Aerodynamic work estimate

The aerodynamic damping of LPT airfoils can be reproduced using linearised aerodynamic models (Panovsky and Kielb, 2000). The aerodynamic damping in the present work has been computed using a well-validated linearised Navier-Stokes solver known as Mu2s2T−L (Corral et al., 2003). The main hypothesis is that the flow can be decomposed into a mean flow, which is computed by a steady nonlinear RANS solver (Burgos et al., 2011; Corral et al., 2017) plus an unsteady periodic perturbation, which is Fourier transformed in time and solved using a linearised frequency-domain Navier-Stokes solver.

The aerodynamic work at low-reduced frequencies (Corral and Vega, 2016a,b; Vega and Corral, 2016), such as those present in LPT flutter, can be expressed as

where ρa is the air density at the rotor exit, Ue is the exit velocity, S is the rotor surface area, c is the chord of the blade and δa is the vibration amplitude of the rotor. The sub-index a is added to remark that this is the characteristic macroscopic displacement of the airfoil, which should not be confused with the displacement in the rotor fir-tree attachement δ.

The dimensionless aerodynamic work-per-cycle W~a depends on the reduced frequency k=ωc/Ue, the Inter-Blade Phase Angle, IBPA, and the airfoil loading, which is represented by the lift coefficient, CL. Other parameters, such as the Reynolds or Mach numbers, play a minor role. The actual dependence of the minimum aerodynamic damping Wamin=minW~a(IBPA) with k is linear at low reduced frequencies (Corral and Vega, 2016a; Vega and Corral, 2016). Flutter severity increases with the loading form but in a more indirect way than with the CL [see (Corral and Vega, 2017)]. The main impact of the Mach number is frequently related to an increase in the dynamic pressure (i.e. Ue∝Me) but its influence in dimensionless terms is small. However, the impact of the Mach number on the pressure coefficient distribution along the blade and the lift coefficient, CL(Me), gives rise to a slight change in the dimensionless aerodynamic work [see (Vega and Corral, 2017)].

Therefore, the minimum aerodynamic damping and the exit Mach number are related as

though it must be stressed that, technically speaking, the sole true dependence on the Mach number, i.e. compressibility, is through the CL and ρa.

Mechanical work estimate

An accurate estimate of the mechanical damping is paramount to determining the rotor blade vibration amplitude and the measured unsteady pressure. Material damping is much smaller than the dry friction in the rotor blade attachment, and can be disregarded safely in the modelling. Dry friction is a nonlinear mechanism responsible for the vibration amplitude saturation in unstable rotor blades (Corral and Gallardo, 2014). In this work, a heuristic approach used previously to predict the vibration amplitude of realistic rotor blades (Corral and Gallardo, 2006, 2008) will be employed to predict firstly the final saturated vibration amplitude of an aerodynamically unstable rotor and then the unsteady pressure perturbations generated due to the vibration. The primary hypothesis is that the normal stresses in the fir-tree contact area due to the pull load are very high. Therefore, the relative displacement between the rotor blade and the disk attachment is small.

Sellgren and Ollofson’s micro-slip model (Sellgren and Olofsson, 1999) assumes the contact surface has asperities and a certain roughness. The model relates the tensional state in the contact with the macroscopic displacement using laws different from Coulomb’s. The model describes analytically the hysteresis curves of the tangential force in the contact, F, during the preloading as

where the dimensionless displacement in the attachment is δ~∗=δ∗/δc, and δc=μE′λ/4G′ is the characteristic length of the micro-slip, μ is the friction coefficient, N is the normal load, E′ and G′ are the composite elasticity and shear modulus respectively, and λ the contact penetration length. It may be shown that, if the hysteresis loop is symmetric, i.e.: F∗=FU=FL and δ∗=yU=yL, the work per cycle is

where F~∗=F∗/μN=1−(1−δ~∗)5/2 is the dimensionless tangential force. If δ~∗≪1, or in other words, if the tangential force F≪μN then it may be shown that

The scaling of the mechanical work with an exponent greater than two is essential for the stabilization of aerodynamically unstable rotors.

Estimate of the vibration amplitude

The vibration amplitudes that appear in the aerodynamic damping, δa, [see Equation (3)], and the mechanical work, δ∗, [see Equation (6)] are not only different but have different orders of magnitude. While the former relates to the macroscopic rotor blade displacement, the latter, δFT=δ∗, relates to the relative displacement between the rotor blade and the disk in the fir-tree attachment. An approximate relationship between both to close the problem was provided in (Corral and Gallardo, 2006, 2008), but it is summarised here for completeness.

Since the magnitude of the displacements in the fir-tree and and rotor blade tip are very different, it is convenient to express the mechanical work as a function of the alternate tangential force in the attachment, F~∗, since it is easier to link with the macroscopic displacement. If δ~∗≪1 then Equation (6) reduces to F~∗=(5/2)δ~∗, and Equation (7) becomes

However, if δ~∗∼O(1), Equation 5 must be injected into Equation (6) to obtain W~m=W~m(F~∗).

The next step is to relate the alternate tangential force or stress at the attachement to the blade vibration. If the rotor blade is modelled as beam clamped in the attachment then the alternate moment, M is

where ma is the rotor blade mass, L the height of the blade including the shank, and ωN the natural frequency of the blade. Yξξ(0) is a constant that depends on the beam boundary conditions at the attachment. This relation can be obtained too from a FEM. If the internal dynamics of the attachment is neglected (Corral and Gallardo, 2006) then this momentum has to be balanced by the vertical alternate reaction in the fir-tree. It can be shown that

where aFT is the half-width of the fir-tree, α the inclination angle of the contact surface, Ac, and nlob the number of lobes. The above expression allows to close the model except from a constant that is derived from experimental data (Corral and Gallardo, 2006).

Estimate of the unsteady pressure component due to flutter

In this work, flutter severity is measured using unsteady pressure transducers. It is, therefore, paramount to estimate the component of the unsteady pressure associated with flutter. In the low-reduced frequency limit, i.e. k≪1, the characteristic unsteady pressure on the airfoil can be estimated as pc′∝ρeUe2(δa/c). It has been shown that the vibration amplitude δan−2∝Wa′/Wm′, where n=2.5 or 3 depending on whether the vibration amplitude in the attachment is small or large respectively. The component associated with the dry friction, Wm′ depends mainly on geometric factors, the natural frequency of the blade, and the shaft speed. For a given rotor blade and shaft speed, Wm′ is constant and hence δan−2∝Wa′. The blade vibration amplitude scales [using Equation (4)], as δan−2∝F(Me2). Finally, if W~a is deemed approximately constant in the Mach number range of interest, the unsteady pressure scales approximately

The main conclusion is that the sensitivity of the unsteady pressure due to the exit Mach number is very high.

Wind-tunnel and rig description

An isolated shrouded high aspect ratio LPT rotor blade was tested in the CTA high-speed wind tunnel during 2022 (Gallardo et al., 2022). The rotor blade used in this second experimental campaign is similar to that described in Corral et al. (2019) and tested in 2011. The aerodynamic shape and the two-lobe fir-tree attachment with the disk used in both experimental campaigns are the same, but the blade count was reduced from 146 to 144 to ease the grouping of blades in packets. The tip-shroud of the rotor blade was designed to accommodate a rear magnet to excite the rotor in a separate in-vacuum test to characterise the damping of the bladed-disk. The two rotor blades are very similar from a dynamic point of view. A more thorough description of the test can be found in Corral et al. (2019) and Gallardo et al. (2022).

The experiment uses an isolated bladed rotor subject to the incoming conditions of the facility VIGVs, which are located more than ten chords upstream of the testing article. Therefore, the rotor blade is considered isolated and subject only to the weak perturbations associated with the blade-passing frequency of the upstream VIGVs, whose chord is about 50% longer than that of the rotor blade. The boundary layer generated at the inlet duct is sucked one chord upstream of the rotor to avoid unrealistic effects in the secondary flow vortex system. A general layout of the wind tunnel can be seen in Figure 1(left).

Figure 1.

Layout of the CTA wind-tunnel used for the experimental campaign (left), time and circumferential averaged Fourier transform of the unsteady pressure signals obtained for Ω=103%, ρ=100% and three different levels of exit Mach number (Me=0.675, Me=0.775 and Me=0.825) (center) and modeshape of the first flap mode (right).

The instrumentation used to measure aerodynamics includes total pressure and total temperature rakes located at the inlet and outlet of the rotor. Inlet and outlet boundary conditions of the row were acquired using fast-response five-hole miniaturized probes that are radially and azimuthally traversed across a distance equivalent to several rotor pitches. Blade Tip-Timing (BTT) optical probes were installed in the outer casing, pointing radially to the rotor to measure the vibration amplitude of every individual blade. Finally, unsteady pressure transducers (Kulite XT-190M) were flush-mounted in the outer casing, one chord downstream of the rotor blade.

Unsteady pressure measurements

Unsteady pressure transducers are flush mounted in the outer casing, one chord downstream of the trailing edge of the rotor blade. Each pressure transducer records the unsteady pressure signal for 40 seconds with a sampling frequency of 25.6 kHz, which is much higher than the range of interest, that in this case, oscillates between 500 and 2,000 Hz. Pressure signals are segmented into one-second windows, Fourier transforms in time are performed on each window, and their Fourier coefficients are time-averaged. Afterward, Fourier transforms of different pressure transducers located at different circumferential positions are averaged to obtain the mean representation of the signal’s harmonic content.

The test matrix of the experimental campaign includes different intentional mistuning patterns, forcing mechanisms (using magnets on the tip-shroud of the blade), a wide range of rotational speeds, densities and three levels of exit Mach number. However, the present study has been carried out for the baseline configuration without magnets and without intentional mistuning patterns. Three different exit Mach numbers at different rotational speeds have been analyzed. Figure 1(center) shows the harmonic content of the unsteady pressure transducers for Ω=90%, the baseline level of density (ρ=100%), and three different exit Mach numbers (Me=0.675,0.775and0.825).

The acoustic responses can be divided into different families depending on the source of the excitation. Figure 1(center) shows five families of peaks sorted from low to high frequency corresponding to facility noise, non-synchronous vibration whose source has not been identified, together with a strong forcing and its harmonics, flutter signals as a result of the blade vibration due to different unstable nodal diameters, and the blade-passing frequency.

Since the present paper aims to study the evolution of flutter peaks with the exit Mach number, the frequency range shown below is limited to the frequencies of interest for flutter, coinciding with the Doppler-shifted frequency of the unstable nodal diameters. Then, the vibration frequency in the stationary frame of reference is

in physical terms, the frequency separation between different flutter peaks is Ω. One of the main advantages of unsteady pressure sensors over BTT measurements is that they can easily identify the unstable NDs of the flutter signals by their associated Doppler-shifted frequency.

Aerodynamic design

The rotor blade aerodynamic design is representative of an aeronautical low-speed LPT. The total deflection is about 100 deg, the lift coefficient of the blade sections is around 1, and the exit Mach number is high subsonic. In particular, the design intent of the 70% span section is given in Table 1, for more information, the interested reader is referred to Corral et al. (2019) and Gallardo et al. (2022).

Figure 2(left) displays the isentropic Mach number, Mis, of the 70% section and 90% of the nominal shaft speed, Ω, for three different pressure ratios. The section is located out of the secondary flow region. The airfoil operates at a slight positive incidence at these speeds, inducing a front-loaded distribution. At high Mach numbers (M_e = 0.775 and 0.825), the Mis and Cp distributions exhibit an overshoot on the suction side since the rotor blade was designed for a lower Mach number (Me≃0.7). Another interesting and well-known behaviour is that the front loading of the blade is reduced when the Mach number is increased [see Figure 2(right)].

Figure 2.

Isentropic Mach number (left) and pressure coefficient (right) distribution at 70% height and Ω=90%.

Aeroelastic design

The tested bladed disk consists of a disk stiff enough to ensure a small participation of the disk, except perhaps in the first four NDs. Figure 1(right) shows the first mode of the blade, which is a flap mode. It is well-separated in frequency for all the ND from the second mode to avoid interactions among them. The natural frequency of the first mode is about 186 Hz at design conditions, increasing slightly with the rotational speed.

The reduced frequencies of the first mode, k=ωc/Ue, based on the axial chord and exit velocity of the 70% section, at Ω=90% are k = 0.112, 0.105 and 0.092 for M_e = 0.675, 0.775 and 0.825, respectively. The slight decrease in the reduced frequency is mainly associated with the increase in the exit velocity. The critical damping ratio, ξ, and the dimensionless aerodynamic damping are related by:

where μ=ρaS/m is the mass ratio, and m is the modal mass. The impact of the increase of k in the critical damping ratio with the Mach number, through W~a[k(Me),CL(Me)] is negligible since the variation in k is about 20% in the range of interest, and the CL or Swiffel numbers are C_L = 1.195, 1.182 and 1.180 for M_e = 0.675, 0.775 and 0.825, respectively.

The critical damping ratio, ξ, of the first mode for Ω=90%, computed using a linearized frequency domain solver (Corral et al., 2003), is displayed in Figure 3(left). It can be seen that the flap mode exhibits several highly unstable nodal diameters. This situation is frequently seen in aeronautical cantilever LPT rotor blades and the values of k and ξ fully representative of an aeronautical LPT rotor blade (Corral and Gallardo, 2014). The change of the aerodynamic damping with the incidence (shaft speed) is moderate (not shown here for the sake of brevity), whilst ξ is proportional to the facility density level, as it was shown in Corral et al. (2019).

Figure 3.

CDR and CDR scaled with the exit static pressure and the inverse of Me2 normalised with the factor for Mref=0.625 for Ω=90% (N = 2,500 rpm).

Figure 3(right) displays the critical damping ratio ξ scaled with the inverse of the Mach number function, F(Me2), given in Equation (4), as a function of the ND for different Mach numbers. It can be seen that despite the change in the steady blade loading due to the Mach number [Figure 2(right)] and the change in the reduced frequency, the three damping curves collapse into a single one. A similar behaviour was reported by Corral and Vega (2016b) but with a simpler Me2 scaling on the aerodynamic influence coefficients. The matching between the three curves is remarkable validating the expression 4 to scale the aerodynamic damping for different pressure ratios.

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