Vee-tail conceptual design criteria for commercial transport aeroplanes

Alejandro SANCHEZ-CARMONA,Cristina CUERNO-REJADO

Faculty of Aerospace Engineering,Department of Aircraft and Spacecraft,Technical University of Madrid(Universidad Polte′cnica de Madrid),Madrid 28040,Spain

KEYWORDS Commercial transport aeroplane;Conceptual aircraft design;Tail design requirements;Unconventional tail design;Vee-tail

Abstract Vee-tail configuration is an unconventional tail configuration for commercial transport aviation,the use of which could suppose reductions on CO2emissions.The conceptual design criteria have been selected inspired on the certification requirements established by the aviation regulation in force.They are static stability in cruise,control after Critical Engine Failure(CEF),control in crosswind landing,and trim in these three conditions.The study is carried out through a combination of semi-empirical techniques and Vortex-Lattice Methods(VLM)and thus this analyses the consequences of applying these criteria to a reference aeroplane substituting its conventional tail by a parametrised Vee-tail configuration.The Vee-tail is de fined by four parameters:span,root chord,taper ratio and dihedral angle.The results of the study establish a relation between the parameters in order to accomplish the proposed conceptual design criteria.To sum up,minimum and maximum limits are obtained for dihedral angle depending on the combination of the rest of parameters.In addition,a design restriction to the yawing trailing-edge tail control is reached when the results are analysed,demonstrating that its minimum size must be between 60%and 80%of the half-span of the tail,depending on the Vee-tail geometry.

1.Introduction

The majority of governments around the world are concerned about air pollution emissions.1-4On concrete,the aviation is also taking actions to combat climate change.In fact,the International Civil Aviation Organisation(ICAO)agreed in October 2016 on a resolution to address CO2emissions from international aviation as of 2021.5This resolution establishes a scheme to offset around 80%of the emissions above 2020 levels during the period 2021-2035.6This should allow for improvements including how this schema contributes to the objectives of the Paris Agreement.7One possible way to reduce CO2emissions is to decrease the fuel consumption of commercial aeroplanes by improving engines,reducing their weight or changing the aircraft configuration.8,9This last option can improve aerodynamic efficiency and consequently reduce transport emissions.Aeroplane configurations have hardly changed throughout history.The conventional configuration is composed of high aspect ratio wings,a slender fuselage,two stabilizing surfaces situated in the rear end of the fuselage and engines positioned under the wing or at the rear of the fuselage.10Many unconventional configurations contemplated in the open literature suggest radical changes to the aircraft appearance such as joined-wing, flying wing or box-wing.8,11-13Alternatively,it may be more valuable to study unconventional configurations which suggest small changes in aircraft appearance as these would be easier to implement in commercial aviation.For example,the designers can only act in the empennage.The rear-end of the aircraft contributes around 20%to the aeroplane drag,so a new tail configuration which reduces this drag would result in a more environmentalfriendly aircraft.In response to that,a new conceptual design tool for unconventional tail configurations needs to be developed to aid designers in analysing the potential benefits and drawbacks of these configurations in the first stages of the design process.

The simplest unconventional tail configuration which could be installed in commercial transport aeroplanes is Vee-tail configuration.12,14This configuration is more often employed in the Remotely Piloted Aircraft Systems(RPAS) field in aircraft such as the Northrop Grumman Global Hawk or the General Atomics MQ-9 Reaper.In spite of that,a mass produced manned Vee-tailed aeroplane exists today:the Beechcraft Bonanza Model 35,a six-seated general aviation aeroplane,was introduced in 1947.Vee-tail aircraft present potential benefits from the standpoint of empennage drag and weight,because this tail configuration is composed of two surfaces instead of the three surfaces of conventional tails.However,some studies deal with the consequences of the use of this configuration in the controllability of the RPAS.A comparative study between conventional tails and Vee-tails of directional stability derivatives has been performed through wind tunnel tests at a low Reynolds number.15,16This study is thought suitable for unmanned aircraft because of the Reynolds number.Analogous studies deal with a comparison of the whole stability derivatives of a RPAS using Computational Fluids Dynamics(CFD)techniques,also for low Reynolds number.17,18Finally,this configuration has been studied also from the standpoint of dynamics models for RPAS taking into account different failure models with the objective to develop faulttolerant autopilots for this kind of aircraft.19

Furthermore,some works study the possibility of using Vee-tails in commercial aviation.The results of the EU funded NACRE project indicate that this configuration could present benefits in reductions of empennage drag but not in weight.12Also,the effects of varying the dihedral angle of Vee-tails have been studied for use in a blended wing-body aeroplane.20In this case,the results have been reached by modelling the aerodynamic forces and moments with Vortex Lattice Method(VLM).As a result of the limitations of this kind of tool,this study has only been carried out for air speeds lower than 50 m/s.

Despite the amount of research carried out regarding the Vee-tail,none of these studies establish a procedure for designing these tails in conceptual stages of the design process.Some authors indicate design guidelines for these tails,but they suggest designing the Vee-tail just considering the horizontal and vertical projections of the surface as a conventional tail.21In these stages,a rapid design tool is desirable because the aeroplane geometry is not yet fixed,so a low computational time of calculations to obtain results is necessary to allow the designer to implement changes in a short period of time.The price to pay for the use of these rapid design tools is the loss of precision,where the error is bounded around 10%.11Accuracy will be a mandatory requirement in more advanced stages of the design process.In this framework,because the available knowledge of conventional tails is far more extensive than that of others,there are already several methods for designing conventional tails with these requirements.Some of them are based on correlations and dimensionless parameters such as the volume coefficient.11,21-23These procedures need information about/on several similar aircraft in order to obtain valid results.This is a drawback to using these kinds of methods for unconventional tail configurations because there are no similar aircraft with the same tail configuration with the same operations.The conceptual design tool valid for Vee-tail is multidisciplinary because it encompasses both aircraft performances and structural aspects,so Multidisciplinary Design Optimisation(MDO)techniques are applicable.This design practise has already been used in some other studies in the aviation field.In fact,it has been applied to optimise an aircraft design to reduce CO2emissions24and also to unconventional tail configurations,25among others.

Notwithstanding the previous studies about conceptual aircraft design for unconventional tail configurations,there is a crucial aspect that has not been taken into account,the regulation in force.It is possible to establish a design procedure based on the roles and the certification requirements of the rear end surfaces.Basically,there are four criteria applicable to tails:guarantee longitudinal and lateral-directional controllability in cruising conditions,have enough trim capacity to balance the aircraft,assure the control of the aircraft if the critical engine fails and, finally,allow landing in crosswind conditions.11,26These criteria require that estimations of some stability derivatives are made to study each condition.The stability derivatives of conventional tails can be estimated through semi-empirical equations.27,28Furthermore,some studies present corrections to these classical semi-empirical methods with the aid of CFD software.29The drawback is that these semi-empirical equations are restricted to conventional tails.In addition,other techniques can be used in order to analyse the static and dynamic behaviour of the aircraft such as the bifurcation and continuation methods.30,31These methods allow the designer to analyse the behaviour of the aircraft in trim and unbalanced conditions depending on the value of only one design parameter.

Thus,it is possible to conclude that the knowledge of the design of a Vee-tail is not mature yet,especially the viability of the usage of this configuration in commercial transport aviation.Because of that,the goal of this paper is to include the airworthiness regulations in force nowadays applicable to a Vee-tailed commercial transport aeroplane in a conceptual design tool.The schema followed by this tool is slightly different from that which has previously been used in conceptual design because of the lack of similar aeroplanes necessary to build statistical methods or re fine semi-empirical correlations.First of all,a reference aircraft with a conventional tail configuration is selected in order to analyse its behaviour towards the criteria proposed to be studied,static stability in cruise,controlling after Critical Engine Failure(CEF),controlling in crosswind landing,and trimming in these three conditions(cruise,climb and land).This aeroplane is taken from the Central Reference Aircraft Data System(CeRAS)database,32which is thought to give support to the scientific community because it is difficult to find reliable data about commercial aircraft.So,the database will be composed of data for several aeroplanes from every flight segment,in order to allow the community use of the same data,which means comparisons between different studies will be possible.Nowadays,the database contains just one aircraft:CSR-01,with similar performances and geometry to an Airbus A320 or a Boeing 737.Then the next step is to substitute its conventional empennage with a Vee-tail.To do this,the Vee-tail geometry will be de fined through four design parameters:span b,root chord cr,taper ratio λ and dihedral angle Γ.Some previous studies indicate that these parameters are the most relevant to the performances of this kind of tail.14Some other parameters such as the sweep angle,twist angle or the aerofoil will be fixed a priori.In conclusion,the results of the paper will establish the feasible design space of these parameters by applying certification requirements for commercial aviation to Vee-tail configuration.

2.Materials and methods

2.1.Design criteria for unconventional tail

As it has been stated previously,this study focuses on implementing a new conceptual design procedure for unconventional tails.To establish a set of conditions for developing the procedure,compliance with the essential airworthiness regulations related to the flight requirements has been selected as the starting point.The regulations applicable to commercial transport aviation,in the case of large aeroplanes,are CS-25,26in case of the European Union(EU)or FAR-2533for the United States of America(USA).In both regulations,the Subpart B Flight from articles 25.171 to 25.181 deal with aircraft stability.Trim conditions are included in the same subpart,article 25.161.The crosswind velocities are indicated in articles 25.233 and 25.237.Finally,the articles which deal with aeroplane performances and controllability conditions where the critical engine failure condition is involved are articles 25.121 and 25.147.All these articles are applicable to aircraft tail design.From them,it is possible to extract four design criteria:being longitudinally and laterally stable in cruising conditions,controlling the aircraft if the critical engine fails during climbing after taking-off,controlling the aircraft in crosswind land and trimming capability in the three previous conditions.The first criterion will be implemented through comparison with the reference aircraft in terms of static stability derivatives.It is established that the new unconventional tail aircraft must be at least as statically stable as the reference aircraft in a certain cruising condition in order to accomplish the certification requirements.The second and the third criteria indicate that the aircraft must be controllable after the critical engine failure and must be able to land with 90°crosswind of 20 kn(1 kn=1.852 km/h)or 20%of the stall speed,whichever is greater,but needs not exceed 25 kn.The controllability is reached if the resultant bank angle of the aircraft is lower than 5°.In the case of this study,these manoeuvres will be carried out without taking into account the ailerons deflections,which are known as flat manoeuvres,in order to be conservative in the results.34In addition,the crosswind speed considered is 25 kn,which is the boundary established by regulation.Regarding critical engine failure condition,the sideslip angle β is one of the degrees of freedom.If the engine which fails is on the right-hand side of the pilot position,the aeroplane can remain in flight with negative values of sideslip angle by using a smaller tail plane as compared with zero sideslip angle.26Nevertheless,the sideslip angle is not totally free because it is constrained by the maximum bank angle admissible by certification requirements.In order to be conservative,the aircraft will be balanced at zero sideslip angle in this condition instead of selecting the sideslip angle that means the aircraft to fl y with the maximum admissible bank angle.If the resultant bank angle is below the limit,the sideslip angle which means the aircraft to flight at maximum admissible bank angle would be selected.Finally,the last criterion consists in having enough power control to balance the aircraft in every flight condition.In the case of pitching moment,it has been considered that both trailing edge and leading edge high-lift devices are deflected.

The aerodynamic forces and moments acting on the aeroplane will be estimated through a combination of semiempirical and VLM methods.The main reason for the usage of these methodologies is due to the fast-tool requirements for conceptual design stages.In addition,because in future works the aerodynamic forces distribution along the Vee-tail will be necessary to estimate the structural weight of the surface,it has been decided to estimate the aerodynamic forces in the tail using a VLM tool by the name of Tornado.35This is a VLM for linear aerodynamic wing design applications in conceptual design stages.By modelling all lifting surfaces as thin plates,Tornado can solve most of aerodynamic derivatives for a wide range of wing geometries and also estimate aerodynamic forces distribution.With a very high computational speed,Tornado gives the user immediate feedback on design changes,making quantitative knowledge available earlier in the design process.This software is valid for subsonic flow,but it is possible to extend its validity range to high subsonic flow through the Prandtl-Glauert correction.36Moreover,Tornado gives more reliability in the estimation of forces than moments.35In view of that,a combination procedure of semi-empirical and VLM methods has been proposed.The idea is to use Tornado for estimating only the force acting on the tail.This software has already been utilized to estimate aerodynamic force coefficients in Vee-tails and it has been established a validity range up to 45°of dihedral angle approximately.14,37The rest of the terms which contribute to aerodynamic forces and moments are estimated through the semi-empirical methods described by Torenbeek27because they are well-known procedures applicable at these early design stages for commercial transport aeroplanes.Correction factors have been included to take into account the interferences among the tail and the fuselage,the wing and the other tail surfaces,both in longitudinal and lateral derivatives.These interferences have relevant contributions to aerodynamic forces and moments.27,38-41In the case of Vee-tail,the interference with the fuselage has been modelled as in horizontal tails for longitudinal forces and moments and as in vertical tail for lateral ones.The interference among a horizontal tail and a fuselage depends on the geometry of the part of the surface which is inside of the fuselage.27Thus,this same methodology can be applied to Vee-tails.The interference among a vertical tail and a fuselage depends on the vertical tail geometry.40So,it has been decided to consider an equivalent vertical tail obtained by projecting the Vee-tail to the plane of symmetry of the aircraft.This equivalent vertical tail is used as an input to the semi-empirical estimation of the interference among these elements,the vertical tail and the fuselage.In the case of conventional tails,it is also possible to consider the interference among the horizontal and vertical tail surfaces,but this factor has not been included for Vee-tail configuration.In addition,Tornado does not retain non-linear effects for high deflection angles of the controls.In order to improve the reliability of the results,a semi-empirical correction has been proposed which consists in considering an effective control deflection to be the actual deflection multiplied by an effectiveness factor which depends on the deflection angle.39

Before starting with the physical-mathematical model,the unconventional tail configuration needs to be characterised.In the case of the Vee-tail,four geometric parameters have been selected:span,root chord,taper ratio and dihedral angle.The rest of the parameters necessary to fully de fine the configuration,such as sweep angle,twist angle or aerofoil geometry,are fixed a priori because their effects are not as relevant as the previous ones.14In addition,the tail needs to include both yawing and pitching controls.The geometry of these controls will also be fixed a priori.After analysing how these controls are implemented in actual Vee-tailed RPAS,it has been decided to divide each half-wing control in two equal parts.A longitudinal trailing edge control has been installed in the inboard half of the tail and in the outboard part there is a lateral control.In Fig.1 there is a schema of how these controls are set up.In future works,the effect of interchanging the roles between the controls and what will suppose for the Vee-tail design will be accomplished.Finally,the maximum deflections considered for these controls will be based on the reference aircraft data,assuming the yawing control deflection limits for the Vee-tail to be the same as for the rudder of the reference aircraft.Analogously,the maximum longitudinal control deflection for the Vee-tailed aircraft is assumed to be the same as for the pitching control limits of the reference aircraft.

Table 1 Vee-tail geometric parameters range.

The study of Vee-tailed aircraft will be carried out by analysing the effects of the four proposed parameters(span,root chord,taper ratio and dihedral angle)according to the criteria presented previously.These parameters will take values between the limits indicated in Table 1.These boundaries have been established based on previous studies.14Furthermore,it is necessary to indicate some additional hypotheses.The longitudinal and vertical position of the tail is the same as the horizontal tail of the CSR-01.The idea is to examine how much dihedral angle must be given to a horizontal tail in order for it to be able to do without the vertical tail.As an initial solution to the problem,it has been decided not to change the longitudinal position of the Vee-tail because a redesign of the fuselage rear-end would be required.The aerofoil used in the tail is NACA 0012.Future research and optimisation processes could result in changing the aerofoil,but for this study it has been fixed a priori.Another initially selected parameter is the incidence angle.On one hand,it has been considered that the tail has no variable incidence angle because further research is necessary to assure that this system is viable for Vee-tails.On the other hand,in the present study,the incidence angle has been set to zero and,in future extensions of this research,a comprehensive analysis will be carried out in order to check the effect of this parameter.

2.2.Physical-mathematical model

To carry out this study,it is necessary to make use of the longitudinal and lateral equilibrium moments equations.The expression of longitudinal one is indicated below:

where ρ is the air density,V is the aeroplane speed,Swis the wing area,MAC is the mean aerodynamic chord of the wing,qT/q is the dynamic pressure ratio between tail and wing,lTHis the tail moment arm,CLTis the tail lift coefficient,Teis the thrust of the aircraft and zeis the vertical distance between the thrust and the center of gravity.In Eq.(1)the coefficient CmA-Trepresents the pitching moment of the aircraft eliminating the contribution of the tail.This coefficient is estimated through semi-empirical methods.26

where CmTorand Cmairfoilare the aircraft pitching moment contributions of the twist angle distribution of the wing and the aerofoil pitching moment coefficient,respectively;ΔCmHLDand ΔCmfusare aircraft pitching moment contributions of the high-lift devices and the fuselage;xcgand xacare the longitudinal positions of the centre of gravity and the mean aerodynamic chord,respectively;CLis the lift coefficient of the aircraft.

In Eq.(2),the first term corresponds to the contribution of the twist angle distribution of the wing and the second term to the aerofoil pitching moment coefficient.Furthermore,it is necessary to take into account the contribution of the high-lift devices because during climbing after taking-off and landing flight conditions,these are extended.Also,a correction corresponding to the fuselage contribution has been added.Finally,the last term is associated to the moment generated by the weight W,which is supposed to be balanced by the lift of the aircraft.The lift coefficient is determined by the vertical axis forces equation:

In order to reach the necessary lift coefficient CLextracted from Eq.(3),it is necessary to determine the angle of attack α and the longitudinal control deflection δsymand also accomplish the longitudinal moment equilibrium equation.The dependency of the lift coefficient with these parameters and with the high-lift devices deflection can be estimated as follows:

In Eq.(4),the dependencies of each term are indicated,where the lift coefficient without tail contribution CLA-Tonly depends on the angle of attack.The contribution of the high-lift devices ΔCLHLDdepends only on the fl ap deflection δf,in first approximation,and the lift coefficient of the tail CLTdepends on the deflection of the control δsymand also of the angle of attack α.It is remarkable that the angle of attack of the tail does not have to be the same as the angle of attack of main wing because of the downwash gradient caused by the wing situated upstream of the tail.The downwash deflection depends on the angle of attack of the main wing and has been modelled according to classical semiempirical methods.27The lift coefficient of the aircraft without tail contribution and the increment of lift coefficient generated by fl ap deflections are both calculated through classical semiempirical methods.27Thus,taking into account the angle of attack of the tail and the control deflection,the tail lift coefficient can be estimated using the Tornado software.To sum up,the final system of equations to be solved is

?＞The unknowns in Eq.(5)are the angle of attack and the longitudinal control deflection.They cannot be solved analytically because the terms dependent on the tail are calculated by means of Tornado.Hence,it has been decided to solve the system through a Newton method.So,it is necessary to compute the Jacobian matrix of the system,which is estimated using a finite differences scheme.Note that a positive deflection of the elevator is considered if it generates a nose-down pitching manoeuvre of the aircraft.

Following this,the development of the yawing equilibrium moment equation has been carried out.The final expression is

where bwis the wing span and yeis the lateral distance between the critical engine and the aircraft plane of symmetry.The terms in Eq.(6)dependent on the aircraft without tail effects CnA-Thave been estimated using semi-empirical approximations and the contribution of the tail to lateral force CYThas been estimated using Tornado.The main contribution to the lateral moment of the aircraft CnA-Tis generated by the fuselage and its value depends on the sideslip angle.The moment caused by the failure of the critical engine is taken into account in the last term of the previous equation.The value of the lateral force acting at the tail needs to balance this equation.This force can be varied by modifying the lateral control deflection δasym.The sign criterion of the asymmetric control deflection is taken as positive if it generates positive yaw moment.To meet the regulations in force,once the aircraft is balanced laterally,it is necessary to assure that the aircraft does not reach a bank angle higher than 5°,both in critical engine failure and crosswind landing conditions.Fig.2 shows the forces acting in the aircraft when flying with bank angle φ.

Thus,the lateral forces equilibrium equation is

and the vertical forces one is

Again,the lateral force generated by the aircraft without taking into account tail contribution YA-Tis calculated through semi-empirical methods.This term has two contributions:fuselage and dihedral angle of wing,which also depend on the sideslip angle β.The term YTis the lateral force generated by the tail and R is the turning radius of the manoeuvre.Assuming that the bank angle is low,it is possible to simplify the equation,and the final expression for the bank angle is as follows:

Fig.2 Forces in aircraft when flying with bank angle.

Finally,the static cruising stability properties of the aircraft will be characterised through two parameters:Cmαand Cnβ.The first one represents the static longitudinal stability behaviour and the second one the static lateral stability one.These parameters are going to be studied in cruising condition and compared with the reference aircraft.The way to calculate them is easy once the moment equations are known,pitching and yawing respectively.The Cmαis calculated by deriving this equation from the angle of attack around the point which balances the aircraft.Because the tail contribution is estimated by Tornado,the derivative is calculated numerically.The same procedure can be applied to the computation of Cnβ,but using the lateral moments equation and deriving from the sideslip angle.

3.Reference aircraft

3.1.Repository data

First of all,it is necessary to characterise the reference aircraft in order to carry out the goals of this paper.As it has been mentioned in the Introduction,the reference aircraft will be the only aeroplane included in the CeRAS repository:the CSR-01.There,it is possible to find a wide range of information about its geometry,weights and performances.This information is necessary for applying the semi-empirical methods described in Section 2.42Table 2 includes the main parameters obtained from the CeRAS repository used in this study,and Fig.3 shows the planform geometry of the wing.Additionally,the repository also includes information about polar curves depending on the Mach number.These data are useful for estimating the necessary thrust for each flight condition in order to maintain a constant cruise speed.

Table 2 Parameters of CSR-01 airplane.42

Fig.3 CSR-01 wing planform.

3.2.Additional data

There are some necessary data about the CSR-01 that are not included in the database repository,for instance the maximum deflections of the controls.These limits have been estimated using data from similar aircraft.In this case,a flight controls document about A320 has been used as a reference,43which indicates that the elevator deflection δeis in the range[-17°,30°].These boundaries must not be exceeded when longitudinal trimming of the aeroplane is reached by the methodology presented in the previous section.In this case,the design is not feasible.On the other hand,the maximum rudder deflection depends on the flight speed,since the regulation indicates that the maximum force applied by the pilot must be lower than 667 N.26The maximum rudder deflection δrextracted from the same document is represented in Fig.4.This limit applies when the aircraft must be balanced laterally.In this study,there are two conditions:after critical engine failure after taking-off and at crosswind landing.The first occurs at the minimum control speed,which is 13%higher than the stall speed at 1 g.Therefore,the minimum control speed is approximately 94 m/s and,consequently,the maximum rudder deflection is 18°.On the other hand,at landing the speed is supposed to be equal to 1.23 times the stall speed at 1 g.In this condition,the maximum control deflection is 25°.

According to the described methodology,there is a limitation in the cruising condition introduced by Tornado.It is not possible to select a transonic condition for determining the static stability derivatives.Because of this,the flight manoeuvring envelope has been analysed depending on the flight altitude.The manoeuvring speed(VA)has been selected as the cruise speed at the altitude where the Mach number Ma is 0.6.The result of this analysis is that the chosen altitude is 22000 ft(1ft=30.48 cm).Finally,it is also necessary to estimate the aerofoil moment coefficient in order to apply the methodology to the CSR-01.This contribution is estimated with the aid of X-Foil tool.44The weight for this condition is obtained as a representative one for a middle point of the route and takes the value of 66600 kg.Of course,in this condition the fl aps are retracted so the corresponding terms in the procedure are neglected.Additionally,the tail incidence angle has been considered as constant and has been fixed to-0.1°based on repository information.Finally,the interference factors considered for CSR-01 are the following:fuselage/tail surfaces,wing/tail surfaces and horizontal/vertical tails.

Fig.4 CSR-01 maximum rudder deflection estimation.

3.3.Comparison of methodology

A comparison process has been carried out to analyse the reliability of the procedure combining semi-empirical and computational methods.The comparison has been made using Tornado,Torenbeek's method and a combination of both as proposed in the present work.The reference aeroplane with conventional geometry has been selected.It is supposed that Torenbeek's method is adequate for the conceptual design stages because it has been used in many studies and principally for commercial aviation.Two flight conditions have been selected:cruising flight at VAand climbing flight after taking-off.Torenbeek's method consists of the procedure stated previously but also estimates the tail lift with semiempirical equations,like the main wing.As it has been indicated,the performances of the aerofoils have been modelled with the aid of X-Foil software.Tornado's procedure has been computed by introducing the main wing and tail geometries into the software as it is not possible to consider the effects introduced by the fuselage.The results for moment coefficient have been modified in order to take into account the aerodynamic moment of the aerofoils,a term that Tornado does not compute.Finally,the proposed combined methodology consists of estimating the force acting on the tail with Tornado and the rest of the terms with Torenbeek's method,as it has already been explained.Fig.5 shows the results for the two flight conditions studied.On viewing the graphs,Tornado's results do not follow the curves extracted by Torenbeek's method,which is the reference.As a result of this,it has been decided not to estimate the aerodynamic forces and moment of the complete aeroplane with Tornado.Finally,when the proposed method is analysed,the results change substantially.They present more reliability for lift coefficient estimation.In fact,both curves,from Torenbeek's and proposed methods,are almost overlapped.If cruising condition at VAis examined,for which the lift coefficient is approximately 0.5,the angle of attack necessary for reaching that lift coefficient presents a deviation of-0.14%with respect to Torenbeek's method.The same comparison can be made for climbing flight and the deviation is-0.25%.By analysing the moment curves,it is possible to see that in the case of low airspeed,the proposed method better follows the results obtained with Torenbeek's method than for the high subsonic condition.In fact,studying the lift coefficient necessary for flying in climbing condition,the variation of the pitching coefficient with respect to Torenbeek's method is 1.8%,but the deviation in cruising flight is around-12%.Thus,it is possible to conclude that the combined method proposed in this paper is better than using only a VLM for the full aircraft and is very similar to Torenbeek's method,especially in the case of low subsonic airspeeds.

Fig.5 Comparison between Tornado,Torenbeek and proposed method in cruising flight at VAand climbing flight.

Another comparison has been performed in order to analyse if non-linear effects for high angles of control deflection δcare retained with the hypothesis considered in the proposed methodology in this paper.It has been carried on by estimating the lift coefficient at zero angle of attack and varying the trailing-edge control deflection through four procedures:two semi-empirical methods,which are Torenbeek's method and Roskam's method,39Tornado and the combination of Roskam's method and Tornado(proposed method).The results are shown in Fig.6.It is possible to see that Tornado estimates the lift coefficient linearly for the whole range of control deflection values.However,both Roskam's and Torenbeek's methods consider that the control effectiveness is lower for high control deflection angles.So the hypothesis to use an effective angle of control deflection as input for Tornado software obtained to multiply the actual value by the effectiveness of the control included in Roskam's methodology is an improvement to Tornado's aerodynamic coefficients estimation.In fact,the differences between the proposed method and Roskam's or Torenbeek's methods are under 10%in the range of angles studied.

Fig.6 Comparison of control derivatives between Torenbeek,Roskam,Tornado and proposed method for zero angle of attack.

3.4.Application of design criteria to reference aeroplane

The three scenarios considered are tested using data from the reference aircraft.The first scenario is the cruising flight.In this case,only the equations for longitudinal moments and vertical forces(Eq.(5))are used.First of all,it is necessary to estimate the lift coefficient and the moment coefficient of the CSR-01 eliminating the contribution of the tail and depending on the angle of attack.As it has been indicated previously,these coefficients have been obtained using semi-empirical methods and the results are as follows:CLA-T=0.1812+4.876α;CmA-T=-0.073002+0.0719α.The contribution of the horizontal stabiliser is very important especially with respect to the pitching moment coefficient.In fact,when the tail is considered,the sign of the Cmαchanges,so the aircraft becomes stable.The thrust contribution to the pitching moment equation is obtained with the aid of the polar curve included in CeRAS documentation,for the flight conditions of this case.Once all the necessary data are estimated,the system of equations is solved.Table 3 shows the results for the angle of attack and the elevator deflection.The values of elevator deflections are between the limits established by the control system.Furthermore,the static stability derivative estimated around this point is:Cmα=-2.493.

The yawing moment coefficient of the aircraft,without the contribution of the tail,is estimated using Torenbeek's method:CnA-T=-0.1396β.Now it is possible to estimate the lateral-directional static stability:Cnβ=0.1590.This has been obtained through equation of equilibrium of yawing moments,with no deflection of the rudder and no sideslip angle.By analysing the results for both static stability derivatives,it is possible to conclude that the tail accomplishes its role,as the aircraft becomes stable when the contribution of the tail is taken into account.These values are also relevant because it is supposed that the aircraft presents adequate stability behaviour in cruising flight with this tail geometry.Thus,the Vee-tail needs to reach at least these static stability values in order to assure similar stability behaviour to the reference aircraft.

The procedure for the other two flight conditions,climbing and landing,is analogous.The results for balancing the aircraft longitudinally are included in Table 3.In these cases,the fl aps are extended with deflections of 10°for climbing and 35°for landing,and the slats are extended with deflections of 15°and 25°respectively.In addition,the thrust considered in the first case is the maximum at take-off,and in the second case it is zero,because it is a case included in the certification requirements.Again,it is possible to see that the values of pitching control deflections fall into the feasible range of the control.After that,the aircraft needs to be balanced laterallybecause of the asymmetric thrust or the crosswind condition respectively.For both conditions,it is necessary to estimate the lateral force coefficient of the aircraft eliminating the tail contribution.The coefficient depends on the sideslip angle as follows:CYA-T=-0.1893β.In the case of the failure of the critical engine,the aeroplane has remained in flight without sideslip angle,as it has been commented previously.On the other hand,the crosswind landing condition supposes a sideslip angle of 11°,considering 25 kn of crosswind at 90°.The value of rudder deflection needs to be between the limits of the control system,which is accomplished according to Fig.3.The results obtained for both lateral conditions are included in Table 3.

Table 3 Trimming angles of CSR-01.

4.Results

4.1.Pitching moment trimming

First of all,the necessary deflection of symmetric control for each flight condition is analysed.To do this,the equations in Eq.(5)must be solved.Since the tail does not contribute greatly to the lift of the aircraft,only around 10%,the angles of attack obtained in all the combinations of geometric parameters for the different Vee-tails for each flight condition are almost the same.The resultant angles of attack are around 3.6°for cruising condition,0.6°for landing and 1.9°for climbing.In fact,the differences between the values for each combination of geometric parameters are under 0.5%.On the other hand,the deflection of the symmetric control has more remarkable differences.According to the hypothesis explained previously,the control needs to be deflected down in order to balance the aircraft longitudinally in the three flight conditions.This means that without any deflection there is too much downward lift on the tail to balance the aircraft longitudinally in the cruising condition.If the tail dihedral angle increases,this excess of lift is lower because the tail lifts less for the same angle,concretely it is reduced to the cosine squared of the dihedral angle.37Thus,the control deflection necessary to balance the aircraft decreases with higher dihedral angles.This is shown graphically in Fig.7(a).Additionally,this same behaviour is extracted from the other two flight conditions:climbing and landing.The results are shown in Figs.7(b)and(c)respectively.

The behaviour of the control deflection with changes in root chord is more complex to explain.First of all,it is important to highlight that in order to balance the aircraft,each tail needs to generate the same lift while considering the same flight condition.From now on this tail lift will be referred to as goal tail lift.Obviously,the goal tail lift depends on the flight condition.Considering the same flight condition,what changes between different tail geometries is the necessary control deflection.Therefore,the root chord of the tail affects the moment arm and the tail area.This supposes that for the same angle of attack,higher values of root chord suppose lower values of tail lift coefficient to reach the goal tail lift.In addition,the root chord also affects the aspect ratio of the surface,reducing its lift curve slope for higher values of root chord.However,there is also an increase in the value of the pitching control derivative.These two effects come into conflict but the second one is more pronounced.Thus, finally the control deflection necessary to balance the aircraft becomes lower.This behaviour can be seen in Fig.7 for each flight condition studied in this paper.

Fig.7 Analysis of effect of span(b),root chord(cr)and taper ratio(λ)varying dihedral angle Γ of Vee-tail in symmetric control deflection for longitudinal trimming in cruising,climbing and landing conditions.

It is possible to apply analogous reasoning to explain the effect of varying the tail taper ratio.Increasing the taper ratio supposes an increase in the area and the lift curve slope because the aspect ratio augments too.However,in this case the power control derivative barely changes.Hence,if the area and the slope are slightly higher,the necessary deflection to reach the goal tail lift must be higher too.In spite of that,the effect of the taper ratio is not very strong,as it possible to see in Fig.7.

However,the behaviour with span changes is different between climb and cruise,on one side,and land,on the other.In climbing and cruising conditions,the equilibrium lift coefficient of the tail is quite different than in the other two cases due to the effect of the thrust.The thrust generates a pitching moment that aids the aircraft to balance.This means that the goal tail lift in these conditions is different than in the landing condition;in fact,the sign is not the same in both cases.On the other hand,increasing the span supposes that the tail lift curve slope will also increase.Consequently,the tail lift coefficient is higher,in absolute value,for the same angle of attack.So in order to reduce the tail lift to obtain the goal tail lift,in the case of negative lift,it is necessary to reduce the control deflection;and in the case of positive lift,it is necessary to increase it.Thus,for climbing and cruising conditions,increasing tail span implies that the symmetric control deflection increases too.On the other hand,increasing tail span in landing condition supposes that the necessary control deflection becomes lower.This is shown schematically in Fig.8.

4.2.Static stability at cruising condition

Fig.8 Schematic of qualitative behaviour of symmetric control deflection to balance aircraft with changes in tail span in three different flight conditions.

In this section,the static stability derivatives are analysed at cruising condition.The results are shown in Fig.9,for longitudinal and lateral derivatives respectively.In Fig.9 the results of the derivatives for the reference aircraft in the same conditions are indicated by dashed lines.The Vee-tail aircraft must be at least as stable as the reference aircraft.Thus,in the case of longitudinal stability,the possible design zone is situated under the horizontal line called in the legend as Cmαref.However,for lateral stability,the feasible design field sits above the horizontal line called in the legend as Cnβref.As expected,increasing the tail's area supposes more stability.The effect of augmenting the tail dihedral angle is bene fi cial for lateral stability,but it is detrimental for longitudinal stability.For instance,in the case of 20 m of span in the first graph of Figs.9(a)and(b),the static longitudinal stability restricts the dihedral angle from being higher than approximately 45°and the lateral stability from being higher than 40°.This means that the static longitudinal stability is an upper restriction to dihedral angle and the static lateral stability is a lower one.

4.3.Yawing moment trimming at critical engine failure condition

In this condition,the asymmetric control deflection required to null the moment generated by the engine failure introduces a new restriction to the design.The results are shown in Fig.10.The boundary fixed by the maximum control deflection is marked with the horizontal line called in the legend as δdasymMAX.The maximum deflection was determined by the reference aircraft and takes the value of 18°.This restriction supposes a lower limit to the dihedral angle,because for lower dihedral angle the necessary deflection is higher than the maximum admissible.Again,if the area of the tail increases,this restriction allows the design of tails with lower dihedral angles.The last result to be analysed for this flight condition is the bank angle,which has been calculated for all the cases where the maximum control deflection was not achieved.This angle depends on the lateral force generated by the tail.As the force to be balanced is always the same,the thrust of one engine,the necessary tail force is always the same too.Therefore,theoretically,the resultant bank angle is equal to that obtained for the reference aircraft:3.1°.However,the resultant bank angles vary between 2.7°and 3.0°.This is caused by the imprecisions made by the Tornado software and also by the mathematical method used to find the control deflection.In spite of that,every configuration meets the regulations in force,which allows a maximum bank angle of 5°.

4.4.Yawing moment trimming at crosswind landing condition

At crosswind landing condition,the necessary asymmetric control deflection required to balance the aircraft laterally has been determined by solving Eq.(6)for each geometric parameters'combination for the Vee-tail.The results are pictured in Fig.11.The maximum deflection of the control according to landing speed of this condition is also represented in this figure.This boundary is estimated using data from the CeRAS aircraft,as explained previously,and takes the value of 25°.It is both upper and lower limit because it corresponds both to maximum positive and negative deflections.Fig.11 shows that the maximum negative control deflection is hardly reached by any parameter combination.However,the maximum positive deflection bounds the dihedral angle as a lower limit.The behaviour of the control deflection with changes in dihedral angle is as follows:for low dihedral angles the vertical projection of the tail is small enough to need high positive deflections of the asymmetric control in order to balance the aircraft.On the other hand,for high dihedral angles,the vertical tail projection is too big in order to balance the aircraft with maximum negative deflection of the asymmetric control.

Fig.9 Analysis of effect of span,root chord and taper ratio varying dihedral angle of Vee-tail in longitudinal and lateral static stability in cruising condition.

Fig.10 Analysis of effect of span,root chord and taper ratio varying dihedral angle of Vee-tail in asymmetric control deflection for lateral balancing with critical engine failure in climbing conditions.

Fig.11 Analysis of effect of span,root chord and taper ratio varying dihedral angle of Vee-tail in asymmetric control deflection for lateral balancing at crosswind landing conditions.

5.Discussion

On viewing the results obtained for each flight condition,it is possible to compile them in order to analyse which restriction is active depending on the combination of the design parameters.First of all,it has been seen that longitudinal trimming does not introduce any limitation to the Vee-tail design.Only the case of trimming in the landing condition supposes a control deflection near to the maximum,and only for lowest tail areas.This means that the longitudinal control power is excessive,so a redesign of longitudinal tail control is necessary.In this case study,the longitudinal control spanned half of the tail and 25%of the chord.Hence,one possibility is to reduce its span or the percentage of chord.Another variable to take into account is the trim incidence angle.In this study,it has been fixed at zero,but a re fined design will select an angle that supposes not to de fl ect the longitudinal control to trim the aircraft longitudinally for a cruising design condition.The drawback of this is that the design flight condition is Ma＞0.6,which would require the designer to use another aerodynamic model.Also,a variable tail incidence angle system could be considered for the Vee-tail.In spite of these improvements changing the results,it has been shown that longitudinal trimming is not a determining problem when designing a Vee-tail.

However,it is now necessary to discuss the stability derivatives results for cruising condition and asymmetric control deflection for one engine failure during climbing and crosswind landing conditions.In these cases,the established boundaries are eventually reached,so all of them introduce some kind of restriction to the design.Thus,it is possible to determine the relation between the design parameters at the boundary.The proposed problem has four degrees of freedom:span,root chord,taper ratio and dihedral angle,and each active restriction reduces in one degree the freedom of the problem.Therefore,the equations of the boundaries have three degrees of freedom.Because all the restrictions are represented by inequalities,each of these three-dimensional surfaces will divide the design space into two subspaces,in one of which the solution will be possible,but in the others it will not.This means that one of the initial degrees of freedom could be chosen as the dependent variable.By choosing the dihedral angle,the dependency of this variable along the boundaries with the other three parameters is presented.Fig.12 includes the resultant boundaries in the design space when static stability in cruising conditions and critical engine failure are applied.The figure shows that the longitudinal static stability restriction is an upper boundary to the dihedral angle and the lateral static stability one is a lower boundary,as stated previously.Moreover,the critical engine failure during climbing restriction represents a lower boundary too.However,this boundary is inactive because lateral static stability is more restrictive.This establishes the possible design space between the two curves de fined by stability restrictions.This space becomes wider when higher root chords are considered.The fact that critical engine failure restriction is not active indicates that the hypothesis about the size of the asymmetric control has been too conservative.The size could be selected such as the associated boundary will move to reduce the distance to the curve de fined by the lateral static stability derivative.However,it is necessary to study the crosswind landing condition beforehand as it would be more restrictive than controlling the aircraft after critical engine failure one for the asymmetric control redesign.

Analogously,the crosswind landing condition can be analysed in the same way.As it has been explained previously,there are two boundaries introduced by this restriction.The lower limit is generated by the positive maximum deflection of the asymmetric control.The restriction obtained through this condition is inactive,because lateral static stability in cruise is more critical.In fact,the crosswind landing condition is even less restrictive than controlling after critical engine failure.The results are shown in Fig.13.However,the upper limit appears when the maximum negative control deflection is reached.As it can be seen in Fig.13,this restriction is active only for the highest root chord considered and the higher values of span.

Fig.12 Design space taking into account cruising static stability,both longitudinal(Cmαlabel)and lateral(Cnβlabel),and balancing aircraft after critical engine failure(CEF label).

Fig.13 Design space taking into account cruising static stability,both longitudinal(Cmαlabel)and lateral(Cnβlabel),and balancing aircraft in crosswind landing conditions,for maximum negative deflection of control(CwLMAX label)and maximum positive deflection(CwLMIN label).

The results obtained for crosswind landing condition indicate that the procedure to redesign the asymmetric control needs to be applied to critical engine failure condition.This could be concluded because the boundary associated with the crosswind landing condition is below the curve associated to critical engine failure condition,resulting in a bigger control for critical engine failure condition.Thus,Fig.14(a)represents the minimum span basymof the asymmetric control in order to accomplish with controlling after critical engine failure without converting this restriction to an active one.This means that with the size obtained through this procedure,the critical engine failure curve is overlapping the lateral static stability boundary.The redesign has been implemented by simply changing the percentage of span which has a trailing edge device,as it is indicated in Fig.14(b).The percentage of chord has been maintained equal to 25%.The corresponding control sizes represented in Fig.14(a)are referred to the quarter span,which was the previous span of the asymmetric control.Thus,the necessary control size is between 62%and 84%,depending on the tail geometry,of the size considered in the initial hypotheses.

Fig.14 Minimum asymmetric control size(basym)referred to previous control size(b/4)to assure that critical engine failure boundary is on lateral static stability in cruise one and Definition of parameter basym.

Fig.15 Final design space imposing critical engine failure for maximum deflection on lateral static stability in cruise and static stability in cruising and crosswind landing restrictions(The feasible design space is shaded for cr=3.3m and λ=0.28).

The last step is to analyse what happens in crosswind landing condition with this new yawing control Definition.Currently,the size of the asymmetric control for each combination of geometrical parameters is the result obtained in Fig.14.The analysis is focused on the upper limit because the lower limit is less restrictive than the lower limit for critical engine failure,as it has been seen previously.This upper restriction is presented in Fig.15 in conjunction with the other active restrictions.It is possible to see that,with this control size,the crosswind landing condition becomes active and bounds the space design in maximum dihedral angles.In order to clarify which is the feasible design space for each Vee-tail geometry,Fig.15 represents this space in the case of cr=3.3 m and λ=0.28 by shading this region.Of course,if the asymmetric control is bigger than the considered,the critical engine failure restriction becomes inactive and the crosswind landing one moves upwards so it is less restrictive.On the other hand,if the control span is lower,the critical engine failure condition becomes more restrictive than lateral static stability in cruising condition,and crosswind landing curve goes down,so the design space is more reduced.

6.Conclusions

(1)In this paper,an analysis of the application of classical design criteria to an unconventional tail configuration has been presented.These classical criteria are based on certification aspects such as static stability,and aircraft control after critical engine failure or in crosswind landing conditions.The study is focused on the Vee-tail configuration,which has been modelled through four design parameters:span,root chord,taper ratio and dihedral angle.

(2)The analysis has established the feasible design space according to the aforementioned criteria.The dihedral angle has been considered as dependent on the other three variables,so the conclusions obtained are that longitudinal static stability is an upper limit to dihedral angle,but lateral static stability is a lower one.The critical engine failure is also a lower boundary to dihedral angle.Finally,the crosswind landing condition introduces two more boundaries,lower and upper limits respectively.

(3)Furthermore,some restrictions to pitching and yawing controls have been determined.The initial hypothesis was to divide the tail surface in two halves along semi span such as the inboard part with trailing edge device used as longitudinal control and the outboard part which has a trailing edge device as directional-lateral control.This hypothesis is based on the Vee-tail configurations used in the RPAS field,in which the number of aeroplanes with this kind of tail is greater than in other aeroplane categories.

(4)It has been demonstrated that the longitudinal control has enough capability to balance the aircraft in climb after taking-off,cruising and landing conditions.Despite this,deeper studies should be developed in order to optimise the longitudinal control size for assuring controllability in non-stationary manoeuvres.

(5)On the other hand,the yawing control is oversized when the design criteria considered in this study are analysed.Thus,it is possible to determine the minimum size of this control in order to make the boundary introduced by critical engine failure to be as restrictive as the lateral static stability one.This minimum control span is around 20%and 35%lower than that in the initial hypothesis.When considering this asymmetric control span,the upper boundary introduced by crosswind landing condition becomes active and consequently,the design space is more reduced.In spite of that,the feasible design space includes the typical dihedral angles installed in Vee-tailed RPAS.

(6)From the standpoint of the methodology presented in this study,it is possible to conclude that Tornado does not retain non-linear effects caused by high control deflection angles nor does it estimate any interference among the elements of the aircraft.Because of that,it has been demonstrated that a combination of semiempirical design methodologies and Tornado improves the fidelity of the results.The semi-empirical methodologies allows to take into account in the procedure interference factors,non-linear effects,aerodynamic forces and moments generated by the main wing.

The results obtained in this paper are the first step towards developing an optimisation design tool for unconventional tail configurations.In future research,an objective function to be optimised in the final design space established in this paper will be selected.This goal function would be the tail drag,its weight,or even a combination of these two.In response to this,MDO techniques will be used to solve this problem.