Lid-driven Cavity Flow Problem Research Articles

On convergence rate of the augmented Lagrangian algorithm for nonsymmetric saddle point problems

We are interested in solving the system (1) [ A L T L 0 ] [ c λ ] = [ F G ] , by a variant of the augmented Lagrangian algorithm. This type of problem with nonsymmetric A typically arises in certain discretizations of the Navier–Stokes equations. Here A is a ( n , n ) matrix, c, F ∈ R n , L is a ( m , n ) matrix, and λ , G ∈ R m . We assume that A is invertible on the kernel of L. Convergence rates of the augmented Lagrangian algorithm are known in the symmetric case but the proofs in [R. Glowinski, P. LeTallec, Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics, SIAM, 1989] used spectral arguments and cannot be extended to the nonsymmetric case. The purpose of this paper is to give a rate of convergence of a variant of the algorithm in the nonsymmetric case. We illustrate the performance of this algorithm with numerical simulations of the lid-driven cavity flow problem for the 2D Navier–Stokes equations.

Trivariate spline approximations of 3D Navier-Stokes equations

We present numerical approximations of the 3D steady state Navier-Stokes equations in velocity-pressure formulation using trivariate splines of arbitrary degree d d and arbitrary smoothness r r with r > d r>d . Using functional arguments, we derive the discrete Navier-Stokes equations in terms of B B -coefficients of trivariate splines over a tetrahedral partition of any given polygonal domain. Smoothness conditions, boundary conditions and the divergence-free condition are enforced through Lagrange multipliers. The pressure is computed by solving a Poisson equation with Neumann boundary conditions. We have implemented this approach in MATLAB and present numerical evidence of the convergence rate as well as experiments on the lid driven cavity flow problem.

Application of the Krylov Subspace Method to the Incompressible Navier-Stokes Equations

The preconditioned Krylov subspace methods were applied to the incompressible Navier-Stoke's equations for convergence acceleration. Three of the Krylov subspace methods combined with the five of the preconditioners were tested to solve the lid-driven cavity flow problem. The MILU preconditioned CG method showed very fast and stable convergency. The combination of GMRES/MILU-CG solver for momentum and pressure correction equations was found less dependency on the number of the grid points among them. A guide line for stopping inner iterations for each equation is offered.

Element-by-Element Parallel Computation of Incompressible Navier--Stokes Equations in Three Dimensions

Development of a stable finite element model for solving steady incompressible viscous fluid flows in three dimensions is the main theme of the present study. For stability reasons, weighting functions are designed in favor of field variables on the upstream side. For accuracy reasons, it is required that weighting functions be equipped with the streamline operator so that false diffusion errors can be largely suppressed. In the steady-state analysis of Navier--Stokes equations, we adopt the mixed formulation to preserve mass conservation on quadratic elements which accommodate the Ladyzhenskaya--Babuska--Brezzi (LBB) stability condition. To resolve difficulties arising from asymmetry and indefiniteness in the resulting large-size matrix equations, we abandon the elimination-like solution solver because the storage demand exceeds the ability of our hardware to solve for three-dimensional problems. A modern iteration solver, known as the biconjugate gradiant stabilized (BICGSTAB) solution solver, is thus implemented in an element-by-element fashion to effectively alleviate the problem. For performance reasons, the finite element code developed here should be implemented in a hardware environment which is suited to the use of an iterative solver. To this end, our analysis is implemented in shared memory parallel architectures, CRAY C-90 and J-90. We benchmark the parallel computing performance through a lid-driven cavity flow problem and a problem amenable to analytic solution.

Effect of Reynolds number on the eddy structure in a lid-driven cavity

In this paper we apply a finite volume method, together with a cost-effective segregated solution algorithm, to solve for the primitive velocities and pressure in a set of incompressible Navier–Stokes equations. The well-categorized workshop problem of lid-driven cavity flow is chosen for this exercise, and results focus on the Reynolds number. Solutions are given for a depth-to-width aspect ration of 1:1 and a span-to width aspect ratio of 3:1. Upon increasing the Reynolds number, the flows in the cavity of interest were found to comprise a transition from a strongly two-dimensional character to a truly three-dimensional flow and, subsequently, a bifurcation from a stationary flow pattern to a periodically oscillatory state. Finally, viscous (Tollmien–Schlichting) travelling wave instability further induced longitudinal vortices, which are essentially identical to Taylor–Görtler vortices. The objective of this study was to extend our understanding of the time evolution of a recirculatory flow pattern against the Reynolds number. The main goal was to distinguish the critical Reynolds number at which the presence of a spanwise velocity makes the flow pattern become three-dimensional. Secondly, we intended to learn how and at what Reynolds number the onset of instability is generated. © 1998 John Wiley & Sons, Ltd.

An element-by-element BICGSTAB iterative method for three-dimensional steady Navier-Stokes equations

Construction of a stabilized Galerkin upwind finite element model for steady and incompressible Navier-Stokes equations in three dimensions is the main theme of this study. In the time-independent context, the weighted residuals statement is kept biased in favor of the upstream flow direction by adding an artificial damping term of physical plausibility to the Galerkin framework. This upwind approach has significant advantage of seeking solutions free from cross-stream diffusion error. Finite element solutions have been found by mixed formulation, implemented in quadratic cubic elements which are characterized as possessing the so-called LBB (Ladyzhenskaya-Babuška-Brezzi) condition. An element-by-element BICGSTAB solution solver is intended to alleviate difficulties regarding the asymmetry and indefiniteness arising from the use of a mixed formulation for incompressible fluid flows. The developed three-dimensional finite element code is first rectified by solving a problem amenable to analytic solution. A well-known lid-driven cavity flow problem in a cubical cavity is also studied.

NUMERICAL SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE FLOW BY A NEW FORMULATION

A new mathematical formulation, called the pseudovorticity–velocity formulation, of the three-dimensional incompressible Navier–Stokes equations is presented as an alternative to the vorticity–velocity approach. For the model lid-driven cavity flow problem in two and three dimensions, combined with an explicit mixed spectral /finite different numerical scheme the proposed formulation is found to be efficient and very accurate as compared with the results available in the literature. In particular, the simulation results demonstrate an attractive feature of the present formulation compared with the vorticity–velocity approach, namely that the divergence-free condition of the velocity field can always be achieved on a non-staggered mesh.

Numerical experiments with the lid driven cavity flow problem

The centerline velocity profiles obtained from the solution of the two- and three-dimensional representations of the lid driven cavity flow problem are compared for different Reynolds numbers. Two configurations were used in this study: a unit cavity and a cavity with an aspect ratio of 2. The Reynolds numbers ranged from 100 to 5000 for all of the configurations studied. A new method of extending the Jacobi collocation technique called spectral difference is developed in this paper together with a unique computational grid. In addition, an iterative method for solving the pressure problem is also developed. This new numerical method allowed the calculation of three-dimensional Navier-Stokes equations to be performed in computers with very modest computational capabilities such as workstations.

Low and moderate Reynolds number transient flow simulations using space filtered Navier‐Stokes equations

AbstractIn this study, low and moderate Reynolds number flow problems in the laminar range are solved numerically with grids that do not resolve all the significant scales of motion. Spatial averaging or filtering of the Navier‐Stokes equations and Taylor series approximations to the filtered advective terms are used in order to account for the effects of the unresolved or subgrid scales on the resolved scales. Numerical experiments with a transient 2‐D lid driven cavity flow problem, using a penalty method Galerkin finite element code, show that this approach enhances the momentum transfer properties of the numerical solution, eliminates 2Δx type oscillations, and enables the use of coarser grids. The significance and order of the terms that describe the interaction between the resolved and the subgrid scales is studied and the success of the series approximations to these terms is demonstrated. © 1994 John Wiley & Sons, Inc.