Linear Stability of Incompressible Flow Using a Mixed Finite Element Method
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Journal of Computational Physics 139 (2): 243 - 273 (1998)

We analyzed the linear stability for incompressible flow using the mixed finite element method. First, by means of the fractional step finite element method, the two-dimensional base flows were numerically computed over a range of Reynolds numbers, and then they were perturbed with three-dimensional disturbances. By using linear stability and normal mode analysis, we obtained the partial differential equations governing the evolution of perturbation from linearized Navier–Stokes equations with slight compressibility. In terms of the mixed finite element discretization (in which six-node quadratic Lagrange triangular elements with quadratic interpolation for velocities (P2) and three-node linear Lagrange triangular elements for pressure (P1) were used), a nonsingular generalized eigenproblem was formulated from these equations whose solution gives the dispersion relationship between the complex growth rate and the wave number. We examined the stability of lid-driven cavity flow in this study. The stability results obtained from theP2P1elements were compared with those fromP1P1elements. The spatial distribution of the pressure disturbance can be smoothed by using aP2P1element (such as that in the simulation of incompressible flows by the mixed finite element method); however, the choice of elemental type has little influence on the stability results. This study presents stability curves to identify the critical Reynolds number and the critical wavelength of the neutral mode and discusses the instability mechanism. The Taylor–Göertler-like vortices in the cavity were obtained by means of the reconstruction of three-dimensional flows. The stability results and the reconstructed three-dimensional flows were in agreement with the observed ones and the other numerical stability results, respectively.
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