Optimized semi-flexible supersonic nozzle design using exact throat solution with characteristic inversion and Bézier Mach distribution

Supersonic semi-flexible wall nozzles with the ability to continuously vary the Mach number have become a key configuration for advanced supersonic wind tunnels, with their contour design typically based on the Sivells method. However, this method faces two persistent challenges in its application: (i) suboptimal flow field uniformity, particularly during continuous Mach number variation, and (ii) localized Mach number depression (“dimples”) along the nozzle axis. This paper presents an enhanced design methodology to overcome these challenges. The proposed improvements incorporate two key elements: (1) exact analytical solutions for transonic flow to replace the conventional source-flow approximation used in the Sivells method at the throat section, and (2) a Bézier-optimized characteristic method for expansion contour design. Experimental results demonstrated that the improved design method significantly enhanced performance even in a small 0.3 m-scale wind tunnel: flow field uniformity reached 0.0010–0.0053 in fixed-geometry mode (Ma = 1.2–3.0), while dynamic flow uniformity was maintained at 0.002–0.010 in continuous morphing mode (Ma = 1.5–3.0). These research outcomes provide a reliable technical approach for designing supersonic wind tunnel semi-flexible nozzles.