Arbitrary order Taylor expansions of the base flow and its eigenproblem

First order sensitivities and adjoint analyses are widely used today to assess the linear stability of unstable flows. Second order sensitivities have recently helped to increase accuracy. In this work, we calculate high order Taylor expansions of the base flow and its eigenproblem around a scalar parameter for the incompressible Navier-Stokes equations.

Two scenarios are investigated: first, for the two-dimensional cylinder flow, we expand along the inverse of the Reynolds number. We predict the base flow and its leading eigenvalue by Taylor-expanding around Re=47, which is the onset of flapping. With subsequent Taylor expansions, we calculate the base flow accurately until Re=1000, thus establishing a new method to calculate unstable base flows. We discuss the topic of convergence radii by evaluating predicted base flows until Taylor order 40 and show that the method is computationally very efficient.

Second, a small control cylinder is inserted into the computational domain of the cylinder flow for stabilization. For relatively low Reynolds numbers, stabilizing areas are accurately predicted by modeling the small cylinder with a steady forcing and drawing sensitivity maps up to Taylor order 10. The results are validated by incorporating the small cylinder directly into the grid.