**1. Introduction**

A long-standing problem of inland waterways in Europe is the existence of bottlenecks related to the provision of the required navigation parameters. International agreements oblige individual European states to ensure the navigability of international waterways with required parameters [1]. Ensuring a minimum navigation depth of 250 cm for 300 days a year is particularly important for the Danube. Long-term forecasts of the impact of climate change [2,3] assume a further deterioration of the situation, which is mainly related to the phenomenon of drought. This problem directly and negatively affects the use of inland waterway transport. This limits the navigation period during the year. This also has a direct impact on the economics of operating shipping companies [4]. Another negative is the lower attractiveness, stability, and usability of the transport mode compared to others.

The draft of an inland ship can undoubtedly be included among the critical points that significantly affect its usability during the year. As the design of a ship and determination of the design draft are closely related to the tonnage of the ship, it is natural that the limitation of the draft will result in a reduction in the deadweight capacity of the ship. This is not the way.

Propulsion systems intended for inland or coastal navigation vessels are equipped with small propellers due to low under keel clearance (UKC). Due to their small size, it is necessary to operate at high rotational speeds for sufficient thrust of the propeller. In general, this fact results in relatively low efficiency of such systems (combination of small propeller diameter and high revolutions per minute).

Several possibilities to increase the efficiency of the propulsor are presented in [5]. One of the ways to increase propulsion efficiency is the application of half-submerged double propellers. The efficiency of the propulsion can be increased by increasing the diameter of the propeller, taking into account the restricted draft and the reduction in its rotational speed. By using two counter-rotating semi-submerged propellers, their dimensions can be increased by a factor of approximately 2.5. Another way to increase efficiency is the installation of surface-cutting double propellers. This propulsion system increases propulsion efficiency by about 20–40% compared to conventional drives and does not require an independent steering (rudder) system [5].

Another way to increase the availability of inland waterway transport is to adapt the fleet to current and future navigation conditions. The solution may be the optimization of the hull shape. At limited water conditions, the flow around the hull changes compared to the flow in unrestricted water due to the influence of the bottom and banks of the waterway. This leads to increased backflow, stronger squat effects, and changes in the waveform created by the ship's motion. The lower the UKC, the sooner the backflow occurs. This leads to additional immersion, lowering the water level around the ship, and usually, to an increase in wave resistance (in some cases, it may decrease, see [6]) [7].

Several methods have been proposed to estimate the increased resistance in shallow water navigation [6,8–11]. In addition to resistance, the wake fraction and thrust deduction also change [12,13]. There are also additional effects in shallow water on the propeller compared to sailing at unlimited depth [14].

Optimization studies for wave resistance in shallow water have been published [15,16], but due to changes in the wake field and propulsive efficiency, attention should be paid to thrust force when optimizing for shallow water.

The Rotteveel study addresses the effect of water depth on inland ship rudder optimization. This represents the optimization of propulsion power for different water depths using parametric inland ship stern shape, computational fluid dynamics (CFD), and surrogate modeling. Using Pareto fronts, a compromise is proven: the driving force in shallow water can be reduced at the cost of increasing driving force in unrestricted water and vice versa. The results of the study show that the effects of shallow water on effective wake fraction are significant. In addition to the wake fraction and thrust deduction factors, relative rotational efficiency may also vary due to different behavior in shallow water. In the conclusions of this study, it is recommended to focus on the effects of shallow water on propulsion as resistance [7].

One of the solutions that could increase the throughput of the ship through critical sections is changing the propulsion system. Most inland ships are powered by conventional propellers, usually operating in synchronous dual mode [17]. Changing the propulsion system by distributing the propulsion equipment to other suitable locations may result in better throughput of the ship through critical points with restricted draft. With the traditional conventional stern propulsion method, a natural suction from the propellers occurs. This results in a reduction in UKC. Distribution of the propulsion to other places, or its extension, may result in a decrease in suction, which will ensure a greater margin under the hull [18].

This study describes an investigation process aimed at finding an efficient way to propel inland vessels for navigation at restricted draft conditions. It is based on a typical inland cargo vessel hull with dimensions suitable for the Rhine–Main–Danube waterway, which was used for all calculation cases in the CFD analyses.

The Methods section describes that in order to compare the thrust efficiency, four different symmetrical layouts of ideal propulsors were proposed along the hull, with 2, 4, 6, and 8 propulsion units. As a fifth case, the classical two-propeller stern arrangement for reduced draft was also analyzed. A series of CFD simulations was performed with identical conditions and with the same total pressure-generated force on the propulsors. During the calculation, the mass flow values in front of the plane, at the working plane, and behind the plane of the propulsors were recorded, and subsequently, the results were evaluated using Rankine's theory.

In the Results section, the results from numerous CFD analyses are processed and clearly visualized in the form of various comparative figures. In these figures, the differences in the total generated thrust and absorbed power values under the unrestricted and restricted water depth conditions can be seen.

The last two sections discuss and explain the results and provide conclusions, with possible guidance on how to proceed in further investigations.
