A Parametric Study on Air Lubrication for Ship Energy Efficiency

Abstract

With the new target set by the International Maritime Organization (IMO) of zero net emissions of atmospheric gases from maritime vessels by 2050, studies of methods that improve the efficiency of vessels have become highly relevant. One promising method is air injection, which creates a lubricating film between the hull and water, reducing the total resistance. Despite the potential of air injection, there is a lack of studies defining the correlation between key parameters (such as air layer thickness, injection angle, vessel speed, and the number of nozzles) in the method efficiency. Therefore, this study aimed to assess the method’s efficiency through a parametric analysis. The study utilized the OpenFOAM software to analyze the air injection method in the Duisburg Test Case (DTC) hull, a 1:59 scaled container ship. The numerical solution used finite volumes to discretize the conservation equations, RANS (Reynolds-Averaged Navier–Stokes) in the momentum equation, and $\kappa$-$\omega$ SST in the turbulence model. The optimum configuration achieved 14.13% net power savings, while the worst configuration increased the power consumption instead. An analysis of variance (ANOVA) confirmed the relationship between parameters and effectiveness. Therefore, the results showed the importance of adjusting the method’s parameters.

1. Introduction

The transportation of goods by water is considered one of the most efficient methods for moving cargo between different global points. According to [1], in 2021, over 80% of the world’s commercialized products were transported by sea. The cost of this transportation is directly associated with the fuel consumption of the vessels. In 2020 alone, more than 200 million tons of fuel were consumed by sea cargo ships around the world [2]. It is projected that by 2030, there will be a 20% increase in shipping tonne-kilometers [3]. In response, in 2023, the International Maritime Organization (IMO) revised the targets for the net emission of greenhouse gases (GHG) by maritime vessels. The revised reduction targets are as follows: at least 20%, striving for 30%, by 2030, at least 70%, striving for 80%, by 2040, and 100% by 2050, compared to 2008 [4].

The change in shipbuilding must happen today as the lifespan of a commercial ship is more than 25 years, which is currently a crucial period to reach the IMO 2050 target [5]. Following [6], five major areas contribute to the decarbonization of ships: logistics and digitalization, with the potential to reduce GHG emissions by more than 20%; hydrodynamics, with the potential to reduce them between 5% and 15%; machinery, with the capacity to reduce them between 5% and 20%; energy, achieving reductions of 0% to 100%; and post-treatment, enabling reductions of between 0% and 90%. In particular, this study belongs to the area of hydrodynamic performance improvement, using the air lubrication method to reduce GHG emissions. Other techniques in hydrodynamics include anti-fouling and hull geometry optimization.

One traditional approach in hydrodynamics involves applying a treatment on the ship hull to minimize its physical and biological roughness, thus avoiding the growth of barnacles on the hull. Anti-fouling coatings can effectively reduce barnacles on the hull, thereby preventing increased ship resistance [7,8]. On the other hand, studies involving more efficient hull geometries focus on reducing the share of viscous pressure resistance, which leads to a slight reduction in total ship resistance [9]. Another method to reduce the viscous pressure resistance is using hydrodynamic profiles. The study [10] indicates that stern flaps reduce effective power by 6%, resulting in energy savings. Entering the air lubrication category and in line with the hydrofoil solution, some studies explore inducing air by negative pressure, creating microbubbles that lubricate the hull, reducing the viscous resistance and ship resistance [11,12,13,14]. The microbubbles method is also useful for ice-going ships, as it can reduce the ice-induced resistance caused by friction between the hull and ice floes [15]. Still, within the category of air lubrication, some approaches use air cavities that trap the air below the hull forming air layers that reduce the viscous resistance. This method can reduce ship resistance by 5% to 22% [16,17].

In this work, air injection in the bottom of the hull was adopted to reduce the total resistance. The potential to reduce the total resistance and the low changes required in the hull design indicate a promising method for the implementation of a full-scale ship. The air injection method forms a lubricating film that avoids the direct interaction between the hull and water. Thus, the viscous resistance decreases due to differences in air and water properties, such as dynamic viscosity and density. The water dynamic viscosity and density are approximately sixty times and one thousand times greater than the same air properties, respectively. However, this method is susceptible to air leakage, as was observed in a study carried out in [18,19] in which it was verified that for low airflow rates and high velocities in the free stream, the air layer became intermittent and could be undone. Even so, the method performed well, reducing total resistance by 8% to 19% [20,21]. Hence, reducing ship resistance by air lubrication is an effective strategy for decreasing fuel consumption, contributing to GHG emission reduction targets, and improving the competitiveness of the maritime sector.

For many years, the air injection method was considered to be ineffective in reducing ship resistance. This conception always occurred in parallel with the fall in the price of a barrel of oil, because with the price falling, there was not enough economic justification for implementing new technology, and the global concerns for environmental issues in the 20th century were of less prominence. However, at the end of the last century, with the rise in the price of oil and the first studies into lubricating hulls with air injection, the method gained importance in the scientific community. In the 1990s, References [22,23] observed that the efficiency of the method could be improved to commercial levels and that this improvement was directly linked to the amount of airflow injected into the bottom of the hull.

Nevertheless, the studies did not consider the type of airflow regime at the bottom of the hull due to the amount of airflow used. It was not until 2008 that [24] identified that there were three types of regimes in which air flowed close to the bottom surface of the hull: microbubbles, transitional regime, and fully developed air layer. In their study of the effect of the regimes on the viscous resistance of a flat plate, they found that the maximum reduction in viscous resistance due to microbubbles (BDR—bubble drag reduction) was 20%, while the maximum reduction in viscous resistance due to the air layer (ALDR—air layer drag reduction) was over 80%. This result was later confirmed by other important work carried out by [25].

More recently, following the success of the injection method on flat plates with fully developed air layers, studies were conducted on scale models of hulls with low Froude numbers, demonstrating that the air injection method was suitable for large displacement hulls [17,19,26]. The study [20] investigated the savings in propulsive power of a 1:24 scale hull of a bulk carrier in the supramax category, with 200 m and 66,000 DWT, by injecting air through six nozzles in the flat region of the bow bottom. The study found that the power required by the propulsion system was reduced by 8 to 10%. When the energy needed for the air injection required by the method was discounted, the net power saving was 5 to 6%. In a 2018 study [27], the impact of multiple air injection regions at the bottom of the hull was examined. The study was conducted on a 1:33 scale hull belonging to a tanker with a displaced volume of 50,000 ${\mathrm{m}}^3$. The study found that sprayed injections yielded superior results to injections at a single location on the hull.

Aspects of the air injection method have been studied extensively by [16,20,21,26,27,28] and others; however, a lack of studies on the method is observed when analyzing the results without considering the derivative effect of one parameter on the other. Reference [29] shows that the efficiency of the method depends on various parameters related to the air injection method, including the Froude number, Reynolds number, flow rate, injection angle, and air layer thickness. For instance, Reference [28] recently investigated the impact of the air layer thickness on the method; Reference [29] analyzed the effect of the number of injector nozzles on the effect of the method; and Reference [19] studied the effects of parameters linked to the flow, such as fluid velocity and airflow, and the impact of the hydrophobic surface on the stability of the air layer developed in the ALDR category. However, to the best of the authors’ knowledge, essential parameters such as the injection angle, air layer thickness, hull speed, and nozzle quantity have not been correlated in the state-of-the-art literature. Correlating these parameters is important because the reduction in fuel consumption varies as a function of each parameter and the correct evaluation of these changes may provide important insights into the system design. Consequently, inferences can be drawn indicating the optimal configuration.

The central contribution of this work is to provide a more complete analysis of the method by carrying out a parametric study that includes the main design and operational parameters: injection angle, air layer thickness, hull speed, and number of injection nozzles. Another contribution of this study is to show clearly to the reader, both quantitatively and qualitatively, the causal effect of the method’s gain or loss of efficiency on each parameter variation. Finally, the study provides an overview of the energy balance between power saving (due to the gross power saving) and the power required in the compression system for the air injection method.

2. Methodology

The present study employed computational techniques, specifically CFD (Computational Fluid Dynamics), to investigate the efficacy of air injection in reducing viscous resistance in displacement hulls. To do so, the conservation equations that govern the physical problem must be solved. The OpenFOAM [30] software (version 8) was employed, which uses a finite volume formulation to discretize the governing differential equations of the problem. The InterFoam solver was used because it is an OpenFOAM tool capable of dealing with incompressible, viscous, and multiphase flows in both laminar and turbulent regimes.

2.1. Computational Domain

As illustrated in Figure 1, the computational domain is initially filled with water, represented in blue. This water extends up to the level of the Duisburg Test Case (DTC) hull draft. Above the waterline, the domain is comprised of air. Below the hull, the air injection faces (nozzles) are present, with boundary conditions of the type described. The unit of distance, designated by the symbol “L”, is the measure of the total scale length of the DTC, which is approximately 6.28 m.

Figure 1 is illustrative and does not accurately represent the dimensions between the hull and the boundary conditions, i.e., it is not to scale. To save computational resources the domain was divided into two equal parts through the creation of a symmetry plane. The symmetry plane is parallel to the $XZ$ plane and is situated on the longitudinal axis (X-axis) of the DTC hull. Details on the dimensions of the full-scale vessel and the scaled hull are found in [31]. The model speed was determined using an incomplete similarity analysis, which set the same Froude numbers to the model and the full-scale ship.

The solution to the multiphase problem was obtained using the Volume of Fluid (VOF) method, as proposed by [32]. In the VOF method, all the immiscible fluid phases present in a control volume share the same velocity and pressure. The model is predicated on the assumption that the two fluids, air and water, are not diluted within each finite volume. To define the volumes containing water or air for the configuration of the problem under study, the relationship between the phases was defined as a function of the volumetric fraction $\left(\alpha \right)$. This assumption was based on the following considerations: when the cell contained only water, the value was assumed to be one; when the cell contained only air, the value was assumed to be zero; and when both fluids coexisted in the same cell, the value was assumed to be an intermediate value. The average physical properties of the flow density $\left(\rho \right)$ and dynamic viscosity $\left(\mu \right)$ are defined in Equations (1) and (2), respectively.

$$\rho ={\rho }_{wtr}\alpha +(1-\alpha ){\rho }_{air}$$

(1)

$$\mu ={\mu }_{wtr}\alpha +(1-\alpha ){\mu }_{air}$$

(2)

where ${\rho }_{wtr}$ and ${\rho }_{air}$ are the water density and air density, respectively, expressed in $(\mathrm{kg}/{\mathrm{m}}^3)$. ${\mu }_{wtr}$ and ${\mu }_{air}$ are the dynamic viscosity of the water and the dynamic viscosity of the air, respectively, written in $(\mathrm{kg}/\mathrm{m}·\mathrm{s})$.

The flow was considered to be incompressible, and the Reynolds-Averaged Navier–Stokes (RANS) formulation was employed in the turbulent modeling of the problem [33]. Consequently, a time average was calculated for the conservation equations for the mass (Equation (3)) and momentum (Equation (4)).

$${\displaystyle \frac{\partial \overline{u_i}}{\partial x_i}}=0$$

(3)

$${\displaystyle \frac{\partial \left(\rho {\overline{u}}_i\right)}{\partial t}}+\overline{u_j}{\displaystyle \frac{\partial \left(\rho {\overline{u}}_i\right)}{\partial x_j}}=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\mu {\displaystyle \frac{\partial {\overline{u}}_j}{\partial x_j}}-\rho \overline{u_i}\overline{u_j}\right)+F_i$$

(4)

where ${\overline{u}}_i$ and ${\overline{u}}_j$ are the mean velocities $(\mathrm{m}/\mathrm{s})$ in Cartesian coordinates X and Y, respectively. $x_i$ and $x_j$ are the Cartesian coordinates’ positions $\left(\mathrm{m}\right)$, X and Y, respectively. Finally, the t term refers to time $\left(s\right)$, and $F_i$ the source term $(\mathrm{kg}/{\mathrm{m}}^2{\mathrm{s}}^2)$.

The turbulence model employed was the $\kappa$-$\omega$-SST, proposed by [34], recommended for an accurate flow description near walls.

In addition to the transport equations for mass and momentum, an additional transport equation was solved for the volumetric fraction $\left(\alpha \right)$ in the method [32].

$${\displaystyle \frac{\partial \left(\alpha \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\alpha {\overline{u}}_i\right)}{\partial x_i}}=0$$

(5)

2.2. Turbulence Model

The turbulence model used in this study was the $\kappa$-$\omega$-SST, as proposed by [34]. This model is recommended because it accurately describes the flow in the wall and near-wall regions. This is achieved by combining the $\kappa$-$\omega$ and $\kappa$-$ϵ$ models. The $\kappa$-$\omega$-SST model introduces two additional equations to close the turbulent formulation, namely, the turbulent kinetic energy, Equation (6), and the specific dissipation rate, Equation (7):

$${\displaystyle \frac{\partial \rho \kappa }{\partial t}}+{\displaystyle \frac{\partial \rho {\overline{u}}_j\kappa }{\partial x_j}}=P-\beta \rho \kappa \omega +{\displaystyle \frac{\partial }{\partial x_j}}\left[(\mu +{\sigma }_{\kappa }{\mu }_t){\displaystyle \frac{\partial \kappa }{\partial x_j}}\right]$$

(6)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho {\overline{u}}_j\omega \right)}{\partial x_j}}=P-\beta \rho {\omega }^2+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]\hfill \\ & +2\left(1-F_1\right){\displaystyle \frac{\rho {\sigma }_{{\omega }^2}}{\omega }}{\displaystyle \frac{\partial \kappa }{\partial x_j}}{\displaystyle \frac{\partial \omega }{\partial x_j}}\hfill \end{array}$$

(7)

where $\omega$ is the specific dissipation rate $(1/\mathrm{s})$, $\kappa$ is the turbulent kinetic energy $({\mathrm{m}}^2/{\mathrm{s}}^2)$, $F_1$ is the constant responsible for switching the model in the representations $\kappa$-$\omega$ and $\kappa$-$ϵ$. The constants ${\sigma }_{\kappa }$, ${\sigma }_{\omega }$, ${\sigma }_{{\omega }^2}$, and $\beta$ are empirical constants. The production term $\left(P\right)$ and the turbulent eddy viscosity $\left({\mu }_t\right)$ is given by Equations (8) and (9), respectively:

$$P={\tau }_{ij}{\displaystyle \frac{\partial u_i}{\partial x_j}}$$

(8)

$${\mu }_t={\displaystyle \frac{\rho a_1\kappa }{max\left(a_1\omega ,S{F}_2\right)}}$$

(9)

where $a_1$ is the damping coefficient for turbulent viscosity, and $F_2$ is the mixing term for turbulent viscosity, $F_2$ being an empirical constant [35]. The stress rate tensor, $S_{ij}$ $(1/\mathrm{s})$, is given by Equation (10):

$$S_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right)$$

(10)

2.3. Boundary Conditions

Boundary conditions for each face of the computational domain and the ship hull are presented in Figure 1. The DTC hull was assumed to be static, and the water was forced to flow around it.

Table 1. Boundary conditions for the main surfaces in the computational domain.

Face	Boundary Conditions
	Flow	Volume Fraction	Turbulence ($\mathit{\kappa }$-$\mathit{\omega }$)
Atmosphere	Prescribed pressure	Zero derivative *	Prescribed ($\kappa$ = 0.00015 and $\omega$ = 2)
Nozzles	Prescribed flow rate $\left(Q\right)$	Prescribed ($\alpha$ = 0)	Zero derivative *
Hull	No Slip	Zero derivative *	Zero derivative *
Inlet	Prescribed velocity	Prescribed ($\alpha$ = 1)	Prescribed ($\kappa$ = 0.00015 and $\omega$ = 2)
Bottom	Symmetrical	Symmetrical	Symmetrical
Side	Symmetrical	Symmetrical	Symmetrical
Symmetry plane	Symmetrical	Symmetrical	Symmetrical
Outlet	Zero derivative *	Zero derivative *	Prescribed ($\kappa$ = 0.00015 and $\omega$ = 2)

* Relative to normal surface.

shows the flow, volumetric fraction, and turbulence boundary conditions used in the current solution.

The nozzles are imaginary regions at the bottom of the hull. The airflow rate was increased gradually and then kept at a constant level. This is explained in Section 2.8.

2.4. Mesh

The hull’s geometric proportions were kept in the reduced model. The DTC was discretized using OpenFOAM tools [30]. The mesh near the hull was more refined to capture the turbulent effects accurately. SnappyHexMesh was the OpenFOAM application used to merge the mesh with the DTC hull, an important step in studying the air injection method because it allows the creation of new faces as air injection nozzles in the computational domain. Figure 2 shows the mesh with the DTC hull.

2.5. Numerical Solution

A numerical solution was performed with InterFoam, which is the OpenFOAM solver for multiphase flows of two immiscible fluids. For pressure–velocity coupling, the PIMPLE algorithm, a blend of the PISO and SIMPLE algorithms, was used. PIMPLE can correct the velocity to satisfy the conservation of momentum equation and solve the pressure equation [33]. An Euler interpolation scheme was used for the transient regime, and a Linear Gauss scheme was used to approximate the divergent terms under the upwind condition.

Table 2. The main physical properties of study.

Properties	Value
Water density (${\rho }_{wtr}$)	1025 $\mathrm{kg}/{\mathrm{m}}^3$
Air density (${\rho }_{air}$)	1.21 $\mathrm{kg}/{\mathrm{m}}^3$
Water kinematic viscosity (${\nu }_{wtr}$)	$1.0·{10}^{-6}$ ${\mathrm{m}}^2/\mathrm{s}$
Air kinematic viscosity (${\nu }_{air}$)	$1.5·{10}^{-5}$ ${\mathrm{m}}^2/\mathrm{s}$

shows the physical properties of the problem at ${20 }^{°}$C and atmospheric pressure.

2.6. Mesh Independence

A mesh independence test was conducted using the following configuration: a speed of 1.668 $\mathrm{m}/\mathrm{s}$, 8 nozzles, $5^{°}$ of injection angle, and an active airflow rate injection of $9.63·{10}^{-3}$ ${\mathrm{m}}^3/\mathrm{s}$ (this value corresponds to 8 $\mathrm{mm}$ of air layer thickness, see Section 2.8).

Table 3. Independence test of the solution due to the mesh.

Mesh	Cells	Resistance [N]			$\mathit{RD}$ [%]			${\mathit{Y}}^{+}$	Time [h]
		Viscous	Pressure	Total	Viscous	Pressure	Total
M1	781,506	15.84	4.76	20.60	18.85	3.93	14.52	123.49	4.70
M2	1,236,514	19.52	4.58	24.10	1.61	3.62	0.66	63.48	7.42
M3	1,822,576	19.84	4.42	24.26	0.90	1.84	0.41	47.89	11.13
M4	2,506,067	20.02	4.34	24.36	-	-	-	29.33	14.70

contains all the cases. Four different meshes (M1, M2, M3, and M4) were tested, starting with a coarse mesh. The resistance values $\left(R\right)$ to viscous, pressure, and total resistance were used to compare the meshes. The resistance values listed correspond to the entire ship (twice the value for the computational domain) while the number of cells corresponds to the computational domain. The relative difference ($RD$) between the solutions of two subsequent meshes is given by Equation (11), where i ranges from 1 to 3.

$$R{D}_i=\left|{\displaystyle \frac{R^i-R^{i+1}}{R^{i+1}}}\right|\times 100\%$$

(11)

The relative difference decreased as the mesh was refined, but the simulation time increased. Due to the complexity of the problem, a relative difference of less than 2% was considered acceptable. The maximum relative difference between the M3 and M4 mesh was 1.84% for the pressure resistance, so the M3 mesh was considered independent. The $Y^{+}$ value for the independent mesh was 47.89, which fell within the acceptable range for simulations using the $\kappa$-$\omega$-SST model and standard wall functions. According to [36], a $Y^{+}$ value between 30 and 300 is appropriate for these conditions, ensuring accurate near-wall treatment.

2.7. Validation

A large data set of results was available for the DTC hull. These tests were conducted on the standard hull, without air injection. The ship was tested at different speeds and values for the viscous, pressure, and total resistance were obtained. These data are currently being used to validate numerical studies related to this hull [31]. To validate the current study, the results were compared for speeds of 1.335 and 1.668 $\mathrm{m}/\mathrm{s}$, the lowest and highest operating speeds without air injection, as shown in

Table 4. Validation of numeric solution.

	Experimental [31]		Numeric (This Work)		Error %
Velocity [$\mathrm{m}/\mathrm{s}$]	1.335	1.668	1.335	1.668	1.335	1.668
Viscous resistance [N]	17.61	26.43	16.54	24.90	6.07	5.79
Pressure resistance [N]	2.73	5.40	2.52	5.10	7.69	5.56
Total resistance [N]	20.34	31.83	19.06	30.00	6.29	5.75

.

As seen in Table 4, all the calculated errors were below 8%. Given the complexities of the physical problem and the numerical model, these values were taken as acceptable. Thus, it was assumed that the proposed computational model was validated.

2.8. Air Injection

The air injection area was located at the flat bottom of the hull, between the bow and midship. The airflow rate $\left(Q_{air}\right)$ was calculated based on the air layer thickness developed at the bottom of the hull, given by Equation (12). According to patents [37,38], the ideal air layer thickness value is within the 4 to 8 $\mathrm{mm}$ range.

$$Q_{air}=t_{air}{B}_{air}{V}_S$$

(12)

where $t_{air}$ is the air layer thickness at the bottom of the hull $\left(\mathrm{m}\right)$, $B_{air}$ is the width of the air layer $\left(\mathrm{m}\right)$, and $V_S$ is the ship speed $(\mathrm{m}/\mathrm{s})$. The study used 4, 6, and 8 $\mathrm{mm}$ of air layer thickness, and the air layer width was determined by the flat area width available at the bottom of the hull. The DTC hull had a width of 0.722 $\mathrm{m}$, which was used as $B_{air}$.

The injection angle is defined by the vector decomposition of the air injection velocity on the X and Z axes. With the Z axis normal to the plane comprising the flat bottom of the hull, the air outlet velocity at Z directly affects the volumetric air flow rate injected by the nozzle. Thus, by controlling the air outlet velocity at X and Z, it is possible to form a resultant vector with the desired injection angles ($5^{°}$, ${15}^{°}$, and ${25}^{°})$. Thus, the air velocities in the X $\left(V_{ai{r}_X}\right)$ and Z $\left(V_{ai{r}_Z}\right)$ direction are given, respectively, by Equations (13) and (14):

$$V_{ai{r}_X}=\left({\displaystyle \frac{Q_{air}}{A_{noz}}}\right)ctg\left({\gamma }_{inj}\right)$$

(13)

$$V_{ai{r}_Z}={\displaystyle \frac{Q_{air}}{A_{noz}}}$$

(14)

where $A_{noz}$ is the total nozzle injection area $\left({\mathrm{m}}^2\right)$, and ${\gamma }_{inj}$ is the air injection angle. All nozzle injections had the same constant area ($L_{no{z}_X}·L_{no{z}_Y}$); therefore, the total nozzle injection area increased with the number of nozzles, resulting in a decrease in air outlet velocity.

A water levels analysis was conducted on the entire wet area of the DTC hull to introduce air into the flat area at the bottom of the hull. The wetted area was divided into three regions:

• Region I—the flat area at the bottom of the hull where the injection nozzles are located;
• Region II—the flat area at the bottom of the hull without injection nozzles;
• Region III—the entire wetted area with varying water levels.

These three regions are illustrated in Figure 3a. The nozzles are rectangular faces located on the hull mesh, and the airflow is evenly distributed among them. The spatial discretization for the three nozzle quantity configurations (4, 6, and 8) are displayed in Figure 3b, Figure 3c and Figure 3d, respectively.

The nozzles in region “I” were equally spaced. The horizontal distances between the nozzles on the X axis $\left({\Delta }_{no{z}_X}\right)$ and the vertical Y axis $\left({\Delta }_{no{z}_X}\right)$ are given by Equations (15) and (16), respectively.

$${\Delta }_{no{z}_X}={\displaystyle \frac{1.23-L_{{noz}_X}}{\frac{Q_{noz}}{2}}}$$

(15)

$${\Delta }_{no{z}_Y}={\displaystyle \frac{0.361-L_{{noz}_Y}}{\frac{Q_{noz}}{2}+1}}$$

(16)

where $L_{{noz}_X}$ is the length of the injector nozzle on the X axis with a value of 0.01172 $\mathrm{m}$; $L_{{noz}_Y}$ is the length of the injector nozzle on the Y axis with a value of 0.05 $\mathrm{m}$; and $Q_{noz}$ is the number of nozzles injecting air. The fixed values 1.23 and 0.361 are the distances between the origin of region “I” and the origin of the X and Y axes, respectively.

The airflow rate was determined by multiplying the air velocity and the nozzle area. The velocity of the injected air followed a linear function with an angular coefficient $\left(a\right)$ equal to 0.001. This implied that the full airflow rate required to reach the ALDR regime was reached after 1000 s of simulation. Thus, the airflow rate as a function of simulated time $\left(Q_{air}^{\prime }\right)$ was given by Equation (17).

$$\left\{\begin{array}{cc}{Q}_{air}^{\prime }=0\hfill & forT<4000\hfill \\ Q_{air}^{\prime }=Q_{air}(0.001T-4)\hfill & for4000\le T\le 5000\hfill \\ Q_{air}^{\prime }=Q_{air}\hfill & forT>5000\hfill \end{array}\right.$$

(17)

where T represents the current simulation time (s). The gradual introduction of air was intended to assess the decreases in viscosity, pressure, and total resistance, as well as to aid in stabilizing the new transitional regime.

The injection angle, air layer thickness, and number of injection nozzles had 3 different values each. The speed parameter had 2 variations, representing the lowest and highest DTC hull operating speeds. Each simulated case was created by combining these parameters, resulting in 54 tests.

Table 5. Simulated values for each air injection parameter.

Parameters	Values
Velocity	1.335 and 1.668 $\mathrm{m}/\mathrm{s}$
Air layer thickness	4, 6, and 8 $\mathrm{mm}$
Number of injector nozzles	4, 6, and 8 units
Angle of injection	$5^{°}$, ${15}^{°}$, and ${25}^{°}$

displays the value variations for each parameter.

2.9. Power Savings

The goal was to quantify the net power saved with the air injection method. This was achieved by subtracting the power required for air compression from the gross power saving obtained through the air injection method. The compressor efficiency and the total pressure at the nozzles were taken into account. Pressure losses in the air distribution were not included. Considering a polytropic process, the estimated power required by the compressor was given by Equation (18), [39]:

$$P_c={\displaystyle \frac{Q_{air}{p}_1}{{\eta }_c}}\left({\displaystyle \frac{n}{n-1}}\right)\left[{\left({\displaystyle \frac{p_2}{p_1}}\right)}^{{\displaystyle \frac{n-1}{n}}}-1\right]$$

(18)

where ${\eta }_c$ is the compressor efficiency, n is the polytropic coefficient, $p_1$ is the pressure at the compressor inlet (atmospheric pressure) $\left(\mathrm{Pa}\right)$, and $p_2$ is the total pressure at the nozzle outlet $\left(\mathrm{Pa}\right)$.

Centrifugal compressors were suitable for this application due to the required airflow, ranging from 1.93 to 4.82 ${\mathrm{m}}^3/\mathrm{s}$, and the relatively low-pressure ratio required, which varied between $2.47·{10}^3$ and $2.90·{10}^3$ $\left(\mathrm{Pa}\right)$. According to [39], the efficiencies of centrifugal compressors range from 70% to 85%. Consistent with [20], the present study used a compressor efficiency value of 75% and a polytropic coefficient of 1.4 in Equation (18).

According to [40], for hulls with a high block coefficient and single screw, the hull efficiency can exceed 100% (values between 110% to 130%) due to the wake fraction coefficient being larger than the thrust deduction coefficient. Still, in [40], propeller efficiency values ranged from 55% to 70%, while shaft efficiency ranged from 95% to 99%. In this study, the values used were 120% for hull efficiency, 55% for propeller efficiency, and 98% for shaft efficiency, resulting in an approximate value of 65% for total propulsive efficiency $\left({\eta }_T\right)$. Thus, the brake power required by the standard hull ($P_B$) and the hull with air injection ($P_{B,air}$), as well as the brake power saving ($P_{B,sav}$) are given by Equations (19), (20) and (21), respectively:

$$P_B={\displaystyle \frac{R_T{V}_S}{{\eta }_T}}$$

(19)

$$P_{B,air}={\displaystyle \frac{R_{T,air}{V}_S}{{\eta }_T}}$$

(20)

$$P_{B,sav}=P_B-P_{B,air}$$

(21)

where $R_T$ is the total resistance for a standard hull without air injection $\left(\mathrm{N}\right)$, and $R_{T_{air}}$ is the total resistance obtained with the air injection method $\left(\mathrm{N}\right)$. Finally, these values enable the calculation of the percentage power savings, both excluding the air compression power (gross power saving, $P{S}_{gross}$) and including it (net power saving, $P{S}_{net}$), as given by Equation (22) and Equation (23), respectively:

$$P{S}_{gross}={\displaystyle \frac{P_{B,sav}}{P_B}}\times 100\%$$

(22)

$$P{S}_{net}={\displaystyle \frac{P_{B,sav}-P_c}{P_B}}\times 100\%$$

(23)

Given the pressing importance of carbon emissions, it is noteworthy that a certain net energy savings (10%, for instance) typically results in a similar percentage reduction in carbon emissions. This relationship is due to the fuel’s consistent carbon content, meaning each ton burned releases a predictable amount of ${CO}_2$. While propulsion and air compression may utilize power sources with different efficiencies, this generally has a minimal effect on the link between net energy savings and emissions reductions. Typically, prime movers, which power propulsion systems, operate up to 20% more efficiently than auxiliary engines used for air compression. However, because propulsion power demands are substantially higher than those of air compression, it is reasonable to approximate the reduction in carbon emissions by the net energy savings percentage.

3. Results

To assess the impact of air injection on the viscous, pressure, and total resistance, the simulation initially ran without air injection. The transitional regime lasted around 2000 s, followed by a 2000-s wait period. At 4001 s, the gradual injection of air through the nozzles began. Air injection took place gradually over 1000 s, with complete airflow occurring after 5000 s of simulation. The gradual injection process helped stabilize the simulation during the air injection. Around 4200 s, the effect on the resistance due to the method became observable, and the system converged again from 5600 s onwards. The test concluded at 10,000 simulated seconds. Figure 4 illustrates the entire process.

The analysis in Figure 4 corresponds to the configuration with a speed of 1.668 $\mathrm{m}/\mathrm{s}$, eight nozzles, 8 $\mathrm{mm}$ of air layer thickness, and a $5^{°}$ injection angle.

As shown by the yellow curve in Figure 4, the pressure resistance was reduced during the initial air injection process. But, briefly, the pressure resistance was restored to the original value, indicating the independence of pressure resistance with the method. The resulting resistance values were calculated as the average of the resistance values simulated during the last 2000 s, specifically within the interval between 8000 and 10,000 s.

3.1. Gross Power Saving and Net Power Saving

The results are summarized in graphs to analyze the relationship between the injection parameters better. Each graph displays nine simulated configurations. Within each configuration, there are nine results for the gross power saving (shown as a continuous line) and nine results for the net power saving (shown as a dotted line). The gross power savings and net power savings for the speeds of 1.335 and 1.668 $\mathrm{m}/\mathrm{s}$ are depicted in the graphs in Figure 5.

The gross power saving at a speed of 1.335 $\mathrm{m}/\mathrm{s}$ showed patterns of dependence linked to the injection angle, number of nozzles, and air layer thickness parameters. The gain was directly proportional to the number of nozzles and air layer thickness. It was observed by [22,23] that the effectiveness of the method increased as a function of the air layer thickness. As for the injection angle, the gains were lower for larger angles.

The graph in Figure 5a shows that when there were four nozzles, the most significant variations in gross power savings occurred when the injection angle was decreased from ${25}^{°}$ to ${15}^{°}$. This variation was more prominent for the 6 $\mathrm{mm}$ air layer thickness. In Figure 5c, with six nozzles, the gross power savings were not affected by the change in injection angle for the 8 $\mathrm{mm}$ and 6 $\mathrm{mm}$ air layer thickness. However, for the 4 $\mathrm{mm}$ air layer thickness, reducing the injection angle from ${15}^{°}$ to $5^{°}$ was most effective for the gross power saving. In Figure 5e, with eight nozzles, the gains were higher than with four nozzles but lower than with six nozzles.

Observing the gains as a function of the air layer thickness, it can be seen that they could be more than doubled by increasing the air layer thickness, specifically in the case of low speed and few nozzles (Figure 5a). For the other two cases, Figure 5c,e, the enhancement in power savings was around 50% when the air layer thickness went from 4 $\mathrm{mm}$ to 8 $\mathrm{mm}$.

For a speed of 1.335 $\mathrm{m}/\mathrm{s}$, the injection angle was the most relevant parameter for the effectiveness of the air injection method. In all the air layer thickness curves, higher power savings occurred for angles of $5^{°}$. One possible explanation was the fact that the air was directed to flow downstream, almost parallel to a flat area. Therefore, it prevented early leakage from the sides and achieved better air coverage in the stern region.

Regarding the net power savings, Figure 5 illustrates that the energy costs involved in injecting air at a speed of 1.335 $\mathrm{m}/\mathrm{s}$ presented better results for the cases with a 6 $\mathrm{mm}$ air layer thickness. This means that the net power saving achieved by the 8 $\mathrm{mm}$ air layer thickness did not compensate for the extra energy cost required for that flow rate. As a result, gains were achieved for the 6 $\mathrm{mm}$ air layer thickness. Nonetheless, the value of 6 $\mathrm{mm}$ of air layer thickness belongs to the optimal range, 4–8 $\mathrm{mm}$, proposed by [37,38].

The highest gross power saving found for a speed of 1.335 $\mathrm{m}/\mathrm{s}$ was 18.89%, corresponding to the simulation with six nozzles, 8 $\mathrm{mm}$ of air layer, and a $5^{°}$ injection angle. The lowest gross power saving was 2.62% for the simulation with four nozzles, 4 $\mathrm{mm}$ of air layer, and a ${25}^{°}$ injection angle. The better net power saving was 11.83%, which occurred with the configuration of six nozzles, 6 $\mathrm{mm}$ of air layer, and a $5^{°}$ injection angle. The lowest net power saving achieved was −1.32% with the configuration of four nozzles, 4 $\mathrm{mm}$ of air layer, and a ${25}^{°}$ injection angle. A negative power saving indicates an increase in the total energy consumption when comparing the air injection method with the standard hull.

In summary, the gross power savings for the speed of 1.668 $\mathrm{m}/\mathrm{s}$ showed better results than those of 1.335 $\mathrm{m}/\mathrm{s}$. The efficiency of the method increased with the number of nozzles and air layer thickness but decreased when the injection angle increased. The following discussion compares the results for speeds of 1.335 and 1.668 $\mathrm{m}/\mathrm{s}$.

Analyzing the graph in Figure 5b, with four injection nozzles, the most significant variations in power savings were observed again when decreasing the injection angle from ${25}^{°}$ to ${15}^{°}$. However, the higher variation occurred in the air layer thickness curve corresponding to 4 $\mathrm{mm}$. The lower influence of the injection angle in the gains, presented in the six nozzles and 8 $\mathrm{mm}$ air layer thickness curve, occurred at both speeds of 1.335 $\mathrm{m}/\mathrm{s}$ and 1.668 $\mathrm{m}/\mathrm{s}$ (see Figure 5c and Figure 5d, respectively). The graph in Figure 5f, with eight nozzles, followed the same patterns as for the speed of 1.335 $\mathrm{m}/\mathrm{s}$ (see Figure 5e), but the improvement was greater for the 8 $\mathrm{mm}$ air layer thickness curve. Overall, the analysis based on the number of nozzles indicated that for higher speeds and a higher number of injector nozzles, there was an enhancement in the efficiency of the method.

When evaluating the impact of air layer thickness on power savings, similar to the observations at a speed of 1.335 $\mathrm{m}/\mathrm{s}$, it became evident that an increase in air layer thickness resulted in improvements. However, this improvement was less pronounced at a speed of 1.668 $\mathrm{m}/\mathrm{s}$. Comparing Figure 5c with Figure 5d, it is evident that with six nozzles, the gain achieved at a speed of 1.335 $\mathrm{m}/\mathrm{s}$ was higher than that at a speed of 1.668 $\mathrm{m}/\mathrm{s}$ for air layer thicknesses of 6 and 8 $\mathrm{mm}$. This analysis suggests that at a speed of 1.668 $\mathrm{m}/\mathrm{s}$, the method performed better with a larger number of nozzles. This result aligns with the previous study carried out by [27].

The graphs in Figure 5b,f show that the gains for any air layer thickness curves did not change much as a function of the injection angle. By comparing these with the correspondent graphs for a speed of 1.335 $\mathrm{m}/\mathrm{s}$ (see Figure 5a,e), it suggests that at higher speeds, the injection angle had less influence on the efficiency of the method.

For a speed of 1.668 $\mathrm{m}/\mathrm{s}$, the Figure 5b,d also show that the curve for the air layer thickness of 6 $\mathrm{mm}$ achieved better results, as was observed for the speed of 1.335 $\mathrm{m}/\mathrm{s}$. The explanation is the same: the power savings obtained with the 8 $\mathrm{mm}$ air layer did not compensate for the extra energy costs required to increase the flow rate. However, with eight nozzles, Figure 5f, the most gain in net power saving was found for the air layer thickness of 8 $\mathrm{mm}$.

Comparing Figure 5a,b with Figure 5c,d revealed that as speed increased, the difference between gross power saving and net power saving decreased. This indicated that at higher speeds, the method became more efficient, but this depended on the nozzles’ arrangement since the same was not observed for eight nozzles (Figure 5e,f). Higher ship speed was also shown to enhance gross power savings; however, this benefit was sensitive to the nozzle arrangement and air layer thickness, as it was not observed for configurations with six nozzles and air layer thicknesses of 6 mm and 8 mm. This specific nozzle arrangement, combined with higher airflow rates (resulting from increased air layer thickness and ship speed, Equation (12)), led to more pronounced lateral air leaks, which hindered the uniform coverage of the hull bottom. This effect is discussed in Section 3.3.

In summary, the best gross power saving for a speed of 1.668 $\mathrm{m}/\mathrm{s}$ was 19.13%, corresponding to a n injection configuration with eight nozzles, 8 $\mathrm{mm}$ of air layer, and a $5^{°}$ injection angle. The lowest gross power saving was 8.67% for an injection configuration with four nozzles, 4 $\mathrm{mm}$ of air layer, and a ${25}^{°}$ injection angle. These results indicate that the air injection method performs better at higher hull cruising speeds. The highest net power saving found, for a speed of 1.668 $\mathrm{m}/\mathrm{s}$, was 14.13%, corresponding to the configuration with eight nozzles, 8 $\mathrm{mm}$ of air layer, and a $5^{°}$ injection angle. The lowest net power saving was 6.16%, for the four-nozzle configuration, 4 $\mathrm{mm}$ of air layer, and a ${25}^{°}$ injection angle configuration.

Table 6. Results for total resistance, gross power savings, and net power savings.

		Total Resistance (${\mathit{R}}_{\mathit{T}}$), in N Gross Power Savings (${\mathit{PS}}_{\mathit{gross}}$), in % Net Power Savings (${\mathit{PS}}_{\mathit{net}}$), in %
Velocity [$\mathrm{m}/\mathrm{s}$]		1.335			1.668
Angle [°]		5	15	25	5	15	25
Number of Injections Nozzles	Air Layer Thickness $[mm]$
4	4	17.96 5.77 1.83	18.00 5.56 1.62	18.56 2.62 −1.32	26.24 12.53 10.03	26.50 11.67 9.16	27.40 8.67 6.16
	6	16.98 10.91 5.01	17.00 10.81 4.91	17.64 7.45 1.55	25.66 14.47 10.72	25.94 13.53 9.79	26.44 11.87 8.12
	8	16.50 13.43 5.55	16.64 12.70 4.82	17.06 10.49 2.61	25.30 15.67 10.66	25.50 15.00 10.00	26.10 13.00 8.00
6	4	16.60 12.91 8.97	16.94 11.12 7.18	17.02 10.70 6.76	26.16 12.80 10.30	26.60 11.33 8.83	26.90 10.33 7.83
	6	15.68 17.73 11.83	15.76 17.31 11.41	15.90 16.58 10.68	25.26 15.80 12.05	25.80 14.00 10.25	26.26 12.47 8.72
	8	15.46 18.89 11.01	15.54 18.47 10.59	15.60 18.15 10.27	25.04 16.53 11.53	25.20 16.00 11.00	25.38 15.40 10.40
8	4	16.74 12.17 8.23	16.98 11.12 6.97	17.10 10.28 6.34	26.66 11.13 8.63	27.10 9.67 7.16	27.16 9.47 6.96
	6	15.98 16.16 10.26	16.30 14.48 8.58	16.42 13.85 7.95	25.22 15.93 12.19	25.76 14.13 10.39	25.80 14.00 10.25
	8	15.56 18.36 10.48	15.98 16.16 8.28	16.16 15.22 7.34	24.26 19.13 14.13	24.60 18.00 13.00	24.64 17.87 12.86

shows all total resistance values found in this study, as well as the values for the gross power savings and net power savings.

3.2. Analysis of Variance (ANOVA)

An ANOVA can be used to see if there is a relationship between the air injection parameters and the total resistance. This method tests the equality of three or more populations based on the analysis sample variances. Key values within the ANOVA are the F-ratio and p-value. The F-ratio shows the difference between the variance within the group and the variance between the groups. This value is the mean square of each independent variable divided by the mean square of the residuals, i.e., if the F-ratio value is large, the change in the dependent term has its origin in the independent variable [41]. Another important indicator is the p-value, which shows the probability of the alternative hypothesis being supported when in fact the null hypothesis is true. With low values of the p-value (p-value < 0.001), the null hypothesis can be rejected, confirming the alternative hypothesis [41].

For this study, the null hypothesis (the means of all populations are equal) stated that the total resistance was not dependent on changes in the parameters of the air injection method. The alternative hypothesis stated that the total resistance was dependent on the air injection parameters. RStudios [42] software (Version 4.3.1) was used to carry out the analysis of variance.

Table 7. ANOVA results between the air injection parameters and the total resistance.

	Degrees of Freedom	Sum of Square	Mean of the Sum of Squares	F-Ratio	p-Value
Velocity	1	1165.0	1165.0	9652.35	$2·{10}^{-16}$
Number of Injection nozzles	2	7	3.5	28.83	$7.66·{10}^{-09}$
Air layer thickness	2	19.4	9.7	80.57	$9.32·{10}^{-16}$
Injection angle	2	2.9	1.5	12.17	$5.72·{10}^{-05}$

shows the ANOVA results.

Refuting the null hypothesis, the ANOVA confirmed the relationship between the total resistance and the air injection parameters. Hence, the variations in total resistance did not come from the dispersion of the data read in each simulation.

3.3. Air Injection Flow Patterns

To examine the effects of air injection on the hull and its surroundings, the best and worst configurations were selected for each speed. For a hull speed of 1.335 $\mathrm{m}/\mathrm{s}$, the best configuration included six nozzles, 8 $\mathrm{mm}$ of air layer, and a $5^{°}$ injection angle, while the worst configuration consisted of four nozzles, 4 $\mathrm{mm}$ of air layer, and a ${25}^{°}$ injection angle. For a hull speed of 1.668 $\mathrm{m}/\mathrm{s}$, the best configuration was eight nozzles, 8 $\mathrm{mm}$ of air layer, and a $5^{°}$ injection angle, and the worst configuration was four nozzles, 4 $\mathrm{mm}$ of air layer, and a ${25}^{°}$ injection angle. These analyses were conducted under steady-state conditions for a simulated time of 10,000 s. Figure 6 depicts the hull side and the bottom view of the hull for the best configuration at 1.335 $\mathrm{m}/\mathrm{s}$.

In the side view of the best configuration, Figure 6b, it was observed that the air cover at the DTC rear area reached a volumetric fraction of water ranging from 0.5 to 0.8. On the flat part of the hull’s bottom, the ALDR regime covered the entire region with air. In Figure 6a, the red circle indicates a small air leak near nozzles 1A and 1B. An early air leak occurs when the air released by the nozzle travels directly to the surface without forming an air layer at the bottom or on the side of the hull. At the side midship region, the regime reached the ALDR regime. Downstream, the volumetric fraction of water ranged from 0.3 to 0.6.

Figure 7 shows the side and bottom DTC views for the worst configuration at a velocity of 1.335 $\mathrm{m}/\mathrm{s}$.

Figure 7 displays the side and bottom DTC views for the worst configuration. The worst configuration occurred at a speed of 1.335 $\mathrm{m}/\mathrm{s}$, with four nozzles, 4 $\mathrm{mm}$ of air layer, and a ${25}^{°}$ injection angle. In Figure 7b, the air coverage inconsistency was evident at the bottom of the hull, and the ALDR regime occurred only immediately after air injection by nozzles 1A, 1B, 2A, and 2B. There was no air coverage in the flat central region of the bottom of the hull, and two early air leaks were identified on the side, highlighted by the red circles in Figure 7a. In the stern region, high values of volumetric fraction of water were found, between 0.7 and 0.9.

Figure 8 shows the side and bottom DTC views for the best configuration at a velocity of 1.668 $\mathrm{m}/\mathrm{s}$.

The red circle in Figure 8a indicates an air leak through the side midship region. This is possibly due to the relative proximity of the injection lines, accumulating air and resulting in the early air leakage observed at the ends of the hull. Figure 8b shows the air cover almost completely formed downstream. The low values of the volumetric fraction of water at the flat bottom region indicate that the ALDR regime developed. In the stern region, both the bottom and sides showed a volumetric fraction of water ranging between 0.5 and 0.8. However, an air leak occurred and was larger than the best configuration at a speed of 1.335 $\mathrm{m}/\mathrm{s}$.

Figure 9 shows the side and bottom DTC views for the worst configuration at a velocity of 1.668 $\mathrm{m}/\mathrm{s}$.

The worst configuration occurred at a speed of 1.668 $\mathrm{m}/\mathrm{s}$ with four nozzles, a 4 $\mathrm{mm}$ air layer thickness, and a ${25}^{°}$ injection angle. The same configuration resulted in the worst efficiency at a speed of 1.335 $\mathrm{m}/\mathrm{s}$. Figure 9b shows the inconsistency of the air coverage at the bottom of the hull, with the ALDR regime only developing after air injection by nozzles 1A, 1B, 2A, and 2B.

A significant portion, the central flat area, was not covered by air, with a volumetric fraction of water of 1.0 or close. The red circles in Figure 9a highlight two early air leaks on the side, similar to the ones present in the worst configuration at the speed of 1.335 $\mathrm{m}/\mathrm{s}$. In the stern region, volumetric fractions of water between 0.6 and 0.9 were found.

The two best configurations for velocities of 1.335 and 1.668 $\mathrm{m}/\mathrm{s}$ exhibited a single area of premature air leakage, while the least favorable configurations exhibited two areas of premature air leakage. It can be seen that the highest air coverage at the side and bottom hull region occurred precisely in cases where there were fewer early air leaks, implying better results for the method.

To better understand the relationship between the number of nozzles and air layer thickness on the gross power savings (resistance reduction), a further investigation was conducted. Figure 10 and Figure 11 display airflow patterns and air coverage from side and bottom views, respectively, for the lowest and highest speed. In both figures, comparing sub-figures (a), (c), and (e) shows the effect of the number of nozzles, while comparing (b), (d), and (e) indicates the effect of the air layer thickness on air distribution. It is possible to verify that at both speeds, the air beneath the hull covered a significant portion of the submerged area.

The volumetric fraction of water in the flat area of the hull bottom ranged mostly from 0 to 0.4. However, in configurations with only four nozzles (illustrated in Figure 10a and Figure 11a), some areas at the flat bottom of the hull did not receive air lubrication, leading to a water volume fraction of 1.0. When modifying the air layer thickness while retaining the configuration with eight nozzles, there was little change in the air coverage area; however, the volumetric fraction of water was affected, as demonstrated in Figure 10b,d, as well as Figure 11b,d. Another important finding was that the number of injection nozzles significantly affected the coverage of the air layer at the bottom of the hull. This is evident in Figure 10a,c,e, as well as Figure 11a,c,e. Notably, for configurations with six injector nozzles, as shown in Figure 10c and Figure 11c, one observes that at a speed of 1.335 $\mathrm{m}/\mathrm{s}$, the flat region at the hull bottom exhibited a lower volumetric fraction of water compared to a speed of 1.668 $\mathrm{m}/\mathrm{s}$. This observation may explain why the optimal configuration for the speed of 1.335 $\mathrm{m}/\mathrm{s}$ occurred with six nozzles.

The analysis of the nozzle arrangements showed that a more dispersed air injection across a flat area led to better results. In Figure 11a, when the nozzle arrangements covered 32% of the flat area, the maximum power savings achieved was 10.72%. For 38% of coverage, such as in Figure 11c, the power savings improved to 12.05%, and at 44% of coverage, as in Figure 11e, the best power savings reached 14.13%.

4. Conclusions

This study assessed how the air injection parameters impacted the effectiveness of the air lubrication method. The study utilized a scaled DTC hull to determine the gross power saving and the net power saving in the propulsion system. The method provided a maximum gross power saving of 19.13% (neglecting air compression power) and a net power saving of 14.13%. These values were observed for the best configuration, a speed of 1.668 $\mathrm{m}/\mathrm{s}$, a $5^{°}$ injection angle, an 8 $\mathrm{mm}$ air layer thickness, and eight nozzles. The best and worst configurations resulted in a 16.51% difference in gross power saving. The worst configuration was for a speed of 1.335 $\mathrm{m}/\mathrm{s}$, a ${25}^{°}$ injection angle, a 4 $\mathrm{mm}$ air layer thickness, and four nozzles injecting air.

Regardless of other parameters, the gross power saving improved when the injection angle was reduced (closer to the hull). Specifically, a $5^{°}$ injection angle made the method more effective. This trend was also observed in the net power saving curve. Thus, larger angles reduced the efficiency of the method, because the energy required for air injection remained the same and the reductions were decreased.

The use of eight nozzles yielded the best results. They dispersed the air more evenly at the bottom of the hull, creating a more comprehensive ALDR layer and reducing the risk of premature air leaks. Additionally, using eight nozzles resulted in lower dynamic pressure at the air outlet, leading to higher power savings. It is important to note that the pressure resistance showed similar values when comparing the simulations with and without air injection. This indicates that the air injection method does not significantly affect the distribution of viscous pressure in the DTC hull.

The effectiveness of the method increased as a function of the air layer thickness; it was found in this study that at a speed of 1.335 $\mathrm{m}/\mathrm{s}$, a low velocity, the 6 $\mathrm{mm}$ air layer thickness was the most effective relative to power saving. This happened because the resistance reduction benefit did not justify the extra energy needed to supply the air compression system at 8 $\mathrm{mm}$.

In some cases, the air injection method increased the total power required, such as at a speed of 1.335 $\mathrm{m}/\mathrm{s}$ with four nozzles, a 4 $\mathrm{mm}$ air layer thickness, and a ${25}^{°}$ injection angle, for which a negative net power saving was observed. However, at a speed of 1.668 $\mathrm{m}/\mathrm{s}$, no negative net power savings were found. Each parameter of the air injection method significantly impacted its effectiveness, and understanding the relationships between these parameters enabled a more precise determination of the optimal configuration for further projects.

The findings imply that significant energy savings can be achieved, leading to lower fuel consumption and reduced GHG emissions. Given the air leakage tendencies, studying the influence of the inclination on the air injection in the horizontal plane towards the hull symmetry plane is also intended for future work. Assessing various hull designs to establish a general principle for the arrangement and operation of air injection nozzles for container ships is also a future work intention.