## 4 Results  
The Effect of TSS on the BSW, Near-Field Pressure Signature, and Aerodynamic Characteristics of a Body, L/R = 10  
First, let us consider the effect of TSS on the BSW and the aerodynamic characteristics of a pointed AD body for a fairly short body. The angle at the top of the model was \begin{table}
\begin{tabular}{c c c} \hline \hline
**Grid Number** & \(\mathbf{h_{x}}\), \(\mathbf{h_{r}}\) & **Sizes** \\ \hline Grid1 & \(h_{x}=0.002\) & \(3000\times 2000\) \\  & \(h_{r}=0.001\) & \(3000\times 2000\) \\ \hline Grid2 & \(h_{x}=0.004\) & \(1500\times 1000\) \\  & \(h_{r}=0.002\) & \(1500\times 1000\) \\ \hline Grid3 & \(h_{x}=0.00606060606\) & \(990\times 660\) \\  & \(h_{r}=0.003030303\) & \(990\times 660\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Difference grids. Figure 3: Grid independence test: (**a**)—density fields, \(t=1.5\) (superposed). The distance between isochores is \(0.08\); (**b**)—dynamics of the pressure \(p_{t}\) and density \(\rho_{t}\) at the top of the model; (**c**)—profiles of density and pressure for \(r=0.504\) in the BSW. \(16.6^{\circ}\) fore and aft, \(L/R=10\). The dimensional length of the body is presumably 5 m. The parameters used in the simulations are presented in Table 2. The TSS is assumed to occur at the initial stage of the process, at the time \(t_{i}=0.001\), and it is supposed that it has an unlimited duration in time. In Section 4.1, for the simulations, we used Grid 2 (\(h_{x}=0.004\), \(h_{r}=0.002\)) on the computation domains, which was extended to the entire calculation area and contained \(18\times 10^{6}\) nodes (counting the middle point of the stencil). The dynamics of the density fields at M\({}_{\infty}=2\) in isochores under the action of TSS containing 2.5 layers with equal rarefaction parameters in the layers \(\alpha_{j}=0.3\) is presented in red in Figure 4a-c. Here the \(r\)-coordinate of the upper boundary of the TSS is \(R_{s}=0.35R\). For comparison, the dynamics of the density fields without the impact of the TSS are also shown (blue). The density fields are presented for two different time moments, \(t=9.0\) and \(t=20.0\), which reflect unsteady and steady flow modes relative to the flow at the level of \(r=4.0\) (Figure 4a,b). Figure 4c shows the density field during the interaction of TSS with the BSW in an enlarged view. The corresponding density fields for the steady flow mode (\(t=20.0\)) in colors and isochores are presented in Figure 4d,e. It can be seen that, far from the body, disturbances provided by the TSS are practically not introduced into the flow, while near the body, the source introduces noticeable changes in the flow process. In particular, it can be noted that a fragment of the BSW is completely blurred inside the TSS region (see red isochores). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Description** & **Definition** & **Non-Dimensional Value** & **Dimensional Value** & **Normalizing Value** \\ \hline Freestream Max number & M\({}_{\infty}\) & 2.0; 1.5 & & \\ \hline Freestream pressure & \(p_{\infty}\) & 0.2 & 26.5 kPa & \(p_{n}=132.5\) kPa \\ \hline Ratio of specific heats & \(\gamma\) & 1.4 & & \\ \hline Maximal radius of AD body & \(R\) & 0.4 & 0.5 m & \(l_{n}=1.25\) m \\ \hline Length of the AD body & \(L\) & 4.0 & 5 m & \(l_{n}=1.25\) m \\ \hline The width of the layers in & & & \\ considered TSSs & \(r_{s}\) & 0.04 & 0.05 m & \(l_{n}=1.25\) m \\ \hline Number of layers in different TSSs & \(N\) & 2.5; 3.5; 5.5; 7.5 & & \\ \hline Rarefaction parameter in the layer & \(j\) in different TSSs & \(\alpha_{j}\) & 0.15; 0.2; 0.25; 0.3 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristics of the flow and the AD model used in the calculations. Figure 4: _Cont._  
The dynamics of the corresponding profiles of relative pressure \(\Delta p/p_{\infty}\) for \(r=4.0\) are presented in Figure 5. Here \(\Delta p/p=(p(x)-p_{\infty})/p_{\infty}\).