The demand for lighter and more efficient aero engines has significantly grown during the last few years. To address these issues, modern compressors have been conceived with increasingly larger pressure ratios per stage. Natural consequences of these solutions are an aggressive blade loading and high flutter sensitivity, especially for long and slim blades. Fatigue phenomena, due to large vibratory loads, are severely enhanced [1
]. Blade design approaches aiming specifically to provide a reduction in vibratory loads and a delay of the flutter onset were proposed, e.g., in Refs. [3
]. In addition, the aerodynamic performance is significantly affected on such highly loaded blades. Indeed, larger pressure gradients over the blade suction surface may accelerate the stall onset, therefore degrading the overall compressor performance. Several active flow control solutions have been proposed to optimize the aerodynamic response of heavily loaded blades. Tiedemann et al. [5
] investigated experimentally the effect of steady and pulsed blowing through side wall actuators on a linear compressor cascade. They showed that both steady and pulsed jets are effective in suppressing three-dimensional secondary flow phenomena occurring on critically loaded blades. Matejka et al. [6
] employed a synthetic jet located on the side wall at the level of the blade leading edge. A positive effect of such actuation on the flow field and on the loss coefficient was found. Similarly, a beneficial effect of side wall blowing on flow separation was found in Ref. [7
]. Besides blowing, blade morphing has been extensively studied as a means to enhance the aerodynamic response of heavily loaded compressor and turbine blades. Monner et al. [8
] employed piezo actuators to control the blade shape. The resulting deformed blade was found to affect the measured aerodynamic loads significantly. Hammer et al. [9
] investigated different adaptive blade solutions with numerical computations and experimental tests. Specifically, they studied and compared four morphing solutions, i.e., blades with adjustable angle of attack, movable leading edge, Gurney flaps, and adaptive camber. Suman et al. assessed experimentally [10
] and numerically [11
] the effect of shape memory alloys for blade shape adaptation on a heavy-duty automotive cooling axial fan.
Large interest has been latterly addressed toward plasma actuation. The suitability of these actuators stems from their lightness and their almost negligible intrusiveness into the flow field. Additionally, the absence of mechanical parts avoids the risk of structural and operational failures, likely to occur under high centrifugal loads typical of aero engines. Vo et al. [12
] carried out a computational assessment of plasma actuators installed on a transonic axial compressor. The capability of suppressing spike stall inception was assessed by taking into account several locations and strengths of the plasma actuators. The work of De Giorgi et al. [13
] deals with the experimental and numerical characterization of plasma actuators in a compressor cascade. It was shown that properly tuned plasma actuation allows for an effective reduction of pressure losses on highly loaded compressor blades. Akcayoz et al. [14
] performed numerical and experimental assessments of plasma actuators to control corner stall separation on a heavily loaded blade cascade. In this work, realistic operation conditions were simulated. It was found that a combination of plasma actuators placed on the suction surface is a very effective means to control flow separation and in turn pressure losses. Zhang et al. [15
] carried out numerical investigations of plasma actuators to control flow separation on a high-speed compressor cascade. The effects of both endwall and suction side plasma actuators were studied.
The purpose of the present work is to assess numerically the effectiveness of plasma actuators for load alleviation on a compressor cascade. Here, plasma actuators are meant to reproduce qualitatively the effect of the mechanical L-tab investigated in Refs. [16
] for rotorcraft vibration reduction. Specifically, plasma actuation is meant to develop vortical structures and recirculating flow areas similar to those generated by mechanical control surfaces (see, for instance, Ref. [21
]). Plasma actuators are thought to be located on the pressure and on the suction side (PS and SS, respectively) of the three central blades in the targeted cascade. The plasma induced flow is conceived to be opposite relative to the freestream direction, with the specific aim to generate the aforementioned separated flow areas. Numerical computations performed on the clean blade configuration are validated with experimental data achieved on the wind tunnel facility of the Chair for Aero Engines at Technische Universität (TU) Berlin and with literature results [22
] . Constant angle of attack configurations for the blade cascade are assessed first. It is shown that plasma affects significantly both local and resulting loads on the blade, including the pitching moment. At the same time, no remarkable drag rises are encountered. Traveling wave mode simulations are then performed with alternate PS/SS actuation. Within a traveling wave mode, the blades oscillate at the same frequency but at a certain—constant and uniform—phase lag referred to as Interblade Phase Angle (IBPA), see [25
]. In this work, it is found that appropriate phasing the switch between PS and SS actuation allows for effectively manipulating the peaks of unsteady airloads, especially the pitching moment. As a result, mean and oscillating loads can be controlled, and fatigue phenomena can be alleviated. Additionally, the blade aeroelastic stability could also be improved. The manuscript is structured as follows. Section 2
presents the computational geometry, the set up of the simulations, as well as the numerical modeling for plasma. The comparison between numerical results achieved on the clean cascade and experimental measurements on the same geometry is also reported. Section 3
discusses the simulations carried out on the blades at constant angle of attack. The results of time-dependent computations performed at constant angle of attack are shown to prove the steadiness in the mean of the flow, both for the clean and for the plasma equipped configuration. The effects of PS and SS actuation on the airloads and on the flow field are reported. Section 4
describes the results obtained with traveling mode simulations. The triggering of alternate PS/SS actuation is presented first. The affection of plasma on the pitching moment oscillations is discussed as a second instance. Concluding remarks are provided in Section 5
3. Results at Constant Angle of Attack
Simulations with the blades at constant angle of attack are performed on the blade cascade, in order to investigate the affection of plasma actuation on mean loads and flow field. Only results on the central blade of the cascade are discussed in the following. The counterparts on the remaining blades are not reported for brevity purposes. The freestream velocity is 34.36 m/s and the angle of attack is two degrees. The Reynolds number is again ∼350,000.
shows the time history of the airloads obtained with time-resolved simulations at fixed angle of attack. These simulations are meant to show that the flow can be assumed as steady in the mean, when the blades are kept at a constant angle of attack. The time step for the unsteady computations is set to
s and 100 time steps are performed for each period. It is found that all the three airloads converge to a constant value, after a maximum of ∼0.015 s. Therefore, steady state computations can be performed to assess the affection of plasma actuators on the force coefficients and on the flow field.
shows the normalized velocity and vorticity magnitude fields. The velocity is made dimensionless by the freestream velocity and the vorticity is made dimensionless by the ratio of chord over freestream velocity. A detail of the trailing edge is displayed for the clean blade and for the configurations with PS and SS actuation, respectively. A body force of 300 mN/m is applied on the pressure and on the suction side. Both actuations lead to the development of low speed recirculating flow areas, with a consequent modification of local and resulting loads. In particular, PS actuation yields a recirculating flow region on the pressure side trailing edge, in the area where the plasma is modeled. These locally developed vortical structures shift the application point of the Kutta condition downstream of the physical trailing edge. This is approximately equivalent to employing a longer and more cambered airfoil. Therefore, lift augmentations are expected to be a direct consequence of this phenomenon. The recirculating flow area generated by SS actuation causes an induced camber effect that is opposite relative to the SS counterpart. As a consequence, the resulting effect will be a reduction in the developed airloads. The flow fields of Figure 8
show that, as expected, the wake becomes thicker relative to the clean blade counterpart—the light blue low speed region is larger for the actuated configurations. This effect, which is consistent with the results of Refs. [18
], may lead to possible drag increases compared to the baseline blade. Nevertheless, it will be hereinafter shown that no significant drag rises are encountered for both the actuated configurations. It is also worth noting that the recirculating flow area developed on the suction side is less extended, relative to the pressure side counterpart. This is probably due to the fact that higher velocity gradients are encountered on the suction side, near the trailing edge. Therefore, a larger body force is probably required on the suction side, in order to obtain almost equally extended recirculating flow areas on upper and on the lower side of the blade. It will be illustrated that a consistent asymmetry is encountered in terms of local and resulting airloads achieved with PS or SS actuation. On the basis of these results, the traveling wave mode simulations discussed in the following section are performed with a larger body force on the suction side and a smaller one on the pressure side.
shows the pressure coefficient distribution for the clean blade, compared to the configuration with PS and SS actuation, respectively. Consistently with the results achieved in terms of flow field, PS actuation yields a clearly visible negative peak in the plasma region. The opposite occurs for the SS actuation. The most influential effect induced by the actuation can be seen by looking at the pressure distribution upstream the actuation, up to the leading edge. Indeed, SS actuation yields an overall modification of the pressure distribution, characterized by a covered area that is smaller compared to the clean configuration. This causes an overall attenuation of blade loading. The opposite occurs with PS actuation. This latter induces a modification of the pressure distribution, such that the area covered by the curve is significantly larger relative to the clean configuration. As a consequence, an increase in blade loading is expected. It is worth remarking that these results are in qualitative agreement with the assessments of Refs. [16
], where the same effects were obtained with a mechanical L-tab deflected downward or upward, installed on a helicopter blade section. Additionally, the computed flow fields are qualitatively consistent with the Particle Image Velocimetry measurements of Feng et al. [21
] on a low-speed fixed wing section with PS plasma actuation opposite to the freestream direction.
shows an histogram of the aerodynamic loads, computed on the clean, PS- and SS-actuated configurations, at angles of attack ranging from 0 to 12 degrees. Namely, the lift coefficient
, the drag coefficient
and the mid-chord moment coefficient
are reported. As anticipated, actuation yields a modification of the effective camber, opposite for PS and SS plasma, alongside a chord-wise shift of the Kutta condition downstream of the trailing edge. Both PS and SS actuations provide lift and pitching moment modifications, which are also quantitatively comparable to those of Refs. [16
], achieved with a mechanical Gurney flap. In particular, PS actuation allows for increasing the lift coefficient without changing the blade angle of attack, and with no significant drawbacks in terms of drag. The moment coefficient—nose-down positive—increases in magnitude, relative to the non-actuated counterpart. That is, PS actuation has a stabilizing effect on the mean airfoil torsion (see [46
]). On the other hand, SS actuation yields effects that are comparable to those exhibited by upward deflections of control surfaces. Namely, lift is reduced compared to the clean configuration, with potential benefits in counteracting stall phenomena [19
]. A reduction of the pitching moment, relative to the clean configuration, is observed. Again, no major effects are encountered in terms of drag rise. The affection of the actuation on the pitching moment is regarded as particularly important for this work. Within this framework, it is worth remarking that the behavior of pitching moment according to PS/SS actuation is consistent with the results of several works in literature, dealing with mechanical trailing edge control surfaces, e.g., Refs. [19
]. Additionally, the light reductions of drag coefficient relative to the clean blade—encountered with both pressure and SS actuation—are in agreement with other results found in literature on trailing edge devices (see e.g., Refs. [16
]). It is also worth remarking that the present fully-turbulent RANS assessments can’t provide a quantitative evaluation of drag. The results presented here are meant to give only a qualitative overview on the affection of plasma on this quantity.
4. Results of Traveling Wave Mode Simulations
Traveling wave mode simulations are carried out on the clean and on the plasma equipped blade cascade. For these simulations, an alternate PS/SS actuation is imposed over the pitching oscillating cycle of the blade. The frequency of oscillation for the simulations presented hereinafter is f = 19.17 Hz, yielding a reduced frequency based on the semi-chord of
= 0.4597—being the blade chord and
= 19.65 m/s the freestream velocity. The mean angle of attack is set to two degrees and the oscillation amplitude is set to one degree. The motion law of the n
th blade of the cascade reads:
. Notice that these specific simulation parameters have been chosen because a significant rise in the vibratory loads—alongside a drop in the aeroelastic stability—was detected numerically at IBPA =
] and experimentally at IBPA ∼
], on the same cascade in these conditions.
The blade displacement is achieved via a mesh adaptation algorithm implemented within the flow solver. One hundred time steps per oscillation cycle are employed for the numerical computations, yielding a time step size of ∼5.2 s. A total of three periods are simulated, in order to ensure the convergence of the numerical solution. The time histories of the unsteady loads for the plasma equipped cascade are illustrated in the following part of this section. Several IBPAs between −180 deg. and 180 deg. are considered, including degrees.
In order to maximize the actuation effectiveness, the triggering of PS/SS actuation during the oscillation period has to be set properly. To this aim, the phase of unsteady airloads relative to the blade pitching cycle for the clean cascade is evaluated for different IBPAs, at k = 0.4597. This allows to tune the triggering of PS/SS actuation in order to: (i) maximize the airloads when a minimum of the oscillating loads is observed; and (ii) minimize the airloads when a maximum in the oscillating airloads is encountered. Indeed—due to a combination of vortex shedding phenomena and inertial effects caused by the flow acceleration—the unsteady airloads are out of phase relative to the airfoil harmonic motion. The lead/lag phasing of loads is determined by the reduced frequency and by the thickness of the airfoil. A detailed discussion of these phenomena for an isolated airfoil is provided in Ref. [50
]. Additional out of phasing effects may be caused by the mutual interaction of the blades. Figure 11
a,b show the phase of the mid-chord moment and lift coefficient, computed at IBPAs between −180 and 180 degrees. Notice that the same out of phasing has been computed for the different blades of the cascade, at each of the interblade phase angles. Consistently with Ref. [50
] and with Ref. [51
], chapter 8, the computed phase is different for the two force coefficients. Only the phase of the moment coefficient—which has been proved to yield the most detrimental vibratory loads on the targeted cascade [26
]—is used here to tune the actuation triggering. In fact, the main target of the conceived actuation is—at least for the present work—to manipulate the pitching moment. Indeed, the blade bending stiffness is much larger than the torsional counterpart, for the cascade under consideration [22
]. Moreover, the experimental results of Sachs [24
] showed that the torsional mode is the most critical for the cascade under consideration. Therefore, acting on the pitching moment, can have the further beneficial effect of increasing the aeroelastic stability of the cascade. Additionally, because the geometry of the reference cascade is span-wise uniform and meant to represent a blade section, the lift is not an accurate picture of the blade loading at the hub, which should be quantified with three-dimensional bladings. It is also worth noting that the choice of focusing on the pitching moment only and not on the lift is consistent with other works (see e.g., [2
]). These works deal indeed with the unsteady aerodynamics and with the aeroelastic response of linear two-dimensional cascades.
The results at constant angle of attack presented in Section 3
show that a larger body force is required on the suction side, in order to obtain percentage gains similar to the PS counterpart. As a consequence, a body force equal to 75% of the reference 300 mN/m, i.e., 225 mN/m, is applied on the pressure side. On the other hand, a force equal to 150% of 300 mN/m, i.e., 450 mN/m, is applied on the suction side—this value being feasible for modern plasma actuators [38
]. Therefore, the body force on the suction side is double relative to that applied on the pressure side. The body force is modeled as constant during the actuation-on cycle, hence its time history would be a squared wave. In order to avoid undesired peaks in the pressure distribution, an exponential law modulation is applied to the body force amplitude at the very beginning and end of the operating cycle. This allows a smooth transition between the non-actuated and actuated configuration and vice versa. The triggering of the PS/SS actuation is defined on the basis of Figure 11
a. The previous section showed that PS actuation increases the positive nose-down pitching moment. Therefore, PS plasma should be operated when the pitching moment features negative oscillations, relative to the mean value. On the contrary SS actuation, which provides a reduction of the positive nose-up pitching moment, has to be operated when the Cm
oscillations have positive amplitude. Because Figure 11
a shows that the moment is out of phase relative to the motion, this lead/lag has to be accounted for, when setting the PS/SS actuation triggering. For an oscillation law in the form
, the phase of the PS actuation on the n
th blade is
. Consistently, the phase of the SS actuation relative to the blade motion is
shows the time history of lift, drag and moment coefficient on the central blade of the plasma-equipped cascade at IBPA = −51.43 degrees. In this case, four oscillation periods are simulated. Figure 12
a depicts the four simulated periods, whereas Figure 12
b is a blowing up on the second, third and fourth simulated periods, where the reaching of a well-established period regime can be better detected. Specifically, the transient results are deemed quasi-converged when the force coefficient relative difference between two consecutive periods falls below 0.02%. This condition must be achieved for all the force coefficients considered, i.e., lift, drag and moment. It is found that the difference in lift, drag and moment coefficient falls below the fixed threshold already between the second and the third oscillation period. Notice that also Keerthi et al. [2
], who performed numerical computations on a cascade with pitching blades, found that their numerical solution reaches a good periodicity after two periods of oscillation. Namely, three oscillating cycles are simulated for small amplitude oscillations and four oscillating cycles for high amplitude oscillations are simulated in [2
shows the velocity magnitude field, normalized by the freestream velocity, at four different instants of the pitching cycle. A detail of central blade trailing edge is depicted. Analogous results are obtained on the other two plasma-equipped blades of the cascade. The alternate switching of the PS and SS actuation is clearly visible. It is also worth noting that, differently from the assessments at a constant angle of attack, the recirculating flow areas on the pressure and on the suction side are almost equally extended. This is due to the application of a larger body force on the suction side, relative to the pressure side. It will be highlighted hereinafter that now the affection on the positive and negative oscillations in the pitching moment is almost symmetrical. This means that the effects provided by PS and SS actuation are expected to be more similar, when the SS body force is larger than the PS counterpart.
, Figure 15
and Figure 16
show the fluctuation of the mid-chord pitching moment coefficient versus the non-dimensional time, for the last simulated period. Several IBPAs, ranging from −180 to 180 deg., are shown. The time on the x
-axis is made dimensionless by the oscillation period of the blade. An effective reduction of the unsteady peaks on the pitching moment is observed for each of the interblade phase angles. As a result, two major benefits can be achieved: (i) vibratory loads, responsible for fatigue phenomena are alleviated; and (ii) the blade aeroelastic stability is potentially increased, as the energy transfer from the flow to the blade is reduced [23
]. It is worth remarking that plasma seems to have a greater affection on the pitching moment for negative IBPAs than for positive values of the interblade phase angle. According to the convention employed in this work, for IBPA < 0, the pressure wave travels from the pressure side of a blade to the suction side of the neighbouring one. On the contrary, for IBPA > 0, the pressure wave travels from the suction side of a blade to the pressure side of the neighbouring one. A higher body force is employed on the SS for travelling wave mode simulations. Even though the recirculating flow areas generated on PS and SS with asymmetric actuation have comparable extensions, the affection on unsteady loads might be slightly higher on the suction side. This may be one of the causes for the better affection of the actuation for negative IBPAs which—according to [24
]—feature lower aerodynamic damping, and therefore are the most critical. The differences in actuation effectiveness encountered among the negative IBPAs are probably due to local effects that are not detectable when integrating pressure to obtain the resulting loads. In Ref. [2
], Keehrti et al. investigate the effects of the IBPA on the pressure distribution on a cascade of blades oscillating in pitch. The reduced frequency of oscillation is comparable to the ones of this work, therefore the findings of [2
] can be considered valid also for this manuscript. Keehrti et al. [2
] show that, whereas the phase of the pressure distribution along the chord on the suction side is close to zero for all the IBPAs, a large excursion of this quantity is observed on the pressure side. Specifically, they point out that the “average phase lead on the pressure surface varies [with the IBPA] in steps of 45 deg. for every 45 deg. change in [IBPA]” [2
]. That is, the pressure peaks on the pressure side change remarkably their chord-wise location, according to the IBPA. This shifts change the interaction between blade-induced and plasma-induced pressure distribution. As a result, the plasma affection on the phase of the pitching moment varies with the IBPA. Overall, in order to further improve the effectiveness of the actuation for the diverse IBPAs, the amplitude and the phase of the pressure distribution should be accounted for together with the integral loads. This aspect will be taken under consideration and investigated in detail in future works, the present one being an exploratory study on the capabilities of plasma in controlling vibratory loads.
The unsteady lift curves are shown in Figure 17
for IBPA = [−180, −135, 135, 180] degrees. Overall, the lift coefficient is found to be less affected by actuation, compared to the moment, though the reduction in amplitude provided by the actuation is clearly visible. This discrepancy with respect to the moment coefficient is somehow expected because the phase of the lift coefficient for a pitching airfoil is different from the moment coefficient counterpart, consistently with Figure 11
and with Refs. [50
]. Moreover, it is worth recalling that vibratory loads associated to the bending mode are expected to be less influential for fatigue phenomena—as well as for the aerostructural stability—of the targeted cascade compared to vibratory loads in pitch (see again Refs. [22
]). The computed drag coefficient is not reported for brevity purposes. Indeed, the main concern of this work is to show the effect of plasma actuators on unsteady moment coefficient. Computations at constant angle of attack have shown that no significant drag rises have to be expected when employing plasma actuators. Additionally, as already mentioned, an accurate evaluation of plasma effects on the drag coefficient would require anyway more detailed and much more computationally demanding numerical simulations, which lie beyond the purposes of the present work. The assumption of fully turbulent flow, together with the numerical filtering associated with the Reynolds averaging operation, allows for getting only the qualitative behaviour of the drag.