Numerical verification of a non-residual orthogonal term-by-term stabilized finite element formulation for incompressible convective flow problems

Abstract

The modeling of turbulent flows is relevant in many engineering applications and, is an active field of research on numerical methods. Convergence and stability of proposed formulations are crucial to predict transitional flows from laminar to turbulent flows. In this work, a recently developed stabilized finite element formulation is used as a powerful tool to describe such kind of problems. The essential point of the formulation is the time dependent nature of the subscales and, contrary to residual based formulations, the introduction of two velocity subscale components. The theoretical rate of convergence of the method is verified numerically using linear and quadratic equal-order finite element discretizations. To this end, a standard convergence test of L2-norm is presented where the computed solutions are compared when manufactured solutions are imposed at Gauss-point level. Moreover, the Hopf bifurcation is studied for two well-known benchmark problems: flow past a cylinder and the three-dimensional lid-driven cavity flow. For the flow past a cylinder case, the Hopf bifurcation is verified using dynamic subscales and is assessed so that they do not disturb the solution. In particular, a dominant convective problem (Re=4000) is solved using both the quasistatic and dynamic versions of the method, evaluating the performance of each one in the quality of the solution and in the CPU time needed to obtain a converged solution. For the 3D lid-driven cavity flow problem, the Hopf bifurcation is determined using two different boundary conditions, analyzing their effect on the dynamics of the problem and on the thickness and shape of the boundary layer. The final test case is the turbulent 3D lid-driven cavity problem (Re=12000), where velocity profiles are compared with experimental, LES and DNS reference solutions. Additionally, pressure and velocity spectra are shown at certain representative points of the domain, as well as phase diagrams, correlation function graphs, the Poincaré map, and the Lyapunov exponent, typical mathematical tools used in dynamical system analysis. From the results, the method is robust and accurate for all the numerical tests both in viscous dominant problems as in dominant convective ones.