The oblique shock impinging on the supersonic turbulent boundary layer leads to a mixing layer and the emergence of large-scale coherent structures within the interaction zone which leave significant velocity defect and turbulence amplification downstream. In the present study, we investigate the turbulence recovery in the post-shock region by exploiting direct numerical simulation data of the oblique-shock/turbulent boundary layer interaction flow at the incoming Mach number of $2.28$ and the shock angle of $33.2^\circ$, with special attention paid to the contribution of the mixing layer and large-scale structures to flow dynamics. For that purpose, we propose to split the mean velocity, Reynolds stresses and spanwise spectra into a canonical portion that is constructed according to the statistics of canonical turbulent boundary layers, and a mixing-layer-induced portion. We found that the hidden mixing layer grows with the boundary layer thickness and that the induced mean shear and Reynolds stresses decay at different rates. The mean velocity recovers to the canonical profiles at a distance of 13 boundary layer thicknesses downstream where the mixing-layer-induced mean shear ceases to have strong impacts. The recovery of Reynolds stresses requires 10 boundary layer thicknesses in the near-wall region but a much longer streamwise extent in the outer region due to the slow decay of large-scale motions. These large-scale motions superpose on the near-wall turbulence, intensifying the turbulent fluctuations, yet having a trivial impact on the skin friction, for the contribution of the mixing-layer-induced mean shear and Reynolds shear stress are balanced by the advection term. We further establish a simple physical model capable of approximately predicting the streamwise evolution of mixing-layer-induced mean shear and turbulent kinetic energy. This model suggests that the complete recovery of turbulence in the outer region requires a streamwise extent of approximately 50 boundary layer thicknesses.

We demonstrate that in plane Couette turbulence a separation of the velocity field in large and small scales according to a streamwise Fourier decomposition allows us to identify an active subspace comprising a small number of the gravest streamwise components of the flow that can synchronize all the remaining streamwise flow components. The critical streamwise wavelength, $\ell _{x c}$, that separates the active from the synchronized passive subspace is identified as the streamwise wavelength at which perturbations to the time-dependent turbulent flow with streamwise wavelengths $\ell _x<\ell _{xc}$ have negative characteristic Lyapunov exponents. The critical wavelength is found to be approximately 130 wall units and obeys viscous scaling at these Reynolds numbers.

Because the Navier–Stokes equations are dissipative, the long-time dynamics of a flow in state space are expected to collapse onto a manifold whose dimension may be much lower than the dimension required for a resolved simulation. On this manifold, the state of the system can be exactly described in a coordinate system parameterising the manifold. Describing the system in this low-dimensional coordinate system allows for much faster simulations and analysis. We show, for turbulent Couette flow, that this description of the dynamics is possible using a data-driven manifold dynamics modelling method. This approach consists of an autoencoder to find a low-dimensional manifold coordinate system and a set of ordinary differential equations defined by a neural network. Specifically, we apply this method to minimal flow unit turbulent plane Couette flow at $Re=400$, where a fully resolved solutions requires ${O}(10^5)$ degrees of freedom. Using only data from this simulation we build models with fewer than $20$ degrees of freedom that quantitatively capture key characteristics of the flow, including the streak breakdown and regeneration cycle. At short times, the models track the true trajectory for multiple Lyapunov times and, at long times, the models capture the Reynolds stress and the energy balance. For comparison, we show that the models outperform POD-Galerkin models with $\sim$2000 degrees of freedom. Finally, we compute unstable periodic orbits from the models. Many of these closely resemble previously computed orbits for the full system; in addition, we find nine orbits that correspond to previously unknown solutions in the full system.

We study a fifty-year-old problem of fast acoustic streaming, that is, the generation of moderate or large hydrodynamic Reynolds number ($\textit {Re}$) acoustic streaming (or steady flow) by the convection of momentum in an acoustic wave (or another periodic flow), while the latter is simultaneously altered by the former. The intrinsic disparity of length and time scales makes a brute-force solution of the full Navier–Stokes and continuity equations a formidable problem. Circumventing this difficulty, we split the problem into a time-averaged system of equations for the steady flow component and a dynamic system of equations for its quasi-periodic flow counterpart. The latter system of equations is obtained by subtracting the time-averaged Navier–Stokes equation from its original dynamic form, and is rendered a nonlinear wave equation using the continuity equation and an adiabatic connection between density and pressure. The resulting equations are compatible with the theory by Eckart for small $\textit {Re}$ flow, and capture large-$\textit {Re}$ effects. Scaling analysis and a case study show that acoustic streaming is weak and does not contribute to the acoustic wave close to the wave source, relevant to many microfluidic systems. At small $\textit {Re}$, the streaming magnitude is proportional to an inverse Strouhal number, a small quantity in experiments. Moderate and large $\textit {Re}$ render the streaming magnitude comparable to the pre-attenuating periodic flow (or particle velocity of the wave) at approximately a wave attenuation length away from the wave source or further; the wave is altered by the streaming that it generates, and the streaming dominates the flow far from the wave source.

This investigation characterises the time response and the transient turbulence dynamics undergone by rapidly decelerating turbulent pipe flows. A series of direct numerical simulations of decelerating flows between two steady Reynolds numbers were conducted for this purpose. The statistical analyses reveal that rapidly decelerating turbulent flows undergo four coherent, unambiguous transitional stages: inertial (stage I), a dramatic change of sign in the viscous force associated with the decay of the viscous shear stress at the wall together with a mild turbulence decay in the viscous sublayer; friction recovery (stage II), a recovery in viscous force and progressive decay in the turbulent inertia at the near-wall region; turbulence decay (stage III), a balanced decay in both turbulent inertia and viscous force at the near-wall and overlap regions; core relaxation (stage IV), slow turbulence decay at the core region. The FIK identity derived by Fukagata, Iwamoto and Kasagi (Phys. Fluids, vol. 14, 2002, L73–L76) was used to understand further how the flow dynamics influence the time response of the skin friction coefficient ($C_f$). The results show that although $C_f$ plateaus during the fourth stage, the turbulent contribution keeps decaying, undershoots and finally recovers to attain its final steady value. The time evolution of the azimuthal vorticity ($\omega _\theta$) flux reveals that as the flow is decelerated, a layer of negative $\omega _\theta$ is produced at the wall during the flow excursion. As time progresses, this negative vorticity propagates in the wall-normal direction, attenuating the pre-existing vorticity and producing a decay in the turbulence levels.

The purpose of the paper is to identify Mach-number effects on pressure fluctuations $p'$ in compressible turbulent plane channel flow. We use data from a specifically constructed $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$-matrix direct numerical simulation (DNS) database, with systematic variation of the centreline streamwise Mach number $0.32\leqslant \bar {M}_{{CL}_x}\leqslant 2.49$ and of the HCB (Huang et al., J. Fluid Mech., vol. 305, 1995, pp. 185–218) friction Reynolds number $66\leqslant Re_{\tau ^\star }\lessapprox 1000$. Strong $\bar {M}_{{CL}_x}$ effects (enhanced by the increasingly cold-wall condition) appear for $\bar {M}_{{CL}_x}\gtrapprox 2$, for all $Re_{\tau ^\star }$, very close to the wall ($y^\star \lessapprox 15$). Compared with incompressible flow at the same $Re_{\tau ^\star }$, the wall root-mean-square $[p'_{rms}]^+_w$ (in wall-units, i.e. scaled by the average wall shear stress $\bar {\tau }_w$) strongly increases with $\bar {M}_{{CL}_x}$. In contrast, the peak level across the channel, $[p'_{rms}]^+_{PEAK}$, slightly decreases with increasing $\bar {M}_{{CL}_x}$. In order to study the near-wall coherent structures we introduce a new wall-distance-independent non-local system of units, based for all $y$ on wall friction and the extreme values of density and dynamic viscosity, namely, for cold walls $\{\bar {\tau }_w,\min _y\bar {\rho },\max _y\bar {\mu }\}$. The average spanwise distance between streaks, scaled by this length-unit, is nearly independent of $\bar {M}_{{CL}_x}$ at constant $Re_{\tau ^\star }$. Using the in-plane (parallel to the wall) Laplacian $\nabla ^2_{xz}p'$ we find that the $(+/-)\text {-}p'$ wave-packet-like structures appearing inside the low-speed streaks ($y^\star \lessapprox 15$) with increasing $\bar {M}_{{CL}_x}\gtrapprox 2$ are part of a more complex wave system with spanwise extent over several streaks, whose spatial density decreases rapidly with decreasing $\bar {M}_{{CL}_x}$ or increasing $y^\star$. These $p'$ wave packets appear to be collocated with strong $(+/-)$-$v'$ events and could be responsible for compensating towards 0 the negative incompressible-flow correlation coefficient $c_{p'v'}$, with increasing $\bar {M}_{{CL}_x}$ very near the wall.

The linear stability of the axisymmetric steady thermocapillary flow in a liquid bridge made from 2 cSt silicone oil (Prandtl number 28) is investigated numerically in the framework of the Boussinesq approximation. The flow and temperature fields in the surrounding gas phase (air) are taken into account for a generic cylindrical container hosting the liquid bridge. The flows in the liquid and in the gas are fully coupled across the hydrostatically deformed liquid–gas interface, neglecting dynamic interface deformations. Originating from a common reference case, the linear stability boundary is computed varying the length of the liquid bridge (aspect ratio), its volume and the gravity level, providing accurate critical data. The qualitative dependence of the critical threshold on these parameters is explained in terms of the characteristics of the critical mode. The heat exchange between the ambient gas and the liquid bridge that is fully resolved has an important influence on the critical conditions.

This paper deals with the construction of the tip asymptotes for a hydraulic fracture deflating in a permeable elastic medium. Specifically, the paper describes the changing nature of the asymptotic fields during the arrest and recession phases following propagation of the fracture after fluid injection has ended. It shows that as the fracture deflates in the arrest phase, the region of dominance of the linear elastic fracture mechanics tip asymptote $w\sim x^{1/2}$ of the fracture aperture $w$ with distance $x$ from the front shrinks to the benefit of an intermediate asymptote $w\sim x^{3/4}$. Hence only the velocity-independent $3/4$ asymptote is left at the arrest–recession transition. Furthermore, with increasing receding velocity of the front, a linear asymptote $w\sim x$ develops progressively at the fracture tip, with $w\sim x^{3/4}$ again becoming an intermediate asymptote. These universal multiscale asymptotes for the arrest and recession phases are key to determining, in combination with a computational algorithm that can simulate the evolution of a finite fracture, the decaying stress intensity factor during arrest, the time at which the fracture transitions from arrest to recession, and the negative front velocity during recession.

The propulsion of a pitching flexible plate in a uniform flow is investigated numerically. The effects of bending stiffness ($K$), pitching amplitude ($A_L$) and frequency ($St$) on the wake patterns, thrust generations and propulsive performances of the fluid–plate system are analysed. Four typical wake patterns, i.e. von Kármán, reversed von Kármán, deflected and chaotic wakes, emerge from various kinematics, and the $St-A_L$ wake maps are given for various $K$. The drag-to-thrust transitions (DTT) and the wake transitions (WT) between the von Kármán and reversed von Kármán wakes are examined. Results indicate that the WT and DTT boundaries can be scaled by the chord-averaged distance of travel, $\mathcal {L}$, which leads to $\mathcal {L}\times St \approx 1$ and $\mathcal {L}\times St \approx 1.2$, respectively. Further, the resonance mechanism for the performance enhancement is revealed and confirmed in a wide range of parameters. The dimensionless average speed of plate, $\mathcal {U^*}\left (=\mathcal {L}\times St\right )$, is adopted merely to characterize the propulsive performances. For the first time, the $\mathcal {U^*}$-based scaling laws for the thrust and power are revealed in pitching rigid and flexible plates for various $A_L$ and $St$. This study may deepen our understanding of biological swimming and flying, and provide a guide for bionic design.

The oscillatory Kelvin–Helmholtz (K–H) instability of a planar liquid sheet was experimentally investigated in the presence of an axial oscillating gas flow. An experimental system was initiated to study the oscillatory K–H instability. The surface wave growth rates were measured and compared with theoretical results obtained using the authors’ early linear method. Furthermore, in a larger parameter range experimentally studied, it is interesting that there are four different unstable modes: first disordered mode (FDM), second disordered mode (SDM), K–H harmonic unstable mode (KHH) and K–H subharmonic unstable mode (KHS). These unstable modes are determined by the oscillating amplitude, oscillating frequency and liquid inertia force. The frequencies of KHH are equal to the oscillating frequency; the frequency of KHS equals half the oscillating frequency, while the frequencies of FDM and SDM are irregular. By considering the mechanism of instability, the instability regime maps on the relative Weber number versus liquid Weber number (Werel–Wel) and the Weber number ratio versus the oscillating frequency (Werel/Wel–$\varOmega$s2) were plotted. Among these four modes, KHS is the most unexpected: the frequency of this mode is not equal to the oscillating frequency, but the surface wave can also couple with the oscillating gas flow. Linear instability theory was applied to divide the parameter range between the different unstable modes. According to linear instability theory, K–H and parametric unstable regions both exist. However, note that all four modes (KHH, KHS, FDM and SDM) corresponded primarily to the K–H unstable region obtained from the theoretical analysis. Nevertheless, the parametric unstable mode was also observed when the oscillating frequency and amplitude were relatively low, and the liquid inertia force was relatively high. The surface wave amplitude was small but regular, and the evolution of this wave was similar to that of Faraday waves. The wave oscillating frequency was half that of the surface wave.

A planar jet issuing from a fully developed two-dimensional turbulent channel flow is studied, with a focus on the transverse flapping of the jet core. The streamwise and transverse velocities were measured with hot-wire anemometry using an X-type probe. The mean velocity field and the velocity covariances were first characterised to assess the undisturbed flow field. Periodic excitations were introduced from a slot mounted at the channel exit and the coherent fluctuating part of the signal was obtained by using a phase-locked averaging technique, where the periodic initial forcing was used as trigger. This enabled the eduction of the coherent structure associated with the introduced perturbation. Its amplitude was found to be directly proportional to the intensity of the initial forcing and, within a certain range of the initial forcing amplitude, the growth curves were identical as well as the spatial distribution of the extracted fluctuations. Parallel and non-parallel linear stability theory captures qualitatively and quantitatively the features of the educed coherent structure. The existence of the linear mode in the turbulent jet implies that the large-scale perturbations observed in natural (unforced) jets can be regarded as an incoherent set of linear modes.

In this work, we perform weakly nonlinear analysis of the reflection process of a thin oscillating wave beam on a non-critical surface in a fluid rotating and stratified along the same vertical axis in the limit of weak viscosity, i.e. small Ekman number $E$. We assume that the beam has the self-similar viscous structure obtained by Moore & Saffman (Phil. Trans. R. Soc. A, vol. 264, 1969, pp. 597–634) and Thomas & Stevenson (J. Fluid Mech., vol. 54, 1972, pp. 495–506). Such a solution describes the viscous internal shear layers of width $O(E^{1/3})$ generated by a localized oscillating source. We first show that the reflected beam conserves at leading order the self-similar structure of the incident beam and is modified by an $O(E^{1/6})$ correction with a different self-similar structure. We then analyse the nonlinear interaction of the reflected beam with the incident beam of amplitude $\varepsilon$ and demonstrate that a second-harmonic beam and localized meanflow correction, both of amplitude $\varepsilon ^{2} E^{-1/3}$, are created. We further show that for the purely stratified case (respectively the purely rotating case), a non-localized meanflow correction of amplitude $\varepsilon ^{2} E^{-1/6}$ is generated, except when the boundary is horizontal (respectively vertical). In this latter case, the meanflow correction remains localized but exhibits a triple-layer structure with a large $O(E^{4/9})$ viscous layer.

We obtain an optimal actuation waveform for fast synchronization of periodic airfoil wakes through the phase reduction approach. Using the phase reduction approach for periodic wake flows, the spatial sensitivity fields with respect to the phase of the vortex shedding are obtained. The phase sensitivity fields can uncover the synchronization properties in the presence of periodic actuation. This study seeks a periodic actuation waveform using phase-based analysis to minimize the time for synchronization to modify the wake shedding frequency of NACA0012 airfoil wakes. This fast synchronization waveform is obtained theoretically from the phase sensitivity function by casting an optimization problem. The obtained optimal actuation waveform becomes increasingly non-sinusoidal for higher angles of attack. Actuation based on the obtained waveform achieves rapid synchronization within as low as two vortex shedding cycles irrespective of the forcing frequency, whereas traditional sinusoidal actuation requires ${O}(10)$ shedding cycles. Further, we analyse the influence of actuation frequency on the vortex shedding and the aerodynamic coefficients using force-element analysis. The present analysis provides an efficient way to modify the vortex lock-on properties in a transient manner with applications to fluid–structure interactions and unsteady flow control.