Nomenclature

Symbol	Description	Unit
Latin Symbols
\(a\)	Speed of sound	m/s
\(c\)	Blade chord	m
\(c_\mathrm{ax}\)	Blade axial chord	m
\(C_{\mathrm{f}}\)	Diffuser skin friction coefficient	–
\(CR\)	Contraction ratio in the [Benner et al., 2006] loss model	–
\(d_{\mathrm{m}}\)	Turbine mean diameter	m
\(d_{\mathrm{s}}\)	Turbine specific diameter	–
\(F_t\)	Tangential loading parameter in the [Benner et al., 2006] loss model	–
\(H\)	Blade height	m
\(h\)	Static specific enthalpy	J/kg
\(h_{0}\)	Stagnation specific enthalpy	J/kg
\(\Delta h_{\mathrm{ts},s}\)	Total-to-static isentropic specific enthalpy change	J/kg
\(\Delta h_{\mathrm{tt},s}\)	Total-to-static isentropic specific enthalpy change	J/kg
\(\dot{m}\)	Mass flow rate	kg/s
\(N\)	Number of turbine stages	–
\(o\)	Blade opening	m
\(p\)	Static pressure	Pa
\(p_{0}\)	Stagnation pressure	Pa
\(PR\)	Pressure ratio	–
\(r_{\mathrm{h}}\)	Radius at the hub of the blades	m
\(r_{\mathrm{m}}\)	Turbine mean radius	m
\(r_{\mathrm{t}}\)	Radius at the tip of the blades	m
\(s\)	Blade pitch	m
\(s\)	Specific entropy	\(\mathrm{J/kg\,K}\)
\(s_\mathrm{ax}\)	Axial spacing between cascades	m
\(t\)	Blade thickness	m
\(T\)	Static temperature	K
\(T_{0}\)	Stagnation temperature	K
\(t_{\mathrm{max}}\)	Maximum blade thickness	m
\(t_{\mathrm{te}}\)	Trailing edge thickness	m
\(u\)	Blade velocity	m/s
\(v\)	Absolute flow velocity	m/s
\(v_{0}\)	Isentropic velocity (also known as spouting velocity)	m/s
\(w\)	Relative flow velocity	m/s
\(\dot{W}\)	Actual power output	W
\(\dot{W}_{s}\)	Isentropic power output	W
\(Y\)	Stagnation pressure loss coefficient	–
\(Z_{\mathrm{te}}\)	Passage vortex penetration depth at the trailing edge	m
Greek Symbols
\(\alpha\)	Absolute flow angle	\(^{\circ}\)
\(\beta\)	Relative flow angle	\(^{\circ}\)
\(\gamma\)	Heat capacity ratio	–
\(\delta\)	Flow deviation angle	\(^{\circ}\)
\(\delta\)	Diffuser semi-divergence angle	\(^{\circ}\)
\(\delta\)	Boundary layer thickness	m
\(\delta^*\)	Displacement thickness	m
\(\delta_{\mathrm{fl}}\)	Blade flaring angle	\(^{\circ}\)
\(\epsilon\)	Blade wedge angle	\(^{\circ}\)
\(\epsilon_{\mathrm{cl}}\)	Tip clearance gap	m
\(\eta_{ts}\)	Total-to-static isentropic efficiency	–
\(\eta_{tt}\)	Total-to-total isentropic efficiency	–
\(\theta\)	Metal angle	\(^{\circ}\)
\(\Delta \theta\)	Camber angle	\(^{\circ}\)
\(i\)	Incidence angle	\(^{\circ}\)
\(\lambda\)	Hub-to-tip radii ratio	–
\(\mu\)	Dynamic viscosity	Pa,s
\(\xi\)	Stagger angle (also known as setting angle)	\(^{\circ}\)
\(\rho\)	Density	kg/m^3
\(\tau_{\mathrm{w}}\)	Wall shear stress	Pa
\(\phi\)	Flow coefficient	–
\(\phi^2\)	Kinetic energy loss coefficient (\(\phi^2+\Delta\phi^2=1\))	–
\(\Delta\phi^2\)	Complementary of the kinetic energy loss coefficient	–
\(\chi\)	Incidence parameter in the [Moustapha et al., 1990] loss model	–
\(\chi'\)	Incidence parameter in the [Benner et al., 1997] loss model	–
\(\psi\)	Loading coefficient	–
\(\omega\)	Angular velocity	rad/s
Abbreviations
CFD	Computational Fluid Dynamics
EOS	Equation of State
IP	Interior Point (optimization algorithm)
ODE	Ordinary Differential Equation
SQP	Sequential Quadratic Programming (optimization algorithm)
Subscripts
\(0\)	Stagnation state
\(1,2,3,\ldots\)	Flow stations
\(\mathrm{cl}\)	Tip clearance
\(\mathrm{error}\)	Violation of an equality constraint
\(\mathrm{in}\)	Inlet of the cascade or turbomachine
\(\mathrm{inc}\)	Incidence loss
\(m\)	Meridional direction
\(\mathrm{mid}\)	Blade midspan
\(\mathrm{out}\)	Outlet of the cascade or turbomachine
\(\mathrm{p}\)	Profile loss
\(r\)	Radial direction
\(\mathrm{rel}\)	Relative to the rotating frame of reference
\(s\)	Refers to isentropic quantities
\(\mathrm{s}\)	Refers to specific quantitities
\(\mathrm{sec}\)	Secondary loss
\(\theta\)	Tangential direction
\(\mathrm{te}\)	Trailing edge
\(x\)	Axial direction