An Asymptotic Self-Sustaining Process Theory for Uniform Momentum Zones and Internal Interfaces in Unbounded Couette Flow

Meinhart \& Adrian (Phys. Fluids, vol. 7, 1995, p 694) were the first investigators to document that the wall-normal ($y$) structure of the instantaneous streamwise velocity in the turbulent boundary layer exhibits a staircase-like profile: regions of quasi-uniform momentum are separated by internal shear layers across which the streamwise velocity jumps by an O(1) amount when scaled by the friction velocity $u_\tau$. This sharply-varying instantaneous profile differs dramatically from the well-known long-time mean profile, which is logarithmic over much of the boundary layer, and prompted Klewicki (Proc. IUTAM, vol. 9, 2013, p. 69--78) to propose that the turbulent boundary layer is singular in two distinct ways. Firstly, spanwise vorticity and mean viscous forces are concentrated in a near-wall region of thickness $\mathit{O}(h/\sqrt{Re_\tau})$, where $Re_\tau$ is the friction Reynolds number and $h$ is the boundary-layer height. Secondly, in a turbulent boundary layer, spanwise vorticity and viscous forces are also significant away from the wall (outboard of the peak in the Reynolds stress), but \emph{only} in spatially-localized regions, i.e. within the internal shear layers. This interpretation accords with Klewicki\u27s multiscale similarity analysis of the mean momentum balance for turbulent wall flows (J. Fluid Mech., vol. 522, 2005, pp. 303--327). The objective of the present investigation is to probe the governing Navier--Stokes equations in the limit of large $Re_\tau$ in search of a mechanistic self-sustaining process (SSP) that (i) can account for the emergent staircase-like profile of streamwise velocity in the inertial region and (ii) is compatible with the singular nature of turbulent wall flows. Plausible explanations for the formation and persistence of sharply-varying instantaneous streamwise velocity profiles all implicate quasi-coherent turbulent flow structures including streamwise roll motions that induce a cellular flow in the transverse (i.e. spanwise/wall-normal) plane. One proposal is that the large-scale structures result from the spontaneous concatenation of smaller--scale structures, particularly hairpin and cane vortices and vortex packets. A competing possibility, explored here, is that these large--scale motions may be \emph{directly} sustained via an inertial--layer SSP that is broadly similar to the near-wall SSP. The SSP theory derived in this investigation is related to the SSP framework developed by Waleffe (Stud. Appl. Math, vol. 95, 1995, p. 319) and, especially, to the closely-related vortex-wave interaction (VWI) theory derived by Hall \& Smith (J. Fluid Mech., vol. 227, 1991, pp. 641--666) and Hall \& Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178--205) in that a rational asymptotic analysis of the instantaneous Navier--Stokes equations is performed. Nevertheless, in this dissertation, it is argued that these theories cannot account for organized motions in the inertial domain, essentially because the roll motions are predicted to be viscously dominated even at large $Re_\tau$. The target of the present investigation is an inherently multiscale SSP, in which inviscid streamwise rolls differentially homogenize an imposed background shear flow, thereby generating uniform momentum zones and an embedded internal shear layer (or interface), and are sustained by Rayleigh instability modes having asymptotically smaller streamwise and spanwise length scales. The Rayleigh mode is supported by the inflectional wall-normal profile of the streamwise--averaged streamwise velocity. Because the thickness of the internal shear layer varies comparably slowly in the spanwise direction, the Rayleigh mode is refracted and rendered fully three--dimensional. This three--dimensional mode is singular, necessitating the introduction of a critical layer inside the shear layer within which the mode is viscously regularized. As in VWI theory, a jump in the spanwise Reynolds stress is induced across the critical layer, which ultimately drives the roll motions. This multiscale and three--region asymptotic structure is efficiently captured using a complement of matched asymptotic and WKBJ analysis. The resulting reduced equations require the numerical solution of both ordinary differential eigenvalue and partial differential boundary-value problems, for which pseudospectral and spectral collocation methods are employed. Crucially, in contrast to Waleffe\u27s SSP and to VWI theory, the rolls are sufficiently strong to differentially homogenize the background shear flow, thereby providing a plausible mechanistic explanation for the formation and maintenance of both UMZs and interlaced internal shear layers

A self-sustaining process theory for uniform momentum zones and internal shear layers in high Reynolds number shear flows

Many exact coherent states (ECS) arising in wall-bounded shear flows have an asymptotic structure at extreme Reynolds number Re in which the effective Reynolds number governing the streak and roll dynamics is O(1). Consequently, these viscous ECS are not suitable candidates for quasi-coherent structures away from the wall that necessarily are inviscid in the mean. Specifically, viscous ECS cannot account for the singular nature of the inertial domain, where the flow self-organizes into uniform momentum zones (UMZs) separated by internal shear layers and the instantaneous streamwise velocity develops a staircase-like profile. In this investigation, a large-Re asymptotic analysis is performed to explore the potential for a three-dimensional, short streamwise- and spanwise-wavelength instability of the embedded shear layers to sustain a spatially-distributed array of much larger-scale, effectively inviscid streamwise roll motions. In contrast to other self-sustaining process theories, the rolls are sufficiently strong to differentially homogenize the background shear flow, thereby providing a mechanistic explanation for the formation and maintenance of UMZs and interlaced shear layers that respects the leading-order balance structure of the mean dynamics