## Abstract

Boundary Layer Ingestion (BLI) is a concept with the potential to reduce energy consumption needed for aircraft propulsion. For the fan this results in a need to work well in an environment of increased continuous distortion. The objective of this study is to increase the understanding of unsteady aerodynamic forces due to total pressure distortion in a BLI fan. This is done by using computational fluid dynamics to analyze a fan design intended for a BLI installation. Results include low subsonic to transonic operating speeds and distortion in a wide range of wave numbers. The forcing exhibits a significant dependency on the aeroacoustic cut-on/cut-off condition. This is in particular true at part speed where the normalized unsteady force rises sharply before dropping suddenly as the corrected speed or engine order increases. Unsteady pressure in the blade passage is observed to exhibit the same increase in level followed by a sudden drop once the cut-on limit is passed and undamped pressure waves propagating out from the blade row appear. The unsteady forces on the first three modes exhibit different sensitivities to the distortion wavelength but all are affected by the acoustic condition. By comparing results for the reference fan to a similar fan design with a higher blade count the cascade properties of the blade row are found to dominate the interaction. A result of this is that a higher blade count fan may be less affected by the distortion, and be less prone to propagate noise due to low engine order distortion.

## 1 Introduction

The development of novel concepts that integrate the aircraft engine or fan closely with the fuselage emphasizes the importance of understanding the effects of distorted inlet flow. An example of this is BLI (Boundary Layer Ingestion) technology where the goal is to lower fuel consumption by improving propulsion efficiency. The concept for this is to place the fan in a position where it ingests air from wakes and boundary layers with a lower momentum than in the free stream. Increasing the momentum of this airstream then requires less power for the same amount of output thrust. A large study where the potential of using a fuselage tail cone BLI installation was explored can be found in Ref. [1].

A direct consequence of using BLI is that the fan will be exposed to unsteady forces due to the distorted inlet flow. A second potential issue is noise emissions from the fan that could result from the interaction of the distortion with the fan rotor. Aerodynamic forces acting on the blade at resonance conditions are well-known causes for high blade vibration levels and associated HCF (high cycle fatigue) problems in turbomachinery blades and vanes. Design strategies to avoid such problems include modifying the sources of excitation or eigenfrequencies of the blade. Examples of design choices that reduce the sources are to increase the axial separation of blade rows or to change the vane count. In the case of BLI, the distortion depends on the integration of the fan with the fuselage and is part of the concept. Since it is generated outside the fan and varies with operating conditions, it is difficult to affect the distortion by the design of the turbomachine. High levels of forcing from inlet distortion at low engine order (EO) can be expected at all operating conditions. Off-resonance forcing may therefore also need attention as fatigue load cycles accumulate over time. In Ref. [2], results from testing a BLI fan are presented. Findings in this study include high-level blade vibrations due to the distortion. In Ref. [3], a fan blade was also re-designed with respect to distortion in a BLI application. These studies point to the importance of understanding the aerodynamically induced forces due to distortion, in order to minimize design trade-offs that could reduce efficiency.

A BLI fan designed for the integration on a Fokker 100 aircraft tail cone is the starting point for this study (Fig. 1). Performance and details of this design were presented in Ref. [4], and overall fan data for the present study are listed in Table 1.

Diameter | 1.2 m |

Hub tip ratio | 0.5 |

Corrected speed NR | 4812 rpm |

PRtt | 1.3 |

Flow coefficient φ | 0.73 |

Load coefficient, ψ | 0.5 |

Diameter | 1.2 m |

Hub tip ratio | 0.5 |

Corrected speed NR | 4812 rpm |

PRtt | 1.3 |

Flow coefficient φ | 0.73 |

Load coefficient, ψ | 0.5 |

For the fan used in the present study, it was found in Ref. [5] that there is a correlation between the amplitude of radially averaged inlet total pressure for each wave number and the force on the blade at the corresponding engine order. This makes it plausible that forcing can be analyzed using individual incoming waves. It was also found that removing the swirl content at the inlet had a small effect on the blade forcing in this case. The analyses were performed at the aerodynamic design point (ADP) of the aircraft which is an altitude of 10,668 m (35,000 ft.) and a flight Mach number of $Ma=0.74$. The distortion field was taken from a whole aircraft analysis as described in Ref. [4], where the BLI fan was part of the analysis with the fan modeled as an actuator disc.

In Fig. 2, the Campbell diagram for the BLI fan from [4] is shown with the ADP absolute fan speed at 4419 rpm and the SLS speeds marked in the diagram. In Fig. 2, the first 12 EO are plotted as solid lines showing that the five lowest modes are potentially excited by the distortion. It may be noted that the fan was designed with the first mode (1F) above the 2EO line, which is considered a robust choice in view of the uncertainties in the intended application. This differs from the BLI fan tested in Ref. [2] and the BLI fan design proposed in Ref. [3] where the first mode lies below the 2EO line at full speed. In Ref. [5], it was found that for the design condition, significant forcing amplitudes could be expected up to at least 10 EO.

For noise emission, an important contribution comes from the rotor-stator interaction. Here the first design goal is to avoid conditions allowing acoustic waves to propagate out from the fan. Such tonal noise generation has some similarities with aerodynamic forcing in that the blade and vane counts govern possible interactions.

In fan design, a clear goal is to avoid interactions that feed into cut-on waves in operation where noise can be an issue. With a fan operating in distortion, it can be suspected that modes of interaction may become important involving the incoming distorted airstream. In Ref. [6], the well-known Tyler–Sofrin relation $M=Min+kB$ was derived that identifies the possible linear harmonic interactions between the incoming perturbation $Min$ and the blade count ** B**. $Min$ can in this context be considered synonymous with EO. The basic conditions for the propagation of sound waves in 2D as described in Ref. [7] are well understood. 3D conditions for which waves will propagate are more involved and may be very complex. In Ref. [8], the conditions for propagation are derived in the more general annular case with a flow profile across the flow path.

Assessing the effects of widely differing operating conditions, or enabling the calculation of a load spectrum may require a very large number of evaluations of the aerodynamic forcing of the blades. Full 360 transient computational fluid dynamics (CFD) modeling of the unsteady interaction makes it possible to extract unsteady forces acting on a blade. Although such analyses have been possible for some time, with early examples in transonic conditions given in Refs. [9,10], they are still time-consuming. Application of less computationally intensive analysis methods is therefore of interest when performing parametric analyses. It is also of interest to simplify the model in order to more easily identify the different physical mechanisms governing the interaction between inlet total pressure distortion and fan response.

This paper will present results from a full 360 nonlinear URANS model also used in Ref. [5] as well as a simplified sector model based on the linearized Navier–Stokes (LNS) equations in the frequency domain. It will be shown that the LNS model results are in good agreement with the URANS model in terms of transfer of distortion to blade force, which supports the LNS model as a tool to explore the aerodynamic forcing in a wide range of operating conditions and distortion wave numbers.

The aeroacoustic cut-on/off condition will be shown to have a large impact on the blade forcing with a significant peak in force just below the cut-on limit. Using the LNS model a blade force map will be presented for a wide range of speed and inlet distortion wave numbers. This is followed by a study of the effect of blade count on the fan interaction with the inlet distortion. Finally, the blade forces due to the inlet distortion projected onto the first three blade mode shapes are presented for the full range of speed and distortion wave numbers.

## 2 Methods

CFD is generally well established and used today as a tool for assessing unsteady aerodynamics in turbomachinery. In this work, two different CFD modeling techniques will be employed.

### Full 360 URANS Model Description.

First, the commercial CFD package ANSYS CFX version 2021r1 [11] is used for modeling the airstream through a fan from the intake lip to the nozzle with a direct time-stepping method. The computational model includes all 18 fan blades and 46 outlet guide vanes, as shown in Fig. 1 without assumptions of periodicity in time or space. A non-axisymmetric field, extracted from a whole aircraft model, is imposed on the inlet boundary as a variation of total pressure and flow direction. At the outlet boundary, the average static pressure is imposed with a radial equilibrium condition governing the pressure distribution. The equations solved are the compressible URANS (Unsteady Reynolds' Averaged Navier–Stokes) equations. Turbulence is modeled by the *k*–*ω* SST model which is a common choice for turbomachinery flows with this software. Implementation details of the CFD code turbulence models used can be found in the software documentation [11]. The URANS model used here was used also in Ref. [5] with the same mesh and settings to compute the flow at design speed.

In the nonlinear full 360 URANS model, the inlet boundary condition is taken from extracting the flow field over the inlet plane of the fan from the whole aircraft model, as shown in Fig. 1. It can be noted that the total pressure at the inlet plane is symmetric with respect to the vertical axis, which follows from the fuselage symmetry. The total pressure distribution and flow direction at this plane are imposed as boundary conditions on the URANS model. The total temperature at the inlet boundary is constant. On the outlet boundary located at the nozzle plane, the static pressure is specified, and on all walls, no-slip conditions are applied.

The strength of this model lies in its completeness in terms of physical modeling. The model is, however, large and computationally heavy, and it can be difficult to isolate what causes the levels of forcing on a specific mode.

### Linearized Navier–Stokes Method With RANS Model for the Mean Flow.

The second method employed in this study uses the compressible LNS (linearized Navier–Stokes) equations in the frequency domain for the unsteady solution. This method requires a mean flow solution that is computed prior to the LNS solution and then solves the unsteady field for any number of unsteady boundary conditions and frequencies.

The LNS solver is an in-house developed software (called Linnea) used for aeroacoustics and aeromechanics. The spatial discretization used is an upwind biased 3rd order finite volume scheme. This scheme has low dissipation and dispersion errors making it suitable for acoustic propagation problems. Solving the system of equations is made using a Runge–Kutta time marching scheme coupled with the GMRES method. The linearization is made around a mean flow computed as a steady-state solution on the same mesh. For the mean flow in this case the ANSYS CFX solver is used with a *k*–*ε* model for turbulence with standard choices built into the software of using wall functions and the Kato-Launder production term modifier. In Linnea, turbulence is considered by importing the eddy viscosity and turbulent kinetic energy from the mean flow solution. While it can be used with any turbulence model for the mean flow solution the *k*–*ɛ* model used here for the LNS model has been found to work well for simulations near design conditions. The CFD model in this case is simplified by removing the outlet guide vane (OGV) and is extended 1.5 m upstream and 1.5 m downstream from the rotor with straight annular channels. Figure 3 shows typical views of the mesh that consists of a total of approximately 1M nodes with near-wall resolution suitable for the use of wall functions. The tip clearance is not seen in the mesh but the mesh includes the tip gap.

The linearized harmonic model is computationally less demanding which is helpful for computing many cases. Comparing the methods typical numbers found are 100–150 h on 200 CPU:s for one URANS case versus 2–3 h on 12 CPU:s for one LNS case. The LNS model is very suitable to use with a focus specifically for aeroacoustic capabilities. This is an important consideration for fan blade row interaction with the ingested distortion field. An important advantage of the linearized harmonic technique is the availability of very effective non-reflecting boundary conditions. Two varieties of non-reflecting boundaries are used with the model. The model used for most of the analyses and the mapping of a wide range of operating speeds uses a 1D non-reflecting boundary condition that allows waves normal to the boundary to propagate out without reflection. A more advanced 3D non-reflecting boundary condition is used to check that the effects of the remaining reflection do not compromise the conclusions of the study. The 3D absorbing boundary condition is based on solving the generalized eigensystem of the linearized Euler equations following [8], in a parallel mean flow in an annular duct. No assumption of a constant flow field is made, so the actual mean flow variation over the annulus cross section is used. Experience from aeroacoustic calculations has verified that this results in a highly non-reflective boundary condition with an absorption of >40 dB. The boundary condition separates the pairwise occurring acoustic modes from the hydrodynamic entropy and vorticity modes. In rare cases, it can be difficult for the code to correctly separate and apply the different eigenmodes, and in these cases, the distinction is made manually. While the 3D absorbing boundary condition allows for a very good damping characteristic, it is cumbersome to use for mapping out hundreds of cases. In addition to the non-reflective boundary condition, a long inlet annulus is used in order to minimize reflections from the inlet and outlet boundaries.

The scaling with the shaft speed ratio is done to get a radial distribution of swirl angles in the rotating frame that does not vary at different corrected speeds. On the outlet boundary, the static pressure is specified as the boundary condition for the mean flow. Figure 4(c) shows the spanwise distribution of total pressure in the simplified model in Eq. (1) at $\nu =100%$ and the circumferentially averaged total pressure from the full 360 BLI model (Fig. 4(a)). Operating conditions for the mean flow at different speeds are defined by controlling the flow coefficient $\phi =cmU$. In this study, $\phi =0.73\u22120.74$ is achieved for all speeds by adjusting the nozzle back pressure for each individual case.

An example of the total pressure over the inlet by combining Eqs. (1) and (3) for $Min=4$ is shown in Fig. 4(b).

The incoming unsteady total temperature perturbations as well as velocity components in the plane are set to zero on the inlet boundary. On the outlet boundary, no perturbations are imposed.

### Forcing Metrics.

The blades travel through the distorted flow field resulting in unsteady aerodynamic forcing with harmonic frequency content as indicated in the Campbell diagram engine order lines in Fig. 2. In addition to a frequency each eigenmode has a specific shape in which it vibrates. Here the modal analysis is made with ANSYS Workbench 2021r1. The modal analysis is made with the blades fixed to the ground at the root. Omitting the disk results in eigenmode shapes that are independent of the nodal diameter, both in shape and frequency. The disk coupling can be important in general, but in this context, the increased complexity would make it more difficult to interpret the results. Eigenmode displacements used in the analyses are shown in Fig. 5, where the plots show the displacement in the direction of the surface normal over the suction side of the blade.

*p*, and the shear stress tensor $\tau ik$.

*S*. A force coefficient is defined using the time average of the circumferential force for each individual wave number $MandF\theta $:

## 3 Results and Discussion

Results are derived from three separate configurations where the main body of results is drawn from LNS analyses of the fan rotor operating near design incidence. URANS analyses of the entire stage are used for comparison and to verify that the LNS results are relevant to describe the impact of distortion on the blade force. Furthermore, a configuration of the fan rotor where the blade count is doubled to 36 while conserving the cascade solidity is analyzed in the same way as the baseline configuration.

### Mean Flow Results.

In Ref. [4], the steady-state performance of the fan, and the whole aircraft model is discussed for key operating conditions of the fan. A steady-state RANS model is introduced here to provide the mean flow field that is used as input to the LNS solver.

The mesh used for the fan RANS and LNS models is shown in Fig. 3 and comprises a single blade sector with long straight annular ducts extending to the inlet and outlet boundaries. A large range in corrected speed is covered using this model, with analyses done in 5% increments up to 110% of the corrected design speed. The outlet static pressure boundary conditions are chosen such that the flow coefficient is near constant. In Fig. 6, the performance map for the fan is shown, with the working line followed by the present steady-state analysis points used in the LNS analyses (green diamonds), using the rotor pressure ratio. The flow coefficients for the RANS analyses vary from $\phi =0.73\u22120.74$, which corresponds to an incidence angle variation up to 0.5 deg. Red markers in the performance map are the operating points for the URANS analyses. Other lines and points defining the performance map are based on the same dataset as used in Ref. [4]. These are included to help interpret the operating conditions of the fan. Working lines denoted by WL are thus those operating points that the fan follows at varying corrected speed at a constant flight Mach number with a fixed nozzle. ADP and SLS in black markers denote the fan operating points used in the design of the fan. The constant speed lines (40–100%) from the mean line analysis in Fig. 6 help visualize the fan performance map. The very low speeds are included in the data set in order to follow the forcing all the way down to incompressible conditions.

Figure 7 shows the radial total pressure profiles upstream of the leading edge and downstream of the trailing edge for the circumferentially averaged URANS and the RANS models of the fan at $\upsilon =100%$. On the upstream side end wall boundary layers have developed from the inlet boundary condition shown in Fig. 4(c). The profiles match well considering the differences in the computational domains, geometries, and the definitions of the boundary conditions. The URANS model exhibits a slightly thicker boundary layer than the RANS at the shroud and small differences are observable near the hub. For the 36-bladed configuration, only the downstream location is shown since the upstream profile falls on the same line as the baseline RANS model for 18 blades. The 36-blade fan is designed with the intent to be as similar as possible to the 18-blade fan. The blade profiles and the geometric shape of the blade-to-blade passage are retained from the 18-blade fan. The hub and tip radii are kept the same through the blade passage in order to not alter the axial velocity ratio over the fan. The main change to the aerodynamics due to this scaling is that the slope of the end walls through the fan increases for the 36-blade fan. Consequently, a slight difference in the radial profile out from the rotor is observed at the hub, where the total pressure is slightly lower at the exit of the 36-bladed rotor.

At design speed, $\upsilon =100%$, the fan blade tip is transonic and the flow enters the cascade with a supersonic patch on the suction side and a shock ahead of the leading edge as seen in Fig. 8. At 50% span, the suction side exhibits a pocket of marginally supersonic flow, and at the 10% span, the flow is subsonic. There is a positive incidence at the 90% span section and a negative incidence at 10%, which can be seen in the locations of the stagnation points. This affects the flow at the front end of the blade, but in both cases, the flow remains attached.

By time averaging the URANS model, flow and performance data can be calculated and compared to the RANS calculation. The largest difference in the incoming total pressure occurs near the shroud as seen in Fig. 7. The corrected mass flow in the URANS model at $\u03d1=100%$ is 1% lower than in the RANS model and the pressure ratio over the rotor is less than 1% higher. Plotting the Mach number distribution in Fig. 9 in the same way as for the RANS case at 90% span shown in Fig. 8(a) allows a comparison. Small differences are found in the blade-to-blade flow at the early suction side where the flow expands to supersonic velocity. The URANS model exhibits a slightly smoother expansion in this view which could be expected as the blade moves through the distorted field. Differences are smaller yet at lower spans and the steady-state field is considered representative of the aerodynamics of the rotor. The mean flow fields computed from the RANS and URANS models are both characterized by attached flows and the Mach number distributions around the blade are very similar in the two configurations.

Turning to the 36-bladed rotor, this consists of blades with the same shape and solidity as the 18-bladed baseline fan. The intent is to gain an understanding of the effects of blade count on the interaction due to inlet distortion. From comparing the radial profiles in Fig. 7, the largest difference is found near the hub. Showing the 10% section where the largest differences are found in Fig. 10 demonstrates that the Mach number distribution is very similar to the baseline fan (Fig. 8(c)). The total mass flow at 100% speed is 0.2% lower and at 50% speed less than 0.1% lower than for the 18-blade fan. The pressure ratio is 1% higher than for the 18-blade fan due to the lower total pressure observed near the hub in Fig. 7.

### Blade Forcing Levels With Complex Boundary Layer Ingestion Inlet Distortion.

From the full 360 nonlinear model, a correlation of distortion to harmonic forces on the blades and vanes was studied at $\nu =100%$ [5]. In the present work, the same URANS model is used for an additional operating point, $\nu =50%$, as a complement to the LNS model analyses. At $\nu =100%$, the boundary data are taken directly from the whole aircraft simulation in the same way as previously done in Ref. [5] for ADP. The same pattern of distortion using the total pressure variation normalized by the dynamic pressure at the inlet is used for the simulation at $\nu =50%$. Figure 11 shows the corresponding amplitude of the distortion, per engine order evaluated from the full distortion pattern used in the simulation. In Fig. 11, also the force amplitude normalized by the time average of the force, $|F\u2032\theta ,M|/F\xaf\theta $, is shown. This gives a direct measure of the harmonic force per EO spectrum relative to the aerodynamic force producing torque. It can be noted that the force amplitudes correlate qualitatively to the distortion level. The forcing as well as the distortion amplitude at the inlet has the highest levels at 3EO-4EO, and then tapers off for increasing EO, with levels at 12EO and above being an order of magnitude lower than the maximum.

### Comparison of URANS to the Linearized Navier–Stokes Model Forcing Spectrum.

The first objective of this study is to present results demonstrating that the linearized method represents the fan forcing in a useful way. This will be done by comparing the full 360 URANS result with the linearized harmonic LNS model with 1D non-reflecting boundary conditions at the flow boundaries.

In order to bring focus to the interaction of the distortion with the blade row, the blade force shown in Fig. 12 is further normalized with the incoming perturbation amplitude, yielding $Cf\theta $ as defined in Eq. (7). In Fig. 12, the comparison is made for both speed ratios computed with the URANS model. The results show that for both speeds, the correlation is good between the URANS and LNS for the EO with a significant distortion amplitude. This suggests that the distortion field can be treated wave number by wave number in this case, as done in the LNS model. The force coefficient exhibits a higher peak at 50% speed than at 100% speed and also exhibits a sharp drop which will be further examined later in the paper. At 100% speed, a larger discrepancy is observed between LNS and URANS at $Min=5$ than at $Min=4$ and $Min=6$. It can be noted that $Min=5$ is a wave number with low distortion amplitude in the URANS model as seen in Fig. 11, which makes $Cf\theta $ sensitive due to the distortion amplitude normalization. This applies also to $Min>11$ for which the distortion levels taper off as seen in Fig. 11. In contrast, since the LNS model is computed using only one wave number for each simulation, the model gives a cleaner transfer function from distortion to force. The results in Fig. 12 show that the nonlinear and linear models predict the same features in terms of $Cf\theta $ and that there are rapid changes in $Cf\theta $ as a function of the wave number at 50% speed.

### Relation Between Forcing and Acoustic Propagation.

The sharp drop in blade forcing observed in Fig. 12 at $\nu =50%$, and $Min=11$ to 12 could be sensitive to reflections at the boundary condition. Significant changes in forcing level coincide with acoustic modes moving from cut-off wave numbers to cut-on. Waves propagating out from the blade row will reach the boundary at cut-on conditions, and reflections from the boundary could in turn interfere with the solution in the blade passage giving undesirable results. Linearized analyses using fully 3D absorbing boundary conditions [8] are used to check if reflections from the boundary conditions could be a factor in this strong dependency on the acoustic conditions. The 1D and 3D absorbing boundary condition results are compared in Fig. 13. Differences are, however, found to be small and do not affect the result significantly. By this, the results should be interpreted as being due to the aerodynamic interaction of the incoming perturbation with the blade row without secondary interferences.

The unsteady pressure field at 50% speed is in Fig. 14 visualized on the blade surface as well as on a cylindrical surface at 75% span where the interaction of the blade with the aerodynamic wave pattern can be observed. Looking at the response, it is immediately obvious that the interaction for $Min=11$ in Fig. 14(a) is cut off with damped waves in the axial direction whereas for $Min=12$ the solution in Fig. 14(b) is cut on with undamped propagation in the upstream direction. The level of the pressure perturbation for $Min=11$ is higher by factors of 2 or more than for $Min=12$, which accounts for the rapid change in force as shown in Fig. 13. The cut-on mode in the forward direction resulting from the interaction in Fig 14(b) has a periodicity of 6 as predicted by the Tyler–Sofrin relation $M=Min+kB$. The downstream mode in Fig. 14(b) appears visually to be cut on but it is actually cut off with a very weak damping in the axial direction.

A further step may be taken to examine the correlation of the acoustic propagation to the forcing. Calculating the sound power level allows for a quantification of the rate at which energy is expelled from the unsteady pressure field around the blade. The PWL measure as expressed in Ref. [8], “the axial mean value of the area averaged acoustic power,” is made non-dimensional with the reference power $10\u221212W$. The absolute level is, however, not of key importance here. Plotting the PWL and force per wave number in the same graphs in Fig. 15 shows that the drop in forcing at part speed coincides with the transition from cut-off to cut-on on the upstream side of the blade at $\nu =50%$. The drop is not as distinct at $\nu =100%$, and the force tapers off more gradually as the downstream side also becomes cut-on. In Fig. 15(a), it is also confirmed that the downstream acoustic mode for $Min=12$, $\nu =50%$ shown in Fig. 14(b) is cut-off since the PWL is zero.

### Force Map Over a Wide Speed Range.

The simulations are carried out from $\nu =5%$ which is very low subsonic up to $\nu =110%$ where the tip-relative Mach number is supersonic. The aim is to get a broad view of the interaction by extending the range above and well below the intended use.

Using the linearized model a full map of $Cf\theta $ from very low, $\nu =5%$, to high, $\nu =110%$, corrected speeds for all inlet wave numbers, $Min$, up to the blade count, B = 18 is generated (Fig. 16). Increased forcing levels relative to the inlet perturbation occur especially around 50% speed as was seen also in earlier plots. In Fig. 16, the cut-on/cut-off limits for upstream and downstream propagation are overlaid on the map. The 3D propagation condition from [8], based on the same discrete eigenvalue solution as that used in the 3D absorbing boundary condition, is shown as circles. Open circles depict the first found real eigenvalue along the $\nu $-axis for each $Min$, which identifies the mode as a propagating mode. Closed circles is the last point with a complex eigenvalue, representing a cut-off mode decaying exponentially away from the blade. In addition, curves are plotted based on a simpler and more readily computed classic 2D plane wave propagation condition. The 2D conditions are evaluated using the averaged flow conditions over planes approximately one axial chord up- and downstream of the leading and trailing edges, respectively. The circumferential velocity as well as the blade-to-blade pitch, as used in the 2D formulae, is evaluated at 75% span assuming a free vortex velocity distribution yielding the correct mean line circumferential velocity at the evaluation plane. The 2D classic plane wave condition and 3D condition are not identical, but both predict a similar proximity to the region of high response. This clearly indicates that the region of high forcing occurs close to the cut-on/cut-off limit over the entire speed range and that the highest peak occurs approaching the upstream cut-on limit. The acoustic modes that may be generated can be identified using the Tyler–Sofrin relation stipulating that the possible circumferential modes satisfy the relation $M=Min+kB$. The corresponding frequency in the stationary frame is $fs=kBN$ and in the blade relative frame of reference $fr=\u2212MinN$. Here, the important propagating modes both upstream and downstream are specifically found to be the $M=Min\u2212B$ mode, which are modes spinning in the forward rotation direction.

### Effect of Blade Count on Interaction.

The 36-bladed fan that is aerodynamically similar to the 18-bladed fan is created by scaling the blade profile at every radius down by a factor of 0.5. The flow path hub and shroud radii are conserved at the leading and trailing edge so as not to alter the axial velocity ratio of the blade row. The aspect ratio is doubled and the hub and shroud slope angles become steeper as the chord is reduced. The effects on the aerodynamic performance are, as shown earlier, small and the 36-bladed fan can serve for a back-to-back comparison of the effect of blade count on the unsteady aerodynamic response to the incoming perturbations. In the same way as for the 18-bladed fan, the response per wave number is calculated with the linear solver, this time only for the 3 speeds $\upsilon =[10%,50%,100%]$. Figure 17 shows the force coefficients for the 18- and 36-bladed fans as a function of normalized incoming wave number, $Min/B$. With the normalized incoming wave number on the *x*-axis, one can clearly see that the force coefficients $Cf\theta $ are very similar for the two fans. This also means that the peak force associated with the acoustic propagation limit is shifted toward higher incoming wave numbers $Min$ for the 36-bladed fan. The increase in force close to $Min/B=0.6$ at $\upsilon =50%$ is very similar for the two fans. The difference is due to the small differences in the mean flows where the 18-blade fan is very close to becoming cut-on but the 36-blade fan only just has passed to cut-on limit. The good correlation suggests that the blade cascade pitch-to-wavelength ratio dominates the interaction. Since the incoming distortion is not generally affected by the number of blades, this also means that the peak and cut-on limit will occur at a higher $Min$ for a higher blade count. An interesting consequence of this is that for the 36-blade fan the force peaks at $Min=22$ whereas it peaks at $Min=11$ for the 18-blade fan. The implication of this for design is that the peak will more likely occur at engine orders where the distortion amplitudes have tapered off. For noise propagation, the effect is similar, with lower incoming perturbation levels and resulting lower acoustic emissions for a higher blade count.

### Forcing of the First Three Mode Shapes.

Up to this point, the characterization has been made using a generic force without any consideration of the blade vibration modes. Aerodynamic forcing may be assumed unaffected by small amplitude blade vibrations. This means that we can calculate the generalized modal forces onto the modes directly from the same CFD solutions as those already used in the sections above. Using the first three mode shapes as shown in Fig. 5, the generalized force can be calculated as the non-dimensional $Cg$ (Eq. (9)). This gives us a unique map indicating how each mode is affected by the aerodynamics of the interaction with the incoming distortion field (Fig. 18). A notable difference between modes 1 and 3 in Fig. 18 is the location of high forcing. Figure 18(a) shows that Mode 1 exhibits a $Cg$ map resembling that of $Cf\theta $ in Fig. 16. This can be expected as Mode 1 is a bending mode with some torsional content. Mode 2 is predominantly a torsion mode where the largest generalized forces for mode 2 are found at very low $Min$. Mode 3 is more complex, as seen in Fig. 5, with a shape that exhibits bending displacements but also torsional content with large displacements at the leading edge. All three modes have in common that $Cg$ changes rapidly near the transition from cut-off to cut-on.

Figure 19(a) shows the pressure distribution for $Min=3$ where mode 2 is the strongest at $\nu =50%$ in Fig. 18(b). This may be compared to $Min=11$ at $\nu =50%$ (Fig. 14(a)) where mode 1 has its maximum in Fig. 18(a). It is clearly visible that the pressure amplitude in the blade passages is higher for $Min=11$, whereas the distribution along the chord on each side of a blade is more uniform at $Min=11$ than at $Min=3$. For mode 2, the torsion mode, a signed difference in net pressure difference from leading to trailing edge, produces a torque around the twisting axis of the mode. This can be deduced by considering Eq. (8). For a thin blade such as the fan, the displacement of suction and pressure side are virtually the same for a given location *x* and *r*. In the case of mode 1, the displacement does not change sign along the chord and therefore only the difference of the pressure from suction to pressure side becomes important, and some effect of the pressure perturbation is canceled. Figure 19(b) shows the pressure distribution for $Min=9$ which is away from the cut-on limit, but where the Mode 1 generally has a high forcing level. This wavenumber index is specific as it results in an odd-even response on the blades as the blade count is 18. The pressure distribution is very regular with high or low pressure alternating in the flow passages almost but in a similar way along the chord. This effectively generates force on bending modes, but not so much on the torsion mode. The pattern itself with high-low pressures inside the blade passages for $Min=9$ in Fig. 19(b) is relatively similar to $Min=11$ in Fig. 14(a). Noting also the figure scale, however, the levels are significantly lower for $Min=9$.

## 4 Discussion

The results show that a simplification of the complexity in this case (full 360 nonlinear to linearized sector model) retains most of the physics related to the harmonic forces due to inlet distortion. Simplifications include the incoming distortion, linearization of the flow, and removal of the outlet guide vanes. This works well in the present case but may not be universal. In other configurations with higher distortion amplitudes or operating conditions nonlinear effects or effects of stronger vortices in the ingested flow may require other forms of analyses. It is also known that the acoustic impedance from the intake can interfere with the aeromechanics, as in the case of the “flutter bite” described in Ref. [12].

The map of the forces is computed over a wide speed range for total pressure excitations of all engine orders individually up to the blade count *B* = 18. This reveals a strong dependency of the force on the aeroacoustic conditions, given by the corrected speed and the wavenumber. Increasing the running speed, the aerodynamic force rises sharply before dropping down to lower levels as the cut-on condition is reached, and propagating acoustic waves appear. The relative effect on the forcing is found to be strongest at part speed.

## 5 Conclusion

The interaction of total pressure distortion with a fan is studied using numerical analyses. Aeroacoustic interaction is found to have a large effect on the unsteady aerodynamic blade forces. It is shown that in particular, the upstream propagation affects the forcing levels. This is most strongly seen at part speed where the normalized harmonic force rises sharply as the upstream cut-on limit is approached before dropping suddenly as the corrected speed or engine order increases. The unsteady pressure in the blade passage is observed to exhibit the same increase in level followed by a sudden drop once the cut-on limit is passed and undamped pressure waves propagating out from the blade row appear. Intuitively, it could be expected that high noise levels from the interaction would be associated with high forcing levels. The results found here would indicate that high tonal noise escaping from the fan would rather be a sign that the highest levels of forcing have been passed. By comparing the results for a similar fan with a higher blade count, it is found the interaction is dominated by the cascade properties of the fan blade.

## Acknowledgment

The authors wish to thank GKN Aerospace for allowing the publication of this research, and the use of background knowledge on fan design.

## Funding Data

This research was funded by the Swedish National Aeronautics Research Programme NFFP, Grant No. 2019-02759, and GKN Aerospace.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

- $k$ =
harmonic multiplier

- $p$ =
pressure

- $r$ =
radius

- $t$ =
time

- $\nu $ =
ratio of corrected speed to design corrected speed

- $x$ =
distance in axial direction

- $B$ =
rotor blade count

- $C$ =
blade chord

- $L$ =
blade length along leading edge

- $M$ =
wave number resulting from interaction

- $U$ =
blade linear speed at the mean line radius

- $cm$ =
mean meridional velocity

- $di$ =
local displacement vector

- $fi$ =
local force vector

- $fr$ =
frequency in relative frame

- $fs$ =
frequency in stationary frame

- $ni$ =
normal vector

- $Cf\theta $ =
normalized force in the circumferential direction

- $Cg$ =
normalized force projected on a mode

- $DM$ =
normalized distortion wave amplitude

- $Fg$ =
generalized force

- $F\theta $ =
force in the circumferential direction

- $Min$ =
wave number of incoming distortion

- $Ps$ =
static pressure

- $Pt$ =
total pressure in stationary frame

- $Pt,in$ =
total pressure in stationary frame at inlet

- $pt,M\u2032$ =
total pressure perturbation at the inlet boundary

- $EO$ =
engine order

- $Ma$ =
Mach number

- $NR$ =
corrected shaft speed

- $NRADP$ =
corrected shaft speed at ADP

- $PRtt$ =
total-to-total pressure ratio

- $WR$ =
corrected mass flow

- $\theta $ =
circumferential angle

- $\tau ij$ =
shear stress tensor

- $\phi $ =
flow coefficient $cm/U$

- $\varphi i$ =
mode displacement vector

- $\psi $ =
load coefficient $\Delta ht/U2$

- ADP =
aerodynamic design point

- BLI =
boundary layer ingestion

- CFD =
computational fluid dynamics

- EO =
engine order

- GMRES =
generalized minimal residual

- HCF =
high cycle fatigue

- LNS =
linearized Navier–Stokes model

- OGV =
outlet guide vane

- PWL =
acoustic power level

- RANS =
Reynolds averaged Navier–Stokes model

- SLS =
sea-level static

- URANS =
unsteady Reynolds averaged Navier–Stokes model

- WL =
Working Line