FAST ITERATIVE SOLVERS FOR THE DISCRETIZED INCOMPRESSIBLE NAVIER–STOKES EQUATIONS

In this paper some iterative solution methods of the GMRES type for the discretized Navier–Stokes equations are treated. The discretization combined with a pressure correction scheme leads to two different types of systems of linear equations: the momentum system and the pressure system. These systems may be coupled to one or more transport equations. For every system we specify a particular ILU-type preconditioner and show how to vectorize these preconditions. Finally, some numerical experiments to show the efficiency of the proposed methods are presented.     

Hybrid finite element/volume method for 3D incompressible flows with heat transfer

An implicit hybrid finite element (FE)/volume solver has been extended to incompressible flows coupled with the energy equation. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centred finite volume (FV) method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centres and the auxiliary variable at vertices. The Generalized Minimal Residual (GMRES) matrix-free strategy is adapted to solve the governing equations in both FE and FV methods. The presented 2D and 3D numerical examples show the robustness and accuracy of the numerical method.     

A segregated implicit solution algorithm for 2D and 3D laminar incompressible flows

A segregated algorithm for the solution of laminar incompressible, two- and three-dimensional flow problems is presented. This algorithm employs the successive solution of the momentum and continuity equations by means of a decoupled implicit solution method. The inversion of the coefficient matrix which is common for all momentum equations is carried out through an approximate factorization in upper and lower triangular matrices. The divergence-free velocity constraint is satisfied by formulating and solving a pressure correction equation. For the latter a combined application of a preconditioning technique and a Krylov subspace method is employed and proved more effecient than the approximate factorization method. The method exhibits a monotonic convergence, it is not costly in CPU time per iteration and provides accurate solutions which are independent of the underrelaxation parameter used in the momentum equations. Results are presented in two- and three-dimensional flow problems.     

Schwarz domain decomposition for the incompressible Navier–Stokes equations in general co‐ordinates

This paper describes a domain decomposition method for the incompressible Navier–Stokes equations in general co‐ordinates. Domain decomposition techniques are needed for solving flow problems in complicated geometries while retaining structured grids on each of the subdomains. This is the so‐called block‐structured approach. It enables the use of fast vectorized iterative methods on the subdomains. The Navier–Stokes equations are discretized on a staggered grid using finite volumes. The pressure‐correction technique is used to solve the momentum equations together with incompressibility conditions. Schwarz domain decomposition is used to solve the momentum and pressure equations on the composite domain. Convergence of domain decomposition is accelerated by a GMRES Krylov subspace method. Computations are presented for a variety of flows. Copyright © 2000 John Wiley & Sons, Ltd.     

A 3‐D non‐hydrostatic pressure model for small amplitude free surface flows

A three‐dimensional, non‐hydrostatic pressure, numerical model with k–ε equations for small amplitude free surface flows is presented. By decomposing the pressure into hydrostatic and non‐hydrostatic parts, the numerical model uses an integrated time step with two fractional steps. In the first fractional step the momentum equations are solved without the non‐hydrostatic pressure term, using Newton's method in conjunction with the generalized minimal residual (GMRES) method so that most terms can be solved implicitly. This method only needs the product of a Jacobian matrix and a vector rather than the Jacobian matrix itself, limiting the amount of storage and significantly decreasing the overall computational time required. In the second step the pressure–Poisson equation is solved iteratively with a preconditioned linear GMRES method. It is shown that preconditioning reduces the central processing unit (CPU) time dramatically. In order to prevent pressure oscillations which may arise in collocated grid arrangements, transformed velocities are defined at cell faces by interpolating velocities at grid nodes. After the new pressure field is obtained, the intermediate velocities, which are calculated from the previous fractional step, are updated. The newly developed model is verified against analytical solutions, published results, and experimental data, with excellent agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

A convergence accelerator of a linear system of equations based upon the power method

This paper considers the convergence rate of an iterative numerical scheme as a method for accelerating at the post‐processor stage. The methodology adapted here is: (1) residual eigenmodes included in the origin of the convex hull are eliminated; (2) remaining residual terms are smoothed away by the main convergence algorithm. For this purpose, the polynomial matrix approach is employed for deriving the characteristic equation by two different methods. The first method is based on vector scaling and the second is based on the normal equations approach. The input for both methods is the solution difference between two consecutive iteration/cycle levels obtained from the main program. The singular value decomposition was employed for both methods due to the ill‐conditioned structure of the matrices. The use of the explicit form of the Richardson extrapolation in the present work overrules the need to employ the Richardson iteration with a Leja ordering. The performance of these methods was compared with the GMRES algorithm for three representative problems: two‐dimensional boundary value problem using the Laplace equation, three‐dimensional multi‐grid, potential solution over a sphere and the one‐dimensional steady state Burger equation. In all three examples both methods have the same rate of convergence, or better, as that of the GMRES method in terms of computer operational count. However, in terms of storage requirements, the method based upon vector scaling has a significant advantage over the normal equations approach as well as the GMRES method, in which only one vector of the N grid‐points is required. Copyright © 2001 John Wiley & Sons, Ltd.     

A VORTICITY–STREAMFUNCTION FORMULATION FOR STEADY INCOMPRESSIBLE TWO-DIMENSIONAL FLOWS

A vorticity–streamfunction formulation for incompressible planar viscous flows is presented. The standard kinematic field equations are discretized using centred finite difference schemes and solved in a coupled way via a Newton-like linearization scheme. The linearized system of partial differential equations is handled through the restarting linear GMRES algorithm, preconditioned by means of an incomplete LU approximate factorization. The proposed solution technique constitutes a fast and robust algorithm for treating laminar flows at high Reynolds numbers. The pressure field is obtained at a subsequent step by solving a convection– diffusion equation in terms of the stagnation pressure, which presents certain advantages compared with the widely used static pressure Poisson equation. Results are shown for a wide variety of applications including internal and external flows.     

A novel fully implicit finite volume method applied to the lid‐driven cavity problem—Part I: High Reynolds number flow calculations

A novel implicit cell‐vertex finite volume method is described for the solution of the Navier–Stokes equations at high Reynolds numbers. The key idea is the elimination of the pressure term from the momentum equation by multiplying the momentum equation with the unit normal vector to a control volume boundary and integrating thereafter around this boundary. The resulting equations are expressed solely in terms of the velocity components. Thus any difficulties with pressure or vorticity boundary conditions are circumvented and the number of primary variables that need to be determined equals the number of space dimensions. The method is applied to both the steady and unsteady two‐dimensional lid‐driven cavity problem at Reynolds numbers up to 10000. Results are compared with those in the literature and show excellent agreement. Copyright © 2003 John Wiley & Sons, Ltd.     

Using a segregated finite element scheme to solve the incompressible Navier-Stokes equations

In this paper, a segregated finite element scheme for the solution of the incompressible Navier-Stokes equations is proposed which is simpler in form than previously reported formulations. A pressure correction equation is derived from the momentum and continuity equations, and equal-order interpolation is used for both the velocity components and pressure. Algorithms such as this have been known to lead to checkerboard pressure oscillations; however, the pressure correction equation of this scheme should not produce these oscillations. The method is applied to several laminar flow situations, and details of the methods used to achieve converged solutions are given.     

Dirichlet and Neumann boundary conditions for the pressure poisson equation of incompressible flow

In a recent paper Gresho and Sani showed that Dirichlet and Neumann boundary conditions for the pressure Poisson equation give the same solution. The purpose of this paper is to confirm this (for one case at least) by numerically solving the pressure equation with Dirichlet and Neumann boundary conditions for the inviscid stagnation point flow problem. The Dirichlet boundary condition is obtained by integrating the tangential component of the momentum equation along the boundary. The Neumann boundary condition is obtained by applying the normal component of the momentum equation at the boundary. In this work solutions for the Neumann problem exist only if a compatibility condition is satisfied. A consistent finite difference procedure which satisfies this condition on non-staggered grids is used for the solution of the pressure equation with Neumann conditions. Two test cases are computed. In the first case the velocity field is given from the analytical solution and the pressure is recovered from the solution of the associated Poisson equation. The computed results are identical for both Dirichlet and Neumann boundary conditions. However, the Dirichlet problem converges faster than the Neumann case. In the second test case the velocity field is computed from the momentum equations, which are solved iteratively with the pressure Poisson equation. In this case the Neumann problem converges faster than the Dirichlet problem.     

A model for solute transport following an abrupt pressure impact in saturated porous media

A mathematical model is developed of an abrupt pressure impact applied to a compressible fluid with solute, flowing through saturated porous media. Nondimensional forms of the macroscopic balance equations of the solute mass and of the fluid mass and momentum lead to dominant forms of these equations. Following the onset of the pressure change, we focus on a sequence of the first two time intervals at which we obtain reduced forms of the balance equations. At the very first time period, pressure is proven to be distributed uniformly within the affected domain, while solute remains unaffected. During the second time period, the momentum balance equation for the fluid conforms to a wave form, while the solute mass balance equation conforms to an equation of advective transport. Fluid's nonlinear wave equation together with its mass balance equation, are separately solved for pressure and velocity. These are then used for the solution of solute's advective transport equation. The 1-D case, conforms to a pressure wave equation, for the solution of fluid's pressure and velocity. A 1-D analytical solution of the transport problem, associates these pressure and velocity with an exponential power which governs solute's motion along its path line.     

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids.An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling.The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex.For the temporal integration of the momentum equations,an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term.The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM).The momentum interpolation is used to damp out the spurious pressure wiggles.The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure.The classic test cases,the lid-driven cavity flow,the skew cavity flow and the backward-facing step flow,show that numerical results are in good agreement with the published benchmark solutions.     

A staggered spectral element model with application to the oceanic shallow water equations

A staggered spectral element model for the solution of the oceanic shallow water equations is presented. We introduce and compare both an implicit and an explicit time integration scheme. The former splits the equations with the operator-integration factor method and solves the resulting algebraic system with generalized minimum residual (GMRES) iterations. Comparison of the two schemes shows the performance of the implicit scheme to lag that of the explicit scheme because of the unpreconditioned implementation of GMRES. The explicit code is successfully applied to various geophysical flows in idealized and realistic basins, notably to the wind-driven circulation in the North Atlantic Ocean. The last experiment reveals the geometric versatility of the spectral element method and the effectiveness of the staggering in eliminating sprious pressure modes when the flow is nearly non-divergent.     

Jacobian-free Newton-Krylov subspace method with wavelet-based preconditioner for analysis of transient elastohydrodynamic lubrication problems with surface asperities

This paper presents an investigation into the effect of surface asperities on the over-rolling of bearing surfaces in transient elastohydrodynamic lubrication(EHL) line contact. The governing equations are discretized by the finite difference method. The resulting nonlinear system of algebraic equations is solved by the Jacobian-free Newtongeneralized minimal residual(GMRES) from the Krylov subspace method(KSM). The acceleration of the GMRES iteration is accomplished by a wavelet-based preconditioner.The profiles of the lubricant pressure and film thickness are obtained at each time step when the indented surface moves through the contact region. The prediction of pressure as a function of time provides an insight into the understanding of fatigue life of bearings.The analysis confirms the need for the time-dependent approach of EHL problems with surface asperities. This method requires less storage and yields an accurate solution with much coarser grids. It is stable, efficient, allows a larger time step, and covers a wide range of parameters of interest.     

MODELLING OF WAVE PROPAGATION IN THE NEARSHORE REGION USING THE MILD SLOPE EQUATION WITH GMRES-BASED ITERATIVE SOLVERS

The mild slope equation in its linear and non-linear forms is used for the modelling of nearshore wave propagation. The finite difference method is used to descretize the governing elliptic equations and the resulting system of equations is solved using GMRES-based iterative method. The original GMRES solution technique of Saad and Schultz is not directly applicable to the present case owing to the complex coefficient matrix. The simpler GMRES algorithm of Walker and Zhou is used as the core solver, making the upper Hessenberg factorization unneccessary when solving the least squares problem. Several preconditioning-based acceleration strategies are tested and the results show that the GMRES-based iteration scheme performs very well and leads to monotonic convergence for all the test-cases considered.     

On pressure boundary conditions for the incompressible Navier-Stokes equations

The pressure is a somewhat mysterious quantity in incompressible flows. It is not a thermodynamic variable as there is no ‘equation of state’ for an incompressible fluid. It is in one sense a mathematical artefact—a Lagrange multiplier that constrains the velocity field to remain divergence-free; i.e., incompressible—yet its gradient is a relevant physical quantity: a force per unit volume. It propagates at infinite speed in order to keep the flow always and everywhere incompressible; i.e., it is always in equilibrium with a time-varying divergence-free velocity field. It is also often difficult and/or expensive to compute. While the pressure is perfectly well-defined (at least up to an arbitrary additive constant) by the governing equations describing the conservation of mass and momentum, it is (ironically) less so when more directly expressed in terms of a Poisson equation that is both derivable from the original conservation equations and used (or misused) to replace the mass conservation equation. This is because in this latter form it is also necessary to address directly the subject of pressure boundary conditions, whose proper specification is crucial (in many ways) and forms the basis of this work. Herein we show that the same principles of mass and momentum conservation, combined with a continuity argument, lead to the correct boundary conditions for the pressure Poisson equation: viz., a Neumann condition that is derived simply by applying the normal component of the momentum equation at the boundary. It usually follows, but is not so crucial, that the tangential momentum equation is also satisfied at the boundary.     

A preconditioned semi‐staggered dilation‐free finite volume method for the incompressible Navier–Stokes equations on all‐hexahedral elements

A new semi‐staggered finite volume method is presented for the solution of the incompressible Navier–Stokes equations on all‐quadrilateral (2D)/hexahedral (3D) meshes. The velocity components are defined at element node points while the pressure term is defined at element centroids. The continuity equation is satisfied exactly within each elements. The checkerboard pressure oscillations are prevented using a special filtering matrix as a preconditioner for the saddle‐point problem resulting from second‐order discretization of the incompressible Navier–Stokes equations. The preconditioned saddle‐point problem is solved using block preconditioners with GMRES solver. In order to achieve higher performance FORTRAN source code is based on highly efficient PETSc and HYPRE libraries. As test cases the 2D/3D lid‐driven cavity flow problem and the 3D flow past array of circular cylinders are solved in order to verify the accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

Solution of the incompressible mass and momentum equations by application of a coupled equation line solver

This paper describes an iterative technique for solving the coupled algebraic equations for mass and momentum conservation for an incompressible fluid flow. The technique is based on the simultaneous solution for pressure and velocity along lines. In a manner similar to ADI methods for a single variable, the solution domain is entirely swept line-by-line in each co-ordinate direction successively until a converged solution is obtained. The tight coupling between the equations that is guaranteed by the method results in an economical solution of the equation set.     

SENSITIVITY ANALYSIS AND OPTIMIZATION OF COUPLED THERMAL AND FLOW PROBLEMS WITH APPLICATIONS TO CONTRACTION DESIGN

The finite element method and the Newton–Raphson solution algorithm are combined to solve the momentum, mass and energy conservation equations for coupled flow problems. Design sensitivities for a generalised response function with respect to design parameters which describe shape, material property and load data are evaluated via the direct differentiation method. The efficiently computed sensitivities are verified by comparison with computationally intensive, finite difference sensitivity approximations. The design sensitivities are then used in a numerical optimization algorithm to minimize the pressure drop in flow through contractions. Both laminar and turbulent flows are considered. In the turbulent flow problems the time-averaged momentum and mass conservati on equations are solved using a mixing length turbulence model.     

An analysis and comparison of the time accuracy of fractional‐step methods for the Navier–Stokes equations on staggered grids

Fractional‐step methods solve the unsteady Navier–Stokes equations in a segregated manner, and can be implemented with only a single solution of the momentum/pressure equations being obtained at each time step, or with the momentum/pressure system being iterated until a convergence criterion is attained.The time accuracy of such methods can be determined by the accuracy of the momentum/pressure coupling, irrespective of the accuracy to which the momentum equations are solved. It is shown that the time accuracy of the basic projection method is first‐order as a result of the momentum/pressure coupling, but that by modifying the coupling directly, or by modifying the intermediate velocity boundary conditions, it is possible to recover second‐order behaviour. It is also shown that pressure correction methods, implemented in non‐iterative or iterative form and without special boundary conditions, are second‐order in time, and that a form of the non‐iterative pressure correction method is the most efficient for the problems considered. Copyright © 2002 John Wiley & Sons, Ltd.