Finite Element Analysis of the Dynamics of Power-Law Fluid around an Obstacle in a Channel

Abstract

Control of fluid forces is an emerging area of research with numerous engineering applications. The uneven wake behind an obstacle causes undesirable structural oscillations, which can lead to fatigue or structural failure. Controlling the wake phenomena could directly benefit a wide range of engineering applications, including skyscrapers, naval risers, bridges, columns, and a few sections of airplanes. This study is concerned with the time dependent simulations in a channel in presence of an obstacle aiming to compute fluid forces. The underlying mathematical model is based on nonstationary Navier–Stokes equations coupled with the constitutive relations of power law fluids. Because the representative equations are complex, an effective computing strategy based on the finite element approach is used. To achieve higher accuracy, a hybrid computational grid at a very fine level is used. The${\mathrm{\mathbb{P}}}_2-{\mathrm{\mathbb{P}}}_1$elements based on the shape functions of the second and first-order polynomials were used to approximate the solution. The discrete nonlinear system arising from this discretization is linearized by Newton’s method and then solved through a direct linear solver PARADISO. The code validation study is also performed for Newtonian fluids as a special case, and then the study is extended to compute drag and lift forces for other cases of viscosity as described by the power law index. When looking at the phase plot, it can be seen that for the Newtonian case n = 1, there is only one closed orbit after the steady state is reached, whereas for$n=0.5$, there are multiple periodic orbits. Moreover, the effects of shear rate on the drag-lift phase plot are also discussed.