# Rotor design by the vortex-flow method The rotor method turns a uniform supersonic inlet flow into a vortex field, guides that field between circular surfaces, and returns it to uniform flow at the outlet. Each surface contains an inlet transition arc, a circular arc, and an outlet transition arc. The package implements the construction of Goldman and Scullin, with real-fluid properties replacing the perfect-gas relations in the original report {cite:p}`goldmanBlading1968`. Unlike the nozzle construction, the rotor method starts from a supersonic inlet and prescribes Mach numbers on both circular surfaces. Its objective is to produce the required turning and passage shape, rather than to accelerate a subsonic inlet through a nozzle throat. ## Relative frame and thermodynamic states The inputs `P0_rel`, `T0_rel`, and `Q0_rel` specify relative stagnation conditions. The thermodynamic path used in the implementation is ```{math} :label: eq-rotor-relative-energy h+\frac{W^2}{2}=h_{0,\mathrm{rel}},\qquad s=s_{0,\mathrm{rel}}. ``` This is a local blade-section approximation. In a rotating row, the conserved rothalpy is $h+W^2/2-U^2/2$ under the usual adiabatic inviscid assumptions. At constant machine radius, the blade speed $U$ is constant and this reduces to Eq. {eq}`eq-rotor-relative-energy`. The package does not solve the general rotating, three-dimensional flow field or convert a complete stage's absolute inlet conditions into these relative inputs. Four Mach numbers specify the inlet, outlet, lower circular surface, and upper circular surface. The implementation finds each corresponding $W$ on the relative isentrope and evaluates the generalized Prandtl–Meyer angle $\nu(W)$ using Eq. {eq}`eq-generalized-pm`. Typically, the lower surface decelerates the inlet flow and the upper surface accelerates it, while both remain supersonic. ```{admonition} Review note — inversion of rotor Mach targets The rotor helper `_V_of_M` uses a bisection update that assumes Mach number increases with velocity over its search interval. The [isentropic-expansion chapter](isentropic_expansions.md) shows why this need not hold for non-ideal or saturation-crossing paths. Check branch selection before applying the rotor method to those cases. Sharing the nozzle property backend does not establish that every flashing or wet-to-dry rotor input is supported by this inversion. ``` ## Vortex field and circular surfaces An irrotational circular flow satisfies ```{math} :label: eq-vortex-law WR=K. ``` The code uses $K=a_\mathrm{ref}R_\mathrm{ref}$, so its dimensionless vortex radius is ```{math} :label: eq-vortex-radius \widehat R(\nu)=\frac{R}{R_\mathrm{ref}} =\frac{a_\mathrm{ref}}{W(\nu)}. ``` For a smooth sonic reference, $a_\mathrm{ref}=W_\mathrm{ref}$ and $R_\mathrm{ref}$ is the radius of the sonic-velocity streamline in the underlying vortex field. Along either circular wall, $R$ is constant, so $W$, $a$, and $M$ are constant. Across the passage, these quantities vary with radius. The free-vortex law constrains velocity times radius, not local Mach number times radius. The report's relation $M^*R^*=1$ uses $M^*=W/a_\mathrm{ref}$; it must not be read as $(W/a)R^*=1$. Each circular surface is represented in normalized coordinates by $x=\widehat R_w\sin\zeta$, $y=\widehat R_w\cos\zeta$. Its angular extent is obtained after subtracting the turning supplied by the transition arcs from the required total flow turning. ## Outlet angle from continuity For equal blade spacing and span at inlet and outlet, conservation of axial mass flux gives ```{math} :label: eq-rotor-continuity \rho_\mathrm{in}W_\mathrm{in}\cos\beta_\mathrm{in} =\rho_\mathrm{out}W_\mathrm{out}\cos\beta_\mathrm{out}. ``` The implemented turning branch is ```{math} :label: eq-rotor-outlet-angle \beta_\mathrm{out}=-\cos^{-1}\!\left( \frac{\rho_\mathrm{in}W_\mathrm{in}} {\rho_\mathrm{out}W_\mathrm{out}}\cos\beta_\mathrm{in}\right). ``` The argument must lie in $[-1,1]$. A value outside that interval means that the specified states and inlet angle cannot satisfy the assumed passage continuity. Thus `beta_outlet` is derived rather than accepted as an independent target. Continuity is a necessary condition; it does not by itself guarantee a nonintersecting blade for arbitrary surface Mach numbers. If inlet and outlet correspond to the same state on the selected branch, their relative speeds are equal and the rotor has no static enthalpy drop in this model, as in an impulse section. For unequal speeds, $h_\mathrm{in}-h_\mathrm{out}=(W_\mathrm{out}^2-W_\mathrm{in}^2)/2$. A stage degree of reaction additionally requires the stage enthalpy drop, which this isolated section calculation does not supply. ## Marching a transition arc The following equations describe the current numerical construction in its local normalized frame. They make explicit the distinction between a point on a reference characteristic and a point on the blade wall. For each inlet or outlet transition, sample $\nu_i$ uniformly from the circular wall value $\nu_w$ to the corresponding far-field value $\nu_f$. Define $\sigma=-1$ for the lower surface and $\sigma=+1$ for the upper surface. The reference angle and point are ```{math} :label: eq-rotor-reference-point \phi_i=\sigma(\nu_i-\nu_w),\qquad \boldsymbol Q_i=\widehat R(\nu_i) \begin{pmatrix}\sin\phi_i\\\cos\phi_i\end{pmatrix}. ``` The march begins at the wall point $\boldsymbol P_0=(0,\widehat R_w)$, where $\phi_0=0$. The local flow direction is $-\phi$. The previous wall segment and the current Mach segment therefore have slopes ```{math} :label: eq-rotor-marching-slopes m_{w,i}=\tan(-\phi_{i-1}),\qquad m_{c,i}=\tan\!\left[-\frac{\phi_i+\phi_{i-1}}{2} +\sigma\frac{\mu_i+\mu_{i-1}}{2}\right]. ``` The new wall point is the intersection of the line through $\boldsymbol P_{i-1}$ with slope $m_{w,i}$ and the line through $\boldsymbol Q_i$ with slope $m_{c,i}$: ```{math} :label: eq-rotor-wall-march x_i=\frac{y_{i-1}-m_{w,i}x_{i-1}-Q_{y,i}+m_{c,i}Q_{x,i}} {m_{c,i}-m_{w,i}},\qquad y_i=y_{i-1}+m_{w,i}(x_i-x_{i-1}). ``` The dependence on $\boldsymbol P_{i-1}$ makes this a genuine sequential construction. Plotting only $\boldsymbol Q_i$ would plot the reference characteristic, not the transition wall. Increasing `num_points` reduces the turning represented by each straight segment, but extreme input combinations can still require a geometry check. ```{admonition} Figure placeholder — rotor transition construction Show a circular wall joined to an inlet transition arc. Mark the previous wall point $P_{i-1}$, the reference point $Q_i$, the local Mach segment, and their intersection $P_i$. A second panel should show the complete lower and upper surfaces with inlet transitions, circular arcs, and outlet transitions. Use separate line styles for the reference characteristic and the blade wall. ``` ## Orienting the surfaces The code uses the following circular-arc parameters, expressed in the guide's notation: | Surface | $\zeta_\mathrm{in}$ | $\zeta_\mathrm{out}$ | | --- | --- | --- | | Lower | $(\nu_\mathrm{in}-\nu_\mathrm{lower})-\beta_\mathrm{in}$ | $-(\nu_\mathrm{out}-\nu_\mathrm{lower}+\beta_\mathrm{out})$ | | Upper | $(\nu_\mathrm{upper}-\nu_\mathrm{in})-\beta_\mathrm{in}$ | $-(\nu_\mathrm{upper}-\nu_\mathrm{out}+\beta_\mathrm{out})$ | For a local transition point $(x,y)$, the inlet transformation is ```{math} :label: eq-rotor-inlet-transform X=x\cos\zeta+y\sin\zeta,\qquad Y=-x\sin\zeta+y\cos\zeta. ``` The outlet transformation includes reflection: ```{math} :label: eq-rotor-outlet-transform X=-x\cos\zeta+y\sin\zeta,\qquad Y=x\sin\zeta+y\cos\zeta. ``` These signs match the implementation's local frames. The assembled surface traverses the inlet transition from the far field to the circular arc and then the outlet transition back to uniform flow. ## Pitch and finite blade closure Let $L_\mathrm{in}$ and $U_\mathrm{in}$ be the inlet endpoints of the lower and upper passage surfaces. Extend the upper endpoint in the inlet flow direction to the lower endpoint's axial station. The vertical gap is the pitch: ```{math} :label: eq-rotor-pitch t_b=y_{L,\mathrm{in}}-y_{U,\mathrm{in}} -\tan\beta_\mathrm{in}(x_{L,\mathrm{in}}-x_{U,\mathrm{in}}). ``` The reported chord is the distance between the lower surface's inlet and outlet endpoints, and solidity is $c/t_b$. The returned `pressure` and `suction` curves describe the open passage construction. To form a solid blade, the upper surface is translated by one pitch and then by a further offset needed for the finite edge thickness. The target trailing-edge radius is $r_\mathrm{TE}=t_b/\texttt{le\_te\_ratio}$. At the trailing edge, a common tangent direction defines the unit normal $\boldsymbol n=(-\sin\beta_\mathrm{out},\cos\beta_\mathrm{out})$. If $D_0$ is the normal separation before the extra pitchwise translation, the required translation is $(2r_\mathrm{TE}-D_0)/n_y$. The code then intersects one surface tangent with the normal through the other endpoint and uses that normal segment as the diameter of a semicircular closure. The same translation determines the leading-edge radius; it is not specified independently. ```{admonition} Review note — finite-thickness geometry The edge-radius rule is a package construction, not a result prescribed by the original report. `blade_upper` contains the pitch translation, whereas the final `blade` also includes the additional thickness translation. Check the neighboring passage opening after this adjustment before treating the finite blade as an unchanged realization of the ideal vortex flow. ``` All local lengths are multiplied by `r_star` when that dimensional scale is provided. The [rotor example](../examples/rotor_design.md) exports both the assembled blade and the separate passage surfaces. Radial placement is a subsequent [geometric mapping](radial_mapping.md).