Shape sensitivity of eigenvalues in hydrodynamic stability, with physical interpretation for the flow around a cylinder

Flows around or within objects can oscillate at well-defined frequencies. Engineers often want to eliminate these oscillations or alter their frequencies. Current methods, such as suction at the boundaries, can be impractical. The methods presented here control these oscillations by changing the object's shape, which is often more practical. The methods are based on a linear stability analysis about a steady baseflow and use adjoints, computing the sensitivities to all possible deformations in two calculations. We demonstrate the methods, physically interpreting the results for a cylinder flow at Reynolds number 50.

This work shows that deformations affect hydrodynamic oscillations mainly by changing the steady baseflow, rather than by changing the unsteady feedback mechanisms. Deformations strongly affecting the baseflow are shown to strongly affect the oscillations, as expected. In addition, subtle deformations around the separation points are shown to exploit small baseflow changes that have disproportionately large influence on the growth rate. The physical mechanisms behind this is shown to be similar to the well-known phenomenon of 'base bleed'. The method presented here is general and versatile, providing both gradient information and physical insight.