Optimal design of savonius wind turbines using ensemble of surrogates and CFD analysis

Abstract

Current study presents fluid flow analysis using CFD and a surrogate based framework for design optimization of Savonius wind turbines. The CFD model used for the study is validated with results from a physical model in water tunnel experiment. Four variables that best define blade geometry are considered and a feasible design space consisting of different combinations of these variables that provide positive overlap ratio is identified. The feasible space is then sampled with Latin hyper cube design of experiment. Numerical simulations utilizing K-epsilon turbulence model are performed at each point in the Design of Experiments to obtain coefficient of performance and weighted average surrogate (WAS) is fitted to them. Novelty of the current work is the use of WAS for design of savonius turbine. The WAS is an ensemble of surrogates that consists of polynomial response surface, kriging and radial basis functions. Error metrics reveal that WAS performs better compared to any surrogate individually thus avoiding misleading optima and eliminates surrogate dependent optima. WAS is used to explore the design space and perform optimization with limited number of CFD analyses. It is observed that at the optimal profile, there is more power on the rotors and primary recirculation in the immediate downstream of rotor is high, enforcing maximum momentum on turbine.

1 Introduction

Wind is one of the sought after renewable energy sources to combat the energy crisis that the world is facing. Wind turbine is an integral part of wind power generation. Wind turbines are classified as horizontal and vertical axis, based on their axis of rotation. In the recent years, there has been increased interest in vertical axis wind turbines (VAWT) because of their multifaceted advantages such as: independent of the wind direction, less noise, easy maintenance and potential to develop into a standalone roof top solution for a household energy requirement (Al-Bahadly 2009). Among the VAWTs, Savonius wind turbine is a simple structure; requires less starting torque, and literature reports an efficiency of about 30% (Mojola 1985). Perhaps, it provides the best cost to benefit ratio compared to other VAWTs such as Darrius, H-rotor etc. In the context of harnessing wind, often developments are focussed on wind energy in seashores and other windy sites using horizontal axis wind turbines (HAWT). Though considerable amount of improvement have been achieved in HAWT, they are expensive set ups and require an efficient network to transmit and distribute the energy. This calls for expensive infrastructure to provide energy to remote locations. The cost is justified when the demand is large and continuous. Sometimes, energy requirements are discrete and less. In such situations, VAWT can be an attractive alternate.

Researchers have tried to understand the characteristics and increase the performance of Savonius turbine using experimental (Mojola 1985; Modi and Fernando 1989a, 1989b; Fujisawa 1992; Jaohindy et al. 2011), numerical (Altan and Atilgan 2008; Kamoji et al. 2009; Kacprzak et al. 2013) studies or both (Rahai and Hefazi 2008; Mohamed et al. 2011; Mahmoud et al. 2012; Świrydczuk and Doerffer 2012; Zeid et al. 2012; Zhou and Rempfer 2013). In all these studies, researchers have considered a particular parameter such as blade arc angle (BAA) which is the angle measure of circular arc of the blade and investigated its effects on the performance. Since varying one parameter will benefit a particular response but likely to affect another, it is only logical to follow an optimal design approach. Some studies in the literature have dealt with optimal design of VAWT (Modi and Fernando 1989; Zhou and Rempfer 2013). However, it is desirable to develop a comprehensive framework that can perform numerical simulations, conduct optimization, provide designer with information on trades-offs, and be able to incorporate advanced modules like uncertainties (Dhamotharan et al. 2015).

To this end, the current work concentrates on two elements: to develop a validated computational fluid dynamics (CFD) model and an optimization framework to design a Savonius wind turbine. Ways to interpret/visualize the surrogate, metrics to measure the performance of the surrogate and on how to use this information in optimization is also discussed. The overall objective is to increase coefficient of performance (C_p) by identifying the optimal blade profile. The design variables considered for optimization are blade rotation angle (θ), blade arc angle (ϕ), blade arc radius (r) and perpendicular distance between the blades (d). The feasible design space is identified by exploring different profiles spanned by the combination of design variables and choosing the ones with positive overlap ratio (OR) and manufacturable. A Latin Hypercube Sampling (LHS) design of experiment (DoE) is used to sample the feasible design space. For each point in the DoE, CFD analysis is performed to obtain C_p. Surrogate modelling techniques are used to establish the relationship between C_p and the design variables. Here we use a weighted average surrogate (Goel et al. 2007; Viana et al. 2009) model. We also use error metrics that can lead to understanding the goodness of surrogate’s fit to data. Particularly, we use the cross validation error metric (predicted error sum of the squares) to understand the performance of the surrogate. The constructed surrogate was then used in an optimization framework to obtain the optimal C_p. A CFD simulation was carried out to obtain the C_p for the optimal design and it was compared to the C_p obtained from the surrogate. They compared well allowing for the conclusion that the proposed approach is suitable for performing optimization of Savonius wind turbines.

The rest of the paper is organized as follows: review of research conducted in Savonius wind turbine is presented in Section 2. Validation of the numerical approach is presented in Section 3 followed by the description of parametric and CFD model in Section 4. The surrogate modelling technique is discussed in Section 5 followed by the optimization results and discussion in Section 6.

2 Savonius wind turbine

The Savonius wind turbine is essentially a differential drag based device, which at small rotor angles behaves like a slender body and lift starts contributing to the mechanical power (Modi and Fernando 1989). In order to characterize the turbine’s performance, both numerical and experimental studies have been conducted. A review on performance of Savonius wind turbines is available (Akwa et al. 2012). The paper aims at gathering relevant information about the efficiency of Savonius and as a result of the survey, reports differences in the performance characteristics owing to various factors such as operational conditions, geometric and air flow parameters, though the geometry and flow pattern are reproduced. Multiple references are available on the numerical simulation of Savonius turbine. However, none of them discuss model fidelity and details of the numerical model. As noted in (Akwa et al. 2012) there is a disparity in the results presented. Therefore, there is a need to benchmark these simulations.

Though the Savonius turbine’s efficiency is low, owing to its multifaceted advantages researchers investigated different ways in which its efficiency can be increased. Researchers conducted experimental studies to understand the role or effect of parameters on turbine performance. The parameters of interest included geometric variables such as: aspect ratio (AR) (Alexander and Holownia 1978; Modi and Fernando 1989a, 1989b; Kamoji et al. 2009; Mahmoud et al. 2012), overlap ratio (Alexander and Holownia 1978; Mojola 1985; Modi and Fernando 1989a, 1989b; Fujisawa 1992; Menet and Bourabaa 2004; Kamoji et al. 2009; Mahmoud et al. 2012; Świrydczuk and Doerffer 2012; Zeid et al. 2012), BAA (Kamoji et al. 2009), shape factor and noise variables such as velocity, tip speed ratio (TSR) (Mojola 1985; Modi and Fernando 1989a, 1989b; Fujisawa 1992; Al-Bahadly 2009; D’Alessandro et al. 2010; Zeid et al. 2012) and reynolds number (Re) (Alexander and Holownia 1978; Modi and Fernando 1989a, 1989b; Menet and Bourabaa 2004; Kamoji et al. 2009; Mahmoud et al. 2012; Zeid et al. 2012). Some studies included the effect of end plates, number of stages (Mahmoud et al. 2012), number of blades (Mahmoud et al. 2012) etc. The observations are diverse. Optimal performances are reported for: OR varying between 0 to 0.3, high AR, TSR ranging from 0.3 to 1.6. Mahmoud et al. (2012) observed that the two bladed turbine is more efficient that three and four bladed turbine.

Numerical studies were also conducted and used CFD software to study the effect of OR, Re, TSR and presence of shaft (Rahai and Hefazi 2008; Gupta et al. 2009; D’Alessandro et al. 2010; Dobrev and Massouh 2011; Jaohindy et al. 2011; Kianifar and Anbarsooz 2011; Mohamed et al. 2011; Jaohindy et al. 2013; Kacprzak et al. 2013; Nasef et al. 2013; Zachár and Burány 2013). Few researchers performed optimization where the blade design was parameterized (Rahai and Hefazi 2008; Hu et al. 2009; Mohamed et al. 2011). Mohamed et al. (2011) reports that computational expense was the major challenge because they evaluated 210 geometries for a period of one month. They used OPAL (Optimization Algorithms) coupled with genetic algorithm to perform a shape optimization using non uniform splines. Rahai and Hefazi (2008) performed CFD based optimization to obtain an optimal profile with 23% increase in efficiency over traditional profiles. The computational results depend on the numerical model used. Widely used models include K-omega (ω) and K-epsilon (ε) which are two-equation models used to calculate turbulent viscosity. In K- ω model the turbulent viscosity is calculated using kinetic energy, K and the specific rate of dissipation of kinetic energy, ω whereas in K- ε model the turbulent viscosity is calculated using kinetic energy, K and dissipation rate, ε. Jaohindy et al. (2011) investigated a 3D model and observed that K-ω is superior to K-ε. Kacprzak et al. (2013) compared the conventional Savonius rotor with Bach type design and reported the Bach type to be very efficient at higher TSR (0.8). Kamoji et al. (2009) as well recommends using a Bach type blade. Roy and Saha (2015) proposed a modified design of a Bach-type profile and reported improvement in C_p, Coefficient of torque (C_T) and pay-back period of the installation system over various other blade profiles. Menet and Bourabaa (2004) reported a difference in C_p by only 2% between static and dynamic models. Altan and Atilgan (2008) perform experimental and 2-D CFD simulations to evaluate the static torque for a conventional Savonius rotor with and without curtain arrangements. They report similar trends and acceptable difference in the static torque obtained at varying rotor positions for different curtain arrangements. Similarly, Kacprzak et al. (2013) report acceptable comparison between quasi 2-D simulations and experiments that were performed to investigate the power characteristics of classical, bach-type and elliptical Savonius rotors. The surveyed literature is presented in Tables 1 and 2. Current work studies the Bach type blade and conducts only static analysis. In order to address the prohibitive computational cost involved in optimization, we use a surrogate model based approach.

Table 1 Literature review of experimental studies (shaded block denotes parameter discussed in the respective reference)

Table 2 Literature review of numerical studies (shaded block denotes parameter discussed in the respective reference)

3 Validation of numerical approach

The first part of this work focuses on building a computational model equivalent to the conventional Savonius rotor tested in a water tunnel and to reproduce the vorticity contours (note that pattern reproduction is targeted rather than magnitude). Prototype shown in Fig. 1 was built and tested in a water tunnel.¹ The dimensions of the two-blade conventional Savonius rotor are given in Table 3.

Fig. 1

Prototype of Savonius turbine used in water tunnel

Table 3 Dimensions of the conventional Savonius turbine

The water tunnel consists of a tank 2500 mm × 1500 mm with a depth of 150 mm, at one end of which are located 2 sets of aluminium disks which rotate in opposite directions horizontally and create a flow. The flow is guided to the test section where cylinders of different diameters can be placed. The flow rate is adjusted by controlling the rate of rotation of the disks; velocities ranging from 0.01 m/s to 0.2 m/s can be achieved. Since the width of the tunnel is 1500 mm, the resulting blockage is 10%, as Savonius rotor diameter is 160 mm.

The CFD model is built using ANSYS ICEM and FLUENT with water as fluid medium. The layout of the CFD model is presented in Fig. 2. The fluid domain is divided into two parts: the outer domain measures 10D x 15D and the inner circular domain has a diameter of 2D, where D is the diameter of rotor. An interface boundary condition is defined between the inner and outer mesh domain. Enough separation between the geometry and domain boundaries helps to avoid reflections of the fluid from walls. The fluid domain is subjected to a uniform velocity, v of 0.14 m/s similar to the velocity of water in the tunnel. A structured 2 dimensional mesh is generated with about 150,000 quadrilateral elements. A high fidelity mesh is maintained in the vicinity of rotor blades while the mesh is coarse, away from the rotor. The exponential mesh law is followed with a ratio of 1:5 along the rotor blades to the domain boundaries. A mesh convergence study is performed in (Jadhav et al. 2013) using four different mesh sizes to obtain a computationally cost effective model. The results are presented in Appendix 1. The selected mesh size is used for all the simulations in this work. Free slip condition is applied on the outer walls and it is bounded. The effect of blades is not felt beyond 1000 mm in the simulation along the wall normal direction. Therefore, in order to reduce the computational cost, we use a domain of width 1000 mm when the water tunnel has a width of 1500 mm. Also, the flow is fully developed at the vicinity of the blades. Hence, a domain independence study is performed to capture the underlying physics with reference to experimental results in the stream wise direction.

Fig. 2

Computational domain with boundary conditions

The developed CFD model when used in an optimization framework discussed later, needs to be accessed for every point the optimizer visits in the design space. Since the design variables are geometric parameters, for every call, a new geometry need to be meshed. The meshes are generated using an automated script. Though other algorithms are reported to be better in solution capabilities, K-ε is used in the current work because it worked well with the not superior element quality owing to the automated mesh generation. Advantages of K-ε are that it is proven to be stable and numerically robust.² We use the Coupled algorithm (Rhie and Chow 1983) where Navier-Stokes and continuity equations are solved implicitly for calculating velocity and pressure fields. The models are iterated until the values of the residuals converged to a value of 10⁻⁶.

Vortices from water tunnel test and CFD results are presented in Fig. 3. It can be observed that the vorticity pattern in Fig. 3b matches with experimental results in Fig. 3a. The vortex formation around the concave blades of rotating turbine form adverse pressure gradient, which is marked as 3 in Fig. 3a. A large vortex is slowly shed behind the returning blade marked as 1 in Fig. 3a, which produces, low-pressure downstream of the advancing blade and similar phenomenon is observed in Fig. 3b also. Savonius turbine utilizes the difference in coefficient of drag for rotation.

Fig. 3

Validation of numerical model - vortices from a Water tunnel test b Computational study with contour levels for vorticity as −20(2)20

In the current work, we estimate the frequency of vortex shedding and estimate the Strouhal number for the water tunnel test and compare it to the results from the CFD simulations. Strouhal number (St) is a dimensionless quantity defined as

$$ St=\frac{fr\;L}{U_{fluid}} $$

(1)

Where fr is the frequency of vortex shed by the blades, L is the characteristic length which is turbine diameter, and U_fluid is velocity of fluid. L and U_fluid in the water tunnel test and CFD simulation are 1000 mm and 0.14 m/s, respectively. In the water tunnel test, the water is filled with black dye and sufficient amount of aluminium filings are added to clearly observe the vorticity patterns. Water velocity U_fluid is measured by noting the time taken for a floating particle (external material) to traverse a fixed distance in the test region. Then, the frequency of vortex shedding fr is calculated by measuring the time taken to shed a specific number of vortices from the turbine and Strouhal number is evaluated using (1).

Figure 4 shows snapshots of the water tunnel with Savonius turbine, at different time instances when the vortices were observed. The entire test spans for 10.34 s, and there were a total of 3 full vortices. Frequency of vortex shedding is calculated as the ratio of number of vortices to the total time taken and it is found to be 0.29.

Fig. 4

Vortices observed in water tunnel test - Strouhal number calculation

In the numerical model, the time history of the coefficient of lift, C_l was obtained as shown in Fig. 5 . A Fast Fourier Transform is applied to the C_l to obtain the shedding frequency fr and hence St. Table 4 provides a comparison of quantitative results of Strouhal number from experiment and simulation. It can be observed that the results compare well. To this end, we have a validated computer model that can be used for further design. For the rest of the paper, the medium is air. Nevertheless, since the validation was for the fidelity of the computer model, it holds good for the air medium as well and we retain the mesh size.

Fig. 5

Time history of coefficient of lift C_l

Table 4 Comparison of Strouhal number

4 Parametric model and optimization framework

Since literature (Kamoji et al. 2009; Jaohindy et al. 2011; Zhou and Rempfer 2013) shows that Bach type models are more efficient, we focus on Bach type blades in this work. A single stage Bach type Savonius rotor is used for this study. The parametric model of the geometry used is discussed in this section. Figure 6 shows the cross-sectional view of the rotor profile illustrating the design parameters.

Fig. 6

Geometry of Bach-type profile

In this work D is taken as 100 mm. r, θ, ϕ and d are the design variables. The design space spanned by these variables is explored extensively and the feasible design space is obtained. Profiles in which the tangent of one blade projects into the circular arc of other blade and profiles that have a negative overlap are considered infeasible and hence are not included with the design space. A preferable design is one where the arc part of the returning blade is fully exposed to incoming air and have a positive OR. If the tangent part of advancing blade extends into the arc part of returning blade, only a portion of the arc is impacted by air, thus resulting in lesser transfer of momentum. The OR for a bach type turbine is d/D. The geometry of the blade is defined by parameters as shown in Fig. 6. A computational domain similar to the one used for validation is used for this study. The air velocity at the inlet is 8 m/s and the gauge pressure at the outlet is 0 Pa.

First, a feasible design space needs to be identified. Here we identify this space as the one that contain the points with positive OR. Then, a DoE is used to explore the design space. Space filling techniques such as LHS are usually preferred. Upon generating the DoE, CFD analysis is conducted for each point in the DoE to compute the response of interest. Surrogate models are then used to establish black box functions that relate the design space to the response. Here, we use an ensemble surrogate approach. That is, we approximate the responses by an ensemble of surrogates of polynomial response surface, kriging and radial basis functions. Then, the surrogate is used for optimization.

5 Surrogate modelling

As mentioned earlier, optimization involves repeated calls to the CFD model and to reduce the computational expense, surrogate models are widely used. The underlying idea of surrogate modelling (Viana et al. 2009) is to develop a mapping function $\widehat{f}(X)$ given the response data y, obtained for the corresponding design variables X. Since $\widehat{f}$ is an approximation of the actual function f as in (2), it is called the Surrogate.

$$ y=f(X)=\widehat{f}(X)+\Delta $$

(2)

where, Δ comprises of bias and random errors from the approximation.

There are different ways in which $\widehat{f}$ can be obtained. One of the widely used approaches is the Polynomial Response Surface where an m^th degree polynomial is best fitted to the responses obtained at a set of design points. An approximation of the order of m can be written as

$$ \widehat{f}(X)={a}_0+\sum \limits_{i=1}^m{a}_i{x}_i+\sum \limits_{i=1}^m\sum \limits_{j=1}^m{a}_{ij}{x}_i{x}_j $$

(3)

Where a₀, a_i, a_ij are the coefficients that define the correlation between the input X and response y. These coefficients are usually estimated through a simple least squares regression.

Another widely used surrogate model is the Radial basis function whose value depends on the distance from origin or alternately, distance from a center c. Separate functions are assigned a weight and added to approximate the target function. Consider a scalar valued function f, according to the sampling plan X(x⁽¹⁾, x⁽²⁾……x^(p)) yielding responses y. A radial basis function approximation $\widehat{f}$ will be of the fixed form

$$ \widehat{f}=\sum \limits_{i=1}^{n_c}{w}_i\psi \left(\left\Vert x-{c}^{(i)}\right\Vert ={y}^{(j)},j=1,\dots, p\right) $$

(4)

Where c⁽ⁱ⁾ denotes i^th of the p_c basis function centers, p is the number of design points and ψ is a vector containing the values of basis functions. w_i is the weight assigned to i^th basis function. (4) is linear in terms of the weights w_i, thus making them easy to estimate. ‖⋅‖ here denotes Euclidean distance. A unique solution to the weights can be obtained when p = p_c, i.e. the bases actually coincide with the data points. Different basis functions such as linear, cubic, thin plate spline, Gaussian, multiquadric and inverse multiquadric are available to build the regression model.

Kriging, which has gained popularity in the recent times, is a stochastic interpolation method used to estimate the value at a given location as a weighted sum of data values at neighboring points. It uses a basis function of the form

$$ {\psi}^{(i)}=\exp \left(-\sum \limits_{j=1}^k{\vartheta}_j\left|{x}_j^{(i)}-{\left.{x}_j\right|}^{g_j}\right.\right) $$

(5)

where ${x}_j^{(i)}$ is the i^th unknown location and x_j are the neighboring data points that form the input vector X. ϑ_j are the coefficients attributed to the individual distances. A target function is approximated in the form of a linear combination of the basis functions as in (6). Unlike a Gaussian basis function where the exponent is fixed at 2, Kriging allows the exponent, g to vary for each dimension in X. The unknown value has an assumed correlation with the sample values and the sample values are correlated among themselves. This correlation information is readily translated into the variance of the prediction. The coefficients in the linear combination are chosen such that the variance is as small as possible. The reader is referred to (Forrester and Keane 2008) for further details on surrogate modeling.

$$ \widehat{f}=\sum \limits_{i=1}^n{w}_i{\psi}^{(i)} $$

(6)

Selection of surrogate depends on the data and application since their performance is case dependent. Most researchers observed that no single surrogate could act efficiently for all the problems. Therefore, as an alternative to using a single surrogate, several researchers combined multiple surrogates in the form of an ensemble. Weighted Average Surrogate (WAS) used for this study is an ensemble (Viana et al. 2009) of all surrogates discussed above. The individual approximations are weighed upon the individual errors to obtain a single regression with reduced errors and bias. Details of WAS are provided in Appendix 2. Acar (2010) and, Acar and Rais-Rohani (2009) suggests different ways and optimized weight factors to construct an ensemble. Though, these methods can be used, this work uses WAS to demonstrate the framework. Figure 7 depicts the layout or standard procedure one follows in building and analysing an ensemble of surrogates.

Fig. 7

Layout for surrogate modelling

In order to build the surrogate, the design space is usually sampled using a DoE. LHS DoE is used in this study. A LHS design is constructed in such a way that each of the dimensions is divided into equal bins and there is only one point at each bin (Viana 2013). Usually, the location of each point is chosen randomly while the above criterion is satisfied. This guarantees that the design space is explored entirely in a random fashion. 70 LHS samples are generated in the design space. Additional 16 samples are chosen on the boundary of the design space to avoid extrapolation errors. For each design point, profiles are constructed and followed by CFD analysis to evaluate C_T which is then used to compute C_p using (7) where TSR has been taken as 0.9 (Nasef et al. 2013; Roy and Saha 2013)

$$ {C}_p={C}_T(TSR) $$

(7)

Once the C_p is obtained for all points in the DoE, a surrogate needs to be fit. WAS (Viana et al. 2009; Acar 2013) is used here and we use the multiple surrogate toolbox (Viana 2011). The ensemble here consists of Polynomial Response Surface (PRS), Kriging and Radial Basis Functions (RBF). The C_p is approximated as a function of r, θ, ϕ and d. The quality of the surrogate model is measured using error metrics like Predicted Error Sum of Squares (PRESS). To estimate the PRESS, an observation is removed at a time and a new surrogate is fitted to the remaining observations. The new surrogate is used to predict the withheld observation. The difference between the withheld observation and the computed response value gives the PRESS residual for that observation. This process is repeated for all the observations and the PRESS statistic is defined as the sum of the squares of the computed PRESS residuals. The repetitive estimate of PRESS residuals can be obviated by using the following equation.

$$ \mathrm{PRESS}=\sum \limits_{i=1}^n{\left(\frac{er_i}{1-{E}_{ii}}\right)}^2 $$

(8)

Where er_i is the error in i^th iteration of the PRESS estimate, E = X(X^TX)⁻¹X^T. Data points at which E_ii is large will have large PRESS residuals. These observations are considered high influence points. That is, a large difference between the ordinary residual and the PRESS residual will indicate a point where the model fits the data well, but the model built without that point has a poor prediction. A Root Mean Square (RMS) version of PRESS gives a better comparison. The PRESS_RMS is expressed as

$$ {\mathrm{PRESS}}_{\mathrm{RMS}}=\sqrt{\frac{\mathrm{PRESS}}{n}} $$

(9)

Where n is the number of data points. PRESS_RMS is usually compared to mean of the response. Lower the value of PRESS_RMS, the better is the fit. R² values vary between 0 and 1. Larger the R² value, better is the fit. Predicted R² can be obtained using PRESS.

$$ {\mathrm{R}}_{\mathrm{predicted}}^2=1-\left(\frac{\mathrm{PRESS}}{\sum \limits_{i=1}^n{\left({y}_i-\widehat{y}\right)}^2}\right) $$

(10)

Kriging approximation is influenced by its parameters, the regression and correlation polynomial. In the current work, the kriging approximation is run for different combinations of regression and correlation models and the one with the best PRESS error is selected. The selected Kriging model has a first order regression polynomial and a correlation polynomial of type cubic. Similarly, RBF was tested for different combinations of its parameters and the one with the best PRESS error was built with ‘Bi Harmonic’ type basis function. 3rd order polynomial was used for PRS. The error metrics for the different surrogates fit to the 86 responses is presented in Table 5. It can be readily observed that WAS has better metrics than its constituent surrogates. Since, the ensemble weight factors are selected based on PRESS minimization, WAS superiority is expected. However, the optimization is carried out with the indidual surrogates and also the ensemble.

Table 5 Error metrics for the surrogate models

6 Optimization results and discussions

The formulation for the optimization is as follows:

$$ {\displaystyle \begin{array}{ll} Maaximize& {C}_p\left(\theta, \phi, r,d\right)\\ {} such\ that& 9.5\le \theta \le 14.45\left(\deg \right);\\ {}& 117.5\le \phi 126.5\left(\deg \right);\\ {}& 40\le r\le 45\left(\mathrm{mm}\right);\\ {}& 17.5\le d\le 25\left(\mathrm{mm}\right);\end{array}} $$

(11)

Where C_p(θ, ϕ, r, d), the coefficient of power is a nonlinear multivariable function and WAS is used to evaluate it. A constrained optimization is carried out based on geometric constraints. A four variable nested axis plot is generated (Fig. 8) over the design space to graphically represent the correlation. The plot has a global coordinate system defined by θ on x-axis and ϕ on y-axis. Each tile has a local nested coordinate frame depicted by r and d on local x-axis and y-axis respectively. It can be observed from the plot that C_p = f(θ, ϕ, r, d) is multimodal with multiple peaks over the design space. The filled contour plot is presented in Fig. 9 with the lower bounds as baseline values. That is, C_p variation is studied by varying two variables while the other two variables are held at their lower bounds. Figure 9 shows that C_p is sensitive to changes in all four variables. Sequential quadratic programming based optimization is carried out to obtain an optimal profile. The optimal profile obtained is subjected to a CFD analysis and the results are compared with the ones estimated from the surrogates as shown in Table 6. It can be observed that the C_p corresponding to the optimal design from the surrogate model compare well with the C_p from the CFD model for all the cases. Interestingly, the individual surrogates and the ensemble provide more or less the same optima. A globally good surrogate does not guarantee a better (Acar et al. 2011; Ayancik et al. 2017) optima. They suggest that one of the reasons for this could be that the globally most accurate surrogate model may have large local errors around the optimum. Instead, performing optimization with different surrogate models to get multiple candidate optimum designs, and selecting the candidate with the best performance is a better approach. However, in the current case the PRESS_RMS for the individual surrogates was greater than 10% of the mean of the response C_p while for the WAS, it was less than 2%. This is captured by the predicted R² values presented in Table 5. The individual surrogates perform as well as the WAS, around the optima while there might be large errors at other spots in the design space. The WAS did not perform badly reiterating the point that, even in the worst case it will serve as an insurance estimate against bad predictions.

Fig. 8

Nested axis plot of C_p

Fig. 9

Contour plot for the surrogate model with lower bounds as baseline values

Table 6 Performance at optimum design

A qualitative analysis of the optimal profile is also performed using the pressure and vorticity contours obtained from CFD study. Figure 10 shows the pressure distribution for an initial design and the optimum profile. From the figure it is evident that the pressure is maximum along the stagnation zones. The maximum value of pressure along the blade is less in the optimum design which clearly indicates that the mechanical stress imparted on the blade by the fluid flow is minimum. The pressure gradient is also maximum in the optimum design which indicates that there is more flow which in turn generates more power. Vorticity contours in Fig. 11 indicate the alternate vortex shedding of vortices along the blades. The strength of the vortex along the optimum design is found to be lesser in magnitude when compared with initial design. The higher and lower strength vortices are not observed along the blades. This clearly indicates the flow induced vibration on blade structure is less in the case of optimal design. Figure 12 depicts the vector plot of an initial turbine design and optimum design. The size of the primary recirculation in the immediate downstream of blade is high in the optimum design which enforces more fluid flow towards the blade. This will increase the coefficient of thrust developed by the turbine. The secondary vortex is far downstream in the optimum design when compared with initial design. These discussions show that the optimal profile obtained through the surrogate indeed gives the maximum C_p. The associated discussion on CFD analysis explains the reason for the increase in C_p in the optimal profile. Stream function contours (Fig. 13) depict two prominent vortices, one along the blade and the other downstream to the blade. The optimum design (Fig. 13b) shows the size of the recirculation region downstream of the blade increases when compared with the initial profile. Also the size of the vortex near the blade is reduced for the optimum profile and detached from the blade. This will reduce the mechanical stress induced by the vortex on the blade.

Fig. 10

Pressure contour plots for the Savonius turbine: a Initial profile b Optimal profile

Fig. 11

Vorticity contours with a magnified image in the vicinity of blade end: a Initial profile b Optimal profile

Fig. 12

Velocity vector plots with magnified view of region around the blades: a Initial profile b Optimal profile

Fig. 13

Contours of stream function with magnified view of region around the blades: a Initial profile b Optimal profile

7 Summary

Since optimization using computer models could be computationally expensive, this work proposes a surrogate based optimization framework for the optimization of Savonius wind turbine. First, a water tunnel test for a conventional Savonius rotor is performed followed by a validation study to build an equivalent computer model. Using these specifications, a parametric model of a Bach-type rotor is developed in terms of Blade rotation angle (θ), Blade arc angle (ϕ), radius ® and perpendicular distance between the blades (d). The design space is explored to obtain the feasible design space. A LHS DoE is used to sample the design space. For each point in the DoE, CFD simulations are carried out to obtain the C_p and a weighted average surrogate is fit to the C_p. The transfer function acquired from surrogate modelling approach is subjected to multivariate constrained optimization. The optimal profile thus achieved is observed to have a C_p of 0.30 and is verified with individual CFD simulations for the optimal variables. It is observed that the individual surrogates provide almost the same optima as the WAS inspite of performing much lower in terms of error metrics. Qualitative comparisons are also provided between a random profile and the optimal profile. In summary, a surrogate based scheme is developed to explore design space and perform optimization of a Savonius turbine. This scheme is also capable of including uncertainties module (Dhamotharan et al. 2015) in the future.

Appendices

Appendix 1

The mesh convergence information for the computer model used in the validation study is provided below in Table 7. Two meshes were generated for the same profile with ±10% difference in number of quad elements and the Cp values were compared. It is observed that there is ~ 4% change in Cp value when the mesh size is decreased and ~1% increase when the mesh cells are increased by 10%. Hence further study is performed by increasing the number of mesh cells by 75% and still the observed Cp value increases by ~1%. Therefore, the mesh size corresponding to iteration number 3 is used in the work.

Table 7 Mesh convergence information

Appendix 2

The weighted average surrogate (WAS) is formulated as a weighted sum of the three individual approximation methods. The weights are calculated in such a way that they (a) reflect the confidence in each individual surrogate and (b) filter out adverse effects associated with individual surrogates which represent the sample data well, but predict poorly at designs not included in the sample data. WAS can be expressed as follows:

$$ {\widehat{y}}_{WAS}(x)=\sum \limits_{i=1}^N{q}_i(x){\widehat{y}}_i(x)={Q}^T(x)\widehat{y}(x) $$

(12)

$$ \sum \limits_{i=1}^N{q}_i(x)=1 $$

(13)

where ${\widehat{y}}_{WAS}(x)$ is the predicted response by the WAS model, q_i(x) is the weight associated with the ith surrogate at x and and ${\widehat{y}}_i(x)$ is the predicted response by the ith surrogate. The various functions that could be used for assigning weights for WAS are shown in Table 8. We use the Best PRESS in this work.

Table 8 Different forms of weights for WAS