Introduction

At a depth of water, d with density, ρ of water, under the influence of gravity, g, the pressure P can be defined as follows⁽⁶⁾:

Referring to Fig. 2, consider a simple vertical plate which is submerged in water with the depth, d. The triangles represent the hydrostatic pressure. The forces acting on either sides of this plate (its width, w) are F⁽⁶⁾:

Fig. 2 
Operating principle of VTT

If it is now imagined that the plate moves laterally with velocity, V, the power at the plate available from water flow, Pa, is:

Where A is the cross section area of the plates submerged in water under consideration (m²):

The water wheel converts a part of this power to the shaft’s power, Pshaft⁽¹⁾:

Where CP is the power coefficient of the wheel and represents the ratio between the power output from the wheel and power available in the water. Not all of the available power is converted into the shaft power by the wheel. Leaving water should have sufficient kinetic energy to continue its way outside the wheel. The above equations show that the power output from the wheel depends on the power available in the water and the efficiency of the wheel.

In order to analyze the turbine’s performance at different rotational speeds, the runner’s velocity is normalized by the incoming water velocity and a non-dimensional parameter, called “tip-speed ratio” (λ), is used and defined as follows⁽⁷⁾:

Where R is rotor’s radius (mm), ω is rotational speed (rad/s), Vin is inflow velocity (m/s). Tip speed ratio is related to the efficiency, and in this paper, the study is carried out with a range of TSRs by varying the cut-in velocity, as shown in Table 2.

A three-dimensional model of this turbine and its flow field are generated and numerically discretized in computational simulation. Power performance and characteristics of VTT are calculated using CFD commercial code, ANSYS 14. Meshing is done using ICEM CFD. Hexahedral grids are used and the mesh contains 313,000 nodes for the stator part (inlet, outlet, bedplate, side walls). In all cases, the device’s sizing is maintained. For double row rotor setup, both blade rows are equally divided and have the same width.

Totally, there are four options available for blade arrangement, the same mesh strategy is applied for all cases resulted in relatively similar total number of nodes, about 830,000. Figure 3 illustrates the mesh formation for the stationary frame of the calculation domain. Figures 4 and 5 show the mesh formation for six and twelve-blade turbines, respectively, with two types of blade arrangement (single and double rows).

Fig. 3 
Mesh formation for stator

Fig. 4 
Mesh formation for rotor of 6-blade turbine

The computational domain is shown in Fig. 5. Two ends of the channel (inlet and outlet) are lengthened ten times as many as the runner’s width. This is sufficient for computational simulation and capable of capturing the wake characteristics. Inlet boundary is set to “Cartesian velocity components”, outlet boundary is average static pressure, and opening boundary condition is “Opening pressure and direction”. The blades, hub, channel, bedplate and side walls are all no-slip wall boundaries. The mesh connection between the rotating and stationary frames is general connection interface boundaries, and is set to “Transient rotor stator”. The working fluid are defined as sea water and air at 25 ^oC. Turbulence model is shear stress transport (SST), with turbulence intensity set to 5% for the whole flow-field. Simulation is done in transient by ANSYS CFX Solver. Time step is set depending on the rotational speed to effectively get convergency.

Fig. 5 
Mesh formation for rotor of 12-blade turbine

Fig. 6 
Computational domain

The visualizations of averaged pressure distribution for the six and twelve-blade VTTs with two kinds of blade arrangement at different TSRs are shown in Figs. 7 to 10. These figures present the surfaces of pressure contours which are distributed over the length of the channel from inlet to outlet. In all cases, the view of averaged pressure distribution is oriented from the left (high pressure side, darker color) to the right (low pressure side, lighter color), the area in grey scale illustrates the variance of fluid pressure within the range. The pressure contours plane is marked as four regions: A, B, C, D. Region A illustrates the averaged pressure value distributed from the inlet of the channel to the entrance of rotor, where water is about to contact with the blades. Regions B and C show the averaged pressure distribution of water stream in the spaces between two adjacent blades. And the last one, region D, is where water flows out of rotor and keep flowing to the outlet.

Fig. 7 
Averaged pressure distribution of six-blade at TSR=0.47

Fig. 8 
Averaged pressure distribution of six-blade at TSR=0.94

Fig. 9 
Averaged pressure distribution of twelve-blade at TSR=0.47

Fig. 10 
Averaged pressure distribution of twelve-blade at TSR=0.94

Figures 7 and 8 show averaged pressure distribution for six-blade type at TSR=0.47 and TSR=0.94, respectively. As shown in Fig. 8, at TSR=0.94 (lowest cut-in speed), it is obvious that the difference in pressure color for single row type between region A and region D is relatively more distinguished than that of the double row type. It means that there is more considerable amount of available energy in outflow water passing through the double row type VTT. In the other words, the water current energy is absorbed effectively by the single row turbine in comparison with the double row type for the same number of blades. At the lowest TSR (highest cut-in speed) as depicted in Fig. 7, the color of pressure from region A to region D in either double or single row rotors does not change as much as at TSR=0.94..

Figures 9 and 10 illustrate averaged pressure distribution for twelve-blade turbines at various TSRs. On the contrary to the six-blade arrangements as analyzed above, the twelve-blade single row turbine has a lower capacity of water flow energy than the twelve-blade double row one. Besides that, in all cases of TSR for twelve-blade turbine, the difference in pressure color of the single and double row types between the region A and the rest, especially at TSR = 0.47, is not as clear as that of the six-blade designs at the same TSR.

Figures 11 and 12 show the comparison of water volume fraction, which also discovers the behavior of water flow for six and twelve-blade single row turbines. In these figures, it can be clearly seen that at TSR=0.94 (1 m/s and 10 rpm), the water elevation at all types of VTT has the same rate. However, at TSR=0.47 (2 m/s and 10 rpm), water flow has a tendency to ascend and accumulate abnormally, right in front of the entrance of rotor, consequently the water level rise slightly higher than that to be seen at higher TSRs. It can be explained that due to low revolution of the rotor and is kept constant at 10rpm, at higher TSR (lower inflow velocity) the amount of water running throughout the rotor is more regular and it will reduce the formations of some profitless phenomena, such as back flows, local vortexes, etc. These phenomena will increase the energy losses, prevent the incoming flows from approaching the turbine’s blades, and reduce the power efficiency of the device. For six and twelve-blade double row types, the abnormal surge of water level occurs similarly to the six and twelve-blade single row cases.

Fig. 11 
Water volume fraction for six-blade single row type

Fig. 12 
Water volume fraction for twelve-blade single row type

Power performances of VTT are evaluated by power coefficient Cp and torque coefficient Cq which were calculated and expressed in Figs. 13 and 14, respectively. The figures express Cp and Cq as a function of inflow velocity, and show the performances of two configurations of VTT with different blade arrangement (single and staggered blade rows). The study is carried out with a various TSRs by varying the inflow velocity from 1 to 2 m/s within sufficiently long revolutions of the wheel.

Fig. 13 
Power coefficient as a function of TSR

Fig. 14 
Torque coefficient as a function of TSR

Both graphs of Cp and Cq show the same patterns which are inclined to rise up from the lowest to the highest TSRs. According to the results depicted in Fig. 13, the power coefficient of six-blade single row type achieves the highest value than that of six-blade double row one. At TSR=0.94 (the biggest cut-in speed), this type reaches up to 33.8% Cp which is about eight and a half times as high as the value at TSR=0.47. Conversely, for twelve-blade turbine, the double row type generates the highest of Cp in comparison with the single one. The highest power coefficient absorbed from this turbine obtains 16% at TSR=0.94, meanwhile, the single row type just attains 12 %. In addition, it is obvious that at lower TSRs (higher cut-in speeds), all kinds of VTT show their worst performance since the power extracted from the blades is extremely low as well as the power coefficient. Similar to the power coefficient chart, as shown in Fig. 14 of torque coefficient that is a similar non-dimensional analysis for the torque⁽⁸⁾, the six- blade single row turbine keeps showing its predominant performance comparing to the others at various TSRs. It means that the torque extracted from the rotor’s shaft of this turbine always gets greater than the other kinds of VTT. On the contrary, the tidal current energy conversion capability of the twelve-blade single row type is the smallest compared to the others at the same TSR, as the generated torque coefficient of this kind only achieves 12.6 % which is three times as low as the six-blade single row type, at the greatest cut-in speed (TSR=0.94).

These results in turbine efficiency are reasonable to characteristics of flow patterns (including pressure contours and water volume fractions) analyzed above, where high efficiency is found at the cases with obvious change in pressure of water flow before and after the wheel, and lower water level at higher TSR (lower inflow velocities).

As shown in Fig. 15, it illustrates the variation of torque extracted at the rotor’s shaft due to the water flow acting on the blades at TSR=0.94 for six and twelve-blade double row turbines. The results of six- blade Double row turbine show that when submerged in the water, one blade needs to be inclined with at least 4.5 degrees angle to start contributing to the torque’s generation. The blade contributes by producing torque until 18 degrees; after that, the contribution becomes negative. That means the blade will be active for 13.5 degrees of the 36 degrees inside the water. By adding more blades, more torque generated from the turbine will be improved by contributing to the positive torque⁽¹⁾. The torque’s extraction in one revolution of the wheel (360 degrees rotational angle) for twelve- blade double row type is more continuous and the number of blades takes part into generating torque is more than the other types of VTT.

Fig. 15 
Torque generation in one revolution