An a-posteriori finite element error estimator for adaptive grid computation of viscous incompressible flows

In this thesis, an à-posteriori error estimator is presented and employed for solving viscous incompressible flow problems. In an effort to detect local flow features, such as vortices and separation, and to resolve flow details precisely, a velocity angle error estimator e _&thetas; which is based on the spatial derivative of velocity direction fields is designed and constructed. The à-posteriori error estimator corresponds to the antisymmetric part of the deformation-rate-tensor, and it is sensitive to the second derivative of the velocity angle field. Rationality discussions reveal that the velocity angle error estimator is a curvature error estimator, and its value reflects the accuracy of streamline curves. It is also found that the velocity angle error estimator contains the nonlinear convective term of the Navier-Stokes equations, and it identifies and computes the direction difference when the convective acceleration direction and the flow velocity direction have a disparity.; Through benchmarking computed variables with the analytic solution of Kovasznay flow or the finest grid of cavity flow, it is demonstrated that the velocity angle error estimator has a better performance than the strain error estimator. The benchmarking work also shows that the computed profile obtained by using e_&thetas; can achieve the best matching outcome with the true &thetas; field, and that it is asymptotic to the true &thetas; variation field, with a promise of fewer unknowns. Unstructured grids are adapted by employing local cell division as well as unrefinement of transition cells. Using element class and node class can efficiently construct a hierarchical data structure which provides cell and node inter-reference at each adaptive level. Employing element pointers and node pointers can dynamically maintain the connection of adjacent elements and adjacent nodes, and thus avoids time-consuming search processes.; The adaptive scheme is applied to viscous incompressible flow at different Reynolds numbers. It is found that the velocity angle error estimator can detect most flow characteristics and produce dense grids in the regions where flow velocity directions have abrupt changes. In addition, the e _&thetas; estimator makes the derivative error dilutely distribute in the whole computational domain and also allows the refinement to be conducted at regions of high error.; Through comparison of the velocity angle error across the interface with neighbouring cells, it is verified that the adaptive scheme in using e_&thetas; provides an optimum mesh which can clearly resolve local flow features in a precise way. The adaptive results justify the applicability of the e_&thetas; estimator and prove that this error estimator is a valuable adaptive indicator for the automatic refinement of unstructured grids.