Numerical Study on the Influence of Water Depth on Air Layer Drag Reduction

Abstract

Over the years, air lubrication technology has been widely applied to maritime vessels, demonstrating its significant energy-saving and emission-reducing effects. However, the application of this technology in inland waterway transportation faces unique challenges due to the shallower water depths, particularly during low water periods. Under such conditions, the formation of the air layer and its associated drag-reduction effects may undergo alterations. Conducting research on air lubrication technology in shallow water conditions holds great practical significance for promoting its application in inland waterway vessels. Therefore, a numerical study is undertaken to examine the impact of water depth on Air layer Drag Reduction (ALDR) to promote the use of ALDR technology on inland canal boats with shallow water depths. The object was a specific river-sea direct ship model, and a groove was created at the bottom of the model with air injection. At two distinct speeds, numerical simulations were run for four different depths: deep water, moderate water, shallow water, and ultra-shallow water. A comparative examination of the air layer morphology on the ship bottom and drag reduction was conducted to investigate the impact of water depth on ALDR and confirm the viability of using ALDR technology on shallow-water navigation boats. The results indicate that due to the change in the velocity and pressure fields at the bottom of the ship, the efficiency of drag reduction and the form of the air layer on the ship’s bottom are significantly impacted by variations in water depth in restricted waters. However, the total resistance can still be significantly reduced by setting grooves on the hull with air injected in shallow waterways. Reduced frictional resistance no longer predominates the overall resistance reduction in shallow water; the proportion of the decrease in viscous pressure resistance rises and can reach up to 4.8 times the decrease in frictional resistance. The research confirms the application prospects of this technology on inland waterway transport ships.

1. Introduction

“Air lubrication technology” refers to a technique for jetting or cavitating a stable air-liquid interface onto a ship’s surface to greatly reduce frictional resistance [1]. Air lubrication techniques can be classified into two categories: Bubble Drag Reduction (BDR) and Air Layer Drag Reduction (ALDR), depending on the shape of the gas-liquid interface. Numerous research organizations and academics have carried out substantial and in-depth studies on this technology since McCormick (1973) [2] initially achieved a 40% drag reduction impact through experiments on a rotation body in 1973.

There are two types of research on the lubricating of air. One is the study of the shape of air layers, the mechanisms that reduce drag, and the methods for creating and maintaining the air layer. The main research objects were flat plates and rotating bodies. University of Michigan researchers Sanders [3], Elbing [4], and Winkel E S [5] performed a series of experiments on a sizable flat plate measuring 12.9 m × 3.05 m in a water tank with a maximum Reynolds number of 2.2 × 10⁸. The outcomes demonstrated that, on the big flat plate, the gas existed in two states: bubbles and air layers, with a transition zone between them. The drag reduction impact was noticeable once a stable air layer had developed, with a drag reduction rate of over 80% in the air layer region. However, for smooth flat plates, the airflow rate, the plate’s surface roughness, and incoming flow disturbances all had an impact on the development of a stable air layer. It was discovered that adding grooves to the plate’s bottom aided in the creation of a stable air layer and decreased the necessary jet flow rate.

To investigate the viability of air lubrication procedures on real ships, Kodama [6], Takahito [7], and Kawashima [8] carried out a series of air lubrication experiments on a big flat plate measuring 50 m in length and 1m in breadth. A combined jetting process and certain friction sensors were mounted in various locations at the flat plate’s base. According to the experimental findings, the reduction in drag was more pronounced in the leading edge of the large flat. The drag reduction effect was not significantly affected by the combined jetting technique. It is possible to significantly lower local frictional resistance and improve the drag reduction effect by adding side plates to the sides.

In a towing tank, Wu Hao [9] performed air layer drag reduction studies on a sizable flat plate with dimensions of 5.016 m in length, 1 m in breadth, and 0.12 m in thickness. A system for observing the air layer was built at the bottom of the tank, and friction sensors were installed in specified locations on the flat plate. The study looked at how the distribution of the air layer and resistance at the panel’s base was affected by varied water depths, bottom groove arrangements, combination jetting techniques, and variations in air flow rate.

The design of air layer drag reduction techniques, their efficacy, and the interaction between the air layer and ship navigational features are the subject of another type of research that utilizes scaled ship models or actual ships. Its main objective is to advance the use of air lubrication technology in engineering.

Researchers from the Netherlands conducted a series of experiments in water tanks using two ship models with scale ratios of 1:10.4 and 1:20; the experimental content covers various aspects of ship resistance performance, propulsion performance, maneuverability, and motion performance in waves. The results indicated that net energy savings can be achieved by 3–10% in calm water by air injection; at the same time, the maneuverability will also undergo significant changes [10,11].

In a flowing water tank, Matveev [12,13] carried out experimental research on the technology of gas layer extension. The impact of the step on the stability of the air layer was confirmed by putting up a ship model with a step at the bottom for jet injection while varying the jet flow rate and ship attitude. At the same time, it was suggested that proper hull trim adjustment helped maintain the air layer and that there is an ideal trim angle for various model and step size combinations to produce the best air layer.

A 95000DWT bulk carrier was used in a 1:38 scale air lubrication experiment conducted by Ye [14]. The absolute drag reduction rates were 26.99% and 33.79% for the conditions of design displacement and load displacement, respectively, by air jetting grooves into the bottom of the ship. The effective power and main engine power were decreased by 9.65% and 22.16%, respectively, after conversion to the real ship.

Many nations have verified and applied the use of air lubrication technology on actual ships, building on the research done by numerous academics. The Netherlands, Sweden, Japan, and China have among these nations particularly concentrated on low-speed, large-displacement vessels, with a focus on ocean-going ships. In contrast, Russia has made tremendous strides in the study and use of air lubrication technology in high-speed bubble crafts.

The Netherlands-based company DKgroup [15] has created an air cavity system for actual ships that has produced energy savings of 5–9% on container ships and 10–15% on oil tankers and bulk carriers. The renowned Swedish shipowner Stena Bulk [16] carried out the “Air Max Project” in cooperation with Chalmers University of Technology in Gothenburg and the SSPA Towing Tank. They built a 35-ton test vessel for air layer drag reduction as part of this project and performed experiments on it. The air layer efficiently covered 30% of the ship’s wetted surface area during the testing, resulting in a maximum drag reduction rate of up to 25%.

In 2010, the Mitsubishi Heavy Industries Group tested the micro-bubble air lubrication system in real ship conditions on the NYK-subsidiary and NYK-Hinode modular transport vessels. According to these experiments, the net energy-saving benefit could only be as high as 12% [17].

The Krelov Shipbuilding Research Institute of Russia has been conducting research on aerodynamic drag reduction technology since 1961 and has conducted theoretical and experimental research on various ship types, such as large transport ships, high-speed boats, planing boats, catamarans, and multibody ships. In the field of high-speed boats, since 1981, more than 70 fast boats have successfully adopted aerodynamic drag reduction technology, with maximum speeds of over 30 knots and a maximum speed of 52 knots [18,19].

Inland water transportation makes up a substantial percentage of the shipping sector in addition to sea transportation. The motion response of inland transport ships is relatively low due to the low wind and wave activity in the inland waterway, which is beneficial for the coverage and stability of the air layer. However, the waterway’s width and depth are substantially reduced due to the limited water depth and channel width, particularly during times of low water levels. Shallow waters have several negative navigational effects, which raise the possibility of collision and grounding. For ships using air lubrication, this will also result in changes to the ship’s flow field, which will have an impact on the shape of the air layers and the efficiency of drag reduction.

Jebin Samuvel T [20] studied air lubrication in shallow water circumstances utilizing a 2D flat plate to address this problem. The research uncovered certain trends about how shallow water affects the efficiency of air cushion drag reduction. Mateev [21] also carried out a comparative study on a planning craft with discontinuous steps intended for shallow water navigation, examining the variations in total resistance, trim angle, and heave and pitch motions with respect to the vessel’s speed in both deep and shallow water conditions. In terms of hull design and drag reduction techniques, the research subject, however, differs from inland waterway cargo vessels.

With the development of computational fluid dynamics(CFD) technology, numerical simulation has become an effective means of studying air lubrication technology. FLUENT 18.0 [22], OpenFOAM 1.2 [23], and Star ccm+ 13.02 [21] are currently mainstream numerical simulation tools. Get benefit from the advantages in grid generation technology, batch processing, and macro customization, the application of Star ccm+ in computational fluid dynamics simulation has gradually increased in recent years. In this software, the gas-liquid two-phase flow adopts a fluid volume model (VOF), and a high-resolution interface capture method (HRIC) is proposed for capturing gas and liquid interfaces. This method can generate a suitable method for tracking sharp interfaces when simulating the convective transport of immiscible fluid components. To capture a more accurate gas-liquid interface in a more accurate format [24]

Using a river-sea direct ship model with a groove at the bottom as the subject, a numerical study was conducted to examine the effects of water depth on the air layer morphology of the ship bottom and the effectiveness of drag reduction. This study was based on the validation of numerical calculation methods through water tank experiments. The findings of this study have important ramifications for how ALDR technology is used in inland waterway cargo vessels.

2. Materials and Methods

2.1. The Model

A transport vessel operating in shallow water-restricted waterways was the subject of the research. The vessel model was a flat-bottomed barge with a length-to-width ratio of 5.04 and a width-to-draft ratio of 3.82. It had a square coefficient of 0.846. A 1:10 scale was used to reduce the size of the model. The model’s primary dimensions characteristics are presented in

Table 1. Principal scale parameters of the simulated model.

Parameter	Value	Unit	Parameter	Value	Unit
P	2.195	t	Tf	0.38	m
Loa	6.28	m	Ta	0.30	m
LWL	6.245	m	Tm	0.34	m
B	1.24	m	LCG	3.140	m

, and the model’s three-dimensional view is shown in Figure 1. In Table 1, P represents the displacement, L_oa refers to the overall length of the hull, L_WL represents the waterline length (6.245 m in this case), and B denotes the ship’s width. In a static positive buoyancy state, the model had a forward draft (T_f) of 0.38 m, an aft draft (T_a) of 0.30 m, an average draft (T_m) of 0.34 m, and a center of gravity distance from the vertical line at the stern of 3.14 m (L_cg).

A bottom groove measuring 30 mm in depth was set under the model, which is shown in Figure 2. Figure 3 shows the 3D diagram of the groove and the distribution of the air vents. Nine air vents with a diameter of 10 mm were located in the arc area at the head of the grooves, and the total area of the air inlet is 706.85 mm².

2.2. CFD Methods

2.2.1. Governing Equations Viscous Flow

In this study, the ship model’s resistance was simulated using the RANS codes of a commercially available software, Star ccm+ 16.02. The RANS equation serves as the fundamental equation governing the kinematic and hydrodynamic aspects of viscous flow, encompassing primarily the continuity equation and momentum equation. The explicit formulations of these two equations are presented below:

$$\frac{\partial }{\partial t}\left({\rho }_m\right)+\nabla \cdot \left({\rho }_m{v}_m\right)=0$$

(1)

$$\frac{\partial }{\partial t}\left(\rho u_i\right)+\frac{\partial }{\partial x_j}\left(\rho u_i{u}_j\right)=-\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}\left(-\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right)+\rho f_i+\frac{\partial }{\partial x_j}{\left[\mu \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)-\frac{2}{3}\mu \frac{\partial u_l}{\partial x_l}{\delta }_{ij}\right]}_i$$

(2)

where ρ is the fluid density, μ is the fluid viscosity, p is the static pressure, f_i is the mass force at the unit, δ_ij is the unit tensors, $u_i^{\prime }$ is the pulse of u_i, and u_i is the velocity component of x direction, respectively.

2.2.2. Turbulence Model

The current study adopts the RNG k-ε model as the turbulence model [25]. The equations for turbulent kinetic energy and volume fraction for air and water are delineated below:

$$\left.\frac{\partial }{\partial t}(\rho k)+\frac{\partial }{\partial x_i}\left(\rho k{u}_i\right)=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right)\right]+P_k-\rho \epsilon$$

(3)

$$\frac{\partial }{\partial t}(\rho \epsilon )+\frac{\partial }{\partial x_i}\left(\rho \epsilon u_i\right)=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]+C_{\epsilon 1}\frac{\epsilon }{k}{P}_k-C_{\epsilon 2}\rho \frac{{\epsilon }^2}{k}-R_{\epsilon }$$

(4)

where ${\mu }_t=\rho C_{\mu }{k}^2/\epsilon$, $P_k={\mu }_t{S}^2$, $S=\sqrt{2}{S}_{ij}$, $S_{ij}=0.5\left(\partial u_i/\partial x_j+\partial u_j/\partial x_i\right)$, $R_{\epsilon }=C_{\mu }\rho {\eta }^3\left(1-\eta /{\eta }_0\right){\epsilon }^2/k\left(1+\beta {\eta }^3\right)$, $\eta =Sk/\epsilon$, $C_{\mu }=0.0845$, ${\eta }_0=4.38$, β = 0.012, $C_{\epsilon 1}=1.42$, $C_{\epsilon 2}=1.68$, turbulent number for k and ε are ${\delta }_k=1.39$, ${\delta }_{\epsilon }=1.3$.

2.2.3. VOF Model

The two-phase flow of gas and liquid adopts the Volume of Fluid (VOF) model, and its governing equations are shown in Equations (5)~(8). The mass conservation equation is given by:

$$\frac{\partial }{\partial t}{\displaystyle {\int }_V{a}_i}dV+{\displaystyle {\oint }_A{a}_i}v\cdot d\alpha ={\displaystyle {\int }_V\left(S_{a_i}-\frac{a_i}{{\rho }_i}\frac{D{\rho }_i}{Dt}\right)}dV-{\displaystyle {\int }_V\frac{1}{{\rho }_i}}\nabla \left(a_i{\rho }_i{v}_{d,i}\right)dV$$

(5)

where α is the surface area vector, ν is the mixture (mass-averaged) velocity, ν_d,i is the diffusion velocity, S_ai represents the custom source term for phase i, and D_ρi/D_t denotes the Lagrangian derivative with respect to phase density ρ_i.

The total mass conservation equation for all phases is given by Equation (6):

$$\begin{array}{cc}\frac{\partial }{\partial t}{\displaystyle {\int }_V\rho }vdV+{\displaystyle {\oint }_A\rho }v\otimes v\cdot d\alpha & =-{\displaystyle {\oint }_{A}p}Id\alpha +{\displaystyle {\oint }_{A}T}\cdot d\alpha +{\displaystyle {\int }_V\rho }gdV\\ & +{\displaystyle {\int }_V{f}_b}dV-{\displaystyle \sum _i{\displaystyle {\int }_V{a}_i}}{\rho }_i{v}_{d,i}\otimes v_{d,i}\cdot d\alpha +{\displaystyle {\int }_V{S}_i^a}dV\end{array}$$

(6)

In Equation (6), p represents pressure, I is the unit tensor, T is the stress tensor, f_b is the volume force vector, and $S_i^a$ represents the phase momentum source term.

Equation (7) represents the energy equation:

$$\begin{array}{cc}\frac{\partial }{\partial t}{\displaystyle {\int }_V\rho }EdV+{\displaystyle {\oint }_A\left[\rho Hv+{\displaystyle \sum _i{\displaystyle {\int }_V{a}_i}}{\rho }_i{v}_{d,i}{H}_i{v}_{d,i}\right]}d\alpha & =-{\displaystyle {\oint }_A{\dot{q}}^{″}}\cdot d\alpha +{\displaystyle {\oint }_{A}T}\cdot vd\alpha \\ & +{\displaystyle {\int }_V{S}_E}dV+{\displaystyle {\int }_V{f}_b}\cdot vdV\end{array}$$

(7)

where E is the total energy, H is the total enthalpy, q″ is the heat flux vector, and S_E represents the custom energy source term.

2.2.4. HRIC Model

The HRIC scheme is primarily used for simulating the convective transport of immiscible fluid components, thus forming a scheme suitable for tracking sharp interfaces. The foundation of this scheme is the normalized variable diagram (NVD) method. In the diagram, ${\alpha }_D$, ${\alpha }_A$, and ${\alpha }_U$ represent the volume fractions of the donor cell, acceptor cell, and upwind cell, respectively, while ${\alpha }_f$ is the volume fraction at the control volume boundary, calculated by the expression:

$${\alpha }_f=(1-{\tilde{\beta }}_f){\alpha }_D+{\tilde{\beta }}_f{\alpha }_A$$

(8)

where:

$${\tilde{\beta }}_f=\frac{{\tilde{\alpha }}_f-{\tilde{\alpha }}_D}{1-{\tilde{\alpha }}_D},{\tilde{\alpha }}_f=\frac{{\alpha }_f-{\alpha }_U}{{\alpha }_A-{\alpha }_U},{\tilde{\alpha }}_D=\frac{{\alpha }_D-{\alpha }_U}{{\alpha }_A-{\alpha }_U}$$

(9)

The estimation of the normalized volume fraction ${\tilde{\alpha }}_f$ is conducted using a mixed upwind-downwind scheme.

$${\tilde{\alpha }}_f=\left\{\begin{array}{ll}{\tilde{\alpha }}_D& {\tilde{\alpha }}_D<0,{\tilde{\alpha }}_D\ge 1\\ 2{\tilde{\alpha }}_D& 0\le {\tilde{\alpha }}_D<0.5\\ 1& 0.5\le {\tilde{\alpha }}_D<1\end{array}\right.$$

(10)

The fluid has a density of 997.56 kg/m³ and a dynamic viscosity of 8.8871 × 10⁻⁴ Pa/s of fresh water at a temperature of 15 °C. The starting computational state is intended to simulate a static, non-jetting condition with a fixed trim and ship hull sinkage due to the minor attitude changes and small wave amplitudes experienced by inland waterway vessels operating in shallow water circumstances (

Table 2. The hull attitude at different sailing speeds.

Fr	Center of Gravity Heave (mm)	Trim Angle (°)
0.034	−0.78	−0.78
0.045	−2.26	−2.26
0.057	−3.71	−3.71
0.068	−5.07	−5.07
0.079	−6.74	−6.74
0.091	−9.57	−9.57

). In the computational domain, the overlapping grid approach is used, and the reference pressure of the flow field is set to zero. The effects of jetting-induced hull attitude fluctuations, free surface wave generation, and gravity are all disregarded. The gas-liquid-free surface is captured using the Volume of Fluid (VOF) model, and the interface is captured using the High-Resolution Interface Capturing (HRIC) technique. The computational parameters are set as follows: a lower Courant-Friedrichs-Lewy number (CFL_l = 0.8), an upper Courant-Friedrichs-Lewy number (CFL_u = 1), a sharpening factor of 0.5, an angle factor of 0, and a time step of 0.002 s.

2.2.5. Computational Domain and Meshing

The computational domain is shown in Figure 4 with a water depth of 0.4 m in shallow water which corresponds to a depth-to-draft ratio of 1.18, and a water depth of one times the ship length in deep water. The outflow portion is set at 2.5 times the ship length, whereas the inflow section and the ship’s lateral side are taken to be one times the ship length. The maximum and minimum grid sizes for the ship’s surface are specified at 10 mm and 5 mm, respectively. On the ship’s bottom surface, a boundary layer grid is applied with a 1 mm initial grid thickness and five layers total. With a growth rate of 1.25, the grid’s thickness increases perpendicular to the ship’s surface. Additionally, a grid refinement with a grid size ranging from 5 mm to 10 mm is used in the region between the ship’s bottom and the channel bed. The grid size in the wake zone is typically 10 mm, and the refined region’s length along the flow direction is set to 0.5 times the ship’s overall length (L_oa). The computational grids and refinement in critical areas are illustrated in Figure 5.

3. Model Test Validation

3.1. Grid Independence

By modifying the grid size while keeping the same grid structure, various grid refinement systems were created to examine the impact of grid size variation on computational output. The number of grids varied from 1.4 million to 6 million in deep water settings and from 2.6 million to 7.9 million grids in shallow water (h/T_m = 1.18) conditions. Without jetting, simulations were run for both the deep water and shallow water conditions. Investigations were done into how varying grid quantities affected resistance. In both deep water and shallow water situations,

Table 3. The effect of the mesh number on the model resistance.

No.	Deep Water			Shallow Water (h/Tm = 1.18)
	Number of Grids (Million)	Total Resistance (N)	Change Rate	Number of Grids (Million)	Total Resistance (N)	Change Rate
1	1.4	18.16	/	2.6	41.36	/
2	1.8	17.70	2.54%	3	40.07	3.12%
3	1.9	17.60	0.56%	3.2	39.86	0.51%
4	2.2	17.51	0.50%	3.7	39.69	0.43%
5	3.5	17.48	0.18%	5.1	39.58	0.27%
6	6	17.46	0.11%	7.9	39.53	0.14%

shows the fluctuation in model resistance about the number of grids with a model velocity (V_m) of 0.878 m/s (Froude number F_r = 0.113). The change rate listed in the table shows how much the resistance value has changed concerning the prior grid layout.

Table 3 shows that when the number of computing grids rises, the computed resistance eventually tends to stabilize in both deep and shallow water. When the total number of grids reaches 2.2 million or greater under deep water circumstances, continued grid expansion leads to a resistance change rate of less than 0.5%. Similar to this, in extremely shallow water depths, the resistance begins to stabilize and shows a variation rate of less than 0.43% concerning the grid amount when the number of grids exceeds about 3.7 million. As a result, grid systems with 2.2 million and 3.7 million grids, respectively, are chosen for further research into situations involving deep water and shallow water.

3.2. Verification by Experiment

At Wuhan University of Technology, model tests were carried out in the towing tank to verify the correctness of the numerical calculations. The towing tank has a total length of 132 m and a width of 10.8 m; the depth can be adjusted as needed, and the maximum depth is 2 m. The experimental model and the computer model were identical. The water depth for the deep water state experiment was set at 1.8 m following the actual circumstances of the towing tank, which corresponds to a depth-to-draft ratio of 5.29 and satisfies the deep water navigation standard established by The World Association for Waterborne Transport Infrastructure (PIANC), Which states that the shallow water effect can be disregarded when the depth to ratio is greater than 3.0 [26].

The experimental water depth for the shallow water state was chosen at 0.4 m, or a depth-to-draft ratio of 1.18, to ensure uniformity between the experimental and computational circumstances. Model resistance tests were carried out at various speeds and water depths while taking into account the two typical states of fully saturated and non-saturated jetting. The approach outlined in the reference [14] was used to determine the saturated airflow rate. Figure 6 presents a photograph of the experimental ship model. The pictures of the model tank experiment are shown in Figure 7. Figure 7a is the schematic diagram of the installation of the model for the tank experiment, and Figure 7b is the schematic diagram for the installation of the jet devices.

For non-jet and fully saturated jet situations under two distinct water depth scenarios,

Table 4. Comparison of simulated and experimental values of resistance under different jetting conditions in deep water.

No.	Fr	Resistance for Non-Jet (N)			Resistance for the Fully Saturated Jet (N)
		Simulated	Experimental	Error	Simulated	Experimental	Error
1	0.045	3.765	3.637	3.52%	2.894	2.83	−2.26%
2	0.057	5.080	5.224	−2.76%	3.838	3.98	3.57%
3	0.068	7.854	7.537	4.21%	5.842	5.59	−4.51%
4	0.079	10.383	10.101	2.79%	7.801	7.29	−7.01%
5	0.091	12.160	12.691	−4.18%	9.580	9.02	−6.21%

and

Table 5. Comparison of simulated and experimental values of resistance under different jetting conditions in shallow water.

No.	Fr	Resistance for Non-Jet (N)			Resistance for the Fully Saturated Jet (N)
		Simulated	Experimental	Error	Simulated	Experimental	Error
1	0.045	5.493	5.351	2.66%	3.984	4.001	0.42%
2	0.057	9.293	9.152	1.54%	6.541	6.123	−6.83%
3	0.068	14.330	14.015	2.25%	9.842	9.741	−1.04%
4	0.079	19.215	20.124	−4.52%	13.854	14.588	5.03%
5	0.091	25.047	24.538	2.07%	17.954	17.299	−3.79%

compare the simulated and experimental values of the total model resistance. As seen in Table 4 and Table 5, the computed values of the total model resistance for the non-jet situation are reasonably close to the experimental findings, with the greatest error being within 5%. The computed values of the total model resistance for the jet condition differ from the experimental values less than those for the non-jet condition, with some examples showing errors of more than 6%. The inability of the model testing to accurately regulate the jet airflow rate, which might vary due to fluctuations in different speeds and residual pressure in the air cylinder, may be the cause of this mismatch.

4. Analysis of the Influence of Water Depth on Air Layer Drag Reduction

Numerical research was done on the target ship model to examine the effects of different water depths on the air layer morphology and efficiency of drag reduction. The World Association for Waterborne Transport Infrastructure (PIANC) guidelines were followed in the selection of water depth variations for this study. Following the recommendations, water depths classified as deep water have a depth-to-draft ratio greater than 3.0, moderately deep water has a ratio between 1.5 and 3.0, shallow water has a ratio between 1.2 and 1.5, and extremely shallow water has a ratio of less than 1.2. The navigational water depth conditions for this investigation, as shown in

Table 6. List of calculation conditions for water depth variation.

No.	h (m)	h/T	Category
1	1.36	4	deep water
2	0.765	2.25	moderately deep water
3	0.459	1.35	shallow water
4	0.4	1.18	extremely shallow water

, were formed based on these categories, with the appropriate water depths being selected within each depth-to-draft ratio range.

4.1. Effect on Air Layer Morphology

For saturated jetting conditions at two different model speeds, F_r = 0.057 and F_r = 0.091, F_r is the velocity Froude number:

$$F_r=\frac{V}{\sqrt{gl}}$$

(11)

where V is speed, g is the gravitational acceleration, and l is the characteristic length of the object.

Figure 8 and Figure 9 show the effects of water depth change on the air layer morphology in the groove and the gas overflow at the stern. The left side of the figure is the nephogram of the gas layer thickness distribution, and the darker the color, the larger the gas layer thickness. The figure on the right is a comparison of the composition of the gas phase and the liquid phase, and the darker the color, the more the gas phase content. The results show that variations in water depth have an impact on air layer morphology in the groove and how the gas overflows from the stern. However, there is no discernible air layer fragmentation or new overflow occurrences, demonstrating that the groove keeps the air layer stable and contained until it reaches the extremely shallow water depth. The groove may still be able to sustain a stable, enclosed air layer, according to this evidence. Gas overflow reduces from both sides of the stern and increases from the mid-profile to the rear as the water depth becomes shallower. The gas generally overflows from the mid-profile to the rear when the water depth reaches the extremely shallow water depth (h/T = 1.18), and localized fragmentation takes place in the middle region of the air layer. This shows that the air layer in the bottom groove is substantially more disturbed by the exceedingly shallow water depth.

The air layer fluctuation and overflow in the groove along the longitudinal section are compared in Figure 10 from extremely deep to extremely shallow water depth conditions. Figure 10 shows that the number of air layer fluctuation cycles decreases as the water depth becomes shallower, which was caused by the increase in the flow velocity near the hull due to shallow water effects.

In terms of air layer morphology and coverage, in the case of extremely deep water, the air layer thickness is considerable, and the trough points of the air layer coincide with the lowest point of the stepped bottom of the groove while the peak points remain somewhat apart from the groove bottom. Without fragmentation or flow overflow from the rear end throughout the longitudinal section, the air layer creates a well-sealed effect against the hull’s incline at the back of the groove.

Under the interference of the bottom of the groove, however, the air layer fluctuation pattern in the groove was changed relative to when the water depth was exceptionally shallow (h/T = 1.18). The distance between the peak points and the bottom of the groove is dramatically shortened toward the front region of the groove, where nonlinear fluctuations are significantly enhanced. Localized air layer fragmentation results from local contact between the peak points and the groove’s bottom. Gas overflows from the back of the groove along the longitudinal section as a result of the air layer failing to form a closed effect with the hull’s inclined slope there.

4.2. Effect on the Drag Reduction

The effects of channel depth variation on drag reduction for two distinct model speeds—F_r = 0.057 and F_r = 0.091—under saturated jetting circumstances are shown in

Table 7. Effect of water depth on total resistance reduction (F_r = 0.057).

h/T	Rt (N)			Resistance Reduction (N)		Reduction Rates (%)
	No Groove	Groove No Jetting	Groove Saturated Jetting	Relative	Absolute	Relative	Absolute
1.18	12.33	14.33	9.84	4.48	2.49	31.33%	20.19%
1.35	10.34	11.30	7.63	3.67	2.71	32.48%	26.21%
2.25	7.17	7.71	5.60	2.11	1.57	27.37%	21.90%
4.0	5.90	6.47	5.01	1.45	0.88	22.57%	15.08%

and

Table 8. Effect of water depth on total resistance reduction (F_r = 0.091).

h/T	Rt (N)			Resistance Reduction (N)		Reduction Rates (%)
	No Groove	Groove No Jetting	Groove Saturated Jetting	Relative	Absolute	Relative	Absolute
1.18	29.08	34.04	22.25	11.79	6.84	34.64%	23.52%
1.35	24.28	26.30	17.41	8.88	6.86	33.76%	28.25%
2.25	16.85	18.27	10.11	8.16	6.75	44.66%	40.06%
4.0	13.89	15.37	10.08	5.28	3.81	34.35%	27.43%

. In the Tables, “No Groove” denotes a model without a bottom groove, “Groove No Jetting” denotes a model with a bottom groove but no air injection, and “Groove Saturated Jetting” denotes a model with a bottom groove and jetting with the saturated airflow rate.

In Table 7 and Table 8, “Relative Reduction” indicates a reduction in resistance when comparing the resistance of “Groove Saturated Jetting” with “Groove No Jetting”, whereas “Absolute Reduction” indicates a reduction in resistance when comparing the resistance of “Groove Saturated Jetting” with “No Groove”. The appropriate ratios of the reduction in resistance are shown by the relative drag reduction rate and absolute drag reduction rate.

As can be seen from Table 7 and Table 8, the overall model resistance rises for all scenarios of “No Groove”, “Groove No Jetting”, and “Groove Saturated Jetting” as the water depth decreases. In comparison to the situation without jetting, the decrease in total resistance brought on by jetting rises initially and then falls as the water depth lowers. The absolute drag reduction rate and relative drag reduction rate both show this pattern.

Depending on the model speed, different water depths correspond to the best drag reduction efficacy. If F_r is 0.057, the depth-to-draft ratio at which the drag reduction is most effective is 1.35; if F_r is 0.091, the ratio at which the drag reduction is most effective is 2.25. However, the absolute and relative drag reduction rates are greater than 30% and 20%, respectively, even in extremely shallow water circumstances. This suggests that even in shallow and extremely shallow water, the jetting drag reduction technique still has a noticeable drag reduction effect.

The impacts of bottom groove jetting on frictional resistance under various water depths are provided in

Table 9. Effect of water depth on friction resistance reduction (F_r = 0.057).

h/T	Rt (N)			Resistance Reduction (N)		Reduction Rates (%)
	No Groove	Groove No Jetting	Groove Saturated Jetting	Relative	Absolute	Relative	Absolute
1.18	4.32	3.98	3.21	0.77	1.11	19.35%	25.69%
1.35	4.49	4.27	3.20	1.07	1.28	25.06%	28.73%
2.25	3.99	3.92	2.93	0.99	1.06	25.26%	26.57%
4.0	3.72	3.68	2.78	0.90	0.93	24.46%	25.27%

and

Table 10. Effect of water depth on friction resistance reduction (F_r = 0.091).

h/T	Rt (N)			Resistance Reduction (N)		Reduction Rates (%)
	No Groove	Groove No Jetting	Groove Saturated Jetting	Relative	Absolute	Relative	Absolute
1.18	10.01	9.33	7.17	2.15	2.83	23.04%	28.27%
1.35	10.41	9.92	6.93	2.99	3.48	30.14%	33.43%
2.25	9.19	9.06	6.39	2.68	2.80	29.58%	30.47%
4.0	8.56	8.52	6.02	2.50	2.54	29.34%	29.67%

, which help to further study the variation of various resistance components. The following observations can be drawn from the tables: (1) The model with a bottom groove but no jetting displays a decrease in frictional resistance in comparison to the “No Groove” example. (2) When there is no jetting, the frictional resistance rises as the water depth drops. However, rather than increasing, frictional resistance reduces when the water depth reaches the extremely shallow water condition (h/T = 1.18). (3) When comparing the situations of “Groove No Jetting” and “Groove Jetting”, it can be seen that the relative decrease in frictional resistance increases as the water depth decreases. However, the magnitude of the relative reduction in frictional resistance starts to diminish when the water depth reaches the extremely shallow water state. This pattern matches the pattern seen in non-jetting cases. It may be caused by the poorer coverage effect of the air layer in extremely shallow water conditions.

The effects of bottom groove jetting on viscous pressure resistance at various water depths are shown in

Table 11. Effect of water depth on pressure resistance reduction (F_r = 0.057).

h/T	Rt (N)			Resistance Reduction (N)		Reduction Rates (%)
	No Groove	Groove No Jetting	Groove Saturated Jetting	Relative	Absolute	Relative	Absolute
1.18	8.01	10.35	6.63	3.72	1.38	35.94%	17.23%
1.35	5.85	7.03	4.43	2.6	1.42	36.98%	24.27%
2.25	3.18	3.79	2.67	1.12	0.51	29.55%	16.04%
4	2.18	2.79	2.23	0.56	−0.05	20.07%	−2.29%

and

Table 12. Effect of water depth on pressure resistance reduction (F_r = 0.091).

h/T	Rt (N)			Resistance Reduction (N)		Reduction Rates (%)
	No Groove	Groove No Jetting	Groove Saturated Jetting	Relative	Absolute	Relative	Absolute
1.18	19.07	24.71	15.08	9.63	3.99	38.97%	20.92%
1.35	13.87	16.38	10.48	5.9	3.39	36.02%	24.44%
2.25	7.66	9.21	3.72	5.49	3.94	59.61%	51.44%
4	5.33	6.85	4.06	2.79	1.27	40.73%	23.83%

. The following conclusions can be drawn from the tables: (1) Viscous pressure resistance increases significantly as water depth decreases, suggesting that the increase in resistance caused by shallow water is mainly attributed to viscous pressure resistance. (2) By contrasting the examples of “Groove No Jetting” and “Groove Jetting”, it can be seen that jetting not only lowers frictional resistance but also lowers viscous pressure resistance. (3) As the water depth falls, the reduction in viscous pressure resistance accounts for a greater share of the overall resistance reduction than the reduction in frictional resistance. The ratio may rise noticeably and reach a maximum of 4.8 times the decrease in frictional resistance (F_r = 0.057, h/T = 1.18).

Figure 11 and Figure 12 show the pressure distribution on the ship model’s surface in jetting and non-jetting states at model speeds of F_r = 0.057 and F_r = 0.091 under various situations. To facilitate comparison, the pressure range displayed is 0 to 3750 Pa.

It is possible to see that the pressure at the bottom of the ship gradually decreases as the water depth becomes shallower. This is because the water flow velocity at the bottom of the ship increases in shallow water conditions, but the sinking of the ship was constrained. There are not many differences between the pressure distribution around the ship’s surface before and after jetting in deep water when comparing the pressure contour maps under different water depth situations. However, jetting causes major changes in the pressure distribution surrounding the ship’s surface in shallow water. The bow pressure decreases, and the stern pressure increases and this trend intensifies as the water depth becomes shallower, what is the main source of the significant reduction in viscous pressure resistance in shallow water.

5. Conclusions

This study used numerical analysis to investigate the effects of changing water depth on a full-formed low-speed ship with a bottom groove at various speeds. The resulting insightful conclusions are as follows:

(1)

The use of a bottom groove in combination with jetting can significantly reduce drag under various water depth circumstances. Both the relative and absolute drag reduction rates exceeded 30% and 20%, even in extremely shallow water conditions, demonstrating the tremendous drag reduction potential of jetting in both shallow and extremely shallow waters.

(2)

Due to the influence of shallow water effects, the velocity and pressure fields at the bottom of the ship have undergone significant changes; therefore, the efficiency of drag reduction and the form of the air layer on the ship’s bottom are significantly impacted by variations in water depth in restricted waters. Lower water depths alter the air layer fluctuation and overflow patterns, which has an unfavorable impact on the stern’s ordered gas overflow. These modifications do not, however, affect how stable the air layer is inside the groove.

(3)

As the water depth becomes shallower, the air layer’s ability to reduce drag exhibits an increasing-then-decreasing trend, demonstrating that there is an ideal drag reduction depth. For various speeds, there are several ideal depths for reducing drag.

(4)

Reduced frictional resistance no longer predominates the overall resistance reduction in shallow water circumstances. Opening a groove with jetting causes noticeable changes in the pressure field surrounding the hull as the water depth becomes shallower. In the overall resistance reduction, the proportion of the decrease in viscous pressure resistance rises and can reach up to 4.8 times the decrease in frictional resistance.