# Numerical Validation of a Vortex Model against ExperimentalData on a Straight-Bladed Vertical Axis Wind Turbine

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## Abstract

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## 1. Introduction

## 2. Experimental Data

**Figure 4.**Allowed variations of the asymptotic wind velocity inside a bin with the steady wind flow conditions.

## 3. Simulation Model

**Figure 5.**The sign convention of the normal and tangential forces. The counter-clockwise direction of the blade azimuth angle θ is defined as positive.

#### 3.1. Vortex Model of the Turbine

**Figure 6.**Flow chart of the vortex model combined with the dynamic stall (DS) model. ${N}_{step}$ is the current time step, and ${N}_{step,max}$ is the maximum number of time steps. ${N}_{step,max}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}12,000$, corresponding to 120 time steps per each of 100 revolutions.

#### 3.2. Dynamic Stall Modeling

**Figure 7.**Normal force coefficient for the pitching NACA0021 airfoil, $\alpha =12+10sin\left(\omega t\right)$, $k=0.06$, $M=0.1$, $c=0.55\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$, where k, M and c are the reduced frequency, the Mach number and the chord length correspondingly.

#### Modification Due to Vortex Shedding

**Figure 8.**Dynamic vortex shedding for a straight-bladed vertical axis turbine operating in a towing tank at the tip speed ratio (TSR) of 2.14, obtained from [32]. a, ${a}^{\prime}$, b and c denote vortices.

## 4. Results and Discussion

**Figure 9.**The normal force at $\lambda =1.84$, $\Omega =40.29\phantom{\rule{0.166667em}{0ex}}\mathrm{rpm}$. The air density and the kinematic viscosity are $\rho =1.25\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ and $\nu =1.42\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$.

**Figure 10.**The normal force at $\lambda =2.26$, $\Omega =45.19\phantom{\rule{0.166667em}{0ex}}\mathrm{rpm}$. The air density and the kinematic viscosity are $\rho =1.25\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ and $\nu =1.42\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$.

**Figure 11.**The normal force at $\lambda =2.85$, $\Omega =50.80\phantom{\rule{0.166667em}{0ex}}\mathrm{rpm}$. The air density and the kinematic viscosity are $\rho =1.27\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ and $\nu =1.39\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$.

**Figure 12.**The normal forces at similar λ and different Ω. The air densities and the kinematic viscosities are ${\rho}_{1}=1.27\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$, ${\rho}_{2}=1.24\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ and ${\nu}_{1}=1.39\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$, ${\nu}_{2}=1.45\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$.

**Figure 13.**The normal forces at the similar λ and different Ω. The air densities and the kinematic viscosities are ${\rho}_{1}=1.28\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$, ${\rho}_{2}=1.24\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ and ${\nu}_{1}=1.39\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$, ${\nu}_{2}=1.45\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$.

**Figure 14.**The normal force at $\lambda =3.74$, $\Omega =65.07\phantom{\rule{0.166667em}{0ex}}\mathrm{rpm}$. The air density and the kinematic viscosity are $\rho =1.24\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ and $\nu =1.44\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$.

**Figure 15.**The normal forces at the similar λ and different Ω. The air densities and the kinematic viscosities are ${\rho}_{1}=1.28\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$, ${\rho}_{2}=1.24\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ and ${\nu}_{1}=1.39\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$, ${\nu}_{2}=1.43\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$.

**Figure 16.**The normal forces for two different sets of data with almost identical λ and Ω. The air densities and the kinematic viscosities are ${\rho}_{1}=1.24\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$, ${\rho}_{2}=1.24\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ and ${\nu}_{1}=1.44\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$, ${\nu}_{2}=1.45\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$.

**Figure 17.**The normal force at $\lambda =4.57$, $\Omega =65.35\phantom{\rule{0.166667em}{0ex}}\mathrm{rpm}$. The air density and the kinematic viscosity are $\rho =1.25\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ and $\nu =1.43\xb7{10}^{-5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$.

#### General Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Dyachuk, E.; Goude, A. Numerical Validation of a Vortex Model against ExperimentalData on a Straight-Bladed Vertical Axis Wind Turbine. *Energies* **2015**, *8*, 11800-11820.
https://doi.org/10.3390/en81011800

**AMA Style**

Dyachuk E, Goude A. Numerical Validation of a Vortex Model against ExperimentalData on a Straight-Bladed Vertical Axis Wind Turbine. *Energies*. 2015; 8(10):11800-11820.
https://doi.org/10.3390/en81011800

**Chicago/Turabian Style**

Dyachuk, Eduard, and Anders Goude. 2015. "Numerical Validation of a Vortex Model against ExperimentalData on a Straight-Bladed Vertical Axis Wind Turbine" *Energies* 8, no. 10: 11800-11820.
https://doi.org/10.3390/en81011800