A biharmonic approach for the global stability analysis of 2D incompressible viscous flows

• First use of ψ–v approach to construct the generalized eigenvalue problem (GEP).
• First stability analysis performed on the 2D cross-lid-driven cavity flow.
• Extensive studies through phase plane analysis, spectral density analysis.
• Extensive studies on structures of the matrices and their effects on computation.
• Drastically reduces the computation time for the GEP for 2D flows.

This manuscript embodies an investigation of stabilities in flows governed by the incompressible Navier–Stokes equations with the recently developed compact scheme by Kalita and Gupta (2010) which was derived by using the biharmonic formulation of the 2-D incompressible Navier–Stokes equations. In the current work, we globally analyze the flow stability utilizing this transient ψ–v approach. Critical parameters are found by constructing a generalized eigenvalue problem resulting from the discretization of the stability equations through the above mentioned scheme. This approach is seen to drastically reduce the CPU time for finding the critical eigenvalues. Three fluid flow problems, namely, the square lid-driven cavity, the two sided cross lid-driven cavity and the flow past an inclined square cylinder have been chosen as test cases. For the square cavity and the flow past an inclined square cylinder, critical parameters found by us are in excellent agreement with the established results while for the two-sided cross cavity, its global stability analysis has been carried out for the first time and new results are obtained. The critical parameters are also reconfirmed by phase plane and spectral density analysis.