A Parametric Study on the Effect of Blade Configuration in a Double-Stage Savonius Hydrokinetic Turbine

Abstract

Ocean energy represents a promising resource for renewable energy generation. Hydrokinetic turbines (HKTs) provide a sustainable method to extract energy from ocean currents. However, turbine efficiency remains limited, particularly in marine environments with low flow velocities. A parametric evaluation of blade configurations is conducted in this study to assess their effect on the power and torque performance of a double-stage drag-based Savonius HKT. Numerical simulations are conducted using the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations with the k-ω SST turbulence model. The numerical model is validated against published data, and analyses on mesh density, domain size, and time step are performed to ensure accuracy. Three blade configurations—(0°, 0°), (0°, 45°), and (0°, 90°) are evaluated under flow velocities of 0.6 m/s, 0.8 m/s, and 1.0 m/s. Results indicate that blade configuration significantly affects turbine performance. The (0°, 0°) configuration performs best at high flow velocity (1.0 m/s), while the (0°, 45°) setup achieves the highest efficiency at 0.6 m/s. The (0°, 90°) configuration performs the least effectively across all conditions. A similar performance trend is observed for the torque coefficient. This study recommends selecting blade configurations based on flow velocity, providing design guidance for double-stage HKTs operating in varying marine conditions.

1. Introduction

The development of renewable energy technologies is accelerating due to the depletion of non-renewable resources and the impacts of climate change [1]. Non-renewable resources like fossil fuel reserves have declined annually as global energy consumption has grown [2]. The International Energy Agency (IEA) reported a 2.2% increase in global electricity demand in 2023. The demand is expected to grow at an average rate of 3.4% annually until 2026 [3]. This rising energy demand has driven the exploration of alternative renewable sources.

Hydropower is a major contributor to global renewable electricity. The IEA reported that hydropower capacity reached 1360 GW in 2023, supplying 17% of global electricity and 60% of all renewable electricity generation [4]. Its technical potential remains high, with significant untapped resources in Africa, Asia, and Latin America [5].

Most hydropower is generated through dam-based systems. These systems control water flow using reservoirs to optimize energy output [6]. However, they create environmental and social concerns such as ecosystem disruption, displacement of communities, and methane emissions from submerged vegetation [7]. Alternatives such as run-of-river hydropower and hydrokinetic turbines (HKTs) provide lower environmental impact.

HKTs generate electricity without altering river flow or requiring large-scale infrastructure. Hydrokinetic energy can also be extracted from river currents, tidal streams, and ocean currents and waves. It has an estimated global potential exceeding 300 GW [8]. Therefore, countries with extensive river networks and long coastlines have considerable hydropower development opportunities [9].

HKTs extract energy from moving water streams [10]. The generated electricity is either transmitted to the power grid or processed through a commutator for battery storage, depending on the application and energy demand [11]. HKTs are classified based on rotor axis orientation relative to the incoming flow [10,12]. Axial flow turbines have a rotational axis that is parallel to the flow direction, while crossflow turbines have an axis positioned at a 90-degree angle to the flow. The axial flow turbines primarily generate rotation through lift forces, whereas the crossflow turbines operate mainly based on drag forces. The classification of HKTs based on the flow orientation and structural configuration are shown in Figure 1.

Crossflow turbines are commonly used in small-capacity power generation, particularly in low-flow environments. These systems require minimal maintenance and allow for electrical components to be positioned above the water surface, making them suitable for various applications. Among them, Savonius turbines are favoured for their simple construction and ease of modification [13]. They are particularly effective at operating at low fluid speeds [14], making them suitable for renewable energy deployments.

Several CFD studies have investigated drag-based hydrokinetic turbines using 2D and 3D simulations. These studies focused on optimizing blade numbers, blade overlap, and duct augmentation for low-flow conditions. The URANS model with SST turbulence has been commonly used due to its balance between computational cost and accuracy. Alizadeh et al. [15] evaluated the influence of blade profile on torque and power output. Muratoglu and Yuce [16] studied ducted Savonius turbines and reported improved performance under low Reynolds numbers. Nag and Sarkar [17] validated their CFD model against experiments and showed reliable power coefficient predictions across a range of tip speed ratios. These works show that CFD can capture key flow features such as vortex formation, pressure drag, and operation under low-velocity marine flow.

Different geometric strategies have been proposed to improve the performance of drag-based turbines, including blade number optimization [10,18,19], endplate size enhancement [20] and augmentation design [21]. However, their application in marine environments remains limited. The global ocean energy potential is 45,000–130,000 TWh per year [22]. Despite this, only 528 MW of ocean energy capacity was installed globally as of 2023 [23], with minimal growth from 2017 to 2023. This reflects the slow adoption of marine hydrokinetic technologies and the need for further research and development.

Multi-stage HKTs have shown a potential to increase energy capture compared to conventional single-stage designs. Kumar et al. [24] demonstrated that increasing the number of stages improves the turbine’s efficiency by enhancing flow interaction and energy extraction. This approach is considered effective in low-flow environments, but few numerical studies have evaluated inter-stage blade configuration under marine flow conditions.

This study is built on the previous work, where a drag-based Savonius turbine was optimized through blade number, blade shape, deflector, and diffuser design [25,26,27,28,29]. The present study extends this design into a two-stage configuration, where each stage carries a set of blades. A parametric evaluation is conducted to assess the effect of blade configuration on the turbine’s power and torque output. Blade configuration is defined as the angular offset between the blades of Stage 1 and Stage 2 in a two-stage turbine. Stage 1 is fixed at 0°, while Stage 2 is rotated by 0°, 45°, or 90°, resulting in three configurations: (0°, 0°), (0°, 45°), and (0°, 90°). Simulations are conducted across different flow velocities to evaluate their power and torque performances. The results support a configuration-based design approach for improving turbine efficiency in low-velocity environmental conditions.

2. The Hydrokinetic Turbine Model

2.1. Single-Stage Hydrokinetic Turbine (HKT) for Numerical Setup Validation

A three-dimensional computer-aided design (CAD) model of an optimised drag-based Savonius HKT is used for numerical setup validation. The model is developed based on previous works by Ng et al. [26]. It consists of a two-blade rotor profile, a deflector, and a diffuser to improve the local flow velocity. The turbine geometry is shown in Figure 2, while the geometrical parameters are labelled in Figure 3 and listed in

Table 1. Geometrical parameters of the HKT model.

Parameter	Dimension	Unit
Rotor Width (Ly)	1.23	m
Overall Turbine Length (Lx)	2.08	m
Rotor Diameter (D)	1.0	m
Vertical gap between diffuser walls and the blades (d1)	0.05	m
Horizontal gap between diffuser walls and the blades (d2)	0.0615	m
Deflector Span (Ld)	0.68	m
Deflector Angle (${\theta }_d$)	37°
Deflector and Diffuser Thickness (td)	0.07	m
Diffuser Inlet Length (Li)	0.25	m
Diffuser Outlet Length (Lo)	0.80	m
Inlet Convergence Angle (${\theta }_i$)	20°
Outlet Divergence Angle (${\theta }_o$)	15°
Rotor Aspec Ratio (AR)	1.23
Turbine Swept Area (A)	1.23	m2

. The turbine geometry adopts the design parameters from Maldar et al. [28].

2.2. Double-Stage Hydrokinetic Turbine (HKT) Configurations for Performance Analysis

A parametric study is conducted to evaluate the effects of blade configuration in a double-stage drag-based HKT. Each configuration consists of two identical rotor stages arranged vertically along a shared shaft. The first stage remains fixed at 0°, while the second stage is rotated by 0°, 45°, or 90° to form three blade configurations: (0°, 0°), (0°, 45°), and (0°, 90°). These configurations are designed to investigate the impact of inter-stage angular offset on power performance. Simulations are carried out under three inlet velocities: 0.6 m/s, 0.8 m/s, and 1.0 m/s.

Figure 4 illustrates the double-stage hydrokinetic turbine model with a 90° angular offset as an example. The primary aim of this study is to investigate the impact of blade-stage configurations on the turbine performance. The illustrated configuration is selected for visual representation.

3. Numerical Methodology

A computational fluid dynamics (CFD) approach is applied using the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations with the k-ω SST turbulence model.

3.1. Governing Equations and Solver Setup

Numerical simulations are performed using the SimScale platform (Version 3.8, 2025 release), which is a cloud-based CFD software built on OpenFOAM solvers [30]. A transient unsteady solver is used with second-order implicit time integration. Turbine rotation is modeled using a sliding mesh with a dynamic interface between rotating and stationary regions. The governing equations are discretized using the finite volume method. Torque was recorded at each time step using a moment control function applied to the rotor surface. Figure 5 presents the transient torque output for TSR = 0.64. The plot shows periodic behaviour after the initial startup. Time-averaged values were extracted from stabilised revolutions and used for the subsequent calculations.

The governing equations describing the conservation of mass and momentum for incompressible flow are given as follows [31]:

$$\nabla ·\stackrel{-}{\mathrm{v}}=0,$$

(1)

$$\rho {\displaystyle \frac{\partial \stackrel{-}{\mathrm{v}}}{\partial t}}+\rho (\stackrel{-}{\mathrm{v}}·\nabla )\stackrel{-}{\mathrm{v}}=-\nabla \stackrel{-}{p}+\mu {\nabla }^2\stackrel{-}{\mathrm{v}}+\rho g,$$

(2)

where

• $\stackrel{-}{\mathrm{v}}$ is the mean velocity vector;
• $\rho$ is the fluid density;
• $\stackrel{-}{p}$ is the fluid pressure;
• $\mu$ is the dynamic viscosity;
• g is gravitational acceleration.

The k-ω SST turbulence model is selected for this study due to its ability to accurately predict near-wall boundary layer behaviour while maintaining the performance in the free-stream region. The model integrates the k-ω formulation near walls with the k-ε model in the outer flow and was originally developed by Menter [32]. The model is suitable for flows involving wall-bounded regions and free-stream interactions, such as those around HKT blades. The governing equations for turbulence transport are expressed as follows [33]:

Turbulent kinetic energy (k):

$${\displaystyle \frac{\partial \left(\rho k\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho U_{i}k\right)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]+P_k+P_b-\rho ϵ+S_k$$

(3)

Dissipation rate $\left(\omega \right)$:

$${\displaystyle \frac{\partial \left(\rho ϵ\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho U_iϵ\right)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{ϵ}}}\right){\displaystyle \frac{\partial ϵ}{\partial x_j}}\right]+{{\displaystyle \frac{\gamma }{v_t}}P}_k+\beta \rho {\omega }^2+{\displaystyle \frac{2\rho \sigma {\omega }^2}{\omega }}\Delta k:\Delta \omega ,$$

(4)

where:

• ${\sigma }_k,{\sigma }_{ϵ}$ are the turbulent Prandtl numbers for k and $ϵ$;
• $P_k$ is the turbulent kinetic energy (TKE) due to mean velocity shear;
• $P_b$ is the TKE due to buoyancy;
• $S_k,S_{ϵ}$ are the user-defined source term.

The turbulence intensity at the inlet is given as follows [34]:

I = 0.16 Re^−1/8,

(5)

where Re is the Reynolds number based on the inlet velocity (U) and characteristic length (D). For this study, U= 1 m/s, D = 1 m, ν = 1.0 × 10⁻⁶ m²/s, and Re = 9.98 × 10⁵ are resulting I = 0.028.

For initialisation, the turbulent kinetic energy (k) and specific dissipation rate (ω) are given as follows:

$$k={\displaystyle \frac{3}{2}}{(U·I)}^2,$$

(6)

$$\omega ={\displaystyle \frac{\sqrt{k}}{C_{\mu }^{0.25}·L}},$$

(7)

where C_μ = 0.09 and L = 0.07D. The turbulent kinetic energy (k) is 1.21 × 10⁻³ m²/s². The specific dissipation rate (ω) is 0.9091 s⁻¹. These values define the initial conditions for the model setup. During the parametric study, these parameters are adjusted based on changes in flow velocity and Reynolds number to ensure accurate performance evaluation.

3.2. Computational Domain and Boundary Conditions

The computational domain is defined through domain size analysis. A single-stage HKT model developed by Ng et al. [26] is used to validate the numerical setup. To minimize blockage effects while ensuring computational efficiency, five domain configurations are tested. Each configuration is expressed in terms of turbine diameter (D), with each subsequent size reduced by 10%. The tested domain dimensions are presented in

Table 2. Domain size configurations.

Domain	Length	Width	Height
D1	10D	5D	5D
D2	9D	4D	4D
D3	8.1D	3.6D	3.6D
D4	7.29D	3.24D	3.24D
D5	6.56D	2.916D	2.916D

.

Torque output and runtime are evaluated for each domain size. The domain size analysis results are presented in

Table 3. Domain size analysis results.

Domain	Runtime (h)	Torque (Nm)	Relative Change (%)
D1	14.17	174.45	-
D2	11.23	170.23	2.42
D3	9.90	172.99	2.48
D4	8.05	173.46	0.27
D5	7.83	176.80	1.93

. The findings indicate that smaller domains reduce total runtime and improve computational efficiency. Marsh et al. [35] mentioned that the domain is considered unaffected by boundary interference if the measured variable, torque in this case, changes by less than 3% with increasing domain size. The results show that all configurations satisfied the 3% performance variation criterion. Therefore, D3 is selected as the reference domain due to its balance between numerical accuracy and reduced computational time efficiency. The relative change (%) is calculated as the difference in torque between two domain sizes and relative to the torque of the preceding domain.

The selected D3 domain is used for all subsequent simulations. The inlet and outlet boundaries are set to be 4.05D measured from the upstream and downstream of the turbine centre, respectively. Several CFD studies on hydrokinetic turbines have adopted a 4D inlet distance measured from the inlet boundary to the turbine model to ensure undisturbed upstream flow. For example, Liu et al. [36] used a 4D inlet distance in their simulations of a horizontal-axis turbine to ensure upstream flow stability. Du et al. [37] also applied a 4D inlet distance in diffuser-augmented turbine simulations to define the free-stream velocity. These studies confirm that the distance between the inlet boundary and the model used in the present study is sufficient to maintain free-stream conditions. Additionally, a similar inlet boundary distance was applied in the validated simulations by Maldar et al. [25,28,29] and Ng et al. [26,27], which used similar turbine profiles and numerical methods.

Figure 6 shows the computational domain and boundary conditions used in this study. A velocity inlet is applied at the inlet boundary, and a zero-gauge pressure condition is imposed at the outlet boundary. The outlet pressure (P) is set to zero, a standard condition in numerical simulations to prevent backflow events [15]. Symmetrical boundary conditions are assigned to the top, bottom, and side faces. A no-slip condition is applied to the remaining faces to maintain zero velocity relative to the wall.

The dimensionless wall distance (y⁺), which determines mesh resolution near the turbine surface, is also evaluated. Due to the complexity of the turbine geometry and rotating motion, the k-ω SST model with wall functions is applied. According to Menter [32], the model can be applied under moderate-to-coarse near-wall resolution in the logarithmic region, provided that appropriate wall treatment is used [34].

The computational domain consists of a stationary zone and a rotating zone. The rotating zone encloses the turbine and rotates over time to capture blade motion. A transient unsteady RANS solver with a sliding mesh approach is used. The interface between the stationary and rotating zones is handled using the Arbitrary Mesh Interface (AMI) method. This method transfers fluid variables between non-conforming mesh interfaces at each time step. It allows for the rotating mesh to move through the static domain without requiring remeshing [38]. This setup enables smooth momentum exchange and resolves time-dependent flow characteristics during turbine rotation.

3.3. Simulation Conditions and TSR Matrix

The simulations are conducted under varying inflow velocities (U) and tip speed ratios (TSRs) to evaluate turbine performance across different flow conditions. Three inflow velocities are considered: 0.6 m/s, 0.8 m/s, and 1.0 m/s. At each velocity, four TSR values are tested: 0.4, 0.6, 0.8, and 1.0.

The rotational speed (ω) for each case is calculated based on the prescribed TSR and inflow velocity using the following equation [39]:

$$\omega ={\displaystyle \frac{\mathrm{TSR}\times U}{R}},$$

(8)

where ω is the rotational speed (rad/s), U is the inflow velocity (m/s), and R is the rotor radius, which is set at 0.5 m in this study. This relationship ensures that the turbine operates at the desired TSR by adjusting the rotational speed accordingly for each simulation case. The dynamic TSR matrix summarizing the corresponding rotational speeds is presented in

Table 4. Dynamic TSR matrix for the HKT simulation cases (rotational speed ω in rad/s).

Inlet Velocity, U (m/s)	Rotational Speed, ω (rad/s)
	TSR = 0.4	TSR = 0.6	TSR = 0.8	TSR = 1.0
0.6	0.48	0.72	0.96	1.20
0.8	0.64	0.96	1.28	1.60
1.0	0.80	1.20	1.60	2.00

.

3.4. Mesh Density Analysis

The computational mesh is generated using the integrated hex-dominant automatic meshing tool provided by the SimScale platform [40]. An unstructured finite volume mesh is created, consisting primarily of hexahedral elements, with tetrahedral and prism cells applied where necessary to capture complex geometries.

A mesh density analysis is performed to ensure accuracy and efficiency. The domain consists of a single fluid region that surrounds the rotating zone. Four refinement regions are applied to control mesh resolution. The base region defines the full domain extent with the coarsest mesh. The far-field region surrounds the outer boundaries and supports a stable inflow and wake dissipation. The close-field region surrounds the rotating zone and is refined to match the interface resolution. The rotating zone encloses the turbine and rotates using a sliding mesh. The interface between the rotating and stationary zones is handled using the Arbitrary Mesh Interface (AMI). The close-field region improves mesh compatibility at the AMI interface and prevents abrupt cell transitions. This refinement structure improves resolution only where needed and maintains solution stability across the domain.

Table 5. Mesh density analysis detail and results.

Refinement Level (RL)	Cell Count (×106)	Torque (Nm)	Relative Change (%)
1	1.2	138.02	-
2	3.3	174.45	26.4
3	4.3	175.43	0.6
4	6.1	171.40	2.3

presents the details and results of the mesh density analysis. Four refinement levels (RLs), ranging from coarse to fine mesh, are tested. The torque output increases significantly from RL 1 to RL 2. The relative change reduces after RL 2, and a stable torque value is observed. A slight drop in torque is noted at RL 4. This behaviour is attributed to improved resolution of local flow separations and wake dissipation. As the mesh becomes finer, the solver captures smaller-scale turbulence structures and boundary layer losses more accurately. This leads to a marginal decrease in predicted torque compared with intermediate mesh levels, which may slightly overestimate performance due to residual numerical diffusion. This outcome is consistent with expected CFD behaviour and does not indicate a loss of accuracy. The torque values remain stable from RL 2 onward. Therefore, it is selected for all subsequent simulations. This level provides sufficient resolution near the turbine while maintaining manageable computational costs. Torque performances across RL 1 to RL 4 are shown in Figure 7.

Figure 8 illustrates the mesh refinement zones at RL 2. The turbine and diffuser mesh define the fluid–solid interface and enforce no-slip conditions. Surface refinement and inflated boundary layers are applied to resolve shear layers and improve near-wall velocity prediction. The flow is not solved inside the solid regions.

3.5. Time-Step Analysis

A time-step analysis is conducted to determine the optimal time step for transient simulation. The validated domain size and refinement level are used to ensure spatial accuracy. Three time steps (0.01 s, 0.005 s, and 0.0025 s) are tested.

Table 6. Time-step size analysis results.

Time-Step	Runtime (hrs)	Azimuthal Increment (°/Step)	Torque (Nm)	Relative Change (%)
0.01	8.05	0.92	173.46	-
0.005	16.20	0.46	179.60	3.54
0.0025	36.67	0.23	183.21	2.01

presents the time-step analysis results, including the corresponding azimuthal angle increments at TSR = 0.8. These are 0.92°, 0.46°, and 0.23° per step, respectively, illustrating the temporal resolution relative to rotor motion.

At lower inflow velocities and TSR values, the turbine operates at low rotational speeds ranging from 0.48 to 2.00 rad/s, as summarised in Table 4. This results in longer physical times required to complete one full rotation, consistent with typical operating characteristics of hydrokinetic turbines, as reported by Kolekar and Banerjee [41].

Torque increases as the time-step decreases, with a 3.54% increase from 0.01 s to 0.005 s and a 2.01% increase at 0.0025 s. Smaller time steps yield slightly higher accuracy but also require significantly longer simulation time. The 0.0025 s simulation takes over four times longer to complete compared with the 0.01 s simulation. The small variation in torque output with different time steps confirms that the simulation results are invariant with respect to the step angle within acceptable CFD accuracy standards. Yin et al. [42] and Luo et al. [43] showed that variations below 5% are acceptable in unsteady simulations, where a medium time step accurately captured rotating flow behaviour with less than 5% change in predicted forces. Therefore, the time step of 0.01 s is selected for the subsequent simulations. The selected time step ensures consistent performance trends while maintaining manageable simulation time.

Recent studies have shown that a time step of 0.01 s is acceptable for rotating turbine simulations. For example, Diego et al. [44] and Bao et al. [45] demonstrated that maintaining azimuthal increments below 1° per step ensures accurate transient torque prediction. In the present study, the 0.01 s time step results in a 0.92° increment per step, which aligns with these standards and supports the selected configuration.

3.6. Performance Parameters

The tip speed ratio (TSR) is a key parameter in evaluating HKT performance. It represents the ratio of the rotational speed of the rotor (ω) to the free-stream velocity (U), which is given as follows [10]:

$$\mathrm{TSR}={\displaystyle \frac{\omega R}{U}},$$

(9)

where:

• ω is the rotational speed (rad/s);
• R is the rotor radius (m);
• U is the free-stream velocity (m/s).

In this study, simulations are executed for at least three complete cycles of the turbine rotation to ensure a steady periodic state with minimal transient effects. The periodic state is identified when torque values stabilised over successive cycles. However, it is observed that only TSR values of 0.4 and above met this convergence requirement. Lower TSR values exhibit prolonged transient effects, leading to unsteady fluctuations in the torque values. As a result, these cases are excluded from further analysis to maintain data reliability. This approach ensures that the evaluated TSR cases reflect the steady-state turbine performance and minimise uncertainties associated with transient behaviour.

The power coefficient (C_p) and torque coefficient (C_t) are essential parameters for analysing the effect of HKT blade arrangement angles on the power output performance. The C_p measures the turbine’s efficiency in extracting energy from the fluid flow [24]. The C_t evaluates the rotational force exerted by the fluid on the turbine blades [46]. These coefficients are defined as:

Power coefficient (C_p):

$$C_p={\displaystyle \frac{P_{output}}{P_{fluid}}}={\displaystyle \frac{\tau \omega }{0.5\rho A{U}^3}},$$

(10)

Torque coefficient (C_t):

$$C_t={\displaystyle \frac{\tau }{0.5\rho A{U}^{2}R}}$$

(11)

where:

• P_output is the power output;
• P_fluid is the power available in the fluid;
• τ is the average peak torque (Nm);
• A is the turbine swept area (m²).

3.7. Model Validation

The HKT model, as shown in Figure 2 and Figure 3, is developed and validated before proceeding with further analysis. The torque (T), power coefficient (C_p), and torque coefficient (C_t) values of the model are compared to the simulation results of Maldar et al. [29] to ensure accuracy. Key input parameters, including a flow velocity of 1 m/s and a tip speed ratio (TSR) of 0.8, are selected based on the study. Validation is performed at TSR = 0.8 only, as it represents a typical operating point for drag-based turbines. The reference study also uses the same TSR for validation, allowing direct benchmarking using a consistent setup.

Table 7. Turbine model validation results.

Parameters	Literature [29]	Present Study	Difference (%)
Constant tip-speed ratio, TSR	0.8	0.8	-
Flow velocity, v (m/s)	1.0	1.0	-
Turbine torque, T (Nm)	167.59	173.46	3.5
Turbine torque coefficient, Ct	0.545	0.564	3.49
Turbine power coefficient, Cp	0.436	0.451	3.44

presents the validation results. According to Basumatary et al. [47], deviations within 6% are considered acceptable for model validation. All results in this study fall within that range, confirming the accuracy and reliability of the numerical model for further analysis.

4. Results and Discussion

4.1. Effects of Double-Stage Turbine’s Blade Configurations on the Torque Performance

The torque performance of the double-stage HKT model was analysed at three blade configurations: (0°, 0°), (0°, 45°), and (0°, 90°). Figure 9, Figure 10 and Figure 11 illustrate the torque variation with respect to the tip speed ratio (TSR) at three flow velocities (0.6 m/s, 0.8 m/s, and 1.0 m/s).

Torque values decrease as TSR increases for all blade configurations and flow velocities. At 0.6 m/s, the (0°, 0°) configuration produces the highest torque, followed by the (0°, 45°) configuration, while the (0°, 90°) configuration records the lowest torque. At 0.8 m/s, the (0°, 45°) configuration generates the highest torque at mid-TSRs. The (0°, 90°) configuration shows improved performance compared with 0.6 m/s, while the (0°, 0°) configuration maintains a consistent trend across TSRs. At 1.0 m/s, the (0°, 90°) configuration produced the highest torque at low TSRs, but its performance declined as TSR increased. The (0°, 0°) configuration maintains higher torque at higher TSRs, while the (0°, 45°) configuration remains stable. To comprehensively evaluate turbine performance, the power and torque coefficients (C_p and C_t) for each blade configuration are analysed in the following sections.

The computed points at the tested TSR values are selected to capture the performance trends of each blade configuration. Although finer TSR intervals could refine the optimal conditions, the current resolution is sufficient for trend evaluation and is consistent with CFD practices in hydrokinetic turbine studies [48].

4.2. Effects of Blade Configuration on Power Coefficient of a Double-Stage Hydrokinetic Turbine

The power coefficient (C_p) is evaluated for each blade configuration at different TSR and flow velocities. Figure 12, Figure 13 and Figure 14 present the C_p variation across TSR values at 0.6 m/s, 0.8 m/s, and 1.0 m/s, respectively.

The (0°, 0°) configuration shows a steady increase in C_p up to a TSR of 0.8, followed by a decline. At 0.6 m/s, the C_p peaks at 0.162 at TSR 0.8 before decreasing at TSR 1.0. At 0.8 m/s, the highest C_p of 0.197 is recorded at TSR 0.8, confirming that moderate flow velocities optimise efficiency. At 1.0 m/s, the C_p reaches its maximum value of 0.252 at TSR 0.8, making this configuration perform best under high flow conditions.

The (0°, 45°) configuration achieves the highest C_p in low-flow conditions. At 0.6 m/s, it records a maximum C_p of 0.230 at TSR 0.8. At 0.8 m/s, C_p remains relatively stable across TSRs, reaching 0.212 at its peak. At 1.0 m/s, C_p increases with TSR and reaches 0.204 at TSR 1.0, indicating improved performance at higher TSRs.

The (0°, 90°) configuration consistently produces the lowest C_p values. At 0.6 m/s, C_p peaks at 0.114 at TSR 0.8 but remains significantly lower than the other cases. At 0.8 m/s, performance remains weak, with a maximum C_p of 0.112. At 1.0 m/s, C_p remains low, and a negative value is observed at TSR 1.0 in the 0.6 m/s case, suggesting energy loss caused by flow separation or dominant drag effects.

To evaluate performance consistency, the mean C_p and standard deviation are calculated for each blade configuration and velocity. The results are presented in

Table 8. Mean and standard deviation of power coefficient.

Blade Configuration	Mean Cp (0.6 m/s)	Mean Cp (0.8 m/s)	Mean Cp (1.0 m/s)
(0°, 0°)	0.126	0.162	0.203
(0°, 45°)	0.179	0.183	0.157
(0°, 90°)	0.108	0.106	0.091
Blade Configuration	Standard Deviation (0.6 m/s)	Standard Deviation (0.8 m/s)	Standard Deviation (1.0 m/s)
(0°, 0°)	0.126	0.162	0.203
(0°, 45°)	0.179	0.183	0.157
(0°, 90°)	0.108	0.106	0.091

. The (0°, 45°) configuration records the highest mean C_p at 0.6 m/s and 0.8 m/s, confirming its efficiency in low to moderate flow conditions. At 1.0 m/s, the (0°, 0°) configuration achieves the highest mean C_p, indicating stable performance in high-velocity environments. The (0°, 90°) configuration consistently exhibits the lowest mean C_p in all cases.

Standard deviation values indicate the variation in C_p across TSRs. The (0°, 45°) configuration shows moderate variability, reflecting fluctuating performance across TSRs. The (0°, 0°) configuration exhibits lower variation in standard deviation, reflecting more stable efficiency. The (0°, 90°) configuration has the lowest standard deviation but also the poorest performance, reinforcing its limited effectiveness.

The maximum C_p values for each configuration are presented in Figure 15. The (0°, 0°) configuration achieves its highest C_p of 0.252 at TSR 0.8 and 1.0 m/s. This configuration is best suited for higher flow velocities. The (0°, 45°) configuration reaches its peak C_p of 0.230 at TSR 0.8 and 0.6 m/s, making it the most efficient setup in lower velocity conditions. The (0°, 90°) configuration records its highest C_p of 0.128 at TSR 1.0 and 0.8 m/s, but its performance remains significantly lower than the other configurations, confirming its poor efficiency across all conditions.

Based on the C_p results, the optimal configuration depends on flow velocity. At 0.6 m/s, the (0°, 45°) setup achieves the highest C_p and mean C_p, making it most suitable for low-flow applications. At 0.8 m/s, both (0°, 45°) and (0°, 0°) show comparable performance. At 1.0 m/s, the (0°, 0°) configuration performs best. The (0°, 90°) configuration shows consistently low C_p values and is not recommended.

4.3. Effects of Blade Configuration on Torque Coefficient of a Double-Stage Hydrokinetic Turbine

The torque coefficient (C_t) is evaluated for each blade configuration at three flow velocities (0.6 m/s, 0.8 m/s, and 1.0 m/s). Figure 16, Figure 17 and Figure 18 illustrate the variation of C_t with TSR for the (0°, 0°), (0°, 45°), and (0°, 90°) configurations.

The (0°, 0°) configuration shows consistent performance across all velocities. At each velocity, C_t peaks at TSR 0.4 and decreases as TSR increases. The maximum C_t values are 0.481, 0.496, and 0.486 at 0.6 m/s, 0.8 m/s, and 1.0 m/s, respectively. This configuration demonstrates strong and stable performance, particularly under moderate to high flow conditions.

The (0°, 45°) configuration exhibits variable performance. At 0.6 m/s, C_t peaks at 0.410 at TSR 0.4. At 0.8 m/s and 1.0 m/s, the peak values shift to TSR 0.6, with C_t reaching 0.416 and 0.411, respectively. These results indicate improved performance at mid-range TSR values under higher flow velocities.

The (0°, 90°) configuration consistently records the lowest C_t values. At 0.6 m/s, the maximum C_t is 0.255 at TSR 0.4. At 0.8 m/s, C_t reaches 0.248 at TSR 0.4. At 1.0 m/s, C_t increases slightly to 0.255 at the same TSR. Despite this improvement, performance remains significantly lower than the other configurations.

Figure 19 presents the maximum C_t values for each configuration across the three flow velocities. The (0°, 0°) configuration achieves the highest peak C_t of 0.496 at 0.8 m/s, confirming its suitability for moderate flow environments. The (0°, 45°) configuration shows good performance at higher TSR values, with a peak C_t of 0.416. The (0°, 90°) configuration performs the least effectively in all cases, confirming its limited energy capture capability.

The C_t trends further support the C_p findings. The (0°, 0°) configuration records the highest C_t at all velocities, especially at 0.8 m/s, indicating strong and consistent rotational performance. The (0°, 45°) configuration performs well at mid-TSRs and higher flow rates. The (0°, 90°) configuration yields the lowest C_t across all conditions.

5. Conclusions

A parametric evaluation of blade configurations is conducted in this study to assess their effect on the performance of a double-stage hydrokinetic turbine. Simulations are performed using the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations with the k-ω SST turbulence model. Mesh density, domain size, and time-step analyses are conducted to ensure simulation accuracy while maintaining computational efficiency. The computational model is validated against published data, with deviations within acceptable limits.

The key findings are summarised as follows:

• The (0°, 0°) blade configuration achieved the highest C_p of 0.252 at TSR 0.8 and 1.0 m/s. It maintained strong performance at higher flow conditions;
• The (0°, 45°) configuration recorded the highest C_p of 0.230 at 0.6 m/s and TSR 0.8, showing its effectiveness in lower flow conditions;
• The (0°, 90°) configuration consistently produced the lowest C_p values across all velocities;
• Torque output decreased with increasing TSR for all cases. The (0°, 0°) configuration shows higher torque at high TSRs, while the (0°, 45°) configuration performed effectively at mid-TSRs. The (0°, 90°) configuration performed the least effectively;
• The maximum C_t of 0.496 is recorded for the (0°, 0°) configuration at 0.8 m/s. The (0°, 45°) configuration achieved its peak C_t of 0.416 at 0.8 m/s. The (0°, 90°) configuration showed the lowest C_t across all cases.

Based on these findings, the (0°, 45°) blade configuration is recommended for low-flow conditions between 0.6 m/s and 0.8 m/s. The (0°, 0°) blade configuration is optimal for higher flow velocity at 1.0 m/s. The (0°, 90°) configuration is not recommended due to its low power and torque performance across all cases.