Steady-state Incompressible Navier-Stokes Equations Research Articles

A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier–Stokes equations

A nonlinear additive Schwarz preconditioned inexact Newton method (ASPIN) was introduced recently for solving large sparse highly nonlinear systems of equations obtained from the discretization of nonlinear partial differential equations. In this paper, we discuss some extensions of ASPIN for solving steady-state incompressible Navier–Stokes equations with high Reynolds numbers in the velocity–pressure formulation. The key idea of ASPIN is to find the solution of the original system by solving a nonlinearly preconditioned system that has the same solution as the original system, but with more balanced nonlinearities. Our parallel nonlinear preconditioner is constructed using a nonlinear overlapping additive Schwarz method. To show the robustness and scalability of the algorithm, we present some numerical results obtained on a parallel computer for two benchmark problems: a driven cavity flow problem and a backward-facing step problem with high Reynolds numbers. The sparse nonlinear system is obtained by applying a Q 1 − Q 1 Galerkin least squares finite element discretization on two-dimensional unstructured meshes. We compare our approach with an inexact Newton method using different choices of forcing terms. Our numerical results show that ASPIN has good convergence and is more robust than the traditional inexact Newton method with respect to certain parameters such as the Reynolds number, the mesh size, and the number of processors.

Efficient solution of the steady‐state Navier–Stokes equations using a multigrid preconditioned Newton–Krylov method

AbstractAn inexact Newton's method is used to solve the steady‐state incompressible Navier–Stokes equations. The equations are discretized using a mixed finite element approximation. A new efficient preconditioning methodology introduced by Kay et al. (SIAM J. Sci. Comput., 2002; 24: 237–256) is applied and its effectiveness in the context of a Newton linearization is investigated. The original strategy was introduced as a preconditioning methodology for discrete Oseen equations that arise from Picard linearization. Our new variant of the preconditioning strategy is constructed from building blocks consisting of two component multigrid cycles; a multigrid V‐cycle for a scalar convection–diffusion operator; and a multigrid V‐cycle for a pressure Poisson operator. We present numerical experiments showing that the convergence rate of the preconditioned GMRES is independent of the grid size and relatively insensitive to the Reynolds number. Copyright © 2003 John Wiley & Sons, Ltd.

Higher‐order finite element discretizations in a benchmark problem for incompressible flows

AbstractWe present a numerical study of several finite element discretizations applied to a benchmark problem for the two‐dimensional steady state incompressible Navier–Stokes equations defined in Schäfer and Turek (The benchmark problem ‘Flow around a cylinder’. In Flow Simulation with High‐Performance Computers II. Notes on Numerical Fluid Mechanics, vol. 52, Hirschel EH (ed.). Vieweg: Wiesbaden, 1996; 547–566). The discretizations are compared with respect to the accuracy of the computed benchmark parameters. Higher‐order isoparametric finite element discretizations turned out to be by far the most accurate. The discrete systems obtained with higher‐order discretizations are solved with a modified coupled multigrid method whose behaviour within the benchmark problem is also studied numerically. Copyright © 2001 John Wiley & Sons, Ltd.

A mixed stream-function and vorticity formulation for axisymmetric Navier–Stokes equations

We present a mixed formulation of the axisymmetric steady-state incompressible Navier–Stokes equations. The new unknowns are the stream function and the vorticity. Spectral methods are used to approximate the continuous problem. We prove that the error estimates between the exact and the discrete solutions are optimal. We use the Glowinski–Pironneau algorithm for the implementation of the discrete problem and give numerical results.

Improved lattice Boltzmann model for incompressible two-dimensional steady flows.

An improved lattice Boltzmann model with single time relaxation has been proposed for incompressible two-dimensional (2D) steady flows. In the improved model, the steady-state incompressible Navier-Stokes equations can be recovered in exact form and the density of the fluid becomes an irrelevant invariant, satisfying the requirement of incompressibility. Exact analytical solutions to the distribution functions of the 2D triangular and square lattice Boltzmann models have been obtained for steady plane Poiseuille flow based on the present scheme. Boundary conditions that can be used to recover exactly such analytical solutions in numerical simulations are proposed.

Finite-element error estimates for the MAC scheme

This paper confirms that the MAC method, applied to solve the steady-state incompressible Navier–Stokes equations, satisfies error estimates of order one. It shows that the finite-difference equations of the MAC method can be derived by combining a velocity–vorticity mixed finite-element method of degree one with an adequate quadrature formula. In the case of a rectangular grid, this interpretation simplifies the proofs of the error estimates because it allows the use of many of the tools of mathematical theory of finite-element methods. Moreover, this finite-element approach has straightforward extensions to higher-degree approximations or approximations on non-distorted meshes of three-dimensional problems.

An exponential decay estimate for the stationary axisymmetric perturbation of Poiseuille flow in a circular pipe

We prove a decay estimate for the steady state incompressible Navier-Stokes equations. The estimate describes the exponential decay, in the axial direction of a semi-infinite circular tube, for an energy-type functional in terms of the axisymmetric perturbation of Poiseuille flow, provided that the Reynolds number does not exceed a critical value, for which we exhibit a lower and an upper bound. Since the motion is considered axisymmetric we use a stream function formulation, and the results are similar to those obtained by Horgan [8], for a two-dimensional channel flow problem. For the Stokes problem our estimate for the rate of decay is a lower bound to the actual rate of decay which is obtained from an asymptotic solution to the Stokes equations. Finally we describe a numerical approach to computing bounds to the energy functionalE(0).

An efficient finite element incompressible Navier-Stokes solver that approximates nonlinear terms with finite differences

A new method for solving the steady-state incompressible Navier-Stokes equations in primitive variable form is presented. The approach uses finite difference operators for the convective terms within the framework of a penalty finite element method. The technique is shown to be computationally very efficient. Indeed, not only does the penalty function method remove one degree of liberty per node, but also the special treatment given to the nonlinear terms yields a global coefficient matrix that is both symmetric and banded. Both storage needs and calculation time are reduced tremendously. Furthermore, for all flow cases tested, the authors have been able to reach a convergent and stable solution up to Reynolds number of the order of those normally attained by “conventional” finite element models using an identical space discretisation. Results are given for the popular benchmark problem of the driven cavity, as well as for a practical application in the Port of Montreal.

An upwind differencing scheme for the incompressible navier–strokes equations

The steady-state incompressible Navier–Stokes equations in two dimensions are solved numerically using the artificial compressibility formulation. The convective terms are upwind differenced using a flux-difference split approach that has uniformly high accuracy throughout the interior grid points. The viscous fluxes are differenced using second-order accurate central differences. The numerical system of equations is solved using an implicit line relaxation scheme. The scheme is applicable to both steady-state and unsteady flow computations. In the current work steady-state applications are emphasized. Characteristic boundary conditions are formulated and used in the solution procedure. The overall scheme is capable of being run at extremely large pseudo-time steps, leading to fasr convergence. Three test cases are presented to demonstrate the accuracy and robustness of the code. These are the flow in a square driven cavity, flow over a backward facing step, and flow around a two-dimensional circular cylinder.

An adaptive multigrid technique for the incompressible Navier-Stokes equations

An automatic adaptive refinement technique has been coupled to the multigrid approach to produce an efficient and stable solution strategy for solving the steady-state incompressible Navier-Stokes equations. Solutions have been obtained for the driven cavity and flow over a backward-facing step, for Reynolds numbers up to 5000 and 800, respectively. The refinement criterion is based on the local truncation error. The solution error is monitored and automatic refinement can continue until it is reduced to a satisfactory level. For driven cavity flow at Re = 1000, the adaptive refinement approach reduced the computer memory and CPU time to 20 and 40% of the requirements of the ''pure'' multigrid method. The primitive-variable formulation of the Navier-Stokes equations is used so the method can be extended easily to three dimensions. Application to other nonlinear elliptic problems is equally possible.

Parallel multigrid solution of the Navier-Stokes equations on general 2D domains

Multigrid methods are distinguished by their optimal (sequential) efficiency and by the fact that all their algorithmical components are fully parallelizable. For this reason, this class of numerical methods is especially attractive for use on parallel (MIMD, local memory) computers. In this paper, we describe a parallel multigrid solver for steady-state incompressible Navier-Stokes equations on general domains which is currently being developed at the GMD. Due to the geometrical generality of the problem, our approach is based on a non-staggered (nodal-point) finite volume scheme on multi-block boundary fitted grids. The typical instability of non-staggered schemes is overcome by suitably modifying the discrete continuity equation without affecting the overall order of consistency. Starting from the most simple Cartesian case, we discuss several possible multigrid approaches to the general 2D-problem. This motivates the basic design decisions of our multigrid solver in regard to both the discretization and the choice of multigrid components (smoothing schemes). Furthermore, the principal technique of parallelization (grid partitioning) is described as well as some fundamental aspects of the implementation (communication library).

Multiple discrete solutions of the incompressible steady-state Navier-Stokes equations

This paper discusses the computation of multiple solutions of various discretizations of the steady state incompressible Navier-Stokes equations. Solution paths (?,R) satisfying the discrete system of equationsH(?,R)=0, where ? represents the discrete flow field andR is the Reynolds number, are computed using a pseudo arc-length continuation procedure. For flows over the back end of an axially symmetric body with a cusped tail in a coaxial circular cylinder, the solution paths often exhibit "hairpin" turning points. Dependence of the paths on the mesh spacing and various selections for the discretizations of the convective and diffusive terms are presented.

A finite-difference scheme for the solution of the steady Navier-Stokes equations

Abstract A new finite-difference scheme of second-order accuracy, free of artificial diffusion, for solving the steady-state incompressible Navier-Stokes equations is presented. The scheme uses a false transient approach with a combination of variable time step size and over-relaxation for the convection and diffusion terms. The driven flow in a square cavity is used as the model problem. The convergence is much better than the conventional Gauss-Seidel type iterative process. Results for Reynolds numbers up to 5000 are presented.

The bid method for the steady-state Navier-Stokes equations

The Biharmonic Driver (BID) method uses direct (non-iterative) linear biharmonic solvers to obtain solutions to the nonlinear two-dimensional steady-state incompressible Navier-Stokes equations. Low Reynolds number recirculating flows are solved in 6–8 non-time-like iterations. Modifications are suggested for flow-through problems, and comparisons are made with other semidirect methods.