A critical-layer framework for turbulent pipe flow

By B. McKeon, Ati S. Sharma. Journal of Fluid Mechanics, 2010

660 citations recorded by the source corpus.

A model-based description of the scaling and radial location of turbulent fluctuations in turbulent pipe flow is presented and used to illuminate the scaling behaviour of the very large scale motions. The model is derived by treating the nonlinearity in the perturbation equation (involving the Reynolds stress) as an unknown forcing, yielding a linear relationship between the velocity field response and this nonlinearity. We do not assume small perturbations. We examine propagating helical velocity response modes that are harmonic in the wall-parallel directions and in time, permitting comparison of our results to experimental data. The steady component of the velocity field that varies only in the wall-normal direction is identified as the turbulent mean profile. A singular value decomposition of the resolvent identifies the forcing shape that will lead to the largest velocity response at a given wavenumber–frequency combination. The hypothesis that these forcing shapes lead to response modes that will be dominant in turbulent pipe flow is tested by using physical arguments to constrain the range of wavenumbers and frequencies to those actually observed in experiments. An investigation of the most amplified velocity response at a given wavenumber–frequency combination reveals critical-layer-like behaviour reminiscent of the neutrally stable solutions of the Orr–Sommerfeld equation in linearly unstable flow. Two distinct regions in the flow where the influence of viscosity becomes important can be identified, namely wall layers that scale with R+1/2 and critical layers where the propagation velocity is equal to the local mean velocity, one of which scales with R+2/3 in pipe flow. This framework appears to be consistent with several scaling results in wall turbulence and reveals a mechanism by which the effects of viscosity can extend well beyond the immediate vicinity of the wall. The model reproduces inner scaling of the small scales near the wall and an approach to outer scaling in the flow interior. We use our analysis to make a first prediction that the appropriate scaling velocity for the very large scale motions is the centreline velocity, and show that this is in agreement with experimental results. Lastly, we interpret the wall modes as the motion required to meet the wall boundary condition, identifying the interaction between the critical and wall modes as a potential origin for an interaction between the large and small scales that has been observed in recent literature as an amplitude modulation of the near-wall turbulence by the very large scales.

Subfields: Computational Mechanics, Environmental Engineering, Global and Planetary Change

View Paper: https://www.semanticscholar.org/paper/6709f3b29775e792c21d2061d4a8f63993885d8b

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Citations in the corpus

Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability (2017, positive): “This viewpoint has been adopted in a number of studies exploiting resolvent analysis (e.g. Farrell & Ioannou 1993; McKeon & Sharma 2010; Towne et al. 2015), and is supported by previous studies comparing parabolized stability equation models for the coherent wavepackets in jets to microphone and…”
High–Reynolds Number Wall Turbulence (2011, background): “Conditional averaging techniques reveal that the signature of the very large scales takes a form resembling large streamwise roll cells, in agreement with the predictions of the most amplified structures from various linear theories (del Álamo & Jiménez 2006, McKeon & Sharma 2010).”
Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels (2013, methodology): “We follow the development of McKeon & Sharma (2010) for turbulent pipe flow, showing that equivalent results are obtained for channels and highlighting the new observations.”
Stability, Receptivity, and Sensitivity Analyses of Buffeting Transonic Flow over a Profile (2015, methodology): “A similar approach has already been used to describe pseudoresonances in a channel-flow configuration [44], turbulent pipe flow [45], Blasius boundary layer [46], and jets [47].”
The engine behind (wall) turbulence: perspectives on scale interactions (2017, methodology): “(3.14) The pressure fluctuations are then recovered via the Poisson equation for pressure, ∇2pk =−2ikxvk +∇ · f k. (3.15) McKeon & Sharma (2010) considered a divergence-free set of basis functions that meet the wall boundary conditions (Meseguer & Trefethen 2003) and thus precluded 817 P1-20 the…”
On coherent structure in wall turbulence (2013, methodology): “This provides a missing piece of the previous work of McKeon & Sharma (2010).”
Coherent structures in wall-bounded turbulence (2017, background): “This resonance is both a linear result (McKeon & Sharma 2010), and an approximate nonlinear one (Hall & Sherwin 2010).”
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis (2017, methodology): “Since the bases are complete, the output can be expanded as ŷ(x, f) = ∞ ∑ j=1 ûj(x, f)σj(f)βj(f), (5.15) where βj(f) = 〈η̂(x, f), v̂j(x, f)〉η (5.16) is the projection of the forcing onto the j-th input mode (McKeon & Sharma, 2010).”
Colour of turbulence (2016, methodology): “…17(d-f), the normalized energy density peaks in the vicinity of the wall-normal location where the turbulent mean velocity, marked by the thick black lines, equals the wave speed; McKeon & Sharma (2010) argued that the emergence of critical layers is because the resolvent norm peaks for c ≈ U(y).”
Conditions for validity of mean flow stability analysis (2016, neutral): “McKeon & Sharma (2010) showed that the first optimal response computed from an SVD around the mean flow dominates the velocity field of the full flow field when the first singular value is significantly larger than all others.”
The lift-up effect : The linear mechanism behind transition and turbulence in shear flows (2014, methodology): “A similar linear model-based description of the scaling and location of turbulent fluctuations in turbulent pipe flow is presented in McKeon and Sharma (2010) and used to understand the behaviour of the very large scale motions.”