Numerical analysis on aerodynamic characteristics of short cylindrical terminal-sensitive bullet

2.1. Description of the simulation model and grid division

Sliding grid technology is used to simulate the rolling motion of short cylindrical TSB. The flow field geometry model and boundary conditions are shown in Fig. 3 (a). When the values are discrete, the entire flow field is divided into inner flow zone and far-field static zone. By giving a specific rotational speed to the inner flow zone, the rolling motion of the bullet body is achieved. The interface boundary is used to connect two zones, and an interpolation method is used to transfer the flow field information at the interface. The diameter of the cylinder in the center of the TSB is $D$, and its length is $L$. The detector is a quadratic prism of $a\times b\times c$, and the outside edge has an inverted arc r. The far-field static zone is a 20D×20D×25D cuboid. The inner flow zone is a sphere with 7D diameter. The distance between the center of the sphere in the inner flow zone and the far-field exit of the downstream is 15D, and the distance from the other five boundaries is 10D. The upstream entrance of the far-field static zone adopts the velocity-inlet boundary, the downstream exit adopts the outflow boundary, and the other four planes and the surface of the bullet body adopt the non-slip wall boundary.

Fig. 3The numerical calculation model of TSB. a) Simulation model of flow field b) Inner and outer region of moving mesh c) y+ distribution of bullet

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b)

c)

The hexahedral structure mesh is used to divide the entire flow field in the far-field static zone and the internal flow zone. The mesh quality is directly related to the accuracy of the solution results. And there is complicated three-dimensional shear separation flow in the boundary layer around the TSB and aerodynamic characteristics such as Magnus effect caused by rotation, so the number of grids in the internal flow zone accounts for more than 78 % of the total number of grids. The total number of flow field grids is 2.9 to 3 million, and the calculated flow field grid division results are shown in Fig. 3(b). In the internal flow zone, the bullet boundary layer is encrypted using an O-type topology. The thickness of the first layer of the wall is taken as 0.003 mm, and the growth rate of the thickness of the mesh extending from the wall surface to the outside is less than 1.2. Thus, $y^{+}$ < 1 on the surface of TSB, and its distribution cloud is shown in Fig. 3(c).

2.2. Numerical verification

Before analyzing the flow characteristic of the rotating short cylinder, the accuracy of the simulation model was verified by comparing with the experimental results of Zdravkovic. The flow pattern (as shown in Fig. 4) of short cylinder obtained from this paper is in good agreement with the experimental results shown in Fig. 2 (c). In the front of the double free end plane, reflux vortex and secondary separation are formed. On the side of the short cylinder, there is a clear separation line. Different from the flow field structure of short cylinder ($L/D$ < 1) with one free end in Fig. 2(a), there is no horseshoe vortex for short cylinder with two free ends. Therefore, the flow field of a short cylinder with two free ends cannot simply be equal to the symmetry of the flow field for a short cylinder with one free end. As shown in Fig. 5, keeping the Reynolds number unchanged, the change law between $Cd$ and $L/D$ obtained in this paper is exactly the same as experimental results. Due to the existence of the supporting device in the experiment, the experimental data was slightly larger than the simulation results in general. It can be seen that this simulation model can be used to solve the flow problem of a short cylinder ($L/D$ < 1) with two free ends.

Fig. 4Flow field structure of the short cylinder

Fig. 5Comparison between present work and EXP [7, 8]

3.1. The analysis of flow field characteristic

Fig. 6 shows the flow structure of rotating short cylinder at $Re$ = 1.74×10⁵, where (a) is a front view, (b) is a top view, and (c) is a graphic model. Compared to the flow pattern of the non-rotating cylinder in Fig. 4, when $\alpha$ = 1, the recirculating region of the free end and the leeward surface and the separation line of the side surface are all significantly deflected in the positive direction of $\omega$. On the side where the fluid is accelerated, the streamline is spirally bent, causing the top vortex to deform into a ‘C’ shape. Fig. 7 shows the flow structure of a rotating short cylinder bullet at $Re$ = 1.74×10⁵, where (a) is the front view, (b) is the side view, (c) is the top view, (d) is the graphic model, and the relative rotational speed $\alpha$ = 1. The flow becomes more unsteady at this time. Compared with Fig. 6, due to the presence of the detector, the deflection angle of the recirculation zone is larger at the free end and the leeward side, and the position of the separation line on the side surface is no longer fixed.

Fig. 6Flow patterns around short cylinder (α = 1)

Fig. 7Flow patterns around bullet (α = 1)

3.2. The analysis of aerodynamic coefficient

According to published experiments and simulation results, the $Re$ number has a direct impact on the structure of the flow field for cylinder. The relationship between drag coefficient and $Re$ has important guiding significance for engineering application. At present, the relationship between the drag coefficient and Re for short cylinder ($L/D$ < 1) with two free ends has not been proposed. For non-rotating short cylinders ($L/D$ = 0.71) with two free ends, the relationship between $Cd$ and $Re$ (as shown in Fig. 8) is obtained by changing the flow velocity $u$. As can be seen from the figure, the average drag coefficient $Cd$ gradually decreases as $Re$ increases. Because there is no boundary layer transition of the infinite cylinder and sphere, the average resistance coefficient $Cd$ of the short cylinder does not decrease rapidly in the critical region. In the non-critical region, the average drag coefficient of the short cylinder is between the infinite cylinder and the sphere. The Reynolds number has a greater influence on the average drag coefficient $Cd$ of the short cylinder than the long cylinder with two free ends.

Fig. 8The relation between short cylinder Cd and Re is compared to infinite cylinder, sphere [4] and long cylinder with two free end [12]

Fig. 9Time histories of cd and cl of TSB (L/D = 0.7)

The time histories of $cd$ and $cl$ for the TSB are shown in Fig. 9 with different $\alpha$. Both $cd$ and $cl$ exhibit a single-periodic vibration with multiple extremes. The waveform of the $cd$ can be summarized as the ‘W’ shape shown in Fig. 9 within a single cycle. That is, there are two local maximum points with same value, and two local minimum points with different values. In a single period, when $\alpha$ > 1, the global minimum point of the cd is located at position 2 in Fig. 10(a); when $\alpha$ ≤ 1, the minimum point of the $cd$ is located at position 4 in Fig. 10(a). Similarly, in a single cycle, the waveform of $cl$ can be summarized as the ‘M’ shape shown in Fig. 10(a). That is, there are two local minimum points with same value, and two local maximum points with different values.

Fig. 10The wave structure of the cd for TSB, and statistical results of aerodynamic coefficients

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b)

The statistical results of the average aerodynamic coefficients of rotating short cylinders and rotating TSB are shown in Fig. 10(b). Due to the presence of the detector, when $\alpha$ = 0, the $Cl$ of the bullet is not zero, and its $Cd$ value is greater than the $Cd$ of the short cylinder. With the increase of $\alpha$, the aerodynamic coefficients of the bullets and short cylinders both increase, and the vibration amplitudes of $cd$ and $cl$ increase rapidly in the form of approximately quadratic functions. The $Cd$ curve of the bullet and short cylinder is approximately linear. From the time history curve of $cl$ in Fig. 9, it can be seen that $cl$ appears negative value with the increase of $\alpha$, that is, the direction of lift changes. Therefore, when $\alpha$ ≥ 0.5, the average lift coefficient $Cl$ of the bullet is less than that of the short cylinder.