# Consequences of Borescope Blending Repairs on Modern HPC Blisk Aeroelasticity

Benjamin Hanschke, Arnold Kühhorn, Sven Schrape, Thomas Giersch

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Benjamin Hanschke, Arnold Kühhorn, Sven Schrape, Thomas Giersch. Consequences of Borescope Blending Repairs on Modern HPC Blisk Aeroelasticity. 17th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISROMAC2017), Dec 2017, Maui, United States. hal-02980991

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# Consequences of Borescope Blending Repairs on Modern HPC Blisk Aeroelasticity

Benjamin Hanschke<fnref n="1" />*, Arnold Kühhorn<fnref n="1" />, Sven Schrape<fnref n="2" />, Thomas Giersch<fnref n="2" />

<figure><img src="image_2.png" /></figure>

ISROMAC 2017

International Symposium on Transport Phenomena and Dynamics of Rotating Machinery

Maui, Hawaii

December 16-21, 2017

## Abstract

Objective of this paper is to analyse the consequences of borescope blending repairs on the aeroelastic behaviour of a modern HPC blisk. To investigate the blending consequences in terms of aerodynamic damping and forcing changes, an exemplary blending of a rotor blade is modelled. Steady state flow parameters like total pressure ratio, polytropic efficiency and the loss coefficient are compared. Furthermore, aerodynamic damping is computed utilising the AIC approach for both geometries. Results are confirmed by SPF simulations for specific nodal diameters of interest. Finally, an unidirectional forced response analysis for the nominal and the blended rotor is conducted to determine the aerodynamic force exciting the blade motion. Fourier transformation of the forcing signal yields to the frequency content as well as the forcing amplitudes. As a result of the present analysis, the amplification of expected blade vibration amplitude is computed.

## Keywords

Aeroelasticity — Compressor Blisk — Blending Repair — Forced Response — Aerodynamic Damping

## INTRODUCTION

To fulfil the steadily increasing demands in the field of aero engines, original equipment manufacturers (OEMs) adopted the blade integrated disk (blisk) design to civil aviation. In contrast to conventional bladed disk assemblies, blisks endure higher rotational speeds and improve aerodynamic efficiency while significantly reducing the weight of modern high pressure compressors (HPCs). Due to missing friction at the blade to disk contact surfaces mechanical damping of blisks is comparably low and an increasing risk for aeroelastic instability (flutter) emerges. Especially in the service sector, detection and repairs of foreign object damages (FODs) on HPC blisks are relevant. To ensure structural integrity of the components all damages - exceeding a size specified by the OEM - are repaired by e.g. borescope blending of the concerned area. In case of critical damage sizes engine strips are necessary to replace the whole component. Regarding the financial aspect of such events [1] it is necessary to define the structural and aeroelastic limits for blending repairs as accurate as possible. Recently published research mainly focuses on structural consequences of blending repairs with regard to amplification caused by mistuning [2, 3] and the effect of blending geometries on high cycle fatigue strength of the component [4]. The aim of this paper is to identify the aeroelastic effects caused by borescope blending repairs in terms of changes in aerodynamic damping and forcing acting on neighbouring blade rows. According to this knowledge, recommendations for acceptable blending procedures can be acquired for the service sector.

## 1. METHODS

CFD results presented within this paper are generated using the Rolls-Royce proprietary code AU3D, which is an unsteady flow and aeroelasticity solver [5, 6]. Unsteady Reynolds averaged Navier-Stokes (URANS) equations are solved utilising the Spalart-Allmaras turbulence model and the code has been successfully validated for a wide range of turbomachinery flows [7]. Semi-unstructured meshes are used to capture the complex geometric features of the aerofoils [8]. Furthermore, finite element (FE) computations provide the mode shapes necessary for the aeroelastic analysis. All subsequent analyses are conducted at an exemplary rotor stage of a modern HPC compressor surrounded by an upstream and downstream positioned stator row.

For a comparison of the nominal design and a blended test case different steady state flow parameters are evaluated.

$$ \pi = p_{t2}/p_{t1} \quad \eta = \frac{\kappa - 1}{\kappa} \frac{\ln(p_{t2}/p_{t1})}{\ln(T_{t2}/T_{t1})} \quad \zeta_l = \frac{p_{t1} - p_{t2}}{p_{t1} - p_1} \quad (1) $$

In eq. (1) $\pi$ denotes the total pressure ratio, $p_{t1}$ the inlet total pressure, $p_{t2}$ the outlet total pressure, $\eta$ the polytropic efficiency, $\kappa$ the adiabatic index, $T_{t1}$ the inlet total temperature, $T_{t2}$ the outlet total temperature, $\zeta_l$ the loss coefficient and $p_1$ the inlet static pressure. Additionally, a comparison of the mass flow averaged outlet flow angle $\alpha_2$ is conducted, to identify deviations between both aerofoil geometries. To save computing time the steady flow analysis setup is composed of 1 blade per blade row with periodic boundary conditions

<footnote n="1">¹Chair of Structural Mechanics and Vehicle Vibration Technology, Brandenburg University of Technology Cottbus - Senftenberg, Cottbus, Germany</footnote>

<footnote n="2">²Rolls-Royce Deutschland Ltd & Co KG, Blankenfelde-Mahlow, Germany</footnote>

<footnote>*Corresponding author: Benjamin.Hanschke@b-tu.de</footnote>
and mixing plane interfaces between the blade rows. This setup represents a worst case scenario, because all blades of the observed stage are modelled with a blending due to the periodic boundaries. Hence, a comparison of steady state performance parameters between both cases should yield conservative results with respect to the application of blendings in real compressors.

## 1.1 Aeroelastic Model

Foundation of the aeroelastic analysis is a coupling of a structural dynamic model with an unsteady dynamic model of aerodynamic forces. The governing aeroelasticity equations derived from this coupling can be written as

$$M\ddot{x} + D\dot{x} + Kx = f_e \quad (2)$$

where M represents the mass, D the damping, K the stiffness matrix, x is the displacement vector and $f_e$ the vector of external forces. By omitting the external forcing, left-hand side of eq. (2) can be solved by a standard FE software and yields to the natural frequencies $\omega_i$ and eigenmodes $\phi_i$ of the system. For further analysis, the external forcing vector is obtained from the pressures occurring in the flow field. Accordingly, $f_e = pn\Lambda$ with the pressure vector p, the unit normal vector on the blade surface n and the application area A. Due to the neglectable mechanical damping of modern HPC blisks, the damping matrix is dropped. Additionally, a modal transformation of the system described in eq. (2) is performed and the governing equations become

$$m_i \ddot{q}_i + m_i \omega_i^2 q_i = f_{aero_i} \quad \text{with} \quad f_{aero_i} = f_i^m + f_i^D \quad (3)$$

Herein $m_i$ is the generalized mass, $\omega_i$ denotes the natural frequency, $q_i$ the modal displacement and $f_{aero_i}$ the aerodynamic forcing. According to Crawley [9] the aerodynamic forcing $f_{aero_i}$ is a simple superposition of motion dependent aerodynamic forces $f_i^m$ and aerodynamic disturbance forces $f_i^D$. Separation of the fluid forces allows an individual consideration of different aeroelastic problems [10]. Negligence of disturbance forces caused by neighbouring blade rows enables flutter analysis and the computation of aerodynamic damping, whereas setting $f_i^m = 0$ leads to the forced response problem.

## 1.2 Aerodynamic Damping Calculations

Two different approaches are considered to calculate the aerodynamic damping of the observed rotor. One is based on a single passage simulation with phase lagged boundary conditions, while the other focuses on the computation of the motion dependent aerodynamic forces $f_i^m$ in a multiple passage model. Blade vibration is modelled by a mesh morphing process at every time step.

**Single Passage Flutter** A basic setup for a single passage flutter (SPF) simulation consists of a single blade with phase-lagged periodic boundaries [11]. Furthermore, Riemann invariants boundary conditions are applied at inlet and outlet

of the passage and the blade executes a sinusoidal motion with the prescribed modal amplitude $\hat{q}$.

$$q(t) = \hat{q} \sin(\omega t) \quad (4)$$

Basic principle of the energy method is to evaluate the work done per cycle by the aerodynamic forces on the blade [12].

$$W_c = \int_{cycle} f_{aero} dq \quad (5)$$

In this context negative work refers to a transfer of energy from the blade to the flow field and vice versa. In the case of a blade to flow field energy transfer, the blade motion itself is damped by the fluid which corresponds to a stable operating condition. In contrast, a positive work per cycle excites the blade motion and aeroelastic unstable conditions (flutter) are present. Aerodynamic damping is derived by a conversion of the logarithmic decrement $\Lambda$ into a damping ratio value D.

$$\Lambda = \frac{-W_c}{2\omega^2 \hat{q}^2} = \frac{2\pi D}{\sqrt{1 - D^2}} \quad \text{with} \quad D \approx \frac{\Lambda}{2\pi} \quad (6)$$

In eq. (6) the damping ratio is obtained by a linearisation of the logarithmic decrement expression. This simplification is a common approach for values $D \ll 1$.

**Aerodynamic Influence Coefficients** Another possibility to determine the aerodynamic damping of a cascade is the method of aerodynamic influence coefficients (AICs). This approach is based on the idea of Hanamura et al. [13] and utilizes a multi passage setup with 1 vibrating and several resting blades. Origin of further explanations is the blade individual formulation of eq. (3) deduced from Crawley [9] using mass normalised eigenmodes.

$$\ddot{q} + \operatorname{diag}\{\omega^2\}q = f_{aero} \quad \text{with} \quad f_{aero} = Lq \quad (7)$$

The AIC matrix L has a circulant structure and each term in the matrix describes a physical correlation.

$$L = \begin{pmatrix} l_0 & l_{-1} & l_{-2} & \cdots & l_{-1} \\ l_{-1} & l_0 & l_{-1} & \cdots & l_{-2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ l_{-1} & l_{-2} & \cdots & l_{-1} & l_0 \end{pmatrix} \quad \text{with} \quad l_k = \frac{f_{aero_k}}{\hat{q}_0} \quad (8)$$

Main diagonal terms $l_0$ represent the force acting on a blade caused by its own motion while the terms $l_1$ and $l_{-1}$ describe the force produced by the vibrating blade on its direct neighbours and so on. Each influence coefficient $l_k$ is easily derived from evaluating the right aligned expression of eq. (8) in a computational fluid dynamics (CFD) simulation. A transformation of the aerodynamic influence coefficients back into the form of travelling wave mode coefficients provides the possibility to compute the aerodynamic damping according to eq. (9).

$$\underline{c}_{\sigma_n} = \sum_{k=0}^{N-1} l_k e^{-j \frac{2\pi k n}{N}} \quad \text{with} \quad D = \frac{-\mathfrak{F}(\underline{c}_{\sigma_n})}{2\omega^2} \quad (9)$$
## 1.3 Forced Response Analysis

For the conducted forced response analysis a single passage multi row setup is used as proposed by Stapelfeldt [14]. The flow domain consists of three single passage meshes (observed rotor and neighbouring stator rows) which are connected with sliding plane boundary conditions at row to row interfaces. Additionally, phase lagged periodic boundary conditions are applied at each blade row. Excitation of the blade row can be monitored by an evaluation of the modal force

$$f_{aero_i}(t) = \sum_{l=1}^{N_n} \phi_{i,l}(A_l n_l) p_l(t) \quad (10)$$

where $\phi_l$ represents the mode shape of interest, $A_l$ is the blade surface share associated with node l and $n_l$ the corresponding unit normal vector. A Fourier transform of the obtained forcing signal reveals the amplitudes and the corresponding frequency of excitation. Knowledge of the forcing amplitude and aerodynamic damping for the observed mode shape allows computation of the resonance amplitude of blade motion. Due to the modal transformation of the aeroelastic system each degree of freedom can be treated as a driven harmonic oscillator and the resonance amplitude is derived according to eq. (11).

$$x_{max} = |\hat{q}_i| \phi_{i,max} = \frac{\hat{f}_{aero_i}}{2\omega_i^2 D} \phi_{i,max} \quad (11)$$

## 2. RESULTS AND DISCUSSION

Before shifting the focus to the numerical computations, a short analytical consideration of the expected blending consequences is performed. The incorporated blending geometry has a width to depth ratio $w_b/d_b = 4$ to prevent excessive stress concentration within this area. Positioning of the centre of blending is done at the location of maximum deflection of the investigated mode shape. An additional criterion used for the blending depth was to retain the nominal mode. Figure 1 opposes both modes and visualises the influence of blending on the mode shape. While preserving the overall shape a significant change in the modal mass and therefore

<figure><img src="image_3.png" /><figcaption>Figure 1. Comparison of the displacement magnitude of the nominal and blended mode shape</figcaption></figure>

the maximum mode shape element $\phi_{b,i_{max}}$ is present for the blended case. For a better understanding of consequences regarding a modal mass change of the investigated mode the dimensionless mass ratio $\mu$ is introduced in eq. (12).

$$\mu = \sqrt{\frac{m_b}{m_n}} = \frac{\phi_{n,i_{max}}}{\phi_{b,i_{max}}} \quad (12)$$

Assuming a linear force dependency from the blade vibration amplitude, identical physical displacements result in identical forces acting on the blade caused by the fluid. This assumption is true for a simple modal mass change of the same mode shape as long as the pressure distribution in the flow field does not change. Regarding the modal forces, vibration amplitudes, travelling wave mode coefficients and as a consequence thereof the aerodynamic damping following relations reveal.

$$\hat{f}_b = \frac{\hat{f}_n}{\mu} \qquad \hat{q}_b = \mu \hat{q}_n \qquad c_b = \frac{c_n}{\mu^2} \qquad D_b = \frac{D_n}{\mu^2 \nu^2} \quad (13)$$

Thereby $\nu = \frac{\omega_b}{\omega_n}$ represents the quotient of the natural frequencies of both modes. For the present case the incorporated blending results in a mass ratio $\mu = 1.424$ and a natural frequency quotient $\nu = 1.028$. According to eq. (13) a decrease of the modal force by $\approx 29.8\%$ and a drop of the aerodynamic damping by $\approx 53.3\%$ is expected for the blended case under the aforementioned circumstances.

For a sufficient capturing of the blending geometry in the CFD mesh a radial refinement at blending position is used. To minimise the influence of different mesh resolutions the nominal mesh is built with the same radial distribution. A direct comparison of both meshes is shown in fig. 2.

<figure><img src="image_4.png" /><figcaption>Figure 2. Comparison of nominal and blended geometry CFD mesh</figcaption></figure>

### 2.1 Steady State Comparison

Evaluation of the steady state flow parameters according to eq. (1) is done based on a front block analysis of the HPC. Total pressure and temperature values necessary for the evaluation of the parameters are obtained by mass flow averaging at inlet and outlet boundaries of the observed blade row. Table 1 summarises the results for both setups. Slight drops of 0.148% for the total pressure ratio and 0.237% for the polytropic efficiency are noticed. Comparing the loss coefficients of the nominal case and the blended geometry a distinct increase of 44.37% is apparent. Since the inlet velocities for both cases are nearly identical the only explanation for this issue is a larger total pressure loss along the blade for the blended
<table><caption>**Table 1.** Steady state flow parameter comparison</caption><thead><tr><th>setup</th><th>π/π<sub>nom</sub></th><th>η/η<sub>nom</sub></th><th>ζ<sub>l</sub>/ζ<sub>l<sub>nom</sub></sub></th></tr></thead><tbody><tr><td>blended</td><td>0.99852</td><td>0.99763</td><td>1.4437</td></tr></tbody></table>

geometry. Moreover, the comparably low deviations from the nominal design may be caused by the spline approach which is used for meshing, whereas sharp-edged geometries in the blended area should cause significantly higher losses.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Relative outlet flow angle</figcaption></figure>

Figure 3 visualises the relative outlet flow angle of the blended blade. Outlet flow angle deviations are negligible outside of the blending and reach a maximum of ~ 4.4% within the blending area. Considering stable operating conditions for the HPC it has to be mentioned, that a change of the incidence angle of more than 4% possibly influences the performance of the downstream stator row. Therefore, it is recommended to check the consequences of incident angle changes on following blade rows to ensure safe HPC operation.

The influence of blending on the velocity field at a cut in the centre of blending is shown in fig. 4. Due to the removed material at leading edge the metal angle slightly differs from the incident flow angle and causes a small separation at the pressure side. In comparison to the nominal velocity field an area of reduced velocity spans along the whole pressure

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Velocity field at blending position</figcaption></figure>

side of the blade. Additionally, this misaligned flow is responsible for a larger acceleration at the transition to the suction side. Moreover, fig. 5 depicts the static pressure dis-

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Relative static pressure of nominal geometry and blending repaired blade</figcaption></figure>

tribution at a layer in the centre of blending. Apparently, the blending influence on static pressure values is present for relative chord length values < 0.4. For higher relative chord values static pressures are identical and no deviation between both aerofoils exist. Steady state flow parameter comparison revealed, that the blending influences the flow field within its application area. Nevertheless, it has to be mentioned that the blending area is comparably small in relation to the blade surface. Hence, observed changes in terms of polytropic efficiency and pressure ratio were small.

## 2.2 Influence on Aerodynamic Damping

Various computations were performed to identify the blending influence on aerodynamic damping of the observed rotor. Firstly, the aerodynamic damping curve of the nominal design case is taken as a basis. Subsequently, an AIC analysis for the blended blade case was done for comparison.

For reasons of complexity, the blended blade case is modelled with every blade blended, which represents a worst case scenario and therefore should lead to conservative results. Figure 6 opposes both damping curves and shows some remarkable consequences. Location of minimum aerodynamic damping is shifted from low forward travelling wave nodal

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Aerodynamic damping curves</figcaption></figure>
diameters to low backward travelling wave nodal diameters. In comparison to the nominal damping curve the blended blades suffer from lower aerodynamic damping for all backward travelling wave modes. Regarding the forward travelling wave modes an increased aerodynamic damping is present for a majority of the nodal diameters. Furthermore, a considerable drop of damping from -12.38% is noticed at the nominal nodal diameter with the highest damping. An explanation for this drop may be obtained by an analysis of the local work done by the fluid to the blade surface for this specific nodal diameter, which is visualised in fig. 7 and fig. 8 for pressure and suction side of the blade. Here, the local work coefficient is defined as the quotient of the work done at each element surface and the corresponding element surface area.

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Work done by aerodynamic forces for interblade phase angle of -102° at blade pressure side</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Work done by aerodynamic forces for interblade phase angle of -102° at blade suction side</figcaption></figure>

Sections coloured in deep blue represent negative work, which corresponds to damping while red areas excite the blade motion. A comparison of pressure sides reveals, that positioning of the blending and a change of the flow field within this area cause a large part of the negative work done at the blade leading edge to vanish. Additionally, lower negative work values are contributed in direct tip area and also at trailing edge near tip areas. Likewise, a certain share of the positive work done at leading edge also decays. Considering

the suction sides of both setups similar effects are perceptible. Negative work areas decrease at leading edge as well as in direct tip area and trailing edge near tip areas. In contrast to pressure side observations no distinct influence is present on blade motion exciting areas. Consequently, an integral consideration of local work yields to a smaller damping induced in the blended case because of lower negative work contributions in several areas of the blade surface. To confirm the results provided by the AIC method, 4 additional SPF computations are performed for inter blade phase angles (IBPAs) of maximum aerodynamic damping of the nominal as well as the blended geometry. For a -102° IBPA backward travelling wave the maximum deviation is 2.39% between both methods for aerodynamic damping calculations. Regarding the maximum damped IBPA of the blended geometry the deviations are even smaller with a maximum of 0.83%. Whilst taking into account, that two different approaches for aerodynamic damping calculations provide coherent results, AIC damping curves are validated by the SPF simulations.

In contrast to the theoretic considerations that predicted a serious drop of the aerodynamic damping due to the modal mass change of the investigated mode just a slight drop is determined. As already highlighted in fig. 4 the blending heavily influences the flow field and those effects partly recover the initial damping magnitude. Additionally, several effects have to be considered for a reasonable evaluation of blending consequences in terms of aerodynamic damping changes. At first, the simplified approach for aerodynamic damping calculations - to model all blades with a blending - not represents common practise in compressor repairs. Nevertheless, this setup should represent a worst case scenario and usual blending repairs, where one blended blade is surrounded by intact neighbours, should not influence the aerodynamic damping as much as the analysed setup. A possible alternative would be the computation of the AICs including one blended and several nominal blades in a cascade. Disadvantageously, this setup would destroy the circulant structure of the AIC matrix and several simulations would be necessary for damping curve computations. Furthermore, a blending repair causes frequency mistuning of the affected blade which directly influences the full assembly mode. Taking account of the mistuning level present in current HPCs it has to be mentioned, that a mistuned assembly mode is rather influenced by the mean value of aerodynamic damping than by the maximum deviation between both setups [15]. For the analysed setup the mean value of aerodynamic damping just drops slightly by -2.95% for a fully blended rotor. In case of a sufficiently small frequency mistuning caused by blending, the change of aerodynamic damping may be insignificant regarding the predicted change in blade vibration amplitude. However, large frequency mistuning of a blended blade may lead to assembly modes where only the blended blade vibrates. Keeping in mind the physical meaning of the different AICs, the force of a single vibrating blade caused by its own motion is described by the coefficient $l_0$. For a localised mode with only one vibrating blade, an aerodynamic
damping derived by solely evaluating $l_0$ represents the value of interest. A detailed consideration of eq. (9) reveals, that picking only the zeroth coefficient yields to the mean value of the aerodynamic damping curve. Even for the worst possible mistuning case of an isolated vibrating blade, the mean value of the damping curve derived from blended blade analysis is assumed to represent a realistic damping value. In the present case the maximum damped IBPA of the nominal geometry could suffer a relative loss of 9.23% compared to the mean value of damping for the blended case.

## 2.3 Influence on Forced Response Behaviour

To investigate the influence of blending repairs on the forced response behaviour a 3 blade row setup is analysed. This setup consists of the observed rotor as well as the upstream and downstream surrounding stator rows. Such a domain enables evaluation of blade row interactions and to compute the aerodynamic force on the rotor caused by the wakes of the upstream positioned stator row as well as the potential field effects of the downstream positioned stator. For a detailed analysis the aerodynamic forcing is recorded for 2 complete rotor revolutions and afterwards Fourier transformed to gain informations about the frequency content as well as the corresponding amplitudes. Figure 9 shows a comparison of the relative modal force amplitudes for the nominal and blended geometry.

<figure><img src="image_11.png" /><figcaption>Figure 9. Line spectra of modal forcing</figcaption></figure>

Hereby, the frequency is normalised by the blade passing frequency of the upstream stator row. Conspicuously, the blended geometry undergoes a much higher modal forcing than the nominal blade. An amplification of more than 80% is observed. This result is especially surprising in connection with the theoretical considerations mentioned in eq. (13) because a slight drop of the forcing was expected due to the modal mass change of the mode shape. According to eq. (10) the only possible explanation is a distinct change in the pressure field which yields to the increased forcing recognised for the blended blade. To visualise this issue fig. 10 and fig. 11 show the maximum local aerodynamic force on the blade surface. Analogous to the local work coefficient observations in the damping section certain differences between both geometries are present. Figure 10 shows, that the blending influence on the flow field is significant, even

<figure><img src="image_12.png" /><figcaption>Figure 10. Maximum local aerodynamic force comparison for blade pressure side</figcaption></figure>

for non vibrating blades. In the nominal setup, a wide area of negative force is in place at the upper third of the blade leading edge. Since the blending is located exactly within this area, the effect of the blending on the local force at this position is severe. The area, which induces negative work in the nominal case, reduces in size and is interrupted by several positive work regions. This is mainly caused by the change of the flow field in the leading edge surroundings and seriously impacts the aerodynamic force affecting the blade. An area of positive modal force is present slightly above the centre of leading edge for both setups. In contrast to the nominal blade, the values reached within this region are somewhat higher for the blended case and an increase of the application surface exists. Besides, a tiny area of negative forcing arises in the blended case, which is not clearly present in the nominal setup. Even close to the leading edge fillet section more positive work is introduced to the blade in the blended case. A second mechanism which yields to a higher forcing of the blended case is visible in the downstream tip as well as trailing edge near tip areas. While the nominal blade shows two areas with vast negative forcing at those positions, the blended case differs. Negative forcing is still present at the initial locations but the magnitude is clearly decreased. Apparently, the misaligned flow at the blending layer highlighted in fig. 4 influences the unsteady pressure fluctuations on the blade surface.

Regarding the suction side comparison of both cases the visible effects mainly refer to the direct surroundings of the blending. In the nominal setup this is a pure area of negative work, whereas the blending clearly reduces the surface area of negative work acting on the blade. Additionally, very small areas of positive work emerge at this position in the blended case. Agreeable to the pressure side observations, the local forcing for the blended case is higher slightly above the centre of leading edge. Regarding the near tip and leading edge near tip areas it has to be mentioned, that there is a bit higher forcing present in the nominal case, while the negative force areas nearly vanish in this areas for the blended case.

Most relevant for the application of a blending to real HPC hardware is structural integrity of the component. A combination of aerodynamic damping values and the com-
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Maximum local aerodynamic force comparison for blade suction side</figcaption></figure>

puted forcing amplitudes allows for the prediction of blade vibration amplitudes expected during operation. Only if the predicted amplitude accomplishes the OEM limits regarding safe engine operations a blending will be approved. For the present case a comparison is conducted between the nominal and the blended blade vibration amplitude. Blade vibration amplitudes are derived according to eq. (11) taking into account the different forcing amplitudes, eigenfrequencies, damping values and maximum mode shape elements of both cases. Evaluation of both expressions yields to an amplitude ratio $x_b/x_n$ of 1.267. An amplification of 26.7% in blade vibration amplitude purely introduced by the application of a blending is a remarkable effect. Neglecting the effects of aerodynamic damping and forcing changes may lead to a positive assessment of a requested blending. Depending on the nominal design excitation the observed amplification may be responsible for serious damages to the blended blade.

### 3. CONCLUDING REMARKS

This paper presented the effects of a borescope blending repair on a compressor in terms of steady state flow parameters, aerodynamic damping and forced response behaviour. Steady state flow analysis revealed, that the influence of the blending on the flow field is mainly present in the radial layers of the blending. Material removal at the leading edge produces a misaligned flow in the blending area, which is responsible for a slight separation at the blade pressure side. Additionally, an area of reduced flow velocity spans along the whole pressure side of the blade. An evaluation of steady state flow parameters showed no significant effects of the blending. Total pressure ratio of the stage dropped by 0.148% and the polytropic efficiency by 0.237%. Only the loss coefficient raised noticeably by 44.37%.

For aerodynamic damping comparison two different approaches are used to validate the results. At first, AIC analyses are conducted for the nominal and blended case. To confirm the damping curves additional SPF simulations were performed at the IBPAs of maximum nominal and blended damping values. Maximum deviation of -12.38% from the nominal case was observed at the original nodal diameter

of maximum damping and the mean value of aerodynamic damping dropped slightly by -2.95%. In general the maximum damped IBPA for the blended case changed from backward travelling waves to forward travelling waves. Reasons for the deviations were explained based on the local work co-efficient changes at the pressure and suction side of the blades. For reasons of complexity the analysed setup consisted of a fully blended rotor and therefore the results represent a worst case scenario and should be conservative. Effects for a single blended blade neighbouring several nominal blades were not investigated.

Finally, the forced response behaviour of both setups was compared. A Fourier transform of the modal force signal revealed the frequency content and the forcing amplitudes for the nominal and the blended geometry. A direct comparison showed a forcing amplification of more than 80% for the blended case. Reason for this significant deviation is a change of the flow field caused by the blending. Local force analysis at pressure and suction side of the blade surfaces unveiled, that certain areas of negative forcing in the nominal case are strongly affected by the blending consequences. Additionally, new regions of blade motion exciting forces emerge and a combination of both effects yield to the higher forcing amplitude for the blended case. To evaluate a blending in terms of safe engine operation not the forcing itself, but the predicted blade vibration amplitude is vital. Taking into account the different modal forcing, damping, maximum mode shape element and eigenfrequencies an amplification of 26.7% is predicted for the blended case. This amplification is a remarkable effect and should be considered for the decision whether a blending is acceptable or not. Furthermore, a blending repair of compressor blades still has to deal with the amplification caused by mistuning the system. A superposition of both effects may be responsible for additional vibration amplification as well. All in all, blending repairs are able to significantly influence the flow field and may cause even higher force and vibration amplitude amplifications than presented within this paper.

### ACKNOWLEDGMENT

This publication is based on the work within the research project VITIV which is funded by the ProfIT program of the federal state Brandenburg and the European Regional Development Fund (Project 80164702). The authors gratefully acknowledge Rolls-Royce Deutschland Ltd. & Co KG for the permission to publish the achieved results.

### ABBREVIATIONS

**AIC** aerodynamic influence coefficient

**blisk** blade integrated disk

**CFD** computational fluid dynamics

**FE** finite element

**FOD** foreign object damage

**HPC** high pressure compressor
**IBPA** inter blade phase angle

**OEM** original engine manufacturer

**SPF** single passage flutter

**URANS** unsteady Reynolds-averaged Navier-Stokes

## NOMENCLATURE

**A** application area (usually blade surface)

**$c_{\sigma_n}$** travelling wave mode coefficient

**$d_b$** depth of the blending repair

**D** damping ratio

**D** damping matrix

**$f_{aero_i}$** aerodynamic force

**$f_i^m$** motion dependant aerodynamic forces

**$f_i^D$** aerodynamic disturbance forces

**$f_e$** vector of external forces

**$\mathcal{Y}(\cdot)$** imaginary part of a complex number

**K** stiffness matrix

**$l_k$** aerodynamic influence coefficient k

**L** aerodynamic influence coefficient matrix

**$m_b$** generalised mass of the blended mode shape

**$m_n$** generalised mass of the nominal mode shape

**$m_i$** generalised mass

**M** mass matrix

**n** unit normal vector

**$p_{ti}$** total pressure at position i

**p** pressure vector

**$q_i$** modal displacement

**$\hat{q}$** modal amplitude

**$T_{ti}$** total temperature at position i

**$w_b$** width of the blending repair

**$W_c$** work done per vibration cycle

**$W^*$** local work coefficient

**x** physical displacement vector

**$\alpha_2$** mass flow averaged outlet flow angle

**$\zeta_l$** loss coefficient

**$\eta$** polytropic efficiency

**$\kappa$** adiabatic index

**$\mu$** modal mass ratio

**$\nu$** natural frequency ratio

**$\pi$** total pressure ratio

**$\phi$** mode shape vector

**$\omega$** natural frequency

**$\Lambda$** logarithmic decrement

**$\bar{\lambda}$** complex value

**$\cdot$** amplitude of a specific value

**$\cdot\cdot$** first time derivate

**$\cdot\cdot\cdot$** second time derivate

## REFERENCES

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