Global analysis of convective instabilities in nonparallel flows
Seminar Room 1, Newton Institute
A new scheme for the global analysis of convective instabilities in nonparallel flows is proposed. The linearized perturbation equations for an incompressible flow are written in a moving frame of reference that travels with the perturbation. In the moving frame, the base flow varies with time. However, at t=0, it is same as the one in stationary frame. Therefore, this analysis, for determining the global convective instability, is valid in an instantaneous sense. A stabilized finite element method is utilized to discretize these equations. A sub-space iteration procedure is utilized to solve the resulting generalized eigenvalue problem. Unlike local analysis, the proposed method gives the global eigenmode and the corresponding growth rate. The scheme is applied to assess the stability of uniform flow past bluff bodies. For the flow past a circular cylinder the critical Re for the onset of convective instability is found to be 4, approximately. The critical Re for the onset of the shear layer instability has been a point of contention. Various estimates have been proposed ranging from Re_c=350 to 2600. The proposed method is applied to find Re_c. To suppress the wake mode, that leads to Karman vortex shedding, flow past one half of the cylinder is studied. The Re_c is found to be ~54. The wake and shear layer modes for a full cylinder are compared to bring out the differences between the two. Also, the connection of the instability at low Re to the shear layer modes at higher Re (=500) is presented. The results are compared with earlier work from local stability analysis. Results are also computed for flow past a flat plate normal to the flow. All the results are in excellent agreement with the direct numerical simulation of the linearized flows equations.