Simon. J. Illingworth

The linearised Navier–Stokes (LNS) equations are employed in channel flow to study linear amplification mechanisms from an input–output point of view. We consider two models: the full LNS system and a simplified model. In the simplified model, we retain only the forcing pathways that pertain to the shear-driven lift-up mechanism. This approach enables individual analysis of the Orr–Sommerfeld and Squire operators as subsystems, revealing the relationship between linear amplification and the linear mechanisms from which it arises. We examine wavenumber regions corresponding to streamwise streaks, oblique waves and Tollmien–Schlichting (TS) waves, linking the underlying mechanisms back to the LNS equations. Analysis is performed for laminar Poiseuille flow, laminar Couette flow and turbulent Poiseuille flow using an eddy-viscosity model. Results indicate that the Orr–Sommerfeld system amplifies regions for streamwise streaks, oblique waves and TS waves, whereas the Squire system only amplifies streamwise streaks and oblique waves. Leading modal structures between the full and simplified models are also compared.

We use Navier–Stokes-based linear models for wall-bounded turbulent flows to estimate large-scale fluctuations at different wall-normal locations from their measurements at a single wall-normal location. In these models, we replace the nonlinear term by a combination of a stochastic forcing term and an eddy dissipation term. The stochastic forcing term plays a role in energy production by the large scales, and the eddy dissipation term plays a role in energy dissipation by the small scales. Based on the results in channel flow, we find that the models can estimate large-scale fluctuations with reasonable accuracy only when the stochastic forcing and eddy dissipation terms vary with wall distance and with the length scale of the fluctuations to be estimated. The dependence on the wall distance ensures that energy production and energy dissipation are not concentrated close to the wall but are evenly distributed across the near-wall and logarithmic regions. The dependence on the length scale of the fluctuations ensures that lower wavelength fluctuations are not excessively damped by the eddy dissipation term and hence that the dominant scales shift towards lower wavelengths towards the wall. This highlights that, on the one hand, energy extraction in wall turbulence is predominantly linear and thus physics-based linear models give reasonably accurate results. On the other hand, the absence of linearly unstable modes in wall turbulence means that the nonlinear term still plays an essential role in energy extraction and thus the modelled terms should include the observed wall distance and length scale dependencies of the nonlinear term.

We analyse the inter-scale transfer of energy for two types of plane Poiseuille flow: the P4U exact coherent state of Park & Graham (J. Fluid Mech., vol. 782, 2015, pp. 430–454) and turbulent flow in a minimal channel. For both flows, the dominant energy-producing modes are streamwise-constant streaks with a spanwise spacing of approximately 100 wall units. Since the viscous dissipation for these scales is not sufficient to balance production, the nonlinear terms redistribute the excess energy to other scales. Spanwise-constant scales (that is, Tollmien–Schlichting-like modes with zero spanwise wavenumber), in particular, account for a significant amount of net energy gain from the nonlinear terms. We compare the energy balance to predictions from resolvent analysis, and we show that it does not model energy transfer well. Nevertheless, we find that the energy transferred from the streamwise-constant streaks can be predicted reasonably well by a Cess eddy viscosity profile. As such, eddy viscosity is an effective model for the nonlinear terms in resolvent analysis and explains good predictions for the most energetic streamwise-constant streaks. It also improves resolvent modes as a basis for structures whose streamwise lengths are greater than their spanwise widths by counteracting non-normality of the resolvent operator. This is quantified by computing the inner product between the optimal resolvent forcing and response modes, which is a metric of non-normality. Eddy viscosity does not respect the conservative nature of the nonlinear energy transfer, which must sum to zero over all scales. Since eddy viscosity tends to remove energy, it is less effective in modelling nonlinear transport for scales that receive energy from the nonlinear terms.

Pulsatile rough-wall turbulent pipe flow is compared against its non-pulsatile counterpart using data obtained from direct numerical simulation. Results are presented at a mean friction Reynolds number of 540 for a set of three geometrically scaled roughness topographies at a single forcing condition, which, based on existing classifications, falls into the current-dominated very-high-frequency regime. By comparing the pulsatile data against an equivalent non-pulsatile dataset (Chan et al., J. Fluid Mech., vol. 854, 2018, pp. 5–33), the key differences (and similarities) between the forced and unforced configurations are identified. A major finding of this study is that the flow in the outer region retains its self-similar functional form under pulsatile rough-wall conditions, and, as a result, Townsend’s outer-layer similarity hypothesis holds for the roughness-forcing combinations considered here. On the other hand, the unsteady cases exhibit a rich array of flow physics in the region beneath the roughness crests not observed in the steady case. These differences are examined using a Moody chart, which encapsulates how the hydraulic properties of pulsatile rough-wall pipe flow differ from their non-pulsatile counterpart.

The linear amplification mechanisms leading to streamwise-constant large-scale structures in laminar and turbulent channel flows are considered. A key feature of the analysis is that the Orr–Sommerfeld and Squire operators are each considered separately. Physically, this corresponds to considering two separate processes: (i) the response of wall-normal velocity fluctuations to external forcing; and (ii) the response of streamwise velocity fluctuations to wall-normal velocity fluctuations. The analysis is performed for both plane Couette flow and plane Poiseuille flow; and for each we consider linear amplification mechanisms about both the laminar and turbulent mean velocity profiles. The analysis reveals two things. First, that the most amplified structures (with a spanwise spacing of approximately is the channel half-height) are to an important degree encoded in the Orr–Sommerfeld operator alone, thus helping to explain their prevalence. Second – and consistent with numerical and experimental observations – that Couette flow is significantly more efficient than Poiseuille flow in leveraging the mean shear to produce channel-wide streamwise streaks.