The Magnus moment characteristics of rotating missiles with Mach numbers of 1.3 and 1.5 at different altitudes and angles of attack were numerically simulated based on the transition SST model. It was found that the Magnus moment direction of the missiles changed with the increase of the angle of attack. At a low altitude, with the increase of the angle of attack, the Magnus moment direction changed from positive to negative; however, at high altitudes, with the increase of the angle of attack, the Magnus moment direction changed from positive to negative and then again to positive. The Magnus force direction did not change with the change of the altitude and the angle of attack at low angles of attack; however, it changed with altitude at an angle of attack of 30°. When the angle of attack was 20°, the interference of the tail fin to the lateral force of the missile body was different from that for other angles of attack, leading to an increase of the lateral force of the rear part of the missile body. With the increasing altitude, the position of the boundary layer with a larger thickness of the missile body moved forward, making the lateral force distribution of the missile body even. Consequently, Magnus moments generated by different boundary layer thicknesses at the front and rear of the missile body decreased and the Magnus moment generated by the tail fin became larger. As lateral force directions of the missile body and the tail were opposite, the Magnus moment direction changed noticeably. Under a high angle of attack, the Magnus moment direction of the missile body changed with the increasing altitude. The absolute value of the pitch moment coefficient of the missile body decreased with the increasing altitude.

Rotating missiles produce the Magnus effect during flight [

Altitude has a great influence on the aerodynamic characteristics of a missile. For a rotating tail-stabilized missile, the Magnus moment is greatly affected by the altitude, leading to a large-angle conical pendulum movement and greatly affecting the flight stability of the missile. The Qinghai-Tibet Plateau in China has an average elevation of over 4000 meters and experiences low air density and low atmospheric pressure. Hence, many missiles with good flight stability cannot meet stability requirements in this plateau [

Zhai et al. and Dang et al. [

The aerodynamic characteristics of tailed rotating missiles have been also investigated through numerical simulations. Yin et al. [

The ballistic altitude of a long-range missile ranges between 50 and 80 km. With increasing altitude, variations in pressure, density, and temperature cause a change of the Reynolds number, which has a direct impact on boundary layer transition and flow separation.

Table

Variation of parameters with altitude.

Altitude (km) | Temperature (K) | Pressure (Pa) | |
---|---|---|---|

0 | 289.10 | 101325 | 1.7940 |

2 | 276.44 | 78535 | 1.7323 |

4 | 263.79 | 60990 | 1.6694 |

6 | 251.13 | 46780 | 1.6050 |

8 | 238.48 | 35391 | 1.5391 |

10 | 226.15 | 26370 | 1.4733 |

12 | 221.46 | 19415 | 1.4479 |

14 | 221.50 | 14265 | 1.4481 |

16 | 221.50 | 10481 | 1.4481 |

18 | 221.50 | 7700 | 1.4481 |

20 | 221.50 | 5657 | 1.4481 |

30 | 226.51 | 1211 | 1.4753 |

40 | 250.33 | 286 | 1.6008 |

60 | 247.06 | 21 | 1.5839 |

Geometric model of the finner.

In the aerodynamic calculation of a rotating missile, the correct prediction of separation flow and boundary layer transition is the key to accurately calculate the Magnus force. Nobile et al. [

The transition in a boundary layer and the vortex viscosity coefficient in a turbulence model are mainly controlled by intermittent factors. The intermittent factor transport equation can be expressed as [

The Reynolds number of transition momentum thickness is a decisive factor for the transition starting point, and the corresponding transport equation can be expressed as [

The generated term of the transport equation can be further written as

The transport equations of the

Three-dimensional (3D) structured hexahedral grids are displayed in Figure

Schematic diagram of slip grids.

The rotation of the missile was realized by the grid movement of the sliding grid area (Figure

Overall grid diagram.

The lateral force coefficient and lateral moment coefficient of the missile under different mesh quantities with the Mach number of 1.5 were calculated in Table

Grid independence verification.

Number of grids | ||
---|---|---|

2.89 M | 0.00872 | −0.19414 |

4.52 M | 0.04035 | −0.17457 |

7.86 M | 0.03957 | −0.17231 |

12.20 M | 0.04002 | −0.17187 |

Time step independence verification.

The lateral force coefficient and lateral moment coefficient of the missile under different turbulence models are exhibited in Figures

Lateral force coefficient.

Lateral moment coefficient.

It is evident that the calculated results of the transition SST model were close to the experimental values. Therefore, the adoption of the transition SST model has a certain credibility.

For rotating missiles with tail fins, the dynamic stability factor (

The dynamic stability condition of a rotating missile with a tail fin can be formulated as

When

For axisymmetric missiles,

When

Generally, three variables determine the stability of a rotating missile—pitch moment coefficient (

Theoretically, the following situation may occur. When a missile is launched at a high angle of launch, transonic conditions tend to occur at the highest point of its trajectory [

Figures

Variation curves of the Magnus moment coefficient with altitude at

Variation curves of the Magnus moment coefficient with altitude at

Figures

Variation curves of the pitching moment coefficient with altitude at

Variation curves of the pitching moment coefficient with altitude at

Variation curves of the stability factor with altitude at

Two important characteristics were noticed in the variation of the Magnus moment characteristics of the missile. First, the direction of the Magnus moment changed with the increase of the angle of attack. Second, at high angles of attack, the direction of the Magnus moment changed with the increasing altitude.

Figure

Variations of the average lateral pressure difference of the missile body along the missile axis at

Figure

Lateral force coefficient curves of the single tail fin at

Figures

Variation curves of the Magnus force coefficient with altitude at

Variation curves of the Magnus force coefficient with altitude at

Cloud maps of the intermittent factor at

Figure

Streamlines at

Figures

Boundary layer thickness and

Boundary layer thickness curves at different altitudes when

The normal force was obtained by calculating the effective angle of attack of the rotating tail fin, and the lift force was projected to the

Now, using the derivative formula of the normal force coefficient of a quadrangle airfoil proposed by Harmon [

Therefore, the lateral force coefficient of the tail fin is

Figure

Cyclic variation curves of the lateral force coefficient of the single tail fin.

Figures

Interference of the missile body with the lateral force coefficient of the tail fin at

Interference of the missile body with the lateral force coefficient of the tail fin at

With the increase of the angle of attack, the lateral force characteristics of the missile body changed because of the interference of the tail fin with the missile body. The lateral force coefficient curve of the missile body was fitted by the following Fourier series.
^{th-} and 8th-order Fourier series, and the corresponding

Comparison of lateral force coefficient curves obtained by different Fourier series.

Figure

Fourier coefficient of the missile body without tail fins.

Fourier coefficient of the missile body with tail fins.

Now, subtracting Figure

Percentage change of the Fourier coefficient.

The Magnus moment characteristics of rotating missiles with Mach numbers of 1.3 and 1.5 at different altitudes and angles of attack were numerically simulated based on the transition SST model. The main observations are presented below.

The Magnus moment direction of the rotating missiles changed with the increase of the angle of attack. When the altitude was low, the Magnus moment direction changed from positive to negative with the increase of the angle of attack. At higher altitudes, the Magnus moment direction changed from positive to negative and then again to positive with the increase of the angle of attack

When the angle of attack was 20°, the interference of the tail fin to the lateral force of the missile body was different from that for other angles of attack, leading to an increase of the lateral force of the rear part of the missile body. With the increasing altitude, the position of the boundary layer with a larger thickness of the missile body moved forward, making the lateral force distribution of the missile body even. Consequently, Magnus moments generated by different boundary layer thicknesses at the front and rear of the missile body decreased and the Magnus moment generated by the tail fin became larger. As lateral force directions of the missile body and the tail were opposite, the Magnus moment direction changed noticeably

Under a high angle of attack, the Magnus moment direction changed with the increase of the angle of attack. The absolute value of the pitch moment coefficient of the missile body decreased with the increasing altitude

The data [[

The authors declare that they have no conflicts of interest.

These data are the variation of aerodynamic parameters of spinning missile with altitude when Mach number is 1.3 and 1.5.