1 Introduction

The ocean waves possess a huge amount of untapped energy and can be sufficient to fulfil the global energy needs. Extensive works were reported to make such energy competitive and economic [1]. Wave energy conversion system (WECS) demonstration plants and operational plants are installed in countries such as Belgium, Canada, China, Denmark, Italy, Korea, Portugal, Spain, Sweden, and the USA [2]. An oscillating water column (OWC) is a WECS that consists of a plenum chamber, entrapped air, and air turbine (Fig. 1) [3]. As the wave enters into the system or leaves the system, the OWC breathes and creates a bidirectional airflow through a duct, which rotates the turbine. Hence, for each wave cycle, the turbine faces approximately sinusoidal air velocity. The turbine has to extract energy from a wide range of flow rates for efficient operation. Hence, the efficiency of an OWC is highly dependent on turbine performance [4].

Fig. 1
figure 1

Oscillating water column with Wells turbine

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The Wells turbine used in an OWC is self-rectifying and produces unidirectional torque although the flow is bidirectional. The turbine consists of a symmetric airfoil, and the chord is aligned perpendicular to the direction of incoming flow (Fig. 1). A detailed study of the performance parameters of the Wells turbine can be found in the study by Raghunathan [5]. Several aerodynamic design modifications for Wells turbine, with guide vane and without guide vane, have been reported to improve the performance [6]. Ample efforts were made to improve the performance of the Wells turbine through various design modifications and optimization [7]. From the literature, it is found that the performance of the Wells turbine is very sensitive to tip clearance and tip leakage flow. The blade tip clearance plays a significant role in the stalling of the Wells turbine. A larger tip clearance value delays the stall and decreases efficiency. A tip clearance of less than 2%C is recommended by Watterson and Raghunathan [8] to avoid performance deterioration. Many authors adopted various methods such as endplate addition [9], tip clearance optimization [10], non-uniform tip clearance [11], duct geometry modification [12], casing treatment [13], and penetrating blade tip [14] to modify the flow characteristics of the tip leakage flow, thereby improving the performance.

The tip leakage loss is a critical factor, which affects the performance of turbomachines significantly [15,16,17,18]. There are various types of blade tip treatments like squealer tip [19, 20], radiused tip [21], contoured tip [22], and tip shape optimization [23], etc., are reported in the literature to minimize such loss. The Wells turbine is self-rectifying in nature, and its blade profiles are symmetric. Hence, the blade symmetry should be retained while adopting these tip treatments. The radiused edged tip blade (RETB) proposed by Bindon [21] mitigates the separation bubble and tip clearance losses in an annular turbine cascade [24]. The RETB streamlines the tip clearance flow and minimizes the entropy generation due to flow mixing and reattachment inside the tip clearance region [22]. The RETB performs better [25,26,27] and minimizes the tip separation relative to a sharp-edged tip blade (SETB) [28]. The tip separation at a high flow rate gives a narrow operating range in a Wells turbine, and RETB can overcome such adverse effects. So far, no work is reported RETB used in any bidirectional flow turbines.

In the present work, the RETB concept is introduced in the Wells turbine, and the problem is solved through numerical analysis. In RETB, both sides of the blade tip edges are modified symmetrically. Different fillet radius and tip clearances are considered, and the performance is compared with the reference blade. The detailed flow analysis showing the performance change in the modified blade is reported.

2 Numerical Formulation

2.1 Reference Turbine

The reference Wells turbine geometry consisting of a symmetric airfoil with zero camber and untwisted blades of 90o stagger having the SET was taken from the work of Torresi et al. [10]. The design parameters are given in Table 1. The SET and RET concepts are shown in Fig. 2. The fillet created between the tip and the span of the blade has a radius of 1%C, 2%C, and 3%C. In this study, the RET was created on the suction side (SS) and the pressure side (PS).

Table 1 Specifications of reference geometry [10]
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Fig. 2
figure 2

Sharp edge tip and radiused edge tip blade

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2.2 Computational Methodology

The numerical results were obtained by solving steady Reynolds-averaged Navier–Stokes (RANS) equations in ANSYS CFX 14.5. The equations account only for the mean flow quantities without resolving the turbulent fluctuations. To resolve the closure issue, the two-equation k-ω SST turbulence model with the automatic wall function was used [29]. The k-w SST turbulence model is a hybrid model that resolves the viscous sublayer region, unlike the k-ε model which uses wall function [30]. It uses the k-w turbulence model in the viscous sublayer and the log layer region and switches to the k-ε turbulence model in the free shear flows and the wake region. It is suitable for solving flows with adverse pressure gradients and flow separation. Moreover, several researchers successfully used the k-w SST model to simulate the Wells turbine problem [13, 31, 32]. Hence, the k-w SST model was selected as the turbulence closure model for the present study.

The turbulence kinetic energy (k) and the specific dissipation rate (w) are obtained from the subsequent transport equations [33]:

$$ \rho \frac{\partial k}{\partial t} + \rho u_{j} \frac{\partial k}{{\partial x_{j} }} = \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\mu + \sigma *\mu_{t} } \right)\frac{\partial k}{{\partial x_{j} }}} \right] + G_{k} - \rho \beta *kw\quad i,j = 1,3 $$
(1)
$$ \rho \frac{\partial w}{\partial t} + \rho u_{j} \frac{\partial w}{{\partial x_{j} }} = \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\mu + \sigma *\mu_{t} } \right)\frac{\partial w}{{\partial x_{j} }}} \right] + \alpha G_{k} - \rho \beta w^{2} \quad i,j = 1,3 $$
(2)

where Gk is the turbulent kinetic energy production.

Constant

σ

σ*

α

β

β*

Value

0.5

0.5

0.56

0.075

0.09

The coefficient and constant of the SST model are as follows:

$$ {\text{The}}\,{\text{turbulent}}\,{\text{viscosity}},\quad \mu_{t} = \frac{{\rho a_{1} k}}{{\hbox{max} (a_{1} wSF_{2} )}} $$
(3)

S and F2 indicate the magnitude of the strain rate and blending function represented by

$$ F_{2} = \tanh (\arg_{2}^{2} ) \,{\text{with}}\,\arg_{2} = \hbox{max} \left( {\frac{2\sqrt k }{{\beta^{{\prime }} wy}},\frac{500\nu }{{y^{2} w}}} \right) $$
(4)

All constants are calculated from the following equation:

$$ \alpha = \alpha_{1} F + \alpha_{2} \left( {1 - F} \right) $$
(5)

Constant

β′

α1

β1

αk1

σw1

Value

0.09

0.555

0.075

0.85

0.5

The continuity and momentum equations for the incompressible, steady, and isothermal fluid flow in the tensor form are given by Eqs. 1 and 2 [30].

Continuity equation

$$ \frac{{\delta u_{i} }}{{\delta x_{i} }} = 0 $$
(6)

Momentum equation

$$ u_{j} \frac{{\partial u_{i} }}{{\partial x_{j} }} = - \frac{1}{\rho }\frac{\partial p}{{\partial x_{i} }} + \frac{\delta }{{\delta x_{j} }}\left( {\gamma \frac{{\delta u_{i} }}{{\delta x_{j} }}} \right) $$
(7)

To ensure second-order accuracy in space, the high-resolution advection scheme was used. It uses a blend factor value of 1 in the regions of low gradients maintaining second-order accuracy and a value of 0 with a first-order accuracy in the regions of high gradients to maintain robustness [30]. The residual convergence criteria for all simulations were kept as 1.0E−5. The solver uses a fully implicit discretization of the equations and employs false timestep to accelerate convergence. ANSYS CFX 14.5 is a coupled solver, and it is much robust and stable than the segregated solver [30]. The simulations were run parallel in the VIRGO supercluster available at the Indian Institute of Technology Madras. It consists of 292 compute nodes, 2 master nodes, and 4 storage nodes. Each node has 64 GB ram, and it is populated with 2 X Intel E5-2670 8 C 2.6 GHz processor. Each simulation took around 20 h to complete, and the double-precision option was enabled for all the simulations.

2.2.1 Meshing and Boundary Conditions

A single blade passage with a cyclic boundary condition was used as the computational domain (Fig. 3). The upstream and the downstream of the computational domain were extended to 4C and 6C, respectively. The x-, y-, and z-axes were taken as the chordwise, streamwise, and spanwise direction, respectively. The rotor rotates in a counter-clockwise direction at 2000 rpm. The Coriolis and centrifugal components were added to the governing equations while using the rotational reference frame method [34]. The discretization of the computational domain was done with unstructured tetrahedral elements. To capture the flow physics near the blade, 20 layers of prism elements were created around the blade with a layer growth ratio of 1.2 (Fig. 3). The first cell height was chosen as 0.011 mm to retain the wall y+ < 1.

Fig. 3
figure 3

Discretized computational domain

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The blade, shroud, and hub were kept as no-slip walls. The inlet and the outlet were fixed as velocity inlet and pressure outlet boundary conditions, respectively. A uniform axial velocity was applied to the inlet, and the relative pressure at the outlet was fixed at 0 atm. The inlet turbulence intensity was taken as 5%. It is described as the ratio of root mean square of fluctuating velocity components (u′) and an average of the mean velocity components (U). It can be expressed as

$$ I = \frac{{u^{{\prime }} }}{U} $$
(8)

where \( u^{{\prime }} = \sqrt {\frac{1}{3}\left( {u_{x}^{{{\prime }2}} + u_{y}^{{{\prime }2}} + u_{z}^{{{\prime }2}} } \right)} \)

$$ U = \sqrt {\left( {\bar{u}_{x}^{2} + \bar{u}_{y}^{2} + \bar{u}_{z}^{2} } \right)} $$
(9)

It is also known as the level or degree of turbulence. Based on the details available in the open literature, the inlet turbulence value was fixed as 5% [5, 35, 36]. Moreover, the lateral side of the computational domain was given periodic boundary condition, to realize the periodic nature of the rotor geometry. The computational domain chosen for the flow analysis is illustrated in Fig. 4.

Fig. 4
figure 4

Computational domain for the flow analysis

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2.3 Performance Parameters of the Wells Turbine

The performance of the Wells turbine is evaluated in terms of the following non-dimensional parameters. They are listed below [10].

  • Flow coefficient (U*)

    $$ U^{*} = \frac{{U_{A} }}{{U_{\text{tip}} }} $$
    (10)

    where \( U_{A} \) is the inlet axial velocity (m/s) and \( U_{\text{tip}} \) is the blade tip velocity (m/s).

  • Torque coefficient (T*)

    $$ T^{*} = \frac{T}{{\rho \omega^{3} R_{\text{tip}}^{5} }} $$
    (11)

    where T is the torque, ρ is the air density (1.225 kg/m3), ω is the angular velocity (209.44 rad/s), and \( R_{\text{tip}} \) is the blade tip radius (m).

  • Pressure drop coefficient (Δp*)

    $$ \Delta p^{*} = \frac{\Delta p}{{\rho \omega^{2} R_{\text{tip}}^{2} }} $$
    (12)

    where Δp is the pressure drop across the inlet and outlet (N/m2).

  • Efficiency (η)

    $$ \eta = \frac{T\omega }{Q\Delta p} $$
    (13)

    where Q is the volume flow rate of air (m3/s).

2.4 Grid dependency and Numerical Validation

The grid convergence was assessed based on the grid convergence index (GCI). The GCI provides the uncertainty caused due to the discretization error. Three grids with a different resolution such as fine, medium, and coarse were selected with a grid refinement ratio of 1.3. The method recommended by Celik et al. [37] was used to calculate the GCI. Table 2 provides the GCI calculation result. It shows that the numerical uncertainty reduces with an increase in grid refinement. The GCI value for the fine and medium grid is 0.438% and 1.928%, respectively. According to Manna et al. [38], the grid is assumed to be spatially converged if the GCI between the successive grids is less than 3%. Hence, the medium grid with 3 million cells was retained for further simulations to reduce the computational cost, which is shown in Fig. 5. Also, to validate the present numerical results, the computations were performed for a wide flow coefficient and compared against the available experimental [39] and numerical results [10, 12, 40] (Fig. 6). The present results follow a similar trend with the existing results. The sudden drop in torque and the efficiency at a flow coefficient (U*) of 0.225 indicates stall.

Table 2 GCI calculation
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Fig. 5
figure 5

Grid independency test

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Fig. 6
figure 6

Numerical validation

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3 Results and Discussion

3.1 Effect of Radius Edge Tip (RET)

To evaluate the effect of RET, three fillet radii (1%C, 2%C, and 3%C) were chosen, and the results are presented in Fig. 7. As the flow coefficient increases, the torque increases up to the stall point and then falls. At a low U*, the RET produces a lower torque. However, a higher U* generates a higher torque, while SET blade stalls (U* = 0.225). On the other hand, a marginal difference in torque is observed between the performance characteristics of the RET blades with different fillet radius. The pressure drop coefficient exhibits a linear relationship with the U*, which is a similar trend reported in the other literature [39]. The RET blade has a lower pressure drop compared to the SET blade. The efficiency of the SET blade drops rapidly at U* = 0.225, while the RET blade gives the efficiency drop at U* = 0.28. In the RET blade, the peak efficiency is lower, but RET with 1%C shows a better efficiency for the entire flow range. The reduced peak efficiency can be attributed to the enlarged tip clearance causing an increase in leakage losses.

Fig. 7
figure 7

Performance comparison of sharp edge and the radius edge tip blades

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Figure 8 shows the blade loading distribution at midspan (50%) and tip (95%). As apparent, the RET blades show reduced blade loading at midspan and tip compared to the SET blade. The area of the pressure coefficient (Cp) indicates the lift force; it explains the reduced torque of RET blades at low U*. Similar behaviour is observed at U* 0.225, where the SET shows higher blade loading in contrast to RET blades. At high U* (= 0.275), the leading edge (LE) minimum suction pressure vanishes in the SET blade and exhibits almost constant SS pressure along the chord that indicates stall. However, the RET blade displays increased LE suction pressure and it is the reason behind the delayed stall and improved torque at higher U*. Hence, it can be concluded that the RET blade reduces blade loading and delays the stall. Figure 9 illustrates the spanwise tangential velocity distribution along with blade LE. The tangential velocity component is related to torque generation. As evident in the figure, the tangential velocity increases from hub to tip for all cases. At U* = 0.075, the SET blade shows relatively higher tangential velocity compared to RET blades and it is in agreement with the results shown in Fig. 6a. The tangential velocity increases with increasing U*. At U* = 0.275, the RET blade shows reduced tangential velocity near the tip that implies stall, whereas the RET blades show increased tangential velocity and delayed stall. Also, the RET blades exhibit similar performance for varying fillet radius.

Fig. 8
figure 8

Blade loading at midspan (left) and tip (right) for different U*

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Fig. 9
figure 9

Spanwise tangential velocity distribution near blade LE

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The inlet velocity triangle for different axial velocities is shown in Fig. 10. The terms V1, W1, and UT denote absolute inlet, relative inlet, and peripheral blade velocities, respectively. The term α1 denotes the angle of attack at the inlet, where the relative flow meets the rotating blade. The angle of attack is a critical factor that affects the stall characteristics and torque generated by the turbine.

Fig. 10
figure 10

Velocity triangle at inlet for different axial velocities (V1a and V1b)

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In general, the torque increases with angle of attack up to a critical value and then decreases due to the blade stall phenomenon. Additionally, as shown in Fig. 10, for a given blade velocity, the angle of attack increases with an increase in axial velocity (V1b > V1a; α1b > α1a). The normalized spanwise distribution of the axial velocity near blade LE is shown in Fig. 11. It can be seen that the axial velocity of the SETB along the span is relatively higher than the RETBs. As explained earlier, reduced axial velocity decreases the angle of attack and it is responsible for the reduced torque at low U* and the delayed stall. The SETB stalls at U* = 0.275 and it shows a higher axial velocity near the tip compared to the RETBs. The reduced blade loading in the case of the RETBs (Fig. 8) is attributed to the decreased axial velocity, and it is responsible for the delayed stall and shift in peak torque towards a higher U*.

Fig. 11
figure 11

Spanwise distribution of axial velocity near blade LE, for different values of U*

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The pressure variation between the PS and SS of the blade tip drives the tip leakage flow. It interacts with the primary flow and increases the secondary losses, which leads to reduced efficiency. It was reported in the literature that the tip leakage flow affects the performance of the Wells turbine significantly [5] [11]. The effect of the radius on the tip leakage flow is analysed using Fig. 12b. The strength of tip leakage flow is minimum at low U* (U* = 0.075), and it is predominant near the TE for all cases. The fillet radius increases the leakage flow in the RET blade. The RET blade shows higher leakage flow at low U* compared to the SET blades, and this explains the improved efficiency of the SET blade at low U*. With the increasing flow rate, the tip leakage flow increases for all cases.

Fig. 12
figure 12

a Flow direction and blade rotation direction. b Tip leakage flow streamlines coloured with turbulence kinetic energy contours

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At a higher U*, the tip leakage flow shifts towards the LE and rolled into a vortex and interacts strongly with the primary flow, unlike the previous cases (U* = 0.075 and 0.225), where the interaction is weak. This leads to the turbine stall, and a similar phenomenon was reported by Dhanasekaran and Govardhan [41]. Moreover, the tip leakage flow emerges as a flat jet [26] in the RET blade and has a weak interaction with the primary flow even at higher U*. On the contrary, it is rolled into a sheet of vortex in the SET blade and interacts strongly with the primary flow. The reason behind the delayed stall of the RET blades can be attributed to a weak interaction between the primary and the tip leakage flow. The delay of stall inception improved the performance of RET blades significantly at a higher U*. Also, with an increase in fillet radius, a decrement in peak efficiency is observed.

4 Conclusions

The effect of the radiused edge tip blade on the performance of the Wells turbine is presented in this work. The numerical investigations were conducted for the entire flow range including the post-stall regime. Based on the analysis, the following conclusions were drawn.

  • The radiused edge tip blades exhibit reduced blade loading at midspan and tip compared to the reference blade. The reduced blade tip loading implies reduced axial velocity and angle of attack at the tip, and it is responsible for the delayed stall and shift in peak torque towards a higher flow coefficient.

  • The radiused edge tip blades showed a weak interaction between the primary and the tip leakage flow compared to the reference blade, at higher flow coefficients and improved the performance.

  • The peak efficiency was reduced, and the stall point was shifted towards the higher flow coefficient.

  • A marginal difference in performance is observed for radiused edge tip blades with different fillet radius.

  • The radius edge blade tip showed a better performance compared to the sharp edge blade. The relative peak torque was improved by 37%, and the relative stall margin was enhanced by 25%.