Engineers who build high-speed wind tunnels face a stubborn trade-off: the contoured nozzles that produce the smoothest test flows are traditionally long and rigidly constrained by empirical design rules, while shorter minimum-length designs suffer from throat geometries that can trigger airflow separation. A new open-source study published in Advances in Aerodynamics proposes a way out, introducing a modular, parametric framework that lets designers specify the nozzle length they need and then computes an axisymmetric contour to match, without abandoning the flow uniformity that tunnel testing demands.
The framework, developed by Pan Shen, Jie Niu, Peixu Guo and Sangdi Gu, is built on the method of characteristics, or MOC, a classical technique for solving supersonic internal flows. Its central innovation is the introduction of what the authors call the Terminal Characteristic Line, or TCL, a mathematically defined boundary that splits the nozzle into two aerodynamically independent regions. Because disturbances in high-speed flow propagate only along characteristic lines, changes made to the wall downstream of the TCL cannot perturb the upstream flow field. This property, which the authors term flow-field isolation, allows each section of the nozzle to be designed, optimized or even swapped out independently, turning nozzle design into a modular exercise rather than a monolithic calculation.
The upstream portion of the nozzle, the initial expansion section, is where the framework departs most sharply from tradition. Classical MOC design methods, notably the systematic approach established by Sivells in the mid-twentieth century, relied on assumed empirical distributions of Mach number along the nozzle axis to shape the contour. The new framework abandons those assumptions entirely. Instead, it starts from Sauer’s analytical approximation of the transonic flow at the throat, treats that sonic initial line as the sole strong boundary condition, and solves the governing characteristic equations directly, treating the wall and the centerline as weak boundaries whose flow properties emerge from the computation itself. The result, according to the authors, is a high-fidelity flow-analysis engine that can evaluate any predefined wall geometry, not just contours generated by the design procedure.
Length control comes from a simple geometric parameter. The throat inlet incorporates a straight-walled initial expansion section of adjustable length and variable inclination angle. When that length is set to zero, the throat flow feeds directly into the damping section, reproducing a minimum-length nozzle as a limiting case. As the length grows, the design moves continuously toward conventional gradual-expansion contours. This single user-defined parameter therefore spans a continuous design space between the two historical extremes, unifying the compactness of minimum-length nozzles with the smooth expansion of classical contoured designs.
The downstream damping section, responsible for cancelling the waves generated during expansion and delivering uniform exit flow, borrows its computational machinery from minimum-length nozzle theory. The authors adapted Zebbiche’s grid construction and wall-point tracing method, originally developed for nozzles with a straight sonic line at the throat, and applied it for the first time to a gradual-expansion nozzle by using the TCL as the left boundary of the damping-section mesh. The complete inviscid contour emerges in three segments: the throat arc, the initial expansion section and the damping section, traced wall point by wall point until the nozzle exit is reached.
The entire procedure is implemented in Python and released on GitHub, complete with a semi-empirical turbulent boundary layer correction based on Sivells’ approach. This correction solves the momentum integral equation for the boundary layer displacement thickness along the wall, applies high-temperature recovery-factor corrections and a curvature-adjusted projection, and offsets the inviscid contour accordingly. The authors are candid about the limits of this correction: comparisons with viscous simulations showed that simply adding the displacement thickness to the inviscid contour is not sufficiently accurate, echoing earlier arguments in the literature, and they position the correction as a conceptual starting point rather than a definitive viscous solution.
Validation relied on two independent computational tools. Inviscid simulations were run in ANSYS Fluent on structured axisymmetric grids, with grid-independence studies confirming that a coarse mesh of 1200 by 60 cells already captured the outlet Mach number distribution. Viscous simulations used Eilmer, an open-source hypersonic flow solver from the University of Queensland, comparing the k-omega and Spalart-Allmaras turbulence models on a baseline Mach 6 nozzle before adopting k-omega for subsequent cases. Against these benchmarks, the MOC predictions along the initial expansion section boundaries matched CFD closely, with average Mach number relative errors below 0.5 percent and flow angle deviations below 0.1 degrees.
The flow-field isolation property of the TCL was tested directly. Two damping-section configurations, one an extended straight wall and the other the designed wave-cancellation contour, were attached to the same fixed initial expansion section of a Mach 6 nozzle. Mach numbers extracted at 29 points along the TCL were nearly identical between the two cases, with a maximum absolute error of 0.20 and an average of 0.067, supporting the claim that downstream wall modifications leave the upstream flow untouched, at least within the transition from a straight wall to a damping section that the authors examined.
Full nozzle designs at target Mach numbers of 6, 8 and 10, all sharing a throat radius of 0.05 meters and stagnation conditions of 15 megapascals and 1200 kelvin, delivered highly uniform inviscid outlets. The Mach 6 nozzle, 3.999 meters long, produced an average exit Mach number of 5.98 with a standard deviation of 0.0209 and an error of 0.33 percent against the target. The Mach 8 design, stretching 8.836 meters, averaged 7.97 with a 0.37 percent error, while the 16.715-meter Mach 10 nozzle averaged 9.90 with a 1.0 percent error and a noticeably thicker boundary layer that shrank the uniform core flow. The authors note that turbulent performance may degrade beyond Mach 10 and that larger dimensions may be needed there.
Length controllability was demonstrated with three Mach 6 nozzles sharing the same throat radius and outlet radius but spanning length-to-diameter ratios from 8.24 to 14.43. All three maintained uniform exit flow close to the design target, with errors of 0.33, 0.33 and 0.16 percent for the short, medium and long configurations respectively. The framework assumes an ideal gas with a constant specific heat ratio of 1.4, a simplification the authors stress is unreliable for high-temperature real-gas effects, though they outline how a modular subroutine could later incorporate variable thermodynamic properties. Within those limits, the study presents a flexible, computationally fast platform for preliminary nozzle design and parametric exploration, one its authors intend to complement, rather than replace, the advanced physical models emerging in modern MOC research.
Subject of Research: A modular method of characteristics framework for designing axisymmetric high-speed wind tunnel nozzles with controllable length
Article Title: A modular method of characteristics framework for controllable-length axisymmetric high-speed wind tunnel nozzle design
Article References: Shen, P., Niu, J., Guo, P., & Gu, S. (2026). A modular method of characteristics framework for controllable-length axisymmetric high-speed wind tunnel nozzle design. Advances in Aerodynamics, 8(1), Article 21. https://doi.org/10.1186/s42774-025-00244-1
Image Credits: AI Generated
DOI: 10.1186/s42774-025-00244-1
Keywords: wind tunnel, nozzle design, method of characteristics, aerodynamics, computational fluid dynamics, hypersonic flow, open-source software, boundary layer correction, flow uniformity, axisymmetric nozzle, Python, Mach number
News Source: Audrey Campbell. (October 29, 2026). Research report: A modular method of characteristics framework for controllable-length axisymmetric high-speed wind tunnel nozzle design. Scienmag.
