The unstable transition of conducting fluid caused by Hopf bifurcation under magnetic field has not been studied. In this work, the numerical method SCM-ACM which combines the spectral collocation method and artificial compressibility method developed by our research group, is used to solve the MHD governing equations directly. The Fourier analysis is employed to calculate the spectral distribution of the velocity oscillation. The first Hopf bifurcation of a conducting fluid from a steady state flow to an unsteady periodic oscillatory flow in a three-dimensional square cavity is studied with a range of Hartmann number . The results show that the magnetic field strongly suppresses the velocity oscillation and significantly increases the critical Reynolds number for the first Hopf bifurcation. With the increase of from 0 to 5, the decay velocity of velocity amplitude increases sharply in a parabolic form. The also increases in a parabolic form, from 1916.6 to 2040.1. However, the velocity oscillations only have a unique dominant dimensionless angular frequency (ω=0.5752) under different . The present research method and results on Hopf bifurcation can provide reference for the engineering design and operation control.
[1] Reynolds O. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels [J]. Philosophical Transactions of the Royal Society A, 1883, 174: 935–982.
[2] 刘清扬, 雷娟棉, 刘周, 等. 适用于可压缩流动的γ-Ret-fRe转捩模型[J]. 航空学报, 2022, 43(08): 327-337.
[3] 孔维萱, 阎超, 赵瑞. 壁面温度条件对边界层转捩预测的影响[J]. 航空学报, 2013, 34(10): 2249-2255.
[4] Feldman Y, Gelfgat A Y. Oscillatory instability of a three-dimensional lid-driven flow in a cube [J]. Physics of Fluids, 2010, 22: 093602.
[5] Kuhlmann H C, Albensoeder S. Stability of the steady three-dimensional lid-driven flow in a cube and the supercritical flow dynamics [J]. Physics of Fluids, 2014, 26: 024104.
[6] Cazemier W, Verstappen R W C P, Veldman A E P. Proper orthogonal decomposition and low-dimensional models for driven cavity flows [J]. Physics of Fluids, 1998, 10: 1685–1699.
[7] Auteri F, Parolini N, Quartapelle L. Numerical investigation on the stability of singular driven cavity flow [J]. Journal of Computational Physics, 2002, 183: 1–25.
[8] Peng Y F, Shiau Y H, Hwang R R. Transition in a 2-D lid-driven cavity flow [J]. Computers & Fluids, 2003, 32: 337–352.
[9] Boppana V B L, Gajjar J S B. Global flow instability in a lid-driven cavity [J]. International Journal for Numerical Methods in Fluids, 2010, 62: 827–853.
[10] Brezillon A, Girault G, Cadou J M. A numerical algorithm coupling a bifurcating indicator and a direct method for the computation of Hopf bifurcation points in fluid mechanics [J]. Computers & Fluids, 2010, 39: 1226–1240.
[11] Lin L S, Chang H W, Lin C A. Multi relaxation time lattice Boltzmann simulations of transition in deep 2D lid driven cavity using GPU [J]. Computers & Fluids, 2013, 80: 381–387.
[12] Gorban A N, Packwood D J. Enhancement of the stability of lattice Boltzmann methods by dissipation control [J]. Physica A, 2014, 414: 285–299.
[13] Lestandi L, Bhaumik S, Avatar G R K C, et al. Multiple Hopf bifurcations and flow dynamics inside a 2D singular lid driven cavity [J]. Computers & Fluids, 2018, 166: 86–103.
[14] Suman V K, Viknesh S S, Tekriwal M K, et al. Grid sensitivity and role of error in computing a lid-driven cavity problem [J]. Physical Review E, 2019, 99: 013305.
[15] Wang T, Liu T. Transition to chaos in lid-driven square cavity flow [J]. Chinese Physics B, 2021, 30(12): 120508.
[16] Shen J. Hopf bifurcation of the unsteady regularized driven cavity flow [J]. Journal of Computational Physics, 1991, 95: 228–245.
[17] Liberzon A, Feldman Y, Gelfgat A Y. Experimental observation of the steady oscillatory transition in a cubic lid-driven cavity [J]. Physics of Fluids, 2011, 23: 084106.
[18] Chang H W, Hong P Y, Lin L S. Simulations of flow instability in three dimensional deep cavities with multi relaxation time lattice Boltzmann method on graphic processing units [J]. Computers & Fluids, 2013, 88: 866-871.
[19] Kuhlmann H C, Albensoeder S. Stability of the steady three-dimensional lid-driven flow in a cube and the supercritical flow dynamics [J]. Physics of Fluids, 2014, 26: 024104.
[20] Anupindi K, Lai W, Frankel S. Characterization of oscillatory instability in lid driven cavity flows using lattice Boltzmann method [J]. Computers & Fluids, 2014, 92: 7-21.
[21] Loiseau J C, Robinet J C, Leriche E. Intermittency and transition to chaos in the cubical lid-driven cavity flow [J]. Fluid Dynamics Research, 2016, 48: 061421.
[22] Gelfgat A Y. Linear instability of the lid-driven flow in a cubic cavity [J]. Theoretical and Computational Fluid Dynamics, 2019, 33: 59-82.
[23] Zhang J K, Cui M, Zuo Z L, et al. Prediction on steady-oscillatory transition via Hopf bifurcation in a three-dimensional (3D) lid-driven cube [J]. Computers & Fluids, 2021, 229: 105068.
[24] Hammami F, Souayeh B, Ben-Cheikh N. Computational analysis of fluid flow due to a two-sided lid driven cavity with a circular cylinder [J]. Computers & Fluids, 2017, 156: 317–328.
[25] Zhang J K, Cui M, Li B W, et al. Performance of combined spectral collocation method and artificial compressibility method for 3D incompressible fluid flow and heat transfer [J]. International Journal of Numerical Methods for Heat and Fluid Flow, 2020, 30(12): 5037–5062.
[26] Zhang J K, Dong H, Zhou E Z. A combined method for solving 2D incompressible flow and heat transfer by spectral collocation method and artificial compressibility method [J]. International Journal of Heat and Mass Transfer, 2017, 112: 289–299.
[27] Chorin A J. A numerical method for solving incompressible viscous flow problems [J]. Journal of Computational Physics, 1967, 2: 12–26.
[28] Canuto C, Hussaini M Y, Quarteroni A. Spectral Methods, Fundamentals in Single Domains [M]. New York: Springer-Verlag, 2006: 85–87.
[29] Yu P X, Tian Z F. A high-order compact scheme for the pure streamfunction (vector potential) formulation of the 3D steady incompressible Navier-Stokes equations [J]. Journal of Computational Physics, 2019, 382: 65–85.
[30] Shu C, Wang L, Chew Y T. Numerical computation of three-dimensional incompressible Navier-Stokes equations in primitive variable form by DQ method [J]. International Journal for Numerical Methods in Fluids, 2003, 43: 345–368.
[31] Albensoeder S, Kuhlmann H C. Accurate three-dimensional lid-driven cavity flow [J]. Journal of Computational Physics, 2005, 206: 536–558.
[32] Bhaumik S, Sengupta T K. A new velocity vorticity formulation for direct numerical simulation of 3D transitional and turbulent flows [J]. Journal of Computational Physics, 2015, 284: 230–260.
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