Buoyancy-driven systems are subject to several types of flow instabilities. To evaluate the performance of such systems it is becoming increasingly crucial to be able to predict the stability of a given base flow configuration. Traditional Modal Linear stability Analysis requires the solution of very large eigenvalue systems for three-dimensional flows, which make this problem difficult to tackle.
An alternative to modal Linear stability Analysis is the use of adjoint solvers  in combination with a power iteration . Such methodology allows for the identification of an optimal disturbance or forcing and has been recently used to evaluate the stability of several isothermal flow systems . In this paper we examine the extension of the methodology to non-isothermal flows driven by buoyancy. The contribution of buoyancy in the momentum equation is modeled through the Boussinesq approximation.
The method is implemented in the spectral element code Nek5000. The test case is the flow is a two-dimensional cavity with differential heating and conductive walls and the natural circulation flow in a toroidal thermosiphon.