Element Method For Incompressible Flows Research Articles

On First-order Formulations of the Least-squares Finite Element Method for Incompressible Flows

To reduce the element continuity requirement for the least-squares finite element method (LSFEM), it is customary to rewrite the Navier–Stokes equations as first-order partial differential equations. In this work, numerical experiments for three-dimensional incompressible flows are carried out by using the LSFEM based on three types of first-order formulations, namely the velocity–vorticity–pressure (VVP) formulation, the velocity–stress–pressure (VSP) formulation and the velocity–velocity gradient–pressure (VVGP) formulation. In addition, two modifications for each formulation are considered. Numerical results indicate that proper problem formulation can significantly reduce computing time and improve the accuracy of numerical solutions.

Numerical study of a piston-driven laminar flow and heat transfer in a pipe with a sudden expansion

The present paper deals with the numerical investigation of piston-driven heat transfer and fluid flow in a cylinder containing a sudden expansion. The calculation procedure employed is based on a control-volume-based finite element method for incompressible flow with a staggered and moving grid and SIMPLER algorithm for pressure-velocity coupling. The numerical results show a good agreement with the experimental data reported in the literature. Results concerning time and space evolution of the thermal and flow fields downstream and upstream of the sudden expansion are presented for different values of the expansion ratio, the initial clearance volume, and the piston velocity. The effects of these parameters on the instantaneous convective heat flux and on the total convected heat are also evaluated at the sudden expansion section.

Piston-driven laminar flow and heat transfer in a plane channel with a sudden expansion

This paper presents a numerical study of piston-driven heat transfer and fluid flow in a plane channel containing a sudden expansion. The numerical method employed is based on a control-volume-based finite element method for incompressible flow with a staggered and moving grid and SIMPLER algorithm for pressure-velocity coupling. The numerical results show a good agreement with the experimental data reported in the literature. Results concerning time and space evolution of the thermal and flow fields are presented for different values of the expansion ratio, the initial clearance volume, and the piston velocity.

Pressure Stability in Fractional Step Finite Element Methods for Incompressible Flows

The objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows that use a pressure Poisson equation. For the classical first-order projection method, it is shown that there is a pressure control which depends on the time step size, and therefore there is a lower bound for this time step for stability reasons. The situation is much worse for a second-order scheme in which part of the pressure gradient is kept in the momentum equation. The pressure stability in this case is extremely weak. To overcome these shortcomings, a stabilized fractional step finite element method is also considered, and its stability is analyzed. Some simple numerical examples are presented to support the theoretical results.

Gauge finite element method for incompressible flows

Gauge finite element method for incompressible flows

Gauge finite element method for incompressible flows

A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J-G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher-order) finite elements. This method can achieve high-order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite element methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright © 2000 John Wiley & Sons, Ltd.

A least-squares finite element method for incompressible flow in stress-velocity-pressure version

In this paper we are concerned with the incompressible flow in 2-D. Introducing additional variables of derivatives of velocity, which are called stresses here, the second-order dynamic equations are reduced into a first-order system with variables of stress, velocity and pressure. Combining the compatability conditions: and the divergence ice condition, we have a system with six first-order equations and six unknowns. Least-squares method performed over this extended system. The analysis shows that this method achieves optimal rates of convergence in the H 1-norm as the h approaches to zero. Numerical experiences are also available.

Piecewise linear approach to the stokes equations in 3-D

Since the beginning of the 1970's, people have developed mixed finite element methods for incompressible flow. The velocity and pressure interpolations are required to satisfy a Ladyshenskaya-Babuska-Brezzi (LBB) condition which precludes many natural elements. Over the past 20 years, most mathematicians and engineers believed this to be necessary. In this research, a least-squares method for three-dimensional (3-D) problems is proposed. This method leads to a minimization problem and thus it is not subject to the restriction of the LBB condition. Piecewise linear elements can be applied for the approximation functions; the simplest and natural elements are easy to program and achieve optimal rates of convergence.

Implementation of implicit finite element methods for incompressible flows on the CM-5

A parallel implementation of an implicit finite element formulation for incompressible fluids on a distributed-memory massively parallel computer is presented. The dominant issue that distinguishes the implementation of finite element problems on distributed-memory computers from that on traditional shared-memory scalar or vector computers is the distribution of data (and hence workload) to the processors and the non-uniform memory hierarchy associated with the processors, particularly the non-uniform costs associated with on-processor and off-processor memory references. Accessing data stored in a remote processor requires computing resources an order of magnitude greater than accessing data locally in a processor. This distribution of data motivates the development of alternatives to traditional algorithms and data structures designed for shared-memory computers, which must now account for distributed-memory architectures. Data structures as well as data decomposition and data communication algorithms designed for distributed-memory computers are presented in the context of high level language constructs from High Performance Fortran. The discussion relies primarily on abstract features of the hardware and software environment and should be applicable, in principle, to a variety of distributed-memory system. The actual implementation is carried out on a Connection Machine CM-5 system with high performance communication functions.

境界要素法を用いた航空機胴体形状の最適空力設計について

A general form of boundary element method for incompressible flow has been developed using Green's function. An integral equation of the perturbation velocity potential is obtained. On the paneldivided surfaces, this results in a set of linear algebraic equations. The pressure distribution on complete aircrafts including wing, body, and tail can be found using present method. Two examples of fuselage design problem for low drag are presented, in which the frictional drag is approximated with the local velocity and the surface area. Flexible modification of fuselage shape is obtained by using spline surfaces. One of the examples shows a practical application of the method for flow separation prediction.

Variable penalty method for finite element analysis of incompressible flow

AbstractA new scheme is applied for increasing the accuracy of the penalty finite element method for incompressible flow by systematically varying from element to element the sign and magnitude of the penalty parameter λ, which enters through ∇.v + p/λ = 0, an approximation to the incompressibility constraint. Not only is the error in this approximation reduced beyond that achievable with a constant λ, but also digital truncation error is lowered when it is aggravated by large variations in element size, a critical problem when the discretization must resolve thin boundary layers. The magnitude of the penalty parameter can be chosen smaller than when λ is constant, which also reduces digital truncation error; hence a shorter word‐length computer is more likely to succeed. Error estimates of the method are reviewed. Boundary conditions which circumvent the hazards of aphysical pressure modes are catalogued for the finite element basis set chosen here. In order to compare performance, the variable penalty method is pitted against the conventional penalty method with constant λ in several Stokes flow case studies.

Flow of a curtiss-bird fluid over a transverse slot using the finite element drift-function method

A finite element method for incompressible flows of a memory fluid is described which is based on macroelements of crossed linear triangels. This allows virtually exact computations of strains in trial velocity fields. Memory is handled by construction of special Gaussian quadrature formulas. The problem of plane flows over slots is studied numerically using the reptational constitutive equation proposed by Curtiss and Bird. With this constitutive equation and an improved nonlinear iteration scheme, convergence difficulties which plagued earlier attempts at modelling viscoelastic flow seem to be avoided. Particular attention is paid to the relationship between first normal-stress difference and pressure difference across the slot as a function of Deborah and Reynolds number. The model predicts that there is a strong correlation between hole-pressure and first normal-stress difference if the inertial contribution to the hole-pressure is accounted for. Nevertheless the correlation differs to some extent from the Higashitani-Pritchard prediction. In our model this is traceable to specific violations of Higashitani and Pritchard's assumptions related to fluid-memory effects.