Beaume Cédric, Chini Gregory P, Julien Keith, Knobloch Edgar
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK.
Department of Mechanical Engineering and Program in Integrated Applied Mathematics, University of New Hampshire, Durham, New Hampshire 03824, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Apr;91(4):043010. doi: 10.1103/PhysRevE.91.043010. Epub 2015 Apr 15.
A reduced description of exact coherent structures in the transition regime of plane parallel shear flows is developed, based on the Reynolds number scaling of streamwise-averaged (mean) and streamwise-varying (fluctuation) velocities observed in numerical simulations. The resulting system is characterized by an effective unit Reynolds number mean equation coupled to linear equations for the fluctuations, regularized by formally higher-order diffusion. Stationary coherent states are computed by solving the resulting equations simultaneously using a robust numerical algorithm developed for this purpose. The algorithm determines self-consistently the amplitude of the fluctuations for which the associated mean flow is just such that the fluctuations neither grow nor decay. The procedure is used to compute exact coherent states of a flow introduced by Drazin and Reid [Hydrodynamic Stability (Cambridge University Press, Cambridge, UK, 1981)] and studied by Waleffe [Phys. Fluids 9, 883 (1997)]: a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force. Numerical continuation of the lower-branch states to lower Reynolds numbers reveals the presence of a saddle node; the saddle node allows access to upper-branch states that are, like the lower-branch states, self-consistently described by the reduced equations. Both lower- and upper-branch states are characterized in detail.
基于在数值模拟中观察到的流向平均（平均）速度和流向变化（波动）速度的雷诺数标度，对平面平行剪切流过渡区域中精确相干结构进行了简化描述。所得系统的特征是一个有效的单位雷诺数平均方程与波动的线性方程相耦合，并通过形式上的高阶扩散进行正则化。通过使用为此目的开发的稳健数值算法同时求解所得方程来计算稳态相干态。该算法自洽地确定波动的幅度，对于该幅度，相关的平均流恰好使得波动既不增长也不衰减。该过程用于计算由德拉津和里德[《流体动力稳定性》（剑桥大学出版社，英国剑桥，1981年）]引入并由瓦莱夫[《物理流体》9, 883 (1997)]研究的一种流动的精确相干态：一种线性稳定的平面平行剪切流，限制在静止的无应力壁之间，并由正弦体力驱动。将下分支态数值延拓到较低雷诺数揭示了鞍结的存在；鞍结允许进入上分支态，这些上分支态与下分支态一样，由简化方程自洽地描述。对下分支态和上分支态都进行了详细表征。
