For steady-state simulations, only the passage averaged values are exchanged at the rotor/stator interface and thus periodic boundary conditions can be applied in the circumferential direction, regardless of the blade numbers involved. Unsteady computations, in contrast, require the direct coupling of rotating and stationary grid and the interpolation of the physical variables require that the blade pitch be equal at the coupling interface. The equal pitch often is enforced artificially by scaling the blade count to values that allow for a direct coupling of the rotor and stator domain with modelling only one (or a few) blade passages (e.g. a blade count of 23:45 would then be scaled to 23:46, which allows the application of periodic boundaries using 1 and 2 passages respectively).

If the blade count is not to be scaled, there is the possibility of introducing an extended formulation of the periodic boundary conditions which always allow the computational domain to be restricted to a single blade passage, regardless of the blade count of the blade rows involved. This extended formulation of the boundary conditions and its implementation will be discussed in the following section.

Theory

The standard (meaning direct) periodic boundary condition for single passage computations with equal pitches in the circumferential direction is

\begin{equation} u_L(x,r,\varphi,t) = u_U(x,r,\varphi+\Phi,t) \end{equation}

with \(u_L\) denoting the flow vector at the lower periodic boundary and \(u_U\) the vector at the upper periodic boundary. The distance in radians between both boundaries is determined by the blade pitch angle \(\Phi\).

When unequal blade numbers in single passage stage computations occur, a time shift between lower and upper periodic boundary is introduced and the boundary conditions then extends to

\begin{equation} u_L(x,r,\varphi,t) = u_U(x,r,\varphi+\Phi,t+\Delta t) \end{equation}

The time lag in this relation accounts for the different blade pitches. As the time shift is related to a certain frequency one major assumption of the approach is that the perturbation frequency is known and discrete. For a single stage, the time shift \(\Delta t\) is determined by the number of rotor blades B, the number of stator vanes V and the rotational speed \(\Omega\). The frequency of any perturbation e.g. in the rotor frame of reference is

\begin{equation} f_B = \frac{V}{2\pi}(\Omega_B - \Omega_V) \end{equation}

the resulting inter-blade phase-angle

\begin{equation} \Delta \varphi_B = -2\pi \frac{B-V}{B} sgn(\Omega_B - \Omega_V) \end{equation}

and the resulting time shift between lower and upper periodic boundary is then given by

\begin{equation} t_B = \frac{\Delta \varphi_B}{2\pi f_B}. \end{equation}

In the literature three different methods have been described of integrating the time-lag into the formulation for the periodic boundary conditions; only the approaches which have been proved in the literature to be applicable in a flow solver are listed though:

Erdos and Alzner [10] suggested a direct storage of the flow vector at all periodic and the interface boundary points over a blade passing period. Before exchanging data between the lower and the upper periodic boundary the time lag can be introduced by simply retrieving the corresponding time shifted value from the storage buffer. The method is relatively easy to implement but the storage requirements are rather high compared to other methods.
A more theoretical approach has been suggested by Giles [13] . Instead of storing the time history, the equations to solve are transformed in time using the time-lag. This transformation allows the formal application of (simple) direct period boundary conditions. The advantage of this method is that the actual boundary conditions are kept very simple, a disadvantage is that the calculation is not being carried out in physical time but the results need to be transformed based on a time periodic solution.
A third method has been proposed by He and Gerolymos [17] (Shape Correction Method) (Chrochronic Periodicity). Both authors describe the introduction of the timelag by temporal Fourier-decomposition of the boundary points. When doing the transformation from frequency to time domain the time shift can be applied easily. Also the numerical stability can be controlled easily by the Fourier coefficients. The method is very low memory consuming and offers the possibility of decomposing the perturbations into the portions of different frequencies and thus in principle may be applicable to multi-stage calculations with more than one frequency involved.

Due to its wider application and its potential for multi-stage computations the Shape-Correction Method as it was proposed by He will be summarized in the following.

In order to introduce the time shift into the boundary conditions all flow variables are Fourier-decomposed along the upper and lower passage domain boundaries. A general formulation for a variable number of perturbations i is given by

\begin{equation} u = u_0 + \sum_{i=1}^{N_{per}} u_i' \end{equation}

meaning that an instant flow vector can be written as the time average \(u_0\) and the sum of the portions from other frequencies. The portion of the i-th perturbation then can be reconstructed at time t using the computed Fourier-coefficients by

\begin{equation} u_i' = \sum_{i=1}^{N_{FOU}} \left[ a_{i,n} \sin(2\pi n f_i t) + b_{i,n} \cos(2\pi n f_i t) \right] \end{equation}

with the Fourier coefficients of the n-th harmonic \(a_{i,n}\) and \(b_{i,n}\) being calculated for each perturbation i during the computation using

\begin{eqnarray} a_{i,n} &=& f_i \sum_{i=1}^{N_{per}}(u-R_j)\sin(2\pi n f_i t)\Delta t \\ b_{i,n} &=& f_i \sum_{i=1}^{N_{per}}(u-R_j)\cos(2\pi n f_i t)\Delta t \end{eqnarray}

When calculating the coefficients for the i-th perturbation, the portions of the other frequencies are subtracted, these portions are given by

\begin{equation} R_j = \sum_{i=1}^{N_{per}} \sum_{n=1}^{N_{FOU}} \left[ a_{i,n} \sin(2\pi n f_i t) + b_{i,n} \cos(2\pi n f_i t) \right] \quad (i \ne n) \end{equation}

A general expression for the reconstruction of any flow variable is then given by

\begin{equation} u = u_0 + \sum_{i=1}^{N_{per}} \sum_{n=1}^{N_{FOU}} \left[ a_{i,n} \sin(2\pi n f_i (t+ \Delta t_i)) + b_{i,n} \cos( 2\pi n f_i (t+ \Delta t_i)) \right] \end{equation}

with any perturbation i having its own time shift \(\Delta t_i\).