Shape optimization to suppress the helical vortex breakdown based on linear stability and adjoint theory*

Global instabilities that lead to large-scale periodic fluctuations play a key role in many engineering applications. They arise due to a distinct self-excitation mechanism that is often spatially localized in a narrow region. It is, therefore, often minor changes of the flow-immersed geometry that can have a large impact on the instability. In this work, a shape optimization framework is developed to exploit this property in order to suppress an instability by delaying its bifurcation.

For this purpose, nonlinear simulations of the helical vortex breakdown downstream of an axisymmetric annular swirling flow are conducted with a finite element method. Based on linear stability and adjoint theory, and utilizing the Hadamard theorem, the eigenvalue sensitivity due to shape deformations of the centerbody at the downstream end of the annulus is derived. Using this shape sensitivity, a shape optimization of the centerbody is conducted with the objective to suppress the helical instability by minimizing its growth rate beyond marginal stability. The optimal shape for suppression is found and its physical implications are discussed. This passive flow control approach serves as a first step towards shape optimization of more complex geometries under turbulent flow conditions.