# Flow Modeling over Airfoils and Vertical Axis Wind Turbines Using Fourier Pseudo-Spectral Method and Coupled Immersed Boundary Method

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling

#### 2.1. Mathematical Modeling of Fluid Flows

**x**is the position vector of a point in the Eulerian domain, $p={p}^{*}/\rho $, where ${p}^{*}$ is the static pressure field in (N/m${}^{2}$), $\rho $ is the specific mass of the fluid in (kg/m${}^{3}$) and $\nu $ is the kinematic viscosity of the fluid in (m${}^{2}$/s). The term ${f}_{i}={{f}_{i}}^{*}/\rho $ is used to model the components of any force field applied to the flow, where ${{f}_{i}}^{*}$ is shown in (N/m${}^{3}$). These force fields are due to fluid–structure and fluid–fluid interactions, due to electromagnetic effects, gravitational effects or other physical effects internal to fluid particles.

#### 2.2. Fourier Pseudo-Spectral Method

#### 2.3. Immersed Boundary Method

- Using Equation (16), we calculate the parameter temporary or Eulerian velocity field ${{u}_{i}}^{*}$;
- By the interpolation procedure presented by Equation (19), the information from the Eulerian domain, ${{u}_{i}}^{*}$, is transmitted to the Lagrangian domain. Thus, ${{U}_{i}}^{*}$ is determined;
- ${{U}_{i}}^{FI}$ is determined, that is, the velocity that the immersed boundary must have over the simulated physical time. This velocity is imposed or calculated by some additional mathematical model. For airfoils, ${{U}_{i}}^{FI}=0$. For vertical axis turbine blades, ${{U}_{i}}^{FI}$ is calculated by Equations (25) and (26), described in Section 2.4;
- Using Equation (18), we calculate the Lagrangian force. In general, this step is about the application of Newton’s Second Law on the Lagrangian domain. The boundary condition of the immersed interface, given by ${U}_{FI}$, is now guaranteed in terms of the Lagrangian force;
- Using Equation (12), we propose the distribution of the Lagrangian force for the points of the Eulerian domain. The Eulerian force field ${f}_{i}$ is determined;
- The Eulerian velocity field ${{u}_{i}}^{*}$ is then corrected by the term ${f}_{i}$, using Equation (20).

- Before advancing the time step, ${{u}_{i}}^{t+\Delta t,it}={{u}_{i}}^{t+\Delta t}$, where $it$ is the interaction of multiple direct imposition of force;
- It is interpolated to ${{u}_{i}}^{t+\Delta t,it}$, using Equation (19);
- Obtained from interpolation, the new Lagrangian velocity ${{U}_{i}}^{t+\Delta t,it}$ is replaced in Equation (18). It is calculated as ${F}_{i}^{it}$;
- With the term ${f}_{i}^{it}$, it is corrected to ${{u}_{i}}^{t+\Delta t,it}$, and it is estimated as ${{u}_{i}}^{t+\Delta t,it+1}$,$${{u}_{i}}^{t+\Delta t,it+1}={{u}_{i}}^{t+\Delta t,it}+\Delta t{f}_{i}^{it};$$
- It is updated to $it=it+1$, return to step (1) or advance in time, $t+\Delta t$.

#### 2.4. Mathematical Modeling of Rotary Motion

## 3. Numerical and Computational Modeling

## 4. Validation

#### 4.1. Calculation Domain

#### 4.2. Eulerian Domain Refinement

#### 4.3. Influence of the Number of NIT Interactions of Multi-Direct Forcing

#### 4.4. Influence of Angle of Attack

## 5. Flow over a Vertical Axis Turbine

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- John, D.; Anderson, J. Fundamentals of Aerodynamics, 6th ed.; McGraw-Hill Education: New York, NY, USA, 2017. [Google Scholar]
- Clercq, K.M.D.; de Kat, R.; Remes, B.; van Oudheusden, B.W.; Bijl, H. Aerodynamic Experiments on DelFly II: Unsteady Lift Enhancement. Int. J. Micro Air Veh.
**2009**, 1, 255–262. [Google Scholar] [CrossRef] [Green Version] - Paraschivoiu, I. Wind Turbine Design With Emphasis on Darrieus Concept; Polytechnic International Press: Montreal, QC, Canada, 2002. [Google Scholar]
- Howell, R.; Qin, N.; Edwards, J.; Durrani, N. Wind tunnel and numerical study of a small vertical axis wind turbine. Renew. Energy
**2010**, 35, 412–422. [Google Scholar] [CrossRef] [Green Version] - Castelli, M.R.; Englaro, A.; Benini, E. The Darrieus wind turbine: Proposal for a new performance prediction model based on CFD. Energy
**2011**, 36, 4919–4934. [Google Scholar] [CrossRef] - Li, C.; Zhu, S.; Xu, Y.; Xiao, Y. 2.5D large eddy simulation of vertical axis wind turbine in consideration of high angle of attack flow. Renew. Energy
**2013**, 51, 317–330. [Google Scholar] [CrossRef] [Green Version] - Ribeiro, A.; Awruch, A.; Gomes, H. An airfoil optimization technique for wind turbines. Appl. Math. Model.
**2012**, 36, 4898–4907. [Google Scholar] [CrossRef] - Lee, Y.; Lim, H. Numerical study of the aerodynamic performance of a 500 W Darrieus-type vertical-axis wind turbine. Renew. Energy
**2015**, 83, 407–415. [Google Scholar] [CrossRef] - Marinić-Kragić, I.; Vučina, D.; Milas, Z. Numerical workflow for 3D shape optimization and synthesis of vertical-axis wind turbines for specified operating regimes. Renew. Energy
**2018**, 115, 113–127. [Google Scholar] [CrossRef] - Hansen, J.; Mahak, M.; Tzanakis, I. Numerical modelling and optimization of vertical axis wind turbine pairs: A scale up approach. Renew. Energy
**2021**, 171, 1371–1381. [Google Scholar] [CrossRef] - Chena, J.; Wanga, Q.; Zhanga, S.; Eecenc, P.; Grasso, F. A new direct design method of wind turbine airfoils and wind tunnel experiment. Appl. Math. Model.
**2016**, 40, 2002–2014. [Google Scholar] [CrossRef] - Wang, S.; Zhou, Y.; Alam, M.M.; Yang, H. Turbulent intensity and Reynolds number effects on an airfoil at low Reynolds numbers. Phys. Fluids
**2014**, 26, 1–23. [Google Scholar] [CrossRef] - Venkataraman, D.; Bottaro, A. Numerical modeling of flow control on a symmetric aerofoil via a porous, compliant coating. Phys. Fluids
**2012**, 24, 093601. [Google Scholar] [CrossRef] [Green Version] - Meena, M.G.; Taira, K. Low Reynolds Number Wake Modification Using a Gurney Flap; American Institute of Aeronautics and Astronautics: Washington, DC, USA, 2017; Volume 56. [Google Scholar]
- Ukken, M.G.; Sivapragasam, M. Aerodynamic shape optimization of airfoils at ultra-low Reynolds numbers. Sadhana
**2019**, 44, 130. [Google Scholar] [CrossRef] [Green Version] - Seshadri, P.K.; Aravind, A.; De, A. Leading edge vortex dynamics in airfoils: Effect of pitching motion at large amplitudes. J. Fluids Struct.
**2022**, 116, 103796. [Google Scholar] [CrossRef] - Mateescu, D.; Abdo, M. Analysis of flows past airfoils at very low Reynolds numbers. J. Aerosp. Eng.
**2009**, 224, 757–775. [Google Scholar] [CrossRef] - Kurtulus, D.F. Vortex flow aerodynamics behind a symmetric airfoil at low angles of attack and Reynolds numbers. Int. J. Micro Air Veh.
**2021**, 13, 17568293211055653. [Google Scholar] [CrossRef] - Nguyen, V.D.; Duong, V.D.; Trinh, M.H.; Nguyen, H.Q.; Nguyen, D.T.S. Low order modeling of dynamic stall using vortex particle method and dynamic mode decomposition. Int. J. Micro Air Veh.
**2023**, 15, 17568293221147923. [Google Scholar] [CrossRef] - Ferrer, E.; Willden, R. Blade wake interactions in cross-flow turbines. Int. J. Mar. Energy
**2015**, 11, 71–83. [Google Scholar] [CrossRef] [Green Version] - Ramírez, L.; Foulquié, C.; Nogueira, X.; Khelladi, S.; Chassaing, J.C.; Colominas, I. New high-resolution-preserving sliding mesh techniques for higher-order finite volume schemes. Comput. Fluids
**2015**, 118, 114–130. [Google Scholar] [CrossRef] [Green Version] - Liang, C.; Ou, K.; Premasuthan, S.; Jameson, A.; Wang, Z. High-order accurate simulations of unsteady flow past plunging and pitching airfoils. Comput. Fluids
**2011**, 40, 236–248. [Google Scholar] [CrossRef] - Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A. Spectral Methods: Fundamentals in Single Domains; Springer-Verlag: Berlin, Germany, 2006. [Google Scholar]
- Nascimento, A.A.; Mariano, F.P.; Padilla, E.L.M.; Silveira-Neto, A. Comparison of the convergence rates between Fourier pseudo-spectral and finite volume method using Taylor-Green vortex problem. J. Braz. Soc. Mech. Sci. Eng.
**2020**, 42, 1–10. [Google Scholar] [CrossRef] - Mariano, F.P.; de Queiroz Moreira, L.; Nascimento, A.A.; Silveira-Neto, A. An improved immersed boundary method by coupling of the multi-direct forcing and Fourier pseudo-spectral methods. J. Braz. Soc. Mech. Sci. Eng.
**2022**, 44, 1–23. [Google Scholar] [CrossRef] - Cooley, J.W.; Tukey, J.W. An Algorithm for the Machine Calculation of Complex Fourier Series. Math. Comput.
**1965**, 19, 215–234. [Google Scholar] [CrossRef] - Briggs, W.L.; Henson, V.E. The DFT: An Owners’ Manual for the Discrete Fourier Transform; SIAM: Philadelphia, PA, USA, 1995. [Google Scholar]
- Peskin, C.S. The immersed boundary method. Acta Numer.
**2002**, 11, 479–517. [Google Scholar] [CrossRef] [Green Version] - Wang, Z.; Fan, J.; Luo, K. Combined multi-direct forcing and immersed boundary method for simulating flows with moving particles. Int. J. Multiph. Flow
**2008**, 34, 283–302. [Google Scholar] [CrossRef] - Zhong, G.; Du, L.; Sun, X. Numerical Investigation of an Oscillating Airfoil using Immersed Boundary Method. J. Therm. Sci.
**2011**, 20, 413–422. [Google Scholar] [CrossRef] - Tay, W.B.; Deng, S.; van Oudheusden, B.; Bijl, H. Validation of immersed boundary method for the numerical simulation of flapping wing flight. Comput. Fluids
**2015**, 115, 226–242. [Google Scholar] [CrossRef] - Yang, X.; He, G.; Zhang, X. Large-eddy simulation of flows past a flapping airfoil using immersed boundary method. Sci. China Phys. Mech. Astron.
**2010**, 53, 1101–1108. [Google Scholar] [CrossRef] [Green Version] - Ouro, P.; Stoesser, T. An immersed boundary-based large-eddy simulation approach to predict the performance of vertical axis tidal turbines. Comput. Fluids
**2017**, 152, 74–87. [Google Scholar] [CrossRef] [Green Version] - Posa, A. Wake characterization of coupled configurations of vertical axis wind turbines using Large Eddy Simulation. Int. J. Heat Fluid Flow
**2019**, 75, 27–43. [Google Scholar] [CrossRef] - Mariano, F.; Moreira, L.; Silveira-Neto, A.; da Silva, C.; Pereira, J. A new incompressible Navier-Stokes solver combining Fourier pseudo-spectral and immersed boundary methods. CMES
**2010**, 1589, 1–35. [Google Scholar] - Kinoshita, D.; Padilla, E.L.M.; da Silveira-Neto, A.; Mariano, F.P.; Serfaty, R. Fourier pseudospectral method for nonperiodical problems: A general immersed boundary method for three types of thermal boundary conditions. Numer. Heat Transf. Part B Fundam.
**2016**, 70, 537–558. [Google Scholar] [CrossRef] - Silveira-Neto, A. Escoamentos Turbulentos: Análise Física e Modelagem Teórica, 1st ed.; Composer: Uberlândia, MG, Brazil, 2020. [Google Scholar]
- Pantakar, S.V. Numerical Heat Transfer and Fluid Flow; Hemisphere Publishing Corporation: Washington, DC, USA, 1980. [Google Scholar]
- Stockie, J.M. Analysis and Computation of Immersed Boundaries, with Application to Pulp Fibres. Ph.D. Thesis, Institute of Applied Mathematics, University of British Columbia, Vancouver, BC, Canada, 1997. [Google Scholar]
- Allampalli, V.; Hixon, R.; Nallasamy, R.; Sawyer, S. High-accuracy large-step explicit Runge-Kutta (HALE-RK) schemes for computational aeroacoustics. J. Comput. Phys.
**2009**, 228, 3837–3850. [Google Scholar] [CrossRef] - Takahashi, D. FFTE: A Fast Fourier Transform Package. Available online: http://www.ffte.jp (accessed on 30 September 2022).
- Ilio, G.D.; Chiappini, D.; Ubertini, S.; Bella, G.; Succi, S. Fluid flow around NACA 0012 airfoil at low-Reynolds numbers with hybrid lattice Boltzmann method. Comput. Fluids
**2018**, 166, 200–208. [Google Scholar] [CrossRef] - Kurtulus, D.F. On the wake pattern of symmetric airfoils for different incidence angles at Re=1000. Int. J. Micro Air Veh.
**2016**, 8, 109–139. [Google Scholar] [CrossRef] [Green Version] - Kouser, T.; Xiong, Y.; Yang, D.; Peng, S. Direct Numerical Simulations on the three-dimensional wake transition of flows over NACA0012 airfoil at Re = 1000. Int. J. Micro Air Veh.
**2021**, 13, 1–15. [Google Scholar] [CrossRef] - Tornberg, A.K.; Engquist, B. Numerical approximations of singular source terms in differential equations. J. Comput. Phys.
**2004**, 200, 462–488. [Google Scholar] [CrossRef] - Liu, Y.; Li, K.; Zhang, J.; Wang, H.; Liu, L. Numerical bifurcation analysis of static stall of airfoil and dynamic stall under unsteady perturbation. Commun. Nonlinear Sci. Numer. Simul.
**2012**, 17, 3427–3434. [Google Scholar] [CrossRef]

**Figure 1.**Representation of the Eulerian domain ($\Omega $) and the Lagrangian domain ($\Gamma $), where

**x**is the position vector of any point in the Eulerian domain and

**X**is the position vector of any point in the Lagrangian domain.

**Figure 3.**(

**a**) Top view of the rotor of a three-blade vertical axis turbine: azimuthal angle variation $\theta $. (

**b**) Imposition of the tangential velocity on a Lagrangian point belonging to the blade of a vertical axis turbine.

**Figure 4.**Decomposition of the components of the Lagrangian force: tangential force ${F}_{t}$ and normal force ${F}_{n}$ applied to a Lagrangian point.

**Figure 5.**(

**a**) Components of the forces applied to the blade of a vertical axis turbine, subjected to a flow with free stream velocity ${\mathit{U}}_{\mathbf{0}}$. (

**b**) Components of the forces applied to an airfoil, where $\alpha $ is the angle of attack.

**Figure 7.**Influence of mesh refinement on ${C}_{l}\times {t}^{*}$: (

**a**) ${10}^{\circ}$ and (

**b**) ${16}^{\circ}$. Influence of mesh refinement on ${C}_{d}\times {t}^{*}$: (

**c**) ${10}^{\circ}$ and (

**d**) ${16}^{\circ}$.

**Figure 8.**Order of spatial convergence. (

**a**) Decay as a function of the relative percentage difference of mean ${C}_{d}$ in relation to the work of [42]. (

**b**) Decay as a function of the time average of the ${L}_{2}$ norm of the horizontal Lagrangian velocity.

**Figure 10.**(

**a**) Variation of mean ${C}_{l}$ and ${C}_{d}$ as a function of the number of $NIT$ interactions. (

**b**) Temporal mean of the ${L}_{2}$ norm of horizontal Lagrangian velocity as a function of $NIT$.

**Figure 11.**(

**a**) Mean ${C}_{l}$ variation as a function of attack angle ($\alpha $). (

**b**) Mean ${C}_{d}$ variation as a function of the angle of attack ($\alpha $).

**Figure 12.**Isolines of pressure fields, at ${t}^{*}=80$: (

**a**) $\alpha ={0}^{\circ}$, (

**b**) $\alpha ={10}^{\circ}$, (

**c**) $\alpha ={16}^{\circ}$ and (

**d**) $\alpha ={20}^{\circ}$.

**Figure 13.**Streamlines of mean velocity fields: (

**a**) $\alpha ={0}^{\circ}$, (

**b**) $\alpha ={5}^{\circ}$, (

**c**) $\alpha ={10}^{\circ}$, (

**d**) $\alpha ={16}^{\circ}$, (

**e**) $\alpha ={25}^{\circ}$ and (

**f**) $\alpha ={30}^{\circ}$.

**Figure 15.**Instantaneous vorticity fields $-1\le {w}_{z}c/{U}_{0}\le 1$ over the NACA 0012 airfoil for different $\alpha $ values.

**Figure 17.**(

**a**) Variation of ${C}_{t}$ as a function of azimuth position $\theta $ of a single blade. (

**b**) Variation of ${C}_{n}$ as a function of the azimuthal position $\theta $ of a single blade.

**Figure 18.**Isolines of the absolute velocity field over the turbine in $\theta ={720}^{\circ}$. (

**a**) Present work. (

**b**) [33].

**Table 1.**Influence of mesh refinement: mean coefficients ${C}_{l}$ and ${C}_{d}$, obtained in the interval time ${t}^{*}=[70:100]$, for $\alpha ={10}^{\circ}$.

Mesh | ${\mathit{C}}_{\mathit{l}}$ | ${\mathit{C}}_{\mathit{d}}$ |
---|---|---|

$512\times 256$ | 0.5897 | 0.3670 |

$1024\times 512$ | 0.5264 | 0.2287 |

$2048\times 1024$ | 0.4173 | 0.1881 |

[44] | 0.4184 | 0.1661 |

**Table 2.**Influence of mesh refinement: mean coefficients ${C}_{l}$ e ${C}_{d}$, obtained in the interval time ${t}^{*}=[70:100]$, for $\alpha ={16}^{\circ}$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Monteiro, L.M.; Mariano, F.P.
Flow Modeling over Airfoils and Vertical Axis Wind Turbines Using Fourier Pseudo-Spectral Method and Coupled Immersed Boundary Method. *Axioms* **2023**, *12*, 212.
https://doi.org/10.3390/axioms12020212

**AMA Style**

Monteiro LM, Mariano FP.
Flow Modeling over Airfoils and Vertical Axis Wind Turbines Using Fourier Pseudo-Spectral Method and Coupled Immersed Boundary Method. *Axioms*. 2023; 12(2):212.
https://doi.org/10.3390/axioms12020212

**Chicago/Turabian Style**

Monteiro, Lucas Marques, and Felipe Pamplona Mariano.
2023. "Flow Modeling over Airfoils and Vertical Axis Wind Turbines Using Fourier Pseudo-Spectral Method and Coupled Immersed Boundary Method" *Axioms* 12, no. 2: 212.
https://doi.org/10.3390/axioms12020212