Direct numerical simulations (DNS) of temporally evolving shear layers have been performed to study the entrainment of irrotational flow into the turbulent region across the turbulent/non-turbulent interface (TNTI). Four cases with convective Mach number from 0.2 to 1.8 are used. Entrainment is studied via two mechanisms; nibbling, considered as vorticity diffusion across the TNTI, and engulfment, the drawing of the pockets of the outside irrotational fluid into the turbulent region. The mass flow rate due to nibbling is calculated as the product of the entrained mass flux with the surface area of the TNTI. It is found that increasing the convective Mach number results in a decrease of the average entrained mass flux and the surface area of the TNTI. For the incompressible shear layer the local entrained mass flux is shown to be highly correlated with the viscous terms. However, as the convective Mach number increases, the mass fluxes due to the baroclinic and the dilatation terms play a more important role in the local entrainment process. It is observed that the entrained mass flux is dependent on the local dilatation and geometrical shape of the TNTI. For a compressible shear layer, most of the entrainment of the irrotational flow into the turbulent region due to nibbling is associated with the compressed regions on the TNTI. As the convective Mach number increases, the percentage of the compressed regions on the TNTI decreases, resulting in a reduction of the average entrained mass flux. It is also shown that the local shape of the interface, looking from the turbulent region, is dominated by concave shaped surfaces with radii of curvature of the order of the Taylor length scale. The average entrained mass flux is found to be larger on highly curved concave shaped surfaces regardless of the level of dilatation. The mass fluxes due to vortex stretching, baroclinic torque and the shear stress/density gradient terms are weak functions of the local curvatures on the TNTI, whereas the mass fluxes due to dilatation and viscous diffusion plus the viscous dissipation terms have a stronger dependency on the local curvatures. As the convective Mach number increases, the probability of finding highly curved concave shaped surfaces on the TNTI decreases, whereas the probability of finding flatter concave and convex shaped surfaces increases. This results in a decrease of the average entrained mass flux across the TNTI. Similar to the previous works on jets, the results show that the contribution of the engulfment to the total entrainment is small for both incompressible and compressible mixing layers. It is also shown that increasing the convective Mach number decreases the velocities associated with the entrainment, i.e. induced velocity, boundary entrainment velocity and local entrainment velocity.

The unstart phenomena in a model scramjet with a free stream Mach number of 4.5 were investigated at an arc-heated hypersonic wind tunnel. High-speed schlieren imaging and high resonance frequency pressure measurements were used to capture the flow features during the unstart process. Three unstart conditions were tested: (i) a low-enthalpy free stream with mass loading, (ii) a high-enthalpy free stream with mass loading and (iii) a high-enthalpy free stream with mass loading and heat release. It was revealed that the unstart threshold and the time from the onset to the completion of unstart depended strongly on the mass loading rate and the heat exchange. The negative heat addition (cooling) significantly increased the threshold of mass flow rate triggering unstart. The decrement of the mass flow rate threshold for unstart was observed in the presence of heat release by combustion. The observed transient and quasi-steady behaviours of the unstart shockwave system and the jet motion were similar in all of the test conditions. On the other hand, at the lip of inlet model, the unstart shockwave under the cold free stream condition exhibited a relatively steady behaviour while severe oscillatory flow motions of the jet and the unstart shockwave were observed in the combustion-driven unstart process. The different unstarted flow behaviours between the three flow conditions were explained using a simplified one-dimensional flow choking analysis and use of the Korkegi criterion.

The formation of concentrated vortices like tornadoes and tropical cyclones in rotating fluids is of much interest in atmospheric flows. It is shown by direct numerical simulation that the selective decay of inviscid invariants leads to concentration of vorticity in a destabilized vortex. By selective decay we mean here that the circulation of the mean flow decays faster than the angular momentum or energy. Initially localized disturbances are superimposed onto the two-dimensional flattened Taylor–Green vortices to trigger the elliptic instability. In the later stage of nonlinear evolution of the disturbance circulation decays faster than angular momentum and energy, giving rise to a sharp peak in the vorticity distribution of the mean flow. During the selective decay vortex pairs reconnect and eventually annihilate at the cell boundaries of the Taylor–Green vortices. By evaluating the weight function of the inviscid invariants it is shown that the loss of angular momentum is much smaller than that of circulation when vorticity is lost at the cell boundary by reconnection or annihilation. Thus the reconnection and subsequent annihilation of vortex pairs is responsible for the selective decay and concentration of vorticity. The relevance of the mechanism to previous experiments and general cases is also discussed.

Nonlinear evolution of long-crested wave groups can lead to extreme interactions with coastal and marine structures. In the present study the role of nonlinear evolution in the formation of extreme run-up events on a vertical wall is investigated. To this end, the fundamental problem of interaction between non-breaking water waves and a vertical wall over constant water depth is considered. In order to simulate nonlinear wave–wall interactions, the high-order spectral method is applied to a computational domain which aims to represent a two-dimensional wave flume. Wave generation is simulated at the flume entrance by means of the additional potential concept. Through this concept, the implementation of a numerical wavemaker is applicable. In addition to computational efficiency, the adopted numerical approach enables one to consider the evolution of nonlinear waves while preserving full dispersivity. Utilizing these properties, the influence of the nonlinear wave evolution on the wave run-up can be examined for a wide range of water depths. In shallow water, it is known that nonlinear evolution of incident waves may result in extreme run-up events due to the formation of an undular bore. The present study reveals the influence of the nonlinear evolution on the wave run-up in deep-water conditions. The results suggest that extreme run-up events in deep water may occur as a result of the disintegration of incident wave groups into envelope solitons.

Direct numerical simulations were performed to investigate the effect of severe contamination on interfacial gas transfer in the presence of isotropic turbulence diffusing from below. A no-slip boundary condition was employed at the interface to model the severe contamination effect. The influence of both Schmidt number ($Sc$) and turbulent Reynolds number ($R_{T}$) on the transfer velocity ($K_{L}$) was studied. In the range from $Sc=2$ up to $Sc=500$ it was found that $K_{L}\propto Sc^{-2/3}$, which is in agreement with predictions based on solid–liquid transport models, see e.g. Davies (1972, Turbulence Phenomena, Academic). For similar $R_{T}$, the transfer velocity was observed to reduce significantly compared with the free-slip conditions. The reduction becomes more pronounced with increasing Schmidt number. Similar to the observation for free-slip conditions made by Theofanous et al. (Intl J. Heat Mass Transfer, vol. 19 (6), 1976, pp. 613–624), the normalized $K_{L}$ in the present no-slip case was also found to depend on $R_{T}^{-1/2}$ and $R_{T}^{-1/4}$ for small and large turbulent Reynolds numbers, respectively.

The fundamental difference between continuous modes of the Orr–Sommerfeld/Squire equations and the entrainment of free-stream vortical disturbances (FSVD) into the boundary layer has been investigated in a recent paper (Dong & Wu, J. Fluid Mech., vol. 732, 2013, pp. 616–659). It was shown there that the non-parallel-flow effect plays a leading-order role in the entrainment, and neglecting it at the outset, as is done in the continuous-mode formulation, leads to non-physical features of ‘Fourier entanglement’ and abnormal anisotropy. The analysis, which was for incompressible boundary layers and for FSVD with a characteristic wavelength of the order of the local boundary-layer thickness, is extended in this paper to compressible boundary layers and FSVD with even shorter wavelengths, which are comparable with the width of the so-called edge layer. Non-parallelism remains a leading-order effect in the present scaling, which turns out to be more general in that the equations and solutions in the previous paper are recovered in the appropriate limit. Appropriate asymptotic solutions in the main and edge layers are obtained to characterize the entrainment. It is found that when the Prandtl number $\mathit{Pr}<1$, free-stream vortical disturbances of relatively low frequency generate very strong temperature fluctuations within the edge layer, leading to formation of thermal streaks. A composite solution, uniformly valid across the entire boundary layer, is constructed, and it can be used in receptivity studies and as inlet conditions for direct numerical simulations of bypass transition. For compressible boundary layers, continuous spectra of the disturbance equations linearized about a parallel base flow exhibit entanglement between vortical and entropy modes, namely, a vortical mode necessarily induces an entropy disturbance in the free stream and vice versa, and this amounts to a further non-physical behaviour. High Reynolds number asymptotic analysis yields the relations between the amplitudes of entangled modes.

Double-diffusive lock-exchange gravity currents in the fingering regime are explored via two- and three-dimensional Navier–Stokes simulations in the Boussinesq limit. Even at modest Reynolds numbers, for which single-diffusive gravity currents remain laminar, double-diffusive currents are seen to give rise to pronounced small-scale fingering convection. The front velocity of these currents exhibits a non-monotonic dependence on the diffusivity ratio and the initial stability ratio. Strongly double-diffusive currents lose both heat and salinity more quickly than weakly double-diffusive ones, and they lose salinity more quickly than heat, so that the density difference driving them increases. This differential loss of heat and salinity furthermore results in the emergence of strong local density maxima and minima along the top and bottom walls in the gate region, which in turn promote the formation of secondary, counterflowing currents along the walls. These secondary currents cause the flow to develop a three-layer structure. The late stages of the flow are dominated by currents flowing oppositely to the original ones. Three-dimensional simulation results are consistent with the trends observed in a two-dimensional parametric study. A detailed analysis of the energy budget demonstrates that strongly double-diffusive currents can release several times their initially available potential energy, and convert large amounts of internal energy into mechanical energy via scalar diffusion. Scaling arguments based on the simulation results suggest that even low Reynolds number double-diffusive gravity currents are governed by a balance of buoyancy and turbulent drag.

A nonlinear theory for thermoacoustic waves in a gas-filled, narrow channel and pore subject to an axial temperature gradient is developed based on the fluid dynamical equations for an ideal gas. Under the narrow-tube approximation, three small parameters are introduced as asymptotic parameters, one being the ratio of a span length to a typical thickness of the thermoviscous diffusion layer, another the ratio of the typical propagation speed of thermoacoustic waves to an adiabatic sound speed and the final parameter is the ratio of the typical magnitude of a pressure disturbance to uniform pressure in a quiescent state. No thermal interaction between the gas and the solid wall is taken into account on assuming that the wall has a large heat capacity. Using the three small parameters, the fluid dynamical equations are approximated asymptotically to be reduced to a single nonlinear diffusion wave (advection) equation for an excess pressure. All field variables are determined consistently in terms of the excess pressure so as to satisfy the boundary conditions on the wall. Supposing a time-periodic solution to the equation derived, the mean value of the excess pressure over one period is examined. It is shown that while the mean vanishes in the linear theory, it decreases monotonically due to nonlinearity. It is also shown that mean values of the shear stress and the heat flux at the wall, as well as those of the vector fields of the mass and energy fluxes representing, respectively, acoustic and thermoacoustic streaming, are expressed in terms of the mean values of the products of the spatial and/or temporal pressure gradients, which are reduced to the spatial derivatives of the mean pressure.

Noise generation mechanisms for a perfectly expanded supersonic Mach number $M=1.8$ turbulent jet impinging on a $45^{\circ }$ inclined plate are investigated for a Reynolds number of $1.6\times 10^{6}$ employing a large-eddy simulation. Excellent comparisons with experimental acoustic far-field measurements and pressure measurements on the impingement plate are obtained. Two local maxima are identified in the far-field overall sound pressure levels in the $75^{\circ }$ and $120^{\circ }$ observer directions, which are associated with different noise generation mechanisms. The peak frequencies in the spectra with Strouhal numbers of $St=0.2$ for $75^{\circ }$ and $St=0.5$ for $120^{\circ }$ match the experimental measurements. The jet-impingement region generates pressure waves that propagate predominantly in the $120^{\circ }$ observer direction. The noise generation in this region is attributed to vortex stretching and tearing during shear-layer impingement, and shock oscillations that are induced by the motion of downstream convected vortical flow structures. The second peak in the overall sound pressure distribution at $75^{\circ }$ is associated with noise sources located in the wall jet. The noise generation in the wall jet is associated with supersonically convecting large-scale coherent flow structures that also interact with tail shocks in the wall jet causing large localized pressure fluctuations. Strongly coherent flow structures are identified by applying proper orthogonal decomposition (POD) to the unsteady flow field. The frequency characteristics of the most energetic POD modes are distinctly different based on which energy norm is chosen. The most energetic entropy-based POD modes contain a peak frequency of approximately $St=0.4{-}0.6$, while the most energetic turbulent kinetic-energy-based POD modes appear to be dominated by lower-frequency content. The causality method, based on Lighthill’s acoustic analogy, is used to link the acoustic noise signature to the relevant physical mechanisms in the source region. A differentiation is made between the application of normalized and non-normalized cross-correlation functions for noise source identification and characterization.

The vibration of bubbles can produce intense microstreaming when excited by ultrasound near resonance. In order to study freely oscillating bubbles in steady conditions, we have confined bubbles between the two walls of a silicone microchannel and anchored them on micropits. We were thus able to analyse the microstreaming flow generated around an isolated bubble or a pair of interacting bubbles. In the case of an isolated bubble, a short-range microstreaming occurs in the channel gap, with additional in-plane vortices at high amplitude when Faraday waves are excited on the bubble periphery. For a pair of bubbles, we have observed long-range microstreaming and large recirculations describing a ‘butterfly’ pattern. We propose a model based on secondary acoustic Bjerknes forces mediated by Rayleigh waves on the silicone walls. These forces lead to attraction or repulsion of bubbles and thus to the excitation of a translational mode in addition to the breathing mode of the bubble. The mixed-mode streaming induced by the interaction of these two modes is shown to generate fountain or anti-fountain vortex pairs, depending on the relative distance between the bubbles.

This paper analyses the response and receptivity of the hypersonic boundary layer over a wedge to free-stream disturbances including acoustic, vortical and entropy fluctuations. Due to the presence of an attached oblique shock, the boundary layer is known to support viscous instability modes whose eigenfunctions are oscillatory in the far field. These modes acquire a triple-deck structure. Any of three elementary types of disturbance with frequency and wavelength on the triple-deck scales interacts with the shock to generate a slow acoustic perturbation, which is reflected between the shock and the wall. Through this induced acoustic perturbation, vortical and entropy free-stream disturbances drive significant velocity and temperature fluctuations within the boundary layer, which is impossible when the shock is absent. A quasi-resonance was identified, due to which the boundary layer exhibits a strong response to a continuum of high-frequency disturbances within a narrow band of streamwise wavenumbers. Most importantly, in the vicinity of the lower-branch neutral curve the slow acoustic perturbation induced by a disturbance of suitable frequency and wavenumbers is in exact resonance with a neutral eigenmode. As a result, the latter can be generated directly by each of three types of free-stream disturbance without involving any surface roughness element. The amplitude of the instability mode is determined by analysing the disturbance evolution through the resonant region. The fluctuation associated with the eigenmode turns out to be much stronger than the free-stream disturbances due to the resonant nature of excitation, and in the case of acoustic disturbances, to the well-known amplification effect of a strong shock. Moreover, excitation at the neutral position means that the instability mode grows immediately without undergoing any decay, or missing any portion of the unstable region. All these indicate that this new mechanism is particularly efficient. The boundary-layer response and coupling coefficients are calculated for typical values of parameters.

We report on a study, employing direct numerical simulations, of the turbulent/non-turbulent interface of a wake in a stably stratified fluid. It is found that thresholds for both enstrophy and potential enstrophy are needed to identify the interface. Using conditional averaging relative to the location of the interface, various quantities of interest are examined. The thickness of the interface is found to scale with the Kolmogorov scale. From an examination of the Ozmidov and Kolmogorov length scales as well as the buoyancy Reynolds number, it is found that the buoyancy Reynolds number decreases and becomes of order 1 near the interface, indicating the suppression of the turbulence there by the stable stratification. Finally the overall rate of loss of energy due to internal wave radiation is found to be comparable to the overall rate of loss due to turbulent kinetic energy dissipation.

The focus of this study is to analyse the tip region of a hydraulic fracture, for which a fluid flow inside the crack transitions from the laminar to the turbulent regime away from the tip. To tackle the problem, a phenomenological formula for flow in pipes has been adapted to describe flow in a fracture through the concept of a hydraulic diameter. The selected model is able to capture laminar, turbulent and transition regimes of the flow. The near-tip region of a hydraulic fracture is analysed by focusing on steady propagation of a semi-infinite hydraulic fracture with leak-off, for which the aforementioned phenomenological formula for the fluid flow is utilized. First, the distance from the tip within which a laminar solution applies is estimated. Then, expressions for asymptotic solutions that are associated with fully developed turbulent flow inside the semi-infinite hydraulic fracture are derived. Finally, the laminar zone size and the asymptotic solutions are compared with the numerical solution, where the latter captures all regimes of the fluid flow.

The mean velocity profile (MVP) in thermally stratified atmospheric surface layers (ASLs) deviates from the classic logarithmic form. A theoretical framework was recently proposed (Katul et al.Phys. Rev. Lett., vol. 107, 2011, 268502) to link the MVP to the spectrum of turbulence and was found to successfully predict the MVP for unstable stratification. However, the theory failed to reproduce the MVP in stable conditions (Salesky et al.Phys. Fluids, vol. 25, 2013, 105101), especially when ${\it\zeta}>0.2$ (where ${\it\zeta}$ is the atmospheric stability parameter). In the present study, it is demonstrated that this shortcoming is due to the failure to identify the appropriate length scale that characterizes the size of momentum transporting eddies in the stable ASL. Beyond ${\it\zeta}\approx 0.2$ (near where the original theory fails), the Ozmidov length scale becomes smaller than the distance from the wall $z$ and hence is a more stringent constraint for characterizing the size of turbulent eddies. An expression is derived to connect the Ozmidov length scale to the normalized MVP (${\it\phi}_{m}$), allowing ${\it\phi}_{m}$ to be solved numerically. It is found that the revised theory produces a prediction of ${\it\phi}_{m}$ in good agreement with the widely used empirical Businger–Dyer relation and two experimental datasets in the stable ASL. The results here demonstrate that the behaviour of ${\it\phi}_{m}$ in the stable ASL is closely linked to the size of momentum transporting eddies, which can be characterized by the Ozmidov scale under mildly to moderately stable conditions ($0.2<{\it\zeta}<1-2$).

An analytical model is proposed for flow around a plane bubble rising steadily in a channel. The fluid surrounding the bubble is assumed to be inviscid, and the flow is assumed to be irrotational. The resulting potential flow is sought as the sum of a uniform stream and a source in the unit strip. Explicit expressions for the bubble shape and rise velocity, and for the streamlines, are given as functions of the dimensionless surface tension. The predictions are in good agreement with existing experimental and numerical results.