哈特曼边界层的初级稳定性分析

导电流体在法向外置磁场的作用下,在贴近壁面处会形成哈特曼边界层.哈特曼边界层的稳定性研究对电磁冶金过程和热核聚变反应冷却系统等相关设备的设计和运行都有着十分重要的意义.本文采用非正则模态稳定性分析方法,对两无限大绝缘平行平板内导电流体流动的稳定性进行了研究.通过在时间上迭代求解扰动变量的控制方程组和伴随控制方程组,获得了在磁场作用下初级扰动的增长情况及其空间分布形式,分析了磁场强度对最优扰动增长倍数G_max、最优展向波数β_opt和最优时刻t_opt的影响,并考察了上下两个哈特曼边界层之间的相互作用.结果表明,最优初始扰动的空间分布形式为沿着流场方向的漩涡,关于法向方向对称或者反对称.当哈特曼数Ha较大时(Ha>10),对称漩涡和反对称漩涡形式的初始扰动增长倍数基本相等;上下两个哈特曼边界层可以认为是彼此独立的,不会相互影响,此时最优扰动增长倍数G_max与局部雷诺数R的平方成正比,相应的最优展向波数β_opt和最优时刻t_opt均正比于哈特曼数Ha.当哈特曼数Ha较小时(Ha<10),反对称漩涡形式的初始扰动更为不稳定,其增长倍数大于对称漩涡的增长倍数,且上下两个边界层之间存在着一定的相互作用,并对整个流场的稳定性产生一定的影响. 

THE PRIMARY STABILITY ANALYSIS OF HARTMANN BOUNDARY LAYER

A thin Hartmann boundary layer will be formed near the wall when the conducting fluid is passing through a vertical magnetic field.The property of Hartmann boundary layers is vital to design and operation of equipments in electromagnetic metallurgy and thermonuclear fusion cooling system.This stability problem is investigated by non-modal stability analysis method.Through solving the governing equations of disturbances and adjoint field variables iteratively, the amplification and spatial distributions of primary perturbations are obtained.The effects of magnetic field on the optimal perturbation amplification G_max, spanwise wavenumber β_opt and time t_opt are analyzed, and the interaction between two opposite Hartmann boundary layers is considered as well.Results indicate that the optimal initial perturbations are in the form of streamwise vortices, which is symmetric or antisymmetric with respect of the normal direction.When the Hartmann number Ha is larger(Ha>10), the initial perturbations of symmetric and antisymmetric vortices are amplified equally, and the two opposite Hartmann boundary layers can be considered as independent from each other.In this case, the dependence of optimal perturbation amplification G_max on the square of local Reynolds number R is obtained, and the corresponding optimal spanwise wavenumber β_opt and time t_opt are proportional to the Hartmann number Ha.When the Hartmann number Ha is smaller(Ha<10), the antisymmetric vortices are more unstable, and the perturbation amplifications G_max is larger than that of symmetric vortices.There is still a kind of interaction between the two opposite Hartmann boundary layers to influence the stability of the flow field.