Yufei ZHANG, Chongyng YAN, Hixin CHEN, Yuhui YIN

a School of Aerospace Engineering, Tsinghua University, Beijing 100084, China

b Key Laboratory of Icing and Anti/De-icing, China Aerodynamics Research and Development Center, Mianyang 621000, China

KEYWORDS Drag reduction;Large Eddy simulation;Relaminarization;Riblet;Sweep angle

Abstract This paper studies the riblet drag reduction effect for an infinite swept wing under a low Reynolds number using a large-eddy simulation. The results show that the drag reduction ratio is not linear under different sweep angles.The maximum drag reduction ratio in this study is 9.5%for a wing with a 45°sweep angle.The local surface streamline angle and turbulence quantities are calculated to analyze the drag reduction mechanism. The results demonstrate that the riblets considerably suppress the Reynolds stresses above the wing upper surface, while the turbulence kinetic energy in the near wake is increased. A possible relaminarization phenomenon is observed at the middle part of the wing. Quasi-two-dimensional flow structures are observed near the wall, and a peak frequency is considered as the dominant frequency of the region.

1. Introduction

Friction is an important component of the source of drag for an aircraft. It constitutes a proportion of approximately 40-50%of the total drag of a typical civil aircraft when cruising.1Complex motion of the turbulence structure of a high Reynolds boundary layer is the primary cause of highfriction drag.2Laminar technologies, including natural laminar flow, laminar flow control or hybrid laminar flow, are usually applied for the reduction of friction drag.3However,laminar flow is hard to maintain for a wing with a large sweep angle because of the cross flow transition.4A riblet is a passive flow control method for friction drag reduction that has been studied for decades.5,6It can be used to effectively depress turbulence fluctuations and reduce the Reynolds stresses.7,8The viscous drag reduction ratio can reach as high as 10% with an optimal riblet shape and size on a flat plate9.When a riblet film is installed on an airfoil, the drag reduction ratio is even higher.Caram and Ahmed10observed a 13.3%drag reduction on an airfoil at a Reynolds number of 2.5×105. Sundaram et al.11reported a 16% viscous drag reduction at a 6° angle of attack; accordingly, the total drag is reduced by 13%. At transonic speed, an approximately 12% drag reduction ratio was observed for a supercritical airfoil.12A stirring drag reduction of 40% was reported by Li and Wang for a delta wing.13The adverse pressure gradient on an airfoil or a wing has shown drag reduction benefits due to the riblets.11

The riblet size is highly dependent on the Reynolds number of the flow. The optimal riblet peak-to-peak distance s+(superscript‘‘+”means normalized by wall unit)is approximately 15-20,14which is roughly dozens of microns for a typical Reynolds number for a civil aircraft. The manufacturing and maintenance of riblets are intractable problems for application on a real aircraft because of the small riblet size involved.This problem has been gradually relieved with the development of coating technology. Stenzel et al.15developed a paint method for riblet manufacturing and tested it on an in-service flight for 12 months. The result demonstrated a considerable drag reduction ratio and good durability. Researchers16-18further validated the drag reduction efficiency of the paint-riblet by a wind tunnel experiment or flight test. The application of a riblet on civil aircraft will be progressed with paint-riblet or advanced coating19technology.

Modern civil aircraft adopt a supercritical wing with a sweep angle to enable higher flight speeds.20The sweep angle induces cross flow along the spanwise direction. Many researchers noticed a degeneration of the drag reduction efficiency when the riblet is not installed along the flow direction.Walsh and Lindemann21reported that the riblet drag reduction for a flat plate is almost unchanged when the yaw angle(the misalignment angle of the riblet and the flow direction)was less than 15°,while no drag reduction was measured when the yaw angle was increased to 30°.Gaudet22found that a drag penalty arose at the supersonic condition when the riblet was installed with a misalignment of larger than 30°. McLean et al.23also reported that the drag reduction effect was reduced when the riblet was installed with a misalignment in the flight test of the T-33 airplane. Coustols24,25tested the riblet yaw angle up to 22.5° on an airfoil and demonstrated that higher yaw angle has weaker friction reduction. Hage et al.26carried out detailed testing of the drag reduction degeneration for different riblet shapes on a flat plate with yaw angles of up to 20°and found that a riblet with a smaller s+is less sensitive to the yaw angle. Sloshing,27which is a lateral movement of fluid between riblets that produces a motion normal to the wall and induces additional momentum exchange, is considered to be the underlying cause of the decay in the drag reduction.26Koeltzsch et al.carried out a detailed review28of the yaw angle effect of a riblet. The drag reduction ratio becomes zero at a yaw angle of approximately 20-35°. The critical yaw angle for losing efficacy depends on both the riblet height h+and the riblet spacing s+,i.e.,the geometry of the riblet.Benschop and Breugem29reported that a blade riblet geometry is more susceptible to yaw angle than other geometries, as the critical yaw angle for losing efficacy is lower.Sundaram et al.30tested riblet drag reduction for a wing with a sweep angle of 25°.They reported that the yaw angle of the riblet varies with the streamwise location on a swept wing as well as the angle of attack, both of which affect the drag reduction efficiency.

Although many researchers have studied the drag reduction efficiency of a riblet under a yaw angle, most of these studies have been based on experiments and on flat-plate geometries.The drag reduction mechanism for an airfoil or a wing is quite different from that for a flat plate. For example, the drag reduction ratio on a wing is far greater than that on a flat plate.The mechanism for drag reduction degeneration under a yaw angle is not well explained.Zhang and Yin31numerically studied riblet drag reduction for both channel flow and an airfoil Eppler E374.32They found that the riblets lost efficacy in the channel when the riblets yawed at 30°; however, a good drag reduction effect was demonstrated for an infinite wing with a sweep angle of 30°.

Numerical simulation is a useful tool to study the interaction between a turbulent flow structure and a riblet under a yaw angle. Especially, when a riblet is installed on a swept wing, the complex interaction between the adverse pressure gradient, surface curvature and cross flow will strongly influence the drag reduction effect. In this paper, the riblet drag reduction effect for infinite wings with different sweep angles is numerically investigated based on a Large-Eddy Simulation(LES).An isosceles triangle riblet installed on four wings with sweep angles ranging from 0°to 45°is numerically tested.The flow structures and drag reduction mechanism are also analyzed.

2. Numerical methods and computational settings

2.1. Numerical methods

The numerical simulation in this study was carried out based on a low-dissipation finite volume code. Implicit large-eddy simulation was applied for turbulence flow computation,which means no explicit subgrid scale model was adopted,and the dissipation effect of turbulence was simulated by numerical dissipation of the spatial discretization scheme. A three-step Runge-Kutta method was applied for time integration.This numerical method has been well validated by several basic flows in our previous study,33and successfully applied for study of riblet drag reduction for both channel flow and the flow around an airfoil.31,34The numerical scheme is not our focus in this paper; consequently, a separate validation case is not presented here.

2.2. Computational grid

A low speed Eppler E37429was adopted as the baseline airfoil for the infinite wings.Four different sweep angles,0°,15°,30°and 45°, were numerically studied. Fig. 1(a) shows the airfoil configuration.Because the Reynolds number computed in this paper is low,a numerical trip with a width of 0.01c is placed at x/c=0.13 to ensure flow transition from laminar to turbulence. The trip is implemented by steady suction and blowing.31The red spots and purple spots in Fig. 1(a) are applied for suction and blowing, respectively. The riblets are installed from x/c=0.3 to x/c=0.99.The spanwise width of the computational domain is 0.05c. A periodic boundary condition is applied in the spanwise direction; consequently, the case is equivalent to an infinite wing.As shown in Fig.1(a),two coordinate systems are used to report the computational results in the following text. The xyz coordinate system is a Cartesian coordinate system, in which the x direction is parallel to the airfoil chord line. The ξηζ coordinate system is also a rectangular coordinate system, while the ζ direction is parallel to the wing leading edge. The computational domains of the x and y directions are [-20c, 20c] and [-16c, 16c], respectively.The computational grid is spanwise extruded from an airfoil grid.Fig.1(b)and(c)show the x-y plane grid near the leading and trailing edges. The grid has 1321×181 points in the circumferential direction and wall-normal direction. The grid has a C-type topology. It has 241 points in the spanwise direction. The total grid cell count is 57 million for each computational case. Fig. 1(b) and (c) are plotted at every two points in both the circumferential and wall-normal directions.The first layer height of the wall normal direction is 1.0×10-4c for the smooth wing case. When riblets are installed on the wing,the first layer height varies with the spanwise direction, as shown in Fig. 1(d). The grid is refined near the riblet tip and coarsened in the riblet valley because of the different flow velocities near the riblet tip and valley.The riblet tip-to-tip distance is 0.0025c, and the height is 0.002c. Twenty riblets are computed in the spanwise direction.

Fig. 1 Airfoil configuration and computational grid.

2.3. Computational settings

According to the inviscid flow theory, wings with different sweep angles are equivalent based on a cosine law,35as shown in Eqs. (1)-(4),which provides a transformation method from an airfoil to a swept wing. The Λ is the sweep angle and t/c is the relative thickness of the wing. To make the results comparable under different sweep angles, the flow conditions are transformed from Ma=0.2,Re=2×105,α=3°for an airfoil with a 0°sweep angle,as shown in Table 1.The dimensionless timesteps for the riblet cases under different sweep angles are also listed in the table. As the minimum grid space is little changed due to variation of the relative thickness, the dimensionless timesteps are decreased with sweep angle.

3. Numerical results

In this study,a two dimensional flow with a coarse grid is first computed to obtain an initial flow field.The initial flow field is then interpolated to a three-dimensional grid to accelerate the computation. More than twenty time units (tU∞/c) are computed on the fine grid, with the final ten time units being used to determine the flow-field statistics. A total of approximately 2.5-3.0 million computational steps are used for the cases studied, which requires approximately 4×105core hours for each case on an intel cluster with a clock frequency of 2.4 GHz.

Table 1 Flow conditions for different sweep angles.

3.1. Aerodynamic coefficients

In this section,the aerodynamic coefficients for different sweep angles are presented. Fig. 2 shows the histories of the lift and drag coefficients (CLand CD) for the smooth wing and ribletcovered wing under a sweep angle of 15°.The force coefficients show fluctuations, while the averaged values show convergence.

The averaged lift and drag coefficients for the different sweep angles are shown in Fig. 3. The data for both the smooth wing and riblet-covered wing are presented in the figure.With increasing sweep angle, the lift coefficient monotonically decreases because the pressure coefficient is proportional to cos2Λ according to the cosine law.The scaled lift coefficient in Fig.3(a)scales the computed lift coefficient by 1/cos2Λ.The scaled lift coefficient demonstrates a little decrease with sweep angle, while the relative differences for the 0° and 45° sweep angles are less than the value of 8% obtained for the smooth wing and 5% for the riblet-covered wing, respectively. These findings demonstrates that the cosine law is a useful transformation method for comparing different sweep angles, which makes the present comparison for different sweep angles reasonable. The drag coefficient has a more complex tendency.The total drag for both the smooth wing and riblet-covered wing is first increased at a sweep angle of 15° but then gradually decreases,which is mainly caused by the pressure drag.In contrast, the variation in the friction drag under different sweep angles is very small.

The effect of the riblet on aerodynamic forces is quite considerable. The lift coefficient is increased by the riblet, while the drag coefficient is decreased. The drag reduction arises from the decrease in the friction drag; however, the pressure drag is increased by the riblets.The total drag reduction ratios for the four sweep angles are 6.0%, 5.0%, 8.5% and 9.5%,respectively. The riblet shows the best drag reduction ratio for a sweep angle of 45° because the increment in the pressure drag is small.

3.2. Mean flow fields

In this subsection, the mean flow fields for the four sweep angles are presented.The correlation in the spanwise direction and the wall normal Δy+are first shown to illustrate that the grid resolution is sufficient.Then,the averaged flow quantities are presented.

Fig.5 shows the first layer Δy+in the wall normal direction averaged along the spanwise direction for the 15° swept wing.The first layer Δy+exceeds 1.0 only at the very leading edge of the wing, where the flow is laminar and the boundary layer is extremely thin. The first layer Δy+is less than 1.0 when x/c>0.01, which illustrates the fact that the grid resolution is sufficient for LES. When the riblet film is installed on the wing(x/c>0.3), the first layer Δy+is decreased significantly.

Fig. 6 shows the wall pressure coefficients Cpfor the 15°swept wing. The data for both the smooth and riblet-covered wings are presented. The riblets have a little influence on the pressure distribution. A small pressure bump occurs at the starting location of the riblets (x/c=0.3). Then, the pressure coefficient is a little shifted by the riblets at 0.3

Fig. 7 shows the pressure distributions for riblet-covered wings with different sweep angles. The pressure coefficients are scaled by the factor 1/cos2Λ.The data show that the scaled pressure coefficients match well with each other. The sweep effect has very little effect on the pressure distribution. This result confirms that the scaled lift coefficients in Fig.3(a)show a quite limited variation with the increasing sweep angle.

Fig. 2 Histories of lift and drag coefficients under a 15° sweep angle.

Fig. 3 Force coefficients for different sweep angles.

Fig. 4 Spanwise correlations for the four smooth wings with different sweep angles at x/c=0.9.

Fig. 5 Δy+ for the first grid layer of 15° swept wing.

Fig. 6 Pressure distributions for a smooth wing and riblet wing at a 15° sweep angle.

Fig. 7 Scaled pressure distributions for different sweep angles.

Fig. 8 Surface streamlines and local flow angle contours for wings with different sweep angles.

As described in the introduction part, the flow direction is important for the drag reduction effect of riblets. When the riblets are yawed larger than 30°, the aerodynamic drag will be increased by the riblets.21To observe the flow direction under different sweep angles, Fig. 8 presents the surface streamlines and local streamline angle β contours for different sweep angles. The computation domain in the spanwise direction is duplicated to clearly show the streamlines and contours. The streamline angle is computed by using β=arctan(u/w) and plotted in units of degrees. As shown in Fig. 8(a),the streamline angle at the front part of the smooth wings(x/c<0.4) is negative because the flow perpendicular to the leading edge at the front part is accelerated,while the flow parallel to the leading edge is unchanged according to the sweep wing theory.In contrast,the streamline angle near the trailing edge is positive because the flow perpendicular to the trailing edge is decelerated. The flow angle for the most part of the smooth wings is less than 15°, while the flow angle near the trailing edge (x/c>0.97) is larger than 25°. According to the results of Walsh and Lindemann,21the present riblet installation region (0.30.3. The surface streamlines travel along with the riblets, which are parallel to the incoming flow direction. The cross flow near the wall is suppressed by the riblets. However, the cross flow arises when leaving the riblets at the trailing edge (x/c>0.99) and the streamline angle is large.

Fig. 9 Wall friction coefficients for wings with different sweep angles.

Fig.9 shows the wall friction coefficients Cffor the different wings. The wall friction is averaged through the ζ direction.The shear stresses either on the riblet tip or in the valley are all averaged. At the upper surface, the wall friction shows a strong jump near the trip location, which illustrates the fact that the numerical trip applied in the computation is effective.The friction of the smooth wing is increased at 0.4

Fig.10 shows the averaged velocity profiles at two wall normal line segments at x/c=0.5 and 0.9.The Fig.10(a)presents the wall parallel velocity in the x-y plane,and Fig.10(b)shows the velocity in the z-direction. The ‘‘SM_00” in the figure denotes a smooth wing with a 0° sweep angle, while the‘‘RB_00” refers to a 0° sweep wing covered with a riblet film.It is clear that the velocity gradient is decreased at the bottom of the profile. The tendency is similar under different sweep angles. The spanwise velocity w is negative at x/c=0.5 and positive at x/c=0.9. No matter the sign of the velocity, the absolute value of the spanwise velocity w is significantly decreased by the riblets when Δy/c<0.006.

The velocity of a sweep wing can be decomposed to components that are perpendicular and parallel to the leading edge.Fig. 11 shows the decomposed velocities in the ξ (perpendicular to the leading edge) and ζ (parallel to the leading edge)directions. With increasing sweep angle, the velocity component perpendicular to the leading edge is decreased and the component parallel to the leading edge is increased. The deviation in the profile caused by the riblets is below Δy/c<0.005 at x/c=0.5 and is approximately Δy/c<0.010 at x/c=0.9.The height of influence increases along the streamwise location, while the amplitude of influence weakens.

3.3. Turbulence flow quantities

Fig. 13 presents the maximum turbulence kinetic energy in the wall normal direction along the streamwise direction. The distributions for the 0°to 30°sweep angles are quite similar,as the TKE first increases along the streamwise direction and then decreases. A strong adverse pressure gradient in the middle part of the wing enhances the turbulence structures. When riblets are installed on the wing, the maximum TKE is decreased at 0.31.0, which can also be observed in Fig. 12.

Fig.14 shows the root-mean-square of the velocity fluctuations and the Reynolds shear stress in the xyz coordinate at two streamwise locations.The tendency of the effect of the riblets is similar for different sweep angles.The rms of the velocity fluctuations is equal to the square root of the Reynolds normal stress. The Reynolds stresses are significantly decreased at x/c=0.5 and slightly depressed at x/c=0.9. This effect is particularly clear for vrmsand wrms. The shear stress is important for friction generation. It is also considerably decreased at x/c=0.5,which confirms the presence of the wall friction pit in Fig. 9.

3.4. Unsteady flow characteristics

Fig. 15 presents the instantaneous spanwise vorticities Ωzfor the wings with different sweep angles. The left column shows data for the smooth wings, and the right column shows data for the riblet wings. For a 0° sweep wing, the smooth wing shows clear irregular vortex structures at 0.4

Fig. 10 Averaged velocity profiles for the smooth and riblet cases.

Fig. 11 Averaged velocity profiles in the ξ and ζ directions.

Fig. 12 Turbulence kinetic energy contours for wings with different sweep angles.

Fig.16 shows the streamwise vorticities for wings with a 45°sweep angle. At x/c=0.8, the streamwise vorticities are clearly decreased by the riblets. Moreover, the size of the riblets is smaller than that of the vorticities,which means the high friction induced by the vortex only occurs on the tip of the riblets.However,the vortex structures for the riblet-covered wing become larger than those for the smooth wing in the wake(x/c=1.05). The size of the vortex structure for the ribletcovered wing is also bigger. The vortex in the boundary layer of the riblet-covered wing is less developed than that for the smooth wing because of the relaminarization phenomenon;consequently, the vortex behavior in the wake may act as a flow with a lower Reynolds number, with the scale of the vortex larger than that for the smooth wing.

Fig. 13 Maximum turbulence kinetic energy above upper surface along streamwise direction.

Fig. 14 Root-mean-square of velocity fluctuations and the Reynolds shear stress.

Fig. 17 shows the power spectrum density of the pressure fluctuations Φppnormalized by the free stream parameters,which is computed by the fast Fourier transform method.37Two sampling locations for the four sweep angles are shown in the figure. At the high frequency part (fc/U∞>20), the fluctuations are suppressed by the riblets at both x/c=0.5 and 0.9. The suppression effect at x/c=0.5 is significant.However, a spectral bump is observed at approximately fc/U∞=10 for all riblet-covered cases at location x/c=0.5.This observation might be connected to the relaminarization phenomenon shown in Fig. 15. Fig. 18 shows the turbulence structures characterized by the iso-surface of the spanwise vorticity Ωz. The turbulence structures for the smooth wings are small and abundant, while the riblets greatly reduce the small structure. An interesting phenomenon is that a quasi-twodimensional spanwise flow structure can be observed near the wall for the riblet-covered cases in the vicinity of x/c=0.5,as shown in Fig.18,which is similar to laminar separation bubble behavior. According to the interval of the quasi-2D structure and local flow velocity, the frequency of the flow structure is coincidence with the peak frequency in Fig. 17.

Fig. 15 Instantaneous vorticities for wings with different sweep angles.

Fig. 16 Instantaneous streamwise vorticities for wings with a 45° sweep angle.

Fig. 17 Power spectrum density of the pressure fluctuations.

Fig.18 Turbulence flow structures on the upper surfaces(iso-surface of spanwise vorticity Ωzc/U∞=40,colored by streamwise vorticity Ωxc/U∞).

4. Conclusions

This paper studies the effect of riblet drag reduction on infinite wings with four different sweep angles using a large-eddy simulation.The upper surface of a low speed Eppler E374 airfoil is applied for the riblet study. The work can be summarized as follows:

(1) The flow conditions for different sweep angles are transformed by a cosine law,which makes the results comparable to those obtained for an unswept wing. The lift coefficient and pressure coefficient scaled by 1/cos2Λ match well under different sweep angles. The tendency of the friction drag reduction induced by riblets is similar for different sweep angles. However, the increasing trend for the pressure drag caused by the riblets is more complex, as the largest pressure drag is found for a sweep angle of 15°. A pressure bump found to occur on the upper surface that is induced by the riblets may be the cause of the pressure drag.

(2) Although the yaw effect of the riblets can result in decay of the drag reduction effect, the surface streamline and local flow angle for the smooth cases show that the local yaw effect is small except for the trailing edge of the wings. The local flow angles across most of the wing area are less than 20°, which leads to reasonable drag reduction due to the riblets and results in a wide range of sweep angles. When a riblet film is installed, the local flow near the wall follows the streamwise direction.

(3) Statistical results obtained for the velocity profiles and turbulence quantities demonstrate that the most effective region for turbulence suppression is 0.3

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos.: 91852108 and 11872230), and Open Fund of Key Laboratory of Icing and Anti/De-icing of China(No.: IADL20190201).