Ka4-55 Series Based Toroidal Propeller Design

2.1 Propeller geometry and design parameters

A toroidal propeller consists of a front and a rear blade set, both derived from the Kaplan Ka4-55 series profile. The front and rear blades are designed with an Expanded Area Ratio (EAR) of 0.55. Both blades have identical chord lengths of 62.2 mm, and this blade is connected at its tips through a parabolic rake geometry, forming a continuous closed-loop shape [12]. The rear blade is rotated by a 15° generator line angle relative to the front blade to improve alignment with the flow path and simplify the tip integration.

The skew distribution applied to the front blade follows the pattern used in the B-Series propellers, with a total skew angle of 20.44°. The rake is applied parabolically, with a maximum rake angle of 30°, providing a smooth axial displacement between the front and rear blades. The detailed drawings of the front and rear blades are presented in Figures 1 and 2, respectively. A three-dimensional view of the blades before connection is shown in Figure 3, while the complete toroidal configuration is illustrated in Figure 4. These geometric adjustments support the structural connection at the blade tips and assist in shaping the propeller into an integrated, looped configuration.

Figure 1. Front side toroidal propeller design

Figure 2. Back side toroidal propeller design

Figure 3. Front and back toroidal propeller design

Figure 4. 3D model of toroidal design

The tips of the blades are truncated at r/R = 0.8, enabling a smooth structural connection between front and rear blade tips. This configuration forms the defining feature of the toroidal design: a seamlessly joined blade loop intended to reduce tip vortex and enhance propeller efficiency. The pitch ratio (P/D) of the front blade is fixed at 1.0, while the rear blade uses a P/D ratio of 1.25. The pitch distribution is kept identical to that of the original Kaplan series.

The left side of Figures 5-7 corresponds to the upstream (inflow) region, while the right side represents the downstream (outflow) region. The geometry resulting from the intersection of the front and rear propeller foils in the toroidal configuration exhibits a distinct arrangement, as illustrated in Figures 5-7. In this design, the rear propeller blades are not positioned in line with the front propeller blades. Consequently, the downstream flow generated by the front propeller is not directly utilized by the rear propeller in a conventional tandem manner. Additionally, the spacing between the foils increases progressively toward the upper region of the blade. This configuration is intentionally applied to create a smoother connection between the blade tips, ensuring a continuous and seamless transition for the toroidal propeller shape. This gradual variation in spacing is aimed at minimizing flow separation and optimizing the overall hydrodynamic performance of the toroidal propeller system.

Figure 5. Toroidal foil profile at 0.2 R

Figure 6. Toroidal foil profile at 0.4 R

Figure 7. Toroidal foil profile at 0.8 R

2.2 CFD setup

In modeling turbulent fluid flow around a propeller, the Reynolds-Averaged Navier–Stokes (RANS) equations are commonly employed to capture the variations in flow velocity, pressure, and vorticity [13, 14]. These equations, derived from the Navier–Stokes formulation, consider the advection and diffusion mechanisms that significantly influence the temporal change of velocity distribution around the propeller.

$\frac{\partial u_i}{\partial x_i}=0$                     (1)

$\begin{gathered}\frac{\partial u_i}{\partial t}+\frac{\partial u_i u_j}{\partial x_i}=-\frac{1}{\rho} \frac{\partial P}{\partial x_i}+\frac{\partial}{\partial x_j}\left(\frac{\mu \partial u_i}{\partial x_j}\right) +\frac{\partial}{\partial x_i}\left(\mu \frac{\partial u_i}{\partial x_j}\right)-\frac{\partial}{\partial x_j}\left(-\rho \overline{u_i^{\prime} u_j^{\prime}}\right)\end{gathered}$                     (2)

Advection accounts for the movement of fluid itself, while diffusion represents the effects of fluid viscosity. As noted in the formulation:

$\frac{\partial}{\partial x_j}\left(\mu \frac{\partial u_i}{\partial x_j}\right)+\frac{\partial}{\partial x_k}\left(\mu \frac{\partial u_i}{\partial x_k}\right)$                     (3)

These terms describe how changes in momentum and velocity gradients occur due to internal fluid stresses. Additionally, the pressure gradient term, $-\frac{\partial p}{\partial x_j}$ plays a critical role in driving the flow dynamics.

The k-omega turbulence model is a widely used RANS-based approach that solves two additional transport equations for turbulent kinetic energy (k) and specific dissipation rate (ω), representing turbulence energy and its dissipation. The dissipation rate is defined as ε = β·kω, with β = 0.09. Variants such as the standard, BSL, and SST k-omega models offer improved performance for specific flow conditions [15].

$\epsilon=C_\mu k \omega$                     (4)

Transport equation for turbulent kinetic energy (k) is given as follows:

$\begin{gathered}\frac{\partial(\rho \epsilon)}{\partial t}+\frac{\partial\left(\rho U_i \epsilon\right)}{\partial x_i}=\frac{\partial}{\partial x_j}\left[\left(\mu+\frac{\mu_t}{\sigma_\epsilon}\right) \frac{\partial \epsilon}{\partial x_j}\right]+\frac{\gamma}{v_t} P_k -\beta \rho \omega^2+\frac{2 \rho \sigma_\omega^2}{\omega} \Delta k: \Delta \omega\end{gathered}$                     (5)

where,

$\Delta k: \Delta \omega=\frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}$                     (6)

The term,

$\frac{2 \rho \sigma_\omega^2}{\omega} \Delta k: \Delta \omega$                     (7)

The hydrodynamic performance of the toroidal propeller was analyzed using CFD simulations under steady-state conditions. The flow was assumed to be incompressible, and the governing equations used were the RANS equations. Turbulence effects were modeled using a two-equation model capable of handling adverse pressure gradients and flow separation, which are critical phenomena around marine propellers [16-19].

Figure 8. Domain size

The computational domain was created in accordance with the guidelines provided by the International Towing Tank Conference (ITTC) for open water propeller simulations. The upstream distance from the propeller to the inlet boundary was set to at least 2D, the downstream outlet extended to 6D, where D is the propeller diameter. The radial boundary distance from the axis of rotation to the domain wall was set to 3D, ensuring minimal blockage and reflecting realistic open water conditions (Figure 8) [20].

An unstructured mesh was generated with local refinement around the propeller blades and wake region. Mesh independence studies were conducted to verify that further mesh refinement did not significantly influence the simulation results. Boundary conditions included a uniform inflow velocity, a fixed rotational speed for the propeller, and a static pressure outlet to emulate an open water environment.

To ensure the reliability of the numerical method, simulation results for the benchmark Kaplan Ka4-55 propeller were compared against available open water test data. The comparison focused on KT, KQ, and open water efficiency. The CFD predictions showed acceptable accuracy, with deviations under 5% for KT and 10% for KQ [13, 20], which are consistent with ITTC-recommended tolerances for numerical validation in open water propeller analysis.

2.3 Propeller geometry and design parameters

The characteristics of ship propellers can be expressed in non-dimensional form using the advance coefficient (J), thrust coefficient (K_T), torque coefficient (K_Q), and efficiency (η) [21, 22]. These three coefficients are used to construct performance curves that illustrate how the propeller operates under various ship operating conditions. Variations in the K_T and K_Q curves with respect to changes in the advance ratio provide insights into the propeller’s performance in relation to ship speed and propeller rotation. Each type of propeller has its own unique performance curve characteristics; therefore, the study of propeller characteristics cannot be generalized to all shapes or types of propellers [23].

$J=\frac{V_a}{n D}$                      (8)

$K_T=\frac{T}{\rho n^2 D^4}$                      (9)

$K_Q=\frac{Q}{\rho n^2 D^5}$                      (10)

$\eta=\frac{J K_T}{2 \pi K_Q}$                      (11)

In this context, T is thrust (N), Q is torque (Nm), ρ is water density (kg/m³), n is rotational speed (rad/s), and D is propeller diameter (m).

4.1 Segmented blade analysis

Based on the results, the toroidal simulation provides efficiency but has a wider range of advanced coefficients. However, the main problem is that the torque required by the propeller reaches almost 2 times greater than the torque of the Ka4-55 propeller. This makes the overall toroidal have no advantages. Therefore, further analysis is needed regarding the cause of this propeller becoming more labor-intensive than its original form.

The performance of the Ka4-55 propeller is getting worse than that of the Ka4-55, so the propeller is made into several segments as shown in Figure 11. From this segment, it is known that the root part of the propeller does not have a significant impact on the thrust of the propeller. So that this part functions more as a propeller blade amplifier. When the tip section is modified in such a way, the potential thrust and torque values will also change.

Figure 11. Propeller segmentation

The results in Tables 5-7 of the open water simulation showed an interesting performance pattern while leaving anomalies, especially in the rear toroidal blade in low advance coefficient conditions (J = 0.2 and 0.4). Under these conditions, the rear blade toroidal produces a negative thrust coefficient (KT) value of up to -0.0026, indicating an obstacle force compared to thrust. This statement is supported by the pressure distribution shown in Figure 12, where the back propeller exhibits negative values and significantly lower pressure compared to the front propeller. This phenomenon is amplified by the torque coefficient (10KQ) of the rear blade, which is close to zero or even negative, which generally occurs due to unstable flow or local stalls in certain areas of the blade. This situation is inversely proportional to the tip blade of toroidal, which shows a significant increase in performance compared to the conventional propeller Ka4-55, both in terms of KT, 10KQ, and efficiency in the entire J range, especially at the high advance coefficient.

Table 5. KT propeller simulation results (1 blade)

Table 6. 10KQ propeller simulation results (1 blade)

Table 7. Propeller efficiency (1 blade)

Figure 12. Pressure distribution of 0.375 r/R

The condition of the rear blade that experienced negative performance at low J is suspected to be caused by a suboptimal flow interaction between the front and rear blade, so that the rear blade receives the remaining flow (wake) with too high an angle of attack. The initial hypothesis states that the equal pitch distribution between the front and rear blades and the P/D = 1.25 configuration may cause the rear blade to overload in low J conditions. Therefore, further evaluation of pressure distribution, wake flow patterns, and possible pitch or angle of attack adjustments on the rear blade is needed to optimize performance at low advance coefficients without sacrificing efficiency advantages at high J.

4.2 Flow visualization and interpretation

From the visualization image of the X-axis relative flow velocity, the difference in speed distribution between the conventional Ka4-55 propeller and the toroidal propeller in both radial positions can be seen. At the position r/R = 0.75, as shown by Figures 13 and 14, the difference becomes more pronounced. The Ka4-55 propeller still shows a sharp, symmetrical wake tip around the blade tip. In contrast, in toroidal, two wake streams converge in the blade tip area due to the integration between the blades, with a larger wake area and a low-speed flow (blue-green color) spreading more backwards. This can indicate a more intense wake interaction and a potential vortex tip reduction due to the splicing of the blade tip.

Figure 13. Ka4-55 0.75 r/R

Figure 14. Toroidal propeller 0.75 r/R

At the r/R = 0.375 position, the flow around the Ka4-55 propeller shows a relatively uniform velocity distribution around the blades as described in Figures 15 and 16, with the wake area behind the propeller being concentrated. Meanwhile, in toroidal, there are two separate wake distributions due to the presence of integrated front and rear blades, where the wake or turbulence from the front blades slightly interacts with the rear-blade, resulting in a wider and more complex distribution of speed.

Figure 15. Ka4-55 0.375 r/R

Figure 16. Toroidal propeller 0.375 r/R

The authors would like to thank Marine Manufacturing and Design (MMD) Laboratory of Marine Engineering, Sepuluh Nopemebr Institut of Technology for the Software License.