Ballistic and Aerodynamic Characteristics Simulation for Trajectory Correction Projectile

method of theoretical analysis and numerical calculation. These findings are considered instrumental in the trajectory correction technology research and engineering application.


Introduction
With the continuous development of weapons and military equipment, fundamental changes in state-of-the-art war strategy have been observed, including the wide use of smart bombs, intelligent missiles, and precision-guided ammunition [1][2][3][4]. Current requirements for weaponry include both precision strikes on military installations and effective reduction of collateral damage to civilians and civilian infrastructure. Missiles have high hit accuracy, but their production and maintenance are very expensive, their number is limited, and their targets should have high value to substantiate their expenditure. A new kind of ammunition with relatively low cost, high precision, and damage efficiency is required to satisfy the modern war demands [5][6][7]. A probable solution is furnished by trajectory correction projectiles (TCPs). Noteworthy is that trajectory correction projectiles and missiles belong to two different precision attack categories, with the following fundamental differences. Missiles hit the targets directly by ballistic correction and require shooting with unfailing accuracy, while TCPs reduce the original shooting dispersion and improve the hit probability through several ballistic corrections. They have to hit a small-radius circular area with the target located in the center [8][9][10].
Using a 57 mm projectile as a design platform, trajectory correction projectiles with different aerodynamic layouts were designed. Combining the method of theoretical analysis with numerical simulation, the aerodynamic characteristic, static stability, exterior ballistic characteristic, and firing density of projectile were studied. Research results can provide technical support for the development of ballistic correction projectiles.

Geometrical model
The projectile platform was based on SOCBT (secant ogive cylinder boat-tail) rotational stability projectile. To achieve the goal of trajectory correction, a drag-controlling device was installed in the projectile arc part. Specific parameters of the projectile platform were as follows: 230 mm in length, a diameter of 57 mm, a stern rake angle of 7 , and geometry, is shown in Fig. 1.
The trajectory correction projectile took the form after installing the drag-controlling ring in the arc part as shown in Fig. 2. The distance between the installed dragcontrolling ring and the top of projectile is 19.5 mm.   Table 1.
The maximum radius with the unfolded drag-controlling ring was 27.1 mm, being less than the projectile ra-dius of 28.5 mm. It was conducive to the flight stability of projectiles. This implied that the TCP design is reasonable. The pre-treatment used the Gambit software. The computational domain length and width were equal to the 20-fold projectile length and ten-fold projectile diameter, respectively. Due to the model's axial symmetry, the calculation domain covered only half of the model. The calculation domain diagram is depicted in Fig. 7, while the meshing quality report is summarized in Table 2.  The maximum volume grid quality was 0.764, being less than the FLUENT fluid software quality ceiling grid of 0.97 [11]. Therefore, it can be used for numerical simulation. The adopted computational procedure used a single-equation Spalart-Allmaras model [12][13] for turbulence, which has been widely used for solving the transport equation with the eddy viscosity, providing good results for the wall limit flow problem and inverse pressure gradient of the boundary layer problem. It is commonly applied to such aerodynamic problems of aircraft as streamlined flow around the airfoil profile, airflow field analysis, etc.

Aerodynamic numerical simulation
The numerical simulation utilized the following settings of the initial conditions according to references [14][15].
1. The grenade deflection angle and the angle of attack were 0°in the calculation, using density solver, Colin-Gaussian function gradient calculation method based on the nodes, and the single-equation Spalart-Allmaras turbulence model.
2. The WALL condition was a no-slip adiabatic viscous solid wall; the boundary condition was the far-field outer cylinder surface pressure.
3. Turbulence specification method used a turbulent viscosity ratio of 10.
4. The fluid physical properties were those of an ideal gas; viscosity was assessed via the Sutherland law, with an air density of 1.176674 kg/m 3 . According to the fluid mechanics' theory, the dynamic viscous coefficient was expressed as follows: where: μ is the dynamic viscous coefficient at the corresponding temperature; βa=1.458×10 -6 kg/(s·m·K 1/2 ) and Sutherland Constant Ts = 110.4 K. At an initial temperature T=15 ℃ or 288.15 K, we get the μ0N=1.786×10 -5 kg/(m·s·K 1/2 ). 5. Reference length and area of the calculation model were L = 0.057 m and S = 0.0026 m 2 , respectively, at pressure P = 101325 Pa.
6. The aerodynamic convergence conditions determined the convergence, without setting the residual of each equation convergence criteria.
7. The flux types selected were the ROE-FDS Flux difference method for low Mach numbers and the AUSM method for high ones, which combination could accurately capture the shock wave and had good convergence.
8. Considering regional fluid's severe changes in the fluid mechanics' calculation, such as shock wave in the calculation of surface and its movement, the adaptive grid technology was used to improve the calculation precision.  Table 3. The drag coefficient ratio was defined as the ratio between the drag coefficient values of the unfolded and folded drag-controlling rings. The respective results are summarized in Table 4.

Static stability analysis
According to the theory of external ballistics [16], when the angle of attack exists, the flight attitude of the projectile is shown in Fig. 12.
Symbols p and c correspond to positions of pressure center and mass center, respectively. While h is the distance between pressure center and mass center, and l is the projectile length.
According to the theory of flight stability, the static stability reserve was defined as: The flying projectile should meet the requirement that the pressure center is behind the center of mass and the static stability reserve is greater than 10%, ensuring a stable flight.
Under the conditions of flight speed is 3Mach and attacking angle between -2° to +2°, a numerical simulation of the flying flow field of the projectile has been taken. The mean value of static stability reserve of each projectile model has been obtained, which is given as follow. Under the same conditions, the static stability reserve of the TCPs was increased by 2.03%, 4.37% and 7.30%, respectively, compared with the original projectile.

Calculation of external ballistics
To evaluate the correction ability of the correction mechanism, the particle trajectory calculation program was selected, which had a sufficient precision of shooting range and height recognition.

Trajectory model of the projectile of centroid
According to the external ballistics theory, variation of air temperature, air pressure, and air density with altitude follows certain rules. Under the artillery standard meteorological conditions, the provisions are given as follows.
The definitions of subscript are as follows: 0 is zero altitude; N is standard value; P is air pressure expressed as: ρ is air density expressed as: cs is the speed of sound and expressed as: where: π(y) is air pressure function; τ(y) is virtual temperature function; y is the height of projectile trajectory; k is air specific heat ratio and Rd is the gas constant. Furthermore, the expressions of air pressure function and virtual temperature function are given as follows:  To effectively simplify the problem and grasp the regularity and characteristics of flying projectile motion, establishing the projectile external ballistic model based on the following basic assumptions. Assumption 1. Standard meteorological conditions, calm wind, and no rain. Assumption 2. There is no mass eccentricity in the projectile; the centroid of the whole projectile is kept at the same point after the nose deflection, and the plane is exactly symmetric.
Assumption 3. Changes in the coriolis inertial force, gravity acceleration, and latitude changes are neglected.
Assumption 4. Changes in the earth's curvature and gravity acceleration and the changes in height are neglected; gravity acceleration is g = 9.80m/s 2, and it is applied in the vertical direction to the ground. Assumption 5. No projectile spinning is assumed (i.e., the Magnus force, moment, damping moment, and angular moment in the empennage are neglected); the projectile is assumed to fly in the fore-and-aft plane.
The centroid motion kinematics equation, are given as follows: , where: x is a firing range; y is the height of projectile trajectory and θ is trajectory angle.
where: dt dV is the tangential acceleration. Based on the above four equations, Eqs. (12) - (15) can be derived: .
The initial value setting: t = 0; θ = θ0; x = y = 0 and The four-stage Runge-Kutta method was applied in the simulations, due to its high accuracy and easy implementation in designing the program. A self-developed ballistic calculation program was written based on the above statement for the ballistic calculation of trajectory correction projectiles.

Ballistic calculation results
The basic parameters of ballistic calculations were as follows: a diameter of 0.05715 m, a weight of 0.66 kg, an initial velocity of 1050 m/s, a firing angle of 40°.
The ballistic calculation results of the original projectile model (M0), which contained no drag-controlling mechanism are summarized in Table 7. The external ballistic characteristics of the original M 0 projectile and the trajectory correction projectiles M 1, M 2, and M 3 can be reflected by the relationship between the shooting range and the shooting height, as shown in Fig.  13. Fig. 13 Relationship of firing range and height

Correction ability calculation
The correction ability was defined as ΔX = X0 -XC, where ΔX is the correction shooting range; X0 is the firing range without the drag-controlling device and XC is the firing range with the operating drag-controlling device. The particular results of the calculated range corrections are listed in Table 8. To analyse the ground density of the projectile, the Monte Carlo method [17] is used to simulate projectile firing under certain initial conditions that listed in Table 9.
The expressions of distance and direction middle deviation are as follows: where: n is the number of projectiles in one group (equal to 5000 in the simulation).
The statistical results on distance and direction intermediate deviations of the projectile dispersion are listed in Table 10. The results show that the firing densities of M 1, M 2 and M 3 exceeded that of M0 by 18.01%, 10.91% and 3.91%.

Conclusions
This study used numerical simulations and engineering calculations to assess the aerodynamic characteristic, static stability, exterior ballistic characteristic and firing density of several trajectory correction projectile(TCPs). And the following conclusions could be drawn: 1. TCPs with different aerodynamic configurations were designed based on the 57mm diameter rotation stability projectile. The numerical simulation of the flight flow field for different projectiles was performed via the FLUENT software, yielding the aerodynamic parameters for different Mach numbers.
2. According to the external ballistics theory, the static stability reserves of each projectile were calculated, showing that the static stability of TCPs was improved by 2.03%-7.30% compared with the original projectile.
3. According to the artillery standard meteorological conditions, the projectile centroid kinematics dynamics and motion equations of the projectile's mass center under the ground rectangular coordinate system were derived. The numerical simulation results of projectile external ballistic motion indicated that the correction value of shooting range of the TCPs under study reached 1457.69 m.
4. The calculation of the firing density of each TCP under study was executed via the Monte Carlo method under the test conditions with 5000 repeated times. Its results indicated that the firing density of TCPs with the proposed improvement increased by 3.91 %-18.01 %.