The Lift Enhancement Effect of a New Fluidic Oscillator on High-Lift Wings

Abstract

Fluidic oscillators have emerged as a prominent topic of research in the field of flow control, owing to their broad sweep range and enhanced control efficiency. However, the underlying mechanisms governing the operation of fluidic oscillators remain poorly understood, and the effect of oscillation frequency on flow control performance has yet to be conclusively determined. In this study, a novel fluidic oscillator is proposed that achieves frequency decoupling by replacing the conventional feedback channel with synthetic jets, thereby enabling modulation of oscillation frequency at a constant momentum coefficient. When applied to a high-lift airfoil, results show that at a momentum coefficient of 14.1%, the lift coefficient increase achieved under F⁺ = 1 control outperforms that under F⁺ = 10 by more than 0.3. This finding suggests the presence of an optimal frequency for fluidic oscillators, which maximizes their flow control effectiveness. Notably, this optimal frequency is unaffected by variations in the momentum coefficient. A deeper analysis of the fluidic oscillator’s working principle reveals that periodic oscillations dominate the turbulent kinetic energy and Reynolds shear stress, driving enhanced chordwise momentum exchange. This increased energy transfer strengthens the boundary layer’s resistance to separation, effectively mitigating flow detachment and improving lift enhancement. Finally, the periodic flow field on the surface of the high-lift airfoil under fluidic oscillator control was examined. It was observed that, at low frequencies, the fluidic oscillator effectively controls the shedding of separation vortices, ensuring that the frequency of vortex shedding aligns with the oscillation frequency of the fluidic oscillator. This alignment likely contributes to the superior lift enhancement observed under low-frequency conditions.

1. Introduction

As an emerging aerodynamic technology, high-lift systems are recognized for their ability to enhance lift while reducing drag, presenting significant potential for optimizing aerodynamic efficiency and lowering fuel consumption. However, these systems are susceptible to boundary layer separation, particularly under high angles of attack and large control surface deflections. As a result, delaying flow separation has become a critical area of focus in aerodynamic research for improving the performance of high-lift configurations.

Flow control technologies can be broadly classified into two main categories: active and passive control. Passive flow control, exemplified by vortex generators [1,2,3], is widely used in fluid dynamics due to its simple design and the lack of need for additional energy input. However, a limitation of passive control is its inability to autonomously adapt to changing operational conditions, which can lead to suboptimal performance under certain circumstances.

Active flow control technologies encompass a variety of techniques, including blowing and suction [4,5], plasma actuators [6,7,8,9,10], and synthetic jets [11,12,13], among others. In this context, the fluidic oscillator [14,15,16,17] has emerged as a prominent solution. It converts steady jets into oscillating flows by exploiting inherent flow instability mechanisms, thus eliminating the need for moving components in the actuation process. The absence of electromechanical, piezoelectric, or complex driving systems not only enhances energy efficiency but also significantly extends the operational lifespan of these devices. These inherent advantages have positioned fluidic oscillators at the forefront of contemporary aerodynamic research. Figure 1 illustrates the fundamental mechanism of fluidic oscillators.

Emile K. Oshima et al. [18] applied fluidic oscillators to three-dimensional swept wings and demonstrated that the lift augmentation capabilities of the oscillators are closely linked to the strength of vortex lift. Moreover, their study revealed that fluidic oscillators are effective in mitigating the static instability of swept wings, significantly reducing the likelihood of aircraft stall. Similarly, Chengze Wang et al. [19] investigated the use of fluidic oscillators in S-shaped inlets, showing that oscillatory jets can reduce flow losses, enhance the uniformity of the inlet flow field, and inject additional energy into low-energy regions of the flow. This contributes to the effective suppression of flow separation.

Florian Ostermann et al. [20] conducted experimental investigations into the characteristics of low-frequency oscillating jets. Their findings revealed that the velocity of oscillating jets rapidly decays while demonstrating a significant entrainment rate. In a subsequent study, Ostermann et al. [21] explored the complex interaction between a fluidic oscillator and the surrounding external flow dynamics. The study highlighted that at lower Strouhal numbers, spatially oscillating jets exhibit effects similar to those of temporally varying vortex-generated jets, resulting in the formation of a time-evolving vortex-like flow structure. In contrast, at higher Strouhal numbers, the cross-flow fails to synchronize with the jet’s motion, leading to the formation of a quasi-stable wake in the downstream region of the jet flow field.

Kim et al. [22] applied fluidic oscillators at the 20% chord location of a NACA0015 airfoil and found that these devices effectively suppressed leading-edge flow separation, resulting in a maximum lift coefficient increase of 45%. However, they also identified that the size of the fluidic oscillators imposed limitations on their installation position. To address this challenge, the researchers developed a new fluidic oscillator [23] with a curved outlet to overcome the geometric constraints associated with its size. When applied to flow control, this redesigned oscillator demonstrated even better lift enhancement performance compared to previous designs. Furthermore, they investigated the effects of the oscillator’s curvature and pitch angle on airfoil performance [24]. Similarly, Alexander Spen et al. [25] developed a new curved fluidic oscillator and applied it to a NACA0018 airfoil. Their study revealed that when the oscillator was positioned near the leading edge, it effectively suppressed flow separation. Additionally, relocating the installation point further aft significantly increased the lift coefficient.

Daniel J. Portillo et al. [26] conducted a modal decomposition of fluidic oscillators and experimentally assessed their oscillation frequency, sweep angle, and modal structure at various lengths. Their findings revealed that the sweep angle was inversely proportional to the flow coefficient and directly proportional to the size of the oscillator. Similarly, Pengcheng Yang et al. [27] employed numerical simulations to investigate three fluidic oscillators of different sizes. They observed that as the oscillator size decreased, its sweeping characteristics underwent significant alterations, with the sweep angle decreasing by more than 10°. This phenomenon was attributed to the frictional forces acting on the upper and lower walls of the fluidic oscillator.

Additionally, Xiaodong Chen et al. [28,29,30] applied fluidic oscillators to control the wake vortices behind an inclined circular cylinder, exploring the complex three-dimensional interactions between the fluidic oscillator and the wake vortices.

Currently, extensive research has been conducted on fluidic oscillators, with significant attention given to their intrinsic characteristics and diverse applications in flow control, fuel combustion, and other engineering fields. These studies primarily aim to assess the feasibility and advantages of fluidic oscillators in practical scenarios. However, the underlying working mechanism of fluidic oscillators in flow control remains unclear. Furthermore, fluidic oscillators exhibit specific frequency characteristics during operation, which are influenced by variations in inlet flow rate and oscillator size. This variability makes it challenging to determine whether an optimal frequency exists that could maximize the performance of fluidic oscillators during operation.

To address this issue, this study focuses on a novel fluidic oscillator design previously proposed [31], which effectively decouples the oscillator’s frequency from factors such as momentum coefficient and size. In this work, the feasibility of the fluidic oscillator is first validated through both numerical simulations and experimental methods. Subsequently, the oscillator was applied to a three-dimensional high-lift airfoil to investigate the effect of oscillation frequency on lift enhancement, and the flow field was decomposed into periodic motion and turbulent components using velocity decomposition to study their impact on the flow field. Finally, explore the working mechanism of fluid oscillators.

2. Numerical Simulation Methods

2.1. Turbulence Equation

In this study, the SST k-ω turbulence model was chosen, and its equations are shown below.

$${\displaystyle \frac{\partial \left(pk\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho u_{j}k\right)}{\partial x_j}}={\tau }_{ij}{\displaystyle \frac{\partial u_i}{\partial x_j}}-{\beta }^{\ast }\rho \omega k+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_k\mu \right){\displaystyle \frac{\partial k}{\partial x_j}}\right]$$

(1)

$${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho u_j\omega \right)}{\partial x_j}}={\displaystyle \frac{\gamma }{{\mu }_t}}{\tau }_{ij}{\displaystyle \frac{\partial u_i}{\partial x_j}}-\beta \rho {\omega }^2+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]+2\left(1-F_1\right){\displaystyle \frac{\rho {\sigma }_{{\omega }^2}}{\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}$$

(2)

$$\gamma ={\displaystyle \frac{{\beta }_i}{{\beta }^{\ast }}}-{\displaystyle \frac{{\sigma }_{\omega }{\kappa }^2}{\sqrt{{\beta }^{\ast }}}}$$

(3)

$$F_1=\tanh \left({\varphi }_1^4\right)$$

(4)

$${\varphi }_1=\min \left[\max \left({\displaystyle \frac{\sqrt{k}}{0.09\omega y}},{\displaystyle \frac{500\mu }{\rho y^2\omega }}\right),{\displaystyle \frac{4\rho k}{{\sigma }_{\omega 2}{D}_{\omega }^{+}{y}^2}}\right]$$

(5)

$$D_{\omega }^{+}=\max \left[2\rho {\displaystyle \frac{1}{{\sigma }_{\omega 2}}}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial k}{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}},{10}^{-20}\right]$$

(6)

The expression for the model turbulent viscosity is shown in Equations (7) to (9):

$${\mu }_t={\displaystyle \frac{\rho a_{1}k}{\max \left(a_1\omega ,\Omega F_2\right)}}$$

(7)

$$F_2=\tanh \left({\arg }_2^2\right)$$

(8)

$${\arg }_2=\max \left(2{\displaystyle \frac{\sqrt{k}}{0.09\omega y}},{\displaystyle \frac{500\mu }{y^2\omega }}\right)$$

(9)

In Equations (1) to (9), ${\tau }_{ij}$ represents the Reynolds stress tensor, $\mathsf{\Omega }$ is the vorticity tensor, $\mu$ is the laminar viscosity, ${\mu }_t$ is the turbulent viscosity, and y is the distance to the wall. The model constants are as follows: $a_1$ = 0.31, ${\beta }^{\ast }$ = 0.09, $\kappa$ = 0.41, ${\sigma }_{k1}$ = 0.85, ${\sigma }_{\omega 1}$ = 0.5, ${\sigma }_{k2}$ = 1.0, ${\sigma }_{\omega 2}$ = 0.856.

2.2. Post-Processing Methods for Mean Flow Characteristics

A.

Averaging operators

In this part, we describe the definitions of averaging operators. Let f represent an instantaneous physical quantity, which is the function of time t and space $r=(x,y,z)$. The total and phase averaging operators are defined as $\overline{•}$ and ${\langle •\rangle }_{\phi }$, respectively:

$$\overline{f}=\underset{T_{all}\to \infty }{\lim }{\displaystyle \frac{1}{T_{all}}}{\displaystyle {\int }_{t_0}^{t_0+T_{all}}f(t,t)dt}\simeq {\displaystyle \frac{1}{K}}{\displaystyle \sum _{i=0}^{k-1}f(t_0+i\Delta t,r}),(\Delta t=T_{all}/K)$$

(10)

$${\langle f\rangle }_{\phi }=\underset{N_{\phi }\to \infty }{\lim }{\displaystyle \frac{1}{N_{\phi }}}{\displaystyle \sum _{n=0}^{N_{\phi }-1}f(t_{\phi }+nT,r)}\simeq {\displaystyle \frac{1}{N_{\phi }}}{\displaystyle \sum _{n=0}^{N_{\phi }-1}f(t_{\phi }+nT,r)},(T=1+F^{+},0\le \phi \le 2\pi ,t_0\le t_{\phi }<t_0+T)$$

(11)

where $t_0$ represents a start time of the averaging procedure; $T_{all}$ denotes the averaging period; $\Delta t$ is identical to the computational time step; $\phi$ denotes the phase angle between 0 and $2_{\pi }$ (in this study, $\phi$ is associated with the actuation frequency of the fluidic oscillator); $N_{\phi }$ indicates the maximum number of ensembles; $t_{\phi }$ denotes the time that corresponds to the phase angle $\phi$; and T is the time period that is characterized by a specific flow motion, e.g., the actuation period T = 1/F⁺ in this study. “F⁺” signifies the dimensionless frequency. Equation (12) is the definition of F⁺. Accordingly, the averaging operator ${\overline{•}}^{\phi }$ is also defined as

$$F^{+}=f\ast c/U$$

(12)

$${{\displaystyle \overline{{\langle f\rangle }_{\kappa }}}}^{\phi }={\displaystyle \frac{1}{2\pi }}{{\displaystyle {\int }_0^{2\pi }\langle f\rangle }}_{\phi }d\phi \simeq {\displaystyle \frac{1}{M}}{\displaystyle \sum _{m=0}^{M-1}{\langle f\rangle }_{{\phi }_m}}$$

(13)

where f is the oscillation frequency, c is the chord length of the wing, and U is the external inflow velocity.

Where M denotes the total number of discrete phases (the present paper adopts 10 segments for each period.) Note that the total averaging operator $\overline{•}$ is theoretically (but not numerically) the same as the total phase-averaging operator ${\overline{•}}^{\phi }$, although they are separately used in this study.

B.

Phase decomposition

In this subsection, we consider a physical quantity $f$, which is a function of time $t$. We will conduct a phase decomposition to extract coherent flow structures based on the period that corresponds to the actuation frequency F⁺. For this purpose, the instantaneous physical quantity $f(t)$ is decomposed into overall average $\overline{f}$; phase fluctuation $\tilde{f}$; and turbulent fluctuation $f^{″}$ as follows:

$$f(t)=\overline{f}+f^{\prime }(t)=\overline{f}+{\tilde{f}}_{\phi }+f^{″}(t)={\langle f\rangle }_{\phi }+f^{″}(t)$$

(14)

Therefore, the following decomposition holds:

$$\overline{f^{\prime }{g}^{\prime }}={{\displaystyle \overline{{\tilde{f}}_{\phi }{\tilde{g}}_{\phi }}}}^{\phi }+\overline{f^{″}{g}^{″}}$$

(15)

$${\langle f^{\prime }{g}^{\prime }\rangle }_{\phi }={\langle {\tilde{f}}_{\phi }{\tilde{g}}_{\phi }\rangle }_{\phi }+{\langle f^{″}{g}^{″}\rangle }_{\phi }$$

(16)

Equation (17) is the definition of ${{\displaystyle \overline{{\tilde{f}}_{\phi }{\tilde{g}}_{\phi }}}}^{\phi }$. The expression for $\overline{f^{″}{g}^{″}}$ is Equation (18). The expressions on the right side of Equation (15) are Equations (19) and (20).

$${{\displaystyle \overline{{\tilde{f}}_{\phi }{\tilde{g}}_{\phi }}}}^{\phi }={\overline{{\langle f\rangle }_{\phi }{\langle g\rangle }_{\varphi }}}^{\phi }-\overline{f}\overline{g}$$

(17)

$$\overline{f^{″}{g}^{″}}=\overline{fg}-{\overline{{\langle f\rangle }_{\phi }{\langle g\rangle }_{\phi }}}^{\phi }$$

(18)

$${\langle {\tilde{f}}_{\phi }{\tilde{g}}_{\phi }\rangle }_{\phi }={\langle f\rangle }_{\phi }{\langle g\rangle }_{\phi }-{\langle f\rangle }_{\phi }\overline{g}-\overline{f}{\langle g\rangle }_{\phi }+\overline{f}\overline{g}$$

(19)

$${\langle f^{″}{g}^{″}\rangle }_{\phi }={\langle fg\rangle }_{\phi }-{\langle f\rangle }_{\phi }{\langle g\rangle }_{\phi }$$

(20)

3. Set-Up

3.1. Fluidic Oscillator

In this article, the selected fluidic oscillator model is shown in Figure 2, and its geometric parameters and boundary condition settings are identical to those studied previously [31]. However, since this article studies the three-dimensional structure of fluidic oscillators, a thickness of 3 mm is selected. Unlike traditional fluidic oscillators, synthetic jets are used to replace the feedback channel of traditional fluidic oscillators, and the frequency of synthetic jets is used to replace the frequency characteristics of traditional fluidic oscillators, thereby achieving decoupling of the frequency and momentum coefficient geometry of fluidic oscillators.

3.1.1. Experimental Setup of Fluidic Oscillator

As shown in Figure 3, the experimental setup for the fluidic oscillator is illustrated, with the model of the fluidic oscillator and the locations of the monitoring points shown in Figure 2. The monitoring points are placed on the upper side of the fluidic oscillator outlet, rather than along the centerline, because the oscillatory jet generated by the fluidic oscillator produces periodic sweeping flows as it exits the nozzle. This causes the jet to cross the centerline twice. Therefore, placing the monitoring points along the centerline would detect the harmonics of the fluidic oscillator’s oscillation frequency, rather than the fundamental oscillation frequency of the jet itself.

Data measurements were conducted using a hot-wire anemometer, which can measure velocities in the range of 1 m/s to 150 m/s. The hot-wire anemometer has excellent dynamic characteristics, with a maximum frequency response of up to 50 kHz, making it suitable for frequency measurements of high-frequency fluidic oscillators. To control the mass flow rate of the fluidic oscillator, a TSK621 mass flow meter (Beijing Jierui Shengxin Technology Co., Ltd, Beijing, and China) was used. The mass flow meter is connected to the inlet of the actuator through plastic tubing, and its flow can be controlled via computer control. The specific flow rate can be displayed on the computer or on the screen of the mass flow meter. The working fluid for this experiment is air.

The synthetic jet actuator is selected as a single-diaphragm, dual-chamber piezoelectric synthetic jet actuator, with each chamber connected to the synthetic jet inlet of the fluidic oscillator. A sinusoidal signal with a specific amplitude and frequency is generated by a signal generator, amplified by a voltage amplifier, and input to the synthetic jet actuator. The input signal is continuously monitored in real time using a digital oscilloscope.

3.1.2. Numerical Simulation of Fluidic Oscillator

The mesh of the fluidic oscillator generator is shown in Figure 4, and the hexahedral mesh is drawn by FLUENT MESHING v.22.0.(Ansys Corporation, Pittsburgh, and America). The left boundary is set as the flow inlet, the right boundary is set as the pressure outlet, the position of the synthesized jet is set as the velocity inlet, and the other settings are set as no-slip boundary conditions.

URANS equation and the SST k–ω turbulence model are solved using the commercial computational fluid dynamics (CFD) software, Ansys Fluent v22.0. The working fluid is air. A pressure-based, time-dependent solver with a constant time step of t = 10⁻⁵ s is employed, and the second-order discretization for pressure and the second-order-upwind formulation for density, momentum and turbulent kinetic energy are adopted. During the data acquisition process, due to the relatively long duration required for collecting low-frequency data, we chose to collect data for 20,000 steps, or 0.2 s, to ensure that the fluidic oscillator’s internal jet undergoes at least five complete oscillation cycles.

In the selection of velocity for the synthetic jet, insights from the study of conventional dual-feedback sweeping jet actuators revealed that the velocity field within the feedback channel can be represented using sinusoidal or cosine functions. Appendix A presents the velocity variations within the feedback channel of a conventional fluidic oscillator, which validates this phenomenon. Thus, for the waveform of the synthetic jet, a sinusoidal function was chosen.

The velocity variation between synthetic jets A and B is defined as follows: Equation (21) represents the velocity of synthetic jet A, and Equation (22) represents the velocity of synthetic jet B. In these equations, “$v_0$” stands for the amplitude.

$$f\left(x\right)=v_0\sin \left(2\pi F^{+}t\right)$$

(21)

$$f\left(x\right)=-v_0\sin \left(2\pi F^{+}t\right)$$

(22)

Due to the fact that the previously studied fluidic oscillator model was two-dimensional, the model studied in this paper is three-dimensional. In order to verify the impact of the mesh on the numerical simulation results, grid independence verification was conducted under operating conditions with oscillation frequencies of 40 Hz and 800 Hz, respectively. The specific grid size and quantity are shown in

Table 1. The number of grids of Fluidic oscillator.

Grid Name	Number of Cells	Y+ (the Outlet of the Fluidic Oscillator)
Fluidic oscillator-mesh1	1,300,000	5
Fluidic oscillator-mesh2	2,600,000	1.5
Fluidic oscillator-mesh3	3,700,000	0.8

. In the flow field inside the fluidic oscillator, the velocity is maximal at the outlet throat due to the combined effects of the pressure gradient and viscosity. Therefore, in the study of Y+, this location is chosen as the reference point.

Figure 5 shows the comparison of velocity at the same monitoring point for three different grids and the experimental results under control frequencies of 40 Hz and 800 Hz. Although there are some numerical differences between the simulation results and the experimental data, the overall trends are consistent, which validates the feasibility and reliability of the numerical simulation results. At 40 Hz and 800 Hz, the velocity curves for the three meshes are nearly identical. Further spectral analysis of these velocity curves reveals that the oscillation frequency matches the control frequency, confirming the feasibility and reliability of this strategy for three-dimensional fluidic oscillators. The reason for the appearance of two peaks at 40 Hz is due to a disturbance observed in the velocity curve. This disturbance manifests as periodic peaks in the velocity data, although the amplitude of these peaks is relatively small. In the FFT analysis, this disturbance is treated as one cycle, which leads to the appearance of two peaks. However, the energy level of this disturbance is lower than that of the 40 Hz frequency. Therefore, for the purpose of this frequency selection, the primary frequency is identified as 40 Hz. The disturbance is shown in Figure 6.

3.2. High-Lift Wings

In this study, to investigate the effect of the fluidic oscillator’s frequency on the lift enhancement of high-lift configurations and to explore its working mechanism, a high-lift configuration model was selected. The basic geometry of the model is shown in Figure 7. The airfoil cross-section is based on the CHN-T1 swept wing, a standard domestic model. It comprises three main components: a leading-edge slat, a main wing, and a trailing-edge flap. The trailing-edge flap is connected to the main wing through a step, with a step height of 1.88 mm, and the flap is deflected at an angle of 30°.

3.2.1. Experimental Setup of High-Lift Wings

As shown in Figure 7, the high-lift airfoil experimental model has a chord length of 240 mm, a span length of 400 mm, and an incoming flow velocity of 20 m/s. Endplates are installed at both ends of the airfoil to maintain the two-dimensional characteristics of the high-lift airfoil and minimize the influence of wingtip vortices. The experiment was conducted in the 1-m wind tunnel at Nanjing University of Aeronautics and Astronautics. The wind tunnel test section has a cross-sectional dimension of 1.5 m × 1 m, which meets the blockage ratio requirements for this study. The maximum wind speed in the tunnel is 30 m/s, and the turbulence intensity in the core region of the test section is 0.08%. For aerodynamic force measurements in this study, the wind tunnel balance used was the ATI balance (TSI Inc, Shoreview, MN, USA), which can measure forces in the X, Y, and Z directions as well as moments (Mx, My, Mz) around these axes.

3.2.2. Numerical Simulation of High-Lift Wings

Figure 8 shows the distribution of the high-lift airfoil and the fluidic oscillator on the high-lift airfoil in the numerical simulation study. The overall chord length of the wing is 240 mm, and the wingspan is 70 mm. The fluidic oscillator is positioned on the main wing, forming an angle of 15° with the freestream direction. The oscillator has a thickness of 3 mm and is connected to the step between the trailing-edge flap and the main wing through an adapter. This arrangement allows the fluidic oscillator to effectively control the flow separation over the trailing-edge flap, thereby achieving efficient lift enhancement.

The left boundary of the calculation domain is defined as the velocity inlet, and the right boundary is defined as the pressure outlet. The inlet velocity is V∞ = 20 m/s. Speed direction horizontally to the right. The boundary conditions and numerical simulation methods for the oscillating jet device are consistent with the calculations mentioned above.

The grid for the high-lift wing is shown in Figure 9. In this study, a structured mesh is chosen as the computational grid.

In the numerical simulation process, grid independence verification was first conducted for the wing model. The specific grid settings are shown in the

Table 2. The number of grids of high-lift wing.

Grid Name	Number of Cells	Y+
mesh1	1,500,000	3.2
mesh2	3,100,000	1.1
mesh3	7,500,000	0.6

, and Figure 10 illustrates the variation of the lift coefficient with the angle of attack for the high-lift airfoil. By comparing the results, it can be observed that, under the same Reynolds number, all grids exhibit similar flow characteristics, with stall occurring at an angle of attack of 15°, which aligns with the trend observed in the experimental results. Although there is some numerical discrepancy, considering the influence of experimental conditions and simulation parameters, such errors are within an acceptable range. As the grid resolution increases, the lift coefficient slightly increases, though the difference is minimal. Therefore, for computational efficiency and accuracy, a medium-resolution grid is selected for the simulations in this study. Figure 11 shows the velocity flow field of the wing at an angle of attack of 10°. It can be observed that at this angle of attack, a large separation zone has already formed at the trailing edge flap, while the main wing and leading-edge slat do not experience separation. Therefore, this study primarily focuses on investigating the impact of the fluidic oscillator’s frequency and momentum coefficient on the lift enhancement under this operating condition, as well as analyzing its working mechanism. The frequency and momentum coefficient of the fluidic oscillator used in this study are shown in

Table 3. Computational cases.

Case Description	$\mathbf{Input}\mathbf{Momentum}\left({\mathit{C}}_{\mathit{\mu }}\right)$	F+
Noncontrolled	/	/
Controlled (weak input)	7.2%	0.5, 1, 2, 5, 10
Controlled (medium input)	14.1	0.5, 1, 2, 5, 10
Controlled (strong input)	21.88%	0.5, 1, 2, 5, 10

below. The equation for calculating the momentum coefficient ($C_{\mu }$) is given by Equation (23).

$$C_{\mu }={\displaystyle \frac{m{U}_{jet}}{q{A}_{ref}}}={\displaystyle \frac{({\rho }_{jet}{U}_{jet}n{A}_{nozzle})U_{jet}}{(0.5{\rho }_{\infty }{U}_{\infty }^2)A_{ref}}}=2{\displaystyle \frac{{\rho }_{jet}{A}_{nozzle}{U}_{jet}^2}{{\rho }_{\infty }{U}_{\infty }^2{A}_{ref}}}$$

(23)

U_jet refers to the average velocity at the outlet of the fluidic oscillator, while U_∞ denotes the freestream velocity. The reference area (A_ref) is calculated by multiplying the chord length with the active span.

4. Results

As shown in Figure 12, the lift enhancement effect of the fluidic oscillator on a high-lift wing under different momentum coefficients is demonstrated. Comparison reveals that, under momentum coefficient control, the fluidic oscillator is able to increase the lift coefficient of the high-lift wing. However, it does not effectively suppress flow separation. This is because, at high angles of attack, the separation point is located at the leading edge, while the fluidic oscillator is positioned at the trailing edge flap, thus unable to control the overall flow separation of the wing. Nevertheless, it still provides a notable lift enhancement effect. Furthermore, it is observed that the frequency plays a crucial role in the lift enhancement performance of the fluidic oscillator, with F⁺ = 1 providing the best lift improvement.

At a momentum coefficient of 7.2%, the lift enhancement effect of the fluidic oscillator under different frequency controls is not as significant. However, by comparison, it is observed that the lift enhancement effect is the worst when F⁺ is 10. When the momentum coefficient is 14.1%, comparing the lift coefficients under different frequency controls reveals a more noticeable difference in lift at angles of attack less than 15°. Specifically, when F⁺ is 1, the lift enhancement effect is the best. Compared to F⁺ = 10, the lift coefficient can increase by more than 0.3. When F⁺ is 0.5 or 2, the control effects are similar, and when F⁺ is 5 or 10, the lift coefficients are almost the same. This suggests that there is indeed an optimal frequency for achieving the best lift enhancement effect during oscillating jet control, with high frequencies yielding the poorest control performance for this airfoil type.

This finding supports the idea that in traditional fluidic oscillators, increasing the momentum coefficient or decreasing the size may worsen the frequency control performance, leading to suboptimal efficiency. When the momentum coefficient increases to 21.88%, it is again found that F⁺ = 1 provides the best lift enhancement, confirming the existence of an optimal frequency for fluidic oscillator control that is independent of momentum coefficient changes. This further validates the notion that, in traditional fluidic oscillators, the frequency may not always remain at its optimal control state during operation.

As shown in Figure 13, the lift enhancement effect at a 10° angle of attack varies with the momentum coefficient. It can be observed that the lift enhancement is proportional to the momentum coefficient; as the momentum coefficient increases, the lift enhancement improves. However, at the same momentum coefficient, comparing the lift enhancement effects under different frequencies reveals that frequency is also a key factor influencing the lift enhancement performance.

As shown in Figure 14, under a 10° angle of attack and a momentum coefficient of 14.1%, the flow fields under the control of F⁺ = 0.5, 1, and 10 are presented. It can be observed that during the fluidic oscillator control process, the separation region is significantly reduced, which demonstrates the feasibility and effectiveness of fluidic oscillators in flow control.

Regarding the control mechanism of the fluidic oscillator, turbulent kinetic energy (TKE) is analyzed in this study. As shown in Figure 15, it is found that the fluidic oscillator increases the turbulent kinetic energy in the rear flap area, resulting in more intense velocity fluctuations at the rear flap and enhanced energy transfer. This helps optimize the turbulent structure, improving the efficiency of energy transfer, and promotes stronger mixing in the boundary layer, which enhances the wing’s resistance to flow separation and improves lift enhancement.

However, comparing the turbulent kinetic energy flow field under different frequency controls, it can be seen that with increasing frequency, the turbulent kinetic energy region becomes smaller. This suggests that as the frequency increases, energy exchange on the wing surface weakens. In the case of F⁺ = 1, the turbulent kinetic energy generation point occurs at the leading edge of the rear flap, and compared to F⁺ = 10, it is shifted forward and covers a larger area. This could be the primary reason for the differences in control effectiveness between high and low frequencies. Additionally, two monitoring points were set to track the time-dependent velocity variations in the rear wing and the external flow field of the wing. The specific locations of these monitoring points are shown in Figure 16.

Figure 17 and Figure 18 show the frequency spectra at the two monitoring points under different frequency controls. At low frequencies (F⁺ = 0.5 or 1), the frequencies of these two monitoring points align with the oscillation frequency. This demonstrates that, during the operation of a low-frequency fluidic oscillator, it can effectively control the rear flap and the external flow field of the wing, ensuring that the frequency of the separated vortices matches the oscillation frequency. However, at high frequencies (F⁺ = 5 or 10), the frequencies of the two monitoring points do not align with the oscillation frequency. Additionally, the observed frequencies at these points are significantly lower than the oscillation frequency. This indicates that at high frequencies, the fluidic oscillator is unable to fully control the rear flap and the separated vortices. This discrepancy likely explains the poorer performance of the fluidic oscillator at higher frequencies.

Subsequently, we analyzed Reynolds stress by decomposing it into two components: periodic fluctuations and turbulence. This decomposition allowed us to investigate the contributions of these components to the overall Reynolds stress. As shown in Figure 19, the Reynolds stress and its components are presented for momentum coefficients of 14.1% under F⁺ = 1 and F⁺ = 10 conditions. The results reveal a similar trend across different frequencies. The left column in the figure displays the dimensionless Reynolds stress, while the middle and right columns illustrate the dimensionless periodic fluctuations and turbulence components, respectively.

It is evident that periodic fluctuations dominate the Reynolds stress near the surface, while the contribution of the turbulence component to the Reynolds shear stress is negligible. This finding demonstrates that the periodic flow induced by the fluidic oscillator is the primary factor contributing to improved flow control. The periodic motion generated by the fluidic oscillator leads to intense momentum exchange in the chordwise direction of the airfoil, which transitions the surface flow into a turbulent state. This turbulence increases the energy required for flow separation, effectively suppressing separation and enhancing flow control efficiency.

In the previous flow field analysis, it was observed that periodic fluctuations are the dominant factor contributing to turbulent kinetic energy and Reynolds shear stress. To further investigate, the periodic shear stress field of the fluidic oscillator was analyzed by decomposing it into periodic fluctuation ($-{\langle {\tilde{u}}_{\phi }{\tilde{w}}_{\phi }\rangle }_{\phi }/U_{\infty }^2$) and turbulence components ($-\langle u^{″}{w}^{″}\rangle /U_{\infty }^2$). As shown in Figure 20, the periodic shear stress field under a control frequency of F⁺ = 0.5 is presented. The streamlines are represented by the ${\langle u\rangle }_{\phi }$ and ${\langle v\rangle }_{\phi }$ velocity components, illustrating the periodic shear stress distribution across the flow field.

Observing the flow field reveals that at a phase of 1/10, when the main jet of the oscillating jet sweeps through the section, the separation vortex near the trailing-edge flap nearly vanishes, leaving only a small, barely visible vortex near the fluidic oscillator’s outlet. This small vortex facilitates entrainment, allowing the high-energy flow from both the free stream and the fluidic oscillator to enter the boundary layer, effectively suppressing flow separation.

At a phase of 3/10, as the main jet moves away from the section, separation vortices begin to reappear and grow in size. By the phase of 7/10, the main jet has completely moved out of the section, and the separation vortex has fully developed, covering the entire surface of the trailing-edge flap. At 9/10 of the cycle, the main jet begins to return to the section, and part of the jet re-enters the section. This reintroduction of energy into the boundary layer suppresses flow separation and reduces the size of the separation vortex. When the main jet fully enters the section, the flow field resembles that of the 1/10 phase, initiating another periodic cycle of vortex evolution. This periodic behavior explains why the frequency spectrum of the monitoring points aligns with the oscillating frequency of the fluidic oscillator. Notably, at the 9/10 phase, unlike other phases, the shear stress generated by the turbulent component is not negligible. This indicates that, at low frequencies, turbulent components can play a significant role during certain phases, though periodic fluctuations remain the dominant factor throughout.

As shown in Figure 21, the periodic shear stress flow field under F⁺ = 1 demonstrates that periodic fluctuations dominate the shear stress, with no noticeable contribution from turbulent separation to the shear stress, unlike the case under F⁺ = 0.5. This confirms that at this frequency, periodic fluctuations play a dominant role in shear stress, while the impact of turbulent components can be neglected.

Additionally, similar to the flow field under F⁺ = 0.5, the variation of the separation vortex near the trailing-edge flap is closely linked to the sweeping process of the fluidic oscillator. This explains why the frequency spectrum aligns with the oscillation frequency. However, due to the shorter sweeping period and higher frequency at F⁺ = 1, the maximum size of the separation vortex when the oscillating jet moves away from the section is smaller compared to F⁺ = 0.5. This is reflected in the lift coefficient, where the minimum value of its periodic fluctuation is higher, resulting in a greater lift coefficient increment under this frequency.

However, higher frequency does not always lead to better performance. As shown in Figure 22, the periodic shear stress flow field under F⁺ = 10 reveals that the separation vortex does not exhibit periodic variation with the frequency. This is because the short sweeping period at high frequencies prevents the separation vortex from undergoing significant changes before completing a cycle. As a result, the frequency of the separation vortex does not align with the oscillation frequency of the fluidic oscillator.

This discrepancy is evident in the flow fields controlled by fluidic oscillators at F⁺ = 5 and F⁺ = 10, where the frequency spectrum far from the oscillator does not match the oscillation frequency. This also explains why the chordwise momentum exchange of shear stress and turbulent kinetic energy is less frequent, which may contribute to the lower lift coefficient increment observed under high-frequency control.

5. Conclusions

In this study, to investigate the influence of fluidic oscillator frequency on the lift enhancement of high-lift wings and to explore the working mechanism of fluidic oscillators, a type of oscillator capable of decoupling frequency and momentum coefficient was selected for analysis and applied to a high-lift wing. The findings are as follows:

1.

Effect of Frequency on Lift Coefficient:

By studying the variation of lift coefficient for high-lift wings under different frequencies and momentum coefficients, it was observed that at low momentum coefficients, the impact of frequency differences on lift enhancement was not significant. However, F⁺ = 10 consistently exhibited the poorest lift enhancement, while F⁺ = 0.5, 1, and 2 produced comparable effects. As the momentum coefficient increased to 14.1%, a notable difference in lift enhancement emerged under various frequency controls. At this momentum coefficient, F⁺ = 1 achieved the best lift enhancement, increasing the lift coefficient by more than 0.3 compared to F⁺ = 10. When the momentum coefficient was further raised to 21.88%, although the lift enhancement effect under different frequencies diminished, F⁺ = 1 still demonstrated the optimal lift improvement. This confirms that there exists an optimal frequency for fluidic oscillators to achieve maximum lift enhancement.

2.

Analysis of Flow Fields under Different Frequencies:

Investigations into the flow fields controlled by oscillators of different frequencies revealed that in terms of turbulent kinetic energy, low-frequency control led to a noticeable increase in turbulent kinetic energy, with the region of increased energy diminishing as the frequency rose. The enhanced turbulent kinetic energy significantly intensified energy transfer at the trailing-edge flap, improving the energy distribution in the boundary layer and further suppressing flow separation.

3.

Shear Stress Field Analysis:

Analyzing the shear stress fields of high-lift wings under different frequency controls, the shear stress was decomposed into periodic fluctuations and turbulent components. It was found that periodic fluctuations dominated the shear stress field under fluidic oscillator control, while the influence of turbulent components was negligible.

4.

Periodic Analysis of Shear Stress Fields:

In low-frequency cases, the size of separation vortices varied periodically according to the oscillation frequency of the fluidic oscillator, which explains why the flow field frequency matched the oscillation frequency. However, at F⁺ = 0.5, the extended period allowed the separation vortices to grow larger, which could explain the relatively lower lift coefficient. In contrast, under high-frequency control, the size of separation vortices in the flow field exhibited minimal variation, leading to a mismatch between the flow field frequency and the oscillation frequency. This suppressed energy transfer along the chordwise direction of the wing, likely contributing to the reduced lift enhancement effect of high-frequency oscillators.

In summary, this study demonstrates that there exists an optimal frequency for fluidic oscillators to achieve superior lift enhancement, and the working mechanism is closely linked to the interplay of turbulent kinetic energy and periodic fluctuations in the shear stress field.