A robust control framework is used to investigate a streamwise constant projection of the Navier Stokes equations for plane Couette flow. Study of this streamwise constant model is motivated by both numerical and experimental observations that suggest the prevalence and importance of streamwise and quasi-streamwise elongated structures. Small-amplitude Gaussian noise forcing is applied to a two-dimensional, three-velocity component (2D/3C) model to describe its response in the presence of disturbances, uncertainty and modeling errors. A comparison of the results with Direct Numerical Simulation (DNS) datauddemonstrates that the simulations capture salient features of fully developed turbulence. In particular, the change in mean velocity profile from the nominal laminar to the characteristic “S” shaped turbulent profile. The application of Taylor’s hypothesis shows that theudmodel can also reproduce downstream information in the form of large-scale coherence resembling numerically and experimentally observed flow features. The 2D/3C model is able to generate “turbulent-like” behavior under small-amplitude stochastic noise. The laminar flow solution is globally stable, therefore transition to turbulence in this model is likely a consequence of the laminar flow solution’s lack of robustness in the presence of disturbances and uncertainty. In fact, large disturbance amplification is common in both this model and the linearized Navier Stokes equations.ududPeriodic spanwise/wall-normal (z–y) plane stream functions are used as input to develop a forced 2D/3C streamwise velocity equation. The resulting steady-state solution is qualitatively similar to a fully turbulent spatial field of DNS data. Both numerical methods and a perturbation analysis confirm that the momentum transfer that produces a “turbulent-like” mean profile requires a nonlinear streamwise velocity equation.ududA system theoretic approach is used to study the amplification mechanisms that develop through the 2D/3C nonlinear coupling in the streamwise velocity equation. Theudspanwise/wall-normal plane forcing required to produce each stream function is computedand used to define an induced norm from this forcing input to the streamwise velocity. This input-output response is used to determine the energy optimal spanwise wavelength (i.e.,the preferential spacing) over a range of Reynolds numbers and forcing amplitudes.
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机译：鲁棒的控制框架用于研究平面Couette流的Navier Stokes方程的流向恒定投影。数值和实验观察都推动了对这种水流常数模型的研究，这些数据表明了水流和准水流拉长结构的普遍性和重要性。将小振幅高斯噪声强迫应用于二维三速度分量（2D / 3C）模型，以描述其在存在干扰，不确定性和建模误差的情况下的响应。将结果与直接数值模拟（DNS）数据进行比较表明，模拟捕获了完全形成的湍流的显着特征。特别是平均速度曲线从标称层流到特征“ S”形湍流曲线的变化。泰勒假说的应用表明， udmodel还可以以类似于数字和实验观察到的流动特征的大规模相干形式来再现下游信息。 2D / 3C模型能够在小幅度随机噪声下生成“类似湍流”的行为。层流解决方案是全局稳定的，因此，在存在扰动和不确定性的情况下，层流解决方案缺乏鲁棒性的结果可能是此模型中向湍流的过渡。实际上，在此模型和线性化Navier Stokes方程中，大的扰动放大都是常见的。 ud ud周期性的翼展方向/壁法线（z-y）平面流函数用作输入，以开发强制2D / 3C流向速度方程。所得的稳态解决方案在质量上类似于DNS数据的完全湍动的空间场。数值方法和扰动分析都证实，产生“湍流状”平均轮廓的动量传递需要非线性的气流速度方程。 ud udA系统理论方法用于研究通过2D / 3C形成的放大机制流速度方程中的非线性耦合。计算产生每个流函数所需的 u跨度/壁法向平面强迫，并将其用于定义从该强迫输入到流向速度的诱导范数。该输入-输出响应用于确定在雷诺数和强制振幅范围内的能量最佳翼展方向波长（即优先间距）。
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