## 3 Theoretical Formulation of Turbulent Gas Flow Modeling  
This section presents the mathematical framework developed to simulate the compressible turbulent gas flow that takes place in the gas-assisted thermal cutting process. The governing equations were solved using the commercial CFD solver ANSYS Fluent v19.0 which uses the finite volume method [24]. The flow regime is assumed to be steady, therefore steady-state simulations were performed. For pressure-velocity coupling, a coupled algorithm is employed. Turbulence is a three-dimensional phenomenon and a two-dimensional simulation approach often leads to non-physical results. Therefore, three-dimensional flow is considered for this study. The solution domain is subdivided into a finite number of contiguous control volumes and conservation equations are applied to every control volume. The equations used for turbulent flows are obtained from those of the laminar flows using the time averaging procedure commonly known as Reynolds averaging. Thus, the flow is assumed to be governed by the compressible Reynolds-averaged Navier-Stokes (RANS) equations, and an appropriate turbulence model is applied for closure of the RANS  
Figure 3: Schlieren experiment: (**a**) Schematic diagram of Schlieren method. (**b**) Experimental setup for Schlieren method. equations to simulate all the averaged unsteadiness. The governing equations are solved in generalized coordinates and in conservative form. Computation fluid dynamics (CFD) tools offer several turbulence models ranging from algebraic to linear and nonlinear two-equation turbulence models. For a simple viscous flow, an algebraic model does the job well because the turbulent viscosity is determined by a local function. Whereas for more complex viscous flow features such as shear layer and regions of separated flow, the two-equation turbulence model with second-order closure needs to be employed. Several studies related to the high-pressure nozzle flow [8; 17; 25] have employed two-equation turbulence models, namely \(k\)-\(\epsilon\) and \(k\)-\(\omega\) turbulence models, and have experimentally verified the prediction of shock wave structure and their position. [PERSON] et al. [8] employed a standard \(k\)-\(\epsilon\) turbulence model for numerical modeling of the gas-assisted laser cutting process at various stagnation pressures and successfully verified their numerical simulation result with the Schlieren method. The authors in [17] used the renormalization group (RNG) \(k\)-\(\epsilon\) turbulence model to study the phenomena of shock wave that is induced by a supersonic impinging jet emanating from a straight nozzle onto a substrate with varying inclined angle and verified their numerical results with experimental visualization using shadowgraph imaging. [PERSON] et al. [25] investigated five turbulence models, namely standard \(k\)-\(\epsilon\) turbulence model, extended \(k\)-\(\epsilon\) turbulence model, realizable \(k\)-\(\epsilon\) turbulence model, shear stress transport (SST) \(k\)-\(\omega\) turbulence model, and Reynolds stress model (RSM), over a wide range of nozzle pressure ratios to demonstrate their numerical accuracy in predicting the turbulent gas flow in rocket nozzle with complex nozzle wall geometry and found the SST \(k\)-\(\omega\) turbulence model outperformed all other turbulence models. From their assessment, they concluded that the \(k\)-\(\epsilon\) turbulence models perform well in most of the cases related to high-pressure gas flow; however, for the near-wall flow problems, the SST \(k\)-\(\omega\) turbulence model gives better results. In this study, the simulation model consists of gas flow through a narrow slot within the cut kerf. Thus, the flow experience narrow wall flow condition. Hence, four turbulence models, namely standard \(k\)-\(\epsilon\), realizable \(k\)-\(\epsilon\), RNG \(k\)-\(\epsilon\), and standard \(k\)-\(\omega\) turbulence models were assessed in terms of their agreement with the experimental results.