Sustainable Marine Structures

Numerical Simulations of the Pitching Airfoil by Using Discrete Vortex Method

Peng Ren

Department of Engineering Mechanics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200030, China;
Key laboratory of Hydrodynamics (Ministry of Education), Shanghai Jiao Tong University, Shanghai, 200030, China

Ke Lin

Department of Engineering Mechanics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200030, China;
Key laboratory of Hydrodynamics (Ministry of Education), Shanghai Jiao Tong University, Shanghai, 200030, China

Jiasong Wang

Department of Engineering Mechanics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200030, China;
Key Laboratory of Hydrodynamics (Ministry of Education), Shanghai Jiao Tong University, Shanghai, 200030, China;
SJTU-Sanya Yazhou Bay Institute of Deepsea Science and Technology, Sanya, Hainan, 572024, China

DOI: https://doi.org/10.36956/sms.v5i2.944

Received: 5 September 2023; Revised: 15 October 2023; Accepted: 25 October 2023; Published: 31 October 2023

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Abstract

This paper presents a two-dimensional discrete vortex method that uses the vortex growing core model to simulate the unsteady force and the wake patterns of the pure pitching airfoil efficiently and accurately. To avoid the random fluctuation caused by the random walk method, a vortex growing core method is used to simulate the viscous diffusion motion. In addition, the vortices fall off randomly on the body surface. Referring to the experimental configurations of Mackowski and Williamson (2015), a good agreement is achieved through the comparisons between the present simulation results and the experimental results, including the mean force coefficients, oscillation amplitude and wake patterns. It shows that the two-dimensional discrete vortex method can be used to predict the mechanical behavior and wake patterns on the pitching airfoil motion.

Keywords: Discrete vortex method; Pitching motion; Airfoil; Thrust

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References

[1] Jones, K.D., Bradshaw, C.J., Papadopoulos, J., et al., 2005. Bio-inspired design of flapping-wing micro air vehicles. The Aeronautical Journal. 109(1098), 385-393.

[2] Jones, K., Duggan, S., Platzer, M. (editors), 2001. Flapping-wing propulsion for a micro air vehicle. 39th Aerospace Sciences Meeting and Exhibit; 2001 Jan 8-11; Reno. p. 126.

[3] Read, D.A., Hover, F.S., Triantafyllou, M.S., 2003. Forces on oscillating foils for propulsion and maneuvering. Journal of Fluids and Structures. 17(1), 163-183.

[4] Cheng, Q., Liu, X., Ji, H.S., et al., 2017. Aerodynamic analysis of a helical vertical axis wind turbine. Energies. 10(4), 575.

[5] Lewthwaite, M.T., Amaechi, C.V., 2022. Numerical investigation of winglet aerodynamics and dimple effect of NACA 0017 airfoil for a freight aircraft. Inventions. 7(1), 31.

[6] Nguyen, M.T., Balduzzi, F., Goude, A., 2021. Effect of pitch angle on power and hydrodynamics of a vertical axis turbine. Ocean Engineering. 238, 109335.

[7] Von Karman, T., 1935. General aerodynamic theory-perfect fluids. Aerodynamic Theory. 2, 346-349.

[8] Garrick, I.E., 1936. Propulsion of a Flapping and Oscillating Airfoil [Internet]. Available from: https://ntrs.nasa.gov/api/citations/19930091642/downloads/19930091642.pdf

[9] Koochesfahani, M.M., 1989. Vortical patterns in the wake of an oscillating airfoil. AIAA Journal. 27(9), 1200-1205.

[10] Bohl, D.G., Koochesfahani, M.M., 2009. MTV measurements of the vortical field in the wake of an airfoil oscillating at high reduced frequency. Journal of Fluid Mechanics. 620, 63-88.

[11] Godoy-Diana, R., Aider, J.L., Wesfreid, J.E., 2008. Transitions in the wake of a flapping foil. Physical Review E. 77(1), 016308.

[12] Schnipper, T., Andersen, A., Bohr, T., 2009. Vortex wakes of a flapping foil. Journal of Fluid Mechanics. 633, 411-423.

[13] Young, J., Lai, J.C., 2004. Oscillation frequency and amplitude effects on the wake of a plunging airfoil. AIAA Journal. 42(10), 2042-2052.

[14] Chandravanshi, L.K., Chajjed, S., Sarkar, S. (editors), 2010. Study of wake pattern behind an oscillating airfoil. Proceeding of the 37th National & 4th International Conference on Fluid Mechanics and Fluid Power; 2010 Dec 16-18; Chennai. p. 16-18.

[15] Wu, Q., Huang, B., Wang, G., 2016. Lagrangian-based investigation of the transient flow structures around a pitching hydrofoil. Acta Mechanica Sinica. 32, 64-74.

[16] Lin, H., Vezza, M., Galbraith, R.M., 1997. Discrete vortex method for simulating unsteady flow. AIAA Journal. 35(3), 494-499.

[17] Sarkar, S., Venkatraman, K., 2006. Numerical simulation of thrust generating flow past a pitching airfoil. Computers & Fluids. 35(1), 16-42.

[18] Andersen, A., Bohr, T., Schnipper, T., et al., 2017. Wake structure and thrust generation of a flapping foil in two-dimensional flow. Journal of Fluid Mechanics. 812, R4.

[19] Priovolos, A.K., Filippas, E.S., Belibassakis, K.A., 2018. A vortex-based method for improved flexible flapping-foil thruster performance. Engineering Analysis with Boundary Elements. 95, 69-84.

[20] Chorin, A.J., 1973. Numerical study of slightly viscous flow. Journal of Fluid Mechanics. 57(4), 785-796.

[21] Yamamoto, C.T., Meneghini, J.R., Saltara, F., et al., 2004. Numerical simulations of vortex-induced vibration on flexible cylinders. Journal of Fluids and Structures. 19(4), 467-489.

[22] Meneghini, J.R., Saltara, F., de Andrade Fregonesi, R., et al., 2004. Numerical simulations of VIV on long flexible cylinders immersed in complex flow fields. European Journal of Mechanics-B/Fluids. 23(1), 51-63.

[23] Lin, K., Wang, J., 2019. Numerical simulation of vortex-induced vibration of long flexible risers using a SDVM-FEM coupled method. Ocean Engineering. 172, 468-486.

[24] Park, W.C., 1989. Computation of flow past single and multiple bluff bodies by a vortex tracking method [Ph.D. thesis]. Minneapolis: University of Minnesota.

[25] Mackowski, A.W., Williamson, C.H.K., 2015. Direct measurement of thrust and efficiency of an airfoil undergoing pure pitching. Journal of Fluid Mechanics. 765, 524-543.

[26] Qian, L., Vezza, M., 2001. A vorticity-based method for incompressible unsteady viscous flows. Journal of Computational Physics. 172(2), 515-542.

[27] Clarke, N.R., Tutty, O.R., 1994. Construction and validation of a discrete vortex method for the two-dimensional incompressible Navier-Stokes equations. Computers & Fluids. 23(6), 751-783.

[28] Wang, L., Yeung, R.W., 2016. Investigation of full and partial ground effects on a flapping foil hovering above a finite-sized platform. Physics of Fluids. 28(7).

[29] Walther, J.H., 1994. Discrete vortex method for two-dimensional flow past bodies of arbitrary shape undergoing prescribed rotary and translational motion. Den Polytekniske Læreanstalt: Lyngby.

[30] Spalart, P.R., 1983. Numerical simulation of separated flows. Stanford University: Stanford.

[31] Taylor, G.K., Nudds, R.L., Thomas, A.L., 2003. Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. Nature. 425(6959), 707-711.

[32] Laitone, E.V., 1997. Wind tunnel tests of wings at Reynolds numbers below 70 000. Experiments in Fluids. 23(5), 405-409.