To identify laminar/turbulent transition paths in plane Couette flow, a variational formulation incorporating a restricted nonlinear (RNL) system that retains a single streamwise Fourier mode, is used. Considering the flow geometry originally used by Monokrousos et al. (2011) and Duguet et al. (2013) and the same Reynolds numbers ($Re$), we show that initial perturbations obtained by RNL optimizations exhibit spatial localization. Two optimal states are found with comparable initial energy levels above which the flow structure evolves to turbulence. It is found that this level is twice that of the minimal threshold energy which has been obtained using the full nonlinear equations (Duguet et al. (2013)). Especially, the $Re$ dependence of energy thresholds is studied within a RNL optimization framework for the first time, with evidence for a $O\left(R{e}^{-2.65}\right)$ scaling close to the one found using the full Navier–Stokes equations ($O\left(R{e}^{-2.7}\right)$). The first state is obtained for a short target time. It is symmetric with respect to the mid-plane $y=0$ and spanwise localized. For a long target time, the optimal appears to be localized in both spanwise and wall-normal directions. The mechanisms highlighted within the scope of nonlinear nonmodal theory (Kerswell (2018)): Orr mechanism, oblique wave interaction, lift-up, streak breakdown, localized pocket of turbulence and turbulence spreading, are also observed in the RNL simulations. Although greatly simplified, the RNL system provides a good approximation of these different fundamental mechanisms. The analysis gives then some insight into the potential of RNL optimizations for estimating $Re$ scaling laws and routes to turbulence for shear flows.

为了识别平面Couette流中的层流/湍流过渡路径，使用了包含受限非线性（RNL）系统的变分公式，该系统保留了单个流式傅里叶模式。考虑到Monokrousos等人最初使用的流动几何形状。（2011年）和Duguet等人。（2013）和相同的雷诺数（$\mathrm{[R}Ë$），我们显示了由RNL优化获得的初始扰动表现出空间局限性。发现具有最佳初始能量水平的两个最佳状态，高于该初始能量水平，流动结构演变为湍流。发现该水平是使用完全非线性方程式获得的最小阈值能量的两倍（Duguet等人（2013））。特别是$\mathrm{[R}Ë$ 首次在RNL优化框架内研究了能量阈值的依赖性，并为 $Ø\left(\mathrm{[R}{Ë}^{-2。\mathrm{65岁}}\right)$ 缩放比例接近使用完整的Navier–Stokes方程找到的比例（$Ø\left(\mathrm{[R}{Ë}^{-2。7}\right)$）。在较短的目标时间内获得第一状态。相对于中平面对称$ÿ=0$并沿展向定位。对于较长的目标时间，最佳值似乎位于翼展方向和壁法线方向。在非线性非模态理论（Kerswell（2018））范围内强调的机制：在RNL模拟中也观察到了Orr机制，斜波相互作用，抬升，条纹破坏，湍流局部化和湍流扩散。尽管已大大简化，但RNL系统提供了这些不同基本机制的良好近似。通过分析，可以深入了解RNL优化的潜力，以进行估算$\mathrm{[R}Ë$ 调整剪切流的湍流定律和路线。