**3. Results and Discussion**

The resulting values of mass flow increase, generated thrust, and absorbed power are shown in Figures 6–11, where their relative values are represented for an easier comparison and evaluation.

In the following comparative line graphs, the resulting total values are shown, where every curve represents a specific hull/propulsion layout. The absolute values of layout A in deep water at a mean speed of 4 m/s have been chosen as the reference unity values to calculate the relative values.

**Figure 6.** Total mass flow rate, deep water (comparison). Source: Authors.

**Figure 8.** Total generated thrust, deep water (comparison). Source: Authors.

**Figure 9.** Total generated thrust, shallow water (comparison). Source: Authors.

**Figure 10.** Total absorbed power, deep water (comparison). Source: Authors.

**Figure 11.** Total absorbed power, shallow water (comparison). Source: Authors.

Rapid changes and some problematic areas can be identified easily in these graphs. In general, a significant fall in performance indicates intensive air intake into the propulsor. The occurrence of such ventilation can be checked visually in the graphical output from the CFD analysis.

In the figures showing the restricted water conditions, a special region can be localized around the sailing speed of 3 m/s. All the curves have a "wave trough" at this area, and this is a hot candidate for a speed value characteristic for this specific hull and water flow field.

It can also be seen that the absence of the propulsion tunnel (and fins) has a much more significant impact on the performance of hull/propulsion layout A (traditional twin-screw) under the restricted sailing conditions.

To show which propulsion locations are prone to have a strong air intake from the free water surface, a different type of figure has been introduced.

The examples in Figures 12 and 13 represent the resulting generated trust and absorbed power values for hull/propulsor layout E (8 propulsors) on a per-propulsor basis. For these two special graphs, the absolute values of the aftmost propulsor at speed 2 m/s in shallow water have been chosen as the reference unity values.

In this example case, the critical propulsor position is No. 1, which is located in the aftship zone of the hull (aftmost side propulsors, see Figure 14). Intensive ventilation of the propulsion is typical for this location also in other layouts. The performance of propulsor No. 4, located in the foreship zone (foremost side propulsors), seems to be quite unstable; in some layouts, air intake is also present.

Both propulsion positions are typically located at the curved (shaped) hull endings, where the water flow speeds up quickly, causing a pressure drop and a wave trough. Therefore, the air intake occurs much easier in these locations than in the straight and long midship area.

The following comparative figures (Figures 15 and 16) show the thrust and power curves of three distributed propulsion layouts (C, D, E) in restricted and unrestricted water depth conditions. The thrust and power of layout A at mean speed 4 m/s in deep water have been used as unity values.

**Figure 13.** Propulsion power, shallow water (layout E). Source: Authors.

**Figure 14.** View of the aftship air intake zone (Layout A, shallow water). Source: Authors.

**Figure 15.** Thrust of distributed systems in deep and shallow water conditions. Source: Authors.

**Figure 16.** Power of distributed systems in deep and shallow water conditions. Source: Authors.

Generally, after comparing the hydrodynamical performance in Figures 6–11, 15 and 16 of ideal propulsors under both water depth conditions, it can be stated that the distributed propulsion systems consisting of 4+ (preferably 6 or 8 or more) units produce noticeably higher thrust effects in shallow water sailing than the traditional aft end layouts. Under restricted conditions, the thrust increase between two distributed layouts with a different number of propulsors is higher in contrast to deep water sailing, where differences in performance are not so significant.

The aim of this investigation was not to evaluate the absolute values of hydrodynamic quantities obtained from CFD analyses, but rather, to find new ways and possible principal solutions for shallow water propulsion systems. Due to the large number of CFD simulations to be performed, some simplifications had to be made on the computational domain in order to keep the computing time at a reasonable level. For the same reasons, the real propulsion units have been represented with ideal propulsors (actuator discs).

However, the accuracy reached in this way by CFD calculations is sufficient only if the relative quantity values are needed for comparison purposes. The resulting main performance parameters of all the examined propulsion layouts have been compared properly, so it was possible to filter out the most promising solutions. The comparison of the results from the restricted and unrestricted water depth conditions has clearly shown the differences in the courses of their graphs.

To determine the effect of mesh quality, a dependence study has been performed for case E in shallow water. For the new analyses, the CFD mesh of the entire computational domain has been refined by halving the average element sizes. With these several times larger amounts of elements, the interval of sailing speeds from 2 to 6 m/s has been analyzed and the results have been compared with the values obtained from the original coarser mesh. The result of this comparison is shown in Figure 17 as percentage difference of the new values from the reference values. The graph shows that in the operating range of the vessel, the maximum difference value is close to 2.5%, which is fully acceptable for comparison purposes.

**Figure 17.** Percentual comparison of the original and the refined mesh results. Source: Authors.

In the technical literature, the expressions "distributed propulsion system" or "multiple propeller ship" refer to ships which are equipped with more than three propulsors (propellers) in general, located in the aftship area and arranged in different geometric patterns. Based on the lack of available publications concerning side-mounted propulsions, it can be assumed that we have given a new meaning to the expression "distributed propulsion".

Similar R&D and experiments have been performed in the past for another reason, i.e., for better longitudinal distribution of thrust power in terms of the propeller load and cavitation. The best-known cases are overlapping propellers and tandem propellers (their modern versions are twin propeller azimuths and pods), which have also been implemented on experimental ships. Overlapping propellers each have their own shaft line, while the tandem system is mounted on a single common shaft. However, all these propulsions are located in a traditional place under the stern part of the ship (aft end layout), which is a significant difference from the distribution along the side of the ship that we have investigated. In addition, these systems tend to appear as integrated propulsion units due to their small distance from each other. Based on the significant differences and the poor documentation of the above listed cases, they have not been mentioned in the work as reference cases or comparative bases.

For a final validation of the working method, it was not possible to make a comparison of our CFD-based results with other published results from earlier investigations either made by computer simulations or by model tests made in towing tanks. No usable publication reporting a similar approach studying longitudinal distribution of propulsion units has been found up to now.

#### **4. Conclusions**

The analysis results have confirmed that shallow water vessels driven by distributed (side-mounted) propulsion systems can be operated efficiently, eliminating the unwelcome side effects of the traditional stern-mounted propeller systems. The preliminary assumptions have been substantiated, at least at the simulation level.

Technical implementation of the propulsors was beyond the scope of this work. Standard propeller-based propulsion units are probably not usable because of their tendency to take in the air from near the free water surface. Rather, different, more suitable propulsors should be employed, or a completely new concept should be developed for this special purpose.

This was the first step of investigations around distributed propulsion systems, bringing purely principal solutions to the problem. In the next step, the most promising hull/propulsion layouts should be further examined at a higher level by performing a series of complex and demanding CFD analyses. The most appropriate are layouts E and D (8 and 6 units), but for shorter ship hulls, layout C (4 units) could be useful as well. In this phase, a new side-mounted type of propulsion unit should be preliminarily designed. The purpose of the third phase should be the validation of the results from the previous two phases. Preferably, a towing tank test should be performed, examining the real hull-propulsion interactions.

The main contribution of this work is that it shows a possible way how the more efficient vessels of the future intended for restricted navigation depth could be developed. Such vessels could also operate on waterways where the ships with a traditional hull and propulsion can no longer do so, either for technical or economic reasons. Further R&D tasks should lead to the development of special propulsors and hulls optimized for their best interaction in restricted but also in unrestricted conditions of navigation.

**Author Contributions:** Conceptualization, L.I., T.K. and M.J.; methodology, L.I. and M.J.; software, L.I.; validation, L.I., T.K. and M.J.; formal analysis, M.J.; investigation, L.I., T.K., M.J. and V.L.; resources, M.J. and V.L.; data curation, T.K.; writing—original draft preparation, L.I.; writing—review and editing, M.J.; visualization, T.K. and V.L.; supervision, T.K. and M.J.; project administration, T.K and M.J.; funding acquisition, T.K. and M.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Institutional research of the Grant system of the Faculty of Operation and Economics of Transport and Communications, University of Zilina. This research is also the result of the Project VEGA No. 1/0128/20: Research on the Economic Efficiency of Variant Transport Modes in the Car Transport in the Slovak Republic with Emphasis on Sustainability and Environmental Impact, Faculty of Operation and Economics of Transport and Communications, University of Zilina, 2020–2022.

**Conflicts of Interest:** The authors declare no conflict of interest.
