# Validation of a Modal Work Approach for Forced Response Analysis of Bladed Disks
## 3 Results and Comparisons
### LPT Bladed Rotor Test Case
#### 3.2.4 Unsteady Multi-Row Analysis  
The unsteady multi-row analysis allows investigating the interaction between adjacent rows and calculating the unsteady pressure field on the rotor blade surface to be applied in the harmonic analyses. All the computations were performed using the two-equation \(k-\omega\) turbulence model in [PERSON]'s high Reynolds formulation. Standard aerodynamic boundary conditions were applied at the inlet and outlet of the computational domain. Inlet conditions were defined by the span-wise distributions of circumferentially averaged total pressure, total temperature and flow angles, while at the outlet, the static pressure span-wise distribution was imposed. The average total temperature and pressure imposed at the inlet are 331.5 K and 76.2 kPa, respectively, as the stage was tested in a cold flow rig. At the stage outlet, a mean static pressure of 41.1 kPa was imposed, leading to a static temperature of 286.3 K.  
The unsteady multi-row analysis was performed using a dual time-stepping method and the full annulus approach. A reduced computational domain was defined by 1 IGV blade, 18 stator blades, 21 rotor blades representing 1/7 of the entire wheel. To monitor the computation periodicity, a lift coefficient based on the blade surface pressure was stored at each time step for each profile and had to become periodic to ensure the unsteady computation convergence. The rotor lift coefficient is plotted in Figure 9 for two arbitrarily selected rotor profiles that experience the same time history with a time shift related to their different tangential position. The same unsteady lift time history was experienced by all the 147 rotor profiles with different time shifts. The number of peaks depends on the number of stator blades (and IGV profiles) because the rotating rows undergo the effect of the fixed distortions and vice versa. Indeed, in addition to the 18 peaks caused by stator wakes, it is evident that the presence of a single higher peak due to the IGV wake convected through the stator row and impinged the rotor at EO = 7. The instantaneous entropy contours on a blade-to-blade surface are shown in Figure 10 to highlight the typical interaction between adjacent rows due to the wakes. The IGV wake passes through the stator and impinges the rotor blades, adding to the stator wake distortion (EO \(=\) 126) a further distortion with EO \(=\) 7. This causes the single highest peak in the lift coefficient shown in Figure 9.  
Figure 9: Unsteady lift coefficient for two different rotor profiles during 1/7 of the rotor revolution. A run-time temporal DFT on the rotor profiles was activated during the TRAF unsteady run to decompose the time-varying fluctuations at \(\text{EO}\,=\,7\) and \(\text{EO}\,=\,126\), which correspond to the first BPF of IGV and stator, respectively. Then, the unsteady pressure distribution described by complex Fourier coefficients field acting on the rotor blade surface was circumferentially decomposed in space to extract the rotating perturbations that can excite the corresponding traveling wave mode shapes of the bladed rotor. The in-house post-processor extracts the pressure perturbation at circumferential order \(\ell\,=\,+7\) from the \(\text{EO}\,=\,7\) and \(\ell\,=\,-21\) from the harmonic at the \(\text{EO}\,=\,126\), accounting for the aliasing phenomenon. Considering the rotor rotational direction, the circumferential order values were converted to nodal diameters to distinguish between forward and backward traveling waves in ANSYS, so the pressure perturbation is referred to \(\text{ND}\,=\,-7\) backward for \(\text{EO}\,=\,7\) and \(\text{ND}\,=\,+21\) forward for \(\text{EO}\,=\,126\). Figures 11 and 12 show the amplitude and phase of the unsteady pressure perturbation circumferentially decomposed for \(\text{EO}\,=\,7\) and \(\text{EO}\,=\,126\), respectively.