A finite element approximation of the unsteady two-dimensional Navier–Stokes equations

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Abstract

In this paper a penalty finite element solution method for the unsteady Navier–Stokes equations for two-dimensional incompressible flow is described. The performances of the Euler implicit (EI) and the Crank–Nicolson (CN) time integration methods are analysed. Special attention is payed to the undamped pressure oscillations which can occur when the Crank–Nicolson integration rule is used in combination with the penalty function method. Stability and convergence properties are illustrated by means of the computation of fully developed oscillating flow between two flat plates. Furthermore, the von Karman vortex street past a circular cylinder is computed to demonstrate the behaviour of the time integration schemes for a more complicated flow. It is concluded that the EI method has its advantages over the CN method with respect to the damping of numerical oscillations. However, for flows with an important convective contribution, where physically originated oscillations may be present, the CN method is preferable.
Original languageEnglish
Pages (from-to)427-443
Number of pages17
JournalInternational Journal for Numerical Methods in Fluids
Volume6
Issue number7
DOIs
Publication statusPublished - 1986

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Crank-Nicolson
Finite Element Approximation
Navier-Stokes Equations
Oscillation
Time Integration
Oscillating flow
Euler
Incompressible flow
Circular cylinders
Integration rule
Penalty Function Method
Flat Plate
Finite Element Solution
Implicit Method
Vortex flow
Circular Cylinder
Stability and Convergence
Damping
Incompressible Flow
Convergence Properties

Cite this

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title = "A finite element approximation of the unsteady two-dimensional Navier–Stokes equations",
abstract = "In this paper a penalty finite element solution method for the unsteady Navier–Stokes equations for two-dimensional incompressible flow is described. The performances of the Euler implicit (EI) and the Crank–Nicolson (CN) time integration methods are analysed. Special attention is payed to the undamped pressure oscillations which can occur when the Crank–Nicolson integration rule is used in combination with the penalty function method. Stability and convergence properties are illustrated by means of the computation of fully developed oscillating flow between two flat plates. Furthermore, the von Karman vortex street past a circular cylinder is computed to demonstrate the behaviour of the time integration schemes for a more complicated flow. It is concluded that the EI method has its advantages over the CN method with respect to the damping of numerical oscillations. However, for flows with an important convective contribution, where physically originated oscillations may be present, the CN method is preferable.",
author = "{Vosse, van de}, F.N. and A. Segal and {Steenhoven, van}, A.A. and J.D. Janssen",
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language = "English",
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}

A finite element approximation of the unsteady two-dimensional Navier–Stokes equations. / Vosse, van de, F.N.; Segal, A.; Steenhoven, van, A.A.; Janssen, J.D.

In: International Journal for Numerical Methods in Fluids, Vol. 6, No. 7, 1986, p. 427-443.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - A finite element approximation of the unsteady two-dimensional Navier–Stokes equations

AU - Vosse, van de, F.N.

AU - Segal, A.

AU - Steenhoven, van, A.A.

AU - Janssen, J.D.

PY - 1986

Y1 - 1986

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AB - In this paper a penalty finite element solution method for the unsteady Navier–Stokes equations for two-dimensional incompressible flow is described. The performances of the Euler implicit (EI) and the Crank–Nicolson (CN) time integration methods are analysed. Special attention is payed to the undamped pressure oscillations which can occur when the Crank–Nicolson integration rule is used in combination with the penalty function method. Stability and convergence properties are illustrated by means of the computation of fully developed oscillating flow between two flat plates. Furthermore, the von Karman vortex street past a circular cylinder is computed to demonstrate the behaviour of the time integration schemes for a more complicated flow. It is concluded that the EI method has its advantages over the CN method with respect to the damping of numerical oscillations. However, for flows with an important convective contribution, where physically originated oscillations may be present, the CN method is preferable.

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JO - International Journal for Numerical Methods in Fluids

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