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This study demonstrates an active flow control for deflecting a direction of wake vortex structures behind a NACA0012 airfoil using an active morphing flap. Two-dimensional direct numerical simulations are performed for flows at the chord Reynolds number of 10,000, and the vortex pattern in the controlled and noncontrolled wakes as well as the effect of an actuation frequency on the control ability are rigorously investigated. It is found that there is an optimum actuation-frequency regime at around
*F ^{+}* = 2.00 which is normalized by the chord length and freestream velocity. The wake vortex pattern of the well-controlled case is classified as the 2P wake pattern according to the Williamson’s categorization [
1] [
2], where the forced oscillation frequency corresponds to the natural vortex shedding frequency without control. The present classification of wake vortex patterns and finding of the optimum frequency regime in the wake deflection control can lead to a more robust design suitable for vortex-induced-vibration (VIV) related engineering systems.

When a body is placed in the vortex-dominated flows such as wakes of cylinders or airfoils at a nonzero angle of attack (AoA), the body experiences a fluctuating lift force and vibrates due to the unsteady vortex motion interacting with the body itself, which is known as a vortex-induced vibration (VIV) and regarded as an important problem ranging from a design of civil engineering structures to a flutter constraint of aircraft wing designs [

In the present study, we focus on the active flow control for deflecting a vortex shedding direction in the wake such that the wake-body interaction does not occur. There were many studies for classifying the wake vortex pattern behind oscillatory cylinders or pitching airfoils in the forced motion. Koochesfahani has experimentally investigated the wake vortex structures behind a pitching airfoil of the chord Reynolds number 12,000 by changing the amplitude, frequency, and shape of the oscillation waveform [

As such, the forced oscillation can drastically change the wake vortex structures, and thus it is expected that the vortex shedding direction can be appropriately deflected by controlling the vortex-vortex interaction in the wake, which leads to more robust engineering design for avoiding VIV phenomena. This study numerically investigates the wake deflection control for a two-dimensional NACA0012 airfoil at AoA = 5 deg. in the chord Reynolds number 10,000. In specific, the aft-portion of the airfoil is deformed in the direction perpendicular to the flow in a periodic motion, which we call the active morphing flap hereinafter. The primary objective of this study is to clarify whether the active morphing flap can effectively control and deflect the vortex shedding direction in the wake. Furthermore, the effect of actuation frequency on the control ability and associated wake-vortex structures is discussed. To the best of our knowledge, the classification of vortex patterns in such wake deflection control had never been focused with respect to the Williamson’s map [^{6} - 10^{7})), the main focus of the present study is to investigate the modification of two-dimensional vortex structures which could be involved as large-scale flow motions in practical engineering problems. The rest of this paper is organized as follows: Section 2 specifies the problem settings; Section 3 describes the numerical methodology including a grid and validation studies; Section 4 discusses the results, and Section 5 concludes the paper.

In this study, the flow over a NACA0012 airfoil is considered for the wake deflection control, where the chord Reynolds number and AoA are set to be 10,000 and 5 deg., respectively. The aft-portion of the airfoil is assumed to be smoothly deformed as an active morphing flap with a length of l and amplitude of A (A is defined at the trailing edge). ^{ }^{+} = ωh/(2πU). F^{ }^{+} is so-called the actuation frequency which is widely used in the relevant studies [_{s}(t), z_{s}(t)) represent the coordinates of a grid point on the airfoil surface at time t. The deformation of the airfoil surface is defined as follows.

x s ( t ) = x s ( 0 ) , z s ( t ) = w 1 z s ( 0 ) + { z s ( 0 ) + w 2 A sin ( ω t ) } ( 1 − w 1 ) , w 1 = 1 2 { 1 + k w tanh ( s / R − 1 / 2 ) } , w 2 = 1 − cos ( π 2 x s ( 0 ) − x 1 x t e − x 1 ) , s = ( x s ( 0 ) − x t e ) 2 + ( z s ( 0 ) − z t e ) 2

The parameters k_{w} and R control a deformation gradient and set to be k_{w} = 5.0 and R = 0.9h. (x_{te}, z_{te}) represents the coordinate of the original trailing edge and is defined as (x_{te}, z_{te}) = (h, 0). x_{1} indicates the location where the deformation starts, which is set to be x_{1} = h − l in this study. Equations above indicate that the airfoil surface deforms only vertically in time, and the weight functions of w_{1} and w_{2} provide a smooth deformation of the airfoil surface with keeping the amplitude of A in the trailing-edge movement. Note that the equations for the surface deformation above can be also applied to determine the deformation of the entire grid. The present study mainly focuses on the basic capability of the wake deflection control under forced oscillation and associated wake vortex modification. Therefore, the forced motion affecting the wake vortex structure is hoped to be as simple as possible, and thus we do not adopt a conventional pitching airfoil that may strongly oscillate leading-edge separated shear layers or a rigidly-oscillating trailing-edge flap that may cause a sharp perturbation in the flow passing over the kinked airfoil surface in the vicinity of the flap.

To clarify the effect of F^{ }^{+} on the wake deflection control, an induced flow momentum is kept constant throughout this study. C_{μ} is so called the momentum coefficient [

C μ = ρ s a c t 2 l a c t ρ U 2 h ,

where s_{act} is a root mean square (RMS) value of the trailing edge velocity; l a c t is the representative length scale of the morphing flap; ρ is the freestream density. s_{act} and l a c t are defined as follows:

s a c t = A ω / 2 , l a c t = D a c t , D a c t = A l / 2 .

In this study, the momentum coefficient is fixed as 0.01, which indicates the momentum induced by the flap can be regarded as 1% of that of the freestream and is relatively small compared to the existing study [

Name | Amplitude [A/h] | Frequency [F^{ +}] |
---|---|---|

case 0 | -Not controlled- | |

case 1 | 0.01 | 8.00 |

case 2 | 0.02 | 4.00 |

case 3 | 0.03 | 2.00 |

case 4 | 0.04 | 1.59 |

case 5 | 0.06 | 1.00 |

case 6 | 0.10 | 0.50 |

The wake deflection control is analyzed by a direct numerical simulation (DNS) of the two-dimensional (2D) compressible Navier-Stokes equations in this study. The governing equations are expressed in the body-fitted coordinate system, and the spatial derivatives of the convective and viscous terms, metrics, and Jacobian are evaluated by a sixth-order compact scheme with a tenth-order spatial filtering with a filtering coefficient 0.495 [^{1/2} (or 0.12% of the chord length), which is of DNS resolution. The number of grid points is 362,103 (=1203 × 301) for the airfoil grid and 324,742 (=638 × 509) for the refined wake grid. The simulation starts from the uniform flow. When the controlled simulation is performed, the morphing flap starts oscillation after 1 flow pass over the chord length, and the simulation continues for 10 flow passes to reach quasi-steady state. Then, a time average is taken for 10 flow passes.

Next, the validation study is performed. ^{ }^{+} = 2.00) results in approximately the same mean flow as the 2D simulation. _{p}) at AoA = 5 deg. Four different grids with an ultra-coarse to fine density (9,605 to 686,485 points) were examined, which shows an excellent convergence. The 3D simulation with the extruded fine grid also corresponds to the 2D results. The fine 2D grid is hereafter used in this study.

First, the noncontrolled case (case 0) is discussed. In _{p} (pressure coefficient) at (x/h, z/h) = (2.0, 0.0) with respect to a non-dimensional frequency (Strouhal number: St) based on the chord length and freestream velocity. There are clear peaks at St = 1.0 and its harmonic and higher modes. In the present Reynolds number, a single vortex generated from the separated shear layer in the suction surface (at around x/h = 0.75) directly becomes a clockwise vortex in the P mode wake. Therefore, it is expected that the shear layer instability on the suction surface strongly affects the formation of the P mode wake in addition to the wake instability. For this reason, the active morphing flap that directly oscillates the separated shear layer in the aft-portion of the airfoil is expected to have a strong effect on the wake modification.

Next, the controlled cases are discussed. ^{ }^{+} = 8.00 in ^{ }^{+} = 4.00 and 2.00 in ^{ }^{+} = 4.00 and 2.00. Interestingly, the peak value of the RMS of w’ becomes slightly stronger in the F^{ }^{+} = 4.00 case than F^{ }^{+} = 2.00, and thus the F^{ }^{+} = 2.00 case should be preferable in terms of decreasing the VIV if a body is located in the wake. It is noteworthy that vortex patterns seem to be different between these two cases although the peak locations of velocity fluctuation are similarly modified. In the low frequency cases (F^{ }^{+} = 1.59, 1.00, 0.50), the wake deflection is not achieved. In specific, the w’-RMS of F^{ }^{+} = 1.00 and 0.50 broadly distribute in the vertical direction, which corresponds to larger vertical motions of the wake vortices. As such, the capability of the wake deflection control strongly depends on the actuation frequency. The frequency needs to be around F^{ }^{+} = 2.00 for the effective control, which corresponds to the shedding frequency of each single vortex in the P-mode wake without control.

The previous paragraph discussed the effect of actuation frequency on the capability of the wake deflection control. Next, we focus on the vortex pattern in the noncontrolled and controlled cases of F^{ }^{+} = 8.00, 2.00, and 1.59. ^{ }^{+} = 8.00) does not show a clear synchronized pattern but small vortices coalesce into a large-scale structure, which is called “C” in [^{ }^{+} = 2.00, the wake vortex is split into two directions. One of them comprises a series of strong vortex pairs and is horizontally shedding downstream; the other line consists of a pair of small clockwise and weak counter-clockwise vortices. This is approximately categorized as “2P” (or “P + S”), which typically appears in a fundamental lock-in flow field [^{ }^{+} = 1.59) shows the wake structure consisting of single vortex and a pair of counter-rotating vortices (P + S). Such a low frequency regime is not controllable although the clear synchronized P + S mode is observed. To summarize, in the present flow condition, the wake vortex pattern is originally the P mode, and then it is transient to P + S, 2P, and C modes as F^{ }^{+} increases.

Finally, the PSD of C_{p} is discussed in ^{ }^{+} = 8.00, the PSD of controlled case (red line) becomes broadly higher than that of the noncontrolled case (black line). The higher frequency modes (St > 10) are enhanced broadly associated with a few peaks at harmonic modes of the input frequency such as St = 8, 16, and 32. This indicates that the control using a high-frequency input results in enhancement of nonharmonic modes as well as harmonics of the input frequency, and thus no clear synchronized structure appears. On the other hand, the effectively controlled case (F^{ }^{+} = 2.00, ^{ }^{+} = 1.59 case shows peaks of PSD at the input frequency and its harmonics; however the peak at St = 1.59 does not originally exist in the noncontrolled case so that the strong lock-in phenomena unlikely occurs. As such, the well-controlled case (F^{ }^{+} = 2.00) introduces vortex perturbation close to the natural shedding frequency, which results in the clearly synchronized wake-vortex structures, and the wake deflection control is achieved.

This paper has demonstrated the wake deflection control around a NACA0012 airfoil at the chord Reynolds number of 10,000 with an active morphing flap. A series of two-dimensional DNS was performed, and the vortex pattern in the controlled wake as well as the effect of the actuation frequency on the control ability has been rigorously investigated. It is found that there is an optimum actuation frequency regime around nondimensional frequency of F^{ +} = 2.00, which produces the 2P wake pattern based on Williamson’s categorization [

The present study was partly supported by JSPS KAKENHI (Grant Number K19K152050). This work was also in part supported by Council for Science, Technology and Innovation (CSTI), Cross-ministerial Strategic Innovation Promotion Program (SIP), “Materials Integration” for revolutionary design system of structural materials (Funding agency: JST). Numerical simulations of this study were performed on the Supercomputer system “AFI-NITY” at the Advanced Fluid Information Research Center, Institute of Fluid Science, Tohoku University.

The authors declare no conflicts of interest regarding the publication of this paper.

Abe, Y., Konishi, T. and Okabe, T. (2020) Numerical Simulation of Wake Deflection Control around NACA0012 Airfoil Using Active Morphing Flaps. Journal of Flow Control, Measurement & Visualization, 8, 121-133. https://doi.org/10.4236/jfcmv.2020.83007