1 Introduction

Blade flutter in turbines and compressors represents a serious problem for designers and operators of large turbomachines. Lack of experimental data constrains faster progress in this area. Flutter of turbine and compressor blades is defined as flow-induced vibration where the vibration frequency is not integer multiple of rotational frequency. A similar phenomenon, non-synchronous vibration (NSV), occurs at higher incidence angles and higher reduced frequencies [1, 2].

Flutter in turbomachines appears due to the interaction of unstable aerodynamics and bending or torsional vibrating mode [3]. Growth of oscillation amplitudes that may lead to blade fatigue failure occurs when aerodynamic damping is negative, i.e. when the work per oscillation cycle is positive. When flutter occurs, the oscillation of blades in bladed disk assembly are not in phase but are shifted. This shift is expressed by the Inter Blade Phase Angle (IPBA) and the travelling flutter wave can be observed. The initiation of flutter travelling waves is not yet fully explained [4].

In the past years, a considerable number of studies that focused on wind turbine flutter have been published, e.g. [5,6,7,8]. With the growing number of wind and solar power plants, turbines in standard power plants must be operated in wider range of conditions to compensate the fluctuation of supply by renewable sources. The turbines are then operated in off-design conditions and flutter in low-pressure stages, where long and slender blades are installed, can occur [9]. The problem of flutter becomes even more severe due to modern design of compressor blisks with negligible mechanical damping and increased structural coupling [2].

As flutter is an aeroelastic phenomenon, the structural and aerodynamic properties of the blades can be tuned to avoid flutter. Structural parameter of high importance is the modeshape. The reduced frequency \(k = \omega b / u\), where \(\omega = 2 \pi f\) is the circular frequency, b the semi-chord of the blade and u is flow velocity, is both a structural and aerodynamic parameter. An important aerodynamic property is the incidence angle - flow separation can occur when the incidence angle is deflected to off-design conditions which can lead to stall flutter [3].

Many attempts to provide “flutter free” design of blades were carried out. Improvement of the flutter stability of a fan blade was presented by Stapelfeldt and Vahdati [10]. They suppressed flutter by altering the radial distribution of stagger angle which led to lower efficiency. As an alternative approach they reduced flutter by drawing bleed air from the casing above the tip of the blade. It is known that well-tuned symmetric designs contribute to the fluid–structure coupling [2]. In 2000, Nowinski [11] demonstrated benefits of blade mistuning. Mistuning has beneficial effect on blade flutter but unwanted effect on forced response—due to mistuning the forced-response oscillations are present in wider range of frequencies. Therefore, it is required to perform optimization on bladed disk for both forced response and blade flutter. Practical optimization method was proposed by Kaneko [12] or Raeisi [13]. Vanti et al. [14] considered numerical procedure to optimize both efficiency and flutter-free design of the compressor blade-row. In 2018, Corral et al. [15] experimentally validated the benefits of intentional alternate mistuning of blades by flutter free tests. They showed that mistuning can suppress flutter and also modulate vibration amplitude. Geometric optimization of the low-pressure turbine blade airfoil to enhance stability of the turbine was realized by Peeren and Vogeler [16] and the design recommendations were given. Tani et al. studied possibility of flutter suppression by change of torsion axis position [17].

Although the development of numerical methods and experimental testing was intensive in the past decade, the problem of blade flutter has not yet been fully resolved. There is a need to further improve aeroelastic design tools to prevent self-excited vibrations. CFD predictions of aeroelastic instabilities are possible but not always reliable, especially for transonic flow regimes [2]. Numerical simulations have to be supplied by experimental testing.

Even with improvements in the measurement techniques it is not possible to carry out flutter research on real turbines [18]. Simplified experimental test setups have to be employed. Experimental test setups for flutter measurements can be (a) linear cascades and (b) annular cascades. Annular cascades are more realistic, but linear cascades offer easier instrumentation of measuring techniques. The motion of the test object can be flow-induced (free flutter testing) or controlled (controlled flutter testing). In case of free flutter testing, the flow conditions are modified until flutter develops. Controlled flutter testing is often used in research. Oscillations of the blades are prescribed and the aerodynamic response is measured. From these data, aerodynamic damping can be evaluated [3].

A new test facility for controlled flutter testing was developed in the Institute of Thermomechanics of the Czech Academy of Sciences in cooperation with Technical University of Liberec, see Fig. 1. This advanced experimental setup was developed upon experience gained while working with the NASA Transonic Flutter Cascade [19]. The cascade in a transonic suck-down wind tunnel consists of five blades with a simplified flat geometry (see Fig. 2). The rotational oscillation of the middle blade can be controlled by a mechanical drive. The oscillation of the middle blade is monitored using a laser triangulation sensor, and the pressure distribution over the blade is measured using miniature unsteady pressure transducers. Detailed description of the test facility can be found in [20]. Using this facility, an experimental study investigating the limits of applicability of the quasisteady approximation was carried out in [21].

Fig. 1
figure 1

New test facility for forced flutter research

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A similar transonic linear cascade test rig was constructed at the Royal Institute of Technology (KTH) by Glodic et al [22]. The oscillations of blades in a traveling wave mode are induced by piezoelectric actuators. These actuators allow oscillation frequencies \(f=1{-}2.5\, {\textrm{kHz}}\) which corresponds to reduced frequencies \(k=2-4\). Another transonic linear cascade was developed in Indian Institute of Science [23]. The oscillations of central blade are driven by barrel cam mechanism up to frequency 250 Hz.

Subsonic linear cascade for controlled flutter measurement is present at University of West Bohemia in Pilsen [18]. Annular test rigs for controlled flutter measurement were developed at KTH (subsonic annular section) [24] and at École Polytechnique Fédérale de Lausanne (annular, up to \(M= 1.4\)) [25]). Test rigs for controlled flutter testing that the author is aware of are summarized in Table 1.

Table 1 Test rigs for controlled flutter measurements
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Tall slender blades of modern turbomachines have first flap and torsional eigenfrequencies up to 500 Hz. Therefore, the forced oscillation of blades in the current test facility has to be in identical frequency range. This produces high inertial forces acting on the blade, and therefore it is necessary to take into account the elastic deformation of the blade. Consequently, the influence of the elastic blade deformation onto the flow field and its two-dimensionality has to be assessed. Knowledge of inertial and aerodynamic loading is also necessary for the structural design of the blades in this experimental facility.

This study utilizes experimental and computational modal analysis to determine the eigenfrequencies and damping of the blade, providing insights into its dynamic behaviour (Sect. 2—Modal analysis). In Sect. 3 Transient structural analysis is performed to obtain the deformed shape of the blade under high-frequency oscillations. Section 4CFD analysis compares the flow around the deformed blade with that around the ideally rigid (undeformed) blade, evaluating the influence of elastic deformation on the flow field. These combined techniques contribute to a better understanding of the structural and aerodynamic aspects of the blade cascade for controlled flutter testing, facilitating the development of robust test blades that accurately simulate real-world conditions and enhance our understanding of flow-induced vibrations in turbomachinery.

Fig. 2
figure 2

Geometry of the blade

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2 Modal analysis

In the current experimental setup, where the middle blade undergoes forced oscillation, the natural frequencies of the blade need to be known. When the oscillation frequency approaches resonance, the structural stresses could outreach the material strength, or the elastic deformations of the blade might exceed the geometric limits given by the dimensions of the sidewall slots. The eigenfrequencies of the blade can be quite easily obtained by computational modal analysis. However, the numerical simulation of stresses and deformations during forced oscillation is dependent on the damping, especially near resonance. The damping of the blade can be identified only experimentally, because the overall damping is dependent on the internal damping of the material, geometry of the structure and friction damping. For that reason, free experimental modal analysis was performed. The structure in free modal analysis is unconstrained, placed on the soft pad with damping an order of magnitude higher than measured structure.

2.1 Computational modal analysis

Two types of computational modal analyses are performed: free and fixed. The equation of motion for an undamped system, expressed in matrix notation using the linear elastic material model assumptions, is given by

$$\begin{aligned} {[}M]\ddot{u} + [K]u = 0 \ , \end{aligned}$$
(1)

where [M] is the mass matrix, u is the displacement vector and [K] is the structural stiffness matrix. For a linear system, free vibrations will be harmonic of the form:

$$\begin{aligned} u = \varphi _i \cos (\omega _i t) \end{aligned}$$
(2)

where \(\varphi _i\) represents the mode shape of the ith natural frequency (eigenvector), \(\omega _i\) is the ith natural circular frequency in radians per second, and t is time. This leads to an eigenvalue problem which may be solved for up to n values of \(\omega \) and n eigenvectors \(\varphi \) satisfying equation (1), where n is the number of DOFs. The main dimensions of the blade are: length \(l =341\,{\textrm{mm}}\), width \(w = 120\,\textrm{mm}\) and thickness (as well as radius of leading and trailing edge) \(t =5\,\textrm{mm}\). Material of the blade is high-strength stainless steel with Young’s modulus \(E = 210 \,\textrm{GPa}\), Poisson ratio \(\nu = 0.3\) and density \(\rho = 7850\,\text{kg \,m}^{-3}\).

Free modal analysis of single blade was done for comparison with free experimental modal analysis. The fixed modal analysis was performed to evaluate the eigenfrequencies of the blade in the test rig. The boundary conditions for fixed modal analysis for displacement \(\textbf{u}\) were \(\textbf{u} \cdot \textbf{n} = 0\) in the bearings, where \(\textbf{n}\) is unit outer normal, and \(\textbf{u} = 0\) in the hexagonal shoulder. For the free modal analysis there are no boundary conditions, just the six rigid-body motion modes with zero natural frequencies have to be removed from further analysis. For both free and fixed modal analyses, no loads are applied. The governing equations were solved numerically in ANSYS Mechanical on a mesh consisting of 590,000 elements (see Fig. 3).

Fig. 3
figure 3

Detail of mesh for computational modal and structural transient analyses. (Color figure online)

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Oscillation frequency of the blades in the test rig should not be near the eigenfrequency of the first torsional eigenmode to avoid resonance. In the fixed computational modal analysis, the first eigenfrequency (torsional eigenmode) is 270 Hz and the eigenfrequency of the second (bending) eigenmode is 541 Hz. The results of computational modal analysis are summarized in Table 2 and corresponding eigenmodes are depicted in Fig. 4.

Table 2 Results of computational free and fixed modal analysis
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Fig. 4
figure 4

First (torsional), second (bending) and third (2nd torsional) eigenmodes from fixed computational modal analysis

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2.2 Experimental modal analysis

The experimental modal analysis was performed in the free configuration, which allows measurement of blade out of the test rig, see Fig. 5. Therefore, the constraints are not considered, which means that the frictional damping is neglected. This simplification keeps vertical displacements, predicted by the transient structural simulation, on the safe side.

The blade was arranged on two flexible elements and a mesh of 15 measuring points was prepared. Measurement apparatus consisted of impact hammer Brüel &Kjaer 8206-002 and the miniature accelerometer Brüel &Kjaer 4394 which was placed in point 14 (grey dot in Fig. 5). In every measuring point, five measurements (impacts) were performed and the average transfer function was computed. The modal parameters (eigenfrequencies, modal damping and eigenmodes) of the system were determined by regression calculation in the frequency range up to 1600 Hz.

Fig. 5
figure 5

Measurement setup (left) and comparison of torsional eigenmode from free computational (top) and experimental (bottom) modal analysis

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Experimentally determined first torsional eigenfrequency (2nd overall) is 711 Hz compared to the computational 694 Hz, which is suitable agreement. Measured modal damping ratio for the first torsional eigenmode is 0.03%. Results are summarized in Table 3.

Table 3 Results of experimental and computational free modal analysis: first torsional (second overall) eigenfrequency and modal damping ratio
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Experimental modal analysis provides modal damping ratio \(\xi \), which is used to compute constants \(\alpha \) and \(\beta \) for the Rayleigh damping model used in the structural transient simulation. In this simulation, the damping matrix \(\textbf{B}\) is estimated as

$$\begin{aligned} {\textbf {B}} = \alpha {\textbf {M}} + \beta {\textbf {K}}. \end{aligned}$$
(3)

\({\textbf {M}}\) and \({\textbf {K}}\) are the mass and stiffness matrices and constants \(\alpha \) and \(\beta \) are given by

$$\begin{aligned} \alpha = 2 \xi \frac{1}{\omega _1 + \omega _2}, \qquad \beta = 2 \xi \frac{\omega _1 \omega _2}{\omega _1 + \omega _2}, \end{aligned}$$
(4)

where \(\omega _1 = 2 \pi f_1\), \(\omega _2 = 2 \pi f_2\) are angular eigenfrequencies, \(f_1 = 313 \, \textrm{Hz}\) and \(f_2 = 711 \, \textrm{Hz}\) are eigenfrequencies obtained from experimental free modal analysis.

3 Transient structural analysis

Structural transient analysis was performed to obtain the deformed shape of the blade due to inertial forces. The equation of motion for damped system expressed in matrix notation is given by:

$$\begin{aligned} {[}M]\ddot{u} + [C]\dot{u} + [K]u = {F(t)} \ , \end{aligned}$$
(5)

where [M] is the mass matrix, u is the displacement vector, [C] is the damping matrix and F(t) is load vector.

The boundary conditions for displacement \(\textbf{u}\) were

$$\begin{aligned} \textbf{u} \cdot \textbf{n} = 0 \end{aligned}$$
(6)

in the bearings. In the hexagonal shoulder, harmonic rotational displacement

$$\begin{aligned} \phi (t) = \phi _{0} \hspace{0,1cm} \textrm{sin} (2 \pi f t) \end{aligned}$$
(7)

was prescribed. The traction boundary condition for all other surfaces is

$$\begin{aligned} \mathbf {\sigma } \cdot \textbf{n} = 0, \end{aligned}$$
(8)

where \(\mathbf {\sigma }\) is the stress tensor.

Same material constants as in Sect. 2.1 are employed. Linear elastic material model was used. The damping of the structure was described by Rayleigh damping model (3). Computational mesh is identical as in Sect. 2.1. The timestep was \(t = 0.0001 \, \textrm{s}\). A computational time of \(T= 0.1\,\textrm{s}\), i.e. 20 periods of oscillation, was necessary in order to pass the initial transient and reach steady-state oscillations.

The governing equations are solved numerically using the Finite Element Method in ANSYS Mechanical by implicit solver based on Newmark time integration method on a mesh consisting of 590,000 elements (same as in Sect. 2.1, see Fig. 3). Mesh convergence study was performed and the observed parameter was average local von Mises stress (Fig. 6 left). The results do not significantly vary if mesh element size is lower than \(h = 1.5 \, \textrm{mm}\) (Fig. 6 right). The element size \(h = 1.25 \, \textrm{mm}\) was used for calculations. Anticipated frequency of forced oscillation \(f=200\,\textrm{Hz}\) and angular amplitude \(\phi _0 = 1^{\circ }\) were selected.

Fig. 6
figure 6

Area for evaluation of average local von Mises stress (left). Average local von Mises stress as a function of mesh size (right)

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3.1 Results of the transient analysis

Figure 7 shows the temporal evolution of the vertical displacement at the rear end of the leading edge (see point A in Fig. 2), evaluated both for the idealized rigid and real elastic blades. The oscillations of the elastic blade at time \(t=0.08 \, \textrm{s}\) are assumed to be stabilized at the maximum value of vertical displacement circa 3 mm. Out of this value, 1 mm is due to rigid rotation, and 2 mm corresponds to elastic deformation caused by inertial forces.

Fig. 7
figure 7

Temporal evolution of the vertical displacement of rigid and elastic blade

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Fig. 8
figure 8

Shape difference (scaled) between rigid (grey) and elastically deformed (blue) blade

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The deformed geometry of the blade obtained by transient structural analysis, which is further used in the CFD simulation, was taken at two times. For the positive incidence angle of the blade the time is \(t_1 = 0.0937\,\textrm{s}\), which corresponds to rotation angle of the hexagonal shoulder \(\phi = 1^{\circ }\). And for negative incidence angle of the blade the time is \(t_2 = 0.096\,\textrm{s}\), corresponding to rotation angle of the hexagonal shoulder \(\phi = -1^{\circ }\), see Fig. 7.

4 CFD analysis

The goal of the CFD simulations is to evaluate the influence of the blade elastic deformations on the flow field. The difference of the geometry of the rigid and elastic blade twisted due to inertial forces is depicted in Fig. 8. The flow field is numerically computed for the blade motion frozen in four limit configurations which are summarized in Table 4. Configurations are combination of positive/negative incidence angle and rigid/elastic behaviour of blade.

Table 4 Computed configurations of the blade
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4.1 Geometry and mesh

Similarly, as in [26], the cascade (see Fig. 10) has a slope of \(31.5^{\circ }\) and pitch \(74.52\,\textrm{mm}\). The span of the blade exposed to air flow is \(160\,\textrm{mm}\). The computational mesh is composed of hexahedral elements in the free stream, with a structured layer capturing the boundary layer of the blades. The values of y+ on the surface of the blades range between 3–6. The refinement is not realized towards the sidewalls, therefore the side boundary layers are not well resolved. Even with these compromises necessary to keep the computational cost of the simulation reasonable, the mesh consists of up to 16,000,000 elements. The mesh dependence was investigated, see Fig. 9. Based on this analysis, the “Fine mesh”, shown in Fig. 10, was selected for all simulations. The computation took 15 h on a workstation with 8 cores Intel Xeon W-2245 3.90 GHz.

Fig. 9
figure 9

Influence of the computational mesh on the isentropic Mach number distribution over the chord of blade 3 in NEG-RIG configuration

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4.2 Mathematical model

The airflow is modeled as a steady flow of a compressible ideal gas described by compressible Navier–Stokes equations:

$$\begin{aligned} \nabla \cdot (\rho \textbf{u})= & {} 0, \end{aligned}$$
(9)
$$\begin{aligned} \nabla \cdot (\rho \textbf{u} \otimes \textbf{u})= & {} \rho \varvec{f} + \nabla \cdot \varvec{\tau }, \end{aligned}$$
(10)
$$\begin{aligned}{} & {} \nabla \cdot ( E \textbf{u}) = \rho \varvec{f} \cdot \textbf{u} + \nabla \cdot (\varvec{\tau } \textbf{u}) + \rho q - \nabla \cdot \textbf{q} \end{aligned}$$
(11)

where \(\rho \) is the density, \(\textbf{u}\) is the velocity vector, t is time, \(\varvec{f}\) is density of external volume forces, \(\varvec{\tau } = -pI + \lambda (\nabla \cdot \textbf{u})I + 2 \mu \textbf{D}\) is the stress tensor, p is pressure, I is identity tensor, \(\lambda = - \frac{2}{3} \mu \) is second viscosity coefficient, \(\mu \) is dynamic viscosity coefficient, \(\textbf{D}\) is rate of deformation tensor, E is the total energy per unit volume, q is volume heat source and \(\textbf{q}\) is the heat flux vector.

Air is considered as an ideal gas with specific heat capacity \(c_p = 1005\,\text{J.kg.K}^{-1}\), molar mass \(M = 28.966 \, \text{kg.kmol}^{-1}\) and dynamic viscosity \(\mu = 1.81 \times 10^{-5} \, \text{Pa} \cdot \text{s}\). Simulating a suction-type wind tunnel with atmospheric entrance conditions, the boundary conditions at the inlet are the total pressure \(p_{tot} = 100,000\,\textrm{Pa}\), total temperature \(T_{tot} = 300\,\textrm{K}\), turbulence intensity \(T_u = 2 \, \%\), turbulent viscosity ratio \(\frac{\mu _{T}}{\mu } = 10\), zero velocity gradient \(\frac{\partial \textbf{u}}{\partial \textbf{n}} = 0\) and zero heat flux \(\frac{\partial T}{\partial \textbf{n}} = 0\). Two flow regimes were investigated. In order to reach the outlet isentropic Mach number \(M_{2isA} = 0.5\) and \(M_{2isB} = 0.7\), the static pressure at the outlet boundary condition is set to \(p_{2A} = 84,302 \, \textrm{Pa}\) and \(p_{2B} = 72,092 \, \textrm{Pa}\), respectively (see Tab. 5). The backflow boundary conditions consist of: turbulent intensity \(T_{ub} = 5 {\%}\), turbulent viscosity ratio \((\frac{\mu _T}{\mu })_b = 10\) and total temperature \(T_{totb} = 300 \, \textrm{K}\). For the velocity, zero gradient is enforced by extrapolating the velocity values from the interior nodes. On the top, bottom and side walls and blades, the no-slip boundary condition \(\textbf{u} = \textbf{0}\) is specified, further pressure gradient \(\frac{\partial p}{\partial \textbf{n}} = 0\) and heat flux \(\frac{\partial T}{\partial \textbf{n}} = 0\).

Table 5 Boundary conditions at the outlet
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Fig. 10
figure 10

Geometry and mesh of the 3D computational domain and detail of the mesh refinement near the blade

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4.3 Numerical solution

The governing equations (9), (10) and (11) are solved numerically using the Finite Volume Method by software ANSYS Fluent 2021 R2. The computational setup was: implicit density based solver, k-omega SST turbulence model with log-law not employed. The solver turns on wall functions in wall-adjacent cells where y+ < 11.2. Zero initial conditions were set and full multigrid initialization was employed as a pivotal step in preparing the initial conditions for the simulation. Discretization schemes of the second order were used: second order upwind discretization for the convective term, turbulent kinetic energy and specific dissipation rate, and least squares scheme for the diffusion term. Euler implicit discretization in time was used and CFL (Courant–Friedrichs–Lewy) condition is used to compute the time step. The computations were stopped when residuals of continuity, energy, turbulent kinetic energy, specific dissipation rate and velocities were lower than \(1\times 10^{-5}\).

4.4 Validation of the CFD model

The CFD model was validated using the experimental data obtained during wind tunnel measurements for outlet isentropic Mach number \(M_{2is} = 0.5\). Experiments were carried out in test section depicted in Fig. 1. For these measurements, all blades were kept stationary. The experimental results were compared to the CFD calculations with idealised rigid blade in plane P2 (see Fig. 12). In the experimental setup, the static pressure is measured in 10 positions along the blade chord. Further details about the measurement setup can be found in [21].

Isentropic Mach numbers are computed from measured static pressures using (12). Comparison of CFD results for idealised rigid blade and experiment for outlet Mach number \(M_{2is} = 0.5\) are presented in Fig. 11. The comparison between the CFD and experimental data revealed a very good agreement on the upper side of the blade. There is slight yet acceptable difference between CFD and experiment on the lower side of the blade.

Fig. 11
figure 11

Comparison of CFD results with measurements for \(M_{2is} = 0.5\)

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4.5 CFD results

The results are presented in terms of isentropic Mach number

$$\begin{aligned} M_{is} = \sqrt{\frac{2}{\gamma - 1} \left[ \left( \frac{p}{p_{tot}} \right) ^{\frac{1 - \gamma }{\gamma }} - 1 \right] } \end{aligned}$$
(12)

where p is static pressure, \(p_{tot}\) is the total pressure at the inlet and \(\gamma = C_p / C_v = 1.4\) is the specific heat ratio. Isentropic Mach numbers are evaluated in three planes perpendicular to the blade cascade. Sections P1 a P3 are 10 mm from the side walls and section P2 is in the mid-span, see Fig. 12. The local incidence angles for the elastic blade in these planes are \(\alpha _{P3} = \pm \, 2.37^{\circ }\), \(\alpha _{P2} = \pm \, 3.16^{\circ }\) and \(\alpha _{P1} = \pm \, 3.65^{\circ }\). The evolution of incidence angle along blade span of elastic blade is shown in Fig. 13. The incidence angle for the rigid blade is caused only by rotation and is constant, \(\alpha _{P1} = \alpha _{P2} = \alpha _{P3} = \pm \, 1^{\circ }\).

Fig. 12
figure 12

Location of planes for result evaluation

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Fig. 13
figure 13

Evolution of the incidence angle along the blade span due to elastic deformation (see also Fig. 8)

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4.5.1 Positive incidence angle results

Figure 14 shows comparison of the flow fields between POS-EL and POS-RIG cases (\(\phi = 1^{\circ }\)) in plane P2 for outlet isentropic Mach number \(M_{2is} = 0.5\).

Elastic configuration with higher incidence angle of the middle blade induces faster airflow in the higher interblade channel and slower airflow in the lower interblade channel. Although the incidence angle of the elastic blade varies along the span of the blade, the isentropic Mach number is only slightly distorted.

In both cases POS-EL and POS-RIG, the flow is mostly subsonic. Only, in the POS-EL case, the higher incidence angle of the middle blade causes small supersonic area shortly downstream of the leading edge of fourth blade of cascade.

Fig. 14
figure 14

Isentropic Mach number distribution in the elastic (top) and rigid (bottom) case in section cut P2 in rotation \(\phi =1 \) for \(M_{2is} = 0.5\)

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The difference between the elastic and rigid blade in positive incidence angle for outlet Mach number \(M_{2is} = 0.7\) is significantly larger. The results in the form of isentropic Mach number are depicted in Fig. 15, sonic lines are highlighted in black. In the POS-RIG configuration, there is a large supersonic area in the interblade channel between blade 1 and 2, compared to smaller supersonic area in the case of elastic blade. Opposite phenomenon is visible in the interblade channel between blade 3 and 4. There is larger supersonic area in the POS-EL case due to the higher incidence angle of blade 3 and thus narrower interblade channel.

Fig. 15
figure 15

Isentropic Mach number distribution in the elastic (top) and rigid (bottom) case in section cut P2 in rotation \(\phi = 1\) for \(M_{2is} = 0.7\). Sonic lines highlighted in black

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4.5.2 Negative incidence angle results

Flow fields in plane P2 for cases NEG-EL and NEG-RIG for outlet isentropic Mach number \(M_{2is} = 0.5\) are depicted in Fig. 16. Higher incidence angle in the NEG-EL case induces faster airflow in the lower interblade channel. The difference of airflow velocity in the lower and higher interblade channel of the middle blade is more significant than in the cases of positive incidence angle (POS-EL, POS-RIG).

In both NEG-EL and NEG-RIG cases there are supersonic areas near the leading edge of the middle blade. In the elastic case the supersonic area is slightly larger. This is in contrast to the POS-EL case where supersonic area was present only on the leading edge of the fourth blade.

Fig. 16
figure 16

Isentropic Mach number distribution in the elastic (top) and rigid (bottom) case in plane P2 in rotation \(\phi = -1\) for \(M_{2is} = 0.5\)

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Figure 17 shows comparison of the flow fields between NEG-EL and NEG-RIG cases (\(\phi = -1^{\circ }\)) in plane P2 for outlet Mach number \(M_{2is} = 0.7\). There is only very little difference in the field of the isentropic Mach number between elastic and rigid blade.

Fig. 17
figure 17

Isentropic Mach number distribution in the elastic (top) and rigid (bottom) case in section cut P2 in rotation \(\phi = -1\) for \(M_{2is} = 0.7\). Sonic lines highlighted in black

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Profiles of the isentropic Mach number along the blade chord are plotted in Fig. 18. For outlet Mach number \(M_{2is}=0.5\), there is considerable difference between the elastic and rigid blade in all planes P1, P2 and P3 due to the higher local incidence angle in the elastic case. The changes of the isentropic Mach number profiles in planes P1, P2 and P3 can be hardly observed. Bigger difference is visible in the case of negative incidence angle (compared to the positive incidence angle) in the isentropic Mach number behind the leading edge between rigid and elastic blade.

For outlet Mach number \(M_{2is}=0.7\) and positive incidence angle, the insentropic Mach number profiles are considerably different for the EL and RIG case (see Fig. 19). The same was also clearly visible in Fig. 15, which shows change of locations of supersonic areas between elastic and rigid blade. In case of negative incidence angle and outlet Mach number \(M_{2is}=0.7\), the isentropic Mach number profiles of rigid and elastic case are the most similar. Clearly the flow field in the channel is aperiodic and changing inclination of the middle blade affects this aperiodicity. Periodicity is better with positive inclination of the middle blade and worse with negative inclination of the middle blade.

Fig. 18
figure 18

Comparison of Isentropic Mach numbers along the blade chord for \(M_{2is} = 0.5\). Top—POS-RIG and POS-EL, bottom—NEG-RIG and NEG-EL

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Fig. 19
figure 19

Comparison of Isentropic Mach numbers along the blade chord for \(M_{2is} = 0.7\). Top—POS-RIG and POS-EL, bottom—NEG-RIG and NEG-EL

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4.5.3 Aerodynamic moments

Total force and moment acting on the middle blade was computed from the flow variables provided by the CFD simulations. Inertial moments were evaluated by structural transient simulations for the elastic case. In case of rigid blade, the inertial moments were estimated by calculation for a flat plate, see Tab. 6.

The aerodynamic moments for POS-RIG, NEG-RIG and NEG-EL are 2–3 times higher for outlet isentropic Mach number \(M_{2is}=0.7\) than for \(M_{2is} = 0.5\). The faster flow \(M_{2is}=0.7\) led to approx. two times lower aerodynamic moment than for \(M_{2is}=0.5\) in the POS-EL case. The highest aerodynamic moment \(10.26 \, \textrm{Nm}\) was observed in NEG-EL configuration. Nevertheless, for the frequency of oscillation \(f = 200\,\textrm{Hz}\), the aerodynamic moments are at least ten times lower than inertial moments due to the high frequency oscillation. Inertial moments increase quadratically with the oscillating frequency. The situation is more complicated for the aerodynamic moments. For subsonic flow, they increase with flow velocity and incidence angle. But the value of aerodynamic moment is non-trivial for transonic and supersonic flow regimes.

This difference between aerodynamic and inertial moments justifies the uncoupled approach of flow field computations along the “frozen” blade. Performing full fluid–structure simulations in this case would not help to obtain more reliable data. Even with this simplified model it can be assumed that the inertial forces are more severe from the point of stress induced in the middle blade.

Table 6 Comparison of aerodynamics moments acting on the middle blade for oscillation frequency \(f = 200\,\textrm{Hz}\)
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5 Conclusion

A new blade cascade for controlled flutter testing at transonic speeds was developed. The middle blade in the cascade is exposed to high-frequency oscillations to simulate blade flutter which induces high inertial forces in the blade. For the measurements it is important to know how the elastic deformation of the blade due to inertial forces affects flow field along the blade span - if the flow along the span remains two dimensional. Evaluation of the influence of elastic deformation on the flow field in the blade cascade for controlled flutter measurement was performed by combination of measurements and numerical simulations.

Free experimental modal analysis was performed to estimate damping of the structure. Fixed computational modal analysis provided eigenfrequencies of the blade to avoid resonance during measurements. Free experimental and computational modal analysis was compared for validation which gave good agreement. The deformed shape of the blade was obtained by structural transient simulation at peak positive and negative rotation angle \(\phi = \pm \, 1^{\circ }\) after stabilization of the initial transient oscillations. The flow fields in the blade cascade for outlet isentropic Mach number \(M_{2is} = 0.5\) and \(M_{2is} = 0.7\) were computed for four configurations - positive and negative incidence angle for rigid (idealized) and elastic (real) blade. The flow fields in the elastic configurations were affected by higher incidence angles, but surprisingly only very weakly. Significant difference between elastic and rigid blade was observed only for positive incidence angle and outlet isentropic Mach number \(M_{2is} = 0.7\). For the given Mach numbers and given cases of either elastic or rigid blade, the flow fields remain nearly two-dimensional along the blade span.

The structural and aerodynamic moments acting on the blade were computed. Moments induced by inertial forces due to high-frequency oscillation are by an order of magnitude larger than moments caused by the fluid flow. In this case the stresses in the oscillating blade are mainly induced by inertial forces.

Experimental data for a lower isentropic Mach number \(M_{2is} = 0.5\) was available and CFD model was validated, demonstrating good agreement, especially on the upper side of the blade.

The loading conditions of blades in the experimental setup for forced flutter testing were determined, providing valuable insights for blade design. The computed structural and aerodynamic moments clearly demonstrate that inertial forces due to high-frequency oscillation induce significantly larger moments than those caused by fluid flow for analysed flow regimes. This highlights the importance of considering inertial forces in blade design to ensure the blade’s ability to withstand high-frequency oscillations and accurately simulate real-world conditions. These conclusions will guide the development of robust test blades that can cover a wider frequency range, effectively simulating the flow-induced vibrations experienced by real turbomachine blades. In particular, exploring the use of fiber composite materials in blade design offers the potential for improved strength-to-weight ratio and enhanced damping properties, further enhancing the test blade performance and accuracy.