# A Review of Advanced Air Lubrication Strategies for Resistance Reduction in the Naval Sector
## 4 Mathematical Models
### Models on ACDR  
In [71; 72], [PERSON] applies a method of hydrodynamic discrete sources for two-dimensional modelling of stepped planing surfaces. The focus is on modelling the deformations of the water surface, wetted hull lengths, and pressure distribution for hulls at various attitudes and Froude numbers, including configurations with pressurized air cavities.  
The mathematical model utilizes the Bernoulli equation and hydrodynamic point sources distributed along a horizontal line. It assumes a steady, irrotational, incompressible, and inviscid flow (Figure 21).  
The model calculates the water surface elevations, source intensities, and pressure distributions on the hull by solving Equations (6)-(8).  
\[C_{p}=(p-p_{0})/\left(\rho u_{0}^{2}/2\right)=-\sigma \tag{6}\]  
\(p_{0}\) is the upstream pressure, \(p\) is the pressure on the water surface, and \(u_{0}\) is the upstream velocity. \(\sigma=(p_{0}-p)/\left(\rho u_{0}^{2}/2\right)\) is the ventilation number, defined to be zero on the free surface and nonzero on the surface of a pressurized cavity.  
\[u^{\prime}(x_{i}^{c})=\frac{1}{2\pi}\sum\
olimits_{j}\frac{q_{j}}{x_{i}^{c}- x_{j}^{s}} \tag{7}\]  
\(x_{i}^{c}\) and \(x_{j}^{s}\) are positions of the \(i\)-th collocation point and the \(j\)-th source with strength \(q_{j}\).  
\[\frac{1}{2}(\frac{q_{i}}{\Delta x_{i}}+\frac{q_{i-1}}{\Delta x_{i-1}})=-2u_{0} \frac{y_{i}^{s}-x_{i-1}^{s}}{x_{i}^{s}-x_{i-1}^{s}} \tag{8}\]  
\(\Delta x\) is the cell size and \(y^{s}\) is the water surface elevation at a source location.  
The model also considers the lift and drag coefficients, providing insights into the hydrodynamic performance of different hull configurations, including those with open and pressurized cavities.  
Figure 21: (**a**) Schematic of a two-dimensional stepped planing hull with main geometrical parameters. (**b**) Simplistic view of the staggered arrangement of sources (circles) and collocation points (crosses). (Ref. [71] reproduced under a CC BY-NC-ND 4.0 DEED license).  
The paper presents validation examples and parametric calculations for single-step and multi-step hulls, demonstrating the model's effectiveness in capturing the complex interactions between hull geometry, air cavities, and water flow dynamics.  
In [73], the objective is to estimate the loads experienced during the water entry of air-cavity hull sections. It includes drop tests with rigid two-dimensional hull sections simulating air-cavity hulls. These tests varied the platform's vertical position to observe the impact force reduction.  
A mathematical model based on simplified added mass theory is developed. It considers forces on the hull sides (akin to semi-wedges), compression of trapped air in the cavity, and water-air interaction. The model accounts for actual gravitational forces, hull buoyancy, and the force due to adiabatic compression of air in the cavity. The aim was to predict water entry loads, especially assessing how cavity height variations impact force reduction.  
The maximum acceleration can be calculated as in Equation (9).  
\[a_{max}=\frac{m_{ad,p}}{m+m_{ad,p}}\cdot\frac{4|v_{0}|c_{w}}{b}\sqrt{\theta} \tag{9}\]  
\(m_{ad,p}\) is the plate-added mass (which depends on the water density, hull section length, cavity width, wetted half-beam of the wedge, and deadrise angle), \(v_{0}\) is the plate entry velocity, \(c_{w}\) is the speed of sound in water, and \(\theta\) is retrieved from empirical correlation.  
The model is validated with experimental data, showing a reasonable correlation with the test results, particularly regarding the timing of peak acceleration occurrences.