# Penalty Finite Element Applications to Flow Problems

• Der-Liang Young
• Wey-Bin Ni
Conference paper
Part of the Lecture Notes in Engineering book series (LNENG, volume 43)

## Abstract

This paper focuses on the appropriate elements to analyze two-dimensional internal and external flow problems by the penalty finite element method. The governing equations are the incompressible Navier-Stokes equations. Four different elements were tested to study some two-dimensional flow problems, such as Couette flow, square cavity flow, circular eddy flow, and flow past a circular cylinder. The four elements used are T6, T3M, Q4, and Q4M respectively. The first two are triangular elements, and the remains are quadrilateral elements.

It was found that for the low Reynolds number linear cases, there are no major computational differences for the four elements due to the flow simplicity. However, only the Q4 element can accurately predict the flow structures of the high Reynolds number flows. For the flow past a circular cylinder, the salient characteristics of Karman vortex streets are vividly simulated, such as the instability, vortex shedding, and pressure oscillation. It is further observed that the time integrating parameter will play an important role in the flow simulation. The velocity field will be most accurately predicted while the pressure is not, when the central (Crank-Nicolson) difference scheme is adopted. On the other hand, the reversed conjecture is true when backward difference scheme is used instead.

### Keywords

Vortex Vorticity Advection Incompressibility

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