CFD Analysis of Hull Separation and Vessel Speed on the Hydrodynamic Performance of Catamaran Trash Skimmers

2.1. Research Object / Predmet istraživanja

The design used for the trash skimmer boat is a catamaran with a demi hull flat side outward (FSO) configuration. The main dimensions of this ship were obtained from previous research by Wulandari et al. [20]. Table 1 and Figure 1 display the principal dimensions of the ship and a 3D model of the trash skimmer boat, which serves as further supporting data.

Table 1. The main dimensions of the trash skimmer boat.

Tablica 1. Glavne dimenzije broda za prikupljanje otpada

Figure 1 Trash skimmer boat 3D design

Slika 1. 3D prikaz broda za prikupljanje otpada

This investigation included altering the separation-to-length hull ratio and operational speed to determine optimal conditions for operating the trash skimmer boat. This research will replicate the operation of the trash skimmer boat in restricted water environments, such as beaches or rivers, to get results that closely resemble the actual operating circumstances. Table 2 presents the modifications and specifics of the setting to be developed in the present study.

Table 2 Variations in simulation conditions.

Tablica 2. Varijacije u uvjetima simulacije

2.2. Numerical Approach / Numerički pristup

The governing equations for fluid flow are based on the principles of mass, momentum, and energy conservation. These fundamental equations form the foundation for the Reynolds-Averaged Navier-Stokes (RANS) approach used in this study to model turbulent flows. The mass conservation equation, also known as the continuity equation, ensures the incompressibility of the flow and is expressed as

[TeX:] \nabla \bullet U = 0$\nabla •U=0$ (1)

Where [TeX:] U$U$ is the velocity vector, the momentum conservation equation describes the fluid's momentum exchange due to pressure, viscous forces, and turbulence, and it is given by the Reynolds-Averaged Navier-Stokes (RANS) equation:

[TeX:] \frac{\partial\left( \rho U \right)}{\partial t} + \nabla \bullet \left( \rho\text{UU} \right) = - \nabla p + \nabla \bullet (\mu\left( \nabla U + \left( \nabla U \right)^{T} \right)) + F$\frac{\partial \left(\rho U\right)}{\partial t}+\nabla •\left(\rho \mathrm{\text{UU}}\right)=-\nabla p+\nabla •(\mu (\nabla U+{\left(\nabla U\right)}^T))+F$ (2)

Here, [TeX:] p$p$ is the pressure, [TeX:] \mu$\mu$ is the dynamic viscosity, and [TeX:] F$F$ represents external forces. The computational fluid dynamics (CFD) software NUMECA was utilized to conduct the simulations with the turbulence model of [TeX:] k - \omega$k-\omega$ Shear Stress Transport (SST). Several key equations form the basis of its calculations [21–24]. The transport equation for the turbulent kinetic energy ([TeX:] k$k$), describes how turbulent kinetic energy changes over time, influenced by production, dissipation, and diffusion. This equation is expressed as follows:

[TeX:] \frac{\partial k}{\partial t} + U_{j}\frac{\partial k}{\partial x_{j}} = P_{k} - \beta^{*}\text{ωk} + \frac{\partial}{\partial x_{j}}\lbrack\left( v + \sigma_{k}v_{t} \right)\frac{\partial k}{\partial x_{j}}\rbrack$\frac{\partial k}{\partial t}+U_j\frac{\partial k}{\partial x_j}=P_k-{\beta }^{*}\mathrm{\text{ωk}}+\frac{\partial }{\partial x_j}[(v+{\sigma }_k{v}_t)\frac{\partial k}{\partial x_j}]$ (3)

Here, [TeX:] P_{k}$P_k$ Represents the production of turbulent kinetic energy, reflecting the transfer of energy from the main flow to turbulence. Meanwhile, [TeX:] \beta^{*}\text{ωk}${\beta }^{*}\mathrm{\text{ωk}}$ represents the rate of dissipation of turbulent energy.

The second equation is the transport equation for the specific dissipation rate ([TeX:] \omega$\omega$), which also considers production, dissipation, and diffusion, with the addition of a particular function that allows the transition between the [TeX:] k - \omega$k-\omega$ and [TeX:] k - \varepsilon$k-\epsilon$ models:

[TeX:] \frac{\partial\omega}{\partial t} + U_{j}\frac{\partial\omega}{\partial x_{j}} = {a\frac{\omega}{k}P}_{k} - \text{βω}^{2} + \frac{\partial}{\partial x_{j}}\left\lbrack \left( v + \sigma_{\omega}v_{t} \right)\frac{\partial\omega}{\partial x_{j}} \right\rbrack + 2(1 - F_{1}\frac{\sigma_{\omega 2}}{\omega}\frac{\partial k}{\partial x_{j}}\frac{\partial\omega}{\partial x_{j}}$\frac{\partial \omega }{\partial t}+U_j\frac{\partial \omega }{\partial x_j}={a\frac{\omega }{k}P}_k-{\mathrm{\text{βω}}}^2+\frac{\partial }{\partial x_j}\left[(v+{\sigma }_{\omega }{v}_t)\frac{\partial \omega }{\partial x_j}\right]+2(1-F_1\frac{{\sigma }_{\omega 2}}{\omega }\frac{\partial k}{\partial x_j}\frac{\partial \omega }{\partial x_j}$ (4)

The function [TeX:] F_{1}$F_1$ this equation plays a crucial role in the blending process between the characteristics of the [TeX:] k - \omega$k-\omega$ and [TeX:] k - \varepsilon$k-\epsilon$ models, allowing the [TeX:] k - \omega$k-\omega$ SST model to handle various flow conditions more effectively.

Next, the turbulent viscosity ([TeX:] v_{t}$v_t$) is calculated using the following relation:

[TeX:] v_{t} = \frac{a_{1}k}{max(a_{1}\omega,\text{SF}_{2})}$v_t=\frac{a_{1}k}{max(a_1\omega ,{\mathrm{\text{SF}}}_2)}$ (5)

This relation ensures that the turbulent viscosity adaptively adjusts to the flow conditions, particularly in regions with high-velocity gradients.

Finally, the production of turbulent kinetic energy ([TeX:] P_{k}$P_k$) is calculated as:

[TeX:] P_{k} = v_{t}(\frac{{\partial U}_{i}}{{\partial x}_{j}} + \frac{{\partial U}_{j}}{{\partial x}_{i}})\frac{{\partial U}_{i}}{{\partial x}_{j}}$P_k=v_t(\frac{{\partial U}_i}{{\partial x}_j}+\frac{{\partial U}_j}{{\partial x}_i})\frac{{\partial U}_i}{{\partial x}_j}$ (6)

This equation explains how energy from the main flow is transferred to turbulence through flow deformation.

The [TeX:] k - \omega$k-\omega$ SST model, with these equations, enables more accurate flow simulations, especially in handling flow phenomena such as separation and reattachment [25]. The combination of the [TeX:] k - \omega$k-\omega$ approach near walls and the [TeX:] k - \varepsilon$k-\epsilon$ approach in the free stream region provides better predictive capabilities for pressure distribution and skin friction. Therefore, the use of the [TeX:] k - \omega$k-\omega$ SST model in this study provides a solid foundation for analyzing complex turbulent flows [26].

In cases of multiphase flow, where the flow consists of two or more immiscible fluids (e.g., air and water), the Volume of Fluid (VOF) method is used to track the fluid interfaces. The VOF method tracks the volume fraction of each phase within each computational cell and is governed by the following equation.

[TeX:] \frac{\partial(\alpha_{i}\rho)}{\partial t} + \nabla \bullet \left( \alpha_{i}\text{ρU} \right) = 0$\frac{\partial ({\alpha }_i\rho )}{\partial t}+\nabla •\left({\alpha }_i\mathrm{\text{ρU}}\right)=0$ (7)

Here, [TeX:] \alpha_{i}\rho${\alpha }_i\rho$ represents the volume fraction of phase [TeX:] i$i$, [TeX:] \rho$\rho$ represents water density, and [TeX:] U$U$ represents the velocity vector. This method is essential for accurately capturing the behavior of phase interfaces, such as the free surface of water, in ship hull simulations.

In the context of space, the computational domain is partitioned into a set of control volumes, sometimes known as a mesh. Similarly, the time dimension is split into a limited number of discrete time steps [27]. The determination of timesteps is based on the recommendation of the ITTC standard formula [28] in equation (5), where [TeX:] L$L$ represents the ship's length overall (LOA) and [TeX:] U$U$ represents the ship's velocity. This yields timestep values in the range of [TeX:] 1.15 \times 10^{- 3} - 1.98 \times 10^{- 3}$1.15\times {10}^{-3}-1.98\times {10}^{-3}$, as shown in Table 3.

[TeX:] \mathrm{\Delta}t = 0.001\sim 0.01\ \frac{L}{U}$\Delta t=0.001\sim 0.01\frac{L}{U}$ (8)

Table 3 Timestep of each speed variation

Tablica 3. Vremenski korak za svaku varijaciju brzine

2.3. Meshing and Boundary Condition / Umrežavanje i granični uvjeti

Figure 2 illustrates the computational domain's boundary conditions and meshing, following the ITTC guidelines [28]. The length of the domain extends from −2.5L to 1L, based on the length between perpendiculars (L), with the zero reference point located at the stern and draft of the vessel. The width was established as 1.5L, and the water depth was selected to simulate river conditions, resulting in a domain depth of 0.62L. The inlet, exit, and sidewall boundary conditions were uniformly assigned the same free-stream far-field velocity. The boundary conditions for the top wall were defined by a fixed pressure, while for the bottom wall, we applied a wall function to simulate the effects of river conditions. The ship surface was modeled with no-slip boundary conditions. For computational efficiency, the simulation was performed on only half of the hull. The normal velocity and gradient variables were set to zero at the symmetry plane condition.

Figure 2 Boundary conditions and computational domain in conjunction with the simulation model

Slika 2. Granični uvjeti i računalna domena u kombinaciji sa simulacijskim modelom

The simulation used an adaptive grid refinement technique to accurately resolve the air-water interface, which is crucial for analyzing wave dynamics. For real-time flow physics simulations, our method dynamically altered mesh resolution to prioritize directional refinement at the free surface to capture wave creation and propagation. The refinement procedure used flow-derived tensor fields to identify places requiring localized grid modifications, allowing targeted subdivision of cells along wave-aligned orientations and coarsening non-critical areas. This method precisely resolved transient wave heights and interface phenomena without refinement zones, decreasing computing costs.

The wall distance (Y+) is utilized in areas where the turbulence is significantly influenced by the viscosity of the wall. The dimensionless distance from the first grid node to the wall surface, Y+, is calculated using the local viscous length scale [25]. According to the 2011 International Towing Tank Conference (ITTC) guidelines, the Y+ value should fall within the range of 30–100 [25,29]. The calculation can be performed using the formula (6) as follows:

[TeX:] Y^{+} = \frac{U_{*}\gamma}{\upsilon}$Y^{+}=\frac{U_{*}\gamma }{\upsilon }$ (9)

Where [TeX:] U_{*}$U_{*}$ Represents the friction velocity at the closest wall, [TeX:] \gamma$\gamma$ denotes the distance to the nearest wall, and [TeX:] \upsilon$\upsilon$ signifies the local kinematic viscosity of the fluid. Figure 3 presents the Y+ value of the ship, which generally ranges between 20 and 40.

Figure 3 Wall Y+ on catamaran trash skimmer boat at Fr = 0.28 on (a) bottom view, (b) side view

Slika 3. Vrijednosti Y+ na katamaranu za prikupljanje otpada pri Fr = 0,28: (a) pogled odozdo, (b) bočni pogled

3.1. Grid Independence / Neovisnost mreže

The Grid Independence test is performed to achieve an ideal equilibrium between total ship resistance and cell quantity, as seen in Figure 4. The numerical results were achieved using varying quantities of elements. The mesh was globally refined in successive iterations (e.g., coarse → medium →, fine) by uniformly reducing the base cell size across the domain while maintaining the structured topology. This approach ensures consistency in resolving flow features such as boundary layers, free-surface waves, and wake regions, which are critical for ship simulations. The disparity in the quantity of cell numbers is roughly 1.5 to 2 times more than the prior estimate [29]. This is attributable to the constraints of the computer processor used in the solver computation [26,30]. The methodology for the grid convergence index uses the error concept as delineated in Formula 7.

[TeX:] \varepsilon = \left| \frac{f_{1}{- f}_{2}}{f_{1}} \right| \times 100\%$\epsilon =\left|\frac{f_1{-f}_2}{f_1}\right|\times 100\%$ (10)

Where ε denotes the relative approximation error between [TeX:] f_{1}$f_1$ (finer mesh) and [TeX:] f_{2}$f_2$ (coarser mesh). Moreover, the error tolerance or variance between successive results is less than 10%. The grid independence analysis indicates that the ideal simulation of the Trash Skimmer Boat, conducted with [TeX:] 1.08 \times 10^{6}$1.08\times {10}^6$ grids, exhibits a total drag force of 56.1 N at Fr 0.28, accompanied by an optimal resistance variation of 6%.

Figure 4 Graph of grid independence illustrating the relationship between cell number and ship resistance (R_T).

Slika 4. Grafikon neovisnosti mreže koji prikazuje odnos između broja ćelija i otpora broda (RT)

3.2. Total Ship Resistance / Ukupni otpor broda

The investigation into the resistance of the trash skimmer catamaran was conducted using CFD models with varying distances between the hulls (S/L = 0.2, 0.3, 0.4, and 0.5). Figure 5 depicts the correlation between the force in the x-direction and the physical time of the simulation at 15 seconds. The catamaran simulation initiates with a distinct adjustment phase and thereafter stabilizes. Initially, the force fluctuates rapidly as the flow surrounding the boat stabilizes into a pattern. Subsequent to this early phase, the fluctuations diminish significantly, resulting in a more level curve. At the conclusion of the graph, the line approaches a horizontal orientation, indicating that the force remains relatively constant over time.

Figure 5 Force in the x direction (Fx) of CFD trash skimmer boat simulation at S/L 0.2, Fr 0.28 with the function of physical time simulation

Slika 5. Sila u x smjeru (Fx) pri S/L 0,2, Fr 0,28 s funkcijom vremenske simulacije u CFD prikazu broda za prikupljanje otpada

Figure 6 illustrates the relationship between the Froude number and the overall resistance of the vessel, where the Froude number quantifies the influence of vessel speed on resultant resistance. At low Froude numbers, changes in the distance between the demi hulls result in minimal variation in resistance outcomes, a finding consistent with previous research [31]. This indicates that hull separation has negligible influence on total resistance at lower speeds, as frictional resistance dominates the hydrodynamic profile under these conditions. At these speeds, frictional resistance is the dominant factor and is largely unaffected by the interaction between the hulls.

However, as the Froude number increases, a more noticeable difference emerges in resistance based on the demi hull spacing. Vessels with a more significant demi hull distance (S/L = 0.5) experience a more substantial increase in resistance compared to those with smaller spacings. Conversely, the model with an S/L ratio of 0.2 exhibits the least resistance as the Froude number grows. The resistance at higher speeds follows an order from highest to lowest based on the S/L ratio.

Figure 6 Total ship resistance (R_T) for all model variants as a function of the ship's Froude number.

Slika 6. Ukupni otpor broda (RT) za sve varijante modela kao funkcija Froudeova broja

Viscous and pressure forces contribute to ship resistance, as seen in Figure 7. The pressure force (FxP), measured at about 91.1 N, is the primary factor that determines the overall resistance of the catamaran at a Froude number of 0.28. The sum of the pressure and viscous components is measured at 95.3 N, which equals the total force (Fx). Although the force is small and inconsequential, the viscous force (FxV) is much less, contributing just 4.0 N. This effect suggests that, instead of frictional forces caused by the water's viscosity, the ship's hull encounters higher resistance due to pressure distribution on the surfaces.

Figure 7 Resistance component of CFD simulation results of trash skimmer boat at Fr 0.28 at S/L 0.2

Slika 7. Sastavnice otpora prema rezultatima CFD simulacije broda za prikupljanje otpada pri Fr 0,28 i S/L 0,2

3.3. Velocity Contour / Kontura brzine

The simulation results of velocity contours on a trash skimmer boat equipped with a conveyor, conducted in a river condition domain, reveal hydrodynamic phenomena influenced by the ratio of the distance between the hulls (S/L) and the vessel's speed (Vs) are depicted in Figure 8. The river condition significantly affects the flow patterns, as the limited water depth amplifies the effects of total ship resistance, leading to stronger interactions between the flow and the hull surface, especially in the vicinity of the hull [32].

In the area around the hull, flow acceleration is observed, marked by a color gradient from green to yellow, indicating higher velocities of 2.0–3.0 m/s due to the flow interaction with the hull. However, within the space between the hulls, the relative velocity decreases, as shown by the color gradient from blue to green (0.5–1.5 m/s). This reduction in speed is mainly due to pressure buildup in front of the conveyor, which obstructs the flow path and causes deceleration in the demi hull region. In river conditions, this effect is more pronounced as the restricted water depth intensifies the deceleration and turbulence near the hull.

At lower S/L ratios (0.2 and 0.3), reduced hull separation enhances the Venturi effect, accelerating flow around the hulls and increasing wake turbulence behind the conveyor. This results in a more elongated and turbulent wake, leading to higher total ship resistance. Conversely, at higher S/L ratios (0.4 and 0.5), more significant hull separation reduces flow interactions between the hulls, promoting flow stability and a smaller, more uniform wake, which lowers resistance and improves hydrodynamic efficiency. The effect of this stabilizing flow pattern is more potent in river conditions due to the limited space for water movement between the hulls.

The velocity contour patterns demonstrate that both the S/L ratio and vessel speed significantly affect flow behavior and total ship resistance in general. River conditions further amplify the effects of hull separation, as the confined space exacerbates flow deceleration and turbulence. A larger S/L ratio results in a more consistent flow, a smaller wake, and reduced resistance, while a smaller S/L ratio accelerates flow but increases turbulence and drag. The conveyor's presence, which obstructs the flow path between the hulls, increases deceleration, particularly at higher speeds and in river conditions. Therefore, optimizing the S/L ratio while considering the effects of river conditions is crucial for maximizing hydrodynamic efficiency, especially during high-speed operations, as it directly influences both flow dynamics and vessel performance.

Figure 8 Visualization of velocity contour obtained through CFD simulation results of trash skimmer boat with S/L variations ranging from 0.2 to 0.4 at three distinct speeds: 3, 4, and 5 knots.

Slika 8. Vizualizacija konture brzine dobivene rezultatima CFD simulacije broda za prikupljanje otpada s varijacijama S/L od 0,2 do 0,4 pri trima različitim brzinama: 3, 4 i 5 čvorova

3.4. Wave Elevation / Visina valova

The simulation results for the trash skimmer catamaran, equipped with a conveyor situated between the hulls, demonstrate the influence of varying speeds and hull spacing (S/L ratios of 0.2, 0.3, 0.4, and 0.5) on wave elevation, as seen in Figure 9. Conducted in a river condition domain, these simulations reveal how limited water depth can accentuate specific hydrodynamic effects, especially at higher speeds.

At lower speeds (3 and 4 knots), the wave patterns show minimal changes across different S/L ratios, with wave crests reaching up to 0.06 meters above the draft. This indicates that at these speeds, the conveyor's effect on wave elevation is negligible, and the vessel experiences a relatively constant flow with no significant variations in wave height due to hull spacing. The river conditions do not appear to alter these wave patterns significantly at lower speeds.

However, at higher speeds (5 knots), a significant increase in wave height is observed, with the maximum wave height reaching 0.26 meters over the ship's draft. At elevated velocities, the interaction between the water flow and the conveyor becomes more pronounced, leading to more substantial wave fluctuations and heightened turbulence. River conditions contribute to this effect by limiting the vertical space for wave formation, amplifying the intensity of the waves and turbulence near the hulls [33]. The wave patterns show a clear distinction between the areas before and after the conveyor, indicating increased energy dissipation and turbulent flow.

The impact of the S/L ratio becomes more pronounced at 5 knots. Smaller S/L ratios (0.2 and 0.3) produce larger wave amplitudes before the conveyor, while larger S/L ratios (0.4 and 0.5) result in more stable and uniform wave configurations. Despite these differences, the wave height remains significantly elevated at higher speeds, regardless of hull spacing, suggesting that the conveyor's flow disruptions become more prominent as speed increases. River conditions exacerbate these effects, as the restricted depth limits the flow dynamics and intensifies the wave patterns.

In summary, while changes in the S/L ratio have minimal effects on wave height at lower speeds, they become more significant at higher velocities, where increased hull separation stabilizes the waves. The presence of the conveyor between the hulls increases wave height and turbulence, particularly in river conditions, where the limited depth amplifies these effects. To optimize hydrodynamic efficiency, it is crucial to consider both vessel speed and proper hull spacing, especially in confined environments such as river conditions, where these factors are magnified.

Figure 9 Wave elevation was determined by the CFD simulation results of the trash skimmer boat with S/L variation of (a) 0.2, (b) 0.3, (c) 0.4, and (d) 0.5

Slika 9. Visina valova određena je rezultatima CFD simulacije broda za prikupljanje otpada s varijacijama S/L od (a) 0,2, (b) 0,3, (c) 0,4 i (d) 0,5