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

Abstract

Control of uid forces is an emerging area of research with numerous engineering applications. ­e uneven wake behind an obstacle causes undesirable structural oscillations, which can lead to fatigue or structural failure. Controlling the wake phenomena could directly bene t a wide range of engineering applications, including skyscrapers, naval risers, bridges, columns, and a few sections of airplanes. ­is study is concerned with the time dependent simulations in a channel in presence of an obstacle aiming to compute uid forces. ­e underlying mathematical model is based on nonstationary Navier–Stokes equations coupled with the constitutive relations of power law uids. Because the representative equations are complex, an e ective computing strategy based on the nite element approach is used. To achieve higher accuracy, a hybrid computational grid at a very ne level is used. ­e P2 − P1 elements based on the shape functions of the second and rst-order polynomials were used to approximate the solution. ­e discrete nonlinear system arising from this discretization is linearized by Newton’s method and then solved through a direct linear solver PARADISO. ­e code validation study is also performed for Newtonian uids 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 e ects of shear rate on the drag-lift phase plot are also discussed.