This paper proposes a novel idea of Goldstein’s virtual boundary method which improves the calculation of the feedback forcing term and extends the applicability of this immersed boundary method. The original virtual boundary method includes the time integration of velocity deviation, therefore confining this method to time-dependent Navier-Stokes (N-S)equations with a severe limitation of time steps for the explicit scheme. In contrast, this paper calculates the feedback forcing by the sum of velocity deviation in iteration to avoid time dependent parameters. Thus, the improved method is not only suitable for the unsteady implicit scheme, but can couple with the steady solver without any time-dependent terms. To verify this improved method, this paper simulated the flow past a stationary cylinder, the inline oscillation of a cylinder in a fluid at rest, a flapping ellipse wing and a stationary sphere. All results agree well with previous numerical results, verifying the accuracy of the present method. We come to the conclusions that the feedback force is dependent on the velocity deviation during iteration, and that the present method can couple with the implicit algorithm for unsteady flows as well as the steady Navier-Stokes solver, indicating wider applicability of the present method for extensive flow problems.
[1] NOACK R W, BOGER D A, KUNZ R F, et al. Suggar++:An improved general overset grid assembly capability[C]//19th AIAA Computational Fluid Dynamics. Reston:AIAA, 2009.
[2] PESKIN C S. Numerical analysis of blood flow in the heart[J]. Journal of Computational Physics, 1977, 25(3):220-252.
[3] MITTAL R, IACCARINO G. Immersed boundary methods[J]. Annual Review of Fluid Mechanics, 2005, 37:239-261.
[4] SOTIROPOULO F, YANG X L. Immersed boundary methods for simulating fluid-structure interaction[J]. Progress in Aerospace Sciences, 2014, 65:1-21.
[5] KIM W, CHOI H. Immersed boundary methods for fluid-structure interaction:A review[J]. International Journal of Heat and Fluid Flow, 2019, 75(1):301-309.
[6] GOLDSTEIN D, HANDER R, SIROVICH L. Modeling a no-slip flow boundary with an external force field[J]. Journal of Computational Physics, 1993, 105(2):354-366.
[7] SAIKI E M, BIRINGEN S. Numerical simulation of a cylinder in uniform flow:Application of a virtual boundary method[J]. Journal of Computational Physics, 1996, 123(2):450-465.
[8] 李秋实, 徐飞, 李志平. 一种包含运动边界的高精度流场数值计算方法[J]. 航空学报, 2014, 35(7):1815-1824. LI Q S, XU F, LI Z P. A numerical method for simulating flow involving moving boundaries with high order accuracy[J]. Acta Aeronautica et Astronautica Sinica, 2014, 35(7):1815-1824(in Chinese).
[9] WANG Z, DU L, ZHAO J S, et al. Structural response and energy extraction of a fully passive flapping foil[J]. Journal of Fluids and Structures, 2017, 72:96-113.
[10] LI X J, ZHAO R G, ZHONG C W. Novel immersed boundary-lattice Boltzmann method based on feedback law[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2012, 29(2):179-186.
[11] HUANG W X, SHIN S J, SUNG H J. Simulation of flexible filaments in a uniform flow by the immersed boundary method[J]. Journal of Computational Physics, 2007, 226(2):2206-2228.
[12] SHOELE K, ZHU Q. Leading edge strengthening and the propulsion performance of flexible ray fins[J]. Journal of Fluid Mechanics, 2012, 693:402-432.
[13] GONG C L, HAN J K, YUAN Z J, et al. Numerical investigation of the effects of different parameters on the thrust performance of three dimensional flapping wings[J]. Aerospace Science and Technology, 2018,84:431-445.
[14] XU S, WANG Z J. An immersed interface method for simulating the interaction of a fluid with moving boundaries[J]. Journal of Computational Physics, 2006, 216(2):454-493.
[15] HAN J K, ZHANG Y, CHEN G. Effects of individual horizontal distance on the three-dimensional bionic flapping multi-wings in different schooling configurations[J]. Physics of Fluids, 2019, 31(4):1903-1919.
[16] 胡国暾, 杜林, 孙晓峰. 基于浸入式边界法的叶栅颤振数值模拟[J]. 航空学报, 2015, 36(7):2269-2278. HU G D, DU L, SUN X F. Numerical simulation of an oscillating cascade based on immersed boundary method[J]. Acta Aeronautica et Astronautica Sinica, 2015, 36(7):2269-2278(in Chinese).
[17] ZHONG G H, DU L, SUN X F. Numerical investigation of an oscillating airfoil using immersed boundary method[J]. Journal of Thermal Science, 2011, 20(5):413-422.
[18] DU L, SUN X F, VIGOR Y. Generation of vortex lift through reduction of rotor/stator gap in turbo machinery[J]. Journal of Propulsion and Power, 2015, 32(2):1-14.
[19] WANG L, CURRAO G, HAN F, et al. An immersed boundary method for fluid-structure interaction with compressible multiphase flows[J]. Journal of Computational Physics, 2017, 346:131-151.
[20] GUO X X, YAO J K, ZHONG C W, et al. A hybrid adaptive-gridding immersed-boundary lattice Boltzmann method for viscous flow simulations[J]. Applied Mathematics and Computation, 2015, 267(9):529-553.
[21] YANG X L, ZHANG X, LI Z L, et al. A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations[J]. Journal of Computational Physics, 2009, 228(20):7821-7836.
[22] LEE C. Stability characteristics of the virtual boundary method in three-dimensional applications[J]. Journal of Computational Physics, 2003, 184(2):559-591.
[23] SHIN S J, HUANG W X, SUNG H J. Assessment of regularized delta functions and feedback forcing schemes for an immersed boundary method[J]. International Journal for Numerical Methods in Fluids, 2008, 58(3):263-286.
[24] ANSYS FLUENT. Release 14.0, theory guide[M]. Ansys Inc, 2011.
[25] 袁瑞峰. 气体动理论格式的浸入边界法研究[D].西安:西北工业大学, 2015. YUAN R F. The immersed-boundary method for gas-kinetic scheme[D]. Xi'an:Northwestern Polytechnical University, 2015(in Chinese).
[26] HAMA R, MELDI M, FAVIER J, et al. A pressure-corrected immersed boundary method for the numerical simulation of compressible flows[J]. Journal of Computational Physics, 2018, 374:361-383.
[27] WANG S Z, ZHANG X. An immersed boundary method based on discrete stream function formulation for two- and three-dimensional incompressible flows[J]. Journal of Computational Physics, 2011, 230(9):3479-3499.
[28] WU J, SHU C. Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications[J].Journal of Computational Physics, 2010, 228(6):1963-1979.
[29] LEE D S, HA M Y, KIM S J, et al. Application of immersed boundary method for flow over stationary and oscillating cylinders[J]. Journal of Mechanical Science and Technology, 2006, 20(6):849-863.
[30] DÜTSCH H, DURST F, BECKER S, et al. Low-Reynolds-number flow around an oscillating circular cylinder at low Keulegan-Carpenter numbers[J]. Journal of Fluid Mechanics, 1998, 360:249-271.
[31] WANG Z J. Computation of insect hovering[J]. Mathematical Methods in the Applied Sciences, 2001, 24:1515-1521.
[32] JOHNSON T A, PATEL V C. Flow past a sphere up to a Reynolds number of 300[J]. Journal of Fluid Mechanics, 1999, 378:19-70.
[33] WANG Y, SHU C, YANG L M, et al. An immersed boundary-lattice Boltzmann flux solver in a moving frame to study three-dimensional freely falling rigid bodies[J]. Journal of Fluids and Structures, 2017, 68:444-465.
[34] WANG Y, SHU C, TEO C J, et al. An efficient immersed boundary-lattice Boltzmann flux solver for simulation of 3D incompressible flows with complex geometry[J]. Computers & Fluids, 2016, 124:54-66.
CJA