Partial Oscillation Flow Control on Airfoil at Low Reynolds Numbers

Abstract

Among the critical factors contributing to the decline in the aerodynamic performance of near-space aircraft under low Reynolds number conditions, a significant one lies in the occurrence of laminar separation bubbles forming on the wings. Within the scope of this investigation, the primary research methodology adopted involves utilizing an unsteady numerical simulation technique rooted in a spring-smoothed dynamic grid system. This study meticulously examines the aerodynamic attributes and flow patterns exhibited by an airfoil undergoing partial oscillation, thereby elucidating the underlying mechanisms through which such oscillations lead to enhanced lift and diminished drag forces. The outcomes of this research reveal that the imposition of partial oscillation engenders a noteworthy augmentation of 4.9% in the lift coefficient of the airfoil, concurrent with a substantial diminution of 15.3% in its drag coefficient when juxtaposed against the non-deforming counterpart. The oscillation frequency exerts a profound influence on both the onset location of transition and the extent of the laminar separation bubble’s development. As the oscillation frequency escalates, it follows an initial ascending trend in the lift coefficient of the airfoil, followed by a subsequent decline, whereas the drag coefficient exhibits an initial decrement prior to a rising tendency, thus indicating the existence of an optimal frequency point where the airfoil achieves its most favorable aerodynamic characteristics. It is observed that the flow control effects are optimally pronounced when the region subjected to partial oscillation is proximate to the airfoil’s leading edge or situated precisely at the centroid of the laminar separation bubble.

1. Introduction

Near-space aircrafts (e.g., solar-powered aircrafts and stratospheric airships) have many advantages, such as high operational altitude and long endurance. This class of aircraft holds considerable potential for applications in intelligence gathering, surveillance, and reconnaissance activities, thereby constituting a pivotal developmental trajectory within the future realm of aerospace engineering [1,2]. Nevertheless, the laminar separation bubble (LSB) emerges as a ubiquitous flow feature in near-space aircraft operating at low Reynolds numbers (≤10⁶) [3,4,5], a phenomenon that severely compromises the lift-to-drag ratio and stands as a paramount challenge impeding further advancements in near-space aviation technology. Consequently, the efficacious manipulation of LSB phenomena occurring on airfoils under low Reynolds number conditions constitutes a pivotal factor in enhancing the overall aerodynamic efficiency of near-space aircraft, hence necessitating focused research efforts in this domain. Research findings indicate that the implementation of partial oscillation exerts a substantial influence on the aerodynamic attributes and flow patterns associated with the airfoil, thereby altering its performance characteristics profoundly. Accordingly, the implementation of partial oscillation has emerged as a burgeoning area of inquiry within active flow control strategies applied to airfoil design [6,7].

To mitigate the detrimental impact of LSB on airfoils under low Reynolds number conditions and thereby enhance their aerodynamic properties, extensive research endeavors have been dedicated to exploring various active flow control methodologies [8,9]. Typical active flow control methods prevent the formation of LSB by exciting the K-H (Kelvin Helmholtz) instability in the separated mixing boundary layer or the T-S (Tollmien Schlichting) wave in the laminar boundary layer upstream of the separation bubble [10,11]. For the partial oscillation flow control of airfoil, the present achievements of research focus on the following two aspects:

One aspect is the aerodynamic characteristics of flexible airfoil. Research indicates that the application of a flexible material coating on the wing’s surface engenders an aeroelastic coupling interaction with the fluctuating flow field, thereby leading to vibrations. These vibrations, under suitable circumstances, have the capacity to influence the fluid dynamics structure and ultimately deliver an enhanced lift and reduced drag performance [12]. Song [13] conducted a wind tunnel experimentation to investigate the aerodynamic properties of a biomimetic bat wing, revealing that the synergistic interaction between the airflow field and the structural design significantly contributes to the postponement of stall occurrence on the membrane wing. Gordnier [14] conducted an investigation into the interplay between the aerodynamic characteristics and structural behavior of membrane wings operating at low Reynolds numbers. The findings indicate that the temporal variation in membrane wing surface displacement exerts a substantial influence on the unsteady airflow around the wing, particularly at elevated angles of attack. This phenomenon not only retards the onset of stall but also enhances the wing’s lift coefficient under such high-angle conditions. From previous studies, it has been ascertained that employing a membrane surface under suitable conditions can lead to the enhanced aerodynamic performance of airfoils. However, overall, the method of flow control utilizing self-excited vibrations in flexible airfoils is fraught with numerous unpredictable and uncontrollable factors, rendering its efficacy challenging to forecast and, consequently, making it difficult to implement practically.

Another significant facet involves the application of active oscillatory flow control on the surface skin. The advancement in high-performance piezoelectric materials technology has paved the way for a novel approach where the active oscillation of the wing’s skin can be leveraged to enhance its aerodynamic performance. Pal [15] was among the first to implement a high-frequency oscillation methodology to regulate the boundary layer separation phenomenon along a cylindrical surface, successfully reducing the drag force acting upon the cylinder by a significant margin of 12.4%. Regarding the application of active flow control via oscillation on airfoils, Jones [16] ingeniously utilized piezoelectric composite materials to effectuate periodic deformations along the airfoil’s upper surface. The research demonstrates that the frequency of oscillation exerts a substantial influence on the airflow patterns and stall attributes of the airfoil. Edward [7] employed electroactive polymers to induce periodic deformation within specifically designated discrete areas on the wing surface. Through the use of Particle Image Velocimetry (PIV) in a wind tunnel setting, he investigated the effects of such periodic deformation on LSBs. His findings revealed that when the frequency of the induced deformation approximates that of the K-H instability, the tendency for laminar separation can be significantly attenuated. In comparison with the inherent self-vibrations exhibited by flexible airfoils, the active oscillation of the airfoil’s surface material demonstrates enhanced controllability and broader potential for practical application.

Currently, research predominantly focuses on the impact of localized oscillations at predetermined positions or full-surface oscillations on the aerodynamic attributes of wings. Most of these studies concentrate on the lift characteristics of wings under high angles of attack. Research into the effects of the placement of localized oscillations and the impact of oscillation patterns on the drag characteristics of a wing, particularly at its cruise angle of attack, remains insufficiently explored. On the basis of previous studies, a partial oscillation mechanical model with different chord positions and different oscillation frequencies is established in this study. The effects of oscillation position and frequency on the lift and drag characteristics and flow structure of the airfoil near the cruise angle of attack are studied, and the mechanism of the lift increasing and drag reducing of partial oscillation is revealed. The reasonable flow control method of partial oscillation is obtained, which can effectively improve the aerodynamic performance of the airfoil.

2. Research Methods

2.1. Numerical Simulation Method

In the present study, the Reynolds-averaged Navier–Stokes (RANS) equations have been rigorously solved, with the utilization of an unsteady numerical methodology to emulate the dynamics of the flow field. The inviscid terms have been discretized utilizing a second-order upwind scheme, while the viscous terms have been handled with a second-order central difference scheme. In the temporal advancement process, we have adopted the dual-time stepping method [17].

Given that the Reynolds number of the airfoil under scrutiny in this study falls within the range of 10⁵, categorizing it as a low Reynolds number regime, the transition from laminar to turbulent boundary layer must be meticulously accounted for in the CFD [18]. The turbulence modeling approach employed in the numerical simulations of this study is the γ-Re_θ transitional model, which has been developed by Langtry and Menter [19]. This particular model integrates the empirical relationships governing transition phenomena with the Shear Stress Transport (SST) k-ω turbulence model, thereby combining the strengths of both approaches. As a result, it has emerged as a widely adopted transitional modeling solution in contemporary practice. In comparison to the SST turbulence model, the γ-Re_θ transition model incorporates two supplementary transport equations: one for the local transition Reynolds number $\overline{R}{e}_{\theta t}$ and another for the intermittency factor γ. These equations extend the model’s capacity to capture transitional flow phenomena. $\overline{R}{e}_{\theta t}$ serves as a predictive criterion for determining the onset location of transition, whereas the intermittency factor γ is employed to simulate the flow dynamics within the transition zone. The governing transport equation for the intermittency factor γ is expressed as follows [19]:

$$\frac{\partial \left(\rho \gamma \right)}{\partial t}+\frac{\partial \left(\rho U_j\gamma \right)}{\partial x_j}=P_{\gamma 1}-E_{\gamma 1}+P_{\gamma 2}-E_{\gamma 2}+\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_{\mathrm{t}}}{{\sigma }_{\gamma }}\right)\frac{\partial \gamma }{\partial x_j}\right]$$

(1)

The equation of the local transition Reynolds number $\overline{R}{e}_{\theta t}$ is as follows:

$$\frac{\partial \left(\rho \overline{R}{e}_{\theta \mathrm{t}}\right)}{\partial t}+\frac{\partial \left(\rho u_j\overline{R}{e}_{\mathsf{\theta }\mathrm{t}}\right)}{\partial x_j}=P_{\theta \mathrm{t}}+\frac{\partial }{\partial x_j}\left[{\sigma }_{\theta \mathrm{t}}\left(\mu +{\mu }_{\mathrm{t}}\right)\frac{\partial \overline{R}{e}_{\theta \mathrm{t}}}{\partial x_j}\right]$$

(2)

In Equations (1) and (2), $P_{\gamma 1}$ and $E_{\gamma 1}$ are source terms of transition; $P_{\gamma 2}$ and $E_{\gamma 2}$ are source terms of the relaminarization.

Menter et al. [20] emphasized that to ensure the precise simulation of both laminar and transitional boundary layers, it is essential to maintain the y+ value closely proximate to 1 during computations under conditions of low Reynolds numbers. The H-type 2D structured grid is used in this study. The chord length of the airfoil is denoted by c. The computational domain extends its boundaries to be 20 times the chord length (c) upstream of the airfoil’s leading edge and 25 times the chord length downstream of the trailing edge. Moreover, the top and bottom boundaries of this domain are set at a uniform distance of 15 times the chord length from the airfoil’s chord line. The grid for the numerical simulation is shown in Figure 1.

To validate the accuracy of the aerodynamic properties of an airfoil at low Reynolds numbers using the grid configuration and unsteady CFD approach employed in this research, a numerical simulation has been performed on the NACA 0009 airfoil and its outcomes are juxtaposed against the results from wind tunnel experiments. The model of the wind tunnel experiment had a 12 in chord and 33.625 in span, and the lift, drag and angle of the attack measurements were corrected for solid blockage, wake blockage and streamline curvature effects [21].

In order to align with experimental conditions, during the computations of the validation case, the airfoil was stationary and a two-dimensional model was employed. The calculation Reynolds number is 4.0 × 10⁴, and the time step of unsteady calculation is 0.005 s.

The lift curves obtained from the calculation and wind tunnel experiments are shown in Figure 2. As illustrated in the figure, at low Reynolds numbers, laminar separation bubbles form on both the upper and lower surfaces of the airfoil, with the length of these bubbles varying with the angle of attack. Consequently, the lift coefficient curve of the airfoil under low Reynolds number conditions exhibits marked differences compared to that of a typical high Reynolds number airfoil, namely, the presence of nonlinear behavior within a small angle of attack range. Furthermore, the computational outcomes align well with experimental data, thus validating the accuracy of the numerical methodology and grid configuration employed in this study for simulating unsteady flows under low Reynolds number conditions.

2.2. Grid Deformation Method

Given the minuscule nature of the airfoil deformation in this study, a dynamic grid methodology rooted in spring smoothing algorithms has been implemented to facilitate the localized deformation of the airfoil surface [22]. Among other methodologies, the spring smoothing technique notably decreases computational workload while enhancing calculation efficiency. Based on the spring smoothing hypothesis, the conceptual framework treats each pair of nodes as being interconnected by hypothetical ideal springs, represented through grid lines. The initial spacing of grid nodes prior to boundary deformation and movement establishes the stable state of the grid. The prescribed displacement at the boundary nodes leads to a force that is directly proportional to the spring displacement. According to Hooke’s law, the force on the grid node is as follows:

$${\mathit{F}}_i={\displaystyle \sum }_j^{n_i}{k}_{ij}\left(\mathsf{\Delta }{\mathit{x}}_j-\mathsf{\Delta }{\mathit{x}}_i\right)$$

(3)

where $\mathsf{\Delta }{\mathit{x}}_i$ and $\mathsf{\Delta }{\mathit{x}}_j$ are the displacements of node i and the adjacent node j; $n_i$ is the number of nodes connected to node i; and $k_{ij}$ is the elastic coefficient of the spring between node i and node j.

The resultant force on the nodes connected by all springs must be zero in the equilibrium state, so the iterative equation is as follows:

$$\mathsf{\Delta }{\mathit{x}}_i^{m+1}=\frac{{\displaystyle \sum _j^{n_i}{k}_{ij}\Delta {\mathit{x}}_j^m}}{{\displaystyle \sum _j^{n_i}{k}_{ij}}}$$

(4)

where m is the number of iterations.

Given that the boundary displacements are known after updating the positions at the boundary nodes, Equation (3) can be solved using the Jacobi iterative method. Upon the convergence of the solution, the position is updated as follows:

$${\mathit{x}}_i^{n+1}={\mathit{x}}_i^n+\Delta {\mathit{x}}_i^{\mathrm{converged}}$$

(5)

where n + 1 and n are used to represent the next time step and the current time step, respectively.

2.3. Water Tunnel Test Method

In this investigation, the water tunnel test and the numerical method are employed to examine the aerodynamic characteristics of the rigid airfoil. The test apparatus is a low-speed water tunnel located at Beihang University. The test section of the water tunnel measures 16 m in length with a cross-sectional configuration that is a rectangular shape of 1.2 m by 1 m. The velocity of water is 0.1 m/s–1.0 m/s, and the turbulence intensity is 0.27–0.45%.

To ensure the preservation of two-dimensional flow characteristics at the model’s midsection, measures were taken to minimize the impact of three-dimensional flow. Consequently, the experimental model comprised a chord-length-equal segment with a rectangular planform. Additionally, isolation plates were installed on either side of the wing segment to segregate the boundary layer along the tunnel walls, thereby mitigating the three-dimensional effects emanating from the wingtips and suppressing spanwise flow. The angle of attack of the model can be precisely controlled through an integrated crank-link mechanism. The schematic diagram of the water tunnel experimental system is depicted in Figure 3. In this experiment, the staining solution technique is employed to visualize the flow pattern over the airfoil. Multiple stain solution injection ports are strategically positioned on both the upper and lower surfaces of the model, each of which is interconnected via flexible hoses to the stain solution injection system. The water tunnel test system and model of the airfoil are shown in Figure 4.

3. Calculation Model

This study focuses on a particular type of high-lift laminar flow airfoil as its subject of investigation. This airfoil boasts a relative thickness of 15.3%, with its maximum thickness occurring at 30.7% of the chord length. Moreover, it features a relative camber of 7.0%. The calculation model depicting partial oscillations on an airfoil is illustrated in Figure 5. The chord length of the airfoil is denoted as c, and the span of the flexible surface measures 0.1c. The distance between the center of the partial flexible surface and the leading edge of the airfoil is 0.1c, 0.2c…0.7c.

The displacement of the oscillating surface can be articulated as follows:

$$\Delta y=A\sin (2\pi ft)\sin (\pi d_c/l)$$

(6)

where $\Delta y$ is the oscillation displacement perpendicular to chord direction; l is the length of the oscillation surface; $d_c$ is the chord distance between the oscillation point and the starting position of oscillation; A is the amplitude; f is the frequency of oscillation; and t is the time.

Both the fixed and oscillating boundaries of the airfoil surface employ no-slip wall boundary conditions, wherein the fixed boundary remains stationary throughout the computation. The nodes on the oscillating boundary, adhering to Equation (6), undergo a periodic up-and-down motion within the two-dimensional plane, with grid deformation managed by the spring smoothing method described in Section 2.2. Should the oscillation cycle be denoted as T, Figure 6 presents the varying shapes of the flexible surface on the airfoil across distinct time points within a single cycle.

During the deformation process of the airfoil surface, variations in grid density may occur, potentially impacting the computation results. To further investigate the impact of grid distribution variations on computational outcomes, considering the deformation at position P1 as an example, under the condition of amplitude A = 0.001c where the local skin deformation reaches its lowest point, the grid becomes notably sparse. To address this, the refinement of the grid in the direction perpendicular to the wall surface was performed at this instance, ensuring that the post-refinement grid distribution matches that prior to deformation. The comparison of the grid distribution near the oscillation surface before and after refinement is depicted in Figure 7.

Calculations were performed under the conditions of Re = 3 × 10⁵, an angle of attack of 3° and with the surface held stationary, using both refined and initial unrefined grids. The results are summarized in

Table 1. Calculation results of refined grid and initial grid.

	CL	CD	K
Refined grid	1.2347	0.01737	71.0823
Initial grid	1.2342	0.01750	70.5257
Relative error	−0.0405%	0.748%	−0.783%

.

The computational results presented in Table 1 reveal that the aerodynamic coefficient errors obtained from the initial, unrefined grid are minimal, all falling within a 1% margin, compared to those derived from the refined grid. Consequently, it is inferred that during the oscillation of the airfoil surface, due to the small amplitude of oscillation, the variation in grid distribution is negligible. Thus, the computational errors introduced by the grid distribution changes can be disregarded.

4. Results and Discussion

Initially, under the condition where the Reynolds number based on the airfoil chord is Re = 3 × 10⁵, the laminar separation performance of the rigid airfoil is meticulously examined through both experimental testing and computational analysis. Subsequently, the aerodynamic properties of an airfoil with a partially oscillating section are computed to investigate the impact of such oscillations on the airfoil’s behavior.

4.1. The Performance of Laminar Flow Separation on the Rigid Airfoil

An unsteady numerical simulation approach has been employed to investigate the laminar separation bubble on a rigid airfoil, complemented by a corresponding water tunnel test conducted under identical conditions.

At a Reynolds number of 3 × 10⁵ and an angle of attack set at 3°, the structural configurations of the laminar separation bubble, as derived from both the water tunnel testing and numerical simulations, are depicted in Figure 8. The results of the water tunnel testing reveal a distinct flow pattern where the streamline along the leading edge of the airfoil initially appears as a narrow and straight streak, indicative of a laminar flow regime. It is further elucidated that the flow field in the central region of the model maintains its two-dimensional characteristics, remaining largely unaffected by three-dimensional effects. Subsequently, this streak broadens and darkens, signifying the onset of laminar flow separation followed by the emergence of reverse flow. Thereafter, the staining line exhibits chaotic behavior, marking the transition to turbulent flow. Ultimately, the evidence of backflow dissipates, and the fluid once again adheres to the airfoil surface. The LSB is characterized as the zone stretching between the laminar separation point and the point of reattachment. As illustrated in Figure 8, the LSB phenomenon on the airfoil at low Reynolds numbers is distinctly observable through the water tunnel testing, and the numerically simulated structure of the LSB aligns well with the experimental findings.

Figure 9 illustrates the time-averaged positions of laminar separation and turbulent reattachment. In the given figure, LS denotes the position of laminar separation, while TR represents the location of turbulent reattachment. The region encapsulated between the laminar separation (LS) point and the turbulent reattachment (TR) point constitutes the laminar separation bubble. As evidenced by this figure, it is apparent that at low Reynolds numbers, a distinct laminar separation bubble exists on the upper surface of the airfoil. In comparison to the experimental outcomes, the calculated errors are sufficiently minute, thereby validating the high accuracy of the unsteady numerical simulation method employed in this study for predicting the locations of laminar separation and turbulent reattachment at low Reynolds numbers. The calculation result shows that when the angle of attack is 3°, the time-averaged length of the laminar separation bubble is 22.2%c. The substantial extent of the laminar separation bubble suggests a significant impact on the airfoil’s aerodynamic performance.

4.2. Influence of the Oscillation Frequency

In this segment, we examine the impact of oscillation frequency on the airfoil’s aerodynamic properties through numerical simulations. The oscillating surfaces are positioned at P = 0.1c and P = 0.6c, with an amplitude of 0.001c. Based on the unsteady computations of the rigid airfoil, it is revealed that the frequency of the flow field around the rigid airfoil is approximately 5.5 Hz. Consequently, this study elects to investigate oscillation frequencies in the vicinity of the rigid airfoil’s flow field frequency, with a range of 1–10 Hz, which corresponds to Strouhal numbers St = 0.23–2.3.

Figure 10 depicts the time-averaged lift coefficient (C_L), drag coefficient (C_D) and lift-to-drag ratio (K) for varying oscillation frequencies, where “rig” refers to the rigid airfoil within the illustration.

Figure 10 demonstrates the significant impact of oscillation frequency on the aerodynamic properties of the airfoil. When the oscillation position is set at 0.1c, it is observed that as the oscillation frequency increases, the lift coefficient (C_L) of the airfoil initially rises before declining, whereas the drag coefficient (C_D) experiences a decrease followed by an increase. Consequently, the lift-to-drag ratio of the airfoil exhibits an ascending and then descending trend. At the oscillation frequency of f = 6 Hz, the lift-to-drag ratio attains its peak value. Relative to a rigid airfoil, partial oscillation leads to a 4.9% enhancement in the lift coefficient, a 15.3% reduction in the drag coefficient and, correspondingly, a substantial boost of 23.8% in the lift-to-drag ratio. At an oscillation position of 0.6c, similar trends in variation are noted for the airfoil’s C_L, C_D and K as the frequency increases, paralleling the scenario when the oscillation position is at 0.1c. It is noteworthy that when the frequency hits f = 4 Hz, the lift-to-drag ratio attains its peak value. Under these conditions, the partial oscillation leads to a 4.9% boost in the lift coefficient, a 15.2% reduction in the drag coefficient and thereby a significant 23.7% enhancement in the lift-to-drag ratio of the airfoil.

The aforementioned research demonstrates that partial oscillation can lead to a substantial decrease in the airfoil’s drag coefficient. To delve into the underlying mechanisms of this drag reduction, the total drag on the airfoil is partitioned into its pressure drag and viscous drag components. At an oscillation position of 0.6c, Figure 11 depicts the time-averaged pressure drag coefficient (C_Dp) and viscous drag coefficient (C_Dv) for varying frequencies. This figure reveals that, relative to the rigid airfoil, the pressure drag coefficient of the oscillating airfoil diminishes, while its viscous drag coefficient experiences an increase. These data highlight that as the oscillation frequency rises, the pressure drag coefficient of the airfoil initially dips before climbing back up, whereas the viscous drag coefficient follows a pattern of initially increasing and then subsequently decreasing. At the point where f = 4 Hz, the viscous drag coefficient peaks, registering an 11.6% boost, whereas the pressure drag coefficient hits its lowest point, witnessing a notable drop of 23.3%. Taking into account the fluctuations in both pressure and viscous drags, the overall drag coefficient for the oscillating airfoil sees a reduction of 15.2% in comparison to its rigid counterpart.

Figure 12 illustrates the position of transition and the extent of the LSB along the airfoil under varying oscillation frequencies, with the oscillation position fixed at 0.6c. Figure 12a depicts that for the rigid airfoil, the transition position is located at 0.656c; however, when subjected to oscillation, this transition point shifts forward. At a frequency of f = 4 Hz, the transition position shifts to 0.602c. In Figure 13, the friction coefficient C_f along the upper surface of the airfoil is depicted. Evidently from the illustration, partial oscillation induces an earlier transition effect, thereby causing a shift in the frictional characteristics on the airfoil’s surface. As gleaned from Figure 13, the C_f value upstream of the oscillation zone remains unaffected by the partial oscillation. Conversely, within the downstream area of the oscillation, C_f values escalate due to the oscillation and rise above those of the rigid airfoil. This phenomenon constitutes the primary driver behind the fluctuation in the viscous drag coefficient.

Figure 12b illustrates that the length of the LSB on a rigid airfoil amounts to 0.215c, and the implementation of partial oscillation significantly reduces this LSB dimension. At a frequency of f = 4 Hz, the LSB length reaches its minimum at 0.185c, demonstrating a 14.0% reduction in comparison to the length of the LSB on the rigid airfoil. From Figure 13, one can discern the changes in the LSB, where the regions marked by C_f values lower than zero denote the LSB’s location. Figure 14 presents the precise outline of the LSB on the airfoil. Combining insights from Figure 13 and Figure 14, it becomes evident that the oscillation does not have a significant influence on the onset of laminar separation; however, it visibly causes the reattachment point to advance. Consequently, the LSB shortens, which in turn results in a reduction in the pressure drag coefficient.

4.3. Influence of the Oscillation Position

In this segment, we examine the impact of oscillation position on the aerodynamic properties of the airfoil, maintaining a constant oscillation frequency of f = 5 Hz and an amplitude of A = 0.001c throughout the study. The aerodynamic attributes of the airfoil subject to various oscillation positions are illustrated in Figure 15. This graphical representation highlights that when the oscillation position is at P = 0.1c and P = 0.6c, the airfoil experiences a reduction in its drag coefficient, concurrent with a significant boost in both the lift coefficient and the lift-to-drag ratio. Specifically, when P = 0.1c, there is a 4.87% increase in C_L, a 15.29% decrease in C_D and a remarkable 23.79% augmentation in K, all in comparison to the rigid airfoil. With P = 0.6c, there is a notable augmentation of 4.94% in C_L, a substantial reduction of 15.15% in C_D and a considerable surge of 23.68% in K. From the foregoing analysis, it is evident that the changes in the aerodynamic coefficients exhibit similar patterns when P = 0.1c and P = 0.6c, thus substantiating that partial oscillations at both 0.1c and 0.6c exert comparable flow control effects.

From Figure 15, it is discerned that the time-averaged location of laminar separation lies at 0.506c, while the time-averaged reattachment point is situated at 0.721c. Consequently, the oscillation position at 0.6c is proximate to the centroid of the LSB on the airfoil. The research demonstrates that when oscillation occurs in close proximity to the leading edge or the central region of the LSB, the enhancement in lift and the reduction in drag resulting from partial oscillation are most pronounced.

Figure 16 presents time-averaged streamline patterns and velocity distributions for the oscillating airfoil at various positions. Time averaging entails computing the temporal mean of flow field parameters over multiple cycles, yielding a result that represents the averaged conditions. This approach serves as a pivotal technique widely adopted in the study of unsteady flows at low Reynolds numbers [23].

From this figure, it is discernible that multiple LSBs exist across the upper surface of the rigid airfoil within the span of 0.506c to 0.721c. Under a Reynolds number of 3 × 10⁵, the cumulative length of these LSBs measures 0.215c. At P = 0.5c, a significant extent of laminar separation persists. However, when P assumes values of 0.1c and 0.6c, multiple smaller separation bubbles coalesce into a single larger bubble within the laminar separation region on the upper surface of the airfoil, resulting in a visibly reduced separation area. On the other hand, these velocity contours reveal that when the oscillation positions are at 0.1c and 0.6c, the acceleration of airflow over the upper surface of the airfoil is reduced compared to the rigid airfoil case and the scenario where oscillation occurs at 0.5c. Concurrently, the formation of low-velocity regions indicative of recirculation is diminished. These observations suggest that local oscillation at positions 0.1c and 0.6c exert a controlling effect on the laminar separation bubble over the upper surface of the airfoil.

The pressure distribution on the upper surface of both the rigid airfoil and airfoils with surface oscillation at different locations is depicted in Figure 17. As illustrated by the distribution curves of pressure coefficients, remarkable fluctuations in pressure coefficients are observed over the separation bubble region on the upper surface of the rigid airfoil, a phenomenon attributed to the presence of multiple vortex structures in this area. However, when oscillation occurs at positions 0.1c and 0.6c, the pressure fluctuations within the separation bubble region are notably mitigated. This further suggests that oscillation at these two specific locations exerts a certain degree of control over the laminar separation bubble.

The mechanism underlying the aforementioned phenomena is as follows: When the oscillation position is at 0.1c, proximate to the airfoil’s leading edge, the oscillation serves to efficiently infuse energy into the boundary layer, influencing a broader downstream area, thus effectively managing the LSB. In contrast, when the oscillation occurs at 0.6c, situated within the LSB itself, it directly supplies energy into the low-velocity airflow within the separation bubble, thereby efficaciously controlling laminar separation.

At P = 0.1c and P = 0.6c, Figure 18 depicts the peak displacement of the oscillating surface, denoted as △y_m, alongside the aerodynamic forces recorded at distinct instants across two complete oscillation cycles. This figure demonstrates that as the airfoil undergoes partial skin deformation, both its lift and drag coefficients exhibit periodic fluctuations. Over the course of one oscillation cycle, the lift coefficient of the airfoil surpasses that of its rigid counterpart, and, concurrently, the drag coefficient is found to be less than that of the rigid airfoil. Therefore, the introduction of partial oscillation enhances the aerodynamic performance of the airfoil at low Reynolds numbers. The magnitude of the aerodynamic coefficients when the position P = 0.6c exceeds that observed at P = 0.1c, suggesting that the influence of oscillation on the LSB is more pronounced at the former position. Consequently, the unsteady nature of the aerodynamic forces is more evident when P is 0.6c.

Illustrated in Figure 19 are the varying shapes of the LSB on the upper surface of the airfoil at distinct moments within a single oscillation cycle when P = 0.6c. From the figure, it becomes evident that the LSB assumes diverse configurations at different instants during the oscillation cycle. This cyclical transformation of LSB shapes directly contributes to the periodic fluctuations in the aerodynamic forces exerted upon the airfoil.

5. Conclusions

(1) The application of partial oscillation on the airfoil skin significantly enhances its aerodynamic attributes. Relative to a rigid airfoil configuration, the lift coefficient rises by 4.9%, the drag coefficient reduces by 15.3% and consequently, the lift-to-drag ratio escalates by 23.8% due to this oscillatory effect.

(2) In comparison with a non-oscillating rigid airfoil, the pressure drag coefficient of the oscillating airfoil diminishes, whereas the viscous drag coefficient increases. As the oscillation frequency rises, the lift coefficient initially ascends but subsequently declines, while the drag coefficient first drops and then rises.

(3) The chordwise position of the oscillating surface plays a pivotal role in influencing the airfoil’s performance and flow field properties. The most advantageous placement for the partial oscillation surface is in proximity to the airfoil’s leading edge and at the center of the LSB. Under such configurations, the partial oscillation effectively mitigates laminar separation over the airfoil, thereby improving its overall lift-to-drag performance characteristics.

This study confines its investigation to two-dimensional analyses of localized oscillation on airfoil surfaces, inherently possessing certain limitations. Future endeavors will extend to explore flow control research involving the three-dimensional oscillation of wing surfaces. The intent is to delve deeper into the synergistic interactions between three-dimensional vortex formations, spanwise flow, wingtip vortices and wing surface oscillation under three-dimensional conditions, thereby advancing the understanding of these intricate dynamics. On the other hand, this paper conducts a study on the local oscillation of airfoils using numerical simulation methods. Nevertheless, experimental investigations into the local oscillation of the airfoil have not yet been undertaken, and thus, the numerical simulation outcomes regarding the local oscillation of the airfoil have not yet been validated through experimental verification. Future work will encompass an extended study into wind tunnel experiments on the local vibrations of airfoils, with the objective of validating their flow control efficacy.