Bernd Beirow
email: beirow@b-tu.de

Felix Figaschewsky

Arnold Kühhorn

Alfons Bornhorn

Vibration Analysis of an Axial Turbine Blisk with Optimized Intentional Mistuning Pattern

Keywords:

Aiming to limit the forced response of an axial turbine blisk for ship Diesel engine applications efforts have been made to increase the aerodynamic damping contribution for the most critical modes. In this regard the potential of intentional mistuning is investigated since it offers the opportunity to ensure a safe operation without a severe loss of aerodynamic performance. Genetic algorithms have been chosen to derive an optimized mistuning pattern. In order to keep the manufacturing effort within a limit only two possible blade geometries are allowed which means that an integer optimization problem has been formulated. For the purpose of demonstrating the benefit of the intentional mistuning pattern found, two blisk prototypes have been manufactured: One with and another one without employing intentional mistuning for purposes of comparison. Furthermore, this offers the opportunity for an experimental determination of mistuning being really manufactured and other modal properties as well. The experimental data basis is employed to update structural models which are well suited to demonstrate the forced response reduction under operational conditions.

INTRODUCTION

It is well known in the turbomachinery community that unavoidable variations of mechanical properties from blade to blade, which are denoted as mistuning, can cause frequency splitting and severe magnifications of the forced response compared to the regular design with identical blades. On that account researchers have been occupied with this problem for many years. In this regard the work of Whitehead [START_REF] Whitehead | Effect of Mistuning on the Vibration of Turbomachine Blades Induced by Wakes[END_REF] deserves a special mentioning, since it allows for simply calculating a very conservative limit of the maximum response magnification, which only depends on the number of blades. Several years later, Martel and Corral [START_REF] Martel | Asymptotic Description of Maximum Mistuning Amplification of Bladed Disk Forced Response[END_REF] derived an enhanced and less conservative limit by taking into account the modal coupling within the blade mode family considered. Most recently Figaschewsky and Kühhorn [START_REF] Figaschewsky | Analysis of Mistuned Blade Vibrations Based on Normally Distributed Blade Individual Natural Frequencies[END_REF] presented a further modification for calculating the worst expectable forced response by additionally considering the mistuning strength. Their approach is covering the before mentioned limits [START_REF] Whitehead | Effect of Mistuning on the Vibration of Turbomachine Blades Induced by Wakes[END_REF] [START_REF] Martel | Asymptotic Description of Maximum Mistuning Amplification of Bladed Disk Forced Response[END_REF] as special cases. Petrov and Ewins [START_REF] Petrov | Analysis of the Worst Mistuning Patterns in Bladed Disk Assemblies[END_REF] as well as Chan [START_REF] Chan | Variability of Blade Vibration in Mistuned Bladed Discs[END_REF] did apply optimization algorithms to find mistuning patterns yielding the worst forced response. However, the maximum forced response given in the majority of publications (e.g. [START_REF] Bladh | Dynamic Response Prediction for a Mistuned Industrial Turbomachinery Rotor Using Reduced-Order Modeling[END_REF] and [START_REF] Judge | Experimental Investigations of Mode Localization and Forced Response Amplitude Magnification for a Mistuned Bladed Disk[END_REF] ) is calculated by employing probabilistic methods [START_REF] Bladh | Dynamic Response Prediction for a Mistuned Industrial Turbomachinery Rotor Using Reduced-Order Modeling[END_REF] or really measured mistuning patterns [START_REF] Judge | Experimental Investigations of Mode Localization and Forced Response Amplitude Magnification for a Mistuned Bladed Disk[END_REF] . Commonly, the maximum response magnification take values clearly below those computed in [1]- [5] .

Regarding fundamental blade modes, in particular low engine order excitations (LEO) are well known to cause forced vibrations of blade integrated disks in turbocharger applications. Hence, LEOs have to be considered as a relevant source of fatigue. Recently, efforts have been spent to mitigate the forced response by means of employing intentional mistuning patterns. This has proved to be a promising way by both Petrov [START_REF] Petrov | Reduction of Forced Response Levels for Bladed Discs by Mistuning: Overview of the Phenomenon[END_REF] and Schoenenborn et al. [START_REF] Schoenenborn | Contribution to Free and Forced Vibration Analysis of an Intentionally Mistuned Blisk[END_REF] if first the dedicated travelling wave mode of the tuned counterpart is weakly damped compared to most of the other travelling wave modes and second that sufficiently high differences are existing between the maximum and the minimum of inter blade phase angle-dependent aerodynamic modal damping ratios. Similar results are achieved in [START_REF] Beirow | Optimization-Aided Forced Response Analysis of a Mistuned Compressor Blisk[END_REF] for a high pressure compressor blisk, where a mistuning pattern has been derived by employing genetic algorithms which effects a forced response reduction down to a level of 52 % compared to the perfect design with identical blades.

In this paper, genetic algorithms are applied as well to design an optimized intentional mistuning pattern for an axial turbine blisk with 41 blades namely to reduce the forced response of the fundamental flap mode clearly below that of the tuned counterpart without severely increasing the response of higher modes. Subset of nominal system mode (SNM) [START_REF] Yang | A Reduced-Order Model of Mistuning Using a Subset of Nominal System Modes[END_REF] models are employed for that purpose in which the aeroelastic coupling is considered by means of aerodynamic influence coefficients (AIC) as suggested by Kahl [START_REF] Kahl | Aeroelastic Effects of Mistuning and Coupling in Turbomachinery Bladings[END_REF] . Two prototypes have been manufactured to prove the suitability of the approach: A first one with the optimized intentional mistuning pattern (Fig. 1) and another one with theoretically identical blades. The necessary differences of blade alone frequencies from blade to blade in case of the intentionally mistuned blisk have been accomplished by intended geometry variations. Since additional random mistuning due to the process of manufacture is unavoidable, disparities in modal properties are expected between design intention and really manufactured blisk. That is why experimental analyses are carried out in order to identify the differences between intended designs and really manufactured blisks in terms of determining frequency based mistuning. Finally, updated structural models are used for numerical forced response analyses in order to prove the robustness of the intentional mistuning pattern with respect to additional random mistuning.

BASIC VIBRATION ANALYSES

Basic numerical analyses have been carried out by means of a finite element sector model (Fig. 2) with approximately 300.000 DOF in order to get both an overview about the basic natural vibration characteristics of the blisk with identical blades according to the original design and to prepare the input for SNM-models to be employed later on. Exemplary computational results (Fig. 3) are showing two typical blisk modes which are denoted as cyclic symmetry modes (CSM) 2 and 5 in compliance to the number of nodal diameter lines, which are separating sectors of identical size. If mistuning is present, the cyclic symmetry will get lost and the sectors will differ. As far as the original characteristic of the shape is still recognizable, such a mode is denoted as modified cyclic symmetry mode (MCSM). A further distinction of modes is affecting the blades, which are most sensitive with respect to possible forced responses due to aerodynamic excitation. It is common to group blade modes (Fig. 4) in blade mode families (BF). The natural frequencies of modes grouped in a BF are commonly differing in dependence on the modal coupling.

The Campbell diagram shown in Fig. 5 gives an impression about the most substantial resonance crossings of blade modes and engine orders (EO). The consideration of both rotational and temperature effects yield a slight decrease of the blade modes (solid black) with increasing engine speed. Preceding numerical analyses [START_REF] Figaschewsky | Probabilistic Analysis of Low Engine Order Excitation Due to Geometric Perturbations of Upstream Nozzle Guide Vanes[END_REF] have shown that small perturbations of the nominal NGV-geometry are severely contributing to the forcing of the fundamental first blade mode (M1). Hence, the LEO-crossing occurring at 90% speed, namely EO6, is highlighted in Fig. 5 and considered in more detail in this paper.

OPTIMIZATION OF INTENTIONAL MISTUNING PATTERN

Aiming to find an optimized intentional mistuning pattern for a turbine blisk regarding the relevant EO 6 excitation of the first mode (M1) a minimization of the greatest possible forced response is formulated as objective function. In order to keep the manufacturing effort within a limit only two possible blade geometries are allowed which means that an integer optimization problem is formulated. Hence, the dimension of the design space takes a value of 2 41 or about 2.2*10

12 respectively. Due to the great number of design variables and the non-linear objective function it is hardly possible to find the global minimum. That is why genetic algorithms have been employed for this pure deterministic optimization with an in-house code which is based on SNMmodels.

According to the results of the optimization one blisk prototype has been manufactured with two different blade geometries, another one with actually identical blades. In consequence, two different blade frequencies are theoretically expected in case of the intentionally mistuned blisk, which are exemplarily shown for the first flap mode as relative frequency deviation in Fig. 6a (blue curve) whereas identical blade frequencies should appear for the tuned blisk (Fig. 6b). However, in reality deviations compared with the design intentions are unavoidable (Fig. 6, red curves), e.g. caused by wear of the milling tool or allowable tolerances in general. For details regarding the identification of real mistuning please refer to section 3.1.

EXPERIMENTAL ANALYSES 3.1 Identification of Frequency Mistuning

Real frequency mistuning patterns are experimentally determined for both prototypes applying blade by blade tests according to a patented approach introduced by Kühhorn and Beirow [START_REF] Kühhorn | Method for Determining Blade Mistuning on Integrally Manufactured Rotor Wheels[END_REF] . In detail miniature hammer excitation is employed for each blade of the blisk step by step. Simultaneously the vibration velocity response of the same blade which is currently excited is recorded non-intrusively by means of laser Doppler vibrometry (LDV). The hammer itself is fixed in a holding device and released by means of electromagnetic control, which enables impacts at the exactly same position and with identical intensity as well (Fig. 7). The blisks are put on foam rubber while testing in order to accomplish approximately free support conditions. Additional mass detuning is employed for the purpose of isolating blade dominated frequencies in the frequency response functions (FRF) by putting individual masses on every blade except the one which is currently excited. In this manner, the slightly disturbed cyclic symmetry due to mistuning is completely destroyed and modal decoupling of blade dominated frequencies is effected by choosing both the quantity and the positioning of the additional mass detuning in a skilful manner depending on the blade mode to be excited. Further efforts are carried on with regard to find an ideal position of excitation which usually represents a compromise between the frequency range to be excited, the mode of interest and the flexibility of the blade. Finally, blade dominated frequencies have to be read out of the FRFs yielding frequency mistuning distributions. Later on these mode-individual mistuning patterns are used as input information for updating structural models.

Frequency mistuning patterns derived in the aforementioned way are displayed in Figs. 8 and9 for the blade families 1, 2, 3, 5 and 7 for both the blisk with and without intentional mistuning. In general, the mistuning distributions are differing from mode to mode. Starting with the intentionally mistuned blisk and the first blade mode on which the optimization had been focused on, indeed deviations are occurring but the intended mistuning pattern is still largely visible (Fig. 6a). The greatest mismatch can be found at blade 40 and its direct neighbors. The distributions of modes 2 and 3 are quite similar (Fig. 8). Although the pattern of mode 1 is still clearly recognizable at mode 5 increasing differences are appearing. In particular, a drop of the frequency level from blade 34 to 35 is conspicuous, which becomes extreme for mode 7 (~3 %), where the overall shape of the mode 1 distribution cannot be recognized any more. In case of the blisk without intentional mistuning similarities regarding the same modes are hardly visible. However, as expected the mistuning strength takes significantly lower values as compared with the intentionally mistuned blisk.

Model Updating

The experimental mistuning data is processed for the purpose of updating structural models. SNM-models have been chosen since they are featuring an extreme low number of DOF and allow for a very simple consideration of blade mistuning. However, SNM-models need to be updated individually for each blade mode k. Adjusting the Young's modulus E k,i of every blade i according to the measured distributions of relative blade dominated frequency deviations is a well established technique in this regard [START_REF] Hönisch | Modelling of Industrial Blade Integrated Disks (Blisks) with Regard to Mistuning[END_REF] . According to the solution of Kirchhoff-plate's differential equation namely

i k, i k, E f (1)
it is assumed that the connectivity between deviations of blade dominated frequencies and blade Young's modules of blade i and BF k can be approximated by

i k, i k, E f ∆ ≈ ∆ 2 1 . ( 2
)
If required an additional correction of possible mean frequency shifts can be provided by adjusting the global structural density.

Experimental Modal Analyses

The accomplishment of experimental modal analyses is focused on both the determination of modal data for validating the SNM-models by means of mode shape involutes and to provide modal damping ratios for later forced response analyses in dependence on state variables such as the air density. Hence, the experiments are carried inside a pressure vessel starting at technical vacuum (1 mbar) up to room conditions (1013 mbar). Deviating from the mistuning identification procedure described before piezoelectric excitation of the blisks (Fig. 1) at the disk and LDV response measurement at the blades is applied to derive FRF without any additional mass detuning. Subsequently, MDOF fitting methods have been employed to extract modal ratios.

Examples of experimentally determined FRF are given in Fig. 10 for the BF 3 region of both blisk prototypes. Although the dominance of particular MCSM-peaks is kept, peak frequencies and peak characteristics are clearly differing. In particular the range between MCSM 1 and 2 is significantly varying due to an increasing number of localized mode shapes appearing for the intentionally mistuned blisk (Fig. 10a). The peak labeled with 'lok' is dedicated to the mode shape involute shown in Fig. 15b.

The acoustic impedance Z F (Eq. 3) has been chosen regarding the assessment of the dependency of modal damping ratios and natural frequencies on ambient conditions since it allows for considering the effect of temperature apart from ambient pressure. In this regard the impact of ambient air generally yields linearly increasing modal damping ratios with increasing acoustic impedance (Fig. 11a). The greatest damping ratios and gradients as well are surprisingly found for BF 3, where the modal damping at 1013 mbar (Z F = 413.48 kg/s/m² at ϑ º 20°C) is 10 times greater than the value at technical vacuum (1 mbar). Comparably low magnifications ranging between 2 and 5 are found for BF 1 and 2. This can be explained by an increasing contribution of aero-acoustic interaction between adjacent blades due to their lower natural frequencies [START_REF] Beirow | Simplified Determination of Aerodynamic Damping for Bladed Rotors. Part 1: Experimental Validation at Rest[END_REF] .

In addition very moderately decreasing natural frequencies are noticeable with increasing pressure or acoustic impedance, respectively due to the contribution of co-vibrating air masses as exemplarily shown for MCSM 0 in Fig. 11b. As found before for the modal damping ratios the greatest change occurs for BF 3.

T R T R p Z S S F ⋅ ⋅ ⋅ ⋅ = κ
(3)

Model Validation

SNM-models have been updated for BF 1, 2 and 3 for both the intentionally mistuned and the actually tuned blisk. The validation of SNM-models is accomplished by comparing computed mode shape involutes with those derived from experimental modal analyses. All in all Figures 12131415are showing a largely satisfying match for a selection of modes, in particular those representing BF 1 (Figs. 12, 13). This applies to both, modes which are similar to modes expected for the tuned reference (MCSM) and strongly localized mode shapes as seen in Figs. 12b and13b. Some characteristics found in the curves are worth to be considered in a deeper discussion. The fluctuations appearing between blades 23 and 30 (Fig. 12c, BF 1 / MCSM 1) are most likely directly correlated with the underlying intentional mistuning distribution (Fig. 6a), which features alternating blade dominated frequencies in the same region. In this regard it has to be mentioned that the frequency splitting caused by mistuning takes the greatest value within BF 1. A likewise bumpy characteristic of a mode shape involute is also found for BF 3 / MCSM 2 (Fig. 15c) which again features a great frequency splitting of dedicated natural frequencies.

FORCED RESPONSE ANALYSES AND ROBUSTNESS

Forced response analyses are focused on an EO 6 excitation the fundamental mode at 90 % speed as indicated in the Campbell diagram (Fig. 5). The aeroelastic coupling between blades is considered by employing AICs, which have been derived from a CFD-simulation regarding the motion of one reference blade in an originally stationary flow field. Following the AICs are incorporated into the SNM, which is only valid for one BF and the operational conditions presumed during the computation of AICs. Small deflections and linear conditions are required within the AIC-approach, in which the intensity of aeroelastic coupling depends on the blade distance. By formulating an eigenvalue problem based on SNM including AICs, aerodynamic damping ratios in dependence on the inter blade phase angle (IBPA) or the travelling wave mode (TWM) respectively can be calculated (Fig. 16). The run of the damping curve is promising with respect to take advantage of mistuning for increasing the resulting aerodynamic damping since it meets the target that both the damping dedicated to the TWM to be excited in case of a tuned blisk (TWM -6) is weakly damped compared to most others and a large difference between maximum and minimum damping becomes apparent. The ratio takes a value of more than 10 in the case under consideration:

max ζ a / min ζ a = 10.17 (4)
For the forced response analysis four cases are distinguished:

• Ideally tuned blisk • 'Tuned' blisk as manufactured • Intentionally mistuned blisk according to optimization • Intentionally mistuned blisk as manufactured Results regarding the worst ODS of each case are revealed in Fig. 17. The value γ denotes the maximum magnification of the forced response compared to the tuned reference. As can be seen Fig. 17a the ODS dedicated to the maximum tuned response takes a pure sine shape. The sine shape is appearing slightly disturbed for the really manufactured 'tuned' blisk (Fig. 17b) and a moderate increase of maximum magnification is indicated (γ max = 1.10). For the intentional mistuning pattern obtained from optimization the maximum forced response will be tremendously decreased down to a level of 29.4% (γ max = 0.294, Fig. 17c) compared to the tuned counterpart (100%). This is a consequence of an increase of the resulting aerodynamic damping ratio by a factor greater than 4 which in turn is explainable with the superposition of the forced response by several TWM instead of only the 6 th one as it would occur in case of a tuned blisk (Fig. 18). Since the TWM -6 is dedicated to an aerodynamic damping ratio close to the minimum (Fig. 16), nearly all other TWM participating to the response are contributing additional damping. The resulting aerodynamic damping [START_REF] Beirow | Modal Analyses of an Axial Turbine Blisk with Intentional Mistuning[END_REF] can be computed by

which takes a more than 4 times greater value for the optimized mistuning pattern compared to the tuned reference (Table 1). In case of the intentionally mistuned blisk as manufactured even more damping has been activated due to the TWM contribution. Nevertheless, a part of the response reduction gets lost (γ max = 0.376) since the response magnifying effect of unavoidable random mistuning is working against the effect of enhanced damping. Additional forced response analyses have been carried out for purposes of comparison in which again EO 6 excitations are considered but the aerodynamic damping is kept constant corresponding to the tuned blisk (ζ a,norm = 0.698). The results shown in Fig. 19 and Table 1 (last column) as well are indicating that without reducing the resulting aerodynamic damping no reduction of the maximum forced response beyond that of the tuned reference is computed but a general increase. Indeed the difference for the really 'tuned' blisk is small whereas a strong magnification of the response is found for the blisks with intentional mistuning (IM). For the optimized design the magnification takes a value of 55.4 % and completely different ODS are computed (Figs. 17c and19c). In contrast, more similar ODS are calculated for the IM as manufactured both featuring a localization at blade 27. The maximum magnification of the response does not exceed Again considering IBPA-dependent damping the robustness of the gain achieved by intentional mistuning is analysed. For this purpose Monte Carlo simulations have been carried out allowing a particular amount of additional Young's modulus variations of each blade accomplished by the use of uniformly distributed random numbers. For a first study a one percent random variation is chosen which conforms to a frequency variation between +/-0.5% (Eq. 2). Figure 20 indicates the maximum displacement magnification (EO 6 at 90 % speed) considering 5000 samples. The greatest magnification is found at blade 7 with γ max = 0.480, which indeed exceeds the maximum of the optimized intentional mistuning design (γ max = 0.294), however clearly remains below the tuned reference (γ max = 1). The 95% percentile takes a maximum value of 0.328 at blade 25. Even if an additional random mistuning of 2% or even 4 % Young's modulus variation is allowed in a second and third analysis with again 5000 samples each, the 95% percentiles do not exceed γ max = 0.358 at the same blade (Fig. Hence, the optimized intentional mistuning pattern proves to be robust towards unavoidable additional random mistuning.

CONCLUSIONS

Genetic algorithms have been employed to identify an optimal intentional mistuning pattern for an axial turbine blisk yielding a minimized forced response in consequence of LEO. The tremendous response minimization (EO 6, 90% speed, BF 1) down to about 30% compared to the tuned reference is mainly achieved by taking advantage from enhancing the aeroelastic interaction in terms of reducing the resulting aerodynamic damping. Both large differences between the IBPA-dependent aerodynamic damping ratios and a small damping ratio of the regularly excited TWM are basic requirements for the success of the approach.

In order to keep the effort for a technical implementation of the mistuning pattern within limits, only two different blade geometries have been admitted for accomplishing the mistuning in this integer optimization problem. With the objective to prove both suitability and robustness of the approach, two blisk prototypes have been manufactured: A first one with the optimized intentional mistuning pattern, a second one without any intentional mistuning. By means of blade by blade measurements the real mistuning could be identified in terms of blade dominated frequencies, which has been subsequently used to update SNM models. These models have been validated by comparing numerical and experimental results of mode shape involutes. Within forced response analyses it could be shown that the effect of unavoidable additional mistuning hardly reduces the gain achieved by the impact of intentional mistuning. In addition, the robustness of the mistuning pattern could be proved by means of Monte Carlo simulations allowing for superimposing additional random mistuning even greater than the intentional mistuning itself.

It could be shown within further experimental investigations carried out for the blisks at rest inside a pressure vessel that the modal damping ratios are linearly increasing with increasing pressure of the ambient air. Modal damping ratios found at 1013 mbar take values which are up to 10 times greater than those determined at technical vacuum. Contrary a slight decrease of natural frequencies has been proved with increasing pressure due to the impact of covibration air mass.

ACKNOWLEDGMENTS

The research was conducted as a part of the joint research program "SEABLISK". The work was supported by the Bundesministerium für Wirtschaft und Energie (BMWi) under FKZ 03SX335B. The authors gratefully acknowledge MAN Diesel & Turbo SE for their support and permission to publish this paper. The responsibility for the content lies solely with its authors.

NOMENCLATURE

Figure 1 .

 1 Figure 1. Blisk Prototype Inside Pressure Vessel

Figure 2 .

 2 Figure 2. FE-Sector Model

Figure 3 .

 3 Figure 3. Magnitude Plots of CSM 2 and 5 (BF 1)

Figure 4 .

 4 Figure 4. Blade Modes 1, 2 and 3

Figure 5 .

 5 Figure 5. Campbell Plot

Figure 6 .

 6 Figure 6. Frequency Mistuning of Fundamental Flap Mode, a) Intentionally Mistuned and b) Tuned[[START_REF] Beirow | Modal Analyses of an Axial Turbine Blisk with Intentional Mistuning[END_REF]

Figure 7 .Figure 8 .

 78 Figure 7. Frequency Mistuning Test with Response Measurement at LE Tip

Figure 9 .

 9 Figure 9. Real Mistuning of Different Blade Modes (Blisk without Intentional Mistuning)

Figure 10 .Figure 11 .

 1011 Figure 10. Experimentally Determined FRF BF 3 at 1013 mbar, a) Intentionally Mistuned and b) Tuned

Figure 12 .Figure 13 .Figure 14 .Figure 15 .Figure 16 .

 1213141516 Figure 12. Measured and Calculated Mode Shape Involutes at LE Tip (BF 1, Intentionally Mistuned Blisk)

Figure 17 .Figure 18 .

 1718 Figure 17. Maximum ODS (EO6, BF 1, 90% Speed, Aerodynamic Damping as Given in Fig. 16). a) Tuned, b) 'Tuned' as Measured, c) Optimum Intentional Mistuning and d) Intentional Mistuning as Measured

γFigure 19 .

 19 Figure 18. TWM-Development of Maximum ODS Given in Fig. 17)

Figure 20 .

 20 Figure 20. Maximum Displacement Magnification (EO 6, BF 1, 90 % Speed, Designed Intentional Mistuning Superimposed by 1% Random Mistuning)

Table 1 .

 1 Increase of damping ratios and response magnification

	Blisk	γ	ζ a,res	ζ a,res	γ (ζ a = 0.698)
				/ζ a,res,tuned
	____________________________________________________________________________________________________________________________________________________________________________________________________
	Tuned (ref.)	1.000 0.698 1	1
	'Tuned' (real)	1.100 1.520 2.18	1.157
	IM (opt.)	0.294 2.867 4.11	1.554
	IM (real)	0.376 3.230 4.63	1.298
	____________________________________________________________________________________________________________________________________________________________________________________________________


Jiahuan Cui

Yunfei Ma

Bryn Ubald

Paul Tucker

O Tatin

Multi-fidelity simulation for a transonic compressor with inflow distortions

Keywords:

e low and high fidelity methods are used to study a transonic fan case. is transonic fan case features a distortion generator upstream of the rotor. e low fidelity methods (immersed boundary method and immersed boundary method with smeared geometry) show promising results against hi-fi simulation (unsteady RANS) and experiments.

e flow in the tip region, especially the interaction between the distorted inflow and tip leakage flow, are investigated in detail. It is found that the tip leakage flow is the dominant flow structure near the casing. It is strong enough to sustain even at the exit of the stator. Apart from analysing the tip leakage flow structures, the stalling mechanism is also explored. As mass flow decreases, more rotor blades are contaminated by the distorted flow. A rotating cell starts to develop shortly a er all the rotor blade passages are contaminated. e rotating cell is found to be swept in circumferential direction at the % speed of the rotor.

INTRODUCTION

Flow distortions in a modern aero engine system are o en induced due to the integration of different components. Inflow distortions can reduce the performance and the stall margin of a compression system [ ]. Nacelle stall and boundary layer ingestion are two common scenarios where inflow distortions are interacting with compressors/fan. In these two scenarios, inlet distortion mainly affects the tip region of the compressors/fan.

Flow in the tip region of the rotor is highly complex and unsteady [ , ]. Due to the clearance between the rotating blade tip and the stationary casing, several vortices are formed. As demonstrated by You and Moin [ ] and also illustrated in figure , tip vortex system consist of ) tip-leakage vortex, ) induced vortex, and ) tip-separation vortex. e tip-leakage vortex is the dominant one.

For a transonic fan/compressor, the tip leakage vortex is strongly affected by the shock. e shock intersects the trajectory of the tip leakage vortex. As the flow coefficient is reduced, the tip leakage vortex strengthens and moves upstream [ ]. When the tip leakage vortex cannot overcome the adverse pressure gradient, the breakdown of the tip leakage occurs. As a result, the blockage in the tip region increases, which further reduces the velocity. As less mass flow travels through the region near the casing, the blade loading decreases. If the mass flow decreases further, the tip leakage vortex starts to travel tangentially towards the adjacent blade leading edge[ ]. is is also termed leading edge tip clearance flow spillage. At this condition, the rotating e stalling process becomes more complicated in the presence of a distorted inflow. Such highly turbulent and complex flow is very challenging to be predicted accurately, even for the state of the art of computational fluid dynamics[ ]. In the current study, a transonic fan (the Darmstadt case) will be used to investigate the interaction between the inflow distortion and the downstream rotating blades. e Darmstadt case is featured by a distortion generator (DG) located two axial chord upstream of the compressor. e DG is a ached to the case and it is designed to generate a large separation to represent a stalled engine nacelle [ ]. e current paper will investigate the general flow features in the tip region affected by the inlet distortion. A range of different levels of fidelity modelling strategies will be used to study different aspects of the flow. In addition, the stalling mechanism under the influence of inlet distortion will be explored.

. CASE SETUP

e Darmstadt transonic compressor case is used in the current study.

e case itself has been studied by many other researchers [ , , ]. It consists of rotor blades and stator blades. e spool speed is , rpm. is gives a total pressure ratio around . at the design condition.

To investigate the inlet distortion on the performance and stability margin of the downstream compressor, a distortion generator is a ached to the casing with a distance of two axial chord length upstream of the rotor leading edge. e height of the distortion generator is designed to be % of rotor span. Such a distorted inflow is large enough to influence the rotor tip region. e computational domain used in the current study is shown in figure . Depending on the fidelity level of the simulation, the blades and distortion generator are modelled by different methods.

High fidelity simulation. A range of unsteady RANS simulations are carried out to study the detailed flow features at different total pressure ratio along the performance line, including the stall point. Figure shows the surface mesh of the blades and the distortion generator. In total, around million cells are used to resolve the whole domain, with million cells for the rotor blades and million for the stator. On average, there are around . million cells for each blade passage.

Low fidelity simulation. At the design stage, the ability to account for the effects of the fan on the flow upstream is extremely beneficial, especially for a short-intake design at the high angle of a ack.

e low fidelity method can accurately capture the potential filed that created by the fan with a cost only a fraction of the hi-fi simulations. e current study performed several low fidelity simulations with the distortion generator resolved by the immersed boundary method (IBM) and the blades by immersed boundary method with smeared geometry (IBMSG). Details of these methods are giving in the Numerical Methods section.

. NUMERICAL METHODS

e numerical solver used by the current study is a Rolls-Royce in-house CFD (Computational Fluid Dynamics) code -HYDRA [ ]. HYDRA is a second order finite volume RANS (Reynolds Average Naiver-Stoke) solver. e solver has been used to a ack a range of changeling problems in aero engines [ , , ]. Here, the IBMSG and IBM are developed within the HYDRA framework. A brief summary of these two methods are given below in this section.

. IBM

IBM is used to avoid meshing the complicated DG geometry in the low fidelity simulation. In the currently study, the feedback force method [ ] is used to calculate the force terms:

f = α ˆt 0 (v -u)dt + β(v -u) ( )
where, v is the intended velocity on the boundary (zero in the current study) and u is the actual local velocity on the boundary. e force term is added to the NS equations so that the actual local velocity is driven to the intended boundary velocity as the solution is converging. e constants, α and β, can affect the convergence speed.

. IBMSG e immersed boundary method with smeared geometry is used to develop a low-order compressor/fan model. is loworder model is useful in cases where calculations for fully resolved blades are extremely expensive. For example, at the design stage of an engine nacelle, sometimes the effects of the compressor/fan on the flow over the nacelle can not be ignored. Under this circumstance, a low-order compressor/fan model is highly valuable. Equation is the basic NS equation with some extra terms for IBMSG.

∂ ∂t ˆΩ λqdΩ+ ˛A λFdS = ˆΩ λ f n dΩ+ ˆΩ λ f p dΩ+ ˆΩ BdΩ ( )
where q are the primitive variables. F are the sum of inviscid and viscous terms. f n are the force terms normal to the blade camber line. f p are the force terms parallel to the blade camber line. λ is the blockage factor. B are the extra terms due to blockage. e detailed description and calculation for the each term can be found in [ ].

.

RESULTS AND DISCUSSIONS

. Validation.

To obtain the characteristic line from numerical simulations, the pressure at the outlet was increased step by step, from the peak efficiency (PE) condition to the near stall (NS) condition. Figure shows the characteristic line at the % speed for experiments (solid line), low fidelity simulations (dots), and high fidelity simulations (squares). In general, both low and high fidelity simulations agree with the experimental results. Although the choking mass flow and the pressure ratio at near stall condition are slightly ( -%) different between the numerical results and the experiments, these differences are marginal and do not compromise the conclusion of the current study. pressure ratio is lower. e distortion generator creates a large separation, which dissipates energy and results a lower total pressure region downstream.

is lower pressure region is convected downstream. At the both rotor and stator exits (figure (b,c)), the lower pressure region is still visible. e distorted region is also extended in circumferential direction at the downstream. As the blade travels from right to le in the figure, at the exit side of the distorted region (the le side of the lower pressure region), the total pressure is increased slightly. is is due to the rotor incidence variation. More details are given in section . .

In figure , the total pressure ratio is plo ed at three different span locations: near hub, mid span and near case.

e effects of the inflow distortion extends to the whole span at the exit of the stator. Both the URANS and IBMSG captures the distorted region, though the averaged total pressure in the IBMSG calculation is higher at the mid span and lower near the case.

. General flow features.

In this and the following sections, results from URANS will be studied in more detail. Figure shows the Q-criterion coloured by Mach number in the region between % -% span. To increase the visibility into the blade passage, only % of span is shown. Also, the annular cascade is transformed to a linear one. e y coordinate in figure is the circumferential distance and the z coordinate is the radius. To calculate the circumferential distance, a fixed radius at . % of span is used rather than the local one. is is to make sure that the blades at the lower and upper bound of the y coordinate are not leaned.

Flow within the separated region immediately downstream of the DG. e distortion generator creates a large separation upstream of the rotor. e separated flow constantly sheds vortices. is results in a highly unsteady region downstream of the DG. In this region, the velocity is lower than in the undistorted region. As a result, the rotor blades experience a variation of the incidence in the circumferential direction.

Since the static pressure in the distorted region is lower, the vortices formed at the circumferential end of the DG converge towards the centre (see figure ). As a result, the circumferential velocity increases at the entry side of the distorted region and decreases at the exit side. Hence, the incidence is higher at the exit and lower at the entry side.

is is the reason why the total pressure is higher at the exit in figure (b,c). Similar results can be found in section . . Flow structures within the rotor blade passage. Upstream of the rotor, the leading edge shocks create vortices on the casing. ese vortices are relatively small compared to the tip leakage vortex. Similar to the findings in the literature, all the three main vortices -induced vortex, tip-leakage vortex and tip-separation vortex are captured in the simulation. However, in the distorted region, the tip leakage vortex breaks down immediately a er the shock. A er the rotor exits the distorted region, the flow recovers to the normal state. Around three/four blade passages are significantly affected by the distortion generator. However, as the mass flow . Flow at peak efficiency and near stall points.

A er understanding the general flow features in the blade passage, the difference between the flow at the peak efficiency point and the near the stall point will be examined. Only the region near the casing is studied here.

Figures and show the flow at % of span for the peak efficiency point and near stall point, respectively. All the quantities are based in the absolute frame. As mentioned before, for both operating points, the distorted flow converges towards the middle. is leads to an increase of incidence at the exit side and an decrease at the entry side of the distorted region. As a result, higher entropy is observed at the exit side in the rotor blade passages (figure (c)). At the peak efficiency point, most entropy is generated in the separated region downstream of the distortion generator. Around four rotor blade passages are contaminated, where entropy generation rate is higher. Consequently, around five stator passages are contaminated. Other loss sources, such as shocks, wakes and secondary flows, can also identified in the figure (d). Among them, wakes are the second dominant source of the entropy generation. Interestingly, the interaction of the rotor wakes and the stator suction surface is also a large source of entropy generation.

At the near stall point, only entropy and entropy generation rate are shown. Note that to highlight the entropy generation within the blade passages, the colouring levels are different from that in figure . At the near stall condition, the axial flow velocity is lower. e convection time for the distorted flow to travel through the rotor is longer. Hence, more rotor blade passages (around ) are contaminated by the distorted flow. Other entropy generation sources also increase due to a higher pressure gradient in the rotor and stator. At this condition, all the boundary layers on the suction surface of the stator are separated, except for the three passages at the entry side of the distortion.

is is, again, due to the lower incidence and lower pressure rise at the entry side. e separated boundary layer is also a large source of entropy generation at near stall condition.

. Stalling process.

As mass flow decreases further, more blade passages will be contaminated. Eventually, the compressor would be fully stalled. Figure

shows the radial velocity signals at % of the span upstream of the rotor. At t 0 , the mass flow rate is around . kg/s. At this mass flow rate, rotor passage between -1 radian to radian are contaminated, where the radial velocity is oscillating more than that at -3 radian to -1 radian. As the mass flow decreases, the contaminated region extends wider in the circumferential direction. At t 1 , the mass flow rate drops to around . . Almost all the rotor blade passages are contaminated. Shortly a er all the blade passages are contaminated, a rotating cell starts to develop at t 2 . It rotates along the circumferential direction at the % of the rotor rotating speed. As the rotating cell starts to develop, the mass flow rate drops faster.

. CONCLUSIONS e current paper demonstrated the applications of low and high fidelity simulations for a transonic fan. e low fidelity methods (IBM and IBMSG) showed promising results against hi-fi simulation (URANS) and experiments. ese low order methods are useful in designing a coupled system where mesh resolved blades are computationally expensive.

For hi-fi simulations, the detailed flow features near the casing were analysed. It was found that the tip leakage flow was stable and strong enough to sustain even at the stator exit. Apart from the analysis of flow structures, the stalling mechanism was also investigated. When mass flow was decreased, more rotor blades are contaminated by the incoming distorted flow. As mass flow reduced to the near stall condition, separation can be observed over most of the stator blade suction surface. A rotating cell started to develop shortly a er all the rotor blade passages were contaminated.

e rotating cell was swept around at the % speed of the rotor.

Figure .

 . Figure . Sketch of the tip vortex system.

Figure .

 . Figure . Computational domain.

Figure .

 . Figure . Mesh for unsteady RANS calculations.

Figure .

 . Figure . Characteristic line for Darmstadt case at % speed; line -experiments, dots: low fidelity simulation (IBM for DG and IBMSG for blades), squares: high fidelity simulations (URANS)

Figure .

 . Figure . Total pressure ratio (π) in circumferential direction at (a) S , (b) S and (c) S .

Figure .

 . Figure . Q-criterion isosurface contoured by absolute Mach number. To have a clear view of the flow structure within the blade passage, the annular cascade is transformed to a linear one as illustrated in the embedded frame.

Figure . Figure .

 .. Figure . Flow at % span at PE point; (a) Mach number, (b) total pressure, (c) entropy, and (d) entropy generation rate.

Figure .

 . Figure . Stalling process: radial velocity signals at the % of span upstream of the rotor tip.

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  Fabian Wartzek, Felix Holzinger, Christoph Brandste er, and Heinz-Peter Schiffer. Realistic inlet distortion pa erns interacting with a transonic compressor stage. In Advances in Simulation of Wing and Nacelle Stall, pages -. Springer, . [ ] F Wartzek, C Brandste er, F Holzinger, and HP Schiffer. Response of a transonic compressor to a massive inlet distortion. In European Turbomachinery Conference, Madrid, Spain, Mar, pages -, . [ ] Barthmes Niehuis Reinhard Haug, Jakob Philipp. Full annulus unsteady cfd simulations on effects of inflow distortions in a transonic axial compressor stage. In International Gas Turbine Congress Tokyo. .

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ACKNOWLEDGMENTS

e authors would like to acknowledge the insightful discussion and input from Dr. Rob Watson, Dr. James Page, Dr. Yangwei Liu and Mr. Callum Mantell.


Dipl.-Ing Matthias Teich

Prof. Dr -Ing Martin Lawerenz

Numerical Simulation of the Rotating Instability in an Annular Compressor Cascade Test Rig

. Although the main characteristics of the rotating instability are captured by the simulation, there are some differences concerning the quantitative parameters, which can be related to the applied total pressure profile at the inlet.

NOMENCLATURE

INTRODUCTION

The basic objectives of designing modern turbo compressors are high efficiency, low costs and a stable and safe operating range. These requirements are connected with highly mechanically and aerodynamically loaded blades. Thereby, the flow is dominated by complex three-dimensional, unsteady processes, which, among other things, are mainly caused by relative motion and boundary layer interaction between blade and side-wall. These so called secondary flows have a significant impact on the stability limit and the operating range. The objective of many experimental and theoretical investigations is to gain a deeper understanding of those phenomena.

Well known flow phenomena which are associated with approaching or crossing the surge line are rotating stall or surge.

In addition to these effects, there is a rotating flow pattern at stable operating condition previous to rotating stall, which was investigated by severeral authors [START_REF] Kameier | Rotating Blade Flow Instability as a Source of Noise in Axial Turbomachines[END_REF][START_REF] Mailach | Rotating Instabilities in an Axial Compressor Originating From the Fluctuating Blade Tip Vortex[END_REF][START_REF] Weidenfeller | Time Resolved Measurements in an Annular Compressor Cascade with High Aerodynamic Loading[END_REF][START_REF] Beselt | Strömungsinduzierter Schall in Turbomaschinen -Die Rotierende Instabilität[END_REF][START_REF] Hermle | Experimentelle Untersuchung der rotierenden Instabilität im Axialverdichtergitter[END_REF]. In the past, different terms have been used to describe such phenomena (tip clearance flow unsteadiness, fluctuating tip clearance flow, small rotating stall, part-span stall, high frequency stall or rotating instability). Since 1990 the term rotating instability, which has been promoted especially by Kameier [START_REF] Kameier | Experimentelle Untersuchung zur Entstehung und Minderung des Blattspitzen-Wirbellärms axialer Strömungsmaschinen[END_REF], Kameier & Neise [START_REF] Kameier | Rotating Blade Flow Instability as a Source of Noise in Axial Turbomachines[END_REF] and Baumgartner [START_REF] Baumgartner | Non-Engine Order Blade Vibration in a High Pressure Compressor[END_REF], has been established. (For the sake of completeness it should be noticed that the term rotating instability has already been used before by Mongeau [START_REF] Mongeau | Experimental study of the mechanism of sound generation by rotating stall in centrifugal turbomachines[END_REF] and Bent [START_REF] Bent | The Influence of Discharge Configuration on the Generation of Broadband Noise in Centrifugal Turbomachinery[END_REF]).

The RI can be described as a self-induced multipassage rotating flow phenomenon at high incidence within a wide Mach number range, which rotates in circumferential direction from the suction side to the pressure side and has a characteristic signature in the frequency spectrum, which is illustrated in figure 1. It is an aerodynamic local flow instability at globally stable conditions and is located near the leading edge at the hub for stators and for rotors at the casing. Concerning the radial extend, it can be specified as a part-span phenomenon.

The rotating instability has been studied experimentally and numerically by several authors and could be identified as the source of noise ( [START_REF] Kameier | Rotating Blade Flow Instability as a Source of Noise in Axial Turbomachines[END_REF][START_REF] Kameier | Experimentelle Untersuchung zur Entstehung und Minderung des Blattspitzen-Wirbellärms axialer Strömungsmaschinen[END_REF]), nonsynchronous blade vibrations ( [START_REF] Baumgartner | Non-Engine Order Blade Vibration in a High Pressure Compressor[END_REF], [START_REF] Thomassin | Blade Tip Clearance Flow and Compressor Nonsynchronous Vibrations: The Jet Core Feedback Theory as the Coupling Mechanism[END_REF]), and a preliminary indicator for rotating stall [START_REF] Vo | Role of Tip Cleanace Flow in Rotating Insabilities and Nonsynchronous Vibrations[END_REF].

Typical for the RI spectra is a broadband enhancement of the amplitudes in the lower frequency region with equidistant peaks. Van Rennings describes this shape as a comb-like pattern [START_REF] Van Rennings | Modelling of the Dynamics and Acoustic Emission of Rotating Instability in an Annular Compressor Cascade[END_REF]. Each peak is connected with a circumferential mode of a different order. In experimental studies of Hermle [START_REF] Hermle | Experimental Analysis of Rotating Instability Using Hot Wire Triggered PIV and Wavelet Transform[END_REF][START_REF] Hermle | UNSTEADY SEC-ONDARY FLOW IN AN ANNULAR COMPRESSOR CASCADE[END_REF], circumferential mode orders of m = 1 up to a mode order of m = 3 could be identified in the frequency range between 61 to 135 Hz for M a = 0.34 and between 65 to 234 Hz for M a = 0.4 (see figure 1). Pardowitz et. al. [START_REF] Pardowitz | Rotating Instability in an Annular Cascade: Detailed Analysis of the Instationary Flow Phenomena[END_REF] conducted unsteady pressure measurements in the upstream section of a similar annular compressor cascade and observed rotating instability modes up to an order of m = 13 in a frequency range between 200 to 900 Hz. nvestigations of Mailach [START_REF] Mailach | Rotating Instabilities in an Axial Compressor Originating From the Fluctuating Blade Tip Vortex[END_REF] confirmed mode order of half the number of blades in the frequency spectrum at about 30% of the first blade passing frequency (1. BPF), which corresponds to 300 Hz.

The assumption that the unsteady clearance flow is the main reason for the RI ( [START_REF] Schrapp | Breakdown of the Tip Clearance Vortex in a Rotor Equivalent Cascade and in a Single-Stage Low-Speed Compressor[END_REF], [START_REF] März | An Experimental and Numerical Investigation Into the Mechanisms of Rotating Instability[END_REF], [START_REF] Inoue | Effect of Tip Clearance on Stall Evolution Process in a Low-Speed Axial Compressor Stage[END_REF]) was rebutted by Beselt [START_REF] Beselt | Influence of the Tip Clearance Size on Rotating Instability in an Axial Compressor Stator[END_REF]. He could verify the presens of RI in an annular compressor cascade with and without a hub clearance. Thus, the clearance flow is not a compellable criterion for the rotating instability. This discovery is in contradiction with the existing explanations ( [START_REF] Mailach | Rotating Instabilities in an Axial Compressor Originating From the Fluctuating Blade Tip Vortex[END_REF], [START_REF] Schrapp | Breakdown of the Tip Clearance Vortex in a Rotor Equivalent Cascade and in a Single-Stage Low-Speed Compressor[END_REF]) of the RI source mechanisms, which emphasize the clearance flow as a necessary element for RI. Today, there is no sufficient theory for explaining the source mechanisms of this flow instability.

Regarding this background, the presented work contributes to the investigation of rotating instability with numerical methods. Therefore, a compressor cascade of an annular test rig, which exhibits RI at specific conditions, is numerically investigated to clearify the question if RI can be simulated with URANS and/or DES and of which quality the main RI-characteristics are.

INVESTIGATED TEST RIG

The investigated test rig is an annular compressor cascade at the Section of Turbomachinery, University of Kassel, which is run in a closed loop. The air supply is ensured by a radial compressor with a maximum pressure ratio of 1.8 and a maximum mass flow of 8 kg/s. The annular compressor cascade consists of three main sections, figure 2. At the first section, an inlet guide vane (IGV), which consists of 15 blades, turns the incomming flow by 55 • . Downstream, very close to the IGV, 15 additional adjustable symmetric profiles allow further variation (±20 • ) of the flow angle. In order to ensure a mixed out flow state, a flow path with a length of 580 mm exists between the first section and the investigated axial compressor stator cascade (2nd section). Further downstream of the compressor cascade, six struts with a NACA0020 profile are installed. The examined compressor stator consists of Hermle [START_REF] Hermle | UNSTEADY SEC-ONDARY FLOW IN AN ANNULAR COMPRESSOR CASCADE[END_REF] and Weidenfeller [START_REF] Weidenfeller | Time Resolved Measurements in an Annular Compressor Cascade with High Aerodynamic Loading[END_REF] at this test rig, it is known, that the RI occurs at an incidence of i ≥ 12.3 • at all investigated Mach numbers (M a = 0.2 to M a = 0.5). Especially at M a = 0.4 the RI is clearly visible in the frequency spectra. For this reason the investigated operating point for the numerical simulations is at a Mach number of M a = 0.4 at an incidence of i = 12, 3 • . This corresponds to a Reynolds number of about Re = 440000. The Mach number and the incidence of the operating point are related to mid span.

Numerical Methodology

The CFD-Software Fine TM /Turbo was used to study the unsteady flow field of the annular compressor cascade at the specific investigated condition, where RI is expected.

Numerical Scheme

The numerical flow solver is based on the compressible timedependent three dimensional Navier-Stokes equations for blockstructured grids. A cell centered finite volume approach is used for the discretization in space with a central scheme for the viscous and inviscid fluxes. In order to get a stable method, a Jameson type [START_REF] Jameson | Numerical Solutions of the Euler Equations by Finite Volume Methods using Runge-Kutta Time-Stepping Schemes[END_REF] damping is applied with 2nd and 4th order derivatives of the conservative variables for the numerical dissipation term. Concerning the time discretization, an explicit multistage Runge-Kutta method is applied in combination with a dual time stepping approach proposed by Jameson [START_REF] Jameson | Time dependent calculations using multigrid, with applications to unsteady flow past airfoils and wings[END_REF]. By introducing a pseudo-time step to the time dependent Navier-Stokes equations, for each physical time step a steady state problem is solved in a pseudo time until a given number of subiterations n sub or a residual reduction of at least three orders of magnitude is reached.

The turbulence is considered in a statistical sense by using the Reynolds-averaged method and by a hybrid RANS/LES approach -a DES-method, [START_REF] Spalart | Comments on the Feasibility of LES for Wings, and on a Hybrid RANS/LES Approach[END_REF]. Based on the turbulent viscosity approach from Boussinesq, a one equation turbulence model of Spalart Allmaras [START_REF] Spalart | A One-Equation Turbulence Model for Aerodynamic Flows[END_REF] was applied to model the unknown components of the Reynolds stress tensor for URANS and DES. The DES is a hybrid approach, which uses a RANS formulation for near-wall regions and acts in a LES-like manner for the other regions. This is done by replacing the wall distance d in the destruction term of the Spalart Allmaras turbulence model by a new length scale d, which is dependent on the local grid size ∆ and the wall distance

d = min(d, C DES ∆). (1)
C DES is a DES-parameter with a typical value of 0.65. Accelaration techniques, such as implicit residual smoothing, local time stepping and multigrid, which result in a faster and stable solution, are being applied.

Computational Grid

The computational grid of the full cascade is shown in figure 3 in the lower part. For the unsteady simulations the grid covers all 17 blade passages and extends in the axial direction from 0.65 chord length upstream of the leading edge (LE) of the stator vanes to 1.8 chord length downstream the trailing edges (TE). One blade passage consists of nine blocks and contains about 1.17 million grid points, whereas an O-type grid was generated around the blade surrounded by four Htype grids. The depicted plains MP1 and MP2 indicate the position of experimental measurements. 21 points have been used to discretize the clearance gap at the hub in spanwise direction. The size of the wall nearest grid cells was set to 0.0012 mm to ensure an average value of y + ≤ 1. Thus, the boundary layer is resolved within the viscous sublayer and

Boundary Conditions and Numerical Set Up

Radial inflow and outflow profiles for the investigated operating point were measured between 10% and 90% of the channel height by a pneumatic five hole probe mounted 15 mm upstream of the blade LE and 20 mm behind the blade TE [START_REF] Hermle | Experimental Analysis of Rotating Instability Using Hot Wire Triggered PIV and Wavelet Transform[END_REF]. Area averaged radial profiles of the measurements were used as boundary conditions for the numerical set up. The radial distribution of the area averaged total pressure p t and the inflow angle α 1 were specified between 10 und 90% of the span, whereas in the near wall regions the 1/7 power law ist applied to ensure a reasonable physical distribution. As it is depicted in figure 3, there is a certain distance between the inlet and measurement plain MP1. If one would use the experimentally measured profiles at MP1 as boundary counditions at the inlet, this would lead to wrong results. For this reason, a correction of the total pressure distribution at the inlet is used to ensure an almost perfect match between experiment and numeric at MP1, which could be achieved, see figure 4 a).

As boundary conditions at the outlet a static pressure of p = 93825 Pa has been prescribed at mid span, which was taken from the measurements. Concerning the chosen turbulence model, one has to specify the turbulent kinetic energy k and the turbulent dissipation rate ǫ at the inlet. These quantities were estimated by the following equations. B The averaged velocity in measurement plain MP1 leads together with turbulent intensity of 2% to a tubulent kinetik energy of 10.30 m 2 /s 2 for the analysed operating point. The following equation can be used to calculate the turbulent dissipation rate ǫ using C µ = 0, 09, the ratio of dynamic viscosity and turbulent dynamic viscosity η/η T with a typical value of 1/50 and a reference value for the density ρ ref .

ǫ = C µ η η T ρ ref k 2 η = 12646.38 m 2 /s 3 (2)
The chosen time step is dependent on the investigated phenomenon, the applied method (URANS or DES) and the grid size. The highest frequency of the RI at the investigated compressor stator cascade for M a = 0.4 and i = 12.3 • is around 233 Hz [START_REF] Hermle | UNSTEADY SEC-ONDARY FLOW IN AN ANNULAR COMPRESSOR CASCADE[END_REF]. In order to resolve one period by 20 time steps a physical time step size of ≈ 2.15 • 10 -4 s is required. Concerning the grid size, a DES time step sensitivity study of one blade passage simulations was done by the authors [START_REF] Teich | Beitrag zur numerischen Simulation selbsterregter Strömungsinstabilitäten in Axialverdichtern[END_REF].

To resulve at least 75% of the turbulent kinetic energy spectrum, a time step of at least ∆t = 1.0 • 10 -5 s has to be applied for a DES computation on a grid with about 1.17 million cells per passage. An additional URANS time step sensitivity study was not carried out in [START_REF] Teich | Beitrag zur numerischen Simulation selbsterregter Strömungsinstabilitäten in Axialverdichtern[END_REF], since DES requirements are stronger. For this reason, the chosen time step is set to ∆t = 1 • 10 -5 s for the URANS and DES computation in combination with 100 subiterations for each time step to guarantee a converged solution. A good initial solution for the unsteady computations is provided by a steady RANS simulation with 7 000 iterations. To get well resolved frequency spectra a lot of time steps had to be performed. All in all 18 364 iterations were carried out for the DES computation and 11 432 iterations for the URANS, whereas the first 2 000 iterations of both computations are neglected with respect to the analysis, since it is known that there is a transient at the beginning of every unsteady computation.

All simulations were conducted at the Lichtenberg-Cluster at TU Darmstadt, which allows parallel computation with up to several hundred processor units. For the presented unsteady computations of the compressor stator cascade up to 128 CPU units of the type Intel R Processor E5-2680 v3 were used.

RESULTS

Validity of Numerical Model

In order to assess the validity of the numerical model, computed mass flow as well as radial profiles of flow quantities upstream of the leading edge at MP1 and downstream of the trailing edge at MP2 are compared to experimental results.

The mass flow of URANS, DES and experiment are listed in table 3. Both numerically calculated mass flows are time averaged for all simulated time steps (≈ 20 000) and correspond well to the experiment, with URANS slightly worse.

The standard deviation σ indicates that the whole cascade is in a globally stable operating point.

In figure 4 the experimental and numerical (URANS and DES) results for i = 12 • show an overall good agreement, especially the distributions at the inlet plain MP1 are well described by the numerical results, see figure 4 a), b) and c). Due to a dominant hub corner stall, which exists even at the design point [START_REF] Weidenfeller | Time Resolved Measurements in an Annular Compressor Cascade with High Aerodynamic Loading[END_REF] and extends to around 70% the channel height, the flow can not follow the blade geometry. Thus, the outflow angle α 2 is larger in this region, which means less flow turning. This can be seen in figure 4 f) for the experiment and for both computations. However, there are still some discrepancies between the simulations and the experiment at MP2 for 0.1 < r/h < 0.4. Due to the high deviation of the flow at this axial position, the probe measurement could have been out of the calibrated range Concerning the flow angle α 2 the URANS computation predicts a much lower angle compared to DES and the experiment. Due to higher dissipation in the URANS formulation, the development of complex flow structures, like a hub corner stall, is inhibited. Thus, flow is less disturbed and the outflow angle α 2 shows a reduced underturning. The comparison illustrates, that Fine TM /Turbo is capable to predict the main flow features with both numerical methods (URANS, DES). Considering the complex flow field, especially at MP2, the results agree fairly well. Additionally, the validity of the experimental results is limited, due to the interaction of the 5-hole probe with the hub, probe size relative to gradients in the hub region and the calibration procedure itself.

Unsteady Pressure Field

To examine the flow field of the simulations, the static pressure signals of 360 circumferential distributed virtual sensors at eight different radial locations (0%, 10%, 20%, 40%, 60%, 80%, 90% and 100% channel height) for ten different axial positions are used, which leads to 28 000 points to tap the pressure fluctuations, see figure 5. RI can be detected by analysing the unsteady pressure field in front of the leading edge. Hence, only the axial position 3 is taken into account, which corresponds to the axial position of the used pressure sensor in the experiment, [START_REF] Hermle | Experimental Analysis of Rotating Instability Using Hot Wire Triggered PIV and Wavelet Transform[END_REF]. In figure 6 This kind of data visualisation is very useful to get an idea of what the flow field is like and to show differences of both numerical methods. That is why only a fraction of the whole simulation time is shown. As one can see in the first row of figure 6, there is a rotating pattern for DES and URANS at the hub. It is rotating into positive circumferential direction, which means from suction side to pressure side, compare figure 5. The rotating flow pattern is much more obvious for the URANS simulation. There are almost straight lines over the whole circumference. Concerning the DES computation, the pressure pattern is more scattered and smeared. In contrast to the URANS simulation, more lines can be identified over a fraction of the circumference going also into positive circumferential direction but with different gradients. With the help of equation ( 4), the frequency of a rotating pattern can be estimated, if the flow pattern is clearly visible. Thus, a frequency of about f rot,URANS ≈ 285 Hz can be calculated for the URANS. Estimating frequencies of the DES with equation (4) is difficult. Due to the smeared flow structure it is not easy to measure correct values for ∆ϕ and ∆t. A closer look of single pressure signals is given in the last row in figure 6. Three signals at ϕ =10 (red), 40 (green) and 70 • (blue) are displayed and shifted against each other to keep them apart. The URANS signals (right figure) are very smooth compared to the DES (left), which shows more fluctuations.

f rot = ω 2π = ∆ϕ ∆t2π (4)
With increasing channel height, the rotating flow pattern seems to disappear in both computations, see figure 6 second and third row. In contrast to the static pressure field at the hub 17 equidistant vertical lines of higher static pressure can be seen for URANS and DES. These lines are connected to the blade number of the cascade and are caused by an upstream effect of these blades. Due to a higher Mach number and a higher velocity for 50% and 100% channel height, the flow is slowed down in front of the blades, which results in a higher static pressure.

To gain more insight, a spectral analysis of the time dependent pressure fluctuations for all three channel heights is performed for URANS and DES and compared with the experiment by means of the Power-Spectral-Density function (PSD).

The PSD function of two different signals leads to the Cross-Power-Spectral-Density function(CPSD, G xy ), which is used to get information about dominant frequencies and the phase angle Φ xy , see equations ( 5) and [START_REF] Mailach | Rotating Instabilities in an Axial Compressor Originating From the Fluctuating Blade Tip Vortex[END_REF].

G xy (f ) ∼ = 1 T X * (f )Y (f ) (5)
Φ xy (f ) = arctan Im[G xy (f )] Re[G xy (f )] (6) (7)
The distance between the experimentally used pressure sensors is about 22 • (see [START_REF] Hermle | Experimental Analysis of Rotating Instability Using Hot Wire Triggered PIV and Wavelet Transform[END_REF]), which is also set as the circumferential distance for calculating an averaged CPSD based on URANS and DES data. Since the RI is a phenomenon distributed over the entire circumference, it is justified to calculate an averaged CPSD based on all 360 positions of URANS and DES. For each pair with a circumferential distance of 22 • in the computations, a CPSD is performed, summed up and divided by the number of pairs to get the averaged CPSD and also an averaged phase angle. The CPSD is combined with a hanning-window and a zero padding is applied to obtain smoother spectra.

The results are presented in figure 7. In the upper figure the CPSD of URANS, DES and the experiment are shown. First of all, a comb-like pattern can be seen for DES and URANS, whereas the DES shows up to four and the URANS and experiment up to three peaks. The dominant frequencies are listed in table 4. Due to the limited frequency resolution of the numerical results, which are about 6.11 Hz for the DES and 10.56 Hz for URANS, and the usage of a window-function, the identification of dominant frequencies in the numerical data is also limited and may lead to false conclusions. Thus, it is possible, that especially within the URANS computation, the comb-like pattern is only the result of the low frequency resolution as well as the usage of a window-function, which produces side slopes. Taking into account the very smooth pressure fluctuation in figure 6 (last row, right figure) of the single pressure fluctuations with almost one frequency, this consideration seems to be comprehensible.

The comb-like pattern is shifted to higher values for both computations compared to the experiment, with a larger difference between experiment and URANS. Regarding the amplitudes of the CPSD the left y-axis in figure 7 belongs to the DES and the experiment whereas the right y-axis to the URANS spectra. The match for the DES is very well, whereas the CPSD-level of the URANS is up to 15 times higher compared to the experimental values. As it was mentioned earlier, RI is a phenomenon located near the hub with a radial extent of about 30%, [START_REF] Hermle | Experimental Analysis of Rotating Instability Using Hot Wire Triggered PIV and Wavelet Transform[END_REF]. This can also be seen in the numerical data.

A decrease of dominant (RI-)frequencies for increasing chan- nel height within both computations can be seen. In the lower part of figure 7 phase angle spectra are shown. The match between DES and experiment as well as between URANS and experiment is qualitatively very good. The direction of the phase angle indicates a propagation from suction to pressure side, which is in accordance with the findings based on figure 6. In the relevant frequency range the phase angle is higher for both computations compared to the experiment. With the help of the phase angle spectrum and the distance of the sensors, one can additionally calculate the mode order m and the rotational speed ω. Based on equation ( 4), the rotational speed ω (or the rotational frequency f rot ) can directly be calculated, whereas the relation Φ xy = ∆t2πf is used to replace ∆t, which yields to

ω = ∆ϕ2πf Φ xy . (8)
It is known, that for the rotating instability each dominant frequency peak is related to a mode order, which can be derived by evaluating equation ( 9), where ∆ϕ is the 21.18 • for the experiment and 22 • for the computations. Φ xy can be calculated by using equation [START_REF] Mailach | Rotating Instabilities in an Axial Compressor Originating From the Fluctuating Blade Tip Vortex[END_REF].

m = Φ xy ∆ϕ . (9)
The resulting rotational speeds as well as the mode orders are listed in table 4. Additionally, the circumferential speed can be found in the last column. The mode orders of the experiment are between 1 and 3 and lower compared to those of the numerical results. Within the DES, mode orders from 4 up to 7 are calculated. Evaluating equation ( 9) for the URANS computation results in mode orders of about 7. Concerning the rotational speed ω and the cricumferential speed, there are also differences between the numerical and experimental results. c u varies from 32 up to 39 m/s within the experiment, whereas these velocities are slightly smaller for URANS and DES, which means that this phenomenon rotates slower in circumferential direction.

The presented numerical results in front of the leading edge indicate that a rotating flow phenomenon, similar to the rotating instability can be simulated with a DES method. Results based on an URANS simulation are worse in terms of quality and quantity compared to the experiment, which may be caused by the frequency resolution as well as the method itself. The results of a DES seem to be more accurate and realistic. Based on an analysis in the frequency domain, dominant frequency peaks with a comb-like pattern can be detected at the hub within the DES. The influence of these dominant peaks decreases with increasing channel height. Compared to experimental data the relevant frequency range as well as quantitative values of the DES, like mode order and rotational speed, show some smaller differences, which could have been caused by the DES method itself or the the used grid and applied time step. The authors assume, that the main reason for the differences are the applied boundaries. Especially at the inlet near the hub the development of a RI-like pattern is very sensitive to the applied conditions, [START_REF] Teich | Einfluss der Zuströmrandbedingungen auf die Ausbildung rotierender Strukturen in einem hoch belasteten Axialverdichtergitter[END_REF]. When the gradient of the total pressure up to a channel height of 15% is increased a little bit, the comb-like pattern is shifted to lower frequencies and the resulting mode orders are closer to those of the experiment, see figure 8 upper graph.

CONCLUSION

Three-dimensional unsteady Reynolds-averaged Navier Stokes as well as Detached-Eddy-Simulations of a compressor stator cascade were performed in order to investigate a rotating flow pattern, which is called rotating instability (RI). The results show that the development of a rotating flow structure is very sensitive to the appplied method. It can be concluded, that a DES method can be used to simulate the rotating instability, which was confirmed by an analysis in the time and frequency domain. The typical broadband enhancement at a low frequency range with discrete peaks, which are related to different mode orders could be verified. The direction of propagation as well as the radial extent are in agreement with the experimental data. The differences between experimental and numerical data are referred to quantitative characteristics. Thus, dominant frequencies are shifted to higher values in the computation and the resulting mode orders are increased. Recent results have shown an fairly good match, when the total pressure profile at the inlet is slightly increased. Beside the work of van Rennings [START_REF] Van Rennings | Numerical simulation and modelling of sound generated by rotating instabilities in an annular compressor cascade[END_REF] this is the first time, that the simulation of the rotating instability phenomenon has been verified by experimental data to a large extent.

Figure 1 .

 1 Figure 1. Experimentally observed rotating instability in the frequency spectra of two pressure signals with a circumferential distance of about 42.36 • , [4].

Figure 2 .

 2 Figure 2. Annular compressor test rig

Figure 3 .

 3 Figure 3. Computational grid with inlet (blue) and outlet (red) plains as well as measurement plain MP1 (green) upand MP2 (grey) downstream of the compressor stator cascade. Mesh details of the leading and trailing edge are shown in the upper part.

Figure 4 .

 4 Figure 4. Comparison of the numerical (DES=red, URANS=green) and experimental (blue) radial distribution of a) total pressure p t,1 , b) Mach number M a 1 , c) flow angle α 1 at the inlet plain MP1 and d) total pressure p t,2 , e) Mach number M a 2 and f) flow angle α 2 at the outlet plain MP2 for M a = 0.4 and i = 12, 3 • .

Figure 5 .

 5 Figure 5. Location of the virtual sensors from front view (left) and from above view (right). For the sake of clarity, in the left figure only 24 sensors out of 360 are illustrated.

Figure 6 .

 6 Figure 6. Row 1-3: Time dependent static pressure difference ∆p of 360 measurement points at 0% (first row), 50% (second row) and 100% (third row) channel height for DES (left) and URANS (right). Row 4: Additional single static pressure fluctuation at the hub for a circumferential position of ϕ = 10 (red), 40 (green) and 70 • (blue).

Figure 8 .

 8 Figure 8. Cross-Power-Spectral-Density (upper) and phase angle (lower) for experimental (black) and numerical (DES=red) data with an increased total pressure distribution at the inlet near the hub. The right axis in the upper figure belongs to the CPSD of the DES.

Table 1 .

 1 Geometric details of the investigated annular compressor stator cascade

	Geometric Compressor Stator Cascade Details
	chord length	ℓ c	50 mm
	hub radius	r H	110 mm
	casing radius	r C	165 mm
	aspect ratio		1.1
	solidity at midspan	ℓ c /t	0.99
	tip clearance	s	0.8%•ℓ c
	inlet flow angle, design point	α 1,DP	43 •
	outlet flow angle, design point α 2,DP	21 •
	stagger angle	β s	26 •
	number of blades	z	17

• . This cascade is a scaled version of blades first used by Gallus and Mertens

[START_REF] Gallus | Michungsvorgänge I, Abschlussbericht Vorhaben Nr. 531. Technischer Bericht[END_REF]

. Further geometric details are summarized in table 1. From experimental work of

Table 2 .

 2 Quality criteria of the discretized domain no wall function has to be used in this region. The resulting grid quality criteria are listed in table 2.

	grid quality criteria
		min./max.	∅
	aspect ratio	1992.50	230.59
	orthogonality	31.09 •	83.15 •
	expansion ratio	2.96	1.37
	y +	1.2	< 0.65

Table 3 .

 3 Comparison of the mass flow for i = 12.3

				∆ ṁ
	URANS	4.21	0.038	2,43%
	DES	4.14	0.046	0,73%
	EXP	4.11	-	-

the total pressure p t , the absolute Mach number M a and the absolute flow angleα are presented for the investigated operating point on both axial positions, MP1 and MP2. All radial profiles are also time and circumferential averaged with at least 700 time steps. The comparison of method ṁ [kg/s] σ [kg/s] • and M a = 0.4.

Numerical Simulation of the Rotating Instability in an Annular Compressor Cascade Test Rig -8/10

		600					15000
	/Hz]						12500
	CPSD [Pa 2	400					7500 10000
		200					5000
							2500
		0					0
		200					200
		100					100
	]
	Φ[ •	0					0
		-100					-100
		-200					-200
				200		400
					f [Hz]
	Figure 7. Cross-Power-Spectral-Density (upper) and phase
	angle (lower) for experimental (black) and numerical
	(DES=solid lines, URANS=dashed lines) data for different
	channel heights (red=0%, green=50% and blue=100%) in
	front of the leading edge.
			55.16	25.16	1.18	291.76	32.09
		EXP	120.8	44.34	2.09	362.56	39.88
			174.89	69.20	3.26	337.92	37.17
			163.58	95.11	4.32	237.74	26.15
		DES	177.67 103.37 205.89 121.80	4.47 5.54	237.58 233.66	26.13 25.70
			225.63 157.33	7.15	198.23	21.80
			251.42 175.80	7.99	197.69	21.75
	URANS	275.21 -162.86 -7.40	233.59	25.69
			298.99 -141.73 6.44	291.61	32.01

method f [Hz] Φ xy [ • ] m [-] ω [rad/s] c u [m/s]

Table 4 .

 4 Phase angle Φ xy , mode order m and rotational speed ω as well as circumferential speed c u of dominant frequency peaks for the experiment and both computations.

ACKNOWLEDGMENTS

All calculations for this research were conducted on the Lichtenberg high performance computer of the TU Darmstadt. The authors would like to thank for the opportunity to conduct the extensive numerical investigations, and for the kind support and advice by the staff.