Numerical Study of Supersonic Exhaust Systems Off Design

The operational efficiency of supersonic convergent-divergent (Laval) nozzles is highly sensitive to variations in the Nozzle Pressure Ratio (NPR) during transient flight trajectories. This paper presents a comprehensive numerical study evaluating the aerodynamic behavior, shock-induced flow separation boundaries, and thrust performance of a fixed cubic polynomial supersonic exhaust nozzle across a broad range of NPRs (2.0 to 9.0) at a target design Mach number of Me = 2.0. A parametric algorithm was developed in MATLAB to define the core nozzle contour, which was subsequently validated through high-fidelity two-dimensional (2D) axisymmetric density-based CFD simulations in ANSYS Fluent utilizing the k-ω Shear Stress Transport (SST) turbulence model. The numerical results reveal that operating below the nominal design threshold (NPR < 6.80) forces the nozzle into heavily over-expanded regimes. In these conditions, strong adverse pressure gradients trigger boundary layer separation and internal oblique shock trains, resulting in a critical thrust degradation of up to 70.17% at NPR = 2.0 due to severe pressure drag penalties and internal flow recirculation. Conversely, transitioning into the under-expanded regime (NPR = 9.0) yields a 32.06% increase in net thrust over the design baseline (26772.45 N), driven by elevated internal wall pressures and secondary external Prandtl-Meyer plume expansion. Furthermore, the systematic tracking of off-design conditions identifies a sharp operational inflection point at NPR = 4.50, above which the supersonic jet achieves complete geometric attachment and predictable quasi-linear thrust behavior. These insights offer robust predictive data and critical aerodynamic boundaries for the flight control and optimization of next-generation aerospace propulsion systems operating across transient altitudes.