由於對個人可再生能源解决方案的需求不斷增長，以及它們在城市地區的適用性，垂直軸風力渦輪機(VAWTs)最近獲得了關注。此外，垂直軸風力渦輪機結構緊湊、安靜、且易於安裝在建築物內。它們還可以接收來自任何方向的風，在城市地區裡常見的紊流條件下表現良好。為了預測和研究VAWTs的空氣動力學表現，需要一個高效的流體求解器，因此，開發了一個Fortran 3-D流動求解器，通過利用有限體積法和直接施力沉浸邊界法(DFIB)來模擬VAWTs。對於空間項的離散化，採用二階中央差分法和對流動力學的二階二次上游插值(QUICK)方法。時間項的離散化採用了Adam-Bashforth方法，然後使用逐次超鬆弛法(SOR)來解壓力泊松方程，Smagorinsky-Lilly的紊流模型用於大渦流模擬(LES)以解決紊流尺度問題。

流動求解器也被混和OpenMP/MPI架構並行化，以最佳速度求解數值問題，該流動求解器在基準層流和紊流問題上得到了驗證。本求解器用於進行多個案例研究，在第一項研究中，通過在二維層流狀態下將流動求解器與遺傳算法(GA)相結合，最佳化了Darrieus型VAWT葉片的形狀。研究了兩種情況，一種是靜態機翼，另一種是垂直軸風力渦輪機中的旋轉機翼。NACA 0012翼型用作VAWT的第一個翼型橫截面，所提出的方法成功地模擬了流場中的旋轉葉片。 結果表明，與原始翼型相比，採用該方法生成的最佳化風力機葉片效率提高了5.61\%。

在第二項研究中，開發了一種基於使用DFIB模型之固體體積(VOS)函數的新算法，以模擬薄型無體積剛體在流固界面處的流動。所提出的方法使用靜止和旋轉的Savonius風力渦輪機作為薄型剛體，在紊流中進行了測試。驗證結果表明，使用DFIB模型與新算法相結合，可以成功地模擬不可壓縮紊流通過薄型無體積剛體時的單向和雙向流固耦合。

第三項研究模擬了紊流中的三維Savonius轉子，以調查方位角增量和渦輪轉數對不同葉尖速比(TSRs)下模擬結果精度的影響，然後在不同的TSRs(0.576、0.80和1.2)下進行模擬，以檢查方位角增量的影響。確定獲得靜態穩定模擬結果所需的渦輪機轉數，並可將其用作收斂標準。深入探討了不同TSRs下，方位角增量與性能特徵之間的關係，結果發現，要達到靜態穩定狀態需要渦輪機轉八圈。此外，Savonius 轉子氣動學表現的預測受到方位角增量很大的影響。最後，使用Q準則、渦度場和瞬時壓力來研究流體與轉子之間的相互作用，並且將葉尖處產生的渦流強度與功率係數值相關聯。

Vertical axis wind turbines (VAWTs) have recently gained interest as a result of the growing demand for individual renewable energy solutions and their suitability in urban areas. In addition, VAWTs are compact, silent, and simple to install in buildings. They can also receive wind from any direction and perform well in turbulent wind conditions, which are common in urban areas. An efficient flow solver is required to predict and investigate the aerodynamic performance of VAWTs. Thus, a Fortran 3-D flow solver was developed that simulates VAWTs by utilizing the finite volume and direct-forcing immersed boundary (DFIB) methods.
For the discretization of spatial terms, second-order central difference and second-order quadratic upstream interpolation for convective kinetics (QUICK) schemes were employed. The temporal term discretization was accomplished using the Adam-Bashforth scheme. The successive overrelaxation (SOR) method was then used to solve the pressure Poisson equation. Smagorinsky-Lilly's model for turbulent flow is used in large eddy simulation (LES) to resolve turbulence scales. The flow solver was also parallelized with a hybrid OpenMP/MPI architecture to solve numerical problems at an optimal speed.

The flow solver was validated on benchmark laminar and turbulent flow problems. The present solver was used to conduct multiple case studies. In the first study, the shape of a Darrieus-type VAWT blade was optimized using by combining the flow solver with genetic algorithms (GA) in a two-dimensional laminar flow regime. Two scenarios were investigated, one with a static airfoil and the other with a rotating airfoil in a vertical-axis wind turbine. A NACA 0012 airfoil served as the VAWT's first airfoil cross-section. The proposed method successfully simulated the rotating blades in the flow field. The results showed that compared to the original airfoil, the optimized wind turbine blade generated using this method had an efficiency improvement of 5.61\%.

In the second study, a new algorithm based on the volume of a solid function (VOS) using a DFIB model was developed to simulate flows at the fluid-structure interface for thin and volumeless rigid bodies. The proposed method was tested in turbulent flow using a stationary and rotating Savonius wind turbine that functions as a thin, rigid body. The validation results demonstrated that one-way and two-way fluid-structure interactions in incompressible, turbulent flows through thin, volumeless rigid bodies could be successfully simulated by the DFIB model combined with the new algorithm.

The third study simulated a three-dimensional Savonius rotor in turbulent flow to investigate the effect of azimuthal increment and the number of turbine revolutions on the accuracy of simulation results at various tip speed ratios (TSRs). The simulations were then conducted at various TSRs (0.576, 0.80, and 1.2) to examine the effect of azimuthal angle increment. The number of turbine rotations required to attain statically steady state simulation results was determined and can potentially be used as a convergence criterion. The relationship between azimuthal angle increment and performance characteristics at different TSRs was thoroughly explored. The results found that achieving a statically steady state needed eight turbine revolutions. Furthermore, the prediction of the aerodynamic performance of the Savonius rotor was highly influenced by azimuthal angle increment. Finally, using the iso-surfaces of the Q-criterion, the vorticity field, instantaneous pressure, and torque were used to study the interaction between the fluid and the rotor. The intensities of vortices generated at the blade tip were then related to the power coefficient value.

[1] Saillard, E. Static/Dynamic Analyses for Validation and Improvements of Multi-Model
HPC Applications. PhD thesis, Universite de Bordeaux, France, 2015. ´
[2] Masramon, E. M. Experimental study of flow through a Savonius wind Turbine. PhD
thesis, Universitat Politecnica de Catalunya. Escola Superior d’Enginyeries, Spain, 2015. `
[3] Martorell Pol, P. A. Numerical study of flow through a savonius wind turbine. B.S.
thesis, Universitat Politecnica de Catalunya, Spain, 2015. `
[4] Irabu, K. and Roy, J. N. Study of direct force measurement and characteristics on blades
of savonius rotor at static state. Experimental Thermal and Fluid Science, 35(4):
653–659, 2011. doi:10.1016/j.expthermflusci.2010.12.015.
[5] Hansen, J. T., Mahak, M., and Tzanakis, I. Numerical modelling and optimization of
vertical axis wind turbine pairs: A scale up approach. Renewable Energy, 171:
1371–1381, 2021. doi:10.1016/j.renene.2021.03.001.
[6] Rezaeiha, A., Montazeri, H., and Blocken, B. Towards accurate cfd simulations of
vertical axis wind turbines at different tip speed ratios and solidities: Guidelines for
azimuthal increment, domain size and convergence. Energy Conversion and
Management, 156:301–316, 2018. doi:10.1016/j.enconman.2017.11.026.
[7] Simao Ferreira, C., Van Kuik, G., Van Bussel, G., and Scarano, F. Visualization by piv ˜
of dynamic stall on a vertical axis wind turbine. Experiments in fluids, 46(1):97–108,
2009. doi:10.1007/s00348-008-0543-z.
[8] Yu, J., Leu, T.-S., and Miau, J.-J. Investigation of reduced frequency and freestream
turbulence effects on dynamic stall of a pitching airfoil. Journal of Visualization, 20(1):
31–44, 2017. doi:10.1007/s12650-016-0366-6.
[9] Amet, E., Maˆıtre, T., Pellone, C., and Achard, J.-L. 2d numerical simulations of
blade-vortex interaction in a darrieus turbine. Journal of fluids engineering, 131(11),
2009. doi:10.1115/1.4000258.
[10] Bianchini, A., Balduzzi, F., Rainbird, J. M., Peiro, J., Graham, J. M. R., Ferrara, G., and
Ferrari, L. An experimental and numerical assessment of airfoil polars for use in
darrieus wind turbines—part ii: Post-stall data extrapolation methods. Journal of
Engineering for Gas Turbines and Power, 138(3), 2016. doi:10.1115/1.4031270.
[11] Ferreira, C. S., Madsen, H. A., Barone, M., Roscher, B., Deglaire, P., and Arduin, I.
Comparison of aerodynamic models for vertical axis wind turbines. In Journal of
Physics:Conference Series 524 012125, Copenhagen, Denmark, 2014.
doi:10.1088/1742-6596/524/1/012125.
[12] Ghasemian, M., Ashrafi, Z. N., and Sedaghat, A. A review on computational fluid
dynamic simulation techniques for darrieus vertical axis wind turbines. Energy
Conversion and Management, 149:87–100, 2017. doi:10.1016/j.enconman.2017.07.016.
[13] Tahani, M., Babayan, N., Mehrnia, S., and Shadmehri, M. A novel heuristic method for
optimization of straight blade vertical axis wind turbine. Energy Conversion and
Management, 127:461–476, 2016. doi:10.1016/j.enconman.2016.08.094.
[14] Shigetomi, A., Murai, Y., Tasaka, Y., and Takeda, Y. Interactive flow field around two
savonius turbines. Renewable energy, 36(2):536–545, 2011.
doi:10.1016/j.renene.2010.06.036.
[15] Tescione, G., Ragni, D., He, C., Simao Ferreira, C., and van Bussel, G. Near wake flow ˜
analysis of a vertical axis wind turbine by stereoscopic particle image velocimetry.
Renewable Energy, 70:47–61, 2014. doi:10.1016/j.renene.2014.02.042. Special issue
on aerodynamics of offshore wind energy systems and wakes.
[16] Kinzel, M., Araya, D. B., and Dabiri, J. O. Turbulence in vertical axis wind turbine
canopies. Physics of Fluids, 27(11):115102, 2015. doi:/10.1063/1.4935111.
[17] Abkar, M. and Dabiri, J. O. Self-similarity and flow characteristics of vertical-axis wind
turbine wakes: an les study. Journal of Turbulence, 18(4):373–389, 2017.
doi:10.1080/14685248.2017.1284327.
[18] Strom, B., Polagye, B., and Brunton, S. L. Near-wake dynamics of a vertical-axis
turbine. Journal of Fluid Mechanics, 935, 2022. doi:10.48550/arXiv.2103.00407.
[19] Bausas, M. D. and Danao, L. A. M. The aerodynamics of a camber-bladed vertical axis
wind turbine in unsteady wind. Energy, 93:1155–1164, 2015.
doi:10.1016/j.energy.2015.09.120.
[20] Maeda, T., Kamada, Y., Murata, J., Furukawa, K., Yamamoto, M., et al. Effect of
number of blades on aerodynamic forces on a straight-bladed vertical axis wind turbine.
Energy, 90:784–795, 2015. doi:10.1016/j.energy.2015.07.115.
[21] Tian, W., Song, B., VanZwieten, J. H., and Pyakurel, P. Computational fluid dynamics
prediction of a modified savonius wind turbine with novel blade shapes. Energies, 8(8):
7915–7929, 2015.
[22] Peng, H. and Lam, H. Turbulence effects on the wake characteristics and aerodynamic
performance of a straight-bladed vertical axis wind turbine by wind tunnel tests and
large eddy simulations. Energy, 109:557–568, 2016. doi:10.1016/j.energy.2016.04.100.
[23] Roy, S. and Ducoin, A. Unsteady analysis on the instantaneous forces and moment arms
acting on a novel savonius-style wind turbine. Energy Conversion and Management,
121:281–296, 2016. doi:10.1016/j.enconman.2016.05.044.
[24] Rezaeiha, A., Kalkman, I., and Blocken, B. Effect of pitch angle on power performance
and aerodynamics of a vertical axis wind turbine. Applied Energy, 197:132–150, 2017.
doi:10.1016/j.apenergy.2017.03.128.
[25] Mukesh, R., Lingadurai, K., and Selvakumar, U. Airfoil shape optimization using
non-traditional optimization technique and its validation. Journal of King Saud
University-Engineering Sciences, 26(2):191–197, 2014.
[26] Della Vecchia, P., Daniele, E., and D’Amato, E. An airfoil shape optimization technique
coupling parsec parameterization and evolutionary algorithm. Aerospace Science and
Technology, 32(1):103–110, 2014. doi:10.1016/j.ast.2013.11.006.
[27] Li, X., Yang, K., Bai, J., and Xu, J. A new optimization approach to improve the overall
performance of thick wind turbine airfoils. Energy, 116:202–213, 2016.
doi:10.1016/j.energy.2016.09.108.
[28] Ma, N., Lei, H., Han, Z., Zhou, D., Bao, Y., Zhang, K., Zhou, L., and Chen, C. Airfoil
optimization to improve power performance of a high-solidity vertical axis wind turbine
at a moderate tip speed ratio. Energy, 150:236–252, 2018.
doi:10.1016/j.energy.2018.02.115.
[29] Islam, M., Ting, D. S., and Fartaj, A. Aerodynamic models for darrieus-type
straight-bladed vertical axis wind turbines. Renewable and Sustainable Energy Reviews,
12(4):1087–1109, 2008. doi:10.1016/j.rser.2006.10.023.
[30] Carrigan, T. J., Dennis, B. H., Han, Z. X., and Wang, B. P. Aerodynamic shape
optimization of a vertical axis wind turbine using differential evolution. ISRN
Renewable Energy, 2012:16, 2012. doi:10.5402/2012/528418.
[31] Howell, R., Qin, N., Edwards, J., and Durrani, N. Wind tunnel and numerical study of a
small vertical axis wind turbine. Renewable Energy, 35(2):412–422, 2010.
doi:10.1016/j.renene.2009.07.025.
[32] Chan, C., Bai, H., and He, D. Blade shape optimization of the savonius wind turbine
using a genetic algorithm. Applied Energy, 213:148–157, 2018.
doi:10.1016/j.apenergy.2018.01.029.
[33] Zhu, W. J., Shen, W. Z., and Sørensen, J. N. Integrated airfoil and blade design method
for large wind turbines. Renewable Energy, 70:172–183, 2014.
doi:10.1016/j.renene.2014.02.057.
[34] Zhang, Q. and Hisada, T. Analysis of fluid–structure interaction problems with
structural buckling and large domain changes by ale finite element method. Computer
Methods in Applied Mechanics and Engineering, 190:6341–6357, 2001.
doi:10.1016/S0045-7825(01)00231-6.
[35] Zhong, J. and Xu, Z. A reduced mesh movement method based on pseudo elastic solid
for fluid-structure interaction. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 232(6):973–986, 2018.
doi:10.1177/0954406217700177.
[36] Aldlemy, M. S., Rasani, M., Ya, T., and Ariffin, A. Dynamic adaptive mesh refinement
of fluid-structure interaction using immersed boundary method with two-stage
corrections. Scientia Iranica, 26:2827–2838, 2019. doi:10.24200/sci.2018.50347.1650.
[37] Liao, C.-C., Chang, Y.-W., Lin, C.-A., and McDonough, J. Simulating flows with
moving rigid boundary using immersed-boundary method. Computers Fluids, 39(1):
152–167, 2010. doi:https://doi.org/10.1016/j.compfluid.2009.07.011.
[38] Liao, C.-C. and Lin, C.-A. Simulations of natural and forced convection flows with
moving embedded object using immersed boundary method. Computer Methods in
Applied Mechanics and Engineering, 213-216:58–70, 2012.
doi:https://doi.org/10.1016/j.cma.2011.11.009.
[39] Liao, C.-C. and Lin, C.-A. Influences of a confined elliptic cylinder at different aspect
ratios and inclinations on the laminar natural and mixed convection flows. International
Journal of Heat and Mass Transfer, 55(23):6638–6650, 2012.
doi:https://doi.org/10.1016/j.ijheatmasstransfer.2012.06.073.
[40] Liao, C.-C. and Lin, C.-A. Transitions of natural convection flows in a square enclosure
with a heated circular cylinder. Applied Thermal Engineering, 72(1):41–47, 2014.
doi:https://doi.org/10.1016/j.applthermaleng.2014.02.071.
[41] Liao, C.-C. and Lin, C.-A. Mixed convection of a heated rotating cylinder in a square
enclosure. International Journal of Heat and Mass Transfer, 72:9–22, 2014.
doi:https://doi.org/10.1016/j.ijheatmasstransfer.2013.12.081.
[42] Chang, P.-H., Liao, C.-C., Hsu, H.-W., Liu, S.-H., and Lin, C.-A. Simulations of laminar
and turbulent flows over periodic hills with immersed boundary method. Computers
Fluids, 92:233–243, 2014. doi:https://doi.org/10.1016/j.compfluid.2013.10.043.
[43] Liao, C.-C., Hsiao, W.-W., Lin, T.-Y., and Lin, C.-A. Simulations of two
sedimenting-interacting spheres with different sizes and initial configurations using immersed boundary method. Computational Mechanics, 55(6):1191–1200, 2015.
doi:10.1007/s00466-015-1157-y.

[44] Singh, K., Michelin, S., and De Langre, E. Energy harvesting from axial fluid-elastic
instabilities of a cylinder. Journal of Fluids and Structures, 30:159–172, 2012.
doi:10.1016/j.jfluidstructs.2003.12.004.
[45] Farhat, C. and Lakshminarayan, V. K. An ale formulation of embedded boundary
methods for tracking boundary layers in turbulent fluid-structure interaction problems.
Journal of Computational Physics, 263:53–70, 2014. doi:10.1016/j.jcp.2014.01.018.
[46] Antoine, G. O., de Langre, E., and Michelin, S. Optimal energy harvesting from
vortex-induced vibrations of cables. Proceedings of the Royal Society A: Mathematical,
Physical and Engineering Sciences, 472(2195):20160583, 2016.
doi:10.1098/rspa.2016.0583.
[47] Costarelli, S. D., Garelli, L., Cruchaga, M. A., Storti, M. A., Ausensi, R., and Idelsohn,
S. R. An embedded strategy for the analysis of fluid-structure interaction problems.
Computer Methods in Applied Mechanics and Engineering, 300:106–128, 2016.
doi:10.1016/j.cma.2015.11.001.
[48] Sarkar, P., Ghigliotti, G., Franc, J.-P., and Fivel, M. Fluid–structure modelling for
material deformation during cavitation bubble collapse. Journal of Fluids and
Structures, 106:103370, 2021. doi:10.1016/j.jfluidstructs.2021.103370.
[49] Iaccarino, G. and Verzicco, R. Immersed boundary technique for turbulent flow
simulations. Appl. Mech. Rev., 56(3):331–347, 2003. doi:doi.org/10.1115/1.1563627.
[50] Peskin, C. S. Flow patterns around heart valves: a numerical method. Journal of
computational physics, 10:252–271, 1972. doi:10.1016/0021-9991(72)90065-4.
[51] Huang, W.-X. and Sung, H. J. An immersed boundary method for fluid–flexible
structure interaction. Computer methods in applied mechanics and engineering, 198
(33-36):2650–2661, 2009. doi:10.1016/j.cma.2009.03.008.
[52] Luo, H., Dai, H., de Sousa, P. J. F., and Yin, B. On the numerical oscillation of the
direct-forcing immersed-boundary method for moving boundaries. Computers & Fluids,
56:61–76, 2012. doi:10.1016/j.compfluid.2011.11.015.
[53] Tian, F.-B., Dai, H., Luo, H., Doyle, J. F., and Rousseau, B. Fluid–structure interaction
involving large deformations: 3d simulations and applications to biological systems.
Journal of computational physics, 258:451–469, 2014. doi:10.1016/j.jcp.2013.10.047.
[54] Liska, S. and Colonius, T. A fast immersed boundary method for external
incompressible viscous flows using lattice green’s functions. Journal of Computational
Physics, 331:257–279, 2017. doi:10.1016/j.jcp.2016.11.034.
[55] Wang, L., Currao, G. M., Han, F., Neely, A. J., Young, J., and Tian, F.-B. An immersed
boundary method for fluid–structure interaction with compressible multiphase flows.
Journal of Computational Physics, 346:131–151, 2017. doi:10.1016/j.jcp.2017.06.008.
[56] Xu, L., Tian, F.-B., Young, J., and Lai, J. C. A novel geometry-adaptive cartesian grid
based immersed boundary–lattice boltzmann method for fluid–structure interactions at
moderate and high reynolds numbers. Journal of Computational Physics, 375:22–56,
2018. doi:10.1016/j.jcp.2018.08.024.
[57] Huang, W.-X. and Tian, F.-B. Recent trends and progress in the immersed boundary
method. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of
Mechanical Engineering Science, 233(23-24):7617–7636, 2019.
doi:10.1177/0954406219842606.
[58] Wang, L., Tian, F.-B., and Lai, J. C. An immersed boundary method for
fluid–structure–acoustics interactions involving large deformations and complex
geometries. Journal of Fluids and Structures, 95:102993, 2020.
doi:10.1016/j.jfluidstructs.2020.102993.
[59] Mittal, R. and Iaccarino, G. Immersed boundary methods. Annu. Rev. Fluid Mech., 37:
239–261, 2005.
[60] Ji, C., Munjiza, A., and Williams, J. A novel iterative direct-forcing immersed boundary method and its finite volume applications. Journal of Computational Physics, 231(4):
1797–1821, 2012.
[61] Horng, T. L., Hsieh, P. W., Yang, S. Y., and You, C. S. A simple direct-forcing immersed
boundary projection method with prediction-correction for fluid-solid interaction
problems. Computers and Fluids, 176:135–152, 2018.
doi:10.1016/j.compfluid.2018.02.003.
[62] Goldstein, D., Handler, R., and Sirovich, L. Modeling a noslip flow boundary with an
external force field. Journal of computational physics, 105(2):354–366, 1993.
doi:10.1006/jcph.1993.1081.
[63] Lai, M.-C. and Peskin, C. S. An immersed boundary method with formal second-order
accuracy and reduced numerical viscosity. Journal of computational Physics, 160(2):
705–719, 2000. doi:10.1006/jcph.2000.6483.
[64] Kim, J., Kim, D., and Choi, H. An immersed-boundary finite-volume method for
simulations of flow in complex geometries. Journal of computational physics, 171(1):
132–150, 2001.
[65] Silva, A. L. E., Silveira-Neto, A., and Damasceno, J. Numerical simulation of
two-dimensional flows over a circular cylinder using the immersed boundary method.
Journal of Computational Physics, 189(2):351–370, 2003.
doi:10.1016/S0021-9991(03)00214-6.
[66] Fadlun, E., Verzicco, R., Orlandi, P., and Mohd-Yusof, J. Combined immersed-boundary
finite-difference methods for three-dimensional complex flow simulations. Journal of
computational physics, 161(1):35–60, 2000. doi:10.1006/jcph.2000.6484.
[67] Noor, D. Z., Chern, M.-J., and Horng, T.-L. An immersed boundary method to solve
fluid–solid interaction problems. Computational Mechanics, 44(4):447–453, 2009.
[68] Chern, M.-J., Shiu, W.-C., and Horng, T.-L. Immersed boundary modeling for
interaction of oscillatory flow with cylinder array under effects of flow direction and
cylinder arrangement. Journal of Fluids and Structures, 43:325–346, 2013.
doi:10.1016/j.jfluidstructs.2013.09.022.

[69] Chern, M.-J., Kuan, Y.-H., Nugroho, G., Lu, G.-T., and Horng, T.-L. Direct-forcing
immersed boundary modeling of vortex-induced vibration of a circular cylinder.
Journal of Wind Engineering and Industrial Aerodynamics, 134:109–121, 2014.
doi:10.1016/j.jweia.2014.08.015.
[70] Chern, M.-J., Noor, D. Z., Liao, C.-B., and Horng, T.-L. Direct-forcing immersed
boundary method for mixed heat transfer. Communications in Computational Physics,
18:1072–1094, 2015. doi:10.4208/cicp.151214.250515s.
[71] Vaziri, N., Chern, M.-J., and Horng, T.-L. Simulation of dynamic stall using
direct-forcing immersed boundary method at low reynolds number. Aircraft
Engineering and Aerospace Technology, 2018. doi:10.1108/AEAT-05-2017-0128.
[72] Chern, M., Lu, G., Kuan, Y., Chakraborty, S., Nugroho, G., Liao, C., and Horng, T.
Numerial study of vortex-induced vibration of circular cylinder adjacent to plane
boundary using direct-forcing immersed boundary method. Journal of Mechanics, 34
(2):177–191, 2018. doi:10.1017/jmech.2017.55.
[73] Raza, S. A., Chern, M.-J., Susanto, H., and Zhou, Y.-H. Numerical investigation of the
effects of a small fixed sphere in tandem arrangement on viv of a sphere. Journal of
Wind Engineering and Industrial Aerodynamics, 206:104368, 2020.
doi:10.1016/j.jweia.2020.104368.
[74] Chern, M.-J., Goytom Tewolde, D., Kao, C.-C., and Vaziri, N. Vertical-axis wind
turbine blade-shape optimization using a genetic algorithm and direct-forcing immersed
boundary method. Journal of Energy Engineering, 147:04020091, 2021.
doi:10.1061/(ASCE)EY.1943-7897.0000741.
[75] You, C.-S. On two immersed boundary methods for simulating the dynamics of
fluid-structure interaction problems. PhD thesis, National Central University, 2016.
[76] Salih, S. Q., Aldlemy, M. S., Rasani, M. R., Ariffin, A., Ya, T. M. Y. S. T., Al-Ansari, N.,
Yaseen, Z. M., and Chau, K.-W. Thin and sharp edges bodies-fluid interaction simulation
using cut-cell immersed boundary method. Engineering Applications of Computational
Fluid Mechanics, 13(1):860–877, 2019. doi:10.1080/19942060.2019.1652209.
[77] Wang, Y., Sun, X., Dong, X., Zhu, B., Huang, D., and Zheng, Z. Numerical
investigation on the aerodynamic performance of a novel vertical axis wind turbine with
adaptive blades. Energy Conversion and Management, 108:275–286, 2016.
doi:10.1016/j.enconman.2015.11.003.
[78] Bhuyan, S. and Biswas, A. Investigations on self-starting and performance
characteristics of simple h and hybrid h-savonius vertical axis wind rotors. Energy
Conversion and Management, 87:859–867, 2014. doi:10.1016/j.enconman.2014.07.056.
[79] Gonzalez Madina, F., Guti ´ errez, A., and Galione, P. Computational fluid dynamics ´
study of savonius rotors using openfoam. Wind Engineering, 45(3):630–647, 2020.
doi:10.1177/0309524X20924958.
[80] Modi, V. and Fernando, M. Unsteady aerodynamics and wake of the savonius wind
turbine: a numerical study. Journal of Wind Engineering and Industrial Aerodynamics,
46:811–816, 1993. ISSN 0167-6105. doi:10.1016/0167-6105(93)90357-T.
[81] Chauvin, A., Botrini, M., Brun, R., and Beguier, C. The evaluation of the power
coefficient of a savonius rotor. Academie des Sciences Paris Comptes Rendus Serie
Sciences Mathematiques, 296(11):823–826, 1983.
[82] Akwa, J. V., da Silva J´unior, G. A., and Petry, A. P. Discussion on the verification of the
overlap ratio influence on performance coefficients of a savonius wind rotor using
computational fluid dynamics. Renewable energy, 38(1):141–149, 2012.
doi:10.1016/j.renene.2011.07.013.
[83] Kacprzak, K., Liskiewicz, G., and Sobczak, K. Numerical investigation of conventional
and modified savonius wind turbines. Renewable energy, 60:578–585, 2013.
doi:10.1016/j.renene.2013.06.009.
[84] Shaheen, M., El-Sayed, M., and Abdallah, S. Numerical study of two-bucket savonius
wind turbine cluster. Journal of Wind Engineering and Industrial Aerodynamics, 137:
78–89, 2015. ISSN 0167-6105. doi:10.1016/j.jweia.2014.12.002.
[85] Alom, N. and Saha, U. K. Performance evaluation of vent-augmented elliptical-bladed savonius rotors by numerical simulation and wind tunnel experiments. Energy, 152:
277–290, 2018. doi:10.1016/j.energy.2018.03.136.
[86] Dobrev, I. and Massouh, F. Cfd and piv investigation of unsteady flow through savonius
wind turbine. Energy Procedia, 6:711–720, 2011. doi:10.1016/j.egypro.2011.05.081.
[87] Kang, C., Liu, H., and Yang, X. Review of fluid dynamics aspects of
savonius-rotor-based vertical-axis wind rotors. Renewable and Sustainable Energy
Reviews, 33:499–508, 2014. doi:https://doi.org/10.1016/j.rser.2014.02.011.
[88] Jaohindy, P., McTavish, S., Garde, F., and Bastide, A. An analysis of the transient forces
acting on savonius rotors with different aspect ratios. Renewable Energy, 55:286–295,
2013. doi:10.1016/j.renene.2012.12.045.
[89] Ferrari, G., Federici, D., Schito, P., Inzoli, F., and Mereu, R. Cfd study of savonius wind
turbine: 3d model validation and parametric analysis. Renewable energy, 105:722–734,
2017. doi:10.1016/j.renene.2016.12.077.
[90] Nasef, M., El-Askary, W., Abdel-Hamid, A., and Gad, H. Evaluation of savonius rotor
performance: Static and dynamic studies. Journal of Wind Engineering and Industrial
Aerodynamics, 123:1–11, 2013. doi:10.1016/j.jweia.2013.09.009.
[91] Daroczy, L., Janiga, G., Petrasch, K., Webner, M., and Th ´ evenin, D. Comparative ´
analysis of turbulence models for the aerodynamic simulation of h-darrieus rotors.
Energy, 90:680–690, 2015. doi:10.1016/j.energy.2015.07.102.
[92] Patil, R., Daroczy, L., Janiga, G., and Th ´ evenin, D. Large eddy simulation of an ´
h-darrieus rotor. Energy, 160:388–398, 2018. doi:10.1016/j.energy.2018.06.203.
[93] Li, C., Zhu, S., lin Xu, Y., and Xiao, Y. 2.5d large eddy simulation of vertical axis wind
turbine in consideration of high angle of attack flow. Renewable Energy, 51:317–330,
2013. doi:10.1016/j.renene.2012.09.011.
[94] Lei, H., Zhou, D., Bao, Y., Li, Y., and Han, Z. Three-dimensional improved delayed
detached eddy simulation of a two-bladed vertical axis wind turbine. Energy Conversion
and Management, 133:235–248, 2017. doi:10.1016/j.enconman.2016.11.067.

[95] Trivellato, F. and Raciti Castelli, M. On the courant–friedrichs–lewy criterion of
rotating grids in 2d vertical-axis wind turbine analysis. Renewable Energy, 62:53–62,
2014. doi:10.1016/j.renene.2013.06.022.
[96] Balduzzi, F., Bianchini, A., Maleci, R., Ferrara, G., and Ferrari, L. Critical issues in the
cfd simulation of darrieus wind turbines. Renewable Energy, 85:419–435, 2016.
doi:10.1016/j.renene.2015.06.048.
[97] Shahizare, B., Nazri Bin Nik Ghazali, N., Chong, W. T., Tabatabaeikia, S. S., and
Izadyar, N. Investigation of the optimal omni-direction-guide-vane design for vertical
axis wind turbines based on unsteady flow cfd simulation. Energies, 9(3):146, 2016.
doi:10.3390/en9030146.
[98] Rezaeiha, A., Kalkman, I., and Blocken, B. Cfd simulation of a vertical axis wind
turbine operating at a moderate tip speed ratio: Guidelines for minimum domain size
and azimuthal increment. Renewable energy, 107:373–385, 2017.
doi:10.1016/j.renene.2017.02.006.
[99] Davidson, L. Fluid mechanics, turbulent flow and turbulence modeling. Citeseer, 2015.
URL http://www.tfd.chalmers.se/œlada/MoF/.
[100] Smagorinsky, J. General circulation experiments with the primitive equations. Monthly
Weather Review, 91(3):99–164, March 1963.
doi:10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2.
[101] Chorin, A. J. The numerical solution of the navier-stokes equations for an
incompressible fluid. Bulletin of the American Mathematical Society, 73(6):928–931,
1967.
[102] Leonard, B. P. A stable and accurate convective modelling procedure based on quadratic
upstream interpolation. Computer Methods in Applied Mechanics and Engineering, 19
(1):59–98, June 1979. doi:10.1016/0045-7825(79)90034-3.
[103] Blackwell, B. F., Feltz, L. V., and Sheldahl, R. E. Wind tunnel performance data for
two-and three-bucket Savonius rotors. Sandia Laboratories Springfield, VA, USA, 1977.
[104] Ross, I. J. Wind tunnel blockage corrections: an application to vertical-axis wind
turbines. PhD thesis, University of Dayton, 2010.
[105] Sutherland, I. E., Sproull, R. F., and Schumacker, R. A. A characterization of ten
hidden-surface algorithms. ACM Computing Surveys, 6(1):1–55, 1974.
[106] Sobieczky, H. Parametric airfoils and wings. In Recent Development of Aerodynamic
Design Methodologies: Inverse Design and Optimization, pages 71–87. Vieweg Teubner
Verlag, 1999. doi:10.1007/978-3-322-89952-1 4.
[107] Holland, J. Adaptation in natural and artificial systems: an introductory analysis with
application to biology. MIT press, 1975.
[108] Goldberg, D. E. and Holland, J. H. Genetic algorithms and machine learning. Machine
Learning, 3(2):95–99, 1988.
[109] Herrera, F., Lozano, M., and Sanchez, A. M. A taxonomy for the crossover operator for ´
real-coded genetic algorithms: An experimental study. International Journal of
Intelligent Systems, 18(3):309–338, 2003.
[110] National Center for High-performance Computing, Taiwan, May 2021. URL
https://www.nchc.org.tw.
[111] G¨unel, O., Koc¸, E., and Yavuz, T. Cfd vs. xfoil of airfoil analysis at low reynolds
numbers. In 2016 IEEE International Conference on Renewable Energy Research and
Applications (ICRERA), pages 628–632, Birmingham, UK, 2016. IEEE.
doi:10.1109/ICRERA.2016.7884411.
[112] Morgado, J., Vizinho, R., Silvestre, M., and Pascoa, J. Xfoil vs cfd performance ´
predictions for high lift low reynolds number airfoils. Aerospace Science and
Technology, 52:207–214, 2016. doi:https://doi.org/10.1016/j.ast.2016.02.031.
[113] Drela, M. Xfoil subsonic airfoil development system, 2001. URL
web.mit.edu/drela/Public/web/xfoil.
[114] Baluja, S. and Caruana, R. Removing the genetics from the standard genetic algorithm.
In ICML, pages 38–46. School of Computer Science, Carnegie Mellon University, 1995.
[115] Franke, J. and Frank, W. Large eddy simulation of the flow past a circular cylinder at at
ReD=3900. Journal of wind engineering and industrial aerodynamics, 90(10):
1191–1206, 2002. doi:10.1016/S0167-6105(02)00232-5.
[116] Raza, S. A., Irawan, Y. H., and Chern, M.-J. Effect of boundary conditions and domain
size on the turbulent flow characteristics over a circular cylinder. Journal of the Chinese
Institute of Engineers, 44(7):659–672, 2021. doi:10.1080/02533839.2021.1940295.
[117] Sheldahl, R. E., Blackwell, B. F., and Feltz, L. V. Wind tunnel performance data for
two-and three-bucket savonius rotors. Journal of Energy, 2(3):160–164, 1978.
doi:https://doi.org/10.2514/3.47966.
[118] Norberg, C. Effects of reynolds number and low-intensity freestream turbulence on the
flow around a circular cylinder. Chalmers University, Goteborg, Sweden, Technological
Publications, 87(2):1–55, 1987.
[119] Cardell, G. Flow past a circular cylinder with a permeable splitter plate. California
Institute of Technology, 1993.
[120] Breuer, M. Numerical and modeling influences on large eddy simulations for the flow
past a circular cylinder. International Journal of Heat and Fluid Flow, 19(5):512–521,
October 1998. doi:10.1016/S0142-727X(98)10015-2.
[121] Kravchenko, A. G. and Moin, P. Numerical studies of flow over a circular cylinder at
ReD=3900. Physics of Fluids, 12(2):403–417, January 2000. doi:10.1063/1.870318.
[122] Sidebottom, W., Ooi, A., and Jones, D. A parametric study of turbulent flow past a
circular cylinder using large eddy simulation. Journal of Fluids Engineering, 137(9),
2015. doi:10.1115/1.4030380.
[123] Rajani, B., Kandasamy, A., and Majumdar, S. Les of flow past circular cylinder at re=
3900. Journal of Applied Fluid Mechanics, 9(3), 2016.
doi:10.18869/ACADPUB.JAFM.68.228.24178.
[124] Lourenco, L. M. and Shih, C. Characteristics of the plane turbulent near wake of a
circular cylinder. A particle image velocimetry study. (private communication), first published in Beaudan and Moin (1994), 1993. URLhttps://ci.nii.ac.jp/naid/10009520643/en/

[125] Parnaudeau, P., Carlier, J., Heitz, D., and Lamballais, E. Experimental and numerical
studies of the flow over a circular cylinder at reynolds number 3900. Physics of Fluids,
20(8):085101, 2008. doi:10.1063/1.2957018.
[126] Ong, L. and Wallace, J. The velocity field of the turbulent very near wake of a circular
cylinder. Experiments in fluids, 20(6):441–453, 1996. doi:10.1007/BF00189383.
[127] Le, T. Q., Lee, K.-S., Park, J.-S., and Ko, J. H. Flow-driven rotor simulation of vertical
axis tidal turbines: A comparison of helical and straight blades. International Journal of
Naval Architecture and Ocean Engineering, 6(2):257–268, 2014.
doi:10.2478/IJNAOE-2013-0177.
[128] Satrio, D., Utama, I. K. A. P., et al. The influence of time step setting on the cfd
simulation result of vertical axis tidal current turbine. Journal of Mechanical
Engineering and Sciences, 12(1):3399, 2018.
[129] D’Ambrosio, M. and Medaglia, M. Vertical axis wind turbines: History, technology, and
applications. Master’s thesis, Halmstad University, Sweden, 2010.
[130] Kerikous, E. and Thevenin, D. Optimal shape and position of a thick deflector plate in ´
front of a hydraulic savonius turbine. Energy, 189:116157, 2019.
doi:10.1016/j.energy.2019.116157.
[131] Genc, M. S., Kemal, K., and Acikel, H. H. Investigation of pre-stall flow control on
wind turbine blade airfoil using roughness element. Energy, 176:320–334, 2019.
doi:10.1016/j.energy.2019.03.179.
[132] Chowdhury, A. M., Akimoto, H., and Hara, Y. Comparative cfd analysis of vertical axis
wind turbine in upright and tilted configuration. Renewable Energy, 85:327–337, 2016.
doi:10.1016/j.renene.2015.06.037.
[133] Hand, B. and Cashman, A. Aerodynamic modeling methods for a large-scale vertical
axis wind turbine: A comparative study. Renewable Energy, 129:12–31, 2018.
doi:10.1016/j.renene.2018.05.078.