On the Mathematical Modelling of Adaptive Darrieus Wind Turbine

Darrieus wind turbines are experiencing a renewed interest for their application in decentralized power generation and urban installation. Much attention and research efforts have been dedicated in the past to develop as an efficient standalone Darrieus turbine. Despite these efforts, these vertical axis turbines are still low in efficiency compared to the horizontal axis counterparts. The current architecture of the turbine and their inherent characteristics limit their application in low wind speed areas as confirmed experimentally and computationally by past research. To enable and extend their operation for weak wind flows, a novel design of Adaptive Darrieus Wind Turbine (ADWT) is proposed. The hybrid Darrieus Savonius rotor with dynamically varying Savonius rotor diameter based on the wind speed enables the turbine to start, efficiently operate and stop the turbine at high winds. As the wake of Savonius rotor has a profound impact on the power performance of the combined rotor, the wake of two buckets Savonius rotor in open and closed configuration is reviewed. The current study aims to develop an analytical model to predict the power coefficient and the influence of other design parameters on the proposed design. The formulated analytical model is coded in python, and the results are obtained for the 10 kW rotor. Parametric analysis on the chord length and the diameter of the closed Savonius rotor is performed in search of an optimized diameter to maximize the annual energy output. Blade torque and the rotor torque are evaluated with respect to azimuthal angle and compared with conventional Darrieus rotor. The computed results show that peak power coefficient of ADWT is 13% lower than the conventional Darrieus rotor at the rated wind speed of 10 m/s.


Introduction
Renewable energy sources are increasingly popular for their emission-free power generation.The tremendous increase in the wind turbine installation is expected to continue in the future with the current worldwide installation of 468 GW [1].
The advancement in other renewable sources such as solar energy and inexpensive storage system led to the growth of decentralized power generation or distributed energy generation.Reduction in transmission losses and the immediate response to the local power demand are the notable advantages of distributed energy generation [2].There is renewed interest in the development of Vertical Axis Wind Turbines (VAWTs) for their simple design, Omni directionality and east of maintenance [3].Straight bladed Giromill or H-rotor turbines are preferred than other VAWTs, especially for their efficiency.Regardless of the above-said merits, the startup issues [4] and the lack of aerodynamic power regulation at high winds are immediate hurdles in their development.Multiple strategies, field tests, and computational studies spread all over the literature to address these two critical issues.
The past attempts contribute significantly for the enhancement of startup characteristics, yet one design has not been singled out as an implementable solution for both startup and over speed regulation without affecting the performance of the Darrieus turbine at higher Tip Speed Ratio (TSR).Rotors with cambered airfoils are found to generate higher starting torque at low wind speed compared to symmetric airfoils [5].Albeit, the performance at high wind speed is curtailed due to increased drag in the downstream, the reduction in peak power coefficient is small compared with an overall increase in the annual energy output.New airfoils are designed to extend the fatigue life of the blade and to reduce the manufacturing cost of the blades [6] especially for urban turbines, where the flow is characterized by highly turbulent and multidirectional [7].
Conventional airfoils are modified by incorporating a cavity to enhance the lift at low Reynolds number (Re) [8].The computational study on the trapped vortex airfoil shows significant improvement in starting torque, yet the performance deteriorates at higher TSR [9].Recent proposals such as blade pitching, trailing edge flaps and morphing blades addresses both the starting issues and the over speed regulation, but the cost of precise mechanical actuators and the complex sensing elements limits its commercial application.Increasing the solidity by increasing the number of blades enhances the starting torque, but the performance at higher TSR will decrease due to the blade wake interaction from the preceding blade [10].Blades with trailing edge flaps are proposed as a potential solution for low wind startup and to aerodynamically regulate the rpm of the rotor.The solution is practically complex and the improvement in stating characteristics is not attractive as they are not able to sustain the rotation.An elegant, low-cost solution is still in search of the question that is hovering around the development of Darrieus wind turbine for decades.The current study attempts to provide a solution by proposing a novel Darrieus rotor.The remaining part of the

Working Principle of Adaptive Darrieus Wind Turbine
Of several solutions that are discussed above, hybrid Darrieus and Savonius turbine is a potential candidate that be redesigned to improve the low wind speed performance and over speed regulation.The conventional hybrid Darrieus turbine integrates a Savonius rotor to a common shaft with the Darrieus rotor.The strategy is that the high torque generated by the Savonius rotor accelerates the rotor to higher TSR.Similar to other concepts, the hybrid Darrieus-Savonius also suffers from poor performance when the Darrieus rotor accelerates beyond 1.
The optimum TSR for a two-bladed Darrieus rotor lies between 3 to 5 [11] and the optimum TSR for Savonius rotor is 1.The Savonius rotor tends to generate resistive torque and in fact energy must be expended to rotate Savonius rotor for the TSR above 1.The mismatch between the optimum TSR for the two rotors severely degrades the performance at higher TSR.Practically, a conventional hybrid Darrius-Savonius rotor will not accelerate beyond 1.5 resulting in iota of improvement in annual energy output.Hence a novel design has been put forward to minimize the influence of Savonius rotor beyond TSR 1.The strategy is to transform the Savonius rotor into a shape that leaves minimum wake downstream without any resistive torque.Two bucket Savonius rotor can be transformed into a nominal cylinder if they are able to slide.The wake behind the downstream is axisymmetric with minimum width compared to other shapes.
The wake width and the kinetic energy imbibed dictate the performance of the rotor and decelerate the rotor when it rotates beyond the rated rpm.The construction and the mechanical arrangement are less complex making this concept commercially implementable.

Analytical Model of ADWT in Open Configuration (Open Savonius)
The

Mathematical Model
And the equilibrium induced velocity is ( ) With e V as the input velocity for the downstream half-cycle of the rotor the induced velocity at the end of the streamtube is ( ) The relative velocity for the for the upstream half-cycle of the rotor, , is given by the expression ( ) where X r V ω = represents the local tip speed ratio.The general expression for the angle of attack is ( ) By equating the blade element theory and the momentum equation for each stream-tube ( ) where up f is the function that characterizes the upwind conditions cos sin sin cos The turbine Reynolds number will be ( ) ( ) For each blade in the upstream position, the non-dimensional force coefficients as functions of the azimuthal angle θ are given by ( ) ( ) By integrating for the entire blade ( ) The average half cycle of the rotor torque produced by N/2 of the N blades is given by: ( ) The average torque coefficient will be: Thus, the power coefficient for the upstream half can be written as Similarly for the downstream half cycle.The relative velocity for the for the downstream half-cycle of the rotor, π 2 3π 2 θ < < , is given by the expression ( ) represents the local tip speed ratio.The general expression for the angle of attack is ( ) By equating the blade element theory and the momentum equation for each stream-tube ( ) ( ) ( ) ( ) The downstream torque coefficient is given by For the Savonius rotor as shown in the Figure 2, the power coefficient can deduced as follows Suppose pressure difference on retreating side is Assuming advancing side contributes to negative torque through drag ( ) Average power p is obtained by integrating torque from 0 to Normalized power coefficient can be given as ( ) Let's assume V − is equivalent velocity or relative velocity, is v is absolute velocity.
Consider retreating side: p P ∆ value is known.Solving the integral, ( ) Similarly, for the advancing side: Again, resolving the component will yield the equation of the torque for advancing side.Combining all the parts together the final equation for the p C is given as

Analytical Model of ADWT in Closed Configuration (Cylinder)
The power coefficient under the influence of Savonius rotor in closed condition can be derived by treating it as a nominal cylinder placed in a steady and homogenous wind flow.Even this simplified assumption gives rise to complex flow wake structures.The objective of the model is to predict the wake width and the velocity deficit due to the presence of cylinder.The stream tubes that are influenced by the wake width are identified and the input velocity is modified with the velocity deficit calculated during the iteration.However there will be axisymmetric flow acceleration on either side of the wake which is not accounted.
The cylinder can be considered as non-rotating as the cylinder TSR is low, and the flow field displayed by both rotating and non-rotating flow for low rpm of 80 ~ 100 is similar.The current approaches and the assumptions will lead to the development of an analytical model that can be well integrated into the existing subroutine coded in python.

Wake of Savonius Rotor in Closed Configuration (Cylinder)
A wake boundary occurs between two fluids carrying different momentum along with their flow.The momentum deficit in the particular region of unidirectional flow is highly unstable and gives rise to the zone of turbulence mixing layer downstream at a point where the two streams meet for the first time.Though at far downstream the static pressure tends to equalize within the flow and wake, the velocity or the momentum deficit continues to travel along with the flow.
Detailed studies by the previous researchers have emitted a very important conclusion which relates the cylinder wake, its growth to the ratio of rotational to rectilinear speed ratio.The experimental studies by Prandtl [13], Dfaz [14] gave the most vital results which suggested that the eddies behind the cylinder in terms of Karman vortex street disappears at higher rotational to rectilinear speed ratio.One of the most dominating reasons for this is, for the low values of rotational speed ratio (TSR), the vortices are shed in the alternate fashion behind the cylinder.The cylinder Re dictates the wake pattern behind the cylinder and the wake structure can be established by defining the Re.Alternating eddies are formed and moves along the direction of rotation progressively decrease until the TSR reaches 1.After that it starts dissipating completely transferring the turbulent kinetic energy to large eddy structures leaving behind constantly growing wake.In the case of VAWTs in combination with the central shaft the similar kind of pattern is observed.The flow structure behind the cylindrical shaft rotating at TSR greater than 1, creates a uniformly growing wake without breaking into vortex structures.So for the mathematical modeling, it is safe to assume that the wake structure behind the central shaft is in accordance with the

Mathematical Model
The above equation is the governing partial differential equation For cylinder wake which is relatively thick we cannot consider Assuming, X is considerably large should be an even function (symmetric wake) V must be an odd function (asymmetric) in order to get the solution for go- verning differential equation.
Drag prediction from Boundary layer assumptions Ud2 can be neglected from Comparing it with conventional drag equation From Prandtl's mixing length theory we know that Let For two dimensional Equating above expressions Introducing equation of wake width variation into drag equation We derived previously Using variable differential form ( ) We know from 2-D incompressible flow equations From Prandtl mixing length theory We introduce Put this function again in differential equation to get value of ( ) And from momentum equation of experimental data the constant m ∆ yields the value equivalent to 10 ∆ ( ) According to experiments by H. Reichardt ( ) Substituting 0 y = in the d U equation then the velocity deficit is at central region which practically explains the maximum deficit velocity since the distribution is close to Gaussian distribution.
From Boundary layer theory and momentum equation after neglecting small terms we get For small Reynolds number of 10 4 , the following result holds true ( ) ( )

Results and Discussion
The developed mathematical model is applied to 10 kW ADWT and conventional straight bladed Darrieus turbine.The dimensional details of the configurations are listed in Table 1, and the various configurations are shown schematically in Figure 3.The results are evaluated for the starting characteristics, blade and rotor torque and power coefficient variation.Though the starting characteristics does not reflect the actual starting conditions, it will provide insight into the low Re behavior of cylinder.Another possible configuration that may be of interest for the current study is the height of the Savonius rotor with respect to the height of the Darrieus.It is envisaged that half-length of the Savonius rotor may have minimal influence on the performance of the Darrieus rotor.On the negative aspect, the half-length Savonius may induce uneven loading on the Darrieus blade on the downstream half when it enters the wake zone of the cylinder, yet the advantage that can be reaped in the high TSR is attractive to incite an investigation.Blade torque and the rotor torque values are evaluated for the above-said architectures and compared with the conventional Darrieus turbine.the starting torque will be generated by Savonius rotor if the ADWT is in open condition.

Blade Torque and Rotor Torque
The normal and the tangential forces are plotted as a function of azimuthal angles shown in the Figure 5.The well-established pattern of double peak is displayed by both conventional and ADWT rotor.The non-dimensional force coefficients T F and N F are computed from the predicted normal and tangential force.The normal force coefficients are lesser in the downstream half than the upstream half.The tangential force and normal forces are predicted for both full and half-length Savonius rotor.From the figure it is evident that the presence of cylinder increases the AoA resulting in the decrease of lift and increase of drag.
The maximum tangential force is obtained at 0˚ as 900 N. The tangential force difference between conventional Darrieus and the ADWT is lesser than the difference in normal force.The dynamic stall has significant influence on the normal force.

Power Coefficient and Torque Coefficient
The power coefficients are evaluated for various diameters and the conventional Darrieus rotor.The diameters are investigated for the full length and half-length Savonius rotor including the blade tip loss as shown in Figure 6(a) and Figure 6(b) respectively.The power coefficient curve follows the same trend for all the diameters investigated and the difference in their magnitudes is diminutive.A maximum reduction in the power coefficient of 5% is reported for the cylinder of 2000 mm compared to the conventional Darrieus rotor.The peak power coefficient achieved by the conventional Darrieus rotor is 0.42 at TSR 2.5.Also, the difference between the half-length and the full-length Savonius rotor is almost negligible.Hence full-length Savonius rotor is preferred aerodynamically from the starting perspective.The power coefficient is anticipated to reduce drastically as explained in past literature, but in reality, the reduction is negligible.The potential reason is a strong correlation between the wake width, AoA with respect to azimuthal position.The same trend is followed in the torque coefficient prediction for half and full length Savonius rotor.A maximum t C of 0.17 is achieved at 2.2 TSR for conventional Darrieus whereas for the configuration with cylinder the maximum t C is less than 0.16.Hence it can be concluded from the power coefficient and torque coefficient prediction that the diametrical ratio between the Darrieus rotor and the Savonius rotor in closed configuration can be as high as 1:0.5 with acceptable loss in power coefficient.

Experimental Verification of Analytical Results
To validate the derived analytical model, the computed results are compared against the experimental results.An analytical model of the closed condition can be predicted close to the reality, experimentation has been carried out with dif- Figures 1(a)-(c).At low wind speed Darrieus rotor torque ( ) d M ve = + and the analytical model of the ADWT in open configuration is similar to the conventional hybrid Darrieus -Savonius rotor.In order to simplify the development of analytical model, the Savonius buckets are arranged in line with the blades of Darrieus rotor.The simplified configuration for ADWT is with two Savonius buckets without an overlap mounted on the same axis with the Darrieus rotor with two blades.The velocity diagram of the ADWT is shown in the Figure 2.
Airfoil coefficients L C and C are obtained from the wind tunnel test or from literature and interpolating for local Reynolds number and the local angle of attack.Defining the blades local Reynolds number as b Re for local conditions given by b Re Wc V ∞ =

2
60) Journal of Power and Energy Engineering v u ′ ′ + are turbulent velocity components.Also rate of increase of width 'b' of mixing zone is proportional to transverse velocity v′ .; velocity deficit along Y direction is considered to be proportional to transverse velocity max U b

Figure 4 .
Figure 4. (a) Rotor torque for half length Savonius; (b) Cp vs TSR for various chord lengths; (c) Cp vs TSR at low wind speed.

Table 1 .
Dimensional details of the investigated rotor.