#
Numerical Assessment of Virtual Control Surfaces for Load Alleviation on Compressor Blades^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Modeling

#### 2.1. Computational Geometry and Flow Solver

^{®}(Version 17, Ansys, Canonsburg, PA, USA).The domain extends approximately two chords both upstream the leading edge and downstream the trailing edge. In order to reproduce correctly the main features of the boundary layer, a targeted O-grid is realized around the airfoil. The O-grid features 100 elements in the direction normal to the airfoil and 410 elements on the airfoil surface. Refinements are introduced in the area where the plasma is modeled, as well as on the leading edge and on the trailing edge. The height of first cell in the boundary layer is set to keep the y${}^{+}$ equal to one, for the Reynolds number of ∼10${}^{5}$, based on the airfoil chord and on the freestream velocity. One block of the grid is realized specifically for the wake region. In this block, the grid is kept finer, compared to the other far field areas. This allows for capturing accurately the modifications in the airfoil circulation—and in turn on lift and moment—induced by the blade motion. With the aim to maximize the quality of the numerical solution, the elements of the boundary layer and of the wake are all kept approximately squared.

^{®}is employed for the viscous computations on the present cascade. The advection terms are solved with a high resolution scheme, whereas a second order backward scheme is employed for the transient terms. The flow is assumed as fully turbulent, and the system of Reynolds-Averaged Navier Stokes (RANS) equations is closed with the SST k-$\omega $ Menter turbulence model [27]. Wall functions are employed close to the solid walls. In [28], the performance of different turbulence models in describing fully turbulent flows around oscillating airfoils in pre-stall and in deep-stall conditions was investigated. The contents shown in [28] shows that the k-$\omega $ SST model [27], the Spalart–Allmaras [29] model and the Baldwin–Barth [30] model are those that better capture the unsteady flow physics associated with airfoils in oscillating motion. The k-$\omega $ SST model [27] is employed here because it is appropriate to describe the flow around oscillating airfoils and because it allows to get a good accuracy both close to the walls and in external flow. This twofold capability is especially desired for the present application. Indeed, a good capturing of the plasma-induced effects close to the walls, as well as of the interaction between the neighbouring blades, is needed here. The SST k-$\omega $ Menter model combines the advantages of the k-$\u03f5$ and k-$\omega $. By means of a blending function—unitary close to the wall and zero elsewhere—it is possible to switch between the k-$\omega $ and the k-$\u03f5$ models. Namely, the k-$\omega $ model, performing well close to the walls (chapter 11, [31]) is used near the blades. Conversely, the k-$\u03f5$, which is more accurate for describing the external flow (chapter 11, [31]), is employed for the rest of the fluid domain. Notice that the k-$\omega $ SST Menter model is used in other numerical works dealing with turbomachinery cascades. Akcayoz et al. [14] used the same solver and the same modeling to perform RANS computations on a compressor cascade equipped with plasma actuators, employed to control corner stall separations. Keerthi et al. [2] used Ansys CFX

^{®}to carry out fully turbulent RANS computations closed with the k-$\omega $ SST Menter model on a linear cascade featuring blades oscillating in pitch.

^{®}. With this model, the displacements applied on domain boundaries or in subdomains are diffused to other mesh points by solving the diffusion equation $\nabla \times ({\Gamma}_{\mathrm{disp}}\nabla \delta =0)$, $\delta $ being the displacement relative to previous mesh locations and ${\Gamma}_{\mathrm{disp}}$ the mesh stiffness. This quantity determines the entity of the motion in different regions of the domain. In this work, the stiffness is defined as the reciprocal of the cell volumes, so that the mesh deforms less where cells are smaller, e.g., close to the boundary layer. The diffusion equation is solved at the start of every time step of the simulation. Notice that the displacement diffusion model preserves the element distribution of the initial mesh. That is, if the initial mesh is relatively fine in certain regions of the domain, e.g., the boundary layer, then it will remain relatively fine also after solving the displacement diffusion equation. Further details on the algorithm can be found in [32,33].

#### 2.2. Plasma Modeling

^{−4}m see Ref. [13]. Therefore, actuators could be installed also on quite thin trailing edges, by realizing appropriate hollows on the blade surface, both on the pressure and on the suction side. Optimization studies need to be performed to assess the optimal geometrical orientation of actuators. The first experimental studies envisaged in the near future will feature plasma actuators parallel to the chord and covering the whole span of the targeted linear cascade (see [22]). Successive studies are thought to be carried out on a rotating annulus, in order to investigate whether the transverse flow arising from the rotation requires orienting the actuators with a certain angle relative to the chord direction. There are different possible approaches to fit the cables into the blades. The selected configuration depends on the geometrical arrangement of the actuators along the blade span. In works to come, the optimal geometrical configuration for the actuators to control the blade loads will be assessed. If the actuators will cover the entire span, the cables will be connected at the level of the blade root, where there will be less problems in terms of space. Dedicated hollows will be realized to locate the cables. When employing plasma actuators on the compressor rotor, the transmission of the signal from the fixed to the rotating frame will be realized by means of devices like those produced by Jordil Technic Sárl, which have already been used in [44] for a piezo-actuated aero engine blisk. An alternative, which would avoid the transmission of electrical signals from a fixed to a rotating system, was proposed by Iwrey in [45]. Iwrey suggested exploiting the rotation to generate the required voltage, by using electromagnetic effects. The concept of [45] is targeted to plasma actuators employed to control the tip clearance flow on gas turbine engines. Therefore, it is suitable also for the application of the present work. Iwrey proposed locating one or more magnets on the casing of the engine and a magnet with a solenoid on each of the plasma actuated blades. The solenoid is connected to the electrodes of the plasma actuator. Due to the relative rotation of the blade and of the casing magnets, a voltage is generated on the solenoid. This voltage can indeed be used to feed the actuators, if an appropriate signal modulation is applied. A further path, which is currently pursued by the authors in cooperation with the University of Salento, consists of developing micro controllers and micro generators, small enough to be located in targeted hollows, realized internally to the blades. In order to avoid a possible interference of the electrodes on the pressure side with those on the suction side, proper electromagnetic shielding will be employed between the two actuators. This shield can be realised by means of e.g., metal screens, metal foams or metal sheets.

#### 2.3. Grid Sensitivity Study and Comparison with Experimental Data

_{P}is displayed. A steady state configuration is considered, with angle of attack of 2 degrees, and freestream velocity of 34.36 m/s, yielding a chord/freestream velocity based Reynolds number of ∼350,000. A very good agreement is found between the numerical computations and the experiments, including the leading edge and the trailing edge area of the blade.

## 3. Results at Constant Angle of Attack

## 4. Results of Traveling Wave Mode Simulations

_{m}oscillations have positive amplitude. Because Figure 11a 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 $\alpha =2+sin(2\pi \mathrm{f}\phantom{\rule{0.166667em}{0ex}}t+\mathrm{n}\times \mathrm{IBPA}\frac{\pi}{180})$, the phase of the PS actuation on the nth blade is ${\varphi}_{\mathrm{PS}}={\varphi}_{{C}_{m}}+\pi $. Consistently, the phase of the SS actuation relative to the blade motion is ${\varphi}_{\mathrm{SS}}={\varphi}_{{C}_{m}}$.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

c | blade chord, [m] |

CFD | computational fluid dynamic(s) |

C_{d} | drag coefficient, [-] |

C_{P} | pressure coefficient, [-] |

C_{l} | lift coefficient, [-] |

C_{m} | moment coefficient, at the mid-chord if not differently specified, [-] |

C_{mc/2} | mid-chord moment coefficient, [-] |

f | blade oscillation frequency, [Hz] |

k = $2\pi \mathrm{f}\phantom{\rule{0.166667em}{0ex}}\mathrm{c}/2{\mathrm{U}}_{\infty}$ | reduced frequency, [Hz] |

IBPA | interblade phase angle, [deg.] |

PS | pressure side |

Re = U_{∞}c/$\nu $ | Reynolds number, [-] |

SS | suction side |

t | time, [s] |

T | blade oscillation period, [s] |

U | flow velocity, [m/s] |

U_{∞} | freestream velocity, [m/s] |

x | horizontal spacial coordinate, [m] |

y | vertical spacial coordinate, [m] |

$\alpha $ | blade angle of attack, [deg.] |

$\varphi $ | phase angle, [deg.] |

$\nu $ | air kinematic viscosity at 20 ${}^{\circ}$C, [m^{2}/s] |

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**Figure 2.**Detail of the computational grid around one blade (

**top**), and blowing up on the trailing edge area (

**bottom**).

**Figure 3.**Schematic of a plasma actuator with indicative sizes according to [13].

**Figure 5.**Distribution of pressure coefficient C

_{P}. Experiments and CFD on the clean configuration; Re ∼350,000, $\alpha =2$ degrees.

**Figure 6.**Real and imaginary part of unsteady pressure distribution. Experiments and CFD on the clean configuration; Re ∼350,000; IBPA = 0 deg.; $\alpha =2+sin(2\pi \mathrm{f}\phantom{\rule{0.166667em}{0ex}}t+4\times \mathrm{IBPA}\frac{\pi}{180})$ degrees; $\mathrm{f}=5.25$ Hz; $\mathrm{t}$: time.

**Figure 7.**Time history of unsteady lift ${C}_{\mathrm{l}}$, drag ${C}_{\mathrm{d}}$ and mid-chord moment coefficient ${C}_{{\mathrm{m}}_{\mathrm{c}/2}}$ at constant angle of attack of two degrees; Re ∼350,000.

**Figure 8.**Trailing edge detail of velocity magnitude, normalized by the freestream velocity, and of z-vorticity magnitude, normalized by the chord/freestream velocity ratio, for the clean cascade and with PS/SS actuation; Re ∼350,000; $\alpha =2$ degrees.

**Figure 9.**Pressure coefficient distribution on the clean blade and with PS/SS actuation; Re ∼350,000; $\alpha =2$ degrees.

**Figure 10.**Lift, drag and mid-chord moment coefficient on the clean blade and with PS/SS actuation; Re ∼350,000; $\alpha \in \left[0\phantom{\rule{0.222222em}{0ex}}12\right]$ degrees.

**Figure 11.**Phase of mid-chord moment coefficient and of lift coefficient at different IBPAs; Re ∼195,000; $\alpha =2+sin(2\pi \mathrm{f}\phantom{\rule{0.166667em}{0ex}}t+4\times \mathrm{IBPA}\frac{\pi}{180})$ degrees; $\mathrm{f}=19.17$ Hz, $\mathrm{T}=1/\mathrm{f}$.

**Figure 12.**Time history of lift, drag and moment coefficient for the complete simulation (

**a**) and for the last three periods (

**b**); Re ∼195,000; IBPA = −51.43 deg.; $\alpha =2+sin2\pi \mathrm{f}\phantom{\rule{0.166667em}{0ex}}t+4\times \mathrm{IBPA}\pi /180$ deg.; $\mathrm{f}=19.17$ Hz, $\mathrm{T}=1/\mathrm{f}$.

**Figure 13.**Trailing edge detail of velocity magnitude, normalized by the freestream velocity, at four time instants of the oscillation cycle; plasma actuation on; Re ∼195,000; IBPA = −51.43 deg.; $\alpha =2+sin2\pi \mathrm{f}\phantom{\rule{0.166667em}{0ex}}t+4\times \mathrm{IBPA}\pi /180$ deg.; $\mathrm{f}=19.17$ Hz, $\mathrm{T}=1/\mathrm{f}$.

**Figure 14.**Time history of the mid-chord pitching moment coefficient ${\mathrm{C}}_{{\mathrm{m}}_{\mathrm{c}/2}}$ on the central blade for the last simulated period; Re ∼195,000; $\alpha =2+sin(2\pi \mathrm{f}\phantom{\rule{0.166667em}{0ex}}t+4\times \mathrm{IBPA}\frac{\pi}{180})$ degrees; $\mathrm{f}=19.17$ Hz, $\mathrm{T}=1/\mathrm{f}$; $\mathrm{IBPA}=[\pm 45,0]\phantom{\rule{0.222222em}{0ex}}\mathrm{degrees}$.

**Figure 15.**Time history of the mid-chord pitching moment coefficient ${\mathrm{C}}_{{\mathrm{m}}_{\mathrm{c}/2}}$ on the central blade for the last simulated period; Re ∼195,000; $\alpha =2+sin(2\pi \mathrm{f}\phantom{\rule{0.166667em}{0ex}}t+4\times \mathrm{IBPA}\frac{\pi}{180})$ degrees; $\mathrm{f}=19.17$ Hz, $\mathrm{T}=1/\mathrm{f}$; $\mathrm{IBPA}=[\pm 90,\pm 51.43]\phantom{\rule{0.222222em}{0ex}}\mathrm{degrees}$.

**Figure 16.**Time history of the mid-chord pitching moment coefficient ${\mathrm{C}}_{{\mathrm{m}}_{\mathrm{c}/2}}$ on the central blade for the last simulated period; Re ∼195,000; $\alpha =2+sin(2\pi \mathrm{f}\phantom{\rule{0.166667em}{0ex}}t+4\times \mathrm{IBPA}\frac{\pi}{180})$ Hz, $\mathrm{T}=1/\mathrm{f}$; $\mathrm{IBPA}=[\pm 180,\pm 135]\phantom{\rule{0.222222em}{0ex}}\mathrm{degrees}$.

**Figure 17.**Time history of the lift coefficient ${\mathrm{C}}_{\mathrm{l}}$ on the central blade for the last simulated period; Re ∼195,000; $\alpha =2+sin(2\pi \mathrm{f}\phantom{\rule{0.166667em}{0ex}}t+4\times \mathrm{IBPA}\frac{\pi}{180})$ Hz, $\mathrm{T}=1/\mathrm{f}$; $\mathrm{IBPA}=[\pm 180,\pm 135]\phantom{\rule{0.222222em}{0ex}}\mathrm{degrees}$.

**Table 1.**Percent difference between the aerodynamic loads computed on the reference grid and the counterparts achieved on the finer and on the coarser mesh; clean cascade, with PS the actuated cascade and SS the actuated cascade; Re ∼350,000; $\alpha =2$ degrees.

Grids–Airloads | $\mathbf{\Delta}$C_{l} % | $\mathbf{\Delta}$C_{d} % | $\mathbf{\Delta}{\mathbf{C}}_{{\mathbf{m}}_{\mathbf{c}/2}}$ % | |
---|---|---|---|---|

CLEAN BLADE | reference/finer | 0.1 | 0.4 | 0.1 |

reference/coarser | 0.2 | 0.6 | 0.2 | |

BLADE WITH PS PLASMA | reference/finer | 0.7 | 1.2 | 0.2 |

reference/coarser | 9.4 | 41.2 | 3.8 | |

BLADE WITH SS PLASMA | reference/finer | 0.4 | 0.2 | 0.3 |

reference/coarser | 7.5 | 4.3 | 6.4 |

**Table 2.**Percent difference between the aerodynamic loads computed on the reference grid and the counterparts achieved on the finer and on the coarser mesh; clean cascade, with PS the actuated cascade and SS the actuated cascade; Re ∼350,000; $\alpha =8$ degrees.

Grids–Airloads | $\mathbf{\Delta}$C_{l} % | $\mathbf{\Delta}$C_{d} % | $\mathbf{\Delta}{\mathbf{C}}_{{\mathbf{m}}_{\mathbf{c}/2}}$ % | |
---|---|---|---|---|

CLEAN BLADE | reference/finer | 0.049 | 0.13 | 0.45 |

reference/coarser | 0.059 | 0.16 | 0.074 | |

BLADE WITH PS PLASMA | reference/finer | 0.7 | 3.1 | 0.4 |

reference/coarser | 2.6 | 12.5 | 6.1 | |

BLADE WITH SS PLASMA | reference/finer | 0.2 | 1.4 | 0.3 |

reference/coarser | 3.2 | 21.0 | 3.6 |

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**MDPI and ACS Style**

Motta, V.; Malzacher, L.; Peitsch, D. Numerical Assessment of Virtual Control Surfaces for Load Alleviation on Compressor Blades. *Appl. Sci.* **2018**, *8*, 125.
https://doi.org/10.3390/app8010125

**AMA Style**

Motta V, Malzacher L, Peitsch D. Numerical Assessment of Virtual Control Surfaces for Load Alleviation on Compressor Blades. *Applied Sciences*. 2018; 8(1):125.
https://doi.org/10.3390/app8010125

**Chicago/Turabian Style**

Motta, Valentina, Leonie Malzacher, and Dieter Peitsch. 2018. "Numerical Assessment of Virtual Control Surfaces for Load Alleviation on Compressor Blades" *Applied Sciences* 8, no. 1: 125.
https://doi.org/10.3390/app8010125