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TRACE User Guide
TRACE Version 9.6.1
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TRACE User Guide
TRACE Version 9.6.1
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Aerodynamic damping for a structural mode.
\[ \delta = -\frac{\Re W_\mathrm{cyc}}{2E} \]
Aerodynamic excitation (in the context of forced response analysis).
\[ \epsilon = \frac{|W_\mathrm{cyc}|}{2E} \]
Aerodynamic excitation (in the context of forced response analysis) scaled by the AmplificationFactor used during the mapping process of the eigenmode displacements. The scaled excitation is the aerodynamic excitation with respect to the unscaled eigenmode displacements.
\[ \epsilon_{scaled} = \epsilon \cdot \mathrm{AmplificationFactor} \]
Aerodynamic stiffness for a structural mode.
\[ \kappa = \frac{\Im W_\mathrm{cyc}}{2E} \]
Aerodynamic stimulus.
\[ S = W_\mathrm{cyc} / \int_\Gamma |\Psi^H \widehat{(p\vec n)}_0| dS \]
Aerodynamic stimulus with respect to the signed mean aerodynamic forces.
\[ S_{\mathrm{signed}} = W_\mathrm{cyc} / |\int_\Gamma \Psi^H \widehat{(p\vec n)}_0 dS| \]
Work per cycle exerted by the aerodynamic forces for an eigenmode \(\Psi\).
\[ W_\mathrm{cyc} = -i\pi \int_\Gamma \Psi^H \widehat{(p \vec n)}_\omega dS \]
\[ [W_\mathrm{cyc}] = \N\m \]
\(L^1\)-norm of the work per cycle and area.
\[ \left\|dW_\mathrm{cyc}/dS \right\|_{L^1} = -i\pi \int_\Gamma | \Psi^H \widehat{(p \vec n)}_\omega | dS \]
\[ [\left\|dW_\mathrm{cyc}/dS \right\|_{L^1}] = \N\m \]
Part of the modal work per cycle which is due to the mean pressure.
\[ W_\mathrm{cyc,mp} = -i\pi \int_\Gamma \widehat p_0 \Psi^H \widehat{\vec n}_\omega dS \]
\[ [W_\mathrm{cyc,mp}] = \N\m \]
Work per cycle and area exerted by the aerodynamic forces for an eigenmode \(\Psi\).
\[ dW_\mathrm{cyc}/dS = -i\pi \Psi^H \widehat{(p \vec n)}_\omega \]
\[ [dW_\mathrm{cyc}/dS] = \kg/\s^{2} \]
\[ - \dot{m} K_{\abs} \]
\[ \kg \, \m^2 \, \s^{-2} \]
Relation between outflow and inflow throughflow surface area.
\[ A_{\outflow}/A_{\inflow} \]
\[ C_\mathrm{ars} = A_\outflow \, \Omega^{2}/(4 \pi^2) \]
\[ [C_\mathrm{ars}] = \m^2/\s^2 \]
Diffusion coefficient - loading criterion for compressors. \(s/c\) is the mean pitch/chord ratio (hub, tip) of the current blade row.
\[ DF = 1 - v_{\outflow}/v_{\inflow} + |v_{\theta,\outflow} - v_{\theta,\inflow}| /(2 \, v_{\inflow})* s/c \]
\[ C_{\Delta s} = \log(p_{\tot,\abs,\outflow,\is}/p_{\tot,\abs,\inflow})-\log(p_{\tot,\abs,\outflow}/p_{\tot,\abs,\inflow}) \]
Axial component of flow coefficient.
\[ \Phi_x = (v_{x,\inflow}+v_{x,\outflow})/|2\Omega r_{\mean}| \]
Loss coefficient after Scholz (1).
\[ \omega_{\Scholz 1} = \log(p_{\outflow,\is}/p_{\outflow})/\log(p_{\tot,\outflow}/p_{\inflow}) \]
Loss coefficient after Scholz (2). \(h_{\outflow,\is}\) is computed from the absolute total pressure and enthalpy at inlet and the static pressure at outlet.
\[ \omega_{\Scholz 2} = (h_{\outflow}-h_{\outflow,\is})/(\tfrac 1 2v_{\outflow}^2) \]
\[ c_p = (p \, - \, p_{\Reference})/(p_{\tot,\Reference}-p_{\Reference}) = \frac{\Delta p} {q_{\Reference}} \]
Total pressure loss coefficient in relative frame of reference based on dynamic pressure. Compressors: reference is inlet, turbines: reference is outlet
\[ \omega = \begin{cases} \frac{p_{\tot,\inflow}\ -\ p_{\tot,\outflow}}{p_{\tot,\Reference} -\ p_{\Reference}} = \frac{\Delta p_{\tot}}{q_{\Reference}} & \text{in general} \\ \frac{p_{\tot,\outflow}\ -\ p_{\tot,\inflow}} {p_{\tot,\Reference}\ -\ p_{\Reference}} & \text{for compressors (stages, components or analysis volumes)} \end{cases} \]
Total pressure loss coefficient in absolute frame of reference based on dynamic pressure. Compressors: reference is inlet, turbines: reference is outlet
\[ \omega_{\abs} = \begin{cases} \frac{p_{\tot, \abs, \inflow}\ -\ p_{\tot, \abs,\outflow}}{p_{\tot, \abs, \Reference}\ -\ p_{\Reference}} = \frac{\Delta p_{\tot}}{q_{\Reference}} & \text{in general} \\ \frac{p_{\tot, \abs, \outflow}\ -\ p_{\tot, \abs,\inflow}}{p_{\tot, \abs, \Reference}\ -\ p_{\Reference}} & \text{for compressors (stages, components or analysis volumes)} \end{cases} \]
Total pressure loss coefficient in relative frame of reference normalized using the theoretic dynamic pressure.
\[ \omega_{\mathrm{theoretic}} = \frac{p_{\tot,\inflow}\ - p_{\tot,\loc}}{p_{\tot,\inflow}\ - p_{\outflow}} \]
Total pressure loss coefficient in relative frame of reference based on dynamic pressure. Compressors: reference is outlet, turbines: reference is inlet
\[ \omega_{\tot} = \begin{cases} \frac{p_{\tot, \inflow}\ -\ p_{\tot,\outflow}}{p_{\tot, \Reference}} = \frac{\Delta p_{\tot}}{p_{\tot, \Reference}} & \text{in general} \\ \frac{p_{\tot,\outflow}\ -\ p_{\tot,\inflow}}{p_{\tot,\Reference}} & \text{for compressors (stages, components or analysis volumes)} \end{cases} \]
Total pressure loss coefficient in absolute frame of reference based on total pressure. Compressors: reference is outlet, turbines: reference is inlet
\[ \omega_{\tot, \abs} = \begin{cases} \frac{p_{\tot,\abs,\inflow}\ -\ p_{\tot,\abs,\outflow}}{p_{\tot,\abs,\Reference}} = \frac{\Delta p_{\tot}}{p_{\tot, \abs, \Reference}} & \text{in general} \\ \frac{p_{\tot,\abs,\outflow}\ -\ p_{\tot,\abs,\inflow}}{p_{\tot,\abs,\Reference}} & \text{for compressors (stages, components or analysis volumes)} \end{cases} \]
Isentropic total pressure loss coefficient in relative frame of reference based on dynamic pressure. Compressors: Reference is inlet, turbines: Reference is outlet
\[ \zeta_{\tot, \is} = \frac{p_{\tot,\outflow,\is}\ -\ p_{\tot, \outflow}}{p_{\tot,\Reference}\ -\ p_{\Reference}} = \frac{\Delta p_{\tot, \is}}{q_{\Reference}} \]
Isentropic total pressure loss coefficient in absolute frame of reference based on dynamic pressure. Compressors: reference is inlet, turbines: reference is outlet
\[ \zeta_{\tot, \abs, \is} = \frac{p_{\tot,\abs,\outflow,\is}\ -\ p_{\tot,\abs,\outflow}} {p_{\tot,\abs,\Reference\ -\ p_{\Reference}}} = \frac{\Delta p_{\tot, \abs, \is}}{q_{\abs,\Reference}} \]
Total pressure loss coefficient in relative frame of reference based on dynamic pressure. Compressors: Reference is inlet, turbines: Reference is outlet
\[ \zeta_{\tot} = \frac{p_{\tot,\inflow}\ -\ p_{\tot,\outflow}}{p_{\tot,\Reference}\ -\ p_{\Reference}} = \frac{\Delta p_{\tot}}{q_{\Reference}} \]
Total pressure loss coefficient in absolute frame of reference based on dynamic pressure. Compressors: reference is inlet, turbines: reference is outlet
\[ \zeta_{\tot, \abs} = \frac{p_{\tot,\abs,\inflow}\ -\ p_{\tot,\abs,\outflow}}{p_{\tot,\abs,\Reference}\ -\ p_{\Reference}} = \frac{\Delta p_{\tot, \abs}}{q_{\abs,\Reference}} \]
Wall skin friction coefficient.
\[ c_f = \tau_\mathrm{wall}/(p_{\tot,\Reference}-p_{\Reference}) \]
\[ c_{v,\mean} = \frac{|\Omega| r_{\mean}}{\sqrt{2 |w_{\htot}|}} \]
\[ c_{v,\mathrm{ratio}} = \sqrt{\frac{1}{2}|\Omega| r_{\rotor}/ |v_{\theta,\abs,\outflow,\rotor} - v_{\theta,\abs,\inflow,\rotor}|} \]
Work coefficient or enthalpy rise coefficient. \(r_{\mean}\) is the arithmetic mean of the mass averaged radius at in- and outlet.
\[ \psi = w/(\Omega r_{\mean})^2 \]
Alternative definition for work coefficient.
\[ \psi_2 = 2\psi \]
Isentropic work coefficient or isentropic enthalpy rise coefficient.
\[ \psi_{\is} = \eta_{\is,\noleak} \psi \]
Alternative definition for isentropic work coefficient.
\[ \psi_{\is,2} = 2 \psi_{\is} \]
Blade loading criterion for turbines.
\[ C_{\Zweifel} = \Zweifel \, n_{\blades} \, c_{\x} \]
\[ [C_{\Zweifel}] = \m^2 \]
Turbomachinary coordinate which consists of the mean radius and local theta coordinate.
\[ r_{mean}\Theta = (r_{\LE}+r_{\TE})/2\cdot\Theta \]
\[ [r_{mean}\Theta] = \m \]
Simple blade loading criterion for compressors.
\[ DH = v_{\outflow}/v_{\inflow} \]
Magnitude of grid point displacement (difference of coordinates)
\[ \delta xyz = \sqrt{ \Re \{\delta x\}^2 + \Re \{\delta y\}^2 + \Re \{\delta z\}^2 + \Im \{\delta x\}^2 + \Im \{\delta y\}^2 + \Im \{\delta z\}^2 } \]
\[ [\delta xyz] = \m \]
Dimensionless wall distance computed from the distance \(d\) to the nearest viscous wall. The friction velocity is given by \(u_\tau = \sqrt{\tau_w / \rho}\) with the wall shear stress \(\tau_w\).
\[ y^+ = \frac{\rho u_\tau}{\mu} d \]
Isentropic efficiency defined for rotors, stages, groups, turbomachines.
\[ \eta_{\is} = \begin{cases} \frac{\Delta h_{\tot,\is}}{\Delta h_{\tot}} = \frac{h_{\tot,\abs,\outflow,\is}\ -\ h_{\tot,\abs,\inflow}}{h_{\tot,\abs,\outflow}\ -\ h_{\tot,\abs,\inflow}} & \text{for compressors} \\ \frac{\Delta h_{\tot}}{\Delta h_{\tot,\is}} =\frac{h_{\tot,\abs,\outflow}\ -\ h_{\tot,\abs,\inflow}}{h_{\tot,\abs,\outflow,\is}\ -\ h_{\tot,\abs,\inflow}} & \text{for turbines} \end{cases} \]
EfficiencyIsentropicHWoLeak Isentropic efficiency defined for rotors, stages, groups, turbomachines.
\[ \eta_{\is,\leak} = \begin{cases} \frac{P_{\htot,\is,\leak}}{P_{\htot,\leak}} & \text{for compressors} \\ \frac{P_{\htot,\leak}}{P_{\htot,\is,\leak}} & \text{for turbines} \end{cases} \]
\[ \eta_{\is,\mleak} = \begin{cases} \frac{P_{\htot,\is,\mleak}}{P_{\htot,\mleak}} & \text{for compressors} \\ \frac{P_{\htot,\mleak}}{P_{\htot,\is,\mleak}} & \text{for turbines} \end{cases} \]
Isentropic efficiency defined for rotors, stages, groups, turbomachines.
\[ \eta_{\is,\noleak} = \begin{cases} \frac{\Delta h_{\tot,\is}}{\Delta h_{\tot}} = \frac{h_{\tot,\abs,\outflow,\is}\ -\ h_{\tot,\abs,\inflow}}{h_{\tot,\abs,\outflow}\ -\ h_{\tot,\abs,\inflow}} & \text{for compressors} \\ \frac{\Delta h_{\tot}}{\Delta h_{\tot,\is}} =\frac{(h_{\tot,\abs,\outflow} - h_{\tot,\abs,\inflow})}{(h_{\tot,\abs,\outflow,\is} - h_{\tot,\abs,\inflow})} & \text{for turbines} \end{cases} \]
Isentropic efficiency based on swirl (rotors, stages, groups, turbomachines).
\[ \eta_{\is,\swirl} =\begin{cases} \frac{P_{\htot,\is,\leak}}{P_{\swirl}} & \text{for compressors} \\ \frac{P_{\swirl}}{P_{\htot,\is,\leak}} & \text{for turbines} \end{cases} \]
Isentropic efficiency including the total pressure loss in the diffusor.
\[ \eta_{\is,\noleak,\Diff} = \frac{\Delta h_{\tot}}{\Delta h_{\tot,\is}} = \frac{h_{\tot,\inflow}\ -\ h_{\tot,\outflow}} {h_{\tot,\inflow}\ -\ h_{\tot,\outflow,\is}} \]
with \(h_{\tot,\outflow,\is} = f\left(p_{\tot, \abs, \Diff,\outflow}\right)\)Isentropic efficiency including the total pressure loss in the diffusor.
\[ \eta_{\is,\leak,\Diff} = \frac{P_{\htot,\leak}}{P_{\htot,\is,\leak}} \]
with \(P_{\htot,\outflow,\is} = f\left(p_{\tot, \abs, \Diff,\outflow}\right)\)Polytropic efficiency.
\[ \eta_{\pol} = \begin{cases} \frac{\kappa-1}{\kappa} \log\left(\frac{p_{\tot,\abs,\outflow}}{p_{\tot,\abs,\inflow}}\right) / \log\left(\frac{h_{\tot,\abs,\outflow}}{h_{\tot,\abs,\inflow}}\right) & \text{for compressors} \\ \frac{\kappa}{\kappa-1} \log\left(\frac{h_{\tot,\abs,\outflow}}{h_{\tot,\abs,\inflow}}\right) / \log\left(\frac{p_{\tot,\abs,\outflow}}{p_{\tot,\abs,\inflow}}\right) & \text{for turbines} \end{cases} \]
\[ \begin{cases} \rho \left( e + \frac 1 2 v^2\right) & \text{for translational configurations}\\ \rho \left( e + \frac 1 2 (v^2 - (r\Omega)^2\right) & \text{for rotational configurations} \end{cases} \]
\[ \N \m^{-2} \]
\[ \rho \left( e + \frac 1 2 v_\abs^2\right) \]
\[ \N \m^{-2} \]
\[ h = e + \frac p {\rho} \]
\[ [h] = \m^2/\s^2 \]
Relative stagnation enthalpy.
\[ h_{\tot} = h + \frac 1 2 v^2 \]
\[ [h_{\tot}] = \m^2/\s^2 \]
Absolute stagnation enthalpy.
\[ h_{\tot,\abs} = h + \frac 1 2 v_\abs^2 \]
\[ [h_{\tot,\abs}] = \m^2/\s^2 \]
Relative total enthalpy.
\[ [h_\tot] = \m^2/\s^2 \]
Entropy of fluid.
\[ s = c_v(\log(\kappa p)-\kappa\log(\rho)) \]
\[ [s] = \m^2/(\K\s^2) \]
\[ v_{\theta, \abs} / (r\Omega) \]
Relative pitchwise flow turning based on \(\alpha_{\Theta,\MTU}\). Turbine: no absolute value !
\[ |\alpha^*_{\Theta,\outflow} - \alpha^*_{\Theta,\inflow}| \]
\[ \degree \]
Absolute pitchwise flow turning based on \(\alpha_{\Theta,\MTU,\abs}\). Turbine: no absolute value !
\[ |\alpha^*_{\Theta,\abs,\outflow} - \alpha^*_{\Theta,\abs,\inflow}| \]
\[ \degree \]
Relative pitchwise flow turning based on \(\alpha_{\z}\). Turbine: no absolute value !
\[ |\alpha_{\z,\outflow} - \alpha_{\z,\inflow}| \]
\[ \degree \]
Absolute pitchwise flow turning based on \(\alpha_{\z,\abs}\). Turbine: no absolute value !
\[ |\alpha_{\z,\abs,out} - \alpha_{\z,\abs,in}| \]
\[ \degree \]
Radial flow turning based on \(\alpha_r\).
\[ |\alpha_{r,out} - \alpha_{r,in}| \]
\[ \degree \]
Pitchwise flow turning in relative frame of reference. Turbine: no absolute value !
\[ |\alpha_{\Theta,out} - \alpha_{\Theta,in}| \]
\[ \degree \]
Pitchwise flow turning in absolute frame of reference. Turbine: no absolute value !
\[ |\alpha_{\Theta,\abs,out} - \alpha_{\Theta,\abs,in}| \]
\[ \degree \]
Specific work of a rotor divided by the specific work of a contol volume (stage, component, volume)
\[ w_{\rotor_i} / w_{\mathrm{comp}} \]
Velocity of rotating frame of reference at the reference radius, e.g., the mass averaged radius.
\[ \Omega r \]
\[ \m/\s \]
Frequency shift of an eigenmode due to aerodynamic stiffness \(\kappa\).
\[ \delta f = \frac{\kappa f}{2\pi} \]
\[ [\delta f] = 1/\s \]
Mach number in relative frame of reference.
\[ M = v / a \]
Mach number in absolute frame of reference.
\[ M_{\abs} = v_{\abs} / a \]
Isentropic Mach number.
\[ M_{\is} = \sqrt{\frac{2}{\kappa-1} \, [(\frac{p_{\tot,\Reference}}{p})^\frac{\kappa-1}{\kappa}-1]} \]
with \(T_{\tot,\Reference}=T_{\rot} + \frac{v^2}{2c_p}\) and \(p_{\tot,\Reference}=p_{\Reference}(\frac{T_{\tot,\Reference}}{T_{\Reference}})^\frac{\kappa}{\kappa-1}\).Meridional Mach number.
\[ M_{\mer} = \frac{v_\mer}{a} \]
\[ \dot{m} = - \rho \mathbf{v} \cdot \vec n \, dS \]
\[ [\dot{m}] = \kg/\s \]
Mass flow corrected via ISA conditions: \(p_{\Reference}\) = 101325 Pa, \(T_{\Reference}\) = 288.15 K.
\[ \dot{m}_\mathrm{ISA} = \dot{m} \, \sqrt{\frac {T_{\tot,\abs}}{T_{\Reference}}} \, \frac{p_{\Reference}}{p_{\tot,\abs}} \]
\[ [\dot{m}_\mathrm{ISA}] = \kg/\s \]
Mass flow defect.
\[ \zeta = \frac{|\dot{m}_{\inflow}|\ -\ |\dot{m}_{\outflow}|}{|\dot{m}_{\inflow}|} \]
Reduced mass flow rate.
\[ \dot{m}_{\red} = \dot{m} \, \frac{\sqrt{T_{\tot}}}{p_{\tot}} \]
\[ [\dot{m}_{\red}] = \m\s\K^{\frac{1}{2}} \]
\[ [|\dot{m}|] = \kg/\s \]
Mixing loss computed as total pressure loss. \(p_\tot\) and \(p\) refer to the average reference value under consideration (e.g. flux average), \(\overline{p}_\tot^W\) is the so-called work average, cf. [8].
\[ \omega_{mix} = \frac{\overline{p}_\tot^W\ -\ p_\tot}{p_\tot\ -\ p} \]
Meridional velocity density ratio.
\[ MVDR = (\rho v_{\mer})_{\outflow} \, / \, (\rho v_{\mer})_{\inflow} \]
Radial component of the momentum density.
\[ \rho v_r \]
\[ \kg \cdot \m^{-2}\cdot \s^{-1} \]
Circumferential component of the relative momentum density.
\[ \rho v_\theta \]
\[ \kg \cdot \m^{-2}\cdot \s^{-1} \]
Circumferential component of the absolute momentum density.
\[ \rho v_{\theta,\abs} \]
\[ \kg \cdot \m^{-2}\cdot \s^{-1} \]
Axial component of the momentum density.
\[ \rho v_x \]
\[ \kg \cdot \m^{-2}\cdot \s^{-1} \]
Number of segments across annulus.
Pitch to chord ratio for turbomachines and linear cascades.
\[ \frac{s}{c} \]
Overall power output accounting for all bleeds and leaks.
\[ P_{\htot,\leak} = \dot{m}_{\outflow} \, h_{\tot,\abs,\outflow} \, - \, \dot{m}_{\inflow} \,h_{\tot,\abs,\inflow} \, + \, \dot{m}_{\bleed} \,h_{\tot,\abs,\bleed} \, - \, \dot{m}_{\cool} \,h_{\tot,\abs,\cool} \]
\[ [P_{\htot,\leak}] = \W \]
Overall power output accounting for all main bleeds and leaks.
\[ P_{\htot,\mleak} = \dot{m}_{\outflow} \, h_{\tot,\abs,\outflow} \, - \, \dot{m}_{\inflow} \,h_{\tot,\abs,\inflow} \, + \, \dot{m}_{\mbleed} \,h_{\tot,\abs,\mbleed} \, - \, \dot{m}_{\mcool} \,h_{\tot,\abs,\mcool} \]
\[ [P_{\htot,\mleak}] = \W \]
Overall power output based on main inlets and outlets only.
\[ P_{\htot,\noleak} = \dot{m}_{\outflow} \, h_{\tot,\abs,\outflow} \, - \, \dot{m}_{\inflow} \, h_{\tot,\abs,\inflow} \]
\[ [P_{\htot,\noleak}] = \W \]
Overall {isentropic} power output accounting for all bleeds and leaks.
\[ P_{\htot,\is,\leak} = (\dot{m}_{\outflow} \, - \, \dot{m}_{\cool}) \, h_{\tot,\is,\outflow} \, - \, \dot{m}_{\inflow} \, h_{\tot,\inflow} \, + \, \dot{m}_{\bleed} \, h_{\tot,\is, \bleed} \, + \, \dot{m}_{\cool} \, h_{\tot,\is, \cool} \]
\[ [P_{\htot,\is,\leak}] = \W \]
Overall {isentropic} power output accounting for all main bleeds and main leaks.
\[ P_{\htot,\is,\mleak} = (\dot{m}_{\outflow} \, - \, \dot{m}_{\cool}) \, h_{\tot,\abs,\is,\outflow} \, - \, \dot{m}_{\inflow} \, h_{\tot,\abs,\inflow} \, + \, \dot{m}_{\mbleed} \, h_{\tot,\abs,\is, \mbleed} \, + \, \dot{m}_{\mcool} \, h_{\tot,\abs,\is, \mcool} \]
\[ [P_{\htot,\is,\mleak}] = \W \]
Overall isentropic power output {without} accounting for bleeds and leaks.
\[ P_{\htot,\is,\noleak} = \dot{m}_{\outflow} \, h_{\tot,\is,\outflow} \, - \, \dot{m}_{\inflow} \, h_{\tot,\inflow} \]
\[ [P_{\htot,\is,\noleak}] = \W \]
\[ P_{\red,\leak} = \frac{P_{\htot,\leak}}{p_{\tot,\inflow,\abs}\sqrt{T_{\tot,\inflow,\abs}}} \]
\[ [P_{\red,\leak}] = \W/(\Pa \sqrt{\K}) \]
\[ P_{\swirl} = \Omega (\dot{m}_{\outflow} K_{\outflow, \abs} + \dot{m}_{\inflow} K_{\inflow, \abs}) \]
\[ [P_{\swirl}] = \W \]
Normalized pressure.
\[ p^* = \frac{p}{p_{\tot,\inflow}} \]
Static pressure ratio.
\[ \pi = \begin{cases} \frac {p_{\outflow}} {p_{\inflow}} & \text{for compressors} \\ \frac {p_{\inflow}} {p_{\outflow}} & \text{for turbines} \end{cases} \]
Stagnation or total pressure in {relative} frame of reference.
\[ p_{\tot} = p\left(1+\frac{\kappa-1}{2}M^2\right)^\frac{\kappa}{\kappa-1} \]
\[ [p_{\tot}] = \Pa \]
Stagnation or total pressure in {absolute} frame of reference.
\[ p_{\tot,\abs} = p\left(1+\frac{\kappa-1}{2}M_\abs^2\right)^\frac{\kappa}{\kappa-1} \]
\[ [p_{\tot,\abs}] = \Pa \]
Stagnation or total pressure ratio in {absolute} frame of reference.
\[ \pi_{\tot,\abs} = \begin{cases} \frac{p_{\tot,\outflow,\abs}}{p_{\tot,\inflow,\abs}} & \text{for compressors} \\ \frac{p_{\tot,\inflow,\abs}}{p_{\tot,\outflow,\abs}} & \text{for turbines} \end{cases} \]
Stagnation or total pressure ratio in {relative} frame of reference.
\[ \pi_{\tot} = \begin{cases} \frac{p_{\tot,\outflow}}{p_{\tot,\inflow}} & \text{for compressors} \\ \frac{p_{\tot,\outflow}}{p_{\tot,\inflow}} & \text{for turbines} \end{cases} \]
Relative total pressure.
\[ p_{\tot,\rot} = p\left(\frac{T_{\tot,\rot}}{T}\right)^{\frac{\kappa}{\kappa - 1}} \]
\[ [p_{\tot,\rot}] = \Pa \]
Static to total pressure ratio in absolute frame of reference.
\[ \pi_{\mathrm{s},\tot,\abs} = \frac{p}{p_{\tot,\abs}} \]
\[ \rho_h = \frac{h_{\rotor,\outflow}\ -\ h_{\rotor,\inflow}}{h_{\outflow}\ -\ h_{\inflow}} \]
\[ \rho_p = \frac{p_{\rotor,\outflow}\ -\ p_{\rotor,\inflow}}{p_{\outflow}\ -\ p_{\inflow}} \]
\[ \rho_v = -\frac{v_{\theta,\rotor,\outflow}\ +\ v_{\theta,\rotor,\inflow}}{\Omega \left(r_{\outflow}\ +\ r_{\inflow}\right)} \]
Relative arc length of a point on a spatial curve.
\[ s_{\rel} = \frac{s}{S} \]
where \(S\) denotes whole arc lengthRelative channel height in S3 direction.
\[ H_{\rel} = \frac{h}{H} \]
Coordinate \(x_c\) in chordwise direction.
\[ c^* = \frac{x_c \ - \ x_{c,\LE}}{c} \]
Coordinate \(x - x_{\LE}\) in axial direction normalised with axial chord length \(c_{x}\).
\[ c_{\x}^* = \frac{x \, - \, x_{\LE}}{c_{\x}} \]
Relative chord in terms of radial coordinate \(r\).
\[ c_{r}^* = \frac{r \, \ -\ \, r_{\LE}}{r_{\TE} \, \ -\ \, r_{\LE}} \]
Relative radial coordinate.
\[ r_{\rel} = \frac{r\ -\ r_{\min{}}}{r_{\max{}}\ -\ r_{\min{}}} \]
Relative mass flow rate across blade span - h=0: hub, h=H: shroud
\[ \dot{m}_{\rel} = \frac{\dot{m}\left(h\right)}{\dot{m}\left(H\right)} \]
With \(\Delta s_{\mer} = \sum_\LE\sqrt{\Delta x^2 + \Delta r^2} = \sum_\LE\sqrt{(x - x_{\LE})^2 + (r - r_{\LE})^2}\) and \(S_{\mer} = \sum_\LE^\TE \Delta s_{\mer}\)
\[ s_{\mer}^* = \frac{\Delta s_{\mer}}{S_{\mer}} \]
Relative pressure \(p/p_{\steady}\) in terms of the pressure field from the steady solution \(p_{\steady}\).
\[ p_{\rel} = \frac{p}{p_{\steady}} \]
Relative temperature \(T/T_{\steady}\) in terms of the temperature field from the steady solution \(T_{\steady}\).
\[ T_{\rel} = \frac{T}{T_{\steady}} \]
Relative meridional velocity \(v_{\mer}/v_{\mer,\steady}\) in terms of the meridional velocity field from the steady solution \(v_{\mer,\steady}\).
\[ v_{\mer,\rel} = \frac{v_{\mer}}{v_{\mer,\steady}} \]
Reynolds number based on inflow conditions, used primarily in compressors.
\[ Re = Re_\inflow = \frac{\rho_{\inflow} \ c\ v_{\inflow}}{\mu_{\inflow}} \]
Reynolds number based on outflow conditions, used primarily in turbines.
\[ Re = Re_\outflow = \frac{\rho_{\outflow} \ c\ v_{\outflow}}{\mu_{\outflow}} \]
Frame velocity of rotating system.
\[ r \Omega \]
\[ \m/\s \]
Rotational speed of relative system in rounds per second.
\[ \frac \Omega{2\pi} \]
\[ 1/\s \]
Reduced wheel speed of rotor.
\[ \frac{\Omega}{2 \pi \sqrt{T_{\tot,\abs,\inflow}}} \]
\[ 1/(s \sqrt{\K}) \]
Boundary layer shape factor: displacement thickness / momentum thickness.
\[ H_{\12} = \frac{\delta_\1}{\delta_\2} \]
Boundary layer shape factor: energy thickness / momentum thickness.
\[ H_{\32} = \frac{\delta_\3}{\delta_\2} \]
In POST and in the 2D-TRACE.cgns of the unstructured solver of TRACE, the absolute value of the shear stress will be used. In structured TRACE, the direction is determined from the x-component of the tangential velocity.
\[ \tau_w = \mu\frac{\partial u_{||}}{\partial n} \]
\[ [\tau_w] = Pa \]
Ratio of specific heats or heat capacity ratio.
\[ \kappa = \frac{c_p}{c_v} \]
Specific work of turbomachine or turbomachine component.
\[ w = \frac{P_{\htot,\leak}}{\left|\dot{\m}_\inflow\right|} \]
\[ [w] = \J/\kg \]
Specific work based on swirl.
\[ w_{\swirl} = \Omega\left({K_{\abs,\inflow}}- {K_{\abs,\outflow}}\right) \]
\[ [w_{\swirl}] = \J/\kg \]
Flow swirl in absolute frame of reference.
\[ K_{\abs} = r v_{\theta,\abs} \]
\[ [K_{\abs}] = \m^2/\s \]
\[ T_{\rot} = T_{\tot}-\frac{v^2}{2c_p} \]
\[ [T_{\rot}] = \K \]
Total temperature ratio in absolute frame of reference.
\[ \tau_{\abs} = \begin{cases} \frac{T_{\tot, \abs, \outflow}} {T_{\tot, \abs, \inflow}} & \text{for compressors} \\ \frac{T_{\tot, \abs, \inflow}} {T_{\tot, \abs, \outflow}} & \text{for turbines} \end{cases} \]
Total temperature in relative frame of reference.
\[ T_{\tot} = T + \, \frac{v^2}{2c_p} = T\left(1+\frac{\kappa-1}{2}M^2\right) \]
\[ [T_{\tot}] = \K \]
Total temperature in absolute frame of reference.
\[ T_{\tot, \abs} = T + \, \frac{v_{\abs}^2}{2c_p} = T\left(1+\frac{\kappa-1}{2}M_\abs^2\right) \]
\[ [T_{\tot, \abs}] = \K \]
Difference in total temperature in absolute frame of reference.
\[ \Delta T_{\tot, \abs} = T_{\tot, \outflow, \abs} \, - \, T_{\tot, \inflow, \abs} \]
\[ [\Delta T_{\tot, \abs}] = \K \]
Relative total temperature.
\[ T_{\tot,\rot} = T + \frac{v_{\rot}^2}{2c_p} \]
\[ [T_{\tot,\rot}] = \K \]
Total temperature ratio in relative frame of reference.
\[ \tau = \begin{cases} \frac{T_{\tot, \outflow}} {T_{\tot, \inflow}} & \text{for compressors} \\ \frac{T_{\tot, \inflow}} {T_{\tot, \outflow}} & \text{for turbines} \end{cases} \]
Boundary layer displacement thickness.
\[ \delta_1 = \int_0^\delta {\left(1-\frac{\rho(y) u(y)}{\rho_0 u_0}\right)dy} \]
\[ [\delta_1] = \m \]
Boundary layer displacement thickness incompressible formulation.
\[ \delta_{1, \mathrm{inc}} = \int_0^\delta {\left(1-\frac{u(y)}{u_0}\right)dy} \]
\[ [\delta_{1, \mathrm{inc}}] = \m \]
Boundary layer energy thickness.
\[ \delta_3 = \int_0^\delta \frac{\rho(y) u(y)}{\rho_0 u_0} {\left(1-\left(\frac{\rho(y) u(y)}{\rho_0 u_0}\right)^2\right)} dy \]
\[ [\delta_3] = \m \]
Boundary layer momentum thickness.
\[ \delta_2 = \int_0^\delta {\frac{\rho(y) u(y)}{\rho_0 u_0} {\left(1 - \frac{u(y)}{u_0}\right)}} dy \]
\[ [\delta_2] = \m \]
Boundary layer momentum thickness incompressible formulation.
\[ \delta_2 = \int_0^\delta {{\frac{u(y)}{u_0}} {\left(1 - {\frac{u(y)}{u_0}}\right)}} dy \]
\[ [\delta_2] = \m \]
Turbulent dissipation rate.
\[ \omega = \frac{\epsilon}{k} \]
\[ [\omega] = 1/\s \]
Turbulence level or turbulence intensity of fluid with turbulent kinetic energy \(k\) and velocity magnitude \(v\).
\[ Tu = \frac{\sqrt{\frac {2} {3} k}}{v} \]
Turbulence level or turbulence intensity of fluid with turbulent kinetic energy \(k\) and velocity magnitude \(v_\abs\) in absolute frame of reference.
\[ Tu_{abs} = \frac{\sqrt{\frac {2} {3} k}}{v_\abs} \]
Turbulent length scale corresponding to turbulent kinetic energy \(k\) and specific turbulent dissipation rate \(\omega\) (as used in Wilcox and Menter \(\omega\)-equation).
\[ L_T = \frac{\sqrt{k}}{\omega} \]
\[ [L_T] = \m \]
Azimuthal or pitchwise flow angle.
\[ \alpha^*_{\Theta} = 90^\degree - \arctan\big({\frac{v_{\theta}}{v_{m}}}\big) = 90^\degree - \alpha_{\Theta} \]
\[ [\alpha^*_{\Theta}] = \degree \]
Absolute azimuthal or pitchwise flow angle.
\[ \alpha^*_{\Theta,\abs} = 90^\degree - \arctan\big({\frac{v_{\Theta,\abs}}{v_{m}}}\big) = 90^\degree - \alpha_{\Theta,abs} \]
\[ [\alpha^*_{\Theta,\abs}] = \degree \]
Azimuthal or pitchwise flow angle with respect to the axial velocity.
\[ \alpha_{z} = 90^\degree - \arctan\big({\frac{v_{\theta}}{v_{x}}}\big) \]
if \(v_x > 0\).\[ [\alpha_{z}] = \degree \]
Absolute azimuthal or pitchwise flow angle with respect to the axial velocity.
\[ \alpha_{\z,\abs} = 90^\degree - \arctan\big({\frac{v_{\theta,\abs}}{v_{x}}}\big) \]
if \(v_x > 0\).\[ [\alpha_{\z,\abs}] = \degree \]
Radial flow angle.
\[ \epsilon = \arctan\big({\frac{v_{r}}{v_{x}}}\big) \]
if \(v_x > 0\).\[ [\epsilon] = \degree \]
Radial flow angle.
\[ \epsilon_{\cyl,\abs} = \arctan\big({\frac{v_{r}}{v_{abs}}}\big) \]
\[ [\epsilon_{\cyl,\abs}] = \degree \]
Radial flow angle.
\[ \alpha_r = \arctan\big({\frac{v_{r}}{v_{x}}}\big) \]
if \(v_x > 0\).\[ [\alpha_r] = \degree \]
Azimuthal or pitchwise angle defined as angle between the velocity and the meridional velocity.
\[ \alpha_{\theta} = \arctan\big({\frac{v_{\theta}}{v_{\mer}}}\big) = 90^\degree - \alpha^*_{\theta} \]
\[ [\alpha_{\theta}] = \degree \]
Absolute azimuthal or pitchwise flow angle defined as angle between the absolute velocity and the meridional velocity.
\[ \alpha_{\theta,\abs} = \arctan\big({\frac{v_{\theta,\abs}}{v_{\mer}}}\big) = 90^\degree - \alpha^*_{\theta,\abs} \]
\[ [\alpha_{\theta,\abs}] = \degree \]
Angle between y- and x-component of velocity vector in relative frame of reference.
\[ \alpha_{y} = \arctan\big(\frac{v_y}{v_x}\big) \]
\[ [\alpha_{y}] = \degree \]
Angle between y- and x-component of velocity vector in absolute frame of reference.
\[ \alpha_{y,\abs} = \arctan\big(\frac{v_{y,\abs}}{v_x}\big) \]
\[ [\alpha_{y,\abs}] = \degree \]
Angle between z- and x-component of velocity vector.
\[ \alpha_{z} = \arctan\big(\frac{v_z}{v_x}\big) \]
\[ [\alpha_{z}] = \degree \]
Norm of the vector of the meridional velocity \(v_{\mer}\).
\[ \left|{v_{mer}}\right| = \sqrt{v_{\ax}^2 + v_{\rad}^2} \]
\[ [\left|{v_{mer}}\right|] = \m/\s \]
Magnitude of relative velocity vector.
\[ v = |\mathbf{v}| \]
\[ [v] = \m/\s \]
Magnitude of absolute velocity vector.
\[ v_\abs = |\mathbf{v}_\abs| \]
\[ [v_\abs] = \m/\s \]
Relative velocity magnitude
\[ v_{\rot} = |\mathbf{v}| \]
\[ [v_{\rot}] = \m/\s \]
Speed of sound.
\[ a = \sqrt{\kappa R T} \]
\[ [a] = \m/\s \]
Velocity component in pitchwise direction
\[ [v_{\theta}] = \m/\s \]
Wheel velocity component in pitchwise direction
\[ v_{\theta, rot} = v_{\theta, \abs} - \Omega r \]
\[ [v_{\theta, rot}] = \m/\s \]
Eddy viscosity normalised by local molecular viscosity.
\[ \frac{\mu_{T}}{\mu} \]
Molecular fluid viscosity.
\[ \mu = \rho \nu \]
\[ [\mu] = \kg/(\m\s) \]
Vorticity vector and its norm
\[ W_{norm} = \mathbf{W} = \nabla \times \mathbf{v} = \left( \begin{array}{cc} \frac{\partial v_z}{\partial y} -\frac{\partial v_y}{\partial z}\\[0.1cm] \frac{\partial v_x}{\partial z} -\frac{\partial v_z}{\partial x}\\[0.1cm] \frac{\partial v_y}{\partial x} -\frac{\partial v_x}{\partial y} \end{array} \right)\\ W_{norm} = \left|\mathbf{W}\right| = \sqrt{W_x^2 + W_y^2 + W_z^2} \]
\[ [W_{norm}] = 1/\s \]
Reduced work.
\[ W_{\red} = \begin{cases} \frac{h_{\tot,\abs,\outflow}\ -\ h_{\tot,\abs,\inflow}}{T_{\tot,\abs,\inflow}} & \text{for component}\\ \frac{w}{T_{\tot,\abs,\inflow}} & \text{for stage} \end{cases} \]
\[ [W_{\red}] = \J/(\kg\K) \]
Loading criterion for turbines.
\[ \Zweifel = \lvert\frac{\dot{m}_{\inflow} K_{\inflow,\abs}\ +\ \dot{m}_{\outflow} K_{\outflow,\abs}}{(p_{\tot,\inflow}\ -\ p_{\outflow}) A_{\airfoil}}\rvert \]
\[ [\Zweifel] = \m \]
1.9.1