Investigation on Aircraft-store Compatibility Criteria of External Store Separation


 The evaluation of aircraft-store compatibility on external store separation is a key issue in the separation system of vehicle design. Firstly, the aircraft-store compatibility criterion of an external store separation is put forward, and then the criterion is converted to an unequal relationship between velocity and acceleration in vertical displacement and pitch angle based on the constant force assumption, which is validated by the test result of wing pylon finned store model (WPFS). The three-dimensional compressible Reynolds average N-S equation and rigid body six-degree-of-freedom motion equation (6-DOF) are solved by using unstructured dynamic overlap grid technology, to obtain the kinematic parameters of the external separation. Finally, the most dangerous point M on the tail of the external store is selected to verify the aircraft-store separation criterion. The results show that the kinematic parameters of the most dangerous point M on the tail wing of the store fall in the safe separation area, which means that the complete separated process is safe.


Introduction
Reducing the launching cost of space load is an unremitting pursuit in the development of the transportation system between heaven and earth for many countries [1][2]. Reusable Launch Vehicle (RLV), as a transportation platform for space loads, has the remarkable characteristics of low cost, flexibility and conventional airport launch. RLV can be used not only as a low-earth orbit load transportation platform, but also as a combat weapon platform to carry out combat tasks such as precision strike and high-point reconnaissance, which is the key development direction of major aerospace countries [3][4]. The United States, Russia, Japan and many European countries have put forward various conceptual schemes around the Single Stage to Orbit (SSTO) [5] and Two Stage to Orbit (SSTO) [6] modes of RLV. The SSTO reusable launch vehicle scheme can separate the completed launch vehicle from the space load at an appropriate time and return to the ground, which reduces the consumption of engine propellant and decrease launch cost [7]. According to the state-of-the-art conditions, the reusable aircraft with two-stage orbit is more suitable for the round-trip transportation between heaven and earth [8].
The RLV system mainly consists of a first-stage carrier and a second-stage orbiter, in which the carrier is an aircraft that can return to the ground. The installation modes of the orbiter and carrier can be divided into external, internal, captive-on-top and towed [9][10]. The external reusable launch vehicle with wing or fuselage scheme has been relatively mature, which has been adopted by many countries in the world at present [11]. Two stage vehicles are usually still in the atmosphere when it is separated and the second-stage orbiter is usually separated from the carrier by ejection force and aerodynamic force [12].
There are serious aerodynamic interference problems such as shock wave/shock wave, shock wave/boundary layer and shock wave/vortex in the separation process between the orbiter and carrier, which will directly affect the aerodynamic characteristics of the orbiter. If the design of separation system is unreasonable, the collision between the orbiter and carrier may occur, which results in the failure of the carrier separation task [13].
A great deal of research has been done on the sep-aration and evaluation methods of separation systems of RLV at home and abroad. Wang [14] has carried out an experimental study on aerodynamic interference existing in the separation process of multi-body system of hypersonic vehicle. Wang found that there is a complex shock wave interference between booster and reentry vehicle in the separation process and it is concluded that the essence of aerodynamic interference is caused by shock wave interference. Wu [15] conducted a wind tunnel experiments on a typical reusable launch vehicle, accurately simulated the separation motion between two-stage aircraft and measured the aerodynamic characteristics of multi-body interference. Sickles [16]  In order to evaluate and predict whether the external store can be safely and acceptably separated from the carrier-aircraft, it is necessary to establish a separation reference coordinate system and a store coordinate system that are fixedly connected with the store.
The separation reference coordinate system is an inertial coordinate system, which is mainly used to de- The positive Y is given by right-hand system.
The separate reference coordinate system is a moving coordinate system connected to the carrier-aircraft and moves with the aircraft. The origin of the separate reference coordinate is a point on the upper surface of the aircraft, which is on the same line as the origin of the store coordinate system. For obvious difference, it is staggered in Fig. 1. At the initial time of the store separation, the three axes of the separation reference coordinate system and store coordinate system are parallel to each other.

Criterion of compatibility for external store separation
Rudy [19] pointed that the initial half second can be considered as an safe separation event. In the motion of pitch direction, the absolute value of pitch angle cannot be greater than 9 deg when the store reaches the 10 body-diameter long. Then, the criteria for evaluating the separation compatibility of external store in this paper can give as follows:

Vertical displacement and pitch angle
The vertical displacement () Zt and pitch angle () t  of the separated store are given as follows [21]: Similarly, it is assumed that the attitude-angle of the external store separation near the aircraft is controlled by the aerodynamic moment on the missile at the initial release time, and remains unchanged in a very short time (constant moment assumption), and the formula (4) can be changed into the following form: Because of the assumption that the aerodynamic force and moment are constant in a short time, By substituting formulas (5) and (6) into criterion (1) and (2), the criterion for evaluating the separation compatibility of external store is as follows: The safety separation of external store is a sufficient condition for separation compatibility. Therefore, criterion (7) must meet at first. The safety separation area of external store after the introduction of criterion (7) is shown in Fig. 2. As seen from Fig. 2, the separation-region of vertical displacement for the external store can be divided into four regions (I~IV). (1) For region I, the safety separation of external store has enough margin since the relative velocity 1  and acceleration 2  between external store and the aircraft are large at the time of initial separation. That is to say, when the relative velocity 1  and acceleration 2  of the store and aircraft fall in region I, the external store can safely separate from the interference flowfield of the aircraft. (2) For region II, when the relative velocity 1  is large enough but the relative acceleration 2  is small or even negative, as long as the relative velocity 1  is greater , the separation process of the external store can be considered as safe. (3)

Mathematical proof of constant force assumption
The exact expression of vertical displacement () Zt can be written as follows: Where m is the mass of the store, () Ft is the time-varying force acting on the store.
The time-varying force () Ft can be make a Taylor series expansion at 0 t  as follows: Therefore, the expression of vertical-displacement is: Here 0 V is the initial ejection-velocity of vertical direction, 0 Z is the vertical displacement at 0 t  .
It can be seen in equation (11) that the effect of variable force on vertical displacement is mainly reflected in the lower quadratic term of time since the separation time is very short.

Verification of constant force and moment assumption
In order to verify the validity of constant force and moment assumption, the simplest way is to compare the real motion trajectory with that obtained by the constant force assumption. In this paper, a wing pylon finned store (WPFS) [22] is used to verify the con- F of the whole separation process. It can be seen from Fig. 4 (a) that the vertical displacement Z obtained by using the initial and average normal force assumptions is almost coincident. The vertical-displacement obtained by the constant force assumption is smaller than the real motion trajectory.
The simulation trajectories gained by initial and average normal force assumptions are very close to the actual trajectory, but the relative error is getting larger and larger as time goes on (as shown in Fig. 4(b)).
The relative error caused by the initial normal force assumption is larger than that of the initial average force assumption, but the maximum relative error is not more than 11%, which is acceptable within a certain error range. Therefore, the constant force assumption in store separation is acceptable, which can give a rapid assessment of the safety of store separation.
where W is a conserved variable, c F is in-viscid flux, v F is viscous flux.

Motion equation of six-degrees-of-freedom
According to the theoretical mechanics, the motion of rigid-body can be divided into two kinds of motion: centroid translation and centroid rotation. The dynamic equations of centroid translation and rotation of store are given without deduction.
In inertial coordinate system, the translational dynamic equation of mass center of store is given by Where m is the mass of the store,

Dynamic unstructured overlapping grid
In the process of dynamic separation of aircraft-store, there is relative motion among multiple bodies, which need dynamic grid technology to simulate the dynamic separation motion among multiple bodies. Unstructured overlapping grid technology combines the characteristics of unstructured grid adapting to complex shape and overlapping grid dealing with large relative motion, which not only reduces the difficulty of grid generation for complex shape, but also improves the ability of dealing with large relative motion. Therefore, this paper adopts dynamic unstructured overlapping grid method technology.
Specific steps of dynamic unstructured dynamic grid algorithm: (1) first carry out grid division on aircraft and store respectively, overlapping and nesting the carrier flow field grid and the orbiter flow field grid, determining the nesting relationship of overlapping boundaries, and realizing the information transmission between the flow fields of two computational domains; (2) Input initial and boundary conditions to solve the steady flow field of the aircraft and store as the initial value for solving the unsteady flow field;

Computational grids and conditions
Fig . 6 shows the schematic diagram of the overlap between the aircraft (wing) grid and the store grid at the initial time. Fig. 6 Overset grid for vehicle in initial moments Table 1 shows the calculation parameters. The separation of store is divided into two stages: (1) the stage of ejection force acting on the store (stage A), in which the store is acted by ejection force, aerodynamic force and self-gravity; (2) The stage of free flight (stage B), in which the store is acted by self-gravity and aerodynamic force.   Fig. 7 (c) that the external store appears nose-up pitch angle in stage A, which is mainly because the nose-up pitching moment produced by the rear ejection force is greater than the nose-down pitching moment produced by the front ejection force. In stage A, although the nose of the store move upward, the head of the store will not col-lide with the carrier-wing and the separation of the external store is safe at this stage. In stage B, the store appears nose-down pitch angle, which indicates that the pitching moment of aerodynamic force on the store is negative. The maximum value of pitch angle requires the in inequality (2), as shown in Fig. 7 (c).

Numerical calculation results
As the store moves downward in vertical direction, the four tailing wings are the most likely parts to collide with the wing since the tails of store will raise up.
It can be seen from Fig. 7 (d) that the rolling angular velocity of the store in stage B hardly changes.
Therefore, the point M on the tail when the ejection force disappears is selected as the characteristic point.
If the point M does not collide with the wing, the whole rigid-body will not collide with the wing.

Safety separation assessment verification
Where C a r is the acceleration of the CG,  r is the instantaneous angular acceleration when the rigid body rotates around the centroid.
Substituting the motion parameters of the center-of-gravity in numerical calculation into the formula (15) and (16) Fig. 2, the store can be safely separated from the carrier with a certain safety margin.

Conclusion
The safety evaluation of store separation is a key technical problem in the separation system of aircraft. This safety separation criterion can be used not only for safety separation evaluation of external store, but also for the separation evaluation of captive-on-top and internal store embedded orbiters.