Experimental and numerical study on the performance and flow pattern of different Savonius hydrokinetic turbines with varying duct angle

Abstract

In this study, three different types of Savonius rotors viz. Savonius hydrokinetic turbine (SHKT), modified Savonius hydrokinetic turbine (MSHKT) and helical Savonius hydrokinetic turbine (HSHKT) are compared based on the performance. The analysis is done experimentally as well as numerically, where experimental domain ceases. Performance of rotors is also evaluated with and without applying duct as an augmentation technique in the flow channel. Experimentally, MSHKT and HSHKT produce 2.74% and 9.04% more energy than SHKT for 1 ± 0.2 m/s. Uncertainties in TSR, C_p and C_t of rotor are 3.05%, 4.39% and 5.35% for the experiment. It is found that HSHKT has better performance than others. Whereas, with duct, performance of HSHKT improves 48.08% more energy than SHKT. The insight of the hydrodynamic behavior considering wake formation, flow separation and vortex formation of the stream flow surrounding to the rotor is also explained using velocity contour, velocity vector and pressure plot. Inlet velocity of 1 ± 0.2 m/s increases by 27.5%, 28%, 29%, 32% and 34%, respectively, for duct angle 20°, 23°, 26°, 29° and 32°. Simultaneously, low-pressure zone increases which leads to extend the formation of the vortex far from rotor and helps to generate higher Cp for HSHKT as 28.63%, 16.16%, 43.01%, 50.13% and 82.32%, respectively.

1 Introduction

Studies reveal that, in 2018, across one billion people failed to access electricity which is the requisite of modern sustenance (IEA 2018; Global energy statistical yearbook 2018; The World Bank Data 2018). Renewable energy can play a vital role in solving the problem. It is reliable, socio-economic, and it has a diminutive impact on the environment. Studies show that in present days, renewable energy has the fastest growth in electricity production, out of which hydro-power is going to contribute 16% of the electric generation globally by 2023 (REN21 2018; Nag and Sarkar 2018). Hydrokinetic turbines (HKT) are the future of the micro-hydropower plant and the subsidiary elements of renewable energy. Hydrokinetic energy utilizes the kinetic energy of the flowing water in the canals, river, tidal current and ocean waves. According to the Bentz theory, maximum power extracted by the rotors from the free stream is 59.3% of the available kinetic energy of the free stream. These types of turbines are also known as low-pressure and ultra-low head turbines (Rauh and Seelert 1984; Garrett and Cummins 2008). However, the maximum power limit exceeds when the free stream velocity is utilized an utmost, and proper selection of HKTs is done with suitable modifications (Vennell 2013; Cuerva and Sanz-Andrés 2005). In general, classification of HKTs is based on the type of flow. Turbine axis which is parallel to the stream flow is called axial flow turbine and if aligned orthogonally to the stream flow is called cross-flow turbine. Axial flow turbines have variation based on a number of blades viz. two-bladed turbines, three-bladed turbine, and multi-bladed turbine. However, cross-flow turbines are also classified further based on their design viz. Savonius turbine, Darrieus turbine, Gorlov turbine and H-Darrieus-type turbine (Khan et al. 2009; Vermaak et al. 2014; Guney and Kaygusuz 2010; Yuce and Muratoglu 2015). The Savonius turbine is one of the basic types of cross-flow turbine, commonly used because of its simple design, relatively low operating speed, good starting ability, easiness of fabrication and installation (Golecha et al. 2011; Wahyudi et al. 2015). In addition, hydrokinetic turbines are classified based on axis alignment, viz. vertical axis turbine and horizontal axis turbine. This vertical axis Savonius hydrokinetic turbine (SHKT) is the conventional one and has low efficiency. To improve the performance of this vertical axis hydrokinetic turbine, certain design modifications are adopted (Table 1), such as variation in aspect ratio, overlap ratio, twisted blade angle (Kumar and Sarkar 2016a, b). Design modification of the Savonius turbine produces the following types of turbine viz. helical Savonius hydrokinetic turbine (HSHKT) and modified Savonius hydrokinetic turbine (MSHKT). After modification of parameters, SHKTs are generally compared with the conventional Savonius rotor. Moreover, following augmentation techniques are also adopted for better performance of turbine viz. variation in the channel width, application of duct, nozzle and deflector plates in both upstream and downstream, several numbers of deflector plates with systematic positions as well as angular arrangement, multi-stage turbines and arranging the turbines in array (Kumar and Saini 2017; Gaden and Bibeau 2010; Kumar and Sarkar 2016a, b, 2017). Maldonado et al. (2014) showed that implementing augmentation deflector in upstream of the Savonius rotor may increase 20% flow velocity.

Table 1  Modifications of Savonius hydrokinetic turbine design parameters and channel parameters for performance enhancement

Several comparative studies were conducted on different hydrokinetic turbines based on their performances. Talukdar et al. (2018) proved experimentally that HSHKT attains higher starting ability and higher coefficient of power (C_p) compared to SHKT because of its blade design modification on chord length, aspect ratio and angle of twist. do Rio et al. (2013) suggested a better rotor model by optimizing blade length and twist angle for better performance. Kamoji et al. (2009) have done design modification by removing the central axis shaft from MSHKT and found that it has higher C_p than normal MSHKT. Harries et al. (2016) experimented on novel drag driven vertical axis tidal turbine and SHKT. It is found that C_p and C_t of SHKT increase with increasing blockage ratio. Shimokawa et al. (2012) experimentally showed that nozzle increases the C_p, the coefficient of torque (C_t) and reduces flow separation from the blade profile, which produces less negative lift (at the returning blade) and reduces the downstream height of water. As a result, performance of the SHKT increases. Sarma et al. (2014) showed a detailed study of the velocity distribution of stream flow surrounding the blades which provide the insight of the design of the SHKT. Some study clarified on disturbances and vortex interactions of the stream flow surrounding the rotor blade (Alom and Saha 2018; Wahyudi et al. 2015), without mentioning about the performance parameters of the turbine or did not compare with other turbine rotors.

Besides the expensive experimental approach, numerical analysis became popular as an alternate method to explore the performance of HKTs. To replicate the behavior of HKT, researchers preferred the computational fluid dynamics (CFD) simulations in ANSYS CFD-CFX software (Hosseini and Goudarzi 2018; D’Alessandro et al. 2010). It produces more insight into flow separation, wake characteristics, vortices and very less discrepancy in the performance of the turbine (Bianchini et al. 2017; Behrouzia et al. 2015). Kumar and Saini (Kumar and Saini 2017) suggested the k–ε turbulence model for the CFD analysis for the HKT. Elbatran et al. (2017) designed different types of the duct and evaluated their performance on SHKT computationally. They used SST k–ω turbulence model to explore boundary layer separation and reported that 78% of the gain in power is observed when the duct is implemented. Driss et al. (2014) compared the performance of ducted nozzle mounted SHKT with the conventional SHKT using the k–ε turbulence model and validated the same results with experiments. This numerical study reported vortices distribution longitudinally to the fluid flow surrounding the SHKT. However, maximum vorticity is obtained across the SHKT, just downstream of the advancing blade. Moreover, the wake zone is found in the downstream of the SHKT. Birjandi et al. (2013) found that flow stream angle of attack at the upstream of SHKT increases flow separation; as a result, the lift force reduces (at the advancing blade). Simultaneously vortex also increases, which leads to the reduction in the C_p and C_t, and hence performance decreases. Gorle et al. (2016) illuminate the effect of the blade–vortex interaction across the turbine which restricts the rotor rotation.

Following research gaps are not found during literature review (Table 1): (1) effect of adjustable blade on the performance of Savonius hydrokinetic turbine, (2) influence of more than three numbers of blades on the performance of the rotor, (3) role of wake and vortices on the performance of Savonius hydrokinetic turbine, (4) performance comparison of different Savonius hydrokinetic rotors, (5) optimized number of staging and its positioning considering same aspect ratio, (6) rotor simulations with changing aspect ratio in view of turbulence intensity influence, etc.

In view of efficient renewable energy generation, a comparative performance study of different Savonius turbine rotors is a pertinent problem. Open literature does not provide any clear idea of flow phenomenon happening around the blades influencing the rotor performance. Therefore, the aim of this study is set to explore the performance of different Savonius hydrokinetic turbine rotors with and without the application of duct, numerically as well as experimentally. Simultaneously hydrodynamic effects are also also explained surrounding the rotor during operation. Further, a comparison study may be done to evaluate the performance of the rotors, within the experimental limit and as well as beyond the experimental limit with the help of a simulation study. Moreover, Savonius rotor hydrodynamics, duct angle effect, vortices and wake characteristics on SHKT may be explained. Therefore, novelty of this study is to conduct a comparative study of the performance of three different Savonius rotors with the detailed description of the flow phenomenon surrounding the blades, numerically with experimentally, with and without the influence of channel modification parameter like duct augmentation.

2 Methodology

In this study, numerical and experimental analyses are done to evaluate the performance of the Savonius hydrokinetic turbine (SHKT), helical Savonius hydrokinetic turbine (HSHKT) and modified Savonius hydrokinetic turbine (MSHKT) with and without implementing augmentation technique like a converging–diverging duct. Both the analyses are performed considering the distance of the rotor (L) from the initiation of duct convergence as 4 times the diameter of the rotor (D). Performance of the rotor is mainly decided based on the power generation capacity of the rotor. Fluid flow pattern surrounding the rotor and energy conversion is mainly governed by the hydrodynamic theory of rotor.

2.1 Hydrodynamic theory of rotor

Flow pattern surrounding the rotor in operation is shown in Fig. 1. Top view of the rotor shows that the flow passing through the rotor transmits kinetic energy to the advancing blade. Whereas, the front view shows the total specific energy in the stream flow strikes at the advancing blade and leaves with the reduction in the depth of the stream flow.

Fig. 1

Notations of SHKT, MSHKT and HSHKT design and stream flow pattern

At the upstream of the rotor, the advancing blade experiences flow velocity; while at the concave surface, stagnation pressure develops. At the tip of the advancing blade, flow separation takes place due to the low pressure. As a consequence, a pressure difference occurs in between the upstream and downstream of the rotor, which helps to rotate the turbine. Due to the pressure difference, vortices form at the upstream of the advancing blade. This spinning motion of the fluid particles is carried by the wake in downstream. Simultaneously, at the downstream of the rotor, flow separation occurs at the tip of the advancing blade, towards the direction of flow. This creates negative pressure near the downstream of the returning blade. Meanwhile, advancing flow strikes the returning blade also which is convex in geometry, along the flow direction, with positive pressure. As long as the pressure difference exists, the turbine rotates. Simultaneously, the Coanda effect is also observed, which has the tendency to make the fluid particle stay attached to a convex surface. This fluid particle experiences high pressure due to the adjacent fluid layer and low pressure on the inner part of the fluid particle attached to the convex surface. Due to the pressure difference, fluid particle experiences a centripetal force which causes an increase in the torque. Moreover, due to the high-pressure zone on the concave side of the upstream and low-pressure zone on the convex side of the downstream, the advancing blade experiences drag force and returning blade experiences negative drag force. The overall resultant of the drag force provides the rotation to the rotor.

In general, wake formation carries vortices which lead to the vortex shedding. As the flow impinges to the body, it generates first stagnation point (higher pressure) at the upstream of the rotor, second stagnation point (moderate pressure) is just after downstream at the rotor and third stagnation point (low pressure) is at downstream away from the rotor. Due to pressure difference at the second and third stagnation points, the flow has a tendency to proceed along the flow direction. However, simultaneously, vortices influence the flow to turn towards the upstream direction due to circulation. As long as the pressure difference at the second and third points is higher, the above-said effect on the fluid gets reduced and it obstructs the flow to be returned towards the rotor. Therefore, the distance between the second and third stagnation point is preferred to be higher to avoid any obstruction to torque generation by the rotor.

On the other hand, if the volume of the fluid striking the blade is considered, it is clear from Fig. 2a that one part of the fluid is striking in the advancing blade directly, and another part is striking the returning blade. The part striking the returning blade again divides into two halves. One half is gliding over the returning blade and leaves the rotor. Another half is joining the fluid flow striking the advancing blade which results in stagnation pressure. This creates positive pressure, which is responsible for the rotation of fluid. The position of the rotor blade in Fig. 2a is 90° along the flow direction. As the swept area of the rotor is maximum at 90°, it exerts maximum drag force in the advancing blade and produces maximum power. Figure 2b shows the developments of power over one complete rotation of the rotor from 0° to 360°. Power generation follows a sinusoidal curve from 0° to 360°. Moreover, because of the implementation of the duct, the angle of attack of the free stream at the tip of the advancing blade became higher, which helps to increase the intensity of the power.

Fig. 2

Power generation at a different angle of the rotor

In order to carry out the performance analysis, both numerical and experimental studies are conducted. The details of the methodology are described in a flowchart in Fig. 3. For numerical analysis, ANSYS CFD-CFX is used to develop flow simulation and to study the behavior of the different HKT rotor under different operating conditions. This study complies 3D modeling of flume and rotor, and mesh generation of the 3D model. Further, initial and boundary conditions are applied in the setup for CFX and the results are obtained for different operating conditions. Whereas, for the experimental investigation, an experimental setup is designed and fabricated.

Fig. 3

Flowchart of methodology

The experimental setup consists of the flume, different HKT rotors, measuring sensors viz. flow meter, dynamometer, non-contact-type tachometer, etc. In the setup, turbines and all the measuring devices can be moved to x–y–z-direction using mounting trolley while performing the experiments. Values obtained in the experiment are used to validate the results of the numerical study. Whereas, the numerical analysis predicts some results, which are beyond the capacity of the experimental setup. Details of the numerical study are discussed as follows.

2.2 Numerical study

To study the performance of SHKT, whole experimental setup and their analysis are replicated in the computational domain in ANSYS CFD-CFX. In general, to solve turbulence flow problems, a proper model has to be selected based on the Reynolds number, flow geometry and flow separation. In this study, the model is adopted as the flow solver to simulate the unsteady incompressible flow of fluid accounting turbulence and viscosity. The k–ε model is preferably used to solve the flows involving rotational body and boundary layer phenomenon (Launder and Spalding 1983). Moreover, k–ε model has capability to solve the problems involving the fluid flow through the proximity of the complex geometries, such as turbine blades (Sarma et al. 2014; Driss et al. 2014). Savonius turbine blades have sharp corners and curved edges. This study tries to represent flow pattern surrounding the blades in rotating condition. Thus, k–ε model is opted in this study for the computational simulation in ANSYS software.

In this study, for optimum computation time in ANSYS, the shear stress transport is adopted considering 5% of turbulence intensity. Turbulence intensity gives the proximate results of the turbulence models and also used for flow separation with the different pressure gradient. In the computational analysis, the flume is defined as a stationary domain. Whereas, fluid is considered as flowing domain and rotor is enclosed with the cylindrical domain provided with the rotational feature. The input of the rotational domain is the angular velocity of the rotor which varies as per the varying loading conditions. These domains are further meshed fully and provided inflation layer on the rotor blade. This interaction between rotor and fluid is studied in respect of variation in velocity and wake characteristics.

Grid independence test for three different rotors is done at different refinement levels. It is tested by assessing the power output by the rotor for different mesh densities. The details of the selected mesh size corresponding to the adopted refinement level 8 for the different rotors are given in Fig. 4 and Table 2. According to the different rotor shape, the variation in the blockage area is covered on the flume. Hence, fluid domain meshes, respectively, to each rotor. Tetrahedral unstructured mesh has been provided for further simulation.

Fig. 4

Variation of power (W) with the different refinement levels

Table 2 Mesh size details of different types of rotor

In the present study simulation, initial and boundary conditions are kept the same for all the cases. At the inlet of the fluid domain, the initial condition is provided as the uniform velocity of 1 m/s, 2 m/s and 3 m/s for atmospheric pressure conditions, i.e., 0 atm. Boundary conditions are provided as a no-slip wall considering smooth roughness. Outlet conditions are provided as 0 atm distributed uniformly which replicates the proximity of the solution. Besides that, some more necessary inputs are implemented, which are shown in Table 3.

Table 3 Detailed options selected for the simulation in ANSYS CFX

From the above numerical study, power and torque are obtained for each time step with the provided number of turbine rotation simulation. Through this analysis, the hydrodynamic flow field surrounding the HKT is examined. It is observed with the help of a velocity contour plot and velocity vector plot which shows the velocity magnitude and direction throughout the flume. This study is extended to show the behavior of different types of HKTs in the presence of stream flow. Moreover, this study could also explore the behavior of HKT rotors after implementation of augmentation techniques, which can further be validated through experiment.

2.3 Experimental study

The experimental setup was fabricated for analyzing the performance of different HKT rotors (Fig. 5a). The setup is mainly constituted by a flume (rectangular channel), which is further simulated numerically considering the conditions mentioned in Table 3 (Fig. 5b). Reading torque and power at different flow condition is the main objective of this setup, which is done by a dynamometer (Fig. 5c). The cross-sectional dimensions of the open rectangular channel are 6 m × 0.6 m × 0.7 m in which Savonius rotors are placed (Fig. 5d, e). Here, rotors are shown in just immersed condition as water is muddy and recycled water is used for the study; however, experiments are done in full immersed condition. The Savonius rotor is fixed with a frame movable in x-, y- and z-direction (Fig. 5f). Moreover, multiple rotors could be mounted in the flume. The two 10-hp pumps are installed in the setup to supply water to the flume. Two valves are connected separately with each pump to control the discharge. This provides surface fluid flow velocity at the flume test section within a range of 0–2 m/s. The flow is measured by flow meter capable of moving in the x–y–z-direction in the flume. The rotational speed of the rotor is measured by non-contact type tachometer attached with the movable trolley which holds HKT rotor. Rotor torque is measured by the help of dynamometer mounted in the same trolley.

Fig. 5

Experimental setup and its layout for simulation in ANSYS

For the present study, hydrokinetic rotors are designed based on the design specification mentioned in Table 4. Description of the nomenclature is shown in Fig. 1 and photographs of manufactured rotors are shown in Fig. 5a. These rotor blades are fabricated from aluminum plates having a 2-mm thickness (t), end plates which cover the axial direction of the rotor are of the same material having a thickness (t_end) of 10 mm and a central shaft having a 10-mm diameter (D_s) made up of aluminum. SHKT is designed alike of two semicircular buckets which are placed such that the concave part forms ‘S’ in the shape. The difference in forces subjected due to the fluid medium on convex and concave blades produces drag force; this drag force induces rotation of the hydrokinetic turbine. Despite the low efficiency of SHKT, it is favored as it is simple, compact, easy to install and has low manufacturing cost. The design of SHKT is such that it is independent of flow direction. It rotates omnidirectionally or rotates in a clockwise and anticlockwise direction along the rotor axis. The dimensions used for the rotor diameter (D) are 200 mm; end plate diameter (D_o) is 220 mm and its thickness is 10 mm; the height of the rotor (H) is 180 mm; aspect ratio (AR) is 0.9.

Table 4 Design specification of Savonius hydrokinetic turbine

HSHKT has an almost similar specification to the SHKT, the only difference is that the bucket or blade is twisted at any angle along the axis. In this study, HSHKT dimensions remain the same as of SHKT with the same aspect ratio and the angle of twist (θ) is taken as 90° (Fig. 1, Table 4). Through this turbine, the impact of the flow velocity is uninterruptedly exerted positive drag force due to the twisted bucket or blade.

MSHKT has similar design specification as the previous two turbines discussed above. This type of Savonius turbine is basically designed for better performance regarding losses in vortices. The dimensions of rotor diameter, height, angle of twist and aspect ratio remain the same such as 200 mm, 180 mm, 0° and 0.9°, respectively. The overlap distance (a) of both the blades is 15 mm apart considering center as the turbine center axis. The specifications of all the above discussed rotors are shown in Table 4.

Description of the flume is shown in Fig. 5e; also the position of the rotors inside the flume is shown. Inside flume, converging–diverging duct augmentation techniques are used to reduce the fluid flow area of the channels shown in Fig. 5h. In this study, duct angle is kept constant at 29°. However, the duct has the capability to change its angle 20°–32°. The experiment and numerical analyses are performed considering the distance of the rotor (L) from the initiation of duct convergence as 4D. For, without duct condition, same distance has been maintained (Fig. 5d). Moreover, in this flume, their are arrangements to set single and double deflector in the flume bed, to provide more flow in the leading edge of the blade of the HKT rotor. The rectangular channel also has provision to provide slope by tilting the flume from 0° to 5°; however, in this study flume is kept horizontal.

For measuring all the necessary parameters like surface velocity, rotor speed, rotor torque, rotor shaft power, electric power, etc., various devices are installed and connected through the display panel board as shown in Fig. 5g. Panel board consists of all the electrical connections including the output of the sensors. It consists of sensor indicators viz. digital torque indicator, digital speed indicator, and a digital watt indicator. Separately, flow meter and rope brake dynamometer is arranged to measure flow velocity and rotor torque, respectively. The measuring range and uncertainties in measuring devices are provided by the manufacturer. The transducer is attached or circuited to measure torque is strain gauge having a measuring range of the digital torque indicator is 0–6 kg-m having an uncertainty of 1% of full scale. The digital speed indicator works with the help of inductive proximity sensor, measurement of the turbine rotation are in the range of 0.1–3000 rpm associated with the uncertainty of 0.5% of full scale. The digital watt indicator measures electric power generated through the dynamometer and the digital reading is displayed on the panel. The uncertainty which occurs is 0.5% of the full scale.

To measure the free stream velocity (V), the flowmeter is used. The spindle of the flow meter carries cup wheel resting freely on the bearing. The magnetic-type sensor which counts on the one pulse per revolution is attached to the wheel circuited to the microprocessor with alphanumeric LCD. This flowmeter measures in the range of 0–9.999 m/s within the response time of 0–99 s. The calculated uncertainty of the flow meter is 0.1 ± 0.02 m/s.

In the experimental setup, rope brake dynamometer is installed to measure torque (T_rotor) as shown in Fig. 5c. Torque corresponding to varying rotor speed can be calculated by this instrument. However, sensors are there to measure the torque and power directly. This measurement device is simple to install and less sensitive to vibrations and alignment of the turbine shaft. The rope brake dynamometer consists of one spring balance which is connected to one end of the rope which passes through the pulley connected with the turbine shaft and another end of the rope is connected to the load. Initially, HKT is rotated at no load conditions. Thereafter, the mechanical loads are applied for different loading conditions. Rotation of the HKT is controlled by the dead weight of the rope brake dynamometer with a certain error in the experiment.

Out of the above, surface stream velocity (V) can be measured and further another design parameter like torque (T) can be calculated. However, the rotor shaft torque can be measured in this setup directly also along with shaft power (P_rotor). Thereafter, other performance parameters will also be calculated with these measured values.

2.3.1 Design parameters

The available torque at the rotor due to the interaction of velocities in upstream and downstream can be expressed as follows.

$$T = 0.5\rho AR\left( {V_{1}^{2} - V_{2}^{2} } \right)$$

(1)

The available power due to the free stream flow is expressed as follows.

$$P = 0.5\rho AV^{3} \times 4a\left( {1 - a} \right)^{2} ,\;a = \left( {V_{1} - V_{2} } \right)/V_{1} ,$$

(2)

where V is the mean free stream velocity, R is the radius of the rotor, ρ is the density of the water (= 1000 kg/m³), A is the projected area of the rotor, V₁ and V₂ are the upstream and downstream velocity of the flume.

$$T_{\text{Rotor}} = (W_{\text{TS}} - W_{\text{SS}} ) \times g \times (r_{\text{p}} + d_{\text{r}} ),$$

(3)

The power generated by the rotor is termed as,

$$P_{\text{Rotor}} = \frac{{2\pi N \times T_{\text{rotor}} }}{60},$$

(4)

where W_TS is the tension in tight side, W_SS is the tension in slack side, $r_{\text{p}}$ is the radius of the pulley, $d_{\text{r}}$ is the diameter of the rope, g is the acceleration due to gravity (= 9.81 m/s²), ω is the angular velocity of the rotor and N is the number of rotation of the rotor per minute.

The torque generated using rope-brake dynamometer is denoted as rotor torque (T_Rotor).

2.3.2 Performance parameters

Tip speed ratio (TSR) is the ratio of the speed of the blade at its tip to the free stream velocity of the water.

$${\text{TSR}} = \omega R/V.$$

(5)

Coefficient of power (C_p) is the ratio of the power developed by the turbine rotor to the actual power available for the rotor.

$$C_{\text{p}} = \frac{{P_{\text{rotor}} }}{{\rho RHV^{3} }}.$$

(6)

Coefficient of torque (C_t) is related to the torque developed by the turbine rotor.

$$C_{\text{t}} = \frac{{P_{\text{rotor}} }}{{\rho R^{2} HV^{2} }}.$$

(7)

Reynolds number (Re) is the ratio of the inertial force of the fluid medium to the viscous force.

$$\text{Re} = \frac{{\rho VR_{\text{C/H}} }}{\mu },$$

(8)

where

$$R_{\text{c/h}} = A_{\text{c/s}} /P_{\text{wetted}} ,$$

(9)

$${\text{Aspect ratio }}\left( {\text{AR}} \right),{\text{AR}} = H/D.$$

(10)

Froude no. (Fr) is a dimensionless quantity that describes different flow regimes of open channel flow. It is the ratio of inertial forces and gravitational forces. V is the stream velocity and D is the characteristic diameter.

$${\text{Fr}} = \frac{V}{{\sqrt {gD_{\text{C/H}} } }}.$$

(11)

When Fr = 1, flow is considered as critical; if Fr  >  1, flow is Supercritical/shooting/rapid/torrential; if Fr  < 1, flow is considered as Subcritical/tranquil/streaming flow.

$$D_{\text{C/H}} = \frac{{A_{\text{C/S}} }}{{T_{\text{width}} }},$$

(12)

$${\text{Blockage ratio }}\left( {\text{BR}} \right),{\text{BR}} = \frac{{{\text{Area}}\,{\text{covered}}\,{\text{by}}\,{\text{the}}\,{\text{rotor}}}}{{{\text{Projected}}\,{\text{flow}}\,{\text{area}}\,{\text{in}}\,{\text{the}}\,{\text{water}}\,{\text{channel}}}} = \frac{HD}{{H_{\text{W}} W}},$$

(13)

where R is the radius of the rotor, H is the height of the rotor, D is the diameter of the rotor, R_C/H is the characteristics radius, A_c/s is the cross-sectional area of the flow, P_wetted is the wetted perimeter of the rotor, U is the free stream velocity (m/s), D_C/H is the characteristic diameter (m), T_width is the top width of the channel, A_C/S is the cross-sectional area of the flow, H_W is the height of the flowing water, W is the width of the channel, ω is the angular velocity of the rotor, R is the radius of the rotor, ρ is the density of the water (= 1000 kg/m³).

Turbulence intensity is an important parameter for the computational simulation design. The turbulent intensity of the flow at the specified point can be determined as follows.

$${\text{TI}} = \frac{1}{V} \times \sqrt {\frac{1}{3}\left\{ {u_{x}^{2} + u_{y}^{2} + u_{z}^{2} } \right\}} ,$$

(14)

where u_x, u_y and u_w are the fluctuation of the velocity in the x-, y- and z-direction, respectively.

Values obtained in the experiment are used to validate some of the results of the numerical study. In addition to this, uncertainty analysis of the results is performed to find out an error range of the data obtained from experiments. Further, the numerical analysis predicts more results, which are beyond the capacity of the experimental setup.

2.3.3 Uncertainty analysis

According to the Moffat uncertainty analysis, the relative error is expressed as based on literature (Moffat 1982, 1988).

Let f is the function which can be expressed as follows.

$$f = X_{1}^{{a\,_{1} }} \times X_{2}^{{a_{\,2} }} \times X_{3}^{{a\,_{3} }} \ldots \times X_{n}^{{a\,_{n} }} .$$

(15)

Outcomes having uncertainty is represented as follows.

$$\omega_{\text{R}} = f\,\, \times \left[ {\left\{ {\left( {\frac{{\sum {a_{1} } }}{{x_{1} }} \times \omega_{1} } \right)^{2} + \left( {\frac{{\sum {a_{2} } }}{{x_{2} }} \times \omega_{2} } \right)^{2} \ldots + \left( {\frac{{\sum {a_{n} } }}{{x_{n} }} \times \omega_{n} } \right)^{2} } \right\}^{{\frac{1}{2}}} } \right],$$

(16)

where X₁, X₂, X₃ …X_n are independent sensitive coefficients, ω₁, ω₂, ω₃… ω_n be the uncertainties in the independent variables, a₁, a₂, a₃…a_n power coefficients of variables.

For the comparative performance of the SHKT, MSHKT and HSHKT rotors in the flume, different independent parameters are measured. Therefore, uncertainty analysis is conducted to find out the relative error on each performance parameter.

3 Result and discussion

In this study, a comparative assessment of the performance of three different types of turbines viz. SHKT, HSHKT and MSHKT is done experimentally and computationally both. These rotors are experimentally tested with and without applying duct in the flume at different turbulent conditions (Re varies from 2.36 × 10⁵ to 3.54 × 10⁵). Whereas, the study is simulated in ANSYS and extended for higher turbulent conditions (Re varies from 2.36 × 10⁵ to 7.08 × 10⁵), which is beyond the experimental capacity. Both experimental and simulation data are compared at a particular case (i.e. V = 1 m/s) for validation purpose. Front, top and side views of the simulations of SHKT with the ducted flume for V = 1 m/s are shown in Fig. 6a–c. Here, photographs of the rotor are taken as partially submerged condition, to show the flow pattern downstream of the rotor; however, performance results are done for fully submerged conditions (Fig. 5a). It characterizes the velocity profile along the z–x plane. The free stream velocity at the upstream before the ducted area is less than 0.911 m/s. While water passes through the converging duct, it attains a velocity 1.095 m/s. Further, in downstream end velocity reduced to 0.730 m/s. Thus, the free stream velocity enhances available power by striking the advancing blade of the rotor with higher velocity while the duct is installed. Velocity vector at V = 1 m/s is shown in Fig. 6e at 60 rpm and TSR = 0. 691. Velocity vector for V = 2 m/s and 3 m/s is shown in Fig. 6g and h.

Fig. 6

Different stream flow phenomenon in the computational and experimental domain for SHKT with the duct

Whereas, experimental output for a similar condition is shown in Fig. 6d and f. Although in this case, rotors are shown in partially submerged condition. However, results for fully submerged rotor conditions are used for comparison. The free stream velocity is 1 ± 0.2 m/s and the rotational speed of the rotor is recorded as 51 rpm. In Fig. 6d, flow passed rotor is shown and it is also found that the hydrodynamic nature of the free stream velocity and surrounding to the SHKT is very much similar to the Fig. 6a. The figure shows a sudden decrement in the water depth at the downstream of the rotor. This indicates total specific energy, i.e., kinetic head and potential head at upstream are converted into the rotational speed of the SHKT. Therefore, the available power of the free stream at the upstream of the rotor is consumed by SHKT and specific energy at downstream reduces. As a result, free stream velocity is also reduced at the downstream of the SHKT.

When the flow of fluid strikes the advancing blade, it experiences a drag force. As a result, positive pressure forms at the upstream of the advancing blade, which helps to rotate the turbine. Simultaneously, at the downstream of the rotor, flow separation occurs at the tip of the advancing blade, towards the direction of flow. As a result, vorticity, i.e., spinning motion of the fluid appears, just downstream of the central part of the rotor and returning blade (Fig. 6f). This creates negative pressure near the downstream of the returning blade. Meanwhile, advancing flow strikes the returning blade also, along the flow direction, with positive pressure. Differences between these pressures are responsible for the rotation of the rotor. As long as negative pressure at the returning blade is more than positive pressure, the turbine rotates. While in the returning blade, the free stream flow impinges on the convex surface which distributes the flow towards the concave surface of the advancing blade and away from the rotor. However, because of the implementation of the duct, the angle of attack of the free stream at the tip of the advancing blade became higher, which helps to overcome the problem to an extent. In Fig. 6i–k, striking of the free stream at a different angle of SHKT blades is captured and free stream pattern surrounding the rotor is captured. When rotor advancing blade is making 0° along the free stream, then wake formation past the rotor is minimum. Whereas, at 90° position of rotor advancing blade creates maximum wake formation, which indicates maximum energy loss in the stream flow.

To evaluate the relative error among measured variables and dependent variables, uncertainty analysis is performed. The uncertainties of the derived variables are calculated based on the theory described in Eqs. (15) and (16). The same for this experimental setup takes the form as Eqs. (17)–(19). The uncertainties in TSR, C_p and C_t of rotor are found 3.05%, 4.39% and 5.35%, respectively.

$$\omega_{\text{TSR}} = \sqrt {\left( {\frac{R}{V} \times \omega_{\omega } } \right)^{2} + \left( {\frac{ - \;\omega R}{{V^{2} }} \times \omega_{\omega } } \right)^{2} } ,$$

(17)

$$\omega_{{C_{\text{p}} }} = \sqrt {\left( {\frac{{\omega_{{P_{\text{rotor}} }} }}{{\rho RHV^{3} }}} \right)^{2} + \left( {\frac{{ - \;3V^{2} \times \omega_{V} }}{{\rho RHV^{3} }}} \right)^{2} } ,$$

(18)

$$\omega_{{C_{\text{t}} }} = \sqrt {\left( {\frac{{\omega_{{P_{\text{rotor}} }} }}{{\rho R^{2} HV^{2} }}} \right)^{2} + \left( {\frac{{ - \;2V^{ - 1} \times \omega_{\text{V}} }}{{\rho RHV^{2} }}} \right)^{2} } .$$

(19)

3.1 Comparison of Savonius HKT, modified Savonius HKT and helical Savonius HKT

A summary of experimental and simulation results obtained for the three different turbines without duct and with duct is shown in Table 5. In Table 4, maximum power (P_max) and maximum torque (T_max) of different turbines are shown at different free stream velocity. Validation of the numerical analysis is done by the experimental analysis at V = 1 ± 0.2 m/s, for the same initial and boundary conditions.

Table 5 Maximum power (W) and maximum torque (Nm) of the three different turbines without duct and with duct by varying velocity

Numerical simulations show that SHKT generates P_max of 1.36 W and T_max of 0.21 Nm at 1 m/s free stream velocity without duct condition. Experiments validated the fact with P_max of 1.44 W and T_max of 0.24 Nm for the same condition. Percentage errors between numerical and experimental studies of 1 m/s velocity for all the rotors are depicted in Table 4. Percentage calculations are done considering numerical study results as reference/base. Percentage error calculations are done by taking absolute difference of the experimental study and numerical study compared by numerical study (Table 5). Results in Table 1 show the percentage error values of P_max and T_max are below 6% and 19%, respectively. However, percentage error values of $C_{{{{\text{p}}_{ \rm{max} }} }}$ and $C_{{{{\text{t}}_{ \rm{max} }} }}$ are below 2.95% and 7.69%, respectively, in reference to Figs. 9, 10 and 11. Thus, numerical studies are considered to be validated with experimental results. In the experimental analysis for the same condition, MSHKT generates 3.47% more P_max and 4.17% more T_max; whereas, HSHKT generates 9.72% more P_max and 8.33% more T_max, respectively. From this discussion, it is found that experimentally HSHKT produces more energy than MSHKT and SHKT. When the duct is implemented, the maximum power and torque production are increased by 24.31% and 29.17% more for MSHKT. In the same condition, the maximum power and torque production are increased by 32.71% and 41.67% more for HSHKT (Table 5). In the numerical study, similar observations are found for predicted values of P_max and T_max for free stream velocity of 2 m/s and 3 m/s, which are shown in Table 5. It is also observed that without duct condition, for higher free stream velocity, P_max and T_max of a rotor are obtained at a higher value of rotational speed. When the duct is employed in the channel, P_max and T_max of the same rotor will be obtained against the further higher value of rotational speed.

To the study, the insight of the hydrodynamic behavior of the rotor velocity vector (Fig. 7) and pressure plot (Fig. 7) are simulated through ANSYS-CFX. The analyses are done for two different cases, with the implementation of the duct (23° converging angle) and without the implementation of the duct. Velocity ranges with the color band and vector lines for the above-said phenomenon are shown in Fig. 7. Pressure plots are also shown using the color band. In general, uniform pressure is observed throughout the flume (Fig. 8). When free stream in the upstream interacts with the rotor, stagnation pressure (red in Fig. 8) develops at the upstream of the concave surface of the blade at a low velocity (deep blue in Fig. 7). Simultaneously, at the tip of the advancing blade, flow separation takes place due to the low pressure (green in Fig. 8) (i.e., at high velocity, green in Fig. 7). At the downstream of the rotor, a vortex (here circulation Г > 0, therefore, rotation in an anticlockwise direction) is generated at low velocity (blue in Fig. 7), where pressure is also found comparatively low (green in Fig. 8). Moreover, at the upstream of the returning blade, the high-pressure region is found near the convex surface of the returning blade (red and brown in Fig. 8), because of the conflict of rotor rotation direction and free stream velocity movement direction. Whereas, at the downstream of the returning blade, comparatively low-pressure region is obtained (cyan in Fig. 8). Here velocity gets reduced (blue in Fig. 7). Thus, the pressure difference in upstream and downstream regions, near the rotor, also causes rotor rotation (streamline in Fig. 7). Simultaneously, a vortex is formed in the downstream of the rotor. It is also preferable to extend the formation of the vortex far from the rotor for smooth operation, which is happened in case of HSHKT compared to others (streamlined vortex in Fig. 7e). From the figures, it is observed that, for all three different rotors, at the downstream, low-velocity range (blue in Fig. 7b–f) extends while duct is implemented. Specially, with the implementation of duct, this phenomenon is aggravated in case of HSHKT, where vortex formations almost disappear near the rotor (Fig. 7f). Simultaneously, distance of separation from the rotor towards the downstream direction is also increased, which is observed from vector contour. It indicates that due to the implementation of the duct, the influence of vortex generated in the downstream has less effect on the torque generation of the rotor. Flow pattern similar to Fig. 7a and c has been reported corresponding to approximately the same free stream velocity for without duct condition (Talukdar et al. 2018; Kumar and Saini 2017). Moreover, from the pressure plot, it is observed that high-pressure zone at the upstream of returning blade is getting reduced in the following order SHKT > MSHKT > HSHKT. It is to be noted that a higher amount of the pressure at upstream of the returning blade creates hindrance in the smooth rotation of the rotor. Therefore, it is obvious that as per design, better performance of the turbine can be obtained in the following order SHKT < MSHKT < HSHKT. Moreover, due to the implementation of the duct, the same is getting reduced for each type of rotors compared to without duct condition. Therefore, it can be declared that the duct helps to improve rotor performance. On the other way, it can be told that as the blockage ratio is increased due to duct implementation, from 0.086 to 0.150, the performance of the rotors is also increased.

Fig. 7

Velocity color band with a vector contour plot of different rotors with and without duct condition at 2 m/s free stream velocity, N = 120 rpm and TSR = 0.691

Fig. 8

Pressure developed surrounding the different rotors with and without duct condition at 2 m/s free stream velocity, N = 120 rpm and TSR = 0.691

3.1.1 Variation of C_p and C_t with respect to TSR for comparing SHKT with duct and without duct

A performance characteristic of the turbine is explained using the coefficient of power (C_p), coefficient of torque (C_t) and tip speed ratio (TSR). The performance of SHKT with duct and without duct is shown in Fig. 9a–d. A comparative study is conducted between SHKT with duct and SHKT without duct by numerically at the free stream velocity 1 m/s, 2 m/s and 3 m/s. The study is validated experimentally at 1 ± 0.2 m/s free stream velocity. An experimental study is performed initially at no load condition and after each observation mechanical load is increased through rope brake dynamometer. The maximum C_p (i.e. $C_{{{{\text{p}}_{ \rm{max}} } }}$) computed numerically at 1 m/s free stream velocity for without duct condition is 0.069 and maximum C_t (i.e. $C_{{{{\text{t}}_{ \rm{max} }} }}$) is 0.096 at a TSR value of 0.691. This result is validated through experiment for 1 ± 0.2 m/s free stream velocity. In this case, $C_{{{{\text{p}}_{ \rm{max} }} }}$ and $C_{{{{\text{t}}_{ \rm{max} }} }}$ are found 0.073 and 0.111, respectively, at 0.656 TSR. Here, TSR ranges from 0.184 to 0.887. Validation is performed after implementing the duct also, which is shown in Fig. 9c, d. It is found that TSR values are almost independent of the application of the duct. However, overall C_p and C_t increase due to the application of the duct. This suggests that as mainly C_p and C_t increased due to the contraction of inlet area which increases the striking possibility of the flowing water at the advancing blade.

Fig. 9

Effect of different flow velocity on the coefficient of power (C_p), the coefficient of torque (C_t) with respect to tip speed ratio on a SHKT without duct, b SHKT with duct

3.2 Variation of C_p and C_t with respect to TSR for comparing MSHKT without duct and with duct

Similar exercise for MSHKT shows that, for 1 ± 0.2 m/s free stream velocity, enhanced $C_{{{{\text{p}}_{ \rm{max} }} }}$ and $C_{{{{\text{t}}_{ \rm{max} }} }}$ of 2.74% and 3.61% are found for SHKT having without duct condition (Fig. 10a–d). Whereas, if the duct is implemented, the increase of performance is observed as 23.8% $C_{{{{\text{p}}_{ \rm{max}} } }}$ and 28.2% $C_{{{{\text{t}}_{ \rm{max} }} }}$ more than SHKT without duct condition. Here, TSR ranges from 0.219 to 1.094. It is observed that at comparatively low TSR, rotors are performing well.

Fig. 10

Effect of different flow velocity on the coefficient of power (C_p), the coefficient of torque (C_t) with respect to tip speed ratio on a MSHKT without duct, b MSHKT with duct

3.3 Variation of C_p and C_t with respect to TSR for comparing HSHKT without duct and with duct

Similar comparison study has been performed for the HSHKT rotor for 1 ± 0.2 m/s free stream velocity. Here, the overall increment in C_p and C_t are observed (Fig. 11a–d). It is found that $C_{{{{\text{p}}_{ \rm{max}} } }}$ and $C_{{{{\text{t}}_{ \rm{max}} } }}$ are 9.04% and 7.56% more than SHKT without duct condition. For the same rotor, if the duct is implemented, the increment of $C_{{{{\text{p}}_{ \rm{max} }} }}$ and $C_{{{{\text{t}}_{ \rm{max}} } }}$ is 48.08% and 40.36%, respectively. The experimental result signifies that due to the twisted shape of the rotor blade, it experiences more drag force caused by the flow of fluid for a prolonged period of time. Therefore, C_p and C_t of HSHKT are higher among all the rotors. Performance prediction is also considered numerically for 2 m/s and 3 m/s. For these cases, also increase in C_p and C_t with similar trend curves is found.

Fig. 11

Effect of different flow velocity on the coefficient of power (C_p), the coefficient of torque (C_t) with respect to tip speed ratio on a HSHKT without duct, b HSHKT with duct

It is observed from the numerical analysis that for 2 m/s free stream velocity, $C_{{{{\text{p}}_{ \rm{max}} } }}$ is increased in the order of 21.74%, 20.83% and 14.47% and $C_{{{{\text{t}}_{ \rm{max}} } }}$ is increased in the order of 27.08%, 21.70% and 21.60% for SHKT, MSHKT and HSHKT, respectively. Whereas, for 3 m/s free stream velocity, $C_{{{{\text{p}}_{ \rm{max} }} }}$ is increased in the order of 36.23%, 33.33% and 44.74% and $C_{{{{\text{t}}_{ \rm{max} }} }}$ is increased in the order of 43.75, 38.68% and 49.60% for SHKT, MSHKT and HSHKT, respectively. From the above discussion, it can be suggested that power generation capacity at the same flow velocity of HSHKT is higher in small amount than SHKT and MSHKT, while considering nearly the same aspect ratio. Instead of higher power generation in Helical Savonius hydrokinetic turbine, the coefficient of power is insignificant changes are observed at the same range of tip speed ratio. Further, a study is conducted to analyze the flow phenomenon around HSHKT with varying duct angle (20°–32°).

3.4 Flow phenomenon around HSHKT corresponding to varying duct angle

In Fig. 12, analysis of flow behavior around HSHKT with different duct angle varying from 20° to 32° is shown which is validated experimentally also. The experiment and numerical analyses are performed considering the distance of the rotor (L) from the initiation of duct convergence as 4D. It is observed that while the duct angle increases, the inlet velocity is increased just after the converging section of the duct. For 20°, 23°, 26°, 29° and 32°, an inlet velocity of 1 ± 0.2 m/s increases by 27.5%, 28%, 29%, 32% and 34%, respectively. Significant vortex formation is observed at the downstream of the rotor for lower duct angle (Fig. 12a). When duct angle increases, low-pressure zone formation increases which leads to extend the formation of the vortex far from the rotor (Fig. 12c), and finally disappears (Fig. 12e). Thus, the rotation increases and generates more power. Variation of velocity upstream to the duct, average velocity just downstream of convergences of duct, near the rotor, C_p and C_t are shown in Fig. 12g, h using error graph. In this condition, C_p of HSHKT improves by 28.63%, 16.16%, 43.01%, 50.13% and 82.32% for duct angle of 20°, 23°, 26°, 29° and 32° compared to without duct condition of HSHKT. Thus, to improve the performance of HSHKT, higher duct angle is preferred.

Fig. 12

Flow phenomenon around HSHKT and C_p, C_t variation with different duct angle

4 Conclusion

In this paper, performances of the Savonius hydrokinetic turbine (SHKT), modified Savonius hydrokinetic turbine (MSHKT) and helical Savonius hydrokinetic turbine (HSHKT) are evaluated and compared for without and with duct condition. Both the numerical and experimental studies are conducted for the performance analysis. The experimental study validated the performance of the rotor for the free stream velocity of 1 m/s. Whereas, the numerical study extended for 2 m/s and 3 m/s free stream velocity.

• Among all the turbines discussed above, HSHKT generated more power than the other two turbines. The experiment shows that HSHKT without duct generates 1.58 W of P_max and 0.26 Nm of T_max at free stream velocity 1 ± 0.2 m/s. Performance prediction is conducted for the high velocity with the help of numerical analysis. It is observed that for 2 m/s free stream velocity, $C_{{{{\text{p}}_{ \rm{max} }} }}$ is increased in the order of 21.74%, 20.83% and 14.47% and $C_{{{{\text{t}}_{{\rm max} }} }}$ is increased in the order of 27.08%, 21.70% and 21.60% for SHKT, MSHKT and HSHKT, respectively. Whereas, for 3 m/s free stream velocity, $C_{{{{\text{p}}_{ \rm{max} }} }}$ is increased in the order of 36.23%, 33.33% and 44.74% and $C_{{{{\text{t}}_{ \rm{max} }} }}$ is increased in the order of 43.75%, 38.68% and 49.60% for SHKT, MSHKT and HSHKT, respectively.

• Based on the experimental analysis, HSHKT rotor with duct shows an increment of 48.61% in power and 41.67% in torque as compared with SHKT without duct at same free stream velocity.

• For the HSHKT rotor with duct, the peak value of $C_{{{\text{p}}_{ \rm{max} } }}$ and $C_{{{\text{t}}_{ \rm{max} } }}$ is 0.108 and 0.156 at TSR 0.714

• Velocity range and vector contour show that, for all three different rotors, at the downstream, low-velocity range extends while duct is implemented. Simultaneously distance of separation from the rotor towards the downstream direction is also increased, which is observed from vector contour. It indicates that due to the implementation of the duct, the influence of vortex generated in the downstream has less effect on the torque generation of the rotor. Thus, when the blockage ratio is increased due to duct implementation, from 0.086 to 0.150, the performances of the rotors are also increased.

• When rotor advancing blade is making 0° along the free stream, then wake formation past the rotor is minimum. Whereas, the 90° position of rotor advancing blade creates maximum wake formation, which indicates maximum energy loss in the stream flow.

• When increasing duct angle, inlet velocity of 1 ± 0.2 m/s increases by 27.5%, 28%, 29%, 32% and 34%, respectively, for duct angle of 20°, 23°, 26°, 29° and 32°. Simultaneously low-pressure zone increases which leads to extend the formation of the vortex far from the rotor. Thus, the performance of the HSHKT is improved at higher duct angle.

In the above section, conclusions of the study are presented. The study shows that HSHKT perform best out of the three Savonius rotors. Duct angle increases turbine performances. Flow phenomenon study shows uniqueness of power generation from different types of Savonius hydrokinetic rotors and influence of duct augmentation. Simultaneously, role of wake characteristics and vortex formation towards power generation is well explained. Further, study related to rotor design modifications like implementation of adjustable blade, multistaging, influence of aspect ratio, etc. may be considered as the suitable future scopes.