Coupled CFD/RBD Modeling for a BasicFinner Projectile with Control

Ke Xi, Chao Yan, Yu Huang

(1. Research Institute of navigation and Control Technology, Beijing 100089, China 2. School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China)

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Coupled CFD/RBD Modeling for a BasicFinner Projectile with Control

Ke Xi1,*, Chao Yan2, Yu Huang2

(1. Research Institute of navigation and Control Technology, Beijing 100089, China 2. School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China)

This paper introduces the development and application of an advanced time-accurate multidisciplinary technique for the prediction of unsteady aerodynamics and flight dynamics of maneuvering projectiles. The virtual flight simulation of a basic finner projectile has been investigated through coupling solving procedure of unsteady Navier-Stokes equations, rigid-body six degree-of-freedom motion equations and control law. The flow solver uses a finite-volume method based on structure grid with dual time stepping. Additionally, a feedback controller is required for the control surfaces to attain the commanded deflections, and the overset Chimera method is used to simulate relative motions. The predictions show that the gains for the controller must be correctly selected to obtain the proper response. Simulation results show that the virtual flight simulation platform developed here is capable of solving the complicated unsteady flows with moving boundary, has a strong promotion to engineering application.

Computational Fluid Dynamic, Basic finner, Virtual flight simulation, Unsteady flow, Feedback control

0 Introduction

Agility and maneuverability are always the basic requirements of advanced air combat weapons. In order to meet the requirements of future air combat, aircraft design has been pushed towards the expansion of flight envelopes into extreme conditions (high angle of attack) and aggressive maneuvers with high angular rates, involving strongly nonlinear aerodynamic phenomena, when the asymmetric vortical flow would dominate the flow field. The vehicle would typically operate in the post-stall regime of the lifting surfaces. The dynamic and time-dependent effects of the nonlinear flow characteristics, including flow separation, vortex shedding, and vortex lag effects, contribute significantly to the behavior and maneuvering capability of the vehicle. The basis of conventional flight mechanics models is a coefficient based des-cription of the aerodynamics, with the coefficients (stability derivatives) being obtained from experiment or computational fluid dynamics (CFD), and it is incompetent to describe some rather complex aerodynamic phenomena which incorporates significant hysteresis. There is a need

to develop a high-fidelity model which is able to replicate the actual free flight. In the past 30 years, with the rapid growth of the computational resources, CFD has achieved a significant progress and CFD is considered to be very close to its maturing stage in steady flow computation at designed conditions. The next target of CFD is numerical simulation of maneuvering vehicles through multidisciplinary computations [1].

In principle CFD can produce the aerodynamic inputs but also, through a coupling procedure with rigid body dynamics (RBD) and control, can simulate a rigid motion response directly. CFD based virtual flight simulation is such a technology which integrates multidisciplinary design method. The basic concept of virtual flight test which originates from wind tunnel is that the vehicle will move freely, the control surfaces will be able to real time control according to the flight control systems demand, and the aerodynamic loading and motion parameters are obtained simultaneously during the simulation. The CFD based virtual flight simulation has the potential to provide detailed fluid dynamic understanding of the unsteady aerodynamics processes involving the maneuvering flight of modern air combat weapon systems. This multidisciplinary design technology will be able to explore the coupling mechanism of the aerodynamics/flight dynamics, resulting in reduced development costs and higher performance weapons.

A lot of progress has been made recently with the coupled CFD/RBD procedure for prediction of unsteady aerodynamics and flight dynamics of projectiles [2-6]. Control actions and their aerodynamic phenomena have also been extensively studied. The most remarkable subjects are the effect of different actuation schemes, such as jet actuators [7], elastic deformation [8] and motion of control surfaces. Observing independent studies that relate control and RBD to aerodynamics, the natural step is to link them all in a comprehensive, integrated application [9-11]. With the increasing capacity of computers, coupling of CFD with motion dynamics and control can represent an important tool for the design of an aircraft’s geometry and its guidance and control algorithms, as it can be employed to obtain aerodynamic models, evaluate performance during control actions and compute virtual fly-outs.

With this in mind, this paper presents efforts towards such high-fidelity flight dynamic models. The intention here is not to investigate complex flight mechanics behavior, but to describe the development of a tool which can be used for this purpose. The methodology herein proposed uses CFD to estimate the transient flow field around the vehicle; uses the vehicle’s equations of motion to calculate its movement in 3-dimensional space; and a controller which uses the body’s motion state to command a deflection of its control surfaces. This information is then fed back to the CFD domain to restart a new flow field and advance in the simulation of a fly-out.

1 Computational Methodology

1.1 Computational Fluid Dynamics

In this study, the MI-CFD flow solver was used to compute the flow field around the vehicle. MI-CFD is developed under the leadership of Prof. Yan at Beihang University [12]. MI-CFD is a second-order, cell-centered, finite-volume Navier-Stokes (N-S) flow solver. The complete set of three-dimensional time-dependent Reynolds-Averaged Navier-Stokes (RANS) equations is solved in a time-accurate manner for simulations of unsteady flow fields.

The conservation form of the 3-D dimensionless, unsteady, compressible, RANS equations in body-fitted coordinates is as follows:

(1)

whereQis the vector of conservative variables,F,GandHare the convection flux vectors,FV,GVandHVare the viscous flux vectors,tstands for time,Re∞theReynoldsnumberoffreestream.

MI-CFDiscapableofusingseveraldifferentfluxschemes,timeschemes,limitersandturbulencemodels.Inthisstudy,theRoe’sFDSschemeisused,aswellastheminmodlimiter.Two-equationshear-stress-transport(SST)turbulencemodelisadoptedforthecomputationofturbulentflows.Dualtime-steppingisusedtoachievethedesiredtime-accuracy.Open-MPparallelstrategyisem-ployedtospeedupcomputation.Thedetailscanbefound,forinstance,inRef.

[13].

1.2 Chimera Technique

The chimera technique, as implemented in MI-CFD, provides the capability to simulate large amplitude control-surface deflections. The overset grid gets rid of the restriction of structured grids’ topology and thus, for arbitrary complex geometries, grids with different topologies can be generated independently. The technique is based on creating overlapping grids to handle a time-dependent relative motion of different geometries. In the overlapping grids concept, there are two major procedures involved in establishing intergrid communications. The first procedure is hole cutting (automatic or manual). For the cases presented in this paper, automatic hole cutting is used. The second procedure is the identification of interpolation stencils, which involves a search of donor cells for all intergrid boundary points. The search algorithm is based on a state-of-the-art alternating digital tree (ADT) data structure [14].The ADT data structure optimizes the search operations to O(lgN/lg2) as opposed to the total number(N) of elements in the computational domain. For the moving multi-body simulation presented in this paper, an efficient method which is known as dynamic overset is used since the overset grid assembly should be performed at every time step.

1.3 Coupled CFD/RBD/GNC Procedure

A coupled CFD/RBD procedure is needed for the simulation of virtual fly-out of projectiles. The CFD/RBD simulation combines a rigid six degree-of-freedom (DOF) flight dynamic model with a three dimensional, time-accurate CFD simulation. This combination is formed by allowing the RBD dynamic equations to be integrated forward in time, while the aerodynamic forces and moments are generated by CFD. The reference frames used for the CFD/RBD simulation are the standard inertial and body frames, and are depicted in Fig.1 below. The inertial frame,i, is fixed to the ground, and oriented withyipointingupwards.Thebodyframeisfixedtothecenterofmassoftheprojectile,andisorientedsuchthatxbpointsthroughthenoseoftheprojectileandtheotherunitvectorsformaright-handedsystem.TheprojectileorientationisdefinedbytheuseofEuleranglesinthestandardaerodynamicsequenceofrotations.Thus,therigidbodystatecontaintheinertialprojectilecenterofmasslocation, (x,y,z), the Euler angles, (φ,θ,ψ),thebodyframecomponentsoftheprojectilecenterofmassvelocity(u,v,w), and the body frame components of the projectile angular velocity (p,q,r).

Fig.1 Reference frame definition.

The RBD equations used here are the standard equations for the center of mass of a rigid projectile. They are given in the equations below.

where the forces (Fx,i,Fy,i,Fz,i)andthemoments(Mx,My,Mz)intheaboveequationsrepresentthetotalforcesandmomentsappliedtotheprojectile.Thenonlineardifferentialequationscanbesolvedusingafourth-orderRunger-Kuttamethod.Thedual-EulermethodisusedtoovercomethesingularityofEulerequation[15].

Forsimulationsofguidedcontrolmaneuvers,acoupledCFD/RBD/GNCprocedurehasbeendevelopedbyintegratingGNCpartintothecoupledCFD/RBDmethod.Theaddedelementofthiscouplingistheguidance,navigation,andcontrolaspect(GNC).TheGNCcapabilityemploysafeedbackcontroller.Asthecontrolalgorithmisnottheprimarybasisofthisresearchasimpleproportional-integral-derivative(PID)controllerisused.PIDcontrollerisalinearcontrollerbasedonthesimpleestimationofthepast,presentandfuturemessage.Duringtheflight,thecontrolofthevehiclecanbeachievedinmanyways,e.g.,withcontrolsurfacesorpulsejets.Inthepresentpaper,onlytheformercontrolmechanismisstudied.

Fig.2depictstheschematicofthePIDcontroller.αeisthedesiredstate,αtisthereal-timestate,δristhecontrolsurfacedeflection.PIDcontrollerworksviasomefundamentalcontrolelementssuchassummers,comparators,gainsandintegrators.Ithastheform

(3)

Wheree(t)=αe-αtisthedeviation,u(t)isthecontrolvariable,Kp,KiandKdarethecontrolgains.

Fig.2 Schematic of the PID controller.

InthecoupledCFD/RBD/GNCprocedure,theaerodynamicforcesandmomentsarecomputedateverysingletimestepintheCFDsolverandtransferredtotheRBDparttopredictthenewflightstates.Theflightcontrolsystem(FCS)providesthecontrolvariablesbasedonagivenFCSdesign.TheFCSdesignallowsformodelingofbothcontrolledandprescribedmotions.Thecontrolstrategyusesthevehicle’smotionstatevariablesasfeedback,andmustdefinethedesiredattitudes.Controlalgorithmisworkedtodefinethecommandeddeflectionineachcontrolsurface.TheoutputofRBDstateandthecontrolvariablesaretransferredtotheCFDflowsolverwhichthencomputestheaerodynamicforcesandmomentssubjecttotheseRBDstateandcontrolvariables.

ThecouplingoftheCFD/RBDprocedureisalooseone.Fig.3illustratesthemethodology,whichcanbesummarizedasfollows:

Fig.3 Diagram of the coupled process.

1)Withtheinitialmotionoftheobject,thesteady-stateflowfieldiscomputedthroughCFD.

2)Aerodynamicforcesandmomentsareobtainedthroughintegration.

3)RBDmodelissolvedandnumericallyinte-gratedonetimestepforwardtofindthenewmotionstate.

4)Controlalgorithmisexecutedtodefinethecommandedanglesineachcontrolsurface.

5)Oversetgridisusedtoaccountforrelativemotionandthedomain’snewboundaryconditionsaredefined.

6)Dual-time-steppingmethodisusedtoadvancethetimesteptoobtaintheupdatedflowfield.

7)Returntostep2andrepeatuntildesiredphysicalsimulationtimeisreached.

2 Computational Model

The test case for the current simulations is the Basic Finner projectile. The model is a generic missile configuration which consists of a cone-cylinder with four rectangular fins (see Fig.4) [16]. The length of the projectile is 10 calibers and the diameter is 30 mm. The conical nose is 2.84 calibers long and is followed by a 7.16 caliber cylindrical section. Four fins are located on the back end of the projectile. Each fin is one caliber long, and has a thickness of 0.08 calibers. The center of gravity (CG) was located at 5.5 calibers from the nose of the projectile. Table 1 shows the physical properties of the model.

Fig.4 Schematic of the basic finner.

A structured overset computational mesh (see Fig.5) was generated for the projectile. In general, most of the grid points are clustered in the boundary-layer, fins, and the wake regions. The boundary layer spacing near the wall was selected in order to achieve ay+value of 1.0. The mesh was generated using the Gridgen V15.17 grid-generation software developed by Pointwise Inc. The main body and four rectangular fins were gridded separately and this significantly reduced the restrictions on grid generation. The grids are then overset with each other using a Chimera method. The total number of grid points in this case is approximately 5 million. This Chimera procedure requires proper transfer of information between the main body mesh and the fins meshes. However, the advantage is that the individual grids are generated only once and the Chimera procedure can then be applied repeatedly as required during motion. There is no need to generate meshes at each time step during the maneuvers.

Fig.5 Overset grid of the model.

PhysicalpropertiesCalueLength/m0．3Diameter/m0．03Mass/kg1．58Ixx/(kg·m2)1．92×10-4Iyy，Izz/(kg·m2)9．785×10-3CG(fromnose)/m0．165

3 Results and Discussion

This paper pursues the high-fidelity assessment of the flight dynamics. Therefor a virtual flight simulation platform coupling CFD and RBD including control surface motion capabilities was developed. Demonstration of the coupled CFD/RBD technique has been achieved using an angle of attack control maneuver. The intention here is not to investigate complex flight mechanics behavior, but to describe the development of the tool, so we are only dealing with longitudinal motion. Besides, the translational motion is constrained.

Typically, a solution obtained in the steady-state mode serves as the starting solution for the coupled CFD/RBD run. The coupled run starts to deflect the fins in accordance with the control law or the controller. CFD results are obtained in a time-accurate manner.

For the computations presented here, the initial state of the projectile is fixed at zero degrees angle of attack, a height of 10 km and a flight velocity of Mach 2. The initial deflection angle of the elevator is supposed to be zero degree. The initial conditions represent the initial state of the projectile prior to a commanded maneuver.

3.1 Open-loop Simulations

Generally, the flying qualities include both controllability and stability. First we simulate the open-loop response of the projectile for different types of rudder control laws. We can have a comprehensive knowledge of the dynamic characteristics of the projectile from these simulations. After data are acquired, parameter identification techniques can be applied to the motion history to extract aerodynamic coefficients.

Given the four rudder control laws:

δe(t)=5°,t≥0

(4)

(5)

(6)

(7)

Formula(4)through(7)eachstandsforastep-rudder,adoubletrudder,atrapezoidalrudderandaharmonicrudder.Fig.6showsthecontrolsurfacesdeflectionsandpitchanglehistoriesoftheprojectileopen-loopsimulations.Thesimulationtimetakesonly1.5seconds.

TheequationofmotionfortheopenloopsimulationscanbewrittenasshowninEq.(8).

(8)

Thesecond-ordercharacteristicequationis

(9)

Eq.(8)reflectstheinherentdynamicbehaviorsoftheprojectile.TherighthandsideofEq.(8)whichstandsfortheroleoftheflightcontrolsystemwillchangethedynamicbehaviorsoftheprojectile.Whentherighthandsideisnotzero,namely,theflightcontrolsystemtakeseffect,theautonomoussystemchangesintonon-autonomoussystem.Thedynamicbehavioroftheprojectileisdeterminedbytheinherentcharacteristicsandcontrolsystemtogether.

(a) Step-rudder

(b) Doublet rudder

(c) Trapezoidal rudder

(d) Harmonic rudderFig.6 The response to different types of rudder deflections.

Forthestep-rudder,theanalyticalsolutionofthegoverningequation(Eq.(8))canbewrittenas:

(10)

Where

αtrim=-Cmδeδe/Cmα

(11)

Itusuallytakestheovershoot,peaktime,transitiontimeandoscillationfrequencytoassessthedynamicquality.Ingeneral,gooddynamicqualityrequiresnoobviousovershootoroscillationintransitionprocess,andtheprojectilequicklyreachesthedesiredstate.

Fig.6(a)showstheresponseofastep-rudder,whichrepresentsanextremecasewhenthevehicleisundergoingrapidmaneuverandtherudderanglerateisinfinity.Fig.6(b)showstheresponseofadoubletrudder,whichrepresentstheinstantaneousinterferencethevehicleencountered.Fig.6(c)showstheresponseofatrapezoidalrudder,whichrepresentsaslowmaneuvering.Ascanbeseenfromthefigure,theovershootisobviouslylessthanthatofstep-rudder.Fig.6(d)showstheresponseofaharmonicrudder,whichrepresentstheidealizedrepeatedmodificationprocess.Allthecomputationsindicateabothstaticallyanddynamicallystableconfiguration.Fromthesesimulations,wecanobtainthekeyfeaturesoftheprojectilesoastobepreparedfortheclosedloopsimulation.

3.2 Closed-loop Simulations

For a actual flight, the autopilot or flight control system will send control commands according to the flight state in real time, to make the vehicle stabilize in scheduled trajectory or maneuver to the desired state. Therefore, a feedback controller is needed. A simple proportional derivative control law is used for feedback control.

The elevator deflection angle is given by the equation below.

δe(t)=Kp(αd-α(t))+

(12)

ThecontrolgainsKp,KiandKdarechosenexperimentallytoobtainadesirabletimeresponseofthesystem.

Inreality,thecontrolfinsarelimitedinbandwidth.Assumethatthecontrolfinsreceivecontrolsignalseverytruddertime. And the physical time step used in CFD calculations is far less thantrudder. Again suppose that the deflections of the control surfaces are continuous and smooth duringtruddertime, rather than a step rudder. The smooth formula is as follows:

δ(t)=Δδ·(10λ3-15λ4+6λ5)

λ=t/trudder-int(t/trudder)∈[0, 1]

(13)

WhereΔδis the total deflection angle duringtruddertime.

Based on PID controller, a closed-loop angle tracking system was constituted to simulate a commanded maneuver.

The control command was that the projectile maneuvered from the initial state to five degrees angle of attack, and stabilized in this position.

The equation of motion for the closed-loop simulation is the same as Eq.(8), butδeisdeterminedasshowninEq.(12).

SubstitutingEq.(12)intoEq.(8)yields

(14)

AsEq.(14)explicitlyhasthetimevariable,itbecomesanon-autonomoussystem.Thedynamiccharacteristicofthevehicleisdeterminednotonlybytheinherentpropertiesbutalsobythecontrolsystem.Forthestaticallyordynamicallyunstablevehicle,thecontrolgainscanbeadjustedtomakethevehiclestable.

Thesignificantbenefitofthecoupledapproachisthattheinstantaneousaerodynamicforceandmomentarecomputedtodeterminetheresponseoftheprojectile.Thepredictedaerodynamicloadincludesalltherelevantinstantaneouseffects,includingcontrolsurfacesdeflections,deflectionrate,wakeandallotherinterferenceeffects.Asthesimulationresultsaresensitivetothecontrolgains,theparametricstudiesneedtobeperformedtodecidetheoptimalchoice.

Fig.7showstheresponsesfordifferentparameters.Theparametersarelistedontopofthefigures.Ontheleftsidearethetimehistoriesofangleofattack(blacksolidline)andelevatordeflectionangles(reddashdottedline),andtherightsideisthepitchrateversusangleofattack.Kp,Kdandtruddervary from case to case. The integral gain is not important since the system contains little steady state error, so the integral gain maintains as a constant.

As shown in Fig.7, initially the angle of attack is zero, starts to increase with the deflections of control surfaces. Under the effect of the controller, the system brakes in advance to prevent overshoot. The pitch rate begins to reduce while the angle of attack continues to increase. Finally, the projectile reaches the desired state after several adjustments. The phase portrait shows a stable spiral point structure. In general, good dynamic quality requires no significant overshoot or oscillation, and short transition time. We can compare the results of case A with case B to find how the derivative gain affects the dynamic behavior of the projectile. In the two cases, all the parameters are same exceptKd. We can find that the overshoot for case B (with smallerKd) is a little larger than the one for case A, and the maximum pitch rate follows the same tendency. In fact, the derivative gain helps the system brake in advance to prevent overshoot, and it works like damping coefficient, and it’s related to dynamic stability derivative. Also we can compare the results of case C with case D to find how the proportional gain affects the dynamic behavior of the projectile. From the two cases, we can find that the overshoot for case C (with largerKp) is a little larger than the one for case D, and the maximum pitch rate follows the same tendency. Besides, the transition time is a little longer for case C. The proportional gain is related to steerage. The speed of response can be faster by increasing the proportional gain, but the response can also be oscillatory or even diverge if the proportional gain is too large. From the results of case A and D, we can find that the overshoot is almost the same for the two cases, but the maximum pitch rate of case A is larger than that of case D. Obviously, larger pitch rate would be needed to reach the state in shorter time. But it is not a good idea to choose a value close to the natural frequency, or it will cause oscillation or even divergence.

Fig.7 The responses for different parameters, angle of attack and elevator deflections histories, pitch rate vs. angle of attack.

Fig.8 shows the computed pitching moments as a function of time. Results obtained from all four cases are compared to each other. The maximum moment shows similar behavior with the maximum pitch rate.

To demonstrate the capability of the platform for the prediction of high angle of attack conditions, a new simula-tion is conducted. The new control command is that when the projectile reached five degrees angle of attack, and stabilized, then maneuvers to fifteen degrees angle of attack, and stabilized in the new position.

Fig.9 shows the responses of the projectile under new command. Fig.10 shows the phase portrait. As can be seen from the figures, high frequency oscillation appears when the projectile maneuvers from five degrees angle of attack to fifteen degrees angle of attack. This is probably caused by the nonlinear characteristic of the flow field due to the vortices flow at high angle of attack.

Fig.8 Computed aerodynamics pitching moments as a function of time.

Fig.9 The responses under new command.

4 Conclusion and Future Work

This paper describes the development and application of a time-accurate coupled CFD/RBD multidisciplinary technique for prediction of un-steady aerodynamics and flight dynamics of maneu-vering projectiles.

Using this multidisciplinary approach, the time-dependent response of the Basic Finner projectile has been simulated. Simulation results show that the virtual flight simulation platform is capable of solving the complicated unsteady flows with moving boundary, can provide the data for the unsteady aerodynamic modeling, indicate the stability of aircraft and test flight control law, has a strong applicability to engineering application. Work is

currently on-going to simulate the full degrees of freedom of the virtual fly-out.

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0258-1825(2016)02-0182-08

帶反饋控制的基本帶翼外形的耦合氣動/運動數(shù)值模擬研究

席 柯1,*， 閻 超2， 黃 宇2

(1. 中國兵器工業(yè)導(dǎo)航與控制技術(shù)研究所， 北京 100089; 2. 北京航空航天大學(xué) 航空科學(xué)與工程學(xué)院， 北京 100191)

介紹了一種先進(jìn)的、時間精度的多學(xué)科數(shù)值模擬平臺的開發(fā)和應(yīng)用，該平臺可用于預(yù)測飛行器機動過程中的非定常氣動特性和飛行力學(xué)特性。通過耦合求解非定常Navier-Stokes方程、剛體動力學(xué)方程及控制律，對基本帶翼外形進(jìn)行了虛擬飛行數(shù)值模擬研究。求解器采用基于結(jié)構(gòu)網(wǎng)格的有限體積法，時間方法采用雙時間步方法。使用反饋控制器獲得舵偏控制指令，重疊網(wǎng)格方法用來模擬物體間的相對運動。模擬結(jié)果顯示控制器的增益必須正確選擇才能獲得正確的響應(yīng)。仿真結(jié)果表明本文發(fā)展的虛擬飛行數(shù)值模擬方法能夠預(yù)測包含動邊界的復(fù)雜非定常流動，具有較強的工程適用性。

計算流體力學(xué);基本帶翼外形;虛擬飛行;非定常流動;反饋控制

V211.3

A doi: 10.7638/kqdlxxb-2016.0005

*Research Assistant; cauchy86@163.com

format: Xi K, Yan C, Huang Y. Coupled CFD/RBD modeling for a basic finner projectile with control[J]. Acta Aerodynamica Sinica, 2016, 34(2): 182-189.

10.7638/kqdlxxb-2016.0005. 席柯， 閻超， 黃宇. 帶反饋控制的基本帶翼外形的耦合氣動/運動數(shù)值模擬研究(英文)[J]. 空氣動力學(xué)學(xué)報, 2016, 34(2): 182-189.

Received： 2015-12-15; Revised：2016-02-10