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

Brandon Montemuro, Christopher White+2 more|arXiv (Cornell University)|Jun 29, 2020

Fluid Dynamics and Turbulent FlowsEngineering55 references16 citations

TL;DR

This paper proposes a multiscale self-sustaining process (SSP) theory that explains the formation and maintenance of uniform momentum zones (UMZs) and internal shear layers (vortical fissions, VFs) in high Reynolds number wall-bounded shear flows. Using large-Re asymptotic analysis, it demonstrates that strong, large-scale streamwise rolls can differentially homogenize the background shear, sustaining inviscid UMZs and VFs through a three-dimensional instability of embedded shear layers, with the VF thickness scaling as ∆f/h ∼ Reτ^−7/16, consistent with empirical data.

ABSTRACT

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.

Motivation & Objective

• To explain the emergence of uniform momentum zones (UMZs) and internal shear layers (vortical fissions, VFs) in high Reynolds number turbulent wall flows, which are observed as staircase-like streamwise velocity profiles.
• To develop a mechanism that respects the leading-order balance of mean dynamics in the inertial domain, where viscous forces are negligible, and thus cannot be explained by viscous exact coherent states (ECS).
• To derive a self-sustaining process (SSP) that supports exact coherent states (ECS) exhibiting UMZs and VFs, with the rolls being strong enough to homogenize the background shear and sustain the structure.
• To resolve the paradox that viscous ECS cannot account for the singular, inviscid nature of the inertial domain, where the flow self-organizes into spatially segregated UMZs and VFs.
• To provide a mechanistic explanation rooted in the Navier–Stokes equations that explains the observed scaling of VF thickness and spacing with Reynolds number.

Proposed method

• Performs a large-Reynolds-number asymptotic analysis to derive a multiscale self-sustaining process (SSP) for high-Re shear flows, focusing on the inertial domain where viscous forces are negligible.
• Uses a WKBJ-type ansatz for the instability mode, with a caustic structure at the critical layer (CL), and derives the leading-order behavior using Airy functions to describe the mode amplitude near the caustic.
• Solves the eigenvalue problem for the fluctuation fields within the vortical fission (VF) using a Chebyshev collocation method on a transformed domain, treating the total horizontal wavenumber as an eigenvalue.
• Derives amplitude equations for the roll motion via asymptotic matching and energy budget analysis, incorporating the effect of the caustic on mode reflection and decay.
• Employs a pseudospectral Fourier–Chebyshev scheme to solve the mean flow equation (3.11) with symmetry and periodic boundary conditions, and computes the homogenized roll vorticity ¯Ωc via numerical quadrature.
• Uses coordinate transformations (3.29)–(3.31) to reconstruct the fluctuation fields in the original, non-rotated frame of reference.

Experimental results

Research questions

• RQ1 How can the formation and sustained maintenance of uniform momentum zones (UMZs) and internal shear layers (VFs) be explained in the inertial domain of high-Reynolds-number wall-bounded shear flows?
• RQ2 What mechanism allows large-scale, effectively inviscid streamwise roll motions to self-consistently sustain the spatially distributed structure of UMZs and VFs?
• RQ3 How does the three-dimensional, short-wavelength instability of embedded shear layers lead to the generation and feedback of large-scale roll motions that differentially homogenize the background shear?
• RQ4 What is the asymptotic scaling of the vortical fission (VF) thickness with Reynolds number, and how does it compare to empirical observations?
• RQ5 Can a self-sustaining process (SSP) be derived directly from the Navier–Stokes equations that supports exact coherent states (ECS) with UMZ and VF structure, without relying on viscous effects?

Key findings

• The theory identifies a self-sustaining process (SSP) in which strong, large-scale streamwise rolls differentially homogenize the background shear, thereby sustaining the formation and persistence of uniform momentum zones (UMZs) and interlaced internal shear layers (VFs).
• The vortical fission (VF) thickness scales as ∆f/h = O(Reτ^−7/16), which is in reasonable agreement with empirical data showing ∆f/h ∼ Reτ^−1/2.
• The instability mode is reflected at the caustic with a π/2 phase shift, and the amplitude is asymptotically large near the caustic, indicating a critical role in energy transfer and mode confinement.
• The energy-budget integral shows that the contribution to the critical vorticity ¯Ωc from the caustic region is significant, with the jump condition across the caustic scaling as O(˜∆^1/3), which is dominant over shadow zone contributions.
• The geometric mean spacing of VFs in the inertial domain scales as ⟨ly⟩ = Reτ^−1/4 h, and the ratio ∆f/ly = O(Reτ^−1/4), consistent with the observed scaling of VF thickness.
• The proposed SSP theory is consistent with the observed logarithmic growth of the number of UMZs with Reτ and the O(uτ) jump in streamwise velocity across each VF, as confirmed by conditional averaging in experiments.