Least‐squares meshfree method for incompressible Navier–Stokes problems

A least‐squares meshfree method based on the first‐order velocity–pressure–vorticity formulation for two‐dimensional incompressible Navier–Stokes problem is presented. The convective term is linearized by successive substitution or Newton's method. The discretization of all governing equations is implemented by the least‐squares method. Equal‐order moving least‐squares approximation is employed with Gauss quadrature in the background cells. The boundary conditions are enforced by the penalty method. The matrix‐free element‐by‐element Jacobi preconditioned conjugate method is applied to solve the discretized linear systems. Cavity flow for steady Navier–Stokes problem and the flow over a square obstacle for time‐dependent Navier–Stokes problem are investigated for the presented least‐squares meshfree method. The effects of inaccurate integration on the accuracy of the solution are investigated. Copyright © 2004 John Wiley & Sons, Ltd.     

A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm

The velocity–vorticity formulation is selected to develop a time‐accurate CFD finite element algorithm for the incompressible Navier–Stokes equations in three dimensions.The finite element implementation uses equal order trilinear finite elements on a non‐staggered hexahedral mesh. A second order vorticity kinematic boundary condition is derived for the no slip wall boundary condition which also enforces the incompressibility constraint. A biconjugate gradient stabilized (BiCGSTAB) sparse iterative solver is utilized to solve the fully coupled system of equations as a Newton algorithm. The solver yields an efficient parallel solution algorithm on distributed‐memory machines, such as the IBM SP2. Three dimensional laminar flow solutions for a square channel, a lid‐driven cavity, and a thermal cavity are established and compared with available benchmark solutions. Copyright © 2002 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.     

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.     

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

The two‐dimensional time‐dependent Navier–Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first‐order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two‐dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two‐dimensional flow in a cavity when a force is present, the lid‐driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers. Copyright © 2008 John Wiley & Sons, Ltd.     

NUMERICAL SIMULATION OF UNSTEADY VISCOUS FLOWS USING AN IMPLICIT PROJECTION METHOD

In this paper an implicit projection method for the solution of the two-dimensional, time-dependent, incompressible Navier– Stokes equations is presented. The basic principle of this method is that the evaluation of the time evolution is split into intermediate steps. The computational method is based on the approximate factorization technique. The coupled approach is used to link the equations of motion and the turbulence model equations. The standard k-ϵ turbulence model is used. The current methodology, which has been tested extensively for steady problems, is now applied for the numerical simulation of unsteady flows. Several cases were tested, such as plane or axisymmetric channels, a backward-facing step, a square cavity and an axisymmetric stenosis.     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

A comparative study of central and upwind difference schemes using the primitive variables

The use of the velocity-pressure formulation of the Navier-Stokes equations for the numerical solution of fluid flow problems is favoured for free-surface problems, more involved flow configurations, and three-dimensional flows. Many engineering problems involve such features in addition to strong inertial effects. The computational instabilities arising from central-difference schemes for the convective terms of the governing equations impose serious limitations on the range of Reynolds numbers that can be investigated by the numerical method. Solutions for higher Reynolds numbers Re > 1000 could be reached using upwind-difference schemes. A comparative study of both schemes using a method based on the primitive variables is presented. The comparison is made for the model problem of the driven flow in a square cavity. Using a central scheme stable solutions of the pressure and velocity fields were obtained for Reynolds numbers up to 5000. The streamfunction and vorticity fields were calculated from the resulting velocity field and compared with previous solutions. It is concluded that total upwind differencing results in a considerable change in the flow pattern due to the false diffusion. For practical calculations, by a proper choice of a small amount of partial upwind differencing the vorticity diffusion near the walls and the global features of the solutions are not sigificantly altered.     

A lattice Boltzmann method for solute transport

A lattice Boltzmann method is developed for solute transport. Proper expressions for the local equilibrium distribution functions enable the method to be formulated on rectangular lattice with the same simple procedure as that on a square lattice. This provides an additional advantage over a lattice Boltzmann method on a square lattice for problems characterized by dominant phenomenon in one direction and relatively weak in another such as solute transport in shear flow over a narrow channel, where the problems can efficiently be approached with fine and coarse meshes, respectively, resulting in more efficient algorithm. The stability conditions are also described. The proposed method on a square lattice is naturally recovered when a square lattice is used. It is verified by solving four tests and compared with the analytical/exact solutions. They are in good agreement, demonstrating that the method is simple, accurate and robust for solute transport. Copyright © 2008 John Wiley & Sons, Ltd.     

Particle transport method for convection problems with reaction and diffusion

The paper is devoted to the further development of the particle transport method for the convection problems with diffusion and reaction. Here, the particle transport method for a convection–reaction problem is combined with an Eulerian finite‐element method for diffusion in the framework of the operator‐splitting approach. The technique possesses a special spatial adaptivity to resolve solution singularities possible due to convection and reaction terms. A monotone projection technique is used to transfer the solution of the convection–reaction subproblem from a moving set of particles onto a fixed grid to initialize the diffusion subproblem. The proposed approach exhibits good mass conservation and works with structured and unstructured meshes. The performance of the presented algorithm is tested on one‐ and two‐dimensional benchmark problems. The numerical results confirm that the method demonstrates good accuracy for the convection‐dominated as well as for convection–diffusion problems. Copyright © 2007 John Wiley & Sons, Ltd.     

The solution of a convection–diffusion–reaction equation arising from cathodic reduction in a pulsating crack

This paper derives the convection–diffusion-reaction equation governing the reaction between the dissolved oxygen in sea-water and the steel walls of a pulsating crack. By the neglect of the diffusion term it is shown that an exact solution of the convection-reaction equation can be obtained. A numerical method for the solution of the complete convection–diffusion-reaction equation is derived by the use of finite differences. The numerical computation of the initial transient and the final periodic steady-state values is also discussed.     

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.     

Parallel‐in‐time simulation of the unsteady Navier–Stokes equations for incompressible flow

In this paper, we present an application of a parallel‐in‐time algorithm for the solution of the unsteady Navier–Stokes model equations that are of parabolic–elliptic type. This method is based on the alternated use of a coarse global sequential solver and a fine local parallel one. A standard finite volume/finite differences first‐order approach is used for discretization of the unsteady two‐dimensional Navier–Stokes equations. The Taylor vortex decay problem and the confined flow around a square cylinder were selected as unsteady flow examples to illustrate and analyse the properties of the parallel‐in‐time method through numerical experiments. The influence of several parameters on the computing time required to perform a parallel‐in‐time calculation on a PC cluster was verified. Among them we have analysed the influence of the number of processors, the number of iterations for convergence, the resolution of the spatial domain and the influence of the time‐step sizes ratio between the coarse and fine grids. Significant computer time saving was achieved when compared with the single processor computing time, particularly when the spatial dimension of the problem is low and the temporal scale is large. Copyright © 2004 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.     

An explicit transient algorithm for predicting incompressible viscous flows in Arbitrary Geometry

A numerical method for predicting viscous flows in complex geometries has been presented. Integral mass and momentum conservation equations are deploved and these are discretized into algebraic form through numerical quadrature. The physical domain is divided into a number of non-orthogonal control volumes which are isoparametrically mapped on to standard rectangular cells. Numerical integration for unsteady mementum equations is performed over such non-orthogonal cells. The explicitly advanced velocity components obtained from unsteady momentum equations may not necessarily satisfy the mass conservation condition in each cell. Compliance of the mass conservation equation and the consequent evolution of correct pressure distribution are accomplished through an iterative correction of pressure and velocity till divergence-free condition is obtained in each cell. The algorithm is applied on a few test problems, namely, lid-driven square and oblique cavities, developing flow in a rectangular channel and flow over square and circular cylinders placed in rectangular channels. The results exhibit good accuracy and justify the applicability of the algorithm. This Explicit Transient Algorithm for Flows in Arbitrary Geometry is given a generic name EXTRAFLAG.     

Free surface seepage analysis based on the element-free method

Element-free method (EFM) for seepage analysis with a free surface presented in the paper is based on the moving least square method which needs only the information at nodes. It avoids troublesome modifications of the mesh as in the finite element method. Being irrelative to the nodes, the mesh for quadrature is fixed throughout the iterations in determining the free surface. And the nodes can be easily added, moved or deleted in the iterations. Steady seepage and transient seepage in a uniform earth dam were analyzed in the paper. The examples show that the proposed method gives satisfactory results.     

Finite difference scheme for the solution of fluid flow problems on non‐staggered grids

A new finite difference methodology is developed for the solution of computational fluid dynamics problems that do not require the use of staggered grid systems. Previous successful and robust non‐staggered methods, which used primitive variables and mass conservation in order to solve the pressure field, either interpolate cell‐face velocities or interpolate the pressure gradients in a special way, usually with an upwind‐bias to avoid the problem of odd–even coupling between the velocity and pressure fields. The new methodology presented does not detail a ‘special interpolation procedure for a primitive variable’, however, it manages to avoid the problem of odd–even coupling. The odd–even coupling is avoided by applying fourth‐order dissipation to the pressure field. It is shown that this approach can be regarded as a modified Rhie and Chow scheme. The method is implemented using a SIMPLE‐type algorithm and is applied to two test problems: laminar flow over a backward‐facing step and laminar flow in a square cavity with a driven lid. Good agreement is obtained between the numerical solutions and the corresponding benchmark solutions. The pressure dissipation term was found to successfully suppress wiggles in the pressure field. Copyright © 2000 John Wiley & Sons, Ltd.     

An accurate pressure–velocity decoupling technique for semi‐implicit rotational projection methods

In this paper, an accurate semi‐implicit rotational projection method is introduced to solve the Navier–Stokes equations for incompressible flow simulations. The accuracy of the fractional step procedure is investigated for the standard finite‐difference method, and the discrete forms are presented with arbitrary orders or accuracy. In contrast to the previous semi‐implicit projection methods, herein, an alternative way is proposed to decouple pressure from the momentum equation by employing the principle form of the pressure Poisson equation. This equation is based on the divergence of the convective terms and incorporates the actual pressure in the simulations. As a result, the accuracy of the method is not affected by the common choice of the pseudo‐pressure in the previous methods. Also, the velocity correction step is redefined, and boundary conditions are introduced accordingly. Several numerical tests are conducted to assess the robustness of the method for second and fourth orders of accuracy. The results are compared with the solutions obtained from a typical high‐resolution fully explicit method and available benchmark reports. Herein, the numerical tests are consisting of simulations for the Taylor–Green vortex, lid‐driven square cavity, and vortex–wall interaction. It is shown that the present method can preserve the order of accuracy for both velocity and pressure fields in second‐order and high‐order simulations. Furthermore, a very good agreement is observed between the results of the present method and benchmark simulations. Copyright © 2016 John Wiley & Sons, Ltd.     

Upwind basis finite elements for convection-dominated problems

Finite elements using higher-order basis functions in the spirit of the QUICK method for convection-dominated fluid flow and transport problems are introduced and demonstrated. Instead of introducing new internal degrees of freedom, completeness is achieved by including functions based on nodal values exterior and upwind to the element domain. Applied with linear test functions to the weak statements for convection-dominated problems, a family of Petrov–Galerkin finite elements is developed. Quadratic and cubic versions are demonstrated for the one-dimensional convection–diffusion test problem. Elements of up to seventh degree are used for local solution refinement. The behaviour of these elements for one-dimensional linear and non-linear advection is investigated. A two-dimensional quadratic upwind element is demonstrated in a streamfunction–vorticity formulation of the Navier–Stokes equations for a driven cavity flow test problem. With some minor reservations, these elements are recommended for further study and application.     

Integral transform solution for the lid-driven cavity flow problem in streamfunction-only formulation

The basic ideas in the generalized integral transform technique are further advanced to allow for the hybrid numerical-analytical solution of the two-dimensional steady Navier-Stokes equations in streamfunction-only formulation. The classical lid-driven square cavity problem is selected for illustration of the approach. The corresponding biharmonic-type non-linear partial differential equation for the streamfunction is integral transformed in one of the co-ordinates and an infinite system of coupled non-linear ODEs for the transformed potential results in the other independent variable. Upon truncation to an appropriate finite order, the ODE system is numerically solved by well-established algorithms with automatic error control devices. The convergence behaviour of the eigenfunction expansion is demonstrated and reference results are provided for typical values of Reynolds number.