Thermoacoustic-hydrodynamic interactions in the wake of a cylinder embedded in a Rijke tube

In isolation, the Rijke tube and cylinder wake represent canonical, well-understood examples of thermoacoustic and hydrodynamic instabilities, respectively. However, many compressible flows, such as those found in high-enthalpy boundary layers and afterburners, feature thermoacoustic—hydrodynamic interactions that cannot be understood simply as a sum of these two elementary parts. This study analyzes the linear and nonlinear dynamics of a combined Rijke tube—cylinder wake system in order to elucidate how these basic instability modes interact to enrich the system behavior. Leveraging nonlinear time-domain simulations and numerical bifurcation analysis, we model the compressible flow over a heated cylinder embedded within a Rijke tube. We find independent neutral curves associated with both thermoacoustic and hydrodynamic instabilities that divide the parameter space into four regions characterized by the instability of neither, either, or both modes. These curves intersect at a double-Hopf point, where weakly-nonlinear analysis allows us to precisely extract the coupling strengths among the instabilities and their harmonics as well as infer the spatial structure of the components underpinning these couplings. Finally, the weakly-nonlinear structures are used as a basis on which to project the true nonlinear dynamics in the regime where both modes are unstable, enabling us to extract and interpret the underlying thermoacoustic—hydrodynamic interactions.