Spurious Pressure Modes Research Articles

Prediction of flow patterns in the polyurethane foaming process by numerical simulation considering foam expansion

AbstractWe present a foam flow model for flow pattern prediction in the polyurethane foaming process that considers the flow caused by foam expansion due to bubble generation. The flow pattern in the cavity is calculated by taking into account the initial volume shape and the rate of foam expansion of a poured reaction mixture. The validity of the model is investigated by applying it to the Hele‐Shaw formulation, which assumes that the cavity is very thin compared to its width. The predicted flow patterns generally agree closely with experimental results measured in panel‐type molds. A three‐dimensional simulation is further developed based on the Stokes flow. The numerical formulation is performed by the control volume based‐finite element method with an equal‐order velocity‐pressure formulation that does not exhibit spurious pressure modes. The flow patterns predicted by the three‐dimensional simulation agree very closely with the experimental results. In particular, it clearly expresses the edge effect, in which the flow slows down at the edge, which is ignored in the Hele‐Shaw formulation. This numerical simulation is suitable for rapid determination of the pouring position, volume of the mixture, and the air vent position.

HP DISCONTINUOUS GALERKIN APPROXIMATIONS FOR THE STOKES PROBLEM

We propose and analyze a discontinuous Galerkin approximation for the Stokes problem. The finite element triangulation employed is not required to be conforming and we use discontinuous pressures and velocities. No additional unknown fields need to be introduced, but only suitable bilinear forms defined on the interfaces between the elements, involving the jumps of the velocity and the average of the pressure. We consider hp approximations using ℚk′–ℚk velocity-pressure pairs with k′ = k + 2, k + 1, k. Our methods show better stability properties than the corresponding conforming ones. We prove that our first two choices of velocity spaces ensure uniform divergence stability with respect to the mesh size h. Numerical results show that they are uniformly stable with respect to the local polynomial degree k, a property that has no analog in the conforming case. An explicit bound in k which is not sharp is also proven. Numerical results show that if equal order approximation is chosen for the velocity and pressure, no spurious pressure modes are present but the method is not uniformly stable either with respect to h or k. We derive a priori error estimates generalizing the abstract theory of mixed methods. Optimal error estimates in h are proven. As for discontinuous Galerkin methods for scalar diffusive problems, half of the power of k is lost for p and hp pproximations independently of the divergence stability.

Behaviour of isoparametric quadrilateral family of Lagrangian fluid finite elements

AbstractThis paper presents the formulation of both the consistent and inconsistent four‐, eight‐ and nine‐noded isoparametric quadrilateral fluid finite elements that are based on Lagrangian frame of reference. The mesh locking phenomenon due to simultaneous enforcement of twin constraints, namely the incompressibility and irrotationality constraints, is studied in detail. The study shows that the characteristic of the locked fluid elements is that it always generates numerous spurious acoustic (volume change) modes upon the enforcement of rotational constraints. That is, the rotational constraints change the character of certain volume change modes. The study further reinforces the necessity of rotational constraints in not only identifying the spurious pressure modes, but also in reducing the computational effort for determining the eigenvalues and eigenvectors. It is found that all fully integrated inconsistent models exhibit locking behaviour. However, the inconsistent eight‐ and nine‐noded elements, integrated with full integration of volumetric stiffness and one point integration of the rotational stiffness matrices, gives excellent performance, although they do not pass the inf–sup test. The four‐ and nine‐noded consistent models are found to give locking free performance while their eight‐noded counterpart exhibited locking behaviour. The study shows that only consistent nine‐noded element models pass the inf–sup test. The utility of these elements in the coupled fluid–structure interaction problem is also demonstrated. Copyright © 2002 John Wiley & Sons, Ltd.

Generalized mixed variational principles and solutions of ill-conditioned problems in computational mechanics: Part I. Volumetric locking

Although the finite element method (FEM) has been extensively applied to various areas of engineering, the ill-conditioned problems occurring in many situations are still thorny to deal with. This study attempts to provide a high-performing and simple approach to the solutions of ill-conditioned problems. The theoretical foundation of it is the parametrized variational principles, called the generalized mixed variational principles (GMVPs) initiated by Rong in 1981. GMVPs can solve many kinds of ill-conditioned problems in computational mechanics. Among them, four cases are investigated in detail: the volumetric locking, the shear locking, the inhomogeneousness and the membrane locking problems, composing four parts of the study, Part I–IV, respectively. This paper is Part I, wherein a GMVP specially suited to the nearly incompressible and incompressible materials is constructed. A derived condition to overcome the rank deficiency of the global matrix of FEM for the perfect incompressibility is put forward. Based on these, a new approach, named the order-one shooting method, is proposed to avoid the spurious checkerboard pressure modes produced by certain FEM formulations.

A stabilized incremental projection scheme for the incompressible Navier–Stokes equations

AbstractIt is well known that any spatial discretization of the saddle‐point Stokes problem should satisfy the Ladyzhenskaya–Brezzi–Babuska (LBB) stability condition in order to prevent the appearance of spurious pressure modes. Particularly, if an equal‐order approximation is applied, the Schur complement (or, as called some times, the Uzawa matrix) of the pressure system has a non‐trivial null space that gives rise to such modes. An idea in the past was that all the schemes that solve a Poisson equation for the pressure rather than the Uzawa pressure equation (splitting/projection methods) should overcome this difficulty; this idea was wrong. There is numerical evidence that at least the so‐called incremental projection scheme still suffers from spurious pressure oscillations if an equal‐order approximation is applied. The present paper tries to distinguish which projection requires LBB‐compliant approximation and which does not. Moreover, a stabilized version of the incremental projection scheme is derived. Proper bounds for the stabilization parameter are also given. The numerical results show that the stabilized scheme does indeed achieve second‐order accuracy and does not produce spurious (node to node) pressure oscillations. Copyright © 2001 John Wiley & Sons, Ltd.

Conservation Properties of Unstructured Staggered Mesh Schemes

Classic Cartesian staggered mesh schemes have a number of attractive properties. They do not display spurious pressure modes and they have been shown to locally conserve, mass, momentum, kinetic energy, and circulation to machine precision. Recently, a number of generalizations of the staggered mesh approach have been proposed for unstructured (triangular or tetrahedral) meshes. These unstructured staggered mesh methods have been created to retain the attractive pressure aspects and mass conservation properties of the classic Cartesian mesh method. This work addresses the momentum, kinetic energy, and circulation conservation properties of unstructured staggered mesh methods. It is shown that with certain choices of the velocity interpolation, unstructured staggered mesh discretizations of the divergence form of the Navier–Stokes equations can conserve kinetic energy and momentum both locally and globally. In addition, it is shown that unstructured staggered mesh discretizations of the rotational form of the Navier–Stokes equations can conserve kinetic energy and circulation both locally and globally. The analysis includes viscous terms and a generalization of the concept of conservation in the presence of viscosity to include a negative definite dissipation term in the kinetic energy equation. These novel conserving unstructured staggered mesh schemes have not been previously analyzed. It is shown that they are first-order accurate on nonuniform two-dimensional unstructured meshes and second-order accurate on uniform unstructured meshes. Numerical confirmation of the conservation properties and the order of accuracy of these unstructured staggered mesh methods is presented.

Two-dimensional turbulent viscous flow simulation past airfoils at fixed incidence

The present work discusses the computation of the time-mean, turbulent, two-dimensional incompressible viscous flow past an airfoil at fixed incidence. A new physically consistent method is presented for the reconstruction of velocity fluxes which arise from discrete equations for the mass and momentum balance. This closure method for fluxes makes possible the use of a cell-centred grid in which velocity and pressure unknowns share the same location, while circumventing the occurrence of spurious pressure modes. The influence of several turbulent models is investigated. The models involve either an algebraic eddy viscosity or determine the eddy viscosity from transport equations. The method is tested on the following airfoils in pre-stall or post-stall conditions: NACA4412, AS-240 B, GAW-(1) airfoil, for which documented experimental data are available.

Elimination of spurious pressure and kinematic modes in biquadratic nine‐node plane element

AbstractIn two‐dimensional penalty finite element analysis of incompressible materials, Q9/3P, the 9‐node quadrilateral with three assumed quasi‐pressure modes, is the most popular element not bothered by spurious pressure. In constructing the penalty matrix of Q9/3P, it is necessary to form a 3×18 matrix and a 3×3 symmetric matrix. The inverse of the symmetric matrix is then post‐ and pre‐multiplied by the 3×18 matrix and its transpose, respectively. By employing a rank subtraction technique, a new and more efficient implementation scheme is devised for the penalty matrix. Same as the conventional Q9 element, when Q9/3P is fully integrated, it becomes expensive and too stiff. On the other hand, there are two spurious kinematic modes should the element be sub‐integrated. In the proposed Q9/3P element, the two mechanisms will be annihilated by judiciously chosen higher‐order assumed stress modes in conjunction with a modified Hellinger‐Reissner functional. It will be demonstrated that the element is of good accuracy and high efficiency.

Computation of unsteady laminar viscous flow past an aerofoil using the CPI method

AbstractThe present work is a discussion of the computation of unsteady laminar incompressible two‐dimensional viscous flow past complex geometries. A new physically consistent method is presented for the reconstruction of velocity fluxes which arise from the mass and momentum balance discrete equations. This closure method for fluxes makes possible the use of a cell‐centred grid in which velocity and pressure unknowns share the same location while circumventing the occurrence of spurious pressure modes. The method is validated using the impulsively started cylinder problem for circular or aerofoil‐like shapes at several Reynolds numbers of the order of 103–104.

Incompressible flow calculations with a consistent physical interpolation finite volume approach

The computation of incompressible three-dimensional viscous flow is discussed. A new physically consistent method is presented for the reconstruction for velocity fluxes which arise from the mass and momentum balance discrete equations. This closure method for fluxes allows the use of a cell-centered grid in which velocity and pressure unknowns share the same location, while circumventing the occurrence of spurious pressure modes. The method is validated on several benchmark problems which include steady laminar flow predictions on a two-dimensional cartesian (lid driven 2D cavity) or curvilinear grid (circular cylinder problem at Re = 40), unsteady three-dimensional laminar flow predictions on a cartesian grid (parallelopipedic lid driven cavity) and unsteady two-dimensional turbulent flow predictions on a curvilinear grid (vortex shedding past a square cylinder at Re = 22,000).

Chebyshev collocation method and multidomain decomposition for the incompressible Navier‐Stokes equations

AbstractThe two‐dimensional incompressible Navier‐Stokes equations in primitive variables have been solved by a pseudospectral Chebyshev method using a semi‐implicit fractional step scheme. The latter has been adapted to the particular features of spectral collocation methods to develop the monodomain algorithm. In particular, pressure and velocity collocated on the same nodes are sought in a polynomial space of the same order; the cascade of scalar elliptic problems arising after the spatial collocation is solved using finite difference preconditioning. With the present procedure spurious pressure modes do not pollute the pressure field.As a natural development of the present work a multidomain extent was devised and tested. The original domain is divided into a union of patching sub‐rectangles. Each scalar problem obtained after spatial collocation is solved by iterating by subdomains. For steady problems a C1 solution is recovered at the interfaces upon convergence, ensuring a spectrally accurate solution.A number of test cases have been solved to validate the algorithm in both its single‐block and multidomain configurations.The preliminary results achieved indicate that collocation methods in multidomain configurations might become a viable alternative to the spectral element technique for accurate flow prediction.

Low‐order macroelements for two‐ and three‐dimensional Stokes flow

AbstractThe use of low‐order elements for approximating fluid flow is attractive because all the elemental contributions can be quickly and easily obtained. One of the drawbacks is that low‐order elements often give rise to spurious pressure modes or incompatible velocity and pressure approximations. In this paper linear velocity and linear pressure elements are described for both two‐ and three‐dimensional flow that always produce stable solutions provided the elements are assembled into simple macroelements following easily used rules. Some examples of this idea are given for Stokes flow and compared with another popular low‐order method. © 1993 John Wiley & Sons, Inc.

Compatibility conditions for the Stokes system discretized in 2-D Cartesian domains

The maximum number of independent numerical constraints that can be imposed on the velocity field is determined by identifying the compatibility conditions which arise between the discretized incompressibility constraint equations and the velocity boundary conditions. This number does not depend on the chosen numerical scheme and defines the maximum number of pressure modes that are numerically accessible. For the (Navier-)Stokes problem discretized in 2-D Cartesian domains, it is shown that, at most, N-8 pressure modes can be evaluated, N being the number of modes chosen for each velocity component. The eight out of access pressure modes are the so-called spurious pressure modes.

The Treatment of Spurious Pressure Modes in Spectral Incompressible Flow Calculations

Algorithms for the transient and steady state simulation of incompressible Newtonian and non-Newtonian flows are described for the primitive variable formulation of the governing equations. Spectral approximations are used for the spatial discretization. Attention is given to the satisfaction of the incompressibility constraint and the determination of the pressure. Spurious pressure modes are removed by means of a singular value decomposition. The corresponding velocity field is divergence free at all the collocation points. Numerical results are presented for Newtonian flow in a regularized driven square cavity and for non-Newtonian flow in a planar channel and in a journal bearing for a realistic range of material parameters.

Spectral method solution of the Stokes equations on nonstaggered grids

Abstract The Stokes equations are solved using spectral methods with staggered and nonstaggered grids. Numerous ways to avoid the problem of spurious pressure modes are presented, including new techniques using the pseudospectral method and a method solving the weak form of the governing equations (a variation on the “spectral element” method developed by Patera). The pseudospectral methods using nonstaggered grids are simpler to implement and have comparable or better accuracy than the staggered grid formulations. Three test cases are presented: a formulation with an exact solution, a formulation with homogeneous boundary conditions, and the driven cavity problem. The solution accuracy is shown to be greatly improved for the driven cavity problem when the analytical solution of the singular flow behavior in the upper corners is separated from the computational solution.

Finite-Element Analysis of Turbulent Flow in Annular Exhaust Diffusers of Gas Turbine Engines

A finite-element model of the turbulent flow field in the annular exhaust diffuser of a gas turbine engine is developed. The analysis is based on a modified version of the Petrov-Galerkin weighted residual method, coupled with a highly accurate biquadratic finite element of the Lagrangian type. The elemental weight functions in the finite-element formulation are so defined to ensure upwinding of the convection terms in the flow-governing equations while reverting to the conventional Galerkin’s definition for all other terms. This approach is equivalent to altering the integration algorithm as the convection terms in the element equations are derived, with the exception that the latter technique is tailored for low-order elements of the linear and bilinear types. Numerical results of the current analysis indicate that spurious pressure modes associated with this type of inertia-dominated flow are alleviated while the false numerical diffusion in the finite-element equations is simultaneously minimized. Turbulence of the flow field is modeled using the two-layer algebraic turbulence closure of Baldwin and Lomax, and the eddy viscosity calculations are performed at variably spaced points which are different from those in the finite-element discretization model. This enhances the accuracy in computing the wall shear stress and the inner/outer layer interface location. The computational model is verified using a set of experimental data at design and off-design operation modes of the exhaust diffuser in a commercial gas turbine engine. Assessment of the results in this case is favorable and, as such, provides evidence of the model capability as an accurate predictive tool in the diffuser detailed design phase.

3‐D flow analysis of non‐newtonian viscous fluids using “enriched” finite elements

Abstract‘Enriched’ element, Q1+Po, and ‘standard’ element, Q1Po, are compared for the simulation of 3‐dimensional flow of non‐Newtonian fluids. Several 3‐dimensional polymer flow problems are analyzed. The pressure field obtained by using Q1Po elements suffers from spurious pressure modes. For complex flows, depending upon the flow geometry and the boundary conditions used, Q1+Po elements may fall to simulate even the velocity field. Q1Po elements, which satisfy the Babuska‐Brezzi condition, give accurate velocity and pressure distributions for all the problems analyzed here.

An equal-order velocity-pressure formulation that does not exhibit spurious pressure modes

An equal-order velocity-pressure formulation that does not exhibit spurious pressure modes

Old and new finite elements for incompressible flows

AbstractThis paper presents some seemingly new elements for the computation of two and three‐dimensional incompressible flow. We want to obtain elements satisfying the Babǔska‐Brezzi condition for mixed methods and thus introducing no spurious pressure modes (cf. Sani et al.1). In order to present clearly the advantages and disadvantages of our new elements we compare them on a qualitative basis with more standard ones. Of particular importance for incompressible flow is the size and shape of vortices that can be produced by the elements. We shall try to describe this as precisely as possible. The conclusion is that the elements introduced here should be quite competitive on a cost/precision scale.

The cause and cure (!) of the spurious pressures generated by certain fem solutions of the incompressible Navier‐Stokes equations: Part 2

AbstractThe spurious pressures and ostensibly acceptable velocities which sometimes result from certain FEM approximate solutions of the incompressible Navier‐Stokes equations are explained in detail. The concept of pressure modes, physical and spurious, pure and impure, is introduced and their effects on discretized solutions is analysed, in the context of mixed interpolation and penalty approaches. Pressure filtering schemes, which are capable of recovering useful pressures from otherwise polluted numerical results, are developed for two particular elements in two‐dimensions and one element in three‐dimensions. The automatic pressure filter associated with the penalty method is also explained. Implications regarding the effect of spurious pressure modes on accuracy and ultimate convergence with mesh refinement are discussed and a list of unanswered questions presented. Sufficient numerical examples are discussed to corroborate the theory presented herein.