion processes

We propose two timestep selection algorithms, based on feedback control theory, for finite element simulation of steady state and transient 2D viscous flow and coupled reaction–convection–diffusion processes. To illustrate performance of the schemes in practice, we solve Rayleigh–Benard–Marangoni flows, flow across a backward‐facing step, unsteady flow around a circular cylinder and chemical reaction systems. Numerical experiments confirm that the feedback controllers produce in some cases a very smooth stepsize variation, suggesting that robust control algorithms are possible. These experiments also show that parameter selection can improve timesteps when co‐ordinated with the convergence control of non‐linear iterations. Further, computational cost of the selection procedures is negligible, since they involve only storing a few extra vectors, computation of norms and evaluation of kinetic energy. Copyright © 2004 John Wiley & Sons, Ltd.     

Incompressible Roe‐like scheme for turbulent flows

In the current study, numerical investigation of incompressible turbulent flow is presented. By the artificial compressibility method, momentum and continuity equations are coupled. Considering Reynolds averaged Navier–Stokes equations, the Spalart–Allmaras turbulence model, which has accurate results in two‐dimensional problems, is used to calculate Reynolds stresses. For convective fluxes a Roe‐like scheme is proposed for the steady Reynolds averaged Navier–Stokes equations. Also, Jameson averaging method was implemented. In comparison, the proposed characteristics‐based upwind incompressible turbulent Roe‐like scheme, demonstrated very accurate results, high stability, and fast convergence. The fifth‐order Runge–Kutta scheme is used for time discretization. The local time stepping and implicit residual smoothing were applied as the convergence acceleration techniques. Suitable boundary conditions have been implemented considering flow behavior. The problem has been studied at high Reynolds numbers for cross flow around the horizontal circular cylinder and NACA0012 hydrofoil. Results were compared with those of others and a good agreement has been observed. Copyright © 2012 John Wiley & Sons, Ltd.     

A simplified circular function–based gas kinetic scheme for simulation of incompressible flows

In this paper, the circular function–based gas kinetic scheme (GKS), which is often applied for simulation of compressible flows, is simplified to improve computational efficiency for simulation of incompressible flows. In the original circular function–based GKS, the integral domain along the circle for computing conservative variables and numerical fluxes is usually not symmetric at the cell interface. This leads to relatively complicated formulations for computing the numerical flux at the cell interface. As shown in this work, for incompressible flows, the circle at the cell interface can be approximately considered to be symmetric. As a consequence, the simple expressions for calculation of conservative variables and numerical fluxes at the cell interface can be obtained, and computational efficiency is greatly improved. In the meanwhile, like the original circular function–based GKS, the discontinuity of conservative variables and their derivatives at the cell interface is still kept in the present scheme to keep good numerical stability at high Reynolds numbers. Several numerical examples, including decaying vortex flow, lid‐driven cavity flow, and flow past a stationary and rotating circular cylinder, are tested to validate the accuracy, efficiency, and stability of the present scheme.     

Fourth‐order finite difference scheme for the incompressible Navier–Stokes equations in a disk

We develop an efficient fourth‐order finite difference method for solving the incompressible Navier–Stokes equations in the vorticity‐stream function formulation on a disk. We use the fourth‐order Runge–Kutta method for the time integration and treat both the convection and diffusion terms explicitly. Using a uniform grid with shifting a half mesh away from the origin, we avoid placing the grid point directly at the origin; thus, no pole approximation is needed. Besides, on such grid, a fourth‐order fast direct method is used to solve the Poisson equation of the stream function. By Fourier filtering the vorticity in the azimuthal direction at each time stage, we are able to increase the time step to a reasonable size. The numerical results of the accuracy test and the simulation of a vortex dipole colliding with circular wall are presented. Copyright © 2003 John Wiley & Sons, Ltd.     

A multigrid approach for steady state laminar viscous flows

In this paper, we use the laminar viscous flow in a lid‐driven cavity as an example to describe and verify a numerical scheme for non‐linear partial differential equations. The proposed scheme combines a new analytical method for strongly non‐linear problems, namely the homotopy analysis method, with the multigrid techniques. A family of formulas at different orders is given. At the lowest order, the current approach is the same as the traditional multigrid methods. However, our high‐order scheme needs a fewer number of iterations and less CPU time than the classical ones. Copyright © 2001 John Wiley & Sons, Ltd.