## 2 Theoretical Foundations of CFD
### Principles, Mathematical Models, and Numerical Methods in CFD
DNS provides the highest modeling accuracy by solving all turbulence scales without approximations. As a result, it is mainly used in fundamental research, such as modeling bubble dynamics in frying processes or predicting liquid behavior in microflows. Its advantage lies in the lack of simplifications, making it the most precise method, though its enormous computational demands limit its industrial applications [22]. A critical challenge in CFD simulations is turbulence modelling, as most industrial applications involve turbulent flows. Turbulence models are necessary to approximate the effects of unsteady and chaotic fluid motion, as the direct resolution of turbulence on a fine scale is computationally prohibitive. The most commonly used turbulence models are as follows:  
* Reynolds-averaged Navier-Stokes (RANS): This approach averages the Navier-Stokes equations to model the flow in steady-state or near-steady conditions. The RANS equations include additional terms to account for the effects of turbulence, often represented by eddy-viscosity models. The \(k-\varepsilon\) and \(k-\omega\) models are widely adopted, where k represents the turbulent kinetic energy and \(\varepsilon\) (or \(\omega\)) represents the dissipation rate of turbulent energy. These models offer a balance between accuracy and computational efficiency [23]. * Large eddy simulation (LES): LES resolves large-scale turbulent structures (large eddies) while modelling the smaller, more chaotic eddies through subgrid-scale models. This approach is computationally more expensive but provides more detailed information, and is especially useful in cases requiring high accuracy for transient or unsteady flows [24]. * Direct numerical simulation (DNS): DNS solves the Navier-Stokes equations without any turbulence modelling, directly resolving all scales of motion. This method is the most accurate but extremely demanding in terms of computational resources, limiting its application primarily to fundamental research rather than industrial processes [24]. CFD simulations in the food industry frequently involve multiphase flow models, as many food processing operations (for example, mixing, emulsification, or phase separation) involve interactions between multiple phases, such as solid-liquid, liquid-gas, or immiscible fluids. Common approaches for modeling multiphase flow include the following:  
* Volume of fluid (VOF): This method tracks the interface between immiscible phases, such as oil and water, by solving a transport equation for a phase fraction. VOF is commonly used to simulate processes like emulsion formation, phase separation, or boiling [25]. * Eulerian-Eulerian Model: This method is used for modeling continuous phases where both phases are treated as interpenetrating fluids. This model is often applied in situations where the interaction between multiple fluids (for example, liquid and gas) is important [26]. * Eulerian-Lagrangian Model: This approach models one phase (typically discrete particles or droplets) using Lagrangian tracking, while the continuous phase is modeled using the Eulerian method. It is useful for simulating phenomena like slurry flows, solid-liquid suspensions, or droplet dynamics in spray drying [27]. Furthermore, heat transfer models are essential for accurately simulating thermal processes in food engineering. In many thermal food processing operations, such as baking, frying, or pasteurization, the interaction between heat conduction, convection, and radiation governs the efficiency and uniformity of heat transfer. Conjugate heat transfer (CHT) models allow for the coupled simulation of heat transfer between solids and fluids. In these models, the heat transfer in the solid phase (for example, food material) and the surrounding fluid phase (for example, air or oil) is solved simultaneously, ensuring that the boundary conditions are consistent across both phases [28].