A postprocessing mixed finite element method for the Navier–Stokes equations

A fully discrete postprocessing mixed finite element scheme is considered for solving the time-dependent Navier–Stokes equations. In the PP method, we only consider a non-linear equation in the coarse-level subspace and a linear problem in the fine-level subspace. The analysis shows that the PP scheme can reach the same accuracy as the standard Galerkin method with a very fine mesh size h by an appropriate choice of H. Numerical examples are provided that confirm both the theoretical analysis and the corresponding improvement in computational efficiency.     

New stabilized finite element method for time‐dependent incompressible flow problems

A new stabilized finite element method is considered for the time‐dependent Stokes problem, based on the lowest‐order P₁?P₀ and Q₁?P₀ elements that do not satisfy the discrete inf–sup condition. The new stabilized method is characterized by the features that it does not require approximation of the pressure derivatives, specification of mesh‐dependent parameters and edge‐based data structures, always leads to symmetric linear systems and hence can be applied to existing codes with a little additional effort. The stability of the method is derived under some regularity assumptions. Error estimates for the approximate velocity and pressure are obtained by applying the technique of the Galerkin finite element method. Some numerical results are also given, which show that the new stabilized method is highly efficient for the time‐dependent Stokes problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Stream function finite element solution of Stokes flow with penalties

A finite element method for solution of the stream function formulation of Stokes flow is developed. The method involves complete cubic non-conforming (C⁰) triangular Hermite elements. This element fails the patch test. To correct the element and produce a convergent method we employ a penalty method to weakly enforce the desired continuity constraint on the normal derivative across the inter-element boundaries. Successful use of the method is demonstrated to require reduced integration of the inter-element penalty with a 1-point Gauss rule. Error estimates relate the optimal choice of penalty parameter to mesh size and are corroborated by numerical convergence studies. The need for reduced integration is interpreted using rank relations for an associated hybrid method.     

Finite element analysis of axisymmetric free jet impingement

A numerical algorithm to determine the impingement of an axisymmetric free jet upon a curved deflector is presented. The problem is considered within the potential flow theory with the allowance of gravity and surface tension effects. The primary dependent variable is the Stokes streamfunction, which is approximated through finite elements using the isoparametric Hermite Zienkiewicz element. To find the correct position of the free boundaries, a trial-and-error method is employed which amounts to solving a boundary value problem (BVP) for the Stokes streamfunction at each iteration step. An efficient method is proposed to solve this BVP. The algorithm to find the correct position of the free boundaries is tested by computing the impingement upon an infinite disc and a hemispherical deflector. To confirm the correctness of the solution, each problem has been solved using several different mesh gradings. A comparison between the Zienkiewicz and the other standard C⁰ finite elements is also given.     

The influence of the stabilization parameter on the convergence factor of iterative methods for the solution of the discretized Stokes problem

This paper discusses the influence of the stabilization parameter on the convergence factor of various iterative methods for the solution of the Stokes problem discretized by the so-called locally stabilized Q1-P0 finite element. Our objective is to point out optimal parameters which ensure rapid convergence. The first part of the paper is concerned with the dual formulation of the problem. It gives the theoretical precision and practical developments of our stabilized context Uzawa-type algorithm. We assert that the convergence factor of such a method is majored independently of the mesh size by a function of the stabilization parameter. Moreover, we point out that there exists an optimal value of this parameter that minimizes this upper bound. This gives a theoretical justification of pre-existing numerical results. We show that the optimal parameter can be determined a priori. This is a key point when the method has to be implemented. Finally, we base an interpretation of the iterated penalty method numerical behaviour on some theoretical results about the minimum eigenvalue of the stabilized dual operator. This algorithm involves a penalty parameter and a stabilization parameter and we discuss a strategy for choosing optimal parameters. The mixed formulation of the problem is dealt with in the second part of the paper, which proposes several preconditioned conjugate-gradient-type methods. The indefinite character of the problem makes it intrinsically hard. However, if one chooses a suitable preconditioner, this difficulty is overcome, since the preconditioned operator becomes positive definite. We study the eigenvalue spectrum of the preconditioned operator and thereby the convergence factor of the algorithm. In contrast with the two previous formulations, we show that this convergence factor is majored independently of the stabilization parameter. More precisely, we point out convergence factors comparable with those obtained for Poisson-type problems. Finally, we present a variant of the latter method which uses our so-called macroblock-type preconditioner. A comparison with the simple case of diagonal preconditioning is addressed and the improved performance of the macroblock-type preconditioner is evidenced. Various 2D numerical experiments are given to corroborate the theories presented herein.     

Efficient computation of mean drag for the subcritical flow past a circular cylinder using general Galerkin G2

General Galerkin (G2) is a new computational method for turbulent flow, where a stabilized Galerkin finite element method is used to compute approximate weak solutions to the Navier–Stokes equations directly, without any filtering of the equations as in a standard approach to turbulence simulation, such as large eddy simulation, and thus no Reynolds stresses are introduced, which need modelling. In this paper, G2 is used to compute the drag coefficient c_D for the flow past a circular cylinder at Reynolds number Re=3900, for which the flow is turbulent. It is found that it is possible to approximate c_D to an accuracy of a few percent, corresponding to the accuracy in experimental results for this problem, using less than 10⁵ mesh points, which makes the simulations possible using a standard PC. The mesh is adaptively refined until a stopping criterion is reached with respect to the error in a chosen output of interest, which in this paper is c_D. Both the stopping criterion and the mesh‐refinement strategy are based on a posteriori error estimates, in the form of a space–time integral of residuals times derivatives of the solution of a dual problem, linearized at the approximate solution, and with data coupling to the output of interest. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite element solution of the equations governing the flow of electrolyte in charged microporous membranes

Electrical double-layer effects are unimportant in flows through porous media except when the Debye length k^?1 is comparable in magnitude with the pore radius a. Under these conditions the equations governing the flow of electrolyte are those of Stokes, Nernst-Planck and Poisson. These equations are non-linear and require numerical solution. The finite element method provides a useful basis for solution and various algorithms are investigated. The numerical stability and errors of each scheme are analysed together with the development of an appropriate finite element mesh. The electro-osmotic flow of a typical electrolyte (barium chloride) through a uniformly charged cylindrical membrane pore is investigated and the ion fluxes are post-computed from the numerical solutions. The ion flux is shown to be strongly dependent on both zeta potential and pore radius, ka, indicating the effects of overlapping electrical double layers.     

Nonlinear corrector for Reynolds-averaged Navier-Stokes equations

The scope of this paper is to present a nonlinear error estimation and correction for Navier-Stokes and Reynolds-averaged Navier-Stokes equations. This nonlinear corrector enables better solution or functional output predictions at fixed mesh complexity and can be considered in a mesh adaptation process. After solving the problem at hand, a corrected solution is obtained by solving again the problem with an added source term. This source term is deduced from the evaluation of the residual of the numerical solution interpolated on the h/2 mesh. To avoid the generation of the h/2 mesh (which is prohibitive for realistic applications), the residual at each vertex is computed by local refinement only in the neighborhood of the considered vertex. One of the main feature of this approach is that it automatically takes into account all the properties of the considered numerical method. The numerical examples point out that it successfully improves solution predictions and yields a sharp estimate of the numerical error. Moreover, we demonstrate the superiority of the nonlinear corrector with respect to linear corrector that can be found in the literature.     

Approximation of the Stokes eigenvalue problem on triangular domains using a stabilized finite element method

Meccanica - In this paper, we consider a stabilized finite element method for the approximation of the Stokes eigenvalue problem on triangular domains. The method depends on orthogonal subscales...     

A coupling strategy based on anisotropic mesh adaptation for solving two‐fluid flows

We propose a coupling strategy for solving efficiently bifluid flows based on the Stokes equations. Our approach relies on a level set formulation of the interface‐capturing problem, and involves a finite element discretization for the fluid resolution, the method of characteristics for solving the advection of the interface and the anisotropic mesh adaptation of the computational domain in the vicinity of the interface for better accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Adaptive finite element method for an optimal control problem of Stokes flow with L2‐norm state constraint

An adaptive finite element approximation for an optimal control problem of the Stokes flow with an L²‐norm state constraint is proposed. To produce good adaptive meshes, the a posteriori error estimates are discussed. The equivalent residual‐type a posteriori error estimators of the H ¹‐error of state and L²‐error of control are given, which are suitable to carry out the adaptive multi‐mesh finite element approximation. Some numerical experiments are performed to illustrate the efficiency of the a posteriori estimators. Copyright © 2011 John Wiley & Sons, Ltd.     

TWO-GRID ERROR ESTIMATES FOR THE STREAM FUNCTION FORM OF NAVIER-STOKES EQUATIONS

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Anisotropic adaptive stabilized finite element solver for RANS models

Aerodynamic characteristics of various geometries are predicted using a finite element formulation coupled with several numerical techniques to ensure stability and accuracy of the method. First, an edge‐based error estimator and anisotropic mesh adaptation are used to detect automatically all flow features under the constraint of a fixed number of elements, thus controlling the computational cost. A variational multiscale‐stabilized finite element method is used to solve the incompressible Navier‐Stokes equations. Finally, the Spalart‐Allmaras turbulence model is solved using the streamline upwind Petrov‐Galerkin method. This paper is meant to show that the combination of anisotropic unsteady mesh adaptation with stabilized finite element methods provides an adequate framework for solving turbulent flows at high Reynolds numbers. The proposed method was validated on several test cases by confrontation with literature of both numerical and experimental results, in terms of accuracy on the prediction of the drag and lift coefficients as well as their evolution in time for unsteady cases.     

Is the steady viscous incompressible two-dimensional flow over a backward-facing step at Re = 800 stable?

A detailed case study is made of one particular solution of the 2D incompressible Navier–Stokes equations. Careful mesh refinement studies were made using four different methods (and computer codes): (1) a high-order finite-element method solving the unsteady equations by time-marching; (2) a high-order finite-element method solving both the steady equations and the associated linear-stability problem; (3) a second-order finite difference method solving the unsteady equations in streamfunction form by time-marching; and (4) a spectral-element method solving the unsteady equations by time-marching. The unanimous conclusion is that the correct solution for flow over the backward-facing step at Re = 800 is steady—and it is stable, to both small and large perturbations.     

A stabilized finite element method for the Stokes problem based on polynomial pressure projections

A new stabilized finite element method for the Stokes problem is presented. The method is obtained by modification of the mixed variational equation by using local L² polynomial pressure projections. Our stabilization approach is motivated by the inherent inconsistency of equal‐order approximations for the Stokes equations, which leads to an unstable mixed finite element method. Application of pressure projections in conjunction with minimization of the pressure–velocity mismatch eliminates this inconsistency and leads to a stable variational formulation. Unlike other stabilization methods, the present approach does not require specification of a stabilization parameter or calculation of higher‐order derivatives, and always leads to a symmetric linear system. The new method can be implemented at the element level and for affine families of finite elements on simplicial grids it reduces to a simple modification of the weak continuity equation. Numerical results are presented for a variety of equal‐order continuous velocity and pressure elements in two and three dimensions. Copyright © 2004 John Wiley & Sons, Ltd.     

Footbridge between finite volumes and finite elements with applications to CFD

The aim of this paper is to introduce a new algorithm for the discretization of second‐order elliptic operators in the context of finite volume schemes on unstructured meshes. We are strongly motivated by partial differential equations (PDEs) arising in computational fluid dynamics (CFD), like the compressible Navier–Stokes equations. Our technique consists of matching up a finite volume discretization based on a given mesh with a finite element representation on the same mesh. An inverse operator is also built, which has the desirable property that in the absence of diffusion, one recovers exactly the finite volume solution. Numerical results are also provided. Copyright © 2001 John Wiley & Sons, Ltd.     

A projection scheme for incompressible multiphase flow using adaptive Eulerian grid: 3D validation

A three‐dimensional finite element method for incompressible multiphase flows with capillary interfaces is developed based on a (formally) second‐order projection scheme. The discretization is on a fixed (Eulerian) reference grid with an edge‐based local h‐refinement in the neighbourhood of the interfaces. The fluid phases are identified and advected using the level‐set function. The reference grid is then temporarily reconnected around the interface to maintain optimal interpolations accounting for the singularities of the primary variables. Using a time splitting procedure, the convection substep is integrated with an explicit scheme. The remaining generalized Stokes problem is solved by means of a pressure‐stabilized projection. This method is simple and efficient, as demonstrated by a wide range of difficult free‐surface validation problems, considered in the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

Superconvergence of nonconforming finite element penalty scheme for Stokes problem using L 2 projection method

A modified penalty scheme is discussed for solving the Stokes problem with the Crouzeix-Raviart type nonconforming linear triangular finite element. By the L ² projection method, the superconvergence results for the velocity and pressure are obtained with a penalty parameter larger than that of the classical penalty scheme. The numerical experiments are carried out to confirm the theoretical results.     

An h-adaptive solution of the spherical blast wave problem

Shock waves and contact discontinuities usually appear in compressible flows, requiring a fine mesh in order to achieve an acceptable accuracy of the numerical solution. The usage of a mesh adaptation strategy is convenient as uniform refinement of the whole mesh becomes prohibitive in three-dimensional (3D) problems. An unsteady h-adaptive strategy for unstructured finite element meshes is introduced. Non-conformity of the refined mesh and a bounded decrease in the geometrical quality of the elements are some features of the refinement algorithm. A 3D extension of the well-known refinement constraint for 2D meshes is used to enforce a smooth size transition among neighbour elements with different levels of refinement. A density-based gradient indicator is used to track discontinuities. The solution procedure is partially parallelised, i.e. the inviscid flow equations are solved in parallel with a finite element SUPG formulation with shock capturing terms while the adaptation of the mesh is sequentially performed. Results are presented for a spherical blast wave driven by a point-like explosion with an initial pressure jump of 10⁵ atmospheres. The adapted solution is compared to that computed on a fixed mesh. Also, the results provided by the theory of self-similar solutions are considered for the analysis. In this particular problem, adapting the mesh to the solution accounts for approximately 4% of the total simulation time and the refinement algorithm scales almost linearly with the size of the problem.     

Effect of blockage on critical parameters for flow past a circular cylinder

The effect of location of the lateral boundaries, of the computational domain, on the critical parameters for the instability of the flow past a circular cylinder is investigated. Linear stability analysis of the governing equations for incompressible flows is carried out via a stabilized finite element method to predict the primary instability of the wake. The generalized eigenvalue problem resulting from the finite element discretization of the equations is solved using a subspace iteration method to get the most unstable eigenmode. Computations are carried out for a large range of blockage, 0.005?D/H ?0.125, where D is the diameter of the cylinder and H is the lateral width of the domain. A non‐monotonic variation of the critical Re with the blockage is observed. It is found that as the blockage increases, the critical Re for the onset of the instability first decreases and then increases. However, a monotonic increase in the non‐dimensional shedding frequency at the onset of instability, with increase in blockage, is observed. The increased blockage damps out the low‐frequency modes giving way to higher frequency modes. The blockage is found to play an important role in the scatter in the data for the non‐dimensional vortex shedding frequency at the onset of the instability, from various researchers in the past. Copyright © 2005 John Wiley & Sons, Ltd.