A generalised quasilinear (GQL) approximation (Marston et al., Phys. Rev. Lett., vol. 116, 2016, 104502) is applied to turbulent channel flow at $Re_\tau \simeq 1700$ ( $Re_\tau$ is the friction Reynolds number), with emphasis on the energy transfer in the streamwise wavenumber space. The flow is decomposed into low- and high-streamwise-wavenumber groups, the former of which is solved by considering the full nonlinear equations whereas the latter is obtained from the linearised equations around the former. The performance of the GQL approximation is subsequently compared with that of a QL model (Thomas et al., Phys. Fluids, vol. 26, 2014, 105112), in which the low-wavenumber group only contains zero streamwise wavenumber. It is found that the QL model exhibits a considerably reduced multi-scale behaviour at the given moderately high Reynolds number. This is improved significantly by the GQL approximation which incorporates only a few more streamwise Fourier modes into the low-wavenumber group, and it reasonably well recovers the distance-from-the-wall scaling in the turbulence statistics and spectra. Finally, it is proposed that the energy transfer from the low- to the high-wavenumber group in the GQL approximation, referred to as the ‘scattering’ mechanism, depends on the neutrally stable leading Lyapunov spectrum of the linearised equations for the high-wavenumber group. In particular, it is shown that if the threshold wavenumber distinguishing the two groups is sufficiently high, the scattering mechanism can be completely absent due to the linear nature of the equations for the high-wavenumber group.

Continuing from Part 1 (Hernándezet al.,J. Fluid Mech., vol. 936, 2022, A33), a generalised quasilinear (GQL) approximation is studied in turbulent channel flow using a flow decomposition defined with spanwise Fourier modes: the flow is decomposed into a set of low-wavenumber spanwise Fourier modes and the rest high-wavenumber modes. This decomposition leads to the nonlinear low-wavenumber group that supports the self-sustaining process within the given integral length scales, whereas the linearised high-wavenumber group is not able to do so, unlike the GQL models in Part 1, which place a minimal mathematical description for the self-sustaining process across all integral scales. Despite the important physical difference, it is shown that the GQL models in this study share some similarities with those in Part 1, i.e. the reduced multi-scale behaviour and anisotropic turbulent fluctuations. Furthermore, despite not being able to support the self-sustaining process in the high-wavenumber group, the GQL models in the present study are found to reproduce some key statistical features in the high-wavenumber group solely through the ‘scattering’ mechanism proposed by previous studies. Finally, using the nature of the GQL approximation, a further set of numerical experiments suppressing certain triadic nonlinear interactions are carried out. This unveils some key roles played by certain types of triadic interactions, including energy cascade and inverse energy transfer in the near-wall region. In particular, the inhibition of inverse energy transfer in the spanwise direction leads to suppression of the near-wall positive turbulent transport at large scales.

The computation of invariant solutions and the visualisation of the associated state space have played a key role in the understanding of transition and the self-sustaining process in wall-bounded shear flows. In this study, an extension of this approach is sought for a turbulent flow which explicitly exhibits multi-scale behaviour. The minimal unit of multi-scale near-wall turbulence, which resolves two adjacent spanwise integral length scales of motion, is considered using a shear stress-driven flow model (Doohan, Willis & Hwang J. Fluid Mech., vol. 913, 2021, A8). The edge state, 26 travelling waves and two periodic orbits are computed, which represent either the large- or small-scale self-sustaining processes. Given that the spanwise length scales are not widely separated here, it could be envisaged that turbulent trajectories visit these solutions in the state space. Considering the intra- and inter-scale dynamics of the flow, numerous phase portraits are examined, but the turbulent state is not found to approach any of these solutions. A detailed analysis reveals that this is due to the lack of scale interaction processes captured by the invariant solutions, including the mean–fluctuation interaction, the energy cascade in the streamwise wavenumber space and the cascade-driven energy production discovered recently. There is a single solution that resembles turbulence much more than the others, which captures two-scale energetics and a scale interaction process involving energy feeding from small to large spanwise scales through the subharmonic sinuous streak instability mode. Based on these observations, it is conjectured that the state space view of turbulent trajectories wandering between solutions would need suitable refinement to model multi-scale turbulence, when each solution does not represent multi-scale processes of turbulence. In particular, invariant solutions that are inherently multi-scale would be required.

Transverse rectangular microgrooves with local machining flaw are investigated using direct numerical simulations from the viewpoint of suppressing the second-mode instability in a Mach 6 hypersonic boundary layer. Two types of machining flaw are modeled for the local deformation of microgrooves, i.e., chamfered and protruded ones. The effect of locally deformed microgrooves on the boundary-layer stability is studied by superposing the second-mode disturbance to the base flow of the boundary layer. The investigated unstable second mode is 400 kHz, and the microgrooves are installed in its synchronous region. The results show that compression and expansion waves are generated on the surface of the deformed microgrooves, and microscale separation bubbles are formed at the rear edge of the deformed microgrooves. With the increase of local deformation height of microgrooves, the suppression effect of the second-mode disturbance gradually increases. As the deformation height reaches 0.09 mm, the amplitude inhibition rate of locally deformed microgrooves on the second-mode disturbance reaches 44.7%, compared to that of the regular one.

This paper describes the design of a class of three-dimensional (3D) smooth humps mounted on a yawed cone model for a high-precision compressible Navier–Stokes solver. The influence of isolated 3D humps with different heights on the linear stability of streamwise vortices over a yawed cone with a $7°$ half-angle at a $6°$ angle-of-attack, free-stream Mach number of 6, and unit Reynolds number of $1.0\times {10}^7$ /m is investigated. The results show that smooth humps induce stationary streaks inside the streamwise vortices over a yawed cone, effectively increasing the energy of the inward vortex and inhibiting the evolution and development of the outward vortex. Stability analysis shows that the induced streaks enhance the inner mode instability, suppress the outer mode instability, and reduce the dominant frequency of the outer mode. The stability properties of the hump-induced streaks are studied, and the physical process of the transformation between the new streak instability mode and the original boundary layer mode is revealed. The eN method based on global stability theory is used to predict the transition location along the centerline on the leeward side of the cone. It is found that configurations with hump heights of 0.153 and 0.267 times the local boundary layer thickness result in a transition delay of ∼3% and ∼11%, respectively.

A set of several exact coherent states in plane Couette flow is computed under spanwise wall oscillation control, with a range of wall oscillation amplitudes and periods $({A_w}, T)$ . It is found that the wall oscillation generally stabilises the upper branch of the equilibrium solutions and achieves the corresponding drag reduction, while it influences modestly the lower branch. The stabilisation effect is found to increase with the oscillation amplitude with an optimal time period around ${T^{+}} \approx 100$ . The exact coherent states reproduce some key dynamical behaviours of streaks observed in previous studies, while exhibiting the rich coherent structure dynamics that cannot be extracted from a phase average of turbulent states. Visualisation of state portraits shows that the size of the state space supporting turbulent solution is reduced by the spanwise wall oscillation, and the upper-branch equilibrium solutions become less repelling, with many of their unstable manifolds being stabilised. This change of the state space dynamics leads to a significant reduction in lifetime of turbulence. Finally, the main stabilisation mechanism of the exact coherent states is found to be the suppression of the lift-up effect of streaks, explaining why previous linear analyses have been so successful for turbulence stabilisation modelling and the resulting drag reduction.