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.

Abstract

The spectral energetics of a quasilinear (QL) model is studied in uniform shear turbulence. For the QL approximation, the velocity is decomposed into a mean averaged in the streamwise direction and the remaining fluctuation. The equations for the mean are fully considered, while the equations for the fluctuation are linearised around the mean. The QL model exhibits an energy cascade in the spanwise direction, but this is mediated by highly anisotropic small-scale motions unlike that in direct numerical simulation mediated by isotropic motions. In the streamwise direction, the energy cascade is shown to be completely inhibited in the QL model, resulting in highly elevated spectral energy intensity residing only at the streamwise integral length scales. It is also found that the streamwise wavenumber spectra of turbulent transport, obtained with the classical Reynolds decomposition, statistically characterizes the instability of the linearised fluctuation equations. Further supporting evidence of this claim is presented by carrying out a numerical experiment, in which the QL model with single streamwise Fourier mode is found to generate the strongest turbulence for L x /L z = 1 ∼ 3, consistent with previous findings (L x and L z are the streamwise and spanwise computational domains, respectively). Finally, the QL model is shown to completely ignore the role of slow pressure in the fluctuations, resulting in a significant damage of pressure-strain transport at all length scales. This explains the anisotropic turbulence of the QL model throughout the entire wavenumber space as well as the inhibited nonlinear regeneration of streamwise vortices in the self-sustaining process. * x,z,t = 1,(2.15a) * 10 −3 and −Φ uv * 10 −4 . This is in sharp contrast to the spectra of QL model. First of all,

In this paper, we investigate the receptivity of a supersonic boundary layer to impinging acoustic waves. Unlike previous studies of acoustic receptivity, where the sound waves have phase speeds comparable with or larger than the free-stream velocity $U_{\infty }$, the acoustic waves here have much slower ($O(R^{-1/8}U_{\infty })$) phase velocity, and their characteristic wavelength and frequency are of $O(R^{-3/8}L)$ and $O(R^{1/4}U_{\infty }/L)$ respectively, compatible with the triple-deck structure, where $L$ is the distance to the leading edge and $R$ the Reynolds number based on $L$ and $U_{\infty }$. A significant feature of a sound wave on the triple-deck scale is that an $O(\unicode[STIX]{x1D700}_{s})$ perturbation in the free stream generates much stronger ($O(\unicode[STIX]{x1D700}_{s}R^{1/8})$) velocity fluctuations in the boundary layer. Two receptivity mechanisms are considered. The first is new, involving the interaction of two such acoustic waves and operating in a boundary layer over a smooth wall. The second involves the interaction between an acoustic wave and the steady perturbation induced by a wavy wall. The sound–sound, or sound–roughness, interactions generate a forcing in resonance with a neutral Tollmien–Schlichting (T–S) wave. The latter is thus excited near the lower branch of the neutral curve, and subsequently undergoes exponential amplification. The excitation through sound–sound interaction may offer a possible explanation for the appearance of instability modes downstream of their neutral locations as was observed in a supersonic boundary layer over a smooth wall. The triple-deck formalism is adopted to describe impingement and reflection of the acoustic waves, and ensuing receptivity, allowing the coupling coefficient to be calculated. The two receptivity processes with the acoustic waves on the triple-deck scale are much more effective compared with those involving usual sound waves, with the coupling coefficient being greater by a factor of $O(R^{1/4})$ and $O(R^{1/8})$ in the sound–sound and sound–roughness interactions, respectively. A parametric study for both the reflection and coupling coefficients is conducted for representative T–S waves, to assess the influence of the streamwise and spanwise wavenumbers, and the phase speed (or frequency) of the acoustic wave.

Continuing from Part 1 (Hernández et 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.

METHODS:4 groups of 10 SPF chicken each were included: Positive control, vaccinated subcutaneously with a commercial ND vaccine, negative control -non-vaccinated- and two more groups were vaccinated with ND+CTB, one subcutaneously and the fourth group was vaccinated via intranasal. Vaccination at 3 wks of age and challenge 21 days post-vaccination. Sera were collected before vaccination and in a weekly manner during 6 wks post-vaccination in order to evaluate serological response and UHI titers. Mortality and clinical signs were evaluated during 21 days post-challenge. RESULTS. Chickens vaccinated subcutaneously with ND-CTB reached the highest antibody titers when compared to the other experimental groups followed by chickens vaccinated with the commercial vaccine, then the intranasally vaccinated group and then the negative control. The group immunized subcutaneously with CTB also showed an earlier onset of serological response against ND and the duration of immunity was similar to the positive control. Mortality post challenge could be observed in the negative control as well as in intranasaly immunized birds, where all birds died as opposed as in groups of the positive control and the subcutaneously vaccinated, where all the birds survived. CONCLUSIONS: CTB seemed to contribute towards an earlier onset of humoral response in the ND model

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