Notation and coordinate conventions

The same thermodynamic symbols are used throughout the guide. The rotor chapter uses relative velocity, while the nozzle and stator chapters use stationary-frame velocity. Geometric symbols are kept distinct from thermodynamic quantities.

Flow and thermodynamic quantities

Symbol	Definition	Units
\(p\), \(T\), \(\rho\)	Static pressure, temperature, and density	Pa, K, kg m\(^{-3}\)
\(h\), \(s\)	Specific enthalpy and entropy	J kg\(^{-1}\), J kg\(^{-1}\) K\(^{-1}\)
\(h_0\), \(s_0\), \(p_0\), \(T_0\)	Stagnation properties in the frame under consideration	Corresponding SI units
\(V\)	Stationary-frame speed, \(\sqrt{u^2+v^2}\)	m s\(^{-1}\)
\(u\), \(v\)	Velocity components in the nozzle \(x,y\) frame	m s\(^{-1}\)
\(W\)	Rotor-relative speed	m s\(^{-1}\)
\(a\)	Local equilibrium speed of sound	m s\(^{-1}\)
\(M\)	Mach number, \(V/a\) or \(W/a\)	Dimensionless
\(Q\)	Vapor mass fraction (quality)	Dimensionless
\(\alpha_\ell\), \(\alpha_v\)	Liquid and vapor volume fractions	Dimensionless
\(c_p\), \(c_v\)	Specific heat capacities	J kg\(^{-1}\) K\(^{-1}\)
\(\gamma\)	Heat-capacity ratio \(c_p/c_v\)	Dimensionless
\(G\)	Mass flux, \(\rho V\)	kg m\(^{-2}\) s\(^{-1}\)
\(\dot m'\)	Mass flow rate per unit span	kg m\(^{-1}\) s\(^{-1}\)
\(\theta\)	Nozzle flow angle from the positive \(x\) axis	rad
\(\mu\)	Mach angle, \(\sin^{-1}(1/M)\)	rad
\(\nu\)	Generalized Prandtl–Meyer angle	rad
\(\beta\)	Rotor-relative flow angle from its axial direction	rad
\(\lambda_\pm\)	Characteristic slopes, \(\tan(\theta\pm\mu)\)	Dimensionless

Subscripts \(\ell\) and \(v\) denote saturated liquid and vapor. Subscripts \(\mathrm{in}\) and \(\mathrm{out}\) denote inlet and outlet. A subscript \(\mathrm{ref}\) denotes the reference state of the Prandtl–Meyer integral; it can be a smooth sonic state or the downstream state of a flashing transition. For the latter, \(V_\mathrm{ref}\) and \(a_\mathrm{ref}\) need not be equal.

Geometry and frames

Symbol	Definition
\(x,y\)	Nozzle streamwise and transverse coordinates; the symmetry axis is \(y=0\)
\(y_t\), \(w_t\)	Throat half-height and full width, \(w_t=2y_t\)
\(R_t\), \(R_d\)	Radii used for the initial-line and downstream wall constructions
\(t_b\)	Blade pitch
\(c\), \(c_a\)	Chord and axial chord
\(r_\mathrm{TE}\)	Trailing-edge radius
\(\chi\)	Stator outlet metal angle used in the pitch relation
\(R\), \(R_\mathrm{ref}\)	Vortex radius and its dimensional reference scale
\(\widehat R\)	Normalized vortex radius \(R/R_\mathrm{ref}\)
\(\zeta\)	Rotation parameter for a rotor circular arc
\(\xi,\eta\)	Chordwise and pitchwise coordinates passed to the radial map
\(r,\varphi\)	Radius and azimuth after radial mapping

The nozzle upper wall has positive \(y\), with positive \(\theta\) turning away from the axis. The characteristic family \(C_+\) has slope \(\tan(\theta+\mu)\), and \(C_-\) has slope \(\tan(\theta-\mu)\). These definitions determine the signs of the compatibility relations.

Stator coordinates are rotated during assembly: the returned profile has its chordwise extent mainly along its local \(y\) direction and its pitch along \(x\). The rotor profile instead uses local \(x\) as the axial direction and \(y\) as the pitchwise direction. Neither frame should be inferred from the orientation of a plot. The radial mapping explicitly selects the relevant axes.

Relation to package names and units

Theory quantity	Package argument or result	Convention
\(p_0,T_0,Q_0\)	P0, T0, Q0	Pa, K, and a fraction from 0 to 1
\(p_\mathrm{out}\)	p_back	Pa; a target for designing a new nozzle
\(M_\mathrm{out}\)	Noz_Mach, design_Noz_Mach	Dimensionless
\(y_t,R_t,R_d\)	y_t, rho_t, rho_d	Meters; rho_t and rho_d are geometric radii, not densities
\(\theta,\mu\)	Point fields theta, alpha	Radians; the code’s alpha is a Mach angle
\(\chi\)	metal_angle_out	Degrees
\(t_b\)	pitch	Millimeters for stator geometry
\(\beta_\mathrm{in},\beta_\mathrm{out}\)	beta_inlet, beta_outlet	Degrees at the rotor API boundary
\(R_\mathrm{ref}\)	r_star	Meters when specified; otherwise rotor coordinates are dimensionless

design_nozzle returns coordinates in meters. Stator geometry functions accept these coordinates, convert them internally, and return millimeters. Rotor coordinates are dimensionless unless r_star is supplied. Radial mapping requires its input coordinates and radii to use compatible length units.

In the source papers, \(s\) can denote pitch, \(\alpha\) can denote Mach angle, and \(M^*\) can denote velocity normalized by the sonic speed. Here \(s\) is reserved for entropy, \(\mu\) for Mach angle, and normalized velocity is written explicitly.