aerospace
Article Unsteady Simulation of Transonic Buffet of a Supercritical Airfoil with Shock Control Bump
Yufei Zhang *, Pu Yang, Runze Li and Haixin Chen
School of Aerospace Engineering, Tsinghua University, Beijing 100084, China; [email protected] (P.Y.); [email protected] (R.L.); [email protected] (H.C.) * Correspondence: [email protected]
Abstract: The unsteady flow characteristics of a supercritical OAT15A airfoil with a shock control bump were numerically studied by a wall-modeled large eddy simulation. The numerical method was first validated by the buffet and nonbuffet cases of the baseline OAT15A airfoil. Both the pressure coefficient and velocity fluctuation coincided well with the experimental data. Then, four different
shock control bumps were numerically tested. A bump of height h/c = 0.008 and location xB/c = 0.55 demonstrated a good buffet control effect. The lift-to-drag ratio of the buffet case was increased by 5.9%, and the root mean square of the lift coefficient fluctuation was decreased by 67.6%. Detailed time-averaged flow quantities and instantaneous flow fields were analyzed to demonstrate the flow phenomenon of the shock control bumps. The results demonstrate that an appropriate “λ” shockwave pattern caused by the bump is important for the flow control effect.
Keywords: transonic buffet; large eddy simulation; shock control bump; supercritical airfoil
Citation: Zhang, Y.; Yang, P.; Li, R.; Chen, H. Unsteady Simulation of 1. Introduction Transonic Buffet of a Supercritical Modern civil aircraft usually adopt a supercritical wing and operate at transonic Airfoil with Shock Control Bump. speed [1]. Shockwaves appear at both the cruise condition and the buffet condition for Aerospace 2021, 8, 203. a supercritical wing or airfoil [2]. The buffet, which is a self-sustained shock oscillation https://doi.org/10.3390/ induced by periodic interactions between a shockwave and a separated boundary layer [3], aerospace8080203 is a key constraint of the flight envelope of a civil aircraft [4]. The transonic buffet phenomenon has been studied for decades. It was first observed Academic Editor: Sergey B. Leonov by Hilton and Fowler [5] in a wind tunnel investigation. When a buffet occurs, the shockwave location and the lift coefficient, as well as the turbulent boundary layer thickness Received: 28 June 2021 after the shockwave, change periodically [5]. Iovnovich and Raveh [6] classified transonic Accepted: 22 July 2021 Published: 26 July 2021 buffets into two types: one is where a shock oscillation appears on both the upper side and lower side of the wing, while the other is where an oscillation only exists on the upper
Publisher’s Note: MDPI stays neutral side. The latter is more common for a supercritical wing of a civil aircraft [7]. Pearcey with regard to jurisdictional claims in et al. [8] studied the relation between the flow separation near the trailing edge and the published maps and institutional affil- onset of a buffet, which is widely used in the determination of the buffet onset boundary [3]. iations. When the shockwave strength is weak, which is the case for a supercritical airfoil, the Kutta wave is believed to play an important role in the shockwave oscillation [9,10]. A self- sustained feedback model of a shockwave buffet was proposed by Lee et al. [10]. The model illustrated that when the pressure fluctuation of the shockwave propagates downstream of the trailing edge, the sudden change in the disturbance forms a Kutta wave propagating Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. upstream, and the shockwave is made to oscillate by the energy contained in the Kutta This article is an open access article wave [10]. The Kutta wave, which propagates by the speed of sound [10], is also considered distributed under the terms and a sound wave generated at the trailing edge [11]. Chen et al. [12] further extended the conditions of the Creative Commons feedback model to more types of shockwave oscillations. Crouch et al. [13] found that Attribution (CC BY) license (https:// both the stationary mode and oscillatory mode of instability occur on an unswept buffet creativecommons.org/licenses/by/ wing, and that the intermediate-wavelength mode has a flow feature of the airfoil buffeting 4.0/). mode.
Aerospace 2021, 8, 203. https://doi.org/10.3390/aerospace8080203 https://www.mdpi.com/journal/aerospace Aerospace 2021, 8, 203 2 of 18
Experimental [14,15] and numerical investigations [16–18] have been widely used in buffet studies in recent years. The transonic buffet characteristics of an OAT15A supercriti- cal airfoil were experimentally measured [14] and widely adopted as a validation case for numerical study. The unsteady Reynolds-averaged Navier–Stokes (RANS) method [16] and detached eddy simulation [17] can resolve the large-scale pattern of a transonic buffet and determine the main frequency of the shockwave oscillation, while a large eddy simulation (LES) [18] can provide abundant details of the interaction between the shockwave and turbulence structures inside the boundary layer. The buffet onset boundary determines the flight envelope of a civil aircraft [19]. Sup- pressing the shockwave oscillation and improving the buffet onset lift coefficient are vital for modern civil aircraft design. Vortex generators [20] and shock control bumps (SCBs) [21] are two common passive flow control devices used for transonic buffets. The vortex gener- ator has been intensively studied and widely adopted on Boeing’s civil aircraft to control the transonic buffet [22]. However, SCBs are still receiving considerable research attention and are not yet practical on civil aircraft. SCBs can be classified into two-dimensional (2D) bumps and three-dimensional (3D) bumps [23]. The 2D bump is identical along the spanwise direction, while the 3D bump can have different shapes in the spanwise direction. The basic control mechanism of a 2D SCB is a combination of the compression wave and expansion wave near the shockwave [24]. A supercritical airfoil usually has a weak normal shock on the upper surface. If a bump is placed at the shockwave location, a compression wave can be formed when the flow moves upward from the bump and weakens the normal shock. Expansion waves downside of the bump are also beneficial for pressure recovery after the shockwave. With the compression and expansion waves, a “λ” shockwave structure is formed instead of a normal shock on the airfoil [25], meaning the wave drag can be reduced. As the benefit comes from the interaction between the shockwave and the bump, the drag reduction effect is relevant to the bump location and height. Sabater et al. [26] applied a robust optimization on the location and shape of a 2D SCB and achieved a drag reduction of more than 20% on an RAE2822 airfoil. Jia et al. [27] found that a 2D SCB can reduce the drag coefficient at a large lift coefficient; however, it increases the drag coefficient at a small lift coefficient. Adaptive SCBs [28,29], which change shapes according to the shockwave location, provide drag reduction effectiveness under a wide range of flow conditions. Lutz et al. [30] optimized adaptive SCBs and found the drag curve shows significant improvement compared to the initial airfoil. The control mechanism of a 3D SCB includes not only the “λ” shockwave structure formed by the bump but also the vortical flow induced by the bump [31], which is similar to a small vortex generator [32]. The streamwise vortex induced by the edge of the 3D SCB is beneficial for the mixing of high-energy outer flow and low-energy boundary layer flow, consequently suppressing the flow separation. A 3D bump with a wedge shape [33,34] or hill shape [31,35,36] can effectively induce a vortical flow after the bump and change the shockwave behavior. SCBs have shown great potential for buffet control. However, there are also some issues mentioned in the literature. One of the concerns is that the control efficiency is sensitive to the shock location [37]. This might decrease the performance at the off-design condition. An adaptive SCB can extend the effective range of shock control; however, an adaptive SCB might increase the structure complexity and maintenance cost [38]. From the numerical method perspective, the steady [39,40] or unsteady [32,41] RANS method has been widely used in the study of buffets or in the shape design of SCBs for supercritical airfoils. However, the control effect of SCBs is rarely studied by high-fidelity turbulence simulation methods such as the RANS–LES hybrid method or the LES method. As stated before, a buffet is a natural unsteady turbulent flow that includes periodic shockwaves and a thickened turbulent boundary layer under a high Reynolds number. A large eddy simulation can resolve the small-scale turbulent structures of the boundary and the generated acoustic waves of the tailing edge [18], which is a superior method to study the unsteady interaction mechanism between the shockwave and the SCB. Aerospace 2021, 8, 203 3 of 18
In this paper, the transonic buffet of an OAT15A supercritical airfoil with an SCB was numerically studied by a wall-modeled large eddy simulation (WMLES). The numerical method was validated by the experimental data of the OAT15A airfoil. Several 2D SCBs were adopted to test the control effect of the unsteady flow. The flow fields of both the nonbuffet case and buffet case were analyzed. To the authors’ knowledge, this is the first study to use a large eddy simulation on the SCB of a supercritical airfoil. The time-averaged flow quantities, periodic flow patterns and pressure fluctuations were compared for the airfoil without or with an SCB.
2. Numerical Method and Validation 2.1. Numerical Method In this paper, a wall-modeled large eddy simulation was adopted to compute the flow field of an OAT15A supercritical airfoil. The solver was developed based on a finite-volume CFD code [42,43]. The inviscid flux was computed based on a blending of an upwind scheme and a central difference scheme [44,45]. The time advancing method is a three- step Runge–Kutta method. Vreman’s eddy viscosity model [46] was used to model the subgrid-scale turbulence motion. Based on the estimation of Choi and Moin [47], the grid 13/7 requirement of a wall-resolving large eddy simulation is proportional to ReL , and L is the character length of the flow field; in contrast, the grid requirement of a wall-modeled simulation is proportional to ReL. In this study, the computational cost of a wall-resolving LES was too high because the Reynolds number is 3 × 106. Consequently, an equilibrium stress-balanced equation was solved to model the energy-containing eddies in the inner turbulent boundary layer [44,45]. The wall-modeled equation was solved on an additional one-dimensional grid with 50 grid points to obtain the velocity profile of the inner layer of the boundary layer. The shear stress provided by the wall-modeled equation was applied as the boundary condition of the Navier–Stokes equation. The velocity of the upper boundary of the wall-modeled equation was interpolated from the third layer of the LES grid. The numerical method is not the focus of this paper. Consequently, detailed numerical schemes and formulas are not presented here. Readers can refer to our previous work on high-Reynolds separated flows [44,45].
2.2. Computational Grid The computational grid used in this paper was a C-type grid, which was generated by an in-house grid generation code that solves Poisson’s equation. The computational domain was [−18c, 20c] in the x direction and [−20c to 20c] in the y direction (c is the chord length). Although the bump shape was two-dimensional, we carried out a three- dimensional simulation because of the three-dimensional nature of the turbulence. The spanwise length was 0.075c, which is larger than the WMLES in the study by Fukushima et al. [18]. The grid had 1945 points in the circumferential direction and 261 points in the normal direction. It had 51 points evenly located in the spanwise direction. The trailing edge thickness was 0.005c, and a grid block with 21 × 51 × 249 points was located on the trailing edge region. The total grid number was 26.2 million. Figure1a shows the grid of the x–y plane of the baseline OAT15A airfoil. The figure is plotted by every two points of the grid. Figure1b shows the wall grid near the leading edge of the airfoil. The first layer height of the grid was 2.0 × 10−4c. The increasing ratio in the normal direction was less than 1.1. The first grid layer can be located in the logarithmic layer of the turbulent boundary layer because the wall-modeled equation can provide the shear stress boundary condition of the wall. When the SCB was installed on the airfoil surface, the wall surface was deformed to simulate the effect of the bump while keeping the grid topology and grid number the same. Aerospace 2021Aerospace, 8, x FOR2021 ,PEER 8, x FOR REVIEW PEER REVIEW 4 of 18 4 of 18
Aerospace 2021, 8, 203 4 of 18 When theWhen SCB thewas SCB installed was installed on the airfoil on the surf airfoilace, surfthe ace,wall thesurface wall wassurface deformed was deformed to sim- to sim- ulate theulate effect the of effectthe bump of the while bump keeping while keepingthe grid thetopology grid topology and grid and number grid numberthe same. the same.
(a) Surface(a) gridSurface in the grid x– iny planethe x– y plane (b) Wall (gridb) Wall near grid the near leading the edgeleading edge Figure 1. Computational grid of the baseline OAT15A airfoil. Figure 1.Figure Computational 1. Computational grid of gridthe baseline of the baseline OAT15A OAT15A airfoil. airfoil.
2.3. Validation2.3. Validation of the Baseline of the BaselineOAT15A OAT15A Airfoil Airfoil 2.3. Validation of the Baseline OAT15A Airfoil The baselineThe baseline OAT15A OAT15A airfoil was airfoil adopted was adopted as the validation as the validation case. The case. flow The condition flow condition The baseline OAT15A airfoil6 was adopted as the validation case. The flow condition was Ma = 0.73 and Re = 63.0 × 10 . [14] The nondimensional time step was ΔtU∞/c = 1.37 × was Ma = 0.73 and Re = 3.0 × 10 . [14] The6 nondimensional time step was ΔtU∞/c = 1.37 × was−4 Ma = 0.73 and Re = 3.0 × 10 .[14] The nondimensional time step was ∆tU /c = −4 10 . Approximately 180 time units (tU∞/c) were computed for each case, and the ∞last 100 10 . Approximately−4 180 time units (tU∞/c) were computed for each case, and the last 100 1.37 × 10 . Approximately 180 time units (tU∞/c) were computed for each case,5 and the time units were applied for flow field statistics. It cost approximately5 2.0 × 10 core hours time unitslast 100were time applied units werefor flow applied field forstatistics. flow field It cost statistics. approximately It cost approximately 2.0 × 10 core 2.0 hours× 10 5 core for each forflow each case flow on ancase Intel on ancluster Intel with cluster a 2.0 with GHz a 2.0 CPU. GHz CPU. hours for each+ flow case+ on an Intel cluster with a 2.0 GHz CPU. The Δx+ Theand ΔΔ+xy+ andof the Δy +first of thegrid first layer grid were layer collected were collected to demonstrate to demonstrate the grid theresolu- grid resolu- The ∆x and ∆y of the first grid layer were collected+ to demonstrate+ the grid resolu- tion of thetion present of the computation.present computation. Figure 2a Figure shows 2a the shows Δx+ and the Δ+xy+ andof the Δ+ yLES of gridthe LES collected grid collected tion of the present computation. Figure+ 2a shows the ∆x and ∆y of the LES grid collected by an angleby an of angleattack of α attack= 2.5°. αΔ =y +2.5°. was◦ Δapproximatelyy+ was approximately 50 on the 50 upper on the surface upper andsurface 80 on and 80 on by an angle of attack α = 2.5+ . ∆y was approximately 50 on+ the upper surface and 80 the lower surface, while+ Δx was approximately twice+ the Δy . The first grid point in the the loweron thesurface, lower while surface, Δx whilewas approximately∆x+ was approximately twice the Δ twicey . The the first∆y+ grid. The point first gridin the point in wall normalwall directionnormal direction was located was atlocated the loga at rithmicthe loga layerrithmic of layera turbulent of a turbulent boundary boundary layer, layer, the wall normal direction was located at the logarithmic+ layer of a turbulent boundary which satisfieswhich satisfiesthe requirement the requirement for a WMLES. for a WMLES. The Δy+ Theof the Δy first of the+grid first layer grid of layerthe wall- of the wall- layer, which satisfies the+ requirement for a WMLES. The ∆y of the first grid layer of the modeled equation+ (Δy wm) was always less than 1.0, as shown in Figure 2b. This is suffi- modeledwall-modeled equation (Δ equationy wm) was( ∆alwaysy+ ) less was than always 1.0, less as shown than 1.0, in asFigure shown 2b. in This Figure is suffi-2b. This is cient to describe the viscouswm sublayer of the turbulent boundary layer. The spanwise cor- cient tosufficient describe tothe describe viscous the sublayer viscous of sublayer the turbulent of the boundary turbulent layer. boundary The spanwise layer. The cor- spanwise relation [43] of the pressure fluctuation was computed to test the spanwise domain. As relationcorrelation [43] of the [43 pressure] of the pressurefluctuation fluctuation was computed was computed to test the to testspanwise the spanwise domain. domain. As As shown in Figure 3, the correlations of both x/c = 0.4 and 0.8 damp fast in the spanwise shownshown in Figure in Figure3, the 3correlations, the correlations of both of x both/c = 0.4x/ cand= 0.4 0.8 and damp 0.8 dampfast in fastthe inspanwise the spanwise direction, which demonstrates that the present spanwise domain size (0.075 c) is adequate. direction,direction, which whichdemonstrates demonstrates that the that present the present spanwise spanwise domain domain size (0.075 size c (0.075) is adequate.c) is adequate.
+ + + (a) Δx+ and(a) Δxy+ andof the Δ yLES of thegrid LES grid (b) Δy+ of (b the) Δ ywall-modeled of the wall-modeled grid grid + + 6 FigureFigure 2. FigureΔx+ 2.and∆ 2.x Δ +Δyandx+ of and ∆they Δ+ firstyof of the grid the first firstlayer grid grid and layer layer the and wall-m and the the wall-modeledodeled wall-m gridodeled (Ma grid grid= 0.73, (Ma (Ma =Re 0.73,= = 0.73, 3.0 Re × Re 10 = 3.0=6, 3.0α× = ×2.5°).10 106, ,α α= =2.5 2.5°).◦). Figure4 shows the lift and drag coefficient histories at three different angles of attack. When a buffet does not occur (α = 2.5◦), the lift and drag coefficients have oscillations with small amplitudes. Then, the amplitude of oscillation is increased at α = 3.0◦. When α = 3.5◦, a clear periodic oscillation pattern appears on both the lift and drag coefficients, which demonstrates that a buffet occurs.
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FigureFigure 3. 3. Spanwise Spanwise correlation correlation of of the the pressure pressure fluctuations. fluctuations.
FigureFigure 4 4 shows shows the the lift lift and and drag drag coefficient coefficient hi historiesstories at at three three different different angles angles of of attack. attack. WhenWhen a a buffet buffet does does not not occur occur ( (αα = = 2.5°), 2.5°), the the lift lift and and drag drag coefficients coefficients have have oscillations oscillations with with small amplitudes. Then, the amplitude of oscillation is increased at α = 3.0°. When α = 3.5°, small amplitudes. Then, the amplitude of oscillation is increased at α = 3.0°. When α = 3.5°, aa clearclear periodicperiodic oscillationoscillation patternpattern appearsappears onon bothboth thethe liftlift andand dragdrag coefficients,coefficients, whichwhich Figure 3. Spanwise correlation of the pressure fluctuations. demonstratesFiguredemonstrates 3. Spanwise thatthat correlation aa buffetbuffet occurs.occurs. of the pressure fluctuations. Figure 4 shows the lift and drag coefficient histories at three different angles of attack. When a buffet does not occur (α = 2.5°), the lift and drag coefficients have oscillations with small amplitudes. Then, the amplitude of oscillation is increased at α = 3.0°. When α = 3.5°, a clear periodic oscillation pattern appears on both the lift and drag coefficients, which demonstrates that a buffet occurs.
FigureFigure 4. 4. Lift Lift and and drag drag coefficient coefficientcoefficient histories histories of ofof the thethe OAT15A OAT15AOAT15A airfoil. airfoil.
TheThe lift lift coefficient coefficientcoefficient was was time-averaged time-averaged for for co comparisoncomparisonmparison with withwith the thethe experimental experimentalexperimental data, data,data, as asas shown in Figure5. The experimental data were extracted from [ 48]. There were two series shownshown in in Figure Figure 5. 5. The The experimental experimental data data were were extracted extracted from from [48]. [48]. There There were were two two series series of experimental data in the reference. One was from the S3MA wind tunnel, and the other of experimental data in the reference. One was from the S3MA wind tunnel, and the other6 was from the T2 wind tunnel. The Reynolds number in that experiment was 6.0 × 10 6, waswas fromfrom thethe T2T2 windwind tunnel.tunnel. TheThe ReynoldsReynolds numbernumber inin thatthat experimentexperiment waswas 6.06.0 ×× 10106,, whichFigure is 4. higher Lift and than drag the coefficient present histories computation. of the OAT15A Although airfoil. the flow condition was slightly whichwhich isis higherhigher thanthan thethe presentpresent computation.computation. AlthoughAlthough thethe flowflow conditioncondition waswas slightlyslightly different from the present computation, the time-averaged lift coefficients of the present differentdifferent fromfrom thethe presentpresent computation,computation, thethe titime-averagedme-averaged liftlift coefficientscoefficients ofof thethe presentpresent computationThe lift werecoefficient located was between time-averaged the two seriesfor comparison of experimental with the data. experimental data, as computationcomputation werewere locatedlocated betweenbetween thethe twotwo seriesseries ofof experimentalexperimental data.data. shown in Figure 5. The experimental data were extracted from [48]. There were two series of experimental data in the reference. One was from the S3MA wind tunnel, and the other was from the T2 wind tunnel. The Reynolds number in that experiment was 6.0 × 106, which is higher than the present computation. Although the flow condition was slightly different from the present computation, the time-averaged lift coefficients of the present computation were located between the two series of experimental data.
FigureFigure 5. 5. Comparison Comparison of of the the computed computed lift lift coeffi coefficientcoefficientcient with withwith experimental experimentalexperimental data datadata [48]. [[48].48].
TheThe mean mean pressure pressure coefficient coefficientcoefficient was waswas compared comparedcompared with withwith the the experimental experimental data data of of [14].[[14].14]. As AsAs shownshown in in Figure Figure 66, 6,, thethe pressure pressure coefficientscoefficientscoefficients atat at threethree three differentdifferent different anglesangles angles ofof of attackattack attack coincidecoincide coincide wellwell well with the experimental data, except that the shockwaveshockwave location is slightly downstream ofof with the experimental data, except that the shockwave location is slightly downstream of thethe experimental experimental data. data. Figure 5. Comparison of the computed lift coefficient with experimental data [48].
The mean pressure coefficient was compared with the experimental data of [14]. As shown in Figure 6, the pressure coefficients at three different angles of attack coincide well with the experimental data, except that the shockwave location is slightly downstream of the experimental data.
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(a) α = 2.5° (b) α = 3.0° (a) α = 2.5° (b) α = 3.0°
(c) α = 3.5° (c) α = 3.5° 6 Figure 6. Comparison of pressure coefficients at different angles of attack (Ma = 0.73, Re = 3.0 × 10 ).6 FigureFigure 6.6. ComparisonComparison ofof pressurepressure coefficientscoefficients atat differentdifferent angles angles of of attack attack (Ma (Ma = = 0.73,0.73, ReRe == 3.03.0 ×× 10106).). Figure 7a shows the mean Mach contour at α = 3.5°. It is clear that a low-speed region FigureFigure 77aa showsshows thethe meanmean MachMach contourcontour at at α = 3.5°. 3.5◦. It It is is clear clear that that a a low-speed low-speed region appears near the trailing edge of the airfoil. The mean shockwave is also smeared because appearsappears near the trailing edge of the airfoil. The The mean mean shockwave shockwave is also smeared because of the periodic oscillation of the shock location. Figure 7b demonstrates the root mean ofof the periodic oscillation of the shock location.location. Figure 77bb demonstratesdemonstrates thethe rootroot meanmean square (RMS) of the pressure fluctuations. It is normalized by the dynamic pressure of the squaresquare (RMS) (RMS) of of the the pressure pressure fluctuations. fluctuations. It Itis normalized is normalized by bythe the dynamic dynamic pressure pressure of the of freestream condition. The maximum pressure fluctuation appears near the shockwave lo- freestreamthe freestream condition. condition. The Themaximum maximum pressure pressure fluctuation fluctuation appears appears near near the theshockwave shockwave lo- cation. Figure 8 compares the RMS of the streamwise velocity fluctuation with the exper- cation.location. Figure Figure 8 compares8 compares the theRMS RMS of the of stre theamwise streamwise velocity velocity fluctuation fluctuation with the with exper- the imental data of the S3Ch wind tunnel [49]. The values in Figure 8 are dimensional, and imentalexperimental data of data the ofS3Ch the S3Chwind windtunnel tunnel [49]. The [49]. values The values in Figure in Figure 8 are8 dimensional,are dimensional, and the contours have the same legend. The contours of the computation and experiment are theand contours the contours have have the same the same legend. legend. The Thecont contoursours of the of computation the computation and andexperiment experiment are quite consistent in both shape and value, which validates that the present computational quiteare quite consistent consistent in both in both shape shape and and value, value, which which validates validates that that the the present present computational computational method is reliable. methodmethod is reliable.
(a) Mach contour (b) RMS of pressure fluctuation (a) Mach contour (b) RMS of pressure fluctuation Figure 7. Mean flow field and pressure fluctuations (Ma = 0.73, Re = 3.0 × 106,6 α = 3.5°).◦ FigureFigure 7.7.Mean Mean flowflow fieldfield andand pressurepressure fluctuations fluctuations (Ma (Ma = = 0.73,0.73, ReRe == 3.03.0 ×× 106,, αα == 3.5°). 3.5 ).
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(a) (b) (a) (b) Figure 8. Comparison of streamwise velocity fluctuation (Ma = 0.73, Re = 3.0 × 106, α = 3.5°). (a) Experimental data (taken Figurefrom [49]), 8.8. Comparison (b) presentof ofcomputation. streamwisestreamwise velocityvelocity fluctuationfluctuation (Ma (Ma = = 0.73, 0.73, Re Re = = 3.0 3.0× × 101066, α = 3.5°). 3.5◦). ( (aa)) Experimental Experimental data (taken fromfrom [[49]),49]), ((bb)) presentpresent computation.computation. Figure 9 further compares the wall pressure fluctuation at α = 3.5°. The experimental ◦ data Figurewere extracted9 9 further further comparesfromcompares [14]. thetheThe wallwall computed pressure pressure lo fluctuation cationfluctuation of the at at peak αα= = 3.53.5°. fluctuation. The experimental is slightly datadownstream, were extracted which fromis consistent [[14].14]. The with computed the pre locationlossurecation coefficient of the peak distribution fluctuationfluctuation (Figure is slightlyslightly 6c). downstream,However, the whichpeak valueis consistent of the present with the comp pre pressuressureutation coefficient coefficient is close to distribution the experiment, (Figure which 66c).c). However,means that the the peak primary value shock of the buffet presentpresent phenomen computationcomputationon is well is closeclose captured toto thethe by experiment,experiment, the computation. whichwhich meansThe motivation that the primaryof the present shock buffet paper phenomen phenomenonwas to studyon isthe well effect captured of the shockby the control computation. bump. TheThe motivationmotivationvariation caused ofof the the present bypresent the paperbump, paper was rather was to to study th studyan the the effectaccurate effect of the ofshockwave the shock shock control location,control bump. bump. is Theour variationThefocus variation in the caused following caused by the by sections. bump, the bump, rather Consequently, rather than the than accurate thethe numericalaccurate shockwave shockwave method location, is location,validated is our focus is to our be in focusthepracticable. following in the following sections. Consequently,sections. Consequently, the numerical the numerical method is validatedmethod is to validated be practicable. to be practicable.
6 Figure 9. Comparison of wall pressure fluctuationfluctuation (Ma(Ma == 0.73, Re = 3.0 × 1010,6 ,αα == 3.5°). 3.5◦ ). Figure 9. Comparison of wall pressure fluctuation (Ma = 0.73, Re = 3.0 × 106, α = 3.5°). 3. Numerical Study of Shock Control Bump 3. Numerical Study of Shock Control Bump 3.1. Shape of the Shock Control Bump 3.1. Shape of the Shock Control Bump InIn thisthis section,section, aa two-dimensionaltwo-dimensional SCBSCB isis numericallynumerically testedtested andand comparedcompared withwith thethe baselineIn this configuration.configuration. section, a two-dimensional Several bumps withSCB isdifferent numerically heights tested and and locations compared are analyzed. with the ThebaselineThe geometriesgeometries configuration. ofof thethe bumpsbumps Several areare bumps controlledcontrolled with different byby thethe Hicks–HenneHicks–Henne heights and functionfunctionlocations [[27], 27are], asanalyzed.as shownshown f x h c Theinin EquationEquation geometries (1).(1). of TheThe the bumpbump bumps functionfunction are controlled f((xBB) isis superposedsuperposedby the Hicks–Henne onon thethe baselinebaseline function airfoil.airfoil. [27], ash// cshown isis thethe relative height of the bump, x /c is the central location and l is the bump length. Figure 10 inrelative Equation height (1). of The the bump bump, function BxB/c is the f(x Bcentral) is superposed location and onB thelB is baseline the bump airfoil. length. h/c Figure is the relativeshows10 shows the height the geometries geometries of the bump, of bumpsof bumps xB/c withis withthe different central different location parameters. parameters. and FourlB isFour the bumps bumpsbump are length.are computed computed Figure in 10thisin showsthis paper, paper, the as showngeometries as shown in Figure inof Figurebumps 10a. The10a. with lengthsThe diff lengthserent of the parameters. of bumps the bumps are Four 0.2 care. bumps The 0.2 firstc. Theare three computedfirst bumps three are all located at xB/c = 0.55, which is the shockwave location of the baseline OAT15A inbumps this paper, are all as located shown atin xFigureB/c = 0.55, 10a. whichThe lengths is the of shockwave the bumps location are 0.2c .of The the first baseline three h c bumpsairfoil.OAT15A Theare airfoil. all relative located The heightsrelative at xB/ heightsc/ = of0.55, the h /cwhich first of the three isfirst the bumps three shockwave bumps are 0.004, arelocation 0.004, 0.008 0.008 andof the 0.012. and baseline 0.012. The fourth bump is located slightly upstream xB/c = 0.50 and has the same height as Bump 2. OAT15AThe fourth airfoil. bump The is located relative slightly heights upstream h/c of the xfirstB/c = three 0.50 andbumps has are the 0.004, same 0.008height and as Bump0.012. Figure 10b shows the geometries of the airfoil superposed with the bumps. The2. Figure fourth 10b bump shows is located the geometries slightly upstreamof the airfoil xB/ csuperposed = 0.50 and has with the the same bumps. height as Bump 2. Figure 10b shows the geometries of the airfoil superposed with the bumps. h 4 π(x/c − xB/c) π f (xB) = · sin + , xB/c − lB/2 ≤ x/c ≤ xB/c + lB/2 (1) c lB 2
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h ππ()xc− x c =⋅4 B + − ≤ ≤ + f ()xxclxcxclBBBBB sin , 2 2 (1) clB 2 The computational settings are the same as the baseline OAT15A airfoil in Section 2. Aerospace 2021, 8, 203 The computational grids of the bump configurations are deformed from the baseline grid,8 of 18 which keeps the computation appropriate to compare with the baseline OAT15A airfoil.
(a) Shapes of four different bumps (b) Geometries of the airfoil superposed with bumps
FigureFigure 10. 10. TheThe geometries geometries of of the the shock shock control control bumps. bumps.
3.2. Time-AveragedThe computational Characteristics settings of are the the Airfoil same with as theBumps baseline OAT15A airfoil in Section2. TheThe computational aerodynamic grids performance of the bump of configurations the airfoil with are bumps deformed was fromcomputed the baseline and com- grid, which keeps the computation appropriate to compare with the baseline OAT15A airfoil. pared with the baseline airfoil. The nondimensional time step is ΔtU∞/c = 1.37 × 10−4. The flow3.2. Time-Averagedfield of the baseline Characteristics airfoil is of used the Airfoil as the with initial Bumps condition of the bump cases. More than 130 time units (tU∞/c) are computed for each case. The last 80 time units are used for The aerodynamic performance of the airfoil with bumps was computed and compared time averaging to obtain the aerodynamic coefficients. Six cases were computed, which with the baseline airfoil. The nondimensional time step is ∆tU /c = 1.37 × 10−4. The flow are α = 3.5° for all four bumps and α = 2.5° for Bump 2 and Bump∞ 4. field of the baseline airfoil is used as the initial condition of the bump cases. More than Figure 11 shows the histories of the lift and drag coefficients in the computation. It is 130 time units (tU /c) are computed for each case. The last 80 time units are used for time obvious that the periodic∞ oscillation of the lift and drag coefficients at α = 3.5° is sup- averaging to obtain the aerodynamic coefficients. Six cases were computed, which are pressed by Bumps 2, 3 and 4, while Bump 1 has very little effect on the lift oscillation. It α = 3.5◦ for all four bumps and α = 2.5◦ for Bump 2 and Bump 4. can be seen that the bumps may have a negative influence on the lift coefficient. The lift Figure 11 shows the histories of the lift and drag coefficients in the computation. It is coefficient is more or less decreased at α = 3.5°, while it is significantly decreased at the obvious that the periodic oscillation of the lift and drag coefficients at α = 3.5◦ is suppressed nonbuffet condition α = 2.5°. Moreover, the drag coefficient is also influenced by the by Bumps 2, 3 and 4, while Bump 1 has very little effect on the lift oscillation. It can be seen bumps. The drag coefficient at α = 2.5° is notably increased, as shown in Figure 11d. that the bumps may have a negative influence on the lift coefficient. The lift coefficient is moreThe ortime-averaged less decreased values at α =of 3.5the◦, aerodynamic while it is significantly coefficients decreased are listed atin theTable nonbuffet 1. The relativecondition variationsα = 2.5 ◦(in. Moreover, percentage) the in drag the co coefficientefficients iscompared also influenced with the by baseline the bumps. configu- The rationdrag coefficientare also listed at α in= 2.5the◦ parenthesesis notably increased, of the table. as shown A tendency in Figure of the11d. bumps is that the lift coefficientThe time-averaged is consistently values decreased of the aerodynamic by the four bumps coefficients at both are α listed = 2.5° in and Table α 1=. 3.5°. The Therelative lift coefficient variations at (in α percentage) = 2.5° is decreased in the coefficients by approximately compared 5%, with and the it baselineis decreased configu- by approximatelyration are also 1% listed to 5% in theat α parentheses = 3.5°. With of increasing the table. bump A tendency height of(Bumps the bumps 1, 2 and is that 3), the the liftlift drop coefficient becomes is consistentlyincreasingly decreasedsignificant. by If thethe fourlocation bumps of the at bump both α is= moved 2.5◦ and upstreamα = 3.5◦ . (BumpsThe lift 2 coefficient and 4), the at liftα =decrement 2.5◦ is decreased caused by by the approximately bump becomes 5%, larger. and it The is decreased influence of by theapproximately bump on the 1% drag to 5% coefficient at α = 3.5 is◦. Withdistinct increasingly different bump under height the (Bumps nonbuffet 1, 2 and and buffet 3), the conditions.lift drop becomes The drag increasingly coefficient significant.is increased If at the α = location 2.5°, and of it the is bumpreduced is movedat α = 3.5°. upstream Con- sequently,(Bumps 2 the and lift-to-drag 4), the lift ratio decrement is increased caused at the by thebuffet bump case becomes α = 3.5°, except larger. for The the influence highest bumpof the (Bump bump on3), thewhile drag the coefficient lift-to-drag is distinctlyratio is decreased different underat the nonbuffet the nonbuffet case and α = buffet 2.5°. Consequently,conditions. The the drag bump coefficient is beneficial is increased for the buffet at α = case 2.5◦ and, and might it is reducedbe harmful at α for= 3.5the◦ . nonbuffetConsequently, case, thewhich lift-to-drag coincides ratio with is increasedour previous at the study buffet based case onα =Reynolds-averaged 3.5◦, except for the Navier–Stokeshighest bump computation (Bump 3), while [27]. theThe lift-to-dragpitching mo ratioment is is decreased also listed atin thethe nonbuffettable. The caseref- erenceα = 2.5 points◦. Consequently, of the moment the bump are x/c is beneficial= 0.25 and for y the= 0. buffetThe negative case and value might means be harmful a nose- for downthe nonbuffet moment. case, It can which be seen coincides that the with absolute our previous values of study the moment based on are Reynolds-averaged all decreased by theNavier–Stokes SCBs; however, computation the variations [27]. are The not pitching significant. moment is also listed in the table. The reference points of the moment are x/c = 0.25 and y = 0. The negative value means a nose-down moment. It can be seen that the absolute values of the moment are all decreased by the SCBs; however, the variations are not significant.
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(a) Lift coefficient at α = 3.5° (b) Drag coefficient at α = 3.5°
(c) Lift coefficient at α = 2.5° (d) Drag coefficient at α = 2.5° Figure 11. Lift and drag coefficients of the baseline airfoil and airfoil with bumps. Figure 11. Lift and drag coefficients of the baseline airfoil and airfoil with bumps. A more interesting phenomenon shown in Table 1 is that regardless of whether the Tablelift 1. andTime-averaged drag are increased aerodynamic or decreased, coefficients the of theRMSs airfoil of withthe coefficients bumps. are all suppressed by the bumps. With increasing bump height, the suppression effect of the lift oscillation is RMS of Angle of more significant.Lift RMSIn the of present Lift computatioDrag n, Bump 2 has the Lift-to-Draghighest incrementPitching of L/D α Configuration Drag Attack Coefficient Coefficient Coefficient Ratio Moment = 3.5° and a good suppression effect on the shockwaveCoefficient oscillation. The L/D of the buffet case is increased by 5.9%, and the RMS of the lift coefficient fluctuation is decreased by Baseline 2.5◦ 0.919 0.006884 0.03009 0.000724 30.54 −0.1335 67.6%; consequently, it is considered a good SCB design, with an appropriate height and airfoil ◦ 3.5 location.0.968 0.037572 0.04823 0.003737 20.07 −0.1262 Bump 1 0.965 0.027828 0.04634 0.003107 20.82 −0.1252 (h/c = 0.004, 3.5◦ Table 1. Time-averaged aerodynamic coefficients of the airfoil with bumps. (−0.3%) (−25.9%) (−3.9%) (−16.8%) (+3.7%) (+0.8%) xB/c = 0.55) Angle of Lift Coeffi- RMS of Lift Drag Coeffi- RMS of Drag Lift-to- Pitching − BumpConfiguration 2 ◦ 0.866 0.004527 0.03174 0.000662 27.28 0.1260 2.5 Attack − cient − Coefficient cient −Coefficient −Drag Ratio Moment (h/c = 0.008, ( 5.8%) ( 34.2%) (+5.5%) ( 8.6%) ( 10.6%) (+5.6%) 2.5° 0.919 0.006884 0.03009 0.000724 30.54 −0.1335 x /c = 0.55) 0.953 0.012188 0.04484 0.001569 21.25 −0.1229 BBaseline airfoil 3.5◦ 3.5° (−1.5%)0.968 (−67.6%)0.037572 (−7.0%)0.04823 (−58.0%)0.003737 (+5.9%)20.07 (+2.6%)−0.1262 BumpBump 3 1 0.965 0.027828 0.04634 0.003107 20.82 −0.1252 3.5° 0.917 0.008297 0.04642 0.001343 19.75 −0.1184 (h/c(h/c = =0.004, 0.012, xB/c = 0.55)3.5 ◦ (−0.3%) (−25.9%) (−3.9%) (−16.8%) (+3.7%) (+0.8%) (−5.3%) (−77.9%) (−3.8%) (−64.1%) (−1.6%) (+6.2%) xB/c = 0.55) 0.866 0.004527 0.03174 0.000662 27.28 −0.1260 2.5° − BumpBump 4 2 ◦ 0.870(−5.8%) 0.003852(−34.2%) 0.03395(+5.5%) 0.000549(−8.6%) 25.63(−10.6%) (+5.6%)0.1271 2.5 − − − − (h/c(h/c = =0.008, 0.008, xB/c = 0.55) ( 5.3%)0.953 ( 44.0%)0.012188 (+12.8%)0.04484 ( 24.2%)0.001569 ( 16.1%)21.25 (+4.8%)−0.1229 3.5° xB/c = 0.50) 0.925(−1.5%) 0.016171(−67.6%) 0.04559(−7.0%) 0.002168(−58.0%) 20.29(+5.9%) −(+2.6%)0.1193 3.5◦ Bump 3 (−4.4%)0.917 (−56.9%)0.008297 (−5.5%)0.04642 (−42.0%)0.001343 (+1.1%)19.75 (+5.5%)−0.1184 3.5° (h/c = 0.012, xB/c = 0.55) (−5.3%) (−77.9%) (−3.8%) (−64.1%) (−1.6%) (+6.2%) A more interesting phenomenon shown in Table1 is that regardless of whether the lift 0.870 0.003852 0.03395 0.000549 25.63 −0.1271 2.5°and drag are increased or decreased, the RMSs of the coefficients are all suppressed by the (−5.3%) (−44.0%) (+12.8%) (−24.2%) (−16.1%) (+4.8%) Bump 4 bumps. With increasing bump height, the suppression effect of the lift oscillation is more (h/c = 0.008, xB/c = 0.50) 0.925 0.016171 0.04559 0.002168 20.29 −0.1193 ◦ 3.5°significant. In the present computation, Bump 2 has the highest increment of L/D α = 3.5 and a good(−4.4%) suppression (− effect56.9%) on the( shockwave−5.5%) oscillation.(−42.0%) The L/D(+1.1%) of the buffet(+5.5%) case
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Aerospace 2021, 8, x FOR PEER REVIEW 10 of 18 is increased by 5.9%, and the RMS of the lift coefficient fluctuation is decreased by 67.6%; consequently,Figure 12 itshows is considered the time-averaged a good SCB Mach design, contour with of an the appropriate four bumps. height Bump and 1 location.has the lowestFigure Figureheight, 12 12 meaning shows shows the the time-averaged time-averaged shock pattern Mach is Mach similar co contourntour to the of the ofbaseline thefour four bumps. configuration bumps. Bump Bump 1 inhas Figure the 1 has 7a.thelowest A lowest clear height, height,“λ”-type meaning meaning shock the is shock the formed shock pattern near pattern is thesimilar isshockwave similar to the baseline to foot the baselineof configuration Bump configuration 2 and in BumpFigure in4. However,Figure7a. A7 a.clear theA “ clear“λλ”-type” shock “λ ”-typeshock of Bump is shock formed 4 is is stronger formednear the than nearshockwave that the of shockwave Bumpfoot of 2, Bump which foot 2 ofresultsand Bump Bump in 2more 4. and liftBumpHowever, loss 4.and However, lessthe “dragλ” shock the reduction. “λ of” shockBump Bump of4 is Bump stronger 3 has 4 the is than stronger highest that of thanbump Bump that height. 2, of which Bump As resultsshown 2, which in in more Figure results 12c,inlift more the loss bump lift and loss less pushes and drag less reduction.the drag first reduction. shock Bump upstream 3 has Bump the highest 3and has forms the bump highest a secondheight. bump Asshock shown height. on inthe As Figure bump; shown consequently,in12c, Figure the 12bumpc, the pushes bumpaerodynamic the pushes first performanceshock the first upstream shock of upstream thisand bumpforms and isa secondworse forms than ashock second the on others. shockthe bump; on the bump;consequently, consequently, the aerodynamic the aerodynamic performance performance of this ofbump this is bump worse is than worse the than others. the others.
(a) Bump 1 (b) Bump 2 (a) Bump 1 (b) Bump 2
(c)( cBump) Bump 3 3 (d)) BumpBump 4 4
Figure 12. Time-averaged Mach contour of the four bumps (Ma = 0.73, Re = 3.0 × 106,6 α = 3.5°). FigureFigure 12.12. Time-averagedTime-averaged MachMach contourcontour ofof thethe fourfour bumpsbumps (Ma(Ma == 0.73,0.73, ReRe == 3.03.0 ×× 10106, ,αα == 3.5°). 3.5◦). Figure 13 presents the pressure coefficients of the airfoil with bumps. For the drag FigureFigure 1313 presentspresents the the pressure pressure coefficients coefficients of of the the airfoil airfoil with bumps. For For the the drag drag reduction cases (α = 3.5°), the strong shockwave of the baseline configuration is divided reductionreduction cases ((αα == 3.53.5°),◦), thethe strong strong shockwave shockwave of of the th baselinee baseline configuration configuration is divided is divided into into two weak shockwaves. In contrast, for the drag increment cases (α = 2.5°), the shock- intotwo two weak weak shockwaves. shockwaves. In contrast, In contra forst, the for dragthe drag increment increment cases cases (α = ( 2.5α =◦ ),2.5°), the shockwavethe shock- wave is also divided, but the strength of the second shockwave is even stronger than that waveis also is divided, also divided, but the but strength the strength of the of second the second shockwave shockwave is even is strongereven stronger than that than of that the of the baseline airfoil. ofbaseline the baseline airfoil. airfoil.
(a) α = 3.5° (b) α = 2.5°
(a) α = 3.5° (b) α = 2.5° 6 FigureFigure 13. 13.Time-averaged Time-averaged pressure pressure coefficients coefficients for for the the airfoil airfoil with with bumpsbumps (Ma(Ma = 0.73, Re Re = = 3.0 3.0 × ×1010). 6). Figure 13. Time-averaged pressure coefficients for the airfoil with bumps (Ma = 0.73, Re = 3.0 × 106).
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Figure 14 shows the time-averaged wall friction coefficient of the airfoil. It is worth Figure 14 shows the time-averaged wall friction coefficient of the airfoil. It is worth Aerospace 2021, 8, x FOR PEER REVIEWnoting that the wall friction is computed by the wall-modeled equation. The lower11 of 18 surface noting that the wall friction is computed by the wall-modeled equation. The lower surface is hardly influenced by the upper surface bumps, meaning only the lower surface cf of the baselineis hardly airfoil influenced is plotted by the in theupper figure. surface The bumps, wall friction meaning coefficient only the of lower the upper surface surface cf of the is baseline airfoil is plotted in the figure. The wall friction coefficient of the upper surface is stronglyFigure affected 14 shows by the the bumps. time-averaged The friction wall firstfricti declineson coefficient when of approaching the airfoil. It is the worth bump and strongly affected by the bumps. The friction first declines when approaching the bump thennoting increases that the on wall the friction bump, is which computed coincides by the withwall-modeled the tendency equation. of the The flow lower acceleration surface on and then increases on the bump, which coincides with the tendency of the flow accelera- theis bumphardly (Figureinfluenced 12). by the upper surface bumps, meaning only the lower surface cf of the tionbaseline on the airfoil bump is plotted (Figure in 12). the figure. The wall friction coefficient of the upper surface is strongly affected by the bumps. The friction first declines when approaching the bump and then increases on the bump, which coincides with the tendency of the flow accelera- tion on the bump (Figure 12).
FigureFigure 14.14. WallWall friction friction coefficient coefficient of ofthe the baseline baseline airfoi airfoill and andairfoil airfoil with withbumps bumps (Ma = (Ma0.73, =Re 0.73, = 3.0 × 106, α = 3.5°). Re = 3.0 × 106, α = 3.5◦). Figure 14. Wall friction coefficient of the baseline airfoil and airfoil with bumps (Ma = 0.73, Re = 3.0 × 106, α = 3.5°). FigureFigure 15 15presents presents the velocity the velocity profile andprofile Reynolds and stressesReynolds inside stresses the shockwave/boundary inside the shock- wave/boundary layer-induced flow separation region. Two locations at x/c = 0.6 and 0.8 layer-inducedFigure 15 flow presents separation the velocity region. profile Two and locations Reynolds at x/cstresses= 0.6 inside and 0.8 the are shock- presented. Thearewave/boundary velocitypresented. near The layer-induced the velocity bump (nearx/c flow= the 0.6) se bumpparation is decreased (x/c region. = 0.6) by Two is the decreased locations bumps, asatby x/c shownthe = 0.6bumps, inand Figure 0.8 as shown 15a, whileinare Figure presented. thevelocity 15a, while The profile velocity the velocity becomes near the profile bump plump becomes(x/c at =x/c 0.6)= plumpis 0.8 decreased for at Bumps x/c by = the0.8 1 and bumps,for Bumps 2. The as shown RMS1 and of 2. theThe streamwiseRMSin Figure of the 15a, velocitystreamwise while the fluctuation velocity velocity profile fluctuation is equivalent becomes is plumpeq touivalent the at square x/c to = 0.8 the root for square ofBumps the root Reynolds1 and of the2. The Reynolds normal stressnormalRMS < ofu’u’ stress the>. streamwise The
(a) Time-averaged streamwise velocity (a) Time-averaged streamwise velocity
(b) RMS of streamwise velocity fluctuation (c) Reynolds shear stress
Figure 15. Velocity profile and Reynolds stresses of the airfoil with bumps (Ma = 0.73, Re = 3.0 × 106, α = 3.5°). Figure(b 15.) RMSVelocity of streamwise profile and Reynolds velocity stresses fluctuation of the airfoil with bumps (c) Reynolds (Ma = 0.73, shear Re = stress 3.0 × 106, α = 3.5◦). Figure 15. Velocity profile and Reynolds stresses of the airfoil with bumps (Ma = 0.73, Re = 3.0 × 106, α = 3.5°).
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3.3. Fluctuation Characteristics of the Airfoil with Bumps 3.3. Fluctuation CharacteristicsCharacteristics of the Airfoil with BumpsBumps The fluctuations of the baseline airfoil and airfoil with SCBs are compared in this The fluctuations of the baseline airfoil and airfoil with SCBs are compared in this section.The The fluctuations RMS of the of pressure the baseline fluctuation airfoil andon the airfoil upper with surface SCBs is are shown compared in Figure in this16. section. TheThe RMSRMS ofof thethe pressurepressure fluctuation fluctuation on on the the upper upper surface surface is is shown shown in in Figure Figure 16 .16. It It is clear that the pressure fluctuation of the baseline airfoil is significantly suppressed by isIt is clear clear that that the the pressure pressure fluctuation fluctuation of of the the baseline baseline airfoil airfoil is is significantly significantly suppressedsuppressed byby the bumps. Bump 2 has the smallest RMS value of the pressure fluctuation among the five thethe bumps.bumps. Bump 2 has the smallest RMS value of the pressure fluctuationfluctuation amongamong thethe fivefive cases, which is coincident with the fluctuation of the lift and drag coefficients. Not only cases, whichwhich isis coincident coincident with with the the fluctuation fluctuation of of the the lift lift and and drag drag coefficients. coefficients. Not Not only only the the fluctuation near the shockwave but also the fluctuation near the trailing edge is sup- fluctuationthe fluctuation near near the shockwave the shockwave but also butthe also fluctuation the fluctuation near the near trailing the trailing edge is edge suppressed is sup- pressed by Bump 2. Bump 3 and Bump 4 also have favorable suppression effects on the bypressed Bump by 2. Bump Bump 2. 3 andBump Bump 3 and 4 alsoBump have 4 also favorable have favorable suppression suppression effects on effects the pressure on the pressure fluctuation. fluctuation.pressure fluctuation.
Figure 16. RMS of the pressure fluctuation on the upper surface (Ma = 0.73, Re = 3.0 × 106, α = 3.5°). 6 Figure 16. RMS of the pressurepressure fluctuationfluctuation on on the the upper upper surface surface (Ma (Ma = = 0.73, 0.73, Re Re = = 3.0 3.0× × 10106, α = 3.53.5°).◦). Figure 17 shows the RMS contours of the pressure fluctuations for Bump 2 and Bump Figure 17 shows the RMS contours of the pressure fluctuations for Bump 2 and Bump 4. It isFigure clear that17 showsthe high-fluctuation the RMS contours region of near the the pressure shockwave fluctuations is decreased for Bump when 2com- and 4. It is clear that the high-fluctuation region near the shockwave is decreased when com- paredBump with 4. Itthe is baseline clear that configuration the high-fluctuation (Figure 7b). region The near“λ” shock the shockwave region has isa high decreased pres- pared with the baseline configuration (Figure 7b). The “λ” shock regionλ has a high pres- surewhen fluctuation, compared while with the shockwave baseline configuration above the “λ (Figure” shock7 b).is quite The stable. “ ” shock It is interesting region has sure fluctuation, while the shockwave above the “λ” shock is quiteλ stable. It is interesting thata high the pressureshockwave fluctuation, far away from while the the airfoil shockwave fluctuates above greatly. the This “ ” demonstrates shock is quite that stable. the that the shockwave far away from the airfoil fluctuates greatly. This demonstrates that the expansionIt is interesting wave forms that the near shockwave the leading far edge away region, from weakening the airfoil the fluctuates strength greatly.of the shock- This expansion wave forms near the leading edge region, weakening the strength of the shock- wavedemonstrates and inducing that the unsteadiness, expansion wavewhich forms is a basi nearc principle the leading of the edge desi region,gn of weakeninga supercritical the strengthwave and of inducing the shockwave unsteadiness, and inducing which is unsteadiness, a basic principle which of the is desi a basicgn of principle a supercritical of the airfoil [50]. designairfoil [50]. of a supercritical airfoil [50].
(a) Bump 2 (b) Bump 4 (a) Bump 2 (b) Bump 4 Figure 17. RMS contours of the pressure fluctuations for Bump 2 and Bump 4 (Ma = 0.73, Re = 3.0 × 1066, α = 3.5°).◦ FigureFigure 17. 17.RMS RMS contours contours of of the the pressure pressure fluctuations fluctuations for for Bump Bump 2 2 and and Bump Bump 4 4 (Ma (Ma = = 0.73, 0.73, Re Re = = 3.0 3.0× × 10106, α = 3.5°). 3.5 ). ThreeThree streamwise streamwise locations locations on on the the airfoil airfoil are are sampled sampled to to compare compare the the power power spectral spectral Three streamwise locations on the airfoil are sampled to compare the power spectral densitydensity (PSD) (PSD) of of the the pressure pressure fluctuation, fluctuation, as as shown shown in in Figure 18.18. The The locations locations x/cx/c = 0.5 0.5 density (PSD) of the pressure fluctuation, as shown in Figure 18. The locations x/c = 0.5 andand 0.6 0.6 are are near near the the shockwave, shockwave, and and x/cx/c == 0.8 0.8 is is located located in in the the low-speed low-speed region region near near the the and 0.6 are near the shockwave, and x/c = 0.8 is located in the low-speed region near the trailingtrailing edge. edge. The dimensionaldimensional quantitiesquantitiesin in Figure Figure 18 18 are ar scalede scaled based based on on the the length length of theof trailing edge. The dimensional quantities in Figure 18 are scaled based on the length of theexperiment experiment [14 ].[14]. It is It clear is clear that that a peak a peak frequency frequency appears appears on the on baseline the baseline airfoil, airfoil, which which is the the experiment [14]. It is clear that a peak frequency appears on the baseline airfoil, which isbuffet the buffet frequency frequency of the of baseline the baseline airfoil. airfoil. With theWith SCBs, the SCBs the PSD, the of PSD the of pressure the pressure fluctuation fluc- is the buffet frequency of the baseline airfoil. With the SCBs, the PSD of the pressure fluc- tuationis significantly is significantly reduced reduced at x/c = at 0.5. x/c = The 0.5. buffet The buffet frequency frequency of the of baseline the baseline configuration configu- tuation is significantly reduced at x/c = 0.5. The buffet frequency of the baseline configu- rationis clear is evenclear ateven the at downstream the downstreamx/c = x/c 0.8. = Bump0.8. Bump 1 has 1 thehas lowestthe lowest height, height, meaning meaning the ration is clear even at the downstream x/c = 0.8. Bump 1 has the lowest height, meaning
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the buffet is slightly suppressed by Bump 1, and the buffet frequency is slightly increased. buffet is slightly suppressed by Bump 1, and the buffet frequency is slightly increased. Bump 2 is the most effective SCB because there is almost no frequency peak for the Bump Bump 2 is the most effective SCB because there is almost no frequency peak for the Bump 2 2 configuration. configuration.
(a) x/c = 0.5 (b) x/c = 0.6
(c) x/c = 0.8
6 FigureFigure 18. 18.Power Power spectral spectral density density of of the the pressure pressure fluctuations fluctuations at at three three locations locations (Ma (Ma = = 0.73, 0.73, Re Re = = 3.0 3.0× × 10106, α = 3.5°). 3.5◦).
Figure 1919 shows shows the the streamwise streamwise velocity velocity contours contours of of the the first first grid grid layers layers of theof the baseline base- airfoilline airfoil and and airfoil airfoil with with bumps. bumps. As As the the present present computation computation adopted adopted aa wall-modeledwall-modeled simulation, thethe gridgrid spacingspacing of of the the first first grid grid layer layer is is located located at at∆y Δ+y of+ of approximately approximately 50, 50, as shownas shown in Figurein Figure2a; consequently,2a; consequently, the first the gridfirst layergrid layer of the of LES the has LES a considerablyhas a considerably large flowlarge speed. flow speed. The white The white regions regions in the contoursin the contours of Figure of 19Figure are blanked 19 are blanked when u when< 0.0. Fifteenu < 0.0. instantaneousFifteen instantaneous snapshots snapshots with ∆t =with 1.37 Δc/t U=1.37∞ arec/U plotted∞ are plotted along the along vertical the vertical direction direction in each figure.in each Itfigure. is clear It thatis clear the high-speedthat the high-speed region of region the baseline of theairfoil baseline (Figure airfoil 19 (Figurea) oscillates 19a) periodically.oscillates periodically. When Bump When 1 is installedBump 1 onis inst thealled upper on surface the upper (Figure surface 19b), (Figure the oscillation 19b), the is weakened,oscillation butis weakened, periodical but oscillation periodical still oscillation exists. The still high-speed exists. The regions high-speed of Bumps regions 2, 3 and of 4Bumps are almost 2, 3 and constant 4 are almost over time. constant However, over time. strong However, flow acceleration strong flow regions acceleration appear regions at the bumpappear locations at the bump of Bumps locations 3 and of Bumps 4. Moreover, 3 and an4. Moreover, instantaneous an instantaneous reverse flow (reverseu < 0.0) flow can be(u < seen 0.0) beforecan be Bumpsseen before 3 and Bumps 4 at 0.4 3 Aerospace 2021, 8, x FOR PEER REVIEW 14 of 18 Aerospace 2021, 8, 203 14 of 18 (a) Baseline (b) Bump 1 (c) Bump 2 (d) Bump 3 (e) Bump 4 FigureFigure 19. 19.Instantaneous Instantaneous streamwise streamwise velocity velocity of of the the first first LES LES grid grid layer layer near near the the wall wall (Ma (Ma = = 0.73, 0.73, Re Re = = 3.0 3.0× × 106,, α = 3.5°). 3.5◦). Figure 20 shows thethe instantaneousinstantaneous streamwisestreamwise velocityvelocity contourscontours ofof thethe airfoils.airfoils. TheThe unsteady flowflow separation separation near near the the trailing trailing edge ed isge strong is strong for the for baseline the baseline airfoil. airfoil. With bumpWith installation,bump installation, trailing trailing edge separation edge separation is suppressed. is suppressed. A “λ”-type A “ shockwaveλ”-type shockwave can be seen can near be Bumpseen near 2. A Bump secondary 2. A secondary shockwave shockwave is formed foris formed Bumps for 3 and Bumps 4. 3 and 4. AerospaceAerospace 20212021, ,88, ,x 203 FOR PEER REVIEW 1515 of of 18 18 (a) Baseline (b) Bump 1 (c) Bump 2 (d) Bump 3 (e) Bump 4 6 FigureFigure 20. 20. InstantaneousInstantaneous streamwise streamwise velocity velocity contours contours of ofthe the airfoil airfoil (Ma (Ma = 0.73, = 0.73, Re Re= 3.0 = × 3.0 10×, α10 =6 , 3.5°). α = 3.5◦). Δ=ρρρ − FigureFigure 21 21 shows shows the the instantaneous instantaneous density fluctuationsfluctuations ((∆ρ = ρ − ρAVGAVG) of) of the the baseline base- lineairfoil airfoil and and Bump Bump 2 configuration. 2 configuration. The soundThe sound wave wave propagates propagates from from the trailing the trailing edge, edge, which Aerospace 2021, 8, x FOR PEER REVIEW 16 of 18 Aerospace 2021, 8, 203 16 of 18 which is called the Kutta wave and is clear on the lower side of the airfoil. It is not so clear is called the Kutta wave and is clear on the lower side of the airfoil. It is not so clear on on the upper side because the flow phenomenon is complex, and many different waves the upper side because the flow phenomenon is complex, and many different waves (shockwave, expansion wave, sound wave, etc.) are mixed together. It can be seen from (shockwave, expansion wave, sound wave, etc.) are mixed together. It can be seen from the lower side that the amplitude of the Kutta wave is weakened by the bump, and the the lower side that the amplitude of the Kutta wave is weakened by the bump, and the characteristic wavelength is also changed; consequently, the buffet strength of the airfoil characteristic wavelength is also changed; consequently, the buffet strength of the airfoil is is weakened, which confirms the buffet mechanism explained by Lee et al. [10]. weakened, which confirms the buffet mechanism explained by Lee et al. [10]. (a) Baseline (b) Bump 2 6 FigureFigure 21.21.Instantaneous Instantaneous KuttaKutta wavewave propagating propagating from from the the trailing trailing edge edge (Ma (Ma = = 0.73, 0.73, Re Re = = 3.03.0 ×× 106,, αα == 3.5°). 3.5◦). 4. Conclusions 4. Conclusions This paper adopted a wall-modeled large eddy simulation method to analyze the This paper adopted a wall-modeled large eddy simulation method to analyze the buffetbuffet control control effect effect of of a ashockwave shockwave control control bump. bump. This This work work can canbe summarized be summarized as fol- as lows:follows: (1)(1) The The numerical method method was was validated by by the baseline OAT15A airfoil. With the present wall-modeled equation, the grid requir requirementement of the LES near the wall was greatly greatly reduced.reduced. The The pressure pressure coefficient coefficient and RMS of the velocity fluctuation fluctuation matched well with thethe experiment, experiment, which which validates the relia reliabilitybility of the present computation. (2)(2) Four Four different different two-dimensional bumps bumps were were numerically numerically tested tested and and analyzed. analyzed. The time-averaged aerodynamic coefficients coefficients sh showedowed that that the SCB was beneficial beneficial for the lift-to-draglift-to-drag ratio of of the the buffet buffet condition condition but but decreased decreased the the lift-to-drag lift-to-drag ratio of of the the nonbuffet nonbuffet case.case. The The fluctuations fluctuations of of the the lift lift and and drag drag coefficients coefficients can be gr greatlyeatly suppressed by the bumps.bumps. An An appropriate appropriate “ “λλ”” shockwave pattern pattern is is important important for for the control effect of the SCB.SCB. (3)(3) The The instantaneous instantaneous flow flow characteristics characteristics of of the the airfoil airfoil with with bumps bumps were were compared. compared. The power spectral density density of of the the pressure pressure fl fluctuationuctuation demonstrated demonstrated that that the the buffet buffet fre- fre- quency is almost eliminated by an appropriate bump. The instantaneous flow flow fields fields and KuttaKutta wave explain the flow flow control mechanism of the SCB. Author Contributions: Conceptualization, Y.Z. andand H.C.;H.C.; methodology,methodology, Y.Z.; Y.Z.; software, software, P.Y.; P.Y.; valida- vali- dation,tion, P.Y. P.Y. and and R.L.; R.L.; data data curation, curation, Y.Z.; Y.Z.; project project administration, administration, H.C.; H.C.; funding funding acquisition, acquisition, Y.Z. Y.Z. and H.C.and H.C.All authors All authors have have read andread agreedand agreed to the to published the published version version of the of manuscript. the manuscript. Funding:Funding: ThisThis research waswas fundedfunded by by National National Natural Natural Science Science Foundation Foundation of China,of China, grant grant numbers num- bers11872230, 11872230, 91852108, 91852108, 92052203 92052203 and 91952302.and 91952302. 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