The spontaneous pairing of rolled-up vortices in a laminar jet is investigated as a global secondary instability of a time-periodic spatially developing vortex street. The growth of subharmonic perturbations, associated with vortex pairing, is analysed both in terms of modal Floquet instability and in terms of transient growth dynamics. The article has the double objective to outline a toolset for the global analysis of time-periodic flows, and to leverage such an analysis for a fresh view on the vortex pairing phenomenon. Axisymmetric direct numerical simulations (DNS) of jets with single-frequency inflow forcing are performed, in order to identify combinations of the Reynolds and Strouhal numbers for which vortex pairing is naturally observed. The same DNS calculations are then repeated with an added time-delay control term, which artificially suppresses pairing, so as to obtain time-periodic unpaired base flows for linear stability analysis. It is demonstrated that the natural occurrence of vortex pairing in nonlinear DNS coincides with a linear subharmonic Floquet instability of the underlying unpaired vortex street. However, DNS results suggest that the onset of pairing involves much stronger temporal growth of subharmonic perturbations than that predicted by modal Floquet analysis, as well as a spatial distribution of these fast-growing perturbation structures that is inconsistent with the unstable Floquet mode. Singular value decomposition of the phase-shift operator (the operator that maps a given perturbation field to its state one flow period later) is performed for an analysis of optimal transient growth in the vortex street. Non-modal mechanisms near the jet inlet are thus found to provide a fast route towards the limit-cycle regime of established vortex pairing, in good agreement with DNS observations. It is concluded that modal Floquet analysis accurately predicts the parameter regime where sustained vortex pairing occurs, but that the bifurcation scenario under typical conditions is dominated by transient growth phenomena.

Flow blockage by large wind farms leads to an upward displacement of the boundary layer, which may excite atmospheric gravity waves in the free atmosphere and on the interface between the boundary layer and the free atmosphere. In the current study, we assess the sensitivity of wind-farm gravity-wave excitation to important dimensionless groups and investigate the feedback of gravity-wave-induced pressure fields to wind-farm energy extraction. The sensitivity analysis is performed using a fast boundary-layer model that is developed to this end. It is based on a three-layer representation of the atmosphere in an idealised barotropic environment, and is coupled with an analytical wake model to account for turbine wake interactions. We first validate the model in two-dimensional mode with data from previous large-eddy simulations of ‘infinitely’ wide wind farms, and then use the model to investigate the sensitivity of wind-farm-induced gravity waves to atmospheric state and wind-farm configuration. We find that the inversion layer induces flow physics similar to shallow-water flow and that the corresponding Froude number plays a crucial role. Gravity-wave excitation is maximal at a critical Froude number equal to one, but the feedback on energy extraction is highest when the Froude number is slightly below one due to a trade-off between amplitude and upstream impact of gravity waves. The effect of surface friction and internal gravity waves is to reduce the flow perturbation and the related power loss by dissipating or dispersing perturbation energy. With respect to the wind-farm configuration, we find that gravity-wave-induced power loss increases with wind-farm size and turbine height. Moreover, we find that gravity-wave effects are small for very wide or very long wind farms and attain a maximum at a width-to-depth ratio of approximately $3/2$.

A new set of exact coherent states in the form of a travelling wave is reported in plane channel flow. They are continued over a range in $Re$ from approximately $2600$ up to $30\,000$, an order of magnitude higher than those discovered in the transitional regime. This particular type of exact coherent states is found to be gradually more localised in the near-wall region on increasing the Reynolds number. As larger spanwise sizes $L_{z}^{+}$ are considered, these exact coherent states appear via a saddle-node bifurcation with a spanwise size of $L_{z}^{+}\simeq 50$ and their phase speed is found to be $c^{+}\simeq 11$ at all the Reynolds numbers considered. Computation of the eigenspectra shows that the time scale of the exact coherent states is given by $h/U_{cl}$ in channel flow at all Reynolds numbers, and it becomes equivalent to the viscous inner time scale for the exact coherent states in the limit of $Re\rightarrow \infty$. The exact coherent states at several different spanwise sizes are further continued to a higher Reynolds number, $Re=55\,000$, using the eddy-viscosity approach (Hwang & Cossu, Phys. Rev. Lett., vol. 105, 2010, 044505). It is found that the continued exact coherent states at different sizes are self-similar at the given Reynolds number. These observations suggest that, on increasing Reynolds number, new sets of self-sustaining coherent structures are born in the near-wall region. Near this onset, these structures scale in inner units, forming the near-wall self-sustaining structures. With further increase of Reynolds number, the structures that emerged at lower Reynolds numbers subsequently evolve into the self-sustaining structures in the logarithmic region at different length scales, forming a hierarchy of self-similar coherent structures as hypothesised by Townsend (i.e. attached eddy hypothesis). Finally, the energetics of turbulent flow is discussed for a consistent extension of these dynamical systems notions to high Reynolds numbers.

In a rectangular cross-section wind tunnel, a separated oblique shock reflection is set to interact with the turbulent boundary layer (oblique shock wave/turbulent boundary layer interaction (SBLI)) both on the bottom wall and in the corners formed by the intersection of the floor with the sidewalls. To examine how corner separations can affect the ‘quasi-two-dimensional’ main interaction and by what mechanisms this is achieved, an experimental investigation has been conducted. This examines how modifications to the corner separation affect an $M=2.5$ oblique shock reflection. The nature of the flow field is studied using flow visualisation, pressure-sensitive paint and laser Doppler anemometry. The results show that the size and shape of central separation vary considerably when the onset and magnitude of corner separation changes. The primary mechanism explaining the coupling between these separated regions appears to be the generation of compression waves and expansion fans as a result of the displacement effect of the corner separation. This is shown to modify the three-dimensional shock structure and alter the adverse pressure gradient experienced by the tunnel floor boundary layer. It is suggested that a typical oblique SBLI in rectangular channels features several zones depending on the relative position of the corner waves and the main interaction domain. In particular, it has been shown that the position of the corner ‘shock’ crossing point, found by approximating the corner compression waves by a straight line, is a critical factor determining the main separation size and shape. Thus, corner effects can substantially modify the central separation. This can cause significant growth or contraction of the separation length measured along the symmetry line from the nominally two-dimensional baseline value, giving a fivefold increase from the smallest to the largest observed value. Moreover, the shape and flow topology of the centreline separation bubble is also considerably changed by varying corner effects.

Stochastic roughness is a widespread feature of natural surfaces and is an inherent byproduct of most fabrication techniques. In view of the rapid development of microfluidics, the important question is how this inevitable problem affects the low-Reynolds-number flows that are common for micro-devices. Moreover, one could potentially turn the flaw into a virtue and control the flow properties by means of specially ‘tuned’ random roughness. In this paper we investigate theoretically the statistics of fluctuations in fluid velocity produced by the waviness irregularities at the surface of a no-slip wall. Particular emphasis is laid on the issue of the universality of our findings.

The dynamics of bubble clouds induced by high-intensity focused ultrasound is investigated in a regime where the cloud size is similar to the ultrasound wavelength. High-speed images show that the cloud is asymmetric; the bubbles nearest the source grow to a larger radius than the distal ones. Similar structures of bubble clouds are observed in numerical simulations that mimic the laboratory experiment. To elucidate the structure, a parametric study is conducted for plane ultrasound waves with various amplitudes and diffuse clouds with different initial void fractions. Based on an analysis of the kinetic energy of liquid induced by bubble oscillations, a new scaling parameter is introduced to characterize the dynamics. The new parameter generalizes the cloud interaction parameter originally introduced by d’Agostino & Brennen (J. Fluid Mech., vol. 199, 1989, pp. 155–176). The dynamic interaction parameter controls the energy localization and consequent anisotropy of the cloud. Moreover, the amplitude of the far-field, bubble-scattered acoustics is likewise correlated with the proposed parameter. Findings of the present study not only shed light on the physics of cloud cavitation, but may also be of use for the quantification of the effects of cavitation on outcomes of ultrasound therapies including high-intensity focused ultrasound-based lithotripsy.

This study presents an experimental investigation on the internal flow of a ventilated supercavity using fog flow visualization and particle image velocimetry (PIV) measurements. The ventilated supercavity is generated on a backward-facing cavitator and studied in the high-speed water tunnel at St. Anthony Falls Laboratory. Fog particles are introduced into the supercavity through the ventilation line, and then illuminated by a laser sheet for flow visualizations and PIV measurements. The experiments are performed on the supercavities with two closure types, i.e. the re-entrant jet (RJ) and the twin vortex (TV), under the same water tunnel flow condition but different ventilation rates. The flow visualization revealed three distinct regions within the supercavity, including the ventilation influence region near the cavitator, the extended internal boundary layer along the liquid–gas interface and the reverse flow region occupying a large centre portion of the supercavity. The streamwise and vertical extent of the ventilation influence region, the streamwise growth of the internal boundary layer and the reverse flow within the supercavity are then quantified through PIV flow measurements. Compared to the RJ case, the results indicate that the TV supercavity yields a longer vertical extent of the ventilation influence region, a thinner internal boundary layer and a stronger reverse flow. The internal flow results suggest that at the upstream of the location of the maximum cavity diameter, the gas enters the forward flow (including the internal boundary layer and the forward moving portion of the ventilation influence region) from the reverse flow, while at the downstream of that location, the gas is stripped from the internal boundary layer and enters the reverse flow due to the increasing adverse pressure gradient in the streamwise direction. The above results are combined with visualization results of the supercavity geometry and closure patterns to further explain the influence of gas leakage mechanisms on cavity growth and closure transition. Specifically, visualization of the cavity geometry change during the RJ to TV supercavity transition indicates external flow separation associated with a critical incline angle of the bottom liquid–gas interface at the closure contributes to the onset of RJ closure. The closure visualization shows the coexistence of the toroidal vortex and twin-vortex tubes for the RJ supercavity leads to two gas leakage mechanisms: one associated with the shedding of toroidal vortices ($Q_{RJ}$) and the other due to the gas entrained by the internal boundary layer and leaking from the twin-vortex tubes ($Q_{TV}$). For the RJ supercavity, with increasing ventilation input, due to the reduction of $Q_{RJ}$, the supercavity needs to elongate to increase the gas entrained by the internal boundary layer (i.e. $Q_{TV}$) to balance the ventilation increase. The elongation of the supercavity leads to reduced flow separation, and eventually a transition to the TV supercavity with ventilation above a critical value. For the TV supercavity, $Q_{RJ}$ is absent. An increase of ventilation input can be balanced by the increase of $Q_{TV}$ associated with the widening of the twin-vortex tubes, and therefore, no appreciable elongation of cavity length is observed.

A two-dimensional analysis of the resolvent spectrum of a Mach 1.6 transitional boundary layer impacted by an oblique shock wave is carried out. The investigation is based on a two-dimensional mean flow obtained by a RANS model that includes a transition criterion. The goal is to evaluate whether such a low cost RANS based resolvent approach is capable of describing the frequencies and physics involved in this transitional boundary layer/shock-wave interaction. Data from an experiment and a companion large eddy simulation (LES) are utilized as reference for the validation of the method. The flow is characterized by a laminar boundary layer upstream, a laminar separation bubble (LSB) in the interaction region and a turbulent boundary layer downstream. The flow exhibits low amplitude unsteadiness in the LSB and at the reflected shock wave with three particular oscillation frequencies, qualified as low, medium and high in reference to their range in Strouhal number, here based on free stream velocity and LSB length ($S_{t}=0.03{-}0.11$, 0.3–0.4 and 2–3 respectively). Through the resolvent analysis this dynamics is found to correspond to an amplifier behaviour of the flow. The resolvent responses match the averaged Fourier mode of the time dependent flow field, here described by the LES, with a close agreement in frequency and spatial distribution, thereby validating the resolvent approach. The low frequency dynamics relates to a pseudo-resonance process that sequentially implies the amplification in the separated shear layer of the LSB, an excitation of the shock foot and a backward travelling density wave. As this wave hits back the separation point the amplification in the shear layer starts again and loops. The medium and high frequency modes relate to the periodic expansion/reduction of the bubble and to the turbulent fluctuations at the reattachment point of the bubble, respectively.