Stability analysis of jets in crossflow

Jets in crossflow (JICFs), or transverse jets, are a canonical flow where a jet of fluid is injected normal to a crossflow. The interaction between the incoming flat-plate boundary layer and the jet is dependent on the Reynolds number (Re = v_jD/v), based on the average velocity (v_jet) at the jet exit and the diameter (D), as well as the jet-to-crossflow ratio (R = v_jet/u_∞). Megerian et al. (2007) performed experiments at Re = 2000 and collected vertical velocity spectra along the upstream shear-layer. They observed that the upstream shear-layer transitions from absolutely to convectively unstable between R = 2 and R = 4. Using an unstructured, incompressible, direct numerical simulation (DNS) solver, Iyer & Mahesh (2016) performed simulations matching the experimental setup of Megerian et al. (2007). Vertical velocity spectra taken along the upstream shear-layer from simulation show good agreement with experiment, marking the first high-fidelity simulation able to fully capture the complex shear-layer instabilities in low speed jets in crossflow. Iyer & Mahesh (2016) proposed an analogy to counter-current mixing along the leading edge shear-layer to explain the transition from an absolute to convective instability. In addition, Iyer & Mahesh (2016) performed dynamic mode decomposition (DMD) of the velocity field, which reproduced the dominant frequencies obtained from the upstream shear-layer spectra. In the present work, the stability of JICFs is studied when R = 2 and R = 4 using global linear stability analysis (GLSA) (i.e Tri-Global linear stability analysis), where the baseflow is fully three-dimensional. A variant of the implicitly restarted Arnoldi method (IRAM) in conjunction with a time-stepper approach is implemented to efficiently calculate the leading eigenvalues and their associated eigenmodes. The Strouhal frequencies (St = fD/v_jet), based on the peak velocity (v_jet) at the jet exit and the diameter (D), from linear stability analysis are compared with experiments (Megerian et al., 2007) and simulations (Iyer & Mahesh, 2016). The eigenmodes are analyzed and show evidence that supports the transition from an absolutely to convectively unstable flow. Additionally, the adjoint sensitivity of the upstream shear-layer is studied for the case when R = 2. The location of the most sensitive areas is shown to be localized to the upstream side of the jet nozzle near the jet exit. The wavemaker for the upstream shear-layer is then calculated using the direct and adjoint eigenmodes for case R = 2. The results further justify the absolutely unstable nature of the region near the upstream side of the jet nozzle exit.