## 2 Statement of the Problem and Methodology for the Near-Field Simulation  
The impact of an area of thermally stratified energy deposition (which, for short, hereinafter referred to as thermally stratified energy source--TSS) on a supersonic flow past a pointed cylinder body is considered. The freestream Mach number M\({}_{\infty}\) is equal to 1.5 and 2 (Figure 1). It should be noted that the shapes of AD bodies considered in the research were chosen to achieve an attached BSW and do not apply to any existing aircraft shape.  
The simulation of the flow for evaluating the near-field pressure signature is based on the Euler system of equations in a cylindrical form for a perfect inviscid gas with the ratio of specific heats \(\gamma\) = 1.4:  
\[(\mathbf{U}r)_{t}+(\mathbf{F}r)_{x}+(\mathbf{G}r)_{r}=\mathbf{H}, \tag{1}\]  
\[\mathbf{U}=\Big{(}\rho,\rho u,\rho v,E\Big{)}^{\mathrm{T}},\mathbf{F}=\Big{(} \rho u,p+\rho u^{2},\rho uv,u(E+p)\Big{)}^{\mathrm{T}},\]  
\[\mathbf{G}=\Big{(}\rho v,\rho uv,p+\rho v^{2},v(E+p))^{\mathrm{T}},\mathbf{H}= \Big{(}0,0,p,0\Big{)}^{\mathrm{T}},\]  
\[E=\rho\Big{(}\varepsilon+0.5\Big{(}u^{2}+v^{2}\Big{)}\Big{)}.\]  
The state equation for a perfect gas is used:  
\[\varepsilon=p/\big{(}\rho(\gamma-1)\big{)}\]  
Here \(\rho\), \(p\), \(u\), \(v\) are the gas density, pressure, and velocity \(x\)- and \(y\)-components, \(\varepsilon\) is the specific internal energy. The following normalizing values for the parameters are accepted:  
\[\rho_{n}=\rho_{\infty},p_{n}=5p_{\infty},l_{n}=k_{l}^{-1}R,T_{n}=T_{\infty},u_ {n}=\ (p_{\infty}/\rho_{\infty})^{0.5},t_{n}=l_{n}/u_{n}.\]  
where \(k_{l}\) is the dimensionless value of \(R\).  
The thermally stratified energy source (TSS) was defined as a set of rarefied gas layers located in front of the BSW (Figure 1). The layers in TSS were chosen to have the same width; distances between them were equal to half the layer's width. Inside every \(j\)- layer, the gas density was reduced, \(\rho_{j}=\alpha_{j}\rho_{\infty}\), with the rarefaction coefficient \(\alpha_{j}<1\), \(j=1\div N\), \(N\) is a number of layers in TSS in the calculation area taking into account the cylindrical symmetry of the problem (for example, in the TSS in Figure 1, \(N\) = 3.5). Other parameters of TSS were set equal to their freestream values, \(p_{j}=p_{\infty}\), \(u_{j}=u_{\infty}\), \(\gamma_{j}=0\). So, the temperature inside the layers was increased compared to its freestream value, \(T_{j}=\alpha_{j}{}^{-1}T_{\infty}\). The flow  
Figure 1: The scheme of the calculation domain.  
cases are considered when the \(r\)-coordinate of the upper boundary of the TSS is not greater than \(1.1R\), where \(R\) is the radius of the cylinder part of the AD body.  
For evaluation of the NFPS, we used our own software package based on the complex conservative difference schemes [18]. When constructing these schemes, differential consequences of Equation (1) are used, which makes it possible to obtain the second order of space-time approximation on a minimal stencil, which is the stencil of the [PERSON]'s scheme. So, the grids used are Cartesian and staggered. The boundaries of the body are included in the computational domain without violating the space-time conservation properties. The construction of the complex conservative schemes in the computational domain and near the boundaries of an AD body, as well as numerous test examples, are presented in [18].  
The position of the conical part of the body on the difference grid is shown in an enlarged view in Figure 2. In the calculations, the distance between the stencil nodes at each time level was assumed to be \(2l_{x}\), \(2l_{r}\) (where \(l_{x}\), \(h_{r}\) are the space steps in the \(x\)- and \(r\)-directions).