Theory

The method of characteristics (MoC) constructs a supersonic flow field and the wall that supports it. For nozzle design, the objective is an expansion from prescribed stagnation conditions to a specified exit pressure, with a uniform velocity direction at the outlet. The resulting contour can be incorporated into a stator blade. A related vortex-flow construction produces rotor blade sections that turn an already supersonic inlet flow.

These methods are useful for preliminary design of turbines handling real gases and liquid–vapor mixtures, including organic Rankine cycles, refrigeration, and liquefaction systems. A one-dimensional calculation establishes the required area ratio, but does not determine the wall shape or resolve the expansion waves and their reflections. The MoC supplies this two-dimensional construction.

This guide describes the planar methods implemented in TurboMoC, with particular attention to details that are condensed in the accompanying papers [Cioffi et al., 2026, Cioffi et al., 2026]. The starting point is steady, inviscid, irrotational flow with uniform stagnation enthalpy and entropy. Liquid–vapor states are evaluated under the homogeneous equilibrium model (HEM). The geometry is intended for subsequent CFD analysis. The papers compare the equilibrium MoC with barotropic HEM simulations; this does not make the geometry generator a viscous-flow or turbine-efficiency solver.

Reading the guide

Start with the notation and coordinate conventions, then follow the thermodynamic model through the characteristic equations to the geometry construction. The rotor chapter uses the same thermodynamic and Prandtl–Meyer relations as the nozzle chapters. Radial mapping is a geometric operation applied after either blade section has been generated.

• Notation and coordinate conventions
• Thermodynamic model
• Isentropic expansions and choking
• Method of characteristics
• Nozzle design
• Expansions across saturation boundaries
• Stator blade construction
• Rotor design by the vortex-flow method
• Radial mapping of blade sections

The examples provide executable designs and coordinate exports. Parameter definitions are collected in the API reference.

Note

This is a first draft. Admonitions headed Review note identify source inconsistencies or implementation details to check with the authors. Figure placeholders describe the illustrations still needed; extracted figures retain their original notation, which is reconciled in the captions.