Investigation into Propulsive Characteristics of an Inland Vessel in Confined Waterway

Investigation into propulsive characteristics of an inland vessel in confined waterway Kelvin Eloho*, Vladimir Krasilnikov *, and Benjamin Friedhoff † *SINTEF Ocean AS, Trondheim, Norway, †Development Centre for Ship Technology and Transport Systems, Duisburg, Germany [email protected] 1 Introduction Small to medium inland navigation vessels (CEMT Classes I to IV) are considered as a viable alternative cargo transportation mode to relieve loads on some of the most congested European motorways. In the Horizon Europe project AUTOFLEX (https://autoflex-vessel.eu/), a novel type of autonomous inland cargo vessel is being developed, which can carry out transport services in previously underused confined waterways. Confined waterways, characterized by limited depth, width, or both, significantly affect vessel hydrodynamics. In such conditions, vessels experience increased resistance, along with notable changes in sinkage and trim compared to operation in unbounded deep water. Furthermore, interaction with waterway boundaries affects considerably the inflow experienced by the propulsor. The model‐scale experiments remain the most common approach to predicting ship’s resistance and propulsive characteristics; and the extrapolation procedure based on the ITTC 1978 performance prediction method is commonly employed. However, the validity of this approach in application to inland vessels in shallow water conditions is subject to criticism (Mucha et al., 2017). The Lackenby (1963) method, also described in ITTC (2014), shows limited applicability across diverse hull forms due to its reliance on limited experimental data and thus exhibits constrained applicability across diverse hull forms and operating conditions. The form factor concept originally proposed by Hughes (1954), and refined by Prohaska’s method (ITTC, 2002) separates the wave resistance component (which depends only on Froude number, 𝐹𝑟) and viscous resistance (which depends only on Reynolds number, 𝑅𝑒). The form factor, which is assumed independent on both 𝐹𝑟 and 𝑅𝑒 accounts for the additional viscous resistance attributable to hull shape beyond flat‐plate friction. This very assumption has been subject to criticism in many studies, numerical, as well as experimental, with first systematic results opposing the main form-factor hypothesis presented in ITTC (2008). More recent studies (Zeng et al., 2020) have also demonstrated that, for depth‐Froude numbers 𝐹𝑟ℎ> 0.5422 and non‐slender hulls, the wave resistance dependence on 𝐹𝑟 alone may not hold. Furthermore, the assumed linear dependency of wave resistance coefficient on 𝐹𝑟4 may not be observed for various hull geometries (Korkmaz et al., 2021). In general, scale effects in shallow water are expected to be become more pronounced with decreasing water depth, since at lower depths, boundary layers on the hull and channel floor experience stronger interaction as illustrated by CFD analyses presented in Krasilnikov et al. (2025). At very low depths, the said interaction has appreciable influence on wave systems generated by the ship due to changes in pressure distribution over the hull. With these considerations in mind, it is easy to understand the results presented by Raven (2019) which demonstrated that traditional model‐to‐ship extrapolation methods can overestimate water‐depth effects unless a depth‐dependent form-factor is employed. A revised procedure for correcting full‐scale speed trials to account for water‐depth effects has been proposed, and it is now included in the ITTC guidelines (ITTC, 2017). The primary purpose of the present study is to investigate by means of CFD simulations, the influence of limited water depth on ship’s towing resistance in full‐scale, and to illustrate differences from the equivalent test conditions in model scale. The numerical setup is based on the methodology presented in Krasilnikov et al. (2025). The ultimate objective is to establish a validated framework that aids further investigation of vessel propulsive performance in inland waterways. 2 Description of CFD approach The CFD methodology described in Krasilnikov et al. (2025) and used for investigations in this paper employs a fully parameterized simulation template developed in STAR-CCM+ (v18.06.006), specifically configured for vessels operating in confined waterways. The computational domain is divided into the three regions: • Background Mesh (BM): a stationary block representing the waterway, bounded by no-slip side walls and floor, a velocity inlet 4 Lpp upstream, and a pressure outlet 4 Lpp downstream of ship’s aft perpendicular (AP). The inlet/outlet boundaries are followed by 2 Lpp wave-damping zones enforced to suppress reflections. • Overset Mesh (OM): a body-fitted, moving region surrounding the ship ( ±1.05 Lpp upstream to –0.10 Lpp downstream from AP, ±0.80 B laterally from central plane (CP), –0.30 H to +1.35 H vertically from base). Here and above, Lpp is the ship’s length between perpendiculars, B is the ship’s beam, and H is the ship’s height to upper deck. To improve mass conservation across the overset interface, we employ the least-squares scheme with mass-tracking for multi-phase flow models. • Sliding Mesh (SM) Regions: cylindrical domains enclosing each propulsor (only in self-propulsion runs), rotating at prescribed RPM about the shaft axis and moving rigidly with the OM. Dynamic sinkage and trim of the ship are captured using the Dynamic Fluid–Body Interaction (DFBI) solver with the Equilibrium body motion model. During the simulation, an initial fixed ship position period (0.2 % of target simulation time) is followed by a 20 % release period with linear ramping, after which sinkage and trim angle are adjusted iteratively toward zero net vertical force & pitching moment. A predominantly hexahedral grid is generated by the Trimmer Cell Mesher, supplemented by six prism layers (twelve for extreme shallow-water cases) to enforce Y+ ≈ 50 using the k-ω SST turbulence closure model, which solves the transport equations with linear constitutive relation, with wall functions. However, in full scale (FS), all depths and speeds utilised 12 prism layers, with a target Y+ of around 121 (assumed to be around a stretching factor of 1.2 for 8km/h cases). Local refinement boxes and cylindrical controls enhance resolution near the free surface, in the wake, and around propulsors. Deformation of free surface is resolved using a Volume-of-Fluid (VOF) solver with a blended HighResolution Interface Capturing (HRIC) scheme. The default upper and lower Courant number limits in the HRIC scheme are increased to the values Coₗₒ = 200 and Coᵤₚ = 250 to reduce solution dependency on time step while maintaining interface sharpness. Pressure–velocity coupling employs a SIMPLE algorithm with an AMG pressure solver (max cycles = 100; convergence tolerance = 1e⁻³) and reduced under-relaxation factors to facilitate solution stability in shallow water conditions. The definition of time step follows a general formula Δt = 0.005·(Lpp/V). With this definition of time step, the Courant number on the free surface normally varies between 2 and 5 around the ship, and it reaches 7-8 in the areas where the bow and stern wave systems are formed. For the given ship speed, at the constant time step, Courant number levels in the simulations increase with the reduction of water depth, which is explained by the increase of induced velocities. In full scale, maintaining the Courant number level appears more important for the overall quality of the numerical solution. 3 Extrapolation of model test results to full scale As pointed out in the introduction, traditional extrapolation methods from model scale (MS) to full scale (FS) exhibit notable uncertainties applied to shallow-water conditions. Therefore, the procedure proposed by Raven (2019) is followed to estimate hull towing resistance in full scale based on the results of model-scale experiment. According to this method, the revised, depth-dependent form factor (1+𝑘∗) maintains consistent behaviour at both model and full scale, provided that 𝑇/ℎ ≤ 0.5, where 𝑇 is the ship’s draught and ℎ is the water depth. A standard decomposition of the total resistance into the wave and viscous component is then followed: 𝐶𝑇(Fr, Re) = 𝐶𝑊(Fr) + 𝐶𝑉(Re) with the assumption that 𝐶𝑉(Re, ℎ) = (1+𝑘∗(ℎ)) 𝐶𝑓0(Re), where 𝐶𝑓0 represents the conventional flat-plate friction coefficient according to ITTC 1957 extrapolation line. It is shown that, for moderately shallow water conditions (𝑇/ℎ ≤ 0.5), the ratio representing the relative increase in 𝐶𝑉 due to limited depth, 𝐶𝑉(ℎ) 𝐶𝑉 ∞ , remains essentially unchanged from model-scale to full-scale Reynolds numbers. Consequently, a depth-dependent form factor is defined as follows: 1+𝑘∗(𝑇/ℎ)=𝐶𝑉(ℎ) 𝐶𝑓0 =𝐶𝑉(ℎ) 𝐶𝑉 ∞(1+𝑘∞) where 1+𝑘∞ is the deep-water form factor. It is assumed that, for mentioned range of relative water depth (𝑇/ℎ ≤ 0.5), the depth-dependent form-factor is independent on scale. For practical applications, Raven (2019) provides the regression formula that approximates increase of viscous resistance with water depth: 𝐶𝑉(ℎ) 𝐶𝑉 ∞≈ 1+0.57(𝑇/ℎ)1.79 4 Calculation results in model scale and depth-dependent form-factor In the present work, for CFD vs. EFD comparisons, we use the validation dataset of the M2052 model developed at DST as a part of the test campaign with the four ships having identical bow shapes and different stern and propulsion system configurations (Friedhoff et al., 2019). This model is representative of a CEMT Class Va single-screw inland navigation vessel of the following main dimensions: 𝐿𝑝𝑝 =110 m; 𝐵=11.4 m; max draught 𝑇𝑚𝑎𝑥 =2 m. The vessel’s block coefficient is 𝐶𝐵=0.895 at the volume displacement of 3150 𝑚3. In accordance with model tests, CFD resistance simulations were performed at the scale of 1/16, without rudders, duct and propeller, at the draught T=2.8 m. The comparisons between the model-scale CFD simulations and experimental data are described in detail in Krasilnikov et al. (2025). A summary comparison of the computed and measured towing resistance is presented in Fig.1 for different relative water depths, ℎ/𝑇, and range of Froude number, 𝐹𝑟. In the most relevant speed range between the Froude numbers of 0.065 and 0.105 (speed 8 to 12 km/h in full scale), the agreement between the CFD and EFD results is found to be very close. Discrepancies become larger at higher speeds. In this paper, the numerical results were further examined for the influence of tank wall effects. While the CFD results presented in Fig.1 were obtained in the computational domain corresponding exactly to the width of the testing facility with side walls, selected conditions were reproduced in the domain without the restriction of side boundaries (symmetry plane boundaries were used instead), at the same water depth. For the condition of the lowest water depth, ℎ/𝑇 =1.25, relative difference in mean axial velocity on the transverse (Y-Z) plane at the free surface level produced by the restricted domain and unbounded domain simulations were about 0.2 %, while the respective values of total hull resistance differed by less than 0.5%. These numbers are indicative of minor influence of tank walls, which is an important result for subsequent comparisons with fullscale simulations, which are performed in a laterally unbounded domain. Fig.1: Comparison of hull towing resistance predicted by CFD with the results of model tests at different water depths. Fig.2: Comparison of depth-dependent form factor derived from the CFD calculations in model and full scale with approximation formula by Raven (2019) For the next step, the depth-dependent form factor (1+𝑘∗) was computed in model scale and full scale at the lowest Froude number of 0.068 corresponding to ship speed of 8 km/h in full scale. This exercise involved determining the ratio of total computed viscous resistance to friction resistance derived from the ITTC57 Model-Ship correlation line. Simulations performed at ℎ/𝑇≥4 were assumed representative of deep-water conditions. The obtained values of form-factor are compared with predictions using the empirical relationship from Raven (2019) in Fig.2. In general, the dependencies of form-factor on water depth derived from CFD exhibit a trend consistent with Raven's predictions. Some differences in formfactor estimations obtained in model scale and full scale are also noticed, indicating that form-factor may not be entirely free from scale effect. Additionally, as illustrated in Fig.3, even in this low Froude number case discrepancies are found between the total resistance and viscous resistance component, and they increase with reduction of water depth as the influence of wave making resistance becomes more pronounced. Despite the mentioned differences, the depth-dependent farm-factor derived from 0 0.5 1 1.5 2 2.5 3 3.5 0 1 2 3 4 1+k* h/T CFD(FS) CFD(MS) Raven (2016) CFD simulations is found adequate for the use in extrapolation of model test results to full scale conditions for the comparison with full-scale CFD results. Fig.3: Comparison of the total and viscous resistance components computed at different water depths, for the Froude number 𝐹𝑟 =0.068 (8km/h), full scale 5 Calculation results in full scale and scale effects Fig.4 presents a comparison of ship’s towing resistance computed in full scale with the results of extrapolation procedure using the depth-dependent form-factor estimated in Section 4. The comparison is presented for ship speed of 8 km/h and different water depths. The model test data are used in the extrapolation, and no additional corrections or correlation factors is applied in the prediction. Likewise, full-scale CFD simulations are performed on a bare hull without surface roughness. Fig.4: Comparison of full-scale CFD resistance with extrapolations of model tests data at 8 km/h (𝑭𝒓 =0.068) Fig. 5: Variation of the total resistance and its friction and pressure components with water depth in model and full scale. Ship speed 8 km/h (𝑭𝒓 =0.068). The computed ship resistance exhibits a consistent overprediction compared to the results of extrapolation using the depth-dependent form factor, while maintaining a similar trend in the increase of resistance with reduction of water depth. At the same time, the results of extrapolation without depthdependent form-factor predict a significantly larger increase of resistance at lower water depths. To better understand the mechanisms behind resistance variation, the computed friction and pressure resistance components are shown separately in Fig.5. As relative water depth decreases from deep water (represented as h/T =4) condition to ℎ/𝑇≈ 1.25, full-scale total resistance increases from ≈ 0.0027 to ≈ 0.0036 (+33 %), whereas in model-scale it increases from ≈ 0.0043 to ≈ 0.0069 (+60 %). This difference underscores the amplification of shallow-water effects in model-scale tests compared to fullscale conditions, which may be misleading, if left uncorrected during extrapolation. Even larger deviations will be seen at higher ship speeds. The full-scale frictional resistance, Cf, increases by ≈ 21 % (from 0.0019 to 0.0023) from deep to shallow conditions, while model-scale friction rises by ≈ 11 % (0.0037 to 0.0041). For ℎ/𝑇 > 2.7, the percentage increases in Cf remains similar (≈ 5 %) between scales. However, for ℎ/𝑇 < 2.0, the full-scale increase of Cf outpaces model-scale, revealing that changes in shear stress distribution over the hull at very shallow water conditions are scale dependent. The pressure resistance component, Cp, exhibits even greater sensitivity to depth variation. If in deep water condition, the full-scale Cp≈0.0006 and model scale Cp≈0.0011, then at ℎ/𝑇=1.25, the fullscale Cp ≈ 0.0013 (+117 %) and model-scale Cp≈ 0.0029 (+164 %), respectively. Thus, changes in the pressure resistance with variation of water depth also depend on Reynolds number. It is worth noting that the friction resistance coefficients predicted by CFD in deep water shows a good match with the ITTC57 friction line (0.0019 vs. 0.00188 in full scale, and 0.0037 vs. 0.0037 in model scale). The mentioned dependencies in the variation of the friction and pressure resistance in shallow water conditions on scale (Reynolds number) is explained by the interaction between the boundary layer of ship hull with the channel floor. 0 0.001 0.002 0.003 0.004 1.25 1.79 2.68 Deep h/T Ct, Total (Fr=0.06765) Ct,Viscous (Fr=0.06765) 0 5 10 15 20 25 1.25 1.79 2.68 4.00 R_T (kN) h/T CFD(FS) Extrap. with computed form factor initial extrap. from model test 0 0.002 0.004 0.006 0.008 1.25 1.79 2.68 4.00 h/T Ct(FS) Ct(MS) Cf(FS) Cf(MS) Cp(FS) Cp(MS) Fig.6: Fields of vorticity magnitude at different water depths in model and full scale. Ship speed 8 km/h (𝐹𝑟 =0.068). Left: Model scale. Right: Full scale. Top: 5m depth. Bottom: 7.5m depth. As illustrated in Fig.6, in model scale the boundary layer on ship hull is relatively thicker and more diffused than in full scale. A thicker boundary layer experiences stronger interaction (and hence undergoes stronger changes) with decrease of water depth. At the extreme shallow water conditions, the said interaction also affects generation of ship induces waves, in particular downstream of midship, thus adding a viscous contribution to the potential part of wave making caused by domain confinement. The changes in the dynamic sinkage and trim of the ship with variation of water depth show generally similar trends in model and full scale (see Fig.7). Fig.7: Changes in dynamic position of the ship with variation of water depth in model and full scale. Ship speed 8 km/h (𝐹𝑟 =0.068). Except for lowest water depth, the values computed by CFD are found to be in good agreement with experimental measurements. At ℎ/𝑇=1.25, the value of dynamic trim angle computed in model scale agrees better with the measured data, at a very close value of the dynamic sinkage. It should however be noted that, due to the interaction between the hull boundary layer and channel floor and flow separation that develops at the aftship, the flow pattern around ship hull in extreme shallow water conditions becomes unstable. In such scenarios, one can therefore question validity of the Equilibrium motion model employed in the DFBI solution, that drives the solution to a steady state which may not exist in the reality. As demonstrated experimentally in Friedhoff et al. (2019) and confirmed numerically in Krasilnikov et al. (2025), limitation of water depth has a profound effect on the wake field past ship hull. Naturally, the said effect also depends on scale, and in view of the above discussion about the boundary layer interaction with domain floor, it is stronger in model-scale conditions. As illustrated in Fig.8, with reduction of water depth from 7.5 m to 5.0 m, the axial wake field in model scale becomes heavier (greater flow retardation) on the whole propeller disk area as well as on the sides of the ship. The increase of flow separation is evident from larger zones of swirled flow (vorticity formation), which also move close to the hull surface and become wider due to flow confinement. In full-scale, the increase of wake velocity deficit with depth reduction is smaller, and it is mostly limited to propeller disk, particularly around the location of ship’s central plane. -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 1.00 2.00 3.00 4.00 Sinkage/T h/T CFD(FS) CFD(MS) Exp 0 0.005 0.01 0.015 0.02 1.25 1.79 2.68 4.00 Trim (deg.) h/T CFD(FS) CFD(MS) Exp Model Scale Full Scale 5m 7.5m Fig.8: Changes in nominal wake field with variation of water depth in model and full scale. Ship speed 8 km/h (Fr=0.068). 6 Conclusions The CFD setup for simulation of propulsion performance of inland vessels developed and validated at model scale in Krasilnikov et al. (2025) has been applied to investigate the influence of shallow water conditions on ship resistance in full scale and analyse associated scale effects. Comparisons utilize results from results of model tests performed at DST on a CEMT Class Va single-screw inland vessel. The investigation reveals difficulties with extrapolation of model test results to full-scale using conventional scaling procedures established for deep-water conditions. The key finding is that the influence of limited water depth affects both the friction and pressure resistance components, and changes in these components depend on both the Froude number and Reynolds number. The physical mechanism underlying the mentioned dependencies is the interaction between the hull boundary layer and waterway floor, causing increase and re-distribution of wall shear stress on the hull, changes pressure distribution and alters flow separation pattern at the aftship. At the extreme shallow water conditions, the said interaction also affects generation of ship induces waves, thus adding a viscous contribution to the potential part of wave making due to domain confinement. Since the relative thickness of hull boundary layer varies with Reynolds number, these described changes become scale dependent, rendering the extrapolation procedure based on the conventional deep-water form-factor inadequate. It is shown that using a depth-dependent form factor as suggested in Raven (2019) allows for an improvement between the CFD and EFD prognoses in full-scale, and it can be employed for engineering predictions in absence of CFD results, at least in the range of draught-to-depth ratio (𝑇/ℎ) below 0.5. Nonetheless, viscous CFD simulations remain recommended for the elaboration of depthdependent form-factor and, more generally, for the prediction of resistance and propulsion performance of inland vessels, since the described interaction between the vessel and waterway is also found to have significant impact on the wake field past ship hull. Furthermore, it was noticed that for highly confined conditions (H/T ≤ 1.5), the free-motion DFBI method provides superior performance compared to the DFBI equilibrium approach. Under extreme confinement, the equilibrium solver artificially enforces a steady sinkage and trim, masking inherent unsteady squat dynamics and wake oscillations. However, it is noted that the free-motion method significantly increases computational demands compared to the equilibrium approach when combined with the Overset Mesh technique. This is because the overset interface requires updating at each inner iteration of every time step, resulting in approximately four times longer simulation times for identical physical durations. The actual computational overhead may further increase depending on the number of inner iterations used per time step. Conversely, the free-motion approach preserves these unsteady phenomena, yielding improved convergence of hydrodynamic forces, as demonstrated in the case of H/T = 1.25, where the free-motion simulation was successfully initialized following a DFBI equilibrium run. Acknowledgements The present research is funded by the project AUTOFLEX (AUTOnomous small and FLEXible vessels) under the Grant Agreement 101136257 of the European Union Horizon Europe Programme. 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