Chen Pengfei,Wu Feng,Xu Jinglei,Feng Xudong,Yng Qio

aAviation Key Laboratory of Science and Technology on Aero-Engine Altitude Simulation,China Gas Turbine Establishment,Aviation Engine Corporation of China,Mianyang 621703,China

bDepartment of Power Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

Design and implementation of rigid-flexible coupling for a half-flexible single jack nozzle

Chen Pengfeia,*,Wu Fenga,Xu Jingleib,Feng Xudonga,Yang Qiaoa

aAviation Key Laboratory of Science and Technology on Aero-Engine Altitude Simulation,China Gas Turbine Establishment,Aviation Engine Corporation of China,Mianyang 621703,China

bDepartment of Power Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

The aerodynamic design of a rigid- flexible coupling pro file is the decisive factor for the flow- field quality of a supersonic free jet wind tunnel nozzle,and its mechanic dynamic features are the key for engineering implementation of continuous Mach number regulations.To ful fill the requirements of a free jet inlet/engine compatibility test within a wide simulation envelop,both uniform flow- fields of continuous acceleration and deceleration are necessary.In this paper,the aerodynamic design methods of an expansion wall and machinery implementation plan for the halfflexible single jack nozzle were researched.The pro file control in nozzle flexible plate design was studied with a rigid- flexible coupling method.Design and calculations were performed with the help of numerical simulation.The technique of axial free stretching of the flexible plate was used to improve the matching performance between the designed elasticity pro file and the theoretical one,and the rigid- flexible coupling structure was calibrated by wind tunnel tests.Results indicate that the flexible plate aerodynamic design method used here is effective and feasible.Via rigidflexible coupling design,the flexible plate agrees with the rigid body very well,and continuous Mach number changes can be achieved during the tests.The nozzle’s exit flow- field uniformity meets the requirements of China Military Standard(GJB).

Ⓒ2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

From the research and development history of aero-engines,it is indispensable to simulate an aircraft’s real operating environment in ground tests,especially in recent years,with the continuous development of new fighters.1–3During an aircraft’sacceleration,deceleration,orlarge maneuvering,dynamic performances are always the key point and the major difficulty in the design and assessment of aircraft.Moreover,as the requirements of a wind tunnel are various for testing the operation envelopes of different engines development,the simulation capability of a ground test facility is demanded to be more advanced and flexible.To resolve these dilemmas and to make improvement on an aircraft’s performance,it is crucial to build a jet wind tunnel with continuous variable Mach numbers,4–6which can perfectly reproduce the true flow pattern and dynamic response of a real flow-field at the engine inlet during aircraft maneuvers or acceleration and deceleration operations.7Furthermore,during the development of advanced aero-engines,some key technologies must be studied and verified via free jet tests with a variable Mach number wind tunnel,such as the dynamic responses of turbine-based combined cycle engine performance8and variable cycle engine performance during an operation mode transition.9

Today,free jet testing devices with a flexible nozzle have been widely adopted in Western aviation developed countries.10–12In China,the vast majority of supersonic wind tunnels use a fixed geometrical nozzle with a fixed exit Mach number.13Recently,Ref.14introduced the usage of a domestic advanced full-flexible nozzle,and Ref.15researched the performance of a half-flexible multi-jack nozzle.However,no literature about design and calibration of half-flexible single jack nozzles has been reported yet.Compared with the aforementioned two flexible nozzles,a half-flexible single jack nozzle has the following advantages.(1)Due to the single jack structure,the Mach number can be adjusted quickly to simulate an aircraft’s dynamic incoming flow during acceleration and deceleration operations.(2)Without the limitation of supporting collaborative arrangement of multiple points,various nozzle driven modes can be adopted,and even be set outside of a wind tunnel,so as to provide possibilities of attitude simulation.(3)The simple mechanical structure of a half-flexible nozzle lets it cost less than one-tenth of that of a full-flexible nozzle.However,due to the reduction of the jack,the nozzle profile’s aerodynamic performance,mechanical property,regulation performance,and so on,are required strictly,so as to assure that a flexible profile nozzle with an appropriate geometry has equivalent flow-field quality to a fixed nozzle.The key points to ensure the flow-field quality at the nozzle exit are as follows:(1)Compatible design of a rigid-flexible coupling and aerodynamic profile;(2)No concentrated moment is allowed at the end-to-end connection point of the flexible plate during the variation of the Mach number,and the curvature of the rigid-flexible coupling profile should be continuous;and(3)The synchronism and harmony between rigid-flexible couplingmechanism adjustmentand flow-field dynamic response must be ensured.

In this paper,the aerodynamic design and dynamic analysis of a rigid- flexible coupling pro file which was applied to a halfflexible single jack nozzle structure was studied by using a theoretical calculation and numerical simulation method.The flow- field was calibrated via wind tunnel tests,and the design method and flow- field quality were veri fied.

2.Design calculations

2.1.Nozzle design

In 1955,Rose´n first proposed the prototype of an adjustable nozzle.16The half-flexible single jack nozzle profile is composed of a fixed profile plate and a flexible plate with linear thickness,and its principle of regulating its Mach number was introduced in Ref.17.A comparison between the nozzle designed in this paper and a conventional half-flexible single jack nozzle is shown in Fig.1.As can be seen,the present nozzle has the following characteristics:

(1)The longitudinal height compensation improves the axial free deformation at the end of the flexible plate.Rapid regulation of the nozzle’s exit Mach number has been achieved at the expense of the loss of the boundary correction capability at the nozzle end.In addition,coupling performance ofthe rigid-flexible profile is improved.

(2)The rotating mechanism is replaced by an arc rail around the fulcrum N of the fixed plate,in order to resolve technological problems during the transition process,such as subsonic starting or transonic acceleration.

2.2.Elastica inverse problem

The core issue in the design of a flexible nozzle is the elastica inverse problem,also known as the elastic large deflection problem,18which means to ensure a coupled flexible profile to be consistent with a theoretical aerodynamic profile.

For our half-flexible single jack nozzle,the thickness of the plate is increased linearly along the flow direction to ensure that the curvature of the aerodynamic profile changes continuously,as shown in Fig.2.

The change of the flexible plate thickness is given by

where β is the variable thickness factor of the flexible plate,h is the thickness at the flexible plate end,h0is the thickness at the flexible plate starting point,and ξ is a dimensionless relationship,i.e.,x=ξL,in which x is the abscissa of the flexible plate,and L is the total length of the flexible plate.The curvature equation of the flexible plate curve is given by

Fig.1 Airstream models of half-flexible single jack nozzles.

Fig.2 Flexible plate of our half-flexible single jack nozzle.

where M is the bending moments generated at x of the flexible plate,E is the modulus of elasticity of the flexible plate,I is the moment of inertia of the flexible plate,b is the width of the flexible plate,and P is the loading.

Boundary conditions are given by

where ytis the ordinate of the throat,yNis the ordinate of the rotation point N,θAis the angle of the maximum expansion,and ε is the incremental angle of the flexible plate.

According to Eq.(4),the relationship between the variable thickness coefficient β and the flexible plate length L can be obtained.The final parameters of the flexible plate structure are shown in Table 1.Then a simplified bending function y can be obtained according to Eq.(2)as follows:

Table 1 Physical parameters of the flexible plate.

2.3.Coupled dynamic analysis based on Kane equation

In this paper,the maximum displacement of the nozzle throat height reaches 38.7 mm,and this will provoke substantial elasticdeformation oftheflexibleplate.Thecoupling between elastic deformation and the large range movement of multi-body will make dynamic behaviors more complex,and power tempering or other special dynamic phenomena may happen.19As a consequence,it will be bene ficial to study the dynamic characteristics of rigid- flexible coupling with a flexible element.

Based on the Kirchhoff assumption,20a mid-surface model of the flexible plate is established,as shown in Fig.3.The coordinate system O-XYZ is the body-fixed coordinate system.Assuming the flexible plate size as L×b×h,and assuming the deformation displacement vector,from point p0which is a random point on the mid-surface before the deformation to point p which is on the mid-surface after the deformation,as u（u1，u2，u3）,then the velocity vector at point p based on the inertial system can be given by

where V0and ωAare the velocity and angular velocity vectors of the body-fixed coordinate system relative to the inertial coordinate system,respectively;p and u are the position and deformation vectors in the body-fixed coordinate system;andAVpand p are the velocity vectors relative to the floating group.

S，r,and u3are used to describe the deformation displacement,21where S and r are the arc length variables at point p in mid-surface plates,while u3is the coordinate variable along the height direction of the Cartesian coordinate system.Discreting it by assuming modal method,we obtain

where t is the time coordinate,∅1i,∅2i,and ∅3iare the modal functions,N1,N2,and N3are the interception orders of each modal,and q1i（t）,q2i（t）,and q3i（t）are the coordinate of each modal.

According to Kane equation,taking u2=S0,ω=˙θL,ω1=ω3=˙ω1=˙ω3=0,v1=v2=v3=0,and˙v1=˙v2=˙v3=0 as boundary conditions for the rotating rectangular plate,the simplified system dynamic equation can be given by

Fig.3 The deformation description of flexible plate Intermediate flat.

S0is the initial displacement of the flexible plate,ω is the angular velocity,ω1,ω3are the angular velocity of each vector,˙ω1,˙ω3are the first-order derivative of ω1and ω3,θ is the rotational angle of the flexible plate,˙θ,¨θ are the first-order derivative and second-order derivative of θ,v1,v2,and v3are the velocity of each vector,˙v1,˙v2,and˙v3are the first-order derivative of each velocity,q1j,q3jare the coordinate of each modal,˙q1i,˙q3jand¨q1j,¨q3iare the first-order derivative and secondorder derivative of q1i,q3jand q1j,q3irespectively,MTmijis the unit mass,Xmiis the mass of each vector,T=1，3,m=1，3,are corresponding element stiffness,and G is the shear modulus of the flexible plate.

Therefore,system dynamic equations have been obtained.Given different control laws,such as the flexible plate rotation angular velocity ω,the axial deformation with the variation with time of the flexible plate free end can be obtained.

2.4.Design results

A rigid-flexible coupling simulation model of the profile mechanism has been established,as shown in Fig.4.Combined with the kinetics analysis,movement under a nonlinear load is simulated,and the final profile curve can be obtained,as shown in Fig.5.

As shown in Fig.5,results of rigid-flexible coupling simulation at Mach 1.8 suggest that the end supporting system of the profile mechanism has a great influence on its deformation.Because the end is fixed,a reverse deformation of the connection between the flexible plate and the fixed profile is caused.The design of an axial free motion end of the profile has resolved this problem(axial displacement of 0.84 mm),which makes the real profile match the aerodynamic theoretic profile very well.There is a height difference at the profile exit,which is mainly caused by aerodynamic calculation error and boundary layer correction.Fig.6 is the nozzle’s final wind tunnel profile under Mach 1.0 to 2.0.

Fig.4 Shape mechanisms simulation models based on rigidflexible coupling.

Fig.5 Comparison of rigid-flexible coupling simulation lines.

Fig.6 Half-flexible single jack nozzle lines.

3.Calibration

3.1.Test methods

To verify the design method of half-flexible single jack nozzles mentioned above,several wind tunnel tests have been performed.The model nozzle is 760 mm long,the inlet size is 430 mm×190 mm (height×width), the exit size is 190 mm×130 mm(height×width),and the designed Mach number range is from 0.3 to 2.0.Owing to the limitation of the gas capacity of the altitude test facility,the maximum Mach number in the tests is up to 1.8,and the simultaneous temperature was not simulated at this time.

The verification test mainly includes the following parts:

(1)The steady-state calibration of pro file and stress at different Mach numbers.The inlet and outlet pressures,as well as the pro file control parameters are calibrated by a Mach/ flow angularity probe at Mach numbers of 1.1,1.2,1.3,1.4,1.5,1.6,1.7,and 1.8,respectively.The flexible plate stress and aerodynamic pro file are measured simultaneously.Comparing measurements with coupling simulation results,the flexible plate design method of half- flexible single jack nozzles can be verified.A photograph of the flexible plate’s back side with 4 points for strain and static pressure measurements is shown in Fig.7,and a photograph of the Mach number calibration test is shown in Fig.8.

(2)Expansion continuous Mach number regulation tests.The flow- field quality at the nozzle exit is measured by Pitot rakes,and the main results are shown as follows.

3.2.Stress results

Fig.9 shows the stress distributions of the flexible plate at the maximum test status(height 11 km,Mach 1.8).Generally,the stress distributions of the upper and lower profiles are consistent with those from the numerical simulation.However,the inlet stress value is about 30 MPa higher than that of simulation,and this may be attributed to the sidewall friction at the throat connecting or the stress increase caused by structural self-gravity.

Fig.7 Photograph of coupling mechanism and measurement arrangement.

Fig.8 Photograph of nozzle test.

Fig.9 Comparison of stress.

3.3.Aerodynamic profile results

The nozzle is assured to be under expansion by adjusting the pressures at the test inlet and outlet.The nozzle profile location can be calculated by the wall static pressure as adiabatic isentropic flow,which is given by

where Maxis the Mach number of measurement point x,Ma is the Mach number of the nozzle exit,Pxis the static pressure of measurement point x,P*is the total pressure,H is the height of the nozzle exit,Hxis the height of measurement point x,Lxis the axial distance between measurement point x and the nozzle throat,Rexis the local Reynolds number of measurement point x,and k is the gas constant.

Fig.10 Comparison of profiles of Ma 1.393,1.604 and 1.810.

Fig.10 shows the comparisons of profiles between the theoretical ones based on the aerodynamic simulation and the measured ones at different Mach numbers.It is suggested that the nozzle flexible plate height matches well with the designed height at the throat.However,the actual height is appreciably lower than the designed one at the nozzle exit,and the difference increases with the Mach number.This is mainly because of the boundary layer correction error.

3.4.Nozzle flow field quality

Fig.11 shows the measured core area Mach number versus time on the section of 5 mm downstream from the nozzle exit,when the Mach number is continuously adjusted from 1.0 to 1.7.It should be specially explained that:(1)As being limited by the measurement capacity constraints and for security reasons,the maximum Mach number in the continuous Mach number regulation test is 1.7;(2)In order to control test costs,only four flow-field points in which the Mach numbers are 1.3,1.4,1.5,and 1.6 respectively are measured in detail;and(3)The continuous Mach number regulation rules adopt a progressive strategy,in other words,based on the measured vacuum pressure and the profile control parameters calibrated in the first step,the Mach number can be adjusted from 1.1 to 1.2,from 1.2 to 1.3,and so on,until Mach 1.7.

The rigid-flexible coupling mechanism synchronism response performs well,and the Mach number control error does not exceed 0.03.Mach number uniformity is characterized by the root-mean-square error σ of all the Mach numbers of measurement points in the core region.The smaller the σ is,the more uniform the Mach number is.The definition is given by

where Maijis the Mach number of measurement point(i,j)in the core region,and n is the number of measurement points in the core region.

The root-mean-square error σ of Mach numbers measured in the regulation tests is compared with the standard specification about the quality of wind tunnel flow-field in Ref.22,as shown in Fig.12.Results show that the flow-field quality at the exit of the coupling profile is better than the eligible target of Chinese military standards in the continuous Mach number regulation tests,and some indexes approach or even exceed advanced standards.

Fig.11 Regulation course of the Mach number in succession.

Fig.12 Comparison of the Mach number root-mean-square.

4.Conclusions

A rigid- flexible coupling pro file aerodynamic design method has been presented in the paper,which solved the contradiction between aerodynamic performance,mechanical property,and regulation performance for the coupling pro file of a halfflexible single jack nozzle in supersonic free jet tests.In addition,the dynamic characteristics of a typical case were studied further.The main conclusions can be drawn as follows:

(1)The pro file design method of axial free stretching improved the matching performance of the rigidflexible coupling mechanism,and its theoretical design results and dynamic features were basically consistent with the test results.

(2)Via rigid- flexible coupling design,continuous Mach number regulation has been achieved.The Mach number control error did not exceed 0.03,and the flowfield uniformity at the nozzle exit was better than the eligible target of Chinese military standards in the continuous Mach number regulation tests,with some indexes approaching or even exceeding advanced standards.

Acknowledgments

This research was supported by the National Natural Science Foundation of China(Nos.90916023 and 51176075).

Appendix A.Supplementary material

Supplementary data associated with this article can be found,in the online version,at http://dx.doi.org/10.1016/j.cja.2016.09.002.

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Chen Pengfei is an engine testing engineer and has an M.S.degree.His main research interests are aero-engine altitude simulation technology research and aerodynamic devices design.

Wu Feng is a senior engine testing engineer and has an M.S.degree.His main research interest is aero-engine altitude simulation technology research.

Xu Jinglei is a professor and Ph.D.advisor at Nanjing University of Aeronautics and Astronautics.His current research interests are variable Mach number wind tunnel nozzles.

Feng Xudong is an engine testing engineer and has an M.S.degree.His main research interest is aero-engine altitude test facility Mamp;S technology.

Yang Qiao is an engine testing engineer and has an M.S.degree.His main research interest is uncertainty assessment of aero-engine altitude simulation measurement.

11 August 2015;revised 13 June 2016;accepted 28 June 2016

Available online 20 October 2016

Aerodynamic profile;

Flow-field quality;

Free jet;

Nozzle;

Rigid-flexible coupling;

Variable Mach number;

Wind tunnel

*Corresponding author.Tel.:+86 816 3856382.

E-mail addresses:zhongguoren.cpf@qq.com(P.Chen),wufeng_my@qq.com(F.Wu),xujl@nuaa.edu.cn(J.Xu),363166446@qq.com(X.Feng),kubyka@126.com(Q.Yang).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2016.09.002

1000-9361Ⓒ2016 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/).