Towards Climate Resilient Inland Waterway Vessel Design: Concept of Distributed Thrust for Shallow Water Conditions

16th International Symposium on Practical Design of Ships and Other Floating Structures PRADS 2025 Ann Arbor, MI, USA, October 19th-23rd 2025 Towards Climate Resilient Inland Waterway Vessel Design: Concept of Distributed Thrust for Shallow Water Conditions Richmond Anku1,*, Jeroen Pruyn 1,2and Cornel Thill1 1Delft University of Technology, Delft, the Netherlands 2CoE HRTech, Maritime Innovation, Rotterdam University of Applied Science, Rotterdam, the Netherlands Abstract. Inland waterway vessels are critical to the hinterland transportation network, offering an environmentally friendly alternative to road and rail transport. However, climate change poses significant challenges, such as fluctuating water levels and extreme shallow water conditions that lead to increased resistance and reduced propulsive efficiency. These conditions necessitate innovative design and operational strategies to ensure the efficiency and sustainability of propulsion systems. Given the increase in resistance and risk of propeller emergence in shallow water conditions, this study explores the development of climate-resilient inland vessels, by implementing the distributed thrust concept, where multiple smaller propellers replace conventional single relatively large units, offering superior maneuverability, propeller load distribution, and adaptability to varying water depths and conditions. Utilising stateof-the-art resistance approximation and a robust optimisation method, this research proposes a novel shallow-water model that enables optimal configuration of propeller size, number, and placement, considering key performance metrics such as thrust efficiency and ventilation mitigation, contributing to sustainable inland waterway transportation. Results from a case study demonstrate that the distributed propulsion system can effectively shift the operational threshold for propulsion, extending the navigational capabilities and performance in water depths where conventional design would face limitations. The findings highlight the potential of integrating distributed propulsion with advanced optimisation techniques to address climate-induced challenges while ensuring operational reliability. Key words: shallow water, inland waterway vessel, propulsion, climateresilience, robust optimisation *Correspondence to: r[email protected] 1 1. Introduction Waterborne transport plays a crucial role in hinterland transportation, offering a more environmentally sustainable and cost-effective alternative compared to other transport modalities. However, this competitive advantage is increasingly threatened by extended periods of droughts and floods that result in both low and high water levels [1]. Inland vessels operate in environments already constrained by water depth, channel width, and the presence of bridges and locks, making navigation a key factor in determining the transport efficiency of inland waterway transport. Climate change exacerbates these challenges, undermining the reliability and competitiveness of inland navigation. This also hampers the achievement of objectives outlined in the EU’s NAIADES III action plan [2], which seeks to shift a significant share of freight transport from road to more sustainable modes, such as inland waterways and rail transport. Numerous studies have examined the impact of climate change on inland water transport, with a particular focus on the effects of reduced water depth in navigable fairways. Given the expectation that prolonged droughts will occur more frequently in the future, these conditions are anticipated to have more severe consequences on transport efficiency, safety, and reliability compared to the challenges posed by high water levels or discharge fluctuations [3]. In a recent review of the literature [4], two primary challenges facing inland waterway transport in achieving sustainability and reliability were identified. The first concerns the adverse impact of extreme shallow-water conditions on both hydrodynamic efficiency and cargo-carrying capacity, which impedes efforts to increase the modal share of inland waterway transport, already underutilised. The second relates to the challenges of integrating suitable green energy solutions onboard. This study focuses on addressing the first challenge by improving the propulsion performance of inland vessels to ensure sufficient thrust under extremely low water conditions. By extending the technical tipping point or operational limits, the research aims to enhance the technical resilience and effectiveness of inland waterway transport in adverse environmental conditions. Achieving this requires understanding the critical interaction between the vessel hull, propeller(s), and waterway, which influences the hydrodynamic characteristics and efficiency in extreme waterway conditions. This study advances beyond conventional hydrodynamic analysis by applying computationally efficient optimisation techniques that account for the performance of the worst-case scenario, enabling enhanced operational resilience for these challenging conditions. The rest of this paper is structured as follows; section 2., presents the theoretical framework and background literature relevant to this study, section 3. describes the research methodology, key findings are analysed in section 4. and section 5. concludes with contributions and future directions. 2 2. Theoritical background 2.1. Confined waterway effect Vessel navigating in waterways with restricted width and depth, the flow around it is substantially influenced by the confined environment. In such conditions, the boundary layer around the flat bottom plate becomes thinner, resulting in higher friction resistance. The flow around the hull is accelerated, leading to a pressure reduction according to Bernoulli’s principle. This pressure decrease contributes to the squat effect of the ship [5]. The increase in squat also accounts for increased resistance and loss of speed [6]. [7] summarised the threshold for the effects of waterway cross-section relative to the ship dimension on the hydrodynamic effect in table 1.. The main parameters are the blockage factor (Ac/As), the depth to draft ratio (h/T) and the channel width to beam ratio (W/B). Table 1.: Confinement Effect Parameters [7] Parameters Start of confinement effect Important confinement Highly confined Ac/As50 7–8 4 h/T 15 3–4 1.5 W/B 50–200 10–15 4 In addition to the confinement effect, the flow regime plays a critical role in influencing the increased resistance experienced by ships. Specifically, the flow regime significantly affects the wave-making resistance component of inland vessels [8]. Flow regimes are characterized by the vessel’s speed relative to the water depth, expressed as the depth Froude number (Fnh =V √gh ). As highlighted by [9], the shallow water effect becomes more pronounced as the vessel approaches the critical flow regime (Fnh ≈1). However, this condition is typically not so common for inland vessels, as they are low-speed vessels operating predominantly within the subcritical flow regime. In the design of the propulsion system of the ship, the interactions of the hull-propeller (s)-waterways play an important role in accurately determining the propulsive coefficients (thrust tand wake wfactors). Where the wake fraction coefficient describes actual inflow to the propeller, and the thrust deduction factor describes the additional resistance due to the suction pressure by the propeller(s) in action and is used to determine the thrust required to overcome the resistance. These factors have been theoretically investigated by [10], where it was observed that, reduction in water depth weakens the wake (w) and intensifies the propeller suction (t). In conventional ship propulsion systems, vessels typically rely on single or twin-screw large-diameter propellers. However, this configuration has notable limitations, particularly in shallow-water operations, where the risk of propeller ventilation becomes a significant concern. The loss of propulsion thrust and inefficiencies caused by high propeller loading, resulting from increased resistance, highlight the need for a paradigm shift in the design of propulsion systems for inland waterway vessels. Moreover, when operating in shallow water conditions, the propeller is also highly loaded due to the increase in thrust required to meet the increased resistance. When this happens and, therefore, the propeller is highly loaded, the operating rotational speed will have to be increased to maintain the required thrust. The high rotational speed creates conditions that lower the local pressure on the suction side of the blades, which is lower than the vapor pressure, leading to propeller cavitation. The variation in open-water efficiency is highly dependent on the change in propeller loading, which in essence is related to the thrust deduction factor [11]. Advancements have been made in improving the propulsion efficiency of inland vessels for shallow-water conditions. Rotteveel in [12] investigated the different stern shapes, including tunnels, to improve the inflow to the propeller in shallow water. In recent years, numerical methods using CFD simulation tools have been used to predict and optimise inland vessel propulsion performance. However, these are computationally expensive for early design concept exploration. Hence, models that estimate resistance and propulsion performance based on the vessel’s and the waterway’s specific characteristics are essential during the early design stages. To address these challenges and meet the demands of current and future operations, innovative ap3 proaches to propulsion system design must be considered. To benefit from the available beam of the vessel in the stern region, a propulsion concept called the distributed thrust has notably been investigated by [13], [14]. As highlighted by Hagesteijn [13], one of the key advantages of distributed thrust using multiple smaller propellers is the redundancy it provides. In the event of failure of one or more propellers, the system can maintain functionality, thereby minimizing downtime. Furthermore, distributed thrust allows for the use of smaller, more modular power drive trains instead of larger, centralized systems. For inland vessels, this approach enables the use of easily serviceable and replaceable truck engines as an alternative to specialized marine engines [13], significantly simplifying maintenance and reducing operational costs. 2.2. Robust optimisation In today’s complex technological landscape, integrating diverse disciplines has become essential for addressing engineering challenges. Operations Research (OR), a field of applied mathematics, employs analytical methods to support informed decision-making. A key approach within OR is robust optimization, which specifically focuses on decision-making in uncertain environments by accounting for the worst possible scenarios. The efforts to implement robust optimisation to help in decision making in shipping are not new, recent advancements in maritime transport research have leveraged robust optimization in various applications, such as a robust optimization model for ship traffic scheduling presented by [15] and vessel maneuvering control, as explored by [16]. [17] applied robust optimization for fleet deployment and tactical ship assignments, as well as capacity allocation, to account for the volatility of shipping markets and seasonal variations, ultimately aiming to maximize revenue. In ship design, robust optimization has made significant advancements by enhancing design methodologies while minimizing the computational demands associated with high-fidelity, resource-intensive simulations, such as those in computational fluid dynamics. Additionally, the ship design process is becoming increasingly complex, driven by numerous conflicting design objectives that are mutually interdependent, coupled with increasing number of design or operational constraints and decision variables. Accounting for this increasing complexity poses a challenge commonly known as the so-called ”curse of dimensionality”. As such optimisation models have become integral part of ship design workflow. For example, [18] applied robust design optimization to determine optimal hull geometries and floating system configurations, ensuring adaptability to uncertain design requirements. Considering the inherent uncertainties in the operational conditions of inland vessels, robust optimisation in the authors’ view, provides a powerful framework for designing vessels that are resilient to the worst possible conditions they may encounter throughout their operational lifetime. As demonstrated in this study, applying robust optimisation informed by physics to extreme shallow-water scenarios facilitates the development of propulsion systems that ensure both efficiency and reliability, even under variable water depths and challenging constraints. This approach allows for exploration of resilient vessels that can operate effectively across a range of unpredictable environmental conditions. 4 3. Approach 3.1. Optimisation modelling - adversarial robust optimisation A well-known approach in robust optimisation is the adversarial approach. The main idea of the adversarial approach is to explicitly model the uncertainty by considering the deterministic ”worst-case” scenario [19]. This method seeks to optimise the system’s performance while ensuring it remains robust and effective under the most unfavourable conditions the system might encounter, here-in an inland waterway vessel. The optimization process iteratively adjusts the design or decision variables to minimize the impact of these adversarial uncertainties. This ensures the solution is not only optimal under nominal conditions but also resilient under extreme variations. 3.1.1. resistance •uncertainty sets The uncertainties are modelled as a set Uof possible waterway conditions, and in this case, related to the confinement effects of the waterway. U=u=h T,W B, db h T∈[1.1,10],W B∈[2.0,10], db∈[0.1·W, 1.0·W](1) where: h/T :(water depth-to-draft ratio), W/B :(channel width-to-beam ratio), db:(proximity of vessel to sides of channel wall). •worst case for vessel resistance The worst possible scenario would result in increased resistance as given below; RT=f(Vs, u)|u∈U (2) •resistance estimation Estimating the resistance of the vessel, the state-of-the-art friction correction for shallow water used in this work is adopted from [5]. This is a correction for the ITTC friction line, accounting for shallow-water effect; Cf=0.08468 (log Re −1.631)2· 1 + 1.168 log Re −0.5238 ·h T−1.472!(3) The additional resistance due to the squat effect is calculated from [20] empirical ship squat formula, which is corrected for waterway conditions such as the width of the channel, the slope of the channel, and the proximity of the vessel to the sides. z= 0.0065 ·e5.2·Fnh + (0.95 ·F6 nh −0.065) (4) zfinal =z·(αW·αM·αW′)(5) where Fnh =V √gh is the depth froude number and (αw, αm, α′ w) are correction factors for the width of the waterway, slope of bank, and the proximity of the vessel to sides. 5 3.1.2. propulsion Thrust loss in case of propeller emergence has been assumed to be proportional to the out-of-water area of the propeller disc. Ventilation is a phenomenon that occurs when the propeller operates close to the dynamic free surface and drawing air into the propeller stream or when the blades pierce the free surface, leading to sudden loss in propeller thrust and torque. This becomes more pronounced when the propeller loading is high and the submergence is limited. Load can be distributed among multiple propellers to reduce loading on one or two propellers, which is the focus of this research. To improve submergence below the surface, a smaller-diameter propeller can be used. Hence, to mitigate the risk of propeller ventilation. The thrust loss factor βis given empirically as a function of the shaft submergence ratio h Rgiven by eq. (6). This was proposed by [21], accounting for the combined effects of loss of disk area, wave, and wagener effect. β=(1,if h R≥1.3 1−0.675 ·(1 −0.769 ·h R)1.258,otherwise (6) The shaft immersion ratio, which is modelled as a function of the propeller diameter accounting for hull-propeller blade clearance, is given by eq. (7). h R= 2 Te D−0.65(7) where Teis the vessel draft and Dis propeller diameter. •worse case for propulsion Worse possible scenario for propulsion would be the potential breakdown in thrust given below. TF=β·T(8) where TFand Tare the effective propeller thrust and openwater propeller thrust, respectively. In this study, Wageningen B-series propeller, a conventional propeller series widely used in the design and optimisation of propulsion system, is used. Originally developed through systematic model testing by MARIN [22], this propeller series provides comprehensive open-water performance charateristics for conventional fixed pitch propeller across parametric variations in pitch-diameter ratio, expanded area ratio, number of blades, and advance number. Recent advances have represented the open-water characteristic as high-order multivariate polynomials as a functions of these variables, given in eq. (9) and eq. (10). KT=f1(J, PD, AER, Z) = XCT s,t,u,v(J)s·(PD)t·(AER)u·(Z)v(9) KQ=f2(J, PD, AER, Z) = XCQ s,t,u,v(J)s·(PD)t·(AER)u·(Z)v(10) J=Va n·D(11) The coefficients of the high-order B-series polynomial functions are published in [23] 3.2. MINLP formulation The problem, as formulated, is a mixed-integer non-linear (MINLP) optimisation problem. This is a nonlinear problem because the objective function, which is the open water efficiency, is non-linear, and some constraints, such as the (Kt −Kq), are polynomial functions of the pitch-diameter ratio, expanded area ratio, and number of blades. The thrust loss factor (β) is an empirical hydrodynamic function, as described in reference [3], and acts as a penalty factor that reduces the thrust generated by the propeller. This thrust loss factor increases with the propeller size as discussed in the previous subsection. Consequently, the optimisation process aims to minimise this penalty factor by distributing the thrust across multiple smallerdiameter propellers, thereby reducing the overall impact of the thrust loss factor. Hence, to maximise the 6 benefit from the entire beam (Bmax) of a vessel in the stern region, the constraint illustrated in fig. 1. and defined in eq. (12) must be considered. N X np=1 (D+ (np−1) ·(0.2·D)) ≤Bmax (12) The total thrust produced is modelled to account for the worst-case scenario, where the total propeller(s) thrust (np·T) should overcome the resistance in the worst-case (shallow water condition) RT. np·(T·β)≥RT,(13) The thrust loss factor negatively impacts the total thrust, primarily a function of the submergence of the propeller. This submergence, in turn, depends on the propeller’s size (diameter). The formulation of the optimisation problem is summarised below; maximise:η=J 2π·KT KQ , subject to:np·(T·β)≥RT, N X np=1 (D+ (np−1) ·(0.2·D)) ≤Bmax, KT=f1(J, PD, AER, Z), KQ=f2(J, PD, AER, Z), T=KT·n2·D4, Q=KQ·n2·D5, decision variables: D∈[0.5·Te,1·Te], PD∈[0.7,1.4], AER ∈[0.5,1.0], J∈[0.1,1.6], Z∈[3,6] np∈Z+ (a) propeller spacing (b) propeller submergence Figure 1.: Propeller Configuration Constraints 7 This method estimates the diameter, pitch, number of propeller(s), and efficiency required to achieve sufficient total thrust for inland waterway vessels operating in shallow waters. This program was implemented in Python, with the optimisation model formulated using Pyomo (Python Optimisation Modeling Objects), an open-source Python library designed for modeling a wide range of optimisation problems. Pyomo provides a flexible framework that supports linear, nonlinear, and mixed-integer optimisation, allowing users to interface with various solvers depending on the problem’s requirements. For this specific problem, which is classified as a Mixed-Integer Nonlinear Programming (MINLP) problem, the open source solver BONMIN (Basic Open Source Nonlinear Mixed INteger Programming) was employed. BONMIN is designed to efficiently handle MINLP problems by integrating nonlinear and integer programming techniques. 8 4. Results and Discussions 4.1. Resistance and propulsion Implementing this method on a use case with characteristics in the table table 2., this is the CEMT Class VI, a typical vessel of Rhine class. For this vessel, the resistance it may encounter is decomposed Table 2.: Vessel Characteristics Parameters value Cb[-] 0.86 L[m] 110 B[m] 11.4 Tl[m] 3.5 Te[m] 1.8 Sw[m2] 2974.96 ∇[m3] 3795.2 lcb [m] 55.28 lcg [m] 56.27 into the three main components: the viscous resistance and the wave making resistance and ship squat effects, as illustrated in the 1st,2nd and 3rd frames of fig. 3.. The viscous resistance is made up of the resistance due to friction (Cf) and the resistance due to the form of the vessel (1 + k∗). The wave-making resistance exhibits an increasing trend with water depth, as illustrated in the second frame of fig. 3.. Its dominance remains relatively limited. This is because wave-making resistance is primarily governed by the flow regime, characterized by the depth Froude number (Fnd =V √gh ). Notably, significant wavemaking resistance occurs when Fnd approaches the critical flow regime Fnd ≈1, However, this condition is generally not encountered in inland vessels, as they predominantly operate within the subcritical flow regime (Fnd <0.6). Figure 2.: Ship squat However, resistance due to squat must also be considered, as it arises from the additional ship draft or sinkage when the vessel moves in shallow water. This effect occurs due to a pressure drop beneath the keel, 9 [17] X. Lai et al. “Robust Ship Fleet Deployment with Shipping Revenue Management”. In: Transportation Research Part B: Methodological 161 (2022), pp. 169–196. DOI:10.1016/j.trb.2022. 05.005. [18] P. Gallagher. “Novel Approach to Robust Concept Development for Offshore Floating Systems”. In: International Journal of Maritime Engineering 164.A4 (2023). DOI:10.5750/ijme.v164iA4. 1206. [19] J. Th¨ urauf. “Adjustable Robust Nonlinear Network Design under Demand Uncertainties”. In: (2024). 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