Aerodynamic Characterization of a Ballistic-Correction Bullet

Abstract

Obtaining the aerodynamic parameters of a ballistic-correction bullet is crucial for improving ballistic correction efficiency and enhancing shooting accuracy. In this study, a numerical simulation method is utilized to analyze and calculate the aerodynamic characteristics of the bullet. The focus is on investigating the changes in aerodynamic parameters when the rudder wings are in both unexpanded and expanded states, as well as analyzing the impact of deflection angles on the bullet's aerodynamic characteristics. The simulation results demonstrate that the bullet exhibits favorable aerodynamic performance and static stability when the rudder wings are unexpanded, leading to improved ballistic correction efficiency during flight. Moreover, the results indicate that in the expanded state of the rudder wings, increasing the deflection angle enhances the impact on the bullet's aerodynamic characteristics, resulting in more pronounced changes and stronger correction capabilities. This research provides valuable insights into the aerodynamic characteristics of ballistic-correction bullets.

1 Introduction

After leaving the muzzle, traditional bullets are in an uncontrolled state and are susceptible to various external factors [1], making it difficult to guarantee accuracy [2]. With the development of advanced technologies such as inertial microsystems, new materials, and optoelectronics, ballistic correction techniques widely used in rocket projectiles can also be applied to small-caliber bullets. This makes it possible to achieve guided capabilities for bullets. As a type of smart ammunition, ballistic-correction bullets achieve significantly higher accuracy than conventional bullets through the combined efforts of a guidance section for detection and positioning and an actuation mechanism for controlling the bullet's orientation [3, 4]. For this type of bullet, there have been numerous related studies both domestically and internationally. The University of Gothenburg in Germany has studied a tube-launched self-adaptive ammunition that utilizes piezoelectric ceramic materials to control the deviation of the bullet. Sandia National Laboratories in the United States has designed a tail-fin deflection smart bullet and conducted live-fire tests. Jonathan et al. proposed a new concept for the actuation mechanism of bullets used in light weapons based on the technique of transforming the center of mass. Chen Ying et al. studied the effect of tail wings on micro-guided projectiles using numerical simulations of the external flow field. Liu Wei et al. proposed a folding duck tail mechanism based on small-caliber smart ammunition and demonstrated the reliability of the folding rudder mechanism. Zhou et al. [5] studied the influence of changes in the external shape dimensions of the projectile tail on its aerodynamic characteristics based on computational fluid dynamics. Liu et al. [6] proposed a folding duck tail mechanism based on small-caliber smart ammunition and demonstrated the reliability of the folding rudder mechanism.

Studying the aerodynamic characteristics of ballistic correction bullets allows for more accurate acquisition of the external ballistics performance parameters of the bullets. This, in turn, enables further optimization and calculation of the external ballistics to achieve higher shooting accuracy [7, 8]. With the rapid development of computational fluid dynamics, numerical simulation has become an important tool for studying the aerodynamic characteristics of projectiles and arrows. Liu et al. [9] conducted an analysis of the aerodynamic characteristics of a two-dimensional correction projectile using fluid dynamics methods and examined the influence of rudder deflection angle. They concluded that the lift coefficient is positively correlated with the equilibrium angle of attack and rudder deflection angle. Zhong et al. [10] based on numerical simulation methods, analyzed the lateral aerodynamic effects of differential control surfaces on rotating two-dimensional ballistic correction projectiles. The results indicated a linear relationship between the cross-stream force, yawing moment, and angle of attack. Shao et al. [11], using combined fluid dynamics and dynamics simulation, analyzed the impact of different rudder heights and deflection angles on the correction ability of canard correction mechanisms. They found that the range and elevation of the projectile were positively correlated with the rudder deflection angle. Kou et al. [12], based on fluid dynamics methods, analyzed the influence of canard fins on the aerodynamic characteristics of spinning missiles. They found that canard fins can provide a greater rolling moment to some extent, and the rolling moment is positively correlated with the height of the canard fins. Zhang et al. [13], based on numerical analysis, studied the aerodynamic characteristics of a movable canard on mortar projectiles. The results showed that the control surface force increases with the deflection angle and Mach number. The aforementioned studies mainly focus on large-caliber rocket projectiles and highlight the significance of analyzing aerodynamic characteristic parameters for enhancing projectile performance. Similarly, investigating the external flow field aerodynamic characteristics of small-caliber ballistic correction bullets is also crucial for improving bullet performance.

In this paper, numerical simulation method is adopted to analyze and calculate the aerodynamic characteristics of a ballistic-correction bullet, and the aerodynamic characteristics of the rudder wing in the unexpanded state and the rudder wing in the expanded state are obtained respectively, and the effect of the change of the rudder wing's deflection angle on the aerodynamic characteristics of the projectile is analyzed.

2 Model Building

2.1 Shape of the Bullet

This paper investigates a bullet that can perform ballistic correction, and the overall structure includes a guide head, a circular arc section, a cylindrical section, rudder wings, a contraction and expansion section, tail wings and other parts, and the overall structure is shown in Fig. 1. The rudder wing, as the actuator of the projectile, is located in the cylindrical part of the projectile. Through the expansion of the rudder wing to change the aerodynamic shape of the bullet, so as to realize the ballistic correction. When the rudder wing is not unfolded, the rudder wing can be closed inside the bullet, and the rudder wing will be opened when two-dimensional correction of the ballistic trajectory is required.

Fig. 1

Schematic diagram of bullet structure

2.2 Grid Partitioning and Parameter Setting

Due to the intricate geometry of the bullet model at its tail and the presence of four symmetrically arranged tail fins, the computational domain for this study is partitioned using the Poly-Hexcore mesh. This particular mesh type offers an optimal trade-off between mesh quality and memory usage, yielding comparable accuracy to polyhedral meshes while considerably improving solution speed and demonstrating excellent adaptability to complex geometries. Furthermore, the volume mesh undergoes appropriate refinement in regions characterized by significant variations in flow field or fluid variable gradients, including the bullet head, wing sections, and near-field shock wave areas. To better emulate real-world conditions, a rectangular outer flow field calculation domain is adopted, encompassing 1000 mm ahead of the bullet, 5000 mm behind the bullet, and 1500 mm surrounding the bullet. In order to capture the fluid characteristics throughout the flight process more effectively, a boundary layer with a y+ value of 1 is implemented on the surface of the bullet. The first layer of the boundary layer has a height of 0.0013 mm, and a total of 20 layers are employed, exhibiting a grid growth rate of 1.2, resulting in an approximate total of 6 million grids. The computational domain mesh around the bullet body is shown in Fig. 2.

Fig. 2

Computational grid for ballistic correction bullet

Calculation using three-dimensional compressible flow Reynolds-averaged Navier–Stokes (RANS) equations, with its non-dimensionalized form as follows.

$$ \frac{{\partial {\varvec{Q}}}}{\partial t} + \frac{{\partial {\varvec{E}}}}{\partial x} + \frac{{\partial {\varvec{F}}}}{\partial y} + \frac{{\partial {\varvec{G}}}}{\partial z} = \frac{{\partial {\varvec{E}}_{v} }}{\partial x} + \frac{{\partial {\varvec{F}}_{v} }}{\partial y} + \frac{{\partial {\varvec{G}}_{v} }}{\partial z} $$

(1)

In the formula: Q represents the conservation vector; E, F, G represent the inviscid fluxes in the x, y, and z coordinate directions respectively; E_v, F_v, G_v represent the viscous fluxes in the x, y, and z coordinate directions respectively.

The selection of a turbulence model is crucial for accurately simulating the flow field. In this study, the SST k-ω turbulence model is employed for calculations. This model takes into account the exchange of shear stress within the turbulent flow, resulting in more realistic simulation effects for both near-wall regions and regions far from the wall. The specific equations are as follows:

$$ \left\{ \begin{gathered} \frac{{\partial u_{i} }}{{\partial x_{i} }} = 0 \hfill \\ \frac{{\partial x_{i} }}{\partial t} + u_{j} \frac{{\partial u_{j} }}{{\partial x_{j} }} = - \frac{1}{\rho }\frac{\partial \rho }{{\partial x}}\left( {\upsilon \frac{{\partial u_{i} }}{{\partial x_{j} }}} \right) \hfill \\ \frac{\partial }{\partial t}(\rho u_{i} ) + \frac{\partial }{{\partial x_{i} }}(\rho u_{i} u_{j} ) = - \frac{\partial p}{{\partial x_{i} }} + \frac{\partial }{{\partial x_{j} }}\left( {\Gamma \frac{{\partial u_{i} }}{{\partial x_{j} }}} \right) + S_{i} \hfill \\ \frac{\partial }{\partial t}(\rho k) + \frac{\partial }{{\partial x_{j} }}(\rho ku_{j} ) = \frac{\partial }{{\partial x_{j} }}\left( {\Gamma_{k} \frac{\partial k}{{\partial x_{j} }}} \right) + \overline{G}_{k} - Y_{k} + S_{k} \hfill \\ \frac{\partial }{\partial t}(\rho \omega ) + \frac{\partial }{{\partial x_{j} }}(\rho \omega u_{j} ) = \frac{\partial }{{\partial x_{j} }}\left( {\Gamma_{\omega } \frac{\partial \omega }{{\partial x_{j} }}} \right) + G_{\omega } - Y_{\omega } + D_{\omega } + S_{\omega } \hfill \\ \end{gathered} \right. $$

(2)

In the equation: x_i and x_j represent spatial coordinates (i, j = 1, 2, 3); t denotes time; ρ represents air density; ν is the kinematic viscosity; p stands for pressure; k and ω respectively represent turbulent kinetic energy and turbulent dissipation rate; Г, Г_k, and Г_ω are effective diffusion coefficients for velocity u (v or w), turbulent kinetic energy k, and turbulent dissipation rate ω; $\bar{G}_{k}$ and G_ω are production terms for k and ω; Y_k and Y_ω are dissipation terms for k and ω; D_ω represents the cross-diffusion term; S_i, S_k, and S_ω are user-defined terms for various transport equations.

In the process of numerical computation, both the governing equations and the turbulence model are discretely solved using a second-order upwind scheme. The fluxes are discretized using the Roe-Flux Difference Splitting (Roe-FDS) method, which offers higher numerical simulation accuracy.

The pressure far-field boundary condition is selected for the boundary conditions, and the surface of the body is set as a no-slip wall boundary. The initial condition is set as a far-field inflow at the outer boundary, allowing the flow to start from the outer boundary and move towards the inner boundary.

Other settings are as follows: the solver is selected as a density-based solver using an explicit scheme. The fluid physical properties are defined as an ideal gas, and viscosity is calculated using the three-coefficient Sutherland's law. The relaxation factors are set to their default values.

3 Aerodynamic Characterization of Rudder Wing in Undeployed Condition

To achieve a more accurate simulation of the bullet's aerodynamic characteristics, the flight Mach number of the bullet is deliberately chosen to range from 0.5 to 2.25 Ma, encompassing the entire spectrum of subsonic, transonic, and supersonic regimes. Moreover, the bullet predominantly operates within a near-zero or small angle of attack regime, leading to the selection of 0°, 1°, 2°, 3°, and 4° as the designated angle of attack values. By meticulously examining and comparing the drag coefficient, lift coefficient, and pitching moment coefficient, it becomes possible to discern and comprehend the underlying trends in the evolution of the bullet's aerodynamic properties. The specific configurations and corresponding parameters employed for this investigation are presented in Table 1.

Table 1 Model calculation parameters

3.1 Analysis of Aerodynamic Parameter Characteristics

Figure 3 shows the variation curve of drag coefficient with Mach number and angle of attack under the rudder wing unexpanded state, from which it can be seen that the drag coefficient of the bullet shows a trend of increasing and then decreasing with Mach number, and the drag coefficient grows the most rapidly when the flight speed of the bullet grows from 0.75 to 1.03 Ma, and the drag coefficient reaches the maximum at 1.25 Ma. With the increase of the angle of attack, the drag coefficient also shows an increasing trend, but the change is small.

Fig. 3

Drag coefficient variation curve

Figure 4 shows the variation curve of the lift coefficient with Mach number and angle of attack under the rudder wing not extended state, from which it can be seen that when the angle of attack is 0°, the lift of the bullet is basically 0, and the lift coefficient increases gradually with the increase of the angle of attack, and the approximate linear trend is presented. When the angle of attack is certain, with the increase of Mach number, the lift coefficient shows the trend of increasing and then decreasing, and the lift coefficient reaches the maximum value at 1.03 Ma.

Fig. 4

Lift coefficient variation curve

Figure 5 shows the variation curve of the pitching moment coefficient with Mach number and angle of attack under the rudder wing not extended state, from which it can be seen that when the angle of attack is 0°, the pitching moment of the projectile is basically zero, and with the increase of the angle of attack, the pitching moment coefficient also increases gradually, presenting the same trend as that of the coefficient of lift; when the angle of attack is certain, the pitching moment coefficient shows the tendency to increase and then decrease with the increase of the Mach number, and the maximum value of pitching moment coefficient is reached at 1.03 Ma when the pitching moment coefficient reaches the maximum value. The pitching moment characteristic of the bullet mainly reflects the static stability of the bullet, and the larger the pitching moment coefficient is, the better the static stability of the bullet is.

Fig. 5

Pitching moment coefficient variation curve

In summary, when the rudder wing of the bullet is not extended, the aerodynamic characteristics of the bullet are basically consistent with those of ordinary bullets, but due to the design of the blunter head and tail wings, the bullet has a larger drag coefficient and lift coefficient, and good longitudinal static stability.

4 Aerodynamic Characterization of Rudder Wing in Deployed Condition

Due to the insignificant impact of the variable factors on the respective Mach numbers, we opt for flight velocities of 0.5 Ma, 1.03 Ma, and 1.5 Ma to represent the subsonic, transonic, and supersonic flight regimes, respectively. Subsequently, we conduct simulation calculations on bullets with deflection angles of 1°, 3°, 5°, and 7°, while analyzing the influence of the deflection angle on the aerodynamic characteristics of the bullet.

Figure 6 shows the variation of the cross-stream force coefficient and yawing moment coefficient of the bullet with Mach number and angle of attack for different rudder deflection angles when the rudder wing is subjected to lateral correction. From Fig. 6a, it can be seen that the l cross-stream force coefficient of the bullet decreases and then increases with the increase of Mach number at different rudder deflection angles, and gradually increases with the increase of angle of attack. The special thing is that at the rudder deflection angle of 1°, the correction direction of the bullet is opposite to other directions. From Fig. 6b, it can be seen that the yawing moment coefficient of the bullet first increases and then decreases with the increase of Mach number, and increases with the increase of the angle of attack, and when the rudder deflection angle is 7°, the yawing moment coefficient of the bullet shows a faster decreasing trend with the gradual increase of the angle of attack. The larger the cross-stream force coefficient and yawing moment coefficient, the stronger the lateral correction ability of the bullet.

Fig. 6

Cross-stream force coefficient and yawing moment coefficient variation curves

Figure 7a shows the curves of lift coefficient variation with rudder deflection angle for different Mach numbers when the rudder elevator is expanded. Where the rudder deflection angle 0° corresponds to the case when the rudder is not expanded. From Fig. 7a, different rudder deflection angles have more different effects on the lift of the bullet. When the rudder wing is not expanded, the lift of the bullet is smaller than the rudder deflection angles of +1° and −1° at all three Mach numbers. When the bullet flight speed is 0.5 Ma, and the rudder deflection angle is negative, with the increase of the rudder wing deflection angle, the lift coefficient of the bullet gradually decreases, and the negative lift provided by the rudder wing for the bullet gradually increases, in which the rudder deflection angles of −5° and −7°, the lift coefficient of the bullet is smaller than that of the case of the rudder wing is not expanded, and the bullet has a tendency to fall faster; when the rudder deflection angle is positive, with the increase of the rudder deflection angle, the lift coefficient of the bullet gradually increases, with the increase of the rudder wing deflection angle, the lift coefficient of the bullet gradually increases, with the increase of the rudder wing deflection angle. When the rudder deflection angle is positive, with the increase of rudder deflection angle, the lift coefficient of the bullet gradually increases, and the positive lift provided by the rudder wing gradually increases, and the bullet has an upward trend. When the flight speed of the projectile is 1.03 and 1.5 Ma, the change of the lift coefficient of the bullet has the same trend, when the rudder deflection angle is negative, the rudder deflection angle increases, the lift coefficient decreases, and the rudder provides additional negative lift for the bullet when the rudder deflection angle is −7°. When the rudder wing deflection angle is positive, the rudder deflection angle increases and the lift coefficient decreases, and when the rudder deflection angle is +1°, the rudder wing still provides positive lift for the bullet.

Fig. 7

Lift coefficient variation curve

Figure 7b shows the variation curves of the lift coefficient with the rudder deflection angle at different angles of attack. From the figure, when the angle of attack is 0°, the total lift of the bullet is 0 when the rudder wing is not expanded; when the rudder deflection angle is negative, the rudder wing will provide positive lift at −1° rudder deflection angle, and then the bulled lift decreases as the rudder wing deflection angle increases; when the rudder deflection angle is positive, the rudder wing will provide negative lift at +1° rudder deflection angle, and then the bullet lift increases as the rudder wing deflection angle increases; the bullet lift increases as the rudder wing deflection angle increases. Angle of attack of 2° shows the same tendency of angle of attack of 0°. The bullet at angle of attack of 2° shows the same trend as that at angle of attack of 0°. At angle of attack of 4°, with the increase of rudder deflection angle, the bullet lift shows a decreasing trend, and at rudder deflection angles of −7°, +5° and +7°, the bullet has additional negative lift.

The variation curves of the pitching moment coefficient with the deflection angle of the rudder wing at different Mach numbers when the rudder wing elevator and rudder are deployed are shown in Fig. 8a. As can be seen from the figure, the pitching moment coefficient of the bullet maintains a similar trend at different Mach numbers. The pitching moment coefficient of the bullet gradually decreases when the rudder deflection angle is negative to the maximum angle and deflected to the positive to the maximum angle. The pitching moment coefficient of the bullet increases when the bullet is flying at the speeds of 0.5 and 1.03 Ma without rudder wings expanded, and the pitching moment coefficient of the bullet decreases at the speed of 1.5 Ma. When the deflection angle of the rudder wing is +7°, the pitching moment coefficient is greater than 0. Since the lift is positive at this time, the pressure centre of the bullet is located in front of the centre of mass, and the static stability of the bullet can not be guaranteed at this time, and it is prone to out-of-control phenomenon. In the longitudinal comparison, when the speed of the bullet is 1.03 Ma, the pitching moment coefficient of the bullet is the largest, which corresponds to the better flight stability of the bullet at this time.

Fig. 8

Pitching moment coefficient variation curve

Figure 8b shows the variation curves of the pitching moment coefficient with the rudder wing deflection angle at different angles of attack. The larger the rudder wing deflection angle, the larger the additional pitching moment provided to the bullet, and when the rudder deflection angle is negative, the pitch-down moment of the bullet becomes larger, and when the rudder deflection angle is formal, the pitch-up moment becomes larger. When the angle of attack is 0°, the change of moment provided by the negative rudder deflection is more obvious, and the bullet is in static instability when the rudder wing deflection is positive; when the angle of attack is 4°, the change of moment provided by the positive rudder deflection is more obvious.

In summary, the different deflection angle of the rudder wing will have a greater effect on the force and moment of the bullet, and the larger the deflection angle of the rudder wing, the greater the ability of the rudder wing to correct the ballistic of the bullet, and at the same time, the different deflection angle of the rudder wing will also have a greater effect on the static stability of the bullet.

5 Conclusion

In this paper, for a certain ballistic correction bullet, the aerodynamic characteristics of the bullet are investigated by numerical simulation, the changes of the aerodynamic characteristics of the bullet in the unexpanded and expanded states of the rudder wing are investigated respectively, and the influence of the rudder wing deflection angle on the aerodynamic characteristics of the bullet is analysed, and the conclusions are as follows:

• 1. When the rudder wing is unexpanded, the aerodynamic characteristics of the bullet are similar to those of ordinary bullets, the drag coefficient, lift coefficient and pitching moment coefficient increase and then decrease with the increase of Mach number, and its drag coefficient, lift coefficient is larger, and the stability is better. The results indicate that even without any ballistic adjustments, the bullet still exhibits excellent aerodynamic characteristics, enabling precise targeting and striking of the intended objective.

• 2. The larger the rudder deflection angle is, the stronger the correction ability of the bullet is, and the more obvious the changes in aerodynamic characteristics. When the elevator of the bullet extends/retracts, there are different influences of the rudder deflection angle at different Mach numbers and angles of attack. A positive rudder deflection angle at an angle of attack of 0° will cause the bullet to be statically unstable. The findings suggest that it is necessary to choose appropriate deflection angles for the rudder wings based on the varying flight conditions of the bullet.

• 3. Both the longitudinal and lateral correction mechanisms of the bullet offer sufficient control moments to alter its attitude and achieve ballistic correction. However, it is necessary to adjust the morphology of the rudder wings in order to maximize the efficiency of the corrections based on the flight velocity.

• 4. This paper can provide some reference basis for the study of the aerodynamic characteristics and Correction mechanisms of ballistic-correction bullet.