MHD渠道流弱非线性稳定性分析的有效数值方法
Weakly nonlinear stability analysis of MHD channel flow using an efficient numerical approach
摘要：
本文分析了在横向磁场作用下，由压力梯度驱动的粘性导电液体在两平行壁间流动的弱非线性稳定性。利用非标准的数值方法，计算了线性增长率修正和第一朗道系数，它们在足够强的磁场中随哈特曼数的变化分别为$μ_{1}\sim(0.814-\mathrm{i}19.8)\times10^{-3}\textit{Ha}$和$μ_{2}\sim(2.73-\mathrm{i}1.50)\times10^{-5}\textit{Ha}^{-4}$ .这些系数描述了一个亚临界横向速度扰动的平衡振幅$|A|^{2}=\Re[μ_{1}]/\Re[μ_{2}](\textit{Re}_{c}-\textit{Re})\sim29.8\textit{Ha}^{5}(\textit{Re}_{c}-\textit{Re})$，它存在于雷诺数低于线性稳定阈值$\textit{Re}_{c}\sim 4.83\times10^{4}\textit{Ha}.$我们发现，无论磁场强度的流动保持亚临界不稳定。我们的方法计算朗道系数不同于标准的应用程序的可解性条件的离散化，而不是连续的问题。这使我们能够绕过伴随问题的解决方案和随后的积分定义的内积，这导致在一个显着的简化方法的评价。
英文摘要：
We analyze weakly nonlinear stability of a flow of viscous conducting liquid driven by pressure gradient in the channel between two parallel walls subject to a transverse magnetic field. Using a non-standard numerical approach, we compute the linear growth rate correction and the first Landau coefficient, which in a sufficiently strong magnetic field vary with the Hartmann number as $μ_{1}\sim(0.814-\mathrm{i}19.8)\times10^{-3}\textit{Ha}$ and $μ_{2}\sim(2.73-\mathrm{i}1.50)\times10^{-5}\textit{Ha}^{-4}$. These coefficients describe a subcritical transverse velocity perturbation with the equilibrium amplitude $|A|^{2}=\Re[μ_{1}]/\Re[μ_{2}](\textit{Re}_{c}-\textit{Re})\sim29.8\textit{Ha}^{5}(\textit{Re}_{c}-\textit{Re})$ which exists at Reynolds numbers below the linear stability threshold $\textit{Re}_{c}\sim 4.83\times10^{4}\textit{Ha}.$ We find that the flow remains subcritically unstable regardless of the magnetic field strength. Our method for computing Landau coefficients differs from the standard one by the application of the solvability condition to the discretized rather than continuous problem. This allows us to bypass both the solution of the adjoint problem and the subsequent evaluation of the integrals defining the inner products, which results in a significant simplification of the method.
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