Trajectory optimization of a de flectable nose missile

Zhi-yong Zhng,Qi-zhong Tng,Xio-hui Sun,*,Zhi-hu Chen

aNational Key Laboratory of Transient Physics,Nanjing University of Science and Technology,Nanjing 210094,China

bNorth China System Engineering Institute,Beijing 100089,China

Trajectory optimization of a de flectable nose missile

Zhi-yong Zhanga,Qi-zhong Tangb,Xiao-hui Suna,*,Zhi-hua Chena

aNational Key Laboratory of Transient Physics,Nanjing University of Science and Technology,Nanjing 210094,China

bNorth China System Engineering Institute,Beijing 100089,China

A R T I C L E I N F O

Article history:

De flectable nose missile

Numerical simulation

Maximum range

Maneuverability

Genetic algorithm

The de flectable nose missile has a longer range by de flecting its nose to improve its aerodynamic feature. Based on detached eddy simulation(DES),the supersonic flow fields of a missile with de flectable nose are simulated numerically and its aerodynamic force coef ficients are calculated under the condition of the de flection angles vary from 0°to 8°,angles of attack,0°-8°,and mach numbers,2 to 5.Coupling these aerodynamic coef ficients with the plumb plane ballistic equations,the extended flight range has been calculated.Furthermore,the genetic algorithm(GA)is employed for the solution ofmaximum range of the de flected missile.

?2017 The Authors.Published 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/).

1.Introduction

Flight control of missiles is historically realized by tail-fins or canards.Tail-fin controls are located at the rear of the body,well behind the center of gravity.While canard controls are installed in the warhead[1],and are in fluenced by the head oblique shock.The location ofthe canards in the expansion region along the nose may result in undesirable aerodynamic and thermodynamic effects occurring in the high dynamic pressure environment[2].

In 1946,Goddard proposed another control device,the de flectable nose,to controlmissile flight[3].Through the changing of angles between the missile nose and the body axis,the de flectable nose missile can generate the required control force by air.It has the advantages ofsimple structure,smalladditionalresistance, little in fluence on the flow field of tail and high maneuverability. NASA performed wind tunnel tests on missiles with de flectable nose control[4,5]and investigated the feasibility ofbent-nose conic shapes as hypersonic aerocapture vehicles for planetary missions [6].Current studies on the de flectable nose missile are most concentrated in aerodynamic characteristics and dynamic control. Through controlling the head angle of attack,Liang achieved the goals of real-time correction of the trajectory and improved its hitting precision[7].Gao et al.established a control system based on the concept of de flectable nose and veri fied its ef ficiency in hypervelocity flight[8].Wang et al.,based on N-S equation and k-3 turbulent model,simulated the aerodynamic characteristics of models at different angles ofattack and Mach numbers,and found that,for low angles ofattack,the liftto drag ratio increases with the increase of the angles of attack[9].

Based on previous research,in this paper,we firstly simulates the flow fields of the missile with the de flection angles among 0°-8°,angles ofattack from 0°to 8°,and the Mach number from 2 to 5.Then,the aerodynamic forces are calculated based on the numerical results,and are applied to the plumb plane ballistic equations to investigate the flight stability of the de flectable nose missile.Finally GA is adopted to obtain the maximum range.

2.Numerical methods

2.1.Simulation method

Governing equations employed in this paper is Navier-Stokes equations,as shown below

Since the resistance of missiles caused by the gas viscosity increase at supersonic speeds[10],the surface boundary layer is necessary to be considered in numerical calculation.Large eddy simulation(LES)and direct numerical simulation(DNS)can accurately describe the flow structure of missile.But on the boundary layer,the demand for the grid number and the amount of calculation are relatively large.Reynolds average Navier-Stockes(RANS) has less amount of calculation but cannot accurately simulate the transient flow of turbulence.Therefore detached eddy simulation (DES)is adopted,which uses the RANS for approaching the boundary layer to reduce the demand for the grid number and LES is for the main flow.

In order to effectively catch shock waves near the warhead and tail,two-order AUSM scheme is employed to discrete the convection term.In computational fluid equations,two-order center scheme is used for viscosity term and third-order Runge-Kutta method is employed to discrete the time derivative terms.

The plumb plane ballistic equations are employed for the description of flight of the de flected missile and the 4th-order Runge-Kutta method is adopted for the solution.In the flight calculation,the earth is considered as a still sphere and standard atmosphere modelhas been adopted.The aerodynamic forces and moments of the plumb plane equations are inputted from the DES simulation.

2.2.Computational model

As shown in Fig.1,the computational model is a typical threedimensional de flectable nose missile(N2dB28F240)[11]with the mass of 15.87 kg,diameter of 56.2356 mm.

Fig.2 displays grid partition in x axis section near the tail.OBlock is built and re fined near the boundary layer,and the grid nodes become sparse sequently with its increased distance from the wall.Finally,the number of computationalgrids is about three million after the gird convergence tests.In addition,the left boundary is considered as the incoming flow and all other boundaries are out flow pressure field and the missile surface is considered as nonslip.The details are seen in the reference[10].

3.Results and discussions

Based on the requirement of parameters for solving the plumb plane ballistic equations,the flow fields of a missile,with the de flection angle varying amongβ=0°-8°,angles of attack, α=0°-8°,and the Mach numbers Ma=2-5,have been simulated numerically.

3.1.Aerodynamic characteristics of the de flection missile

Fig.3 presents the typicalpressure contours on the surface and around the missile,at angle of attackα=0°and nose de flection angle ofβ=0°,4°and 8°,respectively.It is nondimensionalized with the characteristic value ofthe pressure P0=73429.16 pa.Itcan clearly be seen that,for the case of(a)α=0°,β=0°,the flow field structure is almost symmetricalwith respect to the axis of missile. At the nose,the oblique shock appears,and the pressure increases signi ficantly around the head and decreases rapidly just behind the shoulder due to the expansion waves.In the middle of the missile there are no shock and expansion waves.The pressure is basically stable.But at the front of the tail,the flow is strongly compressed due to the block oftailfoils,and strong oblique shock waves appear, however,there is a low pressure recirculation region at the bottom of the missile.

For the cases of(b)β=4°and(c)β=8°,the structure ofthe flow field is not symmetrical about the XZ plane.The strength of the shock wave on the windward is much higher than that on the leeward,on the contrary,the expansion wave near the shoulder is much weakened on the windward(Fig.3(b-c)).This is because when the nose de flects upward,the compression ofthe flow on the windward is enhanced and the turning angle around the shoulder increases to make the flow expand quicker than thatofthe leeward. The structure ofthe flow field in the rear ofthe missile is almostthe same for allthe cases,but the intensity of the oblique shock of tail foils(Fig.3(b-c))is weaker than the non de flectable case(Fig.3(a)), therefore,the induced drag is smaller.

The high pressure region formed below the nose causes an additional lift over the forebody surface and makes the pressure center move forward,and also a change in aerodynamic coef ficients.Fig.4 exhibits the relationship between aerodynamic coef ficients of the missile and angle of attacks,in which the solid line represents numerical simulation values and the dotted line indicates the experimental values of Ref.[2].It is clear that both experimentaland numericalresults agree wellwith each other.The drag and lift coef ficients increase with the increase of angle of attack and the de flection angle,but the increase gradient of lift is larger.This is because the de flected nose makes the pressure on the windward side increases while that on the leeward side decreases (Fig.3).The pressure difference causes the increase of the force projected on both x and y axes.The main reason of the extended range of the de flected nose missile is caused by its pitching moment coef ficient.

On the other hand,with the increase of attack of angle,the pitching moment decreases and becomes negative at some points, it shows that the missile is longitudinally stable during flight.But for large attack of angles and the de flection angles,the differences between lift coef ficients of experiments and numerical simulation become large(Fig.4(b)),therefore,in this paper,the attack ofangles during flight should be kept smaller than 8°.

Fig.5 illustrates the relationship between aerodynamic coef ficients ofthe missile and Mach number with the nose de flection β=4°,angle of attackα=0°,4°,and 8°.It can be seen that,for supersonic flow,with the increase of Mach number,the drag coef ficient decreases,the lift is basically stable.The pitching moment increases slowly forα=0°,4°,and the smaller its attack of angle,the larger its value.

Table 1 Landing state of de flected missile with different de flection angle.

3.2.Trajectory optimization of the de flected nose missile

To simulate the flight trajectory,we take above de flected nose missile as the example.Substitute the calculated aerodynamic coef ficients into the 6 DOF ballistic equations to simulate the trajectories of the missile numerically with different de flection angles. We set the de flection starts at the end point of the boost phase,at this time,t≈100s,the body velocity is 1689 m/s,pitch angle is about 50°and angle of attack is almost zero.

Table 1 displays the landing state of de flected missile after de flecting the nose atthe highestpointofthe trajectory.Fig.6 is the missile trajectory with different de flection anglesβ=0°,3°and 5°. By comparison it is found the range increases progressively with the increase of de flection angleβ.Whenβ=5°,the growth rate is signi ficantly increased.Itillustrates the de flectable nose controlcan signi ficantly enhance the range ofthe missile but the landing Mach number and pitch angle both decrease.In addition,ifβis over 6°, the angle of attack willexceed 8°.Therefore this condition willnot be considered.

Typical pro files of angle of attack and pitching moment coef ficient are displayed in Fig.7.It is clear that the variation of pitching moment is opposite to that of the angle of attack,and acts as the righting moment and keeps the body stable.When the angle of attack increases,the pitching moment decreases,they reach at the extremum values at the same time,then,they vary contrarily again. Both the angle of attack and pitching moment have the same oscillation cycle.

Figs.8 and 9 shows the curves of attack ofangle and lift-to-drag ratio during the flight with different de flection angleβ=0°,3°and 5°.Itis clear thatboth attack ofangle and lift-to-drag ratio have the same variation tendency with different de flection angles.When β=0°,the lift-to-drag ratio are almost equal to 0°during all the flightdue to the very smallangles ofattack.Whenβ=3°or 5°,after the highest point of trajectory,the angle of attack is greater than 0°and the lift coef ficient increases.From Fig.9,the lift-to-drag ratios of the de flected nose missile are large.This is why the range ofde flected nose missile is longer than thatofnormalmissile.

However,the de flected nose makes the pressure center move forward with the increased fluctuation ofangle of attack and leads to the decrease of flight stability.

We use the projection of overload to the velocity coordinate system(Ox3y3)to discuss the overload.Nx3is the tangential overload and Ny3refers to the normal overload.In fact,a large tangential overload means a large tangential acceleration and the velocity of the missile is changed quickly.Moreover,if the missile has a large normaloverload,it is quicker to modify the direction of the flight.Therefore,the larger the overload of the missile is,the better its maneuverability becomes.As displayed in Fig.10,with the increase of the de flection angle,the tangentialoverload decreases, but its normal overload increases.Since the growth rate of thenormal overload is far larger than the reduction rate of the tangential overload, the de flected missile has better maneuverability.

Table 2 Results of the optimization.

To investigate the superiority ofthe de flected missile,two cases, the normal and the de flected missiles,are considered to optimize its trajectory.The former is to search an optimallaunch angle for a maximum range.The latter is to search the optimal launch angle and the variation of de flecting angle with time to maximize the range.Since the relationship between objective function and variables is nonlinear,and the objective function is discontinuous,an optimization algorithm which can adapt to the nonlinearity and discontinuity is needed.

Genetic algorithm is ideal for the optimization of our range problems and as it is a probabilistic search algorithm,each case is calculated 5 times to achieve the minimum fitness value as expressed in Eq.(1),where R is the range ofthe missile.In addition, the population size and the number of generations are taken to be 50 and 500,respectively.And the cross-over probability and mutation probability are chosen to be 0.7 and 0.1,respectively.

Table 2 shows the results ofoptimization.The range ofde flected missile increases from 156863m to 190840m,its extended range is 21.66%higher than that of the normal missile,therefore,the de flected nose missile is obviously advantageous in extending the flight range.

4.Conclusions

By combining computational fluid dynamics and plumb plane ballistic motion equations,the aerodynamic characteristics, extended range and maneuverability of a de flected missile are investigated.The optimized ranges of both normal and de flected missiles are obtained with the use of genetic algorithm.Following conclusions are concluded.

1)The coupling of DES and plumb plane ballistic equations can be used to investigate numerically the flow characteristics and aerodynamic forces of the supersonic de flected missiles.

2)The de flected missile has larger lift,drag and pitching moment coef ficients,but the pressure center moves forward,therefore its flightstability is decreased.With the increase ofthe de flected angle,its ability to change the flight direction is obviously improved.

3)The de flected missile flies at high angle ofattack with its lift-todrag ratio far larger than that of the normalmissile and it has a longer range.Through the genetic algorithm,the de flectable nose control can make the maximum range increase 21.66%.

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18 December 2016

*Corresponding author.

E-mail addresses:Zhangzhi900720@163.com(Z.-y.Zhang),tang_qz@163.com (Q.-z.Tang),huizi123717@163.com(X.-h.Sun),chenzh@njust.edu.cn(Z.-h.Chen).

Peer review under responsibility of China Ordnance Society.

http://dx.doi.org/10.1016/j.dt.2017.05.001

2214-9147/?2017 The Authors.Published 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/).

Received in revised form 3 May 2017

Accepted 10 May 2017

Available online 10 May 2017