# Trailing Edge Flap Effects on Dynamic Stall Vortex and Unsteady Aerodynamic Forces on a Pitching Airfoil

Authors: Shi-Long Xing; He-Yong Xu; Wei-Guo Zhang
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

This study focuses on the effect of the upward deflection of trailing edge flap (TEF) on the strength and trajectory of dynamic stall vortex (DSV) around a pitching airfoil by means of numerical simulations based on unsteady Reynolds-averaged Navier-Stokes (URANS). The effect of the upward deflection of the TEF on the unsteady aerodynamic forces due to DSV is analyzed. The numerical simulation method for large mesh deformation is constructed. Radial basis function- (RBF-) based mesh deformation algorithm, as well as Laplacian and optimization-based mesh smoothing algorithm, is adopted to ensure the mesh quality in flow field simulations. The results reveal that the upward deflection of the TEF can reduce the peaks of drag and pitching moment coefficients. Although the maximum lift coefficient of the airfoil is slightly reduced, its maximum drag and pitching moment coefficients are significantly reduced by up to 34.8% and 31.8%, respectively. The vorticity transport behavior in a planar control region during the DSV formation and detachment is analyzed. It is found that the TEF can change the development process of the DSV. The upward deflection of the TEF reduces the vorticity flux from the leading edge shear layer, which causes the circulation of the DSV and the translational velocity of the vortex center to decline. The peaks of the unsteady aerodynamic forces on the airfoil induced by the DSV are reduced. The upward deflection of the TEF plays the role of alleviating the pitching moment load. The longer TEF can result in a better control effect. The bigger the upward deflection angle of the TEF, the better its control effect.


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## Body

## 1. Introduction

Dynamic stall refers to a kind of flow phenomenon in which the stall is delayed beyond the static stall angle of attack when an airfoil is experiencing unsteady motion [1]. The dominant feature characterizing dynamic stall is the formation and detachment of DSV when the airfoil angle of attack exceeds the static stall angle. The formation and convection of the DSV cause an overshoot of lift and a delay in stall. The moment stall occurs when the DSV moves rapidly towards the trailing edge, causing the pitching moment coefficient to increase sharply [2]. For a conventional helicopter in forward flight, the angle of attack of the retreating blade is higher than that of the advancing blade, maintaining the torque balance of the airframe. The angle of attack of the retreating blade could exceed the static stall angle of attack at a high advance ratio and large thrust coefficient, which results in the occurrence of dynamic stall. The occurrence of dynamic stall can lead to excessive blade pitching link loads and limit the flight envelope of the helicopters [3]. Kufeld et al. [4] analyzed the maneuver data obtained from the flight tests of a UH-60A under the NASA/ARMY Airloads Program and the relationship between these maneuvers and the UH-60A operational requirements. The flight test data displayed that the pull-up maneuver resulted in considerable pitch-link loads encountered in the flight test program. From an examination of the aerodynamic loads, it is apparent that these high loads were a consequence of the dynamic stall that occurred on the outboard portion of the retreating blade, and this dynamic stall behavior was intimately related to the control system and blade torsional flexibility. Due to these drawbacks brought about by the occurrence of dynamic stall, many researchers have come up with multiple solutions to this problem.Several control strategies have been proposed to mitigate or eliminate the negative effects of DSV. Martin et al. used the wind tunnel test and numerical simulation to investigate a variable droop leading edge (VDLE) airfoil. The wind tunnel test and numerical simulation results showed a dramatic decrease in the drag and pitching moment associated with severe dynamic stall when the VDLE concept is applied to the Boeing VR-12 airfoil. The results also showed an elimination of the negative pitch damping observed in the baseline moment hysteresis curves [5, 6]. Although the VDLE concept can reduce the drag and pitching moment, there are practical difficulties in helicopter engineering application because of the severe environment there. Le Pape et al. proposed a kind of active dynamic stall control based on deployable leading edge vortex generators [7]. The active device would be activated only during the retreating side for dynamic stall flight conditions so as to avoid drag penalties on the advancing blade. The wind tunnel test results validated the effectiveness of the devices to delay static stall and alleviate dynamic stall penalties. The static stall angle was delayed by 3 deg. The negative pitching moment peak was reduced by up to 60% for dynamic stall. Matalanis et al. investigated the dynamic-stall-suppression capabilities of combustion-powered actuation on a high lift rotorcraft airfoil (the VR-12, [8]). The flow mechanisms responsible for stall suppression were investigated using particle image velocimetry. The test results demonstrated that the combustion-powered actuators could improve the stall behavior of the airfoil at Mach numbers up to 0.4. The cycle-averaged lift coefficient during the dynamic stall process increased up to 11% at Mach 0.4. Xu et al. analyzed the effect of a coflow jet on the dynamic stall of a wind turbine airfoil [9]. The results displayed that the dynamic stall can be greatly suppressed. The relevant energy consumption analysis showed that the coflow jet concept controlled dynamic stall economically. Zhao et al. investigated numerically the effects of synthetic jet control on unsteady dynamic stall over rotor airfoil [10]. The numerical results displayed that the dual-jet could improve control efficiency more obviously on dynamic stall of rotor airfoil with respect to the unique jet. Visbal et al. proposed a high-frequency control concept for dynamic stall mitigation targeting the natural instabilities of laminar separation bubble (LSB) to delay the formation of DSV [11–13], which is worthy of further research. The TEF-based active control technology has a promising prospect for engineering applications [14]. Feszty et al. [15] analyzed the effect of the TEF on the mitigation of large negative pitching moment and negative aerodynamic damping caused by DSV. The authors thought that the trailing edge vortex was responsible for the large negative pitching moments and corresponding negative aerodynamic damping. Gerontakos and Lee [16] investigated the effect of the TEF on the dynamic loads of an oscillating airfoil. In the wind tunnel tests, the TEF deflections for different flap actuation start times, durations, and amplitudes were taken into consideration. The test results indicated that the TEF motion did not affect the formation and separation of the DSV. The deflection duration of the TEF affected the maximum lift of the airfoil. Krzysiak and Narkiewicz [17] investigated the unsteady aerodynamic loads on an airfoil with a TEF deflecting with different frequencies. The effect of phase delay between the airfoil angle of attack and the flap deflection at the beginning of the motion was considered. The TEF oscillations resulted in an increase of maximum lift coefficient when the angle of attack of the airfoil and the TEF deflection increased simultaneously. Raiola et al. [18] investigated a kind of airfoil unsteady loads control strategy utilizing a TEF for wind turbines. The wind tunnel test data revealed that the unsteady loads can be mitigated via a linear actuation of the TEF based on instantaneous aerodynamic load measurements. Samara and Johnson [19] concluded that the TEF had not controlled the formation of the DSV. Nevertheless, the TEF was capable of reducing the size of the DSV. It is recommended to dynamically pitch the TEF out of phase with the airfoil pitching to reduce the peaks of the lift and pitching moment coefficients and reduce the negative aerodynamic damping that can lead to stall flutter.In conclusion, these research findings have proved that the TEF is capable of reducing the peak of the pitching moment and alleviating the corresponding loads. However, there is a lack of research on which flow variables are changed by the upward deflection of the TEF and how these flow variables affect the unsteady aerodynamic forces of the airfoil. In the present work, the effect of the upward deflection of the TEF on the unsteady aerodynamic forces is discussed. The influence of the upward deflection of the TEF on the development process of the DSV, as well as the vorticity transport behavior, is analyzed.


## 2. Detail of the TEF

Figure1 displays a sketch illustrating the TEF degree of freedom in the current study. An airfoil is pitched about the quarter chord axis. c represents the airfoil chord length. The relative chord length of the TEF is 0.3c. The deflection angle δ is negative when the TEF deflects upwards. Figure 2 shows the geometry shapes of the airfoil at different deflection angles. In the numerical simulations, when the TEF deflects upwards, the trailing edge point of the airfoil rotates around the TEF rotation axis. Meanwhile, the corresponding upper and lower surfaces of the TEF are scaled. The gap between the airfoil and the TEF is ignored, which is convenient for mesh deformation processing.Figure 1
A sketch illustrating the TEF degree of freedom.Figure 2
Geometry shapes of the airfoil at different deflection angles.


## 3. Methodology

### 3.1. Numerical Simulation Method

ANSYS Fluent 15.0 flow simulation software is used to perform CFD simulations in the present work. The two-dimensional density-based solver is employed in the simulations. The one-equation Spalart-Allmaras turbulence model, well known as S-A, is applied for the closure problem. The pressure far-field boundary condition is employed along the far-field boundary. Along the airfoil surface, a no-slip adiabatic condition is employed. The finite-volume approach is employed to discretize the governing equations. The convection term is spatially discretized using the AUSM+-up scheme, and the Jameson second-order center scheme is used to calculate the viscous flux. In order to simulate the unsteady characteristics of dynamic stall, the dual-time method is used for physical time marching [20]. To accurately compute the unsteady aerodynamic forces of the airfoil in the pitch oscillation process, one pitching motion cycle is equally divided into 630 physical time steps, and the subiteration convergence criterion is that the residual value is less than 10−5.Boer et al. [21] firstly applied RBFs to the mesh deformation algorithm, widely used due to its strong mesh deformation capability. In the current research, the RBF-based mesh deformation algorithm is employed to calculate the mesh deformation. The basic principle of RBF interpolation-based mesh deformation algorithm is that RBFs interpolate the displacement of structural boundary nodes, and the boundary displacement effect is smoothly distributed to the whole computational domain by the constructed RBF sequence. The mesh deformation calculation consists of two main steps. Firstly, the weight coefficient equation of the airfoil surface node is solved according to the motion profile of the airfoil and the TEF, and then, the computational domain mesh is updated. The RBF-based mesh deformation algorithm shows very strong mesh deformability. However, this algorithm generates many poor-quality meshes in certain conditions, such as the TEF at a large deflection angle. Thus, besides the RBF-based mesh deformation algorithm, the dynamic mesh algorithm includes a combined Laplacian and optimization-based mesh smoothing algorithm developed by Canann et al. [22] to guarantee the quality of meshes under all conditions. The dynamic mesh calculation is achieved by the DEFINE_GRID_MOTION macro in ANSYS Fluent user-defined function (UDF).


### 3.2. Dynamic Stall Vortex and Vorticity Transport Analysis

To study the effect of the upward deflection of the TEF on the DSV evolution, it is necessary to introduce a quantitative analysis method to analyze the strength of the DSV. The circulation of the DSV is calculated as a line integral of velocity around the closed boundary of the chordwise planar control region within the flow domain:(1)Γ=∮Cv∙ds.In order to better understand the effect of the upward deflection of the TEF on the development process of the DSV, a vorticity transport analysis used in [23] is conducted in the current research. This analysis method utilizes the properties of the leading edge shear layer to represent the rate of change in DSV circulation. The planar control region is shown in Figure 3. In order to accurately calculate the vorticity flux, Boundary 1 of the planar control region is located at a distance of 0.05 times the chord length from the leading edge, avoiding the influence of the secondary vortex. Boundary 2 is located at a distance of 0.85 times the chord length from the airfoil chord line. The purpose of this setting is to make Boundary 2 locate in the inviscid region through which no vorticity is convected. The vorticity flux only leaves the planar control region through Boundary 3 when the DSV detaches from the airfoil, convenient for analyzing the vorticity flux. To calculate the circulation of the DSV and the secondary vortex more accurately within the planar control region, Boundary 3 is located at a distance of 0.9 times the chord length from the leading edge to avoid the effect of the vorticity of the trailing edge vortex. Boundary 4 is the airfoil surface between Boundary 1 and Boundary 3 at where the vorticity flux is generated primarily due to the streamwise pressure gradient [24].Figure 3
Planar control region used in the vorticity transport analysis.The rate of change of circulation about a closed curveC, bounding the planar control region fixed in a non-inertial coordinate system, is given by
(2)dΓdt=ddt∮Cv∙ds=∮C∂v∂t∙ds.For a fluid with constant properties, substitute the Navier-Stokes equation into Equation (2). In the present work, assuming that the acceleration term has no effect on the vorticity flux creation. Therefore, the acceleration-related integral disappears. The pressure integral vanishes since the effect of density change is ignored. The relevant integral of the streamwise pressure gradient is used to approximate the vorticity generated on the airfoil surface (Boundary 4) due to the viscous effects. So the vorticity flux for the planar control region can be expressed as
(3)dΓdt=−∮Cv∙nCωzds+1ρ∫Boundary4∂p∂xdx.To calculate the vorticity fluxes in Equation (3), the spatial integrals are computed using the trapezoidal rule. Equation (3) is integrated in time for the presentation of results in terms of circulation, and the temporal integration is executed using the trapezoidal rule as well. Besides the circulation of the DSV, the position of its vortex core is also very important to the unsteady aerodynamic forces exerted on the airfoil. The method based on a dimensionless scalar function Γ1 developed by Graftieaux et al. [25] is used to find the vortex center of the DSV in the present work.


## 3.1. Numerical Simulation Method

ANSYS Fluent 15.0 flow simulation software is used to perform CFD simulations in the present work. The two-dimensional density-based solver is employed in the simulations. The one-equation Spalart-Allmaras turbulence model, well known as S-A, is applied for the closure problem. The pressure far-field boundary condition is employed along the far-field boundary. Along the airfoil surface, a no-slip adiabatic condition is employed. The finite-volume approach is employed to discretize the governing equations. The convection term is spatially discretized using the AUSM+-up scheme, and the Jameson second-order center scheme is used to calculate the viscous flux. In order to simulate the unsteady characteristics of dynamic stall, the dual-time method is used for physical time marching [20]. To accurately compute the unsteady aerodynamic forces of the airfoil in the pitch oscillation process, one pitching motion cycle is equally divided into 630 physical time steps, and the subiteration convergence criterion is that the residual value is less than 10−5.Boer et al. [21] firstly applied RBFs to the mesh deformation algorithm, widely used due to its strong mesh deformation capability. In the current research, the RBF-based mesh deformation algorithm is employed to calculate the mesh deformation. The basic principle of RBF interpolation-based mesh deformation algorithm is that RBFs interpolate the displacement of structural boundary nodes, and the boundary displacement effect is smoothly distributed to the whole computational domain by the constructed RBF sequence. The mesh deformation calculation consists of two main steps. Firstly, the weight coefficient equation of the airfoil surface node is solved according to the motion profile of the airfoil and the TEF, and then, the computational domain mesh is updated. The RBF-based mesh deformation algorithm shows very strong mesh deformability. However, this algorithm generates many poor-quality meshes in certain conditions, such as the TEF at a large deflection angle. Thus, besides the RBF-based mesh deformation algorithm, the dynamic mesh algorithm includes a combined Laplacian and optimization-based mesh smoothing algorithm developed by Canann et al. [22] to guarantee the quality of meshes under all conditions. The dynamic mesh calculation is achieved by the DEFINE_GRID_MOTION macro in ANSYS Fluent user-defined function (UDF).


## 3.2. Dynamic Stall Vortex and Vorticity Transport Analysis

To study the effect of the upward deflection of the TEF on the DSV evolution, it is necessary to introduce a quantitative analysis method to analyze the strength of the DSV. The circulation of the DSV is calculated as a line integral of velocity around the closed boundary of the chordwise planar control region within the flow domain:(1)Γ=∮Cv∙ds.In order to better understand the effect of the upward deflection of the TEF on the development process of the DSV, a vorticity transport analysis used in [23] is conducted in the current research. This analysis method utilizes the properties of the leading edge shear layer to represent the rate of change in DSV circulation. The planar control region is shown in Figure 3. In order to accurately calculate the vorticity flux, Boundary 1 of the planar control region is located at a distance of 0.05 times the chord length from the leading edge, avoiding the influence of the secondary vortex. Boundary 2 is located at a distance of 0.85 times the chord length from the airfoil chord line. The purpose of this setting is to make Boundary 2 locate in the inviscid region through which no vorticity is convected. The vorticity flux only leaves the planar control region through Boundary 3 when the DSV detaches from the airfoil, convenient for analyzing the vorticity flux. To calculate the circulation of the DSV and the secondary vortex more accurately within the planar control region, Boundary 3 is located at a distance of 0.9 times the chord length from the leading edge to avoid the effect of the vorticity of the trailing edge vortex. Boundary 4 is the airfoil surface between Boundary 1 and Boundary 3 at where the vorticity flux is generated primarily due to the streamwise pressure gradient [24].Figure 3
Planar control region used in the vorticity transport analysis.The rate of change of circulation about a closed curveC, bounding the planar control region fixed in a non-inertial coordinate system, is given by
(2)dΓdt=ddt∮Cv∙ds=∮C∂v∂t∙ds.For a fluid with constant properties, substitute the Navier-Stokes equation into Equation (2). In the present work, assuming that the acceleration term has no effect on the vorticity flux creation. Therefore, the acceleration-related integral disappears. The pressure integral vanishes since the effect of density change is ignored. The relevant integral of the streamwise pressure gradient is used to approximate the vorticity generated on the airfoil surface (Boundary 4) due to the viscous effects. So the vorticity flux for the planar control region can be expressed as
(3)dΓdt=−∮Cv∙nCωzds+1ρ∫Boundary4∂p∂xdx.To calculate the vorticity fluxes in Equation (3), the spatial integrals are computed using the trapezoidal rule. Equation (3) is integrated in time for the presentation of results in terms of circulation, and the temporal integration is executed using the trapezoidal rule as well. Besides the circulation of the DSV, the position of its vortex core is also very important to the unsteady aerodynamic forces exerted on the airfoil. The method based on a dimensionless scalar function Γ1 developed by Graftieaux et al. [25] is used to find the vortex center of the DSV in the present work.


## 4. Results and Discussions

### 4.1. Verification of Numerical Method and Mesh Convergence

The dynamic stall characteristics of an airfoil are computed in order to verify the ability of the numerical method and analyze the mesh convergence in this section. For modern helicopters, the tip Mach number is normally about 0.7. When the helicopter is at a high-speed forward flight, the Mach numbers for a helicopter airfoil are from near 0 (to account for reverse flow) to about 1. For dynamic stall of helicopters, the representative Mach number at the retreating blade is about 0.3 [2]. The representative freestream condition is chosen in the present work so that analyze the effect of the TEF on dynamic stall process. In the present work, the SC1095 airfoil is employed. The airfoil is considered at a freestream Mach number of 0.302 and chord-based Reynolds number of 3.89×106. The airfoil is pitched about its quarter-chord axis according to the following expression:
(4)αt=α0+αm∙sinωt−π2,where α0 and αm are the mean angle of 9.92 deg and the pitch oscillation amplitude of 9.90 deg, respectively [26]. For airfoils in pitching oscillation, the reduced frequency κ is usually used to describe the unsteady motion, which is defined as
(5)κ=ωc2V∞,whereV∞ is the freestream velocity. The reduced frequency κ=0.148 is employed.Appropriate mesh scale is essential in capturing the right flow physics, especially when it involves unsteady aerodynamic phenomenon. To assess the effects of spatial resolution, three sets of hybrid unstructured meshes with different mesh densities are employed in the numerical simulations to characterize the dynamic stall. Some details of these meshes, denoted as coarse mesh to refined mesh, are summarized in Table1. Along the airfoil surface, a no-slip adiabatic condition is employed. The airfoil trailing edge is rounded with a very small circular arc in order to facilitate the use of quadrilateral meshes. The airfoil surface mesh nodes in these three sets of meshes are sequentially encrypted. Mesh refinement in the streamwise direction is concentrated on the airfoil upper surface in the cause of more effectively capturing the unsteady boundary layer separation and DSV formation. The expected y+ is less than 1. Along the far-field boundary, located 100 chords away from the airfoil, the pressure far-field boundary condition is employed. The total number of mesh cells in these three sets of meshes is approximately 22,500, 66,500, and 131,000, respectively. Figure 4 shows the distribution of the medium mesh.Table 1
Computational mesh parameters.
ParametersCoarse meshMedium meshRefined meshNodes on the airfoil280550820First layer spacing1.0×10−6c1.0×10−6c0.5×10−6cIncreasing ratio1.121.101.08Far-field radius100c100c100cTotal number of cells2.25×1046.65×10413.1×104Figure 4
Medium mesh used in the present study.
(a)(b)The comparison between the calculated and test values of the airfoil lift, drag, and pitching moment coefficients is given in Figures5, 6 and 7. The test data is taken from the experimental study of dynamic stall on advanced airfoil sections made by McCroskey et al. [26]. Good agreement is achieved between the solutions on the three sets of meshes. The differences between the lift, drag and moment coefficients for the three sets of meshes are slight. Figure 5 shows that the max lift coefficient of the numerical simulation is lower than that of the wind tunnel test. The hysteresis loop profile of the lift coefficient of the numerical simulation is basically the same as that of the wind tunnel test. The max drag coefficient of the numerical simulation is lower than that of the test. The peak of the pitching moment coefficient of the numerical simulation is smaller than that of the wind tunnel test. But the moment stall is accurately predicted, meaning that the movement process of DSV is accurately predicted. The differences between the test values and the numerical results are acceptable. It is evident that the numerical simulation method used can effectively simulate the delay phenomenon of the airfoil aerodynamic forces during the dynamic stall. The medium-scale mesh is sufficient to ensure calculation accuracy. Unless noted otherwise, the medium-scale mesh is employed to describe the physical aspects of the dynamic stall.Figure 5
Comparison of lift coefficients between the numerical and test data.Figure 6
Comparison of drag coefficients between the numerical and test data.Figure 7
Comparison of pitching moment coefficients between the numerical and test data.


### 4.2. Effects of the Upward Deflection of the TEF on the Dynamic Stall Process

Figure8 shows the motion profiles of the airfoil and the TEF. The sinusoidal movement of the TEF in deflection motions is shown in Figure 8. T represents the duration time of one pitching cycle. t/T denotes the dimensionless time during one pitching cycle. As shown in the figure, the TEF starts to deflect upwards when t/T is equal to 0.4 (i.e. the angle of attack is 17.96 deg). When t/T is equal to 0.5, namely, the maximum angle of attack, the deflection angle of the TEF reaches the minimum of -20 deg. That is, the TEF deflects upwards by 20 deg. The geometry shapes of the original and TEF airfoils at different t/T is shown in Table 2.Figure 8
Motion profiles of the airfoil and the TEF.Table 2
Geometry shapes of the original and TEF airfoils at differentt/T.
t/TAoAδOriginal airfoilTEF airfoil0.00.02°00.417.96°00.4519.35°-100.519.82°-200.5519.35°-100.617.96°01.00.02°0Figures9, 10 and 11 display the comparison of lift, drag, and pitching moment coefficients between the original and TEF airfoils. Unless noted otherwise in the following sections, the ‘original’ represents the original airfoil, and the ‘TEF’ represents the airfoil with the TEF. The numerical results reveal that the TEF has the ability to reduce the peaks of the drag and pitching moment coefficients. The maximum drag coefficient of the TEF airfoil is 34.8% less than that of the original airfoil. The peak of the pitching moment coefficient of the TEF airfoil is greatly reduced by up to 31.8%, compared with that of the original airfoil. Through the comparison of the pitching moment coefficient curves (Figure 11), it is found that the area of the clockwise loop in the pitching moment curve for the TEF airfoil is smaller than that of the original airfoil. That means that the torsional aerodynamic damping of the TEF airfoil is increased due to the upward deflection of the TEF, and the possibility of stall flutter occurrence is reduced. But one problem appears. The TEF airfoil enters the stall state early because of the upward deflection of the TEF, and its maximum lift coefficient is reduced by 15.4%. The lift augmentation effect caused by DSV is also reduced. The application of the TEF brings the benefit of alleviating the pitching moment load and also the disadvantage that the maximum lift coefficient of the airfoil is reduced. Consequently, it is necessary for practical applications to compromise between the maximum lift coefficient and the pitching moment load alleviation by adjusting certain design parameters to achieve the optimal design solution.Figure 9
Comparison of lift coefficients between the original and TEF airfoils.Figure 10
Comparison of drag coefficients between the original and TEF airfoils.Figure 11
Comparison of pitching moment coefficients between the original and TEF airfoils.In Figures9, 10 and 11, the x-axis variable is the angle of attack that changes nonlinearly over time in a sinusoidal pitching motion. This type of figure is convenient to analyze the relationship between the angle of attack and the aerodynamic force coefficients and the relevant torsional aerodynamic damping. It is difficult to intuitively analyze the development process of unsteady aerodynamic forces during the pitching cycle in the time domain. As shown in Figure 12, the unsteady aerodynamic force coefficient curves with t/T as the x-axis variable is introduced to make the analysis more intuitive. Figure 12(a) shows the comparison of the lift and drag coefficients between the original and TEF airfoils in the time domain, and Figure 12(b) displays the corresponding pitching moment coefficient curves. The pink region between t/T equals 0.4 and 0.5 represents the period in which the TEF deflects upwards from 0 deg to -20 deg. Similarly, the gray region between t/T equals 0.5 and 0.6 represents the period in which the TEF deflects downwards from -20 deg to 0 deg. Figure 13 shows the spatiotemporal distributions of the pressure coefficients on the upper surface of the two airfoils. In this plot, the abscissa represents the chordwise location, and the ordinate corresponds to the dimensionless time during one pitching cycle.Figure 12
Comparison of aerodynamic force coefficients between the original and TEF airfoils in the time domain: (a) lift and drag coefficients and (b) pitching moment coefficients. The pink region represents that the TEF deflects upwards. The gray region represents that the TEF deflects downwards.
(a)(b)Figure 13
Spatiotemporal contours of pressure coefficients on the airfoil upper surfaces: (a) original airfoil and (b) TEF airfoil.
(a)(b)Whent/T is less than 0.3, i.e., the angle of attack is less than 13.0 deg, the lift coefficients of the two airfoils increase gradually with the angle of attack. Meanwhile, the pitching moment coefficients change slightly. The corresponding flow fields remain in the attached state. When t/T is 0.34, i.e., the angle of attack reaches 15.27 deg, and the peaks of pressure coefficient of the airfoils exceed -10 due to the relative higher angle of attack, which brings about the formation of LSB in the vicinity of the airfoil leading edge (Figure 14). As a result of the appearance of the LSB, the skin friction coefficients at part of the upper surface of the two airfoils are negative values at the angle of attack of 15.27 deg. As the airfoil pitches up, the flow acceleration around the leading edge increases, causing a higher peak suction as well as a stronger adverse pressure gradient across the LSB. As shown in Figure 13, the area of the low-pressure region on the upper surface that the pressure coefficients are less than -4 continues to enlarge as the angle of attack increases. The lift coefficients also increase accordingly. With a further increase in the angle of attack, the LSB becomes bigger and moves towards the trailing edge. When the angle of attack is 17.65 deg, the bursting of the LSB occurs due to the increasing adverse pressure gradient (Figure 13). Following the bursting of the LSB, the DSV emerges and moves towards the trailing edge. The moment stall occurs, as shown in Figure 12(b).Figure 14
Comparison of coefficient distributions at the angle of attack of 15.27 deg upstroke: (a) pressure coefficients and (b) skin friction coefficients on the upper surface.
(a)(b)Whent/T equals 0.4, i.e., the angle of attack is 17.96 deg, and the TEF starts to deflect upwards. From now on, there is a clear difference between the aerodynamic force curves of the original and TEF airfoils. The lift coefficient of the original airfoil continues to increase before the maximum angle of attack. The pitching moment coefficient of the original airfoil increases rapidly due to the DSV moving towards the trailing edge. The corresponding coefficient distribution curves in the process are shown in Figures 15 and 16. In contrast to the original airfoil, the TEF airfoil at the angle of attack of 18.07 deg, i.e., t/T equals 0.404, comes up to its maximum lift coefficient, which is 15.4% smaller than that of the original airfoil. The stall angle of attack is reduced from 19.82 deg to 18.07 deg, a drop of 1.75 deg. When t/T goes from 0.404 to 0.45, the reason why the lift coefficient of the TEF airfoil drops so quickly is that the pressure on the lower surface of the TEF is lower than that of the upper surface of the TEF, which results in a negative lift. This is caused by the rapid upward deflection of the TEF. The representative pressure coefficient curve is shown in Figure 16(a). Whereafter, the lift coefficient of the TEF airfoil increases slightly, which is attributed to the lower pressure on the TEF upper surface induced by the stronger DSV. The DSV detaches from the airfoil when the airfoil starts to pitch down. The trailing edge vortex is induced by the detachment of the DSV, as shown in Figures 13 and 16(c) and 16(d). After the DSV detaches, the airfoil is stalled until the reattachment occurs when the angle of attack is much lower than the static stall point. When the angle of attack is down to 6.81 deg, the flow on the airfoil surface is restored to the attached state. During the pitching motion process, the upward deflection of the TEF reduces the pressure difference between the upper and lower surfaces of the airfoil induced by the DSV, causing the peak of the pitching moment coefficient to decline. Thus, the TEF has the function of alleviating the pitching moment load.Figure 15
Comparison of coefficient distributions at the angle of attack of 18.07 deg upstroke: (a) pressure coefficients and (b) skin friction coefficients on the upper surface.
(a)(b)Figure 16
Comparison of pressure coefficient distributions on the airfoil surface at different angles of attack.
(a)(b)(c)(d)


### 4.3. Analysis of the Circulation and Vortex Center Trajectory of the DSV

In the previous section, the effect of the upward deflection of the TEF on the unsteady aerodynamic forces of the airfoils in the dynamic stall process has been discussed. The relevant results reveal that the peak of the pitching moment coefficient is reduced markedly by the upward deflection of the TEF, alleviating the pitching moment load. This section describes the event in further detail, emphasizing the evolution of the DSV that results in the overshoot of the lift and the delay in stall. How the upward deflection of the TEF affects the strength and motion trail of the DSV is discussed.Figure17 shows the normalized circulation of the DSV and secondary vortex against dimensionless time in the planar control region. Figure 18 shows that the normalized vorticity contours at different times during one pitching cycle for the two airfoils. In these figures, the solid white lines represent the contour line that the normalized vorticity is -10, used to compare the size of the DSV qualitatively. When the airfoils are at a relatively low angle of attack and the deflection angle of the TEF is zero, the boundary layer thickness increases with the angle of attack. More vorticity is generated near the airfoil due to viscous effects. So the circulation in the planar control region increases slowly with the angle of attack. When the angle of attack is 15.27 deg, the LSB appears on the upper surface of the airfoils, as shown in Figure 14(b). The scaling of the LSB continues to enlarge, and the circulation in the planar control region augments rapidly. As shown in Figure 13, when the angle of attack is 17.65 deg, the LSB bursting occurs, resulting in the formation of the DSV. The DSV moves towards the trailing edge of the airfoil, and the circulation in the planar control region increases sharply. As the airfoils pitch up, the DSV fed by the leading edge shear layer continues to grow in both size and strength. When t/T is 0.4, i.e., the angle of attack is 17.96 deg, and the TEF starts to deflect upwards. When t/T is 0.45, i.e., the angle of attack is 19.35 deg, and the deflection angle is -10.0 deg; the circulation in the planar control region of the TEF airfoil is reduced by the upward deflection of the TEF, compared with that of the original airfoil. The upward deflection of the TEF can reduce the circulation in the planar control region, namely, can reduce the strength of the DSV. As the airfoils pitch up, the strength of the DSV continues to enlarge. The growth of the circulation of the TEF airfoil is relatively slower than that of the original airfoil. When t/T is 0.5, i.e., the angle of attack is maximum, and the deflection angle is -20.0 deg, the leading edge shear layer and the DSV is weakly linked for the original airfoil (Figure 18(c)). In contrast to the original airfoil, the connection between the leading edge shear layer and the DSV is weaker for the TEF airfoil (Figure 18(d)). When the connection between the leading edge shear layer and the DSV is weak, it means that the DSV no longer accumulates vorticity from the leading edge shear layer [27]. Due to the lack of vorticity supply from the leading edge shear layer, the strength of the DSV decreases rapidly. The DSV detaches from the leading edge shear layer and passes the trailing edge (Figures 18(e) and 18(f)). The circulation in the planar control region drops rapidly.Figure 17
Comparison of the normalized circulation in the planar control region during one pitching cycle for the two different airfoils. The pink region represents that the TEF deflects upwards. The gray region represents that the TEF deflects downwards.Figure 18
Normalized vorticity contours at different times during one pitching cycle for the two airfoils: (a, c, e) original airfoil and (b, d, f) TEF airfoil. The solid white line represents the contour line that the normalized vorticity is -10.In addition to the strength of the DSV, the distance between the vortex core and the airfoil is also one of the factors that determine the pressure on the airfoil surface. In the present study, the position of the vortex center is used to represent the relative location between the vortex core and the upper surface of the airfoil. Figure19 shows the vortex center trajectories over the dimensionless time interval 0.4<t/T<0.55 for the original and TEF airfoils. The vortex center trajectories of the two airfoils are different. The upward deflection of the TEF has a significant influence on the DSV evolution. The vortex center trajectory of the TEF airfoil is nearly the same as that of the original airfoil before t/T is equal to 0.45, except for the slower translational velocity of the vortex center. When t/T is greater than 0.45, the deflection angle of the TEF is always smaller than -10 deg, which brings the bigger negative camber of the airfoil. Figures 20(a) and 20(b) show the streamlines and Mach number contours at the maximum angle of attack for the two airfoils. The position of the vortex center for the two airfoils is different. It is proved that the vortex center of the DSV is farther away from the airfoil upper surface due to the upward deflection of the TEF. Thus, the pressure difference between the lower and upper surfaces induced by the DSV decreases. When t/Tequals 0.55, the streamlines and Mach number contours of the two airfoils are shown in Figures 20(c) and 20(d). It is found that the vortex center of the original airfoil continues to move towards the trailing edge, and the vortex center of the TEF airfoil moves towards the leading edge, which is consistent with the corresponding curve in Figure 19. In the pitching motion process, the upward deflection of the TEF not only reduces the circulation of the DSV but also makes the vortex center farther from the upper surface of the airfoil, resulting in a smaller pressure difference between the upper and lower surfaces.Figure 19
Vortex center trajectories shown in the airfoil coordinate system over the dimensionless time interval0.4<t/T<0.55 for the two airfoils.Figure 20
Streamlines and Mach number contours at different angles of attack for the two airfoils: (a, c) original airfoil and (b, d) TEF airfoil.


### 4.4. Vorticity Transport Analysis in the Planar Control Region

In this section, the terms of Equation (3) are evaluated for the planar control region shown in Figure 3 so as to understand the relative importance of each vorticity transport mechanism in governing the strength and size of the DSV. A comparison of the integrated vorticity flux of each boundary during the deflection duration of the TEF for the original and TEF airfoils is shown in Figure 21. Because the vorticity in the leading edge shear layer, i.e., the feeding shear layer, is the primary contributor to DSV growth [28], we focus on analyzing the vorticity flux through Boundary 1. As t/T increases from 0.4 to 0.5, the integrated vorticity fluxes on Boundary 1 for the two airfoils continue to increase. The maximum slope (maximum shear layer flux) occurs at t/T equals 0.4. The integrated vorticity flux on Boundary 1 of the TEF airfoil is smaller than that of the original airfoil due to the upward deflection of the TEF during the pitchup process. Figures 22 and 23 show the streamlines and Mach number contours around the leading edge at t/T is 0.45 and 0.5 for the two airfoils. In these figures, the solid white lines represent the contour line that the normalized vorticity is -10, same as Figure 18, used to compare the thickness of the feeding shear layer qualitatively. When t/T equals 0.45, the thickness of the feeding shear layer for the TEF airfoil is thinner, and the flow velocity in the feeding shear layer is slower. Likewise, when t/T equals 0.5, i.e., the angle of attack is the maximum, the vorticity flux from the feeding shear layer of the TEF airfoil is smaller than that of the original airfoil, as shown in Figure 23. It is proved that the upward deflection of the TEF reduces the vorticity flux from the feeding shear layer, which causes the circulation of the DSV to decline. In addition, Figure 21 displays that the positive secondary vorticity through Boundary 4 due to the streamwise pressure gradient effects is approximately half of that through Boundary 1 from the feeding shear layer flux during the pitchup motion. The magnitude of the net circulation in the planar control region is significantly reduced by the positive secondary vorticity generated due to the presence of the DSV itself. So the magnitude of the DSV circulation is consistently higher than that of the planar control region due to the positive secondary vorticity contained within the control region, as previously demonstrated for laminar flows [23] and turbulent flows [12].Figure 21
Comparison of the negative integrated vorticity fluxes during the deflection duration of the TEF for the original and TEF airfoils.Figure 22
Streamlines and Mach number contours around the leading edge att/T=0.45 for the two airfoils: (a) original airfoil and (b) TEF airfoil. The solid white line represents the contour line that the normalized vorticity is -10.Figure 23
Streamlines and Mach number contours around the leading edge att/T=0.5 for the two airfoils: (a) original airfoil and (b) TEF airfoil. The solid white line represents the contour line that the normalized vorticity is -10.


### 4.5. Effect of the Relative Chord Length of the TEF

Figure24 shows the airfoil profiles at minimum deflection angle for different relative chord lengths of the TEF. 0.3c represents that the relative chord length of the TEF airfoil is 0.3 times the airfoil chord length and so on. The bigger relative chord length of the TEF causes the bigger negative camber of airfoil at the same deflection angle. Figures 25(a)–25(c) illustrate the comparison of aerodynamic force coefficients at different relative chord lengths. As shown in these figures, the longer the relative chord length of the TEF, the smaller the maximum lift coefficient of the airfoil. The longer the relative chord length of the TEF, the smaller the absolute value of the peaks of the drag and pitching moment coefficients. It proves that the longer TEF has a stronger ability to alleviate the pitching moment load. For wind tunnel tests or engineering applications, an actuator is needed to make the TEF deflected. The required power of the actuator is partly determined by the hinge moment of the TEF. Thus, the hinge moment of the TEF during the deflection motion process needs to be analyzed. Figure 25(d) shows the hinge moment coefficient curves over time for different relative chord lengths. The ‘Original’ represents the hinge moment generated on the original airfoil surface corresponding to the TEF with 0.3 times chord length, equivalent to the deflection angle always being zero during the pitching motion process. It is found that the peak of the hinge moment coefficient of the original airfoil is bigger than that of the TEF airfoil with 0.3c. The reason is the upward deflection of the TEF makes the circulation of the DSV smaller (Figure 17), and the vortex center of the DSV farther away from the trailing edge of the airfoil (Figure 19), causing the pressure difference between the upper and lower surfaces to decrease. The peak of the hinge moment coefficient of the TEF airfoil is smaller than that of the original airfoil. But when t/T is equal to 0.527, i.e., the angle of attack is 19.68 deg, and the hinge moment coefficient of the original airfoil is smaller than that of the TEF airfoil with 0.3c. The reason for this situation is that a trailing edge separation vortex appears at the trailing edge. The emergence of the trailing edge separation vortex causes the bigger pressure difference as well as the hinge moment, as shown in Figures 13 and 16(d). The shorter the relative chord length of the TEF, the smaller the hinge moment coefficient, meaning that the required power of the actuator is lower. Conversely, the longer the relative chord length of the TEF, the better the effect of the pitching moment load alleviation. For practical applications, it is necessary to compromise between the better load alleviation and the lower required power to obtain the optimal design solution.Figure 24
Schematic of airfoil shapes for different relative chord lengths of the TEF at the deflection angle of -20 deg.Figure 25
Comparison of aerodynamic force coefficients for different relative chord lengths of the TEF: (a) lift coefficients, (b) drag coefficients, (c) pitching moment coefficients, and (d) hinge moment coefficients.
(a)(b)(c)(d)


### 4.6. Effect of the Minimum Deflection Angle of the TEF

Figure26 shows that the airfoil profiles at different minimum deflection angles. The relative chord length of these TEFs is 0.3c. -20° represents that the minimum deflection angle of the TEF airfoil is -20 deg and so on. As shown in Figure 27, the upward deflection of the TEF can reduce the peak of the drag and pitching moment coefficients, which is consistent with the conclusions in previous sections. The upward deflection of the TEF causes the negative camber of the airfoil to increase. The bigger the upward deflection angle of the TEF, the better the effect of the pitching moment load alleviation. In Figure 27(d), the legend curve ‘Original’ represents the hinge moment generated on the original airfoil surface corresponding to the TEF with 0.3 times chord length, equivalent to the deflection angle always being zero during the pitching motion process. The larger the upward deflection angle of the TEF, the smaller the corresponding hinge moment. However, compared with the change of the relative chord length of the TEF, the change of the upward deflection angle has less effect on the peak of the hinge moment of the TEF.Figure 26
Schematic of airfoil shapes at different minimum deflection angles of the TEF.Figure 27
Comparison of aerodynamic force coefficients at different minimum deflection angles of the TEF: (a) lift coefficients, (b) drag coefficients, (c) pitching moment coefficients, and (d) hinge moment coefficients.
(a)(b)(c)(d)


## 4.1. Verification of Numerical Method and Mesh Convergence

The dynamic stall characteristics of an airfoil are computed in order to verify the ability of the numerical method and analyze the mesh convergence in this section. For modern helicopters, the tip Mach number is normally about 0.7. When the helicopter is at a high-speed forward flight, the Mach numbers for a helicopter airfoil are from near 0 (to account for reverse flow) to about 1. For dynamic stall of helicopters, the representative Mach number at the retreating blade is about 0.3 [2]. The representative freestream condition is chosen in the present work so that analyze the effect of the TEF on dynamic stall process. In the present work, the SC1095 airfoil is employed. The airfoil is considered at a freestream Mach number of 0.302 and chord-based Reynolds number of 3.89×106. The airfoil is pitched about its quarter-chord axis according to the following expression:
(4)αt=α0+αm∙sinωt−π2,where α0 and αm are the mean angle of 9.92 deg and the pitch oscillation amplitude of 9.90 deg, respectively [26]. For airfoils in pitching oscillation, the reduced frequency κ is usually used to describe the unsteady motion, which is defined as
(5)κ=ωc2V∞,whereV∞ is the freestream velocity. The reduced frequency κ=0.148 is employed.Appropriate mesh scale is essential in capturing the right flow physics, especially when it involves unsteady aerodynamic phenomenon. To assess the effects of spatial resolution, three sets of hybrid unstructured meshes with different mesh densities are employed in the numerical simulations to characterize the dynamic stall. Some details of these meshes, denoted as coarse mesh to refined mesh, are summarized in Table1. Along the airfoil surface, a no-slip adiabatic condition is employed. The airfoil trailing edge is rounded with a very small circular arc in order to facilitate the use of quadrilateral meshes. The airfoil surface mesh nodes in these three sets of meshes are sequentially encrypted. Mesh refinement in the streamwise direction is concentrated on the airfoil upper surface in the cause of more effectively capturing the unsteady boundary layer separation and DSV formation. The expected y+ is less than 1. Along the far-field boundary, located 100 chords away from the airfoil, the pressure far-field boundary condition is employed. The total number of mesh cells in these three sets of meshes is approximately 22,500, 66,500, and 131,000, respectively. Figure 4 shows the distribution of the medium mesh.Table 1
Computational mesh parameters.
ParametersCoarse meshMedium meshRefined meshNodes on the airfoil280550820First layer spacing1.0×10−6c1.0×10−6c0.5×10−6cIncreasing ratio1.121.101.08Far-field radius100c100c100cTotal number of cells2.25×1046.65×10413.1×104Figure 4
Medium mesh used in the present study.
(a)(b)The comparison between the calculated and test values of the airfoil lift, drag, and pitching moment coefficients is given in Figures5, 6 and 7. The test data is taken from the experimental study of dynamic stall on advanced airfoil sections made by McCroskey et al. [26]. Good agreement is achieved between the solutions on the three sets of meshes. The differences between the lift, drag and moment coefficients for the three sets of meshes are slight. Figure 5 shows that the max lift coefficient of the numerical simulation is lower than that of the wind tunnel test. The hysteresis loop profile of the lift coefficient of the numerical simulation is basically the same as that of the wind tunnel test. The max drag coefficient of the numerical simulation is lower than that of the test. The peak of the pitching moment coefficient of the numerical simulation is smaller than that of the wind tunnel test. But the moment stall is accurately predicted, meaning that the movement process of DSV is accurately predicted. The differences between the test values and the numerical results are acceptable. It is evident that the numerical simulation method used can effectively simulate the delay phenomenon of the airfoil aerodynamic forces during the dynamic stall. The medium-scale mesh is sufficient to ensure calculation accuracy. Unless noted otherwise, the medium-scale mesh is employed to describe the physical aspects of the dynamic stall.Figure 5
Comparison of lift coefficients between the numerical and test data.Figure 6
Comparison of drag coefficients between the numerical and test data.Figure 7
Comparison of pitching moment coefficients between the numerical and test data.


## 4.2. Effects of the Upward Deflection of the TEF on the Dynamic Stall Process

Figure8 shows the motion profiles of the airfoil and the TEF. The sinusoidal movement of the TEF in deflection motions is shown in Figure 8. T represents the duration time of one pitching cycle. t/T denotes the dimensionless time during one pitching cycle. As shown in the figure, the TEF starts to deflect upwards when t/T is equal to 0.4 (i.e. the angle of attack is 17.96 deg). When t/T is equal to 0.5, namely, the maximum angle of attack, the deflection angle of the TEF reaches the minimum of -20 deg. That is, the TEF deflects upwards by 20 deg. The geometry shapes of the original and TEF airfoils at different t/T is shown in Table 2.Figure 8
Motion profiles of the airfoil and the TEF.Table 2
Geometry shapes of the original and TEF airfoils at differentt/T.
t/TAoAδOriginal airfoilTEF airfoil0.00.02°00.417.96°00.4519.35°-100.519.82°-200.5519.35°-100.617.96°01.00.02°0Figures9, 10 and 11 display the comparison of lift, drag, and pitching moment coefficients between the original and TEF airfoils. Unless noted otherwise in the following sections, the ‘original’ represents the original airfoil, and the ‘TEF’ represents the airfoil with the TEF. The numerical results reveal that the TEF has the ability to reduce the peaks of the drag and pitching moment coefficients. The maximum drag coefficient of the TEF airfoil is 34.8% less than that of the original airfoil. The peak of the pitching moment coefficient of the TEF airfoil is greatly reduced by up to 31.8%, compared with that of the original airfoil. Through the comparison of the pitching moment coefficient curves (Figure 11), it is found that the area of the clockwise loop in the pitching moment curve for the TEF airfoil is smaller than that of the original airfoil. That means that the torsional aerodynamic damping of the TEF airfoil is increased due to the upward deflection of the TEF, and the possibility of stall flutter occurrence is reduced. But one problem appears. The TEF airfoil enters the stall state early because of the upward deflection of the TEF, and its maximum lift coefficient is reduced by 15.4%. The lift augmentation effect caused by DSV is also reduced. The application of the TEF brings the benefit of alleviating the pitching moment load and also the disadvantage that the maximum lift coefficient of the airfoil is reduced. Consequently, it is necessary for practical applications to compromise between the maximum lift coefficient and the pitching moment load alleviation by adjusting certain design parameters to achieve the optimal design solution.Figure 9
Comparison of lift coefficients between the original and TEF airfoils.Figure 10
Comparison of drag coefficients between the original and TEF airfoils.Figure 11
Comparison of pitching moment coefficients between the original and TEF airfoils.In Figures9, 10 and 11, the x-axis variable is the angle of attack that changes nonlinearly over time in a sinusoidal pitching motion. This type of figure is convenient to analyze the relationship between the angle of attack and the aerodynamic force coefficients and the relevant torsional aerodynamic damping. It is difficult to intuitively analyze the development process of unsteady aerodynamic forces during the pitching cycle in the time domain. As shown in Figure 12, the unsteady aerodynamic force coefficient curves with t/T as the x-axis variable is introduced to make the analysis more intuitive. Figure 12(a) shows the comparison of the lift and drag coefficients between the original and TEF airfoils in the time domain, and Figure 12(b) displays the corresponding pitching moment coefficient curves. The pink region between t/T equals 0.4 and 0.5 represents the period in which the TEF deflects upwards from 0 deg to -20 deg. Similarly, the gray region between t/T equals 0.5 and 0.6 represents the period in which the TEF deflects downwards from -20 deg to 0 deg. Figure 13 shows the spatiotemporal distributions of the pressure coefficients on the upper surface of the two airfoils. In this plot, the abscissa represents the chordwise location, and the ordinate corresponds to the dimensionless time during one pitching cycle.Figure 12
Comparison of aerodynamic force coefficients between the original and TEF airfoils in the time domain: (a) lift and drag coefficients and (b) pitching moment coefficients. The pink region represents that the TEF deflects upwards. The gray region represents that the TEF deflects downwards.
(a)(b)Figure 13
Spatiotemporal contours of pressure coefficients on the airfoil upper surfaces: (a) original airfoil and (b) TEF airfoil.
(a)(b)Whent/T is less than 0.3, i.e., the angle of attack is less than 13.0 deg, the lift coefficients of the two airfoils increase gradually with the angle of attack. Meanwhile, the pitching moment coefficients change slightly. The corresponding flow fields remain in the attached state. When t/T is 0.34, i.e., the angle of attack reaches 15.27 deg, and the peaks of pressure coefficient of the airfoils exceed -10 due to the relative higher angle of attack, which brings about the formation of LSB in the vicinity of the airfoil leading edge (Figure 14). As a result of the appearance of the LSB, the skin friction coefficients at part of the upper surface of the two airfoils are negative values at the angle of attack of 15.27 deg. As the airfoil pitches up, the flow acceleration around the leading edge increases, causing a higher peak suction as well as a stronger adverse pressure gradient across the LSB. As shown in Figure 13, the area of the low-pressure region on the upper surface that the pressure coefficients are less than -4 continues to enlarge as the angle of attack increases. The lift coefficients also increase accordingly. With a further increase in the angle of attack, the LSB becomes bigger and moves towards the trailing edge. When the angle of attack is 17.65 deg, the bursting of the LSB occurs due to the increasing adverse pressure gradient (Figure 13). Following the bursting of the LSB, the DSV emerges and moves towards the trailing edge. The moment stall occurs, as shown in Figure 12(b).Figure 14
Comparison of coefficient distributions at the angle of attack of 15.27 deg upstroke: (a) pressure coefficients and (b) skin friction coefficients on the upper surface.
(a)(b)Whent/T equals 0.4, i.e., the angle of attack is 17.96 deg, and the TEF starts to deflect upwards. From now on, there is a clear difference between the aerodynamic force curves of the original and TEF airfoils. The lift coefficient of the original airfoil continues to increase before the maximum angle of attack. The pitching moment coefficient of the original airfoil increases rapidly due to the DSV moving towards the trailing edge. The corresponding coefficient distribution curves in the process are shown in Figures 15 and 16. In contrast to the original airfoil, the TEF airfoil at the angle of attack of 18.07 deg, i.e., t/T equals 0.404, comes up to its maximum lift coefficient, which is 15.4% smaller than that of the original airfoil. The stall angle of attack is reduced from 19.82 deg to 18.07 deg, a drop of 1.75 deg. When t/T goes from 0.404 to 0.45, the reason why the lift coefficient of the TEF airfoil drops so quickly is that the pressure on the lower surface of the TEF is lower than that of the upper surface of the TEF, which results in a negative lift. This is caused by the rapid upward deflection of the TEF. The representative pressure coefficient curve is shown in Figure 16(a). Whereafter, the lift coefficient of the TEF airfoil increases slightly, which is attributed to the lower pressure on the TEF upper surface induced by the stronger DSV. The DSV detaches from the airfoil when the airfoil starts to pitch down. The trailing edge vortex is induced by the detachment of the DSV, as shown in Figures 13 and 16(c) and 16(d). After the DSV detaches, the airfoil is stalled until the reattachment occurs when the angle of attack is much lower than the static stall point. When the angle of attack is down to 6.81 deg, the flow on the airfoil surface is restored to the attached state. During the pitching motion process, the upward deflection of the TEF reduces the pressure difference between the upper and lower surfaces of the airfoil induced by the DSV, causing the peak of the pitching moment coefficient to decline. Thus, the TEF has the function of alleviating the pitching moment load.Figure 15
Comparison of coefficient distributions at the angle of attack of 18.07 deg upstroke: (a) pressure coefficients and (b) skin friction coefficients on the upper surface.
(a)(b)Figure 16
Comparison of pressure coefficient distributions on the airfoil surface at different angles of attack.
(a)(b)(c)(d)


## 4.3. Analysis of the Circulation and Vortex Center Trajectory of the DSV

In the previous section, the effect of the upward deflection of the TEF on the unsteady aerodynamic forces of the airfoils in the dynamic stall process has been discussed. The relevant results reveal that the peak of the pitching moment coefficient is reduced markedly by the upward deflection of the TEF, alleviating the pitching moment load. This section describes the event in further detail, emphasizing the evolution of the DSV that results in the overshoot of the lift and the delay in stall. How the upward deflection of the TEF affects the strength and motion trail of the DSV is discussed.Figure17 shows the normalized circulation of the DSV and secondary vortex against dimensionless time in the planar control region. Figure 18 shows that the normalized vorticity contours at different times during one pitching cycle for the two airfoils. In these figures, the solid white lines represent the contour line that the normalized vorticity is -10, used to compare the size of the DSV qualitatively. When the airfoils are at a relatively low angle of attack and the deflection angle of the TEF is zero, the boundary layer thickness increases with the angle of attack. More vorticity is generated near the airfoil due to viscous effects. So the circulation in the planar control region increases slowly with the angle of attack. When the angle of attack is 15.27 deg, the LSB appears on the upper surface of the airfoils, as shown in Figure 14(b). The scaling of the LSB continues to enlarge, and the circulation in the planar control region augments rapidly. As shown in Figure 13, when the angle of attack is 17.65 deg, the LSB bursting occurs, resulting in the formation of the DSV. The DSV moves towards the trailing edge of the airfoil, and the circulation in the planar control region increases sharply. As the airfoils pitch up, the DSV fed by the leading edge shear layer continues to grow in both size and strength. When t/T is 0.4, i.e., the angle of attack is 17.96 deg, and the TEF starts to deflect upwards. When t/T is 0.45, i.e., the angle of attack is 19.35 deg, and the deflection angle is -10.0 deg; the circulation in the planar control region of the TEF airfoil is reduced by the upward deflection of the TEF, compared with that of the original airfoil. The upward deflection of the TEF can reduce the circulation in the planar control region, namely, can reduce the strength of the DSV. As the airfoils pitch up, the strength of the DSV continues to enlarge. The growth of the circulation of the TEF airfoil is relatively slower than that of the original airfoil. When t/T is 0.5, i.e., the angle of attack is maximum, and the deflection angle is -20.0 deg, the leading edge shear layer and the DSV is weakly linked for the original airfoil (Figure 18(c)). In contrast to the original airfoil, the connection between the leading edge shear layer and the DSV is weaker for the TEF airfoil (Figure 18(d)). When the connection between the leading edge shear layer and the DSV is weak, it means that the DSV no longer accumulates vorticity from the leading edge shear layer [27]. Due to the lack of vorticity supply from the leading edge shear layer, the strength of the DSV decreases rapidly. The DSV detaches from the leading edge shear layer and passes the trailing edge (Figures 18(e) and 18(f)). The circulation in the planar control region drops rapidly.Figure 17
Comparison of the normalized circulation in the planar control region during one pitching cycle for the two different airfoils. The pink region represents that the TEF deflects upwards. The gray region represents that the TEF deflects downwards.Figure 18
Normalized vorticity contours at different times during one pitching cycle for the two airfoils: (a, c, e) original airfoil and (b, d, f) TEF airfoil. The solid white line represents the contour line that the normalized vorticity is -10.In addition to the strength of the DSV, the distance between the vortex core and the airfoil is also one of the factors that determine the pressure on the airfoil surface. In the present study, the position of the vortex center is used to represent the relative location between the vortex core and the upper surface of the airfoil. Figure19 shows the vortex center trajectories over the dimensionless time interval 0.4<t/T<0.55 for the original and TEF airfoils. The vortex center trajectories of the two airfoils are different. The upward deflection of the TEF has a significant influence on the DSV evolution. The vortex center trajectory of the TEF airfoil is nearly the same as that of the original airfoil before t/T is equal to 0.45, except for the slower translational velocity of the vortex center. When t/T is greater than 0.45, the deflection angle of the TEF is always smaller than -10 deg, which brings the bigger negative camber of the airfoil. Figures 20(a) and 20(b) show the streamlines and Mach number contours at the maximum angle of attack for the two airfoils. The position of the vortex center for the two airfoils is different. It is proved that the vortex center of the DSV is farther away from the airfoil upper surface due to the upward deflection of the TEF. Thus, the pressure difference between the lower and upper surfaces induced by the DSV decreases. When t/Tequals 0.55, the streamlines and Mach number contours of the two airfoils are shown in Figures 20(c) and 20(d). It is found that the vortex center of the original airfoil continues to move towards the trailing edge, and the vortex center of the TEF airfoil moves towards the leading edge, which is consistent with the corresponding curve in Figure 19. In the pitching motion process, the upward deflection of the TEF not only reduces the circulation of the DSV but also makes the vortex center farther from the upper surface of the airfoil, resulting in a smaller pressure difference between the upper and lower surfaces.Figure 19
Vortex center trajectories shown in the airfoil coordinate system over the dimensionless time interval0.4<t/T<0.55 for the two airfoils.Figure 20
Streamlines and Mach number contours at different angles of attack for the two airfoils: (a, c) original airfoil and (b, d) TEF airfoil.


## 4.4. Vorticity Transport Analysis in the Planar Control Region

In this section, the terms of Equation (3) are evaluated for the planar control region shown in Figure 3 so as to understand the relative importance of each vorticity transport mechanism in governing the strength and size of the DSV. A comparison of the integrated vorticity flux of each boundary during the deflection duration of the TEF for the original and TEF airfoils is shown in Figure 21. Because the vorticity in the leading edge shear layer, i.e., the feeding shear layer, is the primary contributor to DSV growth [28], we focus on analyzing the vorticity flux through Boundary 1. As t/T increases from 0.4 to 0.5, the integrated vorticity fluxes on Boundary 1 for the two airfoils continue to increase. The maximum slope (maximum shear layer flux) occurs at t/T equals 0.4. The integrated vorticity flux on Boundary 1 of the TEF airfoil is smaller than that of the original airfoil due to the upward deflection of the TEF during the pitchup process. Figures 22 and 23 show the streamlines and Mach number contours around the leading edge at t/T is 0.45 and 0.5 for the two airfoils. In these figures, the solid white lines represent the contour line that the normalized vorticity is -10, same as Figure 18, used to compare the thickness of the feeding shear layer qualitatively. When t/T equals 0.45, the thickness of the feeding shear layer for the TEF airfoil is thinner, and the flow velocity in the feeding shear layer is slower. Likewise, when t/T equals 0.5, i.e., the angle of attack is the maximum, the vorticity flux from the feeding shear layer of the TEF airfoil is smaller than that of the original airfoil, as shown in Figure 23. It is proved that the upward deflection of the TEF reduces the vorticity flux from the feeding shear layer, which causes the circulation of the DSV to decline. In addition, Figure 21 displays that the positive secondary vorticity through Boundary 4 due to the streamwise pressure gradient effects is approximately half of that through Boundary 1 from the feeding shear layer flux during the pitchup motion. The magnitude of the net circulation in the planar control region is significantly reduced by the positive secondary vorticity generated due to the presence of the DSV itself. So the magnitude of the DSV circulation is consistently higher than that of the planar control region due to the positive secondary vorticity contained within the control region, as previously demonstrated for laminar flows [23] and turbulent flows [12].Figure 21
Comparison of the negative integrated vorticity fluxes during the deflection duration of the TEF for the original and TEF airfoils.Figure 22
Streamlines and Mach number contours around the leading edge att/T=0.45 for the two airfoils: (a) original airfoil and (b) TEF airfoil. The solid white line represents the contour line that the normalized vorticity is -10.Figure 23
Streamlines and Mach number contours around the leading edge att/T=0.5 for the two airfoils: (a) original airfoil and (b) TEF airfoil. The solid white line represents the contour line that the normalized vorticity is -10.


## 4.5. Effect of the Relative Chord Length of the TEF

Figure24 shows the airfoil profiles at minimum deflection angle for different relative chord lengths of the TEF. 0.3c represents that the relative chord length of the TEF airfoil is 0.3 times the airfoil chord length and so on. The bigger relative chord length of the TEF causes the bigger negative camber of airfoil at the same deflection angle. Figures 25(a)–25(c) illustrate the comparison of aerodynamic force coefficients at different relative chord lengths. As shown in these figures, the longer the relative chord length of the TEF, the smaller the maximum lift coefficient of the airfoil. The longer the relative chord length of the TEF, the smaller the absolute value of the peaks of the drag and pitching moment coefficients. It proves that the longer TEF has a stronger ability to alleviate the pitching moment load. For wind tunnel tests or engineering applications, an actuator is needed to make the TEF deflected. The required power of the actuator is partly determined by the hinge moment of the TEF. Thus, the hinge moment of the TEF during the deflection motion process needs to be analyzed. Figure 25(d) shows the hinge moment coefficient curves over time for different relative chord lengths. The ‘Original’ represents the hinge moment generated on the original airfoil surface corresponding to the TEF with 0.3 times chord length, equivalent to the deflection angle always being zero during the pitching motion process. It is found that the peak of the hinge moment coefficient of the original airfoil is bigger than that of the TEF airfoil with 0.3c. The reason is the upward deflection of the TEF makes the circulation of the DSV smaller (Figure 17), and the vortex center of the DSV farther away from the trailing edge of the airfoil (Figure 19), causing the pressure difference between the upper and lower surfaces to decrease. The peak of the hinge moment coefficient of the TEF airfoil is smaller than that of the original airfoil. But when t/T is equal to 0.527, i.e., the angle of attack is 19.68 deg, and the hinge moment coefficient of the original airfoil is smaller than that of the TEF airfoil with 0.3c. The reason for this situation is that a trailing edge separation vortex appears at the trailing edge. The emergence of the trailing edge separation vortex causes the bigger pressure difference as well as the hinge moment, as shown in Figures 13 and 16(d). The shorter the relative chord length of the TEF, the smaller the hinge moment coefficient, meaning that the required power of the actuator is lower. Conversely, the longer the relative chord length of the TEF, the better the effect of the pitching moment load alleviation. For practical applications, it is necessary to compromise between the better load alleviation and the lower required power to obtain the optimal design solution.Figure 24
Schematic of airfoil shapes for different relative chord lengths of the TEF at the deflection angle of -20 deg.Figure 25
Comparison of aerodynamic force coefficients for different relative chord lengths of the TEF: (a) lift coefficients, (b) drag coefficients, (c) pitching moment coefficients, and (d) hinge moment coefficients.
(a)(b)(c)(d)


## 4.6. Effect of the Minimum Deflection Angle of the TEF

Figure26 shows that the airfoil profiles at different minimum deflection angles. The relative chord length of these TEFs is 0.3c. -20° represents that the minimum deflection angle of the TEF airfoil is -20 deg and so on. As shown in Figure 27, the upward deflection of the TEF can reduce the peak of the drag and pitching moment coefficients, which is consistent with the conclusions in previous sections. The upward deflection of the TEF causes the negative camber of the airfoil to increase. The bigger the upward deflection angle of the TEF, the better the effect of the pitching moment load alleviation. In Figure 27(d), the legend curve ‘Original’ represents the hinge moment generated on the original airfoil surface corresponding to the TEF with 0.3 times chord length, equivalent to the deflection angle always being zero during the pitching motion process. The larger the upward deflection angle of the TEF, the smaller the corresponding hinge moment. However, compared with the change of the relative chord length of the TEF, the change of the upward deflection angle has less effect on the peak of the hinge moment of the TEF.Figure 26
Schematic of airfoil shapes at different minimum deflection angles of the TEF.Figure 27
Comparison of aerodynamic force coefficients at different minimum deflection angles of the TEF: (a) lift coefficients, (b) drag coefficients, (c) pitching moment coefficients, and (d) hinge moment coefficients.
(a)(b)(c)(d)


## 5. Conclusions

The flow mechanism of the airfoil dynamic stall load alleviation based on the TEF is investigated numerically, employing an unsteady Reynolds-averaged Navier-Stokes-based simulation approach. RBF-based mesh deformation algorithm, as well as Laplacian and optimization-based mesh smoothing algorithm, is utilized to ensure the mesh quality in flow field simulations during the pitching motion of the airfoil. The SC1095 airfoil at a freestream Mach number of 0.302 and chord-based Reynolds number of3.89×106 is considered. The effect of the upward deflection of the TEF on the unsteady aerodynamic forces due to DSV is analyzed. The results reveal that the upward deflection of the TEF can reduce the peaks of the drag and pitching moment coefficients. Although the maximum lift coefficient of the airfoil is slightly reduced, its maximum drag and pitching moment coefficients are significantly reduced by up to 34.8% and 31.8%, respectively. On this basis, the unsteady flow process is studied in detail, with emphasis on characterizing the circulation and vortex center trajectory of the DSV and the vorticity flux from the leading edge shear layer. Because of the upward deflection of the TEF, the leading edge shear layer of the TEF airfoil is thinner than that of the original airfoil, and the flow velocity near the leading edge of the TEF airfoil is lower than that of the original airfoil. The vorticity flux through Boundary 1 for the TEF airfoil is smaller than that of the original airfoil. Compared with the original airfoil, the circulation of the DSV for the TEF airfoil is smaller, which brings the lower peak of the pitching moment coefficient. The effects of the relative chord length and minimum deflection angle of the TEF on the dynamic stall process are studied. It is found that the shorter the TEF, the bigger the maximum lift coefficient of the airfoil. The longer the TEF, the smaller the absolute value of the peak of the pitching moment coefficient. The larger the upward deflection angle of the TEF, the better the effect of the pitching moment load alleviation.In summary, the fundamental reason why the upward deflection of the TEF can alleviate the pitching moment load is that the upward deflection of the TEF increases the negative camber of the airfoil, which reduces the vorticity flux generated in the leading edge shear layer. The fewer the vorticity flux from the leading edge shear layer, the smaller the circulation of the DSV, the farther the vortex center from the upper surface of the airfoil, which causes the peak of the pitching moment coefficient induced by the DSV to decrease. As a consequence, the goal of the pitching moment load alleviation is achieved by the upward deflection of the TEF.



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*Source: 1674074-2022-03-12.xml*