C HAPTER –3
3.2 Computational methodology
The flow around the SSWTs is assumed to be unsteady and turbulent, operating at a free stream wind speed of 6.2 m/s. The dynamic simulations are carried out by assigning a certain rotational rate (ωs in rad/s) to the rotating zone (Figure 3.1) to predict the performance of the turbine in terms of torque coefficient (CT) and power coefficient (CP) with respect to TSR.
Hence, an artificial dynamic condition is formed as in the real case of dynamic electrical loading. The overall diameter (D) and thickness of SSWT models are taken as 209 mm and 0.63 mm, respectively. Initially, 2D simulations are carried out on a conventional SSWT without any blade overlap, and the trend of CP variation is compared with the experimental trends of Blackwell et al. (1977).
The turbulent flow over the Savonius-style wind turbine is resolved applying the finite volume method into the computational domain and solving the unsteady RANS equations.
The governing mass conservation and momentum equations are represented in Eqs. 3.1 and
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3.2. In mass conservation equation, ui indicates the mean flow velocity and uiindicates the fluctuating velocity due to turbulence, and x indicates the direction of flow. In momentum equation, t is the time, p is the mean pressure, and represent the dynamic viscosity and density of air, respectively.
0
i i i
u
x u (3.1)
j i j j j
i i
j j i
i uu
x x x
u x
p x
u u t
u
1 2
(3.2)
By applying the RANS equations, the fluctuating and mean terms of velocity and pressure increases the number of unknown terms as compared to number of equations. Thus, turbulence models are introduced to solve the unknown terms of RANS equations (Appendix:
A).
In the present study, various turbulence models such as standard k-ε turbulence model, realizable k-ε turbulence model, k-ω turbulence model, and shear stress transport (SST) k-ω turbulence model are used and the most suitable model is selected for the further analysis.
The details of the computational methodology are discussed in the subsequent sections.
3.2.1 Description of the computational domain
The dimensions of the computational domain as shown in Figure 3.1 are given in multiples of the turbine diameter. The size of the computational domain is selected in such a way that the results are not affected by the boundaries of computational domain. The computational domain includes a circular region with rotating mesh bounded turbine. This circular region (interface) has a dimension of 2D. The turbine models are placed at a distance 3D from the upper and lower sides of the domain. At the inlet of the domain, an inlet velocity of 7 m/s is given, whereas, at the outlet of the domain, the pressure is considered equivalent to atmospheric pressure. The sides of the computational are assumed as symmetrical planes.
3.2.2 Details of the domain discretization
The computational domain is spatially discretized by quad cells to solve the equations using finite volume method. It consists of 2 parts: the first part is fixed mesh region and the other is
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Figure 3.1: Schematic diagram of the computational domain
rotating mesh region as shown in Figure 3.1. The rotating mesh is bounded by the interface of both the parts. The mesh generation in the computational domain is shown in Figure 3.2. The finite volume discretization transforms the differential equations into a set of algebraic equations, which is solved by iterative methods. Thus, the solution is heavily dependent on the number of discretized volumes. For grid convergence, time (Δt) and space (Δx) have been varied and CFL (=V.Δt/ Δx) close to 1 is ensured with Δt = 0.001 and Δx = 0.0001. Grid independent test has been carried out (Figure 3.3) and a total of 358542 quad cells have been taken after satisfactory grid refinement.
Figure 3.2: Generated mesh in the computational domain Boundary layer mesh
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Figure 3.3: Grid independence test
Fine boundary layers are adjusted near the blade surfaces with a first layer thickness of 0.00004 m and the increment ratio of 1.12. Figure 3.4 shows the y+ variations at different time intervals corresponding to half rotation of the turbine. An average y+ value of less than one is ensured throughout the simulations in order to capture the flow properties near the blade surfaces.
3.2.3 Details of the solver
A pressure based transient FVM solver, ANSYS Fluent is used to discretize the equations.
The spatial discretization of the conservative equations is treated with 2nd order upwind scheme and the temporal terms of the conservative equations are discretized using 2nd order fully implicit temporal scheme. Good solution stability is ensured through the pressure- velocity coupling with the SIMPLE method (Semi Implicit Linked Equations). For the solution iteration, the time step size and the number of iterations per time step are taken as 0.001 and 20, respectively.
3.2.4 Calculation of performance coefficients
As the simulation is carried out with the unsteady flow assumptions, the performance coefficients are averaged to capture a more accurate time averaged value. The simulations are run for 10 rotational cycle of the turbine. Figure 3.5 shows the variations of torque coefficients at different time intervals. It is observed that after the initial period, it follows almost a cyclic path, and hence, averaged to give a more accurate value.
0.36 0.37 0.38 0.39 0.40 0.41
0 100000 200000 300000 400000 500000 CT
Number of cells
TSR = 0.75
± 6.27%
± 0.64%
± 1.87%
± 2.76%
± 0.91% ± 0.36%
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Figure 3.4: Y+ on the turbine blades at different turbine positions
3.2.5 Selection of turbulence model
As the flow field around a SSWT is turbulent in nature, the selection of turbulence models plays a vital role to acquire the desired solution. However, each of the turbulence models has their own merits and demerits. In this research, standard k-ε turbulence model, realizable k-ε turbulence model, k-ω turbulence model, and SST k-ω turbulence model (Appendix: A) are used to predict the performance of conventional SSWT without any overlap (Figure 3.6). The velocity inlet is given as 7 m/s. The trend of averaged CP is compared with the experimental trend of Blackwell et al. (1977). The SST k-ω turbulence model is found to be suitable for
0.0 1.0 2.0 3.0 4.0
-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12
Y+
x-directional position (m) Advancing blade
Returning blade θ= 10°
0.0 1.0 2.0 3.0 4.0
-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12
Y+
x-directional position (m) Advancing blade Returning blade θ= 40°
0.0 1.0 2.0 3.0 4.0
-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12
Y+
x-directional position (m)
Advancing blade Returning blade θ= 70°
0.0 1.0 2.0 3.0 4.0
-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12
Y+
x-directional position (m) Advancing blade
Returning blade θ= 100°
0.0 1.0 2.0 3.0 4.0
-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12
Y+
x-directional position (m)
Advancing blade Returning blade θ= 130°
0.0 1.0 2.0 3.0 4.0
-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12
Y+
x-directional position (m) Advancing blade Returning blade θ= 160°
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further analysis of the turbine as compared to other turbulence models. It is due to fact that the SST k-ω turbulence model is a blended model of k-ω and k-ε turbulence models, comprising the benefits of near wall and free stream capabilities, respectively. The prediction capability of the SST k-ω turbulence model is also observed in other recent numerical investigations (Abraham et al., 2011; Plourde et al., 2012; Abraham et al., 2012; Kacprzak et al., 2013). With the present computational methodology, the simulation data show an almost similar trend to those of experiments. However, it is observed that due to two-dimensional flow assumption, the numerical results marginally over predicted the experimental data.
Figure 3.5: Variation of CT at different time intervals
Figure 3.6: Validation and comparative study of various turbulence models