Bifurcations in penetrative Rayleigh-Bénard convection in a cylindrical container

doi:10.1007/s10483-019-2474-6

Applied Mathematics and Mechanics (English Edition) ›› 2019, Vol. 40 ›› Issue (5): 695-704.doi: https://doi.org/10.1007/s10483-019-2474-6

Bifurcations in penetrative Rayleigh-Bénard convection in a cylindrical container

Chuanshi SUN, Shuang LIU, Qi WANG, Zhenhua WAN, Dejun SUN

Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China

收稿日期:2018-07-29 修回日期:2018-10-19 出版日期:2019-05-01 发布日期:2019-05-01
通讯作者: Dejun SUN E-mail:dsun@ustc.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 11572314, 11621202, and 11772323) and the Fundamental Research Funds for the Central Universities

Bifurcations in penetrative Rayleigh-Bénard convection in a cylindrical container

Chuanshi SUN, Shuang LIU, Qi WANG, Zhenhua WAN, Dejun SUN

Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China

Received:2018-07-29 Revised:2018-10-19 Online:2019-05-01 Published:2019-05-01
Contact: Dejun SUN E-mail:dsun@ustc.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 11572314, 11621202, and 11772323) and the Fundamental Research Funds for the Central Universities

摘要/Abstract

摘要： The bifurcations of penetrative Rayleigh-Bénard convection in cylindrical containers are studied by the linear stability analysis (LSA) combined with the direct numerical simulation (DNS) method. The working fluid is cold water near 4℃, where the Prandtl number Pr is 11.57, and the aspect ratio (radius/height) of the cylinder ranges from 0.66 to 2. It is found that the critical Rayleigh number increases with the increase in the density inversion parameter θ_m. The relationship between the normalized critical Rayleigh number (Ra_c(θ_m)/Ra_c(0)) and θ_m is formulated, which is in good agreement with the stability results within a large range of θ_m. The aspect ratio has a minor effect on Ra_c(θ_m)/Ra_c(0). The bifurcation processes based on the axisymmetric solutions are also investigated. The results show that the onset of axisymmetric convection occurs through a trans-critical bifurcation due to the top-bottom symmetry breaking of the present system. Moreover, two kinds of qualitatively different steady axisymmetric solutions are identified.

关键词: impact torsional buckling, elastic cylindrical shell, perturbation analysis, initial imperfection sensitivity, bifurcation, linear stability analysis (LSA), convection

Abstract: The bifurcations of penetrative Rayleigh-Bénard convection in cylindrical containers are studied by the linear stability analysis (LSA) combined with the direct numerical simulation (DNS) method. The working fluid is cold water near 4℃, where the Prandtl number Pr is 11.57, and the aspect ratio (radius/height) of the cylinder ranges from 0.66 to 2. It is found that the critical Rayleigh number increases with the increase in the density inversion parameter θ_m. The relationship between the normalized critical Rayleigh number (Ra_c(θ_m)/Ra_c(0)) and θ_m is formulated, which is in good agreement with the stability results within a large range of θ_m. The aspect ratio has a minor effect on Ra_c(θ_m)/Ra_c(0). The bifurcation processes based on the axisymmetric solutions are also investigated. The results show that the onset of axisymmetric convection occurs through a trans-critical bifurcation due to the top-bottom symmetry breaking of the present system. Moreover, two kinds of qualitatively different steady axisymmetric solutions are identified.

Key words: impact torsional buckling, elastic cylindrical shell, perturbation analysis, initial imperfection sensitivity, bifurcation, convection, linear stability analysis (LSA)

中图分类号:

76E20
34C23

Chuanshi SUN, Shuang LIU, Qi WANG, Zhenhua WAN, Dejun SUN. Bifurcations in penetrative Rayleigh-Bénard convection in a cylindrical container[J]. Applied Mathematics and Mechanics (English Edition), 2019, 40(5): 695-704.

参考文献

[1] VERONIS, G. Penetrative convection. Astrophysical Journal, 137, 641-663(1963)
[2] ZHANG, K. and SCHUBERT, G. Penetrative convection and zonal flow on Jupiter. Science, 273, 941-943(1996)
[3] OLSON, P. and AURNOU, J. A polar vortex in the Earth's core. nature, 402, 170-173(1999)
[4] GUBBINS, D., THOMSON, C. J., and WHALER, K. A. Stable regions in the Earth's liquid core. Geophysical Journal International, 68, 241-251(1982)
[5] AHLERS, G., GROSSMANN, S., and LOHSE, D. Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Reviews of Modern Physics, 81, 503-537(2009)
[6] LOHSE, D. and XIA, K. Q. Small-scale properties of turbulent Rayleigh-Bénard convection. Annual Review of Fluid Mechanics, 42, 335-364(2010)
[7] MOORE, D. R. and WEISS, N. O. Nonlinear penetrative convection. Journal of Fluid Mechanics, 61, 553-581(1973)
[8] LARGE, E. and ANDERECK, C. D. Penetrative Rayleigh-Bénard convection in water near its maximum density point. Physics of Fluids, 26, 094101(2014)
[9] HU, Y. P., LI, Y. R., and WU, C. M. Rayleigh-Bénard convection of cold water near its density maximum in a cubical cavity. Physics of Fluids, 27, 034102(2015)
[10] LI, Y. R., OUYANG, Y. Q., and HU, Y. P. Pattern formation of Rayleigh-Bénard convection of cold water near its density maximum in a vertical cylindrical container. Physical Review E, 86, 046323(2012)
[11] LI, Y. R., HU, Y. P., OUYANG, Y. Q., and WU, C. M. Flow state multiplicity in Rayleigh-Bénard convection of cold water with density maximum in a cylinder of aspect ratio 2. International Journal of Heat and Mass Transfer, 86, 244-257(2015)
[12] HU, Y. P., LI, Y. R., and WU, C. M. Aspect ratio dependence of Rayleigh-Bénard convection of cold water near its density maximum in vertical cylindrical containers. International Journal of Heat and Mass Transfer, 97, 932-942(2016)
[13] HUANG, X. J., LI, Y. R., ZHANG, L., and WU, C. M. Turbulent Rayleigh-Bénard convection of cold water near its maximum density in a vertical cylindrical container. International Journal of Heat and Mass Transfer, 116, 185-193(2018)
[14] MA, D. J., SUN, D. J., and YIN, X. Y. Multiplicity of steady states in cylindrical Rayleigh-Bénard convection. Physical Review E, 74, 037302(2006)
[15] ALONSO, A., MERCADER, I., and BATISTE, O. Pattern selection near the onset of convection in binary mixtures in cylindrical cells. Fluid Dynamics Research, 46, 041418(2014)
[16] WANG, B. F., MA, D. J., CHEN, C., and SUN, D. J. Linear stability analysis of cylindrical Rayleigh-Bénard convection. Journal of Fluid Mechanics, 711, 27-39(2012)
[17] GEBHART, B. and MOLLENDORF, J. C. A new density relation for pure and saline water. Deep Sea Research, 24, 831-848(1977)
[18] VERZICCO, R. and ORLANDI, P. A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. Journal of Computational Physics, 123, 402-414(1996)
[19] HUGUES, S. and RANDRIAMAMPIANINA, A. An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 28, 501-521(1998)
[20] KNOLL, D. A. and KEYES, D. E. Jacobian-free Newton-Krylov methods:a survey of approaches and applications. Journal of Computational Physics, 193, 357-397(2004)
[21] DOEDEL, E. and TUCKERMAN, L. S. Numerical Methods for Bifurcation Problems and LargeScale Dynamical Systems, Springer Science and Business Media, New York, 453-466(2012)
[22] VAN DER VORST, H. A. Bi-CGSTAB:a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 13, 631-644(1992)
[23] LEHOUCQ, R. B., SORENSEN, D. C., and YANG, C. ARPACK Users' Guide:Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, Siam, Philadelphia (1998)
[24] WANG, B. F., WAN, Z. H., GUO, Z. W., MA, D. J., and SUN, D. J. Linear instability analysis of convection in a laterally heated cylinder. Journal of Fluid Mechanics, 747, 447-459(2014)

Bifurcations in penetrative Rayleigh-Bénard convection in a cylindrical container

Bifurcations in penetrative Rayleigh-Bénard convection in a cylindrical container

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	C. G. PAVITHRA, B. J. GIREESHA, M. L. KEERTHI. Semi-analytical investigation of heat transfer in a porous convective radiative moving longitudinal fin exposed to magnetic field in the presence of a shape-dependent trihybrid nanofluid[J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(1): 197-216.
[2]	M. HAMID, M. USMAN, Zhenfu TIAN. Computational analysis for fractional characterization of coupled convection-diffusion equations arising in MHD flows[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(4): 669-692.
[3]	B. J. GIREESHA, M. L. KEERTHI. Effect of periodic heat transfer on the transient thermal behavior of a convective-radiative fully wet porous moving trapezoidal fin[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(4): 653-668.
[4]	A. V. ROŞCA, N. C. ROŞCA, I. POP. Three-dimensional mixed convection stagnation-point flow past a vertical surface with second-order slip velocity[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(4): 641-652.
[5]	Zhimin CHEN, W. G. PRICE. Secondary steady-state and time-periodic flows from a basic flow with square array of odd number of vortices[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(3): 447-458.
[6]	Peng WANG, Shaopu YANG, Yongqiang LIU, Pengfei LIU, Xing ZHANG, Yiwei ZHAO. Research on hunting stability and bifurcation characteristics of nonlinear stochastic wheelset system[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(3): 431-446.
[7]	Shuguang LI, M. I. KHAN, F. ALI, S. S. ABDULLAEV, S. SAADAOUI, HABIBULLAH. Mathematical modeling of mixed convective MHD Falkner-Skan squeezed Sutterby multiphase flow with non-Fourier heat flux theory and porosity[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(11): 2005-2018.
[8]	Lele WANG, Liang WANG, Yueting ZHU, Zhanli LIU, Yongtao SUN, Jie WANG, Hongge HAN, Shuyi XIANG, Huibin SHI, Qian DING. Nonlinear dynamic response and stability analysis of the stapes reconstruction in human middle ear[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(10): 1739-1760.
[9]	A. MOSLEMI, M. R. HOMAEINEZHAD. Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(1): 141-158.
[10]	Yaping YAN, Shuang WU, Kaiyuan TIAN, Zhenfu TIAN. Numerical simulation for 2D double-diffusive convection (DDC) in rectangular enclosures based on a high resolution upwind compact streamfunction model I: numerical method and code validation[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(9): 1431-1448.
[11]	Zhaodong DING, Kai TIAN, Yongjun JIAN. Electrokinetic flow and energy conversion in a curved microtube[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(8): 1289-1306.
[12]	Yaode YIN, Demin ZHAO, Jianlin LIU, Zengyao XU. Nonlinear dynamic analysis of dielectric elastomer membrane with electrostriction[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(6): 793-812.
[13]	Lei LI, Hanbiao LIU, Jianxin HAN, Wenming ZHANG. Nonlinear modal coupling in a T-shaped piezoelectric resonator induced by stiffness hardening effect[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(6): 777-792.
[14]	N. A. ZAINAL, R. NAZAR, K. NAGANTHRAN, I. POP. Slip effects on unsteady mixed convection of hybrid nanofluid flow near the stagnation point[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(4): 547-556.
[15]	S. HUSSAIN, T. TAYEBI, T. ARMAGHANI, A. M. RASHAD, H. A. NABWEY. Conjugate natural convection of non-Newtonian hybrid nanofluid in wavy-shaped enclosure[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(3): 447-466.