论文标题
横向磁场下的导管流中的准两维扰动
Quasi-two dimensional perturbations in duct flows under transverse magnetic field
论文作者
论文摘要
受Moresco \＆Alboussière（2004，J。FluidMech。）的实验的启发，我们研究了液体金属流动在矩形，电绝缘导管中具有稳态均匀均匀均匀磁场的稳定性。洛伦兹力倾向于消除沿磁场的速度变化，从而导致垂直于磁场的墙壁附近的Hartmann边界层的准二维基底流动，并在墙壁附近的墙壁附近平行于该场。由于洛伦兹的力也强烈反对沿磁场方向依赖性的扰动的生长，因此我们用Sommeria \＆Moreau（1982，J。流体机械）模型表示流动，这是一种二维浅水模型，具有线性摩擦的二维浅水模型，用于Hartmann层的效果。 该模型的简单性使得在广泛的参数范围内研究准维度扰动的稳定性和瞬态生长是可能的，直到高磁场的极限，其中雷诺数基于shercliff层厚度$ re re re/h^{1/2} $成为唯一相关的参数。 Tollmien-Schlichting波是最线性不稳定的模式，其进一步不稳定的模式$ H \ GTRSIM 42 $。对于$ re/h^{1/2} \ gtrsim 48350 $，该流量是线性不稳定的，对于$ re/h^{1/2} \ sillesim 65.32 $，稳定稳定。在这两个界限之间，非模式的准二维扰动经历了显着的瞬态增长（比纯粹的2D Poiseuille流动的2到7倍，并且以$ re $的较高临界值的速度为多）。在高磁场的极限下，与此短暂增长相关的最大增益$ g_ {max} $随着$ g_ {max} \ sim（re/re_c）^{2/3} $而变化，并在时间$ t_ {gmax} \ sim（re_c）（re_c）^for流式3. {1/3} $ crateber wavermber的时间$ t_ {gmax} \ sim（gmax} \ sim sim（re_c） 稳定。
Inspired by the experiment from Moresco \& Alboussière (2004, J. Fluid Mech.), we study the stability of a liquid metal flow in a rectangular, electrically insulating duct with a steady homogeneous transverse magnetic field. The Lorentz force tends to eliminate velocity variations along the magnetic field, leading to a quasi-two dimensional base flow with Hartmann boundary layers near the walls perpendicular to the magnetic field, and Shercliff layers near the walls parallel to the field. Since the Lorentz force strongly opposes the growth of perturbations with a dependence along the magnetic field direction too, we represent the flow with Sommeria \& Moreau's (1982, J. Fluid Mech.) model, a two-dimensional shallow water model with linear friction accounting for the effect of the Hartmann layer. The simplicity of this model makes it possible to study the stability and transient growth of quasi-two dimensional perturbations over an extensive range of parameters up to the limit of high magnetic fields, where the Reynolds number based on the Shercliff layer thickness $Re/H^{1/2}$ becomes the only relevant parameter. Tollmien-Schlichting waves are the most linearly unstable mode, with a further unstable mode $H \gtrsim 42$. The flow is linearly unstable for $Re/H^{1/2}\gtrsim 48350$ and energetically stable for $Re/H^{1/2}\lesssim 65.32$. Between these two bounds, non-modal quasi-two dimensional perturbations undergo significant transient growth (between 2 and 7 times more than in the case of a purely 2D Poiseuille flow, and for much more subcritical values of $Re$). In the limit of a high magnetic field, the maximum gain $G_{max}$ associated to this transient growth varies as $G_{max} \sim (Re/Re_c)^{2/3}$ and occurs at time $t_{Gmax}\sim(Re/Re_c)^{1/3}$ for streamwise wavenumbers of the same order of magnitude as the critical wavenumber for the linear stability.
