Low-dimensional dynamics of near-wall turbulence

Wall-bounded turbulent shear flows are perhaps the last area in ‘classical’ incompressible turbulence in which there still are open questions about basic physical mechanisms. There are two competing conceptual models. In the first one, wall turbulence is just a modification of ordinary shear turbulence occuring when the latter is a near wall, and is therefore dependent on the prior existence of an outside turbulent flow. In the second, it is an essentially different phenomenon which coexists with the outer flow and merges into it when the distance from the wall is large enough. Jiménez (1999) took the latter view and argued that the dynamics of near-wall turbulence is essentially different from the Kolmogorov (1941) mechanism. While the latter is fundamentally isotropic and the energy is dissipated locally by cascading to smaller length scales, wall-bounded flows are intrinsically inhomogeneous and anisotropic, and a substantial part of their energy diffuses from the wall into the outer flow, increasing, rather than decreasing, its length scales in the process. In this paper we examine the dynamics of the structures of the viscous and buffer layers in very simplified situations in which their interaction with the outer flow is severely restricted. Even natural flows scale in this region approximately in wall units, defined in terms of the kinematic viscosity ν and of the friction velocity uτ = (ν∂yU), where U is the mean velocity profile. In that approximation, and if we admit that near-wall turbulence is not a just a modification by the wall of the outside turbulent flow, only local quantities such as dimensionless distance to the wall, y = uτy/ν, should matter, while global parameters such as the Reynolds number of the flow should be irrelevant. In the reduced systems considered here, the outer flow is effectively removed, and the scaling in wall variables should be strict. No bulk Reynolds number may be relevant because no bulk turbulent flow exists. Since the local wall-normal Reynolds numbers are low, we may expect quasi-laminar structures whose behavior can be understood deterministically. In this sense this region corresponds to the Kolmogorov viscous range of isotropic turbulence, but we will see that, while the latter is a sink for the energy cascading from the larger scales, the structures studied here are not only self-sustaining, but actually export energy to the rest of the flow. The organization of this paper is as follows. The energy balance is briefly examined in the next section. The wall region and the numerical experiments undertaken to isolated it are described next. The results are then discussed, with emphasis on the low-dimensional behavior of the structures in the simplified flows and on how they evolve into a fully turbulent flow once the constraints are removed.