# Investigation of the Inverse Magnus Effect on a Rotating Sphere in Hypersonic Rarefied Flow
## 3 Problem Description
### Computational Settings and Parameters  
Since there is a prescribed motion of the wall surface, the DSMC solver must incorporate the moving wall boundary condition. To be more specific, the linear velocity of a point on the sphere should be considered in two modules of the DSMC code: (1) the reflection of the gas molecules on the surface and (2) the sampling of incident and reflected momentum/energy for every surface element. First, for the reflection model (e.g., diffuse reflection), its input and output variables should be regarded as the molecular velocities relative to the hitting point on the sphere. Therefore, the velocity of the post-reflection molecule should by calculated by  
\[\mathbf{c}=\mathbf{V}_{\rm w}+\mathbf{c}_{\rm re} \tag{1}\]  
where molecular velocity \(\mathbf{c}_{\rm re}\) is the output of the reflection model, and \(\mathbf{V}_{\rm w}\) is the velocity of the hitting point on the moving wall. Another treatment for the moving wall is related to the sampling procedure for the calculation of surface pressure, friction, and heat flux. When sampling the incident and reflected momentum and energy, the molecular velocity relative to the wall (i.e., \(\mathbf{c}-\mathbf{V}_{\rm w}\)) should be used. The coordinate system, the computational domain, and the grid are demonstrated in Figure 5. The \(x\)-axis is aligned with the freestream velocity \(\mathbf{V}_{\infty}\), and the direction of the \(z\)-axis is the same as the angular velocity vector \(\mathbf{\omega}\). The origin of the coordinates is placed at the center of the sphere. Note that the flow is symmetric about the \(xy\) plane, and therefore only half of the space (\(z>0\)) needs to be considered. The computational domain is a hemisphere with a radius \(R_{d}=7R\). The body-fitted grid contains 108,000 hexahedral cells, and the minimum cell size is about 4 mm. At the initial time, the flow field is assumed to be uniform and in equilibrium with the parameters of the freestream. A total of 10.8 million simulation particles are assigned to the grid cells, so there are 100 particles per cell after initialization. The time step size is set to \(\Delta t=4\times 10^{-6}\) s, and cumulative sampling starts from \(t=4\) ms. It takes 20,000 flow-field samplings to obtain low-noise DSMC results for macroscopic quantities. \begin{table}
\begin{tabular}{c c} \hline \hline
**Flow Conditions** & **Values** \\ \hline number density \(n_{\infty}\) & \(1.189\times 10^{19}\) m\({}^{-3}\) \\ freestream velocity \(V_{\infty}\) & 2800 m/s \\ freestream temperature \(T_{\infty}\) & 195 K \\ wall temperature \(T_{\rm w}\) & 300 K \\ diameter of sphere \(D\) & 0.2 m \\ accommodation coefficient \(\alpha\) & 1 \\ freestream Mach number \(M_{\infty}\) & 10 \\ Knudsen number \(Kt_{\infty}\) & 0.51 \\ rotation parameter \(W\) & 0.00, 0.25, 0.50, 0.75, 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Conditions for the freestream and the wall in the study of the inverse Magnus effect. The above computational parameters for DSMC simulations are summarized in the second column of Table 2.  
In order to verify that the DSMC results are independent of the grid and the time step, a refined grid and a reduced time step size (the last column in Table 2) are used to redo the DSMC simulation of the sphere flow under the condition of \(W=0\). Figure 6 compares the DSMC results using the standard and the refined grids. Flow variables along Line 1 and Line 2, marked in Figure 5, are extracted for comparison. The temperature profiles along Line 1 are plotted in Figure 6a, and the velocity profiles along Line 2 are shown in Figure 6b. The comparisons indicate that the DSMC simulations with different computational parameters yield almost indistinguishable solutions. Thus, in the remainder of this paper, the DSMC parameters all follow the standard settings.