In many engineering applications it is desirable to have a steady flow field that is stable to perturbations. This can be described through a global stability analysis. This is an eigenvalue problem which gives a series of mode shapes (the eigenfunctions) and their growth rates and frequencies (the eigenvalues).

This thesis calculates and interprets the shape sensitivity of the eigenvalue of a global stability analysis. The shape sensitivity allows quick calculation of the change in a mode’s growth rate and frequency when the flow geometry is changed. The shape sensitivity is calculated using an adjoint method. This is derived both for flows governed by the incompressible Navier–Stokes equations and also for flows governed by the URANS equations using the Spalart–Allmaras and k-ω turbulence models. Using the laminar and turbulent flow past a cylinder as a model problem, the shape sensitivities of the laminar and turbulent vortex shedding modes are presented. For both cases, control of the eigenvalue by shape deformation is shown to occur primarily by changing the steady base-flow and not by changing the mode’s unsteady feedback. A technique for finding the deformations with the greatest influence on the base-flow is then demonstrated. For the laminar flow, this shows that the shape sensitivity of the growth rate and frequency is primarily due to a single deformation that causes large widespread base-flow changes. This deformation increases the growth rate but decreases the frequency, leading the shape sensitivities to have similar shapes but opposite signs.

The thesis then applies this framework to the cyclone separator. This is an industrial application which has been shown to have a helical instability that reduces the cyclone’s performance. Helical and double-helical modes with similar Strouhal numbers to those seen in experiment are found. As with the cylinder, control of the eigenvalue by shape deformation is shown to occur primarily by changing the steady base-flow and not by changing the mode’s unsteady feedback. The shape sensitivities are shown to be concentrated at the cyclone’s cone tip and vortex finder. Small changes of these parts of the geometry are shown to cause large widespread changes to the base-flow. Finally, the shape sensitivities are used in a gradient based method to show that small changes in the cyclone geometry can significantly reduce the growth rate of the unstable modes. This is done in conjunction with a separation metric to show that the stability of the flow can be improved without reducing separation performance.

The techniques demonstrated in this thesis for finding the shape sensitivity of the eigenvalue and for identifying influential deformations can be easily applied to a wide range of different flow geometries, governing equations and cost functions. This highlights the benefits of the use of adjoint-based methods in engineering design.