## 1 Introduction  
An important aspect of the safe operation of any underground structures for various purposes is the prevention of fires and effective fire management [1]. A particularly high level of danger is posed by fires that occur in long tunnels and branched mining systems, where the objects of fire (transport, conveyor, ventilation equipment, etc.) are capable of releasing large amounts of harmful substances into the environment. In addition to this, high gas concentrations and temperatures near the fire object can lead to strong thermal convection, which not only complicates the ventilation of underground spaces but also makes it unstable [2, 3]. Fires in tunnels and mine workings are fairly common phenomena, and have been repeatedly noted in the scientific literature [4, 5]. As [PERSON] [6] rightly noted, even if all reasonable precautions are taken, there will always be at least a few plausible scenarios that could lead to a major underground fire in any mine. Therefore, important areas of research are not only minimizing the risks of underground fires, but also the development of approaches and methods for localizing and eliminating the consequences of fires [7, 8]. The development of new approaches and methods for localizing and eliminating fires in tunnels and mine workings requires an understanding of the patterns of development of underground fires, as well as their influence on the distribution of heat and mass transfer parameters of air flows in tunnel systems or mine workings. The analysis of such patterns has, to date, been largely studied for subway tunnels and other transport systems. The works presented in the literature describe both experimental studies of the dynamics of air flow parameters in the presence of powerful heat sources [9; 10; 11] and theoretical calculations using modern models and numerical methods. When implementing the latter, three-dimensional models of heat and mass transfer processes in individual sections of tunnels are usually utilized [12; 13]. When applied to mines, researchers also use one-dimensional models of mine ventilation networks [14; 15]. This is important for assessing the potential gas contamination of the mine as a whole as a result of a long-term fire, as well as for developing measures for the evacuation of miners [16]. One of the most important issues when analyzing fires in tunnels and mine workings is to understand the direction of movement of the hot gas-air mixture. [PERSON] and [PERSON] [17] identified the possible stages of loss of stability of the gas-air flow in tunnels and the occurrence of reverse flows due to gas or thermal convection:  
1. Unidirectional flow;
2. Partial return flows;
3. Completely reverse flows. The authors noted that the occurrence of one or another stage depends on the mean air velocity in the cross-section of the mine workings. The concept of critical air velocity, which is key in determining the stages of development of return air flows during a fire, is discussed in this study. Another important parameter is the critical heat release rate (HRR) of the fire, which is usually associated with the critical air velocity by one or another empirical relationship. For example, [PERSON] and [PERSON] [9] proposed the following dependence, obtained experimentally from a model bench simulating a tunnel:  
\[V_{c}=\left\{ \begin{array}{l}0.35q^{*0.3},\ q^{*}\leq 0.12\\ 0.35,\ q^{*}>0.12\end{array} \right. \tag{1}\]  
where \(q^{*}=Q/\left(\rho_{0}C_{p}T_{0}g^{1/2}H^{5/2}\right)\) is the dimensionless HRR of the fire; \(Q\) is the dimensional HRR of the fire (W); \(\rho_{0}\) is the inlet air density (kg/m\({}^{3}\)); \(T_{0}\) is the inlet air temperature (\({}^{\circ}\)C); \(C_{p}\) is the specific heat capacity of air (J/(kg\({}^{\circ}\)C)); \(H\) is the tunnel height (m); and \(g\) is the gravitational acceleration (m/s\({}^{2}\)).