# A Review of Advanced Air Lubrication Strategies for Resistance Reduction in the Naval Sector
## 3 Numerical Studies  
### CFD Simulations on MBDR  
Different studies have focused on the numerical simulation of the drag reduction via microbubbles. These studies differ in the turbulence model, which can vary between Reynolds-averaged Navier-Stokes (RANS), large eddy simulation (LES), and direct numerical simulation (DNS). The RANS model has been used in [PERSON] et al. [43], [PERSON] et al. [44], [PERSON] et al. [45], [PERSON] et al. [46], and [PERSON] et al. [47]. [PERSON] et al. [43] use an Eulerian-Eulerian two-fluid model in OpenFOAM for its simulations. The approach involves an Eulerian-Eulerian two-fluid model, used for both the continuous liquid phase and the dispersed bubble phase. It considers the interaction of microbubbles with the fluid flow around an axisymmetric body. The standard k-\(e\) model is applied to the continuous phase, while a separate turbulence model within OpenFOAM is used for the dispersed phase (gas). The Interfacial Area Transport Equation is used to calculate bubble sizes based on the solution of the transport equation for interfacial area density. The model accounts for bubble coalescence and break-up dynamics. [PERSON] et al. [44] present a comprehensive study using an Eulerian-Eulerian two-fluid model with the Multiple-Size Group (MUSIG) approach. An Eulerian-Eulerian framework is used to model both the liquid and bubble phases. The MUSIG model resolves a wide range of bubble sizes, taking into account bubble coalescence and break-up dynamics. Standard k-\(e\) and Shear Stress Transport (SST) models are used for turbulence modelling in the liquid phase, while a zero-equation turbulence model is used for the microbubbles. The results are validated against experimental data, emphasizing the capability of the MUSIG model to accurately represent bubble dynamics in microbubble-induced drag reduction scenarios. [PERSON] et al. [45] focus on evaluating the drag reduction effects of microbubbles on container ships. The simulations are performed by solving RANS equations in the form of the realizable k-\(e\) model, considering varying Froude numbers and air flow rates to assess the drag reduction effect. [PERSON] et al. [46] discuss the drag reduction characteristics of microbubbles on an underwater vehicle during its pitching movement. The study utilizes a Population Balance Model (PBM) to numerically simulate the aggregation and breakup of microbubbles under different ventilation parameters and motion characteristics. A key aspect of the study is the numerical method established to address the turbulent flow, incorporating different turbulence models. Specifically, the turbulence is modelled using the Shear Stress Transport (SST) k-\(\omega\) model, which is part of the Reynolds-averaged Navier-Stokes (RANS) methodology. The results indicate that microbubbles significantly reduce drag, especially when the vehicle is surrounded by them during pitching. This reduction varies with the pitching frequency and angle, highlighting the complex interplay between vehicle dynamics and microbubble behaviour. To validate the numerical model used in the study, the authors conducted water tunnel experiments. These experiments used a self-designed apparatus along with a high-speed camera and a six-component force balance to measure the dynamic loads on the model underwater vehicle during its pitching movements. By comparing the numerical results with these experimental observations, particularly focusing on the drag and lift coefficients under different conditions (e.g., varying microbubble diameters and ventilation volumes), the paper confirms that the numerical simulations align well with the experimental results. This correlation adds credibility to the effectiveness of the computational model, particularly in its use of the SST k-\(\omega\) turbulence model. [PERSON] et al. [47] present a comprehensive numerical investigation into the microbubble drag reduction on an axisymmetric body submerged in water, particularly focusing on an underwater vehicle.