## 3 Mathematical Modeling
\begin{array}{l}\frac{\partial q_{\rm{i}}}{\partial t}+\frac{1}{\gamma} \frac{\partial\left(Iq_{\rm{i}\xi}\right)}{\partial t_{\xi}}+\frac{1}{\gamma} \frac{\partial\left(Iq_{\rm{i}\xi}\right)}{\partial t}=S_{q}+\frac{\partial q _{\rm{i}}}{\partial t}\left(C_{\rm{i}}fD-\frac{3}{3}\omega div\vec{V}-\omega^{2 }\right),\\ \frac{2\partial\omega}{\partial t}+\frac{1}{\gamma}\frac{\partial\left(Ip_{ \rm{i}\xi}\omega\right)}{\partial t_{\xi}}+\frac{1}{\gamma}\frac{\partial \left(Ip_{\rm{i}\xi}\omega\right)}{\partial t_{\xi}}=S_{\omega}+\rho\left(C_{ \rm{i}}\left(C_{\rm{i}}D-\frac{3}{3}\omega div\vec{V}\right)-C_{\rm{i}}\omega ^{2}\right),\\ \qquad\qquad\qquad\qquad S_{q}=\frac{1}{\gamma}\frac{\partial\left\{\mu_{\rm{ T}}\left(\xi_{q}^{2}+\xi_{q}^{2}\right)\xi_{q}+\mu_{\rm{T}}\omega\right\} \left(I\omega_{\rm{i}}+\mu_{\rm{t}}\right)\eta_{\rm{t}}}{\partial t_{\xi}}+ \\ \qquad\qquad\qquad\qquad\qquad\frac{1}{\gamma}\frac{\gamma^{2}\left\{\mu_{\rm{ T}}\omega\right\}\left(I\omega_{\rm{i}}+\mu_{\rm{t}}\right)\omega_{\rm{i}}+\mu_{\rm{T}} \omega\left(I\omega_{\rm{i}}+\mu_{\rm{t}}\right)\eta_{\rm{t}}}{\partial t_{\xi }}+\\ \qquad\qquad\qquad\qquad\qquad S_{\omega}=\frac{1}{\gamma}\frac{\partial\left\{ \mu_{\rm{T}}\omega\right\}\left(I\omega_{\rm{i}}+\mu_{\rm{t}}\right)\omega_{ \rm{i}}+\mu_{\rm{T}}\omega\left(I\omega_{\rm{i}}+\mu_{\rm{t}}\right)\omega_{ \rm{i}}}{\partial t}+\\ \qquad\qquad\qquad\qquad\qquad+\frac{1}{\gamma}\frac{\partial\left\{\mu_{\rm{T }}\omega\right\}\left(I\omega_{\rm{i}}+\mu_{\rm{t}}\right)\omega_{\rm{i}}+\mu _{\rm{T}}\omega\left(I\eta_{\rm{t}}^{2}+\mu_{\rm{t}}^{2}\right)\omega_{\rm{i} }}{\partial t}+\alpha\frac{\mu_{\rm{T}}}{\mu}\left\{\xi_{q}\frac{\mu_{\rm{T}} }{\mu_{\rm{t}}}+\eta_{\rm{t}}\frac{\mu_{\rm{T}}}{\mu_{\rm{t}}}\right\}}{ \mu_{\rm{T}}\omega_{\rm{i}}}+\\ \qquad\qquad\qquad\qquad\qquad\qquad\mu_{\rm{T}}\
eq\mu_{\rm{m}}+\mu_{\rm{t}},\ \mu_{\rm{T}}\omega_{\rm{i}}=\mu_{\rm{m}}+1,3\mu_{\rm{t}},\ \mu_{\rm{t}}=C_{\rm{i}}f(n)\rho\frac{\mu}{ \mu_{\rm{T}}},\end{array}  \tag{3}\]  
where \(q\) is the pseudo-velocity, \(\omega\) is the pseudo-vorticity, \(f(n)\) is the wall function introduced in order for the model to correctly describe the flow parameters in a laminar sublayer formed on solid streamlined surfaces, and \(n\) is the normal distance from the point under consideration to the nearest surface. To calculate the radiation fields, the radiation transfer equation was used, presented in the form of a system of equations for the multigroup diffusion approximation:  
\[\frac{1}{J}\frac{\partial\left(Iq_{i\xi}\right)}{\partial\xi}+\frac{1}{J} \frac{\partial\left(Iq_{i\eta}\right)}{\partial\eta}+\chi_{i}cU_{i}=4\chi_{i} \sigma_{i}T^{4},\ \frac{c}{3}\frac{\partial U_{i}}{\partial\xi}+\chi_{i}q_{i\xi}=0,\ \frac{c}{3}\frac{ \partial U_{i}}{\partial\eta}+\chi_{i}q_{i\eta}=0, \tag{4}\]  
where \(U_{i}(y,z,t)\) is the density of radiant energy in the \(i\)-th spectral group, \(\chi_{i}\) is the spectral absorption coefficient. Instead of the diffusion approximation (\(D_{I}=c/3\chi_{i}\)) of the radiation transfer equation, the performed calculations use the functions of flux limiters [17]: \(\lambda(\alpha)=1/\sqrt[n]{1+\alpha^{m}}\). Then the diffusion coefficient can be represented as follows: \(D_{I}=c\lambda/3\chi_{i}\), where \(\alpha=|\
abla U_{i}|/(3\chi_{i}U_{i})\). At \(\alpha\rightarrow\infty\), the radiation flux is \(q_{i}=-D\
abla U_{i}\to cU_{i}\), and at \(\alpha\to 0\) the radiation flux is equal to the flux \(q_{i}=-D\
abla U_{i}\rightarrow-c\
abla U_{i}/3\chi_{i}\) of the diffusion approximation:  
The value \(\sigma_{i}\) is calculated by the formula [18]:  
\[\sigma_{i}=2k^{2}\pi[f(\alpha_{i+1})-f(\alpha_{i})]\left(c^{2}h^{3}\right), \alpha_{\rm{i}}=h\
u_{i}/kT,\]  
\[f(\alpha)=\alpha^{3}(1/3-\alpha/8+\alpha^{2}/62.4)\ \text{for}\ \alpha<2,\]  
\[f(\alpha)=6.4939\ -\ \text{e}^{-\alpha}\ (\alpha^{3}+3\alpha^{2}+6\alpha+7.28)\ \text{for}\ \alpha\geq 2.\]  
The numerical calculation method is detailed in [15; 19; 20].