Shear stress-driven flow: the state space of near-wall turbulence as

Abstract

An inner-scaled, shear stress-driven flow is considered as a model of independent near-wall turbulence as the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}\rightarrow \infty$. In this limit, the model is applicable to the near-wall region and the lower part of the logarithmic layer of various parallel shear flows, including turbulent Couette flow, Poiseuille flow and Hagen–Poiseuille flow. The model is validated against damped Couette flow and there is excellent agreement between the velocity statistics and spectra for the wall-normal height $y^{+}<40$. A near-wall flow domain of similar size to the minimal unit is analysed from a dynamical systems perspective. The edge and fifteen invariant solutions are computed, the first discovered for this flow configuration. Through continuation in the spanwise width $L_{z}^{+}$, the bifurcation behaviour of the solutions over the domain size is investigated. The physical properties of the solutions are explored through phase portraits, including the energy input and dissipation plane, and streak, roll and wave energy space. Finally, a Reynolds number is defined in outer units and the high-$Re$ asymptotic behaviour of the equilibria is studied. Three lower branch solutions are found to scale consistently with vortex–wave interaction (VWI) theory, with wave forcing localising around the critical layer.