## 1 Introduction  
Thermal cutting is one of the most important production processes and is widely employed in industrial processes such as welding, assembling, and riveeting. Thermal cutting primarily means the use of energy in various forms to cut virtually any shape from iron and nonferrous materials out of sheets or large slabs. Among various thermal cutting processes, plasma arc cutting, and laser cutting are widely used in the metal producing and metalworking industries such as structural engineering, machine fabrication, energy equipment manufacturing, construction, aerospace, shipbuilding, nuclear power plant, battery, and automotive industries. Plasma arc cutting is a fabrication process that employs superheated, ionized gas funneled through a plasma torch to heat, melt and, ultimately, cut electrically conductive material into custom shapes and designs. On the other hand, the laser cutting process employs a highly concentrated beam of light that is focused on the cutting material to melt, burn, or vaporize the material at the point of focus. Nonetheless, both of these thermal cutting processes use a high-pressure assist gas to remove molten or vaporized material from the cut kerf [1]. The plasma cutting process is suitable for a wide range of metal materials, including structural steel, alloy steel, aluminum, and copper, and can cut through material thicknesses ranging between 5 mm to 150 mm for standard machines, although the range can vary depending on the plasma model [2; 3; 4]. Laser cutting is capable of precise control and high-speed processing, and is widely used in ultra-precision fields such as semiconductors. The application of high-power lasers is gradually expanding with the development of advanced technologies in fields such as automobiles, shipbuilding, cladding, and additive manufacturing processes. High-power lasers are widely used not only for welding but also for cutting structures [5]. In particular, the importance of laser cutting in nuclear decommissioning has been increasing recently because laser cutting has a smaller kerr width and higher cutting quality compared to plasma and flame cutting. Both plasma arc cutting, as well as laser cutting processes, can be widely used in multiple industries, and these cutting methods are the most applicable to thermal cutting processes. During the thermal cutting process, the kinetic energy of the high-speed gas from the nozzle provides a mechanical force that ejects the melt and protects the nozzle from spatter [6]. The quality of the cutting can be evaluated from the kerr width, kerr angle, deposition of dress, level profile, and so on. The intensity of cutting speed, nozzle stand-off distance, and air pressure are key factors affecting the quality of the cut, among others [7]. The stand-off distance, which represents the allowable distance between the nozzle tip and workpiece, is mainly determined by the pattern and uniformity of the gas flow through the nozzle exit. Nevertheless, these parameters are still empirically determined [6]. For the gas flowing out of an axisymmetric nozzle to an ambient pressure surrounding of 1 bar, the jet at the exit becomes supersonic if the upstream total reservoir pressure exceeds 1.89 bar [8]. This is typically the case in most thermal cutting processes. In such conditions, the flow follows a Prandtl-Meyer expansion at the corner of the nozzle. These expansion waves are eventually reflected as compression waves from the constant pressure jet boundary. When the ratio of exit-to-ambient pressure is high, the compression waves of the same family intersect each other, and shockwaves start to appear as these waves coalesce. Further downstream of the flow, a shock develops in the form of a longitudinally curved surface of revolution. This shock is incident into the axis of the flow, and its slope and intensity continuously increase as the shock approaches the axis. Thus, regular reflection from the axis of flow is not possible and the appearance of Mach shock configurations appears [9].