At sufficiently high Reynolds numbers, shear-flow turbulence close to a wall acquires universal properties. When length and velocity are rescaled by appropriate characteristic scales of the turbulent flow and thereby measured in \emph{inner units}, the statistical properties of the flow become independent of the Reynolds number. We demonstrate the existence of a wall-attached non-chaotic exact invariant solution of the fully nonlinear 3D Navier-Stokes equations for a parallel boundary layer that captures the characteristic self-similar scaling of near-wall turbulent structures. The branch of travelling wave solutions can be followed up to $Re=1,000,000$. Combined theoretical and numerical evidence suggests that the solution is asymptotically self-similar and exactly scales in inner units for Reynolds numbers tending to infinity. Demonstrating the existence of invariant solutions that capture the self-similar scaling properties of turbulence in the near-wall region is a step towards extending the dynamical systems approach to turbulence from the transitional regime to fully developed boundary layers.

中文翻译：

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渐近高雷诺数下湍流边界层近壁区域的自相似不变解

在足够高的雷诺数下，靠近壁的剪切流湍流获得通用特性。当长度和速度通过湍流的适当特征尺度重新缩放，从而以\emph {内部单位} 测量时，流动的统计特性变得独立于雷诺数。我们证明了平行边界层的完全非线性 3D Navier-Stokes 方程的贴壁非混沌精确不变解的存在，该方程捕获了近壁湍流结构的特征自相似标度。行波解的分支可以跟进到 $Re=1,000,000$。结合理论和数值证据表明，对于趋于无穷大的雷诺数，该解是渐近自相似的，并且在内部单位中精确缩放。