Algorithms for adaptive mesh refinement using a residual error estimator are proposed for fluid flow problems in a finite volume framework. The residual error estimator, referred to as the ℜ-parameter is used to derive refinement and coarsening criteria for the adaptive algorithms. An adaptive strategy based on the ℜ-parameter is proposed for continuous flows, while a hybrid adaptive algorithm employing a combination of error indicators and the ℜ-parameter is developed for discontinuous flows. Numerical experiments for inviscid and viscous flows on different grid topologies demonstrate the effectiveness of the proposed algorithms on arbitrary polygonal grids.

A modified error indicator is developed to study the supersonic turbulent flow over a backward-facing step. In the Cartesian coordinate system, the unsteady Navier-Stokes equations with a low-Reynolds-number k−ε turbulence model are solved. The modified error indicator, in which the unified magnitude of substantial derivative of pressure and unified magnitude of substantial derivative of weighted vorticity magnitude are incorporated, is applied to treat the mesh refining. To assess the present approach, the transonic turbulent flow around an NACA 0012 airfoil is performed. Based on the comparison with the experimental data, the accuracy of the present approach is confirmed. According to the high-resolutional result on the adaptive mesh, the structure of backstep corner vortex, expansion wave and oblique shock wave is distinctly captured.

A modified error indicator and an extended locally implicit scheme with anisotropic dissipation model on quadrilateral-triangular mesh are developed to study the supersonic flow over a backward-facing step. In the Cartesian coordinate system, the unsteady Euler equations are solved. The modified error indicator, in which the unified magnitude of density gradient and unified magnitude of gradient of vorticity magnitude are incorporated, is utilized to treat the new node spacing of mesh remeshing. To assess the accuracy of the extended locally implicit scheme with anisotropic dissipation model on quadrilateral-triangular mesh, two flow calculations which include the oblique-shock reflection at a wall and transonic flow around an NACA 0012 airfoil are performed. Based on the comparison with the related numerical and experimental data, the accuracy of the present approach is confirmed. According to the high-resolutional result on the adaptive mesh, the structure of backstep corner vortex, expansion wave and oblique shock wave is distinctly captured.