Instability analysis and drag coefficient prediction on a swept RAE2822 wing with constant lift coefficient

Zhenrong JING,Zhngfeng HUANG,b,*

aDepartment of Mechanics,Tianjin University,Tianjin 300072,China

bState Key Laboratory of Aerodynamics,China Aerodynamics Research and Development Center,Mianyang 621000,China

Instability analysis and drag coefficient prediction on a swept RAE2822 wing with constant lift coefficient

Zhenrong JINGa,Zhangfeng HUANGa,b,*

aDepartment of Mechanics,Tianjin University,Tianjin 300072,China

bState Key Laboratory of Aerodynamics,China Aerodynamics Research and Development Center,Mianyang 621000,China

Available online 20 April 2017

*Corresponding author at:Department of Mechanics,Tianjin University,Tianjin 300072,China.

E-mail address:hzf@tju.edu.cn(Z.HUANG).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2017.03.002

1000-9361©2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Swept wing is widely used in civil aircraft,whose airfoil is chosen,designed and optimized to increase the cruise speed and decrease the drag coefficients.The parameters of swept wing,such as sweep angle and angle of attack,are determined according to the cruise lift coefficients requirement,and the drag coefficients is expected to be predicted accurately,which involves the instability characteristics and transition position of the flow.The pressure coefficients of the RAE2822 wing with given constant lift coefficients is obtained by solving the three-dimensional Navier-Stokes equation numerically,and then the mean flow is calculated by solving the boundary layer(BL)equation with spectral method.The cross-flow instability characteristic of boundary layer of swept wing in the windward and leeward is analyzed by linear stability theory(LST),and the transition position is predicted by eNmethod.The drag coefficients is numerically predicted by introducing a laminar/turbulent indicator.A simple approach to calculate the lift coefficients of swept wing is proposed.It is found that there is a quantitative relationship between the angle of attack and sweep angle when the lift coefficient keeps constant;when the angle of attack is small,the flow on the leeward of the wing is stable.when the angle of attack is larger than 3°,the flow becomes unstable quickly;with the increase of sweep angle or angle of attack the disturbance on the windward becomes more unstable,leading to the moving forward of the transition position to the leading edge of the wing;the drag coefficients has two significant jumping growth due to the successive occurrence of transition in the windward and the leeward;the optimal range of sweep angle for civil aircraft is suggested.

©2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Constant lift coefficients;

Cross-flow instability;

Drag coefficients;

Linear stability theory;

Swept RAE2822 wing

1.Introduction

Ever since the famous Reynolds’experiment,an enormous amount of attention has been paid to the laminar-turbulent transition because of both of its scientific interests and its engineering benefits.The frictional drag is found to be about 50%of the total drag in the modern aircraft,1so the flow on the aircraft is expected to be laminar flow as far as possible,especially for the aircraft with swept wing.In order to increase the flight speed and overcome the sonic barrier when the flight speed approaches to the speed of sound,swept wing is widely used in the modern civil aircraft.Due to the sweep angle and the pressure gradient,the velocity normal to the potential flow direction appears to be nonzero,leading to the occurrence of cross-flow.Different from the instability characteristic of boundary layer on a flat plate,which is viscid instability,the instability of cross-flow is inviscid instability due to the existence of inflection point in the cross-flow profile.2The secondary instability appears as the disturbance reaches a certain big amplitude and leads to the modification of the mean flow.The instability of cross-flow plays a key role in the breakdown process of laminar-turbulent transition on swept wing.There are two kinds of unstable waves in the cross-flow,namely stationary wave and travelling wave,while the transition is induced mainly by the stationary wave as the turbulence in the real flight environment of civil aircraft is very small.3

Much effort has been made to understand the mechanism of the instability of cross-flow through experiment(EXP),linear stability theory(LST),parabolized stability equation(PSE)and direct numerical simulation(DNS).Runyan et al.4analyzed the data of 34 selected flight test cases on the F-111 TACT airplane and studied the instability of cross-flow and the Tollmien-Schlichting(T-S)disturbance.They found that laminar flow could extend a relatively large region on the leeward(typically from 21%to 58%).Nitschke-Kowsky and Bippes5studied the cross-flow instability features on a swept flat plate by adding a displacement body to induce a pressure gradient,and found that both the stationary and travelling instability waves observed in the experiment can be correctly predicted by LST.Dagenhart and Saric6measured the shape,wavelength and frequency of disturbance on the windward boundary layer flow on a 45°swept wing with-4°angle of attack,and found that the wavelengths observed in the flow visualization studies are smaller than that predicted by LST by approximately 20%.Haynes7used the nonlinear parabolized stability equation(NPSE)to investigate the saturation and nonlinear stage of the boundary layer(BL)flow over an NLF(2)-0415 swept wing at-4°angle of attack,and his results showed excellent agreement with experimental and theoretical results.Malik et al.8performed the two-dimension eigenvalue calculation to study the secondary instability and found that all the secondary instability modes can be classified into two families:zmode andymode.White and Saric3studied experimentally the secondary instability modes for cross-flow and observed the so-called type-I and type-II modes,which correspond tozmode andymode,respectively.Zuo et al.9investigated the cross-flow instability on swept wing using sublimation method and found that both T-S instability and cross-flow instability can happen on swept wing depending on the Reynold’s number and roughness.Sun and Huang10studied the effect of the sweep angle on the stability and transition on the NACA0012 wing and found that the strength of the cross-flow reaches its peak value when the sweep angle is in the range of 40–50°.The articles of Bippes11,Saric et al.12,13provide a comprehensive overview of various aspects about cross-flow instability.

In order to predict the transition position,Smith and Gamberoni14and Van Ingen15developed eNmethod based on the linear stability theory independently,which has been found to be quite successful concerning the transition position on a flat plate,and has been extended to compressible flow and cross-flow.But eNmethod has been widely criticized since many important aspects of the transition process,such as receptivity,non-parallel effect,nonlinear interaction,and secondary instability,are neglected.As for cross-flow instability,the predicted growth rate and the final transition location usually show great discrepancies with experimental or numerical results.Nonetheless,just as Arnal and Casalis16stated there is no other practical method presently available for industrial application.eNmethod is still used by many researchers to predict the transition position.Dagenhart and Saric6predicted the transition position experimentally in the windward boundary layer flow on a 45°swept wing at a small angle of attack,and found that the maximum theoreticalNfactors for travelling cross-flow at transition range from 8.5 to 9.1,which is in an agreement with the prediction of LST,while the correspondingNfactor for the dominant stationary cross-flow vortices are in the range of 6.4–6.8.Malik et al.8studied the secondary instability of cross-flow vortices and the transition of swept wing boundary-layer,and found that theNfactor correlation based on secondary instability growth rates may yield a more robust criterion for transition onset prediction.Huang et al.17studied the cross-flow instability and transition location on the boundary layer of a swept wing and found that transition occurs in the leading edge of the wing and the wavelength of the inducing wave is about 2 mm.Arnal et al.18studied the effect of suction on the transition position;they found that theNfactor is about 6–10,and theNfactor can be reduced significantly by suction.Jing et al.19studied the effect of the small angle of attack on the stability and transition in a swept wing,and found that transitionfirstly occurs in the windward with theNfactor about 6.Xia and Chen20found that transitional phenomena can be reasonably demonstrated by implementing γ-Reθttransition model into computational fluid dynamics(CFD)simulation,and the results are significantly improved compared with full laminar or full turbulent simulation.Xu et al.21established a transition model by combining linear stability theory and shear strain transport(SST)turbulence model,and the model was validated by experiments for several swept wings.Song et al.22integrated the eNmethod based on LST into Reynolds-averaged Navier-Stokes(RANS)equations solver to calculate the drag more accurately and minimize the drag by changing the shape of the airfoil.The aim of instability analysis and transition prediction of crossflow is to control the flow and reduce the drag coefficients.Joslin23summarized the laminar flow control(LFC)techniques and the results of flight tests conducted by Boeing and Airbus,and pointed out that suction technique can successfully delay the onset of transition.Friederich and Kloker24investigated the effect of suction on swept wing by DNS and found that not only the primary cross-flow instability but also the secondary instability is controlled,leading to a significant delay of transition.Other LFC techniques,such as cooling wall25and roughness26are also studied.

The instability of cross-flow,the transition prediction and the laminar flow control have been investigated for many years.While in the design and optimization process of civil air-craft,the lift coefficients has been establishedfirstly based on the flight requirements,then the parameters of the swept wing,such as sweep angle and angle of attack,are determined according to the cruise lift coefficients,and drag coefficients is expected to be predicted accurately,which involves the instability characteristics and transition position of the flow.In the preliminary wing design,the data are often taken from catalogues or handbooks,whose data usually based on the two dimensional wind-tunnel measurements.The effect of sweep angle is then taken into account by using cosine law.27However,this is not justified since the presence of sweep angle would reduce the Mach number normal to the leading edge.Therefore,the aerodynamic characteristics are changed.As a result,this method is usually of poor accuracy.So in this paper,we propose a corrected relationship among the lift coefficients,the sweep angle,and the angle of attack,study the cross-flow instability on a swept RAE2822 subcritical wing with constant lift coefficients,calculate the drag coefficients accurately with the predicted transition position,and discuss the reasonable sweep angle or angle of attack in the design process of civil aircraft wing.

The structure of the paper is organized as follows.In Section 2,a brief description of numerical methods is presented,including pressure coefficients calculation,mean flow calculation,linear stability theory,eNmethod,drag coefficients calculation and parameters chosen.In Section 3,a simple approach to calculate the lift coefficients of swept wing is proposed,the cross-flow instability is analyzed and the transition position is predicted,and the drag coefficients is predicted by combining LST,eNmethod and numerical simulation.Section 4 summarizes the main conclusions.

2.Numerical method

2.1.Aerodynamic parameters calculation

In order to obtain the aerodynamic characteristics of the RAE2822 wing,finite volume method(FVM)was applied to solving the Navier-Stokes equation,which takes the following conservative form:

whereUis the conservative variable,x,y,zare the Cartesian coordinate,Ei(i=1,2,3)represents the convective terms and theEvi(i=1,2,3)viscous terms.The convective terms were split by Steger-Warming method and discretized by a 3rdorder weighted essentially non-oscillatory(WENO)scheme.The viscous terms were calculated by a 3rd order center scheme.The governing equation was integrated fully and implicitly in time,using a lower-upper Symmetric-Gauss-Seidel(LU-SGS)method.The code was verified by experiment data conducted by Cook et al.28,which has been widely used to test and validate the code and turbulent model in CFD.29,30The parameters of the 9th case in the experiment of Cook et al.28are the Mach numberMa=0.730,the chord Reynolds numberRe=6.5 × 106and the angle of attack α =3.19°.Due to the existence of transition trip at the leading edge,most of the flow in the experiment is turbulent,so the Spalart-Allmaras(S-A)turbulent model was used in our simulation.

The correction effect of the wind tunnel on the original data suggested by Abid was also adopted in our result.

Fig.1 shows a comparison of pressure coefficientsCpbetween the present calculation and the experimental result,in which there is a good agreement in most of the domain,wherecis the chord length.The corresponding lift coefficientsCLand the drag coefficientsCDare listed in Table 1,in which the result simulated by Baldwin-Lomax model is also given for comparison29.The lift coefficients of simulation is a little smaller than that of experiment while the drag coefficients has an opposite result.The difference of turbulent model in the simulation leads to a little difference of the lift coefficients and the drag coefficients,while our drag coefficients is closer to the experimental result.

2.2.Mean flow calculation

The boundary layer(BL)equation was solved by using spectral method to calculate the mean flow.Compared to DNS of the full Navier-Stokes equation,the BL equation has many advantages to calculate the mean flow:(A)it is efficient to obtain the mean flow,because the BL equation is parabolic and hence can be solved by a marching method,while DNS is dominated in time and space and requires a long computation time and a high simulation cost;(B)it has high accuracy,because not only the quantities of the mean flow but also their derivatives are required by the LST and the results of LST are sensitive to the first and second order derivatives.The BL equation can be solved by spectral method to ensure the second order derivatives of the quantities being naturally continuous,while the precision of the second-order derivatives in DNS depends on the difference schemes;(C)it is feasible in the calculation of the swept wing with curvature and has been widely used.

Table 1 Comparison of lift coefficients CLand drag coefficients CD.

The quasi-three dimensional compressible BL equation was proposed by Pruett31and can be written as

where ξ is the stream-wise direction coordinate,and η the normal direction coordinate,qthe curvature in boundary layer equation,Prthe Prandtl number,μ the viscosity coefficients,F,V,G,θ are the variables in boundary layer equation,Cc1,Cc2,Cx1,Ce1,Ce2the coefficients in boundary layer equation.The meanings of other terms and symbols can be found in the research of Pruett.31The coefficients in Eq.(2)are determined by the boundary layer edge quantities and can be calculated through the pressure coefficientsCp,which can be obtained with little effort.The equation is parabolic in the streamwise direction and can be solved by marching method,in which the derivatives at stream-wise direction are approximated by a 3rd-order backward finite difference except for the beginning two steps.At each marching station,the original problems are equal to a set of ordinary boundary value problems and are solved by Chebyshev collocation method.More details can be found in the paper of Pruett.

Ourcodewasverifiedbythecaseofsweptcylinder,forwhich DNS has been done DNS by Balakumar and King.32The profiles of cross-flow and density obtained by two methods are shown in Fig.2,in which a good agreement between our result andtheDNSresultcanbe seen,wherewtisthecross-flowvelocity,u∞the freestream velocity,ρ the density,ρ∞the freestream density,~ω the angle of location on the swept cylinder.

2.3.Disturbance equation

Although DNS needs a lot of time and computation,it can provide all the information about the flow field.We performed DNS for one case in order to compare the results with LST and PSE.The DNS code is based on the so-called ‘disturbance equation”.Suppose any instantaneous flow state can be divided into the basic component and the disturbance component:

whereu,v,ware the velocity components in Cartesian coordinate,Tthe temperature,φ the disturbance vector,the over bar denotes the basic state quantities while the prime,the disturbance quantities.Stability or instability is then delineated according to whether the disturbances decay or grow.Substituting Eq.(3)into the Navier-Stokes equation and subtracting the basic flow equation,disturbance equation can be obtained,which shares the similar conservative form with the Navier-Stokes equation and can be solved by standard numerical method.

2.4.Linear stability theory

Since the disturbances in the cruising environment of civil aircraft are usually very small,the transition on the swept wing is always natural transition,whose early stage can be analyzed by LST.Under the local parallel assumption,the disturbance can be written in the normal mode form:

where φ is the eigenfunction vector, α′the streamwise wavenumber of disturbance,β the spanwise wavenumber of disturbance,ω the frequency of disturbance.Substituting the disturbance in the normal mode into the linearized Navier-Stokes equation yields a set of ordinary differential equations,which arefurthersimplified into thewell-known Orr-Sommerfeld(O-S)equation.The latter forms,along with the homogeneous boundary conditions,an eigenvalue problem,as

The eigenvalue problem is to find the complex eigenvalue α′and the corresponding eigenvector φ for a given β and ω with the mean flow φ and its derivatives in the normal direction at arbitrary locationx.

2.5.eNmethod

The most common method to predict the transition position in the aviation industry is eNmethod,which is a semi-theoretical and semi-empirical method.The theoretical principle of eNmethod is LST,which can provide the growth rate-(imagine part of streamwise wave number)to quantitatively describe the instability of the mean flow.Based on the growth rate,theNfactor in eNmethod can be calculated by

wherex0is the location where the disturbance begins to grow or a reference location.

Fig.3 shows theNfactor curves obtained by LST for the stationary disturbance along the cylinder,in which the results given by Balakumar and King32are also presented for comparison.It can be seen that the curves of our results for different span-wise wave lengths are almost overlapped with those of Balakumar and King.32

2.6.Parabolic stability equation

Unlike DNS,which needs to be solved in the whole domain,the parabolic stability equation developed by Herbert and Bertolotti33can be solved simply by marching downstream,therefore much resource and time can be saved by this method.We only present the basic idea of PSE theory here.More comprehensive description of the method can be found in the researches of Haynes7and Chang.34

The disturbance is assumed to be periodic in both time and span-wise direction,and then it can be then written as Fourierseries expansion:

whereNzandNtare the modes numbers retained in the truncated series.The shape function ψ（x，y）in Eq.(7)is now a function of both normal direction and streamwise direction while the φ（y）in Eq.(4)is only dependent on normal direction.Unlike LST,α′in PSE is dependent on the stream-wise location in order to preserve the history effect of the disturbance.By neglecting the secondary derivative of the shape function in stream-wise direction,we arrived at a set of parabolic partial differential equations.

2.7.Drag coefficients calculation

Drag coefficients is one of the key parameters in the aircraft design and optimization,which is difficult to calculate due to the uncertainty of the transition position.Usually an artificial transition position is given and fixed in the calculation of drag coefficients,before which the flow is assumed laminar while after that the flow is set to be turbulent.We combined the linear stability analysis and transition prediction into the drag coefficients calculation by five steps:(A)calculating the pressure coefficients by solving the Navier-Stokes equation with the assumption of laminar flow at first;(B)calculating the mean flow by solving the boundary layer equation with the pressure coefficients;(C)performing linear stability analysis to obtain theNfactor based on the laminar part of the mean flow;(D)introducing an indicator determined by theNfactor and a thresholdNtto distinguish the laminar/turbulent part and control the source term in S-A turbulent model;(E)recalculating the mean flow until reaching a statistically stationary drag coefficients.

2.8.Parameters chosen and configuration model

The study object is selected to be swept RAE2822 subcritical wing,whose chord isc=1 m.The air parameters are those at the cruising height for the civil aircraft,namely the heightH=104m.Two different cases are considered in this paper.For case 1,a relatively small lift coefficientsCL=0.24 and a Mach numberMa=0.7 are selected.This would permit us to calculate a large range of sweep angle from Λ =25°to Λ =50°.Case 2 is more close to real flight condition,namelyCL=0.5 andMa=0.8.

The configuration model and coordinate systems are illustrated in Fig.4.The angle of attack α is based onzaxis.Sweep angle is based on thezcaxis.xtis parallel with the local potential flow direction andztis perpendicular to the potential flow(cross-flow direction).The adiabatic and non-slip boundary condition is applied to the surface of wing.Since the mean flow is treated as laminar flow,it is calculated without any turbulent model.Due to the variation of the lift coefficients with both the angle of attack α and the sweep angle Λ,we fixed the sweep angle Λ firstly,and then the angle of attack α was adjusted to ensure the relative error to the constant lift coefficientsCLbeing less than 0.1%.

3.Results

3.1.Pressure coefficients

Fig.5 shows the distribution of pressure coefficientsCpfor the two cases.The step size of sweep angle for case 1 is 5°and case 2 is 4°.The two cases show a similar trend as the sweep angle increases.When the sweep angle Λ is small,the curve of pressure coefficients in the leeward of the wing appears to be a ‘rooftop’over a half of the chord.The positive pressure gradient in the windward becomes weaker as the sweep angle Λ increases,while an adversary pressure gradient appears at the leading edge in the leeward with the increase of the sweep angle Λ.The peak of-Cpat the leading edge in the leeward is so-called ‘suction tip’,which should be avoided in the wing design.35In the case ofMa=0.8,shock wave appears when Λ=38°.

3.2.Curves of constant lift coefficients

It has been proven by experiments and classic lift line theory that the lift coefficientsCLvaries linearly with the angle of attack α when the α is small.According to the simple sweep theory,the Mach number normal to the leading edge is reduced toMan=Ma∞cosΛ due to the presence of the sweep angle Λ.Based on the former two facts,the conventional relationship between the lift coefficientsCL,the angle of attack α and the sweep angle Λ can be written as36

whereCL0is the lift coefficients of the wing when the sweep angle Λ =0°and the angle of attack α =0°,kthe lift slope of lift curve with angle of attack α =0°.In the conventional relationship of lift coefficients of Eq.(9),the coefficientsCL0and the slopekare assumed to be a constant.

In the case ofMa=0.7,more lift coefficientsCL=0.26,0.28 and 0.30 are calculated.Fig.6 shows the variation of the lift coefficientsCLwith the angle of attack α for different sweep angles Λ.It can be seen that the coefficientsbis nearly a constant and the normalized lift coefficientsCL/cos2Λ does appear a linear dependence on the angle of attack α for a given sweep angle Λ,while the slopekvaries with the sweep angle Λ.Because the main idea of swept wing is to reduce the normal Mach number and then the compressibility of the flow,the lift slopekshould be affected by the sweep angle Λ.Furthermore,in Fig.6 it can be seen that the lift slopekseems to be linear with the sweep angle Λ.If the slopekcan be expressed as linear function of sweep angle Λ,namely,

the conventional lift coefficients of Eq.(9)can be corrected to be

wherea,dandbare constants determined by experimental or numerical results.Based on our simulation results for case 1,the constants fitted by the leastsquare method area=-0.0011,b=0.0255,andd=0.1725.

Fig.7 shows the contours of the lift coefficients in the plane of sweep angle and angle of attack obtained by the correction of the lift coefficients in Eq.(10),in which the results of the conventional relationship of lift coefficients in Eq.(8)and the predictions by our numerical method for different lift coefficients are also given for comparison.The correction of the lift coefficients in Eq.(10)and the predictions by our numerical method agree quite well,while a significant difference between the conventional relationship of lift coefficients in Eq.(8)and the predictions by our numerical method can be seen in Fig.7.

Eq.(8)determines the quantitative relationship among the lift coefficients,the sweep angle and the angle of attack,and can be used to calculate the lift coefficients with an arbitrary swept or angle of attack easily.For real engineering application,lift is a more interested quantities instead of lift coefficients.It can be obtained by integrating the lift coefficients with the area of the wing.For finite wing,the infinite-span assumption is a good one except for the region near wing root and tip.It is expected that the propose method should give an accurate result for finite wing as well.However,more study for finite wing is needed in the future.

Because the angle of attack has a determined relationship with the sweep angle for a given lift coefficients,only the effects of sweep angle under a constant lift coefficients are presented in this paper.

3.3.Base flow and instability analysis

The base flow for stability analysis is calculated by solving boundary layer equations afterCpis obtained.It should be noted that when the adversary pressure gradient reaches certain value in the leeward,flow separation happens.Since boundary layer equation is parabolic,the calculation would be divergent when the adversary pressure gradient is too strong.For case 2,the shock wave and adversary pressure gradient which appears when Λ =38°make the boundary layer equation fail.So we added a case Λ =36°instead.

Fig.8 presents the profile of cross-flow atx/c=0.1 for the two cases with different sweep angles Λ.The two cases show similar trends with the variation of sweep angle.It can be seen that the peak value of the cross-flow and its location in the windward increase as the increase of the sweep angle,indicating that the sweep angle enhances the instability of cross-flow in windward and leads to the growth of the boundary layer thickness.In the leeward the profile of cross-flow decreases from a positive value to a negative one as the increase of the sweep angle due to the effect of negative pressure gradient.However,the boundary layer thickness for case 2 is smaller than case 1 due to a larger Mach number.

When the sweep angle is small,the absolute peak value of cross-flow in the windward is larger than that in the leeward.The instability of cross-flow belongs to inflection instability,and the strength of instability of cross-flow strongly relates to the peak value of the cross-flow.Therefore,the instability in the windward is stronger than that in the leeward when the sweep angle is small,indicating that transition would occur firstly at the windward for a small sweep angle.When the sweep angle is small,the corresponding angle of attack under a constant lift coefficients is also small.Because the angle of attack in the experiment of Runyan et al.4is small,the conclusion that laminar flow can hold on for a long region in the leeward is consistent with our analysis.While when the sweep angle is big enough or the angle of attack is larger than 3°,the absolute peak value of cross-flow in the leeward becomes larger than that in the windward,indicating that the instability of cross-flow will play a dominate role.

Fig.9 shows the neutral curves in thex-ω plane with different sweep angles.The span-wise wave-number β=2 for case 1 and β=3 for case 2.For both cases,the neutral frequencies become larger as the increase of sweep angle,indicating that more disturbances become unstable for a larger sweep angle.However,the unstable region for case 2 is much smaller in chord-wise due to the small sweep angle of case 2.In the leeward,unstable disturbances cannot be found when the sweep angle is small,and begins to appear until the sweep angle reaches 50°for case 1 and 36°for case 2.Different from the neutral curve in the windward,in which the frequency of the neutral curve decays in the direction of chord,the neutral curve in the leeward appears to be a horizontal line in most regions for both cases.

Fig.10 presents the distribution of the span-wise wavenumber for the most unstable stationary waves.Case 2 has an overall larger span-wise wave number,but the unstable region is restricted to bex＜0.1 when Λ is less than 30°.The two cases show a similar trend for wave-number distribution.In the windward,the span-wise wavenumber of the most unstable stationary wave decreases monotonously with the increase of the sweep angle and the distance to the leading edge.In the leeward,the wave-numbers also become smaller as the position moves downstream.When Mach numberMa=0.8 and Λ=36°,there is a rapid change for β,which is a result of rapid-changing pressure gradient and base flow.

Fig.11 compares the results of DNS,LPSE and LST for two different span-wise wave numbers.Since DNS is quite resource and time-consuming,it is only performed for the caseMa=0.7 and Λ=45°.It can be clearly seen that DNS and LPSE results agree quite well,while the amplitude calculated by LST is much smaller than that by DNS or LPSE.

3.4.Transition prediction

Fig.12 shows theNfactor in eNmethod of the stationary wave for different sweep angles.For case 1,it can be seen that theNfactor atx/c=0.3 in the windward increases from 1 to 16 as the sweep angle increases from 25°to 50°.In the leeward,all of the disturbances decay in the whole region when the sweep angle is less than 45°,while theNfactor can reach 12 when sweep angle is 50°,indicating that the negative pressure gradient has a strong effect on the instability of the flow in the leeward with large sweep angle.For case 2,when the sweep angles are smaller than 30°,theNfactors could not exceed 4.When the sweep angle equals 36°,theNfactor can reach 10 for windward and 8 for leeward.

Fig.13 shows the transition position predicted by eNmethod with the threshold valueNt=7.For case 1,transition occurs firstly in the windward when sweep angle Λ=35°,and the transition position moves to the leading edge with the increase of sweep angle.The transition happens aroundx/c=0.12 when the sweep angle equals 50°for the leeward.For case 2,the transition happens when sweep angle is larger than 36°in the windward,and the position moves forward as well with the increase of sweep angle.For the leeward,transition happens aroundx/c=0.30 when Λ =36°

3.5.Drag coefficients prediction

Fig.14 shows the variation of drag coefficientsCDof laminar/-turbulent flow predicted by our combination simulation.The distribution of drag coefficients of full laminar flow is also given in dashed line for comparison.For Mach numberMa=0.7 case,another set of data forCL=0.30 is also given.It can be seen that with the increase of sweep angle,the drag coefficients of full laminar flow decrease slightly and monotonously as sweep angle becomes larger.However,theCDof laminar/-turbulent flow has two significant jumping growth.When the sweep angle is smaller than 27.5°and the corresponding angle of attack is less than 1°for both lift coefficients,transition does not occur in the whole domain and the flow on the wing is full laminar flow.As the sweep angle increases from 27.5°to 32.5°,transition occurs firstly in the windward and the flow in the windward is turbulent flow after the transition point,leading to the first jump of drag coefficients.When the sweep angle is larger than 42.5°and the corresponding angle of attack is bigger than 3°,another transition occurs in the leeward and the flow in both the windward and leeward is laminar/turbulent flow,resulting in the second jump of drag coefficients.The larger the constant lift coefficients is,the smaller the sweep angle or the angle of attack is required for the occurrence of transition.

For Mach numberMa=0.8 case,when the flow is full laminar,there is a minimum drag coefficients around Λ=32°in the considered sweep angle range.The small increase ofCDafter Λ=32°might be related to the large angle of attack.For laminar/turbulent flow,because the sweep angles for transition appearing on both sides of the wing are close to each other,there is no ‘step” forCDas in the laminar flow.

Based on the result of caseMa=0.7,the sweep angle should be smaller than 25°on the concept of drag coefficients reduction,while the cruise Mach number is difficult to increase for a small sweep angle.When the sweep angle is larger than 25°but smaller than 35°,the transition position in the windward is sensitive to the sweep angle.And sweep angle larger than 40°leads to the occurrence of another transition in the leeward,which will greatly increase the drag coefficients.For case 2,a sweep angle smaller than 30°would be sufficient for cross-flow instability not being amplified for both sides of the wing.It should be noted that this study focuses on the general trend of instability and drag variation instead of any specific values.Different wings might exhibit different stability characteristics even for the same flight condition.The determination of sweep angle needs to take other factors in consideration in real engineering design.

4.Conclusions

In this paper,the stability analysis of cross-flow,the prediction of transition and the simulation of drag coefficients were performed on a swept RAE2822 wing with constant lift coefficients.

(1)An efficient,accurate and feasible approach is adapted to calculate the mean flow by solving the BL equation with spectral method through the pressure coefficients obtained by solving the three-dimensional Navier-Stokes equation numerically,and the results are verified by the experimental result of Cook et al.28and the simulation results of Abbas and Balakumar.

(2)A simple approach to calculate the lift coefficients of swept wing was proposed.Based on the classic lift line theory and the simple sweep theory,the lift coefficients has a quantitative relationship with the angle of attack and the sweep angle,in which the slope of the curve of normalized lift coefficients with the angle of attack is a constant.Based on our observation,we build a linear expression between the slope with the sweep angle and correct the conventional relationship of lift coefficients.The correction of the lift coefficients and the predictions by our numerical method agree quite well.

(3)The instability characteristics of the cross-flow were analyzed and the transition position was predicted for different sweep angles and lift coefficients.It is found that with the increase of sweep angle or angle of attack,the disturbance in the windward becomes more unstable,leading to the moving forward of the transition position to the leading edge of the wing.When the angle of attack is small,the disturbance in the leeward of the wing is stable,while when the angle of attack is larger than 3°the disturbance becomes unstable rapidly,leading to the occurrence of transition in the leeward.

(4)The linear stability analysis and transition prediction were combined into the drag coefficients calculation by introducing a laminar/turbulent indicator,which is not an arbitrary parameter but determined by theNfactor and a thresholdNt.The source term in S-A turbulent model would be set to be zero in the laminar flow region indicated byNt.For the case withMa=0.7,the drag coefficients of laminar/turbulent flow have two signif icant jumping growth due to the successive occurrence of transition in the windward and the leeward.TheCDfor theMa=0.8 case has only one jumping growth because transition happens at nearly the same sweep angle in the leeward and windward.The selection of sweep angle for swept wing design was also discussed.

Acknowledgements

The authors thank Prof.Xuesong WU of Imperial College London and Prof.Jisheng LUO of Tianjin University for valuable discussions.The authors also thank Lei ZHAO of Tianjin University for help.This study was co-supported by the National Natural Science Foundation of China (Nos.11332007,11672351),the Tianjin Natural Science Foundation of China(No.15JCYBJC19500),the Hebei Natural Science Foundation of China(No.A2015105073),and an open fund from the State Key Laboratory of Aerodynamics of China(No.SKLA201601).

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22 April 2016;revised 21 August 2016;accepted 7 December 2016