Abstract
<title>Abstract</title> <p>The design of quiet supersonic nozzles is commonly performed within a two-dimensional framework, assuming either axisymmetric configurations or spanwise-homogeneous planar flows. While this approximation is appropriate for axisymmetric geometries, planar nozzles are inherently three-dimensional due to the presence of lateral sidewalls. This work assesses the robustness of two-dimensional optimization strategies through a combined two and three-dimensional stability analysis. A stability-based optimization, exploiting linear stability theories, is first conducted in a two-dimensional framework targeting the suppression of first-mode and Görtler instabilities. The resulting configurations are then evaluated using a three-dimensional laminar base flow including sidewall effects. The results show that two-dimensional predictions remain accurate along the top wall, whereas sidewall regions exhibit additional instability mechanisms induced by secondary flow mechanisms generating crossflow-type disturbances not captured in the planar model. Wall temperature control is shown to mitigate 1 instability growth on the top-wall and partially reduce crossflow amplification along the sidewalls. These findings demonstrate that, although two-dimensional optimization is effective for top-wall stability, three-dimensional effects must be considered for reliable nozzle design.</p>