# Investigation of the Inverse Magnus Effect on a Rotating Sphere in Hypersonic Rarefied Flow
## 1 Introduction
The pressure differences form a lift acting on the cylinder/sphere, which is also known as the Magnus force. Under normal conditions, the direction of the Magnus force \(F_{\text{M}}\) is the same as the direction of \(V_{\infty}\)\(\times\)\(\omega\) (\(V_{\infty}\) is the freestream velocity and \(\omega\) is the angular velocity of the rotating body). This Magnus effect in low-speed continuum flows is usually employed to explain the curved trajectory of a spinning ball in ball games [12]. In addition, it is also applied to ship propulsion [12], as well as high-lift devices for aircraft [13]. The inverse [PERSON] effect occurs under rarefied flow conditions; however, the direction of the lift is the same as \(\omega\)\(\times\)\(V_{\infty}\), which is contrary to the Magnus force. This phenomenon cannot be explained by continuum flow theory. Indeed, the gas-kinetic theory and the related numerical methods must be used in order to understand the inverse Magnus effect. In the free-molecular regime, the theoretical formula for the inverse Magnus force was firstly obtained by [PERSON] [18]. Then, [PERSON] et al. [19] and [PERSON] et al. [14] extended [PERSON]'s work from different points of view. The theories in [14; 18; 19] are consistent, but they are all limited to the free-molecular regime. In the transition regime between continuum and free-molecular regimes, there is no analytical solution for the flow over a rotating cylinder/sphere. Hence, the study of the inverse Magnus effect must resort to numerical approaches. On the basis of the full Boltzmann equation, the direct simulation Monte Carlo (DSMC) method is the dominant and reliable simulation technique for hypersonic rarefied flows [20; 21]. Recently, [PERSON] et al. [15; 22] carried out a systematic numerical investigation of hypersonic rarefied flows past a rotating cylinder, using two-dimensional (2D) DSMC simulations. In these works, the inverse Magnus force exerted on the cylinder was observed. Furthermore, the influences of different flow conditions on the inverse Magnus effect were studied. The influencing factors include the Mach number, the Knudsen number, the wall accommodation coefficient, and the rotation parameter of the cylinder. [PERSON] et al. [15; 22] only considered the 2D planar flow over a rotating cylinder. On the other hand, the flow over a rotating sphere should be three-dimensional (3D) and may involve richer flow phenomena than its 2D counterpart. Additionally, the quantitative laws for the sphere and cylinder flows can also be different. In this paper, a 3D DSMC solver is used to simulate the air flow past a spinning sphere with a flight velocity of 2800 m/s and a flight altitude of 100 km. The formation of the inverse Magnus force is explained based on the DSMC results. Additionally, the influences of the rotation rate on the flow field and aerodynamic characteristics are investigated. Such a study on the inverse Magnus effect will facilitate the understanding of the difference between rarefied and continuum flows. In engineering applications, the current study will also be useful for predicting the trajectory of space debris and exploring flight in the upper atmosphere with a high lift-to-drag ratio. The remainder of this article is organized as follows. In Section 2, the numerical method and its validation are presented. Then, Section 3 describes the physical problems and the numerical treatments. The numerical results are discussed in detail in Section 4. For different rotation rates of the sphere, the 3D flow patterns, the surface quantities, and the aerodynamic forces are compared and analyzed. Finally, some conclusions are drawn in Section 5.